Advances in Atomic and Molecular Physics, Volume 24

Advances in

ATOMIC A N D MOLECULAR PHYSICS

VOLUME 24

CONTRIBUTORS TO THIS VOLUME N . G. ADAMS A. CROWE

W. HAPPER R. J. KNIZE G. LEUCHS

MICHAEL A. MORRISON D. SMITH S. J. SMITH

z. wu

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited bv

Sir David Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE Q U E E N S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 24

@

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Publishers

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Copyright 0 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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LIBRARY OF CONGRESS CATALOG CARD NUMHER: 65-18423 ISBN 0-12-003824-2

PRINTED IN T H E UNITED STATES OF AMERICA 88899001

9 8 7 6 5 4 3 2 1

Conten ts

The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions

D.Smith and N . G. A(1utn.v 1. Introduction

1

11. Principles and Experimental Aspects of the SIFT Technique 111. SIFT Studies of Ion-Neutral Reactions: Some Illustrative

Results IV. The Variable-Temperature Selected Ion Flow Drift Tube, VT-SIFDT V. VT-SIFDT Studies: Some Illustrative Results VI. Concluding Remarks References

3 23 33 39 44 46

Near-Threshold Electron-Molecule Scattering Micliael A . Morrison 52

I. Introduction and Overview 11. Basic Quantum Mechanics of Low-Energy Electron-Molecule

Scattering 111. Approximate Collision Theories and Their Woes near Threshold IV. The Behavior of Near-Threshold Cross Sections- Explained

V. Beyond the Born-Oppenheimer Approximation: Special Methods for Near-Threshold Scattering VI. Variations on an Enigma: Threshold Structures in Vibrational Excitation Cross Sections VII. Conclusions and Conundrums Acknowledgments Appendix A: Notation and Nomenclature Appendix B: Recent Reviews of Electron-Molecule Scattering References c

56 15 88 115

131 143 144

144 150 151

Contents

Vi

Angular Correlation in Multiphoton Ionization of Atoms S. J . Smith and G. Leuchs 157

I. Introduction 11. Theory of Photoelectron Angular Distributions 111. Experimental Methods

IV. Applications to Atomic Structure and Dynamics V. Conclusions Acknowledgments References

161 174 183 216 217

218

Optical Pumping and Spin Exchange in Gas Cells R. J. Knize, Z .

Wid,and

W. Happer

I. Introduction 11. Optical Pumping 111. Description of Spin-Polarized Atoms in Optical Pumping Experiments IV. Detection of Polarized Atoms V. Spin-Transfer Collisions VI. Relaxation VII. Frequency Shifts VIII. Applications References

224 224 229 -

238 243 254 257 265

Correlations in Electron-Atom Scattering A . Crowe I. Introduction 11. Excitation 111. Ionization Acknowledgments References

269 270 292

Index

323

317 317

7

Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin. N. G. ADAMS, Department of Space Research, University of Birmingham, Birmingham, B 15 2TT, England ( 1 ) A. CROWE, Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 1 NN, North Ireland (269) W. HARPER, Department of Physics, Princeton University, Princeton, New Jersey 08544 (223) R. J. KNIZE, Department of Physics, Princeton University, Princeton, New Jersey 08544 (223)

G. LEUCHS, Max-Planck-Institut fur Quantenoptik and Sektion Physik der Universitat Munchen, D-8046 Garching, Federal Republic of Germany (157)

MICHAEL A. MORRISON, Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahama 73019 and Joint Institute of Laboratory Astrophyscis, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309 (5 1)

D. SMITH, Department of Space Research, University of Birmingham, Birmingham, B 15 2TT, England ( 1 ) S. J. SMITH, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309 (157) Z. WU, Department of Electrical Engineering, Columbia University, New York, New York 10027 (223)

This Page Intentionally Left Blank

ADVANCES IN ATOMIC A N D MOLECULAR

PHYSICS. V O L

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THE SELECTED ION FLOW TUBE (SIFT): STUDIES OF ION-NEUTRAL REACTIONS D. SMITH und N . G. A D A M S Deparrmtw t f S p i w Rtwiircli Uniivrsity of Birmingliuni Birminylimn BIS 2TT, England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . 11. Principles and Experimental Aspects of the SIFT Technique . , , . A. The SIFT Chamber and SIFT 1n.jector. . . . . . . . . . . B. The Flow T u b e . . . . . . . . . . . . . . . . . . . C. The Detection System . . . . . . . . . . . . . . . .

.

D. Flow Dynamics and the Determination of Rate Coefficients . , SIFT Studies of Ion-Neutral Reactions: Some Illustrative Results . The Variable-Temperature Selected Ion Flow Drift Tube, VT-SIFDT VT-SIFDT Studies: Some Illustra~iveResults . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

.

.

,

,

111.

IV. V. VI.

. . . . . . . . .

. . .

. . . .

.

.

. . .

I 3 6 12 16 18

23 33 39 44 46

1. Introduction Among the techniques available for studying the kinetics of ionic reactions in the gas phase, none are more reliable and productive than the fast How tube techniques, the flowing afterglow (FA), and the selected ion flow tube (SIFT). The enormously successful FA was first developed in the early sixties by Ferguson, Fehsenfeld, and Schmeltekopf in Boulder, Colorado, to study ion-neutral reactions under truly thermal energy conditions (Ferguson et ul., 1964, 1969). Subsequently, several FA were constructed in laboratories around the world and exploited to determine the rate coefficients and to identify the product ions for a large number of reactions of positive ions and negative ions with atoms and molecules (e.g., see Albritton, 1978). Notable achievements of the many F A studies were the characterization of different reaction types and the accumulation of sufficiently accurate critical kinetic data for comparison with theoretical predictions (Bohme, 1975; Fehsenfeld, 1975). The elucidation of the ion chemistry of the terrestrial ionsphere (largely by the Boulder group) was a direct consequence of the development 1 Copyright t 19x8 hy Academic Preas. Inc All rights o r rcpradu'tion in any form rearrved ISBN O-I2-W3X24-2

2

D.Smith and N . G. Adurns

of the F A (e.g., see Ferguson et ul., 1979). That the FA could be operated at temperatures as low as 80 K and as high as 900 K permitted the study of the reactions of weakly bound cluster ions, which are readily dissociated at room temperature, and the determination of the temperature dependences of rate coefficients over a wide temperature range. It is worthy of note that the parallel development of the ion cyclotron resonance (ICR) technique and its exploitation also added much to the rapid advancement in the understanding of the mechanisms of ion-neutral reactions. The ICR technique and its use to study ion-neutral reactions have been discussed in several reviews (e.g., see McIver, 1978a, 1978b; Aue and Bowers, 1979). Notwithstanding the great success of the FA, which is still being successfully used to study ion-molecule reactions (DePuy and Bierbaum, 1981; DePuy, 1984), it does have limitations and these are best explained after outlining the basic principles of the technique. In the FA, ionization is created in the upstream region of a fast-flowing carrier gas (usually helium) generally by some sort of electrical discharge through the carrier gas. Thus, for example, in helium a discharge plasma is created consisting of He’ ions and electrons together with metastable atoms. These species are convected downstream by the carrier gas and thus an afterglow plasma is created in which the charged and neutral species are kinetically thermalized. Reactant gases can be added to the thermalised plasma and their interactions with the He’ ions, the electrons, and the metastable atoms can be studied at precisely defined kinetic temperatures. The standard diagnostic technique in the F A is a downstream mass spectrometer that is used to monitor both the rate of loss of the primary (He’) ions and the appearance of the product ions of the reaction as a function of the flow rate of the reactant gas into the afterglow. The rate coefficient for the reaction can readily be determined from such observations since the dynamics of these flow reactors are now understood and well documented. (See Section IID.) To study the reactions of positive ions other than He+, a “source gas” with which the He’ ions can react to form a different “primary” ion is added to the Hef plasma. Reactant gases are then added to the plasma at a position downstream of the source gas addition point and the reactions of these new primary ions are studied as described before. Further sequential gas additions can be made upstream of the reactant gas addition point, creating a variety of primary reactant ions. Negative ion reactions can also be studied by adding electron attaching gases to the afterglow. This extraordinary flexibility of the technique is a major reason for its great success. However, the very fact that ion source gases have to be introduced into the afterglow to create the primary ions is a limitation of the technique because the primary ions very often react rapidly with their parent gases. This can greatly complicate, and may indeed prevent, the identification of the ion products of the reaction under study and may in some cases even result in erroneous rate coefficients.

THE SELECTED ION FLOW TUBE (SIFT)

3

This is a particularly severe problem in the study of, for example, hydrocarbon ion reactions which are so involved in the synthesis of interstellar molecules (Herbst and Klemperer, 1973; Dalgarno and Black, 1976; Smith and Adams, 1981b). It was largely to avoid this type of problem that the SIFT technique was conceived and developed (Adams and Smith, 1976a, 1976b) and with which this chapter is mainly concerned. The basic principle of the SIFT technique and its advantages over the F A have been discussed previously (Smith and Adams, 1979). However, considerable developments and extensions to the technique have been made during the past few years and more than 20 SIFT apparatuses have now been built and are being exploited in laboratories around the world. So it is appropriate to review again the basic principles of the SIFT technique and discuss recent developments. The SIFT (particularly the variable-temperature SIFT or VT-SIFT) allows a much wider range of reaction types to be studied than is possible with the FA. These studies are extending the knowledge of reaction kinetics and are providing critical kinetic and thermodynamic data for use in modelling real plasmas as diverse as laser plasmas (Nighan, 1982), etchant plasmas (Turban, 1984), planetary atmospheres (Huntress, 1974; Thomas, 1974; Wayne, 1985), and interstellar gas clouds (Dalgarno and Black, 1976; Smith and Adams, 1981b). The major thrust of this chapter is in Section 11, which is concerned with the principles and experimental aspects of the VT-SIFT technique. In Section 111, we present some results illustrating the versatility of the technique and the great variety of ion-neutral reaction processes that can be studied using it. More detailed discussions of the results of SIFT studies have been given elsewhere (Smith and Adams, 1984; DePuy, 1984; Adams and Smith, 1983, 1984b; Viggiano and Paulson, 1984). In Section IV, we describe the most recent extension of the technique- the development of the variable-temperature, selected ion flow dr$r tube (the VT-SIFDT), which extends the applicability of the VT-SIFT to studies of ion-neutral reactions at supratherma1 ion energies. With this development, ion-neutral reactions can be studied over appreciable ion-neutral centre-of-mass energies at any temperature within a wide temperature range. Illustrative examples of ion-neutral reactions studied using the VT-SIFDT are given in Section V. In Section VI some future uses of the VT-SIFT and VT-SIFDT techniques are mentioned.

11. Principles and Experimental Aspects of the SIFT Technique The SIFT technique is a natural extension of the FA technique. It builds on and extends the versatility of the FA for the study of ion-neutral reactions under truly thermal conditions. The basic idea of the SIFT is as follows. Ions

4

D.Smith and N . G . Adums View from Roots

D,,,

\SIFT 1 injector

-

e te ct ion system -i

1

50cm

FIG.1. A schematic diagram of a typical SIFT apparatus illustrating its major features. The SIFT chamber, the flow tube (typically 100 cm long and 8 cm in diameter) and the detection system are shown here and are illustrated in greater detail in Figs. 2.4, and 5, respectively. Details of the SIFT injector are given in Fig. 3. The profiles of the reactant gas flows into the carrier gas stream are illustrated for a simple axial port (a), a radial port (b), and a “ring” port (c). (For discussion of these, see text.)

(positively or negatively charged) are created in an ion source containing an appropriate source gas (see Fig. 1). The ions then enter a quadrupole mass filter that can be set to pass ions of a given mass-to-charge ratio and these mass selected ions are injected at low energy via a small aperture into a flow tube along which they are convected by a fast-flowing carrier gas at a pressure of typically 0.5 Torr. The ions thermalise (but see Section III for qualification) in collisions with the carrier gas atoms or molecules and, thus, a thermalised ion swarm reaches a downstream reaction zone. Ultimately, some of the ions arrive at a small orifice in a nose-cone section of a mass spectrometer housing located at the downstream end of the flow tube. Here the ions are sampled, mass analysed, and then detected using ion-counting techniques. Reactions of these thermalised “primary” ions can be studied by introducing reactant gases into the reaction zone of the flow tube and observing the reduction in the primary ion count rate as a function of the reactant gas flow rate in a manner similar to that adopted in FA studies. (See Section 1I.D.) The vital difference between the SIFT and FA techniques is now apparent. In the SIFT, the ions are created in a remote ion source and not in the carriergas-reactant-gas mixture. Thus, crucially, the ion source gas is excluded from

THE SELECTED ION FLOW TUBE (SIFT)

5

the carrier gas and therefore from the reaction zone and so reactions between the primary ions and their source gas cannot occur. Such reactions cannot therefore confuse studies of the reactions of the primary ions with added reactant gases. This simple development has enormously extended the application of fast flow tube techniques allowing the reactions of virtually any ion that can be injected into a carrier gas (even at very low currents) to be studied with stable gases, vapours, and even radical species over a wide temperature range (as illustrated in Section 111). A major advantage over the FA technique is that accuratc ion product distributions can be determined (Adams and Smith, 1976b) due not only to the elimination of primaryion-source-gas reactions and to the exclusion from the flow tube of excited (metastable) species that originate in the electrical discharge in the FA, but also to the fact that a method has been devised to calibrate the downstream mass spectrometer-detection system to account for the inevitable discrimination that occurs between ions of different masses (Section IIC). Another difference between the FA and SIFT that should be stressed is that the FA is essentially a plasma medium in which positive ions, electrons and/ or negative ions can coexist whereas only a swarm of positive ions or negative ions exists in the SIFT. Thus, in the FA, ions are lost along the flow tube by ambipolar diffusion, which is a faster process at a given pressure than is the free diffusive loss of ions that takes place in the SIFT. This has to be appreciated when the flow dynamics are considered in relation to determinations of rate coefficients and ion product distributions (Section IID). The main sections of a typical SIFT apparatus are shown in the overall line diagram in Fig. 1. It consists of three major sections: (i) the ion source and injection mass spectrometer (mass filter) chamber, often called the “SIFT chamber,” (ii) the flow tube, and (iii) the downstream detection system. Surrounding the flow tube is a vacuum jacket for thermal insulation of the flow tube during high- and low-temperature studies. Sections (i), (ii), and (iii) are each coupled to a diffusion pump for cleaning and outgassing prior to operation. During operation, the SIFT chamber and the detection systemsections (i) and (iii)-arc continuously diffusion pumped and the flow tube is coupled to a large Roots-type pump of sufficient pumping speed to cause the carrier gas to flow at a velocity of about lo2 m s - at a pressure of about 0.5 Torr. Central to the operation of the SIFT is the carrier gas injection system, the “SIFT injector,” which is the interface between the SIFT chamber and the flow tube. The design of the injector allows the flow tube to be operated at a relatively high pressure while connected via a relatively large aperture to the SIFT chamber. In Fig. I , the injector and SIFT chamber are coaxial with the flow tube, a design that has been adopted by some workers. In the original Birmingham SIFT (Adams and Smith, 1976a), the SIFT chamber was coupled at an angle to the flow tube to avoid a direct line-of-sight to the

6

D.Smith uncl N . G. Adurns

channeltron detector in the detection system. (This approach has some other advantages, as seen in Section IIA.) In the latest Birmingham double SIFT, two such SIFT chambers are coupled to the flow tube (See Fig. 4). The essential practical details of each major section of the SIFT and the various SIFT injectors that have been used will now be discussed. Also in this section the flow dynamics of the SIFT will be considered together with the method of analysis of experimental data used to obtain rate coefficients and ion product distributions. A. THESIFT CHAMBER AND

THE

SIFT INJECTOR

Since the development of the original SIFT apparatus by the Birmingham group, there have been further interesting and important developments in Birmingham and elsewhere, especially with respect to the SIFT injector. However, the basic idea remains as it was first conceived. Positive or negative ions are generated in some form of ion source that is coupled to or contained within the main SIFT chamber in which is located a quadrupole mass filter (Fig. 2). The chamber is pumped usually by a four-inch or six-inch diffusion pump (suitably trapped) that must maintain the pressure below Torr in the analysis region of the quadrupole mass filter for effective operation. The SIFT chamber is coupled to the flow tube by the SIFT injector plate at the centre of which is an orifice (the ion injection orifice) of diameter typically 1 mm (but which can be as large as 5 mm in combination with a suitable injector, as will be discussed). Ions pass from the SIFT chamber into the flow tube via this orifice. Electrostatic lenses are often included to focus ions from the ion source into the quadrupole mass filter and (after mass selection) to optimise the ion current flowing through the injection orifice. These major features are shown in Fig. 2. Various types of ion source have been used in SIFT experiments, the choice depending on the ion types required. The simplest to use are low-pressure sources because they do not offer excessive gas loading to the diffusion pump and therefore they can be built directly into the SIFT chamber. Low-pressure sources are open-structure devices in which single electron-neutral collisions generate the ions. By carefully controlling the electron energy, some selectivity can be achieved in the production of excited states of ions, dissociation of molecular ions can be prevented or promoted, specific excited states of some ions (Glosik et a/., 1978; Tichy et al., 1979; Adams et al., 1980b) and doubly charged ions (Adams et al., 1979a) can be produced, etc. Reference is made to specific examples in Section 111 where some illustrative results of SIFT experiments are presented. High-pressure (enclosed) ion sources can also be placed directly into the SIFT chamber as long as the apertures in the ion source through which electrons enter and ions leave do not allow an excessive

THE SELECTED ION FLOW TUBE (SIFT)

7

Electro tatic

i o n source

0i f f usion Pump FIG.2. The general features of a SIFT chamber. Ions are generated an ion source (in this example, it is a low-pressure electron impact source located in the main chamber) and focussed into a quadrupole mass filter by an electrostatic lens system. After leaving the mass filter, they enter a second lens system used to optimise the ion current flowing through the injection orifice. The diffusion pump maintains the pressure below Torr in the chamber against the Row of ion source gas and any carrier gas that backflows through the injection orifice. The SIFT injector is located inside the vacuum vessel to which it is coupled via thin stainless steel bellows for further thermal insulation and to allow for thermal expansion and contraction during the heating and cooling of the Row tube.

flow of gas into the chamber. These sources, the most successful of which have been designed by J. F. Paulson and F. Dale at Air Force Geophysics Laboratory (AFGL), are valuable for generating complex (cluster) ions. (See the review by Viggiano and Paulson, 1984, and the references therein.) Flowing afterglow sources have also been used, but these obviously require an additional fast pump and they must be coupled to the SIFT chamber by a small orifice (preferably in a nose-cone) to minimise the flow of carrier gas into the SIFT chamber from the relatively high-pressure afterglow (Smith and Adams, 1980a; Mackay et ul., 1980; Van Doren, 1986). The great advantage of the flowing afterglow ion source is in its well-known chemical versatility by which a wide variety of ions can be formed under controlled conditions (e.g., at low temperatures). Low-pressure microwave discharge sources have been used to provide large currents of positive ions (Adams and Smith, 1976a), and surface ionization sources have been used to produce alkali metal ions (Smith ef al., 1983). The Birmingham double SIFT apparatus allows two different ion sources to be used simultaneously. This has been particularly valuable in the studies of hydrogen atom reactions. (See Section I11 of this chapter and Adams and Smith, 1985.)

8

D.Smith and N . G. Adurns

A standard quadrupole rod arrangement energised by r.f. and d.c. potentials is used as the mass filter. Typically the rods are 10-16 mm in diameter and 10-20 cm long, but smaller rods have been used in some apparatuses. Brubaker filters have also been used on some injection quadrupoles in attempts to improve ion focussing at the entrance and exit of the mass filter (Glosik et al., 1978; Mackey et a1.,1980). To improve the focussing of ions from the ion source into the mass filter, an octopoie device has been included in one SIFT-type apparatus, and this has also been used to prepare secondary ions by allowing ion-molecule reactions to occur in the octopole (Lindinger and Smith, 1983). The secondary ions are then mass filtered in the usual manner by the quadrupole. The electrostatic lens located between the mass filter exit and the ion injection orifice shown in Fig. 2 have been dispensed with in some cases by positioning the mass filter exit aperture very close to the injection orifice. However, this results in intense r.f. fields at a position of increased pressure (due to some carrier gas back flow) that can cause collisional breakup of weakly bound ions (such as cluster ions). It is extremely useful if the ion injection orifice disc is electrically insulated from its supports. This allows the mass-selected ion current to be sampled by this electrode, which greatly facilitates the optimization of a particular ion species from the ion source and the focussing of the ion beam into the mass filter. Depending on the efficiency of the ion source for particular ion types, ion currents to this disc are usually within the range to lo-’ Amps, although currents as large as Amps have been obtained with rare gas atomic ions. By careful focussing of the ion beam, and depending on the ion energy and the diameter of the injection orifice, as much as 10% of this current can be focussed through the orifice and into the carrier gas. For atomic ions, an ion injection energy in the laboratory frame, Elab,of about 20 eV is appropriate. However, for molecular ions an important consideration is that the centre-of-mass interaction energy with the carrier gas, E,,, should be less than the dissociation energy of the ion. In this regard, helium or hydrogen are the favoured carrier gases because of their small masses since E,, = Elabx [ M J M , + M,)] where M , and Mi are the masses of the carrier gas and the injected ion, respectively (e.g., see McDaniel, 1964). Thus, for example, for a hydrocarbon ion (with C-H bond energies of about 4 eV) of mass, say, 16 amu injected into helium carrier gas, Elabhas to be less than 20 eV in order to minimise collisional dissociation of the molecular ion. For more weakly bound low-mass molecular ions or for heavier carrier gases such as argon or nitrogen, then Elabmay need to be as little as 1 eV or so. For such small El&,, the injected ion currents are considerably reduced. Nevertheless, this has not prevented the injection of adequate currents of many weakly bound ions, including large cluster ions. (See Sections 111 of this chapter as well as Smith et al., 1980; Viggiano and Paulson, 1984. Note also that an increased ion mass acts to relax the restriction on A final interesting

9

THE SELECTED ION FLOW TUBE (SIFT)

point to note is that at a given Elab,larger mass ions are more effectively focussed through the relatively small injection orifice. This we presume is due to the smaller excursions of the more massive ions from the axis of the quadrupole mass filter (Smith and Adams, 1979). The essential requirement of a SIFT injector is that it must facilitate the injection of molecular ions into a flowing carrier gas at sufficiently low energy to prevent collisional dissociation and at the same time minimise the backstreaming of carrier gas into the SIFT chamber. This was first accomplished by the Birmingham group using the injector shown in Fig. 3(a). The essential features of this injector are as follows. The carrier gas enters the flow tube via a circular array of 12 holes, each of 1 mm diameter, located on a (a)

,Bellows A r r a y of holes

/

It

II II II I1

I1

lens

II

Y

1cm

1 Carrier

gas

inlet

,Annular slit

\

Quadr upole, mass filter

II II I1 II

1cm

1 Carrier gas inlet

II I1

FIG. 3. (a) A schematic representation of the Birmingham SIFT injector in which the carrier gas enters the flow tube via an array of small holes. Note that the mass filter is separated from the injection orifice by a lens system. (b) A schematic representation of the NOAA-York injector in which the carrier gas enters the flow tube via an annular slit. Note that in this example, the mass filter is located very close to the injection orifice.

10

D.Smith and N . G. Adorns

circle of radius 1 cm that is concentric with the ion injection orifice. The injection orifice plate is set back from the plane containing the array of holes by about 1 cm. At the carrier gas flow rates required for fast flow tube studies (both FA and SIFT), which are typically 100 Torr 1 . C ' (of helium), the carrier gas emerges from the injector holes at supersonic velocity in a direction away from the ion injection orifice. This limits the gas flow into the SIFT chamber and thus the diffusion pump is able to maintain the pressure in the chamber at a sufficiently low level to allow the quadrupole mass filter to operate even at flow tube pressures of one Torr or so. For static gas in the flow tube at this pressure, the SIFT chamber diffusion pump is overwhelmed by gas flow through the ion injection orifice and the quadrupole ceases to operate. With this type of injector, ion injection orifices with diameters of up to 3 m m have been successfully used. For orifices larger than this, the pressure in the SIFT chamber becomes prohibitively high for normal carrier gas flows. Using three SIFT apparatuses incorporating this type of injector, the Birmingham group has studied some thousands of reactions, many involving weakly bound ions. Some illustrative results of these studies are given in Section 111. The obvious success of the Birmingham SIFT has stimulated some users of flowing afterglows to convert them to SIFT apparatuses. This resulted in the design of a second type of SIFT injector by the National Oceanic and Atmospheric Administration (NOAA) group at Boulder, Colorado (Howorka et d.,1979) and independently and concurrently by the York University, Toronto group (Mackey et al., 1980). The essential features of this new design are shown in Fig. 3(b). The carrier gas enters the flow tube via an annular (slit) aperture which, again, is concentric with the ion injection orifice but quite close to it. The diameter of the annular slit is about 9 mm in both the NOAA in the York designs. The width of the annular slit in the York design is 0.025 mm; the NOAA group chose to investigate the effectiveness of two slit widths, these being 0.41 mm and 0.025 mm (designated NOAA I and 11; Dupeyrat et al., 1982) and they have compared the effectiveness of the NOAA I and NOAA I1 designs and the Birmingham design as venturi (or aspirator) inlets. For this they measured the ratio of the pressure in the downstream region of the flow tube (PI) to that in the SIFT chamber ( P 2 ) when the SIFT chamber was isolated from its diffusion pump for a range of carrier gas flow rates though the three types of inlet systems. Thus P , / P , was taken as an indicator of the effectiveness of the venturi but more importantly as an indicator of the backstreaming rate of carrier gas into the SIFT chamber. It was found that the NOAA I1 design (and hence, by inference, the York design) resulted in a significantly enhanced P , / P , compared to the Birmingham and the NOAA I designs. Hence with the NOAA I1 design, less backstreaming of carrier gas occurred and, with the diffusion pump operating, the SIFT chamber was maintained at lower pressures. This explains why

THE SELECTED ION FLOW TlJBE (SIFT)

I1

the York group was able to use an ion injection orifice of 5 mm without “swamping” the SIFT chamber with carrier gas. The explanation for this improved performance of the NOAA I1 and York designs over the Birmingham and NOAA I designs is plain. The much smaller total cross-sectional area of the narrow annulus for a given gas flow rate demands a higher gas velocity at the injection point. However, there is a price to pay for this advantage. It was noticed by Mackay et al. (1980) that the rate coefficient for a gas kinetic reaction as determined in their SIFT apparently decreased significantly as the carrier gas flow rate through their SIFT injector was increased towards that required to establish the normal operating flow tube pressure ( ~ 0 . Torr). 4 This they attributed to the onset of turbulent flow along the flow tube for flow rates through their SIFT injector above a critical value. To avoid this problem, they cleverly introduced a fraction of the carrier gas through a series of 100 holes, each of 1 mm diameter, which were uniformly distributed over the SIFT injector plate while maintaining a carrier gas flow below the critical value through the SIFT (venturi) injector. This combination of supersonic and subsonic gas injection apparently solved the problem. This type of approach has also been used by C. H. DePuy and V. M. Bierbaum and their colleagues at Boulder, Colorado (Van Doren, 1986) but in their case, the two components of the injector were both annular. Further investigations of the gas flow patterns from the SIFT injector have been carried out by Dupeyrat rt a/. (1982). They have installed the Birmingham, NOAA 1, and NOAA I1 injectors into a wind tunnel and explored the flow patterns of the gas emerging from these injectors by ionizing the gas and by photographing the luminous plasma jet so formed. These experiments showed that, as the flow rate of gas increased, the cell structure typical of a hypersonic jet persisted for increasing distances away from the gas injector. Clearly, for narrower gas entry apertures-and for greater gas velocitiesthis effect will be more severe. The proper explanation of the improvement in the behaviour of the York experiment when a fraction of the carrier gas was introduced at subsonic speeds is probably that the cells of shocked gas originating at the annular injector were broken up before reaching the reaction zone of the flow tube by the relatively slow moving gas injected subsonically through the array of holes. This effect would not have been so apparent in the experiments of Mackay rt a / . (1980) if their flow tube had been longer, thus giving the gas a longer time to settle into laminar flow. Recent studies by Miller et al. (1984) have shown that, for a SIFT injector of the NOAA-II-type co-axial with the main flow tube, the region of nonlaminar flow extended some 50 cms down the flow tube. This was particularly serious at low temperatures (90K) at which the mean velocity of the ion swarm in the upstream half of the flow tube was measured to be some four times greater than that in the downstream

12

D.Smith und N . G. Adums

region! If this effect persists even to a relatively small degree into the reaction zone, it can introduce errors into the derived rate coefficients. Such effects have not been observed to anything like the same degree in the Birmingham SIFT. This can be attributed on the one hand to the injector design that, while not being such an effective venturi as the NOAA 11 design, does not generate shock cells so readily, and on the other hand because the carrier gas enters the flow tube from the injector into a side arm that is at an angle to the main flow tube. This “bend” in the flow tube would help to destroy any shock cells that may be present and hence reduce the settling down distance (i.e., the distance necessary to achieve laminar flow). Nevertheless, the YorkNOAA-I1 injector design positioned co-axial with the main flow tube has been preferred by some SIFT users (Howorka et al., 1979; Mackey et al., 1980; King et a1.,1981; Knight et al., 1985) in spite of the potential problems just outlined and of its more complicated construction as compared to the Birmingham injector. What is the best design for a SIFT injector? It seems probable that the Birmingham design with appropriately small diameter holes would perform rather similarly to the York-NOAA-I1 design, although this has yet to be tried. If this were so, the modified Birmingham design would be preferred because of its simplicity of construction. T o avoid severe shock cell effects, which would result from introducing all the carrier gas through these small diameter holes, a second array of larger diameter holes on a greater pitch circle diameter could be included and coupled to an independent reservoir through which a portion of the carrier gas could be introduced (rather like the York design). This approach is being adopted by J. J. Grabowski (1986) in the Chemistry Department at Harvard. Positioning of this injector arrangement on a side arm angled to the main flow tube would also tend to reduce the settling down distance along the flow tube. With this design, a relatively large ion injection orifice could be used as in the York SIFT. However, it should be noted that a large ion injection orifice could lead to significantly large flows of ion source gas from the SIFT chamber into the flow tube with its inherent drawbacks. This could be minimised by separately pumping the ion source and coupling it to the SIFT chamber via another orifice, thus minimising the pressure of ion source gas in the SIFT chamber.

B. THE FLOWTUBE The stainless steel flow tubes in all the F A and SIFT apparatuses used to date have similar dimensions, typically 100-cm length and about 8-cm diameter (see Fig. 4). Variable temperature versions (VT-SIFT) include a copper jacket ( % 3 mm thick) tightly strapped to the flow tube. Onto this

#/discharge Microwave

--.

'ran

-I

c W

1 50 cm FIG.4. A schematic of the Birmingham variable-temperature SIFT that has two ion injection systems. The flow tube is situated inside the vacuum jacket for thermal insulation. Clamped to the main body of the flow tube is a 3mm-thick copper jacket (dashed in the figure) to which are clamped several ohmic heaters and to which are brazed copper tubes (through which refrigerant liquids flow, e.g., liquid nitrogen) and stainless steel tubes through which the carrier gas flows prior to being heated or cooled before its entry into the flow tube via either or both of the SIFT injectors (Fig. 3(a)). Reactant gases pass through other similar stainless steel tubes. The entire flow tube including the copper and stainless tubing is surrounded by aluminium foil which acts as a radiation shield (represented by the dashed lines). The reactant gases enter the flow tube via two ring ports. The wider axial port located upstream allows radical species to be introduced into the flow tube following their production in a microwave discharge.

m

-

W

14

D. Smith und N . G. Adams

jacket are brazed copper pipes through which refrigerant liquids can be flowed for low-temperature operation. Ohmic heaters are also clamped to the jacket for high-temperature operation. The large thermal conductivity of the copper ensures that temperature gradients and hot and cold spots are minimised and that the inside surface of the flow tube is at a uniform temperature. Thermocouples are positioned around the outside of the flow tube and also in the carrier gas to monitor the temperature. A typical spread of +2K is achieved along the reaction zone in the Birmingham SIFT at a mean temperature of SOK. At 500K the spread is f 8 K . Also brazed to the copper jacket are the stainless steel pipes that transport the carrier gas via the SIFT injector into the flow tube. This ensures that the temperature of the carrier gas is equilibrated with the walls of the flow tube prior to its entry into the flow tube. The pipes that carry reactant gases into the flow tube are similarly coupled to the copper jacket, although precooling of the reactant gas is generally not required because of the relatively small flow rates used. (Indeed, precooling is undesirable for condensible vapours since at low temperatures it may result in a reduction in the partial pressure of the vapour.) The flow rates of the carrier and reactant gases are measured in the usual way by using calibrated capillary tubes, commercial flow meters, or the pressure drop in a calibrated volume. The flow tube has to be thermally isolated from its surroundings for high- and low-temperature operation and this is achieved, as previously mentioned, by enclosing it in a vacuum jacket. Decoupling of the flow tube from the vacuum jacket is achieved using stainless steel bellows in the upstream section (these bellows also permit thermal expansion and contraction of the flow tube) and by thinning the stainless steel flow tube in the vicinity of the downstream region as indicated in Fig. 5. The complete flow tube, including the bellows and those copper and stainless steel pipes, that are inside the vacuum jacket, is wrapped with aluminum foil which acts as a radiation shield. With this arrangement, temperatures as low as SOK can be achieved (using liquid nitrogen as the refrigerant) for only a modestly low pressure of Torr inside the vacuum jacket. Heat flow to and from the downstream mass spectrometer housing is minimised by thinning the stainless steel nose-cone that supports the sampling orifice. (See Fig. 5.) This ensures that these surfaces are closely equilibrated at the temperature of the walls of the flow tube via radiative coupling and thermal contact with the carrier gas. Three distinct types of reactant gas inlet ports have been used in F A and SIFT apparatuses and each is illustrated in Fig. 1. The on-axis type (a) was used in the first FA (and in some SIFT) experiments and is sometimes still used. The end of the pipe is generally shaped (or bent) to direct the gas downstream. This results in a positive “end correction,” E, to the reaction length, z (the distance between the gas entry port and the downstream

THE SELECTED ION FLOW TlJBE (SIFT) Ceramic

15

Thin s t a i n l e s s

Insulator

steel

I 1 I I

t ,

-

spectrometer

Mass spectrometeVchanneLtron housing

lOcm

FIG.5. The SIFT detection system. Ions enter the sampling orifice, which is located in a very thin disc (see text), and are then accelerated into the lens system by the electric field established by applying only a low potential to the top hat electrode, which is located very close ( 11 mm) to the orifice disc (shown clearly in the enlargement of the orifice region). The ions are focussed into the mass spectrometer and, after exiting, are detected by the channeltron multiplier. The complete mass spectrometer-channeltron housing is electrically insulated from its supports so that it can be maintained at a different potential from the grounded flow tube when the apparatus is operated as a VT-SIFT. (See Section IV and Fig. 10.)The gas inlet port located just upstream of the sampling orifice permits a monitor has to be added to the carrier gas stream. (See text.)

sampling orifice), i t . the effective reaction length is ( z + c). Depending on the carrier gas flow rate and the pressure, c varies between about 4 and 10 cm, which for a value of z of 50 cm is quite significant and must be accounted for in rate coefficient determinations (Section IID). If the reactant gas is directed initially in the upstream direction (counterflow) via these axial ports, then the E is much smaller. In the first SIFT (Adams and Smith, 1976a, 1976b) and in most flow-drift tube experiments (McFarland er ul., 1973a), the reactant gas was introduced via a series of holes around the walls at a fixed axial position along the flow tube. (The “radial” ports (b) are illustrated schematically in Fig. 1.) For these radial ports, I: is negative and again varies with the pressure and flow rate of the carrier gas in a similar fashion (and is similar in magnitude) to that for the on-axis, downstream-pointing ports. Thus, for these radial ports, the effective reaction length is ( 2 - E )and so I: is proportionately more significant for a given

16

D.Smith und N . G. Adurns

z than it is for the on-axis ports. The “ring ports” (c) illustrated in Figs, 1 and 4, which were first used in the Birmingham flowing-afterglow-langmuirprobe (FALP) apparatus (Alge et al., 1983), have now been adopted in many SIFT apparatuses since they result in very small E when the contraflow reactant gas technique is used. Over the range of pressures used in SIFT experiments, E varies only from about + 1 cm to - 1 cm for these ring ports (over the normal range of carrier gas flow rates), which is hardly significant for a typical z of 40 to 50 cms. Hence ring ports are rapidly becoming the ports of choice in FA, SIFT, and SIFDT apparatuses. In order to ensure that E does not vary with reactant gas flow rate, a constant but small proportion of the carrier gas is used to flush the reactant gas through the port. This has the added advantage of reducing the time taken for the reactant gas flow to stabilize, especially during lowtemperature operation or when the reactant is a condensible vapour. The end correction, E , can be determined by measuring a reaction rate coefficient using two or more identical reactant inlet ports positioned at different z and by varying the reactant gas flow rate. (See Section IID.) Alternatively, a fixed reactant gas flow rate and a movable port (varying z ) can be used to determine rate coefficients, which eliminates the need for a determination of E (DePuy and Bierbaum, 1981), although this approach is hardly practical for variable-temperature SIFT apparatuses because of the presence of the vacuum jacket. Reactant gas ports are usually coupled to the flow tube via side arms. (See Fig. 4.) Strategically positioned side arms also allow the ready addition as required of optical viewpoints, special inlet ports, etc. All flanges are coupled to the flow tube by copper gasket seals to allow high- and lowtemperature operation. In the Birmingham VT-SIFT, a reactant gas port of 5 m m internal diameter has been included that also passes through the vacuum jacket wall, to allow the flow of radical species (created in a microwave discharge) into the flow tube. Thus, some reactions of H and D atoms have been studied over the temperature range available to the VT-SIFT. (See Section 111.) Also included in the Birmingham VT-SIFT is a drift tube section which created the first VT-SIFDT. (See Section IV.)

c. THEDETECTION SYSTEM The detection system consists of an ion-sampling orifice, an electrostatic lens system that focusses ions into a quadrupole mass spectrometer, and, finally, an ion-detector-amplifier that is invariably a “channeltron” multiplier device. The sampling orifice is located at the centre of an electrically conducting disc (usually molybdenum because it is relatively inert to

THE SELECTED ION FLOW TUBE (SIFT)

17

chemical attack) located at the apex of a cone forming part of the housing that contains the mass spectrometer-detection system (Fig. 5). Since the carrier gas flow velocity is only a small fraction ( 1/10) of the sound speed in the gas, no special shaping of the nose cone is required; a cone angle of 90 degrees is commonly used. The detection orifice disc should preferably be electrically insulated from the nose cone. A small bias voltage can then be applied to the disc relative to the flow tube walls to counteract contact potential differences. Also the ion current to the disc can be measured and, hence, the ion current through the orifice can be obtained. Thus, the efficiency of the complete detection system can be determined (that is, the collective transmission efficiency of the lens, the quadrupole mass spectrometer, and the detection sensitivity of the channeltron multiplier) for ions of different massto-charge ratios. By this method, account can be taken of mass discrimination in the detection system, which is essential, of course, if accurate ion product distributions are to be determined for ion-molecule reactions (Adams and Smith, 1976b). Mass discrimination is particularly severe when the quadrupole mass spectrometer is set for high resolution and when the ion products have greatly differing masses. The facility to measure the ion current to the detection orifice disc also helps in the optimisation of the ion current injected upstream into the carrier gas. In order that the carrier gas flow rate into the detection system should not be too great, the sampling orifice must be suitably small. Thus the orifice diameter is typically 0.1-0.3 mm, which in conjunction with a 4-inch diffusion pump ensures that the pressure in the detection system housing remains below about lo-’ Torr for a typical helium carrier gas pressure of x0.5 Torr. To avoid ion-sampling problems (due to contamination of the orifice, which can result in charge buildup and even physical blockage), the orifice must in no way resemble a narrow channel, i.e., it must be in a very thin disc and the diameter-to-depth ratio must be large ( > 10). This is effectively achieved by eroding the disc in the region of the orifice to the shape shown in Fig. 5. Charge buildup is inhibited and ion collection into the detection system is enhanced if appreciable extraction electric fields can be established in the vicinity of the sampling orifice. This can be achieved for only small extraction potentials using the “top hat” electrode shown in Fig. 5 with which the spacing between the orifice and the first element of the lens system can be made about 1 mm. Hence for an extraction potential of only one volt on the top hat relative to the orifice disc, a field strength of about 10 V.cm-’ is established. This results in appreciable field penetration into the flow tube, which acts to focus ions into the detection system and minimises effects due to charge buildup at the orifice. Very small extraction voltages are necessary in the region of the detection system where the gas pressure is largest in order to avoid accelerating weakly bound ions to energies such that they can fragment

-

D.Smith and N . G. Adums

18

on collision with carrier gas atoms. This collisional breakup of weakly bound (cluster) ions can be easily demonstrated in SIFT experiments by observing the currents of such ions and their ion fragments as the potentials on the electrodes in the vicinity of the sampling orifice are varied. Generally, the channeltron multiplier is located in an off-axis position to avoid direct line-of-sight between its entrance aperture and the ion source of an on-axis SIFT injector to minimize photon background counts. For an offaxis SIFT injector such as the Birmingham arrangement, the channeltron can be on-axis; we have noted that this gives an improvement of about a factor of 3 in the collection efficiency of the channeltron. In some apparatuses the channeltron is contained in a separately pumped chamber but this is not necessary unless the SIFT is to be operated at pressures well in excess of 1 Torr or with a large sampling orifice. With the sampling orifice and top hat arrangement shown in Fig. 5 and with an on-axis channeltron, the ion transmission efficiency for low resolution of the quadrupole can approach 100%.

D. FLOWDYNAMICS AND THE DETERMINATION OF RATE COEFFICIENTS The basic experimental approach to the determination of the rate coefficient, k, for the generalized reaction

A+

+ B -+k

C++D

(1)

is straightforward. The current (usually the ion count rate) of A + ions at the downstream detector is monitored as a function of the flow rate of the reactant gas, B, into the flow tube. Thus k is defined by the first-order rate equation

d [ A + ] - d[A'] - ~i = -k[A+][B] aZ dt

~~

~

where the square brackets refer to concentrations of A + and B in the reaction zone of the SIFT and ui is the average ion flow velocity. (The term describing the diffusive loss of A + is not included in this simple analysis but is included in the more detailed analysis below.) When [ B ] 9 [A'], as it invariably is in SIFT experiments, then Eq. (2) can be integrated to give k as

k

= "S(m)In[A+] Z

(3)

where S is the slope of the plot of In [ A ' 3 against [ B ] and z is the reaction length. (Note that in practice z must be modified to include any end

T H E SELECTED I O N FLOW T U B E (SIFT)

19

correction, E . ) Clearly z/vi defines the reaction time. Thus if oi can be determined and [ B ] can be related to the flow rate of B into the flow tube, then k can be readily obtained. To determine these parameters requires an understanding of the flow dynamics of the neutral carrier gas and of the ion swarm; then [ B ] and ui can be calculated. Since ui can also be measured directly, it can be compared with calculated values, as will be discussed. As previously stated, the flow tubes for both FA and SIFT apparatuses are reasonably standard in dimensions, being typically 100 cm long and 8 cm in diameter. A Roots pump of speed typically 400 to 600 Is-' is used to establish the carrier gas flow and pressures of ~ 0 . 5Torr are necessary to ensure that sufficient ions arrive downstream (i.e., that diffusive loss is sufficiently small) to give a measurable ion current to the detector. This combination of tube size and pumping speed results in flow velocities of about lo2 ms-' and, hence, ion residence times in the flow tube of the order of 10 ms. At the typical operating pressure, the mean free path of gas atoms in the carrier gas is small compared to the diameter of the flow tube and so the flow is viscous (or Poiseuille) flow with some slip near to the walls. Thus an approximately parabolic radial profile of the axial velocity, ug(r), is established in the flow tube, which is given by Ferguson et ul. (1969) and Bolden et ul. (1970) as (I-$+;) ug(r)= 2on 1

+ -4as

(4)

where a is the flow tube radius, u n is the bulk flow velocity of the gas, and s is a pressure-dependent slip coefficient. (A knowledge of ug(r) is necessary for calculating ui, as will be discussed). The s in Eq. (4) is commonly related via the expression s = s'/p to the pressure independent slip coefficient s', which depends only on the particular gas and the wall material. Brown and DiNardo (1946) have measured s for a wide variety of gases flowing down glass tubes and Adams et (El. (1975) have measured s'/a for helium flowing down a glass tube. No measurements have been reported for stainless steel tubes. From Eq. (4) the slip velocity at the walls u(a) = 4u,s/a ( 1 + 4 . ~ 4 ~ ) Using the measured values of s'/a at 300K for helium-glass, Adams et al. (1975) have shown that u(u) is -0.40, at 0.1 Torr and - 0 . 0 7 ~at ~ 1.0 Torr. Thus, under normal operating conditions of a SIFT or a F A (i.e., at 0.5 Torr), a near parabolic radial velocity profile is to be expected with a few percent slip at the walls. It is unlikely that the material of the walls will make much difference to u(a) as long as the walls are reasonably smooth (to better than a neutral mean free path in the gas). Clearly it would be unwise to operate a

'.

20

D.Smith

and N . G. Aciams

SIFT at pressures near to 0.1 Torr where u(a) is large, especially if a calculated value of ui has to be used. At such low pressures, the flow begins to become molecular and the pressure gradient along the tube becomes prohibitively large. Clearly this introduces a significant gradient in both the carrier gas and the reactant gas number densities and complicates the derivation of rate coefficients. We have observed such effects for pressures below 0 . 2 Torr at 300K and therefore avoid operating below this pressure. At 0.5 Torr the pressure drop along the reaction zone in the SIFT is only a few percent of the mean pressure. uo is determined from a measurement of the flow rate of carrier gas into the tube, Q, using the relationship uo = (Q/nazp) where p is the flow tube pressure as measured in the centre of the reaction zone. The calculation of ui is more complicated because the ions are destroyed at the walls and hence a radial ion number density profile exists (unlike the radially independent number density of the carrier gas). However, the instantaneous radial velocity profile of the ions will be the same as that of the carrier gas atoms, i.e., ug(r).The majority of the ions are located near to the axis of the flow tube where their instantaneous velocity is greatest (and approximately equal to 2u0, i.e., twice the carrier gas bulk velocity). It is therefore to be expected that the mean ui will exceed uo. How then can the mean ui be calculated? Adams et al. (1975) have calculated ui for ions in a flowing afterglow plasma by averaging their velocities over the radial ion number density profile ni(r) thus:

It is to be expected that in the downstream region of a F A or a SIFT, the so-called fundamental mode of diffusion for ions (and electrons in the FA plasma) will be established with ni(r) being closely described by the wellknown Bessel function radial distribution. By including this ni(r) and ug(r)in Eq. (5) and performing the integration, Adams et al. (1975) showed that ui 1 $I,. They also made detailed measurements of ui in their FA using a pulsed Langmuir probe technique and found that the measured ui/uo were consistent with their predicted value over a range of carrier gas pressures and electron number densities in the plasma. Much subsequent experimental work by the Birmingham group has shown that uJuo at 300K is 1.45 increasing to 1.65 at 80K in both their FALP apparatus and their SIFT apparatus. That the measured values at 300K somewhat exceed the value of 1.33 predicted by Eq. (5) is partly accounted for by the neglect of axial diffusion in the calculation of ui (using Eq. (5)), which would increase ui by about 2 for typical operating conditions (as will be discussed). It is gratifying to note that Ferguson et al. (1969) using a fuller theoretical

21

T H E SELECTED ION FLOW TUBE (SIFT)

approach previously found that iii/u,, varied between 1.4 and 1.6 over a range of experimental conditions appropriate to their flowing afterglow apparatus. However, it should also be noted that Miller et al. (1984) have measured somewhat larger vi/u,, in their variable temperature SIFT-namely 1.68 at 300K and 2.1 at 90K, the latter exceeding the expected maximum on-axis value of vi/uo = 2.0. This strongly suggests that the ion density was excessively peaked on the axis of their flow tube (i.e., that fundamental mode diffusion had not been established) although even this could only give the maximum value of 2 for ui/u,, if the carrier gas flow had become laminar. Whatever the cause of these higher values, the rate coefficients for several well-known reactions, as measured by Miller et al. (1984) using their high values of ui/v0, agree well with values obtained by other workers. Hence it is clearly advisable to measure ui/u,, in each SIFT apparatus and not to rely entirely on theoretical predictions. When the value of ui for a given apparatus has been obtained under appropriate conditions of temperature and pressure, then it (together with the experimentally determined s and v,,) can be used in Eq. (3) to determine k. An alternative, more rigorous approach to the determination of k in a SIFT is to solve the complete continuity equation for the loss of ions:

B

A

C

The terms A , B, and C are respectively the radial diffusive, axial diffusive, and reactive contributions to the loss of A ' , and Di is the free diffusion coefficient for the A' ions. (This equation is identical to that appropriate to a F A except that Di has replaced D,, the ambipolar diffusion coefficient.) If term B is neglected (which results in only a 27; error in ui and hence in k at a typical pressure of 0.5 Torr), the equation can be solved analytically by two different approaches: (i) by assuming that ui(r) is effectively independent of r (i.e., ion plug flow applies averaged over the reaction zone) and (ii) by equating ui(r) to ug(r). Both approaches yield a solution of the same analytic form. (See Bolden et ul., 1970; Adams et al., 1975.) [ A + ] , = [ A ' ] , , exp

-

(7)

[A'],, and are the number densities of A + at the point of entry of B and at the detection system sampling orifice, respectively, while A is a constant that is only weakly dependent on pressure. The first term in the bracket accounts for the diffusive loss of A + along the reaction zone, which is

22

D.Smith and N . G. Adams

independent of [ B ] as long as [B]is much less than the carrier gas number density. Clearly a measurement of [ A ’ ] , (and hence the ion count rate of A + at the detection system) as [B]is varied can provide a value for k (as before) if the constant r is known. Using approach (i) just discussed, it turns out that r = uo/ui and so experimentally determined values of r can be used. Adopting approach (ii), Eq. ( 6 ) has been solved using the Sturm-Liouville equation by Bolden et al. (1970) and using the Graetz-Nussell equation by Adams et al. (1975). Both approaches indicate that r is given by

);

(Y 1 +

r=

The more rigorous approach of Bolden et al. (1970) indicates y = 1.26 and hence that r-’ = uJuo = 1.45 at 0.3 Torr and 1.50 at 1.0 Torr at 300K using the measured values of s‘/a for helium in a glass flow tube (Adams et af., 1975). These predicted uJuo are very close to the values measured experimentally by the Birmingham group but are significantly smaller than those measured by Miller et al. (1984). All things considered, a value of uJu0 = 1.5 0.1 seems most appropriate for typical SIFT conditions, i.e., for helium carrier gas at 0.5 Torr and at 300K. This should not however disguise the fact that ui/vo generally shows a weak increase with increasing pressure and with decreasing temperature. So in order to minimize errors in deriving rate coefficients, it is desirable to measure ui (and uo) under the conditions of each SIFT experiment. In the earliest FA measurements, the relatively large error figure of f 30 was wisely placed on the derived rate coefficients for ion-molecule reactions, an error figure that was largely due to uncertainties in the flow dynamics. With the improved understanding of the flow dynamics (especially using measured values of ui in the SIFT), and if careful attention is given to the accurate determination of reactant gas flow rates and end corrections, then rate coefficients can be determined to better than 20 % at room temperature. At temperatures above and below room temperature, care has to be taken to minimise temperature gradients in the reaction zone of a VT-SIFT and proper account must be taken of this in error estimations. A further practical point is worthy of mention, especially when operating at low temperatures. The raw experimental data can be acquired much more rapidly and with greater reproducibility if the reactant gases are “flushed” into the flow tube by a stream of the carrier gas. This minimises the time taken to establish a constant flow rate of the reactant gas into the flow tube, particularly for those gases and vapours that are prone to absorb on the pipeline surfaces.

THE SELECTED ION FLOW TUBE (SIFT)

23

111. SIFT Studies of Ion-Neutral Reactions:

Some Illustrative Results The many rate coefficients and ion product distributions that have been determined in SIFT apparatuses refer in all cases to reactants that are translationally thermalised at the carrier gas temperature. The reactant ions can be created in a variety of ion sources (as previously indicated) under conditions variously conducive to the production of ground state ions, electronically excited ions, vibrationally excited ions, and vibronically excited ions as required. Using the filter gas technique (to be discussed), ions in particular excited states can be selectively removed from the ion swarm in the flow tube and the reactions of the remaining ions studied without confusion. Similarly, the monitor gas technique (to be discussed) can be used to distinguish between the different states of excitation of a given ionic species. Different isomers of ions can also be identified in a similar way according to their different reactivities. Forward and the corresponding reverse reactions for near-thermoneutral reactions can be studied over a range of temperatures, thus providing critical thermodynamic data, zero point energy differences, bond energies, etc. Cluster ions can be produced in ion sources containing polar (condensible) molecules and injected into SIFT apparatuses held at temperatures well below the condensation temperature of the gas from which the ions are formed; their reactions can then be studied with other less condensible gases. That the rate coefficients for both binary and ternary ionic reactions can be determined over a wide temperature range under wellcontrolled conditions and with good accuracy means that these data are valuable tests of theoretical predictions of rate coefficients. A very large amount of data is available that demonstrates the just mentioned features of SIFT apparatuses. It is not our objective here to present all that has been achieved but rather to present a few results that illustrate the capabilities of the SIFT technique. Detailed results can be obtained from the cited references to research papers and other review articles. One of the most appealing features of a SIFT is that ions in recognisable series can be selectively injected into the flow tube and their reactions studied with a range of gases and vapours. Thus, for example, detailed studies have been carried out of the reactions of positive ions in the series CH:(n = 0 to 4; Smith and Adams, 1977a, 1977b, 1978; Adams and Smith, 1977,1978), C,H,f ( n = 0 to 7; Adams and Smith, 1977; Mackay et al., 1980), H,CO+ ( n = 0 to 3; Adams er al., 1978), NH: ( n = 0 to 4; Adams et al., 1980a) and H,S+ ( n = 0 to 3; Smith et al., 1981b). Members of each of these series are separated in mass by one amu and some are relatively weakly bound. They are therefore useful tests of the SIFT injection system since unit mass resolution is required

D.Smith and N . G. Adums

24

1

0

, 5 10 H2flow rate

I

15 20 ( T o r r I s-')

1 25 x ~ O - ~

H 2 f l o w rate (Torr

lil)x 1 0 3

FIG.6. (a) A typical SIFT data set, in this case for the reaction of NH' with H, at 300 K. The rate coefficient for the reaction is derived from the slope of the linear semilogarithmic decay plot of the N H + count rate versus the H, flow rate. (b) This illustrates the standard approach to determining ion product distributions. The percentages of the count rates of each new ion species obtained from the data in (a) are plotted as a function of the reactant gas flow rate, and the true percentage ion product distribution is that at zero reactant gas flow. Clearly NH: is not a product of the primary reaction; it is produced in the secondary reaction of NHf with H, (data reproduced from A d a m et al., 1980a).

at a low injection energy (to prevent collisional dissociation of the injected ion on the carrier gas) and with a sufficient injected ion current. An example of the data obtained from a typical SIFT experiment is shown in Fig. 6. Figure 6(a) gives the data for the NH' + H, reaction that were obtained in a wider study of the reactions of the NH: series of ions (Adams et a/., 1980b). Note the linear decay on the semilog plot of the NH' ion count rate over three decades from which the rate coefficient, k , for the reactitpn is obtained according to Eq. (7). Note also the appearance of H:, NH:, and NH: ions, which are all energetically possible products of this reaction. Thus: NH'

+ H,-+H: + N +NH;

+H

? NH: + He

(94 (9b) (94

Channels (9a) and (9b) are normal exothermic binary channels whereas (9c) is a ternary association channel. However, Fig. 6(b), which is a plot of the percentage of each product ion at each reactant gas flow, indicates that

25

T H E SELECTED ION FLOW TUBE (SIFT)

channel (9c) does not occur since, at zero reactant gas flow, and NH; count rate is zero. Thus NH: is the product of the secondary reaction between NHl(89%), i.e., the percentage of each ion obtained by extrapolation of the approach (due to Adams and Smith, 1976b) is now standard for determining ion product distributions in SIFT experiments. As can be seen from Fig. 6(b), the true primary ion product distribution for reaction (9) is H l ( l 1 and NH:(89”/,), i.e., the percentage of each ion obtained by extrapolation of the ion product distribution lines to zero H, flow rate. An example of a reaction for which there are three primary products is the C t + NH, reaction, the data for which are reproduced in Fig. 7. Note that the NHd ion appearing in Fig. 7(a) cannot be a product of the primary reaction and this is confirmed by

x)

-

:loor L

a

10

H~CN+ n

-

n

n

8 60uO 2

5

0

NH

3

10

~

.-A-

15

F l o w r a t e ( T o r r I ;I’

20x10

LO-

o v

-30c

c

CN+

W

a

NH3 f l o w r a t e ( T o r r I s-’l

FIG.7. (a) SIFT data for the reaction of C + with N H , at 300 K indicating three possible ion products: NH;, HCN ’, and H,CN+. (b) The usual product distribution plot (analogous to Fig. 6(b)) accounting for all new ions observed including NH:, which can only be a product of u secondary reaction. (Indeed, it is seen to be so.) (c) The product distribution plot after eliminating the NH;. a procedure that provides ii more accurate ion product distribution (taken from Smith and Adams. 1977b).

26

D.Smith and N . G. Adams

Fig. 7(b). Therefore NH,f has been excluded from the ion product distribution in Fig. 7(c) and this improves the accuracy of the extrapolation of the ion product percentages to zero reactant gas flow rate and, hence, the accuracy of the derived ion product distribution. Much information has been obtained concerning the nature of ionmolecule reactions from studies of ions in recognisable series (Adams and Smith, 1983). Generally speaking, the recombination energies of the ions decrease with increasing degrees of hydrogenation and this is reflected in their decreased reactivity. (An excellent example of this phenomenon can be seen in the results of a study of the reactions of the NH; series; see Adams et al., 1980a.) Ordered series of negative ions are not so readily generated and no systematic studies have been made to date. However, an increasing number of negative ion reactions are being studied in SIFT experiments, notably by Paulson and colleagues (e.g., Viggiano and Paulson, 1984) and DePuy and Bierbaum (e.g., DePuy, 1984), as ion sources are developed to generate sufficient currents of negative ions. Adequate currents of the ions in the series such as those just referred to can be generated using low-pressure and high-pressure electron impact ion sources. An important question that must always be asked is "what is the state of excitation of the ions so generated?" Linear decay curves such as those shown in Figs. 6(a) and 7(a) are generally indicative of a pure ground state ionic species, unless the derived k is equal to the collisional k when the presence of excited ions cannot be ruled out. Under these circumstances, it is necessary to investigate whether any excited ions are present and this is usually done by introducing into the SIFT a gas that can only react (usually by charge transfer) with the excited species if present and not with the ground state species. This approach is commonly referred to as the monitor gas technique. (See Glosik et al. 1978.) Low-pressure sources are most likely to generate excited ions whereas high-pressure sources can be exploited to relax ions to their ground states by collisional quenching. (This has been nicely demonstrated in the case of H: ions; Adams and Smith, 1981a). However, generally speaking, it is obvious from curvature on the primary ion decay curves if excited ions are present in the flow tube. Indeed, conditions can be created in the ion source to produce excited ions and this then offers an opportunity to study their reactions. The vibrational states of many ions, notably diatomic positive ions, are very metastable as are certain electronic states of both atomic and molecular positive ions. Such metastable excited ions may survive multiple collisions with the carrier gas atoms and reach the reaction zone. A graphic example illustrating the presence of vibrationally excited N l ions in a SIFT is given in Fig. 8, which shows the decay curve for the reaction of the N: with Ar. The charge transfer reaction of N:(v = 0) with Ar is sufficiently endothermic that at 300K, the reaction is very slow.

27

THE SELECTED ION FLOW TUBE (SIFT)

I 0

I

5

I

I

10

15

20

Ar f l o w rate (Torr I s4) x103 FIG. 8. SIFT data for the reactions of N i ions (generated in a low-pressure ion source) with Ar at 300 K. The decline in the N; count rate at low Ar flows is due to the rapid reaction of the N$(r = I ) component of the N: swarm. The linear plot from which the rate coefficient for the N:(v = 1) + Ar reaction is derived is obtained by subtracting the relatively unreactive N:(v = 0) count rate (dashed line) from the total N i count rate. The rate coefficient for the N$(v = 0) + Ar reaction is derived from the much slower decline of the N: count rate at larger Ar flows (taken from Smith and Adams, 1981a).

However, the reaction is exothermic when the N: is vibrationally excited, i.e. for N:(v 2 1). Thus the curvature on the decay curve in Fig. 8 is attributed to the presence of both N:(v = 0) (the unreactive component) and N:(v 2 1). The vibrationally excited (reactive) component represents some 40 % of the N: ions, and detailed studies have shown that it is essentially all N:(v = 1) (Smith and Adams, 1981a; Lindinger et a/., 1981). A straightforward analysis of the decay curve provides values of k for the reactions of both N:(v = 0) and N:(v = 1) with Ar atoms at 300K. Thus: N:(v N:(v

+ Ar + Ar' + N, : k ,< = 1) + Ar A r + + N, : k = 4 x = 0)

-+

cm3 SKI, lo-'' cm3 s -

(IOa)

'.

(lob)

The k for reaction (lob) is about half of the collisional limiting value. Thus Ar is an effective monitor gas for N:(v = 1 ) and it can be used as such in the determination of the k for the quenching reaction: N:(v

=

1)

+ N,

-+ N:(v

= 0)

+ N,.

(1 1)

This is achieved by introducing a steady flow of Ar into the reaction zone of the SIFT as near as practically possible to the downstream sampling orifice

28

D. Smith and N . G . Adams

(see Fig. 5) to destroy all the N,'(v = 1). The Ar' ions so produced are then monitored as N, is added as a reactant gas to the SIFT in the usual way. Thus, the Ar' count rate decreases with increasing addition of N, due to the occurrence of reaction (1 1). In this way, k for this reaction has been measured to be - 6 x lo-'' cm's-' at 300K (Smith and Adams, 1981a). Using this monitor gas technique, detailed studies of the quenching rates of NO+(v) and O:(v) with several quenching gases, X, have been carried out (Bohringer et al., 1983a, 1983b; Federer et al., 1985a). These quenching reactions have been shown to proceed via vibrational predissociation of the (NO' X)* and (0;.X)* excited intermediate complexes, and the vibrational predissociation rate coefficients have been estimated. This has required measurements of the ternary associate rate coefficients for the production of NO+ . X and 0; . X complexes, most of which have been provided by SIFT studies. (This work has been thoroughly reviewed by Ferguson (1986).) Generally speaking, the reaction rates for two ionic components at a given mass will not be as different as those illustrated in Fig. 8 for reaction (10). Of course the decay curves will be nonlinear but, unlike that in Fig. 8, the curves will usually have more gradual changes of slope. An example of this is shown in Fig. 9 for the reactions of Kr+ ions with H, (Adams et al., 1980b). The curvature is explained by the presence of both of the spin-orbit states of Kr+-i.e., Kr+('PII2) and Kr+('P3,,). The linear part of the decay curve at large H, flows provides a value of the k for the Kr+(2Plj2)+ H, reaction. That for the Kr+(2P3j2) H, reaction is obtained from the data points at low H, flow by standard analytical procedures. An alternative approach is to use a "filter gas" to remove one of the two species from the ion swarm in the upstream region of the flow tube. For example, the addition of N,O upstream prevents the K I - + ( ~ P ~ from / , ) reaching the reaction zone leaving only the Kr+(2P,j2).The addition of H, downstream now results in a linear decay plot (see Fig. 9) from which the k for the Kr'(2Plj2) + H, reaction can be accurately determined. Kr+(*P,,,) can be produced as the only species in the flow tube by injecting CO' ions and reacting them with Kr atoms. Using these techniques, the reactions of the two spin-orbit states of both K r + and Xe+ have been studied with a number of gases. The interesting result is that the lowest energy state (the 3 state) for both Kr' and Xe' is generally more reactive than the higher energy state (Adams et al., 1980b). These monitor gas and filter gas techniques have been used to great effect to study the reactions of the ground states and metastable electronic states of several ions, including O:(X211,, a411u), NO+(X'C,, a'&-) and O'(4S, ,D,' P ) , with a number of gases (Glosik et al., 1978; Tichy et a!., 1979). Similarly, the reactions of the ground and metastable electronic states of the rare gas doubly charged ions-e.g., Xezf('P, 'D,,'So)-have been studied with a number of gases (Adams et al., 1979a; Smith et al., 1979, 1980a; +

+

T H E SELECTED ION FLOW TUBE (SIFT)

0

I

H

29

I

1 2 f l o w rate (Torr I s-’)

I

2 FIG.9. SIFT data for the reactions of K r + with H, at 300K. The curved plot represents the decay of a mixture of Kr+(’P,/J and K r f ( 2 P 3 , z ) .The linear plot of lesser slope refers to data obtained when N,O filter gas is added to the flow tube, which selectively removes Kr+(’P,,,) so the slope of the line provides a value for the rate coefficient for the Kr+(’P,,,) + H, reaction. The steeper linear plot represents the data obtained for the reaction with H, of Kr+(zP3/2), which has been selectively produced in the CO ’ + Kr reaction. Note that the slope of the curved plot at the larger H, flows is, as expected, the same as that for the line of lesser slope (taken from A d a m ef al., 1980b).

Adams and Smith, 1980). Both single- and double-charge transfer are observed in these reactions. From studies of the reactions of these doubly charged ions with the rare gas atoms, a correlation has been noted between the rate coefficient for single charge transfer and the internuclear separation at which the electron transfer occurs (Smith et al., 1980a). Monitor and filter gas techniques are also being used to distinguish between isomeric forms of certain ions. For example, of the two isomers HCO’ and C O H f , only the COH’ can proton transfer to CH, and N, and so either of these gases can be used as a monitor gas for COH +. This approach has been used to determine the isomeric content of the mass 29 amu ion swarm (Wagner-Redeker et a/., 1985; McEwan, 1986). Similarly, the linear and cyclic forms of the C,H: ion have been characterised by their differing reactivities with several gases (Smyth et al., 1982; Smith and Adams, 1987a). For some years there has been a growing interest in the nature and reactivity of ion clusters, e.g., H 3 0 + ( H , 0 ) , , not only because they represent

D.Smith and N . G. Adams

30

a bridge between gas and the liquid phases (Smith et al., 1980b), but also because they are the dominant ions in the lower terrestrial atmosphere (Narcisi and Bailey, 1965; Smith et al., 1981a). The SIFT provides the opportunity to study the reactions of individual cluster ions (i.e., with a given n ) without interference from higher- or lower-order clusters or interference due to the presence of parent neutral gas or vapour. Thus, the cluster ions can be formed in a high-pressure ion source or a flowing-afterglow ion source (Smith and Adams, 1980a; Paulson and Dale, 1982) and injected into the carrier gas at low energy to avoid fragmentation of the weakly bound clusters. The first cluster ion reactions to be studied in this way were those of H,O+(H,O),-, with D,O (Smith et ul., 1980b). These essentially thermoneutral reactions proceed with rate coefficients close to their collisional-limiting values and, more interestingly, total scrambling between the H and D atoms occurs among the products. Thus, for example, in the simplest reaction in the series: H,O+

+ D,O

+ H,O H2DOf + HDO

+HD20+ -+

(1 2b)

two product ions H D 2 0 + and H,DO+ result, the H,DO+/HD,O+ ratio being 2 in accordance with simple statistics. Similar results were obtained for the D 3 0 + + H,O reaction and for the reactions of the higher-order clusters ( n = 1 to 3). The implications of these most interesting results have been discussed in detail elsewhere (Smith et al., 1980b). Ligand switching can also be readily studied, e.g., H,O+(H,O),

+ CH,CN

-+

H+(H,O), .CH,CN

+ H,O

(1 3)

Reaction (13) is very fast at 300K, indicating that CH,CN is more strongly bound to H + ( H 2 0 ) , than is H,O (Smith et al., 1981a). Clearly such studies provide information on relative ligand bond strengths in cluster ions. It is via reactions such as (13) that the mixed clusters of H 2 0 and CH,CN are formed in the stratosphere (Arnold et al., 1978; Arijs et al., 1980). Some notable SIFT experiments have been performed using negative cluster ions of the kind OH-(H,O), that were produced in a high-pressure electron impact ion source (Pauslon and Dale, 1982; Henchman et al., 1983). Such difficult experiments on negative ions at thermal energies could not have been contemplated prior to the inception of the SIFT. The results of these and other SIFT experiments on negative ion clusters have been reviewed by Viggiano and Paulson (1984). A further, most valuable application of the VT-SIFT is in the study of the forward and reverse rate coefficients of reactions as functions of temperature.

THE SELECTED ION FLOW TUBE (SIFT)

31

From such studies not only kinetic but also thermochemical data relating to ionic reactions can be obtained. Detailed studies of isotope exchange in ion-molecule reactions have been carried out in which the forward (k,) and reverse (k,) rate coefficients have been measured as a function of temperature for reactions such as:

and CH:

+ H D % C H 2 D + + H, k,

(1 5 )

k, and k , can be measured quickly under identical conditions of carrier gas temperature and pressure; therefore the relative k can be very accurate. The k, and k, differ because of differences in the zero point energies of the species, the difference between k , and k , being most marked at low temperature. From the variation in k,/k, (i.e., the equilibrium constant for the reaction) as a function of temperature, the enthalpy and entropy changes in the reactions can be determined. A large number of such reactions have been studied. (See the reviews by Adams and Smith, 1983; Smith and Adams, 1984.) It is interesting to note that reaction (14) is considered to be responsible for the observed enrichment of 13C in interstellar CO (Smith and Adams, 1980b). Similarly, reaction (15) is an important step in the fractionation of D into some interstellar molecules (Smith et ul., 1982a). The VT-SIFT has recently been used to study the reactions of some positive ions with H and D atoms, including the reactions: HCO+ + D

2DCO+ + H k,

k, and k, have been determined for reaction (16) (and the corresponding reactions involving N 2 H +and N,D+ ions) at 300K and at 120K (Adams and Smith, 1985). The Birmingham double SIFT facility was especially valuable in these studies because the additional SIFT injector system could be used to determine the H-atom (or D-atom) concentration in the flow tube by injecting F- or C1- ions and observing their reaction with the H-atoms. (Fand C1- are unreactive with H2.) Reaction (16) is considered to be involved in the fractionation of D into DCO' in dense interstellar clouds (Dalgarno and Lepp, 1984). These kinds of isotope exchange studies are providing critical data on bond strengths (Smith et ul., 1982b) and zero point energies (Henchman et al., 1982) in ions. Some similar studies involving negative ions

32

D.Sniitli Lind N . G. Adams

have also been carried out (Grabowski et al., 1983; DePuy, 1984) including some elementary reactions such as: O H - -t H D P O D -

+ H,

(17)

The VT-SIFT will be greatly exploited for further studies of isotope exchange for many years. Measurements of k , and k, can also be used to accurately determine differences in the proton affinities (PA) of molecules. For example, the measurement of k, and k , for the reactions; 0,H'

+ H,

P H:

+ 0,

(18)

at 300 K and 80 K have permitted values of the enthalpy change (AH) and the entropy change (AS) in the reaction to be determined. Thus it has been shown that the PA of H, exceeds that of 0, by 0.014eV (since AH = PA(0,)-PA(H,)), and the measured value of AS verified that 0 2 H + has a triplet electronic ground state (Adams and Smith, 1984a). Similar measurements have accurately determined the difference in the P A of C O and HCl (Smith and Adams, 1985a). The SIFT technique is also well suited to the study of ternary ion-molecule association reactions, particularly those that are relatively slow. For these reactions, the exclusion of the ion source gas is crucial to prevent binary reactions of the ions with their parent gas from swamping the slower loss of the ion by the association reaction. However, the range of carrier gases that can be used, and, hence, the variety of stabilizing third bodies in the association reactions are limited to perhaps He, Ar, H,, 0,, N,, and CO, on the grounds of cost. (Note that the ions must not react with the carrier gas.) Nevertheless, much data have been obtained with a sufficient accuracy to profitably compare with recent theories describing ion-molecule association (as will be discussed). Of special note are the studies of CH: association reactions, such as: CH:

+ H, + He+CH:

+He

(19)

In practice, the effective binary rate coefficient, k;", is determined for reactions such as (19) at several carrier gas number densities [He]. The ternary association rate coefficient, k , , is then obtained from the slope of a plot of kzff against [He] at sufficiently small [He] such that the plot is linear and passes through the origin of coordinates. When this is not the case and the data do not refer to the low-pressure limit (i.e., when pressure saturation is evident as is sometimes the case at low temperatures in the SIFT, especially for very rapid association reactions), then a different analytical procedure has to be adopted (Neilson et al., 1978; Adams and Smith, 1987; also see Section V). Measurements have been made of k , for the reactions of CH: with H 2 ,

T H E SELFCTED ION FLOW T U B E (SIFT)

33

N,, O,, and C O over the temperature range 80-550 K (Adams and Smith, 1981b). The measurements indicate that k , varies with temperature according to a power law, i.e., k , = T-", where n N 2.5 in accordance with theoretical predictions for a polyatomic ion associating with diatomic molecules (Bates, 1979, 1980; Herbst, 1979, 1980). A more rapid variation of k , with temperature was observed for the CH,' CO, association reaction and this has been ascribed to the thermal excitation of the bending vibrational modes in the CO, as the temperature increases. This phenomenon is also apparent in the recent SIFT studies of other ternary association reactions that have been studied in J. F. Paulson's laboratory (Viggiano, 1986). Much stimulus has been brought to the study of ternary association reactions because they can provide estimates of the k for the analogous radiative association reactions that are considered to be so important in the synthesis of interstellar molecules. (The radiative analogue of reaction (19), i.e., CH: H, -+ CH: hv, has now been observed directly in the laboratory; Barlow et al., 1984.) Details of these studies of positive ion association reactions and their relevance to interstellar chemistry have been given in several review articles (e.g., see Smith and Adams,. 1985b; Herbst, 1985; Adams and Smith, 1986). Some interesting SIFT studies of negative ion ternary association reactions have also been carried out. Often these reactions are very rapid, a case in point being the reaction of F - with WF, in a helium carrier gas (Viggiano et al., 1985). Detailed studies of the association reactions of F - , C1-, and Brwith BCI, and BF, that were initially carried o u t in a flowing afterglow (Babcock and Streit, 1984) have recently been repeated as a function of temperature in a VT-SIFT (Babcock, 1986). The unusual form of the variation of the effective binary rate coefficient with carrier gas pressure has been interpreted as evidence for the parallel occurrence of radiative association and collisional association in these reactions.

