Spectroscopy of Low Temperature Plasma

Vladimir N. Ochkin Spectroscopy of Low Temperature Plasma Related Titles R. Hippler, H. Kersten, M. Schmidt, K.H. Sch...

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Vladimir N. Ochkin Spectroscopy of Low Temperature Plasma

Related Titles R. Hippler, H. Kersten, M. Schmidt, K.H. Schoenbach (Eds.)

Low Temperature Plasmas Fundamentals, Technologies and Techniques 2008 ISBN: 978-3-527-40673-9

R. d'Agostino, P. Favia, Y. Kawai, H. Ikegami, N. Sato, F. Arefi-Khonsari (Eds.)

Advanced Plasma Technology 2008 ISBN: 978-3-527-40591-6

B.M. Smirnov

Physics of Ionized Gases 2001 ISBN: 978-0-471-17594-0

Vladimir N. Ochkin

Spectroscopy of Low Temperature Plasma

WILEY-VCH Verlag GmbH & Co. KGaA

The Author Prof. Vladimir N. Ochkin P.N. Lebedev Physical Institute Russian Academy of Sciences Moscow, Russia [email protected] Translation Dr. Sergey Kittell Moscow, Russia

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover Picture A powerful pulsed corona discharge. Photography by G. Mesyats and S. Rukin, Ekaterinburg, Russia

Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de ¤ 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Composition Da-TeX Gerd Blumenstein, Leipzig Printing Strauss GmbH, Mörlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN: 978-3-527-40778-1

In memory of my wife Clara

VII

Contents

Preface

XIII

References

XVI

1

1

Plasma as an Object of Spectroscopy General Notions 1

1.1

The Concept of Low-Temperature Plasma. Diagnostics Problems 1 Equilibrium Plasma 6 Energy Distribution of Particles 6 Law of Mass Action. Neutral and Charged Particle Densities Heat Emission. Kirchhoff’s Law 11 Models of Equilibrium and the Associated Parameters 14 Local Thermal Equilibrium (LTE) Model 14 Partial Local Thermal Equilibrium (PLTE) Model 16 Model of Coronal Equilibrium (MCE) 20 Collisional-Radiative Model (CRM) 21 Optical Spectrum and Plasma Parameters 22 References 25

1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4

2

2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 2.4 2.5 2.5.1

7

Basic Concepts and Parameters Associated with the Emission, Absorption and Scattering of Light by Plasma 27 Photometric Quantities. Remarks on Terminology 27 Spectral Line Proﬁle 31 Lorentz Broadening 32 Doppler Broadening 38 Joint Action of Natural, Doppler and Collision Broadening 41 Absorption in Lines 44 Emission in Lines. Optical Density Manifestations 46 Emission and Absorption in Continuous Spectrum 50 ff Bremsstrahlung Emission 52

VIII

Contents

2.5.2 2.5.3 2.5.4 2.5.5

2.6 2.6.1 2.6.2

3

3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.1.4 3.4.1.5 3.4.2

ff Bremsstrahlung Absorption 54 fb Recombination Emission 55 Absorption Cross Section in bf Photoionization 57 Emission and Absorption of Radiation in the Case of Joint Action of the ff Bremsstrahlung and fb Recombination Mechanisms 58 Scattering of Light 60 Thomson Scattering on a Free Electron 60 Scattering on a Bound Electron 63 References 63 Emission, Absorption and Scattering Techniques for Determining the Densities of Particles in Discrete Energy States 67 Emission Techniques 67 Identiﬁcation of Spectra 67 Absolute Measurements 68 Emission of Extended Inhomogeneous Sources 71 Absorption Techniques Using Classical Emitters 75 Absorption Against the Background of Continuous Spectrum 75 Line Absorption 78 Self-Absorption of Multiplet Lines 83

Absorption Spectroscopy Using Tunable and Broadband Lasers 84 On the Advantages of Laser Sources Over Their Classical Counterparts in Direct Absorption Measurements 84 On the Noise Limitation of Sensitivity 86 Diode Laser Spectroscopy in the IR Region 88 Nonstationary Coherent Effects in Absorption Measurements 92 Use of the Classical Multipass Absorption Cells 95 Intracavity Absorption 95 Measuring Absorption from the Attenuation of Light with Time 99 Indirect Methods for Measuring Absorption of Laser Light 103 Induced Fluorescence 104 General Characteristic 104 Fluorescence Excitation by Continuous-Wave and Pulsed Laser Light 107 Induced Fluorescence Saturation and Decay 109 Induced Fluorescence Quenching and Taking Account of this Process 112 Restrictions Imposed by the Plasma’s Own Glow 118 Optogalvanic Spectroscopy 121

Contents

3.4.2.1 3.4.2.2 3.5 3.5.1 3.5.2 3.5.3 3.5.4

IX

The Use of the Optogalvanic Effect to Measure Light Absorption in Plasma 121 High-Resolution Optogalvanic Spectroscopy 124 Multiphoton Processes. Raman Scattering 129 Two-Photon Absorption 130 Spontaneous Raman Scattering 133 Stimulated Raman Scattering 135 Coherent Anti-Stokes Scattering 137 References 143

4

Intensities in Spectra and Plasma Energy Distribution in the Internal and Translational Degrees of Freedom of Atoms and Molecules 147

4.1

Doppler Broadening, Velocity Distribution of Particles, Neutral Gas Temperature 147 Remarks on the Processing of Line Proﬁles 148 Registered and True Proﬁles 148 Predominantly Doppler Broadening Regions 149 Recovery of the Form of the Velocity Distribution of Particles 150 Examples of Abnormal Doppler Broadening and Nonequilibrium Velocity Distributions of Neutral Particles in Plasma 151 Excitation and Relaxation of Atoms and Molecules with Nonequilibrium Velocity in Interactions with Heavy Particles 154 Source Function 154 Relaxation of the Average Kinetic Energy of Particles with a Finite Lifetime 155 Relaxation of the Form of the Velocity Distributions of Particles in the Case of Large Deviations from Equilibrium and Finite Lifetime 159 On the Determination of the Gas Temperature from the Doppler Broadening of the Lines Emitted by Atoms and Molecules Excited by Electrons 161 Spectroscopic Manifestations of the Motion of Ions in Plasma 164 Distribution of Molecules Among Rotational Levels 167 On the Isolation of the Boltzmann Ensembles in the Bound State System of Particles 167 Distributions of Molecules Among Rotational Levels in an Electronic State with a Long Lifetime 170 Electron Impact Excitation of the Electronic–Vibrational– Rotational (EVR) Levels of Molecules 174 Observations and General Considerations 174

4.1.1 4.1.1.1 4.1.1.2 4.1.1.3 4.1.2 4.1.3 4.1.3.1 4.1.3.2 4.1.3.3

4.1.4

4.1.5 4.2 4.2.1 4.2.2 4.2.3 4.2.3.1

X

Contents

4.2.3.2 4.2.4 4.2.4.1 4.2.4.2 4.2.5 4.2.5.1 4.2.5.2 4.2.5.3 4.3 4.3.1 4.3.1.1 4.3.1.2 4.3.1.3 4.3.2

4.3.3 4.3.4 4.3.5 4.3.6 4.4

5

5.1 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1

Experimental Determination of the Electron-Impact-Induced Changes in the Rotational States of Molecules in Plasma 177 Excitation of EVR Levels by Heavy Particles 181 OH Radical. Violet Bands 181 N2 Molecules. Second Positive System 183 On Gas Temperature Measurements in the Presence of Parallel Molecular Rotation Excitation Channels 184 Extension of the Form of Distribution of the Hot Molecules to the Region of Low Rotational Levels 185 Spectral Resolution 185 Effect of the Conditions Occuring in Plasma on the Rotational Temperature of the Hot Group 188 Line Intensities in the Vibrational Structure of Spectra and Distributions of Molecules Among Vibrational Levels 191 Elements of Vibrational Kinetics. Vibrational Energy and Temperature 191 Harmonic Oscillator Approximation 192 Effect of Anharmonicity 194 Diatomic Molecular Mixture and Polyatomic Molecules 197 Vibrational Temperature and Distribution Measurements by Absorption Spectroscopy Techniques 199 Emission Methods in the IR Region of the Spectrum 206 Combinations of Emission and Absorption Techniques. Spectrum Inversion 211 Raman Scattering 216 Determination of the Vibrational Temperatures of Molecules in the Electronic Ground States from Electronic Transition Spectra Distribution of Particles Among Electronic Levels 224 References 227 Measuring Concentrations of Atoms and Molecules General 235

235

Determining Atomic Concentrations by Absorption Techniques 237 Neutral Unexcited Atoms 237 Metastable Atoms 249 Low-Multiplicity Positive Ions 266 Determination of Molecular Concentration by the Absorption Method 268 Probabilities of Optical Transitions in Diatomic Molecules 269

220

Contents

5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 5.4 5.5 5.5.1 5.5.2

Determination of Diatomic Molecular Concentrations from Absorption on Electronic Spectrum Lines 272 Determination of Molecular Concentration from Absorption in Vibrational–Rotational Spectra 278 Absorption of Radiation by Diatomic Molecules in Metastable Electronic States 281 Absorption of IR Radiation by Polyatomic Molecules 281 Absorption of Radiation by Molecular Ions 284 Actinometric Methods 288 Negative Ions 297 Concentration Measurements 298 Absorption of Light by the H− Ions in Hydrogen LTE Plasma References 303

XI

301

6

Spectral Methods of Determining Electronic and Magnetic Fields in Plasma 307

6.1

Determination of Electric Fields from the Spontaneous Emission of Radiation by Atoms in Plasma 312 Hydrogen-Like Atoms 312 Non-Hydrogen-Like Atoms 318 Laser Stark Spectroscopy 322 Stark Spectroscopy of Atoms 323 Laser-Induced Fluorescence of Polar Molecules in Electric Field 329 Multiphoton Excitation of Atoms 334 Coherent Four-Wave Stark Scattering Spectroscopy 337 Magnetic Field Investigations 342 Measurements Based on the Faraday Effect 342 Spectral Methods 343 References 346

6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.3.1 6.3.2

7

7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.4 7.5

Determination of the Parameters of the Electronic Component of Plasma 351 Interferometry 351 Stark Broadening of Spectral Lines 356 General 356 Plasma Microﬁelds 357 Linear Stark Effect 358 Quadratic Stark Effect 364 Truncation of Spectral Series of Hydrogen-Like Atoms 367 Intensities in Continuous Spectrum 371 Scattering of Light on Electrons 374

XII

Contents

7.5.1 7.5.2 7.5.3 7.5.4 7.5.5 7.6

Scattering of Light by Randomly Moving Electrons (Thomson Scattering) 375 Manifestation Regions of the Thomson and Collective Scattering Mechanisms 377 Scattered Spectrum and Plasma Parameters (Direct Problem) 379 Determination of Plasma Parameters from Scattered Spectra (Inverse Problem) 381 Limitations of the Method, Sensitivity and Examples 385 Some Remarks on Measurements from Intensities in Line and Band Spectra 391 References 393 397

8

Some Information on Spectroscopy Techniques

8.1 8.1.1 8.1.2 8.1.3 8.1.3.1 8.1.3.2 8.2 8.2.1 8.2.2 8.2.3 8.2.3.1 8.2.3.2

Characteristics of Optical Materials. Main Relations 398 Reﬂection at an Interface 398 Dispersion of the Optical Properties of Materials 399 Transmission and Reﬂection of Thin Films 400 Metal Films 400 Dielectric Films 402 Spectral Instruments 406 Slit Instruments 409 Interferometers 414 Spectral Instruments with Interference Modulation 425 Fourier(-Transform) Spectrometers 425 Interference Spectrometers with Selective Amplitude Modulation (ISSAM) 430 Raster Spectrometers 431 Acousto-optic Spectrometers 433 Gas-Discharge Light Sources 439 Illumination Engineering Quantities 439 Gas Discharges in an Envelope (Lamps) 441 Continuous-Discharge Lamps 441 Pulsed-Discharge Lamps 448 Open Light Sources 456 Continuous-Discharge Sources 456 Pulsed-Discharge Sources 456 Photodetectors 460 Parameters 461 Sensitivity 461 Noise 462 Effective and Ultimate Sensitivity 463 Inertia 464

8.2.4 8.2.5 8.3 8.3.1 8.3.2 8.3.2.1 8.3.2.2 8.3.3 8.3.3.1 8.3.3.2 8.4 8.4.1 8.4.1.1 8.4.1.2 8.4.1.3 8.4.1.4

Contents

8.4.2 8.4.2.1 8.4.2.2 8.4.2.3 8.4.2.4 8.4.2.5 8.4.3 8.4.3.1 8.4.3.2 8.4.3.3 8.4.3.4

Main Types of Single-Element Detectors 464 Thermal Detectors 464 Photoelectric (Quantum, Photonic) Detectors with Extrinsic Photoeffect 466 Photoelectric Detectors with Intrinsic Photoeffect 470 Photoemulsion 471 Comparative Characteristics of Single-Element Detectors Multielement and Distributed Photodetectors 477 Spatial Resolution 477 Photographic Detectors 478 Image Converter and Intensiﬁer Tubes 479 Charge-Coupled Detectors 480 References 484

Appendix A Statistical Weights and Statistical Sums

A.1 A.2 A.3 A.4 A.4.1 A.4.2

473

487

Statistical Weight of Energy Levels in Atoms and Ions 487 Statistical Weight of Electronic States in Molecules 488 Statistical Weight of Vibrational Levels of Molecules 488 Statistical Weight of Rotational Levels of Molecules 489 Statistical Sum of Atoms and Ions 492 Statistical Sum of Molecules 492 References 495

Appendix B Conversion of Quantities Used to Describe Optical Transition Probabilities in Line Spectra 497 References 497 Appendix C Two-Photon Absorption Cross Sections for Some Atoms and Molecules in the Ground State 499 References 503 Appendix D Information on Some Diatomic Molecules for the Identiﬁcation and Processing of Low-Temperature Plasma Spectra 505

D.1

D.1.1 D.1.2 D.2

Brief Information from Molecular Spectroscopy – Designations of States and Transitions, Coupling Types, Selection Rules, General Spectrum Structure 505 General Rules 508 More Particular Rules 509 Nitrogen N2 , N2+ 513

XIII

XIV

Contents

D.2.1 D.2.2 D.2.3 D.2.3.1 D.2.3.2 D.2.4 D.2.4.1 D.2.4.2 D.2.5 D.2.6 D.2.6.1 D.3 D.3.1 D.3.2 D.3.3 D.3.3.1 D.3.3.2 D.4 D.4.1 D.4.2 D.4.3 D.4.4 D.4.4.1 D.4.4.2 D.4.5 D.4.5.1 D.4.5.2 D.4.6 D.4.6.1 D.4.6.2 D.5 D.5.1 D.5.2 D.5.3 D.5.3.1 D.5.3.2 D.6 D.6.1 D.6.2 D.6.3 D.6.3.1

Electronic States, Electronic Transition Systems (Bands) 513 Molecular Constants of the Ground and Combining States 513 Second Positive (2+ ) System 513 Vibrational Structure of the C3 Π(v )–B3 Π(v ) Transition 517 Rotational Structure 517 First Positive (1+ ) System 519 Vibrational Structure of the B3 Π g (v )–A3 Σ+ u ( v ) Transition 519 Rotational Structure 519 First Negative (1− ) System 522 2 + Vibrational Structure of the B2 Σ+ u ( v )–X Σ g ( v ) Transition 522 Rotational Structure 522 Carbon Oxide CO 526 Electronic States, Electronic Transitions 526 Molecular Constants of the Ground and Combining States 526 ˚ ¨ Bands System B1 Σ+ –A1 Π 526 Angstr om Vibrational Structure 526 Rotational Structure of the B1 Σ+ –A1 Π Bands 526 Hydrogen H2 and Deuterium D2 528 Electronic States, Electronic Transitions 528 Molecular Constants of the Ground and Combining States 528 Ortho- and Para-Modiﬁcations 529 Fulcher-α Bands System d3 Πu –a3 Σ+ 530 g Vibrational Structure 531 Rotational Structure 531 I1 Π g –B1 Σ+ u Transition 533 Vibrational Structure 533 Rotational Structure 535 1 + G1 Σ+ g –B Σu Transition 537 Vibrational Structure 537 Rotational Structure 537 Nitrogen Oxide NO 541 Electronic States, Electronic Transitions 541 Molecular Constants of the Ground and Combining States 541 γ System (195–340 nm) 541 Vibrational Structure 541 Rotational Structure 542 Cyanogen CN 543 Electronic States, Electronic Transitions 543 Molecular Constants of the Ground and Combining States 545 Violet System 546 Vibrational Structure 546

Contents

D.6.3.2 D.7 D.7.1 D.7.2 D.7.3 D.7.3.1 D.7.3.2 D.8 D.8.1 D.8.2 D.8.3 D.8.3.1 D.8.4 D.9 D.9.1 D.9.2 D.9.2.1

Rotational Structure 546 Carbon Radical C2 548 Electronic States, Electronic Transitions 548 Molecular Constants of States 548 Swan Bands System 548 Vibrational Structure 548 Rotational Structure 550 CH Radical 551 Electronic States, Electronic Transitions 551 Molecular Constants of the Ground and Combining States 553 B2 Σ− –X2 Π Transition 555 Rotational Structure 556 C2 Σ+ –X 2 Π Transition 556 Hydroxyl Radical OH 558 Electronic States, Electronic Transitions 558 Molecular Constants of the Ground and Combining States 560 Rotational Structure 560 References 566

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra of Diatomic Molecules 569 E.1 Singlet Transitions 570

E.1.1 E.1.2 E.2 E.2.1 E.2.2 E.3 E.3.1 E.3.1.1 E.3.2 E.3.3 E.4 E.5

ΔΛ = 0 Transitions 570 ΔΛ = ±1 Transitions 570 Doublet Transitions 570 2 X–2 X, ΔΛ = 0 Transitions 571 2 X–2 Y, ΔΛ = ±1 Transitions 572 Triplet Transitions 573 3 X–3 X, ΔΛ = 0 Transitions 574 Dipole-Forbidden Branches 576 3 X–3 Y, ΔΛ = ±1 Transitions 577 3 –3 Δ, ΔΛ = ±2 Transitions 579 ∑ Remarks on the Normalization of Rotational Line Intensity Factors 580 On Symbolic Notation 581 References 582 1 X–1 X, 1 X–1 Y,

XV

XVI

Contents

Appendix F Measurement of the Absolute Populations of Excited Atoms by Classical Spectroscopy Techniques 583

Yu. B. Golubovskii References 595 Appendix G General Information for Plasma Spectroscopy Problems 597 G.1 Physical Constants 597 G.2 Atomic Values 598

G.3 G.4 G.5 G.6

Correspondence between Spectral and Traditional Energy Measurement Units 598 Electrical Units 599 Units from Molecular Kinetics 599 Quantities from Gas-Discharge Physics 600 References 600 Index 603

Appendix H Optical Constants of Materials∗ H.1 Transmission 611 H.2 Refractive Indices 635 H.3 Reﬂection 638 References 651

∗

611

Appendix H is availible on: www.wiley-vch.de/publish/en/books/bysubjectEE00/ISBN3-527-40778-2

XVII

Preface Low-temperature plasma is a widespread state of matter. Understanding its properties is important not only from purely scientiﬁc standpoint in terms of studying natural space and atmospheric phenomena, but also in connection with the technical problems involved in the development of plasma and luminescent light sources, current switches, electronic tubes, welding devices, analytical instruments and systems, a wide range of power plants, and so on. The existence domains of plasma and its parameter ranges in both nature and technical devices are surprisingly wide. They are, in fact, only restricted by the very notion of low-temperature plasma as a medium containing an incompletely ionized gas phase, the object as a whole being electrically neutral. The term “low-temperature” refers to the fact that incomplete ionization of the plasma-forming medium is possible with the mean energies of motion of the various species of particles (in the steady-state case) being lower than, or on the order of, the ﬁrst ionization potential. Such a situation is typical, for example, of various kinds of ﬂame. In widespread use are devices based on various types of gas discharge. As far as the totality of processes occurring therein is concerned, discharges in monoatomic gases are the preferred and simplest form, though even such remain far from being completely understood. The presence of molecular components increases the number of the internal degrees of freedom of the plasma particles, this leads to their multiform interactions involving redistribution of energy within and among them, and results in transformations making the atomic and molecular composition of the plasma fairly complex and the plasma itself chemically active. Interest in studying such complex systems regularly receives fresh stimuli. The general trend in the evolution of current plasma investigations and plasma technologies is manifest in the advancement into domains of essentially nonequilibrium states. These are realized in plasma media differing in nature.

XVIII

Preface

Rapidly evolving is physical electronics, including quantum electronics. There is a substantial class of gas-discharge lasers operating by transitions in atoms and molecules, including unstable atoms and molecules directly generated in plasma [1, 2]. Many chemical processes occur in plasma, both in the gas phase and in plasma-surface interactions. As such, an important branch of chemistry – plasma chemistry – has formed [3–5]. Current achievements in this area have been achieved on account of the possibility of concentrating the energy deposited in plasma in the desired degrees of freedom of the particles of interest and carrying out selective reactions on a characteristic time scale comparable with the relaxation times of the excited internal motions. The plasma formed upon the rapid ﬂow of gases around bodies entering the atmosphere is also characterized by time-variant “responses” of the translational motion of particles and their internal motions to perturbations and so forth. The many years of worldwide experience in low-temperature plasma investigations has revealed so vast a ﬁeld of methods, phenomena and applications that cataloguing the vast number of the monographs devoted to this subject is a job in itself. Additionally, a great many national and international conferences are regularly held on aspects of the topic. The recent attempt to represent the current state of the subject has manifested itself in the form of the multivolume Encyclopedia of LowTemperature Plasma [6] whose publication is as yet unﬁnished. Generally speaking, the description of nonequilibrium plasma should be based on the physical, energy level kinetics whereby identical (from the standpoint of the classical chemistry) particles in different quantum states should be discriminated by their properties. But such an approach proves impossible to realize, to be anywhere complete or work well even for atomic plasma, which forces one to resort to simpliﬁed models [6], and with molecular plasma the difﬁculties multiply. It is, therefore, natural that the development of low-temperature plasma physics and technology will be largely governed, both nowadays and in the near future, by the capabilities of experimental investigations, followed by theoretical analyses, generalizations and predictions. In this situation, the problem of in-depth diagnostics is especially pertinent. On the other hand, the development of diagnostics means inevitably comes up against the need to study the entire aggregate of the elementary processes occurring in plasma and to choose adequate models substantiating the diagnostic techniques. In other words, the evolution of the notions of the properties of plasma and the development of the methods for its experimental investigations are inseparably connected.

Preface

An important, if not central, position in the group of diagnostic methods is occupied by spectroscopy. It is not out of place to recall in this connection that it was precisely the studies into the spectroscopy of ﬂames conducted as far back as the century before last that stimulated the creation of the quantum theory. The results of this theory, in turn, decisively inﬂuenced the development of optics and spectroscopy in general, and as applied to plasma in particular, so that the development of plasma physics and technology developed throughout the twentieth century hand in hand with the creation of the physical fundamentals of spectroscopy and spectroscopic techniques. This also explains the fact that the fundamental principles in practically all books on spectroscopy and the structure of atoms and molecules (see, e.g., [8–15]) are to a large measure illustrated by examples from the spectroscopy of low-temperature plasma. Plasma spectroscopy has long since become an independent avenue of scientiﬁc exploration, within whose framework an intensive volume of data is amassed, which necessitates their periodical generalization. This occurs in various forms. The most general and established points are set forth in popular scientiﬁc literature (e.g., [16]) and included in textbooks of general physics, optics and plasma physics (e.g., [17–20]), monographs (e.g., [3, 4, 7, 21–24]) and topical collections of papers (e.g., [25–27]). It turns out, however, that the current status of a number of divisions of plasma spectroscopy under discussion in the literature, and this for a fairly perceptible period of time at that, have been far from sufﬁciently well presented in generalized form. This is true in the ﬁrst place of the dynamically developing studies on the spectroscopy of nonequilibrium plasma in general and molecular plasma in particular. Inadequately systematized are the practice and prospects of laser techniques, though individual aspects of these problems have been considered in the pertinent, rather small, sections of books [3, 4, 28] and in some collections of papers (see, e.g., [3, 4, 6, 25, 28–30]). And the dedicated monographs (e.g., [31– 33]), though they have lost none of their value over the years that passed since their publication, embrace only the problems relating to the initial stage of application of lasers to plasma investigations. The disjointed character of the materials makes it difﬁcult for one to correlate the capabilities of the classical and laser spectral diagnostic techniques. The objective of the present book is to partly ﬁll these gaps as regards the application of the up-to-date experimental spectroscopic techniques to nonequilibrium atomic-molecular low-temperature plasma. Individual general methodological problems are considered only so far as is necessary to understand the position and capabilities of spectroscopy in plasma investigations, deﬁne the basic notions, obtain information

XIX

XX

Preface

from spectroscopic measurements, and make its physical interpretation. The book also gleans factual data necessary or convenient for practical plasma spectroscopy. The book is supplied with a detailed table of contents, which obviates the need to additionally explain its structure. I express my profound gratitude to my colleagues at the Department of Optics of Low-Temperature Plasma at the P. N. Lebedev Physical Institute of the Russian Academy of Sciences, namely, S. N. Tskhai for his assistance in the preparation of the manuscript and I. I. Sobelman, V. N. Kolesnikov, and S. Yu. Savinov for their examination of the manuscript and useful comments. I would also like to note that Sects. 6.1, 6.2, and 7.2 were written using review materials prepared in collaboration with V. P. Gavrilenko and S. N. Tskhai, for which I also render thanks. I also give my thanks to Yu. B. Golubovskii who wrote Appendix Appendix F to this book. I am very grateful to my teacher Prof. N. N. Sobolev (1914–1995), one of the founders of plasma spectroscopy, under whose inﬂuence my scientiﬁc interests were being formed. V. N. Ochkin

References

1 A.M. Prokhorov, Ed. Handbook of Lasers (in Russian). Moscow: Sov. Radio, 1–2 (1978). 2 Ch. Rodes, Ed. Excimer Lasers. Heidelberg: Springer Verlag (1979). 3 V.D. Rusanov and A.A. Fridman. Physics of Chemically Active Plasma (in Russian). Moscow: Nauka (1984). 4 D.I. Solovetsky. Chemical Reaction Mechanisms in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1980). 5 Yu.A. Kolbanovsky. Analysis of the Principal Notions and Postulates of the Arrhenius Chemical Kinetics (in Russian). Preprint INKhS No. 1. Moscow (1971); L.S. Polak. Principal Propositions and Statements of the Classical Chemical Kinetics (in Russian), ibid., p. 8. 6 V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), I–IV. Moscow: Nauka (2000).

7 L.M. Biberman, V.S. Vorobyev, and I.T. Yakubov. Kinetics of Nonequilibrium Low-Temperature Plasma (in Russian). Moscow: Nauka (1982). 8 G. Herzberg. Atomic Spectra and Atomic Structure. New York (1944). 9 G. Herzberg. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules, 2nd ed. N.Y.: D. van Nostrand, (1951). 10 G. Herzberg. Spectra of Diatomic Molecules. New York (1939). 11 G. Herzberg. The Spectra and Structure of Simple Free Radicals. London: Cornell University Press (1971). 12 S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). 13 I.I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Fizmatgiz (1963).

Preface 14 A. Mitchel and M. Zemansky. Resonance Radiation and Excited Atoms (in Russian). Moscow: GTTI (1937). 15 M.A. El’yashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 16 L.A. Artsimovich. Elementary Plasma Physics (in Russian). Moscow: Gosatomizdat (1963). 17 N.D. Papaleksi. A Course of Physics (in Russian), V. 2. Moscow: OGIZ (1943). 18 N.A. Kaptsov. Gas-Discharge Physics (in Russian). Moscow: P OGIZ (1947). 19 R.V. Pol. Optics and Atomic Physics (Russian translation), Moscow, Nauka (1966), 552 p. 20 Yu. P. Raizer. Gas-Discharge Physics (in Russian). Moscow: Nauka (1982). 21 H.R. Grim. Plasma Spectroscopy. N.Y.: Mc Graw Hill (1964). 22 H.R. Grim. Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). 23 J. Bekeﬁ. Radiative Processes in Plasma (Russian translation), Moscow, Mir Publishers (1978), 438 p. 24 A. Hohshim, Ed. Kinetic Processes in Gases and plasma (in Russian). Moscow: Atomizdat (1971).

25 W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). 26 R. Huddlestone and S. Leonard. Plasma Diagnostic Techniques. New York (1965). 27 S.E. Frish, Ed. Spectroscopy of GasDischarge Plasma (in Russian). Leningrad: Nauka (1970). 28 V.K. Zhivotov, V.D. Rusanov, and A.A. Fridman. Diagnostics of Nonequilibrium Chemically Active Plasma (in Russian), Moscow, Energoatomizdat (1985), 216 p. 29 V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460 (2002). 30 N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1985). 31 L.A. Dushin and O. S. Pavlyuchenko. Laser Investigations of Plasma (in Russian). Moscow: Atomizdat (1968). 32 A.N. Zaidel and G. V. Ostrovskaya. Laser Methods for Plasma Investigations (in Russian). Leningrad: Nauka (1977). 33 L.N. Pyatnitsky. Laser Plasma Diagnostics (in Russian). Moscow: Atomizdat (1976).

XXI

1

1

Plasma as an Object of Spectroscopy

General Notions

1.1 The Concept of Low-Temperature Plasma. Diagnostics Problems

The optical and spectroscopic methods used in plasma investigations are based on the general laws governing the interaction between electromagnetic radiation and matter; consequently, many plasma diagnostics problems can be quite similar to those arising in the study of gases. There are, however, important differences. Plasma is speciﬁc; that is, it contains charged particles in quantities sufﬁcient to make the given element of space quasineutral. This condition can be considered a deﬁnition of plasma, according to which the size of a plasma object should exceed what is known as the Debye screening radius rD = (kB Te Ti /4πeei (ne Te + ni Ti )) /2 , 1

(1.1)

where ne , e are the concentration and charge of electrons (in cm−3 and CGS units), ni , ei , the same for ions, Te , Ti are the temperatures of the electrons and ions (in kelvins), and kB = 1.38 × 10−16 erg/K is the Boltzmann constant. Figure 1.1 shows dependence (1.1) subject to the condition that Te = Ti and kB Te is expressed in electron-volts. The physical mechanism responsible for this quasineutrality is the expulsion of the ﬂuctuation-induced excess charge to the periphery of the plasma object by the electrostatic forces associated with this charge. Each charged particle interacts with the collection of the other charged particles, the interaction energy (assuming that the ions are singly charged) being w ∼ e2 /rD . Further, it follows from the quasineutrality requirement that the given element of space can be regarded as a plasma object only on a time scale longer than the characteristic time τp determined by

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1 Plasma as an Object of Spectroscopy

Figure 1.1 Debye radius and electron density in plasma at various temperatures.

the plasma (Langmuir) frequency ωL : τp−1 < (1/2π)ωL ∼ 104 ne/2 , 1

Hz (with ne expressed in cm−3 ).

(1.2)

During the course of times t > τp , electrons execute numerous oscillations about their equilibrium position, and so, although the medium averages out as quasineutral, plasma is characterized by the presence of ﬂuctuating ‘statistical’ electric ﬁelds. From what has been said above one can clearly see the difference between a heated gas and plasma. The presence of a noticeable number of charged particles in plasma leads to their interaction both with one another and with uncharged particles, thus, causing a wide range of various kinds of excitation. Plasma is classiﬁed by its existence domain conditions as hot (completely ionized), strongly ionized, and weakly ionized [1, 2]. Hot plasma consists of electrons and nuclei of the elements of the plasma-forming gas. If we use the symbol ι to denote the ratio between the density of electrons and the density N of neutral particles in the parent plasma-forming gas, that is, ι = ne /N, then in hot plasma ι > 1. Strongly ionized plasma contains in addition to electrons singly and multiply charged ions, and so ι ≈ 1. In weakly ionized plasma, the number of neutral particles ex-

1.1 The Concept of Low-Temperature Plasma

ceeds that of charged ones, so that ι < 1. These, of course, are obviously not strict limits, and ι ≈ 10−2 –10−4 is conventionally adopted to discriminate between strongly and weakly ionized plasma [1]. Distinction can also be drawn on the basis of the character of interaction between particles in plasma. In hot plasma, bound electronic states do not manifest themselves in interactions, the plasma-forming gas particles are practically deprived of their individual properties, and the plasma emission spectrum is continuous. In strongly ionized plasma, it is the long-range Coulomb forces, independent of the individual properties of atoms and ions, that govern the collective effects and transfer processes, but the internal structure of heavy particles (atoms, ions) can be seen in the emission spectrum and collision kinetics. In weakly ionized plasma, deﬁning is the processes of interaction of neutral particles with one another and with their charged counterparts. These processes largely govern the transfer processes, collision kinetics, and radiative properties of plasma. Let us deﬁne the notion of low-temperature plasma as a uniﬁed concept for strongly and weakly ionized plasma, wherein the manifestations of the individual properties and energy structure of heavy particles are essential. The ensuing restriction on the existence domain of low-temperature plasma is that the energies of free motion of particles, as well as the energies of their internal degrees of freedom, should not perceptibly exceed the ﬁrst ionization potential. The presence of molecules in plasma has a substantial effect on the plasma’s properties. Molecules can be contained as admixtures in the plasma-forming monoatomic gases, or they can be formed from atoms or ions in the plasma. Even if the parent gas consists entirely of molecules, their compounds and fragments are formed in plasma, including oneatomic particles. Therefore low-temperature plasma always contains both atoms and molecules. Nevertheless, plasma is customarily referred to as atomic if the proportion of molecules in the parent gas does not exceed ca. 10−3 (see, e.g. [1]). But in a general sense, low-temperature plasma is atomic-molecular. Let us specify one more parameter of importance in limiting the range of plasma objects considered in this book, namely γ. The last quantity separates ideal plasma from its nonideal counterpart and is expressed as the ratio between the mean potential interaction energy of the plasma particles, Ep , and their mean thermal energy Ek , that is, γ = Ep /Ek . We will deal with ideal plasma wherein γ 1. This means that free neutral and charged plasma particles (those not bonded into atomic-molecular structures) do not interact while in motion, as is the case with the ‘ordinary’ ideal gas. Strongly ionized plasma is characterized by Coulomb

3

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1 Plasma as an Object of Spectroscopy

Figure 1.2 Degree of ionization and electron concentration in some plasma objects. ECR – electron cyclotron resonance; ICP – inductively coupled plasma.

interactions, and deviations from the ideal become apparent, for example, in thermal plasma with a temperature of 1 eV at concentrations ne , ni > 1017 –1018 cm−3 . The decisive role in weakly ionized plasma is played by charge-dipole interactions, and the ideality condition can be violated at neural particle concentrations N > 1019 –1021 cm−3 . The present book is devoted to the spectroscopy of ideal low-temperature plasma. For the sake of brevity, the term ‘plasma’ as used below has the limited meaning as described above, unless otherwise speciﬁed. The limitations on the subject of study notwithstanding, the range of objects pertaining to this concept remains, as before, highly diverse. Figure 1.2 illustrates the properties of the electronic component of the plasma of some laboratory and natural objects [1, 3, 4]. The objects are arranged on a plane where the abscissa axis is the electron concentration and the ordinate axis, the degree of ionization.

1.1 The Concept of Low-Temperature Plasma

As distinct from the contact (probe) methods, the capabilities of spectral and optical methods are very wide in respect of both the range of parameters being measured and the extent of localization of nonperturbative measurements – from cosmic to ca. λ3 , where λ is the radiation wavelength. At the same time, they are capable of extremely high time resolution (up to ca. 10−14 s with modern laser spectroscopic techniques). This also explains the colossal variety of optical and spectral plasma diagnostics objects, capabilities and techniques. Furthermore, it is necessary to take account of the fact that plasma in both natural and technical objects always borders on the regions of space charge and neutral plasma-forming gas that are essential for the maintenance of plasma and form a uniﬁed and, as a rule, open system. The local character of the optical techniques also make it possible to diagnose structures corresponding to such systems. Typical plasma diagnostics problems include, the determination of the quantitative and qualitative chemical composition of plasma; deduction of both the total energy balance and energy distributions among various plasma particles and their states in quantum and continuous spectra; ﬁnding the parameters of charged particles; establishment of the structure of electric and magnetic ﬁelds. Spectroscopy plays an important part in investigations into those elementary processes whose realization in many applications is the purpose of the development of plasma devices (light generation, plasma chemical processes, etc.). The main advantage of spectroscopy, also in terms of its use in plasma investigations, is it’s high selectivity with respect to various energy states of particles. The extraordinarily highly developed energy-state structure of plasma becomes apparent in its optical spectra. But the ‘cost’ for so high an information capacity from spectroscopic measurements is the fact that it is precisely the interpretation of the experimental results that proves to be the most important, and frequently the most difﬁcult, aspect of spectroscopic investigations. The scale of these difﬁculties depends largely on the kind of equilibrium in each particular plasma object. In conditions of strong excitation of individual particles and their ensembles, typical of plasma, the number of parameters adequately describing the state of both the object itself and its optical spectrum is determined by the degree of deviation of the plasma from the state of thermodynamic equilibrium.

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1.2 Equilibrium Plasma

Plasma existing in a state of thermodynamic equilibrium is called equilibrium plasma. No matter how plasma is produced and sustained, the notion of temperature, T, has one and the same meaning in describing the energy distribution of particles in continuous and discrete energy spectra, the relation between the densities of neutral and charged particles of different chemical species and light emission and absorption spectra. By virtue of such universality, the number of equations describing a thermodynamically equilibrium (TE) plasma is comparatively low. No plasma fully ﬁtting this deﬁnition can exist in nature, in engineering, or in a laboratory. But its description is important, as it enables one to introduce fundamental concepts and relations and, based on the analysis of the conditions of their satisfaction or violation, to construct various models as approximations of the TE plasma [2, 5, 6]. 1.2.1 Energy Distribution of Particles

The distribution of the densities N of particles of species s in the magnitude of the velocity v in a continuous spectrum is given by the Maxwell distribution function formula f (v) = dN/( Ndv); dN = N f (v)dv. The distribution function f (v) expresses the proportion of the total density N of particles that falls within the velocity interval dv: f (v) = 4π

M 2πkB T

3/2

Mv2 v exp − 2kB T 2

,

(1.3)

where M is the particle mass. For the distribution of the velocity components along a speciﬁed direction z in an isotropic object, we have f (vz ) =

M 2πkB T

1/2

Mv2z . exp − 2kB T

(1.4)

The distribution of the particles among the discrete levels of the energy spectrum is described by the Boltzmann formula gk ΔEkl Nk , (1.5) = exp − Nl gl kB T where ΔEkl is the energy difference between the levels k and l, and gk and gl are the statistical weights of the levels. Formula (1.5) gives the relative population densities of the levels k and l. The thermodynamic

1.2 Equilibrium Plasma

equilibrium condition presumes that (1.5) is applicable to all the levels k and l and is universal for all the bound states of the system. Subject to these conditions, the density of particles on a speciﬁed level can also be expressed in terms of the total density N of particles of the given species, provided that ΔEk0 is reckoned from the ground-state level l = 0: Nk gk ΔEk0 . (1.6) = exp − N Qin kB T Here Qin is what is known as the internal statistical sum (sum over the bound states) of the particle that determines the relation between the densities of the excited and unexcited particles: Qin =

1 Σg N , N k k k

which, given equilibrium condition (1.5), corresponds to ΔEk0 . Qin = Σ gk exp − kB T k

(1.7)

(1.8)

Energy is calculated from the position of the lowest level of the bound state, which requires special attention in the case of molecules where the lower level fails to coincide with the minimum of the potential energy curve. Knowledge of statistical weights and statistical sums (sums over states) is necessary to solve many problems associated with the thermodynamics and statistics of various kinds of systems. A typical problem in spectroscopy in general, and the spectroscopy of plasma in particular, is to ﬁnd the total density N of particles from the measured density Nk of a limited number of quantum states. This is done using (1.6), and requires knowledge of the statistical sum. Finding statistical sums for polyatomic molecules is a problem, often a rather involved one, in its own right. More information about the statistical weights and sums frequently used in spectroscopy can be found in Appendix A. 1.2.2 Law of Mass Action. Neutral and Charged Particle Densities

The absorption and liberation of energy in plasma make its composition different from that of the parent plasma-forming gas [7, 8]. As a result, chemical changes take place in the parent plasma-forming gas, and charged particles emerge. The analysis of the composition of plasma under equilibrium conditions can be based on the well-known law of mass action.

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Let particles in a system undergo a series of transformations from Ar into Bq and vice versa during the course of n reactions:

∑ νrm Ar ↔ ∑ νqm Bq , r

(1.9)

q

where the stoichiometric coefﬁcients νrm and νqm show how many particles of species Ar and Bq are consumed or produced in each reaction event m, 1 ≤ m ≤ n. In a system of steady-state composition, the relationship between the particle densities is characterized by a set of the so-called equilibrium constants Km that depend on temperature: −1 νqm νrm = K m ( T ). (1.10) ∏ Bq ∏ [ Ar ] q

r

The square brackets here denote particle concentrations. Law of mass action, (1.10) (see e.g. [5, 8–10]), expresses the equilibrium constants in terms of the statistical sums of the particle-reagents and the energy defect of the reactions: −1 ΔEm νqm νrm , Q exp Km ( T ) = ∏ Qr ∏ q kB T r q (1.11) ΔEm = ∑ Erm − ∑ Eqm . r

q

The statistical sums Q entering into expression (1.11) are the total sums including both the translational and internal statistical sums, Q = Qtr Qin (see Appendix A). For the dissociation reaction of a polyatomic molecule, AB ↔ A+B, expression (1.11) reduces to [A][B] Q Q E = Kd ( T ) = A B exp − d [AB] QAB kB T (1.12) Q Q E 3 ≈ 1.89 × 1020 (μAB T ) /2 A,in B,in exp − d . QAB,in kB T Here Ed is the dissociation energy of the molecule AB and μAB its reduced mass in atomic units. An important case of reactions (1.9) for plasma, is the ionization reaction A↔ A+ + e , for which equilibrium constant (1.12) has the form 3 2QA+ ,in (2πme kB T ) /2 [A+ ] ne IA exp − = Ki ( T ) = [A] QA,in kB T h3 (1.13) Q + I 3 ≈ 4.7 × 1015 T /2 A ,in exp − A . QA,in kB T

1.2 Equilibrium Plasma

Here IA is the ionization energy (potential) of the atom (ion). For a free electron, Qe,in = 2 (two spin orientations). Formula (1.13) is known as the Saha formula. It is easy to generalize the Saha formula to the case of ions of higher multiplicity z by simple substitution of A(z−1) for A and A(z) for A+ . Note also, that in the plasma physics literature use is sometimes made of the so-called Saha-Boltzmann formula that combines formulas (1.13) and (1.6) to replace the total density N = [A] in the equilibrium constant by the population Nk = [A(k)] of an individual level (see, e.g. [11, p. 132]). Sometimes this proves convenient, namely, in plasma spectroscopy problems where the populations of the excited levels are easier to measure than the total density of particles of certain species. Let us use this last particular case of the law of mass action with respect to ionization to clarify the following circumstances: • Where several particle species are present, Saha relations (1.13) are written down for each of them. Ions of all types should be in equilibrium with the total concentration ne of the electrons produced as a result of ionization of all the particles. Owing to the strong (exponential) ionization potential dependence in expression (1.13), particles of lower ionization potential prove, as a rule (accurate to within the difference between the statistical sums involved), the main source of electrons and the corresponding ions. • Direct use of tabulated values of the ionization potential IA (Appendix H) in ionization equilibrium relations (1.13) is not entirely correct. This is due to the Coulomb interaction of charges in the Debye sphere (Section 1.1) with an energy of ca. e2 /rD and the manifestation of the ‘so-called’ ionization potential reduction effect in plasma. For the ideal plasma under consideration, this effect is, as a rule, not very strong, though at high temperatures it may cause deviations. The reduction of the ionization potential by an amount of ΔIA ≈ 1 eV is reached at electron concentrations of ca. 1019 cm−3 [5, 12], and in different gases this occurs at different temperatures. To illustrate, the authors of [7, p. 188] analyzed as an example air of normal density at a temperature of 105 K. Owing to the extreme heating of the air, the ionization potential reduction effect increased the degree of ionization by 14 %. To calculate the concentrations of neutral and charged particles, one should solve a system of equations for the law of mass action. Figure 1.3 presents examples of calculations of the neutral- and charged-particle composition of an atmospheric-pressure plasma in (a) air, (b) argon, (c) nitrogen, and (d) hydrogen (in particles per cm−3 ) [5] and in (e) water vapor (in partial pressures, dyn/cm−3 ) [12].

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Figure 1.3 Chemical composition of equilibrium plasma at atmospheric pressure as a function of temperature: (a) air; (b) argon; (c) nitrogen; (d) hydrogen; (e) water vapor.

1.2 Equilibrium Plasma

To characterize the depth of the change components of the plasmaforming gas undergo as a result of chemical transformations (charge transformations included), use is frequently made of the parameter α – the degree of dissociation, ionization, conversion, and so on – expressed as the ratio of the change in the initial concentration of the component of interest to the total concentration of the products formed. For example, if plasma contains predominantly singly charge ions, the degree of ionization of the component A is αi = ne /([A]+ne ). If αi 1, the concentration [A] at the same temperature is close to the initial concentration [A0 ], and, considering the equality of the electron and ion concentrations, it is evident from (1.13) that Ki = [ A ] < α i > 2 ;

< αi >∼ [A]− /2 exp{− IA /2kB T }. 1

(1.14)

The degree of ionization in the ideal plasma grows rapidly with temperature and more slowly as the gas pressure is reduced. 1.2.3 Heat Emission. Kirchhoff’s Law

Plasma has another important characteristic, namely, the energy (frequency) distribution function Nν of the photons propagating in it. Photons have a whole spin (bosons) and obey the Bose–Einstein statistics. If radiation and matter have one and the same temperature T, then, according to the Bose–Einstein statistics, −1 hν −1 . (1.15) Nν = exp kB T A direct consequence of this distribution is that plasma radiation under thermal equilibrium conditions has a continuous spectrum coinciding with that of black-body radiation (black light). The volume spectral radiation density is described by the Planck formula −1 hν 8πhν3 exp uν = −1 , (1.16) kB T c3 where ν is the radiation frequency. The integral volume radiation density is given by the Stefan–Boltzmann law u=

∞

uν dν = aT 4 ,

(1.17)

0

where a = 7.56× 10−15 ergcm−3 deg−4 . Black light is in equilibrium with the other forms of matter in plasma. This does not mean, however, that

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the total radiant energy density u is equal to the energy density of the other particles. To illustrate, the kinetic energy of particles moving in a volume of 1 cm3 is Ekin = (3/2)kB TN0 , where the Loschmidt number N0 = 2.7× 1019 cm−3 , and so the ratio Ekin /u = 0.74 × 1018 T −3 will become equal to unity at a temperature around a million Kelvin. It is precisely the quantities u and uν that are of prime interest in studying the processes occurring in plasma. But what is measured in actual experiments is not the internal light energy of the object under study, but the fraction p of the light ﬂux coming from the object that is picked up by the detecting instrument. The quantity I associated with p is called the radiant intensity of the object (for more detailed information about terminology, see Section 2.1). It is deﬁned as the absolute value of the energy ﬂux density (Poynting vector) of radiation: I = |S| ,

S=

c [E × H] , 4π

(1.18)

where E and H are the electric and the magnetic ﬁeld vector of the light wave. Within a solid angle of dΩ the radiation transports through an element ds of surface an energy of I · dΩ · ds · cos θ (θ is the angle between the normal to ds and the axis of the cone with the cone angle dΩ) in a unit of time. When related to the spectral interval dν, it is referred to as the spectral intensity Iν . Spectral intensity can be expressed in terms of the volume spectral density

Iν dΩ , q = c/nν , q −1 hν 2qhν3 exp − 1 dν. Iν dν = kB T c3 uν =

(1.19) (1.20)

As far back as the nineteenth century, Gustav R. Kirchhoff analyzed, based on the general principles of thermodynamics, the relationship between the abilities of heated matter to emit light (spectral emissivity ε ν , the power of the emission of a volume element dV into a solid angle of dΩ is given by ε ν dV dΩ) and to absorb it (absorption coefﬁcient χν ). According to Kirchhoff, the intensity of radiation propagating in a medium √ with a refractive index of n = ημ [13] is related to the spectral emissivity and the absorption coefﬁcient by the formula Iν εν = = ρ ν ( T ), n2ν χν n2ν

(1.21)

ρν ( T ) being a universal function (the same for all media). No expression for this function was known at the time. It was determined later, and, as can be seen from a comparison between expressions (1.20) and (1.21),

1.2 Equilibrium Plasma

it is described (accurate up to a numerical factor) by the Planck formula ρν ( T ) = Iν ( T ). Apart from being historically important, Equation (1.21) proves convenient for a number of estimates (see, e.g. Sections 1.3.1, 4.3.4). If one considers the luminous ﬂux issuing from plasma into an external medium with a refractive index of n , then, generally speaking, one should take into account reﬂection at the interface, I/n2 = I /(n )2 [13]. When analyzing radiation intensities, the variation of the refractive index of low-temperature plasma from unity can, as a rule, be disregarded (unless it is the subject matter of a special study, see Section 7.1) and reﬂection from the walls of the plasma object container can be independently allowed for. For the integral radiation ﬂux, we have (see also Section 2.1) p = (c/4)u = (ca/4) T 4 = σT 4 ,

σ = 5.67 × 10−5 erg/cm2 · s · K4 . (1.22)

Relations (1.16) and (1.20) are identically referred to as the Planck formulas, and (1.17) and (1.22), the Stefan–Boltzmann formulas. They differ pairwise by dimension factors, which should be taken into consideration when discriminating between cases where the volume densities or the ﬂuxes (luminosities, intensities) of the emission of the object under study are dealt with. The form and dimensions of the quantities described by formulas (1.16), (1.20) and (1.17), (1.22) bespeak their independence of both the density and the composition of the plasma material. This is a consequence of the detailed balancing principle whereby, in particular, the emission of a photon under absolute equilibrium conditions is counterbalanced by the reverse process – the absorption of an identical photon. The increase in the density of atoms at a speciﬁed temperature simultaneously leads to the increase of the frequency of both these processes. Radiation is absorbed in the element of space where it is formed. For this reason, the TE plasma is optically dense. This is true of all radiation wavelengths. Therefore, photons emitted by one particle can be absorbed by another, and, as a result of a multitude of such processes, lose information about the nature of the elementary emitter. If the ‘leakage’ of the radiation being analyzed, though present, is so weak that the disturbance of the thermodynamic equilibrium in any element of the plasma volume is negligibly small; then, neither the integral nor spectral characteristics of this radiation can provide any information other than that on the plasma temperature. For one to establish, for example, the chemical and charge composition of plasma by means of the relations presented in Section 1.2, sole data on the radiation parameters alone is

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inadequate. Independent information is in this case necessary about the density and composition of the parent plasma-forming gas.

1.3 Models of Equilibrium and the Associated Parameters

The parameters featuring in the description of the completely thermodynamically equilibrium (TE) plasma in relations in Section 1.2, namely, the densities of neutral, ion, and electron components and temperature, are usually considered among the main plasma parameters. If the thermodynamic equilibrium conditions are violated, they continue to try to preserve the terms, but now in a restricted sense. For example, if the spatial homogeneity of plasma is disturbed, one can talk in general not about the density of particles of a given species, but rather about its local value. If, on account of interaction with external ﬁelds, charged particles move faster than their neutral counterparts but both obey Maxwellian velocity distribution (1.3), they talk about the temperatures of the respective particles, and so on. Thus, as the deviation from thermodynamic equilibrium increases, the hierarchy of deﬁnitions of the parameters grows in complexity and becomes increasingly arbitrary with respect to the basic one. If the number of such parameters is not very great, and their interrelations can be traced for a certain class of objects, they are referred to as plasma equilibrium models (see, e.g. [6, 12, 14]). In this sense, the reference is not to the equilibrium of the system as a whole, but rather the set of partial equilibria between individual subsystems in the phase space of the states of plasma. In principle, it would be more correct to call such models plasma non-equilibrium rather than equilibrium models, but the latter term has become established. In addition to the above TE plasma model, the following plasma models are most prominent in the literature: local thermal equilibrium, partial local thermal equilibrium, coronal approximation, and collisionalradiative models. We will restrict ourselves to the qualitative description of these models, aiming principally all at explaining the prevalent (though not always stringent) terminology and referring the reader to [6, 14–16] for details. 1.3.1 Local Thermal Equilibrium (LTE) Model

The disturbance of the enclosed nature of the system brings it out of the state of thermodynamic equilibrium. The reason being a violation of the detailed balancing principle, whereby each process is counterbalanced

1.3 Models of Equilibrium and the Associated Parameters

by its opposite. To illustrate, distributions (1.3) and (1.5) are formed upon collisions of particles with one another. In the simple case of two bound states k, l, the detailed balancing results in the condition Nk wkl = Nl wlk ,

(1.23)

where w is the frequency of the energy-state excitation and decay events, the relation between wkl and wlk being established by expression (1.3) and the structure of the energy spectrum of the particles. If the plasma volume is conﬁned, and energy and mass exchange with the surrounding space takes place at its boundaries, the balance of the arrival and departure of particles in the given energy state will be affected by their departure (arrival) from the given element of the plasma space. The frequencies of the latter events will be determined by the transfer processes. The LTE model generalizes the cases where the deviations from the state of thermodynamic equilibrium are comparatively not very great and is based on the following two basic assumptions: (1) The density of the plasma particles is high enough to ensure that in each plasma volume element (small compared to the overall geometric size of the plasma object, but adequate for statistical description) detailed balancing (1.23) and relations (1.3), (1.4) hold approximately true. In such a scenario, the nonclosed (noninsulated) condition of plasma manifests itself in the observation that when one goes from one element to another, the temperature and density values can vary, but do so identically for all the particles. In other words, the functional form of the distributions of all the particles in the phase space coordinates (energy, momentum, chemical composition) remains the same, but as the spatial coordinate varies so do the distribution average values. (2) No equilibrium conditions are imposed on radiation. It is only required that the frequency of the radiative processes causing radiation to leave the space element under consideration be small in comparison with the frequency of the collision-induced processes. Radiation is no longer ‘black light’, relations (1.16), (1.17) do not hold and plasma is not optically dense (the emission and absorption of a photon are separated in space), at least for the whole spectrum. The latter is of principal importance, because LTE plasma, in contrast to its TE counterpart, provides information not only about temperature (see Section 1.2), but also about the densities of particles and their spatial distributions. In essence, this is precisely what the plasma spectroscopy techniques are based on.

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Given these assumptions, use can be made of the Kirchhoff law (1.21) to estimate the intensity of radiation propagating in the x-direction from plasma that is homogeneous in this direction and has a geometric thickness of L and a ﬁnite optical density of χν L. In accordance with the deﬁnitions of Section 1.2.3, we write down dIν = −χν Iν + ε ν . dx

(1.24)

This equation describes the transfer of radiation in an absorptive medium (see also Section 3.1.3). If we add nν = 1 in (1.21) and assume that the medium is isotropic, then Iν ( L) =

εν (1 − e− χ ν L ). χν

(1.25)

In accordance with Kirchhoff’s law, regardless of what the equilibrium conditions in the plasma under consideration, the ratio χε νν in (1.21) is the black-body radiation intensity. In other words, the spectral intensity of the radiation of plasma of ﬁnite optical density will be lower than that for black bodies. And it is only in the limit of high values χν L 1 that it will reach the intensity of ‘black’ radiation with the temperature of the plasma-forming gas. As distinct from the case of TE plasma, radiation intensity and absorption measurements here provide information about the concentrations of the emitting (absorbing) particles (see Sections 2.3, 2.4 below). 1.3.2 Partial Local Thermal Equilibrium (PLTE) Model

The PLTE model corresponds to the further increase in importance (compared to the LTE model) of the roles of transfer processes and external effects in comparison with that of collisions. This tendency grows stronger as the plasma density is reduced. It is assumed in this case that temperature values in relations (1.3), (1.5), (1.13) can differ. Furthermore, it is supposed that some of these relations can fail to hold true for all particles of all species simultaneously. However, at least one of the relations should hold true at least for some species of particles. For example, it may turn out that owing to the effect of an external electric ﬁeld the temperature for ions in relation (1.3) fails to match that for neutral particles; consequently, the electron distribution cannot be described by this formula, and the temperature values in (1.5) and (1.13) coincide neither with each other, nor with the temperature for ions in (1.13). Nevertheless, such serious disturbances of the ‘harmony’

1.3 Models of Equilibrium and the Associated Parameters

of the TE model notwithstanding, singling out the great enough, but limited number of statistical ensembles of particles characterizing the state of plasma proves quite expedient in practice. The presence of a set of different (partial) temperatures is a characteristic feature of the PLTE approximation. The singling out of ensembles of particles in the phase space of velocities, potential energies, ionization and dissociation states, and so on, has quite clear physical meaning allowing one to orientate oneself within the applicability limits of such an approximation. It is based on the difference between the times of relaxation – establishment of stationary distributions within the limits of different degrees of freedom of different species of particles – and, as a consequence, the difference between the respective average energies and temperatures. In accordance with what has been said above, it is reasonable to introduce the concepts of electron, ion and neutral particle temperatures to describe their velocities by means of relation (1.3); rotational, vibrational and electronic level temperatures to describe the distributions of particles among the corresponding levels by means of relation (1.5); ionization temperatures in relation (1.13) to relate together the densities of ions of one and the same chemical species, but of different multiplicities, and so on. In a number of cases the PLTE model is understood in an even wider sense. Namely, a partial temperature can be introduced even if the densities of particles are described by the thermodynamic equilibrium formulas for only some of the states within the limits of the desired degree of freedom and not for all of them. With this approach, it would sufﬁce if the majority of the states match this description. For example, it is assumed in [3] that the states of the continuous velocity spectrum are described by the PLTE model if formula (1.3) describes a distribution section in the range from 5 to 10 times the average value over the entire distribution. Beginning with the early work [16], many authors have assigned to the PLTE model such cases where the Boltzmann distribution (1.4) holds true for a group of excited levels in atoms, except for the ground-state levels. Such an approach is justiﬁed by the fact that the energy gaps between the ground and the ﬁrst excited levels usually exceed the energy difference between the excited levels. Figure 1.4 shows the character of transition from the PLTE to the LTE model occurring in hydrogen plasma as its electron density ne grows higher (for calculation see [16]; see also [12]). The quantities a and bn characterize the deviation of the populations N of the levels with principal quantum number n from their populations N ∗ in the LTE state. For a, the number n = 1 and

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1 Plasma as an Object of Spectroscopy

Figure 1.4 Change over from plasma in the PLTE model to that in the LTE model. Hydrogen.

for bn , it takes on the values n = 2, 3, 4 . . . a = N/N1∗ ,

bn = ( Nn /N1 )/( Nn∗ /N1∗ ).

In the electron density range ne = (1014 –1018 ) cm−3 and electron temperature range (4-64) × 103 K, the values of bn are close to one another, and for the levels n, n > 1, the ratios Nn /Nn ≈ Nn∗ /Nn∗ , that is, the excited levels obey the Boltzmann distribution, though with a temperature different from the equilibrium value. The populations of these levels are smaller than the equilibrium populations, while the ground state level, with n = 1, is overpopulated throughout the electron density range up to 1018 cm−3 . At ne > 1018 cm−3 the entire ensemble of levels is described by the uniﬁed equilibrium distribution (1.5). By virtue of these circumstances, when discussing the parameters of plasma within the framework of the PLTE model, use is frequently made of some special auxiliary terms that clarify the meaning of the very concept of temperature in each particular case. If formula (1.5) holds true, as in the above-considered case [12], only for a group of excited levels but fails to describe the ground state, it is referred to as the ‘distribution temperature’. But if the parameter T in expression (1.5) describes the population of a level (or a group of levels) relative to that of the ground state level, it is then called the ‘excitation temperature’. To describe the distributions of molecules in a system of vibrational or rotational states, the respective distribution temperatures are referred to as ‘vibrational’ or ‘rotational’. The parameter T that gives the observed electron concentra-

1.3 Models of Equilibrium and the Associated Parameters

Figure 1.5 Temperatures in the plasma of an Ar-H2 arc discharge.

tion, a formula similar to the type of the Saha formula, (1.13) (for more details, see [6, 12]) is called the ‘ionization temperature’, and so on. Figure 1.5 presents the results of measurements [16] of various temperatures in the plasma of an arc discharge in argon with an admixture of hydrogen. The excitation (Texc ), distribution (Td ), ionization (Ti ), neutral gas (Tg ), and electron (Te ) temperatures were measured by independent methods and referred to the axis of the arc. At electron densities ne < 1015 cm−3 (which corresponded to an arc current of I < 2 A) substantial differences are observed between the temperatures, and this corresponds to the PLTE model. At ne > 5× 1015 cm−3 (I > 12 A) practically all the temperatures coincide, which indicates the transition to the LTE state. Similar measurements [17] for an arc in helium with additions of neon and hydrogen showed that the LTE state was reached at ne > 5× 1016 cm−3 . The substantial reduction of the critical value of ne in plasmas where the H2 molecules constitute an insigniﬁcant admixture to the inert gas, in comparison to the case of hydrogen plasma (Figure 1.4), is believed to be due to the intense intermixing of the block of excited levels upon collisions with the inert gas atoms [17, 18]. This view is also supported by the coincidence of the excitation temperature of such levels with the translational temperature Tg of the neutral atoms (Figure 1.5). The authors of [16, 17, 19] have suggested a simple criterion for the determination of the limiting energy Ek of an atomic level above which the levels combine into a single Boltzmann ensemble with a uniﬁed tem-

19

20

1 Plasma as an Object of Spectroscopy

perature equal to the electron temperature Te in an optically thin plasma with a low degree of ionization and predominant role of collisions: ne ≥ 1014 Te/2 ( Ek − I )3 , 1

(1.26)

where I is the ionization potential, ne is expressed in cm−3 , and Ek , I, and Te are expressed in electronvolts. If in expression (1.26) Ek = E1 for the ground level, this then corresponds to the LTE condition. However, by virtue of the assumptions made it does not cover the ions whose energy structure is different from that of the parent atoms (for more details, see [12]). 1.3.3 Model of Coronal Equilibrium (MCE)

This approximation is so called because it is used to describe the conditions obtaining in the solar corona. This is a rareﬁed plasma with an electron density of 106 –1010 cm−3 and an average electron energy of 0.1– 1 keV. The deviation from thermodynamic equilibrium here is so great that practically no processes obey the detailed balancing principle. In this case, (i) for all optical transitions, the plasma is optically thin, (ii) particles are excited by direct electron impact from their ground state and also in cascade radiative processes, (iii) particles are only ionized by direct electron impact from their ground state, (iv) the excited states get deactivated on account of spontaneous emission of radiation, and (v) ions are neutralized by way of photorecombination. By virtue of conditions (iii) and (v), ( Z −1)

N ( Z − 1 ) n e Si

= N ( Z ) ne α ( Z ) ,

or ( Z −1)

N (Z) /N (Z−1) = Si

/α(Z) ,

(1.27)

where Si and α are the ionization and recombination rate constants, respectively. Equality (1.27) is called the coronal ionization equilibrium relation. It does not include the electron density, but in contrast to the Saha ionization equilibrium formula (1.23), the right-hand side of (1.27) depends on the particle species in a more complex manner than simply as a function of the ionization potential. The calculation of these coefﬁcients for some speciﬁc cases can be found in [19, 20]. By virtue of conditions (i), (ii), (iii) and (iv) the population balance of the kth level of a particle

1.3 Models of Equilibrium and the Associated Parameters

has the form (Z)

dNk dt

=

(Z) N1 ne

∞

(Z)

o1k (ve ) f e (ve )dve (1.28)

ve,t

−∑ l k

(Z) Nl Akl

+

∑

mk

(Z) Nm Amk ,

where σ is the excitation cross-section, f e (ve ) is the distribution function of the electron velocities ve (ve,t is the threshold electron velocity), and A is the frequency of emission events. Approximate analytical expressions for S, α in (1.27) and for the integral in (1.28) with the Maxwellian distribution function f e (ve ) (1.3) can be found in [12, p. 47]. Apart from its utilization in the spectroscopy of solar corona, the coronal equilibrium model has been analyzed for possible applicability to denser plasmas, such as the tokamak and beam discharge plasmas. It transpired that if the average electron energies were high enough, Ek > 10 eV, relation (1.18) would hold at ne ≤ 1012 cm−3 . Further, the level populations of light atoms could be described by means of relation (1.19) at up to ne ≤ 1014 cm−3 , if one considered only the ground level (k = 1) and the ﬁrst excited resonance level (k = 2) [6]. Since electron–electron collisions at so high ne values under stationary conditions provide, as a rule, for the maxwellization of the electron velocity distribution, so that relation (1.3) holds true, the use of (1.28)) enables one to spectroscopically determine the average electron velocity. This is a borderline case between the MCE and the PLTE model, where one can simultaneously introduce the partial temperature Te characteristic of the PLTE and use the simple MCE formula (1.28) with a small number of levels that permits of the inversion of the problem, namely, to ﬁnd the parameters of plasma from its radiation (Section 1.4). 1.3.4 Collisional-Radiative Model (CRM)

This name was given to the method used to describe plasma with a strongly disturbed equilibrium, and for which none of the above models could be used. The term ‘model’ in this situation has a very arbitrary meaning. For the case in hand it means the complete level-by-level description of particles in combination with the kinetic equations for the continuous energy spectrum and electrodynamics equations for ﬁnding the ﬁeld. The population balance of the quantum levels of the particles of each particular chemical species is described by general kinetic equations

21

22

1 Plasma as an Object of Spectroscopy

of the type dNk = dt

∑ ( Nl νlk − Nk νkl ) + ∑ Sαk + Trk . l

α

(1.29)

Here ν are the frequencies of the collisional and radiative transitions between the levels, S is the source function describing the creation and annihilation of the particles Nk accompanied by the formation of particles of a different chemical species (chemical reactions, ionization, recombination, and others), and the term Trk is responsible for the transport of particles. There are no general, compact relations to determine plasma parameters in conditions corresponding to the collisional-radiative model; moreover, the meaning of the parameters themselves, based on the introduction of statistical ensembles of particles, should be clariﬁed in each particular case. Naturally all the models considered above are particular cases of the CRM. The questions pertaining to the kinetics of nonequilibrium lowtemperature plasma have been considered comprehensively enough by Biberman and co-workers [2].

1.4 Optical Spectrum and Plasma Parameters

Modern spectroscopy has at its disposal a vast array of experimental potentialities applicable to plasma diagnostics purposes. Spectroscopic techniques can be classiﬁed into groups on the basis of their common characteristics. A distinction is customarily made between emission (emission spectroscopy), absorption (absorption spectroscopy), and scattering (Rayleigh, Tomson, Raman, etc., spectroscopy) techniques. Speciﬁc to each of these techniques are both the spectra involved and the methods to obtain them. The emission spectroscopy techniques use the intrinsic plasma radiation. The absorption and scattering spectroscopy techniques require additional sources of transmitted radiation or an optical system to return into the object under investigation its own radiation. All spectra can be either continuous or line, which can be judged after passing light through the spectral instrument, detecting it, and excluding the distortions caused by this procedure in the true spectrum. The localization of the source of the spectrum in the object of interest is of great importance. The shape of the spectrum can be affected by many factors, including the motion of plasma as a whole, internal mass and radiation transfer, oscillations and instabilities and the presence of external and in-

1.4 Optical Spectrum and Plasma Parameters

trinsic plasma electromagnetic ﬁelds. These inﬂuences are manifested in the structure and intensity of the spectral components, and in their polarization properties. Despite such a wide a variety in terms of their implementations and speciﬁc features, all spectroscopic techniques provide information about the density of states of particles in energy spectrum intervals bound by optical transitions. However, since the shape of the spectrum is inﬂuenced by numerous of additional factors, partly mentioned above, one more consideration is important. Namely, the problem of spectroscopic diagnostics is, in principle, an inverse problem in the mathematical sense. Consider, for example, a case where stationary plasma is investigated from its intrinsic radiation (emission spectroscopy). For the polarizationstate-averaged intensity I (understood as the experimentally measured light power) of the spectral region in the frequency interval dν, we have I (ν) dν = G I [ν, i (ν, r), k (ν, r), Φ].

(1.30)

Here i and k are the coordinate-dependent local intensity and absorption coefﬁcients, respectively. The measured radiation intensity and its spectral distribution are further inﬂuenced by the instrumental factors Φ. The operator G I determines the concrete functional form of (1.30). The forms of i (ν, r) and k (ν, r) are governed not only by the probability of optical transitions, but also by the set Xl of local plasma characteristics. i (ν, r) = Gi (ν, Xl (r));

k (ν, r) = Gk (ν, Xl (r)).

(1.31)

In this case ‘plasma characteristics’ refers not only the populations of the emitting and absorbing states bound by the optical transition of interest, but also the distributions of all particles in the discrete and the continuous energy spectrum. The local characteristics of plasma are in turn governed by its local parameters Pl,m (r) (temperature, chemical and charge composition, electric and magnetic ﬁelds): Xl (r) = G plm ( Pl,m (r)).

(1.32)

These relations should be supplemented by conditions similar to the initial and boundary ones, namely, the composition and pressure of the parent plasma-forming gas, the geometry of the plasma object, and the power deposited in the plasma. If the quantity being measured is the plasma radiation intensity and the unknowns are the plasma parameters, the procedure for ﬁnding the latter corresponds to operations inverse with respect to those described

23

24

1 Plasma as an Object of Spectroscopy

by formulas (1.30), (1.31) and (1.32), that is, 1 i (ν, r) = G − I [ ν, I ( ν )dν, k ( ν, r), Φ ],

Xl (r) = Pl,m (r) =

Gi−1 (ν, i (ν, r)), Xl (r) 1 G− plm ( Xl (r)).

=

Gk−1 (ν, k (ν, r))

(1.33) (1.34) (1.35)

The above relations are not independent and make up a system of nonlinear equations. The solution of inverse problems is, as a rule, more difﬁcult than that of direct ones, and what is more, it is not always unique. In addition, there is the standard problem of the correctness of the inverse problem, deﬁned as the stability of its solution against perturbations of the arguments. These are rather general problems [20, 21] arising in spectroscopy [22] and speciﬁcally in plasma spectroscopy [23, 24]. Directly determining plasma parameters solely from measured plasma radiation intensities, followed by mathematical processing, thus proves practically insurmountable. In this situation, it is required that such simpliﬁcations should be found as would not distort the result. It is extremely desirable to ‘disengage’ the problems in the chain of equations (1.33)–(1.35). To illustrate, great difﬁculties arise in link (1.33) as a result of the fact that plasma radiation emitted in some elementary volume can be absorbed in another (nonlocal effect associated with the ﬁnite optical density), and to take it into account requires solution of the radiation transfer problem. This is manifest in the presence of the absorption coefﬁcient k (ν, r). Simpliﬁcation can be achieved, for example, on account of the fact that this coefﬁcient can be measured by an independent method. After all, according to relations (1.34), both the emission and the absorption of radiation in one and the same elementary volume are governed, though differently, by the same characteristics. Use can also be made of other experimental methods, local as far as the staging of the very measurements is concerned. The case of optically thin plasma with k (ν, r) = 0 is even simpler. In such cases, link (1.33) can be treated as an independent diagnostics stage involving the ﬁnding of i (ν, r) using spectral deconvolution procedures (allowance for the factor Φ) and tomographic techniques (ﬁnding the local radiation intensity proﬁle from radiation intensities averaged over several view axes). Thereafter use is made of procedure (1.34) to ﬁnd the densities of the energy states of the emitting (absorbing) particles and their distributions in the energy spectrum. To this end, it is necessary to take a series of measurements of the intensities of a number of lines corresponding to transitions between various energy levels of the bound states and intensity distributions in the continuous spectrum regions responsible for the free motion of various species of particles. Information will also be necessary

References

regarding the probabilities of transitions and elementary luminescence excitation events (e.g. photorecombination). And ﬁnally, stage (1.35) leads to the sought-for parameters if use is made of one of the equilibrium models described in Sections 1.2 and 1.3. And this is not an easy choice, for the model is not known beforehand. For this reason, the independent comparative measurements taken in different spectra are of great importance. Diagnostics is drastically simpliﬁed if the object under study is a thermodynamically equilibrium plasma. In this class of objects, all parameters are uniquely interrelated and so it is sufﬁcient to measure one of them by the most accessible and reliable method. For example, it sufﬁces to measure absolute integral ﬂux (1.22) to determine the temperature and then calculate the rest of the parameters via the formulas of Section 1.2. What is not trivial in this case is to make sure that the object under study actually complies with the TE plasma model. Of importance here is the experience gained in experiments conducted under closely similar conditions. Additional investigations, for example, into the spectral radiance of plasma, sometimes prove useful, and such are usually undertaken for the sake of certainty. The above example shows that the spectroscopic plasma diagnostics is a laborious and multipronged research tool. But the difﬁculties involved are repaid a hundredfold by the uniqueness and wealth of both the ﬁnal and intermediate information, information that cannot be obtained by any other diagnostic technique. It is obviously very important to have an extensive system of complementary experimental spectroscopic methods. It is also very important to store and generalize experience in the application of these methods and their combinations to a wide class of plasma objects. And it is exactly these problems that the subsequent sections of this book are devoted to. References

1 V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), 1, pp. 1–3. Moscow: Nauka (2000). 2 L.M. Biberman, V.S. Vorobeyov, and I.T. Yakubov. Kinetics of Nonequilibrium Low-Temperature Plasma (in Russian). Moscow: Nauka (1982). 3 B.M. Smirnov. An Introduction to Plasma Physics (in Russian). Moscow: Nauka (1982). 4 V.M. Lelevkin, D.K. Otorbaev, and D.C. Schram. Physics of Non-

Equilibrium Plasmas. North-Holland (1992). 5 Yu.S. Protasov and S.N. Chuvashev. Low-Temperature Plasma. Basic Properties. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 1, pp. 1–190. Moscow: Nauka (2000). 6 V.N. Kolesnikov. Spectroscopic Plasma Diagnostics in the VUV Region. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Rus-

25

26

1 Plasma as an Object of Spectroscopy

7

8

9

10

11 12 13

14 15

sian), 2, pp. 491–507. Moscow: Nauka (2000). V.D. Rusanov and A.A. Fridman. Physics of Chemically Active Plasma (in Russian). Moscow: Nauka (1984). D.I. Solovetsky. Chemical Reaction Mechanisms in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1980). V.P. Glushko, Ed. Thermodynamic Properties of Individual Substances (in Russian). Moscow: Nauka (1962). Ya.B. Zeldovich and Yu.P. Raizer. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena (in Russian). Moscow: Nauka (1966). Yu.P. Raizer. Gas-Discharge Physics (in Russian). Moscow: Nauka (1982). W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). M.A. Leontovich. An Introduction to Thermodynamics (in Russian). Moscow-Leningrad: GITTL (1950). H.R. Grim. Plasma Spectroscopy. N.Y.: Mc Graw-Hill (1964). H.R. Grim. Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997).

16 R.W.P. McWirter and A.G. Hearn. Proc. Roy. Soc., 82, p. 641 (1963). 17 V.N. Kolesnikov. Arc Discharge in Inert Gases (in Russian). Trudy FIAN SSSR, 30, p. 66 (1964). 18 F. Burhorn, R. Wienecke. Zs. Phys. Chem., 215, p. 285, (1960). 19 R. Huddelstone and S. Leonard, Ed. Plasma Diagnostic Techniques. New York (1965). 20 A.N. Tikhonov and V.Ya. Arsenin. Methods for Solving Ill-Posed Problems (in Russian). Moscow: Nauka (1986). 21 V.A. Morozov. Methods for Solving Unstable Problems (in Russian). Moscow: Moscow State University Press (1967). 22 V.V. Lebedeva. Optical Spectroscopy Techniques (in Russian), pp. 343–372. Moscow: Moscow State University Press (1977). 23 N.G. Preobrazhensky and V. V. Pikalov. Unstable Plasma Diagnostics Problems (in Russian). Novosibirsk: Nauka (1982). 24 V.V. Pikalov. Plasma Tomography. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), II, pp. 563–569. Moscow: Nauka (2000).

27

2

Basic Concepts and Parameters Associated with the Emission, Absorption and Scattering of Light by Plasma

2.1 Photometric Quantities. Remarks on Terminology

Quantitative spectroscopy is based on measurements of luminous ﬂux parameters. Photometry is the measurement of radiant energy parameters. Let us recall the most important of these parameters [1, 2]. The medium is characterized by the radiant energy density u – electromagnetic ﬁeld energy per unit volume, (1.16), (1.19). What is usually measured in experiments are the characteristics of radiation that has left the conﬁnes of its source. This may be the intrinsic radiation of the object under study, radiation passed through the object, or radiation originated upon the passage of light through the object. The energy emitted in a unit of time across the entire surface into the external space is called the radiant power or radiant ﬂux p of the source. For a more detailed description of the spatial angular characteristics of the radiant ﬂux, use is made of additional differential parameters. That part of the radiant ﬂux which falls upon a unit surface of the source boundary is referred to as radiant emittance r: r = dp0 /ds,

(2.1)

where ds is the surface element and dp0 is the radiant ﬂux issuing from it into the half-space. The radiance of the source is that part of the radiant ﬂux issuing from a unit surface, which propagates within the solid angle ΔΩ = π whose axis makes an angle of i with the normal to the surface: b = dp/(ds dΩ cos i ).

(2.2)

The radiance b, in addition to the radiant emittance r, characterizes the anisotropy of the radiant ﬂux, for it depends on the observation angle, b(i ). If radiation is isotropic, which is the case with sources in the state of

28

2 Basic Concepts and Parameters

thermodynamic equilibrium (TE) (black body) and sources having rough surfaces, one can then easily ensure, by integrating with respect to solid angles, that radiant emittance and radiance are related (Section 1.2): r = πb.

(2.3)

The aim of the further detailed elaboration of the energy parameters of radiant ﬂuxes is to take account of the spectral distribution of radiation. Special quantities are introduced for the spectral densities of these parameters, namely, uν for the radiant density, rν for radiant emittance, bν for radiance, and pν for radiant ﬂux (power), which are referred to the spectral intervals dν and associated with the respective integral quantities: u=

∞

uν dν, 0

r=

∞

rν dν,

b=

0

∞

bν dν, 0

p=

∞

pν dν.

(2.4)

0

For example, the radiant power p of a source can be expressed in terms of the spectral density of radiance by integrating over the boundary surface, angles, and frequencies: p=

bν (ν, i ) cos i ds dΩ dν.

(2.5)

s,ν,Ω

By virtue of the tradition established in spectroscopy, the concept of intensity I is frequently used for radiant energy characteristics. This concept has already been introduced in Section 1.2.3. as the average value of the Poynting vector. Although, in practice, the concept of intensity can be understood in differing senses, it is usually used to refer to the power of light incident upon the detector. For this reason, light intensity is a quantity similar to radiance, provided that the latter is measured in relative units. The same is true of its spectral density Iν . If the value of intensity is expressed in erg · s−1 · cm−2 = 10−7 W · cm−2 , which is also frequently done, this then corresponds to the radiant ﬂux (power) calculated in terms of radiance via (2.5) within the limits of s, ν, Ω, I set by the geometry of the experiment and the spectral instrument used. The task of absolute measurements is often eased in practice by comparing the intensities of light coming from the object being studied and from a standard source, with the measurement geometry being kept unchanged (Section 3.1.2). In such a case, one refers to the absolute intensities, though this corresponds fully to radiant power (2.5). The use of relative intensities proves convenient in dealing with line spectra, when the relative radiant powers of different spectral lines, Iik : Ilm : Inp . . . ,

2.1 Photometric Quantities. Remarks on Terminology

are measured under the same experimental conditions (with correction made for the spectral sensitivity of the detector). Intensities are proportional to the square of the electric ﬁeld strength of the light wave, I ∼ E2 . If E0 is the ﬁeld in the vicinity of the source, then I ∼ E02 for a plane wave or I ∼ E02 R−2 for a spherical wave, where R is the distance between the detector and the light source. The relation between the intensity (power density) and the mean-square ﬁeld strength of the wave is numerically given by the formula (2.6) E = 19 I [W/cm2 ], V cm−1 . It should be noted that to establish important relations between the energy parameters (radiance, radiant emittance, radiant intensity, etc.) of the radiation being detected and its volume density u in plasma spectroscopy can be a difﬁcult enough problem to which there are two important exceptions. One is the thermodynamically equilibrium (TE) plasma wherein this relation is established by the expression (Section 1.2) bν = (c/4π)uν .

(2.7)

And the other is optically thin plasma wherein radiation produced in one volume element is not absorbed in another. In that case, for example, we have the following relation for the spectral density of the radiant intensity: Iν ∼ c

uν (r) dν dV,

(2.8)

V

where the integral is taken over the volume determined by the observation geometry. If the inhomogeneity of uν (r) is substantial, its local values are restored by tomographic methods, measuring the radiant intensity in different observation directions. If, in the general case, plasma is nonequilibrium, inhomogeneous and has a ﬁnite optical density, the problem of tomography is supplemented by that of radiation transfer. These are difﬁcult inverse problems for which simpliﬁcations can, sometimes, be accomplished by using combinations of emission, absorption and scattering spectroscopy techniques or by introducing additional prior information. In the spectroscopic literature photometric quantities are usually measured in physical units: u[erg · cm−3 , J · cm−3 ]; uν [erg · cm−3 · s, J · cm−3 · s]; p[W, erg · s−1 ]; pν [W · s, erg]; r[W · cm−2 , erg · cm−2 · s−1 ]; rν [W · cm−2 · s, erg · cm−2 ]; b[W sr−1 · cm−2 , erg · sr−1 cm−2 · s−1 ]; bν [W · sr−1 · cm−2 · s, erg · sr−1 · cm−2 ].

29

30

2 Basic Concepts and Parameters

Let us dwell brieﬂy upon one more question of photometry, namely, whether it is possible to increase radiance by transforming light beams with the aid of optical systems. The theory of classical optical instruments answers this question in the negative; that is, the radiance of the image cannot be higher than that of the source. A general proof of this statement for systems of an arbitrary number of optical elements, on the basis of the Lagrange–Helmholtz principle, can be found, for example, in [3] and a number of other books on optics. The gist of the matter at issue is that reducing the image can increase the radiant ﬂux density in the image plane. However, this is attained by increasing the solid angle, wherein the image emits radiation in the direction opposite to the ﬂux that forms it. And in actual fact, the radiance of the image turns out to be lower than that of the source owing to the loss of light in the optical system. However, as shown in [4] (see also commentaries in [5]), such a situation, although ﬁtting the general principles of thermodynamics, holds only for passive systems free from energy exchange between the light beams themselves and between the beams and the system. A wellknown example of a deviation from these conditions is the laser, wherein the pumping light beam from a classical light source is transformed into the narrow laser beam; moreover, it’s radiance can exceed that of the pumping source. It should also be noted [4] (see also commentaries in [5]) that to increase the radiance of light beams use need not necessarily be made of inverted laser media. Modern laser beam transformation methods can increase radiance by using scattering schemes, reducing radiation pulse duration, and so on. The inﬂuence of the angular and spectral factors on the photometric parameters being measured is illustrated in Table 2.1. We assume that the laser beam divergence is of diffraction character, the beam diameter at the exit from the laser is 1 mm, the laser radiation wavelength is λ = 0.5 μm, so that dΩ ca. 10−6 sr, the laser power is 1 W and the spectral width is dν ca. 106 Hz. The solar radiation power incident on 1 cm2 of the Earth’s surface (solar constant) is IE = p/(4πR2ES ) = 0.135 W · cm−2 , the angular size of the Sun is 4.7 × 10−3 rad, the distance from the Earth to the Sun is RES = 1.5 × 1013 cm, and the solar spectrum width is dν ca. 1015 Hz. Although the power of the sun over that of the laser is colossal, laser radiation has a great advantage over its solar counterpart in terms of radiance characteristics. Photometry also uses the so-called illumination engineering units, which are based on a comparison with the standard emitters and allowance for the speciﬁc features of vision. Although these units are rarely used in spectroscopy, let us nevertheless cite for reference pur-

2.2 Spectral Line Proﬁle Table 2.1 The comparison of radiation parameters for sun and low power laser.

Sun Laser

Total ﬂux

Radiance, b,

Spectral radiance, bν ,

(power), p, W

W · cm−2 · sr−1

W · cm−2 · sr−1 · c

4 × 1026

2 × 103

2 × 10−12

108

1

102

poses that for a wavelength of 555 nm, which corresponds to the maximum sensitivity of the human eye, a power of 1 W corresponds to 683 lumens [lm]. For other wavelengths, the luminous ﬂuxes (powers) p [W] and f [lm] are related by the visibility function Φ(λ) of the eye: f (λ) = 683Φ(λ) p(λ). For more details, see Chapter 8.

2.2 Spectral Line Proﬁle

Traditionally, the term spectral line refers to a spectral region corresponding to a transition between the bound states of particles. Even if the optical thickness of the object of interest is small, an actual spectral line always has a ﬁnite width within whose limits the relationship between intensity (radiance, ﬂux, etc.) and frequency ν is described by some distribution function ϕ(ν). For absorption, the function ϕ(ν) describes the spectral dependence of the absorption coefﬁcient (Section 2.4). The form of ϕ(ν) is called the spectral line proﬁle. The mechanisms responsible for the broadening of spectral lines and formation of their proﬁles may be associated with the interactions of particles with one another, external ﬁelds and radiation. These mechanisms are described in the majority of books dealing with general questions of spectroscopy and atomic physics (see, e.g. [6–12]), and it is in accordance with these mechanisms that the different types of line broadening are classiﬁed. In the following we present the basic data and deﬁnitions necessary for the further presentation of the material and for simple estimation purposes. First let us recall the methodological scheme that is most frequently used (and cited in the literature) when considering this general question of spectroscopy. It is based on various versions of the solution of the problem on the spectrum of an atomic oscillator whose oscillation can be written down in the form ⎧ ⎫

t ⎨ ⎬ f (t) ∼ exp −i(2πν0 t + χ(t ) dt ) . (2.9) ⎩ ⎭ −∞

31

32

2 Basic Concepts and Parameters

Here ν0 is the unperturbed frequency and χ(t) is the frequency shift caused by the external factors acting on the oscillator. If, in addition, the oscillator moves translationally, the ﬁeld of the light wave emitted by it will be ⎧ ⎫

t

t ⎨ ⎬ χ(t ) dt + k v(t ) dt ) , (2.10) E(t) ∼ exp −i(2πν0 t + ⎩ ⎭ −∞

−∞

where v(t ) is the velocity of the oscillator at the instant t and k is the wave vector. The presence of phase shifts, η (t) =

t

χ(t ) dt + k

−∞

t

v(t ) dt ,

(2.11)

−∞

disturbs the monochromaticity of the radiation. As a rule, we are concerned with conditions where the pressure, temperature, chemical composition and the state of ionization equilibrium of plasma change little on the time scale of collisions or natural decay. In this case, the functions η (t) and E(t) describe random processes. The spectrum ϕ(ν) = ϕ (ν − ν0 ) (spectral density of random process) can be found via the correlation function Φ(τ ): 1 ϕ (ν − ν0 ) = π

∞

exp {i2π(ν − ν0 )τ } Φ(τ ) dτ,

(2.12)

0

where 1 Φ(τ ) = lim T →∞ T

T/2

E∗ (t) E(t + τ ) dt = E∗ (t) E(t + τ )

− T/2

(2.13)

= exp {−iη (τ )} . In this case, the spectrum ϕ(ν) = ϕ (ν − ν0 ), being the result of averaging of random processes, is no longer random, but a regular function. As follows from (2.12), the characteristic decay time τ of the function Φ(τ) 1 (correlation time) determines the spectral line width Δν ∼ 2πτ and corresponds in the physical sense to the time it takes for the phase incursion η (τ ) ≈ 1 to accumulate. 2.2.1 Lorentz Broadening

t If the oscillator is at rest, the phase shift η (t) = −∞ χ(t ) dt is due to collisions with other particles or to natural decay. With this broadening

2.2 Spectral Line Proﬁle

mechanism (natural decay or particle interactions causing the emission of radiation to discontinue), the distribution, for example, of the spectral line intensity Iν = I0 ϕL (ν) has the form ϕL ( ν ) =

(γ/2)2 . 4π2 (ν − ν0 )2 + (γ/2)2

(2.14)

The quantity γ represents the decay constant. For natural decay upon transition between the upper and the lower level with the respective lifetimes τu and τl , γn = γu + γl = τu−1 + τl−1 .

(2.15)

While on the subject of natural decay, one should bear in mind not only the radiative transitions associated with the levels u and l, but also other spontaneous processes (e.g. autoionization, predissociation) that affect the lifetimes τu and τl . If τc is the time interval between quenching collisions (transitions between levels, strong phase perturbations), then γc /2 = τc−1 = Np vσ ,

(2.16)

where Np is the density of the perturbing particles, v is their average velocity relative to the particle being perturbed, and σ is the broadening cross section. Insofar as both natural and collision broadening act independently, then γ in expression (2.15) is γ = γn + γc ,

(2.17)

and the width of the proﬁle (2.14) at half maximum is ΔνL = γ/2π.

(2.18)

Spectral line proﬁle (2.9) is known under several names. One of them is associated with Lorentz’s name, for it was Lorentz who was the ﬁrst to describe this proﬁle for broadening by collision. Because of the presence of the term with (ν − ν0 )2 in the denominator of expression (2.14), this proﬁle is frequently referred to as the dispersion proﬁle. With both natural and collision broadening mechanisms, each of the emitting (absorbing) particles is equally responsible for the formation of the spectral line proﬁle. For this reason, the broadening resulting in the formation of proﬁle (2.14) is often called homogeneous in the literature. Expression (2.14) derived by Lorentz in 1905 has since been repeatedly discussed. The history of this problem has been described in detail, for

33

34

2 Basic Concepts and Parameters

example, by Frish [6] and Sobelman [7]. Without going into details, one can single out two main paths taken by the theory of broadening in respect of interaction of particles. The ﬁrst is the so called collision theory and is, in many ways, a perfection of Lorentz’s theory. Lorentz believed that the cause of broadening was the disruption of the emission process by collision, whereas prior to such a collision the atomic oscillator suffered no perturbations. Collision theory does not touch upon the question as to the magnitude of the quenching (broadening) cross section σ , which was taken (groundlessly) to be equal to that of the gas-kinetic cross section. This approach was further developed by Lorentz and Weisskopf who suggested that broadening was due not only to the disruption of emission, but also to the perturbation of the phase of the oscillator. This made it possible, at least in principle, to discriminate between the gas-kinetic and broadening collision cross sections. Further, this work also introduced the concept of the so-called Weisskopf radius ρ0 , equal to the minimal ﬂight distance for which the phase perturbations η are large, η > η0 ≈ 1, and equivalent to the effective radius of the atom-emitter in respect of the quenching process, σ = πρ20 . Simple formula were obtained for ﬁnding ρ0 . For example, if the constant Cm characterizes the interaction potential energy, ΔW = − hCm r −m ,

(2.19)

then ρ0 =

Cm αm vη0

1 m −1

,

(2.20)

where v is the relative velocity, different m values correspond to different types of interaction (m = 2, perturbation is caused by electrons or ions and the particle being perturbed suffers the linear Stark effect, m = 4, the same for the quadratic Stark effect, m = 6, van der Waals interaction, m = 3, resonance perturbation by particles equivalent to those being perturbed), and √ αm = πΓ((m − 1)/2)/Γ(m/2), (2.21) where Γ is the gamma function (Γ( x + 1) = xΓ( x ), Γ(1) = 1, Γ(1/2) = π1/2 ), that is, n=

2

3

4

5

6

αn =

π

2

π/2

4/3

3π/8

2.2 Spectral Line Proﬁle

As in the Lorentz theory, emission of radiation over the entire free path length, now determined by the quantity ρ0 , suffers no perturbations, and so expression (2.14) undergoes no changes. Lindholm, who considered not only strong, but also any weak phase perturbations, took the next step in the development of the collision theory. This involved taking account of perturbations in long transits, which produced a qualitatively new result – the emergence not only of the broadening, but also of the shift of the spectral line: ϕ L (ν) =

(γ/2)2 . [2π(ν − ν0 ) − Δ]2 + (γ/2)2

(2.22)

The line shift δν = (2π)−1 Δ. One can introduce, in addition to the broadening cross section σ , the concept of the shift cross section σ − Δ = Np vσ . One can assume, with a certain arbitrariness, that transits at distances shorter than the Weisskopf radius, ρ < ρ0 , cause the line to broaden, while those at distances ρ > ρ0 cause it to shift. Formulas to estimate the magnitudes of the line broadening ΔνL and shift δν in interactions (2.19) with different m values can be found in [6, p. 468], [7, p. 498], for example: m=

2

4

6

δν/Np =

0

1.56C4/3 v1/3 2

0.47C6/5 v3/5

ΔνL /Np =

π2 C22 v−1

1.82C4/3 v1/3 2

1.3C6/5 v3/5

1.15

2.8

ΔνL /δν =

2

(2.23)

2

In simple cases, the constants C2 and C4 can be calculated or determined from measurements of the linear and quadratic Stark effect, respectively. For the interaction of hydrogen-like atoms at relatively low principal quantum numbers n with electrons, C2 is about n(n − 1) cm2 s−1 . Typical values of C4 for various atoms and levels range between 10−15 and 10−12 cm4 s−1 (sometimes up to 10−10 cm4 s−1 ). It follows from approximate estimations (more exact calculation is possible for atoms with one valence electron [7, p. 498]) and analysis of experimental data that the constant C6 ∼ (10−30 –10−32 ) cm6 s−1 . The case with m = 3 corresponds to line broadening in the parent gas, where resonance excitation exchange takes place among identical particles, and where broadening observed experimentally signiﬁcantly exceeds the broadening of the same lines caused by interaction with foreign particles. Since the interacting oscillators are independent of each other, their oscillation phases are disturbed in interaction, which leads to the Lorentz

35

36

2 Basic Concepts and Parameters

broadening. For not very high pressures, the half-width of the spectral line proﬁle is given by the formula [13, 14] ΔνL /Np = (2e2 /3πme ν0 ) f ,

(2.24)

where ν0 is the center frequency of the line, me is the electron mass and f is the oscillator strength of the transition (see Section 2.3). For resonance lines, f ≈ 1. Let us present for convenience the same line half-width estimates expressed in terms of wavelengths. If we relate them to the density of particles of the broadening gas, denoting them as Λ = ΔλL /Np , we will then have Λ ≈ 10−21 nm · cm−3 for the van der Waals interaction, Λ ≈ 10−20 nm · cm−3 for the resonance interaction and Λ ≈ (10−16 –10−17 ) nm · cm−3 for the Stark interaction. The subsequent reﬁnements of the collision broadening theory were undertaken by Vainshtein and Sobelman [15, 16] who considered the ¨ problem in a quantum-mechanical formulation. They solved a Schrodinger equation wherein the interaction potential was dependent on time during the course of interaction (nonstationary theory). The method and analysis results were commented in detail in the books by Frish [6] and Sobelman [7]. It was found that with the formula for the line proﬁle in the form of (2.22) being retained, the line broadening and shift cross sections should be corrected by factors depending on the relative velocity v of the colliding particles. Concrete calculations were made for the case of quadratic Stark effect with m = 4: σ∗ = σ J ( β),

σ∗ = σ J ( β),

β = 2C4/2 (ΔW ) /2 h¯ − /2 v−2 . 1

3

3

(2.25)

Here ΔW is the distance between the level participating in the transition and the nearest level associated with the dipole transition under consideration (only one of the levels responsible for the emission of the spectral line being studied is assumed to become excited). The curves of J ( β) are presented in Figure 2.1. The case where m = 4 in plasma corresponds to the broadening of the spectral lines of non-hydrogenlike atoms by electrons and ions (Section 7.2.3). In nonequilibrium lowtemperature plasma with typical ion velocities vi ≈ 105 cms−1 , β > 1 and J ( β), J ( β) ≈ 1, that is, the result coincides with that of the Lindholm nonstationary theory. In comparison, for electrons with velocities ve ≥ 107 cm · s−1 , β 1 and J ( β), J (β) 1, that is, the correction is substantial. When account is taken of the nonstationary character of interaction, it transpires that the line width-to-shift ratio ΔνL /δν is, in the general case, not constant and coincides with Lindholm’s result only in the limiting cases β > 1 and β < 10−4 . This conclusion is conﬁrmed experimentally.

2.2 Spectral Line Proﬁle

Figure 2.1 Correcting factors J , J .

The second approach to the consideration of the question of line broadening due to interaction of particles is the statistical theory. Here the shift of the frequency of the oscillator is treated as a result of the constant (quasistatistic) action of all the surrounding particles on it. As in the collision theory, the result should obviously depend on the character of particle interaction (2.19) and their density. However, the form of the statistical line proﬁle ϕS (ν) = Iν /Np proves substantially different: 3/m 3/m Cm 4πCm . (2.26) − m ϕS (ν) = 3+m exp r¯ (ν − ν0 ) m(ν − ν0 ) m Here r¯ is the average distance between particles. The greatest frequency shift is caused by the nearest particles with small r values. Therefore, to describe the line wing, the exponential factor in (2.26) can be considered equal to unity: ϕS (ν) =

3/m 4πCm

m(ν − ν0 )

3+ m m

.

(2.27)

The physical cause of line broadening in both the collision and the statistical theory is the same; namely, the interaction with the surrounding particles, and so the mismatch between line proﬁles (2.22) and (2.26) is due to the difference between the domains of applicability of these approximate theories. The result of the joint analysis [7] is that if we introduce the frequency detuning parameter m 1 m −1 v ∗ Δ = , (2.28) Cm αm m

37

38

2 Basic Concepts and Parameters

the frequency region (ν − ν0 ) Δ∗ in the vicinity of the center frequency ν0 of the unperturbed line is better described by the collision theory, whereas where the frequency detuning is great, (ν − ν0 ) Δ∗ , the statistical theory holds true. Accordingly, the terms ‘collision region’ and ‘statistical wing’ are used in the literature. The statistical wing can be located either on the long-wave or the short-wave side of the line center, depending on the sense of the shift of the terms under the effect of the surrounding particles. Of course, both these regions contribute to the integral line intensity. In the region of moderate neutral gas pressures ρ0 Np1/3 1, the major contribution is from the collision region. The contribution from the statistical wing becomes perceptible at pressures of tens and hundreds of atmospheres, provided that line broadening is caused by a foreign neutral gas. In plasma spectra, however, whose line broadening depends on electrons and ions, the contributions from these regions can be similar. The broadening of spectral lines by charged particles in plasma will be considered in greater detail in Section 7.2. 2.2.2 Doppler Broadening

The motion of an emitting particle along the observation beam z with a velocity of vz (vz c, where c is the velocity of light) leads to the Doppler shift of its radiation frequency by an amount of ν0 vz /c, where ν0 is the radiation frequency of the particle at rest (vz = 0). The observed radiation frequency is ν = ν0 +

vz ν0 , c

vz =

ν − ν0 c. ν0

(2.29)

Let the distribution of particles in the z-velocity component be given by the function f (vz ). With this broadening mechanism, the spectral line proﬁle will be ν − ν0 c ϕD ( ν ) = , (2.30) f c ν0 ν0 that is, it will be directly governed by the z-velocity distribution function of the particles. The relationship between ϕD (ν) and the absolute velocity distribution function f (v) of particles usually used in the kinetic theory of gases is given by the integral relation ϕD ( ν ) =

∞ |vz |

K (v, ν) f (v) dv.

(2.31)

2.2 Spectral Line Proﬁle

The form of the kernel K (v, ν) depends on the velocity anisotropy of the emitting particles and should be deﬁned concretely for each particular case. Anisotropy in expression (2.31) is allowed for by the presence of the frequency ν that in its turn depends on the observation direction relative to the given direction in the object under study. A simpler case that is often encountered in practice (speciﬁcally in plasma spectroscopy) is an isotropic medium with a Maxwellian velocity distribution of the emitting particles. In that case, K (v, ν) = 1/(2v), the vz velocity distribution is given by formula (1.4), and so, accordingly, we get the well-known Doppler line proﬁle 1/2 M c Mc2 (ν − ν0 )2 ϕD ( ν ) = exp − . (2.32) ν0 2πkB T 2kB T ν02 The proﬁle is of Gaussian shape, and its width at half maximum is determined, with given ν0 and M, by the temperature T of the emitters: 2ν0 (2 ln 2)kB T ΔνD = , (2.33) c M or, on replacing the mass M of the particles by their molar mass μ, we get ΔνD = 7.16 × 10−7 ν0 ( T/μ) /2 , 1

(2.34)

and in terms of wavelength, ΔλD = 7.16 × 10−7 λ0 ( T/μ) /2 . 1

(2.35)

Expression (2.32) is sometimes used with the variable u = 2(ln 2)1/2 (ν − ν0 )/ΔνD substituted for the frequency ν, which allows the line proﬁle to be described in the following, convenient, form: ϕD (u) = π− /2 exp(−u2 ). 1

(2.36)

Cases where the velocity distributions of the particles are other than Maxwellian are considered in Section 4.1. Some cases of anisotropic media with Maxwellian particle velocity distributions were treated in [17, 18]. For example, when observing the emission of particles produced upon collisions of ions with a surface in the direction z of their predominant escape, f (vz ) = const · vz

∞ |vz |

f (v) dv, v2

(2.37)

39

40

2 Basic Concepts and Parameters

and in the x-direction normal to z, f (v x ) = const

∞

|v x |

f (v) 2 1 (v − v2x ) /2 dv. 2 v

(2.38)

When observing the emission of molecular dissociation fragments in an electron beam at energies close to the threshold values in the beam propagation direction z, f (vz ) =

const · v2z

∞ |vz |

f (v) dv, v3

(2.39)

and in the x-direction normal to z, f (v x ) = const

∞

|v x |

f (v) 2 (v − v2x ) dv. v3

(2.40)

The applicability of the parent formula (2.30) is restricted. It is implicitly supposed in this formula that the particle does not change its velocity within its emission time and, therefore, only one frequency appears in it. But such changes can occur as a result of collisions and entail, quite similarly to the above-considered phase disturbance process, the disturbance of the coherence of the oscillators. One can explain the latter by means of expression (2.10), if one looks beyond the possibility of disturbance of the oscillation harmonicity of the oscillator and assume that the sole role of collisions is to change its translational velocity, that is, η (t) = k

t

v(t ) dt

(2.41)

−∞

and E(t) ∼ exp

⎧ ⎨ ⎩

⎛

−i ⎝2πν0 t + k

t

−∞

⎞⎫ ⎬ v(t ) dt ⎠ . ⎭

(2.42)

In this case the reasoning is as in Section 2.2.1. The quantities v(t ) and E(t) are random, and the Doppler spectrum (i.e. the spectrum associ ( ν − ν ) will be ated with the change of the emitter velocity) φD (ν) = φD 0 determined via the correlation function Φ(τ ) = exp {−ikr(τ )} ,

r( τ ) =

τ −∞

v(t ) dt

(2.43)

2.2 Spectral Line Proﬁle

Here r(τ ) is the displacement of the oscillator in the time τ. The ‘width’ of the correlation function Φ(τ ) will be determined in this case by its λ decay time τ, within which the atom will move a distance of 1k = 2π , where λ is the radiation wavelength. If particles moving with an average velocity of v¯ undergo no velocity changes within the time it takes for them to move this distance, that is, if their free path length is L= √

v¯ ¯ 2N vσ

= √

1 2Nσ

λ , 2π

(2.44)

λ then τ = 2π v¯ and, accordingly, the usual Doppler broadening occurs owing to the frequency shift ΔνD ∼ λ1 v¯ = ν0 vc¯ . Any effect hampering λ and thus limiting the the free motion of the emitters for the distance 2π λ time τ in comparison with τ = 2πv¯ will lead to the broadening of the correlation function Φ(τ). This in turn will lead to the narrowing of the spectrum φD (ν) (Dicke narrowing [7, 19]), with the line proﬁle becoming disperse. For the visible region of the spectrum, λ ∼ 10−4 –10−5 cm, and at a pressure of 1.333 hPa (1 Torr) L ∼ 10−2 –10−3 cm, so that condition (2.44) is satisﬁed at pressures up to approximately atmospheric value. But in the infrared region it can be violated at lower pressures. The comprehensive analysis of such collisions [7] shows that they lead to the narrowing of the central part of the line proﬁle, the Dicke effect [19], which becomes disperse. The degree of the pressure-induced narrowing of the Doppler line proﬁle in the region 2πL λ can be approximately estimated by means of the factor 2πL/λ. In the region of still higher pressures, the Lorentz broadening masks the Dicke narrowing effect. The narrowing effect was studied in detail by laser spectroscopic techniques. For example, in the study of the absorption lines of H2 O in the 6.3 μm - region [20], when the pressure of the buffer gas (Ar, Xe) was increased to 133.320 hPa (100 Torr) (partial H2 O pressure 2.666 hPa (2 Torr)), the line was observed to narrow from its normal Doppler width of 170 MHz to 110 MHz and then grow wider again owing to collision broadening as the pressure was further raised.

2.2.3 Joint Action of Natural, Doppler and Collision Broadening

Comparing what has been said in Sections 2.2.1 and 2.2.2, one can note that the common cause of line broadening is the phase displacement either of the oscillator itself (the Lorentz mechanism) or of the emitted wave as a result of the motion of the oscillator as a whole (the Doppler

41

42

2 Basic Concepts and Parameters

mechanism). In the general case where both mechanisms are operative, the spectral line proﬁle is determined via the correlation function Φ(τ ) = exp {−i[η (τ ) + kr (τ )]} .

(2.45)

In the case of statistical independence of the phase terms η (τ ) and kr (τ ), Φ(τ ) = exp {−iη (τ )} exp {−ikr (τ )}

(2.46)

the total spectrum φtot (ν) is the convolution of the Lorentz and the Doppler line proﬁle. If condition (2.44) is satisﬁed and the particle velocity distribution is Maxwellian and isotropic, the convolution yields the following expression for the spectral line proﬁle [7]:

exp −(v v0 )2 dv γ , (2.47) ϕtot (ν) = (ν − ν0 − Δ − ν0 v/c)2 + (γ/2)2 4π5/2 v0 where γ = γn + γc (2.13) and v0 = (2kB T/M)1/2 . To ﬁnd the proﬁle requires numerical integration. If data are available on the lifetimes of the levels in the course of natural decay and on collision broadening and shift cross sections, such calculations present no difﬁculties with a personal computer (direct problem). In spectroscopic experiments, it is more frequently required to ﬁnd the contributions from the individual broadening mechanisms from the known total spectrum (inverse problem), which is much more difﬁcult to do, speciﬁcally because of the multiparameter character of the problem. The small line shift and limited contribution from the statistical wing obtained within a wide range of lowtemperature plasma conditions at relatively low pressures (see above), allow one to make some simpliﬁcations. Namely, it is assumed that the line proﬁle is formed mainly by the Lorentz (together with natural) and Doppler mechanisms. In that case, proﬁle (2.47) is reduced to the wellknown Voigt proﬁle (traditionally designated by H):

∞ exp −y2 a H ( a, u) = dy. (2.48) π a2 + ( u − y )2 −∞

Here u is the same as in expression (2.36) and a = (ln 2) /2 1

ΔνL . ΔνD

To make the calculations more equivalent notations, including

∞ 1 exp − ax − H ( a, u) = √ π 0

(2.49) convenient, use is sometimes made of 1 2 x cos(ux ) dx 4

(2.50)

2.2 Spectral Line Proﬁle Table 2.2 Voigt proﬁle half-widths [21]. ΔνL /ΔνD

ΔνL /Δνv,observ.

ΔνD /Δνv,observ.

ΔνL /ΔνD

ΔνL /Δνv,observ.

ΔνD /Δνv,observ.

∞

1

0

0.84

0.552

0.656

12.01

0.993

0.083

0.78

0.527

0.675

6.01

0.972

0.162

0.72

0.500

0.694

4.00

0.941

0.235

0.66

0.472

0.715

3.00

0.904

0.301

0.60

0.442

0.736

2.70

0.886

0.327

0.54

0.410

0.758

2.40

0.863

0.359

0.48

0.375

0.780

1.80

0.794

0.441

0.42

0.338

0.804

1.50

0.742

0.494

0.36

0.299

0.829

1.20

0.672

0.559

0.30

0.257

0.855

1.14

0.655

0.574

0.24

0.212

0.882

1.08

0.637

0.589

0.18

0.164

0.910

1.02

0.618

0.605

0.12

0.113

0.939

0.96

0.597

0.622

0.05

0.050

0.984

0.90

0.575

0.639

0

0

1

and others [21, 22]. In the limits a = ∞ and a = 0, expression (2.50) describes the Doppler and the Lorentz line proﬁle, respectively. The integral H ( a, u) is also computed numerically. Detailed tables and graphs are published in practically all books on spectroscopy (e.g. [1–6] and others). Since H ( a, u) is a function of only two parameters, the above problem of discriminating between the contributions from the different broadening mechanisms proves somewhat simpler than in the general case of proﬁle (2.47). If the temperature of the emitting gas particles is known from independent measurements (ΔνD is known) or if the broadening cross section of the line under study is known (ΔνL is known), the lacking parameter can be found from the measured Voigt proﬁle halfwidth Δνv,observ. , for example, by means of Table 2.2 [21] (cited from [22, p. 134]) presented below. A great number of papers and reviews have been devoted to calculations similar to those in [21]. To illustrate, Bakshi and Kearney [23] presented more detailed data, than those listed in Table 2.2, including the Doppler-to-collision width ratio in the Voigt proﬁle not only at its half-height, but also for various fractions of its height. These data make it possible to discriminate between the contributions more accurately and reliably.

43

44

2 Basic Concepts and Parameters

2.3 Absorption in Lines

The reduction of the intensity of light, whose spectrum falls within a narrow interval of frequencies (ν, ν + dν) or wavelengths (λ, λ + dλ), on passage through a homogeneous medium of length l and u → l atomic transition line (u stands for the upper level) proceeds exponentially by the Bouguer–Lambert–Beer (BLB) law I (ν, z = l ) = I (ν, z = 0) exp{−χlu (ν)l }.

(2.51)

The quantity χlu (ν) is called the spectral absorption coefﬁcient, and the spectral absorption index χlu (ν)l is a measure of the spectral optical density of the medium. The various versions of the absorption method for determining concentrations of particles are based on the well-known relationship between integral absorption coefﬁcient χlu = χlu (ν) dν

(2.52)

and the product Blu Nl [6, 24]: g Nu χlu = (hνul /c) Blu Nl 1 − l , gu Nl

(2.53)

where νul is the center frequency of the emission line proﬁle, Blu is the Einstein coefﬁcient for absorption, and Nl and Nu are the particle concentrations on the lower and the upper level, respectively. Similarly, use can be made in expression (2.53) of the Einstein coefﬁcients Aul for the spontaneous emission of radiation and Bul for the stimulated emission of radiation: Blu =

gu B , gl ul

Aul =

3 gl 8πhνlu Blu . gu c3

(2.54)

g

u If gul N Nl 1 and the stimulated transitions are unimportant, the integral χlu directly yields the quantity Blu Nl . Along with the Einstein coefﬁcients, use is also made of the dimensionless oscillator strengths

f lu = (πe2 )−1 me hνlu Blu =

gu me c3 Aul , 2 gl 8π2 e2 νlu

(2.55)

where me and e are the electron mass and charge, respectively. In terms of the wavelength λ Aul = 6.66 × 1013

gl 1 f , gu λ2 lu

(2.56)

2.3 Absorption in Lines

provided that λ is measured in nanometers and Aul , in s−1 . Relations (2.55) and (2.56) determine the oscillator strength in absorption. The oscillator strength for emission is f ul = − f lu gl /gu . Apart from the fact that the oscillator strengths are dimensionless, some conveniences of using exactly these strengths for the quantitative characterization of the absorption (and emission) of radiation are due to the availability of some universal relations, speciﬁcally sum rules, for them. The most general and universal among them is the Thomas–Reiche–Kuhn sum rule

∑ f mn = z ,

(2.57)

m

where z is the number of electrons in the system (atom, molecule) and summation is taken over all the states of the radiative and absorptive transitions and all the states of the inner shells and continuous spectrum. Expression (2.57) in so general a form is difﬁcult to apply in practical measurements, but approximate versions of this sum rule frequently prove useful. To illustrate, for a single electron outside of the closed shells (the so-called optical electron), whose transitions are studied most frequently [7, 24],

∑ f mn ≈ 1 .

(2.58)

m

Expressions for the absorption coefﬁcient often include the so-called line strength Sul , which is associated with Aul and is the same for both the absorption and emission of radiation: Sul = Aul

3hλ3 gu , 64π4

Sul = Slu .

(2.59)

A summary of the relations between the quantities characterizing the probabilities of optical transitions is presented in Appendix B. Ignoring the stimulated emission of radiation, also practised is the notion of the absorption cross section, when the absorption coefﬁcient refers to a single particle on the lower level [6, 7, 24]: σlu (ν) = χlu (ν)/Nl .

(2.60)

The relationship between the integral absorption coefﬁcient (2.52) and the spectral absorption coefﬁcient for the center frequency of the line proﬁle, namely, χlu (ν = νlu ) = χ0,lu , can also prove useful. The form of this relationship depends on the actual line broadening mechanism [6]. In

45

46

2 Basic Concepts and Parameters

the case of Doppler broadening, 1 π 1/2 χ0,lu ΔνD , 2 ln 2 1/2 gu c2 Aul gl Nu ln 2 , = Nl 1 − 2 Δν π gl 4πνlu gk Nl D

χlu = χ0,lu

(2.61)

where ΔνD is the Doppler width at half maximum of proﬁle (2.33). In terms of wavelength: if λ is measured in nanometers, Alu in s−1 , Nl in cm−3 , and χ in cm−1 , then, ignoring the stimulated emission of radiation, χ0,lu = 1.23 × 10−33

gu Aul 4 λ N. gl ΔλD lu l

(2.62)

In the case of Lorentz broadening, χlu = (π/2)χ0,lu ΔνL , χ0,lu

gu c2 Aul = N gl 4π2 νul ΔνL l

g Nu 1− l gu Nl

,

(2.63)

where ΔνL is Lorentz line width (2.18). If collision broadening prevails over natural, then ΔνL = 4σlu Np (kB T/M) /2 , 1

is the cross section of the line-broadening collisions and N is where σlu p the density of the broadening-producing particles of mass M. With the measurement units being the same as in the case of expression (2.62),

χ0,lu = 0.8410−33

gu Aul 4 λ N. gl ΔλL lu l

(2.64)

It is important from the standpoint of diagnostics that, according to expression (2.51), the measurement of the relative change in the intensity of light passing through the object under study yields the absolute value of the absorption coefﬁcient and, correspondingly, the density of the absorbing states (ignoring the stimulated transitions).

2.4 Emission in Lines. Optical Density Manifestations

In the tradition mentioned in Section 2.1, the radiation power of the source is in spectroscopy usually called intensity (although this is, strictly speaking, not so). For this reason, we will hereinafter follow the established practice and take the intensity I to mean a quantity proportional to

2.4 Emission in Lines. Optical Density Manifestations

the power p, and talk about the actual relationship between I and p only when dealing with absolute measurements. With this reservation, by the intensity of a line of ﬁnite spectral width we will mean the integral radiation power due to the entire line proﬁle. In this case, ignoring stimulated transitions, the intensity of radiation emitted by an elementary volume dV of plasma in the u → l transition is v Iul dV = Aul hνul Nu dV.

(2.65)

If measurements are taken in absolute measure, Aul Nu will then be determined in the same measure. This makes the emission method of spectroscopy a powerful yet simple diagnostic tool. However, the simplicity of such measurements often proves limited when working with extended objects of ﬁnite optical density. In such situations, account should also be taken of the absorption and re-emission of light. A spontaneously re-emitted quantum may have a different frequency, polarization, and propagation direction, and it may dissipate in quenching processes. The totality of such nonlocal processes in the phase space is considered within the framework of the general radiation transfer problem [25, 26]. In a number of particular cases, re-absorption within the limits of the light source itself, that is, self-absorption, reabsorption (for inhomogeneous objects) can be taken into account comparatively easily. This is true, ﬁrst of all, for objects that are homogeneous in the observation direction. The light of a homogeneously glowing plasma column of length L observed along its axis can serve as an example. In such a case, one can 1 ( ν ). Consequently, introduce the concept of intensity per unit length, Iul 1 ( ν )dz is the intensity of radiation emitted from an elementary length Iul z + dz of the plasma column in the frequency range ν, ν + dν within the limits of the line proﬁle. If absorption with a coefﬁcient of χlu (ν) takes place there, light emitted from this elementary length will leave the col1 ( ν ) exp { − χ ( ν )( L − z )}dz. umn with an intensity of dIul (ν, z, L) = Iul lu Integrating with respect to z, we get the following expression for the intensity of the column: 1 Iul (ν) = Iul (ν)

1 − exp {−χlu (ν) L} . χlu (ν)

(2.66)

If the optical density is low (χlu (ν) L 1), intensity grows higher in proportion to the length of the column: 1 Iul (ν) = Iul (ν) L.

(2.67)

47

48

2 Basic Concepts and Parameters

By contrast, where the optical density is high, intensity is no longer dependent on the extension of the column: 1 (ν)/χlu (ν). Iul (ν) = Iul

(2.68)

It also follows from expression (2.66) that self-absorption leads to distortion of the emission line proﬁle. Since absorption at the center of the proﬁle is stronger than in the proﬁle wings, the proﬁle becomes ﬂattened near the center, so that its effective width increases in contrast to the case of optically thin plasma. The total spectral line intensity is obtained by integrating expression (2.66) with respect to frequency: 1 Iul = Iul LS∗ (χ0,lu L),

(2.69)

1 is the line intensity per unit length in the absence of selfwhere Iul absorption (the so-called primary intensity [6, 27]), the result depending on the actual broadening mechanism. For the Doppler line proﬁle,

S∗ (χ0,lu L) = S(χ0,lu L) – the Ladenburg and Levi function. For the Lorentz line proﬁle, S∗ (χ0,lu L) = S (χ0,lu L) – the Ladenburg and Reiche function. The functions S and S are tabulated (see, e.g. [6, 27]). Although the present-day availability of computers and mathematical software makes the computation and integration of expression (2.66) with respect to frequency a fairly routine procedure, the results of the early calculations of the functions S and S are convenient to use in practice. These data are listed in Table 2.3. Where optical densities are low (χ0,lu L < 3), the functions S and S are similar. The same, consequently, can be said about the case of mixed broadening with the Voigt proﬁle. Direct numerical computations of selfabsorption in the case of mixed broadening for a wider range of optical densities yield functions similar to the van Held ‘growth curves’ (Section 3.2.1). If self-absorption is present, an emitted photon leaves the plasma with a probability other than unity, exciting the particles neighboring the emitting one and thus retarding the radiative decay of the ensemble of the excited states. For this reason, when dealing with plasma of ﬁnite optical density, use is frequently made of the effective lifetime approximation for the emitting state, introducing the concept of the effective spontaneous radiative decay rate A∗ul = Aul Θ( x ),

(2.70)

2.4 Emission in Lines. Optical Density Manifestations Table 2.3 Ladenburg–Levi (S) and Ladenburg–Reiche (S ) functions. S (L-R)

χ0,lu L

S (L-R)

χ0,lu L

S (L-L)

S (L-L)

0

1

1

2

0.674

0.1

0.976

0.964

2.4

0.635

0.556 0.507

0.2

0.952

0.933

3

0.586

0.45

0.4

0.909

0.872

4

0.524

0.372

0.6

0.87

0.818

5

0.477

0.316

0.8

0.835

0.768

6

0.44

0.276

1

0.802

0.725

7

0.41

0.246

1.2

0.772

0.683

8

0.386

0.222

1.4

0.744

0.646

9

0.365

0.202 0.186

1.6

0.719

0.616

10

0.347

1.8

0.696

0.584

20

0.249

where Θ( x ) is the probability that a photon emitted at the point x will leave the plasma, which is related to the coefﬁcient of absorption at the line center, χ0,lu . The value Θ( x ) ≈ 1 corresponds to an optically thin (for the given transition) medium. For a homogeneous medium in the form of a half-space in the case of disperse line proﬁle, √ Θ( x ) = 3 πχ0,lu x

−1

,

(2.71)

and in the case of Doppler line proﬁle, Θ( x ) =

4χ0,lu x

π ln(χ0,lu x )

−1 .

(2.72)

For a layer of thickness L, in the case of disperse line proﬁle, Θ( x ) =

√ 3 πχ0,lu x −1 + 3

πχ0,lu ( L − x )

−1 .

(2.73)

For a point on the axis of along cylinder of radius R, in the case of disperse line proﬁle, −1 , Θ(0) = 2 χ0,lu R

(2.74)

and in the case of Doppler line proﬁle, Θ (0) =

4χ0,lu R

π ln(χ0,lu R)

−1 .

(2.75)

49

50

2 Basic Concepts and Parameters

The effective lifetime approximation holds subject to the condition that the densities of the emitting and absorbing particles are smooth functions of distance. This in turn means that the characteristic variation distance of these densities is longer than the average free path length of the photon. Thus, this approximation holds better for media of high optical density. Comparison with the numerical solutions of the radiation transfer equation [26] that formulas (2.71)–(2.75) give satisfactory results at χ0,lu x, χ0,lu R ≥ 3.

2.5 Emission and Absorption in Continuous Spectrum

As already stated, the emission line proﬁle of optically dense plasma can substantially differ from the line proﬁle of the ‘primary’ radiation for a thin layer. As the density χ0,lu L grows higher, the intensity of the line center gets saturated, reaching, according to Kirchhoff’s law of spectral radiation, the intensity of a black body with its temperature equal to the ‘population temperature’ of the upper and the lower level (see Section 1.3). The line proﬁle broadens, the overlapping of adjacent lines is possible, and, as the density grows still higher, a continuous spectrum is formed, or separates sections of such a spectrum. Such spectra, line spectra by origin as they are, can be described on the basis of the relations presented in the preceding section [6, 11, 12, 27, 28]. Another class of spectra differs from the preceding one by the fact that even the ‘primary’ radiation for some types of transition has a continuous spectrum in a wide frequency range. Such spectra originate if one or both (the upper and the lower) states of the transition and have continuous regions of allowed energy states. If only one of the states is discrete, the transition is then called bound–free (bf ). If both the states are continuous, they then talk about free–free (ff ) transitions. The conventional character of the distinction of this class of spectra is obvious, because, on the one hand, the motions of particles in the state of continuous energy spectrum are not free in the true sense of the word and, on the other hand, the states of discrete spectrum also have a ﬁnite width. For this reason, the notions of line and continuous spectra are more likely to be distinguished by the mechanisms dominating in the formation of the frequency dependence proﬁles of absorption and emission of radiation in optical transitions. For line spectra, these are the Doppler, natural, and collision broadening mechanisms that disturb the coherence of the radiation of the oscillators.

2.5 Emission and Absorption in Continuous Spectrum

Figure 2.2 Free–free (ff ), free–bound (fb), and bound–free (bf ) transitions.

In terms of bound–free transitions it is the absorption of a photon resulting in the ionization (photoionization; the inverse process – electronion recombination) or dissociation (photodissociation; the inverse process – photorecombination) of atoms or molecules. The same term is applied to spectra due to the absorption of light resulting in the detachment of an electron from a negative ion (photodetachment; the inverse process – photoattachment). Included in this class are also transitions occurring in excimer (exciplex) molecules wherein the excited state is bound and the ground state is repulsive. The bound–free transitions are also called the ‘recombination continuum’. Free–free transitions correspond to the emission (absorption) of light dependent upon a change in the velocity of a free electron during the course of its collision with ions or neutral atoms and also under the effect of an external electromagnetic ﬁeld. Figure 2.2 schematically illustrates all these transitions. The semiclassical theory of the ff, bf, and fb transitions rests upon the early calculations by Kramers [29] and forms the basis for the analysis and interpretation of continuous spectra [11, 12, 22, 30, 31]. A very good and concise exposition of these questions can be found in [32–34]. In the

51

52

2 Basic Concepts and Parameters

following sections we present individual data of use in practical plasma spectroscopy. 2.5.1 ff Bremsstrahlung Emission

To describe ff spectra, use is made of the concept of the differential emission cross section dependent on the electron velocity ve : ff

ff

dσω (v) =

dσω dω, dω

(2.76)

where ω = 2πν. The cross section determines the power emitted by a single electron in the frequency interval ω, ω +dω into the full angle 4π in a medium with a density of N atoms or ions (the electron–electron interaction causes no bremsstrahlung): ff

ff

δWω = hωNve dσω (ve ).

(2.77)

Here, and elsewhere in the expressions of this section, appears the electron velocity ve , owing to the fact that the velocity of heavy particles is much lower. After Kramers, for the electron-ion interaction (the reader can become acquainted with the consistent solution of this problem in the book by Landau and Lifshits [35, 36]), we have Z 2 e6 16π ff dσω (ω ) = √ dω, 3 3 m2e v2e c3 h¯ ω

(2.78)

where Z is the ion charge. The intensity (power) of bremsstrahlung emitted by 1 cm3 of plasma in the frequency interval dω is obtained by integrating expression (2.77) over electron velocities with the distribution function f (v) normalized to unity. The minimal velocity with which the electron can emit a quantum of h¯ ω is ve∗ = (2¯hω/me )1/2 ff

Iω =

∞

h¯ ωNne f (ve )dσ (ve ) dve .

(2.79)

ve∗

For a Maxwellian velocity distribution function f (ve ), the result of averaging is [34] Z2 Ni ne ff h¯ ω ff Iω = C √ dω , g exp − (2.80) kTe Te 1 16 2π /2 e6 1 C= = 1.08 × 10−38 erg · cm3 · K /2 . 1/2 3/2 3 3 3 me c k B

2.5 Emission and Absorption in Continuous Spectrum Table 2.4 The values of the factor gff as a function of the dimensionless parameter x = ω/ω ∗ .

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

gff

∞

2.01

1.61

1.34

1.13

0.97

0.81

0.68

0.53

0.36

0

Formula (2.61) also includes the factor gff referred to as the Gaunt factor. It renders the result of the quasiclassical analysis more precise and is calculated by quantum-mechanical methods. Various authors present different formulas and graphs for gff (for more detail, see [7]). If we denote by ω ∗ = mv2e /2¯h = Ek /¯h the limiting frequency that can be emitted by an electron with a kinetic energy of Ek , the values of the factor gff as a function of the dimensionless parameter x = ω/ω ∗ can be seen below in Table 2.4 [33] (computed in the Born approximation). The emission spectrum is determined by the exponential term. Formula (2.80) gives the spectral intensity of light emitted in an isotropic manner (thanks to averaging over a great number of electron-ion pairs) into the full angle 4π, which corresponds to the total power (Section 2.1) from 1 cm3 of plasma. The same results can also be used for calculating the intensity of transitions caused by collisions with neutral atoms. In case, when calculating cross section (2.78), the Coulomb scattering cross section σC should be replaced by the transport cross section σt . The Coulomb cross section and its velocity dependence depend on the approach distance rC at which the electron kinetic energy is equal to the interaction potential energy: 2 = me v2e /2 = Ze2 /rC ; σC = πrC

4πZ2 e4 . (me v2e )2

(2.81)

The ratio between the bremsstrahlung cross sections for electron scattering by ions and neutral atoms is ff

dσω,n ff

dσω,i

=

πa20 Z2 σt

m v2

2IH Ek

2 .

(2.82)

e e Here Ek = is the electron energy, IH = 2π2 e4 me /h2 = 2 − 19 21.760 × 10 J(13.6 eV) is the ionization potential of the hydrogen atom, a0 = h2 /(4π2 me e2 ) is the Bohr radius, σt = σC (1 − cos θ ), and cos θ is the average scattering cosine. With the typical ratio σt /πa20 ∼ 1– 10, electrons with an energy of a few electron-volts emit, on a per-heavyparticle basis, 10–100 times more when scattered by ions than by neutral particles. That is, the bremsstrahlung emission (and absorption)

53

54

2 Basic Concepts and Parameters

intensity in the case of electron scattering by ions is comparable with that in the case of scattering by neutrals when the degree of ionization δ ∼ 10−1 –10−2 . In weakly ionized plasma, δ ≤ 10−3 , the absence of multiply charged ions is typical, and electron bremsstrahlung of neutral particles is predominant. 2.5.2 ff Bremsstrahlung Absorption

The above data on bremsstrahlung emission can also describe bremsstrahlung absorption, if use is made of the detailed balancing principle for reciprocal processes, bearing in mind that the total balance of the ff processes is accounted for by spontaneous emission and absorption and stimulated emission of radiation. To do so, we introduce, in addition to the deﬁnition of bremsstrahlung emission cross section (2.76), the ff concepts of the coefﬁcients of true bremsstrahlung absorption, aω , and ff stimulated bremsstrahlung emission bω , which describe the absorption and stimulated emission under the effect of an external radiation for a single interacting electron-ion (atom) pair. The spectral power of radiation in the frequency interval ω, ω + dω incident through an elementary surface and within an elementary solid angle of dΩ is Iω dωdΩ. Bearing in mind the law of conservation of energy me v e me v2e = + h¯ ω , 2 2 2

(2.83)

and the Einstein relations [33, 34] bω ( v e ) = ff

ve ff a ω ( ve ), v e

(2.84)

π2 c2 v e dσω (v e ) , dω ω2 we ﬁnd from Kramers formula (2.78) that bω ( v e ) = ff

ff

Z 2 e6 16π3 ff a ω ( ve ) = √ 2 3 3 me c¯hω 3 ve

.

(2.85)

(2.86)

Conversion from the true (elementary) absorption to the spectral abff ff sorption coefﬁcient χω is carried out by multiplying aω into ne N and averaging over electron velocities. In the case of Maxwellian velocity

2.5 Emission and Absorption in Continuous Spectrum

distribution function f (ve ), Z2 ne Ngff √ , cm−1 , (2.87) ω 3 Te 1 16π2 2π /2 e6 1 C1 = = 1.45 × 1010 cm5 · s−3 · K− /2 . 1 3 3 3 me/2 ckB /2 h¯ ff

χω = C1

By analogy with expression (2.80), the Gaunt correction factor gff is introduced here as well. 2.5.3 fb Recombination Emission

In this process, photorecombination, a freely moving electron is captured into a bound atomic state with a negative energy of En , emitting in the process a light quantum h¯ ω: me v2e /2 = | En | + h¯ ω.

(2.88) fb

The photorecombination capture cross section σn can be obtained in the quasiclassical approximation, by analogy with expression (2.78), with the energy region extended to cover negative energies of the bound states. For hydrogen-like atoms, this leads to the expression [34] e10 Z4 16π fb . σn = √ 3 3 c3 h¯ 4 me v2 ωn3

(2.89)

The spectral power emitted by electrons moving with velocities in the interval Sv, v + dv on their capture onto the level n in 1 cm3 of plasma is given by fb

fb

Iωn = h¯ ωNi ne σn f (ve )ve dve .

(2.90)

For a Maxwellian velocity distribution function f (ve ), Z2 ne Ni 2xn h¯ ω fb exp { xn } exp − dω, Iωn = C √ n kB Te Te where xn =

me e4 Z 2 2¯h2 n2 kB Te

=

IH Z2 kB Te n2

=

En kB Te

(2.91)

and C is the same constant as in

expression (2.80). Thus, for the same reasons (reduction of the electron density with energy in a Maxwellian velocity distribution) as in the case of bremsstrahlung emission, the intensity of the recombination emission of radiation decreases exponentially with frequency. However, in the case of photorecombination this relates to the capture of an electron into a bound

55

56

2 Basic Concepts and Parameters

Figure 2.3 Spectra of ff and fb transitions.

state of ﬁxed n. Emission of radiation of the same frequency ω can also occur upon capture onto other levels differing in En , provided that condition (2.88) is satisﬁed. Therefore, the intensity of the spectrum is found by summation of (2.91) over all possible values of n: fb

Iω =

∞

∑∗ Iωn . fb

(2.92)

n

the lower limit of sum, n∗ , is determined by the condition | En∗ | ≤ h¯ ω. For different frequencies of the recombination emission spectrum, the intensity is contributed to by different number of capture levels. As frequency grows, the number of such contributing levels increases, which is manifest in the ‘sawtooth’ shape of the spectrum, as schematically illustrated by Figure 2.3. The dashed lines in the ﬁgure indicate the contribution from individual summands (2.91), and the solid ones from their sums (2.92). Typically at not very high electron temperatures it is levels with adjacent values of n that are mainly summed.

2.5 Emission and Absorption in Continuous Spectrum

2.5.4 Absorption Cross Section in bf Photoionization bf

The cross section σn for photoionization as a process inverse to photorecombination can be obtained from expression (2.89), subject to the detailed balancing conditions with due regard for the conservation of energy (2.88). This leads to what is known as the Milne formula [30] bf

σn =

gZ g Z −1

h¯ 2 ω 2 fb σn . 2 ( me ve c )

(2.93)

Here gZ , gZ−1 are the statistical weights of the ions of the corresponding multiplicities. For a neutral atom Z = 1, and for a hydrogen-like atom, the statistical weight of the level n is gZ−1,n = 2n2 , gZ = 1, whence it follows that 8π e10 me Z4 bf . σn = √ 3 3 c¯h6 ω 3 n5

(2.94)

If ωn is the limiting frequency for photoionization from the level n, ωn = | En |/h, | En | = me e4 Z2 /(2h2 n2 ), then σn = 7.9 × 10−18 bf

n ω n 3 cm2 · Z2 ω

(2.95)

With the frequency rize, starting at the threshold value, the photoionization cross section ∼ ω −3 . The numerical coefﬁcient in expression (2.95) corresponds to the maximum photoionization cross section of the unexcited hydrogen atom (n = 1, Z = 1, ω1 /ω = 1). The experibf mental value σ1 = 6.3 × 10−18 cm2 agrees well with expression (2.95). Since expressions (2.93) and (2.89) are both obtained on the basis of the quasiclassical approximation through generalization of formula (2.78) for bremsstrahlung emission, then, according to the premises of this approximation, the greater the value of n, the closer the approximation compared to the exact quantum-mechanical solution. As in the case of bremsstrahlung emission, the appropriate Gaunt correction factors g f b and gb f are introduced into the quasiclassical formulas for photorecombination and photoionization, respectively. These quantities here are also of the order of unity and depend only weakly on the electron energy.

57

58

2 Basic Concepts and Parameters

2.5.5 Emission and Absorption of Radiation in the Case of Joint Action of the ff Bremsstrahlung and fb Recombination Mechanisms

In deﬁning formulas (2.79) and (2.90) for the spectral intensities of bremsstrahlung and recombination radiation, respectively, no restrictions have been imposed as to the equilibrium state of plasma. To obtain their compact analytical modiﬁcations, (2.80) and (2.91), the electron velocity distribution has been assumed, for simplicity, to be Maxwellian. Therefore, these relations are, generally speaking, also valid under nonequilibrium conditions. However, the Maxwellian electron velocity distribution approximation holds in a relatively wide range of plasma conditions corresponding to the LTE and PLTE models Section 1.3 and, consequently, expressions (2.80) and (2.91) are important on their own account. Spectra (2.80) and (2.91) have common exponential and pre-exponential factors. For comparison, Figure 2.3 also shows a bremsstrahlung spectrum, the intensities for each of the spectra being given in different relative units. Considering expression (2.92), the actual ratio between the spectral intensities of the recombination and bremsstrahlung radiation (accurate up to the Gaunt factors) fb

Iω ff

Iω

∞

=∑

n∗

2xn n

exp { xn }

(2.96)

should be calculated with due regard for the ‘teeth’ corresponding to different values of n. As evident from Figure 2.3, in the region of relatively not very high frequencies of transitions involving the capture of electrons onto the excited atomic levels with n∗ 1, the ‘teeth’ are clustered. Then, according to expressions (2.91) and (2.92), xn∗ ≈ | En∗ |/kB Te ≈ h¯ ω/kB Te and summation in (2.96) can be replaced by integration, which is tantamount to the ‘smoothing’ of this region of the recombination spectrum. Further, taking into consideration the fact that, according to the deﬁnition of the quantity xn in expression (2.91), dxn = −2 xnn dn, the intensity ratio in the smoothed region is given by fb

Iω

ff Iω

≈

xn∗ 0

dxn exp { xn } ≈ ex − 1,

x=

h¯ ω . kB Te

(2.97)

In the region h¯ ω < 0.7kB Te , a greater contribution comes from recombination radiation, and at h¯ ω > 0.7kB Te , from bremsstrahlung. The question of the absorption spectrum in the case of joint action of the ff and bf mechanisms is solved similarly. In this case, it is the absorp-

2.5 Emission and Absorption in Continuous Spectrum

tion coefﬁcients that are summed: ff

bf

χω = χω + χω .

(2.98) bf

The difference being, the coefﬁcient χω is in its turn found through summation of the photoionization cross sections extended over the levels of the atom: bf

χω =

∞

∑∗ Nn σn , bf

(2.99)

n

where the level populations Nn serve the purpose of the weight factors. To obtain analytical formulas for absorption, it is necessary to know these factors. The solution of such a problem is usually achieved under the assumption of equality between the level excitation temperatures and the ionization and electron temperatures [33, 34] (Section 1.2). This corresponds to the plasma LTE model. Note, however, that the ensuing narrowing of the applicability region of the formulas, in comparison with those for emission, does not, in many cases, become critical. Characteristic of highly excited levels is their close connection with the continuum, so that the equality between the excitation and electron temperatures can remain valid over a wide range of plasma parameters. The ‘smoothing’ procedure is also valid for this group of levels, and so the relation between the contributions from the fb and ff absorption proves to be the same as in the case of emission (2.99). To conclude, we note that we have only touched upon questions relating to continuous bremsstrahlung and recombination spectra for hydrogen-like atoms. The results for these cases are obtained in a most simple form; nonetheless, they truly reﬂect the gist of the matter for multielectron atoms and singly charged negative ions. The deviations from Kramers’ theory are due to both quantum-mechanical effects (usually allowed for by the Gaunt correction factors) and the departure of the external electron-atom interaction potential from the Coulomb potential. The reader can become better acquainted with questions relating to continuous spectra for multielectron atoms in [7, 11, 12, 22, 30, 33], for negative ions in [37–39] (see also Section 5.3.6), and for excimer molecules in [40]. A detailed theoretical analysis and bibliography of the photodissociation, photorecombination and bremsstrahlung spectra of molecules and molecular ions can be found in the review [41].

