Advances in Atomic, Molecular, and Optical Physics, Volume 34

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 34

Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Planck-hstitut fur Quantenoptik Garching bei Munchen Germany

Editorial Board P. R. BERMAN New York University New York, New York K. DOLDER The University of Newcastle-upon-Tyne Ne wcastle-upon-Tyne England M. GAVRILA F. 0.M. Instituut voor Atoom- en Molecuulfysica Amsterdam The Netherlands M. INOKUTI Argonne National Laboratory Argonne. Illinois S . J. SMITH Joint Institute for Laboratory Astrophysics Boulder, Colorado

Founding Editor SIR DAVIDBATES

Supplements 1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

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

Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANCK-INSTITUT FUR QUANTENOPTIK GERMANY

Volume 34

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ACADEMIC PRESS

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This book is printed on acid-free paper.

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Copyright 0 1994 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.

Academic Press, Inc. A Division of Harcourt Brace & Company

525 B Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NWI 7DX International Standard Serial Number: 1049-25OX International Standard Book Number: 0- 12-003834-X PRINTED IN THE UNITED STATES OF AMERICA 94 95 9 6 9 7 98 9 9 B B 9 8 7 6

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3 2 1

IN MEMORIUM-PROFESSOR SIR DAVID R. BATES 1916-1994 We are greatly saddened to hear that Sir David passed away on January 5, 1994. Sir David was the founding editor of the series and served as coeditor until his retirement in 1993 with Volume 31. Volume 32 was a special volume published in his honor. Sir David was a Fellow of the Royal Society, and was knighted for service to science in 1978.

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Contents

xi

CONTRIBUTORS

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xi11

PREFACE

Atom Interferometry C. S. Adams, 0. Carnal, and J. Mlynek I. 11. 111. IV. V. V1.

Introduction General Principles Beam Splitters Applications of Atom Interferometry Atom Interferometers Outlook References

1 3 12 14 19 30 31

Optical Tests of Quantum Mechanics R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg I. 11. 111. IV. V. VI. VII. VIII. IX.

Introduction: The Planck-Einstein Light-Quantum Hypothesis Quantum Properties of Light Nonclassical Interference and “Collapse” Complementarity The EPR “Paradox” and Bell’s Inequalities Related Issues The Reality of the Wave Function The Single-Photon Tunneling Time Envoi References

36 38 42 47 51 56 61 69 76 80

Classical and Quantum Chaos in Atomic Systems Dominique Delande and Andreus Buchleitner I. Introduction 11. Time Scales-Energy Scales 111. Spectroscopy IV. Wavefunctions: Localization and Scars

vii

85 94 97 109

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Contents

Vlll

V. Dynamics VI. Conclusion References

115 121 121

Measurements of Collisions between Laser-Cooled Atoms Thad Walker and Paul Feng I. Introduction 11. Collisions in Optical Traps: General Considerations 111. Collisions of Ground State Atoms

IV. Collisions Involving Singly Excited States V. Collisions Involving Doubly Excited States References

125 128 136 142 161 169

The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas J. E. Lawler and D. A. Doughty I. Introduction 11. Theory of the Stark Effect

Ill. Electric Field Mapping Based on the Stark Effect in Atoms IV. Electric Field Mapping Based on the Stark Effect in Molecules V. Conclusion References

171 173 179 196 204 205

Polarization and Orientation Phenomena in Photoionization of Molecules N. A. Cherepkov I. Introduction 11. Spin Polarization of Photoelectrons Ejected from Unoriented Molecules

III. IV. V. VI.

Photoionization of Oriented Molecules Circular and Linear Dichroism in the Angular Distribution of Photoelectrons Optical Activity of Oriented Molecules Conclusions References

207 209 222 228 243 245 246

Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss E. C. Montenegro, W. E. Meyerhof; and J. H. McGuire I. Introduction 11. Theory 111. Comparison with Experiment

IV. Conclusion References

250 254 280 295 297

Contents

ix

Indirect Processes in Electron Impact Ionization of Positive Ions D. L. Moores and K. J. Reed 1. Introduction 11. Basic Ideas: The Independent Processes Model

III. Theory 1V. Comparison of Theoretical and Experimental Data V. Conclusions References

301 305 311 324 421 422

Dissociative Recombination: Crossing and Tunneling Modes David R Bates I. Introduction 11. Upper Limit to Rate Coefficient 111. Crossing Dissociative Recombination IV. Tunneling Dissociative Recombination V. Signature of Polyatomic Ion Dissociative Recombination References INDEX CONTENTS OF VOLUMES IN THIS SERIAL

427 433 434 46 1 419 48 1

487 499

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Contributors

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

C. S. ADAMS (l), Fakultat fur Physik, Universitat Konstanz, 78434 Konstanz, Germany DAVIDR. BATES (427), Department of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast, BT7 lNN, Northern Ireland

ANDREASBUCHLEITNER (85), Laboratoire de Spectroscopie Hertzienne de l'Ecole Normale, Universite Pierre et Marie Curie, F-75252 Paris Cedex 05, France 0.CARNAL (l),Fakultat fur Physik, Universitat Konstanz, 78434 Konstanz, Germany N. A. CHEREPKOV (207), State Academy of Aerospace Instrumentation, Petersburg, Russia

R. Y. CHIAO(35), Department of Physics, University of California, Berkeley, California 94720 DOMINIQUE DELANDE(85), Laboratoire de Spectroscopie Hertzienne de l'Ecole Normale, Universite Pierre et Marie Curie, F-75252 Paris Cedex 05, France D. A. DOUGHTY(171), Corporate Research and Development Center, General Electric Company, Schenectady, New York 12301 PAUL FENG(129, Department of Physics, University of WisconsinMadison, Madison, Wisconsin 53706 P. G. KWIAT( 3 9 , Department of Physics, University of California, Berkeley, California 94720

J. E. LAWLER(171), Department of Physics, University of WisconsinMadison, Madison, Wisconsin 53706 J. H. MCGUIRE(249), Department of Physics, Tulane University, New Orleans, Louisiana 701 17

xi

xii

Contributors

W. E. MEYERHOF (249), Department of Physics, Stanford University, Stanford, California 94305 J. MLYNEK (l), Fakultat fur Physik, Universitat Konstanz, 78434 Konstanz, Germany

E. C. MONTENEGRO (249), Departmento de Fisica, Pontificia Universidade Catolica do Rio de Janeiro, Rio de Janeiro 22453, Brazil D. L. MOORES(301), Department of Physics and Astronomy, University College London, London WC 1E 6BT, United Kingdom K. J. REED(301), High Temperature Physics Division, Lawrence Livermoore National Laboratory, Livermore, California 94550

A. M. STEINBERG (359, Department of Physics, University of California, Berkeley, California 94720 THADWALKER(129, Department of Physics, University of WisconsinMadison, Madison, Wisconsin 53706

Preface

Sir David R. Bates served as Senior Editor of Advances in Atomic, Moleculan, and Optical Physics since the series’ inception in 1965. His original co-editor was Immanuel Estermann, followed by myself with Volume 10 in 1974. Sir David retired from his editorship with Volume 31 in 1993. Volume 32, published in 1994, was a special volume honoring Sir David, co-edited by Alexander Dalgarno and myself. Sir David passed away in January, 1994. Subsequent to Sir David’s retirement, Prof. Dr. Herbert Walther, Director of the Max-Planck-Institut fur Quantenoptik at Garching and Professor of Physics at the University of Munich, has joined me as co-editor. I am very happy to welcome him to this series. He brings to it vast experience in many aspects of atomic, molecular, and optical physics, and wil1 certainly help to broaden its scope, especially in the many rapidly developing areas of modern atomic physics and quantum optics. I look forward to his participation in this important series, and with particular pleasure to working with him to ensure that our Advances volumes continue to occupy respected positions in the scientific literature. Benjamin Bederson

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 34

ATOM INTERFEROMETRY C. S. ADAMS, 0. CARNAL,‘ and J. MLYNEK Fakultat fur Physik, Universitat Konstanz, Konstanz, Germany

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Phase Evolution of an Atomic Wave . . . . . . . . . . . . . . . . B. Atomic Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Spontaneous Emission and Coherence . . 111. Beam Splitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. A. Microfabricated Structures . . . B. Photon Recoil . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Applications of Atom Interferometry . . . . . . . . . . . . . . . . . . A. Inertial and Gravitational Effects . . . . . . . . . . . . . . . . . . B. Tests of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . C. Properties of Atoms . . . . . . . . . . . . . . . . . . . . . . . . V. Atom Interferometers . . . . . . . . . . . . . . . . . . . . . . . . A. Young’s Double Slit . . . . . . . . . . . . . . . . . . . . . . . B. Three-Grating Interferometer . . . . . . . . . . . . . . . . . . . . C. Optical Ramsey Interferometer . . . . . . . . . . . . . . . . . . . D. Atom Interferometry Using Stimulated Raman Transitions . . . . . . . E. Atom Interferometry Using Static Electric and Magnetic Fields . . . . . VLOutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Introduction . .

. .

. . .

11. General Principles . .

,

,

,

,

1 3 4

6 8 12 12 13 14 14 18 19 19 19

21 22 26 29 30 31 31

I. Introduction The essential character of a wave is the property of interference. The wave nature of light was established by Young’s demonstration of interference using a double slit in 1802. Interferometry with light soon developed into an important tool for precision measurement. Soon after de Broglie’s suggestion in 1925 that massive particles can behave as waves, diffraction of both electrons (Davisson and Germer, 1927; Thomson, 1927) and atoms (Estermann and Stern, 1930) was demonstrated. However, some time passed before interference between spatially separated particle beams was observed. In 1954 Marton et al. demonstrated the first massive particle interferometer by diffraction of electrons from thin metal films. A similar three-grating interferometer for neutrons based on Bragg diffraction from crystals was realized by Rauch et al. (1974). Current address: Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California 91 125.

1

Copyright 0 1994 by Academic Press,Inc.

All rights of reproduction in any form reserved. ISBN 0-12-003834-X

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C. S. Adams, 0. Carnal, and J. Mlynek

Progress on interferometry with atoms was hindered by the short de Broglie wavelength and because, unlike neutrons, atoms do not penetrate through matter. However, recently an explosion of activity has occurred in atom interferometry that was stimulated by two main developments: First, progress in microfabrication technology now permits the production of structures sufficiently fine to diffract thermal atomic beams to significant angles. Second, the development of intense tunable lasers allowed rapid progress in techniques to manipulate the trajectories of neutral atoms by means of light forces. In 1991 atom interferometers based on diffraction from microfabricated structures were reported by Carnal and Mlynek (1991) and Keith et al. (1991). Not long after, Riehle et al. (1991) applied the technique of optical Ramsey excitation to demonstrate the Sagnac effect for atoms, and Kasevich and Chu (1991) reported a high sensitivity to gravitational fields using the Ramsey technique in an atomic fountain. Atomic interference has also been observed by nonadiabatic passage of an atomic beam through separated regions with static electric (Sokolov and Yakolov, 1982) and magnetic (Robert et al., 1991) field gradients. The considerable potential of atom interferometry for precision measurement of gravitational or inertial effects (Kasevich and Chu, 1992) and atomic properties (Weiss et al., 1993a) has been demonstrated. In addition, the first experiments to exploit the spatial separation of an atomic wave function have been reported (Ekstrom, 1993). The rapid development of atom interferometry raises the question of the advantages of atoms compared to electrons or neutrons. Precision measurement with electron interferometry is difficult due to stray fields and long-range Coulomb interactions with the walls of the vacuum chamber. However, atoms are neutral and therefore much less sensitive to perturbations by static electromagnetic fields. Whereas neutron interferometry requires a particle accelerator or nuclear reactor, atomic beams are relatively easy to produce. There are many species of atoms with a considerable range of properties and masses. Atoms have a complex internal structure that can be probed and modified by means of resonant laser light or static electromagnetic fields. The internal structure allows precision measurement by spectroscopic techniques. The interaction between excited atoms and the vacuum field induces spontaneous relaxation. This is a dissipative process that permits cooling of the atomic system. For atom interferometry, laser cooling is useful for the preparation of slow atomic sources. As demonstrated by Weiss et al. (1993a), a Ramsey-type interferometry based on slow atoms in an atomic fountain has a sensitivity a million times higher than that of a conventional atomic beam. In the interferometer itself, spontaneous emission is undesirable because it degrades the atomic coherence. This chapter is organized as follows: In Section I1 we present an elementary treatment of the underlying principles of atom interferometry. The phase evolution of an atomic de Broglie wave and the coherence of an

ATOM INTERFEROMETRY

3

atomic beam are discussed. A beam splitter is a key component in an interferometer. The main techniques used to split an atomic beam coherently are considered in Section 111. In Section IV we discuss possible applications of atom interferometry. The second part of the chapter concentrates on existing experiments in atom interferometry. Atom interferometers that have been realized are described, and experiments that use these devices are discussed. The chapter concludes with a summary and outlook. Other reviews of atom interferometry may be found in Levy (1991), Helmcke et ai. (1992), and Pritchard (1993).

11. General Principles In quantum mechanics a wave function lY) is introduced to describe the statistical properties of an ensemble of similarly prepared systems. The wave function contains information about the internal state and the external motion of the atomic ensemble. For a discrete spectrum of internal states, a pure state may be written as

where Ic,) and In) are the center of mass and the internal wave function of state n. Two classes of interference may be distinguished: interference between different components of the center-of-mass wave function (scalar interference) and interference between different internal states (spinor interference). An example of spinor interference is the Ramsey technique of separated oscillatory fields (Ramsey, 1956): The first interaction generates a superposition of internal states that follow different paths in Hilbert space and interfere in the second interaction. Knowledge of the internal state between the two interaction regions destroys the interference, analogous to “which path” information in a double-slit experiment (see Section 1I.C). However, whereas Ramsey fringes may be interpreted classically as the precession of a dipole moment or spin, interference between spatially separated paths has no classical analogue. For optical Ramsey excitation (see Section V.C), the distinction between scalar and spinor interference is less clear cut, because the larger photon recoil can result in spatial separation of the paths. This has led to some debate over what qualifies as an atom interferometer, particularly because, at least until now, no experiment based on the optical Ramsey technique has exploited the spatial separation. The wave function is characterized by an amplitude and a phase. Whereas

4

C. S. Adams, 0. Carnal, and J. Mlynek

most experiments measure amplitudes, interferometry measures phase. Two factors determine the sensitivity of an interferometer, the magnitude of the induced phase shift and the accuracy of the phase measurement. For some applications, for example, an atom gyroscope (Section IV.A), the sensitivity can be increased by increasing the area of the interferometer. The detection of atoms follows Poissonian statistics; therefore, the phase uncertainty for a measurement time t is 1

sq=F where F is the atomic flux. Compared to conventional optics, atomic fluxes are relatively low. Consequently, the sensitivity of an atom interferometer is often limited by counting statistics.

A. PHASE EVOLUTION OF AN ATOMIC WAVE The evolution of the wave function is described by the Schrodinger equation:

where H = T f I! T is the kinetic energy operator, and V represents an external perturbation. In an interferometer, I/ may represent the beamsplitting process or a selective interaction with one arm. Processes that do not conserve the atomic flux, for example, diffraction from a slit, cannot be represented by a Hermitian operator. In this case, the interaction with the slit may be thought of as a state preparation, in which the state subsequently evolves according to the Schrodinger equation. It is often convenient to express the internal states in terms of the interaction eigenstates. In this basis, the potential energy operator is diagonal but the kinetic energy operator may contain nondiagonal elements that give rise to transitions between the eigenstates. However, if the time dependence of the external perturbation is slow compared to the characteristic time scale for the internal evolution, the off-diagonal terms may be neglected. This is commonly referred to as the adiabatic approximation. In this case, the vector Schrodinger equation decouples into independent scalar equations for each eigenstate. In the following we limit the discussion to one internal state. An elegant approach to the phase evolution in an atom interferometer is provided by the path-integral approach to quantum mechanics (Feynman

5

ATOM INTERFEROMETRY

and Gibbs, 1965). The path-integral wave function is

where L[r(t)] is the Lagrangian of the atomic system and r is the path. The plane wave limit of (4)is

[ jr -

$(r, t) = exp i

(k dr - E dt/A)

1

(5)

where E is the sum of kinetic and potential energy and the k-vector of the atomic beam for a static potential V(r) is defined by the dispersion relation k(r) =

/=

If the internal state evolves adiabatically, E is a constant. However, potential may induce transitions to other internal states. In this case, internal energy changes with time, and it is important to keep track of phase evolution due to the internal energy. The phase shift produced by a static potential V(x) relative to unperturbed plane wave is

the the the the

where kg = 2mE/hZ.Note that Acp depends on E, that is, the phase shift is dispersive. In contrast, a purely time-dependent potential V ( t ) causes a phase shift

that is nondispersive. This is known as the scalar Aharonov-Bohm effect

(Aharonov and Bohm, 1959). A time-dependent potential changes only the phase velocity of the de Broglie wave. Thus a combination of purely time- dependent and spatiallydependent potentials can be used to shift the envelope of an atomic wave packet without disturbing the phase. This is similar to the principle of the Wien filter used in electron optics,except that in this case a combination of scalar and vector potentials is used (see, e.g., Nicklaus and Hasselbach, 1993).

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C. S. Adams, 0. Carnal, and J. Mlynek

Returning to the static potential, if V ( x ) << E the longitudinal velocity u is approximately constant2 and the spatial dependence of the potential may be replaced by the explicit time dependencet = x/v. In this case, Eq. (7) reduces to

Thus the phase shift is often interpreted as an energy shift V lasting a time at. The dispersive property of the static potential now appears through the different interaction times for different velocity components.

B. ATOMICCOHERENCE The preceding discussion was based on a plane wave treatment. However, in practice, the atomic beam will have a finite velocity distribution. This more general case can be characterized by a state operator

where 1akI2represents the velocity distribution of the beam and Ik) denotes the basis eigenvectors. If the source emits a constant atomic flux, p = 0, then CP9HI

=0

(1 1)

and p and H share a complete set of common eigenvectors.The eigenstate of the free-space Hamiltonian H in position representation are plane waves ( x l k ) = eikx.If the plane waves or momentum eigenstates have a well-defined phase relationship, the atomicensemble may be described by a wave packet. A continuous beam experiment cannot distinguish between a coherent superposition of plane waves (wave packet) and an incoherent superposition as in Eq. (10). Therefore, we are free to choose whether we use the more general state operator or the wave packet description. Bear in mind, however, that the wave packet picture is based on an assumption that has no justification. In position representation, the state operator (10) becomes p(x19x2) = (X,lPlX,> P

=

J dklak12exp[ik(xl - x2)]

'The assumption of constant longitudinal velocity is analogous to the paraxial approximation in classical optics.