+

+

+

IV. The Variable-Temperature Selected Ion Flow Drift Tube, VT-SIFDT Flowing afterglow and SIFT experiments provide kinetic data for ionneutral reactions in the thermal range of interaction energies, i.e., up to about 0.1 eV. Ion beam experiments provide cross sections for these reactions at energies in excess of a few tenths of an eV in most cases. (Exceptionally sophisticated beam experiments operate at somewhat lower energies; Ervin and Armentrout, 1985.) Clearly, it is desirable to bridge the energy gap

34

D.Smith and N . G. A d a m

between thermal energies and the smallest beam energies and this can be achieved using drift tubes (McFarland et al., 1973b; Lindinger and Smith, 1983). A most important advance toward this goal was the development of the flow-drift tube (FDT) in the NOAA laboratories in Boulder, Colorado (McFarland et al., 1973a) as a natural extension of their flowing afterglow technique. In the FDT, the downstream half of the flow tube (which has overall dimensions similar to a standard flowing afterglow) was made up of a series of narrow ring sections (7.5 mm thick) clamped together via O-rings to form a vacuum tight cylindrical tube, the sections of which were electrically insulated from each other. By applying suitable potentials to each of the rings, a uniform electrostatic field can be established along the axis of the flow tube. Thus, ions can be constrained to drift through the carrier gas by the action of this impressed electrostatic field, which increases their laboratory energy and, significantly, their interaction energy with the carrier gas atoms and with any reactant gas. Clearly therefore, by varying the magnitude of the electrostatic field, E (or, more correctly, the value of E / N , where N is the number density of the carrier gas atoms), reactions can be studied over a range of energies. In the original NOAA FDT, the ionization was created in the carrier gas in a field-free upstream region and both positive ions and electrons (and sometimes negative ions) were convected towards the drift field region by the carrier gas where charged particles of only one sign were allowed to continue their passage down the flow tube according to the direction of the electric field. (Details of operation are given in McFarland et al., 1973a).This apparatus has been used very successfully to study many ionneutral reactions and to determine the mobilities of many ion species in helium as a function of E / N (e.g., see McFarland et al., 1973a, 1973b, 1973c; Lindinger and Albritton, 1975). However, the creation of the ions in the flow tube has all the disadvantages referred to previously in relation to the flowing afterglow technique (Section 11). Therefore it was a natural development to incorporate a SIFT-type injector in the FDT, creating a selected ion flow drift tube (SIFDT), enormously increasing the versatility of the apparatus (Howorka et al., 1980). Similar SIFDT apparatuses have been built and are being exploited by the Universities of Aberystwyth, Colorado, and Innsbruck (Jones et al., 1981; Grabowski et al., 1983; Federer et al., 1985b). The results obtained from the various SIFDT experiments have been summarized in a recent review (Lindinger and Smith, 1983). Also included in this review is an account of the theory of operation of the SIFDT, which will also be summarized in this chapter shortly. The SIFDT apparatuses just referred to can only be operated at room temperature. However, there are obvious advantages in being able to vary the carrier gas temperature in SIFDT experiments. For example, the separate influences of temperature and ion energy on the course of ion-neutral

THE SELECTED ION FLOW TUBE (SIFT)

35

reactions could be studied and also the reactions of cluster ions that are only readily formed in low-temperature carrier gas could be studied as a function of ion energy to mention only two of many advantages. Hence the Birmingham group has successfully developed a variable-temperature selected ion flow drift tube (VT-SIFDT), a schematic of which is given in Fig. 10. A drift tube section, consisting of some SO metal rings of diameter 60 mm and width 9 m m held 1 mm apart by ceramic insulators, is inserted into a standard VT-SIFT. This reduces somewhat the diameter of the downstream region of the flow tube and so a field-free extension to the drift section is included as shown in Fig. 10 to ensure that the carrier gas flow is settled to laminar flow prior to the drift field reaction region. A separate wire connects each ring to a hermetically sealed multiply connector (which is itself sealed into the wall of the flow tube cownstream) via which the potentials are applied to the rings. In this VT-SIFDT, the upstream end of the drift tube is at ground (flow tube) potential and the downstream end and the complete mass spectrometer-detection system is at a high potential. Since the channeltron multiplier is electrically decoupled from this potential and the pulse amplifier is capacitively coupled to the channeltron, then this arrangement of drift tube potentials presents no practical problems. However, because of the close proximity of the ring electrodes to the grounded flow tube, the maximum potential that can be used ( 2350 volts) is less than can be used in the room temperature SIFDT apparatuses. In practice this is not a severe limitation and in no sense does it outweigh the advantage of the variabletemperature feature of the VT-SIFDT. The two reactant gas inlet ports (see Fig. 10) are inserted into the drift field via slotted holes in the metal rings to which they are electrically connected to minimise field distortion. The VTSIFDT is routinely operated over the temperature range from 80 to 600 K and is providing unique data (Smith et ul., 1984a, 1984b; Adams et ul., 1985; Viggiano et ul., 1985; Adams and Smith, 1987; Smith and Adams, 1987a, 1987b). To determine the k for a reaction, an appropriate potential difference is established across the drift region, which may be all of the ring sections (a relatively weak E ) or any fewer number of rings (if a larger E is required), and the experiment then proceeds in the normal manner for SIFT experiments. However, the presence of the E field has several consequences, the most obvious in practice being that considerably more reactant gas is required to reduce the primary ion count rate sufficiently for an accurate k to be obtained. This is largely because the ion residence time, ti, in the reaction region (the reaction time) is considerably reduced because of the drift velocity, ud, imparted by the field. Clearly, to calculate a rate coefficient, ti must be known. Fortunately, since for all but the smallest E / N , the u , ( = p E , where p is the ionic mobility) is the major component of the total flow velocity of the ions u i ( = v d + component due to carrier gas flow), then ion

D.Srnith and N . G. Adurns

36

I

F i e l d free--Electrostatic d r i f t tuberegion I-e x t e n s ion

(b)

I

1I R e a citnalnett g a s I I eramic insulator

R i n g electrodes o m i t t e d h e r e for c l a r i t y

r i n g electrodes

port

FIG. 10. (a) A schematic of the flow-drift tube section of the VT-SIFDT showing the arrangement of the stainless steel rings that are used to establish the uniform electric field in the dowstream reaction zone. The continuous 20-cm upstream section of steel tubing (the field-free extension of the drift tube) ensures that the carrier gas Row becomes laminar prior to its entry into the reaction zone. (b) An cnlargement of the part of the drift tube region near to the upstream ring port. This shows how the metal rings are supported by the machinable ceramic bars that locate the rings and insulate them from each other and from the Row tube walls. The insulated ring ports fit through slots in the metal rings and are maintained at the same potential as the slotted rings by connecting wires.

THE SELECTED I O N FLOW TUBE (SIFT)

37

“plug flow will prevail. Thus the instantaneous axial ion velocity will be radially independent and the simple analysis of the data represented by Eq. (3) will be valid. The mobilities of some ions in helium have been measured over appreciable ranges of E / N at 300K and use can be made of the published tables (McFarland et a/., 1973a; Lindinger and Albritton, 1975; Ellis et ul., 1976, 1978) to obtain p and hence to determine ud and then 0,. However, such data are only available for some ions and so it is often necessary to measure u, directly. In the FDT and the SIFDT that developed from it, a special upstream “shutter” grid was included that could be opened and closed with voltages pulses to gate the ions on and off. This facility together with a grid just in front of the detection orifice could be used to measure drift velocities (McFarland et al., 1973a) and this has allowed ionic mobilities to be measured very accurately. In the VT-SIFDT, no grids are included and drift times are measured by simply modulating the ion swarm by applying a short-duration, low-voltage pulse to one of the rings of the drift tube and determining the arrival time of the disturbance at the detection system using multiscalar techniques. The finite transit time of ions in the detection system is determined by applying the pulse separately to two or more rings and treating the time in the detection system as a “time correction” in the usual way. In this manner u, (and hence p ) can be determined so rapidly and accurately that it is hardly worthwhile to rely on published mobility data even when they are available. It is especially advisable to measure 11, at small E / N when ud = uo (the carrier gas flow velocity) for which the radial velocity profile of the ions departs from plug flow and for which it is difficult to estimate the magnitude of the contribution to u, due to the carrier gas flow. It is also important to directly measure tii when relatively large fractions of reactant gas are present in the carrier gas since this will modify the mobility of the ions from that appropriate to the pure carrier gas. In fact, in some circumstances it may be necessary to measure u, for each reactant gas flow rate. Following the above procedure, it a straightforward matter with the VT-SIFDT to determine k values for ion-neutral reactions as a function of both temperature and E / N . However, it is obviously desirable to relate the k to the ion-reactant gas centre-of-mass energy, E,,, which can be obtained by the following procedure. Wannier (1953) has shown that the mean kinetic energy, E , , of an ion drifting through a gas at a velocity lid is given by ”

E,

=

:M,u:

+ $ M c v ; + jk, T

(20)

where Mi and M , are the masses of the ion and the carrier gas atoms (or molecules), respectively, k, is the Boltzmann constant, and Tis the carrier gas temperature. Equation (20) is only valid if E , 9 (3/2)k, T. The validity of this expression for E , for atomic ions has been demonstrated by the experimental

38

D.Smith und N . G. Aciams

work of McFarland et ul. (1973b) and the theoretical work of Skullerud (1973) and Viehland et al. (1974). (A detailed discussion of this is given by Albritton et al., 1977.) Now Ei, is given by

where MI is the mass of the reactant gas and 62 and 17: are the mean square velocities of the ion and reactant gas particles, respectively obtained from (1/2)Mi$ = E , and (1/2)M1$ = (3/2)kbT. Combining these with Eq. (20) and Eq. (21), we obtain

So from a measurement of u d , a value for Ei, can be obtained. Hence the derived k for an ion-neutral reaction can be related to E,,. The internal energy of the reactant neutral molecules is, of course, independent of E / N and only depends on the temperature. However, what is the internal energy of reactant molecular ions prior to their reaction with the reactant gas? The ions undergo many collisions with the carrier gas particles in the electrostatic field region prior to reaction and, hence, their rotational and vibrational state populations will depart from the zero field thermal distributions. If the ions undergo sufficient collisions with the carrier gas particles, then their internal energies will equilibrate at the ion-carrier gas centre-of-mass energy, Ei,, which can be calculated from an expression similar to Eq. (22) except with MI replaced by M,. This has been discussed theoretically by Viehland et al. (1981) and by Viehland (1986). It is expected that the rotational state population distribution in the molecular ions will rapidly equilibrate with Ei, and, hence, an ion rotational temperature, T,,,, can be ascribed according to = $kbqo, (for nonlinear polyatomic ions) and some experimental evidence is available that supports this conclusion (Adams et al., 1985; Adams and Smith, 1987). Concerning vibrational excitation, it has been argued that this is inefficient in drift tube experiments using helium carrier gas at the E / N typical of those attainable in such experiments, although a good deal of evidence is available that indicates that in argon, carrier-gas vibrational excitation is facile (Ferguson, 1984). However, Federer et al. (1985b) using a SIFDT have shown that Nf and 0: ions can be vibrationally excited in helium carrier gas even at modest E / N . This process is relatively slow and equilibrium among the v = 0 and v = 1 states of the N: and 0; ions is barely achieved in the available length of the drift field except at the highest E / N . Also, when vibrationally excited N: and 0: ions were injected into the SIFDT, collisional quenching of the ions occurred such that the v = 1

T H E SELECTED I O N F1.OW TUBE (SIFT)

39

population reduced and the v = 0 population increased, thus approaching an equilibrium population (as is required by detailed balance). In view of this work, vibrational excitation of ions-especially polyatomic ions-must be considered in SIFDT experiments even when helium carrier gas is used. In the next section, we refer to some VT-SIFDT data for the association reactions of CH: ions demonstrating that both rotational and vibrational excitation of this ion species occur quite efficiently in helium.

V. VT-SIFDT Studies: Some Illustrative Results Many of the room-temperature FDT and SIFDT studies carried out prior to 1983 have been reviewed by Lindinger and Smith (1983). In this section, we shall first mention some interesting results obtained recently using roomtemperature SIFDT apparatuses to illustrate the kind of work that can be done using the technique. However, our major purpose here is to illustrate via a few examples the features and enormous potential of the new VT-SIFDT technique with which ion-neutral reactions can be studied in one apparatus both as a function of EIN and of temperature. This has opened up a new area of reaction kinetics in which the separate influences of translational energy and internal energy on ion-neutral reactions can be investigated. The value of drift tube techniques in the study of endothermic ion-neutral reactions has been appreciated for many years. Traditional drift tubes suffer from the simultaneous presence of ion source gas and reactant gas in the drift reaction region with the consequential difficulties of interpreting data. (Determining ionic reaction products is particularly difficult.) The SIFDT avoids these problems and much beautiful data have been obtained on the influence of translational energy on both slow exothermic reactions and on reactions in which endothermic reaction channels are seen to open up as the translational energy is increased. (See Lindinger and Smith, 1983, for several examples). Studies of molecular ion reactions in both helium and argon carrier gases have revealed the influence of vibrational excitation of the ion on the course of ionic reactions. Such complications do not arise of course for atomic ions and so experiments with atomic ions are generally simpler and can thus provide quite accurate and readily interpretable data. A case in point is the recent SIFDT study of the reaction: N++H,-+NH++H.

(23)

By measuring the k for this reaction as a function of the N f - H 2 centre-ofmass energy, Ei,,at a carrier gas temperature of 300 K, it has been shown that the reaction is only 11 meV endothermic. By combining this endothermicity

40

D. Smith and N . G . Adams

with the accurately known recombination energy of N t and the bond energy of H,, the precise recombination energy of NH' has been derived and, hence, the absolute proton affinity of N atoms has been deduced as 3.531 eV. This reaction is involved in the synthesis of NH, in cold interstellar clouds (Adams et al., 1984). A reaction considered to be important in the shocked regions of interstellar clouds is Cf+H,+CHf+H.

(24)

This reaction is endothermic by 0.4eV and so cannot occur in cold gas. However, at the elevated interaction energies that can occur in interstellar MHD shocks where the C + ions are accelerated to high velocities (through cold gas), then reaction (24) can proceed. The SIFDT offers a teasonable simulation of these conditions and SIFDT studies of reaction (24) have shown that the k increases approximately exponentially as the C f - H , interaction energy (Eir) increases towards the threshold energy for the reaction (0.4 eV). The interstellar implications of this result have been discussed by Adams et al. (1984). Similar studies have been carried out of the reactions of SH; ions ( n = 0 to 3) with H, and H atoms, reactions that also are considered to be important in shocked interstellar gas (Millar et al., 1986). A room-temperature SIFDT has been used to study negative ion-molecule reactions (DePuy, 1984) to see how the chemistry is modified by translational energy increase. These studies have shown that endothermic negative ion reactions can be driven by increasing E,, in the SIFDT. For example, the H-D exchange reaction

DO-

+ CH,CH,

+ HO-

+ CH,CHD

(25)

is endothermic by -0.5 eV but it can readily be initiated in the SIFDT at modest values of E/N. Also the reaction of D,N- ions with CH,CH, results only in H-D exchange at zero field in the SIFDT but at increased translational energies the reaction: D,N-

+ CH,CH,

+ CH,CH-

+ NHD,

(26)

is dominant, producing the vinyl anion CH,CH-. Thus an opportunity is provided to study the reactions of the CH,CH- ion, which is otherwise difficult to create (DePuy, 1984). The extra dimension of temperature variability offered by the VT-SIFDT extends the range of the technique greatly. Although it is to be expected that increasing either the carrier gas temperature or E/N will influence ion-neutral reactions in a similar way, it must be appreciated that there is no direct equivalent between the temperature change, AT, and the energy change, AEir, which produce a given change in the k of a reaction, i.e., AE,, cannot be directly equated to (3/2)k,AT. Clearly this is because the internal energy of a

THE SELECTED ION FLOW TUBE (SIFT)

41

molecular reactant gas is dependent only on the carrier gas temperature and not on Eir.Also, as we have mentioned previously, the rotational states of the ions will usually be in equilibrium with Ei, although in general the vibrational states will not. A proper appreciation of these points is essential if VT-SIFDT data are to be correctly interpreted. These points are well illustrated by the VT-SIFDT study of the reaction: 0;

+ CH,

+ CH,Oi

+ H.

(27)

This reaction has received considerable attention over several years. I t has been studied in SIFTS (Smith et nl., 1978) and flow-drift tubes (Dotan et ul., 1978). Recently the k has been measured over the extraordinarily wide temperature range of 20 to 560 K in a collaborative CRESU-VT-SIFT study (Rowe et al., 1984). This reaction is remarkable in that k decreases from the large value of 4.7 x lo-'' crn3sK1 at 20K towards its minimum value of 5 x 10- l 2 cm3s-l near 300 K and then increases again as the temperature is further increased. The dependence of k on T below 300 K (i.e., k = T - 1 . 8 ) is indicative of a k controlled by the lifetime of the intermediate complex (O,CH,)' * in which rearrangement occurs prior to dissociation to the final CH3O; + H products. Using the VT-SIFDT, the k for reaction (27) has been studied as a function of Eirat fixed carrier gas temperatures of 80, 200, and 300 K (Adams et al., 1985). The change in k with increasing Eirmirrored the change with increasing 17: For example, at a carrier gas temperature of 200 K, an increase in Eircaused k to first decrease to the minimum value and then to increase with further increase in Eir. However, the interesting result is that, regardless of the carrier gas temperature, the A T and the AEir required to produce the same change in k are related by dEir/kbATN 8. This is because the internal states of the reactants (especially the CH,) are not equilibrated to Eir prior to the reaction and so a large fraction of the Ei, must be utilized to excite the rotational and vibrational energy states within the long-lived (O,CH,)+* complex. This complex has 15 vibrational degrees of freedom and, if all vibrational degrees of freedom were energy equivalent, one might expect that, rather than 8, AEir/khATwould be 10. (That is, 1 5kbT/((3/2)khT), where the (3/2)kbT is compounded from the two rotational degrees of freedom in the 0: and one translational degree of freedom since, classically, vibrational degrees of freedom store khTwhile translational and rotational degrees of freedom store (1/2)kbT). It has been suggested that this discrepancy (between 10 and 8) is because the high-frequency C-H and 0 - H stretching modes in the complex do not contribute effectively to energy storage. (For further discussion, see Adams et al., 1985). A similar but more superficial VT-SIFDT study has been made of the reaction of Br- with WF, in which it was found that AEir/kbAT2 34. (Viggiano et al., 1985). This is in good agreement with expectations because, in this case, one might expect that

D. Smith and N . G.Adums

42

AEir/kbATshould be 36-that is, 18kbT/((1/2)k,T) where (1/2)kbTis due to one translational degree of freedom and the (WF,Br-)* complex possesses 18 vibrational degrees of freedom. Another most interesting application of the VT-SIFDT is in the study of ternary association reactions. As mentioned in Section 111, VT-SIFT studies have been carried out of the association reactions of CH: with several diatomic molecular gases in helium carrier gas with the finding that, in agreement with theoretical predictions, the ternary rate coefficients, k,, vary with temperature as k, N T-2.5. What is to be expected if the k, are measured as a function of Ei, in the VT-SIFDT? The result of such a study for the reaction: CH: + N , + H e + C H : . N , + H e

(28)

at a carrier gas temperature of 80 K is shown in Fig. 11. As can be seen, the plot of log k, versus log Eiris not linear but tends towards linearity at high E,. However, the very interesting point is that the plot of log k, versus log E,, is very linear with a slope of - 1.6, that is k 3 N E L This can be compared with the temperature variation of k, for reaction (28), i.e. k, N T-2.7(Adams

'.,.

[

Helium carrier g a s temperature =BOK

rn

t-

10

I

I

20

I I 1 1 \1 1 l 40 60 80 100 Eic ,Eir (meV) I

FIG.11. VT-SIFDT data for the association reaction of CH; with N, in helium carrier gas at a temperature of 80 K. Plotted in log-log form is the ternary association rate coefficient, k , versus the centre-of-mass energy of the CH; both on the helium carrier gas ( E i c )and on the N, reactant gas ( E J . The linearity of the k , plot versus E i , and the slope of the line indicating that k , = E , 1.6 are in accordance with theoretical expectations (Adams and Smith, 1987). The curvature of the plot of k , versus Ei, demonstrates, in accordance with theory, that such a plot is not physically justified. (For discussion, see text.)

THE SELECTED ION FLOW TUBE (SIFT)

43

and Smith, 1981b). The power law relationship involving E,, rather than E , , can be explained if the following assumptions are made: (i) that the temperature, KO,,of the reactant CH: ions is equilibrated with Eic-i.e., (3/2)k,K0, = E,,-and (ii) that the temperature of the excited intermediate complex (CH: .N,)* is controlled by E,,. Assumption (i) is reasonable because the CH: undergoes many collisions with helium atoms (prior to a collision with an N, molecule), which allows equilibrium to be attained among the rotational states. Assumption (ii) is justified because the reactant gas molecule (N,) is much more massive than the carrier gas atom (He) and hence the energy, E,,, of the C H l - N , collision is much greater than E,,. Thus from a consideration of the partition functions of the reactants and of the intermediate complex (similar to the approach taken for the truly thermal reactants by Bates, 1979, 1980 and Herbst, 1979, 1980), it has been shown that the appropriate variable for reaction (28) is E,, and that k , should vary as E L which is in good agreement with the experiment (Adams and Smith, 1987). When the mass of the reactant gas is less than that of the carrier gas, as for example in the reaction of CH; with H, in He carrier gas, then it can be shown that k , should vary as E , 1 . 5 and this is also a good agreement with the results of the VT-STFDT experiments. The studies of relatively slow association reactions-such as reaction (28)-as functions of E,, and E,, are facilitated by operating with the carrier gas in the VT-SIFDT at low temperatures at which the zero field k , is relatively large. For much faster reactions such as the CH: + C O association reaction, accurate experiments can be carried out at higher carrier gas temperatures and at sufficiently high E,, that CH: ions are vibrationally excited in collisions with the helium carrier gas. This is manifest as a more rapid decrease of k , with Ei, than is predicted by the simple theory previously referred to, which ignores effects due to vibrational excitation in the reactant ion (Adams and Smith, 1987). Of course, theory could be developed to account for vibrational excitation and the VT-SIFDT could be exploited to provide the corresponding experimental data. It is well known that vibrational excitation of ions occurs more readily when argon is the carrier gas in drift tubes, which was also demonstrated in the recent SIFDT studies of Adams and Smith (1987). At a helium carrier gas temperature of 80 K and at zero field the CH: + C O reaction is somewhat “pressure-saturated.” (See Adams et al., 1979b.) Then it is usual to plot the reciprocal of k;ff against the reciprocal number density of helium to obtain an estimate of k , (rather than use the k;ff versus helium number density approach as is adopted in the low-pressure regime-see Section 111). An alternative approach to determining the true k , is to measure the apparent k , as a function of E,, at a fixed carrier gas pressure. The log(apparent k 3 ) versus log E,, plot is curved at low E , , under

’,’,

44

D.Smith and N . G. Adams

pressure-saturated conditions but becomes linear at higher Ei,. Extrapolation of the linear part of the plot to the zero-field of E , , then provides the true value of k , . (See Adams and Smith, 1987.) This approach has been used to determine a k , for the very rapid CH; + HCN association reaction (Smith and Adams, 1987b), which is in good agreement with the value determined at low pressure in an ICR experiment (Kemper et al., 1985). A VT-SIFDT study has also been reported of the N: N, + NZ collisional association reaction (Smith et d., 1984a) in which the collisional excitation and dissociation of the NZ ion in the field was also investigated. The use of VT-SIFDT apparatuses to study collisional breakup of ions under controlled conditions can provide a wealth of data on bond energies of cluster ions, on ion structures, etc. This approach has been used to obtain information on the structure of the CH,Ol product ion of reaction (27) (Rowe et a/., 1984; Barlow et a/., 1986). The VT-SIFDT technique is only just beginning to be exploited; clearly it will provide an enormous amount of data on ionic reactions a t suprathermal energies and it will be invaluable for studying the importance of energy partition in ion-neutral reactions.

+

VI. Concluding Remarks The value of the VT-SIFDT technique in the study of ion-molecule reactions at thermal energies is clear from the foregoing sections. Only a few illustrative results have been presented from the many SIFT experiments that have been carried out in recent years. Further details and the results of many more SIFT studies can be found in the references to research and review papers cited in this chapter. As of 1987 some 20 or so SIFT apparatuses are being exploited in various laboratories around the world, so new research papers are continuously being published. Yet the basic technique has hardly been developed since the original SIFT of 1976. The growing desire to investigate ion-molecule reactions at lower temperatures and higher pressures will most certainly mean that further developments of the technique will occur. In particular, new injectors will be developed to operate at higher flow tube pressures. A very recent development in the laboratory of C. H. DePuy and V. M. Bierbaum in Boulder exploits a 10-inch diffusion pump to evacuate the SIFT injection chamber together with a 6-inch diffusion pump to separately pump the injection quadrupole mass filter (Van Doren et ul., 1987). This has resulted in greatly enhanced injected ion currents even from a flowing-afterglow ion source. Such is the efficiency of this ion source-injector combination that more-than-adequate currents of ions containing rare

THE SELECTED ION FLOW TLJBE (SIFT)

45

isotopes (e.g., l80) can be injected into the flow tube from the flowingafterglow ion source using gases that have not been isotopically enriched! Smaller SIFT apparatuses will be constructed that can be cooled to temperatures below 80 K without prohibitive expense, perhaps down to 20 K by cooling with liquid neon, although such low temperatures place more restrictions on the range of reactant gases that can be used. However, data from such experiments would be invaluable in obtaining a greater understanding of the ion chemistry at low temperatures (e.g., the chemistry of interstellar gas clouds). SIFT apparatuses are also being used as sources of kinetically and internally relaxed ions for spectroscopic studies (Moseley, 1985). The greater ion densities that will be produced in flow tubes as the SIFT injectors are improved will open up the possibility of studying the light emitted from thermal energy ion -molecule interactions, a very desirable advance indeed ! The VT-SIFDT is a most exciting development. Some VT-SIFT apparatuses in other laboratories are now being converted to VT-SIFDT apparatuses according to the Birmingham design (Section IV). The attraction of studying ion-molecule reactions as a function of both temperature and collision energy in one apparatus is clear. A few examples of the first results obtained using the VT-SIFDT have been mentioned in Section V. Exploitation of these apparatuses is just beginning. The relative importance of temperature and collision energy on a few binary reactions and ternary association reactions of both positive and negative ions has been studied already and many more reactions will be studied. Also vibrational excitation of molecular ions in collision with carrier gas atoms, collisional dissociation of cluster ions, and electron detachment from molecular negative ions-to mention just a few phenomena -can be studied under controlled conditions. The combination of low carrier gas temperature and the drift field facility is particularly valuable since large, weakly bound cluster ions can be formed in the low-temperature gas and then further association-ligand switching-collisional dissociation can be studied as the collision energy of the ions is varied. The potential of both the VT-SIFT and VT-SIFDT is enormous, not only for fundamental studies of ionic and molecular physics and chemistry, but also in the provision of critical kinetic and thermodynamic data for modelling media as diverse as planetary atmospheres, interstellar gas clouds, laser plasmas, and surface etchant plasmas. These are among the reasons for the growth in the number of SIFT apparatuses and why they will continue to be developed and exploited for years to come. They are enjoyable to operate and very reliable, while the flow of useful data is large and sufficiently accurate to allow comparison with theoretical predictions. What more could experimental physicists, chemists, and theoreticians desire'?

46

D.Smith and N . G. Adams REFERENCES

Adams, N. G., and Smith, D. (1976a). h i . J. Mass Spectrom. Ion Phys. 21, 349. Adams, N. G., and Smith, D. (1976b). J. Phys. B. 9, 1439. Adams, N. G., and Smith, D. (1977). Chem. Phys. Leiis. 47, 383. Adams, N. G., and Smith, D. (1978). Chem. Phys. Lefts. 54, 530. Adams, N. G., and Smith, D. (1980). Ini. J. Mass Specfrom, Ion Phys. 35, 335. Adams, N. G., and Smith, D. (1981a). Ap. J. 248, 373. Adams, N. G., and Smith, D. (1981b). Chem. Phys. Letis. 79, 563. Adams, N. G., and Smith, D. (1983). In “Reactions of Small Transient Species” (A. Fontijn and M. A. A. Clyne, eds.), pp. 311-385. Academic Press, New York. Adams, N. G., and Smith, D. (1984a). Chem. Phys. Leils. 105, 604. Adams, N. G., and Smith, D. (1984b). In “Swarms of Ions and Electrons in Gases” (W. Lindinger. T. D. Mark and F. Howorka, eds.), pp. 194-217, Springer-Verlag, Vienna. Adams, N. G., and Smith, D. (1985). Ap. J. (Lerters). 294, L63. Adams, N. G., and Smith, D. (1986). In “Astrochemistry” (M. S . Vardya and S. P. Tarafdar, eds.), pp. 1-18. Reidel, Dordrecht. Adams, N. G., and Smith, D. (1987). Int. J . Mass. Spectrom. Ion Process. 81, 273. Adams, N. G., Church, M. J., and Smith, D. (1975). J. Phys. D . 8, 1409. Adams, N. G., Smith, D., and Grief, D. (1978). Int. J. Mass Spectrom. Ion Phys. 26,405. Adams, N. G., Smith, D., and Grief, D. (1979a). J. Phys. B. 12, 791. Adams, N. G., Smith, D., Lister, D. G., Rakshit, A. B., Tichy, M., and Twiddy, N. D. (1979b). Chem. Phys. Letts. 63, 166. Adams, N. G., Smith, D., and Paulson, J. F. (1980a). J. Chem. Phys. 72, 288. Adams, N. G., Smith, D., and Alge, E. (1980b). J. Phys. B. 13, 3235. Adams, N. G., Smith, D., and Miller, T. J. (1984). Mon. Not. R. Asir. Soc. 211, 857. Adams, N. G., Smith, D., and Ferguson, E. E. (1985). Ini. J. Mass Spectrom. Ion. Procmc. 67,67. Albritton, D. L. (1978). Atom. Data Nucl. Data Tables 22, I . Albritton, D. L., Dotan, I., Lindinger, W., McFarland, M., Tellinghuisen, J., and Fehsenfeld, F. C. (1977). J. Chem. Phys. 66, 410. Alge, E., Adams, N. G., and Smith, D. (1983). J. Phys. B. 16, 1433. Arijs, E., Nevejans, D., and Ingcls, J. (1980). Nature 288, 684. Arnold, F., Bohringer, H., and Henchen, H. (1978). Geophys. Res. Letts. 5, 653. Aue, D. H., and Bowers, M. T. (1979). In “Gas Phase Ion Chemistry” (M. T. Bowers, ed.), Vol. 2, pp. 1-51. Academic Press, New York. Babcock, L. M. (1986). Private communication. Babcock, L. M., and Streit, G. E. (1984). J. Phys. Chem. 88, 5025. Bates, D. R. (1979). J. Phys. B. 12, 4135. Bates, D. R. (1980). J. Chem. Phys. 73, 1OOO. Barlow, S. E., Dunn, G. H., and Schauer, M. (1984). Phys. Rev. Letis. 52, 902. Barlow, S. E., Van Doren, J. M., DePuy, C. H., Bierbaum, V. M., Dotan, I., Ferguson, E. E., Adams, N. G., Smith, D., Rowe, B. R., Marguette, J. B., Dupeyrat, G., and Durup-Ferguson, M. (1986). J. Chem. Phja. 85, 3851. Bohme, D. K. (1975). In “Interactions between Ions and Molecules” (P. Ausloos, ed.), pp. 489-504. Plenum Press, New York. Bohringer, H., Durup-Ferguson, M., Ferguson, E. E., and Fahey, D. W. (1983a). Pfanei. Space Sci. 31, 483. Bohringer, H., Durup-Ferguson, M., Fahey, D. W., Fehsenfeld, F. C., and Ferguson, E. E. (1983b). J . Chem. Ph,vs. 79, 4201. Bolden, R. C., Hemsworth, R. S., Shaw, M. J., and Twiddy, N. D. (1970). J . Phys. B. 3, 45.

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Brown. G. P.. and DiNardo. A. ( 1946). J. :Ipp/. P/iI..s. 17. X02. Dalgarno. A., and Black, J. H. (1976). Rep/. Prog. Phy.t 39, 573. Dalgdrno, A., and Lepp, S. (1984). Ap. J. (Lerters) 287, L47. DePuy, C. H. (1984). I n “Ionic Processes in the Gas Phase” (M. A. Almoster Ferreira, ed.), pp. 227-241. Reidel, Dordrecht. DePuy, C . H., and Bierbaum, V. M. (1981). Acc. Chem. Res. 14, 146. Dotan, I., Fehsenfeld, F. C., and Albritton, D. L. (1978). J . Chem. Phys. 68, 5665. Duperyrat, G., Rowe, B. R., Fahey, D. W., and Albritton, D. L. (1982). I n / . J. Mass Specfrom. I o n Phys. 44,I. Ellis, H. W., Pai, R. Y., McDaniel, E. W., Mason, E. A,, and Viehland, L. A. (1976). Atom. D u i a Nucl. Dutu Tables. 17, 177. Ellis, H. W., McDaniel, E. W., Albritton, D. L., Viehland. L. A,, Lin, S. L., and Mason, E. A. (1978). Atom. D a t a Nucl. D a t a Tables. 22, 179. Ervin. K. M., and Armentrout, P. B. (1985). J . Chem. Phvs. 83, 166. Federer, W., Dobler, W., Howorka, F., Lindinger, W., Durup-Ferguson, M., and Ferguson, E. E. (1985a). J. Chem. Phys. 83, 1032. Federer. W., Ramler, H., Villinger, H., and Lindinger, W. (1985b). Phys. Reti. Lerl.7. 54, 540. Fehsenfeld. F. C. (1975). I n “Interactions between Ions and Molecules” (P. Ausloos, ed.), pp. 387-412. Plenum Press, New York. Ferguson, E. E. (1984). I n “Swarms of Ions and Electrons in Gases” (W. Lindinger, T. D. Mark, and F. Howorka, eds.), pp. 126-145. Springer-Verlag, Vienna. Ferguson. E. E. (1986). J . Ph.w Chem. 90, 731. Ferguson. E. E., Fehsenfeld, F. C., Dunkin, D. B., Schmeltekopf, A. L., and Schif, H. 1. (1964). Planer. Space Sci. 12, 1169. Ferguson. E. E., Fehsenfeld, F. C., and Schmeltekopf, A. L. (1969). A h . At. Mol. Phys. 5, I. Ferguson, E. E., Fehsenfeld, F. C., and Albritton, D. L. (1979). I n “Gas Phase Ion Chemistry” (M. T. Bowers. ed), Vol. I , pp. 45-82. Academic Press, New York. Glosik, J., Rakshit, A. B., Lister, D. G., Twiddy, N. D., Adams, N. G., and Smith, D. (1978). J . Phys. B. 11. 3365. Grabowski, J. J. (1986). Private communication. Grabowski, J. J., DePuy, C. H., and Bierbaum, V. M. (1983). J . Amer. Chem. SOC.105, 2565. Henchman, M. J., Smith, D., and Adams, N. G . (1982). I n t . J . Mass Spectrom. I o n Phys. 42, 25. Henchman, M. J., Paulson, J. F., and Hierl, P. M. (1983). J . Amer. Chem. SOC.105, 5509. Herbst, E. (1979). J . Chem. Phys. 70, 2201. Herbst, E. (1980). J . Chem. Phys. 72, 5284. Herbst. E. (l985), I n “Molecular Astrophysics: State of the Art and Future Directions” (G. H. F. Diercksen. W. F. Huebner, and P. W. Langhoff, eds.), pp. 237-254. Reidel, Dordrecht. Herbst, E., and Klemperer, W. (1973). Ap. J . 185, 505. Howorka, F., Fehsenfeld, F. C., and Albritton, D. L. (1979). J . Phys. B. 12,4189. Howorka, F., Dotan. I., Fehsenfeld. F. C., and Albritton, D. L. (1980). J . Chem. Phys. 73, 758. Huntress, W. T. Jr. (1974). A h . Arom. Mol. Phys. 10, 295. Jones, T. T. C., Villinger, H., Lister, D. G., Tichy, M., Birkinshaw, K., and Twiddy, N. D. (1981). J . Phys. B. 14, 2719. Kemper, P. R., Bass, L. M., and Bowers. M. T. (1985). J . Phys. Chem. 89, 1105. King, G . H., Maricq, M. M., Bierbaum, V. M., and DePuy, C. H. (1981). J . Amer. Chem. SOC.103, 7133. Knight. J. S., Freeman, C. G., McEwan, M. J., Adams, N. G.,and Smith. D. (1985). Inr. J . Muss Spectrom. I o n Process. 67, 317. Lindinger, W., and Albritton, D. L. (1975). J . Chem. Phys. 62, 3517. Lindinger, W., and Smith. D. (1983). I n “Reactions of Small Transient Species’’ (A. Fontijn and M. A. A. Clyne. eds.), pp. 387 456. Academic Press, New York.

48

D.Smith rmd N . G. Adams

Lindinger. W., Howorka, F., Lukac, P., Kuhn, S., Villinger, H., Alge, E., and Ramler, H. (1981). Phys. Rev. A23, 2319. Mackay, G. I., Vlachos, G. D., Bohme, D. K., and Schiff, H. I. (1980). lnt. J . Mass Spectrum. lun Phys. 36, 259. McDaniel, E. W. (1964). “Collision Phenomena in Ionized Gases,” pp. 2-6. Wiley, New York. McEwan, M. J. (1986). Private communication. McFarland. M., Albritton, D. L., Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1973a). J . Chem. Phys. 59, 6610. McFarland, M., Albritton, D. L., Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1973b). J . Chem. Phys. 59, 6620. McFarland, M., Albritton, D. L., Fehsenfeld, F. C., Ferguson, E. E., and Schmeltekopf, A. L. (1973~).J . Chem. Phys. 59, 6629. McIver, R. T. Jr. (1978a). Rev. Sci. Instrum. 49, 11 1. McIver, R. T. Jr. (1978b). In “Kinetics of Ion Molecule Reactions”(P. Ausloos, ed.), pp. 255- 270. Plenum Press, New York. Millar, T. J., Adams, N. G., Smith. D., Lindinger, W., and Villinger, H. (1986). Mon. Not. R . Astr. SOC. 221, 673. Miller, T. M., Wetterskog, R. E., and Paulson, J. F. (1984). J . Chem. Phys. 80, 4922. Moseley, J. T. (1985). I n “Photodissociation and Photoionization” (K. P. Lawley, ed.), “Adv. Chem. Phys.,” Vol. LX, pp. 245-2953, Wiley, Chichester, England. Narcisi, R. S., and Bailey, A. D. (1965). J . Geophys. Res. 70, 3687. Neilson, P. V., Bowers, M. T., Chau, M., Davidson, W. R., and Aue, D. H. (1978). J . Amer. Chem. Soc. 100, 3649. Nighan, W. L. (ed.) (1982). “Applied Atomic Collision Physics,” Vol. 111, “Gas Lasers.” Academic Press, New York. Paulson, J. F., and Dale, F. (1982). J . Chem. Phys. 77, 4006. Rowe, B. R., Dupeyrat, G., Marquette, J. B., Smith, D., Adams, N. G., and Ferguson. E. E. (1984). J . Chem. Phys. 80,241. Skullerud, H. R. (1973). J . Phys. B. 6, 728. Smith, D., and Adams, N. G. (1977a). In/.J . Mass Spectrom. lon Phys. 23, 123. Smith, D., and Adams, N. G. (1977b). Chem. Phys. Letts. 47, 145. Smith, D., and Adams, N. G. (1978). Chcrn. Phys. Letts. 54, 535. Smith, D., and Adams, N. G. (1979). I n “Gas Phase Ion Chemistry” (M. T. Bowers, ed.), Vol. 1, pp. 1-44. Academic Press, New York. Smith, D., and Adams, N. G. (1980a). J . Phys. D. 13, 1267. Smith, D., and Adams, N. G. (1980b). Ab. J . 242,424. Smith, D., and Adams, N. G. (1981a). Phys. Rev. A23, 2327. Smith, D., and Adams, N. G. (1981b). In/. Revs. Phys. Chem. 1, 271. Smith, D., and Adams, N. G. (1984). In “Ionic Processes in the Gas Phase” (M. A. Almoster Ferreira, ed.), pp. 41 -66. Reidel, Dordrecht. Smith, D., and Adams, N. G. (1985a). Ap. J . 298, 827. Smith, D., and Adams, N. G. (1985b). In “Molecular Astrophysics: State of the Art and Future Directions” (G. H. F. Diercksen, W. F. Huebner, and P. W. Langhoff, eds.), pp. 453-469. Reidel, Dordrecht. Smith, D., and Adams, N. G. (1987a). In/.J . Mass Spectrum. Ion Process. 76, 307. Smith, D., and Adams, N. G. (1987b). In preparation. Smith, D., Adams, N. G., and Miller, T. M. (1978). J . Chem. Phys. 69, 308. Smith, D., Grief, D., and Adams, N. G. (1979). Int. J . Mass Spectrom. Ion. Phys. 30, 271. Smith, D., Adams, N. G., Alge, E., Villinger, H., and Lindinger, W. (1980a). J . Phys. B. 13, 2787. Smith, D., Adams, N. G., and Henchman, M. J. (1980b). J . Chem. Phys. 72,4951. Smith, D., Adams, N. G., and Alge, E. (1981a). Planet. Space Sci. 29, 449.

T H E SELECTED ION FLOW TUBE (SIFT)

49

Smith, D., Adams, N. G., and Lindinger, W. (1981b). J. Chern. Phys. 75,3365. Smith, D., Adams. N. G.. and Alge, E. (l982a). A p Smith. D., Adams, N. G . , and Alge, E. (1982b). J . Ckem. Phys. 77, 1261. Smith. D., Adams, N. G., Alge, E., and Herbst, E. (1983). Ap. J . 272,365. Smith. D.. Adams, N. G., and Alge, E. (1984a). Chern. Phys. Lerts. 105, 317. Smith, D., Adams. N. G., and Ferguson, E. E. (1984b). Inr. J . Mass Specrrom. Ion Process. 61, 15. Smyth, K. C., Lias. S. G., and Ausloos, P. (1982). Combustion Sci. Techno/. 28, 147. Thomas, L. (1974). Radio Sci. 9, 121. Tichy, M.. Rakshit, A. B., Lister, D. G., Twiddy. N. D., Adams, N. G.. and Smith, D. (1979). Int. J . Mass Specfrotn. / o n Phys. 29. 23 1. Turban. G . (1984). Pure .4ppl. Chem. 56, 21 5. Van Doren, J. M., Barlow, S. E., DePuy, C. H., and Bierbaum, V. M. (1987). /nt. J . Muss Specrrom. Ion Process. 81,85. Viehland. L. A. (1986. Chem. Phps. 101, I . Viehland. L. A., Mason, E. A., and Whealton, J. H. (1974). J . Phys. B. 7,2433. Viehland, L. A,, Lin, S. L., and Mason, E. A. (1981). Chem. Phys. 54, 341. Viggiano. A. A. (1986). J . Chem. Pliys. 84, 244. Viggiano, A. A,, and Paulson, J. F. (1984). In “Swarms of Ions and Electrons in Gases” (W. Lindinger, T. D. Mirk. and F. Howorka, eds.), pp. 21 8 -240. Springer-Verlag, Vienna. Viggiano, A. A,, Paulson, J. F., Dale, F.. Henchman, M. J., Adams, N. G.,and Smith, D. (1985). J . Phys. Chem. 89, 2264. Wagner-Redeker, W.. Kemper, P. R., Jarrold, M. F., and Bowers, M. T. (1985). J . Chem. Phys. 83, 1121. Wannier, G. H. (1953). Bell. Sysr. Tech. J . 32, 170. Wayne, R. P. (1985). “Chemistry of Atmospheres.” Clarendon Press, Oxford, England.

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL 24

NEAR- THRESHOLD ELECTRON-MOLECULE SCATTERING MICHAEL A. MORRISON Deparrrnent of Phwics nnd Astronomj. University q/' Oklahomtr Normun. Okluhomu 73019

I . Introduction and Overview . . . . . . . . . . . . . . . . . .

I1. Basic Quantum Mechanics of Low-Energy Electron-Molecule Scattering . . A. Preliminaries: Getting Rid of the Molecular Electrons . . . . . . . B. Electron-Molecule Scattering in the LAB Frame . . . . . . . . . C . Electron-Molecule Scattering in the BODY-Frame . . . . . . . . D . Frame Transformations . . . . . . . . . . . . . . . . . . VII . Approximate Collision Theories and Their Woes near Threshold . . . . A. The Fixed-Nuclear-Orientation Approximation and the BFVCC Method . B. The Fixed-Nuclei Approximation and the Adiabatic-Nuclear-Rotation Method . . . . . . . . . . . . . . . . . . . . . . . C . The Born-Oppenheimer Perspective and the Adiabatic-Nuclear-Vibrational Method . . . . . . . . . . . . . . . . . . . . . D . The Adiabatic-Nuclei Scattering Amplitude. . . . . . . . . . . E . Breakdown of Adiabatic-Nucleus near Threshold . . . . . . . . . IV. The Behavior of Near-Threshold Cross Sections Explained . . . . . . A. Qualitative Answers: Threshold Laws . . . . . . . . . . . . . B. Threshold Expansions: Modified ElTective-Range Theories . . . . . V. Beyond the Born-Oppenheimer Approximation: Special Methods for NearThreshold Scattering . . . . . . . . . . . . . . . . . . . . A . Weak-Scattering Approximations . . . . . . . . . . . . . . B. Methods Based in Part on the Born Approximation . . . . . . . . C. Approximate Nonadiabatic Scattering Theories . . . . . . . . . D . Off-Shell T-Matrix Methods . . . . . . . . . . . . . . . . VI . Variations on an Enigma: Threshold Structures in Vibrational Excitation Cross Sections . . . . . . . . . . . . . . . . . . . . . . A . Provocative Experimental Results . . . . . . . . . . . . . . B. Much Ado about Polar Systems . . . . . . . . . . . . . . . C . The Virtual-State Mechanism in Nonpolar Systems . . . . . . . . VII . Conclusions and Conundrums . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . Appendix A: Notation and Nomenclature . . . . . . . . . . . . Appendix B: Recent Reviews of Electron-Molecule Scattering . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

52 56 56 58 65 69 75 76 17

79 81 82 88

un

100 115

116 117 119 127 131 131 134 141

143 144 144 150 151

51

.

C'opynghl I 19x8 by Acddernic Prrsa Inc All rights 01 reproduction in any form reserved ISBN 0-12-IX)3824-2

52

Michael A . Morrison Whoever wants to see a brick must look at its pores, and must keep his eyes close to it. But whoever wants to see a cathedral cannot see it as he sees a brick. This demands a respect for distance. Jose Ortega Y Gasset

I. Introduction and Overview Low-energy electron-molecule scattering is rich in fundamental physics and fascinating phenomena. It is a field whose applications range from pollution control to astrophysics (Phelps, 1980). To the experimentalist, the dense maze of energy levels of the target and the difficulty of generating a cooperative, well-defined beam of low-energy electrons of known energy make measurements of cross sections particularly difficult. And to the theorist, the complex structure of the target, the indistinguishability of the projectile from electrons in the target, and the nonspherical character of the interaction potential make calculation of these cross sections no less difficult. Indicative of the level of activity and enthusiasm in this field is the extensive coverage it has received in recent books, symposia, and reviews. (Annotated lists of these sources appear in the reviews by Morrison, 1983, and Csanak et al., 1984. Appendix B updates these lists.) But none of these articles has focussed on collisions near threshold-perhaps because only very recently have experimental and theoretical technologies advanced to the state that this energy regime can be accurately studied. In the parlance of electron-molecule scattering, a “low-energy electron” is one whose incident kinetic energy E , is less than about 10 eV. Whether such a scattering event qualifies as a “near-threshold collision” depends not on E o , but on the energy of the electron after the collision (i.e., in the exit channel). If the electron excites the target from an initial state which we’ll denote vo (see 9 ILA), with energy E ~ to, a final state v with energy E,, then the final-state energy of the projectile is E , - ( E , - E,). For a “near-threshold collision,” this energy is very small. (See Fig. 1.) At energies near zero for elastic scattering and a few meV above threshold for inelastic processes, the calculation, understanding, and measurement of cross sections encounter unique challenges and bizarre phenomena. In this chapter, we examine some of the mysteries of this domain and progress in penetrating them. This chapter unabashedly emphasizes theory. But measurements have played a major role in the development of this theory. And the interplay of theoretical and experimental research in the near-threshold domain is particularly vital because of the extensive use by theorists of model potentials

NEAR-THRESHOLD ELECTRON-MOLECULE SC’ATTERING

53

->

aJ

w

z W

J

t 0

EO-(E”

- €o)

2

I

R

(a)

FIG.1. A near-threshold electronically elastic ro-vibrational excitation o,j, an incident electron with energy E , .

+ rj

induced by

(see $ VI). Most recent experimental advances in this field have followed from improved energy resolution in crossed-beam apparatuses (cf. Kochem et al., 1985a, 1985b) and from increased sophistication in the transport analysis of swarm data (cf. Crompton, 1983). These advances-and the results they have yielded-will thread their way throughout the latter half of this chapter. The first major section (9 11) contains yet another assay of the essential quantum theory of electron-molecule scattering. This theory, an understanding of which is crucial to a full appreciation of the problems posed by nearthreshold scattering and the many and varied strategies for solving them, is presented in a compact and, I hope, illuminating way that draws on state space notation and concepts more than is usual in this field. Nevertheless, readers not inclined to formal theory may find this material a bit thick; they are cheerfully invited to skip $11 and I11 and peruse other parts of this chapter (especially 9 IV and VI), where the density of kets, bras, and jargon is lower. Others may find the guide to nomenclature and notation in Appendix A helpful. Most recent theoretical advances are reformulations of the basic collision theory of 5 11. On the other hand, the calculation of near-threshold cross

54

Michael A , Morrison

sections poses special problems. These matters will not be discussed here; this chapter is not designed for the computer aficionado. (See, however, articles in Rescigno et al., 1979, and the review by Buckley et al., 1984.) Instead, I've chosen to emphasize the physical reasons for the failure near threshold of standard approximate scattering theories (#111) and the many new, innovative strategies theorists have devised for coping with this special domain (5 V and VI). Most electron-molecule cross sections act quite sensibly as the final-state energy goes to zero at threshold; the behavior of such data as the momentumtransfer and vibrational-excitation cross sections for H,, N,, and CO in Figs. 2 and 3 can be qualitatively understood by threshold laws and quantitatively represented by threshold expansions (a IV). But this energy regime is also littered with anomalies-strange behavior that does not conform to the expectations of conventional theory. A few tantalizing examples are shown in Figs. 2 and 3. The momentum-transfer cross section for e-CO, in Fig. 2 is huge compared to cross sections for electron scattering from other nonpolar targets. And the near-threshold behavior of the e-HBr vibrational-excitation cross section in Fig. 3b contrasts strikingly with the more typical cross sections in Fig. 3a. These and other oddities will be discussed shortly. I shall focus on elastic scattering and on rotational and vibrational excitation-all within the ground electronic state of the target. (See Fig. 1.) I o3

Io2 N ..

E

0

'D

10

'0

L

I

K

b

\

I

L 10-'1

1

I

0. I

0.2

I

I

i I

0.3 0.4 0.5 E N E RGY (eV 1 FIG. 2. Experimental momentum-transfer cross sections for various molecules: H, (Crompton et al., 1969a, 1969b); N, (Phelps and Pitchford, 1985); CH, (Haddad, 1985); CO, (Lowke, Phelps, and Irwin, 1973). All cross sections were derived from swarm data via transport analysis (Huxley and Crompton, 1962, 1974). 0

55

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

0.5

P

-

1 v = O - I

0.4

N

E U

‘0 0.3

--

I

0

i0.2 0

b’

0.I

0

I 0.2

I

I 0.4

0.6

I 0.8

I

I .o

I .2

ENERGY ( e V 1

40

b

-

I

I

I

0.2

0.4

I

I

3c

N

E

U

‘0

20

0 Y

>

f0 c b C

0.6

E N E RGY

0.8 (eV 1

I .o

I .2

FIG.3. Experimental vibrational-excitation cross sections for various molecules. (a) “Normal” cross sections: CO (Petrovic, private communication, 1986); N, (Crompton, private communication, 1986); H, (Gibson, 1970). (b) The anomalous v,, = 0 + u = 1 cross section for e-HBr scattering (Rohr, 1977a).

56

Michael A . Morrison

Very recently, exciting new research has been published on electronic excitation, but I have excluded this work because it does not emphasize the near-threshold domain. I have also excluded, with some regret, rearrangement collisions such as dissociative attachment. Impressive progress has been made in understanding such processes, both at and above threshold, and their omission from this chapter reflects practicality rather than preference.

11. Basic Quantum Mechanics of Low-Energy Electron-Molecule Scattering A. PRELIMINARIES: GETTING RID OF THE MOLECULAR ELECTRONS

I shall describe the stationary states of the target by their Born-Oppenheimer approximates (Born and Oppenheimer, 1927). That is, each state is represented by the product of an adiabatic electronic state la), which is labeled by a set of quantum numbers a and which depends on the nuclear geometry only parametrically, and a nuclear state lav), which is labeled by v.l The electronic and nuclear states, the electronic energy &C)(R)for internuclear separation R , and the Born-Oppenheim approximation to the total molecular energy, E,,, are obtained by solving the electronic and nuclear Schrodinger equations separately. Letting 2;)denote the molecular electronic Hamiltonian and 2:)the nuclear Hamiltonian (and including the potential energy of internuclear repulsion V(")(R)in the electronic energy), we write the equations describing the target states in this approximation as a ) = &f)(R)Ia ) ,

[T'"' + &;'(R)]Iav) = E,Icv).

(la) (1b)

The nuclear kinetic energy T(") is just the sum of the rotational and vibrational terms, which we shall denote by 2:)and T(").The vibrational Hamiltonian 2:)is just T(")+ &',)(R). Each electronic ket la) is antisymmetric under pairwise electron interchange, in accordance with the Symmetrization Postulate. Stationary states of the electron-molecule system in the entrance and exit channels-the asymptotic free states-are labeled by an electron wave vector

' For example, for the ground state of a homonuclear a, = X'Z:,f and the nuclear quantum numbers v = u, j ,

diatomic molecule, such as H,,

m, correspond, respectively, to the

vibrational Hamiltonian A?:' and to the square and projection on the quantization axis of the rotational angular momentum j .

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

57

k,, and by quantum numbers a and v, e.g., I k,,, av). The stationary scattering states I k,,, av ) are related to these asymptotic free states via the Mdler operator (Taylor, 1972); i.e.,

+

lk,,, av+)

=

a+Ik,,,av).

(2)

The Hamiltonian JV of the system includes operators for the kinetic energy of the projectile, T,; the target molecule, Y,; and the Coulomb interactions between the projectile and the electrons and nuclei of the target, I/coul. The molecular Hamiltonian is the sum of the electronic and nuclear Hamiltonians in (I), so the system Hamiltonian is 2

=

T,

+ %’: + T(”)+ P ) ( R ) + Vcoul.

(3) Finally, the time-independent Schrodinger equation for a state of total energy E is (JV

-

E)Iko, a O v o + )

=

0.

(4)

The solutions of this equation define the physical scattering amplitude from which cross sections are calcuated. (See 3 II.B.l.) Since we are not considering electronic excitation, we can simplify Eq. (4) by projecting out of the scattering ket the ground electronic state of the target, IN,) (Cf. Burke 1979, 1980.) Formally, we do so by expanding this ket in the complete set of eigenkets of X;), { l a ’ ) } . Using the bound-free antisymmetrizer .d to ensure antisymmetry under interchange of the projectile and each bound electron, we write Ik,,

X ~ V , + ) =

.dxla’)(a’la’; k,,

CC,V,+).

(5)

a’

If we truncate this sum at the ground state in the set of coupled equations that result when we substitute Eq. (5) into Eq. (4) (and drop the now-redundant label a, from the scattering ket), we obtain the “reduced” Schrodinger equation that will be the focal point of this review, viz.,

[T, + .HE’

+ T(”)+ EE’(R) + y,,

-

El1 k,,

vo

+ ) = 0.

(6)

The molecular electrons have not, of course, completely disappeared; their influence is felt through the interaction potential V,,, in Eq. (6). This operator is the sum of the static potential K,, which is equal to the Coulomb interactions Vcoulaveraged over the ground electronic state of the target, and a nonlocal exchange operator V e xi.e., ,

v,”, = v,, +

= (a01

Vou,l~o+ ) -fex.

(7)

The exchange operator in Eq. (7) appears because of the action of the antisymmetrizer &. For a closed-shell target with N,,,, doubly occupied

58

Michael A . Morrison

(single-particle) molecular orbitals I ti), V e x can be written in terms of the (nonlocal) electron-electron repulsion operator V"' as

In coordinate space (and atomic units), the repulsion and exchange operators have the more familiar forms (r'(Vee(r") =

1 ~

Ir'

-

r''] '

The exchange operator induces integral terms in the (otherwise differential) scattering equations one solves in electron-collision calculations (e.g., 4 II.B.4) and hence leads to considerable grief.2

B. ELECTRON-MOLECULE SCATTERING IN THE LAB FRAME The most familiar reference frame in which to formulate scattering theory is the space-fixed laboratory reference frame of Fig. 4. This frame is defined by orthogonal unit vectors (2x,, &,, e*,.), and the 2,, axis is usually chosen to lie along the wave vector of the incident electron, k,. This is not, however, the most useful reference frame for calculations of electron-molecule scattering processes, so in 4 II.C.1 I shall introduce an alternative: the B O D Y frame. 1. The Physical Boundary Conditions

In coordinate space (using primed coordinates to denote the LAB frame) the stationary scattering state I k,, v, + ) becomes the wave function Lylko,Vo(r', R). For excitation from initial state v, to final state v, the lab-frame scattering amplitude Lf(k:, v + kb, v,) appears in the asymptotic form of this function. Since we have eliminated the molecular electrons from the Schrodinger equation (5 II.A), this boundary condition contains only the nuclear wave functions z,(R): Lylko,Vo(r',R) I'

-

a3

I

eik,r'

(24-

3/2

+ rlLf(k:,

Fiko'r'zVo(R)

V

1

v + kb, V,)~,,(R).

(10)

* Since the focus of this chapter is scattering theory-how to solve Eq. (6)-little more will be said concerning the calculation and modelling of the interaction potential. For reviews of this important topic, see 9: 1I.G of Lane (1980) and 9: 11.4 of Morrison (1983).

59

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

FIG.4. LAB (primed axes) and BODY (unprimed axes) coordinate systems

The sum in the second term runs over all energetically accessible (i.e., open) channels. The final-state wave vector is k\ = k v P , so Lf(k\, v c k' vo) depends on the angular coordinates of the projectile after the collision. From this amplitude we could construct differential, integral, and momentum-transfer cross sections, e.g., 0 3

where i,,.io is the cosine of the angle between the incoming and outgoing wave vectors of the projectile. In practice, however, one rarely calculates a scattering amplitude or uses Eqs. ( I 1 ) directly; rather, one computes cross sections from various scattering matrices.

2. The LAB- UCAM Representation Although the plane-wave representation of Eq. (10) is closely allied to the situation in the laboratory, it is somewhat awkward for theoretical calculations. To conveniently describe the collision theoretically, we introduce the rotational angular momentum 3 of the nuclei and the orbital angular momentum 1 of the electron.

Michael A . Morrison

60

TABLE I

REPRESENTATIONS OF ELECTRONM O L E C ~ JCOLLISlON LE THEORY Representation

Scattering state

Wave function

T matrix

We shall first consider an uncoupled angular momentum (UCAM) LAB representation, in which asymptotic free states are defined by eigenvalues of the complete set of commuting operators that includes the kinetic energy of the projectile, the vibrational Hamiltonian, and the square and projection on 2z,ofiand i: { T,, P:), I z , I z . , j 2 , j,.). The quantum numbers defined by the last five of these operators are v, I, m,j , and mj;when convenient, we'll encapsulate u, j , and mj into the composite symbol v. The probability amplitude for a transition between initial and final states IE, voIom0) and IE, vlm) is the T matrix LTVlm,volomo = ( E , vIm(TIE, voI,m0). Important scattering quantities in the LAB-UCAM representation-and others we shall meet shortly-are summarized in Table L3 One can easily switch from the LAB-UCAM representation to the planewave representation and thereby relate LTylm, volomo to the scattering amplitude in Eq. (10). For an electron with energy E , = k;/2, the transformation between the plane-wave and angular-momentum representations is

The normalized plane-wave and angular-momentum eigenfunctions in the LAB frame (in atomic units) are (Taylor, 1972)

( r ' l k , ~= (2.rr)-3/2eik:.r' (r'lE,,, Im)

=

if J - ' i j ( k

nk,r'

'

"

(13a)

r')Yy(t'),

A variety of conventions and normalizations are used in the literature of electron-molecule scattering. For consistency and ease of reference, I have adopted the conventions of Taylor (1972) and, where possible, the notation of Lane (1980) and Morrison (1983). And for simplicity, the theory in 0 I1 is formulated for collisions with a linear target. For a detailed look at the generalization to polyatomic systems, the reader is referred to the review by Gianturco and Jain (1986) and references therein.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

61

where j,(k,r‘)is the Ricatti-Bessel function. Hence the transformation coefficient in (12) is ( E v , lmlki,) = k;

1’2Yy*(i:).

(14)

We apply this transformation to the plane-wave asymptotic states 1 k;, ti) by introducing a Born-Oppenheimer factorization of these kets (an excellent approximation for the ground electronic states of nearly all molecules), lkl, 1))

=

Ik:,)lv).

(15)

Using a similar factorization of the asymptotic angular momentum kets I E, ulm), we write the desired transformation in terms of the coefficients (14); viz.,

Now, to relate the LAB-UCAM T matrix LT,,lm.volomo to the physical T matrix Lt(k:,v c kb, vo) and thence to the scattering amplitude Lf’(kL,v

+

kd, vg)

=

-(2n)’ Lt(k:, v + kb, vo),

(17)

we apply closure of the angular momentum basis ( ( E ,l m ) ) and Eq. (16), obtaining4 Lf(k:, v

+

kb, vo)

=

-

(2n)’ 7---C C Yy(i,,) k,,

YT*(io). (18)

LTvlm.vol~rm,,

ko

im ioiw

3. The L A B - C A M Representation

At this juncture we could proceed to solve for the T matrix in the UCAM representation, but this tack ignores a fundamental feature of the electronmolecule system: because Kn, is not spherically symmetric, neither j,. nor l z , is conserved. But the total angular momentum J’=J + i is conserved. So it behooves us to change to a representation in which J z and J z . replace j z ,and l z , in the defining set of operators (Table I ) : the LAB coupled angular momentum (LAB-CAM) represrntution (Arthurs and Dalgarno, 1960). The transformation between the UCAM and CAM representations is easily effected using the Clebsch-Gordan series5

Ij l J M ) C ( j l J ; m j m M ) ,

Ijlmjm) =

(19)

J

If the LAB z axis is chosen to lie along kk. then m0 = 0 in this and subsequent equations. The conventions of Morrison and Parker (1987) are used throughout this review. See also Biedenharn and Louck (1981).

62

Michael A . Morrison

where the (real) Clebsch-Gordan coefficient C(j l J ; m j m M ) is ( j l J M Ijlmjm) and M = m m j . For example, the UCAM T matrix LTvlm,volomo can be regained from the CAM T matrix LT.Ejl,vojolo via

+

LTvlm,volomo

=

CJ C U ~ Jm; j m W

mjomo

LT.Gjl, vojoIo C(j, 1 0 ~ ;

(20)

(Because the system is axially symmetric, its Hamiltonian does not contain J z , ; so the LAB-CAM T matrix does not depend on M.) Using Eqs. (20) and (18) we can express the scattering amplitude (17) in terms of the LAB-CAM T matrix, viz.,

LfK, v

+

kb, v o ) =

x c(jlJ; m j m ~L~{,jI.vOjo,O )

c(j0~,J; mjomoM)Y~;*(ho).

(21)

Expressions for the various cross sections (11) in terms of LT~jI,vojolo follow from Eq. (21); see Arthurs and Dalgarno (1960), Chandra (1975), and Lane (1980).

4. The LAB-CAM Scattering States, Coupled Equations, and Boundary Conditions Equation (21) implies a relationship between the scattering state vo ), with coordinate space representation LYko,vo(r’, R), and the LAB-CAM ket 1 E , u o j o i o ; J M ), with wave function L Y ~ ~ ~ o J oR). lo(r’, Indeed, the transformations (1 6 ) and (19) provide the coefficients of the superposition of (degenerate) LAB-CAM states that obeys the boundary conditions (lo), viz.,

I k,,

+

L ~ k o , v o ( rR) ’, =

+

1 C 2 C(j,l,,J; mjom,M) y;”,o*<&,> L%~ojolo(r’,R). Jk, J lomu

(22) The sum over J in Eq. (22) uncouples the rotational and orbital angular momenta, while the sums over 1, and m, mix partial waves so that the proper plane-wave factors in (10) emerge. Central to the calculation of cross sections for electron-molecule collisions are the LAB-CAM radial scattering equations. To derive these equations, we expand I E, u, j o l o ; J M + ) in a basis appropriate to its representation. The LAB-CAM basis { I u j l J M ) } , the complete set of eigenkets of the operators

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

63

TABLE 11 EXPANSION BASFSOF ELFCTRON MOLECULF COLLISION THEORY Basis

Operators

Variables

Representation

lujmjlm)

.xz’,j2,j z . , 12, Iz. X‘P’, 12, j2,J 2 . J,.

i’. R i‘,R i, R i. R

LAB-UCAM LAB-CAM BODY: I BODY: I1

I p ) = (ujUM) 14) = ( u l A J M ) It) = IIAJM)

Rf’,12, I;, J 2 , J ; . J z , 1 2 , I,, JZ, J,, J ; .

(X)n“,12, j 2 , J 2 , J,.}, does the job. In coordinate space, the LAB-CAM basis functions (r’, RlujlJM) are denoted by @ ,?(’;: R). For a linear molecule, the nuclear wave function is the product of rotational and vibrational eigenfunctions Yjm,(8)and &’(R), and the LAB-CAM basis functions are

@ ,?(‘::

C(jIJ;m j m M ) Y y ( R ) Y r @ ‘ ) .

R ) = cpl’’’(R)

(23a)

mJm

The coupled angular functions in (23a) are usually denoted by gip(?‘, 8) and these basis functions are written as @ ,?(’;:

R ) = (p~’(R)”Yj:M(F’,A).

(23b)

Properties of the LAB-CAM basis set-and of others yet to appear-are summarized in Table 11. Letting p = (ujlJM) be the LAB-CAM channel index, we can write the expansion of the LAB-CAM ket in its basis in the compact form

When this expansion is written in coordinate space, the coefficient ( P I E , p o + ) becomes the normalized radial scattering function6

To derive radial scattering equations, we merely insert (24a) into the Schrodinger equation for the LAB-CAM ket,

[ T , + T‘”’+E:IP,’(R)+~ , + . I , , - E ] l E , p o + ) = o ,

(25)

’The normalization of this function has been chosen to conform to that of the corresponding free function in Eq. (13b). The normalization factor drops out of the coupled equations (29).

64

Michael A . Morrison

to wit:

CT, + T(")+ EZ)(R) + v,, - E l =

-1KxlP')(P'lE>

c IP'>(P'lE,

PO+)

P'

(26)

PO+).

P'

We now reduce (26) to radial form by projecting out (PI, using the Clebsch-Gordan series to uncouple J and 1, and exploiting the orthonormality of the target states. After some algebra, there appear coupled equations that include the channel energies, which are defined in terms of the targetstate energy E , as ik?

= E - E,,

(27)

and coupling matrix elements (in state space and configuration space)

+

c [Vp,,,(r'>+ .yp,,,(r?l Lup,,p,(r') 0. =

P'

(29)

The static matrix elements Vp,,,(r')in Eqs. (29) are usually evaluated from the coefficients of the expansion of the static potential in Legendre polynomials of the angle 6 between r' and the internuclear axis R,

K,(r', R) =

1

u;(r', R)P,(cos 0).

(30)

The resulting matrix elements can be written VP.,,(r'>=

c .M.jLj'r; J)(d?IG'tvY), i.

(3 1)

where the coupling factors f,(,jl, j'l'; J ) are combinations of Clebsch-Gordan and Racah coefficients (see Lane and Geltman, 1967) and the matrix element implies integration over R . The exchange matrix elements in (29) are more troublesome; because cx is nonlocal, each exchange term is an integral. We can conveniently write these terms by introducing the exchunge kernel K p ,pj(r',r"), viz., %,p,(r')Lup,, p,,(r') =

KP,P.(r', r")Lup,,po(r") dr".

(32)

N EA R-THR FSHOLD ELECTRON- MOLECU LE SCATTERING

65

For a closed-shell target with No,,doubly-occupied single-particle orbitals &(r; R), the exchange kernel is

KPJr’, r”) = r‘r“

1

i= 1

s

1

mi(?’,R)ci(r’; R) 1 r‘ - r”1 (:(r”;R)Qp,(i”,R) dP’ di” dR. ~

(33) The LAB-CAM scattering matrix LSijr,l,ojolo appears, at long last, in the boundary conditions on the radial functions in Eqs. (29),

(34) The S matrix is related to the (on-shell) T matrix of Eq. (21)-the matrix element of the transition operator between asymptotic free states I p o ) and Jp)-by the usual relation LSp,po = S P q p n- 27ti LTp,po.

(35a)

This T matrix can usefully be written as an integral over the radial scattering functions. The resulting form incorporates both boundary conditions and radial equations (see Chap. 4 of Rodberg and Thaler, 1967), i.e.,

C. ELECTRON-MOLECULE SCATTERING IN THE BODY FRAME The LAB-CAM representation is less intuitive physically but more useful theoretically than the plane-wave representation. The BODY representation is one more step removed from the laboratory, but it too is valuable to the theorist (Chang and Fano, 1972). But before introducing the BODY representation, we must consider the BODY reference frame. 1. The BODY Rt?firencr Frunie

The BODY frame is relatcd to the LAB frame via a simple rotation of coordinates in S3;as shown in Fig. 4, the rotation angles are chosen to align the BODY z axis with the principal symmetry axis of the target (for a diatomic molecule, the internuclear axis). (We shall use unprimed coordinates for BODY variables.)

66

Michael A . Morrison

Body-fixed reference frames have long been used in molecular structure calculations (cf. McWeeny and Sutcliffe, 1968); in electron-molecule theory they enable us to make physically motivated simplifying approximations to the collision formalism (9 111). The particular choice 2, = l? in Fig. 4 is made because under certain conditions, the projection of i on l? is (to a good approximation) a collision constant. (See Lane, 1980, and Morrison, 1983.) The Euler angles that rotate the LAB frame into the BODY frame are just the angles of the symmetry axis in the LAB frame:7 /? = 8, and CY = q R .The solution of the Schrodinger equation (6), expressed in BODY coordinates, is the wave function (r, R I ko

3

"0

+ ) = BYko,

R).

(36)

Eigenkets in the two frames are related by the unitary rotation operator W ( a , p , y ) that corresponds to the LAB-to-BODY rotation in !R3; for example, eigenkets of the position operator are related by (see, for example, Morrison and Parker, 1987) ~ r )= ~ ( a/j,, y)lr')

= e-ivR'z'e-it'Rly'

I r'>.

(37)

2. The BODY Representation

The fundamental difference between the BODY and LAB formulations of electron-molecule collision theory is the complete set of commuting operators that defines their asymptotic free states (Table 11). Instead of the projections of i and y o n the LAB z axis, the BODY representation { T,, a?:), 12, I,, J 2 , J,, J,.} includes the projections of these operators on the BODY z axis. Hence the eigenvalue A of I , appears (instead of rn) in the BODY asymptotic k e h 8 The transition matrix in the BODY representation connects asymptotic free states corresponding to these BODY operators, and it may not be immediately apparent how this matrix is related to the LAB scattering amplitude of Eq. (10). From Eq. (21) we can derive an expression for this amplitude provided we can get from the BODY wave function to the LABCAM T matrix LTij,,v o j o l o . This transformation will be the subject of § 1I.D. Here we shall use the mathematical machinery of tj1I.B to derive radial scattering equations for the BODY representation.

' The third Euler angle y is arbitrary. so we choose it to be zero.

'

The operator J,. is included in the BODY representation to ensure completeness of the defining set of operators. Strictly, we should also label channels in this representation by the eigenvalue of J , ; but for a target whose ground state has X symmetry-the only case considered here-the eigenvalues of J , and I, are equal.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

67

3. The BODY Basis I : Including):%'

Our first basis for expansion of the BODY states IE, uol,Ao; J M + ) is defined by a set consisting of the vibrational Hamiltonian and the relevant angular momentum operators, {Hz), 12, l:, J 2 , J , , J , . } . The set of configuration-space functions corresponding to the basis kets I v l A J M ) is complete in i and R; each function is the product of a vibrational function q$)(R), a spherical harmonic with eigenvalue A along the quantization axis &, and a symmetric-top rotational function (normalized on the unit sphere dR = Sin OR doR d q R )

We shall use q = ( v l A J M ) as the BODY channel index for this basis and denote the corresponding wave number by k,. The BODY basis functions are therefore (r, R ( q ) = (r, R J v l A J M )= q!")(R)RJMA(fi)Yt(i).

(39)

4. The BODY Scattering States and Coupled Equations

By expanding the BODY scattering state I E , uoloAo;J M + ) in the basis { 14)) we can introduce the (normalized) BODY radial scattering functions

The mathematical form of the coupled radial equations of the BODY formulation is quite similar to that of the LAB-CAM equation (29), with the crucial difference that it includes matrix elements of the rotational Hamiltonian X i ) ,i.e.,

where the channel energy is k;/2 in state Iu).

=

E

- E,>,

with c , the expectation value of

68

Michael A . Morrison

5. The BODY Basis II: Excluding X:)

A widely used alternate B O D Y basis excludes the vibrational Hamiltonian: { I 2 , l,, J 2 , J,, J,.}. The corresponding basis functions are labeled by the channel index t = (1AJM).The set of these “second” BODY basis functions, (r, R l t )

=

(r, RIIAJM)

=

RJMA(R)Yt(i),

(42)

is complete in i and O R , q R ,but not in R. So when the B O D Y scattering ketIE, uoloAo;J M + ) is expanded in the basis {It)}, the resulting radial functions are labeled by the initial vibrational quantum number uo and depend on R, i.e.,

(IAJMIE, uoloAo;J M + )

=

r

The coupled equations obtained from this basis are

(43)

The static matrix element in (44) is V t l , ( r ;R ) = (/A I ytI l’A) =

&j!

uy(r, R ) C(rA1; Ao)

c(rn/;00). (45)

The structure of the exchange matrix element V & , ( r ; R ) is similar to the LAB-CAM matrix element (32). Solution of the coupled equation (44) is rendered complicated by the dependence of the radial function on r and R . The latter dependence can be removed by modelling the target as a rigid-rotator (RR) with internuclear separation R frozen at its equilibrium value R e . This approximation introduces several simplifications in Eq. (44): the vibrational kinetic energy disappears, the electronic energy becomes c ~ n s t a n t and , ~ the radial function no longer depends on uo. If we further make the jixed-nuclear-orientation ( F N O ) approximation (9 III.A), i.e., we assume that for the duration of the collision the molecule does not rotate, then terms involving the rotational Hamiltonian also disappear from (44), taking with them the coupling of radial functions with different values of A. In this approximation, the radial functions become ’Nu&o(r;Re), and Eqs. (44) collapse to those of the widely used body-frame fixed-nuclei (BF-FN) theory (5 1II.B). We can measure E from CE/(Re)and drop the electronic energy from subsequent equations.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

69

The relationship of the coupled equations of the two BODY representations-Eqs. (41) and (44)-becomes clear if we expand %{”,(r,R) in the complete set of vibrational wave functions [ &)( R ) ) , R Jvo

~ i A . i , ~ i \ , , ( r ,R )

=

C ‘ W m .onioi\r,(r)l(PI,V’(R).

(46)

I’

If we substitute this expansion into Eq. (44) and call upon the orthonormality of the vibrational functions, we regain Eq. (41). Whether we choose BODY basis I or BODY basis 11, a problem arises if we attempt to extract a transition matrix from the asymptotic form of a solution to either set of BODY coupled equations. The troublemaker is the matrix element of &$’ in Eq. (44), and the problem is that this matrix element is independent of Y and hence does not vanish as r 4 03. The resulting asymptotic rotational coupling prohibits defining a BODY S matrix via boundary conditions analogous to Eq. (34), because even in the asymptotic limit, the BODY radial functions are coupled. One way out of this difficulty is to make the aforementioned FNO approximation, which eliminates the offending coupling terms from (44). Another, more accurate gambit is the rotational frame transformation (5 II.D), by which we can switch back to the LAB-CAM representation at any desired radius r. Having done so, we continue propagating the solution of Eqs. (29) into the asymptotic region and there extract LT;IjI,l,l,jolo, from whence cometh cross sections. D. FRAME TRANSFORMATIONS The essential difference between the LAB-CAM and BODY representations is the presence in the former ofj’ and in the latter of 1,. In this difference is to be found the key to transforming scattering functions and matrices between these representations. The frame transformation theory, which was originally presented by Chang and Fano (1972), provides valuable insight into the physics of electron-molecule scattering (see Fano, 1970a, and Morrison, 1983) and an approach to developing approximation methods for collisions (§ 111). Chang and Fano identify rotational and vibrational frame transformations; 1’11 discuss the latter in $ II.D.4. The rotational frame transformation (RFT), for example, simultaneously accomplishes two things: ( I ) The RFT “rotates” the BODY angular momentum state //A) of the electron, which is an eigenket of I:, into an eigenket of lz. with the same eigenvalue (A), viz., 1/12)’ = N ( P R , dR)IlA),

(47a)

Michael A . Morrison

70

TABLE 111

THECOUPLED RADIALFUNCTIONS OF ELECTRON-MOLECULE THEORY Theory

Radial function

Coupled eqns.

Channel energy"

Measured from &E(R,).