59

60

2 Basic Concepts and Parameters

2.6 Scattering of Light

When high-intensity light passes through plasma, scattering effects can occur giving rise to new radiation that propagates at angles greater than the diffraction angle relative to the incident direction. The speciﬁc character of this phenomenon, quite general as it is, is conditioned in plasma by the presence of free charges and excited particles. A vast body of literature devoted to the questions of light scattering by plasma already exists, including that forming part of the general manuals on low-temperature plasma diagnostics [25, 31, 42]. 2.6.1 Thomson Scattering on a Free Electron

If the electron at the point r = 0 is in the electric ﬁeld E(r, t) = eE0 exp { − i(ω0 t − k0 r)}

(2.100)

of a plane wave (k0 is the wave vector, ω0 is the circular frequency, and e is the unit polarization vector), it executes an oscillatory motion with the acceleration v˙ (t) = e

e E0 exp { − iω0 t}. me

(2.101)

In the case of nonrelativistic motion (see the remarks following formulas (2.107), (2.110)), the oscillating electron produces in the wave zone the ﬁeld Es (r, t) =

e (v˙ × n) × n, c2 r

(2.102)

where n is the unit vector (see Figure 2.4) in the propagation direction of the radiation from the oscillating electron (scattered radiation with the wave vector ks ). Using the quantity v˙ from expression (2.101), we get Es (r, t) =

1 e2 E0 exp { − iω0 t}(e × n) × n. r me c2

(2.103)

Given that the quantity me c2 = r0 = 2.8 × 10−13 cm is the classical elece tron radius and |(e × n) × n| = sin θ (Figure 2.4), we ﬁnd that the amplitude of the scattered ﬁeld is 2

QEs =

r0 E0 sin θ. r

(2.104)

2.6 Scattering of Light

We deﬁne the scattering cross section dσ as the ratio between the lightoscillation-period-averaged powers (intensities) of the radiation scattered into the solid angle dΩ and the incident radiation: dσ = ( Is /I0 )r2 dΩ.

(2.105)

Since Is ∼ Es2 , I0 ∼ E02 (Section 2.1), then, considering expression (2.104), the effective cross section for the scattering of light by the electron in the direction θ into the solid angle dΩ is dσ = r02 sin2 θ = σe . dΩ

(2.106)

The effective cross section is independent of frequency. Integrating over the full solid angle, we obtain what is known as the Thomson scattering cross section for a polarized incident light wave σTh =

8π 2 r ≈ 6.65 × 10−25 cm2 . 3 0

(2.107)

(Classical expression (2.107) is a particular case of the Klein–Nishina formula ⎧ 2¯hω ⎨ 8π 1 − + . . . , h¯ ω me c2 3 me c2 2 (2.108) σKN = r0 ⎩π mc2 ln 2¯hω + 1 , h¯ ω me c2 2 h¯ ω m c2 e

The rest energy of the electron is me c2 = 817.600 × 10−16 J (511 keV), and so the nonrelativistic approximation in the optical region is justiﬁed). One can see from expressions (2.105), (2.106) and Figure 2.4. that the propagation direction of the scattered radiation is determined by the vector (e × n) × n, its electric ﬁeld vector lies in the same plane as the vectors e and ks and its intensity varies as sin2 θ . If the incident wave is unpolarized, it is then necessary to average expression (2.106) over the angles θ, as a result of which it turns out that dσ 3 = σTh (1 + cos2 Θ). dΩ 4

(2.109)

Here Θ is the scattering angle, cos θ = cos φ sin Θ. In the direction Θ = π/2 the scattered radiation is linearly polarized, whereas in the directions Θ = 0 and Θ = π it is unpolarized. The angle function R(Θ) = 34 (1 + cos2 Θ) is called the Rayleigh phase function. We now assume that the scattering electron has an initial velocity. Owing to the Doppler effect, the electron responds to the incident wave as it would to a wave of shifted frequency. The frequency shift is deﬁned by

61

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2 Basic Concepts and Parameters

Figure 2.4 Vectors of the incident wave and the wave scattered on a free electron.

expression (2.29) with vz = vk0 , where vk0 is the electron velocity projection onto the vector k = ks − k0 , ks being the wave vector of the scattered radiation. The Doppler shift of the electron radiation frequency observed in the scattered light corresponds to the velocity equal to the electron velocity projection vks onto the propagation direction of the observed (scattered) radiation with the wave vector ks . In the ﬁnal analysis, the total Doppler shift of the scattered light frequency relative to the frequency of the incident light is Δω = (k · v).

(2.110)

(The frequency-independent long-wavelength Compton shift ΔλC = 0.048 sin2 (θ/2), is usually negligibly small in comparison with the shifts due to thermal motion.) Here k = ks − k0 , that is, the frequency shift is determined by the velocity vector projection onto the direction of the difference vector and not onto the propagation direction of either incident or scattered radiation. As a rule, the frequencies of the incident and the scattered radiation differ but little, ks ≈ k0 , so that

|k| = k ≈ 2

Θ Θ ω0 4π sin sin . = c 2 λ 2

(2.111)

The presence of the Doppler shift in scattering on a free electron is widely used in determining the parameters of the electronic component of plasma and will be considered in greater detail in Section 7.5.

References

2.6.2 Scattering on a Bound Electron

The motion of an electron bound in an atom in the ﬁeld of a polarized plane wave is described by the well-known expression d e d2 r + γ r + ωR2 r = eE0 exp { − iω0 t}, dt me dt2

(2.112)

where γ is the decay constant and ωR is the eigenfrequency of the electron oscillations. With this equation of motion, the scattering cross section is given by ω04 dσ = r02 sin2 θ 2 . dΩ (ωR − ω02 )2 + γ2 ω02

(2.113)

When ω0 ωR , the cross section corresponds to that for scattering by a free electron, (2.106). The case where ω0 ωR corresponds to the Rayleigh scattering with its characteristic wavelength dependence ∼ 1/λ4 : dσ ω0 4 = r02 sin2 θ . (2.114) dΩ R ωR When ω0 ≈ ω R , resonance scattering (resonance ﬂuorescence) ocurrs. The classical cross section of this process ω2 dσ = r02 sin2 θ 20 dΩ γ

(2.115)

can exceed cross section (2.106) for scattering from a free electron. The resonance ﬂuorescence techniques in plasma spectroscopy are considered in Section 3.4.1. References

1 V. Demtreder. Laser Spectroscopy. Basic Principles and Techniques (in Russian). Moscow: Nauka (1985). 2 V.V. Lebedeva. Optical Spectroscopy Techniques (in Russian), pp. 343–372. Moscow: Moscow State University Press 1977). 3 G.G. Slyusarev. On the Possible and Impossible in Optics (in Russian). Moscow: Fizmatgiz (1960).

4 I.I. Sobelman. Once More on the Possible and Impossible in Optics (in Russian). UFN, 113, No. 4, pp. 701– 705 (1974). 5 B.I. Stepanov. An Introduction to Modern Optics. Photometry. On the Possible and Impossible in Optics (in Russian), pp. 177–250. Minsk: Nauka i Tekhnika (1989).

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2 Basic Concepts and Parameters 6 S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). 7 I.I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Fizmatgiz (1963). 8 A. Mitchel and M. Zemansky. Resonance Radiation and Excited Atoms (in Russian). Moscow: GTTI (1937). 9 M.A. El’yashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 10 S.E. Frish, Ed. Gas-Discharge Plasma Spectroscopy. Leningrad: Nauka (1970). 11 H.R. Griem. Plasma Spectroscopy. N.Y.: Mc Graw-Hill (1964). 12 H.R. Griem. Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). 13 V.S. Fursov and A.A. Vlasov. ZhETF, 6, p. 750 (1936). 14 V.S. Fursov and A.A. Vlasov. ZhETF, 9, p. 783 (1939). 15 L.A. Vainshtein, I.I. Sobelman et al. DAN SSSR, 90, p. 757 (1953). 16 L.A. Vainshtein, I.I. Sobelman et al. Optika I Spektroskopiya, 6, p. 440 (1959). 17 T.T. Karasheva, D.K. Otorbayev, V.N. Ochkin et al. Doppler Broadening of Spectral Lines and Velocity Distributions of Excited Atoms and Molecules in Nonequilibrium Plasma. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1985). 18 G.N. Polyakova and A.I. Ranyuk. Recovering Velocity Distributions of Excited Particles from the Doppler Broadening of Spectral Lines. (in Russian). Preprint KhFTI, No. 81–1. Kharkov (1981) 19 R. Dicke. Phys. Rev., 82, p. 472 (1953). 20 R.S. Eng, A.R. Calava, T.C. Hartman. Appl. Phys. Lett., 21, p. 303 (1972). 21 J.T. Davies and J.M. Vaughan. Astrophys. J., 137, p. 1302 (1963). 22 W. Lochte-Holtgreven , Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). 23 V. Bakshi and R.J. Kearney. New Tables of the Voigt Function. JQSRT, 42, No. 2, pp. 11–115 (1989)

¨ 24 A. Unsold. Physik der Sternatmosph¨aren. Berlin (1938). 25 V.V. Pikalov and T.S. Melnikova. Plasma Tomography (in Russian). Novosibisrsk: Nauka (1995). 26 L.M. Biberman. V.S. Vorobyev, and I.T. Yakubov. Kinetics on Nonequilibrium Low-Temperature Plasma. Moscow: Nauka (1982). 27 G. Bkeﬁ. Radiation Processes in Plasmas. New York: Wiley (1969). 28 N. N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1985). 29 H.A. Kramers. Phil. Mag., 46, p. 836 (1924). 30 N.N. Sobolev, Ed. Optical Pyrometry of Plasma (in Russian). Moscow: IL (1960). 31 V.K. Zhivotov, V.D. Rusanov, and A.A. Fridman. Diagnostics of Nonequilibrium Chemically Active Plasma (in Russian). Moscow: Energoatomizdat (1985). 32 V.N. Kolesnikov. Spectroscopic Plasma Diagnostics in the VUV Region. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 491–507. Moscow: Nauka (2000). 33 Ya.B. Zeldovich and Yu.P. Raizer. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena (in Russian). Moscow: Nauka (1966) 34 Yu.P. Raizer. Gas-Discharge Physics (in Russian). Moscow: Nauka (1982). 35 L.D. Landau and E.M. Lifshits. Quantum Mechanics (in Russian). Moscow: Fizmatgiz (1963). 36 L.D. Landau and E.M. Lifshits. Quantum Mechanics (in Russian). Moscow: Nauka (1974). 37 H. Massey. Negative Ions. London: Cambridge University Press (1976). 38 B.M. Smirnov. Negative Ions (in Russian). Moscow: Atomizdat (1978). 39 A.V. Eletsky and B.M. Smirnov. Negative Ions In Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 1, pp. 250–260. Moscow: Nauka (2000).

References 40 Ch. Rhodes, Ed. Excimer Lasers. Heidelberg: Springer Verlag (1979) (1981). 41 V.S. Lebedev, L.P. Presnyakov, and I.I. Sobelman. Radiative Transitions in the H2+ Molecular Ion (in Russian). UFN, 173, No. 5, pp. 491–510 (2003).

42 M.I. Pergament. Plasma Diagnostics from Scattered Radiation (in Russian). In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 569–572. Moscow: Nauka (2000).

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Emission, Absorption and Scattering Techniques for Determining the Densities of Particles in Discrete Energy States

3.1 Emission Techniques

Historically, emission techniques served as the basis for the development of optical spectroscopy in general and plasma spectroscopy in particular. The classical emission methods are the easiest to implement experimentally and reduce to intensity measurements in spontaneous emission spectra. The term ‘classical’ stresses the fact that the measurement techniques used have no connection with lasers, but rather utilize ﬁlters, prisms, diffraction gratings and their combinations. These techniques and their application capabilities are described in detail in many books [1–4]. The connection between the intensity of spectral lines and the density (concentration, population) of the excited particles emitting these lines is given by relation (2.65). 3.1.1 Identiﬁcation of Spectra

The process of excitation of the own glow of plasma is, as a rule, not remarkable for high selectivity, as to different molecular states even in nonequilibrium conditions. That‘s why the plasma emission spectra have a well-developed structure conditioned by the optical transitions starting from the excited states of several species of particles simultaneously. Once the spectrum is recorded, the wavelength scale calibrated and the intensities compensated for the spectral sensitivity of the detectors, the transmission of the spectral instrument and the materials used in the optical system, the problem arises as to relate the spectral lines with certain particles and their quantum states responsible for the transitions. For diagnostics tasks, well-studied and structurally simple spectra of atoms and small molecules are, wherever possible preferable.

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3 Emission, Absorption and Scattering Techniques

Insofar as the energy structure of such particles is well known, the identiﬁcation problem is of a technical rather than principal character, but its practical solution requires sufﬁciently detailed information about the individual features of the spectra. It is known from experience that comparing the appearance of recorded spectra with individual spectra or their fragments proves very useful in this respect. This reﬂects the inherent human correlation and pattern recognition abilities. Despite the fact that the relative intensities of lines strongly depend on the conditions prevailing in plasma (which, generally, forms the foundation of many diagnostic techniques), the groupings of sets of lines and bands are frequently so speciﬁc that they can be reliably identiﬁed even among a great number of other lines. The distinction of such survey spectra also makes it possible to render more precise the calibration of the wavelength scale and narrows the problem of identiﬁcation of the rest of the fragments of the total spectrum, especially where the spectral resolution is high. It should be stated, however, that in their original form such spectra as serve as the initial material in spectroscopic investigations are rarely published and when then selectively in individual atlases, reports, qualiﬁcation works, and so on, and so, as a rule, are inaccessible for the wider community. As for atomic spectra the situation is somewhat simpler, for the schemes of atomic terms, with the main transitions and corresponding wavelengths indicated, (the so-called Grotrian diagrams) are frequently presented in books on the general questions of spectroscopy. The diagrams for many atoms can be found in the reference book [5]. At the present time the most detailed and systematized selections of such diagrams are presented in the books [6–8] specially devoted to this subject. The situation regarding the availability of exemplary molecular spectra is more difﬁcult. One of a few exceptions is the book by Pearce and Geidon [9] which it‘s age presents photographic survey spectrum records of a number of simple molecules in plasma. Sadly, on account of and short run of its publication, the book has already become a rarity. What is more, some difﬁculties arise in using the book because of the long-outdated detection technique. For practical reasons, Appendix D presents a collection of spectra of some atoms and diatomic molecules that appear in plasmas differing in origin and which are frequently used in plasma diagnostics and related procedures. 3.1.2 Absolute Measurements

When reference is made to the absolute measurements of intensity, meant is the measurement of the power of light emitted by the object under

3.1 Emission Techniques

study (Section 2.1). As applied to such measurements in plasma, the simplest situation is when the following conditions are satisﬁed: • the spectral instrument used picks out a wholly isolated line of the u → l transition whose probability is known, • the plasma under study is optically thin, • the illumination conditions of the entrance aperture (a slit, interferometer aperture, etc.) are such that the plasma (or some its part) can be treated as a point source, that is, all its elements contribute equally to the signal being detected. In such a case, the number N of particles at the upper level u is deﬁned by formula (2.65), provided that the signal is measured in absolute energy units. For this purpose it is necessary to calibrate the entire system, including illumination optics, the spectral instrument itself, detector and the registering system for the transition wavelength. The calibration is usually carried out by comparing signals from the plasma and a standard source. When taking measurements, one should ensure that the light being registered from the standard source and the plasma under study are being emitted in one and the same direction and into the same solid angle, which can be attained, for example, by means of the scheme shown in Figure 3.1 [3]. Here E is the standard source and P, the plasma object being studied. Sources E and P are arranged symmetrically about the axis of the system (lenses L1 and L2 ) illuminating the entrance aperture Sin of the spectral instrument. By using tilting mirror M, lens L1 can be uniformly illuminated by either source. Lens L2 catches the entire light coming from lens L1 and produces the real image of the plane containing lens L1 in the plane of entrance aperture Sin . If, as assumed above, the plasma has a line spectrum, while the spectrum of the standard source is, as usual, continuous, account should be taken of the characteristics of the spectral instrument and the method of detecting radiation. Schematically the measurements are reduced to the following. For the sake of deﬁniteness, let the spectral instrument be a monochromator whose entrance and exit apertures are uniformly illuminated slits of equal height and with a thickness of Sin and Sout , respectively. Use is made of the photoelectric light detection method, with the size of the detector (designated as Det in Figure 3.1) exceeding that of exit slit, Sout . In this case, the detector signal measuring the intensity Iul of the plasma radiation is Wul = ηul Iul Sin,ul ,

(3.1)

69

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3 Emission, Absorption and Scattering Techniques

where Sin,ul is the width of the entrance slit and ηul is a factor allowing for the illumination geometry, loss of light in the optical elements, detector sensitivity and the characteristics of the electronic equipment used. When registering the radiation of the standard source with an intensity spectral density of Iν , the detector signal is WE = ηE Sin,E

Iν dν .

(3.2)

ΔνE

Integration here is over the frequency interval falling within the width of the exit slit. If the spectral density Iν remains practically unchanged in this frequency interval ΔνE , it then can be factored outside the integral sign. We also assume that the dispersion D of the spectral instrument is linear on this frequency interval, so that ΔνE = DSout,E , and the width of the spread function ΔA < ΔνE , as is also the width of the spectral line, Δνul < ΔνE . Taking the ratio between signals (3.1) and (3.2), we get the following expression for the spectral line intensity being measured: Iul = Iν

WE ηE Sin,E DSout,E . Wul ηul Sin,ul

(3.3)

The above assumptions usually hold true. To exclude the effect of the factors ηul and ηE , which are difﬁcult to control separately, one can use the illumination scheme shown in Figure 3.1 and measure the signal ratio so as to, ensure that the transitions frequency νul falls within the interval ΔνE , that is, with the dispersing elements in the same position and Iν = Iν,ul . To eliminate the differences due to the possible nonlinearity of the registration system, the intensities of the signals from the plasma and the standard source should be made as commensurable as possible by appropriately adjusting the widths of the entrance apertures Sin,E and Sin,ul . If these conditions are satisﬁed one then can put ηul /ηE = 1 in formula (3.3). Thus, the absolute line intensity Iul can be deduced from the ratio between the signals from the plasma and standard source; the spectral density Iν in the standard source is known in absolute measure, and the factors remaining in formula (3.3) after exclusion of ηul /ηE are known. As a rule, the standard sources used for comparison purposes are secondary standards (in relation to the primary standards that reproduce to the greatest degree of blackbody radiation) certiﬁed at metrology laboratories. The most widespread are tungsten ﬁlament lamps (visible region of the spectrum) and deuterium (200–400 nm) and hydrogen (120–200 nm) discharge lamps [1–3, 10].

3.1 Emission Techniques

Figure 3.1 Absolute intensity measurement scheme [2].

3.1.3 Emission of Extended Inhomogeneous Sources

Formula (2.65) relating the intensity of a spectral line to the density of states in the emitting state, is only locally valid. And it is only valid for homogeneous media where simple relationships (2.69), (2.70) exist between the ‘primary’ light intensity per unit length (Section 2.4), which are not distorted by either self-absorption or reabsorption and for the total light intensity over the observation beam. For inhomogeneous media, the retrieval of the intensities I 1 from a series of measurements taken along different observation lines (projections) is the task of the optical tomography of plasma [1, 2, 11, 13, 14]. The tomographic problem is formulated on the basis of the radiation transfer equation [15], with observation at a frequency ν conducted in direction L with the current coordinate l: dIν (l ) = Iν1 (l ) − χν (l ) Iν (l ). dl

(3.4)

Here Iν1 (l ) and χν (l ) are the local values of the intensity and absorption coefﬁcient, respectively. If we restrict ourselves to the two-dimensional case and introduce, in accordance with Figure 3.2, the coordinate axes x and y, the angle ξ between the observation direction and the x-axis, and the distance p along the normal from the observation line to the origin of

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3 Emission, Absorption and Scattering Techniques

Figure 3.2 Coordinate axes for two-dimensional tomography.

coordinates, the solution of (3.4) can be written down in the form [13] Iν ( p, ξ ) =

∞

dl Iν1 (l ) exp − χν (l ) dl .

(3.5)

−∞

To reconstruct the spatial distributions Iν1 (l ) and χν (l ) it is necessary to use data arrays for the integral quantities Iν ( p, ξ ), measured along different observation lines. An array wherein p varies, while ξ remains unchanged, is called a projection. To realize such a procedure proves a difﬁcult multifactor problem, even if simplifying assumptions are made. For example, if plasma is optically thin and χν (l ) = 0, expression (3.5) corresponds to the Radon integral transformation R: Iν ( p, ξ ) =

∞

" ! dl Iν1 ( x, y) ≡ R Iν1 ( x, y) .

(3.6)

−∞

To ﬁnd Iν1 ( x, y), transformation (3.6) should be reversed: Iν1 ( x, y) = R−1 { Iν ( p, ξ ) + ε} ,

(3.7)

where ε are random measurement errors (noise). The presence of noise greatly complicates the problem of tomographic reconstruction, and the

3.1 Emission Techniques

Figure 3.3 Coordinate axes in reconstructing the intensity proﬁle in an axially symmetric source from chord measurements.

proper allowance for noise proves very important, and is generally typical of inverse problems (Section 1.4). Increasing the number of projection measurements and simplifying the geometry of the object under study can improve the reliability of the results. The capabilities of spectral-optical tomography have been most thoroughly investigated and tested on the important class of axially symmetric plasma objects. In that case, the local intensity isolines are circles, so that Iν1 ( x, y) = Iν1 (r ), where r is the modulus of the radius vector. The projections Iν ( p, ξ) become independent of the angles ξ. When taking projection measurements along directions parallel to the y-axis (Figure 3.3), the intensities are summed up along the chords (−y0 , y0 ). If the plasma is optically thin, Radon transformation (3.6) reduces to the Volterra integral equation of ﬁrst kind, also known as the Abel equation: Iν ( x ) = −

1 π

R r

Iν1 ( x, y) dy = 2

R 1 Iν (r )r dr

√

x

r2 − x2

.

(3.8)

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3 Emission, Absorption and Scattering Techniques

The solution (reversion) of this equation also bears the name of Abel: Iν1 ( x, y)

=

Iν1 (r )

1 =− π

1 d =− πr dr

R r

R r

dIν ( x ) dx √ dx x2 − r2

Iν ( x ) √

x dx x2 − r2

(3.9)

.

This particular case illustrates the above-mentioned sensitivity of the solution to the accuracy of the integral measurements, because Iν ( x ) should be integrated with a singularity present in the lower limit. To illustrate, in the example presented in [1, 2], the subjective construction of the ﬁtting curve by the experimenter for an array of Iν ( x ) values measured with up to 20% deviations form the true values, results in up to 400% errors in the retrieval of Iν1 ( x ), which corresponds to a 20-fold enhancement of the error in the initial data. The retrieval of local intensity values form projection measurements requires care and experience in model problems. The numerical solution of (3.6) with the use of statistical regularization methods [13, 14] is considered optimal today. With the experimental errors (noise) being normally distributed, which is typically the case, these methods can provide for error of retrieval within the limits of 10–15 %. It should be said, however, that such a result can only be attained when using additional (prior) information about the distribution function of the sought-for quantities. These may be quite general physical assumptions, for example, smoothness, nonnegativeness, the presence of a certain number of extremes and so on. Reversion procedures (3.7), (3.9) are only valid for optically thin plasma, and this fact should be veriﬁed. For nonequilibrium plasma, this should preferably be done experimentally by measuring absorption at the same frequencies and in the same directions as those used in the projection measurements of the integral intensities. If the optical thickness of plasma fails to meet condition χν L < 1, the problem of emission tomography becomes increasingly complicated, and a more detailed discussion than that presented here is required. As applied to dense cylindrical plasmas in the state of local thermal equilibrium (LTE), the Abel reversion procedure is considered in [11]. For more details see Appendix F.

3.2 Absorption Techniques Using Classical Emitters

Figure 3.4 Absorption measurements: (a) with a point probe source; (b) by the two-tube method.

3.2 Absorption Techniques Using Classical Emitters

The measurement of the absorption of light passing through a homogeneous object gives, with the transition probability known, the absolute value of the difference in population between the levels bound by the u → l transition (Section 2.3). This attractive property has no connection with the restrictions as to the equilibrium conditions in plasma and forms the basis of numerous versions of the absorption method. 3.2.1 Absorption Against the Background of Continuous Spectrum

In such a case, radiation source E probing plasma P (Figure 3.4a) has a continuous spectrum. The spectral instrument with entrance slit Sin separates out of the continuous spectrum a frequency interval of Δν, wherein the absorption line of the plasma belongs. Within the limits of the interval Δν the intensity of the continuous spectrum changes little, and so only one line is separated, that is, Δναβ > Δν, where Δναβ is the distance to the next absorption line in the spectrum. On the other hand, in order to ensure that it is exactly the integral absorption ((2.36), (2.37)) that is being measured, it is necessary that the condition Δν > Δνul , where Δνul is the absorption line half-width, should be satisﬁed. As a rule, this condition can be satisﬁed, even when using a spectral instrument of moderate dispersion, by selecting an appropriate section of the spectrum. Some cases where the lines cannot be resolved will be considered in Section 3.2.2. Insofar that the plasma emits radiation in the same transition as is involved in absorption measurements, detector Det in Figure 3.4 is simultaneously illuminated by plasma P and probe radiation source E. In order

75

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3 Emission, Absorption and Scattering Techniques

not to ﬁx the light from the plasma, use can be made, for example, of a method whereby a light modulator (chopper) (designated by M in Figure 3.4) is placed between the plasma and the probe radiation source and synchronous (timed) detection is used in the registration system. However, one should make sure in that case that the detector and registration system characteristics are linear in the actual range of total intensities, as the presence of constant illumination biases the operating point of the system. To relate the absorption measured via this scheme and the level populations, use is made of the concept of the total absorption AG [16]: AG = Δν

I0 − I = ΔνAL , I0

(3.10)

where I0 and I are the intensities of light from source E at the entrance to and exit from the plasma, respectively, and the dimensionless quantity AL is the so-called absorption function (in some popular literature sources, e.g. [3, 17, 18], the deﬁnitions of its quantity differ by the factors π/2). We assume that the plasma segment of length l that is being probed is homogeneous. The function A L is calculated by integrating with respect to frequency the intensity spectral densities I0 (ν) and I (ν) over the entire line proﬁle. With the above assumptions for the width of Δν, AG =

(1 − exp {−χlu (ν)l })dν.

(3.11)

Δν

In the case of Doppler broadening, 1 π Δν χ lS(χ0,lu l ), AG = 2 ln 2 D 0,lu

(3.12)

and in the case of Lorentz broadening, AG =

π Δν χ lS (χ0,lu l ). 2 L 0,lu

(3.13)

The notation in expressions (3.12) and (3.13) is the same as in formulas (2.51), (2.61), (2.62), and (2.69). In experiment the quantities I0 , I, Δν and l, are measured, the functions S and S are tabulated (Table 2.3), and the line widths depend on the conditions occuring in plasma (lifetimes of the levels, temperature and pressure (Section 2.2)). Thus, one ﬁnds ﬁrst AL and AG and then χ0,lu , and the level populations are subsequently given by formulas (2.61) and (2.63). In the case of mixed (Voigt) broadening, the quantities AG as functions of χ0,lu l have no simple analytical expressions and so are calculated numerically. The calculation results presented in graphic form are called

3.2 Absorption Techniques Using Classical Emitters

Figure 3.5 Absorption as a function of optical density at various Voigt parameters a.

the Van Held growth curves. They are shown in Figure 3.5 for several values of the Voigt parameter a (2.49). These curves at a = 0 and a = ∞ (straight line) represent expressions (3.12) and (3.13), respectively, as particular cases. Beginning with the work of Ladenburg and Reiche [16], the quantity AG is sometimes called the equivalent line width. If we have a line with a rectangular proﬁle of width Δ∗ νlu , and if absorption is χ0,lu l = ∞ within its bounds and χ0,lu l = 0 beyond them, the integral in expression (3.11) is AG = Δ∗ νlu , that is, the total absorption is numerically equal to the width of this model line. The total absorption can similarly be expressed in terms of wavelength instead of frequency. In terms of the level populations, obviously with the absorption coefﬁcient given, the concentration sensitivity of measurements by this method grows with increasing length l of the optical path. One can see from the deﬁnition of total absorption (3.10) and formulas (3.12) and (3.13) that with χ0,lu l given, the quantity AL determined directly from the experimentally measured intensities I0 and I decreases with increasing ratios Δν/ΔνD,lu and Δν/ΔνL,lu . This corresponds to the reduction of the relative intensity being measured and in the ﬁnal analysis also limits the sensitivity of measurements taken against the background of a continuous spectrum. To take measurements at extremely

77

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3 Emission, Absorption and Scattering Techniques

small Δν Δνul and determine the absorption coefﬁcient at the center of the line proﬁle is, in principle, possible. But this possibility is also difﬁcult to realize. High demands are imposed on spectral resolution, and the level of the signal detected from source E is sharply reduced. Other versions of the absorption method have been developed to overcome these difﬁculties. 3.2.2 Line Absorption

With this method, the quantity Δν is speciﬁed not by the spectral instrument but by the probe radiation source, with a line spectrum coinciding, at least in part, with the absorption spectrum of the plasma. The role of the spectral instrument is limited to the separation of individual components of the line spectrum of the source. Technically and methodically this approach can be realized in several ways. For instance, • the probe radiation source may be physically identical to the plasma under study. If the plasma sources are gas-discharge tubes, the method is then referred to as the two-tube method (Figure 3.4b); • no additional probe radiation source is used. The plasma under study itself plays its part. If the plasma is homogeneous, those portions furthest removed from the detector serve as a distributed source. Self-absorption of light takes place (Section 2.4). To make use of the quantitative relations between the intensity at a ﬁnite optical density and the absorption coefﬁcient, measurements should be taken with different lengths of the plasma column. If this is a gas-discharge tube, one of its electrodes is made movable, or several electrodes are used and the discharge is initiated between different pairs of electrodes; • if a mirror is placed instead of source E (Figure 3.4a), the role of the probe radiation source will then be played by the virtual image P of plasma itself; • the absorption sensitivity of the above scheme can be improved by placing the tube between two mirrors, one of them (on the detector side) being semitransparent (multipass absorption cell). Measurements are taken with the totally reﬂecting mirror open and shut. This method has found widespread application in spectral analysis (metallurgy, chemical industries, medicine, geology, etc.), where it is also known as the atomic absorption method. Atomic-absorption spectrometer setups are intended for mass-elemental analysis and include spectral

3.2 Absorption Techniques Using Classical Emitters

instruments, sets of line spectrum sources, and atomizers. Substances in atomizers are in a state of thermal equilibrium, and the relation between the concentration of atoms and absorption is established by way of calibration. Modern setups are automated and manufactured in production quantities. This is a special area, which we will not touch upon, because physical investigations into nonequilibrium plasma require, as a rule, less specialized, more ﬂexible approaches. These and some other versions of the absorption method are considered in [17]. Instead, we will restrict ourselves to the analysis of the ﬁrst of the above-listed methods, namely, the two-tube method (Figure 3.4b), from where it will be easier to consider the other methods. In this case, the subject of the measurement is alone the intensity of light in the given transition from the tube of length l1 . The absorption function AL is then found in accordance with formula (3.10) from the two measurements, I0 and I. The quantity I0 can be measured with the tube of length l2 not ﬁlled with the gas and the quantity I, in the presence of plasma in this tube. If the plasmas in both tubes are homogeneous along the observation beam, then, using formula (2.66) and integrating with respect to frequency, we get I0 =

∞ 1 Iul (ν) 0

I=

χlu (ν)

∞ 1 Iul (ν) 0

χlu (ν)

1 − e−χlu (ν)l1

dν,

(3.14)

(1 − e−χlu (ν)l1 )e−χlu (ν)l2 dν

(3.15)

and for AL , we may write

∞ AL = 1 +

0

1 (ν) Iul χlu (ν)

1 − e−χlu (ν)l2

∞ 0

dν −

1 (ν) Iul χlu (ν)

∞ 0

1 (ν) Iul χlu (ν)

1 − e−χlu (ν)(l1 +l2 )

dν .

1 − e−χlu (ν)l1 dν (3.16)

As noted in Section 2.4, the integrals appearing in expression (3.16) are expressed in terms of the functions S∗ , so that AL = 1 +

l2 S∗ (χ0,lu l2 ) − (l1 + l2 )S∗ (χ0,lu (l1 + l2 )) . l1 S∗ (χ0,lu l1 )

(3.17)

According to expression (2.69), S∗ = S for the Doppler line proﬁle and S∗ = S for the Lorentz proﬁle. The case of mixed broadening requires

79

80

3 Emission, Absorption and Scattering Techniques

Figure 3.6 Optical density dependence of the absorption function in the two-tube method at various tube length ratios n.

direct numerical integration. One can introduce the ratio n = l1 /l2 and use it as a parameter. Figure 3.6 presents some results from calculating AL (χ0,lu l1 , n) by formula (3.17) for the Doppler line proﬁle, S∗ = S, at several values of n. One can see that the optimal choice of this parameter in planning the experiment depends on the range of the expected optical densities. In the case where the tubes are of the same length, l1 = l2 = l, and n = 1, formula (3.17) has the form S∗ (2χ0,lu l ) . (3.18) AL = 2 1 − ∗ S (χ0,lu l ) Table 3.1 lists numerical values of absorption function (3.18) for the Doppler (AL ) and the Lorentz (AL ) broadening. Information about the type of broadening is important for quantitative measurements of the absorption coefﬁcient and subsequently the level populations, if the required accuracy of the absolute values of χ0,lu l is required to be better than a factor of ca. 2. Listed in the bottom row of Table 3.1 are the values of the absorption function A0L = 1 − exp (χ0,lu l ) when the illuminating source emits a monochromatic line whose frequency corresponds to the maximum of the absorption line of the plasma, χ0,lu . It can be seen that in all the three cases, relative changes in the line strength of the source probing

3.2 Absorption Techniques Using Classical Emitters Table 3.1 Values of the absorption functions. χ0,lu l

0.1

0.2

AL

0.066

0.130

0.238

0.329

0.404

0.469

0.673

0.806

AL

0.049

0.09

0.163

0.225

0.278

0.319

0.445

0.527

A0L

0.095

0.181

0.33

0.451

0.551

0.632

0.865

0.982

0.4

0.6

0.8

1

2

4

the plasma at a ﬁxed value of χ0,lu are comparable with one another to an accuracy no worse than up to the above factor. Therefore, the use of neither a high-resolution spectral instrument (e.g. interferometer), nor an illuminating source whose emission line is narrower than the absorption line of the plasma, will result in any appreciable difference in the concentration sensitivity. It is important to note, however, that the minimum measurable absorption depends on the noise characteristics of both the illuminating radiation and the measuring system. This question will be considered in greater detail in Section 3.3. The above-mentioned working relations (3.16) and (3.17) have been obtained on the assumption that measurements are taken for an isolated singlet line. But it frequently transpires that spectral lines possess a ﬁne structure unresolved in the experiment. The following three possibilities can be realized in this case: (i) the true proﬁles of the individual components of the muiltiplet barely overlap (Δνul Δναβ ), but cannot be resolved by the spectral apparatus used, (ii) the proﬁles completely overlap (Δνul Δναβ ), and (iii) the proﬁles overlap in part (Δνul ≈ Δναβ ), cases (i) and (ii) being particular with respect to this one. The expressions for AL are obtained by summation of expressions (3.14) and (3.15) over the multiplet components α and β [17, 19]. The expressions for cases (i) and (ii) can be found in [17]. Ochkin and coworkers [19] have considered the third case under the assumptions that (a) the broadening of the multiplet components is the same and (b) the 1 ratio between the ‘primary’ integral intensities Iuα,lβ , as well as that between the integral absorption coefﬁcients χlβ,uα , is governed by the ratio between the respective transition strengths (2.42), these being the same for both emission and absorption: Suα,lβ = Slβ,uα = Sα,β , 1 1 : . . .Iuα,lβ : . . . = χl1,u1 : . . .χlβ,uα : . . . = S11 : . . .Sα,β : . . . Iu1,l1

(3.19)

Physical assumption (b) means that the level splitting responsible for the multiplet structure of the line is small in comparison with the average thermal energy of the particles and, as a result of collisions, the relative populations of the ﬁne-structure levels are determined by their statistical weights.

81

82

3 Emission, Absorption and Scattering Techniques

In the general case (Δνul ≈ Δναβ ) in the two-tube version with identical tubes, l1 = l2 = l [19], ∑

AL =

∞

α,β 0

∑

ϕαβ (ν)Sαβ ∑ ϕαβ (ν)Sαβ

α,β

∞

α,β 0

ϕαβ (ν)Sαβ ∑ ϕαβ (ν)Sαβ

1 − exp

2

−χl1u1 l ∑

α,β

1 − exp

−χl1u1 l ∑

α,β

α,β

Sαβ S11

ϕαβ (ν)

dν

Sαβ S11

ϕαβ (ν)

. (3.20) dν

In accordance with assumption (a), the following designations of the form factors of the line proﬁles are used here to make the notation more compact: 1 1 Iuα,lβ (ν) = Iuα,lβ ϕαβ (ν),

χlβ,uα (ν) = χlβ,uα ϕαβ (ν),

ϕ(ν) dν = 1 .

The expressions for A L in cases (i) and (ii) are simpler. To illustrate, if the unresolved multiplet components do not overlap (Δνul Δναβ ), then ! "2 ∞ S 1 − exp −χl1u1 ϕαβ (ν) Sαβ l dν ∑ AL =

α,β 0 ∞

∑

α,β 0

11

!

S

1 − exp −χl1u1 ϕαβ (ν) Sαβ l 11

.

"

(3.21)

dν

The integrals can be replaced, as before, by the Ladenburg–Levi, Ladenburg–Reiche functions S∗ (χ0,lβ,uα ) (not to be confused with the designations for the transition strengths that contain subscripts). For example, the absorption function for doublets is ⎤ ⎡ S∗ (2χ0,l1u1 l ) + SS22 S∗ 2 SS22 χ0,l1u1 l 11 11 ⎦. (3.22) AL = 2 ⎣1 − S22 ∗ S22 ∗ S (χ0,l1u1 l ) + S S S χ0,l1u1 l 11

11

If the lines overlap (Δνul Δναβ ), the absorption function AL is calculated by the formulas for singlet lines, (3.16) and (3.17), but the absorption coefﬁcient should be found by way of summation: χlu (ν) =

Sαβ

∑ S11 χlβuα ϕαβ (ν).

(3.23)

α,β

Let us present as an illustration some examples of absorption function calculations for the A2 Σ–X2 Π resonance electronic–vibrational– rotational transitions in the OH hydroxyl radical, made in [19, 20] in connection with investigations into water vapor discharges. Because of

3.2 Absorption Techniques Using Classical Emitters

the spin splitting of the rotational levels of the excited state A2 Σ, the spectrum contains, in addition to the six main band branches P1 , P2 , Q1 , Q2 , R1 and R2 , four satellite branches P12 , Q12 , Q21 and R21 , which gives rise to narrow spectral doublets P1 and P12 , Q1 and Q21 , Q2 and Q12 , and R1 and R21 (see Appendix D). Under the conditions of the measurements [6, 7], these doublets were not resolved by the spectral instrument, but the lines did not overlap, which points to the applicability of formula (3.22). For the given example, the quantity χ0,l1,u1 denotes the absorption coefﬁcient at the center of the main line of the doublet, and S11 and S22 are the strengths of the main and the satellite line, respectively (since use is made of their ratio, it is equal to the ratio of the rotational intensity factors (Appendix E)). Table 3.2 lists the values of AL for the range 0.1 < χ0,l1,u1 < 3 in the case of Doppler broadening. The symbol Q1 (1), for example, denotes the doublet Q1 and Q21 corresponding to the transition with the lower rotational level K = 1 (K is the quantum number of the total momentum of the molecule less the spin), P2 (5) – the doublet P2 and P12 with K = 5, and so on (cf. the values of AL for singlet lines in Table 3.1). Table 3.2 Values of absorption functions for doublets in the rotational structure of the A2 Σ–X2 Π transition in the OH hydroxyl radical. AL χ0,l1u1

Q1 (1)

Q1 (5)

Q1 (10)

Q2 (7)

Q2 (8)

P2 (7)

P2 (10)

0.1

0.057

0.059

0.062

0.06

0.061

0.056

0.059

0.2

0.114

0.119

0.126

0.118

0.12

0.113

0.119

0.4

0.212

0.216

0.23

0.222

0.226

0.205

0.217

0.6

0.3

0.298

0.318

0.307

0.312

0.282

0.293

0.8

0.363

0.364

0.385

0.372

0.38

0.344

0.36

1

0.42

0.416

0.44

0.427

0.436

0.392

0.414

1.2

0.47

0.46

0.492

0.475

0.483

0.436

0.46

1.6

0.548

0.527

0.566

0.546

0.553

0.505

0.527

2

0.617

0.592

0.634

0.607

0.616

0.563

0.587

2.3

0.657

0.623

0.667

0.64

0.648

0.596

0.618

2.6

0.696

0.648

0.701

0.668

0.678

0.627

0.647

3

0.728

0.673

0.726

0.694

0.705

0.656

0.674

For more detailed tables, see [19, 20]. 3.2.3 Self-Absorption of Multiplet Lines

Under the same assumptions as made in deriving relations (3.20)–(3.22), one can use formula (2.69) to take into consideration the self-absorption

83

84

3 Emission, Absorption and Scattering Techniques

of multiplet lines [19]. In this case, the ‘primary’ intensity is determined by summation over the multiplet components: 1 Iul =

1 . ∑ Iuαlβ

(3.24)

α,β

It is also necessary to apply summation to expressions (3.14) and (3.15). If νul Δναβ , self-absorption is calculated by formula (2.69) for singlet lines, the effective value of χ0,lu L being directly found from absorption measurements. In the case of separate components (Δνul Δναβ ) of a multiplet not resolved in measurements, ∗ Iul 0 Iul

=

1+ S∗ (χ0,l1u1 l ) +

S22 S11

S22 ∗ S11 S

S22 S11 χ0,l1u1

.

(3.25)

∗ = I 1 l is the intensity of the doublet in the absence of selfHere Iul ul 0 is the intensity being measured, that is, the intensity absorption and Iul with self-absorption taken into account.

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Plasma spectroscopy has materially extended its capabilities, thanks to the use of frequency-tuned lasers. A great number of techniques and schemes have been developed to date to attain lasing in a wide spectral region. A general idea of the capabilities of the spectral tuning of laser frequency, based on the use of various active media in combination with nonlinear optics techniques, can be gleaned from Figure 3.7 [21, 22]. The scheme presented in the ﬁgure lays no claim to completeness, for laser technology is evolving relentlessly. 3.3.1 On the Advantages of Laser Sources Over Their Classical Counterparts in Direct Absorption Measurements

The merits of replacing the classical sources with lasers are most obvious in schemes with direct measurement of the intensity changes radiation suffers in passage through the object under study, when these changes obey the Bouguer–Lambert–Beer (BLB) law (2.51). Compared to the classical sources lasers offer perceptible beneﬁts in a number of aspects, in particular: • Spectral widths as narrow as Δν ≤ 10−4 cm−1 are comparatively easy to attain with lasers, whereas with the classical source-

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Figure 3.7 Spectral tuning regions of various lasers [19, 20].

monochromator combinations this presents a certain difﬁculty. In some cases, the use of tunable lasers allows one to abandon the classical spectral instruments altogether, and in a number of others (as a rule, because of the mode structure of laser radiation), it proves sufﬁcient to use the most simple monochromators for coarse ﬁltering purposes, even in measurements with high spectral resolution. • In comparison with the classical line absorption methods (Section 3.2.2), the use of frequency-tuned lasers makes it possible to unify and automate measurements, thus eliminating the problem of selecting a radiation source with its spectrum matched to the absorption spectrum of the object under study, in addition to the problem of having to select for self-absorption in the source. As already noted, the use of a narrow-band source alone cannot provide for any radical improvement of sensitivity. But in the laser, narrow spectral width is combined with high-directivity radiation (or, as in some semiconductor lasers with small aperture, for example, radiation of perceptible but diffraction divergence can easily be collimated). This allows multipass measurements increasing the optical thickness χlu l. • In many cases the high power and narrow directional diagram of laser radiation avoids the need for taking account of the own glow of the plasma object being probed.

85

86

3 Emission, Absorption and Scattering Techniques

• The high brightness and power of laser radiation are important in the study of nonstationary objects, when the spectrum should be recorded in a short time as in the case of fast frequency scanning (see Section 3.3.3 below). Absorption over a short time interval can be registered with acceptable signal to noise ratio by means of a photographic ﬁlm, photodetector arrays, charge-coupled devices, and so on. 3.3.2 On the Noise Limitation of Sensitivity

The noise factor limits the sensitivity of direct absorption measurements. Let us consider a laser emitting a light beam of power P and cross-sectional area A that is totally collected on the detector. The laser power density is I = P/A. In the ideal case, the minimum detectable absorption-induced power variation ΔPmin is determined by the quantum ﬂuctuations of the signal (shot noise). In the case of photoelectric detection and Poisson statistics of photoelectrons, this minimum detectable variation is given by [23, 24] ' Phνlu ΔPmin = ξ . (3.26) ΔtηD Here Δt is the detection time and ηD is the quantum yield of the photodetector. The set of factors ξ ≈ 1 depends on how the acceptable signal to noise ratio is determined during actual measurements, and so we will omit it in the subsequent estimates. Since the laser power can be high enough, then, generally speaking, account should be taken of the saturation effect–redistribution of the level populations of the particles of the object under study. Where absorption is low and stimulated transitions are disregarded [23], ΔP = I A

σlu Nl L, 1 + IIs

(3.27)

where L is the length of the absorbing object, σlu is the absorption cross section, Nl is the density of particles at the level l (lower), and Is is the saturation power density. If we assume that the quantity ΔP in expression (3.27) is equal to ΔPmin , then, considering expression (3.26), the minimum detectable total number of particles at the level l in the laser beam region, Nmin = Nl AL, will be ' I Phνlu 1 + Is Nmin = . (3.28) ΔtηD Iσlu

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers The least value of Nmin is attained at I = Is : ' 2 Ahνlu Nmin = . σlu ΔtηD Is

(3.29)

The saturation power density for a two-level systems is expressed in terms of the time τul it takes for the system to decay (relax) from the state u to the state l as Is = hνlu /(σlu τul ), and to estimate sensitivity, use can be made of the relation ' Aτul . Nmin = 2 ΔtηD σlu

(3.30)

(3.31)

To convert from the number of particles at the main absorbing level to the total number of particles, one should use in expressions (3.28)–(3.31) not , but N the quantity Nmin min = Nmin Qin , where Qin is the inner statistical sum (1.7). However, in this case, the condition that the form of distribution of the particles among the levels should not be disturbed by radiation is to be satisﬁed. To illustrate, if it turns out, when determining the concentration of atoms from resonance transitions, that σlu ≈ 10−12 cm2 , τul ≈ 10−8 s, all atoms are in the ground state, and Qin ≈ 1, Δt ≈ 1 s, then Nmin ≈ 102 . For the concentration of molecules summed over all levels, sensitivity is reduced to the typical value Nmin ≈ 108 . This results mainly from the increase of the typical values of τul and Qin to ca. 10−4 s and ca. 102 , respectively, in the case of molecules. As noted, the use of an irradiation regime with power density close to the saturation value maximizes sensitivity. This, however, presents certain problems due to the nonlinear character of the relationship between ΔP and P, which makes laser power variations difﬁcult to take into account. For this reason, it is more convenient to use a regime with I Is [24], and with a slightly poorer sensitivity. According to expres sion (3.28), for I = 10−2 Is , the quantity Nmin increases ﬁve-fold. The absolute values of the laser power remain in this case high, and the power and spectral brightness of lasers substantially exceed those of the classical sources. To illustrate, the typical saturation intensity for resonance transitions in atoms is Is ≈ 10 W · cm−2 . With the laser beam diameter being equal to 1 cm2 , the corresponding laser power is 0.1 W, which is many times in excess of the powers the classical sources typically have following the necessary monochromatization (Section 2.1). Laser power being high, the response of the photodetector is strong, and so its intrinsic thermal noise has no critical effect on measurements.

87

88

3 Emission, Absorption and Scattering Techniques

Figure 3.8 Spectral ranges of diode lasers and absorption regions of some molecules [23, 24].

This is especially important in the infrared region of the spectrum. With modern detectors having ηD ≈ 0.4 and a detection ability of D ∗ ≈ 1010 cm · Hz · W−1 , thermal noise in measurements at λ ≈ 10 μm can be neglected at P > 1 mW. (For more detail, see [22–24].) 3.3.3 Diode Laser Spectroscopy in the IR Region

This widely spread process uses current-pumped diode lasers as probe radiation sources. The probe radiation spectrum can be varied from the visible to the far IR region by selecting appropriate laser diodes differing in component composition. Figure 3.8 illustrates typical spectral regions and regions of characteristic molecular spectra covered by various laser diodes [25, 26]. There are both continuous-wave and pulsed diode lasers. Lasers based on the A2 B6 and A3 B5 compounds operate in the near IR region at room

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Figure 3.9 Block diagram of a diode laser spectrometer.

temperature. The operation of longer-wave lasers based on the A4 B6 compounds requires deeper cooling. Figure 3.9 presents a characteristic block diagram of a pulse-periodic diode laser spectrometer [27]. The laser frequency is controlled by varying the pump current and the cooling temperature of the diode. The lasers used in [27] are built around A4 B6 diodes whose temperature is maintained in the range 20– 90 K. To separate laser modes and effect coarse frequency control, use is made of a classical diffraction monochromator. The active thermal stabilization system typically provides for the long-term stabilization of the diode temperature accurate up to ca. 10−2 K, and the temperature sensitivity of the laser frequency is ca. 1 cm−1 · K−1 , which, combined, ensures a spectral resolution of ca. 10−2 cm−1 . To raise the resolution to ca. 10−4 cm−1 , use is made of either an additional optical system for the synchronous triggering of the electronics [22] or a fast-acting digitalprocessing oscilloscope. It should be noted, however, that owing to the variability of the mode composition of diode laser radiation, the frequency tuning of this type of spectrometer is not continuous, but piecewise continuous. By varying the pump current, a single laser diode of a given component composition can be made to cover a region of 200–300 cm−1 , the closely adjacent

89

90

3 Emission, Absorption and Scattering Techniques

zones of truly continuous tuning amounting to 1–10 cm−1 . The complete coverage of the spectral range is attained by varying both the pump current and the cooling temperature of the diode. This presents certain difﬁculties in operating the spectrometer, but is offset by the high spectral resolution, which is especially important in the spectroscopy of highly excited molecules. The above can be illustrated by the very important example of the CO2 molecule having four normal vibrational modes. The wavelength regions corresponding to the spectra of these modes are around 4 μm for antisymmetric vibrations, 8 μm for symmetric vibrations and 15 μm for doubly degenerate deformation vibrations. The most obvious way to study the energy distributions of molecules in these modes is to take spectral measurements in these wavelength regions. For antisymmetric and deformation modes, use should be made of absorption methods, and for symmetric ones, combination (Raman) scattering techniques. However, if high-resolution equipment is available an alternative method proves more advantageous. It is expedient to use not only the dipoleallowed absorption transitions in the fundamental bands, but also sequential and compound transitions, for example, those near the fundamental band of antisymmetric vibrations, v1 v2 v3 − v1 v2 (v3 + 1), where v1 , v2 and v3 are the vibrational quantum numbers for the symmetric, deformation and antisymmetric modes, respectively. The rotational structures of the spectra turn out to be strongly overlapped, but they contain information about a great number of vibrational–rotational states in a narrow spectral interval determined by anharmonicity. Figure 3.10 shows a 0.7 cm−1 wide fragment of such a spectrum in a glow discharge that reﬂects transitions associated with 15 combination states, including isotopic modiﬁcations. The resolution of the spectrum is 10−4 cm−1 , and ﬁxed are the actual line proﬁles of the vibrational–rotational transitions. In the simple single-beam measurement version (Figure 3.10), the minimum measurable (with an accuracy no worse than 10%) values of the absorption index χl typically range between 10−2 and 10−3 . The use of a double-beam differential optical measurement system allows quantitative absorption measurements at a level of ca. 10−4 . To reduce the minimum measurable absorption to ca. 10−5 , use is made of continuous-wave diode laser spectrometers with various modulation schemes: • modulation of the diode laser frequency and amplitude (pumpcurrent modulation) [23, 28], • Stark or Zeeman modulation of a molecular transition frequency [29, 30],

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Figure 3.10 Absorption spectrum fragment of CO2 in a discharge. Digital recording diode spectrometer. Inset: absorption coefﬁcient for one of the lines on an enlarged frequency scale. Table 3.3 Particles present in small quantities in plasmas. Unstable radicals

Positive ions

Negative ions

Excited particles

AsH, BF, CCl, CF, CN, CS, FO, NCl, NF, NS, OH, OT, PO, PH, PCl, PN, PS, PF, SF, SCl, SiH, SO, SiN, BO2 , C2 D, CD2 , CH2 , CF2 , C2 O, C2 H, FCO, FO2 , HCO, HO2 , DO2 , NF2 , CD3 , CF3 , CH3 , CH2 F, NO3 , ClBO, FBO, HBO, S2 O, HBNH

Ar+ , ArH+ , CF+ , CCl+ , HCl+ , NO+ , NeH+ , SH+ , CO2+ , ClH2+ , D3+ , DCO+ , DN2+ , H3+ , HBr+ , H2 D+ , HD2+ , HCO+ , HCS+ , NH2+ , H2 O+ , N2 H+ , HCNH+ , H3 O+ , SH3+

C2− , OH− , SH− , ClHCl− , FHF− , FDF− , N3− , NCO− , NCS−

I, Kr, D2 , N2

• electric-ﬁeld modulation of the velocity of ions (Doppler modulation) [31–33]. Such high-sensitivity measurements are important for various plasma objects, including those in plasma-chemical reactors. Table 3.3 lists some particles present in small quantities in plasmas that are detected by means of diode laser spectrometers [26].

91

92

3 Emission, Absorption and Scattering Techniques

Some practical problems and examples of using diode laser spectrometers in plasma spectroscopy will be considered in Sections 4.2, 4.3, and 5.3. 3.3.4 Nonstationary Coherent Effects in Absorption Measurements

The direct absorption measurement methods under discussion rely on the applicability of the BLB relation (2.51). The use of laser radiation in a number of cases requires that this premise should be analyzed. High intensity laser radiation of high intensity can change the equilibrium conditions of the system being studied and redistribute the level populations of particles. This circumstance has already been taken into consideration (see, e.g. (3.27)) within the framework of the formal applicability of the BLB law by relating together the absorption coefﬁcient and radiation intensity by means of the saturation intensity factor (absorption nonlinearity). Another important radiation property – coherence – has become especially signiﬁcant in connection with the use of lasers. Coherence is the existence of a correlation between the phases of several periodic processes. They may be correlated in space and/or in time. The space coherence of radiation is the existence of a correlation between the phases of a light wave at different points in space. The distance between the most widely spaced points at which this correlation exists is the coherence length. Time coherence is the existence of a correlation between the phases of a light wave at a single point in space. The time for which such a correlation exists is the coherence time that can be estimated from the spectrum width Δνr as Tc ≈ (Δνr )−1 .

(3.32)

The combination of the high intensity and space-time coherence of laser radiation can distinctly manifest itself in spectroscopic measurements [34]. The cause and character of such manifestations is qualitatively as follows. The particles of a medium usually emit or absorb light independently of one another and their radiation phases are not correlated. If the coherence length of the radiation passing through the medium is less than the interatomic distance, the emission and absorption events occur stochastically as usual. And it is precisely in these conditions that the BLB law is valid. But if the radiation is coherent on a large scale, it can induce a high-frequency space-coherent polarization and thus impose

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers Table 3.4 Values of Einstein coefﬁcients and ΩR , σ2 values. Transition

Na

K

OH

3 P −3 S 3/2 1/2

4 P −4 S 3/2 1/2

2 Σ −2

C2 Π

3 Π −3

CH Π

2 Δ −2

CN Π

2 Σ −2

Σ

λ,nm

589

767.6

306

520

431.5

400

A, s−1

0.63+8

0.39+8

1.2+6

8.3+6

2+6

1.7+7

Ω R , s−1

9.83+10

1.15+11

5.08+9

2.96+10

1.1+10

2.74+10

σ2 , A2

185

207

188

on the particles certain emission-absorption phases in a macroscopic region. Naturally, other random processes, speciﬁcally collisions and spontaneous emission of radiation, will disturb this phasing. The rate of disturbance of the phase correlation is the reciprocal to the phase relaxation time T2 (also known as the transverse relaxation time). The disturbance prevails if T2 is less than the coherence time. The above conditions pertaining to the manifestation of the induced coherence may be expressed by the inequalities [34, 35] ΩR =

μij E0 1 1 > ≈ Δνr , 2¯h T2 Tc

(3.33)

where μij is the dipole moment of the i ↔ j transition in resonance with the frequency of the incident light wave and E0 is its electric ﬁeld amplitude. The quantity on the left-hand side of inequality (3.33) deﬁnes the frequency of the quantum transitions a particle is induced to make by the incident light, which is also called the Rabi frequency ΩR . Table 3.4. lists typical values of ΩR for some atomic and molecular transitions in the visible region of the spectrum at a light intensity of 1 mWcm−2 . Also listed are the Einstein coefﬁcients Aij of the spontaneous radiation (0.63+8 means 0.63 × 108 ) [36]. The interaction of coherent radiation with matter gives rise to numerous effects (quantum beats, photon echo, optical nutation, induced transparency, etc.). It constitutes a vast independent avenue of fundamental and applied science. The consistent macroscopic description of a system of a great number of particles interacting with a radiation ﬁeld is given statistically by means of the density matrix formalism [34, 37, 38]. When using quantitative absorption spectroscopy methods, one should estimate the region of conditions wherein coherent effects can invalidate the BLB law. This can occur under ‘transient’ conditions of pulsed interaction between light and matter on a time scale comparable with the phase relaxation times T2 . These times are governed by a combination of several processes, such as collisions accompanied by the deactivation of the i, j states, thermal motion (Doppler shift), and spontaneous emission

93

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3 Emission, Absorption and Scattering Techniques

of radiation. The collisional dephasing cross sections σ2 are, as a rule, large in comparison with the cross sections of many other elementary processes. In this respect, the situation is quite similar to the one considered in the case of collision broadening of spectral lines (Section 2.2.1). Listed in the bottom row of Table 3.4 are the values of σ2 in conditions of air-acetylene ﬂame at a gas temperature of ca. 2000 K. The cross-sections are seen to be perceptibly greater than those for the other processes, elastic collisions included (σ ≈ 10 A2 ). To roughly estimate the region of possible coherence manifestations at room temperature, one can take the effective collisional dephasing time to be approximately one order of magnitude shorter than the time between elastic collisions T2 ∼ 10−8 /P

(Torr), s.

(3.34)

At pressures below 1.333 hPa (1 Torr) and with the Einstein coefﬁcient A ≈ 108 s−1 , spontaneous decay competes with collisions in the dephasing process. We will touch upon these questions in Section 3.3.5 below, when discussing the ﬂuorescence techniques for measuring absorption. Transient processes can be observed when measuring absorption by means of pulsed lasers of ﬁxed frequency, or when the optical resonance is rapidly passed. In the latter case, this may result from rapid variation of either laser frequency in scanning or, in contrast, atomic transition frequency, for example, in the case of Stark or Zeeman effect. Figure 3.11 illustrates two cases of ‘fast’ and ‘slow’ scanning of the Doppler absorption line proﬁle of the vibrational–rotational transition of the CO2 molecule in the region of 4.3 μm [39]. In cases (a) and (b), the rates of change of the frequency of the pulsed diode laser used (PbS1− x Sex , x = 0.14) are dν/dt = 105 cm−1 s−1 and dν/dt = 106 cm−1 s−1 , respectively. The CO2 gas pressure is 1.333 hPa (1 Torr), the Doppler width is ΔνD = 4.4 × 10−3 cm−1 (8.3 × 108 Hz), and the time constant of the recording equipment is 2 ns. Thus, the resonance interaction time ΔνD /(dν/dt) equals 4 × 10−8 s in the former case and 4 × 10−9 s in the latter one. The line proﬁle recorded in the former case is a Doppler proﬁle with absorption at the center ca. 90%. Evident in the latter case are substantial proﬁle distortions and oscillations. At certain instants the ampliﬁcation of the incident radiation occurs. The physical cause of this effect is the emission of light by the light-ordered dipoles in the direction of the incident light and in phase with it. The calculations of the absorption dynamics made in the same work [39] using the density matrix formalism agree well with experimental results.

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Figure 3.11 (a) ‘Slow’ and (b) ‘fast’ scanning of a CO2 absorption line proﬁle.

3.3.5 Use of the Classical Multipass Absorption Cells

The BLB relation (2.51) includes the product χL. To measure small χ, it is expedient use long probing lengths L. Such techniques are well known in the classical spectroscopy. They use optical schemes wherein light repeatedly passes through the object under study in opposite directions at slightly different angles. White [40] developed a workable scheme that came to be known as the White cell. The ever-increasing application of lasers with well-collimated beams in absorption spectroscopy has stimulated the further development of such schemes. The effective optical path in a typical modern 50 cm long cell intended for use with lasers as probe radiation sources in the near IR region (the reﬂection coefﬁcient of copper mirrors equals 0.987 for a wavelength of 1.65 μm) is around 100 m (200 passes) [41]. For the current status of this method and new approaches to the design of multipass absorption cells, see review [42]. Naturally, in such a process information about the spatial distribution of χ within the limits of the cell is lost. In addition to the development of this technique, other approaches were developed that allowed the effective light-matter interaction length to be materially increased. We will consider them in the next two sections. 3.3.6 Intracavity Absorption

The intracavity laser spectroscopy (ICLS) technique is an elaboration of the method described above. Its high sensitivity is also based on the repeated passage of light through the absorbing substance under study, but the substance now is placed inside the cavity of a laser with an ampli-

95

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3 Emission, Absorption and Scattering Techniques

Figure 3.12 Block diagram of an intracavity laser spectrometer: 1 – object of study; 2 – active laser element; 3 – cavity mirrors; 4 – monochromator; 5 – radiation detector; 6 – registering circuit.

ﬁcation band whose spectral width exceeds the spectral fragment being studied (Figure 3.12). The bandwidth is determined by the homogeneous broadening of the lasing transition and the presence of numerous cavity modes. The absorption of light by the medium under study causes additional frequency-selective losses in the laser cavity. The principal difference between this system and the classical multipass absorption cell is that the loss of light on the mirrors is in this case compensated for by its ampliﬁcation (within the limits of the ampliﬁcation band) in the active medium, whereas it is exactly this loss that limits the optical efﬁciency of multipass cells with external light sources. However, the intensity distribution among the cavity modes in the active medium of intracavity laser spectrometers should not be distorted as a result of absorption in the object being studied. This requirement imposes a constraint on the width of the lines of interest; namely, it must be small in comparison with the homogeneous broadening of the laser line. This technique is described in detail in original papers, reviews and monographs [23, 43–48, 50]. Absorption in the ICLS technique is deﬁned by the relation I (ν, t) = I0 (ν, t) exp { − χlu (ν)ct}.

(3.35)

Here I (ν, t) is the laser radiation spectrum at the instant t after the start of lasing, I0 (ν, t) is the radiation spectrum at the same instant in the absence

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

of the absorbing medium under study, χlu (ν) is the absorption coefﬁcient at the frequency ν, and c is the velocity of light. Relation (3.35) expresses the BLB law (2.51) for linear absorption, the only difference in notation being that the length of the optical path is expresses as the product of the velocity of light and the lasing time, Leff = ct. The absorption coefﬁcient for a narrow spectral line, χlu = (1/ct) ln( I/I0 ),

(3.36)

can be found from the ratio between the radiation intensity at the frequency of the absorption line and that at a nearby frequency outside the absorption line, I/I0 . If we take I/I0 = 1/e then k = 1/ct. With the lasing time t = 300 μs, subject to registration are the lines with the absorption coefﬁcient k ≈ 10−7 cm−1 , and with t = 3 ms, those with k ≈ 10−8 cm−1 , which corresponds to an effective absorbing layer 1000 km thick. This substantially exceeds the effective lengths presented in the preceding section for the classical multipass absorption cells, even if the external radiation source used is a laser. Note, however, some limitations in using relation (3.35) in the given method. The sensitivity of the ICLS technique grows with increasing laser pulse duration, but even when passing on to continuous-wave regime (t → ∞), sensitivity remains ﬁnite and governed by spontaneous noise [44]: χlu,min = γ/(c, M ),

(3.37)

where γ is the inverse photon lifetime in the cavity and M is the average number of photons in the mode. With the typical values γ = 3 × 107 s−1 and M = 3 × 107 , the quantity kmin ≈ 3 × 10−11 cm−1 . It is precisely this estimate of the maximum sensitivity of the technique that is adhered to by many authors. The lasing time with which such a sensitivity is attained is t ≈ M /γ ≈ 1 s. At t < ( M /γ), relation (3.35) holds true, and at t > ( M /γ), the development of spectral dips ceases. One should also bear in mind that the time t is not always determined by the laser pulse duration and can be shortened owing to the laser mode lifetime being ﬁnite as a result of nonlinear interactions, four-wave mixing (FWM) in particular. In a number of cases constraints are imposed by the Rayleigh scattering (RS) and /or spontaneous emission (SE). When selecting a particular laser medium and its operating conditions, one should take these circumstances into consideration. Various types of lasers are currently used for the ICLS purposes. The ﬁrst experiments were conducted using Nd+ -glass lasers operating in the region of 1.055–1.067 μm with a pulse duration of ca. (1–10) μs, which

97

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3 Emission, Absorption and Scattering Techniques Table 3.5 Parameters deﬁned the ICLS sensitivity. Laser type

t, ms

Leff , km

Δν, Hz

Limiting factor

Dye

230

70000

45

Titanium-sapphire

4.5

1300

100

Diode

0.13

40

3

SE

Fiber

0.43

130

1500

RS

Nd+ -glass

12

3600

300

t

Color-center

0.4

120

3000

t

5 × 10−6

0.0015

200

t

Optical parametric oscillator

FWM FWM, RS

corresponded to an effective absorption length of Leff ≈ 300–3000 km. Various color-center lasers cover the near- and mid-infrared regions of the spectrum: LiF: F2+ (0.86–0.99 μm), LiF: F2− (1.1–1.28 μm), NaF: F2 F3− (0.98–1.4 μm), NaCl: F2+ (1.48–1.56 μm), KCl: LiF (2.6–2.7 μm). Titaniumsapphire lasers cover the region of 0.67–1 μm, and Cr3+ : YAG lasers, 1.38– 1.55 μm. Of interest among the recently tested laser media is Co: MgF2 (1.6–2.5 μm). But it is the dye lasers, which cover the entire visible region of the spectrum and are capable of a sensitivity at the level of the above theoretical limit, that are ﬁnding the most widespread application in the ICLS technique. More detailed information can be found in [46]. Table 3.5 lists some parameters governing the sensitivity of the ICLS technique using various types of lasers. Included are the laser pulse duration t, effective absorption length Leff , and the spectral resolution Δν, as well as the dominant factor limiting the sensitivity of the technique [47]. In order not to adversely affect the resolution determined by the laser mode interval, one should use for recording purposes a spectrograph of high enough quality (the resolving power in practical laboratory ICLS setups is ca. 106 ). Figure 3.13 presents as an example [49] the absorption spectrum of metastable singlet oxygen in the Q1 branch of the (0,0) band of the O2 ( a1 Δ g − b1 Σ+ g ) quadrupole transition (spontaneous transition probability 0.0017 s−1 ). The observations were made in the afterglow of a high-frequency discharge in oxygen at a pressure of 2.533 hPa (1.9 Torr). With the Co: MgF2 laser pulse duration equal to 0.22 ms, the measured absorption coefﬁcients were of the order of 10−7 cm−1 , and the O2 ( a1 Δ g ) concentrations, of the order of (2 – 5)×1014 cm−3 . Naturally, as in the case of classical multipass absorption cells, the measurements in the ICLS technique are averaged over the caustic of the propagating radiation.

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Figure 3.13 Intracavity absorption spectrum of metastable oxygen on the O2 (a1 Δ g – b1 Σ g ) quadrupole transitions in the afterglow of an RF discharge.

3.3.7 Measuring Absorption from the Attenuation of Light with Time

The BLB law (2.51), (3.35) can also be understood to mean that light, while passing through a medium, loses intensity with time, since (assuming that the refractive index n = 1) L = ct. Comparatively recently, and thanks to the development of frequency-tuned lasers capable of short pulse duration, high-quality optical mirrors, and low-inertia detectors and electronics, it has been possible to turn this theory into measurable fact. Most widespread is the technique known as the cavity ringdown spectroscopy (CRDS), wherein successively repeated passages of light between highly reﬂective mirrors are registered (Figure 3.14). Pulsed laser radiation is introduced into the absorption cell where it passes back and forth over and over again. Some radiation passes through the mirrors and is received by the detector. On each passage round the cell, the laser pulse loses some of its energy, not only because of the transmission loss in the mirrors, but also due to absorption in the medium between them. The proportion of energy lost by the pulse is (1 − R) for transmission and A = (1 − exp { − χct}) for absorption. If the mirrors have one and the same reﬂectance, the amount of intensity lost by the pulse in a time

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3 Emission, Absorption and Scattering Techniques

Figure 3.14 Signal detected on the passage of a light pulse through a cell with highly reﬂective mirrors.

Δt is ΔI =

I (1 − R ) A tr 2

Δt,

(3.38)

where tr = 2L/c is the round-trip time and L is the length of the absorption cell. At A, (1 − R) 1, the pulse intensity drops exponentially with time: I (t) = I0 exp { − 2(1 − R + A)t/tr },

(3.39)

which is recorded as the envelope of a series of pulses from the detector. For the empty cell, A = 0 the characteristic pulse decay time is τ0 = L/(c(1 − R)). Modern technologies are capable of manufacturing mirrors with the reﬂectance R > 99.99 %, and so the pulse decay can be ‘slow’ enough and convenient to analyze. For example, for an empty 50 cm long cell with R = 99.99 %, the pulse decay time τ0 = 17 μs and the effective number of passages of light round the cell is ca. 5000. The absorption coefﬁcient A is found from expression (3.39) by comparing the pulse decay times for the empty and the ﬁlled absorption cell: 1 1 τr . (3.40) − A= τ τ0 2 Here the pulse decay time for the ﬁlled cell is τ = L/(c(1 − R + A)).

(3.41)

3.3 Absorption Spectroscopy Using Tunable and Broadband Lasers

Expression (3.40) for determining absorption by the light attenuation method does not include the intensity of the probe radiation source. To a ﬁrst approximation, therefore, the inevitable intensity ﬂuctuations in the laser pulse sequence only weakly affect the accuracy and sensitivity of measurements. The minimum measurable optical density is

(χL)min = (1 − R)[(τ0 − τ )/τ0 ]min .

(3.42)

In the decay time range 10−5 –10−6 s, to attain a measurement accuracy of 1 % is no big problem and, as a consequence, absorption at χ ∼ (10−6 – 10−8 ) cm−1 can be measured with cells 10–100 cm long. The probe radiation frequency does not appear explicitly in relations (3.38)–(3.41). It is presumed that the spectrum of the probe radiation source is ﬁxed with respect to the absorption spectrum. It is also presumed that the absorbing medium is homogeneous and occupies the entire space between the mirrors. In the more general case, the pulse decay time in a partially ﬁlled absorption cell should be written in the form L ), τ (ν) = ( (3.43) d c |ln R(ν)| + χ(ν, x ) 0

where d is the size of the absorbing region along the x-axis of the cell. If the medium is homogeneous, and R ≈ 1 and depends but weakly on frequency, then τ (ν) =

L . c [(1 − R) + χ(ν)d]

(3.44)

Because of the frequency dependence χ(ν), the set of the decay times measured at different laser frequencies will determine the absorption spectrum of the substance under study if the quantity 1/cτ is plotted as a function of frequency. The resonance part will, thus, be located on a ‘pedestal’ (1 − R)/d high. The above-described scheme of high-sensitivity direct absorption measurements is ideologically very simple and is gaining recognition. One should, however, keep in mind some requirements: • Use should be made of high-quality mirrors. With the reﬂectance of the mirrors being high, the cell transmission is low (ca. 10−8 or 1 - R ≈ 10−4 ), consequently the combination of the laser power and, the detector sensitivity parameters should ensure reliable signal detection. The reverse, advantageous aspect, is that the relatively not very high power within the cell does not, as a rule, cause absorption saturation.

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3 Emission, Absorption and Scattering Techniques

Figure 3.15 Fragment of the absorption spectrum for the X2 Π(v = 0)–A2 Δ(v = 0) transition of the CH radical registered by the CRDS method.

• The time between successive laser pulses should be longer than the decay time τ0 . • In the above, we have disregarded the presence of the intrinsic cell frequencies (modes). In this respect, the situation is similar to the one considered in the preceding section for the ICLS technique. To provide for the longitudinal and transverse mode quasicontinuum and to suppress intermode beats against the signal decay background, the aperture a of the absorption cell should not be too small (the Fresnel number NF = a2 /λL > 1). • The spectral width of the pulsed laser radiation should exceed the mode interval, and the coherence length should be less than the length of the absorption cell. When using pulsed lasers, laser-pumped dye lasers with typical pulse durations ca. 10−8 s, pulse repetition frequencies ≤ 102 Hz, and spectral widths ca. 0.1–1 cm−1 , the above requirements are fulﬁlled with the absorption cell geometries in actual use. Figure 3.15 presents as an example a fragment of the absorption spectrum for the X2 Π(v = 0)–A2 Δ(v = 0) transition of the CH radical in the plasma of an expanding cascade arc stream in an argon-acetylene mixture (typical total CH densities (0.5 –

3.4 Indirect Methods for Measuring Absorption of Laser Light

0.8)×1010 cm−3 [51]. Under the conditions of this experiment, the spectrum width (0.4 cm−1 ) of the dye laser used exceeded both the intermode intervals and the Doppler width (0.16 cm−1 ) of the absorption lines. In this example, the size of the plasma along the axis of the absorption cell was ca. 20 cm, so that χ ≈ 2 × 10−6 cm−1 . The development of the CRDS technique is associated, in particular, with the allowance for and use of the resonance properties of the absorption cell. Experiments show that the use of narrow-band lasers in combination with the tuning of either the laser frequency or the intrinsic absorption cell frequency makes it possible to measure absorption at a level of 10−10 cm−1 [52]. More detailed information about the various schemes and implementations of the CRDS technique can be found in [53, 54].

3.4 Indirect Methods for Measuring Absorption of Laser Light

The absorption methods considered in Sections 3.1 and 3.2 are based on measuring the intensity changes that light suffers on passage through the medium under study. These direct methods are also known as transmission or absorption-transmission methods. Their great advantage is due to the possibility they provide of taking direct measurements of the absolute densities of the absorbing states of particles without having to resort to any additional measurements, provided that light causes no saturation of the absorptive transition. Additionally, no knowledge is required as to the population and decay mechanisms of the levels bound by the transition. An alternative is the measurement of the variation of one or another parameter of the medium caused by the light passing through it. In contrast to the direct linear absorption techniques, use here is made of phenomena depending on the behavior of the excited state. This may be, for example, the heating of the object of study via upon radiationless quenching. In that case, one can measure temperature changes (e.g. via optico-calorimetric, optico-thermal and refraction methods) or the associated density variations (optico-acoustical methods). The transmitted light can excite luminescence in the object (ﬂuorescence methods), effect direct ionization of particles, or indirectly change the ionization balance in plasma (photoionization methods). A more detailed list of such methods and a discussion of their physical bases can be found in [23]. Such techniques prove more efﬁcient if they make use of some effect not inherent in the object of interest in the absence of irradiation.

103

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3 Emission, Absorption and Scattering Techniques

Figure 3.16 Transitions in a three-level particle excited at the frequency ν12 of the 1–2 transition. Solid arrows indicate radiative transitions and the dashed ones, radiationless transitions.

They are known as no-background methods. But such a requirement frequently proves very difﬁcult to meet in the case of plasma with its numerous intrinsic processes and phenomena – heating, luminosity, oscillations, conduction, and so on. Nevertheless, some of the abovementioned techniques have found successful application in plasma spectroscopy. In the following sections we will discuss some of these methods, for all those using laser-induced ﬂuorescence and the optogalvanic effect, achieved with the use of frequency-tuned lasers. 3.4.1 Induced Fluorescence 3.4.1.1 General Characteristic

If the frequency of light transmitted through the object under study from an external source corresponds to the absorption line ν12 of a particle, the radiative decay of the excited state in the three-level scheme (Figure 3.16) is accompanied by the emission of a quantum of either the same frequency, ν21 = ν12 , or of a different frequency, ν23 . In the former case, reemission is called resonance ﬂuorescence and in the latter, nonresonance ﬂuorescence (scattering). In the current discussion, we will refer to them collectively as induced ﬂuorescence. The solid arrows in Figure 3.16 correspond to radiative transitions and the dashed ones, to their radiationless counterparts. Investigations into induced ﬂuorescence en-

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.17 Localization of the induced ﬂuorescence signal.

joy a long history (references to earlier works can be found in [18]). The further development of these investigations and the ensuing applications of induced ﬂuorescence techniques have owed largely, and nowadays practically exclusively, to the use of lasers, so that the term laser-induced ﬂuorescence (LIF) has become standard. The use of collimated beams makes it possible to localize the induced ﬂuorescence region within a volume of Vf = l A, where A is the crosssectional area of the beam and l is the size of the ﬂuorescence region along the beam, determined by the observation angle (Figure 3.17). A very important (if not the major) advantage of the LIF technique is its high sensitivity. Its threshold value is easy to estimate by following the reasoning of Section 3.3.2, the only difference being that subject to measurement in this case is not the small change in the power of the incident radiation, ΔPmin , but the minimum ﬂuorescence power Pf,min . If the power of the laser beam is P, the power density I = P/A, and the absorption coefﬁcient χ, the power of ﬂuorescence from the observation volume is then Pf = I Alχ( I ).

(3.45)

105

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3 Emission, Absorption and Scattering Techniques

The optimal laser power density corresponds to the saturation value (Section 3.2): I = Is = hν12 /(2σ12 τ2 ),

(3.46)

where τ2 is the lifetime of level 2. In that case, the absorption coefﬁcient is reduced by half: χ = σ12 ( N1 − N2 ) = σ12 ( N10 − N20 )/2 ≈ σ12 n/Vf .

(3.47)

It is also supposed here that in the absence of irradiation N10 N20 , N1 − N2 ≈ N.n = N/Vf is the number of particles in the ﬂuorescence region being observed. Then hν12 hν σ n l = 12 n. Pf = A 12 (3.48) 2σ12 τ2 2Al 4τ2 The minimum detectable number of particles, nmin , depends on the minimum detectable ﬂuorescence power Pf,min : nmin = 4τ2 Pf,min /hν12 . Maximum sensitivity is attained if the minimum detectable ﬂuorescence power at the resonance frequency, for example, ν21 , is limited by the quantum noise power Pn of the detector with an efﬁciency of ηD (cf. expression (3.26)): Pf,min = Pn = hν21 BD /ηD ,

(3.49)

−1 where BD is the detector bandwidth (BD is the detection time). Also advantageous is a situation where the lifetime of the excited level is mainly governed by the radiation loss τ2−1 ≈ A21 . A realistic estimate should also contain the geometrical factor determining the efﬁciency of ﬂuorescence transmission to the detector, ηg = Ω/4π (Figure 3.17). With these remarks, for the threshold sensitivity at the ﬂuorescence frequency ν21 , the quantity ν21

nmin =

4BD ( η ηg ) − 1 , A21 D

(3.50)

whence the potentialities of the LIF technique are evident. The presence of the factor (ηD ηg )−1 , whose value in practice can be as high as ca. (102 – 104 ), can be easily offset by the quantity BD /A21 , for example, at a typical detection time of 1 s and Einstein coefﬁcient of 108 s−1 . The meaning of the compensation is that during detection the same particle can be

3.4 Indirect Methods for Measuring Absorption of Laser Light

repeatedly involved in the excitation-ﬂuorescence cycle, and, generally speaking, detection is possible at nmin < 1. The above-considered case of resonance ﬂuorescence can be reduced to the two-level scheme, where the presence of other levels and the associated dissipative processes can be taken into account in the form of appropriate corrections. This case corresponds to the fastest cycle, but in practice it gives rise to one more limitation, in addition to the ones considered above; namely, that due to the background scattering of the exciting radiation, which frequently becomes the dominant factor. If ﬂuorescence is observed at a frequency other than the laser frequency, for example, at a frequency of ν23 (Figure 3.16), the background limitations are removed, thanks to spectral separation, but the more involved ﬂuorescence cycle is slowed down, which in turn impairs sensitivity. The above reservations notwithstanding, even in the early experiments [55, 56] single atoms and ions were detected. On the whole, the two-level description sufﬁciently conveys the main features and tendencies of the LIF technique . We will, therefore, consider the important speciﬁcs of the ﬂuorescence process kinetics in the twolevel approximation, making reference to the important limitations, as may be necessary. 3.4.1.2 Fluorescence Excitation by Continuous-Wave and Pulsed Laser Light

In the text above, we have already used the balance of the population and decay rates of the energy levels of particles, from which it followed that high sensitivity is attained on account of high excitation power saturating the absorptive transition. For this reason, it often proves expedient to use high-power pulsed lasers. In addition, with synchronous detection, this makes it easier to discriminate between the ﬂuorescence signal and various kinds of noise. Moreover, the time evolution of the signal in the case of excitation by a short laser pulse allows one to judge the character of the processes occurring in the object under study, provided that one correctly interprets the ﬂuorescence kinetics. It has already been noted in Section 3.3.4 that in such cases account should also be taken of the possibility of manifestation of coherent processes if the interaction of light with the oscillators of the medium occurs on a short time scale. Therefore, we will ﬁrst touch upon the question as to the applicability of the level population balance equations in the case of light-medium interaction, which has already been partly discussed in Section 3.3.4. A more general description is the density matrix formalism [34–38, 57], the diagonal elements ρii of the matrix describing the populations of the energy

107

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3 Emission, Absorption and Scattering Techniques

levels and the nondiagonal ones, ρij , expressing the measure of concordance of the phases of the wave functions corresponding to these energy eigenvalues. To describe the transition of the system from its coherent to noncoherent state under the effect of external factors, we introduce the characteristic time T2 of the exponential decay of ρij . We assume that • the atomic system is the two-level type, • the light wave of circular frequency ω is plane, • the particles are at rest (the Doppler dephasing is disregarded), • the variation of the intensity of light occupies a time of τp exceeding the light oscillation period. Under the assumptions made, the equations for the elements of the density matrix were derived in [58]. For a single diagonal and a single nondiagonal element, having deﬁned, in accordance with the assumptions made, the slowly varying amplitude σ21 , we get ρ21 = σ21 exp{−iωt}

(3.51)

and averaged over the interval Δt τp : ∂σ21 = σ21 [i(ω − ω0 ) + Q21 ] + iΩR (t)(ρ11 − ρ22 ), ∂t ∂ρ22 ∗ = −iΩR (t)(σ21 − σ21 ) − ( A21 + Q21 )ρ22 . ∂t

(3.52) (3.53)

= T −1 is the dephasHere ω0 is the central resonance frequency, Q21 2 ing (transverse relaxation) rate, Q21 is the collisional population depletion rate of upper level 2, A21 is the spontaneous radiative decay rate of upper level 2, the superscript ∗ denotes complex conjugation and ΩR (t) = μE(t)/2¯h is the Rabi frequency (3.33). Consider conditions under which (3.52) and (3.53) are reduced to the ordinary balance equations with time-independent rate constants. Equation (3.53) for the population of excited level 2 assumes the traditional form if the ﬁrst term on its right-hand side describes the ﬁeld-induced absorption and stimulated emission processes. With the proviso that the rate constants of these processes are time-independent, one should assume that the quantities σ21 have already reached their steady-state values, and so the left-hand side of (3.52) is equal to zero. To calculate the population ρ22 , one should substitute expression (3.51) into (3.53) and use the imaginary part of σ21 : ( ) Q 21 . (3.54) Im(σ21 ) = ΩR (t)(ρ11 − ρ22 ) (ω − ω0 )2 + Q 221

3.4 Indirect Methods for Measuring Absorption of Laser Light

The form of this dispersion expression corresponds to the statement made in Section 3.3.4 that the transverse relaxation rate can be directly determined from the stationary line proﬁle (ﬂuorescence line proﬁle in the given case). Allowance for the dephasing due to the Doppler shifts reduces (3.54) to the form corresponding to the Voigt resonance line proﬁle [58]. For ρ22 , we get ∂ρ22 = 2πΩ2R (t) ϕ(ω )(ρ11 − ρ22 ) − ( A21 + Q21 )ρ22 . (3.55) ∂t Equation (3.55) includes the Rabi frequency and, correspondingly, the transition dipole moment. Let us express in terms of this quantity the Einstein coefﬁcients [35], assuming that the statistical weights of the levels are equal: B = B12 = B21 =

8π μ2 . 3 h¯ 2

(3.56)

Considering also that the light intensity I ∼ E2 , so that Ω2R ∼ μ2 E2 , we rewrite (3.55) in the form common with balance equations: ∂N2 = const BI ( N1 − N2 ) − ( A21 + Q21 ) N2 , (3.57) ∂t where the populations ρ11 and ρ22 are replaced by the more traditional designations N1 and N2 . Thus, the balance equations resorted to for the interpretation of ﬂuorescence signals hold true when phase relaxation comes to an end after the intensity of the external ﬁeld has changed, that is, on a time scale exceeding the reciprocal width of the static line proﬁle (Section 3.3.4). 3.4.1.3 Induced Fluorescence Saturation and Decay

Let us assume that the balance equations are applicable and consider the two-level system of levels 1 and 2 presented in Figure 3.16. The amount of molecules at levels 1 and 2 in the excitation and ﬂuorescence zone (with the diffusive escape of the excited particles disregarded), n1 = N1 /Vf and n2 = N2 /Vf (3.47), and the total number of particles are conserved: n0 = n1 ( t ) + n2 ( t ).

(3.58)

Also, let the spectral width of the exciting light exceed the line width of the 1 → 2 transition. The ﬂuorescence intensity If ∼ A12 n2 and, in notation of Figure 3.16, ∂n2 = B12 I (t)n1 (t) − ( B21 I (t) + A21 + Q21 )n2 (t) ∂t

(3.59)

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3 Emission, Absorption and Scattering Techniques

or, considering (3.58), ∂n2 = n0 B12 I (t) − n2 (t)[ a2 + b2 I (t)], ∂t a2 = A21 + Q21 , b2 = B12 + B21 . Continuous-Wave Excitation

(3.60) (3.61)

The left-hand side of (3.60) is equal to zero,

the intensity I = I0 , n2 = n0 B12 I0 ( a2 + b2 I0 )−1 .

(3.62)

At low exciting radiation powers, a2 b2 I0 , the ﬂuorescence intensity If ∼ n2 = n0 I0 B12 /a2

(3.63)

is proportional to the radiation power and the total density of particles at the absorbing level and, with the Einstein coefﬁcients speciﬁed, inversely proportional to the quenching rate. At high powers, a2 b2 I0 , there takes place complete saturation, n2,s g1 = n1,s g2 , n2,s = n0 B12 /b2 = n0 (1 + g1 /g2 )−1 ,

(3.64)

and the ﬂuorescence intensity is only determined by the density of particles and level degeneration. The radiation power at which the population of the upper level is half that at complete saturation, n2,s,1/2 = n2,s /2 (saturation power) is, as seen from expressions (3.62) and (3.64), I0,s = a2 /b2 .

(3.65)

If the excitation time is commensurable with or less than the derivative on the let-hand side of (3.60) should be conserved. For the sake of deﬁniteness, let the excitation pulse have a nearly square shape. The intensity rise and fall times should be, from coherent phenomena considerations, greater than the reciprocal width of the transition line, but sufﬁciently small in comparison with the pulse duration, so that the ﬁnite duration of the pulse edges can be neglected: I0 at 0 < t < t0 I (t) = (3.66) 0 at t > t0 Pulsed Excitation

a2−1 , b2−1 ,

In that case, the solution of (3.60) will have, at 0≤ t ≤ t0 , the form n2 (t) = n0 B12 I0

[1 − exp {−( a2 + b2 I0 )t}] , a2 + b2 I0

(3.67)

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.18 Fluorescence intensity variation under pulsed excitation.

and at t > t0 , n2 ( t ) =

n0 B12 I0 [exp { a2 t0 } − 1] exp {− a2 t} . a2

(3.68)

These solutions at t0 = 5 ns, a2 = 109 s−1 , b2 I0 = a2 , B12 = b2 /2, and g1 = g2 are graphically shown in Figure 3.18 [59]. Under the conditions of the ﬁgure, the characteristic times for the ﬂuorescence rise (with reference to the 1/e level) after the start of excitation, tr = ( a2 + b2 I0 )−1 , and decay after its ﬁnish, td = a2−1 , amount to 0.5 ns and 1 ns, respectively. By analogy with expression (3.65), the concept of saturation power is also introduced in the case of pulsed excitation. However, here the quantity I0,s is no longer so simply related to the quenching rate as in the case of continuous-wave excitation, but also depends on the duration and shape of the excitation pulse. These questions are analyzed, with due regard for the ﬁniteness of the ﬂuorescence signal detection time, for example, in [60]. Another circumstance that should be taken into account in induced ﬂuorescence experiments is associated with the fact that even where the measurements are ‘rigidly’ localized the conditions at different points of the observation volume may be dissimilar as a result of spatial inhomogeneity. This is true of both continuous-wave and pulsed excitation. As the exciting radiation power is raised, the effective volume at which saturation exists increases on account of the ‘wings’ in the cross section

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of the laser beam. For Gaussian beams, these questions are considered in [61–63]. In principle, the two-level scheme allows the main qualitative, but also in many cases, quantitative features of the induced ﬂuorescence kinetics to be conveyed. Of course, adding complexity to the scheme of the energy terms makes it possible to describe the induced ﬂuorescence of atoms and molecules under actual conditions in greater detail. However, the number of balance equations is increased, and the analytical solutions of the system are not always possible or convenient for the physical interpretation of the experiment. Even for the three-level scheme of Figure 3.16 it is only the limiting cases classiﬁed by the relation between the level-to-level transition rates that are treated analytically. More complex schemes are practically always analyzed numerically. A great many works, mostly early ones, have been devoted to these questions. Their review can be found in [23]. A later bibliography can be found, for example, in [64, 65]. 3.4.1.4 Induced Fluorescence Quenching and Taking Account of this Process

Expressions such as formulas (3.60)–(3.68) describing the relationship between the induced ﬂuorescence intensity and the population of the lower level (for the two-level scheme with a dipole-allowed transition in the visible region of the spectrum and practically full particle density) include the quantity a2 (3.61), dependent on the frequency Q21 of the collisional quenching of the excited level. In contrast to A21 , the quantity Q21 depends on the conditions prevailing in the object under study. At elevated particle densities the quenching process can be important in comparison with spontaneous decay. Two techniques are utilized to asses such. One is to introduce corrections for quenching by means of known quenching rates (frequencies). The other is to choose such measurement schemes as would allow the quenching effect in the object of interest to be excluded or taken into account experimentally. Determination of the Decay Rates of Excited Levels by the Induced Fluorescence Method With the results of measurements of induced ﬂu-

orescence decay at one’s disposal, one can use formulas (3.60)–(3.68) to determine the quantities A21 and Q21 . These results can be obtained both by observing the ﬂuorescence decay directly and by measuring stationary ﬂuorescence intensities. In either case it is necessary to vary the pressure of the gas whose particles are responsible for quenching. Examples of induced ﬂuorescence decay measurements are presented in

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.19 Decay of the ﬂuorescence of Ar (450.4 nm) at various pressures of the quenching molecules CH4 : 1 – 0 Pa; 2 – 90 Pa; 3 – 180 Pa; 4 – 310 Pa.

Figure 3.19 [66]. The ﬁgure illustrates the decrease of the induced ﬂuorescence intensity with time following cessation of the excitation of argon (excitation from a metastable state at a wavelength of 750.4 nm, emitting state 2p1 ) at various pressures of the quenching CH4 gas. In accordance with formula (3.68), the slopes of the curves on the semilog scale determine the decay times td = a2−1 . The quenching frequency Q21 = kQ [Q] is governed by the particle density [Q] of the quenching gas and the appropriate quenching rate constant kQ : a2 = A21 + kQ [Q].

(3.69)

Representing a2 as a function of [Q] or, at ﬁxed temperature, of the quenching gas pressure PQ (the so-called Stern–Fulmer plots), one can ﬁnd k Q from the slope of the plots. Such plots for the quenching of the 2p1 , 2p5 , 2p6 , and 2p8 ) states in Ar by the CH4 gas are presented in Figure 3.20 [66]. If quenching by the own gas can be disregarded, the extrapolation of a2 to PQ = 0 yields the radiative decay rate A21 of the corresponding states. For the above plots at T = 300 K, the constants k Q amount to (9.3, 6, 3.4, and 7.4) ×10−10 cm3 · s−1 , and the effective radiative decay rate for the entire group of states is A21 = (4 ± 0.5) × 107 s−1 . The rate constants for the collisional quenching of induced ﬂuorescence can also be found from the variation of continuous-wave ﬂuorescence as a function of the quenching gas pressure by means of relations (3.61)–(3.63). For example, at low exciting radiation powers, one can see

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Figure 3.20 Decay frequencies of the Ar(2 p1 2 p2 2 p6 2 p8 ) states as a function of the pressure of the quenching molecules CH4 . Temperature 300 K.

from relation (3.63) that when A21 = kQ [Q], the ﬂuorescence intensity is half that at PQ = 0. Figure 3.21 presents the results of measurements [66], showing the pressures of various gases at which the induced ﬂuorescence intensity of the Ar line at λ = 750.4 nm is halved. Methods to Suppress the Effect of Collisional Quenching Saturation of the Induced Fluorescence Signal That the need to take account of the col-

lisional quenching of the induced ﬂuorescence when measuring particle densities can be obviated is evident from relations (3.62), (3.64) and (3.65) for high exciting radiation powers. The necessary excitation conditions are not very difﬁcult to estimate. Let, for example, the line widths of both the exciting radiation and absorptive transition be Δν = 0.1 cm and let quenching dominate, Q21 = 103 A21 . If I0 = (cAΔν)−1 and saturation condition (3.64) is satisﬁed at I0 > 103 A21 /B21 , the power of the continuous-wave exciting radiation is then P [W] > 1.2 × 104 A [cm2 ]. At P = 100 W the radiation should be focused into a region less than 8 × 10−3 cm−3 in size, that is, the focal spot radius should be ≤ 0.5 mm. Such parameters are easy to realize. When using this method in practice, one should ensure that the assumptions made are adhered to:

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.21 Pressures of various quenching gas particles at which the induced ﬂuorescence intensity of the Ar line at 750.4 nm is halved. Temperature 300 K.

• As already noted, as the exciting radiation power is increased, the nonuniform intensity distribution over the cross section of the beam causes the region of saturated induced ﬂuorescence to enlarge, and so the observation geometry should provide for the necessary localization. • Two-level approximation. For real particles (especially molecules), each of the ‘levels’ belongs in a block of states with some distribution of particles. Under excitation by a high-power narrow-band radiation, the rate of phototransitions can be commensurable with that of transitions between these states (cross-relaxation). In this case, the saturation condition can be difﬁcult to attain, and when interpreting the relationships between the induced ﬂuorescence intensity and parameters of the object under study, one should use relaxation models. For this latter reason, the use of this method in practice is limited to simple atomic systems. Excitation of States Characterized by Fast Controlled Radiationless Decay An excitation scheme is selected in which the upper level strongly

interacts with the continuous spectrum (predissociation, autoionization (Figure 3.22)). If the total density of the parent particles barely changes as a result of dissociation or ionization during the action of the exciting light, and the inverse processes of recombination can be disregarded, the

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Figure 3.22 Fluorescence excitation in conditions of interaction between the upper level and the continuous spectrum (i – ionization, d – dissociation).

system can then formally be treated as a two-level system subject to condition (3.58). Let us ﬁrst consider the effect of the 2 → d transition whose rate Q2s is determined by the spontaneous intramolecular process. In the quasistationary case, the induced ﬂuorescence intensity during the excitation and observation period t > ( A21 )−1 + ( B21 )−1 + ( B12 )−1 + ( Q21 )−1 + ( Q2s )−1

(3.70)

is expressed in the form of relation (3.62) wherein a2 is replaced by a2 = A21 + Q21 + Q2s . If Q2s A21 + Q21 and a2 ≈ Q2s , the induced ﬂuorescence intensity ceases to depend on the collisional quenching rate Q21 : n0 = n2 ( Q2s + b2 I0 )( B12 I0 )−1 .

(3.71)

The ‘price’ for this independence is the sharp reduction of the induced ﬂuorescence intensity. This possibility has been intensely studied for the OH radical that plays an important part in plasma chemical reactions [22, 23]. This radical suffers predissociation in the OH(A2 Σ, v = 3) state because of its interaction with the OH(4 Σ− ) repulsive state. To illustrate, when OH is excited at a wavelength of 248 nm in conditions typical of atmospheric-pressure ﬂames, OH(X2 Π, v = 0 → A2 Σ, v = 3), A21 ≈ 106 s−1 , Q21 ≈ 109 s−1 , Q2s ≈ 1010 s−1 . Similar measurements for H2 O and O2 were analyzed in [69]. When taking such measurements, particular attention should be given to the validity of the two-level assumption. If, for example, the power of the exciting laser radiation is raised to B12 I > τrot (τrot is the rotational relaxation time) in order to compensate for the loss of the induced ﬂuorescence intensity, this will disturb the distribution of the molecules among the rotational levels of OH(X2 Π, v = 0 ). This in turn will make the ﬂuorescence intensity dependent on the conditions in plasma, on account of the ﬁniteness of τrot [70].

3.4 Indirect Methods for Measuring Absorption of Laser Light

An alternative possibility to involve a fast, controllable loss of particles in the excited state 2 is associated with the use of the 2→ i photoprocesses (photoionization, photodissociation). To this end, it is convenient to use an extra laser whose frequency is off resonance with regard to the 1→2 transition, but is sufﬁcient to move the particle from state 2 into the continuous spectrum. In this case, the exciting and additional laser powers, I0 and I2i , respectively, are controlled independently, which makes it possible to separate their contributions to the 2→ i process. Formula (3.62) again holds true, with a2 replaced by a2 = A21 + Q21 + Q2i , where Q2i ∼ I2i σ2i is the rate of the 2→ i photoprocess and σ2i is the corresponding cross section. One can see from expression (2.95) for the photoionization cross section that the frequency of the additional laser should be selected near the threshold value. If Q2i A21 + Q21 , then, by analogy with expression (3.71), n0 = n2 ( Q2i + b2 I0 )( B12 I0 )−1 .

(3.72)

In principle, this method also makes it possible to determine n0 at low additional laser powers, without imposing the condition that Q2i A21 + Q21 . To this end, one should take measurements at two different values of I2i under the same conditions in plasma and exclude with their aid Q21 from a2 and, accordingly, from expression (3.62). It will then be possible to avoid great losses of the induced ﬂuorescence intensity. One should bear in mind, however, that from measurement accuracy considerations, the quantity Q2i should nevertheless be comparable with A21 + Q21 . The method is easy to extend to pulsed measurements (see [71, 72] for details). Further, to raise the particle to state 2 from state 1 use is sometimes made of multiphoton excitation. In that case, one or several photons can effect both ﬂuorescence excitation and the 2 → i transition, the powers I0 and I2i being mutually dependent. Such a situation is considered in [66, 72]. Fluorescence Excitation by Short Laser Pulses The availability of lasers capable of short pulse durations, t0 ∼ 10−12 –10−10 s, and the appropriate fast registration techniques also allows minimizing of the effect of quenching on measurements by the induced ﬂuorescence method. For the sake of simplicity, let ﬂuorescence be excited by a square pulse and the temporal behavior of the induced ﬂuorescence intensity be described by formulas (3.67), (3.68). At t > t0 and t0 a2 = A21 + Q21 ,

n2 (t) = n0 B12 I0 t0 exp(− a2 t).

(3.73)

If we extrapolate expression (3.73) to the instant the pulse starts, t → 0, and denote this value by n˜ 2 (0), the particle density measured will then

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Figure 3.23 Time evolution of the ﬂuorescence of the OH radical. Excitation on the R2 (10 ) line of the OH(A−X(1 −0 ) ) band, emission on the P2 (11 ) line of the same band: 1 – registered intensity; 2 – smoothed curve; 3 – registered shape of the exciting laser pulse; 4 – curve following elimination of the instrument function (laser pulse) and extrapolation to the instant t = 0 [60].

be independent of the quenching rate: n0 = n˜ 2 (0)/( B12 I0 t0 )

(3.74)

This technique corresponds to the early proposals [73] that have been further developed by many authors [64, 67, 74]. Figure 3.23 shows as an example an experimental record of the ﬂuorescence signal from the OH radical in an atmospheric-pressure ﬂame [64]. The exciting pulse duration is 80 ps. Shown are the temporal behavior of the induced ﬂuorescence intensity, the response curve of the recording apparatus and the extrapolated curve. When using this approach, one should consider the possibility of development of the above-described coherent effects on a time scale of the order of the phase relaxation time. For this reason, the extrapolation of the induced ﬂuorescence decay curve should be carried out for its section beyond this time interval. The method will then allow one to determine the collisional quenching rate of the induced ﬂuorescence. 3.4.1.5 Restrictions Imposed by the Plasma’s Own Glow

The ﬂuorescence induced in a real plasma object by an external radiation source exists against the background of the plasma’s own glow, which limits the possibility of detecting weak induced ﬂuorescence signals. Let

3.4 Indirect Methods for Measuring Absorption of Laser Light

us present the most simple estimates of the region of this limitation, assuming once more that the atomic particle of interest is a two-level type and the plasma itself is optically thin [75]. For the experimental scheme of Figure 3.17, the detected power of the plasma’s own (background) radiation emitted on the 2→1 transition is Pbgr =

1 A hν ΩV N2 , 4π 21 12

(3.75)

where V is the plasma volume contributing to the background signal and N2 is the population of the emitting level in the absence of the external irradiation. The population ratio hν12 N2 g2 , (3.76) = exp N1 g1 kB Texc where Texc is the excitation temperature (Section 1.3). Let the intensity of the exciting radiation cause complete saturation of the transition, that is, the leveling-off of the populations per unit statistical weight: N2∗ g1 = N1∗ g2

(3.77)

Proceeding from the condition of conservation of the total number of particles, N2∗ +N1∗ = N2 + N1 , and expressions (3.75)–(3.77), we get " ! 1 + ( g1 /g2 ) exp k hνT12exc N2∗ B = . (3.78) N2 1 + ( g1 /g2 ) Considering the geometrical factor, the induced-ﬂuorescence-to background-intensity ratio is N ∗ − N2 Vf Pf . = 2 Pbgr N2 V

(3.79)

The measurement locality condition requires that Vf < V. Therefore, to measure the induced ﬂuorescence against the background, it is necessary that N2∗ > N2 and, accordingly, hν12 > kB Texc .

(3.80)

In LTE plasmas, the excitation temperature is equal to the local equilibrium temperature. In the more complex partial equilibrium conditions (Section 1.3.2), the temperature Texc does not, as a rule, exceed the temperature of certain subsystem, usually the electron temperature. When induced ﬂuorescence is excited in the visible and longer-wavelength regions of the spectrum, condition (3.80) holds true for plasma temperatures of no more than a few electron-volts. This is a principal restriction,

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and so the use of special techniques for detecting induced ﬂuorescence can only lessen to a limited extent the inequality. As can be seen from expression (3.40), the problem of diagnosing highly luminous plasmas can be solved by going over to the short-wave region of the spectrum. However, the problem of developing tunable lasers covering this region currently cannot be considered to have been solved satisfactorily. Worthy of note in this connection are the recent suggestions (see e.g. [80]) as to the use of excitation sources other than lasers, in particular synchrotron X-ray radiation. On the whole, as regards the method under discussion, synchrotron radiation is similar to laser radiation, but in a different region of the spectrum. It has a small spectral width, is capable of frequency tuning and can be collimated. The authors of [80] have used synchrotron radiation in the photon energy range 64.000 × 10−16 J (40 keV)–160.000 × 10−16 J (100 keV) (0.3– ˚ with a monochromaticity of Δλ/λ ≈ 10−3 , ca. 10−5 in beam 0.12 A), divergence, and 1 × 1 mm2 in beam cross section. The measurement locality ca. 1 mm3 . In that case, condition (3.40) formally holds true at excitation temperatures up to 108 –109 K, which embraces practically all the actual plasma objects. The measurement schematic is also identical to that presented in Figure 3.17, although, of course, the techniques used to register the exciting radiation and induced ﬂuorescence intensities are typical for the X-ray region. To monitor the synchrotron radiation intensity, use is made of an ionization chamber, and the induced ﬂuorescence intensity is measured by means of a germanium detector. The spectra of ﬂuorescence induced by so ‘hard’ a radiation correspond to atomic transitions involving the inner electron shells. Figure 3.24 shows the spectrum of ﬂuorescence induced by synchrotron radiation in the central zone of a high-luminance halogen lamp (T ≈ 6000 K, pressure P ≈ 10 atm) in a region of 44.800 × 10−16 J (28 keV)–108.800 × 10−16 J (68 keV), the excitation being effected by photons 110.240 × 10−16 J (68.9 keV) in energy. The broad peaks at 76.160 × 10−16 J (47.6 keV), 87.680 × 10−16 J (54.8 keV), and 103.360 × 10−16 J (64.6 keV) are due to the Compton scattering of the synchrotron radiation by the lamp bulb, and the narrower peaks are the characteristic iodine, cesium and dysprosium lines. The results of such measurements help study the spatial distributions of elements in plasma, which necessitates absolute calibration against a cell of known particle density. Another speciﬁc feature of X-ray-induced ﬂuorescence is the opportunity it provides to investigate plasmas behind cavity walls opaque to radiation in the spectral region traditional for the laser-induced ﬂuorescence techniques.

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.24 Fluorescence intensity of a halogen lamp under excitation with synchrotron radiation (111.680 × 10−16 J (69.8 keV)).

3.4.2 Optogalvanic Spectroscopy 3.4.2.1 The Use of the Optogalvanic Effect to Measure Light Absorption in Plasma

Optogalvanic spectroscopy is based on the optogalvanic effect (OGE) – the change of the electrical characteristics of plasma on the passage of light through it. The physical cause of the OGE is the redistribution of the level populations of atoms or molecules upon the absorption of light. As a result of various radiative and/or collisional and/or collective processes, the densities of charged particles, their mobilities and energies change, which reﬂects on the ionization balance and conductivity of the plasma. The change in the electrical properties of the plasma thus occurs as a result of processes secondary to absorption, the wavelength dependence of the magnitude of the effect being of resonance character. In the ﬁnal analysis, the optogalvanic effect is registered by the change of the current i or voltage U in the electric circuit of the gas-discharge plasma (Figure 3.25) or by means of special probes inserted in the plasma, including plasma of nonelectric origin. The ﬁrst theories concerning such an absorption detection method were presented as early as the 1920s, but their extensive practical de-

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Figure 3.25 Block diagram of optogalvanic measurements: 1 – discharge; 2 – voltage source; 3 – capacitor; 4 – laser; 5 – modulator; 6 – phase-sensitive detector; 7 – recording device; 8 – resistor.

Figure 3.26 Absorption spectra of a discharge in an He-Ne mixture: A – direct registration of the intensity variations of a tunable laser radiation; B – optogalvanic registration.

velopment only started with the advent of frequency-tuned lasers to be used as probe radiation sources. The sensitivity of this method has been found to be frequently in excess of that of the direct absorption techniques. The physics of the optogalvanic effect and its applications in plasma spectroscopy have been considered in the book [77] (among the latest publications, see [78]), and therefore we will restrict ourselves in this section only to a qualitative description, some simple relations given without derivation and individual illustrations. Figure 3.26 presents as an example the absorption spectra of a discharge in an He-Ne mixture, obtained by (i) the ordinary intensity measurement method and (ii) the optogalvanic spectroscopy technique. Along with its high signal to noise ratio, the optogalvanic spectrum has a speciﬁc feature that distinguishes it from the ordinary optical spectrum. While all the lines in the optical spectrum are due solely to light absorption, different lines in the optogalvanic spectrum can be characterized by ΔU values differing in sign. In describing the formation mechanism of the optogalvanic spectrum, one can single out two general stages. First, it is necessary to discover how light absorption causes the density or energy of the charged par-

3.4 Indirect Methods for Measuring Absorption of Laser Light

ticles to change. And secondly, is it necessary to associate this change with the change in the macroscopic characteristics of the object under study and the electrical parameters of the discharge circuit. Until now, the basic physics of the optogalvanic spectroscopy of nonthermal plasma has been best developed for the glow discharge, in which the variations of the electrical parameters of the positive column are of small perturbation character. One can demonstrate that the current variations Δi in the discharge circuit are in this case proportional to the lightinduced changes in the ﬁeld intensity in the positive column, ΔE, for both the normal and the abnormal discharge [77]: Δi ∼ −ΔE.

(3.81)

When discharges are irradiated with light in the visible or ultraviolet regions of the spectrum that correspond to the electronic transition regions in atoms or molecules, the mechanisms of the optogalvanic effect are, as a rule, associated with the lowering of the ionization threshold by electron impact or with the variation of the chemical ionization rate. In the case of resonance light absorption in the region of the infrared vibrational– rotational molecular spectra, the mechanism governing the variation of the electrical parameters of the discharge circuit is associated with the thermal expulsion of the gas upon conversion of the energy of the internal molecular motion into heat (vibrational–rotational–translational relaxation). The intensities of optogalvanic spectra can be substantially increased by irradiating the electrode regions of the discharge, because even an insigniﬁcant change in the rate at which electrons are knocked out of the cathode (even the excited particles) has a noticeable effect on the discharge current. The optogalvanic spectroscopy of the electrodeless high-frequency discharge is speciﬁc in that the discharge circuit impedance is a complex quantity, and the measure of the optogalvanic effect is usually taken to be the light-induced change of the speciﬁc energy deposition ΔQ. The theoretical interpretation of the spectrum agrees well with experiments for discharges with a low density of the plasma-forming gas, even in the fairly simple ambipolar diffusion model. The registration of optogalvanic spectra in thermal plasmas (ﬂames, for example) is carried out by means of extra electrodes inserted in the plasma or by probing it with a microwave ﬁeld. The variations in the absorption of a weak microwave ﬁeld in probing are due to the lowering of the ionization threshold. In such a case, as distinct from nonthermal plasmas, the optogalvanic spectra are easier to interpret, because the ionization rate constant here obeys the Arrhenius theory, and so there is no

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need to resort to kinetic schemes for interpretation. When applying the ﬁeld across the additional electrodes, account should be taken of the polarization of charges and their separation, which speciﬁcally affects the formation dynamics of the optogalvanic signal and its voltage saturation (the ‘two-cloud’ model). High-sensitivity optogalvanic spectroscopy uses the so-called thermionic diodes that are essentially ion traps. In the simplest case, it comprises a cathode (hot wire) and an anode a few millimeters distant from it, usually in the form of a cylinder around the cathode. The cathode emission produces a negative space-charge region. Electrons can overcome this potential barrier, and so a current exists in the diode circuit without any voltage being applied, provided that the electrode materials have different work functions. If an ion enters the space-charge region, the barrier is lowered, which increases the current through the diode. In modiﬁed diode versions, the signal to noise ratio in the optogalvanic spectra is more than 102 times that in the glow discharge. 3.4.2.2 High-Resolution Optogalvanic Spectroscopy

The majority of the purely optical linear and nonlinear laser spectroscopy methods can include the optogalvanic detection technique. The latter often proves preferable, thanks to its high sensitivity and signal to noise ratio. The intra-Doppler optogalvanic saturation and two-photon absorption spectroscopy (two-photon Lamb dip optogalvanic spectroscopy) can be implemented by the so-called intermodulation optogalvanic spectroscopy (IMOGS) method (Figure 3.27). The beam of a frequency-tuned laser is split into two beams that are then made to cross at an angle close to 180◦ . The beams with the intensities I1 and I2 are amplitude modulated at two frequencies, ω1 and ω2 , respectively. When the laser frequency is tuned to the center of the Doppler line proﬁle, both beams simultaneously interact with one and the same group of atoms whose velocity projection on the beam direction is close to zero. The intensity of the optogalvanic spectrum lines is Δi = Cn0 (( I1 + I2 ) − a( I1 + I2 )2 ),

(3.82)

where C is the proportionality constant, n0 is the population of the absorbing level, the coefﬁcient a characterizes saturation, ns = n0 (1 + a( I1 + I2 )),

(3.83)

and ns is the saturated population in the dip region. When registration is carried out at the sum frequency ω1 + ω2 , the linear background is cut off

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.27 Measurement scheme involving detection of the intra-Doppler optogalvanic saturation signal (IMOGS): 1 – discharge; 2 – laser; 3 – chopper with two systems of slits in the disk; 4 – phase-sensitive detector; 5 – resistor; 6 – voltage source.

and it is the saturation signal alone that is being registered. Figure 3.28a presents as an example an intra-Doppler optogalvanic spectrum of helium in a glow discharge. The signal to noise ratio turns out to be two orders of magnitude higher than in the case of direct optical registration. Similarly, the two-photon absorption optogalvanic spectroscopy (TOGS) using counter-running photons shows a higher sensitivity compared with the traditional ﬂuorescence detection technique. Speciﬁcally, this opens up practical possibilities for the spectroscopy of transitions whose lower levels are other than ground or metastable ones and are, therefore, only weakly populated. To illustrate, Figure 3.28b shows a typical TOGS signal of 20 Ne on the 3s1 [1/2], J = 1 → 5d1 [3/2], J = 1 transition whose lower level has a lifetime of 1.5 ns (use is made here of frequency modulation with which the synchronous detector picks up the derivative of the signal of Lorentzian shape). An alternative is narrow optogalvanic resonances in the case of intermodulation of two counter-running beams of polarized radiation. As in the IMOGS technique, use here is made of counter-propagating beams crossing in a limited region of plasma. However, subject to modulation in this case is not the amplitude, but polarization (circular or linear) of one or both beams, the so-called polarization intermodulation excitation (POLINEX). A typical experimental scheme is presented in Figure 3.29. The polarization modulator can use a Pockels cell or some other device with a birefringent medium. The optogalvanic signal is formed as a response to the difference between the two values the absorption coefﬁcient assumes when the counter-running beams have ﬁrst opposite and then like circu-

125

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3 Emission, Absorption and Scattering Techniques

Figure 3.28 Examples of intra-Doppler optogalvanic spectra records. A He-Ne discharge: (a) intermodulation (IMOGS) spectrum of helium. The asterisks indicate artefacts typical of saturated absorption spectra; (b) two-photon absorption (TOGS) spectrum of neon.

Figure 3.29 POLINEX scheme: 1 – laser; 2 – polarizer; 3 – quarter-wave plate; 4 – beam splitter; 5 – mirrors; 6 – polarization modulator; 7 – discharge; 8 – phase-sensitive detector; 9 – high-voltage ampliﬁer.

3.4 Indirect Methods for Measuring Absorption of Laser Light

Figure 3.30 Absorption of neon (1s5 -2p2 , 588.2 nm) in an electrodeless RF discharge: (a) – POLIMEX technique; (b) – IMOGS technique.

lar polarization parameters. In the POLINEX technique, the atoms that change, as a result of collision, their projections on the quantization axis (usually coincident with the beam direction) cease to participate in the signal formation process. Thus, they can no longer make an undesirable contribution to the ‘pedestal’ typical of the Lamb dip spectroscopy with amplitude modulation, no matter what the detection method used. This advantage is illustrated by Figure 3.30. Worthy of note among the other high-resolution laser plasma and gas spectroscopy techniques for which the optogalvanic detection, on the whole, offers practical advantages as regards the background noise and sensitivity (though this, of course, also depends on the plasma object in hand and the speciﬁcity of the problem being solved) are the levelcrossing spectroscopy, double resonance spectroscopy, and multiquantum and stepwise excitation spectroscopy [77]. Through the efforts of numerous researchers, we currently have an extensive pool of knowledge relating to various optogalvanic spectroscopy techniques. Let us present for reference purposes the list [77] of atoms (Table 3.6) and molecules (Table 3.7) whose optogalvanic spectra have been studied in various plasma objects.

127

128

3 Emission, Absorption and Scattering Techniques Table 3.6 Atoms studied by optogalvanic spectroscopy. H(1,5) Li(3,5) Na(3-6) K(3-5) Cu(3,5) Rb(4,5) Ag(5) Cs(1,2,4,5) Au(5)

Be(4) B(5) Mg(4,5) Al(5) Ca(3-5) Sc(5) Ga(3-5) Sr(1-3) Y(3,5) Cd(5) In(3) Ba(1,3-5) La(3) Hg(1) Tl(5)

Sm(3)

Eu(3,4)

Zr(3) Sn(5)

He(1,3) Ne(1-3) Ar(1-3) V(5) Cr(5) Mn(3-5) Fe(5) Co(3,5) Kr(1,2) Mo(3) Xe(2)

Pb(5)

Bi(5)

O(5)

Ti(5)

Ni(5)

U(3) Tu(5)

Yb(3,4) Lu(5)

Note. The numerals in parentheses indicate the object in which the optogalvanic effect was studied: 1 – DC discharge tube, 2 – RF discharge, 3 – hollow cathode, 4 – thermionic diode, 5 – ﬂame, 6 – rareﬁed gas

Table 3.7 Molecules studied by optogalvanic spectroscopy.

Molecules

Spectral region, nm

Notes

Cs2

620–650

In2 , Yb2

390–660

Diode. First observations of molecular optogalvanic spectra Diode. Electronic transitions in excimers and exciplexes. Hybrid resonances of excitation of atoms via intermediate dissociative states of molecules

360–630

Flame. Identiﬁcation of new transitions

563–615

DC discharge. 1+ Letbetter system and bands (c11 Π → a11 Σ) RF discharge. Doppler-free resolution RF discharge. Identiﬁcation of new bands Hollow cathode. Double optical resonance. Identiﬁcation of new bands DC discharge. Identiﬁcation of new transitions between the 2p3 Π states and states near the dissociation limit Low-current DC discharge. B→X bands, Dopplerlimited resolution Moderate-resolution spectra Diode. Determination of the electron afﬁnity energy D3 Π →A3 Π bands Vibrational–rotational transitions in the active medium of the CO laser DC discharge. First observations of free radicals

CsKr CsAr LaO YO ScO N2 N2 N2 N2

598 585–605 595–615

H2

573–610

I2

520–630

I2

575–610

I2 CO

532 640–660

CO

550

3.5 Multiphoton Processes. Raman Scattering Table 3.7 (continued).

Molecules

Spectral region, nm

Notes

NH2 NH2

570–615 596–605

NH2 NH2

580–610 570–600

NO2 HCO

6200 580–620

DC discharge. Doppler-limited resolution RF discharge DC discharge. RF discharge Diode IR laser. DC discharge RF discharge in CH3 CO. Studies into predissociation rates A→X bands DC discharge, hollow cathode, RF discharge. N2 O laser. Doppler-free resolution. Optico-microwave resonance.

CN NH3 NH3 NH3 NH3 He2

643–682 9500

CO2

585–588

D2 O H2 CO SO2 H2 S H2 O SF6

9300–10800 9700–9100

9500 580–630 11250–9200

Diode IR laser. DC discharge RF discharge RF discharge. Discretely tuned CO2 laser. Rydberg transitions Vibrational–rotational transitions in the active medium of the CO2 laser RF discharge. Discretely tuned CO2 laser. Identiﬁcation of transitions, search of transitions for optically pumped far-IR lasers

10600

(0,0) bands of the x2 Σ →B2 Σ transition. First observations of optogalvanic spectra of molecular ions

N2 O N2+

390

CO+ NO PO

490 270–317 302–334

(0,0) band of the x2 Σ →B2 Σ transition Flame. Multiphoton ionization Flame. Multiphoton ionization

3.5 Multiphoton Processes. Raman Scattering

The quantum state of a particle can change on its interaction with several photons. If the change proceeds by way of successive single-quantum processes via intermediate levels, this then corresponds to the stepwise absorption or cascade emission of radiation. Multiphoton interaction is such an interaction, in which several photons simultaneously participate in a single elementary act. Figure 3.31 illustrates some examples of (a) stepwise and (b), (c) multiphoton processes [1, 2].

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3 Emission, Absorption and Scattering Techniques

Figure 3.31 Processes of interaction between a particle and several photons: (a) stepwise ﬂuorescence excitation; (b) twophoton absorption accompanied by ﬂuorescence excitation; (c) spontaneous Raman scattering in the Stokes (s) and antiStokes (as) components. Levels: solid – real; dashed – virtual; l – lower; u – upper; f – ﬁnal; k – a real level near the virtual level; v and i – intermediate.

3.5.1 Two-Photon Absorption

This process corresponds to Figure 3.31b. The particle is excited to rise from the state l to the state u by the photons h¯ ω1 and h¯ ω2 of light waves with the wave vectors k1 and k2 , unit electric ﬁeld vectors e1 and e2 , and intensities I1 and I2 . The particle velocity vector is v. The level v is an artiﬁcially introduced level (the so-called virtual level) and the level k is one of the real levels. The presence of these levels makes the probability of the particle being in the virtual state v other than zero. If Rlk and Rku are the dipole moment matrix elements for the l → k and k → u transitions, respectively, the excitation probability B˜ lu is then given by [22, 79] B˜ lu = !

γlu

" [ωlu − ω1 − ω2 − v(k1 + k2 )]2 + (γlu /2)2 * + , *2 * (Rlk e1 )(Rku e2 ) (Rlk e2 )(Rku e1 ) ** * × *∑ + * I I * k (ωkl − ω1 − k1 v) (ωkl − ω2 − k2 v) * 1 2

(3.84)

3.5 Multiphoton Processes. Raman Scattering

The ﬁrst factor describes the spectral line shape of the two-photon transition. It coincides up to the factor γlu /2 (in [22], use is made of the areal normalization of the line proﬁle) with the shape of the line proﬁle (νlu = (ω − ωlu,0 )/2π) (2.14) of the one-photon transition with the central frequency ωlu,0 = ω1 + ω2 + v(k1 + k2 ) for the moving particle and with the homogeneous line width γlu . By integrating with respect to the particle velocity, one obtains (again accurate up to the normalization factor) the expression for the Voigt proﬁle (2.47), (2.48). The second factor characterizes the probability of the two-photon process. From expression (3.84) one can see the main distinctions of the two-photon transition in comparison with its one-photon counterpart are: • The line width depends on the direction of the incident light waves. If both waves propagate in one and the same direction, the Doppler component proportional to |k1 + k2 | is at its maximum. When the waves propagate counter to each other, k1 = −k2 , the Doppler component vanishes altogether, and it is precisely this fact that forms the basis for the Doppler-free spectroscopy (see Section 3.4.2.2). • Summation in the second factor is extended over all the real levels k, but it can be seen from the structure of the denominators that the main contribution comes from the levels in the vicinity of the virtual one. This also suggests a method, optimal from the standpoint of the magnitude of absorption, for selecting the frequencies ω1 and ω2 . Of course, the fact that the quantity h¯ ω1 in Figure 3.31b is indicated by the down arrow and the quantity h¯ ω2 , by the up arrow is absolutely arbitrary. Their interchanging has no effect on the result of the multiquantum process occurring in a single stage. This is evident from expression (3.84); namely, the probability of the process is the square of the sum of both probability amplitudes. • All two-quantum processes are subject to the same selection rules: B˜ lu = 0 if Rlk = 0 and Rku = 0, that is, each of the states k i must be bound to the levels l and u by allowed one-photon transitions, and it is required that the pairs l, k and u, k should be of unlike parities and the pair l, u, of the same parity. • The products (Rlk e1 ) and (Rku e2 ) depend on the polarization of the laser beams. By appropriately selecting the polarization of the beams, one can excite various u states. For example, two

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3 Emission, Absorption and Scattering Techniques

co-propagating light waves of right-hand polarization cause twophoton atomic transitions with ΔL = 2 (s-d,. . . ), whereas counterpropagating beams of the opposite circular polarization results in transitions with ΔL = 0. To calculate the probability of simultaneous absorption of two photons in accordance with expression (3.84) requires knowledge of the matrix elements R. And it is this which presents the main problem. For this reason, to ﬁnd this quantity in practice, use is made of semiempirical formulas in addition to the necessary computations. By analogy with the classical one-photon absorption (Section 2.3), wide use is made of the notion of the two-photon absorption cross section as the absorption coefﬁcient in the Bouguer–Lambert–Beer law (2.51) related to the density of absorbing particles, (2.60). The analogy is naturally not complete. Insofar as the radiation being absorbed comes simultaneously from two sources with frequencies ω1 and ω2 , to preserve the analogy one should take as one’s choice of the source the absorption whose radiation is described in the form of expression (2.51), though, of course, the result is independent of this choice. Assume that this is the radiation of frequency ω2 . In that case, the attenuation of the radiation in the course of propagation along the z-axis can be described both in terms of absorption coefﬁcient (cross section) (2.51), (2.60) and in terms of the probability B˜ lu (3.84): dI2 (ω2 ) = h¯ ω2 B˜ lu ( Nu − Nl ) = I2 σ(ω2 )( Nu − Nl ), dz

(3.85)

where (Nu − Nl ) is the difference in population between the upper and the lower state. Although the absorption cross section σ (ω2 ) = B˜ lu h¯ ω2 /I2

(3.86)

deﬁned in this way has the dimension of a ‘classical’ cross section (2.60), [cm2 ], it now depends on intensity. In the majority of experiments on the two-photon excitation, use is normally made of a single laser source, and so both photons have the same frequencies, ω2 = ω1 = ω, and the same intensities, I1 = I2 = I. If no account is taken of the recoil energy (kv) in the denominator of the second part of expression (3.84), one then can write the following expression for cross section (3.86): σ=

(2π)3 h¯ ωg(ω ) | Mlu |2 I, (h¯ c)2

Mlu = 2 ∑ k

|μkl | |μku | , h¯ (ωkl − ω )

(3.87)

where g(ω) is the line form factor (the ﬁrst factor in expression (3.84)) and μij = (Rij e). One can see from the structure of the second factor

3.5 Multiphoton Processes. Raman Scattering

in (3.84) and expression (3.87) that the greatest contribution to the sum comes from the level k nearest to the virtual level v, and therefore use is frequently made of a single term instead of the sum. Calculations and measurements performed for a number of atoms and simple molecules have yielded the characteristic values of σ to be ca. 10−35 I, [cm2 ]. If the laser radiation intensity (measured in [Wcm2 ]) is I ≈ 109 Wcm2 , the quantity σ ≈ 10−26 cm2 (one-photon absorption cross section ∼ 10−16 cm2 ). Since in expression (3.84) B˜ lu ∼ I 2 , the two-photon absorption cross section is frequently deﬁned as the quantity σ (which is also referred to as the two-photon absorption coefﬁcient or rate), σ = σ/I , [cm4 W−1 ],

(3.88)

if I is measured in [Wcm−2 ], or σ = h¯ ωσ , [cm4 · s],

(3.89)

if by intensity the number of photons passing in 1 s through an area of 1 cm2 is meant. The characteristic values of σ are − 50 − 46 4 ca. (10 − 10 ) cm · s. The quantity σ is sometimes measured in the non-SI units called goepert-mayers (GM) in honor of the author of the ﬁrst theoretical consideration of the two-photon excitation process (1 GM=10−50 cm4 · s). It should ﬁnally be noted that at present the two-photon absorption cross section in the literature is most frequently taken to be the quantity σ expressed in [cm4 ]. In accordance with expression (3.87), it is independent of the line form factor g(ω) and is related to σ by the relation

σ = σ g(ω ) G (2) , [cm4 ],

(3.90)

where G (2) is the photon statistics factor: G (2) = 1 for coherent radiation and G (2) = 2 for noncoherent one. The characteristic values of the cross section σ deﬁned in this way are ca. 10−36 –10−30 cm4 . Data on the two-photon absorption cross sections of a number of atoms and molecules for various laser wavelengths are tabulated in Appendix C. 3.5.2 Spontaneous Raman Scattering

The spontaneous Raman scattering can be considered as the two-photon process of inelastic scattering of a photon, h¯ ω, by a molecule (atom), as

133

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3 Emission, Absorption and Scattering Techniques

a result of which both the photon frequency and the internal particle energy change simultaneously: M( El ) + h¯ ω → M( E f ) + h¯ ω .

(3.91)

When ω = ωS < ω, the process is called the Stokes Raman scattering and when ω = ωaS > ω, it is referred to as the anti-Stokes Raman scattering (Figure 3.31c). If the virtual state Ev = El + h¯ ω in the scattering cycle coincides with one of the eigenstates of the particle (the level k ≡ v), the process is then called the resonance Raman scattering The spontaneous Raman scattering intensity in a unit solid angle is dσ Iss = Nl I lss dΩ , (3.92) dΩ l f L dσ )l f is the differential spontaneous Raman scattering cross secwhere ( dΩ tion, lss is the length of the object of interest along the exciting beam, from which scattering is being registered, Nl is the population of the initial state, and IL is the incident laser radiation intensity. The Raman scattering cross section depends on the magnitude of the polarization tensor matrix element and is also proportional to ω 4 . Expressions for calculating cross sections can be found in [22, 79]. In accordance with above general rules for two-photon processes, the initial and the ﬁnal state should be of the same parity. Speciﬁcally, this condition is met by the vibrational levels (within the limits of a ﬁxed electronic state) of homonuclear diatomic molecules, between which transitions are allowed on scattering, but forbidden in the dipole approximation for absorption and emission (the so-called alternative selection rule). If the initial and the ﬁnal state are vibrational levels, then, for example, for the Stokes component, v → v + 1, we have

Iss ∼ Nv (v + 1)(ω − ( Ev+1 − Ev )/¯h)4 IL .

(3.93)

For the anti-Stokes component, the spontaneous Raman scattering intensity is proportional to the population of the excited state v + 1 and is usually much weaker than the Stokes intensity. The absolute values of the nonresonance spontaneous Raman scattering cross sections are small, σl f ≈ 10−30 cm2 , and so the spontaneous Raman scattering spectroscopy requires high-power sources. At present, these are, practically always, laser sources. For rareﬁed media such as gases and plasma, the use of high-power pulse-periodic lasers proves dσ optimal. Table 3.8 lists the values of the relative cross section ( dΩ )l f for a number of molecules, when the states l and f correspond to the ground and the ﬁrst excited vibrational level of the electronic ground

3.5 Multiphoton Processes. Raman Scattering

state and scattering takes place on excitation by light with a wavelength of 514 nm. Conversion to the absolute values can be made using the results of measurements taken in [80] for the Q-branch of N2 with the band center at 2335 cm−1 on excitation by light differing in frequency ν0 (wavelength λ0 ). In the excitation wavelength range 330–700 nm, the following relation holds true: dσ = (5.05 ± 0.1) × 10−48 (ν0 − 2331 cm−1 )4 cm6 · sr−1 . (3.94) dΩ Q,N2 For one of the strong lines of the Ar+ laser, λ0 = 514.8 nm, the quantity dσ ( dΩ )Q,N2 = 4.3 × 10−31 cm2 · sr−1 . It is precisely this value that the data listed in Table 3.8 are normalized to. Table 3.8 Relative values for eraser sections [80]. Molecule

N2

Band

2331 1555 4156 2886 2143 1877 1388 3652

Cross section 1

O2

1

H2

3.4

HCl

3.2

CO

0.8

NO

0.4

CO2 H2 O NH3 Cl2 CCl SF6

1.1

3.4

3334

554 459

775

6.4

2.3

3.9

6.2

More detailed cross section data, as regards the lists of both molecules and excitation wavelengths, can be found in [80]. 3.5.3 Stimulated Raman Scattering

In the text above, it has been assumed that the intensity of the scattered light is low and has but an insigniﬁcant effect on the scattering substance. As the intensity IL of the incident (pumping) laser radiation is increased, the situation changes. The intensity of the scattered radiation, principally the Stokes component, also grows higher, and one should take into consideration the interaction of the particles with both the incident laser radiation of frequency ωL and the Stokes radiation of frequency ωS . The particles interact with each light wave, which leads to parametric interaction involving energy exchange between the incident and the scattered wave. This can give rise to an intense directional radiation at the Stokes frequency. This process is called the stimulated Raman scattering. It has been described in detail in a number of monographs, including [22, 79]. The origination of the anti-Stokes stimulated Raman scattering and the direction of its propagation are governed by the so-called synchronism (momentum conservation) condition: 2kL = kaS + kS ,

(3.95)

135

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3 Emission, Absorption and Scattering Techniques

Figure 3.32 (a) Vector and (b) energy level diagrams for the antiStokes stimulated Raman scattering process.

where kL , kaS , and kS are the wave vectors of the pumping, anti-Stokes, and Stokes radiation, respectively. The vector and energy level diagrams for the anti-Stokes stimulated Raman scattering are presented in Figure 3.32. Here k1 = kL , k2 = kS , and k3 = kaS . From the microscopic standpoint, this is a four-photon process wherein the particle absorbs two photons and emits two photons in a single act, its state remaining unchanged. As distinct from the spontaneous Raman scattering, the antiStokes signal in the stimulated Raman scattering can be sufﬁciently intense, because there is no need to populate the excited state on account of internal processes (thermal etc.). Moreover, optimal conditions exist when as many particles as possible reside precisely at the lower level in the stimulated Raman scattering process. The stimulated Raman scattering process is of threshold character in IL , because for the scattered light wave to propagate through the medium, its ampliﬁcation should exceed attenuation. In this case, the Stokes scattering of photons forms a wave at the frequency ωS with the wave vector kS . Obviously the intensity of this wave and of the stimulated Raman scattering process as a whole will depend on the conditions prevailing in the scattering medium. This greatly limits the applicability of the stimulated Raman scattering scheme in spectroscopy. The presence of the threshold and nonlinearities in the pumping intensity and the population density of the initial level mean that the dynamic measurement range turns out to be very narrow, and only the most populated states are active in scattering. To determine their unperturbed population from their intensities in the stimulated Raman scattering spectra presents in this situation a certain problem. Inasmuch as we consider in this book only those methods which bear on practical plasma spec-

3.5 Multiphoton Processes. Raman Scattering

Figure 3.33 CARS process: (a) block diagram of the spectrometer: 1 and 2) – lasers of frequencies ω1 and ω2 ; M and DM – plain and dichroic mirror, respectively; L – lens; P – object; S – spectral instrument; D – detector; R – registration system; (b) collinear vector diagram; (c) energy level diagram.

troscopy, the only reason for our brief description of the stimulated Raman scattering is the possibility of implementing its spectroscopic version by way of some modiﬁcation of its scheme. We will dwell on this point in the next section. 3.5.4 Coherent Anti-Stokes Scattering

A logical development of the stimulated Raman scattering scheme, which makes the Raman scattering technique convenient for spectroscopic applications, is the coherent anti-Stokes Raman scattering (CARS) method (Figure 3.33). In the CARS method, use is made, in addition to the pumping wave of frequency ωL = ω1 , of a second, external pumping wave of frequency ω2 = ωS , that is, the functions of the Stokes radiation of the stimulated Raman scattering are now performed by the additional laser radiation ω2 whose intensity is controlled independently. What is important is that with this organization the anti-Stokes signal ωaS = ω3 can be made strong enough without reaching the stimulated Raman scattering threshold. The anti-Stokes radiation is coherent and satisﬁes condition (3.95) with kL = k1 , kS = k2 , and kaS = k3 . The vector diagram shown in Figure 3.33 corresponds to collinear propagation of the pumping waves ω1 and ω2 . Responsible for the CARS process is the term χ(3) of the expansion of the light-induced polarization in powers of the light wave electric ﬁeld intensity. The quantity χ(3) is called the third-order susceptibility and is a fourth-rank tensor with 81 components [79]. For locally isotropic media, with which gases and plasma can, as a rule, be classed, only three

137

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3 Emission, Absorption and Scattering Techniques

Figure 3.34 CARS signal intensity as a function of the light-medium interaction length: 1 – phase mismatch Δk = 0; 2 – Δk=0.

of them will be independent. The CARS signal intensity in the plane pumping waves approximation is I3 =

256π4 ω32 4 2 c n ( ω1 ) n ( ω2 ) n ( ω3 )

,2 + * * * (3) *2 2 2 sin(Δklint /2) χ I I l . * * 1 2 int Δklint /2

(3.96)

Here I1 and I2 are the intensities of the pumping waves ω1 and ω2 , respectively, n is the refractive index with due regard for dispersion, Δk is the phase mismatch, Δk = k3 − 2k1 + k2 , and lint is the light-medium in2 , teraction length. In the case of exact phase-matching (Δk = 0), I3 ∼ lint when Δk = 0, I3 is a periodic function of lint reaching its ﬁrst maximum in a length of lc = π/Δk, called the coherence length (Figure 3.34). With such an interaction length, the scattering intensity is a factor of (2/π)2 less than in the case of exact phase-matching. Satisfying the condition ω3 = 2ω1 − ω2 still cannot ensure exact phase-matching because of the dispersion (nω) of the medium [81], lc =

πc ( ω1 − ω2 ) 2

2

∂n ∂2 n + ω12 2 ∂ω ∂ω

−1 ,

(3.97)

though in gases and plasma, phase-matching holds over a greater length, on account of the lower dispersion. The values of lc for some gases under normal conditions are listed in Table 3.9 [81], provided that the pumping frequency detuning (1/2π)(ω1 − ω2 ), cm−1 coincides with the frequency of the Raman-active vibrational band of the molecule.