7

ATOM INTERFEROMETRY

In an interference experiment, the atomic beam is split and recombined. The final state may be written as

If)

= 1x1)

+ 1x2)

(13)

where x1 and x2 are the final positions of the two components. The interference pattern is given by

(f I P l f ) For x1 = x - I, x2 = x tribution

+ I,

(14)

and a one-dimensional Gaussian velocity dis-

the cross term is p(x - I, x

+ I ) = exp( - oil2)exp(i2kJ)

(16)

The same result may be derived from the wave packet description. To illustrate the physical significance of the velocity distribution ha,/m, consider the double-slit interferometer shown in Fig. 1. The double slit is illuminated by a well-collimated atomic beam with a finite longitudinal velocity distribution. For convenience we assume that the properties of the atomic beam may be modeled by a wave packet. On axis, the paths are

I

-

FIG.1. Double-slit interferometer illuminated by an atomic beam with a finite longitudinal velocity distribution. The atomic beam is described by a wave packet. The imperfect overlap of the wave packets leads to a decrease of the fringe visibility off axis.

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C.S. Adams, 0. Carnal, and J. Mlynek

equal and perfect overlap exists between the wave packets coming from each slit. At a position y in the detector plane, the wave packets are displaced by f 1 = d/2Ly. The atomic intensity as a function of position in the detector plane follows from (14) and (16):

The only measurable quantities are the spacing and the visibility of the interference fringes. The visibility as a function of the displacement between the wave packets x is ~ ( x= ) exp( - azx2)

(18)

The visibility defines the longitudinal coherence length of the atomic beam 1/ok. Note that, whereas the width of the wave packet expands continuously with time due to dispersion, the coherence length is determined by the state preparation and is time independent. Interference is only observed if the center-of-mass separation of the wave packets 21 is less than the coherence length l/ok.Spatial overlap arising from dispersion alone is not a sufficient condition to observe interference. Dispersion arises due to the unitary time-evolution operator U ( t ) = e-iHt/h,which does not change the interference term

A discussion of these points followed the first interferometric measurement of the coherence length of a neutron beam (Kaiser et al., 1983; Klein et al., 1983). C. SPONTANEOUS EMISSION AND COHERENCE

In this section, we consider the influence of spontaneous emission on the external atomic wave function. This is a useful example to illustrate some points about coherence and “which-path” information in atom interferometry. In addition, for interferometers based on light-atom interactions (see Section V.C), it is useful to know the influence of spontaneous emission on the interference pattern.

9

ATOM INTERFEROMETRY

On spontaneous decay the center-of-mass wave function acquires a phase shift, which depends on the direction of the emitted photon, that is,

I + i ) 0 I e> +

eikL.

l+i>

0 IS>

(20)

where is the center-of-mass wave function of the atom before spontaneous emission; g and e denote the ground and excited state respectively; r is the atomic position; and k, is the wave vector of the emitted photon. The center-of-mass wave function and the state of the emitted photon form an entangled state (see also Section IV). To illustrate the effect of this pause shift, consider the specific example of a double-slit interferometer (Sleator et al., 1991; 1992). We assume that the interferometer is illuminated by a plane atomic wave and that every atom decays immediately after the double slit as shown in Fig. 2. If we select atoms that emit a photon in a specific direction 0, then for a slit separation d, the wave function after spontaneous emission can be written

t j = '[exp(

-i-+cosB)+,. kd

+ exp(iycos0)+2]

(21)

Jz where 1 and 2 refer to an external wave function for the two paths. Note that a transverse recoil produces a longitudinal phase shift. Following the same procedure as in Section II.B, the intensity distribution for those atoms that emit in a direction 0 is

I

L-,,L-,-+-l FIG.2. Effect of spontaneous emission in a double-slit interferometer. The interference pattern without spontaneous emission is shown dashed. By selecting atoms, which emit a photon in a direction 0, an identical interference pattern is produced but displaced by dcosO/I times the fringe separation.

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C. S. Adams, 0. Carnal, and J. Mlynek

where z is the position on the screen, k denotes the wave vector of the atomic beam, and L is the distance between the screen and the double slit. An interference pattern with perfect visibility is observed. The only effect of the spontaneous emission is to displace the fringe pattern by d cos 811 times the fringe separation. If we repeat the experiment but select a different photon direction, we again see a perfect interference pattern, but now the fringes are shifted relative to our previous result due to the 8 dependence in (22). The total interference pattern (uncorrelated to the photon direction) is given by the sum of the intensity distributions associated with each photon direction. Assuming an isotropic photon distribution, the total intensity is

COS k (a]);

The sum is sometimes referred to as an incoherent sum (the crucial point is that intensities, not amplitudes, are summed). The interference pattern has a visibility given by a sinc function as shown in Fig. 3. If the slit separation 1

0.8

z

0.6

m

0.4

4

u, >

0.2

0

-0.2 0

1

2

3

4

SLIT SEPARATION d (units of h) FIG.3. Effects of spontaneous emission on double-slit interference pattern. The visibility of the interference fringes as a function of slit separation is shown. If the slit separation is much larger than the wavelength d >> 1, the fringes are completely smeared out. However, for d I interference is still observed.

-

ATOM INTERFEROMETRY

11

is much larger than the wavelength d >> I , the fringes are completely smeared out. However, for d I interference is still observed. At first glance, it may be surprising that the visibility changes sign. The first visibility is related to the coherence of the atomic beam (see Section 1I.B). For this reason, if the details of the atom-field correlation are neglected, it is appropriate to discuss the effects of spontaneous emission as a reduction of the degree of coherence. Spontaneous emission is frequently referred to as an incoherent process. However, the term incoherent is rather misleading. For example, spontaneous emission may be used to increase the coherence of an atomic beam using laser cooling. The influence of spontaneous emission is identical to the effect of finite slit width [in this case k, is replaced by ka/L in (23), where a is the width of the entrance or detector slit]. Spontaneous decay may be thought of as a localization of the atomic position. The relationship between the momentum and spatial pictures follows directly from the uncertainty principle, that is, the momentum diffusion of 2hk, is equivalent to a real space localization of A/2. If the decay occurs in front of the double slit, the interference pattern can be interpreted as a measurement of the spatial correlation function of an atomic wave function after spontaneous emission. The double slit essentially samples the real space wave function of the incident atomic beam. It follows from (23) that the real space wave function after spontaneous emission has the form of a sinc function. Instead of observing the direction of the emitted photon, one may measure the position. A large solid angle lens or Heisenberg microscope may be used to image the region near the double slit onto a detector. In this case, the interference pattern correlated with the photon detection is given by (23) with the integral taken over the solid angle of the lens. High spatial resolution is obtained at the expense of information about the photon direction. It follows that the fringe visibility is linked to the “which-path” information provided by the emitted photon. In principle, the position of the atom can be determined with a resolution of I/2. If d >> I , we can determine the path, and the complementarity principle demands that the interference fringes be washed out. The physical mechanism causing the loss of interference is recoil; the momentum recoil associated with any process that gives sufficient information about the position will destroy the interference. As described earlier, a transverse recoil is equivalent to a longitudinal phase shift of the external wave function. Recently Scully et al. (1991) claimed to have found a scheme that could provide which-path information without recoil, the implication being that the loss of interference is a result of complementarity rather than the uncertainty principle. Their scheme involved a double-slit interferometer with a micromaser detector placed at each slit. However, Storey et al. have shown (1993) that any position measurement necessarily induces a longitudinal phase shift and hence N

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C.S. Adams, 0. Carnal, and J. Mlynek

recoil. In conclusion, the uncertainty principle and not complementarity is more fundamental.

111. Beam Splitters Beam splitters are a key component in at atom interferometer. Coherent beam splitters for atoms can be realized using diffraction, for example, from crystalline surfaces or microfabricated structures, the interaction with static or optical electromagnetic field gradients and photon recoil. In this section we concentrate on the two main techniques used for atom interferometry, diffraction from microfabricated structures and photon recoil.

A. MICROFABRICATED STRUCTURES The diffraction of matter waves is described completely by the solution of the Schrodinger equation with the appropriate boundary conditions at the diffracting obstacle. Because matter waves and electromagnetic radiation are described by the same Helmholtz equation, the diffraction of light and atoms are essentially equivalent. Thus in the Fraunhofer limit, the atomic diffraction pattern is given by the Fourier transform of the aperture transmission function. Beam splitters based on microfabricated structures may be divided into wave front splitting (e.g., the combination of a single and a double slit as in a Young’s double-slit interferometer) or amplitude splitting (e.g., diffraction from a transmission grating). The drastic loss of atomic intensity accompanying wave front splitting techniques makes gratings more attractive. Diffraction of atoms from a transmission grating is illustrated schematically in Fig. 4. Free-standing gratings with periods as small as 200 nm have been produced (Ekstrom et al., 1992). In this case, the momentum splitting between GRATING I I

ATOMICBEAM

I

I I

FIG.4. Schematic illustration of the diffraction of atoms by a transmission grating.

ATOM INTERFEROMETRY

13

diffraction orders is larger than for optical techniques.However, for effusive or supersonic atomic sources the diffraction angles are still only around 100p a d . A disadvantage of diffraction gratings is the relatively low efficiency; the maximum intensity of the first diffraction order is only about 10% of the incoming intensity. The advantages are that they are passive elements,require little maintenance,and can be used with any atomic species or even molecules.

B. PHOTON RECOIL One important class of atom interferometers (see Sections V.C and V.D) is based on the possibility that we can split and deflect atomic beams using photon recoil. A photon carries a momentum Ak,; therefore, if a two-level atom with an initial center-of-mass momentum p absorbs a photon, the excited atom has a momentum p hk,. The recoil may also be used to split an atomic beam. A n/Zpulse (see Allen and Eberly, 1975) excites an initial state 19, p) into a superposition state

+

1 ---(lg,p) - ie-'+LIe,p

@

+ h kL))

(24)

where &. is the phase of the light field. As the excited state component carries the photon momentum, the excitation process behaves as a beam splitter (Fig. 5). Similarly, for multilevel atoms a stimulated Raman pulse using counterpropagating laser beams can excite a coherent superposition of two ground hyperfine levels I1,p) and 12,p + 2hkJ (Kasevich and Chu, 1991). The use of stimulated Raman transitions has the advantage that the split beams are both in the ground state and therefore not threatened by

XI2

puIse FIG.5 . Schematic illustration of an atomic beam splitter based on photon recoil. The atomic state is labeled by Ii,n) where i denotes the internal state and n is the transverse momentum in units of hk,. A n'2-pulse excites the initial state Ig,O) into a superposition of Ig,O) and ) e ,1>. As the excited state carries the photon momentum, the two components become spatially separated.

14

C. S. Adams, 0. Carnal, and J. Mlynek

spontaneous emission. Once the superposition state has been created, the splitting angle may be increased by applying a sequence of n-pulses (Weiss et al., 1993).

IV. Applications of Atom Interferometry Atom interferometry is a relatively young field, so it is interesting to speculate about possible applications. Three categories may be distinguished: measurement of inertial and gravitational effects, tests of quantum mechanics, and measurements of the properties of atoms. In this section we consider the potential capabilities of atom interferometry and make comparisons with existing technology.

A. INERTIAL AND GRAVITATIONAL EFFECTS I . Gravity In a matter wave interferometer, the interference fringes fall due to gravity. For example, consider a double-slit interferometer oriented vertically as shown in Fig. 6.3 In the detector plane, the fringes fall by a distance

where u is the particle’s longitudinal velocity and u is the initial vertical velocity required for the trajectories to pass through the same point in the double slit as with g = 0. The displacement of the fringes may be written as a phase shift Acp = 2nh/D, where D = Ad,L/d is the fringe spacing. Thus the gravitational phase shift is Acp=-

2ngA

where A = Ld is the area of the interferometer. Young’s double-slit interferometer is a special case, in that, the splitting between the arms does not depend on the atomic momentum. For interferometers based on diffraction or photon recoil, area A is approximately A,,L2/a, where a is the grating

’A similar discussion based on a bi-prism can be found in Pritchard (1993).

ATOM INTERFEROMETRY

15

k----L---k---L---+ FIG.6. Illustration of the effect of gravity in an atom interferometer. The fringes fall by a distance gLz/u2,where u is the atomic velocity. The dashed line shows the interference fringes with g = 0.

period or the optical wavelength, respectively. In this case, (26) becomes

and the phase shift is independent of the mass of the test particle. The same results are obtained by adding a potential term mgh to the Schrodinger equation and integrating the phase shift for the two paths. The wavelength of the particle beam and the velocity are related via the de Broglie relation, which involves the constant A. The appearance of g and A in the phase shift (26) led some authors to describe the effect as grauity-induced quantum interference. However, because quantum mechanics is only invoked to relate particle velocity to a wavelength, the effect of gravity in matter wave interferometry is no more surprising or significant than the role of any other force. The shift of the interference fringes due to gravity was first observed in a neutron interferometer by Colella et al. (1975) (commonly referred to as the COW experiment). A sensitivity of lo-’ g was reported. Compared to neutrons, atom interferometry offers a higher sensitivity to gravity due to the possibility of precise velocity measurements using the internal structure of the atom. As a result, although a COW-type experiment with atoms is feasible, a higher sensitivity may be achieved using a different approach. The idea is to measure the rate of change of the Doppler shift of a falling atom

C. S. Adams, 0. Carnal, and J. Mlynek

16

(Kasevich and Chu, 1991). For a fall time t, the Doppler shift is +kLgt2, where k, is the k-vector of a laser tuned to the atomic resonance. The resolution of the measurement is limited by the linewidth of the transition. For a linewidth of 1 Hz, a fall time of 1s (Doppler shift 100MHz for an optical transition) and assuming an atomic flux of 108/s, a sensitivity of 10- l 2 g is expected. A number of technical difficulties must be solved in order to achieve this level of accuracy: First, any velocity measurement is made relative to some fixed point; therefore, high sensitivity requires an inertially stabilized reference. Second, if the initial velocity of the atom is not well defined, the resolution is limited to the Doppler-broadened linewidth; even for lasercooled atoms the Doppler broadening is > lo3Hz. The solution is to use an interferometrictechnique that measures a velocity change and is insensitive to the absolute velocity (see Sections V.C and V.D). Finally, a l-Hz measurement requires the laser to be stable < 1 Hz for a period longer than the measurement time. An elegant way around this problem is to use stimulated Raman transitions (Kasevich et al., 1991).In this case, only the difference between two optical transitions must be stable, which is much easier to achieve experimentally (see Section V.D). An interferometer based on stimulated Raman transition with a sensitivity of 3 x lo-' g has been demonstrated by Kasevich and Chu (1992) (see Section V.D). Improvements in the experiment should lead to a sensitivity of lO-'Og/J, z where t is the measurement time. Precision measurement of gravity has a wide range of commercial and fundamental applications, for example, geological surveying, measurement of seismic activity, prediction of earthquakes, and monitoring the mean level of the oceans. Fundamental applications include tests of the weak equivalence principle or local Lorentz invariance. Local fluctuations in gravity are dominated by the tides and have an amplitude of 2 x 10-7g (Goodkind, 1988). The next largest component arises from changes in atmospheric pressure (3 x 10-l0g/rnbar). Both of these effects have a characteristic frequency spectrum and can be easily subtracted from the signal. The residual gravity signal has fluctuations of ~ l O - ~and g is sensitive to vertical motion of the earth's crust (3 x lO-'g per centimeter of elevation), the local distribution of mass (even a small physicist at a distance of l m produces a gravitational shift of 5 x lO-'Og), oscillations of the inner core of the earth, excitation of the earth's vibrational modes by gravitational waves, and other gravitational effects such as a universally preferred frame. The most accurate absolute gravimeter measures the acceleration of a falling corner cube by means of laser interferometry (Faller and Marson, 1988). A resolution of a few parts in lo-' has been achieved. A higher sensitivity of < lO-'Og has been demonstrated by a superconducting gravimeter in which a superconducting test mass is suspended in a magnetic field (Goodkind, 1988). If atom interferometry attains a projected sensitivity of < 10- ' g, a new region of gravitational measurement will be opened.