The prime on IlA)' reminds us that A in this ket is the eigenvalue of the projection of i on the LAB z-axis, and 9L,,(qR,6,) is the Wigner rotation matrix-the representation of the rotation operator in the angular momentum basis. (2) The RFT changes from the LAB-UCAM to the LAB-CAM representation, using the Clebsch-Gordan series (19). We can apply the rotational (and vibrational) frame transformation to BODY radial functions (or, once certain approximations have been introduced, to T matrices)." But before embarking on this program, we gather together the rapidly proliferating scattering functions in Table 111.

1. Derivation of the Rotational Frame Transformation The effect of the RFT on the LAB-CAM and BODY basis kets can be expressed in terms of its matrix elements A$ as

I u l A J M ) = 1 I ujlJM)A$,

[LAB + BODY]

(484

j

IujlJM) =

1~ u ~ A J M ) A ~ A ,[BODY

4

LAB]

(48b)

A

Using Eqs. (47a) and (48a) we can write A$ as1 A$

=

(jlJMl&llAJM)r.

(49)

To derive an explicit form for AiJf,we uncouple J and i, using the inverse of the Clebsch-Gordan series (19); induce the desired rotation, using Eq. (47b); l o Chandra and Gianturco (1974) have suggested that the Wigner R matrix (Wigner and Eisenbud, 1947) be used as the transformation variable-an idea that was subsequently implemented by Chandra (1977) in a study of e-CO scattering. Because the RFT leaves untouched the vibrational states, we have omitted vibrational quantum numbers from this equation.

NEAR-THRFSHOLD ELECTRON-MOLECULE SCATTERING

71

and simplify the result, using the orthonormality of the kets Ilm) and the integral of three Wigner matrices (Rose, 1957). Some algebra follows; then we obtain (the real numbers) A$

=

diJ+ 2j

+

1

C(jlJ; O M ) .

For example, we can easily relate the LAB-CAM angular functions %!?(?’, ff) of Eq. (23b) to the BODY basis functions (42) by merely writing Eq. (48b) in configuration space, viz., .Yj, (?,

0 JM

a) = 1 A;AYf(P)RJMA(R).

(RFT)

(51)

A

2. The Radial Frame Transformation The RFT (48) relates basis kets, asymptotic free states, and hence (through the M ~ l l e roperators) stationary scattering states in the LAB and BODY representations, e.g., IE, uoIoAo;J M + )

IE, uojolo;JM+)Af$o.

=

(52)

jo

From this result we can develop a RFT of the normalized BODY radial functions. Projecting the basis state (ulAJMlout of Eq. (52), we obtain (ulAJMIE, uoloAo;J M + )

=

1 (ulAJMIE, uojol,;

JM+)A;ku.

(53)

jn

We now use Eqs. (48) to transform into the LAB-CAM representation. In terms of (unnormalized) radial functions, the resulting equation leads to L J

ut,jl.

JI R J A j A U,~A,

oojolo(r’)=

,,o~o~o(r)A~$o. (radial RFT)

(54)

AAo

Equation (54) can be implemented at any desired transformation radius rir)to switch from the BODY to LAB-CAM representations. This important strategy (Chang and Fano, 1972) has been applied to electron scattering from C O and H,. (See 8 II.D.5.)

3. The Asymptotic Rotational Frame Transformation

In practice, application of the RFT (54) is much easier if one waits until the asymptotic region, where the radial wave functions have settled down to their boundary conditions and one can transform scattering matrices. But, as noted at the end of 0 ILC, we can formally define a BODY S matrix only if we

72

Michael A . Morrison

neglect terms involving #$’ in the coupled equations (41). In this “fixednuclear-orientation approximation” ($ III.A), the BODY radial function becomes FNou~,uofo(r), which satisfies the boundary conditions

If we replace the BODY radial function in (54) with FNou~f,uo,o(r) and let the radius of the RFT rlr)-+ co,we find that this equation directly transforms the S matrix, Lsijf,uojolo

Jf FNO A

=

AjA

Suf,uolo

A!%.

(56)

A

In neglecting 2;)in the coupled equations (41)? we have implicitly assumed that the energies in the entrance and exit channels are independent of the rotational quantum number j, i.e., ‘ k 2u j

-

m l k,, 2.

(degeneracy of rotational target states)

(57)

This assumption is also implicit in the asymptotic RFT, for Eq. (56) cannot reinstate the dependence of the channel energies on j. This inherent limitation in methods that derive from the fixed-nuclear-orientation approximation is primarily responsible for their demise at near-threshold energies. (See 5 1II.E.) The assumption (57) is not, however, brought into play if we apply the RFT to FNOuul,uofo(r) A at some finite r, propagate Lu~jI.uojolo(r’) into the asymptotic region, and there extract L T i j fu,o j o l o . 4. The Vibrationol Frame Transformation ( V F T )

Thus far we have used the frame transformation strategy to treat the rotational motion, working in a basis that includes the vibrational Hamiltonian. This idea can usefully be extended to the vibrational degree(s) of freedom of the molecule by using the expansion (46) of Bu:;PfoAo(r,R)in vibrational states. Projecting out of Eq. (46) a particular vibrational function cp!,“)(R),we obtain a canonical transformation (see $ 52 of Davydov, 1965) from the continuous variable R to the discrete index u, R

J %A,

u ~ f o A o (= ~)

IOrn

q?)*(R) BUi?~foAo(r’ R)R2 dR.

(58)

In principle, this transformation could be used to transform the solutions of Eq. (44)at some radius rp) into values of Btd~ll\,unlol\o(r) that we could use to initialize the integration of Eq. (41) for r > ri”).In practice, however, Eq. (58)

73

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING TABLE IV SAMPLE APPROXIMATION? A N D RFPRFSFNTATIONS F O R FRAMF TRANSFORMATION THFORI€S Region

Representation

Near target ( r < r , ) Far from target ( r > r , )

BODY LAB-CAM

Approximation Neglect lu.;' and/or X Neglect $ e x

c)

is further simplified by implementing an approximate Born-Oppenheimer factorization of Bu:;p,oAo(r, R) [analogous to Eq. (191 to yield an approximate transformation of the BF-FN radial function FNut,o(r;R). (This approximation will be discussed further in gII1.C.)

Having performed the VFT, we can apply the RFT to enter the LAB-CAM representation, where we march forward into the asymptotic region.' Introduction of the BODY representation and the attendant frame transformations would be merely a further complication of an already complicated theory but for the fact that in various regions of space, approximations can be made that greatly simplify the scattering equations. These approximations, which are summarized in Table IV, amount to neglect of various terms in the Hamiltonian near to or far from the target. They can intuitively be understood in terms of the frame transformations discussed in this section. (Introductions to this perspective on electron-molecule collisions have been provided by its originator, Fano, 1970a, and by Morrison, 1983.) In fact the basic idea underlying the approximation methods in Q I11 is to use in each region of space a formulation that most closely reflects the dominant influences on the scattering function. For our purposes, the BODY representation offers a gateway to understanding these methods (4 111). Moreover, the BODY formulation is essential to understanding why these methods come to grief near threshold (Q 1II.E) and how theorists have tried to repair the damage. Before turning to applications of the theory (as approximated in Table IV), we note that Greene and Jungen (1985b) have developed the VFT beyond

'

" V o Ky Lan et al. (1979) have argued that the accuracy of this procedure would be improved by retaining the diagonal matrix element of .W;' in the scattering equations for this function. (Compare Eqs. (41) and (63) to follow.)

74

Michael A . Morrison

these approximations, allowing the nuclear motion to occur (in a BornOppenheimer state) in the inner region. (See § V.C.3.)

5. Applications of Frame Transformation Theory (with Approximations) Frame transformations are an invaluable conceptual tool. But the few existing applications of this procedure bring into question their practical utility. At the heart of an implementation of the RFT is the choice of the transformation radius rjr) (or, for a VFT, r?). The key question is: how should one choose these radii? In practice, of course, one should not have to rely on comparing approximate cross sections with those from an exact (LAB-CAM) treatment; the availability of the latter obviates the need for the former. Indeed, use of frame transformations in day-to-day scattering calculations seems feasible only if there exists a wide, energy-independent band of radii at which the transformation leads to accurate cross sections. In any case, strong sensitivity of the approximate cross sections to the transformation radii poses a serious problem. The first true frame-transformation calculations were those of Chandra (1977) on rotational excitation of C O (see sIII.E.1 of Lane, 1980) and of Weatherford and Henry (1977) on ro-vibrational excitation of H,. There was in Chandra’s results the suggestion of an unexpected sensitivity of the RFT cross sections to r?, but the clearest evidence of such sensitivity was discovered in the most recent study of this method, that of Jerjian and Henry (1985). These authors investigated the accuracy of the ro-vibrational frame transformation (59), in which the BF-FN radial functions FNu&o(r; R ) are evaluated at E,, and of an “energy-modified frame transformation” that follows from ideas proposed by Nesbet (1979) (gV.C.1). In the latter procedure, the scattering matrix was assembled from blocks, each of which corresponds to a pair of vibrational quantum numbers u,, and u. Jerjian and Henry calculated each block by frame transforming the BF-FN radial function calculated at the energy ( k v o k J 1 l 2 ,making the further approximation of neglect of the separation of rotational energy levels. This variant, the “numerical inefficiency” of which is noted by the authors, was a physically motivated attempt to extend the range of validity of the frame-transformation procedure to near-threshold collisions. The need for some such fix-up rests in Jerjian and Henry’s use of Eq. (59), which incorporates the F N approximation in the inner region, rather than Eq. (58), which is exact. (But some such stratagem is essential; use of the latter equation would be at least as complicated as setting r r ) = I:”) = 0 and solving the problem entirely in the LAB-CAM formulation.) For the e-H, system, Jerjian and Henry used a model potential based on a free-electron-gas exchange potential (Hara, 1967) and the nonadiabatic

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

75

polarization potential of Lane and Henry (1968), and they solved the BF-FN and LAB-CAM coupled equations via a noniterative integral-equations method, obtaining cross sections o , , , ~ ~ ,of ~ -high , , . ~ numerical precision. Their results are both surprising and disconcerting. Examination of the dominant contributor to the ro-vibrational cross section gO1+ (see Eqs. (84) below) as a function of the transformation radius r, z rir)= T i v ) revealed anomalous oscillatory structure for r, < 4.0 a , and the inability of the frame transformation to produce cross sections that agree with those of the LAB-CAM calculations. For example, if k , was used in Eq. (59) at E , = 1.5eV, no value of r, > 0 gave the correct cross section; if k, = JkT,k,,was used, only two discrete radii gave agreement of approximate and LAB-CAM cross sections. In practice, the trouble with the frame transformation in this implementation seems to arise from the fact that the energy at which the BODY radial functions are calculated is ill-defined; no choice can fully remove the resulting ambiguity in matching the BF-FN and LAB-CAM radial functions at the transformation radius. (Several crafty ways around this problem will be discussed in 4 V.) If the findings of Jerjian and Henry on e-H, hold for other systems, then use of this procedure (in the context of the approximations of Table IV) will require determining r , at each energy by matching approximate cross sections to their exact LAB-CAM counterparts-not a practically feasible method.

111. Approximate Collision Theories and Their Woes near Threshold As noted at the end of §11, one can devise collision theories that are computationally and conceptually simpler than the exact LAB and BODY treatments by making physically motivated approximations in these representations. The approximations to the radial frame transformations in Table IV, for example, are predicated on the dominance of q,, near the target and of if:) in the far region. In this section we are interested in even simpler approximations: a class of methods based on the adiabatic-nuclei (AN) approximation, which in essence asserts that the scattering function responds adiabatically to changes in the nuclear geometry. (See 9: IV of Golden et al., 1971 and 4 1I.D for references.) Adiabatic-nuclei methods have been reviewed extensively by others in recent years, so we shall proceed quickly to their key equations, focussing on the relationship of AN methods to the LAB-CAM and BODY theories and emphasizing features that are particularly relevant to their precipitous breakdown near threshold.

76

Michael A . Morrison

A. THE FIXED NUCLEAR-ORIENTATION APPROXIMATION AND THE BFVCC METHOD In 8 II.C.5, we mentioned the fixed-nuclear-orientation (FNO) approximation, in which 2;)is neglected in the electron-molecule Hamiltonian (3) (cf. Burke and SinFaiLam, 1970). In this approximation, the BODY scattering state no longer depends on J or M , and it satisfies the Schrodinger equation

[T,

+ 2:)+ V,, + T,- El IE,

A

~010;

+ ) = 0.

(60)

The corresponding wave function FNoyl~,volo(r, R ) does not depend on I?, because in effect the FNO approximation “freezes” the orientation of the molecule for the duration of the collision. To derive radial scattering equations in the F N O approximation, we expand the scattering ket of Eq. (60) in a basis of eigenkets of {A?:),P,I,}. (See Table V.) Or, more simply, we merely drop terms including 2;) from the BODY coupled equations (41), ~ b t a i n i n g ’ ~

Equations (61) constitute the hody-jirame vibrational close-coupling (BFVCC) method. They accurately account for the dynamic interaction of the motion of the projectile and the vibrational motion of the nuclei-but not the rotational motion, because in the F N O approximation the scattering electron responds to changes in the orientation of the nuclei adiabatically. To calculate ro-vibrational cross sections from a BFVCC T matrix, one must apply the RFT to that matrix. (See Eq. (56).) This transformation can also be viewed as an application of the adiabatic-nuclear-rotation (ANR) approximation (5 1II.B). From the radial equations (61) and the boundary conditions (55), we see that the BFVCC T matrix can be expressed as (compare the LAB-CAM equation (35b))

The BFVCC method was originally proposed by Chandra and Temkin (1976a, 1976b, 1976c) as a “hybrid theory” and was applied to resonant e-N, collisions. (See also Choi and Poe, 1977a, 1977b, and Schneider 1976.) It has l3

In this chapter, I have neglected the dependence of I u ) on the rotation quantum number j .

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

77

TABLE V

APPROXIMATIONS TO Approximation

Representation

THE

BODY

REPRESENTATION

Ket

Basis

Method

also been used to calculate near-threshold e-H, cross sections by Morrison et al. (1984b). Expressions for diverse cross sections in terms of the BFVCC T matrix can be found in Lane (1980) and references therein. As noted in Table 111, the channel energies in the BFVCC method are f k f = E - E ” . i.e., this method tacitly neglects the separation of the rotational sublevels of each target vibrational manifold (Eq. (57)). This assumption is not valid for rotational excitation near threshold (tj 1II.E). B. THEFIXED-NUCLEI APPROXIMATION A N D THE ADIABATIC-NUCLEAR-ROTATION METHOD One step beyond the FNO approximation is neglect of the vibrational Hamiltonian. This is the ,fixed-nuclei ( F N ) approximation, to date the most widely used of the scattering theories I have discussed. In methods based on this approximation, such as the adiabatic-nuclear-rotation (ANR) method, the entire nuclear geometry is “frozen,” and the nuclear motion is treated (if at all) adiabatically (Chase, 1956; Temkin and Vasavada, 1967; Hara, 1969; Henry and Chang, 1972). In the FN approximation, the scattering ket of Eq. (60) simplifies further to IE,, I,; A + ) , where we have replaced E by the “body energy” E , = k,2/2, about which more will be said. The F N approximation to the BODY representation is physically equivalent to the rigid rotator (RR) approximation to the LAB-CAM theory of 9 II.B.3: in either approximation, the internuclear separation is fixed at its equilibrium value R e . The difference between the two formulations is that in a LAB-CAM RR study, the effects of the rotational dynamics on the scattering function are treated exactly,14 but in studies based on the R R approximation in the BODY frame, these dynamic effects are treated adiabatically. l 4 The R R approximation considerably simplifies one’s computational chores, for it requires calculation of the interaction potential and solution of the scattering equations only at a single internuclear geometry. One does, however, pay a price; for example, the RR approximation introduces error in low-energy elastic and rotational-excitation e-H cross sections of roughly 10% (Morrison and Saha, 1986). Moreover, because the RR static potential is singular at the nuclei, the number of partial waves required to converge the scattering calculation may be artificially large.

78

Michael A . Morrison

The body-frame fixed-nuclei (BF-FN) approximation can be considered a composite of the FNO and R R approximations in the BODY representation. Of course, one learns nothing about vibrational excitation from a BF-FN calculation; but (§ 1II.D) one can calculate approximate cross sections for this scattering process by applying the VFT asymptotically to F N transition matrices. Expansion of the BF-FN scattering ket JE,, I,; A ) in the appropriate basis, {IIA)} (Table V) leads to the (comparatively) simple coupled radial equations

+

+ C [Vtl,(r; R e ) + v & ( r ; Re)] FNU;?,lo(r;Re) = 0.

(63)

I’

The BF-FN T matrix FNTtl,(Re) = ( E , , 1; A1 P I E b , 1,; A) that we extract from the asymptotic behavior of FNuj’lo(r;Re) can be expressed as an integral,

(64) This transition matrix is evaluated at the body energy E,, the energy at which we solve Eqs. (63). In BF-FN theory, all dependence on target-state quantum numbers has vanished from the coupled equations, the radial functions, and hence the T matrix. There results an ambiguity in defining the body energy: should E , be taken equal to the projectile’s energy in the entrance (Chase, 1965) or exit (Chang and Temkin, 1970) channels, or to some other value, such as the geometric mean of these two possibilities. (See 5 V.D.)’’ This problem plagues applications of A N methods to near-threshold collisions, but can be tackled in ways to be described in § V. We can calculate F N approximations to total differential and integral cross sections (for fixed R ) directly from the BF-FN T matrix, e.g.,

As in the BFVCC method ($III.A), we obtain approximate (RR) rotational-excitation cross sections from the BF-FN T matrix via the RFT, i.e.,I6 yil,jolo(Re) = C

FNTftlo(Re)A;2.

(66)

A

l 5 Bottcher (1969) and Nesbet (1979) have considered the formal relationship between the FN and LAB formulations of electron-moleculecollision theory. See 5 V.C. 1. l6 I shall use script letters to denote AN approximations to various LAB-CAM T matrices.

NEAR-THRESHOLD ELECTRON-MOLECIJLE SCATTERING

79

In this result, we see the RFT carrying out the transformations listed at the beginning of 4 1I.D: rotating the BODY ket of the electron into the corresponding ket in the LAB frame, and transforming from the LABUCAM to the LAB-CAM representation. The approximate LAB-CAM T matrix (66) is the central quantity of the adiabatic-nuclear-rotation (ANR) method. Once obtained, this matrix can be substituted into standard expressions for integral or differential cross sections to calculate ANR approximates to these quantities, such as aYJj(Lane, 1980; Shimamura, 1984). The sum of these approximate cross sections, for a particular j,, is equal to the total integral cross section of Eq. (65), i.e., (Chang and Temkin, 1969)

From formal considerations, one can show that in the ANR approximation, this sum is independent of the initial statej,. Morrison et al. (1984a) have investigated the extent to which the independence of the total integral cross sections on j, hinges on the AN approximation, finding it to hold to an excellent approximation LAB-CAM and experimental cross sections. (See also Lane and Geltman, 1967.) One need not base an implementation of the FN approximation on the orbital angular momentum basis { I l A ) ) . Clark (1979, 1984) has proposed that for scattering from a polar molecule with dipole moment d(R), one should use a “dipole-adapted basis.” (cf. 5 2 of Fabrikant, 1979.) Arguing that the S matrix is most meaningfully defined relative to a state of the electronmolecule system in the absence of all interactions other than the dipole, Clark proposes that the BODY wave function in the FN approximation PNY‘,b(r; R e ) be expanded in a complete set defined by I , and the “effective angular momentum operator,” which (in atomic units) is defined as 1’ - d ( R ) cos 0. The resulting S matrix can easily be transformed back to a basis of spherical harmonics for application of frame transformations and calculation of cross sections. Clark (1984) avers that this formulation will greatly simplify the scattering problem- by reducing the range and strength of channel coupling-and that in this representation “ a body-frame electronpolar-molecule collision is intrinsically no more difficult to describe than a nonpolar case.” C. THEBORN-OPPENHEIMER PERSPECTIVE A N D THE ADIABATIC-NUCLEAR-VIBRATIONAL METHOD The derivation in 9: 1II.B of the ANR T matrix (66) via frame transformation is quick and easy, but it may obscure the physical essence of the

80

Michael A . Morrison

approximation. This dimension of A N methods becomes clear when they are considered as extensions to continuum states of the bulwark of molecular structure theory, the Born-Oppenheimer approximation (Shugard and Hazi, 1975). Formally, the A N R method amounts to assuming that the scattering operator commutes with the angular coordinates k Thus, two approximations are implicit in this method." 1. The rotational dynamics of the nuclei can be separated from those of the projectile. , 2. The energy spacing of the target rotational states, cj - E ~ is~negligible compared to the scattering energy, i.e., k3/2 x ki/2. (Compare to Eq. (57))

In the language of Born-Oppenheimer theory, these approximations are written as a factorization of the BODY scattering ket (for a rigid rotator) in the entrance and exit channels,

I E, I0

JM +)

IE, 1Ao; J M + )

IEb,

IO; A O

% IEb,

+ ) I JMAO),

1; A o + ) I J M A o ) .

(68a) (68b)

The scattering kets on the right-hand sides of Eqs. (68) are those of BF-FN theory evaluated at the body energy; in replacing E with Eb in the initial and final states, we have invoked the assumption of target-state degeneracy in addition to Born-Oppenheimer separability of these factorizations. Because A is a collision constant in the F N approximation, we have set A = A, in (68b). (Lane, 1980, shows in 4 1I.E how to proceed from Eqs. (68) to the ANR T matrix (66).) Such factorizations can be applied to the full scattering ket IE, uoloAo; J M ) (by further separating the vibrational state 1 u o ) ) to derive the full AN T matrix, or to the BFVCC ket 1 E, oolo; A +) to derive the A N approximation for vibrational excitation. The latter gambit is clearer, for it lets us ignore (for the moment) the rotational motion. For example, the Born-Oppenheimer factorization of the BFVCC ket of 4 II1.A for the entrance channel is

+

IE, uo1o; A + > x IE,,

10;

A+)luo).

(69)

To derive the resulting approximation to the BFVCC T matrix, we merely introduce factorizations of the asymptotic free states in the definition of this quantity, viz., FNOTA v~,voio=

( E , 01; AI TIE, u o b ; A> % ( ~ P ~ " I ~ ~ T ~ ~ , I ( (70) P~)).

I' Ficocelli Varracchio (1979a, 1981) has probed the theoretical basis of the AN approximation using field theoretic techniques (see 5 V.D.1) and has applied this formulation to heavyparticle scattering (Ficocelli Varracchio and Celiberto, 1982).

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERlNG

81

We have again invoked target-state degeneracy by setting the energies in the entrance and exit channels equal (to Eb). Equation (70) is the central result of the adiabatic nuclear vibration (ANV) method (Faisal and Temkin, 1972; Temkin and Sullivan, 1974). When Eq. (70) is combined with the ANR approximation-via an asymptotic RFT (# 1II.B)-there results the full AN approximation to the LABCAM T matrix, 0-J J

C A;,!, (cP~. I

1;N

v)

ujl.uojolo

1

(71)

Ttl,)Iq:;))A;:A.

A

In considering near-threshold ro-vibrational excitation, we must keep in mind that the full AN method tacitly assumes all target states to be degenerate; i.e., it does not take into account the energy spacing c U j - c , , ~ ~ ~ , assuming instead that

+kij z f k i . (target-state degeneracy) (72) As we shall see in # III.E, this assumption is far more drastic than that of rotational target-state degeneracy (Eq. (57)), because the thresholds for vibrational excitation are considerably larger than those for rotational excitation. (Cf. Herzberg, 1950.) This inherent deficiency in the AN approximation for very low energy collisions has stimulated the development of approximate scattering theories that are intermediate between the LFCC and AN methods-by, for example, calculating an approximate T matrix off the energy shell. We'll look at some of these methods in 9 V.D. D. THEADIABATIC-NUCLEI SCATTERING AMPLITUDE The BF-FN T matrix FNT" ( R ) , like its LAB-CAM counterpart, is related via partial-wave expansions to a scattering amplitude. This amplitude, F N f ( k , ic k,,i,,; R ) , appears in the asymptotic form of the BF-FN wave R ) , which is just the configuration-space projection of the function (plane-wave) scattering state I k, + ). Because the Born-Oppenheimer factorizations of # 1II.C have eliminated from the scattering equations all reference to target coordinates and quantum numbers, this wave function depends only on r, and its boundary conditions have the refreshingly simple form

-

eikbr

1

'"Ykb(r; R ) r - m (2n)-3/2[eikh'r+ r F N f ( k , it k,&; R ) . ~~

(73)

To see how the BF-FN scattering amplitude and T matrix elements are related, we use the BF counterpart of the transformation (1 2) between planewave and angular-momentum eigenkets to derive

82

Michael A . Morrison

Less transparent is the relationship between the BF-FN scattering amplitude of Eq. (74) and the LAB amplitude in Eq. (10). This relationship is the heart of the AN approximation (Chase, 1956). We first rotate FNf(kbi.c kbkO;R ) into the LAB frame using the rotation operator of Eqs. (47):

FNf(kbi'+ kbko; R ) = -

(24* ~

kb

11 1 @ , ~ c p ~ ,w y v )

FNTtl,(R)Y;loo*(lb)~~oA(cp.,

6.1.

A lm lorno

(75) We next follow the prescriptions of the asymptotic frame transformations, sandwiching the amplitude (75) between target states (vl and Iv,) to obtain the elegant and simple approximation

"f(kv, V + ko,

VO) Z ( V l

FNf(kbi'+ kbko; R)IV,).

(76a)

We can use Eq. (50) to write this fundamental relationship in terms of the BF-FN T matrix elements, viz., L AN

f (kv,v + kb, v,)

= -

(2.)2 k, ~

11 1 Y;n(i')C(jlJ;rnjrnM)AiJ,:

J A Im lorno

x (cp!,")( FNT&oI c p ~ ) ) A ~ & Cj,( 1,J;

rnjorn, M)Y?*(k,). (76b)

(Equation (76b) is precisely what we get if we apply the asymptotic RFT and VFT directly to FNTt,,(R),uncouple j and i in the resulting expression, and substitute the result into Eq. (18))

E. BREAKDOWN OF ADIABATIC-NUCLEUS NEAR THRESHOLD

It comes as no surprise that the AN approximation, which presumes the scattering function to be insensitive to the nuclear dynamics and the initial and final target states to be degenerate, fails near threshold. A "slow" electron (with a small incident energy E , = k6/2) sees not a frozen nuclear geometry but a dynamic system; and the perturbation of its scattering function by X',")may well be nonadiabatic. Moreover the projectile energy in the exit channel of a near-threshold inelastic collision is very small; yet, the assumption of target-state degeneracy (Eq. (72)) sets this energy equal to that of the entrance channel. Thus it is not the incident energy so much as the final-state energy that determines how badly AN methods break down.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

83

The Born-Oppenheimer energies of a diatomic molecule in the ground electronic state a,, relative to the electronic energy Eg)(R), can be simply expressed using the simple-harmonic-oscillator model for vibrations and the rigid-rotator model for rotations. The resulting expression includes two molecular constants: the fundamental vibrational frequency wmOand the rotational constant B,, (Radzig and Smirnov, 1985), i.e., E"j

* E,) +

Ej

= co,,(u

+ 4) + Ba"j(j + 1).

(77)

The quantum numbers u and j each increase in integral steps from 0 (for a molecule with a C ground electronic state). The separation between initial and final target states is just Acv,,,,,= ~ , , j ~,,j,.

(78)

So the limit of near-threshold scattering can be expressed as

$ k i -+ Ac,~,,. (79) Typical threshold energies for selected excitations of selected molecules appear in Table VI. In the next section, we'll discover that the AN T matrix elements of Eq. (71) do not obey the correct threshold laws in the limit (79); indeed, AN cross sections calculated according to the original methodology of Chase ( 1 956), Temkin and collaborators, and Hara (1979) d o not go to zero at threshold. This problem is a consequence of unavoidable ambiguity in the definition of the body energy E , in Eq. (63). The prescription in Chase's derivation is E, = E , , which leads to inelastic AN cross sections that are nonzero at threshold. Various fix-ups have been proposed. The most widely used of these is the suggestion of Chang and Temkin (1970) that all AN cross sections be multiplied by the ratio of the final- and initial-state wave numbers, k,/k,. This TABLE VI

THRESHOLD ENERGIES OF. SELECTED MOLECULAR EXCITATIONS Molecule

Threshold energy (meV)

Excitation

H2

00 + I ?

=

=

02

01 - 0 3 00- 10 00- 12

HF HCI HBr CO,

L',) =

73,s 515.6 557.5

0 + I' = 1

1'0 -- o - r l ' = l

1'0 -

l',)

=

o+I~=l

000 --t 1'

=

44.1

100

49 1

358 317 I72

84

Michael A . Morrison

ad hoc gambit does indeed force inelastic AN cross sections to go to zero in the limit (79). But the AN T matrix elements still do not exhibit the proper dependence on k , in this limit. Alternatives to the choice E , = E,, such as E , = E , (Chang and Temkin, 1970) and E , = (Nesbet, 1979; Norcross and Padial, 1982) give zero cross sections at threshold and improve somewhat the approach to zero, but still do not give T matrix elements that obey the correct threshold laws. The breakdown of the AN approximation was discussed as early as 1970 by Chang and Temkin, who, in a discussion of the ANR approximation for e-H, scattering (in the RR approximation), argued that this method would be valid if

a

E, > 1.65.

(80)

'Eiio

This useful qualitative criterion left open the question of how accurate the method was when this condition was (or was not) satisfied. This question remained unanswered until 1984, when quantitative assessments of the validity of the ANR and ANV approximations first appeared. By carrying out analogous LFCC and AN calculations, using identical (model) interaction potentials and identical, stringent numerical criteria (cross sections converged to better than 1 %), Morrison et al. (1984a, 1984b) determined the percent error introduced into calculations of various excitation cross sections o,,,, of H, by the AN approximation. These percentages are illustrated for selected excitations in Fig. 5. This analysis shows that the AN approximation for rotation introduces serious but not devastating errors near threshold. For example, the criterion (80) predicts validity of the ANR approximation for the pure rotational excitation 0 4 2 above E , = 73 meV and for 1 + 3 above 120 meV. The data of Morrison et al. (1984a) show that the percent error in the ANR cross section for 0 + 2 at 73 meV is 17 % and for 1 --t 3 at 120 meV is 32 %." Other dimensions of the breakdown of the ANR approximation for e-H, have been studied by Ficocelli Varracchio and Lamanna (1983, 1984), whose work will be discussed in 9 V.D.I. As Fig. 5 attests, the AN approximation for e-H, scattering fails most spectacularly for vibrational excitation; ANV integrated cross sections for this process are in error by hundreds of percent even at energies equal to several times threshold. Such excitations therefore provide a useful environment to consider which of the assumptions of these methods is responsible for their calamitous demise near threshold. This is a subtle question, and the The ANR cross sections used to calculate these percentages have been multiplied by the ratio k,/k,. Had this not been done, the percent errors would be far larger.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

+

30

0 0 .>

0 -

o‘-oo;

I’

800 0 .-

.

I 1

of 20

2

/

II

I I

-600

I I

(L

IT

-400 10

W

c3

(3

a Iz

-200

W W

-

(L

w o

a

k

0

rx W

U 0 (L

0 U W

85

a F z W W

(L

W

-

ENERGY ( e V ) FIG.5. Percent error in pure rotational (00 02: solid curve) and ro-vibrational (00 --+ 12: dashed curve) e-H, cross sections calculated using the adiabatic-nuclei (AN) approximation. These percentages were calculated using cross sections from AN and lab-frame close-coupling studies based on identical interaction potentials. (See Morrison et al.. 1984a, 1984b.)

answer, insofar as it is known, will emerge only later in this chapter, when we examine applications of scattering methods that relax one or more of these assumptions (9: V). But for a preview, let’s consider the strange case of vibrational excitation of CO,. This molecule is distinguished by being one of the few nonpolar targets whose vibrational-excitation cross section-for the symmetric-stretch mode (100)-exhibits a pronounced spike near threshold (See Fig. 2).19 Structure in c~~~~~~~~ was first predicted by Morrison and Lane (1979), whose ANV cross sections for this excitation revealed a sharp rise as E , -+ 163 meV, and was subsequently seen in the crossed-beam measurements of Kochem et al. (1985b). But in this energy range the accuracy of the ANV cross sections is subject to uncertainty, because the ANV approximation breaks down as the energy approaches the region of the rise. Subsequently, Whitten and Lane (1982) performed e-CO, calculations that account for the nuclear dynamics nonadiabatically, applying two-state BFVCC theory using a model interaction potential. These calculations were extended to four vibrational states by Kimura and Lane (1986), whose l 9 Cadei et al. (1977) and Cvejanovit et al. (1985) have reported threshold spectra that exhibit strong excitation of very high vibrational levels in CO,. See also Andric et al. (1983).

86

0.10

0.15

0.20

0.25

0.30

0.35

0.40

ENERGY ( e V ) FIG. 6. Cross section for excitation of the symmetric-stretch vibrational mode (100) of CO,: experimental data (Kochem et al., 198517); four state body-frame vibrational close-coupling (BFVCC) (solid curve; Kimura and Lane, 1986); BFVCC with target-state degeneracy (dotted curve; Whitten and Lane, 1982); nonadiabatic resonance theory (dashed curve; Estrada and Domcke, 1985). (The ANV calculations of Morrison and Lane, 1979, give results indistinguishable from those of BFVCC with target-state degeneracy.)

BFVCC cross sections are shown in Fig. 6. The solid and dashed curves in this figure are BFVCC results, so both incorporate the effects of the nuclear Hamiltonian on the scattering function, effects that could not influence the ANV results. But in calculating the results of the dotted curve, the initial and final vibrational states were assumed to be degenerate ( A E , , ,=~ 0), while for the solid curve they were not. This figure shows that it is the assumption of target-state degeneracy, not solely neglect of the influence of ):&‘ on the scattering function, that is responsible for the failure of the ANV approximation to reproduce the spike seen in the measured cross sections. This finding is understandable in view of the conjecture (Morrison, 1982; Estrada and Domcke, 1985) that the structure in c~~~~~~~~ is due to a virtual state in the fixed-R e-CO, interaction potential (see 4 VLC), for a virtual state would strongly perturb the final-state wave function of the electron, which would then be acutely sensitive to the exit-channel energy. A similar situation obtains in scattering from the polar molecule HF, whose vibrational-excitation cross sections also exhibit a striking nearthreshold spike. (See Fig. 7.) Gauyacq (1 983), in his “effective-range approximation” (see gVI.B.3), makes the ANR assumption (which, because the

87

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

a

b

0.20

1

I

I

I

I

e-HF

0.15 -

v = 0-c 2

\

-

I

I

E

\

W

\

E

'0 - 0.10>

t 0

>

b 0.05 0

,ooy

'---- - I

I

----____ I

I

FIG.7. Cross sections for vibrational excitation of (a) the u = 1 and (b) the 1' = 2 states of HF. Experimental data: open circles (Rohr and Linder, 1976). Adiabatic-nuclei theory: longdashed curve (Gauyacq, 1983). Energy-modified-adiabatic approximation: dotted curve (Gauyacq, 1983). Nonadiabatic effective-range approximation: solid curve (Gauyacq, 1983). BFVCC: short-dashed curve (Rudge, 1980).

88

Michael A . Morrison

rotational constant B,, of H F is quite small, should be excellent) but treats coupling of the vibrational and electronic motion nonadiabatically in the region of the target. Comparison of his nonadiabatic and ANV cross sections (Fig. 7) vividly demonstrates the inability of the ANV approximation to represent the physics of these near-threshold excitations: no hint of the peak in the data of Rohr and Linder (1976) is seen in the ANV uo = 0 + u = 1 cross section, and the values of this cross section are seriously in error even far above threshold ( A E ~ ,=, 491 meV). It is important to close this discussion of AN methods by noting their success at higher energies. Indeed, the enormous computational simplifications that ensue from these approximations have opened up the study of rotational and vibrational excitation of molecules, allowing the investigation of systems as diverse as CH,, H,O, and NH,. (Cf. Jain and Thompson, 1983a, 1983b; see also references in Gianturco and Jain, 1986.) Consequently the temptation to use these methods near threshold, where massive coupling of partial waves and target states prohibits full LAB-CAM or BFVCC calculations, is almost irresistible. But the studies examined in this section strongly argue that in this energy regime, AN methods should be used with care in studies of rotation and perhaps not at all in studies of vibration. Alternative theories, simpler than LAB-CAM but more accurate than ANV, are clearly required (9 V).

IV. The Behavior of Near-Threshold Cross Sections- Explained The behavior of near-threshold cross sections can be qualitatively explained by general, very simple laws (9 1V.A). And more quantitative information concerning scattering quantities near theshold can be obtained from analytical expansions of these quantities in powers of k,, such as effective-range theories (5 1V.B). These laws and expansions are very useful to practicing experimentalists and theorists, and several are gathered together in this section.

A. QUALITATIVE ANSWERS:THRESHOLD LAWS Most of these threshold laws can be derived by applying to integral equations for the radial functions (such as (35b), (62), and (64)) the First Born approximation. Thus, in the LAB-CAM formulation, we can deduce the dependence of LTij,,uojolo on k, by replacing Lu~zj,,,,oojolo(r’) in Eq. (35b) by the

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

89

radial function for a free wave in the entrance channel. (See Morrison et al., 1984a, and references therein.)

5 1I.C of

1. Potentiul Scattering Many threshold laws for electron-molecule scattering are similar to those for single-channel scattering from a spherically symmetric potential V ( r ) . Potential scattering is usually discussed in terms of partial waves (see Burke, 1977), and one can easily derive threshold laws for partial-wave scattering amplitudes, phase shifts, and thence for various cross sections as the energy E approaches the sole threshold, E = 0. (See Table VII.) These threshold laws may be altered by the presence in V ( r )of a long-range tail: V(r)+ r - 8 ,

r + m.

(81)

Further complications arise if a pole of the (single-channel) S matrix lurks in the complex energy plane near E = 0; such a pole, if near the real axis, can drastically alter the behavior of the phase shift and cross section. (See 5 IV.A.4.)

2. Electron Scattering ,from a Nonpolar Molecule Inelastic cross sections for scattering from a target with internal structure may exhibit interesting behavior at several thresholds-due, for example, to branch points in the multichannel S matrix (Nesbet, 1975). Moreover, longrange tails in the potential and near-threshold poles of the S matrix may strongly influence the behavior of near-threshold cross sections. In general, the asymptotic form of the electron-molecule interaction I/", of Eq. (7) is a multipole expansion in which appear such molecular properties as TABLE VII THRESHOLD LAWSFOR SINGLE-CHANNEL FROM A SPHERICALLY-SYMMETRIC SCATTERING POTENTIAL

Potential V ( r ) Finite-range Exponentially bounded r-' for I < +(s - 3)

./I

- k2'

$,(mod r )

-

k2lfl

Michael A . Morrison

90

the permanent dipole and quadrupole moments, d(R) and q(R), and the spherical and nonspherical polarizabilities, ao(R)and a,(R), i.e.,

+ higher order terms.

(82)

As usual, the r-, dipole interaction causes special problems, which we’ll avoid (until 0 IV.A.3). The long-range terms (82) do not invariably affect threshold laws. In an investigation of electron-atom scattering, Bardsley and Nesbet (1973) proved that the threshold laws for inelastic transition matrix elements are unaltered by the presence of the long-range quadrupole and polarization potentials; e.g., in the LAB-CAM theory (Morrison et al., 1984a)

just as though Vn,was of finite range.” To determine the threshold behavior of the corresponding cross section ovo-Lv, we must consider the decomposition of this quantity into contributions from various total angular momenta J and pairs of (coupled) partial waves (I, Lo). Since 2 is a collision constant, this cross section can be written m

The orbital angular momentum 1 is not conserved, so partial waves are coupled in each term in Eq. (84a), IT

IJ+jol

diojo+uj= kZj(2jo + 1) l o = I1 J-jol

l=IJ+jl

1

(25 + 1)I LTijI,uojolo12* (84b)

l=IJ-jl

To illustrate the use of Eqs. (83) and (84), let’s consider pure rotational excitation j , = 0 -+j = 2 of H, (Lane and Geltman, 1967; Morrison et al., 1984a). As kj = approaches zero, the J = 1 term dominates Eq. (84a). This term, in turn, is dominated by the ( I = 0, I , = 2) (i.e., “ d -+ s”) term in Eq. (84b). Consequently, scattering in the exit channel is predominantly s-wave (1 = 0), and Eq. (83) implies that goes to zero at threshold like kj. This conforms to the prediction of the quadrupole Born approxima-

Jm

2oThis threshold law can be violated if two degenerate channels are coupled by the quadrupole interaction (Fabrikant, 1974); but even if this occurs, the total cross section obeys Eq. (85).

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

91

tion (Gerjuoy and Stein 1955a, 1955b) and to experimental data. (See the analysis of Chang, 1970.) In general, then, inelastic electron-(nonpolar) molecule cross sections obey the threshold law

where 1 is the order of the dominant partial wave in the exit channel. In contrast, elastic elements of the T matrix-and hence elastic cross sections-are altered by the long-range potential (82). If we denote the dominant long-range power by sd, then (Bardsley and Nesbet, 1973; Morrison et al., 1984a)

For a nonpolar system at near zero energy, elastic scattering is usually controlled by the spherical polarizability term [ - a,/2r4] in Eq. (82). So sd = 4, and Eq. (86) predicts LTkOl+

VOjOlO

ko-0

i

ki k,

I, I,

+ 1 > 0 ;. =

I

=

0

(LAB-CAM)

(87)

Since a spherical interaction cannot couple channels with different partial waves, the sum in Eq. (84b) is dominated by 1, = I = 0, and we conclude (see Fig. 8)”

-

constant.

CT,,~+~~~ ko-0

The elastic threshold law (88) is particularly relevant to the approximation methods of 4 111. Buried in the AN T matrix (71) is the BF-FN matrix FNTA l * l o ( R )evaluated at the body energy E , . In the BF-FN formulation ( 5 III.B), all channels are degenerate, so all elements of this transition matrix obey a threshold law analogous to Eq. (87).

This behavior, in turn, controls approximate cross sections, such as C T ~ ; ! ~ , as the scattering energy approaches a threshold E,,, because the frame transformations Eq. (66) (RFT) and Eq. (71) (VFT) do not alter the energy dependence of the transition matrix. For example, if the conventional choice To discover how u,,~,,,, approaches its zero-energy limit and what that value is, we must turn to the more sophisticated expansions of S1V.B. (See Eq. (98a).)

Michael A . Morrison

92

vj = 00

-c

12

-

-

-

00-00

0.05

0.10

0.I5

0.20

-

0.25

W A V E N U M B E R (a.u.1 FIG.8. e-H, cross sections for elastic scattering and various excitations as a function of final-state wave number k,,,. (Data from calculations described in Morrison et al., 1984a, 1984b.)

k, = k , is made, then all elements of the ANV T matrix, ~ ~ j , , o a j o l oremain , finite at E , , resulting in a nonzero cross section C T ~ ~at! ,threshold. The alternative proposed by Chang and Temkin (1970), k, = k,, yields ANV T matrix elements that go to zero at threshold as k,, which is correct only for elastic elements. Another choice-one that has been widely used of late-derives from the energy-modified-adiabatic (EMA) approximation of Nesbet (1979) (4 V.C.l): set the body energy equal to the geometric mean of the projectile energies in the entrance and exit channels. With this choice, k, = the threshold law becomes

Jkvko,

which is correct for inelastic elements with I = 0. Since these elements dominate the inelastic cross section near threshold, Eq. (90) is the most likely of these alternatives to yield correct near-threshold cross sections. This choice, however, gives the wrong energy in the entrance channel. The threshold A E , , , ~is not the only one where ovo+vexhibits strange behavior. If the scattering energy is large enough, more than one channel is open and threshold structures can appear in various cross sections. One consequence of the unitarity of the multichannel S matrix is that each threshold is a branch point of this matrix (Goldberger and Watson, 1967). If s-waves dominate the final-state wave function for v, -,v, then the branch point at this energy may induce “threshold anomalies”-cusps or rounded

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

93

steps-in cross sections for other processes, such as lower-lying excitations and elastic scattering. This phenomena has been extensively studied for e-He scattering near the threshold for the 1’s 23S excitation (Nesbet, 1975, 1981). For an example from electron-molecule scattering, consider the threshold gZ (961 meV) for the vibrational excitation uo = 0 + u = 2 in e-HF scattering. The cross section for I I ~ )= 0 -+ u = 1 exhibits a rounded step at this energy (Rohr and Linder, 1976; Fabrikant 1977b, 1978). Similarly, g r , o = O + v = has a cusp at the threshold for the uo = 0 + u = 3 excitation. Gauyacq (1983) has attributed the structure in oo-.l at E~ to a nuclear-excited Feshbach resonance-a temporary state of the electron-target complex in which the latter is in excited state u2 and the energy of the former lies just below c Z . Such resonances were first proposed by Gauyacq and Herzenberg (1982), who showed that they can arise from motion of a virtual-state pole (of the BF-FN S matrix) near zero energy. This motion is a nonadiabatic effect of the coupling of the projectile to the nuclear vibrations. (See 0 V1.B.) Threshold anomalies caused by branch points in the S matrix may be very small and may occur over an extremely narrow energy range, making them all but unobservable. Moreover, the extent to which a branch point at the threshold for a new channel affects the cross section for a channel that is already open depends on the strength of the coupling of the channels in question. So not every electron-molecule cross section exhibits a threshold anomaly at every available threshold, nor do different systems exhibit the same structures. For example, B ~ for + e-HCI ~ (Rohr and Linder, 1975) does not show the cusp at the uo = 0 + 1) = 3 threshold that is seen in the corresponding cross section for e-HF scattering. Threshold laws for the sum of the eigenphases are extremely useful in electron-molecule scattering theory. The eigenphases yl: are the multichannel analogues of phase shifts in potential scattering theory. (See 4 10.3 in Newton, 1982.) Their sum is a compact, convenient way to represent the scattering information buried in the multichannel S matrix. In electron-molecule studies, the most often reported eigenphase sum is that of BF-FN theory” (Hazi, 1979). --f

Since in this formulation all channels are degenerate, the corresponding eigenphase sums St,,,( R ) , which depend on the electron-molecule symmetry 2 2 This quantity is labeled by the index i rather than (he partial-wave order I becatisc the coupling of partial waves characteristic of electron-molecule systems makes it impossible to uniquely associate an eigenphase with a partial wave. Near threshold, however, a single partialwave often dominates the larger eigenphases.

Michael A . Morrison

94

I Io-2

I

1

I

I 1 1 1 1 1 l

I

I I I Ill1

to-‘

I

I

I

I I I I I

10

EN E R GY (eV 1 FIG.9. Theoretical total cross sections in the fixed-nuclei approximation for electron scattering from F, (Morgan and Noble, 1984). H , (Morrison et al., 1984a), Li, (Padial, 1985), and CO, (Morrison et al., 1977).

and on R, are “elastic.” They can be determined from the corresponding S matrix FNSA(R)as e2ia&m(R)= det FNSA(R). (91b) Threshold laws for the eigenphase sum follow from the laws that govern the behavior of this S matrix as k b + O . For example, from Eq. (89) we conclude that the C, BF-FN eigenphase sum goes to zero as

and hence that the FN total cross section goes to a constant at zero energy. (See Eq. (102a).) This behavior is illustrated for several nonpolar molecules in Fig. 9.

3. Electron Scattering ,from u Polur Molecule The threshold behavior of scattering quantities for a polar system is quite different from those of a nonpolar system. An extreme example is the BF-FN eigenphase sum, which in the presence of a sufficiently strong long-range r - 2 tail diverges at zero energy (Norcross and Collins, 1982). This pathological behavior also infects the FN total cross section (for any nonzero dipole

95

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

moment d) and disappears only if molecular rotations are explicitly taken into account in the scattering formulation. Even if d is not large enough to induce such aberrant behavior in 6tum(R), the threshold dependence on energy of this quantity-and of various cross sections-is much more complicated than for a nonpolar molecule. (See, for example, Fabrikant 1983a, 1983b.) This problem has been elucidated by Clark (1984), who has shown that Sa,,(R) can be written as the sum of a term 6"(d), which depends only on the dipole moment and the symmetry of the state, and a term that incorporates effects due to the short-range interaction. As the body energy approaches zero, the latter term vanishes (modulo IL), leaving the simple result23

6tum(R)= 6"(d)

at E , = 0.

(93)

Additional long-range interactions (Eq. (82)) complicate the energy dependence of Sa,,,(R) near threshold but do not alter its zero-energy value (93). It would seem, then, that one could use tables of 6"(d), such as those provided by Clark (1984), to verify the results of elaborate ab initio BF-FN calculations for various polar systems. To do so, however, one must have eigenphase sums at energies near enough to zero for Eq. (93) to hold. This leads to a central question regarding threshold laws: how near to threshold must the scattering energy be for the law to be valid? To illustrate the problem, we shall call upon the 2 e-HCI eigenphase sums determined by Padial and Norcross (1987). In their BF-FN calculations, a near--Hartree-Fock target wave function was used to determine the static potential, exchange effects were treated exactly, and polarization was included by a model correlation-polarization potential (O'Connell and Lane, 1983). In Table VIII, their eigenphase sums for a very small body energy ( E b = 1.36 x 10-4eV) are compared to the predictions of Clark's analytic theory. Remarkably, even at this low an energy the ab initio eigenphase sums do not agree with the predicted values. 4. Threshold Behavior in the Presencr. oj a Complex Pole of the S Matrix

The effects on an inelastic cross section of a pole of the S matrix near the corresponding threshold can be stunning. Many have speculated, for example, that such a pole may be responsible for the threshold spikes observed in vibrational-excitation cross sections for several systems ($ VI). And the anomalously large e-CO, momentum-transfer cross section at zero energy (Fig. 2) appears to be due to a near-threshold pole ($ V1.C). 23 Clark (1984) notes that if d is large enough, Eq. (93) does not apply, in which case the eigenphase sum diverges as the logarithm of l/E, as E , approaches zero.

96

Michael A . Morrison TABLE VIII NEAR-ZERO z EIGENPHASE S U M S FOR e-HCI SCATTERING R

d(RY

(ao)

(ea,)

Calculated" E, = Ryd

Predictedb E,=O

2.2 2.4087 2.6 2.8 3.0

-0.439 -0.488 -0.535 -0.587 -0.640

0.24 0.37 1 .oo 0.10 0.31

0.18 0.24 0.29 0.39 0.69

Padial and Norcross (1986). *Clark (1984).

Poles of the S matrix come in three varieties, depending on their location in the complex k plane (Taylor, 1972). Letting k , denote the pole location, we have (see Fig. 10)

k,

=

&, !liy

resonance

k , = +iy kP

bound-state

= - .ly

virtual-state

where /land y are real. In a study of the theory of multichannel threshold structures in electronatom scattering, Nesbet (1977) showed how a simple s-wave pole near a Irn k

B O U N D STATE Re k

V I R T U A L STATE

+ RESONANCE

r

+ RESONANCE

FIG.10. Poles of the fixed-nuclei S matrix in the complex k plane: a bound-state pole (cross), a virtual-state pole (star), and a (double) resonance pole (plusses).

NEAR-THRESHOLD ELECTRON-MOLECIJLE SCATTERING

97

threshold E,, can induce a threshold peak in the inelastic cross section m , ( , + , . (See also Domcke, 1981.) As threshold is approached, this cross section obeys the law (94a) Hence for small 7 , the inelastic cross section rises sharply (as k , ) from zero at k , = 0, then falls just as abruptly (as l/k,). If the pole is near the elastic threshold, its effect is felt on ovo+vo via the slightly different threshold law

Equations (94) describe the behavior of the cross section if the multichannel S matrix has a pole near the threshold c,. In electron-molecule scattering, a more complicated situation may arise: the electron-molecule potential may support an s-wave pole of the BF-FN S matrix 'NSo(R) near k , = 0. If so, two consequences-one expected, the other perhaps not-result. First, in accordance with Eq. (94b), the FN total cross section approaches an anomalously large (but finite) value at zero energy. Second, this pole may also influence one or more inelastic cross sections near their thresholds, which may be far from E , = 0. This phenomenon is due to the very small kinetic energy of the electron in the exit channel, regardless of the value of E,. [But the AN approximation (with k, = k,) equates the projectile's energies in the entrance and exit channels, so such phenomena are not seen in AN inelastic cross sections. (See 6 TILE.)] Thus, Nesbet (1977) has shown how a virtual-state pole in FNSA(R)can give rise to structures at the thresholds for low-lying vibrational excitations (9: VI.B.2). Domcke (1983b) has published useful examples of the effects of such a pole in single-channel scattering calculations from a model potential. The BF-FN eigenphase sum is also strongly influenced by a near-threshold pole. For example, in Fig. 11 we show the Xg eigenphase sums for three nonpolar systems: e-H,, e-N,, and e-CO,. The e-CO, eigenphase sum illustrates the stunning effects of a near-zero pole. A virtual-state or bound-state s-wave pole near E , = 0 causes the BF-FN eigenphase sum to rise or decrease sharply near Lero. The rate of change of 6&,(R ) under these circumstances is determined explicitly by the location of the pole, for (with y > 0) virtual state tan 6:",(R)

(95)

=

- -,

bound state

98

Michael A . Morrison 1.01

8

-

W

I

I

1

I

I

0.5--

-0.75-

- 1.0

0

I 0.05

I

0.10

I

0.15

I

0.20

I

I0

0.25

W A V E N U M B E R (o.u.1

b

I .5

'CI

I

I

I

I

I

I.0-

r a

-

2 -1.0-

W W

w

-1.5-

- 2.0 I 0

I 0. I

I

I

0.2

0.3

I

0.4 W A V E N U M B E R (a.u.1

I

0.5

0.6

FIG. 1 1 . Eigenphase sums from body-frame fixed-nuclei calculations on various molecules. (a) Z, symmetry eigenphase sums for nonpolar systems: Li, (Padial, 1985); CO, (Morrison, 1982); H, (Gibson and Morrison, 1984); CH, (Jain, 1986); N, (Morrison and Collins, 1978). (b) Z eigenphase sums for polar systems: HCI (Padial and Norcross, 1987); HF (Rescigno et al., 1982); CO (Salvini et al., 1984).

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

a

99

0.8

e

- 0.6 -

- HCI

L

z

3

cn w 0.4

cn

a I a

z W

0 0.2

W

0

0

0.05

0.10

0.15

0.20

0.25

WAVE N U M B E R ( o . u . )

b

4'0 e

- HCI

W

0

W

1.5-

I .o

0

0.05

0.10

0.15

0.20

0.25

0.30

WAVE N U M B E R ( o . u . ) FIG.12. Body-frame fixed-nuclei eigenphase sums for e-HCI scattering in the Z symmetry at various internuclear separations ( R shown in 0,) from Padial and Norcross (1987). (a) R < R , = 2.41 a,. (b) R > Re.

100

Michael A . Morrison

A virtual-state pole corresponds to the plus sign in Eq. (95), so such a pole causes S&,,(R) to rise sharply, as does the e-CO, eigenphase sum in Fig. 1la. In contrast, a bound-state pole corresponds to the minus sign, so it causes S:,,,(R) to decrease precipitously as k, increases from zero. A remarkable example of these types of behavior in a single system are the e-HCl eigenphase sums calculated by Padial and Norcross (1987). The behavior of the Z eigenphase sum for R I R e (Fig. 12a) very near k, = 0 conforms to the plus sign in Eq. (95) and is thus consistent with the hypothesis of a virtual-state pole in FNSo(R)just below the real k , axis. (See, however, the critique by Domcke, 1981, of this hypothesis in cj Vl.B.3.) As R changes, so does the strength of and hence the location of the pole it supports. The behavior of Sfu,(R) for R > 2.8 a , (Fig. 12b) near zero energy contrasts sharply with that seen in Fig. 12a; the variation with k , of S:,,,(R) for large R is consistent with the minus sign in Eq. (99, which suggests that perhaps, as the molecule stretches, the pole moves through the origin, becoming a bound-state pole. [For (much) more on this system, see 9: VI.]

vnt

B. THRESHOLD EXPANSIONS: MODIFIED EFFECTIVE-RANGE THEORIES Unlike the threshold laws of 9 IV.A, which provide mainly qualitative insight, effective-range theory (ERT) is a practical tool useful in the quantitative analysis of experimental and theoretical results. The basic idea is to expand (analytic) scattering quantities in powers of the electron wave number, thereby developing expressions for cross sections in terms of (at most) a few energy-independent parameters. (See cj 11.2 of Newton, 1982.) The most famous such parameter is the s-wave scattering length A , which is defined (for scattering from a central potential) in terms of the s-wave phase shift 6, as A

= -1im

1 ~

tan S,(k)

=

-

k-0

Since ERT expansions are valid over a range of low energies, the short-range parameter(s) can be determined by fitting to experimental or theoretical data at the high end of this range-if such data are available. Most familiar are the ERT expansions of very low energy partial-wave phase shifts and cross sections for a local, finite-range, spherically symmetric potential. (See cj 12-4 of Taylor, 1972, and (j 11.6 of Joachain, 1975.) These expansions require modification if, as in electron-atom scattering, the interaction potential has a long-range tail (O'Malley et al., 1961); an extension of

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

101

ERT to a system with an infinite-range potential is called a modified cfli?criverange rheory (MERT). Further modification is required if, as in electronmolecule scattering, this potential is nonspherical (O’Malley, 1964). The latter complication need not always be taken into account, however, and some electron-molecule scattering data are amenable to analysis using the MERT expansions of electron-atom scattering. MERT expansions are widely used to extrapolate scattering data to energies so low that their measurement or calculation is infeasible. But these expansions are also useful for checking the consistency of diverse experimental and/or theoretical studies. For example, Chang (1974b, 1981) used MERT to analyze e-H, cross section data from two very different sources: the transport analysis by Crompton et al. ( 1 969a, 1969b) of swarm data, which yielded the momentum-transfer cross section, and the time-of-flight measurements by Ferch et al. (1980), which produced the total integral cross section. Both experiments predict a scattering length of A = 1.27 f 0.01 a,, a result that is consistent with the theoretical value 1.26 a , , extracted via MERT from the total cross sections of Morrison et al. (1984b). In 0 IV.B.l we gather together some of the simpler and more useful MERT formulae and discuss their applicability to nonpolar electron-molecule systems. In 0 TV.B.2 we confront the considerably more complicated MERT for polar systems and provide a guide (Table IX) to key equations in the literature. More esoteric aspects of threshold expansions appear briefly in the last two sections of this chapter.

1. MERTfor a Nonpolar System The asphericity of the electron-molecule interaction potential enters the derivation of a MERT expansion only via the long-range terms of Eq. (82). If, at low energy, the dominant influence on the scattering quantity being expanded is the spherical [I-ao/2r4] term, then MERT expansions for electron-atom scattering may be applicable (O’Malley, 1963, 1965, 1966, 1971; O’Malley et al., 1961, 1962). Recently, theorists have elucidated the conditions under which these atomic MERT expansions arc valid for electron-molecule systems and what to do if they aren’t. One seeking to expand the BF-FN eigenphase sum 6$”,(R) can use the atomic MERT expansion of the corresponding partial-wave phase shift (Chang, 1981) provided the T matrix is (to a good approximation) diagonal in 1. (The more nearly spherical the molecule, the more likely is this condition to be fulfilled.) For example, the C, eigenphase sum can be represented by the MERT expansion of the tangent of the s-wave phase shift if FNTxg(R) is diagonal and if @:,(R) is predominantly s-wave in character. The resulting

102

Michael A . Morrison

expansion in the body-wave number (to order k:) contains two parameters: the s-wave scattering length A and the “modified effective range” p : 2 4 71 4 tan hf&,,(R)= - A k , - aok; - a,Ak; In k , 3 3 ~

~

+ O(k2).

[I

=

01

(974

MERT expansions of eigenphase sums for higher symmetries are simpler, because low-energy scattering in these symmetries is strongly influenced by the centrifugal barrier (Fano, 1970b). For example, if 1 = 1 dominates the Xu or nueigenphase sum, then the p-wave atomic MERT formula, in which appears the p-wave scattering length A , , is appropriate: tan h&,,(R)

71

= - a,k;

15

- A,k;

+ O(k2).

[1

=

1)

(97b)

For still higher partial-wave order, the centrifugal potential so dominates the scattering that no short-range parameters appear in the atomic MERT expansions. If 1 > 1 denotes the dominant partial wave in a symmetry (with A > l), then the corresponding eigenphase sum can be expanded as25 tan h&,,(R)

=

(21

+ 3)(21 + 1)(21 - 1) a,ki + O(k2).

[ I > 1) (97c)

Most useful to the experimentalist are MERT expansions of cross sections. If the off-diagonal elements of the T matrix are small, then the elastic integral cross section for incident energy k , may be well represented by the atomic MERT expansion (Thompson, 1966; Chang, 1981) ~ v o + v o=

271 471 A 2 + - a,Ak, ( 3

+ 83 a,A2ki In k , + Bk; + O(k&

)

-

(98a)

where

B

1

= - n2a;

9

+ 83 a , A 2 In (1.23JG) + 23 71&A3 -

-

2 nai/’A 3

--

-

A4

+ A3p. (98b)

24 To simplify the equations of this section, I have suppressed the R-dependence of molecular properties such as GL” and A . Equation (97c) may be accurate for the higher symmetries of a strongly nonspherical system for which the short-range interaction renders Eq. (97a) a poor fit to the Z, eigenphase sum.

’’

103

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

Similarly, the elastic momentum transfer cross section can be expanded as 47t

a,Ak,

8

+ 3 a , A 2 k i In k , + Cki -

where

- A4

+ A3p + A , A ,

(99b)

and the elastic differential cross section by davo*vo ~-

dR

= A’

+ na,Ak,

sin

8 2 ~

+ 38 a,A’ki ~

In k ,

+ O(kg).

(100)

To illustrate the use of MERT in electron-molecule scattering, we show in Fig. 13 a fit of the e-N, differential cross sections at 0.1 eV measured by Sohn et al. (1986). Validity criteria for the use of these atomic MERT expansions with electron-molecule data could be derived only once theorists had extended

0.6

\

I

I

1

1

1

I

0.5-

(u

E

V

‘D

0.4

-

I

=0 0.3-

c

e - N2 v=o-0

0

0

20

40

60

80

100

120

140

S C A T T E R I N G ANGLE ( d e g ) FIG. 13. Elastic differential cross section for e-N, scattering at E = 0.1 eV. Experimental data (open circles) (Sohn et al., 1986). MERT fit using Eq. ( 1 0 ) (solid curve). (Adapted from Fig. 3 of Sohn ct al., 1986.)

104

Michael A . Morrison

v,,.

MERT to incorporate the anisotropy of O’Malley (1964) considered the modifications required by a nonspherical potential; assuming a long-range form that accommodates the quadrupole and induced polarizability terms in Eq. (82), he derived an expansion of the LAB-FN scattering amplitude (Eq. (75)) to order k,. (See Eq. (3.21) of O’Malley, 1964, and Eq. (2) of Fabrikant, 1984.) Subsequently Chang (1974b, 1981) and Fabrikant (1984) derived MERT expansions of scattering quantities more closely allied to the results of theory and experiment. By considering corrections to the atomic MERT equation (98) due to anisotropic terms in the long-range potential, Chang (1974b) determined a handy empirical rule for the validity of this equation for electron-molecule cross sections at energies below a few tenths of an eV:26 1q/AI I0.5.For example, using the values of the quadrupole moment and scattering length at equilibrium, we find this ratio to be 0.39 for H, and 2.6 for N,. Analysis of scattering data for these systems (Chang, 1981; Fabrikant, 1984) shows that, indeed, the atomic MERT formulae (98) and (99) are accurate (to better than 2 %) for e-H, cross sections at energies below 0.1 eV, but for e-N, scattering near zero energy, the anisotropic long-range terms may be important. (See (Eqs. (102).) For most systems, one must correct Eqs. (98)-(100) to take account of anisotropic long-range interactions. Chang (198 1) and Fabrikant (1984) have derived MERT expansions for this eventuality; the work of the latter includes expansions of elements of the BF-FN scattering matrix that will be especially useful to theorists studying near-threshold collisions. Fabrikant’s strategy, which was first implemented for a spherical potential (Fabrikant, 1973, 1979), is based on an (imaginary) partitioning of space by a spherical boundary at radius r = r o .The effect on scattering quantities of the interaction in the inner region is represented by one or more short-range parameter^.^' In the outer region, Fabrikant solves (via series expansion) the BF-FN integral equations (63). Interior and exterior solutions are then matched at r = ro via the BF-FN R matrix (Wigner and Eisenbud, 1947), which at r = ro is defined as FN A

U

(Yo,

R) =

FN

&A ( R ) &

.

FN A

I! ( r , R ) r=ro

Effects of the potential anisotropy are apparent in the resulting MERT formulae. For example, the expansion of the F N total cross section-which 26 This criterion is predicated on the assumption that the near-zero elastic cross section being fit is free of anomalous structures such as those caused by near-zero resonances [i.e., it would not apply t o the total e - 0 , cross section (Chang, 1974b)l. In its separation of effects due to long- and short-range interactions, this MERT bears a kinship to quantum defect theory. (See the review by Greene and Jungen, 1985a.)

’’

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

105

will be useful in the analysis of crossed-beam experiments where rotational excitations cannot be resolved-is given by2*

and the total differential cross section by doFN = A2 dR

~

+ 454 q 2 + ~

+ O(k;).

(102b)

These expansions of FN cross sections predict a more complicated energy dependence than do the atomic MERT formulae. Indeed, the full expressions for FN integral and momentum-transfer cross sections-Eqs. (10) and ( I 1) of Eb In E,, and Eb. Fabrikant (1984)-contain terms that depend on Two of Fabrikant’s BF-FN results will be particularly useful to theorists. First, he derived an expansion of the (s-wave-dominated) C, eigenphase, the leading term of which is

fib,

(The higher-order corrections to Eq. (103), too complicated to present here, can be found in Eqs. (13)-(15) of Fabrikant, 1984.) Of equal value is the MERT expansion of the BF-FN reactance matrix, FNK”(R). (I have thus far avoided the K matrix, even though it is widely used in the theoretical trenches. Its popularity derives from the fact that this matrix corresponds to real, rather than complex, boundary conditions on the radial scattering function of Eqs. (63). The K matrix is simply related to the T matrix via the Heitler equations. (See Chap. 9 of Rodberg and Thaler, 1967.)) This expansion contains several constants--, d , e, and Z&,,-that can easily be programmed (see Fabrikant, 1984); its form illustrates the dependence of this crucial scattering quantity on k , and on the molecular parameters in the long-range terms of &,, i.e.,

FNKtf,(R) = &b[-

’’

ASfOS~~Oqcl

To obtain Fabrikant’s expression for the momentum-transfer cross section, simply replace the factor of 1/3 by 2/5 in the third term of this equation. (Fabrikant notes that the resulting expansion disagrees with Eq. (45) of Chang, 1974b.)

106

Michael A . Morrison

2. MERT for A Polar System The complexity attendant on extending MERT to polar systems is only partly due to the presence in Vnt of a long-range F 2 tail. Considerable practical difficulties are posed, for example, by the need to account for molecular rotations. For example, an implementation of MERT in the LABCAM formulation (Fabrikant, 1978) entailed 10 short-range parameters, requiring numerous additional assumptions lest the theory be reduced to mere curve fitting. Recent advances by Clark (see 4 IV.A.3) and by Fabrikant (1983a, 1984) have led to more judicious ways to cope with polar targets. Nonetheless, the theory of near-threshold expansions for such systems remains complex and of limited validity. Fabrikant’s research has emphasized the influence of the long-range dipole term on the final-state scattering function; the only additional long-range term that has been incorporated is the spherical one. Thus, at its most sophisticated (Fabrikant, 1979), the interaction is represented by the model “inner region”

r Ir,,

.

(105)

As usual, the effects of the short-range interaction Kr(r) are encapsulated into one or more short-range parameters. Fabrikant has also considered vibrational excitation, both in a LAB formulation (1978) and in the ANV approximation (1980). In both cases, the MERT formulae for integral and differential ro-vibrational cross sections are based on the assumption that the coupling between the initial and final vibrational states, Iu,,) and Iu), is weak. Hence these expansions are not applicable to cross sections that exhibit near-threshold structures, such as those measured by Rohr and Linder (cf. the reviews by Rohr 1979a, 1979b, 1979c) for scattering by the hydrogen halides. (See 4 VI.) One of the most interesting conjectures to arise from this research is that of “dipole resonances” in vibrational cross sections-resonances that are caused by the long-range dipole tail. Fabrikant (1978) has shown that, provided molecular rotations are properly taken into account, such resonances appear in MERT cross sections for systems with dipole moments greater than the “critical value” d,, = 1.625 debye, such as H,O (d = 1.8473 debye). But these are clearly not the near-threshold phenomena observed by Rohr and Linder. For one thing, they are shape resonances and hence require a centrifugal barrier ( I > 0);consequently, their angular distributions are not isotropic. Moreover, they are very narrow, typical halfwidths being less than 0.5 meV (Fabrikant, 1984). This property renders them very difficult to observe; it is not possible at present to ascertain if they are an artifact of the parameterized model interaction potential (105).

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

107

In order to avoid the proliferation of short-range parameters that plagues MERT for polar systems in the LAB-CAM representation, Fabrikant has presented a dual formulation for systems with dipole moments in the range 1.625 < d < 3.79 (debye). For scattering at an energy that is large compared to the rotational spacing of the target, Fabrikant (1983a) has developed expansions based on the ANR approximation (9: 1II.B). For energies comparable to the target spacing, Fabrikant ( 1983b) has proposed an alternative theory that incorporates molecular rotations via a frame transformation (6 III.D.l) similar to that proposed by Clark (1979). In either case, the resulting expansions contain only a single short-range parameter. If a bound state of the electron-target system exists, this parameter can be determined from the electron affinity of the target. (See also Rudge, 1978a, 1978b, 1978c.)The advantage of this stratagem is that it frees the short-range parameter from dependence on a particular theoretical model of the interaction in the inner region or on experimental cross sections. The disadvantage is that for many molecules of interest, the electron affinity is very small and hence extremely difficult to measure or calculate accurately: e.g., for HCN, it is less than eV; for H F and H,O, there are no bound states of the negative ion, and the (extremely small) electron affinity can only be determined formally (via the Born-Oppenheimer approximation). Application of Fabrikant’s theory to such cases is problematic, for even the shape of the resulting MERT cross sections shows great sensitivity to the shortrange parameter. Thus, as Fabrikant (1983a) has noted, “the applicability of [MIERT . . . in the present form to molecules with small dipole moments is essentially limited” by neglect of quadrupole and dipole interactions and by the necessity to determine a parameter (with respect to which the cross sections may be quite sensitive) by appeal to molecular data that is very difficult to determine experimentally or theoretically. Nevertheless, for some molecules (e.g.. LiF) the polar MERT provides useful expansions; a guide to the key formulae of this theory appears in Table IX. Before leaving polar systems, I should note that Singh (1970) and Petitjean et al. ( 1984) have determined scattering lengths for e-CO scattering, obtaining A = - 1 . 2 and ~ ~A = +0.7a0, respectively. A negative scattering length for this system is, however, inconsistent with experimental data such as the momentum-transfer cross sections of Haddad and Milloy (1983), which show no evidence of a Townsend minimum. (See 9: IV.B.4.) Moreover, the theoretical C eigenphase sums of Salvini et al. (1984), which were obtained in calculations using the R-matrix method with an ab initio interaction potential, decrease smoothly from IL at k = 0 (see Fig. 1lb)-behavior that corresponds to a positive scattering length. Sohn et al. (1985) measured vibrationally elastic (and inelastic) e-CO cross sections at energies well below the 1.8 eV resonance. At these energies, this

Michael A . Morrison

108

TABLE IX

KEYEQUATIONS FOR KEYSCATTERING QUANTITIES FROM MERT FOR POLARSYSTEMS Formulation

Reference"

Scattering quantity

Equation

LAB (RR)

1978

0""+0

(22)

RFT

19836

All references are from papers by I. I. Fabrikant.

cross section is dominated by the rotationally elastic (Aj = 0) contribution (Jung et al., 1982). Sohn et al. (1985) compared measured elastic angular distributions at 0.165 eV to MERT differential cross sections for Aj = 0 based on the theory of Chang (1981) (§IV.B.l)-see Fig. 14. Because Chang's theory does not allow for the dipole term in the long-range potential (82), the MERT differential cross section in this figure does not show the observed rise at small angles. The First Born dipole approximation to the Aj = k 1 contribution to this differential cross section (Takayanagi and Itikawa, 1970) does exhibit this rise; but the dipole moment of C O is so small (d = 0.044 ea,) that this approximation fails at scattering angles greater than about 15", where the differential cross section is sensitive to other interactions than the dipole.

3. MERT about a Pole Considering the dramatic changes wrought in the C, eigenphase sum (and consequently ) in::a by a near-threshold pole of the BF-FN S matrix-as illustrated in Fig. 12-it should come as no surprise that the range of validity

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

-

0.8

I

I

I

109

I

I

L

v)

\

N

E =0.165eV

E 0.6

V

2 I 0 -

-

c: 0.4

0

'0 \

>

t

00.2 b

U

-

. .. . . . .'/. I

0

. . . . . . . . . . . . . . . . .. . . . .

I---r--l

___I___

S C A T T E R I N G ANGLE ( d e g ) FIG. 14. Elastic differential cross section for e-CO scattering at E = 0.165 eV. Experiment (open circles) (Sohn et al., 1985). MERT fit (solid curve). First Born approximation using the dipole interaction only (dashed curve). First Born approximation using the quadrupole interaction only (dotted curve). (Adapted from Fig. 1 of Sohn et al., 1985.)

of the MERT expansions (98)-(104) is affected by such a singularity. (See

9 6.6 of Goldberger and Watson, 1967.) If the pole resides on the imaginary k, axis at k, = iy, then for k, sufficiently near zero the (s-wave-dominated) C, eigenphase sum obeys Eq. (95), so the scattering length (96) becomes A = l/y. Thus the signatures of a near-zero bound- or virtual-state pole are a huge (positive or negative) scattering length and a large finite zero-energy cross section, c$:(kb

=

4n 0) = - -. 1)

(pole at k,

= +iy)

(106)

For k, very near zero, these observations are consistent with the MERT expansions of 9: 1V.B.1. But as k, increases, the pole makes itself felt in terms of higher order in k b ; and if, for example, Eq. (97a) is used for d:",,,(R), a large number of such terms may be required to obtain a good fit over even a small energy range. A more efficacious way to deal with a pesky near-threshold pole is to expand not about l i b = 0 but about k, = k,. For example, O'Malley et al. (1961) have developed such an expansion of the s-wave phase shift in electron-atom scattering. When their theory is applied to the tangent of

Michael A . Morrison

110

C ~ ; ~ , ( Rthe ) , resulting expansion, which includes the (energy-independent) “modified effective range” p , is

tan S:“,(R)

=

- k,A

1 p2ki 2

- -

+ O(k:).

In contrast to Eq. (97a), this expansion lacks terms in k i or k i In k,.29 4. The Ramsauer-Townsend Effect

Among the most striking near-threshold anomalies are Ramsauer-Townsend (RT) minima. Total and momentum-transfer cross sections at and below the energy of the minima can be used in MERT expansions to yield very accurate values for the scattering length. Classic examples of this analysis are those for electron-rare gas atom systems; the heavier rare gases (Ar, Kr, Xe) exhibit pronounced RT minima (O’Malley, 1963; O’Malley et al., 1962; Golden, 1966; O’Malley and Crompton, 1980). The scattering lengths for systems that exhibit the RT effect are negative (e.g., for e-Ar scattering, A = - 1.45ao). The corresponding s-wave phase shifts exhibit the characteristic shape on display in Fig. 15: a rise from nn at k = 0 (where n is a nonnegative integer), followed by a precipitous drop; the phase shift therefore crosses nn at some small but nonzero value of k. In electron-atom scattering, a minimum appears in the cross sections if the energy of this crossing is low enough that the contributions of all other partial waves to the cross section are negligible. In electron-molecule scattering, these remarks pertain to the Xg BF-FN eigenphase sum, with the usual caveat that the cross section must be dominated by s-wave elements of the T matrix, which must be diagonally dominant. This requirement is nearly always satisfied at very low energies. The RT effect comprises two phenomena: a Rarnsauer minimum in the total cross section and a Townsend minimum in the momentum-transfer cross section. These two minima do not necessarily occur at the same energy: in e-CH, scattering, for example, the minimum in the ototoccurs at 0.40 eV (Ferch et al., 1985) and in nmomat 0.29 eV (Haddad, 1985). If a Ramsauer minimum in o,,, occurs at a very low energy Emin= kiiJ2, then a rough estimate of kmincan be obtained from the first two terms in Eq. (98a), viz., kmjn = -

3 A 2na0

(minimum in ntol).

