Advances in Atomic and Molecular Physics, Volume 5

Advances in ATOMIC AND MOLECULAR PHYSICS VOLUME 5

CONTRIBUTORS TO THIS VOLUME A. BEN-REUVEN C. D. H. CHISHOLM R. J. S. CROSSLEY A. DALGARNO H. G. DEHMELT HOLLY THOMIS DOYLE F. C. FEHSENFELD E. E. FERGUSON F. R. INNES

ROY H. NEYNABER A. L. SCHMELTEKOPF 0. SCHNEPP

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS THE QUEEN'SUNIVERSITYOF BELFAST BELFAST, NORTHERN IRELAND

Immanuel Estermann DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL

VOLUME 5

@) 1969 ACADEMIC PRESS New York London

COPYRIGHT 0 1969, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W l X 6BA

LIBRARY OF CONGRESS CATALOG CARDNUMBER : 65-18423

PRINTED IN THE UNITED STATES OF AMERICA

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

A. BEN-REUVEN, The Weizmann Institute of Science, Rehovot, Israel (201) C. D. H. CHISHOLM, Department of Chemistry, The University, Sheffield, Yorkshire (297) R. J. S. CROSSLEY, Department of Mathematics, University of York, York, England (237) A. DALGARNO, Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts (297) H. G. DEHMELT, Department of Physics, University of Washington, Seattle, Washington (109) HOLLY THOMIS DOYLE, Harvard College Observatory, Cambridge, Massachusetts (337)

F. C. FEHSENFELD, Environmental Science Services Administration, Boulder, Colorado (1) E. E. FERGUSON, Environmental Science Services Administration, Boulder, Colorado ( I )

F. R. INNES, Air Force Cambridge Research Laboratories, Bedford, Massachusetts (297)

ROY H. NEYNABER, Space Science Laboratory, General Dynamics/ Convair, San Diego, California (57) A. L. SCHMELTEKOPF, Environmental Science Services Administration, Boulder, Colorado (1)

0. SCHNEPP, Department of Chemistry, University of Southern California, Los Angeles, California (1 55) V

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Foreword This serial publication is intended to occupy an intermediate position between a scientific journal and a monograph. Its main object is to provide survey articles in fields such as the following: atomic and molecular structure and spectra, masers and optical pumping, mass spectroscopy, collisions, transport phenomena, physical and chemical interactions with surfaces, gas kinetic theory. It is the aim of the editors to select articles which combine a rigorous yet understandable introduction to their subject, a summary of past work with emphasis on progress in the last five years, and a sufficiently complete bibliography to enable the reader to locate quickly relevant details in the literature. Suggestions for topics to be covered in the future will be appreciated.

D. R. BATES I. ESTERMANN

Belfast, Northern Ireland Haifa, Israel

vii

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Contents V

LISTOF CONTRIBUTORS FOREWORD CONTENTS OF PREVIOUS VOLUMES

vii xiii

Flowing Afterglow Measurements of Ion-Neutral Reactions E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf I. Historical Introduction 11. General Experimental Aspects of the Flowing Afterglow Technique 111. The Flow Analysis IV. Data Reduction V. Production of Reactant Species VI. Optical Spectroscopic Studies VII. Temperature-Variable Flowing Afterglow Studies VIII. Some Miscellaneous Results IX. Summary References

1 4 14 35 36 46 46 50 52 55

Experiments with Merging Beams Roy H . Neynaber I. 11. 111. 1V. V. VI. VII. VIII.

51 59 62 80 89 100 105 106 107

Introduction General Principles Ion-Neutral Reactions Ion-Ion Reactions Neutral-Neutral Reactions Electron-Ion Reactions Current or Very Recent Studies Concluding Remarks References

Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy H . G . Dehmelt 3. Manipulation and Investigation of Stored Charge 4. Spectroscopic Experiments Relying on Spin Exchange with Polarized

Atomic Beam

5 . Spectroscopic Experiments Based on other Collision Reactions

6. Spectroscopic Line-Shifts and -Broadening ix

109 124 142 149

CONTENTS

X

7. Conclusion Errata for Part I References

152 153 153

The Spectra of Molecular Solids 0. Schnepp I. Lattice Vibrational Spectra 11. Intramolecular Vibrational Spectra 111. Spectra of Solid Hydrogen

References

155 176 187 197

The Meaning of Collision Broadening of Spectral Lines: The ClassicalOscillator Analog A . Ben-Reuven I. Introduction 11. The Fourier-Transform Method 111. Impact Damping

IV. Complex Oscillators V. Statistical Broadening VI. Resonance Broadening References

201 204 210 217 221 228 234

The Calculation of Atomic Transition Probabilities R. J . S . Crossley I. Introduction 11. General Formulas for the Dipole Approximation

III. Approximate Wave Functions: General Considerations IV. V. VI. VII. VIII.

Criteria for Calculation Variational Wave Functions Semiempirical Methods Perturbation Treatments Sum Rules, Bounds, and Variational Principles IX. Summary References

231 243 248 255 257 267 273 279 28 1 288

Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAs'lfpq C . D. H . Chisholm, A . Dalgarno, and F. R. Innes I. Introduction 11. The Calculation of cfp 111. Applications of Two-Particle cfp

IV. Description of Tables References

297 301 308 309 3 34

CONTENTS

xi

Relativistic Z-Dependent Corrections to Atomic Energy Levels Holly Thomis Doyle I. Introduction 11. The Relativistic Z-Dependent Theory 111. Irreducible Tensor Expansions of the Electrostatic and Breit Interaction

Operators IV. Antisymmetrization V. Reduction of Matrix Elements to Matrix Elements between One- and TWOElectron States VI. Results, Comparisons, and Conclusions References

AUTHORINDEX SUBJECTINDEX

331 342 352 357 363 369 412 415 425

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Contents of Previous Volumes Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos Electron Affinities of Atoms and Molecules, E. L. Moiseiwitsch Atomic Rearrangement Collisions, E. H. Eransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Partly and J . P. Toennies High Intensity and High Energy Molecular Beams, J. E. Anderson, R. P. Andres, and J . B. Fenn AurnoR INDEX-SUBJECT INDEX

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W . D. Davison Thermal Diffusion in Gases, E. A . Mason, R. J. Munn, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W . R. S. Carton The Measurement o f the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer Mass Spectrometry of Free Radicals, S.N. Foner AUTHOR INDEX-SUBJECT INDEX

Volume 3

The Quanta1 Calculation of Photoionization Cross Sections, A. L. Srewarr Radiofrequency Spectroscopy o f Stored Ions. I: Storage, H. G. Dehmelr Optical Pumping Methods in Atomic Spectroscopy, E. Eudick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Worf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood AUTHOR INDEX-SUBJECT INDEX xiii

xiv

CONTENTS OF PREVIOUS VOLUMES

Volume 4 H. S. W.Massey-A Sixtieth Birthday Tribute, E. H . S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D . R. Bates and R . H. G . Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P . A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I . C. Perciual Born Expansions, A . R. Holr and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0 . Mohr Recent Measurements on Charge Transfer, J. B. Hasred Measurements of Electron Excitation Functions, D. W. 0 . Heddle and R . G . W. Keesing Some New Experimental Methods in Collision Physics, R. F. Srebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Searon Collisions in the Ionosophere, A. Dalgarno The Direct Study of Ionization in Space, R.L. F. Boyd AUTHORINDEX-SUBJECTINDEX

FLOWING AFTERGLOW MEASUREMENTS OF ION-NEUTRAL REACTIONS E . E . FERGUSON. F. C . FEHSENFELD. and A . L . SCHMELTEKOPF Environmental Science Services Administration Boukfer. Cotorado

. .

1 I Historical Introduction ............................................ I1 General Experimental Aspects of the Flowing Afterglow Technique ........ 4 A Overall Description ............................................ 4 B Ion Sources.................................................... 5 C The Afterglow .................................................. 7 D Measurement and Calibration of Flow and Pressure ................ 9 E Ion Sampling and Detection ..................................... 10 111 The Flow Analysis................................................. 14 A . The Simple Model ............................................. 15 B Radial Diffusion ............................................... 16 C Nonuniform Velocity Profile ..................................... 18 D Axial Diffusion ................................................ 22 E Axial Velocity Gradient and Slip Flow ............................ 24 F. Inlet Effects ................................................... 26 G Conclusions concerning the Analysis .............................. 32 H . Test of the Assumptions......................................... 33 IV Data Reduction ................................................... 35 V Production of Reactant Species...................................... 36 A Reactant Ion Production ........................................ 36 B Neutral Reactant Production .................................... 40 C. Role of Impurities ............................................. 44 VI Optical Spectroscopic Studies ....................................... 46 VII. Temperature-Variable Flowing Afterglow Studies ..................... 46 VIII Some Miscellaneous Results ........................................ 50 IX . Summary ........................................................ 52 References ....................................................... 55

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I Historical Introduction This article presents a discussion of the flowing afterglow technique . This technique has been developed by the authors in the U.S. Department of Commerce Research Laboratories (formerly NBS. now ESSA) in Boulder. Colorado. starting in 1963. primarily for the quantitative measurement of ion-molecule reaction rate constants . 1

2

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

Flowing systems had been widely used in the measurement of neutral reactions by chemists for many years (Schiff, 1964, Kaufman, 1964 and references cited therein). These flow systems ordinarily had slower flow rates and smaller tube diameters. Consequently, charged species did not persist very far downstream and accordingly were not observed in these experiments. The precurser to the first ESSA flowing afterglow system was actually a large diameter, fast-pumped, glass afterglow tube utilized for optical spectroscopic studies in the NBS-Washington Laboratory of H. P. Broida (Schmeltekopf and Broida, 1963). Visual examination of the recombination light clearly showed that charged species existed for a meter or more down a 10-cm diameter tube with gas velocity lo4 cm/sec. It was a logical step to place a mass spectrometer at the end of such a tube and then add neutral reactants into the flowing stream in order to study reactions. The first quantitative reaction rate measurements in the ESSA flowing afterglow system were for atomic helium ions reacting with 0,and N, . They werereportedin 1964(Fergusonetal., 1964).In thecourse ofthesemeasurements we learned that Sayers and Smith (1964), at a Faraday Society Discussion in Edinburgh, reported measurements on the same reactions using stationary afterglow techniques. The flowing afterglow results (for which a factor of 2 accuracy was claimed at that time) agreed within 24 and 31 % of Sayers and Smith's results. Therefore, it was clear from the beginning that the flowing afterglow technique had sufficient capability for quantitative measurements to merit further development. In other words, the initial direct comparison of flowing afterglow measurements with measurements obtained by other methods indicated the general validity of the flowing afterglow method prior to the analysis of the rather complex hydrodynamics. Subsequent analysis has also established this validity. In addition to the example just cited, a number of other comparisons of the flowing afterglow results with those obtained by other techniques have followed, e.g., by Warneck (1967) for a number of positive ion reactions, and by Moruzzi et al. (1968) for several negative ion reactions. It was not required that the flowing afterglow technique yield very accurate (e.g., 10%) rate constants in order to be useful. In the field of ionospheric chemistry, which has been the primary area of application by the present authors, there were many important reactions whose rate constants were entirely unknown. Order-of-magnitude rate constant data could quite drastically alter aeronomical theories and factor of 2 rate constants were as accurate as could be effectively utilized in most cases due to the comparable, or greater, uncertainties in neutral atmospheric composition, ionizing fluxes, ion compositions, and other factors entering ionospheric analysis. We still encounter situations in which yes-no answers on reactions lead to valuable chemical data, e.g., relative electron affinities of molecules, bond dissociation energies, etc.

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FLOWING AFTERGLOW MEASUREMENTS

3

It was quickly apparent that the flowing afterglow technique had a substantially wider versatility than other techniques for measuring ion-neutral reactions at thermal energies. For example, ions could be reacted with unstable neutral species. In 1965, the reaction N2+ 0 -+ NO+ + N was reported (Ferguson et af., 1965a). This is still the only reported measurement of this important reaction, which is a major N,’ loss process in the Earth’s ionosphere. Most aeronomists believed in 1965 that this very fast reaction did not occur, and aeronomy has been significantly influenced by this single measurement. Also, more control of reactant states was possible using the flowing afterglow technique than with other experimental methods. As an example, it is possible in many cases to guarantee that all reactants are in their ground electronic and vibrational states. It has even been possible to measure reactions of several ions with controlled vibrationally excited N, . These are the first such laboratory measurements of ion-molecule reaction rate constants as a function of neutral reactant vibrational temperature (Schmeltekopf et al., 1968). Recently rate constants for reactions of ions with electronically excited oxygen molecules have been measured (Fehsenfeld er af., 1968). Because of the above advantages, sufficient incentive existed to develop and exploit the flowing afterglow technique for ion-molecule reaction studies. There have been steady advances in the experimental technology of the flowing afterglow technique and also in the rather detailed analyses of some of the problems associated with extracting quantitative rate constants from observed changes in ion concentrations. The present article is intended to be a fairly comprehensive compilation of these experimental aspects and to present the detailed mathematical analysis of the system. Several reviews have concentrated more on the experimental results obtained, the consequent theoretical implications and generalizations, and the applications to ionospheric physics problems (Ferguson et al., 1965b; Ferguson, 1967, 1968, 1969). In Section 11, experimental details are discussed in sequence down the flow tube: from the “ nonchemical” aspects of ion production (B), the afterglow (C), the flow measurements (D), and the ion sampling (E), all of which are preceded by a brief overall description (A) designed to give the broad picture before immersing the reader in detail. In Section 111, a detailed discussion of the gas flow and hydrodynamic analysis is given. More space is given to this topic then any other. In part this is because much of the analysis is recent and has not been presented elsewhere, in part because the subject matter unfortunately is somewhat complex and hard to condense, and perhaps most importantly because the tractability of the hydrodynamics is the essential key to the utility of the fast-flowing afterglow system for accurate quantitative rate measurements. Indeed, it seems likely that the potential complexity of the hydrodynamics discouraged previous workers from

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E. E.Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

attempting to use fast-flow systems for quantitative reaction rate studies. Section IV is a brief discussion of the data analysis, including our recent addition of on-line computing capability which speeds data acquisition and reduces operator error. Section V is a discussion of the chemical aspects of ion production (differentiated from direct electronic ionization, as in Section I1,B) and neutral production, including excited states and unstable species. Further sections describe the application of the flow tube to spectroscopic studies of reactions and a temperature variable system which obtains rate constants from 80 to 600°K and which is proving to be useful for three-body reaction studies. 11. General Experimental Aspects of the Flowing Afterglow Technique A. OVERALL DESCRIPTION

The earliest ESSA flow tube was a Pyrex tube about 1 m long and 8 cm inside diameter. The glass tube allowed some estimation of the flow properties of the afterglow by visual observation. This was extremely helpful in the early period. The spatial distribution of certain light-producing reaction products was qualitatively apparent to the eye and gave an indication of such parameters as the distance needed for the added neutral reactant to fill the tube, etc. This point is illustrated by the color photographs of Bass and Broida (1963). A recent version of the flow tube, constructed of stainless steel, is shown in Fig. 1. The metal tube is substantially more versatile.

FIG.1. Pictorial representationof the flowing afterglow tube.

FLOWING AFTERGLOW MEASUREMENTS

5

-

A fast gas flow (u lo4 cmlsec) is established in the tube by a Roots-type pump backed by a mechanical forepump. This pumping system can maintain a 500 liter/sec pumping speed at pressures from to 10 Torr. The principal, or buffer, gas normally has been helium. With a helium flow of 180 atm cm3/sec and at maximum pumping speed, the helium pressure in the tube is typically about 0.4 Torr. This pressure has been varied from 0.1 to 5 Torr by varying the helium flow or the pumping speed. Ions are produced by an excitation source and are carried down the tube in the flowing gas. A wide range of reactant ion species, both positive and negative, can be produced by the addition of a relatively small concentration (compared to the helium) of a suitable gas. This gas can enter with the helium through the excitation region or be injected into the helium afterglow immediately after the excitation source. These ions are carried down the tube past a neutral reactant port where this reactant is added in measured amounts. In the region following the neutral addition port, reactions between the reactant ions and the neutrals take place. The reaction region is terminated by the sampling orifice of a mass spectrometer. The ions are sampled, mass analyzed, and counted. The variation of an ion signal as a function of the reactant concentration leads quantitatively to the reaction rate constant. When the reaction leads to a product ion, it is also observed in the mass spectrometer. Reaction rates may be determined in a flow tube by ways other than variation of the neutral gas addition. One alternative is to establish a fixed neutral reactant gas flow and vary the gas velocity.

B.

ION

SOURCES

In the first tubes a microwave discharge was used as the primary ionization source. Small microwave cavities of the Evenson type (Fehsenfeld et al., 1965a) were slipped over a converging-diverging (deLaval nozzle) section of a small quartz tube preceding the large glass flow tube. While the microwave source was clean and easy to use, it had one distinct disadvantage. The ratio of ultraviolet photons to ions produced by the microwave discharge was very high. These photons are mainly the n'P+ 1 'S series and principally the (2'P + 1 ' S ) 584 A line. This line is resonantly absorbed by neutral helium. Since the lifetime of the resulting excited atom is much longer than the time between the emission and absorption of the photon, these resonant photons spend most of their time absorbed. The resulting excited atoms can be handled experimentally like metastable atoms, as will be detailed later. Since the discharge was produced in the deLaval nozzle (approximately Mach one), the doppler shift given to the photons by the sonic gas flow moved the center of the emission line one doppler half-width to the blue side of the

E. E.Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

6

absorption line. A fraction of the photons were then nonresonant and could thus traverse the length of the flow tube and produce reactant ions near the mass spectrometer sampling hole. This could lead to the deduction of erroneously low reaction rate constants, since the reactant ions produced at this location would not have the full time to react with the neutral reactant. However, pulsing the discharge and gating the ion detector later (-6-12 msec) would eliminate this prompt spurious photoionization signal. Ion arrival-time spectra illustrating these phenomena were published by Goldan et al. (1966). These phenomena would only arise in reaction studies for which the reactant ion was the ion of a neutral species present in a significant concentration in the system. The only instance where ultraviolet photons introduced an error in a published rate constant concerned the reaction N 2 + + O2 + 02+ N, and this was rather quickly discovered and corrected by the pulse technique (Fehsenfeld et al., 1965b). In an attempt to decrease the photon to ion ratio, the microwave discharge was replaced by a dc discharge consisting of a large cylindrical cathode near the glass tube wall and a small grounded wire anode in the center of the tube. This type of discharge produces a lower photon to ion ratio. Moreover, this discharge was in the slow-flowing gas stream, so that the 584 A photons could now be absorbed by the background gas. For this type of discharge however, higher-lying states are excited. Now, the photons from excited states of the He+, primarily 304 A, became a problem, since the He' density was so low that little absorption took place. Therefore, the pulsed mode was still used with the dc discharge but was not as important as it had been for the microwave discharge. Compared to the microwave discharge, the cylindrical cathode resulted in a factor of 20 reduction in the downstream photoproduction of reactant ions. The dc discharge had other advantages over the microwave discharge as well. Its geometrical configuration was more compatible with the detailed flow analysis, and it was more easily incorporated in metal flow tubes, which were soon found to be advantageous. The dc discharges had some serious drawbacks, however. Observable quantities of cathode material were sputtered during operation. Positive ions of the cathode metals and negative ions of the cathode metal oxides were detected in the mass spectrometer. The sputtered metals also had deleterious effects on optical windows, electrical insulation, etc. A rather substantial improvement in excitation sources became possible with the development of the barium zirconium oxide filament by MacNair (1967). This filament can be operated in an oxidizing atmosphere for many hours and is ideally suited to flpw tube use. By employing electron energies at which the electron impact ionization cross section is a maximum ( x 100 eV), the downstream photoproduction of reactant ions is about a factor of 200 less than that produced by the microwave discharge. With this ion source,

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FLOWING AFTERGLOW MEASUREMENTS

7

practically no photon effects have been detected of ion effects); consequently, the pulse method need no longer be employed. The use of an electron emitting filament has greatly increased the versatility of the system. By varying the emission current, the electron density can be varied between wide limits. By varying the gun voltage the electron energy can be controlled. This, in turn, can be used to selectively control ion production. For example, if an electron energy less than 24.6 eV is used in helium, metastable helium atoms will be produced, but no helium ions. If then N, is added to this afterglow, only N,' ions will be produced for further reaction studies (with no N + ions). In addition, the electron emission filament has a much higher efficiency (ions per watt) than the discharges and thus supplies less heat to the gas. Finally, the electron gun is much cleaner than the dc discharge.

C. THEAFTERGLOW Over the years of operation of the flowing afterglow a certain amount of investigation has been carried out on the nature of the helium afterglow, since this was the initial medium for most of the ion-molecule studies. The electron density, electron temperature, gas temperature, and helium metastable concentrations were determined under a variety of conditions. For this purpose, a quartz tube replaced the Pyrex tube, so that ultraviolet spectroscopic and microwave interferometric studies could be carried out. The electron density and temperature were determined by absolute intensity measurements of the He I n 'P + 2 ' S series up to levels near the continuum at 2600 A (e.g., n 19). At levels close to the ionization limit (those within ~ 5 / kT, 2 of the limit), a Boltzmann distribution of levels is maintained by electron collisions, so that the relative line intensities yield T,, which is generally close to 300°K a few centimeters after the excitation region. Absolute intensities were used in conjunction with the Saha equation to determine the electron density n, , The electron density determined was verified by microwave interferometer measurements at two frequencies, X band (9 GHz) and V band (50 GHz). These measurements represent the lowest electron densities and electron temperatures measured by this spectroscopic technique as far as we know. Since the measurements did not enter our ion reaction rate determinations in any critical way we have never refined or published these results. A typical Saha plot is shown in Fig. 2. The neutral gas temperature has also been measured spectroscopically, both by doppler line width and He, rotational intensity distribution measurements and found to be equal to the wall temperature. The He(2 'S) and He(2 'S)densities were measured by absorption of the helium 3889 and 5016 A lines. Measurements have shown that molecular helium ions are formed primarily from the atomic helium ions in the tube, leading to a He,+/He+ ratio

8

E. E. Fergiison, F. C.Fehsenfeld, and A. L. Schmeltekopf

FIG.2. Sample of the results obtained for the determination of electron densities and temperatures in a flowing helium afterglow. The statistically weighted upper state density of the n3P states of helium is plotted versus the separation in reciprocal centimeters, from the series limit, where P = 0.5 Torr, flow = 220 atm cm3/sec, and t = 600 psec.

of -0.1 at the mass spectrometer in very pure helium at 0.4 Torr and 300°K. This is in good agreement with theoretical prediction using the known rate He (Beaty and Patterson, 1965). At high pressures for He' + 2He + He,' this ratio rapidly increases, and Hez+ becomes the dominant ion. Under such conditions He,' reactions with neutrals have been measured. At 80"K, in the ESSA temperature-variable afterglow tube, He,' is found to be the dominant ion in the helium afterglow at pressures greater than -0.2 Torr. This weakly bound ion is not observable at 300"K,consistent with its properties as reported by Patterson (1968). The primary ion loss process in the helium afterglow is ambipolar diffusion to the tube walls, except possibly at very low electron densities (n, < lo7 cm-j). Argon has also been extensively used as a buffer gas in the flowing afterglow tube. Its afterglow properties are similar to those of helium, but have been much less studied. However, its hydrodynamic properties are different and must be given due consideration.

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FLOWING AFTERGLOW MEASUREMENTS

9

The use of the filament source gives the following conditions in the helium afterglow. For 100-eV electrons the only species significantly present are He+, He(2 3S),and He,'. The photon density is very low, as discussed above, and He(2 'S)atoms are not present because superelastic collisions by electrons convert the He(2'S) to He(Z3S). The ions and electrons thermalize to near gas temperature in the first few centimeters. When high currents are used in the ionizer, the electron density midway down the tube can be as high as 2 x 10" ~ m - In ~ .this case photons, metastable states, and highly excited Rydberg states are produced by recombination, as well as He' by He(2 'S) + He(2 3 S )+ He' + He + e. Usually much lower ionization currents are used so that the electron density is =lo9 cm-3, where the above effects are not large enough to be important. In fact, even electron densities less than lo7 cm-3 can be used a t which point one must begin to worry about the plasma condition still holding. At sufficiently low ionization voltages one produces only metastables and, since there are then no electrons present for superelastic collisions, both 2 ' S and 2 3 S metastables are present. One of the most serious problems associated with the use of the flowing afterglow for quantitative measurements can be a definite knowledge of where the reactant ions are being produced. It must be established in some way that there are no significant sources of reactant ions past the point at which the neutral reactant gas is added. Since the introduction of the electron impact ion source the concern here is no longer photoionization, but reactant ion production by secondary processes such as ion-neutral reactions, electron attachment, .Penning ionization, etc. The actual determination of whether there are sources of reactant ions in the reaction zone is a problem specific to the ion. Ordinarily this determination is made by varying either the flow of the reactant ion parent gas or the electron density and observing whether the rate constant is affected by these changes. In the absence of downstream sources of reactant ions, a semi-log plot of the reactant ion current versus added reactant gas flow is fairly linear, as will be noted later, and a strong curvature is usually an indication of reactant ion production in the reaction zone. D. MEASUREMENT AND CALIBRATION OF FLOW AND

PRESSURE

As will be shown in Section 111, the reaction rate constant is dependent on the reactant gas flow Q and the square of the buffer gas flow velocity u. The latter is calculated from the measurements of the downstream pressure and the buffer gas flow. Consequently, if accurate reaction rate constants are to be obtained, these two gas flows, as well as the downstream pressure, must be measured accurately. Finally, in many instances it is convenient to be able to measure the flow of the reactant ion parent gas.

10

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

The downstream pressure is measured by a Baratron capacitance manometer. The calibration of this gage is checked periodically by an untrapped McLeod gage. Three Baratron gages in use in this laboratory have been compared and the overall agreement at 0.376 Torr is 1 % . The flow of the buffer gas, the reactant ion parent gas, and the reactant gas are measured by a method which depends on gas viscosity. In this technique the flow elements are so designed that within the specified range of operation the flow through the elements is viscous. Thus, the flow of gas through the element is proportional to the pressure difference across the element and the average pressure in the flow element. In actual practice the pressure at the input to the flow element is used instead of the average pressure. The slight nonlinearity thereby introduced is negligible. The input pressure is maintained at 760.0 Torr. The three channels of gas addition are shown in Fig. 3. In the buffer gas channel the flow element is designed to accommodate a maximum flow of 300 atm cm3/sec of helium. Typical buffer gas flows are on the order of 180 atm cm3/sec. In the other two channels, a combination of three elements can be used to allow gas flows to be measured over a range extending from 0.001 to 10 atm cm3/sec of helium in full scale ranges of 0.1, 1.0, and 10.0 atm cm3/sec. In order to relate the output of the pressure transducers to absolute gas flows, calibration procedures have been devised. The input pressure transducers PIand P, are calibrated by a water manometer (range: 780 to 740 Torr & 0.05 Torr). The relatively large buffer gas flows permit calibration of the AP3 transducer by a commercial precision wet-test meter (0.2%). The smaller flows of the reactant ion parent gas and reactant gas are calibrated by measuring the rate of pressure decrease in known volumes V , and V, . The details of these calibration procedures are too lengthy to be discussed here, but can be extracted from Fig. 3 by careful study. After calibration, the accuracy of the flow measurements is better than 0.5%. The short-term stability of the apparatus (during a set of measurements) can be as good as 0.02% . Periodic checks of the calibration are routine.

E.

ION SAMPLING AND

DETECTION

The sampling orifice is located in the center of the blunt end of a truncated cylindrical cone. Other nose-cone geometries were tried with less success. The distance between the front of the nose cone and the end of the flowing tube is about equal to the flow tube diameter. Because of this positioning of the sampling port, it is expected that the flowing plasma near the front of the nose cone will characterize the central portion of the steady state flowing plasma just upstream from the end of the flow tube. The flowing gas is

760.0 Torr Lme

FIG.3. Detailed schematic representation of the flowing afterglow system indicating the equipment used to measure and calibrate the flows, the mass spectrometer, the timing system for pulsed operation, and the on-line data scanning system.

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12

E. E.Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

smoothly separated by the conical sides of the nose cone and subsequently exhausted into the pump. This has been qualitatively confirmed by the appearance of the nose cone following a series of experiments on metal ions. The metal ions were produced by ionizing metal atoms evaporated from a furnace located at the input end of the flow tube. Because of the very high accommodation coefficient of the atoms on room temperature surfaces they are deposited on the walls of the flow tube and surface of the nose cone. The metal atoms on the nose cone were distributed in such a way that when it was illuminated concentric interference rings appeared on the surface of the nose cone. The heaviest coating was located on the blunt end of the nose cone and decreasing concentrations were found on the conical walls. The nearcoaxial nature of these rings indicated that the flowing gas was uniformly intercepted. To reduce possible sampling problems arising from the edges of the sampling orifice, this hole is made large, typically 0.5-1.5 mm. The thickness of the edges of the sampling orifice has not been found to be critical for thicknesses between 0.025 and 0.25 mm. Generally, the thickness of the orifice has been made less than one-fourth the diameter of sampling orifice. The electron density in the afterglow region near the sampling orifice is, in general, lo7 cm-3 or greater. Accordingly, the Debye length of the ions is, in most instances, less than the diameter of the sampling orifice. Since the pressure in the afterglow region is of the order of 0.4 Torr, the mean free paths for neutrals and ions are also less than the diameter of the sampling orifice. For these reasons we expect ions to be carried convectively by the buffer gas through the sampling hole, in much the same way as they are carried down the main flow tube. All our experience supports this view. The pressure in the chamber following the sampling orifice is approximately Torr. In this region the motions of the ions are determined by the local fields rather than by collisions. We have found that the most severe sampling problems have been due to deflection of the ions by surface charge on the inside surface of the nose cone and on the front surface of the quadrupole case. To minimize this, these surfaces are cleaned periodically, depending on system use. This cleaning consists of electrocleaning followed by a distilled water rinse. It has been our experience that neither ultrasonic cleaning nor vapor degreasing effectively cleans the surface contamination on these metal surfaces. In the heated system cleaning is supplemented by baking. It was found that reliable sampling, especially when oxygen atoms were in the system, could not be obtained when the material of the sampling orifice was gold, stainless steel, copper, brass, or rhodium. The only two materials which have been found to give reliable results are molybdenum or carbon, the latter in the form of a coating prepared from an alcohol suspension of graphite. Because it is more stable, molybdenum is now used. We do not

FLOWING AFTERGLOW MEASUREMENTS

13

understand why molybdenum works better than other metals. For example, when the sampling orifice was in a gold disk, a drop in the sampling efficiency of lo4 has been observed in the presence of oxygen atoms. With clean molybdenum no change is observed. The nose-cone potential En,is typically near ground potential while the potential of the quadrupole case E,,, is 2-6 V. Variations in these potentials are found not to affect the measured reaction rate constants. The quadrupoles have usually been 10 cm long, typically giving a resolving power of about 120 for full width at half-peak height. They are scanned from 1 to 200 amu, using a variable frequency, fixed voltage, power supply, or from 1 to 400 amu, using a fixed frequency, variable voltage power supply. The frequency-scanning quadrupole has the advantage that the resolving power and mass peak height are independent of mass. The voltage-scanning power supply with fixed ac to dc voltage ratio allows a rapidly scanned display of the mass spectrum on an oscilloscope. The rapid scan facilitates the determination of the qualitative behavior of ions in the system under varying conditions. Moreover, the quadrupole spectrometer using the voltage scan has a resolving power as large as 900 for a full width at half-peak height. With very high resolving power, Eqc must be reduced to less than 0.5 V to prevent some peak broadening due to the velocity acquired by the ions before their entrance into the mass spectrometer. The mass spectrometer chamber is differentially pumped by a 6 in. oil diffusion pump separated from the chamber by a water-cooled baffle and a sorbent trap. The multiplier chamber is pumped by a 4 in. oil diffusion pump and is similarly trapped. No evidence of oil backstreaming has been observed. to The detection system, which has input currents ranging from lo-'' A, normally utilizes pulse counting techniques. Pulse counting has the following advantages over the conventional electrometer detector : (1) easier to obtain a given accuracy; (2) facilitates use of digital analysis on the output; (3) readily adaptable to repetitive pulsed operation; and (4) simplifies detection circuitry to measure positive or negative ions. Points (3) and (4) are the principal reasons for using pulse counting on the ESSA system. The pulse amplifiers and counters have a pulse pair resolution of lo-* sec so that counting rates of lo6 counts/sec can be used with essentially no correction for pulse overlap. The discriminator is set for the plateau condition so that amplifier noise is rejected but essentially all of the pulses from the multiplier cathode are accepted. It has been noticed that nitrogen and perhaps other gases tend to reduce the gain of the multiplier slightly. Therefore, it is important to operate in the plateau condition so that slight changes in gain which could result from the reactant gas addition are not reflected in signal changes. The cathode of the multiplier is operated 1500 V attractive

14

E. E.Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

above the exit of the quadrupole. Since a further increase in this voltage has no effect on the signal, it is felt that all ions leaving the quadrupole are collected and counted. The most important concern with regard to the sampling and detection system is that the sampling efficiency be constant, that is, independent of the reactant ion density. Since the calculated rate constant is derived from the decline of the reactant ion signal as a function of the addition of reactant gas, it must be known that no change in overall sampling is occurring while the reactant gas is added. (As discussed earlier changes did occur when oxygen atoms were added before the molybdenum sampling hole was used.) Certain tests may be used to establish that the sampling efficiency is constant. For positive ions, when no further reactions of the product ions occur and when recombination and diffusion of the product ions are taken into account, the product ion signal should balance the initial reactant ion signal. The same is true for negative ion reactions with negative ion products. For associative-detachment reactions of negative ions, persistent, non reactive impurities (e.g., C1- and NOz-) have been monitored for constancy during experiments. Since the molybdenum sampling orifice has been used, no evidence of change in sampling has been observed except when the system is dirty, and a change in sampling is now taken as evidence of a dirty system. 111. The Flow Analysis

The analysis used to compute a reaction rate constant from the variation of ion signals as a function of the addition of a reactant gas is described in this section. To be reasonably accurate, this analysis must take into account many properties o f the flow. In order to describe the complete analysis it is instructive to first discuss an extremely simple model and then describe modifications of this simple model by the inclusion of more realistic flow properties. The analysis of the simple model is described in Section III,A. In Section III,B, the simple model is extended to include radial diffusion, Also contained in this section is a brief discussion of the analytical similarities of the various experimental approaches to the afterglow experiment. Section III,B concludes with the introduction of a generalized form of the simple solution obtained in Section III,A which will be used for the inclusion of the various flow properties into the analysis. Section III,C describes the effect produced by the inclusion of a parabolic velocity profile. In Section IKD, axial diffusion is introduced with a short discussion of the diffusion coefficients which are used in these calculations. Section III,E considers the pressure and velocity gradients of the flowing gas and incorporates these effects into the rate calculation. Section Il1,F is concerned with the effects

FLOWING AFTERGLOW MEASUREMENTS

15

associated with the perturbation of the reactant ion distribution by the localized addition of reactant gas. Section III,G reviews the conclusions derived from the analysis. Finally, Section III,H describes an experimental test of the accuracy of the assumed flow properties of this system.

A. THESIMPLEMODEL The original analysis of the flowing afterglow data used an oversimplified but mathematically tractable model. This model makes a suitable starting point for the discussion since it is easy to visualize and the further detailed analysis can be viewed as a series of modifications of the simple model. Neglecting diffusion, the decrease in the reactant ion A' in the reaction zone as a consequence of the addition of neutral reactant B, due to the reaction A+

+B +C+ +D

is given by

-dCA+l - - k[A'][B], dt where k is the binary rate constant for the reaction and [A'] and [B] are the concentrations of A+ and B. If the axis of the tube is designated as the z axis and the flow velocity is assumed to be constant and directed along the z axis, then the independent variable t in (2) may be changed to z. Equation (2) may be rewritten as uo -= - k[A+][B], a2

(3)

where z is measured from the reactant gas inlet and uo is the constant flow velocity. If [B] p [A'] for all z 2 0, then (3) may be integrated to yield

where [Ao'] is the concentration of A' prior to the introduction of the reactant gas and 1 is the length of the reaction zone. Now assume that B is initially introduced uniformly over the cross section of the tube so that

CB1 = 0, z
=

- Qkl/nu2uO2.

(8)

16

E. E. Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

Since I and a are dimensions of the flow tube and uo is the independently measured average gas flow velocity, the rate constant can be calculated from the reduction of the reactant ion signal by the reactant gas. Thus

k, = -(na'vo2/Ql) ln([A']/[A,+]) (9) where k, is used to designate the rate constant calculated assuming this simple model. B. RADIAL DIFFUSION To include radial diffusion, a loss term is added to (3) v O X.4' 7 1= D A -1 -a r- a[A'] r ar ar

(

) - (k[A'][B]

0

for z < O for z 2 0

(10)

where D A is the diffusion coefficient for A'. As was the case in the preceding section, the neutral concentration is assumed to be uniform. No azimuthal variations are considered; [A'] is allowed both radial and axial variations. In the solution of (10) it is assumed that [A'] = 0 at r = a, that all modes of [A+ J are continuous at z = 0, and that the interaction region is of infinite extent in either direction along z. When z < 0, A' is attenuated by diffusion alone and the ion distribution decays according to the relation

[A"]

CC

J,(li r/U) eXp[ - &'DA Z/a'Uo]

i= 1

(1 1)

where Jo(ljr/a) is the ith mode of the zero order Bessel function and ,Ii is the ith root of Jo . Since A1 is much smaller than the other Aithe higher modes will be rapidly attenuated compared to the fundamental mode. Thus, an initial distribution composed of a combination of modes will quickly relax to the fundamental mode as it proceeds down the tube. Therefore, the distribution of [A'] will be assumed to be fundamental mode at z = 0, i.e., by the time the ions have reached the reaction zone. The solution of (10) for z 2 0 yields the following expression for the concentration of ions on the axis of the flow tube, attenuated by both radial diffusion and reaction

where [Ai+] is the axial concentration of A + at z = 0. If [Ao'] is the concentration of A' at the end of the reaction region when attenuated by diffusion alone, then the relative loss of ions by reaction would be identical to that from (8) and the rate constant would still be determined according to (9).

FLOWING AFTERGLOW MEASUREMENTS

17

Equation (12) provides a useful basis for the discussion of the various ways in which rate measurements are currently obtained in flowing afterglows. In the ESSA experiment the only parameter which is usually varied is the rate of addition of reactant gas Q. In this method of performing the experiment, the first term representing the ion diffusion loss remains constant. Thus, at the end of the reaction zone ( z = I), the relative decline in the ion signal due to the introduction of the reactant gas is given by (9). In another method, the reactant gas flow is fixed and the reactant gas inlet is made movable, thus z is varied. It is possible to design a movable inlet so that its movement does not significantly change the diffusion losses in the tube. Therefore, the diffusion-loss term in (12) remains a constant. The reaction-loss term is a function of the reaction zone length, the now-movable distance between the reaction zone inlet port, and the sampling orifice. Hence, the relative change in the reaction ion signal due to the movement of the gas inlet is given by (9). Since the change in pumping speed alters the background gas pressure and thereby the gas flow velocity, a reaction rate may also be measured by changing the pumping speed. The diffusion coefficient of A + is inversely proportional to the background gas pressure which, so long as the buffer gas flow is held constant, is inversely proportional to the flow velocity. Thus, the diffusion coefficient of A + is directly proportional to the flow velocity. The net result is that the diffusion-loss term in (12) is once again a constant. Consequently, as in the two preceding cases, the relative change in the ion signal due to the variation in the flow velocity is still given by (9). Certainly, as more details of the gas flow dynamics are taken into account, these three methods are no longer strictly equal. However, even from the solution of the complex transport equations presented later there is little to a priori recommend one method of measurement over another. There are other ways to perform these experiments. These methods combine variations in both the diffusion and the reaction attenuations. It is probably advisable, however, to make reaction rate measurements in such a fashion that diffusion may in the first approximation be neglected. As more realistic flow dynamics are taken into account it is no longer possible to calculate a rate constant from the simple expression of (9). It is still possible, however, to present the results of the analysis in a relatively simple form. To do this, (9) is rewritten in a form which is modified to include corrections obtained from calculations using more realistic gas dynamical models:

k=

+ Ci a i > ( D ~ a 2 0 0 2 / QW[A+I/[&+l), 0

(13)

where the cli and jl are rate correction factors. Each correction factor represents an individual process which separates the results of the sophisticated

18

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

analysis from those of the simple case. The physical processes associated with the various terms in the a series of correction factors are: (1) ctl is the correction for the nonplanar velocity profile of the flowing gas; (2) a2 is the correction for axial diffusion; and (3) a3 is the correction for inlet effects, i.e., a3 refers to the correction associated with the initial source distribution of reactant gas and the perturbation produced on the initial ion distribution by the presence of the neutral gas inlet port. The correction factor pertains to the pressure gradient produced by the flowing gas. This particular correction does not originate from the transport equation which shall be solved later ; therefore, it is considered separately from the ct corrections. In flowing afterglow experiments, reaction rate constants are determined by the reduction of the relative ion signal as a result of the variation of one parameter on the right side of (9). In the ESSA experiment the variable parameter is usually the reactant gas flow. Accordingly, in the following discussion, the values of other parameters will be fixed and the reactant flow will be variable. General expressions have not been derived for the correction factors. All the terms in the ct expansion depend on the velocity profile of the flowing gas. Thus, all the higher-ordered cli are functions of a l . Likewise, due to coupling of diffusion and reaction, all of the aiwill be interdependent to a certain extent. Therefore, the ai should be taken only as indications of the magnitude of the respective effect and must be applied together.

C. NONUNIFORM VELOCITYPROFILE The gross gas flow properties of the flowing afterglow system are determined by the buffer gas, since the relative concentration of the reactant gas introduced into the system is typically only of the order of a few tenths of a percent or less. Under typical operating conditions the Knudsen number (the ratio of the mean free path of the buffer gas atom to the radius of the flow tube) is about 0.01 and the Reynolds number is 17. [ R = 1.41 x MQb/uq,where M is the atomic weight, Qb is the flow (atm cm3/sec), and q is the viscosity (poise) of the buffer gas.] Accordingly, the developed flow should be almost laminar with no more than a few percent slip at the wall due to molecular flow. The distance required for the buffer flow to fully develop the characteristic parabolic profile is given by d = 0.227 aR (Dushman and Lafferty, 1962). For helium at room temperature with a flow rate of 180 atm cm3/sec, d is 14 cm. As an experimental check, the pressure differential measured between points in this tube has been found to agree quite well with calculations based on laminar flow [see (24)]. As a consequence, the form of the velocity profile can be expressed:

u(r)= 2u0[1

- (r/a)’],

(14)

19

FLOWING AFERGLOW MEASUREMENTS

where oo is the average velocity. This average flow velocity (or bulk flow velocity) is defined as 00

=

Qb(atmcm'lsec) 760(Torr/atm) T("K) P(Torr) 273.16("K) ' na2(cmz)

(15)

where P is the pressure in the flow tube. This is an obvious definition; nevertheless, the velocity plays such an important role in determining the rate constant that an explicit statement seems worthwhile. To include the radial velocity profile, (10) can be altered slightly to read

where v(r) is expressed by (14) for the case at hand. Equation (16) is still separable, but because of u(r), the radial part is no longer Bessel's equation. Consequently, its solution is obtained by using both analytical and numerical methods. In the absence of reaction (z < 0), A + is attenuated by diffusion alone. It is no surprise that this solution is similar to that of the zero-order Bessel equation, which was discussed in connection with diffusion in the plane flow in Section II1,B. The similarity can be seen in Fig. 4, which plots the I .o

0.9

0.8 FIG.4. The computed fundamental mode ion distribution in the flowing afterglow for two different assumed velocity distributions in the absence of reactions. The relative ion density is plotted against tube radius. The curve labeled normal Bessel distribution is obtained by assuming a plane velocity profile while the curve labeled fundamental mode distribution in flowing system was obtained assuming the characteristic velocity profile for the flowing gas.

0.7 v)

z

#

0.6

z

0 0.5

5

W

Fundamental Mode

0.4

0.3 0.2 0.I 0

xis

RADIUS (cm)

4 cm

fundamental mode with plane flow and the fundamental mode with parabolic flow (with 10% slip at the walls) against tube radius.

20

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

In any event, neglecting reaction, each of the modes will be attenuated by diffusion following the relation [A']

=

1ai(r)exp[ -yi2D,z/a2u,], i

(17)

where the ai(r)are the profiles of the various modes and the y i are characteristic constants, analogous to the ,Ii. The values of the yi2 for the first five modes are listed in Table I along with the corresponding values of the Ai2 TABLE I

DIFFUSION ATTENUATION CONSTANTS

CHARACTERISTIC

Y'2 ArZ Mode (Parabolic flow) (Plane flow) ~

1 2 3 4 5

3.66 22.30 56.96 107.59 174.20

5.78 30.47 74.89 139.03 222.93

associated with Eq. (1 1). Two observations can be made concerning these values. In the first place, since y1 is much smaller than the other coefficients, the higher-order modes will be attenuated quite rapidly compared to the lowest mode. Thus, a distribution composed of a combination of modes will quickly relax to the fundamental mode as it proceeds down the tube. In the second place, the coefficients for the diffusion attenuation of a particular mode in parabolic flow are smaller than the corresponding term for plane-flow modes. This difference originates from the fact that the peak ion distribution is located in regions where the velocity is maximum. Thus, the ions are transported more rapidly and suffer less diffusion loss. This latter effect is also reflected in the reaction loss. The chief difference between this case for parabolic flow and the plane-flow case, however, lies in the fact that the eigenmodes which make up the radial profile are not the same for z > 0 as for z < 0. Thus, the radial profile may differ in the two regions. Physically this means that when the velocity profile is parabolic, even though the reactant density is uniform, the reaction can distort the ion distribution. In other words, coupling between reaction and diffusion is thereby introduced. Taking these facts and keeping in mind the continuity condition on A' at z = 0, the following description of the flowing afterglow is reached. For z < 0 (no reaction), a fundamental mode distribution compatible with a parabolic velocity profile exists for A'. For z > 0, the fundamental mode

FLOWING AFTERGLOW MEASUREMENTS

21

profile for A' is changed due to the inclusion of reactions. The greater the degree of reaction loss of A', compared to diffusion loss, the more acute will be this distortion. Since the fundamental modes in the two regions are not identical, higher-order modes will be required to maintain continuity at z = 0. The larger the reaction-loss term, the larger the amplitudes of the higher-order modes must be to maintain continuity. However, these higher-order modes are rapidly damped by diffusion relative to the fundamental mode so that the .A' profile in the reaction zone soon relaxes to a fundamental mode compatible with both the velocity profile and reaction. In order to put these observations in a proper perspective, it is useful to introduce the ratio of the rate of loss of ions per unit distance down the flow tube due to reaction to that due to diffusion:

First, consider the case where 5 4 1, i.e., diffusion dominated. In this limit, reaction is not capable of distorting the ion profile, and the numerical solution of (1 6) is

Thus, the reaction rate constant is given by (13) with a1 = 0.626. The correction factor is positive in this case because the parabolic velocity distribution effectively transports the ions down the tube at a more rapid rate. Thus, the reactant gas has less effective reaction time with the ion. Now consider the situation where the reaction attenuation becomes comparable or greater than the diffusion loss ( 5 b 1). The fundamental mode in the reaction zone is now significantly different from that upstream as is shown in Fig. 5. Here a1 has been calculated as a function of [A']/[Ao']. For a set of typical values (helium flow = 167.5 atm cm3/sec, D , = 2580 cm2/sec, a = 4 cm, and I = 42 cm), the results are shown in Fig. 6. Note that a1 increases only slightly as [A']/[Ao'] declines. This is due to the combination of two opposed effects. On the one hand, the reaction introduces higher-order diffusion modes which enhance diffusion loss. This enhanced loss would, tend to reduce al. The source of the other effect can be seen from Fig. 5. Note that as reactions are included, the fundamental mode distribution becomes even more peaked at the center where the flow velocity is high. These ions are transported more rapidly down the tube and therefore have less reaction time. This tends to increase a l . The combination of these two effects almost balance; consequently, the variation in a, is not large. In fact, for a 2 order of magnitude decline in [A']/[Ao+] the change in the rate correction, 1 + tll, is only about 1 % .

E. E.Ferguson,F. C.Fehsenfeld,and A. L. Schmeltekopf

22 I.o

0.9 0.8

>

0.7

5

z

0.6

FIG.5. Plot of relative ion density versus tube radius with and without reaction. The curve with no reaction assumes diffusive loss alone, while in the other curve the reactive loss is four times greater than the diffusive loss.

z

'

0.5

W

L

0.4

J

w

a 0.3 0.2 0.I

C

4cm

RADIUS (cm)

Ql

0.63 0.6211 I I I

0.7 0.5

I

'

0.3 0.2

1 1 1 1 I I

I

I

0.1 0.07 0.05 0.03 0.02

I 0.01

FIG.6. Plot of alversus [A+]/[A,+].

Besides this small reaction dependence, a1 also depends on pressure. As the pressure is reduced the slip flow due to molecular diffusion increases. Thus, the velocity profile more nearly resembles the plane profile and this reduces al. In practice, however, the pressures used in the ESSA tubes are large enough so that the pressure dependence of 1 + a1 is less than 1 % .

D. AXIALDIFFUSION In the absence of reaction, radial diffusion produces a gradient in the

axial direction. This gradient produces a transport of ions by diffusion. As

FLOWING AFTERGLOW MEASUREMENTS

23

reactions are added, this gradient becomes steeper so that the rate of axial diffusion increases. It is the purpose of this section to investigate the contribution of axial diffusion. In the solution of (lo), a unique diffusion coefficient DA is assumed to exist for A'. Since the flowing afterglow experiments are performed with moderate electron densities (n, 5 lo'), the diffusion of the positive ions and electrons will be ambipolar. Now, in general, there will be a mixture of positive ions present in the actual experiment. This is certainly true after a reactant gas has been added and the product ion is substituted for the reactant ion. It has been shown that in the first approximation when there is a mixture of charged particles composed of several types of positive ions and electrons, all of which are at gas temperature, the'diffusion of one type of positive ion is not influenced by the presence of the other types (Oskam, 1958). Thus, for a situation in which there are several types of positive ions and electrons, it is permissible to use a unique ambipolar diffusion coefficient for each species. The ambipolar diffusion coefficients of the ions are taken to vary as T for a constant gas density. This temperature dependence corresponds to a pointcharge, induced-dipole interaction. A detailed discussion of how the temperature dependencies originate from the nature of the interaction is given ' , by McDaniel(l964). For all ions diffusing in helium, except for He' and H the product of the ambipolar diffusion coefficient and pressure at 300°K and 1 Torr has been taken to be 900 cm2 Torr/sec. This very simple expression for the diffusion coefficient arises because of the low mass and nonreactivity of helium and argues strongly for its use as a buffering agent. In order to include axial diffusion, the transport equation becomes 0,

z
r (20) where u(r) is given by (14). As in the preceding sections, [B] is assumed to be uniform in the reaction region. A numerical solution is obtained for (20). For corresponding conditions this solution is compared with the solution of (16). By this comparison the rate correction factor u2 for axial diffusion is obtained. The values of a2 as a function of [A']/[Ao'] are shown in Fig. 7. As noted, the various curves correspond to variation in the values of the diffusion parameter y1 DA/uoa' and to differing reaction lengths 1. A typical operating condition would be one in which D A = 2580 cm2/sec, P = 0.350 Torr, helium flow = 167.3 atm cm3/sec, and a = 4 cm. This corresponds to Y1DA/a2v0 = 0.075 in Fig. 7. A different value of the diffusion parameter might be obtained by varying the pumping speed or the buffer gas flow.

24

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

[A*l/[A:l

FIG.7. Plot of a2 versus [A+]/[A,+]. Solutions were obtained for several values of the diffusion parameter, yIDA/uO d,and for reaction lengths of 40 and 85 crn.

Since an increase in either the diffusion attenuation or the reaction attenuation increases the axial ion density gradient, the axial diffusion correction factor must be positive and must increase as [A+]/[A,'] decreases. Moreover, for a fixed [A']/[Ao'], the ion density gradient is larger for a shorter reaction length; thus, a2 must be larger. The ratio a2/cll is typically 0.1 or less.

E. AXIALVELOCITY GRADIENT AND SLIPFLOW As discussed earlier in Section HI$, the buffer gas flow in the ESSA tube is predominantly viscous with a few percent slip (molecular). The viscous conductance is F,(cm3/sec) = (103na4/8q1)P, (21) where q is the gas viscosity in poise and P is the average gas pressure (in Torr) over the length 1. The molecular conductance is F,(cm3/sec) = 30.48 x 103(a3/l)(T/M)'/2, (22) where T is the gas temperature in degrees Kelvin and M is the atomic weight of the buffer gas atom (Dushman and Lafferty, 1962). The total conductance can be obtained by combining (21) and (22) in the fashion

F = F,,+ZF, (23) where Z is a pressure-dependent term varying between 0.81 and 1.00, as tabulated by Dushman and Lafferty (1962). The axial pressure gradient follows from (23) AP 0.760Qb - -Torr (24) Az cm - (na4P(z)/8q) 30.48Za3(T/M)'/2'

( )

+

FLOWING AFTERGLOW MEASUREMENTS

25

The background gas pressure Po is measured at a point in the reaction zone. The pressure at all other positions, i.e., the axial pressure profile, is then obtained from (24). This axial pressure profile and (15) lead to the average axial velocity ij(z) at any point along the tube. To combine the contributions of slip and viscous flow on the radial dependence of the axial velocity, it is assumed that these two contributions act independently; therefore, P(T,

z) = E(Z){(2F"/F)(I- r'/u2)

+ Z(F,/F)}.

In order to check (25) pitot-tube measurements have been made to determine the velocity profile. Only relative velocity profiles were measured because the pitot tubes which could be used for these measurements had Reynolds numbers less than one. In the low-pressure flowing gas of the afterglow tube, the proper dimensions required by pitot tube theory ( R > 1) were not possible. Thus, an absolute velocity determination from the measurements made with the pitot tube would require large corrections which are not known with sufficient accuracy. However, the results which were obtained, and which should be correct insofar as measuring the profile, show a parabolic profile with an indication of some slip flow at the wall, as predicted by (25). The solution of the transport equation which results from the above discussion is discussed in the next section. It is worthwhile, however, a t this point to discuss how these variations will affect the physical properties of the solution. In the solution of (20) the diffusion terms are inversely proportional to velocity. However, the diffusion coefficients which appear in the solution of this equation are inversely proportional to pressure. This means that, except for effects associated with the small changes in the velocity profile, the diffusion properties are not changed by the pressure gradient. However, the variation of gas pressure (hence average flow velocity) is reflected in the rate of loss of the reacting ions, since A[A+]/Az cc l / v o 2 . Referring to (15), this means that if the average velocity is computed from a pressure measured at the beginning of the reaction region, the average velocity obtained will be too small. Accordingly, the rate constant calculated on this basis will be too small. The opposite result would follow if the pressure were measured at the downstream end of the reaction zone. In practice to avoid this problem, the pressure measurement is made at a point centrally located in the reaction zone. Consequently the p correction represents a " reference point " correction factor to be used only when the pressure is measured at some point other than the center of the reaction region. Figure 8 shows the values for p which are required if points other than the center are used in the measurement. In these calculations the assumed buffer flow was 180 atm cm3/sec, the viscosity of the buffer gas is 1.877 x lov4P. This viscosity corresponds to that of helium at 300°K.For the curve labeled

E. E. Ferguson,F. C.Fehsenfeld, and A. L.Schmeltekopf

26

P 0.900

0

I

1

10

20

I

1

30 40

Z

I

FIG8. The /3 correction factor is plotted as a function of the location at which the background gas pressure is measured. The correction factor is plotted for two pumping speeds (i.e., two background gas pressures). Buffer gas flow = 180 atm cm3/sec; gas viscosity = 1.877 x P; curve I, P0=0.390 Torr; curve 11, Po = 1.560 Torr.