-

'

17

ATOM INTERFEROMETRY

2. Rotations In 1913 Sagnac showed that the phase in a ring interferometer is sensitive to rotation. For light, the Sagnac effect produces a phase shift Acp=-

8xAn IC

where A is the area of the interferometer, is the rotation frequency, and 1 is the optical wavelength. For massive particles with velocity u, the Sagnac phase shift becomes 8zAR 4mAn Arp=-=1dBV h This expression indicates that enhanced sensitivity to rotations can be achieved using massive particles (assuming A does not depend on m; see the previous section). The Sagnac effect for de Broglie waves was first observed by Werner et al. (1979) using a neutron interferometer. In their experiment A 10-3m2, I,, 10-l0m, and u/c lop5.Thus the Sagnac phase shift was a factor lo9 larger than for an optical interferometer with the same area. For optical interferometers, the disadvantage of zero rest mass can be more than offset by a large area, for example, a fiber ring interferometer may have an effective area of up to 108m2. In commercial ring laser gyros high sensitivities =7 x rads- is (typically 5 x lO-'rads-' or 10-4nearthwhere nearth the rotation frequency of the earth), without the need for large areas, are achieved by the introduction of an active medium inside the interferometer (Chow, et al., 1985).In this case, the phase is determined extremely accurately by measuring the beat frequency between counterpropagating laser modes. The Sagnac effect for atoms has been demonstrated by Riehle et al. (1991) using the optical Ramsey technique. In this case, one measures the relative Doppler shift between different points in an atomic beam, which is rotated. In the experiment, the sensitivity was limited to ~ l o - ~ r a d s -by ' the stability of the laser and the relatively small enclosed area (about 6x m2). The area can be increased through the use of cold atoms. For parameters applicable to a cold atom interferometer (Weiss et al., 1933a), a sensitivity to rotations of 5 x 10- l 3 rad s- or lo-' Rearth should be possible. This is considerably better than the precision of laser gyros but still has to be demonstrated. Precise rotation sensors are important in a wide range of applications, from inertial guidance systems for aircraft to tests of metric theories of gravitation. General relativistic effects are typically extremely small. For example, the geodetic and Lense-Thirring precessions, for an orbit 650 km above the earth, are 9 x 10-'3rads-' and 6 x 10-14rads-', respectively (Thorne, 1988). The mechanical gyroscope under development as part of the

-

-

-

'

'

18

C. S. Adams, 0. Carnal, and J. Mlynek

Stanford Gravity Probe B experiment is designed to have a sensitivity in a drag-free frame of 10-l7‘rad s - l (Everitt, 1988). This example illustrates an important point: Individual atoms do not necessarily offer a higher sensitivity to inertial effects than bulk masses. At least for the foreseeable future, microscopic probes will not offer a better test of rival theories of gravitation than macroscopic objects such as binary pulsars. B. TESTS OF QUANTUM MECHANICS Matter wave interferometry with electrons and neutrons has allowed many interesting demonstrations of quantum mechanical effects. A good example is the demonstration of the role of potentials in quantum mechanics: In 1959 Aharonov and Bohm showed that an infinite solenoid piercing an electron interferometer induces a phase shift proportional to the enclosed flux. What is surprising is that there is a measurable effect even though the electrons propagate in a region where the field is zero and therefore never experience a force. The analogous effect produced by a time-dependent scalar potential (see Section 1I.A) was also discussed by Aharonov and Bohm (1959). The vector Aharonov-Bohm effect has been observed in an electron interferometer by Tonomura et al. (1982). A neutral particle analogue of the Aharonov-Bohm effect based on the interaction of a magnetic dipole with a line charge was demonstrated using neutron interferometry by Cimmins et al. (1989). Another interesting application of neutron interferometry as a test of quantum mechanics is the demonstration of the sign reversal of a fermion’s wave function after a 27t rotation (Badurek et al., 1983). Matter wave optics and interferometry provide a precision test of the linearity of quantum mechanics. For example, it has been suggested that in order to counteract the unbounded spreading of the wave function, the Schrodinger equation should contain an additional term of the form f( I t,b 1.)’ Bialynicki and Mycielski (1976) showed that in order to preserve the separability of quantum mechanics, the nonlinear term must have the form b ln(a3I t,b 1.)’ Measurements of the Lamb shift put an upper limit on the coefficient b < 10- l o eV. Matter wave optics offer a more stringent test: Precise measurements of neutron diffraction from an edge have shown that b < 3 x (Gahler et al., 1981). The interactions involved in atom and neutron interferometry are fundamentally different, predominantly electromagnetic for atoms and nuclear for neutrons. Therefore, comparison between the results of neutron and atom interferometry may be thought of as a test of the universality of quantum mechanics. However, apart from repeating the experiments mentioned, atom interferometry opens some other possibilities not easily accessible with neutrons or electrons. Atom interferometry is ideally suited to investigate entangled states. One example of an entangled state resulting from sponta-

ATOM INTERFEROMETRY

19

neous decay was discussed in Section 1I.C. Entangled states that persist after atom and field become spatially separated provide a means to study nonlocality or to perform delayed-choice experiments. For example, an atom passing through a high-Q cavity creates an entanglement between the atom and the cavity field that can survive long after the atom crossed the cavity (Haroche et al., 1992). The transfer of a quantum state to the cavity leads to a macroscopic quantum state or Schrodinger cat state. Such experiments are ideally placed to investigate the rather ill-defined border between the quantum world where probability amplitudes interfere and the macroscopic classical world where probabilities simply add. Interesting related topics include “which-path” experiments (Sleator et al., 1991, 1992; Scully et al., 1991; Storey et al., 1993) and measurement-induced interference (Storey et al., 1992). OF ATOMS C. PROPERTIES

Atom interferometry allows precise measurement of the properties of atoms themselves. For example, the phase shifts induced by static electric fields may be used to measure the atomic polarizability (Pritchard, 1993; Rieger et al., 1993) or as a test of charge neutrality. Atom interferometry based on optical Ramsey excitation allows precise measurement of A/m,,, as demonstrated by Weiss et al. (1993a). In collision experiments, interferometry provides additional phase information about the scattering process. An atomic clock based on an atomic fountain has been used to determine the scattering cross section of ultracold atoms (Gibble and Chu, 1993). Similar experiments could be performed to determine the sign of the atomic scattering length at low temperature. Such a measurement has important implications for the possibility of observing Bose-Einstein condensation. A measurement of the real and imaginary parts of the scattering amplitudes has recently been demonstrated by Ekstrom (1993). Finally, atom interferometers may be a useful tool for investigating the phase shifts associated with atom-surface interactions.

V. Atom Interferometers A. YOUNG’SDOUBLE SLIT In 1991 Carnal and Mlynek demonstrated an atom interferometer using the simple Young’s double-slit configuration. The interferometer was similar to that shown in Fig. 6. A beam of metastable helium atoms was collimated by

C. S. Adams, 0. Carnal, and J. Mlynek

20

a 2-pm slit to produce coherent illumination of a double slit (slit width 1pm and spacing 8 pm). The double-slit interference fringes were recorded by scanning a detector 0.64m downstream. The count rate was enhanced using a grating with the same periodicity as the interference pattern for the detector aperture. A typical experimental result is shown in Fig. 7. The fringe visibility was up to 60% and decreased off axis due to the velocity distribution of the atomic beam (the number of observable fringes is approximately equal to the velocity ratio of the beam v/Av 15). A major problem of the experiment was the low count rate; a phase shift of 1/3rad would require a measurement time of up to 10min. The period of the interference pattern may be increased by using slower atoms. In 1992 Shimizu et al. (1992a, b) performed a Young’s double-slit experiment with cold atoms. Metastable neon atoms were released from a magneto-optical trap placed vertically above a double slit. The slit spacing and width were 6 &id 2 pm, respectively. The interference pattern was observed using a microchannel plate detector placed below the double slit. As the atoms were accelerated by gravity, the period of the interference fringes depended on the vertical distance to the detection plane. Periods in the 100-pm range were observed. In addition Shimizu et al. demonstrated the deflection of the interference pattern by the cylindrical electric field produced by a charged wire placed next to the double slit (Shimizu, 1992b). The advantages of the double-slit interferometer are the conceptual

-

h

.-t

E

I

I

0

Detector position FIG.7. Atomic interference pattern produced by a double slit. The interference pattern was detected using a grating with period of 8pm as the detector aperture. The solid line is a calculated diffraction pattern taking the intensity and visibility as fit parameters. The dashed curve is a computer interpolation through selected points. BG indicates the detector background.

21

ATOM INTERFEROMETRY

simplicity and the high spatial confinement of the interferometer paths. For example, this could be useful to measure the phase shift induced by atomsurface potentials. A disadvantage is that the beam separation cannot be increased without considerable loss of intensity. B. THREE-GRATING INTERFEROMETER An alternative scheme for an interferometer based on microstructures is the three-grating arrangement demonstrated by Keith et al. (1991) (see Fig. 8). A three-grating interferometer is achromatic, which is convenient in atom optics due to the normally broad atomic velocity distribution. In the experiment, a sodium atomic beam was produced by supersonic expansion with an argon carrier gas (A = 16pm, v/Av 9). The beam was collimated with two slits. The beam diameter was -20pm. In recent work (Pritchard, 1993), gratings with a period of 200nm were positioned at intervals of 0.6 m. The transverse position of the gratings was actively stabilized using an optical three-grating interferometer mounted on the same translation stages. Interference fringes were recorded as a function of the transverse position of the second grating by measuring the count rate with a 25-pm hot wire placed behind the third grating. The fringe amplitude was 820 counts/s, allowing a phase sensitivity of 15mrad for a 1-min measurement time. The spatial separation of the zero and first diffraction order at the middle grating was 55 pm-more than twice the beam diameter. An interaction region consisting of a 10-pm copper foil (septum) supported between two electrodes was inserted between the two paths. By applying electric or magnetic fields to one arm, Ekstrom (1993) measured the dc polarizability

-

-\; I l l I

COLLIMATION

I

SLITS

1 I I I

I I

I I

I ISEPTUM I

I I

HOTWIRE

I DETECTOR

FIG.8. Schematic of a three-grating interferometer. The first grating diffracts the atomic beam (the zero and + 1 orders are shown). The diffraction orders are again diffracted by a second grating, leading to a spatial overlap and interference. A third grating is required to monitor the interference pattern because in practice the detector is larger than the fringe period. The septum allows selective interaction with one path.

22

C. S. Adams, 0. Carnal, and J. MIynek

of sodium and observed interference revivals due to the rephasing of different magnetic sublevels. In addition, by introducing an inert gas into one arm, the real and imaginary parts of the scattering amplitudes could be determined. This is an interesting application of atom interferometry because it provides additional information that is not accessible in a conventional scattering experiment. Up to now, these are the only experiments in atom interferometry that exploit the spatial separation of the paths.

C. OPTICAL RAMSEY INTERFEROMETER The Ramsey technique of separated oscillatory fields has been an important tool in atomic physics for more than 30 years (Ramsey, 1956). In the rf or microwave region, a two-zone configuration is used. In the optical region, the photon recoil is sufficiently large that it becomes necessary to redirect the atomic trajectories to restore spatial overlap. This is achieved by repeating the usual Ramsey configuration with a second pair of laser beams propagating in the opposite direction. Optical Ramsey fringes were first demonstrated by Bergquist et al. (1977). An interesting related phenomena is the grating stimulated echo (Mossberg et al., 1979). A thorough treatment of Ramsey interference and echo phenomena in terms of a billiard-ball model was recently undertaken by Friedberg and Hartmann (1993). In 1989 Borde pointed out that the Ramsey fringes could be interpreted as atomic interference. A schematic illustration of the interferometric interpretation of four-zone Ramsey excitation is shown in Fig. 9. There are two possibilities for an incoming ground state to be split and recombined at the fourth interaction. These paths correspond to the high- and low-frequency recoil components and may be interpreted as two distinct Mach-Zehnder interferometers. Optical Ramsey fringes are observed by passing an atomic beam through four traveling-wave laser beams. The intensity of the laser beams is adjusted such that each interaction is equal to a n/2-pulse (see Section 1II.B). The first n/2-pulse excites a superposition state

1 -(lg,p)

Jz

- ie-'"Ie,p

+ hk,))

where 41 is the phase of the laser field and p and hk, are the atomic and photon momenta, respectively. Energy and momentum conservation require

-+ho,= P2 2m

(P + Rk,) 2m

+ ho,

ATOM INTERFEROMETRY

23

t+-d+D+d+

FIG.9. Interferometric interpretation of optical Ramsey excitation of an atomic beam by four traveling laser fields. The atomic state is labeled by li,n) where i denotes the internal state and n is the transverse momentum in units of hk,. The first interaction 1 excites the initial state 1g,O) into a superposition of Ig,O) and le,l). A further three interaction regions lead to a sequence of splittings. There are two possibilities for an incoming ground state to be split and recombined at the fourth interaction. These paths, shown by thin and thick lines, correspond to the high- and low-frequency recoil components, respectively.

where o0is the transition frequency of the atom. This reduces to

kL.P -m - A - ore, where mrec = hk2/2m is known as the recoil frequency. Thus the excited state component acquires a momentum kick along the direction of motion equal to

h

hk, = -(A - o,,, - k,u,) v*

(33)

Thus after propagating a distance x ~ ,the ~ ,excited state component acquires a phase shift kxX1.Z = (A

- Ore, - kZV,)X1.2/~,

(34)

relative to the ground state. At the second interaction both components are resplit. For the low-frequency recoil component (thick line in Fig. 9) the excited state component returns to the ground state 1 le,p + hkL) +-(ie’”’Jg,p)

fi

+ le,p + hkL))

(35)

The ground and excited state components in the two paths interfere. However, the phase difference between the paths depends on the transverse

C. S. Adams, 0. Carnal, and J. Mlynek

24

velocity u, leading to a Doppler dephasing between different plane waves. For typical atomic beam parameters, the Doppler dephasing term dominates, leading to a smearing out of the Ramsey fringes. For fringes to be observed the Doppler dephasing term

where Au, is the transverse velocity spread of the beam. In a wave packet picture, this condition is equivalent to the requirement that the width of the wave packet Az h/rnAu, must be larger than the spatial splitting between the paths u,x1,2/u,. Hence, the fact that interference is only observed if there is spatial overlap of the split beams is identical to the requirement that all plane wave components experience approximately the same phase shift. In an optical Ramsey experiment, the paths are brought to overlap and hence the Doppler dephasing is cancelled by repeating the usual two-zone Ramsey configuration with a second pair of laser beams propagating in the opposite direction. Between zones 2 and 3, there is no additional phase displacement between the two paths. At zone 3, the upper path absorbs a photon but this time from the opposite direction [i.e., k, -,- k, in (5.4)]. The relative phase shift accumulated between zones 3 and 4 is (A - wIec k , ~ , ) x ~ , ~ / uThe , . total phase shift is

-

+

where $L = $2 - 41+ 44 - 4 is the total phase accumulated due to the ~ , is spatial overlap phase of the laser in each interaction. If x l V 2= x ~ ,there at zone 4 and the transverse velocity dependence cancels. For part of the interferometer path the atom is in the excited state; therefore, long-lived optical transitions must be used to avoid spontaneous emission. Ramsey-type interferometers have been reported using calcium (z = 0.4ms) by Riehle et al. (1991) and later using magnesium (T = 4.6ms) by Sterr et al. (1992). The interference fringes are observed by measuring the intensity of the excited state component after the fourth zone, for example, by monitoring the fluorescence. The excited state intensity oscillates between the high- and low-frequency recoil components as a function of the phase shift between the paths. The intensity of the excited state component for one output is

(A - wIeC)+ $ L where d = x1,2= x ~ , It~ .follows that the interference pattern may be recorded by varying the detuning or the phase of the laser beams. An example of interference fringes for calcium (Helmcke, 1992) is shown

25

ATOM INTERFEROMETRY

in Fig. 10. Only a few fringes are visible due to the broad longitudinal velocity distribution of the beam. Between zones 2 and 3, the two interferometers are labeled by different internal states; therefore, either of the recoil components can be suppressed by depopulation or deflection using a second laser (Riehle et al., 1992; Sterr et al., 1992). Figure 10(b) shows the effect of suppressing the low-frequency recoil component by pumping the excited state population between zones 2 and 3 to a metastable level. Between zones 1 and 2, or 3 and 4, the paths are labeled by different internal states. This allows selective interaction with only one path even though the beams may overlap spatially. For example, by applying an

600 In

c. .-

C f

0

0

50

100

150

offset frequency (kHz)

; 1000 c

C

.-a In

offset frequency ( k H t I FIG.10. Interference fringes observed by optical Ramsey excitation using a calcium atomic beam. In (b) the low-frequency recoil component was suppressed by pumping the excited state population between zones 2 and 3 to a metastable level. [After Helmcke et al. (1992).]

26

C. S. Adams, 0. Carnal, and J. MIynek

electric field between zones 3 and 4, the difference in the dc polarizabilities of the 's and 3p states in magnesium has been measured (Rieger et al., 1993). Phase shift-induced ac Stark shifts have also been measured (Riehle et al., 1992; Sterr et al., 1992). Riehle et al. (1991) have observed the Sagnac effect for atoms by rotating their atomic beam apparatus. Figure 11 shows the shift of the interference fringes due to rotation. The measured frequency shift was in agreement with (29).

D. ATOMINTERFEROMETRY USINGSTIMULATED RAMANTRANSITIONS The sensitivity of any measurement based on atom-light interactions is limited by the transition linewidth. To achieve a high sensitivity, narrow transitions are required. However, the probing of narrow transitions requires ultrastable lasers. A way around this problem introduced by Kasevich et al. (1991) is to use stimulated Raman transitions. Velocity-selective stimulated Raman transitions are driven between two hypefine levels in the ground state by counterpropagating laser beams [see

I

-40

-20

I

I

1

0

20

40

frequency (kHz1 FIG.11. The Sagnac effect for atoms. The sequence of curves shows the frequency dependence of the optical Ramsey fringes with the apparatus rotated with an angular frequency (a) R = -0.09/s, (b) R = O.OO/s, and (c) R = 0.09/s. [Adapted from Riehle et al. (1991).]

ATOM INTERFEROMETRY

27

Fig. 12(a)]. Both lasers are detuned by the same amount from the optical transition frequency. Only the frequency difference and not the absolute frequency of the laser beams must be stable. A suitable difference frequency is conveniently generated using an rf oscillator. As the initial and final levels are both ground states, the linewidth of the transition is extremely narrrow. In practice, the linewidth of the Raman transition is limited only by the measurement time. In an atom interferometer, stimulated Raman transitions combine the advantages of the long life-time of an rf transition with the large photon recoil of an optical transition (Kasevich and Chu, 1991). The interferometer is similar to the Ramsey-type configuration discussed earlier. One difference is that because a two-photon transition is used, a three-pulse sequence n/2-n-n/2 is sufficient to cancel the transverse velocity dependence [Fig. 12(b)]. For an atom initially in state 11,p), the first n/2-pulse excites a superposition state 1

+

,p) - ie -i4112,p hk,,,))

(39)

-

where kerf= k, - k, 2k, is the effective k-vector of the Raman transition. The second pulse flips the nternal and momentum state of the two paths

The third pulse remixes. The phase difference between the paths is - 24, + &. For the three-pulse configuration, no atoms are lost into other paths.

a

A

b

AL

11)

I

FIG.12. Atom interferometer based on stimulated Raman transitions between two hyperfine levels 11) and 12). (a) The Raman transition is excited by two counterpropagatinglasers, both with the same detuning A from the atomic resonance. (b) The interferometer is formed by a n/2-x-x/2 pulse sequence. The atomic state is labeled by Ii,n), where i denotes the internal state and n is the transverse momentum in units of hk,.

C.S. Adams, 0.Carnal, and J. Mlynek

28

A second important innovation introduced by Kasevich et al. (1989; 1991) is the use of an atornicjountain. The fountain geometry combined with the effectively infinite lifetime of the Raman transition allows long interaction times; consequently, an interferometer sensitivity 6 orders of magnitude higher than in a conventional beam-type experiment can be achieved (Weiss et al., 1993a). The higher sensitivity is illustrated by Fig. 13, which shows the interference fringes for the conventional Ramsey configuration (four 71/2pulses). In contrast to optical Ramsey fringes for a two-level transition and an atomic beam (Fig. lo), the recoil splitting is clearly resolved and many fringes are visible. A further advantage of the atomic fountain geometry is that the laser beams can be arranged perpendicular or parallel to the atomic trajectories. Kasevich and Chu used the parallel or longitudinal geometry to observe gravitationally induced phase shift in a sodium atomic fountain (Kasevich and Chu, 1991, 1992). As the atoms fall they move out of resonance with the laser due to the Doppler shift. If the laser pulses are detuned to compensate for the Doppler shift, then the phase shift between the two components is

41 - 2 4 2 + 4 3 = -

keff9At2

(41)

where At is the time between the pulses. For At = 0.05s and a total integration time of 2000s, the phase could be determined with an accuracy of 3 x which gives a sensitivity to gravity of 3 x 1OP8g.This accuracy level approaches the resolution of state-of-the-art gravimeters using the falling corner cube-technique (Faller and Marson, 1988). Thus atom interferometry starts to become interesting for gravity tests and geophysical applications. Optical Ramsey excitation using stimulated Raman transitions can also

2

0 4-

0 U II

LL

Frequency (kHz1 FIG.13. Optical Ramsey fringes recorded using stimulated Raman transitions in a cesium atomic fountain. The width of the Ramsey fringes is 500Hz.[After Weiss et al. (1993b).]