(108a)

2 9 Paul (1980) has also discussed alternative expansions of the s-wave phase shift that may, in the presence of a near-zero pole, provide better fits than standard MERT results.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

-0.2 I 0

I

I

0.I

0.2

WAVE NUMBER

111

C 3

(O.U.)

FIG.15. What the Ramsauer-Townsend does to phase shifts and eigenphase sums: the swave phase shift for e-Ar scattering (solid curve) (Bell et al., 1984); the A , (spherical symmetry) eigenphase sum for CH, (dashed curve) (Jain, 1986); the "experimental" s-wave phase shift for e-CH,, as determined by Sohn et al (1987) via a fit of experimental cross sections (circles).

A corresponding Townsend minimum in omom at ELi,, = k',$,/2 gives

kLin = -

5 A 477Cro

(minimum in omom).

(108b)

In any event, MERT expansions can be used to accurately determine the scattering length, the position of a Ramsauer or Townsend minimum, or the momentum-transfer cross section from the measured total cross section. The most exhaustively studied example of the RT effect in electronmolecule scattering is in e-CH, ~cattering.~'Happily, some of the recent activity on this system has focussed on the near-threshold region ( E < 1.0 eV). A convenient summary of experimental work on this system prior to 1985 can be found in Table I of Curry et al. (1985), useful data compilations in Trajmar et al. (1983), and a review of theoretical research in Gianturco and Jain (1986). Of immediate concern are the time-of-flight measurements of Ferch et al. (1985) and of Lohmann and Buckman (1985, 1986), and the crossed-beam measurements of Sohn et al. (1987). The latter used a crossedbeam spectrometer in an apparatus with a reported energy resolution of 30 Wada and Freeman (1981) have suggested that RT minima are present in e-H, and e-N, cross sections; this notion has been disputed by Crompton and Morrison (1982).

Michurl A . Morrison

112 6

1

0

I

I e- CH4

N

t

E

0 4

'0 I

-

0

1 L

0

\

c 2

\

\ \ \

n -

0

.... . . ..-. -. *

0.I

0.2

I

I

0.3

0.4

0.5

ENERGY (eV1 FIG. 16. The total cross section for e-CH, scattering near and below the Ramsauer minimum. Experimental data: open circles (Buckman and Lohmann, 1986); solid circles (Ferch et al., 1985). Theoretical results: long- and short-dashed curves from Jain (1986) with two cutoff radii in a model polarization potential: rc = 1.175 a, and rc = 1.241 a, respectively; solid curve from continuum multiple scattering calculations of BIoor and Sherrod (1986); dotted curve from Gianturco et al. (1987) using orthogondized free-electron-gas exchange potential.

better than 20 meV (full-width at half maximum). The total cross sections from these experiments are in good agreement. (See Fig. 16.) The nearly spherical character of methane makes this system an ideal candidate for application of the atomic MERT equations of$ IV.B.l. Ferch et al. (1 985) used these equations to analyze their experimental data, extracting (from a multiparameter fit) a scattering length of A = - 2.475 a,. This value is consistent with the data of Sohn et al. (1987) and the momentum-transfer cross sections of Haddad (1985). Like other threshold structures, Ramsauer-Townsend minima are especially appealing to theorists because they are very sensitive to the representation of the interaction potential in a calculation-e.g., to modelexchange or polarization potentials. This sensitivity is strikingly illustrated in Fig. 16, where several recent theoretical total cross sections are compared to measured data. The best agreement for this cross section in the vicinity of the minima is obtained by Jain (1986), who solved the BF-FN scattering equations (for a polyatomic system) using a Hartree-Fock static field, a model-exchange

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

113

potential, and a heuristic polarization potential that was adjusted so as to reproduce measured cross sections. Jain’s eigenphase sum for the dominant, spherical A , symmetry crosses zero at 0.48 eV, quite near the zero at 0.45 eV in the s-wave phase shifts extracted by Sohn et al. (1987) from their differential cross sections. Yet Jain’s scattering length (-3.41 ao) differs from that of Ferch et al. (1985), and although his differential cross sections are qualitatively similar to those of Sohn et a]. (1987), they do not consistently show the shift of the minimum to higher energies with decreasing angle seen in the experimental study. In a related study, Gianturco et al. (1987) replaced the heuristic polarization potential used in Jain (1986) by a correlation-polarization model (OConnell and Lane, 1983; Padial and Norcross, 1984a, 1984b), obtaining a model potential that is free of adjustable parameters. The resulting cross sections, however, do not agree with experiment below 2.0 eV; their Ramsauer minimum occurs at an energy significantly higher than experiment. (See Figs. 16 and 17.) Yet they do agree well with experiments at energies above those in Fig. 16. The sensitivity of the total cross section to polarization effects is illustrated by the two cross sections from the study of Jain (1986) shown in Fig. 16. The calculations that produced these results differed only in their choice of cutoff radius, the adjustable parameter in the model polarization potential. An even more pronounced sensitivity to exchange effects was evident in the theoretical cross sections of Gianturco et al. (1987). Perhaps the moral of these comparisons is that the complex interplay of various influences on electronmolecule cross sections can change drastically as energy varies over a rather narrow range. A degree of unanimity has been achieved concerning olol, all recent measurements agreeing with one another (and with results obtained by Ramsauer and Kollath, 1930). The situation regarding the momentum transfer cross section for e-CH, is less clear. The determination of this quantity by transport analysis of swarm data has recently been considered by Haddad (1985), who has demonstrated the failure for this system of the widely used two-term approximation in solving the Boltzmann equation (Crompton 1969; Huxley and Crompton, 1962, 1974); use of a multiterm solution in unfolding momentum-transfer and inelastic cross section from measured swarm data is crucial. At present, however, insufficient data (from single collision experiments or theory) is available for the transport analysis to uniquely determine an accurate set of cross sections for this system, a situation that also plagues the analysis of e-CO, scattering (Haddad and Elford, 1979). Recent progress notwithstanding, the present state of theory, as illustrated by the figures in this section, is far from satisfactory.

Michael A. Morrison

114 a

1.0

2 '0

-c

0.4

U \

b

U

0.2

0 SCATTERING ANGLE ( d e g )

b

SCATTERING ANGLE ( d e g )

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING 0.8 -

I I5

7 e

- CH4

E = 0.8e V

0.4

4%

0

30 60 90 120 150 S C A T T E R I N G ANGLE ( d e g )

180

FIG. 17. Elastic differential e-CH, cross sections at (a) 0.2 eV; ( b ) 0.6 eV; and (c) 0.8 eV. Experimental data (open circles) from Sohn et al. (1987). Theoretical results (solid curves) from Jain (1986) and (open circles) from Bloor and Shrrrod (1986).

V. Beyond the Born-Oppenheimer Approximation: Special Methods for Near-Threshold Scattering For all their usefulness, the MERT formulae of 9: 1V.B are far from an ah initio theory. The quest for scattering theories that are free of dependence on measured cross sections, of the computational demands of a full LAB-CAM calculation, and of the inaccuracies of AN approximations has led in recent years to a number of innovative methods. Each has its own realms of applicability and limitations, its own advantages and disadvantages; perhaps the one thing these diverse theories have in common is that all are so new that none has been thoroughly tested. Space prohibits all but a cursory survey of these methods and a glimpse of their implementations. We shall begin with a nod to the most famous of nearthreshold approximations: the First Born approximation (FBA). This approximation has been joined with more sophisticated theoretical formulations in the methods of $ V.B. Even more sophisticated are theories that either

116

Michael A. Morrison

directly incorporate nonadiabatic effects (8 V.C) or that remove the assumption of target-state degeneracy by calculating an adiabatic scattering matrix off the energy shell (6 V.D).

A. WEAK-SCATTERING APPROXIMATIONS For both polar and nonpolar systems, the FBA offers a quick, easy (and sometimes highly inaccurate) way to compute electron-molecule cross sections. Implementations of this first-order perturbative theory (see Chap. 9 of Taylor, 1972) have been reviewed by Lane (1980, Q H.2), by Itikawa (1978, 4 4.1), and by Norcross and Collins (1 982); additional useful equations can be found in appendixes to papers by Chandra (1977) and Morrison et al. (1 977, 1984a). Little more need be said. The FBA for nonpolar systems-with the quadrupole interaction (Gerjouy and Stein, 1955a) or, additionally, with induced polarization terms (Dalgarno and Moffett, 1963)-may give accurate cross sections for pure rotational excitation (or de-excitation) j , +j = j, k 2 very near threshold (Chang, 1974a). For polar systems, this approximation suffers from additional limitations (Clark, 1977). Still, for polar systems, the FBA with, say, a point dipole potential can provide accurate small-angle differential rotational excitation cross sections forj, - j = j , 1 scattering from molecules such as LiF (Collins and Norcross, 1977, 1978). An improvement on the FBA is the unitarized Born approximation, which, as its name implies, enforces unitarity of the S matrix (Itikawa, 1969; Padial et al., 1981); this variant is extremely easy to implement and simplifies the inclusion of additional long-range interactions such as the quadrupole, which are likely to be particularly important if the polarity of the target is small. Even if the FBA yields inaccurate integral or differential cross sections for a particular excitation, it may give accurate S-matrix elements for high partialwave orders. For these elements, the centrifugal potential limits distortion of the scattering function to the region far from the target, where the interaction potential is suitably weak. This happy fact is exploited, for example, in the “angular frame-transformation method,” which has been used in polarmolecule scattering (see discussion and references in Q II.B.2 of Norcross and Collins, 1982, and in Q II.3.d of Morrison, 1983) and in a variant proposed by Clark (1979) of the radial frame-transformation method. The distorted-wave method, the other important weak-scattering approximation, takes into account some of the distortion ignored in the FBA. But of late this method has not been applied to electronically elastic scattering. (For early applications, see Q 1I.H. 1 of Lane, 1980.)

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

B. METHODSBASEDIN PART ON

THE

117

BORNAPPROXIMATION

The philosophy of using the FBA wherever it is accurate in more sophisticated scattering theories underlies two approximation methods for rotational transitions. These methods have quite different goals: the multipole-extracted adiabatic-nuclei (MEAN) approximation (9 V.B. 1) facilitates calculation of differential cross sections; the scaled adiabatic-nuclear-rotation (SANR) method (9 V.B.2) approximately corrects the ANR scattering matrix. 1. A MEAN Method f o r Calculating Diflerential Cross Sections The calculation of differential cross sections-no easy chore under the best of circumstances-is rendered extremely time-consuming for low-energy electron-molecule collisions by the need to include T-matrix elements for a very large number of coupled partial waves. (Cf. Morrison, 1979.)This cross section can be expanded in Legendre polynomials as

The expansion coefficients A,(v, --* 11) are the troublemakers; they are given by a complicated expression involving products of T-matrix element^.^' These matrix elements could be obtained from the asymptotic form of the LAB-CAM scattering function (see Eq. (34)) or, via an asymptotic frame transformation ($ 11.D.3)from BF-FN or BFVCC T matrices. In any event, I , T-matrix elements of low partial-wave order (small 1 and I,) for small , dominate A,(v, -+ v); as iincreases, elements of higher order become important. Happily, not all of these elements must be calculated using the full power of close-coupling theory. Distortion of elements of the radial scattering function with large partialwave order occurs far from the target, so the corresponding T-matrix elements may be given to high accuracy by the FBA. But in this region of space, the nuclear Hamiltonian may play an important role in this distortion, so one should not further impose on these matrix elements the ANR approximation. Rather, one should approximate high-order elements by their LAB-CAM FBA counterparts. Precisely the reverse situation obtains for low-order elements, which come from pieces of the scattering function that penetrate the molecular core and hence cannot be represented accurately 3 1 The reader will be spared exposure to the full expression for the ro-vibrational differential cross section. See Eq. (20) of Arthurs and Dalgdrno (1960). Chandra (1979, 11 of Lane (1980). and Appendix A of Norcross and Padial (1982).

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Michael A . Morrison

by the FBA. Because V,,, is strong in this region, these elements can be obtained via the ANR approximation-except very near a rotational threshold. Building on earlier work by Crawford and Dalgarno (1971) on polar systems in the LAB-CAM formulation and by Fabrikant (1976), Norcross and Padial (1982) implemented this viewpoint in their MEAN approximation method. To exploit the simplicity of the closed form for the LAB-CAM differential cross section in the FBA, they approximate the cross section (109) as the sum of the LAB-CAM FBA to this cross section and a correction term, i.e.,

The correction term is a sum over A (u la Eq. (109)) from zero to a finite-and. one hopes, small-value Amax. Each term in this sum is the difference of the coefficient A,(v, --t v) as calculated from ANR T matrices (based on BF-FN close-coupling calculations using the full interaction potential) and the LABCAM FBA approximate to this coefficient (using only long-range potential terms). This correction thus replaces small-l terms in the LAB-CAM FBA differential cross section, which are not correct, with their close-coupling counterparts, which are. Initial tests of the MEAN procedure for e-CO scattering (Norcross and Padial, 1982) and for vibrationally elastic e-HCl collisions (Norcross and Padial, 1982) proved its value as a computational tool for polar systems, and the method was subsequently applied to vibrationally elastic and inelastic e-HCl collisions (Padial et al., 1983, and Padial and Norcross 1984a, 1984b, respectively) and to e-HCN scattering (Jain and Norcross, 1985). As formulated by Padial and Norcross, this method treats the nuclear dynamics approximately (in the low-order partial waves) and hence should not be accurate near a rotational threshold. Nevertheless, we have examined the MEAN method because in calculating near-threshold differential cross sections, it is vital to ensure that Eq. (109) be fully converged, a condition that requires a huge number of partial waves. The MEAN procedure is an easily implemented way to mitigate the consequent computational problems. This strategem can be adapted to the near-threshold energy regime by simply using LAB-CAM T-matrix elements to calculate the expansion coefficients for A < A,, in the correction term in Eq. (1 10). For example, Morrison et al. (1984b) found in calculating very low energy e-H, cross sections that Amax = 15 guarantees convergence of ro-vibrational differential cross sections at all angles to better than 2%.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

119

2. A SA N R Way to Calculate Rotutional-Excitation Cross Sections In the scaled adiabatic-nuclear-rotation theory (Feldt and Morrison, 1984) the FBA is used in a radically different way. This method seeks to correct the near-threshold behavior of ANR cross sections. (See 9; IV.A.2.) It differs from the fix-up proposed by Chang and Temkin (1970)-multiplying these cross sections by the kinematic ratio kv/k,-in that it acts directly on inelastic ANR matrix elements, ensuring that all such elements go to zero at threshold according to the correct threshold law, Eq. (83), rather than Eq. (90). The SANR procedure exploits the fact that, whatever its other failings, the LAB-CAM FBA scattering matrix does obey the correct threshold laws, Eqs. (83) and (86). The method requires two additional (computationally trivial) steps: calculation of the BF-FN and LAB-CAM K matrices in the FBA, and frame transformation of the former (u la Eq. (66)). Then the SANR approximation to the exact LAB-CAM K matrix is calculated by simply multiplying each element of the ANR K matrix by the ratio of the corresponding LAB-CAM FBA element to the frame-transformed ANR FBA element. Tests by Feldt and Morrison (1984) of the SANR method for e-H, scattering (in the RR approximation) show that for energies near threshold, this method greatly improves the accuracy of integrated and (especially) differential cross sections based on BF-FN T matrices. Not surprisingly, as the energy increases ( E 2 1.0 eV), the inadequacy of the FBA causes the scaling ratio to inject inaccuracies in these cross sections. Moreover, the inadequacy of the FBA for vibrational excitation (Lane, 1980, and references therein) does not augur well for the extensibility of the SANR gambit to this scattering process. Nonetheless, the computational simplicity of this correction argues for its use whenever scattering calculations must venture into the energy regime very near a rotational threshold.

SCATTERING THEORIES C. APPROXIMATENONADIABATIC The essential premise of the AN approximations of $111 is separability of the dynamics of projectile and nuclei, as manifested mathematically by the absence of S‘,“’and the target quantum numbers v from the BF-FN scattering equations (63). That these dynamics are not truly separable is implicit in the (exact) equation (29) of the LAB-CAM formalism-buried in the coupling of “uijl,vojolo(r’) to other radial functions “~i.~.~., uojo,,,(r‘).The goal of the methods of this section is a formulation that incorporates nonadiabatic effects approximately.

120

Michael A . Morrison

1. The Energy-Modt$ed-Adiabatic Approximation

Exploring the formal relationship between the BF-FN scattering equations and those of the BFVCC theory ($ IILA), Nesbet (1979) developed an approximation to the latter that is the basis of the energy-modijied adiabatic (EMA) method. In the BODY representation, the (fully adiabatic) FN scattering equation is (9 1II.B)

This equation can be made formally equivalent to the BFVCC scattering equation (60) by replacing the body energy E, with the operator E - A?:), where in the FNO approximation3' Z',") is just %$) + EZ'(R). The BFVCC wave function FNoY$,volo(r, R ) is equal to the product of a solution of the BF-FN scattering equation (with the aforementioned replacement) and the vibrational eigenfunction cp$)(R) (Nesbet, 1979). Hence the BFVCC S matrix can be expressed exactly in terms of a "fixed-nuclei'' S operator that depends explicitly on the nuclear motion, i.e.,

The comparison of Eq. (1 12) to the A N approximation, in which FNS"(R) would be evaluated at the body energy E , (see Eq. (70)), suggests an approximation that may be very good if the dependence of this matrix on R is weak: replace each element of the scattering operator in Eq. (112) by the corresponding BF-FN matrix element FNStlo(R)evaluated at the geometric mean of the initial- and final-state energies, i.e., for the excitation vo + v at Em, = J(E - %)(E - Go).

(1 13)

Nesbet proposes evaluating the reactance matrix FNoK"(R) via this prescription and from it the S matrix. Because the reactance matrix is guaranteed to be symmetric, the resulting EMA S matrix is guaranteed to be unitary. But the rows and columns of the reactance matrix do not necessarily correspond to physical channels, so this matrix must be assembled in "blocks," each block corresponding to an initial and final vibrational quantum number, and each block evaluated at the appropriate energy (1 13). (See $ 1II.E.) 32 Because Nesbet focused on vibrational excitation, he worked entirely within the FNO approximation (gI11.A). The essential strategy of the EMA can easily be extended to rovibrational excitation, developing approximations not to the BFVCC S matrix but to the full LAB-CAM S matrix, either by relaxing the FNO approximation or by using an asymptotic rotational frame transformation (5 II.D.3). See, for example, 5 1I.D of Morrison et al. (1984).

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERlNG

121

This prescription has several advantages for near-threshold scattering. Most important, it ensures that certain crucial elements of the. scattering matrix obey the correct threshold law. In particular, elements of the EMA transition matrix-which can be obtained from the elements of * via an asymptotic rotational frame transformation (Eq. (56))-with s-waves in the final state go to zero as LT;ljO,,,ojolok;'' as k , + 0. (See Eq. (83).) In its full form, the EMA method suffers from a practical disadvantage. For each block of the EMA reactance matrix, one must solve the BF-FN equation (63) at the energy prescribed by Eq. ( 1 13); doing so could require a large number of time-consuming calculations. For this reason, the most widely used implementation of the EMA approximation is a variant that is consistent with the spirit (if not the letter) of Nesbet's method (see, however, Jerjian and Henry, 1985): evaluate all elements of the EMA reactance matrix from BF-FN matrices calculated at the geometric mean (1 13) f o r the excitation of interest. For example, to study t'" = 0, j , = 0 + u = 1, j = 3, we would solve the BF-FN equations at E,,, = ( k I 3 k 0 , ) / 2 .This scheme requires only a single scattering calculation [for each required symmetry (A) and internuclear geometry ( R ) ]for each excitation. This implementation of the EMA method has been used by Norcross and Padial (1982), and its accuracy has been investigated by Miindel et al. (1985), by Morrison et al. (1984b), and by Gauyacq (1983). The latter used it in conjunction with the zero-range-potential method to calculate the vihrational-excitation cross section o o + l for e-HF scattering; his results (Fig. 7) exhibit the observed threshold spikes (Rohr and Linder, 1975), structures that are not present in the corresponding ANV calculations. This point has also been demonstrated by Fabrikant (1985b).

-

2. A Nonadiabatic Rrsonance Theor))f o r Vibrational Excitation The EMA approximation enables one to use results of BF-FN scattering calculations to evaluate inelastic cross sections in regimes where a conventional AN treatment would be invalid. This virtue is shared by the nonudiabatic resonance ( N A R ) theory that has been developed over the past decade by Domcke and coworkers. (Cf. Domcke and Cederbaum 1977, 1980, 1981; Domcke et al., 1979; Domcke and Miindel, 1984, 1985.) But at the operational level, the NAR and EMA methods are quite different. To derive the NAR formalism (Domcke and Cederbaum, 1977), one uses many-body optical-potential theory (Bell and Squires, 1959) and the Feshbach projection operator formalism to transform the many-electron, nonadiabatic Schrodinger equation of the electron-molecule system into the equivalent (but by no means trivial) one-particle problem of solving for

122

Michael A . Morrison

bound and continuum states of the nuclei. Domcke and his collaborators have focussed primarily on the effect of shape resonances on vibrational excitation (and dissociative attachment) cross sections, but they have also applied the NAR theory to the study of near-threshold structures in e-CO, and e-HCI scattering. (See 4 VI.) Moreover, this theory is of concern here because the problem it tackles-nonadiabaticity-is central to near-threshold scattering. At the heart of the NAR method is the interpretation of a short-lived resonance state as a discrete state of the negative ion embedded in the continuum of the neutral molecule plus electron; such a state is assumed to exist for molecular geometries near equilibrium. (Cf. Feshbach, 1958; Bardsley, 1968a, 1968b; Chen, 1966.) [For example, a discrete *Xustate of H; is embedded in the continuum of the X’C,f ground state of H, at an energy around 5 eV (Moiseyev and Corcoran, 1979; McCurdy and Mowrey, 1982); strong coupling of this discrete state to scattering states gives rise to resonant enhancement of e-H, cross sections at energies from 2 to 4eV (Mundel et al., 1985).] In the NAR theory, the electronic wave function of this state is not treated as adiabatic (i.e., in the Born-Oppenheimer approximation), but is assumed to vary slowly with R ; such a state is sometimes referred to as diabatic. The potential V,,, causes the discrete state to interact with scattering states in the continuum. This interaction shifts the energy of the discrete state, &;(R), by an amount A(E, R). In addition, the state acquires a width UE, R). The NAR collision theory looks quite unlike anything we have seen so far, for it is cast in terms of a (Lippmann-Schwinger) scattering equation in the Hilbert space of the nuclear motion. The quantum physics of the bound and projectile electrons is buried in the aforementioned many-electron discrete and continuum states, which appear in the Hamiltonian H that represents nuclear motion in the resonance state. To see how such a resonance state can influence vibrational excitation, we must look at the NAR equation for this process.33 In the transition amplitude for the excitation uo + u, coupling of the discrete state to the continuum is represented by electronic matrix elements in the entrance and exit channels, V, and V,,, respectively. The contribution from the resonance state to the cross section for this excitation is (Domcke and Mundel, 1984)

3 3 Domcke and collaborators neglect rotational degrees of freedom; i.e., they invoke the FNO approximation.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

123

The Hamiltonian that governs the nuclear motion in the discrete state, H in Eq. (1 14), is the sum of the vibrational kinetic energy and the complex optical potential

Po,,= E,(R) + A [ R , E - iil-[R, E

-

-

H:’

Xp’ -

-

t$)(R)]

&$)(R)].

(115)

The Hamiltonian H is, therefore, non-Hermitian and energy-dependent. The first two terms in Eq. (1 15) are the potential energy of the neutral and of the discrete state; Domcke and coworkers parameterize these functions, using either a simple-harmonic-oscillator (Domcke and Cederbaum, 1981) or a Morse potential (Miindel and Domcke, 1984) to represent the nuclear motion. The sum of the last two terms in is an operator in the Hilbert space of the nuclear motion called the level s k f r operator. The nonadiabatic character of this operator is evident in its dependence on the nuclear kinetic energy. Having arrived at the level shift operator, we return to threshold. Near threshold, both terms in the level shift operator depend strongly on the energy, implying a highly nonadiabatic collision process (Domcke and Cederbaum, 1977). In applications to date of the NAR theory, the dependence of l- and A on R and on the linear momentum of the nuclei has played a crucial role in near-threshold excitations. For example, in an early application of this theory, Domcke et al. (1979) used a parameterized model to calculate excitation cross sections and demonstrated that a single discrete state coupled to the continuum could induce very sharp threshold spikes (as well as a broad resonance several tenths of an eV above threshold) and that neglect of the nonadiabatic (operator) nature of the level shift in Eq. ( 1 15) caused these spikes to vanish and the magnitude of the near-threshold cross section to drastically diminish. This work is representative in that Domcke and collaborators extensively use phenomenological models (Mundel and Domcke, 1984; Mundel et a]., 1985) to simplify the solution of the scattering equations and, in various works, implement a variety of physically motivated assumptions to make tractable computations based on Eq. ( 1 15)-assuming, for example, that only a single discrete state exists, that the resonant state is purely s-wave, or that the effects of rotational dynamics are negligible (Domcke and Cederbaum, 1981). The resulting easily evaluated expressions for excitation cross sections include parameterized forms of key quantities such as the energies of the neutral and discrete states and the electronic matrix elements V,. In a study of e-HCI scattering (sec 5 VI.B.4), for example, Domcke and Mundel (1985) used seven such parameters, determining them by reference to the extensive BF-FN scattering calculations of Padial and Norcross.

cp,

124

Michael A . Morrison

vopt

Even with a parameterized model of the collision, evaluation of is no easy matter, so in recent work (see, for example, Domcke and Miindel, 1985) the non-local-level shift operator is approximated by a separable expansion. This gambit transforms the Lippmann-Schwinger equation for the stationary scattering states into a more tractable set of linear algebraic equations. Thus far we have emphasized “resonance” ~ c a t t e r i n g .But ~ ~ in many near-threshold scattering processes, such as excitation of the symmetric stretch mode of C O , (Estrada and Domcke, 1985), background scattering plays a role comparable to that due to a pole of the S matrix. To deal with such problems, Domcke and collaborators have melded the NAR method with the Feshbach Projection Operator Method (POM) (Domcke, 1983; Berman et al., 1983a, 1983b, 1983c, 1985). This merger also facilitates the use of scattering data from ab initio BF-FN scattering calculations to determine the potential energies of the neutral and discrete states and the electronic coupling matrix elements V,. (These data can also be calculated from first principles; see Hazi, 1983, and Berman and Domcke, 1984a, 1984b.) Feshbach’s formulation (Feshbach, 1962) is a bulwark of quantum theory. Its extension to nuclear dynamics (Chen, 1966; O’Malley, 1966; Bardsley, 1979) leads to a rigorous, unique decomposition of the transition amplitude for a nuclear excitation into the sum of a background (BG) term, which varies slowly with energy, and a rapidly varying (RV) term, which includes effects due to poles of the S matrix, viz., Lt(kl, v t kb,

vO) =

Ltb,(k:, v + kb, v 0 )

+ Lt,v(ki,,v + kb, v0).

(1 16)

In the FN limit, this separation translates into an equivalent decomposition of the BF-FN eigenphase sum, G,(R)

= X,(R)

+ S&(R).

(1 17)

This separation is illustrated in Fig. 18b (Estrada and Domcke, 1985), where very low energy BF-FN e-CO, eigenphase sums in the Zg symmetry (Morrison, 1982) are decomposed la Eq. (1 17).

3. Vibrational Excitation via an Eigenchannel R-Matrix Method In 1985, Greene and Jungen introduced a third approach to determining nonadiabatic excitation cross sections from ab initio BF-FN scattering data. These authors confronted directly the fundamental ambiguity that plagues AN methods: the indeterminacy of the body energy E , in the BF-FN scattering equation (1 1 1). (See 4 1II.B.) Their collision theory-for vibrational excitation within the FNO approximation of 9 1II.A)-is based on the 34 Strictly speaking, the use here of the term resonuncr is something of a misnomer, for the NAR method also applies when threshold structures are caused by a virtual-or bound-state pole of the S matrix. See Domcke (1983a) and Estrada and Domcke (1985).

a

L

2

1.5

=)

m

I

I

:-

I

W v)

a

I

a 0.5

z W

c3

w

-0.5

-

1 0

. I

5 2

1 3

4

ENERGY ( e V )

b

1.2

0.9

-

T)

0.6

0 L

0.3 3

v,

w

v,

o

a

I

a

z

-0.3

W

2 -0.6 w - 0.9 - I .2

0

, 0.05

0.10

0.15

0.20

0.25

ENERGY ( e V ) FIG. 18. Breakdown of eigenphase sums (solid curves) into rapidly varying (dotted curves) and slowly varying (dashed curves) components Li lu Eq. ( I 17). (a) C, eigenphase sums for H, (Berman et al., 1985) and for N, (Berman and Domcke, 1984b). (h) Z, eigenphase sums o f Morrison (19x2). (Figure 18b adapted from Estrada and Domcke. 1985.) I25

126

Michael A . Morrison

eiyenchunnel R-matrix method (ERM) of Fano and Lee (1973) and brings to bear on electron-molecule scattering the technology of quantum defect theory. (See the review by Greene and Jungen, 1985a.) The ERM can be considered a generalization of the vibrational frame transformation (VFT) theory of Table IV; both methods partition space into inner and outer regions, treating the collision dynamics in the two regions d i f f e r e n t l ~ . But ~ ~ while the VFT assumes that in the inner region, the dependence of the scattering function on the nuclear separation R is parametric-i.e., that the nuclei are frozen-the ERM allows for motion of the nuclei in a Born-Oppenheimer state.36 The potential energy curve of this ( N , + 1)-electron state can be determined from BF-FN scattering data, such as the reactance matrix F N K " ( R )As . implemented by Greene and Jungen, the ERM accurately includes nonadiabatic effects in the outer region via quantum defect procedures, which are also used to impose the asymptotic boundary conditions. The formalism of Greene and Jungen reduces to the EMA approximation if FNK"(R)is weakly dependent on the body energy and to the ANV method in the limit that this matrix is independent of E,. But the ERM approximation to the BFVCC scattering matrix can also cope with conditions where the BF-FN scattering matrix depends strongly on the energy-circumstances that render invalid the ANV and EMA approximations. Thus far, the ERM has been applied only to resonant e-N, scattering (Greene and Jungen, 1985b). This acutely nonadiabatic collision process provides a splendid test of a nonadiabatic resonance collision theory. It also permits a considerable simplification of the ERM formalism, for near this resonance (i.e., from about 2.0 eV to 3.0 eV), the vibrational-excitation cross sections are overwhelmingly dominated by the d-wave contribution in the Il, e-N, symmetry. In practical terms, this means that the ERM can be formulated in terms of a d-wave phase shift, for the BF-FN eigenphase sum of Eq. (91a) simplifies as Gft.,(R) % $z,. Because the ERM allows the nuclei to move-albeit adiabatically-in the inner region and accurately accounts-via quantum defect procedures-for coupling of the motions of the projectile and the nuclei in the outer region, it 35 Like the NAR method of$ V.C.2, the ERM approach is philosophically closely related to the R-matrix theory for Vibrational excitation as formulated by Schneider et al. (1979a); see also Burke and Robb (1975), Schneider (1975, 1977), Schneider and Collins (1984), and Burke et al. (1983). The essential difference is that the R-matrix method uses several adiabatic electronic states in the inner region (see, for example, Schneider et al., 1979b), assuming them to be noncrossing, while the ERM method uses a single diabatic state, which is chosen to satisfy the correct conditions at the boundary. '' In this sense, the ERM is quite similar in spirit to the erective range approximation of Gauyacq (1983). (See 5; VI.B.3.) Both methods take advantage of the (approximate) adiabaticity near the nuclei.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

127

should be well suited to near-threshold scattering. Even a simplification such as Eq. ( 1 17) may pertain to such collisions, which are often dominated by s-waves in the C, symmetry. T-MATRIXMETHODS D. OFF-SHELL The vibrational frame transformation theory of Greene and Jungen ($ V.C.3) uses the eigenchannel R-matrix method to deal with the arbitrari-

ness implicit in the definition of the body energy. A different way to remove this indeterminacy is to venture off the energy shell. By so doing, we can retain an adiabatic picture of the collision without assuming degeneracy of the initial and final target states. Detailed studies of the AN approximation (Morrison et al., 1984a, 1984b) strongly suggest that this assumption is primarily responsible for its breakdown near threshold. (See 8 1II.E.) The F N momentum-space T matrix in the BODY reference frame, F N t ( k , i k b i o ; R), is defined an the energy shell; that is, the energies of the initial and final asymptotic states are equal. So when this scattering quantity (which is proportional to the scattering amplitude of Eq. (74)) is used to calculate excitation cross sections via a transformation into the LAB frame, the resulting physicul T matrix [Eq. 76b)l is also “on-shell,” in the sense that the energies in the entrance and exit channels are equal. This is the assumption of target-state degeneracy ($ 111). It is in force no matter what we use for E , if we base our calculation on an on-shell BF-FN scattering matrix. And it is seriously in error near threshold. But we need not so restrict ourselves. The on-shell T matrix is just a portion of the matrix representation of the transition operator (see 9 3-b of Taylor, 1972); the other elements constitute the ofjshell T matrix. At some cost in computer time, going off-shell can markedly improve the accuracy of near-threshold cross sections over those from AN calculations. The idea is simple. Rather than approximate the LAB excitation amplitude Lt(k:, v t k6, v o ) by the AN expression (see Eq. (76a).) +-

which, speaking perturbationally, is accurate to second order in the nuclear Hamiltonian 2:’, we approximate this amplitude by the matrix element of an off-shell FN amplitude Lf(+’(k:,,v t kb, v o ; R ) , to wit:

Lt(k:,, v +- kb, v o )

FZ

(vlLt‘+)(k;,,v + kb, v o ; R)Iv,).

( 1 19)

The off-shell amplitude in this matrix element is related to the off-shell BF-FN amplitude by a simple coordinate rotation (Eqs. (47)).

128

Michael A. Morrison

This idea is implicit in Chase's original derivation of the A N amplitude (Chase, 1956), but its promise for electron-molecule scattering was first pointed out by Shugard and Hazi (1975), who generalized both off- and onshell adiabatic theories to allow for electronic excitation. Much later, in discussing the NAR method (9V.C.2), Miindel et al. (1985) suggested approximating the background T matrix by the matrix element of an off-shell FN matrix. These authors also examined the accuracy of an off-shell AN approximation for resonant vibrational excitation in H at energies above 1.OeV. The problem is to translate this idea into a workable method for day-today scattering computations. We cannot use directly the BF-FN scattering equation (1 1l), for it puts us firmly on the energy shell. In this section we shall explore two quite different off-shell approximations, one formulated in terms of the Lippmann-Schwinger equation for the T matrix, the other in terms of the BF-FN radial scattering functions FNufL,(r; R).

1. A Field- Theoretic Of-Shell Adiabatic Theory

In a series of papers, Ficocelli Varracchio has studied scattering by molecules using a field theoretic description of the scattering process. This work has included new developments in the representation of polarization effects (Ficocelli Varracchio and Lamanna, 1986) and studies of heavyparticle collisions (Ficocelli Varracchio and Celliberto, 1982). Here we are concerned with his reformulation of electron-molecule scattering theory (Ficocelli Varracchio 1979a, 1981a, 1981b; Ficocelli Varracchio and Lamanna 1983, 1984). In principle this formalism encompasses a full treatment (within the Born-Oppenheimer approximation) of the perturbations of the scattering function by the nuclear Hamiltonian. But to date, it has been applied only in lowest order, where it reduces to an off-shell generalization of the AN approximation. Treating X z )perturbationally, Ficocelli Varracchio has developed a series of corrections to the AN T matrix (118). The nuclear motion is taken into account through the medium of a nuclear relaxation operator c, which is defined as

In terms of this operator, the LAB momentum-space T matrix of ( I 18) can be written as Lt(k:, v + ko, vo)

=

( ~ ( e 'LtFN(k:+- kb; R)(v,).

(121)

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

129

A full implementation of this expression would represent a considerable computational challenge, for in all but the lowest-order approximation (e‘ = i), it calls for differentiation of the LAB-FN T matrix with respect to the nuclear coordinates R and the incident energy E , . But in lowest-order Eq. (121) reduces to an off-shell adiabatic T matrix.37 The further approximation of target-state degeneracy reduces this off-shell T matrix to the usual on-shell ANV amplitude (1 18). Operationally, Ficocelli Varracchio works directly with the Lippmann Schwinger equation for the off-shell BF-FN T matrix (see Eq. (16.25a) of Joachain, 1979, solving the coupled equations obtained when this equation is decomposed into partial waves. Using a quadrature approximation, he converts this integral equation to a set of (complex) linear algebraic equations that can be solved using standard matrix inversion methods. This formulation incorporates exchange effects exactly. Ficocelli Varracchio and Lamanna (1984) have applied this theory (in the lowest-order approximation) to rotational excitation of H, in the rigidrotator approximation. At present, a precise assessment of this approximation is hindered by the lack of benchmark LAB-CAM cross sections based on the interaction potential used in these off-shell calculations. Nonetheless, comparison of their results for ~ o o + 0 2to corresponding on-shell (ANR) cross sections clearly show the off- and on-shell adiabatic cross sections behaving quite differently near threshold. 2. A First-Order Nondeyenerutr Adiabatic Approximation An alternative formulation of the idea of basing calculations of excitation cross sections on an off-shell scattering quantity is the first-order nondeyenerate adiahutic (FONDA) approximation (Morrison, 1986). Physically, this method is equivalent to the lowest-order approximation to Eq. (121); i.e., it retains the Born-Oppenheimer factorization of the system state function, while ensuring that the entrance- and exit-channel energies are correct. Operationally, the FONDA differs markedly from the formulation of Q V.D. 1, in that it is carried out using the BF-FN radial functions (the solutions of Eq. (63)), the numerical and physical features of which have been the subject of exhaustive study. (See the review by Buckley et al., 1984.) The FONDA approximation has been tested (in the rigid-rotator approximation) against fully converged LAB-CAM cross sections for rotational excitation of H, using identical interaction potentials (Morrison, 1986). These comparisons show a dramatic increase in accuracy over the ANR ” Ficocelli Varracchio and Celiberto ( 1982) have evaluated higher-order correction terms generated by the nuclear relaxation operator for atom-molecule collisions.

Michael A . Morrison

130 0.0 I 6

-=;

I

I

I

I

I

I

... .. . . . .. . . 1

I

I

0.012

\

(‘*0

C

.->

0.008

+

e -H2

.-0

j=0+2

0

>

2 0.004

E = 6 5 meV

I 20

0

I I I I 1 I I 40 60 80 100 120 140 160 180 SCATTERING ANGLE ( d e g )

FIG.19. Differential cross section for the pure rotational excitation of H, by a 65-meV electron as calculated using various theories discussed in the text: lab-frame close-coupling theory including rotational and vibrational states (solid curve) and in the rigid-rotator approximation (long-short-short dashed curve) (5 1T.B); adiabatic-nuclear-rotation theory (dotted curve) (5 1II.B); energy-modified-adiabatic approximation (long-dashed curve) (5 V.C); first-order nondegenerate adiabatic approximation (short-dashed curve) (5 V.D.2); First Born approximation (long-short curve) (5 V.A). All but the solid curve were calculated in the rigidrotator approximation with Re = 1.402 a,.

approximation: the percent error introduced into by the FONDA approximation ranges from a maximum of 5 % (at an incident energy 3 meV above threshold) to less than 1 (at and above 0.1 eV). In Fig. 19, differential cross sections for this excitation at 0.065 eV are compared with their counterparts calculated using the LAB-CAM theory and several other approximations we have discussed. Both off-shell theories discussed in this section are based on the post form of the off-shell T matrix (Taylor, 1972), Lt ( + )

(kk, v

+

kb, v o ; R) = ( k ; , vI v n t [ k b v0+). ,

(122)

Ficocelli Varracchio (1 985) has suggested that a more accurate approximation might be obtained if this and the prior forms were unified into a “completely off-shell T matrix.” To date this idea has not been put to the numerical test.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

131

VI. Variations on an Enigma: Threshold Structures in Vibrational-Excitation Cross Sections The methods of § V must be tested on a variety of well-understood systems to be fully evaluated-their strengths and weaknesses discovered, their regions of applicability determined. But once this assessment is complete, a mystery awaits their attention: threshold spikes. Study of this phenomenon is of interest from at least two vantage points: fundamental science and the sociology of atomic physics. To adequately cover the research and controversy that has swirled around these striking structures would take another chapter the size of this one. In the present section, we’ll consider the nature and extent of the phenomena, the theoretical issues involved, and some of the mechanisms that have been proffered to explain them. A. PROVOCATIVE EXPERIMENTAL RESULTS In the mid 1970s, while measuring differential cross sections for e-HCI scattering at very low energies using a crossed-beam apparatus, Rohr and Linder (1975) observed an anomaly. The meV energy resolution of their apparatus permitted clear separation of energy-loss peaks due to vibrational excitation; and these spectra revealed, above each excitation threshold q,,an enormous spike. Similar structures subsequently appeared in measurements on other (polar) hydrogen halides, HBr and HF-and then, astonishingly, in excitation cross sections for the Raman-active, totally symmetric (nonpolar) vibrational mode of SF,. To date, threshold spikes have been reported in the eight systems in Table X.38 Certain features characterize near-threshold excitation cross sections for these systems. Their peaks are very narrow and occur very near threshold. In e-HF scattering, for example, a peak of width less than 30 meV occurs in no+ a mere 10 meV above threshold (491 meV). And the cross section for excitation of the stretching modes (100 and 001) of H,S, with thresholds at 324 meV and 326 meV, shows a narrow peak at about 420 meV. These peaks tower above the excitation cross section at higher energies. For example, for the e-HCI system attains a peak value of roughly 20 A2, more than five times its value at other energies. In some cases, the peak ~ , an order of magnitude larger than the low-energy elastic values of C T ~ , ~ - are 38 Similar structures have been observed in studies of dissociative attachment. See, for example, Abouaf and Teillet-Billy (1977) and Teillet-Billy and Gauyacq (1984).

Michael A . Morrison

132

TABLE X THRESHOLD SPIKES Molecule

I N VIBRATIONAL

EXCITATION CROSS SECTIONS

d(debye)

Group

Primary references"

1.819 1.108 0.827 1.86 0.978 0 0 0

C,,

Rohr and Linder (1976) Rohr and Linder (1975, 1976), Ziesel et al. (1975) Rohr (1977a, 1978a), Azria et al. (1980) Seng and Linder (1974, 1976), Rohr (1977b) Rohr (1978b) Kochem et al. (1985a) Rohr (1980), Sohn et al. (1983) Rohr (1977~)

C ,, C, C, C,

D,, 0, ~~

See also the reviews by Linder (1977) and by Rohr (1979a, 1979b, 1979~).

cross section. This behavior is unusual, to say the least: vibrational-excitation cross sections tend to be considerably smaller than their elastic counterparts. In fact, for most molecules ovo+vis not small only when it is enhanced by a shape resonance (Schulz, 1973, 1976). But the measured angular distributions in the vicinity of a near-threshold peak scupper the conjecture that the cause is a shape resonance, at least in the conventional sense. Such a resonance requires a centrifugal barrier ( I > 0) in the effective potential for the resonant symmetry, but the differential cross sections at energies near a peak are isotropic, implying s-wave scattering. In fact, these cross sections remain isotropic even at energies far above the peak. (See Fig. 20.) Thus, da,,,/dQ for e-HCl is flat from theshold (357 meV) to about 4.0 eV. It is worth taking a moment to appreciate just how odd this behavior is. Differential cross sections for vibrationally elastic and inelastic scattering from polar systems are supposed to be forward-peaked-as predicted by the First Born approximation. (Cf. Norcross and Collins, 1982.) AIthough this approximation rarely yields accurate quantitative values for da,,,,v/dR, it does predict their observed shape: at small enough angles, they usually exhibit the familiar E;'(l - cos O)-' dependence of the Born differential cross section for a polar target. (Cf. Itikawa, 1978.) This behavior is illustrated in Fig. 20a by differential cross sections for e-CO scattering. The uo = 0 + u = 1 cross sections of Fig. 20a contrast starkly to those for e-HCl in Fig. 20b. The differences displayed in these figures suggest that different physical mechanisms may be responsible for the two scattering processes. Certainly the most easily explicable of these phenomena is the strong forward peaking of the differential elastic cross sections. This feature, and the observed increase of the integral elastic cross section as the scattering energy approaches zero, are probably due to simple, direct dipole scattering.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING a

5

I

I

I

I

I33

I

L

In \

E = 1.8 eV

E V

!!

-

e - CO

4-

N

3-

-

c: 2 -

-

I

0 -

-0 \

>

<

4

0

b’

-

I -

v=O+l

U

w

0. 0

w

I

I

I

I

I

20

40

60

80

I00

I20

SCATTERING ANGLE ( d e g )

b

-

0 0 +

4

0 2oo.. -7

e-HCI E.3eV

“ I 2

b

U

150

0

LL

20

40

60

80

100

120

2

ANGLE ( d e g ) FIG.20. Cross sections for elastic scattering and vibrational excitation ( L ’=~ 0 -+ 11 = 1) of (a) CO (at 1.8 eV) and (h) HCI (a! 3.0 eV). Experimental data in (a) (open and solid circles for elastic and inelastic scattering. rcspectively) from Sohn et al. (1986) and in (b) from Rohr and Linder (1975). Solid curves in both figures are numerical fits lo the experimental data. (Figure 20b adapted from Fig. 3 of Rohr and Linder, 1975.)

134

Michael A . Morrison

But some other mechanism seems to be at work in the excitation cross sections, a mechanism that washes out the forward peaking expected of a dipole i n t e r a ~ t i o n . ~ ~ Close examination of the references in Table X yields a sprinkling of subtle curiosities. For example, threshold spikes have not appeared in cross sections for nonpolar diatomics (such as H, and N,), weakly polar diatomics (such as CO and NO), or strongly polar diatomics (such as LiF, CsF, and KI). In fact, these structures are seen in polar systems only if the dipole moment of the target resides in a fairly narrow range rather near the “critical” value d,, = 1.625 debye-the minimum value required for a stationary dipole to bind an electron (Wallis et al., 1960). Another curiosity: in some systems, the threshold spike is accompanied by a broad shape resonance at higher energies. Such companion structures are seen, for example, in e-HC1 scattering (a Z’ resonance at roughly 2.5 eV) and in e-H,O scattering (a 2 A , resonance at about 6 eV). But other systems, such as e-HF, show no trace of an accompanying shape resonance. Still others, such as e-H,O, show companion resonances only in certain vibrational modes.40 B. MUCHADO ABOUT POLAR

SYSTEMS

The calculation of excitation cross sections for systems as complicated as those in Table X at a level beyond the AN approximation is so formidable a task that even the most sophisticated efforts to date rely on parameterized model potentials, focussing their computational labors on accurately treating the nuclear dynamics in the electron-molecule system. The panoply of theoretical studies on near-threshold spikes is suggested by Table XI-which includes only primary reference^.^' Excluded, for example, are the large number of BF-FN calculations that have provided scattering data to some of these dynamic calculations. (See, for example, Padial et al. 1983, on e-HC1; Rescigno et al., 1982, on e-HF; and Gianturco and Thompson, 1977, 1980, on e-H,O and e-H,S.) 39 The Born approximation may be useful in trying to discriminate between various e.g., between a direct mechanism, such as the variation of d explanations for structure in with R , and an exotic mechanism, such as a virtual state. Linder (1977) has discussed the use of the FBA as a diagnostic tool in this context. 40 In e-HBr, the original experiments (Rohr, 1977a) were interpreted as showing threshold peaks in uvo-vfor u from zero to five. Subsequent analysis by Azrria et al. (1980) showed that only for u = 1 and u = 2 are these peaks in the vibrational cross section; the spikes in u,,-~. for example, are manifestations of Br- ions with near-zero kinetic energy ieleased by dissociative attachment. 4 1 Research prior to 1983 has been reviewed in p 1V.B of Norcross and Collins (1982) and in 5 111 of Thompson (1983) and will be de-emphasized here.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

135

TABLE XI THEORETICAL RESEARCH O N THRESHOLD SPIKESCIRCA 1980 Theory Virtual state

Nonadiabatic coupling (nonresonant) Close-coupling Final-state interaction R-matrix MERT Nondissociating ion states Nonadiabatic coupling (resonant)

Systern(s)

Primary references

HCI

Dube and Herzenberg (1977). Nesbet (1977), Kazansky (1982, 1983) Morrison (1982), Estrada and Domcke (1985) Gauyacq and Herzenberg (1982), Teillet-Billy and Gauyacq (1984) Gauyacq (1983) Rudge (1980), Whitten and Lane (1982) Kirnura and Lane (1987) Gianturco and Rahman (1977) Fabrikent (19R5a) Fabrikant (1977a, 1978) Taylor et al. (1977), Segal and Wolf (1981) Domcke et al. (l979), Dorncke and Mundel (1985), Domcke and Cederbaum (1980, 1981)

CO, HCI HF HF

co, HCI HCI HCI HCI HCI

Underlying these studies is the quest for a physical mechanism to explain the anomalous behavior of the systems in Table X. Of course, one need not invoke exotic mechanisms to explain large near-threshold cross sections; for example, such behavior is often seen in oL,o-vif the transition dipole moment (uld(R)lu,) is large (e.g., for the infrared-active modes of CO,). Alas, such simple explanations are incompatible with other experimental results, such as isotropic angular distributions. And so theorists have sought more exotic explanations. 1. The Issues at Hand

From the papers identified in Table XI, one can distill a handful of issues around which controversy has swirled. (I shall couch these questions in the context of e-HCI scattering, but these matters pertain, in whole or part, to other systems in Table X.) (1) Do there exist, just above the potential energy curve of the ground state of HCI, bound states of HCI- that have a finite lifetime at finite R but that dissociate to states that do not exist? If so, could these states be responsible for the threshold spikes in o,,,,,? (See 9 VI.B.2.) (2) Is there a virtual-state, bound-state, or resonance pole of the BF-FN S matrix near zero energy? If so, could its presence explain the observed structures? (See VI.B.3 and VI.B.4.)

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Michuel A . Morrison

(3) What role, if any, does the long-range K 2 dipole tail in b, (see Eq. (82)) play in the formation of these structures? And if this role is crucial, how can one explain the presence of these anomalies in (say) c ~ for~ e-CH,? ~ +

The third issue will run like an undercurrent through the rest of our discussion, so more needs to be said at the outset. Naively, one might suspect from the isotropy of the observed inelastic angular distributions that the role of the dipole interaction is minimal. Still, perhaps because threshold spikes first appeared in several polar systems, early speculation as to their origin emphasized the dipole field. This interaction plays a significant role, for example, in the final-state interaction theory of Gianturco and Rahman (1977), where the influence of the dipole interaction on the (s-wave) scattering function in the exit channel is responsible for the enhancement of.,,,,~c According to this theory, the observed feature depends on the magnitude of the dipole moment d-a point that did not set well with the appearance of spikes in, say, e-CH,. On the other hand, a large number of model studies downgrade or ignore completely the dipole tail. For example, the model studies of e-HCl scattering by Kazansky (1982), using the zero-range-potential (ZRP) approximation, and by Teillet-Billy and Gauyacq (1984), using an extension of this approximation, neglect the dipole field. The latter authors based this assumption on earlier calculations of e-HF scattering by Gauyacq (1983) (9 III.E), in which the dipole field was found to play little role in determining the shape of integral vibrational-excitation cross sections. (See also Fabrikant, 1978.) Yet, threshold spikes appear in these calculations, a consequence of short-range influences. Short-range interactions also play the predominant role in the model calculations of Dub6 and Herzenberg (1977), which are based on a virtualstate enhancement of the scattering function inside the molecular charge cloud. (See pi VI.B.3.) Yet, analysis by Domcke (1981) suggested that a dipole field-which, important or not, is present in these systems-precludes the possibility of a virtual state. And Domcke and Mundel(1985), in perhaps the most sophisticated treatment to date of e-HC1 scattering (pi VI.B.4), conclude that “the threshold peaks are ... a consequence of the polar nature of HCl.” This particular issue, at least, does not intrude on contemplation of spikes in nonpolar systems, to which we shall turn in pi V1.C.

2. Nondissociating Negative-Ion States

One of the earliest and most controversial explanations of the e-HC1 spikes was propounded by Taylor et al. (1977). In configuration interaction

~

N E A R - T H R E S H O L D E L E C T R O N - M O L E C U L E SCATTERING

137

(CI) calculations using the stabilization method (Fels and Hazi, 1971; Taylor and Hazi, 1970) several apparently stable roots of C symmetry appeared very close and parallel to the ground-state electronic energy curve of the neutral-roots that disappeared as R -+ w (Goldstein et al., 1978). Similar roots were found in the H F continuum in stabilization calculations by Segal and Wolf (1981). Using six such curves, Taylor et al. (1977) were able to qualitatively explain data for a plethora of scattering processes. This work was subsequently criticized by Krauss and Stevens (1981), who argued that these roots were, in fact, unstable artifices that appeared stable only because the basis set used in the stabilization calculations was insufficiently flexible. Nesbet (1977) also criticized the interpretation of Taylor et al., contending that the negative ion states they discussed were actually spurious manifestations (due to limitations inherent in the stabilization method) of a virtual state in the e-HCl continuum near zero energy. (See $ VI.B.3.) Subsequently Bettendorf et al. (1983) investigated the nature of these roots. Based on fully variational, large-scale CI calculations for the neutral and the negative ion, they concluded that "there seems to be no compelling reason to look upon any of the HF- and HCl- diffuse states with potential curves which parallel that of the neutral ground state as anything but typical freeelectron states represented in a discrete functional basis."42 Regarding the threshold spikes, Bettendorf et al. suggest that nonadiabatic effects due to a pole of the S matrix, as in the resonance or virtual-state models (9 VI.B.3 and VI.B.4) are more likely responsible. (See also Nesbet, 1981.) 3. The Virtual-State Conjecture

In $ IV.A.4 we saw that a virtual-state pole of the S matrix near zero energy can profoundly alter the near-threshold behavior of the eigenphase sum and cross section. Nesbet (1977) and Dub6 and Herzenberg (1977) first proposed that a virtual state might play an important role in vibrational excitation. Nesbet (1977), using fundamental analytic properties of the multichannel S matrix, showed that if the operator FNS&o(E- .#"',",R ) of Eq. ( 1 12) is weakly dependent on R (see Nesbet, 1979), then a virtual-state pole in the BF-FN scattering matrix near E , = 0 for a range of values of R about equilibrium can produce structure in C T , ~ at ~ ~each , vibrational threshold E,,. that intersects the neutral potential energy curve inside this range. (See also Domcke, 1981, and Gauyacq and Herzenberg, 1982.) Subsequent uh inilio C1 calculations on the bound states of HCI- by ONeil et al. (1987) produced potential energy curves that differ in detail from those of Bettendorfet al. (1983). These results do not, however, alter the quoted conclusions of the earlier study.

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Michael A. Morrison

Just such a virtual state plays a crucial role in the model calculations by Dubt and Herzenberg (1977) of crvo+v for e-HCl. The formulation used in this chapter-and subsequently adopted by Gauyacq and collaborators (see 0 VI.B.4)-extends the zero-range-potential approximation to incorporate long-range interactions. In this method, which Gauyacq calls the effectiverange approximation (ERA), space is partitioned into inner and outer regions, and the effects of the latter on scattering functions are encapsulated in a boundary condition on the logarithmic derivative of this function at the boundary.43 This logarithmic derivative is expanded in powers of R - Re, and the parameters in this expansion are fitted to experimental cross sections (Dube and Herzenberg, 1977) or to ab initio FN scattering data (Teillet-Billy and Gauyacq, 1984). The ERA assumes that a single pole of resides near zero on the imaginary k, axis. Moreover, in applications to e-HCl scattering, the further assumption was made that the final-state scattering function is spherically symmetric (s-wave), an assumption that is inconsistent with the conventional picture of a shape resonance. Unlike resonance localization, a virtual-state enhancement of the final-state probability density in the near-target region, which in the model of Dube and Herzenberg causes the near-threshold spikes, is not accompanied by a time delay.44 A challenge to the virtual-state conjecture for polar systems was raised by Domcke and Cederbaum (1981) and Domcke (1981), who showed that the presence of a long-range r - * tail in Fn,fundamentally alters the analytic properties of the BF-FN S matrix. This alteration “excludes the possibility of a virtual state” (Domcke and Cederbaum, 1981) by forcing the pole off the negative imaginary k, axis. In response, Herzenberg and Saha (1983) and Herzenberg (1984a) have argued that the physics of near-threshold e-HCl scattering nonetheless conforms to the theoretical picture of enhancement due to a virtual state and that the formal definition of such a state should be extended to incorporate this case. And threshold structures do appear in the model cross sections of Dubk and Herzenberg (1977). (See Fig. 21.) I shall return to the virtual-state model in 8 VLC, where I discuss systems where this ambiguity regarding the nature of the pole does not arise.

FNsA(R)

4. Applications of the N A R and ERA Methods

The NAR method and its underlying assumption-that a single discrete state of the projectile-target complex, embedded in and interacting with the 43 In the zero-range-potential approximation, this “boundary” is taken to be r = 0. See the discussion and references in the review by Kazansky and Fabrikant (1984). 44 For a detailed pedagogical discussion of this model, the reader is encouraged to peruse Herzenberg (1984a, 1984b).

139

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

a

ENERGY ( e V )

2 .o

b

N

E

I

1

1

.. * .

I e - HCI v.0-2

.* . .

I .5-

-

'i*-:

-

I.0-

1..

.

0

*.

0

'.O

-

I: 0.5 -

*-..

ID

I: n

0 . ..

*.

I *

I: I:

0

I

' 1: 1

*...

-.

/ &

\ \ \(

. ..

.----/ 0

/

/

*..

-----A.

0-

I

I

I

I

EN ERGY ( e V 1 FIG.21. The threshold spikes in vibrational excitation e-HCI cross sections for (a) v = 1 and (h) u = 2. Experimental data (circles) from Rohr and Linder (1975): theoretical data: virtualstate (dotted curve) (Dube and Herzenherg, 1977); nonadiahatic resonance theory (short-dashed curve) (Domcke and Miindel, 1985); effective-range approximation (solid curve) (Teillet-Billy and Gauyacq. 1984); body-frame VibrdtiOnd close-coupling theory ( I > = 1 only) (long-dashed curve) (Norcross and Padial, 1987).

140

Michael A . Morrison

continuum, becomes a short-lived resonance state-have been discussed in 6 V.C.2. Domcke and coworkers have applied this method to e-HC1 scattering in an increasingly sophisticated series of calculations (see Table XI) and have demonstrated the ability of this model to reproduce qualitatively the features observed in vibrational excitation of HCI: spikes near threshold and the accompanying broad enhancements at a few eV. Unlike its predecessors, the study by Domcke and Miindel(l985) does not determine its adjustable parameters by recourse to experimental cross section data. Rather, these (seven) parameters are determined via a least-squares fit to the ab initio BF-FN X eigenphase sums of Padial and Norcross. The model of the nuclear vibrations used in this study is also improved by parameterizing Eg(R) and E,(R) using Morse potentials rather than a simple harmonic oscillator. The optical potential (1 15) is approximated by a separable expansion, but nuclear dynamics in the electron-molecule complex are still treated nonlocally, a feature of the formalism that Domcke and Miindel characterize as “absolutely essential.” In this model, the dependence of the decay width T(E,, R ) on energy is critical to the threshold behavior of CT,,,,. Domcke and Miindel parameterize this width, introducing parameters A , B, C , and D in a way that is consistent with the threshold law for scattering by a (subcritical) dipole potential :

These model calculations reproduce both the threshold spikes in go+ (see Fig. 21) and C T and ~ the ~ broad ~ shape resonance near 3 eV. Domcke and Miindel ascribe the spikes to a bound ’Z’ state of HCI- near the neutral X’X’ potential curve of HCI-an explanation that echoes those of Taylor et al. (1977) (9 VI.B.2) and of Teillet-Billy and Gauyacq (1984) (to be discussed). In this model, the dipole tail plays an important role, stabilizing the negative ion state for a range of internuclear separations near equilibrium. But Teillet-Billy and Gauyacq (1984) calculate e-HCl cross sections that also show the desired structure (Fig. 21) using the ERA (gVI.B.3) and neglecting completely the dipole interaction. And, because it assumes the pole of the S matrix to be on the imaginary wavenumber axis, their method precludes the existence of a short-lived resonance state of the e-HCl complex. Both studies agree that energy exchange between the electron and the nuclear vibrations, which enhances C T , ~ , ~ near threshold, takes place via a nonadiabatic interaction. This nonadiabatic coupling is carried by parameters in either the logarithmic boundary condition (in the ERA) or in the level shift function (in the NAR method). Both authors note that their calculated cross sections exhibit high sensitivity to these parameters.

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

141

5. Where’s the Mechanism .?

The studies surveyed in this section usefully explore possible explanations of the spectacular near-threshold effects discussed in 4 VT.A, and all have contributed to the evolution of a quantum theory of near-threshold scattering. But the necessary implementation of simplifying assumptions and the use of parameterized potentials that often entail adjustment to experimental data render their findings regarding the physical mechanism at work in threshold structures less than d e f i n i t i ~ e .Resolution ~~ of this mystery may await exhaustive ah initio scattering calculations of a sort that are infeasible with present-day computers. Nonetheless, significant progress along the road to such calculations has been made by parameterizing ah initio fixed-nuclei scattering data. (Cf. the discussions of the NAR and ERA methods in Q V and VI.) But when that time comes, these studies have shown the way: such calculations will have to be based on a theory that incorporates nonadiabatic effects (or an off-shell theory such as those discussed in 5V.C) and will probably entail effects arising from poles of the BF-FN S matrix. MECHANISM I N NONPOLAR SYSTEMS C. THEVIRTUAL-STATE One of the better understood threshold peaks appears in the cross section for excitation of the symmetric-stretch mode (100) of carbon dioxide. Recent experiments by Kochem et al. (1985b) using a high-resolution (25 meV) crossed-beam apparatus produced accurate differential and integral cross sections for elastic scattering and vibrational excitation of all three fundamental modes of CO,. These data clearly show isotropy of do,,,, loo/dQ at low energies and a strong peak in o o o o + l oatoabout 250 meV, quite near the 172-meV threshold for this excitation. These findings are consistent with the virtual-state conjecture, as are the anomalously large increases in clotand omomas E , + 0. (See, for example, Ferch et al., 1981; Lowke et al., 1973; Szmythkowski and Zubek, 1977.) On the theoretical front, two independent and formally quite different studies, both based on BF-FN scattering data (Morrison, 1982) have located a virtual-state pole of the BF-FN S matrix. Analysis of very low energy C, eigenphase sums (see Fig. I l a ) using a MERT expansion about k, (see Q IV.B.4 and Eq. (107)) led to the discovery of a pole on the negative imaginary k, axis at k , = -0.1620i (Morrison, 1982). More recently, Estrada 4 5 For example, both Dorncke and Mundcl (1985) and Teillet-Billy and Gauyacq (1984) assume that near threshold, e-HCl scattering is s-wave-dominated at all R, an assumption that recently has been called into question by new BF-FN scattering data of Padial and Norcross (1987) and by the ab initio CI calculations of ONcil et al. (1987).

142

Michael A . Morrison

and Domcke (1985) used these BF-FN eigenphase sums to analytically continue FNS”(R) into the complex k , plane and found a pole at k , = -0.12793. Both poles correspond to a large negative scattering length, which is consistent with all the aforementioned experimental data. Estrada and Domcke used the NAR method (4V.C.2) to calculate ~oooobtaining impressive agreement with the measured cross sections of Kochem et al. (1985b). (See Fig. 6.) It is noteworthy that a large background contribution to 6f&,(R,) (see Fig. 18) characterizes scattering due to a virtual state but not due to a resonance (Nesbet, 1981; Domcke, 1983a). The virtual-state effects are contained in the rapidly varying term in Eq. (1 17). Estrada and Domcke further showed that as R increases from equilibrium, the pole moves up the imaginary k, axis, eventually becoming a bound-state pole. This behavior, in turn, is reflected in the variation of 6&(R) with R (Morrison and Lane, 1979); the slow but nonnegligible variation with R of this eigenphase sum conforms to Nesbet’s condition for a virtual-state pole in the F N limit causing threshold structures in vibrational excitation cross sections. But carbon dioxide has three fundamental vibrational modes: the Ramanactive symmetric stretch mode we have been discussing and the infraredactive bending and asymmetric stretch modes. Integral cross sections for the latter modes, which were also measured by Kochem et al. (1985b), are comparable in magnitude to a o o o ~ l o o - b u for t a completely different reason. The transition dipole moment of these infrared transitions is large enough to give rise to the observed cross sections via a simple, direct dipole mechanism; this explanation is supported by the angular distributions for the infraredactive modes, which are strongly peaked in the forward direction. A molecule that in many respects is analogous to CO, but that does not exhibit any of the anomalies that characterize this system is acetylene (C,H,). Acetylene is also linear and nonpolar in its (Dmh)ground electronic and vibrational state. It has two Raman-active (nonpolar) symmetric stretch modes, with thresholds at 418 meV and 245 meV; but unlike the excitation cross sections for the symmetric stretch mode of CO,, those for C,H, are negligibly small (on the order of cm2) at energies below about 2 eV (Kochem et al., 1985a). In toto, the experimental data for this system shows no evidence of the influence of a near-zero pole of the S matrix. On the other hand, data for electron scattering from methane (see references in Table X) show abundant evidence of the influence of a pole, paralleling remarkably the qualitative behavior of e-CO, scattering. Thus, e-CH, experiments have revealed anomalously large total and momentumtransfer cross sections at very low energies, isotropic differential cross sections for excitation of the Raman-active vibrational modes, theshold

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

143

spikes in cross sections for excitation of these modes, a companion shape resonance at higher energies (in CH,, a T, resonance at about 5 eV; in CO, a nuresonance at about 4 eV)-all features characteristic of the influence of a virtual state. To date, however, no one has mounted an expedition to find the hypothesized pole of the e-CH, BF-FN S matrix. Like CO,, methane is nonpolar, so it is at least reasonable to speculate that a similar mechanism may be operative in the two systems. But it is by no means clear that this mechanism is at work in polar systems such as e-HCl scattering. Indeed, one could summarize the current state of our theoretical understanding of threshold spikes in electron-molecule scattering with the following conjecture: for nonpolar systems, the interaction potential supports a virtual state that strongly influences integral, momentum-transfer, and Raman-active vibrational-excitation cross sections. For polar systems, an s-wave pole-near threshold, but not on the imaginary k axis-with properties similar to those of a conventional virtual-state pole (no trapping or timedelay of the projectile) may give rise to large threshold spikes and isotropic angular distributions. This is, of course speculation; the proof of the pudding will probably require more extensive, accurate, fully ab initio scattering calculations, especially on polar systems such as e-HCI.

VII. Conclusions and Conundrums After the dense theory of 4 I1 and 111, the elaborate methodologies of (i V, and the subtle arguments of (i VI, it may seem almost churlish to assert that near-threshold electron-molecule collisions are simple. But there is a beautiful underlying simplicity in the behavior of near-threshold cross sections and their physics-as explained, for example, by the threshold laws of (i IV-a simplicity that I hope has not gotten lost. Advances in experimental technology have given us glimpses of a hitherto unobserved realm, where there abound oddities like the near-threshold spikes of (i V1.A. Advances in ab inifio computer calculations, which have been somewhat slighted in this chapter, promise quantitative tests and extensions of these measurements, and, together with model-potential studies such as those in $ VI.B, will likely provide answers to nagging problems that remain. Solving these problems will require considerable effort (a) to develop theoretical and numerical methods for calculating nearthreshold cross sections in nonpolar and, especially, polar systems-methods that avoid the numerical pitfalls of the exact formulations of (i 11, the inaccuracy of the standard approximations of $111, and the reliance on

144

Michael A . Morrison

parameterized model potentials of 5 VI. Such methods must be carefully and extensively tested against benchmark data for systems that are amenable to more rigorous calculations; (b) to resolve the multitude of discrepancies-among the results of different experiments and between experimental and theoretical data-that lurk throughout the literature of near-threshold scattering (see, for example, Crompton and Morrison, 1986); (c) to discover the underlying physical mechanisms behind phenomena such as the threshold structures of § VI. None of these tasks is likely to be easy, but all are fascinating. And their accomplishment is sure to deepen our understanding of the fundamental quantum physics of electron-molecule collisions.

ACKNOWLEDGEMENTS

I owe an enormous debt of gratitude to an enormous number of friends and colleagues whose help, advice, and data have contributed to this chapter. Thanks to all who generously sent me preprints, unpublished data, and detailed answers to specific questions about their work --especially to Drs. David W. Norcross, E. Ficocelli Varracchio, J. E. Bloor, A. Herzenberg, I. I. Fabrikant, J. P. Gauyacq, Robert W. Crompton, Zoran PetroviC, Lee A. Collins, Wolfgang Domcke, and Neal F. Lane, as well as to members of Prof. Helmut Ehrhardt’s group at Kaiserslautern and members of my research group at the University of Oklahoma. For a thoughtful, detailed critique of this manuscript in preliminary form I am indebted to Dr. Steve Alston. Thanks also to the Joint Institute for Laboratory Astrophysics (JILA) for giving me access to an office, peace and quiet, and library facilities during the summer of 1986. Indeed, this chapter would not have existed at all had it not been for the tireless efforts of Dr. Jean Gallagher and her staff in the Information Center at JILA, who provided invaluable technical and, when necessary, moral-support. Finally, my special thanks to my wife, Mary, who put up with me during the months this chapter was in gestation. This work was supported by grant (PHY-8505438) from the National Science Foundation.

Appendix A: Notation and Nomenclature Jargon, acronyms, and symbol-laden notation permeate the literature of electron-molecule scattering theory-including, alas, this chapter. This appendix is intended as a guide to the most important symbols and terms introduced in it. Along with a few defining words, each term or symbol comes equipped with the number of the section in which its definition (though not necessarily first reference) can be found.

I45

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

NOMENCLATURE

AN ANR ANV BF-FN BFVCC CAM EMA ERM ERT FBA FN FNO FONDA MEAN MERT NAR RFT RR SANR UCAM VFT

adiabatic nuclei (approximation) adiabatic nuclear rotation (method) adiabatic nuclear vibration (method) body-frame fixed-nuclei (method) body-frame vibrational close-coupling (method) coupled angular momentum (representation) energy-modified adiabatic (approximation) eigenchannel R-matrix method effective-range theory First Born approximation fixed nuclei (approximation) fixed nuclear orientation (approximation) first-order nondegenerate adiabatic (approximation) multipole-extracted adiabatic-nuclei (approximation) modified effective-range theory nonadiabatic resonance (theory) rotational frame transformation rigid rotator (approximation) scaled adiabatic-nuclear-rotation (method) uncoupled angular momentum (representation) vibrational frame transformation

ij 1ll.D

9: 1II.A 0 I1I.C ij 1II.B ij I1I.A

6 II.B.3

9 v.c.1

ij V.C.3

01v.b 9 V.A 9 1II.B. 9III.A 9 V.D.2 4 V.B.l ij 1V.B 4 v.c.2 4 I1.D 0 1II.B 5 V.B.2 ij II.B.2 4 II.D.4

NOTATION Variables, Wavenurnbers, and Reference Friimes

r

R R

LAB reference frame B O D Y reference frame electron spatial variables (LAB frame) electron spatial variables ( B O D Y frame) coordinates of molecular axis (LAB frame) angular coordinates of molecular axis (LAB frame) wave number of electron (entrance channel) wave number of electron (exit channel)

1I.B 1I.C.1 I1.B 1I.C

I1.B I1.B

IT

146 kP

Y

Michael A . Morrison location of pole of BF-FN S matrix IV.A.4 location on imaginary k , axis of pole IV.A.4

Operators, Energies, and Hamiltonia kinetic energy of electron electron-molecule interaction potential sum of two-particle bound-free Coulomb interactions static interaction potential (nonlocal) exchange potential molecular Hamiltonian electronic Hamiltonian of target nuclear Hamiltonian of target rotational Hamiltonian vibrational Hamiltonian Hamiltonian of the electron-molecule system (Born-Oppenheimer) electronic energy of the target energy of a ro-vibrational target state v incident energy of projectile

internuclear separation spherical polarizability nonspherical polarizability permanent dipole moment permanent quadrupole moment

1I.A

Eq. (7)

SS.A

S1.A Eqs. (9) 1S.A 1I.A 1S.A sr.c.5 II.C.5 Eq. (3) 1I.A I Fig. 1

everywhere Eq. ( 8 2 ) Eq. (82) Eq. ( 8 2 ) Eq. ( 8 2 )

Various and Sundry Angular Momenta and their Eigenjiunctions orbital angular momentum of S1.B electron spherical harmonic of order 1 (LAB Eq. (13b) frame) spherical harmonic of order 1 (BODY Eq. (39) frame)

NEAR-THRESHOLD ELECTRON-MOLECULE SCATTERING

rotational angular momentum of molecule rotational wave function rotational wave function for D,, molecule total angular momentum .i = 7 + 1

147

1I.B Eq. (38) II.B.3

II.B.3

Physicui Kets und Diverse Waiw Functions

lk") Iko, v o )

Ik,,

Yo

Lyk".