50 60

I, the pressure at the center of the reaction region was 0.390 Torr, while for I1 the pressure was 1.560 Torr. The smaller pressure is a typical operating pressure, and at this pressure there is a considerable gradient across the reaction zone. Thus, the average velocity changes by about 5 % from the midreaction zone velocity. This results in about a 10% variation in fl about the reference value of 1.00. As the pressure is increased the correction is reduced. For the higher pressure shown in Fig. 8, the variation in is less than 1 %. F. INLETEFFECTS In the actual flowing afterglow experiment the initial neutral concentration is not uniform. The neutral reactant is introduced through an inlet port, diffuses into the flow stream, and eventually achieves a uniform distribution. Since the concentration of reactant neutral varies down the tube, concentration gradients in A + are produced as a result of the reaction with B. These gradients modify the transport of ions by diffusion and consequently further couple diffusion and reaction. This process depends on both the distribution of B and its concentration. Thus, these effects vary with the amount of reactant gas introduced. In order to correct for these “inlet ” effects, the ci3 factor is computed. With the introduction of this inlet effect, the model is essentially a complete description of the flowing afterglow. The family of transport equations for this model is

where u(r,z) is given by (25), R i is the number density of each reactant (positive ion, negative ion, electron, or neutral), and D i is the appropriate diffusion

FLOWING AFTERGLOW MEASUREMENTS

coefficient for Ri . The term 6, contains all reaction " sinks '' are two-body reactions of the form

"

27

sinks" for Ri . These

or three-body reactions of either the form

where M is the density of the buffer gas, or

such that ai = ai, + ai, + ai, 3 . Since a sink for one reactant is a source for another, we require ai = 0. The diffusion of all reactant species is determined by the interaction with the buffer at operating pressure. Thus, all the transport properties in (26) are determined for all points in the tube and are used as a table of values for the point-by-point solution of (26). The solution of (26) is done on a computer and is based on an explicit finite difference method. The reader will note that the axial diffusion term D i a2Ri/az2is missing in (26). This is because axial diffusion gives rise to a second difference term in the axial coordinate which must be dropped in order to obtain a " well-posed question " for the difference calculation. In analytical solutions of partial differential equations with given initial conditions the solutions which are unbounded are neglected as unreasonable. However, in the difference approach, these solutions cannot be selectively suppressed. Thus, the terms which give rise to them must be modified or dropped. Since in Section III,D it was shown that the correction factor for axial diffusion was small, it is expected that the magnitude of this type of correction will not significantly change in this case. Therefore, the axial diffusion correction a2 will simply be added on here as a perturbation correction. In the solution, a net size in the z direction is determined by the ,,,,Ju(r, z),,,~"and where Ar is stability criterion A z cc ( A r 2 / 2 0 , where o = Di, the net size in the radial direction. The term Di, is the maximum diffusion coefficient for any reactant; u(r, z),,,~" is the minimum value assigned to the velocity. In order to reduce computation time, one chooses u(r, z),~" as large as possible without exceeding a tolerable error. It was found that u(r, z),,,~" could be as large as 10% of uo before an error of 1 % was reached. The net size in the r direction was obtained by successively decreasing the net size until the maximum total error between the difference calculation and the exact solution for a simple model where an exact solution is possible, was less than

28

E. E. Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

2 %. It was found in practice that in the first 2 cm following the introduction

of the reactant gas a net size of 1/90th the tube radius was required. Following this region, a net size of 1/18th the radius of the tube was found to be adequate. The following list of assumptions have been made in these calculations. 1. The concentration of ions, electrons, and metastables are zero at the tube wall, while neutral reactant concentrations have a vanishing normal derivative at the wall. 2. All particles are isothermal at the wall temperature. 3. A unique diffusion coefficient exists for each reactant. 4. The neutral diffusion coefficients in helium vary as T112at constant gas density. [This corresponds to a hard-sphere scattering model (McDaniel, 1964).] The product of the diffusion coefficient and pressure for all neutral reactants (except H2 and atomic hydrogen) is taken to be 570 cm2 Torr/sec at 300°K and 1 Torr.

Most experimental conditions lead to a simplified form of (26) in which only one type of neutral reactant R, is present and is made to greatly exceed the density of all other reactants in the reaction zone. Thus, 6, = 0; i.e., the loss of reactant molecules in reaction with the low concentration reactants (ions, metastable atoms, etc.) can be neglected. Moreover, by virtue of their small concentrations, the binary reaction losses between any combination of the other reactants usually can be neglected so that ki, Ri R j = 0, i, j # 1. However, more complex systems than this have been analyzed, for example, where dissociative recombination is important or where metastable atoms react with metastable atoms. These calculations, however, will not be described since the present discussion is intended to point out some of the more important conclusions concerning the dependence of the computed rate constant on the assumed transport properties of the flowing afterglow. Only a simple reaction of the form A + B + C + + D will be discussed. To initiate the calculation, the fixed parameters of the system ( I , a, QHe, P o , D A , D , , Qk), are supplied along with the initial distribution of the reactant ion and the initial distribution of the reactant neutral source. In practice, it is impossible to know these initial experimental distributions exactly. For this reason, two combinations of initial ion and neutral distributions are considered. These represent extreme cases. The actual experimental distributions should be somewhere between these extremes. In the first case, the reactant neutral issues from a point source and the initial ion distribution is in the fundamental mode for the parabolic flow. This case represents the extreme situation in which the initial distribution of reactant neutral produces the maximum attenuation of the axial ion distribution and the reactant inlet tube produces no perturbation in the initial ion concentration. The initial

+

29

FLOWING AFTERGLOW MEASUREMENTS

ion distribution associated with this case is plotted in Fig. 9 and is labeled “ fundamental mode distribution in flowing system.” In the second case, the reactant neutral issues from an axial source 1 cm in diameter while the initial Inner Boundory for

FIG.9. Initial ion distributions used in the explicit finite difference solution of (26). Relative ion density is plotted against tube radius. The curve labeled fundamental mode distributionin the flowing system pertains to the fundamental mode assuming a characteristic velocity profile while the curve labeled 1-8 coaxial ion distribution pertains to a fundamental coaxial mode between coaxial cylinders of 1- and 8-cm diameter.

3

0.7

0.6

5’ P

0.5 Fundanentot Mode

0.4

RADIUS (cm)

ion distribution is a fundamental coaxial mode between the inner wall of the flow tube and the outer wall of the reactant inlet. This initial ion distribution is also plotted in Fig. 9 and is labeled “1-8 coaxial ion distribution.” A l-cmdiameter inlet was chosen because this represents the maximum inlet tube size which is used in the ESSA experiments. This second arrangement thus represents the other limit where the reactant inlet tube produces the maximum possible perturbation while the reactant source represents the maximum initial dispersal of reactant neutrals. The distortion of the reactant ion distribution by the point source injection of the reactant neutral, and the subsequent relaxation of this perturbation, is shown in Fig. 10. The rate of addition of reactant neutrals was taken to be 7.56 x 1019 molecules/sec, and the reaction rate constant 5 x lo-’’ cm3/sec. In these calculations uo = 8 x lo3 cm/sec, D,P = 900 cm2 Torr/sec, D , P = 570 cm2 Torr/sec, and Po = 0.390 Torr. Upon introduction of the reactant gas from a point source, the ion distribution in the center of the tube is drastically attenuated as shown in the profile plot which corresponds to 0.1 cm. The subsequent curves indicate, .however, that as the reactant neutral is dispersed the ion profile fills in until at 10 cm there is no longer a depression in the center of the ion distribution. At 10 cm, diffusion averaging has almost

E. E. Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

30 IO ,W

Init iol Distribution

cm

Axis

FIG.10. Diffusion redistribution of the ions following the point source introduction of the reactant neutral. The curves were obtained from a solution of Eq. (26). In these calculations the characteristic velocity profile for the flowing system was used and the initial ion distribution was assumed to be fundamental mode compatible with this velocity profile. For comparison the ion profile at 10 cm in the presence of diffusion attenuation alone is also plotted.

4 cm

RADIUS (crn)

rectified the enhanced reaction depletion of the axial ions. The profile corresponding to the ion distribution at 10 cm with noisu'id alone is included for comparison. The persistence of the perturbation with distance down the tube would be increased by lowering the ion or neutral diffusion coefficient or increasing the flow velocity. The resulting t13 correction for the nonperturbing point source of reactant gas is plotted in Fig. 11. It was assumed that the flow profile was parabolic andthat the initial ion distribution was fundamental mode with uo = 8 x lo3 cm/sec, DAP = 900 cm2 Torr/sec, D, P = 570 cmz Torr/sec, and Po = 0.390 Torr. This set of calculations was done for reaction lengths of 10, 30, and 60 cm. In each case t13 is negative. This is due to the fact that the reactant gas injected in this manner is more effective at quenching the axial ion distribution than is the uniformly distributed reactant. The physical reasons for this increased efficiency of a point source are twofold. In the first place, the point source produces larger than average concentrations of reactant gas in regions of larger than average ion concentration. Second, the reactions distort the fundamental mode distribution of the reactant ion and thereby increase diffusion losses. A second feature of the curves in Fig. 11 is that the absolute value of the correction coefficient associated with the point source decreases with a decreasing ratio of [A']/[Ao'] and approaches zero in the limit. This

31

FLOWING AFTERGLOW MEASUREMENTS

-

00'111

1

1.0 0.7 0.5

1

I

0.3

1 1 1 1

I

I

I

I

0.1 0.070.05 0.03

[A+]

I

QOI

/ [Ad

FIG.1 I . The aj correction factor for point source reactant gas addition plotted against [A+]/[A,+] for three reaction lengths. The characteristic velocity profile of the flowing gas was assumed along with an initial ion distribution compatible with this velocity profile.

decrease is due to the fact that the initial addition of reactant produces a large attenuation of the axial ion distribution. Additional reactant gas is consequently less effective in attenuating the ion signal. Finally, it is to be noted that a3 increases as the reaction length is decreased. This arises because diffusion is less able to smooth the initial perturbations prior to ion sampling for the shorter lengths. The calculations of the last paragraph were repeated for the other inlet condition limit, namely the 1-8 coaxial reactant ion distribution. The experimental conditions were otherwise unchanged. The correction factor computed for this case is plotted for'reaction lengths of 20 and 60 cm, and the results are shown in Fig. 12. A comparison of Figs. 11 and 12 indicates that

-0 -0.4

-0.3b

-O.*r

-o.L--L--J [!GI 01.0 0.7 0.5

0.1 0.07 0.05 0.03

0.3

[A+]

0.01

/

FIG.12. The a3 correction factor associated with a 1-cm-diameter reactant gas addition port plotted against [A+]/[A,+] for two reaction lengths. The characteristic velocity profile of the flowing gas was assumed along with an initial ion distribution which is the fundamental coaxial mode between the reactant gas inlet and the main flow tube.

32

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

for the shoyter reaction lengths, the clj correction for the nonperturbing point source is larger than for the extended coaxial inlet and is about the same for the longer lengths. This is what one would expect. Since the presence of the coaxial inlet has quenched the axial ion distribution, the reactant neutral is initially added into regions of low ion density. Thus, a larger than average neutral distribution occurs in a region in which the ion distribution is initially smaller than average. Consequently, for the same addition of reactant gas, the attenuation of ions reaching the sampling port of the mass spectrometer is relatively less for the extended coaxial inlet than for the nonperturbing point source. For the longer reaction lengths, this effect has damped out.

G. CONCLUSIONS CONCERNING THE ANALYSIS The reader should be cautioned that the correction factors of the preceding parts are not universal correction factors, but apply to the specific conditions of P o , a, I, D, , D , , and v,, which were assumed. No general expressions have been derived which could be applied to every system. To show the reader the net effect of these calculations Fig. 13 is presented

FIG.13. Relative decrease in the reactant ion signal as a function of Qkl for four cases. The realistic model contains all the corrections of Section 111. For these two examples a fundamental mode initial ion distribution compatible with the characteristic velocity profile and a point source introduction of reactant neutral were assumed.

and shows the relative decrease in the reactant ion signal as a function of Qkl for four cases. In each of these curves uo = 8 x lo3 cmlsec, Po = 390 Torr, a = 4 cm. The simple model given by (8) is linear in this semilog plot. For the simple model it is not necessary to specify the reaction length since in the absence of coupling between reaction and diffusion the curves for all reaction lengths are identical The curve for the parabolic velocity profile

FLOWING AFTERGLOW MEASUREMENTS

33

model computed from (16) appears to be linear in this plot; nevertheless, there is a slight curvature due to the coupling of reaction and diffusion, as discussed in Section 111,C. This curve was calculated assuming D,P = 900 cm2 Tom/ sec and 1 = 40 cm. Because of the coupling of reaction and diffusion, the curves for other reaction lengths would be different. However, the difference would be extremely slight. Finally two curves, labeled the realistic models, were calculated for reaction lengths of 20 and 40 cm and included all of the effects mentioned in Sections II1,B-E. These curves were calculated for the point source inlet condition of Section III,F and for D,P = 570 cm2 Torr/ sec. Neither of these two curves is linear, although the curvature for the 20-cm reaction length is much greater. The shorter reaction zone lengths place a much more severe requirement on the accurate specification of both ion and neutral initial conditions since for such short lengths diffusion averaging has not taken place. In addition, for shorter reaction lengths the axial diffusion correction becomes so large that the approach used here to solve the transport equation is no longer valid. Because of these difficulties, the ESSA reaction rate measurements have never been made with I less than 40 cm. The reaction rate constant for the simple model is interpreted from the slope of the line. Due to the curvature obtained for the realistic model, no unique slope exists. Consequently, there is no straightforward method of obtaining a rate constant, as in the simple model. For large values of Q k and in the absence of axial diffusion the curves for the realistic model will parallel the curve for the model of Section III,C (parabolic velocity profile). As a result of the calculations it may be concluded that the two curves for the simple model and the parabolic velocity profile model represent the limits between which all practical cases lie, i.e., 15 35 cm and R 5 5 (a x 4 cm). Finally, a point which should be reemphasized is the fact that the initial reactant ion distribution at the reactant neutral inlet port is not precisely known and hence cannot be incorporated into a model. However, the calculations in Section III,F show that, for reasonable reaction zone lengths (1 > 40 cm), the error thereby introduced is small (a few percent, at most). H. TESTOF THE ASSUMPTIONS To test the correctness of the model for the hydrodynamics and diffusion, the following analysis was made and was compared to an appropriate experiment. The analysis was used to calculate the behavior of a diffusing ion sheet in the afterglow tube. A thin sheet of ions was generated at z = 0 and t = 0 by a pulsed beam of electrons incident perpendicular to the axis of the tube. The pulse width was characteristically 100 psec. For ions other than those of helium a suitable source gas was added with the flowing buffer. Thus, the

34

E. E.Ferguson, F. C.Fehsenfeld, and A. L.Schmeltekopf

diffusive properties of various ions, negative as well as positive, were tested in helium. The diffusing ion sheet was carried downstream by the flowing gas. The sampling port of a mass spectrometer was located on the axis downstream. The detector was gated with a width of 100 psec; thus, an ion arrivaltime spectrum could be recorded. In the analysis (Jarvis, 1968) it is assumed that the ions obey the transport equation XA'1 at

XA'1 + u(r) aZ =

DA

V2[A']

- k[A+][B],

and it is assumed that the distribution of B is uniform. Equation (27) is solved using integral transforms. Detailed calculations have been made only for uniform radial initial ion distributions, although other initial distributions can be evaluated as easily. The reader will note that (27) does not include the axial pressure gradient. However, the p correction calculated in Section II1,E can be directly applied. Figure 14 shows a comparison of an experimental arrival-time spectrum

FIG. 14. Plot of an experimental arrival-time spectrum (solid points) and a calculated spectrum (open points) for a flow length of 113 cm. The normalized He+ intensity is plotted against arrival time.

ARRIVAL TIME (m sec)

(solid points) and the calculated spectrum (open points) for the values DA = 2400 cm*/sec, uo = 8 x lo3 cm/sec, and flow distance of 113 cm. The agreement shown was typical for all gases and flow conditions tested. Thus, it is felt that diffusion and hydrodynamics have been handled correctly.

FLOWING AFTERGLOW MEASUREMENTS

35

IV. Data Reduction

This section expresses the mechanics with which a reaction rate constant is determined from raw experimental data using the analysis of Section 111. An on-line computer facilitates this reduction, as well as quick and accurate data acquisition. In a typical reaction rate study the fixed parameters are entered: the viscosity of the buffer gas, the viscosity and flow range of the reactant ion parent gas, the viscosity and flow range of the reactant gas, the calibration factors of the various flow elements, the downstream pressure, the length of the reaction region, the diameter of the flow tube, and the system temperature. During a typical run, the flow rate of the neutral reactant is varied and the count rates of reactant, product, and impurity ions are recorded. In addition to these variables, the flow rates of the reactant ion parent gas and the buffer gas, although nominally fixed during the typical run, are recorded because of the sensitivity of the results to their variations. Additional digital channels are available for recording such parameters as optical absorption, optical emission, and special temperatures and pressures, which are needed in certain measurements. The outputs of all digital channels (see Fig. 3) are coded and sent into a teletypewriter, which makes both a typed copy of the data and a punched paper tape. Upon completion of a run, the latter is used to feed the data into the computer for reduction. The computer automatically converts the raw input data into the appropriate physical parameters. The rate of addition of reactant and source gas, and the flow of the buffer gas are computed in standard units. The gas temperature and pressure, together with the computed buffer gas flow, are used to obtain the average flow velocity. Moreover, certain corrections to the detected ion signal can also be made. These corrections include removal of background counts and compensation for drift and changes in the diffusion losses of ions. The background count is not associated with the ion signal. The subtraction of these extraneous pulses (generally from 0.1 to 1 count/sec for positive ions and from 1 to 10 counts/ sec for negative ions) is only important for small ion signals. To make the correction for drift, the reactant ion current that exists before the introduction of the reactant gas is used as a reference, and during the course of a run the reactant gas will be occasionally removed to check for drift. If changes of 5 % have occurred, the run is aborted and the source of the drift is found and eliminated. For small changes, a linear interpolation of the signal is made. A change in the diffusion loss may occur in certain slow two-body reactions (k 2 cm3/sec) or in three-body reactions because relatively large quantities of reactant gas are required to produce a significant change in the reactant ion signal. Under these circumstances, the assumption that the

36

E. E. Ferguson, F. C, Fehsenfeld, and A . L. Schmeltekopf

reactant ions are diffusing only in helium may not be valid. This produces data from which the deduced reaction rate constant is too small. In certain cases, it is possible to circumvent this problem by monitoring the behavior of an ion which is nonreactive and whose intensity then reflects the perturbation of the ion transport properties by the added gas. The computer now has the reactant ion signal (corrected as described above, if necessary) as a function of the rate of addition of the reactant neutral (particles per second). Also available to the computer is a curve like that in Fig. 13 (realistic model), which predicts the decline in the reactant ion signal as a function of Qk (particles cm3/sec2). There are several ways to obtain the reaction rate constant from these two curves. The simplest and most frequently used is the following. First, a specific decline in the reactant ion signal is chosen (in practice, a decade) and the measured Q associated with this decline is noted. Then, for the same decline, the value of Qk is obtained from the calculated curve. The reaction rate k follows. The dependence of k on the extent of the specified decline can, of course, be tested easily. For an ideal case, this simple method fully utilizes all of the analysis of Section 111. However, experimental data is often less than ideal, e.g., fluctuations in reactant flow (particularly for unstable neutrals) and reactant ion signal (especially for low signal levels). Consequently, it is often difficult to determine accurately the Q at which the specific decline of the ion signal occurs. Thus, it is useful to have a more general method for determining the reaction rate constant. Perhaps the best such method is to let the computer select the value of k which gives the best fit of the analytical curve (Section 111) to the experimental data. In addition to the possibility of a more accurate value of k , this method allows easy comparison of the curvatures of the experimental data and analytical curve, thereby providing a check for possible downstream reactant ion sources, etc. V. Production of Reactant Species A. REACTANT ION PRODUCTION 1. Electron Impact The most straightforward method of ion production is direct ionization of atoms or molecules by electron impact or discharge source. This has the advantage that the ions are produced in a localized region well removed from the reaction zone. For some purposes, this has the relative disadvantage that large flows of reactant ion parent gas (- 10 atm cm3/sec) are required because

FLOWING AFTERGLOW MEASUREMENTS

37

of relatively low values of electron impact ionization cross sections. For example, 0' and 0,' can be produced by electron impact on O,, or 0' can be produced by electron impact on CO or other gases, if 0,' would be a confusing ion to have present during 0' reaction studies. In a similar manner, many negative ions can be produced by dissociative attachment, e.g., NH3+e+NHz- +H, (28) H2 + e+ . H- +H,

(29)

or dissociative ionization, COz + e - + O - +CO+.

(30) Although electron impact is the most straightforward method of ion production, the following techniques are most commonly used. 2. Secondary Reactions A substantial gain in efficiency can be achieved when reactions between the reactant ion parent gas and primary excited or ionized species of the buffer gas can be used to produce the ions of interest. Examples are, He(2w

+ Nz

+.

+ + He

N2+ e

He++CO+C+ +O+He Ar++02+02++Ar.

Penning reactions, such as (31), can be used to produce the parent ions of most gases. On the other hand, helium ions dissociatively ionize most small molecules as in (32): N + from N, and NO, 0' from 0, , C' from CO, etc. Argon is sometimes used as a buffer gas and (33) produces 0,' without O', when that is desirable. With fast reactions [such as (31)-(33) where k 2 lo-'' cm3/sec], only small quantities of the reactant ion parent gas are required, e.g., 0.1 to 1 atm cm3/sec. It is essential that sufficient quantities be added, however, to assure that reactant ion production is completed before the gas flow enters the reaction zone. It is usually possible to ascertain experimentally that this condition is satisfied. When ions are produced in the reaction zone, they do not of course have the full reaction time which is assumed in the data analysis, as discussed in Section 11. Negative reactant ions can be produced in several ways. For example, the most convenient method for producing NO- has involved the addition of N,O through the discharge with the helium buffer gas. The reaction sequence is

+

+

N 2 0 e +. 0- N2

(34)

0-+NzO+NO- +NO.

(35)

followed by

38

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

Alternatively, NO- can be formed by direct electron attachment, NO + e

+ M -+NO-+ M,

(36 ) as can 02-and most negative ions of stable neutrals. Negative ions can also be formed by charge transfer, e.g. and When substantial uncertainties exist, such as the location of ion formation or its vibrational state, it is useful to produce the ion several ways and compare rate constants so obtained. 3. Metal Zon Production

The flowing afterglow system has proven to be useful for reaction studies involving metal ions. The ions Fe', Mg", Ca', K",and Na" have all been produced and their two-body reactions with O3 and three-body reactions with O2 studied. These are the first rate constant measurements of metal ion reactions at thermal energies. The metal ions are produced by vaporizing the corresponding metals into the buffer gas stream from a small, electrically heated, furnace placed in the afterglow tube (Ferguson and Fehsenfeld, 1968). The buffer gas was argon (rather than the usual helium). The argon was bombarded by 14 V electrons producing argon metastable atoms. These, in turn, were used to Penning ionize the metals, i.e., N

ArM+M+Ar+M++e.

(39)

In order to produce MgO" for reaction studies, the reaction

+ Mg

+0 +

MgO+

(4W +Mg+ 0 3 (40b) was utilized. About one-fifth of (40) went into channel a. In a like manner, Si' and SiO' have been made. 02+

-+

4. Reactant Zon States

It was shown quite early in the flowing afterglow studies that certain atomic ions, e.g., O', could safely be assumed to be in their ground electronic states. This is because the time for superelastic electron collision deexcitation (i.e., O"*+ e + 0"+ e + KE) can be made short compared to the flow time between ion formation and ion reaction (Fehsenfeld et al., 1965b). While theoretical deexcitation cross sections are available only in special cases (such

FLOWING AFTERGLOW MEASUREMENTS

39

as O+*), the magnitudes of these cross sections are not expected to ever be so low that this experimental situation cannot be achieved. With the electron impact ion source, this point can be checked experimentally by varying the emission current (and thereby the electron density) and ascertaining that measured rate constants are independent of electron density. This has been ' + N, + NO+ + N. done for certain critical reactions, for example 0 In the case of molecular ions, the question of vibrational state distribution arises, since collisions cannot be depended upon to be sufficiently effective to cause rapid vibrational relaxation. In the case of N,', it has been possible to determine that the reactant ion is in its ground electronic and ground vibrational states. This is due to the fact that in the flow tube N,' ions are produced principally in a radiating state by the Penning process, He(2 3S)+ N, + N2+B'Xu+ + e + He(1 '9. The radiation B 'Xu+ +X 2Xg8+ (first negative system) has been observed spectroscopically and analysis shows that the resulting ions largely radiate into the ground vibrational and electronic states before reaching the reaction zone. (Spectroscopic studies will be discussed further in Section VI.) This situation is of course not general and ordinarily one cannot determine the vibrational distribution spectroscopically. (N,' has been studied more extensively in the ESSA system because of its importance in the ion chemistry of the Earth's atmosphere.) The FranckCondon factors for N, are such that electron impact ionization would also yield ground state N,'. Clearly, it is possible to produce ground state molecular ions alone by electron impact by keeping the ionizing electron energy sufficientlylow. In practice, however, this may run into severe intensity problems. In the case of molecular negative ions, the question of reactant ion states is more severe. Most molecular negative ions do not have metastable electronically excited states, so that the electronic state will not often be a question; however, the vibrational state distribution has so far been unknown. For none of the negative ion production processes is the vibrational distribution of the resulting ion well known. Even if it were, unless the ion is formed in the ground state, the unknown extent of vibrational deexcitation between the time of formation and reaction would be a complication. There are several possible procedures for studying effects of the vibrational state of the ion on reaction rate constants. For example, in some cases reactant ions can be produced several different ways. This has been done for the reaction 0,- + 0 + 0, e, which was found to be independent of the manner of 0,- formation. Such a finding strongly implies either that the ions are largely ground state in both cases, or that the reaction rate is relatively insensitive to the vibrational state of the negative ion. In either case, one has a reasonably confident knowledge of the ground state ion reaction rate constant.

+

40

E. E. Ferguson, F. C. Fehsenfeeld, and A . L. Schmeltekopf

Alternatively (or additionally) one could vary the flow tube length or gas pressure in such a way as to increase the negative ion lifetime between formation and reaction. This would cause more vibrational relaxation, unless the vibrational relaxation efficiency were unexpectedly low for the negative molecular ion (i.e., -g for reasonably attainable circumstances). The corresponding variation (or lack of variation) in a particular rate constant would then suggest dependence (or independence) on mode of ion formation. B. NEUTRAL REACTANT PRODUCTION

I . Neutral Reactant States As mentioned earlier, the neutral reactant is added downstream in the flowing afterglow system; therefore, it is not subject to excitation or dissociation. As a consequence, the neutral reactant is known to be in a Boltzmann distribution of electronic, vibrational, and rotational states at the tube wall temperature. This situation is not generally true of other experimental methods. The flowing afterglow system is not plagued by the confusion of concurrent reactions, such as

which occur in mass spectrometer ion source reaction rate studies and are difficult to unravel. By the same token, charge-transfer reactions are easily studied, e.g., co2++ 0 2 + 0 2 + + coz. (43) In conventional methods, such as mass spectrometer ion sources or stationary afterglows in which both the 0, and CO, are subject to the same ionizing conditions, the lower ionization potential of 0, would cause it to be selectively ionized. Therefore, most of the observed 0,' would be due to primary ionization rather than the charge-transfer reaction (43). For this reason very few thermal energy charge-transfer rate constants had been determined by conventional methods prior to the flowing afterglow development. Also, it has been possible to measure several ion reactions with excited state neutral reactants. For example, rate constants for the ions He' and 0' have been measured in reaction with N,, where the N, vibrational temperature is varied from 300 to 6000°K (Schmeltekopf el a/., 1968). The neutral N, is vibrationally excited up to 6000°K by a microwave discharge in a quartz side tube through which the N, is added. The vibrational temperature is controlled by varying the distance between the microwave cavity and the exit end of the tube. The resulting vibrational temperature is measured

41

FLOWING AFTERGLOW MEASUREMENTS

spectroscopically as described below in Section VI and in detail in the paper of Schmeltekopf et al. (1968). (The 0 ' + N,(T,) + NO' + N result is shown in Fig. 15.) Other ions could similarly be studied in reaction with N, and

1

1 1000

I

I

I

2000

3000 Tv

I 4000

I

I SO00

I

I 6000

(OK)

FIG. 15. Rate constant for the reaction O+ nitrogen vibrational temperature.

+ Nz

-+ NO+

+ N as a function of the

E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schmeltekopf

42

it is expected that similar measurements could be made on other vibrationally excited molecules. Recently, negative ion reactions with electronically excited 0, molecules have been carried out. Specifically the reactions and have been measured (Fehsenfeld et al., 1969). A microwave discharge in oxygen is known to produce up to 20% 02('A8), more or less independently of small variations in certain impurities. A discharge also produces a somewhat lesser concentration of atomic oxygen, which must be removed for the purposes of studying reactions (44) and (49, since 0,- 0 + O3 + e and 0- 0 40, e have large rate constants (Fehsenfeld et al., 1966). This is accomplished by having a glass wool plug coated with mercury oxide in the oxygen tube after the microwave discharge. It is known that the mercury oxide surface destroys atomic oxygen without destroying 02('Ag) molecules (Elias et al., 1959). The O,('AJ concentrations are determined by measuring the intensity of the O,('A$ -+ 02(3q,-) transition at 1.27 p with a PbS detector. This technique is limited to ions which do not react rapidly with ground state 0,, since the concentration of ground state 0,greatly exceeds that of 02('A8). Reactions (44) and (45) are endothermic for ground state 0, in place of 02('Ag).

+

+

+

2. Chemically Unstable Neutrals Another of the marked advantages of the flowing afterglow system is the thus-far unique capability for measuring rate constants for reactions of ions with chemically unstable species at thermal energy. Atoms such as N, 0, and H, and radicals such as OH, and molecules like O3and NO,, which pose some problems in measurement and handling, have been studied. Such measurements have been especially useful for the understanding of certain geophysical and astrophysical situations where such reactions occur in nature. Nitrogen atoms are produced by the discharge of N, and are measured by the titration reaction N + N O + N, + 0 so that the quantitative measurement of NO flow yields the quantitative N flow. The reactions with N atoms so produced are necessarily carried out in the presence of a large excess of N, ([N,] 102[N]) so that only ions which do not effectively react with N, can be studied. An example is the reaction 0,' N + NO' + 0. It is easy to demonstrate (simply by not discharging and dissociating the N,) that the NO does not occur at an observexothermic reaction 02'+ N, 4NO' able rate. N

+

+

43

FLOWING AFTERGLOW MEASUREMENTS

The same NO titration scheme can be used to generate a known concentration of 0 atoms, i.e., titration of the N + NO + N, + 0 reaction to the end point where the 0 atom production then equals the measured NO addition. Again the reaction is carried out in an excess of N, , so that only reactions of ions not reacting with N, can be measured. An important example is N2++ 0 + NO' + N, the main N,+ loss process in the earth's ionosphere. The symmetric charge transfer N2' N, + N, Nb+ has no observable effect in the afterglow system. The H atom production is by a microwave discharge of H, and measurement is by the titration reaction H + NO, + OH NO. The reaction H - + H 4 H, + e has been measured in this way (Schmeltekopf et ul., 1967). This reaction is of interest since it is the controlling loss process for H- ions in the solar photosphere; H- controls the sun's opacity in a wide region of the visible and ultraviolet spectrum (Branscomb, 1968). This reaction also has the only rigorously calculable rate constant of any ion-molecule reaction, and it is gratifying to note that experiment and theory are in agreement here. In this method excess H2 is present so that only ions which do not readily react with H2 can be studied. This eliminates such interesting reactions as 0 ' + H + H + + 0 from consideration, since 0 ' H, +OH+ H has a rate constant of 2 x lo-' cm3/sec at 300°K (Fehsenfeld et al., 1967). The H + NO2 + OH NO titration has been used to produce OH as a neutral reactant for certain ion reactions (Ferguson et ul., 1969a). Ozone plays a key role in the earth's negative ion chemistry and metal ion (meteor ion) chemistry and both metal and negative ions have been reacted with ozone in the ESSA flowing system afterglow. Ozone is produced in a commercial ozonizer which discharges oxygen to produce up to 5 % ozone in the oxygen stream. In the metal ion studies molecular oxygen does not interfere, since the metal ions studied do not react exothermically with oxygen in binary reactions. The same procedure can be applied for many negative ion reactions as well, since molecular oxygen does not often interfere. For example, a very important process for controlling the earth's negative ion chemistry is

+

+

+

+

+

+

-

0 2 - +03+0,-

+ 0 2

(46)

which has been measured in the ESSA flow tube and for which 0,would not interfere. It is generally true in problems of nature that for ions which can , the react with 0,,there will be little interest in their reactions with 0 3 since O3 to 0,ratio in nature is ordinarily small. The ozone-oxygen mixture was fed from the ozonizer, through an optical absorption cell, and thence into the reaction zone of the flow tube. The ozone flow was measured by determining the oxygen flow rate, cell pressure, and ozone concentration, the latter by absorption of the 2537 A Hg line in a double-beam absorption device.

44

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

The addition of O2 can be avoided when it is necessary or desirable to study 0, reactions without 0, present, and our earliest ozone studies were carried out this way, The ozonizer effluent was carried through a U-shaped tube filled with silica gel which was maintained at dry ice and acetone temperature, The ozone was trapped on the silica gel where it was readily visible due to its bright blue color. Oxygen is not appreciably trapped on the silica gel and it was pumped off. When the ozone was to be used for reaction studies it was eluded off of the silica gel by an inert carrier gas; in our case helium was used. The direct use of the oxygen-ozone mixture from the ozonizer is not only a great deal more convenient and time saving, but also somewhat safer than storing of ozone in the laboratory. As is well known, ozone is quite corrosive to some substances. We found it necessary to replace rubber 0 rings (or their fragmentary remains) in our pump lines with synthetic Viton 0 rings, for this reason.

c. ROLEOF IMPURITIES The possible role of impurities in confusing or falsifying flowing afterglow data merits discussion here. It is perhaps surprising to many members of the current generation of experimentalists in this field, schooled in ultrahigh vacuum technology, that impurity problems associated with such large gas flows are not completely ruinous. Why this is indeed not the case is discussed below. There are two ways that impurities can be deleterious to a rate constant measurement in the flowing afterglow technique. The first concerns reactant ion production, while the second directly enters the rate constant measurement. With regard to reactant ion production, the relevant impurities are those which are introduced with the buffer gas and the reactant ion parent gas and those released from the walls of the flow tube. It is apparent that these impurities can produce a problem if their concentration is sufficient to seriously deplete the reactant ions before addition of the neutral reactant gas. As an example, with the helium buffer sufficiently contaminated by nitrogen, the helium ions would be destroyed before reaching the mass spectrometer sampling port. A reactant ion source is not considered satisfactory unless the reactant ion is the dominant ion in the afterglow. This is generally not a problem because sufficiently pure buffer gases and reactant ion parent gases can be obtained in the required bulk quantities. Furthermore, the impurities from the polished, stainless steel tube walls are relatively quickly purged to tolerable levels by the flowing gases. Another problem which may arise from these impurities is the production of an impurity ion with the same mass as the reactant ion. Fortunately, this has not proven to be a problem so far.

FLOWING AFTERGLOW MEASUREMENTS

45

By far the most serious impurity problem is in the case where impurities are added with the neutral reactant gas. Such impurities, if present in a constant mixing ratio in the neutral reactant gas, could lead to a systematic decrease in the reactant ion signal which would mimic a reaction with the added neutral reactant. Thus, this impurity source has the potential of leading to error in the measured rate constant. The difficulties posed by these impurities are highly dependent on the reaction involved. For fast reactions, the problem of purity is not serious. For reactions which occur on nearly every collision (as many ion reactions do) a 1 % impurity in the neutral reactant could at most lead to a 1 % error in rate constant. For extremely slow reactions, the impurity problem can however be quite serious. The ability to measure an extremely slow reaction depends, obviously, on the limit of purification of the neutral reactant gas and this property of gases varies widely. It also depends on the nature of the impurity. If a trace impurity reacts with the ion but produces a clearly distinguishable product ion, as would usually be the case, then the interference may possibly be allowed for. As an illustration of the measurement of small rate constants, it has been possible to establish that several exothermic reactions have rate constants less than IO-" cm3/sec, e.g., Oz+ + N z + N O + + N O

(47)

and 0s-

+ Nz +NOz- + NO.

(48) This is a demonstrated dynamic range of lo6 for measured rate constants in the flowing afterglow system, which exceeds that of any other experimental technique so far for rate constant measurements. However, the ability to cover this wide range is not general. A positive result on (47) could have been accepted only if the concentration of NO in the N, was less than 1 ppm because NO reacts rapidly with 0,' to produce NO'. Aside from the specific disadvantages noted above, impurities may be simply a nuisance or, on the other hand, may be quite helpful. An example of the former is the general contamination of commercial sodium with potassium so that the product ion NaO' of the reaction

(49) is masked by the K + impurity of the same mass. An advantage of a nonreactive impurity has been mentioned earlier. An example was C1- in negative ion reaction studies. A final point to make in connection with the role of impurities in the flowing afterglow system is that a mass scan of the ions is always carried out before and after neutral reactant addition as a part of the experiment. Thus, difficulties associated with impurities are not likely to go undetected and lead Na+ + 0 3 + N a O + +02

46

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

to error. We know of no case in which an impurity has caused an erroneous rate constant to be deduced. There have been a few instances where the nature of the product ion was obscured, but this of course can also happen by virtue of secondary reactions in completely pure gases. VI. Optical Spectroscopic Studies The flowing afterglow system is rather well suited to spectroscopic investigations since an ion-molecule reaction can be caused to take place in a fairly large volume, fixed in time, in front of a spectrometer slit. Spectroscopic studies are necessarily much more complicated than ion-molecule reaction rate measurements but of course yield much more detailed information about reaction processes, in favorable circumstances. As mentioned earlier, one practical utilization of spectroscopy has been the determination of the state of the reactant ion. An example is the production of N,+ by He (2 3S)+ N, + N 2 + B+ e + He, followed by N2+B-+ N 2 + X hv (first negative system). Absolute intensity measurements of the first negative system show that most of the N2+ ions are produced in the B state. Study of the vibrational distribution shows that most of the product N2+X ground state ions are in the v = 0 vibrational level (Schmeltekopf et al., 1968). Hence N,+ reactions, using the N2+ so produced, can be safely assumed to be largely reactions of N,+X 'Zg+, u = 0. Nitrogen can be vibrationally excited by a microwave discharge and its temperature measured spectroscopically. The vibrationally excited N, can be ionized into the N2+B state by either high energy electrons or He(23S), both of which give vertical transitions. Observations of the relative intensities of the bands of the subsequent N,+B -+ N2+Xtransition, the transition probabilities of these bands, and the Franck-Condon factors connecting N,X to N,+B yield the vibrational temperature of the N, (Schmeltekopf et al., 1968). One particularly interesting spectroscopic result obtained so far (Albritton et al., 1968) involved simultaneous study of the second negative system (N,+C2C,+ -+ N,+X2Cg+)and the N+/N2+ ratio produced in the reaction He+ + N, --t N,+C 'Xu+ He, followed by either N 2 + C+ N,+X + hv or N,+C --t N + N + . By making this study as a function of N, vibrational temperature, it was possible to determine in which vibrational level of the C state predissociation occurs and to determine the ratio of predissociative to radiative lifetimes for this vibrational level.

+

+

VII. Temperaturevariable Flowing Afterglow Studies Recently a flowing afterglow system has been constructed in which the gas temperature is variable from 82 to 600°K.This allows the determination of

FLOWING AFTERGLOW MEASUREMENTS

47

reaction rate constants as a function of temperature (Dunkin et al., 1968). This information is of interest in ionospheric physics and also adds additional dataupon which thermal energy reaction rate theory can build. Also three-body reactions, whose rate constants are greatly enhanced at low temperatures, have been studied as a function of temperature. The marked tendency of ions to cluster at low temperature has led to the observation of many previously unobserved ion clusters in this system, at -82"K,e.g., He3+, Ar3+, N5+, N6+,N 7 + , N8+,N9+,Ar202+,Ar04+, etc. A schematic diagram of this apparatus is shown in Fig. 16.This stainless Downstream Pressure Measurement Line

&Copper

Heat Shields

Stainless Steel TLbe with Copper Heal Sink

Ceramic Insulators

4" Diff. PU-mP

6" Diff. Pump Gas Lines: Healer, Thermocouple. . and Filament Leads

FIG.16. Temperature-variableflowing afterglow apparatus.

steel reaction tube is also approximately 8 cm inside diameter and 100 cm long. The reaction tube is surrounded by, and is, in intimate contact with a copper heat sink on which are mounted heaters, cooling lines, gas lines, and thermocouples. A thin polished copper heat shield is wrapped over the tube and heat sink. Polished copper heat shields are also mounted from the outside wall of.the insulating vacuum jacket. This vacuum jacket is pumped by a trapped 4 in. oil diffusion pump. The tube temperature is measured with three chromel-alumel thermocouples which are placed near the ion source, the reactant inlet, and the mass spectrometer sampling port. Each thermocouple goes to a separate proportional servo temperature controller which controls heaters in its respective region to eliminate temperature gradients which would otherwise occur due to nonuniform heat losses. Figure 17 shows the ions present in a relatively clean helium afterglow at 80°K.This is the first mass spectrometric observation of He3+ whose existence was inferred earlier by Patterson (1968). Figure 18 shows the same afterglow ions, sampled with increasing drawout potential on the sampling

E.E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

48

+/+-+

Mass 16

Mass 28

'O3I

L

lo*o

80

160

240

L

lo*o

1

2

3

4

5

6

7

NOSE CONE VOLTAGE (volts)

HELIUM FLOW (atm d s e d

FIG.17. Variation of ion signals present in a rrlatively clean helium afterglow at 80°K as a function of the rate of helium flow. The experiment was performed at maximum pumping speed. Consequently this is essentially a plot of the ions as a function of pressure from 0.17 to 0.42 Torr.

FIG.18. Variation of ion signals present in a relatively clean helium afterglow at 80°K as a function of the sampling port potential. The experiment was performed at a helium flow of about 90 atm cm3/sec. The decline of the He3+ signal with the increase in potential is taken as an indication of the weak binding of the He3 cluster. +

port. The relative decrease of the He,' signal shows that it is only weakly bound. This ion did not exist in measurable concentrations at 300°K. The existence of many cluster ions at low temperature is due in part to the increased three-body rate constants for ion association at these temperatures. This has the consequence that such three-body reactions are readily measured in the flowing afterglow and indeed as a function of temperature. Table I1 gives some data of this kind recently obtained. Figure 19 shows the saturations of two ion association reactions (Bohme et af., 1968) in which the reactions N2+ + N 2 + H e + N 4 + + H e

(50)

and Oz+-t02fHe+04++He (51) become independent of helium pressure at 80'K. These are the first observations of ion association reactions carried out through the region from third order to second order kinetics. This data leads to valuable kinetic information

49

FLOWING AFTERGLOW MEASUREMENTS

TABLE I1 THREE-BODY RATECONSTANTS MEASURED IN THE ESSA FLOWING AFTERGLOW SYSTEM

Reaction N2++N2+He+N4+ +He N + + N Z + H e + N 3 + +He Oz++ 0 2 + H e + 0 4 + + H e O+ + N 2 + H e + N 2 0 + * + H e 0- COz +He + C 0 3 He Ar+ +Ar+He-,Arz+ + H e

+

+

+ + + + + +

+

0- Nz He + N 2 0 - + He Mg+ + 0 2 + A r + M g 0 2 + +Ar Fe+ O2 Ar + F e 0 2 + + Ar Ca+ 0 2 Ar + C a 0 2 + Ar

Temp. (“K)

Rate constant (cm6/sec)

82 280 82 280 82 82 280 82 290 82 300 300 300

1.2(-28) 1.9( -29) 7.2(-29) 8.6(-30) 3.1( -29) 5.4( -29) 1.5(-28) 1.6( - 30) -1.3(-31) 1.3(- 30) -2.5(-30) 1.0(-30) 6.6( -30) N N

Helium Pressure (Torr)

FIG.19. Plot of the apparent two-body rate constant versus helium pressure for two ion association reactions. The change from three-body kinetics to two-body kinetics shown is taken as an indication of the saturation of these reactions.

for example, the lifetimes of the initially formed Na* and 0 : . (Bohme et al., 1968). The “saturation” has not yet been observed at 300”K, presumably because N i * and 0: * lifetimes become much shorter at higher temperatures so that the saturation pressure is higher than we have so far been able to attain. Figure 20 shows the rate constants for several reactions as a function of temperature (Dunkin ef a/., 1968; Ferguson ef al., 1969b). The decrease in rate constant for O++Nz+NO+ + N (52)

E. E. Ferguson, F. C.Fehsenfeld, and A. L. Schmeltekopf

50

-

fs

10'0

c

c

2 u)

FIG.20. Rate constants for several reactions as a function of temperature.

c

0

V

c 0)

16"

0

a

4

IO-~~~

200

400

Temperature

600

(OK)

with increasing temperature has been explained as being due to the shorter NzO+*intermediate complex lifetime at higher temperatures (Ferguson et al., 1969b). The products NO' and N provide the most exothermic channel for NzO'* decomposition, The longer the NzO'* lifetime, the more complete will be the mixing of internal vibrational modes, and mode mixing favors the exothermic decomposition channels because of their greater phase space volume. The temperature independence of the He' N2reaction rate constant from 300 to 600°K is consistent with other studies in which the He+ kinetic energy is varied as well as a reasonable (albeit naive) model for such fast reactions. Accordingly, it seems unlikely that systematic errors are introduced in the flow tube behavior or analysis due to the temperature variation. Also, the 0 ' + 0, -+ Oz+ + 0 measurements in Fig. 20 are in excellent agreement with measurements of Smith and Fouracre (1968) for the temperature range 185-567°K.

+

Vm. Some Miscellaneous Results One kind of negative ion reaction has no positive ion analog, namely, the process of associative detachment A-+B+AB+e (53)

FLOWING AFTERGLOW MEASUREMENTS

51

where either A or B may be diatomic or polyatomic. The measurement of such reactions raises a possibility of a new kind of experimental error which must be considered. The reaction (53) changes the electron density. Accordingly, during a reaction rate measurement which involves measuring the A- ion signal sampled from the weak afterglow plasma as a function of [ B ] , it is conceivable that the ion sampling efficiency might depend on electron density and therefore might also vary. Since the change in A- signal due to addition of B is interpreted as being due entirely to reaction (53), such an effect would obviously introduce error. In cases where (53) did not occur at all and the entire change in the A- current was due to sampling efficiency variations, the error could be a very gross one indeed. Ironically, the presence of certain impurity ion signals have demonstrated the absence of this error. It has been typical, especially with the early glass tubes, to find persistent impurity ion signals, particularly C1- and, when the glass had been cleaned with HF, F-. When these impurity ions cannot react exothermically with the reactant gas, the observed constancy of their signals demonstrates that no change in sampling occurs. For example, Fig. 21

-z P 0 V W

u)

\

u)

L

i

3

FIG.21. Ion current versus nitric oxide flow reaction 0-+ NO NO, + e. --f

8

v

+ z

W

a a

3

V

z

0

NO FLOW (RELATIVE SCALE)

illustrates the data for the reaction 0-+NO+N02+e.

(54)

Note that the C1- signal does not significantly vary, while the 0-signal drops by a factor of 600 due to (54).

52

E. E. Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

Another peculiarity of the associative-detachment reactions is that no detectable charged product results (in the experimental arrangements so far employed). Very often, as in (54), associative detachment is the only exothermic reaction possible and there is no ambiguity in the identity of the products. When two or more exothermic channels exist, e.g., 02-fH-tOH-fO

+Hot

+ e,

care must be taken in specifying the products. In this example, the presence or absence of OH- will in principle settle this question. However, the branching ratio cannot be accurately determined because the products of only one of the branches are detected. A further complication arises when these products can undergo further reactions. For example, if OH-

+ H-+H20+e

(56)

is fast, one might incorrectly infer that only (55b) had occurred. We have discussed the comparison of flowing afterglow rate constant measurements with other experimental techniques. This, of course, is desirable no matter how carefully the analysis of an experiment is done, because one should always be concerned that some relevant factor has been overlooked in the analysis. Alternatively, one can compare experiment and theory if cases exist where theory is well grounded. One ion-neutral reaction does appear to exist, the associative-detachment reaction H-

+ H +H, + e,

(5 7)

for which theory appears to be very secure and fairly accurate (Dalgarno and Browne, 1967). Our preliminary measurement of (57) (Schmeltekopf et af., 1967) agrees with the theoretical calculation. Unfortunately, a factor of 2 uncertainty was considered possible because of uncertainty in the measurement of the H atom concentration. This measurement will be eventually improved. IX. Summary

We have described the experimental and analytical techniques which have been developed for flowing afterglow applications to the quantitative study of ion-neutral reaction processes. The discussion has concentrated on the experience and results obtained in the authors’ laboratory in Boulder. Similar flowing afterglow tubes are either under construction, or are now being operated in other laboratories and it can be expected that the broad potential of this technique for supplying ion-neutral reaction rate data will be well exploited in the future.

FLOWING AFTERGLOW MEASUREMENTS

53

A flowing afterglow system in the laboratory of H. I. Schiff at York University in Toronto, Canada, has recently been used to measure a number of reactions of H3” with a variety of gases (Burt, 1968).Anotherflowingafterglow system has recently been constructed at the University of Pittsburgh and reaction rate constants obtained for the charge-transfer reaction of N,’, 0, +, and NO’ with sodium atoms (Farragher et al., 1969). These results are of great importance for meteor ion chemistry. The present article has concentrated on experimental and analytical methods rather than the application of results to the understanding of ionneutral chemistry. This emphasis was chosen because most of the results have been, or are being reported, in the literature; whereas, the rapidly developing methodology has not been adequately described previously. Tables I1 and 111 give a sampling of flowing afterglow rate constant results as an illustration of the method’s versatility. In order to complete this discussion with a proper perspective, we would like to very briefly summarize some of the broad trends in ion-neutral reaction behavior which have been discovered in the past few years, primarily as a consequence of the application of the flowing afterglow technique. It has been found on the basis of very many examples, with almost no exceptions, that exothermic charge-transfer from atomic or molecular ions to molecular neutrals at thermal energy is invariably fast. Most other techniques for the measurement of ion-molecule reaction rate constants are inherently unsuited for the study of an ion reacting with a neutral where the neutral has the lower ionization potential, and sufficient data for this generalization did not exist prior to the flowing afterglow results. Theoretical prediction, at least for the small molecules of aeronomical interest, was that thermal energy chargetransfer would often be slow. Several dozens of reactions of this type have been measured in the ESSA flowing afterglow system. Furthermore, the same generalization is true for negative ion charge-transfer reactions. This generalization is valid when no competing atom-rearrangement reactions occur and might be stated: “When charge transfer to a molecular neutral is exothermic at 300”K, a fast reaction occurs (fast meaning k > lo-” to lo-’’ cm3/sec).” Charge transfer reactions of negative ions have sometimes been useful in establishing relative electron affinities of molecules, which are often difficult to measure. Positive ion charge-transfer reactions have, on occasion, been useful in establishing relative ionization potentials of molecules, generally known or better measured in more direct ways. Associative-detachment reactions were first measured in the flowing afterglow system and most of the presently available data come from this source. This exploits the flowing afterglow capability of reacting unstable neutrals with negative ions, which is the most common situation in which associative detachment is exothermic. Certainly, reactions with unstable neutrals have

E. E. Ferguson,F. C.Fehsenfeld, and A. L. Schmeltekopf

54

TABLE 111 EXAMPLES OF BINARY ION-NEUTRAL REACTIONS MEASURED IN FLOWING AFTERGLOW SYSTEMS

Temp. ("K)

Reaction

+

O+ Nz 4NO+

+N

80 300 600 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300

O+ +NZ(v=O)+NO+ + N v=l v=2 v=3

Oz-

+ 0 + 0 3+ e

03-

f

0 2 -

+ + +

+

coz + co3- + 0 0 3 +o,- + oz

2

+ +

Fe+ 0, + FeO+ 0, MgO+ 0 + Mg+ 0 2 SiO+ O+Si+ 0, O z ++ N a + N a + + 0 2 NO+ + N a + N a + + N O Nz+ + N a + N a + + N z H3+ +CO+COH+ +Hz NOH+ + N O + N O + + N O H H3+ Hz0 + H30+ Hz Hez+ NZ+ N z + + H e Oz- Oz('A,) -+ 2O2 e 0- Oz('A,) -+ O3 e H- H +Hz e Oz+ Nz +NO+ + N O

+

+ +

+ + +

+

+

+ + +

600 300

a

Rate constant (cm3/sec) 1%-

12) 1 2 - 12)

Source a U

5.0( - 13)

U

12-12) 1.2(-12) 5.0( - 1 1 ) 12-10) 2.S( - 10) 4.0(- 10) 4.0(- 10) I .S( - 10) 1 .O( - 10) -2.0(-10) 6.7( - 10) 7.0( - I I ) 5.8( - 10) 1.4( -9) 7.0( 10) 2.9(- 9) -6.0(-10) -2.O(-10) 3.0( - 10) 13-9) < l.O(- 15) < 1 .O(- 15) < 1.O( - 15)

a a a a a

-

~

N

11

a U

a a b b b c c C

a a n U

a a

a

ESSA results.

' Farragher et al. (1969), University of Pittsburgh results. Burt (1968), York University results.

the most applied interest. Examples are Oz-

+ 0 + 0 3+ e

in aeronomy and perhaps gas discharges, and H-

+H+Hz +e

(59)

in astrophysics. Based on a little over a dozen cases so far, the above generalization can be extended to include associative-detachment reactions. Only two exceptions to this are known: 0- + Nz - + N z O+ e + 0.21 eV, k< cm3/sec (60)

FLOWING AFTERGLOW MEASUREMENTS

55

and OH-

+N

HNO

+ e + 2.4 eV,

k < lo-" cm3/sec.