29

ATOM INTERFEROMETRY

be operated in the more traditional geometry with four n/2-pulses (see previous section). Weiss et al. (1993a) used this configuration to measure the frequency splitting between the high- and low-frequency recoil components for cesium. The recoil splitting was enhanced by a sequence of n-pulses sandwiched between the second and third interaction. The spatial separation between the two paths was a few millimeters. The measurement provided a value of h/mc,, where mcs is the mass of a cesium atom, accurate to lo-’. Combined with knowledge of the Ryberg constant and electron-atom mass ratio, this result provides a QED insensitive measurement of the fine structure constant a.

E. ATOMINTERFEROMETRY USINGSTATIC ELECTRIC AND MAGNETIC FIELDS Interference between different internal states may be induced by nonadiabatic passage of an atomic beam through separated regions with a static field gradient. This idea was first implemented by Sokolov and Yakovlov (1982), who observed interference between 2sIl2 and 2pIl2 states in hydrogen by rapid passage of a beam through electric field gradients. The principle is illustrated schematically in Fig. 14. Hydrogen atoms in the 2s1,2 state enter a region of strong electric field. The initial state is projected onto the interaction eigenstates 11) and (2), which are superpositions of 2sIl2and 2p1,2.When the field turns off, the interaction eigenstates are projected back onto the initial states. Interference fringes are obtained by measuring the

z

Lyman-a

FIG.14. Schematic illustration of atomic interference induced by nonadiabatic passage through an electric field. Hydrogen atoms in the 2s,,, state enter a region of strong electric field. The initial state is projected onto the interaction eigenstates 11) and 12). When the field turns off, the interaction eigenstates are projected back on to the bare states 2s,,, and 2p,,,.

30

C. S. Adams, 0. Carnal, and J. Mlynek

Lyman-ct photons emitted due to the decay of the 2p1,2 state as a function of the electric field. The experiment was used to determine the Lamb shift and hyperfine splitting in hydrogen (Sokolov, 1989). A similar experiment using a magnetic field gradient was performed by Robert et al. (1991). A schematic diagram of the experiment (referred to as the longitudinal Stern-Gerlach interferometer) is shown in Fig. 15. The incoming hydrogen beam is polarized by a transverse magnetic field, which selects the F = 1, mF = 0, 1 levels. A coherent superposition of magnetic sublevels is prepared by nonadiabatic passage through a magnetic field orthogonal to the polarizing field. In the central region, the magnetic sublevels experience different potentials due to a longitudinal field. Finally, the sublevels are remixed and a particular state is selected by an analyzer. Interference fringes are obtained by measuring the Lyman-ct photons emitted due to the decay of the 2p,,, state as a function of the longitudinal magnetic field. The experiment is analogous to the propagation of linearly polarized light through a birefringent medium. The longitudinal SternGerlach interferometer has been used to observe a topological phase shift produced by a coil (Miniatura et al., 1992).

VI. Outlook For matter wave interferometry, atoms have some important advantages over other particles. The sensitivity of electron interferometry is limited by long-range Coulomb interactions with the walls of the vacuum chamber. Neutrons have the disadvantage of being difficult to produce. For these

FIG.15. Schematic illustration of atomic interference induced by nonadiabatic passage through a magnetic field gradient. A beam of hydrogen atoms is polarized by a transverse magnetic field gradient (P). The initial state is projected onto the interaction eigenstates by nonadiabatic passage through a field gradient (Ml). The eigenstates acquire different phase shifts in a region with a longitudinal field and then are projected back onto the bare states in a second mixing region (M2). Finally, the population of one sublevel is monitored by an analyzer (A).

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reasons, the prospects for atom interferometry to find widespread applications are more promising. In a relatively short time period, atom interferometry has reached some notable milestones: The first experiments to exploit the spatial separation of an atomic wave function have been reported. The potential for precision measurement of gravity and inertial effects has been demonstrated using an atomic fountain interferometer. In addition, it has been shown that atom interferometry has considerable potential for precision measurement of atomic properties and fundamental constants. These are just the first experiments in what is likely to be widespread application of atom interferometry in diverse areas of science. For a practical measuring device, miniaturization is an important consideration. Two promising approaches toward the realization of compact and portable interferometersare the technique of optical Ramsey excitation in vapor cell traps using diode lasers or that of a single-crystal interferometer based on diffraction from crystalline surfaces. A single-crystal interferometer would combine high stability and compactness. Difficulties revolve around the stringent demands on surface quality and purity. However, given the current level of vacuum technology and knowledge of atom-surface processes, the attractive goal of an atom interferometer based on crystal diffraction is within reach.

Acknowledgments This work was partly supported by European Community and the Deutsche Forschungsgemeinschaft.

References Aharonov, Y.and Bohm, D. (1959). Phys. Rev. 115,485. Allen, L., and Eberly, J. H. (1975). Optical Resonance and Two-Level Atoms. Wiley Interscience, New York. Badurek, G., Rauch, H., and Summhammer, J. (1983). Phys. Rev. Lett. 51, 1015. Bergquist, J. C., Lee, S. A., and Hall, J. L. (1977). Phys. Rev. Lett. 38, 159. Bialynicki, I., and Mycielski, J. (1976). Ann. Phys. ( N Y ) 100, 62. Bordb, Ch. J. (1989). Phys. Lett. A 140, 10. Carnal, 0. and Mlynek, J. (1991). Phys. Rev. Lett. 66, 2689. Chow, W. W., Gea-Banacloche, J., Pedrotti, L. M., Sandars, V. E., Scleich, W., and Scully, M. 0. (1985). Rev. Mod. Phys. 57, 61. Cimmins, A., Opat, G. I., Klein, A. G., Kaiser, H., Werner, S. A,, Arif, M., and Clothier, R. (1989). Phys. Rev. Lett. 63, 380. Colella, R., Overhauser, A., and Werner, S. A. (1975). Phys. Rev. Lett. 34, 1472. Davisson, C. J. and Germer, L. H. (1927). Phys. Rev. 30, 705.

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Ekstrom, C. R. (1993). Ph.D. Thesis, MIT, unpublished. Ekstrom, C. R., Keith, D. W., and Pritchard, D. E. (1992). Appl. Phys. B 54, 355. Estermann, I., and Stern, 0. (1930). Zeits. Physik 61, 95. Everitt, C. W. F. (1988). In: Near Zero: New Frontiers in Physics (Fairbank, J . D., Deaver, Jr., B. S., Everitt, C. W. F., and Michelson, P. F., eds.), W. H. Freeman and Co., New York. Faller, J. E., and Marson, I. (1988). Metrologia 25, 49. Feynman, R. P., and Gibbs, A. R. (1965). Quantum Mechanics and Path Integrals. McGrawHill, New York. Friedberg, H. and Hartmann, S. (1993). Phys. Rev. A 48, 1446. Gahler, R., Klein, A. G., and Zeilinger, A. (1981). Phys. Reo. A 23, 1611. Gibble, K., and Chu, S. (1993). Phys. Rev. Lett. 70, 1771. Goodkind, J. M. (1988). In: Near Zero: New Frontiers in Physics Fairbank, J . D., Deaver, Jr., B. S., Everitt, C. W. F., and Michelson, P. F. eds.), W. H. Freeman and Co., New York. Haroche, S., Brune, M., and Raimond, J. M.(1992). Appl. Reo. B 54, 355. Helmcke, J., Riehle, F., Witte, A,, and Kisters, Th. (1992). Phys. Scripta, T40, 32. Kaiser, H., Werner, S. A,, and George, E. A. (1983). Phys. Rev. Lett. 50, 560, and 51, 1106. Kasevich, M., and Chu, S. (1991). Phys. Rev. Lett. 67, 181. Kasevich, M., and Chu, S. (1992). Appl. Phys. B 54, 321. Kasevich, M., Riis, E., Chu, S., and DeVoe, R. G. (1989). Phys. Rev. Lett. 63, 612. Kasevich, M., Weiss, D. S., Riis, E., Moler, K., Kasapi, S., and Chu, S. (1991). Phys. Rev. Lett. 66,2297. Keith, D. W., Ekstrom, C. R., Turchette, Q. A., and Pritchard, D. E. (1991). Phys. Rev. Lett. 66,2693. Keith, D. W., Schattenburg, M. L., Smith, H. I., and Pritchard, D. E. (1988). Phys. Rev. Lett. 61, 1580. Klein, A. G., Opat, G . I., and Hamilton, W. A. (1983). Phys. Rev. Lett. 50, 563. Levy, B. (1991). Phys. Today 44, 17. Marton, L., Arol Simpson, J., and Suddeth, J. A. (1954). Phys. Reo. 90,490. Miniatura, Ch., Robert, J., Gorceix, O., Lorent, V., Le Boiteux, S., Reinhardt, J., and Baudon, J . (1992). Phys. Rev. Lett. 69, 261. Mossberg, T. W., Kachru, R. K., Whittaker, E., and Hartmann, S. R. (1979). Phys. Rev. Lett. 43, 851. Nicklaus, M., and Hasselbach, F. (1993). Phys. Reo. A 48, 152. Pritchard, D. E. (1993). In: Atomic Physics I3 (Walther, H., Hansch, T. W., and Neizert, B., eds.), AIP, New York. Ramsey, N. F. (1956). Molecular Beams. Oxford University Press, Oxford. Rauch, H., Treimar, W., and Bonse, U. (1974). Phys. Lett. A 57, 369. Rieger, V., Sengstock, K., Sterr, U., Moller, J. H., and Ertmer, W. (1993). Opt. Commun. 99, 172. Riehle, F., Kisters, Th., Witte, A., Helmcke, J., and BordC, Ch. (1991). J. Phys. Rev. Lett. 67, 177. Riehle, F., Witte, A., Kisters, Th., and Helmcke (1992). J. Appl. Phys. B 54, 333. Robert, J., Miniatura, Ch., Le Boiteux, S., Reinhardt, J., Bocvarski, V., and Baudon, J. (1991). Europhys. Lett. 16, 29. Sagnac, M.G. (1913). Comptes Rendus Acad Sci. 157, 1410. Englert, B.-G., and Walther, H. (1991). Nature 351, 111. Scully, M. 0.. Shimizu, F., Shimizu, K., and Takuma, H. (1992a). Phys. Rev. A 46, R17. Shimizu, F., Shimizu, K., and Takuma, H. (1992b). Jpn. J. Appl. Phys. 31,46. Sleator, T., Carnal, O., Faulstich, A., and Mlynek, J. (1992). In: Quantum Measurements in Optics (Tombesi, P., and Walls, D. F., eds.), Plenum, New York. Sleator, T., Carnal, O., Pfau, T., Faulstich, A., Takuma, H., and Mlynek, J. (1991). In: Laser Spectroscopy X (Ducloy, M., Giacobino, E., and Camy, G., eds.), World Scientific, Singapore. Sokolov, Yu. L. (1989). In: The Hydrogen Atom (Bassani, G. F., Inguscio, M., and Hansch, T. W., eds.), Springer, Berlin. Sokolov, Yu. L., and Yakovlov, V. P. (1982). Sou. Phys. J E T P 56, 7.

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Sterr, U., Sengstock, K., Miiller, J. H., Bettermann, D., and Ertmer, W. (1992). Appl. Phys. B 54, 341. Storey, P., Collett, M., and Walls, D. (1992). Phys. Rev. Lett. 68, 472. Storey, P., Tan, S., Collett, M., and Walls, D. (1993). Submitted to Nature. Thomson, G . P. (1927). Nature 120, 802. Thorne, K. S. (1988). In: Near Zero: New Frontiers in Physics (Fairbank, J. D., Deaver, Jr., B. S., Everitt, C. W. F., and Michelson, P. F., eds.), W. H. Freeman and Co., New York. Tonomura, A., Matsuda, T., Fukuhara, A., Osakabe, N., Sugita, Y.,and Fujiwara, H. (1982). Phys. Rev. Letf. 48, 1443. Weiss, D. S., Kasevich, M., Young, B. N., and Chu, S. (1993b). In: Atomic Physics I3 (Walther, H., Hansch, T. W., and Neizert, B., eds.), AIP, New York. Weiss, D. S., Young, B. N., and Chu, S. (1993a). Phys. Rev. Lett. 70, 2706. Werner, S. A., Staudenmann, J., and Colella, R. (1979). Phys. Rev. Lett. 42, 1102. Young, T. (1802). Phil. Trans. Roy. SOC.12, 387.

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OPTICAL TESTS OF QUANTUM MECHANICS R . Y. CHIAO. P . G. KWIAT, and A. M . STEINBERG Department of Physics University of California Berkeley. CA

I. Introduction: The Planck-Einstein Light-Quantum Hypothesis I1. Quantum Properties of Light . . . . . . . . . . . . . . A . Vacuum Fluctuations: Cavity QED . . . . . . . . . . B. The Down-Conversion Two-Photon Light Source . . . C. Squeezed States of Light . . . . . . . . . . . . . .

D. Quantum Nondemolition . . . . . . . . . . . . . . 111. Nonclassical Interference and “Collapse” . . . . . . . . . A . Single-Photon Interference and Berry’s Phase . . . . . B. The Energy-Time Uncertainty Principle . . . . . . . . C . Collapse Effects in Fluorescence . . . . . . . . . . . D . Two-Photon Interference . . . . . . . . . . . . . . IV. Complementarity . . . . . . . . . . . . . . . . . . . A . Quantum Eraser . . . . . . . . . . . . . . . . . . B. Vacuum-Induced Coherence . . . . . . . . . . . . . C . Suppression of Spontaneous Parametric Down-Conversion V. The EPR “Paradox” and Bell’s Inequalities . . . . . . . . A . Generalities . . . . . . . . . . . . . . . . . . . . B. Polarization-Based Tests . . . . . . . . . . . . . . C. Nonpolarization Tests . . . . . . . . . . . . . . . D . Loopholes and Loophole-Free Tests . . . . . . . . . VI. Related Issues . . . . . . . . . . . . . . . . . . . . A . Information Content of a Quantum . . . . . . . . . . B. Cryptography . . . . . . . . . . . . . . . . . . . C . Nonlocality without Inequalities . . . . . . . . . . . VII. The Reality of the Wave Function . . . . . . . . . . . . A . Test of the de Broglie Guided-Wave Theory . . . . . . B. Bohm’s Deterministic Model of Quantum Mechanics . . C . Application of Bohm’s Theory to Tunneling Times . . . D. Status of the Wave Function in Measurement . . . . . E. Observable Effects of “Empty Waves” . . . . . . . . . VIII. The Single-Photon Tunneling Time . . . . . . . . . . . A . An Application of EPR States: Dispersion Cancellation . B. Single-Photon Propagation Time Measurements . . . . C. Tunneling Time in a Multilayer Dielectric Mirror . . . . D . Interpretation of the Tunneling Time . . . . . . . . .

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IX. Envoi . . . . . . . . . . . . . A. Phase Operators . . . . . . . B. Nonlinear Quantum Mechanics C. Issues in Causality . . . . . . D. Conclusion . . . . . . . . . Acknowledgments . . . . . . . . References . . . . . . . . . . .

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I. Introduction: The Planck- Einstein Light-Quantum Hypothesis Quantum mechanics began with the solution of the problem of blackbody radiation by Planck’s quantum hypothesis: In the interaction of light with matter, energy can only be exchanged between the light in a cavity and the atoms in the walls of the cavity by the discrete amount

E = hv,

(1)

where h is Planck’s constant and v is the frequency of the light. Einstein in his treatment of the photoelectric effect reinterpreted this equation to mean that a beam of light consists of particles (light quanta) with the preceding energy. Hence Einstein changed the meaning of E in Eq. (1) from that of the energy exchanged between atom and light to the energy intrinsic to the particles inside the light beam itself, independent of its interaction with atoms. Later, the name photon was given to these light quanta. The Compton effect supported this particle viewpoint of light by demonstrating that photons carried momentum, as well as energy. Therefore, the waveparticle duality of quanta made its first appearance in connection with the properties of light. It would seem that the introduction of the concept of the photon as a particle would necessarily also introduce the concept of locality into the quantum world, because particles are point-like entities and, hence, are seemingly local in nature. However, in light of observed violations of Bell’s inequalities, exactly the opposite seems to be the case. Here we review some recent results in quantum optics that elucidate nonlocality and other fundamental issues in quantum mechanics. In spite of the successes of quantum electrodynamics, and of the standard model in particle physics, in which the photon, the Wand Z bosons, and the gluon are viewed as the fundamental particles that mediate the electromagnetic, weak, and strong forces, respectively, there is still considerable resistance in parts of the optics community to the concept of the photon as

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a particle. Many papers have been written to try to explain all optical phenomena, including the photoelectric effect, semiclassically, that is, with the light viewed as a classical wave, and the atoms treated quantum mechanically (Barut and Dowling, 1990 Clauser, 1972; Jaynes, 1973). Let us begin by reviewing some quantum optics phenomena that exclude this semiclassical viewpoint. In an early experiment, Taylor reduced the intensity of a thermal light source in Young’s two-slit experiment until, on the average, there was only a single photon passing through the two slits at a time. He then observed a two-slit interferencepattern that was in no way different from that observed for a more intense classical beam of light incident on the two slits. He interpreted this to mean that a single photon must have passed through both slits simultaneously, so that it is commonly believed that this experiment demonstrated, in Dirac’s words (Dirac, 1958), that “each photon then interferes only with itself.” However, a coherent state, no matter how stronglyattenuated, always remains a coherent state, and a classicalwave explanation of Taylor’s experiment therefore always remains a possibility. Because a thermal light source can be modeled as a statistical ensemble of coherent states, a stochastic classical wave model of Taylor’s experiment,in which classical wave amplitudes are smoothly divided between the two slits, in conjunction with a semiclassical theory of the photoelectriceffect (Lamb and Scully, 1969),yields complete agreement with the observations.Consequently,the concept of the photon need never be invoked in the explanation of this experiment, and the claim that this experiment demonstrates quantum interference of individual photons is unwarranted. This weakness in Taylor’s experiment can be removed by the use of nonclassical light sources. In particular, two-photon light sources, combined with coincidence detection, allow the production of single photon (n = 1 Fock) states with high certainty. The first such experiment was performed by Clauser (Clauser, 1974), in which two photons were produced in an atomic cascade within nanoseconds of each other. These photons impinged on two beam splitters, and were then detected in coincidence by means of four photomultipliers placed at all possible ports of these beam splitters. A simplified version of this experiment was performed by Grangier, Roger and Aspect (Fig.l), replacing one of the beam splitters and its two detectors with a single detector (Grangier et al., 1986). This is related to the Hanbury-Brown-Twiss experiment, which can be understood classically (Hanbury-Brown and Twiss, 1958), but in this case, the difference between the predictions of a classical wave theory and the particulate (photon) picture become clear: Any classical wave amplitude is smoothly divided between the transmission and reflection output ports of the beam splitter, but a single photon is indivisible and cannot be so divided. Grangier et al. introduced an anticorrelation parameter for the system

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R Y. Chiao, P.G. Kwiaf, and A.M. Steinberg

FIG.1. Coincidence detection and a pair-source allow one to confirm the indivisibility of a photon incident on a beam splitter.

where N , , , is the rate of triple coincidences between detectors D,, D, and D,; N , , is the rate of double coincidences between D, and D,; N , , is the rate of double coincidences between D, and D,; and N , is the total rate of detections by D,. It has been demonstrated (Aspect et al., 1989; Chiao et al., 1993) from Schwarz’s inequality that a 2 1 for any classical wave dividing smoothly at the beam splitter. By contrast, the indivisibility of the photon leads to strong anticorrelations between D, and D,, making a, in principle, arbitrarily small. In agreement with this quantum-mechanical picture, Grangier et al. observed a 13-standard-deviation violation of the inequality, corroborating the notion of the “collapse of the wave packet” proposed early on by Heisenberg in connection with the behavior of particles at a beam splitter (Heisenberg, 1930).