+)

"R )

XY0(R)

cpl"(R)

single-particle (asymptotic) planewave state asymptotic free state of electronmolecule system scattering state of system electron-molecule wave function (LAB frame) ro-vibrational nuclear wave function vibrational wave function

Eq. ( 1 3a) I1.A

1I.A II.B.l I1.B. 1 II.B.4

In the LAB Representation

LAB scattering amplitude LAB (physical) T matrix LAB-UCAM scattering state LAB-UCAM asymptotic free state LAB-CAM asymptotic free state LAB-CAM scattering state Basis for LAB-UCAM expansion Basis for LAB-CAM expansion (24c) Basis functions for LAB-CAM expansion system wave function in the LAB-UCAM formulation system wave function in the LAB-CAM formulation coupled angular functions of LAB-CAM theory normalized LAB-CAM radial function unnormalized LAB-CAM radial function LAB-CAM T matrix LAB-UCAM T matrix

II.B.1 Eq. (17) II.B.2 II.B.2 II.B.3 II.B.4 II.B.3 Eqs. (23) Eqs. (23) III.B.2 IlI.B.3 Eq. (23b)

Eq. (24b) Eq. (24b) Eq. (35a) Eq. (20)

148

Michael A . Morrison

In the BODY Representation the BODY energy the BODY wave number k , = asymptotic free state in BODY representation BODY scattering state system wave function in BODY representation Basis I for expansion of BODY scattering ket Basis I1 for expansion of BODY scattering ket normalized BODY radial function (Basis I) unnormalized BODY radial function (Basis I) normalized BODY radial function (Basis 11) unnormalized BODY radial function (Basis 11) eigenphases for electron-molecule scattering eigenphase sum for symmetry A at R e

fi

Body-Frame Vibrational Close-Coupling Theory (6 1II.A) IE, ~ 0 1 0 A ; +) FNO A , ,

U" I ,v 0 l o ( r )

FNO

A

Tul,"010

scattering state (FNO approximation) unnormalized radial function BFVCC T matrix

Body-Frame Fixed-Nuclei Theory (DII1.B) IE,, 10; A + )

FNYkb(r; Re) FN A

U r ,1 0 k

F N ~ A R,) ( FN%lo(Re)

Re)

scattering ket (FN approximation) FN wavefunction in the BODY representation at Re unnormalized BF-FN radial function BF-FN T matrix typical element of the BF-FN T matrix

NEAR-THRESHOLD ELECTRON-MOLECLJLE SCATTERING

FNf'(k,i

c

kbio;Re)

FN scattering amplitude in BODY frame

Frame Transjormation Theory &(PR,

operator that rotates LAB quantities into the BODY frame matrix element of the rotation operator rotational frame transformation radius of rotational frame transformation radius of vibrational frame transformation

eR)

g!nA((PR*

OR)

A% rir)

ri")

Adiabatic Nuclei Theory T matrix in ANR approximation T matrix in full A N approximation Diverse Cross Sections

differential cross section for excitation v o -+ v integral cross section for excitation vo + v

integral momentum-transfer cross section for excitation vo + v total cross section in fixed-nuclei approximat ion

Lt'+'(kl, v

+-

kb,

"0;

(s-wave) scattering length energy of discrete state in NAR theory shift of energy of discrete state in NAR theory width of discrete state in NAR theory R) (physical) off-shell T-matrix element (LAB frame)

149

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Appendix B: Recent Reviews of Electron-Molecule Scattering New reviews and collections of articles about electron-molecule scattering seem to appear almost daily. The reader interested in delving further into this field can find extensive annotated lists of many of these in Morrison (1983) and Csanak et al. (1984). Useful for their coverage and currency are a tutorial introduction (Morrison, 1983) as well as reviews by Norcross and Collins (1982) of collisions in polar systems, by Trajmar et al. (1983) of experimental data, by Burke (1980) and by Lane (1980) of the entire field, and by Gianturco and Jain (1986) of polyatomic systems. The scope of these articles is quite broad, and they fill many gaps in the coverage of the present chapter. In addition, these are several book-length collections of articles on electron-molecule scattering. Of the many useful articles in the two-volume set “Electron-Molecule Interactions and Their Applications,” ed. L. G. Christophorou (New York: Academic, 1984), the reviews by Csanak et al. on elastic scattering, by Trajmar and Cartwright on excitations, and by Hasted and Mathur on resonances-all of which appear in Volume I-may be of particular value to readers interested in low-energy collisions. This volume also includes current reviews of rearrangement processes such as dissociative attachment. More detailed articles on many of the topics covered in the present chapter can be found in the collection Electron-Molecule Collisions,” ed. I. Shimamura and K. Takayanagi (New York: Plenum, 1984); especially recommended are the introduction to the field by Takayanagi and articles by Shimamura on rotational excitations, by Herzenberg on vibrational excitations, by Compton and Bardsley on dissociative collisions, and by Buckley et al. on the computational techniques widely used by theorists in pursuit of accurate cross sections. Finally, and at a more technical level, five collections of workshop papers can be recommended. For discussions of computational techniques, see Electron-Molecule and Photon-Molecule Collisions,” ed. T. Rescigno, V. McKoy, and B. Schneider (New York: Plenum, 1979). For in-depth surveys of applications and experimental techniques, see “Electron-Molecule Scattering,’’ ed. Sanborn C. Brown (New York: Wiley, 1979). For overviews of current research, see “Electron-Molecule Collisions and Photoionization Processes,” ed. V. McKoy, H. Suzuki, K. Takayanagi, and S. Trajmar (Deerfield Beach, Florida: Verlag Chemie International, 1983); “Wavefunctions and Mechanisms for Electron Scattering Processes,” ed. F. A. Gianturco and G. Stefani (New York: Springer-Verlag, 1984), and “Swarm Studies and Inelastic Electron-Molecule Collisions,” ed. L. C. Pitchford, V. McKoy, A. Chutjian, and S. Trajmar (New York: Springer-Verlag, 1986). “

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Rohr, K., and Linder, F. (1975). J . Phys. B 8. L200. Rohr, K.. and Linder, F. (1976). J . Phys. B 9,2521. Rose, M. E. (1957). “Elementary Theory of Angular Momentum.” Wiley. New York. Rudge, M. R. H. (1978a). J . Phys. B 11, 1503. Rudge, M. R. H. (1978b). J . Phys. B 11, 2221. Rudge, M. R. H. (1978~).J . Phys. B 11, 1497. Rudge, M. R. H. (1980). J . Phys. B 13, 1269. Salvini, S., Burke, P. G., and Noble, C. J. (1984). J . Phys. B 17, 2549. Schneider, B. I. (1975). Phys. Reu. A 11, 1957. Schneider, B. I. (1976). Phys. Rev. A 14, 1923. Schneider, B. I. (1977). Chem. Phys. Lett. 51, 578. Schneider, B. I., and Collins, L. A. (1984). Phys. Rev. A 30,95. Schneider, B. I., Le Dourneuf, M., and Burke, P. G. (1979a). .I. Phys. B 12, L365. Schneider, B. I., Le Dourneuf, M., and Lan, V. K. (1979b). Phys. Ree. Lett. 43, 1926. Schulz, G. J. (1973). Rev. Mod. Phys. 45, 423. Schulz, G. J. (1967). In “Principles of Laser Plasmas”, (G. Bekefi, ed.), Chapter 2. Wiley, New York. Segal, G. A,, and Wolf, K. (1981). J . Phys. B 14, 2291. Seng, G., and Linder, F. (1974). J . Phys. B 7,L509. Seng, G., and Linder, F. (1976). J . Phys. B 9, 2539. Singh, Y . (1970). J . Phys. E , 3, 1222. Shimamura, I. ( I 984). In “Electron-Molecule Collisions” (1. Shimamura and K. Takayanagi, eds.), Chapter 2. Plenum Press, New York. Shugard, M., and Hazi, A. (1975). Phys. Reti. A 12, 1895. Sohn, W., Jung, K., and Ehrhardt, H. (1983). J . Phys. B 16, 891. Sohn, W., Kochem, K.-H., Jung, K., Ehrhardt, H., and Chang, E. S. (1985). J . Phys. B 18,2049. Sohn, W., Kochem, K.-H., Scheurlein, K.-M, Jung. K., Ehrhardt, H. (1986). J . Phys. B 19, 3625. Sohn, W., Kochem, K.-H., Scheurlein, K.-M., Jung, K., Ehrhardt, H. (1987). J . Phys. B 19, 3625. ~ 10, L31. Szmytkowski, C., and Zubek, M. (1977). J . P h j B Takayanagi, K., and Itikawa, Y . (1970). Ado. At. Mol. Phys. 6, (1970), 105. Taylor, H. S. and Hazi, A. U. (1970). Phy. Reo. A, 1, 1109. Taylor, H. S., Goldstein, E., and Segal, G . A. (1977). J . Phys. B 10, 2253. Taylor. J. R. (1972). “Scattering Theory.” Wiley, New York. Teillet-Billy, D., and Gauyacq, J. P. (1984). J . Phys. B 17, 4041. Temkin, A., and Faisal, F. H. M. (1971). Phys. R ~ LAZ 3, 520. Temkin, A., and Sullivan, E. C. (1974). Phys. Reu. Lett. 33, 1057. Temkin, A,, and Vasavada, K. V. (1967). Phys. Rm. 160, 109. Thompson, D. G. (1966). Proc. Roy. Soc. A294, 160. Thompson, D. G . (1983). Adv. At. Mol. Phys. 19, 309. Trajmar, S., Register, D. F., and Chutjian, A. (1983). Phys. Rept. 97, 221. Vo Ky Lan, Le Dourneuf, M., and Schneider, B. I. (1979). In “ICPEAC, Ilth,” p. 290. NorthHolland, Amsterdam. Wada, T., and Freeman, G . R. (1981). Phys. Rev. A. 24, 1566. Wallis, R. F., Hermann, R., and Milnes, H. W. (1960). J . Mol. Spectrosc. 4, 51. Weatherford, C. A., and Henry, R. J. W., (1977). Can. J . Phys. 55, 442. Whitten, B. L., and Lane, N. F. (1982). Phys. Reu. A 25, 3170. Wigner, E. P., and Eisenbud, L. (1947). Phys. Rev. 72, 29. Ziesel, J. P., Nenner, I., and Schulz, G. J. (1975). J . Chem. Phys. 63, 1943.

ADVANCES IN ATOMIC' A N D MOLECULAR PHYSICS. VOL 24

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION OF ATOMS S. J. SMITH Joinr Institute fiir Lahorulory Astrophysics Unioer.siry of'Colorado und Nuriortul Burn114o f Srundurds Boulder. Coiorudo 80309-0440

G. LEUCHS M a s Plunck lnslirui fur Quunti~noprikund Sekrion Physik der Unioersirur Miindim D-8046 Gurching. Wesi Germony

I. Introduction . . . . . . . . . . .

.

. . . , . 111. Experimental Methods . , . . . . . . , , . . . A. Optical Excitation . . . . . . . . . . . . . . B. Measurement of Angular Distribution , . . . . . , C. Detection of Spin Polarization , . , . . . . . , D. Electron Energy Analyzers . , . . . . . , . . . E. Data Analysis . . . . . . . . . . . . . . IV. Applications to Atomic Structure and Dynamics . . . . A. One-Photon Excitation Plus Ionization . . . . , . B. Two-Photon Sequential Excitation Plus Ionization . . ,

.

11. Theory of Photoelectron Angular Distrihutions . . .

C. D. E. F. G. H.

. . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . .

.

,

,

.

. . . .

. . ,

. . . . . . . Two- and Three-Photon Resonant Excitation Plus Ionization . . . Higher Multipoles in the Bound-Continuum Transition. . . . . . Nonresonant Multiphoton Ionization . . . . . . . . . . . . Perturbations o f the ti 1 . 3 D 2Rydberg Series in Barium . . . . . . Perturbations of the 6sns 'So Rydherg Series in Barium. . . . . . The 5d7s ID, State in Atomic Barium . , . . . . . . . . . . . .

.

1. Other Doubly Excited Configurations in the Alkalmc Earths . . J. Autoionizing Resonances in Alkaline Earth Atoms , . . . . K . Laser-Intensity Dependence in Resonant Multiphoton Ionization V. Conclusions . , . . , . . . . . . . . . . . . . . . Acknowledgments . . . . . . , . . . . . , . , . . . References . . . . . . . . , . . . . . , . . . . . .

. .

. . .

157 161 174 . 174 . 176 . 178 , 180 . 182 . 183

. . .

.

. . . . . . . . . . . . . .

. .

183

189 , 192 . 194 .

.

195

201 204 207 209 210 212 216 217 2 18

I. Introduction

in

This chapter IS primarily concerned with the study of angular correlation multiphoton ionization in atomic systems. The topic of one-photon I57

,

19XX hy A L . i l l r m i L Pres, I n L Copyrighi 411 rlghlr 01 ~ r p n d u ~ l i o111 n an) lorm rmerved l \ B N 0 I?(K)3X?J 2

S. J. Smith and G.Leuchs

158

ionization angular distributions from randomly oriented atomic systems was comprehensively reviewed in a pair of articles by A. F. Starace (1982) and J. A. R. Samson (1982), covering theoretical and experimental work through 1978. The reader is referred to those papers and the references contained therein. However, this discussion will be restricted to the multiphoton case, which was not considered by Starace or Samson, and which has been the subject of increasing interest since 1978 through the development of experimental techniques based on the use of polarized laser beams to resonantly and sequentially excite and photoionize atoms in an atomic beam in high vacuum. Some of this development has been reviewed by Leuchs and Walther (1984). In the simple case of single-photon photoionization from an ensemble of atoms randomly oriented in space, treated in the electric dipole approximation, the angular correlation involves only the axis of polarization of the radiation field and the direction li= (k/k) of the momentum vector k of the photoelectron. For linearly polarized radiation, the direction of linear polarization 2 may be used as the reference axis, and the direction I? may be represented in terms of spherical polar coordinates 0 and CD where cos 0 = P.2. Figure 1 illustrates these vector relationships. The angular distribution then takes the form (Bethe and Salpeter, 1957; Cooper and Zare, 1968, 1969)

I(R)

=

I(@, CD)

= I(@)

=

TOTAL 471

c1

+ /3P,,(C0S 011

where the axial symmetry with respect to 2 is explicitly recognized. The Legendre polynomial is P,, = [($) cos2 0 - (1/2)] and the distribution is commonly written in the form:

I(@)

= c1

+ /3 cos2 0.

This distribution is shown by Cooper and Zare (1968, 1969) to apply to photoionization by one photon, from any atomic state for which all orbital magnetic substates ( M L )are equally populated, or for photoionization from any electron s-orbital (1 = 0) in the central field approximation. The correlation becomes more complex, and consequently more interesting, for multiphoton processes: the correlation then involves the states of polarization of more than one photon. Furthermore, some of these photons may be resonant with transitions between bound states. The case of k', -1 photon excitation, where these ,,No- 1 polarized photons are resonant with successive steps between stationary states leading from an isotropic ground state to a highly excited intermediate state, from which the electron is photoionized by the photon, is of particular interest. If electron spins in the initial state are assumed to be randomly oriented and if the spin

Nt

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

159

f'

X

FIG. 1. Coordinate system showing the representation in spherical coordinates of the wave vector k of the photoelectron and of the radius vector r of the wave function of the electron. (k is to be distinguished from K, the wave vector of the photon.)

polarization of the outbound electron is not measured, the correlation may be described uniquely by the following experimental parameters: (1) the alignment of an atom in the "intermediate" state (this alignment being traceable to the polarizations of one or more photons involved in the stepwise excitation process), (2) the polarization of the ionizing radiation field, and (3) the direction of the k vector of the outbound photoelectron. In this case the angular distribution of photoelectrons requires a more general formulation than Eq. (1). I t becomes an expansion in spherical harmonics (Yang, 1948) .AC

I(@, @)

=

+ 2K

c c

IJ2K.M Y2KSM(@3

@I

Pa)

K=O M=-ZK

where the expansion terminates with , 1 I , V ofor a multiphoton ionization process of order N o . If all the photons are linearly polarized along the same axis, axial symmetry is preserved and one may write '

.n

I(@, @I = I(@)

=

c I&

0 Y2K.

(2b)

K=O

as in the case of multiphoton ionization of an alkali or alkaline earth atom from the ground state ( M , = f 1/2 or M , = 0, respectively). This is equivalent to a description of one-photon ionization of an atom in a definite alignment M , = & 1/2 or 0, since A M , = 0 for each of the .NA0- I absorbed

S. J. Smith and G. Leuchs

160

photons in linearly polarized light. The parity of the excited state is n = ( - l ) N o - ' , since a photon has odd parity; and the orbital angular momentum L I N o - 1 is the sum of A L for the ..Vo - 1 successive transitions, these being determined by spectroscopic selection of the path of sequential excitation. On the other hand, the highest term in the expansion Eq. (2a) is that for which JV is one larger than the difference between the number of AJ = + 1 and AJ = - 1 transitions in the spectroscopically selected route to the intermediate state. The process in a light alkali atom 3S,,,

4

3P,,,

-,continuum

is represented by an expansion to Y40, but the excitation and ionization sequence 3S1/, -+ 3P,,, -+continuum

is correctly represented by an expansion to only Y20.Examples from atomic sodium ionization measurements corresponding to these two paths to the continuum are shown in Fig. 2. S

P

D

s';;i

intermediate

sodium

FIG.2. (a) Level diagram illustrating two-photon ionization through the 3'P,/, level of atomic sodium into a superposition of the S and P continuum states (dipole selection rule, AL = k 1). The 3'P,/, hyperfine structure, coherently excited, introduces a time dependence to the angular distributions as discussed in Section IVA of this chapter. (b) Polar plots of photoelectron angular distributions obtained in two-photon photoionization through the 3'P,,, and 3 , P , , , states. The crosses ( x ) are measured points (Hellmuth rt al., 1981) and the solid lines are best fits of spherical harmonic expansions (Eq. (2b)). The larger anisotropy obtained using the 3'P,,, intermediate state is evident.

ANGULAR CORRELATION IN M U L T I P H O T O N IONIZATION

161

In principle, the representation of angular distributions of photoelectrons is a quite straightforward exercise in angular momentum algebra for the case of an intermediate state that is a pure stationary state in the single-particle picture. The interesting physics is largely in the occurrence of deviations from this picture. It becomes more interesting when correlated electron motion is taken into account, for example. In this case, the angular distribution measurement comprises a tool for characterizing angular momentum properties of mixed states. Examples of this are found in photoionization from states in a one-electron Rydberg series in barium that are coupled to nearby doubly excited states, resulting in perturbations of the Rydberg series, as are strikingly evident in the dependence of the angular distributions on principal quantum number (Matthias et a]., 1983; Leuchs and Smith, 1985); and states consisting of coherently excited superpositions of hyperfine levels, for which the angular distributions evolve in time (quantum beats) (Strand et al., 1978; Hansen et al., 1980; Leuchs et al., 1979). A very different case of interest arises with the introduction of additional spherical harmonic components due to the intrusion of processes of higher order in K . T where K is the wave vector of the incident radiation (e.g.. a significant contribution from a quadrupole term; Leuchs et al., 1986) or of processes of higher order in intensity (e.g., a second-order contribution to the stepwise excitation and ionization processes at very high intensities of the laser field; Leuchs et al., 1985). In this chapter, the primary emphasis is on the experimental method and on the status of experimental research, for example, on properties of intermediate states, based on angular distribution measurements. However, it is appropriate to summarize the theory of photoelectron angular distributions and some of the principal recent contributions to it by various authors.

11. Theory of Photoelectron Angular Distributions The comprehensive and very readable review of the theory of photoionization by Starace (1982) summarizes developments through the end of 1978. That review followed a decade of progress, beyond photoionization calculations based on independent-particle models, into theories capable of taking into account correlations in the dynamical behavior of outer electrons. Some significant successes have come out of this, but, as Starace noted, the area of successes is limited and the investigation of effects of correlation on electron dynamics is in its infancy. This is certainly true with respect to the theory of photoelectron angular distributions. The multichannel quantum defect theory (MQDT) (Seaton, 1983), which incorporates short-range correlation

162

S. J. Smith and G . Leuchs

effects, has been referenced extensively (Aymar, 1984) in connection with measurements of photoelectron angular distributions of the alkaline earths, but there has not been much direct application, even of this theory, to the calculation of angular distributions. Rather, experimentally determined parameters of angular distributions have often been compared to predictions of independent particle models as a basis for inferring the nature of electron dynamical correlations by identifying components of the resulting “mixed” states (Kim and Lambropoulos, 1984). Starace’s review deals with single-photon ionization of isotropic states in dipole-allowed transitions in the low-photon energy (nonrelativistic) regime. A number of authors have worked on the extension to various higher-order cases. We outline the one-photon theory and cite further developments into the multiphoton case. We consider the description of the photoionization processes on the basis of a Hamiltonian

H = HA

+ Hin,

(3)

where

represents a nonrelativistic, spin-independent, N-electron atomic system with nuclear charge 2 ; and

represents the interaction of the atom with a radiation field. Here 2 is the direction of photon polarization, and hw is the photon energy. The ionization field is treated classically, and the dipole approximation has been invoked (eiK‘r 2 1). By use of time-dependent perturbation theory, one arrives at an expression for the differential photoionization cross section

where $; is the wave function of the continuum electron with wave vector k, and $o is the initial state of the N-electron atom. In the “length” formulation the differential cross section may be written (Schlicher et al., 1984)

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

163

Here we take $o = R(r)Y(O,cp) as the independent particle solution of Schrodinger's equation in spherical polar coordinates. (A convenient axis of quantization in the photoionization problem is the axis of polarization of the radiation field.) More specifically Go may be any stationary state with good quantum numbers n, I, and m such that $0

(8)

= II/nlm = Rn/(r)I;m(d,

CP)?

where the I;, are spherical harmonics. The dependences on polar angle (0) and azimuthal angle (cp) may be separated:

where the Pr(cos 0) are unnormalized associated Legendre functions. The continuum wave function I/; with the appropriate asymptotic behavior for the photoionization case is a superposition of an incoming spherical wave and an outgoing plane wave

where 8' is the angle between k and r. This wave function has axial symmetry with respect to k and may be expanded in Legendre polynomials Pi (cos 0') x

I/;

-

=

1 arPi(cosO')Rkr I=n

with the following asymptotic behavior for large radii r+m

a,

C a;p,(cos @)(2jkr)- fI e i [ k r - ( W 2 ) 1 . eih 1

-

C.C.},

(11)

i=o

where C.C.represents the complex conjugate of the first term in brackets. Similarly, the term ei"" in Eq. (10) may be expanded

Using Eq. ( 1 2 ) in (10) and comparing the coefficient of the outbound spherical wave with that in Eq. ( 1 l), one obtains u; = i'(21+ l ) ~ ~ ' so !, x,

I/;

cc

1 i'(21+

l)r-'"P,(cos 8')Rkr(r)

r=o

and, finally, we use the spherical harmonic addition theorem

(13)

S. J. Smilh and G . Leuchs

164

This yields a final state wave function (Starace, 1982) 1 $;(r) = p

+I

a,

C C 1=0 m =

i1e-i6’Rkl(u>Ylm(P)Y;F,(I;)

( 1 5a)

-I

in which r^ and i may be referenced to the axis of quantization, as indicated in Fig. 1. We may write

Application of Eq. (7) then involves integration over F(0, cp) and gives the photoelectron angular distribution as a function of i(0,@) in this approximation. Under the dipole operator r, and by projecting onto the initial state $, = $nlm we obtain an expression (for the one-electron case) 8 . ($nlm[rl$;)

K

C ak”m’i”e-i6~, nlm Y;,m@, @)

(16)

I‘m’

where primes distinguish the final (continuum) state. The number of nonvanishing terms on the right side of Eq. (16) is reduced to two by invoking selection rules on 1 and m.The dipole-selection rule holds in this approximation,

A1=1-1’=+1. Furthermore, we restrict the present discussion to the experimentally convenient case of ionization with linear polarized light. Most recent rneasurements of photoionization angular distributions have used this method, in which the angular distribution is measured in the plane transverse to the laser beam. The axis of quantization is determined by the direction of the electric vector 2. The angle 0 is then the angle between the axis of quantization and the axis of the photoelectron detector. The angle 0 may be varied by rotating the electric field vector with a half-wave plate, while keeping the detector fixed in laboratory space. This geometry will be described in the next section. With this experimental configuration, the interaction operator in the dipole approximation is proportional to 6 . r = Y cos 0. Since this has no cpdependence, all terms on the right-hand side of Eq. (16) will vanish except those for which Am = m

-

m’

= 0.

Then Eq. (7) leads to 2.($nlm/rl$;) oc [a:i:l*m ~ 1 + ’ ( e - i 6 ~ + ~ ) Y I *@)f l , m ( ~ ,

+ a:iL-lsm i ’ - l ( e - i ~ l - l ) Y ~ - l , m(D)]( O ,

(17)

165

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

where the coefficients ~i:;,:.~ depend on the radial integrals

[&(r)Rk,l

f

1(r)r2 dr,

as well as on integrals over H and q. In the particular case I = 0 (the electron is initially in a pure s state) or for any case rn = I, the second term in Eq. (17) is omitted. The angular distribution, from Eq. (7), for these cases has the forms

and, more generally,

For 1 > 0 and M < I, two channels are open and Eq. (7) involves the absolute square of Eq. (17),

k It 1 . m * k . l - l , m - [anin anlm

@)x-,,,,(@,

Yl*,I.fll(O,

+ C.C.],

@)ei@l+l-hJ-”

where the interference term can be written

-

k,l+l.m *k.l-l.m

-2Ianrm

anlm

l.lYi++~.m(@~@ ) X - l , m ( @ *

@)Icos(~ , +, -6I-1).

Thus, the angular distribution measurement is unique in that it contains information about the scattering phase. This information is lost in measurement of the total photoionization cross section, since the interference term vanishes upon integration over 0 and @ due to orthogonality of the Equation (19) can be reduced by applying the identity

x,m.

v

v L...

to the products

I Y , * , . m ( @ , @)Iz>

YT+l,m(Q-

Y +l , m ( @ ,

@)K-i,m(@>

and

@)Y,*_1 . m ( @ , @I.

Each product yields terms Y,,(@,cD). The second 3:j symbol in Eq. (20) vanishes unless I , f 2 + L is even and satisfies the triangle rule. Since I , + I ,

+

166

S. J. Smith and G. Leuchs

+

is even for every case, L must have even values L II , I,. Then Eq. (19) easily reduces to the form given in Eq. (2b). Since M = 0, the angular distribution can be represented in terms of Legendre polynomials P,(cos O),

The point should be emphasized that our discussion has been directed to the case of one-photon ionization of an intermediate state (n, 1, m), where n, 1, and m may have any single-integer allowed values. It may be an excited state, with some specific angular momentum quantum numbers. We have assumed that photoionization is accomplished with linearly polarized light and that the dipole approximation holds. We have neglected electron spin and also any angular momentum or spin coupling between the core and the photoelectron. The z-component of the dipole matrix element is nonvanishing only if m=m', and the azimuthal dependence of the wave functions ei*@ will disappear in every term in the $*z$ product. The photoelectron angular distribution is then independent of a. In general, the intermediate state is not so simple, but may be a coherent mixture of states (n, I, m). This situation readily occurs if the intermediate state Ynlmis not a ground state: (1) It may be a superposition of orbital configurations due to spin-orbit coupling, (2) It may be a state perturbed through configuration interaction, (3) It may be a superposition of states not spectroscopically resolved by the radiation field that induced excitation from the ground (or other lower) state, or (4) It may be a superposition of states resulting from choices of different axes of quantization in the excitation and ionization steps.

The process of resonant multiphoton excitation to reach the intermediate state, which may involve absorption of one or more photons of polarized light, generally produces a nonequal distribution of populations of the magnetic substates, m, of the intermediate state. As a result, the intermediate state will have some degree of alignment. In each such case, the angular distribution calculation requires averaging over all initial states, taking account of relative phases between states that mix. However, the final state may be more complicated than that just considered due to coupling between the outbound electron and a nonspherical core. In certain cases, primarily photoionization of ground-state atoms into the low continuum, spin-orbit coupling for the outbound electron may also be important. In such cases, the initial and/or final states must be recoupled in terms of orbital quantum numbers.

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

167

The total photoionization amplitude is then obtained by summing over final states for each initial orbital. With the squaring of the sum of the individual amplitudes to obtain the final expression for the differential cross section, cross-terms are generated that, in general, do not vanish in the differential cross section. This crucial phase information is accessible through angular correlation studies. On the other hand, the total cross section is obtained from the differential cross section by integrating over 4n solid angle. In this case the cross terms vanish due to orthogonality of the spherical harmonics. No phase information is retained. This fact is significant motivation for the study of angular correlations. The excitation-ionization ladder illustrated in Fig. 2 provides a simple example of the way in which an angular distribution from photoionization into more than one channel works out. Figure 2 illustrates two-step photoionization of atomic sodium. The choice of the two routes to the continuum may be made by tuning a laser into resonance with one or the other transition. A second laser, linearly polarized parallel to the first, is used to photoionize. Here, we assume that electron spins are not measured and are isotropic in the ground state (32Slj2).Because of spin-orbit coupling, J is a good quantum number in the intermediate state. The selection rules are A L = & 1 ; AJ = 0, f 1; and, with linear polarized light, A M , = 0 and AML = 0. Therefore, the two possible intermediate states are 32P1i2,* 1 , 2 and 32P,,2. l , 2 . Either intermediate state may be represented in terms of orbital functions, using Clebsch-Gordan coefficients. It is instructive to follow the development of these two simple cases. First, we consider photoionization of an intermediate state, nPlj2.+ l,z, which is represented as a superposition of orbitals

$(n2Pli2.1 / 2 ) $(n2P1,2, -

112)

+ ) + ( $ P 2 Y l l ( Rcp)I - >I CP)I+ ) + (f>'iz Yio(6, CPI - >I

R , l C - ( 3 ) ' i 2 Y l o ( ~cp)I ,

R,IC-(;)''~ Yi,

i(o9

(22a) (22b)

where I + ) and I - ) are orthonormal electron spin functions. In the calculation of the matrix elements connecting these states to the continuum, only a few terms in the continuum wave expansion, Eq. (1 5), need be considered. This is due to selection rules AL = 1 and AM,- = 0. The new element is the presence of the terms in Y, , ( O , cp) and Yl l ( B , cp), which arise from spin-orbit coupling and which connect to the continuum under the same rules. The continuum terms that may be reached from the n2P,,,, 1i2 state are therefore ~

1

$Loo =

k"z e - i 6 " R k o ( ~ ) y o oCP)*( ~ ~Yo,(@, @)I + >.

(23a)

168

S. J. Smith and G . Leuchs

and

Evaluation of the matrix elements of the operator, z = r cos 0, is straightforward and leads to corresponding bound-continuum transition ampiitudes

for continuum electrons with spin functions I

+ ), and

for continuum electrons with spin functions I-). We assume that the 32P,,,, + 112 substates are reached from the 3's1/2, It 1 / 2 ground substates, with light linearly polarized on the z-axis. Then, if the ground state is isotropic, so will be the 3 2 P , i , state: 32P1/2. and 3'P1/2* - are populated equally. The corresponding bound-continuum transition amplitudes arising from the 3 2 P , i 2 .- substate are

,,,

for continuum electrons with spin functions

I - ), and

+

for continuum electrons with spin functions I ). Since the states M J = rf- 1/2 are incoherently populated in the ground state, one has to add the differential cross sections and not the amplitudes to obtain the cross section. In each case, M j = rf- 1/2, the differential cross section is obtained by summing the absolute squares of U + and U - . Equations (24a, b) yield the same angular distribution as do Eqs. (24c, d). This can also be deduced by symmetry considerations. As long as the electron spin is not measured, the cross terms (e.g., U+U') involving opposite photoelectron spins vanish (see Section 111). The only cross terms that appear

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

I69

involve Yo,(@, a) and YzO(O, @), which have no @dependence. The resulting angular distribution is

IO + J 45

[d:

~~

-

2dod2 C O S (-~So)] ~

Y20(@,

a)

(25b)

which reduces to

a do

(J'112.

N

&(d;

+ 2di) + $ [ d $

-

2dod2 C O S ( ~ , - SO)]P~(COS 0)(25~)

where P,(cos 0 )= ( $ cos2 0 - i)and we have used d, and d, to represent the radial integrals from Eqs. (24a) and (24b), respectively. We may write CTOTAL

4n

2d: - 4d,d, d;

C O S ( ~ ,-

+2 4

6,)

1

P,(cos 0 ) . (25d)

We see that Eqs. (25) have a dependence consistent with the measurements shown in Fig. 2, for photoionization of atomic sodium from the 3 2 P , i 2 state. Although the second and third terms in Eq. (25a) individually lead to terms in C O S ~0 [or P,(COSO)], these exactly cancel, leaving a term in cos2 0 [or P,(cos O)] representing the strength of the multipole asymmetry. This is to be compared with the case of excitation and photoionization of the 32P3/2. state, using parallel linearly polarized lasers. This, in contrast to the 32P,,, case, is a nonisotropic intermediate state. Since A M J = 0 at each step, the M , = +3/2 substates remain unpopulated. We may write wave functions $(n

'P3/2,1/2)

$(n 2P3,,. - I / 2 )

a Rni[(5)"2Y10(f19 v)I+ ) +

= Rnl[(f)"2

YI - l ( R

Yl

1(03v)I-

>I

v)l+ ) + (5P2Y**(&9)l-

>I.

(26a) (26b)

These differ from Eqs. (22) only in the coefficients, so that the relevant terms in the continuum wave expansion are just those licted in Eqs. ( 2 3 ) .Equations

S. J. Smith and G . Leuchs

170

(24) are easily modified accordingly, and again the angular distribution has only the one cross term. This leads to

Again using Eq. (20) we obtain

which reduces to da

(P3/2,* 112)

N

6 d i + &d;

+

d:P,(cos

+ +[$d$ - 4dod2 C O S (-~G~)]P~(cos ~ 0) 0)

(27c)

+

where P,(cos 0 )= c0s4 0 - 9 cos2 0 g. Approximately this form, dominated by the P,(cos 0 ) dependence, is found in the measured angular distribution (Fig. 2b) of photoionization from the atomic sodium 32P312 state, excited directly from the ground state by linearly polarized light, but some modification occurs due to hyperfine coupling, ignored in Eq. (27c). This is discussed in Section IVA. We note that apart from a factor of two, Eqs. (25a) and (27a) differ only in the coefficient of the third term. By the addition to da/dn(P3,,. of the term

the forms of the cross sections would become identical. I t is easy to show that Eq. (28) exactly corresponds to the cross section for photoionization of the P3,2,+ 3 / 2 states into continuum states i2e-i62Y2, @)I k). q)YT,

-

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

17 1

The differential cross section for photoionization of the isotropically populated P 3 1 2 state is identical in form to that of the P I / , state and larger in the ratio of statistical weights (21 + 1). Much physical insight can be gained by viewing higher-order processes in the same elementary framework in which we have described the twophoton process. The three-photon ionization of atomic sodium through the n 2 D 5 , 2 ,312 states with parallel linearly polarized lasers can be described as one-photon ionization of the M , = k 1/2 states, using wave functions $(’D3/2,1/2)

N

-

$(2D3/2. - ~ / 2 )

$(2D5,2.1/2) $(2D5/2,

R n ~ ( r ) [ - ( $ ) ~Y20(07 ” CP)~ + ) + ($>‘” Y21(09 CP)~ ->I Rn2(r)C-(%)L’2Y2. 1(0>CP)I+) ~

( $ ) ” 2 Y ~ ~ ( 0cP)I-)I ,

>I Ae, CP)I+ ) + ( $ ) “ 2 ~ 2 0 ( 0 ,cp)I - )I.

(29a) (29b)

R.z(r>C($)”2Y,,(R cp)l+ ) + ($P2 Y21(6, cp)l-

(29c)

2 l/2y RnZ(rIC(3) 2. -

(29d)

:?#

The remainder of the calculation proceeds as for the preceding two-photon case, but here the resulting forms analogous to Eqs. (25b) and (27b) include even spherical harmonics to P , (cos 0 )and P,(cos O), respectively, representing the higher angular anisotropy acquired through absorption of more photons. These forms show up in the measured distributions shown in Fig. 3.

S

P

D

\v2

F

intermediate

Jv2

limit

P

, I

32 p3/2. I/2

32si/2

sodium

Flci 3. (a) Level diagram illustrating two-photon resonant sequential excitation, threephoton ionization of atomic sodium. The angular distribution obtained depends on the choice between 3 ’ P , , , and 3’P,,, first excited states. (b) Polar plots of photoelectron angular distributions obtained in three-photon photoionization of the 20 ’D states (fine-structure not resolved) through the 3’P,,, and 3 2 P , , , states. The crosses ( x ) a r e measured points (Leuchs and Smith, 1983), and the solid lines are best fits oTa spherical harmonic expansion (Eq. (2b)).

S. J. Smith und G . Leuchs

172

The form of the angular distribution for one-photon ionization with linearly polarized light

do dR

-=

I(@)

= 'JTOTAL ~

4n

[l

+ ~P,(cosa)],

-1 I p I2

(30)

is well established for one electron in a central potential. This follows from symmetry principles (Yang, 1948) if the atom is randomly oriented. In a landmark paper, Cooper and Zare (1 968) demonstrated the generality of this form for the many-electron system, including the case of photoionization from individual fine structure levels. A large number of authors have elaborated on the problem. (See, for example, the reviews by Starace, 1982, and by Manson and Starace, 1982, and references therein.) Early interest in the theory of angular distributions for multiphoton processes developed in connection with angular correlation in nuclear processes (Goerzel, 1946; Yang, 1948; Abragam and Pound, 1953). More recently, with the advent of lasers, interest turned to atomic photoionization processes. Zernik (1964) calculated angular distributions from two-photon nonresonant ionization of the 2 s metastable state in atomic hydrogen, and nonresonant two-photon processes were briefly considered by Cooper and Zare (1969) and by Tully et al. (1968). Gontier and Trahin (1971) and Gontier et al. (1975) Calculated differential cross sections for ionization of atomic hydrogen with two to eight photons. Early observations of multiphoton-produced angular distribution effects carried out by E. J. Robinson and his colleagues (Fox et al., 1971) stimulated theoretical investigations by P. Lambropoulos (1972a, 1972b), by Jacobs (1972), by Arnous et al. (1973), and by Mizuno (1973). Jacobs formulated a one-photon theory in terms of the spherical tensor components of polarization density matrices (Fano, 1957), giving formal explicit expressions for photoelectron angular distributions and spin polarizations as functions of the polarization states of target atom and incident photons. In this paper, he used this technique to develop the case of the unpolarized target, but this was the first general density matrix treatment for the photoionization angular distribution problem. It went beyond the work of Cooper and Zare (1968, 1969) in two important respects: it included coupling between the photoelectron and a nonspherical residual ion; and it considered the possibility of initial atom populations distributed nonuniformly over the magnetic substates. Thus, it could be used to treat problems involving &",-photon ionization through an intermediate state reached by N o - 1 photon resonant excitation with polarized light, which puts the intermediate state in some alignment that can be specified from a knowledge of photon polarizations and application of obvious dipole selection rules. In his subsequent

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

173

paper, Jacobs (1973) treated the .h',-photon nonresonant ionization case as a function of polarization states of the target atom and incident radiation. In the same year M. Lambropoulos (1972) worked out the density matrix formalism for resonant two-photon ionization with polarized light in the central field approximation and then applied it to angular distributions from photoionization of atomic sodium and titanium. (See also M. Lambropoulos and Berry, 1973.) This was the first attempt to incorporate a resonant twophoton process of this sort in a comprehensive density matrix formalism. This formalism was further developed in R. S. Berry's group, notably by Strand (1979), and applied to resonant two-photon ionization of atomic sodium. In this work particular attention was paid to relaxation of the initial angular anisotropy due to hyperfine coupling. This was of interest in the case of excitation of the 2P,,, state of atomic sodium by pulsed lasers, which coherently excited a superposition of hyperfine levels. Other references to early theoretical work on angular distributions for multiphoton ionization of order two and higher are given by P. Lambropoulos (1976). Kollath ( I 980) applied the formalism developed by Jacobs (1972) to the problem of photoionization from excited states of cesium and discussed the use of angular distribution measurements as a tool for determining photoionization transition amplitudes. Klar and Kleinpoppen (1982) applied the density matrix formalism to the case of polarized targets and polarized radiation, as discussed earlier by Jacobs (1972) but here taking into account the nuclear spin effects. They emphasized that for a properly specified polarized target, measurement of the angular distribution of photoelectrons may permit determination of all transition amplitudes (magnitudes and phases) without measuring the state of the photoion or the spin polarization of the photoelectron. Laplanche et al. (1986a) gave a systematic analysis of theoretical methodology based on use of the density operator. Their unified treatment, based on the formalisms of Jacobs (1972, 1973), emphasized the unity of the angular distribution and differential electron spin orientation problems. In a companion paper (Laplanche et al., 1986b), they applied the formalism to a comprehensive description of two-photon ionization of cesium. Earlier work had dealt with the relationship of photoelectron spin polarization produced with circularly polarized light in multiphoton processes, but only in terms of polarization of the total photoelectron current. P. Lambropoulos (1973) was the first to develop the theory to address multiphoton processes. Granneman et al. ( 1976, 1977) compared theoretical predictions with measured values for multiphoton ionization via the cesium 7'P,,,. 3,2 states. See also Nienhuis et al. (1978) and Parzynski (1980). Other recent theoretical works have extended the theory to include effects of laser power density (Dixit and Lambropoulos, 1981, 1983; Geltman and

174

S. J . Smith and G. Leuchs

Leuchs, 1985; Parzynski, 1984, 1985) and laser linewidth effects (Dixit and Lambropoulos, 1983). Some of these and others will be mentioned in connection with discussions of experimental work.

III. Experimental Methods A. OPTICAL EXCITATION In the preceding section, photoelectron angular distributions were expressed in spherical coordinates @ and 0 referenced to the axis of quantization conveniently taken as the direction of the electric vector i? of ionizing radiation assumed to be linearly polarized. Furthermore, we have assumed that each step in a multiphoton excitation process is accomplished with radiation linearly polarized along the same axis as is the ionizing radiation. This configuration is useful, for example, for the study of perturbed systems as discussed in Section I. It maximizes the alignment of the intermediate (highest bound excited) state and results in the photoelectron angular distribution in which the contributions of the highest multipole are differentiated most effectively. In the dipole approximation, assumed valid, the dependence in the selection rule AMJ = 0 holds and there is no azimuthal (0) angular distribution. All information is concentrated in the dependence on the polar angle 0 as seen in Eq. (21) and a measurement is accomplished in a plane transverse to the axis of propagation of the radiation (laser beam). If, however, the polarizations of the different lasers inducing the stepwise excitation and ionization are not parallel to each other, additional terms must be included in the expression for the angular distribution (Strand et al., 1978; Hansen et al., 1980; Chien et al., 1983; Siegel et al., 1983). For example, if, in a two-photon ionization process, the linear polarization of the exciting laser is rotated by some angle r] with respect to that of the ionizing laser, axial symmetry is broken. Additional magnetic substates of the intermediate state are populated and the alignment achieved is intermediate between the isotropic case and the maximum alignment achieved with r] = 0. The additional terms may contain information in useful form from measurements conveniently accomplished in the transverse plane (fixed 0). Siegel et al. (1983) discussed the use of nonparallel linear laser polarization. In the dipole approximation, the unknowns in the description of an aligned (nonspherical) atom may be taken as the ratio of the radial matrix elements for the two allowed bound-continuum channels, the relative phases of the two channels, and the absolute value of one radial matrix element. The latter is determined by an absolute measurement of total or differential cross sections,

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

175

leaving two unknowns to be determined from angular distribution measurements. From the discussion in Section I (see Eq. (2b)) these two unknowns can be determined experimentally using parallel (a = 0) linear laser polarizations if the expansion in spherical harmonics that fits the experimental data has at least three nonzero coefficients, I(@)

=

B” YO” + B? Y?, + B4 y40.

The lowest coefficient is used for normalization. In general, if the measurement is accurate, the other two coefficients from a fit to any one angular > 2 in Eq. distribution measurement are sufficient. For high anisotropy (2b)) the added terms provide useful redundancy. As Siegel et al. (1983) noted, measurements at other fixed values of a( fO) may be used in the same way. However, the anisotropy, hence the information content achieved, is lower, in general. Comparison of results at different values of q (it., 0‘ and 90’) may be used to isolate unknowns and simplify the analysis. An interesting variation, described by Siegel et al. (1983), consisted of fixing the ionizing laser polarization at 0 = 0” (ix., pointing towards the detector) and rotating the polarization of the exciting laser (with fixed @ = 0’). The “angular distribution” obtained then coincides, in the independent particle model, with the angular part of the electron density distribution of the aligned orbital of the valence electron. Figure 4 presents important generalizations of the multiphoton ionization process to include several possible excitation schemes. In the resonant excitation case, (a), which is emphasized in this chapter, there is the freedom (-81.

(0)

FIG.4. (a) Resonant (sequential) multiphoton ionization (several intermediate resonances, several lasers). (b) Resonant multiphoton ionization (one intermediate resonance, one laser). (c) Nonresonant multiphoton ionization (one laser).

176

S. J . Smilh and C. Leuchs

of using different laser polariLations for each step. In addition, the laser intensity can be chosen such that only one laser induces measurable ionization. Also, when pulsed lasers are used, a time delay may be introduced between the laser pulses exciting the resonant intermediate states and the one inducing ionization, such that the high intensity of the ionizing laser does not perturb the atoms during excitation. If the main purpose of the investigation is to study the atomic structure, this excitation scheme is well suited. The option of time-delaying the ionizing with respect to the exciting pulse can also be used to study the temporal evolution of the intermediate state. This dynamical behavior may be due to spontaneous decay, collisional redistribution (M. Lambropoulos and Berry, 1973), or quantum beats (Strand et al., 1978; Leuchs et al., 1979). The cases shown in Figs. 4b and 4c, respectively corresponding to resonant and nonresonant multiphoton ionization with only one laser, do not allow for this freedom, but they may be of interest for detailed studies of the atom-field interaction. There the theoretical analysis will necessarily be more complex. Especially in case (c), there are a large number of different routes to the continuum that have to be summed. Therefore, the theoretical calculation requires the knowledge of a correspondingly large number of atomic wave functions. However, for multiphoton ionization far from any resonant intermediate state, there is a simple rule regarding the shape of the photoelectron angular distribution in photoionization with linearly polarized light. In the case of ionization, from an 1 = 0 ground state with an odd number of photons, the angular distribution will be zero at 0 = 90" to the polarization direction and it will be nonzero for ionization with an even number of photons (Gontier et al., 1975; P. Lambropoulos, 1976). In the preceding discussion the use of 0 and @ to specify angular coordinates of the photoelectron trajectory was intended to provide a clear differentiation with respect to spatial coordinates of wave functions, 6' and q. Throughout the remainder of the chapter 6' and q will be used in place of 0 and Q, to describe photoelectron angular distributions. OF ANGULAR DISTRIBUTION B. MEASUREMENT

The experimental study of angular correlations in the photoionization of excited (intermediate) states reached by a one-photon excitation process was pioneered by R. S. Berry and his colleagues at the University of Chicago (Edelstein et al., 1974). They based their work on rotation of the electric vector with a 4 2 (half-wave) retarding plate, and this has been the basic technique in most of the multiphoton angular correlation measurements carried out subsequently at Chicago and elsewhere. An alternative technique

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

177

is based on the use of double Fresnel-rhomb prisms, which provide equivalent relative phase shifts through total internal reflections. Many of the measurements described in Section IV were carried out in a geometry that is spatially fixed, save only the axes of linear polarization of the radiation which are rotated by external means in a plane transverse to the axis of the laser beam. Thus, possibilities for systematic effects inherent in mechanical movement are avoided. Angular distribution measurements must be carried out in a low-density environment in which electrons with energies of a fraction of an eV and higher will have mean-free-paths of at least several meters, and in which space charge effects will be minimized. Good shielding from electrostatic fields of any kind is essential. Ambient electric and magnetic fields must be reduced to 50.1 V/cm and 5 lo-* gauss T), respectively, to maintain approximately straight-line trajectories. A well-defined source volume is needed to achieve good angular resolution. These requirements are addressed by crossing a well-collimated low-density atom beam in a high-vacuum apparatus, at right angles with one or more laser beams. The laser beams are conveniently used in a coaxial configuration. A schematic representation is given in Fig. 5. If the azimuthal dependence of photoelectron angular distributions is to be studied, the spatially fixed geometry can no longer be used. Then the angle cp between the direction of electron detection and the direction of laser propagation has to be varied. However. relative motion of surfaces close to

laser beams

laser 1 polarization I

/I

electron multiplier

FIG.5. Schematic representation of crossed beam configuration used for photoelectron angular correlation studies (Hellmuth et al., 1981).

178

S. J. Smith and G . Leuchs SECONDARY ELECTRON MULTIPLIER

FIELD FREE INTERACTION

GROOVE FOR SAPPHIRE BALLS

FIG.6 . Schematic representation of apparatus used to measure azimuthal dependence of photoelectron angular distributions (Leuchs et al., 1986).

the interaction region could possibly introduce some pdependence to the trajectories of the low-energy photoelectrons, owing to contact potentials or surface charges. In order to minimize such effects, Leuchs et al. (1986) used a copper box cylindrically symmetric with respect to the atomic beam to define the field-free interaction region (Fig. 6). This copper box and the electron detector-spatially fixed with respect to each other-were mounted on a rotatable ring. The laser beams were carefully aligned in a plane normal to the axis of rotation. No systematic effect on the electron signal correlated with the motion of the assembly was found in the experiment.

c. DETECTION OF SPIN POLARIZATION In Section IT, the angular distributions were calculated by computing the probability of finding the continuum electron with spin up I ) and with spin down 1 - ) and adding the two. If one wants to go a step further and measure the spin polarization, the observable quantity will be the projections S , , S,, or S , of the electron spin onto the x-, y-, or z-axis. Before describing the measurement of the spin polarization, we shall address the question of how to relate the coefficients of the two component spin wave functions

+

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

179

to the three components of the electron spin, S,, S,, and S,. The amplitudes U , , contain the spherical harmonics describing the angular distribution. The density matrix for this spin system is

The operators describing the orientation of the spin (1/2) in three-dimensional space are represented by Pauli’s matrices cX=

(1

-a),

and a, =

(:

-3.

To obtain the expectation values S,, S,, and

S,, the traces of the products 01 the Pauli matrices and the density matrix have to be evaluated

These are the probabilities to find an electron with spin polarization along the x-, y-, and z-axes, respectively. The probability to find an electron with any polarization is

This relation has been used in Section 11 to calculate the angular distribution of photoelectrons from 3’P states of alkali atoms. For the example of photoionization out of the 32P1iz.l i z state of sodium, I U + + I U - l 2 contains an interference term between the different partial waves of the unbound electron that is proportional to the cosine of the difference of the corresponding scattering phases, cos (;i2 - 6,) [Eqs. ( 2 5 ) ] . The advantage of measuring the spin polarization in addition to the angular distribution of the photoelectron is the additional new information available. Using Eqs. (24a-c) to evaluate S, [see Eq. (32b)], one finds interference terms containing the sine of the phase difference, sin(6, - h,,). This fact was recognized by Kollath (1980) in connection with an important experiment (Kaminski et al., 1980) in which the authors emphasized the necessity of measuring the electron spin to decrease the ambiguity in interpreting the bound-free transition. The standard way to measure spin polarization is by use of Mott scattering, which relies on the dependence of large-angle scattering of polarized electrons from a solid target on relative orientation of spin and

S. J. Smith and G. Leuchs

180

orbital angular momentum (Kessler, 1976). This leads to a spatial anisotropy in the differential scattering cross section that can be measured to an accuracy of 5 to 10 % (Hodge et al., 1979; Gray et al., 1984). The anisotropy is largest for material for which the spin-orbit coupling is strong. Interesting new developments in this field have been reported by Celotta and coworkers (Unguris et al., 1986).

-

D. ELECTRONENERGYANALYZERS

A standard tool to measure the kinetic energy of electrons is the cylindrical or spherical analyzer. (See Fig. 7.) In the region of a few eV of electron energy, the resolution can be as low as a few meV. Spherical analyzers are widely applicable for both pulsed and continuous electron sources. A potential disadvantage is that electrons of different energy cannot be measured simultaneously, so some information is lost. In the case of pulsed sources, the electron energy spectrum may be analyzed by using the time-of-flight technique. After production of the photoelectrons in a well-defined small volume during a short time interval, the electrons drift through a field-free region and separate according to their different velocities. For a drift length of 1 m and a pulse duration of 10 ns, an energy resolution of 10 meV can be achieved. An advantage of this method is that all electron energies can be measured for each pulse. Furthermore, it effectively discriminates against most electrons that originate in secondary processes at various surfaces in the apparatus. The time-of-flight technique is rather simple. Even the experimental setup shown in Fig. 6, which had not been designed to allow for energy spectroscopy, could be used for a crude time-of-flight measurement of the electron energy. The field-free drift region effectively has the length of little more than the radius of the copper box (- 5 cm). With an ionizing laser pulse of 15-11s duration, the energy resolution is -250 meV at 1 eV. Figure 8 shows a time-of-flight spectrum taken with the setup shown in Fig. 6. The 23'D,-Rydberg state of barium was ionized using 1.06-pm Nd:YAG laser radiation. Depending on whether the residual barium ion is in its ground state or in one of the 5d 2D3,2,5,2fine structure states, it will have a kinetic energy of 1.16 eV, 0.4 eV, or 0.5 eV. The two lower energy peaks (larger time of flight) are not resolved in Fig. 7. In measurements carried out by Mullins et al. (1985~)using an excimer laser ( 5 4 s pulse ) to pump a dye laser, the short ionizing pulse resulted in improved time-of-flight resolution permitting resolution and independent angular distribution measurements for the D3,2.5,2 continua.

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

18 1

S. J. Smith and G.Leuchs

182

-al3

ln 0.

Ba 23'0,

delay time Ins1 FIG.8. Representation of a low-resolution time-of-flight spectrum for photoionization from the 23 ID,-state of barium with 1.06-pm Nd:YAG laser radiation. The peaks at 180 ns and 210 ns correspond to photoionization into the 6s continuum and into the 5d continuum, respectively. The small signal beyond 250 ns is due to direct photoionization from the 6s6p 'PI excited state by short-wavelength spontaneous background emission from the blue dye laser tuned to the ' P I + ID, transition, part of the sequential excitation process.

E. DATAANALYSIS In the analysis the photoelectron angular distribution data have to be reduced from the set of count rates measured at each angle setting to a Few anisotropy parameters. The standard procedure is to perform a least-squares fit of the theoretical function (Eqs. (la), (21), or (38a)) to the data using, e.g., the curve-fitting routine given by Bevington (1969), which handles any nonlinear function. For a comparison of the fit parameters and theoretical predictions, it is essential to have reliable and meaningful error bars for the fit parameters. This, however, may be a problem if the parameters used in the fit are correlated, which means that the deviation of one parameter from its optimum value can be partially compensated for by adjusting one or several of the other parameters. This is analogous to defining a point in space using a nonorthogonal coordinate system. In order to avoid numerical problems, it is advantageous to choose the fit function such that the parameters are not correlated. An expansion in even powers of cost? as in Eq. (la) is, therefore, not desirable. The Legendre functions P,(cos 8) are orthogonal on the full sphere, so that

:1

d8 ~ o z n d sin q 0 P,g(cos@P,(cos 8) = 0,

(34)

if n # m. Thus, Eq. (21) seems to be the right function to use in the fitting routine, because the coefficients of orthogonal functions are necessarily uncorrelated. However, in most experiments, the angular distributions of

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

183

photoelectrons are measured in one plane only. As a result, the set of angles for which data are obtained consists of discrete values out of a linear interval and not the full sphere. Hence, the best function for the fitting routine is an expansion in functions .f.(6) (that fulfill the orthogonality relation on the linear interval [0, n]:

jozf,(O)f,(d) d6

= 0,

for n # m.

(35)

This is exactly the problem solved by Fourier and the functions are f,(@ = cos(n6) resulting in a fit function for angular distribution N

I(@

=

c A,,

cos(2KB).

K=O

The standard deviations of the uncorrelated fit parameters A , , are determined by the second derivative of x2 with respect to the fit parameter (Leuchs and Smith, 1982), where x 3 is the sum of the squared deviations between the data and the corresponding values of the fit function. The angular position of the peak of the angular distribution is determined by the initial adjustment of the direction of laser polarization and by the interaction and detection geometry. The combination of these effects may result in some inaccuracy, so that the fitting routine should allow for the angle of symmetry 6,, being a fit parameter:

If this modification is not necessary, the fit function I ( 6 ) depends linearly on the fit parameters and a linear least-squares fitting routine may be applied. Furthermore, if the acceptance angle of the detector is too large, one has to correct the anisotropy parameters A,, determined by the fit for the finite angular resolution before comparison with theory.

IV. Applications to Atomic Structure and Dynamics A. O N E - P H O T O N EXCITATION P L U S IONIZATION

The first atomic system studied experimentally in resonant excitation plus ionization was in titanium (Edelstein et al., 1974), chosen because of the coincidence of the 3d24s2a3F, -+ 3d24s4px3C, transition with the energy of a

184

S. J . Smith and G . Leuchs

photon from a molecular nitrogen pulsed laser, 29,652 cm-' (337.1 nm). Photoionization of the x3G, state was accomplished with a second photon from the 20-ns-duration laser pulse. The laser radiation was linearly polarized using a calcite prism, and the polarization was rotated using a revolving half-wave (A/2) plate. Photons were counted at six positions 0' of the A/2 plate from 0" through 90°, corresponding to polar angles (e, i )= 0 = 20' from 0 to 180". The measurements on titanium demonstrated the feasibility of the technique, but the complexities of the structures of Ti and Tif precluded a definitive interpretation of the results. Berry's group turned to atomic sodium, obtaining much more definitive results using a two-step ionization process requiring diferent laser frequencies for the excitation and ionization processes (Duncanson et al., 1976). They studied the 32S,,,2,+ 3 2 P , , 2 . 3 1 2+ continuum processes discussed in preceding sections, exciting either of the 32s1/23 32P,/2,3,2 transitions with a 4-ns FWHM (full width at half maximum) pulse from a dye laser tuned to the chosen transitions. The dye laser was pumped with a 10-ns FWHM nitrogen laser pulse at 337 nm, part of which was split off, delayed for 5 ns, and used as the photoionizing radiation. This scheme guaranteed synchronization of the two pulses and the delay ensured that the excitation and ionization processes were sequential. Measurements carried out on ionization through the 32P1/t state showed good agreement with the anticipated distribution for an isotropic state, Z(0) = tl + /Icos' 0. Measurements through the 32P3j2state were carried out with q = O", 65", and 96", and a methodology was worked out for using the information to obtain values for the radial dipole matrix elements do and d , to the s and d continua and the relative phase shift between these two partial waves (6, - do), represented in Eq. (27c) for the case q = 0. Strand et al. (1978) recognized the dominant role played by hyperfine coupling in the 32P3i2intermediate state. Atomic sodium has a nuclear spin I = 3/2, leading to hyperfine levels F = 0, 1,2, and 3 with intervals 15 MHz, 34.1 MHz, and 58.9 MHz, respectively (Walther, 1976). Thus, the hyperfine structure is confined to a spectral interval that is small compared to the spectral width of a nitrogen laser pulse (-GHz), and the hyperfine levels are coherently excited. Cross-terms deriving from this superposition evolve in time at a significant rate compared to the -5-ns delay in arrival of the ionizing radiation, and this leads to marked effects on the measured angular distributions. A comprehensive theoretical treatment of the problem of angular correlation in photoionization from the 32P3j2state in Na, including the effects of hyperfine interaction, was described by Hansen et al. (1980). Strand et al. (1978) and Hansen et al. (1980) published the same set of general equations describing the angular distributions obtained by the two-step process,

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

185

including the case of the axes of polarization of the two coaxial laser beams not being parallel, but being oriented at an arbitrary relative angle q in the transverse plane. Their equations also included parameters ~ ( 0and ) ~ ( 2that ) contain all information concerning pulse shapes and the dynamics and coherence of the intermediate state, including the hyperfine interaction. These equations were used to interpret pronounced time dependences observed in the photoelectron angular distributions as a function of time delays of up to 8 ns between the arrival of the exciting and ionizing laser pulses, hyperfine interaction induced “quantum beats.” However, the equations (Hansen et al., 1980) are found to be inconsistent with the earlier work of Cooper and Zare (1968) and are therefore in error. A revised version of these equations was later published by Chien et al. (1983), and further corrections have been provided by Berry (1986). Leuchs et a]. (1979) repeated the quantum beat experiment at Munich using a similar experimental configuration, but using only one angle q = 0” (linear polarization parallel). In this case the angular distribution can be represented as

where the time dependences introduced by the hyperfine coupling are included in the coefficients. Angular distributions were measured using optical delays of up to 38 ns between exciting and ionizing pulses, allowing the observation of beat structure directly attributable to the hyperfine intervals. The coefficient /14(t) is plotted in Fig. 9 and its Fourier analysis in Fig. 10, Peaks at 59.8 MHz and 95.3 MHz correspond to hyperfine intervals Av,, and Av3, + A Y ~ , respectively. , These values are in agreement with earlier measurements of the hyperfine structure (Walther, 1976). Av2, and Av,, are unresolved in the peak at 24 MHz. No evaluation of radial integrals or phase difference in the bound-continuum radial integrals ( p + sand p + d ) was attempted. Chien et a]. (1983). in connection with a study of the two-photon ionization of atomic lithium through the 2*P,/, and 2,P,,, states, a system identical except for radial matrix elements and hyperfine intervals to that for ionization through the 3,P,,, and 32P,i2.states of atomic sodium, derived general equations for the angular distribution

-

I ( @ , 0) c,,P,,(cos

+ C,,P,,(cos

0 ) + C,,P,,(COS 0)cos 2(p

0 ) + C,,P,,(COS 0)cos 4

+ c,,P,,(cos

6)

S. J . Smith and G.Leuchs

186

.

0

20

10

30

10

delay time Ins]

FIG.9. Dependence of the coefficient /14(t) (see text) on the time elapsed between arrival of the laser pulse exciting the 3'P3,, state of atomic sodium and the arrival of the ionizing radiation pulse (Leuchs et al., 1979).

-77-1 I I

0

50

1

FIG. 10. Fourier spectrum of

1

I

I

I00 /14(t)

I

I

I

I150

frequency [ M H d

shown in Fig. 9.

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

187

where the coefficients can be expressed (Berry, 1986) as

C,,

+ (2 - 3 sin’ q)x(2, t)]df, + [4x(O, t ) + 9(2 3 sin’ q)x(2, t)]d:) = f { [ - 4 ~ ( 0t ,) + 2(3 sin’ q 2)x(2, t)] x d,d, cos(&, - 6,) + [ 2 ~ ( 0 , ?+ ) ($ - 3 sin’ q)x(2, f)]d:) (38b) = f [ - d , d , cos(6, - 6,) + 4 dtlx(2, t ) sin(2q)

C,,

= &(+ d:

C,,

= g(2-

C,,

=

& sin(2q)d:~(2, t )

C,,

=

sin2 q d:x(2, t )

Coo = &{[2x(O,t)

-

C,,

-

sin’ q)x(2, t ) 3 sin’ q ) d : ~ ( 2 t, )

This set of equations reduces to Eq. (27c) and to consistency with the Cooper-Zare formula by addition of a term for the ionization of an equal (P3,’.c 3 , 2 ) population as in the discussion of Eq. (28). Chien et al. (1983) obtained values for the s - d phase difference cos(b, - 6,) and for the ratio of the radial matrix elements in the two channels of d,/d,. [No reanalyses of these or the corresponding results for atomic sodium (Strand et al., 1978; Hansen et al., 1980) have yet been carried out.] Quantum beat measurements were carried out with optical delays between exciting and ionizing pulses of 0, 20, and 40 ns. Chien et al. (1983) have incorporated parameters ~ ( 0t ), and ~ ( 2t,) in Eqs. (38) (see also M. Lambropoulos, 1972; Strand et al., 1978; Hansen et al., 1980) in which various time-dependent effects are incorporated. If spontaneous decay and hyperfine interactions are of interest, time-dependent functions W(0,t) and W(2,f) are linear combinations of terms of the form - iioFP,f , - rr ,where mFF,= ( E F - E F f ) / hdescribes the precession frequencies of the hyperfine populations, and is the lifetime of the intermediate state. The duration of the laser pulses is taken into account by integrating over the intensities I ( t , ) and I ( t , ) of the exciting and ionizing pulses: ~

(39) - x,

The expression can be modified to take into account other incoherent decay processes (e.g., collisional deexcitation). A different form of angular correlation, but logically an integral part of a comprehensive, unified treatment of angular correlations (Laplanche et al., 1986a, 1986b) is the production of spin-polarized electrons by polarized

188

S. J . Smith and G . Leuchs

radiation. Granneman et al. (1976, 1977) and Kaminski et al. (1979) investigated, experimentally and theoretically, the production of spin polarization in the total photoelectron current in resonant two-photon ionization of cesium, using circularly and linearly polarized radiation. Another important early measurement, carried out by Kaminski et al. (19801, serves to emphasize the potential of angular correlation measurements for obtaining information complementary to that obtained by other methods. The authors pointed out that a set of four measurements would serve as the “complete” photoionization experiment, one that “allows complete determination of the transition matrix elements describing the photoionization process: ( I ) the total photoionization cross section for n light: oto,(n), (2) the ratio of total cross sections for circular (a) and linearly (n)polarized light: R = ~,,,~(a)/a~,,~(n), (3) the angular distribution of photoelectrons I ( @ for 71 light, and (4) the spin polarization Pl(@ of photoelectrons perpendicular to the symmetry plane for n light.” Their conceptually simple experiment was also unique in its treatment of electron spin. In all other photoelectron angular distribution measurements, the electron spin is not measured and is averaged out in the analysis. A flashlamp pumped dye laser was used to photoionize atomic cesium in a twophoton process resonant with the 72Slj2+ 7*P,,,, 3 j 2 transitions. The light was linearly polarized and the electric vector rotated by a rotation of the polarizer about its axis, rather than with a A/2 retarder (presumably leading to an angle-dependent modulation of light intensity). Photoelectron currents were normalized to light intensity at each angle. Kaminski et al. (1980) showed that angular correlation measurements on the photoelectron current and spin in a fixed geometry, rotating only the electric vector of the ionizing radiation, were sufficient to determine the relative phase of the s- and d-wave matrix elements. Values of total cross section-and of the ratio of cross sections for circular and linearly polarized light-were already in the literature. These values of 6, - h d now “complete” the measurement. They obtained 6, - 6 , = 127.4”for the process through the 7*P,,, state where hyperfine coupling is not an issue. Measurements for the 72P3j2 state were complicated by the hyperfine interaction combined with the long duration of the flashlamp pumped dye pulse. The importance of considering the hyperfine interaction was later demonstrated clearly by Compton et al. (1984). Siege1 et al. (1983) carried out angular distribution measurements in photoionization of the neon atom 3 p 3 D , state that were unique in several respects. They were carried out with continuous wave, rather than pulsed, lasers; and they relied on optical pumping to achieve maximum orientation of the 3 D 3 intermediate state of neon produced by absorption of linearly polarized photons from a randomly oriented 3s 3 P , metastable state of neon.

A N G U L A R CORRELATION I N MULTIPHOTON IONIZATION

189

In this excitation, a 3s orbital is promoted to a 311 orbital, a situation highly analogous to the resonance transition in atomic sodium, but additional pumping is required to reorient the originally randomly aligned core so that full alignment of the 3 D 3 state is achieved. The angular distributions obtained were, nevertheless, characterized by independent particle model behavior and were fit by expansion of Eq. (2b) terminating at Yd0. Ions produced were in the P3,’ state to the virtual exclusion of the PI/, state. Values of parameters obtained were dJd0 = 1.96 &- 0.08 from measurements with q = 0 and d,/d, = 1.88 &- 0.15 from measurements with 4 = 90”. In the former case they obtained cos (6, - 6,) = 0.95 0.04 and in the latter cos (6, - 6,) = 1.00:;::;. A series of investigations of photoelectron angular correlations has been carried out at Oak Ridge by R. N. Compton and his colleagues (Compton et al., 1984; Christian et al., 1984; Dodhy et al., 1985, 1986). These have included investigations of a variety of effects in cesium and rubidium, some of which will be discussed in subsequent sections. I n cesium they carried out measurements for two-photon ionization via the 7p 2 P , / 2 . 3 , 2and 8 p 2 P , , L , 3 1states 2 (Compton et al., 1984), as well as for a three-photon ionization via a twophoton resonance with the 8d 2 D 5 1 23 ,,2 states. A hemispherical energy analyzer permitted separation of fine structure states. Calculated distributions that incorporated hyperfine coupling effects were in reasonable agreement with the observations. Dodhy et al. (1986) measured two-photon ionization in rubidium using a one-photon resonance with the 4d ’ D state, which involves a quadrupole transition from the ground state. This was investigated as a function of laser power and in comparison with off-resonance angular distributions. The first and second “above-threshold” ionizations were also observed. The effects of fine-structure mixing were evident in these comparisons.

+

B. TWO-PHOTON SEQUENTIAL EXCITATION

PLUS [ONIZATION

The experimental study of angular correlations in resonant three-photon ionization processes was initiated at the University of Munich (Riedle, 1979; see also a review by Leuchs and Walther, 1984). The experimental arrangement is illustrated in Fig. 5. The atoms of a sodium beam were irradiated by two laser beams originating from two different dye lasers pumped by the same nitrogen laser. The two dye-laser beams were linearly polarized in the same direction. The common direction of polarization could be rotated by means of a 142 plate. In this way, the angle 0 between the direction of emission of the electrons and polarization direction could be changed. The interaction region between the laser and atomic beams

190

S. J, Smith and G. Leuchs

was electrically shielded. The electrons were detected with an angular resolution of 0.35 rad. Beyond the aperture defining the acceptance angle of the detection system, the electrons were accelerated and detected by means of an electron multiplier. The excitation scheme used for the three-photon ionization is shown in Fig. 3. The first laser beam (v,), was tuned either to the 32s1,2-32P1/2 or 32S1,2-32P3i, transition of the sodium atom (wavelengths 589.5 nm and 588.9 nm, respectively). The second laser (v2) performed a resonant excitation to the n2D fine structure states. The duration of the dye laser pulses was -4 ns and the output power between 10 and 50 kW. The spectral width of the laser was about 0.05 A. In this way, the 'Plj2 and 2P,it levels could be excited separately whereas the n2D states were unresolved since the fine structure splitting was rather small. The photoionization was performed by either laser v1 or v2 or by both. More advanced versions of this three-photon resonant ionization experiment were developed at the Joint Institute for Laboratory Astrophysics in Boulder (Leuchs et al., 1983; Leuchs and Smith, 1983). The version currently in use (Leuchs and Smith, 1985) is represented in Fig. 11. This was based on the use of a pulsed Nd : YAG laser instead of on a pulsed nitrogen laser as were the earlier versions. The Nd: YAG laser radiation is conveniently doubled and tripled, offering intense radiation at 355 nm and 532 nm as well as at the fundamental 1.064 pm. This laser has, in addition, a great advantage in reliability and stability, but it has a long pulse duration ( - 9 ns), which is a definite disadvantage in the case of sequential ionization involving the atomic Na 3 2 P 3 , 2 state, for example, in which the angular distribution evolves rapidly due to hyperfine interaction. On the other hand, for measurements in atomic barium, in which hyperfine interaction is absent, system reliability was the key to obtaining large amounts of data on angular distributions from a number of highly excited states. In this measurement, the atom beam, the ionizing laser beam, and the detection axis form a mutually perpendicular configuration, intersecting at the center of an electrostatically shielded ground-potential box in which the magnetic field is nulled ( 5 T ) ,and background gas pressure is less than Pa. The vertical atom beam, from a diffusive source, has a density lob atoms/cm3 in a 1-mm-diameter beam width with -5 milliradian divergence. Photoelectrons from the beam intersection, emitted into a 12"-half angle cone along a fixed axis, drift through a field-free space 6 cm long, are accelerated ( - 100 eV), and are focused into a 16-stage secondary-emission multiplier. The photoelectron collection efficiency is The laser beam waists, at the atom beam, are 200-400 pm in diameter, defining a reaction volume containing lo2 to lo3 atoms. The 6-ns dye laser pulses, superposed in time and space, have a common propagation axis

-

-

-

--

-

-

191

ANGULAR CORRELATION IN MULTIPHOTON LONIZATION

l06pm

HARMONIC

410 - 4 2 0 n m

c

FIELD NULLING COILS

\,-----. '\

,

I

ATOM B E A M 589+4 I5nm

I

I-&ACCELERATION A N D FOCUS _I CASCADE E ? J, L EC T RO N M U LT I PLI E R

'\,,

..4,__l_-'

L-

s IG NA L PROCESSING FIG. 1 1 . Schematic representation of the apparatus used by Leuchs and Smith (1985). L~~

offset at an angle of 1.7" in the horizontal plane to that of the 1.06-pm ionizing radiation to permit use of a separate train of polarizing optics. The two beams at 500-600 nm and 420 nm can be combined and processed in a single optical train using a highly achromatic six-plate quartz and MgF, 11/2( + 172) retarder to rotate the linear polarization axis with respect to the detection axis. In order to ensure complete linear polarization, the A/2 plate is followed by a high-quality Glan-Thompson prism P' (rejection ratio 5 A plano-convex lens (focal length 1 m) is used to focus the I-mm diameter laser beams into the atom beam.

-

-

-

192

S. J. Smith and G . Leuchs

The 212 plate and the Glan-Thompson polarizer are synchronously rotated about their common axis using a geared-down stepping motor operated under computer control. The polarization vector is rotated in 10" increments, for 25 laser pulses (10 per second) at each position. It is critically important to eliminate displacement and deviation of the laser beams by these rotating elements, so that the beam waists remain fixed in space at the atom beam intersection. This is accomplished by careful alignment and by the use of pairs of crossed wedges ( W , and W,) to compensate for residual wedge in each of the rotating elements. With these precautions, wobble of the beam waists can be reduced to < 10% of waist diameters, which eliminates adverse effects. The 1.06-pm ionizing radiation, delayed by 15 ns, is processed by its own similar optical train. Here the only rotating element is the A/Z pfate that shows no discernable wedge effects. The intensity of the beam from dye laser I is normally sufficient to saturate the resonance transition ( - 100-W peak power for sodium or barium) and the intensity of the laser I1 beam is controlled (-kW peak power) to minimize power broadening of the intermediate state. Laser I1 intensity and the intensity of 1.06-pm radiation are also controlled to limit the photoelectron detection rate to 50.3 counts per laser pulse. This permits the use of counting techniques and minimizes dynamic range and linearity problems in the signal-processing equipment. These rates may be normalized to measured total photoion production per pulse, and they may be corrected for two- and more-electron events using experimentally verified Poisson statistics for the photoelectron counts. The maximum correction applied is 15 %. This technique was applied to measurements of angular correlations in atomic sodium. (See, for example, Leuchs and Smith, 1983.) This atomic system is highly predictable since there are no significant core interactions or perturbing states. The only complicating factor is the well-known (Strand et al., 1978) hyperfine interaction in the 3'P3,, state that introduces time dependence to the angular distribution. Leuchs and Smith demonstrated that the methodology and analysis can be refined to a level in which coefficients of spherical harmonic (or trigonometric) expansions can be determined to a few-percent accuracy, and the cutoff term K = .A'* in Eq. (2b) accurately determined on the basis of statistical significance of coefficients of higher terms. Further measurements carried out in Boulder and in Munich with this technique are described in subsequent sections.

-

-

c. TWO- AND THREE-PHOTON RESONANTEXCITATION P L U S IONIZATION Feldman and Welge (1982) carried out the first measurements of angular correlations from intermediate states reached by two- and three-photon resonant processes of the type illustrated in Fig. 4b, as opposed to the sequential resonant excitation processes illustrated in Fig. 4a. Furthermore,

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

193

the latter measurements were the first in which the intermediate state was above the first ionization threshold (autoionizing resonance), with spontaneous emission of a photoelectron. These early experimental results showed some asymmetries in the angular distributions that were artifacts of the experimental method. More refined measurements by this group gave symmetric distributions (Otto, 1985). For the two-photon resonant processes (Fig. 4b), the theoretical prediction (Arnous et al., 1973; P. Lambropoulos, 1980) is analogous to that for the sequential process (Fig. 4a). This was a one-laser measurement using very high intensities ( 5 2 5 GW cm-2) of 5-6-ns pulses, in the wavelength range 560- 576 nm, to reach and photoionize the intermediate states in atomic strontium 5s5d ID,,5s5d 'DZ,( 5 ~ ) '3P0,and ( 5 ~ ) ' 3 P , . Least-squares fits to the trigonometric form of the expansion

gave values of pi that agreed well with theoretical predictions (Kim and Lam bropoulos, 1984). Measurements were carried out with both linear ( n ) and circular (0) polarizations at the 5s5d D , resonance. A semiquantitative comparison of results with theoretical predictions for angular distributions for three-photon ionization of He ' S (Olsen et al., 1978) was useful. The predicted angular distribution for 7r and lr polarizations were

'

da(.rr) = k(B2 cos2 t1 + /I4 C O S ~0 + fi6 COS' dR

6)

(41a)

where the angles 6' are measured with respect to the direction of linear polarization and to the direction of optical propagation, respectively. They obtained a value of 2 for the ratio of the n and lr cross sections. In a unique experiment (Pratt et al., 1985), atomic carbon was prepared in the 3 P , ground state and in the ID, first excited state by UV multiphoton dissociation of carbon tetrachloride. In each case, the atom was excited to a higher state by way of a two-photon resonant absorption, which was followed by ionization in the same YAG-pumped pulsed dye laser beam. The 'D, excited state was taken through the 3p 'So isotropic excited state that must yield a photoelectron distribution I(f1) r x 1 + b 2 P 2 (cos 6). The 3 P , (isotropic) state was taken through the 3 p 3 D , state that, in principle, could retain alignment from the two-photon resonant excitation. However, terms in P , and P6 that might have been expected are found to be small. I t is shown that the single-particle model does not apply at all here. The excited electron couples to the core and in both cases a -P,(cos 0) distribution results

194

S. J. Smith and G . Leuchs

although with very different values of flz. Measurements were also carried out with circularly polarized light. Nagano et al. (1982) used the two-photon resonant, one-photon ionization method to obtain photoelectron angular dependences from five excited states of atomic iron. Multiphoton ionization spectra were obtained with exciting laser and ionizing laser polarizations parallel (q = 0") and perpendicular (q = 9W), and the results at these two angles were used to infer fl parameters. The data did not permit evaluation of the distributions in terms of higher multipoles. Sato et al. (1984), in the same group, used a similar technique to study angular distributions from Rydberg states of Xe and Kr. The states studied are hydrogenlike with ionic cores in ' P 1 / 2 , 3 / 2 fine structure states, and they are reached by a three-photon resonant process. One-photon photoionization angular distributions of the Xe 5d [3+]' J = 3 state, two-photon ionization of the Xe 6s [l $1' J = 1 state, and two-photon ionization of the Kr 5s [lt]" J = 1 state were measured. As in the measurements of Siege1 et al. (1983) for neon, only the ' P 3 / , ion is produced from photoionization of the first and third of these states, but the 6s [l $I0J = 1 state produces both ' P 3 / 2 and ' P l j 2 states, possibly because of a resonance at the fourth photon level. Otherwise, the single-particle picture seems to be valid. Kruit et al. (1983b) have also described angular distributions for threephoton resonant, five- and six-photon photoionization of the 6 s [ 1 +]'J = 1 state in Xe, obtaining a distribution containing even multipoles through P,(cos 0) for the five-photon case. Christian et al. (1984) carried out measurements of angular distributions from two-photon resonant three-photon ionization of atomic cesium, through the 6d 2D3/z.and 6d 2 D 5 / 2states, as well as a one-photon resonant, state. The results were generally in three-photon ionization via the 6 p agreement with theory. In their investigation they used a mass spectrometer and found, in addition to ions produced by purely atomic processes, ions produced by one-photon excitation from the 6 p 'PljZ state that must have been a product of dissociation of molecular cesium. One case involved excitation of the 6 p ' P l i 2state to the 7p 'P3/2 state, a quadrupole transition. The results appear to show effects of strong hyperfine coupling.