(61) Qualitatively, this generalization is readily understandable in terms of a model involving rapid autodetachment of the negative ion complex formed in the collision (Ferguson, 1968). This has the implication that studies of associative-detachment reactions can be expected to lead to useful information on molecular negative ion potential curves, which are generally rather inaccessible to study. It has clearly been of value in understanding ion-neutral reactions, to simply have a great deal of reaction rate constant data available. The capability of providing a large amount of data (and the application of this data to a variety of applied problems) has probably been the major contribution of the flowing afterglow technique in these first few years of its application. It is apparent, however, that the more detailed studies now possible (e.g. temperature dependence of rate constants, dependence on vibrationally and electronically excited states, emission spectra of ion-molecule reactions, etc.) will yield an increasingly detailed picture of ion-neutral reaction mechanisms. In this way, the flowing afterglow technique should complement other methods such as crossed-beam studies which lead rather directly to reaction mechanism details. --f

ACKNOWLEDGMENTS The authors would like to express appreciation to their colleague Dr. D. L. Albritton for a very substantial contribution to the writing of this manuscript. Thanks are also due to Mrs. Alice Keysor for typing the manuscript. This work has been supported in part by the Defense Atomic Support Agency.

REFERENCES Albritton, D. L., Schrneltekopf, A. L., and Ferguson, E. E. (1968). Bull. Am. Phys. SOC.13, 212. (To be published.) Bass, A. M., and Broida, H. P. (1963). J . Res. Narl. Bur. Std. A 67, 319. Beaty, E. C., and Patterson, P. L. (1965). Phys. Rev. 137, A346. Bohrne, D. K., Dunkin, D. B., Fehsenfeld, F. C., and Ferguson, E. E. (1968). J. Chem. Phys. 49, 5201. Branscornb, L. M. (1968). Intern. Cont. Electron. At. Collisions, Sth, Leningrad, 1967, Joint Inst. Lab. Astrophys., Univ. of Colorado, Boulder, Colorado. Burt, J. A. (1968). Ph. D. Thesis, York Univ., Toronto, Canada. Dalgarno, A., and Browne, J. C. (1967). Astrophys. J. 149, 231.

56

E. E.Ferguson,F. C.Fehsenfeld,and A. L.Schmeltekopf

Dunkin, D. B., Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1968). J. Chem. Phys. 49, 1365. Dushman, S., and Lafferty, J. M. (1962). “Scientific Foundation of Vacuum Technique,” 2nd ed. Wiley, New York. Elias, L., Ogryzlo, E. A., and Schiff, H. I. (1959). Can. J . Chem. 37, 1680. Farragher, A. L., Peden, J. A., and Fite, W. L. (1969). J. Chem. Phys. 50,287. Fehsenfeld, F. C., Evenson, K. M., and Broida, H. P. (1965a). Rev. Sci. Znstr. 36,294. Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1965b). Planetary Space Sci. 13, 919. Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1966).J. Chem. Phys. 45,1844 Fehsenfeld, F. C., Schmeltekopf, A. L., and Ferguson, E. E. (1967). J. Chem. Phys. 46, 2802. Fehsenfeld, F. C., Albritton, D. L., Burt, J. A., and Schiff, H. I. (1969). Can. J. Chem. 47, No. 8. Ferguson, E. E. (1967). Reo. Geophys. 5, 305. Ferguson, E. E. (1968). Advan. Electron. and Electron Phys. 24,l. Ferguson, E. E. (1969). Can. J. Chem. 47, No. 8. Ferguson, E. E., and Fehsenfeld, F. C. (1968) J. Geophys. Res. 73, 6215. Ferguson, E. E., Fehsenfeld, F. C., Dunkin, D. B., Schmeltekopf, A. L., and Schiff, H. I. (1964). Planetary Space Sci. 12, 1169. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D,, Schmeltekopf, A. L., and Schiff, H. I. (1965a). Planetary Space Sci. 13, 823. Ferguson, E. E., Fehsenfeld, F. C., Goldan, P. D., and Schmeltekopf, A. L. (1965b). J. Geophys. Res. 70, 4323. Ferguson, E. E., Fehsenfeld, F. C., and Schmeltekopf, A. L. (1969a). Adoun. Chem. Ser. 80, 83. Ferguson, E. E., Bohme, D. K., Fehsenfeld, F. C., and Dunkin, D. B. (1969b).J.Chem.Phys. (in press). Goldan, P. D., Schmeltekopf, A. L., Fehsenfeld, F. C., Schiff, H. I., and Ferguson, E. E. (1966). J. Chem. Phys. 44,4095. Jarvis, S. (1968). J. Res. Natl. Bur. Std. B 72, 135. Kaufman, F. (1964). Ann. Geophys. 20, 106. MacNair, D. (1967). Rev. Sci. Znstr. 38;124. McDaniel, E. W. (1964). “Collision Phenomena in Ionized Gases.” Wiley, New York. Moruzzi, J. L., Ekin, J. W., and Phelps, A. V. (1968). J. Chem. Phys. 48, 3070. Oskam, H. J. (1958). PhilZips Res. Repts. 13, 401. Patterson, P. L. (1968). J. Chem. Phys. 48, 3625. Sayers, J., and Smith, D. (1964). Discussions Furaday SOC. 37, 167. Smith, D., and Fouracre, R. A. (1968). Planetary Space Sci. 16, 243. Schiff, H. I. (1964). Ann. Geophys. 20, 115. Schmeltekopf, A. L., and Broida, H. P. (1963). J. Chem. Phys. 39, 1261. Schmeltekopf, A. L., Fehsenfeld, F. C., and Ferguson, E. E. (1967). Astrophys. J. 148, L155. Schmeltekopf, A. L., Ferguson, E. E., and Fehsenfeld, F. C. (1968). J. Chem. Phys. 48, 2966. Warneck, P. (1967). J. Geophys. Res. 72, 1651.

EXPERIMENTS WITH MERGING BEAMS RO Y H . N E YNABER Space Science Laboratory, General DynamicslConuair San Diego, California I. Introduction.. ................................................... 11. General Principles ................................................

111. Ion-Neutral Reactions ............................................ A. Symmetric-Resonance Charge Transfer .......................... B. Charge Rearrangement ........................................ IV. Ion-Ion Reactions ................................................ A. Symmetric-Resonance Charge Transfer .......................... B. Mutual Neutralization ......................................... C. Radiative Recombination. ...................................... V. Neutral-Neutral Reactions ........................................ A. Charge Transfer and Ionization ................................. B. Rearrangement.. ............................................. VI. Electron-Ion Reactions. .......................................... A. Theory ...................................................... B. Dissociative Recombination ................................... VII. Current or Very Recent Studies ................................... VIII. Concluding Remarks. ............................................ References .....................................................

57 59 62 62 72 80 80 83 87 89 90 ,100

.lo0 100 .lo1 .lo5 .lo6 .lo7

I. Introduction The merging beams (mb) technique was originally developed to study twobody collisions in the energy range from a few tenths to several electron volts (henceforth called the low energy region) because other experimental methods could not be used in this range. Typical problems with these other methods include low ion intensity because of space charge effects, poor energy resolution, and a low upper limit to the energy (i.e., about 0.3 eV) of neutrals generated from thermal sources. The mb technique consists of two beams traveling in the same direction along a common axis. Various other names that have been given to the method include superimposed beams, overtaking beams, and confluent beams. The laboratory energy of each beam is typically several kiloelectron volts with an energy spread of a few electron volts, whereas the relative energy of the beams in the c.m. system is in the low energy region. The energy spread 57

58

Roy H. Neynaber

at this relative energy is generally just a few percent. The use of high laboratory energies minimizes space charge difficulties. The method can also be used to study collisions at energies outside of the low energy region, e.g., at thermal energies and at several hundred electron volts. The mb technique is also of interest in the low energy region and above because it can be used as a method, with good energy resolution, for studying reactions between two, general labile species. In ion-neutral studies up to the present, crossed beams have been used where the neutral species is labile. This conventional method is advantageous when the neutral species can be obtained from a thermal source (e.g., 0 from an rf source or H from a high temperature oven) since beam fluxes from such a source are relatively large. However, a more comprehensive series of free radicals can be obtained by the more general technique of neutralizing the parent ion by charge transfer. For reactions involving many of these species, the mb technique may be the only method for measuring cross sections because larger signals (arising from the availability of more intense beams and longer interaction regions) can be obtained for a given interaction energy. Other merits of the mb technique include easy collection of products for total reaction cross section measurements, and relative ease in detecting all products of reaction. Davis and Barnes (1929a,b, 1931), Barnes (1930), and Webster (1930) merged beams of electrons and alpha particles many years ago. More recently, Cook and Ablow (1959, 1960) suggested the use of superimposed beams for the study of collisions between two heavy particles. The possibility of a very low interaction energy, coupled with a high energy resolution was first pointed out by Trujillo et al. (1963, 1965). The technique can be used to investigate ion-neutral, ion-ion, neutralneutral, and electron-ion reactions. Merging beams studies that have been conducted of such processes will be discussed. Since the method (employment of the concept of low relative energy coupled with high energy resolution is assumed) is in its infancy, the total number of such experiments is small. An earlier review of the subject has been written by the author (Neynaber, 1968). It will be noted, as the discussion proceeds, that an mb apparatus is relatively complicated. Its configuration is tied closely to the type of reaction for which it is to be applied, and this configuration is generally quite inflexible. Although some of the machines to be discussed have been used to study more than one type of process, it would be safe to say that none of them is a universal apparatus for the study of all the above types of reactions. The development of merging beams has ushered in the concept of the study of two-body collisions by the intersection of two beams at a small angle. This method is characterized by properties, as one might expect, that are in between those of crossed and merging beams. The method has been called inclined beams.

EXPERIMENTS WITH MERGING BEAMS

59

There will be no further discussion of inclined beams except to comment briefly on the contemplated work of two groups of scientists. M. F. Harrison and R. Rundel of the Atomic Energy Laboratory at Culham are constructing a machine in which two beams will intersect at 20”. They plan to study H + + H- + H + H at interaction energies of about 200 eV to 30 keV. An advantage of using inclined instead of merging beams for this experiment is that the parent ion of each product can be identified. However, for the same laboratory energy of the reactants, lower interaction energies could be achieved in an mb experiment. Harrison and Rundel also plan to study H + + H- + H(2S) + H and H C + H - + H + H + e using the inclined beams method, J. Schutten and colleagues of the FOM Institute for Atomic and Molecular Physics, Amsterdam, are also constructing an inclined beams machine in which the angle of intersection will be 10”. Included in their plans are experiments to measure cross sections for multiple ionization in neutralneutral, ion-neutral, and ion-ion collisions. The inclined beams method will also enable them to identify the parent ion of each product for all reactions studied. In addition, the mechanics of bringing about the interaction of two charged beams of the same polarity is easier for inclined than for mb systems.

+

11. General Principles

A low interaction energy (i.e., relative energy in c.m. coordinates) can be obtained with an mb system because the difference in beam velocities can be made small even though high energy beams are employed and because the difference of the laboratory energies of the beams is many times larger than the interaction energy. Analytical expressions are presented below. If we assume that all particles in the two beams move along parallel lines (ideal case), then the interaction energy W is

w = t p ( c , - u$,

(1)

where p is the reduced mass of the system, and u1 and u2 are the laboratory velocities of the particles in each beam. We may also express W a s follows:

where m, and m 2 are the masses, and El and E, are the laboratory energies of the particles in each beam. For simplicity, let m, = m, = m. The general conclusions reached for this case apply equally well when m, # m 2 .For equal masses, Eq. (2) becomes

60

Roy H. Neynaber

When the absolute value of the difference in beam energies, AE, is small compared to these energies and E is the average of the two beam energies, it can be shown that W w AE ' / 8 E .

(4)

Now define D = AE/ W. The term D is an energy deamplification factor. We have (again for small A E ) D = A E / W z 8 E / A E + 1.

(5)

Equation ( 5 ) indicates that the interaction energy is small compared to the difference of the laboratory energies of the particles for large laboratory energies. As an example, let El = 5000 eV and E, = 5100 eV. Then AE = 100 eV, and from Eq. (5), D w 400 and W w 0.25 eV. The concept of a deamplification factor can also be applied to the more realistic case of two beams each having an energy distribution rather than being monoenergetic. From Eqs. (4) and (5) it can be shown that a perturbation 6 E in AE gives approximately the following perturbation 6 Win W :

6 W/W x 2 6E/ W D =2 6E/AE.

(6)

Assuming a reasonable energy spread, i.e., full width at half-maximum, of 1.5 eV for each beam (6E = & 1.5 eV), we obtain an approximate spread in W of 6 W = 10.0075 eV, or an uncertainty in W of A 3 %. Therefore, because

of the deamplification factor for an mb experiment, good resolution for the interaction energy can be obtained. Similarly, small, random fluctuations in El or E, result in negligible perturbations in W. Equation (6)rigorously applies to rectangular energy distributions. If the distributions are Gaussian, the expression for 6 W/W should be divided by Therefore, for Gaussian distributions, Eq. (6) conservatively estimates the fractional perturbation in W . We have previously assumed that the particles in the two beams have parallel paths. In a realistic case, there is nonparallelism, and transverse velocities exist. This could result in an average energy of interaction Wfor the realistic case that is different from W , and could cause an uncertainty in the measured cross section. Thus it is necessary to take proper account of transverse velocity components, or to ensure that the largest of these is insignificant. The mathematical formalism for extracting a cross section from an mb measurement in which transverse velocities exist appears formidable (Cook and Ablow, 1959, 1960). To circumvent this task it seems desirable to eliminate particles with large transverse velocity components and to keep only those particles with sufficiently small transverse velocity components. The undesirable particles can be removed by passing the beams through

fi.

EXPERIMENTS WITH MERGING BEAMS

61

two collimating apertures. Small, residual transverse components exist after the passage of the beams through these apertures. For the case when the distance I between the collimating apertures is equal to the length of the interaction region (i.e., that region in which cross section measurements are made), when AE is small compared to the beam energies, and when m, = m2, we have derived the following expression for the upper bound of the fractional difference between Wand W : ( W - W ) /W c 0.54(Ed/AEl)2,

(7)

where W is the energy of interaction averaged over all possible angles of intersection of the beams and over all points in the interaction region, and d is the diameter of the collimating apertures. Applying Eq. (7) to the previous numerical example and choosing the values d = 0.25 cm and 1 = 25 cm, we obtain an upper bound on the percentage difference of approximately 13.5%. Similar studies have been made for the effect of the residual transverse components on the measured cross section. An energy dependence of the cross section must be assumed to make the computations. For a resonant charge transfer process at low energy the effect has been computed and, for the conditions above, the cross section for the realistic case is different from that of the ideal case by less than 6 % . As W increases, these percentage differences due to the residual transverse components become even smaller for a given E. An experiment might consist of determining the number of particles per second S which has been generated in the interaction region by the desired reaction. The expression relating the cross section Q in square centimeters per particle to S is (again assuming equal masses)

Q = (S/I)(EiEz/mV1",

(8)

where

s,

I = J1J 2 d x d y d z , and the integral is performed over the volume of the beams in the interaction region; z is along the axis of the beams in the interaction region; J1 and J, are the beam flux densities in particles per square centimeter per second for the respective particle energies El and E2 in ergs. Since the integrand is nonzero only where the beams overlap, I is called the overlap integral. Also, W is in ergs, and rn in grams. The mb technique should be useful for measuring total reaction cross sections even when the collisions are accompanied by appreciable c.m. scattering since the maximum solid angle for reaction products is much smaller than that for conventional experiments. This is because the kinetic

62

Roy H . Neynaber

energy in the center of mass is much smaller than the laboratory energy of the reactants. For example, with a kinetic energy in the center of mass of several electron volts and initial reactant energies of several kiloelectron volts, the products are confined to a cone whose vertex angle is a few degrees. Finally, the reaction products are easily detected by secondary electron emission because they have large laboratory kinetic energies.

III. Ion-Neutral Reactions A. SYMMETRIC-RESONANCE CHARGE TRANSFER 1. Double Source Experiment Symmetric-resonance charge transfer experiments in Ar have been conducted with a double source apparatus (Neynaber et al., 1967a,b). Relative cross-section measurements were made in a range of W from 0.1 to 20 eV. In addition, absolute cross section measurements were made for W = 0.3 and 100 eV. In Section III,A,2, similar experiments in H and D will be discussed in which only a single source was used. a. Description of Apparatus. The double source apparatus (Trujillo et al., 1966) was designed to study the Ar reaction by observing product ions. The process is Arc(El) Ar(Ez)--+Ar(EA +Art(.&) (9) where the E's represent the laboratory energies of the particles in a given interaction region. Outside of the interaction region, the laboratory energy of the particles in each primary beam was Eo = E, = 3000 eV. For E, > E l , as was the case, El = (3000 - A E ) eV, where AE was defined in Section 11. A schematic diagram of the apparatus is shown in Fig. 1. The sources were of a low-pressure, oscillating-electron-bombardment type (Carlston and Magnuson, 1962) which produced beams with a spread of about 1.5 eV. An Ar' (3000 eV) beam from source 1 was merged with a mechanically chopped Ar (3000 eV) beam. The Ar beam was obtained by passing Ar' from source 2 through a charge-transfer cell containing H, . It was assumed that negligible energy was lost in the near-resonant charge transfer process and that the energy and energy spread of the Ar beam emerging from the cell were the same as that of the Ar' beam entering the cell. An electric field between the condenser plates that follow the cell was used to remove Ar ' which did not undergo charge transfer. The superimposed beams were then collimated to eliminate large transverse velocities and were passed into a decelerating-accelerating system containing the interaction region. The energy difference between Ar and Ar' that was necessary to

+

MOVABLE DETECTOR APERTURE

FIG.1. Schematic diagram of douole source, mb apparatus for studying Ar+

+ Ar -+ Ar + Ar+. Apertures are not to the scale shown.

64

Roy H . Neynaber

obtain the desired interaction energy W was established by raising the potential of the interaction region by an appropriate amount AE/e, where e is the magnitude of the electronic charge. The primary ion beam was thereby decelerated to (3000 - A E ) eV at the entrance to the interaction region and accelerated to 3000 eV at its exit. A schematic diagram of the apparatus for achieving this is shown in Fig. 2. Argon ions produced by the desired process GRID ,4

LFIRST COLLIMATING APERTURE

FIG.2. Schematic diagram of deceleration-acceleration apparatus for double source, mb system. For details of the geometry see Neynaber et al. (1967a). The grid structure and spacing between grids were chosen to achieve a compromise between maximum transmission, minimum electric field penetration, and uniformity of electric fields. Grids 2 and 3 are electrically connected together. Grids 3 and 4 and the electrostatic screen are attached to the movable detector assembly (see Fig. 1 ) and could be translated in all directions.

in the interaction region were accelerated at its exit to an energy equal to (3000 + AE) eV. After the merged beams left the interaction region, they entered the demerging magnet. The function of the magnet was to separate fast neutrals from primary and product ions. The magnet was not designed with a resolution that could separate the primary from the product ions. Other undesired particles were prevented from reaching the detector, a Bendix electron multiplier, by employing several components of a detector assembly (shown in Fig. 1 within the dotted box). One of these components was a retarding grid which was at a potential that would allow product ions to pass through but which would retard both the primary ions and ions formed by other processes. Another component was a low-resolution, hemispherical, electrostatic energy analyzer. Its purpose was reduction of ion noise, which is defined later. The sweep plates following the hemispherical analyzer were not incorporated in this experiment. Finally the output of the multiplier was fed into a lock-in amplifier. The detector assembly was supported on a carriage which could be moved along a set of rails parallel to the axis of the interaction

EXPERIMENTS WITH MERGING BEAMS

65

region and in directions perpendicular to this axis. The purpose of the movable assembly was to permit profiles of each primary beam to be taken along the interaction region to obtain information for calculating Zof Eq. (8). b. Absolute Cross Section Determinations. To obtain an absolute value of the neutral current for determining absolute cross sections, the secondary electron emission coefficient y of the “dirty” A1 plate (part of the neutral beam monitor shown in Fig. 1) for neutrals must be known. It was assumed that y was the same for ions and neutrals of the same energy. Utterback and Miller (1961) have obtained experimental evidence that the y’s are equal to within 20 % at energies above several hundred electron volts for N, on gold surfaces that were not clean. Haugsjaa et al. (1968) have found similar evidence for Ar above 140 eV on a “dirty” gold surface. Other investigators have also reported equal coefficients for energies above 100 eV (Berry, 1948; Rostagni, 1934). With this assumption, an absolute measurement of the neutral current was obtained. The y for ions was measured using ions from source 2, but without gas in the cell, with grounded condenser plates at the exit of the cell, and with the merging and demerging magnets turned off. The ratio of ion currents at the A1 plate with extraction and with suppression of secondary electrons was equal to y + 1. For absolute cross section measurements, J1 and J2 were measured at a number of points in space to permit I to be evaluated numerically. The flux densities were determined by measuring the current under investigation after passing it through an aperture of known diameter. c. Noise. A typical signal arising in the interaction region was about A (the primary beams were each several hundred nanoamperes). Two sources of noise were important. One of these depended only on the presence of the primary ion beam while the other depended only on the presence of the primary neutral beam. These are designated as ion noise and neutral noise, respectively. The ion noise was partially dependent, whereas the neutral noise was totally dependent, on background pressure. Since only the primary neutral beam was chopped, the output from the lock-in amplifier for neutral noise was coherent while that for ion noise was random. It was not surprising that these sources of noise existed since the ratio of desired signal to primary beam current was The effect of ion noise was reduced by using suitable integrating times. The coherent, or neutral, noise was probably due to stripped neutrals from the primary beam. The least energy the primary particles could lose in stripping reactions was their ionization energy. The voltage of the retarding grid could thus be set to minimize the neutral noise without reducing the desired signal. The voltage was (2985 + AE/e) V. The existence of excited neutrals in the beam would probably result in more neutral noise than for a ground state primary beam since excited particles would have larger stripping

66

Roy H . Neynaber Neynaber

cross sections. Since the neutral noise could be obtained separately, it could be subtracted from the signal which appeared with both beams present. There was evidence that excited species existed in the Ar beam when the gas in the charge transfer cell was Ar, since for a given neutral intensity and grid potential the neutral noise was larger than when the gas in the cell was H, . Accordingly, H, was used. d. Results and Discussion. The potential of the interaction region was raised to that AE/e appropriate for the desired W . From the total output of the lock-in amplifier, neutral noise and an anomalous output that was dependent on the presence of both beams when AE/e was zero were subtracted. The net result S was proportional to a relative Q. Each cross-section measurement at W was accompanied by a measurement at 1 eV. From these measurements and Eq. (8), the square root of the ratio of the cross section at W , Qw, to the cross section at 1 eV, Q , , was computed. These ratios, (Qw/Q1)''2, are shown 1.3

I

I

I

r

I

I 1 1 1 1

I

I

l

1

l

l

l

l

I

FIRSOV ;RAPP AND FRANCIS

: *z+s-d: ;-

PRESENT EXPERIMENT

OI 0.9o.8-

0.7

'

>

-

a..

-_

POPESCU IOVITSU AND IONESCU -PALLAS 1

1

I

I

1

I l l

1

I

I

I

1

1 1 1

-2

1 I

FIG.3. (a) (QW/QI)'12vs W for symmetric-resonance charge transfer in Ar. Each dot represents a value obtained for a single measurement; a dot accompanied by a number 2 means that two measurements resulted in the same quantity. Crosses indicate arithmetic averages of dots. The standard deviation of the slope of the experimental line confines the extremities of that line to the space between the arrows. (b) Square root of the absolute cross section for symmetric-resonance charge transfer in Ar.

EXPERIMENTS WITH MERGING BEAMS

67

in Fig. 3(a). The theoretical results of Firsov (F) (1951), Popescu-Iovitsu and Ionescu-Pallas (PI) (1960), and Rapp and Francis (RF) (1962) are included in the figure. The slopes of the F and R F lines are almost equal. The error bar is about f6 % and represents the standard deviation for measurements at 10 eV. Uncertainty in (Qw/Q1)1/2at W = 0.1 eV due to the anomalous signal, end effects of the decelerating-accelerating system, and residual transverse velocity components of the beams in the interaction region is less than 8 % . At higher W the uncertainty is even smaller. The experimental line is made to pass through ( Qw/Q,)”* = 1 at 1 eV. Its slope was obtained from the crosses (properly weighted) by the method of least squares. From Fig. 3a it is noted that a straight line appears to fit the crosses. The extremities of the F and RF lines fall well within the limits of the extremities of the experimental line. The end points of the PI line fall outside these limits. Absolute Q measurements at 0.3 eV were made using the technique described above and also a slightly different one (Trujillo et al., 1966). (This latter method was also used to measure an absolute Q at 100 eV.) The results were compatible and gave an average Q at 0.3 eV of 47.1 A’ with an estimated total error of + 25 and - 21 % . The error excludesthe possibility of metastable reactants, about which little can be said. This value of Q together with the experimental line in Fig. 3a were used to obtain the experimental line in Fig. 3b. The R F line is in agreement with the experimental results, the F and PI lines are not. The average value of the cross-section measurements at 100 eV is 18.1 A2 with a total error (again excluding the possibility of metastable reactants) of + 18 and - 13% . It can be shown that this cross section is compatible with the results in Fig. 3b. Table I is a compilation of some representative theoretical and experimental cross sections for the resonant charge transfer of Ar at W = 100eV. The TABLE I CROSSSECTIONSAT W = 100 eV FOR RESONANT CHARGE TRANSFER OF Ar Experimental or theoretical

Result

Reference Potter (1954) Ghosh and Sheridan (1957) Popescu-Iovitsu and Ionescu-Pallas (1960) Cramer (1 959) Rapp and Francis (1962) Hasted (1951) Kushnir ef al. (1959)

E E T E T E E

19.9 20.9 23.0 23.2 24.0 27.0 30.8

(A2)

68

Roy H. Neynaber

values for the mb experiment agree with Potter’s and Ghosh and Sheridan’s within the mutual errors of the experiments. The experimental cross sections listed in Table I were obtained using beam-gas techniques. Their accuracy depends linearly upon absolute pressure measurements performed with a McLeod gage. These measurements were made prior to the discovery of a systematic error in the normal use of McLeod gages, i.e., the Ishii-Nakayama effect (Ishii and Nakayama, 1962; Rothe, 1964). This error would result in measured cross sections that are too large by, perhaps, 10-30 % . Systematic errors in absolute Q determinations for the mb experiment could be due to excited species in either primary beam (the magnitude of this error is difficult to estimate), the determination of y for neutrals (-5 to + lo%), and the determination of the overlap integral I (+ 10 %). An estimate of the composite systematic error (excluding errors due to excited states) is then - 5 to + 14 ”/,. Excited species could be more easily controlled through the use of electron-bombardment ion sources which operate at lower pressure and in which only single collisions between electrons and atoms occur (Stebbings e t a / . , 1966). The determination of y for neutrals could be made by the alternate method of measuring the heat input to a thermopile from the neutral beam; then the assumption that the y for ions and neutrals is the same would not be necessary. Finally, the overlap integral could be more accurately and readily determined by incorporating automatic or semiautomatic equipment to permit J , and J , to be measured at many more points in the interaction region.

2. Single Source Experiment Belyaev er a/. (1966, 1967a,b) have conducted mb studies of the symmetricresonance charge transfer processes H f + H -+ H + H + and D + D -+ D + D + in a range of W from 5 to 100 eV using a single source apparatus. A mixed atom-ion beam was obtained through partial charge transfer of a single ion beam from the source. In the interaction region the ion beam passed completely inside the atomic beam. a. Description ofApparatus. Figure 4 is a schematic diagram of the apparatus. The anode of the ion source (which is described as an oscillating type operating in a longitudinal magnetic field) was maintained at a potential E/e = + 1 kV with respect to the system. Protons (deuterons) were extracted from this source and focused by lens 2 into a 90“, second-order spatial focusing, monochromator magnet 3. About 10 % ofthese ions were neutralized when the beam passed through the charge transfer cell 5, which contained COz . Some of the resultant atoms were left in the metastable 2 s state. After collimation of the mixed atom-ion beam, the metastable atoms were con-

+

EXPERIMENTS WITH MERGING BEAMS

69

FIG.4. Schematic diagram of single source, mb apparatus. (1) ion source; (2) extracting lens; (3) monochromator magnet; (4) quadrupole lenses; (5) charge transfer cell; (6, 9-13, 15) circular apertures; (7) condenser array; (8) region of high vacuum; (14) collision chamber; (16, 23) electron suppressors; (17) Faraday cylinder; (18) electrometer; (19) analyzing sector magnet; (20,22) adjustable holes; (21 ) cylindrical condenser; (24) calibrating Faraday cylinder; (25) galvanometer; (26) detector; (27-30) diffusion pumps; (31) titanium pump; (Sl, S,) switches.

verted to the ground state during their passage through a n electrostatic field produced by the condenser array 7. The electrodes of this array were divided into three sections, the field in the center section being opposite to that in the end sections ( S , in position a). The ratio of the fields in the sections was chosen so that the ion beam, after passing through the electrodes, returned to the extension of its previous trajectory. The angular divergence of the beam entering the collision chamber (interaction region) 14 was determined by the circular apertures 6 and 9 and was 1'20'. Aperture 9 was placed at the focus of magnet 3. In the collision chamber the energy spread of the ion beam was about 4 eV, and the ion current was approximately A. Because of the discharge voltage of the source, the energy spread of the ion beam at the exit of the source was considerably larger than it was in the interaction region. The laboratory energy and energy spread of the atom beam just outside the collision chamber were presumably about the same as for the ion beam since near-resonance charge transfer took place in cell 5. The collision chamber was cylindrical. At the ends were circular apertures I 1 and 12. The curved surface consisted of fine wires parallel to the axis of the cylinder. The chamber was maintained at an appropriate potential, AE/e, and was used in much the same way as the decelerating-accelerating system previously described for the double source experiment.

70

Roy H. Neynaber

The weakly focusing electrostatic lens formed by apertures 9-11 at the entrance to the collision chamber guaranteed passage of the ions inside the atomic beam along the entire length of interaction. Ions formed in the collision chamber (including stripped primary neutrals) passed through an energy analyzer consisting of a sector magnet 19 and a cylindrical condenser 21, which were tuned for ions of energy E + AE. Primary particles were rejected, but the resolution of the analyzer was not sufficiently good to eliminate neutral noise. The ions that passed through the analyzer were measured at the detector by counting scintillations induced by secondary electrons. These electrons were ejected from a metal plate by the impinging ions and accelerated to 15 keV. To minimize neutral noise, the Torr. pressure in the collision chamber was maintained at about 2 x 6 . Measurements. Cross sections were determined by applying Eq. (8); S was equal to the ion counts per second when the primary ion beam was not deflected away from the collision chamber by the condenser array 7 ( S , in position a ) minus the counts per second when the beam was deflected ( S , in position 6). Neutral noise was eliminated by taking this difference. The ion current corresponding to this difference was in the range lo-’’ to A. In determining the overlap integral, it was assumed that the flux density of the neutral component of the mixed beam was uniform everywhere in the interaction volume, i.e., that volume of space swept out by the narrower ion component of the beam. Under this assumption, the integral was equal to the product of the currents of both components in the interaction region, the length of the region, and the inverse of the cross sectional area of the neutral beam in the region. The ion current through the collision chamber was measured at the Faraday cup 17 ( S , in position a). The bottom of this cup was removable. The neutral current was determined by (1) measuring the current of secondary electrons from the bottom of the cup ( S , in position b) produced by the ion current and that produced by the neutral current and (2) measuring, in a separate experiment, the ratio of the secondary electron emission coefficients of the same metal as the bottom of the cup for ions and atoms at an energy of 1 keV. The ratio was approximately unity for protons and hydrogen atoms and for deuterons and deuterium atoms. An average cross sectional area of the atomic beam was determined from geometrical considerations. The length of the interaction region was obtained from the potential distribution along the axis of the collision chamber. This distribution was measured using an electrolytic bath. c. Results and Discussion. Cross sections as a function of Ware shown in Fig. 5. Included are the results of the single source mb experiments, of other experiments (Fite et al., 1960; Fite et al., 1962; McClure, 1966), and of theoretical studies (Smirnov, 1964; Dalgarno and Yadav, 1953). Belyaev

71

EXPERIMENTS WITH MERGING BEAMS INTERACTION ENERGY, wd (eV)

10

10 6

n

5

N

ms

4

rl I

0

a

3 U

2

1

loo

101

lo2

10

lo4

INTERACTION ENERGY, Wp(eV)

FIG.5 . Cross sections for charge transfer of protons in hydrogen atoms (1, 3-7) and deuterons in deuterium atoms (2) as a function of interaction energy (W, for proton experiment and W, for deuteron experiment). ( I , 2) single source, mb results; (3) Fite et al. (1960); (4) Fite et al. (1962); ( 5 ) McClure (1966); (6) theoretical curve of Smirnov (1964); (7) theoretical curve of Dalgarno and Yadav (1953). The values of W, and W dassociated with any point on the graph correspond to the same relative velocities of particles in the H H and D D reactions.

+

+

et al. (1967a,b) indicate that the uncertainty in the mb cross sections was caused mainly by statistical error in the measurement of S of Eq. (8) and by the presence of energy and angular spread in the interacting beams. It is noted that for the same relative velocity, the mb cross sections for the H + + H and D + + D reactions are the same. This is expected on theoretical grounds. Using crossed beams to study the same reactions (where proton energies are in the range from about 200 to 1500 eV), Fite et al. (1958) do not achieve this result. Figure 5 shows quite good agreement between the various experimental results. Agreement of experiment with theory is best at the higher energies. Resolution in W could be improved in this experiment by using a source that emitted ions with a smaller energy spread. A source like those used in the double source experiment would probably suffice. The aperture arrangement at the entrance and exit of the collision chamber is not the most effective means of achieving uniform decelerating and accelerating fields and of minimizing field penetration into the chamber. Grids such

72

Roy H . Neynaber

as those used in the system shown in Fig. 2 would be better and would probably allow measurements to be made at considerably smaller W. A disadvantage of the single source experiment (in addition to the fact that the reactants must be of the same species) is the inability to use modulation techniques effectively. Much of the statistical error in the measurements probably arises from ion noise. This could be largely eliminated as a source of trouble if the neutral beam were modulated and phase and frequency-sensitive detection methods were used. The difficulty of modulating the neutral beam without modulating the ion beam is obvious. Finally, a better determination of the overlap integral could be obtained if profiles of the atom and ion components of the beam could be determined in the interaction region. The single source experiment is another good example of the potential of the mb technique. With the method, cross sections for the H + + H and D + + D reactions have been obtained at lower W than have been achieved previously.

B. CHARGE REARRANGEMENT

+ H,+H,+ + H study of H,+ + H, H 3 + + H has been made with merging beams

1. H2+

A + (Neynaber and Trujillo, 1968) and represents the first application of the technique to charge rearrangement (ion-molecule) reactions. a. Experimental. Except for one modification, the apparatus used for this study is identical to that employed in the investigation of symmetric-resonance charge transfer in Ar (see Fig. 1). An H2+beam from source 1 was merged with a mechanically chopped H, beam. The energy of the particles in each beam was 3000 eV. The H, beam was obtained by passing H2+ from source 2 through a charge-transfer cell The energy difference between H2+and H, that was necessary containing H2. to obtain the desired interaction energy W was established by raising or lowering the potential of the interaction region by an appropriate amount AE/e. The product ions H,+ were then separated from the reactants by the demerging magnet. Other undesired particles were prevented from reaching the detector by employing the retarding grid and the hemispherical electrostatic energy analyzer tuned for the passage of the H,'. Before the addition of a set of sweep plates after the electrostatic analyzer (these plates were not used in the apparatus employed for charge-transfer studies and represent the one modification of that apparatus), the multiplier output was fairly sensitive to small changes in the fields of the demerging magnet and electrostatic analyzer. Such changes could cause small movements

EXPERIMENTS WITH MERGING BEAMS

73

of the H3+ beam over the multiplier face, and assuming gain irregularities over the face, could result in output variations. The plates were designed to sweep the H,' beam over an area on the multiplier face and so smooth output variations caused by gain irregularities. The use of these plates did solve the above problem. A retarding potential curve for the product ions could be obtained for each W. From the plateaus of these curves relative cross sections could be derived. For positive and negative AE/e's associated with the same W, ion currents at plateaus had the same value, within experimental error. The energy resolutions of the demerging magnet and electrostatic analyzer were such that, for a given AE/e, and therefore, W , H3+ of all possible energies would be passed for fixed settings of these detector components. Because of transverse velocities, H3+ was formed along path lengths of the merged beams where the reactants had the same energy. Therefore, from this mechanism, H 3 + was continually being generated outside the interaction region and, when AE/e = 0, inside as well. If the potential of the retarding grid in the detector were sufficiently low, these ions would be detected. When AE/e # 0, the detector signal was due to H3+ formed inside the interaction region at energy W, and, if the retarding grid were at a sufficiently low potential, to H3+ formed outside. The contribution from H3+ outside could be measured independently by applying any AE/e that would result in no measurable signal from inside the interaction region. For this purpose it was determined that the AE/e associated with W = 10 eV, which will be labeled AE'/e (AE'/e = +470 or -510 V), was satisfactory. The contribution associated with W due to H 3 + formed inside the interaction region was the difference between signals measured with the potential of this region at AE/e and at AE'/e. The retarding potential curve for a given W could be derived from this difference. Each cross-section measurement at W was accompanied by a measurement at 1 eV so that ratios of cross sections, Q,/Q,, were obtained. It was not necessary to take measurements for complete retarding potential curves (i.e., curves for a succession of retarding potentials beginning with potentials sufficiently large to result in no measurable signal to potentials well into the region of the plateau) to derive a given ratio. Only a single measurement on each plateau was required, and, in general, this was the technique followed. Data for a given cross-section ratio could be obtained in a few minutes. Absolute cross-section measurements could be made by the technique discussed for the Ar experiment. b. Results and Discussion. The mb results for Qw/Ql are shown in Fig. 6. Included are data for positive and negative AE/e's associated with the same W. To give an idea of the accuracy of ratios designated by crosses, the error for the ratio at 0.1 eV is estimated to be - 11 to + 6 % , At 3 eV the standard

74

Roy H . Neynaber

0.1

-

1

INTERACTION ENERGY W (aV) 0.25 0.5 I 1

--

1

1

2 3 4 7 I

1 ,

v

PRESENT EXPERIMENT

w-l/z

-

-

( e v I-'''

+

+Hz+ H 3 + H. Each dot represents a FIG.6. Qw/Ql vs W-'/' and W for H2+ value obtained for a single measurement; a dot accompanied by a digit means that number of measurements resulted in the same quantity. Crosses indicate arithmetic averages of dots.

deviation is about 18%. Also shown in Fig. 6 is a W - ' / 2 straight-line fit to the data for 0.1 eV < W < 1 eV. This line has the energy dependence predicted by Gioumousis and Stevenson (GS) (1958). For W 2 1.5 eV the ratios fall below the line. Perhaps a spectator-stripping model (Henglein, 1966) could be used to explain this fall-off. The values of the crosses in Fig. 6 are given in Table 11. Also included are the ratios Qw/Ql for a tandem mass spectrometer experiment by Giese and Maier (GM) (1963), and for a single-stage mass spectrometer experiment by

75

EXPERIMENTS WITH MERGING BEAMS

TABLE I1 RATIOOF CROSS SECTION AT

W (ev) 0.1 0.2 0.3 0.4 0.6 1.o 1.5 2.0 2.5 3.0 4.0 5.0 7.0 10.0 12.0 13.5

mb experiment QwlQi

3.22 2.45 2.04 1.45 1.oo 0.69 0.53 0.33 0.20 0.15 0.07 0 0

w,Qw, TO CROSS SECTION AT 1 ev, Ql Giese and Maier (1963) (QwlQih

1.63 1.39 1.oo 0.65

0.44 0.29 0.19 0.08 0.04

-

Reuben and Friedman (1962) (QwlQih~

’

1.35 1.oo 0.78 0.63 0.52 0.45 0.35 0.29 0.28 0.27 0.18 0

Ratios taken from curve labeled Q = 0 of Giese and Maicr (1963).

Ratios are the result of an analysis of the Reuben and Friedman (1962) data as outlined by Giournousis (1966).

Reuben and Friedman (RF)(1962). The mb results are in better agreement with the GM ratios than with the RF values. An absolute measurement was made at 1 eV with the result that Q , = 12 A2 with an estimated error of -26 to + 37 %. Considerably more reliability should be placed on the ratios Q,/Q, than on the absolute value of Q , as evidenced by the estimated errors indicated above. This is primarily because the absolute value of Q , depended upon measurements of the transmissions of the grids in the decelerating-acceleratingsystem, the secondary electron coefficient of the ion collecting plate in the neutral beam monitor, the primary ion and neutral beam shapes, the relative positions of the primary beams, and the gain of the detector assembly (Trujillo et al. 1966). These quantities were subject to rather large errors. The determinations of ratios did not depend on these quantities. The value of Q,, subject to its rather large uncertainty, overlaps the value of 15AZpredicted bytheGS theory. The experimental errors associated with Q,/Q, became larger as W increased because the Q , became smaller and were more difficult to measure.

Roy H. Neynaber

76

Experience with crossed beams, on the other hand, has indicated that the cross section errors become larger as beam energies decrease. This is attributed to beam intensities (which largely dominate such errors) becoming smaller with decreasing energy. From Fig. 6 it appears that the best fit to the data between 0.1 and 1 eV is a smooth curve that is concave downward. The possibility that this curve represents an actual departure of the cross-section ratio from a linear dependence on W-'12 should not be discounted. There is concrete evidence from signals at an extremely low energy (i.e., W 6 0.03 eV) that the quantity (QW/Ql)W'I2 is from 15 to 30% less than the corresponding value given by the slope of the W -'I2 fit in Fig. 6 . In other words, the W fit shown does not apply at this low energy. These signals arise when AE/e = 0. Transverse velocities account for the interaction energy. An upper bound to the energy has been calculated as 0.03 eV. Uncertainty in the energy prevents a determination of the cross-section ratio. Wolf (1968) has used a modified phase-space theory to calculate cross sections for the process under discussion. He predicts a curve of cross-section ratio versus W that has the same general shape as a smooth curve that fits the mb data. It should be noted that the states of the mb reactants are unknown. If excited states did exist, cross sections for ground state reactants could be obtained, in principle, by applying corrections to the quoted values. The percentage correction to the value of Ql would probably be considerably larger than the percentage correction to the cross-section ratios. Chupka et al. (1968) find that for W less than a few electron volts the reaction cross section decreases with increasing vibrational excitation of H,' ; whereas at higher W a mechanism for which the cross section increases with increasing vibrational energy of the ion becomes more prominent. The existence of such excitation in the mb experiment would therefore result in a smaller Q, than would be obtained from just ground state H,'. If the GS theory applies at 1 eV, electronically excited H, primaries would be expected to result in a Q, larger than would be extracted from measurements with a pure, ground state neutral beam. This follows from the larger polarizability of an excited neutral.

-'',

2. Na

+ 02++ NaO' + 0

The existence of NaO' has been established (Rol and Entemann, 1968) through the use of merging beams of Na and 0,'. The failure to observe NaO' in a flowing afterglow experiment is mentioned by Fite (1968) in a paper in which he discusses the merits of the mb technique. In the mb experiment, a Na beam of 2810 eV and an 02' beam with an

EXPERIMENTS WITH MERGING BEAMS

77

energy ranging from 3953 to 4354 eV were merged, resulting in W's ranging from 0.05 to 5 eV, respectively. The NaO' ions were formed with nearly the same velocity in the laboratory system as the reactant particles, corresponding to a laboratory energy near 4765 eV, which was much higher than the energy of any primary beam particles. The cross section Q for the reaction Na 02+--t NaO+ + 0 was determined through the use of Eq. (8). This Q as a function of Wand W-l" is shown in Figs. 7 and 8, respectively. For each W the maximum of the NaO' distribution corresponded to a considerably higher relative c.m. energy W of the products than calculated from the spectator-stripping model. According to this model, W' =kW, where k is a constant. For this system, k = 0.3. The apparatus used for these measurements was similar to that shown in Fig. 1. However, the energy distribution of product ions was measured directly with an electrostatic hemispherical condenser instead of indirectly with a retarding grid. Good resolution was obtained by passing the ions through a decelerating lens system before they entered the hemispherical analyzer. Figure 9 is a schematic diagram of the apparatus. The Na beam was produced by symmetric charge transfer of Na' in a vapor cell. The Na' was formed from source 1 by emission from a heated glass (prepared from a mixture of NaOH, SO,, and A120,) and was subsequently focused and collimated. The 0 ' was produced in an electron

+

INTERACTION ENERGY, W (eW

FIG.7. Q versus W for Na + O2

--f

NaOt + 0.

78

Roy H. Neynaber

w-1/2, (e") - 1 0

FIG.8. Q versus W-'/'for Na

+ 02+-+NaO+ + 0.

bombardment-type ion source 2 using approximately 40-eV electrons. Only single collisions between electrons and 02+ occurred in the source. A fraction (Turner et al., 1968a) of the OZf ions was formed in a metastable electronic state (411,) which has a lifetime longer than the transit time through the apparatus. In order to determine whether the measured Q was for the reaction of Na with ground state or excited state 02+, the fraction of metascurrent) by lowering the energy of tables was lowered (as was the total 02+ bombarding electrons. The Q was negligibly affected, and therefore may be considered as characteristic of the reaction of Na with ground electronic state 02+. The vibrational and rotational states of the Oz+ were unknown. The intensity of the 2810-eV Na beam was determined by measuring the secondary electron current in a Faraday cup and assuming that the secondary electron emission coefficient for neutral atoms is the same as that measured for ions at the same energy. The NaO+ signal was measured with a Bendix Model 306 electron multiplier. The current was calculated assuming that the where gain for NaO' was equal to the gain for Na" plus half the gain for OZt, the Na+ and Oz+gains were each measured at the NaO+ velocity (Kaminsky, 1965). A transverse component of relative velocity introduced a mean relative energy of about 0.03 eV when the beam velocities had the same magnitude. Thus the fractional uncertainty in the energy, A W/W , becomes rather large for small W.

SOURCE 1

SOURCE 2

FIRST COLLIMATING APERTURE

SECOND COLLIMATING APERTURE

DEFLECTION PLATES

\

\

NEUTRAL BEAM MONITOR ELECTROSTATIC DEMERGER

I

MOVABLE

I RETARDING

._ I

I

I I

I

I I I

I

I DECEL-ACCEL I

I LOCK-IN MAGNET

-

PHOTOCELL LAMP

PLATE

I

A

I

I 1

i

ELECTRON MULTIPLIER

I

WHEEL BEAM FLAG

10 cm

/

FOCUSING LENS

FIG.9. Schematic diagram of rnb apparatus for studying Na

/

HEMISPHERICAL ANALYZER

I

~ARADAY CUP

+ 02+ NaO+ + 0. Apertures are not to the scale shown. --f

80

Roy H . Neynaber

IV. Ion-Ion Reactions

A. SYMMETRIC-RESONANCE CHARGE TRANSFER Brouillard and Delfosse (1967, 1968) and Brouillard (1968) used an mb method in conjunction with delayed-coincidence techniques to make a preliminary cross section measurement for the symmetric-resonance charge transfer process He+ + HeZ+-+ He2+ + He+. Their measurements were made at W’s in the kiloelectron volt range and show the usefulness of merging beams for studying reactions at considerably higher energy than in experiments previously discussed. Figure 10 is a schematic diagram of the apparatus. The beam, a composite of He’ and HeZ+,was generated in the PIG source 1 with an energy spread of about 100 eV. It was accelerated at 2 by a voltage in the range 5 to 20 kV,

FIG.10. Schematic diagram of apparatus for studying He+ He2++ H e 2 + He+. ( I ) source; (2) accelerating and focusing region; (3) energy analyzer (electric deflector); (4-7) collimators; (8, 9) current probes and valves; (10-14) basic vacuum (diffusion pumps); (15, 16) high vacuum (sorption); (17) ultra high vacuum (ion pump); (18) ultra high vacuum (titanium sublimation); (19) collision chamber; (20) magnetic shielding; (21) magnetic analyzer; (22) Faraday cups for primary beam currents; (23) “cleaning ” deflectors; (24-25) magnetic quadrupole lenses; (26) phosphor; (27) 30-kV detector electrode.

+

+

EXPERIMENTS WITH MERGING BEAMS

81

focused at 2, and analyzed at 3 by a low-resolution electric deflector. After collimation, it entered the collision chamber 19. The diameter of the beam in this chamber was 1 mm and the current was about lo-' A (He'). The collision chamber was a 5-cm-long cylinder whose potential was negative with a magnitude equal to one-tenth of the accelerating voltage. The pressure in the chamber was lo-'' Torr. Collimators 6 and 7 were chosen so as to accept collisions with a deflection angle of less than about 5 x rad. After approximately 90" deflection in the magnetic analyzer 21, the primary ions were collected on Faraday cups 22. The exchanged ions left the magnetic analyzer through slits, passed through " cleaning" deflectors 23 and quadrupole lenses 24-25, and finally impinged on the negative 30-kV detector electrodes 27. The detectors were photomultipliers whose light input was obtained from scintillating phosphors 26. The phosphors, in turn, were excited by secondary electrons emanating from the detector electrodes. The " cleaning" deflectors swept out ions formed from primary ions whose energies had been slightly degraded by collisions with the residual gas in the region of acceleration after the source. Figure 11 shows ion trajectories in the magnetic analyzer. The source of the composite H e f , He2' beam is labeled S. The collision chamber, at negative potential V , , is shown by the dotted box. The primary beams followed trajectories 3 and 4 in the analyzer and were collected on Faraday cups P+ and P+ +. Ions created by the desired charge transfer process in the collision chamber at A , for example, followed trajectories 1 and 6 and were collected on detectors D and D . Ions created by charge transfer (at B ) or ionization (at C ) through interaction of the primary beam with residual gas followed paths 5 and 2, respectively, and did not reach the detectors. The potential on the collision chamber permitted discrimination between ions created by charge transfer within the chamber and those created outside. It also accurately defined the length of the interaction region. This region was maintained at very high vacuum to reduce noise, i.e., ion current produced from residual gas. Since noise was still larger than the desired signal, delayedcoincidence techniques were used. The desired He' ion triggered the $art input of a time-to-pulse-height converter; the He2+ triggered the stopfIn a subsequent pulse height analysis, the desired signal appeared in well-defined channels, whereas the noise was spread over all channels. The channel address yielded an unambiguous identification of the desired event. Bates and Boyd (1962) and Brouillard (1968) have calculated the cross section for

+

He+

++

+ He2++He2+ +He+

using the impact parameter version of the Coulomb-Born approximation. They neglected momentum transfer effects. Their results, which are naturally

Roy H . Neynaber

82

.'. '.

'\

I

D++ PULSE HEIGHT ANALYZER

t TIME-TO-PULSEHEIGHT CONV. START A

P+

STOP

t

+

FIG.11. Ion trajectories in the magnetic analyzer of the apparatus for studying He+ HeZ++ H e Z + +He+. (S)source at positive potential V,; (dotted box) interaction region at negative potential V,; (+) primary He+ beam; (++) primary Hezf beam; (3, 4) trajectories of undisturbed primary beams; (1, 6) trajectories of HeZ+and He+, respectively, created at A by the desired charge transfer process; (2) typical trajectory of He2 created by ionization of primary He+ in residual gas at C ; ( 5 ) typical trajectory of He+ formed by charge transfer of primary HezC in residual gas; (P++,P+) Faraday cups to collect primary beams; ( D + + , D + ) product ion detectors. The curved, dotted lines associate trajectories in the magnetic analyzer with the process responsible for ion production. +

FIG.12. Q versus 2 W.Circles (solid curve) are predicted values from a semiclassical treatment by Brouillard (1968). Crosses are values from a theory by Bates and Boyd (1962).

EXPERIMENTS WITH MERGING BEAMS

83

in accord, are shown in Fig. 12. Brouillard also calculated the differential cross section for the process. A preliminary, experimental determination of the cross section Q has been made at W = 1.65 keV with the result that Q = 1 x 10-16cm2 (-30%, +SO%). It is noted from Fig. 12 that the theoretical Q is about 5.6 x cm2. This value is outside the limits of error for the experimental Q . The quoted error includes estimates of the uncertainty in the cross sectional area of the primary beam (which is assumed to be uniform in the collision chamber), in the angular acceptance and transmission of the system for the desired species, and in the detector efficiencies. The errors associated with angular acceptance and transmission seem to be the most serious of those included in the estimate. There are plans to modify the apparatus in order to reduce these errors (Brouillard and Delfosse, 1968). Another error, which is not included in the above estimate, arises from the existence of H 2 + in the primary beam. It is not sufficiently resolved from the primary He2+ and would result in an underestimation of Q . This problem is being attacked by using He3 instead of He4 (Brouillard and Delfosse, 1968). Additional results with improved accuracy are expected in the near future from this experiment.

B. MUTUALNEUTRALIZATION The ion-ion mutual neutralization cross section, Q, was measured for N + + 0- + N + 0 by an mb technique (Aberth et a/., 1968). Preliminary studies of similar reactions have also been made by Aberth et al. (1967). The measurements were made over a range of W from 0.1 to 86 eV with an uncertainty in W of 0.1 eV due to transverse velocities. The results represent the first direct cross-section measurements for this type of reaction in which the energy and particle identity are specified. A schematic diagram of the complete apparatus is shown in Fig. 13. The duoplasmatron sources introduced an energy spread in the primary beams (whose laboratory energies were in the kiloelectron volt range) of about 3 eV (Aberth and Peterson, 1967). The beams were superimposed with a merging magnet and then entered the interaction chamber (see Fig. 14). After traveling 30 cm in superposition, the beams were separated by electrostatic deflection. The neutral particles formed by ion-ion neutralization, as well as those formed by electron stripping and capture reactions of the beam ions with the background gas, continued along the superimposed beam direction and were detected by secondary electron emission from a stainless steel surface. The ion-ion neutralization products were separated from those due to beambackground interaction by chopping the primary beams at different frequencies and using a lock-in amplifier to detect the difference frequency.

84

Roy H. Neynaber

FIG.13. Schematic diagram of mb apparatus for studying mutual neutralization.

FIG.14. Schematic diagram of interaction chamber of mb apparatus for studying mutual neutralization.