11. Quantum Properties of Light A. VACUUMFLUCTUATIONS: CAVITY QED The preceding considerations necessitate the quantization of the electromagnetic field, which in turn leads to the concept of vacuum fluctuations. As with the hypothesis of the photon, there are difficulties with this idea, such as the infinite zero-point energy of an unbounded universe. These difficulties have led some researchers to attempt to dispense with this concept altogether, along with that of the photon, in every explanation of electromagnetic interactions with matter. Of course, it is impossible to explain phenomena such as spontaneous emission and the Lamb shift without some kind of fluctuating electromagnetic fields, but surprisingly one can go a long way toward explaining all of these effects with an ad hoc ambient classical electromagnetic noise filling all of space (Milonni, 1994). In particular, the Casimir effect (the attraction of two uncharged parallel

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conducting plates in the vaccum) is often cited as evidence for the existence of vacuum fluctuations, although it would also be possible to explain this semiclassically, in terms of dipole forces between electrons in each of the plates undergoing zero-point motion and their images in the other plate. Cavity quantum electrodynamics (QED) provides additional motivation for the quantization of the electromagneticfield. A host of nonclassical phenomena, such as the suppression and enhancement of spontaneous emission from atoms by cavities and optical resonators (which show that boundary conditions play an important role in determining the size of the vacuum fluctuations that an atom sees), are most easily understood via quantization of the electromagnetic field. These effects are not reviewed here because they have already been extensively reviewed elsewhere (Hinds, 1991).

B. THE DOWN-CONVERSION TWO-PHOTON LIGHTSOURCE The quantum aspects of electromagnetism are made more striking in experiments with a two-photon light source, in which two highly correlated photons are produced in spontaneous parametric down-conversion, or parametric fluorescence (Burnham and Weinberg, 1970; Harris et af., 1967; Klyshko, 1967). In this process, an ultraviolet “pump” photon produced in a laser (an argon ion laser at a wavelength of 351 nm in our experiments) spontaneously decays inside a crystal with a x”’ nonlinearity (we used a potassium dihydrogen phosphate, or KDP, crystal) into two red photons of nearly equal energy (near a wavelength of 702 nm in our experiments). Thus this process is the time-reversal of second harmonic generation. The two red photons, conventionally called the signal and the idler, are highly correlated. Their emission times have been measured to be within femtoseconds of each other. They are produced in a decay process that is fundamentally no different at the quantum level from a radioactive decay process like that of fission: The pump photon can be thought of as the parent, and the signal and idler photons as its daughters. Energy and momentum are conserved in this process: hw, = ha1+ hw,

hk, x h k ,

+ Ak,

(3)

where h a , is the energy of the parent photon, and hw, and ho, are the energies of the daughter photons. Similarly, hk, is the momentum of the parent photon, and hk, and hk, are the momenta of the daughter photons; k, and k, sum to k, to within an uncertainty given by the reciprocal of the

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crystal length. These conservation laws imply that conjugate (i.e., “twin”) signal and idler photons are tightly correlated not only in their times of emission, but also in their energies and in their momenta. A rainbow of colored cones is produced around an axis defined by the direction of the UV beam, with photons always emitted in correlated pairs on opposite sides of the axis (Fig. 2). Due to the fact that there are many ways to partition the energy of the parent photon, each daughter photon has a broad spectrum, and hence a narrow wave packet in time. However, the sum of the two daughter photons’ energies is extremely well defined, since they must add up to the energy of the extremely monochromatic parent laser photon. Thus the diyerence in their arrival times and the sum of their energies can be simultaneously known to high precision. These characteristics prove important in several of the experiments described later.

C. SQUEEZED STATES OF LIGHT

The creation of conjugate photon pairs in spontaneous down-conversion is closely related to the process of squeezed light production (Rarity et ul., 1992). This becomes clear from the inspection of the quadrature-squeezed vacuum state (Kimble and Walls, 1987; Walls and Reid, 1986):

which corresponds to a vacuum state transformed by the creation (atat) and destruction (aa) of photons two at a time. When the gain arising from parametric amplification becomes large in the down-conversion crystal, a transition from spontaneous to stimulated emission occurs, and squeezed states of light are produced.This gain is dependent on the phase of amplified light relative to the phase of the pump light beam. Hence, vacuum fluctuta-

FIG.2. Conical emissions of down-conversion from a nonlinear crystal. Photon energy depends on the cone opening angle, and conjugate photons lie on opposite sides of the axis, e.g., the inner “circle” orange photon is conjugate to the outer “circle” deep-red photon, etc. Energy and momentum are conserved in this two-photon decay.

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tions in one quadrature are squeezed, but stretched in the other quadrature, in such a way as to preserve the minimum uncertainty principle product of these fluctuations. That is, the fluctuations develop a periodicity at twice the optical frequency; this is a direct consequence of the fact that the photons are produced in pairs. In this way, the fluctuations in the squeezed quadrature can be reduced below the standard quantum limit observed for a vacuum state or a coherent state. For a review of quadrature-squeezed states and their applications, see Kimble and Walls (1987). A simpler type of squeezing (Yamamoto et al., 1986) involves preparation of states with a well-defined photon number, that is, lacking the Poisson fluctuations of the coherent state. The possibility of producing such states demonstrates that “shot noise” in photodetection should not be thought of as merely the result of randomness in a semiclassical model, but as representing real properties of the electromagnetic field, accessible to experimental control.

D. QUANTUM NONDEMOLITION The uncertainty principle between number of quanta N and phase beam of light

4

of a

implies that to know the number of photons in the beam of light exactly, one must give up all knowledge of the phase of the wave. This is a manifestation of the particle wave complementarity principle. In theory a quantum nondemolition (QND) process is possible (Braginsky, 1989; Yamamoto et al., 1986); Without annihilating any of the light quanta, one can count them. It might seem that this would make possible successive measurements on noncommuting observables of a single photon, in violation of the uncertainty principle; it is the unavoidable introduction of phase uncertainty by any number measurement that prevents this. A gedanken example of QND is as follows: The Aharonov-Bohm effect can cause a phase shift of an electron wave function that arises from the time-varying vector potential associated with a nearby light beam; this phase shift can, in principle, be used to count photons in this beam (Chiao, 1970; Lee et al., 1992). Because the photons in such a light beam are not annihilated, this is a QND process. Another QND scheme uses the intensity-dependent index of refraction (the Kerr nonlinearity) to detect the optical phase shift on a probe beam arising from the index change caused by the passage of a signal beam in a number state (Imoto et al., 1985; Kitagawa and Yamamoto, 1986). Still other QND schemes use Rydberg atoms to give indirect information concerning the number of photons in a

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microwave cavity through which they have passed (Haroche et al., 1992; Walther, 1992). A review of recent work is given in Roch et al. (1992).

111. Nonclassical Interference and “Collapse” A. SINGLE-PHOTON INTERFERENCE AND BERRY’S PHASE

We turn now to several interference experiments performed at the one- and two-photon level. The simplest of these was performed by Grangier et al. (1986), using the cascade source discussed in Section I. One of the photons was directed to a “trigger” detector, while the other, thus prepared in an n = 1 Fock state (Clauser, 1974), was sent through a Mach-Zehnder interferometer. The detections at the output ports of the interferometer were registered in coincidence with the trigger photon, and the visibility of the fringes measured to be greater than 98%. Dirac’s statement that a single photon interferes with itself is thus verified. As an extension of these considerations, we have performed an experiment that demonstrates the existence of a Berry’s phase at the single-photon level (Kwiat and Chiao, 1991). Berry’s phase is a geometrical phase shift, which a quantum system acquires after a round-trip in the state space of the system, for example, by its adiabatic following of slow, circuital parameter changes in the system Hamiltonian. This phase shift has also been observed in classical optics (Chiao, 1989), and a controversy has resulted as to whether one should interpret it as classical or quantum [see references in Kwiat and Chiao (1991)l. Using the correlated photons from parametric down-conversion, we have done an experiment that excludes any classical explanation of the observed Berry’s phase (Fig. 3). As in the Grangier experiment, one of our photons is directed to a trigger detector, and the other is directed into an interferometer. In one arm of the interferometer, two quarter-wave plates (one fixed, the other rotatable) cycle the polarization of the light, generating a specific type of Berry’s phase known as Pancharatnamt phase. The state space in this case is the Poincark sphere describing all polarization states. The phase acquired after a round-trip through the wave plates is equal to minus one-half of the solid angle subtended by the state-space trajectory with respect to the center of the sphere. We have ruled out all classical wave theory explanations of these effects by using a beam splitter after the output port of the Michelson interferometer to measure the anticorrelation parameter of Grangier et al. (Section I). We found a = 0.08 f 0.04, which clearly violates any classical wave prediction. Therefore, although this same Berry’s phase shift also

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\I fixedQ1

I==IF3

FIG.3. Apparatus to measure Berry’s phase for single photons, where Q1 and 4 2 are quarter-waveplates. The energy-timeuncertainty relation and wave function collapse were also studied by investigating the effect of various filters before the detectors (Kwiat and Chiao, 1991).

appears at the classical level, it is necessary to view it, and all other Berry’s phases in optics, as a geometric effect that originates at the quantum level, but survives the correspondence principle limit into the classical level.

B. THEENERGY-TIME UNCERTAINTY PRINCIPLE The energy-time uncertainty principle

is one of the fundamental aspects of quantum mechanics that we have tested in recent two-photon interference experiments at Berkeley. The motivation for this study arises from the fact that the energy-time uncertainty principle arises differently from the momentum-position uncertainty principle in the standard formulation of quantum mechanics, because time is not an operator. Unlike the case of momentum and position, where the fundamental commutator [x,p,] = ih exists, from which one can derive in the usual way the momentum-position uncertainty principle, the corresponding commutator for time and energy [ t , H ] = ih does not exist, since the time

44

R Y. Chiao, P.G. Kwiai, and A.M. Steinberg

operator t does not exist. This difficulty arises because the energy H is a positive definite operator, whereas the momentum p x can range from negative to positive infinity. Since the t operator, if it existed, would be the infinitesimal generator of translations of either sign in energy (just as H is the generator of translations in time), it could generate arbitrarily large negative energies. Thus in the time-dependent Schrodinger equation, instead of being an operator, time is a c-number parameter, as is the case for position as well in relativistic quantum field theory (Aharonov and Bohm, 1961). We studied the energy-time relation using the correlations in down- conversion. To account for the correlated, nonlocal effects in quatum theory, and in response to the Einstein-Podolsky-Rosen paper, Schrodinger (1980) introduced the concept of entangled states. An entangled state is a nonfactorizable sum of product states, for example, the Bohm singlet state, 1

IS = 0) = - { l t J

fi

112)

-

If2)

11Jl

(7)

which represents the fact that two spin-1/2 daughter particles decaying from a spin-zero parent will always be correlated with each other, no matter how far apart they are from each other at the moment of detection. The mathematical fact that such a sum is nonfactorizable corresponds to the physical fact that the probabilites of detection of these particles are never independent of each other and, hence, that nonlocal correlations are intrinsic to this system. The energy correlations in down-conversion of a parent photon of sharp energy E , can be similarly described: 12 photons ) =

JoEo

dE, dE, 6(E, - E , - E,) A(E,,E,) lEl)&)

(8)

where A(E,,E,) is the probability amplitude for the production of two photons of energies El and E 2 . The meaning of this entangled state of energy is that after the measurement of the energy of one of the photons, which results in a definite value El, there is an instantaneous collapse of the state of the system to the state IE,)IE, - El). This implies an instantaneous increase of the width of the wave packet of photon 2. We have investigated these issues in a modification of the single-photon Berry’s phase experiment described earlier (Fig. 3). An interference filter F1 was used to measure sharply the frequency (and hence the energy) of trigger photon 1, whereas the Michelson interferometer was used to measure the width of the wave packet of conjugate photon 2. After the trigger photon passed through the filter of narrow width AE and was detected, we observed that photon 2 of the pair collapsed into a broad wave packet of a duration At h/AE, upon coincidence detection. This is a nonlocal effect in that the photons can be arbitrarily far away from each other when the collapse occurs. Again, the anticorrelation parameter u was measured and found to

OPTICAL TESTS OF QUANTUM MECHANICS

45

violate any classical wave description by 9 standard deviations, thus ruling out all classical wave theory explanations of these collapse effects. In our experiment, two collapses are actually occurring: The first occurs inside the Michelson interferometer, where the wave packet size of photon 2 discontinuously changes upon the detection of the “remote” photon 1; the second collapse occurs after the last beam splitter, where, due to its indivisibility, photon 2 is either transmitted or reflected, but not both. The first collapse is temporal in nature, in that the uncertainty in the position of photon 2 in time is changed after the collapse occurs; whereas the second collapse is a spatial one, in that the uncertainty in the position of photon 2 in space is changed.

C . COLLAPSE EFFECTS IN FLUORESCENCE Two other studies of collapse have been provided by examining fluorescence in three-level atomic systems. One of them (Itano et al., 1990) is generally known as the quantum Zeno e$ect, although it is sometimes described as the watched pat efect, which comes closer to the true meaning. The precursor to this effect was the observation of quantum jumps (Nagourney et al., 1986). In the quantum jumps experiment, a strong laser field is used to check for a particle’s presence in the ground state (via resonance fluorescence), while a second field couples the ground state to a metastable state. Because the atom spends some time in the ground state and some time “shelved” in the metastable one, the fluorescence abruptly and stochastically turns on and off, seeming to indicate observable quantum jumps of the atom from state to state. In the related watched pot experiment, a transition is suppressed by the action of a laser field, which is periodically used to check whether the transition has occurred yet. If this check occurs frequently enough, it is most likely to collapse the atom back into its original state, preventing a sufficient dipole from building up to allow the transition to progress. The more frequently one turns on the probe laser, the less likely a transition is to take place. Both experiments tend to be interpreted in terms of collapse of the wave function, although this interpretation has been the source of much controversy (Fearn and Lamb, 1992). It has been shown (Frerichs and Schenzle, 1991) that no collapse postulate is necessary in order to understand them. [Recall Mott’s demonstration (1983) that quantum mechanics leads to the prediction of particle tracks in cloud chambers as jointprobability distributions, without the invocation of collapse.] Their interest lies in the fact that they relate the collapse picture directly to observable time evolution. While this evolution can be formally described in other ways (and should thus not be thought of as a proof of collapse), the ease of their interpretation in terms of collapse is a strong argument for the usefulness of the Copenhagen viewpoint.

46

R. Y. Chiao, P.G. Kwiat, and A.M. Steinberg

D. TWO-PHOTON INTERFERENCE

We have so far discussed two-photon experiments in which one of the photons is used solely as a trigger for the other one, which is then prepared very nearly in an ideal n = 1 Fock state (Hong and Mandel, 1986). An early experiment to demonstrate two-photon interference using the down-conversion light source was performed by Ghosh and Mandel(l987). They looked at the counting rate of a detector illuminated by both of the twin beams. No interference of the two beams was observed, because although the sum of the phases of the two photons emitted in parametric fluorescence is well defined (by the phase of the pump), their &Terence is not. However, when Ghosh and Mandel looked at the rate of coincidence detections between two such detectors whose separation was varied, they observed high-visibility interference fringes. Whereas in the standard two-slit experiment, interference occurs between the two paths a single photon could have taken to reach a given point on a screen, in this case it occurs between the possibility that the signal photon reached detector 1 and the idler photon detector 2, and the possibility that the reverse happened. This experiment provides a manifestation of quantum nonlocality, since at a null of these coincidence fringes, the detection of one photon at the first detector excludes the possibility of the detection of the other photon at the second detector. Such interference becomes clearer in the related interferometer of Hong et al., (1987), which will play an integral part in several of the experiments discussed later. A simplified schematic of this interferometer is shown in Fig. 4. The identically polarized conjugate photons from a down-conversion crystal are directed to opposite sides of a 50-50 beam splitter, such that the

nonlinear crystal FIG.4. Simplified setup for a Hong-Ou-Mandel interferometer (Hong et a!., 1987). Coincidences may result from both photons being reflected or both being transmitted. In a modification of this scheme, a half-wave plate in one arm of the interferometer (at point A) serves to distinguish these otherwise interfering processes, so that no null in coincidences is observed. Using polarizers before the detectors, one can “erase”the distinguishability, thereby restoring interference (Section 1V.A).