D. HIGHER MULTIPOLES I N THE BOUND-CONTINUUM TRANSITION In the discussion so far it has been assumed that the electric dipole operator dominates the ionization process and that higher multipoles can be neglected. This should be true as long as the wavelength of the ionizing radiation is small compared to the atomic dimension. Therefore, one would envision deviations from the electric dipole selection rule not only for

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

195

decreasing wavelength (Bothe, 1924; Krause, 1969) but also for increasing atomic dimension. Since the dimension of the charge distribution of a Rydberg-electron increases as nz (the square of the principal quantum number), higher multipoles could show up when ionizing out of a Rydberg state. Since the magnetic dipole matrix element is independent of the atomic dimension, the first higher multipole operator expected to appear in the interaction operator is the electric quadrupole one, which is of the form ( E . r)(K.

r),

where E is the polarization vector and K the wave vector of the ionizing light. The rotational symmetry, which is characteristic for the angular distribution of electrons obtained under the electric dipole approximation, is broken through the appearance of the wave vector in the interaction operator. Therefore, the effect of an electric quadrupole contribution to the boundcontinuum transition will be detectable through the appearance of some dependence in the angular distribution. An experiment designed to search for an electric quadrupole contribution was carried out by Leuchs et al. (1986). They measured the photoionization current dependence on the azimuthal angle rp for resonant three-photon ionization of atomic sodium via the 13 'D,,,Rydberg state. Theexperimental setup is described at the end of Section IIIB. Pulsed dye lasers were used to populate the 13 'D3,Z state and the second harmonic of a Nd:YAG laser was used to induce the ionization. The significant deviation from rotational symmetry found experimentally was 2.5 which corresponds to four standard deviations. This value compared favorably with a calculation based on quantum-defect theory. From theoretical considerations, it turns out that the size of the quadrupole contribution cannot be increased by increasing the principal quantum number n, as long as the orbital angular momentum quantum number I is not changed (Leuchs, 1984). The main contribution to the quantum mechanical matrix element comes from a region close to the ionic core, much smaller than the spatial extension of the Rydberg electron wave function (GiustiSuzor and Zoller, 1987). Therefore, the appearance of higher multipoles will depend sensitively on the I- and not on the n-quantum number. As a result, ionization of atoms in low-angular momentum states with low-energy photons (up to a few eV), and likewise the corresponding continuumcontinuum transitions, will always be dominated by the electric dipole operator.

*-

x,

E. NONRESONANT MULTIPHOTON IONIZATION Nonresonant multiphoton ionization (Fig. 4c) was a subject of intense interest in the early days of multiphoton ionization (see, for example, Voronov and Delone, 1965; Bakos, 1974; Delone, 1975), but there were no

S. J. Smith and G . Leuchs

196

systematic experimental investigations of angular correlations until Fabre et al. (1981) investigated the nonresonant six- and seven-photon ionization of xenon to the ' P , , , continuum. This investigation followed, and was motivated by, first observations of above-threshold ionization (Agostini et al., 1979, 1981). See, also, a theoretical discussion by Gontier et al. (1980). The experimental apparatus, illustrated in Fig. 7, again is based on use of a 4 2 retarding plate to rotate linear polarization in a laser beam. The radiation was frequency-doubled Nd:YAG radiation in 15-11s pulses. Energy per pulse was -200 mJ but the power density was not described. Photoelectrons were produced in a field-free region and accepted by an electrostatic electron energy analyzer, this being the first use of such an analyzer in angular correlation measurements, an innovation needed to separate electrons that have received energy from a seventh photon from those electrons ionized by only six photons. Figures 12(a) and (b) show the resulting angular distributions. These were fit to expansions in cos (2NB) for which the leading terms should be N = 6 and 7, respectively. However, it appeared that terms above N = 4 gave a negligible contribution. The six-photon ionization shows a secondary maximum, but the ionization with absorption of a photon in a free-free transition showed none. The ionization at 0 = 90" is not zero even for an odd-order ionization, as it should be according to Gontier et al. (1975) and P. Lambropoulos (1976). Leuchs and Smith (1982) measured the angular distribution of five-photon nonresonant photoionization of atomic sodium. Although their apparatus did not include an electrostatic energy analyzer, peak laser intensities, estimated at 1.4 x 10" W/cm', were believed to be below the threshold for generation of continuum-continuum transitions in sodium. Again, a distribution nonzero at 8 = 90" was obtained. In this experiment the multipole expansion, carried out in the equivalent representation cos ( 2 N Q contained significant contributions through N = 5 (one standard deviation) and nonsignificant fits for N > 5, in accord with the dipole transition picture. Dixit (1983) calculated the nonresonant five-photon ionization in atomic sodium, using a truncated summation over intermediate states, at several photon frequencies including the Nd:YAG fundamental frequency used in the measurement of Leuchs and Smith (1982). He obtained a structured angular distribution with maximum values at B = 0 and 180" but it was otherwise dissimilar to the unstructured experimental results just described. Hippler et al. (1983) measured the six- and seven-photon ionization of xenon, the same system as studied earlier by Fabre et al. (1981), but here the two transitions to the 'PI,, continuum were also studied, four angular distributions in all. Energy per pulse was approximately 25 mJ, which was lower than that used by Fabre et al. (1981). A lower target gas pressure was also used to minimize shifting and broadening of the photoelectron spectrum

-

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

197

a

0.21

0

b

,

,

,

,

,

,

,

0 3 (degl

-40

-80

,

,

(a)

80

LO

1.0+

0.8 -

yl c C

-

0.6 -

L h

ee

-

-- 0 4 C

P

L

A

-

0.2

-

01

1

-80

1

1

-40

1

1

0 3 Idegl

I

I

40

I

I

80

FIG. 12. (a) Angular distribution of photoelectrons for six-photon nonresonant photionizdtion. The points are experimental values. The solid curve is fitted (Fabre et al., 1981).(b) Angular distribution of photoelectrons for nonresonant seven-photon above-threshold ionization of xenon (Fabre et al., 1981).

198

S. J. Smith and G. Leuchs

due to space charge, an important experimental limitation. Here the sevenphoton distributions were small at # = 90", and they showed considerable structure. These results showed that a theoretical model based on six-photon ionization plus a free-free transition seriously underestimates the probability for absorption of an extra photon. Again, in this work the pulse-power density was not estimated. Petite et al. (1984) carried out a comprehensive theoretical and experimental investigation of angular correlation in above-threshold ionization (ATI), using cesium since it presents a much more tractable theoretical problem than does xenon, and provides greater experimental convenience. They worked at intensities in the range of lo9 to 10l2 W cm-2, and their efforts

FIG.13. Theoretical angular distributions for four-photon nonresonant ionization of atomic cesium for different intensities (in W cm-*). [Curve 1 : 1 x lo9; curve 2: loLo;curve 3: 10"; curve 4: 2 x 10" curve 5: 3 x 10" (Petite et al., 1984)l.

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

I99

emphasize the importance of intensity-dependent effects, notably by light shifts and high-order coupling of nonresonant bound states. Theoretical angular distributions for four-photon ionization of cesium illustrate the intensity sensitivity (Fig. 13) that can occur. The less sensitive five-photon distribution is shown in Fig. 14. In the experimental work, 54-ps pulses of 1.064-pm radiation were used. Short pulses are required to see intensity dependences because of the ease with which cesium is ionized. The maximum pulse energy was 100 mJ and the cross sectional area at the beam waist was 1.5 x l o p 5cm'. The time- and spatial-dependence of the laser radiation was considered in the analysis of the experimental results. The experimental results were in generally good agreement with the theoretical predictions. For intensities above 10" W cm-', the integrated (over space and time) angular distributions do not change with intensity because of saturation. Some discrepancies were observed in the comparison of theory with measured angular distributions, indicating the great sensitivity of the angular correlation method as a tool for these investigations. Dodhy et al. (1985) measured angular distributions for nonresonant twophoton ionization of cesium and rubidium, just above the ionization threshold, using an apparatus described by Compton et al. (1984). The pulse

-

FIG. 14. Theoretical angular distributions for the five-photon nonresonant ionization of atomic cesium for two intensities: 10' W cm-' (solid line), 10" W cm12 (dashed line) (Pctite et al., 1984).

200

S. J. Smith and G . Leuchs

duration was 5 ns and the power density 10' W/cmZ.Approximate agreement was obtained with angular distributions calculated using Sturmian functions and using the Hartree-Fock procedure. With intensities a factor of 10 higher, three-photon above-threshold ionization was studied. The theory predicts a predominately cos' 6' distribution, but the observed distributions contained higher-order contributions and did not go to zero at 6' = 90". P. Lambropou10s and Tang (1986) attributed some of the principal features of this work to spin-orbit coupling effects and discussed several other aspects of the comparison with theory. Humpert et al. (1985) extended their earlier work (Hippler et al., 1983) and have studied experimentally the photoelectron angular distributions from a series of the first five AT1 peaks in xenon, using linearly polarized 1.064-pm radiation and 50 mJ/pulse. They find photoelectron emission strongly peaked at 0 and 180" with a small amount of structuring in the sectors around 90" and 270". The sharpness of the peaks along the direction of linear polarization of the radiation field increases with the number of above-threshold photons, in a manner qualitatively consistent with ideas derived from theoretical concepts of the idealized process of multiphoton excitation and ionization within the dipole approximation. However, the xenon case is too complex for a quantitatively accurate theoretical prediction as a basis for comparison. Dodhy et al. (1986) obtained angular distributions for the first and second AT1 peaks in atomic rubidium, using 516.43-nm and 526-nm radiation. The first wavelength was resonant with the 5s * S -+ 4d '0 quadrupole transition and the second was an off-resonance transition. In contrast to the results obtained by Humpert et al. (1989, the second AT1 peak was nearly isotropic, much more so than the first AT1 peak which contained significant components of the Y,, as well as Yzn. Of particular interest was the fact that the quadrupole resonance had little effect-the on-resonance and off-resonance measurements did not differ dramatically. Kruit et al. (1983a) introduced the concept that "pondermotive potential" is an important factor in AT1 processes. A free electron of charge q and mass m immersed in an electromagnetic wave of frequency o and amplitude En picks up an average kinetic energy of qzE2,/4m02 associated with its oscillatory motion in that field. Electrons crossing a light beam scatter elastically as though from a potential of magnitude q Z E ~ / 4 m o Z called , the "pondermotive potential." The case of weakly bound electrons is more complicated and controversial. Pondermotive effects on angular distributions of photoelectrons in AT1 of xenon by 1.604-pm light in the intensity range 0.5-5 x lox3W/cm2 have been explored experimentally by Freeman et al. (1986). They show that scattering of photoelectrons from the pondermotive potential due to the intense optical fields required for AT1 may play the

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

20 1

dominant role in determining the observable photoelectron angular distribution. The available experimental results for angular distributions of photoelectrons resulting from nonresonant multiphoton ionization, and in particular from above-threshold ionization, do not appear to provide very definitive tests of the basic theoretical concepts (P. Lambropoulos, 1972a, 1972b; Gontier et al., 1975; Rzaiewski and Grobe, 1985) from which angular distributions might be derived for ideal systems. Rather, the experiments seem to demonstrate the limitations imposed by nonideal aspects of the systems accessible to study: space charge effects, departures from the nonresonant ideal and related power dependences, and the intrusion of the dynamic interaction between the weakly bound electron and the intense radiation fields required by these experiments. It seems plausible that, as the experimental artifacts are better understood, more definitive experimental work in theoretically tractable systems may appear in the not too distant future.

F.

PERTURBATIONS OF THE n

1*3D2 RYDBERGSERIES IN BARIUM

As indicated in this chapter’s Introduction, the study of photoelectron angular correlations is particularly interesting in connection with the influence of a perturbing state on an otherwise entirely predictable series. Photoelectron angular distributions yield detailed information on the intermediate state as well as on the ionization process itself, through admixtures to the angular distributions characteristic of the perturbing state. In atomic barium, a doubly excited configuration, 5d7d ( J = 2, S = 0) occurs near the n = 26 level in the 6snd ‘ . 3 D , Rydberg series (Fig. 15) and results in strong state mixing (Aymar and Robaux, 1979). Leuchs et al. (1983) carried out an extensive series of measurements of angular distributions of photoelectrons from states in the nD, series reached by resonant two-photon excitation, using essentially the apparatus represented in Fig. 6. Polar plots of these measurements are presented in Fig. 16. Included in Fig. 16 are angular distributions of photoelectrons that leave the residual ion in its ground state (6s continuum) and of photoelectrons that leave the ion in its first excited state (5d continuum). These are the only continua accessible by 1.06-pm ionizing radiation, and they give rise to two velocity groups of photoelectrons, at 1.16 eV and 0.64 eV, respectively. These groups’ arrival times at the detector are separated by -60 ns, so they are easily resolved by use of 30-11s gates in the signal-processing electronics. The fine structure of the 5d continuum ( J = 3/2, 5/2) is not resolved, however.

-

-

S. J. Smith and G. Leuchs

202

*lo4cm-’

-I

cm

4 19

4 18

4 17

FIG. 15. Energy-level diagram illustrating the levels and continua relevant to angular series in atomic barium. correlation studies of perturbations in the n 1.302

-

Measured angular distributions are included over the range n = 19 to 30. In a few cases the small triplet peaks were sufficiently well resolved to permit separate measurements. At n = 19, for example, the level is effectively unperturbed and the angular distribution has the characteristics that one would obtain by calculations analogous to those for atomic sodium outlined in Section 11. Here, however, in L-S coupling, the bound-state wave functions would be constructed using electron-spin functions

n

1

Jz CI + >I

~

1

Jz

~

-

> - I - >I + >I (singlet)

I

1+>1+> [I + >I - > + I - >I +>I (triplet)

I->I->

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

203

FIG 16. Measured photoelectron angular distributions ( x ) for barium Rydberg states LO the 6s and 5d continua, distinguished by electron time-of-Right. Photoelectron angular distributions for the 5d7d doubly excited perturber are also included. The top half of the figure, corresponding to a residual ion in the ground state, exhibits mainly singlet-triplet mixing. The bottom half, for the residual ion excited, shows lower photoelectron anisotropy since the excited ions retain .some alignment. The solid lines are least-squares fits of expansions in Legendre polynomials (Leuchs et al., 1983).

in combination with orbital functions, to obtain expressions analogous to Eqs. (27) and (29). Leuchs (1984) has demonstrated that principal characteristics of the distributions are determined by leading terms in the spherical harmonic expansions. Figure 17 compares three characteristic distributions, including the two at n = 19 from Fig. 16, with polar plots of squares of spherical harmonics. As the principal quantum number approaches n 26, the effect of mixing by the perturbing state becomes evident. In a simple analysis (Leuchs et al., 1983; Matthias et al., 1983; Zoller et al., 1984) the mixed wave function can be represented as

-

/6snd, J = 2)

= Z,l6snd

'D,)+ Z,16snd 'D,) + Z,)5d,,2, 7d3,2,J

= 2)

(43)

where the perturber is represented in jj-coupling. The mixing coefficients Z , , Z , , and Z, are obtained from a multichannel quantum defect theory (MQDT) analysis. This wave function can then be used in a calculation of the

S. J . Smith and G . Leuchs

204

8 1

3 SO

D2

:i 1

D2

b)

lYlO1

2

2

l

2

~

FIG. 17. (a) Polar diagrams of electron angular distributions in three-photon ionlzation of atomic barium via the 6 ~ 1 5 s' S o , 6 ~ 1 9 sI D 2 , and 6 ~ 1 9 s3 D , states. All linear polarizations were vertical. (b) Polar plots of absolute squares of spherical harmonics.

differential cross section, Eq. ( 7 ) , to obtain an expansion in spherical harmonics of the form of Eq. (2a), which terminates with /36,0. Since the continuum is reached in a structureless region, the coefficients contain direct information about the aligned n 1 3 3 Dstates. 2 For example, for ionization into the 6s continuum

where R: and R: are radial matrix elements for singlet and triplet ionization to anf-wave photoelectron.

G. PERTURBATIONS OF THE 6sns 'So RYDBERGSERIES IN BARIUM The interaction of the 6sns 'So Rydberg series with another doubly excited perturbing state, 5d7d 3 P 0 , provides a particularly interesting example of the unique properties of the angular correlation technique for investigating configuration interactions. Leuchs and Smith (1985) have used the techniques described in the preceding sections (see Fig. 11) to measure photoelectron angular distributions in this series, from n = 15 to n = 20 as well as that of the 5d7d perturber that lies between the n = 17 and n = 18 members of the series. Figure 18 shows resonant three-photon ionization spectra in the vicinity of these states. The 'So peaks are interspersed among the larger peaks from the 1 * 3 D series, 2 but are well separated and can be clearly resolved.

~

~

l

A N G U L A R CORRELATION I N MULTIPHOTON IONIZATION 6snr ISo n =21 I

20

19

18

I

I

I

15

16 1

17

205

I

5d \d 3P0 6snd lo2

20

19

18

17

16

15

I

I

I

I

I

1

14 I

3 ~ 2I

I

5d7d 'F2

__ W A V E LENGTH

FIG. 18. Resonant three-photon ionization of barium via the 6sns ' S o stales in the range from n = 15 to 21, for two different laser intensities: (a) I) = 0 , I z lo6 W/cm'; (b) I ) = 45", I 2 lo8 Wicm'. Saturation effects are obvious in finewidth and intensity ratios. Note that spectrum (b) was recorded with a lower detector sensitivity than (a) (Leuchs and Smith, 1985).

Again, the barium atom was excited to the 6sns IS, series through the 'PI level by two synchronously pulsed dye lasers tuned to the appropriate transitions. Ionization was accomplished with 1.06-pm Nd:YAG radiation. Laser beams were all linearly polarized along the same axis. Figure 19 shows polar plots of the measured angular distributions and the solid curves represent a fit of the spherical harmonic expansion Eq. (2a). In this case, since in the independent-particle model the electron wave functions are spherically symmetrical, the distributions have the form of Eq. ( la)

206

S. J. Smith and G . Leuchs 6sns'So n = I5

16

18

17

20

19

I

S d n d 3P0

n :?

FIG.19. Polar diagrams of photoelectron angular distributions in three-photon ionization of barium via high-lying J = 0 intermediate states. The arrow indicates the linear polarization direction of the laser radiation. The solid line represents the least-squares fit of I ( 0 ) to the data ( x ) (Leuchs and Smith, 1985).

and all information about the effect of the perturbation resides in the coefficients B2. A plot of these is shown in Fig. 20. The level designations shown in Fig. 18 are derived from the work of a number of authors; they are based on different physical properties and interpreted with MQDT. The level designated as the 5d7d perturber is that with the shortest lifetime (Aymar et al., 1982), the smallest diamagnetic shift

2 0 15s

-

v

16s

1

17s

c

18s

c

19s 20s 21s

1

1

1

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

207

(Fonck et al., 1977), the largest admixture of the pure 5d7d configuration (Aymar et al., 1978), and the largest isotope shift (Neukammer et al., 1982). The surprise here is that it is the 6.~18sconfiguration for which ,4, takes the extreme value, not the perturbing state. This relates to a unique property of the angular correlation technique: the observed angular distributions are sensitive to interferences between different ionization channels. Thus angular distributions contain information about relative signs of the admixtures of the perturber. The other properties cited enter only though the squares of admixture coefficients. For MQDT analysis, Aymar et al. (1978) have found that in the region around n = 18, the states in this series are well represented by the expression

I $)

= Z l I6sns ' S o )

+ 2, I 5dn'd ,Po) + Z , I 5dn'd

lS,).

(45)

Leuchs and Smith (1985) showed that a simple analysis (ignoring the third term, which represents an admixture 5 8 74) was sufficient for prediction of the most important observed features. It led to a set of relative signs for the admixture coefficients Z , and Zz and to ionization amplitudes, which accurately reproduced the measured angular distributions. The fact that the 5d7d state is not the most isotropic in the series (smallest value of fi,) is seen to be due to a change in the sign of 2 , when passing from the perturber (5d7d ' P o ) to the 6 ~ 1 8 s' S o state. The sign change is also responsible for some interesting effects in the ionization spectra (Fig. 18): the low line intensity for ionization from the 6 ~ 1 8 s'So state, which is much smaller than that from the 5d7d ,Po state. I t results from the localized nature of the electrostatic quadrupole interaction matrix element. A value was obtained for the interaction energy I ( V ) I = 12.7 cm ' and values were obtained for amplitudes and relative phases of the bound continuum transitions from the different states. A large amplitude was found for the transition from a pure 5d7d, J = 0 configuration to the 6s,,,,kpl,, partial wave, indicating spinorbit coupling and strong channel mixing in the continuum. Thus, this experimental study, on a system with a specific well-localized interaction with a perturbing state, demonstrates the potential of angular correlations as a unique source of information about mixed states. ~

H. THE5d7s I D , STATEI N ATOMICBARIUM Another rather informative demonstration of the value of investigating angular correlations was carried out by Mullins et al. (1985b) at the University of Chicago. The 5d7s ID, doubly excited state was prepared by resonant two-photon excitation through the 6s6p 'PI state, using 553.7-nm and 635.5-nm radiation from two dye lasers pumped by a pulsed N,-laser.

208

S. J . Smith and G. Leuchs

This intermediate state was then ionized by a delayed ( - 8-ns) component split offfrom the 635.5-nm beam. Three ion core states are accessible in this measurement: 6s 2S1,2,5d 2D,,2, and 5d ' D S i 2 . A literal interpretation of the configuration would lead one to expect that photoionization to the 5d core states would correspond to detachment of a spherically symmetrical s electron, into a continuum p-wave. The angular distributions would then be 1 + 0 P,,(cos 8), according to Eq. (I). Here the photoelectrons corresponding to the two fine structure components of the 5d core could be resolved by time-of-flight techniques. Angular distribution measurements, carried out for several values of q, are shown in Fig. 21.

-

0

712

6

n o

TI2

77

e

FIG.21. Photoelectron angular distributions corresponding to the process 6s' IS, 2 6s6p ' P , 2 5d7s ' D , 2 5d zD,,,,5,2 + k as a function of the angle q between the polarization directions of the tollinear laser beams. The solid curves represent least-squares fits. The individual curves are normalized to the same number of total counts. The three curves on the lefthand side correspond to the production of the j = 3/2 ion core; those on the right-hand side correspond to the production of the j = 5/2 ion core (Mullins et al., 1985b).

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

205)

TABLE 1 T H ELEAST-SQUARE$ VALUES OF THr COFTFICIFNTS FOR THF LFGFNDRF POLYNOMIAL 1 + C , , P , , + C,,,P,,, + C,,,P,, O B T A I N F FROM D O F THE q = 0" AND 90 ANGULAR DISTRIBUTIONS POR THE THRFF FITTING ION CORE STATES IN THE PHOTOIONIZATION OF THE 5d75 If), STATE Ot ATOMIC BARIUM (MULLINS FT AL., 1985b) lon core 6s 2 S , , , 5d ' D , / , 5d ' D , / ,

c,,,

C,,

'1

0" 90"

-0.10 f 0.12 - 1.44 f 0.05

0" 90

0.04 f 0.04 0.69 f 0.04

0"

0.51 f 0.04 0.75 f 0.10

90"

C,,,

1.20 f 0.12 1.29 f 0.08

0.70 f 0.02 -0.66 f 0.02

0.59 0.06 0.80 f 0.05

0.01 0.08 -0.1 1 f 0.06

-0.32 f 0.03 0.33 f 0.03

0.1 1 f 0.04 -0.05 0.06

-

~

*

*

Coefficients obtained from fits of the expansion in Legendre polynomials 1 CzoP,, C40P40 C,,P,, are given in Table I. The first interesting observation was the large branching ratio to the 6s core, 437& which Mullins et al. (1985b) attribute to interactions in the Sd7s 'D, intermediate state, not to continuum-state interactions. The angular distributions are also consistent with a large admixture of 6snd 'D, to the 5d7s 'D,state. The second point is the appearance of large coefficients C4, and C,, in the photoionization to the 5d core, showing that the Sd7.s configuration is not dominant. These angular distributions seem to be consistent with a large contribution from the 5dnd channel. Mullins et a]. (198Sb) cite these properties as contradictory to the predictions of MQDT analysis (Aymar and Robaux, 1979) and as evidence of the limitations of MQDT for the lowerlying states. In any case, they demonstrate quite clearly the capability of the angular correlation measurements to provide characterization of mixed states beyond that provided by the usual configuration labeling.

+

+

+

I. OTHER DOUBLY EXCITEDCONFIGURATIONS IN THE ALKALINE EARTHS The group at Chicago has also carried o u t angular correlation measurements for a number of doubly excited configurations of low-lying orbitals in the alkaline earth atoms. For two-electron systems, the role of electron correlations can be important to structural and dynamical properties of atoms. This has been found to be the case for helium. While the ground state

210

S. J. Smith and G. Leuchs

conforms closely to the independent-particle model, certain doubly excited continuum states (Madden and Codling, 1963; Cooper et al., 1963) have constants of motion with characteristics of linear triatomic molecules (Kellman and Herrick, 1980; Herrick and Kellman, 1980). These doubly excited states of alkaline earths would seem to be an intermediate case for investigating the ranges of validity of the independent particle model and the collective model. Angular correlations should provide basic data to support relevant theoretical development in this important area. Such surveys have been carried out for resonant two-photon ionization through the states Ca(4s4p) ' P ; , Sr(5s5p) ' P ; , Sr(5s5p) 3P?, Ba(6s6p) 'Py, and Ba(6s6p) 'P7 (Mullins et al., 1985a); through the barium states (5d6p) 3 P ; , (5d6p) 'D:, and (6s6p) ' P y (Mullins et al., 1985~);and by resonant three-photon ionization, the states of atomic barium 6s8s 'So, 6pz 3P0, 6p2 3P1,6s7d ID,,and 6 p z 3 P , (Hunter et al., 1986). The data give branching ratios to different ion core continua. In addition, best fits of theoretical expressions similar to Eqs. (31) and (32), to measured angular distributions were used to obtain d,/dd, the ratios of radial matrix elements; (6, - 6 d ) , the relative phases for the two channels usually open to the continuum; and the sign of the product (ddd,) cos (6, - 6,). These fits were based on use of electron configurations believed to be dominant according to MQDT theory or results of other types of measurements. Consideration of individual processes yields considerable information about the nature and extent of mixing. Failure of the assigned configuration may be evident in the branching ratios or in departures from predicted angular distributions. In general, the data suggest that in many cases there is strong configuration mixing and possibly mislabeling of configurations. The details are beyond the scope of this chapter. It is sufficient to note that the angular distributions measured in these surveys provide important new tests for existing theoretical treatments such as MQDT, and they may provide impetus for the development of new theoretical methods for dealing with the region of transition from that in which the independent particle model is appropriate to that in which electron correlations are dominant.

J. AUTOIONIZINGRESONANCES IN ALKALINEEARTH ATOMS The preceding sections have dealt with angular correlations from bound intermediate states, but the study of angular correlations from autoionizing states is also an interesting tool for state characterization. The first such distributions were published by Feldman and Welge (1982). These were fourphoton processes in strontium using laser radiation at several wavelengths from 558 nm to 563 nm, corresponding to J = 3 electronic configurations at 563, 562, 559.8, and 559.2 nm, and to configurations more specifically

ANGULAR CORRELATION IN M U L T I P H O T O N IONIZATION

21 1

identified, 4s(’D5,,)4f[$]y and 4d(2D5,2)4J’[i]y at 560.3 and 558 nm, respectively. Expansion coefficients were obtained, but the interpretation was tentative. T. J. Gallagher and his colleagues have explored the measurement of angular correlations as one of a number of tools for examining the properties of autoionizing states. Sandner et al. (1983) studied angular distributions of electrons ejected from the (6pjru,,,), autoionizing states of atomic barium, excited from the spherically symmetric 6sn.s ‘ S o bound states. Autoionization was to the Ba+ 6s,5d, and 6 p states. Energy levels and the processes involved are shown in Fig. 22. The 6 ~ 1 5 s* S o state, chosen as an example, is reached by sequential excitation using two dye lasers. A third dye laser is required for the next step, in which the “inner” 6s electron is promoted to 6p, while the 15s electron is a “spectator.” Since excitation is from a symmetric state, only the linear polarization of the third laser needs to be rotated to obtain the angular distribution. This amounts to electric dipole excitation of an isotropic system and Eq. ( 1 ) may be used to predict the angular distribution. Energy analysis was used to separate electrons ejected into different continua. In this work the measurements were carried out only for 0 = 0’’ and 0 = 90”. Kachru et al. (1 985) published a more comprehensive investigation of this process, including an MQDT analysis. Here the linear polarization was rotated in 15”steps. One of the results of the analysis is that these Ba I states autoionize predominantly to excited states of the ion, a fact of interest in connection with development of autoionization lasers.

5d3/2, 512

FIG.22. Energy-level diagram for the excitation of the Ba(6p,,, IS,s,,,),= ,. Continua above the B a t 6s and Sd states are shown, as well as the laser pumping steps that are indicated by single-ended arrows. Electrons of 2.5 and 1.7 eV are ejected (Sandner et al., 1983).

212

S. J. Smith and G. Leuchs

At Freiburg, Sandner et al. (1986), using time-of-flight electron spectroscopy, continued this line of investigation, concentrating on the energy range between the Ba' 6p,,, and 6p,,, fine structure levels. They found that the P-parameter (Eq. (1 )) is strongly energy-dependent, a fact in contradiction with conventional MQDT. They show that this behavior may be attributed to a previously unknown shape resonance 5d14fin neutral barium. Feldmann et al. (1986a) have used three-photon ionization, with photons in the wavelength range 281 to 285 nm and intensities lo9 W/cm2, to search for a predicted autoionizing resonance in xenon (Gangopadhyay et al., 1986). Measured angular distributions of photoelectrons show a definite resonance behavior at the wavelength of the predicted resonancc, in contrast to the total ionization cross section, which shows no such structure.

-

K. LASER-INTENSITY DEPENDENCE I N RESONANT MULTIPHOTON IONIZATION The experiments discussed previously generally utilize pulsed lasers. These have the advantage that several dye lasers can be used to provide synchronous pulses tuned to the bound-bound transitions needed for sequential stepwise excitation; and they are usually required to provide radiation intensities necessary to obtain easily measurable ionization currents from highly excited bound states. (Ionization cross sections are generally proportional to n- 3.) The question of the influence of these intense laser fields, which are commonly on the scale of MW/cm2, on the structure of the atom under study becomes very significant. (Corresponding issues arising in connection with nonresonant multiphoton ionization were discussed in Section IVE and involve a much higher scale of radiation intensities than those of interest here.) This problem of ac-Stark effects was addressed theoretically by Dixit and Lambropoulos (1981, 1983). They illustrated the problem by considering the processes in atomic sodium

which involve a two-photon resonance to the 40,,, and 40,,, states as well as two channels to the continuum ( p andfwaves). At low laser intensities, loworder perturbation theory is valid. At higher intensities, the two boundbound channels, as well as the two bound-continuum channels, can saturate at different intensity levels, thus altering the interchannel interference term and the resulting angular distribution. Furthermore, the 4 0 states and the ground state are subject to light shifts. The 3s-40 interval may be changed

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

21 3

and, also, the D,,, - D,,, fine structure interval. Thus, the angular distributions may change due to “intensity tuning” of the available channels with respect to the laser frequency. A density matrix treatment was developed in which all saturation and related effects were taken into account. Also considered were the effects of laser bandwidth. The results, calculated for a 5 4 s monochromatic laser pulse, are shown in Fig. 23 for a two-photon resonance with the 3S,,,-4D,,2 transition. Figure 24 shows how the total ionization probability, P, and probabilities for H = OC’ (PI,)and 19 = 90” ( P , ) vary with power. I t is seen that at very high laser intensities, the angular distribution tends to a limiting form different from that at low intensities. These predictions were investigated by Ohnesorge et al. (1984). They used pulse amplification of a cw beam from an argon-ion pumped ring dye laser. The three-stage Nd:YAG pumped dye amplifier was capable of output powers up to 20 mJ with about 130-MHz bandwidth in 6-11s pulses. Angular distributions (Fig. 25), measured with mean pulse intensities ranging from 0.6 MW/cm2 to 530 MW/cm2, were qualitatively consistent with the predictions of Dixit and Lambropoulos (1981, 1983) although direct comparison was complicated by questions of spatial and temporal homogeneity of the pulses. A more quantitative comparison of the experimental results with a calculation taking into account the ground state hyperfine structure (Geltman and Leuchs, 1985) confirmed the ability of the theory to predict the main features of the effects of intensity on angular dependence. In addition to saturation and light shifts, Raman coupling between the D,,,- D,,, was also included in the theoretical calculations. However. the

0.5 1.0 3.0 4.0 5.0 50.0 510 FIG.2 3 . Photoclectron angular distributions calculated for a two-photon resonant threephoton ionization of sodium. The laser intensities (in MW/cmz) are shown under each plot. Other parameters are laser duration f = 5 nsec, its handwith h = 0, and its frequency ti). (0 IS such that 2w = “ i f D I - to,s, 2 . The arrow indicates the direction of light polarization with respect to which f l is measured. The radial scale is the same for all the plots (Dixit and Ldrnbropoulos. 1983).

S. J . Smith and G. Leuchs

214

10-4

* -

loTloS

107

1OD

1P

D

I (W/cml)

FIG.24. Multiphoton ionization probability as a function of the laser power. P , Pll,and P , correspond, respectively, to the total ionization probability and the ionization probabilities in a direction parallel (0 = 0") and perpendicular (0 = 90") to the direction of light polarization. Solid curves correspond to a monochromatic field and the dashed curves to a field having full width at half maximum 0.002 cm-' (Dixit and Lambropoulos, 1983).

0.6

1.2

0 2.9

9

2L

130

530

FIG.25. Experimental photoelectron angular distributions for photoionization by 579-nm radiation, at seven intensities, by way of the 4'0,,, level in atomic sodium. The solid lines are fits to the data (Ohnesorge et al., 1984).

ANGULAR CORRELATION I N MULTIPHOTON IONIZATION

21 5

effect of the Raman coupling was essentially negligible as compared to that of the other two processes. This situation is changed when excitation of the intermediate state and ionization are separated in time. Then one deals essentially with a bound state that coupled to the continuum by absorption of one photon. Since there is no sharp resonance involved in the bound-continuum transition, ac-Stark shifts should play no role. Also, depletion of the bound state should not affect the angular distribution. However, if there are states nearly degenerate with the initially populated state, then an intensity dependence of the angular distribution may be found because of stimulated Raman coupling between these degenerate states. Such an experiment was performed by Leuchs et al. (1985). Sodium atoms of a thermal beam were excited to the n2D,,, state using two pulsed dye lasers. The ionization was induced by a delayed 1.06-pm Nd:YAG laser pulse not overlapping in time with the dye laser pulses. All laser beams were linearly polarized along a common axis. The angular distributions were measured for n = 7 and n = 8. At a low laser intensity of 0.2 MW/cm2, the photoelectron angular distributions observed for n = 7 and n = 8 were essentially the same, whereas for an intensity of 50 MW/cm2, the distributions changed drastically and were quite different for n = 8 and iz = 7 (Fig. 26). The reason for this difference is that Raman coupling from 'D,,,to 'D,,,

I

I (MW/cmz) FIG.26. 7'0,,, and 8zD,,, intermedlate states, aligned by excitation with linearly polarized lasers, were photoionized by 1.06-pm radiation at diff'erent levels of power density. At 0.2 MW/cm', the distributions are similar. At higher power densities, observed changes in angular distributions are attributed to Raman coupling with the D,,,states (Leuchs el al., 1985).

216

S. J. Smith and G. Leuchs

is enhanced by the 4p-resonance, the virtual state of the Raman process being above and below the 4p-state, for n = 7 and n = 8, respectively. This results in a change of the sign of the Raman coupling matrix element, leading to the drastically different shapes of the distributions for the higher intensity. Although the effect of this Raman coupling appears clearly in the angular distribution, it is difficult to observe this coupling when only the total ionization cross section is measured. It is interesting to note that the Raman coupling transfers as much as 12% of the initial 'D,,,-state population to the 2D,,,state. Since the intensities used were only moderate for pulsed laser beams, this effect may have been present in other experiments. Grobe et al. (1986) worked out a model describing migration of populations from low4 to high-l states of the same principal quantum number by the Raman coupling mechanism. This process may account for anomalous behavior of photoelectron angular distributions observed in resonant three-photon ionization of atomic sodium at high (n 35) principal quantum numbers, discussed by Leuchs and Smith (1983) as possibly providing evidence of breakdown of the dipole approximation, a suggestion since discarded. An effect of Raman coupling was also found by other groups when studying ac-Stark shifts in multiphoton ionization of mercury (Reif et al., 1984) and when studying light polarization effects in three-photon ionization of barium (Kelly et al., 1986). In all these studies the first experimental results have been unexpected and quite puzzling since, intuitively, the one-photon bound-continuum transition was thought to dominate all higher-order processes. However, the experiments have shown that the three-photon process-stimulated Raman coupling plus ionization-may compete effectively with the one-photon process. As discussed previously, most excited-state photoionization studies have, so far, concentrated on investigating either atomic structure effects like configuration mixing or effects of high laser intensity, using different theoretical approaches in each case. Based on recent theoretical efforts by GiustiSuzor and Zoller (1987), there is now hope that a unified description may be found by an extension of multichannel quantum defect theory.

-

V. Conclusions In the preceding sections, recent developments in the field of angular distributions of photoelectrons in multiphoton ionization of atoms have been addressed. We hope to have shown that photoelectron angular distribution studies make accessible information about atomic structure or radiationatom interaction that is not readily available otherwise. The examples

ANGULAR CORRELATION IN MULTIPHOTON IONIZATION

21 7

discussed include the study of the dynamics of excited atomic states resulting from angular momentum coupling or, in multielectron systems, also from electron correlations. In any case, measuring angular distributions is advantageous because cross terms involving ionization amplitudes to different continuum states are retained. These same cross terms vanish in the total ionization cross section due to the orthogonality of the continuum wave functions. On the one hand, these cross terms depend on the relative scattering phases of the various partial waves of the outbound electron. On the other hand, and similarly important, the signal-to-noise ratio for weak partial waves is greatly enhanced when there is a dominating strong partial wave. This could be used, e.g., to look for an interesting wavelength dependence of a partial wave too weak to show up in the total cross section (Aymar et al., 1976). It was essential in the observation of the quadrupole contribution to the boundcontinuum transition (Section IVD). There the size of the cross term was 2.5 7; of the total signal, whereas the angle-integrated ionization cross section is changed through the appearance of the quadrupole contribution only by the square of this value ( < The second area where photoelectron angular distribution measurements have had some impact includes the study of atoms in strong light fields. As shown, angular distributions allow for sensitive investigations of the ac-Stark effect. They also have helped to identify the importance of stimulated Raman coupling to degenerate nearby states, an effect that is related to the observation of laser-induced resonant intermediate states when ionizing with two or more laser beams having different wavelengths (Feldmann et a]., 1986b). Within the past few years, the potential of angular correlation studies in connection with the different multiphoton ionization mechanisms represented in Fig. 4 has gained wide recognition. An interesting variety of applications have appeared in the recent literature, and the prospect is for a continuing interest in this approach. It is our hope that this summary of developments in this relatively new field will encourage and support this interest.

ACKNOWLEDGMENTS

The many contributions and the unending patience of Leslie Haas, Gwendy Lykke, and Lorraine Volsky of the Joint Institute for Laboratory Astrophysics Scientific Reports Office,are gratefully acknowledged. One of us (SJS) acknowledges financial support from the National Bureau of Standards for this project.

218

S. J. Smith and G. Leuchs REFERENCES

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL 24

OPTICAL PUMPING AND SPIN EXCHANGE IN GAS CELLS R . J . KNIZE. Z . WU. and W. HAPPER Depurtment of'l'hysics Princeton Universit.y Pririceron New Jersqy 08544

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1. Introduction . . . . . . . . . . . . . . . . . . . . . I1. Optical Pumping . . . . . . . . . . . . . . . . . . . . A . Basic Principle . . . . . . . . . . . . . . . . . . . B. Depopulation and Repopulation Pumping . . . . . . . . . C . Optical Pumping Sources . . . . . . . . . . . . . . . 111. Description of Spin-Polarized Atoms in Optical Pumping Experiments IV . Detection of Polarized Atoms . . . . . . . . . . . . . . A. Transmission Monitoring . . . . . . . . . . . . . . . B. Fluorescence Monitoring . . . . . . . . . . . . . . . C . Radioactive Atoms . . . . . . . . . . . . . . . . . D . Nuclear Magnetic Resonance . . . . . . . . . . . . . V . Spin-Transfer Collisions . . . . . . . . . . . . . . . . A. Spin Exchange . . . . . . . . . . . . . . . . . . . B. Metastability Exchange . . . . . . . . . . . . . . . C. Charge Exchange . . . . . . . . . . . . . . . . D . Chemical Exchange. . . . . . . . . . . . . . . . . E . Penning Ionization . . . . . . . . . . . . . . . . . VI . Relaxation . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . B. Binary Collisions . . . . . . . . . . . . . . . . . . . C. Three-Body Relaxations . . . . . . . . . . . . . . . D. Wall Collisions and Coatings . . . . . . . . . . . . . E.Diffusion . . . . . . . . . . . . . . . . . . . . . F. Inhomogeneous Magnetic Fields . . . . . . . . . . . . 0. Radiation 'L'rapping and Coherence Narrowing . . . . . . . H . Excited State Depolarization . . . . . . . . . . . . . VII . Frequency Shifts . . . . . . . . . . . . . . . . . . . . A . Collisional Shifts . . . . . . . . . . . . . . . . . . . B. Light Shifts . . . . . . . . . . . . . . . . . . . . . VIII . Applications . . . . . . . . . . . . . . . . . . . . . A . Precision Atomic Measurements . . . . . . . . . . . . B. Frequency Standards . . . . . . . . . . . . . . . . C . Fundamental Physics Tests Using Optical Pumping Techniques . D . Magnetometers . . . . . . . . . . . . . . . . . E . Polarized Targcts and Sources . . . . . . . . . . . . . F . Spin-Polarized Fusion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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Copyright I " 1988 by Academic Press Inc All rights o l reproduction in m y form reserved ISBN 0-12-003824-2

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R. J. Knize, Z. Wu, and W. Happer

1. Introduction Since A. Kastler (Kastler, 1950) first pointed out the possibility of producing strongly nonequilibrium population distributions among the spin sublevels of appropriate atoms by the scattering of resonant light, “optical pumping” has been widely used in laboratories across the world to investigate fundamental questions of physics and to construct useful devices. Some advantages of optical pumping methods are the relative simplicity of the apparatus, the high signal-to-noise ratios, and the very narrow magnetic resonance linewidths that can be observed, often with linewidths of one Hertz or less. The widespread availability of lasers in recent years has broadened the capabilities of optical pumping methods by providing orders of magnitude more light (thereby making possible the pumping of much larger and denser samples) and by offering a light source with a much more finely controlled spectral bandwidth than can be obtained from conventional resonance lamps. The most widespread uses of optical pumping methods in fundamental science have been in very precise spectroscopic studies of optically pumped atoms and in studies of spin relaxation and spin-transfer mechanisms. One of the most interesting results has been the fairly detailed picture that is now available about the spin interactions of alkali-noble gas van der Waals molecules. Optical pumping methods continue to find new uses in nuclear physics for producing spin-polarized radioactive atoms or spin-polarized targets, and there has been recent interest in the use of optical pumping to make spin-polarized fuel for use in controlled thermonuclear fusion experiments. Optically pumped devices are well entrenched in the technological world; optically pumped magnetometers are commercially available and optically pumped frequency standards are used in many satellites. In this chapter, we give a brief overview of the current status of optical pumping. We have not attempted to be exhaustive and we refer the reader to earlier reviews (Kastler, 1957; Carver, 1963; Happer, 1972; and Balling, 1975) for a more complete survey of the literature. We hope, however, that the readers of this chapter will be able to get a good feeling for the basic results and future potential of optical pumping methods.

11. Optical Pumping A. BASIC PRINCIPLE

Optical pumping is a method to produce population imbalances among the ground or excited state sublevels of atoms, molecules, etc. by the

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interaction between the atomic system and light (Kastler, 1950). From the point of view of thermodynamics, optical pumping is a process in which entropy is transferred from the atomic system to the photons. The basic idea is illustrated by the following simple example (Fig. 1). Consider a system of atoms with a 2S11,ground state and a excited state. (An alkali atom with no nuclear spin would be such a system.) The atoms are illuminated by a 6’ resonance radiation propagating along the direction of a static magnetic field. The selection rule Am = + 1 prohibits the atoms in the + 1/2 ground state sublevel from absorbing photons. However, ground state atoms in the - 1/2 sublevel can absorb a photon and make a transition to the + 1/2 sublevel of the excited state ’PI,,. Due to spontaneous emission, the atoms in the excited state 2P11,will decay to either of the two sublevels of the ground state. The + 1/2 sublevel of the ground state will therefore eventually be more populated than the -1/2 sublevel, thus producing a polarization of the electronic spin. The optical pumping of the spin system sketched in Fig. 1 can be described by rate equations for the ground state populations. We denote the number densities of the k 1/2 sublevels of the ground state by N * and we see that the rate equations describing Fig. 1 are N-1,2

=

-2RN-11,

Y + $RN-l,2 + j(N1’2

- N-112)

(la>

We have introduced the “spin relaxation rate” y to describe the transfer of atoms between the k 1/2 sublevels due to collisions etc. We assume there is no transfer of population between the excited state sublevels before the excited atoms decay. The mean pumping rate R is the product of the photon flux (photons/cm2sec) and the absorption cross section oo of an unpolarized atom. We assume that R is much smaller than the excited state decay rate to

m=-1/2

m=1/2

FIG.1. Optical pumping of an alkali atom (neglecting hyperfine structure) usmg circularly polarized D , light.

R. J . Knize, Z . Wu, aitd W. Happer

226

the atoms spend almost all of the time in the ground state. The solutions to Eqs. (1 a) and (1 b) for an initially unpolarized vapor are easily seen to be

+

N l 1 2 N - l , 2 = N = constant,

1

2~

( N I I 2- N - 1 1 2 )= (S,)

(2a)

=

It is characteristic that the optical pumping process is described by exponential transients that involve both the optical pumping rate R and one or more relaxation rates y. It is also characteristic that certain linear combinations of ground state sublevel populations such as (S,) in Eq. (2b) have an important significance, and it is often useful to write the optical pumping equations directly in terms of appropriate observables such as (S,) and N instead of sublevel populations. For example the Eqs. (2a) and (2b) can be written as dN - = 0, dt

Real atoms often have many more than two ground state sublevels. More than two rate equations may be needed to describe optical pumping in these systems. If one takes into account the nuclear spin I and assumes the ' 6 pumping light has a broad spectral line profile (compared to the ground state hyperfine separation-see section KC.), the ground state sublevel F = mF = I + 1/2 will become more populated than all the others, thus giving rise to a polarization in both electronic and nuclear spins at low magnetic fields. Optical pumping can also be achieved using linear polarized light as illustrated in Fig. 2. Light with a sufficiently narrow spectral profile can be used for hyperfine pumping as shown in Fig. 3, m=O

--m=-l

m=O

J-l

m=+I

FIG. 2. Depopulation pumping using linearly polarized light.

OPTICAL PUMPING AND SPIN EXCHANGE

2pl/2-

221

F‘= 2 F’= I

t

Y

In the preceding discussion, we have assumed that the mean pumping rate R is uniform throughout the cell. This is possible only if the vapor is optically thin so the absorption of the pumping light is small. For sufficiently intense sources of light, it is possible to pump optically thick vapors. When the pumping light is first turned on, it is absorbed in one optical depth A at the input face of the cell. The illuminated atoms are optically pumped in a time (n/R)-’ where n is the number of photons needed to fully polarize an atom. We may assume that the fully polarized atoms no longer absorb light so the light can penetrate more deeply into the cell. Consequently, an optical pumping “shock wave” moves into the cell at a velocity u = AR/n (Bhaskar et al., 1980). In our model system (Fig. l), n is easily shown to be 3 / 2 (n = 2 R / N st N d t when y = 0). For a more realistic situation of complete mixing (i.e., large amounts of a quenching gas such as N , are used to suppress fluorescence), one finds n = 1. In the preceding examples, any spin polarization of the atomic vapor generated by the pumping light is along the direction of the static magnetic field. This is called longitudinal pumping. Transverse optical pumping occurs when the pumping light generates an atomic polarization along a direction perpendicular to the static magnetic field. For efficient transverse pumping, the light must be modulated at the difference frequency (Em- E J h of the sublevels with energies Em and En describing the transverse polarization. Optical pumping can also be used to create population imbalance among the rotational sublevels of a molecule. This is the basis of polarization spectroscopy (Wieman and HCnsch, 1976), which finds applications in molecular spectroscopy. Many examples of optically pumped lasers and masers exist.

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B. DEPOPULATION AND REPOPULATION PUMPING In Fig. 1 we have seen that the population changes in the ground state sublevels are caused by two different mechanisms. In the first mechanism, called depopulation pumping, the pumping light removes the atoms from different ground state sublevels at different rates. The term - 2 R N - , , , in Eq. (la) describes depopulation pumping from the sublevel - 1/2. The population imbalances among the excited state sublevels can be partially transferred to the ground state sublevels during the spontaneous decay of the excited state. This second mechanism is called repopulation pumping. The terms ( 4 / 3 ) R N - and ( 2 / 3 ) R N - ,/, in Eqs. (la) and (lb) describe repopulation pumping. Although both depopulation and repopulation pumping are present in Fig. 1, they do not necessarily have to be present at the same time. For example, only depopulation pumping occurs in Fig. 2. Since depopulation pumping relies on the different pumping rates out of different sublevels, the efficiency of depopulation pumping depends on the polarization and spectral profile of the pumping light. For example, no depopulation pumping of the ground state alkali atom can occur if the two D lines have the same intensity and polarization. Repopulation pumping rates are proportional to the polarization of the excited state and therefore any depolarization mechanism of the excited state will degrade the polarization produced by repopulation pumping just as any depolarization mechanism of the ground state will decrease the polarization produced by depopulation pumping. However, it is often true that depolarization cross sections for the optically pumped state are quite small compared to depolarization cross sections for the excited state involved in the optical pumping cycle. For example, for alkali atoms in helium buffer gas, the ground state (S-state) depolarization cross section is on the order of cm2, while the excited (P-state) depolarization cross section is on the order of 10-15cm2, a difference of 10 orders of magnitude. In such cases, depolarizing collisions with buffer gas atoms can almost completely eliminate polarization due to repopulation pumping while having little effect on the polarization due to depopulation pumping. For diamagnetic atoms with a ' S o ground state and nonzero nuclear spin (such as Hg, Cd, and Zn), depopulation pumping is possible only for narrow line excitation (i.e., different hyperfine components in the pumping light have different intensity). However, efficient repopulation pumping can occur for both broad and narrow line excitation provided that the hyperfine intervals of the excited states are comparable to or larger than the natural width, and no collisional depolarization of the excited state or depolarization due to radiation trapping occurs (Lehmann, 1967).

OPTICAL PUMPING AND SPIN EXCHANGE

229

C. OPTICAL PUMPING SOURCES The most common light sources that are used in optical pumping experiments are either resonance lamps or cw lasers. Compared to lasers, lamps have the advantage of being quiet (shot-noise limited) and stable. Lasers, on the other hand, can give much more spectral intensity. For example, the intensities of D lines from a typical alkali resonance lamp are about 1 mW, while the typical output of a dye laser is a few hundred milliwatts. In addition, the output of the lamp is nearly isotropic, while lasers are highly directional. Therefore, in order to achieve large polarization, lasers are often used as a pumping source, while to achieve high signal-to-noise ratios, lamps are used as a probe source. The line profile of the pump source is also an important factor in optical pumping experiments. The D , and D , lines of alkali resonance lamps can easily be separated by use of an interference filter except for sodium and lithium lamps, for which Lyot filters are needed (Kopf et al., 1969). Thus, a dye laser is especially convenient for sodium and lithium. The spectral width of a multimode laser is usually several tens of GHz, which is much larger than the hyperfine splitting of the most ground state atoms. Therefore the intensities of the two hyperfine components in the pumping light are approximately equal and all the hyperfine states are pumped. A single-mode dye laser has a spectral width smaller than the hyperfine splitting of most alkali atoms and can be used in hyperfine pumping. By using separated isotopes in lamps or filter cells, it is also possible to use resonance lamps for hyperfine pumping. The isotopes 85Rband 87Rbare especially useful for this purpose (Davidovits and Knable, 1964).

111. Description of Spin-Polarized Atoms in Optical

Pumping Experiments An atomic state of nonzero spin is said to be polarized if the populations of the different spin sublevels are not the same. When the atoms are in equilibrium at a temperature T, the relative population pmmof a spin substate Im} of energy Em will be

230

R. J . Knize, Z . Wu, and W . Hupper

where the partition function is

z = 1exp( m

g).

For convenience we shall choose the zero of energy such that the mean energy is zero, i.e.

CErn=O m

The sum on m in Eqs. (5) and (6) extends over all energy sublevels of the state in question. The population imbalances implied by Eq. (4) for a given state are quite small at room temperature, typically a few parts per thousand for electron spin sublevels and a few parts per million for nuclear spin sublevels. Nevertheless, these small population imbalances are enough to carry out conventional experiments on nuclear magnetic resonance spectroscopy and electron spin resonance spectroscopy. At low temperatures on the order of 1 K or less and at magnetic fields of some tens of kilogauss, it is possible to attain nearly complete electron spin polarization in thermal equilibrium, and much of this electron spin polarization can be transformed into nuclear spin polarization by nonequilibrium methods such as dynamic nuclear polarization (Abragam, 1961) or chemical recombination of electron-spin-polarized hydrogen (Cline et al., 1981.) By optical pumping it is possible to obtain nearly complete spin polarization at room temperature and even at much higher temperatures. The spin sublevels, whether degenerate or not, can be artificially maintained far from thermal equilibrium. To fully describe experiments with optically pumped atoms, it is necessary to know both the population pmrnof the spin sublevels I m) and the coherences prnnbetween the sublevels 1 rn) and In). To see how these quantities come about, consider a collection of N atoms. If the atoms are weakly interacting with their surroundings, we may imagine that each atom is described by its own wave function I $ i ) where i = 1, 2,. . . ,N . Experiments almost always measure the mean value of some atomic observable-for example, the z component of the electron spin S,. Then the mean value of S, is

OPTICAL PUMPING AND SPIN EXCHANGE

where the density matrix

prim

23 1

is defined by

Note that the density operator for atoms in thermal equilibrium is simply

where H is the spin Hamiltonian of the atom [we assume TrH = 0 as implied by Eq. (6)] and the partition function 2 is given by Eq. (5). The coherences pnm(m# n ) between energy eigenstates are zero in thermal equilibrium. In the presence of optical pumping, the density matrix can be much more complicated than the simple thermal equilibrium value Eq. (9). Although it is possible to completely specify the density matrix by listing all values of the population pmmand coherences pnmrthere are good reasons to specify the density matrix in terms of certain linear combinations of the populations and coherences. We may think of these linear combinations as expectation values of certain observables Q’ (Happer, 1972), i.e.,

The coefficients used to construct the linear combination of prim are the matrix elements of the observable. In many practical cases, the eigenobservable can be chosen to be an orthonormal set so that we may write the density matrix as P

=

c i

(Qi>Qi

From the form of Eq. (lob), we see that the contribution from Q’ to the population of the sublevel rn is (Qi)QLm. There are two kinds of observables Qi that are especially important: eigenobservables and “signal” observables. The time evolution of certain atomic observables may be especially simple and can consist of exponentially damped oscillations (Bouchiat, 1963). Such observables, which we shall call eigenobservables, are completely analogous to the normal modes of a classical system of elastically bound masses, to the electromagnetic resonances of a microwave cavity, or to the heat diffusion modes of a cooling body. For example, the eigenobservables of an alkali atom with nuclear spin I = 3/2 under the influence of electron randomizing collisions are shown in Fig. 4 along with the corresponding relaxation rates in terms of the characteristic electron randomization rate 1 /T.

2D

0

OPTICAL PUMPING AND SPIN EXCHANGE

233

The signal observed in an experiment is a function of some signal observable. For example, the cross section o for the absorption of broadline D1 radiation by electron-spin-polarized alkali atoms with no nuclear spin is an observable with the expectation value (Dehmelt, 1957; Bouchiat, 1965)

(.> = a,(l

-

2S,(S,)).

( 1 1)

Here sz is the mean photon spin of the light, which is assumed to propagate along the z direction. The cross section is not one of the eigenobservables of Fig. 4, but is a linear combination of three eigenobservables, the unit operator ( I z ) , and an observable (Q,), which is defined by Q,

=

Sz - + I z ,

(12)

Thus, if an optically pumped alkali vapor with nuclear spin I = (3/2) is allowed to relax to its unpolarized state under the influence of electronrandomizing collisions, the photon absorption cross section (.will ) decay with two exponentials 1 - 2s,(Q,), exp to the unpolarized value ao. Equations (1 1) and (13) show the significance of special observables in the description of optical pumping experiments. The photon absorption cross section for broadline D, light, which can be directly measured from experiment, depends on the signal observable S , . In its turn, S, is a linear combination of two eigenobservables Q, and I,, each of which undergoes simple exponential decay with the rates (1/T) and (1/8T), respectively. Similar phenomena occur for coherent observables, that is, linear combinations of pmn, except that the decay is an exponentially damped oscillation at the Bohr frequency

IV. Detection of Polarized Atoms Optical pumping can be detected in many different ways based on the various properties of the polarized vapor. In this section, we shall examine each of these methods briefly.

234

R. J. Knize, Z. Wu, and W. Hupper

A. TRANSMISSION MONITORING Optical pumping can be monitored by observing the changes in the properties of the transmitted light. The beam used to probe the vapor does not necessarily have to be the pumping beam. In many cases, a separate weak probing beam is used that does not cause any appreciable optical pumping. Some commonly used types of transmission monitoring are absorption monitoring and birefringence monitoring.

1. Absorption Monitoring

This is the method most frequently used in practice. It is based on observing the change in the intensity of the transmitted light. The mean photon absorption cross section depends on the polarization of the absorbing atomic state and on the polarization and frequency spectrum of the light (Bouchiat and Grossetete, 1966; Mathur et al., 1970). Thus, the intensity of the transmitted light depends on the degree of polarization of the vapor. The circular dichroism method is an important variant of absorption monitoring for spin-polarized vapors. Unpolarized light passes through the optically pumped vapor and acquires some elliptical polarization as a result of the differential absorption of the 0' and 6 - components. The spin polarization of the vapor can be determined from the magnitude of this elliptical polarization. It is possible to detect the hyperfine population imbalance (1.S) with unpolarized light that does not have the same spectral intensity for absorption out of the two ground state multiplets of angular momenta F = I f (1/2). This hyperfine absorption monitoring is especially useful for experiments with alkali vapors with no spin polarization, where ( I . S ) relaxes as a single exponential. Spin exchange cross sections can be determined in a particularly simple way from such experiments.

2. Birefringence Monitoring A spin-polarized vapor is birefringent. It has two eigenpolarizations; light with either of these two polarizations will not change its polarization after passing through the vapor. Light with any other polarization will generally have its polarization changed by the vapor. Detection of this change of polarization is the basis of birefringence monitoring (Gozzini, 1962; Mathur et al., 1970). In birefringence monitoring, the optical frequencies are sufficiently detuned from resonance that there is negligible absorption of the light.

OPTICAL PUMPING AND SPIN EXCHANGE

235

In most optical pumping experiments, the two eigen polarizations are right and left circular polarizations. For the sake of clarity we shall confine ourselves to this particular case in the following discussion. Linearly polarized light after passing through the vapor will have its plane of polarization rotated. The angle through which the plane of polarization is rotated is called the Faraday rotation angle. The physical origin of the rotation of the polarization plane lies in the difference of indices of refraction of the sample for the right and left circularly polarized light. The Faraday rotation angle A8 is given by the following expression

A8

=x

(n’+ - n’-)l-

1

where n‘+ (n’L) is the real part of the index of refraction for right (left) circularly polarized light, 1 is the path length through the vapor, and , Iis the wavelength. The difference in n + and n - can be caused by the following two mechanisms : (i) External magnetic field H (diamagnetic Faraday rotation). Under an external magnetic field H , the c’ and 0 - absorption lines will be separated by an amount proportional to the field H as shown in Fig. 5(a). According to the Kramers-Kronig relation, the index of refraction curves for c + and cwill also be separated by an amount that depends on H . Therefore, for small H , the difference n’+ - n’- is proportional to H .

frequency

frequency

frequency

frequency

(a)

(b)

FIG. 5 . The real ( n ‘ ) and imaginary ( n ” ) parts of the indices of refraction for right ( n , ) and left ( n - ) circularly polarized light. (a) Diamagnetic Faraday rotation. (b) Paramagnetic Faraday

rotation.

R. J. Knize, Z . Wu, und W. Happer

236

(ii) Population difference in the ground state sublevels (paramagnetic Faraday rotation). In order to separate out the effects of the external field H , we assume that the field H is very weak and the splitting between (T+ and 6 lines is negligible. Assume the ground state sublevel 1/2 is more populated than the - 1/2 sublevel (see Fig. 1). The 0- absorption curve will be larger in magnitude than the 6’ curve. Again using the Kramers-Kronig relation, we find a nonzero value for n’+ - n’-, which in this case is proportional to the population difference between the ground state sublevels as shown in Fig. 5(b). Due to the small splitting between (T’ and 6 , the diamagnetic Faraday rotation angle is usually much smaller than the paramagnetic Faraday rotation angle. For example, for Rb metal vapor of number density 10” ~ r n - the ~ , diamagnetic Faraday rotation angle is of the order of 1p rad G - ’ cm-’ at a detuning 1A from the D , line. At the same Rb number density and detuning, the paramagnetic Faraday rotation angle is of the order of 1 m rad cm- for a Rb polarization of 10%.Therefore, unless the field is very high ( 2 lo3 G), one can neglect the contribution to the rotation angle from the external field. Absorption monitoring is based on the real absorption of photons while birefringence monitoring is a result of virtual absorption of photons. In other words, birefringence monitoring is determined by the real part of the index of refraction of the vapor while the absorption monitoring is determined by the imaginary part. The fact that the absorption of the probe beam is negligible for the off-resonant light used in birefringence monitoring makes it a very useful method for optically thick vapor.

+

B. FLUORESCENCE MONITORING Light absorbed by spin-polarized ground state atoms can be radiated as fluorescence. Information about the polarization of the ground state can be obtained from the fluorescent light. Fluorescent monitoring is closely related to transmission monitoring. If less light is transmitted because of a change in the ground state polarization, there will be more fluorescence from the excited atoms. For example, in the model discussed in connection with Fig. 1, the intensity, F , of fluorescent light, emitted over 47r steradians per unit volume, would be

F

= 2 R N - , , , = R N ( l - 2(S,))

(16)

According to Eq. (2b), ( S , ) will grow from zero to a final value of R(3y 2R)-’ within a few time constants after the pumping light is turned

+

OPTICAL PUMPING AND SPIN EXCHANGE

237

on. The value of ( S , ) can be determined from the experimentally measured transient fluorescence, F , and its initial value according to the simple formula

Fluorescence monitoring is especially useful for optically thin vapors where transmission monitoring signals are small. Fluorescence monitoring is also useful when the probe light has a very broad spectral profile so that most of the light does not interact with the vapor and can be a troublesome background for transmission monitoring signals. There are some disadvantages of fluorescence monitoring. Quenching gases such as N, are often used to eliminate fluorescence and thereby stop the harmful effects of radiation trapping. Fluorescence-monitoring signals are weak or nonexistent in such systems. The fluorescence radiation is spread over a large solid angle and practical detectors normally intercept only a few percent of this radiation. For optically thick vapors, most of the fluorescence is emitted from the face of the cell through which the pumping light enters. Because of the disadvantages just mentioned, it is often preferable to use transmission monitoring to study optically pumped vapors.

C. RADIOACTIVEATOMS A technique using radioactive atoms to detect optical pumping has been used by several groups (Cappeler and Mazurkewitz, 1973; Bonn et al., 1975; Calaprice et al., 1985). This method is based on the anisotropy of the pr ay emitted by the polarized radioactive atoms. The detection of y-ray is often more sensitive and has smaller background noise than the detection of optical photons, particularly in samples containing large amounts of other isotopes of the element being studied.

D. NUCLEARMAGNETICRESONANCE Standard nuclear magnetic resonance (NM R ) techniques are sometimes used to monitor optical pumping of dense noble gases (Bhaskar et al., 1982). Atoms are optically pumped in a longitudinal static magnetic field. The precessing magnetization induced by a resonant magnetic field is detected with a sensitive bridge circuit. NMR techniques are especially useful in measuring the absolute polarization of the optically pumped vapor.

238

R. J. Knize, Z . Wu, and W . Happer

V. Spin-Transfer Collisions In addition to directly polarizing an atom by optical pumping as described in Section 11, it is possible to polarize atoms by spin-transfer collisions. This ability to polarize species for which direct optical pumping might not be possible has greatly increased the number of spin systems that can be investigated. Many types of spin-transfer collisions are possible. Some examples are spin exchange between ground state atoms, metastability exchange involving ground and metastable atoms, charge transfer between ions and atoms, chemical exchange between atoms and molecules, and Penning ionization involving polarized metastable atoms.

A. SPIN EXCHANGE

Spin-exchange collisions will always be present in any polarized vapor and can be very important, especially for paramagnetic atoms such as alkali atoms. A spin-exchange collision between two zSl,zatoms A and B can be represented by the equation

A(T)+B(l)+A(l)+B(t)

(18)

where the arrows indicate the direction of the electron spins. The physical origin of the interaction responsible for spin exchange between alkali atoms or hydrogen atoms can be found in the singlet (V,) and triplet (K) potential energy curves of the molecular system A B as shown in Fig. 6 . These potential energy curves can be represented by V ( r ) = V,(r)

+ S,. SBVl(r)

(19)

where S, and S, are the spin operators of atoms A and B. This interaction potential couples the electron spins, which can cause an exchange of the spins during the collision. The total spin S = S, + S, commutes with S, .S,. Thus the interaction [Eq. (19)] will conserve the total spin even though individual spins can flip. There are two common ways to calculate spin-exchange cross sections. Purcell and Field (1956) introduced a classical impact parameter approach. They calculated the phase shift, cp, between the singlet and triplet components of the electron spin functions that accumulates during a collision. This phase shift can be written as

OPTICAL PUMPING AND SPIN EXCHANGE

239

-5t

FIG.6. Interaction energy of separation r (from Carver, 1963).

two hydrogen atoms

as a function of the internuclear

and can be thought of as the rotation angle of the spins S, and S, around S. The probability that an atom will change its spin is given by sin’ ((p/2). Thus, the spin-exchange cross section is given by a = jsin’( ;)2nr

dr.

Since the difference between the singlet and triplet interaction potentials can be significant at many Bohr radii (Fig. 6), the resultant spin-exchange cross sections for two alkali atoms are relatively large, a 2 x 10- l 4 cm’. These spin-exchange cross sections are an order of magnitude larger than gas kinetic cross sections and many orders of magnitude larger than a simple estimate of spin-relaxation cross sections derived from magnetic dipoledipole interactions. Wittke and Dicke (1956) used a partial wave analysis to derive the spinexchange cross section. Their result can be written as

-

where k is the wave constant and Sf and Si are the 1 wave phase shifts for the singlet and triplet potentials. Numerical evaluation of Eq. (22) leads to results similar to the impact parameter approach. Spin-exchange collisions can also transfer angular momentum between electrons and nuclei (Bouchiat et al., 1960). The best understood example of

R. J. Knize, Z. Wu, and W. Happer

240

such transfer occurs in collisions between alkali atoms and noble gases with nonzero nuclear spins such as 3He or 12’Xe. A large fraction of the electron spin is lost to the rotational angular momentum of the colliding pair because of the spin-rotation interaction, discussed in more detail in section VI.C., and only a small fraction, typically less than 10% of the spin, is transferred from the electron to the nucleus. This is in marked contrast to the case of electron-electron spin exchange discussed previously where no more than 1 % of the spin is lost to rotational angular momentum during a spin-transfer collision. Herman (1965) estimated the binary spin-exchange cross sections between Rb-3He and Rb-”Ne to be cm2 and lopz3cm2, respectively. Limited experimental results are of similar magnitude for these collisions. The formation of van der Waals molecules can be important in spin transfer between alkali atoms and the heavy inert gases (Ar, Kr, Xe) and will be treated in more detail in the next section. To examine the effect of spin exchange on optical pumping, we shall go back to the case of two S1,2 atoms, A and B , and neglect nuclear spin as discussed in section 1I.A. In this case, the evolution of the system can be described by S , = (Sz), and S , = (SZ),,

ds, dr ~

=

R (1 - 2S,) 3

~

-

y, s,

+ ua[B](S,

-

S,),

In these equations we assume that only the A atoms are optically pumped by the light with a mean rate R and that both spins relax exponentially (see the next section) at rates y, and ye, respectively. The last terms in the equations describe spin exchange. Solutions to Eqs. (23a) and (23b) are shown in Fig. 7 for a spin system that is exposed to pumping light at t = 0. Both S , and S , increase exponentially to limiting values. At a later time, T, S , is forced to zero, for example, by the application of a resonant radiofrequency field to flip the spins of the atom B. The spin, S,, decays exponentially to a new steady state value as a result. Thus, changes in S , can be detected as corresponding changes in S,. This is the basis for spin-exchange spectroscopy (Dehmelt, 1958a, 1958b) where the magnetic resonance spectra of atoms such as H , D, T, N , and P (which cannot be conveniently optically pumped directly) are recorded by their spin exchange with an optically pumped alkali vapor (Anderson et al., 1960a, 1960b; Holloway et al., 1962). In the case of a transient species such as electrons or H atoms produced by a discharge, Eqs. (23a) and (23b) will be complicated by a time-evolving concentration of the species B.

24 I

OPTICAL PUMPING AND SPIN EXCHANGE

__-.

-.-

._.-.

$04

-

1

a

1

l-

N -

1

1L

702

1

0

I

L1.

z a v)

I

I 0.0

,

C )O

,

,

1

,

0 05

,

,

I

1

1

1

1

1

1

1

0 I5

0.10

1

1

I

1

1

1

1

1

0.20

T I M E (sec) FIG.7. Solutions to theequations(23a) and (23b). The values for the various parameters are taken as follows: R = 100 photons sec- '; = = 1 sec- I ; oa[B] = 10 sec- I ; and ocr[A] = IWsec-'. ;is

B. METASTABILITY EXCHANGE It was mentioned in the previous section that the spin-exchange cross sections for binary collisions between the alkalis and the lighter inert gases (3He, "Ne) were very small. For this reason the technique of metastability exchange has been widely used to polarize 3He. Franken and Colegrove (1958) showed that metastable helium atoms could be optically pumped. The metastable helium atoms were produced by a weak discharge in helium gas at a few Torr pressure which also serves as a buffer gas. The metastable state was polarized with 10,830 A resonance light pumping the transitions 2'SS,-23P,. The ground state atoms were shown to acquire nuclear polarization from metastability exchange collisions (Walters et al., 1962), He 23S,( 1)

+ He 1 'So( 1)

-+

He 1 'So( 7 )

+ He 23S,( 1)

(24)

where 7 denotes the nuclear spin. The metastability exchange cross section for room temperature helium is 7 x 10p16cmZand decreases at lower

R. J. Knize, Z . Wu, and W. Hupper

242

temperatures. Optical pumping with a 1 .OS-p laser together with metastability exchange collisions has been used by Leduc et al. (1983) to produce large ) spin polarized 3He. densities (- 10l8~ m - of~ nuclear

C. CHARGE EXCHANGE Mitchell and Fortson (1968) showed that it is possible to produce Rb' ions with polarized nuclei. They optically pumped rubidium atoms in a discharge and the ions were polarized by charge-exchange collisions, W T ) + Rb+(l)+Rb+(t) + W l )

(25)

where the arrows represent the nuclear spin as before. These collisions were used to measure the magnetic moments of the Rb' and Cs+ 'So ground states. The measured charge-exchange cross sections for Rb+-Rb and Cs+-Cs were 7.1(1.5) x 10-l4cm2 and 8.0(3.0) x 10-I4cm2, respectively.

D. CHEMICAL EXCHANGE Gupta et al. (1974) showed that cesium dimers Cs, present in an optically pumped cesium vapor acquired a nuclear polarization. It is believed that the dimers were polarized by chemical exchange collisions,

where a cesium atom with nuclear spin up replaces one with the opposite spin in the Cs, molecule. The chemical exchange cross section is estimated to be about 10- l4 cm2. Other dimers have been studied using chemical exchange. E. PENNING IONIZATION Schearer (1969) developed another technique to electronically polarize excited states of ions (Cd+, Sr', Zn'). Metastable helium atoms were optically pumped and the ions were produced and polarized by Penning ionization,

He23&( t

t ) + Cd

'So(t l ) + H e ' S d T

1) + Cd+ ,05,,(t ) + e(T)

(27)

where the arrows represent the electron spins. Ion polarizations of 10%were ~ . is no obtained at ion concentrations of the order of 10'o-lO1l ~ m - There reported measurement of the relevant cross section.

OPTICAL P U M P I N G AND SPIN EXCHANGE

243

VI. Relaxation A. INTRODUCTION

The relaxation of spin-polarized atoms to their equilibrium value is a very important process in optical pumping. From a practical viewpoint, the relaxation rates will determine the characteristics of the observed signal or even if it can be seen at all. For example, it was shown in early experiments that the use of a diamagnetic buffer gas would substantially reduce the relaxation rate of alkali spins (Fig. 8) and would lead to narrower hyperfine and Zeeman transition linewidths. If optical pumping is utilized as a method of production of polarized atoms for other applications, then a reduction of the relaxation rates will increase the polarization that can be achieved for a given density. From a fundamental viewpoint, the measurement and interpretation of relaxation rates yield information about atomic collisions and the interaction potentials between atoms. The relaxation of polarized atoms can be caused by many possible mechanisms. Some of the more common include collisions with other atoms or molecules, wall collisions, spatial diffusion, radiation trapping, and spin exchange. The optical pumping process and, in some cases, a resonant radiofrequency field can also be thought of as relaxation mechanisms. A major problem encountered in relaxation measurements is that even though the results might appear simple, the interpretation of what was measured and the corresponding physics can be difficult. There will be in general more than one relaxation mechanism present in any experiment. Even if one relaxation mechanism dominates, the observed signal can consist of a sum of several exponentials (Bouchiat, 1963) and in some cases nonexponential relaxation has been observed (Dupont-Roc et al., 1968). I t is 360

-0 ._ -._zao

320

"7

E

w ; . 240 C

'= 200 . " O O ~ / - * ]

s

rr" 160 120

i

1

I

./

I

,

.

...

10 Neon pressure

, ,

,

100

(Torr)

1000

FIG.8. Pressure dependence of the relaxation rate ol polarized cesium atoms when both diffusion to the walls and collisional relaxation are present (from Franz and Liischer, 1964).