3.5 Multiphoton Processes. Raman Scattering Table 3.9 Typical values for the coherence lengths. Molecule

CO2

CO2

O2

NO

CO

N2

H2

Band, cm−1

1288

1388

1556

1877

2143

2331

4155

210

180

140

98

75

63

20

lc , cm

As the pressure is reduced, the length lc increases because of the reduction of the refractive index. As can be seen from the above table, even under normal conditions the length lc ≈(10–102 ) cm, frequently allows one to use in practice the following relation for complete matching: 2 . This often proves all the more justiﬁed because, at I ∼ I 2 I , I3 ∼ lint 3 1 2 the pumping laser beams are focused in the object under study, and the interaction region is thus localized along the propagation direction of the (collinear) beams. To illustrate, the numerical calculations made in [81] have demonstrated that for collinear Gaussian beams of diffraction divergence and the same diameter d in the plane of the focusing lens, three fourth of the CARS signal intensity is generated in a cylindrical volume with a diameter of w0 and a length of 6l f : w0 = 4λ f /πd;

l f = πw02 /2λ.

(3.98)

Here w0 is the beam diameter at the focus of a lens with a focal length of f . If, for example, the pumping wavelength λ is equal to 53 μm (second harmonic of the Nd−YAG laser) and the laser beam diameter is 1 cm, then, with f = 10 cm, w0 ≈ 6 × 10−4 cm and 6l f ≈ 7 × 10−2 cm, and when f = 50 cm, w0 ≈ 3 × 10−3 cm and 6l f ≈ 0.18 cm. Obviously the size of the CARS signal generation region in the direction normal to the laser beams is signiﬁcantly smaller than that along the beams. However, even in this case the typical values of the ‘longitudinal’ size 6l f are small in comparison with those of the coherence length lc (Table 3.9). But if 6l f > lc , one should then take lc to serve as the interaction length lint . There are ways to more strictly limit the longitudinal size of the interaction region by using a noncollinear geometry with specially selected convergence angles of the pumping beams. The BOXCARS scheme [82] can serve as an example. It satisﬁes phase-matching condition (3.95) with noncollinear interaction. In this scheme, the pumping beam ω1 is split into two beams that are then forced to converge at an angle of 2α, while the beam ω2 is oriented at an angle of θ, so that the scattering direction is determined by the angle φ (Figure 3.35). Considering, for generality,

139

140

3 Emission, Absorption and Scattering Techniques

Figure 3.35 Vector diagram for the noncollinear BOXCARS scheme.

dispersion, these angles satisfy the conditions n2 ω2 sin θ = n3 ω3 sin φ, n2 ω2 cos θ + n3 ω3 cos φ = 2n1 ω1 cos α,

(3.99) (3.100)

all the beams not necessarily being arranged to lie in a single plane. Practice shows that when using lens optics with identical radii, the BOXCARS scheme (and its further modiﬁcations [83]) provides for the longitudinal localization of lint (even with small angles α ≈ 5◦ ) within a region several times smaller than 6l f in the collinear scheme [84]. Another practical convenience attained with the BOXCARS scheme is that the pumping and scattering signals are separated not only in the spectral instrument (as is the case with the collinear scheme), but also in space. The latter factor becomes important in studying spectra with small Raman shifts, such as purely rotational molecular spectra. The ‘price’ for these conveniences is the reduction (by a factor of 5–10) of the Raman radiation intensity. If one considers a single Raman-active transition l → f of frequency ωl f , disregarding the effect of the nonresonance factors [79], the quantity χ(3) in (3.96) can be expressed in terms of the stimulated Raman scattering cross section (3.86): ( Nl − N f ) dσ 1 4 χ (3) = n c , (3.101) 1 dΩ l f ωl f − ω1 + ω2 − iΓl f 2¯hn2 ω24 where Nl and N f are the populations of the states l and f and Γl f is the half-width at half-maximum of the spontaneous Raman scattering line. Let us compare the capabilities of the spontaneous Raman scattering (SRS) and coherent anti-Stokes Raman scattering (CARS) techniques. Following the authors of [85], in order to estimate the orders of magnitudes of the Raman signals, we put in expression (3.96) Δk = 0, that sin(Δkl /2) is, sin c2 ( Δkl2int ) = [ Δkl int/2 ]2 = 1, n1 = n2 = n3 = 1, and, in accorint dance with expression (3.98), assume that lint = l f . Since the geometrical factors for the propagation of the Raman signals differ substantially between the SRS and CARS experiments, to make our comparison conve-

3.5 Multiphoton Processes. Raman Scattering

nient, we will convert, as in Section 3.3.2, from the pumping and Raman radiation intensities I [Wcm2 ] to their powers P [W]. In the case of CARS, it follows from expression (3.98) that the Raman radiation power P3 = I3 π(w0 /2)2 , and, assuming that λ1 ≈ λ, we reduce expression (3.96) to the form P3 =

210 π4 ω32 ** (3) **2 2 *χ * P1 P2 . c4 λ2

(3.102)

In expression (3.101) for χ(3) , we take it that all the particles are concentrated at the level l, that is, N ≈ Nl N f , and so, if the resonance condition ωl f = ω1 − ω2 is satisﬁed, P3 ≈

8πcω1 ω3 h¯ ω24

4 (

N Γl f

dσ dΩ

)2 P12 P2 .

lf

(3.103)

To estimate the SRS signal power within the full solid angle Ω = 4π, we will assume that scattering is excited by the same laser of frequency ω1 and power P1 as in the case of CARS. Expression (3.103) will then deﬁne the spontaneous Raman scattering signal power from lSRS = 1 cm of the object under study in the direction along the laser beam, and the CARSto-SRS signal power ratio will be P3 ≈ PSRS

8πcω1 ω3 h¯ ω24

4

N 4πΓ2l f

∂σ ∂Ω

P1 P2 .

(3.104)

lf

The CARS process becomes progressively more efﬁcient than its SRS counterpart as the density of particles and the powers of the laser sources grow. In practice, the laser powers in the CARS experiments are limited ﬁrst of all (but not only) by saturation effects, when the four-photon process leads to a change in the population N f of the level f in comparison with Nl . No universal estimation is possible here, for its results depend, among other reasons, on the individual relaxation properties of the medium. If, for example, one considers the widespread investigations into molecular vibrational processes, lasers with P1 ≈ 100 mJ, P2 ≈ 5 mJ, a pulse duration of ca. 10−8 s, and a low pulse repetition frequency (ca. 10–100 Hz) cause no appreciable saturation at gas pressures of ca. 1.333 hPa (1 Torr)–13.332 hPa (10 Torr). With the typical focal spot diameter w0 ≈ 10−2 cm, the above laser powers correspond to the intensities I1 ca. 1011 Wcm2 and I2 ≈ 5 × 109 Wcm2 . At λ1 = 532 nm (second harmonic of the YAG laser), molecular vibration frequency (ω1 − ω2 )/2π = 2000 cm−1 , characteristic values Γl f ≈ 10−1 cm−1 and

141

142

3 Emission, Absorption and Scattering Techniques

dσ/dΩ ≈ 10−30 cm2 · sr−1 , and particle densities determined by the pressure p at room temperature, the CARS-to-SRS signal power ratio is P3 /PSRS ≈ 10−5 P1 [W ] P2 [W] P[Torr].

(3.105)

With the above laser powers P1 P2 ≈ 5 × 1013 W2 and the pressure P = 1.333 hPa(1 Torr), the CARS signal is more than 8 orders of magnitude more powerful than the SRS signal. This estimate for actual experiments should, however, be considered inﬂated, because formulas (3.96), (3.102),and (3.103) have been obtained in the plane wave approximation for radially homogeneous beams. Ratios (3.101) observed experimentally are 10–100 times smaller. Further, SRS measurements require no strict focusing, and so the laser power is to a much lesser extent limited by saturation. For reasons of availability, pulse-periodic signal storage conditions and optics resistance, the SRS experiments can use lasers with an energy of up to ca. 10 J at a pulse duration of ca. 10−8 s. However, even such a deﬂation of estimate (3.105) leaves the CARS measurements at a great advantage. Similar estimates made in [79] for the minimum detectable Raman signals, determined by the detector noise, have shown that in scattering, for example, at the vibrational transition frequency of H2 , the CARS technique offers advantages over its SRS counterpart even at pressures p >0.001 Pa (0.01 mTorr). To conclude the discussion of the capabilities of the CARS technique, as developed in the mid-1980s, note that its application requires a fairly detailed analysis in the interpretation of the spectra, including consideration of the character of line broadening and the presence of a nonresonance background when the particles of interest constitute only a small admixture to the buffer gas [79]. The CARS signal can rise sharply (by several orders of magnitude) as a result of accidental resonances, when the laser frequencies used happen to coincide with electronic molecular transition frequencies (the so-called resonance enhanced CARS (RECARS)). This circumstance is being successfully used in studying radicals in plasma at a particle density level of 1010 –1011 cm−3 [86].

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3 Emission, Absorption and Scattering Techniques 27 A.V. Dem’yanenko, I.I. Zasavitsky, V.N. Ochkin et al. Investigation into the Distribution of the CO2 Molecules among Vibrational–Rotational Levels in a Glow Discharge by the Pulsed Diode Laser Spectroscopy Technique (in Russian). Kvantovaya Elektronika, 14, No. 4, pp. 851–859 (1987). 28 E. Hinkley, Ed. Laser Monitoring of the Atmosphere. Heidelberg: Springer (1976). 29 W.H. Weber, D.H. Leslie, and C.W. Peters. J. Molec. Spectr., 89, No. 1, pp. 214–222 (1981). 30 V.M. Krivtsun, Yu. A. Kuritsyn, E.P. Snegirev et al. ZhPS, 43, No. 4, pp. 571–576 (1985). 31 C.S. Gudeman, M.H. Begemann, J. Pfaff et al. Phys. Rev. Letts., 50, No. 10, pp. 727–731 (1983). 32 N.N. Haese, F.S. Pan, and T. Oka. Phys. Rev. Letts., 50, No. 20, pp. 1575– 1578 (?). 33 F. Pan and T. Oka. Radial Distribution of Molecular Ions in the Positive Column of DC Glow Discharge Using Infrared Diode-Laser Spectroscopy. Phys. Rev., A36, No. 5, pp. 2297–2310 (1987). 34 J. Steinfeld, Ed. Laser and Coherent Spectroscopy. New York: Plenum Press (1978). 35 N.V. Karlov. Lectures on Quantum Electronics (in Russian). Moscow: Nauka (1983). 36 J.W. Daily. Coherent Optical Transient Spectroscopy of Flames. Appl. Opt., 18, No. 3, pp. 360–367 (1979). 37 A.M. Meitland and M. Dann. An Introduction to Laser Physics (Russian translation). Moscow: Nauka (1978). 38 A. Yariv. Quantum Electronics. New York: Wiley (1975). 39 I.I. Zasavitsky, M.A. Kirimkulov, A.I. Nadezhdinski, V.N. Ochkin et al. Nonstationary Coherent Effects in the Rapid Recording of Absorption Spectra (in Russian). Optika i Spektroskopiya. 65, No. 6, pp. 1198–1202 (1988). 40 J.U. White. JOSA, 32, p. 285 (1942). 41 A.I. Nadezhdinski, A.G. Berezin, S.M. Chernin et al. Spectrochimica Acta, 55a, p. 2083 (1999).

42 S.M. Chernin. Development of Multipass Matrix System. Journ. of Modern Optics, 44, No. 4, pp. 619–632 (2001). 43 L.A. Pakhomycheva, E.A. Sviridenkov, A.F. Suchkov, L.V. Titova, and S.S. Churilov. Lasing Spectrum Structure of Lasers with Inhomogeneous Laser Line Broadening (in Russian). Pis’ma ZhETF, 12, p. 43 (1970). 44 E.A. Sviridenkov and S.N. Sinitsa, Ed. Intracavity Laser Spectroscopy (in Russian). Proc. of SPIE, 3342 (1998). 45 S.F. Luk’yanenko, M.M. Makogon, and S.N. Sinitsa. Intracavity Laser Spectroscopy (in Russian). Novosibirsk: Nauka (1995). 46 V.N. Ochkin and N.A. Raspopov. Intracavity Plasma Spectroscopy. In: V.E. Fortov, Ed. Encyclopedia of low-Temperature Plasma (in Russian), 2, pp. 583–586. Moscow: Nauka (2000). 47 V.M. Baev, T. Latz, and P.E. Toshek. Laser Intracavity Absorption Spectroscopy. Appl. Phys., B69, pp. 171–202 (1999). 48 V.S. Burakov and S.N. Raikov. Intracavity Laser Spectroscopy: Plasma Diagnostics and Spectral Analysis (in Russian). ZhPS, 69, No. 4, pp. 425–447 (2002). 49 V.S. Pazyuk, Yu. P. Podmarkov, N.A. Raspopov, and M.P. Frolov. Registration of the O2 (a1 Δg ) Singlet Oxygen by the Intracavity Laser Spectroscopy Technique from Absorption in the a1 Δg –b1 Σg+ Transition (in Russian). Kvantovaya Elektronika, 31, No. 4, pp. 363–366 (2001). 50 . M.P. Frolov and Yu. P. Podmarkov. Intracavity Laser Spectroscopy (in Russian). Moscow: MFTI (2003). 51 R. Engeln, R.J.Y. Letourneur, M.G.H. Boogaards et al. Detection of CH in an Expanding Argon/Acetylene Plasma Using Cavity Ring-down Absorption Spectroscopy. Chem. Phys. Letts., 310, pp. 405–410 (1999). 52 D. Romanini, A.A. Kachanov, and F. Stoeckel. Chem. Phys. Letts., 270, p. 538 (1997). 53 G. Berden, R. Peeters, and G. Meijer. Cavity Ring-down Spectroscopy: Experimental Schemes and Applications. Reviews in Physi-

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4

Intensities in Spectra and Plasma Energy Distribution in the Internal and Translational Degrees of Freedom of Atoms and Molecules

4.1 Doppler Broadening, Velocity Distribution of Particles, Neutral Gas Temperature

The temperature of neutral gas is one of the fundamental characteristics of weakly ionized plasma that directly or indirectly affects practically all the processes taking place therein. It determines the densities of particles and their geometrical proﬁles, the rates of physical and chemical processes, the directions and intensities of heat ﬂows, reduced electric ﬁeld strengths, mean electron energies, and so on. The translational degrees of freedom of neutral particles are, as a rule, one of the main reservoirs of the energy deposited in plasma. To illustrate, our analysis of numerous investigations shows that in gas-discharge plasma, the proportion of electric energy spent in heating neutral gas particles amounts to 0.2–0.6 for discharges in inert gases, 0.5–0.7 in N2 , 0.4–0.9 in CO, 0.8–0.9 in CO2 , H2 , and CCl4 , 0.7–08 in SF6 , and so on. In inhomogeneous plasma these proportions also depend on coordinates and result from many elementary processes that frequently prove very difﬁcult to consider theoretically. Therefore, gas temperature measurements constitute an important branch of plasma diagnostics. Among the spectral methods that are currently frequently used is the measurement of the shape and width of the Doppler-broadened spectral lines emitted by plasma. As is the case with many other methods, this process originated from investigations into equilibrium rareﬁed hot gases and plasma and was a later extension to the domain of nonequilibrium conditions. Such a translation was based on considerations of fast (on the order of the time between gas kinetic collisions) establishment of a Maxwellian velocity distribution of atoms and molecules. It should be noted, however, that such reasoning cannot be used in full measure, especially in emission spectroscopy, when the lifetime of the upper level of the optical transition in hand frequently matches the maxwellization

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time. And it is exactly this typical situation that receives primary consideration in this section. The shape of the Doppler-broadened line proﬁle and the velocity distribution function of particles are related by relations (2.30), (2.31). For an isotropic medium with a Maxwellian velocity distribution of particles, the line proﬁle and gas temperature are deﬁned by formulas (2.32) and (2.33), respectively. Should even a single premise out of those used in deriving the above relations (isotropy, Maxwellian velocity distribution, condition (2.44)) be invalid, we will then call the Doppler broadening abnormal, in accordance with the terminology of [1, 2]. It has been repeatedly observed under various nonequilibrium discharge conditions, in recombining laboratory plasma, ﬂames, ionosphere, and so on. In this section, we will give it primary attention and note the possibilities of diagnostics under such conditions. First, however, let us recall the general points for the analysis of experimentally registered Doppler line proﬁles. 4.1.1 Remarks on the Processing of Line Proﬁles 4.1.1.1 Registered and True Proﬁles

In line spectra of plasma at moderate densities and power inputs, the widths of line proﬁles are usually small, and to study them, use is made of high-spectral-resolution techniques. Taking account of the distortions introduced by the spectral instrument frequently proves to be important. To achieve this, the concept of the instrument (spread) function a(ν) is introduced, whose funtion is to register the monochromatic component of the spectrum as the proﬁle a(ν) [3]. The relationship between the true proﬁle ϕ(ν) and its experimentally observed counterpart ϕe (ν) is given by the so-called ‘convolution’ equation

∞

a(ν − ν ) ϕ(ν )dν = ϕe (ν) = ϕe0 (ν) + ε(ν).

(4.1)

−∞

Here account is taken of the fact that (i) the registered signal contains the noise component ε(ν) with zero mean and (ii) the instrument function remains unchanged within the limits of the line proﬁle and is normalized:

∞ −∞

a(ν) dν = 1.

(4.2)

4.1 Doppler Broadening, Velocity Distribution of Particles

If the width (at half-maximum) of the instrument function is small in comparison with that of the true proﬁle, so that a(ν − ν ) can be replaced by the delta function, then, according to (4.1), ϕe (ν) = ϕ(ν). This holds true, for example, when the line proﬁle is being investigated using narrow-band tunable lasers in the visible [4] or IR [5] regions of the spectrum. But in the general case, to ﬁnd the true line proﬁle requires solution of integral equation (4.1). This is an inverse, mathematically ill-posed (after Hadamard) problem. It can be solved exactly if the exact form of a(ν) is known and noise is absent. Otherwise the standard solutions may be highly in error or diverge altogether. Therefore, special error-minimization methods are being developed for such problems. Speciﬁcally worthy of note among them are the modiﬁed Fourier and iteration statistical regularization methods. These methods, as well as the methods of ﬁnding the form of the instrument function, are considered, as applied to the analysis of spectral line proﬁles, in [1]. 4.1.1.2 Predominantly Doppler Broadening Regions

When solving most practical plasma spectroscopy problems, one can neglect the natural broadening, because it is governed, except in a few special cases (autoionization [6], predissociation [7, 8]), by the radiative decay, and even for strong transitions with a level lifetime of ca. 10−8 s, its ˚ magnitude in the visible region of the spectrum amounts to ca. 10−4 A − 4 − 1 (4 × 10 cm ). If such a neglect is unjustiﬁed, the natural broadening can be taken into account by following the recommendations of Section 2.2.3. In the case of normal Doppler broadening, the isolation of the Lorentz component from a mixed proﬁle (2.47) can also be carried out by the methods indicated in Section 2.2.3. However, most simple estimates can help to isolate those regions of conditions, wherein the line proﬁle can be considered broadened predominantly because of the normal Doppler effect. If the collisional broadening cross-section σ (2.16) is measured in cm2 and the effective particle collision frequency is expressed in terms of the molar mass μ [g mole−1 ], pressure P [Torr], and temperature T [K], the Lorentz width ΔνL [Hz] will the be given by ΔνL = 1.99 × 1023 Pσ (μT )− /2 . 1

(4.3)

Generally speaking, the cross-sections σ depend on the particle species and the optical transition at hand. Information about them can be found, for example, in the review [9]. The corresponding magnitudes of the optical collision diameters of various atoms and molecules in various regions of the spectrum, from the violet to the microwave, can differ by several times. For estimation purposes, the authors of [6] recom-

149

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.1 Estimation of normal Doppler and Lorentz broadening conditions. Shaded is the frequency range of the visible region of the spectrum.

mend taking the optical collision diameter at ca. 5 × 10−8 cm (σ ca. 2 × 10−16 cm2 ). Using this value and formulas (4.3) and (2.34), one can write down the following condition for the equality of the Lorentz and Doppler widths: P ≈ 1.8 × 10−15 ν0 T.

(4.4)

Figure 4.1 shows lines corresponding to this condition. In the UV and visible regions of the spectrum, the Doppler broadening dominates even at fairly high pressures, whereas in the IR and microwave regions the role of the collisional broadening becomes substantial at pressures as low as a few torrs, and even a few fractions of a torr. 4.1.1.3 Recovery of the Form of the Velocity Distribution of Particles

If one neglects the natural and collisional broadening, or solely isolates the Doppler broadening, one then will be able to obtain from the shape of the line proﬁle the form of the radial velocity distribution function f (vz ) of the emitting or absorbing particles by formulas (2.29) and (2.30), as well as that of their absolute velocity distribution function f (v) by formula (2.31). The procedure of obtaining f (v) obviously presumes the differentiation of expression (2.31), which is one more stage of solution of the inverse mathematical problem, in addition to the separation of the instrumental distortions. Even if one takes it that the kernel K (v, ν) is

4.1 Doppler Broadening, Velocity Distribution of Particles

known exactly, the presence of actual noise and errors in measuring the proﬁle ϕe (ν) again requires application of the above-mentioned mathematical processing techniques. As applied to the spectroscopic problem, these techniques, including the methods using Fourier transforms, regularization methods, smoothing spline function methods, and model solution testing methods, are described in [1, 10]. 4.1.2 Examples of Abnormal Doppler Broadening and Nonequilibrium Velocity Distributions of Neutral Particles in Plasma

Let us present as an example the results of studies into the spectral line ˚ proﬁles of the (33 P0,1,2 –33 S1o ) transition in the oxygen atom at λ = 8446 A in a glow discharge in an O2 -Ar mixture [1]. The proﬁles of all the three lines coincided. Figure 4.2 presents the area-normalized proﬁles (curves 1 through 6) of the lines at various gas pressures. The proﬁles were recorded by means of a scanning Fabry–Perot cavity standard. The measurements are referred to the discharge axis, and the instrument distortions are eliminated. It can be seen that as the pressure is raised, the proﬁle narrows. Proﬁle 7 is calculated for the normal Doppler broadening in the case of Maxwellian velocity distribution of the emitters with a temperature of 330 K corresponding to the gas temperature at a pressure of 0.267 hPa (0.2 Torr). Figure 4.3 presents a family (curves 1 through 5) of the velocity distribution functions of the O(3 P0,1,2 ) atoms, obtained by processing the line proﬁles of Figure 4.2 under the assumption of isotropy with the kernel K (v, ν) = 1/v in expression (2.31). Also shown here is the theoretical curve 6 for the Maxwellian distribution at T = 330 K. All the curves are constructed in relative units and normalized to a single area. The experimental distributions differ, even qualitatively, from the Maxwellian distribution, and so the concept of the distribution temperature cannot be used. The presence of excited atoms moving with velocities much in excess of those for the bulk of the gas is associated by the authors of [1] with the exothermal character of the excitation reaction occurring on collisions of the oxygen atoms in the ground state O(23 P) with metastable argon atoms, O(23 P) + Ar(3p5 4s) → O(33 P) + Ar(1S0 ) + ΔE,

(4.5)

so that some of the energy ΔE is converted into the translational energy of O(33 P). Another example is presented by the line proﬁles in the rotational structure of the electronic–vibrational bands of the second positive sys-

151

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.2 Proﬁles of the 8446-A˚ spectral line of the oxygen atom. Discharge in an O2 −Ar (1:9) mixture. Discharge tube diameter 2 mm, tube wall temperature T = 300 K, discharge current 20 mA. Gas pressures: 1 – 0.267 hPa (0.2 Torr); 2 – 0.667 hPa (0.5 Torr); 3 – 1.333 hPa (1 Torr); 4 – 4.000 hPa (3 Torr); 5 – 13.332 hPa (10 Torr); 6 – 19.998 hPa (15 Torr). Proﬁle 7 is calculated for the normal Doppler broadening in the case of Maxwellian velocity distribution of the emitters with a temperature of 330 K.

tem of N2 (C3 Π−B3 Π). The experiments [1] were conducted with a discharge in an N2 -He mixture at a pressure of 0.667 hPa (0.5 Torr)–6.666 hPa (5 Torr), with the discharge tube being cooled with liquid nitrogen. Figure 4.4 shows the velocity distributions of the N2 (C3 Π) molecules in the vibrational state v = 0 residing on the rotational levels N = 3 and N = 26. The distributions differ. The molecules on the level N = 3 are distributed by a distribution close to Maxwellian at the gas temperature. But the molecules residing on the level N = 26 have velocity distributions with average energies decreasing with increasing pressure, as is the case with the above example of oxygen. Thus, there occurs a peculiar velocity sorting of the excited particles, depending on the energy of their bound state. In the given case, the molecule gains in velocity on the population of its N2 (C3 Π) state by way of the quenching of the more highly excited state N2 ( E) by the ground-state molecule N2 (X1 Σ): N2 ( E) + N2 (X1 Σ) → N2 (C3 Π, v , N ) + N2 (X1 Σ) + ΔE.

(4.6)

4.1 Doppler Broadening, Velocity Distribution of Particles

Figure 4.3 Velocity distributions of excited O(3 3 P0,1,2 ) atoms. For conditions, see Figure 4.2.

Figure 4.4 Velocity distributions of excited N2 (C3 Π, v = 0) molecules. Discharge in an N2 −He (1:10) mixture. Discharge tube diameter 20 mm, discharge current 20 mA, tube wall cooled to 77 K, temperature on the axis 150 K. Gas pressures: 1 and 6 – 0.667 hPa (0.5 Torr); 2 – 1.333 hPa (1 Torr); 3 – 2.666 hPa (2 Torr); 4 – 4.000 hPa (3 Torr); 5 – 6.666 hPa (5 Torr). Curves 1 through 5 for N = 26, curve 6 for N = 3, curve 7 – Maxwellian distribution at T = 150 K.

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4 Intensities in Spectra and Plasma Energy Distribution

In the experimental conditions of [1], the N2 (C3 Π) molecules of high N values are predominantly excited in reaction (4.2), whereas their counterparts of low N values are excited on collisions with electrons. It is exactly this fact that explains the difference between the velocity distributions of the molecules differing in N . Experiments with discharges in an N2 −Ar mixture, where surplus energy is liberated in the reaction N2 (X1 Σ) + Ar(3p5 4s) → N2 (C3 Π, v , N ) + N2 (X1 Σ) + ΔE,

(4.7)

yield similar results and lead to the same conclusions. 4.1.3 Excitation and Relaxation of Atoms and Molecules with Nonequilibrium Velocity in Interactions with Heavy Particles

It is obvious from what has been said above that the excitation of electronic states in atoms and molecules is accompanied by disturbances in their velocity distributions. The distributions are established under the effect of two factors. First, excitation gives rise to particles with the velocity distribution function f s (v) (source function). Secondly, owing to relaxation processes, the function f s S(v) is transformed in the course of the lifetime of the excited particles to the form f (v) that is determined in experiment. 4.1.3.1 Source Function

Based on the general kinetic equation, Wipple [11] derived equations for the velocity distribution functions of the products of a biomolecular reaction, with the initial velocity distributions of the parent particles and the energy defect of the reaction being speciﬁed. With a number of simplifying assumptions made as to the interaction potentials of the particles, he obtained analytical expressions for the distribution functions f s (v)S of the particles being produced. The result for the solid ball model was generalized by the authors of [1] to the case where also the internal degrees of freedom (vibrational and rotational) of the parent particles are subject to excitation, and calculated the source functions for reactions (4.6) and (4.7). Figure 4.5 presents as an example a theoretical absolute velocity distribution function f s (v) of the N2 (C3 Π, v = 0, N = 26) particles produced in reaction (4.7). It was assumed in this case that the N2 (X1 Σ) molecules resided in their vibrational ground state v = 0 and were distributed among their rotational levels by a Boltzmann distribution at a rotational temperature of T = 15 K equal to the gas temperature. It follows from the energy and momentum conservation laws that in the limit T → 0 there must form in the velocity space a delta distribution

4.1 Doppler Broadening, Velocity Distribution of Particles

Figure 4.5 Source function for reaction (4.5), v = 0, N = 26. Metastable Ar(3 P2 ) atom. 1 and 2 – most probable velocities at T = 150 K and T = 0 K.

with the most probable velocity v T →0 = 2ΔE/(mN2 (1 + mN2 /mAr )),

(4.8)

where mN2 and mAr are the masses of the particles and ΔE is the energy difference between the states Ar(3 P2 ) and N2 (C3 Π, v = 0, N = 26). Relaxation collisions with the colder, unexcited particles smooth out the distributions of the electronically excited particles, similar to that of Figure 4.5, and give rise to low-velocity particles. It is important to take into consideration the fact that the relaxation process can only be spectroscopically observed during the course of the lifetime of particles in a ﬁxed emitting state. Let us now consider this matter in more detail. 4.1.3.2 Relaxation of the Average Kinetic Energy of Particles with a Finite Lifetime

Let us make some simple assumptions: • The molecules of interest interact like solid balls. • Molecules with nonthermal velocity distributions constitute only a small admixture to the equilibrium buffer gas that does not get heated in the course of relaxation.

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4 Intensities in Spectra and Plasma Energy Distribution

• In all relaxation stages, the absolute velocity distributions f (v) are approximated by Maxwellian distributions with the same most probable velocities. Consider the time variation of the average kinetic energy E¯ N of the particles N in the thermostat of particles M with a density of n M : dE¯ N = −n M v˜ N,M σN,M Δ E¯ N dt,

(4.9)

where v˜ N,M is the mean relative velocity of the particles N and M, Δ E¯ N is the average energy lost by the particle N in a single collision, and σN,M is the cross section for the gas kinetic collisions between the particles N and M. The relation between v˜ N,M and the mean velocity v¯ N of the particles N, as well as their average energy loss, were established by Deschampes and Ricard [12] to be given by 1/2 m N TM v˜ N,M = v¯ N +1 , (4.10) m N TN Δ E¯ N = μ E¯ N (1 −

TM ), TN

μ=

8 mN mM . 3 ( m N + m M )2

(4.11)

Considering these relations, the relaxation equation can be re-written in the form 1/2 mN ¯ dE¯ 1 EN + E¯ M − N = 4n M σN,M (3πm N )− /2 μ ( EN − E M ). (4.12) dt mM Note that if deviations from equilibrium are small, E¯ N ∼ E¯ M , (4.10) becomes the well-known Landau–Teller relaxation equation with the constant translational–translational relaxation time τT−T (see, for example, [13]):

−

dE¯ N 1 = ( E¯ N − E¯ M ), dt τT−T τT−−1T

= 4n M σN,M ( E¯ M /3πm N )μ

mN +1 mM

1/2

(4.13) .

The solution of (4.12), subject to the initial condition E¯ N (t = 0) = E¯ 0N , has the form m m [c · exp {t/τT−T } + 1]2 E¯ N (t) = E¯ M 1 + N − N E¯ M , (4.14) m M [c · exp {t/τT−T } − 1]2 mM ( 1/2 1/2 ) m m N N E¯ M c= + E¯ M 1 + E¯ 0N + mM mM ( 1/2 1/2 )−1 m m N N E¯ M · E¯ 0N + − E¯ M 1 + . mM mM

4.1 Doppler Broadening, Velocity Distribution of Particles

Figure 4.6 Relaxation of the mean kinetic energy E¯ N . N = N2 (C3 Π, v = 0, N = 26), the thermostat buffer gas M = Ar at T = 150 K, E¯ N (t = 0) = 2250 K. 1 – variation of E¯ N ; 2 – exponential variation according to expression (4.11); 3 – thermostat temperature.

Figure 4.6 illustrates the time dependence E¯ N (t) for N = N2 , M = Ar, 0 ¯ EN2 = 2250 K (which corresponds to the translational energy of the nitrogen molecules in excitation by reaction (4.7)), and E¯ Ar = 150 K. One can see that relaxation here proceeds rapidly, and faster than that described by the exponential curve with the constant time τT−T characterizing the process in the case of small deviations from equilibrium, (4.13). The energy E¯ N2 decreases by a factor of e within t = 0.5τtt , and within t = 3τtt , E¯ N2 ≈ E¯ Ar . In the above experiments, emission line proﬁles provide information on the stationary velocity distributions of particles. In comparing theoretical and experimental results, consideration should be given to the fact that the emitting particles have formed, generally speaking, at different instants with respect to the observation (probing) moment, and so their residence times in the excited state are statistically dissimilar. Therefore, when using (4.14) to analyze experimental results, it is necessary to carry out averaging over the lifetime τr of the excited state. The probability that the particle will emit a photon within the period from t to t + dt is

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.7 Mean kinetic energy of the N2 (C3 Π, v = 0, N = 26) molecules as a function of the density of an N2 -Ar (1:9) gas mixture. Data points – experiment. Discharge tube diameter 20 mm, discharge current 20 mA, liquid nitrogen cooling. Curves – calculation by formula (4.12) with the cross section σN,M , cm2 : 1 – 4.1 × 10−15 ; 2 – 2 × 10−15 ; 3 – 8 × 10−15 . N = N2 , M = Ar.

given by the formula [14] dW (t) =

1 −t/τr e dt. τr

(4.15)

Averaging (4.14), according to (4.15), (4.12), we get 1 E¯ N (n M , σN,M ) = τr

∞ 0

t ¯ EN (n M , σN,M , t) exp − dt. τr

(4.16)

Comparing the results of calculations by (4.16) with experimental data on the Doppler broadening, one can determine the effective cross sections σN,M of the N–M collisions resulting in the relaxation of the translational energy of the emitting particles. Figure 4.7 shows an example of agreement between the results of measurements and calculations by formulas (4.14), (4.16) for reaction (4.7) with v = 0 and N = 26, presented in the form of the gas density dependence of the mean kinetic energy of the nitrogen molecules in an N2 −Ar (1:9) gas mixture. The best ﬁt is attained at σN,M = (4.1 ± 1.5) × 10−15 cm2 , N = N2 , M = Ar. The

4.1 Doppler Broadening, Velocity Distribution of Particles

calculation results are seen to be sensitive to the choice of the collision cross section. In the given example, the cross section is close to that for the gas kinetic collisions of the particles in the electronic ground states, 0 σN = 4.3 × 10−16 cm2 [15]. In principle, such an agreement might 2 ,Ar not exist, for the effective diameter of particles increases with the growth of their excitation. To illustrate, similar measurements taken in [1] for a discharge in an N2 –He mixture yielded σN,M = (5.8 ± 0.5) × 10−15 cm2 , 0 notwithstanding the fact that σN,M = (4.1 ± 1.5) × 10−15 cm2 , N = N2 , M = He [15]. Note that these values substantially exceed those recommended for use in estimating the Lorentz broadening for atoms (Section 4.1.1.3). 4.1.3.3 Relaxation of the Form of the Velocity Distributions of Particles in the Case of Large Deviations from Equilibrium and Finite Lifetime

The theoretical analysis of the alterations of the form of the velocity distributions f (v) of particles is more complex than the above-considered ¯ subject to the conproblem of the variations of the average energy E, dition that the form of the velocity distribution of particles remains unchanged (canonical approximation). To ﬁnd f (v), it is necessary to solve the Boltzmann kinetic equation. As applied to the spectroscopic problems under consideration, when the deviations from equilibrium are large and the masses of the relaxing particles are comparable with those of the particles of the thermostat, the use of numerical calculation schemes is a must. To interpret the abnormal broadening, the authors of [1] have considered two approaches. The ﬁrst involves solving a nonstationary kinetic equation and lifetime averaging of the states bound by the optical transition under study, via analogy with expression (4.16); however, with respect to the distribution function. The latter is physically justiﬁed if the lifetime is independent of velocity, as is the case, for example, with radiative decay. In the second approach, it is a stationary kinetic equation with excited particle sources that is solved. Both approaches yield results that agree well with both one another and with experimental results. Figure 4.8 shows the velocity distributions of the O(33 P) atoms excited by metastable argon atoms in process (4.5) for two gas pressure values. The experimental conditions correspond to those of Figure 4.2. The calculations are made in the solid ball model. If, summarizing what has been said in Section 4.1.3, we again turn to the practical question as to the determination of the gas temperature in nonequilibrium plasma from the Doppler widths, we then should note that

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Figure 4.8 Velocity distributions of oxygen atoms in a discharge in an O2 -Ar (1:9) gas mixture. For conditions, see Figure 4.2. Data points – measurements from the Doppler broadening, and curves – calculation at σN,M = 4.1 × 10−15 cm2 . N = O(3 3 P), M = Ar.

• the character of the translational motion of electronically excited particles does not always corresponds to that of the bulk of unexcited particles; • this dissimilarity, which manifests itself in the abnormal Doppler broadening, is governed by the exothermal excitation mechanisms of the emitting states and translational relaxation rates; • the greatest deviations from the normal Doppler proﬁle are possible for those lines excited in nonresonance interactions of heavy particles in plasma of lowered density, when the number of collisions with unexcited particles occurring during the lifetime of the emitting state is insufﬁcient for the completion of the translational relaxation; • if the excitation mechanism of the spectrum is unknown, it is necessary to take comparative temperature measurements from the lines of various electronic transitions and various particles. A summary of the experimental investigations into the abnormal Doppler broadening in heavy particle interactions, undertaken up to 1985, can

4.1 Doppler Broadening, Velocity Distribution of Particles

be found in [1]. It can be supplemented with the later references contained in [16, 17]. 4.1.4 On the Determination of the Gas Temperature from the Doppler Broadening of the Lines Emitted by Atoms and Molecules Excited by Electrons

Direct electron impact is an important, and often the main, excitation channel of electronic states in atoms and molecules. When an electron collides with an atom, their velocities change: v = v −

me ( v e − ve ) , M

(4.17)

where v and v are the velocities of the atom prior to and after the collision, respectively, ve and ve the same for the electron, and M and me are the masses of the atom and the electron, respectively. Of most interest are the cases where ve ve and ve ≈ ve , where v is the absolute velocity. The former case is realized in low-temperature plasma, when, as a rule, the average electron energy is lower than the excitation threshold of the electronic states and the main part in their excitation is played by the electrons near the threshold. The latter case is characteristic of excitation by a beam of fast electrons with velocities ve vae (vae is the velocity of the atomic electrons). In the former case, the vector ve can be neglected, and it then follows from expression (4.17) that the relative change of the absolute velocity of an atom on collision with an electron with an energy close to the excitation threshold value is Δv me ve . (4.18) = v M v √ If v = vp = 2kB T/M is the most probable thermal motion velocity and ve = vte is the electron velocity corresponding to the excitation threshold, then Δw =

Δv me vte = √ . p v 2kB TM

(4.19)

Table 4.1 lists the values of this quantity (in %) for two temperature values in the excitation of selected atomic and molecular states of importance in plasma diagnostics. One can see from this table that despite the mass difference, the velocity of an atom can appreciably change on collision with an electron.

161

162

4 Intensities in Spectra and Plasma Energy Distribution Table 4.1 The values Δw in (4.19). Atom, molecule, excited state

Excitation threshold, J (eV)

vet × 10−8 , cm · s−1

Δw, % at 100 K

Δw, % at 300 K

H(n−4) H2 (d3 Π) D2 (d3 Π) He(3 d1 D) N2 (C3 Π) CO(B1 Σ)

20.400 × 10−19 J (12.75 eV) 22.192 × 10−19 J (13.87 eV) 22.208 × 10−19 J (13.88 eV) 36.912 × 10−19 J (23.07 eV) 17.648 × 10−19 J (11.03 eV) 17.232 × 10−19 J (10.77 eV)

2.12 2.21 2.21 2.85 1.97 1.88

90 66 47 60 16 15

52 38 27 35 9 9

The quantity Δw = Δv/Δvp increases as the mass of the atom is decreased, the excitation threshold grows higher and the gas temperature is reduced. Excitation by fast electrons (ve ≈ ve ) has been well studied. The problem can be solved using the Born approximation, and its solution is described in [18]. The main role is played by collisions accompanied by the scattering of electrons through small angles, the maximum momentum transferred being pmax ≈ me vae . This result is physically quite obvious: the transfer of a momentum in excess of that of the atomic electrons is accompanied by ionization. For this reason, formula (4.19) remains valid for the evaluation of the relative change of the velocity of atoms excited by a beam of fast electrons. To describe the Doppler proﬁle of a line excited by electron impact, it is necessary to ﬁnd the distribution of the excited particles along the z-axis of observation. Let this distribution be Maxwellian prior to the interaction with electrons (see (1.4)): 1 vz 2 f M (vz ) = √ p exp − p . (4.20) v πv After collision with an electron the z-velocity of the atom is vz . Let the lifetime of the excited atoms be less than the mean free time and let the particles suffer no collisions. In that case, the distribution of the emitting atoms will have the form f (vz )

=

∞

f M (vz ) f exc (vz , vz ) dvz ,

(4.21)

−∞

where the function f exc (vz , v z ) describes the distribution of the excited atoms along the v z velocity component, provided that the velocity of atoms prior to collision with an electron was vz . With the electron velocity distribution being isotropic, the function f exc (vz , v z ) can be related to

4.1 Doppler Broadening, Velocity Distribution of Particles

the change Δv of the absolute velocity by means of formula (4.17): 1 , if |vz − vz | ≤ Δv , (4.22) f exc (vz , vz ) = 2Δv 0, if |vz − vz | > Δv . Substituting expressions (4.19) and (4.22) into (4.21), we get 1 f (vz ) = √ 2 πΔvvp

vz +Δv vz −Δv

vz 2 dvz . exp − p v

(4.23)

One can see from comparison between expressions (4.20) and (4.23) that the velocity distributions of the excited and unexcited atoms are dissimilar, distribution (4.23) being, generally speaking, not Maxwellian. Note that distribution (4.23) is obtained on the assumption that the electron velocity distribution is isotropic, that is, in conditions typical, in a local sense, of low-temperature plasma. −ν0 Let us now introduce the quantity ω˜ = νΔν e , where ν and ν0 are, as in D expressions (2.22)–(2.26), the current and the central frequency, respece is, for normalization convenience, the half-width of the tively, and ΔνD Doppler proﬁle at the 1/e level, in contrast to the more frequently used full width at half-maximum ΔνD in expression (2.27). In that case, it follows from expressions (4.19) and (4.23) that the spectral line proﬁle is described by the formula 1 ˜ erf(ω˜ + Δw) − erf(ω˜ − Δw) dω, ϕ(ω˜ )dω˜ = 4Δw

∞

ϕ(ω˜ ) dω˜ = 1.

−∞

(4.24) But, if the velocity projection on the observation axis remains unchanged during the course of excitation, the line proﬁle will then be the normal Doppler (Gaussian) proﬁle given by expression (2.36) with the quantity ω˜ substituted for u. This occurs in the above-mentioned example of excitation by a collimated electron beam, observed in a direction normal to the beam axis. Figure 4.9 shows line proﬁles normalized to unit area. Proﬁles 2 and 3 described by expression (4.24) are wider than Gaussian proﬁle 1. Their width increases with increasing Δw. For Δw = 0.3, the difference from the Gaussian width is around 3%, for Δw = 0.6, 13%. Figure 4.10 presents the dependence of the systematic error ΔT/T in determining temperature from the line widths of some atoms and molecules on the true gas temperature, with no account taken of this additional broadening factor.

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.9 Spectral line proﬁles: 1 – Gaussian proﬁle; 2 and 3 – proﬁles (4.24) at Δw = 0.3 and Δw = 0.6, respectively.

If use is made of the lines of relatively heavy particles (N2 CO), the error at the gas temperatures actually occurring in plasma conditions is small. For light atoms and molecules, the error increases. For example, it exceeds 10% in the case of hydrogen atom at T < 550 K, H2 at T < 300 K, He at T < 250 K and D2 at T < 150 K. The results of systematic measurements of the gas temperatures in cryogenically cooled discharges correspond to the calculation data presented in [1]. Based on the data in Figure 4.10, one can estimate approximately the absolute, temperatureindependent and easy to allow for systematic measurement error due to this effect: ΔT = 55 K for H and ΔT = 30 K for H2 . To summarize, we can say that the measurement of the Doppler widths of the electron-impact-excited lines, excepting those of the abovementioned light atoms and molecules at low temperatures, provides for the accuracy of determination of the neutral gas temperature of plasma acceptable for most practical problems. 4.1.5 Spectroscopic Manifestations of the Motion of Ions in Plasma

Electric ﬁelds in plasma cause a directed motion of charged particles therein, which gives rise to the Doppler shift of the lines emitted by the ions. Furthermore, owing to the phenomenon of the resonance charge exchange with the conservation of the velocities of the atoms and ions

4.1 Doppler Broadening, Velocity Distribution of Particles

Figure 4.10 Systematic error in determining neutral gas temperature from the line widths of (1) H, (2) H2 , (3) D2 , (4) N2 , and (5) CO as a function of temperature.

interacting in the own gas [19], there also occurs a directed motion of the neutrals, also accompanied by a line shift. These manifestations and their uses for plasma diagnostics purposes are described in the review [20]. Considered therein are cases where the ions are raised to the emitting state by direct electron impact from the ground state of the atoms. The velocity of the ion at the instant it is excited is assumed to coincide with the velocity of the atom. As can be seen from the preceding section, such an assumption can be considered justiﬁed in the case of comparatively heavy particles at temperatures above the room value. The higher excitation threshold of ions, compared to that of the bound states of neutrals, has no fundamental impact on the situation. The velocity of an excited ion changes under the effect of electric ﬁeld and manifests itself in the spectrum for the lifetime of the excited state. If the z-component of the ion velocity vz changes in a time of t in comparison with the z-component vz0 of the velocity of the atom being excited, vz = vz0 + (eE/M)t (the ﬁeld E is directed along the z-axis, M is the mass of the atom), the distribution of the emitting particles in vz (coincident with the intensity distribution in the Doppler proﬁle ((2.29), (2.30)) can be found by averaging over the lifetime τr , with due regard for expression (4.15). If the distribution of atoms in vz0 is Maxwellian, the following expressions are then valid for the average velocity projection v¯ z0

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4 Intensities in Spectra and Plasma Energy Distribution

and its mean square value [20]: v¯z = v¯ z0 + τr

eE , M

eE 2 . v¯2z = v¯2z0 + 2v¯ z0 + 2 τr M

(4.25) (4.26)

When the ion is excited from the ground state of the atom, v¯ z0 = 0 and, with the lifetime of the emitting state known, one can ﬁnd from the line shift the electric ﬁeld E by (4.25). If the excitation of the upper level of the optical transition of the ion under consideration takes place from its ground state, and its lifetime signiﬁcantly exceeds the de-excitation time, so that v¯ z0 τr eE M , measurements of the line shift and half-width in accordance with expressions (4.25) and (4.26) provide information about the average velocity and energy of the ions in the ground state. Assuming next that the velocity distribution of the unexcited ions is Maxwellian and can be characterized by the temperature Ti , one can relate this temperature to the temperature Tg of the neutral gas particles and the drift velocity vd of the ion A+ amidst the particles B of mass M by the approximate formula [21] Ti = Tg + ( MB v2d (A+ ))/3kB .

(4.27)

The simplicity of formula (4.24) relies on the knowledge of the drift velocity vd , and to ﬁnd it is a seperate problem requiring special measurements or solution of a kinetic equation. Another approach to the description of the average energy (temperature) of ions can be based on the average kinetic energy relaxation model described above in Section 4.1.3.2. In that case, τr in formulas (4.15) and (4.16) refers to the lifetime τi of the ion. This approach was used, in particular, when interpreting the results of measurements of the Doppler broadening and line shift of metal ions in the plasma of a vacuum arc [10]. Figure 4.11 presents the results from determination of the mean energy of the Cu+ ions in the N2 buffer gas from the line proﬁles of their transitions, with their kinetic energy deviating from the equilibrium value by more than 16.000 × 10−19 J (10 eV). The measurement results (averaged over six spectral lines) are well described by formula (4.12). The values of the collision cross section and the ion lifetime in the plasma are used as ﬁtting parameters and are 0 σN,M = (4.1 ± 1.5) × 10−15 cm2 (N = Cu+ , M = N2 ) and τi = 10−5 s (i = Cu+ ). Important in such methods of investigation into the motion of ions in the ground state is the assumption that within its de-excitation time the excited ion in the emitting state suffers no collisions entailing a change in its velocity. For this reason, these methods are mainly applicable to low-pressure plasmas.

4.2 Distribution of Molecules Among Rotational Levels

Figure 4.11 Mean kinetic energy of the Cu+ ion in a vacuum arc [10]. Buffer gas – nitrogen ( M = N2 ). Data points – measurements from abnormal Doppler broadening (averaged over 6 transitions). Curve – calculation by (4.12).

4.2 Distribution of Molecules Among Rotational Levels 4.2.1 On the Isolation of the Boltzmann Ensembles in the Bound State System of Particles

In classifying nonequilibrium plasma models in Section 1.3, we introduced the concept of partial temperatures occuring within the limits of individual groups of bound states of particles. This is a very helpful method that allows one, if permissible, to substantially reduce the number of parameters describing the state of plasma. In this case, Boltzmann, formula (1.5) holds true for a limited, but frequently large, number of levels. As a result, makes it possible to describe the energy stored in the isolated degrees of freedom of the internal motions of particles in terms of the ‘excitation temperatures’. Let us explain this thesis using as an example a nonisothermal mixture of particles of species A and B with densities N A , N B , masses m A and m B and Maxwellian velocity distributions at temperatures TA and TB , respectively. Let the particles A have n energy levels with populations Ni (1 ≤ i ≤ n). When particles collide, transitions i j with the rate

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4 Intensities in Spectra and Plasma Energy Distribution

constants k ijA,B take place between the levels of the particles A. The particle (A or B) participating in the collision with the given particle A, which causes the i↔j transitions therein, are considered as a structureless particles. Under stationary conditions, 0=

dNi = dt

∑

j =i

-

Nj ( N B k Bji + N A k jiA ) − Ni ( N B k ijB + N A k ijA )

.

− Ni Ai + Φi .

(4.28)

Here i, j = 1, 2 . . . n, Ai is the rate of transition of the particles from the level i to states beyond the n levels being considered, and Φi is the rate of population of the level i from the external source. We also assume that the rates of the collision-induced i↔ j transitions exceed the rates of exchange with the external systems: N A,B k ijA,B ,

N A,B k ji Ai , Φi /Ni .

(4.29)

In that case, a homogeneous system (4.28) has the following unique so/ / / / B B B A A A lution for Ni /Nj (det / N (k ji − k ij ) + N (k ji − k ij )/ = 0, which is easy to ascertain by adding together the rows): N B k ijB + N A k ijA Nj = B B . Ni N k ji + N A k jiA Using the detailed balancing principle, we get (see also [22, 23]) ! ΔE " ! ΔE " N B k Bji exp − k T ij + N A k ijA exp − k Tji Nj gi B AB B A = , Ni g j N B k Bji + N A k jiA

(4.30)

(4.31)

where g is the statistical weight, ΔEij is the energy difference between the levels i and j, kB is the Boltzmann constant and TAB = ( TA m B + TB m A )(m A + m B )−1 . Distribution (4.31) is, generally speaking, not a Boltzmann distribution, as deﬁned by formula (1.5), but becomes such with a temperature of TAB = T, at TA = TB = T. Let the particles A be atoms or molecules and B electrons. In that case, TAB ≈ TB . Typically in conditions of weakly ionized molecular plasma NA /NB ≈ 105 –109 , and, for estimation purposes, we put kijA,B ≈ v¯ A,B σ¯ A,B , v¯ B ≈ 103 v¯ A , σ¯ A ≈ σ¯ B ≈ 1016 cm2 . Also, if the energy difference between the levels under consideration is sufﬁciently small and exp −ΔEij /kB TB ≈ 1, then expression (4.31) turns into (1.5) with T = TA . This means, with the assumptions made and the typical parameters used, a Boltzmann distribution occurs in the system of levels at a

4.2 Distribution of Molecules Among Rotational Levels

temperature equal to the temperature of the translational degree of freedom of the heavy particles. Plasma thus corresponds to the LPTE model (Section 1.3) as regards the existence of partial temperatures – the two postulated translational temperatures, namely, the gas temperature TA and the electron temperature TB , and the level excitation temperature TA . The physical meaning of the results of this estimation is simple enough: collisions of the particles A with one another and with the electrons B cause an effective population redistribution in the system of levels. The fact that the temperature of the distribution formed coincides with the kinetic temperature of the heavy particles A is explained by their substantially greater density. The conditions adopted as regards the ‘compactness’ of the structure of levels and large, close to the gas kinetic, cross sections of the levelto-level transitions caused by collisions with heavy particles are met by the rotational energy levels of molecules in a ﬁxed electronic–vibrational state. It is an account of these considerations that is led to the spectroscopic methods of determining the gas temperature TA = T of a nonequilibrium plasma from the relative populations of the rotational levels of molecules. The excitation temperature is in this case referred to as the rotational temperature Tr = T. If the system of rotational levels (usually denoted by the quantum numbers J, N or K, see Appendix D) belongs to the upper state of the optical transition of interest, the intensity of the rotational spectrum component is F( J ) S J J , I J J = const ν4J J exp − (4.32) kB T where νJ J is the frequency of the J → J transition, F ( J ) is the rotational term, and S J J is the intensity factor in the rotational structure (Appendix E). By measuring the relative intensities of the rotational lines, one can determine from the slope angle α of the straight line ( ) const I J J ln = f ( F ( J )) (4.33) ν4J J S J J the gas temperature: kB T = cot α

(4.34)

At the same time, from expressions (4.29)–(4.31) one can see the limitations of such an approach. In particular, condition (4.29) is well satisﬁed for molecules in the electronic–vibrational ground states. It is evident from expression (4.28) that the populations of excited electronic states

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with a short radiative lifetime (Ai is reduced to the Einstein coefﬁcient) are governed by the excitation mechanisms, Ni = Φi /Ai

(4.35)

and the possibility of introducing the excitation temperature (rotational temperature TA = Tr ) depends on the speciﬁcity of Φi . Account should also be taken of the fact that for light molecules of large rotational constant the requirement that the energy level structure should be compact in comparison with kB T is extended over a limited number of rotational levels, which may invalidate the assumptions made as to the electronic ground states of the molecules. The above considerations and the qualitative character of the estimates indicate the need for more experimental and theoretical investigations of rotational distributions under nonequilibrium conditions. Along with the determination of the gas temperature of plasma from the Doppler line proﬁles, the identiﬁcation of the rotational temperature with the gas temperature is one of the main methods of plasma thermometry, and it is important for one to understand its capabilities and limitations. 4.2.2 Distributions of Molecules Among Rotational Levels in an Electronic State with a Long Lifetime

Condition (4.29) for the validity of the above simple classical estimates presupposes, in particular, that the lifetime τi = 1/Ai of the electronic– vibrational state to which the given rotational level ensemble belongs is long enough for the stationary rotational distribution to establish. Such are the ground and metastable states of molecules. The use of a single rotational distribution establishment time and the high frequency of the i ↔ j transitions in their turn presuppose that the separation of the adjacent levels i = N and j = N+1 is less than the translational energy, ΔEN,N +1 ≈ 2Bv N kB T (N is the serial number of the rotational level and Bv is the rotational constant). In that case, there forms a Boltzmann rotational distribution. For most molecules in the ground state this proves valid under a wide range of experimental conditions, up to large N, which can be judged from the spectrum line intensities observed. The typical rotational distribution establishment time is not very much longer than the time between gas kinetic collisions. For example, according to the data from various authors [24, 25], the number Z of collisions in the own gas required for the stationary rotational distribution to set in at temperatures close to the room value are 5–20 for N2 and O2 , 10–15 for CO2 and CH4 , and 5–8 for SF6 and CCl4 .

4.2 Distribution of Molecules Among Rotational Levels

Figure 4.12 Rotational structure of a CARS spectrum of the N2 (X1 Σ, v = 1 → v = 2) band. Nitrogen discharge in a 36 mm dia. tube, pressure 12.665 hPa (9.5 Torr), discharge current 50 mA: (a) – spectrum view; (b) – relative populations of the rotational levels as a function of N ( N + 1). Rotational temperature 600 K [26].

Figure 4.12 shows as an example a CARS spectrum of nitrogen in a glow discharge [26]. The rotational level populations NN determined from the line intensities by means of formulas (3.90)–(3.92) are presented in the same ﬁgure in the form of the relationship between ln( NN /g N ) and the quantity F ( N )/Bv = N ( N + 1) proportional to the rotational energy in the state N2 (X1 Σ, v = 1) (see Appendix D (D.2), Bv = 1.47 cm−1 is the rotational constant). The linearity of the relationship on these coordinates bespeaks a Boltzmann distribution with a temperature of Tr = 600 K in the given case. The authors of [26] identify this temperature with the gas temperature, Tr = T. Figure 4.13 presents similar rotational distributions in the ground state of the hydroxyl radical, OH(X2 Π, v = 0, Bv = 18.5 cm−1 ), obtained in [6] by the line absorption method (Section 3.2.2). The rotational temperatures coincide with the gas temperatures, which were checked by independent measurements. The distribution formation picture becomes more involved if the separation of nonequidistant rotational levels is equal to or greater than the average translational energy, ΔEN,N +1 ≥ kB T. The classical estimates

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.13 Rotational level populations of the OH(X2 Π, v = 0) molecules as a function of the energy of the rotational term. Measurements from the lines of the branches: squares – Q1 ; circles – R2 ; triangles – P1 . 1 and 2 – water vapor pressures 0.667 hPa (0.5 Torr) and 5.333 hPa (4 Torr), respectively. Discharge in a water-cooled tube 20 mm in diameter, discharge current 15 mA.

made above are invalid in this region. For molecules the type of N2 , CO and O2 , this is true of levels of small N at temperatures of 1–20 K, for example, in conditions of fast ﬂows of rareﬁed gases and plasma [28]. But for lighter molecules, such as H2 , D2 and HD, with a large rotational constant, such a situation can also occur in gas discharges. To illustrate, for H2 (X1 Σ) the quantity Bv ≈ 60 cm−1 , and additionally, the ortho- and para-modiﬁcations, which are difﬁcult to intermix by collisions, alternate with the changes of the serial number of the rotational level (Appendix A). In such a situation the stationary rotational distribution can differ from the Boltzmann distribution even at small N. The actual form of the distribution depends on the relation between the rates of level excitation by electrons and heavy particles, diffusion, and so on. Figure 4.14 shows the rotational distributions of the H2 (1 Σ, v = 0) molecules in the head of the stratum of a liquid-nitrogen-cooled discharge in hydrogen at various gas pressures. The measurements were taken by the CARS method [29]. Different pressures correspond to different gas temperatures, and the latter were measured with a thermocouple and determined from the Doppler broadening, the results being in good agreement. The slopes of curves 1 through 4 in the ﬁgure correspond

4.2 Distribution of Molecules Among Rotational Levels

Figure 4.14 Rotational level populations of the H2 (X1 Σ, v = 0) molecules as a function of N ( N + 1). Discharge in H2 , discharge tube diameter 20 mm, discharge current 40 mA, liquid nitrogen cooling. Pressures: 1 – 0.667 hPa (0.5 Torr); 2 – 1.333 hPa (1 Torr); 3 – 2.666 hPa (2 Torr); 4 – 5.333 hPa (4 Torr). Measurements taken in the head of the stratum. Straight lines correspond to Boltzmann distributions with temperatures T of (1) 145 K, (2) 180 K, (3) 230 K, and (4) 280 K.

to these temperatures. At temperatures of 180 K and 230 K the distributions are practically Boltzmann, with the same rotational temperatures. At T = 145 K a slight deviation is seen toward the overpopulation of the levels, which increases as the level serial number grows larger. More substantial deviations are observed in a cooled discharge in an H2 –He mixture (Figure 4.15). The difference in rotational distribution between the discharges in H2 and H2 –He at the same temperature was explained by the authors of [29, 30] as being due to the differences in the average energy and density of electrons. This conclusion was drawn on the basis of comparison between experimental measurements and the results of the theoretical analysis [31]. As a result, recurrence relations were suggested for the populations of the rotator levels subject to concurrent excitation by heavy particles and electrons, the parameters of the neutral and the electron component being speciﬁed phenomenologically. The current status of the theoretical investigations into the excitation of molecular rotations in the electronic ground state is described in the reviews [32, 33].

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.15 Rotational level populations of the H2 (X1 Σ, v = 0) molecules as a function of N ( N + 1). Discharge in an H2 −He (1:3) gas mixture, discharge tube diameter 20 mm, discharge current 40 mA, liquid nitrogen cooling. Pressure 0.667 hPa (0.5 Torr). Straight line – Boltzmann distribution at T = 145 K.

4.2.3 Electron Impact Excitation of the Electronic–Vibrational–Rotational (EVR) Levels of Molecules 4.2.3.1 Observations and General Considerations

A great number of works (see [33–35]) have been devoted to the studies of the processes of excitation of the electron shells of molecules by electrons. Most detailed information is obtained from experiments with electron beams propagating in a gas or with crossing electron-molecular beams. In such experiments, molecules predominantly reside in the electronic–vibrational ground state, and to determine the excitation cross sections of the individual vibrational levels of excited electronic states, it sufﬁces to make the energies of the beam electrons monochromatic up to ca. 0.160 × 10−19 J (0.1 eV)–0.480 × 10−19 J (0.3 eV) which ﬁts well with the capabilities of detection of the ensuing ﬂuorescence [36]. The characteristic scale of the rotational structure is ca. 10−3 eV, and populated in the ground state are a substantial number of rotational levels, which makes the cross sections of the individual rotational transitions difﬁcult to determine. Traditionally one restricts oneself to the reasoning that the light electron cannot impart any substantial momentum

4.2 Distribution of Molecules Among Rotational Levels

to the molecule. In terms of quantum numbers, this corresponds to the selection rule ΔN ≈ 0. Hence it follows that if the populations of the rotational levels in the ground state are distributed by a Boltzmann distribution, the following relation then holds true between the rotational temperatures TR and TR of the ground and the excited state: TR Bv = TR Bv

(4.36)

This standpoint was stated, in particular, in the well-known monograph [37]. However, research into the ﬁne structure intensities of the electronic spectra of homonuclear molecules excited by electrons suggests that this rule does not hold, at least not in the strict sense. For transitions from short-lived excited states, this would lead to the alternation of line intensities in the electron-impact-excited spectrum in accordance with the effect of the nuclear spin on the level statistics, which is actually not the case [38]. One should, therefore, apply some restrictions, other than the selection rule indicated above, on the magnitude of the moment of momentum transferred. Speciﬁcally, for the purpose of studying gas ﬂows with the aid of a probing electron beam, an analogy with optical probabilities is discussed in [39]. It corresponds to the Bethe–Born dipole approximation for the ‘generalized oscillator strength’ equal at high probe electron energies (ca. 1 keV) to the optical oscillator strength (for dipole-allowed transitions) [23]. The subsequent experimental investigations into the excitation of molecules by electrons have shown that this model also encounters difﬁculties. Spectroscopic measurements at low gas temperatures have revealed a substantial excess of the rotational temperatures over the gas temperatures and a tendency of the rotational distributions toward deviation from the Boltzmann distribution [40]. Similar tendencies, and at temperatures close to the room value, were also noted in [19, 20], the magnitude and character of the differences being dependent on the electron energy, even when greatly in excess of the excitation threshold. Thus, the general physical question as to the possible changes of the rotational state of a molecule whose electronic states are being excited by electrons is of practical importance in the diagnostics of plasma and gas ﬂows. While on the subject of the theoretical works, note that the problem of the excitation of individual rotational levels of a molecule following the change of its electronic structure as a result of interaction with an electron is exceptionally difﬁcult to solve consistently by the quantum mechanical methods. And to use model representations, including those described above, requires clear applicability criteria. The difference of opinions on this issue is due to this fact.

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4 Intensities in Spectra and Plasma Energy Distribution

One of the ﬁrst estimates of the possible changes of rotational states in the process under discussion was made by Sakharov [22] on the basis of the classical considerations with the impact parameter equal to the size of the molecule. Using the nitrogen molecule as an example, he indicated the change of the rotational quantum numbers could be as great as ΔN ≈ 5–7. However, its character being decidedly qualitative, this estimate failed to ﬁnd further uses. In connection with the experiments on the electron-beam excitation of the H2 molecules in a triode, the authors of [43] tried to theoretically analyze the process within the framework of the united atom model at electron energies in excess of the threshold value. It followed from their calculations that ΔN ≈ 0, which contradicted the observations of the radiation emitted on transitions from levels of large N. They suggested that either the model was too crude, or the population of the emitting state in H2 (d3 Π) also depended on secondary processes, the latter being preferred. Further, attention is drawn to the theoretical works [44–47] that use the adiabatic approximation. It is assumed that the nuclei of the molecule have not enough time to shift during the course of its collision with the partner, and the problem is broken down into two stages. First scattering on the molecule with ﬁxed nuclei and then their motion. Such an approach was suggested by Chase [48], when solving the problem on the excitation of nuclei is considered by nucleons, and was later on extended to collisions with electrons [49], the electronic state being considered ﬁxed. In [44], the latter limitation was overcome and the problem on the electron excitation of electronic–vibrational–rotational (EVR) levels was considered in general form. The small parameter used was the ratio between the collision time and the characteristic time of the internal motions of the molecule. However, attention was given in [47] to the fact that in the case of molecule-electron collision the presence of an extra small parameter, namely, the ratio between the masses of the electron and nuclei, although related to the ﬁrst, allows one to expect that the domain of applicability of the adiabatic approximation can be extended to the region of near-threshold electron energies, where the electron velocity is low and the collision time grows longer. Based on this approximation and the scattering T-matrix technique, analytical expressions were obtained in [44, 47] for the excitation cross sections of the EVR levels. In the above works, these formulas are presented for various combinations of coupling between the moments of the initial and the ﬁnal electronic state of the molecule. Let us consider the formula for the case where the spin-orbit interaction is weak, the multiplet structure is not resolved in the spectrum and it makes sense to talk only about the excitation cross

4.2 Distribution of Molecules Among Rotational Levels

section referred to the index r of multiplicity: σN N =

∑ σJ J = J

= (2N + 1) ∑ r

[ Qr + Qr∗ ]

N Λ

r Λ − Λ

N −Λ

2

(4.37) .

Written in the big parentheses here is the Wigner 3j-symbol, J is the quantum number of the total angular momentum, Λ is the quantum number of the projection of the electron orbital moment on the axis of the molecule, the doubly primed quantities relate to the ground state and the primed ones to the excited state and Q is the electronic matrix element independent of the rotational quantum numbers. The presence of the two sets Qr and Qr∗ reﬂects the contribution from two parts, one dependent on the spin of the electron being excited and the other independent of it. When excitation is accompanied by a change in the spin of the molecule, Qr = 0, and when no such change takes place, Qr∗ = 0. When Λ = Λ , expression (4.37) turns into the one obtained in [49] for the excitation of molecular rotations entailing no changes in the electronic state. 4.2.3.2 Experimental Determination of the Electron-Impact-Induced Changes in the Rotational States of Molecules in Plasma

The concept of measuring changes in the angular momentum of molecules in collisions with electrons can be explained by means of expression (4.37). This formula is easy to interpret when the initial state is a Σ one (Λ = 0). In that case, as one can easily ascertain considering the properties of the three-j numbers, for transitions from the lowermost rotational state with N = J = 0, the sum over r contains a single term, r = N , and σ0,N coincides with the corresponding partial cross sections. The interpretation in the case where Λ = 0 is similar. The cycle of works [47, 50, 51] describes investigations into gas discharge spectra as follows. Subject to study were the spectra of the hydrogen molecule of large rotational constant in discharge conditions for which information was available on the rotational distributions in the electronic ground state (Section 4.2.2). In liquid-nitrogen-cooled lowcurrent discharges, populated in ortho- H2 (X1 Σ, v = 0, N – odd) proved only the level N = 1. The rotational distributions in the excited electronic states H2 (d3 Π, I1 Π) were determined from the line intensities in the rotational structure of the emission spectrum of the EVR transitions d3 Π-a3 Σ, I1 Π–B1 Σ. Present in the excited Π states of both the ortho- and para-modiﬁcations are levels with all possible N = 1, 2, 3, . . .. At the same time, one can discriminate between the ortho- and para-

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Figure 4.16 Rotational excitation rate constants as a function of ΔN for (1) X1 Σ-d3 Π and (2) X1 Σ–I1 Π transitions in hydrogen.

molecules in the excited states with Λ = 0 in spectroscopic experiment, owing to the selection rules for optical transitions (Appendix D). The short radiative lifetime (ca. 10−8 s) of the excited states corresponds to condition (4.35), for the absence of redistribution among the levels of these states as a result of collisions with atoms and molecules at low (0.133 hPa (0.1 Torr)–13.332 hPa (10 Torr)) gas pressures. The electronic states were excited by direct electron impact, which was conﬁrmed by the current and pressure dependences of the line intensities in the spectra. Thus, experimentally determined in accordance with expression (4.35) were the population rate constants of the rotational states as result of averaging the excitation cross sections over the electron velocities, Ai = a1,1+ΔN = ve σN =1,N . The measured values of a1,1+ΔK are presented in Figure 4.16. For different excited electronic states, these constants also differ somewhat. On the whole, it is evident that most probable are transitions with the changes ΔN = 0 and 1, although, in contrast to the optical selection rules, electronic excitation can be accompanied by changes in the angular momentum with ΔN = 2–5, but with lowering probability. It is obvious from the staging of the experiments [47, 50, 51] that they can help determine only the vector of the values a1,N . To ﬁnd the complete matrix a N ,N , it is necessary to resort to additional means, including the use of expression (4.37). If we assume that the energy depen-

4.2 Distribution of Molecules Among Rotational Levels

dences of σN ,N with different N , N are close to one another or coincide, then a N ,N = ve σN ,N ≈ v¯e σ¯ N ,N , where v¯e is the mean electron velocity and σ¯ N ,N is the effective cross section of the process, which coincides up to a constant with the cross section in expression (4.37). In that case, a N ,N ∼ σN N and this expression can be applied to their relative values. For example, in the case of excitation of the H2 (d3 Π) state with a weak spin-orbit interaction, one should put Qr = 0 in expression (4.37), and the set Qr∗ can be found from comparison with the experimental data for a1,N . The calculations made in [47] for the excitation of this state have shown that the matrix a N ,N can be recovered to a good approximation under the assumption of the ‘symmetry’, a N ,N +ΔN = a N ,N −ΔN ,

(4.38)

and ‘similarity’, a N ,N +ΔN1 a1,1+ΔN1 = a1,1+ΔN2 a N ,N +ΔK2 ,

(4.39)

of its elements. The matrix obtained can help calculate the distributions among the EVR levels, provided that the rotational distributions in the electronic ground state are known. If the latter are Boltzmann distributions, the EVR distributions can be calculated for different temperatures and compared with the corresponding experimental data. Such comparisons are presented in Figure 4.17. In these examples, the distributions of the H2 (d3 Π, v = 0) state among EVR levels were determined from the spectrum of the Fulcher series (Appendix D). Used in the calculations were expressions (4.38) and (4.39) and curve 1 of Figure 4.16. The gas temperature was determined independently by the CARS technique and from the Doppler broadening of the lines of atomic and molecular hydrogen. The calculated and measured distributions are in good agreement. At a temperature close to the room value the EVR distributions are other than the Boltzmann distribution, in contrast to their counterparts in the H2 (X1 Σ) ground state (Figure 4.14). As the gas temperature T grows, the rotational distribution of the molecules in the H2 (d3 Π) state gradually becomes of Boltzmann character. At T = 800 K it is practically a Boltzmann distribution, and the rotational temperature Tr = 410 K is related to the gas temperature (equal to the rotational temperature of the ground state) by relation (4.36), that allows for the difference between the rotational constants (Appendix D). The ‘dubbing’ of the rotational distribution in the electron impact excitation of molecules is accurate to relation (4.36) only in the case where kB T Bv .

(4.40)

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.17 Rotational distributions of the H2 (d3 Π, v = 0) molecules. H2 discharge in a 20-mm-dia. tube, discharge current 30 mA, pressure 0.667 hPa (0.5 Torr). 1 and 2 – T = 300 K; 3 and 4 – T = 800 K. 1 and 3 – orthohydrogen; 2 and 4 – parahydrogen. Circles – measurements; crosses – calculation using a N N .

This should be taken into consideration when determining the gas temperature from the relative intensities of spectral lines. As for the stringency of the above inequality, one can note, for example, that if one draws straight line (4.33) by the least squares method through the data points describing the rotational level populations as a function of the energy of the rotational terms on semilogarithmic coordinates, the gas temperature determined, with due regard for expression (4.36), from the slope of this line by formula (4.34) will differ from its true value by less than 10% at kB T > 7Bv [47]. If inequality (4.40) holds true, and this is, as a rule, the case with relatively heavy molecules at not very low temperatures, the actual form of the matrix a N ,N is not very important and use can be made of the approximation N ≈ 0. This is conﬁrmed by all experience gained in determining gas temperatures from the relative intensities of lines in the rotational structure of the electronic emission spectra of molecules, but only subject to the condition that the spectra are excited by direct electron impact.

4.2 Distribution of Molecules Among Rotational Levels

4.2.4 Excitation of EVR Levels by Heavy Particles

Whereas the effects associated with the imposition of a moment of momentum by an electron on a molecule during the course of excitation of its electronic shell are manifest in a very limited region, the excitation of molecular rotations by heavy particles can be much more pronounced. Substantial deviations of the molecular rotation energies from the equilibrium values have been repreatedly noted in the works on the spectroscopy of nonequilibrium chemical reactors, laser media and gas ﬂows (see, for example, [37] on the spectra of C2 , CH and CuH, [52] on those of CN and NH, references cited in [38] and others). The situation is quite similar to that discussed in Section 4.1 on the nonthermal translational motion of atoms and molecules and results from the redistribution of the energy defect in nonresonance interactions. Let us consider it for some examples typical of plasma spectroscopy. 4.2.4.1 OH Radical. Violet Bands

Numerous works have been devoted to this object, including the early investigations [53, 55–57]. Figure 4.18 shows the behavior of the quantity Z=const · IN N /[ν4 (S N N + SN N )] describing, accurate up to a small correction for reabsorption in measurements on the OH(A2 Σ–X2 Π) transition, the distribution NN /g N of the OH(A2 Σ) molecules among rotational levels in a discharge in water vapor at a pressure of 0.667 hPa (0.5 Torr) [38]. The measurements were taken from the (0,0) and (1,1) vibrational bands. Curve 1-2-3 corresponds to the vibrational state v = 0 and curve 1 -2 -3 , to the state v = 1 (because of the slight difference between the rotational constants Bv =0 and Bv =1 , the scale of x is slightly distorted for curve 1 -2 -3 ). The curves consist of three characteristic sections. Sections 1, 1 and 2, 2 are almost linear, the rotational temperatures determined from their slopes by formula (4.34) differ sharply and are T1 ≈ T1 ≈ 490 K, T2 ≈ 10000 K, and T2 ≈ 9000 K, respectively. Sections 3, 3 are characterized by a faster, as compared with Sections 2, 2 , depopulation of the levels. Such distributions are of a sufﬁciently general character and were observed in a number of ﬂames [58–60], in RF [54] and pulsed [55] discharges, and in the photodissociation of water molecules [61]. Those groups of molecules, which form ensembles with high rotational temperatures, are called ‘hot’, while the groups responsible for the lowtemperature section, ‘cold’. The various speculations about the causes of formation of such distributions were analyzed in [16]. In gas discharge conditions, the hot radicals are produced in the dissociation of the H2 O

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Figure 4.18 Rotational distributions of the OH(A2 Σ, v ). Curve 1-2-3 – v = 0; curve 1 -2 -3 – v = 1. Water vapor discharge in a 20-mm-dia. water-cooled tube, discharge current 20 mA. Dashed curve – with correction made for predissociation.

molecules via an intermediate state: H2 O(X1 A1 ) + e → H2 O(B1 A1 ) + e → OHhot (A2 Σ) + H + e.

(4.41)

Molecules with fast rotations are formed on account of excessive deformation vibrational energy, because in the H2 O(X1 A1 ) conﬁguration the angle αHOH between the O−H bonds is equal to 104◦ 27’, while in the H2 O(B1 A1 ) conﬁguration, it is 180◦ . At the same time, the excitation from them ground state occurs: OH(X2 Π) + e → OHcold (A2 Σ) + e.

(4.42)

The OH(A2 Σ, v , N ) levels of high N values are populated practically exclusively by the molecules of the hot group in process (4.41), whereas the levels with not very high N are excited simultaneously in processes (4.41) and (4.42). Since these are independent mechanisms, one can isolate the cold molecules by subtracting from the full distribution that part which corresponds to the hot molecules. This procedure, described in [27, 38], leads to the conclusion that the cold group proves to be distributed by the Boltzmann distribution with the gas temperature (see also Section 4.2.4.2 below). The population of the molecules of groups 3 and 3 decreases rapidly with increasing N , this being due to the ﬁniteness of the excessive de-

4.2 Distribution of Molecules Among Rotational Levels

formation vibrational energy on the one hand and the development of predissociation on the other. The dashed curve adjoining curve 1-2-3 indicates populations calculated on the assumption that predissociation is absent. 4.2.4.2 N2 Molecules. Second Positive System

This is one of the spectroscopically best-studied systems (C3 Π–B3 Π transition, Appendix D) that is currently used for plasma spectroscopy purposes and speciﬁcally for gas temperature measurements. But even in this case, some effects requiring special analysis can also occur under nonequilibrium conditions. Curve 1 in Figure 4.19 describes the rotational distribution of the N2 (C3 Π, v = 0) molecules, determined from line intensities in the rotational structure of the (0,0) emission band of nitrogen in a discharge tube with cooled walls [62]. On the whole, this distribution is similar to the one described in the above example with hydroxyl. Also present here are cold and hot groups of molecules. Straight line 2 relates to the cold group isolated by subtracting the hot group from the full distribution. The temperature determined from the slope of line 2 proves equal to the gas temperature, which is conﬁrmed by its coincidence with the results of measurements taken from the emission ˚ ¨ system of X CO (line 3) and from the rotaspectrum of the Angstr om tional distribution of the CN radicals in the X2 Σ, v = 0 ground state (line 4), determined from line absorption (line 4) and thermocouple measurements. In this example, the hot molecules result from the quenching of the highly excited molecules N2∗∗ + M → N2 (C3 Π)

(4.43)

by heavy particles. The cold group is formed on the electron impact excitation of the molecules in the ground state [62]: N2 (X1 Σ) + e → N2 (C3 Π) + e.

(4.44)

Similarly, and even more pronounced, nonequilibrium features in rotational states are also observed in the case of nonresonance excitation of the N2 (C3 Π) molecules by metastable atoms: N2 (X 1 Σ) + M∗ → N2 (C 3 Π) + M

(4.45)

The case with excitation by the atoms M∗ = Ar(3p5 4s) was studied in detail in [38].

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Figure 4.19 Rotational distributions of molecules. Discharge in a CO−N2 −He (1:2.5:10) gas mixture, pressure 1.333 hPa (1 Torr). Discharge tube diameter 20 mm, discharge current 5 mA. 1 – N2 (C3 Π, v = 0); 2 – cold group of N2 (C3 Π, v = 0); 3 – CO(B1 Σ, v = 0); 4 – CN(X2 Σ, v = 0). The temperature determined from the slope of straight lines 2, 3, and 4 is T = 130 K.

4.2.5 On Gas Temperature Measurements in the Presence of Parallel Molecular Rotation Excitation Channels

The above examples show that despite the essentially non-Boltzmann distributions of the emitting molecules among the rotational levels of the excited electronic–vibrational states, the rotational temperature of the ground state (and, accurate up to usually small deviations discussed in Sections. 4.2.2 and 4.2.3, the gas temperature) can be determined, provided that one succeeds in isolating from the general distribution the group of molecules excited by direct electron impact. Assuming that the distribution is of a two-temperature character, one should extrapolate the hot Boltzmann distribution to the region of low rotational energies, proceeding from the fact that the rotational level population is NN /g N = Nv x (1/Qr,c ) exp − F ( N )/kB Tc . + (1 − x )(1/Qr,h ) exp − F ( N )/kB Th .

(4.46)

4.2 Distribution of Molecules Among Rotational Levels

Here Tc and Th are the rotational temperatures of the cold and hot groups, respectively, x is the proportion of the cold-group molecules and Qr,c and Qr,h are the rotational statistical sums of the cold and hot ensembles, equal to kB Tc /Bv and kB Th /Bv , respectively, provided that Bv kB Tc,h (Appendix A). The appropriateness, reliability and accuracy of this approach depend on a number of conditions. 4.2.5.1 Extension of the Form of Distribution of the Hot Molecules to the Region of Low Rotational Levels

The temperature of the hot distribution must be substantially higher than the gas temperature. The greater part of this distribution is then distinctly observed in the region of high rotational levels. In the region of low rotational quantum numbers, the cold and hot distributions overlap. That extrapolation is possible is conﬁrmed empirically (see above and also [27, 38, 51, 62]) and by calculations within the framework of the so-called statistical model. The model presumes a random character of energy redistributions in the system in the case of strong particle interaction [63]. The rotational distributions of the hot molecules in the examples under consideration were calculated in [64], with due regard for the results obtained in the works [65] on the decay of an electronically excited water molecule and [66] on the decomposition of an excited four-atom complex ABCD into the fragments AB and CD. The surplus reaction energy due to the degree of freedom under consideration is determined from the experimentally observed levels with maximum excitation. In the given case, these are the rotational levels with the maximum observed N values in the OH(A2 Σ, v = 0) and N2 (C3 Π, v = 0) states formed during the course of decomposition of the (H–OH)∗ and (N2 –N2 )∗ complexes. Figure 4.20 presents comparison between experimental results and calculations [64] for nitrogen. The circles in the ﬁgure represent experimental data points and curves 1 and 2, results of calculations by the statistical theory with two values of the theory parameter, ρ = 0.1 and ρ = 0.25 (ρ2 is the ratio between the moments of inertia of the fragments and the complex). 4.2.5.2 Spectral Resolution

To separate the groups, it is important that the spectral resolution and dynamic range of measurements are sufﬁcient to register individual rotational components of the spectrum. It is not uncommon that adjacent vibrational bands overlap. For example, in the 2+ system of nitrogen, the R branch of the (0,0) band, which is convenient for diagnostics in other respects, is free from superposition on the (1,1) band only in the re-

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Figure 4.20 Rotational distributions of the N2 (C3 Π, v = 0) molecules. Circles – experiment; lines – calculation by the statistical model at (1) ρ = 0.25 and (2) ρ = 0.1.

gion of N < 29, although, given high enough resolution (Appendix D), lines with N ≤ 50 can also be observed. For gas temperature measurements, the least favorable case is one where the spectral region being resolved is restricted for experimental conditions to the range wherein the densities and emission intensities of the hot and cold molecules are commensurable. Since the groups are formed by different mechanisms, the proportion between the densities of their pertinent molecules can depend on the conditions in plasma. In the example with hydroxyl, the threshold of the ﬁrst stage of reaction (4.41) amounts to ca. 16.000 × 10−19 J (10 eV), and in reaction (4.42), ca. 6.400 × 10−19 J (4 eV), and so the ratio between the rates of these reactions is sensitive to the velocity distribution of the electrons. Figure 4.21 h /N c when helium illustrates the behavior of the density ratio G = NOH OH is admitted to a discharge in water vapor (0.667 hPa (0.5 Torr)). It changes because of the alteration of the average electron energy, the amount of the cold molecules increases with helium pressure, while the amount of the hot ones remains practically unchanged [67]. The results of calculating the ratio G, using probe measurement data for the electron velocity distributions and spectral measurement for the concentrations of the hydroxyl molecules in the electronic ground state [67], agree well with the spectral data and are also presented in the ﬁgure.

4.2 Distribution of Molecules Among Rotational Levels

Figure 4.21 Ratio G between the densities of the hot and cold groups of the OH(X2 Π, v = 0) molecules as a function of the pressure of helium: squares – spectral measurements; circles – calculation from the results of probe measurements of electron velocity distribution and spectral measurements of the concentration of the OH(X2 Π) molecules.

By contrast, in the case of N2 (C3 Π) molecules, the ratio G remains practically constant under various discharge conditions. As demonstrated in [38, 62], the excited state N2∗∗ in conditions of gas discharge is the N2 (E3 Σ) state with a long radiative lifetime. The electron impact excitation thresholds of the N2 (C3 Π) and N2 (E3 Σ) states differ by a mere 1.120 × 10−19 J (0.7 eV), their magnitudes being ca. 17.600 × 10−19 J (11 eV) and ca. 18.720 × 10−19 J (11.7 eV), respectively (Appendix D). On the other hand, this ratio can vary considerably during the course of the evolution of the pulsed discharge. Figure 4.22 presents the distributions of the N2 (C3 Π, v = 0) molecules among rotational levels in a liquidnitrogen-cooled discharge, determined from the time-resolved spectra of the 2+ system. The gas pressure is 1.333 hPa (1 Torr), current pulse duration ca. 10 μs with a rise time of ca. 5 μs, and maximum current ca. 1 kA. Observed within 300 ns of the start of the current pulse is only the cold distribution with a temperature of 130 K, that is due to direct electron impact excitation. 7.5 μs after the start of the pulse the rotational distribution of the molecules differs from the Boltzmann one; namely, hot molecules appear produced in more inertial heavy particle interactions.

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Figure 4.22 Rotational distribution of the N2 (C3 Π) molecules in a pulsed discharge: 1 and 2 – 300 ns and 7.5 μs after the start of the pulse, respectively.

Spectral temperature measurements, even if taken with high spectral resolution, but without sufﬁciently high time resolution, would be very difﬁcult to interpret in the given conditions. 4.2.5.3 Effect of the Conditions Occuring in Plasma on the Rotational Temperature of the Hot Group

If the lifetime of molecules in a radiative excited electronic state is shorter than the time between particle collisions, their distribution among rotational levels is deﬁned by condition (4.35), that is, by the excitation mechanism of each individual level. The excitation rate Φi is in this case formed as the sum of the rates of, generally, different processes. The cold group reproduces, accurate up to the above-discussed corrections (4.36)– (4.39), the distribution of the ground-state molecules with a temperature equal to the gas temperature (except for the special cases described in Section 4.2.2). The hot molecules form a statistical ensemble with a temperature determined by the energy defect of the elementary heavy particle interaction act and not equal to the gas temperature. To illustrate, in our examples, for OHhot (A2 Σ, v = 0) in process (4.41), Th ≈ 10000 K and for N2 (C3 Π, v = 0) and M∗∗ = N2 (E3 Σ), Th is ≈ 2100 K. When the pressure of the plasma-forming gas in raised, the effect of particle collisions can come into play. If electronically excited molecules

4.2 Distribution of Molecules Among Rotational Levels

collide most frequently with particles in the electronic ground states, and the cross sections for transitions between rotational states are close to the gas kinetic ones, which is typical of low-temperature plasma, the initial rotational distribution can be modiﬁed as a consequence of the gas pressure variation. To describe such transformations is a complex problem, for the system is of essentially multi-level character and the relaxation of rotational motions must be analyzed simultaneously with the translational relaxation. The cross sections of both processes are comparable, and deviations from equilibrium can be sizeable (see Section 4.1). When roughly describing experimental data, one can assume that in the course of transformations of rotational and translational distributions the cold and hot particles behave as independent ensembles with individual rotational and translational temperatures (canonical approximation). If the rotational constants Bv and Bv of the molecule in the ground and excited electronic–vibrational states are approximately the same, collisions have no effect on the rotational distribution of the cold molecules, because even at a low gas pressure, p → 0, their rotational temperature coincides with the kinetic temperature of the thermostat of unexcited molecules, Tc ( p = 0) ≈ T. For the hot molecules under stationary conditions, one can write down the phenomenological relation [68] Th ( p) − T = [ Th ( p = 0) − T ]/(1 + αp).

(4.47)

The physical meaning of the coefﬁcient α can be explained by comparing expression (4.47) with the Landau–Teller general relaxation equation (for small deviations, cf. (4.10) and (4.11) Th (t) − T = [ Th (t = 0) − T ] exp { − t/τ }

(4.48)

which describes the variation with the characteristic relaxation time τ of the temperature of a rotational ensemble under the effect of collisions. In spectroscopic experiments, the system is observed (on the average) at a time τl of relaxation process, where τl is the radiative lifetime of emitting state. It follows from comparison between expressions (4.47) and (4.48) that α = τl /τ1 , where τ1 is the relaxation time at a unit pressure (p = 1.333 hPa(1 Torr)): 1 τ 1 1+ l p . = (4.49) Th ( p) − T Th ( p = 0) − T τ1 Considering that τl−1 = τr−1 + τq−1 , where τr and τq are the lifetimes determined by radiation and quenching on collisions, one should distinguish between two limiting cases. If τl = τr , the dependence of 1/[Th ( p) − T] on p is linear, the slope of its plot being (τr /τ1 )/[Th ( p = 0) − T]. Figure 4.23 presents the results

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Figure 4.23 Effect of collisions on the rotational temperature.

obtained by Bleekrode [68] who studied the variation of line intensities in the rotational structure of the Swan bands (see Appendix D) as a function of the gas pressure in an oxyacetylene ﬂame. Molecules with fast rotation in the C2 (A3 Π) state were produced in chemical reactions (for mechanisms, see, for example, [69]). The function is linear at pressures up to 66.660 hPa (50 Torr). The slope of its plot corresponds to the rotational relaxation time indicated, τ1 = 3 × 10−6 s at τr = 8 × 10−7 s. Naturally this quantity is not at all universal and depends on the excitations conditions of the spectra. For example, the value of τ1 for this C2 state in conditions of CO gas-discharge plasma was found to be 3 × 10−5 s [69]. For this reason, to determine the gas temperature by means of extrapolation (4.46), one should use the value of Th determined directly in the experiment being conducted. If the lifetime of a given electronic–vibrational state is generally governed by collisional quenching, then τl = τq , τq = τq1 /p, where τq1 is the quenching-controlled lifetime at a pressure of p = 1.333 hPa(1 Torr). In that case, the parenthesized term in expression (4.49) will be (1 + (τq1 /τ1 )) and the rotational temperature Th is independent of pressure. If, in addition, the quenching cross section considerably exceeds the rotational relaxation cross section and τq1 τ1 , then under stationary conditions Th ( p) = Th ( p = 0). In the latter case, the rotational temperature of the hot ensemble is constant and is governed by the elementary excitation process. And to determine the gas temperature in this case one can use this value of Th , no matter what the gas pressure. It is exactly this situation that occurs in our example with the hydroxyl molecule. The cross section of the rotational relaxation of the molecules in the OH(A2 Σ,

4.3 Line Intensities in the Vibrational Structure

˚ 2 , whereas the v = 0) state on collisions with water molecules is ca. 1 A quenching cross section of electronic states on the same collisions is inde˚ 2 [38]. pendent of the serial number of the rotational level and is ca. 70 A

4.3 Line Intensities in the Vibrational Structure of Spectra and Distributions of Molecules Among Vibrational Levels 4.3.1 Elements of Vibrational Kinetics. Vibrational Energy and Temperature

In plasma models with partial equilibrium, use is made, in addition to the translational, rotational and other temperatures, of the concept of vibrational temperature characterizing the store of plasma energy in the vibrations of molecules. In principle, to explain this concept, one should resort, as in the case of rotational temperatures, to the balance conditions (the type of stationary conditions (4.28)) of molecules in various vibrational states and try and isolate those ensembles of states, whose densities are related by a small number of parameters, in particular the partial temperature. A vast body of literature is devoted to the problems of vibrational kinetics (for example, [13, 24, 25, 70, 71]), but we will restrict ourselves to the simplest data necessary for the purposes of the optical plasma spectroscopy. When averaged over velocities and rotational levels, a molecule can be treated as an oscillator, generally speaking, an anharmonic one. An important feature of molecular interaction is that the direct transformation of the vibrational energy therein into the translational energy (VT exchange) is accompanied by effective exchange between vibrational quanta (V-V exchange). The population balance of the vibrational level with the serial number n in V-T and V-V processes is described by the following system of equations: dNn dNn dNn = + , (4.50) dt dt VT dt VV dNn (4.51) = Z ∑ Pmn Nm − ∑ Pnm Nn , dt VT m m Z dNn sl ls (4.52) = ∑ Qmn Ns Nm − ∑ Qnm Nl Nm . dt VV N m,s,l m,s,l Here m, n, s, l = 0, 1, 2, . . ., s = l, m = n, N = ΣNn is the total number of particles, Z is the frequency of gas kinetic collisions, Pnm is the probability

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4 Intensities in Spectra and Plasma Energy Distribution

of the n → m V-T transition on collisions, with the vibrational quantum energy converted into kinetic energy and Qsl mn is the probability of the V-V exchange on collisions between two molecules at the levels m and s, accompanied by their transition to the levels n and l, respectively. The presence of the V-V exchange makes the system of equations (4.50)–(4.52) nonlinear. It can be considered under various simplifying assumptions. Let us note the two most important ones. 4.3.1.1 Harmonic Oscillator Approximation

If anharmonicity is disregarded, the following rules hold true: Pnm = 0 for

m = n ± 1,

Pn+1,n = (n + 1) P10 , ⎧ 10 ⎪ ⎨(m + 1)sQ01 , Qsl m(s + 1) Q01 mn = 10 , ⎪ ⎩ 0,

(4.53) (4.54)

n = m + 1, l = s − 1 n = m − 1, l = s + 1 .

(4.55)

n = m ± 1, l = s ± 1

The following relations for the probabilities of the processes involving molecules at the lower vibrational levels ensue from the detailed balancing principle: 01 Q10 01 = Q10 ;

P01 = P10 e−θ ;

θ=

hν10 , kB T

(4.56)

The kinetic gas temperature dependences of the probabilities are approximated by the expressions 2 ln P10 ∼ −(μν10 /T ) /3 , 1

−1 Q10 01 ∼ C1 T + C2 T ,

(4.57)

wherein ν10 is the oscillation frequency, C1 and C2 are constants allowing for the contributions from the short- and long-range interaction potential components, respectively and μ is the reduced mass of the colliding particles. When use is made of relations (4.53)–(4.56), the system of equations (4.50)–(4.52) is reduced to the balance equation ! " . dNn = ZP10 (n + 1) Nn+1 − (n + 1)e−θ + n Nn + ne−θ Nn−1 dt ! ¯ ) Nn−1 + ZQ01 (4.58) 10 ( n + 1)(1 + α " − (n + 1)α¯ + n(1 + α¯ ) Nn + nα¯ Nn−1 , where α¯ is the number of vibrational quanta per molecule on the average, α¯ =

1 N

∑ lNl . l

(4.59)

4.3 Line Intensities in the Vibrational Structure

If one multiplies expression (4.58) by hν10 n and makes summation with respect to n, one will then get the Landau–Teller equation (see, for example, [13] and comments to (4.12)), for the relaxation of the vibrational energy Ev = ΣEn Nn (t), dEv Ev − Ev0 = dt τv

(4.60)

with the characteristic relaxation time . −1 , τv = ZP10 (1 − e−θ )

(4.61)

where Ev0 is the equilibrium energy corresponding to the Boltzmann distribution of the particles among vibrational levels with the temperature equal to the kinetic gas temperature. Hence . t . (4.62) Ev (t) − Ev0 = Ev (t = 0) − Ev0 exp − τv n The same result will be obtained if the quantity dN (the second dt VV

term) in (4.58) is formally omitted, since the resonance exchange of quanta of a harmonic oscillator with equidistant levels has no effect on the total vibrational energy and τv = τVT . In practice, the situation with the V-V exchange ‘switched off’ is realized if the molecules of interest constitute only a small admixture to the gas of particles the exchange with which is either strongly impeded (molecules with substantially differing quanta) or altogether impossible (monoatomic gas). The presence of the V-V exchange is important for the formation of the form of the vibrational distribution of molecules in the course of variation of the energy store. In most cases of practical interest (see, for exam01 ple, [13, p. 94]), the relation Q10 01 = Q10 P01 holds true, in virtue of which the relaxation process is characterized by two characteristic times τVV ∼ 1/ZP10 and τVT ∼ 1/ZQ10 01 . In this case, the form of the particle distribution at each stage of the ‘slow’ V-T relaxation is governed by the ‘fast’ V-V exchange and can be established by solving (4.58) with the term

dNn dt

(the ﬁrst term) omitted: nhν10 . Nn = N0 exp − kB Tv VT

(4.63)

Here Tv is the vibrational temperature which is, generally speaking, different from the gas temperature and determines the vibrational energy store: + , −1 hν10 −1 . (4.64) Ev = hν10 N exp kB Tv

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Distribution (4.63) is called quasistationary; during the course of the V-T relaxation, it ‘adapts itself’ to the current vibrational energy store (4.62). To get the truly stationary distribution of molecules among vibrational levels, it is necessary to include in (4.50) energy sources: ˜n dN dNn = + Wn . dt dt

(4.65)

Positive sources may be, for example, photoexcitation or exothermal chemical reactions. A most important energy source in plasma is the excitation of molecular vibrations by electrons. Negative sources include emission of radiation, vibrational energy transfer into the internal degrees of freedom of third particles, and so on. But if the V-V exchange rate exceeds the rate of creation or annihilation of vibrational energy quanta under the effect of external sources, the stationary vibrational distribution is then Boltzmann distribution (4.63). 4.3.1.2 Effect of Anharmonicity

The vibrational levels of a real molecule are not equidistant owing to the anharmonicity ΔE(Appendix D): En = E1 n − ΔEn(n − 1)

or

En − En−1 = E1 − 2(n − 1)ΔE,

(4.66)

where E1 = hν10 . Anharmonicity is, as a rule, not very great (typically ΔE/E1 ≈ 10−2 ), but at large n it can impact on the course of relaxation processes. Rules (4.53)–(4.55) for the probabilities P and Q cease to be stringent and should be corrected by introducing correction factors dependent on the quantum numbers, gas temperature and magnitude of anharmonicity. Figure 4.24 illustrates the exemplary calculations of the probabilities Pn,n−1 and Q01 n,n−1 made in [13] for the N2 molecule. Because of the resonance defect increasing with the difference n − m, the exchange probabilities Q grow slower, and the probabilities P, faster, with increasing n than in the case of harmonic oscillator. As with the harmonic oscillator, for the lower levels, Pn,n−1 Q01 n,n−1 and this relationship changes. The situation being what it is, to describe the relaxation process in terms of quasistationary distributions makes no sense, for the separation of the V-V and V-T processes on the principle that their characteristic times are different is baseless. In order to use this convenient approach to the utmost, they restrict themselves to those levels for which P Q. As can be seen from Figure 4.24, the number of such levels can be large, especially at moderate gas temperatures. Disregarding, as before, multiquantum transitions, one can use (4.50) without consideration of the V-T processes in the same way as when obtaining solution (4.63). But

4.3 Line Intensities in the Vibrational Structure

Figure 4.24 Probabilities Pn,n−1 (solid curves) and Q0,1 n,n−1 (dashed curves) as a function of the serial number n of the vibrational level for the N2 molecule at various temperatures.

in this case, however, the detailed balancing principle used to establish the relations between the probabilities of the mutually inverse V-V exchange acts shows that these probabilities, in contrast to those deﬁned by expressions (4.55) and (4.56), are different: ( En+1 − En ) − ( Em+1 − Em ) +1 m+1,m . (4.67) Qm,m = Q exp − n+1,n n,n+1 kB T Performing the operations indicated in the preceding item, we obtain the following result for the vibrational distribution, which differs from expression (4.63): , + E1 ΔE T Nn = N0 exp −n . (4.68) − ( n − 1) kB T1 kB T 0 Here, as above, T is the kinetic gas temperature and T1 = Ek 1 ln N N1 B is the population temperature of the ﬁrst excited vibrational level Section 1.3). Distribution (4.68) is named the Treanor distribution after the author who predicted it. When (n − 1) kΔET k ET1 , it turns into distriB B 1 bution (4.63). The greater the anharmonicity and the lower the temperature, the stronger the differences. Distribution (4.68) is schematically

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Figure 4.25 Vibrational distributions: 1 – Treanor distribution; 2 – distribution allowing for V-T processes.

illustrated by curve 1 in Figure 4.25. It has a minimum for the level with the serial number n0 : n0 =

E1 T 1 + . 2ΔE T1 2

(4.69)

One should, of course, take into consideration the fact that in the region of large n > n0 ignoring of the V-T processes, as in deriving expression (4.68), is improper, especially if the molecules are surrounded by an atomic gas, and so these processes should be allowed for. This complicates equations (4.50) and to solve them requires resorting to numerical methods. Reviews of the main trade-off simpliﬁcations permitting simple analytical solutions in reasonable agreement with numerical computations can be found speciﬁcally in [13, 71]. A typical result of such calculations is illustrated by curve 2 in Figure 4.25. Up to the level nB < n < n0 the distribution is practically of the Boltzmann type coinciding with that deﬁned by formula (4.63) (the dashed line with the slope angle α1 ). In the interval nB < n < n0 , the distribution differs from the Boltzmann distribution and is described competently by Treanor relation (4.68) (curve 1). The number n0 is deﬁned by expression (4.69) and corresponds to conditions wherein the probabilities of the V-V and V-T processes draw closer together (Figure 4.24). At n > n0 a ‘plateau’ region occurs, where the V-V and V-T processes are simultaneously operative and approximation (4.68) proves inadequate. At n ≈ n∗ the V-T processes become predominant and the populations of the vibrational levels decrease rapidly as their serial number grows larger. Based on the form

4.3 Line Intensities in the Vibrational Structure

of the vibrational distribution, one can judge the possibility and advisability of using such for the description of the three temperature values determined, by analogy with expression (4.34), from the slope angles α1 , α2 , and α3 : T1,2,3 =

1 cot α1,2,3 . kB

(4.70)

The temperature T1 has been deﬁned above by expression (4.68). As a rule, the density of molecules at the levels n > n0 is low, Nn>n1 N1 , and T1 determines an important characteristic of the system, namely, the vibrational energy store, as deﬁned by expression (4.64) with Tv = T1 and νnm = ν10 . As can readily be appreciated, this circumstance also emphasizes the importance of the zeroth, harmonic, approximation. Since at n > n∗ a fast energy exchange takes place between vibrations and translational motion, the temperature T3 is close to the gas temperature. The partial temperature T2 has no such simple meaning and is useful as a formal parameter in describing the processes occurring in the system of levels n0 < n < n∗ , which is important, for example, in analyzing the partial inversion conditions in lasers. More detailed theories also allow for the presence of multiquantum transitions, vibration pumping sources, radiative transitions, and so on, but the qualitative picture of the distributions of the type described by curve 2 in Figure 4.25 is, on the whole, retained. 4.3.1.3 Diatomic Molecular Mixture and Polyatomic Molecules

If a gas contains two molecular species, A and B, which exchange vibrational quanta, the process is referred to as the V-V exchange to supplement the V-V and V-T processes. In the group of the lower levels, n A < nBA , n B < nBB (4.69), one can apply the harmonic approximation to each oscillator and introduce the vibrational temperatures TvA , TvB . If the frequencies of such oscillators are related together as pν A ≈ qν B , where p and q are small integers, the most probable transitions in the V-V exchange are those, for which A B − Em ≈ ElB − EsB , Em

(4.71)

where |m − n| = p, |s − l | = q. Following the above reasoning, one can A,B demonstrate [13] that under stationary conditions (t > τVV , τVV ) the vibrational temperatures are related by pE1A qE1B pE1A − qE1B = − . kB T kB TvB kB TvA

(4.72)

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In a polyatomic molecule, various kinds of vibrational motion (vibrational modes) are characterized, as a rule, by different frequencies that can be associated with individual oscillators. Because of the difference in frequency, when the oscillators collide with one another, is favoured, the V-V exchange, as is the case with diatomic molecules, and the partial vibrational temperatures of the modes set in. The exchange of quanta between vibrational modes is similar to the V-V exchange of quanta between colliding diatomic molecules. All oscillators are in the state of V-T interaction with the translational motion. Because of the difference between the V-V and V-T exchange probabilities for different modes and the possible difference between their excitation rates, the vibrational temperatures can differ substantially between modes. If the V-V and V-V process for all modes are faster than the V-T relaxation and the action of external sources, the relationship between the vibrational temperatures is similar to expression (4.72). The generalization of expression (4.72) to an arbitrary number of modes and consideration of their anharmonicity and speciﬁc features of the Fermi interaction between degenerate vibrations can be found in [13, 71]. The above results from the consideration of the processes of interaction of molecular vibrations with one another and with other degrees of freedom, have been obtained on the assumption that the electronic state of the molecules is ﬁxed and its lifetime exceeds all the characteristic times of the V-V, V-V , and V-T processes. This assumption limits, though not by very much, the use of theoretical results in practice, for, as a rule, and in conditions of low-temperature plasma, the overwhelming majority of molecules are in their electronic ground states. They determine the vibrational energy store and play an important part in the general picture of plasma processes. The concept of vibrational temperatures of a block of low-lying levels frequently proves very productive in revealing the energy balance of nonequilibrium systems and plasma in particular. Note also that the theoretical quantitative calculations of vibrational distributions and temperatures require information about a great number of elementary process probabilities in multilevel systems and their gas-temperature dependences, and also consideration of the external sources. Therefore, the development of the theory of molecular vibrational processes relies heavily on experimental investigations.

4.3 Line Intensities in the Vibrational Structure

4.3.2 Vibrational Temperature and Distribution Measurements by Absorption Spectroscopy Techniques

The frequencies νlu of molecular vibrations fall within the IR region of the spectrum, and so the technique of tunable narrow-band IR radiation sources proves most effective in absorption spectroscopic studies into the vibrational distributions of dipole molecules in the electronic ground states. Let us use the integral line absorption coefﬁcient χlu (2.51), (2.52) and express the Einstein coefﬁcient in terms of the line strength Slu (2.59). For a given vibrational–rotational transition, it is usually (Appendix D) represented in the form Slu = gl gu | Rv v |2 S J J ,

(4.73)

where | Rv v |2 is the squared dipole moment matrix element of the vibrational transition and S J J is the line force of the rotational transition (the ¨ Honl–London factor). We denote the level population per unit statistical weight (Appendix A) as n = N/g, nm = nm ( J, v), and then 8π3 nu nl νlu | Rv v |2 S J J 1 − χlu = (4.74) 3hc nl or . −1 8π3 χlu νlu | Rv v |2 S J J nl − nu = ≡ χlu Zlu . (4.75) 3hc Here and hereinafter in Sections 4.3.3 through 4.3.6, to denote vibrational levels, we use, along with the letters n, m, p, q,. . . , the symbol v adopted in spectroscopy. This is done in order to avoid, in writing formulas, confusing the standard designation for the level population per unit statistical weight with the serial number of the level. For the J − J = 0 ± 1 dipole transitions and also in the case of strong vibrational excitation, the quantities nl and nu relating to the vibrational levels v and v can be commensurable and sometimes even inverted in some regions of vibrational levels. For this reason, relations (4.74) and (4.75) cannot, by themselves, help obtain from the measured absorption coefﬁcients the individual level populations. But if it proves possible to measure absorption for a series of vibrational–rotational transitions, with their rotational numbers varying successively, one will then be able to write down a chain of equations (4.75): ni − nk = χik Zik , nk − nl = χkl Zkl , ..................... nm − nn = χmn Zmn .

(4.76)

199

200

4 Intensities in Spectra and Plasma Energy Distribution

Summing Equations (4.76), we get ni − n n =

n

∑ χ pq Zpq .

(4.77)

i

If the number of transitions in the cascade is large enough, one can neglect the population n of the uppermost level and thus determine ni and, successively, the populations of the rotational levels of various vibrational states, up to i − 1. The propriety of this neglect of the population nn is checked by limiting n to various values. If the quantity ln(nk ) is a linear function of the energy of the rotational term, its slope angle determines the rotational temperature (see relation (4.34)). The vibrational level populations are determined as usual: Nv = ni ( J ) Qr gv ,

(4.78)

where Qr is the rotational statistical sum and gv takes account of the possible degeneration of the vibrational states (Appendix A). The authors of [72] have analyzed the possibilities of simplifying the measurement and processing scheme. If the vibrational temperature Tv is not very high and the rotational temperature Tr is not very low, the latter can be determined from the slope of ln(χ/Z ) plotted as a function of the energy of the rotational term within the limits of the given vibrational band. The above-mentioned limitations on the temperatures correspond to the condition hc Nv Qr (v ) (4.79) |ΔG | 1, G = exp Fv ( J ) − Fv ( J ) Nv Qr (v ) kB T over the entire variation range of J under study. Here Fv ( J ) are the rotational terms within the limits of the given vibrational state. The acceptability of such an approximation is ascertained by the fact of the linearity of ln(χ/Z ) established as a result of processing experimental data. Figure 4.26 illustrates some examples of the diode laser spectroscopic measurements (Section 3.3.3) of (i) the rotational distributions of the CO2 molecules for four different vibrational states and (ii) the vibrational distributions of the same molecules [72]. The laser line width was substantially narrower than the width of the absorption line proﬁle, and frequency scanning was used to measure the integral absorption coefﬁcient for the individual rotational components of the v1 v2l v3 - v1 v2l v3 transitions, where v1 , v2 , and v3 are the vibrational quantum numbers for the symmetric, deformation and antisymmetric modes, respectively, and l is the quantum number characterizing the level degeneracy of deformation vibrations. The measurements were taken in the active medium of

4.3 Line Intensities in the Vibrational Structure

Figure 4.26 Distributions of the CO2 molecules among energy levels on the axis of a discharge in a BeO ceramic capillary 2 mm in diameter. CO2 −N2 −He (1:1:8) gas mixture, pressure 79.992 hPa (60 Torr), discharge current 9 mA: (a) rotational distributions in various vibrational states; (b) vibrational distributions.

201

202

4 Intensities in Spectra and Plasma Energy Distribution

a miniature waveguide CO2 laser. Note (Section 3.3.3) that the vibrational distribution was ﬁxed in the energy range 14000 cm−1 , whereas the spectral region used for the purpose was less than 100 cm−1 wide in the 4.5-μm wavelength region. In this case, populations were measured for some 1000 rotational and 30 vibrational levels. The sensitivity of measurements in the given example is ca. 108 cm−3 for the rotational state. This corresponds to the absorption ΔI/I ≈ 10−2 at an optical path length of 8 cm. Figure 4.26a demonstrates Boltzmann distributions of the molecules among rotational states with a temperature of Tr = 530 ± 10 K, which coincides with the gas temperature, determined independently. One can see from Figure 4.26b that there are two vibrational temperatures. One of them is determined by the slope of the dashed lines drawn through the points denoting the populations of levels differing successively by a quantum of symmetric or deformation vibrations, Δv1 = 1 or Δv2 = 1. This temperature coincides with the rotational and the gas temperature, which indicates a close connection between these vibration modes and the translational motion. The other vibrational temperature (solid lines) is determined by the relative populations of vibrational levels differing by a quantum of antisymmetric vibrations, Δv3 = 1, and is T3 = 2040 ± 20 K. The high measurement accuracy allows one to see a weak tendency towards overpopulation, compared to the Boltzmann value, for the levels with v3 > 4. This is deﬁnitely due to anharmonicity, and therefore the above value of T3 was determined from the slopes of the lines with v3 ≤ 4. When determining level populations from the integral absorption coefﬁcients, there is no need for one to take into consideration the line broadening mechanism (Section 2.3). However, if measurements are taken at a ﬁxed frequency within the limits of the absorption line proﬁle, account should be taken of the form factor by means of the formulas of Section 2.2. Such a situation occurs, for example, when measuring absorption using discretely tunable gas lasers as radiation sources. The form factor will be easier to take into account if the same molecules as the lasing ones are studied, because the measurement results can, as a rule, be referred to the center of the absorption line proﬁle with reasonable accuracy (though this should nevertheless also be taken into consideration), even if the molecule under study resides in conditions differing from those of the laser medium. Lasing has been attained on many hundreds of vibrational–rotational transitions in molecules that can be used for plasma diagnostics purposes, and frequently the subject of study are under nonequilibrium conditions (CO2 , CO, N2 O, CS2 , hydrogen halides and others). The implementation of measurements and the processing of the results are, on the whole, similar to those described above, if one uses

4.3 Line Intensities in the Vibrational Structure

Figure 4.27 Vibrational distributions of the CO molecules. Active medium of a CO laser. Circles – measurements; solid line – calculation.

instead of expression (4.74) the formula for the absorption coefﬁcient at the center of the line proﬁle, relating the absorption coefﬁcient values obtained by relations (2.61), (2.63), or making numerical calculations for the Voigt proﬁle (Section 2.2.3). Figure 4.27 presents as an example the vibrational distribution of CO molecules in a gas discharge, studied in [73] with the use of a discretely tunable CO laser with a dispersive cavity. The solid curve presented in the same ﬁgure demonstrates the capabilities of the up-to-date calculations allowing for the V-V and V-T exchange processes, radiative decay, multiquantum transitions, electron impact vibration excitation and a number of other details. Note, however, that when processing the results of measurements of the coefﬁcients of absorption (ampliﬁcation) on vibrational–rotational transitions over a wide range of vibrational quantum numbers, the use of the harmonic oscillator approximation to approximate Rv v is inadmissible. The results of more exact calculations or special measurements are required. Some examples will be presented in the next section. The staging of the experiment and data processing can be changed if the problem to be solved is not the elaboration of detailed, level-bylevel distributions, but the determination of vibrational temperatures on the assumption that such exist in the conditions under study. This approach is widespread and theoretically justiﬁed, at least for the group of the lower levels determining the vibrational energy store (Section 4.3.1.2, Figure 4.25). Let us illustrate this issue with the example of probing plasma with radiation of low intensity (causing no absorption saturation) on CO2 laser transitions. This technique has been used in many works,

203

204

4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.28 To the measurement of the distributions of the CO2 molecules among vibrational–rotational levels and their temperatures from absorption (ampliﬁcation) on laser transitions: (a) lower vibrational levels and laser transitions: R – traditional,

H – hot, and S – sequential; (b) vibrational temperatures T2 and T3 and the gas temperature T as a function of the discharge current in a CO2 −N2 −He (1:1:8) gas mixture in a tube 11.5 mm in diameter at a pressure of 6.666 hPa (5 Torr).

mainly to study various CO2 laser media (see, for example, [74, 75]. Figure 4.28a presents a schematic diagram of the lower vibrational levels of the CO2 molecule, indicating the laser transitions of the main (traditional), ‘sequential’, and ‘hot’ bands. The relative populations and the corresponding temperatures are found from combinations of the absorption (ampliﬁcation) coefﬁcients for various vibrational–rotational transitions. One can, for example, use ratios between the absorption coefﬁcients for the line centers of the sequential bands, χ0,s , hot bands, χ0,h and traditional bands, χ0,r , at a ﬁxed J. The difﬁculty associated with the fact

4.3 Line Intensities in the Vibrational Structure

that transitions in different bands can end up at levels of even or odd J is overcome by averaging the absorption coefﬁcients for adjacent rotational components of the band. By using formulas (4.74), (4.78), assuming (in calculating vibrational level energies) that the rotational constants are the same for all the vibrational levels of interest and considering the dependence of the dipole moment matrix elements on the vibrational quantum numbers to be the same as for the harmonic oscillator,

| Rv,v+1 |2 = (v + 1) | R0,1 |2 ,

(4.80)

we get χ0,s hν = 2 exp − 3 χ0,r kB T3 χ0,h hν2 . = exp − χ0,r kB T2

,

(4.81)

(4.82)

The rotational temperature is taken to be equal to the gas temperature and is determined by ﬁtting the dependence on J of the absorption coefﬁcients measured by formula (4.74). Figure 4.28b presents as an example the results of determining by this technique the vibrational temperatures T3 and T2 and the rotational (gas) temperature T as a function of the discharge current in the active element of a CO2 laser. As in the example of measurements by the diode spectroscopy technique, the temperature of the deformation mode here differs only slightly from the gas temperature. The use of thermal and other wide-band probe radiation sources for the absorption IR spectroscopy of vibrational–rotational transitions necessitates application of high-resolution spectrometers. To illustrate, the authors of [76] measured vibrational temperatures in an RF CF4 discharge by the method of absorption of light from an incandescent element. The spectrum was analyzed with a Fourier spectrometer with a resolution of ca. 0.12 cm−1 . This made it possible to use vibrational band sequences of the mν2 - ν3 intermode transitions at m = 0, 1, 2 and 3, spaced ca. 2 cm−1 apart in the working spectral region ca. 1280 cm−1 wide, and measure the vibrational temperature T2 of the ν2 mode. The rotational structure of the bands was not resolved in this experiment, and so they compared the theoretical and experimentally observed shapes of the bands in the region of the R-branch that was free from overlapping with the adjacent branches. Under the conditions of the experiment [76], the temperatures Tr and T2 measured by this method were 350 and 400 K, respectively.

205

206

4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.29 Fragment of the IR emission spectrum of the N2 O molecules in a discharge [77,78]: (a) experiment; (b) calculation. Bands: 1 – 000 1-000 0, 2 – 000 2-000 0, 3 – 000 3-000 2, 4 – 011 1-011 0, 5 – 011 2-011 1, 6 – 022 1-022 0, 7 – 100 1-100 0, 8 – 020 1-020 0, 9 – 031 1-031 0, 10 – 022 2-022 1, 11 – 033 1-033 0, 12 – 111 1-111 0, 13 – 020 2-020 1, 14 – 011 3-011 2.

In absorption spectroscopic techniques, the probe radiation can be collimated, which localizes measurements across the beam, but the results of measurements taken along the beam are averaged. 4.3.3 Emission Methods in the IR Region of the Spectrum

Compared with the measurement and processing of electronic transition spectra in the visible region of the spectrum, the analysis of vibrational–rotational IR spectra is complicated by a number of circumstances, primarily, (i) the lower resolving power of the standard spectral instruments, (ii) the limited sensitivity of the detectors, combined with the lower spontaneous radiation intensities (ca. ν4 , see expressions (2.48), (2.38)), and (iii) the generally high population of the vibrational–rotational levels and the associated reabsorption. The pertinent investigations are conducted both with a high spectral resolution, revealing both vibrational bands and their rotational components, and without the rotational structure and even groups of vibrational bands being revealed. In the latter case, one usually resorts to comparisons between the experimental and model spectra with variable parameters (direct problem). The advantage of high-resolution techniques is beyond doubt. These use high-aperture modulation instruments such as Fourier spectrometers

4.3 Line Intensities in the Vibrational Structure

or interference spectrometers with selective amplitude modulation (ISSAM) [77]. The work by Farrenq [78] exempliﬁes such an approach (see also [79]). He studied the vibrational–rotational distributions of the N2 O molecules in a glow discharge in an N2 O−N2 (1:25) mixture at a partial N2 O pressure of 0.187 hPa (0.14 Torr). The spectral resolution (ISSAM) was ca. 0.03 cm−1 in the studied spectral range, 2100–2300 cm−1 . The measurement scheme was calibrated against blackbody radiation (Section 3.1.2). Figure 4.29 presents a fragment of the emission spectrum of the molecule. The processed experimental results are shown in Figure 4.30a in the form of relationships between the observed intensities I ∗ /i (i = ν4J J g J S J J ) and the quantity m(m + 1) proportional to the energy of the rotational terms (m = J for the P branch and m = J +1 for the R branch). The results are presented on a semilogarithmic scale, the relationships are generally nonlinear, because the intensities I ∗ are not corrected for reabsorption. The latter is allowed for by formula (2.69) for the Doppler broadening. The quantity S∗ depends on the absorption coefﬁcient (Table 2.3), hence on the form of the vibrational–rotational distribution, and so reabsorption was taken into account by an iteration scheme. For the individual bands corresponding to transitions ending on excited vibrational levels, the relationships were almost linear. Their slopes were used to determine Tr as a ﬁrst approximation (the 022 1–022 0 transition band give Tr = 475 K), and the temperatures of the symmetric and deformation modes were taken to be equal to Tr , T1 = T2 = Tr . These temperatures were used to take account of reabsorption in all the bands, and the procedure was repeated successively. The convergence criterion was the linearity of distributions with one and the same slope, as shown in Figure 4.30b. Thereafter, the total populations of the vibrational levels were determined, as well as their correspondent vibrational temperatures. In the conditions of this experiment [78], T1 = T2 = Tr = 350 K, and T3 = 1320 K. The practical possibility of following such a procedure depends on the experimental conditions. Of speciﬁc importance is the homogeneity of the object under study along the observation direction, and important for the convergence of the iterations is the choice of the ﬁrst approximation. Such investigations are by their nature rather laborious. Generally, and as demonstrated by the above example, allowance for the effect of the ﬁnite optical density is more important for the group of lower vibrational levels. The form of the distribution in the region of higher levels can be recovered from the results of intensity measurements, provided that the actual dependence on J of the probability of optical transitions between vibrational states is taken into account. In that case, as already noted in the preceding section, it is important to

207

208

4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.30 Line intensities in the rotational structure of the IR bands of the N2 O molecules [77,78]: (a) experiment; (b) processing with due regard for reabsorption. Table 4.2 Einstein coefﬁcients Av+1,v , s−1 , for some molecules in the electronic ground states. v

CO

HF

DF

HCl

0

33.4

191.35

54.4

34.6

1

64.5

340.21

97.3

59.4

2

92.9

DCl 9.53 17.2

445.34

129

74.7

23

3

118

517.6

151

81.1

27

4

142

582

163

79.7

29.3

5

164.3

581.94

167

71.9

30.1

6

182

568.14

163

59.8

29.4

7

200.1

537.95

153

45.2

27.5

give due consideration to the effect of anharmonicity that limits the applicability of formula (4.80). Table 4.2 lists the Einstein coefﬁcients of single-quantum transitions for some diatomic molecules [80]. With such measurements, the vibrational level populations are only determined in relative units, because to relate them to the total particle concentration requires calculation of the vibrational statistical sum (see expression (1.6) and Appendix A) that the populations of the lower levels are contributing to. And the extrapolation of the populations to the

4.3 Line Intensities in the Vibrational Structure

region of lower levels calls for a certain care and necessitates additional investigations. To avoid these problems, use can be made of the approach wherein intensities are measured at multiquantum transition frequencies (harmonics). Insofar as the optical transition probabilities Av ,v at v − v > 1 for relatively low-lying levels are low in comparison with the probabilities Av,v−1 , reabsorption is considerably weaker. As v grows higher, the role of anharmonicity increases and the probabilities of two- and three-quantum transitions become comparable with the probability of single-quantum transitions. However, the tendency towards the vibrational level populations decreasing with increasing vibrational quantum number being typical, the problem of allowance for reabsorption in registering harmonic spectra is removed for the entire block of vibrational levels. And while this convenience in the region of small v numbers is attained at the cost of reduced intensities, as v increases, the probabilities of transitions with v − v > 1 become equal to, or even higher than, the single-quantum transition probabilities. Table 4.3 lists the Einstein coefﬁcients Av ,v of the main two- and three-quantum vibrational transitions in the CO(X1 Σ) molecule [80]. The values listed are calculated for the rotational quantum number J = 0. The accuracy of the absolute values is estimated by the authors of [80] (from comparison with the relative probabilities measured by other authors) at ca. 8% for v ≤ 15 and ca. 25% for v = 35 through 37. The slight differences between the values of Av+1,v for the CO molecule listed in Tables 4.1 and 4.2 and the data presented by various authors are within the limits of this accuracy. Figure 4.31 presents the spectrum of vibrational transitions of the CO molecule with v − v = 2 and 3 in the positive column of a cooled discharge [81]. As one can see, the intensities are comparable. Figure 4.32 presents the results of the investigations [82] by this method into the vibrational distributions of the CO2 molecules in an RF (27 MHz) discharge in a CO2 −He−O2 (1:4:x), x=0, 0.25, 0.45, mixture at a total pressure of 2.400 hPa (1.8 Torr). The distributions were recovered from the harmonic spectra of transitions with v − v = 2 observed in the experiment up to v ≤ 30. The form of the distributions is sensitive to the chemical composition of the plasma-forming gas, though their character in all the cases considered is in qualitative agreement with the results presented earlier (Figures 4.25, 4.27).

209

210

4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.31 Spectrum of vibrational transitions of the CO molecule with Δv = 2 and 3. Discharge in a CO−N2 −He (1:2:10) gas mixture in a liquid-nitrogen-cooled tube 25 mm in diameter at a pressure of 15 Torr and discharge current of 10 mA. The serial number of the upper level grows greater with increasing wavelength. Visible for Δv = 2 are the transitions 2≤ v ≤ 26.

Figure 4.32 Vibrational distributions of the CO molecules. RF (27 MHz) discharge in a tube 18 mm in diameter. Discharge length 70 mm. Power 4.4 W/cm3 . Gas mixture CO−He−O2 (1:4: x). 1, 2, and 3 – x = 0, 0.25, and 0.45, respectively.

4.3 Line Intensities in the Vibrational Structure Table 4.3 Einstein coefﬁcients Av ,v , s−1 for the CO(X1 Σ) molecule. v v − v = 1 v − v = 2 v − v = 3 v v − v = 1 v − v = 2 v − v = 3 0

35.79

1.033

0.014

19

291.192

144.749

25.336

1

68.84

3.065

0.056

20

288.418

154.288

28.918

2

99.225

6.04

0.145

21

284.63

163.567

32.748

3

127.035

9.932

0.297

22

279.91

172.535

36.814

4

152.351

14.656

0.527

23

274.335

181.135

41.1

5

175.231

20.187

0.859

24

267.982

189.319

45.589

6

195.802

26.455

1.307

25

260.928

197.047

50.261

7

214.107

33.394

1.894

26

253.243

204.275

55.092

8

230.263

40.947

2.637

27

245

210.965

60.056

9

244.349

49.045

3.556

28

236.267

217.085

65.127

10

256.46

57.621

4.666

29

227.11

222.602

70.274

11

266.679

66.601

5.984

30

217.594

227.491

75.468

12

275.099

75.924

7.526

31

207.781

231.726

80.678

13

281.812

85.513

9.302

32

197.733

235.287

85.871

14

286.904

95.297

11.326

33

187.508

238.155

91.015

15

290.469

105.207

13.604

34

177.162

240.317

96.076

16

292.592

115.172

16.142

35

166.749

241.759

101.023

17

293.363

125.129

18.944

36

156.322

242.47

18

292.868

135.009

22.01

37

145.931

4.3.4 Combinations of Emission and Absorption Techniques. Spectrum Inversion

In the preceding section we have described, using the work by Farrenq [78] as an example, an empirical technique to take account of reabsorption, based on the criteria of internal consistency and physical reasonableness of the results of processing the radiation intensities measured for a large number of transitions. This technique implicitly corresponds to the selection of a set of absorption coefﬁcients in the Ladenburg function S∗ in expression (2.50). However, as already noted, one can easily imagine a situation where the necessary corrections must be so great that achieving to the above criteria is either difﬁcult or even altogether impossible. An alternative approach may be one wherein the ﬁnite absorption in intensity measurements is taken directly into account by way of a special staging of the experiment. If, as is frequently the case, use is made of a technique of moderate spectral resolution and the emission of vibrational–rotational bands is

211

212

4 Intensities in Spectra and Plasma Energy Distribution

registered in the frequency interval Δν, one can introduce, following expression (2.66), an analog of Kirchhoff relation (1.24), (1.25) averaged over this interval: IΔν ≈ AI 0 (ν0 , T ), (4.83) where A = Δν (1 − e−χν l ) dν is the integral absorption, ν0 is the frequency at the center of the interval Δν and l is the length of the object under study. The quantity I 0 is the average intensity of black body radiation with spectrum (1.15) over the interval Δν, provided that the variations of the Planck function uν in this interval are small [81], the temperature in expression (1.15) being equal to the excitation temperature of the levels of the transition of interest, the vibrational temperature in our case. The quantity A can be calculated or measured. When measuring the temperature of the antisymmetric mode of the CO2 molecule in an expanding ﬂow of a preliminarily heated gas, the authors of [82] calculated the integral absorption in various models of bands and recommended some analytical approximations. Bakhir and co-workers [83, 84] experimentally determined the integral absorption from the growth curves (see Section 3.2.1). Despite the widespread application of these, and closely similar measurement techniques based on the use of expression (4.83), they all suffer from a common problem associated with the individual determination of the intensities of the emission and absorption of radiation. This complicates the procedure and leads to accumulation of errors. The authors of [85] describe the measurement of vibrational temperatures by the so-called spectral line inversion in the IR region of the spectrum that was used earlier to determine the temperatures of equilibrium ﬂames from their spectra in the visible region. The essence of this method is to simultaneously register (and in one and the same solid angle) the intensities of emission from the object being diagnosed and from the light source of continuous spectrum that is used to probe it. Spectrum inversion corresponds to the condition when the total intensity of the probe radiation source and the object together, is equal to the intensity of the source in the absence of the object. This condition means that the intrinsic emission of the object compensates for the reduction in the intensity of the probe radiation resulting from its absorption in the object. This condition can be established on the basis of Kirchhoff’s laws and the Planck formula [86]. It can easily be explained by the following microscopic considerations. Let there be two levels l and u bound by an optical transition and let their populations be Nl and Nu , respectively. The sum of the number of spontaneous transitions per unit time in a unit of volume and the number of stimulated transitions caused by the ﬁeld of the probe radiation

4.3 Line Intensities in the Vibrational Structure

with the black body spectrum uνT is Nu Aul + Nu Bul uνT . The number of absorption acts is Nu Bul uνT . If spectrum inversion takes place, then Nu Aul + Nu Bul uνT = Nu Bul uνT

(4.84)

or, considering also the condition that the solid angles are equal, uνT hνul ( Nu Bul − Nl Blu )

Ω Ω = Nu Aul hνul , 4π 4π

(4.85)

where Ω is the solid angle. Substituting uνT in form (1.16), we get expression (1.5), Nu /Nl = ( gu /gl ) exp { − hνul /kB T },

(4.86)

what is, inversion takes place when the relative population of the levels u and l corresponds to the excitation temperature equal to the temperature of the probe radiation source. The expression for uνT can also be used where the probe radiation source is other than a black body. The temperature T in this case is the luminance temperature [85]. The inversion condition depends neither on the transition probability nor on the line broadening mechanism. Therefore, if there are several pairs of levels whose relative populations are described by expression (4.86), the inversion conditions are the same for all. However, for the problem of measuring vibrational temperatures being discussed here, if the rotational structure of the vibrational bands is not resolved in the experiment, account should be taken of the fact that where the vibrational and rotational temperatures are different, the spectrum inversion conditions differ between different lines in the rotational structure. If, for example, the temperature of inversion on the P branch lines is TP , inversion on the R branch lines at a rotational (gas) temperature of T corresponds to the vibrational temperature [85, 87] TR =

hνJ,J −1 TP T . hνJ,J +1 T + 2Bv (2J + 1) TP

(4.87)

If measurements are taken on lines of the same branch and the temper ature of inversion is T12 on the J1,v − J2,v transitions and T34 on the J3,v − J4,v ones, the relation between them has the form 1 1 hν12 = T34 T12 hν34 (4.88) Bv J3 ( J3 + 1) − J1 ( J1 + 1) − J4 ( J4 + 1) + J2 ( J2 + 1) 1 . + hν34 T

213

214

4 Intensities in Spectra and Plasma Energy Distribution

When observations are carried out on several lines simultaneously, the effective inversion temperature Θ differs from the true vibrational temperature corresponding to inversion on the Q branch. The difference ΔΘ depends on the choice of the spectral region and can be taken into account. If ϕ(ν) is the transmission band of the spectral instrument used, the spectrum inversion condition has the form

∑

J ,J

ϕ(νJ ,J )

× NJ ( A J ,J +1 + A J ,J −1 + B J ,J +1 uνJ ,J +1 Θ + B J ,J −1 uνJ ,J −1 Θ ) =

∑

J ,J

ϕ(νJ ,J )( NJ +1 B J +1,J uνJ +1,J + NJ −1 B J −1,J uνJ ,J −1 Θ ) (4.89)

Mikaberidze [87] calculated the corrections ΔΘ as applied to measurements for the CO and CO2 molecules. With the spectral limitation used (interference ﬁlters with a half-width of ca. 0.07 μm in the range 4.2–4.7 μm and dispersion ﬁlters with a half-width of ca. 2 μm in the region of 15 μm), the value of ΔΘ/Θ is no more than 2–3%. Another correction to ΔΘ/Θ is associated with anharmonicity, when, as follows from expression (4.68), the temperatures differ between different pairs of levels: Tv+1 = ( E1 − 2ΔEv)

E1 2ΔE v − T1 T

−1 .

(4.90)

As demonstrated in [87], this correction to Θ for ﬁnding T1 under the conditions of this experiment, is also not very great: (Θ - T1 )/T1 ≤ 3–4%. Although the temperatures Tv at v > 1 can differ perceptibly from T1 , the absolute populations of high-lying vibrational levels are small in comparison with the population of the level v = 1 and their contributions to the measurement results diminish rapidly if the center of the transmission band φ(ν) is close to the center frequency of the v = 1 → v = 0 transition. Their smallness notwithstanding, these contributions can be taken into account. Figure 4.33 presents a schematic diagram of the experiments [85, 87] on the inversion of molecular IR bands. Probe light source 1 (a cavity in a resistance-heated graphite rod) is projected by mirror M1 into discharge tube 2. Mirror SM2 projects the images of the probe source and discharge onto the window of photoreceiver 3. The probe-cum-discharge radiation is chopped by chopper disk 5 driven by synchronous motor 6. The disk has two windows covered by interference (dispersion) ﬁlters 7. One of the ﬁlters is tuned to the frequency of the molecular vibrational transition of interest and the transmission band of the other ﬁlter lies beyond the

4.3 Line Intensities in the Vibrational Structure

Figure 4.33 Schematic diagram of spectrum inversion experiments. For designations, see text.

(emission) absorption bands of the discharge. The light ﬂows alternately, passing through the different ﬁlters to reach the photoreceiver whose signals are fed to ampliﬁer 8 and thence to bridge circuit 9 which is provided with a double-channel synchronous detector. Timing is effected by the signals generated by photoresistor 11 illuminated by lamp 12. Circuit 9 operates in such a way that the signals from radiation passing through the different ﬁlters are fed to the different arms of the bridge. The amount of unbalance in the bridge circuit is registered by circuit 10. Mirrors M1 and M2 are concave, long-focus, metallic-surfaced type. Mirrors are preferable to lenses as they are free from chromatic aberrations. For condition (4.85) to be satisﬁed, it is necessary that the solid angle Ωd of the light beam from the discharge is not in excess of the angle Ωs of the beam from the probe light source, otherwise the inversion temperatures will be overread. This is attained by limiting the size of mirror M2 . Figure 4.34 presents the results of measuring the vibrational temperatures T3 and T2 of the CO2 molecule in conditions of the active medium of a CO2 laser. Following this example, we can compare the results of measurements taken by various authors and by various techniques. We see the same trends as in the results presented above. The temperature T3 substantially exceeds T2 , whereas the latter practically coincides with the gas temperature. The spectrum inversion method is fairly simple and reliable. The temperatures measured under the stationary conditions of [85, 87] are limited (≤ 2500 K) by the maximum heating temperature of source 1. To measure higher temperatures, use can be made, for example, of powerful high-brightness capillary discharges in dense gases. But this requires a thorough calibration of the luminance temperatures in the IR region of the spectrum [88].

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4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.34 Vibrational temperatures of the CO2 molecule in a CO2 –N2 -He (2:1:8) discharge. Pressure 5.333 hPa (4 Torr). Discharge tube diameter 20 mm. 1 through 3 – measurements by the spectrum inversion technique: 1 – T3 , 2 – T2 , 3 – T2 in the mixture with 0.667 hPa (0.5 Torr) H2 O added. 4 – gas temperature (calculation; 5 – rotational temperature measured from the 2+ system of nitrogen; 6 – gas temperature measured form the ampliﬁcation of the laser signal.

4.3.5 Raman Scattering

The possibilities of studying vibrational distributions by the Raman scattering technique are evident from formulas (3.86), (3.90). The scattering cross sections being small, use should preferably be made of probe radiation sources in the form of high-power lasers. To facilitate registration, use is usually made of visible lasers, because high-sensitivity photomultipliers are available that at frequencies characteristic of molecular vibrations are suitable for both the Stokes and anti-Stokes components. To study stationary objects, use can be made of CW lasers in combination with weak signal accumulation techniques. Pulsed lasers are suitable for both stationary and pulsed objects, and pulse-periodic lasers used in conjunction with synchronous signal detection also prove effective. To improve sensitivity in the spontaneous Raman scattering (SRS) scheme, the laser beam is made to repeatedly pass through the object of interest by means of mirrors. Where the scattering signal accumulation time is long enough, the SRS scheme can be successfully used to recover vibra-

4.3 Line Intensities in the Vibrational Structure Table 4.4 Values of the factors f (v), s a (v), sb (v). v

0

1

3

7

10

13

f (v)

1

1.01

1.04

1.09

1.14

1.18

sa (v)

1.05

1.09

1.19

1.38

1.52

1.66

sb (v)

1.28

1.45

1.81

2.51

2.94

14

16 1.2

1.71

tional distributions. For example, by accumulating 2500 counts one can establish the distribution of nitrogen molecules in a pulsed discharge at an N2 pressure of 359.964 hPa (270 Torr) and observe levels with v ≤ 19. The CARS technique enjoys a wider and more efﬁcient application. The technique and its advantages over the SRS method have been described in Section 3.5.4. Here we will only note its more speciﬁc features as applied to the problem in hand. In accordance with expressions (3.96), (3.101), the intensity of a vibrational spectrum is proportional to (ΔNv )2 , where ΔNv is the difference in population between the levels bound by the Raman transition: ΔNv =

∑ ( Nv,J − Nv+1,J ).