The secondary electron emission coefficient, y, for the products of the desired reaction had to be obtained for absolute cross section determinations. It was considered equal to the average measured values of the secondary electron emission coefficients of the N + and 0-primaries. The overlap integral in Eq. (8) was obtained by assuming that the flux

85

EXPERIMENTS WITH MERGING BEAMS

density of the broad 0 - beam was uniform everywhere in the interaction volume, i.e., that volume of space swept out by the narrow N + beam. This flux density was determined with the use of the variable irises shown in Fig. 14. Because the flux density was not uniform in the interaction volume, an error was introduced into the results. These results are shown in Fig. 15. Aberth et al. (1968) chose to represent 0.1

1

5

10

INTERACTION ENERGY, W (eV) 20 30 40

80

60

100

+

FIG.15. Mutual neutralization rate measurements vs u and W for N C 0-. The solid curve represents an approximate least-squares fit of the data points to an eleventh degree polynomial. Energies associated with points above 30 eV reflect slight corrections to those in the similar figure of Aberth et al. (1968).

their results by a least-squares fit of the data points to an eleventh degree polynomial. This fit (except for some rapid oscillations, which were deemed insignificant and therefore smoothed out, between about 70 and 100% of the maximum speed at which measurements were made) is shown by the solid curve. The absolute accuracy of this curve is estimated to be & 50 % , except at energies below 0.5 eV where QU(u is the relative speed of the reactants) is rising rapidly with decreasing speed and the uncertainty in u becomes relatively large. Much of the error is attributed to uncertainty in y and the 0- flux density. Not only does the assumption of the uniformity of the flux density introduce an error, but uncertainty in the measurements of the density are introduced by hysteresis in the variable iris mechanism (Aberth and Peterson, 1968). Possible major sources of scatter in the data points are beam alignment and focusing problems. The state of excitation of Nf was not determined. On

86

Roy H . Neynaber

the basis of work done by Turner et al. (1968b), however, Aberth et al. are of the opinion that N + was in the ground state. An excited state of an atomic negative ion has never been observed. The general structure of the curve is similar to that calculated for H + H- neutralization by Bates and Lewis (1955) using a Landau-Zener approximation, and suggests that if this approximation is good then the neutralization cross section is relatively insensitive to the details of the atomic structure. Very recently (summer 1968) Aberth and Peterson (1968) measured the + 0,-in a mutual neutralization cross sections for N2+ + 02- and 0 2 + manner similar to that described above. The neutral species resulting from these reactions were not identified. The results are shown in Figs. 16 and 17. The state of excitation of the primary ions and the effect of excitation on the mutual neutralization rates are unknown. The standard deviations of the data from the plotted curves for N + 0-, N,+ + 02-,and 02+ + 02-are 1.56, 1.66, and 2.72 x lo-' cm3/sec, respectively. The general shape of the curve for N2+ + 02-is like that for N f + 0-. The rather sharp increase in Qo for these two curves is not observed for the curve of 02+ +O,-. This characteristic may be unobservable in the latter curve because of the large scatter of the data. There is also the possibility that vibrational and rotational excitation of the ions would obscure this rise. A

+

+

INTERACTION ENERGY, W (eV)

20

0.1

1

5

10

60

80

I

t 0

40

20

0 .o

0.5

1 .o

1

1.5

1

2 .o

RELATIVE SPEED, v(10

2.5 6

100

I

3.0

3.5

m/KC)

+

FIG. 16. Mutual neutralization rate measurements versus v and W for N2+ 02-. The solid curve represents an approximate least-squares fit of the data points to an eleventh degree polynomial.

87

EXPERIMENTS WITH MERGING BEAMS

20

0.1

1

--

20

10

5

40

I

I 0

0

L

OO

-

-

-

-

-

-

60

3

I

1

1

2

-3

comparison of the N + + 0-and N,' + 0,-curves indicates that the low energy rise is more suppressed for the molecular reaction. An improvement in this mb system, which has been recognized by Aberth and Peterson (1968), would be the adaptation of a collision chamber whose potential could be varied and used to create the desired energy difference between the primary beams. The mutual neutralization experiments are fine examples of the use of merging beams for obtaining measurements that could not be made by other techniques.

C. RADIATIVE RECOMBINATION Wiener and Berry (1 968) have constructed an mb system to be used principally for studying radiative recombination of two ions. The system consists of twin accelerating ion sources and a single analytical magnet (TAISSAM) with approximately 180" focusing for both beams. Initial experiments will be conducted with Li' emitted from a surface ionization source. The source of negative ions is similar to that used by Aberth and Peterson (1967). Figure 18 is a schematic diagram of the apparatus. There are two sets of accelerating

88

Roy H . Neynaber

I M

IwLrr, I

!

W

FIG. 18. Schematic diagram of mb apparatus for studying radiative recombination. (S+, S - ) positive and negative ion sources; ( A + , A _ ) accelerating electrodes; ( E + ,E - )

focusing einzel lenses; ( C + ,C - ) collimating apertures; ( M ) magnet pole pieces; ( D ) detector (movable, shown in position to collect both beams); ( W )window above beam area; ( T + ,T - ) typical positive (Li+) and negative (0-) ion trajectories.

electrodes, focusing lenses, and collimating apertures (one set for each beam). The detector is movable for monitoring and determining characteristics of the ion beams. A large window is provided for optical monitoring, which will be done spectrographically in the initial stages of the experiments. Initial experiments are being conducted to test the utility of optical monitoring rather than to study radiative recombination. To this end, mutual neutralization of Li' and 0- will be studied at relative energies in the center of mass of 1 eV or less. Observations will be made of the emission of the lithium red resonance line at 6707 A which immediately follows the reaction Li+('S) + O - ( 2 P )+ Li(2P) + O(3P).Weiner and Berry (1968) estimate that the cross section for this reaction is reasonably large (at least cm2)on the basis of predictions for well-studied mutual neutralization processes. (The 6707 A line should be considerably stronger than emission from radiative recombination.) Since typical ion velocities are about lo7 cm/sec in the laboratory system, while the lifetime of Li(2P) is less than lo-' sec, most of the emission occurs in the immediate vicinity of the interaction region. The

EXPERIMENTS WITH MERGING BEAMS

89

Li+.O- system is advantageous for testing purposes because of a known, large oscillator strength and a sharp, concentrated, and easily identified spectrum. Li+ + H - has been selected as the first system for which radiative recombination will be studied (again at low relative energy in the center of mass). Since several potential curves of this system are now known, it is feasible to calculate the expected emission spectrum as a function of internuclear distance. Therefore, experimental spectra can be compared with theoretical, at least in the adiabatic approximation. Weiner and Berry (1968) explain that, in essence, the energy-selected pair of ions can recombine and emit radiation at any point along the collision trajectory. To a first approximation, emission from an ion pair at a given internuclear distance R, is restricted to the energy separating the right-hand turning point of the lower state at R, and the relative energy of the colliding pair. (This crude description neglects contributions from the left-hand turning points. These may also be important and account can be taken of them but to do so now merely complicates the present discussion.) In this approximation, each wavelength corresponds to a unique internuclear separation, so that the measured spectral intensities only need to be rescaled in order to give the oscillator strength or transition dipole amplitude as a function of internuclear distance. Thus, the continuous free-bound emission provides one probe for the study of electronic wave functions at large internuclear distances, which are in the range where the system is making its transition between behaving like separated particles and like a molecule. V. Neutral-NeutralReactions Neynaber et al. (1969) have used mb techniques to investigate some twobody collisions in each of which both reactants were neutrals. These reactions were Na + 0,+ N a + + 0,(charge transfer) (a) (ionization) Na + 0,+ Na+ + 0,+ e (b)

+ 0, Na+ + 0 + 0Na + 0, + NaO+ + 0Na + 0, NaO + 0 + e Na

+

(dissociative charge transfer)

(4

(4 (el For ground state reactants and products all of the above reactions are endothermic. The heat of the reaction, AH, is positive for an endothermic process. The values of AH for (a), (b), and (c) are 4.71, 5.14, and 8.78 eV, respectively, assuming the electron affinity of 0,is 0.43 eV (Pack and Phelps, 1966) and of 0 is 1.48 eV (Berry et al., 1965). The magnitude of A H is unknown for reactions (d) and (e). The discussion will be largely limited to the first three reactions with only a few words at the end devoted to the last two. +

+

(rearrangement) (rearrangement).

90

Roy H . Neynaber

A. CHARGE TRANSFER AND IONIZATION 1. Apparatus

To study reactions (a), (b), and (c) a retarding potential curve of product ions was obtained for each of several interaction energies, W. These curves yielded information on the laboratory energy distribution of the product ions and cross sections for the processes. To investigate (a) alone, attempts were made to detect 0,-. Although some measurements were made on 0,-, in general the signal-to-noise ratio was too small (for unknown reasons) to extract good data. Most of the information for (a) and all of it for (b) and (c) was obtained by observing Na', and further discussion will be confined to experiments involving detection of this ion. The apparatus used for these measurements was very much like that emH, --f H3' + H and described previously. ployed in the study of H,' Figure 19 is a schematic diagram of the apparatus. An Na' beam from source 1 was obtained by emission from a heated glass made from Na,O, S O , , and A1,0,, The energy spread of particles in the Na' beam was 1.5 eV or less. The Na' beam was then merged in the merging magnet with ,' at 4000eV from an 0,beam. The 0, beam was obtained by passing 0 source 2, which was a low pressure, oscillating electron bombardment source, through a charge transfer cell containing 0,. An electric field between the condenser plates that follow the cell was used to remove 0,' which did not undergo charge transfer. The energy spread of the 0, 'was I .5 eV or less. The energy of Na' was adjusted to give the desired W. After leaving the merging magnet, the superimposed beams passed through a collimating aperture and then a charge transfer cell containing Na vapor. The temperature of this cell was adjusted so that the vapor pressure of Na was optimum for neutralization of the Na' beam. Under these conditions the 0, beam suffered about 35 % attenuation. The Na vapor had negligible effect on the energy distribution of particles in the 0, beam. This conclusion was reached by measuring the energy distribution of 0,- particles in a beam formed by interaction of the 0, beam with background gas. The energy distribution of 0,-was the same for the Na cell at room temperature and at the optimum temperature for neutralization of Na'. The merged beams then passed through an electric field between a set of condenser plates. Charged particles were eliminated by this field. After passage through a second collimating hole, the resultant neutral beams entered the interaction region. This region was surrounded by a device used in previous ion-molecule experiments (Neynaber and Trujillo, 1968; Neynaber et al., 1967a,b) to establish the desired energy difference between the primary

+

FIG. 19. Schematic diagram of apparatus for studying Na-0, collisions. The collimating apertures are 2.5 mm in diameter.

92

Roy H . Neynaber

beams and to accelerate product ions when they left the region. In the present experiment the device was used in conjunction with the retarding grid in the detector assembly to allow product ions formed inside the interaction region to reach the detector but to prevent those formed outside from doing so. This was accomplished by applying an appropriate potential, P, to the device in order to accelerate ions formed inside the region. The potential of the retarding grid R allowed transmission of these ions but prohibited the passage of slower product ions formed outside the region. Since deexcitation of excited states of Na to the ground state can occur by direct, optically allowed transitions or by cascading, it is assumed that the Na beam in the interaction region consisted of only particles in the ground state. The general conclusions of this experiment, however, are independent of this assumption. In this region the O2 beam presumably contained some excited particles. After leaving the interaction region, the reactants and products passed through a 0.874-cm-diameter hole in the aperture plate. Na’ was separated from the reactants by the demerging magnet. Other undesired particles were prevented from reaching the detector (a Bendix multiplier) by employing the retarding grid and a hemispherical electrostatic energy analyzer tuned for the passage of the Na’. The output of the multiplier was fed into a Cary 31 electrometer and then displayed on a strip chart recorder. The experiment was conducted using dc techniques. 2. Kinematics Kinematics for reaction (a) are shown by the Newton diagram of Fig. 20. In this case, the magnitude of the laboratory velocity of Na, Iv, 1, is less than that of 0 2 ,IvzI. General expressions for the magnitude of the c.m. velocity of Na before the collision, JV11, and that of Na’ after the collision, lV31, are given by Eqs. (10) and (11): IVII = ( 2 W ~ )”z/ml, IV3J= [2p( W - AU - A H ) ] 1 / 2 / m l

(10)

(1 1)

where p is the reduced mass before and after the collision, m, is the mass of sodium, and AU is the internal energy of the products minus that of the reactants. If W ’ is defined as the relative kinetic energy in the center of mass after the collision, then W ’ = W-AU-AH. (12) For the “ after collision ” case, three circles are shown for the loci of the tip of the c.m. velocity of Na’. Therefore, points on a given circle indicate

93

EXPERIMENTS WITH MERGING BEAMS

BEFORE COLLISION

'AU
+

FIG. 20. Newton diagram for N a +Or + N a + 02-. Subscripts 1, 2, 3, and c refer to Na, O r , N a + , and the center of mass, respectively. (v) laboratory velocity; (V) velocity in c.m. system; ( A U ) internal energy of products minus that of reactants; ( v ~minimum ~) laboratory velocity of N a + for AU = 0. (Symbols with arrows in the figure are equivalent to boldface symbols in the text.)

different angular scattering of Na'. Each circle represents one of the three possible cases for the conversion of c.m. energy from internal to translational modes and vice versa. For AU > 0, translational energy of the reactants is converted into internal energy of 0,-. Sufficient energy to excite Na' was never available. When AU = 0, all of the internal energy of O2is converted into internal energy of 0,- or all of the reactants and products are in the ground state. When AU < 0, there is a conversion of internal energy of 0, into translational energy of the products. For the case of AU = 0, two directions are shown for the c.m. velocity of Na'. When the direction is the same as that of the c.m. velocity of Na before the collision, we define the c.m. scattering angle to be zero. Note that the minimum laboratory velocity of Na', vim, exists for this condition. The minimum laboratory velocity of Na' for AU = 0 is less than or greater than that for AU > 0 or
94

Roy H. Neynaber

been a source of noise in the study of the AU = O case (which was important because it included the case when all of the reactants and products were in the ground state), then JvlI had to be less than I v , ~ , where v3 is the laboratory velocity of Na'. Figure 20 and Eqs. (10)-(1 1) show that this was the case since IvlI c lv21 and JV31< IVII. Experiments for which IvlI > lvzl could not be conducted because of the noise introduced by stripped Na. Newton diagrams similar to Fig. 20 can be drawn for reactions (b) and (c). The diameter of a circle is proportional to W"". For a given AU the circle diameters are different for reactions (a), (b), and (c) because the AH'S are different. The diameter for reaction (a) is the largest. Kinematic considerations of reaction (b) show that Eq. (11) gives the magnitude of the c.m. velocity of Na' if the kinetic energy of the electron in the center of mass is considered as a contribution to the internal energy of the products of the reaction. With this interpretation, Eq. (12) gives the relative kinetic energy in the center of mass of the products Na' and 02. For reaction (c) and a given AU, Eq. (1 1) gives only the maximum IV, I, which occurs when the velocities (magnitude and direction) of 0 and 0-are the same. Equation (12) then gives the relative kinetic energy in the center of mass for Na' and the 0, 0-complex. 3. Method

As mentioned previously, the experiment primarily consisted of measuring retarding potential curves of Na'. It should be noted that the retarding grid only retards normal components of the Na' velocity. The desired Na' signal, or current I,, was the detector output with both primary beams on minus the sum of the outputs due to each beam separately. The output from a single beam was similar to ion noise, which has been described previously. Extraneous signals caused by one primary beam modulating (attenuating or enhancing) the noise associated with the other beam were investigated by making P = O . This had the effect of eliminating the real signal without altering the noise significantly. For this case, the detector output with both beams on minus the sum of the outputs for each beam by itself was negligible, indicating the absence of any significant modulation. An absolute cross section could be obtained by a technique discussed previously. This technique included measuring the profiles of each primary beam in the interaction region, the gain of the detector assembly, and determining the secondary electron emission coefficient of the neutral beam monitor for both Na and O2. Typical primary beam currents in the interaction region were 0.5 pA for both Na and 0 2 .

EXPERIMENTS WITH MERGING BEAMS

95

4. Results and Discussion

Figure 21 shows Z3 as a function of R for W = 12 eV. (For this case, the laboratory energy of Na was E, = 2408 eV.) The solid curve (except the kink at 2995 V, which is assumed to exist and will be discussed later) is drawn on the basis of statistical evidence for the existence of inflections. These inflections suggest the existence of discrete energy distributions of Na'. The

3.0

-

v)

f 2.0 a K >

2

cm

K

4

(CI

CI

1 .a

0 RETARDING POTENTIAL, R(W

FIG.21. Nat current (I3) vs R for Na-02 collisions at W = 12 eV (El = 2408 eV; P = 476 V). (E,) laboratory energy of Na; (P)potential of interaction region; (E.) minimum

+

+

) laboratory energy of Na+ from reaction Na O2+ Na+ 02- for AU = 0 ; ( E ~ minimum laboratory energy of Nat from reaction Na O2.+Na+ 0 2 e for AU = 0; (E,) minimum laboratory energy of Nat from reaction Na O2 Nat 0 0-for AU = 0. Each dot represents a value obtained for a single measurement; the dot accompanied by the number 2 means that two measurements resulted in the same quantity. The basis for drawing the solid curve is described in the text.

+

+

+ + --f

+ +

data are not sufficiently good to deduce fine details of these distributions. The inflections together with the data points and confidence limits (not shown) only give some indication of energy limits and integrals of the distributions. The solid curve of Fig. 21, which consists of straight lines, is used to represent this information. The curve is visually drawn consistent with the data and

96

Roy H . Neynaber

with plateaus located in the regions of the inflections. The J3's of the plateaus represent the integrals of the distributions. It should be noted that the resolution of the retarding grid is such that a monochromatic beam will appear to have an energy spread of 4 or 5 eV and peak at about 4 eV less than its actual energy. The arrow labeled E, is at Re = 2942 eV (where e is the magnitude of the electronic charge), which is equal to the laboratory energy of Na' associated with v3"' for reaction (a). We will designate this energy as 8, , the minimum laboratory energy for the case AU = O . The arrows labeled &b and E, are analogous to E , , and they refer to reactions (b) and (c), respectively; &b is at Re = 2948 eV and E, at Re = 301 1 eV. The energy E, is calculated under the assumption that the velocities of 0 and 0 - are the same. It can be shown (Neynaber et al., 1969) that scattering of Na' in the center of mass after the collision was predominantly confined to angles less than 90" and that essentially all scattered Na' was detected. At R values less than 2995 V, there is evidence of two Na' distributions. One of these is between the two plateaus and includes the energy E , . This distribution is very narrow, i.e., a few electron volts or less. An analysis of these data (Neynaber et al., 1969) suggests that this energy distribution is associated solely with reaction (a). In addition, it seems likely that AU is very close to zero (Lee,AU < 0.6 eV), and therefore almost all the excess translational energy of the reactants (i.e., W - AH = 7.29 eV) is converted into translational energy of the products with very little going into internal energy. Finally, because of the very narrow energy distribution about E,, the products in the center of mass must be confined to a very small angle about the axis of the interaction region. In particular, Na' in the center of mass appears to be scattered almost completely in the 0" direction. Apparently the process proceeds via a direct, rather than a complex mechanism. The cross section for reaction (a) leading to the formation of Na' in this narrow energy distribution will be called Q, . Consider, now, the second distribution, which is between the plateau to the right of E, and the kink in the high energy tail. It will be assumed that this distribution is the result of reaction (b), that the scattering of Na' in the center of mass and in the vicinity of the peak of the distribution is at 0", and that AU 2 0. Under these assumptions, the lowest energy in this distribution would be equal to & b . Since measurements with the retarding grid shift an actual energy by an Re z 4 eV to the left in such a curve as Fig. 21, E~ would effectively come very close to or actually merge with the energy distribution associated with reaction (a). Figure 21 invalidates these assumptions since it indicates that the distribution is well removed from that of reaction (a) and that its lowest energy is effectively near 2960 eV. However, compatibility could be achieved within the rather flexible limitations imposed by the data

EXPERIMENTS WITH MERGING BEAMS

97

by departing from straight lines between about 2944 and 2963 eV and drawing an inflection between these energies. The introduction of another inflection point at Re 2 2963 eV would result in a well-defined peak at the point. From Eq. (1 1) as applied to reaction (b), the AU associated with this peak (if it were at Re = 2963 eV) would be about 1.3 eV. This means that at 12 eV, reaction (b) is most probable when 1.3 eV, or 19%, of the excess translational energy [i.e., W - AH = 6.86 eV) is converted into either kinetic energy of the free electron, internal energy of 0,, or both. The significance of such a partition of energy is not known. The high energy tail preceding the kink could be explained by the conversion of translational energy into kinetic energy of the electron and internal energy of 0, (the largest AU for the distribution would be about 3.5 eV) and/or by angular scattering of Na'. The cross section for reaction (b) leading to the formation of Na' in the energy distribution discussed above will be called Qb. There is no evidence for the kink at 2995 V in Fig. 21. Small signal-tonoise ratios in the vicinity of the kink make it extremely difficult to obtain such evidence. The straight lines meeting at the kink could be smoothed to form one continuous curve that would fit the data. This could be interpreted in two ways: (1) as simply a longer high energy tail associated with reaction (b) than that described above; and (2) for R > 2995 V, as adistributionresulting from reaction (c) with the peak of this distribution at about E,. The high energy side of this distribution could be explained by assuming a difference in the velocities of 0 and 0 - . The cross section for reaction (c) leading to the formation of Na' in this distribution will be called Q , . If the second interpretation is assumed, then from Fig. 21, Q, : Qb : Q, E 5 : 3 3 : 1. The retarding potential curves for W = 6 , 8, and 10 eV can be interpreted in a fashion similar to that used for W = 12 eV, and at these energies QJQb 1. At W = 10 eV, there were no observable signals for Re greater than or equal to the associated 8 , . If the high energy tail at W = 12 eV for R > 2995 V results from reaction (c), then at 10 eV the absence of signals for Re 2 E, could be due to very small Q , at such energies. At W = 6 and 8 eV there were no high energy tails analogous to those at 12 eV. At 6 and 8 eV there is insufficient energy for the onset of reaction (c). Therefore, for W = 6 , 8, and 10 eV, it will be assumed that Q, = 0. At W = 6 eV (El = 2540 eV and P = 234 V), the retarding potential curve was taken not only for Re > E, = 2877 eV but also for Re down to 2780 eV. Below E , there were two plateaus other than those associated with processes that have been discussed. These indicate the existence of two other processes. From Fig. 20 and Eq. (1 l), it is concluded that reaction (a) with AU < 0 is responsible for this part of the retarding curve. This means that some 0, is

Roy H . Neynaber

98

excited and that there is a conversion of some of this internal energy into translational energy of the products. For W = 4.71 eV (El = 2579 eV and P = 275 V), which is the threshold for reaction (a), E, = 3025 eV. If, indeed, this is the threshold for the process, then no Na' should be observed for Re 2 3025 eV. The largest R for which there was a nonnegligible Na' current was 2970 V. Two plateaus at smaller R were observed; these are further evidence for excited states of O2 and reactions for which AU < 0. Figure 22 shows I3 vs R for W = 6,8,10, and 12 eV. Only the data for those 1.0

I

I

. @

I

- 2.0

I

I

I

I

I

1

I

I

2

a

013% .5-

0

I

I

1

W = 1 2 eV

W = l O eV

2.0 I3 1.0-

1.00

I

2905

I

I

2915

I

2925 R(V)

2920

2930 R(V)

2940

FIG. 22. Na+ current ( I , ) versus R for W = 6, 8, 10, and 12 eV. Symbols have the same meaning as in Fig. 21. At W = 6, El = 2540 eVand P = 234V; at W = 8 eV, El = 2490 eV a n d P = 343 V; at W = lOeV, El = 2448 eV and P = 416V; at W = 12 eV, El = 2408 eV and P = 416 V.

Re values in the vicinity of E, and the plateau immediately to the left of E, are shown. No plateaus were observed for W = 15 and 25 eV. Figure 23 shows relative Po’sas a function of W where Q , is defined as Q, + Q,, Q,. These were obtained by measuring the 13’s of the appropriate plateaus (see Fig. 22) and normalizing the measurements to the same primary beam currents. The Z,s’ for each W were taken one after another as rapidly as possible. Source conditions (except E l ) for all measurements were kept the same. At W = lOeV, an absolute Q , was calculated as 0.1 A’ with an estimated error of - 50 to + 35 % . The primary beam currents used in this calculation were those measured at the neutral beam monitor.

+

EXPERIMENTS WITH MERGING BEAMS I

I

I

I

1

1

99 I

I

0 X

1.

a

If

0

a

-

4

k

a

v)

z 2 > a a

a

+

I 0..

8

-a a

d

0

”=:

t,

; Q d

1;

1 :

1’2

INTERACTION ENERGY, W (eV)

FIG.23. Relative Qoversus W. Qo is the cross section for the formation of Na+ in Na-02 collisions for AU 2 0. Each dot represents a value obtained for a single measurement. Crosses are arithmetic averages of dots. Qo was chosen as unity for the cross at W = 8 eV.

Relative Q, , Qb, and Q, versus W can be obtained from this figure and the known ratios (at given W’s) Q, : Qb : Q, . At 4.71 eV, Qo = Q, x 0.Since the ratios Q, : Qb : Q, are approximately 1 : 1 : 0 at W = 6, 8, and 10 eV, the relative Q, and Qb are the same as the measured relative Q, . The ratio of Q, at 10 eV to Q, at 12 eV is approximately 1.2. The corresponding ratios of Q,’s and Qb’s are about 1.1 and 1.6, respectively. Crossing of potential curves could contribute to the fall-off of Q, and Q b with increasing W above 8 eV. It is not known whether quantitative agreement with the data could be achieved through such a consideration. The threshold energy in Fig. 23 could not be obtained with less than 0.5 eV uncertainty since the signal-to-noise ratio was so small in the vicinity of

Roy H. Neynaber

100

threshold. As a result, the electron affinity for 0, could not be obtained accurately. As mentioned previously, Qo w 0.1 A’ at W = 10 eV. Therefore, at this same energy Q, w Qb w 0.05 A’. (These cross sections apply, of course, when the specific states of the reactants and their abundances are the same as exist in this experiment.) Since the states of the molecular reactant and product are unknown, the experimental results only shed light on the AU of a process. From the results it appeared that AU = 0 for the process with cross section Q, . This reaction is a charge transfer process, and presumably the FranckCondon principle applies. Since the equilibrium internuclear distance for the ground state of 0,and 0,-are nearly equal (Gilmore, 1965), the conversion of ground state 0,to ground state 0,-has a favorable Franck-Condon factor in addition to satisfying the condition that AU = 0. If it is reasonably assumed that ground state 0,to ground state 0,-was the only significant process occurring, then the cross section would be 0.05A2if all the O2 had been in the ground state. If only a fraction of the 0, had been in the ground state, the cross section would be proportionately larger.

B. REARRANGEMENT Reactions (d) and (e) were studied using the same method and procedures as for reactions (a), (b), and (c) except that the components of the detector assembly were tuned for the passage of NaO+ (by using a K + impurity beam from source 1). A signal attributable to NaO+ was obtained. This signal could have resulted from reaction (d) and/or (e). The first experimental evidence for the existence of NaO+ was recently reported by Rol and Entemann (1968). In the present experiment, the curve of the cross section for the formation of NaO+ as a function of Wrose from zero at about 5.5 eV, reached a maximum at 7eV, and became zero again at about 8.5eV. The cross section at W = l eV was estimated to be about 0.004AZ.

VI. Electron-Ion Reactions A. THEORY Because of the mass difference between an electron and an ion, the mb theory for electron-ion studies warrants special consideration. Subscript 1 will refer to the ion and 2 to the electron. From Eq. (2), it can be shown that for a reasonable percentage uncertainty in the ion laboratory energy (i-e., about 0.1 %) and a reasonable uncertainty in the electron laboratory energy (i.e., a few tenths of an electron volt), an approximate expression

EXPERIMENTS WITH MERGING BEAMS

101

for the fractional uncertainty in W, 6 W/ W,depends only upon the uncertainty in the electron energy and is 6 Wl W x 6E,/(E, W)'I2,

(13)

where 6E, is the uncertainty in E2. Of major importance in merging beams studies of electron-ion systems is the achievement of W considerably less than E, . Having done this, one can use a relatively large E, (and still have a small W) and obtain a concomitantly large electron intensity. Therefore, for these systems a deamplification factor, D,, might reasonably be defined as :

D, = E2/W.

(14)

From Eqs. (13) and (14), the following expression is derived :

6 W/W N D','' 6E,/E2.

(15)

In the following examples it will be assumed that the uncertainty in E, is due to the energy spread which we will choose as 0.3 eV, i.e., 6E2 = k0.15 eV. This relatively large spread is compatible with high beam currents of electrons. Such currents will be required for mb studies. Now, if E2=10eV and W = l eV, then GW/Wx k0.05 and 6 W x +0.05eV. If the ion is N 2 + , El w240keV. If W=O.1 eV, then 6 W/W x 50.15, 6 W x kO.015 eV, and El ~ 4 2 keV. 0 Finally, if E, = 1 eV and W=O.leV, then 6WIWwkO.5, 6 W x f 0 . 0 5 e V , and Elw24keV. Therefore, for best resolution, E2should be relatively large. The resolution is quite good for E2 = 10 eV and rather poor for E2 = 1 eV. An upper limit to to E, would be partially dictated by the high voltage requirements for the associated El. It should be noted that the laboratory velocities of the reactants would be relatively high in an electron-ion, mb experiment compared with those for studies involving two heavy particles. Equation (8) indicates that such high velocities would result in relatively small signals. There would be some mitigation from the rather large flux density that presumably could be achieved for the high velocity ions. B . DISSOCIATIVE RECOMBINATION

Hagen (1967, 1968) has measured cross sections for the dissociativerecombination process N 2 + + e + N* +N** (where the stars refer to either an excited or the ground state of N) in a range of W from 0.1 to several electron volts. The experiment consisted of superimposing a beam of N2+ at 1 keV with a beam of electrons whose energy was varied between about 0.1 eV and about 4 eV. Throughout most of the electron energy range, the

Roy H . Neynaber

102

electron velocity was considerably greater than that of N 2 + and, to a first approximation, N2+could be considered at rest with respect to the electron. Except for the fact that the laboratory velocity of N2+was sufficiently high to permit easy detection of the products, the principles of merging beams were not applied in this experiment. It is discussed because, with a slight modification of the apparatus, a true mb experiment could be conducted. In the experiment, N2+ was generated in a directly heated hot cathode bombardment source (see Fig. 24). The extracted ion current was about

SECTION1

j

SECTION

II

SECTION

IlI

NEUTRAL BEAM DETECTOR' SECTION I ION SOURCE

II lII

MASS F I L T E R INTERACTION REGION

PUMP 1500 I l s e c D P

1000 I I s c c 0 P TITANIUM SUBLIMATION PUMP

PRESSURE

2 a IO-'Torr 2 x 10-6Torr

2

x IO-~TOII

FIG.24. Schematic diagram of apparatus for dissociative recombination experiment.

100 PA. The ions were focused and then passed through an rf quadrupole mass filter. Charge transfer of N 2 + with residual gas resulted in a large fast neutral current which was collected in a titanium sublimation pump. The N2+ beam was bent 45" by an electrostatic analyzer and collimated before it entered the interaction region. Figure 25 is a schematic diagram of the interaction region. It consisted of a hollow, cylindrical permanent magnet. Typical magnetic field lines are shown in the figure. The field was uniform at the center of the region. The

103

EXPERIMENTS WITH MERGING BEAMS ELECTRON REPELLER COLLIMATING APERTURE

\

CUP PROBE

.ELECTRON .LECTOR

P

,EXIT APERl‘U RE

ION BEAM

/

OXIDE CATHODE

I

.- -- ,

POLE. P . lFrF

ALNICO V MAGNET

MAGNETIC FIE1.D IC I IUCS 8 . L

FIG.25. Schematic diagram of interaction region of apparatus for dissociative recombination experiment.

electron emitter was an indirectly heated oxide cathode in the shape of a ring and at a potential equal to the desired electron energy. The electrons followed the magnetic field lines and were collected at the other end of the region. The 1.5-mm-diameter electron beam was contained within the collinear ion beam, which was typically 5 mm in diameter. Beam profiles were measured with a small scanning probe. The electron current was about 1 pA per electron volt of energy with an energy independent spread for the electron beam of about 0.2 eV. The perturbation of the ion beam by the magnetic field was negligible. After leaving the interaction region, the N2+ beam was deflected into a collector. Fast neutrals were detected by an electron multiplier. To discriminate against fast neutrals generated by charge transfer of N2+ with the background gas, the electron beam was modulated. The electron energy was continually swept and data were obtained automatically. The electron energy scale was calibrated by measuring the appearance potential of N, . Figure 26 shows the experimentally measured recombination coefficient, u, as a function of Wand the rms deviations associated with random errors. For any W in the range from 0.1 to 0.6 eV the average u = 2 x cm3/sec within f0.5 x lo-’ cm3/sec. Kasner (196?) measured the same u in an afterglow experiment over a range of temperatures from 205 to 480°K. Over this range, u =(2.7 f 0.3) x lo-’ cm3/sec fits the data. The systematic error for the experiment is estimated by Hagen (1968) as + 10 to - 15% . Included in the systematic error are uncertainties arising from the assumption that the ion flux density in the interaction region was always uniform everywhere within the confines of the electron beam, from

104

Roy H . Neynaber

0

1

2

3

4

5

INTERACTION ENERGY, W (eV)

FIG.26. a. versus W for dissociative recombination of N2+-ecollisions. The solid curve is considered the best visual fit to data points obtained at 0.05-eV intervals over the entire range of the curve. Each data point represented an average of continuous measurements over a 0.025-eV interval. Vertical distances between the solid and dashed lines give rms deviations.

absolute determinations of the desired fast neutral current, and from spiraling of the electrons around the magnetic field lines. The spiraling resulted from components of the thermal contribution of the electron velocity and, to a smaller extent, from components of the electric field each being transverse to the magnetic field at the cathode. The primary result of this was an effective increase in the length of the interaction region and a consequent overestimate of a, because this increase was not considered in its evaluation. The existence of a stronger magnetic field at the cathode than along the axis of the interaction region minimized the effect. As an electron proceeded from the cathode through the magnetic field gradient, the ratio of electron velocity transverse to the magnetic field to velocity parallel to the field decreased. From Section VI,A, it is observed that the Hagen system could be converted to a true mb apparatus by the substitution of a several hundred

EXPERIMENTS WITH MERGING BEAMS

105

kiloelectron volt ion-source power supply for the 1 keV supply. Hagen's 6E2= kO.1 eV. This is smaller than that chosen for the merging beams examples, and would result in smaller 6 W's. As shown by these examples, for the same W, better resolution could be obtained in an mb experiment. In addition, W's less than 0.1 eV would be practical in an mb experiment. As a result, a range of W could be achieved which would overlap the W's of the Kasner and Hagen experiments. VII. Current or Very Recent Studies

To make this review of merging beams experiments as up-to-date as possible, a few brief remarks will be made about some very recent studies. Belyaev et al. (1968) have measured cross sections Q for the symmetric C + C C + and N + N N resonance charge transfer processes C' + N + in a range of W from 7 to 100 eV. They used a method similar to their single source, mb technique described in Section 111,A,2. As expected, for each of the processes a straight line will fit their experimental points on a vs In Wplot. Examples of the Q values for the C + + C reaction are about cm2 and 3.5 x cm2 for W=7 and 100 eV, respectively. For 7.5 x the N + + N reaction, Q z 3 . 6 ~ cm2 and 2.3 x cm2 for W = 9 and 100 eV, respectively. Rol and Entemann (1 969) have studied the ion-molecule reactions Na + N O + --* NaO' + N and Na NO' -+ NaN" + 0 at W' s down to 0.05 eV. The method was similar to that described in Section 111,B,2, except that increased sensitivity of the detector to product ions was obtained by using counting techniques. It appears that only metastable states of NO+ were responsible for the observed NaO+ and NaN'. There were no measurable signals due to these product ions when NO+ was entirely in the ground state. Rol and Entemann have also measured W' as a function of W for the Na +NO+ reactions and find that W' is considerably higher than can be explained, for example, by the spectator-stripping model. The high W' effect is more pronounced for the N a + N O + reactions than for the process -+ NaO+ 0 (see Section III,B,2). Na 0 2 + Rozett (1968) is engaged in developing a merging beams system to study collisions of hydrocarbon ions with free radicals. Neynaber and colleagues are studying the reactions Na 0 +Na+ + 0and Na 0 -+ Na' 0 + e in a manner similar to that described in Section V. The results should be easier to interpret than those for the Na + O2 reactions since there will be fewer excited states of 0 than there were of 0 2 .

+

+

+

+

+

+

+

+

+

-+

106

Roy H . Neynaber

VIII. Concluding Remarks From the foregoing discussion, we have seen that the merging beams technique has been successfully used to study several types of reactions in an energy range heretofore inaccessible. The fact that chemical bond and activation energies are in an energy range in which mb experiments can be conducted should stimulate use of the method for studying collision dynamics. In the opinion of Fite (1968) the method can be uniquely used at low energies to investigate reactions having small rate coefficients. He observes that flowing afterglows, for example, while useful in determining large rate coefficients, cannot be used to determine coefficients smaller than about lo-" cm3/sec. This limitation is the result of competing processes and of impurity effects in flowing afterglow experiments. There are, of course, limitations to the mb method and problems associated with it. Obtaining differential cross sections by measuring laboratory angular distributions would be difficult because the maximum laboratory solid angle for reaction products is relatively small in mb experiments. Identification of the states of the reactants and products poses a problem with mb techniques as well as with many other methods for studying two-body collisions. However, excited species in the primary beam can be partially controlled and their abundance studied through the use of electron-bombardment ion sources in which only single collisions between electrons and molecules occur (see Section III,B,2). Turner et af. (1968b) have developed a method for measuring the abundances of specific states of ions in a beam. This method has been used for several ionic species. The ion beam is attenuated by passing it through a cell of gas; abundances are determined from an analysis of the attenuation as ? function of gas pressure in the cell. It is conceivable that a generalization of this method for other ions and neutrals could be applied in the analysis of excited states of particles in an mb experiment. Additional problems with mb experiments include, for example, those associated with making absolute measurements of fast neutral currents and determining the overlap integral It is hoped that solutions to these and other problems will be found with increasing use of the mb technique. ACKNOWLEDGMENTS I wish to thank those people whose work I discussed in this chapter for their assistance in sending me up-to-date reports of their research. I also want to acknowledge useful discussions with E. A. Entemann, R. K. B. Helbing, P. K. Rol, E. W. Rothe, and S. M. Trujillo. I give special thanks to B. F. Myers who kindly read the manuscript and gave many helpful suggestions during its preparation. My work on merging beams has been supported by the Advanced Research Projects Agency through the Office of Naval Research.

EXPERIMENTS WITH MERGING BEAMS

107

REFERENCES Aberth, W., and Peterson, J. R. (1967). Rev. Sci. Instr. 38, 745. Aberth, W., and Peterson, J. R. (1968). Stanford Res. Inst., Palo Alto Calif., Final Rept. for AFCRL Project No. 5710. Aberth, W., Peterson, J. R., Lorents, D. C., and Cook, C. J. (1967). Intern. Conf Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 162-1 65. Nauka, Leningrad. Aberth, W., Peterson, J. R. ,Lorents, D. C., and Cook, C. J. (1968). Phys. Rev. Left. 20,979. Barnes, A. H. (1930). Phys. Rev. 35, 217. Bates, D. R., and Boyd, A. H. (1962). Proc. Phys. SOC.(London) 80, 1301. Bates, D. R., and Lewis, J. T. (1955). Proc. Phys. SOC. (London) A68, 173. Belyaev, V. A., Brezhnev, B. G., and Erastov, E. M. (1966), Zh. Eksperim. i Teor. Fiz. Pis’ma u Redaktsiyu 3, 321 [JETP Letters (English Transl.) 3, 207 (1966)l. Belyaev, V. A., Brezhnev, B. G., and Erastov, E. M. (1967a). Zh. Eksperim. i Teor. Fiz. 52, I170 [Soviet Phys. JETP (English Transl.) 25, 777 (1967)l. Belyaev, V . A,, Brezhnev, B. G., and Erastov, E. M. (1967b). Intern. Conf. Phys. Electron. At. Collisions, 5rh, Leningrad, 1967, pp. 156-158. Nauka, Leningrad. Belyaev, V . A., Brezhnev, B. G., and Erastov, E. M. (1968). Zh. Eksperim. i Teor. Fiz. 54, 1720 [Soviet Phys. JETP (English Transl.) 27, 924 (1968)l. Berry, H. W. (1948). Phys. Rev. 74, 848. Berry, R. S., Mackie, J. C., and Taylor, R. L. (1965). J. Chem. Phys. 43, 3067. Brouillard, F. (1968). Ph.D. Thesis, Univ. of Louvain, Belgium. Brouillard, F., and Delfosse, J. M. (1967). Intern. Conf. Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 159-162. Nauka, Leningrad. Brouillard, F., and Delfosse, J. M. (1968). Private communication. Carlston, C. E., and Magnuson, G. D. ( I 962). Rev. Sci. Instr. 33, 905. Chupka, W. A., Russell, M. E., and Refaey, K. (1968). J. Chem. Phys. 48, 1518. Cook, C. J., and Ablow, C. M. (1959). Stanford Res. Inst., Palo Alto, Calif., AFCRC-TN59-472. Cook, C. J., and Ablow, C. M. (1960). Natl. Meeting Am. Chem. SOC.,137th, Cleveland, 1960, p. 28R (Abstracts). Cramer, W. H. (1959). J. Chem. Phys. 30, 641. Dalgarno, A., and Yadav, H. N. (1953). Proc. Phys. SOC.(London) A66, 173. Davis, B., and Barnes, A. H. (1929a). Phys. Rev. 34, 152. Davis, B., and Barnes, A. H. (1929b). Phys. Rev. 34, 1229. Davis, B., and Barnes, A. H. ( I 931). Phys. Rev. 37, 1368. Firsov, 0. B. (1951). Zh. Eksperim. i Teor. Fir. 21, 1001; as shown in Fig. 12.5 of Hasted, J. B. (1964); “Physics of Atomic Collisions,” p. 417. Butterworth, London and Washington, D.C. Fite, W. L. (1968). Intern. Assoc. Geomagnetism Aeronomy Symp. Lab. Meas. Aeronomic Interest, Toronto, 1968, pp. 59-82. Fite, W. L., Brackmann, R. T., and Snow, W. R. (1958). Phys. Reo. 112, 1161. Fite, W. L., Stebbings, R. F., Hummer, D. G., and Brackmann, R. T. (1960). Phys. Rev. 119, 663. Fite, W. L., Smith, A. C. H., and Stebbings, R. F. (1962). Proc. Roy. SOC.(London) A268, 527. Ghosh, S . N., and Sheridan, W. F. (1957). Indian J. Phys. 31, 337. Giese, C. F., and Maier, 11, W. B. (1963). J. Chem. Phys. 39, 739. Gilmore, F. (1965). J. Quant. Spectry. Radiative Transfer 5, 369. Gioumousis, G. (1966). Lockheed Res. Lab., Palo Alto, California, Rept. No. 2-12-66-4.

108

Roy H . Neynaber

Gioumousis, G., and Stevenson, D. P. (1958). J . Chem. Phys. 29, 294. Hagen, G. (1967). Intern. Conf. Phys. Electron. At. Collisions, 5th, Leningrad, 1967, pp. 165-1 67. Nauka, Leningrad. Hagen, G. (1968). Private communication. Hasted, J. B. (1951). Proc. Roy. SOC.(London) A205,421. Haugsjaa, P. O., McIlwain, J. F., and Amme, R. C. (1968). J . Chem. Phys. 48, 527. Henglein, A. (1966). Advan. Chem. Ser. 58, 63-79. Ishii, H., and Nakayama, K. (1962). Trans. Natl. Vacuum Symp., 8th, Washington, D.C., 1961, Vol. 1 , p. 519. Pergamon Press, Oxford, 1962. Kaminsky, M. (1965). “Atomic and Ionic Impact Phenomena on Metal Surfaces,” p. 319. Springer, New York. Kasner, W. H. (1967). Phys. Reo. 164, 194. Kushnir, R. M., Polyukh, B. M., and Sena, L. A. (1959). Bull. Acad. Sci. U S S R Phys. Ser. (English Transl.) 23, 995. McClure, G. W. (1966). Phys. Rev. 148,47. Neynaber, R. H. (1968). In ‘‘ Methods of Experimental Physics” (L. Marton, ed.-in-chief), “Atomic and Electron Physics: Atomic Interactions” (B. Bederson, and W. L. Fite, eds.), Vol. VII, Part A, pp. 476-486. Academic Press, New York. Neynaber, R. H., and Trujillo, S. M. (1968). Phys. Rev. 167, 63. Neynaber, R. H., Trujillo, S. M., and Rothe, E. W. (1967a). Phys. Rev. 157, 101. Neynaber, R. H., Trujillo, S . M., and Rothe, E. W. (1967b). Intern. Conf Phys. Electron. At. Collisions, 5rh, Leningrad, 1967, pp. 158-159. Nauka, Leningrad. Neynaber, R. H., Trujillo, S. M., and Myers, B. F. (1969). Phys. Rev. 180, 139. Pack, J. L., and Phelps, A. V. (1966). J . Chem. Phys. 44, 1870. Popescu-Iovitsu, I., and Ionescu-Pallas, N. (1960). Zh. Techn. Fiz. 29, 866 [Soviet Phys.Tech. Phys. (English Transl.) 4, 781 (196O)l. Potter, R. F. (1954).J. Chem. Phys. 22, 974. Rapp, D., and Francis, W. E. (1962). J. Chem. Phys. 37,2631. Reuben, B. G., and Friedman, L. (1962). J. Chem. Phys. 37, 1636. Rol, P. K., and Entemann, E. A. (1968). J . Chem. Phys. 49, 1430. Rol. P. K., and Entemann, E. A. (1969). To be submitted for publication. Rostagni, A. (1934). Z. Physik 88, 55. Rothe, E. W. (1964). J. Vacuum Sci. Techol. 1, 66. Rozett, R. W. (1968). Natl. Meeting Am. Chem. Soc. 156th, Atlantic City, 1968, No. 41 (Abstracts). Smirnov, B. M. (1964). Zh. Eksperim. i Teor. Fiz. 46, 1017 [Sooiet Phys. JETP (English Transl.) 19, 692 (1964)l. Stebbings, R. F., Turner, B. R., and Rutherford, J. A. (1966).J . Geophys. Res. 71,771. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1963). Gen. Dyn. Astronautics, San Diego, California, GDA-63-1296. Trujillo, S . M., Neynaber, R. H., Marino, L. M., and Rothe, E. W. (1965). Intern. Con$ Phys. Electron. At. Collisions, 4th, Quebec, 1965, p. 210. Science Bookcrafters, Hastingson-Hudson, New York. Trujillo, S. M., Neynaber, R. H., and Rothe, E. W. (1966). Rev. Sci. Instr. 37, 1655. Turner, B. R., Rutherford, J. A., and Compton, D. M. J. (1968a). J . Chem. Phys. 48, 1602. Turner, B. R., Mathis, R. F., and Rutherford, J. A. (1968b). Conf. Heavy Particle Collisions, Belfast, 1968, pp. 126-130. Utterback, N. G., and Miller, G. H. (1961). Reo. Sci. Instr. 32, 1101. Webster, H. C. (1930). Proc. Cambridge Phil. SOC.27, 116. Wiener, J., and Berry, R. S. (1968). Private communication. Wolf, F. H. (1968). BUN.Am. Phys. SOC.13, 213.

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS 11: SPECTROSCOPY* H . G. DEHMELT Department of Physics. University of Washington Seattle. Washington

.

3 Manipulation and Investigation of Stored Charge ...................... 3.1 Excitation of Ion Oscillations .................................... 3.2 Interaction with Tuned Circuit .................................. 3.3 Ion Cooling .................................................. 3.4 Counting of Ions after Ejection .................................. 4 . Spectroscopic Experiments Relying on Spin Exchange with

Polarized Atomic Beam ............................................ Basic Detection Scheme ........................................ ,He and H2 Magnetic Resonance .............................. Hyperfine Structure of the 3He+ Ion ............................ Electron Spin Resonance ...................................... 5 . Spectroscopic Experiments Based on Other Collision Reactions .......... 5.1 rf Spectra of H, Aligned by Selective Photodissociation .......... 5.2 Bolometric” Detection of the eCyclotron Resonance. ............. 6. Spectroscopic Line-Shifts and -Broadening ............................ 6.1 Doppler Effects ................................................ 6.2 Electric Field Effects .......................................... 6.3 Magnetic Field Effects .......................................... 6.4 Collision Effects .............................................. 7. Conclusion ...................................................... Errata for Part I .................................................. References ........................................................ 4.1 4.2 4.3 4.4

+

.

+

+

109 109 112 119 124 124 124 127 129 140 142 142 148 149 149 150 150 151 152 153 153

.

3 Manipulation and Investigation of Stored Charge 3.1 EXCITATION OF IONOSCILLATIONS Oscillations of an ion contained in any three-dimensional harmonic potential can be excited by homogeneous electric fields alternating at the ion frequency w. and in addition at one of the harmonics 20.. 3w.. 401.. .... if

* Part I. Sections 1 and 2 of this article appear in Volume 3 of this series. 109

H. G. Dehmelt

110

the field is sufficiently inhomogeneous, as is well known. Oscillations along the x and y axes, of course, can be obtained in an analogous manner. Under

excitation by an homogeneous radio-frequency field at o,giving rise to a force f = f o cos(w, t + p) the oscillation amplitude increases linearly with time,

z

= z1 cos(o,

t

+ a) + [ f o / ( 2 m o z ) ]sin(w, t t + p)

(3.1)

neglecting dephasing effects of any kind. It grows exponentially when a harmonic, e.g., 2w, is used, for the field must at least have a nonvanishing gradient. We show this by obtaining an expression for the power P absorbed by the oscillating ion under the action of such a force of amplitude&,

P = (;if,o sin 201, t),, which, withf,,

=foo

(3.2)

+ z dfzo/dzand z = zo sin o,t, zo cos o,t , reduces to

P = dW,/dt cc k zo2

or

dW,ldt = f W z / t Z w .

(3.3)

Since P turns out to be proportional to the energy of the z motion of the ion W,, the quantity t,, denotes the time constant of the corresponding exponential growth or decay also providing a measure of the strength of excitation. If the cage contains a number of noninteracting ions, their center of mass will carry out the same motion as a single particle for excitation at o,. However, in the 20, case the center of mass of the ion cloud will remain at rest even though after an excitation interval t,,, ,under the conditions underlying (3.3) the energy component W,of an ion will have changed by a factor e",

with X = t e x c / t 2 ,

(3.4)

again neglecting dephasing. This can be understood by realizing that two ions oscillating 180" out of phase inside the cage at o,can both still be in phase with the exciting field at 20,. This kind of excitation can also be explained as parametric. Compare, e.g., Landau and Lifshitz (1960). We now apply (3.4) to an ion gas in thermodynamic equilibrium contained in a rf quadrupole trap when subjected to an intense 2E,-excitation pulse short enough so that dephasing effects such as collisions may be neglected. With Videnoting the =3 Wi . The total final total initial average energy we have for the z motion WZi energy may then be expressed as

-

W' = ( 2

+ $e+X + +e-")

F,

v / 3 N &ex

X B 1,

(3.5)

where the actual random phase case has been crudely approximated. Since this energy multiplication process directly affects only the z motion, one does have to rely on ion-ion collisions for redistribution of the energy over the other degrees of freedom. Parametric excitation at 20, would have

111

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

the advantage of directly multiplying jFVi and the total average energy in all degrees of freedom, Wi, may be multiplied directly by simultaneous axial and radial excitation. For the higher harmonics in the presence of corresponding higher terms in the expansion of the inhomogeneous rf field a procedure quite analogous to the one above may be adopted. Besides these very general modes of excitation, another class specific to the rf quadrupole trap must now be discussed. Here excitation by a homogeneous rf fields can, according to (2.27) in Part I of this article, also be expected to occur at frequencies R f W,,2R Wz,etc., of which the first is of special practical interest. As we will show, the motion of the center of charge (2.26) having its dominant component at W,can be effectively excited by a homogeneous rf field El cos(R k GJr, which is very useful when detecting the stored

+= A (

Excitation of coherent ion oscillation

x2+ y2 - 2 ~ ~ )

To ion charge monitor

Atcinic beom

FIG.4. The rf quadrupole-ion trap showing schematicallyelectricalpotential distribution associated with the trapping voltage Uo Vo cos at. The voltage uo cos wot is induced in the resonant circuit by the cooperative motion of the ions and allows their detection. The ion oscillation is excited by an approximately homogeneous alternatingfield at the side-band frequency wo R ; from Schuessler et al. (1969).

+

+

charge by a circuit tuned to W, (see Fig. 4). In our discussion we use the pseudopotential approach (2.14), (2.23):

H . G . Dehmelt

112

For E, we take E, = (E,Z/z,) cos Rt + El cos(R & G,)t where Em is the amplitude of the field on the electrode at x = y = 0. Averaging over one period 2n R,it follows that

+

+

(Ez2),, = Em’Z’/(2zO2) E I 2 / 2 (22ElEm/zo) x (cos2 Rt cos 0,t T cos Rt sin Rt sin 0,t),,

.

(3.6)

Carrying out the average over the last term and taking sin G,t, cos 0 , t as quasiconstants, gives finally, with (2.29),

Here, the first term is the unchanged restoring force while the appearance of the second term demonstrates that except for the amplitude reduction by G,/(RJ?), this case is completely equivalent to sinusoidal excitation at 0,. 3.2

INTERACTION WITH

TUNEDCIRCUIT

3.2.1 Coherent Signals When a charge moves between arbitrarily shaped grounded electrodes, the distribution of field lines that originate at the charge and end on the various electrodes changes. This must be accompanied by a change in the surface charges on a given electrode at which these field lines end, and consequently result in currents to and from the electrode. Since the number of field lines connecting the charge and a given electrode is not affected by the presence of other charges on the electrode or elsewhere in the system, the displacement current reaching the electrode appears to come from an infinite impedance current source provided the motion of the charge is rigidly maintained. The magnitude of the current reaching an electrode is conveniently evaluated from a formula given in the literature (Shockley, 1938; Sirkis and Holonyak, 1966) i = qvE,

(3.8)

where E, is the component of the electric field in the direction of the velocity v with which the charge q moves that would result if the electrode of interest were placed at unit potential, all other electrodes being grounded. Applying (3.8) to a parallel plate capacitor whose plate 1, located at - z o , is connected by a conductor to plate 2 at +z, , we find for motion parallel to the z direction i = (q/2zO)(dz/dt)

(3.9)

neglecting end effects. Formula (3.8) may be derived with the help of energy conservation arguments which we demonstrate here for the simple configuration described by (3.9). Assume plates 1 and 2 to be connected to the negative

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

113

and positive poles of a battery of voltage V . If the positive charge q now moves inside the capacitor at a speed u in the direction of the positive z axis the field will do work on it at a rate uqV/(2zo).Because of energy conservation the battery must furnish an equal power

iv = uqV/(2z,) involving a positive current i = qu/(2zo)to plate 2 in agreement with (3.9). So far then, the moving charge inside the capacitor represents an infinite impedance current source paralleled by the capacity of the plates C. If the motion of the charge is harmonic at w, of amplitude z o , as will be the case in our applications, and it is desirable to develop as high as possible a voltage across the current source, it is advantageous to tune out C by a suitable inductance L, oZzLC= 1. In steady state situations the current source then simply appears to be paralleled by the shunt impedance of the tuned circuit, R, across which it develops a signal voltage of amplitude uo = Rqzo w,/(2zo)*

(3.10)

Measurements of uo , in conjunction with (3.10), may now be used to determine q. However, if more than one ion is in the cage as in almost all practical situations, it is necessary to bear in mind that the energy in an oscillation of the center of mass of the ion cloud once excited will decay and be degraded to disordered motion approximately exponentially with a characteristic phase memory time td = Q,/w,. This is due to various causes such as imperfections of the trap, ion-ion interactions, collisions with residual gas atoms, etc. In praxis, small-signal relative ion resonance linewidth Aw,/w, = Q;' of about 1 % may be realized, and this value will be assumed in the following. In order to obtain a strong response under these circumstances which reflects as closely as possible the absolute magnitude of q and as little as possible the side effects due to variable line shape, anharmonicity, and charge-dependent deformation of the potential well, etc., it appears advantageous to subject the ion cloud to a step function rf pulse at w, which is intense enough to drive the center of mass of the ion cloud against the electrodes at k z o in a time t, 4 Q,/w,, effecting complete neutralization of the ion cloud. If the ion cloud is concentrated in a region small compared to zo , as we will assume for the time being, the forced ion motion is accompanied by an rf current pulse of triangular envelope, duration t,,, and maximum amplitude qw,/2. The response of the tuned circuit now depends strongly on its Q value, Q , . For Q, 4 t,,w,, the signal voltage u faithfully follows the current pulse assuming a peak value uof = Rqw,/2 from which the absolute value of q now may be obtained with fair accuracy. At the same time it is clear that the bandwidth of the detection apparatus must be made the wider the smaller t, is, admitting more and more thermal noise while the peak signal uof remains constant. Consequently, the

H . G . Dehmelt

114

above procedure need not be very suitable when small changes of q are to be detected. Under the constraints inherent in the experiment, it is most convenient to analyze the crucial optimum signal-to-noise conditions on the basis of an energy approach. A constant resonant drive excites a steady state center of mass oscillation of the ion cloud in anrf trap ofenergy W, 4 neD. Degrading of this oscillation leads to an increase in the incoherent ion energy Wi’ at the rate dWi/dt = W,/td.