OPTICAL TESTS OF QUANTUM MECHANICS

47

transmitted and reflected modes overlap. If the difference in the path lengths prior to the beam splitter is larger than the two-photon correlation length (of the order of the coherence length of the down-converted light), the photons behave independentlyat the beam splitter,and coincidencecounts between detectors in the two output ports are observed half of the time-the other half of the time both photons travel to the same detector. However, when the two path lengths from the crystal to the beam splitter are nearly equal, such that the photons’ wave packets overlap at the beam splitter, the probability of coincidences is reduced,in principle to zero ifthe path lengths are identical. One can explain the coincidence null at zero path-length difference using the Feynman rules for calculating probabilities: Add the probability amplitudes of indistinguishable processes that lead to the same final outcome, and then take the absolute square of the resulting amplitude to find the probability of this outcome. The two indistinguishableprocesses that lead to coincidence detection in the preceding setup are both photons being reflected at the beam splitter (with Feynman amplitude r * r ) and both photons being transmitted (with Feynman amplitude t a t ) . The probability of a coincidence detection is then

where we have assumed a real transmission amplitude, and the factors of i come from the phase shift upon reflection at a beam splitter. The possibility of a perfect null at the center of the dip is indicative of a nonclassical effect. Indeed, Mandel (1983) has shown that 50% coincidence-fringe visibility is the maximum possible from classical field predictions. The tendency of the photons to travel off together at the beam splitter can be thought of as a manifestation of the Bose-Einstein statistics for the photons (Fearn, 1990). In practice the bandwidth of the photons, and hence the width of the null, is determined by filters and/or irises before the detectors-widths as small as 5 pm have been observed, corresponding to time delays of only 15 fs (Steinberg et al., 1992b). Consequently, one application of this phenomenon is the determination of various single-photon propagation times, with extremely high time resolution (see Section VIII).

IV. Complementarity A. QUANTUM ERASER The complementary nature of wave-like and particle-like behavior is commonly interpreted as follows: Due to the uncertainty principle, any attempt to measure the position (particle aspect) of a quantum will lead to an

48

R. Y. Chiao, P.G. Kwiat, and A.M. Steinberg

uncontrollable, irreversible disturbance in its momentum, thereby washing out any interference pattern (wave aspect) (Bohr, 1983). The measurement provokes an irreversible reduction of the state vector, irrevocably introducing an uncertainty in the phase. This picture, however, is incomplete; no state reduction, or collapse, is necessary to destroy wave-like behavior, and measurements that do not involve reduction can be reversible in a certain sense. To understand this phenomenon fully, one must view the loss of coherence as arising from an entanglement of the system wave function with that of the measuring apparatus (MA); this is identical to the first step in von Neumann’s measurement theory (von Neumann, 1983), but lacks the step in which the off-diagonal elements of the expanded density matrix are postulated to vanish. Through this entanglement, previously interfering paths can become distinguishable (assuming that the final MA states are orthogonal), such that no interference is observed. This is true even though one may not actually make subsequent measurements on the MA to determine which path actually occurred, i.e., even if one does not look at the result of the MA. Whenever welcher Weg (“which way”) information is available in principle about which possible path occurred, the paths are distinguishable, and no interference is possible. Interference may be regained, however, if one somehow manages to “erase” the distinguishing information, by correlatiqg the results of measurements on the interfering particle with the results of particular measurements on the MA. This is the physical content of quantum erasure (Hillery and Scully, 1983; Scully et al., 199 1). We have reported one demonstration of a quantum eraser, based on the Hong-Ou-Mandel interferometer described earlier (Fig. 4) (Kwiat et al., 1992). A half-wave plate inserted into one of the paths before the beam splitter serves to rotate the polarization of light in that path. In the extreme case, the polarization is made orthogonal to that in the other arm, and the reflection-reflection and transmission-transmission processes become completely distinguishable (because one could use this “label” to tell after the beam splitter which photon had taken which path); hence, the destructive interference that led to a coincidence null does not occur. The distinguishability can be erased, however, by using polarizers before the detectors afer the photons have left the beam splitter. In particular, if the initial polarization of the down-converted photons is horizontal, and the wave plate rotates one of photon’s polarizations to vertical, then a polarizer at 45” before each detector will restore the original interference dip (use of a polarizer in front of only one detector is not sufficient, because the photon traveling to the other detector still carries which-path information). If one polarizer is at 45” and the other at -45”, interference is once again seen, but now in the form of a peak instead of a dip (Fig. 5). It is characteristic of quantum erasers that there are four basic measurements possible on the MA (here the polarization)-two of which yield which-path information,

49

OPTICAL TESTS OF QUANTUM MECHANICS

400

350

100

50

-1110

-1090

-1070 -1050 -1030 Trombone prism position (microns)

-1010

-990

FIG.5. Experimental data and scaled theoretical curves (adjusted to fit observed visibility of 91%) for a quantum eraser with polarizer 1 at 45" and polarizer 2 at various angles. Far from the dip, there is no interference and the angle is irrelevant (Kwiat et a!., 1992).

one of which recovers the initial interference fringes (here the coincidence dip), and one which yields interference antifringes (the peak instead of the dip). In all cases, one must correlate the results of measurements on the M A with the detection of the originally interfering system. The primary lesson of the quantum eraser is that one must consider the total physical state, which in addition to photon spatial wave functions may include the state of the photon polarization or even the state of distant photons or atoms. If the coherence of the measuring apparatus is maintained, then interference may be recovered by correlating results of measurements on the original system with results of particular measurements on the MA.

B. VACUUM-INDUCEDCOHERENCE We now present a somewhat different demonstration of complementarity, involving two down-conversion crystals. In the experiment of Zou et al. (Greenberger et al., 1993; Zou et al., 1991), two nonlinear crystals, NL1 and NL2, are aligned such that the trajectories of the idler photons from each crystal overlap (Fig. 6). A beam splitter acts to superpose the trajectories of the signal photons. The basic interference effect arises between these signal

50

R Y.Chiao, P.G. Kwiat, and A.M. Steinberg

Fro.6. Schematic of setup used in the Zou et al. experiment (1991), with the possible inclusion of additional elements to make it suitable for a quantum eraser. In the absence of these elements, the identically polarized idler photons from either crystal are indistinguishable. Consequently, interference fringes may be observed in the signal singles rate (at detector D,) if any of the path lengths in the interferometer is varied.

photons as any of the path lengths in the interferometer is varied. If the path lengths are adjusted correctly and the idler beams overlap precisely, there is no way to tell, even in principle, from which crystal a photon detected at D, originated-interference results in the signal singles rate at D, (and also trivially in the coincidence rate between D, and DJ. If the idler beam from crystal NL1 is prevented from entering crystal NL2 (or even if the two idler beams are only slightly misaligned), then the interference vanishes, because the presence or absence of an idler photon at Di then labels the parent crystal. It is important to stress that the interference observable in the singles rate at D, is not due to a stimulated emission in NL2 by idler photons from NL1; by using a weak pump beam, the experimenters have effectively ensured that the probability of more than one pair of photons in the interferometer at one time is very small. Note that one could also make the two processes leading to a signal photon being detected at D, distinguishable by inserting a halfwave plate between the two crystals (at point A in Fig. 6), thereby rotating the polarization of idlers from NL1 to be orthogonal to the polarization of idlers from NL2. As in the previous example of a quantum eraser, one could recover interference by using a polarizer before Di (it is not necessary here to use a polarizer before D,) and correlating counts between the two detectors. If one employed fast detectors and a rapidly switchable polarizer, one could perform a delayed-choiceversion, in which the orientation of the polarizer was not chosen until after the signal photon had already been detected. That is, the decision to measure which-path information (particle-like behavior of the photons) or to regain interference (wave-like behavior) could be made after the originally interfering particles were detected'. Of course, only by correlating the measurements at D, and Di 'This is thus an extension of the original delayed-choice discussion by Wheeler (1979) and the experiments by Hellmuth et al. (1987) and Alley et al. (1986), in which the decision to display wave-like or particle-like aspects in a light beam may be delayed until after the beam has been split by the appropriate optics.

OPTICAL TESTS OF QUANTUM MECHANICS

51

could one determine the results of the decision, so there is no possibility of superluminalsignaling. This and other improved quantum eraser schemes are discussed by Kwiat et al. (1994a).

C. SUPPRESSION OF SPONTANEOUS PARAMETRIC DOWN-CONVERSION

A clever modification of the two-crystal experiment has recently been reported by Herzog et al. (Greenberger et al., 1993; Herzog et al., 1994); see Fig. 7. In this case only a single nonlinear crystal is used, but it is used in two directions. A given pump photon may down-convert in its initial right-ward passage through the crystal or in its left-going return trip (or not at all-the most likely outcome). As in the previous experiment, the idler modes from these two processes are made to overlap; moreover, the signal modes are also aligned to overlap. Thus, the left-going and right-going production processes are indistinguishable and interfere. The result is that fringes are observed in all of the counting rates (i.e., the coincidence rate and both singles rates) as any of the mirrors is translated. One interpretation of these results is as a variable enhancement (or suppression) of the spontaneous emission of the down-converted photons. In contrast to the cavity QED demonstrations discussed in Section 1I.A (Hinds, 1991), the distances to the mirrors can be much longer than the coherence lengths of the spontaneously emitted photons.

V. The EPR “Paradox” and Bell’s Inequalities A. GENERALITIES Nowhere is the nonlocal character of the quantum-mechanical entangled state as evident as in experiments involving the “paradox” of Einstein, Podolsky, and Rosen (EPR) (1935) and the related inequality by Bell (1964). Following the modified EPR-argument by Bohm (1983), we consider two photons traveling off back to back, described by the entangled polarization singlet-like state [cf. Eq. (7)]:

where the letters denote horizontal ( H ) or vertical ( V ) polarization, and the

R. Y. Chiao, P.G. Kwiat, and A.M. Steinberg

52

n

FIG.7. Schematic of the Innsbruck experiment to demonstrate enhancement and suppression of down-conversion (Herzog et al., 1994).

subscripts denote the photon propagation direction. This state is isotropic, in that measurement of any polarization component for one of the particles will yield a count with 50% probability; individually, each particle is unpolarized. Nevertheless, if we measure the polarization component of particle 1 in some basis, we can predict with certainty the value of particle 2’s polarization in the same basis, seemingly without disturbing it, since it may be arbitrarily remote. Therefore, according to EPR, one must ascribe an “element of reality” to that component of polarization. The same logic applies for all polarization bases, implying that the polarization is well defined along all directions simultaneously. Of course, a quantum mechanical state cannot specify that much information, and is consequently an incomplete description, according to the EPR argument. An intuitive explanation that was implied by EPR (though not explicitly stated) is that the particles leave the source with definite, correlated properties; these properties are presumably determined by some local “hidden variables” not present in QM. At the level of two entangled particles, a local hidden variable (LHV) theory can be advanced that correctly describes situations of perfect correlations or anti correlations (i.e., measurements made in the same polarization basis). Under these conditions, the choice of an LHV theory versus QM is a philosophical decision, not a physical one. It was only in 1964 that John Bell discovered that QM gives different statisticaz predictions than does any theory based on local realism, for situations of nonperfect correlations (i.e., analyzers at intermediate angles). In particular, he proved that any LHV theory’s predictions for certain coincidence probabilities must satisfy an inequality that certain quantum mechanical predictions violate.

OPTICAL TESTS OF QUANTUM MECHANICS

53

B. POLARIZATION-BASED TESTS Since Bell's inequalities were presented in 1964, they have been experimentally tested many times, and the vast majority of experiments have violated these inequalities (when several supplementary assumptions are made-see discussion later), in support of quantum mechanics. The initial tests were performed with pairs of photons produced via atomic cascades (Freedman and Clauser, 1972; Aspect et al., 1982; Clauser and Shimony, 1978). Unfortunately, the angular correlation of the cascade photons is not very strong, and this weakens the correlations in polarization (due to transversality). In contrast, the strong correlations of the down-converted photons make them ideal for such tests. The first such tests were performed by Shih and Alley (1988), and Ou and Mandel (1988), using setups essentially identical to that already discussed in connection with quantum erasure (Fig. 4). At the condition of equal path lengths, and with the half-wave plate oriented to rotate the polarization from horizontal to vertical, the state of light after the 50-50 beam splitter is:

I$)

1 =p& + i.il VlH2) + iIVlH1) + iIV2H2)l v2>

(11)

which reduces to (10) if one considers only terms that can lead to coincidence detections. A simple calculation reveals that the probability of coincidence detection depends only on the difference of the orientation angles of the two polarizers: P , a sin'(8, - O1). C. NONPOLARIZATION TESTS The advent of parametric down-conversion has also led to the appearance of several nonpolarization-based tests of Bell's inequalities, similar to those first proposed by Horne et al. (1989). In the experiment of Rarity and Tapster (1990), an entanglement of the momenta of down-converted photons is employed (Fig. 8). By use of small irises (labeled A in the figure), only four down-conversion modes are examined: Is, li, 2s, and 2i. Beams 1s and li correspond to one pair of conjugate photons; beams 2s and 2i correspond to a different pair. However, photons in beams 1s and 2s have the same wavelength, as do photons in beams li and 2i. With proper alignment, therefore, after the beam splitters there is no way, even in principle, to tell whether a pair of photons was previously in the ls-li or the 2s-2i paths. Consequently, although the singles rates at the four detectors indicated in Fig. 8 remain constant, the rates of coincidences between detectors display interference.This interferencedepends on the difference of phase shifts induced by rotatable glass plates Pi and P, in paths l i and 2s, respectively,and is formally

54

R. Y. Chiao, P.G. Kwiat, and A.M. Steinberg

FIG.8. Outline of apparatus used to demonstrate a violation of a Bell's inequality based on momentum entanglement (Rarity and Tapster, 1990).

equivalent to the polarization case considered earlier, in which it is the difference of polarization-analyzer angles that is relevant. By measuring the coincidence rates for two values for each of the phase shifters-a total of four combinations-the experimenters were able to violate an appropriate Bell's inequality, after making the standard supplementary assumptions. One interpretation is that the paths taken by a given pair of photons are not elements of reality. A different Bell's inequality proposal, based on energy-time entanglement of the photons, was made in 1989 by Franson (1989), and implemented by several groups (Brendel et al., 1992; Kwiat et al., 1993a; Shih et al., 1993). A schematic of the basic setup is shown in Fig. 9. Each member of a down-converted pair is directed into an unbalanced Mach-Zehnder interferometer, allowing both a short and long path to the final beam splitter. Correlations between detectors at two of the output ports are examined. For sufficiently large interferometer imbalances (i.e., larger than the 60-pm coherence length of the down-converted photons), no interference fringes are present in the detector singles rates. Nevertheless, fringes in the coincidence rate are predicted (as long as the difference in the path-length differences of the interferometers is less than 60 pm), arising from interference of the two indistiguishable processes that could lead to coincidence detection-"shortshort" and "long-long." If the detectors are fast enough, it becomes possible to exclude the background of noninterfering processes in which one photon

FIG.9. Simplified schematic of a Bell's inequality experiment based on energy and time correlations as proposed by Franson (1989).

55

OPTICAL TESTS OF QUANTUM MECHANICS

-*

-I$* = 0'

-I$*

Visib. = 92.8 It 0.9%

700

= 45' Visib. = 90.3 f 0.7%

600

N

-

z-

500

YI

CIS

'1

e, U

400

N

!:

300

.6

200

Ee CI

I00 0 0'

45'

90'

135'

180'

225'

Interferometer 1 phase shift,

FIG.10. High-visibility coincidence fringes in a Franson dual-interferometerexperiment, for two values of the phase in interferometer 2, as the phase in interferometer 1 is slowly varied. The curves are sinusoidal fits.

takes its short path and the conjugate photon takes its long path; the resulting reduced quantum mechanical state [to be compared with (lo)] is

where the letters indicate the short or long path, and the phase is proportional to the sum of the relative phases in each interferometer. The highvisibility fringes arising from (12) lead to a violation of an appropriate Bell's inequality (Brendel et al., 1992; Kwiat et al., 1993a). In particular, sinusoidal fringe visibilities in excess of 90% were observed (Fig. lo), whereas the maximum possible for a local hidden variable model is only 71%. One conclusion of such a violation is that it is incorrect to ascribe to the photons a definite time of emission from the crystal, or even a definite energy, until these observables are explicitly measured. A more general interpretation, applicable to all violations of Bell's inequalities, is that the predictions of quantum theory, which seem to be supported by nature, cannot be reproduced by any completely local theory. As discussed in slightly different ways by Jarrett (1984) and Shimony (1990), it must be that the results of measurements on one of the photons depend on the results for the other, and these correlations are not due to a common cause at their creation.

D. LOOPHOLES AND LOOPHOLE-FREE TESTS No test of Bell's inequalities to date has been incontrovertible, due to several loopholes that reduce the true impact such an experiment might yield. One of these, the angular correlation problem, has essentially been solved by

56

R. Y. Chiao, P. G. Kwiat, and A.M. Steinberg

turning to down-conversion light sources over cascade sources, leaving the fast-switching loophole and the detection loophole. Only one Bell-type experiment, that of Aspect et al. (1982), has made any attempt at all to close the former, but even in that experiment the loophole remains. Although the experiment used rapidly varying analyzers, the variation was not random, and it has been argued that the speed of the polarization switching was not sufficient to disprove a causal connection between the analyzer and the source (Franson, 1985; Zeilinger, 1986). The detection loophole arises from the nonunit detectionefficiency in any real experiment. If the efficiency is sufficiently low, then it is possible for the subensemble of detected pairs to give results in agreement with quantum mechanics,even though the entireensemblesatisfies Bell’s inequalities.Due to the inadequacy of existing detectors,experimentshave so far employed an additional assumption, roughly equivalent to the “fair-sampling” assumption that the fraction of detected pairs is representative of the entire ensemble (Clauser et al., 1969;Santos, 1992).Formerly,it was believed that 83% was the lower efficiency limit needed to close this loophole experimentally.However,Eberhard (1993) has shown that by using a nonmaximally entangled state (i.e., one where the magnitudes of the probability amplitudesof the contributingterms are not equal), one may reduce the detector requirement to 67Y0,in the limit ofno background. At present, we know of only two efforts under way to attempt a loophole-free test of Bell’s inequalities. One relies on the recent development of highefficiency single-photon detectors, with measured efficiencies of 75 %, and the possibility of even further improvements (Kwiat et al., 1993b). Also central to this approach is a novel two-crystal down-conversion arrangement (Kwiat et al., 1994b),in which it is not necessary to discard half of the counts to produce a polarization singlet-like state, unlike most of the previous down-conversion schemes [cf. Eq. (1l)]. A very different proposal has been made by Fry and Li (1991),who for the first time proposed dissociated atoms (mercury dimers) as the correlated particles. The advantage is that detection efficiencies Of 95% are possible by photoionizing the atoms and detecting the photoelectrons.