(4.91)

J

One should pay particular attention to the dependence of a number of quantities in expression (3.101) on the vibrational quantum number v [89]. • Scattering cross section dσv dσ ∼ 0 ( v + 1) f ( v ), dΩ dΩ

(4.92)

where the factor f (v) allows for anharmonicity. The values of this factor calculated at the Morse potential for the N2 (X1 Σ) molecule are listed in the second row of Table 4.4. • For large numbers v, account should be taken of the effect of the vibrational transition frequency variation on the line width Γ. This dependence is weak for collision broadening, and the Doppler broadening is proportional to frequency (2.33). If the spectral width of the laser radiation exceeds the Doppler width, the correction factor in expression (3.101) will be ωv /ω0 . • For transitions between levels differing in the number v, the level population perturbation (saturation) effects can manifest themselves in varying degrees. This fact should be primarily taken into

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4 Intensities in Spectra and Plasma Energy Distribution

consideration in the case of low-pressure gases, where the collision processes responsible for the formation of distributions are insufﬁciently fast for one to completely disregard radiative transitions on the scattering of high-power radiation. The procedure suggested in [89] to take account of saturation, and implemented using the nitrogen molecule as an example, bases on the knowledge in that the effect of saturation is determined experimentally for several lower levels with v ≤ 3–8 at various laser powers. This effect is then calculated for the same levels, the necessary ﬁtting parameters are selected and the calculation is extended to all the levels observed in the experiment. The third row in Table 4.4 lists the values of the correction factor s a (v) to formula (3.96) for the conditions of the experiment [89], namely, a discharge in N2 at a pressure of 2.666 hPa (2 Torr), single-mode laser powers W1 = 80 mJ and W2 = 3 mJ, discharge diameter 20 mm, laser beam diameters 6 mm, lens focal length 710 mm and laser pulse duration 12 ns. The fourth row of Table 4.4 lists the values of this factor in the case of sharper focusing with a lens 500 mm in focal length. • The absolute values of Nv are found by summing expression (4.91) with respect to ΔNv . However great the number of the transitions v → v − 1 observed in scattering, one has either to disregard the population of the uppermost level or to ﬁnd it by way of extrapolation: Nv =

vmax

∑

v =v

ΔNv + Nvmax +1 .

(4.93)

As a result of such an extrapolation, the errors in measuring the populations of the lower levels augment the errors in determining the comparable populations of the levels next to vmax . In the experimental conditions indicated above, a 5% error in determining the ratio Nv=1 /Nv=0 gives rise to an error with a factor of ca. 2 in determining the populations of the high-lying levels with v = 12–14 from their CARS spectra. The case described here exempliﬁes high-sensitivity CARS measurements where vibrational level populations are determined at a level of 1014 cm−3 . At a rotational temperature of T =540±30 K, determined in the same experiment with resolved rotational structure, this corresponds to the rotational level populations Nv=14,J =10 = 6 × 1012 cm−3 , Nv=14,J =2 = 3 × 1012 cm−3 . A CARS measurement sensitivity of Nv=2,J =2 = 3 × 1011 cm−3 was reported in [90] for hydrogen molecules having 3.4 times

4.3 Line Intensities in the Vibrational Structure

Figure 4.35 Radial excitation proﬁle of the ﬁrst vibrational level of nitrogen in a waveguide CO2 laser. RF excitation power 75 W. 1 – without lasing; 2 – with lasing.

as large a scattering cross section and approximately 30 times greater rotational constant. The requirements on the accuracy and sensitivity of CARS measurements are less stringent where it is necessary to determine the populations of a few low-lying vibrational levels and/or T1 or similar quantities. Figure 4.35 presents some examples of measurements of the ratio between the population Nv=1 of the N2 molecule and the total density N of these molecules [91]. The vibrational temperature T1 = 1600 K corresponds to Nv=1 /N = 0.107, and T1 = 2000 K to Nv=1 /N = 0.152. The measurements were taken with a spatial resolution of 0.07 mm between the walls of the active element of an RF excited waveguide CO2 laser. The two proﬁles in the ﬁgure correspond to the cases where lasing is present and absent. In the former case, the density of molecules on the upper laser level CO2 (000 1) decreases, which affects the vibrational energy store of the nitrogen molecules that are in close vibrational exchange relations with CO2 .

219

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4 Intensities in Spectra and Plasma Energy Distribution

4.3.6 Determination of the Vibrational Temperatures of Molecules in the Electronic Ground States from Electronic Transition Spectra

The methods involved here are indirect. Their development being associated, in the main, with the attempts at transferring emission measurements from the IR into the visible or the UV region of the spectrum, wherein the spectral measurement techniques are simpler as regards the necessary instruments, detection methods and allowance for reabsorption. One such technique is the determination of the temperature TvX of the electronic ground state X from the relative intensities of the vibrational bands associated with the C–B transitions between the excited electronic states C and B [92–95]. This assumes that the state C is excited from the state X by direct electron impact and decays in a radiative fashion. In that case, the stationary population of the vibrational bands of the state C is NvC =

ne N0 AvC

∑ ve σvX vC NvX ,

(4.94)

vX

where N0 is the population of the vibrational level vX = 0, ve and ne are the electron velocity and density, respectively, σvX vC is the excitation cross section of the vX − vC transition and AvC is the radiative decay rate of the level vC . It has been demonstrated, both theoretically [96] and experimentally [97, 98], that the electron impact excitation of molecules occurs in accordance with the Franck–Condon principle: σvX vC ∼ qvX vC . If the channel C–B is the main radiative channel, then AvC =

∑ AvC vB ∼ ∑ |Re |2 qvC vB , vB

(4.95)

vB

where |Re | and qvC vB are the dipole moment matrix element of the electronic transition and the Franck–Condon factor, respectively. The excitation rate constant ve σ proves proportional not only to qvC vB , but also to the factor allowing for the value of the electron velocity distribution function f (ε) at the threshold excitation energy ε = ε∗ [95, 99], and so it follows from expression (4.94) that NvC = const ∑ qvX vC vX

ε∗vX vC f ε∗vX vC NvX

(

) −1

∑ |Re | vB

2

qvC vB

,

(4.96)

where ε∗vX vC is the threshold excitation energy and f (ε) is the electron velocity distribution function. Generally speaking, NvC is not a Boltzmann

4.3 Line Intensities in the Vibrational Structure

Figure 4.36 Radial distributions of the vibrational temperatures Tv,X of nitrogen in a discharge in a 23-mm-dia. tube at a discharge current of 30 mA, measured from the spectrum of the 2+ system: 1 – N2 , 2.666 hPa (2 Torr); 2 – CO2 −N2 −He (1:3:6) gas mixture, 9.199 hPa (6.9 Torr).

distribution, but to avoid difﬁculties in solving the inverse problem of recovering NvX from NvC , it is parametrized by introducing the effective temperature TvC (for example, by reducing it to the Boltzmann form by the least-squares method). In that case, the relation between TvC and TvX can easily be calculated. Such calculations, as applied to measurements from the 2+ system of nitrogen, were made in [38, 92, 95, 99]. Figure 4.36 presents some examples of the determination of the temperature TvX of the electronic ground state of N2 (X = X1 Σ, C = C3 Π, B = B3 Π). The dependence of the measurement results on the vibrational temperature of the ground state on f (ε) presents a certain difﬁculty. In some particular cases it can be avoided. For example, the authors of [100, 101] have noted that the matrix of the qvX vC elements for the CO molecule (X = X1 Σ, C = B1 Σ, B = A1 Π) is practically diagonal, that is, ∑ qv vC =vX qvX =vC . X vX ,vC Since the values of the vibrational quanta and anharmonicity of the molecule in the X1 Σ and the B1 Σ state are similar (Table D.5 of Appendix D), the difference between the threshold excitation energies ε∗vX vB can be neglected. In that case, from formulas (4.94) and (4.96) follows the duplication of the vibrational distributions, NvX ∼ NvC , and the equality of the vibrational temperatures, TvX = TvC . Considering the shortness of

221

222

4 Intensities in Spectra and Plasma Energy Distribution

Figure 4.37 Temperatures in a discharge in a CO−N2 −He−Xe−O2 (1:4:15:0.5:0.03) gas mixture: squares – gas temperature (diode laser spectroscopy (DLS)); vibrational temperatures in the electronic ground state, Tv,X : triangles – nitrogen; circles and crosses – carbon monoxide temperatures ˚ ¨ bands, measured by the DLS technique and from the Angstr om respectively.

˚ ¨ bands the lifetime of the B1 Σ state (Table D.5), the B1 Σ–A1 Π Angstr om prove a convenient ‘thermometer’ for determining the vibrational temperature Tv of the electronic ground state of the CO molecule. Figure 4.37 presents the results of 100 vibrational temperature measurements taken in a discharge in a gas mixture by various methods: Tv,CO(X) (diode ˚ ¨ bands), Tv,N2 (X) (CARS). Also shown in the laser spectroscopy, Angstr om ﬁgure are the values of the gas temperature T (diode laser absorption spectroscopy in CO(X)). Independent measurements for CO give wellagreeing results. The relation between Tv,CO(X) and Tv,N2 (X) is found by formula (4.72) at p = q = 1. The applicability of the method is generally limited by the density of heavy particles in plasma. It can be used until collisions of the molecules in electronically excited states within the radiative lifetime result in the redistribution of the vibrational level populations NvC . According to the data of [100], these conditions for the N2 (C3 Π) and CO(A1 Π) molecules are satisﬁed at gas pressures p < 133.320 hPa(100 Torr), though it follows from the data of Figure 4.37 that the method also works at pressures up to 399.960 hPa (300 Torr).

4.3 Line Intensities in the Vibrational Structure

Let us also brieﬂy mention other indirect methods based on the observation of line intensities in electronic transition spectra (see also the references cited in [38, 79, 102]). To determine the temperature Tv of molecules in the ground state, use can be made of the relative line intensities in the vibrational structure of the spectrum of the excited molecular ion formed on collision between the molecule and a metastable atom. The excitation of the short-lived ion occurs in accordance with the Franck–Condon principle, and the further reasoning is basically similar to that expounded above in the section on electron impact excitation. The reaction M( X, vX ) + He(23 S1 ) → M+ (C, vC ) + He + e,

(4.97)

where M = N2 (X1 Σ), CO(X1 Σ), can serve as an example. Some experimental techniques are based on the fact that the resonance levels in alkali metal atoms have relatively low (a few electron-volts) potentials and become populated on collisions with vibrationally excited molecules. This fact was already used in earlier works (see [103]) to determine the temperature Tv of nitrogen molecules in shock waves and gas ﬂows from the excitation temperature lines of sodium atoms trace (‘tinting’). The latter temperature was found by inversion of the resonance spectral lines of the atoms in the visible region of the spectrum. The authors of a number of works (see [79]) describe the determination of the vibrational temperature Tv from the Doppler broadening of the lines of alkali admixtures. This possibility is substantiated by the fact that collisions between alkali atoms and a number of molecular species (including N2 and CO2 ) give rise to intermediate ionic complexes with strong internal interactions. When these complexes decay, the impurity atoms acquire kinetic energy equal to the vibrational energy of their collision partners. The range of conditions satisfying these prerequisites proves, however, to be sufﬁciently narrow: hνv < TvX < hνv ( M1 /M2 )2 , X

X

M1 M2 ,

(4.98)

where hνv is the vibrational quantum of the molecule in the ground X state and M1 and M2 are the masses of the molecule and the alkali atom, respectively. The techniques using small admixtures – indicators – belong in the group of the so-called actinometric methods (for more detail, see Section 5.4). When applying them, one should consider some circumstances in addition to the ones indicated above. If admixtures are introduced artiﬁcially (not as, for example, in magnetohydrodynamic generators), the properties of plasma may be distorted. Since the electronically ex-

223

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4 Intensities in Spectra and Plasma Energy Distribution

cited states of atoms are in resonance with high-lying vibrational levels of molecules, the results of the atomic spectral line intensity measurements can be correlated with the vibrational energy of the electronic ground state of the molecules only if the harmonic oscillator approximation holds true in the region of these levels. And this is true when the vibrational and gas temperatures of the molecules, according to expressions (4.68), (4.90), differ but little.

4.4 Distribution of Particles Among Electronic Levels

The populations of electronically excited states can be quite successfully determined by traditional emission spectroscopy methods in the visible and UV regions of the spectrum (Section 3.1). The relationship between the line intensities in the resolved line spectrum and the populations is given by simple relations (2.48)–(2.50). The populations of metastable levels can also be determined by the classical or laser absorption methods (Sections 3.2 through 3.4 and Chapter 5 below). Without discussing further this aspect of the problem, we note only that the information obtained in this way can be directly used to ﬁnd the radiative loss in the energy balance of plasma. Since the probabilities of optical transitions between electronic levels are, as a rule, high, this contribution may be appreciable, even when the stationary populations of these levels are small. And while considering spectroscopy as a diagnostic method of determining plasma parameters, it should be stated that the establishment of the relationships between the populations of electronic levels and the main plasma characteristics (electron concentration and energy, heating of the neutral component) in the absence of complete or local equilibrium is a fairly difﬁcult problem of plasma theory. Analytically these relationships can only be established within narrow ranges of plasma conditions, notwithstanding the fact that a great many works have been devoted to this problem (see, for example, the references cited in [70, 79, 104–109]). In contrast to the rotational and vibrational ensembles of molecules in long-lived electronic states, it is difﬁcult to describe any perceptible number of electronic levels in terms of a small number of parameters. The notions of the temperatures Te,d , Te,exc introduced by analogy for limited groups of electronic levels (Section 1.3.2) are formal, though useful at the same time, for they reduce the number of the total distribution parameters. To physically interpret such partial parameters requires numerical modeling on the basis of the population balance of individual levels (it is exactly in this way that the curves of Figure 1.4 have been computed).

4.4 Distribution of Particles Among Electronic Levels

Figure 4.38 Distributions of atoms among energy levels: (a) qualitative picture: 1 – equilibrium plasma; 2 and 3 – ionization and recombination conditions, respectively; (b) experiment [109], cesium vapor discharge: 1 – cesium atom density Na = 1.1 × 1013 cm−3 , electron concentration

ne = 4 × 1013 cm−3 , electron temperature Te = 2250 K; 2 – Na = 1.2 × 1014 cm−3 , ne = 6.5 × 1012 cm−3 , Te = 3850 K. The slope of the dashed lines determines Te . Solid line – calculation by the MDA model [69].

The most important exclusion are the groups of levels near the ionization limit. The latter statement can be explained with reference to Figure 4.38 [70, 79]. The ﬁgure presents a qualitative picture for three kinds of distribution. Straight line 1 corresponds to equilibrium. Its slope determines the uniﬁed plasma temperature. In this case, energy is calculated from the ionization limit, and the energies of the bound levels are negative. The serial numbers k of the levels are not related here with concrete quantum numbers. The level populations are given by Boltzmann formula (1.6): Nk,0 = Na,0 ( gk /Qa ) exp { − ( E1 − Ek )/kB T }.

(4.99)

Here Nk,0 and Na,0 are the equilibrium level population and atomic concentration, respectively, gk is the statistical weight of the level and Qa is the internal statistical sum Qin in formula (1.6) for the atom. The line segment in the region of positive energies describes the distribution of free electrons, f e ( E) = ln(ne,0 ( E)/ge ( E)). Here ne,0 is the equilibrium concentration of the electrons and ge ( E), their statistical weight (Appendix A). If Ni,0 is the equilibrium ion concentration, then, by analogy with expression (1.13), the electron concentration is uniquely related to the

225

226

4 Intensities in Spectra and Plasma Energy Distribution

population of any level by the equilibrium constant Kk : ne,0 Ni,0 Ek gk h3 , = Kk = exp Nk,0 kB T 2Qi (2πkB mT )3/2

(4.100)

where Qi is the internal statistical sum of the ion. Assume that the concentration of electrons in plasma has decreased, while their temperature Te remained the same. In that case, the section of line 1 in the region of positive energies in Figure 4.38a will be translated downward. This will also distort the distribution in the region of bound states, and relation (4.100), even though it remains, will only hold for the group of the levels k immediately adjacent to the continuum. Such a deviation from equilibrium is referred to as ionization nonequilibrium – distribution 2. A similar situation will remain in the opposite case, that is, if at the same temperature Te the concentration ne,0 > ne (recombination nonequilibrium), which corresponds to distribution 3. It is these considerations that lie at the root of the determination of the electron temperature from the relative populations in the group of highly excited atomic levels. If spectroscopic measurements give level populations in the absolute measure, the electron concentration can also be determined, when using the effective level excitation temperature (see Section 1.3.2) in the exponential factor of expressions (4.99) and (4.100). For this, one should also know the concentrations of atoms and ions and ﬁnd their statistical sums. Obviously this requires taking detailed, level-by-level measurements or using a theoretical model. As already stated, the latter is possible subject to a great number of assumptions. The most thoroughly developed model is the so-called modiﬁed diffusion approximation (MDA). In this approximation, electronic transitions in the system of excited levels are likened to random wanderings in the form of energy space diffusion. The discreteness of the levels is allowed for additionally, which leads in the ﬁnal analysis to equations similar to the well-known Fokker–Planck equations in ﬁnite difference formulation. The authors of [70] that describes this theory in detail contributed much to its development. The results of the theory can be obtained in analytical form in cases where the level-to-level transition kinetics is governed by electron-atom collisions. We do not present here the rather cumbersome MDA formulas for highly excited levels, for they can be found, along with their application examples, in a number of books, including [70, 79]. The inclusion of radiative decay necessitates combining analytical and numerical calculation results. The requirement that the effect of radiative processes should be small limits the amount of levels under consideration, because radiation intensity decreases with increasing level number as k−3 , while the cross sections of collisional processes

4.4 Distribution of Particles Among Electronic Levels

enlarge as k4 . As a result, the energy interval of the bound states can be divided into the regions E < ER , where the collisional processes dominate, and E > ER , where levels are excited on collisions and depopulated through emission. The boundary between these regions is indicated by the arrow in Figure 4.38a, and to evaluate it, the authors of [70] have suggested using the formula ER = [ne /(4.5 × 1013 )] /4 (kB Te )− /3 , 1

1

(4.101)

where kB Te is expressed in electron-volts and ne , in cm−3 . In the above-indicated applicability region (populations are governed by electron-atom collisions and radiative decay), the MDA theory agrees well with experiment and supports the above qualitative discussion. Figure 4.38b presents the results of measurements [110] of the distribution of Cs atoms in a low-density cesium vapor discharge for two combinations of Na , ne , and Te , as well as those of calculation by the MDA theory for one of them. The MDA calculation not only describes the distribution quite satisfactorily, but also gives such values of ne and Te as coincide with those measured by both spectroscopic and probe techniques. In addition to those indicated above, one more limitation of the theoretical models describing the distribution of particles among electronic levels is the ‘roughening’ of the actual quantum structure of the levels k. Its consideration requires inclusion of information about the probabilities of elementary processes as regards levels differing not only in energy, but also in quantum numbers (principal and orbital spin), and in coupling scheme. This is very difﬁcult to achieve for the majority of atoms, and is especially problematic for the electronic states of molecules. Therefore, the data on the distribution of such particles among electronic states are usually borrowed from experiment and tied to concrete plasma conditions. The temperatures Ted introduced empirically (if possible) for the total distributions among electronic levels differ, as a rule, from the other partial temperatures of PLTE plasma (Section 1.3.2). Figure 4.39 presents an example of the distribution of nitrogen molecules in a group of a few electronic states in a glow discharge [104]. Shown in the same ﬁgure are lines corresponding to Boltzmann distributions at the electron and neutral gas temperatures. The experimental populations of the excited states differ by many orders of magnitude from their counterparts determined from the PLTE distributions with partial electron and gas temperatures. The interpretation and use of the experimental results should be based on kinetics equations, with an as thorough as possible account of the elementary processes. Examples of such investigations can be found in the book by Solovetsky [104].

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Figure 4.39 Distributions of nitrogen molecules among electronic states [103]. A glow discharge in nitrogen at a pressure of 0.933 hPa (0.7 Torr) and current density of 10 mA/cm2 . The slopes of the dashed lines correspond to the electron temperature Te and the gas temperature T . Ted – effective temperature of the experimental distribution.

References

1 T.T. Karasheva, D.K. Otorbaev, V.N. Ochkin et al. Doppler Broadening of Spectral Lines and Velocity Distributions of Excited Atoms and Molecules in Nonequilibrium plasma. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1985). 2 G.N. Polyakova and A.I. Ranyuk. Recovering Velocity Distributions of Excited Particles from the Doppler Broadening of Spectral Lines (in Russian). Preprint KhFTI No. 81–1. Kharkov (1981). 3 V.I. Malyshev. An Introduction to Experimental Spectroscopy (in Russian). Moscow: Nauka (1979). 4 H. Lange and R. Bussiahn. Tunable Diode Laser Absorption Spectroscopy. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma

5

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at Elevated Pressures (in Russian). Proc. SPIE, 4460, pp. 177–187 (2002). S.N. Andreev, V.N. Ochkin, and S. Yu. Savinov. Effect of Temperature on the Collisional Broadening of the Spectral Lines of the CO2 Molecule (in Russian). Kvant. Elektr., 32, No 7, pp. 647–653 (2002). S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). M.A. El’yashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). H.R. Grim. Plasma Spectroscopy. N.Y.: McGraw-Hill (1964); Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). Sh. Chen and M. Takeo. Broadening and Shift of Spectral Lines Due to Extraneous Gases (in Russian). UFN, 66,

References No. 3, pp. 392–474 (1958). 10 V.M. Lelevkin, D.K. Otorbaev, and D.C. Schram. Physics of NonEquilibrium Plasmas. North-Holland (1992). 11 E.J. Wipple, Jr. Theory of Reaction Product Velocity Distribution. J. Chem. Phys., 60, pp. 1345–1351 (1974). 12 J. Deschampes, A. Ricard. Transfert d’excitation dans les m´elanges argonazote. J. Phys., 34, pp. 123–128 (1973). 13 B.F. Gordiets, A.I. Osipov, and L.A. Shelepin. Kinetic Processes in Gases and Molecular Lasers (in Russian). Moscow: Nauka (1980). 14 M.A. El’yashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 15 E. McDaniel. Collision Phenomena in Ionized Gases. New York (1964). 16 C. Barbeau and J. Jolly. J. Phys., D23, p. 1168 (1990). 17 V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), I-IV. Moscow: Nauka (2000). 18 L.D. Landau and E.M. Lifshits. Quantum Mechanics. Nonrelativistic Theory (in Russian). Moscow: Nauka (1974). 19 L.A. Sena. Collisions of Electrons and Ions with Gas Atoms (in Russian). Moscow-Leningrad: Gostekhizdat (1948). 20 V.M. Zakharov and Yu. M. Kagan. On the Motion of ions and Atoms in Plasma. In: S.E. Frish, Ed. Spectroscopy of Gas-Discharge Plasma (in Russian), pp. 291–318. Leningrad: Nauka (1970). 21 D.I. Solovetsky. Temperature of Heavy Particles in Electric-Discharge Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), III, pp. 304–305. Moscow: Nauka (2000). 22 A.D. Sakharov. Excitation Temperature in Gas-Discharge Plasma (in Russian). Izv. AN SSSR, Ser. Fiz., 12, pp. 372–375 (1948). 23 E. Bauer. Methods for Calculating Inelastic Collision Cross-Sections at Low Energies. In: A. Khokhshtim, Ed. Kinetic Processes in Gases and Plasma (in Russian), pp. 327–363. Moscow: Atomizdat (1971).

24 Ya. B. Zeldovich and Yu. P. Raizer. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena (in Russian). Moscow: Nauka (1966). 25 E.V. Stupochenko, S.A. Losev, and A.I. Osipov. Relaxation Processes in Shock Waves (in Russian). Moscow: Nauka (1965). 26 K.A. Vereschagin, V.V. Smirnov, A.V. Bodronosov et al. Local Non-Intrusive Diagnostics of Electron Components of Plasma Glow Discharge in Nitrogen by CARS Spectroscopy. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 111– 121 (2002). 27 V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Determination of the Level Populations in the OH radical by Line Absorption and Emission Techniques with Due Regard for the Unresolved Doublet Structure (in Russian). ZhPS, 26, pp. 900–905 (1977). 28 P.A. Skovorodko and R.G. Sharafutdinov. Kinetics of Rotational Level Populations in a Free Nitrogen Jet (in Russian). PMTF. No. 5, pp. 40–49 (1981). 29 V.N. Ochkin, S. Yu. Savinov, N.N. Sobolev, and S.N. Tskhai. Local Distributions of the H2 (X1 Σ) Molecules among Rotational Levels in a Glow Discharge (in Russian). ZhTF, 58, No. 7, pp. 1283–1290 (1988). 30 S.N. Tskhai. Excitation of the Rotational Structure of Spectra of Diatomic Molecules in Nonequilibrium Gas-Discharge Plasma (in Russian). Ph.D. Thesis FIAN. Moscow: FIAN (1989). 31 B.P. Lavrov, V.N. Ostrovsky, and V.I. Ustimov. On the Formation Mechanism of Nonequilibrium Rotational Level Population of Molecules in Plasma (in Russian). ZhTF, 50, pp. 2072–2088 (1988). 32 N.F. Lane. The Theory of ElectronMolecule Collisions. Rev. Mod. Phys., 52, pp. 29–119 (1980). 33 B.M. Smirnov. Atomic Collisions and Elementary Processes in Plasma (in Russian). Moscow: Atomizdat (1968).

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4 Intensities in Spectra and Plasma Energy Distribution 34 E. McDaniel. Collision Phenomena in Ionized Gases. New York (1964). 35 A.V. Eletsky and B.M. Smirnov. Elementary Processes In Plasma. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), I, pp. 190–266. Moscow: Nauka (2000). 36 V.V. Skubenich, M.M. Povch, and I.P. Zapesochnyi. Excitation of diatomic Molecules on Collisions with Monoenergetic Electrons (in Russian). Khimiya Vysokikh Energii, 11, pp. 116–123 (1977). 37 G. Herzberg. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules. 2nd Ed. N.Y.: D. van Nostrand (1951). 38 V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Mechanisms Responsible for the Formation of Distributions of Electronically Excited Molecules among Vibrational–Rotational Levels in Gas Discharges. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian), pp. 6–85. Moscow: Nauka (1985). 39 E.P. Muntz. Static Temperature Measurements in a Flowing Gas. Phys. Fluids, 5, pp. 80–90 (1962). 40 H. Ashkenas. Rotational Temperature Measurements in Electron-Beam Excited Nitrogen. Phys. Fluids, 10, pp. 2509–2516 (1967). 41 G. Culp and A.T. Stair. Effective Ro˚ tational Temperature of N2+ (3914 A) Excited by Monoenergetic Electrons in a Crossed Beam. J. Chem. Phys., 64, pp. 57–62 (1967). 42 J. Allison, T. Kondon, and R.N. Zare. Laser Induced Fluorescence Measurement of the Nascent Rotational Distribution of N2 (X2 Σ) Formed by Electron Impact on N2 . Chem. Phys. Lett., 64, pp. 202–204 (1979). 43 P. Baltayan and O. Nedelec. Polarization of Molecular Lines by Electron Impact. United Atom and Born Approximations. Atom. and Molec. Phys., 4, pp. 1332–1338 (1971). 44 P.L. Rubin. Rotational Level Population Probabilities in the Electronic Excitation of Diatomic Molecules (in Russian). ZhETF, 65, pp. 1375–1381 (1973).

45 B.P. Lavrov, V.N. Ostrovsky, and V.I. Ustimov. On the Determination of Cross-Sections and Rate Constants for the Electron Impact excitation of Molecules (in Russian). Pis’ma v ZhTF, 5, pp. 366–358 (1979). 46 B.P. Lavrov, V.N. Ostrovsky, and V.I. Ustimov. Rotational Transitions in the Electron Impact Excitation of Electronic States in Molecules (in Russian). ZhETF, 76, pp. 1521–1529 (1979). 47 D.K. Otorbaev, V.N. Ochkin, P.L. Rubin et al. Electron Impact Excitation of Rotational Levels of Molecules in a Gas Discharge. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian), pp. 6–85. Moscow: Nauka (1985). 48 D.M. Chase. Adiabatic Approximation for Scattering Processes. Phys. Rev., 104, pp. 838–842 (1956). 49 Yu. D. Oksyuk. Electron Impact Excitation of Rotational Levels of Diatomic Molecules in the Adiabatic Approximation (in Russian). ZhETF, 49, pp. 1261–1273 (1965). 50 D.K. Otorbaev, V.N. Ochkin, N.N. Sobolev et al. On the Transfer of Substantial Moments of Momenta in the Electronic Excitation of Molecules (in Russian). Pis’ma v ZhETF, 28, pp. 424–429 (1978). 51 A.P. Bryukhovetsky, E.N. Kotlikov, D.K. Otorbaev et al. Electron impact Excitation of Electronic–Vibrational– Rotational Levels of Hydrogen Molecules in Nonequilibrium GasDischarge Plasma (in Russian). ZhETF, 79, pp. 1687–1703 (1980). 52 M.R. Gorbal and M.I. Savadatti. Rotational Structure of CN Emission from Active Nitrogen Flames. JQSRT, 26, No. 5, pp. 457–462 (1981); N.N. Math and M.I. Savadatti. Rotational Distribution in Simultaneously Excited Electronic States of NH. JQSRT, 35, No. 6, pp. 501–502 (1986). 53 O. Oldenberg. On Abnormal Rotation of Molecules. Phys. Rev., 46, pp. 210–215 (1934). 54 H.P. Broida and W.R. Kane. Rotational Intensity Distribution of OH and OD in Electrodeless Discharge through

References

55

56

57

58

59

60 61

62

63

64

65

Water Vapor. Phys. Rev., 89, pp. 1053– 1059 (1953). J.P. Pichamuthy, J.S. Hassler, and H.D. Coleman. Gas-Temperature Measurement in Pulsed H2 O Laser Discharges. J. Appl. Phys., 43, pp. 4562–4565 (1972). T. Horie, T. Nagura, and M. Otsuka. Radiative Collisions between Electronic and Molecular Beams. J. Phys. Soc. Jap., 11, pp. 1157–1170 (1956). H. Meinel and L. Krauss. Uber die Besetzung der Rotationszustande von OH und C2 in Niederdruckplasmen. JQSRT, 9, pp. 443–460 (1969). J.Ferdinand. The Comparison of the Spectroscopic and Probe Electron Temperature Measurements. Phys. Lett., A31, pp. 370–371 (1970). H.P. Broida. Rotation Temperature of OH in Methane-Air Flames. J. Chem. Phys., 19, pp. 1383–1390 (1954). A. Geydon. The Spectroscopy of Flames. London (1957). T. Carrington. Angular Momentum Distribution and Emission Spectrum of OH(A2 Σ) in the Photodissociation of H2 O. J. Chem. Phys., 41, pp. 2012– 2018 (1964). V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Nonequilibrium Distribution of the N2 (C3 Π) Molecules among Rotational Levels in a Gas Discharge (in Russian). ZhETF, 75, pp. 463–472 (1978). E.E. Nikitin and S. Ya. Umansky. Statistical Theory of Bimolecular Reactions. In: B.M. Smirnov, Ed. Plasma Chemistry (in Russian). 1, pp. 8–66. Moscow: Atomizdat (1974). V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Superposition of Rotational Distributions of Molecules in Nonequilibrium Plasma and a Statistical Model of the Decay of Excited Complexes. In: Journal of Brief Communications on Physics (in Russian), No. 2, pp. 9–14. Moscow: FIAN (1980). T. Horie and T. Kasuga. Statistical Model Including Angular Momentum Conservation for Abnormal Rotation of OH Split from Water. J. Chem. Phys., 40, pp. 1683–1686 (1964).

66 M. Kimura, S. Watanabe, and T. Horie. Initial Distributions of Vibrational and Rotational Populations Predicted by Statistical Model for Two Atomic Species Split from Four Atomic Complex. J. Phys. Soc. Jap., 32, pp. 1348–1358 (1972). 67 S. Yu. Savinov, I. Skala, and O. Shtirand. On the Effect of Helium on the Character of the Rotational Distribution of Hydroxyl Radicals in a Glow Discharge. In: Journal of Brief Communications on Physics (in Russian), No. 7, pp. 15–21. Moscow: FIAN (1982). 68 R. Bleekrode. Pressure Dependence of Rotation Intensity Distribution in C2 ‘Swan’ Emission from Low-Pressure Oxyacetylene Flames. J. Chem. Phys. 45, pp. 3153–3154 (1966). 69 K.S. Kini and M.I. Savadatti. Investigation of the C2 (A3 Π−−X3 Π) Band System Excited in a Discharge through Carbon Monoxide. J. Phys. B10, No. 6, pp. 1139–1149 (1977). 70 L.M. Biberman, V.S. Vorobyev, and I.T. Yakubov. Kinetics of Nonequilibrium Low-Temperature Plasma (in Russian). Moscow: Nauka (1982). 71 B.I. Katorgin, A.A. Stepanov, and V.A. Shcheglov. Fundamentals of the Kinetics, Gas Dynamics, and Optics of Nonequilibrium Media (in Russian). Moscow: Moscow Aviation Institute Press (2002). 72 A.V. Dem’yanenko, I.I. Zasavitsky, V.N. Ochkin et al. Investigation into the Distribution of the CO2 Molecules among Vibrational–Rotational Levels in a Glow Discharge by the Pulsed Diode Laser Spectroscopy Technique (in Russian). Kvantovaya Elektronika, 14, No. 4, pp. 851–859 (1987). 73 I.V. Kochetov and D.I. Slovetsky. Kinetics of Vibrationally Excited Molecules. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), III, p. 308. Moscow: Nauka (2000). 74 I.K. Babaev. Investigation of the Absolute Populations of Vibrational Levels in the Active Medium of a CO2 Laser. Ph.D. Thesis. Moscow (1972).

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4 Intensities in Spectra and Plasma Energy Distribution 75 W.J. Witteman. The CO2 Laser (Russian translation). Moscow: Mir Publishers (1990). 76 M. Haverlag, F.J. de Hoog, and G.M.W. Kroesen. Vibrational and Rotational Excitation in a Capacitevely Coupled 13.56 MHz Radio Frequency CF4 Plasma Studied by Infrared Absorption Spectroscopy. J. Vac. Sci. Technol., A9, No. 2 (1991) 77 V.VC. Lebedeva. Optical Spectroscopy Techniques (in Russian). Moscow: Moscow State University Press (1977); Experimental Optics (in Russian). Moscow: Moscow State University Press (1994). 78 R. Farrenq. J. Molec. Spectr., 49, pp. 28–288 (1974). 79 V.K. Zhivotov, V.D. Rusanov, and A.A. Fridman. Diagnostics of Nonequilibrium Chemically Active Plasma (in Russian). Moscow: Energoatomizdat (1985). 80 S.R. Landhoff and C.W. Baushlicher. Global Dipole Moment Function for the X1 Σ Ground State of CO. J. Chem. Phys., 102, No. 13, pp. 5220–5225 (1995). 81 S. Penner. Quantitative Molecular Spectroscopy and Gas Emissivities. Massachusetts (1959). 82 N.N. Kudryavtsev, S.S. Novitsky, and I.B. Svetlichnyi. Experimental Determination of the Temperature of the (001) Level of the Carbon Monoxide Molecule in a Nonequilibrium Flow of a CO2 + N2 + H2 (He) Gas Mixture (in Russian). Fizika Goreniya i Vzryva, 13, No. 2, pp. 205–212 (1977). 83 L.P. Bakhir and V.V. Timanovich. Method for Determining Active Medium Parameters of Flowing-Gas Electric-Discharge CO2 Lasers from Absorption and Emission at the Centers of the 001–000, 101–000 Bands of the CO2 Molecule (in Russian). ZhPS, 52, No. 4, pp. 553–559 (1985). 84 L.P. Bakhir, V.V. Elov, O.M. Kiselev et al. Study of the Energy Parameters of the Active Medium of Closed-Circuit CO2 Lasers from Absorption and Emission in the Region of 4.3 μm (in Russian). Kvant. Elektr., 15, No. 1, pp. 91–100 (1988).

85 A.A. Mikaberidze, V.N. Ochkin, and N.N. Sobolev. Measurement of Vibrational Temperatures of Molecules in the Electronic Ground State by the Spectral Line Inversion Method in the Infrared Region of the Spectrum. The CO2 Laser. JQSRT, 12, No. 4, pp. 169–188 (1972). 86 A. Mitchel and M. Zemansky. Resonance Radiation and Excited Atoms (in Russian). Moscow: GTTI (1937). 87 A.A. Mikaberidze. Optical Pyrometry of the Gas Discharge in Molecular Lasers (in Russian). In: Gas Lasers and their Applications (in Russian), pp. 58–101. Moscow: Nauka (1977). 88 I.I. Galaktionov, T.D. Korovkina, V.D. Mikhalevsky, and I.V. Podmashensky. Measurement of the Temperature and Concentration of CO2 in the Wake of a Shock Wave from IR Bands (in Russian). TVT, 7, No. 1, pp. 85–89 (1969). 89 B. Massabeaux, G. Gousset, M. Lefebvre, and M. Pealat. Determination of N2 (X) Vibrational Level Populations and Rotational Temperatures Using CARS in a DC Low Pressure Discharge. J. Physique, 48, pp. 1939–1949 (1987). 90 M. Pealat, J.P. Taran, M. Bacal, and F. Hillion. Rovibrational Molecular Populations and Negative Ions in H2 and D2 Magnetic Multicasp Discharges. J. Chem. Phys., 82, No. 11, pp. 4943–4953 (1985). 91 T. Kishimoto, N. Wenzel, H. GrosseWilde, and G. Marowsky. Population Dynamics of a CO2 Laser Studied by Coherent Anti-Stokes Raman Scattering. J. Appl. Phys., 69, No. 4, pp. 1872–1881 (1991). 92 M.Z. Novgorodov, V.N. Ochkin, and N.N. Sobolev. Measurement of Vibrational Temperatures in a CO2 Optical Quantum Generator (in Russian). Preprint FIAN No. 172. Moscow: FIAN (1969); ZhTF, 40, pp. 1268–1275 (1970). 93 R. Bleekrode. A study of the Spontaneous Emission from CO2 −N2 −He−H2 Laser Discharge. C-B Emission of N2 . IEEE J. Quant. Electr., QE5, pp. 57–60 (1969).

References 94 S.G. Gagarin, L.S. Polak, and D.I. Solovetsky. Recovery of the Relative Population of Vibrational Levels in Molecules in the Ground Sate from the Vibrational Level Populations in Electronically Excited states. In: Proc. IV All-Union Conference on LowTemperature Plasma Physics and Generators (in Russian), p. 33. Alma-Ata (1977). 95 V.N. Ochkin. To the Question of the Interrelation of the Vibrational Distributions of the N2 Molecules in the X and C States (in Russian). Preprint FIAN No. 102. Moscow: FIAN (1972). 96 H. Massey and E. Burhop. Electronic and Ionic Impact Phenomena. Oxford (1953). 97 V.V. Skubevich. Experimental Study of the Effective Slow-Electron Excitation Cross-Sections of N2 , O2 , and CO (in Russian). Ph.D. Thesis. Uzhgorod (1969). 98 A.J. Williams and J.P. Doering. An Experimental Study of the Low Energy Electron Scattering Spectrum of Nitrogen. Planet and Space Sci., 17, pp. 1527–1537 (1969). 99 A.Kh. Mnatsakanyan. Averaging of Excitation and Absorption CrossSections of Diatomic Molecules from Vibrational Structure (in Russian). Optika i Spectroskopiya, 30, pp. 1015– 1018 (1971). 100 S.N. Andreev, M.A. Kerimkulov, B.A. Mirzakarimov et al. Effect of Collisions on the Distribution of Molecules among Vibrational Levels of Excited Electronic States in a Gas Discharge (in Russian). ZhETF, 101, No. 6, pp. 1732–1748 (1992). 101 O.A. Evsin, E.B. Kupriyanov, V.N. Ochkin et al. Determination of the Vi-

brational Temperature of Molecules in Low-Temperature Plasma from Band ˚ ¨ System of Intensities in the Angstr om CO(B1 Σ−A1 Π) (in Russian). Kratkie Soobshcheniya po Fizike, No. 9–10, pp. 53–58 (1994). 102 V.N. Ochkin. Physical Problems, Tasks, and Methods of the Spectral and Optical Diagnostics of LowTemperature Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 411–424. Moscow: Nauka (2000). 103 N.N. Sobolev, Ed. Optical Pyrometry of Plasma (in Russian). Moscow: IL (1960). 104 D.I. Solovetsky. Chemical Reaction Mechanisms in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1980). 105 V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), I–IV. Moscow: Nauka (2000). 106 H.R. Grim. Plasma Spectroscopy. N.Y.: McGraw-Hill (1964); Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). 107 G. Bekeﬁ. Radiation Processes in Plasmas. New York: Wiley (1969). 108 W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). 109 R. Huddlestone and S. Leonard, Plasma Diagnostic Techniques. New York (1965). 110 E.E. Antonov, Yu. P. Korchevoi, and V.I. Lukashenko. Experimental Investigation into the Nitrogen Quenching of Excited States in the Cesium Atom (in Russian). Teploﬁzika Vysokikh Temperatur, 14, No. 6, pp. 1151–1158 (1976).

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5

Measuring Concentrations of Atoms and Molecules

5.1 General

When measuring particle concentrations in plasma, one should distinguish between the problems of elemental spectral analysis and diagnostics. In the 1930s–1950s, spectral analysis methods were developed in great detail and stimulated the evolution of spectroscopy and its technological, industrial applications. These methods are based on the relation between the intensity of spectral lines and the concentration of particles. The particle concentration is determined through calibration, with the sample under analysis being specially introduced in a plasma object with ﬁxed parameters and with suitable analytic lines selected. With the problem formulated in this fashion, it is precisely the composition of the original sample that is to be determined. If the sample contains molecular components, the molecules suffer dissociation (atomization) in the plasma object, but the degree of ionization of the elements should not be too high. The conditions in the plasma object used for the purpose usually correspond to the state of local thermal equilibrium (ﬂames, electric arcs), and the introduction of the sample to be analyzed only weakly affects the properties of the plasma. Inductively coupled plasma [3] has recently been frequently used as such an object. Plasma diagnostics problems include, in particular, the determination of the species and densities of molecules, atoms and ions in the mutual transformations of particles in plasma under, generally speaking, nonequilibrium conditions. To qualitatively analyze plasma, it is sufﬁcient to identify its emission spectra within as wide as possible region. In practice, quantitative analysis is to a great extend based on the methods for determining the populations of discrete energy levels (Chapter 3). The relation between the populations of the energy states of interest and the total particle density (formulas (2.53), (2.66), (3.92), and others)

236

5 Measuring Concentrations of Atoms and Molecules

is established within the framework of the model used to describe the plasma. To measure the concentration of atoms, it sufﬁces, as a rule, to determine the population of the ground state, because in low-temperature plasma conditions the population of the excited states can be disregarded and internal statistical sum (1.7) can be replaced by the statistical weight of the ground state: Qin,a ∼ g1 (Appendix A). The concentration of molecules, with their system of terms being as developed as it is, is more difﬁcult to ﬁnd. The statistical sum of internal motions is the product of the vibrational, rotational and electronic statistical sums: Qin = Qv Qr Qe

(5.1)

As is the case with atoms, the factor Qe can, for practically all diatomic and polyatomic molecules, be replaced by the statistical weight of the electronic ground state. This is conﬁrmed by numerous studies of molecular distributions among electronic states (see, for example, Figure 4.39). An exception can build some molecules with a multiplet electronic ground state, for example, NO2 , ClO2 and free radicals. In such cases, the populations of the ground state components should be determined experimentally. Some contribution, though not very great, to Qe can sometimes come from metastable states. In this respect, oxygen in the O2 (1 Δ) state, which can accumulate in discharges in amounts of up to a few percent of its concentration in the ground state, can be considered an exception rather than a rule. Under nonequilibrium conditions, the quantities Qv and Qr should, generally speaking, be found by direct summation (1.7). However, as shown in the preceding chapter, molecules in the electronic ground state in many cases form ensembles with vibrational, Tv , and rotational, Tr , temperatures. This is a fairly useful circumstance, for it allows the internal statistical sums to be calculated via substitution of the appropriate distribution temperatures into the formulas of Appendix A. A reservation should, however, be made as to the correctness of calculating Qv in the harmonic approximation. Deviations from the Boltzmann vibrational distributions depend on concrete conditions. If, however, one turns to the actually studied distributions (for example, Figures 4.27, 4.32), calculation by direct summation (1.7) shows that the difference between Qv and its counterpart calculated in the harmonic approximation with the temperature T = T1 (4.68) does not, as a rule, exceed a few percent. This is due to the sufﬁciently rapid, though not exponential, reduction of the populations of the levels numbered one through n0 (4.69). Such differ-

5.2 Determining Atomic Concentrations by Absorption Techniques

ences are usually commensurable with experimental errors, but they are possible and it is advisable to evaluate them as well. Considering what has been said above, to determine the total density of both atoms and molecules in nonequilibrium plasma by spectroscopic technique, it is preferable to pose the problem of measuring the populations of the electronic ground state levels.

5.2 Determining Atomic Concentrations by Absorption Techniques 5.2.1 Neutral Unexcited Atoms

If absorption has been measured by one of the methods described in Chapter 3, the density of the absorbing particles then can be found by the formulas of Section 2.3. To do so, one should know the radiative transition probability. Table 5.1 lists some selected data on the parameters of transitions involving the electronic ground state. Column 1 lists the elements, their serial numbers and the symbol of the ground state level l, column 2 wavelengths λ (in nm), column 3 the upper level u, columns 4 and 5 the statistical weights of the lower and upper levels, respectively and columns 6 and 7 the Einstein coefﬁcients Aul for spontaneous emission (in 108 s−1 ) and the oscillator strengths f lu in absorption, respectively. Tabulated in column 8 are ﬂuorescence wavelengths (if ﬂuorescence is excited on absorption of laser radiation at the frequency of the u → l transition). And column 9 gives information on some other lines due to absorption from the ground state without particularizing the parameters. The data on the transition probabilities were taken from a number of sources. Preference was given to the data bases (if available) of the National Institute of Standards and Technology (NIST, USA [5, 6]). The other sources are indicated in the table. Identiﬁcation of transitions and statistical weights, were taken from [7–10], and ﬂuorescence data from [11]. The data are restricted to transitions whose wavelengths fall within the visible, the near UV and IR regions of the spectrum.

237

2 P◦

2 P◦

2 P◦

2 P◦

3 P◦

3 D◦

14 Si (33 P0 ) 19 K (4 2 S1/2 )

2 S◦

13 Al

2 D◦

1 P◦

(3 2 P01/2 )

1 P◦

2 P◦ 3 2 P◦

2 S◦

2 D◦

12 Mg (3 1 S0 )

(3 2 S1/2 )

11 Na

(2 1 S0 ) 5B ◦ ) (2 2 P1/2

4 Be

2 P◦ 1 1 P◦

2 P◦

3 Li

(2 2 S1/2 )

Upper level, u

Element, ground state, l

220.8 251.4 404.4 766.5 767.6 769.9

394.4 308.2

202.6 285.2

589 589.6

208.9 249.6

234.86

323.27 670.78

λ, nm

1 1 2 2 2 2

2 2

1 1

2 2

2 2

1

2 2

gl

3 3 4 4 6 2

2 4

3 3

4 2

4 2

3

6 4

gu

0.31 0.61 1.2 × 10−2 0.39 0.39 0.38

0.48 0.61

0.84 4.95

0.62 0.62

1.8 1.2

5.56

1.2 × 10−2 0.37

Aul × 108 , s−1

6.8 × 10−2 0.17 6.8 × 10−3 0.62 1.02 0.34

0.12 0.17

0.16 1.81

0.65 0.32

0.24 0.11

1.38

5.5 × 10−3 0.75

f lu

Table 5.1 Atomic transitions involving the ground state and their probabilities.

404.4

298.8

308.2 309.2

285.2

589

234.8

323.3

λ f l , nm [10]

20 lines in the range 232–405 with f lu ca. 10−3 –10−6

243.9 with f lu = 2*10−3

211.8; 213; 219.9; 225.8; 226.3; 236.7; 237.2; 256.8; 265.2 with flu ca. 10−2 –10−3

Weak (flu ca. 10−6 –10−12 ) lines in the range 456–458.

30 lines in the range 243.4–330.3 with f lu ca. 10−3 –10−6

Doublets 208.89, 208.95 and 249.68, 249.17. For transitions standing second, lower level 2 P3/2 with energy 15.3 cm−1 . Aul = 2.1*108 s−1 , f lu = 0.21 and Aul = 2.4*108 s−1 , f ul = 0.11, respectively.

313.0; 313.1; 454.8

Some 20 lines in the range 233.4–323.3 with f lu ca. 10−3 –10−5

Other lines, nm Notes

238

5 Measuring Concentrations of Atoms and Molecules

2 P◦

5 P◦

(4 6 S5/2 )

25 Mn

(4 7 S3 )

24 Cr

6 P◦

6 P◦

7 P◦

7 P◦

7 P◦

4 D◦

4 G◦

4 F◦

3 G◦

3 F◦

3 G◦

3 D◦

3 F◦

3 F◦

2 D◦

2 F◦

2 D◦

(44 F3/2 )

23 V

(4 3 F2 )

22 Ti

(4 2 D1/2 )

21 Sc

1 P◦

1 P◦

1 P◦

20 Ca

(4 1 S0 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

335.2 360.5 425.4 357.9 279.5 403.1

239.9 272.1 422.7 269.3 270.7 390.7 402 294.2 293.3 394.9 318.6 398.2 368.55 305.3 318.4 381.8

λ, nm

7 7 7 7 6 6

1 1 1 4 4 4 4 5 5 5 5 5 5 4 4

gl

7 5 9 9 8 8

3 3 3 2 4 6 4 5 7 3 7 5 7 6 2

gu

1.2 × 10−3 1.62 0.31 1.48 3.7 1.7

1.3 2.4 0.67

0.17 2.7 × 10−3 2.18 0.16 0.31 1.66 1.63 1 9.6 × 10−2 0.48 0.8 0.38

Aul × 108 , s−1

2 × 10−4 0.23 0.1 0.37 0.58 5.5 × 10−2

4.3 × 10−2 9 × 10−4 1.75 8.8 × 10−3 3.4 × 10−2 0.57 0.4 0.13 1.7 × 10−2 6.8*10−2 0.17 8 × 10−2 0.22 0.18 0.95 7.3 × 10−2

f lu

279.5

464.6

296.7

671.7 422.7

λ f l , nm [10]

7 lines in the range 257–402 with f = 10−2 –10−4

Some 15 lines in the range 209–430 with f = 10−1 –10−4

Some 15 lines in the range 209–430 with f ca. 10−2 . 283.8 line is recommended in [11] to measure LIF. Fluorescence at 286.4

230; 242.8; 252; 259.4; 260; 263.2; 264.1; 266.2; 297; 334.1; 335.8; 337; 363.5; 501.4 with f lu = 10−2 –10−3

40 lines in the range 211–638 with f ik ca. 10−1 –10−4 .

215.1; 220.1; 227.5; 239.9 with f ik ca. 10−3 – 10−6 . 254.1; 261.8; 272.2 with f ik ca. 10−4 .

Other lines, nm Notes

5.2 Determining Atomic Concentrations by Absorption Techniques 239

4 G◦

27 Co

3 G◦

28 Ni

4 P◦

(4 4 S3/2 ) 34 Se (4 3 P2 )

33 As

5 S◦

3 S◦

4 P◦

3 P◦

1 P◦

3 D◦

2 S◦

(4 3 P0 )

32 Ge

2 D◦

2 S◦

2 S◦

2 D◦

2 D◦

3 P◦

1 P◦

30 Zn

(4 1 S0 ) 31 Ga ◦ ) (4 2 P1/2

2 P◦

2 P◦

29 Cu

(4 2 S1/2 )

3 F◦

(4 3 F4 )

4 F◦

4 F◦

(4 4 F4/2 )

5 F◦

5 P◦

5 D◦

26 Fe

(4 5 D4 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

213.9 307.6 299.4 287.4 403.3 266 245.0 237.1 204.2 249.8 265.2 197.3 193.8 196 207.4

324.7 327.3

252.28 271.9 296.7 240.7 304.4 352.7 232 301.9

λ, nm

1 1 2 2 2 2 2 2 1 1 1 4 4 5 5

2 2

9 9 9 10 10 10 9 9

gl

3 3 4 4 2 2 4 2 3 3 3 2 4 3 5

4 2

9 7 11 12 10 10 7 7

gu

7.09 3.3 × 10−4 7 × 10−2 1.2 0.49 0.12 0.28 5.7 × 10−2 1.1 0.13 0.85 2 2 3.6 [12] 4.9 [11]

1.39 1.37

2.9 1.4 0.27 3.6 0.19 0.13 6.9 6.4 × 10−2

Aul × 108 , s−1

1.46 1.4 × 10−4 1.1 × 10−2 1.2 0.12 1.3 × 10−2 5 × 10−2 4.8 × 10−3 0.21 3.7 × 10−2 0.27 5.8 × 10−2 0.11 0.12 0.32

0.44 0.22

0.28 0.12 4.4 × 10−2 0.38 2.6 × 10−2 2.4 × 10−2 0.68 7 × 10−3

f lu

534.9

275.4

294.4

307.6

510.5

310.1

340.5

373.5

λ f l , nm [10]

189,0, flu = 0.16.

202.5; 206.2

30 lines in the range 208–515 with f lu = 10−1 –10−3

20 lines in the range 230–363 with f lu = 10−1 –10−3

15 lines in the range 229–353 with f lu = 10−1 –10−3

12 lines in the range 208–386 with f ik = 10−1 –10−3 .

Other lines, nm Notes

240

5 Measuring Concentrations of Atoms and Molecules

1 P◦

(5 6 D1/2 )

(5 3 F2 ) 41 Nb

40 Zr

(2 D3/2 )

39 Y

(5 1 S0 )

38 Sr

296.1 374.2

6 D◦

420.2 780.0 794.8 242.8 293.1 460.7 256.9 294.8 297.5 355.3 407.7 414.3

λ, nm

3 D◦

2 D◦

2 F◦

2 P◦

2 F◦

2 D◦

1 P◦

1 P◦

1 P◦

2 P◦

2 P◦

2 P◦

37 Rb

(5 2 S1/2 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

2

5

2 2 2 1 1 1 1 4 4 4 4 4

gl

4

5

4 4 2 3 3 3 3 4 6 4 6 4

gu

1 [11]

3 [11]

1.8 × 10−2 0.37 0.34 0.17 1.9 × 10−2 2.01 5.3 × 10−2 0.35 0.35 0.23 1.1 1.6

Aul × 108 , s−1

4.2 × 10−2

3.3 × 10−2

9.5 × 10−3 0.67 0.32 4.5 × 10−2 7.3 × 10−3 1.92 1.6 × 10−2 4.6 × 10−2 7 × 10−2 4.4 × 10−2 0.41 0.41

f lu

376.4

301.1

302.2

16.7

420.1

λ f l , nm [10]

More than 20 lines in the range 353– 525 [7]. Prominent at the arc center are strong lines whose intensity is to that of 374 line as: 353.5 – 8.6; 354.4 – 2.1; 371.1 – 1.2; 374.2 – 6.4; 376.5 – 1.3; 411.7 – 1.1; 413.7 – 8.6; 416.8 – 13

More than 30 lines in the range 224–677

4 lines in the range 268–292 7 lines in the range 299–464 with f ik = 10−1 –10−2 . 5 lines in the range 602–880 with f lu ca. 10−2

689.3

334.9; 421.6; 358.7; 359.2

Other lines, nm Notes

5.2 Determining Atomic Concentrations by Absorption Techniques 241

2 F◦

1 P◦

48 Cd

2 S◦

2 D◦

2 D◦

3 P◦

2 P◦

(5 1 S0 ) 49 Ir (5 2 P1/2 )

2 P◦

3 P◦

4 D◦

4 G◦

4 F◦

(4 1 S0 ) 47 Ag (5 2 S1/2 )

46 Pd

(5 4 F9/2 )

45 Rh

5 D◦

5 G◦

5 F◦

7 P◦

(5 5 F5 )

44 Ru

7 P◦

7 P◦

7 P◦

7 P◦

7 P◦

5 P◦

42 Mo

(5 7 S3 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

228.8 326.1 256 303.9 410.2

328.1 338.3

294.4 313.3 317 319.4 379.8 390.3 386.4 287.5 349.9 372.8 312.4 339.7 350.2 349.2 369.2 276.3

λ, nm

1 1 2 2 2

2 2

7 7 7 7 7 7 7 11 11 11 10 10 10 10 10 1

gl

3 3 4 4 2

4 2

7 9 7 5 9 5 7 11 13 11 8 10 10 12 8 3

gu

5.3 4 × 10−3 0.4 1.3 0.56

1.4 1.3

0.37 1.79 1.37 1.53 0.69 0.62 0.62 6.7 [11] 0.87 [12] 0.76 [12] 4.6 × 10−2 0.65 0.43 1.3[11] 0.9 0.62 [11]

Aul × 108 , s−1

1.2 2 × 10−3 7.9 × 10−2 0.36 0.14

0.45 0.22

4.8 × 10−2 0.34 0.21 0.17 0.2 0.1 0.14 0.83 0.19 0.16 5 × 10−3 0.11 7.9 × 10−2 0.28 0.15 0.21

f lu

325.9

326.1

338.3

361.7

328.1

297.7

444.3

λ f l , nm [10]

238.9; 277.5 – 2 P-4 P transitions; 246 – 2 P– 2 S transition

206.1; 207

244.8; 247.6 – 1 S–1 P, 2 D transitions

4 lines in the range 238–344

3 lines of the 5 F – 5 D, 3 F, 3 H transitions in the range 257–274

300.2; 311.2; 315.8; 320.8; 345.6; 346.6 with f lu ca. 10−2

Other lines, nm Notes

242

5 Measuring Concentrations of Atoms and Molecules

1 P◦

2 D◦

57 La

4I

◦ ) (6 4 I9/2 60 Nd (6 2 D3/2 ) 62 Sm (6 7 F0 )

59 Pr

7 D◦

5 F◦

5 K◦

5 H◦

2 P◦

2 D◦

(6 2 D3/2 )

1 P◦

1 P◦

(6 1 S0 )

2 P◦

2 P◦

56 Ba

5 S◦

4 P◦

2 P◦

1/2 )

4 P◦

4 P◦

3 P◦

(5 3 P2 ) 55 Cs (6 2 S1/2 )

52 Te

(5 4 S

51 Sb

3 S◦

3 D◦

3 P◦

50 Sn

(5 3 Po )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

562 492.4 497.6 471.7

307.2 350.1 553.5 357.4 550.1 392.8 495.1

455.5 852.1 894.3

207.3 224.6 254.7 286.3 217.6 231.1 206.8 225.9

λ, nm

9 9 1 1

1 1 1 4 4 4 10

2 2 2

1 1 1 1 4 4 4 5

gl

7 11 3 3

3 3 3 4 4 2 10

4 4 2

3 3 3 3 4 4 6 5

gu

0.67 [11] 0.9 [12] 0.16 0.31

0.41 0.19 1.19 0.88 [12] 0.32 [11] 0.7 [12] 0.84

1.9 × 10−2 0.33 0.29

3.6 × 10−2 1.6 0.21 0.54 1.8 [12] 1.5 [11] 1.8 [12] 0.33 [11]

Aul × 108 , s−1

0.82 0.4 0.18 0.31

0.17 0.1 1.64 0.16 0.14 4.5 × 10−2 0.31

1.2 × 10−2 0.7 0.34

7 × 10−3 0.36 6 × 10−2 0.2 0.17 0.12 0.17 5.6 × 10−2

f lu

562

583.9

472.6 553.5

253

231.1

300.9

λ f l , nm [10]

14 lines in the range 454–663

15 lines in the range 357–754

270.2; 388.9; 413.2; 791.1 with f lu ca. 10−3

361.1; 387.6; 459.3 – 2 S−−2 P transitions with f lu = 10−3 –10−4 441.7 - 2 S–2 D transition

214.3 – 3 P–3 S transitions

202.4; 212.7; 206.8; 217.6 – 4 S–2 P3/2 , 2 P1/2 , 4 5/2 , P3/2 transitions

4P

Other lines, nm Notes

5.2 Determining Atomic Concentrations by Absorption Techniques 243

2)

4 I◦

◦ ) (6 4 I15/2 68 Er (6 3 H6 ) 69 Tm (6 2 F07/2 )

67 Ho

3 H◦

3 H◦

2 F◦

2 F◦

3 H◦

6 I◦

5 H◦

6 H◦

6 H◦

7 P◦

5 F◦

(6 5 I8 )

66 Dy

◦ ) (66 H15/2

65 Tb

(6 9 D

64 Gd

8 P◦

7 P◦

10 G◦

8 P◦

63 Eu

(6 8 S7/2 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

388.3 409.4 410.6 438.6

415.1

419.5 461.2 325.9 405.4

432.7

287.9 333.4 459.4 462.7 466.2 308.7 368.4

λ, nm

8 8 8 8

13

17 17 17 16

16

8 8 8 8 8 5 5

gl

6 6 10 8

11

17 15 19 16

16

10 6 10 8 6 7 7

gu

1 0.9 0.6 4.2 × 10−2

1.8

0.72 8.2 × 10−2 8.5 × 10−3 24 [11]

7.2 [11]

2.8 × 10−2 0.34 1.4 1.3 1.47 1.7 [11] 1.3 [12]

Aul × 108 , s−1

0.17 0.17 0.19 1.2 × 10−2

0.39

0.19 2.3 × 10−2 5.6 × 10−3 5.9

2.02

4.4 × 10−3 4.3 × 10−2 0.55 0.32 0.36 0.34 0.26

f lu

405.4

432.7

313.8

536.1

λ f l , nm [10]

20 lines in the range 374–590

20 lines in the range 333–659

20 lines in the range 395–661

15 lines in the range 372–833

Lines with intensity comparable with that of 432.7 line in the arc [10]: 431.9; 433.8. A weak line at 579.6

More than 20 lines with intensity in the arc [10] exceeding that of 308.7 line, including 301, by 23 times, 307.7 - 7, 309.9 5, 368.4 - 25, 404.5 - 20, 405.5 - 10, 430.6 12, 432.7 - 23, 447.6 – 10. 6 weak lines in the range 457–745

A few tens of lines in the range 270–711

Other lines, nm Notes

244

5 Measuring Concentrations of Atoms and Molecules

(6 5 D0 )

74 W

(4 F3/2 )

73 Ta

(6 3 P2 )

72 Hf

(6 2 D3/2 )

71 Lu

28689◦ ,K 6 D◦ 4 D◦ 2 D◦ 4 D◦ 34342◦ ,K 31323◦ ,K 5 F◦ 7 D◦ 7 F◦

3 G◦ 3 1 P◦ 1 6 D◦

2 D◦

2 F◦

1 D◦

1 D◦

2 P◦

3 P◦

1 P◦

1 F◦

1 S◦

1 P◦

70 Yb

(6 1 S0 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

326 348.5 373.1 391.8 481.2 540.2 291.1 319.2 384.7 466 498.3

246.4 267.2 346.4 398.8 555.6 298.9 308 337.6 356.8 451.9 349.7 377.6

λ, nm

4 4 4 4 4 4 1 1 1 1 1

1 1 1 1 1 4 4 4 4 4 5 5

gl

4 4 6 2 4 2 3 3 3 3 3

3 3 3 3 3 3 6 4 6 4 7 3

gu 0.25 3.8 × 10−2 0.33 1.26 1.6 × 10−2 0.11 3.8 × 10−2 0.38 0.17 6.4 × 10−2 0.46 0.043 9.2 × 10−4 1.5 × 10−3 1.7 × 10−3 2.9 × 10−3 4.2 × 10−3 3.1 × 10−3 9 × 10−2 1.5 × 10−2 5.5 × 10−3 9.8 × 10−3 4.7 × 10−3

5.8 × 10−3 8.5 × 10−3 5.3 × 10−3 2.5 × 10−2 1.2 × 10−2 1.4 × 10−2 7.7 × 10−2 3.2 × 10−2 8.3 × 10−3 10−2 4.2 × 10−3

f lu

0.91 0.12 0.62 1.76 1.1 × 10−2 1.1 [11] 0.18 [11] 2.23 0.59 0.21 0.18 0.20

Aul × 108 , s−1

402.8

λ f l , nm [10]

20 lines in the range 273–541

18 lines in the range 298–714

15 lines in the range 273–574

Other lines, nm Notes

5.2 Determining Atomic Concentrations by Absorption Techniques 245

6 G◦

(6 3 P0 )

82 Pb

(6 3 D3 ) 79 Au (6 2 S1/2 ) 80 Hg (6 1 S0 ) 81 Tl ◦ ) (6 2 P1/2

78 Pt

(4 F9/2 )

77 Ir

33874,◦ K 4 F◦ 3 F◦ 3 F◦ 2 P◦ 2 P◦ 3 P◦ 1 P◦ 2S 2D 2S 3 D◦ 3 P◦

6 G◦

7 F◦

7 F◦

5 P◦

(6 5 D4 )

76 Os

6 P◦

6 P◦

6 P◦

6 D◦

75 Re

(6 6 S5.2 )

Upper level, u

Element, ground state, l

Table 5.1 (continued).

284.9 292.5 295.1 263.97 265.94 293 242.79 267.59 253.6 184.95 258.0 276.79 377.57 217 283.3

280.7 305.87 330.16

289.6 346.05 346.47 345.19

λ, nm

10 10 10 10 7 7 2 2 1 1 2 2 2 1 1

9 9 9

6 6 6 6

gl

10 12 8 10 9 7 4 2 3 3 2 4 2 3 3

7 9 11

8 6 8 4

gu

0.22 0.14 2.8 × 10−2 0.47 [12] 0.81 [12] 2.1 [11] 1.99 1.64 8×10−2 7.5 0.18 1.26 0.62 1.5 0.58

4.5 [11] 0.29 [12] 0.1

0.22 [11] 0.31 [12] 0.25 [12] 0.18 [12]

Aul × 108 , s−1

2.7 × 10−2 2.2 × 10−2 3 × 10−3 4.9 × 10−2 0.11 0.27 0.35 0.18 2.3 × 10−2 1.15 [12] 1.8 × 10−2 0.29 0.13 0.32 0.21

0.41 4.1 × 10−2 2 × 10−2

3.7 × 10−2 7.5 × 10−2 4.5 × 10−2 2.1 × 10−2

f lu

205.3; 202.2 with f ik ca. 10−2

231.6; 238 with f ik ca. 10−2 352.9

405.8

227; 265.6

10 lines in the range 214–352

15 lines in the range 215–380

20 lines in the range 233–442. The intensity of some of them in the arc [10] exceeds that of the 280.7 line, including 290.9, by 4 times, 305.9 - 4, 426.6 – 2

23 lines in the range 230–528. The intensity of some of them in the arc [10] exceeds that of the 289.6 line, including 346, by 130 times, 346.5 – 80, 345.2 – 35, 488.9 –5

Other lines, nm Notes

253.6

304.1

439.1

λ f l , nm [10]

246

5 Measuring Concentrations of Atoms and Molecules

(7 5 L6◦ )

92 U

5M

7N

5M

7N

4P

2D

4P

4P

83 Bi

◦ ) (6 4 S3/2

Upper level, u

Element, ground state, l

Table 5.1 (continued).

206.17 222.8 223.1 306.77 358.49 311.5 358.48 387.1

λ, nm

13 13 13 13

4 4 4

gl

13 15 15 13

4 6 2

gu

0.19 [12] 1.7 [11] 0.18 0.24 [12]

0.89 2.6 2.07

Aul × 108 , s−1

5.3 × 10−2 1.7 [11] 0.18 0.24 [12]

6.6 × 10−2 0.29 0.15

f lu

311.5

472.2

λ f l , nm [10]

14 lines in the range 310–870

227.6; 202.1; 211 with f ik ca. 10−2

Other lines, nm Notes

5.2 Determining Atomic Concentrations by Absorption Techniques 247

248

5 Measuring Concentrations of Atoms and Molecules

Let us, on the basis of these data, estimate the particle densities that can be determined by direct measurement of the absorption of the probe radiation from an external source in the object of interest. Let us assume that absorption relates to the center of a Doppler line and disregard stimulated transitions. Using formulas (2.44), (2.45), (2.40) and expressing the absorption coefﬁcient χ0,lu in cm−1 , the wavelengths λlu and ΔλD in nanometers and the particle density Nu in cm−3 , we get Nl ≈ 1.21 × 1019

1 ΔλD χ . f lu λ2ul 0,lu

(5.2)

For the Lorentz distribution, a formula similar to (5.2) is obtained from expression (2.63). If the line width of the probe source is comparable with the width of the absorption line, or if absorption is measured against the background of a continuous spectrum, formula (5.2) is corrected by means of the relations of Section 3.2. Let us assume that the intensity of light that has passed through an object with a length of L centimeters should range between the limits 0.1 < Ilu ( L)/Ilu (0) < 0.9 in order to ensure sufﬁcient measurement accuracy. According to expression (2.51), this means that 0.1 < χ0,lu L < 2.3. Assume that subject to measurement is the concentration of titanium atoms in a high-current low-pressure discharge like that used in the metal ﬁlm deposition technology [13]. The gas temperature T = 500 K and the gas pressure equals 0.133 Pa (1 mTorr). The absorbing transition wavelength is equal to 398.2 nm, f ul = 0.09 (Table 5.1), ΔλD = 1.5 × 10−3 and the probe source is a hollow-cathode lamp with a narrower line width. The length of the plasma object is 50 cm, that is, 2 × 10−3 cm−1 < χ0,lu L < 4.6 × 10−2 cm−1 . In that case, the optimal region for measurement falls within the interval 2.2 × 109 cm−3 < N < 5.1 × 1010 cm−3 . If measurements are taken at a wavelength of 318.6 nm, f ul = 0.17, then 1.5 × 109 cm−3 < N < 3.5 × 1010 cm−3 . Figure 5.1a shows the results of measurements taken under these conditions (after the data of [13]). They are located near the lower limit of the favorable region indicated above. Of course, this region is determined by the absorption measurement scheme used. For example, when using a double-beam scheme, measurements in the wider dynamic range 0.01 < Ilu ( L)/Ilu (0) < 0.99 are quite possible, and with high-sensitivity methods, such as laser induced ﬂuorescence or optogalvanic techniques (Section 3.4), the minimum measurable absorption can be substantially reduced and the concentration sensitivity can thus be raised by many orders of magnitude. Table 5.2 lists the threshold atomic concentrations (in ngcm−3 ) that can be measured by various absorption methods in an equilibrium ﬂame [14]. Here OG stands for optogalvanic techniques, AB for direct absorption

5.2 Determining Atomic Concentrations by Absorption Techniques

Figure 5.1 Current dependences of the concentrations of titanium atoms and ions in a low-pressure arc (after [12]): (a) Ti, pressure 5× 0.133 Pa (1 mTorr); squares – λ = 398.2 nm, circles – λ = 318.6 nm; (b) Ti+ , λ = 324.2 ; squares – 0.133 Pa (1 mTorr), circles – 1.333 Pa (10 mTorr).

techniqies, EM for emission methods, FLn for ﬂuorescence techniques with excitation from noncoherent light sources and FLl for the same with laser excitation. The detection limit typically ranges from 10−3 to 102 ngcm−3 . In some cases, for example Pb, transitions between excited levels provide for lower detection limits than those from the ground state. This is due to transition probabilities and can be used under equilibrium conditions in the same way as the emission method. In the absence of equilibrium, as already repeatedly noted, to compare between the detection limits in these cases requires additional investigations. In many cases, the optogalvanic detection of absorption proves preferable. With this method, the detection threshold is limited to ca. 105 cm−3 by the electric noise in the ﬂame. This limit has been reached for lithium. 5.2.2 Metastable Atoms

For the metastable states of atoms, optical transitions to lower-lying energy levels are forbidden by the selection rules for dipole radiation. And although the prohibition is, as a rule, softened in higher approximations with increasing serial number of the element [15–17], the radiative lifetime on the atomic scale proves long. In plasma conditions, the lifetime

249

250

5 Measuring Concentrations of Atoms and Molecules

Table 5.2 Detection limits (ngcm−3 ) for atoms in ﬂames. Atom Ag Al Au Ba Bi Ca Co Cr Cr Cs Cu Cu Fe Fe Ga Ga In K Li Li Li Lu Mg Mn Na Na Ni Pb Pb Rb Sc Sn Sn Sr Tl

Wavelength, nm

OG

328.1 309.3 242.8 307.2 306.8 300.7 252.1 298.6 301.8 455.5 282.4 324.8 298.4 302.1 287.4 294.4 303.9 404.4 610.4 639.3 670.8 308.2 285.2 279.5 285.3 589 300.2 283.3 280.2 420.2 301.9 284 286.3 460.7 291.8

1 0.1 1 0.2 2 0.1 0.08 2 2 0.004 100 100 4 2 0.007 0.1 0.006 0.1 0.012 0.4 0.001 0.2 0.1 0.02 0.05 0.01 0.08 0.09 0.6 0.1 0.2 0.3 2 0.4 0.09

AB 1

20 50 1

EM

FLn

FLl

2

0.1

4

1 20000 0.1

5 20

8 3 0.08

2

2

5

1

1

0.1

0.5

1

4 50 30

1 1 0.1 0.8 0.8

5 10 0.4

0.02 0.02 5 1 0.1

8 10

30 0.9

100

0.2

0.1 1

0.5 0.5 0.2 0.4 0.1

5

20

3

2

10

100

10

13

20

100

50

20

20

8

4

5.2 Determining Atomic Concentrations by Absorption Techniques

of the electronic metastable states of particles is normally governed by collisions with the surrounding particles and walls. Collision results are fairly diverse, but selective as regards quantum numbers, both prior to and after interaction, for the energies of these states are usually high in comparison with kB T, so that interaction with the thermal motion is hindered. The concentrations of such particles can be sufﬁciently high and can be measured by the traditional absorption techniques on allowed transitions. Table 5.3 provides information on the parameters of such transitions in atoms where the lower state of the transition is metastable. The table was compiled using the data of [5–9, 12, 15, 18–20]. In a number of cases, the second column indicates not only the symbol of the metastable term, but also its lifetime.

251

14 Si

0)

(3 3 P0 )

12 Mg(3 1 S

(21 S0 )

(11 S0 )

10 Ne

2 He

(1 2 S1/2 )

518.36

4 3 S1

4.339 × 10−19 J (2.712 eV) 2.717 3.054 × 10−19 J (1.909 eV)

3 3D

26.752 × 10−19 J (16.72 eV)

3s [1/2]o (3 3P0o ) (400) 33 P1o (0.002) 33 P2o

3 1 S0 (1.1)

383.230

3p1 [3/2]1 3p1 [1/2]1

26.592 × 10−19 J (16.62 eV)

3s[3/2]2o or (3 3P2o ) (20)

o

3 1P 5 1P

32.992 × 10−19 J (20.62 eV)

2 1 S0 (0.02)

41 P1o

3p[5/2]3 3p[3/2]1 3p[3/2]2

33 P 43 P

31.712 × 10−19 J (19.82 eV)

2 3 S1 (7900)

390.552

626.650 616.359

640.225 621.728 614.306

501.568 396.473

388.865 318.774

656.272 486.13

o 32 P3/2 o 42 P1/2

16.320 × 10−19 J (10.2 eV)

2 2 S1/2 (0.12)

1H

λlu , nm

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Element, ground state, l

Table 5.3 Atomic transitions involving metastable states and their probabilities.

1

5

3

1 1

5 5 5

1 1

3 3

2 2

gl

3

3

5

3 3

7 3 5

3 3

5 5

4 2

gu

0.118

0.57

1.25

0.25 0.146

5.1 × 10−2 6.37 × 10−2 0.28

0.13 7×10−2

9.48 × 10−2 5 × 10−2

0.22 9.7 × 10−2

Aul × 108 , s−1

8.3 × 10−2

0.14

0.46

0.44 0.246

4.39 × 10−2 2.22 × 10−2 0.153

0.15×10−2 5×10−2

6.4 × 10−2 2 × 10−2

0.435 0.1

f lu

252

5 Measuring Concentrations of Atoms and Molecules

(4 1 S

20 Ca

0)

(3 1 S0 )

18 Ar

Element, ground state, l 18.480 × 10−19 J (11.55 eV)

18.752 × 10−19 J (11.72 eV) 3.006 × 10−19 J (1.879 eV) 3.018 × 10−19 J (1.886 eV)

4s[3/2]2o (4 3P2o ) (60)

4s [1/2]0 or (4 3P0o ) (50)

43 P0o

43 P1o (3.5×10−4 )

o

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

5 3 S1 43 D1 43 D2 4 3 P1 6 3 S1

4p[1/2]1 4p[5/2]3 4p[5/2]2 4p[3/2]1 4p[3/2]2 4p1 [3/2]1 4p1 [3/2]2 4p1 [1/2]1 5p[5/2]3 6p[3/2]2 4p[3/2]1 4p1 [3/2]1 4p1 [1/2]1 5 3 S1 4 3 D1 4 3 P1

Excited term u

612.222 443.569 443.496 428.301 395.705

610.272 442.544 428.936

866.794 794.818 772.421

912.297 811.531 801.479 772.376 763.511 714.704 706.722 696.543 420.067 355.43

λlu , nm

3 3 3 3 3

1 1 1

1 1 1

5 5 5 5 5 5 5 5 5 5

gl

3 3 5 5 3

3 3 3

3 3 3

3 7 5 3 5 3 5 3 7 5

gu

0.25 0.34 0.63 0.43 9.8 × 10−2

8.3 × 10−2 0.42 0.57

2.8 × 10−2 0.19 0.12

0.19 0.33 9.6 × 10−2 5.7 × 10−2 0.27 4.5 × 10−3 3.9 × 10−2 6.7 × 10−2 1.03 × 10−2 3 × 10−3

Aul × 108 , s−1

0.14 0.1 0.3 0.2 2 × 10−2

0.14 0.37 0.47

9.5 × 10−2 0.54 0.32

0.14 0.45 9.2 × 10−2 3 × 10−2 0.24 2.1 × 10−3 3 × 10−2 3 × 10−2 3.8 × 10−3 5.5 × 10−4

f lu

5.2 Determining Atomic Concentrations by Absorption Techniques 253

(42 D3/2 )

(4 3 F2 )

(44 F3/2 )

21 Sc

22 Ti

23 V

Element, ground state, l

41 F3o 41 P1o 51 F3o 61 P1o 61 F3o 51 P1o o 44 D3/2

4.334 × 10−19 J (2.709 eV)

2.291 × 10−19 J (1.432 eV) 2.302 × 10−19 J (1.439 eV) 2.317 × 10−19 J (1.448 eV) 1.333 × 10−19 J (0.833 eV) 2.160 × 10−19 J (1.35 eV) 2.200 × 10−19 J (1.375 eV)

3 1 D2 (17)

4 4 F5/2

4 2 G9/2

4 2 G7/2

4 5 F5

4 4 F9/2

o 42 F7/2

o 42 H9/2

55 F4o

o 44 D7/2

o 44 D5/2

5 3 S1 4 3 D2 4 3 D3 5 3 P2 6 3 S1 3 3 P1

3.038 × 10−19 J (1.899 eV)

43 P2o

4 4 F7/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

305.039

320.557

455.55

474.383

474.103

473.765

487.813 671.769 435.508 504.162 410.853 452.694

616.218 445.589 445.478 430.253 397.37 300.92

λlu , nm

10

8

11

10

8

6

5 5 5 5 5 5

5 5 5 5 5 5

gl

8

10

9

8

6

4

7 3 7 3 7 3

3 5 7 5 3 3

gu

0.53

1.3

0.12

9.8

9.1

8.8

0.19 0.12 0.19 0.33 0.9 0.41

0.48 0.2 0.87 1.2 0.18 0.43

Aul × 108 , s−1

5.9 × 10−2

0.25

3 × 10−2

2.6

2.3

2.0

9.4 × 10−2 4.9 × 10−2 7.4 × 10−2 7.6 × 10−2 0.32 7.5 × 10−2

0.15 6 × 10−2 0.36 0.34 2.5 × 10−2 3.5 × 10−2

f lu

254

5 Measuring Concentrations of Atoms and Molecules

26 Fe( 5 D ) 4 4

(50)

25 Mn( 6 S 4 5/2 )

(4 7 S3 )

45 F3o

3.485 × 10−19 J (2.178 eV) 3.499 × 10−19 J (2.187 eV) 1.374 × 10−19 J (0.859 eV) 1.464 × 10−19 J (0.915 eV) 1.523 × 10−19 J (0.952 eV) 1.584 × 10−19 J (0.99 eV)

4 6 D3/2

4 6 D1/2

4 5 F5

4 5 F4

4 5 F3

4 5 F2

45 D4o 54 F5o y5 F6o 45 D3o 45 G5o

3.462 × 10−19 J (2.164 eV)

4 6 D5/2

45 G3o

o 46 F3/2 6 o 4 D3/2

o 46 F5/2 o 46 D5/2

o 46 F7/2 o 46 D3/2

o 46 F9/2 o 46 D7/2

3.429 × 10−19 J (2.143 eV)

3.382 × 10−19 J (2.114 eV)

4 6 D9/2

4 6 D7/2

55 P3o 55 P3o 45 P3o o 46 D9/2 o 46 F11/2

1.506 × 10−19 J (0.941 eV)

4 5 S2

24 Cr

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Element, ground state, l

Table 5.3 (continued).

361.877

375.824

382.588 357.01

526.954 373.486 358.12

326.023 407.941

384.107 408.294

383.436 404.874

382.351 405.554

404.136 380.671

272.650 298.865 520.842

λlu , nm

5

7

9 9

11 11 11

2 2

4 4

6 6

8 8

10 10

5 5 5

gl

7

7

7 11

9 11 13

4 4

6 6

8 4

10 8

10 12

7 7 7

gu

0.73

0.63

0.6 0.68

1.3*10−2 0.9 1.02

0.38 0.38

0.33 0.3

0.43 0.75

0.52 0.43

1.0 0.59

0.75 0.52 0.5

Aul × 108 , s−1

0.23

0.13

0.1 0.16

4.3 × 10−3 0.19 0.23

0.12 0.19

0.11 0.11

0.13 0.12

0.14 0.15

0.26 0.15

0.12 0.1 0.29

f lu

5.2 Determining Atomic Concentrations by Absorption Techniques 255

28 Ni(4 3 F

9/2 )

4)

27 Co(4 4 F

Element, ground state, l

0.691 × 10−19 J (0.432 eV) 0.264 × 10−19 J (0.165 eV) 2.682 × 10−19 J (1.676 eV) 0.677 × 10−19 J (0.423 eV)

b4 4 F9/2

4 3 F3

4 3 F2

a 1 S0

2.922 × 10−19 J (1.826 eV) 0.440 × 10−19 J (0.275 eV)

0.358 × 10−19 J (0.224 eV)

a4 4 F3/2

b1 D2 (1.7) a1 D2

0.278 × 10−19 J (0.174 eV)

a4 4 F5/2

43 D1o

232.997

547.69

208.555 338.057

53 F2o 51 P1o z1 P1o

513.707

231.234

350.228 345.351 340.512

254.425 243.904 241.529

2535.96 2436.66 2414.46

252.897 243.221 241.162

868.862

λlu , nm

51 P1o

o 44 D5/2 4 o 4 F7/2 o 44 G9/2 4 o 4 D3/2 4 o 4 F5/2 o 44 G7/2 o 44 D1/2 o 44 F3/2 o 44 G5/2 4 o 4 D7/2 4 o 4 G11/2 o 44 F9/2 3 4 G3o

45 P3o

3.482 × 10−19 J (2.176 eV) 0.162 × 10−19 J (0.101 eV)

4 5 P3

a4 4 F7/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

5

1

5 5

5

7

10 10 10

4 4 4

6 6 6

8 8 8

7

gl

3

3

5 3

3

7

8 12 10

2 4 6

4 6 8

6 8 10

7

gu

5.3

9.5 × 10−2

0.26

0.13

0.17 0.13

2 × 10−3

8.6 × 10−3 2.6 1.3

0.44

0.12 0.27 0.17

0.16 0.25 0.49

0.13 0.22 0.42

0.2 0.23 0.42

8.8 × 10−3

f lu

5.5

0.8 1.1 1

3 2.8 3.6

2 2.5 3.6

2.8 2.6 3.9

7.7 × 10−3

Aul × 108 , s−1

256

5 Measuring Concentrations of Atoms and Molecules

32 Ge(4 3 P

0)

31 Ga(42 P0 ) 1/2

30 Zn( 1 S ) 4 0

29 Cu( 2 S 4 1/2 )

Element, ground state, l

Excited term u o 42 P3/2 2 Po 3/2 2 Po 1/2 2 Po 3/2 5 3 S1 4 3 D1

4 3 D1 4 3 D2 5 3 S1 4 3 D3 4 3 D2 4 3 D1 5 2 S1/2 4 2 D3/2 4 2 D5/2 4 3 D2 5 3 P0 4 3 D3 5 3 P2 5 3 P1 53 P1o 53 P1o

El , J (eV) 2.222 × 10−19 J (1.389 eV) 2.627 × 10−19 J (1.642 eV) 6.410 × 10−19 J (4.006 eV) 6.448 × 10−19 J (4.030 eV) 0.163 × 10−19 J (0.102 eV) 0.163 × 10−19 J (0.102 eV) 0.110 × 10−19 J (0.069 eV) 0.280 × 10−19 J (0.175 eV) 1.413 × 10−19 J (0.883 eV)

Metastable term l, (τ, s)

4 2 D5/2

4 2 D3/2

43 P0o

43 P1o (2*10−5 )

43 P2o (103 )

o 42 P3/2 (200)

43 P1o (320)

43 P2o (120)

4 1 D2 (6.7)

Table 5.3 (continued).

303.906 326.95

209.426 265.118 275.459

206.865 270.963

417.206 294.424 294.364

334.502 334.557 334.594

330.294 330.258 472.225

468.014 328.233

578.21 276.637

510.554 261.837

λlu , nm

5 5

5 5 5

3 3

4 4 4

5 5 5

3 3 3

1 1

4 4

6 6

gl

3 3

7 5 3

5 1

2 4 6

7 5 3

3 5 3

3 3

2 4

4 4

gu

2.8 0.29

0.97 1.2 1.1

1.2 2.8

1.06 0.27 1.4

1.64 0.4 4.5×10−2

0.67 1.25

0.14 0.9

1.65 × 10−2 3.6 × 10−2

2 × 10−2 0.31

Aul × 108 , s−1

0.23 2.8 × 10−2

8.9 × 10−2 0.13 7.5 × 10−2

0.13 0.1

0.14 0.035 0.27

0.38 6 × 10−2 4 × 10−3

0.11 0.34 0.43

0.14 0.44

4.1 × 10−3 1.1 × 10−2

5.2 × 10−3 2.1 × 10−2

f lu

5.2 Determining Atomic Concentrations by Absorption Techniques 257

2.101 × 10−19 J (1.313 eV) 2.165 × 10−19 J (1.353 eV) 3.608 × 10−19 J (2.255 eV) 3.699 × 10−19 J (2.312 eV) 0.395 × 10−19 J (0.247 eV) 0.502 × 10−19 J (0.314 eV) 15.864 × 10−19 J (9.915 eV)

o 42 D3/2 (13)

o 42 D5/2 (180) 4 2 P1/2

5 3 F4

39 Y( 2 D 3/2 ) 4

40 Zr(3 F ) 2

4 3 P1 (5.9) 4 3 P0 (11.5) 5s[3/2]2o or 53 P2o (85)

4 2 P3/2

53 P1o 51 P1o

3.246 × 10−19 J (2.029 eV)

4 1 S0 (0.46)

0.246 × 10−19 J (0.154 eV)

53 G5o

o 52 F7/2 2 o 5 F3/2

0.105 × 10−19 J (0.0658 eV)

360.119

467.485 410.238

768.52 850.89

769.45 810.44 811.29

5p[3/2]1 5p[5/2]2 5p[5/2]3 5p1 [1/2]0 5p1 [3/2]1o

206.279

53 S1o

16.896 × 10−19 J (10.56 eV)

203.985

53 S1o

278.022

286.044

o 52 P1/2 o 52 P3/2

228.812

234.984 249.291

468.583 422.66

λlu , nm

42 P1/2

o 52 P1/2 2 o 4 P1/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

5s [1/2]0o or 53 P0o (0.5) 4 2 D5/2

36 Kr( 1 S ) 4 0

34 Se( 3 P ) 4 2

33 As(42 So ) 3/2

Element, ground state, l

Table 5.3 (continued).

9

6 6

3 3

5 5 5

1

3

4

2

6

4 4

1 1

gl

11

8 4

1 3

3 5 7

3

3

4

2

4

2 2

3 3

gu

1.4

0.14 3.1

0.33

0.06 0.51

0.14 0.26

3 × 10−2 0.13 0.5

5.6 × 10−2 0.13 0.36 0.49 0.24

0.11

0.10

9 × 10−2

6.8 × 10−2

0.15

0.13 5.6 × 10−3

9.4 × 10−2 0.17

f lu

0.56

1.7

0.78

0.55

2.8

3.1 0.12

9.5 × 10−2 0.21

Aul × 108 , s−1

258

5 Measuring Concentrations of Atoms and Molecules

48 Cd( 1 S ) 5 0

53 P2o 53 F4o 53 F3o 53 D2o 6 3 S1 5 3 D1 6 3 D1 6 3 S1 5 3 D1 5 3 D2

1.536 × 10−19 J (0.96 eV) 5.974 × 10−19 J (3.734 eV) 6.080 × 10−19 J (3.8 eV)

5 3 D2

53 P0o

5 3 P1 (2*10−6 )

o 54 F7/2 o 54 D5/2

1.302 × 10−19 J (0.814 eV)

5 4 F7/2

45 Rh( 4 F 5 9/2 )

0.240 × 10−19 J (0.15 eV) 0.304 × 10−19 J (0.19 eV)

55 S1o 55 P3o 55 P1o 55 G5o

o 56 F11/2

5 3 D3

6 5 F4

44 Ru( 5 F ) 6 5

46 Pd( 1 S ) 4 0

5 5 S2

5 6 D9/2

o 56 F9/2

o 56 F7/2

0.078 × 10−19 J (0.049 eV) 0.138 × 10−19 J (0.086 eV) 0.208 × 10−19 J (0.130 eV) 2.136 × 10−19 J (1.335 eV)

5 6 D5/2

5 6 D7/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

42 Mo( 7 S ) 5 3

41 Nb( 6 D 1/2 ) 5

Element, ground state, l

Table 5.3 (continued).

479.992 346.765 346.62

467.815 340.365 283.69

360.955 342.124

363.470 340.458

352.802 365.799

343.674

334.017 550.649 557.045

405.894

407.973

410.092

λlu , nm

3 3 3

1 1 1

5 5

7 7

8 8

9

5 5 5

10

8

6

gl

3 3 5

3 3 3

7 5

5 9

8 6

11

7 7 3

12

10

8

gu

0.4 0.67 1.36

0.14 0.86 0.28

0.82 0.95

1 1.3

0.85 0.88

0.47

0.12 0.36 0.33

1.3

0.99

0.76

Aul × 108 , s−1

0.14 0.12 0.41

0.13 0.45 0.1

0.22 0.17

0.14 0.3

0.16 0.13

0.10

1.2 × 10−2 0.23 9.2 × 10−2

0.38

0.31

0.26

f lu

5.2 Determining Atomic Concentrations by Absorption Techniques 259

54 Xe( 1 S ) 5 0

51 Sb(54 So ) 1/2

50 Sn( 3 P ) 5 0

49 In(52 Po ) 1/2

Element, ground state, l

63 P0o 63 P2o 53 D2o

0.336 × 10−19 J (0.21 eV) 0.680 × 10−19 J (0.425 eV) 1.709 × 10−19 J (1.068 eV) 3.405 × 10−19 J (2.128 eV)

5 3 P1 (12)

5 3 P2 (16)

5 1 D2 (1)

5 1 S0 (0.12)

15.115 × 10−19 J (9.447 eV)

1.955 × 10−19 J (1.222 eV) 13.304 × 10−19 J (8.315 eV)

6 2 S1/2 5 2 D3/2 5 2 D5/2

0.438 × 10−19 J (0.274 eV)

o 52 P3/2 (10)

o 52 D5/2 (10) 6s[3/2]2o or 63 P2o (150) o 6s [1/2]0

6 3 S1 5 3 D3 6 3 D3

6.314 × 10−19 J (3.946 eV)

5 3 P2

796.734

840.919 823.163

6p[3/2]1 6p[3/2]2 6p[1/2]1

287.792

563.171 452.47

326.234 242.171 380.102

317.505 283.999 242.949

303.412 270.651 235.484

451.13 325.856 325.609

508.582 361.051 298.06

λlu , nm

6 2 P3/2

61 P1o 61 P3o 63 P1o 63 P1o 61 P1o

63 P1o 63 P2o 63 P3o

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

1

5 5

6

1 1

5 5 5

5 5 5

3 3 3

4 4 4

5 5 5

gl

3

3 5

4

3 3

3 7 3

3 5 7

1 5 5

2 4 6

3 7 7

gu

8.6 × 10−3

6.4 × 10−3 0.23

1 × 10−2 0.22 3 × 10−3

0.11

3.4 × 10−2 0.24

0.15 0.18 3.6 × 10−2

6.3 × 10−2 0.21 0.12

6.9 × 10−2 8.4 × 10−2 0.17

0.15 0.04 0.35

0.14 0.42 0.1

f lu

1.8

2.4 × 10−2 0.26

1.6 1.46 0.28

0.69 1.7 0.96

1.5 0.46 1.23

0.96 0.23 1.47

0.62 1.53 0.59

Aul × 108 , s−1

260

5 Measuring Concentrations of Atoms and Molecules

73 Ta( 4 F 6 3/2 )

71 Lu( 2 D 3/2 ) 6

64 Gd(69 D o ) 2

56 Ba( 1 S ) 6 0

Element, ground state, l

4.3073o 4.0361o 3.5411o 3.3253o 6 9 P3

2.261 × 10−19 J (1.413 eV)

0.043 × 10−19 J (0.0267 eV) 0.106 × 10−19 J (0.0661 eV) 0.198 × 10−19 J (0.124 eV) 0.341 × 10−19 J (0.213 eV) 1.576 × 10−19 J (0.985 eV) 1.686 × 10−19 J (1.054 eV) 0.395 × 10−19 J (0.247 eV) 0.398 × 10−19 J (0.249 eV)

5 1 D2

69 D3o

69 D5o

6 4 F5/2

6 2 D5/2

611 F8o

611 F2o

69 D6o

o 66 P5/2

o 62 F5/2

6 9 G8

6 11 D6

6 9 D6

6 9 D6

6 9 D5 6 9 F4

73 F2o 63 P1o 63 P0 63 D1

1.792 × 10−19 J (1.12 eV)

5 3 D1

69 D4o

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

333.78

384.118

440.925

440.314

417.554

405.364

407.870 432.569

371.357

428.31 472.645 582.628 648.291

390.991 599.709 601.947 659.532

λlu , nm

6

6

17

15

13

11

9 9

7

5 5 5 5

3 3 3 3

gl

6

6

17

13

13

13

11 9

7

7 3 3 7

5 3 1 3

gu

5.5 × 10−2 2.2*10−3

1.3*10−2

1.25

0.32

0.11

0.19

0.18 0.12

0.2

0.25 9.2×10−2 0.17 0.39

0.19 0.15 0.25 0.25

f lu

0.25

4.96

1.27

0.42

0.67

0.59 0.43

0.96

0.64 0.46 0.56 0.44

0.49 0.27 1.4 0.39

Aul × 108 , s−1

5.2 Determining Atomic Concentrations by Absorption Techniques 261

74 W( 5 D ) 0 6

Element, ground state, l o 3.919/2 o 5.6419/2 o 4.1011/2 o 3.6867/2 4 Po 3/2 o 3.44511/2 o 3.5679/2 o 4.1177/2

65 P1o 67 D1o 65 P2o 65 D1o 65 F4o 67 D3o 65 P3o 65 D3o 4.6462o 7 Po 4 4.2591o 3.8841o

0.733 × 10−19 J (0.458 eV) 1.115 × 10−19 J (0.697 eV) 1.936 × 10−19 J (1.21 eV) 2.648 × 10−19 J (1.655 eV) 0.331 × 10−19 J (0.207 eV) 0.659 × 10−19 J (0.412 eV) 0.958 × 10−19 J (0.599 eV) 1.138 × 10−19 J (0.711 eV) 0.586 × 10−19 J (0.366 eV) 1.890 × 10−19 J (1.181 eV)

6 4 F7/2

6 4 F9/2

6 6 D1/2

6 6 D9/2

6 5 D1

6 5 D2

6 5 D3

6 5 D4

6 7 S3

6 3 P0

o 3.8523/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

402.879 458.685

289.645 400.875

410.270 421.938

375.792 468.051

383.505 408.833

376.845 505.33

692.738 648.537 503.737

593.976 469.190

364.206 414.789

362.662 317.029

λlu , nm

1 1

7 7

9 9

7 7

5 5

3 3

10 10 10

2 2

10 10

8 8

gl

3 3

5 9

7 7

9 7

5 3

3 3

12 10 8

4 4

12 8

10 10

gu

2 × 10−2 4.2 × 10−3

1.26 0.17

4.9 × 10−2 6.1 × 10−3

1.4 × 10−2 1.4 × 10−2

5.2 × 10−2 4.1 × 10−3

3.5 × 10−2 1.9 × 10−2

1 × 10−2 5.8 × 10−2 4.4 × 10−2

1.6 × 10−2 4.1 × 10−2

5.8 × 10−2 1.8 × 10−2

7.1 × 10−2 8.5 × 10−2

Aul × 108 , s−1

1.5 × 10−2 4 × 10−3

0.113 5 × 10−2

9.6 × 10−3 1.3 × 10−3

3.8 × 10−3 4.6 × 10−3

1.1 × 10−2 6.2 × 10−4

7.4 × 10−3 7.3 × 10−3

8.7 × 10−3 3.7 × 10−2 1.3 × 10−2

1.7 × 10−2 2.7 × 10−2

1.3 × 10−2 3.7 × 10−3

1.7 × 10−2 1.6 × 10−2

f lu

262

5 Measuring Concentrations of Atoms and Molecules

4 6 a F9/2 )

80 Hg( 1 S ) 6 0

79 Au( 2 S 6 1/2 )

77 Ir(

Element, ground state, l

o 66 G9/2

1.254 × 10−19 J (0.784 eV) 1.410 × 10−19 J (0.881 eV) 1.818 × 10−19 J (1.136 eV) 4.253 × 10−19 J (2.658 eV) 7.467 × 10−19 J (4.667 eV)

a4 F7/2

b4 F3/2

7 3 S1 6 3 D3 8 3 D2 6 1 D2 6 3 D2

8.738 × 10−19 J (5.461 eV) 10.725 × 10−19 J (6.703 eV)

3 Po 2

1 Po 1

(6.5)

7 3 S1 7 1 S0 6 3 D2 8 3 S1

7.819 × 10−19 J (4.887 eV)

7 3 S1 6 3 D1 8 3 S1

o 62 P1/2

3 Po 1

63 P0o (1.45)

6 2 D3/2

o 62 P3/2

o 66 F7/2 o 66 F11/2 6 o 6 F9/2 o 5.0099/2 o 4.2017/2

0.562 × 10−19 J (0.351 eV)

b4 F9/2

6 2 D5/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Table 5.3 (continued).

579.066 576.96

546.074 365.014 302.347

435.835 407.781 312.566 289.36

404.656 296.728 275.278

627.817

312.278

357.372

293.464 362.867

322.078 269.423 266.198

λlu , nm

3 3

5 5 5

3 3 3 3

1 1 1

4

6

8

8 8

10 10 10

gl

5 5

3 7 5

3 1 5 3

3 3 3

2

4

10

10 8

8 12 10

gu

0.58 0.24

0.49 1.3 9.4 × 10−2

0.56 4 × 10−2 0.66 0.16

0.49 0.2

0.13 0.36 1.3 × 10−2

0.16 3.3 × 10−3 0.16 2 × 10−2

0.15 0.18 2.08 × 10−2

1×10−2

3.4 × 10−2 0.21 0.45 6.1 × 10−2

1.8 × 10−2

1.3 × 10−2

3.2 × 10−2 5.5 × 10−3

3 × 10−2 6.3 × 10−2 2.7 × 10−2

f lu

0.19

5.4 × 10−2

0.2 2.8 × 10−2

0.24 0.48 0.25

Aul × 108 , s−1

5.2 Determining Atomic Concentrations by Absorption Techniques 263

83 Bi(64 So ) 3/2

82 Pb( 3 P ) 6 0

7 2 S1/2 6 2 D5/2 8 2 S1/2 73 P0o 73 P1o 63 D2o 73 P1o 63 F3o 71 P2o 73 P1o 63 D1o 71 P1o 83 P2o 7 2 P1/2 7 4 P3/2 7 4 P1/2 6 2 D3/2 7 2 P3/2 7 4 P5/2 6 2 D3/2 7 2 P1/2 7 2 P3/2

1.546 × 10−19 J (0.966 eV) 1.550 × 10−19 J (0.969 eV) 2.112 × 10−19 J (1.32 eV) 4.240 × 10−19 J (2.65 eV)

2.266 × 10−19 J (1.416 eV) 3.062 × 10−19 J (1.914 eV) 4.298 × 10−19 J (2.686 eV)

o 62 P3/2 (0.23)

6 3 P1

6 3 P2

6 1 D2

o 62 D3/2 (0.032)

o 62 D5/2 (0.12)

o 62 P1/2 (0.016)

81 Tl(62 Po ) 1/2

Excited term u

El , J (eV)

Metastable term l, (τ, s)

Element, ground state, l

Table 5.3 (continued).

449.93 412.153 359.611

351.085 293.83 302.464

289.798 298.903 472.252

722.897 406.214 357.272 373.993

405.781 280.2 266.315

368.347 363.957 261.417

535.046 351.924 322.975

λlu , nm

2 2 2

6 6 6

4 4 4

5 5 5 5

5 5 5

3 3 3

4 4 4

gl

4 2 4

4 4 6

2 4 2

3 3 3 5

3 7 5

1 3 5

2 6 2

gu

1.51−2 0.16 0.2

6.8 × 10−2 1.23 0.28

1.53 0.55 0.17

9 × 10−3 0.92 0.99 0.73

0.89 1.6 0.71

1.5 0.34 1.9

0.7 1.24 0.17

Aul × 108 , s−1

9.1 × 10−3 4.2 × 10−2 7.7 × 10−2

8.4 × 10−3 0.11 0.12

9.6 × 10−2 7.4 × 10−2 2 × 10−2

4.2 × 10−3 0.14 0.11 0.15

0.13 0.26 7.6 × 10−2

0.1 6.8 × 10−2 0.32

0.15 0.35 1.35 × 10−2

f lu

264

5 Measuring Concentrations of Atoms and Molecules

5.2 Determining Atomic Concentrations by Absorption Techniques

Figure 5.2 Concentrations of metastable atoms, cm−3 . For conditions, see text. (a) Cathode region of a DC atmosphericpressure discharge in helium, discharge current 1 A, L – distance from the cathode: 1 – He(2 1 S0 ), 2 – He(2 3 S1 ). (b) Concentrations of the Ar(3 P2 ) atoms in the discharge of a surface microwave in an Ar−N2 gas mixture. (c) Concentrations of the Xe(3 P2 ) atoms in the discharge of a hollow microcathode: 1 – 26.664 hPa (20 Torr), 2 – 53.328 hPa (40 Torr), 3 – 99.990 hPa (75 Torr).

Figures 5.2a through c present some examples of measuring the concentrations of inert gas atoms in metastable states. Figure 5.2a shows the variation of the concentrations of the He(2 3 S1 ) and He(2 1 S0 ) atoms along the axis of an atmospheric pressure helium discharge with plane electrodes. The positive column diameter is 20 mm and the discharge current is 1 A. Distance is calculated from the cathode. The axial and radial distributions were found by measuring the integral absorption coefﬁcient on the 389 nm (2 3 S1 ) and 501-nm (2 1 S0 ) lines using a continuousspectrum probe source. Figure 5.2b illustrates the variation of the concentration of the Ar(3 P2 ) atoms as a function of the plasma-forming Ar−N gas mix proportion [21](b). The discharge in a 7.5 mm diameter quartz tube was sustained by a 2.4 GHz surface wave at a constant electron density of ne = 1.2 × 1012 cm−3 . The measurements in these experiments were taken by the line absorption method using an auxiliary argon discharge. The variation range of the atomic concentration is 1011 –1013 cm−3 . When using

265

266

5 Measuring Concentrations of Atoms and Molecules

the same method in a DC discharge [18], the minimum detectable concentrations of the Ar(3 P2 ) atoms amounted to 5 × 107 cm−3 , with the optical path length equal to 60 cm. In the work [21](c) devoted to the measurement of the integral absorption coefﬁcient with a view to determining the concentration of the Xe(3 P1 3 P2 ) atoms in a hollow microcathode discharge (discharge length 1.8 mm), use was made of a tunable diode laser. Figure 5.2c illustrates the results of measuring the concentrations of the Xe(3 P1 3 P2 ) atoms (823-nm transition) as a function of the discharge current at various xenon gas pressures. 5.2.3 Low-Multiplicity Positive Ions

In conditions of nonequilibrium low-temperature plasma (Chapter 1), the degree of ionization of atoms and molecules is, as a rule, relatively low, and when studying the charge balance of plasma, ion-molecular reactions and so on, of interest is the behavior of low-multiplicity ions. Table 5.4 provides data on the wavelengths and probabilities of transitions involving such ions in the ground state, which can be used to measure their concentrations. As with Tables 5.1–5.3, this table lists elements and absorbing transitions in the UV though the near IR regions of the spectrum [5–9, 12, 15, 18–20]. Table 5.4 Transitions involving the ground state of ions and their probabilities. Element, ground state l

Upper levelu

λ, nm

gl

gu

Aul × 108 , s−1

f lu

4 Be+ (22 S

2 Po 5/2 2 Po 3/2 2 Po 1/2 4P 3/2

313.063 313.042 313.106

2 2 2

6 4 2

1.15 1.15 1.15

0.51 0.34 0.17

232.469

2

4

7.3 × 10−7

5.9 × 10−8

3 Do 1 2D 3/2 2 Po 5/2 2 Po 3/2 2 Po 1/2 4P 1/2

1083.98 9897.9

1 2

3 4

3.2 13

0.17 0.11

279.79 279.553 280.27

2 2 2

6 4 2

2.67 2.68 2.66

0.94 0.63 0.31

233.44

2

2

1.1 × 10−4

9 × 10−6

2 Po 5/2 2 Po 3/2 2P 1/2

394.52 393.366 396.847

2 2 2

6 4 2

1.48 1.5 1.46

1.04 0.69 0.34

1/2 )

6 C + ( 22 P o ) 1/2 7 N + ( 23 P ) 0 7 N2 + ( 2 2 P ◦ ) 1/2 12 Mg+ (32 S 1/2 )

14 Si+ (32 Po ) 1/2 20 Ca+ (42 S 1/2 )

5.2 Determining Atomic Concentrations by Absorption Techniques Table 5.4 (continued). Element, ground state l

Upper level u

λ, nm

gl

gu

Aul × 108 , s−1

f lu

21 Sc+ (43 D

3 Po 2 3 Fo 2 3 Do 1 3 Po 0 3 Do 2 4 Go 5/2 4 Fo 3/2 2 Fo 5/2 4 Do 1/2 5 Do 1 5 Fo 1 7 Po 3 7 Po 2 6 Do 9/2 6 Fo 11/2 6 Po 7/2 6 Fo 9/2 2D 5/2

356.770 364.279 358.092 336.193 356.770

3 3 3 3 3

5 5 3 1 5

0.35 1.13 1.23 1.17 0.35

0.11 0.37 0.24 6.6 × 10−2 0.11

338.77 324.199 320.344 307.298

4 4 4 4

6 4 6 2

1.09 1.16 2.1 × 10−2 1.6

0.28 0.18 4.8 × 10−3 0.11

268.309 270.522

1 1

3 3

0.34 4.3 × 10−2

0.11 1.4 × 10−2

259.373 260.569

7 7

7 5

2.6 2.7

0.26 0.2

259.936 238.204 234.35 237.374

10 10 10 10

10 12 8 10

2.2 3.8 1.7 0.33

0.22 0.33 0.11 2.8 × 10−2

399.306

6

6

5.2 × 10−9

1.2 × 10−9

2 Po 1/2 2 Po 3/2 2 Po 1/2 2 Po 3/2 3 Po 1 1 Do 1 1 Po 1 2 Po 1/2 2 Po 3/2 2 Po 1/2 2 Po 3/2 3 Do 2 6 Ko 9/2 9P 3 9P 4 9P 5

206.191 202.551

2 2

2 4

3.2 3.3

0.2 0.41

421.552 407.771

2 2

2 4

1.27 1.42

0.34 0.71

420.047 349.609 311.204

1 1 1

3 3 3

2.2 × 10−2 0.35 1.3 × 10−2

1.75×10−2 0.19 5.7×10−3

226.502 214.441

2 2

2 4

3.0 3.0

0.23 0.41

493.409 455.403

2 2

2 4

0.95 1.17

0.35 0.73

408.672

5

5

0.51

0.13

430.358

8

10

0.47

0.16

420.505 412.97 381.967

9 9 9

7 9 11

0.71 0.69 1.27

0.15 0.18 0.34

64 Gd+ (610 Do ) 5/2 65 Tb+ (67 Ho ) 8 66 Dy+ (66 I 17/2 )

8D

333.138

6

8

0.55

0.12

350.917

16

18

1.1

0.34

353.17

18

20

1.4

0.29

67 Ho+ (65 Ho ) 8 68 Er+ (64 H 13/2 )

29412, cm−1

339.898

17

17

0.63

0.11

29641 25592

337.276 390.631

14 14

16 12

1.3 0.28

0.24 5.5 × 10−2

22 Ti+ (44 F

1)

3/2 )

23 V+ (35 D ) 0 25 Mn+ (47 S

3)

26 Fe+ (46 D 9/2 )

28 Ni+ (32 D 5/2 ) 30 Zn+ (42 S 38 Sr+ (52 S 39 Y+ (51 S

1/2 )

1/2 )

0)

48 Cd+ (52 S 56 Ba+ (62 S

1/2 )

1/2 )

57 La+ (53 F

2)

60 Nd+ (66 I

7/2 )

63 Eu+ (69 So ) 4

7/2

7H

17/2

6 Ko 19/2

267

268

5 Measuring Concentrations of Atoms and Molecules Table 5.4 (continued). Element, ground state l

Upper level u

λ, nm

gl

gu

Aul × 108 , s−1

f lu

69 Tm+ (63 Fo ) 4

31927 28875 25980

313.126 346.22 384.802

9 9 9

11 11 7

1.1 0.45 0.5

0.2 0.1 8.6 × 10−2

70 Yb+ (62 S

2 Po 3/2 2 Po 1/2 3 Do 1 2 Po 1/2

328.927 369.419

2 2

4 2

1.8 1.4

0.58 0.29

219.554

1

3

4 × 10−2

4 × 10−2

194.227

2

2

3.43

0.19

1/2 )

71 Lu+ (61 S ) 0 80 Hg+ (62 S

1/2 )

The procedure for measuring ion concentrations by the absorption techniques and the sensitivities of the latter are similar to those in the case of neutral atoms. An example of measuring titanium ion concentration in the plasma of a low-pressure arc discharge by the standard line absorption method in a single probe radiation pass is presented in Figure 5.1b. 5.3 Determination of Molecular Concentration by the Absorption Method

Compared to atoms, molecules have more developed spectra, which makes the problem of determination of their concentrations under nonequilibrium conditions speciﬁc. Experimentally this entails higher demands on the spectral resolution, aperture ratio and dynamic range of the instrumentation. Atomic and molecular spectra are both processed using, in principle, the formulas of Section 2.3. In practice, however, a number of additional circumstances arise in the case of molecules. It is necessary, for example, to know numerous optical transition probabilities. Further, the relationship between the concentration of molecules at the electronic–vibrational–rotational level of experimental interest and the total particle density should be analyzed in sufﬁcient detail, which in the absence of equilibrium is far from trivial. The latter requires investigating the form of the energy-level distribution of the particles and their grouping into a limited number of ensembles and is attained by the methods described in Chapter 4. As regards optical transition probabilities, use is, as a rule, made of an approximation based on the separation of the nuclear and electronic motions in the molecule.

5.3 Determination of Molecular Concentration by the Absorption Method

5.3.1 Probabilities of Optical Transitions in Diatomic Molecules

The methods for calculating and measuring the probabilities of radiative transitions in diatomic molecules and the results obtained have been generalized, and are being systematically updated, in many reviews, monographs and electronic data bases. Among the relatively recent and more thorough ones are the works by Kuznetsova and co-workers [22, 23]. Here we present some basic expressions for the radiative characteristics necessary in using the reference materials available in the literature. The determination of particle concentrations from absorption measurement results is based on the main relations (2.61, 2.62). If one neglects spontaneous emission, then for the integral line absorption on the electronic transition with the lower rotational level J and the upper rotational level J , one can write the expression for the relation between the absorption coefﬁcient and the particle concentration at the lower level as: χ J J =

8π3 νJ J Sˆ N . g J 3hc J J J

(5.3)

Here Sˆ J J is the line strength (compared to expression (2.59), the hat here is introduced to distinguish between the total line strength and the rotational factors presented later in the text), NJ is the density of the absorbing molecules at the lower level and νJ J is the transition frequency. The line strength is in turn expressed in the dipole approximation in terms of the matrix elements of the dipole moment operator Pˆ of the molecule: * *2 Sˆ J J = Sˆ J J = ∑ * Ψ M | Pˆ |Ψ M * , (5.4) M M

where Ψ M are the wave functions for individual Zeeman magnetic states. The following, generally accepted, scheme of composing expressions for radiative characteristics is based on the well-known Born– Oppenheimer approximation on the possibility of representing the function Ψ M as a product of the electronic, vibrational and rotational wave functions. In the case of isotropic nonpolarized radiation, this gives, with the use of formula (5.4), the following expression for the rotational lines of the spin-allowed (ΔS = 0) electronic–vibrational transitions: Sˆ J J = | Ψv | Re (r ) |Ψv |2 S J J = | Rv v |2 S J J .

(5.5)

Here Re is the electronic transition moment, r is the internuclear separation, v and v are respectively the upper and the lower vibrational ¨ quantum number of the transition, S J J is the Honl–London factor characterizing the dependence of the radiative characteristic on the rotational

269

270

5 Measuring Concentrations of Atoms and Molecules

states and Rv v is the electronic–vibrational transition moment. In writing S J J in the form of expression (5.5), the Born–Oppenheimer principle is used but partially; that is, it is only part of the wave function associated with the molecular rotation that is isolated. When this principle is used more consistently, the squared wave function overlap integral qv v = | Ψv | Ψv |2 ,

(5.6)

known as the Franck–Condon factor, is introduced as a factor into the squared matrix element | Rv v |2 . However, as proved in practice and demonstrated by more detailed analyses, the latter procedure is the main source of error in the Born–Oppenheimer approximation. Therefore, although such a separation is made empirically, in so doing one keeps Re dependent (usually weakly) on the vibrational quantum numbers through the intermediary of the internuclear separation:

| Rv v |2 ≈ qv v | Re (rv v )|2 .

(5.7)

This method is called the r-centroid approximation The line strength S J J is the same for both absorption and emission ¨ and is independent of the method used to normalize the Honl–London factors. Also presented in the literature are other characteristics, including those for the description of electronic–vibrational and electronic transitions as a whole. If the transition moments R are expressed in the atomic units a0 e (a0 is the Bohr radius and e is the electron charge), the line strength in a20 e2 = 6.46 × 10−36 CGS units, the Einstein coefﬁcient A in s−1 , the wave numbers in cm−1 , and the radiative lifetime τ in seconds, with the oscillator strengths being dimensionless, then the electronic transition strength Se = (2 − δ0,Λ +Λ )(2S + 1)| Re (r )|2 ,

(5.8)

the electronic–vibrational band strength Sv v = (2 − δ0,Λ +Λ )(2S + 1) | Rv v |2,

(5.9)

band oscillator strength f v v = L1

(2 − δ0,Λ +Λ ) ν | Rv v |2, (2 − δ0,Λ ) v v

(5.10)

line oscillator strength f J J = L1

S J J (2 − δ0,Λ +Λ ) ν | Rv v |2 (2 − δ0,Λ ) J J 2J + 1 ,

(5.11)

5.3 Determination of Molecular Concentration by the Absorption Method

band Einstein coefﬁcient Av v = L2

(2 − δ0,Λ +Λ ) 3 ν | Rv v |2, (2 − δ0,Λ ) v v

(5.12)

line Einstein coefﬁcient A J J = L2

S J J (2 − δ0,Λ +Λ ) 3 , νJ J | Rv v |2 (2 − δ0,Λ ) 2J + 1

(5.13)

radiative lifetime τv− 1 = L2

(2 − δ0,Λ +Λ ) νv3 v | Rv v |2 . (2 − δ0,Λ ) ∑ v

(5.14)

Here L1 = 8π2 mca20 /3h = 3.04 × 10−6 , L2 = 64π4 a20 e2 /3h, the factor ¨ (2 − δ0,Λ +Λ ) is associated with the normalization of the Honl–London factors S J J , the factors (2 − δ0,Λ )(2 − δ0,Λ ) allow for the double degeneracy of the electronic state with Λ = 0 (lambda doubling) and δmn = 1 at m = n and δmn = 0 at m = n. With these deﬁnitions, the relation between the absorption coefﬁcient integrated along the line proﬁle and the molecular concentration at the level J , v of the electronic state m has the form

χ J J (ν) dν = χ J J

line

=

S J J 8π3 Sv v Nmv J νJ J , cm−2 3hc (2 − δ0,Λ +Λ )(2S + 1) 2J + 1

(5.15)

or χ J J =

S J J 8π3 Se (rv v ) , cm−2 . ( a0 e)2 Nev J νJ J qv v 3hc (2 − δ0,Λ +Λ )(2S + 1) 2J + 1 (5.16)

Compared to formula (5.3), the above expression includes as a numerical factor the squared dipole moment a20 e2 in order to allow one to use the value of Se in atomic units and retain the dimension of χ J J (ν) in cm−1 and that of χ J J in cm−2 , when measuring the molecular density N in cm−3 and the wave number ν in cm−1 . The coefﬁcient 8π3 a20 e2 /3hc ≈ 2.7 × 10−18 . If the integral absorption coefﬁcient χ J J or the quantities related to it (Chapter 2) are deﬁned (Sections 3.2 through 3.5), then in order to ﬁnd the concentration Nev J by formulas (5.15), (5.16) it is necessary to use the data available on the quantities entering into them. Sources where these data for diatomic molecules are collected most fully and systematically collected, include:

271

272

5 Measuring Concentrations of Atoms and Molecules

• constants associated with molecular structure [24–26], • oscillator strengths of electronic transitions, Se [22], • tables of Franck–Condon factors qv v [23], • tables of r-centroids [27], ¨ • formulas for calculating the Honl–London factors S J J [28] (Appendix E). Complete data sets are only available for a limited number of molecules, even diatomic ones (Appendix D). 5.3.2 Determination of Diatomic Molecular Concentrations from Absorption on Electronic Spectrum Lines

Table 5.5 below lists some data necessary for taking the measurements under discussion for a number of molecules. In addition to the literature sources indicated above, use was made in compiling the table of individual, original publications and of the modern database updates, speciﬁcally www.chem.msu.ru of the Moscow State University, Department of Chemistry. The ﬁrst column of the table gives the molecules, the states bound by the electronic transition and the bond type (after Hund [26]) for these states. The choice of the molecules and transitions is limited, on the one hand, by the availability of complete data sets, and on the other by the transition spectral range 200–1000 nm. The transition frequencies (wave numbers) are listed in the second column. Tabulated in the third and the fourth column are the constants of the vibrational (ωe , ωe xe ) and the rotational (Be , αe ) structure, respectively, of the terms of the electronic ground state. The ﬁfth column gives the electronic transition strengths Se corresponding to the equilibrium conﬁguration r00 . In making this choice, consideration was given the fact that measurements should preferably be taken for the most populated vibrational level v of the electronic ground state. Listed in the sixth and the seventh columns are the maximum values of the Franck–Condon factors for the levels v = 0 and v = 1, respectively. The eighth column gives the ratios between the spin-orbit splitting constant and the rotational constant, Y = A/Be , if the states involved have an intermediate bond type. ¨ These values are used in calculating the Honl–London factors S J J by the formulas of Appendix E.

Transition frequency ν00 , cm−1

20635

23646 23521 16772

21197

43226

42924 49340

23217 25698 31778 9245 25752 38798

20396

15368

Molecule, transition, bond type

AlO(X2 Σ(b)−−B2 Σ(b))

BO(X2 Σ(b)−−A2 Π)

BaO(X1 Σ(b)−A1 Σ(b))

BeO(X1 Σ(b)−B1 Σ(b))

C2 (X1 Σ(b)−D1 Σ(b))

CF(X2 Π–A2 Σ(b)) -B2 Δ

CH(X2 Π–A2 Δ, -B2 Σ− (b), -C2 Σ+ (b)) CN(X2 Σ(b)-A2 Π, -B2 Σ(b)) CS(X1 Σ(b)-A1 Π(b))

Cu2 (X1 Σ(b)−B1 Π(b))

K2 (X1 Σ(b)−B1 Π(b))

14.457 0.53 1.90 0.017 0.82 0.006 0.1087 0.0006 0.0567 0.000165

2068.59 13.09 1285 6.46 264.5 1.025 92.02 0.2829

0.6413 0.0058 1.782 0.0166 0.3126 0.0014 1.651 0.019 1.8198 0.0176 1.417 0.0184

Be , cm−1 αe , cm−1 Ground state

2858 63

979.23 6.97 1885.69 11.81 669.76 2.02 1487.32 11.83 1854.71 13.34 1308.1 11.1

ωe , cm−1 ωe xe , cm−1 Ground state

= 0.901 = 0.865 = 0.999 = 0.503 = 0.918 = 0.781

0.32±0.02 0.17±0.1 0.29 0.41±0.12 0.93±0.07 0.08

26

1.7

q03 = 0.338 q00 = 0.676

0.84±0.16 0.88

q02 = 0.249

q01 = 0.32

q00 q00 q00 q00 q00 q00

q00 = 0.997

q00 = 0.89

q04 = 0.165

q03 = 0.12

q00 = 0.723

qmax 0,v

0.42±0.05

0.56

0.004± 0.016 1.09

1.12±0.35

Se , at. unit (a0 e)2

Table 5.5 Parameters of electronic transitions involving the electronic ground state in diatomic molecules.

= 0.98 = 0.569 = 0.998 = 0.37 = 0.78 = 0.42

q10 = 0.250

q10 = 0.41

q11 q11 q11 q10 q11 q11

q10 = 0.205 q10 = 0.282

q11 = 0.993

q11 = 0.7

q12 = 0.152

q11 = 0.187

q11 = 0.337

qmax 1,v

27.9/14.46(2 Π) -1.0/14.9(2 Δ)

77.1/1.417(2 Π) 0.76/1.32(2 Δ)

122.3/1.402(2 Π)

Y = A/Be

5.3 Determination of Molecular Concentration by the Absorption Method 273

Transition frequency ν00 , cm−1

20635

17838 14671 15150 14021

16500

45392 44080 52251

32684

31689

38095

34638

42641

Molecule, transition, bond type

AlO(X2 Σ(b)−−B2 Σ(b))

LaO(X2 Σ(b)-B2 Σ(b), -C2 Π)

Li2 (X1 Σ(b)−A1 Σ(b))

MgO(X1 Σ(b)−B1 Σ)

NO(X2 Π–B2 Π, -A2 Σ(b), -C2 Π)

OH(X2 Π−−A2 Σ)

S2 (X3 Σ− (b)−B3 Σ+ (b))

SO(X3 Σ− (b)−A3 Π)

SiF(X2 Π−−B2 Σ(b))

SiO(X1 Σ(b)−A1 Π(b))

Table 5.5 (continued).

725.6 2.844 1149.2 5.6 857.19 4.735 1241.55 5.966

3737.76

351.43 2.61 785 5.1 1904 14.07

979.23 6.97 812.7 2.22

ωe , cm−1 ωe xe , cm−1 Ground state

0.2954 0.0016 0.7208 0.0057 0.5812 0.005 0.7267 0.005

18.91 0.72

0.6726 0.007 0.574 0.005 1.672 0.0171

0.6413 0.0058 0.3526 0.0014

Be , cm−1 αe , cm−1 Ground state

1.3

8.8

5.5

5.5

q01 = 0.255

q02 = 0.241

q00 = 0.58

q0,13 = 0.0668

q00 = 0.907

q0.12 = 0.061 q01 = 0.33 q01 = 0.32

0.078± 0.005 0.079±0.11 0.36 0.042± 0.001

q00 = 0.933

q10 = 0.279

q10 = 0.217

q12 = 0.35

q19 = 0.669

q10 = 0.893

q1.8 = 0.068 q10 = 0.262 q10 = 0.25

q11 = 0.945

q12 = 0.197

q11 = 0.618 q11 = 0.990

q00 = 0.86 q00 = 0.997 q03 = 0.19

q11 = 0.337

qmax 1,v

q00 = 0.723

qmax 0,v

0.8

12.2

5.3 6.08

1.12±0.35

Se , at. unit (a0 e)2

162/0.627(2 Π)

160/0.61(3 Π)

139/16.9(2 Σ) 139/18.5(2 Π)

123/1.7(X2 Π) 31/1.09(B2 Π) 3/2(C2 Π)

221.4/0.34(2 Π)

Y = A/Be

274

5 Measuring Concentrations of Atoms and Molecules

Transition frequency ν00 , cm−1

20635

14163 14096 14019 20408

25556

Molecule, transition, bond type

AlO(X2 Σ(b)−−B2 Σ(b))

TiO(X3 Δ−−A3 Φ)

CO+ (X2 Σ(b)−A2 Π)

N2+ (X2 Σ(b)−B2 Σ(b))

Table 5.5 (continued).

2214 15.16 2207 16.1

979.23 6.97 1009.02 4.498

ωe , cm−1 ωe xe , cm−1 Ground state

1.997 0.019 1.9318 0.0188

0.6413 0.0058 0.5354 0.003

Be , cm−1 αe , cm−1 Ground state

0.97±0.01

0.21±0.06

43.7

1.12±0.35

Se , at. unit (a0 e)2

q00 = 0.651

q01 = 0.43

Q00 = 0.72

q00 = 0.723

qmax 0,v

q12 = 0.406

q12 = 0.36

Q12 = 0.33

q11 = 0.337

qmax 1,v

-117/1.59(2 Π)

49/0.53(3 Δ) 55/0.51(3 Φ)

Y = A/Be

5.3 Determination of Molecular Concentration by the Absorption Method 275

276

5 Measuring Concentrations of Atoms and Molecules

If one has no knowledge about the character of plasma equilibrium, one should then ﬁnd the total particle concentration by direct summation of the concentrations at the individual levels, which is practically unfeasible, the energy spectrum of molecules being as developed as it is. Under local thermal equilibrium conditions, the relation between the concentration Ne,v,J of diatomic molecules at a single vibrational–rotational level of the electronic state e has, according to expression (1.6), the form Ee + Ev + E J ge,v,J exp − , (5.17) Ne,v,J = N Q kB T where Ee , Ev , and E J are the energies of the electronic, vibrational and rotational motions, respectively, ge,v,J is the statistical weight of the level and Q is the statistical sum of the molecule (Appendix A). For the sake of brevity, let us denote Ne,v,J = NJ , J = J . Under partially nonequilibrium conditions (the PLTE model, Section 1.3), one can introduce distribution temperatures in the different degrees of freedom, namely, electronic, Te , vibrational, Tv and rotational, Tr : ga,s EJ 1 Ev Ee I (2J + 1) exp − . (5.18) + + NJ = N Q kB Tel Tv Tr The following replacements are made here: ge,v,J = ga,s I (2J + 1), where ga,s is the nuclear statistical weight of the antisymmetric and symmetric I levels, Q = Qn Qin , where Qn = (2I1 + 1)(2I2 + 1)σ−1 , I1 and I2 are the nuclear spins and σ is the symmetry number (equal to 1 for heteronuclear molecules and 2 for homonuclear ones). For heteronuclear molecules, ga,s I = Qn , and for homonuclear ones, see Table A.6 of Appendix A. Qin = Qel Qv Qr =

∑ Qel ∑ Qv ∑ QJ , el

v

J

Ee , Qel = (2 − δ0,Λ )(2S + 1) exp − kB Tel EJ . Qr = exp − kB Tr

(5.19) Ev Qv = exp − , kB Tv (5.20)

It follows from expressions (5.18) through (5.20) that the molecular concentration determination methods are at a great advantage where the measurement result is the concentration of molecules at a level belonging to the electronic ground state. Of importance for the direct absorption methods is the fact that the ground state levels are, as a rule, populated more than the levels of electronically excited states, and so the absorption coefﬁcients are easier to measure, all other things being equal. It is demonstrated in Section 5.4 below that introducing partial rotational and

5.3 Determination of Molecular Concentration by the Absorption Method

vibrational temperatures by the PLTE model for the electronic ground state proves justiﬁed within a wide range of plasma existence conditions, something which cannot be said of the electronically excited states of molecules. With the vibrational and rotational temperatures present, the corresponding statistical sums, calculated through direct summation by formulas (5.19) and (5.20), are given by simple expressions (A11) and (A13) of Appendix A. Of course, the requirement remains to know the electronic statistical sum Qel , and the introduction of a uniﬁed distribution temperature for the electronic levels is problematic (Section 4.4). However, on a positive note the quantity Qel usually barely differs from the electronic statistical sum of the ground state, this difference in many cases being calculabale with sufﬁcient accuracy to have no harmful effect on the measurement result. For example, the typical rotational and vibrational statistical sums in a low-current low-pressure nitrogen glow discharge (Section 4.4) are Qel ≈ 150, Bv ≈ 2 cm−1 , kB Tr ≈ 300 K, and Qv ≈ 3, hνv ≈ 3400 K, kB Tv ≈ 3000 K, respectively. Assume that the electronic states of the molecule are populated by direct electron impact and depopulated on collisions with electrons, and this in a radiative fashion. In that case, the quantity Qel is principally contributed to not only by the ground state (X1 Σ, 2 S+1=1, δ0,Λ = 1), but also by the ﬁrst metastable state (A3 Σ, 2 S+1=3, δ0,Λ = 1) with an energy of ca. 9.600 × 10−19 J (6 eV). With the electron velocities distributed by a Maxwellian distribution with kB Te ≈ 3.680 × 10−19 J(2.3 eV), the contribution from the metastable state amounts to some 15%: Qel ≈ 1 + 3 exp { − 3} ≈ 1.15. This rough upper-bound estimate takes no account of the deactivation of the metastable molecules on collisions with walls, in the processes of energy transfer to collision partners and so on. The actual proportion of metastable molecules in the above-indicated conditions [29, 30] comes to ca. 10−4 (Figure 4.39). Nevertheless, estimates of this kind should not be completely disregarded. Standing out in this respect is the example of the O2 discharge plasma where the so-called singlet oxygen O2 (1 Δ, 2 S+1=1,δ0,Λ = 0, energy 1.600 × 10−19 J (1 eV)) decays but poorly on the walls and can accumulate, given specially selected discharge conditions, in quantities of (20–30)% of the amount of oxygen in the ground state O2 (X3 Σ, 2 S+1=3,δ0,Λ = 1) (see review [31]). In this case, according to experimental data, Qel ≈ 3 + 2(0.2to0.3)} ≈ 3.4–3.6. The highest measurement sensitivity is attained with absorption at the levels of the electronic–vibrational ground state X, v . The choice of the serial number of the rotational level is governed by the requirements on the measurement sensitivity and dynamic range. The maximum sensitivity, subject to the condition kB Tv < hν01 , is reached with Jmax ), at the

277

278

5 Measuring Concentrations of Atoms and Molecules

Figure 5.3 Partial pressure of the OH radicals as a function of the pressure of the gas added to water vapor, PH2 O = 0.400 hPa(0.3 Torr). Discharge in a 24 mm diameter. tube at a current of 30 mA: 1 – N2 ; 2 – He; 3 – CO2 –N2 -He (1:3:6); 4 – O2 ; 5 – CO2 .

maximum of the Boltzmann rotational distribution: ' kB Tr 1 Jmax = − . 2Bv 2

(5.21)

Figure 5.3 presents the results of measuring the partial pressure of hydroxyl radicals in a glow discharge by the line absorption method on the OH(X2 Π–A2 Σ) electronic transition [32]. 5.3.3 Determination of Molecular Concentration from Absorption in Vibrational–Rotational Spectra

Molecular concentrations can be determined from absorption in the IR region on the lines of vibrational–rotational transitions entailing no change in the electronic state. One can use again formulas (5.15) and (5.9) provided that the electronic state e is ﬁxed. By virtue of expression (5.4), if Ψ are the wave functions of vibrational–rotational states, the line strength is other than zero only if the molecule has a constant dipole moment.

5.3 Determination of Molecular Concentration by the Absorption Method

Table 5.6 lists data on the probabilities of the v = 0 ↔ v = 1 vibrational transitions in some diatomic molecules in the electronic ground states. These data partially overlap the data listed in Table 4.2 (the slight differences between them, however, fall within the error limits indicated in the sources). We present here only a few interrelated quantities (Appendix B), because they are often given separately in the literature, and their deﬁnitions in different works may differ by factors. Given in the ﬁrst column of the table are molecules and their electronic ground states. The second column lists, according to formula (2.60), the values of the absorption coefﬁcient χ01 for the v = 0 → v = 1 transition, integrated over all the branches of the vibrational band. The conditions under which this quantity is measured are usually such that the vibrational level population N1 N0 , stimulated emission (2.61) can be neglected, and N0 corresponds to the known total particle density, N0 ≈ N. These measurements are practically always taken at room temperature in equilibrium gas. For most diatomic molecules, kB T ≈ 210 (cm−1 ) ν10 (cm−1 ), and the neglect of the populations of the excited vibrational levels is justiﬁed to a good precision. In many literature sources, the integral absorption coefﬁcient χ01 is normalized to a partial molecule pressure of 1 atm at a temperature of 296 K to have the dimension [cm−2 · atm−1 ], is denoted by S˜lu [cm−2 · atm−1 ] and is called intensity (see, for example, [12–14]). In other sources, for example the HITRAN data base (see, for example, [15, 16] for description and pertinent references), it is also called intensity, but is normalized −1 −2 ˜ = ˜H to a single molecule, S˜H lu [cm /(molecule · cm )], Slu Slu NL , where NL = 2.7 × 1019 cm−3 is the Loschmidt number. To avoid possible confusion, we note the need to distinguish between the designations of the line and band strengths in formulas (5.8) through (5.16) and those of the absorption intensities (integral absorption coefﬁcients). For this reason, when denoting intensities in this text, we use the tilde (which is not done in the original works, and discrimination by meaning is necessary). Under these conditions, the experimentally determined integral absorption coefﬁcient of the fundamental vibrational band is related, for example, to the oscillator strength by the relation [33] χ01 [cm−2 · atm−1 ] ≡ S˜lu [cm−2 · atm−1 ] = 2.38 × 107 f lu .