(3.1 1)

Defining the neutralization time t,, as the interval in which Wi grows to the value neD and the ions hit the electrodes, it holds that

V,’= neDtd/t,,,

(3.12)

n from here on denoting the total ion number (change from notation used in earlier sections). On the other hand, the coherent ion cloud oscillation induces a tuned circuit oscillation of energy V,given by

w,= ~cnt,o/tzt

(3.13)

where t,, = Q,/G, and tZt stands for the chracteristic single-ion z-motion damping time due to coupling with the LC circuit [cf. (3.19)]. Denoting the signal voltage by S and the rms noise voltage by N, both averaged over the observation internal t,, we have Sz cc W , , and using the longest possible observation interval t, = t,, results in N2 a kTt,,/t,, . It follows then that

(S/N)’

= n2ebtd/(kTt,,)

(3.14)

for the optimum signal-to-noise ratio independent of the t, chosen, provided t,, % t d , f,,. Underlying this formula is the following signal behavior. When the amplitude of the resonant excitation pulse is increased the average signal amplitude S first increases while the optimum observation interval t, = t,, decreases, thus resulting in a larger noise N in such a way that S/N remains constant. Later, when the condition t,, % t d , ttOis beginning to get violated t , approaches the minimum value t,, , the persistence time of the signal in the tuned circuit induced by the ions, and N becomes constant. Simultaneously, S goes through a maximum and then decreases rapidly as the shorter and shorter ion current pulses of constant peak amplitude excite the tuned circuit less and less and S/N deteriorates. Using the expressions (2.24) and (3.19) for e n and t,, , (3.14) may be rewritten (S/N)’

= n2e2QiQJ(8kTC).

(3.15)

It will often be possible to use full traps where, according to (2.39), n cc Dz, . Since it also holds that C a zo , one may now write (3.16)

115

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

which favors deep and, to a lesser degree, large traps. In the actual experiments carried out in the past the ions have not been subjected to a constant frequency-resonant excitation pulse as assumed in the analysis, but rather their resonance frequency 0,has been swept through that of a continuously applied excitation field at oo. Assuming that the traversal is quasistationary and that during its course the ions are driven against the electrodes, we take here for r,,, the distance between half-maximum points of the observed response pattern on a time scale, i.e., the dwell time in the resonance line. This is done more or less automatically by the data processing equipment when the lower part of the rectified response pattern of Fig. 5a together with

2 V/Cm

I

l

l

1

1

1

1

1

1

1

1

1

1

1

(b)

FIG5. (a) Ion number rf signal for = 8 x lo6 3He+ions in the Major trap and (b) thermal noise. Coherent excitation of the ion macromotion at the sideband frequency R wocauses the signal voltage to be induced in the fixed circuit tuned to wo when sweeping the macromotion frequency Gzthrough resonance. Experimental parameters: Vo = 137 peak, Uo= 0 V, i, = 25 mA, re = 20 msec, sensitivity is 2 V/cm, amplification is 2000. For the noise trace the ion macromotion frequency is adjusted to an off-resonance value. Sensitivity: 1 mV/cm, amplification: 2000. The time axis runs from left to right, linear sweep duration 32 msec. From Schuessler ef al. (1969).

+

the noisy base line is chopped off by suitably biasing the diode output and the resulting wave form run into an analog-to-digital converter accepting only a single polarity whose output is stored in a memory.

116

H . G. Dehmelt

The whole question of tuned circuit noise (see Fig. 5b) now denoted by

N,,becomes, important in our experiments only when it is larger than the ion noise Ni associated with the Poisson fluctuations ,/. Good design should aim for

n,

= 8kTC/(e2QiQ,) + n.

in the ion number n. (3.17)

If (3.17) is satisfied, the observed signal-to-noise ratio will be essentially SIN, N

A.

(3.18)

Here Ni refers to the signal fluctuations associated with the Poisson fluctuations expected to occur in the number of ions contained in samples created under identical experimental conditions, and n, is that ion number for which these fluctuations equal those due to thermal noise. 3.2.2 Low-Noise Detection Circuits The implications of formula (3.15) for the design of practical low-noise detection circuits, namely small electrode capacity C and as large as possible Q values for ions and tuned circuit, are clear enough. Here values C = 10 pF, Qi = 100, Q, = 200 may be attainable without undue difficulty yielding unity signal-to-noise ratio for n = 20 ions. To this corresponds the value n, = 400 which should be sufficiently low for many experiments. In addition a number of other problems arise, some of them similar to those encountered in NMR circuits. One of them is the removal of the carrier on which the small ion signal appears when excitation by an rf field of the same frequency as that of the tuned circuit is used. This carrier is often contaminated with noise and hum, and its presence also creates difficulties when small rf ion signals have to be amplified before rectification. A standard way of carrier reduction is by means of bridge circuits (Rettinghaus, 1967), see Fig. 6, which, however, have a tendency to become complex. In the case of ions in an rf quadrupole trap the more elegant procedure of excitation by an rf field at the frequency R k 0,,not W,,as already described in Section 3.1 may be employed, thus completely eliminating any carrier at 8,; see Fig. 5a (Dehmelt, 1968; Schuessler et al., 1969). Other difficulties arise in connection with the application of the trapping rf voltages to the trap electrodes. The high inductance coil connecting the cap electrodes to form the tuned detection circuit acts as a choke to the trapping frequency, and, due to the finite capacity between the caps and the ring electrode usually held at a high rf voltage, causes undesirable voltages at R to appear at the cap. These voltages have been reduced in earlier asymmetric circuits (Major and Dehmelt, 1968) by series resonant shunt circuits which add some undesirable capacity to the detection circuit. The symmetric circuit of Rettinghaus (1967), employing a ferrite core coil of high internal

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

117

Detection clrcuil

1 Amplitude modulolion

FIG.6. Bridge circuit for low-noise detection of stored ionic charge. This circuit employs

excitation of the ion macromotion at the frequency of the center-groundeddetection circuit, w o ;after Rettinghaus (1967).

flux linkage, solves this problem in a more elegant way. However, this coil now

must be able to accommodate the considerable currents at R flowing across the interelectrode capacities. The symmetric arrangement also minimizes the interelectrode capacity appearing across the tuned circuit. Further problems arise in connection with the electron beam used for the ion creation from the residual gas as it may load the R-drive circuit and also cause noise- and dynatron-type excitation of the detection circuit at least during the filling phase of the trapping cycle. The simple asymmetric circuit of Fig. 7 developed at the University of Washington constitutes a workable compromise solution. The side-band excitation voltage at R + coo is applied to a very low impedance circuit resonant at this frequency connecting the cap electrode near the electron gun to ground. The trapping rf voltages at R and the dc bias are applied to the ring electrode. The capacity of the other cap electrode in this configuration to ground is about 36 pF for the Major trap. At the expense of an increase of this capacity by one-third, a low impedance path to ground for the trapping frequency is provided. Thus the effective capacity in the parallel resonant circuit of frequency wo formed with the choke coil mounted immediately adjacent to the trap tube amounts to about 50 pF. The ion signal is

Tuned amplifier 134 kHz

cable '

4- -

CJ

Rectifier

8 11

RF regulator

FIG.7. Asymmetric circuit for the low-noise detection of stored ion charge. The ion macromotion at (3, but excited at the sideband frequency R w o ;from Schuessler et al. (1969).

+

= wo is detected at this frequency

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

119

transmitted to the commercial low-noise amplifier by a second LC circuit, which is also resonant at oo, and is formed by the double-shielded cable of several feet in length and a choke, and has a capacity of about one-tenth the impedance of the elements of the principal circuit. In this way, the electric field energy is concentrated in the trap where the field may interact with the ions, loading of the high impedance circuit by the input impedance of the amplifier is minimized, and undesirable signals at R and R coo are strongly rejected. The necessary dc path from cap to ground is provided by a choke. For this circuit with the values C = 50 pF, Q,= 60, Qi = 100, the value S/N= 8 x lo3 was observed for 8 x lo6 ions in a bandwidth about 10 times as wide as the optimum value (see Fig. 5), which agrees reasonably well with the value S/N= 8 x lo4 given by (3.15). For the critical ion number below which thermal noise dominates the Poisson noise, formula (3.17) gives here n, N lo4.

+

3.3 IONCOOLING 3.3.1 Adiabatic Reduction of Well Depth

The question of the minimum temperature attainable for a small ion sample in a perfect vacuum is closely related to how far geometrical and contactpotential imperfections affecting the trap structure may be reduced in an experimental situation. This has already been touched upon in Sections 2.4 and 2.5. Another approach is to look upon these imperfections as fixed macroscopic soft collision centers which cause heating of the ion gas by the rf drive at R in a similar fashion as considered in Section 2.6. Afurtherimperfection of this type becomes important when the amount of stored charge is large enough to distort the well shape. Specifically, patchiness of the contact potential over the electrodes may be expected to set a lower limit to the well depth D at which practical traps may be operated. Rather than trying to achieve low ion temperatures simply by making D small it becomes of some interest to consider ways by which the energy of an ion gas contained in a relatively deep well may be reduced. Adiabatic reduction of the well depth is one possibility since the quotient WZ/az for a single ion is invariant. This implies that W’/D should also be invariant. As an example we consider a small ion sample stored in a trap of initial depth Di = 100 V whose temperature has been reduced to room temperature by coupling it to a tuned LC circuit (see 3.3.3). When the well depth is now slowly reduced to Df = 1 V the ion temperature should fall to 30°K temporarily. This method is obviously much better adapted to the (U,= 0) rf trap since in order to simultaneously reduce the energy in the x, y , and z motions it suffices to reduce V , , while for the Penning trap both U, and the magnetic field H, have to be reduced in proportion.

-

120

H. G . Dehmelt

3.3.2 Collisional Cooling The initial average energy of ions created in a trap by electron bombardment may be expected to be of the order of the well depth eD. Concentrating on situations in which only a single species is stored and the ion lifetime in the trap T is much longer than t, , the characteristic time for momentum transfer, we expect that considerable cooling of the ion gas should occur due to a process analogous to the evaporation cooling of ordinary liquids. The continuous loss of ions in the energetic tail of the Boltzmann distribution by neutralization at the electrodes has already been touched upon in Section 2.5 as the cause of this process. The equilibrium value to which this evaporation cooling reduces the ion gas temperature T,, may be considerably lower than the trap depth e n in practical cases. For a small trap, 22, = 0.32 cm, containing about 3 x lo3 protons and depth values D = 2.6,4.6, and 10 V it was observed that eD/kT,, N 10 (Church and Dehmelt, 1969). Measurements further showed t , c 300 msec with theoretical estimates from (2.45) as low as 20 msec for D = 2.6 V in the face of experimental lifetimes T = 100 sec. The observation that an ion is able to survive 5000 collisions with its partners provides some justification of our assumptions in Section 2.5 that Q rf heating in the trap due to ion-ion collisions is not very important. Unfortunately, this form of cooling loses its effectiveness very rapidly when e b / k T i becomes too large though it may be possible to counteract this by brief periodic adiabatic reductions of b. Another major weakness of evaporation cooling is the continuous loss of ions. Viscous drag cooling by a very light inert buffer gas briefly mentioned in Section 2.5 does not have this deficiency, but involves, of course, a certain perturbation of the ion state. No temperature measurements on such systems have been reported so far but increases of Hg' ion lifetimes from T =2.5 sec at lo-' Torr to T = 200 sec upon introduction of the ,He buffer gas at pressures around Tom have been observed (Huggett and Menasian, 1965 ; Menasian, 1968). The use of velocity-selected slow beams for viscous drag cooling may result in very low ion gas temperatures, In other favorable situations the atomic beam used for ion polarization may also effect this cooling.

3.3.3 Radiative Cooling and Thermometry Assuming that collisions between the stored ions establish very quickly a thermal distribution of ion energies over all motional degrees of freedom it is also possible to cool the ions by coupling their disordered, e.g., axial motion to a suitably tuned (cold) LC circuit of shunt resistance R which may also act as the thermometer. The following simple model exhibits many of the important features of this process. Consider a harmonically bound ion with chargeq

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

121

and mass m oscillating with frequency w,along thezaxis whichis perpendicular to the faces of an infinite parallel plane capacitor in which it is confined. The capacitor is assumed to have negligible capacitance; its plates, separated by a distance d, are connected together by a large resistor R. The oscillating ion according to (3.9) induces a current i = qu,/d to flow through the resistor, where u, is the instantaneous z component of velocity of the ion. The Joule heating of the resistor exponentially damps the ion’s motion with a time constant t,, = md2/(q2R). (3.19)

This formula follows from the expression for the average loss rate of the energy in the z motion, -dVz/dt

w,.

= (i2R>,, = [q2R/(md2)lm(v,2),, = (l/tzo)

(3.20)

The energy of n harmonically bound ions moving with random phases is damped at the same rate. The effect of replacing the infinite plane capacitor by a quadrupole ion trap, d = 2z,, is to increase t,, somewhat. The condition of negligible capacitance is satisfied when the z oscillation frequency of the ions equals the resonant frequency of the tank circuit. For the purpose of obtaining a measure of the ion temperature, the noise voltage across the tank circuit is amplified, square law detected, and filtered yielding a dc voltage proportional to the noise temperature T, of the tank circuit. When the ions are externally heated or cooled the tank circuit temperature and hence the dc level are changed accordingly. For the purpose of discussing the system in greater detail it is convenient to refer to the block diagram shown in Fig. 8. Grouping together x and y motions of the ion cloud stored in an rf quadrupole trap, a temperature T, and a heat capacity c, is assigned to this perpendicular or p motion and analogous quantities are introduced for the axial or z motion. Applied heoting

1 01

I

wp

Porornetric heating 01 ZW,

Number calibration white mse

Oven

I

p motion

u Noise

thermometer

FIG.8. Block diagram for the analysis of radiative cooling of and temperature measurements on a gas of stored ions. Heat capacities and temperatures for the respective heat reservoirs are listed as well as characteristic times reflecting the coupling between them.

H . G. Dehrnelz

122

The LC circuit is treated as a loss-free harmonic oscillator at temperature T, shunted by a resistance R at the same temperature as the infinite heat reservoir, To. Introducing the three characteristic times t,, , t,, , and t,, reflecting the coupling between the four heat reservoirs, the system may be described by the set of coupled linear differential equations, taking t,, = t,,,

cPdT,/dt = - c,(T, - T,)/t,

+ C p V p - T,)/t,,

- TJ/L k dT,/dt = +nc,(T, -T$)/Z,, - k(T, - T,)/z,, .

c, dTzldt =

- c,(T,

(3.21)

In case of external heat inputs appropriate terms will of course have to be added. The symbols n and k refer to the number of stored ions and Boltzmann's constant. We use these equations here to obtain an expression for q, the relative steady state temperature increase of the tuned circuit which results when the p-motion heat reservoir is forcibly held at a temperature T, # To,

v = (T, - To)/(Tp - To) = (nlncYC1 + (n/nc)l*

(3.22) Here the critical ion number n, and the dimensionless quantities a, /? are defined by (3.23) /? = c, t,,/kt,, . n, = (1 a)/(a/?), c1 = cpt,,/c, f,,, In practically important limit i,, < z,, this simplifies to

+

n, N kt,,/(c, z,,,)

T, N T, N Ti, (3.24) where Ti now stands simply for the ion temperature. For temperature measurements on a cloud containing a reasonably constant number of ions it is most convenient to measure q directly. To this end the ions are brought up to a known high temperature T, by coupling them long enough to the LC circuit to assume its temperature which by excitation with white noise has been raised to T, . Since the ions will retain their temperature long after the noise excitation of the LC circuit has been switched off, and it has cooled down to a value T, near the equilibrium value appropriate for Ti = Th, relations (3.22), (3.24) will then apply. Now n may be determined from q as t,, is simply the characteristic damping time of the LC circuit and t,, may conveniently be obtained by analyzing T, transients following pulse heating of the ion motion. In the already cited experimental work of Church and Dehmelt (1969), values of t,, = 3 sec have been realized. This also made it possible to lower ion temperatures so that values eb/kTi N 30 were reached. The lowest value so far attained in this way was Ti = 900"K, with the LCcircuit at room temperature, but the residual causes of ion heating indicated hereby are not well understood. These values were observed in a background gas of undetermined composition and at the lowest pressure achievable, 3 x lo-" Torr. The lifetime of the cooled ions under these conditions was about 3000 sec and appeared to be limited by ion-molecule reactions. Once this residual ion and

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

123

heating is eliminated the way may be open for cooling stored ions down to liquid helium temperatures simply by cooling the elements of the LC circuit. Our analysis applies essentially also to experiments on electrons stored in a Penning trap (Dehmelt and Walls, 1968), see Fig. 9, where no great experimental difficulties were encountered in bringing down the ion temperature to that of the tuned circuit at room temperature.

Vorion 12” electromcqnet

-

L

‘ Q

Aquadag electrostatic shield

r ,

Electron gun Pyrex

’

-

Recorder

I cm

TOW

detector

miw

pussbond 1-23 kHz OSCll lotor

FIG.9. Apparatus for temperature measurements on a gas of radiatively cooled electrons contained in a Penning trap; from Dehmelt and Walls (1968).

3.3.4 Incoherent Signal-to-Noise Ratio

As in the case of coherent detection discussed earlier it is of crucial importance to analyze the optimum signal-to-noise ratio available in these experiments. We do this for the steady state case in which the ions are held at the practical maximum temperature 3kT, = e B (see Section 2.5). Assuming that the noise voltage of the tuned circuit is amplified by a noiseless amplifier and then rectified by a square-law detector the output will be proportional to the tuned circuit noise temperature T , . The rms fluctuations in the detector output when averaged over the measurement time t,,, will be, taking the proportionality constant as unity, (3.25) In the limit n 4 n, it will be T, 1: To, and the signal S = T , - To will be small and given by kS N neBtf0/(3t,,), (3.26)

H. G. Dehmelt

124 yielding

(S/N)’

N

n2(efjlkT0)’t,,,rto/(9t2J.

(3.27)

This may be compared with the corresponding expression for coherent excitation given earlier in (3.14) to form

(S/N)t,/(S/N>L,N ( e W h ) t mtt0/(9td tzt). (3.28) It is very amusing to point out that this quotient may easily approach unity or even exceed it since it usually holds that elfilkT, N 100, f r o 2! fd and while t , certainly cannot exceed the ion lifetime in the trap, it may in favorable cases approach tzt.Also, when the ion temperature is kept low for the purpose of reducing ion loss, 3kT,/eb -4 1, the corresponding loss in S/Nmay be recouped by increasing f,,, . This mode of operation allows essentially non-destructive observation on the ion gas. The recent experiments on protons by Church and Dehmelt (1969) seem to prove this. For a discussion of an experiment in which the temperature of a stored electron gas is continuously monitored, see Section 4.2.2. Since it holds that T i - T o = S/q, (3.27) also determines the accuracy with which measurements of the ion temperature may be carried out. 3.4 COUNTING OF IONS AFTER EJECTION A very direct way of counting the stored ions is by electron multiplier after they have been ejected from the trap. Ejection through a perforated cap electrode by quickly pulsing it to a negative voltage exceeding the well depth has been used by Dawson and Whetten (1968). These authors claim a collection efficiency of 10% of the ions stored in the trap. Earlier a procedure selectively ejecting only a single species had been developed by Jefferts (1968). Here, the center of the storage well is slightly displaced toward one of the cap electrodes which is built from mesh. A short rf pulse of the appropriate frequency may now be used to excite a large amplitude oscillation of the ionic species of interest and drive them through the mesh electrode whereafter a simple ion optic focuses them on the multiplier (see Fig. 24). As evident from formula (3.17) ejection and counting of all the ions has decided noise advantages when a very small number of ions, n 5 lo3, is to be detected. This type of counting procedure is basically destructive, however, while the inducedcurrent approach may be made nondestructive (see Section 3.3).

4. Spectroscopic Experiments Relying on Spin Exchange with Polarized Atomic Beam 4.1 BASICDETECTION SCHEME The basic detection scheme (Dehmelt and Major, 1962; Major and Dehmelt, 1968) will now be described in its original form developed for the rf

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

125

spectroscopy of He’ ions. Briefly, it relies on orienting a sample of He’ ions, contained in an rf trap (Fischer, 1959; Wuerker et al., 1959), by spin-exchange collisions with a beam of optically pumped Cs atoms and subsequently monitoring changes in the He+ orientation induced by an hfs transition through its effect on the rate of He’ neutralization. The latter depends on simultaneous spin-dependent He+-Cs charge-exchange interactions, and is reflected in the residual number of ions still present at the end of the interaction cycle. In more detail the He’ ions were created in the rf trap from the background gas by electron pulses of 80 msec duration and a repetition rate of 1 sec-’ which was fast compared to the ion lifetime of 2 8 sec in the trap. The stored ions were continuously subjected to a beam of Cs atoms which had been polarized by optical pumping to an electron polarizationP = (s,)/s = 0.5. The resulting collisions caused the stored ions to quickly (in -100 msec) assume a polarization p equal to that of the beam by the reaction

(4.1) and also to be lost from the trap by various processes with a characteristic time of 400 msec. The principal spin-dependent loss process is assumed to be the charge-exchange reaction Cst+He+&-+Csl+He+T

C s + H e + + C s + +He*+AE. (4.2) The predominant channels of this reaction are assumed to be leading to the excited He* levels 2s ‘ S and 2p 3P. The corresponding energy defects are quite small, AEl = 0.1 eV and AE, = -0.25 eV. Approximate values for the corresponding cross sections may be obtained from curves published by Rapp and Francis (1962); see Fig. 10. At an ion energy of 4 eV, they amount to cm2 and QY N 0.3 x cm’. A lower limit for Q,, the QY N 6 x spin-exchange cross section, may also be obtained from these curves by taking the charge-exchange cross section for AE = 0, Q, 2 170 x cm2. In an approximation we consider in the following the ions to be instantaneously polarized by spin exchange and slowly analyzed by the charge-exchange interaction. A charge-exchange collision of a He’ ion with a Cs atom in the m, = + 3 state may be considered as an “observation” of the He+-ion spin state just as if an atom were traversing a Stern-Gerlach magnet, which can only give the result m, = + 3 or -3 for the atom. The angular splittingof the atomic beam in the Stern-Gerlach experiment is here replaced by a differencein the ion loss rate for ions with spins parallel and antiparallel to that of the atoms. To relate these ion loss rates, which may be denoted vp and v, , respectively, to the cross sections Q, and Q 3 introduced earlier, we note first that parallel spins correspond to a pure triplet state, while the wave function for the antiparallel case must be expanded, according to

t.l = Htl + tl) + 3(tl - tt),

(4.3)

126

H . G . Dehmelt

-E

12-

01

lo3

Cs + He+ +Cs+

I

2

+ He*,

I

I

4

7 I$

Q(tJ)

1

-

Q,/2;

1

2

I

I

4 7 ’01 v (crnkec)

Q(ft)= Q3

1

-

1

I

2

4

0, $He+)

I

J

7 loe

-

2 x 10“ cm/sec

FIG.10. Theoretical curves of Rapp and Francis (1962) for charge-exchange cross section Q as a function of the velocity in near-resonant collisions involving Cs, with the energy defect A E as the single essential parameter, and energy levels important in the Cs He+ chargeexchange reaction.

+

into equal parts of triplet and singlet states. Assuming nearly identical translational orbitals for these states, it must follow that vp ~c Q 3

and

Va

( ~ ( Q I+

Q3).

(4.4)

Also, for an ion sample of n ions, a description in terms of spin-state populations is readily obtained for ions having a spin polarization p interacting with atoms of polarization P # p , on recalling that the fractions of the ions having m, = ++ and -4, which may be denoted n , / n and n-In, respectively, are given by

n, =(Nl + A n ,

n- = W ( l

-An.

(4.5)

Similar equations hold for N + and N - of the N atoms in the trap. Now the rate equations for these population numbers are given by

Nn,

=

-n, N , vp - n , N - v,,

Nn- N - vp = -n- - n- N , v,. (4.6)

127

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

Rewriting the population numbers in terms of the spin polarizations p and P, one finds, on adding the two equations,

- dn/dr = (n/To)Cl -pP(AQ/Q)/41

(4.7)

where

T1= (Nv,+ v,).

e

=

(NQ, + 3Q3h

and

AQ = Q,- Q 3 .

This analysis remains valid if Q , and Q3 are interpreted to refer to ion loss from all experimentally important processes, not only charge exchange. The polarization signal S ( p P ) may now be defined as =

Cn(pP) - n(O)I/n(O),

(4.8)

where nbP) and n(0) refer to the number of ions remaining in the trap after an interaction time t i . One may now adjust tito optimize the signal-to-noise ratio, assuming the Poisson fluctuations n1/’(0)of the ion number to be the only important noise source. Maximizing S(pP)n’/’(O) as a function of time yields for the optimum signal S * ( p P ) S*(PP) = (+lpP(AQ/Q),

S* G 1

(4.9)

for t i = 2T0. This result is a very useful relationship which is already applicable to such cases of interest as transitions between the two magnetic substates of ,He+ and the double-quantum transition between the (1, + 1) and (1, - 1) hfs levels of the ,He+ ion. In such experiments, when using intense rf, one obtains p = 0 and the vanishing of the signal. Without rf, a signal S* corresponding to p = P is observed. (See the center resonance of Fig. 16.) 4.2 ,He+

AND

H,+ MAGNETIC RESONANCE

Polarization and magnetic resonance of ,He+ ions were first demonstrated with the vertically mounted glass apparatus housing the trap structure and the Cs beam shown in Fig. 11 (Dehmelt and Major, 1962). The sealed-off system was continuously evacuated by two 15-liter/sec Vacion pumps. The intense Cs beam emanated from a refluxing oven. The beam was polarized by irradiating it with circularly polarized resonance radiation emitted by homemade Cs lamps. The lamps were filled with 1 Torr of argon, contained a few grams of Cs in the seal-off tips at the bottom, and were constant-current excited from a 500-W dc-input oscillator. The whole apparatus was immersed in a magnetic field created by a 150-cm-diam Helmholtz pair providing afield of 5.5 G parallel with the axis of the trap. Dimensions and operating parameters of the trap in this experiment have been discussed in Section 2.8. After an interaction period with the polarized Cs beam of 0.8 sec the residual ions were counted in the manner described in Section 3.2.1 using excitation at wo = 6,.

H . G. Dehmelt

128

Cold trop

rf

auadruDole

Electron gun

I

Cesium resonance

polarizers

lamps and r f excitation coils

Ceslum oven-

To He quartz leak and vaclon pumps

0 10 cm bl-duUJ

FIG.11. Basic setup used in the experimental demonstration of He+ orientation and magnetic resonance showing sealed-off glass apparatus housing ion trap and Cs atomic beam. Also indicated is the optical pumping arrangement for polarizing the Cs atoms. In the hfs experiment on 3He+ the glass apparatus was rotated 45" around its axis; from Dehmelt and Major (1962).

The polarization of the optically pumped Cs beam could be destroyed at will by magnetic resonance disorientation. For this purpose, a coil wound around the tube immediately above the pumping region supplied a rf field at 1.93 MHz. This feature made it easy to experimentally obtain the ion polarization signal S * simply by destroying the beam polarization in alternate interaction

129

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

cycles and comparing the respective residual ion numbers. Numerical values S * = 0.04 were measured by direct observation on the oscilloscope in the face of statistical fluctuations in the counting signal amounting to 0.01. The test pattern shown in Fig. 12, which was obtained by summing up about 40 n

1 I I ! I I I' I ! ! ! 1 ! ! ! ' ! ! i

Pumping light onloff

I

Hcfield on/offl Cs - resononce

On

On

I I

Off

I On I

On

On

[Off

I

On

I I

Off

On

Time

-

FIG.12. He+ polarization as demonstrated by pulse difference integrator output as function of Cs spin polarization. The Cs polarization was on-off modulated by a resonant rf field applied in the beam region during alternate interaction cycles. The essential element of the integrator was a RC circuit with a 100-sec time constant into which alternate signals monitoring the residual ion charge were fed with opposite polarity. A division corresponds to about 30 sec; from Major and Dehmelt (1968). N

individual S* signals by means of an RC circuit, established the absence of spurious effects. The experimental S* signal obtained in the above manner could further be destroyed by applying a rf field at 15.44 MHz in the trap region perpendicular to Ho and the trap axis which destroyed the 4He+ polarization by magnetic resonance. An analogous magnetic resonance experiment has also been carried out on the rotational ground state K = 0, I = 0 of H2+ which is sufficiently similar to ,He+ so that the same apparatus could be used though with greatly improved data handling procedures (Schuessler, 1968). 4.3 HYPERFINE STRUCTURE OF

THE

,He+

ION

The hfs levels of ,He+ in a magnetic field are split according to the BreitRabi formula (Fig. 13), and the following transitions between sublevels are possible :

H.G . Dehmelt

130

I

E

FIG.13. Breit-Rabi diagram of 3He+;from Schuessler et al. (1969).

(a) (b) (c) (d) (e) (f)

00-11 00010 00-1 - 1 11-10 11 er 1 - 1 10-1 - 1

at v, = Av - v, + 6, at v,, = Av 26, at v c = Av v, + 6, at vd = v, 6, at v, = v, (double quantum), at vf = v, - 6.

+ + +

Here Av zz 8665.649 MHz is the absolute value of the zero-field hfs separation, v, is defined by (4.10) 2hv, = 19, + g.rl Po Ho 9

and 6 may be expressed as

6 = V,(V,/AV)C~

- (2gr/g.r)12.

(4.1 1)

All listed transitions were observed in a magnetic field Ho corresponding to V, =

10,010.8 f 0.7 kHz

(about 7.23 G) and

6 = 11,510 HZ which was stabilized by a potassium optical pumping magnetometer. While the field-dependent transitions showed linewidths 6 v of about 2 kHz, due to residual field inhomogeneities and hum, it was possible to reduce that of the only weakly field-dependent b transition to as low as 10 Hz.

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

131

In general, when discussing how transitions between the hfs levels of an ion with nonvanishing nuclear spin affect the polarization signal, it is clearly sufficient, within the approximations of the present treatment, to consider only how these transitions affect the electron spin polarization p . This quantity may be expressed in terms of the population numbers nu,np ,n, , nd , adding up to n for the hfs states of the ,He' ion, a = (1, +I), p = (1, 0), y = (1, - l), 6 = (0, 0), and the electron spin polarization values, for which a look at the expanded wave functions shows p.=+l,

ps=o,

&=-I,

pa=o,

(4.12)

leading to

P

=

+ ~ np p

+ P, n, + pa n a b = (nu - n,)/n.

(4.13)

In the absence of rf transitions and on the basis of the assumed ordering of the cross sections, neglecting at this point the relatively slow ion loss processes, the following rate equations may be obtained for the population numbers : ria = rip =

n y= rid =

-2(1 - P)n, + (1 + P ) n p (1 - P)n, +

(1 - P ) n ,

+ (1 + P h , - 3 9 + (1 + P)n, +n6, (1 -P)rtp-2(1 + P ) n , + ( I -P)n.j, + n p + (1 +P)n,+ -3na.

(4.14)

This result may be arrived at in an elementary way (allowing some insight into the spin-exchange process) which will be outlined in the following. During the spin-exchange collision in which electronic angular momentum is transferred to the ,He+ ions, the nuclei are not immediately affected. Nevertheless, in the long collision-free interval afterwards, the hfs precession in the ,He+ ion does transfer angular momentum from the electron to the ,He nucleus. Since the collision frequency T i 1 of a He' ion with Cs atoms is largely determined by the spin-independent polarization forces, we schematize the collision process as follows. At average intervals T p , the exchange interaction between pairs of fixed He' ions and Cs atoms is switched on for a statisticiilly varying time, long compared with the spin-exchange period. During the interaction period, the ,He+ nuclear wave functions remain unchanged, while the electron spin wave functions evolve in time because they are a superposition of two stationary states, singlet and triplet, with different energies (Purcell and Field, 1956). Thus, for example, an initial wave function ' P i K tfl must be written

yi ~ 2 t t J + C(tJ +JT)

+ e'*(tJ- J t ) L = o t.

where Q=

-(E3

- El)t/h.

(4.15)

132

H . G . Dehmelt

In the term 2 f f l the first two arrows refer to the electronic spins of Cs and 3He+, while the third denotes the nuclear spin of the He ion. An analogous position convention is used in the following. After a time t this becomes, bracketing out the Cs electron spin function,

Y f fC2I-l + lt(1 + e'")l

+ 1m1 - e'">l.

(4.16)

Transforming back to the coupled representation for the He ion, one finds Y, cc ~.[Jz(l - e'">al

+ f[(3 + e")j + (1 - ei")61.

(4.17)

The effect of the hfs interaction which we now switch on is to multiply 6 in quite analogous to the procedure leading to (4.15). this expression by ei4(hfs), However, this phase factor has no effect on the average populations in the states a, p, 6 which we obtain by averaging the squares of the absolute values of the expansion coefficients in (4.17) over CP. We see that as a result of the collision, an ion initially in the ffi state has the relative probabilities 1/4, 5/8, and 1/8 of appearing afterwards in the c(, p, 6 , states, respectively. An analogous procedure is applicable to the other ,He+-ion hfs states and the rates at which ions in an ensemble are scattered in and out of the hfs levels for a general Cs spin polarization P can be written down using the (+)(1 + P ) and (+)(1 - P ) weighing factors for the two Cs electron spin states. In this way the system of Eq. (4.14) may be constructed. It should be noted that, as defined, T, is the mean time between collisions which randomize 0 and for which the probability of spin exchange is only 3. Since Qe is defined in terms of a mutual spin flip, the corresponding T, must be related to T, by T, = 2T,. For simplicity of appearance, Eqs. (4.14) have been written with the time measured in units of 4Te. To demonstrate the usefulness of these equations, we derive the relative signal strength for saturated rf transitions between the hfs levels, e.g., ci and p, under slow passage conditions. Since S a p and p = P in the absence of rf transitions, we define the corresponding relative depolarization signal as (4.18) SLs 1 - (Paa/P)= [n(PP) - n(p,jjP)]/[n(PP)- n(O)]. The transition-reduced ion polarization pas is found from Eq. (4.13) and the solution of the rate equations (4.14)for A,, Ad under the conditions n, = ns and A? = rid = 0. The numerical results for the signals SAP for the limiting cases P e 1 and P = 1 are listed in Table 111, and formula (4.19) is applicable for general P . The equilibrium value p = P postulated earlier also follows easily from (4.14). According to Table 111 it is necessary to take special steps in order to make the system sensitive to the b-transition of primary experimental interest. This is possible by simultaneously saturating the a and c transitions. It is clear that additional application of an intense rf field saturating the b-transition will lead to a reduced ion polarization pabc= 0. We obtain a

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

133

TABLE I11 RELATIVESIGNAL STRENGTH s' OF SATURATED rf TRANSITIONS BETWEEN hfs LEVELS OF 3He+ a Transition

Pg 1

P= 1

(1, +1)-(1, 0) ( I , + l t ( l , -1) (1, O)-(l, -1) ( I , +1)-(0, 0) (1, 0)40, 0) (1, -1)-(0, 0)

0.4 1.0 0.4

417 1.o

0.4 0.0

0.0 4/'7

0.4

0.0 0.0

From Schuessler et a/. (1969).

measure of the expected effect of interest by evaluating the corresponding quantity pOc= 3 for P = 1. This shows that the signal due to the b transition alone may be made as large as one-third of that due to the saturated e transition which exhibits the maximum signal obtainable. In order to avoid frequency-pulling and line-broadening effects believed to affect the b transition in this basic scheme it was modified into the pulse scheme sketched in Fig. 14. (F. rn,)

lr

180' pulse d tronsition

(0,O)

bl Ic

t Field; independent

b tronsition

I I

H, = Constant

Tb 2 T>,>

Td and T, ( Not to %ole

1

FIG. 14. Pulse scheme developed for the purpose of making the ( I , 0)- (0, 0) transition in -,He+ directly observable. The quantities HI,p, and T,are, respectively, the amplitude of the rf field, the ion polarization, and the average time for spin exchange with the polarized Cs beam; from Schuessler et al. (1969).

134

H . G. Dehmelt

The time Tb during which the ions were only excited by an rf field near the b transition under investigation was chosen nearly equal to the spin-exchange time T, introduced earlier. In sufficient approximation the average ion polarization indicated in the figure may now be used in the theoretical expression for the signal S(pP). For the experiments on ,He+, the old 4He+ apparatus was modified in the following ways. In order to feed microwave energy for the IAFI = 1 transitions into the trap a A/2 slot antenna was cut radially into one of the cap electrodes. This antenna was excited by means of a dielectrically loaded wave guide butting against it via a reentrant glass tube. Trap dimensions had been chosen thus that the TE013 mode providing an axial rf field had the appropriate frequency. This field was able to induce both nand CT transitions since the trap axis also had been rotated 45"against the H, field, which was increased to 7.23 G and stabilized by an optically pumped potassium magnetometer (Bloom, 1962). The sensitivity of ,He+ to depolarizing charge-exchange collisions with the parent gas necessitated operation at a lower pressure and the increase of the pumping capacity to 400 liters/sec. This in turn forced an increase in the intensity of the ionizing electron beam by an order of magnitude. Somewhat later the signal-to-noise ratio of the ion-counting signal was appreciably improved by incorporating excitation at w, R instead of at w, ,as described in Section 3.2. Essential experimental parameters are listed in Table IV. In order to carry out the various processes required for the observation of the hfs spectrum according to the preceding analysis experimentally, it became necessary to develop a rather complex machinery, especially since the maximum experimental signal S * obtainable (for the e transition) turned out to be only 0.02 in the presence of background fluctuations of about 0.005 and the weak signals corresponding to the various rf transitions had to be extracted from the background fluctuations. A block diagram of the complete experimental setup is shown in Fig. 15. Its most important features only will be touched upon here. The sequencer provided the basic 1-sec cycle and controlled the various pulses necessary for the handling of the stored ions. The ion-counting signal was transformed into a pulse train by an analog-todigital converter and fed into a multichannel memory whose address system was advanced by one channel after each basic cycle and recycled after a predetermined number of channels. By obtaining the rf power for the hfs transitions from a frequency synthesizer (0-50 MHz) containing a voltage-tunable oscillator controlled by a voltage derived from the address system, it was possible to traverse the transition many times and thereby improve the signalto-noise ratio. While it was feasible to obtain the weak power sufficient for the narrow b transition by direct multiplication from the synthesizer output, a phase-locked klystron operating in the 8-GHz range had to be employed for the a and c transitions. The auxiliary circuit simply had the purpose of locking

+

10

m 4

I36

H. G.Dehmelt TABLE IV EXPERIMENTAL PARAMETERS IN 3He+ hfs EXPERIMENT' Axial dimension Radial dimension dc bias ac trapping amplitude Frequency of trapping rf Detection frequency Axial oscillation frequency Axial well depth Radial well depth Stored charge (3He+) Electron current Electron acceleration voltage Electron pulse duration Coherent excitation amplitude Duration of excitation pulse Excitation frequency Detection sweep amplitude Duration of traversal through 8, resonance Quality factor of detection circuit Quality factor of stored ion cloud Ion lifetime 3He density Cs density at site of trap Ultimate vacuum Ratio of ion number signal to total rms fluctuations with Cs beam present at the end of interaction interval (0.8 sec) and comparing consecutive signals Ratio of ion number signal to total fluctuations without Cs beam Ratio of ion number signal to thermal noise

z o = 25.5 mm

ro = 36.0 mm

u,=ov

Vo = 137 V peak !2=27rx 1 MHz W O = 277 x 134 kHZ G Z = 2 x 126kHz D,= 6.1 V Dr = 3.1 V q = 8 x 106e i, = 10 mA ue=500v 1. = 80 msec Vn+- = 3 v Tn+o= 40 msec ~ 0 + ! 2 = 21134kHz ~ U d = O to -3.6 V t,,,= 4 msec

Q,= 50 100 T = 25 sec n, = 2 x lo8 atoms/cm3 N = 1 x 10' atoms/cm3 p = 1 x 1O-lOTorr

Qi =

From Schuessler et al. (1969).

the klystron to the ultrastable vb frequency 10 MHz. The basic frequency of the synthesizer was provided by a high-quality quartz oscillator whose frequency was continuously compared with WWVB. The low-frequency d, e, and f transitions first seen unresolved by Major (1962) were relatively easily observable. In Fig. 16 is shown a digital-analyzer display of the rf-saturated AF = 0 spectrum of ,He+ obtained by Schuessler et al. (1969), with the Cs optically pumped into the m, = -4 state resulting

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

10.013. I

10,002.3

137

9990.8

kHz -Y

FIG. 16. Digital analyzer display of low-frequency 3He+ spectrum in a field of 7.23 G obtained in a 2-hour run. The amplitude of the rf field for this recording was continuously increased to such a value that no further changes in the heights of the d, e, or f peaks occurred; from Schuessler et a/. (1969).

in a depolarization signal S' greater for the lower frequency (1, - 1)-(1, 0) transition than the (1, O)-(l, + 1) transition. The relatively narrow line in the middle is the double-quantum transition (1, - 1)-(1, + 1). The magnetic field had the value H = 7.23 G, for which the spectrum was centered about 10.00 MHz, and the Paschen-Back-Goudsmit shift amounted to 6 = 1 1.5 kHz. The frequency increases from left to right, and the quantity displayed (plotted downward) is simply n (0.8 sec), the number of stored ions remaining after the interaction interval, averaged over 80 traversals of the spectrum. From these data, and with the help of the expression

P = (2 - 5S/')/(2 - 2S/'),

(4.19)

the valuelPl = IpI = 0.5 may be derived for the Cs and ,He+ polarizations. Analysis of the experimental signal S * = 0.02 on the basis of the formula for S * ( p P ) and the theoretical values of the cross sections for charge exchange show that other ion loss processes were about 10 times more effective than the latter. The field-dependent a- and c-microwave transitions were observed with intensities comparable to those of the d and f transitions as expected. A

H.G. Dehmelt

138

display of the much weaker b transition, first investigated by Fortson et al. (1966) with lower resolution, obtained in an 8-hour run by Schuessler et al. (1969), is presented in Fig. 17. Here the individual data points have been (1.0)4 0 . 0 )

I

FIG.17. Digital analyzer display of the field-independent b transition in 3He+ obtained in an 8-hour run. For clarity the individual data points have been arbitrarily connected by straight lines; from Schuessler et al. (1969).

\

8,665.672:887Hz - Y

arbitrarily connected by straight lines. A study of the theoretical line shape which is expected to exhibit weak side bands because of the pulse scheme employed is still in progress. During all runs undertaken for the determination of the hfs separation Av, vb, and v, were measured simultaneously. Data collected in eight runs are given in Table V together with Av values derived from them by means of formula (4.1 1) for 6. An error larger than twice the maximum deviation has been assigned to the data to allow for possible systematic errors. The final result is AvlS = 8,665,649,867 f 10 Hz. The b-transition is expected to be shifted principally by the second-order Doppler effect, spin exchange with the Cs atoms, and the Stark effect (Schwartz, 1959; Fortson et al., 1964). The following expressions may be obtained for the relative absolute values of these shifts, all being negative, 50.6eD/mc2,

5flQg/'

5(1 5a04Vo2)/(Z6e2z02).(4.20)

and

The corresponding numerical values for the present experiment are - 0.7 x - 3 x 10-13, and - 1 x lo-'' which indicates that at present only the second-order Doppler shift requires further consideration. Magnetic fields arising from the motion of the ions through the electric fields in the trap also need not be considered at this stage. Using a somewhat lower estimate for the average ion energy than assumed above, a correction of + 4 Hz has been applied to the data. The experimental values for the hfs separation AveXpmay first be used to obtain a value for the hfs anomaly which is primarily caused by nuclear structure effects. Here 6 is defined by (4.21) hveXplAvF= 1 6.

+

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

139

TABLE V

SUMMARYOF 3He+hfs €ZI?SIJLTS"*~ Field-independent transition vb: (O,O)-(l, 0) 8 8 8 8 8 8 8 8

a

665 665 665 665 665 665 665 665

612 612 612 612

Line width

881,3 881,2 886,5 886,l

612 881,8

612 884,O 612 883,9 612 880,O

22 28 14 14 10 11 13 10

Double quantum transition v.: (1, l)ti(l, -1)

hfs-splitting

Av

10 011 43 10 010 93 10 010 77 10 010 61 10 010 48 10 010 44 10 010 28 10 010 17 Weightedmeanvalue number

8 8 8 8 8 8 8 8 8

861 864 864 865 861 863 863 860 665 649 863

665 665 665 665 665 665 665 665

649 649 649 649 649 649 649 649

From Schuessler et ul. (1969). Frequencies in Hz.

Considering nucleus and electron as point dipoles it holds that

+

+

+

AvF = (y)Z3azcR, n-3(pN/po)[1 (rn/M)]-3[l (a/27c) 0.328(a2/n2)] x [1 ~(n)(za)']{ 1 - a'Z [($) - In 23) [I r(a3 ;n, Z) r(a2rn/M; n, z ) ] , with B(2) = 17/8. B(1) = 3/2,

+

+

+

In this expression the five brackets reflect, respectively, the reduced mass correction, the anomalous electron magnetic moment correction, the relativistic Breit correction, the radiative correction of Kroll and Pollock, and finally n-dependent higher radiative correction of order a3 and a2m/M.Comparing the numerical value obtained with only the first four corrections

Av,

= 8,667,513,000 &

50,000

HZ

with the experimental value yields

6 = -186 k 6 ppm, in agreement with the value obtained by Novick and Commins (1958) from their experimental result for Avzs. That the nuclear structure corrections are identical in the 1s and 2s states is a consequence of a general property of the radial electronic wave functions R,,(r). In the limit r - 0 the Schroedinger equation h2d2(rR,,)/dr2 - 2rn[(Zez/r)+ E,,]rRn= 0 (4.22)

H . G. Dehmelt

140 reduces to

dR,/dr = ZR,/ao,

(4.23)

Now the Fermi contact term is proportional to the average of Rn2(r)over the nucleus. Expanding this quantity around r = 0 yields, with the above result, Rn2(r)= Rn2(0)(1

+ 2Zr/uo +

*

*).

(4.24)

This shows that the ratio Avls/Av2s should be independent of nuclear structure effects within error limits of order (rnuc,eus/u0)2. This makes it possible to use this experimental ratio to check out the (n-dependent) radiative corrections mentioned earlier and the underlying quantum electrodynamics. Defining

+

~ A v ~ ~ =/ 1A v R~ ~

the latest theoretical value (Zwanziger, 1961 ; Sternheim, 1963) is R = 137.21 f 0.03 ppm of which R, = 133.12 ppm is due to the relatively well-established relativistic Breit correction. Combining the Novick and Commins experimental results with the above yields R = 137.33 & 0.19 ppm where the error is mostly due to that in the experimental AvZs value. Subtracting R , from both values yields the theoretical value for the remaining (now purely quantum electrodynamic) corrections Rqed = 4.09 & 0.03 ppm and the corresponding experimental value Rqed = 4.21 f 0.19 ppm which agree within less than 3%.

4.4 ELECTRON SPIN RESONANCE Graeff et al. (1968) have carried out a spin-resonance experiment' on elmtrons stored in a Penning trap (see Section 2.2) with D, = 15 V, ro = 0.9 cm and employing a magnetic field Ho N 2 kG. Their apparatus, which uses a Na beam polarized to P = 0.93 by a 6-pole-magnet state selector, is shown in Fig. 18. Similarly, as in the He' experiment described in Section 4.1.2, the electrons are polarized quickly by spin exchange with the alkali atoms. However, for the analysis of changes in the electron polarization p a different process is relied upon. Cooling of the electron cloud by the spin-dependent excitation process (Salmona, 1965) e + Na(3S) -+ e Na*(3P) - AE, AE = 2.1 eV, (4.25)

+

' This technique has now also been applied to a direcf determination ofg-2; see Graeff e t a / . (1969).

141

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS Beam

Zero field coil

Electron trap Hot wire detector

Homogeneous mognet

FIG.18. Schematic picture of the atomic-beam apparatus used to polarize trapped electrons; from Graeff et al. (1968).

is used here. Just as the neutralization rate for unpolarized reaction partners in the He' experiment is multiplied by the factor

(Q - PP AQ/4)/Q

(4.26)

when the partners are polarized [see formula (4.7)] thus the excitation and, consequently, the cooling rate associated with the process (4.25) is, for polarized partners, multiplied with the same factor (4.26), the symbols now of course referring to cross sections for excitation. Experimentally, spindependent changes in the cooling rates were observed as follows. The trap, initially held at D, = 15 V, is filled by a short pulse of a 1-pA electron beam with electrons of a presumable initial average energy 5 15 eV. The electrons are allowed to interact with the beam for a cooling period of 1 sec, during which thermalization due to e-e collisions according to formula (2.42), which should be applicable in a rough approximation, may be appreciable since T,* N 1 msec. Assuming the latter, an electron temperature T, may be introduced which drops appreciably during the interaction period. The trap depth is then reduced for 0.5 sec to a lower value, presumably D,' N 4 V which, according to Section 3.3.1, should reduce the electron temperature by less than 20% since only the z motion is affected. How many of the electrons will now be lost in this period will be determined by the size of the Boltzmann tail with electron energies exceeding eD,';cf. formula (2.48). Clearly the number of electrons remaining will reflect the electron temperature T,, and thereby the polarization p . The electrons were counted by means of the procedure outlined in Section 3.2.1 relying on excitation at 0,.A signal reflecting changes induced in p by spin resonance is shown in Fig. 19. It has also been proposed to exploit the spin dependence of elastic collisions between exchange-polarized slow electrons and Na atoms for the analysis of the polarization of the former. However, subsequent experiments in which the

H . G . Dehmelt

142 loo

I-----+----( b )

p\

i

Cyclotron line

\,

r \

\

I)

1

/

\

7' Spin - resonance line

5453.43

0.47

051

5459.74 078

C 32

Frequency (MHz)

FIG.19. Variation of electron-detection signal height as a function of applied microwave frequency showing the cyclotron- and spin-resonance lines of the trapped electrons; from Graeff et al. (1968).

electrons were to be boiled out of a flat Penning trap, D, 5 0.1 V, in a spindependent way by rf heating of the type described in Section 2.6 caused by elastic collisions with the atoms in an optically pumped Na beam were not successful, presumably because of fast spin relaxation due to Majorana flops induced by the cyclotron motion through the insufficiently homogeneous and too-low magnetic field, H , N 50 G, then employed (Dehmelt, 1961).

5. Spectroscopic Experiments Based on Other Collision Reactions 5.1 rf

SPECTRA OF

Hz+ ALIGNEDBY

SELECTIVE PHOTODISSOCIATION

Jefferts (1962) has carried out an experiment in which H2+ ions created from H2 by electron impact and stored in an rf quadrupole trap (Paul et al., 1958) were dissociated by bombarding them with linear polarized photons, the survivors becoming aligned. The rate for the reaction hv

+ H,+ -+ H + H + + kinetic energy

(5-1)

is porportional to 60s' (E,R) where E and R are vectors referring to the electric light vector and the (fixed) molecular axis, respectively. This follows from the fact that the photodissociation arises from an electric dipole transi-

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

143

tion between the 'C, ground state I/I, and the 'Xu first excited repulsive state @u (see Fig. 20). The superposition of the two states occurring during the transition

+

Y ( t )= gI/19 uI/Iuexp( - i AEt/h)

(5.2)

cm

FIG.20. Energy levels of the H2-H2+ system. The vibrational levels are unlabeled but to scale; from Jefferts (1968).

corresponds to an electron cloud pulsating along the @xed) molecular axis whose associated electric dipole moment can only interact with an electric field having a component in this direction (Mulliken, 1939). It is now convenient to represent the broad spectrum of light used for the photodissociation by pulses of radiation much shorter than vibrational or rotational periods.

144

H . G . Dehmelt

When taking into account the molecular rotation one will then expect that for large rotational quantum numbers K, the dissociation rate R will vary with sin2(K, E). The actual molecule will be primarily in states with small K values, and one also has to take into account electronic and nuclear angular momenta which couple with the rotational angular momentum. The resulting hfs states with total angular momentum F, on the basis of the vector modes depicted in Fig. 21, may be labeled ( K , F 2 , F, MF1, and the corresponding dissociation

FIG.21. Vector model for H2+. The strong magnetic electron nuclear interaction couples the total proton angular spin momentum Z and the electron spin angular momentum S to form the resultant F2 which, in turn, by a much weaker magnetic interaction, couples with the rotational angular momentum K to give the total angular momentum F; from Dehmelt and Jefferts (1962).

rates may be expressed as the quantum mechanical averages of sin’ (K, E) with E taken parallel to the z axis A look at Table VI, which gives R values obtained in this fashion, shows that, nevertheless, appreciable structure remains in the dissociation rates. Intense rf transitions between any two sublevels s, t will equalize the populations in these levels on the average and cause the sum of the initial populations a, + a, to decay with the average dissociation rate (R, + R,)/2while without rf the populations in each of the two levels would decay at their respective (different) rates. In this way, the residual ion number remaining after a fixed interaction period is made to reflect the action of the rf transitions (Dehmelt and Jefferts, 1962) quite analogous to the method described in Section 4.1.

145

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

TABLE VI PHOTODISSOCIATION RATESR OF hfs SUBLEVELS OF THE THREE LOWEST ROTATIONAL STATES MOLECULAR IONFOR ELECTRIC LIGHTVECTOR PARALLEL TO STATIC MAGNETIC

OF THE H2

+

FIELD".^

K

2

F2

F

gF

IMI

R

100a4

-215

312 112

0.650 1.350

7.43 0.45

+215

512 312 1/2

0.500 1.100 1.400

13.50 1.23 0.37

+546

+22/45

312 112

1.200 0.800

0.82 4.08

+544

+ 1019

112

1.000

1.83

+378

+2/5

512 312 1/2

0.750 1.050 1.200

4.98 1.50 0.82

-898

-219

312 112

0.750 1.250

4.98 0.67

-973

-219

1/2

1.000

1.83

+2

1/2

1.000

1.83

WF(Mc/sec)

112

--

0

Hfs energies W , ,g factors gF, and residual population numbers a4 after irradiation for four dissociation time constants are listed also. Initial populations a,, have been taken equal to unity. This table is from Dehmelt and Jefferts (1962).