-

-

VI. Related Issues A. INFORMATION CONTENT OF A QUANTUM It is now reasonably well known that it is impossible to “clone” photons. That is to say, given an unknown polarization state of a single photon, it is impossible to create a second photon in the same state, without destroying the first one. If cloning were possible, one could communicate instantaneously over arbitrarily large distances using EPR correlations. If two

OPTICAL TESTS OF QUANTUM MECHANICS

57

separated scientists each receive a particle from an EPR pair, one of them (the “sender”) can choose to measure the spin in any basis. This collapses the “receiver’s” particle into a corresponding spin state, depending on the choice and random outcome of the measurement. Tracing over the state of the sender’s particle to find the effective density matrix for the receiver’s, we always find the density matrix of an unpolarized particle, with no information about which basis the sender used. The message of which measurement the sender made cannot be decoded by the receiver. However, if one or more “clones” of the receiver’s photon existed, a set of polarization measurements could in fact determine the precise polarization of this photon, to an accuracy in bits equal to the number of copies available. Thus to preserve causality (the impossibility of such an instantaneous transfer of information), photon-cloning must be impossible (Glauber, 1986; Wootters and Zurek, 1982). Even though a photon may be in a definite polarization state (described by an angle drawn from a continuous distribution, and thus containing, in principle, an unlimited number of bits of information), no experiment can extract more than 1 bit of information about the polarization of a single photon: In the EPR example, this bit represents not the choice of measurement made by the sender, but that measurement’s random outcome. This makes the recent work by Bennett et al. (1993) on quantum teleportation all the more remarkable. They found that an unknown polarization state (with its, in principle, infinite amount of information)can be “teleported”to a distant lab by means of transmitting a mere 2 bits of classical information, as long as the distant lab ( “ B ) and the one with the particle it wishes to teleport (“A”) each possesses one of a pair of EPR particles. Thus, although one may only extract 1 bit of (normally useless) information from an EPR particle, the perfect correlations may be used to transfer an infinite amount of information, i.e., precise specification of a point in the state space of the particle (the Poincark sphere for a photon or the Bloch sphere for an electron). The basic idea of the scheme is that the joint system in A’s possession, consisting of one particle of the EPR pair and a second particle in the unknown state 14>, has a total of four basis states. (We consider spin1/2 particles for simplicity.) These basis states can be described in terms of the two possible spin projections of each of the two particles, or alternatively in terms of the total angular momentum and spin projection of the joint system. Envision A performing a measurement in the latter basis and finding the system either in the singlet state or in one of the three triplet states. The process works in all four cases, but is easiest to understand for the singlet, which (as alluded to in Section V) corresponds to perfect anticorrelation of the spins of the two particles, regardless of the choice of quantization axis. This anticorrelation implies that upon A’s detection of a singlet state, his member of the EPR pair collapses into a state with spin exactly opposite that of 14 >. But the two EPR particles themselvesare (by definition)in a singlet state:

58

R. Y. Chiao, P.G. Kwiat, and A.M. Steinberg

FIG.11. A simple chart, demonstrating the use of EPR-correlated photon pairs for cryptography. Each collaborator measures the polarization of one of each EPR pair in a linear or circular basis, chosen randomly. Lines 1 and 2 depict their results. The collaborators determine (via pub& communication) for which pairs they used the same measurement basis, and keep only these results (line 3). The correlations ensure that they both now possess the same random binary key (line 4) (Collins, 1992).

then B s particle has spin exactly opposite to A’s, which is to say identical to 14 > . Experimenter A need only send B the 2-bit message that of the four possible outcomes, a singlet state was found, and B knows that she has a perfect copy of the particle A was studying. This “teleportation” scheme works even if neither experimenter had any prior information about the state 14 >. The “no cloning” theorem is not violated, since the state of A’s particle is irrevocably altered by the measurement he performs, leaving only one particle in the original state,

B. CRYPTOGRAPHY As discussed earlier, the EPR schemes cannot be used to send any signals

superluminally;nevertheless,the proposal has been made that these be utilized for a different kind of communication, namely, cryptography. In the “one-time pad” of classical cryptography (Brassard, 1988), two collaborators need to share a secret “key” (a random string of binary digits) in order to encode and decode a message. Such a key may provide an absolutely unbreakable code, provided that it is unknown to any potential eavesdropper. The problem arises in key distribution: Any classical distribution scheme is subject to noninvasive eavesdropping, for example, using a fiber-coupler to tap the line, without disturbing the transmitted classical signal. In the quantum cryptography proposals and demonstrations made to date, the security is guaranteed by using single-photon states (Bennett et al., 1992; Ekert, 1991); some of the schemes employ particles prepared in an EPR-entangled state. [Alternatively, the sender can send photons in a state of random, but dejinite, polarization (Bennett, 1992); equivalently, some schemes use phase (Ekert et al., 1992).] Each collaborator receives one member of each correlated pair, and measures the polarization in a random basis. After repeating the process many times, the two then discuss publicly which bases were used for each measurement, but

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not the actual measurement results. The cases where different bases were chosen are not used for conveyingthe key, and may be discarded,along with instances in which one party detected no photon. In cases where the same bases were used, however, the participants will now have correlated information. From this, a random, shared key can be generated (Fig. 11). As long as single photons are used, any attempt at eavesdropping, even one relying on QND, will necessarily introduce errors due to the uncertainty principle. For if the eavesdropper uses the wrong basis to measure the polarization (phase) of a photon before sending it on to the real recipient, the very act of measuring will disturb the original state.

C. NONLOCALITY WITHOUT INEQUALITIES In the preceding experiments for testing nonlocality, the prediction of local realistic theories and those of standard quantum theory need not be contradictory in situations of perfect correlation. (Here we mean situations in which one can predict with certainty the results of a measurement on one particle, given the outcomes of measurements on other particles of the system; this is the EPR criterion for an “element of reality.”) Indeed, one might even say it was the very fact that LHV theories could intuitively explain such cases that led EPR to conclude that quantum mechanics was incomplete. As Bell showed, the nonlocality of a singlet-like entangled state becomes manifest when statistical tests are made in cases of nonperfect correlation. Recently, Greenberger, Horne, and Zeilinger (GHZ) proved that no local theory is consistent with even the perfect correlations predicted for some systems involving three or more entangled particles (Greenberger et al., 1990; Greenberger et al., 1989; Mermin, 1990). One can thus prove the incompatibility of local realistic theories with quantum mechanics without the need for inequalities.(In practice, their incompatibility with nature can only be tested with inequalities.) A schematic of one version of the GHZ apparatus is shown in Fig. 12. It is essentially a three-particle generalization of the Rarity-Tapster experiment discussed earlier (cf. Fig. 8). The source at the center emits trios of correlated particles, just as a down-conversion crystal produces pairs. Just as the Rarity-Tapster experiment selected two pairs of photons, the GHZ source selects two trios of photons; these are denoted by abc and a’b‘c’. Hence, the state coming from the source may be written

I$)

1

= -((labc)

Jz

+ I a’b’c’))

After passing through a variable phase shifter (e.g., Cp,), each primed beam is recombined with the corresponding unprimed beam at a 50-50 beam splitter. Detectors (denoted by Greek letters) at the output ports of all three

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R. Y. Chiao, P.G.Kwiat, and A.M. Steinberg

FIG.12. A three-particle gedanken experiment to demonstrate the inconsistency of quantum mechanics and any local realistic theory. All beam splitters are 50-50 (Greenberger et al., 1993).

beam splitters allow one to record the occurrence of triple coincidences, involving one detector for each beam splitter. A detailed discussion of this experiment is beyond the scope of this review and is given in Greenberger et al., (1990); here we present a simplified argument to convey the spirit of the GHZ result. Given the state (13), one can calculate from standard quantum mechanics the probability of a triple coincidence as a function of the three phase shifts, assuming ideal optical elements and perfect detectors:

where the plus sign applies for coincidences between all unprimed detectors, and the minus sign for coincidences between all primed detectors. Consider first the case in which all phases are equal to 0; then occasionally (one-eighth of the time) there will be a triple coincidence of all primed detectors. Using what is known as a counterfactual approach, we can ask what would have happened if 4a had been 4 2 instead. We assume that this would not have changed the state from the source, and by the locality assumption, it certainly would not have changed the fact that detectors pand y’ went off. But from (14) probability of a triple coincidence for primed detectors is zero in this case; therefore, we can conclude that if r$o had been 4 2 , then detector a would have “clicked.” The exact same logic implies that if 4bhad been n/2, detector p would have clicked, and if r$c had been n/2, detector y would have

OPTICAL TESTS OF QUANTUM MECHANICS

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clicked. Consequently, if all the phases had been equal to n/2, we would have seen a triple coincidence between unprimed detectors. But according to (14) this is impossible: The probability of triple coincidences between unprimed detectors when all three phases are equal to 4 2 is strictly zero! Hence, if one believes the quantum mechanical predictions for these cases of perfect correlations, it is not possible to have a consistent local realistic model. Lucien Hardy was the first to show that one could, at least in a gedanken experiment, demonstrate nonlocality without inequalities using only two particles. His first proposal relied on electron-positron annihilation (Hardy, 1992a); later he proposed a quantum optical version of the experiment (Hardy, 1992b), relying on down-converted photon pairs. The argument is similar to that for the GHZ case. Central to the arguments used to infer nonlocality in the GHZ and Hardy experiments is the ideality of all components-to satisfy EPRs criterion of reality, one must be able to predict with certainty the results of particular measurements. This is never possible in any real experiment, so the arguments must be modified somewhat, as GHZ (Greenberger et al., 1990), Hardy (1992b), and Eberhard and Rosselet (1994) have discussed. This done, one is left once again with an inequality that can only be violated with high-efficiency detectors (or by making supplementary assumptions). Specifically, detection efficiencies of 91% are needed for the GHZ scheme, and greater than 98% for the Hardy scheme! Moreover, one must still contend with the rapid-switching loophole in any test of nonlocality. Given this, it seems more likely that a loophole-free test of nonlocality would be based on Bell’s inequalities, where the requirements are less stringent. Although demonstrations of nonlocality without inequalities exist only at the gedanken level (and generalizations to real experiments seem to have even more difficult experimental constraints than loophole-free tests of Bell’s inequalities), they are important because of their pedagogical value. In the original EPR argument, it is assumed that quantum mechanical calculations yield correct predictions for the case of perfect correlations. A more intuitive, local model is then espoused, which explains the correlations as being due to elements of reality, essentially arising at the source of the particles. The meaning of the GHZ result, and also the Hardy result, is that one is not even able to explain the perfect correlations of quantum mechanics with a local realistic model.

VII. The Reality of the Wave Function A. TESTOF THE DE BROGLIE GUDED-WAVE THEORY The interpretation of the quantum-mechanical wave function t+h was a subject of no small import to the founders of quantum theory. Although it

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is now common to interpret ij as simply a mathematical tool for calculating probabilities of measurement outcomes, other approaches have been suggested. The guiding wave (sometimes called a pilot wave) model of de Broglie is one such alternative, according to which the wave function describes a real physical wave (de Broglie, 1969). In the case of light, this approach describes both photons and electromagnetic waves guiding them, much like ocean waves guide a surfer. According to one version of the de Broglie model (Croca et al., 1990), these waves are assumed to exist in ordinary three-dimensional space, so that waves associated with different particles may interfere. It is then possible to test experimentally the validity of the picture [at least this version (Holland and Vigier, 1991)] as it leads to predictions different from those of standard quantum theory. An experiment of this type, using down-conversion photons, was performed by Wang et al. (Fig. 13). First consider the singles rate at detector D,: Assuming 50-50 beam splitters and ideal detectors, the signal photons (s) will be detected at D, one-fourth of the time, whereas idler photons ( i ) never make it to this detector. Idler photons can only be detected by D,, and this occurs one-fourth of the time. To arrive at D,, the signal photons may bounce off either BS, or BS,, which act like the mirrors of a Michelson interferometer. Hence, the rate of counts at D, from signal photons depends on the interferometer path imbalance Ax, and the total rate at D, displays fringes with visibility 50% (again, for 50-50 beam splitters). These considerations apply equally to both conventional quantum theory and the de Broglie approach. Differences between the theories are manifested in the rate of coincidences between D, and D,. In the conventional theory the coincidence rate is a constant (equal to one-sixteenth the total rate of down-converted pairs), because there is only one possible way for a coincidence event to occur-idler photon detected at D,; signal photon detected at D,, after passing through beam splitters BS, and BS,. According

uv

.

J

LiO,

I

FIG.13. Simplified schematic of experiment to test one version of a de Broglie guiding-wave theory. All beam splitters are 50-50 (Wang et al., 1991).

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to the version of the de Broglie picture tested here, however, even if a signal photon is detected at D,, the resulting empty signal wave propagating to D, interferes with the idler wave. Because the signal wave depends on Ax, so too does the total guiding wave acting on the idler photon, and coincidence fringes of 50% visibility are predicted. The results of Wang et al. (1991) strongly confirm the quantum calculation, disproving this particular version of the de Broglie theory.

MODELOF QUANTUMMECHANICS B. BOHM’SDETERMINISTIC While de Broglie’s early picture of real waves carrying particles along like surfers fails to explain some of the deeper surprises in quantum mechanics, Bohm’s natural extension of his work fares much better. This extension differs from the simpler de Broglie model in that it sees the wave function as existing in configuration space, not in real space, as it must in order to replicate the quantum mechanics of more than one particle. While it is a realistic model, it is not local. Bohm demonstrated (Bohm, 1952; Bohm and Hiley, 1993; Holland, 1993b) that quantum mechanics is exactly equivalent to a theory of classical particles that undergo deterministic evolution and interact with a “quantum potential” determined by the wave function Y = Reis” (which obeys the Schrodinger equation). In particular, if we assume that the initial distribution of these particles is described by )‘PI2= R 2 as in the Born interpretation of quantum mechanics and a similar relation for their initial momenta, and we assume that they interact with a physical field Y according to the equation of motion mf =

- V [ V ( x ) - (h2/2rn)V 2 R / R ]

then their distribution at later times will agree exactly with the quantum mechanical prediction. This is equivalent to saying that the ensemble of particles, described by the probability distribution R2(x),follows a velocity field

v ( x ) = V S(x)/m The difference between this picture and the Born interpretation is that we need only assume the initial distribution of the particles. As time passes, deterministic evolution of the state of each particle preserves the equivalence between IYIz and the probability density, not as a hypothesis but as a mathematical consequence. The Y-field is not directly detectable, but serves

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merely to “guide” the particle. Each particle has a well-defined position at all times,’ and for this reason the picture is philosophically more palatable to many people, although it is sufficiently unwieldy that most physicists disregard it. Perhaps its most serious problem is that the real trajectories it predicts seem completely unphysical, unless it can be shown that these trajectories lead to some conclusion that can be tested experimentally and are more difficult to extract from standard quantum mechanics. The origin of the model’s idiosyncracy is the fact that (16) has the mathematical form of a fluid-flow equation. The various particle trajectories describing the members of an ensemble may never cross, because the velocity field is a single-valued function of position. How, then, is a two-slit experiment described? If all the trajectories are calculated, it will be seen that none of them crosses the symmetry plane. Quite contrary to our mechanistic intuitions, all the particles that pass through the upper slit “bounce” off the central plane, and contribute to the upper half of the interference pattern. Does this not contradict experiment, in that if we insert a measuring device to record which slit is traversed, the screen records the diffraction pattern of this slit, extending on both sides of the symmetry plane? No, not at all. For the trajectories of Bohm particles are (nonlocal) hidden variables, entirely inaccessibleto measurement. If we introduce some measuring device, this contributes another degree of freedom to the configuration space in which Bohm’s “quantum potential” and thus the velocity field resides; the trajectories may now cross in real space without doing so in configuration space. The Bohm theory is in agreement with our intuitions about the particle trajectories under these conditions. Unfortunately, the conclusion seems to be that the information added by Bohm to our understanding of quantum mechanics is only valid as long as we do not attempt to make any use of it! Perhaps there is some indirect effect of these trajectories that will afford the model some predictive power. As an analogy, we no longer believe in Bohr’s planetary description of the hydrogen atom, yet his model correctly describes features that are less evident in modern quantum mechanics.

c. APPLICATIONOF BOHM’STHEORY TO TUNNELING TIMES One attempt to apply Bohm’s theory has been introduced by Leavens and Aers in the context of the tunneling-time controversy (Leavens, 1990a, 1990b; Leavens and Aers, 1993). It has long been known (Biittiker and *At present, the relativistic generalization of the Bohm model does not ascribe real trajectories to photons in the same way as to electrons (Bohm and Hiley, 1991; Holland, 1993a). However, the phenomena discussed in this paper generally have electronic analogs and involve quasi-free propagation of a sort that should be interpretable in similar terms. We implicitly consider the Bohm theory of the electron throughout this chapter, even in contexts where the experiments have been performed with photons.

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Landauer, 1982; Hauge and Stmneng, 1989; Wigner, 1955) that there are difficulties associated with calculating the duration of the tunneling process, due to the fact that evanescent waves do not accumulate any phase. First of all, the momentum in the barrier region is imaginary, so the most naive semiclassical traversal time is not a real quantity. Second, if one applies the stationary phase approximation to calculate the group delay-which we define as the time of arrival of a wave packet peak at the far side of the barrier, relative to the time at which it would have arrived at the same point had no barrier been present-the insensitivity of the phase to barrier thickness leads to a delay that tends to a constant as the thickness diverges, in seeming violation of relativistic causality (Bolda et al., 1993). As we explain later, we have recently confirmed this striking prediction experimentally, finding effective tunneling velocities as high as 1.7~.It is not trivial to overcome this latter problem in the same way that one deals with superluminal group velocities in classical optics-as pulse reshaping effects (Siegman, 1986). Since the standard interpretation of quantum mechanics does not associate definite trajectories with individual particles, it is tempting to try to define some more meaningful “interaction time” for tunneling, which might accord better with our relativistic intuitions and perhaps even have implications for the ultimate speed of devices that rely on tunneling (Biittiker and Landauer, 1985). Various proposals have been made along these lines, and some experiments have even been done (Landauer, 1989), but by necessity they all consider modified experimental setups. In general, some additional interaction must be introduced to serve as a “clock,” and as we have learned from Bohr (1983), it is only when we “refer only to observations obtained under circumstances whose description includes an account of the whole experimental arrangement” that quantum mechanics can provide an unambiguous description of a phenomenon. One scheme that might describe tunneling times without introducing experimental modifications is the Bohm-de Broglie pilot wave model. As explained earlier, this model does define trajectories for individual particles. Thus one can calculate a distribution of traversal times for the tunneling particles using this model, and in general they do not share the apparent superluminality of the group delay. Like the trajectories for the particles in the two-slit experiment, however, these trajectories display some very odd features. In particular, it is found that all the transmitted particles correspond to those particles that were nearer to the leading edge of the incident pulse; this explains how they can arrive faster than had the incident peak traversed the barrier at c, though they themselves never travel superluminally. This behavior is due to the absence of crossing trajectories in the theory: If a particle that originates at a time t o is transmitted, all particles that originate earlier than to must likewise be transmitted; otherwise, their trajectories on reflection would intersect that of the first particle considered. This all-or-nothing behavior has a very odd consequence. Consider a

R. Y. Chiao, P.G. Kwiut, and A.M. Steinberg

66

3 N

0

4

8

t

12

[lo-16sl

FIG.14. Bohm trajectories for a Gaussian wave packet with E , = 5 eV, Ak = 0.04 k ’ incident on a static barrier Vo = 10 eV (solid curves) and an oscillating barrier (dashed curves). Only the earliest arriving trajectories make it across the barrier. (Reprinted from Leavens and Aers, 1991, 1015-1023, with kind permission from Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 OBW, UK.)

narrow-bandwidth wave packet incident on a tunnel barrier with a 1% transmission. Quantum mechanics and the Bohm model both predict that the group delay does not depend on the precise bandwidth; as long as the bandwidth is small enough for the method of stationary phase to apply, the location of the transmitted peak is essentially constant. But the location of the leading 1% of the incident wave packet (where all of the particles predicted by the Bohm theory to be transmitted are said to originate) is not bandwidth independent. In fact, its initial location (relative to the peak) diverges as the bandwidth of the wave packet tends toward zero. Clearly, for small incident bandwidths, the Bohm trajectories for the transmitted particles must be delayed enormously by the tunnel barrier, because they arrive at the barrier long before the incident peak, but leave shortly after the peak‘s arrival. This is confirmed by direct calculation (see Fig. 14). Although this behavior does not actually violate any principle of physics, it is difficult for many physicists to believe that such oddly behaved trajectories could possess physical significance. The fact that only the particles in the early part of the wave packet are transmitted accords well with intuitions one might have from considerations of classical wave propagation. As discussed in Chu and Wong (1982) and Garrett and McCumber (1970), superluminal group velocities of a classical wave can be understood as a “reshaping” of the leading edge into an early,

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scaled-down version of the incident pulse. In the context of tunneling, we understand this as a sort of coherent transient effect whose times scale depends only on energy differences, not barrier thicknesses. At a physical level, the reflection from a multilayer dielectric, for example, occurs because of coherent multiple reflections between the different layers. Over time, a steady-state field builds up inside the structure, which interferes destructively with the incident field at the exit face and constructively at the entrance face. At early times, before this field reaches its steady-state value (which is critically dependent, of course, on the frequency of the incident field), there is little destructive interference, and a significant part of the wave is transmitted. At later times, there is little transmission. This preferential treatment of the leading edge engenders a sort of optical illusion, shifting the transmitted peak earlier in time.