(5.22)

The relationships with the rest of the fundamental characteristics of the radiative transition probability are established by formulas (5.8) through (5.14). Thus, the third column of Table 5.6 gives the values of the squared matrix element | R01 |2 in atomic units D2 (1 D = 10−18 CGS units). Listed in the fourth and ﬁfth columns are the Einstein coefﬁcients A10 [s−1 ] and the dimensionless oscillator strengths f 01 , respec-

279

280

5 Measuring Concentrations of Atoms and Molecules

tively. The sixth column gives the wave numbers ν10 of the centers of the 0–1 transition bands. Table 5.6 Probabilities of radiative vibrational transitions in the fundamental IR bands of some diatomic molecules. Molecule, ground state

χ01 [cm−2 · atm−1 ]

| R01 |2 [D2 ]

A10 [s−1 ]

f01

ν10 [cm−1 ]

270

1.1×10−2

35

1.1×10−5

2170

OH(X2 Π)

100

2.5×10−3

40.6

4.3×10−6

3737

NO(X2 Π)

120

5.7×10−3

12.3

5×10−6

1904

HF(X1 Σ)

400

8.6×10−3

1.6×10−5

4138

DF(X1 Σ)

220

6.5×10−3

54.5

9×10−6

2998

HCl(X1 Σ)

140

4.1×10−3

34.5

5.7×10−6

2991

DCl()

74

3.1×10−3

9.5

3.1×10−6

2145

HBr()

55

1.8×10−3

10.7

2.3×10−6

2649

CO(X1 Σ)

190

It follows from comparison between Tables 5.5 and 5.6 that the sensitivity of absorption measurements is, on the whole, higher on electronic than vibrational transitions. With the molecular concentration at the vibrational–rotational level J of the ground state being the same, the ratio between the respective integral (over the line proﬁle) absorption coefﬁcients, according to formulas (5.8), (5.15) and (5.16), is χel J J χvib J J

2 el νel J J | Re | qv v S J J = vib * . * *2 Svib νJ J * Rvib v v J J

(5.23)

The superscripts ‘el’ and ‘vib’ here denote the corresponding quantities for electronic and vibrational transitions. In the region of maximum absorption intensity (Table 5.5), for small v , v the factors qv v are ca. 1. Similarly, it follows from calculations by the formulas of Appendix E that vib the ratio Sel J J /S J J is ca. 1. For example, in the case of the OH radical, for the (X2 Π, v , J = 7/2 – A2 Σ, v , J = 5/2) and (X2 Π, v , J = 7/2 vib – X 2 Π, v , J = 5/2) P-branch transitions, the ratio Sel J J /S J J ≈ 0.8, the

quantity q00 ≈ 0.9, and the wave-number ratio νel /νvib ≈ 9. And though the strength of the OH(2 Π – 2 Σ) electronic transition is not very great, Se = 4 | Re |2 ≈ 4 × 10−2 D2 , ratio (5.23) proves to be ca. 30. In practice, however, this does not mean that the IR absorption methods are less effective. The laser spectroscopy techniques developed today (Chapter 3) enable one to register small amounts of absorption in the IR region of the spectrum. Record-high concentration sensitivity is not always a decisive factor in plasma diagnostics, while the IR absorption

5.3 Determination of Molecular Concentration by the Absorption Method

spectroscopy methods are universal enough and are capable of determining the concentrations of several components at once. The mutual complementarity of the methods of absorption on electronic and vibrational transitions is also associated with the fact that electronic transitions involving the ground state in the majority of molecules fall within the VUV region of the spectrum that is inconvenient for diagnostics purposes, and the IR methods are restricted to molecules possessing a dipole moment of their own. 5.3.4 Absorption of Radiation by Diatomic Molecules in Metastable Electronic States

In principle, the determination of the concentration of molecules in metastable electronically excited states from absorption on electronic transitions differ only slightly from the concentration measurements for molecules in the ground state. The difference has already been discussed above, namely, the proportion of the metastable molecules relative to the total amount of the given molecular species is usually small, although the concentrations of the metastable molecules and unexcited radicals in plasma may be comparable. The determination of the concentration of metastable molecules in plasma by this method has been described in many works. For example, the authors of [29, 30] (see also Figure 4.39) measured the concentrations of the N2 (A3 Σ) molecules. Worthy of note is also the work by Pazyuk and co-workers [39] who were the ﬁrst to directly measure the concentrations of the O2 (1 Δ) singlet oxygen molecules (with a radiative lifetime of 45 min) by the intracavity absorption technique (Section 3.3, Figure 3.13). The choice of transitions suitable for measurement in this case being extremely limited, they used the forbidden O2 ( a1 Δg → b1 Σg ) transition with a wavelength of 1.91 μm and an absorption cross section of ca. 7 × 10−22 cm2 . The minimum detectable singlet oxygen concentration amounted to ca. 1014 cm−3 . 5.3.5 Absorption of IR Radiation by Polyatomic Molecules

The use of the IR absorption methods for determining the concentrations of polyatomic molecules is, on the whole, preferable. The principal reason is that the structure of the electronic spectra, and even their fragments, in polyatomic molecules is very complex, frequently smeared because of overlapping and information about the probabilities of electronic transitions is scarce. The measurement of the IR absorption of

281

282

5 Measuring Concentrations of Atoms and Molecules

polyatomic molecules with a small number of atoms is generally quite similar to that of diatomic molecules. But new conditions arise here that require consideration. The presence of several types of vibration in polyatomic molecules means that the structure of their IR absorption spectrum over a wide frequency range is more developed than that of diatomic molecules. As a rule, the rotational constant of polyatomic molecules is smaller than that of their diatomic counterparts of comparable mass. Experimentally these circumstances set high demands on the aperture ratio and spectral resolution of the instrumentation and optical materials used. Essential in the determination of molecular concentrations is the question of the magnitude of the internal statistical sum and its determination precision. This problem arises as early as the stage of experimental determination of the probabilities of vibrational transitions, even under thermodynamic equilibrium conditions, if the vibrational quantum energies of even some modes are commensurable with kB T. Therefore, to ﬁnd the radiative transition probability characteristics independent of the gas density, the integral absorption measurements should be supplemented with calculations of the total internal statistical sums [29, Appendix 1]. Table 5.7 lists data on the integral absorption coefﬁcients for some characteristic IR bands of tri- through heptatomic molecules, measured under equilibrium conditions at room temperature [33]. Table 5.7 Integral absorption coefﬁcients (intensities) S˜lu [cm−2 · atm−1 ] for some vibrational bands of polyatomic molecules. Molecule

Band, ν, cm−1 3716

CO2

N2 O

H2 O

COS

C2 H2 C 2 N2

2349

S˜lu

Molecule

39 2706

Band, ν, cm−1

S˜lu

950

600

NH3

1627

110

3337

20

PH3

2323

260

712

204

3312

241

667

187

1285

383

2223

1650

1595

300

3755

100

DCN

2629

136

2079

2633

CNCl

2214

78

2064

4300

CS2

1523

2520

729

723

SiF4

1031

2635

3287

278

SF6

947

4800

2150

30

CF4

633

4540

HCN

5.3 Determination of Molecular Concentration by the Absorption Method

Figure 5.4 Variation of the concentration of the CO2 molecules as a result of dissociation in a discharge. Conditions are as indicated in Figure 4.26.

¨ The Honl–London factors are calculated by the same relations of Appendix E as used for diatomic molecules, with due regard for the fact that the upper and the lower vibrational level can differ in symmetry (Σ, Π, Δ,. . . ). When measuring the total absorption of polyatomic molecules in nonequilibrium plasma, in order to take account of the proportion of particles at the excited levels one must investigate the distributions among the various degrees of freedom of the internal motions. This can be done by the methods described in Chapter 4. At a great advantage in such a situation are the laser absorption spectroscopy methods of high resolution and aperture ratio. Their uses have in fact been already described in Chapter 4. Let us present for illustration the results of processing the absorption spectra of the CO2 molecules in a discharge when determining their vibrational–rotational distributions and the corresponding temperatures (Figure 4.26). Figure 5.4 shows the dissociation-induced variation of the concentration of carbon dioxide molecules in the active medium of a waveguide CO2 laser as a function of the discharge current [38].

283

284

5 Measuring Concentrations of Atoms and Molecules

5.3.6 Absorption of Radiation by Molecular Ions

The concentrations of molecular ions are determined by the absorption spectroscopy methods on both electronic and vibrational–rotational transitions of the ground state. Table 5.8 gives information concerning the electronic transitions involving the ground state in the widespread molecular ions CO+ and N2+ . Progress in the development of IR laser absorption spectroscopy (Chapter 3) stimulated, beginning in the early 1980s, numerous investigations into molecular ions, their formation mechanisms and their behavior in conditions of low-temperature gas-discharge plasma. Speciﬁcally devoted to these problems were the reviews [40, 41]. Table 5.9 lists, based on the material presented in [40, 41] with minor additions, the di- and polyatomic ions determined and investigated from absorption on vibrational–rotational transitions in gas discharges. The ﬁrst column gives the ions and their electronic ground states. Tabulated in the second column are the vibrational bands involved, in the third column, the probe radiation sources (see Figures 3.7, 3.8) (NM - difference-frequency radiation produced by nonlinear mixing, CCL – color-center laser, DL – diode laser), and in the fourth column are notes and commentaries.

Transition frequency ν00 , cm −1

20408

25556

Molecule, transition l − u, bond type

CO+ (X2 Σ(b)-A2 Π)

N2+ (X2 Σ(b)-B2 Σ(b)) 2207 16.1

2214 15.16

ωe , cm−1 ωe xe , cm−1 ground state

1.9318 0.0188

1.997 0.019

Be , cm−1 αe , cm−1 ground state

0.97±0.01

0.21±0.06

Se , at. units ( a0 e )2

q00 = 0.651

q01 = 0.43

qmax 0,v

Table 5.8 Parameters of electronic transitions involving the electronic ground state in molecular ions.

q12 = 0.406

q12 = 0.36

qmax 1,v

-117/1.59(2 Π)

Y = A/Be

5.3 Determination of Molecular Concentration by the Absorption Method 285

(0,1)

ν1 , CCL, NM, DL

DL

ν2

HCO +

ν2 , ν3

NM

D3+

NM, DL

HeH + (X1 Σ)

H2 D + HD2+

NM, CCL, DL

ν2 , 2ν1 − −ν2 ν1 , ν2 , ν3

H3+

DL DL

(0,1), (1,2)

DL

DL

(0,1), (1,2)

(X1 Σ)

(0,1) – (6 to 7)

DL

NO +

CCl

+

C2– ( X2 Σ)

CF +

CO

DL

(0,1), (1,2)

HCl + (X2 Π3/2 )

(X2 Σ+ )

DL

OH – ( X1 Σ), OD− ( X1 Σ)

+

DL

DL

NM

NM

Probe source

OH + (X3 Σ)

(0,1), (1,0)

(0,1)

(X1 Σ)

+

ArH + (X1 Σ)

NeH

HeH

Bands (l, u)

(X1 Σ)

+

Ions (ground state)

[10, 11]; H35 Cl+ : ωe = 2673.69, ωe xe = 52.54, Be = 9.96

Hollow cathode, velocity modulation; for ν1 : ν01 = 3088.74 cm−1 , B0 = 1.487 cm−1

Discharge 2 m; ν01 = 2910.95 cm−1 , B0 = 33.56 cm−1

Velocity modulation; radial distributions in the discharge (see text)

ωe = 1175, ωe xe = 5

ωe = 1781.04, ωe xe = 11.58, Be = 1.75

AC discharge in helium with CO, N2 admixtures. Absorption in the ground state and on the X2 Σ(v = 0) – A2 Π1/2 (v = 1) electronic transition; velocity modulation; ωe = 2214.2, ωe xe = 15.16, Be = 1.98

35 Cl, 37+ Cl

Rotational transitions (DL); HC band (0,1); velocity modulation; OH− : ωe = 3700, Be = 18.9]; OH− : ωe = 2700, Be = 10.02

Rotational transitions (DL); HC band (0,1); velocity modulation; ωe = 3113.37, ωe xe = 78.52, Be = 18.9

ν01 = 2589.3 cm−1 , B0 = 10.27 cm−1 , ωe xe = 61.6 cm−1 ; ν12 = 2470.5 cm−1 , B1 = 9.9 cm−1 ; concentration, temperature, and mobility measurements (see text)

Discharge 2 m; 20 NeH+ − −ν01 = 2900 cm−1 , B0 = 17.88 cm−1 , ωe xe = 111 cm−1 ; 22 NeH+ − − ν = 2894 cm−1 , B = 17.80 cm−1 0 01

Discharge 2 m, ν01 = 2910.95 cm−1 , B0 = 33.56 cm−1

Notes, commentaries

Table 5.9 Ions determined by IR laser absorption spectroscopy methods in plasma.

286

5 Measuring Concentrations of Atoms and Molecules

ν2

ν3

ν1 , ν3

ν2 ,

ν3

ν1 , ν3

ν1

ν1

ν1 ,

ν3

H2 Cl +

H2 O +

NH2–

H3 O + D3 O +

CH3+

H3 S +

HCO2+

HN2 O +

H2 CN + HDCN +

NH4+

ν4

ν3

FHF –

NH3 D

ν3

HBF +

+

ν2 ,

ν1 ,

CO2+

HCS

ν1 , ν3

H2 F +

+

ν1 ,

ν1 , ν3

ν2

ν3 , ν4

ν3

ν2

ν2 , ν3

Bands (l, u)

HN2+ DN2+

DCO

+

Ions (ground state)

Table 5.9 (continued).

NM

NM, CCL

NM

NM

NM

NM

NM

CCL, NM, DL

CCL

HC

DL

DL

DL

DL

CCL, DL

CCL

CCL, DL

DL

Probe source

Individual lines of the ν3 mode

Velocity modulation; more than 200 lines, ν3 = 3143.14 cm−1 , B3 = 5.8 cm−1

Velocity modulation; for ν2 (C-H bond): ν2 = 3187.86 cm−1

Velocity modulation

Detailed discussion of the spectrum [40, 41]

Velocity modulation; for ν1 : ν01 = 3334.67 cm−1 , B0 = 12.88 cm−1

Velocity modulation; for HN2+ ν1 : ν01 = 3233.95 cm−1 , B0 = 1.541 cm−1 ; for DN2+ ν1 : ν01 = 2636.98 cm−1 , B0 = 1.286 cm−1

Notes, commentaries

5.3 Determination of Molecular Concentration by the Absorption Method 287

288

5 Measuring Concentrations of Atoms and Molecules

The results of the investigations conducted are used both for plasma diagnostics purposes and to gather comprehensive spectroscopic information on molecular ions. In the ﬁrst place, this forms the basis for the determination of the concentrations of ions, their spatial and energy distributions. Let us present as an example some results of investigations carried out in argon discharges where the ArH+ ions were produced as a result of the trace amounts of hydrogen present [42]. To register absorption spectra, use was made of a tunable diode laser. The laser beam passed parallel to the discharge axis. The radial resolution was around 2 mm. Subject to registration were the v = 0, J = 3 → v = 1, J = 2 (2525.475 cm−1 ) and v = 0, J = 3 → v = 1, J = 2(2525.414 cm−1 ) transitions. The dipole moment matrix elements of the vibrational transitions of the ion are large enough (| R10 |2 = 6.25 × 10−2 D2 , | R21 |2 = 1.17 × 10−1 D2 ). Together with the high magnitude of the rotational constant (B0 = 10.46 cm−1 ) limiting the rotational statistical sum, this ensured good measurement sensitivity. Under the given experimental conditions, absorption reached 17% with the path length equal to 2 m. Also measured were the Doppler broadening and shift in order to determine the kinetic temperature (the rotational temperature was taken equal the kinetic) and the ion drift velocity. These data formed the basis for calculating the vibrational temperature and the total internal statistical sum of the ion, necessary for the determination of the ion concentrations. Figure 5.5 shows the radial distribution of the ions in the discharge tube. The shape of the distribution strongly differs from the Besselian form prescribed by the Schottky diffusion discharge theory.

5.4 Actinometric Methods

The direct spectral methods for determining particle concentrations in nonequilibrium plasma from the integral absorption frequently prove difﬁcult to use. For example, particles may not have any IR (dipole) absorption spectrum. Or the electronic absorption spectra fall within the VUV region of the spectrum that presents experimental difﬁculties. The various versions of the nonresonance scattering techniques (Section 3.5) have limited sensitivity. These problems arise, in the ﬁrst place, with atomic radicals that only exist in low concentrations, their chemical activity being high, and vanish outside of plasma, and, in contrast to many molecular radicals, have no strong absorption lines in the spectral regions convenient for measurements.

5.4 Actinometric Methods

Figure 5.5 Radial distribution of the ArH+ ions. Discharge in argon with trace amounts of hydrogen present. Discharge tube diameter 25 mm, discharge current 1.25 A, pressure 7.999 hPa (6 Torr).

In contrast, such particles in plasma may possess intense electronic emission spectra, but as already discussed, to relate the intensities of these spectral lines with the concentration of the unexcited particles is a very intricate problem. But in individual cases proving this relation is possible by way of special indirect measurements. The so-called actinometry technique is one such method. At its root is comparison between the intensities of the spectra of two different particle species. The concentration of the particles of one species, X, is to be determined, and the concentration of the particles of another species, A, is known. The particle of species A is called the actinometer. The emission intensities IX,A of the particles X, A on the ul transitions for an optically thin plasma layer are given by −1 X,A X,A X,A Aul k u NX,A quX,A + (τuX,A )−1 , IX,A = CX,A hνul

(5.24)

where NX,A are the concentrations of the particles X, A in the electronic ground states (these concentrations are assumed to practically coincide with the total concentrations of these particles), k uX,A are the rates of excitation of the upper transition levels u from the ground states on a perparticle basis, τuX,A are the radiative lifetimes of the upper states u, quX,A

289

290

5 Measuring Concentrations of Atoms and Molecules

are the radiationless decay frequencies of the states u as a result of colX,A lisions (quenching), Aul are the Einstein coefﬁcients of the transitions, and CX,A are coefﬁcients governed by the solid angles of the emissions collected, the transmission of the optical system and the detection spectral sensitivity. The sought-for concentration is NX = NA

A AA A A A −1 IX C A νul ul k u q u + ( τu ) . X X X X X I A CX νul Aul k u qu + (τu )−1

(5.25)

If, under low-pressure plasma conditions and at short radiative lifetimes one can deﬁne qu τu−1 , the only problem remaining is to reveal the ratio between the excitation rates of the emitting states. The ﬁrst practical suggestion as to the solution of this problem was made and tested in [43, 44] on the important additional assumption that the upper transition levels were excited by direct electron impact: k u = ne

∞

f (ve )σe,u (ve )ve dve ,

(5.26)

0

where σe,u is the cross section for the excitation of the level u from the ground state g, ve is the electron velocity and f (ve ) is the electron velocity distribution function. The subsequent reasoning is based on the Born approximation for excitation cross section calculations [30]: * *2 gu (5.27) σe,u (ve ) = * Me (ΔEgu )* ϕ(ε e /ΔEgu )qv v . Here Me2 is the maximum excitation cross section, ϕ describes the dependence of the cross section on the electron energy ε e = mv2e /2, ΔEgu is the energy difference between the upper and the ground level (the threshold gu excitation energy), and qv v is the Franck–Condon factor (for the excitation of molecular electronic–vibrational bands). The quantities | Me |2 are calculated in the Bethe–Born approximation in terms of the oscillator strengths f gu of the transitions [30]. For the allowed g − u transitions in atoms, * * * Me (ΔEgu )*2 = 1.48πa2 (Ry/ΔEgu )2 f gu , (5.28) 0 where Ry is the ionization potential of the hydrogen atom and a0 is the Bohr radius. Similar formulas for molecules can be found, for example, in [45–47]. The function ϕ in the same approximation has the form ϕ(ε e /ΔEgu ) = 2.7(ΔEgu /ε e ) ln(ε e /ΔEgu ).

(5.29)

Strictly speaking, this approximation is unjustiﬁed near the threshold, but comparison between the results of calculations by formula (5.28) and

5.4 Actinometric Methods

Figure 5.6 Excitation cross sections (in relative units) of some molecules as a function of the electron energy (in terms of the excitation threshold). (a) Optically allowed and (b) optically forbidden transitions: 1 – average over experimental data (3 through 8), 2 – calculation by for-

mulas (5.27)–(5.29), 3 – H(1 S−2 P), 4 – H2 (X1 Σ–B1 Σ), 5 – H2 (X1 Σ–C1 Π), 6 – H2 (X1 Σ–D1 Π), 7 – CO(X1 Σ – A1 Π), 8 – N2 (X1 Σ-b1 Π), 9 – N2 (X1 Σ − a1 Π), 10 – N2 (X1 Σ – B3 Π), 11 – CO(X1 Σ − a3 Π), 12 – H2 (X1 Σ − b3 Σ), 13 – N2 (X1 Σ – A3 Σ), 14 – N2 (X1 Σ – C3 Π).

experimental data shows that they agree well enough in the case of allowed transitions and ϕ(ε e /ΔEgu ) (5.29) is in fact a universal function, with a maximum around ε e /ΔEgu ≈ 4, depending only weakly on the type of the states g, u and the species of the atom or molecule. Figure 5.6a presents as an example some results of calculating and measuring the excitation cross section as a function of the electron energy (in terms of the threshold excitation energy) for allowed transitions. The situation with the excitation on optically forbidden transitions is somewhat more involved (some examples are presented in Figure 5.6b), though in this case, the excitation cross section maxima are also grouped in the vicinity of ε e /ΔEgu ≈ 1.2–1.6. In low-temperature plasma, the mean electron energy ε¯ e frequently proves less than ΔEgu , and so the maximum contribution to excitation rate constant (5.26) is from the electrons with energies close to the maximum of the excitation cross section. Based on what has been said above, one can assume [43, 44] that if the excitation energies of the emission lines of the particles X and the

291

292

5 Measuring Concentrations of Atoms and Molecules Table 5.10 References for plasma actinometry. X A

O

CO, CO + CO2 , CO2+ F

Ar

[51–53]

[51, 54]

Cl ·

Br

[51]

[51, 52], [58, 59] [60] [54–56]

[51]

[51, 52], [59] [55, 56]

H

N

N2 , N2+ CFx [61]

[51, 54], [56]

CO [43, 44, 48] [48] N2

[51, 52]

[51, 54]

[48]

[48]

[54, 56]

Ne He

[61]

Kr Xe

[62] [62, 63] [62]

actinometer A are nearly equal, the ratio between the excitation rate constants in expression (5.25) should depend only weakly on the distribution f (ve ), and hence on the conditions prevailing in plasma. But for this to happen, the following requirements should naturally be met: (i) the states u in the particles X and A should be excited by direct electron impact, (ii) the collisional deexcitation (quenching) of the levels u during the course of spontaneous decay (and other intramolecular processes like predissociation, autoionization, etc.) should be insigniﬁcant (iii) the threshold excitation energies of the states u in the particles X and A should be nearly equal, and (iv) the g → u transitions in both particle species should preferably (but not strictly necessarily) be of one and the same type (forbidden or allowed). Under real conditions, these requirements are only fulﬁlled (if at all) with a limited accuracy. Nevertheless, wherever it is possible to prove by independent measurements, the actinometric method demonstrates satisfactory results. As already noted, the actinometric method was used in [43, 44] to determine the concentration NO of the oxygen atoms in the study of plasma chemical processes occurring in a carbon dioxide discharge. In this case the actinometer was the stable CO molecule whose concentration was measured chromatographically in samples taken after the discharge was switched off. This technique was later used, in particular, when measuring NO in the active medium of a CO laser [48] and an RF oxygen discharge [49]. Note that the results of measurements taken in [49] correlate well with those obtained in [50] under similar conditions by the electron paramagnetic resonance technique. The actinometric method was most actively used in conditions of plasma chemical reactors when studying the etching of the materials used in microelectronics. Table 5.10 lists some selected references to works on the actinometry of plasma.

5.4 Actinometric Methods

Table 5.11 lists some parameters of the particles A and X that allow one to evaluate the correspondence of their combinations to the prerequisites of the actinometric procedure under its application conditions. Table 5.11 Data to choose the actinometric pairs. u

l

Eu , eV

λul , nm

τu , ns

6s[3/2]2 4p [1/2]0 4p [1/2]1

4p[5/2]3 4s [1/2]1 4s [1/2]1

14.84 13.48 13.33

703.02 750.39 826.45

360 21 60

5p [3/2]2

5s [1/2]1

10.56

826.32

34.1

6s[3/2]2

7p[3/2]2

11

462.43

105.6

3p 2 P3/2

3s 2 P3/2

14.75

703.75

26

4s 4 P5/2

10.4

837.6

36

2p 2 P1/2,3/2

12.09

656.4

17.6

3s 4 P1/2,3/2,5/2

12

742–746

29.6

3s 3 S1 3s 5 S2

10.99 10.74

844.6 777.5

35.1 30

B1 Σ A1 Π

A1 Π X1 Σ

10.78 8.07

412–612 114–280

30 15

X2 Σ

B2 Σ

X2 Σ

5.69

180–315

50

N2

X1 Σ

C3 Π

B3 Π

11.05

281–498

40

CF

X2 Π

B2 Δ

X2 Π

6.1

197–220

20

A, X

g

Ar

3p6

1S

0

Kr

4p6

1S

0

Xe

5p6

1S

F

2p5

Cl

3p5

H

1s 2 S1/2

N

3p3

4S

O

2p4

3P

CO

X1 Σ

CO+

0 2P 3/2 2P 3/2

4p 3d

3/2 0,1,2

3p 3p 3p

4D 7/2 2D 3/2,5/2 4S 3/2 3P 0,1,2 5P 1

Some examples of the application of the actinometric method are illustrated in Figures 5.7 and 5.8. Presented in Figure 5.7 are the results of measuring the axial distribution of the concentrations of oxygen atoms (the actinometer is the CO molecule) in a CO2 discharge in a tube with three metal electrodes arranged at the center and at both ends of the tube [24]. The effective role of the recombination of the atoms on the electrode metal is obvious. Donnely and co-workers [57] studied the etching of an SiO2 coating on electrodes in CF4 /O2 /Ar, NF3 /Ar, and F2 /Ar discharge plasmas. The authors noted a correlation between the etching rate and the density of the ﬂuorine atoms formed in the discharge (the actinometer is the Ar atom) (Figure 5.8b). As regards the method under discussion, note that the dynamic range of concentration measurements in the given example amounted to three orders of magnitude. An attractive feature of actinometry is undoubtedly its technical ease. This method can be used as the basis for the operation of sensors for controlling plasma processes and the like. However, one should be aware that the reliability of actinometric measurements depends entirely on the prerequisites indicated above. Despite the fact that experience with this method has been generally positive, individual cases are known where

293

294

5 Measuring Concentrations of Atoms and Molecules

Figure 5.7 Axial distribution of the relative (to the total concentration of particles) concentrations y0 of oxygen atoms in a CO2 discharge in a 32 mm diameter. tube with three electrodes. P = 1.467 hPa(1.1 Torr), discharge current 30 mA, discharge length 640 mm. Data points – measurements, curve – approximation of the experiment by the kinetic scheme [29].

difﬁculties were encountered. To illustrate, when measuring the concentrations of the CCl radicals in a CCl4 discharge plasma, Doebele and coauthors [64] noted contradictory results. The reason was that the luminescence of the radicals was due, at least in part, to the birth of the excited CCl∗ radicals on the dissociation of CCl2 , CCl3 , and CCl4 . Gottsho and Donnelly [58] studied RF discharges in CF4 /O2 /Ar and Cl2 /Ar. In the former case, actinometric and independent laser-induced ﬂuorescence measurements gave coincident results as to the concentration of ﬂuorine atoms. In the latter case these methods were also found to produce similar results on the concentration of chlorine atoms, except for the concentration dynamics in the near-electrode layer, though when averaged over the electric ﬁeld period, these results were observed to agree well enough. The cause of the differences was that the production of the excited Cl∗ atoms in the cathode layer, when the electrode played the part of the cathode, was strongly affected by heavy particle interaction mechanisms, in particular dissociative attachment. This manifested itself in the abnormal Doppler broadening of the actinometric lines. We have

5.4 Actinometric Methods

Figure 5.8 Etching rate of SiO2 as a function of the concentration of ﬂuorine atoms: open and full circles – CF4 –O2 ; open and full squares – NF3 −Ar; full triangles – F2 −Ar. Full symbols – one of the electrodes coated with silicon. 1 – approximation of experimental data by the kinetic scheme [56].

considered such phenomena in Section 4.1, and they certainly violate the prerequisites of actinometry. In recent years, the ideas of actinometry have been developing along new lines, speciﬁcally involving the use of laser-induced ﬂuorescence (LIF) technique. As already discussed in Chapter 3, the LIF technique belongs in the class of indirect absorption methods. The proportion of the absorbed radiation can be very small and difﬁcult to measure, especially if it is two-photon absorption. For this reason, we correlate the attendant ﬂuorescence with the concentration of the absorbing particles requires calibration. In the case of equilibrium, the appropriate calibration methods have been well worked through (see, for example, [11]), but they are absolutely unﬁt under nonequilibrium conditions. However, if in place of the excitation rates in expression (5.24) the optical pumping rates k u ∼ Ilu σlu (Ilu is the intensity of the radiation exciting ﬂuorescence and σlu , the photoabsorption cross-section) are substituted, the sought-for particle concentration NX is expressed in terms of the actinometer particle concentration NA in the form of expression (5.25), where A σ A / I X σ X . Excitation cross sections for allowed transitions k uA /k uX ∼ Ilu lu lu lu (including resonance ones in the visible and near UV regions of the spectrum) are, as a rule, known (see, for example Tables 5.1 and 5.2 and references cited therein). But for the majority of particles whose transitions

295

296

5 Measuring Concentrations of Atoms and Molecules

Figure 5.9 Two-photon excitation diagrams for the Kr, H, N, Xe, and O atoms.

involving the ground state fall within the far UV region, the two-photon laser-induced ﬂuorescence (TALIF) methods (Section 3.5) prove useful. Such techniques were used in [49, 64–66] to determine the densities of the H, O and N atoms using the Kr and Xe atoms as actinometers. The two-photon excitation of ﬂuorescence in these particles was effected by UV lasers at 224–226 nm (Figure 5.9). The ratios between the twophoton excitation cross sections of the appropriate particles, measured in [63] with an accuracy of some 5% (as estimated by the authors) were as follows: σuKr (5p [3/2]2 = 0.62; σuH (32 D J )

σuKr (5p [3/2]2 = 0.67; σuN (32 D J )

σuXe (5p [3/2]2 = 0.36. σuO (3p3 P J )

The great advantage of the LIF actinometry over the actinometry of spontaneous plasma emission is that it requires no knowledge of the electron velocity distribution. The LIF technique also makes it easier for one to take account of the quenching of the upper states u. Table 5.12 lists data on the quenching rate constants kq (quenching rate q[s−1 ] = kq [cm3 · s−1 ] · Nq [cm−3 ], Nq is the density of the quenching particles) for the states used in the above measurements. These data can be used in the actinometry of not only low-, but also elevated-pressure plasmas [65]. Since the LIF-excitation is carried out in a pulse-periodic regime, its effect can be discriminated against the background of spontaneous plasma emission by way of synchronous detection.

5.5 Negative Ions Table 5.12 Quenching rate constants kq , 10−10 cm3 · s−1 . Quenching

Kr(5p [3/3]2 )

H(3d2 D3/2,5/2 )

N(3p4 S3/2 )

Xe(7p[3/2]2

O(3p3 P0,1,2 )

826.3 nm

656.3 nm

745 nm

426.4 nm

844.6 nm

particle H2

8.44

20.4

1.11

28.5

N2

3.35

20.1

0.41

14

10.9 5.9

O2

6.34

32.6

6.63

20.6

9.3

He

0.78

0.18

0.11

2.33

Ar

1.29

3.93

0.37

3.16

0.007

Kr

1.46

7.15

3.16

5.91

−

Xe

3.78

19.8

7.75

5.05

−

CH4

6.27

25

5.81

0.25

−

−

One should, however, bear in mind that the LIF version of actinometry is technically perceptibly more difﬁcult to implement than spontaneous plasma emission measurements. So far, this fact has restricted the mass-scale application of this technique, but it can serve as a good methodical basis for revealing the applicability conditions of spontaneous emission actinometry, and the rapid progress of laser technology allows one to deﬁnitely talk about the future practical prospects of the method. For information on spontaneous emission actinometry, see also the reviews [13, 67, 68].

5.5 Negative Ions

Many atoms and molecules can capture external electrons and thus form negative ions [69–71]. The difference in energy between the ground states of an atom (molecule) and its negative ions, E0 − E0− = EA , is called the electron afﬁnity energy. This energy must be expended to detach the electron captured. Table 5.13 list the values of EA , eV, for some particles. Table 5.13 Energy of electron afﬁnity. H

O

F

S

Cl

Se

C

I

Au

Pt

0.75

1.46

3.4

2.07

3.61

2.02

1.26

3.07

2.31

2.13

K

Na

Rb

Si

OH

CH

NH

O2

CO3

NO

0.5

0.55

0.49

1.39

1.83

0.7

0.38

0.45

2.8

2.7

297

298

5 Measuring Concentrations of Atoms and Molecules

Typically EA < E0 , and so for negative ions to exist in perceptible amounts, it is necessary that the source of free electrons be sufﬁciently intense and the kinetic temperature low. In the case of equilibrium, this follows directly from the formulas for the law of mass action, (1.9) through (1.13), whereby the concentrations of negative ions can be calculated. Under nonequilibrium conditions, such considerations are, generally speaking, incorrect – the analysis of the elementary process kinetics is necessary, but their qualitative aspect remains valid. Under such conditions, the experimental methods for registering negative ions play an important part. 5.5.1 Concentration Measurements

The spectroscopic method is based on the phenomenon of photodecay (photodetachment) of negative ions: X− + hν → X + e.

(5.30)

This corresponds to a continuous absorption spectrum for bound–free transitions with the limiting frequency hν = E A . The inverse process, photoattachment, X + e → X− + hν,

(5.31)

gives the corresponding continuum in emission. The cross sections for photodecay, σd , and photoattachment, σa , are related by the detailed balancing relation σa =

gi k 2 σ , ga q 2 d

(5.32)

where ga and gi are the statistical weights of the atom and the ion, respectively, k = 2πν/c is the photon wave number, q is the electron wave number, hq/2π = (2mEe )1/2 , gi = (2L− +1)(2S− + 1), ga = (2La + 1)(2Sa + 1), L− and S− are the orbital moment and the spin quantum number, respectively, of the negative ion and La and Sa , the same for the atom. There are theoretical and experimental methods for determining the cross sections of photoprocesses (5.31) and (5.32) [69, 70]. Figure 5.10 presents the cross sections σd for some atoms and molecules [69–74]. The values of σd typically range between 10−18 and 10−16 cm2 . With the density of negative ions being around 1010 cm−3 , the absorption coefﬁcient χ is ca. 10−8 – 10−6 cm−1 . Therefore, when determining the densities of negative ions by absorption techniques, use should be made of their high-sensitivity

5.5 Negative Ions

Figure 5.10 Photodetachment cross sections for some atoms and molecules.

versions (Chapter 3). But even though absorption in this case is so small, one should bear in mind that the probe radiation can cause considerable perturbation, because each absorption act entails the death of a negative ion. A change in the density N − of negative ions is accompanied by the corresponding change in the electron density in the irradiated volume: Δne (t) + N − (t) = N0− (t),

(5.33)

where N0− is the unperturbed density of negative ions. For light with the intensity I and frequency ν [3, 4], σa (ν) I (ν) − dΔne = ( N0 − Δne ), dt hν + ,

σa (ν) Δne (t) = N0− 1 − exp − I (ν) dt . hν

(5.34) (5.35)

If the probe radiation is pulsed, the integral in expression (5.35) is the light pulse energy. The change in the number of electrons due to irradiation can be determined, for example, by independent spectroscopic,

299

300

5 Measuring Concentrations of Atoms and Molecules

Figure 5.11 Excess electron density in photodetachment as a function of the probe radiation energy. Discharge in NF3 . Data points – measurements, curve – approximation by formula (5.35) at N0− = 2.4 × 109 cm−3 [14, 78].

probe, or radio-frequency techniques. In that case, one can ﬁnd the unperturbed density N − by approximating, in accordance with expression (5.35), the pulse energy dependence of the measured values of Δne . An example of such dependence is presented in Figure 5.11. With this measurement method, the energy of light perturbing the charge density can be directly determined from the saturation of this dependence. If one manages to relate the charge current with the electron density (for example, by means of special calibration or proceeding from known electron and ion mobilities), one will then ﬁnd it convenient to perform the measurements by the high-sensitivity optogalvanic spectroscopy technique (Section 3.4.2) Such measurements have been carried out in many works (see, for example, [14, 76–79]). Figure 5.12 illustrates an example of measuring [77] the density of the H− ions relative to the electron density in a high-current low-pressure arc. In this case, the proportion of the negative ions is relatively small – N0− /ne ≈ 10−3 –10−2 at ne in the range 1011 –1012 cm−3 . But entirely different situations also occur. For example, the density of negative ions in ﬂuorine and chlorine containing gas discharges can considerably exceed the electron density, so that N0− /ne ≈ 50 [78, 79].

5.5 Negative Ions

Figure 5.12 Ratio between the density of the H− ions and the electron density as a function of the hydrogen pressure. High-current arc [14, 77].

5.5.2 Absorption of Light by the H− Ions in Hydrogen LTE Plasma

The optical properties of the hydrogen plasma are of special interest. First of all this is due to the fact that negative ions play an important role in the radiation balance of stellar atmospheres. It is a well-established fact, for example, that the H− ions contribute much to the absorption of visible light in the solar photosphere, though their density is low – H− /H ≈ 10−8 . Understandably these densities are difﬁcult to measure directly, but since the state of plasma in such conditions can be described well enough in the LTE plasma model approximation (Section 1.3.1), they can be calculated. And since the hydrogen atom is, in addition, the simplest type, numerous attempts have been made to quantitatively analyze the corresponding absorption spectrum theoretically (see [80]). Here quantum-mechanical and thermodynamic calculations of the spectrum of bound–free (b − f ) and free–free ( f − f ) transitions are performed: H− (1s2 ) + hν → H(1s) + e,

(5.36)

H(1s2 ) + e + hν → H(1s) + e.

(5.37)

Since these are two- and three-particle processes, respectively, it is convenient to express the total absorption coefﬁcient χ = χ(36) + χ(37) in terms

301

302

5 Measuring Concentrations of Atoms and Molecules

Figure 5.13 Rate constants for the absorption of light by negative hydrogen ions at various temperatures.

of the integral photoabsorption rate constant η: χ(ν) = χ(36) (ν)+ χ(37) (ν) = η (ν) NH ne ,

(5.38)

− − χ(36) = σH NH , χ(37) = η (37) NH ne .

(5.39)

The dimensions here are, as usual, χ [cm−1 ] and η [cm5 ]. The calculation results for the rate constants η (λ) at various temperatures are presented in Figure 5.13 (plotted on the y-axis is the constant ξ (λ) = η (λ)[1 − exp(− hν/kB T )], which corresponds to correction for stimulated transitions). It can be seen that the major contribution in the visible region of the spectrum is from b − f transitions (5.36), while in the region λ > λth = 1.644 μm, from f − f ones (5.37). The results [80] also show that absorption in the UV region of λ < 0.3 μm can be contributed to by the H2+ ions, the contribution being considerable at temperatures T < 4000–5000 K. For one to determine the absorption coefﬁcient from the data of Figure 5.13, one should know the densities NH and ne . They can be calculated by formula (1.12) for the law of mass action and its particular case – the Saha equation (1.13). They can also be found with graphical accuracy from the results of such calculations presented in Figure 1.3d. To illustrate, at T = 10000 K and a pressure of 1 atm, NH ≈ 1018 cm−3 , ne ≈ 2 × 1016 cm−3 (see Figure 1.3d), ξ (0.55 μm) ≈ 10−38 cm5 (Figure 5.13), and χ(0.55 μm) ≈ 2 × 10−4 cm−1 . In the region of relatively low temperatures (T < (1.600 × 10−19 J (1 eV)–3.200 × 10−19 J

References

(2 eV), the internal statistical sums in formulas (1.12), (1.13) can be replaced by the statistical weights of the electronic ground states of the + − particles, ge , gH = 2, gH , gH = 1. The binding energy of the hydrogen atom is EH =21.770 × 10−19 J (13.606 eV) and that of the negative ion, EH− = 1.206 × 10−19 J (0.754 eV). Substituting these data into formu+ las (1.12), (1.13) and assuming that ne ≈ NH , we get 0.754 − −23 −3/2 ne NH , exp (5.40) NH = 8.28 × 10 T T 13.606 3 NH . (5.41) n2e = 3.018 × 1021 T /2 exp − T Here T is in electron-volts and N and n, in cm−3 . If we take the temperature T at 6000 K = 0.517 eV and the atomic hydrogen density NH at 1016 –1017 cm−3 , the values typical of the solar photosphere, extrapolation of the data of Figure 5.13 yields ξ(0.55 μm, 6000 K) ≈ 5 × 10−38 cm5 . If − NH = 1016 cm−3 , then ne = 6.46 × 1012 cm−3 , NH = 6.19 × 108 cm−3 , − 9 − 1 χ(0.55 μm, 6000 K) ≈ 3.2 × 10 cm . At the same temperature and − NH = 1017 cm−3 , ne = 2.04 × 1013 cm−3 , NH = 1.95 × 109 cm−3 , and − 7 − 1 χ (0.55 μm, 6000 K)≈ 1.02 × 10 cm . The thickness of the solar photosphere is ca. 3 × 108 cm, and mechanisms (5.36), (5.37), now in addition to others, provide for its high optical density in continuous spectrum. References

1 A.A. Babushkin, P.A’ Bazhulin, F.A. Koroleva et al. Spectral Analysis Techniques (in Russian). Moscow: Moscow State University Press (1962). 2 V.M. Chulanovsky. An Introduction to Molecular Spectral Analysis (in Russian). Moscow-Leningrad: GITTL (1950). 3 M. Thomson and D.N. Walsh. A Manual of Spectrometric Analysis Using Inductively Coupled Plasma (Russian translation). Moscow: Nauka (1988). 4 A.A. Ionin, A.P. Napartovich, and N.N. Yuryshev. Problem of Development of Oxygen-Iodine Laser with Electric Discharge Production of Singlet Delta Oxygen. Proc. SPIE, 4760 (2002). 5 W.L. Wiese, M.W. Smith, and B.M. Glennon. Atomic Transition Probabilities, 1. Washington: NSRDS-NBS-4

6

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(1966); NIST Electronic Data Base (2002). W.L. Wiese, M.W. Smith, and B.M. Miles. Atomic Transition Probabilities, 2. Washington: NSRDS-NBS-22 (1969); NIST Electronic Data Base (2002). W.F. Meggers, C.H. Curliss, and B.F. Scribner. Tables of Spectral Lines Intensities, 32, Pt. 1, 2. Washington: NBS (1961). C.E. Moore. Atomic Energy Levels, Circ. 467, 1, 2. Washington: NBS (1961). C.E. Moore. An Ultraviolet Multiplet Table, Circ. 488, Section 1–5. Washington: NBS (1950–1962). A.R. Striganov and G.A. Odintsova. Tables of Spectral Lines of Atoms and Ions (in Russian. Moscow: Energoizdat (1982).

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5 Measuring Concentrations of Atoms and Molecules 11 M.A. Bolshov. Laser AtomicFluorescence Analysis. In: V.S. Letokhov, Ed. Laser Analytical Spectroscopy (in Russian). Moscow: Nauka (1986). 12 A.G. Zhiglinsky, Ed. Handbook of Elementary Process Constants Involving Atoms, Ions, Electrons, and Photons. St. Petersburg: St. Petersburg State University press (1994). 13 V.M. Lelevkin, D.K. Otorbaev, and D.C. Schram. Physics of NonEquilibrium Plasmas. North-Holland (1992). 14 V.N. Ochkin, N.G. Preobrazhensky, and N.Y. Shaparev. Optogalvanic effect in Ionized Gas. London-Moscow: Gordon and Breach Science Publishers (1998). 15 S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). 16 I.I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Fizmatgiz (1963). 17 M.A. El’yashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 18 N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1985). 19 B.M. Smirnov. Excited Atoms (in Russian). Moscow: Energoizdat (1982). 20 A.A. Radtsig and B.M. Smirnov. Parameters of Atoms and Atomic Ions (in Russian). Moscow: Energoatomizdat (1986). 21 V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460 (2002): (a) V.I. Arkhipenko and L.V. Simonchik. The High-Current SelfSustained Atmospheric Pressure Discharge with Normal Current Density, pp. 1–16; (b) C.M. Ferreira, E. Tatarova, V. Guerra et al. Wave Driven Molecular Discharges as Sources of Active Species, pp. 99–110; (c) H. Lange and R. Bussiahn. Tunable Diode Laser Absorption Spectroscopy for Plasmas at Elevated Pressures, pp. 177–187.

22 L.A. Kuznetsova, N.E. Kuzmenko, Yu. Ya. Kuzyakov, and Yu. A. Plastinin. Optical Transition Probabilities of Diatomic Molecules (in Russian). Ed. by R.V. Khokhlov. Moscow: Nauka (1980). 23 N.E. Kuzmenko, L.A. Kuznetsova, and Yu. Ya. Kuzyakov. Franck–Condon Factors of Diatomic Molecules (in Russian). Moscow: Moscow State University Press (1984). 24 K.P. Huber and G. Herzberg. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules. N.Y.: Van Nostrand (1979). 25 G. Herzberg. The Spectra and Structure of Simple Free Radicals. London: Cornell University Press (1971). 26 G. Herzberg. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules. 2nd ed. N.Y.: D. van Nostrand (1951). 27 J.C. McCallum, R.W. Nichols, and W.R. Jairman. CRESS Spectroscopic Reports. Toronto: York University, No. 1 (1970), No. 2 (1971), No. 3 (1972), No. 4 (1972), No. 5 (1972). 28 I. Kovach. Rotational Structure in the Spectra of Diatomic Molecules. Budapest: Akademiai Kiado (1969). 29 V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Mechanisms Responsible for the Formation of Distributions of Electronically Excited Molecules among Vibrational–Rotational Levels in Gas Discharges. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian), pp. 6–85. Moscow: Nauka (1985). 30 D.I. Solovetsky. Chemical Reaction Mechanisms in Nonequilibrium Plasma (in Russian). Moscow: Nauka (1980). 31 A.A. Ionin, A.P. Napartovich, and N.N. Yuryshev. Problem of Development of Oxygen-Iodine Laser with Electric Discharge Production of Singlet Delta Oxygen. Proc. SPIE, 4760 (2002). 32 V.N. Ochkin and N.A. Shubina. Effect of Water Vapor Admixtures on the Processes Occurring in the Gas-Discharge Plasma of a CO2 Laser (in Russian). Khimiya Vysokikh Energii, 6, No. 1, pp. 26–30 (1972).

References 33 S. Penner. Quantitative Molecular Spectroscopy and Gas Emissivities. Massachusetts (1959). 34 R. Kim. The integral Intensity of Carbon Monoxide Fundamental Band. JQSRT, 30, No. 5, pp. 413–416 (1983). 35 P.K. Falcone, R.K. Hanson, and C.H. Kruger. Tunable Diode Laser Measurements of the Band Strength and Collision Halfwidths of Nitric Oxide. JQSRT, 29, No. 3, pp. 205–222 (1983). 36 L.S. Rothman and L.D.J. Young. Infrared Energy Levels and Intensities of Carbon Dioxide-II. JQSRT, 25, No. 5, pp. 505–524 (1981). 37 A. Goldman, R.R. Gamache, A.Perrin et al. HITRAN Partition Functions and Weighted Transition-Moments Squared. JQSRT, 66, pp. 455–486 (2000). 38 A.V. Dem’yanenko, I.I. Zasavitsky, V.N. Ochkin et al. Investigation into the Distribution of the CO2 Molecules among Vibrational–Rotational Levels in a Glow Discharge by the Pulsed Diode Laser Spectroscopy Technique (in Russian). Kvantovaya Elektronika, 14, No. 4, pp. 841–859 (1987). 39 V.S. Pazyuk, Yu. P. Podmarkov, N.A. Raspopov, and M.P. Frolov. Direct Registration of the O2 (a1 Δg Singlet Oxygen by the Intracavity Laser Spectroscopy Technique from Absorption in the a1 Δg –b1 Σg+ Transition (in Russian). Kvantovaya Elektronika, 31, No. 4, pp. 363–366 (2001). 40 S. Gudeman and R.J. Saykally. Velocity Modulation Infrared Laser Spectroscopy of Molecular Ions. Annual Rev. Phys. Chem., 35, pp. 387–418 (1984). 41 T.J. Sears. Infrared Absorption Spectroscopy of Molecular Ions Using Tunable Lasers. J. Chem. Soc., Faraday Trans. 2, 83, pp. 111–126 (1987). 42 F. Pan and T. Oka. Radial Distribution of Molecular Ions in the Positive Column of DC Glow Discharge Using Infrared Diode-Laser Spectroscopy. Phys. Rev., A36, No. 5, pp. 2297–2310 (1987). 43 L.S. Polak, D.I. Slovetsky, A.S. Sokolov, and T.V. Fedoseeva. Study of the CO2 Dissociation Mechanism

44

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in a Glow Discharge. In: Studies of Nonequilibrium Physico-Chemical Processes (in Russian), pp. 188–226. Moscow: Nauka (1974). L.S. Polak, D.I. Slovetsky, A.S. Sokolov, and T.V. Fedoseeva. Study of the Mechanisms of Physico-Chemical Processes Taking Place in a Carbon Dioxide Glow Discharge. In: Nonequilibrium Chemical Kinetics (in Russian). Moscow: Nauka (1974). H. Massey and E. Burhop. Electronic and Ionic Impact Phenomena. Oxford (1952), (1958). B.M. Smirnov. Atomic Collisions and Elementary Processes in Plasma (in Russian). Moscow: Atomizdat (1968). K. Takajanagi. Progr. Theor. Phys. Suppl., No. 40, pp. 216–248 (1967). S. De Benedictis, F. Cramarossa, and R. d’Agostino. Infrared and Visible Analysis of He-CO and He-CO-O2 Radio Frequency Discharges. J. Appl. Phys., 55, No. 2, pp. 413–424 (1984). H.M. Katsch, A. Tewes, E. Quandt et al. Detection of Atomic Oxygen: Improvement of Actinometry and Comparison with Laser Spectroscopy. J. Appl. Phys., 88, No. 11, pp. 6232–6238 (2000). E.A. Trubacheev, V.N. Ochkin, and Azatyan. Atomic Oxygen Concentrations in the Discharge Plasma of a CO Laser (in Russian). Zh. Kratkykh Soobshchenii po Fizike, No. 3, pp. 3–6 (1973). R. d’Agostino, F. Cramarossa, S. De Benedictis et al. Spectroscopy Diagnostics of CF4 –O2 Plasmas During Si and SiO2 Etching Processes. J. Appl. Phys., 52, No. 3, pp. 1259–1265 (1981). R. d’Agostino, V. Colaprico, F. Cramarossa et al. Proc. VII ESCAMPIG, p. 107 (1984). R.E. Walkup, K.L. Saenger, and G.S. Sewyn. J. Chem. Phys., 84, p. 2668 (1986). G.K. Vinogradov, L.S. Polak, D.I. Sloetsky et al. Proc. XV ICPIG, p. 620 (1981). J.W. Coburn and M. Chen. J. Appl. Phys., 51, p. 3134 (1980). R. d’Agostino, F. Cramarossa, V.

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Colaprico et al. Mechanisms of Etching and Polymerization in Radiofrequency Discharges of CF4 –H2 , CF4 – C2 F4 , C2 F6 –H2 , C3 F8 –H2 . J. Appl. Phys., 54, No. 3, pp. 1284–1288 (1983). V.M. Donnely, D.L. Flamm, W.C. Dautermont-Smith et al. Anisotropic Etching of SiO2 in Low-Frequency CF4 /O2 and NF3 /Ar Plasmas. J. Appl. Phys., 55, No. 1, pp. 242–252 (1984). R.A. Gottsho and V.M. Donnelly. Optical Emission Actinometry and Spectral Line Shapes in RF Glow Discharges. J. Appl. Phys., 56, No. 2, pp. 245–250 (1984). D.L. Flamm, V.M. Donnelly, R.H. Bruce et al. Proc. V Int. Symp. On Plasma Chemistry (ISPC-5), 1, p. 307 (1981). D.E. Ibbotson, D.L. Flamm, and V.M. Donnely. J. Appl. Phys., 54, p. 5974 (1983). A. Ricard. Proc. XVII ICPIG (1985). H.F. Doebele, K. Niemi, and V. Sculzvon Der Gathen. Laser Induced Fluorescence Diagnostics of Atomic Radicals in Non-Equilibrium Plasmas at Elevated Pressure. Proc. LAPD 10, pp. 71–80 (2001). A.V. Filippov, Y.A. Mankelevich, A.F. Pal et al. Spectroscopy, Actinometry and Simulation of a DC Discharge in CO/H2 Gas Mixtures. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 285– 295 (2002). R.A. Gottsho, G.P. Davis, and R.H. Burton. J. Vac. Sci. Technol. A, 1, p. 622 (1983). A. Francis, T. Gans, K. Niemi et al. Measurement of Quenching Coefﬁcients and Development of Calibration Methods for Quantitative Spectroscopy of Plasmas at Elevated Pressures. Proc. SPIE, 4460, pp. 122– 133 (2002). A. Goechlich, T. Kavetski, and H.F. Doebele. J. Chem. Phys. 108, p. 9362 (1998). D.K. Otorbaev. Optical Actinometry of Plasma. In: V.E. Fortov, Ed. En-

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cyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 606–609 Moscow: Nauka (2000). R.A. Gottsho and T.A. Miller. Pure Appl. Chem., 56, p. 189 (1984). H. Massey. Negative Ions. London: Cambridge University Press (1976). B.M. Smirnov. Negative Ions (in Russian). Moscow: Atomizdat (1978). A.V. Eletsky and B.M. Smirnov. Negative Ions in Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 1, pp. 250–260. Moscow: Nauka (2000). J. Taillet. Determination des concentrations en ions negatifs par photodetachment-´eclair. C.R. Acad. Sc. Paris, 269, pp. 52–54 (1969). A. Mandl. Electron Photodetachment Cross Sections of Cl− and Br− . Phys. Rev., A14, No. 1, pp. 345–348 (1976). S.P. Hong, S.B. Woo, and E.M. Helmy. Photodetachment of Thermally Relaxed CO3− . Phys. Rev., A15, No. 4, pp. 1563–1569 (1977). M. Bacal, G.W. Hamilton, A.M. Bruneteau et al. Measurement of H2− in a Plasma Photodetachment. J. de Physique, 40, pp. 791–792 (1979). R.A. Gottsho and C.E. Gaebe. Negative Ion Kinetics in RF Glow Discharges. IEEE Trans. Plasma Sci., PS-14, No. 2, pp. 92–102 (1986). M. Pealat, J.P. Taran, M. Bacal, and F. Hillion. Rovibrational Molecular Populations and Negative Ions in H2 and D2 Magnetic Multicasp Discharges. J. Chem. Phys., 82, No. 11, pp. 4943–4953 (1985). K.E. Grundberg, G.A. Hebner, and G.T. Verduen. Negative Ion Densities in NF3 Discharges. Appl. Phys. Lett., 44, pp. 299–302 (1984). J. Kramer. The Optogalvanic Effect in 13.56 MHz Chlorine Discharge. J. Appl. Phys., 60, pp. 3072–3080 (1986). V.S. Lebedev, L.P. Presnyakov, and I.I. Sobelman. Radiative Transitions of the H2+ Molecular Ion. UFN 173, No. 5, pp. 491–510 (2003).

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Spectral Methods of Determining Electronic and Magnetic Fields in Plasma Electric ﬁeld is a most important parameter governing the properties of plasma. In gas discharges, the electric potential of the external source is the prime cause of both the origination and the maintenance of plasma, and the self-consistent spatial distribution of electric ﬁeld and voltampere characteristics form the basis for the classiﬁcation of discharges. The electric ﬁeld of plasma may be caused by oscillations (Langmuir, ion-acoustic, etc.), absorption of an external radiation or space-charge formation mechanisms (cathode layer). The local electric ﬁeld within the limits of the Debye screening radius rD is due to accidental separation of charges in the course of their statistical motion (see Section 1.1). It is, therefore, necessary to distinguish between electric ﬁelds differing in correlation length with respect to rD . The electric ﬁeld largely determines the kinetics of plasma processes. Most important in plasma physics is the local ﬁeld approximation with which the magnitude of the reduced ﬁeld E/N ( N is the gas density) governs the electron energy distribution. It allows the scaling of such important parameters as transport coefﬁcients, particle excitation and ionization rate constants, and so on by means of E/N. The spectroscopic methods of determining electric ﬁelds are based on the Stark effect – the change of the wave functions and the eigenvalues of the energies of state of atoms, molecules and ions under the effect of an external electric ﬁeld. This affects the ﬁeld dependence of the characteristics (frequency, intensity, polarization) of emitted, absorbed and scattered light, which is exactly what is used to measure the parameters of the electric ﬁeld associated with it. In this section, we use the designation E for time-dependent electric ﬁeld and F for static (quasistatic) ﬁeld. Various Stark spectroscopy versions are considered in Sections 6.1 and 6.2, mainly using the material of the analytic-review publications [1–5]. Figure 6.1 presents a most simple scheme illustrating the effect of electric ﬁeld on spectrum [5]. The groups A and B of states include the sublevels α, α and β, β . The distance between the sublevels α and α is

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.1 Transitions in a system with two isolated levels, α and α .

sorter than the distance from them to any other level, so that the group A can be considered isolated from the other intra-atomic states (two isolated levels approximation). When an atom is placed in an electric ﬁeld, its angular momentum is quantized in terms of the quantum m¯h (m is the magnetic quantum number) along the z-direction of the ﬁled. The action of the ﬁeld results in the shift of the energy levels α, α . The energies of the levels α, α in the electric ﬁeld F may be represented as follows [10]: WαF = 12 (Wα + Wα ) .1/2 , + 12 (Wα − Wα )2 + 4 | γα jα m |z| γα jα m |2 e2 F2 WαF = 12 (Wα + Wα ) .1/2 , − 12 (Wα − Wα )2 + 4 | γα jα m |z| γα jα m |2 e2 F2

(6.1)

where j is the total angular momentum quantum number and γ is the set of the quantum numbers, other than j and m, for the levels α and α . The change in the frequency of the transition α − β , for example, which accompanies the shift of the terms, can serve as a measure of the strength F of the applied electric ﬁeld. In the case of not very strong ﬁelds, where the energy of interaction between the dipole moment and the electric ﬁeld is small in comparison with (Wα –Wα ), x ( F ) ≡ | γα jα m |z| γα jα m eF/(Wα − Wα )| 1, the shifts of the energy levels α and α depend on the ﬁeld strength F in a

6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

quadratic manner: WαF = 12 (Wα + Wα )

(

4 | γα jα m |z| γα jα m |2 e2 F2 + 12 (Wα − Wα ) 1 + (Wα − Wα )2

)1/2 (6.2)

2 2 2

≈ Wα + | γα jα m |z| γα jα m | e F /(Wα − Wα ), WαF ≈ Wα − | γα jα m |z| γα jα m |2 e2 F2 /(Wα − Wα ) When the ﬁeld strength F is increased so that the condition x ( F ) 1 is satisﬁed, the dependence of the displacement of the above levels on F becomes linear: WαF = 12 (Wα + Wα ) .1/2 + 12 (Wα − Wα )2 + 4 | γα jα m |z| γα jα m |2 e2 F2

≈ 12 (Wα + Wα ) + | γα jα m |z| γα jα m | eF

(6.3)

≈ Wα + | γα jα m |z| γα jα m | eF WαF ≈ Wα − | γα jα m |z| γα jα m | eF Only those levels can interact, which have the same m, but differ in parity [6–11], otherwise the matrix element in expression (6.2) equals zero. The same matrix element determines the probability of the α − α dipole transition whose wavelength for closely spaced levels falls within the infrared or the microwave region of the spectrum. It is convenient to express this probability in terms of the oscillator strength f of the transition (when multiplied into the statistical weight g, it is invariant under both emission and absorption of radiation (see formula (2.56))), because these data can be found in the spectroscopic reference literature (see Appendix E and references cited therein) [5, 6]:

| γα jα m |z| γα jα m |2 = | m, 0 | jα m |2 3g f λ2C me c2 /[8π2 (2jα + 1)(Wα − Wα )], (6.4) where λC = 2.43 × 10−10 cm and me c2 = 817.600 × 10−16 J (511 keV) are the Compton wavelength and the rest mass of the electron, respectively, and | m, 0 | jα m | is the Clebsch–Gordan coefﬁcient [6, 10]. There is, however, another circumstance that is attractive from the standpoint of measuring electric ﬁeld parameters. If in the absence of the ﬁeld the α–β and α –β transitions are allowed, whereas the α –β and α– β ones are forbidden in the dipole approximation, the intermixing of the states α, α upon the application of the electric ﬁeld weakens this forbidding. The ratio between the intensities of the “forbidden” and “allowed”

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Figure 6.2 Function R( x ) (6.5).

transitions Iαβ Iαβ

(Wα − Wβ ) Aα β = R ( x ), (Wα − Wβ ) Aαβ

R( x ) =

2x √ 1 + 1 + 4x2

2 .

(6.5)

Here A is the Einstein coefﬁcient and x is deﬁned by formula (6.2). For the ratio between the intensities of the “forbidden” α – β and “allowed” α –β transitions, use can also be made of formula (6.5) wherein the following replacements should be made: α → α , α → α, β → β , β → β. The measuring of these ratios enables one to determine the electric ﬁeld strength F. The graph of the function R( x ) is shown in Figure 6.2. In accordance with this ﬁgure, as the electric ﬁeld strength F is increased so that | γα jα m |z| γα jα m | eF ≈ 3 |Wα − Wα |, the intensity ratio Iαβ /Iαβ in formula (6.5) becomes saturated. This occurs in the same ﬁeld strength region where the ﬁeld dependence of the level shift, (6.1), turns from quadratic into linear. Note that presented in the review [5] is also an expression for the ratio between the intensities of the “forbidden” and “allowed” transitions. It correctly describes the limiting cases where x 1 and x 1. In the intermediate region, however, the use of formula (4) from [5] can lead to errors, and so use should be made of the above formula (6.5). If the transitions differing in the number m are spectrally allowed, relation (6.1) can directly be used to measure F, but if not, account should be taken of the result of averaging over m, observation angle and polariza-

6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

tion of the photons emitted. An important role in this case is played by the ﬂuorescence excitation mechanism. If excitation is effected directly in plasma, for instance, by electrons with a velocity distribution having no pronounced anisotropy, this mechanism is nonselective with respect to m. But if ﬂuorescence is excited by laser radiation, account should be taken of the direction of the polarization vector of this radiation relative to the electric ﬁeld vector F. The relations necessary for performing such averaging operations can be found in [6]. Relations (6.1) and (6.5) mostly prove useful in the qualitative interpretation of the effect of electric ﬁeld on spectrum. As for the quantitative aspect and particular real systems, one should analyze the validity of the assumptions made, speciﬁcally that of the “isolated character” of the levels. Similarly, the spectral methods of measuring the induction B of magnetic ﬁelds are based on the Zeeman effect – the splitting of spectral lines, frequency shift and light polarization under the effect of magnetic ﬁeld. Let us indicate the characteristic scale of values of the intra-atomic electric ﬁeld strength FA and magnetic ﬁeld induction B A : FA = eaB−2 ≈ 5.1 × 109 Vcm−1 = 1.7 × 107 CGS units, B A = Ry/137μB ≈ 1.7 × 107 Gs = 1.7 × 103 T, where aB and μB are the Bohr radius and magneton, respectively, and Ry ≈ 21.760 × 10−19 J (13.6 eV) is the Rydberg constant. The values of FA and B A are very high, but the spectroscopic manifestations of external ﬁelds are also observable at much lower values, especially as the principal quantum number n grows larger. To illustrate, the critical electric ﬁled strength at which an atom gets ionized is Fcr = FA /16n4 ; at n = 10Fcr ≈ 3 × 104 Vcm−1 . The inﬂuence of the internal magnetic ﬁelds associated with the ﬂow of electric current on the properties of plasma manifests itself mainly in the case of heavy currents. This occurs in pulsed high-current discharges, in TOKAMAK- and STELLATOR-type facilities, and so on (see reviews [1– 3]). The electron density here is high, and the heavy energy deposition leads to heating, and so, as a rule, the plasma of such facilities approaches equilibrium condition. Since this book mainly deals with nonequilibrium plasma, emphasis will be on electric ﬁeld measurements. For magnetic ﬁled measurements, see Section 6.3.

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6.1 Determination of Electric Fields from the Spontaneous Emission of Radiation by Atoms in Plasma 6.1.1 Hydrogen-Like Atoms

This case is important for plasma spectroscopy for the following two reasons. 1. The spectrum of the hydrogen (deuterium) atom is very sensitive to the presence of electric ﬁeld. Energy levels in hydrogen levels are degenerate (with neglect of the ﬁne structure) to the quantum number l of the orbital momentum. Therefore, there exist such states φk as there are superposition of states having one and the same principal quantum number n, but differing by the quantum number l. The atom in the states φk has a nonzero average value of the projection of the dipole moment vector d on the z-axis: φk |dz |φk = 0. If the atom resides in a state φk , the electron density distribution is noncentral (as distinct from the case where the atom is in a state with a ﬁxed orbital quantum number l). Assume that the z-axis is along the direction of the electric ﬁeld vector F. ˆ = −dz F, Nonzero in the state φk is the average value of the operator V that is, the operator of the dipole interaction between the atom and the ˆ kk ≡ φk |V ˆ |φk = 0. As a result, even in the ﬁrst perturbation ﬁeld F: V theory approximation and with weak ﬁelds, the splitting of the energy levels of hydrogen-like atoms is linear in the ﬁeld strength F (the linear Stark effect). It is here convenient to describe the effect in the parabolic coordinate system. In this case, the formula for the splitting of the energy levels of hydrogen in the electric ﬁeld F has the form (see, for example, [6–13]): ΔW = (3/2)n(n1 − n2 ) aB eF,

n1 , n2 = 0, 1, 2, . . . , n − 1.

(6.6)

Here n is the principal quantum number and n1 and n2 are the parabolic quantum numbers. The extreme energy sublevels of the Stark multiplet correspond to the case where |n1 − n2 | = n − 1. Their position is deﬁned by the relation ΔW (lat) = ±(3/2)n(n − 1) aB eF. For large numbers n, the relation * * * * (6.7) *ΔW (lat) * ≈ (3/2)n2 aB eF can be obtained from the simple quasiclassical analysis [14]. The interaction between the hydrogen atom and the electric ﬁeld F is maximal when the major semi-axis of the elliptical orbit of the atomic electron is

6.1 Determination of Electric Fields from Spontaneous Atomic Radiation

directed along the vector F. The average numerical value of the radiusvector r drawn from the nucleus of the hydrogen atom to the point of residence of the atomic electron will in turn be maximal in the extreme case of a strongly prolate ellipse degenerating into a line segment. In the latter case, the atomic electron moves in a straight line along the z-axis between the extreme points z = 0 and z = 2a. Here the point z = 0 is the position of the nucleus of the hydrogen atom and a is the major semi-axis. The average length of the radius-vector r is given by 2a dr r/v(r ) rav = 0 2a , (6.8) dr/v ( r ) 0 where v is the velocity of the atomic electron. The velocity v and the length of the radius-vector r of the atomic electron are related by the relation v(r ) = eme− /2 [2/r − 1/( aB n2 )] /2 , 1

1

(6.9)

which follows from the law of conservation of energy. Substituting rela* * tion (6.9) into (6.8), we get rav = (3/2)n2 aB . Since *ΔW (lat) * = erav F, we arrive at relation (6.7). In the spherical coordinate system, the matrix elements determining the dipole interaction between the hydrogen atom and the electric ﬁeld F are given by the formula [6] ,1/2 + 2 3 (n − l 2 )(l 2 − m2 ) nlm |eEz| n(l − 1)m = eEa0 n . 2 4l 2 − 1

(6.10)

The solution of the secular equation wherein the perturbation matrix elements are deﬁned by formula (6.10) yields, even in the ﬁrst perturbation theory approximation, a nonzero level shift. That is, the classical and the quantum-mechanical analysis both lead to the conclusion that the splitting of the energy levels of the hydrogen and hydrogen-like atoms (H-atoms) is linearly dependent on the electric ﬁeld strength F. The linear character of this dependence is also evident from the general expression (6.1). In the case of energy-level degeneracy with respect to l, or weak splitting of energy levels differing in orbital momentum, (Wα − Wα ) ≈ 0, the splitting will be deﬁned by expression (6.3) and be linear in F, no matter how low its value. 2. Another circumstance underlying the practical interest in the Stark effect in hydrogen is the fact that as the principal quantum number n increases (and the presence of lines due to transitions from such levels in emission spectra is characteristic of plasma), all atoms become hydrogenlike (H-atoms) [9–12].

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In the case of linear splitting, the spectrum of transitions between these levels is deﬁned by the expression I (e) (Δω ) =

∑ Gα,β (e) δ(Δω − καβ F).

(6.11)

α,β

In this expression, α and β denote the levels belonging to the upper state A and the lower state B, respectively, the frequency Δω is calculated from the nonshifted transition frequency, Gα,β is the intensity of the α → β Stark component, καβ is the Stark constant for the α → β transition, e is the unit polarization vector of the emitted photon and the delta function δ(Δω − καβ F ) indicates the spectral position of the central frequencies of the α → β components. The values of Gα,β (e) and καβ can be calculated by means of the formulas presented in [6]. The electric ﬁeld strength can be determined using expression (6.11) by comparing the experimentally measured splitting of the spectral line of the H-atom and its Stark components intensities with their theoretically calculated counterparts. According to what has been said above, expression (6.11) proves a ( B)

( B)

good approximation when F > h¯ ωfs d( A) −1 ≡ F ∗ , where h¯ ωfs is the characteristic ﬁne structure splitting of the lower level B of hydrogen and d( A) is the characteristic dipole moment of its upper level A. For the Balmer lines of hydrogen, the value of F ∗ can be roughly estimated at (30/n2 ) kVcm−1 , where n is the principal quantum number of the upper level. It frequently occurs that the Stark components are not resolved because of line broadening and inadequate spectral resolution. The Stark effect in such cases manifests itself as the broadening of the A → B transition line. Figure 6.3 presents as an example the proﬁle of the Hδ spectral line (λ = 410 nm) recorded in the cathode sheath of a hydrogen glow discharge [15] at F ≈ 1200 Vcm−1 , a gas pressure of 0.800 hPa (0.6 Torr) and a discharge current density of 0.8 mAcm−2 . The theoretical curve is plotted with due regard for the Doppler broadening of the individual components and the instrument (spread) function of the spectral instrument used. The inset shows the structure of the Stark spectrum of the Hδ line. The parameter Y characterizes the spectral position of the Stark components at frequencies of Δω = 3¯hYF/(2me e) calculated from the center (Δω = 0) of the line of interest. Under the above experimental conditions [15], the minimum electric ﬁeld strength that could be determined from the Stark broadening of the Hδ line amounted to a few tens of volts per centimeter. Figure 6.4 presents the result of determination of the ﬁeld distribution in the cathode sheath under the above conditions.

6.1 Determination of Electric Fields from Spontaneous Atomic Radiation

Figure 6.3 Measured (data points) and calculated (curve) proﬁles of the Hδ line at F = 1200 Vcm−1 . Inset – calculated spectrum of the π components (Δm = 0).

Figure 6.4 Electric ﬁeld strength in the cathode sheath. Data points – measurements, straight line – approximation.

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The direction of the vector F in plasma can be determined using emission spectrum polarization measurements and the theoretical dependence of Gα,β on e. The “strongest” α → β spectral components with the polarization vector e normal to the electric ﬁeld vector F are located, on average, closer to the nonshifted frequency of the A → B transition than their counterparts with the polarization vector parallel to the electric ﬁeld vector. Let the electric ﬁeld distribution in plasma be described by the function W ( F ) = W ( F, θ, φ), where θ and φ are the angles in the spherical coordinate system that deﬁne the direction of the vector F. The quantity W ( F )d3 F is the probability that the end of the vector F is located in the elementary volume d3 F. Such a situation is possible, for example, when the electric ﬁeld F is due to a plasma turbulence or chaotically moving ions. Integrating (6.11) with respect to the distribution function W ( F ), one can obtain the following expression for the emission spectrum [4]: I (e) (Δω ) =

∑

2π

α ,β 0

+

dφ

π 0

dθ Gα β (e) W (Δω/κα β , θ, φ)/κα β

(6.12)

(e) I0 (Δω ) (e)

where α and β are the states for which κα β = 0 and I0 is the entire spectrum of the nonshifted (central) components (at κα β = 0). Figure 6.5 shows the proﬁle of the Hα spectral line (λ = 656 nm) [16] recorded in the anode region of a high-power gas-ﬁlled diode for two orthogonal polarization directions, one of which coincides with the ion beam direction. One can see the substantial difference in width between these proﬁles, which is due to a “unidimensional” turbulence in the discharge plasma, whose electric vector is normal to the anode surface. The average amplitude of the turbulent electric ﬁeld was estimated at 5 kVcm−1 . The inset in Figure 6.5 shows the structure of the Stark spectrum of the Hα line. The dashed and solid lines correspond to polarization directions normal and parallel to the electric ﬁeld vector F, respectively. Regular quasi-monochromatic electric ﬁelds (microwave ﬁelds, for example) can be determined on the basis of the theory developed by the authors of [17], who demonstrated that the spectrum of the hydrogen atom in the ﬁeld E(t) =E0 cos ωt could be described by the expression I (e) (Δω ) = (e) Sp

+∞

∑

p=−∞

(e)

S p δ(Δω − pω ),

= ∑ Gαβ (e) J p2 (καβ E0 /ω ), α,β

(6.13)

6.1 Determination of Electric Fields from Spontaneous Atomic Radiation

Figure 6.5 Proﬁle of the Hα line under ion-beam excitation. Solid and dashed lines – observations along and across the ion beam, respectively.

where J p (κ ) is a Bessel function. The emission spectrum thus consists of a number of satellite lines whose frequency is shifted by an amount of Δω = pω (p = 0, ±1, ±2,. . . ) from the frequency of the nonshifted (Δω = 0) line of the A → B transition. The electric ﬁeld amplitude F can be determined from the ratio between the intensities of two satellites, S p /Sq , or from the proﬁle of the integral spectrum of a group of satellites in the emission spectrum. Figure 6.6 presents the intensity ratios S1 /S0 and S2 /S1 as a function of the reduced electric ﬁeld strength ε = 3¯h E0 /(2me eω) for the Hα and Hβ lines. The curves are plotted using the numerical calculation results obtained in [18] and correspond to an observation direction normal to the electric ﬁeld direction, with no polarization elements used. Presented in Figure 6.7 is the proﬁle of the Dγ line of deuterium in plasma in the focal region of focused microwave radiation with a frequency of f = 34.8 GHz [19]. In the presence of radiation (curve 1), two satellites can be seen at frequencies Δω = ±ω. The ratio between the intensities of the satellites at the frequencies Δω = ±ω and Δω = 0 in the given case corresponds to the microwave ﬁeld amplitude E0 ≈ 1.24 kVcm−1 . The inset in Figure 6.7 shows the satellite intensity ratio S1 /S0 as a function of the reduced electric ﬁeld strength ε = 3¯h E0 /(2me eω) for the Dγ line, as observed in a direction normal to that of the electric ﬁeld. The spectroscopic diagnostics of the electric ﬁeld Eturb (t) produced by turbulence in a turbulent plasma with strong noise and oscillations,

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.6 Satellite intensity ratios: (a) S1 /S0 for the Hα line and (b) S2 /S1 for the Hβ line as a function of the reduced electric ﬁeld strength.

where Eturb (t) is not a regular ﬁeld, but rather a superposition of a set of independent harmonics of arbitrary phase φj and frequency ω j , was theoretically analyzed in [20] (see also [4]). 6.1.2 Non-Hydrogen-Like Atoms

This case can be analyzed satisfactorily on the two isolated levels assumption made above (Figure 6.1). Let us extend our simpliﬁcation

6.1 Determination of Electric Fields from Spontaneous Atomic Radiation

Figure 6.7 Proﬁles of the Dγ line recorded with the microwave ﬁeld switched (1) on and (2) off. Vertical arrows indicate the positions of the ±ω satellites. Inset – satellite intensity ratio as a function f the reduced electric ﬁeld strength.

further and consider a scheme wherein the upper subsystem A consists of two levels α and α , whereas the lower subsystem B contains but a single level β. We will assume that in the absence of electric ﬁeld the α → β transition is allowed, while the α → β transition is forbidden in the dipole approximation. In the presence of a static electric ﬁeld F, there occurs the intermixing of the wave functions corresponding to the levels α and α , Stark shift (6.1)) and the emergence of the lines of “forbidden” transitions (see expression (6.5)). The emission spectrum of the (α, α ) → β transition may be represented in the form I (Δω ) = Iα δ(Δω − ωαβ − σF ) + I f δ(Δω − ωα β + σF ), Ia = cos2 (η/2), σF = [(1 + tan η ) 2

I f = sin2 (η/2), 1/2

− 1]/2.

η = arctan[2dαα F/(h¯ ωαα )], (6.14)

The frequency ωαα here determines the distance between the levels α and α and dαα is the matrix element of the dipole moment of the atom between the states α and α . The ﬁrst term on the right-hand side of the above equation corresponds to the allowed spectral lines and the second term, to the forbidden ones. Note that in contrast to hydrogen-like atoms, in the limit of weak ﬁelds (|2dαα F/(h¯ ωαα )| 1) the quantity σF describes the energy level shift quadratic in the ﬁeld strength F. In the approximation adopted, one can obtain an expression similar to (6.5) for the ratio between the intensities Ia and If of the allowed and forbid-

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.8 Ratio between the line intensities of the forbidden (41 P-21 P, λ = 491.1 nm) and the allowed (41 D-21 P, λ = 492.2 nm) transition in helium. Radiation polarization parallel (solid line) and normal (dashed line) to the direction of the static electric ﬁeld vector F .

den spectral components. Figure 6.8 presents as an example the ratio between the intensity of the 41 P-21 P forbidden spectral line of helium (λ = 491.1 nm) and that of its 41 D-21 P allowed line (λ = 492.2 nm) as a function of the electric ﬁeld strength F [4]. Let us consider the case of alternating electric ﬁeld E0 cos ωt of a not very high amplitude E0 , so that |dαα E0 /[2¯h(ωαα − ω )]| 1. The emission spectrum for the (α, α ) → β transition will then have the following form [4]: I (Δω ) = δ(Δω − ωαβ + S− δ(Δω − ωα β − ω ) + S+ δ(Δω − ωα β + ω ), (6.15) where S± ={dαα E0 /[2¯h(ωαα ± ω)]}2 . The ﬁrst term on the right-hand side of (6.15) corresponds to the allowed α → β spectral line, while the second and the third term, to the satellites of the forbidden α → β spectral line. As the frequency ω decreases, both satellites tend to the position of the forbidden spectral line, and in the limit as ω → 0, there appears, instead of the two satellites, a forbidden line at the frequency Δω = ωα β . In that case, the spectrum I (Δω) is described by (6.14) at |η | 1. To determine the ﬁeld amplitude E0 , use can be made of the ratio between the intensity of one of the satellites of the forbidden α → β line and that of the allowed α → β line. The frequency ω can be found by measuring the distance between the spectral positions of the two satellites, equal to 2ω.

6.1 Determination of Electric Fields from Spontaneous Atomic Radiation

Figure 6.9 Emission spectra of helium in the neighborhood of the 492 nm line in two regions of plasma. Top – in the propagation region of a microwave radiation with a power of 60 kW; bottom – outside of the microwave region.

This method of studying oscillating electric ﬁelds by means of spectral satellites was ﬁrst suggested by Baranger and Mozer [21] and used for plasma diagnostics purposes by Kunze and Griem [22]. In the case of strong oscillating ﬁelds (|dαα E0 /(2¯hωαα )| > 1), the satellite intensity ratio in the emission spectrum (6.15) becomes a function of the electric ﬁeld amplitude: S− /S+ = f ( E0 ). This function was calculated in the adiabatic approximation by Oks and Gavrilenko [20]. Its usability in the diagnostics of electric ﬁelds E0 cos ωt owes to the fact that the localization volume VE of the ﬁeld E0 cos ωt, from which spectral satellites are emitted, in many cases proves much smaller than the volume V whence the allowed transitions emit. And the use the traditional diagnostics methods based on the measurement of the ratio S± /Ia as a function of E0 in such a situation may lead to errors because of the difference in localization of the corresponding emission regions. Such quasi-local measurements of the ratio S− /S+ as a function of E0 were used in [23] to diagnose a plasma interacting with a strong microwave ﬁeld. Figure 6.9 presents the emission spectra of the (41 D, 41 F)→21 P transitions in helium recorded in this work. The 41 D→21 P transition is allowed, whereas the 41 F→21 P one, forbidden. The proﬁle of the spectral line emitted from the region corresponding to the plasma resonance condition ω ≈ ωL (ωL is the plasma (Langmuir) frequency) features two satellites (S± ) of the forbidden 41 F→21 P line. The electric ﬁeld amplitude found in this way in [23] was E0 = 6.2 ± 0.4 kVcm−1 .

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

The direction of the oscillating electric ﬁeld E(t) = 2−1 [E0 (t) exp(−iωt) + E0∗ (t) exp(iωt)] can be found from the polarization measurements of the satellites of forbidden spectral lines. The polarization state of the ﬁeld ∗ } , where E ( t ) is deﬁned by the elements of the tensor σkk = {E0k E0k av 0k (k = 1, 2, 3) are the rectangular coordinates of the complex amplitude E0 , {. . . }av denote time averaging, and the asterisk symbolizes complex con∗ } , it jugation [24]. To obtain all the elements of the tensor σkk = { E0k E0k av is necessary to take a series of polarization measurements of the emission spectra of both dipole allowed and forbidden lines. This series should include measurements of the emission spectra with both linear and circular polarization. When discussing the classical Stark emission methods, we should note their following merits and usability. First is the availability of a vast amount of published data on the effect of electric ﬁeld on the emission line proﬁles of atoms and ions. Secondly, there are well-developed theoretical approaches taking into account, among other things, the temporal behavior effects of plasma electric ﬁelds. Thirdly, the traditional emission spectroscopy techniques require no intricate experimental instrumentation. Fourthly, emission methods play a key part in astrophysical investigations where the use of the active diagnostics techniques is impossible. At the same time, the Stark methods based on the analysis of the spectra of the intrinsic plasma radiation suffer from a number of limitations. Speciﬁcally, the spatial and temporal resolution of the electric ﬁeld measurements based on the emission methods in many cases prove inadequate. Another restriction is associated with the Doppler and collisional broadening effects that can mask the Stark structure of the emission line proﬁles. Finally, there are “dark” regions in plasma, where no spectral line emission can be observed. These limitations can be overcome in part by laser spectroscopic methods.

6.2 Laser Stark Spectroscopy

Practically all of the speciﬁc features of radiation from frequency-tuned lasers, as compared will the classical ancillary (probe) radiation sources and intrinsic plasma radiation, also prove useful in this ﬁeld of plasma spectroscopy, namely, • the narrow spectrum of laser radiation provides for high resolution in the study of the Stark multiplets in emission spectra;

6.2 Laser Stark Spectroscopy

• high intensity combined with special absorption detection techniques (laser-induced ﬂuorescence (LIF), optogalvanic (OG), intracavity) provide for high sensitivity and selectivity of linear spectroscopy measurements; • laser beam directivity can be used in space-resolved measurements; • use can be made of the nonlinear absorption and emission spectroscopy methods both to improve spatial and spectral resolution (exclusion of the Doppler broadening, see Section 3.5.1) and to suppress collisional effects in coherent multiphoton mixing (Sections 3.5.3 and 3.5.4). 6.2.1 Stark Spectroscopy of Atoms

Electric ﬁeld parameters can be determined by spectroscopic methods using frequency-tuned lasers to excite optical transitions involving atomic levels with large principal quantum numbers (Rydberg atoms). The high sensitivity of the Rydberg atoms to electric ﬁeld is due to their large dipole moment and the small separation of their energy levels of opposite parity [10]. As already noted above, the splitting pattern of such levels is close to that for the hydrogen atom, with the amount of splitting linearly dependent on the ﬁeld strength. Figure 6.10a presents the splitting pattern of the level with n = 11 in helium [5, 25]. On the ﬁgure’s scale, splitting can be seen to perceptibly deviate from being linear only for the 11P level in the region E≤ 200 Vcm−1 (the 11S state is located much lower on the energy scale, beyond the ﬁgure’s ﬁeld). The ﬁxed magnetic quantum number in this example was m = 0. Experimentally such a pattern can be registered when the atom is excited from the 21 S (l = 0, m = 0) metastable state by laser radiation polarized along the electric ﬁeld direction (Δm = 0). Similar illustrations obtained by direct calculation can be found, for example, in [26] for Li (n = 15), [27] for He (n = 19), [7, 8] for Ne (n = 19), and others. Figure 10b shows the intensities of individual spectral components as a function of the electric ﬁeld strength. When registering spectra, one can use, apart from direct absorption measurements, also much more sensitive optogalvanic and ﬂuorescence techniques (Section 3.4). The principal quantum number n of the Stark multiplet is selected as a trade-off between the oscillator strength (ca. n−3 ) sufﬁcient to produce an absorption-associated signal and the magnitude of the Stark effect (ca. n2 ). Another limitation on the value

323

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.10 (a) Splitting of the level with n = 11, m = 0 in helium; (b) relative intensities of the components.

of the number n is due to the interaction with the adjacent states, whose density in the spectrum grows with increasing n. Departure from the scheme with “isolated” levels (Figure 6.1) complicates the calculation and interpretation of splitting and intensities in multiplets, especially so if direct experimental calibrations are difﬁcult to accomplish. The power and polarization of the laser radiation must also be optimized with due regard for the measurement objectives and conditions. If the radiation power is relatively not very high, so that the transition is not saturated, the frequency characteristics and intensities of the components can be directly used to determine the ﬁeld intensity. The measurement region can be localized, for example, by way of stepwise excitation of the Rydberg level via an intermediate state in intersecting laser beams. Figures 6.11a and b present the spectra of Stark multiplets in helium 1 (2 S - 111 P) [5] and neon (2p5 3p3 P1 – 2p5 (2 P1/2 )11d ) [28], obtained with the use of optogalvanic detection. Both experiments were conducted in glow discharges with planar aluminum electrodes. The laser radiation polarized normal to the discharge axis passed at different distances from the cathode into the cathode drop region where the typical electric ﬁeld strength values ranged between 102 and 103 Vcm−1 . In the experimen-

6.2 Laser Stark Spectroscopy

Figure 6.11 Stark optogalvanic spectra in the cathode drop region of glow discharges in (a) and (b) helium (21 S-n1 P) at pressures of 4.666 hPa (3.5 Torr) and 1.667 hPa (1.25 Torr), respectively, and (c) neon (3 P1 −3 P1/2 11d ) at a pressure of 2.666 hPa (2 Torr) [28]. (a) and (b) – n = 11 [4]; (c) – n = (26–35) [5]. The electric ﬁeld strengths in (a) are borrowed with graphical accuracy from [5].

tal conditions of Figure 6.11b, for example, use was made of two dye lasers pumped by a pulsed nitrogen laser. The laser beams intersected at a small angle. One of the dye lasers generated a wavelength of 588.2 nm and excited the neon atom to rise from the 2p5 3s3 P2 metastable state to the 2p5 3p3 P1 state, while the second dye laser, tunable in the vicinity of 439.3 nm, raised it a few nanoseconds later to the 2p5 (2 P1/2 )11d Stark multiplet. The delay is necessary in order to exclude the occurrence of the optogalvanic effect at the ﬁrst excitation step. This exclusion is aided by the fact that the laser radiation used at the ﬁrst step causes no strong optogalvanic effect, for the associative ionization channel proves inoperable, the energy of state being insufﬁcient. In contrast, the second excitation step provides for effective ionization precisely via this channel. The spatial resolution attained in this experiment was around 10−3 cm−3 . The authors of [29] experimentally studied the radial distribution of electric ﬁeld strength in the positive column of a low-pressure glow discharge. The typical ﬁeld strength values in such conditions are

325

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

ca. 10 Vcm−1 . Under such conditions, the individual components of the Stark multiplets in the spectrum overlap because of the Doppler broadening, even for n ≥ 26. Examples of spectra registered at various distances from the axis of discharge in helium at a pressure of 1.667 hPa (1.25 Torr) and discharge current of 1.2 mA in a 6 mm diameter. tube are presented in Figure 6.11c. However, lasers can help determine the broadening of the multiplets as a whole. These data can be simulated by calculating the Stark effect with due regard for the laser radiation width, and the electric ﬁeld strength can then be determined from comparison between the calculated and measured results, as described in Section 6.1.1. For light hydrogen-like Rydberg atoms, the authors of [29] assumed (cf. (6.6), (6.7)) that the distance between the extreme components of spectral multiplets was [9, 10] Δω =

3h (n2 − n2l ) F, 2πme Z u

(6.16)

where nu and nl are the principal quantum numbers of the upper and the lower level of the transition, F is the electric ﬁeld strength in atomic units, and Z = 1 for neutral atoms. One can see from Figure 6.11c that the maximum value nu,max for the transitions observed in the spectrum decreases with increasing F. This is due to the well-known series-limiting Inglis–Teller effect ([9, 10, 30–32]; see also below Section 7.3), whereby the Stark broadening of a hydrogen level becomes comparable with the separation of the neighboring levels: 3eFaB n2 = e2 /aB n3 . This gives the following simple estimate for the series limit: F (Vcm−1 ) = 1.7 × 109 (nu,max )−5 ,

(6.17)

which can be used in ﬁeld strength measurements. The authors of [29, 33] have also taken into consideration the fact that the electric ﬁeld in the conditions of Figure 6.11c is constituted by the constant ﬁeld produced by the source sustaining the discharge, which is of vectorial character, and the statistical ﬁeld whose directions are averaged within the Debye sphere (Section 1.1). By virtue of this fact, to ﬁnd the constant ﬁeld, the numerical coefﬁcient in (6.13) was taken equal to 1.2 × 109 . For more details on the Stark effect in plasma microﬁelds, see Chapter 7. The ﬁeld strengths determined from the broadening and from the limitation of the series are in good agreement. With the typical multiplet widths equal to 2–3 cm−1 and the laser line 0.15 cm−1 wide, the accuracy in determining electric ﬁeld strengths by formula (6.16) was estimated at

6.2 Laser Stark Spectroscopy

1 Vcm−1 at ﬁeld strengths F ≈ 10 Vcm−1 . The ﬁeld strength values are also indicated in Figure 6.11c. The limit of high n values corresponds to the approximation of any atom to the hydrogen scheme of optical transitions, but the characteristics of the Stark effect for multielectron atoms, which are suitable for quantitative measurements, require calculations allowing for such approximation. They grow in complexity as the mass of the atom understudy increases in comparison with that of the light H, He and Ne ones, but still possible to some extent. For example, Gavrilenko and co-authors [34] made such calculations for the argon atom, and the results obtained in this work were experimentally conﬁrmed in [35]. Based on these data, the authors of [34, 36] determined ﬁeld intensities in discharges from argon transitions using LIF registration. In their ﬁrst experiments [34], sensitivity amounted to 140 Vcm−1 for the cathode drop region at an argon pressure of 6.666 hPa (5 Torr). In the later work [37], a sensitivity of 3 Vcm−1 was attained in measuring ﬁeld intensities in a low-pressure 1.333 Pa (10 mTorr)–0.133 Pa (1 mTorr) inductively coupled Ar plasma by a stepwise laser-induced ﬂuorescence method (for more details on the method, see Section 6.2.3 below). If the absorption of the probe radiation is registered by the LIF method, information on the polarization of the ﬂuorescence radiation can also be additionally used to determine the electric ﬁeld strength. The work by Takiyama and co-workers [38] who conducted such investigations in the peripheral layers of TOKAMAK plasma can serve as an example. The n1 D states in helium atoms were excited by laser radiation from the He(21 S) metastable state. The 1 S – 1 D transitions are forbidden in the dipole approximation, but in the presence of an electric ﬁeld the n1 P and n1 D states become intermixed, which weakens the forbidding, and the intensity of ﬂuorescence radiation in the n1 D→21 P transitions can, as already stated in connection with the LIF method, serve as a measure of the electric ﬁled. In this case, however, quadrupole transitions considerably contribute to excitation, so that the ﬂuorescence intensity Iﬂ ∝ n2S ρL [χS ( F ) + χQ ],

(6.18)

where ρL is the power density of the laser radiation, n2S is the population of the metastable state, χS is the ﬁeld-dependent “dipole” absorption coefﬁcient, and χQ is the “quadrupole” absorption coefﬁcient. To discriminate between the contributions from transitions differing in multipolarity, use can be made of their difference in the selection rules for m and, as a consequence, the difference in polarization between the ﬂuorescence radiations excited via the different channels. The ﬁeld strength can be

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

obtained from the relation F = G ( Dp ),

(6.19)

where Dp is the degree of polarization of the ﬂuorescence radiation and G ( DP ) is a known function calculated theoretically. If the polarization of the laser radiation is parallel to the z-axis, the quantity Dp is deﬁned by the relation Dp = ( I z − I y )/( I z + I y ), where I z and I y are the intensities of the ﬂuorescence radiations of z- and y-polarization, respectively. When exciting the 51 D level of helium in [38], the sensitivity of this method amounted to ca. 10 Vcm−1 . The possibility of determining not only the magnitude, but also the direction, of the electric ﬁeld vector is based on the difference between the spectra of laser radiation absorbed in dipole-allowed transitions at different angles between its linear polarization direction and the electric ﬁeld vector F. This follows from the selection rules for the magnetic quantum number m, namely, Δm = 0, when the laser polarization direction is parallel to the electric ﬁeld vector, and |Δm| = 1, when it is perpendicular to the latter. Let the electric ﬁeld vector F lie in the xz-plane, its direction being unknown. To ﬁnd the direction of the vector F, one can measure the intensities Iμx and Iμz of the signals characterizing the absorption of laser radiation on the μ lines of a Stark multiplet for two laser polarization directions – parallel to the x- and the z-axis, respectively. Using the ratio Rμ ( F ) = Iμx ( F )/Iμz ( F ) obtained experimentally and the theoretical dependence of Rμ on the angle θ between the direction of the vector F and the z-axis, one can determine the angle θ, hence the direction of the vector F. Ganguly and Garscadden [39] used this technique for the ﬁrst time to measure the radial distributions of the electric ﬁeld vector magnitude and direction in the positive column of a helium glow discharge. Gavrilenko and co-workers [36] studied the polarization effects in greater detail in an argon glow discharge. In Figure 6.12, the theoretical function Rμ (θ) calculated at F = 3 kVcm−1 is presented for four Stark components of the optogalvanic laser spectrum of the argon transition from the 4s[3/2]2 lower metastable level to the upper level with the principal quantum number n = 9. A good agreement was achieved between the experimental and theoretical optogalvanic argon spectra for various angles θ and various components of the Stark μ-line spectrum [37]. The angle measurement error was estimated at 15 ◦ C–20 ◦ C. Worthy of note is also the method of determining microwave ﬁeld strength from the intensity of ﬂuorescence induced by laser radiation of high power density If a hydrogen-like atom is exposed to the light wave EL = E0L cos(ωL t + φL ) and microwave electric ﬁeld EM = E0M cos(ωM t + φM ), the induced ﬂuorescence spectrum will contain the

6.2 Laser Stark Spectroscopy

Figure 6.12 Function Rμ (θ ) calculated at F = 3 kVcm−1 . Inset: laser optogalvanic spectrum of the 4s[3/2]2 →n = 9 transition in argon for the A, B, C and D transition lines.

resonance frequencies ωL ≈ ω AB + kωM ,

k = 0, ±1, ±2, . . . ,

(6.20)

where ω AB is the frequency of the atomic transition between the lower level B and the upper level A. With this method, one measures the relationships between the ﬂuorescence intensity and the laser radiation intensity and compares them with their theoretical counterparts. The method was suggested by Gavrilenko and Oks [40] and implemented by Polushkin and co-workers [41], who determined the amplitude of a microwave from the intensity of ﬂuorescence induced on the Hα line of hydrogen in the microwave power range 40–60 kW. The results obtained at 45 kW (4.3 kVcm−1 ) and 60 kW (5.5 kVcm−1 ) were veriﬁed by independent measurements. For the theory behind the method, see [42–44]. 6.2.2 Laser-Induced Fluorescence of Polar Molecules in Electric Field

Diatomic molecules, like Rydberg atoms, have closely spaced sublevels of opposite parity. Such closely spaced sublevels – Λ doubling sublevels – are present in states with Λ = 0, where Λ is the projection of the electronic orbital angular momentum on the internuclear axis. These sublevels in a molecule are designated by the letters e and f . By deﬁnition (see [45, 46]), a sublevel e is one whose parity is (−1) J , while that of a

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.13 Fluorescence photoexcitation on the 1 Π-1 Σ transitions. Solid arrows indicate allowed transitions and the dashed ones, forbidden transitions. The hatched circle symbolizes the intermixing of states.

sublevel f is (−1) J +1 , where J is the rotational quantum number of the given molecular energy level. The number J here is taken to be whole. The Λ doubling sublevels in diatomic polar molecules are related by the dipole moment matrix element. Therefore, the spectra of polar molecules in the states with Λ = 0 can be very sensitive to the effect of the electric ﬁeld F. The spectral method of measuring electric ﬁelds in plasma, based on the Stark effect and the laser-induced ﬂuorescence (LIF) of polar diatomic molecules, was suggested Moore and co-workers [47] (see also [48–50]). Laser radiation raises the molecule from its ground state 1 Σ+ to the electronic state 1 Π (Figures 6.1 and 6.13). Each rotational level of the molecule in the 1 Π state is split into two closely spaced sublevels (Λ doublet) bound by a dipole transition. Electric ﬁeld intermixes the wave functions of the Λ-doublet sublevels, so that the structure of the ﬂuorescence spectrum for the transitions involving them is modiﬁed. Assume that a polar diatomic molecule, whose energy level diagram is illustrated in Figure 6.13, is excited by laser radiation on an R( J ) transition. The ex( J +1)

citing radiation line width is greater than the separation Δe f

1Π

of the

state. In sublevels of the Λ doublet for the rotational level J+1 of the the absence of the ﬁeld F, the induced ﬂuorescence spectrum contains the R( J ) and P( J + 2) lines, the Q( J + 1) line being absent. If the ﬁeld

6.2 Laser Stark Spectroscopy

is present, it mixes up the states of the doublet, and the line Q( J + 1) appears in the ﬂuorescence spectrum. When the laser radiation is polarized along the z-axis parallel to the ﬁeld F, the intensities of the R( J ) and the ‘new’ Q( J + 1) line in the LIF spectrum are described by the expressions [47] + , M 2 ( J + 1)2 − M 2 Φ2 z , IQ = ∑ ( J + 1)2 (2J + 1)(2J + 3) 1 + Φ2 M + , ( J + 1)2 − M 2 ( J + 2)2 − M 2 2 + Φ2 z IP = ∑ , (2J + 1)(2J + 3)2 (2J + 5) 1 + Φ2 M (6.21) + , ( J + 1) − M2 ( J + 1)( J + 2) − M2 Φ2 x IQ = ∑ , 2( J + 1)2 (2J + 1)(2J + 3) 1 + Φ2 M + , ( J + 1) − M2 ( J + 2)( J + 3) + M2 2 + Φ2 x IP = ∑ , 2(2J + 1)(2J + 3)2 (2J + 5) 1 + Φ2 M where Φ = 2μFM/[ p( J + 1)( J + 2)]2 , (J for the Σ state), M is the projection of J on F, and μ is the dipole moment of the molecule in the 1 Π state. (Note that the factor 2p is lacking in the expression for Φ in [47]). Summation is extended from − J to J. It is not very difﬁcult to see that the parameter Φ depends on the magnitude of the Λ-splitting (Appendix D, formula (D.6a)) Δ( J +1) = p( J + 1)( J + 2), p being the doubling constant. The superscripts z and x denote the ﬂuorescence observation directions. In view of the strong dependence Φ ∼ J −4 , electric ﬁeld intensities can be determined over a wide dynamic range by varying J. Naturally, excitation can also be affected at the frequencies of the P- or Q-branch transitions. In that case, the application of an electric ﬁeld will supplement the ﬂuorescence spectrum with the SQ- or P- and R-branch lines, respectively. Figure 6.14 presents some fragments of the ﬂuorescence spectrum of the NaK molecule excited on the P(5) line of the X1 Σ, v = 0 →B1 Π, v = 5 transition by laser radiation polarized normal to the direction of the electric ﬁeld differing in intensity [51]. The appearance of lines due to ﬁeld application becomes visible even at F ≈ 5 Vcm−1 (according to the estimates made in [51], the Stark splitting at such ﬁeld strengths is of the order of the natural line width and is totally masked by the Doppler broadening). Let us compare formulas (6.21) with formula (6.5). If one disregards, for the sake of obviousness, summation with respect to M in formulas (6.21), one can then easily obtain the following expression for the ratio between the intensities of the forbidden (Q) and allowed (P) lines in the ﬂuorescence spectrum: IQ Φ2 x2 ∝ ∝ . IP 2 + Φ2 1 + x2

(6.22)

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.14 Fragments of the LIF spectra of the NaK molecule excited to move to the B1 Π (v = 5, J = 4) state from the groundstate level X1 Σ (v = 0, J = 5 ( P(5) line). ( J +1)

Here x = De f F/Δe f , the quantity De f coinciding up to a numerical factor with the matrix element of the dipole moment between the Λdoubling sublevels e and f . It is not very difﬁcult to see from comparison between formulas (6.5) and (6.22) that they are identical in the limiting cases of weak ﬁelds F (x 1) and strong ﬁelds F (x 1). But in the intermediate region (x ≈ 1), where the magnitude of the dipole interaction of the atom (molecule) with the ﬁeld F is of the same order of magnitude as the distance between the closely spaced upper levels, there is a noticeable difference. The reason for this difference is as follows. In the case of formula (6.5), we are dealing with spontaneous radiative transitions from the upper level α. The role of the electric ﬁeld F here is solely to modify the dipole moment matrix elements dαβ and dαβ for the transitions from the level α to the lower levels β and β . As a consequence, the oscillator strengths for the α → β and α → β transitions under consideration, and hence the intensities of the corresponding spectral lines, become dependent on the ﬁeld F. In the case of formula (6.22), we have to contend with laser-induced ﬂuorescence. In this case, the ﬁeld F also modiﬁes the dipole moment matrix elements coupling the lower levels of the electronic state Σ and the Λ-doubling sublevels e and f . This gives rise to two effects. First, the populations of the Λ-doubling sublevels e and f become dependent on the ﬁeld F, for they become populated on laser excitation from the lower electronic state Σ. Secondly, the oscillator strengths for transitions from the Λ-doubling states e and f to the lower levels belonging to the electronic state Σ also become ﬁeld-dependent. Thus, the main cause of the difference between formulas (6.5) and (6.22)

6.2 Laser Stark Spectroscopy

is the fact that formula (6.22) allows for the effect that is not covered by formula (6.5), namely, the dependence of the populations of the upper levels (Λ-doubling sublevels e and f ) on the ﬁeld F. Note, however, the inherent limitation of this method. Formulas (6.21) are obtained on the assumption of purely radiative population and decay of the levels of the 1 Π state. This assumption is only acceptable for lowdensity plasma (and gas) conditions where the collisions of the particles in the 1 Π state with the surrounding particles, leading to transitions between the Λ-doublet levels and the Zeeman levels differing in M, can be neglected. No systematic information is available on the cross sections of such collisions, and so the applicability condition of formulas (6.21) in the form τr < τc (τr and τc are the radiative decay and the gas kinetic collision time, respectively) should be considered approximate (note that the cross section for the quenching of NaK (B1 Π, molecules by inert gas atoms and simple diatomic molecules is, according to the measurements made in [51], σ ≈ (1–3) × 10−14 cm2 ). Maurmann and co-workers [52] have derived formulas similar to (6.17) on the assumption that the Zeeman levels differing in M are equally populated, owing to collisions, within the limits of each of the Λ-levels. In this limiting case, and opposite to the preceding one, the formulas for excitation on the R branch (Figure 6.10), laser polarization parallel to the z-direction of the electric ﬁeld and ﬂuorescence observation direction along the x-axis have the form [52] IQ =

J J +1 . 1 2 2 ( J + 1 ) − M ∑ ∑ 1 ( J + 1)2 (2J + 1)(2J + 3)2 M=−( J +1) M1 =− J . × ( J + 1)( J + 2) + M2 1 − cos β M cos β M1 ,

J +1 J . 1 2 2 ( J + 1 ) − M IP = ∑ ∑ 1 (2J + 1)(2J + 3)3 (2J + 5) M=−( J +1) M =− J 1 . 2 1 + cos β M cos β M1 , × ( J + 2)(3J + 7) − M

(6.23)

where cos β M = (1 + Φ2M )− /2 , 1

ΦM =

2μMF ( J +1)

( J + 1)( J + 2)Δe f

.

(6.24)

In writing down formulas (6.23), (6.24), provision has also been made for the fact that Δ( J +1) can be approximated by different expressions (Table 6.1, Appendix D). Similar intermixing effects can also be caused by collisions with electrons. For this reason, the application ﬁeld of this high-sensitivity

333

334

6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma Table 6.1 Constants of polar diatomic molecules. Molecule

BCl

BH

CS

NaK

Transition

X1 Σ+ –A1 Π

X1 Σ+ –A1 Π

X1 Σ+ –A1 Π

X1 Σ+ –B1 Π (λ ∼ 585 nm)

(λ ∼ 270 nm)

(λ ∼ 430 nm)

(λ ∼ 260 nm)

μ (1 Π, Debye)

0.93

0.58

0.67

2.4

Δ( J +1) (1 Π)(cm−1 )

2.5 × 10−5 J

0.0389

2.5 × 10−4 J

6.5 × 10−7 J

· ( J + 1)

−0.0027(v + 1/2) J

· ( J + 1)

· ( J + 1)

· ( J + 1) (for v = 1)

method is restricted to low gas pressures and degrees of ionization. The role of collisions can be evaluated experimentally. One can, for example, check on the presence of forbidden lines at minimal ﬁeld strengths. If such a check can be made under given experimental conditions, integrating the spatial distributions of F from the anode to the cathode should give the total voltage drop across the electrodes. Measurements made at various plasma gas pressures can also provide useful information. Another difﬁculty is associated with the fact that the choice of molecules complying with the prerequisites of the method within a spectral region suitable for measurements is limited. Only four molecules have so far been used for the purpose. Some information about these molecules is given in Table 6.1. However, despite these restrictions the method has found rather widespread practical application. The molecules listed in Table 6.1 frequently form in plasma chemical reactors implementing etching, surface cleaning and other technologies at low plasma densities. Detailed investigations into electric ﬁelds in reactors, dynamics of sheath layers, and so on have been carried out with high spatial and temporal resolution (see, for example, [56]). 6.2.3 Multiphoton Excitation of Atoms

One of the problems in selecting the working particle in LIF Stark spectroscopy is associated with the fact that the wavelength of transitions from the electronic ground state to excited states in most stable atoms and molecules fall within the UV and VUV regions of the spectrum and so beyond the tuning bands of the standard lasers Figure 3.7). In a number of cases, this difﬁculty is surmounted by multiphoton ﬂuorescence excitation techniques. Booth and co-workers [54] used a (2+1)-photon nonlinear optical excitation scheme. With this scheme (Figure 6.15a), atomic hydrogen was ex-

6.2 Laser Stark Spectroscopy

Figure 6.15 Schematic diagrams of the (2+1)-photon Stark spectroscopy of hydrogen: (a) excitation of Hα with participation of cascade transitions; (b) direct excitation of Hα .

cited from the ground state to the state with n = 2 by UV laser radiation at λ = 243 nm. Absorption of two counter-running photons removed the Doppler limitation on the spectral resolution (Section 3.5.1). The wavelength of the second laser was varied over the range of the Stark multiplet components of the n = 2 → n = 6 transition. The electric ﬁeld F could be determined from the splitting of the level n = 6. The absorption spectrum was registered by the intensity variation of the Hα ﬂuorescence resulting from the n = 6 → · · · → n = 3 spontaneous cascade transitions. Sensitivity in conditions of the glow discharge [54] amounted to some 50 Vcm−1 . To exclude the sensitivity-limiting stage of excitation of the Hα line of interest via spontaneous cascade transitions, Czarnetzki and coworkers [55] developed further the idea of the (2+1)-photon excitation scheme. In this version (Figure 6.15b), two laser photons directly excite the level n = 3 whence the multiplets of the nl = 3 → nu 3 transitions are scanned by the radiation of a second tunable laser. Absorption is ﬁxed by the reduction of the intensity of the Hα ﬂuorescence

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

(ﬂuorescence-dip spectroscopy). This technique allowed the authors to observe states up to nu = 55 (at F = 0). It should be noted, however, that the power of the UV pump lasers should in both cases be limited in order to prevent the fast photoionization of the atoms from the n = 2 or the n = 3 state. On the other hand, the use of unfocused laser beams makes it possible to attain good spatial resolution with a simple optical system and a CCD camera. The given (2+1)-photon excitation scheme has a higher sensitivity, so that the minimum measurable electric ﬁeld strengths prove to be around 5 Vcm−1 . The great number of Rydberg states accessible for observation provides for a wide dynamic range of electric ﬁeld measurements. For the application of the (2+1)-photon excitation scheme in plasma diagnostics, see, for example, [56, 57] and the reviews [42, 58]. Note that the ﬂuorescence-dip detection technique suggested ﬁrst for the Stark spectroscopy of two-photon processes was also used later on in absorption spectroscopy with stepwise excitation schemes. In particular, it was used in the work by Takizawa and co-workers [37], already cited in Section 6.2.1, in determining ﬁeld intensities from the Stark multiplets of atomic argon and can apparently be extended to other cases involved in absorption are particles in metastable states. The optogalvanic detection technique can be used as an alternative to the LIF detection method in the Sub-Doppler Stark spectroscopy. ¨ For example, the optogalvanic effect was used by Grutzmacher and coworkers [59] to register the absorption spectra of the H(1S-2S,3S,3D) lines. The sensitivity in electric ﬁeld strength measurements amounted to some 30 Vcm−1 . To summarize, we can say that the advent of high-resolution laser spectroscopy techniques based on the registration of molecular spectral components with ﬁeld-dependent intensities and spectra of the Stark multiplets of high-lying Rydberg levels in atoms has substantially extended the possibilities of diagnosing electric ﬁelds in plasmas. The common limitations of these techniques are as follows. • All the schemes using the LIF detection method experience difﬁculties in diagnosing plasmas with intense intrinsic luminosity, and so they are used most successfully in studying the complex structure of ﬁelds in the “dark” near-electrode and near-wall regions. • The capabilities of the methods for measuring correlated ﬁelds are restricted by the relatively low electron densities (typically ne ≈ 1013 cm−3 ) by virtue of the state intermixing effects and the presence of static space-charge ﬁelds in plasma.

6.2 Laser Stark Spectroscopy

• As with the methods based on the registration of spectral components that are “forbidden” in the absence of electric ﬁeld, Stark splitting measurements are also limited by the gas pressure at which the broadening of multiplet components becomes comparable with the spacing of levels with neighboring principal quantum numbers n and luminescence quenching comes into play. The typical broadening is Δν/N ≈ 1017 cm2 , energy spacing of Rydberg levels, ca. 105 /n3 cm−1 , number of components, 2n − 1 [60], and at pressures over 13.332 hPa (10 Torr) (N > 3 × 1017 cm−3 ) measurements become problematic. Some of the above problems can be minimized by means of the novel coherent laser spectroscopy methods. 6.2.4 Coherent Four-Wave Stark Scattering Spectroscopy

The scheme of the processes forming the basis of this method results from the evolution of the CARS spectroscopy schemes (Section 3.5.4) and is presented in Figure 6.16 in comparison with them. The schemes of Figures 6.16a and b correspond to the nondegenerate and degenerate CARS processes, respectively. The Stark scattering spectroscopy scheme is illustrated by Figure 6.16c and is essentially the limiting case of the nondegenerate scheme (Figure 6.16a), where the wave being scattered has a frequency of ω3 ≈ 0, which corresponds to the placement of the particle in a (quasi) static electric ﬁeld. The result of scattering is a new wave whose frequency ωΩ is equal to the frequency Ω of a Raman-active molecular vibrational transition. Continuing with the analogy of the phenomenological CARS scheme and assuming that the directions of the polarization vectors of the pumping beams coincide with the electric ﬁeld direction, we can represent the intensity of the radiation induced by the static electric ﬁeld in the case of biharmonic pumping in the form (cf. (3.102), (3.103)) * * * (3) * 2 IΩ ∼ *χΩ * I1 I2 F2 .

(6.25)

Therefore, the electric ﬁeld intensity can be determined from the intensity IΩ . The change from the ﬁelds of optical frequencies to a constant ﬁeld is rather formal, but convenient, for it allows one to use the well-developed CARS theory. The physics of the generation of the IR radiation is simple enough. In the absence of the ﬁeld, radiative transitions within the limits of a ﬁxed electronic state are parity-forbidden, while in its presence other

337

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.16 Schematic diagrams of four-photon transitions: (a) nondegenerate CARS process; (b) degenerate CARS process; (c) generation of IR radiation in an external static electric ﬁeld (ω3 = 0).

electronic states of different parity become “admixed” to the states active in the Raman scattering process [10]. In principle, the generation of scattered radiation at the natural Raman transition frequency is a manifestation of the removal of the forbidding of emission, as in the cases described above in Sections 6.2.1 through 6.2.3. However, there is also a principal difference. The above scattering is a coherent multiphoton process. The scattering particle therein is not ﬁxed in intermediate states with a ﬁnite lifetime, and the role of collisions is drastically reduced. Experimental investigations show that relation (6.25) between the scattered radiation intensity and the electric ﬁeld strength holds at gas pressures up to a few and a few tens of atmospheres [61]. For this reason, the natural ﬁeld of application of the scattering technique to gas discharge investigations is the region of elevated gas pressures. The experimental instrumentation implementing this technique is based on the CARS spectrometer (Figure 3.33) supplemented with an IR radiation registration channel. The experimental setup and measurement method are described, for example, in [42, 61–63]. To determine the direction of the electric ﬁeld vector, it is necessary to ﬁnd the polarization of the scattered radiation [64]. Following the above analogy, we use the CARS theory to express the scattering polarization (Section 3.5.4). For an isotropic medium and three pumping waves, the nonlinear polarization vector can be deﬁned in terms of (3)

the components of the third-order nonlinear susceptibility tensor χijkl = (3)

χijkl (ωΩ ; 0, ω1 , −ω2 ) and the electric ﬁeld vectors of the pumping waves. Omitting the frequency arguments, we may write down the following expression for the polarization vector governing the direction of the elec-

6.2 Laser Stark Spectroscopy

tric ﬁeld vector of the scattered radiation [65]: ! " (3) (3) (3) PΩ (3) = 6FE1 E2 χ1122 e (e1 e2∗ ) + χ1212 e1 (e e2∗ ) + χ1221 e2∗ (e e1 ) , (6.26) where F, E1 , and E2 are the intensities and e, e1 , e2 , the unit polarization vectors of the constant ﬁeld and of the pumping waves ω1 and ω2 , respectively, and χijkl are the nonlinear susceptibility tensor components. The susceptibilities χ(3) are fourth-rank tensors with 81 components, but by virtue of symmetry for isotropic medium, only three of them are independent: χ1111 = χ1122 + χ1212 + χ1221 . For completely symmetrical oscillations, speciﬁcally for the H2 molecule in the electronic ground state, (3)

(3)

χ1212 = χ1221 . In this case, it is only two nonlinear susceptibility tensor components that remain independent. It follows from expression (6.26) that the signal has a maximum intensity when the polarization vectors of all the three ﬁelds are parallel. By rotating the plane of polarization of one of the pumping waves, for example, ω1 , while keeping the polarization of the pumping wave ω2 parallel to the constant ﬁeld, one can determine all the components of the susceptibility tensor. With the pumping wave polarization vectors being collinear (e1 e2 ), the polarization vector of the signal may represented in the form ! " (3) (3) (6.27) PΩ (3) = 6FE1 E2 χ1122 e + 2χ1212 cos ϕ · e1 , where ϕ is the angle between the constant ﬁeld vector e and the pumping wave polarization vectors e1 e2∗ . Thus, the orientation of the constant electric ﬁeld is found from the polarization direction of the IR signal. Use can be made of various measurement procedures [64]. For example, if e1 e2 and the direction of the electric ﬁeld vector F known, one can construct the dependence of the angle α between the polarization vector of the IR signal and the ﬁeld vector on the angle ϕ between the collinear polarization vectors of the lasers and the ﬁled vector (Figure 6.17). Measuring the polarization of the IR signal with an analyzer placed before the radiation receiver allows one to determine the direction of the electric ﬁeld strength vector from the dependence constructed. For the hydrogen molecule, the deviations of the polarization vector of the IR radiation from the constant electric ﬁeld vector are small (according to the (3)

(3)

measurements taken in [64], the ratio χ1122 /χ1212 = 17.3 ± 2.1). The deviations calculated in [42, 64] are shown in Figure 6.17 by the solid curve. The maximum deviation amounts to ca. 3 ◦ C. The deviations calculated in [42] for the nitrogen molecule are also shown in Figure 6.17. As follows from expression (6.26), in the case of e1 e2 e or e1 e2 ⊥ e, the IR signal polarization vector eΩ e. Thus, by taking

339

340

6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.17 Angle α between the polarization vector eΩ of the scattered radiation and the electric ﬁeld vector F as a function of the angle φ between the collinear laser polarization vectors e1 PPe2 and the ﬁeld vector F . Solid curve – calculation for H2 by formula (6.26). Data points – measurements for H2 . Dashed curve – calculation for N2 .

two measurements with the laser beams polarized in two mutually orthogonal planes, one can determine the tangential and normal electric ﬁeld vector components deﬁned in terms of the IR signal intensity as 1 IΩτ ∝ χ1111 Eτ E1 E2 , (6.28) 1 IΩn ∝ χ1122 En E1 E2 . Figure 8.17 illustrates the investigation conducted by van Goor and coworkers [66] into electric ﬁeld in an atmospheric-pressure hydrogen discharge creeping over a ferrite surface (see Section 8.3.3.2 below). The authors studied the mechanism governing the spread of the discharge, for which purpose they measured the ﬁeld in the gap between the propagating cathode- and anode-heading leaders as a function of the time elapsed from the instant the voltage is switched over. Presented are the relations for the tangential and normal electric ﬁeld components and also for the angle of rotation of the ﬁeld vector relative to the discharge propagation axis. To conclude this section devoted to the methods of spectroscopic investigations into electric ﬁelds in plasma and gas, we brieﬂy mention the considerable recent progress in this ﬁeld. More accurate and reliable results have been obtained in the classical emission spectroscopy for diagnostics from the spectra of hydrogen- and

6.2 Laser Stark Spectroscopy

Figure 6.18 Creeping discharge on a ferrite surface. Temporal behavior of (a) the tangential ( Ft ) and normal ( Fn ) electric ﬁeld components and (b) angle of rotation of the electric ﬁeld vector relative to the propagation axis of the discharge leader.

non-hydrogen-like atoms. Novel approaches to the studies of harmonic and nonharmonic plasma oscillations and plasma turbulences have been developed and experimentally veriﬁed; nevertheless, the classical experimental methods still suffer from sensitivity limitations due to inadequate spectral and spatial resolution and Doppler broadening. The use of laser techniques in the Stark plasma spectroscopy has opened up new possibilities for studying the electrical parameters of plasma, providing for electric ﬁeld measurements over a wide dynamic range. The classical and laser Stark spectroscopy methods are evolving concurrently, supplementing one another. Their combined use makes it possible to investigate plasmas over wide regions of pressures and compositions.

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

6.3 Magnetic Field Investigations

In accordance with the general remarks made at the beginning of Chapter 6, we will restrict ourselves to a brief dicussion on the main optical and spectral magnetic ﬁeld measurement methods. 6.3.1 Measurements Based on the Faraday Effect

When light propagates through homogeneous plasma along magnetic ﬁeld lines, the Faraday effect occurs – the rotation of the plane of polarization of the light wave. If the magnetic ﬁeld induction B is measured in gausses, the light wavelength λ in centimeters, the optical path L in plasma in centimeters and the electron density ne in centimeters to the inverse 3rd power, the rotation angle of the plane of polarization of the light wave is (see, for example [67, 68]) φ ≈ 2.6 × 10−17 λ2 Bne L, radians.

(6.29)

It is this relation that is used for magnetic ﬁeld measurements, and it is evident that the attainable measurement sensitivity is not very high. When plasma is probed with visible radiation with a wavelength of λ = 5 × 10−5 cm at B = 104 Gs, ne = 1019 cm−3 , and L = 1 cm, the angle φ ≈ 7 × 10−3 , that is, around half a degree. In principle, the modern methods of measuring the rotation of the plane of polarization are capable of ﬁxing angles φ ≈ 10−7 , but this requires application of highprecision instrumentation [69]. However, what is more important is that the Faraday effect is but one of the mechanisms responsible for the birefringence of plasma, which is associated with its anisotropy produced by the magnetic ﬁeld. Anisotropy of plasma can also be due to its various inhomogeneities. For this reason, increasing the wavelength λ of the light wave, which impairs spatial resolution, will not always improve the measurement sensitivity. Polukhin [67] has indicated the following applicability criteria for the method: • the geometrical optics approximation (invariability of the plasma parameters over the wave length along the radiation propagation line) should be valid • the probe radiation frequency should exceed plasma frequency (1.2), the electron cyclotron frequency ωc = 1.76 × 107 B, Hz, and the collision frequency • any absorption lines in the vicinity of the probe radiation wavelength should be absent.

6.3 Magnetic Field Investigations

In accordance with expression (6.29), the experiment should allow for the measurement of the electron density. Practically, this method proves effective at magnetic inductions B ≥ 104 Gs typical of the conditions of laser plasma, Z-pinch, plasma focus, linear discharge at currents ≥ 103 A and similar plasma objects, when probed with visible radiation. The authors of a number of works on Faraday measurements have used a peculiar combination of contact and optical methods. In such a case, the light beam is passed through a solid-state transducer in plasma. The transducer material is transparent and has a high Verdet constant, so that the rotation of the plane of polarization of the light beam is mainly due to the Faraday effect occurring in the transducer, even if its size is small. Of course, all the general limitations speciﬁc to the contact methods should be overcome. For more details and further references, see [68]. 6.3.2 Spectral Methods

The direct methods are based on the Zeeman effect – the effect of magnetic ﬁeld on the spectrum structure and polarization of radiation. In a magnetic ﬁeld with an induction of B, atomic levels with the total momentum quantum number J get split into 2J + 1 magnetic sublevels whose component energies relative to the energy of the nonsplit level are E( B) (α, J, m) = g(α, J )μ0 mB,

(6.30)

where m is the quantum number of the projection of J on the direction speciﬁed by the ﬁeld (m = ±1, ±2, ±3, . . .), μ0 is the Bohr magneton, g is the Lande factor and α is the set of quantum numbers except for J and m. Level and line splitting patterns can be found in numerous books (see, for example, [9, 10]). The selection rules for k → i transitions are ΔJ = 0, ±1 (except for Jk = 0 → Ji = 0), Δm = 0, ±1. When gi = gk , the number of the components of the Zeeman spectral multiplet is determined by J and α. When gi = gk , this is a triplet corresponding to transitions with Δm = 0 (nonshifted π-component) and Δm = ±1 (symmetrically shifted σ-components). If the observation direction is along the magnetic ﬁeld, in the latter case one observes two circularly polarized σ-components, and if it is across the ﬁeld, all the components are polarized circularly, the π-components being polarized along the ﬁeld, while the σ ones, normal to the ﬁled. Relation (6.30) and the polarization rules lie at the basis of the spectral diagnostics of magnetic ﬁelds.

343

344

6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

Figure 6.19 Line proﬁles of the Ba + line at λ = 614.17 nm.

The magnitude of splitting can be estimated by the formulas for the simple Zeeman effect (disregarding the spin splitting) [6]: Δλ = ±4.67 × 10−5 λ2 B,

Δν = ±4.67 × 10−5 B,

(6.31)

where λ is measured in centimeters, ν in centimeters to the inverse 1st power, and B in gausses. To illustrate, even at as strong a magnetic ﬁeld ˚ which is comparable with as B = 2 × 104 Gs the splitting Δλ = ±0.23 A, the Doppler broadening of the components. Therefore, frequently observed in plasma diagnostics experiments are not separate resolved components, but an integral broadened line proﬁle whose shape depends on the ﬁeld strength. Various approaches are possible to the exclusion of the Doppler broadening, for example, by way of modeling the line proﬁle at a given gas temperature. It may prove helpful to use additional polarization measurements whereby one can discriminate between the contributions from the π- and σ-components. An example of such measurements is illustrated in Figure 6.19. Subject to study in this experiment was the ion Ba + (6p2 P–5d2 D, λ = 614.17 nm) in conditions of a plasma commutator [13, 70]. The ﬁgure shows the line proﬁles of the πand σ-components, discriminated by polarization with the observation direction across the ﬁeld. These components both have equal Doppler widths, and so the inﬂuence of the magnetic ﬁeld can be discerned as the difference in width between the line proﬁles. In the given example, this corresponds to an induction of B = 6 kGs.

6.3 Magnetic Field Investigations

Let us mention one more technique for measuring magnetic ﬁelds, which involves the injection of beams of neutral particles into plasma. This is an indirect method based on the fact that the motion of an atom with a velocity of v in a magnetic ﬁeld gives rise to the dynamic Stark effect (“motional Stark effect”) in the Lorentz electric ﬁeld with the intensity FL = [v×B]/c. For hydrogen-like atoms with the linear Stark effect, the splitting can be estimated (disregarding the splitting of the lower transition level) as [9, 10] ΔωmS ≈ ea0 n2 |[v × B]|/h,

(6.32)

and the ratio between the magnitudes of the motional Stark effect and the Zeeman shift for a nonrelativistic beam is ΔωmS /ΔωZ ≈ n2 vn /αc = (2Ea /Ma c2 ) /2 n2 α−1 sin θ, 1

(6.33)

where α = 1/137, vn is the atomic velocity component normal to the magnetic ﬁeld, Ea and Ma are the energy and mass of the atom, respectively and θ is the angle between the magnetic ﬁeld and beam propagation directions. The motional Stark shift of the components can exceed the Zeeman shift even at not very high energies of the atomic beam injected into plasma either for diagnostics purposes or to heat it. To illustrate, for the Hα hydrogen line, ΔωmS /ΔωZ ≈ 1at Ea ≈ 1.600 × 10−16 J(1 keV). At Ea ≈ 160.000 × 10−19 J(100 eV), the motional Stark splitting is an order of magnitude greater that the Zeeman splitting, which makes this method of measuring magnetic ﬁeld induction more preferable than the direct observations of the Zeeman splitting. For more details and further references, see [13, 71, 72].

345

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6 Spectral Methods of Determining Electronic and Magnetic Fields in Plasma

References

1 V. N. Ochkin, N. G. Preobrazhensky, and N.Y. Shaparev. Optogalvanic Effect in Ionized Gas. London-Moscow: Gordon and Breach Science Publishers (1998). 2 V. P. Gavrilenko. Spectroscopic Methods for Determining Electric Fields in Plasma. In: V. E. Fortov. Ed. Encyclopedia of Low-Temperature Plasma (in Russian) 1, pp. 354–366; Moscow: Nauka (2000). 3 V. P. Gavrilenko. Spectroscopic Methods for Determining Electric Fields in Plasma. In: V. E. Fortov. Ed. Encyclopedia of Low-Temperature Plasma (in Russian) 2, pp. 559–563. Moscow: Nauka (2000). 4 V. P. Gavrilenko, V. N. Ochkin, and S. N. Tskhai. Progress in Plasma Spectroscopic Diagnostics Based on the Stark effect in Atoms and Molecules. In: V. N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 207– 229 (2002). 5 J. E. Lawler and D. A. Doughty. The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas. Advances in Atomic, Molecular and Optical Physics, 34, pp. 171–206 (1994). 6 H. A. Bethe and E. E. Salpiter. Quantum Mechanics of One- and TwoElectron Atoms. N.Y.: Plenum Press (1977). 7 H. R. Griem. Plasma Spectroscopy. N.Y.: McGraw-Hill (1964). 8 H. R. Griem. Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). 9 S. E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). 10 I. I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Fizmatgiz (1963). 11 E. Oks. Plasma Spectroscopy. The Inﬂuence of Microwave and Laser Fields. Berlin: Springer (1995). 12 L. A. Bureeva, V. P. Gavrilenko, and V. S. Lisitsa. An Atom in External Fields, In: V. E. Fortov, Ed. Encyclo-

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pedia of Low-Temperature Plasma (in Russian), 1, pp. 354–366. Moscow: Nauka (2000). V. P. Gavrilenko. Spectroscopic Methods for Determining Magnetic Fields in Plasma. In: V. E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 556–558. Moscow: Nauka (2000). D. R. Inglis and E. Teller, Ionic Depression of Series Limits in OneElectron Spectra. Astrophys. J., 90, pp. 439–448 (1939). C. Barbeau and J. Jolly. Electric Field Measurement in the Cathode Sheath of a Hydrogen Glow Discharge. Appl. Phys. Lett. 58, pp. 237–239 (1991). E. Sarid, Y. Maron, and L. Troyansky. Spectroscopic Investigation of Fluctuating Anisotropic Electric Fields in a High-Power-Diode Plasma. Phys. Rev. E, 48, pp. 1364–1374 (1993). D. I. Blochinzew. Zur Theorie des Starkeffects im Zeitveranderlichten Field. Phys. Z. (USSR), 4, pp. 1364– 1374 (1933). E. A. Oks and Yu. M. Shagiev. Tables of Line Proﬁles of the Balmer Series of Hydrogen in the Field E(t) =E0 cos ωt (in Russian). Preprint No. 77. Gorky: Institute of Applied Physics Press (1983). A. Kamp and G. Himmel. Determination of RF Electric Field Strengths in Deuterium from Satellite Structure of Balmer Lines. Appl. Phys. B. Photophys. and Laser Chem., B47, pp 177–185 (1988). E. A. Oks and VB. P. Gavrilenko. The Quasilocal Principle of Measurement of Electric Fields in Plasma from the Satellites of Forbidden Spectral Lines of Helium (in Russian). Pis’ma v ZhTF, 9, No. 5, pp. 257–260 (1983). M. Baranger and B. Mozer. Light as a Plasma Probe. Phys. Rev. 123, pp. 25–28 (1961). H.-J. Kunze and H. R. Griem. Laboratory Observation of Plasma Satellites on the 21P-41D and 23P-43P Lines of

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Helium. Phys. Rev. Lett., 21, pp. 1048–1052 (1968). M. P. Brizhinev, V. P. Gavrilenko, S. V. Egorov et al. A Method of Quasilocal Measurements of Electric Fields in Plasma from the Satellites of Forbidden Spectral Lines of Helium (in Russian). ZhETF, 85, No. 3, pp. 893– 908 (1983). V. P. Gavrilenko. Spectroscopic Method for Determining the Polarization State of Oscillating Electric Fields in Plasma (in Russian). ZhTF, 103, No. 2, pp. 455–466 (1993). E. A. Den Hartog, D. A. Doughty, and J. E. Lawler. Laser Optogalvanic and Fluorescence Studies of the Cathode Region of a Glow Discharge. Phys. Rev., A38, pp. 2471–2491 (1988). L. A. Bureeva and V. S. Lisitsa. An Excited Atom (in Russian). Moscow: IZDAT (1997). J. R. Shoemaker, B. N. Ganguly, and A Garscadden. Stark Spectroscopic Measurement of Spatially Resolved Electric Field and Electric Field Gradients in Glow Discharges. Appl. Phys. Lett., 52, No. 24, pp. 327–239 (1991). D. A. Doughty, S. Salih, and J. E. Lawler. Two Step Optogalvanic Effect Using Intersecting Laser Beams: a Pinpoint Discharge Diagnostics. Phys. Lett., 103A, pp. 41–44 (1984). B. N. Ganguly and A. Garscadden. Optical Diagnostics of Positive Column DC Discharge by Rydberg Stark Spectroscopy. Appl. Phys. Lett., 46, No. 6, pp. 540–542 (1984). W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). L. M. Biberman, V. S. Vorobyev, and I. T. Yakubov. Kinetics of Nonequilibrium Low-Temperature Plasma (in Russian). Moscow: Nauka (1982). Y. P. Raiser. Physics of Gas Discharge (in Russian). Moscow: Nauka (1982). A. Garscadden. Gas discharge laser diagnostics update. In: J. M. Proud and L. H. Luessen, Eds. Radiative Processes in Discharge Plasmas pp. 467– 494. N.Y.: Plenum Press (1986). V. P. Gavrilenko, H. J. Kim, T. Ikutake et al. Measurement Method for Electric Fields Based on Stark Spec-

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troscopy of Argon Atoms. Phys. Rev., E62, pp. 7201–7208 (2000). H. J. Kim, J. H. Lee, K. W. Wang, and Y. W. Choi. Temporal Behavior of the Wall Voltage in a Surface-Type Alternating Current Plasma Display Panel Cell Using Laser Induced Fluorescence Spectroscopy. J. Appl. Phys., 89, No. 5, pp. 2539–2541 (2001). V. P. Gavrilenko, H. J. Kim, T. Ikutake et al. Method to Measure the Electric Field Vector in an Argon Glow Discharge Using Laser Polarization Spectroscopy. Phys. Rev., E63, pp. 7401–7404 (2001). K. Takizawa, K. Sasaki, and A. Kono. Sensitive Measurements of Electric Field Distributions in Low-Pressure Ar Plasmas by Laser-Induced Fluorescence-Dip Spectroscopy. Appl. Phys. Lett., 84, No. 2, pp. 185–187 (2004). K. Takiyama, T. Katsuta, M. Watanabe et al. Spectroscopic Method to Directly Measure Electric Field Distribution in Tokamak Plasma Edge. Rev. Sci. Instr., 68, pp. 1028–1031 (1997). B. N. Ganguly, A Garscadden. Electric Field Vector Measurement in a Glow Discharge. Phys. Rev., A32, pp. 2544–2545 (1985). V. P. Gavrilenko and E. A Oks. A New Principle of Measuring Microwave Fields Using the Resonance LaserInduced Fluorescence of the Hydrogen Atom (in Russian). Pis’ma v ZhTF, No. 10, pp. 609–610 (1984). I. N. Polushkin, M. Yu. Ryabikin, Yu. M. Shagiev, and V. V. Yazenkov. A Method of Local Measurements of Microwave Fields Using the Resonance Laser-Induced Fluorescence of the Hydrogen Atom (in Russian). ZhTF, 62, p. 953 (1985). V. P. Gavrilenko, V. N. Ochkin, and S. N. Tskhai. Progress in Plasma Spectroscopic Diagnostics Based on Stark Effect in Atoms and Molecules. In: V. N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 207– 229 (2002). E. A. Oks, St. Boddeker, and H.-J. Kunze. Spectroscopy of Atomic Hy-

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drogen in Dense Plasma in the Presence of Dynamic Fields: Intra-Stark Spectroscopy. Phys. Rev., A44, pp. 8338–8347 (1991). V. P. Gavrilenko and E. A. Oks. Enhancement of Spectroscopic Method for Mapping Microwave Fields in Tokamak Plasmas. Rev. Sci. Instr., 70, pp. 363–367 (1999). J. M. Brown, J. T. Hougen, K.-P. Huber et al.. The Labeling of Parity Doublet Levels in Linear Molecules. J. Molec. Spectroscopy, 55, pp. 500–503 (1975). K.P. Huber, G. Herzberg. Constants of Diatomic Molecules (in Russian): Mir (1984). C. A. Moore, G. P. Davis, and R. A. Gottscho. Sensitive, Nonintrusive, InSitu Measurement of Temporally and Spatially Resolved Plasma Electric Fields. Phys. Rev. Lett. 52, pp. 538–541 (1984). J. Derouard and N. Sadeghi. Effect of a Weak Electric Field on the Fluorescence of a Polar Molecule in A1Π Electronic States: E-F Zero Field Anticrossings in NaK(B1Π). Opt. Comm., 57, pp. 239–243 (1986). H. Debontride, J. Derouard, P. Edel et al. Transient Current and Sheath Motion Following the PhotoelectronInitiated Avalanche in DC Glow Discharges. Phys. Rev., A40, pp. 5208– 5219 (1989). Y. Yamagata, Y. Kawano, K. Muraoka et al. In-Situ Calibration for Electric Field Measurements in GlowDischarge Plasmas by Laser-Induced Fluorescence Using the Optogalvanic Technique. Jap. J. of Appl. Phys., 30, pp. 166–170 (1991). J. Derouard, H. Debontride, T. D. Nguen, and N. Sadeghi. Time Resolved Stark Spectroscopy in NaK. Measurement of Permanent Electric Dipole, Radiative Lifetimes, and Collisional Rate Constants. J. Chem. Phys., 90, pp. 5936–5943 (1989). S. Maurmann, V. Gavrilenko, H.-J. Kunze, and E. Oks. Use of Fluorescence Spectroscopy of CS for Measurements of Electric Fields in a Discharge Plasma. J. Phys., D29, pp. 1525–1531 (1996).