An important difference is that the polarized light alone only causes an align-

ment of the sample with equal populations in levels with f MF. The basic apparatus used in the first experiments is shown in Fig. 22. Operating parameters of the trap and detection electronics are similar to those used in the ,He+ experiments. As a measure of the alignment building up during the interaction cycle it was, however, most advantageous to count the H + ions formed during a short period at the end of it. These early experiments established the workability of the new technique by the observation of the strong magnetic resonance at lgFl N 2/5 shown in Fig. 23, which is due to a number of hfs states; see Table VI. Magnetic resonances for all states listed in Table VI, except for the unobservable ones with F = .), were later investigated by Richardson et al. (1968). Analysis of the g,-factor ratios obtained in this work

H . G. Dehmelt

146

Elliptical mirror Mercury arc lamp

Brewster's ongle polorizer

ode surfaces of tube rotational symmetry

Electron gun structu Cathode and heater Glass-to-metal sea

Steady magnetic field H, directed into page

Direction of r f mag field

FIG.22. Diagram of trap and optics. Basic arrangement with which first experiments on alignment and magnetic resonance of H,+ were performed. The trap electrodes were formed by metallizing appropriately shaped regions of the Pyrex interaction tube; from Jefferts (1 962).

67647Hz

61069 HZ

FIG. 23. Unresolved magnetic resonance signal of the ( K F 2 F ) states (1, 3/2, 5/2), (2, 1/2, 3/2), (2, 1/2, 5/2), all having lgFI 2 2/5, of the H2+ ion in a field HO u 115 mG.A point-by-point digital analyzer display obtained by Richardson (unpublished) in 11 hours is shown.

147

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

for the (K, F,, FI states ( 1 , 1/2, 3/21, ( 1 , 312, 3/21, and (1, 312, 5/21 yielded the first value for the electron spin-rotational coupling constant in H,’, d = +32.2 f 5

MHz

since the corresponding interaction causes appreciable mixing of the first two states with F = 3. Somewhat later the much more accurate, but only, absolute values

I~(u

=5

f 1)1 = 30,239 f 1 kHz,

Id(u = 6 f 1)1= 28,092 f 1 kHz

were measured (Jefferts, 1968), u referring to the vibrational quantum numbers, by direct observation of transitions in the 70-MHz region between the levels F = 3 and F = 3 of the K = 2, I = 0 state., In this work the greatly refined apparatus shown in Fig. 24 was used. Theoretical estimates of the coupling constant ranged from d = - 150 MHz (Dalgarno et al., 1960) to d = + 122 MHz (Burke, 1960). blarlzed Light beom%

E

1

U FIG.24. Sophisticated apparatus used in experiments on the spin-rotational fine structure of H,+. Note selective ejection and counting by electron multiplier of H+ and H2+; from Jefferts (1968).

Most of the hfs spectrum for the k = 1 state, which extends into the 20 cm region, has now been observed; Jefferts (to be published).

H. G. Dehmelt

148 5.2

"

BOLOMETRIC " DETECTION OF THE e-CYCLOTRON RESONANCE

Even in a perfect vacuum, one may expect that the energy absorbed by an electron gas from the rf field when the cyclotron resonance is excited will quickly lead to a detectable. increase of its translational temperature via e-e collisions. Formula (2.40) should be valid for electrons in a magnetic field as long as the cyclotron radius is appreciably larger than the average distance between electrons. Monitoring the electron temperature by the techniques described in Section 3.3.3 now provides a convenient " bolometric " method (Dehmelt and Walls, 1968) for the detection of the cyclotron resonance of the electrons. The sensitivity of this method may be judged by a look at Fig. 25,

-

Frequency

-

Frequency

FIG.25. (a) Temperature increase of the electron gas as the exciting frequency is swept through the cyclotron resonance. The sidebands separated by multiples of w,,, , are caused by the magnetron motion through the inhomogeneous microwave field. The temperature scale is correct to approximately 50°K.The slight slope of the base line is due to losses from the initially injected electron sample during the duration of the sweep; which was approximately 4 min. (b) The cyclotron line with (1) and without (2) parametric postmultiplication of the electron-gas temperature. Off-resonance points of (1) and (2) refer to room temperature. The data were obtained with the apparatus shown in Fig. 9 containing about los electrons. From Dehmelt and Walls (1968).

which demonstrates that cylclotron resonance excitation resulting in a temperature increase of only a few degrees Kelvin are easily detectable. The technique may even be capable of detecting the spin resonance of the electrons provided a sufficient relaxation linkage between the cyclotron motion and the spins can be established. Also, in other ion gases favorable relaxation conditions may make rotational, electronic, and hfs spectra observable. It is also possible to employ cyclotron heating of such intensity that the electrons are boiled out of the trap and to detect the resonance by counting the reduced

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

149

electron number (see Fig. 19). Here relativjstic shifts can be kept small by using very shallow traps (Dehmelt, 1961) but may become appreciable in traps several volts deep (Graeff et al., 1968).

6. Spectroscopic Line-Shifts and -Broadening The general situation is in many respects similar to that encountered in the hydrogen maser (Kleppner et al., 1962). However, specific effects arise from the periodic character of the ion motion in the trap and the inevitable electric trapping fields. The discussion of most of these effects may be kept very brief since so far they have been too small for experimental observation. Numerical examples of the more important ones have already been given in Section 4.1.3.

EFFECTS 6.1 DOPPLER Dicke (1953) has shown that a large fraction of the energy radiated from an oscillator, whose motion may be arbitrarily fast, appears in the form of a spectral line of natural width and unshifted frequency as soon as the motion is restricted to a region of dimensions 544.The symbol 1 here refers to the wavelength(s) of the radiation field (free space, cavity) with which the oscilIator is interacting. Similar considerations underly the Mossbauer effect. A simple way of understanding the above result is available by considering an oscillator which is executing an oscillatory motion around the maximum of a standing-wave pattern of a radiation field from which it is absorbing energy. If the above condition holds the moving oscillator will see a radiation which is weakly amplitude modulated at twice the motional frequency and therefore contains, besides the strong, unshifted carrier, weak side bands at twice the oscillatory motional frequency and its harmonics. Consequently, as long as the rf field is essentially constant over the ion sample, the dominant, unshifted carrier will cause an easily identifiable signal. In the presence of nonvanishing gradients, side bands due to z and r motions and also the Larmor precession of the ion orbits in the magnetic field will be expected and have been observed (Schuessler et al., 1969). Unlike the first-order Doppler shift the relativistic second-order Doppler shift does not vanish. The shift is negative, and it holds with formulas (2.18) and (2.19) and the assumption 3kTi 5 e D introduced in Section 2.5 16v/vl = KE/(rnc2) = 3kTi/(mcZ)c eD/(mcZ). (6.1)

In the ,He+ hfs experiment 6v from this cause has been estimated as - 4 Hz, and it is the most important shift. If the ion orbits with their spread in kinetic energies were stable for indefinite periods, one would expect that this shift would be accompanied by a broadening of comparable magnitude. Various

150

H . G. Dehmelt

randomizing effects which are fast compared to the spin-exchange time T, should, however, greatly reduce the broadening through averaging of the kinetic energy. Formula (6.1) clearly indicates the advantage to be gained from the use of low temperatures and heavy ions. 6.2 ELECTRIC FIELDEFFECTS The electric fields used in the trapping of the ions will, of course, always cause a Stark effect and limit the usefulness of the method to ions not exhibiting strong first- or second-order Stark shifts. This is not too serious as by far the most ions only show small second-order effects. Most favorable in this respect appears to be the situation found for the hfs spectra of single-electron atoms in S states. Fortson et al. (1964) have measured for the 0-0 transition of the hydrogen grounds state hfs

6 v / v = -3.4 x ~ O - ~ ~ E [E]~ =, V/cm, (6.2) while for the analogous transition in such a highly polarizable atom as Cs Haun and Zacharias (1957) found only

6v/v = -2.5 x 10-16E2, [El = V/cm. (6.3) A theoretical formula which agrees with the measurements on H and which may be generalized to hydrogenic systems has been given by Schwartz (1959), 6 v / v = -(14.91 ao4)/(Z6e2)E2 [cgs], (6.4) showing a strong inverse dependence on the ionic charge. For the 3He+ hfs this shift is negligible. The electric fields in the trap are, by virtue of the ion motion, also the source of small magnetic fields of a magnitude which, in cgs units, is reduced by the factor v/c. In past experiments this quotient did not In electric fields of the order 1 [cgs] or 300V/cm thiscorresponds exceed to motional magnetic fields of order lo-' G. This is not a negligible value when systems with g factors as large as 2.8 MHz/G are involved. Analogous magnetic ac fields with frequencies equal to the various combinations of multiples of O,,W,,and R might become especially annoying if they coincide with an internal ion frequency. For low energy ions concentrated in the fieldfree center of the trap all trapping field effects are, of course, correspondingly reduced.

FIELDEFFECTS 6.3 MAGNETIC The rf spectroscopy experiments of the University of Washington group have been troubled by problems due to 60 Hz magnetic hum fields of 2.0.5 mG since it was impractical to provide better shielding in these early experiments. Because of the resulting frequency modulation the field-sensitive a, c,

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

151

d, e, andftransitions of 3He+ showed a large number of side bands spaced at 60 Hz and spread over a region =2 kHz wide, the swing of the frequency modulation. Other small-field fluctuations sufficed to wash this pattern out, making it impossible to resolve the individual sidebands. For the chosen value v, = 10 MHz this corresponds to a relative broadening of these lines, 6v,/ v, N 2 x The displacement of the 0-0 transition, 26 N 23 kHz, depends quadratically on v,. This transition would therefore be expected to show a corresponding modulation swing of 4 x or 9 Hz; however, since this value is much less than 60 Hz the sidebands will be weak, and no broadening of the carrier will result from this cause. Because of the quadratic dependence of 6 on the magnetic field, ac fields may, however, cause noticeable dc shifts. In this connection the circular magnetic rf fields associated with the displacement currents from the electric trapping fields at R may become important. In the experiments referred to these were less than 2 mG which under the worst possible circumstances H , = 0, would have led to an average value A6 5 Hz. These rf fields may also cause internal ion transitions when resonant with them. In contrast to hum fields, static inhomogeneities of the magnetic field are much less serious since the ion motion tends to average them out. However, Majorana flops occurring in inhomogeneous magnetic fields when one of the motional frequencies coincides with an internal ion frequency may cause lifetime broadening of the states involved. These flops, which are simply due to the fact that the ion in its own reference frame may now see magnetic rf fields at all of the motional frequencies and various combinations thereof, may even be turned to good use if it is desirable to establish relaxation channels. The usefulness of this procedure may beextended by applying inhomogeneous magnetic rf fields making it possible to induce the flops at various sum and difference frequencies (cf. Section 4.2.2). 6.4 COLLISION EFFECTS All collisions with other atomic systems are potentially capable of shifting and broadening the rf resonance lines of the hyperfine spectrum of the ions under study. The broadening may either be caused by the modification of the hfs interaction during the collision or by collision-induced relaxation effects shortening the lifetime of states involved. We now will briefly discuss in order of decreasing importance some such collision processes, beginning with two resonant ones. Charge exchange with the parent atom He+ + H e + H e + H e +

(6.5)

may terminate the existence of a given hfs state by exchanging the nucleus of a ,He+ ion for a ,He nucleus in another spin state and thereby cause lifetime broadening. It is the process which makes it more difficult to orient ,He+ ions

H , G. Dehmelt

152

than 4He+ ions. To the extent that one may assume the same nucleus will not be involved in two consecutive such collisions without having its spin state thoroughly disoriented in the meantime, no shift will result. The velocity dependence of the cross section is shown in Fig. 10. For low velocities the collision frequency tends to a finite value. The spin-exchange process He+?

+ Csl + H e t l +

Cst

(6.6) is also resonant. It causes lifetime broadening through termination of the electron spin state of the ion (Wittke and Dicke, 1956). Spin-exchange shifts may be estimated as follows. Assuming unpolarized Cs atoms and ,He+ ions, the collisions result in the formation of short-lived complexes of which threefourths are in the triplet state and one-fourth in the singlet state. The hfs associated with the ,He nucleus in the triplet molecular complex is essentially the same as in the free ion while this hfs vanishes for the singlet state. Consequently, the ,He+ ions moving through a Cs beam of particle density will have their hfs turned off for a small fraction, of the time. For f one takes 4/3xr&, the fractional spin-exchange volume, where rSEis one-fourth of defined by nr& = ZQ, . The fractional spin-exchange shift of the hfs separation Av is then approximately given by the quantity$ The above effects may also be caused by other paramagnetic atoms and molecules present in the residual gas such as 0 2 .Spin exchange between ions will vanish as the energy is reduced because of the ion-ion repulsion. With respect to the use of inert buffer gases, cf. Section 3.3.2, collisions of the type

-

(6.7) are also of interest. Experimental evidence shows that the spin states of S-state paramagnetic ions are only moderately more sensitive to such collisions than paramagnetic atoms (Ackermann et al., 1967). Hg+T+He+Hgt&+He

7. Conclusion In the preceding chapters, we have seen that it is possible to carry out highresolution spectroscopy of ions by storage-collision techniques. The resolution achieved for the hydrogen atom by very specialized techniques has so far not been attained in ion experiments. However, it appears that high-resolution spectroscopy of stored ions as a field is only in its infancy. The technique is potentially applicable to all, ground state or metastable, atomic and molecular ions, singlyor multiply charged, and not restricted to S states. For the purpose of orientation, hfs or fs state selection of the stored ions, other reactions than those discussed here are likely to be used in the future. These may include optical pumping of the ions, ion-molecule reactions at low energies, charge exchange including charge exchange of ions with isotopic atoms or molecules

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

153

at low energies, superelastic collisions, electron impact ionization, and electron impact dissociation of molecules and molecular ions. No ultrahigh resolution technique for neutral atoms of comparable generality seems to be available at present. The techniques described may also be useful in collision studies at low and ultralow energies involving polarized ions.

ACKNOWLEDGMENT The author is indebted to his colleagues and collaborators Dr. E. N. Fortson, Dr. H. A. Schuessler,Dr. C. B. Richardson, Dr. T. S. Stein, and Dr. G. H. McCall and to his graduate students S. C. Menasian, D. A. Church, F. L. Walls, and K. Zieher for discussions and for reading the manuscript or parts of it and offering suggestions for its improvement. The manuscript was typed by Miss Karen Haskell.

ERRATA FOR PART I Formulas (2.43) and (2.44) given in Part I of this article should read

Tc = (nmaxhYc*N 3-”l’(~/qmsx)Tc**

(2.43)

On pages 63, 65, and 66 read F??< b, V, = (16r/3)rq3,and q/qms.= 150, respectively. On page 72, read T,= 40 rnsec and t, =0.6 msec instead of T,= 200 msec and t, = 3 msec. Also on page 72, read: Wuerker, R. F., Goldenberg, H. M., and Langmuir, R. V. (1959). J. Appl. Phys. 30,342 and 441.

REFERENCES Ackermann, H., Z u Putlitz, G., and E. W. Weber (1967). Phys. Letters MA, 567. Bloom, A. L. (1962). Appl. Opt. 1, 61. Burke, B. F. (1960). Astrophys. J . 132,514. Church, D., and Dehrnelt, H. G. (1969). J. Appl. Phys. 40. Dalgarno, A,, Patterson, T. N. L., and Sommerville, W. B. (1960). Proc. Roy. SOC.(London) A259, 100. Dawson, P. H., and Whetten, N. R. (1968). J. Vacuum Sci. Techn. 5, 1, 11. Dehrnelt, H. G. (1961). Private communication. Dehmelt, H. G. (1968). Proc. Intern. Symp. Phys. One- and Two-Electron Atoms, Munich, September, 1968.

154

H. G. Dehmelt

Dehmelt, H. G.,and Jefferts, K. B. (1962). Phys. Rev. 125, 1318. Dehmelt, H. G., and Major, F. G. (1962). Phys. Reu. Letters 8, 213. Dehmelt, H. G., and Walls, F. L. (1968). Phys. Rev. Letters 21, 127, Dicke, R. H. (1953). Phys. Rev. 89, 472. Fischer, E. (1959). Z. Physik 156, 1. Fortson, E. N., Kleppner, D., and N. F. Ramsey (1964). Phys. Rev. Letters 13, 22. Fortson, E. N., Major, F. G., and Dehmelt, H. G. (1966). Phys. Rev. Letters 16, 221. Graeff, G., Klempt, -, and Werth, G. (1969). Z. Physik 222,201. Graeff, G . , Major, F. G., Roeder, R. W., and Werth, G. (1968). Phys. Rev. Letters 21,340. Haun, R. D., Jr., and Zacharias, J. R. (1957). Phys. Rev. 107, 107. Huggett, G. R., and Menasian, S . (1965). Private communication. Jefferts, K. B. (1962). Thesis, Univ. of Washington, Univ. Microfilms, Ann Arbor, Michigan. Jefferts, K. B. (1968). Phys. Rev. Letters 20, 39. Jefferts, K. B. (to be published). Kleppner, D., Goldenberg, H. M., Ramsey, N. F., (1962). Phys. Reo. 126, 603. Landau, L. D., and Lifshitz, E. M. (1960). “Mechanics.” Pergamon Press, Oxford. Major, F. G . (1962). Thesis, Univ. of Washington, Univ. Microfilms, Ann Arbor, Michigan. Major, F. G., and Dehmelt, H. G. (1968). Phys. Reu. 170,91. Menasian, S . (1968). Private communication. Mulliken, R. S. (1939). J. Chem. Phys. 7 , 20. Novick, R., and Commins, E. D. (1958). Phys. Rev. 111, 822. Paul, W., Osberghaus, O., and Fischer, E. (1958). Forschungsber. Wirtsch. Verkehrsministeriums Nordrhein-Westfalen No. 41 5 . Purcell, E.M., and Field, G . B. (1956). Asfrophys.J. 124, 542. Rapp, D., and Francis, W. E. (1962). J. Chem. Phys. 37, 2631. Rettinghaus, G. (1967). Z. Angew. Phys. 22,321. Richardson, C. B., Jefferts, K. B., and Dehmelt, H. G . (1968). Phys. Rev. 165, 80. Salmona, A. (1965). Compt. Rend. 260,2434. Schuessler, H. A. (1968). Bull. Am. Phys. SOC.13, 1674. Schuessler, H. A., Fortson, E. N., and Dehmelt, H. G. (1969). Phys. Rev. (to be published). Schwartz, C. (1959). Ann. Phys. 6, 156. Shockley, W. (1938). J. Appl. Phys. 9, 63.’. Sirkis, M. D., and Holonyak, N. H. (1966). Am. J . Phys. 34,943. Sternheim, M. M. (1963). Phys. Rev. 130,211. Wittke, 5. P., and Dicke, R. H. (1956). Phys. Rev. 103, 620. Wuerker, R. F., Shelton, H., and Langmuir, R. V. (1959). J . Appl. Phys. 30, 342. Zwanziger, D. E. (1961). Phys. Rev. 121, 1128.

THE SPECTRA OF I I MOLECULAR SOLIDS 0. SCHNEPP Department of Chemistry University of Southern California Los Angeles. California

I . Lattice Vibrational Spectra ........................................

A . Introduction ................................................ B. General Theory .............................................. C. Intermolecular Potentials ...................................... D . Complete Brillouin Zone Treatments ............................ E. Inorganic Molecular Solids .................................... F. Organic Solids ................................................ G . Intensities of Lattice Vibration Spectra .......................... I1 Intramolecular Vibrational Spectra ................................ A . Introduction ................................................ B. General Theory .............................................. C. Dipole-Dipole Treatments .................................... D . Atom-Atom Interaction Potential Treatments .................... E Applications ................................................ F Intensities of Intramolecular Vibrational Spectra .................. G Induced Spectra .............................................. 111. Spectra of Solid Hydrogen ........................................ A Introduction ................................................ B . Pure Rotational Spectra of Solid Hydrogen ...................... C Vibration-Rotation Spectra of Solid Hydrogen .................. D . Spectra of Cubic Hydrogen .................................... References .....................................................

.

. . . . .

155 155 156 160 160 164 170 174 176 176 177 179 181 182 186 187 187 187 188 191 196 197

.

I Lattice Vibrational Spectra A . INTRODUCTION

Lattice vibrations of molecular solids involve motions of the molecules as a whole. Such vibrations can for many purposes be divided into two types. translational motions in which the centers of mass of the molecules are displaced from their equilibrium positions and librations for which the centers of mass are fixed whereas the molecules perform quasi-rotational motions. The importance of such lattice motions in molecular solids was recognized 155

156

0. Schnepp

early with respect to the x-ray determination of crystal structures (Cruikshank, 1958). More recently it has become clear that the investigation of the lattice dynamics of such solids provides an important source for information concerning intermolecular potentials. The first attempt to calculate librational frequencies from intermolecular potentials in the form of atom-atom interaction terms was made by Dows (1962b) for the ethylene crystal. Some attempts were made to deduce the frequencies of lattice vibrations indirectly from x-ray investigations (Cruikshank, 1958) and from the infrared spectra of molecular solids in the region of intramolecular vibrations where lattice modes have been observed in combinations with the intramolecular vibrational frequencies (Dows, 1959). Direct observations of lattice vibrations was made possible by the development of good Raman techniques which permitted the investigation of small frequency displacements. Kastler and Rousset (1941) investigated naphthalene solid, Fruhling (1950) investigated solid benzene, and the Raman frequency of hexamine (hexamethylene tetramine) was reported by Couture-Mathieu et al. (1951). Recent developments in the Raman techniques by the application of laser excitation has made further progress possible (see Sections I.E, 1.F). In many cases only part of the lattice vibrations can be observed in the Raman spectrum. Therefore, the significant progress made in far-infrared instrumentation has been a further stimulus to this field. During the past five years, commercial spectrometers and interferometers for far-infrared spectroscopy have become available. Anderson et al. (1964) were the first to apply interferometry to the problems of lattice vibrations. These authors investigated the hydrogen and deuterium halides; Walmsley and Anderson (1964) investigated the halogen crystals; and Anderson and Walmsley (1964) reported the far-infrared absorption spectrum of C 0 2 and that of ammonia and hydrogen sulfide (Anderson and Walmsley, 1965). This pioneering work was expanded in the following years by other groups. No comprehensive review on the subject of lattice vibrations of molecular solids is available at this time although it has been briefly treated in reviews dealing with the infrared spectra of molecular crystals (Dows, 1963,1965,1966). On the other hand, a number of good reviews on the lattice motions of ionic and metallic crystals are available (Mitra and Gielisse, 1964; Martin, 1965). The classical source for lattice dynamics is the book by Born and Huang (1 954). Another standard reference is the book by Ziman (1960).

B. GENERALTHEORY To illustrate the treatment, we shall describe the work of Walmsley and Pople (1964) on the intermolecular vibrations of solid C 0 2 in some detail. The displacement of theqth molecule in thepth cell is denoted by r,(p/q) where

THE SPECTRA OF MOLECULAR SOLIDS

157

ct can assume the three designations x,y , z. The spatial direction of the linear molecule is denoted by the unit vector

AucP/q) = A,OcP/q) + Aa(P/q)

(1)

where Aao is the equilibrium value and &(p/q) is the displacement coordinate. Each molecule has only two rotational degrees of freedom, and therefore a redundancy is included if a here is allowed to assume three designations x, y , and z as above. The redundancy condition in this case is obtained in the form ofEq. (21,

which simply signifies that the unit vector has retained its length and only changed its orientation. Next, symmetry coordinates of the crystal space group are formed and this is achieved by a transformation of the molecular displacement coordinates to linear combinations which belong to irreducible representations of the translation group. These symmetry coordinates are of the form r:(q) = N-'"

A:(q) = N-l"

C expC2nik P 1expC2nik

R(p)]ra(p/q) R@)]la(p/q).

(3)

P

Here N is the number of unit cells in the crystal, k is a wave vector, and R@) is the position vector of the pth cell. The use of such coordinates will assure the factorization of the energy problem or the problem of the normal vibration frequencies of the system into problems concerning a particular point in the Brillouin zone designated by the wave vector k. This problem can then be formulated generally in terms of k and solved for any point in the zone. Since, however, the transition probability from the ground state to a single phonon state (excitation of one quantum of lattice vibration) will vanish both for the Raman effect as well as for infrared absorption except for k = 0, Walmsley and Pople treated only this point of the zone usually designated by r. In principle, the vibrational degree of freedom excited will have k equal to the wave vector of the radiation interacting with the solid. However, this wave vector will be very close to zero since the wavelengths of light involved are very long compared with the lattice spacing. For k = 0 further simplification of the problem is possible by making use of the symmetry at this point, which is the full factor group symmetry (Koster, 1957). For solid CO, this factor group is T,,,and therefore symmetry coordinates can be formed which belong to irreducible representations of the factor group by forming the proper linear combinations of k = 0 functions of

0.Schnepp

158

the type given in Eq. (3). The resulting symmetry coordinates are of the following form: (i = 1, . . . , 12) Si = T/,aqr:(q)

2

a.9

where r;(q) and AaO(q) are given by substituting k = 0 in Eq. (3) rao(q) = N-1’2 C r a W q ) , P

naO(q)

= N-”2

C Aa@/q>.

(5)

P

The crystal structure of C 0 2 , as already mentioned, belongs to space group Th6.The molecular centers are located at face-centered cubic (fcc) sites and the molecular axes are oriented parallel to the body diagonals of the cube. There are four such directions and accordingly there are four physically equivalent moelcules in the primitive unit cell. We therefore expect 4 x 3 translational degrees of freedom for every point k in the Brillouin zone. At the center of the zone, k = 0, the three acoustic frequencies are zero and represent the translational motion of the crystal as a whole. Thus we expect nine translational optical modes. Straightforward group theoretical analysis gives the result that this 9 x 9 reducible representation contains the following irreducible representations of the factor group T,,: A,

+ E, + 2T,.

Since the translations x, y , z belong to the representation T,. we expect two absorption bands in the far infrared. To solve the frequency problem, the potential energy in the harmonic approximation (i.e., for small displacements) is expanded up to second-order terms in the displacements. For k = 0 this is best done in terms of those linear combinations of the molecular displacements which form the symmetry coordinates Si given in (Eq. 4). The potential energy then has the following form : 12

where the force constants Fij are the second derivatives of the crystal potential with respect to the symmetry displacements. The frequencies are then given by the roots of the secular equation of dimension 12 IFij - M o 2 hijl = 0

(8)

where M is the mass of the molecule. Since the potential energy has been written in terms of symmetry coordinates, this determinant will break up into smaller blocks, each for an irreducible representation of the factor group. As usual, for degenerate representations, there will be an identical block for

THE SPECTRA OF MOLECULAR SOLIDS

159

each line of the representation. As already pointed out three of the roots of the twelve-dimensional problem will be zero and these can be identified in advance and separated from the problem in order to facilitate its solution. The librational lattice motions are treated in a parallel manner. For a centrosymmetric molecular site, as is the case in the CO, lattice, the symmetry coordinates formed from the translational displacements (4) belong to u representations, whereas the symmetry coordinates formed from the librationa1 displacements are even and belong to g representations. This can be understood if it is considered that a rotational displacement keeping the center of mass of the molecule fixed preserves the center of inversion, whereas the displacement of the center of mass bf the molecule in a translational displacement destroys that center of inversion. As a consequence of this differentiation between the two distinct types of lattice motions the translations and librations are accurately separable at k = 0 for this case. As formulated by Walmsley and Pople the quasirotational displacement coordinates are twelve in number, three components for each molecule in the unit cell. However, there are only two rotational degrees of freedom for a linear molecule and as a result four redundant coordinates have been introduced. It is therefore necessary to modify the procedure in order to take account of this fact. We therefore conclude that the librational motions span a reducible representation of order 8, and this reduces into irreducible representations of the factor group T,, as follows :

EB-t2T,. (9) Accordingly, we expect three Raman lines in the low frequency region. As already stated, the problem is similar to that of the translational motions except for the replacement of the mass of the molecule by the moment of inertia I in Eq. (8) and the problem of redundancies. It should be pointed out, however, that it is not necessary to introduce these redundancies as formulated by Walmsley and Pople. An alternate scheme can be used whereby two angular displacement coordinates are assigned to each molecule and the problem is then straightforward and does not involve any redundancy (Kuan and Schnepp, 1968). This is of particular advantage if the lattice dynamical problem is to be solved through the Brillouin zone. It is appropriate to digress here from the contents of the paper by Walmsley and Pople to point out what steps have to be taken in order to obtain solutions for points k # 0. In this case it is not as advantageous to use symmetry coordinates since the applicable symmetry is greatly reduced from that of the factor group symmetry. For a general point, there is, in fact, no more than trivial syrnmetry C, apart from the translational symmetry of the lattice. We therefore expand the potential energy in displacement coordinates of type (3) and calculate the appropriate force constants on this basis. It must

I60

0.Schnepp

also be remembered that for k # 0 the translational motions are no longer separable from the librations and therefore the problem has to be solved in full. If redundancies are not included, this implies that a secular problem of order 20 must be solved for every point of the Brillouin zone (every value of k). Three of those solutions will be those of the acoustic branches and the remaining belong to the optical branches. C. INTERMOLECULAR POTENTIALS It is now clear that the lattice dynamical problem can be solved once a potential for the crystal is formulated. It has generally been assumed that the potential of the crystal can be expressed as the sum of pair potentials. There has been some discussion of this assumption in the literature, but it is probably valid for molecular solids (Williams et al., 1967; Dymond and Alder, 1968). Two general types of pair potentials have been used in this field. The first, applied by Walmsley and Pople (1964) to the calculation of the k = 0 lattice frequencies of C 0 2 , consists of a central forces term in the form of a 6-12 Lennard-Jones potential and an angle-dependent term for which a quadrupole-quadrupole potential is used. The second type of intermolecular pair potential is expressed as a sum of atom-atom interaction terms between neighboring molecules. This latter type was developed for organic molecules in which case it seemed originally that the hydrogen-hydrogen repulsions were the dominant interactions. The first application of such a potential to the calculation of librational lattice frequencies is due to Dows (1962b), who used an exponential repulsion term in accordance with the approximate quantum mechanical calculation by deBoer (1942). Kitaigorodskii (1966) derived parameters for atom-atom potenials from crystal structures of aromatic systems. Such a potential has the form

- AIR6 + B exp( - CR)

(10)

where R is the interatomic distance of the interacting molecules. Coulson and Haigh (1963) listed parameters for hydrogen-hydrogen potentials and Williams (1967) determined potential parameters for nonbonded H-H, C-H, and C-C interactions from a large number of crystal properties, including structures, elastic constants, and sublimation energies of nine aromatic hydrocarbons. Potential constants determined by other workers in connection with specific applications will be discussed in the following sections. D. COMPLETE BRILLOUIN ZONETREATMENTS Cochran and Pawley (1964) discussed the theory of diffuse scattering of x rays by a molecular crystal, hexamethylene tetramine. In the course of this

THE SPECTRA OF MOLECULAR SOLIDS

161

work they studied the lattice dynamics of this solid, which is body-centered cubic (bcc) with one molecule per primitive unit cell. Both translational and librational degrees of freedom were considered and the lattice dynamics problem solved throughout the zone. Dispersion curves were computed for some symmetry directions and the density of state curve was computed. They parametrized the lattice dynamics problem and used four parameters for nearest-neighbor interactions and four more for next-to-nearest-neighbor terms. However, only three independent observables were available to evaluate these parameters, i.e., two independent elastic constants and the Raman frequency which had been observed at 40 cm-’ (Couture-Mathieu et al., 1951; Cheutin and Mathieu, 1956); this frequency represents the k = O optical libration of the molecule. The study is of great fundamental interest inasmuch as it is the first such investigation and demonstrates the mixing of translational and librational degrees of freedom throughout the zone. This work was followed up by Pawley (1967), who studied the lattice dynamics of naphthalene and anthracene. This author used a specific intermolecular potential which was expressed in terms of sums over atom-atom interaction terms of the form of Eq. (10). The parameters used were those given by Kitaigorodskii (1966). The lattices of naphthalene and anthracene are more complex inasmuch as they are of lower symmetry (space group P2,/u, C,”,), i.e., they are monoclinic crystals with two molecules per unit cell. Since the molecules are not linear, there are three rotational degrees of freedom for each and as a result we have six rotational and six translational degrees of freedom that must be considered at each point in the zone. Contacts up to 5.5 A were included in the calculation, but it was shown that only a 1 % change in the resulting frequencies occurred if the lattice sums were cut off at 5 A distance. Dispersion curves were given only for the [OIO] symmetry direction since this is the only symmetry axis in the crystal, and as a result, for any point along this direction in the Brillouin zone, the modes can be classified as symmetric or antisymmetric with respect to the two-fold rotation (symmetry species A and B, respectively).The results at k = 0 can be compared to known Raman frequencies. In the paper these values are compared to the results of Wilkinson (1 966) and the comparison was judged to be reasonable. No attempt was made in this calculation to adjust the parameters of the potential to give better agreement with measured observables. The results of this calculation and the experimental results of Wilkinson quoted here are included in Table I, which lists the naphthalene lattice vibrations as observed and calculated by different authors. Note that librations are of symmetry species A , and B, and are Raman active. Since the molecules are located at centrosymmetric sites, librations and rotations are separable at k = 0. Translations are of species A , and B, and are infrared active by contrast. The far-infrared data were not available at the time. Two sets of calculated results of Pawley

0.Schnepp

162

TABLE I LAITICEVIBRATIONS OF SOLIDNAPHTHALENE Observed Species

Clh A,

B,

Raman

IR

KandR"

Wb (300°K)

I' (77°K)

127 76 54 109 74 46

146 90 71 124 86 59

120 88 69 141 81 56

A.

B. a

Handsd (300°K)

98 53 66

Calculated Handsd (300°K) 115 85 58 82 70 42 97 54 74

119 86 62 93 77 47 104 58 76

PC (300°K) 134 94 61 126 70 77

139 92 62 129 77 49 88 47 58

Kastler and Rousset (1941). Wilkinson (1966) (from Pawley, 1967). Ito ef al. (1967). Harada and Shimanouchi (1966). Pawley (1967).

are given in Table I. The first set was calculated by assuming rotation of the molecules to be about the principal axes of the molecules whereas the second set is a more correct set, and it was found eventually that in some cases the actual normal mode represents libration of the molecule about axes considerably different from those of the molecule. In other words, the normal modes represent combinations of molecular motions and these are not bound to the molecular frame. The author also pointed out that the calculation shows the symmetric mode in each pair but one of the symmetric and anti-symmetric librational modes ( A g ,B,) to be of higher frequency whereas previously the opposite was assumed (Kastler and Rousset, 1941). Pawley also points out that his calculation neglected interactions between lattice modes and internal molecular vibrations, some of which occur at frequencies as low as 200 cm-', and therefore some serious errors may be expected due to this approximation. The density-of-states curve peaks for naphthalene at about 40, 55, and 80 cm-'. Also elastic constants that had been measured by ultrasonic methods were calculated and found to give a reasonable fit to the experimental data. Reference to Table I indicates that the infrared-active modes of naphthalene as calculated by Pawley are similarly in agreement with the observed values

THE SPECTRA OF MOLECULAR SOLIDS

163

of Harada and Shimanouchi (1966) within the same limits of error of about 10 cm-' as the Raman-active frequencies. It should also be pointed out that the density-of-states curves are probably of importance in determining the combination frequencies of lattice vibrations with internal molecular vibrations in the infrared and with electronic-vibrational spectra in the ultraviolet. For such combinations, k of the lattice vibration need not be zero but only the total lattice vector need give that value (i.e., the sum of the lattice vibration wave vector and that of the other combining degree of freedom). Unfortunately, the limits of error of the calculation are at this time such as not to allow clear differentiation between the k = 0 frequencies and the frequencies at which the density-of-state curve peaks. Most recently, Ron and Schnepp (1968) have undertaken a calculation of the lattice vibrations throughout the Brillouin zone for a-N,. The work was undertaken in an effort to account for large differences in line widths observed in the far-infrared lattice vibration spectrum of this solid. Two types of potential were used in this work. First, a central forces 6-12 Lennard-Jones potential was used. The crystal structure is very similar to that of solid COz already described, i.e., the molecular centers are located at fcc sites with molecular axes along the body diagonals of the cube. There are four molecules per unit cell. The accurate crystal structure was reported to be somewhat different from this structure but the deviations were neglected in this work. When the central force model as described was used, the problem was similar to that of solid argon, which has recently been treated in full (Grindlay and Howard, 1965), except that the frequencies are distributed differently throughout a zone of different structure. The argon crystal is face-centered cubic, but its primitive cell contains only one atom. As a result only acoustic branches occur. The difference between the two cases can best be visualized by considering the dispersion curves in the [lo01 direction. The length of the primitive unit cell is, in the case of argon, half the cubic cell parameter, whereas it is the whole cubic cell parameter in the case of nitrogen. As a consequence, the Brillouin zone is larger in the case of argon; and if we want to modify results of the argon problem to fit the case of nitrogen, we have to terminate the zone halfway and fold the dispersion curves backwards such that the points at the edge of the zone for argon fall at k = 0 for a-N, . The density-of-state curve is unaffected. The same problem was also treated by using an atom-atom potential with inverse 6-12 power interaction terms. The parameters for this potential were those determined in the course of a calculation for k = 0 (Kuan and Schnepp, 1968). The results of this work will be discussed further in the next section. The group theoretical analysis of lattice vibrations in molecular crystals throughout the Brillouin zone has recently been discussed by Chen and Dvorak (1968). They applied their discussion based on the theory of space

164

0. Schnepp

groups to hexamine (hexamethylene tetramine) which was treated by Cochran and Pawley (1964) as discussed above. Chen and Dvorak list symmetry points and symmetry directions in the Brillouin zone and discuss the point groups appropriate to these regions. They show that irreducible representations, the degeneracies, and the types of motion determined by the degree of mixing between translational and librational displacements. The second example discussed by Chen and Dvorak is adamantane, which also has one molecule per primitive unit cell and has an fcc structure. As a third example, these authors chose NH4CI.

E. INORGANIC MOLECULAR SOLIDS 1. Carbon Dioxide As already described, Walmsley and Pop.; (1964) treated solid CO,. They determined the normal frequencies of the translational and librational modes for k = 0. The intermolecular pair potential used has already been referred to and the molecular quadrupole moment is known. They obtained very good agreement between calculated and observed translational modes, but there were considerable discrepancies for the librational frequencies. Table I1 summarizes the frequencies of solid CO, . The T, modes were observed in the far-infrared by Anderson and Walmsley (1964) and by Ron and Schnepp (1967). The librational frequencies have been observed in the Raman spectrum by Ito (1968) and by Cahill et al. (1967a).

2. Solid a-Nitrogen

The far-infrared spectrum of crystalline nitrogen below the phase transition at 36°K (a-phase) has been reported by Anderson and Leroi (1966) and by Ron and Schnepp (1967). The latter authors also performed calculations similar to those by Walmsley and Pople for C 0 2 , using Lennard-Jones potential force constants available in the literature and the experimentally measured quadrupole moment of the molecule. The results are compared to the experimental frequencies in Table 111. The Raman frequencies have been reported by Cahill et al. (1967b) and have been remeasured by Brit et al. (1969). Both these groups agreed inasmuch as they found only two lines, whereas three Raman-active modes are expected. Again, agreement between calculated and observed values was very good for the translational modes but poor for the librations. Relative intensity measurements of the Raman lines coupled with calculations of these intensities indicate that there is accidental coincidence of two of the librational frequencies (Brit et al., 1969). Recently, calculations have been carried out using atom-atom potentials

TABLE I1 LATTICE VIBRATIONS OF SOLID Translational modes

Librational modes

Calculated

Observed -~

Symmetry

W and P'

A and W* Rand s'

Tu (Qz)

74 77 94 113

68 114

E" A.

T.

(Qi)

Intensity ratio; I J I z (R and S ) :

coz

5.4

Observed

Calculated

Symmetry

Wand P

E,

35 48 88

T, T,

I' (Dg) (195°K) 65 83 (1 12)

Lh (170°K)

70 85

124

L (90°K)

I (77'K)

I (4.2"K)

74 91.5 131

72 92 136

72 92 136

-l

2i

; 4

P

s

2.5 P m

Frequencies given in reciprocal centimeters. From Walmsley and Pople (1964), uiz = 4d(o/R)" - (o/Wl E = 3.18

x

ergs,

+Q-Q (T

= 3.72

Walmsley and Pople (1964). Anderson and Walmsley (1964). Ron and Schnepp (1967). Ito (1968). Dows (1959). Cahill et a/.(1967a).

x

cm.

0.Schnepp

166

TABLE 111 LATTICEVIBRATIONS OF %LID

a-NZ n*b

Translational modes Calculatedb

Observed

Symmetry

R and Sc K and Sd

A and Le R and S’

Tu (Qz) Tu Q(d

49 71

48.8 70

Intensity ratio, IJIZ : a

Librational modes Calculatedb

Observed

c, L, LO Symmetry

K and S

BRS’

Eo

27.8 38.4 70.2

-

T, To

33

37

1 f0.2

2.3

Frequencies given in reciprocal centimeters. /LIZ = ~E[(u/R)”- (u/RYI

E = 1.18 X

+Q -Q

ergs, Ron and Schnepp (1967). Kuan and Schnepp (1968). a Anderson and Leroi (1966). Brit et af. (1969). Cahill et ol. (1 967b).

u = 3.71 x

cm.

’

of the 6-12 type (Kuan and Schnepp, 1968). The results are summarized in Table IV, in which they are compared to the new assignments. Recently, St. Louis and Schnepp (1969) found that the two far-infrared absorption bands of u-N, in the far-infrared have very different line widths. The lower frequency line at 48.8 cm-’ has a line width smaller than 0.3 cm-’, whereas the higher frequency line at 70 cm-’ has a half-intensity width of 6 cm-’ (Table IV). This observation was made for a sample condensed in the liquid state in a closed cell which was then further cooled to form solid phase p and then further cooled through the transition temperature to give samples of phase c1 at 15°K. This observation motivated the lattice dynamics calculation referred to in the previous section (Ron and Schnepp, 1968). The aim was to investigate phonon densities of states which would be available for the relaxation of these two optical phonons into two other phonons subject to conservation of total lattice vector k = 0 and to conservation of energy. Translational lattice motions only were considered in this work. The ratio of the appropriate two-phonon state densities was found to be of the order of 20 indicating that the difference in line width of these two optical modes can probably be accounted for in terms of the relaxation of the optical

THE SPECTRA OF MOLECULAR SOLIDS

167

TABLE IV

LATTICE VIBRATIONSOF SOLIDa-Nz'Pb ~ _ _ _ _ _

~~

Translational modes

Symmetry ~~

Librational modes

Calculated

Observed

Calculated

K and S'

S and Sd

Symmetry

K and S

48.6

EP

35.3

Observed

c,L, L8 BRS'

~

Tu (Qz) Tu (Qd

72.5

48.8 (0.3) 70 (6)

Intensity ratio Zl/12(infrared):

1.05

1 f 0.2

I,

33 (1.5)

I = 3.6

To

TP

Intensity ratio (Raman):

35.4 44.8

31 (1.5) I= 1

2.3

3.6

Frequencies given in reciprocal centimeters (with half-intensity widths in parentheses). From Kuan and Schnepp (1968), 4

u12

= 4EC 1(u/&)12- ( O / W I a= 1

ergs, u = 3.387 x Kuan and Schnepp (1968). St. Louis and Schnepp (1969). Cahill et al. (1967b). Brit et al. (1969). E

J

= 0.415 x

cm.

mode into two phonons, as has been assumed for other systems (Kleinman, 1960). Further work is necessary before more definitive conclusions can be drawn.

3. Hydrogen Hardy et al. (1968) have recently discovered far-infrared absorptions due to optical phonons in solid hydrogen and solid deuterium in the low-temperatureordered state below the transition point at about 1.5"K. Below this temperature the crystal structure is face-centered cubic but neutron diffraction studies (Mucker et al., 1966) have shown that the crystal structure is lower than face-centered cubic. Theoretical work (Homma et al., 1967; Raich and Etters, 1968; Ueyama and Matsubara, 1967) predicted that the low temperature structure would involve an ordering of the rotational axis of quantization. The structure would thus be analogous to that of a-N2 except that here the quantization axis for the angular momentum would be [l 111 instead of this

0. Schnepp

168

being the direction of a fixed molecular axis. As a result, the J = 1 rotational state, which is the lowest state for orthohydrogen or for paradeuteriurn, would be split in the axial field of this site, M = 0 being lower relative to M = k 1, and this splitting was predicted to be of the order of 10 cm-'. Thus at temperatures corresponding to energies appreciably lower than this order of magnitude, a rotational ordering would occur since all molecules would then be in the M = 0 substate. Since now four sublattices physically distinct from one another occur, optical phonons are expected. These have, indeed, been found for enriched orthohydrogen and paradeuterium. Three absorptions have been recorded in each case, the lowest being very sharp, i.e., half-intensity width of less than 1 cm-' at 62.2 and 57.4 cm-' for hydrogen and deuterium, respectively. The second absorption line at 80 and 74.5 cm-', respectively, is broader and has a half-intensity width of about 4 cm-' ; and the highest energy line at 93 and 85 cm-'. respectively, has the greatest half-intensity width, which is about 10 cm-'. As discussed in the preceding section, or-N, and CO, have two lattice bands as expected from group theoretical treatment. Solid hydrogen is, however, basically different from all other solids discussed here. If a pair potential derived for gas phase interactions is employed, negative force constants are obtained for the lattice; and as a consequence, imaginary frequencies are found (deWette and Nijboer, 1965; Van Kranendonk and Sears, 1966). It must, therefore, be concluded that the harmonic approximation breaks down for these solids and this is correlated with the fact that the zero-point motions are large in amplitude. It turns out that the zero-point energy is approximately half of the heat of sublimation per molecule. Good agreement was obtained by Hardy et a/., between measured frequencies and those calculated on the basis of a modified potential fitted to measured parameters of the hydrogen solid. It should be noted that more fundamental treatments of this type of solid have recently been described (Nosanow, 1966; Fredkin and Werthamer, 1965) which formulate the treatment of " self-consistent phonons ". However, this treatment has not as yet been applied to solid hydrogen. 4. Miscellaneous Solids

Both Anderson and Leroi (1966) and Ron and Schnepp (1967) have measured the far-infrared absorption spectrum of CO. However, the disorder prevailing in this solid precluded accurate theoretical treatment although Ron and Schnepp carried out frequency calculations. Anderson et a/.,(1964) have reported the far-infrared spectra of hydrogen and deuterium chlorides and bromides at 77°K in the spectral region 20-400 cm - The absorptions were assigned as lattice vibrations and translational and librational modes were distinguished on the basis of frequency changes on

'.

THE SPECTRA OF MOLECULAR SOLIDS

169

deuteration. It was shown by these authors that their results are consistent with a crystal structure containing four molecules per unit cell in a planar array. Some intermolecular force constants were derived from the experimental results. Recently, Arnold and Heastie (1967) have investigated the farinfrared absorption of these solids in different phases, two phases for HCl and three for DBr. The absorptions due to the librational modes broaden faster with rise in temperature than the translational bands. Leech and Peachey (1968) have investigated the frequency distribution of phonons in the low temperature phase of solid HCl. Results of calculations have been reported for two possible space groups. Force constants were obtained by fitting the parameters of an atom-atom potential function to obtain agreement with observed infrared absorption frequencies. The infrared spectrum of crystalline H F and DF have been described by Kittelberger and Hornig (1967). A lattice band was observed near 200 cm-' for both H F and D F and the Raman spectra were found to contain one other lattice mode each. Raman and infrared lines are coincident and this observation has important implications for structural considerations. Cahill et al. (1967b) have reported the low frequency Raman spectrum of N,O at 90 and 170°K. The far-infrared spectra of crystalline acetylenes C,H, and C2D2 have been reported by Anderson and Smith (1966). Two bands were observed in each case and these were assigned as translational lattice modes on the basis of the frequency change on deuteration. Deductions were made relevant to the crystal structure. The results were found to be consistent with a D,,,factor group with two molecules per unit cell on sites of symmetry C Z hThis . work was done at 77°K or below the phase transition at 133°K. Low frequency Raman results were also quoted in this work. The translational lattice modes of the solid halogens Cl, , Br, , and I, have been observed in the far infrared by Walmsley and Anderson (1964). Two low frequency bands were observed in accordance with theoretical predictions on the basis of the known structure. Weak absorption bands, that were assigned to the intramolecular stretching frequencies, were also observed in this work. Reasons for the infrared activity of these modes are discussed. Anderson and Walmsley (1965) described the far-infrared spectra of ammonia, H,S, and their deuterated analogs. Giguere and Chapados (1966) reported the far-infrared spectra of solid H,0, and D,02. All the lattice vibrations predicted by group theoretical analysis were observed. A number of papers have appeared in recent years which treat the optical spectra of ices. Bertie and Whalley (1964a,b) discussed the infrared spectra of two ices. The translational lattice vibrations of orientationally disordered crystals have been discussed by Whalley and Bertie (1967) and the same authors (Bertie and Whalley, 1967) have applied this theory to the infrared spectrum of ices Ih and Ic. The frequency spectrum of hexagonal H 2 0 ice as

170

0. Schnepp

investigated by inelastic neutron scattering has been reported by Prask et al. (1968). F. ORGANICSOLIDS 1. Benzene

Information concerning benzene lattice vibrations known as of this time is summarized in Table V. The low frequency Raman spectra that were reported by Fruhling (1950) were recently reinvestigated at lower temperatures by Ito and Shigeoka (1966a). The far-infrared absorption spectrum of films condensed from the vapor on a cold surface has been recently reported by Harada and Shimanouchi (1967). Harada and Shimanouchi (1966, 1967) also carried out calculations on the basis of intermolecular pair potentials in the form of atom-atom interaction terms of the form of Eq. (10). They first took into account five H-H interaction pairs but found that the calculated frequencies were much lower than those observed. They thereupon included C-H interactions and found that the repulsive exponential terms both for the H-H and C-H interactions were decisive. Five pairs of H-H and ten pairs of C-H interactions were used. The exponential parameters [C in Eq. (lo)] were taken from the work of deBoer (1942) for H-H and from Abe et al. (1966) for C-H. Then the preexponential constants [B in Eq. (lo)] were fitted to the experimental observables, i.e., the Raman and infrared data. In this way good agreement was obtained, particularly for the Raman-active frequencies. The potentials so determined were further tested by application to H-H interactions for intramolecular vibrations in the gas phase. Good consistency was found for CH4 and C6H6. Similarly the C-H potentials were tested by applying them to intramolecular F-matrices where these appear as off-diagonal terms. Good agreement was found for CH,OH, CH,CN, C2H4, C2H6, and C,H6. 2. Naphthalene The lattice dynamics of naphthalene as already described has been discussed by Pawley (1967) and the available information concerning lattice vibrations is compiled in Table I. The early low frequency single-crystal Raman work by Kastler and Rousset (1941) has recently been supplemented by the work of Wilkinson (1966) and the work of Ito et a/. (1967). The farinfrared spectrum of naphthalene has been reported by Harada and Shimanouchi (1966). These authors also performed calculations at k = 0 and found again that they had to use H-C interaction terms in addition to H-H interaction terms. Two sets of calculated results of Harada and Shimanouchi

171

THE SPECTRA OF MOLECULAR SOLIDS

TABLE V

LATTICE VIBRATIONS OF SOLID BENZENE Raman-active modes

Species

0bserved

F" D2h

(-3°C)

Calculated

I and Sb (138°K)

H and Sc (-3°C)

H and Sc

H and Sc

H-H

H-H

H-H C-H

79 57 28

96 71 34

102

120 70 33

121 83 45

131 99 65

76 69 56

91 84 67

106 1 02 82

100

116 86 76

128 96 81

79 57 126

-

126 90

73 60

-

(138°K)

(138°K)

80 55

Infrared-active modes

Species

Observed

Dih

H and Sc (1 40°K)

Calculated

(-3°C)

H and Sc (138°K)

H and Sc (138°K)

58 49 26

68 56 35

86 66 54

85 70

58 34

68 44

79 66

94

57 31

67 38

86 55

94

44 30

56 35

87 49

H and S'

A.

Fruhling (1 950). Ito and Shigeoka (1966a). Harada and Shimanouchi (1966,1967).

0. Schnepp

172

(1966) are given in Table I. The first set was obtained by considering H-H interaction terms only, whereas the second set included C-H interaction terms. The two columns of calculated results by Pawley (1 967) in Table I represent two states of sophistication where in the first stage the librations of the molecules were assumed to occur close to the molecular axes, whereas the second set was obtained by determining the best frequencies for the crystal. In this latter calculation, as already discussed, Pawley found that some of the librations occur about axes which are as much as 30" inclined to the molecular principal axes.

3. Anthracene Pawley (1 967) has discussed the lattice dynamics of anthracene on the basis of Kitaigorodskii (1966) potentials throughout the Brillouin zone. Ito et al. (1967) have tabulated the Raman spectra of anthracene in the low frequency range as obtained by laser excitation. Colombo and Mathieu (1960) have described the optical properties and Raman spectra of solid anthracene. Pawley (1967) quotes Raman data by Wilkinson (1966). Bree and Kydd (1968) have measured the infrared spectrum out to 50 cm-'. 4. Ethylene

Table VI summarizes the information concerning solid ethylene. The even TABLE VI OF SOLD ETHYLENE" LATTICE VIBRATIONS

Species

DZL B3P

B2#

B1 0 B3u B2U

Observed

Calculated

B and Hb [IR(vib)]

B and Rc (far IR)

14 11

-

-

-

-

-

B and Rc Dd

I

90 40

-

-

-

-

75

40 -

-

12 (1.5)

110 (4)

-

100

-

61

-

I1

13

93 34

a Frequencies given in reciprocal centimeters (with half-intensity widths in parentheses). Brecher and Halford (1961 )-infrared combinations. Brit and Ron (1969tfar infrared. Dows (1962b).

THE SPECTRA OF MOLECULAR SOLIDS

173

vibrations have been observed only in combination with intramolecular vibrations by Brecher and Halford (1961). No Raman data are available. The far-infrared spectrum was recently observed by Brit and Ron (1968). Calculations were performed earlier by Dows (1962b) on the basis of atomatom interactions but only for the even vibrations. Brit and Ron (1969) have carried out two sets of calculations. one making use of a Lennard-Jones center-center potential (column I in Table VI) and a second calculation using exponential repulsion terms for the H-H interaction (column IL). Although three infrared-active vibrations are predicted only two have been observed. Brit and Ron used mostly samples prepared by spraying a vapor on a cold surface. However, in one experiment they used a sample prepared as liquid in a closed cell and then cooled to solidification. They observed that the line width was appreciably decreased by this sample preparation, and these line widths are given in parentheses in Table VI. This observation of line-width dependence on sample preparation is in accordance with the observations of St. Louis and Schnepp (1969) for a-nitrogen.