244

R. J. Knize, Z . Wu, and W. Hupprr

very difficult to extract the individual decay rates from a transient signal that is the superposition of several exponentials. For example, consider an experiment that attempts to determine the electron disorientation rate of an alkali vapor due to buffer-gas collisions by measuring optical transients after a change in some parameter (light intensity, magnetic field reversal, etc.). The photon absorption cross section at low magnetic fields will be dependent on the spin polarizations a, and b, of the two hyperfine states a and b, the population imbalance (S.1) between these states, and the respective quadrupole moments Q, and Qb. In order to observe ( S , ) [Eq. ( 1 l)] only, it is necessary that the light be circularly polarized and have equal intensities 1, and I,, for pumping from each hyperfine level. Provided that this condition on the apparatus is met, then the observed optical transient will exhibit two exponentials [Eq. (13)] if alkalibuffer-gas collisions dominate over all other relaxation mechanisms such as diffusion and spin exchange. For an alkali atom of nuclear spin I , the slowest decaying exponential will be a factor of 2/(21 1)2 slower than the electron disorientation rate, T - '. Inclusion of spin exchange will slightly modify this decay rate (Hou et al., 1984). Additional exponential decay curves may also be present if higher spatial diffusion modes are excited. These and related effects were not always carefully considered in past experiments, and some of the quoted relaxation rates are therefore hard to interpret in terms of the fundamental relaxation mechanisms involved. Relaxation mechanisms can be divided into the categories of weak or strong collisions. Weak collisions, such as binary collisions between a Rb atom and a Ne atom, resuit in a small change in the density matrix p and can be represented by a small time-dependent potential V ( t ) .The evolution of the density matrix in the interaction representation is given by perturbation theory,

+

The first term on the right of Eq. (28) is proportional to the ensemble average (P) of the perturbation 8. It can cause a shift in the energy levels of the spin system but it does not cause any relaxation. The relaxation rate described by the second term on the right of (28) will be on the order of (( ~ 2 ) / h 2 where ) ~ , z, is the correlation time of the interaction and ( 8') is the mean squared value of the fluctuating potential. For strong collisions, such as spin exchange, there will be a large change in p per collision. In this case, the evolution of the density matrix is described by: dt

OPTICAL PUMPING AND SPIN EXCHANGE

245

where T is the mean time between collisions and p c is the density matrix after the collision. For Eq. (29) to be valid, it is essential that the collision duration be much shorter than T. The various relaxation rates for strong collisions will all be on the order of T - '. Once the form of the interaction is known, the time evolution of the density matrix can be determined using Eq. (28) or (29). Complications arise because of the multiplicity of states. For a ground state of nuclear spin I and electron spin S, there will be n = (21 1)(2S + 1) sublevels and n2 independent real parameters in the density matrix. To make the problem tractable, the density matrix can be written in terms of observables [Eq. (lo)] or equivalently in a multipole expansion (Happer, 1970). Fortunately, in a given experiment, only a few observables will be detectable and/or nonzero (i.e., I,, Qe, S . I , Qa, and Qb for alkali atoms at low magnetic fields). The use of observables greatly reduces the number of equations. These observables will be either diagonal (representing populations) or off-diagonal (representing coherences between states). In analogy to NMR, relaxation times of the former are frequently written as TI while the latter are designated by T2. It is often possible to describe these many relaxation rates in terms of a few characteristic rates of the fundamental physical relaxation mechanisms. For example, the many relaxation rates in Fig. 4 are all simple multiples of the characteristic electron randomization rate 1/T.

+

B. BINARYCOLLISIONS

An important class of relaxation phenomena is due to binary collisions between a spin-polarized atom and another atom or molecule in the gas phase. These collisions have a duration of a few picoseconds, and therefore the corresponding relaxation rates cannot be affected by readily obtainable laboratory magnetic fields. The contribution of a given species of gas to the characteristic relaxation rates of the spins are proportional to the partial pressure of that gas and independent of the partial pressures of the other types of gas. This is in contrast to the situation for relaxation due to threebody collisions where the rates depend on products of partial pressures. Binary relaxation rates can arise from several different types of interactions. These interactions (excluding spin exchange, which was treated in the previous section) are usually weak so that the density matrix will evolve according to Eq. (28). Also, interactions involving electron spins will in general be stronger than those involving nuclear spins. If the polarized atom possesses hyperfine structure, then collisions with buffer gas atoms or molecules can modify the hyperfine coupling constant.

246

R. J . Knize, Z . Wu, und W. Hupper

The change in the hyperfine interaction of the alkali valence electron spin S with the alkali nuclear spin I is given as follows:

v = 6A(I. S)

(30)

where 6 A is a function of r, the internuclear separation of the colliding pair. The physical origin of this interaction arises from a modification of the wave function of the alkali valence electron. At large distances, van der Waals forces will pull the electron away from the nucleus so that SA is negative. At short distances, 6 A will be positive due to exchange forces (Adrian, 1960). At low magnetic fields, 6A(I. S) will not cause relaxations, but results in a shift of the hyperfine frequency (Arditi and Carver, 1961). At high magnetic fields where I and S decouple, this interaction will induce a relaxation (Franz and Sieradzan, 1981). For atoms with electron spin S > 0, there will be an interaction between the electron spin S and the rotational angular momentum N of the atombuffer-gas pair, I/ = y(r)S.N,

(31)

where y ( r ) is the spin-rotation coupling coefficient. This interaction is well known in molecular physics and is one of the mechanisms responsible for fine structure in molecules. There are two contributions to y(r). The first is the familiar spin-orbit interaction, (1/2)g,pBS p x E/mc. In the case of collision between alkali atoms and the heavier noble gases, Ar, Kr, and Xe, the spinorbit interaction within the noble gas core seems to be the main source of y (Wu and Happer, 1984; Wu et al., 1985; Mackintosh et al., 1985). There is also an interaction between the spin S and the local magnetic field H produced by rotation of the quasi-molecule, gspBS.H which seems to be negligible for most systems except H: (Burke, 1960). If the buffer gas is a molecule such as H,, then there can also be an interaction with the magnetic field produced by the internal rotation of the buffer-gas molecule. There have been no good comparisons between calculations and experiments for these interactions. If both colliding atoms have Sl,, ground states, then there can be a spin-spin interaction between electrons,

-

v = 343s; - S,] where Iz is the spin-spin interaction constant and ( is along the internuclear axis. This interaction is responsible for the fine structure of 3C molecules such as 0,. It has been proposed that this interaction is responsible for relaxation in cesium-cesium collisions,

W f ) + Cs(T>+Cs(T)+ W l )

(33)

OPTICAL PUMPING AND SPIN EXCHANGE

247

where the reported cross section was oe = 2 x 10-16cm2 (Bhaskar et al., 1980). Subsequent measurements (Knize, 1985) for rubidium and potassium yield cross sections 1 x lO-”crn2 and 3 x 10-18cm2, respectively. This spin-spin interaction will be particularly important in the polarization of high-density alkalis where it will set an upper limit to the density that can be highly polarized with fixed light intensity. For collisions between an atom with electron spin S and a buffer gas with nuclear spin K, there will be an interaction

[S.K (K.r)(S.r) r,-3--------b(r)K.S r5

87t . 3

I/= -2y,,pnpB

where gnis the gyromagnetic ratio of the buffer-gas nucleus, and p,, and pLRare the nuclear and Bohr magnetons. Due to the large peak of the electron wave function at the buffer-gas nucleus (Fig. 9), the last term of Eq. (34), representing the contact interaction, dominates. It is this interaction that is responsible for spin exchange between electronic spin of an alkali atom and the nuclear spin of a noble gas atom (Bouchiat et al., 1960). The interaction

t

I

0

I

2o

1

1

4

R (A) FIG.9. Rubidium valence electron wave function in the vicinity of a Xenon atom. The wave function plotted here is the Rb 5s electron wave function after being orthogonalized to all the xenon electron orbitals. The large amplitude of the valence electron wave function at the xenon nucleus is responsible for the spin-exchange interaction between the alkali valence electron and the noble gas nucleus.

248

R. J . Knize, Z. Wu, and W . Happer

(34) is also the dominant spin relaxation mechanism for the polarized nuclei of noble gases with paramagnetic impurities, such as alkali atoms or oxygen molecules.

C . THREE-BODY RELAXATIONS Due to their high polarizability, the heavier noble gas atoms, such as Ar, Kr, and Xe, are able to form van der Waals molecules with alkali atoms. Substantial spin exchange and spin relaxation can occur during the relatively long lifetime of these molecules (Bouchiat et al., 1968, 1971). There are several obvious ways for van der Waals molecules to form. In three-body formation, a nearby third body (for example, a buffer-gas molecule) carries away the energy released by the molecular formation. In radiative molecular formation, which seems to be completely negligible in practice, the energy is carried off by a photon. In resonant two-body molecular formation, a colliding pair with just the right energy can form a quasi-bound molecule by tunneling through the centrifugal barrier. There is no evidence that these quasi-bound molecules are of any practical significance compared to the molecules formed in three-body collisions. In the following, we shall assume that the molecules are formed only by three-body collisions. Nitrogen gas is frequently used to provide a third body. The van der Waals molecules thus formed have very small binding energy (about a tenth of kT) and therefore can break up upon the next collision. However, the molecule's lifetime (- 100 ns for 1 Torr of buffer gas) is several orders of magnitude longer than the duration of binary collisions ( lo-', sec). Since the spin-flip probability is proportional to (Vrlh)', with V being the interaction Hamiltonian and z the time duration of the interaction, the spin-flip probability can be 10" times bigger for a three-body collision than it is for a binary collision. Therefore, even though the three-body collision rate is slower than the binary collision rate by a factor of the order R 3 [ N 2 ] (e.g., for 1 Torr N , ) where R is the nuclear separation in the molecule, the formation of van der Waals molecules is still the main mechanism of spin transfer except for very high third-body pressures ( 100 atm) where the molecular lifetime would be comparable to the duration of a binary collision. The theory of three-body relaxation of alkali electron spin polarization was first given by Bouchiat et al. (1968). Extensions of this theory to account for the nuclear spin of the noble gas were recently developed by Volk et al. (1980) and by Happer et al. (1 984). The spin Hamiltonian is assumed to be N

-

OPTICAL PUMPING AND SPIN EXCHANGE

249

Here A1.S is the hyperfine interaction, y N . S is the coupling between the electronic spin S and the rotational angular momentum N of the molecule, ctK.S is the magnetic dipole interaction between the electronic spin S and the noble gas atom nuclear spin K, and g, , pRS. Bis the Zeeman interaction. Instead of going into the details of the theory, we shall just write down some of the basic results. Under many experimental conditions, the alkali atoms can be described by a spin temperature /?-I, i.e., the density matrix of the alkali atoms can be written as

where the partition function Z is 2

=

Tr ea’F

(37)

and B 4 1. Under these circumstances the spin transfer between the alkali electronic spin S and the noble gas nuclear spin K can be described by the following equations:

where I/T, ( l / T F )is the molecular formation rate per noble gas atom (alkali atom) and 4’s are the spin-transfer coefficients. The general expressions for the spin-transfer coefficients are rather cumbersome (Happer et al., 1984). However, in the limit of very short and very long molecular lifetimes, they are greatly simplified. For example, for very short molecular lifetime t such that

G1

but long enough for

to be still valid, we have

(40)

R. J . Knize, Z . Wu, and W . Happer

250 where

is the Larmor frequency of the valence electron in the external magnetic field B. The factor [ I ] = 21 + 1 in Eq. (42) expresses the slowing down of the relaxation rate due to nuclear spin 1. In alkali a t ~ m - ' ~ ~ pairs, X e the Breit-Rabi parameter y N / a is found to be about 3 (Zeng et al., 1985) and therefore the term containing y (the spin-rotation term) in Eq. (42) contributes approximately an order of magnitude more to the relaxation than the term containing D: (the spin-exchange term). Physically this means that in a collision between a spin-polarized alkali atom and a '29Xe atom, only some 10% of the spin is transferred to the lZ9Xenucleus. The remainder is lost to rotational angular momentum. The experimental studies of van der Waals molecules of alkali atoms (Na, K, Rb, and Cs) with '29Xe and 13'Xe have shown that for these systems the relation A 2 B ( Y N ) B~ a2

(44)

(Zeng et al., 1985) holds in the Hamiltonian H in Eq. (35). This allows one to use perturbation theory to calculate the spin-transfer coefficients, with the ) ~ the expansion parameter. The perturbation parameter x-' = ( ~ l / y N as calculation results are very close to those obtained by exact numerical calculations; the discrepancy is no more than a few percent (Happer et al., 1984). At fixed alkali and noble gas number densities, the dependence of the threebody relaxation rate on third-body pressure p has the following functional form (Volk et al., 1980): ap

1

+ bp2'

(45)

The origin of this functional dependence is that the three-body relaxation rate is the product of the three-body formation rate, which is proportional to p and the spin-flip probability in the molecule, which is proportional to [1/(1 b p 2 ) ] . One can define a characteristic pressure p o as the third-body gas pressure for which the molecular breakup rate z - l is equal to the spin-rotation frequency yN/h. The spin-relaxation rate of '"Xe nuclear spins in Rb and Cs vapors as a function of the dimensionless third-body gas pressure p / p o is found to be well described by a universal (i.e., independent of third-body gases) spin-relaxation function (Ramsey et al., 1983; Hsu et al., 1985; Happer et al., 1984), as shown in Fig. 10.

+

OPTICAL PUMPING AND SPIN EXCHANGE

25 1

Y

cr Ic Q)

Reduced Third-Body Pressure p/po

FIG. 10. The dependence of Rb-induced '"Xe nuclear spin relaxation rate on the thirdbody (N, and He) pressure can be described by a universal spin relaxation function (the solid line) if the third-body gas pressure is measured in units of the characteristic pressure p o . The He data are taken from Ramsey et al. (1983); the N, data are from Volk et at. (1980) and Bhaskar et al. (1983).

Due to the long lifetime of van der Waals molecules, three-body relaxation rates of both alkali electrons and noble gas nuclei depend strongly on the external magnetic field (Bouchiat et al., 1968). The field dependence is Lorentzian. [Strictly speaking, the field dependence at low third-body pressure is not Lorentzian, but the deviation from being Lorentzian is very small (Bouchiat et al., 1968; Happer et al., 1984).] The halfwidth measured in the experiment indicates a molecular lifetime of the order of sec per Torr of third-body pressure. The halfwidth in the limit of zero third-body gas pressure yields the value of the averaged spin-rotation coupling constant y (Bouchiat et al., 1968; Bhaskar et al., 1983). D. WALLCOLLISIONS AND COATINGS Optical pumping is usually done with vapors contained in glass or metal cells. Polarized atoms will relax due to collisions with the cell walls. For alkali atoms, depolarization will occur after only a few wall collisions on untreated walls. For typical cell geometries, this will cause a relaxation rate of about lo3- lo4 sec- '. The utilization of a buffer gas will impede diffusion (see next section) to the walls and reduce this relaxation rate. The wall depolarization

252

R. J . Knize, 2. Wu, and W. Hupper

rates for inert gas atoms are substantially smaller, with a reported lifetime of 9 days obtained for 3He in an aluminosilicate glass cell (Fitzsimmons et al., 1969). The interactions responsible for these wall relaxations are similar to those for binary collisions between polarized atoms and other gas phase atoms, the main difference being the relatively long residence time on the walls as compared to the binary collision time. One method that has been developed to reduce relaxation at the wall is to coat the cell with a chemically inert substance such as a paraffin CnHznc2 (Robinson et al., 1958). A well-prepared wall coating will allow up to lo4 atom-wall collisions before the electron spin of an alkali atom is depolarized. For a typical 6-cm diameter cell, this will produce a rubidium spin lifetime of about 1 sec. (Bouchiat and Brossel, 1966). The dramatically increased spin lifetime resulting from a good wall coating will allow the alkali atoms to sample the magnetic field throughout the cell, thus reducing the magnetic resonance linewidth by motional narrowing. There has been some success in the utilization of wall coatings for inert gases with a reduction of about an order of magnitude obtained in the '29Xe relaxation rate (Zeng et al., 1983).

E. DIFFUSION In uncoated cells containing a buffer gas and polarized alkali atoms, diffusion of the atoms to the walls can be an important relaxation process. An observable Q such as 1 . S, I,, or Q , will be practically zero at the cell walls and its evolution in the absence of pumping will be described by the diffusion equation :

In this equation, D is the diffusion coefficient that will depend on the temperature and pressure and y is the bulk relaxation rate due to processes such as binary collisions. The solution to Eq. (46) will depend on the initial spatial dependence of the polarization and in general there will be an infinite number of diffusion modes. The decay rate for the slowest mode is given by 1 -n2 760 Do--+? T r2 P

-

for a spherical cell of radius r and -1=

T

[n'-+- (2.:5)'] L2

D o 760 p+Y

(47)

OPTICAL PUMPING A N D SPIN EXCHANGE

253

for a cylindrical cell of length L and radius Y. In these equations, Do is the diffusion coefficient at 760 Torr ( D o = DP/760). In practice, it is usually difficult to measure relaxation rates other than the slowest. Masnou-Seeuws and Bouchiat (1 967) developed general solutions for diffusion in optical pumping as functions of the buffer-gas density and the disorientation quality of the walls.

F. INHOMOGENEOUS MAGNETIC FIELDS The presence of inhomogeneity in the applied magnetic field can cause relaxation. The resultant relaxation rate (Schearer and Walters, 1965; Gamblin and Carver, 1965) for spin 1/2 atoms is

where 5, is the time between collisions, 21 is the velocity, oois the Larmor frequency, and the last term is the square of the fractional magnetic field gradient. The physical origin of this relaxation can be thought of as due to the inability of the spin to follow the local magnetic field direction as the atom moves around inside the cell. For typical magnetic fields, evaluation of Eq. (49) shows that the relaxation due to an inhomogeneous magnetic field is generally unimportant except for very slowly relaxing species such as 'He.

G . RADIATIONTRAPPING A N D COHERENCE NARROWING The optical pumping cycle (depopulation and repopulation pumping) described earlier assumed that the absorbed photon was from the incident pumping light and that the excited atom then emitted a photon that left the cell. An important modification of this process occurs when the vapor is optically thick ( > 10l2cm p 3for an alkali vapor) so that the emitted photons are reabsorbed. This radiation trapping will alter the ground and excited states evolution. A quantitative analysis of the effect of radiation trapping on ground state spin polarization for low magnetic fields has not been made, though qualitatively it is known that it degrades the ground state polarization. For high magnetic fields, Tupa et al. (1986) have shown that radiation trapping is not such a severe problem. A remedy for radiation trapping in dense alkali vapors is to use a small amount of a molecular buffer gas such as nitrogen (50 Torr) which will quench the excited state (the quenching cross section 50 A2). This quenching process probably occurs by transfer of energy into vibrational and rotational degrees of freedom of the molecule.

-

254

R. J . Knizr, Z . Wu, and W . Happer

The quenching cross sections for noble gases are small (< 10- l9 cm’) and hence noble gases are not suitable for the elimination of radiation trapping, but they are good for hindering diffusion to the wall. Related to radiation trapping is the phenomena of coherence narrowing (Barrat, 1959). This effect arises from the increased effective lifetime of the coherence of the excited atoms due to the reabsorption of emitted photons. Coherence narrowing is seen in level crossing or optical double resonance where signals involving the excited states become narrower by as much as a factor of three due to the increased coherence lifetime. The effective lifetime of the population of excited atoms can be increased by orders of magnitude by radiation trapping.

H. EXCITEDSTATEDEPOLARIZATION ‘ The excited states produced by depopulation pumping can be strongly affected by depolarizing collisions with a buffer gas. For example, the lowest P,,, excited state of an alkali atom has a depolarization cross section of about 10- l4 cmz for collisions with inert buffer-gas atoms. The depolarization cross section for the lowest P l j Zexcited state of an alkali atom is smaller and depends on the fine structure interval of the lowest P doublet. A typical Plj2depolarization cross section is 10- l 6 cm2 for Cs in an inert gas. This excited state mixing is important in determining the effect of repopulation pumping on the ground state polarization. For alkali atoms pumped with D, resonance light, the ground state polarization will increase if the PI/, sublevels become mixed. On the other hand, for pumping with D, resonance light, the ground state polarization will reverse sign if the P,/, sublevels become mixed. In addition to depolarization within a given J state, there exists the possibility of transfer between fine structure doublets (P1/,-P3/,). These transfer cross sections are strongly velocity-dependent and are a function of the fine structure interval. Atoms with large fine structure intervals (such as cesium) have small transfer cross sections (10- 20- 10- 21 cm2).

VII. Frequency Shifts The magnetic resonance frequencies of optically pumped atoms are often shifted slightly from those of free atoms because of the effects of collisions with other atoms and with photons of the pumping beam. A few examples are discussed in this section.

OPTICAL PUMPING AND SPIN EXCHANGE

255

A. COLLISIONAL SHIFTS

Optically pumped alkali atoms generate an effective magnetic field that can be sensed by other atomic species, for example '29Xe or 'lNe, whose magnetic resonance frequency will thus undergo a small shift. The effective magnetic field sensed by noble gas nuclei is not the magnetic dipole field of the polarized spins. It is caused by the Fermi contact hyperfine interaction between the electron spin of the alkali atom and the nuclear spin of the noble gas atom. Whereas the nuclear-spin relaxation of heavy noble gases in an alkali vapor is dominated by interactions in van der Waals molecules, the frequency shift is mainly contributed by binary collisions between a noble gas atom and an alkali atom, the contribution of van der Waals molecules being about 10%. Binary collisions are so frequent (typically once every millisecond) that although the angle rotated during a binary collision is very small (rad), there is a steady secular accumulation in the angle rotated, which gives a shift in the Lamour frequency. Similar frequency shifts occur in the nuclear magnetic resonance frequency of 'He which is undergoing metastability exchange collisions with optically pumped He 23S, atoms (Dehmelt, 1964). Since the frequency shift coefficient is proportional to ( V ) , the mean perturbation amplitude, and the spin relaxation rate is proportional to ( V 2 ) , the mean squared amplitude [see Eq. (28)], the measurements using the frequency shift can be more sensitive than those based on the relaxation rates when V is small (see Fig. 11).

&--circular

,frequency

dichroism signal

s h i f t signal

r f frequency

FIG.11. For small perturbations V , the measurements using the frequency shift, which is proportional to ( V ) , can be more sensitive than those using the circular dichroism method, which is proportional to ( V 2 ) .

256

R. J . Knize, Z. Wu, and W. Hupper

B. LIGHTSHIFTS The magnetic resonance frequencies for an optically pumped atom are shifted slightly from the magnetic resonance frequencies of an atom that is not exposed to light. For small light intensities, these “light shifts” are proportional to the intensity of the light. There are two kinds of light shifts, shifts due to the virtual absorption of light and shifts due to real absorption of light followed by spontaneous radiative transitions back to the initial state of the atom (Barrat and Cohen-Tannoudji, 1961a, 1961b). The shift due to virtual absorption of light is just the quadratic Stark shift of the atomic energy sublevels in the oscillating electric field of the light. If the electric field of the light at the atom is written as

then by second-order perturbation theory, the light shift in the energy of the ground state sublevel m is

where T r is the spontaneous decay rate of the excited-state sublevel p, D is the electric dipole moment operator of the atom, and the sum extends over all sublevels p of the excited state. For nonmonochromatic light, Eq. (51) should be averaged over the power spectrum of the pumping light. One can see from Eq. (51) that the sign of the light shift reverses when the optical frequency w passes through resonance. The measured light shift due to virtual transition in 199Hgis shown in Fig. 12. The magnitude of the light shift due to virtual transition is typically a few Hz for the weak light from conventional lamps, and it can be a gigahertz or more for light from intense lasers. The light shift due to virtual transition is proportional to the index of refraction of the vapor, and this is why the light shift curve of Fig. 12 has the characteristic shape of an anomalous dispersion curve. Light shifts due to real transitions are a consequence of the partial conservation of coherence during the absorption and reemission of light during the optical pumping cycle. For example, imagine that an atom is precessing about a magnetic field at a Larmor frequency w g in the ground state, when the atom absorbs light it will continue to precess at a frequency we in the excited state. After a mean radiative lifetime, t,, the atom will decay to the ground state and its phase will have advanced, relative to the phase of an excited atom, by an amount (we - w,)t. If the atoms absorb photons at a rate

OPTICAL PUMPING AND SPIN EXCHANGE

I

257

Frequency Shilt

FIG.12. Frequency dependence of the light shift due to virtual transitions. The shift proportional to the index of refraction of the vapor [from Cohen-Tannoudji, 1962).

IS

R, the phase advance of the Larmor precession per optical pumping cycle will lead to a shift due to real transitions on the order of

It is worth noting that the shift due to real absorption of light cannot be bigger than the optical pumping rate R , but the shift due to virtual transition can greatly exceed R.

VIII. Applications In addition to the study of optical pumping as a physical process, there have been many applications involving optical pumping in atomic and other fields of physics. In this section, we shall discuss some of these applications. Due to the limitation of space, we can only highlight a few experiments. The interested reader can find many additional experiments in the literature.

A. PRECISION ATOMICMEASUREMENTS Optical pumping produces a nonthermal population distribution in an atomic vapor. By using a radiofrequency field to drive transitions between

R. J . Knize, Z. Wu, and W. Hupper

258

atomic sublevels, it is possible to measure the energy difference between sublevels to a high precision. In particular, the gyromagnetic g factor and the hyperfine structure of various atoms can be measured to better than a part in a million. Robinson and coworkers (Johnson and Robinson, 1980; Keiser et al., 1977; Tiederman and Robinson, 1977) used optical pumping in a series of experiments to measure the ratio of g factors for Rb, e-, He(23S,) and He+(lSl/z).The rubidium atoms were polarized by optical pumping and the other species were polarized by spin exchange. The application of a resonant radiofrequency field perpendicular to the longitudinal magnetic field will depolarize these species and cause a change in the light transmission through the cell. Figure 13 shows an example of the Zeeman resonances of 'He(lS,,,) and HeQ3S,) that they observed in a l00G magnetic field. From the frequency ratio of these line centers and other related experiments, they were able to obtain gJ(4He+ l2SIiz)/gJ(e-)= 1 - 70.87(30) x

(53)

This 0.3-ppm measurement allows a test of bound state corrections of the electron g factor to 0.5 %. This series of experiments shows that it is possible to use optical pumping for precision g factor measurements on both neutral and charged species. Another high-precision atomic experiment using optical pumping is the measurement of atomic hyperfine structure. As an example, Economou et al. (1977) measured the rubidium45 hyperfine splitting as a function of the

............., I

1

22

I

I

26

I

1

30

FREOUENCY(kHZ)

1

I

I

34

- 280200

1

I

38 kHZ

FIG. 13. Zeeman resonances of He+ (12S,,2) and He (Z3S,)in a 100 Gauss magnetic field. Their respective linewidths are about 0.8 and 0.4 kHz, and their separation is 8.37 kHz (30 ppm) (from Johnson and Robinson, 1980).

OPTICAL PUMPING AND SPIN EXCHANGE

2 :it

3

4

259

5

11G'oG2)

FIG.14. The hyperfine frequency of "Rb as magnetic field H, (from Economou et al., 1977).

ii

function of the square of the external

applied magnetic field and showed that there is a small diamagnetic shift in the hyperfine structure. The applied magnetic field constrains the outer electron, reducing the size of its orbit. This perturbation of the electronic structure increases the probability amplitude of the electron at the nucleus and thus increases the magnitude of the contact hyperfine interaction. Figure 14 shows the observed diamagnetic shift which is in agreement with the result obtained from perturbation theory. By improving the apparatus, they hope to observe smaller effects such as the diamagnetic susceptibility of the nucleus. B. FREQUENCY STANDARDS

As shown in the previous experiment, it is possible to use optical pumping for precise measurement of the hyperfine structure of an atom. The hyperfine splitting measured with such precision can be used as the basis of a frequency standard. Figure 15 (Hellwig, 1984) shows a block diagram of an optically pumped R7Rbgas cell frequency standard. This rubidium frequency standard has three desirable features: a narrow linewidth, a transition frequency that is large, and a high signal-to-noise ratio. In order to produce a population imbalance between hyperfine levels, a *'Rb filter cell is used. This cell preferentially absorbs light which pumps the F = 2 of the 87Rbground state.

R. J. Knize, Z . Wu, and W . Happer

260

MAGNETIC S H I E L D

vCTOR PHOTO

1,

$q

EXCITER

POWER S U P P L I E S FOR LAMP, F I L T E R

CAVITY

INPUT 6.834.685 k H z

h C-F I E L D POWER SUPPLY

FIG. 15. Schematic of a rubidium gas cell frequency standard (from Hellwig, 1984).

The transmitted light will pump atoms from the F = 1 to the F = 2 levels. The associated electronics are used to measure the hyperfine separation and generate an appropriate frequency. A rubidium frequency standard can achieve a stability of better than 10- l 1 per day. The limitations of this gas cell standard arise from buffer-gas shifts, magnetic field inhomogeneities, and light shifts. While the stability of the rubidium frequency standard is not as good as the cesium atomic beam or hydrogen maser, these standards have the advantage of being inexpensive, compact, and lightweight. For these reasons, the rubidium frequency standard is frequently used in satellites. Optical pumping is being used for state selection and detection in cesium atomic beam standards (Hellwig, 1984). It is expected that up to an order of magnitude improvement in stability might be achievable. Optical pumping techniques will also be used in frequency standards based on ion trap technology (Bollinger et al., 1985).

C. FUNDAMENTAL PHYSICS TESTSUSINGOPTICAL TECHNIQUES PUMPING In addition to measurements of atomic structure and collisions, there have been several experiments that use optical pumping and related techniques to examine fundamental laws of physics. In particular, optical pumping techniques have been used to measure parity violation in atoms (Bouchiat and

OPTICAL PUMPING AND SPIN EXCHANGE

26 1

Bouchiat, 1974; Fortson and Lewis, 1984),search for electric dipole moments in atoms (Vold et al., 1984), and test general relativity (Prestage et al., 1985). The standard model for the electroweak interaction predicts a paritynonconserving (PNC) neutral-current interaction between electrons and nucleons (Bouchiat and Bouchiat, 1974). Measurements of PNC have been made on several heavy atoms using optical pumping techniques. The importance of these measurements is that they test the electroweak theory in a regime that is different from that probed by high-energy experiments. Present experiments (Gilbert et al., 1985) have achieved an accuracy for the W boson mass that is comparable to results obtained from high-energy experiments. PNC experiments measure a parity-violation (pv) amplitude by observing its interference with a parity-conserving amplitude. In one type of experiment, the E l transition induced by the PNC interaction interferes with a weak A41 transition, producing a rotation of the plane of polarization of incident linearly polarized light. The rotation angle AOPuis given by Eq. (15). These experiments measure this rotation angle close to the M1 transition where A P " changes sign as the laser is swept through the transition. This allows AOP" to be separated from the much larger non-PNC background. These experiments (Fortson and Lewis, 1984) have been done on Bi and Pb and yield accuracies of about 15 to 20"/, for the PNC amplitude. PNC in heavy atoms has also been measured by observing the interference between the E l transition induced by the PNC interaction and the El transition induced by a static electric field in a highly forbidden A41 transition. This interference term can be isolated since it changes sign with the polarity of the applied electric field. This technique has been used to observe PNC in Cs and T1 (Bouchiat et al., 1982, 1984; Bucksbaum et al., 1981). Recently, these measurements (Gilbert et al., 1985) have been improved for Cs and an accuracy of about 8 % has been obtained for the PNC matrix element. By using optical pumping to produce a polarized Cs beam, they hope to improve these results. In the hope of finding another manifestation of C P violation besides the kaon decay (Christenson et al., 1964), Vold et al. (1984) have used optical pumping to search for a permanent electric dipole moment (edm) of the lz9Xeatom. Rubidium atoms were pumped by a diode laser and lZ9Xeatoms were polarized by spin exchange enhanced by van der Waals molecule formation. A permanent electric dipole moment of the 29Xe nucleus would make the precession frequency of the xenon nuclear spin dependent upon the externally applied electric field. They obtained the null result, d(lz9Xe)= (-0.3 1.1) x e-cm. Improvements in this experiment are in progress and it is hoped that the sensitivity can be improved several orders of magnitude. Observation of a nonzero edm or an improved upper bound

262

R. J . Knize, Z . Wu, and W. Happer

would be particularly important in understanding the origin of T violation in elementary particle interactions. Another test of fundamental physics that utilizes optical pumping is a search for inertial mass anisotropy. Prestage et al. (1985) have used optical pumping of Be’ ions in a magnetic trap to see if the nuclear spin flip frequency depends on the orientation of the Be+ ions in space. Chupp (1985) is starting an experiment using polarized ’lNe (I = 3/2) in a variant of the Hughes-Drever experiments (Hughes et a]., 1960; Drever, 1961).Chupp and Coulter (1985) optically pumped rubidium atoms that polarized the neon by binary spin-exchange collisions. They were able to achieve a neon nuclear polarization of 30% at a density of 2 x l O ” ~ m - ~ Scientists . hope to eventually search for changes in the neon NMR frequency as a function of the angle between the neon nuclear spin and the direction of matter anisotropy in the nearby universe. D. MAGNETOMETERS One of the most important applications of optical pumping is the very accurate and sensitive measurement of magnetic fields. In this section we shall briefly describe the physical principle of operation of a few of the most commonly used optically pumped magnetometers. Self-oscillating magnetometers (Bloom, 1962), which are widely used in space exploration missions and the monitoring of geomagnetic fluctuations, consist of a cell containing alkali metal vapor, which is optically pumped by a resonant light beam (e.g., from an alkali resonance lamp). An oscillating resonant rf magnetic field in the direction of the pumping beam induces a precessing transverse atomic polarization, which in turn modulates the transmitted light. The transmitted light, after being amplified and phase shifted by n/2, is fed into the rf coil. As is well known from the Bloch equations, the transmitted light is phase shifted by 71/2 with respect to the rf field at resonance. Thus the total phase shift in the loop is zero at resonance and one obtains self-oscillation. The locked oscillator magnetometer makes use of the longitudinal component of the atomic polarization rather than the transverse one (Bloom, 1962; Farr and Otten, 1974). The alkali atoms contained in a cell are optically pumped by a resonant D, pumping beam. An oscillating resonant rf field is applied in a direction perpendicular to the pumping beam and is frequency modulated at the reference frequency R of a lock-in amplifier, which is used for the phase detection of the transmitted light. The detector sees the derivative of the absorption signal, i.e., an error signal, which is used to lock the radio frequency oscillator to the center (Larmor frequency) of the error

OPTICAL PUMPING AND SPIN EXCHANGE

263

signal. Instead of frequency modulating the rf field, the error signal can also be obtained by applying a small longitudinal magnetic field oscillating at the frequency R. With a simple feedback loop, the locked oscillator magnetometer can be used to lock the longitudinal magnetic field to a specified value. Another type of magnetometer makes use of the ground state Hank effect (Dupont-Roc et al., 1969; Cohen-Tannoudji et al., 1970). The circularly polarized pumping light propagates in a direction perpendicular to the field H , that is to be measured (transverse optical pumping). To make the detection of the Hank signal more sensitive, an rf field H , cos mt is applied in the same direction as H,. The transmitted light intensity will then be modulated at various harmonics p o of w (parametric resonances). In the case of measuring a very small magnetic field H , , the odd and even harmonic signals have the following dispersion and absorption shape as H , varies around zero, w0 z

~~

1

+ ( m o t ) * sin p o .-

(podd)

(54)

(peven).

(55)

and 1 1 -k

- cospo

(t00.r)~

It is interesting to note that the widths of these resonances are independent of the amplitude of the rf field, in contrast with some of the other magnetometers (e.g., the self-oscillating magnetometers). For a coated glass cell, the relaxation time of the Rb electronic spin can be as long as 1 sec. The widths of the resonance curves can therefore be as narrow as a few microgauss. This, in conjunction with the large signal-to-noise ratio, makes it possible to measure magnetic fields of the order of l o p 9 Gauss. The odd harmonic signal in Eq. (54) is more frequently used because, for moz < 1, the signal is directly proportional to the magnetic field H , . E. POLARIZED TARGETS AND SOURCES Experiments involving polarized targets and sources are becoming important in nuclear and high-energy physics. The utilization of laser optical pumping in the past decade has allowed the development of targets and sources that are frequently more dense and highly polarized than obtainable by conventional techniques. In this and the next section, we shall present three experiments that illustrate the use of optical pumping for production of polarized targets and sources.

264

R. J . Knize, Z. Wu, and W . Hupper

Walters and coworkers (Gray et al., 1983) have used optical pumping to produce a source of spin-polarized electrons. Metastable He atoms formed by a microwave discharge are polarized with a NaF (F:)* color-centered laser. CO, gas is then injected and polarized electrons are generated by Penning ionization. They were able to obtain electrons with a polarization of 80% at a current of 0.05 pA. These performance characteristics are comparable to those obtained using other methods. Optical pumping has also been used to produce a polarized H - source (Mori et al., 1984). Fast protons are injected into an optically pumped sodium target. The sodium cell is located in a 0.6T magnetic field and is pumped with two one-watt circularly polarized single-frequency dye lasers at 598.6nm. A broadband frequency laser at 589.3nm is used to probe the vapor. The polarization of the alkali vapor is determined by the Faraday rotation of this linearly polarized probe laser. They were able to obtain a sodium polarization of 90% at a target thickness of 1 x 1013cm-,. Neutral electron spin-polarized hydrogen atoms are formed by charge exchange between the protons and the polarized sodium atoms. The hydrogen atoms then travel to a second polarized sodium cell where the magnetic field is reversed. The sudden magnetic field reversal transfers the electron spin to the nucleus (Sona, 1967). At the second Na cell, some of the hydrogen atoms capture an electron to form a nuclear spin-polarized H - beam. A beam current of about 30 pA has been obtained with a polarization of 60 %. F. SPIN-POLARIZED FUSION The goal of the worldwide controlled fusion effort is to achieve sufficient plasma density, confinement time, and temperature to fuse isotopes of hydrogen and release a net amount of energy. Kulsrud et al. (1982) pointed out the advantages of using a nuclear polarized fuel in a fusion reactor. In particular, they showed that there can be a significant increase in the fusion cross section and the possibility of controlling the directivity of the products. In order to test and take advantage of these effects, it is necessary to construct a sufficiently intense polarized hydrogen (H, D, T) source that could produce loz1 polarized atoms per second for a steady state reactor or 10,' polarized atoms for present-day pulsed devices (Knize, 1986). The problem encountered in spin-polarized fusion is that these rates are many orders of magnitude greater than present polarized hydrogen sources. One possible source that could be scaled to produce polarized hydrogen at these rates would use optical pumping. An alkali such as rubidium would be polarized with a laser and the hydrogen atoms would become polarized by spin-exchange collisions. A one-watt laser would be capable of producing

OPTICAL PUMPING AND SPIN EXCHANGE

265

about loL8polarized hydrogen atoms per second. Initial experiments have been started in cells containing a buffer gas and a hydrogen polarization of 64:r; has been obtained (Knize and Cecchi, 1985). Future experiments will be done in paraffin wall coated cells to increase the polarization.

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Mathur, B. S.. Tang, H., and Happer, W. (1970). Phys. R ~ P2,. 648. Mitchell, J. K., and Forston, E. N. (1968). P h y s . Re11 Lett. 21, 1621. Mori, X., Ikegdmi, K., Igarashi. Z., Takagi, A., and Fukurnoto, S. (1984). AIP Conf. Proc. No. 117, p. 123. Prestage,J. D., Bollinger, J. J., Itano. W. M., and Wineland, D. J. (1985). Phys. Rev. Lett. 54,2387. Purcell, E. M., and Field, G . B. (1956). Astrophys. J . 124, 542. Ramsey, N., Miron, E., Zeng, X., and Happer. W. (1983). Chem. Phys. Letr. 102, 340. Robinson. H. G., Ensberg, E. S., and Dehmelt, H. G . (1958). Bull. Am. Phys. SOC.3, 9. Schearer, L. D. (1969). Phys. Reo. Lett. 22,629. Schearer, L. D., and Walters, G. K. (1965). Phys. Reo. 139, A1398. Sona, P. G. (1967). Energia Nucleare 14, 295. Tiedman, J. S., and Robinson. H. G. (1977). Phvs. Re<..Lrrt. 39, 602. Tupa, D.. Anderson, L. W.. Huber, D. L., and Lawler, J. E. (1986). Phys. Rev. A 33, 1045. Vold. T. G., Raab, F. J., Heckel, B., and Forston, E. N. (1984). Phys. Rev. Lett. 25, 2229. Volk, C. H.. Kwon, T. M., and Mark, J. G. (1980). Phys. Keu. 21, 1549. Walters, G. K., Colegrove, F. D., and Schearer, L. D. (1962). Phys. Reo. Lett. 8, 439. Wieman, C., and Hansch. Th. W. (1976). Phys. Rev. Lett. 36, 1170. Wittke, J. P.. and Dicke. R. H. (1956). Phys. Rec. 103, 620. Wu, Z., and Happer, W. (1984). “Proceedings of the Workshop on Polarized Targets in Storage Rings,” Argonne National Laboratory, pp. 289-302. (Argonna, Illinois) Wu, Z., Walker, T. G., and Happer, W. (1985). Phys. Reu. Lett. 54, 1921. Zeng, X., Miron, E., Van Wijngaarden. W. A,, Schreiber. D., and Happer, W. (1983). Phys. Lett. 96A, 191. Zeng, X., Wu, Z., Call, T., Miron. E., Schreiber. D., and Happer, W. (1985). Phys. Rea. 31, 260.

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I1

ADVANCES IN ATOMIC A N D MOLEClJLAR PHYSICS. VOL. 24

CORRELATIONS IN ELECTRON-A T O M SCA TTERING A . CRO W E Department of Pure and Applied Physics The Queen's Vniversitv of' BeIJhst Belfasr BT7 INN, Northern Ireland

I. Introduction . . . . . . . . . . . . . . . . . 11. Excitation . . . . . . . . . . . . . . . . . . A. n'P Excitation in Helium . . . . . . . . . . B. n 3 P Excitation in Helium . . . . . . . . . . C. zP Excitation in Atomic Hydrogen . . . . . . . D. P Excitation in Heavy Atoms . . . . . . . . . E. Excitation of D States . . . . . . . . . . . F. Excitation of Molecules . . . . . . . . . . . G. Time Evolution of States . . . . . . . . . . 111. Ionization . . . . . . . . . . . . . . . . . . A. High-Energy Asymmetric Kinematics . . . . . . B. Low-Energy Asymmetric Kinematics . . . . . . C. Threshold Ionization . . . . . . . . . . . . D. Autoionization . . . . . . . . . . . . . . . E. Inner-Shell Ionization and Auger Electron Decay . . Acknowledgements . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

. . . . . .

269

. . . . . . 270 . . . . . . . 272 . . . . . . . 279 . . . . . . .

281 . . . . . . . 283 . . . . . . . 287 . . . . . . . 288 . . . . . . . 289 . . . . . . 292 . . . . . . . 294 . . . . . . . 303 . . . . . . . 306 . . . . . . 310 . . . . . . . 31 1 . . . . . . 3 17 . . . . . . 3 17

I, Introduction This chapter deals with the study of angular and polarisation correlations between outgoing particles following inelastic electron-atom scattering processes. It will be seen that such measurements yield more detailed and fundamental information on the scattering process than more conventional studies involving observation of a single outgoing particle. Experimentally, this type of study was first reported by Ehrhardt et al. (1969) when they observed the angular correlation between scattered and ejected electrons following the electron impact ionization of helium. For excitation, Eminyan et al. (1973) performed the first such study by measuring the angular correlation between the outgoing scattered electron and the photon emitted in the decay of the excited state. This was followed in 1976 by a measurement of the Stoke's parameters of the emitted radiation in an 269 Copyright ( 1988 by Academic Preaa, In' All rights of reprodu'tion in m y form reserved ISBN 0 12-03R24 2

270

A . Crowe

electron-photon coincidence experiment -the polarisation correlation technique (Standage and Kleinpoppen, 1976). Different aspects of early work on these so-called third-generation experiments have previously been reviewed in Advances in Atomic and Molecular Physics by Weigold and McCarthy (1978) and Kleinpoppen (1979). The present review concentrates on some of the more important developments since then for both excitation and ionization processes. The basic experimental methods used have not altered significantly since the publication of these and other reviews (McCarthy and Weigold, 1976; Slevin, 1984) and are not discussed here. Recent experimental advances are outlined, particularly in relation to the new physical information that they yield. There are a number of common features of the angular and polarisation correlation technique irrespective of the process studied. In all cases the scattered electron, es, with a specific energy E , and momentum k , is detected. This defines a specific momentum transfer collision, where the momentum transfer is K = k, - k,, k, being the incident electron momentum. In noncoincidence experiments (e.g., excitation studies where only the decay photons are observed), the results correspond to an integration over all possible momentum transfer collisions. The coincidence technique also defines a scattering plane about which the system has reflection symmetry compared with the axial symmetry about the incident beam direction of a noncoincidence experiment. Once a particular momentum transfer collision has been defined by observation of the scattered electron, detailed information on the collision process and the atomic state formed is obtained from the polarisation of the transfer is K = k, - k,, k, being the incident electron momentum. In directly in a polarisation correlation experiment or sometimes more conveniently from an angular correlation experiment, since the particle’s angular distribution can be related simply to its polarisation. For direct outer-shell ionization, less detailed information is obtained by observation of the ejected electron, except at energies close to threshold where it may be reasonable to assume that only a small number of states with different angular momentum are involved. Nevertheless, the angular correlation between the scattered and ejected electrons is the most sensitive test of ionization theories.

11. Excitation Unlike the measurement of total or differential cross sections where measurement of the various experimental quantities (e.g., electron beam current, atom density, scattered signal) or, alternatively, some normalisation

CORRELATIONS IN ELECTRON-ATOM SCATTERING

27 1

procedure leads directly to an absolute cross section (Heddle 1979), considerable theoretical analysis is required in the interpretation of angular and polarisation correlation measurements to extract the maximum possible physical information. Moreover, the interpretation of correlation measurements must be considered carefully for each excitation-decay process in each atom. The theoretical framework for interpretation of these experiments was originally formulated by Macek and Jaceks (1971) and Fano and Macek (1973) and developed by Blum and Kleinpoppen (1979) and Blum (1981). With the exception of two cases discussed later, the experimental studies have been confined to S - P excitation processes. Figure 1 shows the experimental kinematics and defines the different coordinate systems used in parameterising the data for an excited P-state, shown schematically. The scattering plane is defined by the incident electron of momentum k, and the scattered electron of momentum k,. Most analyses have been performed in the collision frame (xc,y', zc) in which the z-axis is in the incident electron direction and the scattering plane is the x-z plane. Hermann and Hertel (1982a) have argued that more physical insight into the dynamics of the collision is obtained and that the resulting parameters relate more directly to the experimental data in the natural frame (x", y", z") with its z-axis perpendicular to the scattering plane and x-axis in the k, direction. They also considered the atomic frame (x u ,y", z") obtained from the natural frame by a

y: z", za

t

FIG. I . Schematic representation of the angular part or the electron-charge cloud density for an atomic P state relative to the collision, natural, and atomic collision frames of reference.

272

A. Crowe

rotation around the z" = Z" axis through an angle y so that x" is directed along the major symmetry axis of the charge cloud. In general, the anisotropy of the excited P state is characterised by four independent parameters that are functions of the incident electron energy E , and the scattering angle 9,-e.g., alignment and orientation parameters (Fano and Macek, 1973) and state multipoles (Blum and Kleinpoppen, 1979). The parameterisation of Hermann and Hertel (1982a,b), Andersen et al. (1984, 1986), and Andersen and Hertel (1986) is particularly enlightening from the point of view of the physical picture of the excited state charge cloud (Figure 1). They define the parameters y; the charge cloud alignment, Plin= ( A - B)/(A B), the linear polarisation of the charge cloud where A is its length along the X" axis and B its width; poo = H / ( A + B H), where H is its height along the Z" axis; and L, is the angular momentum perpendicular to the scattering plane.

+

+

A. n'P EXCITATION I N HELIUM The theoretical interpretation of this system is most straightforward. The excited state produced can be described as a coherent superposition of degenerate magnetic sublevels (Macek and Jaecks, 1971), i.e.

I$(n'P>) =u+1111)

+ u,110) +

u-111

-

1)

where u M are the excitation amplitudes for the states 1 L M ) . In the absence of spin-dependent forces, the excited atomic state possesses positive reflection symmetry with respect to the scattering plane. The number of independent parameters characterising the excited state is reduced to two, and these, together with the absolute differential scattering cross section 0,completely define the collision. The most commonly used parameterisation of n'P states in helium is the (a,2, x) parameterisation, developed in the collision frame of reference, by Eminyan et al. (1974).

where u M is the differential cross section for excitation of a particular magnetic sublevel

These parameters can be obtained from measurements either of the scattered electron-photon angular correlation or of the polarisation correla-

CORRELATIONS IN ELECTRON-ATOM SCATTERING

273

tion. For the angular correlation, the measured photon intensity I(8, @) is given by Eminyan et al. (1 974):

I(8, @)

-

1 (1 2 ~

-

A)(

+ cos’

1

H - sin2 (I cos 2@) + ,i sin’ 8

+ [A( 1 A)] ‘1’ cos x sin 20 cos cp (1) The parameters II and 1x1 can be determined by fitting this expression to the experimental data. The technique does not permit the sign of x to be determined since it appears as cos x in Eq. (1). Alternatively, in the polarisation correlation technique, measurement of the Stoke’s parameters (PI,P,, P 3 ) perpendicular to the scattering plane yields I and x. The Stoke’s -

parameters are defined by

I . PI

=

I(o0)

I.P2

=

Z(4.5”)

I ’ P,

= I(RHC) -

-

l(90”)

-

I(13.5”)

I(LHC)

where I is the total light intensity in the direction perpendicular to the scattering plane and I(0) is the light transmitted by a linear polariser making an angle 8 with the incident electron beam direction. The linear polarisations P, =2A- 1 and P, give II and

=

- 2 [ 4 1 - A)]’/’ cos x

1x1, while the circular polarisation P,

=

2[,i( 1 - A)]

‘I2

sin x

gives the sign of x. In the natural frame parameterisation, the excited P state can be described by the two independent parameters y defining the direction of the major axis of the charge cloud and L, = u: - u? the angular momentum perpendicular to the scattering plane. In this case the angular correlation measured in the scattering plane and the polarisation correlations measured perpendicular to the scattering plane are simply related to the angular probability density of the excited-state charge cloud. In the polarisation correlation experiment, the photon intensity I,(@ in the +z” direction, measured with a polariser having its transmission axis at an angle 8 with respect to the x” direction is

I,(8)

-

1

+ P,i” cos 2(H

-

y)

274

A . Crowe

+

+

where the linear polarisation Plin= (P: P:)"2. Hence (1 Pli,)/2 and (1 - Plin)/2determine the size of the major and minor axes of the charge cloud with alignment angle y. In the angular correlation experiment, the photon angular distribution measured in the scattering plane is given by r,(e)

-

1 - plincos 2(e - y).

This corresponds to the charge cloud rotated through 90°, with y the direction of the minimum of the angular correlation. Since we are dealing with a transition between two pure states (Standage and Kleinpoppen, 1976; Blum, 1981), the degree of polarisation of the radiation emitted is unity, i.e.:

P

= (P:,

+ P:) = 1

Also, since P , is a measure of the angular momentum L , (with opposite sign), then

Pi"

=

1 + L:,

(2)

and so the angular correlation measurement that determines Plingives only I L, I and y, while the circular polarisation measurement also gives the sign of L,. Since the initial angular correlation measurements of Eminyan et al. (1973, 1974) for excitation of the 2 ' P state of helium, a considerable number of experimental studies have been reported for this system. The most extensive studies have been carried out at an incident electron energy of 80 eV. Three independent measurements of the angular correlations between the scattered electrons and 58.4-nm photons from the decay of this state have been reported for a wide range of electron-scattering angles (Hollywood et al., 1979; Steph and Golden, 1980; Slevin et al. 1980). The recent development of circular polarisation analysers in the vacuum ultra-violet VUV region of the spectrum (Williams, 1983; Westerveld et al., 1985) has renewed interest in these measurements. At 80 eV, Khakoo et al. (1986) have confirmed that the sign of L , changes at a scattering angle of 60 to 70". This is shown in Fig. 2, where the variation of P , ( - L,) is given as a function of the electronscattering angle. The results give the sign of P , consistent with theoretical predictions. The values of I P , I from angular correlation measurements have been plotted so as to be consistent with the directly measured P , values. Unfortunately, the circular polarisation data do not resolve the discrepancy between the data of Steph and Golden (1980) and those of Hollywood et al. (1979) and Slevin et al. (1980) in the region 8, = 80-100". However, since the error bars quoted represent one standard deviation, the discrepancy does not merit the considerable attention it has received in the past.

275

CORRELATIONS I N ELECTRON-ATOM SCATTERING

0'41 I

0

,

l

50

,

l

l

l

100

I

1

1

1

L

I

150

E l e c t r o n s c a t t e r i n g o n g l e ldegi FIG.2. Variation ofcircular polarisation P , with electron-scattering angle at E , = 80 eV lor He(2' P). Polarisation measurements: 0 , Khakoo et al. (1986); m, Williams (1983). IP31 deduced from angular correlations: A, Eminyan et al. (1974); 0. Hollywood et al. (1979); x , Steph and Slevin et al. (1980). Golden (1980); 0, Theory: -First-order many-body theory of Cartwright and Csanak; - - - distorted wavc calculation of Madison. Both private communications to Khakoo el al. (1986) (from Khakoo et a]., 1986).

The circular polarisation data of Khakoo et al. (1986) also contradict the conclusions from the angular correlation work of Beijers et al. (1984) that L , maintains the same sign at all scattering angles for incident electron energies of 50 and 60 eV. Figure 3 shows the observed change of sign at 50 eV. The discrepancy in magnitude at 80" between the polarisation data of Khakoo et al. and the angular correlation data of McAdams et al. (1980) and Beijers et al. ( 1 984) probably confirms the difficulty of extracting 1 L, I from angular correlation data when Plin is close to unity (Andersen et al., 1986), as represented by Eq. (2). The other major development in the study of 2 ' P excitation in helium has been the measurement of angular correlations at incident electron energies approaching the threshold energy at 21.2 eV. Measurements have been reported over a wide range of electron-scattering angles at 29.6eV by McAdams et al. (1980), 30 and 27 eV by Steph and Golden (1983), 29.6 and 26.6 eV by van Linden van den Heuvell et al. (1982), 26.5 eV by Crowe and Nogueira (1982), 24.0 eV by Crowe et al. (1983), and 22.0 eV by Neil1 and Crowe (1984).

276

A . Crowe

I

I

0

20

I

I

I

I

60 80 100 Electron s c a t t e r i n g a n g l e (deg)

40

I

120

FIG.3. As Fig. 2, but E , = 50 eV. Polarisation measurements: 0 , Khakoo et al. (1986). [PSI deduced from angular correlations: 0, McAdams et al. (1980), shown as both positive and negative; 0 , Beijers et al. (1984) (from Khakoo et al., 1986).

The results of these measurements are summarised in Figures 4 and 5 in terms of the 2 and I x I parameters. The data at the lowest energy are consistent with A -+ 1 at threshold as expected from simple angular momentum arguments. Figures 4 and 5 also highlight considerable discrepancies between the Steph and Golden data and other data at 27 eV in that their data are inconsistent with the general trend of the data as a function of energy. It may be significant that parameters obtained in this laboratory at 26.5 eV under conditions where resonance trapping of the radiation occur are consistent with the 27-eV data of Steph and Golden. Their assumption that the onset of nonlinearity of the photon flux measured in a particular direction as a function of helium gas pressure is a sensitive test of the presence of radiation trapping is unlikely to be valid. The most significant observation concerning this low-energy data is the improving agreement with the five-state R-matrix calculation of Fon et al. (1980) as the incident energy is lowered until at 24 eV (Crowe et al., 1983) there is complete agreement. This is not the case for any other calculation when compared with experimental correlation data for any excitation process in any atom.

CORRELATIONS IN ELECTRON-ATOM SCATTERING

277

1

0

0 A

0

0

0 Incident electron energy ( e V 1 FIG.4. Variation of i.as a function of incident electron energy for He(2 ' P ) . Data at electron-scattering angles of 40" are denoted by circles. 60 by squares, 80" by upright triangles, loo" by diamonds, and 120" by inverted triangles. Open symbols are data from this laboratory: Neil1 and Crowe (1984) at 22.0 eV, Crowe et al. (1983) at 24.0 eV, Crowe and Nogucira (1982) at 26.5 eV, and McAdams et al. (1980) at 29.6 eV. Right-hand shaded symbols are from Steph and Golden (1983) (27 eV, 30 eV) and left-hand shaded symbols from van Linden van den Heuvell et al. (1982) (26.6 eV, 29.6 eV). The broken curves are drawn to assist in the association of data at the same electron-scattering angle. The vertical broken line represents the excitation threshold energy (from Neil1 and Crowe, 1984).

With experimental data available over a wide range of electron-scattering angles (8, 5 120') for incident electron energies from within 1 eV of threshold to 80 eV, and for more restricted angular ranges at higher energies, the 2 ' P excitation in helium is the most comprehensively studied system experimentally. By contrast, the 3 ' P excitation has received relatively little attention since the original polarisation correlation work of Standage and Kleinpoppen (1976). This process can in principle be studied by observation of either the 53.7-nm (3 ' P - 1 'S) or the 501.6-nm (3 ' P - 2 IS) photons. Observation of the latter has distinct advantages. At this visible wavelength, circular polarisation measurements can be carried out much more efficiently than in the VUV. The second advantage arises from the difliculty in resolving the

A . Crowe

278 I

1

1

I

I

I

I

I

I

I

I

1 I

22 0

~

I

I

2L

0

1

26 0

I

I

I

I

28 0

30 0

I n c i d e n t e l e c t r o n energy l e v )

'

FIG.5. Variation of 1x1 as a function of incident electron energy for He(2 P) Notation as in Fig. 4 (from Neil1 and Crowe, 1984).

n = 3 singlet states by electron spectroscopy coupled with the difficulty in isolating the 53.7-nm photons from the 58.4-nm photons from the decay of the 2 ' P state. Hence the unresolved VUV signal contains 58.4-nm photons due to cascade from the 3 ' S and 3 'D states to the 2 ' P . Under circumstances where the 3 ' S and 3 'D states have differential cross sections that are not negligible relative to that for the 3 ' P state, then a true angular correlation is not measured for the 3 ' P state by observation of the VUV radiation. The data of van Linden van den Heuvell et al. (1983) suggest that the ratio a(3 'D)/a(3 ' P ) approaches 0.5 at 28.5 eV and BS = 35", differing markedly from the small ratio predicted by Chutjian and Thomas (1975) theoretically. The only difficulty associated with observation of the visible photon is the 40: 1 branching ratio in favour of the VUV line. In addition to the original polarisation correlation work of Standage and Kleinpoppen (1976), electron-VUV photon angular correlations have been reported by Eminyan et al. (1975), Crowe et al. (1981), Gelebart et al. (1982). McAdams and Williams (1982), and van Linden van den Heuvell et al. (1983) over limited ranges of incident energy and scattering angle. However, only recently have extensive studies of the 3 ' P excitation been carried out. Ibraheim et al. (1985) have used the polarisation correlation technique (501.6nm) to study the process at 50, 80, 120, and 160eV for electronscattering angles up to 100". At all these energies the direction of L, changes

CORRELATIONS IN ELECTRON-ATOM SCATTERING

279

sign as was the case with 2 ' P excitation. A t 80 eV the sign change occurs at a scattering angle near 70", close to the 2 ' P position. However, at 50 eV the 3 ' P circular polarisation appears to behave very differently from the 2 ' P case, remaining close to zero over a very wide range of scattering angles. A t lower energies (29.6,26.5 eV) Neil1 and Crowe (1985) have measured angular correlations (501.6-nm photons) for scattering angles up to 120". In this region the data show considerable similarity with 2 ' P data at the same energies above their respective thresholds. B. n3p EXCITATION IN HELIUM Excitation of the 3 3 P state in helium has been studied experimentally by Silim et al. (1985) and Williams and Humphrey (1985) by observation of the 3 3 P -2 3S(388.9-nm) photons in polarisation correlation experiments. These can now be compared with the first-order many-body calculations of Cartwright and Csanak (1986) for n = 2-8 3 P states. The only other calculation is the five-state R-matrix calculation of Fon et al. (1979) for the 2 3 P state. Considerable problems associated with detection of the 2 3 P- 2 3S (1083-nm) photon have so far excluded this state from experimental studies. Excitation of these triplet states is particularly interesting since it can only take place through electron exchange. Hence in contrast to the 'Pexcitation where the scattering amplitudes contain contributions from both direct and exchange processes, the 3 P amplitudes are pure exchange amplitudes. The three Stoke's parameters measured (PI,P *, P 3 ) are related to the scattering parameters in the same way as for the ' P states (Blum and Kleinpoppen, 1979), except that the fine structure of the ' P states causes considerable depolarisation in the time between the excitation and subsequent decay of the excited states. However, the depolarisation factors are readily calculated (Blum and Kleinpoppen, 1979) modifying the linear polarisations P and P , by a factor of 15/41 and P , by 27/41. This increases the difficulty of the experiments over those for the ' P states since, for example, if we consider P the difference I ( O o ) - I(90") for the 3 P state is only 15/41 of that for the equivalent ' P state, lowering the relative statistical accuracy of the ' P data. The experiment is also made more difficult because of the long lifetime of the 3 3 P state, 100 ns, compared with 1 ns for the low-n'P states. This again lowers the relative statistical accuracy of the 3P data since a large random coincidence signal is observed in the broader time window over which the 3 P signal is observed. Figure 6 shows L , and y as a function of electron-scattering angle calculated by Cartwright and Csanak (1986) compared with the data of Silim et al. (1985) at 60 eV. Unfortunately, the large uncertainties associated with

-

-

280

A . Crowe I

0.8

l.O

1

/

30 e V ( n . 3 )

60 e V I n = 2 - 8 ] I

E m

4 I Ill h

-4

-0.2 -

v

-0.8

I

1

- 1 . 0 1 , 0

0

I

20

20

,

I

I

I

,

I

,

]

,

1

J

,

80 100 120 Scattering angle I d e g )

140

160

180

80 100 120 Scattering angle ( d e g )

140

160

180

LO

60

LO

60

FIG.6. (a) The parameter L , for excitation of the He(n3P) states as a function of electron-scattering angle. The broken ( n = 3 ) and full curves (n = 2-8) are the calculations of Cartright and Csanak (1986) for the incident electron energies shown. The shaded region denotes the experimental results of Silim et al. (1985). (b) The parameter y for excitation of the He(n3P) states. Notation as in (a) (from Cartwright and Csanak, 1986).

CORRELATIONS IN ELECTRON-ATOM SCATTERING

28 1

the data, together with the small variations in the predictions of the calculations as a function of energy from 30 to 500eV, prohibit useful evaluation of the calculation. The data of Williams and Humphrey (1985) at 39.7 and 23.5 eV also need to be extended over a much wider angular range to enable meaningful comparisons to be made with theory.

c. ' P EXCITATION IN ATOMICHYDROGEN Although atomic hydrogen, with a single electron, is theoretically the simplest atomic system, analysis of the electron impact S - P excitation in H is less straightforward than the ' S - ' P excitation in helium. Since the H states are doublets, the excitation process is described by both singlet and triplet scattering amplitudes. Ideally, the excitation needs to be carried out with electron spin selection to isolate the singlet and triplet scattering processes. In the absence of this, the observables are bilinear combinations of the scattering amplitudes averaged over initial spins and summed over final spins. The anisotropy of the ' P state has been parameterised by Morgan and McDowell (1975) by (I., R, I ) where

and

where 1 (UM'aM,)

=

4

{3 UM l-aMT*

+ OM,SO&fS*},

uMTand uMs being the triplet and singlet amplitudes associated with each state I L M ) . Experimental studies have been restricted to excitation of the 2 ' P statc by observation of scattered electron-Lyman-cc angular correlations in the scattering plane and analysed using the Morgan-McDowall parameterisation. This measurement yields the ,I. and R parameters, 1 being obtained only from a measurement of the circular polarisation. In the natural frame parameterisation, the linear, P,in, and circular polarisation ( P 3 = - L , ) are now independent-i.e., the 2 P state is described by (y, P,in,L,) (Andersen et al., 1986). Again the fine structure depolarisation must be taken into account (Blum, 1981). The maximum amplitude of the angular correlation is reduced to 30% of that of the equivalent ' P helium correlation, again making the experiment relatively difficult.

-

282

A . Crowe

Only two incident electron energies have been studied systematically over a wide range of electron-scattering angles: 54.4 eV (Williams, 1981; Weigold et al., 1980) and 35.5 eV (Slevin et al., 1982). At 54.4 eV there is agreement between the two experiments for both I and R. Figures 7 and 8 show the A and R data of Williams (1981) compared with some of the recent and apparently better calculations (van Wyngaarden and Walters, 1986). For I , there is good agreement among the calculations shown and with experiment concerning the position of the first minimum at 0, 17", but there are large discrepancies between the theoretical predictions for both the position and depth of the second minimum. The most serious disagreement between theory and experiment occurs for the R parameter for 0,2 70", with the

-

1

1.0 -

1

1

1

1

1

1

1

1

1

1

1

1

1

-

0.8

0.6 A

0.4

0. ;

I

FIG.7. Variation of A with electron-scattering angle at 54.4 eV for the H(22 P ) state. The experimental data are from Williams (1981). multipseudostate close coupling (van Wyngaarden and Walters, 1986); - - -, Theory: -, Is-2s-2p close coupling (Kingston et al., 1976); _ _ _unitarised _, eikonal-Born series (Byron et al., 1985a); _. . ._,exact second Born (Madison et al., 1985) (from van Wyngaarden and Walters, 1986).

CORRELATIONS IN ELECTRON-ATOM SCATTERING

283

Scattering angle Ideg 1 FIG.8. Variation of R with electron-scatterlng angle at 54.4 eV for the H(2' P ) state. Notation as Fig. 7 (Corn van Wyngaarden and Walters, 1986).

measured values being significantly negative while all the calculated values are strongly positive. At 35 eV the data of Slevin et al. (1982) show the same qualitative behavior for both A and R as observed at 54.4 eV. A 12-state close-coupling calculation by Morgan (1982) gives A values consistent with experiment over the complete electron-scattering angular range. However, this same calculation disagrees with the experimental values of R at all angles. It is generally true that no theoretical approach accurately predicts the measured behavior of the alignment and orientation parameters and the differential cross section for any system. The one exception is the low-energy He(2 ' P ) excitation (Crowe et al., 1983) suggesting the need for measurements closer to threshold in atomic hydrogen.

D. P EXCITATION IN HEAVY ATOMS So far we have considered only the excitation of P states in light atoms where LS coupling can safely be assumed. However, for heavier target atoms (e.g., the heavier rare gases), explicit spin-dependent effects during the

284

A . Crowe

collision cannot be ignored. Comprehensive discussions of spin effects in inelastic electron-atom scattering have been given by Hanne (1983) and Blum and Kleinpoppen (1983). The main spin effect that has been considered in relation to electron-photon correlation studies is the "spin-orbit coupling effect," which allows for breakdown of LS coupling in the excited target state but ignores spin-orbit interaction for the continuum electron, i.e., spin flips of scattered electrons are due only to exchange scattering. Consider excitation from a 'So ground state to ' P , and 3 P , excited states with subsequent de-excitation to a 'So state. Although the excited states are designated singlet and triplet, both consist of a mixture of pure states, i.e.:

1%)

=

+',)

+ Bi3p1>

and

I3h)= a13P1)- Blipl) where a2 - p2 = 1 and I ' P , ) and I ,P1) are pure Russel-Saunders states. The lack of full coherence and of positive reflection symmetry in this case means that the four parameters generally required to describe a P state are independent. Hence in the natural frame parameterisation (Hermann and Hertel, 1982b; Andersen and Hertel, 1986), the parameters (7, poo, P,i,,,L,) are independent. The (1,x) parameterisation has been extended by Blum et al. (1980) and da Paixao et al. (1980) to include spin-orbit coupling effects. They define four independent parameters A, f , E, and A as follows:

Blum (1983) has shown that, provided the spin orbit coupling effect as just defined is the only spin effect of importance in the excitation process, then --A

< COS E < 1

(1 -1)-

If the excited state can be regarded as a pure triplet state, then the lower limit applies, while the upper limit denotes, a pure singlet state. All four parameters can be obtained from a combination of linear polarisation, angular correlation and circular polarisation measurements. In the absence of a P , measurement, only 1,cos E, and cos x can be determined where

CORRELATIONS I N ELECTRON-ATOM SCATTERING

285

The only system to be studied completely is the 6 3 P , state of mercury (Zaidi et al., 1980, 1981). They measured polarisation correlations with the photon detector perpendicular to the scattering plane (PI,P,, P 3 ) and the linear polarisation in a direction perpendicular to the incident electron beam at an azimuthal angle of 45". Hyperfine structure effects were eliminated by the use of an isotope cell (Zaidi et al., 1978) which absorbs radiation from isotopes with I # 0. The four parameters A, f , E, and A were determined at incident electron energies of 5.5 and 6.5 eV (threshold energy = 4.9 eV) for electron-scattering angles of So" and 70". The importance of spin effects is clear, both from the large values of E and A obtained and also from the lack of coherence (lPl 0.6). Also, Blum's inequality just presented is not satisfied for the 0, = 70" data, implying that other spin effects, presumably relating to the continuum electron, are important. A consideration of the spin-dependent parameters E and A also allows specific and basic aspects of theoretical models to be evaluated (Bartschat and Blum, 1982). The rare gases, other than helium, fall into this category and have been studied mostly by the angular correlation method, since the photons emitted by the lowest-energy P states lie in the VUV, making polarisation measurements difficult. Studies prior to 1980 analysed assuming the absence of spindependent effects will be ignored. Since then, the most studied system is excitation of the 4p5(2P,,,)5s'P, and 4p5(2P3,,)5s3P, states of krypton. Determination of the three independent parameters A, 1x1, and I E ~ for each state requires electron-photon angular correlations to be measured with two different geometries. McGregor et al. (1982) and Danjo et al. (1985) have used the fact that the angular correlation measured in a plane with azimuthal angle cp = 135" is independent of E, i.e., i, and I x I can be determined from this single correlation. I E ~is then found from an angular correlation measured in the collisional plane. Alternatively King et al. (1985) have measured angular correlations in the scattering plane and over a conical surface out of the scattering plane. Although the extraction of the parameters is less straightforward, this geometry allows both correlations to be measured simultaneously under identical experimental conditions. At 60 eV and 8, I 30°, where data are available from all three groups, there is excellent agreement between that of King et al. ( 1985) and Danjo et al. ( 1 985). There are serious discrepancies with the measured correlations of McGregor ct al. (1982), particularly those in the scattering plane. In all cases studied, no meaningful difference is observed between the results for the ' P , and 3 P , states, which is consistent with each containing nearly equal mixtures of both pure states. The angular correlation parameters for these states of krypton have been calculated by Meneses et al. (1985) using first-order many-body theory. In general, there is poor agreement between the theoretical and experimental results. The comparison between theory and the data of Murray et al. (1988)

-

A . Crowe

286 90

1

9

1

I

I

I

/

t

/

/

60

/

-

/

/

I

/ /

/

30

I

-

I

/ /

/ / I

/ /

u

w

0 -

0'7 \

:;i

\

p

\

-30-

/

\

/

\

/

\

\

/ /

'\-/'

$

9

I

I

-60-

i -90;

I

I

I

20

40

60

for the charge cloud alignment, y, is shown in Fig. 9 for scattering angles up to 50". Angular correlation studies have also been reported by McGregor et al. (1982) and Nishimura et al. (1986) for the lowest 3P state of xenon. An alternative approach has been used (Teubner et al., 1985; Riley et al., 1985, 1986) to study spin-dependent processes in the electron impact excitation of the 3 2P state of sodium. They have studied the coherence of the excitation by measuring the Stoke's parameters P,,P , , and P 3 in polarization-correlation experiments. Unlike the original coherence experiment (Standage and Kleinpoppen, 1976) for the 3 ' P state of helium where the total polarisation IPI = (P: + P : Pi)''' = 1, the unresolved fine and hyperfine structure of the 2 P state reduces P below unity, irrespective of the coherence of the excitation. They have taken account of this postcollision

+

CORRELATIONS IN ELECTRON-ATOM SCATTERING

287

depolarisation by using the reduced polarisation [PI Andersen et al. (1980) defined as

\PI = (P: + P: + P;)1'* where for Na (3 ' P ) , P , = 7.095P,, P , = -7.095P2 and P ,

= 1.794P3 (Teubner et al., 1985). Hence IF( is a measure of the polarisation of the radiation prior to the depoiarisation and, if the excitation process is coherent, then IPI = 1. Excitation of the Na('P) states is also complicated by the presence of both singlet and triplet scattering as discussed for H(*P) excitation. Herman and Hertel (1982a) have shown that IPI = 1 only if exchange scattering is negligible. Hence a measurement of [PI yields information on the role of exchange without the need for a direct spin analysis. Experimental data show that [PI = 1, within experimental error, at 100 eV and for 8, I 10" (Teubner et al., 1985) and at 22.1 eV (0, 5 15") (Riley et al., 1985). Consequently the data can be described by either (A, x) or (y, L,) parameterisations. The spin-flip cross section poo can be determined simply from measurements of the linear polarisations P , and P , where I . P , = I(0") - I(90") is measured in the scattering plane perpendicular to the incident electron beam direction (Hermann and Hertel, 1982b). Riley et al. (1986) have found poo = 0 at a scattering angle of 5" over the energy range 22.1 to 100 eV using this technique. The first study of spin effects by combined use of polarised beams and the electron-photon coincidence technique has been reported by Wolke et al. (1984) for the 6 ' P , state of mercury. The measurements were confined to zero-degree scattering, but even under these conditions, spin effects were shown to be important at low-incident electron energies.

E. EXCITATION OF D STATES The excitation of D states from S states with the subsequent decay to a P state has been discussed by Blum et al. (1978), Nienhuis (1980), van Linden van den Heuvell et al. (1981, 1983), Andersen and Nielsen (1982), Andersen et al. (1983), and Neitzke and Andersen (1984). Considering 'D excitation in helium, the excited state is a coherent superposition of the ( L M ) states, i.e. 2

111/(">>=

1 M=

-2

Positive reflection symmetry in the scattering plane reduces the number of independent parameters describing the D state to four. In the collision frame these have been given by Nienhuis (1980) as A = lao12/a, x = arg ( u ~ / u ~ ) ,

288

A . Crowe

+

p = 21a,lz/a,and II/ = arg (a,/a,) where 0 = laJ2 21a,I2. Van Linden van den Heuvell et al. (1983) have introduced an alternative parameterisation relating to the natural frame of reference. However, the D-Pdipole radiation pattern that is completely determined by P,, P,, P 3 , and P, is insufficient to define the D state completely. This can be attributed to the fact that two charge clouds, one being the mirror image of the other in the (x", z") plane, have the same dipole radiation patterns (Andersen et al., 1983). This ambiguity has been removed by the application of an external magnetic field, in what amounts to the first coincident level crossing experiment, in a heavyparticle excitation study (Neitzke and Andersen, 1984). The only electron impact studies of 3 'D excitation in helium are those of van Linden van den Heuvell et al. (1981, 1983) by measurement of angular correlations in the scattering plane and in a plane perpendicular to the incident electron beam direction. They have shown that the same information is obtained by observation of the 2 'P-1 'S radiation following the 3 '0-2 ' P decay as from the direct observation of this radiation, and they have used this technique experimentally. Unfortunately, the technique suffers from the problem already discussed in relation to the study of 3 'P excitation by observation of VUV radiation without wavelength selection; i.e., in this case the coincidence signal observed also contains contributions from the 3 'P and 3 ' S states. The separate contributions of the three states have been determined by fitting the sum of the exponential decay functions for each state to the measured time spectra making use of the widely differing lifetimes of the three states. The measurements enabled three parameters to be determined at incident electron energies of 28.5, 31.5, and 34.6 eV for a scattering angle of 35". The fourth parameter corresponding to the angular momentum transfer L , can only be obtained from a circular polarisation measurement. Chwirot and Slevin (1985)have used the scattered electron-cascade photon angular correlation technique to study excitation of the 3 ,D,states of atomic hydrogen. Their measurements at 54.4 and 100 eV and scattering angles of 20 and 25" are confined to the scattering plane and do not permit determination of any parameters describing the D state. Nevertheless, the high quality of their angular correlatign data, together with the recent availability of circular polarisation measurebents in the VUV (Westerveld et al., 1985), clearly demonstrates the feasibility of experiments that will allow determination of the parameters available from the dipole radiation pattern.