53 R. A. Gottscho. Glow Discharge Sheath Electric Fields: Negative Ions, Power and Frequency Effects. Phys. Rev., A36, No. 5, pp. 2233–2242 (1987). 54 J. P. Booth, M. Fadallah, J. Derouard, and N. Sadeghi. Electric Field Measurements in Discharges by 2+1 Photon Laser Stark Spectroscopy of Atomic Hydrogen. Appl. Phys. Lett., 65, pp. 819–821 (1994). ¨ 55 U. Czarnetzki, D. Luggenholsher, and ¨ H. F. Dobele. Sensitive Electric Field Measurement by Fluorescence-Dip Spectroscopy of Rydberg States of Atomic Hydrogen. Phys. Rev. Lett. 81, pp. 4532–4595 (1998). ¨ 56 U. Czarnetzki, D. Luggenholsher, and ¨ H. F. Dobele. Investigation on Ionic Processes and Dynamics in the Sheath Region of Helium and Hydrogen Discharges by Laser Spectroscopic Electric Field Measurements. Appl. Phys., A72, pp. 509–521 (2001). ¨ 57 U. Czarnetzki, D. Luggenholsher, ¨ and H. F. Dobele. Space and Time Resolved Electric Field Measurements in Helium and Hydrogen RF-discharges. Plasma Sources Sci. Technol., 8, pp. 230– 248 (1999). 58 V. N. Ochkin, S. Yu. Savinov, S. N. Tskhai, U. Czarnetzki et al. Nonlinear Optical Techniques for Plasma Diagnostics. IEEE Trans. Plasma Sci., 26, No. 5, pp. 1502–1513 (1998). ¨ 59 K. Grutzmacher, A. Steiger, J. Seidel, M. I. de la Rosa, and C. Perez. Local E-Field Measurements Via StarkSplitting of Hydrogen Resonance Lines. Proc. 10th Internat. Symp. on LaserAided Plasma Diagnostics, pp. 487– 495 (2001). 60 V. S. Lebedev and I. L. Beigman. Physics of Highly Excited Atoms and Ions. Berlin: Springer (1998). 61 O. A. Evsin, E. B. Kupriyanova, V. N. Ochkin et al. Measurement of Electric Field Strengths in Gases and Plasma by the CARS Technique (in Russian). Kvant. Elektr., 22, No. 3, pp. 295–299 (1995). 62 A. M. Zheltikov, N. I. Koroteev, A. N. Naumov et al. Measurement of Electric Fields in Plasma Using the Coherent Four-Wave Interaction Po-

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larization Technique (in Russian). Kvant. Elektr., 26, No. 1, pp. 73–76 (1999). S. N. Tskhai, D. A. Akimov, S. V. Mitko et al. Time Resolved Polarization-Sensitive Measurements of the Electric Field in Sliding Discharge by Means of DC Field-Induced Coherent Raman Scattering. J. Raman Spectrosc., 32, pp. 177–181 (2001). D. A. Akimov, A. M. Zheltikov, N. I. Koroteev et al. Coherent Raman Scattering in Molecular Hydrogen in a DC Electric Field (in Russian). Pis’ma v ZhETF, 70, No. 6, pp. 375–379 (1999). S. A. Akhmanov and N. I. Koroteev. Nonlinear Optics Techniques in Light Scattering Spectroscopy: Active Light Scattering Spectroscopy (in Russian). Moscow: Nauka (1981). F. A. van Goor, S. V. Mitko, V. N. Ochkin et al. Arc Discharge Sliding over a Conducting Surface. J Russ. Laser Res., 18, pp. 247–259 (1997). S. N. Polukhin. Measurement of Magnetic Fields in Plasma from the Fara-

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day Effect. In: V. E. Fortov Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 552–555. Moscow: Nauka (2000). Yu. P. Zakharova. Contact Faraday Measurement Techniques for Local Magnetic Field Measurements in Plasma. In: V. E. Fortov Ed. Encyclopedia of Low-Temperature Plasma (in Russian), 2, pp. 555–556. Moscow: Nauka (2000). Yu. V. Bogdanov, S. I. Kanorsky, I. I. Sobelman, V. N. Sorokin et al. Studies of the Impact Broadening of the Hyperﬁne Structure Components of the 648 nm Line of Bismuth by the Faraday Spectroscopy Method. (in Russian). Optika i Spektroskopiya, 61, pp. 446–453 (1986). M. Sarfaty, R. Spitalnik, R. Arat et al. Plasma Phys., 2, p. 2583 (1995). H. R. Griem. Plasma Spectroscopy. N.Y.: McGraw-Hill (1964). H. R. Griem.Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997).

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7

Determination of the Parameters of the Electronic Component of Plasma In view of the importance of data on the concentration and energy distribution of free electrons in plasma and the wide variation range of these parameters, several groups of measuring methods, including probe [1–4] and microwave ones [5, 6], have been and are being developed. In the following chapter we discuss the optical and spectral methods.

7.1 Interferometry

The interferometric methods of measuring the density of matter are well known [7–9]. The speciﬁcity of their application to the diagnostics of low-temperature plasma of low, as a rule, degree of ionization lies in the relatively low density of the material and the need to discriminate between the contributions to refraction (refractive index) from electrons and heavy particles. The refractive index of an ionized gas at a frequency of ω is given by [6–10] n ≈ 1 + 2π

∑ χ j Nj − ne e2 /me ω2

= 1 + n0 + ne .

(7.1)

j

Here the ﬁrst term between the parentheses stands for refraction from heavy particles (atoms, molecules, ions), while the second, for that from electrons. The neutral gas consists of various particle species Nj characterized by the polarizabilities χ j . The refractive indices n0 and ne refer to the neutral and the electronic component, respectively. Formula (7.1) is obtained subject to the following conditions: the probe radiation frequency ω is off resonance with the optical transition (|ω − ωik | Δωik , Δωik is the width of the transition line) and exceeds both the plasma and the electron cyclotron frequency (ω 2 ω 2L = 4πne e2 /me , ω ωe ≈ eB/me c) and also the electron-heavy-particle collision frequency (ω νeff ).

352

7 Determination of the Parameters of the Electronic Component of Plasma

The contribution to refraction from electrons is deﬁned by the expressions ω 2L ; ω2 1 ω 2L e2 λ2 ne − 1 ≈ − =− ne ≈ −4.49 × 10−14 λ2 ne . 2 2ω 2πme c2

( ne )2 = 1 −

(7.2)

In low-temperature plasma, most of the heavy-particle contribution to refraction frequently comes from particles in electronic ground states i = 1. Insofar as the resonance transition frequencies usually fall within the vacuum ultraviolet region of the spectrum (one should bear in mind such important exclusions as, for example, alkali metal atoms and molecular radicals, see Sections 5.2, 5.3), while probing is effected with visible light, λ λ1k . Subject to this additional condition, the polarizabilities and refractive indices for heavy particles are calculated by the Cauchy formulas χ = a + b/λ2 , n0 − 1 = ( A + B/λ2 ) N/NL ,

(7.3)

where NL = 2.687 × 1019 cm−3 is the Loschmidt number. The values of the constants a and b, and also of A and B (the latter at normal conditions – 0 ◦ C, 1013.232 hPa(760 Torr), i.e. N = NL ) for unexcited particles of some gases are listed in Table 7.1 [6, 10]. Table 7.1 Constants for some non-excited atoms.

a, 10−24 cm3

He

Ar

Ne

Kr

Xe

H2

N2

O2

Air

Hg

0.394

2.48

4.04

0.805

1.76

1.58

1.7

5.2

0.206

1.65

b, 10−35 cm−1

0.474

9.26

0.947 17.3

6.2

13.3

8

9.67 118

χ, 632.8 nm

0.207

1.68

0.397

2.52

4.14

0.82

1.8

1.6

1.73

A, 10−5

3.48

27.97

6.66

41.89

68.23

13.58

29.06

26.41

B, 10−4 cm2

0.08

1.56

0.16

2.92

6.92

1.02

2.24

41

5.5

10−24 cm3 28.5

For the visible region of the spectrum, the wavelength dependence of polarizability and refractive index is weak, a b/λ2 , A B/λ2 . Table 7.2 lists the “static” (λ → ∞) values of χ ≈ a for some more atoms, molecules and ions [10]. Disregarding the wavelength dependence of the polarizability of heavy particles and assuming linear dependence of polarizability and refraction on density (Gladstone-Dale law), we may write (for a single particle species) n0 − 1 ≈ πχN = Ca N,

(7.4)

7.1 Interferometry Table 7.2 Optical atomic constants in “static” (λ → ∞) case. H

H–

Li

Li –

Na

Na +

Al

Al +

Cl

Cl +

Cl –

Ar +

0.67

30.7

24.4

118

24.2

0.2

6.8

3.5

2.2

1

3.6

1.1

Ni

Cu

Cu

+

I

–

CO

K2

Li2

CO2

H2 O

21.1

22.3

1.6

7.4

1.9

77

34

2.9

1.4

Particles a, χ, 10−24 cm3 Particles

K

a, χ, 10−24 cm3

43.6

K

+

1

5.4

I

where Ca = 2πχ is the refraction per particle, χ = A/2πN. Such notation is of certain convenience, for formulas (7.2) and (7.3) are both linear in the density of the particles involved. Using the data listed in Table 7.1, formula (7.1) for a hydrogen plasma, for example, may be represented in the form 0.38 × 10−33 NH2 . (7.5) n − 1 = −4.49 × 10−14 λ2 ne + 0.51 × 1023 + λ2 The experimental interferometric schemes for measuring electron density are based on the registration of the phase shift between a light wave passing through plasma and its counterpart passing outside of plasma. For this purpose, the light beam used in the experiment is usually split into two, one passing through the cell containing the plasma object of interest and the other, through a dummy cell of similar design. The two beams are then re-united to produce an interference pattern, the shift of whose fringes is measured to determine the phase change. The phase incursion Δφ is determined by the optical path in plasma from z1 to z2 : ω Δϕ = c

z2

[n( x, y, z) − n0 ] dz

(7.6)

z1

For homogeneous plasma with a length of z2 − z1 = L, the phase incursion is contributed to, according to formula (7.1), from both heavy particles and electrons. If the dummy cell is ﬁlled with the plasma-forming gas, the phase shift, which is the sum of the phase shifts δϕ0 and δϕe due to the change of the densities of the neutral and the electronic component, will in the case a of one-component gas be Δϕ = δϕ0 + δϕe =

2ωLπ c

kΔN −

e2 . Δn e me ω 2

(7.7)

If the dummy cell is vacuumized, then ΔN = N, Δne = ne . The difference in frequency dependence between the electronic and the heavyparticle component in formulas (7.1) and (7.7) allows one to discriminate

353

354

7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.1 Ratio between the phase shifts due to electrons and heavy particles as a function of the degree of ionization of plasma.

between their contributions by using different probe radiation sources with frequencies ω1 and ω2 and the measured phase incursions. For the electronic component, it follows form expression (7.7) that ω12 ω22 cme Δϕ2 Δϕ1 Δne = − . (7.8) ω2 ω1 ω22 − ω12 2Lπe2 Substituting expression (7.8) into (7.7), we obtain the value of ΔN. Relation (7.7) allows one to orient oneself in selecting the probe radiation source. Figure 7.1 presents the ratio between the phase shifts due to electrons and heavy particles as a function of the degree of ionization of plasma. The different wavelengths indicated in the ﬁgure correspond to the typical laser and microwave sources used in plasma diagnostics experiments. In calculating the relationships, use was made of the typical value of polarizability, χ = 10−24 cm3 (see Table 7.1). The lower the degree of ionization of plasma, the longer the advisable probe radiation wavelength that should be used. However, account should be taken of the above requirement that the plasma frequency should be small in comparison with the probe radiation frequency for formula (7.1) to hold true. As these frequencies draw closer together, the probe light weakens. Figure 7.2 presents the phase shifts due to electrons in terms of δϕe /2πL for various probe radiation wavelengths. The different y-axis scales correspond to two cases of organization of measurements. In one of them (left-hand y-axis), the probe radiation passes through plasma

7.1 Interferometry

Figure 7.2 Phase shifts in passing 1 cm of plasma as a function of the concentration of electrons. The lines designated as SP and MP indicate the transmission cutoff in single- and multipass (100pass) cells, respectively.

only singly. In the other case, plasma is placed in a multipass cell (Section 3.3.5) with the effective number of passes equal to 100. The phase incursion per centimeter in plasma is increased the same number of times, but the maximum measurable electron concentration is reduced accordingly because of the weakening of the probe signal. This is indicated by the dashed lines. The weakening is associated with the fact that at an electron density of ne > ncr = mω 2 /4πe2 ≈ 1.2 × 104 ( f [MHz])2 ≈ 1013 (λ[cm])−2 , [cm−3 ] plasma reﬂects the incident electromagnetic radiation with a frequency of ω = 2π f (with a wavelength of λ) [5]. The minimum measurable value of the electron density ne depends not only on the probe radiation wavelength and the optical path length nLeff , but also on the the choice of concrete interferometric scheme and the ways to compensate for the imperfection of the interferometer and phase shift registration method used. These questions are considered, for example, in [7, 9, 10, 13, 14]. When using a triple-mirror interferometer, a homodyne technique for registering phase shifts up to δϕe ≈ 10−6 and a submillimeter probe laser, one can achieve a sensitivity of ne L ≥ 108 cm−2 . In general measurements with δϕe ≈ 10−2 in the submillimeter region, more typical sensitivity values are ne L ≥ 1013 cm−2 , and in ˚ ne L ≥ 1016 cm−2 . the visible region (λ ≈ 5000 A), Numerous techniques have been developed in connection with the problem of interferometric measurements of electron density under dis-

355

356

7 Determination of the Parameters of the Electronic Component of Plasma

cussion, including those with the visualization of interference patterns and photoelectric mixing of the interfering signals, techniques of interferometry in abnormal dispersion regions (in the vicinity of resonance lines), and so on. Holographic methods prove convenient in investigations into spatially inhomogeneous plasma structures. Apart from interferometry, also worthy of note among the other optical schemes for determining ne from the refraction of plasma are the various modiﬁcations of the probe beam deviation measurements. For more details, see, in addition to the references already cited in this section, also [15, 16].

7.2 Stark Broadening of Spectral Lines 7.2.1 General

It has already been noted in Section 2.2 that the speciﬁcity of formation of spectral line proﬁles in plasma results from the interaction of the emitting atoms and molecules with charged particles. We have also noted that important in this respect is not only broadening by electron and ion impact, but also broadening due to the Stark effect in plasma microﬁelds (statistical or quasistatic broadening). On account of the exisiting body of literature on the subject, for example [17–28], in this section we will touch upon theoretical questions only as regards their results, with a view to introducing the basic notions in use and evaluating the conditions relating to the operation of some or other broadening mechanisms in plasma. We will also dwell upon the practical aspects of utilizing the form of the spectral line proﬁle for determining the concentration of charged particles. The intense research activities in this ﬁeld have been motivated by, and at the same time provided for, the considerable progress made in plasma diagnostics. The method of determining the electron concentration ne from the Stark broadening allows measurements over a gigantic range, provided that the choice of the particular spectral line is well founded. To illustrate, in his review [28] Griem noted measurements from ne = 0.1 cm−3 to ne = 1024 cm−3 . In the former case, the matter concerns interstellar medium (Te = 50 K), and measurements are taken in the RF region from transitions between Rydberg levels with the principal quantum numbers n ≤ 740 in hydrogen. The latter example concerns laser plasma (Te ≈ 107 K), and measurements here are taken in the X-ray region from transition lines of highly ionized ions. The photon

7.2 Stark Broadening of Spectral Lines

energies in these examples differ by 11 orders of magnitude. Of great importance in selecting particular spectral lines is to reveal the region of conditions wherein the Stark broadening is not hidden by the other broadening mechanisms, the optical width is small and the localization of measurements allows the plasma of interest to be considered homogeneous. Violation of these conditions does not make the measurements of ne impossible, but complicates the processing of the experimental data. The density nch of charged particles in the method under discussion is determined by comparing between theoretically calculated and experimentally measured line proﬁles. In general, account should also be taken of both the impact and statistical broadening mechanisms. Following the reasoning of Section 2.2.1, one can tentatively distinguish the regions wherein one of the broadening mechanisms prevails. If the aver−1/3 age distance between charged particles, ca. nch , exceeds the perturbing interaction radius ρ0 deﬁned by formula (2.20), particle interactions are of pulsed character and the main part is played by the impact broadening mechanism. In the opposite case, the statistical mechanism is predominant. Therefore, one can take as a criterion the comparison between the magnitude of the dimensionless parameter hm = nch ρ30 and unity. When hm < 1, the impact approximation is valid, and when hm > 1, the statistical approximation holds true. In either case, the perturbation of the energy structure of an emitter by a charged particle manifests itself as the Stark effect. 7.2.2 Plasma Microﬁelds

In the statistical approach, under study is the perturbation of the energy structure of an emitting particle by the electric ﬁeld produced at its location by the aggregate of charged particles surrounding it. Various conﬁgurations of this environment, producing various ﬁelds, are possible. It is supposed that for each spectral component of the ﬁne structure of the α → β transition line the intensity distribution Iαβ (ω ) is deﬁned by the electric ﬁeld distribution (in magnitude) function W ( F ) = W (|F|): Iαβ (ω ) dω = Iαβ W ( F ) dF.

(7.9)

Here Iαβ is the total intensity of the component. If the active ﬁeld does not change the relative intensities of the components, perturbations are independent and the plasma is optically thin, the integral intensity distribution in the line is obtained by summing expression (7.9) over the ﬁne-structure components α and β. To describe the proﬁle of a spectral line, one should (i) know the form of the function W ( F ) and (ii) establish

357

358

7 Determination of the Parameters of the Electronic Component of Plasma

the relation between the magnitude of the ﬁeld and the frequency shift of the components. In this section we will dwell on item (i). The ﬁrst and best known solution of the problem on the intensity distribution of a plasma ﬁeld was obtained by Holtsmark in his early work [29] on the assumption that all particles behave like an ideal gas and that the electron and the ion charge is e. The ﬁeld is quasistationary and is produced by the entire aggregate of chaotically arranged immovable and noninteracting ions (interaction with electrons is of impact character, see below). The Holtsmark distribution function has the form 2 WH (γ) = πγ

∞

. 3 dx exp −( x/γ) /2 x sin x,

(7.10)

0

γ = F/FH ,

FH = 2.603eNe/3 . 2

The values of this function are tabulated (see, for example, [18, 20, 24, 26]). In strong ﬁelds (γ 1), the function WH (γ) ≈ 1.496γ−5/2 , and in weak ones (γ 1), WH (γ) ≈ 4γ2 /(3π). In a large cycle of subsequent works, a number of limitations of the Holtsmark model were removed. Hooper [30] obtained a more general distribution W (γ) allowing for the effects of cross-correlation of ions and the screening of their electric ﬁelds. The deviation of the function W (γ) from the Holtsmark function WH (γ) depends on the dimensionless parameter a = r0 /rD , where r0 = (3/4π)1/3 N −1/3 is the average distance between ions and rD is the Debye radius (1.1). The Holtsmark distribution corresponds to the case a = 0 (or rD → ∞). Figure 7.3 shows the distribution functions of the ion electric microﬁeld for several values of the parameter a, including the Holtsmark function (the case a = 0). The character of the relation between a plasma ﬁeld and the frequency shift produced by it in a spectral line is governed by the atomic structure of the emitting particle. A distinction is made here between the linear and the quadratic Stark effect. 7.2.3 Linear Stark Effect

In this case, the frequency shift of a Stark component is proportional to the ﬁeld. Since the ﬁeld produced by a charge of e at a distance of r is ∼ r −2 , the quantity m in formula (2.19) is then equal to 2, and so Cm = C2 , and the frequency shift is

(ω − ω0 )αβ = Bαβ F,

Bαβ = (C2 )αβ /Ze.

(7.11)

7.2 Stark Broadening of Spectral Lines

Figure 7.3 Distribution functions of the ion electric microﬁeld for several values of the parameter a = r0 /rD . The case a = 0 – the Holtsmark distribution function.

For the statistical model, expression (7.9) will be written down in the form Iαβ ω − ω0 dω. (7.12) W Iαβ (ω ) dω = Bαβ Bαβ The line proﬁle summed over all the Stark components, Iαβ ω − ω0 dω, I (ω ) dω = Ze ∑ W Ze (C2 )αβ (C2 )αβ α,β

(7.13)

can be directly calculated from the known function W (γ) (Figure 7.3) and the relative intensities of the Stark components. The parameter γ in the Holtsmark function (7.10) (and similar corrected functions) contains the ion density that can be determined from comparison between the theoretically calculated and experimentally measured line proﬁles (when making the comparison, account should be taken of the additional broadening factors and instrumental errors). Since the plasma is quasineutral and contains only singly charged ions, the concentration of the ions equals that of electrons. However, one should make sure that broadening here is by ion impact and the statistical analysis is valid. To this end, it may prove useful to make some estimates. Using in formula (2.20) the value of αm at m = 2 for the linear Stark effect, we write down the following expressions for the broadening parameter hm introduced in the preceding Section 7.2.1 in the case of electron

359

360

7 Determination of the Parameters of the Electronic Component of Plasma

and ion impact broadening: m = 2:

h2,e = ne (πC2 /ve )3 ,

h2,i = ni (πC2 /vi )3

(7.14)

Taking as an example the hydrogen atom with the linear Stark effect and the experimental values of C2 , we obtain the following values of the above parameters for the ﬁrst three members of the Balmer series in plasma at a temperature of 1.2 × 104 K [27]: Hα (n = 3 − n = 2)

Hβ ( n = 4 − n = 3)

Hγ (n = 5 − n = 4)

h2,e

1.3 × 10−20 ne

1.0 × 10−19 ne

0.5 × 10−18 ne

h2,i

0.8 × 10−15 ni

0.6 × 10−14 ni

3 × 10−14 ni

One can see that even at moderate densities nch ≥ 1014 it is the quasistatic broadening by ions that is responsible for the formation of the proﬁles of these lines, whereas the same broadening mechanism involving electrons becomes operative at substantially higher electron concentrations. This result is easy to explain physically. The velocities of electrons are high, their times of ﬂight near the atom are short and the perturbations caused by them are of short-term, pulsed character. The heavier ions are more “static”. Note in passing, that the ratio between the values of the parameters h2,ch differing by n agrees well with the dependence of the “constant” C2 on the principal quantum number of the upper transition level: C2 ≈ n(n − 1) (see Section 2.2.1). Figure 7.4 illustrates some calculations [26] of the applicability regions of the statistical approximation for a wider range of the quantum numbers n of hydrogen transition lines in the Lyman, Balmer and Paschen series in plasma at a temperature of 104 K. Indicated are various causes of the limitation of the applicability regions, namely, the Inglis–Teller effect (curves 1; the effect will be discussed in the next section) and h2,e,i limits (curves 2, 3). It can be seen that the contribution to broadening from electrons is only manifest in a narrow region near the upper limit set by the density of the charged particles. Though the boundaries shown in the ﬁgure are rather schematic, they give, on the whole, correct estimates of the quantities. What is important is that the quasistatic broadening mechanism is operative in a sufﬁciently wide region and includes many experimentally realizable parameters. By appropriately selecting a particular series of lines and the serial number of the upper transition level in hydrogen and hydrogen-like atoms, one can attain a wide dynamic range of electron density measurements. The linear Stark effect has been studied most thoroughly in hydrogen lines. As a result of numerous perfections of the theory, practically com-

7.2 Stark Broadening of Spectral Lines

Figure 7.4 Broadening of hydrogen lines in a low-temperature plasma with a temperature of 104 K. The quasistatic broadening regions in the (a) Lyman, (b) Balmer, and (c) Paschen series are hatched. Horizontal hachure – the action of ions (protons), vertical hachure – the

action of electrons. Causes of the limitation of the quasistatic mechanism: 1 – the Inglis–Teller effect; 2 and 3 – minimal electron and ion concentrations, respectively, for the conditions h2,e,i > 1 by formula (7.14) to be satisﬁed; 4 – equality of the Stark and Doppler widths.

plete agreement has been achieved between theory and experiment as regards both the width and the shape of line proﬁles. Detailed tables of the Stark proﬁles and half-width of hydrogen and ionized helium lines in low-temperature plasma can be found in [26]. They reﬂect the state of the art in the mid 1970s. It is precisely these data that are being used today in practical plasma diagnostics. In the later work by Gigosos and Cardenoso [31] devoted to this classical problem, the slight discrepancy remaining in the proﬁle shape near the center was eliminated by consistently allowing for the ion dynamics. Calculations and experiments show that the Stark width at halfmaximum of a hydrogen line may be described as a function of the electron concentration by the simple approximative formula ΔλS = Cs ne/3 . 2

(7.15)

Here CS is a constant weakly dependent on temperature and density. In the case of statistical broadening, the constant CS = C2 and grows greater (∼ n2 ) with increasing principal quantum number. The theory shows that the impact broadening mechanism gives a faster growth (∼ n4 ). In the range T = (5–40) × 103 K and ne = 1014 cm−3 –1017 cm−3 the constant

361

362

7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.5 Widths at half-maximum of the Hα and Hβ lines as a function of the electron density. The temperature of charged particles is 2 × 104 K.

Cs changes by 20–30%. For the Hβ line, ΔλS ≈ (1.8–2.3) × 10−9 ne/3 , nm. 2

(7.16)

Figure 7.5 presents the widths at half-maximum of the Hα and Hβ lines, deﬁned by formulas (7.15) and (7.16), as a function of ne at T = 20000 K. Figure 7.6 shows in greater detail the relative changes of these widths in the above range of parameters [30, 31]. As the electron density varies over the above range, the width of the Hβ line changes approximately from 0.04 nm to 0.5 nm, which can be reliably measured with interference, diffraction and prism spectral instruments. The line has a relatively low excitation potential and is intense enough in plasma glow even if hydrogen is present in trace concentrations. The lower hydrogen transition level is in this case excited and plasma can be frequently considered optically thin. Insofar as the Stark broadening of the Hβ line has also been most thoroughly investigated theoretically and the sensitivity of its width to electron temperature variations is relatively low (Figure 7.6), it is one of the most popular lines in the ne measurements. Measurement practice sows that the characteristic attainable precision is no worse than 5%.

7.2 Stark Broadening of Spectral Lines

Figure 7.6 Ratio Δλ1/2 ( T )/Δλ1/2 ( T0 ) as a function of the temperature of charged particles for the (a) Hα and (b) H β lines at various electron densities ne . Δλ1/2 – proﬁle width at half-maximum. T0 = 2 × 104 K.

363

364

7 Determination of the Parameters of the Electronic Component of Plasma

7.2.4 Quadratic Stark Effect

The broadening and shift of the spectral lines of non-hydrogen-like atoms are quadratic in the magnitude of the applied ﬁeld, and the quantity m in formula (2.20) is equal to 4. In that case, h4,e = ne πC4 /2ve ,

h4,i = ni πC4 /2vi .

(7.17)

As noted in Section 2.2.1, the typical values of C4 are ca. 10−15 – 10−12 cm4 /s. At the characteristic electron and ion velocities of ve = 5 × 107 cm/s and vi = 2 × 105 cm/s, h4,e = 3(10−19 –10−22 )ne ,

h4,i = 0.75(10−17 –10−20 )ni .

Because of the weaker velocity dependence of the broadening parameter h4 , the discrimination between the contributions from the statistical and the impact broadening mechanism by the plasma parameter regions is not so contrasting as in the case of linear Stark effect. At moderate charged particle densities, nch < 1016 cm−3 , the quantities h4,e and h4,i are much less than unity, which means that both electrons and ions broaden spectral lines by the impact mechanism. Since broadening in this case is, according to formula (2.23), Δν4 ∼ v1/3 , the main contribution to the impact broadening comes from electrons, the contribution from ions being ca. (vi /ve )1/3 = (me /Mi )1/6 ≈ 0.15–0.2. The statistical mechanism prevails at charged particle densities of nch > (1017 cm−3 – 1018 cm−3 ). At the same time, account should be taken of the fact that even at moderate densities, the statistical line broadening by ions can be perceptible in the shape of the line proﬁle in its wing. Such a statistical wing is located on the long-wavelength (at C4 < 0) or the shortwavelength (at C4 > 0) side relative to the line center. The boundary of frequency detuning from the center of the line proﬁle, beyond which the effect becomes important is given by parameter (2.28). In the case m = 4 under discussion, this frequency detuning region (in terms of wavelength) has the form λ20 vi/3 . 2πc C1/3 4

λ0 − λ >

(7.18)

4

To illustrate, for the 616.07 nm line of sodium (atomic weight A = 23, C4 = 3.6 × 10−13 cm4 /s, h4,i = 0.006) in plasma (T = 5000 K, ne = 3 × 1015 cm−3 ), the frequency detuning boundary (the long-wavelength one), compared with the Doppler width, is λ0 − λ = 2.75ΔλD . In that case, the asymmetry of the line proﬁle is practically imperceptible. If the values of C4 are great, the effect of the statistical wing can

7.2 Stark Broadening of Spectral Lines

Figure 7.7 Manifestation of the statistical wing. Atomic sodium line proﬁles: 1 − λ = 616.07 nm; 2 − λ = 498.28 nm.

be strong. Under the same conditions, for the 498.28 nm line of sodium (C4 = 4.1 × 10−11 cm4 /s, h4,i = 0.7), we have λ0 − λ = 0.47ΔλD , and the asymmetry of the proﬁle affects the line width being measured. These examples are illustrated by Figure 7.7 borrowed from [18, 33]. The theory of line broadening due to the quadratic Stark effect was analyzed in detail in [17, 18, 26, 28]. In his works [26, 28], Grim presented tabulated data on the results of numerical calculations of the proﬁle shapes and half-width of a number of spectral lines of some atoms, from helium to cesium, in low-temperature plasma at temperatures T = (2.5–80) × 103 K and ne = 1016 cm−3 . According to [20, 25, 32], these data can be approximated and inter- (extra-)polated for various values of T and ne by the formula . 1 −1/2 Δλs,4 = 2 + 3.5 × 10−4 ne/4 α 1 − 0.068n1/6 r 10−16 wne . (7.19) e Te Here α is the so-called ion broadening parameter and w is the electron impact half-width (numerical data tabulated in [26] are given for the half-widths at half-maximum in angstrom units). The second term in formula (7.19) describes the contribution from ions and is frequently small. Therefore, the line width depends only weakly on temperature

365

366

7 Determination of the Parameters of the Electronic Component of Plasma

and grows practically linearly with increasing ne . This provides a simple means of measuring electron concentration. The above formula is applicable at α < 5 × 103 ne−1/4 . Some selected data from [26, Table 4] on the parameters α and w for lines in the visible and near UV regions of the spectrum are listed in Table 7.3. Listed are, as a rule, cases featuring large impact half-width and absence of line overlapping. Table 7.3 Parameters α and w. Atom

˚ λ, A

He

3297

C O Ne Na Mg

T = 5000 K w 20.7

α 0.991

T = 20000 K w 15.9

α 1.209

T = 80000 K w 10.6

α 1.634

4388

7.31

1.405

4.66

1.97

2.76

2.919

4932

0.172

0.091

0.286

0.062

0.337

0.055

4268

0.303

0.107

0.536

0.07

0.64

0.061

4368

0.054

0.036

0.113

0.021

0.151

0.017

3692

0.156

0.012

0.302

0.008

0.373

0.006

5881

0.014

0.035

0.024

0.023

0.034

0.018

3417

0.046

0.046

0.071

0.033

0.082

0.029

6160

0.261

0.088

0.46

0.058

0.576

0.049

8649

4.42

0.137

5.71

0.113

5.65

0.114

4167

1.39

0.171

2.08

0.126

2.1

0.125

0.35

8.75

0.397

6.67

0.487

Al

5785 5557

0.693

0.108

1.23

0.07

1.45

0.062

S

4411

0.385

0.108

0.634

0.074

0.709

0.068

7244

1.73

0.103

2.46

0.079

2.46

0.079

3606

0.217

0.113

0.373

0.075

0.437

0.067

5495

1.86

0.109

2.92

0.078

3.08

0.075

6936

0.554

0.086

0.749

0.068

0.738

0.069

Ar K

11022

10.3

13.4

0.431 0.18

1.31

0.105

0.514

7.44

0.668

1.87

0.2

1.42

0.246

1.62

0.09

1.58

0.092

5512

Cs

3314

17.4

0.52

17.6

0.515

14.5

0.597

3289

27.4

0.624

27.2

0.628

22.1

0.735

4685

2.15

10.6

Ca

The formulas for line half-widths and shifts presented in a number of other works on line proﬁles with the quadratic Stark effect (see references cited in [32, 34]), differ from expression (7.19). The principal error, however, is contained not in the structure of the formulas, but in the data on the Stark constants C4 that determine the line widths. The possibilities of using lines with either the linear or the quadratic Stark effect can be evaluated even by means of simplest formulas (2.23) and (2.25) and the data of Figure 2.1. With this evaluation, according to

7.3 Truncation of Spectral Series of Hydrogen-Like Atoms

what has been said above, one should use ion velocities for the linear effect and electron velocities for the quadratic one: Δν2 /Δν4 ∼ [π2 C22 /vi ][1.82C4− /3 ve/3 J ( β)]−1 . 2

1

(7.20)

As a rule, the linear Stark effect clearly exceeds its quadratic counterpart, except for individual cases of abnormally high values of the constant C4 , and as far as the sensitivity of the ne measurements is concerned, the linear effect is preferable: hydrogen, hydrogen-like ions, transitions involving Rydberg levels [35].

7.3 Truncation of Spectral Series of Hydrogen-Like Atoms

In Section 6.2.1, we discussed the possibility of determining the magnitude of an external electric ﬁeld from the reduction of the number of Rydberg levels manifest in the emission (ﬂuorescence) spectrum (relation (6.17)). In conditions of plasma with a sufﬁciently high electron concentration, the Coulomb interactions between bound electrons and free charged particles leads to the same effect. As the principal quantum number n is enlarged, the Stark splitting and line widths increase, while the spacing of the adjacent levels decreases as n−3 [17, 18]. Beginning with some value of nmax , the levels merge into a continuum (Figure 7.8), this value depends on the concentration of charged particles the distance between which is ca. n1e/3 . The states with nmax,g > n > nmax that form a continuous spectrum are, however, associated with the atomic nucleus and contribute to the internal statistical sum, as distinct from the case where the bound states go over into a continuum at nmax,g as a result of reduction of the ionization potential (Section 1.2.1). When making spectral observations in plasma, the bound–free continuum, as a rule, masks the continuous spectrum boundary associated with the reduction of the ionization potential. The ratio of the frequency shift Δνg of the series limit (as a result of the reduction of the ionization potential) to that resulting from the merging of split and broadened lines, Δνm , for the hydrogen atom is [19] /30 − /2 Δνg /Δνm ≈ 2.5 × 103 nch Te , 7

1

(7.21)

where nch is the concentration of charged particles. The greatest broadening occurs in the case of linear Stark effect in hydrogen and hydrogenlike Rydberg atoms. For these cases and statistical plasma microﬁelds distributed by the Holtsmark distribution (see Section 7.2.2), the value of

367

368

7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.8 Frequency shifts of the limit of a series as a result of reduction of the ionization potential, Δνg , and owing to the merging of split and broadened lines, Δνm .

nmax is determined by the Inglis–Teller formula [20–22] log nch = 23.26 − 7.5 log nmax + 4.5 log Z,

(7.22)

where Z is the nuclear charge. In the case of plasma with ions of unit multiplicity, one should use for nch in formulas (7.21) and (7.22) the sum of the concentrations of electrons and ions at relatively low temperatures satisfying the condition T < 4.6 × 105 /nmax , K.

(7.23)

In this case, the following relation holds true for the electron concentration in hydrogen plasma [19]: −15/2 ne ≈ 0.91 × 1023 nmax .

(7.24)

In the case of the inverse inequality in (7.23), one should use in relation (7.22) the ion concentration ni for nch : nch = ni . In deriving formulas (7.21) and (7.24), the Stark broadening is assumed to exceed its Doppler counterpart. This places the lower limit on the electron concentration being measured, that is, the same limitation as in the case of measurements from the Stark broadening of individual lines. To illustrate, at a gas temperature of T ≈ 103 K the minimum measurable electron concentration is ne ≥ 1013 cm−3 . Relations (7.21) and (7.24) make the determination of the electron concentration from the limit of discernibility of individual lines in a series

7.3 Truncation of Spectral Series of Hydrogen-Like Atoms

of lines for transitions with progressively increasing serial number of the upper level, both simple and convenient. It should be noted, however, that the accuracy of such measurements is not very high, hardly better than up to a factor of 2. This is due, on the one hand, to the limited accuracy of determination of the series limit in the experimental spectrum, and on the other hand, to the limitations of the Holtsmark theory (high temperatures, low densities) and the further simpliﬁcations made by Inglis and Teller. Formula (7.22) is obtained by equating half the energy difference between levels with adjacent numbers n to the Stark splitting in the ﬁeld equal to the average over the Holtsmark distribution 2/3 F = 3.7ench , without allowing for the temperature dependence. These circumstances were analyzed by Vidal [23]. The modernized version developed by Vidal [23] reduces, in short, to the following. The ratio R(n) = Imax /Imin is constructed between the intensities in the minima and maxima of the partially resolved spectrum recorded in the vicinity of the discernibility limit (Figure 7.8). The same ratio is calculated in the quasistatic approximation with due regard for the temperature dependence of the line broadening, for which purpose the following parameter is introduced: ni/6 1

r0 /rD = 0.09

Te/2 1

(cmK) /2 , 1

(7.25)

where r0 is the average distance (in cm) between charged particles, rD is the Debye radius, and Te is the electron temperature in kelvins. The Holtsmark distribution corresponds to the case where r0 /rD = 0 (Section 7.2.2). The second parameter of the theory is Q=

2ARyZ3 /3 5 nch n [1 − (δ/n)]3 2

,

(7.26)

where Ry is the Rydberg constant (1.097 × 105 cm−1 ), δ is the quantum defect, A is a constant weakly dependent on n (near the discernibility limit; at n > 10, A = 4.9 × 1010 cm−1 and nch = ne + ni . For the number n = nmax determined by the Inglis–Teller formula (7.22), the parameter QIT = 3.35. Calculated values of R0 (r0 /rD , Q) are shown in Figure 7.9. In accordance with the recommended procedure, ﬁrst the function R(n) is constructed from the recorded spectrum. As a ﬁrst approximation for ﬁnding the charge density, use is made of the Inglis–Teller formula (7.22). The supposed value of the temperature T is then estimated, and the parameter r0 /rD is found by formula (7.12). As a result of a few iterations and interpolation of the data of Figure 7.9, the parameter Q is selected

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.9 Calculated values of the ratio R0 between the maximum and minimum line intensities in a line spectrum for various parameters r0 /rD and Q.

such that the condition R(n) = R0 (r0 /rD , Q)

(7.27)

is best satisﬁed for a number of values of n. To ﬁnd the reﬁned value of nch , one should again turn to formula (7.26): +

nch

2ARyZ3 = Qn5 [1 − (δ/n)]3

,3/2 .

(7.28)

The estimates made by Vidal [23] and their comparison with experimental results allowed the author to conclude that this procedure ensured an accuracy of no worse than 5% in determining charged particle concentrations, provided that condition (7.23) was satisﬁed and ne < Z9/2 × 1016 cm−3 . In the experiments conducted in [23] with RF discharges in hydrogen and helium, the accuracy of determining electron concentrations in the range ne = (1 – 3.4)×1013 cm−3 amounted to some 2%.

7.4 Intensities in Continuous Spectrum

7.4 Intensities in Continuous Spectrum

The determination of the parameters of the electronic component of plasma from the spectra of the bremsstrahlung (ff ) and recombination (fb) continua is based on the fact that the cross sections of the processes leading to the emission of a continuous spectrum depend on the velocity and intensity of free electrons, on the concentration of charged particles. To ﬁnd ne , one should measure the absolute intensities, and to determine Te it is necessary to measure the frequency dependences of these intensities. The quasiclassical approximation formulas have been presented in Section 2.5. The presence of exponential factors in these formulas allows one to determine the electron temperature from the slope of the frequency dependence of the logarithm of intensity (absorption coefﬁcient) (it has already been noted in Section 2.5 that the electron temperature need not be equal to the other plasma temperatures). This procedure is quite similar to that used, for example, in determining the gas temperature from intensities in the rotational structure of a molecular spectrum (formulas (4.33) and (4.34)). Despite the simplicity of the approach, account should be taken of a number of circumstances. It is necessary to discriminate the regions corresponding to predominantly ff and fb continua, to exclude the effect of the other line, band and continuous spectra. In particular, relations (2.77), (2.80), (2.91) and (2.96) can be of use in discriminating the ff and fb regions of spectra. At electron densities ne < 1015 cm−3 the ff transitions prevail in the IR region of the spectrum, whereas their fb counterparts prevail in the visible and UV regions (Figure 2.3). Measurement practice has shown that the necessary emission measurement sensitivity, along with reasonable accuracy (of a few tens percent), limited as they are by intensities in the continuum as well, are attained at ne l > 1016 cm2 , where l is the length of the luminous zone. The Kramers quasiclassical approximation formulas should be corrected by means of factors that are, generally speaking, also frequency dependent. For hydrogen and hydrogen-like atoms, these are the Gaunt g-factors introduced in Section 2.5, and for multielectron ones, the analogous ξ-factors (sometimes referred to as the Biberman factors). For example, to determine the electron temperature Te from the recombination continuum intensities measured in the frequency interval Δω, we have from expression (2.91), corrected with the ξ-factor, Te =

h¯ kB

+ Δ ln

Δω

ξ (ν,Te ) fb

Iωn

, ,

(7.29)

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.10 Gaunt factors of the recombination continuum of hydrogen-like atoms.

which is similar to what we have in the case of correction with a Gaunt factor. If the form of ξ (ω, Te ) is known, it will be sufﬁcient to take two fb measurements of the intensity Iωn either in a region between two prominent “teeth” of a saw-tooth recombination spectrum or in a region where these “teeth” are suitably smoothed out (see expressions (2.96), (2.97) and Figure 2.3). Having differentiated expression (7.29), one can easily see that the temperature determination accuracy grows as the interval Δω is increased. This interval is selected as a trade-off between the interval where the recombination spectrum is free from other spectra and the reliability of the data available on ξ(ω,Te ). The fact that the frequency dependences of the g- and ξ-factors are not too sharp and have no strong effect on the measurement results within the limits of Δω, frequently proves a favorable circumstance. The results of calculating the recombination gfb -factors for hydrogen-like atoms are presented in Figure 7.10 [20]. Shown here are individual curves plotted for “teeth” corresponding to recombination entailing the formation of the atom in states differing in the principal quantum number. The results of calculating the ξ tot -factors for some atoms [34] are presented in Figure 7.11. Since the spectra of multi-electron atoms are more complex and feature a great number of “teeth”, the calculation results here are averaged over the principal quantum numbers and, further, account is taken of the superposition of the recombination and bremsstrahlung continua. This is indicated by the subscript “tot”. One can see, for example, that in the

7.4 Intensities in Continuous Spectrum

Figure 7.11 Correcting factors ξ tot with Z = 1 for neutral atoms and with Z = 2 for singly charged ions.

Figure 7.12 Continuum of a helium plasma at ne = 5 × 1016 cm−3 . The electron temperature determined from the slope in individual “teeth” is Te = 5 × 104 K. Data points – experiment; lines – calculation.

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7 Determination of the Parameters of the Electronic Component of Plasma

range 0.1–0.3 on the x-axis of Figure 7.11, which corresponds to a spectral range between the near IR to the near UV region of the spectrum, the variation of the factor does not exceed a few tens percent. At the same time, intensities at Te ≈ 1.600 × 10−19 J (1 eV) vary over the same spectral range by a few orders of magnitude. That the Te measurements are correct can be testiﬁed to by the linearity of the frequency dependences fb of ln Iωn and the equality of their slopes for as great as possible number of intervals out of the total continuous spectrum registered. Figure 7.12 shows as an example a spectrum used to measure the electron temperature in a helium plasma [27, 35].

7.5 Scattering of Light on Electrons

In Section 2.6.1, we have described the process of scattering of light by a single electron. The Doppler shift of the frequency of light scattered by a moving electron (see formulas (2.110) and (2.111)) can be used to ﬁnd the velocity distribution of electrons. The integral intensity of scattered light provides information on the electron concentration. Talking exclusively about electrons, we recognize that, as readily seen from formula (2.103), the contribution from heavy ions to the scattering of light incident on plasma is negligibly small. However, the resultant intensity and spectrum of light scattered by an ensemble of electrons cannot always be obtained by simply summing up the intensities of light scattered by the individual electrons. It is, rather, necessary to sum up the vectors of the complex electric ﬁeld strength amplitudes of the scattered light, (2.100) and (2.103). It is not very difﬁcult to understand that if electrons are uniformly distributed within a volume much greater than the wavelength of light (otherwise diffraction should be taken into consideration), one can select pairwise such volume elements as give scattered waves in antiphase, no matter what the observation direction. This means the absence of scattering according to the Huygens–Fresnel secondary-wave interference mechanism providing for the rectilinearity of propagation of light. Scattering is, nevertheless, present, as the amounts of electrons in different, but volumetrically equal, elements of macroscopically homogeneous plasma are not strictly equal because of natural density ﬂuctuations (which is characteristic of any medium) and interaction of charges (typical of plasma). The scattering result thus proves dependent on many parameters, including the spectrum of ﬂuctuations, the character of their phase correlations and observation direction. The solution of the light

7.5 Scattering of Light on Electrons

scattering problem allowing for these factors is described in many books, for example, [6, 7, 9, 10, 16, 36–38]. Let the average number of electrons ¯ ej 1, the deviation from this in each elementary volume j be equal to N value being δNej . In that case, in the absence of the thermal motion of electrons, the total electric ﬁeld of the scattered light at the observation point will be Es ∼

∑ ( Nej + δNej )−iϕj ,

(7.30)

j

where ϕ j are the phases of the scattered light waves and the angle brackets . . . denote statistical averaging. If the number of the elementary cells j in the total volume ΔV forming the scattering signal is great, all the phases will then be present in scattering, and when making summation in expression (7.30), the term with Nej will be equal to zero. Therefore, the scattered intensity Is is determined by the ﬂuctuation terms whose phases differ between different cells: Is =

∑ Ej Ei∗ ≈ ∑ δNei δNej e−i( ϕi − ϕj ) . i,j

(7.31)

i,j

The result of summation (7.31) depends on the character of the correlations of ﬂuctuations in different cells. 7.5.1 Scattering of Light by Randomly Moving Electrons (Thomson Scattering)

If correlations are absent, electrons in the entire scattering volume move randomly, independently of one another, and expression (7.31) then gets simpliﬁed: Is /σe ∼

∑ (δNej )2 = ∑ Nej = Ne = ne ΔV, j

(7.32)

j

that is, the scattered intensity is determined by the total number Ne of electrons in the scattering volume ΔV and the scattering process here is of purely Thomson character. The coefﬁcient of proportionality includes the cross-section for scattering by a single electron, σe (2.106). This result can apparently be obtained by simply summing up the intensities of radiation scattered by the individual illuminated electrons (this reasoning is valid for not very small scattering angles θ > λ/(2πl ), where λ and l are the wavelength of light and the linear size of the scattering region, respectively [10, 36]). The single-electron scattering cross-section (2.105)– (2.107) is frequently replaced by the scattering cross-section per unit volume (for notation, see Section 2.6): dσ = ne σe = ne r02 sin2 θ. dΩ

(7.33)

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7 Determination of the Parameters of the Electronic Component of Plasma

In accordance with expression (2.110), the scattered spectrum is determined by the Doppler effect, the electron velocity projections on the vector k = ks - k0 being important. Introducing the distribution function of the electron velocity projections vk in this direction (see Section 2.2.2), ωs − ω0 , (7.34) dne = f (vk )d vk , f (vk ) = f |k| we ﬁnd that the scattering cross-section per unit frequency interval is d2 σ 1 = σe f (vk ) . dΩdω |k|

(7.35)

Thus, by measuring the form of the spectral proﬁle of the Thomson scattering of light by free, noninteracting electrons, ϕT,e (k, ω ) ∼ f (vk ), ω = ωs − ω0 , one can establish the distribution of electrons in velocity projections. According to (7.34), the function ϕT,e (k, ω ) should be normalized to the electron density ne , and so the latter can be found from the absolute intensities of the incident and scattered radiation. To swap from distribution in velocity projections to that in total velocities (in magnitude) depends on the isotropic properties of the distribution (see Section 2.2.2). In the simplest, but frequently occurring, case of isotropic Maxwellian distribution (1.3) and (1.4) with an electron temperature of Te , the scattered spectrum has a Gaussian form with the center at the incident radiation frequency ω0 : ' m mω 2 dω. (7.36) exp − ϕT,e (k, ω ) dω = ne 2πkB |k|2 Te 2kB |k|2 Te Going over, with due regard for expression (2.111), from the angles θ between the electric ﬁeld vector of the incident radiation and the observation direction to the angles Θ between the wave vector of the incident radiation and the observation direction, we get the following expressions for the half-width of the spectral proﬁle Θ 2kB Te ln 2, (7.37) Δω = 4ω0 sin 2 mc2 Θ 2kB Te Δλ = 4λ0 sin ln 2, (7.38) 2 mc2 (ln 2 = 0.693, mc2 = 5.11× 160.000 × 10−16 J (100 keV)). The electron temperature can thus be found from the half-width of the scattered light proﬁle. For example, in the case of scattering of the second-harmonic radiation of the Nd-YAG laser with a wavelength of λ = 532 nm, the half-width isΔλ = 2.48 nm.

7.5 Scattering of Light on Electrons

7.5.2 Manifestation Regions of the Thomson and Collective Scattering Mechanisms

The situation changes, compared to the one described above, if collective effects are manifest in plasma and electron motions cannot be considered independent. In this case, the scattered intensity and spectrum are determined by the temporal and spatial spectra of electron density ﬂuctuations (7.31). These ﬂuctuations can be schematically divided into two types. One embraces the above-considered ﬂuctuations of freely moving electrons and the other, the ﬂuctuations of electrons whose motion is tied to that of ions. Accordingly, the space-frequency spectrum of scattered radiation is described by the sum of the “electron” and the “ion” component: ϕ(k, ω ) = ϕe (k, ω ) + ϕi (k, ω ),

d2 σ = σe ϕ(k, ω ). dΩdω

(7.39)

The quotation marks (hereinafter omitted) stress the fact that both these components describe the scattering of light by electrons: ϕe by randomly moving electrons and ϕi by those in the form of clouds following the motions of ions and screening their charges. In the general case, the explicit form of expression (7.39) would be very cumbersome, ϕe and ϕi being functions of a set of scalar arguments, ϕe = ϕe (ω, |k| , Te , θ, ne ),

ϕi = ϕi (ω, |k| , Te /Ti , Z, θ, ne ),

that is, the ion component ϕi depends on the same parameters as the electron component ϕe and also on the ion temperature Ti and the ion charge Z. For this reason, it would be convenient to discriminate the regions where the different types of ﬂuctuations contribute the most to scattering. To this end, one should ﬁrst integrate expression (7.39) with respect to frequencies: Φ (k) = Φe (k) + Φi (k)

(7.40)

(this expression is frequently referred to as the form factor). To estimate the contributions from the terms, one may use, if the electron temperature Te and the ion temperature Ti exist, the parameter α suggested by Salpeter [39] in the form of a combination of the length of the scattering vector and the Debye screening radius rD : ' 1 λ ne e2 α= = . (7.41) 4πkB Te |k| rD sin Θ2

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.13 Thomson and collective scattering regions, λ = 694.3 nm: (a) α = 1, various observation angles; (b) various α values for observation angles of 90◦ and 5◦ .

In that case, the terms in expression (7.40) have the following form: Φe (k) =

1 , 1 + α2

Φi (k) =

-

Zα4

. . (1 + α2 ) 1 + α2 1 + Z TTei

(7.42)

At α 1, Φe ≈ 1 and Φi 1 and scattering is contributed to mainly from the electron component. This corresponds to the above-considered case of Thomson scattering by uncorrelated electrons. When α 1, Φe 1 and Φi ∼ Z/(1+ZTe /Ti ) so the scattering signal is formed by “ion” ﬂuctuations. Of course, the discrimination between the regions of plasma parameters depending on which ﬂuctuations are decisive for scattering, can never be more than general. It can be made conventionally by taking α = 1 to be the boundary value. The straight lines in Figure 7.13a show combinations of ne and Te values for α = 1 at various observation angles Θ, considering the scattering of light from a ruby laser (λ = 694.3 nm). The region above the line is due to Thomson scattering, while that below the line through collective scattering. Figure 7.13b shows combinations of ne and Te = Ti for a series of values 0.1 ≤ α ≤ 160 at Θ = 90◦ and Θ = 5◦ . In practice, observation in a direction normal to the incident radiation is convenient from the standpoint of localization

7.5 Scattering of Light on Electrons

Figure 7.14 Form factor components as a function of the parameter α.

of measurements and suppression of parasitic scattering effects. Sometimes it proves convenient to make use of the fact that the width of the Thomson spectrum, (7.37) and (7.38) grows wider with increasing angle Θ. If, however, subject to study are collective effects and ion parameters, then, as evident from the ﬁgure, they are easier to reveal at small observation angles. One more possibility to shift the boundary between the Thomson and collective scattering regions is to use light sources differing in wavelength. For example, replacing the ruby laser with a CO2 laser (λ = 10.6 μm) increases α 15 times. Figure 7.14 presents the components of form factor (7.40) as a function of the parameter α. In intermediate cases where α ≈ 1, both scattering mechanisms contribute to scattering. 7.5.3 Scattered Spectrum and Plasma Parameters (Direct Problem)

In accordance with expression (7.31), the scattered radiation spectrum is calculated from the plasma density ﬂuctuation spectrum (direct problem). Figure 7.15 presents some examples of such calculations for plasma with singly charged ions, Z = 1. In the case of purely Thomson scattering with statistical ﬂuctuations, α 1, only the electron component φe (k, ω ) = φT,e (k, ω ) is present. With a Maxwellian electron velocity distribution, this is a Gaussian pro-

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.15 The φe and φi components of scattered spectra at various plasma parameters, Z = 1. Electron and ion velocities are distributed by Maxwellian distributions with temperatures Te and Ti , respectively. (a) and (b) the electron and the ion component at various values of α and Te /Ti , respectively (ωe = (2kB Te /m)1/2 , ωi = (2kB Ti /M )1/2 [9]); (c total scattered spectrum: 1 − α = 2.17, ne = 2 × 1017 cm−3 , Te = 8 × 10−19 J(5 eV); 2 − α = 1.09, ne = 5 × 1016 cm−3 , Te = 8 × 10−19 J(5 eV).

ﬁle, and it is its half-width that determines, according to expression expressions (7.37) and (7.38), the electron temperature Te . The form of the proﬁle is shown in Figure 7.15a by the curve for α = 0. Plotted on the xaxis are the values of the frequency detuning ω = ωs − ω0 from the laser line, related to ωe = (2kB Te /m)1/2 (Figure 7.15a) and ωi = (2kB Ti /M)1/2 ( M is the ion mass, Figure 7.15b). At α 1 the spectrum is determined by the ion component Φi , because the electron component Φe → 0, no matter what its spectral structure (Figure 7.14). The frequency spectrum φi (k,ω) of the component Φi depends on the character of motion of the ions surrounded by electron clouds. A typical situation is one wherein the ion proﬁle at Te > Ti consists of two components with the central frequencies ω0 ± ωia , where ωia is the ion acoustic oscillation frequency. Considering the difference between the x-axis scales of Figure 7.15a and b, one can conclude that the

7.5 Scattering of Light on Electrons

ion component of the scattered spectrum is much narrower (by approximately a factor of (m/M)1/2 ) than its electron counterpart, even when the structure of its acoustic constituents is expressed. At Te Ti the ion line has a Gaussian proﬁle whose width is governed by the ion temperature deﬁned by formulas (7.37) and (7.38) with the electron mass replaced by the ion mass (Figure 7.15b, the curve for Te /Ti = 0). In intermediate cases α ≈ 1, the relation between the contributions to the scattered spectrum from the electron and the ion component depends on the actual plasma parameters. Figure 7.15a illustrates the transformation the electron component undergoes as the parameter α grows higher (the high-frequency part of the symmetrical proﬁle). The proﬁle breaks down into two components distant from the probe radiation frequency by an amount of ±Ωl . Such a “splitting” is due to the fact that in plasma natural oscillations occur with the frequency ωL = (4πe2 ne /m)1/2 (see expression (1.2)) and which is deﬁned by the well-known dispersion relation [36] 3 Ω2l = ωL2 1 + 2 . (7.43) α As α is increased, the frequency shift approaches the plasma frequency. Thus, the total spectrum in the general case consists of a narrow central structure of ion-acoustic components and two electronic satellites shifted from it for (7.43). Figure 7.15c presents some examples of calculation of scattered spectra in plasma at Te = Ti = 8 × 10−19 J(5 eV) for two combinations of α and ne values. Note also the differences in both the xand y-axis scales between the electron and the ion component. The collective scattering by acoustic waves and plasma oscillations (plasmons) can, in a sense, be likened to scattering by phonons in a solid or by molecular vibrations. For this reason, it is sometimes called combination scattering [37]. 7.5.4 Determination of Plasma Parameters from Scattered Spectra (Inverse Problem)

The determination of the parameters of the charged component of plasma from the spectrum of radiation scattered on electrons is a typical inverse spectroscopic problem with its inherent correctness problems (see Section 1.4). Considering the fact that the scattering cross-sections are small (σe ≤ σTh ≈ 10−24 cm2 ) and the minimization of errors is a rather complex experimental problem, the formal way to solve the inverse problem is only possible in a limited number of cases and usually requires a priori information on the anticipated results. In the ﬁrst

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7 Determination of the Parameters of the Electronic Component of Plasma

place, one should take it, as a postulate, that the physical model used in solving the direct problem is adequate. Thereafter one can adopt a sufﬁciently obvious scheme of comparison between the array of theoretically calculated spectra and their experimental counterparts. In doing so, one should limit the variation ranges of the parameters to be determined and used in calculations, though the required set of calculated spectra would nevertheless be rather bulky. Therefore, additional, physically substantiated assumptions and simpliﬁed formalized theory-experiment comparison procedures become important. With least assumption, one can establish the electron velocity distribution in the region of plasma parameters satisfying the condition α 1. It follows from expressions (7.35) and (7.39) that the electron velocity distribution along the scattering vector governs the intensity distribution in the spectrum. The inverse problem solution, namely, the establishment of the distribution f (v) from the form of the function ϕT,e (k, ω ) has the form [6] ,

+ 2 d ϕT,e ( R) 1 dΩ, (7.44) f (v) = − 2 8π dR2 R=(v · n) where R = ω/k, n = k/k, and integration is extended throughout the solid angle. To recover an arbitrary velocity distribution, it is necessary to use a set of experimental Thomson scattered spectra obtained at different observation angles. If the velocity distribution is isotropic, exhaustive information on it is contained in the distribution in modulus. In that case, it is sufﬁcient to establish the form of the distribution at a single observation angle, and formula (7.44) then reduces to the form dϕ T,e ( R) 1 f (v) = − . (7.45) 2πv dR R=v The difﬁculties arising in solving the inverse problem in such cases are associated with the sensitivity to measurement errors of the result of differentiation of the spectral proﬁle. Consequently, experiments in this area need to be planned and executed with great care. Some references to the works where non-Maxwellian and/or anisotropic distributions were studied by the scattering method can be found in [6, 10], which are devoted to plasma diagnostics by this method. If it turns out that in the same region α , 1 the distribution of intensities in the experimentally measured spectra can be described adequately (within the accuracy of measurement) by a Gaussian function, this means that the diagnostics possibilities here are limited to the measurement of

7.5 Scattering of Light on Electrons

the electron temperature in the Maxwellian electron velocity approximation. The temperature is found from the half-width of the proﬁle by formulas (7.37) and (7.38). In practice, it is exactly such problems that have to be solved most often. One can see from relations (2.105), (2.106), (7.33), and (7.34) that the electron concentration ne can be determined if the ratio between the intensities of the incident laser radiation and the radiation scattered in a unit plasma volume into a unit solid angle in a ﬁxed direction is known. However, in view of the smallness of the scattering cross-section, the characteristic value of this ratio amounts to ca. 1010 –1015 . This causes substantial, though not insurmountable, difﬁculties for the calibration of the measuring system, which affects the accuracy and reliability of the measurement results. More convenient is another method whereby the system is calibrated by comparing the intensities of radiation scattered by the Thomson and the Rayleigh mechanism. The neutral gas atoms subject to Rayleigh scattering to can either be introduced into plasma specially for calibration purposes or, which is typical of low-temperature plasma, be already present as the plasma-forming particles. With measurement geometry remaining the same, the ratio between the Thomson and the Rayleigh scattered intensity is Is,Th (dσ/dΩ)Th Il,Th = , Is,R (dσ/dΩ)R Il,R

(7.46)

where Il,Th and Il,R are the laser radiation intensities in the Thomson and Rayleigh scattering experiments, respectively. What is important is that both types of scattering have, as seen from relations (2.106), (2.113), and (2.114), the same angular characteristics. If, in addition, use is made of one and the same laser, then, at an atomic density of Na , Is,Th σ ne = Th . Is,R σR Na

(7.47)

The Rayleigh scattering cross-section is comparable with its Thomson counterpart and is governed by the polarizability χ of the gas particles: σR =

8π ω 4 2 32π3 (n − 1)2 χ = , 3 c 3λ4 NL2

(7.48)

where NL is the Loschmidt number and (n - 1) is the refraction of the gas at normal conditions. The values of polarizability and refraction are found by formulas (7.1)–(7.3) with coefﬁcients from Table 7.1. By measuring the Thomson and Rayleigh scattered intensities integrated over the spectral proﬁles, one can determine from relations (7.46)–(7.48) the absolute electron density ne .

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7 Determination of the Parameters of the Electronic Component of Plasma

Of interest for low-temperature plasma is also the case where α ≈ 1, when the spectrum depends on a greater number of parameters than in the preceding case (Section 7.5.2). This increases the information value of the spectrum, but obviously complicates the procedure of extracting these parameters. Therefore, one usually proceeds from the assumption that the electron and ion velocities are Maxwellian-distributed. One such procedure is described, for example, in the books [7, 10, 36]. It is based on selecting such a theoretical spectrum as coincides best with its experimental counterpart. In his book [10], Pyatnitsky has presented a large number of such spectra depending on a series of parameters. In order to impart to this array a more compact form, he suggestes a number of methods for scaling by way of compiling combinations of some parameters. To this end, he introduces the dimensionless frequency 1 (7.49) ω˜ e = (2ω/c) sin(θ/2) 2kB Te /m to be used to measure the shift of the maximum of the satellite Ω1 relative to the center and the satellite half-width δΩ: x1 = Ω1 /ω˜ e ,

δx = δΩ/ω˜ e ,

δΩ/Ω1 = δx/x1 .

(7.50)

In the region α ≥ 1, electronic satellites already have sufﬁciently wellpronounced maxima (Figure 7.15a). If the satellite has not yet formed (α ≤ 0.5), δx is then the distance from the center to the point on the proﬁle at which the ordinate is half the maximum. Next, theoretically calculated spectra are used to plot ancillary dependences of δx and x1 on the parameter α (Figure 7.16). The quantity x1 is found from experimentally measured spectra and curve 1 of Figure 7.16 is then used to ﬁnd the value of α, and next curve 2 to determine that of δx. Thereafter the electron density and temperature are found from the relations kB Te =

mc2 8 sin2 (θ/2)

Ω1 ωx1

2 ,

ne =

m 8πe2

α

Ω1 x1

2 .

(7.51)

Kuntse [36] has used a similar procedure to process the ion component φi of the spectrum with a view to determining Ti . Thus, the abovementioned increase in the information value of the spectrum at α ≈ 1 in comparison with that in the case where α 1 consists in that ne and Te can be found from the form of the scattered spectrum without the need to measure absolute intensities. This result is explained by the fact that these quantities are related by dispersion relation (7.43).

7.5 Scattering of Light on Electrons

Figure 7.16 Ancillary dependences for the processing of Thomson scattering spectra in the case α > 1: 1 – to determine α; 2 – to determine δx (see text).

7.5.5 Limitations of the Method, Sensitivity and Examples

Some of the above-mentioned results of the theory of light scatter on plasma electrons were obtained in the weak coupling approximation, that is, on condition that the Debye sphere contains many electrons, 3 1. The limits of this region can be seen from Figure 1.1. ne rD Sensitivity limitations are due to the fact that the scattering crosssection is small in magnitude. When estimating the minimal electron densities for the application of the scattering spectroscopy method, consideration is given to (i) the minimum number of photons that can be detected while meeting the necessary requirements on the spectral and aperture resolution, (ii) the presence of the own glow of the object of study, and (iii) limitations to be imposed on the power and energy density of the laser used in order to avoid excessive disturbances of plasma parameters. Factor (i) is the most “technical”, and it is becoming progressively less demanding as the detectors, registration electronics and signal processing methods are being perfected. The combination of factors (ii) and (iii) is of a more objective character, although here, too, the choice of the laser operating conditions and spatial resolution, as well as the other require-

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.17 The ne and Te regions of applicability of the method of scattering on free electrons. The regions between curves 1 and 1 , 2 and 2 , 3 and 3 , and 4 and 4 correspond to ruby laser powers of 0.5 J, 5 J, 50 J, and 500 J, respectively.

ments ensuing from the statement of the problem, give one some leeway. Therefore, the sensitivity estimates made by different authors can differ somewhat. Figures 7.17 and 7.18 illustrate typical results. Figure 7.17 [6] illustrates the probing of hydrogen (Z = 1) plasma with ruby laser pulses 500 μs in duration. The electron concentration region accessible to the scattering spectroscopy method falls within the range ne ≈ (1010 cm−3 –1019 cm−3 ), depending on the laser pulse energy (0.5 J–500 J) and the electron temperature Te ≈ 103 K–108 K. Also plotted here is the straight line delimiting the Thomson and collective scattering regions, which corresponds to the parameter α = 1 at an observation angle of Θ = 90◦ (see Figure 7.13). The upper limit of the regions is deﬁned by the own glow of plasma. Considered in the calculations was only the bremsstrahlung ff radiation (Section 2.5.1), and so the limit is somewhat overrated. The presence of the upper limit is due to the fact that the scattered intensity grows as ∼ ne , whereas the intensity of the bremsstrahlung (as well as recombination, though) radiation, as ∼ ne ni ∼ n2e . The lower limit was calculated so as to ensure that a single laser pulse used in photoelectric registration produced a minimum of nph.e ≥ 30 photoelectrons in the photomultiplier. Of course, the diagrams presented in the ﬁgure give only an idea as to the order of magnitude of the quantities in question. In an actual experiment, ac-

7.5 Scattering of Light on Electrons

Figure 7.18 Requirements for the power of a CW laser generating at a wavelength of λ = 500 nm in measurements by the Thomson scattering method. Causes of limitations: 1 and 1 – weak signal detection; 2 and 2 – own glow of plasma; 3 – stray laser lighting; 4 – heating of plasma by the radiation. Shaded area corresponds to measurements with a signal integration time of 1 s.

count should be taken of the concrete conditions and the speciﬁcity of the apparatus used. One can see that increasing the laser pulse energy extends the ne region, both above and below. However, there are also limitations. According to the calculations made in [36], allowing only for the bremsstrahlung ff absorption gives estimate of the maximum possible power density of radiation causing no plasma disturbance (107 Wcm−2 –1010 Wcm−2 ). In estimating the upper ne limit set by the own glow of plasma, of prime importance is the laser radiation power at which the scattered radiation power at each instant exceeds the power of the own glow of plasma within the same spectral intervals and solid angles. And the lower limit is determined by the laser pulse energy, because integrator schemes can be used to register weak signals. Since the problem of sensitivity at low electron concentrations is speciﬁc to the application practice of the scattering spectroscopy method in the diagnostics of low-temperature plasma, frequent use is made here of CW and pulse-periodic lasers. Figure 7.18 (borrowed from [7]) presents the results of estimating the ne limits for measurements from the scattering of a CW laser radiation (λ ≈

387

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7 Determination of the Parameters of the Electronic Component of Plasma

500 nm, Θ = 90◦ , spectral resolution Δλ = 0.1 nm, Te ≈ 0.8 × 10−19 J – (−6.4) × 10−19 J (0.5 eV − 4)eV). Under these conditions, the lower limit is determined by both the own glow of plasma and the possibility of detecting weak scattering signals. The shaded area in the ﬁgure denotes the region of operating conditions in experiments with a signal integration time of 1 s. As one can see, at minimal electron concentrations (ca. 1012 cm−3 ) the insufﬁcient signal integration time has to be compensated for by raising the laser power. Limit 3 is due to the stray illumination by the laser radiation resulting from the ﬁniteness of the spectral resolution of the apparatus used, which is most critical to the measurements of the ion component of the scattered spectrum at small observation angles. Limit 4 is determined by the heating of plasma in bremsstrahlung absorption. Practical measurements of the electron component parameters of plasma by the scattering spectroscopy method were started as far back as the early 1960s. By virtue of the above-indicated limitations on sensitivity, most of the measurements were conducted for plasma objects with a sufﬁciently high electron density (ne ≈ 1015 cm−3 –1018 cm−3 )). Classed with such objects are, in particular, various pinches, laser breakdown sparks, high-current arcs, high-power plasmatrons, and so on [6, 7, 9, 10, 16, 36, 37]. Figure 7.19 illustrates an example of determination of the radial proﬁles of ne and Te in the plasma jet of an argon plasmatron [10]. The results of measurements by the scattering spectroscopy method are presented in comparison with the appropriate data obtained by other methods. As the measurement techniques are being perfected, the theoretically estimated lower limits set by sensitivity have recently been being approached. Let us give an example of ne and Te measurements in microdischarges ca. 0.1 mm–1 mm long [40], which is typical of plasma display cells. Such sizes correspond to the breakdown minimum region of the Paschen curves at a gas pressure of ca. 1 atm [41], and it is important to reliably discriminate the signal due to the Thomson scattering on electrons from those due to the Rayleigh and parasitic scattering. Figure 7.20 qualitatively illustrates the spectral distribution of these scattering signals. In conditions of the above-cited work [40], the typical ratio between the intensities of the Thomson and Rayleigh scattering signals measured in a Ne-Ar (1:10) gas discharge at a pressure of P ≈ 267 hPa(200 Torr) amounted to ca. 1:1000. Since the Thomson scattering signal has a wider spectrum than its Rayleigh and parasitic counterparts, the spectral discrimination of these signals is possible, though this places fairly high requirements upon the monochromator to be used for the purpose. To illustrate, Hooper [30] has used a triple monochromator ensuring sup-

7.5 Scattering of Light on Electrons

Figure 7.19 Radial proﬁles of the electron density (solid curves) and temperature (dashed curves) in a plasmatron jet in the exit section of the nozzle. Gas – argon, current 70 A, ﬂow rate 30 g/min. Measurement methods: S – Thomson scattering, CA – absolute intensity of the continuum, LA – absolute intensity of Ar lines, CR – relative intensities in the continuum.

Figure 7.20 Schematic relation between the contributions from the various scattering mechanisms to the scattered spectrum.

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7 Determination of the Parameters of the Electronic Component of Plasma

Figure 7.21 Scattered spectrum of a microdischarge plasma. The laser beam passes at a height of Z = 0.1 mm over the surface of the cathode at a distance of x = 1 mm from the center of the discharge gap.

pression by a factor of ca. 108 for signals deviating by an amount of Δλ ≈ 1 nm from the center of the laser line (532 nm, garnet laser, pulse duration 6 ns, pulse energy 10 mJ, pulse repetition frequency 10 Hz), the possibility remaining of absolute calibration against the Rayleigh scattering signal, (7.48). A photon counting regime is used in conjunction with signal integration over repeated laser pulses locked into step with the discharge pulses. Discharge gap breakdown occurs ca. 0.2 μs after application of the discharge voltage, and the discharge current duration is ca. 0.1 μs. The discharge gap is formed by two strip electrodes on a dielectric surface. Figure 7.21 shows the Thomson scattering spectrum in the form of the number of photons counted 0.32 μs after application of the discharge voltage across the discharge gap per 3000 laser pulses as a function of the spectral detuning (Δλ)2 , nm, from the laser line. A focused laser beam passes at a height of Z = 0.1 mm above the surface of the cathode at a distance of x = 1 mm from the center of the discharge gap. The dashed lines in the ﬁgure show a series of spectra calculated theoretically on the assumption of the purely Thomson scattering mechanism and isotropic Maxwellian electron velocity distribution at different electron temperatures Te . With the scale of abscissas as selected and a logarithmic scale of ordinates, the dependences are, in accordance with expressions (7.36) and (7.45), linear, which is conﬁrmed, within the accuracy

7.5 Some Remarks on Measurements from Intensities in Line and Band Spectra

Figure 7.22 Space-time distributions of (a) Te and (b) ne at a height of Z = 0.1 mm over the surface of the discharge electrodes.

of measurement, by experimental measurements. Figure 7.22 presents the results of measurements of the spatial distributions of electron densities and temperatures and their dynamics in a plasma object formed over the surface of electrodes in the vicinity of a microgap 0.1 mm wide at a height of 0.1 mm as well. The electrodes and the gap are shown schematically on the x-axis. The ranges of the values being measured are Te ≈ (2.4 × 10−19 J (1.5 eV) – 5.6 × 10−19 J (3.5 eV)), ne ≈ 1012 cm−3 – 3 × 1013 cm−3 . Scattering was observed at an angle of Θ = 90◦ , and this conﬁrms the fact that spectra can be processed by the simplest procedure corresponding to the parameter α 1.

7.6 Some Remarks on Measurements from Intensities in Line and Band Spectra

If the upper atomic level j of a radiative transition is excited in plasma by direct electron impact from the ground state 0, the excitation rate depends on the density of electrons and their velocity distribution f (ve ),

391

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7 Determination of the Parameters of the Electronic Component of Plasma

and the population balance of the level is deﬁned by the relation dNj /dt = N0 ne ve σ0j + S j . Here ve σ0j = k0j =

∞ vj

(7.52)

f (ve )σ0j (ve )ve dve , v j is the threshold excitation

energy of the level j and S j is the sum of the rates of the processes other than the electronic excitation and depopulation of the level j. If the latter process reduces solely to radiative decay and the coronal equilibrium model holds true (see Section 1.3.3), then S j = − A j , A j being the sum of the Einstein coefﬁcients for decay to all the lower-lying levels l. Assume next that the principal radiative decay channel is the j →l. transition. In the stationary case, dNj /dt = 0 and the radiation intensity is given by Ijl = − hνjl S j = hνjl A jl Nj .

(7.53)

If a similar situation occurs with the i → k transition, the intensity ratio Ijl /Iik = (k0j /k0i )(νik /νjl )

(7.54)

depends neither on the concentration of electrons and ground-state atoms, nor on the radiative decay probabilities, and is only determined by the ratio between the excitation rate coefﬁcients (constants) of the transitions and that between their wavelengths. The behavior of the excitation rate coefﬁcients, with the integral in expression (7.52) being calculated with various σ (ve ) and f (ve ) approximations, was analyzed in detail in many works (see, for example, the review [41]). These calculations can help determine the quantities k0j (v j ) and the corresponding values f (v j ) of the electron velocity distribution function at the threshold electron velocities for the excitation of the level j. By measuring the integral relative intensities Ij /Ii , one can select such ratio f (v j )/ f (vi ) as describes the behavior of the electron velocity distribution function in the region of the excitation thresholds of the radiating levels. If f (ve ) is a Maxwellian function, the electron temperature Te can then be determined from the ratio f (v j )/ f (vi ) by means of formulas (1.3) and (1.4). If one measures the absolute intensities for transitions starting at a speciﬁc level, one will then be able to ﬁnd the electron concentration, from relations (7.53) and (7.54). The processing scheme for the intensity measurement results can be further simpliﬁed, considering the fact that the functional relations σ (ve ) expressed in terms of the threshold excitation energy are fairly similar, provided that the multiplicity of the level being excited is the same as that of the ground level (Section 5.4, Figure 5.6). Typical of nonequilibrium low-temperature plasma is that the average electron energy is

7.6 Some Remarks on Measurements from Intensities in Line and Band Spectra

lower than the spectrum excitation threshold, the function f (ve ) decreasing exponentially in the region beyond the threshold. Therefore, to determine Te by formulas (1.3) and (1.4), one may put k0j /k0i ≈ max are the maximum excitation crossσjmax v j f (v j )/σimax vi f (vi ), where σj,i sections of the levels. Such measurements are frequently practised (see, for example, [42– 44]). One should, however, pay attention to the fact that this is an indirect method and requires that some other assumptions, apart from the ones indicated above, should be made, namely: • The atom, ion or molecule is excited by direct electron impact. • The decay of the excited states should be radiative. If radiationless decay channels are present, their decay rates should be known and added to the radiative decay rates. • Since the electron velocity distribution parameters are recovered from the intensities of a limited number of lines (bands), an assumption should be made as to the functional form of f (ve ). It is usually a Maxwellian function. Deviations from this form will lead to errors. The greater the excess of the threshold excitation energy over the average electron energy, the more critical is the feasibility of the last condition listed above, responsible for excitation being the “tail” of the function f (ve ), for which the Maxwellian description is most problematic, especially at low electron densities [19, 45, 46]. To minimize this problem, Vereschagin and co-workers [47] suggested using intensities in Raman scattering spectra, instead of electronic transition intensities in the spontaneous emission spectra of plasma. Raman scattering measurements help determine the vibrational and rotational distributions of molecules (Chapter 3). Responsible for the excitation of molecular vibrations and rotations are electrons with relatively low energies (1.600 × 10−19 J (1 eV)–8.000 × 10−19 J (5 eV)) close to the average. By calculating vibrational–rotational distributions of molecules for various f (ve ) and comparing them with their experimentally measured counterparts, one can draw a conclusion about the actual electron velocity distribution. This approach is justiﬁed, provided that one is sure of the reliability of one’s calculations of the vibrational–rotational distributions in the given measurement conditions. Measurements by the method under discussion are often supplemented and checked by those carried out by absorption and scattering techniques [48]. Of course, in each concrete case one should weigh up how voluminous and justiﬁed the additional investigations into the feasibility of the prerequisites of the method actually are.

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References

1 B.V. Alekseev and V.A. Kotelnikov. A Probe Diagnostics Technique (in Russian). Moscow: Energoatomizdat (1988). 2 V.I. Demidov, N.B. Kolokolov, and A.A. Kudryavtsev. Probe Methods to Study Low-Temperature Plasmas (in Russian). Moscow: Energoatomizdat (1996). 3 O.V. Kozlov. An Electric Probe in Plasma (in Russian). Moscow: Atomizdat (1969). 4 D.V. Mozgrin and I.K. Fetisov. Electron Temperature Measurements in Dense Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), II, pp. 476–477. Moscow: Nauka (2000). 5 V.ER. Golant and G.M. Batanov. Microwave Diagnostics of LowTemperature Plasma. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), II, pp. 608–618. Moscow: Nauka (2000). 6 L.A. Dushin. Microwave Interferometers for Plasma Density Measurements in a Pulsed Gas Discharge (in Russian). Moscow: Atomizdat (1973). 7 V.K. Zhivotov, V.D. Rusanov, and A.A. Fridman. Diagnostics of Nonequilibrium Chemically Active Plasma (in Russian). Moscow: Energoatomizdat (1985). 8 M. Born and E Wolf. Principles of Optics. Pergamon Press (1959). 9 A.N. Zaidel and G.V. Ostrovskaya. Laser Plasma Diagnostics (in Russian). Leningrad: Nauka (1977). 10 L.N. Pyatnitsky. Laser Plasma Diagnostics (in Russian). Moscow: Atomizdat (1976). 11 V.P. Shevelko. Photo Processes with Atoms and Ions. Physic Reviews, 20, p. 2. Moscow: Harwood Academic Publishers (2000). 12 S.V. Khvistenko, A.I. Maslov, and V.P. Shevelko. Molecules and their Spectroscopic Properties. Berlin: Springer (1998). 13 A. Garscadden. Gas Discharge Diagnostics Update. In: J.M. Proud and

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L.H. Luessen, Eds. Radiative Processes in Discharge Plasma, pp. 467–494. N.Y.: Plenum Press (1986). A.G. Zhiglinsky and V.V. Kuchinsky. A Real Fabry–Perot Interferometer (in Russian). Leningrad: Mashinostroenie (1983). V. Ya. Nikulin. Shadow, Schlieren, and Interferometric Plasma Diagnostics Methods. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), II, pp. 545–552. Moscow: Nauka (2000). V.N. Ochkin. Physical Problems, Tasks, and Methods of Spectral and Optical Plasma Diagnostics. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), II, pp. 411–424. Moscow: Nauka (2000). S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963). I.I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Fizmatgiz (1963). Yu. P. Raizer. Gas-Discharge Physics (in Russian). Moscow: Nauka (1982). W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). D.R. Inglis and E. Teller. Ionic Depression of Series Limits in One-Electron Spectra. Astrophys. Journ., 90, pp. 439–448 (1939). F. Burhorn and R. Wienecke. Zs. Phys. Chem., 215, p. 285 (1960). C.-R. Vidal. Determination of Electron Density from the Line Merging. JQSRT, 6, No. 4, pp. 461–477 (1966). L.A. Vainshtein, I.I. Sobelman, and E.A. Yukov. Excitation of Atoms and Spectral Line Broadening (in Russian). Moscow: Nauka (1979). L.V. Levshin, Ed. Practical Work in Spectroscopy. Moscow: Moscow State University Press (1976). H. Griem. Spectral Line Broadening by Plasmas. New York: Academic Press (1974). V.N. Kolesnikov. Optical and Spectral Plasma Diagnostics Techniques. In: V.E. Fortov, Ed. Encyclopedia of Low-

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Temperature Plasma (in Russian), II, pp. 490–507. Moscow: Nauka (2000). H.R. Griem. Principles of Plasma Spectroscopy. N.Y.: Cambridge University Press (1997). ¨ J. Holtsmark. Uber die Verbreiterung von Spektrallinien, Ann. Physik, Ser. 4, 58, pp. 577–630 (1919). C.F. Hooper, Jr. Low-Frequency Component Electric Microﬁeld Distributions in Plasmas. Phys. Rev., 165, No. 1, pp. 215–222 (1968). M.A. Gigosos, V. Cardenoso. New Plasma Diagnostic Tables of Hydrogen Stark Broadening Including Ion Dynamics. J. Phys. B., 29, pp. 4795– 4838 (1996). V.P. Gavrilenko, V.N. Ochkin, and S.N. Tskhai. Progress in Plasma Spectroscopic Diagnostics Based on Stark Effect in Atoms and Molecules. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 207– 229 (2002). B.L. Wize. In: R. Huddleston and S. Leonard, Eds. Plasma Diagnostic Techniques. New York (1965). L.G. Diyachkov. Continuous Spectra. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), I, pp. 391–400. Moscow: Nauka (2000). . V.N. Kolesnikov. Arc Discharge in Inert Gases (in Russian). Trudy FIAN SSSR, 30, p. 66 (1964). H.-J. Kunze. Plasma Diagnostics Using Lasers. In: W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). M.I. Pergament. Plasma Diagnostics from Scattered Radiation. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), 2, pp. 569–572. Moscow: Nauka (2000). H.-J. Kunze. Light Scattering. In: J.M. Proud and L.H. Luessen, Eds. Radiative Processes in Discharge Plasma, pp. 467–494. N.Y.: Plenum Press (1986). E.E. Salpeter. Phys. Rev., 122, p. 1663 (1960). K. Muraoka, M. Maeda, K. Uchino, and Y. Noguchi. Laser Thomson Scattering Diagnostics of Non-

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Equilibrium High-Pressure Plasmas as Applied to Excimer Laser Pumping and Plasma Display. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460 (2002). S.E. Frish. Fluorescence of LowPressure Gas-Discharge Plasma. In: S.E. Frish, Ed. Gas-Discharge Plasma Spectroscopy (in Russian), pp. 244–273. Leningrad: Nauka (1970). S. deBenedictis, F. Cramarossa, and R. d’Agostino. Infrared and Visible Analysis of He-CO and He-CO-O2 Radiofrequency Discharges. J. Phys. D: Appl. Phys., D18, pp. 413– 423 (1985). R. d’Agostino, F. Cramarossa, S. deBenedictis, and G. Ferraro. Spectroscopic Diagnostics of CF4 –O2 Plasmas During Si and SiO2 Etching Processes. J. Appl. Phys., 52, No. 3, pp. 1259–1265 (1981). E.A. Kralkina, L.M. Volkova, A.M. Devyatov, and S.F. Shushurin. On the Possibility of Determining Electron Velocity Distribution Function in Plasma from Spectral Line Intensities (in Russian). Vestnik MGU, Se. Fiz. Astron., No. 6, pp. 735–745 (1975). L.M. Biberman, V.S. Vorobyev, and I.T. Yakubov. Kinetics of Nonequilibrium Low-Temperature Plasma (in Russian). Moscow: Nauka (1982). N.A. Kaptsov. Gas-Discharge Physics (in Russian). Moscow: OGIZ (1947). K.A. Vereshchagin, V.V. Smirnov, A.V. Bodronosov, O.A. Gordeev, and V.A. Shakhatov. Local Nonintrusive Diagnostics of Electron Components of Plasma Glow Discharge in Nitrogen by CARS Spectroscopy. In: V.N. Ochkin, Ed. Spectroscopy of Nonequilibrium Plasma at Elevated Pressures. Proc. SPIE, 4460, pp. 111– 121 (2002). S.N. Andreev, M.A. Kerimkulov, B.A. Mirzakarimov et al. Effect of Collisions on the Distribution of Molecules among Vibrational Levels of Excited Electronic States in a Gas Discharge (in Russian). ZhETF, 101, No. 6, pp. 1732–1748 (1992).

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8

Some Information on Spectroscopy Techniques The abilities and success of experiments on plasma spectroscopy and applications of plasma objects as radiation sources, largely depend on the rational choice of the materials and instrumentation to be used for the purpose. Since Newton’s discovery of the visible spectrum in 1666, the physical fundamentals and technology of spectral instruments have turned into a large, well developed and diversiﬁed independent avenue of scientiﬁc exploration. It is not an exaggeration to state that the optical and spectral instrument making industry is one of the most advanced branches of production and business in the world. A vast body of periodic literature, as well as newly published scientiﬁc monographs and courses of the fundamental and applied problems of optics and spectroscopy, are devoted to spectroscopic techniques (see, for example, [1–15]). This chapter is of an ancillary character and is included solely for the readers’ convenience. We will present here the main ideas and information on the possibilities of technical support of the plasma spectroscopy methods described in the preceding chapters. The variety of the approaches used notwithstanding, a typical spectroscopic experiment uses, apart from the object of study, a spectral instrument, an auxiliary radiation source to form light beams, an optical system and a radiation detector. Accordingly, we will only dwell in this chapter on the properties of the widespread and new optical materials (Section 8.1 and Appendix H). Further, we will restrict ourselves to the classical optical instruments (Section 8.2), because some characteristics of the lasers used in plasma spectroscopy have already been discussed, and the more systematic elucidation of this matter will undoubtedly require an individual generalization of the progress made along the lines of [16], which is far beyond the scope of the present book. Section 8.3 considers the spectral characteristics of plasma light sources, and Section 8.4, the parameters of radiation detectors.

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8.1 Characteristics of Optical Materials. Main Relations 8.1.1 Reﬂection at an Interface

Assume that along the z-axis in a medium with a complex dielectric constant of ε there propagates a plane wave with a frequency of ω = 2πν: E(z) = E(0) exp {−iω (t − z/v)} , 2σ ε = ε + i , ν

(8.1) (8.2)

where σ√is the conductivity of the medium. The propagation velocity v = c/ ε , where c is the velocity of light in vacuum. The dielectric constant is related to the refractive index n and the absorption coefﬁcient nκ by the relation ε = (n )2 = (n + inκ )2

(8.3)

Considering expression (8.2), on splitting relation (8.3) into the real and the imaginary parts, we get 2 1 4σ n2 = ε2 + 2 + ε , 2 ν (8.4) 1 4σ2 2 2 2 ε + 2 −ε n κ = 2 ν When the wave is normally incident from a vacuum (air, nair − 1 = 0.0003) upon a smooth surface, the reﬂection coefﬁcient in terms of power is *√ *2 * * * ε − 1* ( n − 1)2 + n2 κ 2 , R = *√ *2 = ( n + 1)2 + n2 κ 2 * * * ε + 1*

(8.5)

that is, the character of reﬂection is completely determined by the dielectric constant of the reﬂecting material. For the ideal dielectric, the conductivity σ = 0 and the dielectric constant is real, ε = ε. At ε > 0 the refractive index is also real, n = n , and from expression (8.5) we have R=

n−1 n+1

2 (8.6)

8.1 Characteristics of Optical Materials. Main Relations

This is a particular case (normal incidence) of the Fresnel formulas. The quantity in exponent in expression (8.1) is purely imaginary: no light is absorbed within the material. For an absorbing medium, reﬂection coefﬁcient (8.5) is contributed to from both the real and the imaginary parts of dielectric constant (8.2) and refractive indices (8.3) and (8.4). The conductivity of metals is high, ε σ/ν, and it follows from expressions (8.2), (8.4), and (8.5) that σ ν 2 2 , R ≈ 1−2 (8.7) n ≈ nκ ≈ ≈ 1− ≈ 1− . ν σ n nκ When the incident radiation frequency is low, the reﬂection coefﬁcient of high-conductivity metals is close to unity. Considering expression (8.7), (8.1) for the propagation of a wave in a medium assumes the form ! ωnκ " ! n " E(z) = E(0) exp −iω (t − z) exp − z , (8.8) c c and describes a wave that is damped by a factor of e while traveling a distance of c/ωnκ. This distance is called the skin layer and is given by δ=

c √

2π νσ

.

(8.9)

For an ideal conductor, σ = ∞, δ = 0, the wave ﬁeld does not penetrate inside, no matter what the ﬁnite frequency, and the reﬂection coefﬁcient R = 1. 8.1.2 Dispersion of the Optical Properties of Materials

The frequency dependence (dispersion) of the complex dielectric constant means that the character of reﬂection and absorption of one and the same material differs between different regions of the spectrum. The well-known dispersion formulas for dielectric permeability and refractive index were derived by analyzing the motion of a charged particle of a medium under the effect of the electric ﬁeld of a light wave incident on it and the relation between the dielectric permeability ε and the polarizability α of the medium (the electric dipole moment induced in the medium by an external electric ﬁeld of unit strength). This relation in the electron theory is given by the Lorentz–Lorenz formula α=

3 ε − 1 . 4πN ε + 2

(8.10)

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8 Some Information on Spectroscopy Techniques

Considering expressions (8.2) and (8.3), the dispersion relations have the form ε = n2 (1 − κ 2 ) = 1 + 4π ∑ i

n2 κ = 2π ∑ i

Ni ei2 (ωi2 − ω 2 ) , mi (ωi2 − ω 2 )2 + γi2 ω 2

Ni ei2 γi ω . mi (ωi2 − ω 2 )2 + γi2 ω 2

(8.11) (8.12)

Here ωi are the resonance frequencies of the oscillators of the medium and γi , their damping. For oscillators corresponding to the oscillations of electrons in an atom, the resonance frequencies correspond to optical transition frequencies. These relations have been analyzed in detail in many books on optics for a number of particular cases of rareﬁed and dense media – gases, liquids, metals and dielectrics (see, for example, [1, 2, 4–7, 17, 18]). When the incident radiation frequency is varied over wide limits, materials exhibit a fairly complex structure of transparency, reﬂection, and absorption regions. The respective quantities are implicitly related by relations (8.2)–(8.5), (8.11) and (8.12). To illustrate, having measured the reﬂection spectrum of a material, one can calculate the dispersion curves for its dielectric permeability, absorption and transmission. The factual reference data on the constants of materials in spectral regions from the ultraviolet to the infrared are presented in Appendix H. 8.1.3 Transmission and Reﬂection of Thin Films

In the given context, the term thin can be applied to ﬁlms whose thickness l is negligible in comparison with the coherence length lc of light incident on them. The coherence length of radiation with a bandwidth of Δν can be estimated at lc ∼ c/Δν. If this condition is satisﬁed, the interference of those beams repeatedly reﬂected from the ﬁlm surfaces can produce important results, and the reﬂection and transmission of ﬁlms can substantially differ from those of a massive optical element made of the same material. 8.1.3.1 Metal Films

When a light wave propagates in the bulk of massive metal for a distance of l much less than skin layer (8.9), l δ, its electric ﬁeld strength E remains practically unchanged. If light is incident on a ﬁlm of the same thickness, the situation with regard to E remains unchanged, and the ﬁeld strength at the ﬁlm surface on the incidence side is equal to that of light leaving the ﬁlm at the exit surface, Eout . However, by ‘the strength

8.1 Characteristics of Optical Materials. Main Relations

of the wave ﬁeld on the ﬁlm surface on the incidence side’ should be meant ‘the sum of the ﬁeld strength Eon of the incident wave and the ﬁeld strength Er of the wave reﬂected by this surface’ that is, Eout = Eon + Er .

(8.13)

Let r and t be the reﬂection and transmission coefﬁcients of the ﬁlm in terms of ﬁeld strength. If one takes account of the fact that on reﬂection from metal the wave ﬁeld suffers a phase jump, r = −|r |, one then can write down, proceeding from the law of conservation of energy, the following expression for the power A lost by the light wave in passage through the metal ﬁlm to induce the conduction current therein: A = 1 − |t|2 − |r |2 = 2|r |(1 − |r |).

(8.14)

One can see from the above expression that absorption is maximum when the reﬂection coefﬁcient in terms of ﬁeld strength |r | = 0.5 or, for absorption in terms of power, R = |r |2 = 0.25, or for transmission, T = |t|2 = 0.25. In that case, the maximum absorption in terms of power is A = 0.5. To calculate the dependences of the reﬂection and transmission coefﬁcients of a ﬁlm on the properties of the metal and its thickness, one should, as already stated above, sum up the ﬁeld strength of the waves propagating in the ﬁlm as a result of repeated reﬂections from its surfaces. At normal incidence, the formulas for the complex amplitudes of these coefﬁcients have the form [6, 7, 19] 1 − exp {−2(1 + i)l/δ} , 2 exp {−2(1 + i) l/δ } 1 − rmet exp {−(1 + i)l/δ} 2 t = (1 − rmet ) . 2 exp {−2(1 + i) l/δ } 1 − rmet

r = rmet

(8.15)

Here rmet is the reﬂection coefﬁcient of a smooth surface of a massive metal body in terms of ﬁeld strength. Figure 8.1 [6, 7] presents the quantities A, R = |r |2 and T = |t|2 , calculated by formulas (8.13) for (i) Rmet = 0.91 and (ii) Rmet = 0.54, as a function of the dimensionless parameter l/δ and also their experimental dependences on the thickness of a silver ﬁlm in the visible region of the spectrum (see Appendix H). Comparison between the results of calculations by formulas (8.13) and experimental measurements shows that the best quantitative agreement occurs in the IR region of the spectrum. This is facilitated by the weaker effect of the granular structure of thin (less than 10 nm thick) ﬁlms, the more justiﬁed use of the Maxwellian approximation for metals and weaker wavelength dependence of Rmet if we are dealing with

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8 Some Information on Spectroscopy Techniques

0 0.5

1 0.91 R,T 0.5

A

a)

R

A

T 0

0 0.5

0.1

1 R,T 0.54 0.5

0 0.5

0.3 l/δ b)

A R T

A 0

0.2

0.5

1.0

1 R,T

A

0.5

R

A

1.5 l/δ c)

T 0

10

20

30 40 l,nm

Figure 8.1 Transmission T , reﬂection R and absorption A of a conductive ﬁlm calculated as a function of the parameter l/δ. (a) R|rmmet = 0.91; (b) Rmet = 0.54; (c) experimental measurements for a silver ﬁlm.

a wideband incident radiation. Nevertheless, in the visible and UV regions of the spectrum, formulas (8.13) qualitatively correctly reﬂect the situation. Apart from being used as coatings for partially transparent optical elements, thin metal ﬁlms ﬁnd application as ‘windows’ in the spectroscopy of various spectral regions, the UV and VUV ones included (see Appendix G). 8.1.3.2 Dielectric Films

Dielectric ﬁlms are used to change the reﬂection and transmission of optical elements. Let a light wave is normally incident from medium 1 on the surface of medium 2, with thin ﬁlm 3 placed at the interface, the materials of all the three media being dielectric. The reﬂection coefﬁcients in terms of ﬁeld strength at the interfaces are given by the Fresnel formulas r13 =

n1 − n3 ; n1 + n3

r31 =

n3 − n1 = −r13 ; n3 + n1

r32 =

n3 − n2 . n3 + n2

(8.16)

8.1 Characteristics of Optical Materials. Main Relations

The reﬂected wave ﬁeld strength ErΣ is expressed in terms of the ﬁeld strength Eon of the wave incident from medium 1 and is found by summation [6, 7, 19]: ErΣ = Eon r13 + t23 r32 t13 t31 exp {−i2βl } (8.17) . 2 t13 t31 exp {−i4βl } + · · · , + t43 r32 where l is the ﬁlm thickness, t3 is the transmission of the layer in terms of ﬁeld strength and t13 , t31 and t32 are the transmissions of the interfaces in terms of ﬁeld strength. The quantities in exponent include the phase term β = 2πν c n3 + δ, where δ is the phase jump at the interface. The total reﬂection coefﬁcient in terms of ﬁeld strength is rΣ =

t2 r32 t13 t31 exp {−i2βl } ErΣ = r13 + 3 2 . Eon 1 − t3 r32 r13 exp {−i2βl }

(8.18)

The transmission coefﬁcient of the structure in terms of ﬁeld strength is tΣ =

EoutΣ t3 t13 t32 exp {−iβl } = . Eon 1 − t23 r32 r31 exp {−i2βl }

(8.19)

The corresponding reﬂection and transmission coefﬁcients in terms of power (intensity) are calculated by formulas (8.18) and (8.19) as RΣ = |rΣ |2 , TΣ = |tΣ |2 . Figure 8.2 [6, 7] shows the reﬂection of a ﬁlm with n3 deposited on glass with n2 = 1.5 as a function of the optical thickness n3 l of the ﬁlm in the case of normal incidence of a wave from vacuum with n1 = 1. In practice, to enhance reﬂection or transmission, use is made of ﬁlms whose optical thickness is a multiple of an odd number of quarter-waves λ0 /4: 1 λ0 , q = 1, 2, 3 . . . n3 l = q − (8.20) 2 2 In that case, if absorption in the ﬁlm is disregarded, t3 = 1, it follows from expressions (8.16), (8.18) and (8.19) that RΣ =

n1 n2 − n23 n1 n2 + n23

2 ;

TΣ =

4n1 n23 n2 . (n1 n2 + n23 )2

(8.21)

The quantity RΣ = 0 if n3 =

√

n1 n2 ,

(8.22)

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8 Some Information on Spectroscopy Techniques

RΣ 0.5

3.0

0.4 0.3 0.2

2.0 1.7

0.1

1.0 and 1.5

0.04 1.4

0.02 0

1.2 1λ 4 0

1λ 1λ 2 0 4 0

λ0

η3 L

Figure 8.2 Reﬂection of dielectric ﬁlms on glass for a wavelength of λ0 . n3 l – optical thickness of the ﬁlm. Numerals on the curves indicate n3 values.

that is, there occurs the so-called ‘blooming’, when the interfering waves reﬂected from the front and rear interfaces of the ﬁlm cancel out. Conversely, RΣ = 1 if n23 n1 n2

or

n23 n1 n2 ,

(8.23)

thus ensuring high, ‘mirror’ reﬂection. The curves at the top of Figure 8.2 correspond to the enhancement of reﬂection, while those at the bottom, to its reduction in comparison with Fresnel reﬂection (8.6) from the surface of glass free from coating (R = 0.04 at n2 = 1.5). When manufacturing bloomed or mirror optics with the use of dielectric ﬁlms, one has to accept that the choice of the dielectric materials meeting the technological and service requirements (solubility, volatility, strength, transparency in the required spectral region, stability in the atmosphere, etc.) is limited. Added to this is the serious problem of adequate satisfaction of equations (8.22) and (8.23). This is especially true of reﬂective optics, the demands on which are increased incessantly (highpower light ﬂuxes, lasers, multipass cavity optics, etc.). These difﬁculties are being overcome mainly by means of ﬁlms consisting of several alternating dielectric layers with high and low refractive indices. The total reﬂection coefﬁcient is raised on account of the interference of the beams reﬂected from each of the layers.

8.1 Characteristics of Optical Materials. Main Relations

R 1 δ 0,5

0 R 1

1

2

3

λ0 / λ δ

0,5

0

1/3

1/2

λ / λ0

1

Figure 8.3 Reﬂection of a 7-layer mirror (ZnS-3 NaFAlF3 ) on glass.

Let a coating consist of a series of dielectric layers of two sorts. Each of the layers has an optical thickness of λ0 /4, n0 is the refractive index of the medium from which the incident radiation originates (e.g. air with n0 = 1), n is the refractive index of the substrate and n1 and n2 are the refractive indices of the ﬁrst (and all the odd, as reckoned from the substrate) and the second (and all the even) dielectric layers, respectively. In that case, the total reﬂection coefﬁcient for an even or an odd number of layers at a ﬁxed wavelength of λ0 is given by the formulas [6, 7] + RΣ,2N = +

RΣ,2N +1

1 − (n/n0 )(n1 /n2 )2N 1 + (n/n0 )(n1 /n2 )2N

,2 ;

1 − (n1 /n0 )(n1 /n)(n1 /n2 )2N = 1 + (n1 /n0 )(n1 /n)(n1 /n2 )2N

(8.24)

,2 .

Figure 8.3 [6, 7] illustrates the numerical calculation of the reﬂection of a mirror made up of 7 layers of ZnS(n1 = 2.3)–3 NaFAlF4 (cryolite, n2 = 1.35) on glass (n = 1.5) as a function of wavelength in λ0 units. In the vicinity of λ0 /λ = 1 and 3 there are high-reﬂectivity regions. Observed between them are maxima of lower reﬂectivity, whose number is equal to that of the mirror layers. As the number of layers is increased, the overall reﬂection grows higher, while the region of wavelengths characterized by high reﬂectivity narrows, which is used in the manufacture of narrow-band mirrors, for example, those of reﬂection ﬁlters. If, how-

405

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8 Some Information on Spectroscopy Techniques

ever, it is necessary to make a wide-band mirror, this can be achieved by superimposing several groups of quarter-wave layers, each constituting a mirror with its reﬂection band shifted relative to those of the mirrors made up by the other groups of layers. An alternative approach is to vary the thickness of the layers near the exact value of λ0 /4. Of course, these and other methods involving the enlargement of the number of layers in the structure of optical elements are subject to limitations of their own, associated with absorption and scattering in the materials used, their strength, and so on, and require development of very complex processes to implement them. Their development has by now made available mirrors with reﬂection coefﬁcients up to RΣ ≈ 0.99999. Table 8.1 lists characteristics of some mirrors manufactured by. Los Gatos Research (the data is as of 2007). The reﬂection coefﬁcients here as given in percent, while the bandwidth Δλ means that at its edges (1 − RΣ (λ0 ± Δλ/2)) = 2(1 − RΣ (λ0 )). Table 8.1 Some reﬂection characteristics of up-to-date mirrors. λ0 , nm

220

340

370

415

488

532

620

690

RΣ , %

99.5

99.95

99.99

99.995

99.995

99.9985

99.995

99.995

Δλ, nm

5

20

25

30

35

50

60

60

λ0 , nm

765

1064

1530

1900

3300

4000

6200

9600

RΣ , %

99.995

99.999

99.9975

99.99

99.97

99.98

99.98

99.99

Δλ, nm

55

90

130

95

400

770

750

800

8.2 Spectral Instruments

Various elements and instruments have been developed to break up light of complex spectrum into its components and/or select spectral regions of interest. They can be classiﬁed, somewhat arbitrarily, in several groups. Table 8.2 provides a convenient summary.

8.2 Spectral Instruments Table 8.2 Classes and types of spectral instrumentation.

Group

Class

Principle characteristic features

Filters. Typical selected region δλ/λ ≈ 10−1 –10−2 , (ca. 10−2 –10−3 for interference ﬁlters) Transmission (10–90)%

Absorbing

Truncation of part of spectrum. Plates of materials with suitable spectral window (Appendix H). Tinted glasses with a transparency cutoff from UV to IR

Reﬂecting

Isolation of a band in the region of reﬂection maximum of materials. Suppression of adjacent regions can be augmented on account of several reﬂections (method of residual rays)

Focal

With radiation focused on a small aperture, transmission is maximal for a certain wavelength, owing to chromatic aberration

Dispersive

Particles of one substance are suspended in another. Transparency is maximal for the wavelength at which the refractive indices of the substances coincide

Total internal reﬂection

When a transparent dielectric is brought a distance of λ to the boundary of the material wherein light propagates, the light wave undergoing total internal reﬂection goes partially into the dielectric

Interference

Single- and multilayer dielectric coatings (Appendix H)

Interferencepolarization

Interference of polarized beams. Uniaxial crystal, polarization analyzer (Wood ﬁlter). A stack of such ﬁlters – Lyot ﬁlter

Spectroscopes

Simplest instruments with visual observation through an eyepiece. Styloscopes – specially designed instruments for steel analysis

Spectrographs

Photographic registration of spectra

Slit instruments

Monochromators Isolation of spectral regions, scanning of spectra, exit slit

407

408

8 Some Information on Spectroscopy Techniques Table 8.2 (continued).

Group

Slitless instruments

Instruments and ﬁlters with cross dispersion

Class

Principle characteristic features

Polychromators

Several exit slits

Spectrometers

Spectrograph and monochromator versions equipped with photoelectron registration systems

Quantometers

Spectrometer versions equipped with polychromators

Spectrophotometers

Double-beam spectrometer versions for measuring absorption spectra on transmission scale

Interferometers

Interference in the case of large optical path difference

Acousto-Optic

Bragg diffraction by acoustic waves in crystals

Fourier spectrometers

Optical path difference modulation in interferometers. Different modulation frequencies correspond to different wavelengths of light

ISSAM

Interference (Fourier) spectrometers with selective amplitude modulation on the desired wavelength interval

Raster

Spectrometer versions wherein the entrance and exit slits are replaced by plates with alternating transparent and reﬂecting zones deposited on them (rasters). For the given wavelength, the exit raster coincides with the image of the entrance one

Filter-spectral in- Preliminary monochromatization, strument stray light suppression. In diffraction instruments, also elimination of superposition in spectra in different diffraction orders Interferometerslit instrument

Elimination of superposition in spectra in different interference orders by selecting orders

8.2 Spectral Instruments Table 8.2 (continued).

Group

Class

Principle characteristic features

Double monochromators

Elimination of stray light. One monochromator serves as a preliminary one for the other. Dispersion addition and subtraction are possible

Interferencedispersion ﬁlters

Wider-band dispersion ﬁlters suppress transparency (reﬂection) side lobes of interference ﬁlters

Scientiﬁc research, plasma spectroscopy in particular, makes use of spectral instruments of all classes and groups, and the success or failureof experiments is dependent on the choice of instrument. Whereas ﬁlters are generally ﬁxed in terms of their parameter elements and structures, the instruments belonging to the other groups are much more versatile as regards the selection and setting of their necessary parameters. There are several main characteristics of instruments. Such characteristics as the working region of the spectrum, light-gathering power (aperture ratio), dispersion and resolving power are common to all instruments, but it would be advisable to dwell upon them in each group of instruments individually for the sake of subsequent comparison. 8.2.1 Slit Instruments

Figure 8.4 presents a schematic diagram of a slit instrument. The entrance aperture (slit) 1 and objective lens 2 with a focal length of f 2 form a collimator. A beam of parallel rays is incident on dispersive block 3 (a diffraction grating or prism). The spectrum is observed in focal plane 5 of lens 4 which in combination with the dispersive block is referred to as camera. The symbol D denotes the focal aperture. The spectrum is a collection of inverted images of slit 1 by light of various wavelengths, magniﬁed (disregarding the magniﬁcation by block 3) approximately f 2 / f 1 diameters. If the width and height of slit 1 are s and h, respectively, and those of the slit image, s and h , then s = s

f2 1 ; f 1 sin ε

h = h

f2 . f1

(8.25)

In the case of monochromator, an ext slit is placed in the focal plane of the camera. The quantity dϑ/dλ is called angular dispersion.

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8 Some Information on Spectroscopy Techniques

ƒ2

ƒ1 D

2 S

1

8 (λ)

λ1

3 4

λ0

ε

5 λ2

Figure 8.4 Schematic diagram of a slit instrument: 1 – entrance slit; 2 – collimating objective lens; 3 – dispersive block; 4 – camera lens; 5 – focal plane of the camera. Wavy line – intensity distribution as a result of diffraction by the entrance slit.

The inverse angular dispersion is dλ/dϑ. The linear dispersion is given by dl dϑ 1 , = f2 dλ dλ sin ε

(8.26)

where l is the distance along the image and ε is the angle between the axis of the camera and the spectrum image plane. The inverse linear dispersion is dλ/dl. The resolving power (resolution) R˜ determines the capability of the instrument to discriminate between two adjacent spectral lines of the same intensity whose wavelengths differ by an amount of δλ: λ R˜ = . δλ

(8.27)

The instrument function A(l ) (spread function) determines the degree to which the instrument distorts the true spectrum. This function describes the intensity distribution in the image of the spectrum, provided that the entrance aperture (slit) is illuminated by monochromatic light. The form of the spectrum registered is determined by the integral convolution of the true spectrum and the instrument function. The reconstruction of the true spectrum from the one observed requires performing an inverse integral transformation, which, mathematically, is an ill-posed problem. The determination of instrument functions and methods to perform inverse integral transformations constitutes the subject of special studies. For reasonable estimates, use if frequently made of the appropriate widths of instrument functions.

8.2 Spectral Instruments

The forming of the instrument function in slit instruments is always contributed to from the diffraction by the focal aperture and the ﬁnite size of the slit. The interval δλsp occupied by the image of the entrance slit in the spectrum image plane is determined by the sum of these contributions: δλsp = δλd + δλsl .

(8.28)

If one uses, in determining the contribution from diffraction, the Rayleigh criterion for the resolution of diffraction images (i. e. two lines are considered to be separately observable if the distance between the principal diffraction maximum of one in the spectrum image and the diffraction zero of the other is no less than the distance between the principal maxima, which approximately corresponds to a 20% dip in intensity), then λ dϑ R˜ = D. = δλd dλ

(8.29)

The resolving power of an instrument cannot exceed this quantity, which is called theoretical resolution. Associated with the diffraction-limited resolution is the notion of the normal slit width. This is deﬁned as an entrance slit of width nn , for which its geometrical image in the spectrum image plane is equal to the width of the principal diffraction maximum: sn = f 1

λ . D

(8.30)

The use of narrower slits will not improve resolution, but will worsen the spectrum registration conditions. Because of diffraction by the entrance slit, only some of the radiation is incident on the focal aperture D (Figure 8.4). With the width of the entrance slit equal to the normal one, the distance between the principal maximum and the nearest zero of the diffraction pattern (the wavy line in Figure 8.4) is, as evident from expression (8.30), equal to the focal aperture. When s < sn , an unjustiﬁable loss of light entering into the instrument occurs. As the slit is widened, s > sn , the ﬂow of light entering (from a unit surface area of the slit) into the camera ﬁrst increases somewhat, on account of the ‘attachment’ of the diffraction maxima nearest to the principal one, and then becomes saturated. Various approximations are applied to describe the effect of the ﬁnite width of the slit in actual use. Since it falls within the range of widths in excess of the normal one, of prime importance are geometrical factors. In the widespread case of a monochromator with equally wide entrance

411

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8 Some Information on Spectroscopy Techniques

and exit slits, at s > sn the diffraction term in expression (8.28) can be disregarded, and in the absence of magniﬁcation in the dispersive block, a triangular form can be adopted for the instrument function. In that case, δλsp =

s dλ f 1 dϑ

(8.31)

The light-gathering power (aperture ratio) of an instrument is a coefﬁcient relating the photometric quantities (Chapter 2) for the light entering the instrument with the image of this light in the spectrum formation region. Depending on the spectrum registration method (photographic, photoelectric, via the exit slit) and the form of the spectrum (line, continuous, combined), of interest for the description of the properties of the instrument can be the relationships between various photometric quantities. Without touching upon the parameters of the actual light source and slit illumination method, we will consider the entrance slit as a homogeneous self-luminescent source with a surface area of hs and luminance of bλ dλ (2.2) and (2.4). It produces a luminous ﬂux that is directed into a solid angle of πD2 /4 f 12 and ﬁlls up the focal aperture D. In this case, the luminous ﬂux in the ‘monochromatic’ spectral interval dλ, which arrives at the focal plane of the camera is Pλ dλ = bλ dλhs

πD2 T. 4 f 12

(8.32)

The factor T stands for the transparency of the optical system of the instrument. In the focal plane of the camera an image is formed with a surface area of s h (8.25) and a ‘monochromatic’ illumination of Eλ =

πD2 sin ε Pλ dλ = bλ dλ T. hs 4 f 22

(8.33)

If the result of detection in the experiment is the image of the entire spectrum (e.g. photography, use of high-resolution video cameras), the image illumination is the decisive factor. Light-gathering power in terms of illumination is LE =

Pλ dλ πD2 sin ε = T. bλ dλh s 4 f 22

(8.34)

When registering a line spectrum of narrow lines, Δλ δλsp , to ﬁnd the image illumination one should replace the quantity dλ in expression (8.33) by Δλ: Eline =

πD2 sin ε Pλ Δλ = bλ Δλ T, hs 4 f 22

(8.35)

8.2 Spectral Instruments

that is, to attain high illumination, one should select a spectrograph with a camera of small relative aperture D/ f 2 . When registering a continuous spectrum, to ﬁnd the image illumination Econt one should sum the ‘monochromatic’ light ﬂuxes Pλ over the spectral width δλsp (8.31) of the slit: Econt =

Pλ δλsp πD2 sin ε s dλ T, = b λ h s 4 f 22 f 1 dϑ

(8.36)

that is, in contrast to line spectra, the image illumination for continuous spectra is affected, among other things, by the angular width of the entrance slit and angular dispersion. When a line spectrum is registered against the background of a continuous one, the ratio between the respective illuminations is Eline Δλ Δλ f 1 dϑ , = = Econt δλsp s dλ

(8.37)

and to better reveal the line component, one should reduce the spectral width of the slit – as long as the condition Δλ < δλsp is satisﬁed. If use is made of a monochromator and the spectrum is registered by individual regions with the aid of a detector placed beyond the exit slit, of importance is the magnitude of the luminous ﬂux arriving at the detector, and the light-gathering power in terms of luminous ﬂux LP =

πD2 Pλ dλ = hs 2 T bλ dλ 4 f1

(8.38)

proves a convenient energy characteristic of the spectral instrument. As in the above-considered case of image illumination, when registering a line spectrum, Δλ δλsp , the luminous ﬂux is summed over the wavelengths within the limits of Δλ: Pline = Pλ Δλ = bλ Δλ

πD2 hs T, 4 f 12

(8.39)

while when registering a continuous spectrum, it is summed over the spectral width δλsp of the slit: Pcont = Pλ δλsp = bλ

πD2 hs2 dλ T. 4 f 12 dϑ

(8.40)

The ratio between the luminous ﬂuxes is the same as that between the image illuminations, (8.37).

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8 Some Information on Spectroscopy Techniques

The working spectral region of slit instruments is determined by the transmission and reﬂection of the optical system and the medium between the light source and the optical elements (Appendix H), the sensitivity of the registering system, and so on. The free spectral range (free dispersion region) of the instrument is deﬁned as the range for which there is no spectral overlapping. If the dispersion of light into a spectrum results from multibeam interference, in the focal plane f the camera overlapping of spectra of different wavelength ranges can occur. Such a situation is typical of slit instruments using a dispersive block in the form of a diffraction grating. Within the free spectral range Δλfree in the spectrum of the order m, no such overlapping with the spectra of adjacent orders m + 1 and m − 1 will take place if the condition Δλfree = λ/m

(8.41)

is satisﬁed. The problem of separating overlapping spectra, is, as a rule, speciﬁc to those of diffraction orders nearest to the working one, because for greatly differing wavelengths there is an additional limitation imposed by the working spectral region of the instrument. Separation can be achieved by way of preliminary, rather crude monochromatization. 8.2.2 Interferometers

A common feature of the instruments of this class is the use of interference at large optical path differences, up to ca. 106 wavelengths. A necessary condition for the observation of interference is that the spatial coherence length of the interacting waves should be no less than the optical path difference. This imposes restrictions on the spectral width of the light source. Interferometers are required to implement the observation of interference without, as far as possible, any additional limitations. Among the well-known types of interferometers (Young, Fresnel, Jamin, Lummer–Gehrke, and others), the schemes and methods based on the Fabry–Perot multibeam-interference interferometer are the most widely used in spectroscopic practice. (For the use of the Michelson interferometer, see Section 8.2.3.) The Fabry–Perot interferometer (FPI) is a plane-parallel layer of optical thickness L bounded by partially reﬂecting surfaces. Figure 8.5 presents an FPI scheme where the layer is bounded by two plates. In situations where the thickness of the layer must be rigidly ﬁxed, the mirrors are deposited on the outside surfaces of a plane-parallel plate, and the structure is then referred to as the Fabry–Perot cavity standard (FPCS). Regardless

8.2 Spectral Instruments

L

P

415

P

R

T

m-2 m-1

φ

1

2

R,T

T3

R,T

3

4

Figure 8.5 Interference pattern formation in the Fabry–Perot interferometer: 1 – light source; 2 – collimator; 3 – camera lens; 4 – focal plane with the interference ring pattern.

of other details, it is essential that the mirrors should be parallel and the material of the layer, optically homogeneous. The system of coherent light beams in the Fabry–Perot interferometer is formed by repeated reﬂections of the light wave incident on it. The light beam from source 1 is formed by collimator 2. The wavefront width of the incident and reﬂected beams can be sufﬁciently wide (usually from a centimeter to a few centimeters), which ensures a wide overlapping zone of coherent waves. The system of beams that have passed through the FPI forms, by means of camera lens 3, the image of the interference result in focal plane 4. The characteristics of interferograms are obtained by summing up the beams, with due regard for the phase change due to the variation of the optical path difference. The phase difference between two adjacent interfering beams that have passed through the FPI, with consideration for their inclined incidence (but without regard for the phase jump at the reﬂecting surface, which is the same for all the beams in the layer (see also Section 8.1.3)), is 2βL = 2

2π L cos ϕ , λ

(8.42)

where the angle ϕ is deﬁned in Figure 8.5. For the transmission function of the FPI with identical mirrors, summation yields a result in the form of the Airy function: TFPI =

(1 − T3

R )2

T 2 T3 . + 4T3 R sin2 ( βL)

(8.43)

Here T and T3 are the transmissions of the mirrors and the layer, respectively, and R is the reﬂection of the mirror coatings. If the mirrors are

416

8 Some Information on Spectroscopy Techniques

√ differ in reﬂection, then R = R1 R2 . The variable in expression (8.43) is the phase term β determined by the wavelength λ and the angle ϕ. In writing down expression (8.43), use is frequently made of the argument γ = βL/π. At T3 = 1 TFPI =

1+

R2

(1 − R )2 . − 2R cos(2πγ)

(8.44)

For a given wavelength, the interference pattern is determined by the set of the angles of incidence of light beams on the FPI, and to obtain as full a plattern as possible, one should use a light source of as large an angular size as possible. If this condition is satisﬁed, a system of equally inclined fringes arises in focal plane 4. If the entrance aperture is round (can be considered as the focal aperture and speciﬁed by the size of the plates or collimator), the loci of the fringes are concentric circles. The transmission maximum of the FPI is attained in the case sin( βL) = 0, that is, 2nL cos ϕ = mλ,

(8.45)

while the transmission minimum corresponds to the optical path difference 1 λ, (8.46) 2nL cos ϕ = m + 2 where the integer m is the interference order. Accordingly, TFPI max =

T 2 T3 , (1 − T3 R)2

TFPI min =

T 2 T3 . (1 + T3 R)2

(8.47)

The interference pattern formed by light reﬂected from the FPI is inverted relative to that formed by transmitted light, provided that no loss occurs in the layer and plates. Transmission does not vanish because of the ﬁniteness of losses and reﬂection of the mirror surfaces (see Figure 8.6). The angular dispersion is found by differentiating expression (8.45): dϕ m , = dλ 2L sin ϕ

(8.48)

or, with m expressed by formula (8.45), dϕ 1 , =− dλ λ tan ϕ

(8.49)

8.2 Spectral Instruments

Figure 8.6 Transmission and reﬂection of a Fabry–Perot interferometer at R = 0.6, T = 0.4, T3 = 1.

and for small angles ϕ, dϕ 1 =− dλ λϕ

(8.50)

that is, it is independent of the thickness of the interferometer. Linear dispersion. If f is the focal length of the camera lens (collimator lens) and l, the radius of fringes or the distance from the fringe pattern (m → ∞), then dl f2 = . dλ λl

(8.51)

One can see from expressions (8.48), (8.51) that as the fringe radius decreases (m is increased), the angular and linear dispersion both grow higher. The instrument function of the FPI in angular units is given by expression (8.43) for monochromatic radiation. By ﬁxing the interference order m in formulas (8.45), (8.57) and using expression (8.48) for the angular dispersion, one can ﬁnd the wavelength difference δλ corresponding to the transmission peak at half maximum, that is, the width of the interference-order-dependent instrument function in terms of wavelength: δλ =

λ 1 − T3 R √ . m π T3 R

(8.52)

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8 Some Information on Spectroscopy Techniques

Resolving power. The Rayleigh criterion used for slit instruments relates to diffraction images of spectra and is, generally speaking, inapplicable to the case of the Fabry–Perot interferometer whose instrument function is close to a dispersion one and has no zeroes characteristic of diffraction patterns. However, if one superimposes two transmission peaks of ﬁxed order m, but shifted through an angle of δϕ corresponding to the quantity δλ from expression (8.52), one will ﬁnd that the combined proﬁle has at its center a dip of ca. 20% irrespective of the order m. This is the same quantity as obtained when using the Rayleigh criterion. For this reason, using the resolution for two wavelengths differing by an amount equal to the width of the instrument function as a criterion for the ultimate resolving power of the Fabry–Perot interferometer allows one to objectively compare between the resolving powers of various instruments. In this case, the resolving power is simply deﬁned by formula (8.52): √ π T3 R λ R˜ = . (8.53) =m δλ 1 − T3 R Free spectral range (region without overlap). If light is not monochromatic, the overlapping of spectra differing in order can be observed in the fringe pattern. The pattern will be free from overlapping if the width Δλ of the light spectrum does not exceed the distance (in terms of wavelength) between adjacent maxima of function (8.43). The region without overlap, Δλ, is found by differentiating expression (8.45) at ﬁxed λ, which yields 2Ln sin ϕΔϕ = λΔm. For adjacent orders, Δm = 1, and using formula (8.49), we get Δλ = Δϕ

dλ λ2 λ = = . dϕ 2L cos ϕ m

(8.54)

Because, usually, L λ, the free spectral range is small and the working order of spectrum, high. If, for example, n = 1, L = 1 cm and λ = 500 nm, then Δλ = 0.0125 nm and m = 4 × 104 . Using expression (8.53), one can rewrite expression (8.54) in the form √ π T3 R Δλ = δλ = δλNeff . (8.55) 1 − T3 R √

The quantity Neff = π1−TT33RR is called the effective number of interfering beams. It is equal to the number of equal-amplitude interfering beams providing for the same resolving power and free spectral range as being provided by the inﬁnite number of beams of decreasing amplitude propagating between the partially transparent mirrors of the Fabry–Perot interferometer. It deﬁnes how many times the distance between adjacent

8.2 Spectral Instruments Table 8.3 Sharpness indices, pattern contrasts and resolving powers of Fabry–Perot interferometers. T3 R

Neff = F

C

R˜ (m = 4 × 104 )

T3 R

14

0.85

19.3

0.9 0.92

30 38

C

R˜ (m = 4 × 104 )

= ηπ/2

= ηπ/2 0.8

Neff = F

9

5.6×105

0.95

39

2.46×106

12.3

7.72×105

0.97

103

65.7

4.12×106

19

1.2×106

0.98

156

99

6.24×106

24

1.52×106

0.99

314

199

1.26×107

61.5

maxima is greater than their width. We have already come across it in the expression for the resolving power. The free spectral range is often expressed in terms of wave numbers, cm−1 , or in terms of frequency, Hz. For n = 1 and small angles, cos φ ∼ 1, Δν =

1 [cm−1 ], 2L

Δν =

c [Hz]. 2L

(8.56)

Quality of interference pattern. To describe the characteristics of interferograms (including those from interferometers of other types and interference ﬁlters), use is made of special parameters. Speciﬁcally, the pattern contrast is the ratio between the maximum and minimum transmission (8.47): 1 + T3 R 2 TFPI max C= = . (8.57) TFPI min 1 − T3 R Considering expression (8.47), transmission function (8.43) can be written down in the form √ 1 2 T3 R . (8.58) , η= TFPI = TFPI max 1 − T3 R 1 + η 2 sin2 ( βL) The coefﬁcient η characterizes the slope of the transmission peaks and coincides up to a factor of π/2 with Neff . Therefore, the quantity Neff is also called the ﬁneness or sharpness factor of the FPI and denoted F. Table 8.3 lists the values of Neff and C for some values of T3 R, as well as the resolving powers for the above example m = 4 × 104 . Light-gathering power. As in slit instruments, with interferometers various photometric parameters, depending on the method used to register spectra. In principle, there is no need for one to limit the entrance aperture in order to realize the spectral characteristics of the FPI, and this provides a great luminous ﬂux gain in comparison with the case where light should be passed through a narrow extended slit at the very entrance to the instrument. For monochromatic radiation uniformly ﬁlling

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up the working surface of the FPI, the illumination E in the focal plane in transmission maxima (8.47) is E = TFPI max BS/ f 22 ,

(8.59)

where B and S are the brightness and surface area of the radiation source, respectively, and f 2 is the focal length of lens 3 (Figure 8.5), TFPI max ≈ 1 being usually the case. The light-gathering power in terms of illumination, L E = E/B = TFPI max S/ f 22 ,

(8.60)

is important in registering fringe pattern images. In the case of photoelectric registration, of importance is the light-gathering power in terms of the luminous ﬂux which is selected in the focal plane of lens 3 by means of a selecting diaphragm with a surface area of SD : L P = L E SD = TFPI max SD S/ f 22 .

(8.61)

The quantity L P depends on the surface area of the selecting diaphragm. The surface area and shape of the diaphragm are selected as a trade-off between the necessary spectral resolution, composition of the radiation to be analyzed, and the free spectral range of the instrument. In the case of a round entrance aperture the interference pattern comprises a series of rings, and so the diaphragm must also be ring-shaped. When it is necessary to select a wavelength interval of δλ, the inner and outer radii of the ring diaphragm can be established by means of the condition for maxima. For small angles, cos ϕ ≈ 1 − ϕ2 /2, and so this condition can be written down as ϕ2 = 2 −

mλ . L

(8.62)

The radii of the diaphragm that limit the wavelength interval λ, λ + δλ being selected are ρλ = f 2 ϕλ and ρλ+δλ = f 2 ϕλ+δλ , and its surface area is f2 mλ 2 2 2 2 m ( λ + δλ ) − = πm 2 δλ. (8.63) SD = π f 2 ( ϕλ − ϕλ+δλ ) = π f 2 L L L From a practical standpoint, ring diaphragms are inconvenient, but one can make use of the fact that interference order is high and the area SD remains practically unchanged within the limits of a comparatively small variation of m. The interference order and dispersion grow higher towards the center of the fringe pattern, and so one can use a diaphragm

8.2 Spectral Instruments

with ρλ+δλ = 0, that is, a hole. Its size providing for the selection of the spectral interval λ, λ + δλ is given by 1 1 √ √ ˜ ρD = SD /π = f 2 mδλ/L = f 2 2δλ/λ = f 2 2/ R. (8.64) If δλ corresponds to the ultimate resolution of the FPI, then ˜ SD = 2π f 22 / R,

(8.65)

and the solid angle delimited by the next aperture is ΩD = SD / f 22 = 2πδλ/λ.

(8.66)

The product ˜ D = 2π. RΩ

(8.67)

thus proves invariant. For the typical FPI base value of L = 1 cm, a wavelength of λ = 500 nm, mirror reﬂection coefﬁcients R = 0.8 and focal distance f 2 = 500 mm, the surface area of the diaphragm is SD ≈ 2.8 mm2 . The spectral ˚ If even a slightly lower resolution ensured in that case is δλ ≈ 0.009 A. ˚ should be ensured with a good diffractionresolution of δλ ≈ 0.01 A −1 , ˚ limited slit interferometer with an inverse linear dispersion of 1 Amm the slit width will then be 0.01 mm, and with the typical height of the slit being equal to 10 mm, its surface area will be 0.1 cm2 . The gain provided by the FPI in the registered luminous ﬂux turns out to be ca. 30. This is a rather typical, but not the highest ratio. Given the resolution and surface area S of the dispersive element, the geometrical factor of ˜ the Fabry–Perot interferometer is characterized as u = SΩD = 2πS/ R, ˜ and in a diffraction-limited instrument it is u ∼ βS/ R, where β is the angular height of the slit. The ratio between these factors is equal to 2π/β. The quantity β depends on the relative aperture D/ f 2 (8.32) and (8.33), typically ranging between 1/5 and 1/50. This gain in the light-gathering power provided by interferometers with axially symmetrical light beams in comparison with slit instruments is known as the Jacquinot advantage. Thus, the Fabry–Perot interferometer is a high-resolution instrument of high light-gathering power. However, utilizing these merits simultaneously and in full measure is only possible when the emission spectrum of the object under study is sufﬁciently narrow, and not in excess of the free spectral range of the instrument. For the above example, Δλ = 0.0125 nm. If it is necessary to have high spectral resolution within a wider region of the spectrum, one should use additional monochromatization in order to avoid overlapping of spectra of different orders. This reduces the light-gathering power of the spectral setup as a whole.

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The above relations and commentaries refer to the so-called ideal Fabry–Perot interferometer. The high spectral characteristics of this instrument presuppose that the mirror surfaces are of good quality and exactly orientated. In practice, both hold true to a ﬁnite precision. The deviations of the FPI from the ideal have been extensively studied. A most comprehensive summary of these investigations can be found in [20], in the following we discuss some of the salient results. To describe deviations from the ideal geometry, they have introduced the dimensionless parameters αi that express in terms of wavelength the maximum amplitude Ai of the defect αi = Ai /λ. This parameter is then introduced in the argument of Airy function (8.43) and (8.44), so that the phase incursion is contained, for example, in the quantity γ in expression (8.44). Since such is usually a defect measuring less than a half-wave, 0 < Ai <λ/2, of importance are only changes in the fractional part of γ, and so it proves sufﬁcient for one to consider the result of the presence of the defect for a single interference order. Defects can, however, differ in character. Let us consider the results of analyzing some of them. The parabolic defect of a mirror is the deviation of its surface from a plane, which has the shape of a parabola with the apex at the center of the mirror. The fractional part of the distance (in terms of wavelength) between the two mirrors of the interferometer varies as t(r ) = t0 + αi r2 − αi /2. Here r is the radius in the polar coordinate system, normalized to the mirror radius, and t0 is the fractional part of the distance between the mirrors at the apex of the parabola facing the other mirror. The latter is assumed to be plane, for its parabolic defect can be included into that of the former mirror. In that case, the argument ψ = 2πγ of the cosine function in expression (8.44) is replaced by ψ = 2π(γ+2αi r2 − αi ). The results of calculating the instrument function of the FPI for some combinations of the reﬂection coefﬁcient R and the parameter αi are presented in Figure 8.7. With αi < 1/2, the spherical defect entails practically the same results as its parabolic counterpart. What is more, the dependences of Figure 8.7 turn out to be universal for deviations from the ideal that one actually occur in practice as owing to inadequate manufacture and adjustment of the mirrors. Worthy of note among them is the skewness of rectangular mirrors, in which case the parameter αi is the maximum amplitude of the wedge between the edges of the mirrors. It also transpires that the dependences of Figure 8.7 can be used as the instrument functions of the FPI with an ideal mirror geometry, but with a limitation by a round diaphragm of radius ρD (8.64) in the focal plane to select a spectral range of δλ (in the text above, we have only presented the sizes ensuring the

8.2 Spectral Instruments

Figure 8.7 Transmission of a Fabry–Perot interferometer in the case of parabolic defect of round mirrors or skewness of rectangular mirrors, or for an ideal FPI with a round exit diaphragm: 1 – R = 0.9, αi = 0.02; 2 – R = 0.8, αi = 0.02; 3 – R = 0.9, αi = 0.1; 4 – R = 0.8, αi = 0.1; 5 – Airy function at R = 0.9.

‘truncation’ of the spectrum beyond this wavelength interval). To this 2 ρD L end, one should put αi = 2λ . f2 The skewness of round mirrors causes changes in the instrument function, as illustrated in Figure 8.8. Here, as in the case of rectangular mirrors, the parameter αi deﬁnes the maximum amplitude of the wedge between the edges of the mirrors. The same curves reﬂect the effect of sinusoidal defects on the instrument function. This corresponds to harmonic deviations of the surface of a mirror from a plane as one goes round the mirror along concentric circles, the amplitude increasing linearly from the center toward the edges (a combination with a wedge). The curves of Figure 8.8 describe this case if αi is the swing of the sine curve at the edge of the mirror. The presence of statistical microdefects on the surface of the mirrors causes the thickness of the FPI to deviate from its average value. In that case, the parameter αi has the meaning of the mean-square deviation for the Gaussian distribution of the sizes (in terms of wavelength) of the defects. The form of the instrument function of the FPI is shown in Figure 8.9.

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8 Some Information on Spectroscopy Techniques

Figure 8.8 Transmission of a Fabry–Perot interferometer in the case of skewness of round mirrors or sinusoidal defect: 1 – R = 0.8, αi = 0.036; 2 – R = 0.9, αi = 0.017; 3 – R = 0.8, αi = 0.072; 4 – R = 0.9, αi = 0.082; 5 – Airy function at R = 0.9.

Figure 8.9 Transmission of a Fabry–Perot interferometer in the case of statistical microdefects present on the surface of mirrors: 1 – R = 0.9, αi = 0.02; 2 – R = 0.8, αi = 0.02; 3 – R = 0.9, αi = 0.1; 4 – R = 0.8, αi = 0.1; 5 – Airy function at R = 0.9.

8.2 Spectral Instruments

Considered in [20] are also a number of other defects and their combinations, including the effects of the homogeneity and coherence of illumination, inertial character of detection and so on. It is obvious from general physical considerations that the presence of defects measuring more than the half-wave, αi > 1/2, will completely destroy the interference pattern. The curves presented above show that the higher the reﬂection coefﬁcient of the mirrors, the more substantial the ‘smearing’ of the instrument function of the FPI. For mirrors with R ≈ 0.9, the effect of the defects manifests itself even at αi > 0.01. The up-to-date processes of manufacturing optical plates and coatings ensure their roughness heights at a level of ca. λ/(500 to 1000). However, even with so high a surface quality, to use mirrors with a reﬂection coefﬁcient in excess of 0.98 to improve the resolving power of the FPI is senseless. 8.2.3 Spectral Instruments with Interference Modulation

Instruments of this class differ from those considered above in that the spectrum is analyzed without the spectral components being separated in space. An approach is implemented here wherein the different optical spectrum frequencies νi acquire different amplitude modulation frequencies Ωi νi . This proves, in principle, possible to realize by using the interference phenomenon. The subsequent separation of the optical frequencies by this attribute is effected by way of harmonic analysis in the low-frequency region. Although these instruments have been actively developed and used over a period of about 50 years, their history is longer. There are a great many books on this subject, including [21–24] and others. The bestknown versions of such modulation instruments are the Fourier spectrometers and interference spectrometers with selective amplitude modulation (ISSAM). 8.2.3.1 Fourier(-Transform) Spectrometers

The optical scheme of the instruments can be based on various interferometer whose optical path difference can be varied. Most widespread in the Fourier spectrometers (FS) is the scheme with the Michelson interferometer (Figure 8.10a). As in the case of the Fabry–Perot interferometer using photoelectric registration, the result of the interference of light issuing from aperture 1 is a series of equally inclined bands (fringes) in plane 2. The interfering light beams pass through a diaphragm placed in this plane and reach the detector.

425

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8 Some Information on Spectroscopy Techniques Φ

4

L

3

1

b)

X

5 Φ

c)

0

2

X 2

a)

0

X

Figure 8.10 Fourier spectrometer. (a) schematic diagram of a Michelson-interferometer-based instrument: 1 and 2 – entrance and exit apertures; 3 – beam-splitting plate; 4 and 5 – mirrors. Photoelectric-registration interference patterns: (b) monochromatic radiation; (c) incandescent lamp.

Moving mirror 5 changes the optical path difference between the beams incident on the detector. Moving the mirror for a distance of x/2 changes the optical path difference by an amount of x. If the radiation is monochromatic, the luminous ﬂux registered by the detector during the course of movement of the mirror is (Figure 8.10b) P ( x ) ∼ b cos2 2πν

x b = (1 + cos 2πνx ). 2 2

(8.68)

Here b is the brightness of diaphragm 1 and ν = 1/λ is the wave number. If the mirror moves uniformly with a velocity of v, the variable part of the luminous ﬂux P is P( x ) = P0 cos 2πνx = P0 cos 2πvtν = P0 cos 2πΩt,

(8.69)

where P0 is the amplitude coefﬁcient and Ω = vν = v/λ is the interference modulation frequency. This frequency proves to be dependent on the radiation wavelength and the translation velocity of the mirror. If, for example, the radiation is produced by a sodium lamp and the mirror moves with a velocity of v = 10−3 cms−1 , then for the yellow doublet line at λ = 589 nm, Ω = 16.9779 Hz, while for the other double component of λ = 589.6 nm, Ω = 16.9606 Hz. The green line of mercury at λ = 546.1 nm will be modulated with a frequency of Ω = 18.3117 Hz. If

8.2 Spectral Instruments

the interferometer is illuminated by all the lines at once, the signal will contain the sum of all the harmonics added with statistical weights corresponding to the brightness of the source at each individual line. In the general case, if the spectrum contains a set of frequencies in the wavenumber interval from ν1 to ν2 , the signal will be registered in the form P( x ) ∼

ν2

bν (ν) cos 2πνx dν,

(8.70)

ν1

that is, the registered signal (interferogram) is the Fourier transform of the optical spectrum bν (ν) (this follows from the fact that the interferogram is the autocorrelation function of the light wave). Figure 8.10c presents an example of the interferogram of the continuous spectrum of an incandescent lamp. The maximum interferogram signal corresponds to the zero optical path difference, when all the separated beams are mutually coherent and reach the detector without delay. This is the so-called ‘point of white light’. The original spectrum can be reconstructed by the inverse Fourier transformation: bν ( ν ) ∼ 2

2L

P( x ) cos 2πνx dx.