5 . Polyethylene The lattice motions in crystalline polyethylene have been discussed by Tasumi and Shimanouchi (1965). These authors calculated the translational lattice modes and compared their calculation with the observation of one far-infrared absorption at 73 cm-' as reported by McKnight and Moeller (1964), Bertie and Whalley (1 964a,b) Frenzel and Butler (1964), and Genzel (1964). The calculated frequency was 76 cm-'. The second expected infraredactive mode has not been found until recently when Dean and Martin (1967) reported a weak absorption line in a sample cooled to 2°K at 109 cm-'. These authors also reported a shift of the previously observed line to 79.5 cm-' at 2°K.The calculated value for the higher frequency mode by Tasumi and Shimanouchi was 105 cm-'. In view of the difference in lattice dimensions expected for the different temperatures, the agreement between calculated and observed frequencies is most satisfactory. Dean and Martin (1967) also reported a weak line at 110 cm-' for paraffin samples at 2°K. Tasumi and Krimm (1967) have discussed the effects of cell dimension changes with temperature on the frequencies of lattice motions. They found that it was possible to account for the observed frequency changes in the lower frequency line on the basis of such dimensional variation. 6 . Miscellaneous Solids

Ito (1964) observed the lattice vibrations of solid methyl iodide by Raman scattering. All six frequencies were observed thus confirming the low symmetry of the crystal structure.

174

0. Schnepp

Ito and Shigoeka (1966b) have investigated the lattice vibrations of solid pyrazine at four different temperatures and have compared these observations with calculations performed on the basis of correlations with the work of Harada and Shimanouchi (1966). Of particular interest is the N-H force constant in this solid since it represents a hydrogen bond interaction. It was found that this force constant is about one-tenth of that of the 0-H hydrogen bond force constant in crystalline formic acid. Very good agreement was achieved between calculated and observed frequencies. Ito et al. (1967) have also tabulated the low frequency Raman spectra of p-dichlorobenzene and p-dibromobenzene. Colombo (1967) has reported the low frequency Raman spectrum of crystalline p-toluidine. Brot et al. (1968) have studied the far-infrared spectra of liquid and crystal tertiary butyl chloride. This study is of interest in connection with rotation in the solid phases and their correlation with the motions in the liquid phase. The lattice vibrations of thiourea and the deuterated compound have been studied by Takahashi and Schrader (1 967). Calculations of these frequencies were performed and were found to be in good agreement with observed frequencies in the Raman and far-infrared spectra. The far-infrared spectra and structure of crystalline hydrazine have been discussed by Baglin et al. (1967). It is found that the crystal structure can be characterized on the basis of these studies. Shimanouchi and Harada (1964) have applied their computational methods for the vibrational degrees of freedom of molecular solids to cyanuric acid, uracil, and diketopiperazine. They also measured the farinfrared spectra between 300 and 90 cm-'. These authors characterized the hydrogen bond force constants in these solids. Temperature dependences of line widths in Raman spectra have been studied by Colombo and Moreau (1967). Such a study for cc-hydroxynaphthalene crystal has been performed by Korshunov et al. (1967).

G . INTENSITIES OF LATTICE VIBRATION SPECTRA Poll and Van Kranendonk (1963) treated the infrared absorption intensities of lattice branches accompanying induced infrared absorption in solid hydrogen. These treatments, as described in Section 111. C.2,, were based on molecular quadrupole-induced mechanisms. Schnepp (1 967) has formulated the theory for the far-infrared absorption intensities of the pure lattice vibrations in molecular solids. For crystals made up of nonpolar molecules, a molecular quadrupole-induced mechanism is again formulated. Thus, the absorption intensity is proportional to the square of the molecular quadrupole moment and the square of the molecular polarizability, and inversely proportional to the tenth power of the lattice constant. The calculation is carried out in full for the space group Pa3 (T:) with four molecules per unit cell suit-

THE SPECTRA OF MOLECULAR SOLIDS

175

able for CO, and approximately applicable to a-N, . Ron and Schnepp (1967) extended this theory to the accurate structure of a-N,, which is P213 (T4). Schnepp (1967) also discussed the intrinsic transition moment which occurs if the molecules making up the crystal have a dipole moment. In this case librational modes have an intrinsic moment which can be calculated. He also considered the derivative of the polarizability with respect to a lattice motion in connection with developing the theory of the quadrupole-induced intensity. This formulation is directly applicable to the calculation of Raman intensities of lattice motions as has also been pointed out previously by Van Kranendonk (1960) in connection with his interpretation of the spectra of solid hydrogen. Ron and Schnepp (1967) found that the intensity theory gave good order of magnitude descriptions of the observations for EN,, C O , , and CO. St. Louis and Schnepp (1969) have measured the absolute intensities of the farinfrared absorption lines of a-N,. They find that the measured intensities are lower than those calculated by a factor of at least 3 and possibly 4. The uncertainty arises from the observation that the intensity is strongly temperature dependent, increasing with lowering of temperature and falling to zero as the temperature approaches the transition temperature between u- and fl-N, at 36°K. The discrepancy between calculated and measured intensities is probably due to errors involved in the multipole expansion of the charge, i.e., in the representation of the molecule as a point quadrupole. David and Person (1968) have investigated the intensities of the farinfrared lattice vibration lines of the halogen crystal. These authors found a very good correlation between the measured absolute intensities and the molecular properties, quadrupole moments, and polarizabilities, in accordance with the theory of Schnepp (1967). Friedrich (1967) has discussed the absolute infrared intensity of the librational mode of crystalline OCS. Since this molecule has a dipole moment the librational intensities can be calculated as discussed above. Without considering the induced intensity, the agreement between calculated and measured intensities here is excellent. Takahashi and Schrader (1967) have discussed the intensities of the lattice vibrations of thiourea. These authors also limited their discussion to the intensity arising from the libration of the polar molecules. No absolute intensity measurements were carried out, however, The assumption made by both Friedrich and Takahashi and Schrader to the effect that the intrinsic intensity due to the dipole moment of the molecule dominates and far exceeds the induced intensity is borne out by the results of Ron and Schnepp (1967) for CO. Although the dipole moment of CO is very small (0.1 D), the intrinsic transition moment is already twice the expected quadrupole-induced moment. This theoretical value is now believed to be high on the basis of the results of St. Louis and Schnepp (1969) for a-N, . Brit and Ron (1969) have discussed the infrared absorption intensities of the translational lattice mode lines of ethylene.

0. Schnepp

176

They found that the measured intensities agreed qualitatively with those calculated from the quadrupole-induced theory of Schnepp (1967). Hardy et al., (1968) have very recently observed the far-infrared-active optical phonon spectrum of solid hydrogen and deuterium in the ordered state, i.e., below the phase transition at 1S"K. The structure of this low temperature phase is believed to be analogous to that of cc-N,. Therefore, the theory for induced absorption intensity of Schnepp (1967) is expected to be applicable. It was indeed found in this case that agreement between experimental and calculated intensities is within a factor of 2 for solid hydrogen, with the experimental intensity being lower, and the agreement was excellent for solid D2 , The superior agreement of experimental intensities with the quadrupoleinduced mechanism theory is to be correlated with the fact that the anisotropic part of the intermolecular potential of hydrogen molecules is very adequately described by a quadrupole-quadrupole interaction term (Van Kranendonk, 1960), whereas such a potential is not applicable to nitrogen (Ron and Schnepp, 1967; Kuan and Schnepp, 1968).

11. Intramolecular Vibrational Spectra

A. INTRODUCTION A number of reviews have been written on this subject during the last decade. The references are as follows: Dows (1966), Dows (1963, 1965), Mitra and Gielisse (1964), Vedder and Hornig (1961). It is largely the object of the present section to bring these reviews up to date. The near-infrared spectrum of solid hydrogen is a topic of such importance and depth that it will be treated in a separate section (Section 111). The original incentive to the development of the field of infrared spectroscopy of molecular crystals was twofold. On one hand it seemed promising to study the absorption spectra of single-oriented crystals in polarized light in order to obtain information helpful for the assignment of intramolecular vibrations. Since the molecules in the crystal are rigidly fixed in space, it was thought that it should be possible to relate the results of polarized light investigations to the symmetry of the vibration observed (Pimentel et al., 1955). It became clear, however, that a good theory of the molecular vibrations on the solid was indispensible, and this was indeed available at the time (Halford, 1946; Hornig, 1948; Winston and Halford, 1949). Dows (1966) has sketched the background theory briefly but with great clarity.

THE SPECTRA OF MOLECULAR SOLIDS

177

B. GENERAL THEORY The ground state wave function for the crystal is given by the product of ground state vibrational molecular functions:

Here q designates the site in the unit cell and p numbers the unit cells. An excited state function for the crystal might be written as a product representing all molecules but one in the ground state and the r, s molecule in an excited state i,

This function describes a localized excitation, localized on the molecule r, s, and does not possess the translational symmetry of the crystal. The correct crystal function belonging to representation k of the infinite translation group is a Bloch-type function

c

. 1 Qr’ = - exp(ik R,)Y;,, . J

N

S

If the crystal structure contains a primitive unit cell with more than one physically distinguishable molecule, i.e., if r as used in Eqs. (12) and (13) has more than one value, then further linear combinations between functions of the type given in Eq. (13) have to be formed in order to diagonalize the crystal problem. A set of solutions are found by diagonalizing the matrix of the crystal problem. A set of solutions are found by diagonalizing the matrix of the crystal problem for every value of k in the first Brillouin zone. It is easily shown that only the states of k = 0 can interact with light, and this selection rule is based on the fact that the wavelength is very much longer than the unit cell dimension. The potential Vis written as a sum whose first term is the sum of the intramolecular potentials and is called Vo and the additional term represents a sum of two-molecule interactions

It is justified in all cases discussed in the literature to treat the sum over two-molecule interaction terms as a perturbation since for molecular crystals the interaction energy is small compared to molecular energies. The perturbed energies obtained by first-order perturbation theory are give by the roots of the secular determinant

IH’ + E(5 - 5,Jl = 0

(15)

0. Schnepp

178

where X is the matrix of the perturbed roots Izi = 4n2cZvi2and

H$ = +ia

1’V,,,;,,,

dr.

u,u

Here + i a is one of the linear combinations of terms (13) which now belongs to one of the irrreducible representations of the factor group of the crystal. In this case, the determinantal Eq. (15) factors into blocks according to the irreducible representations. The first-order solution for the frequency of the ccth component of the multiplet associated with the ith nondegenerate vibration of the free molecule is given by = hcvio

+ D’ + M i a

with

~

‘

9

”

=

c

B a r B a u /(+:,s

+:,u>*

v r , s ; u , v <+y,s

+t,u>

d7.

(19)

u,u

Here, the Barare the coefficients in the symmetry combination 01 of functions where obviously the index r refers to a particular site in the unit cell or, in other words, to a sublattice consisting of translationally equivalent molecules. The term D’is an energy shift term and gives the frequency shift of the vibrational level in the crystal as compared to that in the free moelcule. Equation (18) shows that this is a term analogous to a Coulomb term. On the other hand, the term M i . agiven by Eq. (19) respresents a splitting term and has a different value for each k = 0 component in the crystal exciton band. If there are Z molecules in the primitive unit cell, then there will be Z different k = 0 states. Each such state is characterized by a different symmetry will have a different combination of functions (13) and for each such state Mi.“ value depending on the coefficients Barwhich appear in the symmetry function. The selection rules are then found by considering the final crystal states of a given symmetry. These states are of course classified according to the factor group of the crystal. This group consists of operations as in a point group except that it is coupled with translations and as a result may contain screw axes instead of pure rotation axes and glide planes instead of reflection planes. General methods of determining selection rules in crystals and the formulation of crystal functions have recently been summarized by Kopelman (1976a). This author defines an “interchange symmetry” which is made up of the factor group operations interchanging equivalent crystal sites. Kopelman thus formalized and described methods which have been used by workers in the field. Crystal symmetry coordinates and crystal symmetry have recently been discussed by Oehler and Giinthard (1968). Qri

THE SPECTRA OF MOLECULAR SOLIDS

179

C. DIPOLE-DIPOLE TREATMENTS The vibrational exciton theory was based on the electronic exciton theory of molecular solids (McClure, 1959). In this case, the first attempts at quantitative calculations were made for an intermolecular potential which was expanded in a multipole series retaining the first term, the dipole-dipole term. Thus the transition dipole moment which could be obtained from absorption intensities was used to carry out these calculations. Similarly, Hexter (1962, 1963) attempted to correlate the absorption intensities with the energy differences between the various components CI of crystal absorption lines. Hexter had varying success in these correlations and it seemed that the dipole-dipole interaction is only important in the case of very intense infrared absorption bands. Fox and Hexter (1964) formulated the exciton theory for a cubic symmetry crystal using the dipole-dipole term as interaction potential. For high symmetries the crystal is optically isotropic and only states belonging to the triply degenerate representation of the translations x, y, z are infrared active. Three cases are discussed according to the number of crystal lines to be expected corresponding to a given molecular vibrational exciton. The first case is the so-called one-T case. The space group of solid carbon dioxide is T : and that of carbon monoxide is similar except that here the molecule has no center of symmetry and as a result also the crystal space group is of lower order, namely, it is T4. In both cases there are four molecules per primitive unit cell which have different axis directions in space. For a nondegenerate molecular vibration four k = 0 states are obtained and the reducible representation of these can be reduced by group theoretical techniques to give A,, + T,, or in the case of CO, A T. Only the triply degenerate representation T,, or T is infrared active, as already pointed out. As a result only one line is predicted for the crystal, and this is indeed found experimentally for CO (Ewing and Pimentel, 1961) and for the antisymmetric stretching vibration v 3 of C 0 2 . The energy difference between the two states A and T can be calculated but cannot be observed. Other examples of the one-T case are v3 and v1 of solid N,O, which is isomorphous with solid CO and belongs to the space group T4 with four molecules per unit cell, and v 3 of solid SiF,, which is body-centered cubic (bcc), space group Td3.As already pointed out no splitting is expected from the crystal field theory. However, it is important to note that a crystallite shape dependence, that may produce further splitting is predicted by Fox and Hexter (1964). This shape splitting or shape dependence of the energies is due to the fact that a dipole interaction is assumed as the dominant perturbation mechanism and the resulting dipole-dipole lattice sum does not converge. As a result the energy is a function of the surface shape of the crystallite.

+

180

0. Schnepp

Maki (1961) and Ewing (1962) have indeed observed a shoulder on the high frequency side of the crystal absorption of a-CO. If this structure is indeed due to internally ordered crystals, it must be attributed to the shape splitting and thus confirms this theoretical prediction. No indication of splitting has been observed for solid NzO. The antisymmetric stretching frequency v3 of CO, has been observed by Osberg and Hornig (1952). These authors report a weak satellite but assign this to a reflection phenomenon. The second case considered by Fox and Hexter (1964) is the two-T case encountered when the doubly degenerate vibration v, of COz is treated. Because of the two fold degeneracy of the molecular excited state, eight k = 0 states occur in the solid and these reduce to give the representations E. 2Tu.Here two absorptions are predicted and an energy difference can be observed. The energy difference between the two-T states is given by

+

+?/,

p 2h, a - 3 .

(20)

Here p is the transition dipole moment, a is the cubic lattice constant; and h, is a lattice sum which here has a value of 4.333. The intensities of the two-T components are given by 1, = +(2 T

J3,

(21)

and the intensity ratio between these bands is given by Eq. (22)

I-/Z+ = 1.47.

(22)

Solid COz does, indeed, have two well-resolved peaks in the region of v, with a frequency difference of 5 to 6 cm-' and a relative intensity of 1.37 to 1.49. If the sample is prepared by fast deposition from the vapor on a cool surface, then three additional weak absorptions have been observed within a few reciprocal centimeters of the strong absorption doublet. Fox and Hexter (1964) attempted to fit all these components to the theory which takes into account the dependence of the spectrum on crystallite shape. It was found, however, that not all components could be fitted with respect to energy and intensity distribution. For the two intense components the energy splitting is about 100 % too high as calculated from this dipole interaction theory. On the other hand the intensity ratio is in agreement with the theory within experimental error. The spectrum of the chlorate ion C103- has been reported by Hollenberg and Dows (1960). Small particles that were suspended in a mull were used in the experiment. For v3 a splitting of 17 cm-' was observed, and this is again SO-lOO% from the calculated result for a flat slab-shaped crystallite but fits for a needle shape. For solid N,O only one absorption has been reported for v z , but this spectrum has not been studied at high enough resolution to observe the predicted splitting.

THE SPECTRA OF MOLECULAR SOLIDS

181

The third case considered by Fox and Hexter is the three-T case. This is encountered in phase I1 of solid methane. In this case, as the descriptive term states, three infrared-active components are predicted. There have been some reports of observation of three components but no splitting is predicted in the dipole-dipole approximation for this solid. Also the reported splitting magnitudes are unreasonable for a shape splitting. As a result it is concluded that the splittings observed are due to forces different from electrostatic dipole-dipole interactions. It is concluded that for intense infrared transitions the splitting energies observed are to a large degree accounted for by the dipole interaction approximation. However, this interaction term is far from generally satisfactory. This conclusion is in agreement with previous observations and in particular the same order of magnitude for dipole-allowed and for molecular dipoleforbidden transitions in the crystal (Dows, 1966). The most cited example is that of solid benzene where El, and E,, vibrations exhibit very closely similar splitting energies in the neighborhood of 3 cm-', whereas the former is dipole-allowed in the molecule, and the latter is dipole-forbidden, and therefore the dipole interaction theory would clearly predict decisive differences (Dows, 1966). D. ATOM-ATOM INTERACTION POTENTIAL TREATMENTS Another type of intermolecular interaction potential is expressed in terms of a sum over nonbonded pairs of atoms of neighboring molecules in the crystal. This formulation implies that an intermolecular pair potential can be expressed as a sum of atom-atom interaction terms. DeBoer (1942) showed that the repulsive overlap energy of two hydrogen molecules can be expressed as a sum of four atom-atom exponential terms. This was the result of a very approximate quantum mechanical calculation. Dows (1962) applied this type of potential to solid ethylene. A number of atom-atom interaction potentials have been proposed in the course of recent years, all of which are of the form - AIR6

+ B exp( - CR).

(23)

Kitaigorodskii (1966) reviewed earlier work along these lines. This author proposed sets of constants A , B, and C for Eq. (23) for C-C, C-H, and H-H interaction terms which were derived from the crystal structures of hydrocarbons. Pawley (1967) used the potentials proposed by Kitaigorodskii to calculate the complete lattice dynamics of naphthalene and anthracene. This work is discussed further in Section I concerned with lattice vibrations. Williams (1967) determined potentials of the type of Eq. (23) from the crystal structures and properties of nine aromatic hydrocarbon solids. He determined the C-C repulsive exponential coefficient from a calculation based on the

182

0.Schnepp

interplanar spacing and compressibility of graphite. He also adopted the H-H repulsive exponential coefficient given by deBoer (1942). Altogether 77 properties of nine aromatic hydrocarbon solids were used, including the crystal structures, elastic constants, and energies of sublimation.

E. APPLICATIONS 1. Single-Crystal Studies of Condensed Gases

Halford and his group made a large number of contributions to the study and understanding of infrared spectra of molecular crystals. This group developed a technique for growing single crystals of such substances as ethylene (Brecher and Halford, 1961), cyclopropane (Brecher et al., 1961), acetylenes (Freund and Halford, 1965a), and diborane (Freund and Halford, 1965b). The work on ethylene was particularly profitable inasmuch as it has proved to be difficult to determine the full crystal structure by x-ray techniques.

2. Ethylene The x-ray investigation of solid ethylene determined the position of the carbon atoms only and this skeleton forms a crystal of space group Dii-Pnnm, However Brecher and Halford (1961) were able to decide on the basis of this x-ray structure and the infrared absorption of single crystals in polarized light that the molecular site was centrosymmetric and the splittings observed determined unambiguously that the structure had to be of symmetry C,, . The crystal structure was then unambiguously determined as Cz,-P2,/n. Dows (1962a) also investigated solid ethylene at high resolution and was able to observe the crystal splittings better although he used polycrystalline samples and therefore could not use polarized light techniques. The splittings which he observed were consistent with the space group assignment of Brecher and Halford. Dows (1962b) used the H-H repulsion potential of deBoer to calculate force constants and vibrational frequency splittings by the perturbation methods outlined. He obtained reasonable agreement between calculated and observed values and concluded that the H-H repulsion parameters derived from the experimental splitting energies were consistent both with the potential proposed by deBoer and that of Amdur et al. (1961), which was of the form I/ = 1.44/r6." where the energy as given in electron volts and r is in angstroms.

3. Polyetliylene Tasumi and Shimanouchi (1965) have treated the splitting observed in the infrared absorption spectra of crystalline polyethylene and normal paraffins.

THE SPECTRA OF MOLECULAR SOLIDS

183

These solids are orthorhombic and contain essentially infinitely long polymethylene chains; and the factor group of these solids is D Z h .A UreyBradley force field was used which was based on previous molecular normal coordinate analyses. As intermolecular perturbation potential only short range H-H interactions were included in the calculation. In the infrared spectra doublets are observed for the CH, scissors and rocking vibrations due to intermolecular interaction (Krimm et al., 1956). These splittings are about 10 cm-'. Four H-H distances of two CH, groups were included in the calculation. The force constants were adjusted to fit the observed splittings in the infrared and to fit the Raman data (Brown, 1963). These force constants were compared to force constants calculated from exponential-6 potentials of the type given in Eq. (23) as tabulated in the literature (Coulson and Haigh, 1963) and also to the potential proposed by deBoer (1942). Reasonable agreements were obtained with some of these potentials, the latter being among the successful ones. Tasumi and Krimm (1967) studied the relative importance of transition dipole coupling and intermolecular H-H interactions. They found that the former is of minor importance in determining the crystal field splittings of vibrational lines.

4. Benzene The splittings of the vibrational absorptions of crystalline benzene and naphthalene have been studied by Harada and Shimanouchi (1966) and very recently by Rich and Dows (1968). The infrared absorption spectrum of single crystals of benzene were investigated in polarized light by Zwerdling and Halford (1955). These authors did not know the orientation of their crystal independently but determined the developed plane and the axes in that plane by invoking oriented gas intensity rules (see below). Dows (1966) has described the high resolution spectra of polycrystalline samples which exhibit very distinct and well-resolved splitting patterns. The A,. fundamental which is nondegenerate is expected to give rise to three absorption lines in the infrared and this number of lines is indeed observed. Zwerdling and Halford were able to distinguish to some extent between the different components by the use of polarized light but their sample was too thick to give them clear resolution. The in-plane E l , vibration v,,, which occurs near 1030 cm-' and vl,, which is of symmetry E,, , are seen to exhibit six components each, in accordance with predictions for a degenerate molecular mode. The former vibration is infrared active in the molecule, whereas the latter is forbidden, but in the crystal all u vibrations are allowed although they are considerably weaker. However, as already referred to previously, the splitting is of the same order of magnitude for both these bands, namely, in the neighborhood of 10 cm-' for the whole pattern. Kopelman (1967b) has recently discussed

184

0. Schnepp

Zwerdling and Halford's assignments and interpretation. Harada and Shimanouchi calculated the splitting energies of the benzene vibrations in the solid in order to account for the observations of Hollenberg and Dows (1962). At the time of this work, the crystal structure had only been described at - 3"C, whereas the experimental observations of the splitting were made at 85°K and 155°K. Three sets of atom-atom intermolecular potentials were used. The potential constants were fitted to the vibrational absorption splittings as well as to the known lattice vibrations. The latter have been discussed in Section I. F of this review. It was found that the force constants obtained here fit very well on the curve of the second derivative of deBoer's exponential repulsive H-H potential. Rich and Dows (1968) included twenty H-H contacts, comprising four contacts at each of five different distances between 2.4 and 2.7 A. They used the potential V(ergs) = 1.2 x lo-'' exp( - 3.54R) which is deBoer's potential. These authors get very good agreement for some of the modes, particularly the out-of-plane A,, mode, but the inplane vibrations were somewhat less successful. They also calculated the splitting expected for the totally symmetric breathing mode at 992 cm-' which is Raman active. Four components are predicted all of which would be observable. These components are predicted to range over a frequency of 1.2 cm-'. Raman spectra have been reported by Ito (1965), by Gee and Robinson (1967), and by Ito and Shigeoka (1966a,b). However, no splitting was observed although the investigations included low temperature work down to 2°K (Gee and Robinson, 1967). Similar results were obtained for the inplane degenerate Eze mode at 606 cm-' for which eight components are expected to cover a range of 8.6 cm-'. Gee and Robinson indeed, have observed four lines spread over 6 cm-' in this region. No definitive assignments, however, have been possible to date. Ito and Shigeoka (1966a,b) have reported observing crystal field splittings for v , (e,,) and vg (e,,) in the Raman spectra of benzene and perdeuterobenzene solids.

5 . Naphthalene Harada and Shimanouchi (1966) also considered solid naphthalene but confined their work to the calculation of lattice vibrations in view of ambiguities they felt existed concerning the intramolecular force field in their previous work. Rich and Dows (1968) made an extensive investigation of the splittings of molecular vibrations in solid naphthalene. Four sets of potentials were used, namely those due to Kitaigorodskii (1966), to Williams (1966) (two sets), and to deBoer (1942). They included 156 H-H pairs, 348 H-C pairs, and 192 C-C pairs. These contacts included all nonbonded interactions up to 5A. The authors conclude that the C-C interactions are unimportant but that the

THE SPECTRA OF MOLECULAR SOLIDS

185

C-H interactions must be considered. This conclusion is in agreement with the work of Craig et al. (1965) and Harada and Shimanouchi (1966). Agreement was obtained in the general trends of the splitting energies. Large calculated splittings corresponded to large observed splittings of the order of 10 cm-' and small splittings of the order of 3 cm-' corresponded to small observed splittings but the quantitative agreement is only moderate. As already described, the information concerning g-mode splittings is limited. However, it seems that the factor group splittings for such modes are generally small (Gee and Hanson, 1967). According to the calculations some splittings of g modes are larger than 1 cm-', but generally the predicted values are indeed smaller than those calculated for u modes. The larger splittings observed for out-of-plane vibrations are also reproduced by the calculations. Very significantly, the sign of the splittings, i.e., the polarization of, for example, the higher frequency component was correctly predicted in almost all cases. The spectrum of naphthalene crystal was reinvestigated by Rich and Dows in order to improve on previous work. The best quantitative fit was obtained by using the potential constants of Williams (1966). It is therefore concluded that potentials which are useful in accounting for properties such as elastic constants, coefficients of expansion, and heat of sublimation can also account for the splittings of the vibrational infrared bands. Ito et al. (1967) have recently observed polarized Raman spectra of single crystals of naphthalene, anthracene, p-dichlorbenzene, and p-dibromobenzene using argon laser excitation. They have assigned the Raman-active molecular and lattice vibrations of these solids and have generally found their results to be in agreement with previous assignments. 6. Miscellaneous Studies. Vibrational Assignments The investigation of the spectra of molecular crystals has for some years been a valuable aid in the assignment of molecular vibrations, as already pointed out. Some recent applications of this method will be listed here. Bree and Kydd (1 968) have measured the infrared spectrum of anthracene and have assigned all bands. Loisel (1968a, 1968b) has investigated naphthalene and quinoline. LeRoy and Jouve (1967) and LeRoy and Thourenot (1967) have reported on solid N,O and cyclopentene; 1,3,4-oxadiazole has been investigated by Christensen et al. (1968), urea by Nefeolov and Failkovskaya (1966), pyrimidine by Sobrana et al. (1966), and glycine derivatives by Novak and Cotrait (1966). Dows (1965) has completed the assignment of cylohexana fundamentals on the basis of a solid state investigation. Kyogoku et al. (1967) have investigated the infrared spectra of single crystals of l-methyl thymine, 9-methyl adenine, and their one-to-one complex.

186

0. Schnepp

7. Miscellaneous Studies. Crystal Structures

In cases in which x-ray investigations are incomplete, infrared and Raman spectra of solids have been helpfufto characterize crystal structures. Marzocchi et al., (1966) have investigated the structure and phase transition in CH21, and Kartha (1967) has assigned structures for CH,CI, , CH,Br2 , and CH212. Dows (1965) has partially characterized the low temperature crystal structure of cyclohexane and Marsault and Dumas (1967) have observed changes in the absorption spectrum of 1,4-dioxan. Brunel and Peyron (1967a,b) have obtained interesting results which bear on the crystal structures of HCl and HBr. Information on the crystal structure of ethylene oxide has been deduced by LeBrumant and Maillard (1967), of SO2 by Gerding and Ypenburg (1967), of cyclopentane by LeRoy (1967) of thiophene, furan, and pyrrole by Loisel and Lorenzelli (1967), and of 9,lO-anthraquinone by Pecile and Lunelli (1968). The absorption spectrum of solid HF has been reported by Kittelberger and Hornig (1967), and structural implications of their observations have been discussed by them also. The Raman spectra of solid CH, and CD, were reported by Anderson and Savoie (1956). These authors observed changes in the vibrational spectrum over the temperature range 10-77"K7 which includes all three solid phases of CD, and phases I and 11 of CH, .

F. INTENSITIESOF INTRAMOLECULAR VIBRATIONAL SPECTRA The intensities of the infrared absorptions of molecular crystals have been the subject of some attention. Dows (1966) has discussed this question and has evaluated the validity of the " oriented gas " model. This model assumes that the transition moments of the molecules are unaffected on condensation and therefore the dipole moments of the transition in the condensed phase can be obtained as a vector sum of the oriented molecular transition moments. In some cases, however, large quantitative changes have been observed on condensation and therefore the oriented gas model cannot be accepted as valid. In benzene, for example, one band is weakened by a factor of 3, on condensation, whereas another band is doubled. The out-of-plane A , , band remains practically unchanged on the other hand. The change of intensity is particularly severe for hydrogen-bonded solids and for HCI and solid state intensity is six times that of the free molecule. To date, it has not been possible to account for these considerable intensity changes on crystallization, theoretically, and the problem is still unresolved at this time. On the other hand, Person and co-workers (Cook et al., 1967) have found a number of instances in which the absorption intensity did not change greatly on solidification.

THE SPECTRA OF MOLECULAR SOLIDS

187

Recently, an investigation of the vibrational infrared spectrum of solid naphthalene has been reported by the technique of attenuated total reflection (Yamada and Suzuki, 1967). Single crystals and polarized light were used. This seems to be the first report of the application of this technique to solids, but it has been very successfully employed in the investigation of the absorption intensities of liquids (Crawford et al. 1966).

G. INDUCED SPECTRA Intermolecular forces can induce absorption in the crystal where the molecular vibration is inactive in the vapor. Many examples are known from the study of polyatomic molecules in the crystal state. For example, all u modes of benzene become weakly infrared active by this mechanism (Harada and Shimanouchi, 1966; Rich and Dows, 1968; Zwerdling and Halford, 1956). A number of examples of induced infrared absorptions are now known for solids of homopolar diatomic molecules. The well-studied case of H, is discussed in detail in the next section of this review. In this case the fundamental molecular vibration becomes active because the molecular site is not centrosymmetric in the crystal of hexagonal close-packed structure (Van Kranendonk, 1960; Van Kranendonk and Karl, 1968). The fundamental absorption of O2 in the infrared spectra of u- and 8-oxygen has been reported by Cairns and Pimental (1965). These authors conclude that the absorption appears in violation of selection rules since the 8 phase was known to have the 0, molecular sites centrosymmetric. Since the absorption intensities were similar for the two phases, Cairns and Pimentel concluded that the molecular sites are also centrosymmetric in the u phase. This has since been confirmed by the work of Barrett et al., (1967). The weak transition moment is ascribed to the perturbation by lattice imperfections. Walmsley and Anderson (1964) have reported weak absorptions assigned to the fundamental molecular vibrations in crystals of chlorine, bromine, and iodine. Again, the molecular sites are centrosymmetric and static crystal imperfections or lattice vibration coupling must be responsible for the nonvanishing transition moments. 111. Spectra of Solid Hydrogen A. INTRODUCTION Because of its importance, a separate section is devoted here to solid hydrogen, which has proved to be spectroscopically of great interest. This solid is unique inasmuch as the molecules rotate virtually freely as is evidenced by the fact that the rotational constants remain virtually unchanged on condensation. Rotation in the solid perseveres to very low temperatures because the

0. Schnepp

188

spacing of rotational levels is large (of the order of 350 cm-') compared with intermolecular binding energies. However, intermolecular interaction causes the delocalization of rotational excitation and this gives rise to a rotational exciton band of width 20 cm-'. As a result structure is observed in the spectrum. The solid parahydrogen spectra have been intensively investigated experimentally by Welsh and his group and theoretically by Van Kranendonk and co-workers. No comprehensive review of this beautiful work has been available. The early work has been reviewed by Vedder and Hornig (1961) and, while the present review was in preparation, an updated summary of the theory of the spectra of solid parahydrogen was published (Van Kranendonk and Karl, 1968). This publication covers the frequencies but does not discuss intensities or coupling of the molecular motions with phonons. B. PUREROTATIONAL SPECTRA OF SOLIDHYDROGEN

I . General Theory The theory of the rotational states of solid hydrogen has been described by Van Kranendonk (1959, 1960) and Van Kranendonk and Karl (1968). For a given molecular rotational state J , the 2 N ( 2 J 1) crystal states are described by the Bloch-type functions

+

Jlm'(k)

= N-"'

C exp(ik .Ri) [Om(Ri)f O,,,(Ri + T)]. i

+

(24)

Here m denotes one of the (25 1) substates of the rotational state J and k is the wave vector denoting the irreducible representation of the translation group; R i is the position vector of one of the two molecules in the unit cell i of the hexagonal close-packed crystal and R i + T is the position vector of the second molecule in the same unit cell characterized by Ri. There are N unit cells in the crystal. Since the wavelength of light is large compared to the unit cell dimensions, it can be shown that only the states of k = 0 will appear in the spectrum. Furthermore, since the crystal has a center of symmetry midway between the two molecules of the unit cell, only the even states to an interchange of these two molecules will be Raman active whereas the odd states will be infrared active. Quadrupole-quadrupole interaction between the molecules in the lattice causes the broadening of the molecular J = 2 state into a band which is calculated to have a width of approximately 30 cm-'. However, the experimentally observable k = 0 levels are predicted to form a triplet of levels representing transitions from J = 0 to the states m = 0, + 1 , + 2 of J=2.

THE SPECTRA OF MOLECULAR SOLIDS

189

2. Raman Spectra The pure rotational Raman spectra of solid hydrogen and deuterium were described by Bhatnagar et a/. (1962). For solid parahydrogen the spectrum contains three sharp lines of half-width 0.3 cm-' at 351.84, 353.85, and 355.83 cm-'. The spacings are nearly equal and amount to 2 cm-'. The'Raman spectrum of solid orthodeuterium also contains three narrow lines that are partially resolved and have frequencies 176.8, 179.4, and 182.0 cm-'. In normal hydrogen (25% para and 75% ortho), two asymmetric and broad bands are observed which peak at 353.5 and 586.8 cm-'. Clearly, the higher frequency band is due to ortho-H, . The Raman spectrum of solid HD in this region consists of one broad band centered at 268.7 cm-'. All these features, except the higher frequency band of normal H2,are ascribed to the transition S,(O) (J = 0 to J = u in the ground vibrational state u = 0) of the parahydrogen molecule and the higher frequency band of normal H , represents the transition So(l), ( J = 1 to J = 3, u = 0) of the orthohydrogen molecule. The three components of So(l) observed in the Raman spectrum of solid parahydrogen have been assigned to the states +m+(0) with the different absolute m values 0, f 1 , f 2 (Van Kranendonk, 1959,1960; Van Kranendonk and Karl, 1968). The spacing between these levels calculated on the basis of quadrupolar forces is about 10 % larger than the observed spacing of 2 cm-' The triplet pattern is centered about the level m = k 2 . It is interesting to note that the triplet spacing as experimentally observed is not accurately symmetric. Van Kranendonk considered a general crystal field perturbation of arbitrary intermolecular forces in an ideal lattice and found that even then the spacings are expected to be equal. The deviation of 0.03 cm-' is then attributed to lattice imperfections or lattice vibrations.

3. Infrared Spectra The So(0)line has been measured by Kiss (1959) in the infrared absorption of solid parahydrogen. The measured frequency is 355.6 0.3 cm-'. Van Kranendonk (1960) has given the theory of this absorption. It is interpreted as occurring due to the quadrupole-induced transition moment. The induced dipole moment is assumed to be additive, i.e., p=Cpi i

(25)

where i designates one of the molecules in the crystal and the sum extends over the whole crystal. Investigation of the transition moment. (+o

I clk I \Ir,*(k)>

(26)

190

0.Schnepp

immediately gives the result that transitions are infrared active only to states of k = 0 and those that are odd to the interchange of two molecules in the unit cell, in other words, states of the type $,,,-(O). The symbol pk designates a component of the dipole moment p, Furthermore, quadrupole-induced transitions will only occur from the ground state Jr, to the excited states of in = 2 2 . These transitions are also shown to be polarized perpendicular to the hexagonal axis. The calculated frequency is 355.7 cm-', which is in very satisfactory agreement with the experimental value. This frequency is also in satisfactory agreement relative to the frequency of the Raman lines. The integrated absorption intensity has also been caclulated by Van Kranendonk (1960) on the basis of the quadrupolar induction mechanism. The calculated integrated absorption intensity is 0.77 x sec-' cm3, whereas the experimentally measured intensity is 0.52 x 10-'3sec-'cm3 (Kiss, 1959). The discrepancy of 30 % is ascribed to the sharpness of the absorption line which precluded satisfactory spectroscopic resolution, and thus presumably caused an error in the intensity measurement. It is of interest to note that the lattice sum that enters into the calculations of the integrated absorption intensity would vanish if the crystal structure were face-centered cubic. This property arises from the fact that in this structure the molecular sites would be centrosymmetric, whereas for the hexagonal close-packed (hcp) structure the center of symmetry lies midway between the two molecules i n the unit cell, as already pointed out. Therefore, the observation of the S,(O) line strongly supports the hcp crystal structure (Van Kranendonk and Gush, 1962). Kiss (1959) also measured the infrared absorption due to the excitation of two rotational excitons, i.e., a line assigned to S,(O) S,,(O). The measured sec-' cm3, or of very similar integrated absorption intensity is 0.6 x order as the single-rotational exciton transition. Van Kranendonk (1960) has calculated the quadrupole-induced moment for this transition and has obtained sec-' cm3. The discrepancy is ascribed the theoretical value of 0.49 x to limited experimental accuracy due to the overlap of the phonon branch, which accompanies this transition. It turns out that the coefficients in the expansion of pk are small as compared to those which are responsible for the single-exciton transition moment. However, the lattice sum which occurs in the calculation of the single phonon transition is small due to cancellations caused by symmetry. This lattice sum squared occurs in the expression for the integrated intensity. On the other hand, the expression for the integrated intensity of the double-rotational exciton absorption contains a lattice sum which is the sum of square terms and therefore no cancellations occur. As a result the final values for the two cases are very similar. The pure rotational infrared spectrum of solid hydrogen is not available in the published literature. Van Kranendonk (1960) refers to measurements of frequency and intensity by Kiss (1959) which are contained in a thesis. From

+

191

THE SPECTRA OF MOLECULAR SOLIDS

the theoretical paper of Van Kranendonk (1960) and from a latter paper by Poll and Van Kranendonk (1962), it appears that both single- and doublerotational exciton absorption lines are accompanied by phonon branches which occur due to the coupling of the rotational motion with the lattice vibrations of the solid. Phonon branches will be discussed in connection with the infrared absorption of solid hydrogen in the region of the molecular vibrational fundamental band.

C. VIBRATION-ROTATION SPECTRA OF SOLIDHYDROGEN I . Raman Spectra

Bhatnagar et al. (1962) observed very sharp lines (half-width 0.1 cm-') in the Raman spectrum of solid parahydrogen at 4149.81 and 4485.85 cm-'. These lines were assigned to the Q,(O) and S,(O) transitions, respectively. In solid normal hydrogen also the Q,(l) line has been recorded. The corresponding lines have also been observed in the Raman spectrum of normal deuterium, orthodeuterium, and HD. Van Kranendonk (1959, 1960) and Van Kranendonk and Karl (1968) have developed the theory of the vibrational exciton band and of the vibrationrotation levels of solid parahydrogen. The width of the virbrational band of the Y = 1 molecular level was found to be 2.4 cm-'. The broadening of the molecular level into a crystal band is due to dependence of the intermolecular dispersion forces on the internuclear distance in the molecule. Again, only the k = 0 levels in the band will be active. Also here, analogous to the situation in the rotation band, two such levels exist, one which is symmetric and the other antisymmetric with respect to an interchange of two molecules in the same unit cell. Their energies are given in terms of a parameter E' which is called the vibrational coupling constant. Relative to the band origin the energies of these two states are E+(O)= -68' and E-(O) = 0. The reference energy (E' = 0) is skifted from the gas phase due to isotropic intermolecular forces. This shift has been determined empirically from the S,(O) line of the Raman spectrum. Again, only the state symmetric to interchange of molecules is Raman active. Using the empirical values of the reference energy as described above, the theoretical frequency value corresponds to the observed frequency for the Q,+(O) line of 4149.81 cm-' with the vibrational coupling constant E' = 0.40 cm- The calculated value is 0.50 cm- and this agreement indicates that the main contribution to the vibrational coupling is indeed due to dispersion forces. In accounting for the S,(O) line which appears as a single sharp line in parahydrogen, Van Kranendonk (1959, 1960) considers the coupling between vibration and rotation in the solid. As has already been described, exciton bands are formed which have their origin in the molecular rotational levels (width 20 cm-' for J = 2) and vibrational levels (width 2.4

'.

'

192

0 . Schnepp

cm-’ for u = 1). In addition, there exist 5N degenerate “impurity levels.” These represent the u = 1, J = 2 excitations on the same molecule and these levels lie below the rotational band. These states are localized and do not broaden to form a band. The band states are Raman inactive in the approximation that the polarizability is independent of intermolecular interaction. The inactivity is due to the fact that the band states represent vibrational and rotational excitations on different molecules as opposed to the bound localized states. The bound states are Raman active and give rise to the observed single narrow S,(O) lines. The frequency of the crystal S,(O) line is lowered relative to that of the free molecule by two terms. The first represents a shift due to isotropic forces in terms of a parameter which is empirically determined from the measured frequency. The second term is due to quadrupolar forces and depends on the quadrupole coupling which can be evaluated theoretically and the rotation-vibration interaction constant which is known from the gas phase spectrum. This part of the theory is elaborated upon by Van Kranendonk and Karl (1968). The experimental solid state frequency is 4485.8 cm-’, or it is shifted to lower frequency by 11.7 cm-’. The S,(O) line is also split similar to the S,(O) into three components by the anisotropic forces of the crystal field. By using the splitting parameter determined from the observed splitting of the S,(O) line, the spacing between the lines is calculated to be 0.35 cm-’. It is seen that the splitting here is an order of magnitude smaller than that for the S,(O) lines, as and a result the components have not been resolved by the experiment. Soots et ul. (1965) have studied the Raman spectrum at a series of orthoparahydrogen ratios. The Ql(0) line was found to split into a number of closely spaced lines for ortho concentrations below 0.20. These lines are interpreted as due to localized states with different neighbor interactions; the concentration is too low for coupling between orthomolecules and exciton band formation. On the other hand, a single maximum was observed for higher concentrations of o-H, , which is characteristic of the vibrational exciton of J = 1 species. Rosevear et al. (1967) have found an anomaly in the intensity ratio of the Q,(l) and Q,(O) vibrational lines in the Raman spectrum of solid H, and D, . The anomaly has been found to persist in the liquid state. 2. Infrared Spectra

The infrared vibrational fundamental band of solid hydrogen has been described in detail by Gush et al. (1960). At low resolution the band consists of three absorption bands, each 3040 cm-’ wide and increasing in intensity with frequency. Each of these bands is accompanied by a broad band adjacent to it and extending to higher frequency. The three bands have been assigned as related to the Q, S(O), and S(1)

THE SPECTRA OF MOLECULAR SOLIDS

193

transitions of the molecule. The broad features to higher frequencies of each of these bands have been interpreted as phonon branches. The Q phonon branch is by far the most intense and is much more intense than the narrower Q zero-phonon band. On the other hand, the S phonon branches are much less prominent relative to the respective zero-phonon bands. In addition, two weak features have been observed to higher frequency and these have been assigned as S(0) + S(l) and 2S(O).These are double transitions. The spectrum, as described above, is characteristic of normal hydrogen. Under high resolution, each of the narrower bands can be resolved into a number of components, and these high resolution spectra have been studied as a function of orthopara concentration. The Q band consists of three components, Q(l), Q(O), and a third line of higher frequency. With decreasing ortho composition, the Q(0) line increasingly dominates the Q(1) line as expected, but in addition the total intensity decreases and the whole Q zero-phonon branch disappears in the limit of pure parahydrogen. However, before this happens the three features which compose this branch are narrow and very well resolved at low ortho concentration (2) %. Van Kranendonk (1960) has developed the theory for the infraredfundamental induced band of solid hydrogen. The infrared-active k = 0 state of the vibrational exciton band is that which is antisymmetric to interchange of the two molecules in the unit cell. The pure vibrational transition to this level is designated Q,-(0), and it is predicted to be at 4152.2 cm-' as compared to the observed frequency of 4153.0 cm-'. The intensity is expected to depend on the overlap induction term, but the corresponding lattice sum vanishes for the hcp structure. As a result, the zero-phonon Q branch is infrared inactive in pure parahydrogen. The sharp line corresponding to this transition in solid hydrogen containing 2% orthohydrogen is due to the presence of the orthohydrogen impurity (Sears and Van Kranendonk, 1964). As already described, for such a solid three sharp features are observed. The Two at lower frequency are assigned to Ql(l) and Q,(O) and they are due to the presence of single-orthohydrogen molecules in an environment of parahydrogen. Since the Q,( 1) line is characteristic of the impurity orthomolecules, it remains sharp since the exictation is localized. On the other hand, the Q,(O) transition has a width of about 3 cm-' because of the vibrational exciton band width. The J = 1 level of the orthohydrogen molecules is threefold degenerate and in a perfect hcp lattice this degeneracy is not removed. However, removal of the degeneracy is expected to arise from the large amplitude zero-point vibration of the lattice, which exists in hydrogen due to the low mass relative to the intermolecular forces. This problem has been investigated by Van Kranendonk and Sears (1966). The resulting expected splitting has, however, not been observed. The third Q branch feature at 4155.0 cm-' is ascribed by Sears and Van Kranendonk (1964) to pairs of orthohydrogen

0. Schnepp

194

molecules and the transition is to the u = 1 vibrational exciton band of the parahydrogen crystal accompanied by an orientational transition in the pair of orthomolecules on neighboring sites. The orientational states of the pair of orthomolecules are 9 in number and their degeneracy is expected to be removed by the quadrupole-quadrupole interaction. No structure however, has been seen due to this removal of degeneracy. The integrated intensities of the Q,(O) and Q,( 1) lines have been calculated and agreement with experiment is satisfactory. The S(0) branch of solid normal hydrogen has been observed to consist of three features at high resolution. These are assigned S,(O), Q,(l) + S,(O), and Q,(O) + So(0). In pure parahydrogen the S,(O)line is extremely sharp. The Q,(O) S,(O), on the other hand, is about 20 cm-' wide and has a number of peaks (Gush et al. 1960). Van Kranendonk (1960) has interpreted this spectrum in terms of his theory of rotational and vibrational excitons. The rotation-vibration exciton band is, as already described, about 20 cm- in width. The band states give rise to the transition designated Q,(O) + S,(O) to indicate that the vibrational and rotational excitations are never located on the same molecule and the selection rule k = 0 requires only the sum of the wave vectors of the rotational band and of the vibrational band states to vanish. For this reason the total exciton bandwidth is observed. On the other hand, the transition designated S,(O) is to a bound state in which the rotational excitation J = 2 is localized on the vibrating molecule in the state u = 1. This localized state lies below the exciton band, and the transition to it is therefore a sharp line. Now in the infrared, only the substates m = 1 are active similar to the situation in the pure rotational So(0) transition. In the vibrational Raman spectrum all substates of J = 2 are active, but as already described the splitting is very small. Therefore, it is predicted that the Raman and infrared lines coincide, and this is indeed found experimentally. At very low orthohydrogen concentrations, as mentioned above, the 20-cm- wide Q,(O) + So(0)rotation-vibration band transition exhibits a number of peaks. Gush and Van Kranendonk (1962) have investigated this structure. These authors have considered the partial removal of the 15-fold degeneracy of a pair of molecules, one parahydrogen ( u = 1, J = 2) and the second orthohydrogen ( u = 0, J = 1). This degeneracy is removed by the intermolecular interaction due to the quadrupole-quadrupole terms. It was found that the overall structure of the observed band could be reproduced on the basis of this theory, and further improvement was obtained when the quadrupolehexadecapole term was included in the calculation. Van Kranendonk ( 1 960) calculated the quadrupole induced transition intensity for the S,(O) and Ql(0) + S,(O) transitions. The theory is similar to that developed by the same author for the pure rotational infrared transitions. Again, the integrated absorption intensity of the localized state S,(O) is

+

'

THE SPECTRA OF MOLECULAR SOLIDS

195

obtained in terms of a lattice sum where cancellation due to crystal symmetry occurs. The intensity of the Q,(O) + So(0)transition to the rotation and vibration exciton band states is found to depend on the sum of square terms in the corresponding lattice sum, and as a result no symmetry cancellations occur. sec-' cm3, The total absorption intensity is predicted to be 0.48 x which is in excellent agreement with the experimental value of 0.45 x lo-''. Also, the theoretical ratio of the intensities of these two transitions agree very well with the observed value. As already described above, very intense phonon branches have been observed in the infrared spectrum in the region of the vibrational fundamental (Gush et al., 1960). The Q phonon branch is 200 cm-' wide, and its intensity overshadows that of the zero-phonon branch. In addition, the Q phonon branch remains intense in pure parahydrogen whereas the Q zero-phonon branch disappears. On the other hand, the Raman spectrum does not contain the phonon branches since the polarizability is not affected by molecular displacements. Poll and Van Kranendonk (1962) have described the theory of the phonon branches. The induced transition dipole moments are strongly dependent on intermolecular separation, and therefore the vibration-rotation transitions couple the lattice and give the phonon branches their intensity. On the other hand, it is found that the sharp internal lines are not broadened by lattice vibrations and that their intensities are not appreciably affected. In the theory, coupling between the internal molecular rotational states and the lattice modes are neglected and a product function is used. The integrated absorption intensity is formulated as the square of induced moments which are obtained by summing over the contributions by the molecules in the crystal. For single transitions such as S,(O), Q,(O), cross terms are thus obtained of the type p i p j where i and j refer to different molecules, whereas for double transitions such as Q,(O) + So(0),the integrated intensity is obtained as a sum of square terms. The double sums contained in these expressions are contracted by summing over the complete set of lattice vibration states n, and as a result second-order terms are obtained whose expectation values have to be calculated only over the lattice vibrations of the ground state. The moments are expanded in molecular displacements and finally the total intensity is obtained in terms of the expectation value of the square of molecular displacements in the zero-lattice vibration states. The expectation values are then equated to average square displacements and these quantities are evaluated from both the Einstein and the Debye models for lattice vibrations. It is found that the Ql(0) phonon branch intensity is entirely due to overlap distortion terms. Agreement is obtained within 20%, the Debye model giving a somewhat better fit than the Einstein model. The S(0) phonon branch intensity is found to be due to quadrupolar forces. Again, agreement within 20% is obtained for the lattice vibration models used.

196

0. Schnepp

Crane and Gush (1966) have studied the infrared absorption spectrum of solid D, and of solid HD. In the latter case, now J = 1 species persists to low temperature, and therefore the spectrum is analogous to that of pure parahydrogen. Van Kranendonk and Karl (1968) conclude that agreement between experimental and theoretical values for a wide range of observables justifies the conclusion that the intermolecular potential assumed is well based. This potential consists of a 6-12 Lennard-Jones term and an angle-dependent term, the quadrupole-quadrupole interaction.

D. SPECTRA OF CUBICHYDROGEN Clouter and Gush (1965) first discovered changes in the infrared vibrationrotation spectrum of normal hydrogen occurring at the temperature at which a &type specific heat anomaly had been known to exist, i.e., near 1.5"K. These authors found that the S,(O)and S,(1) lines disappear completely as the sample is cooled through this temperature. As has been described, Van Kranendonk (1960) showed that these lines appear in the infrared absorption due to quadrupolar intermolecular interaction. This mechanism is able to induce a transition moment only as long as the molecular sites are not centrosymmetric, as in the hcp structure (Van Kranendonk and Gush, 1962). Therefore, the disappearance of these absorption lines in Clouter and Gush's work proves that a sturctural change occurs in the solid at the transition temperature. Furthermore, the lower temperature phase must have centrosymmetric molecular sites. Accompanying the disappearance of the S, lines, changes occur in the phonon branches, and an intensity decrease of about 25 % is observed for the zero-phonon Q band. Both of these changes support the above conclusions. The structure of the low-temperature phase of solid n-hydrogen has been established to be face-centered cubic (Schuch and Mills, 1966; Schuch et al., 1968; Mucker et al. 1966). It is also known that the phase transition temperature is very sensitive to orthopara composition and occurs only in samples containing at least 60 % orthohydrogen ( J = 1 species). The disappearance of the So(l) line in the pure rotational infrared spectrum of paraenriched samples of deuterium has also been observed at the phase transition temperature by Hardy et al. (1968). The far-infrared absorption spectra (50-100 cm- ') of enriched orthohydrogen and paradeuterium due to optical phonons has been discussed in Section I. E.3 of this review dealing with lattice vibrations. Briefly, the observation of optical phonon absorptions in the low temperature cubic phase (Hardy et al., 1968) proves unambiguously that the crystal is composed of a number of physically distinct sublattices. The discovery of this spectrum

THE SPECTRA OF MOLECULAR SOLIDS

197

thus confirms theoretical predictions (Homma et al. 1967; Raich and Etters, 1968; Raich and James, 1966) according to which ordering of the rotation in the J = 1 state is expected in the cubic phase. The four fcc sites have axial symmetry at each of the sites. The axial symmetry causes the splitting of the J = 1 state into substates m = 0 and m = f I, the former being lower in energy by about 10-15 cm-’. Therefore, at the temperature of the phase transition, orientational ordering of the molecular rotation takes place. Mucker et al. (1966) reported evidence for a Pa3 crystal structure from a neutron diffraction study of solid paraenriched deuterium.