F. EXCITATION OF MOLECULES

Little attention has been given to the application of angular-polarisation correlation techniques to excitation in molecules due to the considerable

CORRELATIONS IN ELECTRON-ATOM SCATTERING

289

theoretical and experimental difficulties associated with their vibrational and rotational motion. Blum and Jakubowicz (1978) have derived relations between the state multipoles describing the excited state and the Stoke's parameters which are measured experimentally. They show that the amount of information available is dependent on whether or not individual rotational states can be selected experimentally. Only H, has sufficiently large rotational spacings to allow a particular rotational line to be isolated by conventional optical spectroscopy. An experiment to study molecular excitation using the polarisation correlation technique was developed by Malcolm and McConkey ( 1979). They measured the Stoke's parameters P , and P , perpendicular to the scattering plane for excitation of c 'x,,(u' = 0) state of H,. No wavelength selection of the emitted VUV photons was performed and hence transitions to different vibrational and rotational final states, X 'Z; were observed. Considerable polarisation was observed at an incident electron energy of 50 eV and at a scattering angle of 5". Similar measurements over an extended energy and angular range have been reported for excitation of the C i 'C: state of N, (Becker et al., 1983) and the c In,, state of H, and D, (Becker et al., 1984). Although strong variations in both P , and P , were observed as a function of electron-scattering angle for 0, I 20" for all three molecules, a detailed analysis of the data is hindered by the unresolved rotational structure. McConkey et al. ( 1985) have performed a rotationally resolved polarisation correlation study on excitation of the 3n,,(u' = 0, N' = 1) state of H,. This state was chosen as its decay to the a 'Z; (0'' = 0, N" = 1) state lies in the visible region of the spectrum where it could be resolved from other rotational lines using an optical interference filter and where both linear and circular polarisation measurements can be carried out efficiently. Despite these advantages, the experiment proved to be extremely difficult due to low signal and high background rates. Only the linear polarisation P , was found to be significantly nonzero with a value of0.30 f 0.25. With P , = 0.1 5 f 0.14 and P , = -0.01 0.13, then (PI = 0.34 0.25.

G. TIMEEVOLUTION OF ATOMIC STATES In the preceding discussions of the study of excited atomic states by detection of the dipole radiation emitted in their decay, the possibility of observation of time-dependent effects in the period between excitation and decay of the states has been ignored, apart from making allowances for the depolarisation of the radiation due to fine and hyperfine structure where necessary. In the absence of internal fields (e.g., the n ' P states of helium), each of the J L M ) states decays with the same exponential time dependence,

290

A . Crowe

exp( - y t ) , and the only additional information to be obtained from the study of this time dependence is the decay constant y of the excited state (Imhof and Read, 1977). However, if the excited state is perturbed by internal or external fields prior to decay, then the exponential decay is modulated giving the phenomenon of quantum beats. In the cases just considered, the experimental time resolution has been insufficient to observe such effects. An extensive discussion of time evolution of states under the influence of perturbing fields has been given by Blum (1981). The time evolution of states has been studied experimentally using the electron-photon coincidence technique in two cases. Teubner et al. (198 1) have observed “zero-field quantum beats” following excitation of the 3 * P state of sodium. The observed effects are due to the hyperfine structure associated with the J = 3/2 level (Fig. lo), the fine structure splitting being too great to produce observable effects in this case. Figure 11 shows the beat pattern observed together with a fitted function of the form Z(t)

-

A

+ C Bi C O S ( O ~ ~t ,+ qFF,)

(3)

i

where mFF. = (EJF- EJF,)/IZ.The B, contain information on the alignment and orientation of the excited states. However the extraction of this information from the measured data was not possible due to insufficient time F

VFF’ ( MHz)

56.6 35.7 16.7

5.1‘ x l o 5 ( MHz)

J = ‘/z

188.9 -1

FIG. 10. Energy level diagram of the 3 ’ P state in sodium showing the fine and hyperfine structure.

CORRELATIONS IN ELECTRON-ATOM SCATTERING

15

-15

0

29 1

T

5

15 20 25 30 35 ( nanoseconds 1 FIG. 1 1 . A fit of the beat pattern observed by Teubner et al. (1981) to a sinusoidal function of the form in Eq. (3) (from Teubner et al., 1981).

10

Time

resolution. It would certainly be interesting to repeat this experiment with currently available time resolution ( < 1 ns). In the second case where quantum beats have been observed in electronphoton coincidence experiments (Back et al., 1984; Heck and Williams, 1985), decay of the n = 2 states of atomic hydrogen in an external electric field is monitored. Here the origin of the quantum beats is quite different from that for the Na(3 2 P ) case. Electron impact excitation of the n = 2 states of atomic hydrogen does not produce a state with well-defined orbital angular momentum L. Rather, the near degeneracy of the 2 s and 2P states results in the production of a state that must be considered as a coherent superposition of states of different L (Gabrielese and Band, 1977; Band, 1979; Blum, 1981). In the absence of external fields, only coherence between states of the same parity can give rise to time-dependent observable effects. Hence, as has been seen, information is only obtained on the anisotropy of the 2 2 P state. However, application of an external field leads to the mixing of states with opposite parity and to the observation of quantum beats superimposed on the exponential decay curve, from which information on the interference between the coherently excited 2 s and 2P states can be determined. Back et al. (1984) have observed structure in the electron-photon coincidence time spectrum following excitation of the n = 2 states of atomic deuterium by 350-eV electrons in a longitudinal electric field of 250 V cmElectron-scattering angles of 1.5 to 5" were chosen. Unfortunately the data is not of sufficient quality to enable an analysis of the oscillatory structure superimposed on the exponential decay curve. A somewhat similar study has

'.

A . Crowe

292

been reported by Heck and Williams (1985). They observed photons with a particular polarisation perpendicular to the scattering plane whereas Back et al. observed the total photon signal in a direction perpendicular to the incident electron beam in the scattering plane. Both groups have used microchannel plate detectors to achieve time resolutions of -0.6 ns. The quality of the data of Heck and Williams appears to be superior to that of the original data, but further details of their measurements and analysis are awaited.

111. Ionization The electron impact ionisation process e + A + A + + e, + e, has been studied using the angular correlation technique, both to gain insight into the reaction itself and as a tool to study certain aspects of the atom A and ion A + . When the angular correlation between the scattered electron e , and ejected electron e , is observed, then both the energy and momenta of all the electrons involved are defined. Energy conservation then enables an electron with a specific binding energy E to be studied, i.e.: E =

E,

-

E, - E,

where E,, E,, and E , are the incident, scattered, and ejected electron energies, respectively. Strictly, the ejected and scattered electrons are indistinguishable, but the term scattered will be used to describe the fast electron observed at a small scattering angle 0, with respect to the incident electron and ejected, the slower electron, where such conditions exist. Momentum conservation gives the following relationship between the electron momenta:

k,

=

k,

+ k, + q

where q is the recoil momentum of the ion. Again the momentum transfer K = k, - k,, is an important parameter in discussing these studies (Fig. 12). During the 1970s three distinct groups of experiments were developed depending on the magnitude of the momentum transfer K. The low-tomedium-K-range studies were pioneered by Ehrhardt et al. (1969). Using low-energy incident electrons (from 6 eV above the ionization threshold to 600 eV), they have measured angular correlations between the outgoing scattered and ejected electrons under conditions where a slow ejected electron is observed in coincidence with a fast electron scattered through small angles (low-energy asymmetric kinematics). These correlations yield relative tripledifferential cross sections, d3a/dR, dR, dE, which provide a stringent test of theoretical models of electron impact ionization (Joachain and Piraux, 1986).

CORRELATIONS IN ELECTRON-ATOM SCATTERING

293

FIG.12. Momentum diagram for an ionizing electron collision defining the momentum transfer K and the ion recoil momentum q(= -p), the struck electron momentum for a binary collision) in terms of the initial (ko),outgoing scattered (kJ and ejected (k,) electron momenta. For the experiments of Lahmam-Bennani et al., p is scanned as the vector k, describes the circular path shown (from Lahmam-Bennani et al.. 1983b).

The value of K was maximised in a series of experiments initially performed by Amaldi et al. (1969), Camilloni et al. (1972), and Weigold et al. (1973). These are often referred to as (e, 2e) experiments. Here the two outgoing electrons sharing equally the energy available from the higher incident energy are observed at equal angles 8 = 8, = 0, relative to the incident electron-high energy symmetric kinematics. Under these conditions, the collision approximates to that between two free electrons. The preceding momentum conservation equation still holds with - q = p (Fig. 12) now the momentum of the struck electron before the collision. The magnitude of the struck electron momentum is given by (2k,cos 0 - k,)’

-1

+ 4k,” sin228 sin2 cp -

1‘2

where 0 is the outgoing electron angle measured in the scattering plane and cp is the azimuthal angle. Hence, in principle, variation of any of these parameters k,, 8, and cp allows the electron momentum p to be scanned. Two of these possibilities have been used (McCarthy and Weigold, 1976). In the coplanar symmetric geometry, the outgoing angle 8 = 0, = 8, is varied, while in the noncoplanar symmetric geometry, 0, = 8, = 45- and the azimuthal angle cp of one of the electron detectors is varied. In both cases, E, = E , = ( E , - c)/2 is fixed.

294

A . Crowe

If the validity of the plane wave impulse approximation (PWIA) is assumed and electron exchange is ignored, then irrespective of the kinematical arrangement, the triple-differential cross section can be written

where 1 TRlz= 1/4z4K4 is the Rutherford scattering cross section and p&) the spherically averaged electron momentum distribution of the atomic electron. Hence, assuming K remains constant, the triple-differential cross section measured as a function of p gives the atomic electron momentum distribution which is simply related to the position wave function through a Fourier transform. Because the symmetric noncoplanar arrangement maintains K constant as cp is varied, it is to be preferred for structure determination, while the symmetric coplanar geometry is more sensitive to the details of the reaction mechanism. In the third experimental arrangement, a fast forward scattered electron is observed in coincidence with a slow ejected electron following high-energy electron impact ionization, creating conditions in which K 0. The aim of these experiments has been to simulate photon impact processes, making use of the relationship between high energy electron and photon-induced processes (Inokuti, 1971; Kim, 1972). Under these conditions, the tripledifferential cross section can be related to the optical oscillator strength and the process is equivalent to the absorption of a photon of energy (E - I?,). The ease with which the “photon energy” can be varied in the electron experiment, especially at energies corresponding to wavelengths in the ultraviolet, is its main attraction. For this reason it has been referred to as the “poor man’s synchrotron.” This type of experiment and the high-energy symmetric scattering experiments will not be discussed further. Details and analysis of the simulated photon experiments have been given by Hamnett et al. (1976) and references therein, while the high-energy symmetric scattering experiments have been extensively reviewed (McCarthy and Weigold, 1976, 1983; Weigold and McCarthy, 1978; Giardini-Guidoni et al., 1981; and Weigold, 1984). The following discussion of the use of angular correlation techniques in ionizing collisions will concentrate on recent advances in particular areas.

-

A. HIGH-ENERGY ASYMMETRICKINEMATICS

An interesting series of experiments have been reported since 1983 by Lahmam-Bennani et al. at Orsay. In this work a scattered electron of 8-keV energy is observed over a range of small scattering angles 8, (1 to 17”) in

CORRELATIONS IN ELECTRON-ATOM SCATTERING

2Y5

coincidence with an ejected electron of relatively low energy E , in the range 20-600 eV over the angular range 6, = 20 - 135" in the scattering planehigh energy asymmetric coplanar scattering. One of the major assets of these experiments is the ability to study the complete range of momentum transfer collisions just discussed, from the near-dipole ( K -+0 ) type collision, through the intermediate-K (Ehrhardttype) region to the large-K collision enabling electron momentum distributions to be measured, simply by varying the fast-electron-scattering angle, O,, with a suitable choice of the ejected-electron energy, E,, in the range just discussed. Secondly, in view of the high energies of both the incident and scattered electrons, it would be expected that they could be treated in a straightforward way theoretically for all values of K , i.e., they would be represented by plane waves with only the low-energy ejected electron requiring a more sophisticated treatment. Exchange effects would also be expected to be small under these experimental conditions. Lahmam-Bennani et al. (1986) have shown this for large-K collisions. At this incident energy the Bethe sum rule is also assumed to be valid, enabling the results to be put on an absolute scale with an overall accuracy as high as 10% claimed. From an experimental point of view, the use of a scintillation counter to detect the 8-keV scattered electron gives the experiment increased stability relative to lower energy measurements. Details of the experiment have been given by Lahmam-Bennani et al. (1985). Since the double differential ioniLation cross sections fall off rapidly with increasing energy (Eo- I"), one serious disadvantage that might be expected relative to the lower-incident energy measurements used for structure determination is the lower signal rates. However, provided similar K values are observed, the triple-differential cross sections are very similar in magnitude in the different experiments (Daoud et al., 1985). Results and analyses have been reported for ionization of helium with lower ejected-electron energies ( E , = 20 and 46 eV) (Lahmam-Bennani et al., 1984a) and for E , = 100 and 337 eV (Lahmam-Bennani et al., 1983a, Lahmam-Bennani, 1984). The measured triple-differential cross sections for an ejected-electron energy E , = 46 eV and scattering angles 8, of 1, 2,4, and 6" are shown in Fig. 13. Despite the small angular range and the low value of E,, the data show a typical transition from near-dipole ( K + 0) collisions, (a), to a binary electron (large-K) collision, (d), for ejection of an s electron. In general, as in the low-energy asymmetric Ehrhardt-type experiments, the ejected electrons are emitted into two distinct lobes. The one that has its maximum in the K direction and is symmetric about it is generally referred to as the binary encounter peak, while that close to the - K direction is the recoil peak due to reflection of the ejected electron in the potential well of the nucleus.

J

\

I

I

200

Y

\ 4

:

n ........

.O

.O

CORRELATIONS IN ELECTRON-ATOM SCATTERING

297

It is interesting to compare the data of Fig. 13(a) where K = 0.44 au with that expected for photoelectrons ejected by radiation with linear polarisation along K (Hamnett et al., 1976). For s electron ejection, as in helium, one would expect two identical lobes centered about the 8, = k90" directions and with a width (FWHM) of 90". Figure 13(a) shows that these simple dipolar distributions reproduce the data well except that the binary peak is centered about the momentum transfer direction K rather than 8, = 90". Lahmam-Bennani et al. (1984a) demonstrate that the condition 8, = 90" can never be reached in a scattering experiment, and, hence, the discrepancy with the optical situation. Also it is possible that the deviation of the recoil lobe from the - K direction is due to other effects rather than representing the dipole situation. Figures 13(b), (c), and (d) show a number of general trends as a function of increasing 8,(K). The recoil peak direction becomes symmetric about the - K direction and decreases rapidly in intensity relative to the binary peak. A t 8, = 6" an almost pure binary collision is observed. It can also be seen that the binary peak becomes narrower. The relative triple-differential cross sections have been put on an absolute scale by integrating them over all 0, and comparing them with absolute double-differential cross sections for scattering at the same angle 0, with an energy loss ( E , - Es).This assumes the triple-differential cross sections are cylindrically symmetric about the K direction, a valid assumption in the true binary collision region. The double-differential cross section used have been made absolute using the Bethe sum rule (Lahmam-Bennani et a]., 1980). Lahmam-Bennani et al. ( 1983a) have also reported absolute triple-differential cross sections for helium under conditions where E , and K were sufficiently high that momentum distributions p(p) could be obtained within the framework of the plane wave impulse approximation (Eq. (4)) for larger 8,. Figure 14 shows the absolute-differential cross section as a function of scattering angle 6, for E, = 100 eV and 8, = 80". Under these conditions the PWIA becomes valid above an angle 0, z H e , where 8,, is the angle of scattering in a binary electron-electron collision of energy loss ( E , - E,) given by E , = E , cos2 O,,. Qualitative agreement with the data at smaller 0, is obtained if the binary energy of the atomic electron is taken into account (the corrected impulse approximation or CIA). At small scattering angles the data are well represented, particularly the binary peak, by a first Born approximation in which the fast incoming and scattered electrons are represented by plane waves and the slow ejected electron is treated as a Coulomb wave (BCW Fig. 14). Some problems persist in reproducing the recoil peak under these conditions. Corresponding measurements for the 3p shell of argon have been reported by Lahmam-Bennani et al. (1983b). Figure 15 shows the data for an electron

A . Crowe

298 10

-I

IA

I I 8

6 I

3

0

I

N

O 7

-x m -

0

4

2

.

. L.

I

0

I

2

1

I

I

4

e

I

6

I

I

8

I

10

(deg)

FIG.14. Absolute triple-differential cross section for helium as a function of scattering angle 8, for E , = 8124.6 eV, E , = 8000 eV, E , = 100 eV, and 0, = 80". The experimental data of Lahmam-Bennani et al. (1983a) are shown together with the resuslts of PWIA (-.-), the (from Lahmam-Bennani corrected PWIA (. ... .), and Coulomb wave Born calculations (-) et al., 1983a).

energy of lOOeV for scattering angles 8, from 1 to 8" showing generally similar trends to those in helium. However the triple-differential cross section at flS = 7" showing a double-binary lobe symmetric about the K direction is particularly noteworthy. It reflects the p character of the ionized electron and was first observed and discussed by Ehrhardt et al. (1974). The data can be understood by reference to Figs. 12 and 16. Figure 16 shows the electron

CORRELATIONS IN ELECTRON-ATOM SCATTERING

299

(r

(01

186 ICI

6

0'

180'

180' If1

0'

186

0,

180'

FIG.15. Absolute coplanar triple-differentialcross sections (in atomic units) ror the 3 p shell of argon with E , = 8115.8eV, E , = 8000eV. and E , = lOOeV for the scattering angles 0, as shown (from Lahmam-Bennani et al.. 1983h).

momentum distribution for the 3 p orbital of argon, and the PWIA is assumed (Eq. (4)). From Fig. 12 it can be seen that as 0, is varied, p = 0 only when K and k, are equal. For the values of k,, k,, and k, of Lahmam-Bennani et a]., this only occurs for 0, 7", and hence from Fig. 15 a zero-differential cross section would be expected when 0, is in the momentum transfer direction. The double lobe peaks at k 12" with respect to K correspond well with the peak in Fig. 15 at p zz 0.65 au. At all the other values of Q s , the minimum value of p selected by the experimental conditions is greater than p = 0.65 and so a single binary lobe is observed.

-

300 010

A . Crowe '

008 -

006 -

P

-

L

0.

OOL-

002

0

-

OL

08

12

16

2

Ploj')

FIG. 16. The momentum density p ( p ) deduced from the experimental triple-differential cross sections of Lahmam-Bennani et al. (1983b) for the 3p shell of argon with E , = 8115.8 eV, E, = 8000 eV, E , = 100 eV, and 0, = 8" (m), 7"(e), and 4 ' ( ( 0 ) .The solid line is obtained from the Ar3p Hartree-Fock-Clementi wave function (from Lahmam-Bennani et al., 1983b).

The value of the high-energy asymmetric technique for the determination of electron momentum distributions is shown in Fig. 16 where values of p ( p ) calculated from the experimental data using the PWIA (Eq. (4)) are compared with a Fourier inversion of the argon 3p Hartree-Fock-Clementi radial wave function. The largest K data-3.4 au (6, = 8O) and 3.0 au (0, = 7")--are in excellent agreement with the theoretical momentum distribution. On the other hand the invalidity of the PWIA for the lower K data ( K = 1.7 au, 6, = 4") is clear. Measurements for the 2p shell of argon (Lahmam-Bennani et al., 1984b) are more difficult experimentally due to the relatively low cross section. Their interpretation is also complicated by the higher (249-eV) binding energy, particularly for the relatively low ejected-electron energy observed ( 150 eV). Electron correlations are expected to play a significant role and so it is not surprising that impulse approximations do not produce good agreement with the data. Unlike the outer-shell ionization, it is found that the recoil peak persists over the angular range studied (6, I 7"), exceeding the size of the binary peak at small 6, and equal to it at 0, = 7".

CORRELATIONS IN ELECTRON-ATOM SCATTERING

30 1

Ionization of the 2p and 2s shells of neon have been reported by Daoud et al. (1985) and Lahmam-Bennani et al. (1986). For 2p-ionization and electron-scattering angles 0, 2 0,,, the measured triple-differential cross sections are again well reproduced by the PWIA. For 0, < He,, the description of the ejected electron by a Coulomb wave within the framework of the first Born approximation is highly dependent on the adjustable parameter Z , the effective nuclear charge seen by the slow electron, and no single value of 2 reproduces the measured binary and recoil lobes simultaneously. When the PWIA is used to determine the electron momentum distributions from the measured triple-differential cross sections, excellent agreement is obtained with the Hartree-Fock wave function for the 2 p orbital but the experimental 2s values lie about 30 below the corresponding theoretical distribution. This is probably a consequence of the breakdown of the PWIA in this case, since the ejected-electron energy observed is only four times the binding energy. The high-momentum region of the Hartree-Fock electron wave function is characterised by the existence of nodes and secondary maxima for all orbitals except the 1s (Leung and Brion, 1983). Figure 17 shows the Hartree-Fock electron momentum distribution for Ne(2.s) with the first node at p = 4 au. Lahmam-Bennani et al. (1986) have determined experimental electron momentum distributions for both 2s orbitals of neon and the 4s orbital of krypton, over values of p where nodes are expected. The difficulty in these experiments is that the momentum distribution lies mostly within the primary lobe. For the Ne(2s) case, the triple-differential cross section at the second maximum near p = 6 au is more than three orders of magnitude less than at p = 0. To increase the coincidence signal at large p , the solid angles of the electron detectors have been increased and their energy resolutions degraded relative to those in earlier experiments. Unfortunately both of these have the effect of degrading the momentum resolution of the apparatus and, hence, decreasing the depth of any sharp variation in p ( p ) as a function of p. The effect of the momentum resolution used (Ap = 0.45 au) has been investigated by measuring the Ne(2p) momentum distribution. The minimum at p = 0 was still clearly visible and the data were found to be in agreement with a Hartree-Fock wave function convoluted with the experimental momentum resolution. The data of Lahmam-Bennani et al. (1986) for both the Ne(2s) and Kr(4s) cases have been obtained under conditions ( E , = 405 and 598 eV, 0, z d,,) where the PWIA is good, the data being made absolute by normalisation to theory at low p . The Ne(2s) case has been investigated previously by both Leung and Brion (1983) and Dixon et al. (1978) using noncoplanar symmetric geometries with incident electron energies of 1200 and 2500 eV, respectively. Their relative data, when normalised to the theoretical distribution at small p , are in good agreement with each other and with that of

x,

A . Crowe

302 1

N e - 2s

2

ELECTRON

XLX

-ox

MOMENTUM

6

k x a

p laul

FIG. 17. Ne (2s) electron momentum distribution. -from Hartree-Fock-Clementi wave function; _ _ _ _ _ _ from Hartree-Fock-Clementi wave function convoluted with experimental momentum resolution of Lahmam-Bennani et al. (1986). The dotted lines are best fits to the various experimental data: LB, Leung and Brion (1983); Di, Dixon et al. (1978); (l), LahmamBennani et al. (1986) with E , = 405 eV, 0, = 17"; (2), Lahmam-Bennani et al. (1986) with E , = 598 eV, I), = 16" (from Lahmam-Bennani et al., 1986).

Lahmam-Bennani et al. (Fig. 17). At higher values of p , all of the experiments depart from the Hartree-Fock distribution. The data of Leung and Brion are first to show deviations as p increases, followed by that of Dixon et al., indicating a lack of independence on the incident electron energy in these data as p increases. At larger p values, the data of Lahmam-Bennani et al. at both ejected-electron energies and 0, = O,, suggest a secondary maximum in p(p), in qualitative agreement with the Hartree-Fock prediction. For the 4s orbital of krypton, no secondary maximum is observed.

CORRELATIONS I N ELECTRON-ATOM SCATTERING

303

More work is required, both theoretically and experimentally, to understand the discrepancies between theory and experiment for larger p values. The Hartree-Fock wave functions may be inadequate due to strong relativistic and configuration interaction effects. It should be noted that a relativistic Hartree-Fock electron momentum distribution does not exhibit nodes and its secondary maxima are shifted relative to the nonrelativistic case (Mendelsohn and Smith, 1977). It may be that the secondary maximum observed in the triple-differential cross section for the 2s shell of neon at large p is associated with second-order scattering effects predicted by Bryon et al. (1983a) and Pochat et al. (1983). Such effects have been considered for coplanar symmetric (e, 2e) processes in atomic hydrogen and helium. They write the triple-differential cross section as d3a

dR, dR, d E

=

k k = k,

I h l +.fn2(1

lS)l2

(5)

in the case of helium, wheref,, is the first Born approximation to the direct scattering amplitude and?,,( 1 ' S ) is the contribution of the initial 1 ' S state to the second Born amplitude. Figure 18 shows the theoretical triple-differential cross section for helium as a function of 8 = 8, = 8, calculated from Eq. (5) compared with the equivalent first Born approximation for an incident electron energy of 200 eV and E , = E , = 87.7 eV. It can be seen that while the first Born term dominates for scattering angles close to 45" ( p 0), the second term dominates at large 8. The experimental data of Pochat et al. (1983), normalised to the second-order calculation at 6' = 40", is in qualitative agreement with the large-angle behavior predicted by this theory. This interpretation relies heavily on the 8 = 115" experimental point, but it lies more than an order of magnitude above the value predicted by the first Born approximation at that angle. From a physical point of view, the study of highp collisions probes the region close to the nucleus where second-order effects might be expected to be more dominant.

-

B. LOW-ENERGY ASYMMETRIC KINEMATICS Significant advances, both experimental and theoretical, have recently been made in the study of ionization with low-intermediate energy electrons studied under asymmetric (Ehrhardt-type) conditions. Experimentally tripledifferential cross sections for helium have been put on an absolute scale (Jung et al., 1985; Muller-Fiedler et al., 1985). Their technique utilizes the equality between the generalised oscillator strength in the limit of zero-momentum

lo-’

I

I

1

I

I

I

t t

e- + He (1’s) d H e * ( l s l+e- + e -

10-2 r

t lo-’

:

Lu V

c V 4

. c U D U

c

10-5

r

\

10‘6

\

0

I

20

I

I

I

I

LO

60

80

100

\

I

120

CORRELATIONS IN ELECTRON-ATOM SCATTERING

305

transfer with the optical oscillator strength (Inokuti, 1971). They measure the generalised oscillator strength in both the k K directions for a range of values of K beyond the minimum value determined by the chosen kinematics and/or the minimum scattering angle attainable in the experiment. Both sets of data are then fitted by polynomials and extrapolated to the same optical limit where they are put on an absolute scale using the photoionization data of Samson (1976). The second significant advance is the application of the technique to an atomic hydrogen target. (Lohmann et al., 1984, Ehrhardt et al., 1985; Klar et al., 1986.) These new measurements should help significantly in the development of theoretical models for electron impact ionization since both the initial state and final unbound target state wave functions are known. The recent experiments of Ehrhardt et al. (1985) and Klar et al. (1986) giving absolute triple-differential cross sections for atomic hydrogen combine these recent developments and must represent the ultimate study of this type with unpolarized beams. Early calculations of the triple-differential cross sections corresponding to Ehrhardt-type kinematics were based on the first Born approximation and predict the two lobes in the + K direction. A t these lower-incident electron energies ( 2 6 0 0 eV), this is not in agreement with experiments that show both lobes shifted with respect to the axis defined by the momentum transfer vector. They also failed to predict the measured relative intensities of the binary and recoil peaks. More sophisticated first-order theories such as the Coulomb projected Born calculation (e.g., Geltman, 1974) and distortedwave Born approximations (e.g., Tweed, 1980) also reveal large discrepancies with experiment. Recently two independent theoretical approaches have been taken by Joachain and Piraux (1986) and references therein, Byron et al. (1986), and by Klar and Franz (1986), and Klar et al. (1986) for atomic hydrogen and helium. Initial calculations for atomic hydrogen by Byron et al. (1980) were based on the second Born approximation, similar to those outlined for symmetric (e, 2e) processes. More recently they have performed eikonel Born series (EBS) calculations (Byron et al., 1983b, 1984, 1985b) where the direct scattering amplitude is given by fE"S = A 1

+7",

+J;;3>

fG3 being the third term of the Glauber series. They have also taken exchange into account using the Ochkur amplitude:

306

A . Crowe

Figure 19(a) shows the results of this calculation compared with the recent absolute triple-differential cross section data of Klar et al. (1986). The EBS calculation gives good agreement with experiment. For helium the agreement is less satisfactory, possibly due to the wave functions used in the calculation (Byron et al., 1986). Klar and Franz (1986) and Klar et al. (1986) include correlation effects between the outgoing electrons in their calculations. They attribute this postcollision interaction (PCI) to long-range Coulomb and polarisation interactions allowing exchange of energy and angular momentum between the outgoing electrons, and they account for it within the framework of the first Born and Glauber approximations using a semiclassical treatment. The inclusion of PCI in the calculations results in the binary and recoil peaks being shifted to larger angles as observed in the experiment and improves the relative recoil to binary peak intensities. Figure 19(b) shows the results of the first Born and Glauber approximations with PCI compared with experiment under the same conditions as in Fig. 19(a). Both calculations reproduce the peak positions well, but the Glauber approximation also gives impressive agreement with the absolute intensity of both peaks. Since both the Glauber calculation of Klar et al. (1986) and the EBS calculation of Byron et al. (1985b) contain second-order effects, their importance, particularly for larger 8,, appears to be confirmed. On the other hand, PCI effects are clearly important at smaller scattering angles, leading to the conclusion that the inclusion of both effects is necessary to reproduce the experiment results over a range of kinematic conditions. IONIZATION C. THRESHOLD Since the predictions of Wannier (1953), much theoretical and experimental effort has been devoted to the study of electron impact ionization at incident electron energies close to the ionization threshold. With the exception of an earlier coincidence measurement by Cvejanovic and Read (1974), experimental studies have involved the observation of a single particle-e.g., Hammond et al. (1985). Recently Fourier-Lagarde (1984) and Mazeau et al. (1986) have measured coplanar triple-differential cross sections for a range of incident electron energies down to 0.5 eV above the helium ionization threshold, providing a highly sensitive test of threshold ionization theories. The Wannier theory, which is restricted to states with total angular momentum L = 0, predicts that the angular correlation between the two outgoing electrons has a maximum at (0, + 8,) = n with a width varying with excess energy above the threshold E = ( E , - E ) given by Ell4. All combinations of scattered and ejected electron energies at a particular E are predicted

307

CORRELATIONS IN ELECTRON-ATOM SCATTERING

I

-4 --I I I I I

I I I 0

- 0

180

120

60 0 Scattering angle

- 60

-120

-180

(degl

FIG.19. (a) The coplanar triple-differential cross section for ionization of atomic hydrogen EBS as a function of the ejected-electron angle 0, for E , = 250 eV, 8, = 5 eV, and 0, = 8”. calculations of Byron et al. (1985b); _ _ _ first Born approximation, data of Klar et al. (1986) (from Jochain and Piraux, 1986). (b) As (a) except _ _ _ is first Born approximation with PCI, _ _ Glauber approximation with PCI, both due to Klar et al. (1986) (from Klar et al., 1986). _ .

308

A . Crowe

+ E p E, s0.25 eV

E,, E,=0.5 eV

I

I

50

I

f

100

150

e,

FIG.20. The relative coplanar triple-differential cross section for helium as a function of the scattering angle 0, with (0, + 0,) = I.0,E , = E , = 0.5 eV; +, E , = E, = 0.25 eV. The solid line is a fit of Eq. (8) to the data (from Mazeau et al., 1986).

to be equally probable, although a slightly increased probability for E , = E , has recently been predicted (Read, 1984) and observed (Hammond et al., 1985).

Fourier-Lagarde et al. (1984) and Mazeau et al. (1986) have reported a series of triple-differential cross section measurements, each aimed at testing various aspects of threshold ionization theories. Measurements as a function of 8, for (8, 8,) = 180" test whether L # 0 states contribute to the ionization since

+

under these conditions (Stauffer, 1982). Figure 20 shows the measured cross section for E , = E , = 0.5 eV and 0.25 eV. Clearly L # 0 states make a very significant contribution even at these low energies. If it is assumed that only states with L I2 contribute (ISe, ' P o , 3P0, ' D e ,3De),then within the framework of the so-called Wannier-Peterkop-Rau theory (Peterkop, 1971; Rau, 1971), the triple-differential cross section can be written d3a

=

[41 las12 + 3 1 a ~ / 2e ]n p j - 4

In 2[ (0,

dQ dQ, d E

+ 8, - 18Oo)]l)

(7)

8112

where the singlet (aS)and triplet (aT)scattering amplitudes are related simply to the individual singlet- and triplet-state amplitudes and 8,, is the halfwidth (FWHM) of the ejected-electron angular correlation. From Eq. (7) it can be seen that all the LSn states have the common angular correlation factor

0, + 0, - 180°).] exp[ -4 In +-8iiL---

309

CORRELATlONS IN ELECTRON-ATOM SCATTERING

Also from Rau (1976), the common energy dependence of OliZ is given by

O,,,

= f),

and each partial cross section is independent of the sharing of the energy E between E, and E,. Under the coplanar conditions of Fig. 20, the triple-differential cross section takes the form given by Eq. (6) or, more precisely (Stauffer, 1982),

where all the CLSnwith odd ( L + S) = 0. Hence only the ' S e , 3P0, and 'D' states can contribute. From the fit of this expression to the data of Fig. 20, the relative amplitude and phases of these three states were determined. The 3P0 amplitude was found to be near zero. . Similar measurements at higher energies show that all of the three states have the same energy dependence up to an excess energy of 2 eV. For a coplanar geometry with 8,= O,, the contribution of triplet states to the triple-differential cross section is eliminated (Eq. (7)) and hence, since the ' S e and IDe contributions have been determined, only the 'Po contribution and 8,are unknown. The measured coplanar triple-differential cross sections as a function of 8, = 8, are shown in Fig. 21 for E , = Es = 0.25 and 0.5 eV. -go=

680

---go

= 85O

A

I

L .

I

t

60

Es=Ee

= 0.25 eV

I

I

8

I

90

E,

I

120

I

-

8

= E, = 0.5 eV

FiCj. 21. The relative coplanar triple-differential cross section for helium as a function of 0, = 0, for (a) E , = E , = 0.25 eV, (b) E , = E , = 0.5 eV. The crosses are the data of Mazeau et at. (1986) and the full and dotted lines are theoretical fits with 0, = 68' and 85" (see text) (from Mazeau et al., 1986).

3 10

A . Crowe

The theoretical fits to the measurements with 0, = 68" and 85" show that the quality of the data in this case is insufficient to yield a unique value of 8, and hence the ' P o contribution. 8, values predicted theoretically range from 67" to 89". In a third measurement by the Paris group, coplanar triple-differential cross sections for different combinations of E , and E , at a particular excess energy E and scattering angle 0, confirm the independence of the cross section on energy partitioning as predicted. Recent ab initio calculations by Crothers (1986) show remarkable agreement with experiment.

D. AUTOIONIZATION In an electron-electron correlation experiment, if it is arranged that the energy of the scattered electron detected, E , , is such that the energy loss ( E , - E,) in the collision corresponds to the energy of a doubly excited state lying in an ionization continuum, and the ejected electron detected has an energy corresponding to that from the autoionization process, then the technique can yield fundamental information on the autoionizing state. In all the studies previously discussed in this chapter, this choice of conditions has been avoided. In principle, a study of the angular correlations between the scattered and ejected electrons following the excitation of an autoionizing state should yield the alignment of the doubly excited state in a similar way to the electronphoton angular correlations discussed earlier for low-lying excited states. However, electrons from the direct ionization process leaving the ion in the same final state as the autoionization process also contribute to the observed signal, and strong interference between the two processes prevents information on the alignment being readily obtained. Only four such experiments have been reported. The feasibility of the experiment was demonstrated by Weigold et al. (1975) for the (2s2p)'P and (2p')'D states of helium. More recently, further studies have been reported in helium by Pochat et al. (1981, 1982) for the (2sz)'S, ( 2 ~ 2 p ) ~(2s2p)'P, P, and (2p2)'D autoionizing levels as well as by Moorhead and Crowe (1985) for the (2s2p)'P and (2p')'D states. A brief report of an experiment for the 3s3p64p state of argon has been given by Jung et al. (1977). The helium experiments have been analysed in terms of the plane wave Born approximation (Balashov et al., 1973) where the triple-differential cross section is given by

CORRELATIONS IN ELECTRON-ATOM SCATTERING

31 1

where f(k,,K) is the cross section for direct ionization, b(k,,K) is the contribution of the autoionizing state, and a(k,, K) characterises the asymmetry of the resonance due to the interference between the direct and autoionizing processes. E = 2(E, - E J T where E , is the resonance energy and r its width. a and h are simply related to the Fano profile parameter q(E,, 0,) by the relation

b-~- q 2 - 1 a 2q In the helium experiments, coplanar triple-differential cross sections have been measured as a function of the ejected-electron energy E , for a range of 8, at scattering angles 0, (10, 15') and incident electron energies 100-400 eV. The experiments of Weigold et al. (1975) and Pochat et al. (1982) were unable to resolve the (2s2p)'P and (2p')'D states. The subsequent analysis of the data of Weigold et al. assumed the ' D contribution to the observed signal to be negligible while Pochat et al. used sophisticated fitting techniques to extract the a and b parameters for each state. The same procedure was also adopted by Pochat et al. for the unresolved (2s')'S and ( 2 ~ 2 p ) ~states. P The recent higher-energy resolution experiments of Moorhead and Crowe (1985) clearly show that the ' D state cannot be ignored. The coincidence spectrum observed at an incident electron energy E , = 200 eV, 8, = lo", and 0, = 130" is shown in Fig. 22. The 'D and ' P states at 35.30 and 35.54 eV are clearly visible. Under these kinematic conditions, the direct ionization cross section f ( k , , K) is small and only weak interference is observed. Data close to the momentum transfer direction K under the same conditions show a large direct ionization contribution with highly asymmetric resonance profiles. These experiments are still in progress and a full analysis will be presented in the literature.

E. INNER-SHELL IONIZATION

AND

AUGERELECTRON DECAY

Consider processes of the type e-

+A

--t

A + ( J , M ) + es-

+ e;

where the ion A + ( J , M ) has an inner-shell vacancy and can decay by either X-ray A +(J , M ) -+ A +(J", M " ) + hv

or Auger electron emission A + ( J , M ) + A + + ( J ' , M ' ) + eAuger

312

A . Crowe

Ejected Electron Energy

eV)

FIG. 22. The coplanar triple-differential cross section of Moorhead and Crowe (1985) as a function of the ejected-electron energy E , in the region of the ( 2 p 2 ) ' D and (2s2p)'P autoionizing states of helium for an incident energy of 200 eV, 0, = lo", and 0, = 130".

The study of inner-shell ionization processes by measurement of the angular correlation between the scattered and ejected electrons has already been discussed (Lahmam-Bennani et al., 1984b). However, there is potentially more information to be gained from observation of the initial ion decay product in coincidence with the scattered electron, in particular the alignment of A + ( J , M ) . Mehlhorn (1968) showed that provided J > 1/2, then the inner-shell ion A + will be aligned. More recently the theory of scatteredelectron-Auger-electron-X-ray angular correlations and their relation to the alignment of the initial ion state has been discussed by Berezhko et al. (1978). The first experimental study of this type was reported by Sewell and Crowe (1982) for ionization of the 2p shell of argon at an incident electron energy of 1000 eV. For this case, scattered-electron-Auger-electron angular correlations were measured since the fluorescence yield is negligible. Further measurements have been reported by Volkel and Sandner (1983), Crowe (1984), Sewell and Crowe (1984a, b), Sandner and Volkel (1984), LahmamBennani et al. (1984c), and Sandner (1985), all for the 2p ionization of argon. The experimental technique is similar to that for previous electron-electron

CORRELATIONS IN ELECTRON-ATOM SCATTERING

313

correlation measurements, the major difficulty being the low signal rates ( - a few per hour). Berezhko et al. (1978) show that for scattering through an angle O,, the probability of Auger emission in a direction (flA, qA)is given by

where 8, and qA are the polar and azimuthal angles relative to the q = 0 scattering plane with the incident electron beam direction as the z-axis. The ak terms characterise the Auger decay process while the A,, are normalised (Aoo = 1) statistical tensors describing the anisotropy of the initial ion state A + . The &,(OA, qA)are spherical harmonics. For a particular ionizationAuger decay, the values of k and K are limited by parity conservation and symmetry arguments and by the angular momenta of A + and the Auger electron observed. Two cases are of interest in relation to the experiments performed: 2p,,, and 2p3,, ionization in argon. For the 2p,!, case, the angular correlation should be isotropic, consistent with the prediction that a state with J s 1/2 cannot be aligned. This is not an interesting case to study from a physical point of view, but it provides a convenient test of the experiment. Sewell and Crowe (1984) have studied ionization of the 2 p , , , shell of argon under the following conditions

e(1000 eV)

+ Ar

-+

Ar+(2p,,,)-'(~ = 250.5 eV) + e,(5 eV)

+ e,(744.5 eV)

4Ar2+(jP)+ e,L,-M,,, M2,3(3P)(207.2eV).

(1 1)

The measured scattered electron-L,-M,,, M,,,(jP) Auger electron angular correlation is shown in Fig. 23(a) for 0, = 15". Clearly it is not isotropic as predicted. This anisotropy is most likely due to electrons from other processes satisfying the energy conditions in Eq. (1 1) such as e(1000 eV)

+ Ar

-+

Ar+*(&)+ e,(744.5 eV)

+ e,(207.2 eV)

(12)

and/or -+

A'+(&') + e,(744.5 eV) + e,,(207.2 eV)

+ e2,(E2,).

(13)

In Eq. (12) a highly excited, singly charged ion with binding energy -48 eV must be produced, i.e., it must lie in the Ar+ continuum. Alternatively direct double ionization may be responsible (Eq. (1 3)) giving an undetected electron whose energy E,, is dependent on the binding energy E'. Figure 23b shows a measured estimate of the background due to such processes. It can be seen that the correlation is peaked close to the momentum transfer direction, characteristic of scattered-ejected electron angular correlations. Subtraction of the two signals yields, within experimental error, the isotropic correlation shown in Fig. 23c. +

A . Crowe

314

z -

0

u

(c)

T

T

8 -

_--------,

-----

c 1

0

I

=-@-+,

FIG.23. Scattered electron--L,-M,. M 2 ,( 3 P )Auger electron angular correlations for the 2p,,, shell of argon under the conditions in Eq. (1 1) at 0, = 15". Measured correlations for (a) all electrons with this Auger energy and (b) background electrons (Eqs. ( 1 1 ) and (12)). (c) is the subtraction of (b) from (a). The arrows show the K direction (from Sewell and Crowe, 1984a).

The 2p3,, alignment is characterised by three independent statistical tensors A,,, Azl, and A22. In general the measured angular correlation is also influenced by the Auger decay parameter u 2 . However, by choosing an Auger decay that leaves the final ion A + in a ' S o state (Cleff and Mehlhorn, 1974), a2 = 1 and the angular correlation depends only on the three statistical tensors, i.e. : +

w(eA,

(PA)

- (4n/5>"2 K=O.1,2

y2K(eA,

(PA).

(14)

315

CORRELATIONS IN ELECTRON-ATOM SCATTERING

- 0

0 -

o

tu

40 60 ao 100 130 AUGER EMISSION ANGLE

140

(deg

im

1

im

FIG. 24. As for Fig. 23 but for the L , - M , , , M 2 . 3( ' S o ) Auger electron from the 2p,,, ionization of argon under the conditions in Eq. (15) (from Sewell and Crowe, 1984a).

The results for the 2p3,, shell of argon under the following conditions, for 8, = 15" are shown in Fig. 24. e(1000 eV)

+ Ar + A r + ( 2 ~ , / ~' ()E- = 248.6 eV) + e,(5 eV) + eJ746.4 eV) 4Ar2t(1So)+ e,L3-M,,3MZ,3(1S,)(201.1 eV). (15)

In this case, when the background subtraction is made (Fig. 24c), an anisotropic correlation is obtained. The three alignment parameters cannot be uniquely determined from this coplanar data and that of Lahmam-Bennani et al. (1984~)or that over a

316

A . Crowe

conical surface by Volkel and Sandner (1983). However, the data can be used as a direct test of the first Born approximation. If the quantisation axis is taken in the direction of momentum transfer K, then in the first Born approximation AFK = 0 except for K = 0, i.e., the coplanar angular correlation is given by w($A)

-

A$OP2(cos $ A )

where $ A is measured with respect to K. The data of Fig. 24c are clearly inconsistent with the first Born prediction. The scattered- Auger electron coincidence method is also a powerful technique for the study of PCI effects in the Auger electron decay process. Such effects have been investigated extensively in the decay of autoionizing states by observing either the scattered or ejected electron line shape (e.g., Helenelund et al., 1983, and references therein). In this case the energy of the undetected electron is known from energy conservation. In Auger electron decay, the excess energy in the collision is shared between the scattered and ejected electrons, complicating any analysis of the observed Auger electron line shape. However, if the Auger electron line shape is measured in coincidence with a scattered electron of energy E,, then the undetected ejected-electron energy is again known from energy conservation. This technique also permits the study of such effects at high-incident electron energies if one of the outgoing electron energies is chosen to be low. Studies of this type have been reported for 2p ionization in argon by Sewell and Crowe (1984b) and Sandner and Volkel (1984). Figure 25 shows the coincident L , - M 2 , 3M , , 3(3P,.1, 2) Auger line shape measured under the conditions of Eq. ( 1 1) for a scattered electron angle of 16" and an Auger electron angle of 60". The line shape shows the characteristic shift and broadening in energy associated with PCI. The calculation of Niehaus (1977) for photoionization with an excess energy corresponding to the ejected electron in this experiment (5 eV) shows good agreement with the present data. Similar results have been obtained for the L 3 - M , , M 2 ,3( ' S o ) Auger line by Sandner and Volkel (1984). Their data also suggest that the measured line shape shows additional asymmetry under conditions where background processes of the type discussed (Eqs. (12) and (13)) are present. This is consistent with interference between the inner-shell ionization decay process and the background processes and is to be expected since both mechanisms may lead to the same final state, Ar' +('So) in this case. Strong interference effects would invalidate the background subtraction process used in Figs. 23 and 24 and complicate the extraction of information on alignment of the ion A'. Both interference and PCI effects would violate the assumption in Eq. (10) that the ionization and decay processes can be treated independently (Sandner, 1985). Hence further work on alignment in inner-shell

317

CORRELATIONS IN ELECTRON-ATOM SCATTERING

-1

I

'

:I

-

I

I

T I

+

3

-0

U

;r

T T T

T T

I

0'-110

I I

0

I

10

I

2 .o

Relative energy i e V i

FIG.25. Argon L , - M , , , M 2 , , ( ' P ) Auger electron line shape measured at V, = 60' in coincidence with a scattered electron (Vs = 16") under the conditions of Eq. (1 I). The broken curve is the measured noncoincidence Auger electron line shape and the full curve is the theoretical prediction of Niehaus (1977) (from Scwell and Crowe. 1984b).

ionization is required, particularly under conditions where interference and PCI effects are negligible. More detailed studies are also required of the PCI and interference effects themselves and of the background processes that lead to the interference.

ACKNOWLEDGEMENTS

The financial support of the United Kingdom Science and Engineering Research Council (SERC) to enable a range of experimental angular correlation studies across the breadth of this chapter to be carried out in the author's laboratory is gratefully acknowledged, as is the contribution of various SERC-supported research sta& postgraduate students, and departmental technical staff.

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A

Above-threshold ionizations (ATI), 189-201 peaks, 200 Absorption monitoring, 234 ~ 2 3 6 Adiabatic approximation, first-order nondegenerate, 129, 130 Adiabatic cross sections, 129 Adiabatic theory, 128 Adiabatic-nuclear-rotation (ANR) approximation, 76, 79, 81, 84. 107, 108, 117, 118, 129 First Born, 119 cross sections, 84, 119 method, 77-79, 80 scattering matrix, 1 17, 119 theory, 70, 119, 130 Adiabatic nuclear vibration (ANV) approximation, 84-86, 88. 106, 126 cross sections, 85, 88 matrix, 92 method, 79, 81, 88, 126 Adiabatic-nuclei (AN) approximation, 75, 78-82, 84, 85, 97, 115, 119, 120, 126, 128, 134 off-shell, 128 cross sections, 83, 84, 97 methods, 75, 78 81, 87, 88, 124 scattering amplitude, 81, 82 Angular anisotropy, 17 I , 173 Angular correlations, 157-2 17, 269-276, 279, 281, 285, 288, 292, 294, 312-316 Angular distributions, 106, 136. 159-217, 270 measurements of, 176- 178, 180 Angular momentum operator(s), 67, 79 Angular-polarization correlation, 270, 27 I , 288 ANR method, 77-79,80 ANV, see Adiabatic nulcear vibration Approximate nonadiabatic scattering theories, 119- 127

Asymmetric kinematics high-energy, 294-303 low-energy, 292. 303-306 Asymmetric stretch modes, 142 Asymptotic frame transformations, 82, 117 rotational, 81, 82 vibrational, 82 Asymptotic free states, 56. 57, 60, 66, 80 Asymptotic rotational coupling, 69 ATI, 189 201 Atomic collisions, 243 Atomic frame, 271 Atomic photoionization processes, 172 Atomic polarization, 227 Atomic wave functions, 176 Auger decay, 313, 314 Auger electron angular correlation, 3 13, 314 decay, 311-317 emission, 311, 313 line shape, 316, 317 Autoionization, 310, 31 1 Autoionization resonance(s), 193, 210-212

B Background (BG) term, 124 Background scattering, 124 Bessel function radial distribution, 20 Bethe sum rule, 295. 297 BF-FN theory, 68, 80, 93 BFVCC method, see Body-frame vibrational close-coupling method BG term, 124 Binary collisions, 241, 244248, 252. 255, 292, 297 collision region, 297 electron, 295 electron-electron, 297 Binary encounter peak, 295 323

324

INDEX

Binary ionic reactions, 23 Binary peak, 297, 300. 305, 306 Binary rate coefficient, 32, 33 Birefringe monitoring, 234--236 Birmingham double selected ion Row tube, 6, 7, 31 Birmingham flowing-afterglow-langmuirprobe (FALP) apparatuses, 16, 20 Birmingham ion flow tube design, 11, 12.45 selected, 5, 10, 12, 14 variable-temperature selected, 13, 16 BODY basis function(s), 67, 68, 71 BODY channel index, 67,68 BODY coupled equations, 69, 76 Body-fixed reference frame, 66 BODY formulation, of electron-molecular collision theory, 66 BODY frame, 58, 65-69, 77, 79, 127 reference, 65, 66 Body-frame fixed-nuclei (BF-FN) approximation, 78 calculations, 95, 134 eigenphase sums, 94, 97, 98, 101, 124, 126, 142 integral equations, 104 matrix, 91, 120, 121 radial (scattering) functions, 73, 75, 128, 129 reactance matrix, 105, 126 scattering amplitude, 81, 82 scattering calculations, 121, 123, 124 scattering equations, 112, 119, 120, 124, 128 scattering matrix, 104, 126, 127, 137 theory, 68, 80, 93 wave function, 81 Body-Frame vibrational close-coupling (BFVCC) method, 76-78, 86, 87, 139 theory, 70, 120, 139 wave function, 120 BODY radial functions, 69, 70, 71, 72, 75 scattering, 67 BODY representation, 60-78, 120 BODY scattering state(s), 67, 76 BODY wave function, 66, 79 Bohr frequency, 233 Bohr magnetons, 247 Bohr radii, 239

Boltzmann constant, 37 Boltzmann equation, I13 Bond energy of cluster ions, 23,40,44 Bond strengths, 31 Born approximation. see First Born approximation Born differential cross section, 132 Born-Oppenheimer approximation, vi, 79-81, 107, 115-130 energies, 83 factorization, 61, 73, 80, 81, 129 separability, 80 state, 74, 126 theory, 80 Boundary conditions, physical, 58. 59 Bound-bound channels, 212 Bound-bound transitions, 212 Bound-continuum channels, 21 2 Bound-continuum transitions, 194, 195, 207, 215,217 amplitudes of, 168 Bound-state poles, 96, 100, 124, 135, 142 Bound-state wave functions, 202 Breit-Rabi parameter, 250 Buffer-gas collisions, 244

C

CAM, 68-81 Centre-of-mass interaction energy, 8, 38, 42 Charge exchange, 242, 264 Chemical exchange, 242 CIA, 297 Circular dichroism method, 234, 255 Circular polarizations, 273, 274, 275, 277, 279, 281, 284, 288, 289 Clebsch-Gordan coefficient(s), 62, 64, 167 Clebsch-Gordan series, 61, 64, 70 Close-coupling theory, 117, 135 Cluster ions bond energy of, 23,40,44 ligand bond strengths in, 30 weakly bound, 2,45 Coherence narrowing, 253, 254 Collision frame, 271, 272, 287 Collision theory, 53, 75-88, 124 Collisions low-energy, 52, 150

325

INDEX near threshold, 52 in polar systems. 150 Collisional association, 33 reaction, 44 Collisional breakup of ions, 8, 44 Collisional dissociation, 8, 9, 24 Collisional quenching of ions, 26, 38 Collisional relaxation, 243 Collisional shifts, 255 Configuration-space functions, 67 Continuity equation, 21 Continuum-continuum transition(s), 195. 196 Cooper Zare formula, 187 Coplanar angular correlation, 3 16 Corrected impulse approximation (CIA), 297 Correlation-plarization potential, 95 Coulomb interactions, 57 Coulomb projected Born calculation, 305 Coulomb wave, 297, 301 Born calculations, 298 Coupled angular momentum, 61 -81 Coupled radial equations, 78 Coupled radial functions, of electronmolecule theory, 70 CRESU-VT-SIFT study, 41 Crossed-beam experiment measurements, 105. 111 D Decay curves, 26-28, 291 Depolarization cross sections, 228, 254 Depolarizing collisions, 228, 254 Depopulation pumping, 228. 229. 253. 254 Diamagnetic Faraday rotation, 235, 236 Differential cross sections, 59, 105, 108. 113, 116. 117, 119, 130. 131, 132, 141, 162, 167, 168, 172, 270, 272,278 photoionization, f62, I70 Diffusion coefficient. 252, 253 Dipole approximation, 162-166, 174, 200. 216 Dipole radiation, 288, 289 Dipole selection rule, 164, 172. 194 Dissociative attachment. 56. 131. 134 Distorted-wave Born approximation, 305 Distorted-wave method, 116 Double differential ionization cross sections, 295. 297

Dynamic nuclear polarization. 230

E Edm., 261 Effective-range approximation (ERA), 138- 140 methods, 138 Effective-range theory (ERT), 88, 100 1 1 5 Eigenchannel R-matrix method (ERM), 124, 126, 127 Eigenobservables, 23 I , 232, 233 Eigenphase sums, 95, 98, 102, 107. 113, 125. 137, 140 Eikonel Born series (EBS), 282, 305, 306, 307 unitarized. 282 Elastic cross sections differential, 103. 109 integral, 102, 132 low-energy, 77, 131, 132 momentum transfer, 103 Elastic scattering, 52, 54, 91-93, 116, 133, 141 Elastic threshold, 97 Electric dipole approximation, 158, 195 excitation, 21 1 moment (edm), 261 Electron- atom scattering, vi, 90, 96, 100, 101, 109, 110, 269-317 Electron-collision calculations, 58 Electron correlations, 300 Electron-electron correlations, 310, 312 Electron-electron spin exchange, 240 Electronic energy, 68 Electronic excitation, 56, 57, 128 Electronic spin, 225. 226, 247, 249, 263 Electronic wave function, 122 Electron impact excitation, 286. 291 Electron impact ionization. 269, 292, 294, 305. 306 Electron scattering, vi, 54, 71. 89-100, 142 Electron spins, 166. 167, 168, 179, 188. 240, 242, 245, 246, 252, 255, 264 Electron-molecule collisions, 62, 73. I 17, 144 Electron-molecule cross sections, 54, 93, 104, 113, 1 I6 Electron-molecule interaction potential, 97, 101 Electron- molecule scattering, 51 IS0

326

INDEX

Electron-molecule theory collision, 60, 63, 66, 78 coupled radial function of, 70 scattering, 93, 128, 144 Electron momentum distribution, 294, 295, 299, 300, 301, 302

Electron-photon angular correlation, 272, 285, 310

Electron-photon coincidence technique, 270, 287,290, 291

Electron-photon coincidence time spectrum, 29 1

Electron randomization rate, 231, 232, 245 Electron randomizing collisions, 231-233 Electron-rare gas atom systems, 110 Electron scattering, vi, 54, 71, 89-100, 142 Electron spins, 166, 167, 168, 179, 188, 240, 242,245,246,252, 255, 264

functions of, 202, 238 EMA approximation, see Energy-modifiedadiabatic, approximation Endothermic ion-neutron reactions, 39 Energy-modified-adiabatic (EMA) approximation, 87, 92, 120, 121, 126, 130 reactance matrix, 121 transition matrix, 121 Enthalpy changes, 31, 32 Entropy changes, 31, 32 ERA, 138-140 ERM, 124, 126, 127 ERT, 88, 100-115 Euler angles, 66 Exact second Born theory, 282 Exchange kernel, 64, 65 Exchange scattering, 284, 287 Excitation, vi, 270-292 amplitudes, 272 cross sections, 84, 127, 129, 134 decay process, 271 Excited state depolarization, 254 Excited state photoionization, 216

F FA, see Flowing afterglow FALP, 16, 20 Fano profile parameter, 31 1 Faraday rotation, 264 angle, 235 Fast-flow tube techniques, 1, 5

FBA, see First Born approximation FDT, see Flow-drift tube Fermi contact hypefine interaction, 255 Feshbach projection operator method (POM), 124 Field-theoretic off-shell adiabatic theory, 128, 129

Filter gas technique, 23, 28, 29 Final-state interaction theory, 135, 136 scattering function, 106, 138 wave function, 86, 92 wave vector, 59 Final unbound target wave functions, 305 First Born approximation (FBA), 88, 109, 115-119, 130, 132, 134, 296, 297, 301, 303-307, 316 adiabatic-nulcear-rotation, 119 dipole, 108 for nonpolar systems, 116 for polar systems, 116

First-order many-body theory, 275, 285, 286 nondegenerate adiabatic (FONDA) approximation, 129, 130 perturbative theory, 116 Five-photon nonresonant photoionization, 196, 199

Fixed-nuclear-orientation (FNO) approximations, 68, 69, 72, 76-78, 120-124

Fixed-nuclei (FN) approximation, 74, 77-79, 80 cross section, 94, 97, 104, 105 limit, 124, 142 scattering data, 138, 141 scattering equation, 120 transition matrices, 78 Flow-drift tube (FDT), 15, 34, 37, 39, 41 Flow dynamics, 18-22 Flowing afterglow (FA) apparatuses, 14, 16, 19 ion source, 30, 44, 45 technique@), 1-5, 10, 19, 20, 33, 34 Flowing-afterglow- Langmuir-probe apparatuses, 16, 20 Flow tubes, 12-16 Fluorescence monitoring, 236, 237 FN approximation, see Fixed-nuclei, approximation

327

INDEX F N O approximation, see Fixed-nuclearorientation FONDA, 129, 130 Fourier analysis, 185 Fourier inversion, 300 Fourier spectrum, 186 Fourier transform, 294 Four-photon ionization, 198, 199 Frame transformations, 69-75, 79, 91, I19 asymptotic, 82, 117 calculations of, 74 theory of, 69, 72 75 Free-electron-gas exchange potential, 74, 1 12 Frequency shifts, vi, 254-257

excitation, 288 scattering, 80 Heitler equation. 105 High-energy asymmetric kinematics, 294-303 High-energy electron impact ionization, 294 Hilbert space, 122. 123 Hydrocarbon ion reaction(s), 3 Hydrogen atom reactions, 7 Hyperfine absorption monitoring, 234 coupling, 170, 173, 184, 188, 189, 194 population imbalance, 234 pumping, 226, 227, 229 splitting, 258, 259 structures, 258, 259, 285, 286, 289, 290

G I Gauss magnetic field, 258 General relativity, 261 Glauber approximations, 306, 307 Glauber series, 305 Graetz-Nussell equation, 22 Ground state Hanle effect, 263 polarization, 236, 254 populations, 225, 226 spin polarization, 253

H Hamiltonian electron-molecule, 76 molecular electronic, 56, 57 non-Hermitian, 123 nuclear, 56, 57, 86, 117, 127. 128 rotational, 67, 68 system, 57 vibrational, 56, 60, 67, 68, 72, 77 Hanle signal, detection of, 263 Hartree Fock distribution, 302 electron momentum distribution, 301. 303 electron wave function, 301 -303 procedure, 200 static field, 112 Hartree-Fock-Clementi wave function, 300 Heavy particles collisions. 128

ICR, 2,44 lndependent-particle models, 161, 175, 210 Inelastic angular distributions, 136 Inelastic cross sections, 89, 92, 95, 97, 113, 121 Inelastic electron- atom scattering, 269, 284 Inelastic electron-(nonpolar) molecule cross sections, 91 lnelastic scattering, 133 Inertial mass anisotropy, 262 Infrared-active bending stretch modes, 142 Inhomogeneous magnetic fields. 253 Initial state wave functions, 305 Inner-shell ionization, 31 1-3 17 Integral cross section, 59, 116, 119, 141 Interaction Hamiltonian, 248 Interaction potential, 57, 100, 107 Internuclear separation, 29, 68, 246 Interstellar M H D shocks, 40 Ion clusters, 29, 30 Jon cyclotron resonance (ICR). 2, 44 Ionic mobilities, 37 Ionization, vi, 292 317 cross sections, 212, 216, 217 threshold, 180, 193, 199, 292 Ionization-Auger decay, 313 Ion-molecule reactions association, 2, 8, 17, 26. 32, 44, 45 isotope exchange in, 31 negative, 40 rate coefficients for, 22

328

INDEX

Ion-neutral reactions, vi, 1-45 Ion product distributions, 5, 17, 23-26 Ion recoil momentum, 292,293 Ion swarm, flow dynamics of, 19-22, 28 Ion trap technology, 260 Isotope exchange, 31, 32 Isotropic angular distribution, 135, 143 Istropic correlation, 31 3, 314

K Kramers-Kronig relation, 235, 236

L LAB coordinate systems, 59 LAB coupled angular momentum (LABCAM) angular functions, 71 basis functions, 63 channel index, 63 coupled equations, 75 cross sections, 75, 79, 118, 129 equations, 67. 76 First Born approximation, 117, 118 differential cross section, 118 scattering matrix, 119 formulation, 74, 88, 106, 118, 119 radial scattering equations, 62, 65, 117 representation, 60-65, 69-73, 79, 91, 107 rigid rotator study, 77 scattering matrix, 65, 81 theory, 70, 76, 88,90, 130 LAB excitation amplitude, 82, 127 LAB formulation, of electron-molecule collision theory, 66, 78, 106 LAB frame, 58, 65, 66, 82, 127 close-coupling, 85, 130 LAB representation, 71 LAB(RR) formulation, 108 LAB scattering amplitude, 66 Laboratory reference frame, space-fixed, 58-65 LAB-UCAM representation, 59-63, 70, 79 Larmor frequency, 250, 253, 255,256, 262 Larmor precession, 257 Laser-intensity dependence in multiphoton ionization, 212-216

Laser plasmas, 3, 45 Legendre functions, 163, 182 Legendre polynomials, 64, 117, 158, 163, 166, 203, 208 Level shift operator, 123, 124 Ligand bond strengths, in cluster ions, 30 Light shifts, 256, 257, 260 Linear polarizations, 273 285, 287, 289 Lippmann-Schwinger scattering equation, 122, 124, 128, 129 Longitudinal pumping, 227 Long-range Coulomb interactions, 306 Low-energy asymmetric kinematics, 292, 303-306 Low-energy collisions, 52, 150 Low-energy elastic cross sections, 77, 131. 132 Low-energy electron-molecule collisions, 117 Low-energy electron-molecule scattering, vi, 52-75 Low-order perturbation theory, 212

M Magnetic dipole interactions, 249 dipole-dipole, 239 Magnetic fields, 253 Magnetic resonance frequencies, 254 256 Many body optical-potential theory, 121 Many-electron, nonadiabatic Schrodinger equation, 121 Matter anisotropy, 262 MEAN, approximation, 117, 1 18 Metastability exchange, 238-242 collisions, 241, 242, 255 cross section, 241 Modified effective-range theory (MERT), 100-115, 135 cross sections, 106, 107, 108 equation, atomic, 104, 112 expansions, 101 -105, 109-111, 141 formulae, 106, 115 at near-threshold pole, 108 for nonpolar systems, I01 for polar systems, 106-108 Mdler operator, 57 Momentum conservation, 292, 293 Momentum transfer collision, 270, 295

329

INDEX cross section(s), 54, 59, 95, 101, 105, 107, 110-113, 142, 143 direction, 297. 299, 311. 313 Monitor gas technique, 23, 26, 28, 29 Morgan- McDowall parameterisation, 281 Morse potentials, 123, 140 Mott scattering, 179 Multi-pole-extracted adiabatic-nuclei (MEAN) approximation, 117, 118 Multichannel quantum defect theory (MQDT), 161, 203, 206, 210. 212, 216 analysis, 207-21 1 Multichannel threshold structures, theory of. 96 Multiphoton excitation, 166, 174, 200 Multiphoton ionization, vi, 157-217

N NAR, see Nonadiabatic resonance Natural frame parameterisation, 273, 28 I, 284 Near-Hartree-Fock target wave function, 95 Near-threshold adiabatic nucleus, breakdown of, 82 approximations. 115 collisions, 52, 74, 78, 82, 104, 143 cross sections, 53, 54. 88-115. 123, 127, 135, 143 electron molecule scattering, vi, 51 144 excitation, 88, 123 excitation cross sections, 131 expansions. theory of, 106 poles, 95, 97, 108, 109 ro-vibrational excitation, 81 scattering, vi, 51- 144 spikes, 134, 138. 140, 143 structures, 106, 122 Near-zero pole, 97, 109, 110, 142 bound-state, 109 Neutral carrier gas, flow dynamics of, 19-22 NMR, 237. 245 Nonadiabatic collision process. 123. 126 Nonadiabatic coupling theory, 135 Nonadiabatic excitation cross sections, 124 Nonadiabatic resonance (NAR) equation. 122 theory, 86, 121 124, 116, 128, 139, 140, 142 Nonadiabatic scattering theories. 119- 127

Nondissociating negative-ion states theory, 136, 137 Nonpolar electron molecule systems, 94, 98, 101 105, 136, 143 Nonpolar molecule, 89, 94, 95 Nonpolar systems. virtual statc mechanisms in. 141-143 Nonresonant multiphoton ionization, 173. 175, 176, 195-201, 212 Nonspherical potential, 101, 104 Nuclear-excited Feshbach resonance, 93 Nuclear magnetic resonance (NMR) techniques, 237, 245 Nuclear polarization, 241, 242 Nuclear relaxation operator, 128, 129 Nuclear spins, 173, 226, 231, 233, 242, 245. 246, 247, 248, 250, 255, 261 relaxation, 251, 255 Nuclear vibrations, 93, 140 Nuclear wave functions, 58

0 Ochkur amplitude, 305 Off-shell adiabatic theory, 128 130 Off-shell adiabatic T matrix, 129, 130 Off-shell approximations, 128 Off-shell fixed nuclei matrix, 127 Off-shell T-matrix methods, 127- 130 One-electron Rydberg series, 16 1 One-photon excitation, 176, 183 189, 194 One-photon ionization, 158, 159, 162, 166, 171, 172, 194, 216 On-shell adiabatic-nuclear-rotation cross sections, 129 Optical excitation, 174- 176 Optical potential, 123, 140 Optical pumping, vi, 223-263 Orbital angular momentum, 59, 62, 79, 291 Orthonormal electron spin functions, 167 Outer-shell ionization, 270, 300 ~

P PA, of molecules, 32 Paramagnetic Faraday rotation. 235, 236 Parameterized model interaction potential, 106

330

INDEX

Parity-nonconserving (PNC) neutral-current interation, 261 Partial-wave phase shift(s), 100, 101 Pauli matrices, 179 PCI, postcollision interaction Penning ionization, 238, 242, 264 Perburbation amplitude, 255 theory, 162, 244, 250, 256, 259 Phase shift(s), 89 Photoelectron angular correlation, 177, 189, 201 angular distributions, 160-217 theory of, vi, 161-174 anisotropy, 203 detection, 192 spin, 168 spin polarization, 173 Photoionization amplitude, 167 angular distributions, 164, 172 cross section, 165, 188 processes, atomic, 172 theory, 161 transition amplitudes, 173 Photon absorption cross section, 233, 234, 244 Photon angular distribution, 274 Photon polarizations, 162, 172 Photon resonant excitation, 172 Planetary atmospheres, 3, 45 Plane wave Born approximation, 3 10 Planewave factors, 62 Plane wave impulse approximation (PWIA), 294,296, 297, 298,299, 300, 301 Planewave representation, 60, 65 PNC interaction, 261 Poiseuille flow, 19 Poisson statistics, 192 Polarization correlations, 269-273, 277-279, 284-286,289 Polarization density matrices, 172 Polarization potentials, 75, 90, 112, 113 Polarized atoms, 233, 238 Polar-molecular scattering, 116 Polar-systems, 94, 98, 106-108, 118, 132-143, 150 Polyatomic ions, 33, 38, 39 POM, 124 Population imbalances, 224, 227, 244, 259

Postcollision depolarization, 286, 287 Postcollision interaction (PCI), 306, 307, 316, 317 Potential scattering theory, 93 Proton affinities (PA), of molecules, 32 Pulsed Langmuir probe technique, 20 PWIA, see Plane wave impulse approximation

Q Quantum beats, measurements of. 176, 185, 187, 290, 291 Quantum defect theory, 104, 126, 195

R Racah coefficients, 64 Radial frame transformation method, 71, 116 Radial scattering functions, 63, 65, 105, 117 Radiation trapping, 228, 237, 243, 253, 254, 275 Raman-active symmetric stretch mode, 142 Raman-active vibrational-excitation cross sections, 143 Raman-active vibrational modes, 131, 142 Raman coupling, 213, 215, 216, 217 Raman process, 216 Ramsauer minimum, 110-1 15 Ramsauer-Townsend (RT) minima, 110- I15 Rapidly varying (RV) term, 124 Rate coefficients, determination of, 18-22 Reactant gas Bow rate, 4, 22, 37 Recoil peaks, 295, 297, 300, 305, 306 Relaxation, vi, 243-254 rates, 243, 244, 251-255 Repopulation pumping, 228, 229, 253, 254 Reonant multiphoton ionization, 175, 176, 212 RFT, see Rotational frame transformation Ricatti-Bessel function, 61 Rigid rotator (RR), 68 approximation, 77, 78, 84, 119, 129, 130 model for rotation, 83 R-matrix theory, 126, 135 Rotational angular momentum, 56, 59, 62, 240, 246, 250

33 1

INDEX Rotational excitations. 39, 54, 74, 81, 90, 105, 116, 129, 130, 150 cross section(s), 77, 119 Rotational frame transformation(s) (RFT), 69, 70-73. 76, 78, 79, 120, 121 formulation, 108 Rotational threshold, 118, I19 Ro-vibrational cross sections, 75. 76, 106, 117, 118 Ro-vibrational excitation, 74, 81. 120 Ro-vibrational frame transformation. 74 RT minima, 110- 1 1 5 Rutherford scattering cross section, 294 RV term. 124 Rydberg electron wave function, 195 Rydberg series, 201-207

S

Scaled adiabatic-nuclear-rotation (SANR) method, 117, I19 Scattered Auger electron coincidence method, 316 Scattered-ejected electron angular correlations, 313 Scattered-electron- Auger-electron angular correlations, 312 Scattered electron- Auger-electron-X-ray angular correlation theory, 3 I2 Scattered electron-cascade photon angular correlations, 288 Scattering amplitude, 60, 61, 281. 303 functions, 69, 75, 116 theory, 51-144 Schrodinger equations, 56, 57, 58, 63, 66, 76. 121. 163 Second Born amplitude, 303 Second Born approximation, 304 Second-order perturbation theory, 256 Selected flow drift tube (SIFDT), 34-40 apparatus(es), 16, 35, 39 Selected ion Row tube (SIFT), vi, 1-45 apparatus(es), 4, 10, 14, 16, 19, 20, 21, 23, 44,45 detection system, 15 studies of ion neutral reactions, I 45 technique, 3-22 Seven-photon ionization, 180- 196

Simple-harmonic-oscillator model for vibrations, 83 Single-channel scattering, 89, 97 Single-photon photoionization, 158, 162 Singlet scattering processes, 28 I , 287 Six-photon ionization, 180, 196 -198 Spherical harmonics, 203, 204 addition theorem, 163 expansion(s), 171. 203 Spherically-symmetric potential, 89 Spin exchange, vi, 223-263 collisions, 238, 239, 242, 264 cross-sections, 234. 238 -24 I Spin-Rip probability, 248, 250 Spin Hamiltonian, 231, 248 Spin-orbit coupling, 166, 167, 180, 200, 207, 284, 285 Spin-orbit interaction, 246 Spin polarizations, 172, 173, 178- 180, 227. 230, 234, 244 Spin-polarized atoms, vi, 229-233, 243 Spin-polarized electrons, 187, 264 Spin-polarized fusion, 264, 265 Spin relaxation, 224, 248 cross sections, 239 function, 250, 251 rate, 225, 250. 255 Spin-rotation interactions, 240, 248 251 Spin-spin interaction, 246, 247 Spin transfer, 248. 249 coefficients, 249, 250 collisions, vi, 224, 238 242 Stationary scattering states, 57. 58, 71 Stoke’s parameters, 269, 273, 289 Sturmian function(s), 200 Sturm Liouville equation, 22 Swarm data, analysis of, 53, 1 13 Symmetric-stretch vibrational mode, 85. 86. 124, 141 Symmetrization Postulate, 56

T Target-state degeneracy, 80-86. 116, 127, 129 Target-state quantum numbers. 78 Ternary association rate coefficients, 28, 32, 42 Ternary ionic reactions, rate coefficients for, 23,42

332

INDEX

Ternary ion-molecule association reactions, 32, 33, 42, 45 Thermal energy ion-molecule interactions, 45 Three-body collisions, 245. 248 Three-body formation rate, 250 Three-body relaxations, 248-25 1 rates, 250, 251 Three-photon ionization, 171, 189-195, 200, 204, 206, 212, 216 resonant ionization, 190, 194 Three-photon resonant excitation, 192-194, 204, 205,210, 216 Threshold anomalies, 92, 93 expansions, 100-1 15 ionization, 196, 306-3 LO laws, 88-91, 94, 97, 119, 121 spikes, 131 143 structures, 96, 131-144 Time evolution of atomic states, 289-292 Time-dependent perturbation theory, 162, sir also Perturbation, theory Time-of-flight electron spectroscopy, 212 Time-of-flight measurements, 101, 11I, 180, 208 T-matrix methods, off-shell, 127-130 Townsend minimum, 107, 1 10, I 1 1 Transmission monitoring, 234- 236, 237 Transport analysis, 54, 113 Triple-differential cross sections, 292- 3 I2 Triplet scattering processes, 28 I , 287 Two-photon ionization, 160, 171- 174, 185, 188, 189, 194, 210 Two-photon resonance, 189. 212,213 Two-photon resonant excitation, 192 194, 207 Two-photon resonant one-photon ionization, 194 Two-photon resonant sequential excitation. 171 Two-photon resonant three-photon ionization, 213 Two-photon sequential excitation, 189 192 Two-state body-frame vibrational closecoupling theory, 85

U Uncoupled angular moment urn representation, 59-63, 70

V Variable-temperature selected ion flow drift tube (VT-SIFDT), vi, 3, 16,33-45 Variable-temperature selected ion flow tube (VT-SIFT), 3, 12, 15, 16, 22, 30-35,45 Vibrational excitations, 38-45, 54, 80, 81, 84, 85, 87,93, 106, 119, 120-128, 131, 133, 137, 141, 150 cross section(s), 54, 55, 85, 86, 95, 106, 121, 122, 126, 131-143 Vibrational frame transformations (VFT), 70-74, 78, 91, 126 Vibrational kinetic energy, 68, 123 Vibrational quantum numbers, 68, 70. 120 Vibrational wave functions, 69 Virtual-state poles, 96, 97, 100, 109, 124, 135, 137, 142, 143 Virtual-state theory, 135-141

W Wall costings, 251, 252 Wall collisions, 243, 251, 252 Wall depolarization, 251 rates, 251, 252 Wall relaxation(s), 252 Wannier-Peterkop-Rau theory, 308 Wave functions, vibrational, 69 Weakly bound cluster ions, 2, 30, 45 Weakly bound electrons, 200, 201 Weak-scattering approximations, 116 Wigner rotation matrix, 70, 71

Z

Zeeman interaction, 249 Zeeman resonances, 258 Zero point energies, 23, 3 1 Zero-range potential (ZRP) approximation, 136, 138 method. 121