(8.71)

0

The upper limit of integration corresponds to an optical path difference of 2L, when the mirror is moved for a distance of x = L, the factor 2 being due to the fact that the cosine function is even and the interferogram is symmetrical about the zero optical path difference x = 0. Free spectral range. In the Fourier spectrometers, the acquisition and analysis of interferograms involves no spatial separation of the interference orders, and so the problem of free spectral range, so acute, for example, in the Fabry–Perot interferometers, is removed here. The region without overlap actually coincides with the working region and depends on the optics and detectors used. With these elements properly selected, one can investigate spectra from the visible to the far infrared region. One should, however, bear in mind that this possibility depends on the proper choice of the magnitude of the δL ‘step’ in registering the interferogram: δL ≤

1 . 2(νmax − νmin )

(8.72)

This condition stems from the fact that only a discrete Fourier transformation can be applied to a discrete array of signal readings (and it is

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8 Some Information on Spectroscopy Techniques

a (v)

D (x)

1,0

1 2 2L x

0 1 0,2 - 3L 4

- 1L 2

- 1L 4 - 0,2 1/2 L

1L 4

1L 2

3L 4

v

Figure 8.11 Instrument function of a Fourier spectrometer: 1 – without apodization; 2 – apodization with a triangular (inset) function D.

exactly in this manner that the data array for computer processing is formed). If condition (8.72) is violated, the free spectral range in reconstructing the continuous function in this way will be limited because of the superposition of different interference orders. The resolving power of the Fourier spectrometer is determined by its instrument function. As a result of transformation (8.71), the original monochromatic wave acquires a ﬁnite spectral width. This is due to the ﬁniteness of the limits of integration and the ﬁnite optical path difference in the experiment. The limits of integration can formally be made inﬁnite by introducing the function D ( x ) to modify the integrand: bν ( ν ) ∼ 2

∞

P( x ) D ( x ) cos 2πνx dx.

(8.73)

−∞

Figure 8.11 shows the instrument function of the Fourier spectrometer in the case of rectangular and triangular function D ( x ). The choice of the optimal function D ( x ) is referred to as apodization (elimination of the pedestal). In the case of triangular apodization, the width of the instrument function is δν = 1/2 L and the resolving power, ν/δν = 2Lν .

(8.74)

Irrespective of the apodization method used, the resolving power is, as in expression (8.74), proportional to the maximum optical path difference. The light-gathering power of the Fourier spectrometer is determined in the same way as for any spectrometer with centrally symmetric interfering beams. Similar to the FPI, the advantage is retained over slit

8.2 Spectral Instruments

instruments, owing to the geometrical factor – the greater surface area of the exit diaphragm at the same resolution (the Jacquinot advantage). Apart from this principal advantage, another two advantages of a more technical and not so unique character are usually attributed to the Fourier spectrometers. The Fellgett advantage (multiplex effect) is associated with the fact that in instruments with photoelectric registration and spatial separation of optical frequencies, the spectrum is registered in the course of scanning by successively registering its elements. In the Fourier spectrometers, all spectral information continuously comes to the detector throughout the variation time of the optical path difference. This offers an advantage in statistics and improves the signal-to-noise ratio S/N. If one breaks down the spectrum into M intervals, so that each is measured successively in the course of spatial scanning (corresponds to the spectral width of the slit in slit instruments), then, with the recording time of a spectrum with a Gaussian noise being the same, the Fourier-transform spectrometer will afford the following advantage over its space-scanning counterpart:

(S/N )FTS 1 = M /2 . (S/N )S

(8.75)

The same result will be obtained if the time it takes to record a single element in the course of scanning is equal to the time required to record the interferogram. In that case, during the course of scanning the Fourier spectrum can be recorded M times. In practice, such situations occur when working in the medium wave and far IR regions of the spectrum. As one goes over to the visible and, the more so, the UV regions this advantage gradually vanishes. When registering the signal in the Fourier spectrometer, the detector is constantly illuminated by light of all wavelengths, and this increases the photon noise. This effect is more pronounced in short-wave radiation detectors. Further, in modern practice photoelectric detectors registering images, similar to photography, have become increasingly popular. Such detectors will apparently be developed further and used not only in the UV and visible, but also in the IR region. The Connes advantage. The greater the maximum optical path difference that can be attained in a Fourier spectrometer, the greater its advantages. In individual instruments, it reaches a few meters. The key point in spectrum reconstruction in experiment is the precision measurement of the position x of the mirror. The up-to-date high-speed computers and special algorithms allow the spectrum to be reconstructed from the interferogram P( x ), with increasingly ﬁner spectrum detail being revealed in the course of movement of the mirror. To implement this procedure,

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Pierre Connes introduced into the Fourier spectrometer scheme a laser frequency control, based on the counting of the number of interference fringes of its radiation. This made it possible to make the optical path ˚ and difference as great as 106 wavelengths at a discretization error of 7 A ensure a resolving power at a level of ca. 106 . All the advantages are greatest in the IR region. The present-day commercial Fourier spectrometers of medium class from 1 to 10 cm in optical path difference and 10−1 –10−2 cm−1 in resolution allow one to register, for example, spectra in the range from 2 to 30 microns in a time from a few seconds to a few minutes. 8.2.3.2 Interference Spectrometers with Selective Amplitude Modulation (ISSAM)

A schematic diagram of such a spectrometer is presented in Figure 8.12. Use is made here of the same principle as in the traditional Fourier spectrometer, but the Michelson interferometer employs diffraction gratings, 5, instead of mirrors. The interferogram is produced by beams with wavelengths selected by the gratings that propagate along the optical axes of the instrument. These wavelengths satisfy the autocorrelation condition. The spectral range is changed by the concordant rotation of the gratings. Thus, with the position of the gratings ﬁxed, interference modulation is effected within a narrow spectral range, and so there is no need to have and control great optical path differences. The optical path difference is varied by tilting compensating plate 4. The interferogram is simpliﬁed, and the computational part of the spectrum reconstruction problem is rendered easier. The light-gathering power remains as high as in the Fourier spectrometer (the Jacquinot advantage), but the spectral resolution is now governed by the gratings, and the spectrum recording sequence is similar to that in the slit-type diffraction instruments. For the latter reason, the realization of the Fellgett advantage is problematic, especially in the short-wave region, because the problem of the constant stray lighting of the detector and photon noise is incompletely removed here. Practice has also shown that in exchange for the lightening of the technical problems associated with large optical path differences in the traditional Fourier spectrometers, in the ISSAM-type instruments there arise problems of the same level of complexity, associated with the rotation of the gratings and modulator. The ISSAM-type instruments ﬁnd application in scientiﬁc research, but are less widespread than the traditional Fourier spectrometers.

8.2 Spectral Instruments

5 4

3

1

5

2 2

6 Figure 8.12 Schematic diagram of an interference spectrometer with selective amplitude modulation (ISSAM): 1 and 6 – entrance and exit apertures; 2 – lenses; 4 – beam-splitting and compensating plates; 5 – diffraction gratings in an autocollimation arrangement.

8.2.4 Raster Spectrometers

These are analogs of slit instruments wherein narrow spectral slits are replaced by more complex structures with a set of transparent and opaque areas in order to increase the light-gathering power, the total surface area of the transparent areas substantially exceeding the area of the ordinary slits. As with slits, the image of the entrance raster by light of the desired wavelength is constructed in the plane of the exit raster and can be shifted. If a dispersive system is placed between the entrance and the exit raster and other lines (or an arbitrary spectrum) are present along with the selected line, the exit raster allows light of complex spectral and spatial structure to pass through. As the image of the entrance raster is shifted to scan the spectrum, the signal received by the detector placed after the exit raster is modulated, and it is the character of this modulation that is used to reconstruct the radiation spectrum. Raster spectrometers are, in a sense, similar to both slit and ISSAM-type instruments. The difference being that the signal processing in instruments with interference modulation involves the harmonic analysis of the radiation spectrum, whereas in raster-type instruments the radiation spectrum is reconstructed by analyzing spatial harmonics.

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A slit instrument can be treated as a low-pass spatial ﬁlter. With the entrance and exit slits having the same width s, what remains at the exit from the monochromator is the information about the spectrum in the region ν < 1/s. This condition can be satisﬁed with the entrance and exit apertures of different shape. If use is made of a set of entrance and exit masks with holes s and greater in size, the same information can be obtained as with slits with a width of s, but at an advantage as to luminous ﬂux. Proceeding from these ideas, Girard analyzed a number of structures with ﬁgures described by various functions [6]. Figure 8.13 shows a Girard raster in the form of equilateral hyperbolas delimiting transparent and opaque (mirror) zones. When moving the image of the entrance raster relative to the exit one, a complex spatial structure (moire fringe pattern) is formed at the exit. The same ﬁgure presents a schematic diagram of a Girard raster spectrometer. Exit raster 2 is alternately illuminated by light reﬂected or transmitted by entrance raster 1. For a certain wavelength, the bright and dark zones of the image alternately coincide with the transparent and reﬂecting zones of raster 2, and the radiation proves to be completely modulated in intensity. For other wavelengths, the images are displaced because of the dispersion of grating 4, and the modulation depth is reduced. The farther the radiation wavelength from the one being selected, the smaller the period of the moire fringes, and if modulator 4 is used, the fringes are shifted to the right and left for half a period, so that the contribution from them to the intensity modulation turns out to be insigniﬁcant. Light of wavelengths other than those selected causes a constant stray lighting of the detector, which, as in the case of Fourier and ISSAM-type spectrometers, creates some difﬁculties when working in the short-wave region of the spectrum. In accordance with what has been said above, the resolving power of the raster spectrometer depends on the minimum size a of the period of the raster at its edge [22]: λ/δλ = ( Lλ f /1.48a2 )(dβ/dλ),

(8.76)

where L is the width of the raster, f is the focal length of the projection optics, and (dβ/dλ) is the angular dispersion of the grating. Typical of raster-type instruments is the critical character of the quality of the image-forming optics and mechanical stability. The ﬁrst Girard raster spectrometer used hyperbolic rasters 3 by 3 cm2 in area with a period of 0.11 mm at the edge. When a grating with 300 grooves mm−1 (205 by 135 mm2 ) was used, the resolution amounted to 0.1 cm−1 in the range 2000–3000 cm−1 . Compared to slit spectrometers of the same resolving power, the gain in the signal-to-noise ratio came to 130. The time required to record the vibrational–rotational spectrum of CO2 in the region

8.2 Spectral Instruments

a)

b)

c)

3 1

2

5

4

Figure 8.13 Schematic diagram of a raster spectrometer: 1 – entrance raster; 2 – exit raster; 3 – image modulator; 4 – diffraction grating; 5 – detector. (a) Hyperbolic Girard raster; (b) moir e fringe patterns formed when moving the image of raster 1 horizontally in the plane or raster 2.

of 15 μm was reduced by approximately a factor of 50 in comparison with that for a slit-type instrument (20 minutes and 15 hours, respectively). Later on spectrometers with functional rasters of approximately the same capabilities were (and continue to be) manufactured. Rasters of the above type are called functional. There is another type of raster, referred to as coded. Rasters of this type are designed in the form of a set of slits or other transparent and opaque elements whose position in the image-forming plane is speciﬁed in a certain way. When such sets are superimposed and shifted relative to each other, the transmission of the combination is either other than zero or altogether absent. These situations are encoded as 0 or 1. A number of encoding schemes were developed (multiple Golay slits, Hadamar matrices, etc.). For coded rasters, see [22]. 8.2.5 Acousto-optic Spectrometers

This class of slitless instruments makes use of the phenomenon of diffraction in the bulk of a liquid or solid exposed to ultrasound. The acoustic wave propagating in the substance produces local condensation and

433

434

8 Some Information on Spectroscopy Techniques

rarefaction zones that spatially modulate its refractive index and serve as a phase diffraction grating for the light waves propagating through the medium. The phenomenon of the scattering of light by a traveling acoustic wave was discovered and studied as the Brillouin scattering as far back as the 1920s. The sum of all the light scattering events on the individual condensation periods of the traveling acoustic wave can produce a noticeable resultant effect only if the total momentum of all the photons participating in scattering is conserved (the Bragg condition). In other words, the wave vectors of the incident radiation, scattered radiation and acoustic wave must satisfy the condition kin + q = ks

(8.77)

(according to modern terminology, the condition of phase synchronism in a parametric process). Subject to the Bragg condition, the magnitudes of the wave vectors are given by kin = 2πnin /λin ,

ks = 2πns /λs ,

q = 2π/Λ,

(8.78)

where Λ is the acoustic wave length and nin and ns are the refractive indices for the incident and scattered waves λin and λs , respectively. Bragg condition (8.77) deﬁnes the direction of the total diffracted wave, and this direction can be controlled by varying the sound frequency f = vq/2π, where v is the velocity of sound in the medium. This principle used the operation of the acousto-optic (AO) devices for modulating and deviating light beams. To enhance the intensity of the diffracted wave, it is necessary to form an extended grating of many periods Λ = 2πq L, where L is the light-sound interaction region. In practice, with L ≈ 1–10 cm, depending on the material, this corresponds to sound frequencies f ≈ 108 Hz, that is, much lower than the frequencies ν of light waves (ν ≈ 1014 Hz). If the medium is isotropic and nin = ns , then to discriminate between the directions of the diffracted rays differing in wavelength at the exit from the medium (angular dispersion) by the ordinary method of selecting, with a slit, a fragment of the spectrum in the plane of its image, all the rays in the incident light beam must be parallel up to an angular factor of Δθin /θin ∼ (λin ± λs )/λs . This is difﬁcult to achieve in practice, and it was apparently for this reason that the Bragg diffraction method in the acousto-optic version with artiﬁcially induced gratings (as distinct from diffraction by the natural atomic lattice of a crystal in the X-ray region) I for a long time was not discussed as a principle of operation of real spectral instruments. Progress along these lines proved possible when it was recognized that different wavelengths could be better discriminated

8.2 Spectral Instruments

Figure 8.14 (a) Collinear and (b) noncollinear acousto-optic interaction schemes; (c) schematic diagram of an acousto-optic spectrometer. P1 and P2 – crossed polarizers.

(ﬁltered out) not on account of the spatial separation of diffracted waves with almost equal propagation conditions in isotropic media, but rather on account of the more substantial difference in refractive index between the incident and diffracted waves in anisotropic media. This factor becomes clear if the incident and scattered waves differ in polarization, so that their refractive indices nin and ns are actually the ordinary and extraordinary indices no and ne . In that case, for the sake of simplicity we may put λin ≈ λs ≈ λ0 and write down the Bragg condition in the form kin = 2πne /λ0 ,

ks = 2πno /λ0 ,

q = 2π f /Λ.

(8.79)

It is satisﬁed if the vertices of the vectors kin and ks are located on the surfaces of their respective Fresnel ellipsoids, while the vector q closes the triad as before (see Figure 8.14 a,b). If the directions of all the vectors coincide, the scheme is then referred to as collinear (see Section 3.5). The realization of this spectral selection scheme is explained in Figure 8.14 c. The beam of nonmonochromatic light being analyzed passes through polarizer P1 . Assume that the polarization selected corresponds to the ordinary ray. On entering the crystal at an angle to the optical axis, this ray becomes a source of excitation of both ordinary and extraordinary waves polarized at right angles to each other [25]. In the absence of sound wave, the incident ordinary ray does not change its polarization. The cause of change is associated with photoelasticity giving rise to local variations of the Fresnel ellipsoid shape under the effect of sound.

435

436

8 Some Information on Spectroscopy Techniques

The incident radiation is scattered by the sound wave, the scattered radiation leaves the beam aperture and dies out, except for the radiation whose wavelength satisﬁes conditions (8.77), (8.78), and (8.79). And it is exactly thanks to the fact that the acoustic wave vector q, small as it is compared to the light wave vectors, closes the scheme that the conjugate scattered wave derives energy from the incident wave (parametric interaction) and propagates in the speciﬁed direction. At the exit from the crystal there is a polarizer P2 crossed with the entrance polarizer P1 . It selects the radiation of this scattered wave. The energy transformation coefﬁcient and transfer rate depend on the length of the crystal and the ultrasound power. These can be selected so that the incident wave will practically completely give up its energy to the scattered wave at the exit from the crystal. If the birefringence of the crystal is Δn = (ne − no ), it then follows from expression (8.79) that at a sound frequency of f the device selects from the incident radiation spectrum the wavelength λ0 = vΔn/ f = ΛΔn

(8.80)

that can be varied by changing the sound frequency. At Δn = 10−2 (for quartz, SiO2 , Δn = 0.008, and for paratelluride, TeO2 , Δn = 0.06), to select a wavelength of 500 nm, a 5 cm long crystal must contain an ultrasonic grating with 1000 periods. The analysis of the instrument function of the spectrometer shows that the width of the transmission band of the collinear ﬁlter [6, 7], Δν = Δλ/λ2 ≈ 1/LΔn,

(8.81)

is constant all over the spectrum. At Δn = 10−2 and L = 10 cm, the value of Δν amounts to 10 cm−1 , that is, in the visible region of the spectrum the resolving power of the spectrometer is R ≈ 103 . The light-gathering power of the spectrometer is at the same level as that of interference instruments and much higher than in slit instruments. For collinear schemes, the angular aperture is governed by the ratio between the longitudinal and the transverse size of the crystal and usually comes to a few degrees. In noncollinear schemes, the angular factor is higher (up to 10–12 degrees), because the phase synchronism condition is fulﬁlled within a substantial incidence angle interval. The analysis performed in [26–28] shows that the product of the angular factor into the resolving power, RΔΩ = πn2 ,

(8.82)

8.2 Spectral Instruments

Figure 8.15 Emission spectrum of an OSRAM metal-halide lamp registered with an acousto-optic spectrometer.

proves to be invariant for both collinear and noncollinear AO schemes. At n = 2.2 (TeO2 ), this quantity is ca. 15. For noncollinear schemes, it reaches ca. 30. The free spectral range of the AO spectrometers is determined by the working region associated with the transparency of the crystal. It is usually the range 0.21 μm, though there are also projects for the middle IR region [26–28]. Thus, acousto-optic spectrometers are instrument of high light-gathering power and moderate spectral resolution. Important merits of these instruments are their ﬂexible controllability via variation of the operating regime of the sound generator and automated measurement capabilities. Their large angular aperture allows them to be used for the analysis of spatial images of objects in various colors. Figure 8.15 presents an example of the spectrum of an OSRAM metal-halide lamp recorded with an AO spectrometer. To conclude, we present Table 8.4 listing comparative characteristics of typical spectral instruments of various classes and groups. The numerical data are approximate. The data for prism, diffraction and Fabry– Perot instruments (rows 1 through 3) [8] refer to the visible region of the ˚ and camera lenses with a focal length of f ≈ 1 m. spectrum, λ ≈ 5000 A, Those for acousto-optic spectrometers (row 4) also refer to the visible region. The reﬂection coefﬁcient of the interferometer mirrors is ca. 0.92. ˜ characteristic order m Indicated are the number of interfering beams, n, of spectrum, inverse linear dispersion dλ/dl, resolving power R = λ/δλ

437

438

8 Some Information on Spectroscopy Techniques

and resolvable interval δλ and the free spectral range (free dispersion re˚ For the Fourier spectrometers and ISSAM-type gion) Δλ = λ/m, A. instruments, the resolvable interval is retained throughout the working region, and is indicated in cm−1 . The data for the acousto-optic spectrometers (row 6) also refer to the visible region. The table also provides information on the so-called informational pa˜ rameter Q of the instruments, which is composed as the product Q = ΩR ˜ of the resolving power R and the light-gathering power Ω (allowing for the geometrical factor and transmission of the optics) in relative units. Following [8], we take Q = 1 to be typical of diffraction instruments. In that case, for the other spectrometers the parameter Q is the Jacquinot advantage. Table 8.4 Comparison between typical instruments. Instrument

n˜

˚ m dλ/dl R; δλ, A −1 or ˚ Amm δν, cm−1

˚ Δλ = λ/m A

˜ Q = ΩR

1 Prism, total length of prism bases 30 cm, dn/dλ = 500 cm−1

1

10

1.5 × 104 ; ˚ 0.3 A

0

Deﬁned by the working region

0.1

2 Diffraction. Grating 100 mm, 1000 groovesmm−1

105

5

2 × 105 ; ˚ 0.025 A

2

2500

1

3 FPI, Ln = 50 mm

∞, Neff = 30

0.01

6 × 106 ; ˚ 0.0008 A

2 × 0.025 105

4 Fourier spectrometer

2

5 ISSAM

2

6 Acousto-Optic

103

(1 to 106 6 10)×10 ; 1– 10−2 cm−1 (1 to 1–3 10)×106 ; 1– 10−2 cm−1 6 × 104 ; 1 5–20 cm−1

30–300

Deﬁned by the working region

30–300

Deﬁned by the working region

30–300

Deﬁned by the working region

10–30

8.3 Gas-Discharge Light Sources

8.3 Gas-Discharge Light Sources

The possibility of ﬁlling the discharge gap with various gas particles and widely varying their compositions and pressures, as well as the type of discharge using a pulsed or continuous current differing in power, has long been used to develop light sources for domestic, industrial, and scientiﬁc applications. Descriptions of the operating principles, parameters, designs, and operational speciﬁcities of a wide class of light sources can be found in a great many books and reviews, for example [6, 7, 29–33]. We will mainly mention here the existing gas-discharge sources, and also those currently newly developed, for uses in spectroscopy and optical systems. 8.3.1 Illumination Engineering Quantities

The most important characteristics in selecting and comparing between light sources are their spectra and photometric parameters. By established tradition, light source manufacturers characterize them by illumination engineering quantities. As already stated in Section 2.1, the illumination engineering units are the derivatives of the standard unit called the candle or candela, which is 1/60th of the luminous ﬂux emitted by 1 cm2 of the surface of the standard emitter along the normal to it into a unit solid angle. In spectroscopy using radiant energy quantities, this corresponds to the notion of radiant intensity (power). The alternate use of radiant energy and illumination engineering units is extremely widespread. Table 8.5 gives the names of some photometric quantities measured in illumination engineering and radiant units. Presented in some cases are different names of illumination engineering quantities widely used in practice [34]. Table 8.5 Illumination engineering and radiant energy units. Quantity

Illumination engineering units

Radiant units

Luminous intensity

candle (cd) = candela

W · sr−1

Luminous ﬂux

lumen (lm)

W

Quantity of light

lumen(lm) · s

J

Illumination

lux (lx)= lumen · m−2 = 10−4 phot (ph)

W · m−2

Luminance

nit =

=

candela · m−2

10−4

stilb (sb) =

W · m−2 · sr−1 10−4 π−1

lambert (L)

439

440

8 Some Information on Spectroscopy Techniques Table 8.6 Visibility functions. λ, nm

Φ(λ)

Φ (λ)

λ, nm

Φ(λ)

Φ (λ)

420

0.004

0.097

560

0.995

0.329

440

0.023

0.328

580

0.87

0.121

460

0.06

0.567

600

0.631

0.033

480

0.139

0.783

620

0.381

0.0074

500

0.323

0.982

640

0.175

0.0045

520

0.71

0.935

660

0.061

0.0003

540

0.954

0.65

680

0.017

0.00001

555

1

0.405

700

0.004

0

The above quantities are integral over the spectrum. Numerical correspondences between the spectral radiant and illumination engineering quantities are established not by constant coefﬁcients, but by functions. This is associated with the history of evolution of illumination engineering, when luminous quantities were evaluated and compared visually. In order to minimize the subjective factor, the so-called ‘visibility function’ was introduced, averaged over the perceptions of a great many observers, for both daytime, Φ(λ), and twilight, Φ (λ) (because of the change in the vision mechanism from cone to rod vision). Table 8.6 lists these functions [34]. By using the visibility function, one can compare between the spectral (i.e. referring to a narrow spectral interval) illumination engineering and radiant energy quantities. To this end, one should also draw on the experimentally established fact that a 1-lumen luminous ﬂux produced by a standard source in the region of the maximum of the daytime visibility function, λ = 555 nm, amounts to 1/683 W. In that case, for example, the spectral powers p(λ)[W] and p˜ (λ)[lm] are related together by means of the visibility function as p˜ (λ) = 683Φ(λ) p(λ) (see Section 2.1). To convert to integral quantities in illumination engineering units, it is necessary to integrate the spectral quantities with the weight Φ(λ): p˜ (λ) = 683

p(λ)Φ(λ) dλ

(8.83)

Let us present, for reference purposes, some typical values of illumination and luminance in illumination engineering units.

8.3 Gas-Discharge Light Sources Table 8.7 Illumination engineering characteristics in ordinary situations. Illumination from various sources

lx

Sun

Luminance of sources

Sun

nt 1.5 × 109

Direct rays in summer

105

Incandescent lamp ﬁlament

5 × 106

Cloudiness, open space

103

White surface under direct sun rays

3 × 104

By day in a light room

102 Moon

3 × 103

Moon Necessary for reading

2 × 10−1 30

8.3.2 Gas Discharges in an Envelope (Lamps) 8.3.2.1 Continuous-Discharge Lamps

Following [29], we divide continuous-discharge lamps into three types (Table 8.8). Gas-discharge lamps emit light as a result of optical transitions in atoms and molecules. Spectra are excited in a gas discharge. In luminescence lamps, the walls of the discharge chamber are coated with a phosphor. Fluorescence is excited by the discharge glow. In incandescent-electrode lamps, the light source is an electrode heated in a discharge. Listed for comparison in the bottom part of Table 8.8 are the parameters of incandescent lamps. These lamps have a continuous spectrum emitted by a heated element (a ﬁlament, coiled-coil ﬁlament, ribbon, etc.). The difference of this spectrum from Planck’s blackbody radiation spectrum (Chapter 1) is allowed for by introducing an approximate spectrum characteristic called the color temperature, Tc . If all the particles of the emitter have one and the same temperature T and its spectral radiances for two wavelengths are b(λ1 , T ) and b(λ2 , T ), the color temperature Tc is then equal to the temperature of a blackbody whose spectral radiances b0 for the same wavelengths meet the condition b0 (λ1 , Tc )/b0 (λ2 , Tc ) = b(λ1 , T )/b(λ2 , T ). It is usually taken that λ1 = 655 nm, λ2 = 470 nm(the red-blue ratio). In that case, the colors of the emitter and the blackbody will practically coincide in visual perception.

441

Filling, pressure, discharge

Arc, positive column; (a) low pressure, sodium vapor 0.013 Pa (0.1 mTorr); (b) high pressure, sodium vapor ca. 133.320 hPa (100 Torr), neon, xenon, argon additions

Arc; metals, elevated-pressure halides

Lamp

(1) Sodium

(2) Metal-halide

Table 8.8 Continuous-discharge lamps.

Similar to mercury lamp envelopes. Low-volatile elements combine and dissociate as halides in a closed cycle; in the case of alkali metals, the problem of the resistance of walls is eased

Two oxide-coated and an igniter electrodes; chemicaland heat-resistant envelope materials

Design

Varies from line to continuous Figures 8.18, 8.15

ca. 70% of light in 589- and 589.6 nm lines Figure 8.16

Spectrum

(a) gas analysis, polarimetry; (b) wide product range; indoor and outdoor illumination

Record-high, for gas-discharge sources, luminous efﬁciency (300– 400 lmW−1 ) (a) 5 cd; 5 × 104 nt;

50–80 lmW−1

(a) 10–15 W; (b) ca. 103 W

102 –103 W

Optical projection systems; illumination

Speciﬁcities, application

Luminous parameters

Power consumption

442

8 Some Information on Spectroscopy Techniques

Ditto, mercury vapor 10–150 atm

Hydrogen, deuterium, arc, 1.333 hPa (1 Torr)– 133.320 hPa (100 Torr)

(6) Hydrogen, deuterium

Filling, pressure, discharge Arc, glow; mercury vapor ca. (3 to 8)× 0.133 Pa (1 mTorr); inert gas 4.000 hPa (3 Torr)– 5.333 hPa (4 Torr) Arc; argon (a few Torr) for igniting; mercury vapor ca. 0.3–12 atm

(5) Mercury, ultrahighpressure

(4) Mercury, high-pressure

(3) Mercury, low-pressure

Lamp

Table 8.8 (continued).

Figure 8.20; special windows for the VUV region (quartz UV, Al2 O3 , LiF)

Figure 8.17. Short arc with initiation in a spherical thickwalled quartz bulb; capillary with water cooling

Straight twoelectrode quartz discharge tube

Two-electrode quartz discharge tube

Design

Self-reversed resonance lines at 185 and 253.7 nm, 313 and 366 nm; continuous spectrum up to the millimeter region Figure 8.19; as power and pressure are raised, the proportion of continuous spectrum increases; because of self-reversal, UV resonance lines are only slightly manifest 500–170 nm – continuous; 170–90 nmline, Figure 8.21

Mercury atom spectrum

Spectrum

Luminous parameters 103 –104 nt; 80 lmW−1

106 –107 nt 40–60 lmW−1

Spherical 5 × 107 –1.2 × 109 nt; 20–70 lmW−1 Capillary (2 to 4)×108 nt 60–70 lmW−1

Deuterium lamp is 30–50% brighter than the hydrogen one

Power consumption 2–60 W

10–103 W

5 × 101 –3 × 103 W

20–200 W

Secondary luminance standard. Light source in spectrophotometers

Projection and illumination equipment

Luminescence, Raman scattering, antibacterial procedures

Speciﬁcities, application Spectroscopy; antibacterial procedures

8.3 Gas-Discharge Light Sources 443

(9) Spectral, hollow-cathode

(8) Spectral, arc

(7) Spectral, low-pressure

Lamp

Cathode region; inert gas 0.133 hPa (0.1 Torr)–4.000 hPa (3 Torr); microadditions of metalcontaining substances or metals in the case of cathode erosion

Filling, pressure, discharge High-volatile metals (NaKRbHg, etc.), inert gas ca. 1.333 hPa (1 Torr). Heating and excitation in a glow or RF discharge Low-voltage arc (a) argon (a few Torr); dosed amount of metal (b) inert gas (a few Torr) without metal additions

Table 8.8 (continued).

Dismountable (Figure 8.23) and industrial (Figure 8.24)

Auxiliary igniter electrode

Glow discharge in a tube with two electrodes; electrodeless RF discharge ca. 1 GHz

Design

Doppler-broadened resonance lines for the majority of stable atoms

Doppler-broadened atomic lines (a) (Figure 8.22). Hg, Cd, Zn, Tl, Na, K, Rb, Cs (b) Ne, Ar, Kr, Xe

Atomic lines

Spectrum

20–30 W

Power consumption 101 –102 W

Typical of metals 5 cd, 104 nt;

Luminous parameters

Spectral analysis, calibrations. If optics transmits only UV radiation (black-light lamps), can be used to detect phosphors Atomic absorption analysis. Wide range of commercial lamps

Speciﬁcities, application Spectral analysis

444

8 Some Information on Spectroscopy Techniques

Luminescent (13) Mercury, low-pressure

(12) Gas, highand ultrahighpressure

(11) Gas, mediumpressure

(10) Neon, lowpressure (a) anode glow (b) extended

Lamp

Based on (3)

High-power RF discharge in inert gases 13.332 hPa (10 Torr)–133.320 hPa (100 Torr) Arc in inert gases, ﬁlling pressure 10– 20 atm

Filling, pressure, discharge 1.333 hPa (1 Torr)– 2.666 hPa (2 Torr); neon, helium, argon (mercury)

Table 8.8 (continued).

Phosphor on the inside surface of the wall

Similar to straighttube and sphericalbulb mercury lamps, but free from mercury additions

(a) Two closely spaced electrodes; luminescent anode layer; (b) widely spaced electrodes, positive column Capacitive discharge

Design

Phosphor emission bands with superimposed mercury lines (Figure 8.28)

Combination of a line and a continuous spectrum. (Figure 8.27)

Continuousdischarge in the VUV region. (Figure 8.26)

Predominantly neon lines, (Figure 8.25)

Spectrum

20–200 W; typical balance: 1. Visible light 20%, 2. Thermal radiation 25%, 3. Convection and heat conduction 55%

2–5 102 W

Power consumption (a) 10−2 –10 W (b) ca. 102 W

(4 to 8)×103 nt; up to 80 lmW−1

(1.2 to-1.5)×109 nt; 40–45 lmW−1 ; up to 5 × 104 cd; for xenon lamps Tc = (6.1 to 6.3)×103 K

Luminous parameters 102 –104 nt; 1 lmW−1

White-light lamps – widespread use; colored lamps – advertising, greenhouse illumination, back lighting

Speciﬁcities, application (a) allows modulation up to 2 × 104 Hz; voltage indication (b) illumination, advertising

8.3 Gas-Discharge Light Sources 445

Arc; (a) argon 13.332 hPa (10 Torr)–26.664 hPa (20 Torr); (b) ditto, with additions of mercury vapor Incandescent lamps (for comparison) (17) Domestic Bulb, spiral ﬁlament, vacuum or inert gas, nitrogen

(16) Tungsten

Filling, pressure, discharge (14) Arc, glow discharge; With inert gases inert gas 2.666 hPa (2 Torr)–4.000 hPa (3 Torr) With luminous electrodes (15) Point-source Short arc; argon lamps (zirco1 atm nium, thallium, etc.

Lamp

Table 8.8 (continued).

Luminous is the incandescent ﬁlament

20–200 W

2–20 W

Tc = 2600–2800 K

Continuous with maximum at 1 μm; metal and Ar lines

Glowing is a droplet of molten metal on the cathode 0.07– 5 mm in diameter (Figure 8.29) Incandescent is the tungsten cathode heated to 2800– 3200 A ca. 100 W

Ditto, with inert gas lines

Ditto

Power consumption Ditto

Continuous thermal W spectrum with small contributions from Ar and Hg lines

Spectrum

Design

General-purpose, domestic

Practically a point source for testing optical instruments; photometry Ditto

108 nt at a diameter of 0.07 mm and power of 2W ca. 107 nt

(1 to 3)×106 nt 101 –102 cd 8–20 lmW−1

Speciﬁcities, application Ditto

Luminous parameters (1 to 2)×103 nt; 30 lmW−1

446

8 Some Information on Spectroscopy Techniques

(18) Cineprojector, projection (19) Specialpurpose, with a narrow directivity diagram (OSRAM, Halospot)

Lamp

Ditto, with halogen additions

Filling, pressure, discharge Ditto

Table 8.8 (continued).

Ditto

Ditto, coiled-coil ﬁlament

Design

Power consumption 2 × 102 –5 × 103 W

101 –103 W

Tc = 3100–3300 K Tc = 2800–3300 K

Spectrum

Up to 5 × 105 cd 40 lmW−1

Luminous parameters 3 × 107 nt 30 lmW−1 Special lighting

Speciﬁcities, application Illumination equipment

8.3 Gas-Discharge Light Sources 447

448

8 Some Information on Spectroscopy Techniques

8.3.2.2 Pulsed-Discharge Lamps

Important for the operation of pulsed light sources are not only their photometric parameters, such as luminance and luminous intensity, but also their pulse duration and repetition frequency. These lamps are optimized together with their power supplies. Because the mercury lamps are of inertial character due to their vapor-pressure equilibrium dynamics, pulsed-discharge lamps are usually ﬁlled with inert gases. Like their continuous-discharge counterparts, they come in two basic conﬁgurations – tubular and spherical. The characteristic ﬁlling pressure of tubular lamps ranges between 13.332 hPa (10 Torr) and 133.320 hPa (100 Torr), that of spherical lamps being 1–120 atm. The envelope of spherical lamps with a small electrode separation is only required to be strong and transparent enough, its role in the formation of the radiation characteristics being insigniﬁcant. Tubular lamps may be straight, ring-shaped, helical and of special conﬁgurations. Compared to spherical lamps, they feature higher luminous efﬁciency, luminance and emitter locality. The characteristic power deposition in the plasma of pulsed-discharge lamps is, as a rule, much heavier than the power consumption of continuous-discharge lamps. Considering the general tendency (Chapter 1), the degree of deviation of plasma from its equilibrium condition diminishes with increasing pressure and power deposition. Therefore, instead of the full description of the spectrum of the essentially nonequilibrium emission of a continuous light source, for a high-power pulsed source, use is sometimes made of the approximate spectral characteristic called the color temperature, Tc . Table 8.9 lists the luminances and color temperatures of pulseddischarge lamps ﬁlled with various gases [29]. Table 8.9 Luminances and color temperatures of high-pressure pulsed-discharge lamps. Gas, pressure Xenon, 10 atm

Luminance, 1010 nt

Temperature Tc , 104 K

9

2.55

Krypton, 10 atm

11.5

2.84

Argon, 10 atm

14

3.4

Nitrogen, 2.5 atm

20

4.5

Neon, 2.5 atm

19

4.2

8.3 Gas-Discharge Light Sources

lλ, relative units

2

a)

1

0

lλ, relative units

2

b)

1

0

lλ, relative units

2

c)

1

0 Figure 8.16 Sodium lamp spectra [32a]: (a) low-pressure lamps; (b) and (c) high-pressure lamps with color temperatures of 2100 K and 2500 K, respectively.

Figure 8.17 Spherical mercury lamp.

449

lλ, relative units

lλ, relative units

8 Some Information on Spectroscopy Techniques

80

a)

60 40 20 0

300

400

500

600

15

λ, nm b)

10 5 0

300 400

450 500

550

600

λ, nm c)

lλ, relative units

450

00

50

0

400

500

600

700

λ, nm

Figure 8.18 Spectra of metal-halide lamps with various emitting substances [32a]: (a) NaTa, and In iodides; (b) Na. Sc, and Tc iodides; (c) DyHo, and Tm iodides and bromides.

8.3 Gas-Discharge Light Sources

0.4

1014 1128.7 1367.3 1529.5

a)

1014 1128.7 1367.3 1.7 μm

Spectral instrument slit width

b)

1014 1128.7 1367.3 1.7 μm

577 ; 579.1

690.75

546.1 577 ; 579.1

0.5

577 ; 579.1

546.1

546.1

491.6 ; 496

435.8 435.8 435.8

404.7 ; 407.8

404.7 ; 407.8

404.7 ; 407.8

lλ, relative units

1

2

c)

λ, μm

Figure 8.19 Spectra of mercury lamps at mercury vapor pressures of (a) 1 atm; (b) 20 atm; (c) 130 atm. Wavelengths in the ﬁeld of the ﬁgure are in nanometers.

Figure 8.20 Hydrogen lamp: 1 – bulb; 2 – UV window; 3 – anode; 4 – screening cylinder; 5 – discharge hole; 6 – cathode.

451

452

8 Some Information on Spectroscopy Techniques

Figure 8.21 Spectrum of a hydrogen lamp.

Figure 8.22 Spectral lamp: A – bulb; B – discharge tube: 1 and 2 – arc electrodes; 3 – igniter electrode connected via a high-ohmic resistor.

Figure 8.23 Dismountable hollow-cathode lamp: 1 – cathode; 2 – anode terminal; 3 – anode; 4 – seal; 5 – ﬁt ring; 6 – vacuum system union; 7 – window; 8 – adhesive.

8.3 Gas-Discharge Light Sources

Figure 8.24 Sealed hollow-cathode lamp: 1 – cathode; 2 – anode; 3 – ceramic insulator; 4 – centering disks; 5 – window.

Figure 8.25 Non lamp spectrum: 1 – 614 nm; 2 – 640 nm; 3 – 650 nm; 4 – 703 nm.

lλ, relative units

H2 1470Å

Xe 1236Å

Kr 1067Å

Ar

744Å

Ne He

600Å

500 25 20

1000 15

1500 10

9

8

Figure 8.26 Hydrogen and inert gas continua.

2000 7

λ, Å

6 hv, eV

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8 Some Information on Spectroscopy Techniques

Figure 8.27 Spectrum of a high-intensity tubular xenon lamp.

Figure 8.28 Typical spectrum of a gas-discharge luminescent lamp with phosphor excited at mercury lines.

Figure 8.29 Zirconium lamp: 1 – bulb; 2 – anode disk (the cut is made to show the cathode); 3 – pressed zirconium dioxide ZrO2 ; 4 – tungsten cup (the cut is made to show the ZrO2 ﬁller).

0.14

8 × 10−3

6 × 104

5 × 107

Ring-shaped

IFB-300

200

1.5 × 10−5

500

5 × 106

U-shaped

ISK-10

500

2.5 × 10−5

6 × 103

2 × 108

300

0.05

0.18

(b) 1

(b) 3.5 × 10−3

(b) 1.8 × 107

(b) 6 × 108

(b) 1.5 × 104

(b) Helical

Straight

(a) 0.017

(a) 6 × 10−3

(a) 3 × 109

(a) 8 × 104

(a) Spherical

ISP-70

0.15

2.5 × 10−5

(b) 4 × 106

(b) 1011

(b) 160 (a) 3.6 × 107

100

IFK-80000

(b) 0.7

2 × 10−3

(b)

(a) 0.06

(a) 4 × 10−3 6 × 10−6

Straight

ISX-500

(b) 400

(b)

6 × 105

5 × 108

(b)

(a) 1.5 × 106

(a) 1.3 × 109

(a) 2000

0.1

1.2 × 10−3

(a) 106

U-shaped

IFK-2000

2.5 × 105

7 × 108

120

Flash frequency, Hz

Flash duration, s

(a) 1011

U-shaped

IFK-120

Luminous intensity, cd

Luminance, nt

Flash energy, J

(a) 5

Design

Lamp

Table 8.10 Pulsed-discharge lamps.

104

3.6 × 107

1.5 × 108

–

–

–

3.6 × 105

(b) 108

(a) 5 × 106

104

Flash life, ﬂashes

8.3 Gas-Discharge Light Sources 455

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8 Some Information on Spectroscopy Techniques

Table 8.10 presents data on some commercial pulsed-discharge lamps manufactured in Russia. The lamp type designation comprises three letters meaning: I – pulsed, F – photoﬂash, S – stroboscopic, K – compact, P – straight, B – ring-shaped, X – spherical. 8.3.3 Open Light Sources 8.3.3.1 Continuous-Discharge Sources

Discharges glowing in open air played a great role in the development of electric lighting; however, these days they have been practically completely replaced by the more convenient enclosed sources (lamps). Arcs continue as light sources for spectroscopic analysis purposes. The high temperature ((3 – 7)×103 K) of the arc allows to be used simultaneously as an atomizer (decomposition of molecules into atoms) and a means of exciting atomic spectra. The samples under study can be introduced in the arc column or placed on the electrodes. Some inconvenience, especially when analyzing small amounts of substances, is associated with the presence of the lines of the electrode material (third elements) in the spectrum. This difﬁculty, however, can to a large measure be overcome by using arc discharges in gas ﬂows – plasmatrons. Plasmatrons with the so-called inductively coupled plasma (ICP) have proved most effective in spectral analysis. The operating principle of such an analytical source is illustrated in Figure 8.30. The plasma torch is surrounded by a few turns of an induction coil connected to a high-frequency generator. The plasma-forming gas, usually argon, is blown through the torch. Once discharge is initiated by means of an auxiliary electrode, the ﬂow is rendered conductive and starts playing the part of the secondary, relative to the transformer. The sample, in the form of ﬁnely dispersed powder or aerosol, is introduced through a narrow channel on the axis of the torch. While passing through the discharge zone, the sample particles get heated to (8 – 10)×103 K and are atomized, and the atomic spectrum is excited. The spectrum can now be observed in the zone beyond the highly luminous plasma. ICP devices have found widespread application, and are currently being manufactured in the form of automated analytical setups. 8.3.3.2 Pulsed-Discharge Sources

Best known are spark discharges resulting from the electrical breakdown of an insulator in an open space, their giant version occurring in the case of breakdown of the atmospheric air, that is, lightning. A developed form of spark discharge can be considered as a pulsed arc.

8.3 Gas-Discharge Light Sources

Figure 8.30 Plasmatron with inductively coupled plasma: 1 – observation zone; 2 – plasma; 3 – induction coil; 4 – plasmaforming gas; 5 – cooling gas ﬂow; 6 – sample aerosol.

Of greatest interest from the standpoint of spectroscopic applications are discharges along the surface of dielectrics, as well as those initiated by instabilities when electric current ﬂows through conductors or along the surface of partially conducting materials. They are characterized by relatively low breakdown voltages that, in addition, only weakly depend on the pressure and properties of the surrounding medium. The length of such discharges can be great enough, up to 1 m and more. Being a source of high-energy light ﬂashes, they are used for the optical pumping of laser media, in photochemistry and some other applications, and also as standard sources in spectroscopy. The efﬁciency of conversion of electrical energy into light amounts to ca. 30–50%. The radiance and color temperatures of the plasma of such discharges range between 10 and 50 kK. Let us mention some of these discharges. Well known are the so-called creeping discharges. If a small-sized electrode is brought close to the (hypothetically) top face of a dielectric plate and another, large-sized electrode of opposite polarity is applied to the bottom face of the plate, the discharge developing upon application of a high-voltage pulse to the former electrode will then start propagating from it over the top surface, the current circuit being closed by the tracking currents. The charges induced on the surface of the dielectric being of the opposite sense with respect to the charge carriers along the surface, the discharge proves to be pressed to the latter and skims over it, but the spark channels have no strict orientation and branch out. When the currents are high, these channels deform the surface of the dielectric, impressing the propagation paths. These are what is known as the Lichtenberg ﬁgures described by him as far back as 1777. The complex and

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8 Some Information on Spectroscopy Techniques

difﬁcult to control discharge trajectory make such sources inconvenient to apply, and therefore, a series of capacitively coupled electrodes with successive gap breakdown are sometimes pressed into the dielectric to specify the direction for the spark channel to develop. The electric burst of conductors – thin wires and foils – have until recently been a widespread method of initiating high-power emitting discharges. Being around 1 m long, such bursting conductors comprise simple and reliable sources with well-reproducible parameters at an energy of 200–500 Js−1 deposited within 5–10 μs. Their luminance is ca. 2 × 1011 nt. The main disadvantage of such sources is that they are onetime devices. If, as in the preceding case, the discharge vaporizes the material of the body near which it occurs and now develops in the vapor of this material, its spectrum is named the Lyman continuum after the investigator who observed such spectra during the course of a high-power capillary discharge. Based on this principle, high-power radiation sources were developed. To illustrate, the authors of [36] describe a source with a discharge in a capillary 2 mm in diameter and 10 mm long in a hard textolite. At a current of ca.9 × 104 A (3 × 105 Acm−2 ) plasma develops in the capillary with a pressure of up to 500 atm and an electron concentration of 1020 cm−3 , which emits a spectrum close to that of a blackbody with a temperature of 40 000±2000 K. Unfortunately, the spectrum can be reproduced only within 3–4 pulses, because of the deterioration of the capillary, after which it must be replaced. The so-called magnetically-pressed discharge generates a powerful Lyman continuum. The discharge propagates over the bottom of a rectangular recess in a dielectric. A magnetic ﬁeld applied parallel to the bottom of the recess and normal to the discharge current presses the discharge to the bottom, as a result of which the dielectric material is vaporized and enters the discharge zone [37]. An entirely different technical solution to the problem of developing a high-power continuous-spectrum source was suggested and investigated in [38, 39]. It is based on the observation that if a conductive strip is deposited on the surface of the high-ohmic NiZn(Fe2 O4 ) ferrite (resistivity 107 Ω · cm and a current pulse is passed through it, the pulse initiates a powerful emitting discharge, as in the case of bursting wire. An essential difference is that the pulse can be repeated without the conductive strip being deposited anew. This is because the discharge modiﬁes the surface layer of the ferrite. The modiﬁed layer is around 1 μm in thickness has an increased conductivity, ca. 105 Ω · cm, and becomes a fresh discharge initiation element. The next current pulse bursts this layer to initiate discharge and at the same time ‘prepares’ the initiation element

8.3 Gas-Discharge Light Sources

Figure 8.31 Photographs of burst-initiated discharges with speciﬁed trajectory on the surface of ferrite(a) rectangular meander; (b) initials of one of the authors of the work [39].

for the subsequent discharge. The possibility thus arises to effect a pulseperiodic discharge with a pulse resource of ca. (105 –106 ) pulses. The pulses reproduce both the luminous parameters and discharge geometry speciﬁed by the original conductive ‘primer’. It as proved possible to make such a primer not only by metal deposition, but also by much simpler methods. Figure 8.31 presents some examples of burst-initiated discharges on the surface of ferrite in the form of a meander and the initials of one of the authors of [39], which were set by simple penciling with subsequent heating with a small direct current. Two types of such a ‘high-ohmic’ ferrite have been discovered to date, namely, NiZn(Fe2 O4 ) indicated above and LiZn(Fe2 O4 ). Let us mention one more possibility to initiate high-power extended high-luminance discharges creeping over the surface of conductors. The difference of this type of discharge from the creeping discharge over the surface of a dielectric is explained by Figures 8.32a and b. When a dis-

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8 Some Information on Spectroscopy Techniques

Figure 8.32 Creeping discharge (CD): (a) over a dielectric surface (D) (arrows indicate tracking currents) and (b) over a metal surface (M) (arrows indicate conduction currents). A – anode, C – cathode.

charge propagates over the surface of a dielectric (a), the current circuit is closed, as mentioned above, by the tracking currents and when it creeps over the surface of a conductor (b), the circuit is closed by the conduction currents. Obviously the conductor must have a ﬁnite resistance in order not to completely short out the circuit. Among the substances that satisfy this condition, we can mention electrolytes or water containing a small amount of salt [40, 41], carbon and graphite ﬁbers and rods [42], and low-ohmic (ca. 5Ω · cm) NiMn(Fe2 O4 ) and ferrites [43].

8.4 Photodetectors

Photodetectors are intended for detecting electromagnetic radiation and measuring its radiant, temporal, spectral and spatial characteristics. The need to know these characteristics arises when solving many problems not only in spectroscopy, but also in sensing devices, navigation and target detection and tracking systems, computer engineering, security and control systems, and so on. These developments are progressing very dynamically. Nowadays, an investigator designing a spectroscopic experiment can select from a wide range of photodetectors classiﬁed in several groups and types. The operation of any photodetector is based on the conversion of the electromagnetic energy absorbed by its sensor into some other form (thermal, chemical, mechanical, electrical, etc.) which is more convenient to measure directly. There is at present a vast body of literature devoted to various photodetectors (see, e.g. [6–8, 44]), their technical design being constantly improved, and so here we will only restrict ourselves to brief explanations of the physical fundamentals of their operation.

8.4 Photodetectors

When staging a spectroscopic experiment, the choice among different photodetectors should be made on the basis of a comparison between their parameters and the parameters of the devices built around them. Photodetectors are classiﬁed by the physical operating principle of their sensors into two large groups – thermal and quantum (photonic). The thermal photodetectors rely for their operation on the change of some property of their sensor as a result of its temperature variation. To a ﬁrst approximation, they are insensitive to the spectral composition of the radiation being detected. The operation of the quantum photodetectors is based on the photoelectric effect in the electron-photon interaction. This directly causes the electrical characteristics of the sensor to change, and therefore such devices are also known as photoelectric detectors. According to the principle of localization of measurements, photodetectors are divided into single-channel (single-element) and multichannel (multielement) ones. In the former case, there is a single photosensitive element within whose limits the result of the action of light is integrated, no matter what the size of the element. In the latter case, there is a system of independent elements distributed in space. Each group includes a great number of various types of photodetectors implementing different physical light conversion mechanisms (e.g. through variation of electrical resistance on heating, via the effect of light on the spontaneous polarization of ferroelectric materials, etc.). Each type of photodetector can, as a rule, be designed in several different technical versions to optimize the effect. It is, therefore, important to have a limited set of parameters making it possible to compare between different photodetectors. 8.4.1 Parameters 8.4.1.1 Sensitivity

The integral sensitivity is the quantity S equal to the ratio between the variation of some quantity characteristic of the photodetector and that of a photometric quantity associated with the radiation incident on it. For example, the variation of electrical resistance consequent upon a variation of luminous ﬂux, S = ΔR/ΔP [Ω/W], or the variation of photocurrent in photoelectric effect, S = Δi/ΔP [AW−1 ], and so on. Use is most frequently made of the so-called volt-watt characteristic, S = ΔU/ΔP [VW−1 ], collectively characterizing the photodetector and its connecting circuit.

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8 Some Information on Spectroscopy Techniques

The spectral sensitivity Sλ is related to the integral sensitivity by the relation ⎛ ⎞⎛ ⎞ −1 λ

2

S = ⎝Sλm

Sλ Pλ dλ⎠ ⎝683

λ1

∞

Φ(λ) Pλ dλ⎠

.

(8.84)

0

Here Sλm is the maximum value of Sλ , Pλ is the power spectral density of the luminous ﬂux incident on the photodetector from a standard source (a tungsten lamp with a color temperature of Tc = 2850 K), Φ(λ) is the visibility function, 683 lm/W is the conversion factor between radiant and illumination engineering quantities (the standard source is certiﬁed in illumination engineering units, Section 8.3.1) and λ1 and λ2 are the boundary wavelengths for the sensitivity range of the photodetector or the transmission region of the optical system. 8.4.1.2 Noise

The magnitude of an actually measured signal always ﬂuctuates with time. If these ﬂuctuations are not speciﬁed in a regular form, but occur with a random amplitude, phase and frequency, they are then referred to as noise. The origin of some kinds of noise is known. Thermal noise (Johnson noise) is associated with the ﬂuctuations of the number of photons emitted by any body at a temperature of T > 0. Contributing to noise is not only the source of radiation being detected, but also the detector itself, the optical elements used, and so on. Dark noise results from the chaotic thermal motion of electrons. At any instant, the number of electrons moving in a conductor in a given direction is not equal to that of the electrons moving in the opposite direction. Such noise occurring in the photodetectors themselves in the absence of irradiation, is referred to as shot noise, which stresses the discrete character of electric charges. In photodetectors relying for their operation on the intrinsic photoeffect, this noise is called generation-recombination noise. In thermal photodetectors, it results from spatial-temporal temperature ﬂuctuations. Photon noise or radiation noise is due to the ﬂuctuations of the number of photons arriving at the photodetector in a unit of time. The root-meansquare deviation Δn˜ from the average number n of events is ˜ Δn/n ∼ n− /2 . 1

(8.85)

The above types of noise have a broad spectrum of frequencies f , their bandwidths exceeding those of the registering devices. Their spectral density is uniform and is ca. f 0 .

8.4 Photodetectors

Low-frequency noise, in contrast to the above-mentioned types of noise, has its spectral density ca. f −1 . Included in this group are differently named types of noise differing in origin. In photodetectors using the effect of variation of electrical resistance under irradiation, the noise is due to random changes in resistance as a result of various contact and surface phenomena and is called current noise. In photodetectors with photoemission, it results from the ﬂuctuations of the emissivity of the photosensitive element in time and space and is referred to as jitter or ﬂicker noise. The typical frequency region wherein this noise dominates over the other types of noise is f < 100 Hz. 8.4.1.3 Effective and Ultimate Sensitivity

The signal of power PS incident on the photodetector contains a noise component of power PN . The transformed signal at the output of the photodetector includes a useful and a noise component, QS and QN , respectively. The ratio between the signal-to-noise ratios at the input and output of the photodetector is η = ( PS /PN )/( QS /Q N ) .

(8.86)

The ratio P/Q is the reciprocal of sensitivity, and if it is retained for both the useful and the noise component, the photodetector then reproduces the signal-to-noise ratio of the input signal. Such a photodetector is called ideal, and for it η = 1. A real photodetector, of course, has intrinsic noise of its own that adds to the input signal noise, and for such a photodetector, η < 1. Relation (8.86) can be understood to mean that the quantity η indicates how many times the number of photons incident on the ideal photodetector is smaller than that incident on the real photodetector, the signal-to-noise ratio in the registered signal being the same. Based on this interpretation, the ‘imperfection’ coefﬁcient η is sometimes called the effective quantum efﬁciency. The threshold sensitivity Pthr of a photodetector indicates the power of the light signal at its input at which the amplitude of the signal registered at the output is equal to that of the intrinsic noise of the detector. The spectrum of noise being wide, it is expedient, in the case of photoelectric detection, to modulate the light signal with a frequency of f and measure it within a comparatively narrow band of Δ f , thus avoiding the effect of noise in the rest of the spectrum. The criterion for the detectability of a light signal is taken to be the equality between the signal at the output of the photodetector and the root-mean-square noise signal, with the modulation frequency being f and the detection band, Δ f = 1 Hz. ¯ 2 ∼ Δf, If the output signal is measured in the form of the voltage U N 1 ¯ UN ∼ Δ f , then the dimension of the threshold power per unit band

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8 Some Information on Spectroscopy Techniques

is [W · Hz−1/2 ]. This quantity is sometimes called the noise equivalent power (NEP). To certify a photodetector, use is made of a standard light source with a spectrum close to that of a blackbody with a ﬁxed temperature, the temperature of the standard source, the modulation frequency and the detection bandwidth used in the certiﬁcation being indicated in the certiﬁcate. A photodetector is frequently characterized by the reciprocal of Pthr , namely, D = ( Pthr )−1 [HZ /2 W−1 ] , 1

(8.87)

called the detection capability. Since, approximately, Pthr ∼ A1/2 , where A is the light-sensitive area, use is made of the reduced detection capability √ (8.88) D ∗ = A/Pthr . 8.4.1.4 Inertia

The reaction of a photodetector to a light signal is not instantaneous and is characterized by a time constant of τD . During this period the signal at the output of the photodetector reaches a value that differs from its stationary value by an amount of 1/e, what is, 0.63 of the stationary value. Assuming that the rise and fall of the signal are described by exponentials with one and the same quantity t/τD in exponent, the dependence of the sensitivity of the photodetector on the modulation frequency f M , with the signal varying harmonically, is given by the frequency characteristic S( f M ) = 1

S (0) 1 + (2π f M τD )2

,

(8.89)

where S(0) is the stationary value of the signal. At 2π f M τD ≤ 1, the photodetector sensitivity is independent of the modulation frequency, S( f M ) = S(0). In the opposite case, S( f M ) ∼ 1/ f M . At f M = 1/2πτD , S( f M ) = 0.71S(0). 8.4.2 Main Types of Single-Element Detectors

8.4.2.1 Thermal Detectors

Thermoelements make use of the thermoelectric (Seebeck) effect – the development of electromotive force (thermo-emf) on the heating of one of the two junctions of two dissimilar conductors (thermocouple). Thermocouples for metal thermoelements are composed of Cu, Al, Ar, Bi, and

8.4 Photodetectors

constantan wires. Typical thermo-emf values for a single thermocouple equal (5–10) × 102 μm · deg−1 . To increase the emf, two thermocouples are assembled together, and to reduce noise and compensate for the temperature ﬂuctuations of the medium, the assembly is placed in a vacuum, with the adjacent junctions being connected in such a way that only one of the couples is exposed to radiation. In that case, it is only the temperature ﬂuctuations of the surrounding medium that are compensated for, while the radiation remains uncompensated and can be measured. The predominant Johnson noise here is approximately 1.5 times that for a resistor of the same resistance, but thermoelements are always operated at room temperature, for cooling reduces their thermo-emf. The working region of the spectrum is mainly restricted by the optical materials used. Bolometers. The operation of these devices is based on the change of the electrical resistance of their thermoelement on being heated by radiation: ΔRB /RB ≈ β · ΔT. The thermoelement can be made of metal or semiconductor. Semiconductor bolometers are also known as thermistors. Typical for metals is β ≈ 1/T, while for semiconductors, β ≈ −3000/T 2 . The quantity β reaches ca. 0.5%/deg in metal bolometers and ca. 4%/deg in semiconductor ones. A sensitivity as high as β ≈ 5000%/deg can be provided by superconducting bolometers. Superconductivity is attained at the superconducting transition temperature Tst . In niobium nitride, for example, Tst = 14.36 K. Bolometers are connected in a bridge circuit and supplied with direct or alternating current. Predominant in metal bolometers is thermal noise, current noise being prevalent in semiconductor ones. The highest signal-to-noise (S/N) ratio is achieved with the luminous ﬂux modulated with a frequency of 20 Hz. At a modulation frequency of 25 Hz, the S/N ratio is reduced by a factor of 1.2–1.5. Pyroelectric detectors. The operating principle of these detectors is based on the ability of crystals which have no center of symmetry, ferroelectric crystals in particular, to generate surface charges when being deformed. In pyroelectric detectors, the deformation results from heating by radiation. The crystals used include BaTiO3 , LiNbO3 , Seignette salt, (NH2 CH2 OOH)3 · H2 SO4 , and some others. The weak wavelength dependence of sensitivity, typical of thermal detectors, is combined here with a high speed of response, because, as distinct from the other thermal photodetectors, the effect is governed not by the amount, but by the rate of change of temperature. For this reason, pyroelectric detectors operate only with pulsed or modulated light signals. If the thermal conduction time of the crystal is τTh , the optimal signal modulation is that with which the light pulse duration is shorter than τTh , while the pulse period is several times longer than τTh . Pyroelectric detectors come in

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many design versions, with the light-sensitive area ranging from 0.1 to 900 mm2 . They operate in open air and are insensitive to temperature (−60 to +100 ◦ C) and other climatic factors. The Johnson noise is predominant. Optico-acoustic detectors. If the radiation incident on an absorbing surface heats it up, this causes the temperature and pressure of the adjacent gas layer to in crease. An optico-acoustic detector (OAD) is a gasﬁlled chamber with a sidewall in the form of a thin ﬂexible membrane. The radiation to be detected enters into the chamber and is absorbed there by a darkened plate, and the resultant disturbance of the gas ﬁlling the chamber manifests itself as the deﬂection of the membrane. The amount of deﬂection is measured by means of a light beam reﬂected off the membrane, usually in a raster scheme. Trade-off between sensitivity and speed of response is attained by appropriately selecting the absorbing plate and the gas. For good response, it is desirable that the plate should have a low heat capacity and the gas, high heat conductivity, but sensitivity in this case will not be very high. For example, when the chamber of an optico-acoustic detector is ﬁlled with the heat-conducting helium gas, its time constant amounts to ca. 10−3 s, and when it is ﬁlled with the heavy xenon gas, the time constant is ca. 3 × 10−2 s, but the sensitivity of the detector is almost one order of magnitude higher, reaching 4 × 104 VW−1 . Optico-acoustic detectors can be made selective. In that case, radiation is absorbed not by the plate, but by the gas itself, and on resonance lines at that, so that the detector responds only to radiation of the appropriate resonance wavelengths. Predominant in optico-acoustic detectors are mechanical and thermal noise. 8.4.2.2 Photoelectric (Quantum, Photonic) Detectors with Extrinsic Photoeffect

These photodetectors rely for their operation on the classical photoelectric effect – the liberation of electrons by light incident on a photocathode. The crystal lattice of the photocathode material takes no part in the electron-photon interaction, and so the incident luminous ﬂux does not raise the temperature of the detector. The time it takes for a photon to detach an electron is estimated at ca. 10−14 s, and though inertia in actual photoelectric detectors is higher, their time constant, ca. (10−9 –10−10 ) s, is much smaller than in their thermal counterparts. The measure of the incident luminous ﬂux is the current of the emitted electrons. Another difference of photoelectric detectors from thermal ones is their higher wavelength selectivity, owing to the so-called long-wavelength cutoff or photoelectric (photoemission) threshold. By virtue of this fact, photoelectric detectors are used in the region of wavelengths shorter than 1.5 μm.

8.4 Photodetectors

The photocathodes of modern photoelectric detectors are manufactured in the form of semiconductor layers on a metal or dielectric surface. As a rule, the emissive capacity of the cathode is determined by a thin layer of cesium atoms on the surface of this structure. Figure 8.33 presents the wavelength dependences of the quantum yield ηλ (the number of electrons liberated by a single photon) and sensitivity Sλ , mAW−1 , for various photocathodes. Photocathodes can be roughly classiﬁed according to their spectral properties in three groups: (i) infrared (oxide), threshold at ca. 1.3 μm, (ii) ultraviolet-visible (antimony-cesium and multialkali), threshold at 650–850 nm, and (iii) solar blind photocathodes (metals or simple binary compounds, such as, MgF2 , KBr, and CsJ), insensitive to the visible or even near-ultraviolet radiation. The latter are convenient to use in the short-wavelength UV region, considering, in addition, that radiation in the region of λ < 300 nm is absorbed by ozone, and so one can do without sunlight protection. Photocathodes of negative electron afﬁnity (epitaxial ﬁlms doped with semiconductors the type of A I I I BV )) have the highest sensitivity and the most distant photoemission threshold. The simplest photocathode-based device is the photocell. This is, as a rule, a vacuumized spherical glass bulb 20–50 mm in diameter. The photocathode is deposited on the inner surface of the bulb, except for its transparent portion intended to receive radiation. The ring-shaped anode is at the center of the bulb. The time constant of photocells is ca. 10−8 s. As in the other photoelectric detectors, shot noise is predominant here. A coaxial photocell is an improved version of the above device. The anode here is a grid placed parallel to the cathode at a distance of ca. 1 mm from it and having two terminals for connection to a matched coaxial line. At a voltage of ca. 2 kV, the original velocity spread of photons, which affects the time constant of the cell, is now unimportant. The time constant of coaxial photocells ranges between 10−10 and 10−11 s. Photoelectric multipliers (photomultipliers) differ from photocells by the presence of additional electrodes (dynodes). The dynodes (10 and more) are intermediate electrodes between the anode and cathode, their potential being progressively increased in a stepwise fashion. The electron-optical system of the device successively focuses the electrons emitted by the cathode onto the dynodes whereon the electrons multiply as a result of secondary emission, thus increasing the photocurrent by a factor of 106 –107 and improving the sensitivity of the device. The time constant of photomultipliers depends on the time it takes for the electron avalanche to develop and amounts to ca. 10−8 s. The voltage drop across a photomultiplier can be as high as ca. 1 kV, sometimes even higher. The

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Figure 8.33 Spectral sensitivity Sλ and quantum yield η of photocathodes [1]: 1 – Cs3 Sb; 2 – Ag−O−Cs; 3 – Na2 KSb−Cs. Solar blind photocathodes: 4 –Cs2 Te; 5 – Rb2 Te. Dielectric cathodes: 6 – CsI; 7 – KBr. Materials of negative electron afﬁnity: 8 – GaAsPCs; 9 – GaAsCs; 10 – InGaAsPCsO. 1a, 3a, and 5 – Al2 O3 windows; 4, 6, and 7 – LiF windows.

ultimate sensitivity of these devices is affected, apart from shot noise, by the dark current (in the absence of illumination). At low voltages (up to 500 V), the dark current is mainly contributed to by the leakage current. In the region of the working voltages (500–1000 V), the contribution to the dark current and noise comes from the thermionic emission of the photocathode. At high voltages (over 1000 V), the dark current grows in magnitude and instability due to autoelectronic emission and ionic and luminescence feedbacks (the ions and luminescence photons produced in the accelerating gap cause additional emission of electrons from the photocathode). Figure 8.34 [45] shows the threshold sensitivity Pthr of some types of Russian-made photomultipliers as a function of the radiation wavelength. Photomultipliers can be operated in a photon-counting mode. At an electron multiplication factor of ca. 108 , even a single photon yields a measurable current at the output. With amplitude discrimi-

8.4 Photodetectors

Figure 8.34 Spectral sensitivity threshold of some photomultipliers: 1 – SbCsO (PM-70); 2 and 3 – SbNaKCs (PM-51 and PM88); 4 and 5 – SbCs(PM-64 and PM-17); 6,7, and 8 – AgOCs (PM-62, PM-22, and PM-28).

nation used, the measuring circuit can only allow photocurrent pulses to pass whose amplitude is higher than the noise level. Apart from the capability of handling weak luminous ﬂuxes, this mode has the advantage of wider dynamic range (ca. 106 ) and higher relative measurement accuracy (up to 10−6 ), governed by the counting statistics, over the analog mode. Secondary-electron multipliers, also known as channel photomultipliers, differ from photoelectric multipliers mainly by the following two features: (i) the dynode system of secondary-electron multipliers is not discrete, but distributed and (ii) these devices are of open type and can also operate in the air. The distributed dynode system comprises a channel 1–2 mm across coated on the inside with an oxidized photoemissive layer. A high voltage (ca. 500 Vcm−1 ) is applied across the ends of the channel. A photoelectron entering into the channel repeatedly collides with its sidewalls, thus producing an electron avalanche. To reduce feedbacks, the channel is curved. The photoelectron multiplication factor reaches 108 . With the current ampliﬁcation being so high, a space charge is formed at the end of the channel, which makes the signal being measured perceptibly nonlinear. On the other hand, high ampliﬁcation facilitates the operation of the device in a photon counting mode. Channel photomultipliers are convenient to use in the VUV region of the

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spectrum, wherein problems exist with the transmission of the entrance window. 8.4.2.3 Photoelectric Detectors with Intrinsic Photoeffect

The presence of the long-wavelength cut-off for the emission of electrons into the surrounding medium restricts the spectral range of photoelectric detectors relying for their operation on the extrinsic photoeffect. However, electric conductivity in a solid can also develop under the inﬂuence of ‘softer’ photons causing electrons to transit from the bound state in the valence band or from impurity levels in the forbidden band to the conduction band. This process if often called the intrinsic photoeffect. Most suitable for use in the capacity of photodetectors are semiconductors with a narrow band gap. For this reason, semiconductor photodetectors are especially important for the IR region, where the other photoelectric detectors are inoperable. These devices are classiﬁed in two classes – photoresistors and photovoltaic detectors (photodiodes). A photoresistor is a semiconductor chip or ﬁlm connected in a circuit with an emf (usually 10–100 V), similar to a semiconductor bolometer. The difference is that the conduction band in the semiconductor bolometer is ﬁlled with thermal electrons, whereas in the thermoresistor this is an undesirable effect. Therefore, to detect the purely quantum effect in photoresistors of band gap comparable with or smaller than the average energy of the particles at room temperature, these devices are cooled to the temperature of liquid nitrogen and even lower. Because a voltage is applied across the photoresistor, photoelectrons are accelerated and can produce fresh conduction electrons via impact ionization. The photocurrent ampliﬁcation depends on the voltage across the photoresistor and the design of the latter and can reach ca. 105 . The photoelectron multiplication effect makes the signal nonlinear at high luminous ﬂuxes. The time constant of photoresistors ranges from 10−2 –10−6 s for pure semiconductors to 10−6 –10−9 s for doped ones. Their sensitivity is limited by dark (generation-recombination) and current noise. Figure 8.35 presents the detection capability curves of some photoresistors in comparison with those of thermal photodetectors. A photodiode is an element relying for its operation on contact phenomena. It consists of p- and n-type semiconductors with a p − n-type intermediate layer in between. In the absence of external perturbations, electrons and holes in both the p- and the n-region of the system are in equilibrium. When the intermediate layer is illuminated, the photons whose energy is high enough for the intrinsic photoeffect produce extra electron–hole pairs. Under the action of the contact potential difference,

8.4 Photodetectors

Figure 8.35 Detection capability D∗ [cm · Hz1/2 W−1 ] of some thermal photodetectors and photoresistors: 1 – PbS(300 K); 2 – InSb(77 K); 3 – HgCdTe (230 K); 4 =- InSb (300 K); 5 – GeZn (4 K); 6 – GeCu(4 K); 7 – GeAu (77 K); 8 – GeHg; 9 – thermistor (300 K); 10 – thermopile; 11 – LiTaO3 pyroelectric detector; 12 – OAD; 13 – superconducting bolometer.

the electrons diffuse into the p-region, while the holes, into the n-region, which causes an emf to develop. If it is this emf or the photocurrent that is being measured, this operating regime of the photodetector is called the photovalve mode. The element can be connected into a circuit with an external emf, so that the voltage proves to be blocking (with the positive pole being connected to the p-region and the negative pole, to the n-region). In the absence of illumination, in the circuit ﬂows a weak dark current. When the device is illuminated, the induced emf reduces the contact potential barrier, and the current increases. Such an operating regime is referred to as the photodiode mode. The photodiode mode has the advantage of a higher speed of response and better linearity of the signal. The disadvantage is the presence of the dark current and noise produced by it. Most widespread are germanium and silicon photodiodes whose spectral and frequency characteristics, Sλ (in relative units) and S( f M ), respectively, are presented in Figure 8.36. 8.4.2.4 Photoemulsion

Photographic emulsion is the oldest photodetector with a wide variety of merits. This line of registration of light continues to be actively devel-

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Figure 8.36 Spectral and frequency characteristics, Sλ and S( f M ), of (1, 1a) silicon and (2, 2a) germanium photodiodes.

oped and so far has no alternative in some applications. In this section, we will consider it as a single-element photodetector. The central component of photoemulsion is a suspension of AgBr crystals with a band gap of 2.5 eV. The bromine atom is electronegative. An incident photon detaches an electron from Br – and transfers it to the conduction band. The motion of the electron within the emulsion produces a chain of results, ending with the formation of latent image elements in the form of clusters of silver atoms. One can, therefore, classify photoemulsion-based photodetectors with quantum photodetectors with intrinsic photoeffect. The subsequent development operation can be likened to ampliﬁcation, and ﬁxation, to the recording of information for storage. The procedure for measuring the luminous ﬂux in photography is well known [6–8]. The photomaterial is exposed to a narrow light beam to ﬁnd the so-called ‘darkening’ D = log( P0 /P), where P0 and P are the powers the light beam has after passing through exposed and unexposed areas of the material. The darkening is next related to the illumination E and exposure time t. This is done experimentally and the darkening is expressed in the form of a darkening curve whose characteristic shape is shown in Figure 8.37. The linear section of the curve (normal darkening region) may be approximated by the expression D = γ[log H − j] ,

(8.90)

8.4 Photodetectors

Figure 8.37 Darkening D and quantum yield η of photoemulsion as a function of exposure H .

where γ = tan β is the emulsion contrast, H = Et p is the exposure, p is the Schwarzschild constant and j is the inertia of the emulsion. The ﬁgure also shows the quantum efﬁciency η (8.86) of the emulsion, where the part of noise is played by the fog of the unexposed areas. The efﬁciency strongly depends on the exposition and is not very high, reaching a maximum of η ≈ 0.01 at the beginning of the normal darkening region. It can be increased to η ≈ 0.04 by a special treatment. The spectral sensitivity curves of various emulsions can be found in [6–8]. Most photoemulsions require 108 –1010 quanta cm−2 to have a darkening of D = 0.2 in the maximum sensitivity region. This is taken to be the sensitivity threshold. 8.4.2.5 Comparative Characteristics of Single-Element Detectors

Table 8.11 lists the main parameters of single-element detectors.

473

Optics transmission Ditto Ditto

Spectral range, μm

Photocell (300) Coaxial photocell (300)

0.2–1.1

Photoelectric with extrinsic photoeffect

Optico-acoustic (300) Bolometers metal Bi (300) semiconductor (300) Superconducting NbN (15) carbon (2.1) Ge (0.4) Ge and Si (1.2) Pyroelectric

Thermoelement (300)

Thermal

Detector type, (working temperature, K)

20–200 μAW−1

102 –106 VW−1

5–20 V/W 4 × 104 VW−1 107 VW−1 103 –106 VW−1

Integral sensitivity, VW−1 ; AW−1

Table 8.11 Comparison of photodetectors (typical parameters). Maximum sensitivity, μm

10−4 –10−8 10−10 –3 × 10−11

1011 (1000)

109

1.7 × 109 (10) 1.5 × 109 (6) (1 to 3)×108 (10) 4.8 × 109 (360) 4.3 × 1010 (13)

(2 to 3)×10−2 2 × 10−2 2 × 10−6 5 × 10−4 10−2 10−6 10−6 10−9 –10−11

2 × 109 (5)

Detection capability ∗ Dmax W−1 Hz1/2 cm (modulation frequency, Hz)

(1 to 3)×10−2

Time constant τD s

5 × 10−10

Up to 6 × 10−16 Up to 2 × 10−16

Equivalent noise (NEP), W · Hz−1/2 ; dark current ∗) , A, and resistance ∗∗) , Ω

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8 Some Information on Spectroscopy Techniques

CdSe (300) CdS (300) Ge : Au(77) Ge : Cu (4.2) Ge : Hg (30) Ge : Zn (77) Ge : Zn (4.2) InSb (300) InSb (77) PbS (300) PbS (77)

0.4–1.2 0.4–0.9 2–12 3–30 2–15 2–15 7–40 2–7 1–6 0.6–3 0.6–3.2

Cathode Anode∗) 0.5– 1.4 0.3–0.7 0.3–1.1 0.2–1

Photomultipliers, photocathode AgOCs (300) SbCs (300) SbCs(O) (300) SbKNaCs (300)

Photoresistors

Spectral range, μm

Detector type, (working temperature, K)

Table 8.11 (continued).

105 VW−1 105 VW−1

20 kVW−1

1 kAW−1

20–400 AW−1 2–7 kAW−1

40 mAW 0.7 kAW∗) 35 mAW 6 kAW∗) 130 mAW−1 70 kAW−1 ∗) 110 mAW−1 0.7 kAW−1 ∗)

Integral sensitivity, VW; AW−1

0.7 0.6 5.0 25 11 12 36 6.0 5.0 2.1 2.5

0.7 0.4 0.4 0.4

Maximum sensitivity, μm

2 × 1011 1011 2 × 1010 (900) 3 × 1010 (900) 4 × 1010 (800) 3 × 1010 (800) 1010 (800) 5 × 107 (800) 6 × 1010 (900) 3 × 1010 (90) 1011 (90)

1012 –1013 (1000) (1 to 5)×1015 (1000) (1 to 2)×1015 (1000) (3 to 5)×1014 (1000)

10−7 10−8 10−8 10−8

10−2 10−2 3 × 10−8 3 × 10−9 10−9 10−9 10−8 5 × 10−8 10−7 (1 to 4)×10−4 (1 to 4)×10−4

Detection capability ∗ Dmax W−1 Hz1/2 cm (modulation frequency, Hz)

Time constant τD s

5–10∗∗) 2 × 103∗∗) (0.5 to 10)×106∗∗) (0.5 to 10)×106∗∗)

5 × 106 ∗∗) 107 –1010∗∗) 4 × 105∗∗) 2 × 104∗∗) 4 × 104∗∗)

5 × 10−9∗) S3 × 10−9∗) 4 × 10−9∗) 4 × 10−10∗)

Equivalent noise (NEP), W · Hz−1/2 ; dark current ∗) , A, and resistance∗∗) , Ω

8.4 Photodetectors 475

Ge (300) Si (300) InAs (300) InAs (77) GaAs (300) Se-SeO (300)

0.4–1.9 0.5–1.2 1–4.5 1–4 0.3–0.95 0.4–0.8

1–4 1–6 1–6 0.4–1.2 8–13

PbSe (300) PbSe (77) PbTe (77) Tl2 S (300) HgCdTe (77)

Photodiodes

Spectral range, μm

Detector type, (working temperature, K)

Table 8.11 (continued).

10 AW−1 5 AW−1 1 AW−1 3 × 103 VW−1 1 AW−1

106 VW−1

Integral sensitivity, VW; AW−1

1.6 1.0 3.4 3.0 0.85 0.69

3.4 4.5 3.8 0.9 10

Maximum sensitivity, μm

5 × 1012 (90) 3 × 1012 (90) 3 × 109 (90) 1011 (90) 1010 (90) 1011

3 × 109 (90) 1010 (90) 1.4 × 1010 (90) S2 × 1012 (90) 9 × 1010 (900)

4 × 10−6 4 × 10−5 (1 to 3)×10−6 5 × 10−4 10−4

5 × 10−6 10−6 2 × 10−6 10−8 10−3 2 × 10−6

Detection capability ∗ Dmax W−1 Hz1/2 cm (modulation frequency, Hz)

Time constant τD s

10−5∗) 2 × 10−7∗)

5–50∗∗)

107∗∗) 107∗∗) 106∗∗)

Equivalent noise (NEP), W · Hz−1/2 ; dark current ∗) , A, and resistance∗∗) , Ω

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8 Some Information on Spectroscopy Techniques

8.4 Photodetectors

8.4.3 Multielement and Distributed Photodetectors

Spectral instruments built by the spectrograph scheme (Section 8.2) are equipped with sensors capable of spatial resolution in the focal plane of the camera (Figure 8.4). For a long time these were exclusively photographic materials. Since in the 1970s, photoelectric detectors comprising sets of separate sensing elements arranged in strips or matrices have become increasingly popular, a number of other arrangements coming into use later on. 8.4.3.1 Spatial Resolution

The quality of a photographic image is characterized, among other things, by its spatial resolution. The spatial resolving power R denotes the number of discernible image elements per unit of length, usually mm−1 . The total resolving power of an instrument is collectively deter−1 −1 −1 mined by its optical system and detector: ROD = RO + RD . The optical system forms some illumination distribution E( x ) on the surface of the detector elements. It proves more convenient to use not the coordinate frequency, but the spatial frequency ω = 2π/λ, for a random illumination distribution can be represented, through Fourier transformation, as a set of spatial harmonics ω, 2ω, 3ω,. . . with their own amplitudes and phases. The wavelength of the fundamental harmonic is λ = 1/N, where N is the number of wave periods per 1 mm. The distribution in harmonics can be characterized by the contrast K: K = ( Emax − Emin )/( Emax + Emin ) .

(8.91)

Following detection, the contrast can be partially lost for various reasons, such as, the ﬁnite size and noise of the individual photocells in the photoelectric detector, nonlocal character of the photographic process due to the scattering of light by the microcrystals of the photoemulsion and migration of particles during the course of formation of the latent image elements, and so on. If the contrast of the image formed by the optical system at the input of the detector is Kin and that at the output, Kout , the quantity T = Kout /Kin < 1 is called the contrast transfer coefﬁcient, and as a function of the light-induced grating period number density, T ( N ), it is referred to as the frequency-contrast characteristic (FCC) [46]. When experimentally testing a system by means of the images of gratings differing in period, the function T ( N ) allows for the loss of contrast due to imperfections of the optics and detector, as well as that occurring during the course of transfer of the signal from the detector to the recording device as a result of the nonlinearity and noise of the electronics, speciﬁci-

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Figure 8.38 Frequency-contrast characteristic of a photolayer.

ties of the development of the latent image elements in the photographic procedure, and so on. Figure 8.38 shows as an example the FCC of an objective lens – photograph system [47]. The FCC method is convenient, for it describes the quality of the image as regards its fragments independent of size. In contrast to the FCC, the resolving power R bespeaks only the discernibility of the ﬁne details of images, and somewhat conditionally, that as far as the discernibility criterion is concerned. One can, for example, digress from the physical substantiation of the analogy with the Rayleigh criterion for the resolution of diffraction images and set the condition that adjacent images should be considered to be resolved if a 20% intensity ‘dip’ exists between them. Referring this condition to the illumination of detectors, one can see from expression (8.91) that this corresponds to a contrast of K ≈ 0.1. In that case, for the FCC of Figure 8.38, the spatial resolving power is R ≈ 65 mm−1 . 8.4.3.2 Photographic Detectors

Traditional photographic plates and photographic ﬁlms. Since spectral decomposition is effected by the optical system of spectral instruments, spectroscopy mainly uses black-and-white photomaterials. The resolving power R of such materials exposed to white light averages 50– 300 mm−1 and decreases with increasing sensitivity. In the UV region, R is approximately doubled, while in the region of the long-wavelength cut-off, it is reduced by 30–50%. In some cases (e.g. microﬁlm materials), R can be as high as 1000 mm−1 .

8.4 Photodetectors

The many years’ experience with photographic materials and the evolution of the notion that the initial phase of the photoprocess is a quantum intrinsic photoeffect have led to the development of a series of image recording versions in which the development of latent images differs from the chemical processes. Beginning in the 1960s, electrophotography methods were developed, including such well-known versions as xerography and photography on electret materials. The former is based on the surface adsorption of ions from a corona discharge in illuminated areas and the latter, on the induction of polarization in photoconducting areas. These methods have found widespread application, primarily in heliography. In terms of spectroscopic applications, theydis not prove not sensitive enough. At the same time, the so-called thermoplastic techniques were being developed for recording television images [48]. With these techniques, a thin electron beam is used to induce charges on a ﬁlm of soft plastic. The ﬁlm is applied to a thin conductor-electrode. When a voltage is applied to the electrode, the ﬁlm is heated, and the electrostatic forces produced by the charges deform its surface. When the ﬁlm is cooled, the surface relief gets ﬁxed. The relief can be read by using a raster-type optical system or through scanning with a narrow light beam. In the present-day spectroscopic applications, this technique has been improved by placing a layer of photosensitive semiconductor between the plastic ﬁlm and the electrode. As in xerography, charges are drawn from external corona discharges, and the latent image is formed on the semiconductor surface. The resolving power in this case can be sufﬁciently high, up to 3000 mm−1 . For more details, see [6, 7, 49]. 8.4.3.3 Image Converter and Intensiﬁer Tubes

An image converter and intensiﬁer tube is a device intended to transfer images in space and over the spectrum and amplify their luminance. The primary image is formed on a photocathode by an optical system. As a result of extrinsic photoeffect, electrons are emitted from the cathode, their local densities near its surface being proportional to the luminance of the corresponding image areas. The cathode materials are the same as in photomultipliers. The emitted electrons are accelerated toward the anode and excite ﬂuorescence of a luminescent screen (cathode luminescence) in the focal plane of an electron-focusing camera. The luminescence intensity in the case of single-stage conversion is 10 to 100 times the image luminance on the photocathode. In multistage image intensiﬁers, the image transfer cycle can be repeated several times, so that their luminance ampliﬁcation can reach 108 [50]. This is speciﬁcally facilitated by the use of intermediate electron current ampliﬁers in the form

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of channel photomultipliers (Section 8.4.2.2). At present, such built-in ampliﬁers in the form of microchannel plates have proved most popular. A microchannel plate is a round plate 30–60 mm in diameter and 0.3– 1 mm thick built up of a large number (ca. 106 ) of microtubes, each operating on the secondary electron multiplication principle. As distinct from the secondary-electron multipliers wherein ionic feedbacks are suppressed by curving the channel, the tubes in the microchannel plate are inclined with respect to the magnetic force lines of the ﬁeld. Additional feedback suppression can be attained by using a pair of microchannel plates with oppositely inclined tubes placed one after the other (chevron assembly). The use of microchannel plates makes it possible to sharply reduce the size of image converter and intensiﬁer tubes and, more importantly, to practically abandon the complex focusing system and improve spatial resolution. While the typical linear resolution of the predecessor single-stage image converters comes to ca. 30 mm−1 , that of the modern converters employing microchannel plates can be as high as ca. 100 mm−1 . 8.4.3.4 Charge-Coupled Detectors

Early in the 1970s works appeared on the measurements of the magnitude of local photoeffect and its coordinates on the surface of a semiconductor plate, which at once found application in image converters of novel type [51–55]. Their operating principle is associated with the spatial ﬁxation of photoionization-induced charges and their transfer to a measuring device, which is explained by Figures 8.39 and 8.40. Light passes through thin transparent islands of a conductor (polysilicon in Figure 8.39) and a dielectric (silicon oxide in Figure 8.39) on the surface of a semiconductor plate (silicon in Figure 8.39) and is absorbed in the latter. Use is made of a p-type silicon. When a photon enters into silicon, it produces, on account of the intrinsic photoeffect, a fresh electron– hole pair. If a positive potential of +V is applied to an electrode formed by a conductive island (the potential of the adjacent electrodes being −V), the region in silicon opposite this electrode becomes depleted of holes and enriched with electrons. A potential well is thus formed with a charge stored as a result of the photoeffect. If this positive potential is reduced to −V, while the potential of the adjacent electrode is raised to +V, the charge will ﬂow under this electrode. Figure 8.39 illustrates a metal-dielectric-semiconductor microcapacitor system (a MOS structure) in which the top capacitor plates are segmented, whereas the bottom ones are integrated, the coupling between them being controlled by the potentials applied to the top plates. The magnitude of the capac-

8.4 Photodetectors

Figure 8.39 Charge localization in a potential well.

itor charge is determined by the number of the incident photons. By successively switching over the potentials of the top electrodes, one can purposefully transfer the charge towards the edge of the semiconductor and take it out to the input of a measuring instrument. There are various potential switching circuits. The four-phase circuit shown in Figure 8.40 is considered to be optimal. The motion of the charge can be not only unidimensional (strip), but also two-dimensional (matrix). In the latter case, the potential switching logic is more complex, and to restrict the uncontrolled spread of the charge across its transfer direction, the so-called ‘stop-channels’ with elevated concentration of the majority impurity are formed in the semiconductor. Such or similar (three-, two-phase) schemes solve the problem of measuring and ﬁxing the photoeffect. Of course, there are many details of physical and technological character in this scheme, which must be taken into account when considering some or other applications of image-forming devices, including those in television systems, surveillance cameras, spectral facilities, and so on. All these devices are called the charge-coupled devices (CCD) or charge-transfer devices (CTD). The CCD version in which the functions of the photodetector and reading system are separated has a number of advantages over the devices that implement the above principle directly. Here light is detected by an array of photodiodes that in their turn charge MOS microcapacitors. The CCD structure in this case functions as a reading system.

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Figure 8.40 Cross-section of a four-phase shift register of a CCD. Proﬁles of potentials beneath electrodes at instants t1 through t5 and register operation phase diagrams (Ph1 through Ph4).

There are at present a large number of major manufacturers of such devices. Figure 8.41 presents the spectral sensitivities of CCD strips manufactured for spectroscopic applications by the company ‘Elektron’ in Russia and Figure 8.42 typical frequency-contrast characteristics of CCD cameras manufactured by Hamamatsu Photonics. The difference between the curves for different radiation wavelengths is due to the difference in the depth of penetration into silicon.

8.4 Photodetectors

Figure 8.41 Relative spectral characteristics of CCD strips manufactured by the company ‘Elektron’ in Russia in two versions (with a quartz entrance window and a ﬁber optic connecting cable).

Figure 8.42 Typical frequency-contrast characteristics of CCD cameras manufactured by Hamamatsu Photonics.

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References

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18 S.E. Frish. Spectroscopy Techniques (in Russian). Leningrad: Leningrad State University Press (1936). 19 A.N. Zaidel and A.N. Shreider. Vacuum Spectroscopy and its Applications (in Russian). Moscow: Nauka (1976). 20 A.G. Zhiglinsky and V.V. Kuchinsky. A Real Fabry–Perot Interferometer (in Russian). Leningrad: Mashinostroenie (1983). 21 R.J. Bell. Introductory Fourier Transform Spectroscopy. New York: Academic Press. (1972). 22 K.I. Tarasov, Ed. Wide-Aperture Spectral Instruments (in Russian). Moscow: Nauka (1988). 23 L. Mertz. Transformations in Optics. New York (1965). 24 Yu. A. Tolmachev. Novel Spectral Instruments. Operating Principles (in Russian). Leningrad: Leningrad State University Press (1976). 25 N.V. Karlov and N.A. Kirichenko. Oscillations, Waves, Structures (in Russian). Moscow: Fizmatlit (2001). 26 V.I. Pustovoit and V.E. Pozhar. Collinear Diffraction of Light by Sound Waves in Crystals: Devices, Applications, New Ideas. Photonics and Optoelectronics, 2, No. 2, pp. 53–69 (1994). 27 P.J. Treado, I.W. Levin, and I.N. Lewis. Near-Infrared Acousto-Optic Spectroscopic Microscopy: a Solid State Approach to Chemical Imaging. Appl. Spectroscopy, 46, No. 3, pp. 553–558 (1992). 28 D.A. Glenar, J.J. Hillman, B. Saif, and J. Bergstralh. Acousto-Optic Imaging Spectropolarimetry for Remote Sensing. Appl. Opt., 33, No. 31, pp. 7412–24, (1994). 29 G.N. Rokhlin. Discharge Light Sources (in Russian). Moscow: Energoatomizdat (1991). 30 Yu. G. Basov. Pumping Sources for Microsecond Lasers (in Russian). Moscow: Energoatomizdat (1990). 31 A.L. Vasserman. Tubular Xenon Lamps and their Applications (in Russian). Moscow: Energoatomizdat (1989).

References 32 V.V. Lebedeva. Gas Discharges as Light Sources. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma (in Russian), V. Plasma Optics (in print). 33 I.S. Marshak. Pulsed Light Sources (in Russian). Moscow: Energiya (1978). 34 B.I. Stepanov. An Introduction to Modern Optics (in Russian). Minsk: Nauka i Tekhnika (1989). 35 Yu. Yu. Protasov. Gas-Discharge Light Sources. In: V.E. Fortov, Ed. Encyclopedia of Low-Temperature Plasma. Encyclopedic Dictionary (in Russian), 1, pp. 456–4568. Moscow: Yanus (2003). 36 N.N. Ogurtsova and I.V. Podmoshensky. Capillary Discharge as a Plasma Source for Quantitative Investigations. In: A.E. Sheindlin, Ed. LowTemperature Plasma (in Russian). Moscow: Mir Publishers (1967). 37 L.D. Gorshkova, V.A. Gorshkov, and I.V. Podmoshensky. Plasma Production in a Discharge Pressed against a Wall by a Magnetic Field (in Russian). Teploﬁzika Vysokikh Temperatur, 6, No. 6, pp. 1130–1135 (1968). 38 S Kashivabara, K. Watanabe, and R. Fujimoto. Discharge Properties of Formed-Ferrite Plasma Sources. Appl. Phys., 62, No. 3, pp. 787–791 (1989). 39 S.V. Mit’ko, V.N. Ochkin, A.V. Paramonov, and A.P. Shirokikh. FerriteInitiated Discharge. Breakdown Conditions and Application to the Optical Pumping of a J2 Laser (in Russian). Kratkie Soobshcheniya po Fizike, No. 11, pp. 47–49 (1989). 40 S Hesketh. The Propagation of Arcs Over a Water Surface. Proc. VIII Int. Conf. on Phenom. in Ionized Gases, p. 255. Vienna: (1967). 41 B.Kh. Brodskaya. Development of Pulsed Discharges at a Gas-AqueousElectrolyte Interface and Estimation of their Effect on Chemical and Biological Systems (in Russian). Khimiya Vysokikh Energii, 18, No. 5, pp. 458– 464 (1982). 42 E.A. Azizov, E.A. Akhmerov, G.G. Gladysh, and I.P. Shedko. Breakdown of a Gas Gap by a Creeping Arc Discharge (in Russian). Teploﬁzika

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54

55

Vysokikh Temperatur, 22, No. 4, pp. 655–660 (1984). F.A. van Goor S.V. Mit’ko, V.N. Ochkin, A.P. Paramonov, and W.J. Witteman. Arc Discharge Sliding over a Conductive Surface. Journal of Russian Laser Research, 18, No. 3, pp. 147–259 (1997). L.Z. Krikunov. Handbook of Infrared Technology Principles (in Russian). Moscow: Sov. Radio (1978). I.I. Anisimova and B.M. Glukhovsky. Photoelectric Multipliers (in Russian). Moscow: Sov. Radio (1974). A.I. Tudorovsky. Theory of Optical Instruments (in Russian), 2nd Ed., Pt. 2, pp. 222–223. Moscow-Leningrad (1952). A.T. Ashcheulov. Resolving Power. In: Physical Encyclopedic Dictionary (in Russian), 4, p. 329. Moscow: Sovetskaya Entsiklopediya (1965). W.E. Glenn and I.E. Wolf. Thermoplastic Recording. Intern. Sci. Technol., No. 6 (1962). V.A. Barachevsky, G.I. Loshkov, and V.A. Tsekhovsky. Photochromism and its Applications (in Russian). Moscow: Khimiya (1977). A.G. Berkovsky, V.A. Gavanin, and I.N. Zaidel. Vacuum Photoelectric Instruments (in Russian). Moscow: Energiya (1976). C.Sequin and M. Thompsett. Charge Transfer Devices. New York: Academic Press (1975). P. Jespers, F. Van de Wiele, and M. White, Eds. Solid State Imaging. Leyden (Netherlands): Noordhoff (1976). F.P. Press. Photosensitive Charge Coupled Devices (in Russian). Moscow: Radio i Svyaz (1991). V.A. Arutyunov, N.G. Bogatyrenko, and T.A. Kormacheva. Charge Transfer Devices for Spectroscopy. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), Appendices, Ser. B, Low-Temperature Plasma Diagnostics, Reference Materials, VI-1, Pt. 2, p. 348. Moscow: Yanus-K (2007). A.A. Krasnyuk and V. Ya. Stenin. CCD Spectroscopy. In: V.E. Fortov,

485

486

8 Some Information on Spectroscopy Techniques Ed. Encyclopedia of Low-Temperature Plasma (in Russian), Appendices, Ser. B, Low-Temperature Plasma Diagnos-

tics, Reference Materials, VI-2, Pt. 2. Moscow: Yanus-K (2007).

487

Appendix A

Statistical Weights and Statistical Sums Assuming that the internal and translational motions of particles are independent, the expression for the total statistical sum referred to a volume of 1 cm3 , is written down as the product Q = Qtr Qin ,

(A.1)

where Qtr is responsible for the motion of the mass centers of the particles. It is well known from statistical mechanics that 2πMkB T , (A.2) Qtr = h2 where h and kB are the Planck and Boltzmann constants, M is the particle mass and T is the kinetic temperature. The main problem is usually to ﬁnd the internal statistical sum Qin , deﬁned by expression (1.7). Generally speaking, this requires direct summation, which is extremely difﬁcult to do in a multilevel energy structure. A helpful simpliﬁcation can be attained by using expressions (1.6) and (1.8), but in this case it is also necessary to know the statistical weights and the positions of the levels involved. A.1 Statistical Weight of Energy Levels in Atoms and Ions

For atoms and atomic ions, the statistical weight gl determining the level degeneration depends on the quantum numbers L, S and J, the total orbital momentum, total spin and total electron momentum: gl = (2L + 1)(2S + 1)

for the L − S term,

gl = 2J + 1

for the J level.

488

Appendix A Statistical Weights and Statistical Sums

A.2 Statistical Weight of Electronic States in Molecules

The statistical weights of various electronic states in molecules are also determined by the type of the states (for the nomenclature of the states with the symbol 2S+1 Λ, see [1–3] and Appendix D). For diatomic and linear polyatomic molecules, the quantity gl allows for the spin splitting into 2S + 1 components and additional splitting into two components in the states whose quantum number of the electronic momentum projection on the internuclear axis is Λ > 1 (Λ-doubling [1–3]), see Table A.1. Table A.1 Statistical weights of diatomic and linear polyatomic molecules [1–3]. e

1Σ

2Σ

3Σ

1Π

2Π

3Π

1Δ

2Δ

ge

1

2

3

2

4

6

2

4

For nonlinear polyatomic molecules, see Table A.2. Table A.2 Statistical weights of nonlinear polyatomic molecules [1–3]. e

1A

ge

1

1

2A

1

2

3A

1

3

1A

2

2A

1

2

2

3A

3

2

1B 1

2B 1

3B 1

1

2

3

A.3 Statistical Weight of Vibrational Levels of Molecules

In all diatomic and some polyatomic molecules, the vibrational levels v of their electronic states are not degenerate, and so gv = 1. In some polyatomic molecules, depending on their symmetry, some vibrations can be degenerate. If a molecule has l degenerate vibrations, the number of normal vibrations differing in frequency will then be, for linear molecules with Na atoms, nv = 3Na − 5 − l ,

(A.3)

and for nonlinear molecules, nv = 3Na − 6 −

n=l

∑ ( d n − 1)

,

(A.4)

n =1

where dn is the degeneracy multiplicity of the normal vibration. When dn = 1, the vibration and its associated levels are not degenerate. The

A.4 Statistical Weight of Rotational Levels of Molecules

statistical weight of any vibrational level v of a diatomic harmonic oscillator is gv = 1. The statistical weight of the vibrational levels of the nth doubly degenerate vibration of a polyatomic molecule (which is also attributed the sense of an individual oscillator) is gvn = vn + 1, and for a triply degenerate vibration, gvn = (vn + 1)(vn + 2)/2. The total statistical weight of the vibrational state {v1 v2 v3 . . . } of a polyatomic molecule is g{v1 v2 v3 ...} =

n=nv

∏

n =1

( v n + d n − 1) ! . v n ! ( d n − 1) !

(A.5)

The quantity dn is determined by the afﬁliation of the molecule to a certain point symmetry group and its electronic state. These data for concrete molecules can be found in [2–4, 7]. Table A.3 lists selected data on the normal vibration frequencies of simple 3- to 5-atom molecules. The dn values other than 1 are indicated in parentheses next to the vibration frequency values (in cm−1 ). Table A.3 Frequencies and multiplicity of normal molecular vibrations. CO2

CS2

OCS

C2 O

CH2

ν1 1333 664 863 1978 3000 ν2 667(2) 395(2) 520(2) 379(2) 1362 ν3 2350 1535 2055 1074 3200 ν4 SnO2 ν1 745 ν2 230(2) ν3 864 ν4

BH3

AlO2

2623 1096 1125 496 2808(2) 550 1605(2)

CH3

CF3

CF4

C 2 N2

3000 607 3162(2) 1398(2)

1090 701 1259(2) 512(2)

908 435(2) 1282(3) 632(3)

2330 960 845 380(2) 215 1350 502(2)

MgF2

CaOH CrO3

GaOH InOH Be2 O

SiO2

SiH3 1955 996 1999(2) 925(2)

Na2 O CsOH

600 520 620 540 570 800 320 336 310(2) 300(2) 320(2) 165(2) 305(2) 400 60(2) 306(2) 3600 3600 1200 800 3700 980(2) 600 3705 350(2)

A.4 Statistical Weight of Rotational Levels of Molecules

Let us ﬁx the electronic and the vibrational state of a molecule. Its energy structure in that case is determined by the rotational structure of its terms. To calculate the statistical weight g J of a rotational level with the moment of momentum quantum number J, one should take into consideration the presence of the intrinsic spin in the nuclei of the atoms constituting the molecule: g J = gn (2J + 1).

(A.6)

489

490

Appendix A Statistical Weights and Statistical Sums

Here gn is the statistical weight of the nuclei. For polyatomic and heteronuclear diatomic molecules, gn =

1 σ

Na

∏ (2In + 1),

(A.7)

n =1

where In is the nuclear spin of the nth atom, Na is the number of atoms in the molecule and σ is the so-called symmetry number. For heteronuclear diatomic molecules, the quantity σ depends on the afﬁnity of the molecule in the ﬁxed electronic state to a certain point symmetry group (see Table A.4). Table A.4 Symmetry numbers and groups. σ

Symmetry group

σ

Symmetry group

1 2 3 4 5

C1 , Ci , Cs , C∞v C2 , C2v , C2h , D∞h C3 , C3v , C3h , S6 C4 , C4v , C4h , D2 (= V ), D2d , D2h (= Vh ) C5 , C5v , C5h

6 8 12 24

C6 , C6v , C6h , D3 (= S6v ), D3h D4 , D4d (= S8v ), D4h D6 , D6d , D6h , T, Td Oh

Table A.5 gives examples of assigning some simple molecules in the electronic ground states to the numbers σ indicated in Table A.4. Table A.5 Symmetry numbers for some molecules. σ=1

All diatomic heteronuclear molecules and also HDO, HCO, HNO, HO2 , FCO, ClCO, ClCN, HCN, FCN, NCO, COS, CHF, CHCl, CFCl, HBO, HFCO, FBO, HBO2 , HDO2 , N2 O, S2 O, HOCl, NOCl, N3 H NHFCO, FClCO, SOF2 , N2 H4 , C2 H3∗ , C2 HF, CH2 F, CHF2 , CHFClBr, ClBrHC−CHBrCl, HC2 Cl, SiC2 , CX2 YZ(X Y Z −H F C l B r · I), CXYZV(X Y Z V −H F C l B r I ), C2 X(X−F C l O ), C2 X3 Y(X Y −H F C l), GeX2 (X−O F C l S )

σ=2

All homonuclear diatomic molecules and also Al2 O, AlO2 , AlF2 , AlCl2 , Li2 O, C3 O2 , H2 O, D2 O, F2 O, ClO2 , Cl2 O, SO2 , H2 S, SF2 , BeF2 , BeCl2 , NO2 , H2 CO, F2 CO, Cl2 CO, SF4 , CH2 Cl2 , SO2 F2 , C6 H2 Cl2 Br2 , SiX2 (X−O F C l), Si2 C, BX2 (X−O C l F ), B2 O2 , B2 O3 , MgF2 , MgCl2 , CX2 Y2 (X Y −H F C l B r I ), C2 X2 (X−H F N ), LiOH, ZrO2 , PbF2 , N3 , NF2 , PF2 , H2 O2 , D2 O2 , N2 H4∗ , CX2 (X−H F C l B r I O S ), C3 C2 X2 Y2 (X Y −H F C l), SnX2 (X−F C l O S ), PbX2 (X−F C l O S )

σ=3

NH3 , NF3 , POF3 , POCl3 , CX3 (X−H F C l I ), CX3 Y(X Y −H F C l B r I ), CH3 CCl3∗ , SiX3 (X−H F C l B r), H3 BO3∗ ,

σ=4

Al2 O2 , BCl4 , Li2 F2 , Li2 Cl2 , Na2 Cl2 , PtCl4− , C2 H4 , O4 , N2 O4 , C3 H6 , C2 X4 (X−H F C l)

σ=6

AlX3 (X−H F C l), C2 H6∗ , BF3 , BCl3 , GaF3 , InF3 , SO3 , PF5 , PCl5 , C6 H3 Cl3 , H3+

σ=8

S8 , C4 H8 , Be4 O4

σ = 12

C6 H6 , C6 Cl6 , P4 , CX4 (X−H F C l B r), P4 O6 , C(CH3 )4∗ , SiX4 (X−H F · Cl · Br · I), BF4−

σ = 24

SF6 , PtCl6−

A.4 Statistical Weight of Rotational Levels of Molecules

The asterix ∗ indicates molecules possessing internal rotation (see below). Identical atoms in molecules are understood to be isotopically equivalent. Homonuclear diatomic molecules, each of whose atoms has a spin of In , have two sequences of statistical weights for rotational levels with J values differing in parity. They are deﬁned, as before, by expression (A.6), but with different nuclear statistical weights gn : gn1 = In (2In + 1),

(A.8)

gn2 = ( In + 1)(2In + 1).

(A.9)

The decision on which of the above formulas (A.8) or (A.9) should be substituted into formula (A.6) for different (in the parity of J) rotational levels is made with the aid of Table A.6. Table A.6 To the calculation of the statistical weights of rotational levels in homonuclear molecules Electronic state indices Nuclear spin For even J For odd J

+ g integral g n1 g n2

− u half-integral g n2 g n1

− g integral g n2 g n1

+ u half-integral g n1 g n2

Let us recall the meaning of the standard indices used in spectroscopy [1–4, 7] to reﬂect the properties of the wave function with respect to various symmetry operations. They indicate whether the wave function maintains or changes sign on reﬂection in the plane containing the molecular axis (+, −) or on the change of sign of the coordinates of the electrons (g, u). It is appropriate to note here that one and the same symbol, g, is generally used to denote both the statistical weight and the result of a symmetry operation. However, no confusion usually occurs, for the attribute relating to symmetry is indicated as an index in the designation of the term. The symbols s and a, respectively, show whether or not the wave function changes sign on the permutation of nuclei. To denote the electronic states of molecules with zero projection of the orbital moment of the electrons on the internuclear axis, Λ = 0, the indices g, u, + and − (if any) are denoted directly on the symbol Σ. For example, 1 Σ+ , 2 Σ− , 3 Σ− , and so on. If Λ = 0 (Π, Δ,. . . , states) and the Λ-doubling g g u takes place, for each number J of a split rotational level there exist different sets of indices g, u, + and −. By virtue of this fact, as J is successively changed, the upper and lower components of Λ-doublets change their statistical weight from gn1 to gn2 and vice versa as successively. This can be illustrated by Table A.7 which indicates comparative symmetry prop-

491

492

Appendix A Statistical Weights and Statistical Sums Table A.7 Symmetry of rotational levels of Σ-Π-states. Dissimilar nuclei J= Σ+ Σ− Π

0 + −

1 − + (+, −)

2 + − (−, +)

Similar nuclei 3 − + (+, −)

J= Σ+ g Σ+ u Πg Πu

0 (+, s) (+, a)

1 (−, a) (−, s) (+, s) (−, a) (+, a) (−, s)

2 (+, s) (+, a) (−, a) (+, s) (−, s) (+, a)

3 (−, a) (−, s) (+, s) (−, a) (+, a) (−, s)

erties of the rotational levels of hetero- and homonuclear molecules in the Σ (Λ = 0) and Π (Λ = 1) states (for the Π state, J ≥ 1). Speciﬁcally, one can see from formulas (A.6), (A.8), and (A.9) and Table A.7 that in homonuclear molecules every second rotational level is missing. A.4.1 Statistical Sum of Atoms and Ions

In that case, the sum over internal states is the sum over the electronic states of the particles, (1.7) or (1.8) at equilibrium. The values of the statistical weights are given by relations (A.1) and (A.2). The practical problem is to limit the number of the electronic states to be considered. The limiting factor in plasma is the interaction of the charged particles. As stated in Section 1.1, the Debye interaction energy of electrons is e2 /rD , and atoms in the states of lower binding energy become spontaneously ionized. This effect is called the reduction of the ionization potential in plasma. To illustrate, for the hydrogen atom in plasma with an electron concentration of ne ca. 1015 cm−3 and a thermal particle energy of kB T = 1.600 × 10−19 J(1 eV) (T = 11605 K), the ultimate value of the principal quantum number n = 47. Use is also made of a much cruder approximation, provided that the temperature of plasma (only at equilibrium) is below the potential of the ﬁrst excited state. This condition is generally satisﬁed in practice, especially for ions, and the statistical sum then can be replaced, in accordance with expression (1.6), by the statistical weight of the ground state. A.4.2 Statistical Sum of Molecules

If the assumption, made in elucidating formula (A.1), that the different types of motion of particles are independent is extended to their internal degrees of freedom, the expression for the internal statistical sum of a

A.4 Statistical Weight of Rotational Levels of Molecules

molecule will then assume its most frequently used form Qin = Qe Qv Qr ,

(A.10)

where Qe , Qv and Qr are the electronic, vibrational and rotational statistical sums of the molecule. The electronic statistical sum Qe is found by analogy with atoms. The vibrational statistical sum is calculated most often, and with an accuracy acceptable for the problems of low-temperature plasma spectroscopy, using the harmonic oscillator approximation. For diatomic molecules, direct summation by formula (1.8), with due regard for the equidistance of vibrational energy levels, ΔEv0 = vν (here, instead of k, use is made of the generally accepted symbol v to denote the serial number f the vibrational level and ν is the oscillator frequency), and with inﬁnite upper limit, yields −1 , Qv = 1 − exp {−hcν/kB T }

(A.11)

the dimensions being as follows: h[erg · s], ν[cm−1 ], c[cms−1 ], T[deg, K], kB [erg/deg], hc/kB = 1.44. The vibrations of a polyatomic molecule are described in the same harmonic oscillator approximation by a set of harmonic oscillators of normal vibration frequencies νn . Summation by formula (1.8) extended over a ﬁnite number of types of normal vibration yields, with consideration for degeneracy and statistical weights (A.5), [2, 4] Qv =

nv

∏ (1 − Zn )−dn ,

n =1

Zn = exp {−hcνn /kB T } .

(A.12)

When calculating the rotational statistical sum by formula (1.8) with the above-indicated statistical weights, use is most frequently made of the rigid rotator model. For diatomic molecules, Qr =

1 kB T , σ hcBv

(A.13)

where Bv is the rotational constant and σ = 1 for heteronuclear molecules and σ = 2 for homonuclear ones. For polyatomic molecules, ' kB T 3 π Qr = , (A.14) Av Bv Cv hc where Av , Bv , and Cv are rotational constants, not equal to one another, in asymmetric top molecules. In symmetric top molecules, two of these constants coincide. To determine σ, one should use Table A.4.

493

494

Appendix A Statistical Weights and Statistical Sums

Note in conclusion, that the use of the harmonic oscillator and rigid rotator approximations to calculate the appropriate statistical sums provides for small vibration amplitudes of molecules and their small stretching, compared to the internuclear separation, under the action of centrifugal forces. Experience shows that such approximations prove quite satisfactory for molecules with a small number (2–5) of atoms and relatively not very high excitation (heating) level. An important example of violation of these conditions is the class of molecules with internal rotations or torsional vibrations. If the potential barrier preventing the free internal rotation of molecular fragments is high enough, in comparison with the excitation energy (temperature at equilibrium), this corresponds to the case of torsional vibrations, and these can be treated as an additional vibrational degree of freedom in the general scheme and so be properly included in the vibrational statistical sum. If the barrier can be overcome, so that free internal rotation can take place, this motion should then be considered a rotational rather than vibrational degree of freedom and allowed for in the rotational statistical sum. What is more, the symmetry numbers σ will in that case change in comparison with those listed in Table A.4. These questions are considered in [2, 4]. We will restrict ourselves here to some examples of individual molecules. Table A.8 lists the values (in kelvins) of the potential barriers VCH3 and VOH preventing the internal rotation of the CH3 and OH groups. Direct calculations show (e.g. [7]) that failure to allow for the internal rotation in C2 H6 at 300 K results in the reduction of the internal statistical sum Qin by ca. 40 %, and at 1000 K, by a factor of ca. 3.5. Table A.8 Potential barriers for internal rotation. Molecule

C2 H2

CH3 CF3

CH3 CH2 OH

CH3 CHOHCH3

VCH3 , K

1460

1740

1510

1710

5030

2520

VOH , K

HNO3

3520

References

References

1 G. Herzberg. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules. 2nd ed. N.Y.: D. van Nostrand (1951). 2 G. Herzberg. The Spectra and Structures of Simple Free Radicals. London: Cornell University Press (1971). 3 M.A. Elyashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 4 V.P. Glushko, Ed. Thermodynamic Properties of Individual Substances (in Russian). 1–3. Moscow: Nauka (1978).

5 L.A. Kuznetsova, N.E. Kuzmenko, Yu. Ya. Kuzyakov, and Yu. A. Plastinin. Optical Transition Probabilities of Diatomic Molecules (in Russian). Moscow: Nauka (1980). 6 L.A. Kuznetsova and S.T. Surzhikov. Optical Characteristics in the Spectra of Diatomic Molecules. In: V.E. Fortov, Ed. Encyclopedia of LowTemperature Plasma (in Russian), 1, pp. 376–391. Moscow: Nauka (2000). 7 G. Herzberg. Infrared and Raman Spectra of Polyatomic Molecules. New York (1945)

495

497

Appendix B

Conversion of Quantities Used to Describe Optical Transition Probabilities in Line Spectra Table B.1 [1] lists conversion factors for ﬁve quantities, namely, the Einstein coefﬁcients Aul , Bul and Blu , the oscillator strength f lu (in absorption) and the line strength Sul = Slu = S. The subscript u denotes the upper level and l, the lower one. The wavelength λ is expressed in centimeters, Aul , in cm−1 , Bul and Blu , in cmg−1 , and S, in dyn · cm4 .The line strength S has the dimension of the squared dipole moment and is frequently expressed in atomic units e2 a20 , where e is the electron charge and a0 is the Bohr radius. In the CGS system, the quantity e2 a20 equals 6.46 × 10−36 CGS units. The table is presented for dipole-allowed transitions. Similar conversions for quadrupole and magnetic dipole transitions are presented in [2]. References

1 W. Lochte-Holtgreven, Ed. Plasma Diagnostics. Amsterdam: Elsevier (1968). 2 W.L. Wiese, M.W. Smith, and B.M. Miles. Atomic Transition Probabilities.

2. NSRDS-NBS 22, Washington, DC (1969).

l

5.31 × 10−55 gu Bul

3.19 × 10−30 gu λ3 Aul

S=

l

2.5 × 10−25

g

1.50 gu λ2 Aul

gu Bul gl λ

Blu λ

gl Blu gu λ3

5.31 × 10−55 gl Blu

2.5 × 10−25

–

gu B gl ul

1.67 × 10−25 gl B gu lu

Bul λ3

–

1.67 × 1025

f lu =

6.01 × 1024 gu λ3 Aul

g

6.01 × 1024 λ3 Aul

Bul =

Blu =

–

Aul =

Table B.1 Conversion relations between the coefﬁcients for optical transition probabilities.

2.13 × 10−30 gl λ f lu

–

4.01 × 1024 λ f lu

gl f lu gu λ2 g 4.01 × 1024 l λ f lu gu 6.67 × 10−1

–

1 S gu λ3 1 1.88 × 1054 S gu 1 1.88 × 1054 S gl 1 4.7 × 1029 S gl 3.14 × 1029

498

References

499

Appendix C

Two-Photon Absorption Cross Sections for Some Atoms and Molecules in the Ground State Table C.1 lists absorption cross sections for two-photon transitions in a number of atoms and simple molecules [1, 2], obtained experimentally and theoretically. For notation and dimensions, see Section 3.5.1. The cross sections are presented in the same units as in the original works. The formulas relating them together can also be found in Section 3.5.1.

248

192.75 193.49

193

193

230

Xe

N2 O

Kr

H2

Xe

CO

550–680

226

Ba

O

NO

193

248 193

I2

1

4p5 6p

193

Kr

3p3 P

6p2 (3 P2 ) 6s7d(3 D2 )

(2.6 ± 0.4)

(1.4 ± 0.43) × 10−46 cm4 s

7 × 10−30 cm4 3 × 10−29 cm4

[10]

[9]

[7] [8]

(2.9 ± 1.8) × 10−49 cm4 s

[4](b)

[3](a)

(1.5 ± 0.7) × 10−35 cm4

< 3 × 10−34

[3](a)

[3]

B1 Σ +

2.34 × 10−31

2.62 × 10−31

∼ 3 × 10−35

8 × 10−32 10−34

[6](d)

5s[3/2]1

10−34

[3]

6 × 10−31

[3]

10−31

[3] [3]

[3](a)

Ref.

5 × 10−35 cm4 1.7 × 10−35 cm4 2.9 × 10−35 cm4

∼ 109

3 × 106 ∼ 109

10−34

σ (cm4 W−1 ) experiment

4 × 10−34

5 × 10−33

9 × 10−33

σ (cm4 W−1 ) theory

6p[1/2]0 6p[3/2]2 6p[5/2]2

0.806 0.763

∼ 10−3

1 1

0.1

D1 Σ+ u 5p5 6a

4p5 5s

3 × 106 3 × 106

b 1 Σ+ u

A1 Π a3 Π

B1 Σ+ u

k

3 × 106

3 × 106

3 × 106

Δω (MHz)

[5](c)

1.39 1.39

∼1

1 1

1

1

0.01

0.01 ∼ 10−3

1

μ22 D2

(2 ± 0.9) × 10−36 cm4

E, F1 Σ+ g (v = 6)

6p[1/2]0 6p[1/2]2

N2 + O ∗

Xe –

1 Σ+ (1 S+1 S) g Xe ∗ (5p5 6p)

1

E3 Σ + g

193

N2

1 10−3

F1 Σ +

193

CO

1

μ21 D2

193

E, F1 Σ+ g (v = 2, J = 2)

u

H2

λ (nm)

Table C.1 Cross sections for two-photon absorption.

500

Appendix C Two-Photon Absorption Cross Sections for Some Atoms and Molecules

170

F

2 P0 −2 D0 3/2 5/2 2 P0 −2 D0 3/2 3/2 2 P0 −2 D0 1/2 5/2 2 P0 −2 D0 1/2 3/2

3p4 D 3p4 S3/2

μ21 D2

μ22 D2

Δω (MHz)

3s2 P

k

2.7 × 10−36 cm4 2.7 × 10−36 cm4 2.7 × 10−36 cm4 2.7 × 10−36 cm4

[17](k)

[15](i) [16]( j)

(2.8 ± 0.7) × 10−35 cm4 (9.8 ± 2.5) × 10−35 cm4 2.7 × 10−36 cm4

[11](e) [12]( f ,l ) [13]( g,l ) [14](h,l ) [14](h,l )

(2.6 ± 0.8) × 10−35 cm4 (1.8 ± 0.6) × 10−35 cm4 (4.8 ± 2.4) × 10−35 cm4 0.7 −35 cm4 (0.7+ −0.5 ) × 10

×10−35 cm4

Ref.

σ (cm4 /W) experiment

σ (cm4 /W) theory

(b)

Estimates [3] from experimental data. As concluded by the authors of [4], a good enough agreement was obtained with experimental results on ﬂuorescence. (c) Experiments were conducted with a dye and an ArF laser, the photon statistics factors G (2) being taken equal to 1.4 and 2, respectively. The average cross section amounted to (1.2 ± 0.5) × 10−47 cm4 s. (d) The measurement error is estimated at 30%. (e) The quantities G (2) = 2 and g(0) = 0.93944/Δω . For G (2) = 1, the cross section equals (1.319 ± 0.2) × 10−35 cm4 , or D (2.392 ± 0.4) × 10−46 cm4 s, or (2.717 ± 0.5) × 10−28 cm4 /W. ( f ) G (2) = 2, g (0) = 1.81 × 10−11 s, cross sections 4.85 × 10−46 cm4 s, or 5.5 × 10−28 cm4 W−1 . ( g) Compared to [10], measurements were taken with a single-frequency laser to eliminate the uncertainty associated with the photon statistics, G (2) = 1.4 ± 0.2. (h) G (2) = 1.4, g (0) = 1.55 × 10−11 s for the 5p3 P transition. (i ) The product G (2) σ. ( j) Obtained by multiplying the value indicated in the preceding line into the relative cross section for nitrogen, σ (207 nm)/σ (211 nm) = 3.5 [16].

( a)

211 207

4p3 P2,1,0 5p3 P2,1,0

225.6 225.7 226 200.6 192.5

N

u

λ (nm)

Table C.1 (continued).

Appendix C Two-Photon Absorption Cross Sections for Some Atoms and Molecules 501

(2 P01/2 −2 D05/2 )

Calculated from relative measurements of the transition cross sections (2 P03/2 −2 D05/2 ) = 1 ± 0.2, (2 P03/2 −2 D03/2 ) = 0.6 ± 0.2, = 0.9 ± 0.3, (2 P01/2 −2 D03/2 ) = 0.9 ± 0.3. The cross section for the 2 P03/2 −2 D05/2 transition is taken equal to the theoretical one. (l ) Summed over J. The values for the individual J → J transitions can be found in the work cited.

(k)

502

Appendix C Two-Photon Absorption Cross Sections for Some Atoms and Molecules

References

References ¨ ¨ 1 M. Goppert-Mayer. Uber Elemen¨ tarakte mit zwei Quantensprungen. Ann. Phys. (Leipzig) 5, pp. 273–294 (1931). 2 F. Shiga, S. Imamura. Phys. Lett, A, 25A, pp. 706–707 (1967). 3 W.K. Bischel, J. Bokor, D.J. Kligler, and C.K. Rhodes, Nonlinear optical processes in atoms and molecules using rare-gas halide lasers. IEEE J. Quant. Electr., QE15, No. 5, pp. 380–392 (1979). 4 J. Bokor, J. Zavelovich, and C.K. Rhodes. Multiphoton ultraviolet spectroscopy of some 6p levels in krypton. Phys. Rev., A21, No. 5, pp. 1453–1459 (1980). 5 J.D. Buck, D.C. Robie, A.P. Hickman, D.J. Bamford, and W.K. Bischel. Twophoton excitation and excited-state absorption cross-sections for H2 E, F 1 Σ g (v = 6): Measurement and calculations. Phys. Rev., A39, No. 8, pp. 3932–3941 (1989). 6 S. Kroll and W.K. Bischel. Two-photon absorption and photoionization crosssection measurements in 5p6p conﬁguration of xenon. Phys. Rev., A41, No. 3, pp. 1340–1349 (1990). 7 M.D. Di Rosa and R.L. Farrow. Twophoton excitation cross-section of the BX(0, 0) band of CO measured by direct absorption. JOSA B, 16, No. 11, pp. 1988–1994 (1999) 8 J. Burris, T. McGee, and T. McIlrath. Absolute two-photon excitation crosssections in NO. In: Applications of laser chemistry and diagnostics. Proceedings of the Meeting, Arlington, VA, May 3, 4, 1984. Bellingham, WA, SPIE – The International Society for Optical Engineering, pp. 10–16 (1984). 9 N.A. Cherepkov and A. Yu. Elizarov. Two-photon excitation of Ba atoms and absolute measurements of σ(2) . J. Phys. B.: At. Mol. Opt. Phys., 24, pp. 4169–4179 (1991). 10 A.D. Tserepy, E. Wurzberg, and T.A. Miller. Two-photon-excited stimu-

11

12

13

14

15

16

17

lated emission from atomic oxygen in rf plasmas: detection and estimation of its threshold. Chem. Phys. Lett., 265, pp 297–302 (1997). The values are borrowed from Y.L. Huang, and R.J. Gordon. The effect of ampliﬁed spontaneous emission on the measurement of the multiplet state distribution of ground state oxygen atoms. J. Chem. Phys., 97, pp. 6363– 6368 (1992). R.P. Saxon and J. Eichler. Theoretical calculation of two-photon absorption cross-sections in atomic oxygen. Phys. Rev., A34, No. 1, pp. 199–206 (1986). D.J. Bamford, L.E. Jusinski, and W.K. Bischel. Absolute two-photon absorption and three-photon ionization cross-sections for atomic oxygen. Phys. Rev., A34, pp. 185–198 (1986). D.J. Bamford, M.J. Dyer, and W.K. Bischel. Single-frequency laser measurements of two-photon cross-sections and Doppler-free spectra for atomic oxygen. Phys. Rev., A36, No. 7, pp. 3497–3500 (1987). D.J. Bamford, R.P. Saxon, L.E. Jusinski, J.D. Buck, and W.K. Bischel. Twophoton excitation of atomic oxygen at 200.6, 192.5, and 194.2, nm: Absolute cross-sections and collisional ionization rate constants. Phys. Rev., A37, No. 9, pp. 3259–3269 (1988). D.G. Fketcher. Arcjet ﬂow properties determined from laser-induced ﬂuorescence of atomic nitrogen. Appl. Opt., 38, No. 9, 1850–1858 (1999) S.F. Adams and T.A. Miller Twophoton absorption laser induced ﬂuorescence of atomic nitrogen by an alternative excitation scheme. Chem. Phys. Lett., 295, p. 305 (1998). G.C. Herring, M.J. Dyer, L.E. Jusinski, and W.K. Bischel. Two-photon-excited ﬂuorescence spectroscopy of atomic ﬂuorine at 170 nm. Optics Letters, 13, No. 5, pp. 360–362 (1988).

503

505

Appendix D

Information on Some Diatomic Molecules for the Identiﬁcation and Processing of Low-Temperature Plasma Spectra

D.1 Brief Information from Molecular Spectroscopy – Designations of States and Transitions, Coupling Types, Selection Rules, General Spectrum Structure

Diatomic molecules are axially symmetric. The motion of the molecular shell electrons is characterized by angular momenta; the total orbital momentum L and the total spin momentum S. The quantum numbers corresponding to these momenta are L and S. The rotational moment of the molecule (nuclei) is R. The molecular states are classiﬁed according to the magnitudes of the projections of the momenta on the internuclear symmetry axis. Their vector components in the direction of the axis are Λ and Σ and the quantum numbers corresponding to them are Λ and Σ. The projection of the moment R on the internuclear axis is equal to zero, and so no quantum number is introduced for it. The total angular momentum J = R + S + L, its quantum number is J, and the quantum number of its component along the axis is Ω = Λ + Σ. The symbol of the electronic term is 2S+1 ΛΩ . The states with Λ = 0, 1, 2, 3, . . ., are denoted Σ, Π, Δ, Φ, . . . (the Greek letter Σ for Λ = 0 should not be confused with the italicized letter Σ for the quantum number of the spin projection). The letter before the electronic term symbol denotes the sequence of the energies Te of the electronic states. The values of Te are determined from the lowermost level of the electronic ground state. The letter X is used for the ground state. Capital letters are used to denote states having the same multiplicity (2S + 1) as the ground state, while the lower-case ones, for states of a different multiplicity. For example, X1 Σ, A1 Π, B1 Π, a3 Π, b3 Π,. . . The sequence and style of these additional letters can differ between different sources. This is due, in particular, to the historical sequence of the discovery of various states during the course of investigations into

506

Appendix D Information on Some Diatomic Molecules

molecules. Here we adhere to the notation used in the detailed tables of molecular constants [1]. Additional information about the state of a molecule is associated with the properties of its wave function in relation to various symmetry transformations. If the multicomponent wave function remains unchanged on reﬂection of the coordinates of the electrons and nuclei in the plane containing the axis of the molecule, the state described by it is called positive (+), and where it changes negative (−). This is reﬂected by the respective superscript on the right of Λ. If the wave function does not change (changes) sign on he reﬂection of the coordinates of the electrons and nuclei with respect to the center of the internuclear axis, the state is then called even (g) (odd (u)). If the wave function maintains (changes) sign on permutation of two identical nuclei, the state is called symmetric s (asymmetric a) correspondingly. When writing down the electronic state (term) symbol, it is usual not to indicate the entire information on it. For example, the notation B3 Π3/2 means a second excited triplet state with Λ = 1, S = 1, Ω = 3/2, and Σ = 1/2. The notation B3 Πg denotes an even second excited triplet state wherein Ω = 1/2 and 3/2 are possible. (See also Tables A.6 and A.7 of Appendix A.) The generally accepted designation Te – the energy of the electronic term, determined from the lowermost vibrational–rotational level of the electronic ground state. Because of the spin-orbit interaction, the components of a multiplet electronic term prove to be split in energy: Te = T0 + Av ΛΣ.

(D.1)

Here T0 is the position of the mass center of the multiplet and Av is the spin-orbit interaction constant. Electronic terms have both a vibrational and a rotational structure with characteristic groupings on the energy scale. The following designations are used: G (v) the energy of the vibrational level v within the limits of the electronic state, G (v) = ωe (v + 1/2) − ωe xe (v + 1/2)2 ;

(D.2)

Fv ( J ) the energy of a rotational level within the limits of the vibrational level v, Fv ( J ) = Bv J ( J + 1) − Dv J 2 ( J + 1)2 ,

(D.3)

where ωe and ωe xe are vibrational constants and Bv is the rotational constant for the level v, Bv = Be − αe (v + 1/2);

Dv = De + β e (v + 1/2).

(D.4)

D.1 Brief Information from Molecular Spectroscopy

Generally speaking, rotational energy (D.3) should be supplemented with a term allowing for the rotational energy of the electrons, equal to − Bv Λ2 . It remains constant within the limits of the vibrational level and is therefore included in the vibrational energy, so that formula (D.3) remains valid for a simple rotator with Λ = 0. However, the lowermost rotational level in this case proves to be not the one with J = 0, but that with J = Λ. The more detailed classiﬁcation takes account of the character of interactions (couplings) between the motions of the electrons and nuclei. As suggested by Hund (1928) [3], 5 interaction schemes (a, b, c, d, e) are recognized, which correspond to different orders in which the electron momenta and the rotational moment of the molecule are composed. They are described in detail in practically all books on molecular spectroscopy [2–4]. In practice, one encounters cases a and b, less frequently c and d. In case a, the relationships between the velocities of the particles are such that most strongly coupled to the axis are both the total orbital momentum L and the total spin momentum S, which compose a joint momentum whose projection on the axis of the molecule is Ω = Λ + Σ. This momentum as a whole is combined with the rotational moment R of the molecule to form the total momentum J. In this case, the rotation of the molecule as a whole in the coordinate system associated with its nuclei practically does not manifest itself. Even in the ﬁrst relation (assuming the absence of rotations) the spin-orbit interaction leads to the splitting of the electronic terms, (D.1). In this case, the expression for the rotational energy should be supplemented not with the term − Bv Λ2 , but with − Bv Ω2 . As in the general case, this term is included in the vibrational energy, but the lowermost level will then be the rotational level with J = Ω. In case b, the spin momentum is characteristically only weakly coupled to the axis, its projection on the latter is undeﬁned, as distinct from Λ. Therefore, it is the joint momentum N = R + Λ that is formed ﬁrst and then N and S form J = N + S. For the quantum numbers, J = N + S, N + S − 1, . . . | N − S|

(D.5)

Obviously all the Σ-states with Λ = 0 and all singlet states with S = 0 are classed with this case. This is the most widespread coupling type. Common to cases a and b is the weak interaction between the orbital motion of the electrons and the molecular rotation, though its manifestation in the next approximation (D.6) leads to the removal of the Λ degeneracy of the electronic states. Such a ΔEΛ splitting is called the lambda

507

508

Appendix D Information on Some Diatomic Molecules

doubling. For some states, it has the form [6] ΔEΛ = pJ ( J + 1)

for the 1 Π state,

(D.6a)

ΔEΛ = q( J + 1/2)

2

(D.6b)

2

(D.6c)

ΔEΛ ≈ 0

for the Π1/2 state, for the Π3/2 state.

As for the order of magnitude, p ∼ Bv2 /ΔEEE , q ∼ ABv /ΔEEE , where ΔEEE is the energy difference between the state of interest and the next one. By virtue of the lambda doubling, for each split level J there exist different sets of indices +, −, g, and u (see Table A.7 of Appendix A). In case c, the strongest is the spin-orbit interaction. These momenta determine the joint momentum Ja = S + L with the projection Ω on the axis. The total momentum J = Ja + R. The quantum number Λ here is only poorly determined, and so it would be more logical to classify the states belonging to this coupling type not in accordance with Λ, as in cases a and b, but according to Ω, which is written down, for example, as 0+ . However, both designations are frequently retained. For example, the term with an energy of 15769.01 cm−1 for the I2 molecule is designated B3 Π(0+ u ) [1]. This is in keeping with the tradition of atomic spectroscopy, where the designation of the LS-type coupling is retained for the states with the J J-, JL-, and jJ-type couplings. The frequency structure of transition spectra is determined by the selection rules for the quantum numbers. These rules are derived from calculations of the matrix elements of the electric moment operator of the molecule in various approximations as to multiplicity (see e.g. [3, 4]). Here we will restrict ourselves to the most important, dipole approximation. D.1.1 General Rules

For molecules with zero nuclear spin ΔJ = 0, ±1,

except for

J = 0 ← I → J = 0.

If the nuclear spins are other than zero and the total momentum F = J + In , ΔF = 0, ±1,

except for

F = 0 ← I → F = 0.

Parity selection rules

+↔− As regards the latter rules, to avoid confusion, we would like to emphasize that the plus (+) or the minus (−) sign relate to a concrete rotational

D.1 Brief Information from Molecular Spectroscopy

level. For example, in the 1 Σ+ state, the rotational levels of even N are positive, whereas those of odd N, negative (Table A.7; S = 0, J = N). D.1.2 More Particular Rules

For molecules with identical nuclei s←I→a g↔u Additionally for the a-, b-, and intermediate-type couplings ΔΛ = 0, ±1 Σ+ ← I → Σ− ΔS = 0, ΔΩ = 0, ±1, for the a-type coupling ΔΣ = 0, for the b-type coupling ΔN = 0

for transitions, except for

Σ ↔ Σ,

for the c-type coupling ΔΩ = 0, ±1. The above rules hold true if the combining states are of the same coupling type. If the couplings differ, the rules common to both these states then apply. The Hund coupling types describe limiting cases. For real molecules, one can only talk about approximation to some or other scheme. Moreover, the coupling type of the given electronic–vibrational state can change with the serial number of the rotational level, for it is exactly the electronic momenta and nuclear rotation composition hierarchy that forms the basis for the coupling classiﬁcation. The “purity” of the coupling type can be characterized by the ratio between the spin-orbit interaction constant (D.1) and the rotational constant, Y = Av /Bv : for the a-type coupling, Y J ( J + 1)

(D.7)

509

510

Appendix D Information on Some Diatomic Molecules

and for the b-type coupling, Y N ( N + 1).

(D.8)

The states with rotational levels whose serial numbers fall within this interval have an intermediate type of coupling (a, b). The selection rules show only whether the transition matrix elements are equal to zero (in the dipole approximation) or not. To describe line intensities, it is necessary to know the magnitudes of these elements. In this appendix, we will present data on the vibrational matrix elements (Franck–Condon factors) for a limited number of select molecules, for which selection rules are lacking. The data on the line intensity factors for the rotational structure of the electronic–vibrational spectra of various types of transitions are of a more general character and are contained in Appendix E. Selection rules indicate that an electronic–vibrational–rotational spectrum consists of a system of electronic transitions, within which, as the resolution is improved, a structure of components is revealed, on account of transitions between vibrational and rotational levels. The aggregate of the rotational structure fragments within the limits of a transition between the vibrational levels of two electronic states is called a band, hence the name “band” given to molecular spectra. If resolution is high enough, this ﬁne rotational structure of the band will be revealed as well. The arrangement of lines within the band is governed by the selection rules for the numbers describing the quantization of the rotational energy. These are referred to as groups (branches) of spectral lines: ⎧ ⎪ ⎨0 ΔJ = +1 ⎪ ⎩ −1

Q branch, R branch, P branch.

The branch designations are supplemented with the parenthesized value of J = J , where the double prime denotes the lower level of the transition, for example, P(20). To indicate the changes of the other quantum numbers, other than J, which relate to rotational motions, use is made of additional indices and symbols. Figure D.1 is presented as an example to help explain them. The ﬁgure shows transitions between electronic states of purely b-type coupling. If we denote the electronic term components with J = N + 1/2 as F1 and those with J = N − 1/2 as F2 , the four main branches of the b-b-type transitions in emission will be grouped as

D.1 Brief Information from Molecular Spectroscopy

Figure D.1 Rotational structure of b − b transitions.

follows: R1 : F1 ( N ) − F1 ( N ) = F1 ( J = N + 1/2) − F1 ( J = N + 1/2), ΔJ = ΔN = +1 R2 : F2 ( N ) − F2 ( N ) = F2 ( J = N − 1/2) − F2 ( J = N − 1/2), ΔJ = ΔN = +1 P1 : F1 ( N ) − F1 ( N ) = F1 ( J = N + 1/2) − F1 ( J = N + 1/2), ΔJ = ΔN = −1 P2 : F2 ( N ) − F2 ( N ) = F2 ( J = N + 1/2) − F2 ( J = N + 1/2), ΔJ = ΔN = −1 To designate satellite branches with ΔJ = ΔN, use is made of an additional index indicating that the grouping of the satellite branch is close

511

512

Appendix D Information on Some Diatomic Molecules

to those of the R or the P branch: R

Q21 : F2 ( N ) − F1 ( N ) = F2 ( J = N − 1/2) − F1 ( J = N + 1/2), ΔJ = 0, ΔN = +1

P

Q12 : F1 ( N ) − F2 ( N ) = F1 ( J = N + 1/2) − F2 ( J = N − 1/2), ΔJ = 0, ΔN = −1.