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0.Schnepp

Novak, A., and Cotrait, M. (1966). Ann. Chim. (France) 1, 263. Oehler, O., and Giinthard, Hs.H. (1968). J. Chem. Phys. 48,2032. Osberg, W. E., and Hornig, D. F. (1952). J. Chem. Phys. 20, 1345. Pawley, G. S . (1967). Phys. Status Solidi 20, 347. Pecile, C., and Lunelli, B. (1968). J. Chem. Phys. 48, 1336. Pimentel, G. C., McClellan, A. L., Person, W. B., and Schnepp, 0. (1955). J. Chem. Phys. 23. 234. Poll, J. D., and Van Kranendonk, J. (1962). Can. J. Phys. 40, 163. h a s k , H., Boutin, H., and Yip, S. (1968). J. Chem. Phys. 48, 3367. Raich, J. C., and Etters, R. 0. (1968). Phys. Reu. 168,425. Raich, J. C., and James, H. M. (1966). Phys. Rev. Lett. 16, 173. Rich, N., and Dows, D. A. (1968). Thesis, Univ. of Southern California. Ron, A., and Schnepp, 0. (1967). J. Chem. Phys. 46, 3991. Ron, A., and Schnepp. 0. (1968), unpublished. Rosevear, A. H. M., Whiting, G., and Allin, E. J. (1967). Can. J . Phys. 45, 3589. Schnepp, 0. (1967). J. Chem. Phys. 46, 3983. Schuch, A. F., and Mills, R. L. (1966). Phys. Rev. Letters 16, 616. Schuch, A. F., Mills, R. L., and Depatie, D. A. (1968). Phys. Rev. 165, 1032. Sears, V. F., and Van Kranendonk, J. (1964). Can. J. Phys. 42, 980. Shimanouchi, T., and Harada, I. (1964). J. Chem. Phys. 41,2651. Sobrana, G., Adembri, G., and Califano, S., (1966). Spectrochim Acta 22, 1831. Soots, V., Allin, E. J., and Welsh, H. L. (1965). Can. J. Phys. 43, 1985. St. Louis, R. V., and Schnepp, 0. (1969). J. Chem. Phys., in press. Takahashi, H., and Schrader, B. (1967). J. Chem. Phys. 47, 3842. Tasumi, M., and Knmm, S . (1967). J. Chem. Phys. 46,755. Tasumi, M., and Shimanouchi, T. (1965). J. Chem. Phys. 43, 1245. Ueyama, H., and Matsubara, T. (1967). Prog. Thoert. Phys. (Kyoto) 38, 784. Van Kranendonk, J. (1959). Physica 25, 1080. Van Kranendonk, J. (1960). Can. J. Phys. 38,240. Van Kranendonk, J., and Gush, H. P. (1962). Phys. Letters 1,22. Van Kranendonk, J., and Karl, 0.(1968). Rev. Mod. Phys. 40, 531. Van Kranendonk, J., and Sears, V. F. (1966). Can. J. Phys. 44, 313. Vedder, W., and Hornig, D. F. (1961). Advan. Spectry. 2, 189. Walmsley, S. H., and Anderson, A. (1964). Mol. Phys. 7, 411. Walmsley, S. H., and Pople ,J. A. (1964). Mol. Phys. 8, 345. Whalley, E., and Bertie, J. E. (1967). J. Chem. Phys. 47, 1264. Wilkinson, G. (1966). unpublished; quoted by Pawley (1967). Williams, D. E. (1966). J. Chern. Phys. 45, 3770. Williams, D. E. (1967). J . Chem. Phys. 47, 4680. Williams, D. R., Schaad, L. J., and Murrell, J. N. (1967). J. Chem. Phys. 47, 12. Winston, H., and Halford, R. S . (1949). J. Chem. Phys. 17, 607. Yamada, H., and Suzuki, K. (1967). Specfrochim. Acta 23A, 1735. Ziman, J. M. (1960). “Electrons and Phonons.” Oxford Univ. Press (Clarendon), London and New York. Zwerdling, S., and Halford, R. S . (1955). J. Chem. Phys. 23,2221.

THE MEANING OF COLLISION BROADENING OF SPECTRAL LINES : THE CLASSICAL-OSCILLATOR ANALOG* A . BEN-REU VEN The Weizrnann Institute of Science. Rehooot. Israel

I . 1[ntroduction ....................................................... 201 204 I1 m e Fourier-Transform Method ...................................... 4. Radiation Damping ............................................. 204 B Time Correlations and the Spectral Shape Function .................205 C. The Undamped Oscillator ....................................... 208 210 111. [mpact Damping .................................................. A Complete Interruptions ........................................ 210 B. An Interruption Model .......................................... 211 C. Sense-Reverting Collisions ...................................... 212 D Application to Rigid Linear Rotors ............................... 215 IV Complex Oscillators ................................................ 217 A Line Overlapping ............................................... 217 B Collision and Doppler Broadening ................................ 220 V Statistical Broadening .............................................. 221 A . Adiabatic Perturbations .......................................... 221 B. Adiabatic against Impact Pressure Shifts ........................... 224 IC The Method of Moments ........................................ 225 VI Resonance Broadening.............................................. 228 A Resonance Broadening as a Many-Body Effect ...................... 228 B Holtsmark's Model ............................................. 229 IC A Quantum-Mechanical Justification .............................. 231 D The Finite Size of Molecules ..................................... 232 References ........................................................ 234

.

.

.

.

. .

. . .

.

. . . .

.

I Introduction The theory of collision broadening of spectral lines has undergone several metamorphoses since its early beginning in the days of Michelson and *This research has been supported in part by the Air Force Cambridge Research Laboratories through the European Office of Aerospace Research. OAR. United States Air Force. under contract AF 61(052).838 . 201

202

A . Ben-Rewen

Lorentz. The simple model of Lorentz, of an ensemble of classical harmonic oscillators (representing the spectral line) interrupted by randomly occurring collisions, gave way to increasingly more sophisticated theories, using a variety of quantum-mechanical methods.' Most notable advances in the quantum-mechanical theory were the introduction of scattering matrix techniques (Anderson, 1949 ; Baranger, 1958 ; Fano, 1963), thermodynamics of irreversible processes (Fano, 1963), and, most recently, many-body formalism (Ross, 1966; Bezzerides, 1967). A major advance in the theory has been the introduction by Fano (1963) of the Liouville space formalism. In this formalism, the basis for describing the effect of collisions is shifted from energy eigenvalues and eigenfunctions to resonance frequencies and optical transition amplitudes. In this manner, it is duly recognized that the wave functions of the upper and lower levels of an optical transition should not be treated as being perturbed independently, but it is rather their coherent product that matters. A line in the spectrum, associated with the optical transition i - f , is represented in Liouville space by a unit vector l i ) ( f l (which is a Schrodinger operator formed by the product of a bra and a ket vector). The strength and position of the line are closely related to an amplitude (il p a If) (where p a is a component of the dipole moment of the molecular system) and an angular frequency m i / . This description immediately raises association with the classical theory of complex harmonic oscillators. The complex oscillator is represented by a set of normal-mode unit vectors nk, amplitudes Qk, and oscillation frequencies mk . Furthermore, the equations of motion of the Liouville space vectors and of the classical normal-mode coordinates are formally similar. So, the Liouville space formalism is, in a sense, a natural extension of the classical-oscillator models of early collision-broadening theories. This formal similarity should not be pushed too far. It is not correct, for example, to view the oscillator theory as a classical approximation from which the quantum-mechanical theory is obtained by the process of quantization. The quantum operator li)(fl is restricted to a space of two levels only, whereas the field operators obtained by quantization of the harmonic oscillator act in the infinite Hilbert space of the quantum oscillator. The molecular system is not a quantum oscillator. But it is formally similar, in a sense, to a classical oscillator. This analogy may be helpful in providing models with which various collision-broadening effects can be given a simple interpretation, or by which various results can be derived in an easily conceivable manner. The fruitfulness of this approach in stimulating a better underAn exhaustive review of the theoretical work on collision broadening, from the early works of Michelson and Lorentz to the mid-fifties, can be found in the monograph of Breene (1961).

THE CLASSICAL-OSCILLATOR ANALOG

203

standing and creating new concepts may be demonstrated in connection with four apparently unrelated recent developments in the theory. The first occasion to which we refer is the solution to the problem of the collapse of microwave resonance spectra into spectra of a nonresonant type at pressures where damping rates become comparable to the resonance frequencies (Ben-Reuven, 1965b, 1966). In this demonstration of the advantages of the Liouville space formalism, the collapse of the resonance mode is fully accounted for by introducing an additional cross-relaxation parameter 5. The same line shape expression could have been obtained, however, by a simple purely classical model in which 5 represents the rate of a particular type of collision that inverts the motion of the harmonic oscillator. Another occasion is connected with a recent treatment of the effect of collisions on the Doppler broadening of spectral lines (Rautian, 1967; Rautian and Sobel’man, 1967). A simple classical model, treating the various Doppler-shifted frequencies as a band of distinct normal modes, leads to the same integral equations which are the basis of the treatment of Rautian and Sobel’man. The third occasion was brought forward by the realization of the important contribution of the reorientation of molecules by collisions to the broadening and shifting of spectral lines. Although this had already been pointed out by Spitzer (1940), proper consideration of this effect has been made only in recent treatments (Ben-Reuven, 1965a; Gordon, 1966a). This could hardly have happened if classical models were consulted more frequently. Indeed, Gordon (1966a) has shown that cross sections for broadening and shifting of lines due to rotational phase shifts and reorientations can be computed using a completely classical model. One has to assume that the molecule is represented by a classical rotor with a fixed unperturbed rotational frequency equal to the resonance frequency of the spectral line. Given a potential function for the interaction with perturbing molecules, the phase shifts and reorientations of the rotor can be calculated by integrating along the classical collision trajectories. The fourth occasion is connected with the treatment of resonance broadening by Reck et al. (1965). A conjecture concerning the square root density dependence of the line width due to resonance transfer of excitation between similar molecules, obtained under certain conditions, reaffirms in fact the conclusion reached by Holtsmark (1925), using a simple classical model of coupled oscillators. A resort to classical models shows that, at the wings of the line, resonance broadening is after all a binary collision effect, linearly dependent on the density. It seems desirable, therefore, to regain the line of thought of Lorentz and his contemporaries, and rewrite classical-theoretical models for the various types of damping phenomena (such as impact, statistical, or resonance

A . Ben-Reuven

204

broadening) starting from, as much as possible, a common point of view. In doing so, we can exploit the insight provided by quantum theory and the vast experience gained in line shape studies and avoid certain wrong conceptions which have persisted in earlier works.

II. The Fourier-Transform Method A. RADIATION DAMPING

The emission of radiation near a characteristic resonance frequency is associated in the classical theory with an electrically charged oscillator of mass m,charge e, and harmonic angular frequency oo. Let the oscillator be stretched to the amplitude Qo at the time t = 0 and then be left to oscillate freely. Its displacement, when not disturbed, should vary in time according to

where 9(t) =

1 0

when t > O when t < O

is the “step” function. As an accelerated charge, however, it must emit radiation at the rate (cf. Heitler, 1954) G ( t ) = 3eZ(1’/c3.

(3)

Its motion cannot therefore be strictly harmonic ; energy will be radiated at the expense of diminishing the amplitude. As a result, the spectral distribution of the emitted light will have significant components spread continuously over a finite range of frequencies, of the order of magnitude of the rate of dissipation of the oscillator energy, instead of the single component at o = a,,. The oscillator, so to speak, undergoes natural or radiation damping. Provided the damping is weak enough, its effect is introduced by simply adding into the right-hand side of Eq. (1) a factor exp(-Etj, where E is the rate of radiation damping. I n what follows we shall be concerned with another source of damping, namely, collisions of the radiating oscillator with other particles. Collision damping is usually much stronger than radiation damping, and the latter can be neglected in dealing with the motion of the oscillator under the effect of collisions. The presence of natural damping is, nevertheless, essential for understanding the meaning of relaxation of systems of finite size. In relaxation theories it is customary to consider a microscopically small part of the system -one molecule, say-whose periodic oscillations are perturbed by an

THE CLASSICAL-OSCILLATOR ANALOG

205

infinite number of other particles, acting as a thermal reservoir. An aperiodic motion, such as an exponentially damped oscillation, then results. In finite conservative systems, however, multiperiodic motions may still exist with discrete resonance frequencies (compared to the continuous spectrum of a damped oscillator). In the classical theory, these periodic motions are associated with the discrete normal modes of the complex system. A discrete spectrum would mean that (though the chance may be exceedingly small, owing to the complexity of the system) the excitation may periodically return to the same molecule if it cannot eventually escape by such means as spontaneous emission. A closed finite system cannot strictly act as a thermal bath; it must be coupled to the "outer world" to form one. Radiation damping provides for such a link (though it is not the only conceivable one; the system may also interact directly with the surrounding room by collisions of the gas molecules with the walls of the container). The interaction with the radiation field, mathematically expressed by the addition of the damping parameter E , secures a truly aperiodic motion. Periodic motions of the macroscopic system, superimposed on the oscillations of its microscopic part, will then be completely smeared out by natural broadening (or by other means of relaxation of the macroscopic system), to appear as a single smooth line in the spectrum.

B. TIMECORRELATIONS AND THE SPECTRAL SHAPEFUNCTION Light is emitted in ordinary experiments by a macroscopically large number of oscillators, the motion of which is incoherent. We would, therefore, expect the power spectrum per oscillator to be simply related to the Fourier transform of an ensemble average of Q(r), but in such a manner that the incoherence of phase (and the randomness of orientation) of the various oscillators will not make it vanish. The most obvious suggestion is to try and relate the power spectrum to the autocorrelation of eQ(r) in time,

where the brackets stand for the ensemble average. Consider Q ( t ) as an unspecified function of the time and its Fourier transform Q(o).In general, Q(o)=

1; Q(t)e-'"' dt m

and co

Q(t)= (27c)-' -m

Q(co)e'O' do.

206

A . Ben-Reuven

SOY m

Q(t)= -(2n)-1/

oZQ(o)eiat do

(7)

-W

and

[(l(t)lZ= (4~')~'

IIwa*(o')Q ( w ) o Z o f 2exp[i(o - w')t] do dw'. (8)

By transforming back, we have m

[(l(t)I2

Q(f')~zd2

= (4n2)-' lTJTQ(t") -m

- oft"- (o- o')t])do do'dt' dt"

x exp( - i[wf'

= (4n2)-' I f l / Q ( f " ) Q(t"

+ z)w2ofZ

-m

x exp[- i(o - o')(t"- t ) - i o z ] do d w ' d t " d z ,

where z = t'

(9)

- t".

At this point, the ensemble average is introduced. The correlation (Q(t") Q(t" + z)) will then appear in the calculation in the time interval z, given the initial time t". By the ergodic hypothesis of statistical mechanics, averaging over an ensemble is equivalent to averaging over all initial states. The brackets are therefore independent of t", which appears now only in the exponent in (9). Given any well-behaved function f(w),

jm f ( o fexp[i(o ) - o')?"] dt" do'

1-

-m

W

= 2n

f(w') 6 ( o - 0') do'= 2 n f ( o ) , (10)

m

where 6 is the Dirac delta function. Therefore, performing these integrations, the mean power emitted is ( G ( t ) ) = (2e2/3c3)(2n)-'

jm

fm

-m

-

w4(Q(0) Q(z))e-'"'

-m

d o d?. (1 1)

Notice that ( G ( t ) ) is no longer time dependent. It follows that m

( G ( t ) ) = (2/3c3)

0

m4F(o) d o =

/

m

0

G(o)do,

(12)

where F ( o ) = ( e 2 / n )Re( f m ( Q ( 0 ) Q(z))e-'"' d r ] -W

(13)

207

THE CLASSICAL-OSCILLATOR ANALOG

is the so-called spectral shape function, and (14)

C(O) = (2/3c3)04F(w)

is the power emitted in a unit interval of o,as may be seen by considering an ensemble of harmonic oscillators. The latter conjecture is true if the ensemble averaging is made over randomly distributed initial times and phases of the excitation. The absorption coefficient K of a gas of n, similar oscillators per unit volume is related to G(w) by Kirchhoff's law,

K ( 4 = n, G(o)/cU(o), (15) provided the radiation is sufficiently weak so as not to cause appreciable deviation from thermal equilibrium. The term U(w) is the energy density (per unit volume) of blackbody radiation in a unit range of w. In classical physics, (16)

U(W)= w ' k T / n ' ~ ~ ,

where kT is the classical value of the mean energy of each normal mode of the radiation field at the temperature T. If kT is not much larger than hw, kT should be replaced, according to Planck's law, by ho(e"w'kT- I)-', giving K(u) = n , ( 2 ~ ' ~ / 3 h ~ ) ( dI)F(o). ~'~~ (17) In dealing with harmonic oscillations, it is sometimes more convenient to use complex quantities, the time variation of which is described by exp( f.ioot) instead of sine or cosine functions; i.e.,

+

X ( t ) = Q ( t ) i(mwo)-'P(t), Y(t) = Q(t) - i(mwo)-'P(t) = X*(t),

(18)

where Q and P are, respectively, the canonical coordinates and momenta, The use of the ergodic hypothesis enables us to choose the initial phase of the oscillation at will. We choose, therefore, as initial conditions those appropriate to the example of Eq. (I), namely lim X(t) = t-ro

(OQo

when t > 0 , when t c0.

Then, written in terms of X and Y,

F(o)= X F C ( 4

+ Fc( - 41'

where F,(w) = n-'Re(~om8c(r)e-i"' dr

dr).

A . Ben-Reuven

208

The second term in (20) arises from the similar contribution of Y(z), noting that E(o) = X*(-o). (22) F(o)is thus an even function of w . Though only positive values of w are of physical significance, it is formally possible to associate with a term in the “ physical ” region of positive o,around a resonance frequency wo ,a similar term in the “unphysical” region around - a o , with the latter, however, extending a “ tail ” to positive frequencies. As will be illustrated later, each line is composed of “ positive” and “ negative” resonance terms. It follows from (21) that, no matter how X(t) varies with time,

lOmF(o)do

m

=4

F,(o) do = +e’(QoZ) = +@( +0)

=A,

-m

(23)

where by + O we mean the limit as t approaches zero from the positive side. Equation (23) is sometimes referred to as the “ law of spectroscopic stability.”

C. THEUNDAMPED OSCILLATOR Consider, again, the undamped oscillator of Eq. (l), Q(t) = Re{X(t)} = Re{O(t)Qo exp(ioof ) } . The phase-space variables X(t) and Y(t) obey first order differential equations,

(d/dt - iwo)X(t) = d(t)Qo,

(d/dt + io,)Y(t) = S(t)Qo ,

(24)

where the singular source at t = 0 results from the step function O(t). A Fourier transformation of Eqs. (24) gives

+

i(w - wo)%(o)= Qo , i(o oo)P(o)= Qo . (25) The solution of these equations is obtained by a method known from the theory of Green’s functions. To cope with the divergences at w = w o , formed in solving for %(o)and fr(w), we add to wo either + ie or - ie, where E is a positive increment which is ultimately made to vanish. The general solution is an arbitrary linear combination of the two corresponding solutions, I.e.,

where a, + a- = I . The correct combination is determined by the boundary conditions (19), in the following manner: Consider the transformation back to X(t),

209

THE CLASSICAL-OSCILLATOR ANALOG

performed by contour integration in the complex w plane. In case (a) t > 0, the integration path can be closed by a semicircular arc in the upper half-plane (Im w > 0), the contribution of which will vanish as the radius of the arc is made infinitely large. Nonvanishing contributions to the integral in (27) come only from poles encircled by the path, namely at w = coo + ie. Therefore X(t) = a+Qoexp[(iw, - ~ ) t ]

for t > 0.

(28)

In case (b), t < 0, the integration path should be closed in the lower halfplane, encircling the pole at w = wo - ie, with the result

+~ ) t ]

for t < 0.

(29) The minus sign results from traversing the path clockwise. The boundary conditions (19) force us to choose X(t) = - a - Q o exp[(iwo

a - =o. with poles at w Following a similar solution for %?(o), a, = I ,

Q(o)= +[8(0)+ f’(w)]

+

-

(30) = -coo

= frQo{[i(w wo - i ~ ) ] - ’ [i(w

f ie, we get

+ wo - zk)]-’}. (31)

Since, in the absence of collisions, all the oscillators of the ensemble behave in the same manner, F(w) =

(t)(

(w

E

+ + (w + + 6(w + oo)},

- wo)2

= A{6(w - 0 0 )

E2

E

wo)2

+ E2

1

(32)

as E -+ 0, where A = $e2( Q o z ) . The spectrum of the undamped oscillator thus consists of an infinitely sharp line at w = wo . The “negative resonance” term 6(w wo) should be retained, however, since in reality no oscillation will go on undamped. The total power emitted is given by

+

(C) = ImG(w) d o = (w~/3c3)e2Q02. 0

(33)

Notice that E has the right sign necessary to keep F(o)positive. But the sign of E is determined solely by the boundary conditions (19) which are an expression of the condition of causality: Q(t) has a nonvanishing value only after the initial displacement Qo has taken place. It can be shown that similar arguments hold also in the quantum-mechanical case. Thus, e.g., causality turns out to be the reason why, as a linear response to a radiation field, a system in thermal equilibrium always dissipates (absorbs) energy. The same solution would have resulted, with the same sign of E, if radiation damping were retained in the form of a factor exp(-et) added to Q(t).The only difference is that E is then a finite physically meaningful quantity.

A . Ben-Reuven

210

III. Impact Damping A. COMPLETE INTERRUPTIONS

The idea of a sudden interruption of the oscillation by a strong collision originates from the works of Michelson and Lorentz in the earliest days of pressure-broadening theory. Suppose the oscillator is enclosed in a gas of N identical particles, contained in a box of volume V . The gas is supposed to be dilute so that, .to a good approximation, the various particles move in a random, uncorrelated manner, and the time during which each of them interacts with the oscillator is negligibly short compared to the mean time between successive collisions. Let the oscillator be perturbed strongly at each collision so that every phase and orientation are equally probable after the collision. In the process of averaging over the outcoming phase or orientation, this amounts to cutting the amplitude to zero. It is still necessary to average over the random temporal distribution of collisions, which occur at a mean rate y. Let At be a time interval short enough to incorporate, at most, only one collision with the oscillator, yet sufficiently long compared to the duration of the collision. The probability for a collision to occur in At is therefore y At. Suppose the collisions are independent of each other, so that the effect of each collision does not depend on the past history of collisions. In calculating the mean change of O,(t) in the interval t to t + At, we may therefore average first over the collisions in At, for a given X ( t ) , independently of the averaging of X ( t ) itself. Since a single collision changes X by A X = -X,

(34)

the mean change of X in At is Aco,,X(t) = - v X ( t ) At. (35) In a “coarse grain” sense, considering the collisions infinitely short, At may be treated as a differential. The equation of motion of the undamped oscillators, Eq. (24), should then be modified to (d/dt + y - iwo)X(t)= B(t)Qo. (36) Inserting the solution of (36) in (21), the Lorentz line shape

F,(o) = n - ’ e 2 ( Q o 2 )

Y ( w - wo)2

+y2

(37)

is obtained. The full expression for F(w) which follows from (20) and (23), (38) is also known as the Van Vleck and Weisskopf (1945) line shape.

21 1

THE CLASSICAL-OSCILLATOR ANALOG

B. AN INTERRUPTION MODEL

The oscillation need not be completely interrupted by collisions in order to damp the radiation. Other ways by which a collision may affect the oscillator are (a) changing its amplitude, (b) shifting its phase, and (c) reorienting it in space. Explicit reference is made here to linearly polarized oscillations only. Other types of polarization, such as circular, may be expressed as linear out-of-phase combinations of the three components of linear polarization. Let a collision of type s change the amplitude of the oscillation by a factor a,, shift its phase by q,, and reorient it from X to R,X, where R, is an orthogonal rotation matrix (or a complex unitary matrix of unit modulus, if nonlinear polarizations are considered). A collision of that type changes x by AX = (a,eiqsRs- l)X (39) where a,, qs are assumed to be independent of X. If the rate of collisions of type s is v, , then Atoll X( t ) = - v,( 1 - a, e'"R,)X( t) At . (40)

c S

Included in the summation over s is an averaging over an isotropic distribution of the orientation of the collision trajectories in space. The matrix R, then becomes invariant, or scalar, and may be replaced by cos O,, where 8, is the angle by which X is rotated. Consequently, Aco,lX(Q= -(y - id)X(r)Ar,

where y =

c v,(l - a, cos

(41)

9, cos e,),

(42)

S

and 6=

1v,a, sin q, cos 8,.

(43)

S

The resulting line shape is Lorentzian with a shifted peak, F,(o) = n-'eZ(Qo2)

Y

(w - wo - S ) 2

+

72'

(44)

The combined effect of phase shifts and amplitude variations was first calculated by Lindholm (1 942). The classical reorientation effect was not treated properly until the recent work of Gordon (1966a). This reorientation effect, well known to be the major cause of relaxation in nonresonant (or dielectric) relaxation, was unjustifiably neglected for a long time in the study of optical spectra [in spite of Spitzer's (1940) observation of its importance].

212

A . Ben-Reucen

The parameters that distinguish a collision of type s, and the corresponding mean rate v,, have yet to be specified. Whenever the inner structure of the perturbing particles is immaterial, the collision trajectory is completely specified, apart from its orientation, by the relative velocity u and the impact parameter b. The rate of collisions in the range b to b bd and u to u + do, isotropically averaged over orientations, is

+

v, = nb 2nb db vg(u) do,

(45)

where nb is the number density of perturbing particles. Here g(o) is the Maxwellian distribution of velocities at the temperature T, g(u) = 4 n ( r n , / 2 ~ k T ) ~exp( / ~ v-~m , v2/2kT),

(46)

where m, is the collision reduced mass. Equation (44) represents the spectrum of a damped oscillation if y is positive. From (42) it follows that y is positive if c(, < 1 for all collisions. If values of x , greater than unity are allowed, the positive-definite nature is not ascertained, i n general. Therefore, the above model does not guarantee damping. This might be guaranteed by some further specifications of the nature of the collisions and the causes of amplitude variations. This question does not arise in the quantum-mechanical impact theory (cf. Anderson, 1949), where owing to the unitarity of the scattering matrix the oscillations are always damped.

C. SENSE-REVERTING COLLISIONS Damping is frequently associated, by the suggestion of Lorentz (1915 ) , with the existence, on a phenomenological level, of a velocity-dependent friction force. Yet, it was Lorentz himself, in his famous Note 57, who pointed at the difference between the effects of interruptions by collisions and those of the friction force. The resulting line shapes, in the two cases, are different and become similar only when y < oo. Since at the time of Lorentz radio and microwave spectroscopy did not yet exist, and for all practical purposes the two line shapes seemed to coincide, little attention was paid to the difference. It is nonetheless remarkable that this distinction has managed to avoid recognition even after more than two decades of microwave spectroscopy, and so many years after spectra had been measured with y comparable to w o . This failure of recognition led to the use of the wrong line shape expression for the interpretation of the data, and consequently to the discovery of some “shifts” of an exceptional behavior which could not be explained by an impact theory. The most outstanding observations of this kind were made with the inversion spectrum of ammonia in the microwave region (Bleaney and Loubser, 1950; Birnbaum and Maryott, 1953).

213

THE CLASSICAL-OSCILLATORANALOG

A quantum-mechanical derivation of an appropriate expression for microwave line shapes, in the impact approximation, was presented recently (Ben-Reuven, 1956b, 1966). An observation was made there that this line shape can also be obtained classically, if two distinct processes are considered as jointly affecting the oscillator. One process is the interruption by collisions of the type discussed in Section II1,B. The other process is an interruption by collisions that revert the sense of the oscillation, having the same effect as a velocity-dependent friction force. These collisions leave the diplacement of the oscillator unaltered while reverting its momentum, i.e., AQ

= 0,

AP = -2P.

(47)

As before, we average over collisions in a time interval At, AcollQ = 0 ,

AcollP= -2cP At

(48)

c

where is the mean rate of momentum-reverting collisions. This amounts, in the " coarse grain " sense, to adding - 2cP to the time derivative of P. But this is precisely the form in which velocity-dependent friction is phenomenologically introduced into the equations of motion of the oscillator. Let us proceed now to express (47) in terms of the complex variables X and Y . Using ( 1 8), we may write AX

= i(moo)-'AP =

-2i(moo)-'P = - ( X - Y)

(49)

and AY = -i(m~o)-'P = -(Y

- X).

(50)

That is, a sense-reverting collision interchanges X and Y. The resulting equations of motion are (d/dt

+ 5 - ioo)X - cY = h(t)Qo,

ro--?+

(d/dt

+ 5 + iwo)Y - (X = S(t)Qo. (51)

Their Fourier transform is, in matrix notation, i(w

+it)+ i)

(*(coJ=

(:I)*

(52)

The solution follows immediately, by inverting the matrix in Eq. (52):

ro+ 7 )+

+ 2ior

oo2- o2

(53)

Thus, i(o

+ oo)+ 26

X(0) = wo2 - w 2 + 2iwc Q o .

(54)

A . Ben-Reuven

214 from which follow

and

When

C Q oo,and Iw - oolQ oo, as the case is in optical spectra, we get e2( Q o 2 ) F,(w) % 71

(0

c - coo)’

(57)

+ 5’’

which is similar to (44),with C instead of y, and hence the source of confusion between the interruption and the friction models. Equation ( 5 5 ) vanishes whenever oo= 0. But spectra of zero resonance frequency are known to exist; they are caused by reorientation by collisions of otherwise stationary (nonoscillating) dipoles. This process (Debye, 1929), known by various names as nonresonant, orientational, dielectric, or Debye relaxation, is certainly an interruption-type process.’ Since the two models were taken to be equivalent, it led to rejecting the friction model in favor of the interruption model. Having now recognized that the two models represent two distinct processes, it seems most natural to incorporate both of them in the equations of motion. Letting y’ be the rate of interruption-type collisions [Eq. (42)], we then get [d/dt ( 7‘ - i ( ~ 0 S)]X - CY = 6(t)Qo, [d/dt y y l i(w0 6)IY - yx = 6(t)Qo. (58)

+ + + + +

+ +

Again, using Fourier transforms,

(W)= ( K W

-

””;

8) + Y i ( 0

+ wo-[+ 6) + y

)-l(:;),

(59)

where y = y‘

+ y.

(60)

A straightforward calculation yields X(W)

=

i(w

+

+ uo+ 6) + y + C

(o 6)’ - w z + y z

- yz + 2iwy Qo

For recent theoretical treatments of nonresonant absorption and Debye relaxation in gases see Ben-Reuven (1966), Birnbaum (1966),and Gordon (1966b).

THE CLASSICAL-OSCILLATOR ANALOG

215

Hence

(62)

and A 2(Y - C)w2 + 2(Y - C)C(oo + 4z+ Y 2 - C21 F(o) =n [(oo 6 ) 2 - w2 y z - 5 2 3 2 402y2

+

+

+

(63)

is the resulting shape function (Ben-Reuven, 1966). The Van Vleck and Weisskopf shape function (38), with y now representing the full relaxation rate y‘ + C, is approached as a limiting case of (63) when 4 wo 6. In the case wo 6 = 0, however, Eq. (63) attains the Debye shape for nonresonant relaxation,

+

+

A 2y’ F ( o ) =n o z + y’2 ~

where only y’ participates. The quantum-mechanical equivalent of y’ and C has been discussed by Ben-Reuven (1966). The functions X and Y are the classical counterparts of the operators li)(fl and If)(il, respectively.Therefore, theequivalent to collisions which transform X into Y are collisions which cause transitions from i to f and fromfto i ; namely, inelastic transitions between the two levels of the observed line. D. APPLICATION TO RIGIDLINEAR ROTORS

We have so far avoided the question of how to calculate q, ,9,, or a, which appear in the interruption model of Section II1,B. One usually uses quantum theory for the description of internal states of molecules in such calculations. The treatment of the collision problem, having to deal with the many channels of molecular collisions, is very cumbersome, in spite of the many simplifying assumptions ordinarily introduced. It is therefore gratifying to know that, at least in some cases, various line shape parameters may be calculated by integrating the classical equations of motion of the colliding molecules treated as classical particles (Gordon, 1966a,c). Consider, for example, linear molecules as rigid rotors. A linear rotor with a moment of interia Z has the quantized resonance frequencies

o,=hz-’j

o’= 1,2, ...),

corresponding to the transitions between the rotational levels j - 1 and j . We can look upon this rotor as an “angular harmonic oscillator” with the

21 6

A . Ben-Reuven

fundamental frequency h l - ' and its harmonics. Consider as an initial state a particular value of j , so that w j is the classical angular frequency of rotation, and hj is the magnitude of the associated angular momentum, which has a fixed orientation in space. As demonstrated by Gordon (1966a), a rotational phase shift q and an angle describing the reorientation of the angular momentum cector in space may be obtained from the classical equations of motion, given an angle-dependent interaction potential. Gordon, for example, uses dispersion forces for HC1-noble gas collisions, including the loaded-sphere correction (Herman, 1963, 1966). It may be necessary to introduce a few quantum-mechanical modifications, utilizing the correspondence principle. Thus, e.g., the effect of inelastic collisions between rotational states may be introduced by a relation between the quantum transition probability and the classically calculated change in the magnitude of the angular momentum as a result of the collision. In vibration-rotation spectra, it is also necessary to introduce a vibrational phase shift, due to anharmonicities. This may be accomplished by calculating the vibrational perturbation quantum mechanically, and integrating over the time along the classical trajectory (Gordon, 1966a). An interesting analogy to the case of sense-reverting collisions, discussed in the previous section, is encountered in an application of the linear rotor model to the problem of line overlapping in vibration-rotation spectra (Gordon, 1966~).A vibration with an angular frequency w, modulated by a rotational frequency w j gives rise to the doublet of combination frequencies 0, & w j I An inversion of the orientation of the angular momentum by a collision amounts to a change of sign of w j , therefore transferring from one member of the doublet to the other. This is similar to the effect of sensereverting collisions on the w,, " modes " of the classical oscillator. More generally, a collision alters the orientation of j by a certain angle. The final orientation may be looked upon as a superposition of the initial state and the state of inverted momentum, to a degree determined by the reorientation angle. Therefore, reorienting collisions are partially effective as " sense-reverting" collisions. Their outcome is the merging of the doublet into a single line at o,as the rate of collisions is increased. This is essentially Gordon's explanation of the collapse of the P and R branches into a Q branch observed in the vibration bands of compressed CO gas by Vu et al. (1963). A question may be posed regarding the foregoing interpretation. In the previous section the sense-reverting collision model has been treated as a mathematical analog of the collapse of resonance lines into a nonresonant spectrum. The physical processes involved are, however, quite different. In the classical analog, the relevant collisions are elastic ones, in which only the sign of P is inverted. Contrarily, in the quantum-mechanical theory, inelastic collisions connecting the initial and final levels of the line are responsible for

*

THE CLASSICAL-OSCILLATOR ANALOG

217

the collapse. Similarly, in vibration-rotation spectra, inelastic transitions between rotational levels are responsible for the mixing of the P and R branches. Gordon, nevertheless, considers the elastic reorienting collisions as actually causing the mixing of the P and R branches in a model which represents an approximation to the real physical system. This apparent contradiction may possibly be settled by a closer look at the nature of the classical approximation. One should remember that the classical oscillator is an analog of the optical transition between two levels, and collisions affect the two levels simultaneously. Consider, for example, the doublet R(J - 1) and P(J). In order to transfer from the one line to the other, it is necessary to have inelastic rotational transitions in both lower and upper vibrational states. But the energy given to the molecule in the lower state is just compensated by energy taken in the upper state. Since, in the classical limit, there is no way of telling in which state the molecule is, no energy is transferred to the molecule on the average. The manner in which the classical expression results from the quantum formulas and, particularly, how reorienting collisions emerge as the dominant process, still need a rigorous demonstration

IV. Complex Oscillators A. LINEOVERLAPPING

So far we have considered only three-dimensional single-mode harmonic oscillators. Let us now suppose that the oscillator is a complex system, con. sisting of s normal modes with angular frequencies ol, o2, ..., u s Collisions may couple various modes, exciting one mode at the expense of others, in addition to damping individual modes. We then have cross relaxation between overlapping lines. In addition to averaging over the effects of collisions in the ensemble, we shall from now on also average over all possible initial configurations Qo = (Qoi, Qoz

3

-

4

,QoA

(65)

using a distribution function

The distribution of initial configurations is assumed to be independent of the positions and the momenta of the perturbing particles. It is therefore possible, as before, to average first over all collisions, given a fixed initial configuration, and then use (66) to average over the configurations of the oscillating system.

218

A . Ben-Reuuen

Suppose a collision changes the 3s-dimensional vector A’, which describes the configurations of the oscillator at t, into X’. The transformation is described by X’ = S X where S is a complex matrix in our 3s-dimensional space. As in the impact model of Section III,B we assume that S is independent of X itself. It may be simplified by rotational invariance if collisions are isotropically distributed. Then the three Cartesian components will transform independently by the same (now s-dimensional) matrix S. We may, finally, add collisions of the sense-reverting type, but then S will be a 2s-dimensional matrix acting on 2, = ( X d , * *

f

9

x,,,

y,1, *

-

* 9

Yo,>,

(67)

where a = x, y, or z. Let w(a) be the mean rate of collisions of the type n, represented by the matrix S(a). The mean rate of change of 2, by collisions is then

The 2s equations of motion of the components of 2, are, in matrix notation, [dldt

- i(R, + A,)l2,

(69)

= a(t)Qoa,

where Qo, is now Q o ~= ( Q o a i ,

* * * 9

Qoas 9 Q0.1,

. * QoA

and R, is the real diagonal matrix whose eigenvalues are -0,s.

W,1,...,~,s,-Wo1,...,

Here, A, is the complex matrix Ac = i

1w(a)[I e

- S(a)]

Im du 2nbug(u)[I

= in, Jomdb

0

- S(b, u ) ] ,

(70)

where I is the unit matrix. The last equality in (70) holds if S does not depend on the internal state of the perturbing particle. The element i, j of Ac signifies an excitation of thejth mode by a collision, the oscillator having an excited ith mode before the collision. If these transfers of excitation occur with welldefined phase shifts, S will be a complex matrix. Equation (69) is an eigenvalue problem in which we may consider R, as an “unperturbed” Hermitian matrix and A, as a non-Hermitian “ perturbation.” The eigenvalues of R, + Ac are complex; their real parts correspond to shifted resonance frequencies, and their imaginary parts to damping rates (or line widths). The spectral shape function will not be, however, a sum of Lorentzian terms since the transformation that diagonalizes Ro + Ac is not unitary, and Q& will be transformed into a complex quantity.

THE CLASSICAL-OSCILLATOR ANALOG

219

If the elements of A, are small in comparison to the separations between the resonance frequencies w o j , then Ac may be treated by perturbation techniques. It is convenient to split A, into a sum of two matrices, one Hermitian and one anti-Hermitian, Ac = Ac

+ ir,.

(71)

The off-diagonal elements of ir,, treated as a small perturbation, have a peculiar property : instead of repelling two eigenvalues of the unperturbed matrix Roaway from each other, as the off-diagonal elements of an Hermitian perturbation would do, they attract them toward each other, causing two distinct resonance lines to collapse into each other. We had an example of this effect in Section II1,D [Eq. (59)], in which momentum-reflecting collisions, represented by c, act on the 2 x 2 matrix of eigenvalues ooand -coo as an off-diagonal anti-Hermitian perturbation. The result was a "shift" of the two eigenvalues toward each other (or of the resonance frequency toward zero) when 5 is increased by raising the gas pressure, ending up with the nonresonant Debye shape. In general, processes that cause the mixing of any closely spaced resonance modes w1 and w2 do not contribute to the line width in the limit where the cross-relaxation rates are much larger than Iwl - w21 and the two lines completely merge. If, in addition, the two modes have equal amplitudes, we get in this limit a single line with position and width given by

-

00

+ s + ir = +(a,+

02)

+ i c v(0) a

x (1 - ",(a)

+ Sz(41 - 3CSld4 + S21(4119

(72)

where

c v ( M - S 1 2 ( 4 + SZl(d3, U

which is essentially the cross-relaxation rate, is subtracted from the total width. Such behavior is typically observed in the exchange narrowing of nuclear magnetic resonance lines (Abragam, 1961). More generally, a band of N closely spaced resonance frequencies oOf (f= 1, . . . , N), with corresponding amplitudes X a J , will merge by cross relaxation into a single line which, to a first approximation, has the mean position and width

where Q,, is an N-dimensional vector.

220

A . Ben-Reuuen

B. COLLISION AND DOPPLER BROADENING The translational motion of the " oscillator" itself, as a particle of finite mass, has been so far disregarded. Owing to the Doppler effect, similar oscillators moving at different velocities to or from the observer will be seen radiating at different frequencies; their displacement from the resonance frequency at rest is proportional to the longitudinal component of the velocity, u,,

- (un/c))*

(74) A Maxwellian distribution of velocities will therefore show up as a Gaussian distribution of frequencies in the spectrum (neglecting damping). An ensemble of such oscillators may be treated as an ensemble of oscillators of varying resonance frequencies; to different values of un correspond different normal modes with the appropriate value of w D .The Doppler-broadened line may be considered as a continuous band of resonance lines. If only the internal state of the oscillator is affected by collisions, then the internal and translational motions may be treated independently and the resulting shape of an isolated line is a convolution of a Gaussian and a Lorentzian shape. Collisions, however, change u,,. This effect may result in the narrowing of the line width with increasing pressure at sufficiently low pressure, where Doppler broadening still dominates over collision broadening (Galatry, 1961, and references therein). Considering the Doppler-broadened line as a continuous spectrum of eigenvalues, the effect of impacts can be expressed by an equation similar to (69), but with oD(un)

=

(75) being a kernel of an integral over un' (appearing in the equation for the normal mode corresponding to un) instead of a matrix. The result is an integral equation, = Ac(un 9 on')

i[a - ~O(vn)]z,(w;

vn)

-i

I

OD

-m

dun' &(On

u n ' ) z a ( ~ ; )0 ;

= Qou(vn)-

(76)

This equation is essentially equivalent to the one used by Rautian and Sobel'man (1967) in their recent treatment of combined Doppler and impact broadening. Collisions represented by the kernel A, couple the " normal modes," as in the discrete case. By making them collapse into each other, the Doppler-broadened line is narrowed. This behavior is reminiscent of motional narrowing in nuclear magnetic resonance (Abragam, 1961). There the spin-spin coupling takes the place of Doppler broadening in our case, and the temporal variation of the spin-spin interaction is the equivalent of changing the velocity by collisions.

22 1

THE CLASSICAL-OSCILLATOR ANALOG

Let S(un,un’; a) be the velocity-dependent collision matrix, and gn(un)the Gaussian distribution of radiating molecules. Then, with &(On

3

un’)

=i

C [d(un - On’) - S(vn

on’

;a)Iv(c),

d

(77)

we get by application of (73) (assuming the amplitudes are velocity independent), in the limit of high relaxation rate, the impact-broadening result 8,

+ 6 + il = oo+ i C u(a)[1

- ~(a)],

(78)

0

where

represents the effects of collisions averaged over velocities of the radiating molecules.

V. Statistical Broadening A. ADIABATIC PERTURBATIONS Consider, again, the single-mode oscillator, contained with the gas of N identical particles in a large box of volume V. Let us assume now that the gas perturbs the oscillator in an adiabatic manner, that is, the oscillator’s angular frequency at any moment t is given by mo‘W = 0 0

+ Ao(q(09 P(t)),

(80)

where the the perturbation Am depends on the position of the gas in phase space, 4 = (q1,qZ . * * 2 q N ) ,

p = (Pl, P Z

9

3

*

--

Y

PN),

(81)

at that instant. Given the initial configuration q(O), p(O), the displacement of the oscillator at t > 0 will be

X(t) = e(t)Qoexp

(82)

In calculating the correlation function, Eq. (82) has to be averaged with a distribution function f ( q , p) d3Nqd3Np,with q = q(O), p = p(0). For simplicity’s sake, we shall assume that the motion of the various particles is uncorrelated, and that their perturbations are additive, i.e., N

222

A . Ben-Reuven

and

N

c

Pi).

P) =i = l

(84)

The superscripts ( I ) denote single-perturber functions. The single-perturber distribution functionf'" approaches the Maxwell-Boltzmann distribution as lqil increases (the oscillators being at the origin). If the volume Vis sufficiently large, thenf") must be inversely proportional to it in order to remain properly normalized : lim f("(q, p) d3q d3p = V-'g((pl)4np2 d p d3q. Iql-rm

(85)

With the brackets denoting the averaging over the perturbing gas (an averaging over a distribution of initial values of Q may be postponed till the end of the calculation), and using Eqs. (83) and (84), (exp i

1;Am(?') d t )

= (exp i

j: Am(')(?') d t ) N

= (1 - V - ' ( V ( 1 - exp i = exp( - n( V ( 1 - exp i

j: Am(')(t) dt')))n"

1:

Am(')(t) d t ) ) }

where n = N / V is the number density. The last equality in (86) was obtained by applying the binomial theorem in the limit V 4 00 (keeping n finite). The remaining average is independent of the volume, owing to (85), if Am decreases rapidly enough as 191 increases. Equation (86) is still a complicated function of t , and further approximations may be required in order to obtain a simple expression for F(o).The so-called statistical (or static) approximation is contained in the assumption that the motion of the particles is so slow that in a typical time interval zero to t, the configuration of the particles alters only insignificantly. In that case

1;

Am(?') dt'

t

Am(q, p )

(87)

where q = q(O), and p = p(0). Equation (86) attains a particularly simple form if it is furthermore assumed that the perturbation is sufficiently weak to allow expanding the exponential within the brackets in a power series of t. Then, to second order terms, (exp i

1,Am(tf)d t )

= exp{ -n( V [ - i t A d 1 )

+ ft2(Ao(1')2])}.

(88)

THE CLASSICAL-OSCILLATOR ANALOG

223

The spectral shape function obtained by Fourier transformation is Gaussian, i.e., Fc(w)= (In 2/n)”’y-’ e’(Qo2) exp{ -(In 2 ) ( 0 - coo - ~ 5 ) ~ / y ’ > , (89)

with a shift of the peak

6 = n ( V Ad’)),

(90)

and half-width at half-intensity y

=

((2 In 2)n( V(Aw(1))2)}1’’.

(91)

The approximation of weak interactions is a rather poor one since the power series appears in the exponent. To be of any use, it must be valid for time intervals of magnitude comparable to y - l . But then the second order term in (88) is of the order of magnitude of unity, and it is very unlikely that all higher-order terms are much smaller than unity. One usually must resort to better approximations, such as those made by Margenau (1951) (see also Breene, 1961). It has been argued by Margenau and Jacobson (1963) that, given any line shape, as long as it is sharp and symmetrical around its peak, Eq. (90) may still be valid if the perturbation is adiabatic. Suppose the line is sufficiently sharp so that its width is considerably smaller than wo . We may then neglect the term F,(-w) as well as the asymmetry caused by the factors preceding F(o)in (14) or (17). If Fc(w)is symmetrical around the peak then the position of the peak will be given by the mean value of the frequency, defined as Q

W=

j-

m

oFc(w)d ~ / ! : ~ F , ( w ) dw.

(92)

But by the rules of Fourier transformation, it follows from Eq. (82) that (5 =

-iC(4d0@,(01,= +ol@(+O>

+ n(

= o0

V Aw(’)(p, q))

= WO

=0

0

+ 6,

+ ( A N q , PI) (93)

where by t = + O we mean that the zero limit is approached from the positive side. Equation (90) has been used as the starting point for several attempts to calculate infrared line shifts (Buckingham, 1962; Bridge and Buckingham, 1963; Margenau and Jacobson, 1963). One should be cautious, however, when that depends on the state of the oscillator using a distribution function f(’) itself. The calculations may then lead to different values for the shift of corresponding lines in absorption and emission, in plain defiance of Kirchhoff’s law-and of experiment (Jaffe et at., 1963).

224

A . Ben-Reuven

B. ADIABATIC AGAINST IMPACTPRESSURE SHIFTS The consequences of the adiabatic assumption do not involve, in any explicit manner, the duration of the collision though the validity of the assumption may depend on how fast the collision occurs. It is therefore possible to envisage a case in which the conditions allow both the impact and the adiabatic assumptions to hold. The perturber particle may adiabatically change the state of the radiating oscillator, finally leaving it in the same state (same amplitude and oscillation frequency in the classical sense ; same energy eigenstate in the quanta1 sense) as before the collision, with only a shift of its phase. But this change may take place in a relatively short time, compared to the mean time between collisions during which the oscillation continues uninterrupted. If so, we face a difficulty, for the adiabatic assumption leads to Eq. (90), where the line shift depends linearly on the perturbation of the frequency, whereas the impact approximation requires that the line shift depend linearly on the sine of the phase shift. The two results agree only when the phase shifts are much smaller than unity and sin 9 x q . It could be argued that the adiabatic assumption, in the strict sense of Eq. (82), may not be used at all. So, for example, even though an adiabatic collision leaves the oscillator’s amplitude and frequency ultimately unaltered, it may change its orientation. Suppose, nevertheless, that the adiabatic assumption is still valid. Let us review, then, what the respective results of the impact and adiabatic theories, Eqs. (43) and (93), do imply. The impact approximation tells us that the shape function of a single isolated line is Lorentzian around the resonance frequency. The exponentially decaying correlation function is correct only for time intervals appreciably longer than a typical duration of a collision At so that the transient effects of the collision may be neglected. Therefore, the Lorentz shape is correct only in a finite range of frequencies Am 6 l/At. Equation (43) tells us then where the peak of the Lorentz curve is (provided, of course, 6 < Am). Equation (93) provides, in contrast, a measure of the mean frequency, in the calculation of which we have to integrate over the whole line [Eq. (92)]. The two results coincide only if Fc(o) is symmetric around E and its first moment converges rapidly enough when the limits of integration are extended to infinity. But the Lorentz function, which is asymptotically proportional to offers a very poor convergence. Indeed, only the principal value of E , N

P[E] = Nlim +m

I-N

oF,(w) d o / / I m F c ( o )do,

(94)

is well defined. The slightest asymmetric deviation from the Lorentz shape out in the wings may place the mean frequency, calculated by Eq. (92), off the peak. Since the line is Lorentzian only in a restricted range, and practically

THE CLASSICAL-OSCILLATOR ANALOG

225

nothing is known about the shape further in the wings, it may well be that the mean frequency does not represent the position of the peak at all. Reliable measurements of line wings are rarely carried out beyond several line widths away from the peak. The position of the peak is usually measured with comparatively much higher precision. Therefore by " shift," we generally mean either the position of the peak or any related parameter on which F,(w) depends in a simple preestablished manner. With Lorentz shapes, we can measure the shift 6, but we can hardly expect to have a reliable measurement of the mean frequency.

C. THEMETHODOF MOMENTS In order to judge whether or not the line is Lorentzian and whether the position of the peak can be represented by the mean value 0, we resort to Van Vleck's method of moments (Abragam, 1961). The kth moment of the line is defined by

100

Mk

=

m

F,(w)(w - 0)& do

IL

F,(o) do.

(95)

In a Lorentzian line, all integrals with k > 1 diverge. However, we can use 1 = a, so the approximation of truncating the line at the frequencies Iw - 0 that F,(o) vanishes further in the wings. The moments of the line are then finite, and given by k-1

k-3

+

* *

-1

(k even),

(96)

( k odd). In the limit a/y -+ co, we have

W(W2 = b/6)(a/Y)

+

00.

(97)

Therefore, if the calculated moments of a line diverge very rapidly with k, we can quite confidently expect it to be Lorentzian at least in the wings. A typical example of better converging moments is provided by the Gaussian line that we encountered in Section V,A. For a Gaussian given by (89), we have (Abragam, 1961) Ak(k - l)(k

- 3)

1

Mk=(O

(k even) ( k odd),

where A

= (2 In 2 ) - ' I 2 y .

(99)

A . Ben-Reuuen

226

In this case, M 2 is simply A2 and ( 100)

M4/(M2)2= 3.

Let us calculate now the moments of a particular example of adiabatic perturbation. By the rules of Fourier transformation we have, in general, M k = [(- i d/dt)ke-i"'@,(r)],,+ o / @ , ( + ~ ) .

(101)

Applying (101) to the adiabatic approximation (80) through (84), we get Mk

=

(CwL3'(q,p ) - G I k ) ,

( 102)

where (83) is used in averaging over the perturbers. The single-perturber function Am(')can be renormalized by subtracting from it the infinitely small quantity (0- o 0 ) N - ' so that Mk

=

([2

i= 1

Am'l)(qi? pi)]').

(103)

This renormalization amounts to choosing W as the origin on the frequency scale. Equation (103) may be written (in the limit of large N ) as a polynomial in the density n,

The leading term at low densities is the binary collision term

More generally,

where {v,} denotes any set of nonnegative integers obeying the two conditions m

mv, = k ,

m

vm

=j ,

(107)

and the weight of each set is determined by the coefficients Dk,j({V,})= k ! [ n ( m ! ) V m V , ! ] - l . rn

(108)

As a result of our renormalization, the first moment vanishes and, therefore, all terms with v1 > 0 vanish. Thus, for example, M 2 = n ( V(Ao'")2)

( 109)

THE CLASSICAL-OSCILLATOR ANALOG

227

and

M4 = n( V(ACO(’))~) + 3n2( V ( A W ( ~ ) ) ~ ) ~ .

(1 10)

The renormalization correction is infinitely small whenever m > 1, and therefore we can insert back in (109) and (1 10) the orginal A d ’ ) . Consider now the example of the van der Waals gas, with a billiard-ball distribution in which f‘” equals its asymptotic value (85) whenever IqI is larger than the molecular diameter r o , and vanishes at smaller separations ( V is then thefree volume of the gas). The perturbation is of the inverse power type,

A d ’ ) = C(r0/r),F,

(1 11)

where r = 191, the strength of the perturbation at r = r,, is given by C,and F is a geometrical factor (of unit order of magnitude) which contains all angular coordinates. In ordinary dispersion forces, p = 6. The leading term of the kth moment is now

N ( ( A w ) ~ )= (2/rr(k - 1))Ck4,(k)a.

(112)

a= +nro3n

( 1 13)

Here,

is the ratio of the unpenetrable volume around a molecule to the specific volume per molecule in the gas, and

The second term in (110) is of the order of u2 and is therefore negligible in dilute gases, where u < 1.

(1 15)

Equation ( I 12) is similar in form [aside from the numerical factor $,(k)] to the leading term in (96) for even k . Therefore in the extreme case (1 15), the behavior is quite similar to that of a truncated Lorentzian shape, C being usually several orders of magnitude larger than the line width. The odd moments do not vanish, in general, and therefore 0 does not represent the position of the peak. The actual values of the position and width depend on the properties of the line wings, to which the truncation is only a crude approximation. One would expect appreciable deviations from a Lorentz shape at IW - 0 1 2 T-’, where T is a measure of the duration of a collision. Since T is inversely proportional to the relative velocity of the colliding particles, results are generally velocity dependent, in contradiction to ordinary statistical-broadening theories.

228

A . Ben-Reuven

VI. Resonance Broadening A. RESONANCE BROADENING AS A MANY-BODY EFFECT

’

When two similar polar molecules encounter each other, one at the lower level of a resonance line and the other at the upper, they may exchange their excitation states by means of dipole-dipole interaction. The excitation is transferred with well-defined phase and polarization. Therefore, the radiation field cannot distinguish which particle is excited (provided the wavelength is much longer than a typical intermolecular distance). Matters are further complicated by the long-range nature of the dipole-dipole interaction. For example, attempts to calculate pressure shifts using first order perturbation may lead to diverging integrals (Mizushima, 1961). Resonance exchange, so it seems, is therefore a many-body effect. Wherever it predominates, one should study carefully whether it is allowable to analyze it in terms of binary collisions only, as is the case in impact broadening. A binary collision approximation, with the resulting linear dependence of the relaxation parameters on the molecular density, is undoubtedly valid if the coherent resonance transfer is destroyed. As was pointed out by Mead (1966), this is possible if the Doppler broadening exceeds the rate of resonance transfer. Owing to the Doppler shift, the colliding molecules are then in di