The subscripts 21 or 12 indicate the F2 → F1 transition or vice versa. The N-based numbering is convenient for the b-type coupling. However, within the limits of an electronic–vibrational band, wherein the atype coupling becomes established as N is reduced, the designations are formally retained unchanged. As the molecular rotation increases, the Q12 , R12 , P21 and Q21 branches of the a-coupling transitions become the P Q , Q R , Q P and R Q 12 12 21 21 satellite branches of the b-coupling transitions. As before, the symbols P, Q and R denote the selection rules as to J. The R21 and P12 branches of the a-coupling transitions have their ΔN = ±2 and correspond to the S R21 and O P12 branches forbidden for the b-coupling transitions. They are observed at small N values and rapidly lose their intensity as N increases. In the case of a-type coupling, electronic term components are frequently denoted by the lower-case letter f, rather than the capital F. If transitions take place between b-type states, the letter J in the branch symbols is frequently directly replaced by N without primes, the upper and lower electronic terms being distinguished by primes. In this case, N denotes the lower level of the transition, and the branch notations equivalent to those presented above have the form R1 ( N ) = F1 ( N + 1) − F1 ( N )

R2 ( N ) = F2 ( N + 1) − F2 ( N ) P1 ( N ) = F1 ( N − 1) − F1 ( N ) P2 ( N ) = F2 ( N − 1) − F2 ( N ) R

Q21 ( N ) = F2 ( N + 1) − F1 ( N )

P

Q12 ( N ) = F1 ( N − 1) − F2 ( N )

In the text below, we present reference data on the structure of terms and the form of spectra of a number of diatomic molecules that are most frequently used in practical spectroscopic plasma diagnostics. In the tables listing the molecular constants, the values of Te , ωe , ωe xe , Be , αe , and A are given in cm−1 , radiative lifetimes τ, in ns (10−9 s). If not otherwise indicated in the box heading, the values of De are given as De × 106 cm−1 . These data were predominantly borrowed from [1, 6]. A most complete collection of data on the Franck–Condon factors can be found in [7].

D.2 Nitrogen N2 , N2+

D.2 Nitrogen N2 , N2+ D.2.1 Electronic States, Electronic Transition Systems (Bands)

Figure D.2 presents a detailed schematic diagram of the potential curves of the neutral nitrogen molecule and the nitrogen ion. A great number of transitions between these states have been studied. Figure D.3 shows those bearing their own names in the literature. Most frequently used for diagnostics purposes are the following vibrational–rotational transition systems C3 Π–B3 Π – the second positive (2+ ) system and B3 Π–A3 Σ) – the ﬁrst positive (1+ ) system of transitions in the neutral molecule and B2 Σ–X2 Σ) – the ﬁrst negative (1− ) system of transitions in the ion. D.2.2 Molecular Constants of the Ground and Combining States

Table D.1 lists the molecular constants. The nuclear spin of the nitrogen atom is InN = 1. The total spin I of the N2 molecule can be equal to 0, 1, and 2. The ratio between the statistical weight, gs , of the symmetric rotational levels to that, ga , of the antisymmetric ones is gs /ga = 2/1. Even values of I correspond to symmetric rotational levels, while odd ones, to antisymmetric levels (see formulas (A.8), (A.9) and Tables A.6 and A.7). D.2.3 Second Positive (2+ ) System

The transition strength Sˆe = (249 ± 19)(1 − 1.9583rv v + 0.8602)2 , where the quantity rv v ranges between 1.03 and 1.3. For r00 = 1.1843, Sˆe = 3.17 ± 0.24. The oscillator strength f e = 5 × 10−2 .

513

514

Appendix D Information on Some Diatomic Molecules

Figure D.2 Potential curves of N2 and N2+ .

D.2 Nitrogen N2 , N2+

Figure D.3 Some bands and systems of transitions in N2 and N2+ .

515

59619.3

59619.3

89136.9

A3 Σ + u

B3 Π g

C3 Π

0*

25461.4

X2 Σ + g

B2 Σ+ u

N2+

0

Te

X1 Σ + g

N2

State

2419.84

2207

2047.17

1733.39

1460.64

1516.88

ωe

23.18

16.1

28.445

14.122

13.87

12.18

ωe xe

2.075

1.932

1.825

1.637

1.454

1.473

Be

0.024

0.019

0.019

0.018

0.018

0.017

αe

6.17

6.1

5.9

6.1

5.76

De

39.2 *

42.2 *

A

60 *

37 **

5000

*

τ

Table D.1 Molecular constants of the N2 , N2+ ground and combining states.

* τ = τv=0

* 125666 cm−1 relative to N2 (X1 Σ+ g )

* Av=0 = 39; Av=4 = 34.5 τv=0 = 35; τv=2 = 41

* Av=0 = 42.3; Av=12 = 41.3

* 1.3 × 109 for F2 levels; 2.5 109 for F1 , F3 levels

Note

516

Appendix D Information on Some Diatomic Molecules

D.2 Nitrogen N2 , N2+

517

Figure D.4 Fragment of the vibrational structure of a spectrum of the 2+ system of N2 . H2 −N2 plasmatron.

D.2.3.1 Vibrational Structure of the C3 Π(v )–B3 Π(v ) Transition

The Franck–Condon factors are listed in Table D.2 [7]. Table D.2 Franck–Condon factors for C3 Πu –B3 Π g transition. v \v

0

1

2

3

4

5

6

7

8

9

10

11

12

0

4.55-1 3.31-1 1.45-1 4.94-2 1.45-2 3.87-3 9.68-4 2.31-4 5.36-5 1.21-5 2.61-6 5.23-7 9.1-8

1

3.88-1 2.29-2 2.12-1 2.02-1 1.09-1 4.43-2 1.52-2 4.68-3 1.33-3 3.57-4 9.15-5 2.25-5 5.22-6

2

1.34-1 3.35-1 2.3-2 6.91-2 1.69-1 1.41-1 7.72-2 3.32-2 1.23-2 4.12-3 1.27-3 3.69-4 1.03-4

3

2.16-2 2.52-1 2.04-1 8.81-2 6.56-3 1.02-4 1.37-1 9.93-2 5.26-2 2.31-2 8.95-3 3.16-3 1.03-3

4

1.15-3 5.66-2 3.26-1 1.13-1 1.16-1 2.45-3 4.7-2 1.09-1 1.04-1 6.67-2 3.4-2

1.5-2 5.97-3

Note. The notation 4.55-1, for example, means 4.55 × 10−1 .

Figure D.4 [5] presents a fragment of the spectrum of the 2+ system of N2 , with its rotational structure not resolved. Indicated are the wavelengths of the band edges and the values of v , v . D.2.3.2 Rotational Structure

In both electronic states, the quantity Y = Av /Bv ≈ 21–26, and in accordance with formulas (D.7) and (D.8), an intermediate type of coupling

518

Appendix D Information on Some Diatomic Molecules

Figure D.5 Rotational structure of the bands of the 2+ system of N2 .

is realized: the a-type coupling at weak rotation rapidly changes, as rotation increases, to the b-type at J, N ≈ 4–6. To denote spectral lines, it is convenient to use the number N of the lower state. The structure of transitions is shown in Figure D.5. The rotational terms F1 , F2 and F3 relate respectively to the 3 Π0 , 3 Π1 , and 3 Π2 components of the states C and B. By virtue of the lambda doubling, each term is a doublet. The doublet components are designated F and F . According to the selection rules, the following 16 main branches of the 3 Π-3 Π transition are possible:

D.2 Nitrogen N2 , N2+

Pi ( J ), i = 1, 2, 3

Ri ( J ), i = 1, 2, 3

Fi ( J = J − 1 ) − Fi ( J = J )

Fi ( J = J + 1) − Fi ( J = J )

Fi ( J = J − 1) − Fi ( J = J )

Fi ( J = J + 1) − Fi ( J = J )

Qi ( J ), i = 2, 3 Fi ( J = J ) − Fi ( J = J ) Fi ( J = J ) − Fi ( J = J ) Apart from these branches, there can also be satellite bands (not shown in the ﬁgure) with ΔJ = ΔN, for example, Q12 : F1 ( J ) − F2 ( J ); P23 : F2 ( J − 1) − F3 ( J ), and so on. ¨ The Honl–London rotational line intensity factors S J J are calculated by the formulas of Section E.3.1 of Appendix E. The calculations made in [9] show that the use of the general formulas for the intermediate type of coupling and the formulas for the b-b transitions in the case of transitions with J > 8 gives a difference of less than 1% in S J J . Figure D.6 shows a spectrogram of the (v = 0 – v = 0) band. Note that the alternation of the intensities of adjacent triplets observed in the ﬁgure (in the R branch in the given case) is not described by the S J J factors. This effect is due to the speciﬁcities of the elementary processes of excitation of the spectrum and is discussed in Section 4.2.4. To identify lines in the spectral band structures of the 2+ system, we also recommend using the table of rotational line frequencies presented in [8]. D.2.4 First Positive (1+ ) System

The transition strength Sˆe = (92 ± 16)(1 − 1.278rv v + 0.410r2v v + 0.02r3v v )2 , where the quantity rv v ranges from 1.03 to 1.3. For r00 = 1.2536, Sˆe = 0.61 ± 0.1. The oscillator strength f e = 6.1 × 10−3 . D.2.4.1 Vibrational Structure of the B3 Π g (v )–A3 Σ+ u ( v ) Transition

The Franck–Condon factors are listed in Table D.3. D.2.4.2 Rotational Structure

In the A3 Σ, the coupling type is b. Each rotational level corresponds to a certain value of N. The presence of the spin S = 1 splits the levels into three sublevels with J = N, N ±1. In the B3 Π g state, the coupling is of intermediate type, no strict selection rules (see D.1.1, D.1.2) as to N exist,

519

520

Appendix D Information on Some Diatomic Molecules

Figure D.6 Rotational structure of a band sequence of the 2+ system of N2 . N2 −Ar (1 : 50) discharge in a 19 mm-dia. tube at a pressure of 0.667 hPa (0.5 Torr) and a discharge current of 20 mA.

D.2 Nitrogen N2 , N2+

521

Table D.3 Franck–Condon factors for B3 Π–A3 Σ transition. v \v

0

1

2

3

4

5

6

7

8

9

10

11

12

0

4.01-1 3.31-1 1.66-1 6.69-2 2.38-2 7.99-3 2.57-3 8.21-4 2.60-4 8.25-5 2.63-5 8.49-6 2.8-6

1

3.98-1 2.91-1 1.59-1 1.97-2 1.31-1 6.58-2 2.83-2 1.13-2 4.27-3 1.57-3 5.7-4 2.07-4 2.62-5

2

1.62-1 2.74-1 6.88-2 2.21-2 1.25-1 1.43-1 1.01-1 5.61-2 2.72-2 1.21-2 5.15-3 2.12-3 8.6-3

3

3.42-2 2.76-1 9.56-2 1.52-1 5.12-3 4.24-2 1.07-1 1.12-1 8.06-2 4.75-2 2.49-2 1.21-2 5.61-3

4

4.06-3 9.74-2 2.98-1 7.47-3 1.51-1 5.12-2 2.22-3 5.49-2 9.43-2 8.99-2 6.51-2

5

2.68-4 1.64-2 1.69-1 2.42-1 1.05-2 9.52-2 9.43-2 8.78-3 1.41-2 5.99-2 8.2-2 7.44-2 5.45-2

6

1.34-5 1.41-3 3.93-2 2.31-1 1.56-1 5.84-2 3.52-2 1.04-1 3.93-2 2.02-6 2.55-2 5.94-2 7.09-2

7

4.46-7 6.18-5 4.32-3 7.19-2 2.66-1 7.41-2 1.06-1

8

2.79-7 8.72-7 2.22-4 9.95-3 1.12-1 2.69-1 1.99-2 1.28-1 4.81-3 4.44-2 7.7-2 3.27-2 8.25-3

3-3

4-2

2.23-2

8.13-2 6.74-2 1.04-2 4.15-3 3.28-2

9

1.09-7 2.81-8 6.74-6 6.23-4 1.91-2 1.56-1 2.49-1 1.68-4 1.2-1 2.94-2 1.34-2 6.55-2 5.19-2

10

1.04-7 6.3-11 2.17-7 1.62-5 1.41-2 3.26-2 1.99-1 2.05-1 1.03-2 8.95-2 5.92-2 2.5-4 4.2-2

11

6.96-7 2.22-8 5.42-7 2.48-7 1.54-5 2.8-3 5.12-2 2.38-1 1.52-1 3.93-2 5.17-2 7.96-2 6.06-3

12

7.34-8 5.98-9 3.93-8 5.2-10 7.11-7 9.59-5 5.08-3 7.32-2 2.59-1 9.6-2 7.22-2 1.99-2 8.33-2

and transitions are noted with ΔN = 0, ±1, ±2, ±3. As in the preceding case of the 2+ system, it is convenient to use the number N of the lower state A3 Σ to designate spectral lines. Following the recommendations given in [8] and by analogy with the above-described example of doublet transitions (Figure D.1), the branch symbols are differentiated by a pair of indices for the spin components of the upper and the lower state. If the indices coincide, use is made of only one of them, which corresponds to the main branches ΔJ = ΔN. Thus, there are 27 branches: N13

O13 O12 O23

P13 P12 P1 P23 P2 P3

Q12 Q1 Q23 Q2 Q21 Q3 Q32

R1 R2 R21 R3 R32 R31

S21 S32 S31

T31

To calculate the rotational line intensity factors for the main branches, use should be made of the formulas presented in Section E.3.1 of Appendix E. For the satellite branches, the reader should refer to the references cited in Appendix E. Figure D.7 presents a schematic diagram of the energy levels and transitions, and Figure D.8, a spectrogram of the (1, 0) band.

522

Appendix D Information on Some Diatomic Molecules

Figure D.7 Rotational structure of the bands of the 1+ system of N2 .

D.2.5 First Negative (1− ) System

The transition strength Sˆe = (289 ± 3)(1 − 1.631rv v + 0.704r2v v )2 , where the quantity rv v ranges from 0.974 to 1.265. For r00 = 1.097, Sˆe = 0.97 ± 0.01. The oscillator strength f e = 3.8 × 10−2 . D.2.6 2 + Vibrational Structure of the B2 Σ+ u ( v )–X Σ g ( v ) Transition

The Franck–Condon factors are listed in Table D.4. D.2.6.1 Rotational Structure [11, 12]

The Σ states both relate to the b-type coupling, present in the spectrum are the P and R branches (ΔN = ±1). Each line contains 3 spin com-

D.2 Nitrogen N2 , N2+

523

Figure D.8 Rotational structure of the (1,0) band of the 1+ system of N2 . N2 −He (1 : 9) discharge in a 19 mm-dia. Tube at a pressure of 0.933 hPa (0.7 Torr) and a discharge current of 30 mA. Table D.4 Franck–Condon factors for B2 Σtu –x2 Σt g transition. v \v

0

1

2

3

4

5

6

2-5

8

9

10

11

12

0

6.51-1 2.59-1 7.02-2 1.6-2

1

3.01-1 2.23-1 2.86-1 1.32-1 4.27-2 1.14-2 2.7-3 5.86-4 1.18-4 2.24-5 3.93-6 6.29-7 8.75-8

2

4.54-2 4.06-1 5.06-2 2.29-1 1.65-1 7.11-2 2.36-2 6.69-3 1.69-3 3.93-4 8.44-5 2.67-5 3.03-6

3

2.25-3 1.06-3 4.14-1 2.1-3 1.56-1 1.71-1 9.45-2 3.8-2 1.26-2 3.67-3 9.64-4 2.32-4 5.16-5

4

1.45-5 6.93-3 1.66-1 3.79-1 6.73-3 9.29-2 1.57-1 1.09-1 5.24-2

5

4.63-7 3.99-5 1.34-2 2.2-1 3.31-1 2.92-2 4.81-2 1.33-1 1.16-1 6.49-2 2.83-2 1.04-2 3.41-3

6

9.48-9 3.09-6 5.73-5 2.07-2 2.67-1 2.83-1 5.33-2 2.04-2 1.06-1 1.15-1 7.44-2 3.65-2 1.5-2

7

6.4-10 3.42-8 1.13-5 4.93-5 2.79-2 3.07-1 2.41-1 7.24-2 5.94-3 8.08-2 1.08-1 8.05-2 4.41-2

8

4.2-13 5.75-9 4.95-8

9

1.3-12 3.9-12 2.66-8 1.74-8 6.42-5 1.07-6 3.89-2 3.69-1 1.84-1 9.07-2 5.55-4 4.07-2 8.63-2

10

3.9-14 1-11 1.7-10 8.36-8 3.29-8 1.16-4 8.09-5 4.15-2 3.93-1 1.67-1 9.14-2 3.6-3 2.71-2

11

2.9-16 7-13 3.9-11 1.68-9 1.97-7 6.71-7 1.82-4 3.87-4 4.17-2 4.15-1 1.58-1 8.82-2 7.9-3

12

5.3-16 2.5-15 5.6-12 7.8-11 9.18-9 3.59-7 3.52-6 2.51-4 1.09-3 3.93-2 4.34-1 1.55-1 8.2-2

3-5

3.3-3 6.34-4 1.15-4

7

3.28-6 5.03-7 7.03-8 8.42-9 7.5-10

2-2

6.6-3 1.94-3 5.21-4

1.81-5 3.42-2 3.4-1 2.08-1 8.47-2 4.74-4 5.86-2 9.82-2 8.32-2

524

Appendix D Information on Some Diatomic Molecules

Figure D.9 Rotational structure of the bands of the 1− system of N2+ .

ponents according to transitions with ΔJ = 0, ±1. For one of the components (ΔJ = 0), ΔJ = ΔN, and it is denoted, as before, by two indices. The structure of the transition is shown in Figure D.9. Figure D.10 presents the resolved rotational structures of the (0, 0) and (0, 1) vibrational bands. The rotational line intensity factors for the bands should be calculated by the formulas presented in Section E.2.1, with due regard for formula (E.10), of Appendix E.

D.2 Nitrogen N2 , N2+

Figure D.10 Rotational structure of (a the (0,1) band and (b) the (0,0) band of the 1− system of N2+ . RF (35 MHz) N2 discharge in a 41 mm-dia. tube 170 mm in length at a pressure of 66.660 hPa (50 Torr). Power 2.6 kW, rotational temperature 5500 K.

525

526

Appendix D Information on Some Diatomic Molecules

D.3 Carbon Oxide CO D.3.1 Electronic States, Electronic Transitions

The potential curves are schematically shown in Figure D.11a. The bestknown electronic transition systems of the molecule are illustrated in Figure D.11b. D.3.2 Molecular Constants of the Ground and Combining States

The molecular constants of the ground and combining states of the CO molecule are listed in Table D.5. Table D.5 Molecular constants of the CO ground and combining states. State

Te

ωe

ωe xe

Be

αe

De

τ

Note

X1 Σ + g

0

2169.81

13.288

1.931

0.019

6.36

A1 Π

65075.7

1518.2

19.4

1.611

0.023

7.33

10.7 *

* v = 0; for v = 1–6;

B1 Σ+

86945.2

2112.7

15.2

1.961

0.026

7.1

21.8 *

*v=0

A

τ = 9–10.5

˚ ¨ singlet system B1 Σ+ –A1 Π located in Being convenient, the Angstr om the visible region of the spectrum is frequently used in plasma diagnostics. D.3.3 ˚ ¨ Bands System B1 Σ+ –A1 Π Angstr om

The transition strength Sˆe = (0.133 ± 0.025)(1 + 1.745rv v + 0.768r2v v )2 , where the quantity rv v ranges from 1.12 to 1.19. For r00 = 1.177, Sˆe = 0.52 ± 0.09. The oscillator strength f e = 2.1 × 10−2 . D.3.3.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.6. D.3.3.2 Rotational Structure of the B1 Σ+ –A1 Π Bands [3, 9]

˚ ¨ bands system, the spin S = 0 in both the upper and In the Angstr om the lower state, and so the b-type coupling (after Hund) is realized here. A certain value of N is ascribed to each rotational level. The transition structure is shown in Figure D.12. Figure D.13 presents the resolved rota-

D.3 Carbon Oxide CO

Figure D.11 (a) Potential curves of the CO molecule. (b) Some bands and systems of transitions in CO.

527

528

Appendix D Information on Some Diatomic Molecules Table D.6 Franck–Condon factors for B Σ–A Π transition. v \v 0

0

1

2

3

4

5

6

7

8

9

10

11

12

8.9-2 1.82-1 2.1-1 1.83-1 1.34-1 8.71-2 6.21-2 2.94-2 1.59-2 8.35-3 4.29-3 2.17-3 1.09-3

1

2.5-1 1.76-1 3.04-2 4.2-3 5.21-2 9.55-2 1.07-1 9.31-2 7.01-2 4.78-2 3.05-2 1.85-2 1.08-2

2

3.09-1 8.33-3 7.1-2 1.17-1 5.21-2 2.65-3 1.13-2 4.51-2 7.04-2 7.68-2 6.87-2 5.42-2

3

2.19-1 8.32-2 1.23-1 2.69-3 4.34-2 5.22-2 9.61-3 1.36-3 2.03-2 4.37-2

4

9.77-2 2.32-1 1.87-3 9.68-2 6.58-2 2.5-4 3.38-2 6.58-2 8.55-2

5

2.84-2 2.02-1 1.05-1 6.93-2 1.59-2 8.49-2 3.58-2 1.2-4

6

5.4-3 9.01-2 2.28-1 1.15-2 1.12-1 5.61-3 3.96-2 6.57-2

7

6.5-4 2.32-2 1.62-1 1.74-1 8.82-3 8.19-2 4.7-2

8

5

9

0

3.52-3 5.64-2 2.13-1 8.86-2 5.66-2 3-4

1.06-2 1.02-1 2.25-1

10

0

1-5

1.06-3 2.38-2

11

5

5.02-3

tional structure of the (0, 1) vibrational band. The rotational line intensity factors should be calculated by the formulas presented in Section E.1.2 of Appendix E.

D.4 Hydrogen H2 and Deuterium D2 D.4.1 Electronic States, Electronic Transitions

Figure D.14 presents a schematic diagram of some potential curves of the H2 molecule [13]. Well revealed in the spectra of hydrogen-containing 1 1 + plasmas are the electronic systems of the d3 Πu − a3 Σ+ g , I Π g –B Σu , and 1 + 1 + G Σ g –B Σu bands. There is a detailed atlas of transition frequencies in hydrogen [14, 15]. D.4.2 Molecular Constants of the Ground and Combining States

The molecular constants of the ground and combining states of hydrogen are listed in Table D.7.

D.4 Hydrogen H2 and Deuterium D2

˚ Figure D.12 Rotational structure of the ACO bands. Table D.7 Molecular constants of the H2 O ground and combining states. State X1 Σ + g B1 Σ + u a3 Σ + g d3 Π u G1 Σ+ g I1 Π g

Te

ωe

ωe xe

Be

αe

De

A

τ

Note

0 4401.23 121.33 60.853 3.062 4.71 91700 1358.09 20.888 20.015 1.184 1.625

0.8 * * for v = 3–11; τ = 0.8–1

95226 2664.83 71.65 34.261 1.671 2.16 *

10.4

112700 2371.58 66.27 30.364 1.545 1.91 * 0.03

31

*v=0 *v=0

112834 2343.9

55.9

28.4

See text

113142 2259.1

78.4

29.25

See text

D.4.3 Ortho- and Para-Modiﬁcations

The nuclear spin of the hydrogen atom is InH = 1/2, and that of the deuterium atom, InD = 1. The total nuclear spin I of the hydrogen molecule can assume a value of 0 or 1 (the para- and orthohydrogen, respectively), and that of the deuterium molecule can be equal to 0, 1 and 2. In accordance with formulas (A.6) and (A.8) and Table A.6a of Appendix A,

529

530

Appendix D Information on Some Diatomic Molecules

Figure D.13 Rotational structure of the (0,1) band of the ˚ ¨ CO system. CO discharge in a 19 mm-dia. tube at a Angstr om pressure of 1.333 hPa (1 Torr) and a discharge current of 20 mA.

the ratio between the statistical weight gs of the symmetric rotational levels and that ga of the antisymmetric ones is gs /ga = 1/3 for H2 and gs /ga = 2/1 for D2 . The even I values correspond to symmetric rotational levels, while the odd ones correspond to antisymmetric rotational levels [3]. The rotational constant is large, so that the rotational structure of the spectrum can be resolved even with a spectral instrument of moderate dispersion. The vibrational band intensity is found by summation over the rotational line intensities. The electronic ground state X1 Σ+ g is of the b-type coupling. Each rotational level corresponds to a certain value of N. For parahydrogen and the D2 modiﬁcations with I = 0, 2, N is even, while for orthohydrogen and D2 with I = 1, N is odd. D.4.4 Fulcher-α Bands System d3 Πu –a3 Σ+ g

The transition strength Sˆe = 9.9 and the oscillator strength f e = 0.17 are obtained as transition averages from measurements of the lifetime τ on the assumption that Sˆe (rv v ) = const. The relative values Sˆe = const(1 + 0.7511rv v )2 are tabulated.

D.4 Hydrogen H2 and Deuterium D2

Figure D.14 Potential curves of the excited states in the H2 molecule.

D.4.4.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.8 D.4.4.2 Rotational Structure

In the d3 Πu state, the triplet splitting is small in comparison with the distance between rotational levels (A/B ≤ 10−3 ), which corresponds to practically pure b-type coupling. By virtue of the lambda doubling, each rotational level is split into two sublevels. One of them (s) relates to parahydrogen and D2 with I = 0, 2, while the other (a) to orthohydrogen and D2 with I = 1.

531

532

Appendix D Information on Some Diatomic Molecules Table D.8 Franck–Condon factors for d3 Πu –a3 Σ+ g transition. v \v

0

1

2

3

4

5

10

11

2.84-1 4.32-1 1.56-2 5.5-4

4-5

1-5

1.1-4 1.93-3 1.57-2 7.25-2 1.91-1 2.15-1 4.84-1 1.81-2

9-4

0

9.31-1 6.86-2 7.8-4

1

6.41-2 7.98-2 1.35-1 2.26-3

2

4.85-3 1.17-1 6.75-1 1.99-1 4.33-3

3

3.9-4 1.42-2 1.57-1 5.6-1 2.61-1 6.87-3

4

3-5

6

7

8

3-5 8-5

1.71-3 2.72-2 1.84-1 4.57-1 3.2-1 9.72-3 1.7-4

1-5

2-4

4.48-3 4.25-2 1.98-1 3.64-1 3.77-1 1.27-2 3.1-4

6

2-5

7.3-4

8 9

2-5

9.1-3 5.82-2

2-1

2-5

4.1-4 4.17-3 2.38-2 8.33-2 1.74-1 1.58-1 5.34-1 9-5

12

1-5

5 7

9

2-2

9-5

2-5

1.39-3 1.8-4

1.13-3 7.72-3 3.26-2 8.91-2 1.5-1 1.12-1 5.83-1 2.09-2 2.05-3

10

9-5

8.9-4 4.98-3 1.83-2 4.65-2 1.23-1 7.65-2 6.31-1 2.05-2

11

3-5

3.3-4 2.06-3 8.37-3 2.37-2 8.23-2 9.4-2 5.01-2 6.79-1

12

1-5

1.3-4

9-4

3.98-3 1.22-2 4.83-2

7-2

6.66-2 3.16-2

The a3 Σ+ g is also of the b-type coupling. Because of the spin-nuclearrotation interaction, the triplet splitting is exceedingly small. All the components of narrow triplet terms with the same N are of equal symmetry. For parahydrogen and D2 with I = 0, 2, the N values are even, while those for orthohydrogen and D2 with I = 1, odd. In accordance with formulas in Sections D.1.1 and D.1.2 of the present appendix, the selection rules for the d3 Πu − a3 Σ+ g transition are as follows: ΔJ = 0, ±1, except for J = 0 ←P→ J = 0, ΔN = 0, ±1. Thus, nine main branches (ΔJ = ΔN) and ten satellite ones (ΔJ = ΔN) are possible. Figure D.15 presents a schematic diagram of transitions (only the main branches are shown). Since the triplet splitting is small, single lines are registered in transitions with the same N and N , that is, only three branches are observed in the spectrum, namely, P (ΔN = −1), Q (ΔN = 0) and R (ΔN = +1). Since it is only levels of the same symmetry that can combine (see D.1.2) transitions starting at the rotational levels of the d3 Πu state with even N values in parahydrogen and D2 with I = 0, 2 correspond to the Q branch, while those starting at the levels with odd N values to the P and R branches. For orthohydrogen and D2 with I = 1, the situation is quite the reverse. Figure D.16 presents the spectrum of the rotational components of the Fulcher-α bands sequence Δv = 0. The spectrum was registered in a hydrogen discharge at a pressure of 0.667 hPa (0.5 Torr) and a discharge current of 20 mA in a 20 mm-dia. tube with water-cooled walls.

D.4 Hydrogen H2 and Deuterium D2

Figure D.15 Rotational structure of the Fulcher-α bands system of H2 .

¨ The Honl–London factors are calculated by the formulas presented in Section E.3.2 of Appendix E. Considering the above-described properties of combining states, these formulas assume the following simple form for the P, Q and R branches: S N +1,N = N/2,

S NN = (2N + 1)/2,

S N −1,N = ( N + 1)/2.

D.4.5 I1 Π g – B 1 Σ + u Transition D.4.5.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.9.

533

534

Appendix D Information on Some Diatomic Molecules

Figure D.16 Rotational structure of a sequence of Fulcher-α bands of H2 . Hydrogen discharge in a 20 mm-dia. tube at a pressure of 0.667 hPa (0.5 Torr) and a discharge current of 20 mA.

Table D.9 Franck–Condon factors for I Π g –B Σ+ u transition. v \v

0

1

2

3

4

5

6

7

8

9

10

2.3-3

8-4

3-4

1-4

1-4

2-3

9-4

0

4.66-1 3.13-1 1.4-1 5.28-2 1.85-2 6.4-3

1

3.9-1 2.48-2 1.89-1 1.85-1 1.12-1 5.51-2 2.47-2 1.06-2 4.6-3

2

1.26-1 3.6-1 2.64-2 4.82-2 1.3-1 1.25-1 8.45-2 4.78-2 2.48-2 1.24-2 6.2-3

3

1.67-2 2.55-1 2.28-1 9.88-2

4

5-4

9-4

6

1-4

12

5-4

2-4

3.1-3 1.6-3

5.89-2 9.81-2 8.95-2 6.3-2 3.89-2 2.24-2 1.25-2 6.9-3

4.61-2 3.44-1 1.32-1 1.27-1 9.9-3

5 7

8-4

11

1.6-3 5.92-2 7.46-2 6.48-2 4.68-2 3.06-2 1.89-2

7.16-2 4.04-1 9.05-2 1.16-1 2.84-2 1.5-3 2.96-2 5.31-2 5.62-2 4.69-2 3.43-2 2-4 4-4

7.75-2 4.58-1 9.41-2 8.56-2 3.9-2

2-4

1.32-2 3.47-2 4.39-2 4.14-2

9-4

5.39-2 4.97-1 1.49-1 4.65-2 4.4-2

8

5-4

9.4-3 1.14-2 4.68-1 2.76-1 8.6-3 5.47-2

9

2-4

0

1.1-3

5.7-3 2.25-2 3.24-2 2-4

2.7-3 1.61-2

2.22-2 1.09-2 2.88-1 4.36-1 7.4-3 9.19-2 3.3-3 1-4

10

3-4

11

2-4

3.4-3 1.26-2 1.09-1 3.71-2 4.17-1 6.74-2 1.78-1 4.49-2 1-4

12

1-4

0

1.15-2 5.1-3 1.36-1 6.59-2 1.1-1 5.38-2 2.09-1 4.5-3

8-4

6.91-2 2.1-3 1.61-1 9.2-3 1.04-2

D.4 Hydrogen H2 and Deuterium D2

535

D.4.5.2 Rotational Structure

The B1 Σ+ u state is characterized by the b-type coupling. For parahydrogen and D2 with I = 0, 2, the N values are even, while for orthohydrogen and D2 with I = 1, odd. The I1 Π g state is of an intermediate (between b and d) type of coupling. There is a highly excited electron; its interaction with the “frame” of the molecule is weak, and its orbital moment is not quantized along the internuclear axis. As the molecular rotation increases, the quantization axis becomes the molecular rotation axis, that is, the interaction between the electronic motion and nuclear rotation plays here a perceptible part. The severance of the interaction between the orbital moment of the electrons with the axis of the molecule (l-uncoupling) results in the splitting of the I1 term into the Π± sublevels. The splitting is great, of the order of the distance between adjacent rotational levels. The Π+ sublevels with odd N and the Π− ones with even N correspond to orthohydrogen and D2 with I = 1, whereas the Π− sublevels with odd N and Π+ ones with even N are associated with parahydrogen and D2 with I = 0, 2. Standard formula (D.3) for calculating rotational terms is inapplicable in this case. Table D.10 gives the values of the energy (in cm−1 ) of some lower electronic–vibrational–rotational levels of the I1 Π g state [14, 15]. Table D.10 Energies (cm−1 ) of low lying electronic-vibrational levels of I Π g state. N

s

a

s

v=0

a

s

v=1

a

s

116225.76

117878.96

v=2

a v=3

1

112064.91

112127.23

114164.13

114215.95

116106.42

2

274.24

139.61

345.75

244.86

341.14

189.52 118019.29

3

264.09

463.04

371.11

494.49

316.01

487.05

80.35

4

695.70

441.12

669.65

544.96

666.92

486.72

249.58

5

671.1

958.43

767.09

701.73

875.04

6

113283.62

7

285.57

8

953.21

115037.29 115353.78

113666.18

117932.18 56.15 118131.3

960.88 117260.9

715.34

117606.4

The selection rules for the transition: ΔN = 0, ±1. Accordingly, the P-, Q- and R-branch lines are observed in the spectrum. A schematic diagram of the transition is presented in Figure D.17. Figure D.18 shows spectrograms of the (0,0) bands of the I1 Π–B1 Σ and G1 Σ–B1 Σ transitions. Also inapplicable to transitions of this type are the standard formulas of Appendix E for the rotational line intensity factors that are essentially the branching factors. The interaction between the electronic motion and molecular rotation means that each electronic–vibrational–

246.04

536

Appendix D Information on Some Diatomic Molecules

Figure D.17 Rotational structure of the I1 Π–B1 Σ transition in H2 .

rotational level has a lifetime of its own. These lifetimes were measured in [18, 19] by the Hanle level-crossing method in a zero magnetic ﬁeld. The results obtained are listed in Table D.11. Table D.11 Radiative lifetimes of the H2 ( I 1 Π g , v = 0) electronic–vibrational–rotational levels. N

1(ortho)

2(ortho)

2(para)

3(ortho)

4(ortho)

5(ortho)

6(ortho)

τ, ns

36 ± 6

34 ± 6

190

85 ± 10

74 ± 28

120

100 ± 30

D.4 Hydrogen H2 and Deuterium D2

537

D.4.6 1 + G1 Σ+ g –B Σu Transition D.4.6.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.12. 1 + Table D.12 Franck–Condon factors for G1 Σ+ g –B Σu transition.

v \v

0

1

2

3

4

5

6

7

2.4-3

6-4

1-4

8

9

1.6-3

5-4

10

11

12

0

5.12-1 3.17-1 1.21-1 3.68-2 9.7-3

1

3.41-1 4.15-2 2.37-1 2.07-1 1.07-1 4.33-2 1.53-2

2

1.19-1 3.05-1 2.61-2 6.61-2 1.69-1 1.51-1 8.97-2 4.31-2 1.84-2 7.4-3

3

2.45-2 2.43-1 1.38-1 1.22-1

4

2.7-3 8.09-2 2.95-1 2.47-2 1.48-1 3.13-2 9.4-3 7.08-2 1.02-1 9.05-2 6.24-2 3.73-2 2.05-2

5 6 7 8

1-4

1.18-2 1.54-1 2.73-1 6-4

1-4 7-4

5-3

7.33-2 1.3-1 1.15-1 7.46-2 4.06-2 1.08-1 7.75-2 3.3-3

2-2

2.74-2 2.23-1 2.19-1 2.49-2 5.33-2 9.36-2 2.76-2

2-4 2.9-3 2-2

1-4 1.2-3 5-4 9.3-3 4.3-3

6.4-2 8.14-2 7.19-2 5.21-2 4-4

2.63-2 5.58-2 6.51-2

1.3-3 4.63-2 2.81-1 1.66-1 5.44-2 1.73-2 8.15-2 5.08-2 5.6-3

4.9-3 2.87-2

1.6-3 6.29-2 3.29-1 1.33-1 6.98-2 2.5-3 5.87-2 6.01-2 1.92-2 1-4

9

1-3

10

2-4

11 12

7.08-2 3.71-1 1.25-1 6.9-2

0

3.84-2 5.74-2 2.99-2

1-4

6.41-2 4.08-1 1.45-1 5.49-2 1.8-3

5-4

1.1-4 4.09-2 4.29-1 2.04-1 3.08-2 4.1-3 1.83-2 6-4

7.2-3 1.01-2 4.02-1 3.11-1

2.5-2 4.92-2 6-3

D.4.6.2 Rotational Structure

As in the case of I1 Π g state, the coupling in the G1 Σ+ g state is of an intermediate (between b and d) type. In parahydrogen and D2 with I = 0, 2, the N values are even, while in orthohydrogen and D2 with I = 1, the values of N are odd. The position of the electronic–vibrational–rotational levels is also not described by standard formula (D.3). The values of the energies (in cm−1 ) of the lower levels are listed in Table D.13. According to the data presented in [1], the lifetimes of electronic-vibrational-rotational levels are τ (v = 0, N = 1) = 27 ns, τ (v = 0, N = 2, 3) = 39 ns. The selection rules: ΔN = ±1. The P- and R-branch lines are observed in the spectrum. A spectrogram of the (0,0) band is presented in Figure D.18 and a schematic diagram of the transition, in Figure D.19.

7.8-3

538

Appendix D Information on Some Diatomic Molecules

Figure D.18 Rotational structure of the (0,0) bands due to the I1 Π–B1 Σ and G1 Σ–B1 Σ transitions. The recording conditions of the top spectrum are indicated in Figure D.16. The bottom spectrum [16] – radiation of a Model DVS25 [17] spectral hydrogen lamp.

D.4 Hydrogen H2 and Deuterium D2

539

Table D.13 Energies (cm−1 ) of low lying electronic-vibrational-rotational levels of G1 Σ+ g state. N 0

s

a v=0 111797.11

6

112379.49

a

116095.65

85.58

206.23

310.58

911.56 979.71

416.46 565.33

737.01

117839.36 117865.75

297.38

500.14 632.99

s v=3

140.36

172.25 112162.01

7 8

114022.86

885.07

5

a

116156.81

38.36

997.49

s v=2

114036.66

819.78

3 4

a v=1

111804.63

1 2

s

761.06

684.18

Figure D.19 Rotational structure of the G1 Σ–B1 Σ transition in H2 .

118098.46

540

Appendix D Information on Some Diatomic Molecules

Figure D.20 Potential curves of NO, NO + and NO – .

D.5 Nitrogen Oxide NO

541

D.5 Nitrogen Oxide NO D.5.1 Electronic States, Electronic Transitions

A detailed schematic diagram of the potential curves of the neutral nitrogen molecule and nitrogen ions is presented in Figure D.21. Shown are the three most characteristic systems of electronic–vibrational–rotational bands, namely, γ, β and δ ones. The upper states for them are A2 Σ+ , B2 Π and C2 Π, respectively. Their common lower state is the ground state X2 Π. As is generally characteristic of radicals, their transitions fall within the near UV and visible regions of the spectrum, which makes them convenient to use in diagnostics, not only from emission, but also from absorption. The γ system is predominantly used in diagnostics. D.5.2 Molecular Constants of the Ground and Combining States

Table D.14 Molecular constants of the NO ground and combining states. State X2 Π1/2 X2 Π

Te 0

ωe xe

Be

αe

De 0.5

1904.2

14.07 1.67

0.017

0.018 10.2 123.3

1.72

τ

A

Note

1904

14.1

43965.7

2374.3

16.11 1.996 0.019

B2 Π1/2 45913.6

1037.2

7.7

1.092 0.012

4.9

B2 Π3/2 45942.6

1039.8

8.3

1.152 0.012

4.9 Av * ** 2000 * Av = 31.3 + 1.15(v + 1/2)

3/2

A2 Σ+

119.82

ωe

5.4

* 215 * v = 0

** v = 0

D.5.3 γ-System (195–340 nm)

The transition strength Sˆe = (3.08 ± 0.44)103 (1 − 2.8986rv v + 2.7499r2v v −0.8597r3v v )2 , where the quantity rv v ranges between 1 and 1.2. For r00 = 1.109, Sˆe = 0.079 ± 0.011, f e = 2 × 10−3 . D.5.3.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.15.

542

Appendix D Information on Some Diatomic Molecules

Figure D.21 γ, β and δ bands of NO. Table D.15 Franck–Condon factors for A2 Σ–X2 Π transition. v \v 0

0

1

2

3

4

5

6

7

8

9

10

11

12

1.62-1 2.62-1 2.37-1 1.61-1 9.2-2 4.69-2 2.21-2 9.81-3 4.18-3 1.72-3 6.93-4 2.74-4 1.07-4

1

3.3-1 1.07-1 5.54-4 6.93-2 1.32-1 1.33-1 9.89-2 6.19-2 3.44-2 1.77-2 8.55-3 3.96-3 1.78-3

2

2.95-1 1.52-2 1.55-1 7.52-2 6.91-4 3.26-2 8.58-2 1.04-1 8.92-2 6.3-2 3.92-2 2.23-2 1.19-2

3

1.51-1 1.98-1 4.55-2 3.95-2 1.12-1 5.02-2 6.02-4 2.21-2 6.37-2 8.35-2 7.8-2 5.97-2 4.02-2

4

4.89-2 2.41-1 4.33-2 1.24-1 1.46-3 5.7-2 8.62-2 3.24-2 1.12-4 1.83-2 5.14-2 5.96-2 6.82-2

5

1.03-2 1.28-1 2.13-1 1.21-3 1.09-1 4.55-2 5.77-3 6.46-2 6.53-2

2

2.62-2 1.67-2 4.35-2

D.5.3.2 Rotational Structure

The A2 Σ+ state is of the b-type coupling. The ground state X2 Π is on an intermediate (between a and b) type of coupling. To denote rota-

D.6 Cyanogen CN

Figure D.22 Rotational structure of the γ bands of NO.

tional transitions, use is usually made of the number N. The transition structure is shown in Figure D.22. The rotational structure of the spectrum of the (0,0) band is illustrated in Figure D.23a [20] and Figure D.23a and b [21]. The spectra were obtained under laser excitation, which correspondeds to a Boltzmann distribution of the molecules among energy levels at room temperature. The rotational line intensity factors are calculated by the formulas presented in Section E.2.2 of Appendix E.

D.6 Cyanogen CN D.6.1 Electronic States, Electronic Transitions

Some potential curves are presented in Figure D.24. Characteristic electronic systems are the violet (B2 Σ+ - X2 Σ+ , 344–460 nm) and the red (A2 Π - X2 Σ+ , 437–1500 nm) ones. The X2 Σ+ state is the ground state.

543

544

Appendix D Information on Some Diatomic Molecules

Figure D.23 Rotational structure of the (0,0) band of the γ system of NO. Laser ﬂuorescence spectrum of the gas at room temperature.

D.6 Cyanogen CN

545

Figure D.24 Potential curves of CN.

D.6.2 Molecular Constants of the Ground and Combining States

Table D.16 Molecular constants of the ground and combining CN states. State X2 Σ + A2 Π

Te

ωe

ωe xe

Be

αe

De

A

0

2068.59

13.09

1.9

0.017

6.4

26.9*)

12.61

1.71

0.017

5.93

* Av

20.2

1.97

0.023

6.6

9245.28 1812.5

τ

Note *v=0

680 **

* v = 52.64 + 0.036v ** average v = 1–9

B2 Σ+ 25752

2163.9

65.6

546

Appendix D Information on Some Diatomic Molecules

D.6.3 Violet System

The transition strength Sˆe = (1.00 ± 0.008)(1 - 0.03rv v )2 , where the quantity rv v ranges from 0.95 to 1.32. For r00 = 1.1658, Sˆe = 0.93 ± 0.07, f e = 3.6 × 10−2 . D.6.3.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.17. Table D.17 Franck–Condon factors for B2 Σt –X2 Σ+ transition. v \v

0

1

2

3

5.8-3

3-4

4

5

0

9.18-1 7.6-2

1

8.09-2 7.79-1 1.24-1 1.43-2 1.2-3

2

1.2-3 1.42-1 6.75-1 1.55-1 2.39-2 2.6-3

3 4 5 6

2.8-3

6

7

8

5-4

1-4 1-3

5.4-3 2.67-1 4.84-1 1.78-1 5.54-2 9.7-3 2-4

8 9 10 11 12

11

12

4-4

1-4

2-4

4.3-3 2.32-1 5.28-1 1.82-1 4.52-2 7.1-3

7

10

1-4

1.9-1 5.93-1 1.74-1 3.42-2 4.6-3 1-4

9

1-4 1.7-3

3-4

5-3

2.93-1 4.58-1 1.64-1 6.39-2 1.23-2 2.4-3

7-4

3.2-3 3.09-1 4.55-1 1.42-1 7.08-2 1.41-2 3.1-3 5-4 1.7-3

9-4 3.1-3

3.12-1 4.75-1 1.15-1 7.66-2 1.48-2 3.7-3 1-4 5-3

2.96-1 5.18-1 8.15-2 8.28-2 1.35-2 5-3

2.54-1 5.8-1 4.95-2 9.01-2

6.7-3 2.04-2 1.9-1 6.45-1 2.27-2 3-4

6.4-3 4.81-2 1.13-1 6.97-1

Figure D.25 shows a spectrum of the vibrational sequence v - v = 0 with only weakly resolved rotational structure. The spectrum was obtained by recording the glow of a laser breakdown spark [22]. The vibrational and rotational temperatures were ca. 8000 K. D.6.3.2 Rotational Structure

Both Σ states are of the pure b-type coupling. Each rotational level corresponds to a certain N value. The effect of the spin S = 1/2 causes each rotational level in each of the electronic states to split into two sublevels, FS1 ( N ) and F2 ( N ), with J = N ± 1/2. The selection rules, Sections D.1.1, D.1.2: ΔJ = 0, ±1, except for J = 0 ←I→ J = 0, ΔN = ±1. Accordingly, 6 branches are possible, namely, P1 , P2 , P12 ( P Q12 ), R1 , R2 and R21 ( R Q21 ). It is precisely this case that has been used to explain the nomenclature of branches (see Figure D.1). The rotational line intensity factors are calculated by the for-

D.6 Cyanogen CN

Figure D.25 Vibrational sequence in the violet band spectrum of CN. Laser breakdown spark.

Figure D.26 Rotational structure of the (0,0) band of CN. CH2 Cl2 ﬂame with an admixture of excited nitrogen.

mulas presented in Section E.2.1. Figure D.26 shows a high-resolution spectrum of the (0,0) band [23] in a CH2 Cl2 ﬂame with an admixture of rf-discharge-excited and partially dissociated nitrogen.

547

548

Appendix D Information on Some Diatomic Molecules

D.7 Carbon Radical C2 D.7.1 Electronic States, Electronic Transitions

Spectra of the C2 radical are clearly manifest in many astronomical, technical and laboratory plasma objects. A schematic diagram of its potential curves is presented in Figure D.27. Some electronic transition systems bear the names indicated in Figure D.28. The Swan bands of the d3 Π g –a3 Πu transition are located in a convenient blue-green region of the spectrum. These bands are frequently used in plasma investigations. The a3 Πu state is close to the ground state and is perceptibly populated even at moderate temperatures, which makes measurements convenient not only in emission, but also in absorption. D.7.2 Molecular Constants of States

Table D.18 Molecular constants for some c2 -states. State

Te

ωe

ωe xe

Be

αe

De

A

τ

X1 Σ + g

0

1854.71

13.34

1.82

0.018

6.92

a3 Πu

716.2

1641.35

11.67

1.63

0.017

6.44

b3 Σ − g A1 Π u d3 Π g C1 Π g D1 Σ+ u

6434.2

1470.4

11.2

1.5

0.016

6.22

8391

1608.35

12.08

1.62

0.017

6.44

20022.5

1788.2

16.44

1.75

0.016

6.7

34261.3

1808

15.81

1.78

0.018

6.8

32

43239.4

1829.6

13.94

1.83

0.02

7.32

14.6

−15.25

−16.9

170

The nuclear spin of the carbon atom is InC = 0, and so there are no antisymmetric rotational levels (see (A.8), (A.9) and Tables A.6 and A.7). D.7.3 Swan Bands System

The transition strength Sˆe = (33 ± 17)(1 − 0.52rv v )2 , where the quantity rv v ranges between 1.12 and 1.488. For r00 = 1.2937, Sˆe = 3.6 ± 1.7, f e = 3.3 × 10−2 . D.7.3.1 Vibrational Structure

The Franck–Condon factors are listed in Table D.19 [7].

D.7 Carbon Radical C2

549

8

12

Figure D.27 Potential curves of C2 .

Table D.19 Franck–Condon factors for d3 Π g –a3 Πu transition. v \v

0

1

2

3

4

5

1.5-3

2-4

6

7

0

7.21-1 2.21-1 4.76-2 8.8-3

1

2.51-1 3.37-1 2.8-1 9.99-2 2.54-2 5.4-3

2

2.72-2 3.74-1 1.38-1 2.62-1 1.38-1 4.53-2 1.19-2 2.02-2 5.2-3

3 4 5 6 7 8

8-4

0

2-4

0

1-3

2.7-3

6-4

9

10

11

1-4 1.2-3

2-4

6.59-2 4.25-1 4.77-2 2.11-1 1.57-1 6.47-2 7.82-2 2.79-2 8.1-3

2-3

4-4

2.2-3 1.05-1 4.45-1 1.43-2 1.58-1 1.59-1 1.53-1 8.37-2 3.4-2 1.08-2 2.9-3 6-4 2.8-3 1.34-1 4.58-1 4.6-3 1.14-1 8.04-2 1.41-1 8.58-2 3.75-2 1.26-2 3.5-3 2-4

2.9-3 1.49-1 4.83-1 4.2-3 1.03-2 5.68-2 1.32-1 8.16-2 3.89-2 1.35-2 8-4

1-4 2.1-3

1.41-1 5.19-1 5.6-2 3.06-2 3.67-2 1.29-1 7.27-2 3.79-2 1.7-3

1.1-1 5.85-2 5.82-1 7.96-2 1.84-2 1.35-1 5.95-2

550

Appendix D Information on Some Diatomic Molecules

Figure D.28 C2 transition systems: 1 – Mulliken; 2 – DelanderAzambuch; 3 – Philips; 4 – Freighmark; 5 – Fox–Herzberg; 6 – Swan; 7 – Ballik–Ramsy.

Figure D.29 presents a fragment of a spectrum of the vibrational sequence v − v = 1 with weakly resolved rotational structure. This spectrum and the CH spectra presented in the text below were obtained on excitation in a low-pressure oxyacetylene ﬂame [23]. D.7.3.2 Rotational Structure

The structure of the main and satellite branches of the transition is, on the whole, similar to the one shown in Figure D.5 for nitrogen. The difference is that the terms are inverted (A < 0, Table D.19), and the fact that the nuclear spin In of the carbon atom is equal to zero should be taken into account in the rotational level statistics. In both the upper and the lower state, the ratio Y = Av /Bv ≈ 10, and according to (D.7), an intermediate type of coupling is realized: the a-type coupling at weak rotation changes, as rotation is increased, to the b type at J, N ≈ 4–5.

D.8 CH Radical

Figure D.29 Fragment of a vibrational–rotational spectrum of the Swan C2 bands system.

Figures D.30a through c show fragments of a spectrum with resolved rotational structure of the vibrational bands. The rotational line intensity factors should be calculated by the formulas presented in Section E.3.1 of Appendix E. At J, N ≥ 4–5, one can resort to simpliﬁcations (E.27), (E.28). As is the case with the spectrum of the 2+ system of nitrogen (Figure D.6), one can see alternation of the intensities of adjacent rotational components in the spectra of Figure D.30b and c (more distinct for the P-branch lines in these cases), which is also associated with the excitation process (Section 4.2.4). Some examples of calculation of line intensities in the rotational structure of the Swan bands at rotational equilibrium at various rotational temperatures can be found in [24].

D.8 CH Radical D.8.1 Electronic States, Electronic Transitions

Figure D.31 shows some low-lying potential curves [25]. A schematic diagram of the energy levels, including the Rydberg ones, and electronic transitions observed is presented in Figure D.32. The transitions of three

551

552

Appendix D Information on Some Diatomic Molecules

Figure D.30 Rotational structure of a sequence of the Swan C2 bands system.

D.8 CH Radical

553

Figure D.31 Potential curves of CH.

electronic systems in the near UV and visible regions of the spectrum involve the doublet ground state X2 Π. D.8.2 Molecular Constants of the Ground and Combining States

Table D.20 Molecular constants of the CH-ground and combining states. State X2 Π

Te 0

ωe

ωe xe

Be

αe

De (10−4 )

2858.5 63

14.46

0.53

14.5

A2 Δ

23189.8

2930.7 96.65

14.93

0.7

15.4

B2 Σ −

26044

2251

C2 Σ +

31801.5

2840.2 125.9

0.72

15.5

230∗

11.16∗∗ ) 14.6

A

τ

Note

27.95

−1

540 380

* ﬂat potential; ** B1

* 5-4 * N = 1–24

554

Appendix D Information on Some Diatomic Molecules

Figure D.32 Electronic transitions observed in CH.

Detailed high-resolution CH spectra were recorded for a low-pressure oxyacetylene ﬂame. They are presented in the atlas [26], and we use them in part in the text below. Figure D.33a–d present an overview spectrum in the region 445– 305 nm. Falling within this spectral region are also the bands of the 1 Π– 1 Π singlet system of C . One should pay attention to the difference in 2 intensity scale between different fragments in these and subsequent CH spectra. In the text below, we will present in more detail the results for the B2 Σ− –X2 Π and C2 Σ+ –X2 Π transitions whose branch system is simpler than that of the A2 Δ–X2 Π transitions, which often proves preferable in diagnostics.

D.8 CH Radical

Figure D.33 Overview CH spectrum in the range 305–445 nm. Oxyacetylene ﬂame.

D.8.3 B2 Σ− –X2 Π Transition

The transition strength Sˆe = 0.166 ± 0.013 and the oscillator strength f e = 3.2 × 10−3 are obtained as transition averages from measurements of the lifetime τ on the assumption that Sˆe (rv v ) = const. The Franck–Condon factors [7] are listed in Table D.21.

555

556

Appendix D Information on Some Diatomic Molecules Table D.21 Franck–Condon factors for B2 Σ− -X2 Π transition. v \ v

0

1

2

3

4

5

6

7

8

9

10

0

8.65-1 1.14-1 2.01-2 1.73-4 1.12-3 4.88-5 8.86-5 2.39-5 1.84-5 1.06-5 7.28-6

1

1.18-1 5.69-1 1.71-1 1.16-1 8.94-3

2

6.39-3 1.97-1

1.4-1

1.5-2

8.04-5 1.89-3 7.07-5 3.36-4 9.03-5

1.13-1 2.47-1 8.85-2 1.16-1 3.28-2 3.75-2 8.08-3 9.8-3

D.8.3.1 Rotational Structure

The structure of the transition is presented in Figure D.32. The rotational line intensity factors are calculated by the formulas presented in Section E.2.2 of Appendix E. Spectrograms with resolved rotational structures of the (1,1), (0,0) and (1,0) bands are shown in Figure D.34a–e. D.8.4 C2 Σ+ –X 2 Π Transition

The transition strength Sˆe = 0.288 and the oscillator strength f e = 7 × 10−3 are obtained as transition averages from measurements of the lifetime τ on the assumption that Se (rv v ) = const. The Franck–Condon factors [7] are listed in Table D.22. Table D.22 Franck–Condon factors for C2 Σ+ -X2 Π transition. v \v

0

1

2

3

4

5

6

7

8

9

10

0

9.99-1 1.92-4 3.31-4 1.87-6 1.05-7 1.08-7 2.14-7 2.64-7 2.52-7 2.12-7 1.67-7

1

1.83-4 9.98-1 6.05-4 1.29-3 1.01-5

2

3.38-4 7.37-4 9.87-1 7.82-3 3.69-3 1.94-5 2.18-5 1.03-5 9.02-6 8.15-6 7.07-6

3

7.01-6 1.05-3 9.51-3

4

8.81-8 1.15-4 1.52-3 3.78-2 8.64-1 7.34-2 2.29-2 2.99-5 2.66-4 8.67-5 7.06-5

5

2.91-8

6

3.38-8 1.67-7 8.51-5 1.62-3

1.7-4

1.9-1

4.82-1

7

1.32-7

2.3-3

9.2-3

2.89-1 2.25-1 2.45-1 1.98-1 2.21-2

8

1.66-7 6.64-7 3.59-6 1.41-4 1.76-3 7.74-4 4.89-2 3.23-1 3.61-2

9

1.47-7 8.44-7 2.56-6 4.17-5 7.79-4 2.67-3 2.01-3 1.28-1 2.19-1 1.02-2 5.73-2

10

1.08-7 7.66-7 2.26-6 1.65-5 3.12-4 2.06-3 5.03-4 3.04-2 1.74-1 4.34-2 6.92-2

4-6 3-7

9.5-1

3.2-6

1.76-6 1.94-6 1.99-6 1.81-6 1.51-6

2.98-2 9.48-3 7.95-6 8.59-5 3.56-5 2.87-5 2.4-5

5.96-4 7.84-4 9.71-2 7.08-1 1.39-1 5.17-2 7.19-4 7.58-4 1.54-4 1.18-5 5.45-4

2.1-1

1.07-1

5-3 1.9-1

2.22-3 3.01-1

Spectrograms with resolved rotational structures of the (0,0) and (1,1) bands are presented in Figure D.35a–c.

D.8 CH Radical

Figure D.34 Rotational structure of the CH(B2 Σ -X2 Π) bands. Oxyacetylene ﬂame.

557

558

Appendix D Information on Some Diatomic Molecules

Figure D.34 (continued).

D.9 Hydroxyl Radical OH D.9.1 Electronic States, Electronic Transitions

The potential curves of the radical are shown in Figure D.36. The spectra of a wide range of plasma objects containing oxygen and hydrogen (even as small admixtures) in their plasma-forming gas, feature the characteristic violet bands of the OH(A2 Σ+ - X2 Π) transition terminating in the ground state.

D.9 Hydroxyl Radical OH

Figure D.35 Rotational structure of the CH(C2 Σ -X2 Π). Oxyacetylene ﬂame.

559

560

Appendix D Information on Some Diatomic Molecules

Figure D.36 Potential curves of OH.

D.9.2 Molecular Constants of the Ground and Combining States

Table D.23 Molecular constants of the OH ground and combining states. State

Te

ωe

ωe xe

X2 Π

0

3737.76

84.88

A2 Σ+32684.1 3178.86

92.92

De (10−14 )

A

18.91 0.72

19.38

* Av

17.36 0.79

20.39

Be

αe

τ

Note * Av = −139.21 − 0.27v

800

The transition strength Sˆe = (0.706 ± 0.017)(1 − 0.75rv v )2 , where the quantity rv v ranges between 0.8 and 1.2. For r00 = 1.008, Sˆe = 0.042 ± 0.001 and f e = 1.2 × 10−3 . The Franck–Condon factors are listed in Table D.24. D.9.2.1 Rotational Structure

The structure of the A2 Σ+ –X2 Π transition is presented in Figure D.37 (cf. Figure D.22 – inverted term). The coupling realized in the A2 Σ+ state is

D.9 Hydroxyl Radical OH Table D.24 Franck–Condon factors for transition. v \v

0

1

2

0

9.07-1

8.94-1

3.58-3

1

8.6-2

7.14-1

1.86-1

1.32-2

2

1.71-1

5.07-1

2.77-1

2.99-2

3

2.31-2

2.41-1

3.01-1

3.67-1

4

2.8-3

5.12-2

2.83-1

3

4

A2 Σ+ –X2 Π

Figure D.37 Rotational structure of the OH(A2 Σ–X2 Π) transition.

of the pure b type. The X2 Π state is of an intermediate type of coupling: a for weak and b for strong rotation. The transition region is at J, N ≈ 10– 12. In that case, at least for not very high J values, the number N has no direct physical meaning, but is retained for level numbering purposes. For this reason, the ﬁgure illustrating the rotational levels of the states X2 Π3/2 and X2 Π1/2 uses the notation f1 and f2 for the a-type coupling. The selection rules: ΔJ = 0, ±1,

except for

J = 0 ← 1 → J = 0.

ΔN = 0, ±1. The transitions satisfying both these rules form intense main branches, while those meeting only one of them form satellite branches whose intensity rapidly drops with increasing J. The following 12 branches are

561

562

Appendix D Information on Some Diatomic Molecules

possible: O12 ( N ) = F1 ( N − 2) − f 2 ( N ),

J − 1 → J (ΔJ = −1, ΔN = −2);

P1 ( N ) = F1 ( N − 1) − f 1 ( N ),

J − 1 → J (ΔJ = ΔN = −1);

P2 ( N ) = F2 ( N − 1) − f 2 ( N ),

J − 1 → J (ΔJ = ΔN = −1);

P12 ( N ) = F1 ( N − 1) − f 2 ( N ), Q1 ( N ) = F1 ( N ) − Q2 ( N ) = F2 ( N ) − Q21 ( N ) = F2 ( N ) − Q12 ( N ) = F1 ( N ) −

f 1 ( N ), f 2 ( N ), f 1 ( N ), f 2 ( N ),

J → J (ΔJ = 0, ΔN = −1); J → J (ΔJ = ΔN = 0); J → J (ΔJ = ΔN = 0); J − 1 → J (ΔJ = −1, ΔN = 0); J + 1 → J (ΔJ = 1, ΔN = 0);

R1 ( N ) = F1 ( N + 1) − f 1 ( N ),

J + 1 → J (ΔJ = ΔN = 1);

R2 ( N ) = F2 ( N + 1) − f 2 ( N ),

J + 1 → J (ΔJ = ΔN = 1);

R21 ( N ) = F2 ( N + 1) − f 1 ( N ), S21 ( N ) = F2 ( N + 2) −

f 1 ( N ),

J → J (ΔJ = 0, ΔN = 1); J + 1 → J (ΔJ = 1, ΔN = 2).

The rotational line intensity factors are calculated by the formulas presented in Section E.2 of Appendix E. It should be recalled, however, that these standard formulas are obtained with the wave function divided into the electronic, the vibrational and the rotational term. As regards the OH radical, this approximation is limited because of the vibrational– rotational interaction. The effect of this interaction on the transition probabilities was analyzed in [27]. To calculate the rotational line intensity factors with consideration for this interaction, it is necessary to multiply the results of calculations by the above formulas into the coefﬁcients obtained in [27] and listed in Table D.25. The symbol J denotes the rotational level of the X2 Π state, at which the transition comes to an end. Table D.25 Vibrational–rotational interaction correction coefﬁcients for the rotational line intensity factors in the (0,0) and (1,1) bands of the A2 Σ+ –X2 Π transition in the hydroxyl radical OH. J

P

3/2 1

Q

R

P

(0,0) band 0.996 0.99

21/2 0.941 0.917 0.89 31/2 0.864 0.83

Q

R

(1,1) band 0.78

J

P

Q

0.775 0.765 41/2 0.761 0.72

0.725 0.688 0.647 51/2 0.64

R

(0,0) band

P

Q

R

(1,1) band

0.678 0.537 0.476 0.415

0.595 0.55

0.411 0.347 0.285

0.794 0.645 0.593 0.539

Let us present some high-resolution spectrogram fragments of the OH(A2 Σ+ –X2 Π) transition recorded in a low-pressure oxyacetylene ﬂame [28]. Figure D.38 presents an overview spectrogram of the tran3 + sition (the Schuman-Runge bands due to the 3 Σ− u - Σ g transition in O2 partially fall within the same rang). Figures D.39a–f show detailed spec-

D.9 Hydroxyl Radical OH

Figure D.38 Overview spectrum of the OH(A2 Σ–X2 Π) transition in the range 280–380 nA. Oxyacetylene ﬂame.

tra with the lines deciphered, and Figure D.40 presents some fragments of spectra excited in various plasma sources differing in the composition of the plasma-forming gas.

563

564

Appendix D Information on Some Diatomic Molecules

Figure D.39 Rotational structure of the spectral bands of the OH(A2 Σ–X2 Π) transition.

D.9 Hydroxyl Radical OH

Figure D.39 (continued).

565

566

References

Figure D.40 OH spectra under various excitation conditions: (a) oxyacetylene ﬂame; (b) RF (150 MHz) discharge in water vapor at a pressure of 1.333 hPa (1 Torr); (c) RF (150 MHz) discharge in an H2 −Ar (1 : 10) mixture at a pressure of 4.000 hPa (3 Torr).

References

1 K.P. Huber, G. Herzberg. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules. N.Y.: Van Nostrand (1979). 2 G. Herzberg. The Spectra and Structure of Simple Free Radicals. London: Cornell University Press (1971). 3 G. Herzberg. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules. 2nd ed. N.Y.: D. van Nostrand (1951). 4 M.A. Elyashevich. Atomic and Molecular Spectroscopy (in Russian). Moscow: Fizmatgiz (1962). 5 A.M. Monamed et al. A Nitrogen and Nitrogen Plus Hydrogen Plasmatron. Rev. Roum. Phys., 26, No. 2, pp. 135– 141 (1981).

6 L.A. Kuznetsova, N.E. Kuzmenko, Yu. Ya. Kuzyakov, and Yu. A. Plastinin. Transition Probabilities of Diatomic Molecules (in Russian). Moscow: Nauka (1980). 7 N.E. Kuzmenko, L.A. Kuznetsova, and Yu. Ya. Kuzyakov. Franck–Condon Factors of Diatomic Molecules (in Russian). Moscow: Moscow State University Press (1984). 8 G.H. Dieke and D.F. Heath. The First and Second Positive Bands of N2 . In: John Hopkins Spectroscopic Report No. 17. Baltimore, Md. (1959). 9 V.N. Ochkin, S. Yu. Savinov, and N.N. Sobolev. Formation Mechanisms of the Distributions of Electronically Excited Molecules among Vibrational–

References

10

11

12

13

14

15

16

17

18

Rotational Levels in a Gas Discharge. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka, pp. 6–85 (1985). G.H. Dieke. The Molecular Spectrum of Hydrogen and its Isotopes. J. Molec. Spectr., 2, pp. 497–517 (1958). D.R. Churchill, S.A. Hagstrom, and P.K.M. Landshoff. The Spectral Absorption Coefﬁcient of Heated Air. JQSRT, 4, No. 2, pp. 291–321 (1964). F. Cramarossa, G. Ferraro, and E. Molinari. Spectroscopic Diagnostics of R.F. Discharges at Moderate Pressure and Chemical Applications – I. Pure Nitrogen. JQSRT, 14, No. 6, pp. 419–436 (1974). T.E. Sharp. Potential-Energy Curves for Molecular Hydrogen and its Ions. Atomic Data, 2, pp. 119–169 (1979). G.H. Dieke. The Molecular Spectrum of Hydrogen and its Isotopes. J. Molec. Spectr., 2, pp. 497–517 (1958). H.M. Crosswite. The Hydrogen Molecule Wavelength Tables of Gerhard Heinrich Dieke. N.Y.: WilleyInterscience (1972). S.A. Astashkevich. Study of the Nonadiabatic Effects of Perturbations in the Rovibronic Spectra of Hydrogen and Deuterium. Doctor’s Thesis (in Russian). St. Petersburg: St. Petersburg State University Press (2004) K.P. Kureichik, A.I. Bezlepkin, A.S. Khomyak, and V.V. Aleksandrov. Gas-Discharge Light Sources for Spectral Measurements (in Russian). Moscow: Nauka (1988) A.P. Bryukhovetsky, E.N. Kotlikov, D.K. Otorbayev et al. Excitation of Electronic–Vibrational–Rotational Levels in Hydrogen Molecules by Electron Impact in Nonequilibrium

19

20

21

22

23

24

25 26

27

28

Gas-Discharge Plasma. ZhETF, 79, pp. 1687–1703 (1980). D.K. Otorbayev, V.N. Ochkin, P.L. Rubin et al Excitation of the Rotational Levels of Electronic States in Molecules by Electron Impact in a Gas Discharge. In: N.N. Sobolev, Ed. Electronically Excited Molecules in Nonequilibrium Plasma (in Russian). Moscow: Nauka, pp. 6–85 (1985). T.J. McGee, G.E. Miller, J. Burris, Jr. et al. Fluorescence Branching Ratios from A2 Σ(v = 0) State of NO. JQSRT, 29, No. 4, pp. 333–338 (1983). I.C. McDermid and J.B. Laudenslager. Rotational Levels of NO(A2 Σ, v = 0). JQSRT, 27, No. 5, pp. 483–492 (1982). J.O. Hornkohl. Temperature Measurements from CN Spectra. JQSRT, 46, No. 5, pp. 405–412 (1991). R. Bleekrode. Absorption and Emission Spectroscopy of C2 , CH, and OH in Low-Pressure Oxyacetylene Flames. Phillips Res. Rep. N 7, pp. 1–63 (1966). V.K. Zhivotov, V.D. Rusanov, and A.A. Fridman. Diagnostics of Nonequilibrium Chemically Active Plasma (in Russian). Moscow: Eneregoatomizdat (1985). E.F. van Dishoech. J. Chem. Phys., 86, p. 196, (1987). A.M. Bass and H.P. Broida. A Spectrometric Atlas of the Spectrum of CH from ˚ to 5000 A. ˚ NBS Monograph 24. 3000 A Washington D.C. (1961). T. Anketell and R.C. Learner. Vibration Rotation Interaction in OH and the Transition Moment. Proc. Roy. Soc. London A, 301, pp. 355–361 (1967) A.M. Bass and H.P. Broida. A Spectrometric Atlas of the 2 Σ+ −2 Π Transition of OH. NBS Circular 541. Washington D.C. (1953).

567

569

Appendix E

Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra of Diatomic Molecules The number of the rotational components of molecular terms that actually manifest themselves in molecular spectra is many times that of the electronic and vibrational levels. Therefore, it proves very useful that ¨ the line strengths S J J (the Honl–London factors) of these components can be analytically expressed (in the approximation of divisibility of the electronic, vibrational and rotational parts of the total wave function) as functions of quantum numbers. A most detailed summary of such formulas can be found in the monograph [1]. The formulas are presented in the dipole and quadrupole approximations for transitions between electronic terms with a multiplicity of up to 2S + 1 = 7, including those involving violation of the rule ΔS = 0. In the present appendix, we will present the dipole approximation formulas for 2S + 1 ≤ 3, which is a frequent requirement in practical plasma spectroscopy. We have taken account of the technical errors committed in numerous formulas [1], which were noted in [2]. In the tables presented below, use is made of the same designations of the terms, quantum numbers, transition branches and parameters as in Appendix D. The prime relates to the upper state and the double prime, to the lower one. Use is made of the quantum number J of the total momentum, which relates to the upper (J ) and the lower (J ) state for emission and absorption, respectively. In the designations of electronic transitions, it is the upper state that goes ﬁrst. If not indicated by primes, the values of Λ and Ω are the least of those for the combining states. Accordingly, ΔΛ = Λ − Λ. The formulas in this appendix are grouped into sections according to the types of electronic transitions. If necessary, additional designations are introduced and explained at the beginning of each section. Remarks on the normalization of S J ,J are presented in Section E.4. The formulas presented below, except for those of Section E.3.1, are restricted to the main transition branches. The line strengths of the satellite branches, accurate up to the corrections [2], are also presented in [1].

570

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

E.1 Singlet Transitions

These are always b–b-type transitions (Appendix D). E.1.1 1 X–1 X, ΔΛ = 0 Transitions

( J + Λ)( J − Λ) , J Λ2 (2J + 1) = , J ( J + 1) ( J + Λ + 1)( J − Λ + 1) . = J+1

P( J ): S J,J +1 =

(E.1)

Q( J ): S J,J

(E.2)

R( J ): S J,J −1

(E.3)

E.1.2 1 X–1 Y, ΔΛ = ±1 Transitions

ΔΛ = +1Δ,

Λ = −1,

P( J ): S J,J +1 ,

R( J − 1) : S J −1,J −2 ,

Q( J ): S J,J ,

Q( J ) : S J,J ,

R( J ): S J,J −1 ,

P( J + 1) : S J +1,J +2 ,

S J ,J

( J − Λ − 1)( J − Λ) , (E.4) 2J ( J − Λ)( J + Λ + 1)(2J + 1) , (E.5) 2J ( J + 1) ( J + Λ + 1)( J + Λ + 2) . (E.6) 2( J + 1)

E.2 Doublet Transitions

As distinct from transitions between singlet states of the b-type coupling, the coupling type of doublet states can vary. The formulas below are written down in general form for an intermediate (between a and b) type of coupling. The following designations are used in addition to the ones indicated above: u± ( J ) = [Λ2 Y (Y − 4) + 4( J + 1/2)2 ] /2 ± Λ(Y − 2), ! " C ± ( J ) = 1/2 [u± ( J )]2 + 4[( J + 1/2)2 − Λ2 ] . 1

(E.7) (E.8)

E.2 Doublet Transitions

For normal terms with the spin-orbit interaction parameter A > 0, the quantity Y = A/B > 0. For inverted terms with A < 0, the quantity Y in formulas (E.7) and (E.8) should be taken with the minus sign. In the particular cases of pure-type couplings, for the a–a-type transitions, Y J ( J + 1), u+ ( J ) = 2Λ(Y − 2);

u− ( J ) = 0;

C + ( J ) = 2Λ2 (Y − 2)2 ;

C − ( J ) = 2( J − Λ + 1/2)( J + Λ + 1/2),

(E.9)

and for the b–b-type transitions, Y J ( J + 1), u+ ( J ) = 2( J − Λ + 1/2);

u− ( J ) = 2( J + Λ + 1/2);

C + ( J ) = 4( J + 1/2)( J − Λ + 1/2);

(E.10)

−

C ( J ) = 4( J + 1/2)( J + Λ + 1/2). E.2.1 2 X–2 X, ΔΛ = 0 Transitions

P1 ( J ):

S J,J +1 =

( J − Λ − 1/2)( J + Λ + 1/2) 4JC − ( J − 1)C − ( J )

(E.11)

· {u− ( J − 1)u− ( J ) + 4( J − Λ + 1/2)( J + Λ − 1/2)}2 Q1 ( J ):

SJ J =

J + 1/2 2J ( J + 1)C − ( J )C − ( J )

(E.12)

· {(Λ + 1/2)u− ( J )u− ( J )

+ 4(Λ − 1/2)( J − Λ + 1/2)( J + Λ + 1/2)}2 R1 ( J ):

S J,J −1 =

( J − Λ + 1/2)( J + Λ + 3/2) 4( J + 1)C − ( J + 1)C − ( J )

(E.13)

· {u− ( J + 1)u− ( J ) + 4( J − Λ + 3/2)( J + Λ + 1/2)}2 P2 ( J ):

S J,J +1 =

( J − Λ − 1/2)( J + Λ + 1/2) 4JC + ( J − 1)C + ( J )

(E.14)

· {u+ ( J − 1)u+ ( J ) + 4( J − Λ + 1/2)( J + Λ − 1/2)}2 Q2 ( J ):

SJ J =

J + 1/2 2J ( J + 1)C + ( J )C + ( J ) · {(Λ + 1/2)u+ ( J )u+ ( J )

+ 4(Λ − 1/2)( J − Λ + 1/2)( J + Λ + 1/2)}2

(E.15)

571

572

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

R2 ( J ):

S J,J −1 =

( J − Λ + 1/2)( J + Λ + 3/2) 4( J + 1)C + ( J + 1)C + ( J )

(E.16)

· {u+ ( J + 1)u+ ( J ) + 4( J − Λ + 3/2)( J + Λ + 1/2)}2 E.2.2 2 X–2 Y, ΔΛ = ±1 Transitions

ΔΛ = +1

ΔΛ = −1

S J ,J

P1 ( J ): S J,J +1

R1 ( J − 1): S J −1,J −2

( J − Λ − 3/2)( J − Λ − 1/2) 8JC − ( J − 1)C − ( J ) · {u− ( J − 1)u− ( J )

(E.17)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2 Q1 ( J ): S J,J

Q1 ( J ): S J,J

( J + 1/2)( J − Λ − 1/2)( J + Λ + 3/2) 4J ( J + 1)C − ( J )C − ( J ) · {u+ ( J − 1)u+ ( J )

(E.18)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2 R1 ( J ): S J,J −1

P1 ( J + 1): S J +1,J +2

( J + Λ + 3/2)( J + Λ + 5/2) 8( J + 1)C − ( J + 1)C − ( J ) · {u− ( J + 1)u− ( J )

(E.19)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2 P2 ( J ): S J,J +1

R2 ( J − 1): S J −1,J −2

( J − Λ − 3/2)( J − Λ − 1/2) 8JC + ( J − 1)C + ( J ) · {u+ ( J − 1)u+ ( J )

(E.20)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2 Q2 ( J ): S J,J

Q2 ( J ): S J,J

( J + 1/2)( J − Λ − 1/2)( J + Λ + 3/2) 4J ( J + 1)C + ( J )C + ( J ) · {u+ ( J − 1)u+ ( J )

(E.21)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2 R2 ( J ): S J,J −1

P2 ( J + 1): S J +1,J +2

( J + Λ + 3/2)( J + Λ + 5/2) 8( J + 1)C + ( J + 1)C + ( J ) · {u+ ( J + 1)u+ ( J )

(E.22)

+ 4( J − Λ + 1/2)( J + Λ + 1/2)}2

E.3 Triplet Transitions

E.3 Triplet Transitions

The formulas in this section are written down for the general case of intermediate (between a and b) type of coupling. The following designations are used: u1± ( J ) = [Λ2 Y (Y − 4) + 4J 2 ] /2 ± Λ(Y − 2); 1

u3± ( J ) = [Λ2 Y (Y − 4) + 4( J + 1)2 ] /2 ± Λ(Y − 2), 1

(E.23)

C1 ( J ) = Λ Y (Y − 4)( J − Λ + 1)( J + Λ) + 2(2J + 1)( J − Λ) J ( J + Λ); 2

C2 ( J ) = Λ2 Y (Y − 4) + 4J ( J + 1), C3 ( J ) = Λ2 Y (Y − 4)( J + Λ + 1)( J − Λ).

+ 2(2J + 1)( J − Λ + 1)( J + 1)( J + Λ + 1)

(E.24)

For inverted terms with A < 0, the quantity Y should be replaced by −Y and, besides, in the expressions for C1 (J) and C3 (J) the quantity Λ should be replaced by −Λ. In the limiting case of the a–a-type transitions, u1+ ( J ) = u3+ ( J ) = 2Λ(Y − 2);

u1− ( J ) = u3− ( J ) = 0,

(E.25)

C1 ( J ) = Λ (Y − 2 )( J − Λ + 1)( J + Λ); 2

2

C2 ( J ) = Λ2 (Y − 2)2 ; C3 ( J ) = Λ2 (Y − 2)2 ( J + Λ + 1)( J − Λ).

(E.26)

If the combining terms are both of the b-type coupling, u1+ ( J ) = 2( J − Λ);

u1− ( J ) = 2( J + Λ);

u3+ ( J ) = 2( J − Λ + 1);

u3− ( J ) = 2( J + Λ + 1),

(E.27)

C1 ( J ) = 2(2J + 1)( J − Λ) J ( J + Λ); C2 ( J ) = 4J ( J + 1); C3 ( J ) = 2(2J + 1)( J − Λ + 1)( J + 1)( J + Λ + 1).

(E.28)

573

574

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

E.3.1 3 X–3 X, ΔΛ = 0 Transitions

P1 ( J ):

S J,J +1 =

( J − Λ)( J + Λ) 16JC1 ( J − 1)C1 ( J )

· {( J − Λ + 1)( J + Λ − 1)u1+ ( J − 1)u1+ ( J )

+ ( J − Λ − 1)( J + Λ + 1)u1− ( J − 1)u1− ( J )

+ 8( J − Λ − 1)( J + Λ − 1)( J − Λ)( J + Λ)}2 , (E.29) Q1 ( J ):

S J,J =

2J + 1 16J ( J + 1)C1 ( J )C1 ( J )

· {( J − Λ + 1)(Λ − 1)( J + Λ)u1+ ( J )u1+ ( J )

+ ( J + Λ + 1)( J − Λ)(Λ + 1)u1− ( J )u1− ( J ) + 8Λ( J − Λ)2 ( J + Λ)2 }2 , R1 ( J ):

S J,J −1 =

(E.30)

( J − Λ + 1)( J + Λ + 1) 16( J + 1)C1 ( J + 1)C1 ( J )

· {( J − Λ + 2)( J + Λ)u1+ ( J + 1)u1+ ( J )

+ ( J − Λ)( J + Λ + 2)u1− ( J + 1)u1− ( J )

+ 8( J − Λ + 1)( J + Λ + 1)( J − Λ)( J + Λ)}2 , (E.31) P2 ( J ):

S J,J +1 =

4( J − Λ)( J + Λ) JC2 ( J − 1)C2 ( J )

· {1/2Λ2 (Y − 2)(Y − 2) + ( J − Λ − 1)( J + Λ + 1)

+ ( J − Λ + 1)( J + Λ − 1)}2 , Q2 ( J ):

SJ J =

(E.32)

4(2J + 1) J ( J + 1)C2 ( J )C2 ( J )

· {1/2Λ3 (Y − 2)(Y − 2) + (Λ + 1)( J − Λ)( J + Λ + 1)

+ (Λ − 1)( J − Λ + 1)( J + Λ)}2 , R2 ( J ):

S J,J −1 =

(E.33)

4( J − Λ + 1)( J + Λ + 1) ( J + 1)C2 ( J + 1)C2 ( J )

· {1/2Λ2 (Y − 2)(Y − 2) + ( J − Λ)( J + Λ + 2)

+ ( J − Λ + 2)( J + Λ)}2 ,

(E.34)

E.3 Triplet Transitions

P3 ( J ):

S J,J +1 =

( J − Λ)( J + Λ) 16JC3 ( J − 1)C3 ( J )

· {( J − Λ + 1)( J + Λ − 1)u3− ( J − 1)u3− ( J )

+ ( J − Λ − 1)( J + Λ + 1)u3+ ( J − 1)u3+ ( J )

+ 8( J − Λ + 1)( J + Λ + 1)( J − Λ)( J + Λ)}2 , (E.35) Q3 ( J ):

SJ J =

2J + 1 16J ( J + 1)C3 ( J )C3 ( J )

· {(Λ − 1)( J − Λ + 1)( J + Λ)u3− ( J )u3− ( J )

+ (Λ + 1)( J − Λ)( J + Λ + 1)u3+ ( J )u3+ ( J ) + 8Λ( J − Λ + 1)2 ( J + Λ + 1)}2 , R3 ( J ):

S J,J −1 =

(E.36)

( J − Λ + 1)( J + Λ + 1) 16( J + 1)C3 ( J + 1)C3 ( J )

· {( J − Λ + 2)( J + Λ)u3− ( J + 1)u3− ( J )

+ ( J − Λ)( J + Λ + 2)u3−+ ( J + 1)u3+ ( J ) + 8( J − Λ + 1)( J − Λ + 2)( J + Λ + 1)( J + Λ + 2)}2 . (E.37) In formulas (E.29) through (E.37), account is taken of the fact that it is only the Σ+ − Σ+ and Σ− − Σ− transitions (b-type coupling, Appendix D) that are possible in the dipole approximation. However, in the presence of spin-orbit interaction and rotational structure perturbations, this rule can be violated in part. The S J J factors corresponding to this situation are presented below in summary (E.38) for the main branches.

575

P1 ( J + 1)

∑

+

R1 ( J )

−3

Q1 ( J )

∑

−

branches

R1 ( J − 1)

3

P1 ( J )

∑

−

Q1 ( J )

∑

+ −3

3

∑

∑

0

2J ( J +1) 2J +1

0

S J J +,− −3 −,+

∑

P2 ( J )

−3

R2 ( J )

Q2 ( J )

∑

+

−

Transitions 3

3

∑

+

P2 ( J + 1)

Q2 ( J )

R2 ( J − 1)

∑

− −3

branches 3

∑

0

0

0

S J J +,− −3 ∑

−,+ 3

−3

R3 ( J )

Q3 ( J )

P3 ( J )

∑

+

∑

Transitions 3

−3

∑

−

P3 ( J + 1)

Q3 ( J )

R3 ( J − 1)

∑

+

branches 3

∑

∑ −,+

0 (E.38)

2J ( J +1) 2J +1

0

S J J +,− −3

OQ

13 ( J )

SQ

− 1)

+ 1)

31 ( J )

21 ( J

O P ( J) 12

SR

QP (J 21

QR

12 ( J )

1 2(2J +1)

( J −1)( J +1) 2(2J −1)

J ( J +1) 2(2J +3) 21 ( J )

32 ( J

SR

O P ( J) 23

12 ( J

QP (J 32

QR

OP (J 12

23 ( J )

QR

Q P ( J) 21

OR

− 1)

+ 1)

− 1)

+ 1)

( J −1)( J +1) 2(2J +1)

( J +1)( J +2) 2(2J +1)

J ( J −1) 2(2J +1)

J ( J +2) 2(2J +1)

32 ( J )

31 ( J )

Q P ( J) 32

SR

SQ

QR

23 ( J

− 1)

+ 1)

13 ( J ) OP (J 23

OQ

J ( J +1) 2(2J −1)

J ( J +2) 2(2J +3)

1 2(2J +1)

(E.39)

In this case, satellite branches also become manifest. In summary (E.39) below we present those of them, which have nonzero rotational line intensity factors.

3

Transitions

E.3.1.1 Dipole-Forbidden Branches

576

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

E.3 Triplet Transitions

577

E.3.2 3 X–3 Y, ΔΛ = ±1 Transitions

ΔΛ = +1

ΔΛ = −1

S J ,J

P1 ( J ): S J,J +1

R1 ( J − 1): S J −1,J −2

( J − Λ − 1)( J − Λ) 32JC1 ( J − 1)C1 ( J )

· {( J − Λ + 1)( J + Λ)u1+ · ( J − 1)u1+ ( J )

+ ( J − Λ − 2)( J + Λ + 1)u1−

· ( J − 1)u1− ( J ) + 8( J − Λ − 2) · ( J − Λ)( J + Λ)2 }2 ,

Q1 ( J ): S J,J

Q1 ( J ): S J,J

(E.40)

(2J + 1)( J + Λ + 1)( J − Λ) 32J ( J + 1)C1 ( J )C1 ( J )

· {( J − Λ + 1)( J + Λ)u1+ · ( J )u1+ ( J )

+ ( J − Λ − 1)( J + Λ + 2)u1− · ( J )u1− ( J )

+ 8( J − Λ − 1)( J − Λ)( J + Λ) · ( J + Λ + 1)}2 , R1 ( J ): S J,J −1

P1 ( J + 1): S J +1,J +2

(E.41)

( J + Λ + 1)( J + Λ + 2) 32( J + 1)C1 ( J + 1)C1 ( J )

· {( J − Λ + 1)( J + Λ)u1+ · ( J + 1)u1+ ( J )

+ ( J − Λ)( J + Λ + 3)u1− · ( J + 1)u1− ( J )

+ 8( J − Λ)2 ( J + Λ)( J + Λ + 2)}2 , (E.42) P2 ( J ): S J,J +1

R2 ( J − 1): S J −1,J −2

2( J − Λ − 1)( J − Λ) JC2 ( J − 1)C2 ( J )

· {1/2Λ(Λ + 1)(Y − 2)(Y − 2)

+ ( J − Λ + 1)( J + Λ) + ( J − Λ − 2)( J + Λ + 1)}2 , (E.43)

578

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

Q2 ( J ): S J,J

Q2 ( J ): S J,J

2(2J + 1)( J + Λ + 1)( J − Λ) J ( J + 1)C2 ( J )C2 ( J )

· {1/2Λ(Λ + 1)(Y − 2)(Y − 2)

+ ( J − Λ + 1)( J + Λ) + ( J − Λ − 1) · ( J + Λ + 2)}2 , R2 ( J ): S J,J −1

P2 ( J + 1): S J +1,J +2

(E.44)

2( J + Λ + 1)( J + Λ + 2) ( J + 1)C2 ( J + 1)C2 ( J )

· {1/2Λ(Λ + 1)(Y − 2)(Y − 2)

+ ( J − Λ + 1)( J + Λ) + ( J − Λ) · ( J + Λ + 3)}2 , P3 ( J ): S J,J +1

R3 ( J − 1): S J −1,J −2

(E.45)

( J − Λ − 1)( J − Λ) 32JC3 ( J − 1)C3 ( J )

· {( J − Λ + 1)( J + Λ)u3− · ( J − 1)u3− ( J )

+ ( J − Λ − 2)( J + Λ + 1)u3+ · ( J − 1)u3+ ( J )

+ 8( J − Λ − 1)( J − Λ + 1) · ( J + Λ + 1)}2 , Q3 ( J ): S J,J

Q3 ( J ): S J,J

(E.46)

(2J + 1)( J + Λ + 1)( J − Λ) 32J ( J + 1)C3 ( J )C3 ( J )

· {( J − Λ + 1)( J + Λ)u3− · ( J )u3− ( J )

+ ( J − Λ − 1)( J + Λ + 2)u3+ · ( J )u3+ ( J )

+ 8( J − Λ)( J − Λ + 1)( J + Λ + 1) · ( J + Λ + 2)}2 ,

(E.47)

E.3 Triplet Transitions

R3 ( J ): S J,J −1

P3 ( J + 1): S J +1,J +2

( J + Λ + 1)( J + Λ + 2) 32( J + 1)C3 ( J + 1)C3 ( J )

· {( J − Λ + 1)( J + Λ)u3− · ( J + 1)u3− ( J )

+ ( J − Λ)( J + Λ + 3)u3+ · ( J + 1)u3+ ( J )

+ 8( J − Λ + 1)2 ( J + Λ + 1) · ( J + Λ + 3)}2 ,

(E.48)

E.3.3 3 –3 Δ, ΔΛ = ±2 Transitions ∑

In contrast to the above-presented formulas, where Λ denotes the smaller quantity for the two states in calculating the quantities deﬁned by formulas (E.23) and (E.24), in this case Λ = 2. ΔΛ = −2

ΔΛ = +2

S J ,J

P1 ( J ): S J,J +1

R1 ( J − 1): S J,J −1

( J + 1)( J + 2) {( J − 1)u1+ ( J ) 2(2J − 1)C1 ( J ) + 2( J − 2)( J + 2)}2 ,

Q1 ( J ): S J,J

Q1 ( J ): S J,J

( J − 1)( J + 2) { Ju1+ ( J ) 2JC1 ( J ) + 2( J − 2)( J + 2)}2 ,

R1 ( J ): S J,J −1

P1 ( J + 1): S J +1,J +2

(E.49)

(E.50)

( J − 1) J ( J + 2) 2( J + 1)(2J + 3)C1 ( J ) · {( J + 1)u1+ ( J )

+ 2( J − 2)( J + 2)}2 , P2 ( J ): S J,J +1

R2 ( J − 1): S J −1,J −2

Q2 ( J ): S J,J

Q2 ( J ): S J,J

(E.51)

4( J + 1)( J + 2) (Y − 2)2 , (E.52) JC2 ( J ) 4( J − 1)( J + 2)(2J + 1) (Y − 2 ) 2 , J ( J + 1)C2 ( J ) (E.53)

579

580

Appendix E Rotational Line Intensity Factors in the Electronic–Vibrational Transition Spectra

R2 ( J ): S J,J −1

P2 ( J + 1): S J +1,J +2

4( J − 1) J (Y − 2 ) 2 , ( J + 1)C2 ( J )

P3 ( J ): S J,J +1

R3 ( J − 1): S J −1,J −2

( J − 1)( J + 1)( J + 2) { Ju3− ( J ) 2J (2J − 1)C3 ( J ) + 2( J − 1)( J + 3)}2 ,

Q3 ( J ): S J,J

Q3 ( J ): S J,J

P3 ( J + 1): S J +1,J +2

(E.55)

( J − 1)( J + 2) {( J + 1)u3− ( J ) 2( J + 1)C3 ( J ) + 2( J − 1)( J + 3)}2 ,

R3 ( J ): S J,J −1

(E.54)

(E.56)

( J − 1) J {( J + 2)u3− ( J ) 2(2J + 3)C3 ( J ) + 2( J − 1)( J + 3)}2 ,

(E.57)

The rotational line intensity factors for the satellite branches are presented in [1, Table 3.11].

E.4 Remarks on the Normalization of Rotational Line Intensity Factors

So far there is no unanimous opinion about the sum rule for the rotational line intensity factors S J J . A recent review of the different versions suggested in various publications can be found in [3]. In practice, these differences only relatively rarely cause difﬁculties, provided that one uses in measurements combinations of lines belonging in different transition systems and bands. The authors of [3] believe that the most convenient would be a normalization simultaneously meeting the following requirements: ¨ • The Honl–London factors must be the same for both emission and absorption. ¨ • The Honl–London factor for a multiplet line must be equal to the sum of the factors for the individual components of the line. • The normalization must be universal for both spin-allowed and spin-forbidden transitions.

E.5 On Symbolic Notation

These requirements are fulﬁlled by the rule of summation over all electronic–vibrational–rotational transitions:

∑ ∑ ∑ S J J = (2 − δ0,Λ +Λ )(2S + 1)(2J + 1),

(E.58)

∑ ∑ ∑ S J J = (2 − δ0,Λ +Λ )(2S + 1)(2J + 1),

(E.59)

Σ ,Σ v ,v Σ ,Σ v ,v

J

J

The normalizations actually used in the formulas of this appendix are as follows: Transitions

Normalization

Singlet

2J + 1

Doublet

(2S + 1)(2J + 1)

= 2(2J + 1)

(2S + 1)(2J + 1)

= 3(2J + 1)

Triplet 3

X −3 X,3 X −3 Y,

except for 3

∑ + − 3 ∑ − ,3 ∑ − − 3 ∑ +

2(2J + 1)

3

∑−

8(2J + 1)

3

Δ

¨ Thus, the Honl–London factor normalizations adopted in [1, 2] and used in this appendix correspond accurate up to a factor of (2 − δ0,Λ −Λ ) with those recommended in [3].

E.5 On Symbolic Notation

The set of formulas presented in this appendix for dipole-allowed transitions with the “pure” a–a, b–b, a–b, and b–a types of coupling without limitations on S, Λ, and Σ can be written down in the compact form [4]. ¨ This might prove useful in the computer calculation of the Honl–London factors by means of special program packages like “Mathematics”. For the a–a-type transitions, S J J

J = H1 (Λ , Λ , Σ )(2J + 1)(2J + 1) Ω

1 Λ − Λ

J −Ω

2 , (E.60)

where H1 (Λ , Λ , Σ ) =

2 if Λ = Σ = 0, Λ = 1 or Λ = Σ = 0, Λ = 1 1 in other cases

581

582

References

For the b–b-type transitions, S J J = H (Λ , Λ )(2J + 1)(2J + 1)(2N + 1)(2N + 1) 2 2 N 1 J 1 N J × , Λ Λ − Λ −Λ S N N where

H (Λ , Λ ) =

2

if Λ = 0, Λ = 1 or Λ = 0, Λ = 1

1

in other cases

(E.61)

Here S = S = S, because intercombination quantum transitions are not considered. For the a–b- and b–a-type transitions, S J J = H2 (Λ , Λ , Σ )(2J + 1)(2J + 1)(2N + 1) 2 J 1 J × , Λ + Σ Λ − Λ −Λ − Σ where H2 (Λ , Λ , Σ ) =

(E.62)

2 if Λ = Σ = 0, Λ = 1 or (Λ )2 + (Σ )2 = 0, Λ = 0 1 in other cases

In the above formulas, the expressions in parentheses are the 3-j symbols and in braces, the 6-j ones. The formulas for these symbols can be found in many books, [5] for one. The normalization is ∑ J S J J = (2J + 1) H, ∑ J S J J = (2J + 1) H Compact formulas like (E.60) through (E.62) [4] can be found in [6]. References

1 I. Kovach. Rotational Structure in the Spectra of Diatomic Molecules. Budapest: Akademiai Kiado (1969). 2 E.E. Whitting, J.A. Paterson, I. Kovach, and R.W. Nicholls. Computer Checking of Rotational Line Intensity Factors for Diatomic Transitions. J. Mol. Spectr., 47, pp. 84–98 (1973). 3 L.A. Kuznetsova, N.E. Kuzmenko, Yu. Ya. Kuzyakov, and Yu. A. Plastinin. Probabilities of Optical Transitions in Diatomic Molecules (in Rus-

sian). Ed. by R.V. Khokhlov. Moscow: Nauka (1980). 4 P.L. Rubin. Line Intensity Factors in Electronic Spectra of Diatomic Molecules. Opt. i Spektrosk., 20, No. 4, pp. 576–581 (1966). 5 I.I. Sobelman. An Introduction to the Theory of Atomic Spectra (in Russian). Moscow: Nauka (1977). 6 L.D. Landau and E.M. Lifshits. Quantum Mechanics (in Russian), 3rd ed. Moscow: Nauka (1974).

583

Appendix F

Measurement of the Absolute Populations of Excited Atoms by Classical Spectroscopy Techniques Written by Prof. Yu. B. Golubovskii

In plasma spectroscopy, the populations of excited atoms are frequently measured by the classical (other than laser) light emission and absorption methods. At the foreground in that case is the problem of adequate illumination of the spectral instrument and the correct calculation of the luminous ﬂux that in the ﬁnal analysis reaches the detector. The speciﬁcity here is that plasma sources are, as a rule, extensive and volumetric. The luminous ﬂux from volumetric plasma sources is compared with that from standard sources, frequently in the form of ﬂat tungsten-ribbon photometric lamps. The radiation of such a standard source can be char˚ · sr] – the power emitacterized by the surface brightness bλ [Wcm−2 · A ted by a unit surface area into a unit solid angle within a unit wavelength interval in a direction normal to the area – or by the radiance rλ ˚ · sr] – the emission of a unit surface area into a solid angle of [Wcm−2 · A 2π. The relation between brightness and radiance for cosine radiators is well known to be rλ = πbλ . By comparing between the luminous ﬂuxes from plasma and the standard source, which have been made to travel one and the same optical path by means of a tilting mirror, one can deduce the absolute spectral line intensity Ii,k [Wcm−3 ] or the continuum ˚ that is, the power emitted by a unit volume of intensity Iλ [Wcm−3 · A], plasma into a solid angle of 4 π, either integrally, in the spectral line, or within a unit wavelength interval in the continuum. The population Ni [cm−3 ] of the emitting level is then found from the absolute spectral line intensity by the relation Ii,k = Ni Ai,k hνi,k , where Ai,k [s−1 ] is the transition probability and hνi,k is the quantum energy. The traditional scheme of the experiment is shown in Figure F.1. When taking measurements with a high spatial resolution, one should use a good objective lens, corrected for various kinds of aberrations, to project a reduced real image of the volumetric plasma source onto the entrance slit of the spectral instrument. In the plane of the slit is formed a sufﬁciently sharp real image of the source, because the image reduction in the longitudinal direction is squared that in the transverse direction.

584

Appendix F Measurement of the Absolute Populations of Excited Atoms

Figure F.1 Schematic of absolute line intensity measurements: 1 – standard source (a photometric lamp); 2 – tilting mirror; 3 – plasma source (e.g. a discharge tube); 4 – objective lens diaphragm; 5 – objective lens; 6 – spectral instrument; 7 and 8 – entrance and exit slit, respectively; 9 – radiation detector; 10 – image (scaled up) of the plasma object in the slit plane.

Indeed, if one uses the thin lens formula and differentiates, 1 1 1 + = ; x x f

Δx x2 = − 2, Δx x

one can then see that if the image reduction is deﬁned by the ratio − x /x, the ratio between the longitudinal dimensions of the image and the object is reduced squared number of times more (x, x are conjugate points, f is the focal length of the objective lens). The minus signs mean that the image will be reversed and inverted (the near and far sections will change places). We can cite a typical example of measurements of the radiation intensity using the positive column of a discharge 20 mm in diameter and 40 cm long. Let the image of the discharge tube reduced by a factor of 7 be projected onto the slit of the spectral instrument by means of an objective lens 25 mm in focal length. The reality of measurements is that the ﬂat tungsten ribbon of the standard source is projected onto the plane of the slit absolutely sharp (up to the aberrations of the lens), whereas different sections of the volumetric source prove to be out of focus in the plane of the slit, except for the section whose distance from

F Measurement of the Absolute Populations of Excited Atoms

the objective lens coincides with that of the tungsten ribbon. This section is assumed to be coincident with the center of the discharge tube and 200 cm distant from the objective lens, with the given focal length and 7-fold image reduction. Accordingly, the front and rear windows of the discharge tube will be 180 cm and 220 cm distant from the objective lens. In the image plane, they will be 28.2 cm and 29 cm distant from the lens, while the plane of the slit, 28.6 cm remote from it. The front window will be reduced by a factor of 6.38, whereas the rear one, 7.58 times. The image of the discharge tube will look like a conical layer of small thickness, a mere 8 mm, and will be perceived as a sharp image of an object in the form of a circle some 3 mm in diameter. It is expedient to place two crossed slits at the entrance, which will make it possible to isolate from the image the region of interest with a high (no worse than 0.01 mm) accuracy. It will then be possible to isolate from this circle a small area, say 2 mm by 2 mm in size, and set the diaphragm of the objective to approximately 2 cm diameter, so as to match the angular dimensions to the relative aperture of the spectral instrument (1:10 in our example). In that case, the solid angle through which the light from the sources is collected turns out to be small enough. This scheme of illumination of the spectral instrument provides for a sufﬁciently good spatial resolution, which will be discussed later in the text. The prime task facing the experimenter is to correctly calculate the differential luminous ﬂuxes issuing from various elementary plasma volumes and integrate them over that plasma region, whose light passes through the slit of the spectral instrument. The next, simpler step is to calculate the luminous ﬂux from the photometric lamp that has passed through the slit. Taking the ratio between these ﬂuxes, which are proportional to the signals registered from plasma and from the standard source, one can determine the absolute intensity of the spectral line or continuum. This problem can be solved by two equivalent methods. First of all, it is necessary to ﬁnd the locus of points of the volumetric source, from which light enters the spectral instrument. Next, as already mentioned, it is necessary to calculate the luminous ﬂux from an elementary plasma volume, with due regard for absorption in the source, and ﬁnally, integrate over the locus found, allowing for the possible spatial inhomogeneity. The second method consists in calculating the illuminance in the plane of the slit produced by some section of the volumetric source (in the general case, the image will be out of focus), summing up the illuminances from all the sections of the source, and ﬁnally, integrating the total illuminance over the area of the slit. Naturally both these methods yield the same ﬁnal results.

585

586

Appendix F Measurement of the Absolute Populations of Excited Atoms

Figure F.2 To the calculation of the luminous ﬂux entering the spectral instrument through the elementary area on the optical axis in the slit plane.

Let the reduced image of the volumetric source be projected into the slit plane P as shown in Figure F.2. The coordinates are reckoned from the principal planes of the optical system. The plane P, conjugate to P , is projected exactly into the slit plane with a reduction of a /a. Let us isolate an elementary volume dV with the coordinates x, r within the limits of the source in the plane P1 , which is projected into the point x , r in the plane P1 , and ﬁnd the luminous ﬂux that passes from the elementary volume dV through the elementary area dσ of the slit on the optical axis of the system. The luminous ﬂux is captured by the objective lens within the solid angle Ω and, propagating in the image space, forms a uniformly illuminated circle of radius ρ in the slit plane, the center of this circle lying at a distance of r from the optical axis (bottom part of Figure F.1). Obviously, light will pass through the elementary area dσ only if the condition r ≤ ρ is satisﬁed. This condition determines the locus of those points of the source, light from which enters the spectral instrument through the small area dσ . In actual experiments, the solid angles Ω, Ω are small enough, so that the paraxial optics approximation can be used. Using the similitude of triangles and expressing the primed coordinates in terms of the unprimed ones by the rules of geometrical optics, one can write down, accurate up to second-order, terms of the condition for the passage of light through the slit of the spectral instrument in the form x . r ≤ R 1− a

F Measurement of the Absolute Populations of Excited Atoms

Figure F.3 Conical surface cutting out that volume in the plasma source, from which light enters the spectral instrument through the center of the slit (top). Cylindrical volume equivalent to the cone in calculating the luminous ﬂux (bottom).

This expression describes in the space of the object a conical surface with vertex at the point x = a, which rests on a circle of radius R on the surface of the objective lens as on a base (Figure F.3). Since it is only a fraction of the luminous ﬂux, equal to the ratio dσ /πρ2 , that will pass through the elementary area dσ of the slit (which is equivalent to the capture of the luminous ﬂux within a small solid angle dΩ shaded in Figure F.1), the luminous ﬂux gathered from the elementary volume dV turns out to be dF =

IdV dσ Ω( x ) 2 . 4π πρ ( x )

(F.1)

The total ﬂux that has passed through dσ is obtained by integrating over the conical surface. For a homogeneous and nonabsorbing source, we

587

588

Appendix F Measurement of the Absolute Populations of Excited Atoms

have I dσ F= 4π π

2π 0

dϕ

a+ L/2

Ω( x ) R(a− x)/a a− L/2

ρ2 ( x )

0

r dr dx.

(F.2)

Integration with respect to ϕ gives 2π. Considering that Ω( x ) = πR2 /x2 ;

ρ=R

f (a − x) , x(a − f )

we get an interesting corollary, namely, the integrand in x * 2 πR2 Ω( x ) 1 2 ** R( a − x )/a 2 a− f r * = πR = 2 = Ω 2π 2 0 af πρ ( x ) 2 a

(F.3)

is independent of the coordinate x and is equal to the solid angle Ω at which the objective is viewed from the point dσ . Integration with respect to x is in this case reduced to multiplication into the length L of the source, and ﬁnally it follows from formula (F.2) that F = Idσ L

Ω Ω = IdσL , 4π 4π

(F.4)

where dσ = dσ ( a/a ) is the element of the area dσ scaled into the 2 image space with a magniﬁcation of ( a/a ) and Ldσ is the volume of a cylinder of length L and base area dσ. This result shows that, subject to the assumptions made above, the luminous ﬂux gathered from any section of the cone within the limits of the volumetric source is the same. Only small fractions of light, equal to dσ /πρ2 , pass from elementary volumes in the unfocused image, but these volumes are sufﬁciently numerous for these two factors to be completely offset for any section of the volumetric source. Thus, for the illumination scheme under consideration, a cone inscribed in the volumetric source is totally equivalent to a cylinder of length L and base area dσ, the luminous ﬂux from each element of the cylinder being gathered through a solid angle of Ω = πR2 /a2 (shaded region in the bottom part of Figure F.3). The effective diaphragm radius R can be determined either by the diameter of the objective lens or by the relative aperture of the spectral instrument, for the angle Ω can be limited by the collimator of a spectrograph. Figure F.2 shows the region of the volumetric source, whose light passes through the elementary area dσ of the slit, which is located on the optical axis (the vertex of the cone coincides with dσ , point slit approximation). If the slit is of ﬁnite size σ , light passing through the other elementary areas of the slit will be gathered from the cones whose vertices coincide with the images of these areas in the space of the object. To 2

F Measurement of the Absolute Populations of Excited Atoms

Figure F.4 Conical surface cutting out that region of the source, from which light enters the spectral instrument through a ﬁnitesize slit.

calculate the luminous ﬂux that has passed through the slit of ﬁnite size, it is necessary to swing the optical axis from one edge of the slit to the other. The region from which radiation enters the spectral instrument in the case of ﬁnite-size slit is shaded in Figure F.4. This region determines spatial resolution in absolute measurements. One can demonstrate that in this case, analogy also holds between the cone from Figure F.3 and a cylinder with a length of L and a cross-sectional area of σ equal to the 2 total area σ of the slit increased by a factor of ( a/a ) . In actual experimental conditions, spatial resolution is of the order of a millimeter and can be bettered to a few fractions of a millimeter, depending on the size of the slit and the effective diaphragm. As in any physical experiments, the improvement of spatial resolution worsens the signal-to-noise ratio at the output of the recording system, and so a reasonable trade-off decision is required in selecting the parameters of the optical system. With the luminous ﬂux calculation scheme suggested, one can easily introduce corrections for the axial inhomogeneity within the limits of the source if one puts I ( x ) = I0 Ψ( x/L), where I0 is the maximum intensity and the function Ψ describes the relative axial radiation intensity distribution. Obviously the magnitude of the luminous ﬂux is in this case given by Ω F = I0 dσL ψ; 4π

1 ψ= L

a+ L/2 a− L/2

Ψ ( x/L) dx.

(F.5)

589

590

Appendix F Measurement of the Absolute Populations of Excited Atoms

Figure F.5 Luminous ﬂux gathered from an axially symmetric source viewed across the axis.

The radial inhomogeneity in the given illumination scheme can be disregarded at small angular apertures. As applied to an axially symmetric source viewed across the axis (Figure F.5), expression (F.5) assumes the form Ω F (y) = I0 dσR0 ψ(y); 4π

√

ψ(y) = 2

1− y2

0

Ψ(r )dx = 2

(F.6)

1 Ψ(r )dr y

1

r 2 − y2

.

(F.6a)

The relative coordinates x, y, r range between 0 and 1, I0 is the absolute radiation intensity at the center of the source, ψ(y) is the measured relative transverse distribution normalized to unity at the center and Ψ(r ) is the relative radial intensity distribution. To convert from the measured transverse distribution ψ(y) to the true radial distribution Ψ(r ), it is necessary to solve Abel‘s equation (F.6)a by any of the numerous methods developed for the purpose. Spatial resolution can be improved with the aid of the instrument function that describes the broadening of a delta-shaped point source in the course of projection into the plane of the slit. Indeed, the aberrations of the projection optics and the ﬁnite sizes of the slits and effective diaphragms result in instrumental distortions as regards spatial measurements. The instrument function is rather difﬁcult to calculate, but its form can be easily found by replacing the volumetric with by a point one and scanning its image in the actual experimental setup. The instrument function thus obtained should be normalized to unit surface area. In that

F Measurement of the Absolute Populations of Excited Atoms

Figure F.6 Illustrating the effect of instrumental distortions when taking spatial luminous ﬂux measurements for an axially symmetric source viewed (a) along and (b) across the axis. F (r ) and F (r ) – observed and recovered radial distributions for the source viewed along the axis, ψ(r ) – observed transverse distribution, Ψ1 (r ) – radial distribution recovered without allowing for instrumental distortions, Ψ2 (r ) – radial distribution recovered with due regard for instrumental distortions.

case, the measured radial distribution F (r ) will be related to the true distribution F (r ) by the convolution-type equation F (r ) =

F (ρ) A(r − ρ) dρ.

To exclude instrumental distortions, use can be made of the welldeveloped methods for solving ill-posed problems. Allowing for instrumental distortions is especially important when measuring luminous ﬂuxes across the axis in axially symmetric sources (Figure F.5). One should ﬁrst correct the observed transverse distribution for the spatial instrumental distortions and then take the Abelian transformation to get the true radial distribution. One has to deal here with the successive solution of two ﬁrst-order integral equations, which is generally a complex enough problem. Figure F.6 presents some illustrative examples showing what instrumental distortions one can expect when measuring radial line intensity distributions along and across the axis of a discharge. For example, when studying the radial structure of a contracted discharge ﬁlament 3 mm in radius, the radial distribution measured along the axis is F (r ) (Figure F.6a). The instrument function is approximated by a Gaussian curve with a half-width of 1 mm. Following correction for the instrumental distortions, one gets the curve F (r ) which at maximum differs by

591

592

Appendix F Measurement of the Absolute Populations of Excited Atoms

ca. 30% from the measured one. The transverse distribution measured across the axis (Figure F.6b) is ψ(y), and the radial distribution recovered from it without correction made for the instrumental distortions is Ψ1 (r ), while that recovered with due regard for the distortions, Ψ2 (r ). Taking account of instrumental distortions is especially important where plasma glow is concentrated in peripheral regions, for example, in the case of skin effect. In that case, the effect of instrumental distortions in the analysis of the spatial structure of a plasma source can be appreciable. This luminous ﬂux calculation method makes it possible to easily take account of self-absorption within the limits of an axially homogeneous source. To this end, one should multiply expression (F.1) into the probability that quanta will cover the distance from the point with the coordinates x, r to the boundary of the source without being absorbed and then integrate it over the conical surface. The probability that the quanta emitted in a spectral line with an emission proﬁle of ε ν will cover the distance x without being absorbed is w( x ) =

∞ 0

ε ν exp(−k ν x )dν,

(F.7)

∞ where k ν is the absorption line proﬁle, 0 ε ν dν = 1. It proves convenient to move the origin of coordinates to the point x = a. Multiplying expression (F.1) by (F.7) and integrating over the conical surface with due regard for property (F.3), we get

∞ Ω L/2 L dx ε ν exp −k ν −x dν. (F.8) F = Idσ 4π − L/2 2 0 Changing the order of integration and integrating with respect to the coordinate, we have F = IdσL

Ω 4π

∞ εν 0

kν L

(1 − exp (−k ν L)) dν ≡ IdσL

Ω S ( k 0 L ), 4π

(F.9)

where S(k0 L) is the Ladenburg function and k0 is the absorption coefﬁcient at the line center. Expression (F.9) gives in absolute measure the magnitude of the luminous ﬂux that has passed through the slit of the spectral instrument from the volumetric plasma source in the given illumination scheme in the presence of reabsorption within the limits of the source. The Ladenburg function shows how much the luminous ﬂux emitted by the plasma column of ﬁxed length is reduced as the absorption coefﬁcient grows higher. The product of the column length and the Ladenburg function shows how the radiant ﬂux increases with increasing column length at a ﬁxed absorption coefﬁcient. The integrand in expression (F.9)

F Measurement of the Absolute Populations of Excited Atoms

shows how the spectral line proﬁle deforms outside of the source as the optical density is increased. Obviously expression (F.9) becomes (F.4) if one puts k ν → 0 and expands the exponent in the integrand in (F.9) into a series. Passage to the limit of high absorption coefﬁcient values is not so obvious. If one simply neglects the exponent in comparison with unity in expression (F.9), then, assuming similar emission and absorption line proﬁles, one obtains divergence on integrating with respect to frequency between inﬁnite limits. The important point is that despite the great optical thickness near the line center, absorption in the far wings of the line becomes weak, the exponent approaches unity and cannot be omitted when taking the integral in expression (F.9). Actually this means that photons in the line wings can move large distances without being absorbed and leave the plasma volume. One can expunge the divergence if one formally cuts off the spectral line wings at some ﬁxed frequencies. In this hypothetical case, integration over a ﬁnite spectral interval, with the exponent disregarded and the emission line proﬁle normalization taken into account, yields F = Idσ

Ω 1 , 4π k0

and the luminous ﬂux no longer depends on the length of the source. For real line proﬁles, the Ladenburg functions fail to reach saturation. Figure F.7 presents the luminous ﬂux issuing from a plasma column as a function of the column length at a ﬁxed absorption coefﬁcient. To conﬁne the emission of spectral lines within the volume of plasma, it is necessary to introduce stimulated transitions along with the spontaneous ones and to take into account the broadening and overlapping of spectral lines. In this case the intensity in the line center will be equal to the Planck’s blackbody intensity value. Based on expression (F.9), one can construct classical methods for measuring the densities of emitting atoms, which was started by Ladenburg and co-workers and described in many books on plasma spectroscopy. Formula (F.4) for an optically thin source and (F.9) for a source with self-absorption can be used to calculate the absolute intensities of spectral lines. To this end, it is necessary to calculate the luminous ﬂux that has passed through the slit from a standard source to be located at a distance of a from the objective lens, which is attained by means of a tilting mirror. Let S be the surface area of the ribbon ﬁlament of the photo∞ metric lamp and b = 0 bλ dλ, the integral brightness determined from the ribbon temperature speciﬁed in the lamp certiﬁcate, depending on the ﬁlament current. The luminous ﬂux that has been gathered by the objective lens and passed through the slit with a surface area of dσ is

593

594

Appendix F Measurement of the Absolute Populations of Excited Atoms

Figure F.7 Luminous ﬂux as a function of the discharge column length at a ﬁxed absorption coefﬁcient for a Doppler and a Lorentz line proﬁle.

obviously 2 dσ a = bdσ Ω = bdσΩ. S a This expression is tantamount to the statement that the image brightness is equal to the brightness of the object. The luminous ﬂux issuing from the spectral instrument’s exit slit with a width of δl will be dλ δldσΩ ∼ ust Fst = bλ (F.10) dl Fst = bSΩ

where ( dλ dl ) is the dispersion of the spectral instrument and ust is the registering system signal proportional to the luminous ﬂux from the standard source. If the width of the spectral line emitted by plasma is much smaller than the spectral width of the exit slit, the signal upl from the plasma source will be proportional to the luminous ﬂux F that in its turn is determined by the integral line intensity I (formulas (F.4) and (F.9)). If we take the ratio between the luminous ﬂuxes registered from the plasma and the standard source, we will have, on canceling out the geometrical factors identical in both cases, the following simple expressions for calculating the absolute intensities of spectral lines: • for optically thin sources, 1 upl dλ δl , I = 4πbλ dl L ust

(F.11)

References

• for sources with reabsorption, upl dλ 1 δl S(k ν L) I = 4πbλ . dl L ust

(F.12)

When registering a continuous spectrum from an optically thin source, the luminous ﬂux passing trough the exit slit of small spectral width will be dλ Ω δldσL . F = Iλ dl 4π From the ratio between the luminous ﬂuxes from plasma and the standard source we have the following expression for the absolute intensity of the continuum at a wavelength of λ: Iλ = 4πbλ

1 upl . L ust

(F.13)

Thus, formulas (F.11) through (F.13) enable one to calculate the absolute intensities of spectral lines and continua and determine the populations of emitting atoms, and expression (F.9) can help one to get the populations of absorbing atoms when using the classical absorption methods described in Chapters 3 to 5. References

1 Yu. B. Golubovskii, R.V. Kozakov et al. Phys. Rev., E62, p. 2707 (2000). 2 S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz. (1963)

3 S.E. Frish, Ed. Gas-Discharge Plasma Spectroscopy (in Russian). Leningrad: Nauka (1970).

595

597

Appendix G

General Information for Plasma Spectroscopy Problems

G.1 Physical Constants

Velocity of light in vacuum

c = 2.99792 × 1010 cm s−1

Electron charge

e = 4.803 × 10−10 cgse units = 1.602 × 10−19 C

Electron mass

me = 9.109 × 10−28 g

Proton-to-electron mass ratio

Mp /me = 1836

Electron rest mass

ε e = me c2 = 8.185 × 10−7 erg = 817.600 × 10−19 J (511 eV)

Classical electron radius

re = e2 /me c2 = 2.818 × 10−13 cm

Proton mass

Mp = 1.672 × 10−24 g

Atomic mass unit

MA = MO /16 = 1.6598 × 10−24 g

Planck’s constant

h = 6.625 × 10−27 erg · s h¯ = h/2π = 1.054 × 10−27 , erg · c

Compton wave-length

λ0 = h/me c = 2.426 × 10−10 cm

Bohr magneton

μB = 0.927 × 10−20 ergg−1 (force)

Nuclear magneton

μn = 5.05 × 10−24 ergg−1 (force)

Proton magnetic moment

μp (H) = 14.10 × 10−24 ergg−1 (force)

Thomson scattering cross-section

σTh = 8πre2 /3 = 6.65 × 10−25 cm2

598

Appendix G General Information for Plasma Spectroscopy Problems

Number of molecules in 1 mole of substance (Avogadro number) Boltzmann constant

NA = 6.02 × 1023 mole−1 kB = 1.3807 × 10−16 ergK−1 = 1.3807 × 10−23 JK−1 σ = 5.67 × 10−5 erg(cm2 · c · K4 )−1

Stefan–Boltzmann constant

G.2 Atomic Values

First Bohr orbit radius Electron velocity in ﬁrst Bohr orbit Electric ﬁeld in ﬁst Bohr orbit Hydrogen atom ionization potential Rydberg constant Fine structure constant

rB = h2 /4π2 me e2 = 0.529 × 10−8 cm vB = 2πe2 /h = 2.187 × 108 cm · s−1 F = e/rB2 = 5.14 × 109 Vcm−1 IH = e2 /2rB2 = 13.6 eV = 109678.76 cm−1 Ry = IH /h = 2π2 e4 me /h3 = 3.29 × 1015 s−1 α = e2 /¯hc = 7.297 × 10−3 , α−1 = 137.036

The ionization potentials of various atoms and singly charged ions are shown in Figure G.1.

G.3 Correspondence between Spectral and Traditional Energy Measurement Units

Wave

Wavelength

number λ,cm

Energy

Energy

Energy

W, erg

W, eV

W, kcal/ ature mol−1

ν, cm−1 1 cm−1 1 erg

1

1.9857 × 10−16

0.5036

1.9857 × 10−16

1.2397 × 10−4

2.8578

1.4388

6.2426 × 1011

1.4391

×1016 1 eV

8066.63

1 cal/mole 0.3499 1◦ K

0.695

TemperT, ◦ K

0.7246

×1016 12396.8 × 10−8 1.6019 × 10−12 2.8578

0.6949 × 10−16

1.4388

1.3801×10−16

23053

11606.16 0.5034

4.3378 × 10−5 0.8616 × 10−4

1.9863

G.4 Electrical Units

Figure G.1 Ionization potentials of (AI) neutral atoms and (AII) singly-charged ions as a function of the serial number of the element.

G.4 Electrical Units

Charge Current Voltage Resistance Conductivity Capacitance Energy Power

1 C = 3 × 109 cgse units = 6.25 × 1018 electron charges = 9 × 1011 , V · cm 1 A= 1 C/s 1 V = 1/300 cgse units 1 Ω = (1/9) × 1011 cgse units = (1/30) s σ [Ω−1 · cm−1 ] = (1/9) × 1011 σ [s−1 ] 1 F = 9 × 1011 cgse [cm] 1 J = 107 cgse [erg] 1 W = 107 cgse [erg/s]

G.5 Units from Molecular Kinetics Pressure

• standard atmosphere 1 atm = 1.013 × 106 dyn/cm2 (erg/cm)3 = 1.013 × 105 Pa • bar 1 bar = 105 Pa • atmosphere at 0 ◦ C 1 atm = 760 mm Hg = 760 Torr = 1.013 × 105 Pa

599

600

Appendix G General Information for Plasma Spectroscopy Problems

• Torr 1 Torr = 1 mm Hg = 133.3 Pa • Pascal 1 Pa = 7.5 × 10−3 Torr Number of molecules in 1 cm3 at 0 ◦ C and 1 atm (Loschmidt number)

NL = 2.687 × 1019 cm−3

Number of molecules in 1 cm3 at 20 ◦ C and 1.333 hPa (1 Torr)

3.295 × 1016 cm−3

Velocity of particle of relative atomic mass A

v = 1.38 × 106

1

ε[eV]/Acm/s

1 v¯e = 1.45 × 104 1 T [K]/A = 1.56 × 106 T [eV]/A cm/s 1 ve = 5.93 × 107 ε[eV]cm/s 1 v¯e = 6.21 × 105 1 T [K] = 6.71 × 107 T [eV]cm/s

Average thermal velocity of particle Electron velocity Average thermal velocity of electron

σB = πrB2 = 0.88 × 10−16 cm2

Atomic cross-section unit

G.6 Quantities from Gas-Discharge Physics Electric ﬁeld strength (intensity)

• reduced to gas density, E/N 1 Td (townsend) = 10−17 V · cm2 • reduced to gas pressure, E/P [V/(cm · Torr)] = 3.3 × 1016 E/N [V · cm2 ] = 0.33E/N Td 1 1 2 Debye radius rD = kT 1e /4πe ne = 6.88 Te [K]/ne = 742 Te [eV]/ne cm Plasma frequency

ωp = (4πe2 ne /me )1/2 = 5.65 × 104 n1e/2 s−1

Critical electron velocity

ncr = me ω 2 /4πe2 = 1.24 × 104 ( f [MHz])2 cm−3 = 1.11 × 1013 (λ[cm])−2 cm−3

Electronic conductivity

σ=

= Root-mean-square electromagnetic wave ﬁeld strength

e2 ne νc = 2.53 × 108 ne (ω2ν+c ν2 ) , c−1 me (ω 2 +νc2 ) c 2.82 × 10−4 ne (ω2ν+c ν2 ) Ω−1 cm−1 c (νc and ω in s−1 )

E = 19

1

I [W/cm2 ] V/cm

References

References

1 A.A. Radtsig and B.M. Smirnov. Parameters of Atoms and Atomic Ions (in Russian). Moscow: Energoatomizdat (1968). 2 S.E. Frish. Optical Spectra of Atoms (in Russian). Moscow-Leningrad: Fizmatgiz (1963).

3 Yu.P. Raizer. Gas-Discharge Physics (in Russian). Moscow: Nauka (1982). 4 A.S. Yatsenko. Grotrian Diagrams of Neutral Atoms (in Russian). Novosibisrsk: Nauka (1993). 5 A.S Yatsenko. Grotrian Diagrams of Singly-Charged Ions (in Russian). Novosibisrsk: Nauka (1996).

601

603

Index

Λ-doubling sublevels 332, 333

330,

a Abel equation 73 absolute measurements 28, 47 absorption coefﬁcient 12, 24 absorption function 76, 79, 80, 82 absorption spectra of the CO2 molecules in a discharge 283 acousto-optic spectrometer 435, 437 actinometric pairs 293 actinometry technique 289 adiabatic approximation 176, 230 Airy function 415, 422–424 amplitude of a microwave 329 angular dispersion 409, 410, 413, 416, 417, 432 anharmonicity 192, 194, 195, 198, 202, 208, 209, 214, 217, 221 apodization 428 arcs 456 atomic oscillator 31, 34 atomization 235 autoionization 115

b black-body radiation 11, 16 Bohr magneton 343 Bohr radius 53 bolometers 465, 474 Boltzmann formula 6 Boltzmann rotational distribution 170 Born–Oppenheimer principle 270 Bose–Einstein statistics 11 Bouguer–Lambert–Beer (BLB) law 44 bound–free transitions 51 bremsstrahlung 52–54, 57–59 c cavity ringdown spectroscopy 99 characteristic relaxation time 189, 193 charge-coupled devices 481 Clebsch–Gordan coefﬁcient 309 coaxial photocell 467, 474 coherence length 92, 102, 138, 139 coherent anti-Stokes Raman scattering 137, 140

604

Index

coherent laser spectroscopy methods 337 collective scattering 377–379, 381, 386 collisional-radiative model 21, 22 complex dielectric constant 398, 399 Compton shift 62 Compton wavelength 309 concentrations of molecular ions 284 concentrations of polyatomic molecules 281 concentrations of the metastable molecules 281 conductivity of the medium 398 continuous spectrum 6, 11, 24, 45, 50 contrast of the image 477 convolution equation 148 coronal approximation 14 coronal ionization equilibrium relation 20 correlations 374, 375 correlations of ﬂuctuation 375 cosine radiator 583 d Debye screening radius 1 degree of dissociation, ionization, conversion 11 densities of negative ions 298 density matrix formalism 93, 94, 107 density measurements 360, 394 diatomic polar molecules 330 Dicke narrowing 41 diode lasers 88

dipole interaction 312, 313, 332 dipole moment 308, 312, 314, 319, 323, 330–332 discharges along the surface 457 dispersion proﬁle 33 distribution temperature 18 Doppler broadening abnormal 148 Doppler line proﬁle 39, 41, 42, 48, 49 Doppler shift 38, 62 Doppler width 41, 46 dynamic range of electron 360 dynamic Stark effect 345 e effective inversion temperature 214 effective lifetime approximation 48, 50 Einstein coefﬁcient 44 electric ﬁeld 307–333, 335–341, 345–349 electric ﬁeld in an atmosphericpressure hydrogen discharge 340 electron temperature 18, 20 electron, ion and neutral particle temperatures 17 electronic metastable states 251 electrophotography 479 elliptical orbit 312 emission methods 67 exchange between vibrational quanta 191 excitation of the electron shells of molecules by electrons 174 excitation temperature 19

Index

f Fabry–Perot interferometer 414, 415, 418, 421–425, 484 Faraday effect 342, 343, 349 Fellgett advantage 429, 430 Fermi interaction 198 ﬂuorescence decay 112, 118 ﬂuorescence induced by synchrotron radiation 120 ﬂuorescence spectrum of the NaK molecule 331 ﬂuorescence-dip detection 336 focal aperture 409, 411, 412, 416 focal plane 409, 410, 412, 414– 416, 420, 422, 477, 479 four-photon process 136, 141 four-wave mixing 97, 146 Fourier spectrometer 408, 425, 427, 429, 430, 438 Franck–Condon factor 220 free spectral range 414, 418– 421, 427, 428, 437, 438 free–free transition 51 frequency detuning parameter 37 frequency shift 32, 37, 41, 61, 62 frequency-tuned laser 84, 85, 99, 104, 122 Fresnel formula 399, 402 Fresnel number 102 Fulcher series 179 g gas-discharge lamp 441 Gaunt correction factor 55 Gaussian shape 39 Girard raster 432, 433 h ¨ Honl–London factor

199

Holtsmark distribution function 358, 359 hot plasma 2, 3 Hund coupling types 509 hydrogen-like atoms 35, 55, 59 i ideal plasma 3, 9, 11 identiﬁcation of spectra 67 illumination engineering unit 30 illumination engineering units 439, 440, 462 image converter and intensiﬁer tube 479 incursion 353, 355 inductively coupled plasma 235, 303 Inglis–Teller effect 326 instrument (spread) function 148 integral absorption 200, 202, 212 integral absorption coefﬁcient 44, 45 intensities of the satellites 317 interferometric methods of measuring the density 351 intermodulation optogalvanic spectroscopy 124 internal rotation 494 internal statistical sum 7, 8 intracavity laser spectroscopy 95, 144 intrinsic photoeffect 462, 470, 472, 479, 480 inverse problem 23, 24 inversion of molecular IR band 214 ion broadening parameter 365 ion component of the scattered spectrum 381, 388

605

606

Index

ion concentration 268, 288 ion velocities 36 ionization energy 9 ionization potential reduction 9 ionization temperatures 17 j Jacquinot advantage

421, 438

k Kirchhoff’s law 11, 16 Klein–Nishina formula 61 Kramers formula 54 l Ladenburg and Levi function 48 Ladenburg and Reiche function 48 Lagrange–Helmholtz principle 30 lambda doubling 271 Landau–Teller relaxation equation 156 Lande factor 343 laser-induced ﬂuorescence 104, 105, 120 law of mass action 7–9 light intensity 28 light scattering 60 light-gathering power 412, 413, 419–421, 428, 430, 431, 436, 438 limit of discernibility of individual lines 368 linear dispersion 410, 417, 421, 437 linear splitting 314 linear Stark effect 34 local electric ﬁeld 307 local thermal equilibrium 14, 16

Lorentz broadening 32, 36, 41, 46 Lorentz–Lorenz formula 399 Loschmidt number 12 m magnetic ﬁeld measurement method 342 magnetic quantum number 323, 328 Maxwell distribution function 6, 21 measurement of the electron density 343 Michelson interferometer 414, 425, 430 Milne formula 57 modiﬁed diffusion approximation 226 multipass absorption cells 95, 97, 98 multiphoton ﬂuorescence excitation 334 multiphoton interaction 129 multiplet component 81, 82, 84 n natural decay 32, 33, 42 noise component 463 noise equivalent power 464 noise factor 86 non-hydrogen-like atoms 364 nonisothermal mixture of particles 167 nonlinear susceptibility tensor 338, 339 nuclear spin 490, 491 nuclear statistical weight 276 o optical density 50

29, 44, 46–48,

Index

optically thin plasma 20, 24 optico-acoustic detector 466 optogalvanic effect 104, 121– 123, 128, 146 oscillator strengths 44, 45 p parabolic coordinate 312 parity 491 partial equilibria 14 partial local thermal equilibrium 14, 16 partial temperature 17, 21 phase 353–355, 374, 375 phase perturbations 34, 35 phase-matching 138, 139 photocathodes 467, 468 photographic emulsion 472 photoionization 57, 59 photometric parameter 419, 439, 448 photometric quantitie 29 photometric quantity 27 photomultiplier 467–470, 475, 479, 480 photon noise 429, 430, 462 photorecombination 55, 57, 59 photoresistor and photovoltaic detector 470 Planck formula 11, 13 plasma microﬁeld 326 plasma turbulence 316 plasmatron 456 plasmatrons 456 point symmetry group 489, 490 polarizability 351, 352 polarization intermodulation excitation 125 polarization state of the ﬁeld 322 Poynting vector 12

predissociation 116 probability of radiative transitions in diatomic molecul 269 proﬁle homogeneous 33 proﬁle of the π- and σcomponent 344 proﬁle of the Dγ line 317 proﬁle of the Hα spectral line 316 proﬁle of the Hδ spectral line 314 proﬁle of the spectral line 321 pulsed light source 448 pyroelectric detector 465 q quadratic Stark effect 34–36 quality of a photographic image 477 quantitative spectroscopy 27 quantum efﬁciency 463, 473 quantum numbers 487 quantum photodetectors 461, 472 quasineutrality 1 quenching cross section 34 r Rabi frequency 93, 108, 109 radiant emittance 27–29 radiant energy density 27 radiant ﬂux 27, 28, 30 radiation induced by the static 337 radiation transfer 29, 47, 50 Radon integral transformation 72 random processes 32 raster spectrometer 431

607

608

Index

ratio between the Thomson and the Rayleigh scattered intensity 383 Rayleigh criterion 411, 418, 478 Rayleigh scattering 63 recombination continuum 51 reduced ﬁeld 307 reﬂection and transmission of ﬁlm 400 reﬂection coefﬁcient 398, 399, 401, 403–405, 422, 425, 437 refractive index 351, 352 relative intensities of lines in the rotational structure 180 resolving power 409–411, 418, 419, 425, 428, 430, 432, 436– 438, 477–479, 485 resonance ﬂuorescence 63 rotational statistical sum 200 rotational temperature 154, 169–171, 179, 184, 188–190, 200, 205, 216, 218, 230 Rydberg atom 323, 326, 329 s Saha-Boltzmann formula 9 satellite branche 511, 512, 521, 550, 561 satellite intensity ratio 317, 319, 321 saturation power density 86, 87 scattering cross-section per unit volume 375 scattering region 378, 379, 386 scattering signal 375, 378, 388, 390 selection rule 508, 510, 512, 518, 519, 532, 535, 537, 546, 561 sensitivity 414, 461–470, 473– 476, 478

series of dielectric layers 405 sharpness factor 419 shift cross section 36, 42 skin layer 399, 400 slit instrument 409, 410, 432 solar constant 30 solar photosphere 301, 303 source function 154 spectra of a discharge in an HeNe mixture 122 spectral absorption coefﬁcient 44, 45, 54 spectral analysis methods 235 spectral density of radiance 28 spectral intensity 12, 16 spectral line proﬁle 31, 33, 36, 38, 42 spectral proﬁle of the Thomson scattering 376 spectrum of ﬂuctuations 374 spectrum inversion 211–216 spin splitting 488 spontaneous Raman scattering 130, 133, 134, 136, 140, 141 standard sources 583 Stark effect for multielectron atoms 327 Stark multiplet 312, 323, 325, 328, 335 Stark spectroscopy 307, 322, 323, 334–336, 341, 347, 348 statistical sum 7–9 statistical weight 7 statistical weight for rotational levels 491 statistical weight of rotational level of molecules 489 statistical weight of rotational levels 491 statistical weight of the levels 6

Index

statistical weight of the vibrational level 489 statistical wing 38, 42 Stefan–Boltzmann law 11 Stern–Fulmer plots 113 stimulated Raman scattering 135–137, 140 strongly ionized plasma 2, 3 sub-Doppler Stark spectroscopy 336 surface brightness 583 Swan bands 190 symmetry number 490 symmetry operation 491 t thermal noise 462, 465, 466 thermal photodetector 461, 462, 465, 470, 471 thermodynamic equilibrium 5, 6, 14, 15, 20, 25 thermoelement 464, 465 thermoplastic techniques 479 third-order susceptibility 137 Thomas–Reiche–Kuhn sum rule 45 Thomson and collective 377– 379, 386 time coherence 92 tomographic problem 71 torsional vibrations 494 transfer of radiation 16 translational temperature 19 transparency of the optical system 412 Treanor distribution 195, 196 two isolated levels approximation 308 two- and three-quantum transition 209 two-level scheme 107, 112

two-photon absorption cross section 132, 133 two-photon absorption optogalvanic spectroscopy 125 two-photon Lamb dip optogalvanic spectroscopy 124 two-photon transition 131 two-tube method 75, 78–80 types of ﬂuctuations 377 v van Held ‘growth curves’ 48 velocity of the ion 165 Vibrational distributions of the CO molecules 203, 210 vibrational level 174, 191–194, 196, 199, 202, 204, 205, 207, 209, 214, 219, 224, 231, 233 vibrational temperature 191, 193, 199, 200, 202, 205, 212– 214, 219, 221–223, 233 virtual state 130, 134 Voigt proﬁle 42, 43, 48 Volterra integral equation 73 volume spectral density 12 volumetric source 584–586, 588 w weak coupling approximation 385 weakly ionized plasma 2–4 Weisskopf radius 34, 35 White cell 95 width of the principal diffraction maximum 411 widths at half-maximum of the Hα and Hβ lines 362 z Zeeman effect

311, 343, 344

609