D. STATUSOF THE WAVE FUNCTION IN MEASUREMENT The odd behavior of these tunneling wave packets pushes us to reexamine our interpretation of wave packets. Is it true that a particle’s position should never be thought of as more precisely determined than the wave function suggests? Or does the tunneling phenomenon-coupled with our knowledge of causality-imply that location really is an “element of reality,” in that once we see a transmitted photon we can say with certainty that it originated toward the front of its wave packet? These questions raise difficult issues of quantum philosophy. We are accustomed to using the wave function as a calculational tool without pondering such questions, but we have also learned that the wave function is in some sense physical, and should not be regarded merely as some distribution from classical statistics. A recent proposal by Aharonov et al. (1993; Aharonov and Vaidman, 1993) demonstrates that under certain carefully controlled conditions, a singleparticle wave function may be measured. When a state is “protected” from change, for example, by an energy gap, and measurements are performed sufficiently “gently” (i.e., slowly enough that no excitations across the energy gap are likely), one should be able to determine not just the expectation value of position but the wave function at many different positions, without altering the state of the particle. No violation of the uncertainty principle or of the no-cloning theorem (see Section V1.A) arises from this, because the ability to “protect” a state relies on some preexisting knowledge about the state, but it assigns a deeper significance to the wave function, one Aharonov terms ontological, as opposed to merely epistemological. For instance, it would certainly be illustrative to measure a particle’s wave function, allow it to tunnel, and then measure its new wave function, thus removing all questions about how to interpret the superluminal peak motion.

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E. OBSERVABLE EFFECTS OF “EMPTYWAVES” The real (but undetectable) fields of the Bohm model are similar to the “vacuum fluctuations” of quantum field theory. This can be seen particularly well by considering a gedanken experiment recently proposed by Elitzur and Vaidman (1993), although the experiment can be described perfectly well by standard quantum mechanics as well. This experiment consists of a simple single-particle interferometer, a Mach-Zehnder, for example. Particles are injected one at a time into one input port. The path lengths are adjusted so that all the particles leave a given output port (A), and never the other (B). Now we suppose that a “ b o m b is inserted into one of the interferometer‘s two arms. The hypothesis is that this is an infinitely sensitive bomb, such that interaction with even a single photon will cause it to explode. However, there is some chance that the bomb is a dud, that is, that it lacks the sensitive trigger; in this case, incident photons sail through the bomb with no consequences. Unfortunately, classical intuition is quite clear: Any attempt to check for the presence of the trigger involves interacting with it in some way, and by hypothesis this will inevitably set it off, rendering it useless. Elitzur and Vaidman’s insight is that quantum mechanics gives us another option, allowing us some portion of the time to be certain that the bomb is functioning, without setting it off. This point is in some sense complementary to the usual picture of quantum mechanics as denying the reality of any phenomena not directly observed: In contrast to classical mechanics, we can learn something about reality without (in a technical sense) “observing” it. If the trigger is missing, the interferometer functions as before. All incident particles exit output port A. After a sufficient number of attempts, we can be fairly certain that the bomb is a dud. The same would occur classically. But now consider what happens when a functioning bomb is inserted and a photon enters the interferometer. A Mach-Zehnder interferometer begins with a beam splitter, after which the photon has a 50% chance of heading toward the bomb. If this occurs, the bomb explodes, and we are no better off than the classical physicists down the hall. On the other hand, if the photon takes the path without the bomb, there is no more interference, since the fact of the bomb not exploding provides welcher Weg information. Thus the photon reaches the final beam splitter and chooses randomly between the two exit ports. Some of the time, we see a photon at output port B, something which never happened before the bomb was inserted. This immediately tells us that a trigger is in place-even though (since the bomb is unexploded) the trigger has not interacted with any photon. It is the possibility that the trigger could have interacted with a photon that destroys the interference. How can the final state at the detctors depend on what is in a given arm of the interferometer, when we simultaneously know that no particle has gone through that arm? According to the Bohm theory, of course, the

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answer is straightforward: The real Y-field enters both arms, but is incapable of setting off the bomb. At the final detector, Y (which now depends on whether or not the bomb is a dud) “pilots” the particle accordingly. This interpretation is surprisingly similar to one orthodox description of the two-crystal welcher Weg experiment performed by Mandel and described in Section 1V.B. In such a picture, space is permeated by electromagnetic vacuum fluctuations, real (but random) fields that have measurable effects, but do not consist of any photons. These fields are part of the electromagnetic ground state, so no energy can be extracted from them and they will thus never trigger photodetectors such as those considered by Elitzur and Vaidman, according to the treatment originally due to Glauber (1963). At a beam splitter, however, these vacuum fields can interfere with other incident fields (Caves, 1981, 1987) and influence the port of the beam splitter from which a photon will exit. The placement of a functioning bomb in the path of the vacuum fields destroys the coherence at the final beam splitter, on which interference re lie^.^ The choice the photons make at the final beam splitter has thus been affected not by direct action of the bomb on any photons, but by its influence on the random fields permeating the vacuum.

VIII. The Single-Photon Tunneling Time A. AN APPLICATIONOF EPR STATES: DISPERSION CANCELLATION The crucial element in the Einstein-Podolsky-Rosen argument is the observation that while position and momentum are noncommuting operators, the difference of two positions and the sum of two momenta commute (Einstein et al., 1935). In their idealized example, the Wigner distribution

represents a pair of particles, each with completely uncertain position and momentum, the sum of whose momenta is precisely 0 and the difference of whose positions is precisely qo (Bell, 1986). As discussed in Section V.C, this is analogous to the state of the photon pairs emitted in parametric

3Even if the bomb’s trigger is of the QND sort, the number-phase uncertainty relation guarantees this.

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fluorescence, which exhibit simultaneous correlations in time-of-arrival (the analog of position) and in wave vector (the analog of momentum); these correlations exceed by many orders of magnitude what would be allowed by the uncertainty relation for the position and momentum of a single photon. This suggests that apart from its importance in elucidating philosophical questions tied to quantum theory, the EPR state has practical importance in high-resolution timing experiments. Due to the uncertainty principle, in order to perform such experiments, it is necessary to introduce a large energy uncertainty, or bandwidth. Such a large bandwidth means that even normally small dispersive effects, such as those found in glass or even air, can be of great importance, rapidly broadening a pulse. Until recently, it seemed that this trade-off between pulsewidth and bandwidth was unavoidable, leading to a fundamental limit on the resolution of pulse propagation experiments in dispersive media (Franson, 1992).However, one can capitalize on the EPR correlations of down-converted photons to make high-resolution time measurements essentially immune to dispersive broadening (Steinberg et al., 1992ab). The Hong-Ou-Mandel interferometer (see Section 1II.D)can be used to measure delay times and we shall see that in this setup the dispersion is sensitive only to the sum of the two photons’ energies, while the time measurement is sensitive to the diference of the two emission times! Because these are compatible observables, they may be simultaneously specified arbitrarily well, thus preserving measurement accuracy. The effect can be understood intuitively by applying Feynman’s rules for interference more carefully, taking into account once more the fact that outcomes that are in principle distinguishable do not interfere. First of all, recall the discussion of Section III.D, which showed that no coincidences are observed in a perfectly aligned Hong-Ou-Mandel interferometer. The interference effect only occurs as long as the reflection-reflection and transmission-transmission processes are fundamentally indistinguishable, which is only the case if the two photons reach the beam splitter at the same time. To be indistinguishable, the two processes must also “agree” as to what frequency photon is seen by each detector. As can be seen in Fig. 15, this means that in some sense, each interferometer arm samples both of the anticorrelated frequencies, leading to an automatic cancellation of any first-order dispersive broadening. This is why the interferometer can be used for precise time-delay measurements; if a delay greater than the coherence time of about 20 fs is inserted in one arm, the coincidence null vanishes until this delay is compensated, for example, by piezoelectrically translating a mirror by some fraction of a micron. Such measurements can be more than 5 orders of magnitude more precise than would be possible via electronic timing of direct detection events, and in principle better than those performed with nonlinear autocorrelators (which rely, after all, on the same fundamental physics as down-conversion, but do not benefit from a cancellation of dispersion).

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red

v

+

BS-

(b) FIG.15. (a) Simplified setup for a Hong-Ou-Mandel interferometer with a dispersive medium inserted in one arm. (b) Interfering Feynman paths for coincidence detection involve anticorrelated colors traversing the glass.

B. SINGLE-PHOTON PROPAGATION TIMEMEASUREMENTS This interference technique has certain unique features that immediately recommend it for a variety of investigations. As explained in the early sections of this chapter, coincidence counting allows one to study light at the quantum (i.e., single-photon) level, as opposed to most of modern optics, which can generally be described self-consistentlyusing only classical field theory. The Hong-Ou-Mandel interferometeris thus an ideal apparatus for studying the quantum propagation of light, a problem that has been ofgreat theoretical interest; at least until recently, the only quantum theory of light in dispersive media was an ad hoc one (Glauber and Lewenstein, 1991).Our first application ofthe interferometer was therefore to confirm that single photons in glass travel at the group velocity (Steinberg et al., 1992b). While not unexpected, this result is an interesting one at the interpretational level. It suggests that when looking for a microscopic description of dielectrics, it is inappropriate to consider the medium as being polarized by an essentially classicalelectricfield due to the collective action of all photons present, and reradiating accordingly. Linear dielectric response is not a collectiveeffect in this sense, and really involves each photon interfering only with itself (as per Dirac’s dictum) as it is partially scattered from the atoms in the medium. The fact

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that the particle’s propagation is described by a group velocity is an example of wave-particle duality observed in a single experiment.This is a relatively trivial application of the interferometer. More interesting questions arise when we go beyond simple dielectrics to the consideration of the tunneling problem discussed earlier. Tunneling, even more strikingly than the group velocity, is a wave phenomenon. It is also manifestly nonlocal (although not in the strong sense of EPR or of Aharonov-Bohm) in that the final state involves a coherent superposition of two far-separated wave packets. Yet at the quantum level, a particle is detected only at a single position (recall Fig. 14). At the completion of a tunneling “event,” the particle has either been transmitted or reflected, but its position is described by the solution to a wave equation, renormalized to account for this extra bit of information. This would not be so surprising were it not for the superluminality of tunneling. As alluded to earlier, such superluminality is explicable at the classical-field level. In particular, the area of the transmitted pulse is so small that it can be seen as originating entirely in the leading edge of the incident pulse [although this need not be the case in planned experiments (Chiao, 1993; Steinberg and Chiao, 1994)]. See Fig. 14. If we accept, however, the collapse hypothesis of the Copenhagen interpretation, what happens when a given particle is detected on the far side of the barrier (or, for that matter, if a perfectly efficient detector on the near side of the barrier fails to detect it)? Suddenly, this small wave packet is renormalized according to the projection postulate (von Neumann, 1983). At this point, the entire quantum (Am if the particle in question is a photon) appears as a single unit in one wave packet. We can no longer get around the question of superluminality as easily as in the classical case. According to standard quantum theory, the incident wave function is seen as a complete description of the physical state of the photon, not merely an expression of our own personal uncertainty as to the photon’s position. We are led to say that whenever the photon is transmitted, it arrives (on average) earlier than it would have had it travelled always at c. Although this effect depends critically on analytic wave packets with arbitrarily long tails and therefore cannot be used for superluminal signaling [as a signal must correspond to an abrupt change, which can never propagate faster than c (Brillouin, 1960)], it goes against deeply held intuitions about the meaning of causality, which need to be reexamined in this context. [Since the anomaly occurs not for a complete ensemble, but for a subensemble determined by joint preselection (preparation of a Gaussian input state) and postselection (discarding reflected photons), it is an example of what Aharonov and Vaidman (1990) have termed weak measurement (Steinberg, submitted).]

c. TUNNELING TIME IN A MULTILAYER DIELECTRIC MIRROR While some tunneling time experiments have been performed in the past (Landauer, 1989), optical tests offer certain unique advantages (Martin and

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Landauer, 1992). In our experiment, the main advantages include the ease of construction of a barrier, the high resolution of the timing possible, the absence of dissipation, the fact that the photon is intrinsically relativistic, and that the particular barrier we were able to use has very little energy dependence. At a theoretical level, the fact that photons are described by Maxwell's (fully relativistic) equations is an important argument against interpreting superluminal tunneling predictions as a mere artifact of the nonrelativisticSchrodinger equation. At an experimental level, it puts us much closer to a regime in which an apparent violation of causality might be observed (Deutch and Low, in press; Low and Mende, 1991).The fact that our barrier (described later) has little energy dependence means that the transmitted wave packets suffer little distortion, and are essentially indistinguishable from the incident wave packets. It also means that one is denied the recourse suggested by some workers (Dumont and Marchioro, 1993) of interpreting the superluminal appearance of transmitted peaks to mean that only the high-energy components [which traveled faster even before reaching the barrier (Hauge et af., 1987)] were transmitted. In our setup (Fig. 16), the tunnel barrier was a multilayer dielectric mirror with 11 alternating layers of low and high index material, each one quarter wave thick at the design frequency of the mirror, which corresponded to a wavelength of 700 nm in air (Steinberg et al., 1993). Such a periodic structure leads to a photonic bandgap (Yablonovitch, 1993) analogous to that in the Kronig-Penney model of solid-state physics. The gap represents a forbidden range of energies, in which the multiple Bragg reflections will interfere constructively so as to exponentially damp any incident wave. We

-

Trombone prism

P W

Substrate

Multilayer coating "'Y....UJ-.

FIG.16. Apparatus for measuring the single-photon tunneling time (Steinberg et al., 1993).

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worked at 700nm, the center of the gap, where this 1.1-pm thick structure reflected 99% of all incident light. Throughout the bandgap (extending from 600 to 800 nm in wavelength), the transmissivity is relatively flat; outside the gap, it reaches a series of resonant transmission peaks, easily understood as Fabry-Perot resonances, but also analogous to resonant transmission for electrons incident on a square barrier. The analogy with tunneling in nonrelativistic quantum mechanics arises because of the exponential decay of the field envelope within the periodic structure, that is, the imaginary value of what in solid-state physics would be termed the quasimomenturn. The analogy is sufficiently accurate that the same qualitative features arise for the transmission time: For thick barriers (including ours, where kd x lo), it saturates at a constant value of approximately 1.7 fs. This means that 700-nm light transmitted through a mirror should arrive on average 1.9 fs sooner than a similar wavepacket traveling through a micron of air instead of the micron-thick mirror! To measure such a small time difference, we had the multilayer structure coated on one-half of one surface of a high-quality optical flat. This way, as we scanned across the Hong-Ou-Mandel dip, we could slide the coated half in and out of place without affecting any of the alignment or changing the optical path length of the rest of the apparatus. This also enabled us to avoid problems associated with hysteresis when translating the prism that was used to adjust the path-length difference; the reflective coating was slid in and out of place periodically while the prism made a single slow scan, thus forming a pair of coincidence dips. Each dip was subsequently fitted to a Gaussian, and the difference between their centers was calculated. Typical data are shown in Fig. 17. When several such runs were combined, we found that the tunneling peak arrived 1.47 & 0.21 fs earlier than the one traveling through air. In this way, we confirmed that the tunneling delay time was superluminal. Our results excluded the semiclassical time, which corresponds to treating the magnitude of the (imaginary) crystal momentum as a real momentum. This time is of interest mainly because it also arises in a calculation by Buttiker and Landauer (1982) of a critical time scale in problems involving oscillating barriers, which they interpret to mean that this time is a better measure of the duration of the interaction than is the group delay. Our data are consistent with the group delay, and also with the Larmor time as defined by Buttiker (1983). The Larmor time is one of the early efforts to attach a “clock” to a tunneling particle, in the form of a spin aligned perpendicular to a small magnetic field confined to the barrier region. The basic idea is that the amount of Larmor rotation (in the plane perpendicular to the applied field) experienced by a transmitted particle is a measure of the time spent by that particle in the barrier. At midgap, this time agrees with the group delay; both predict superluminal delay times. In other regimes, the two times differ. Most of this difference reflects the fact that the aligned spin component is more easily

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*-.. r

time

-

FIG.17. Coincidence profiles with and without the tunnel barrier, taken by scanning the trombone prism of [Fig. 161, map out the single-photon wave packets. The lower profile shows the coincidences with the barrier; this profile is shifted by - 2 fs to negative times relative to the one with no barrier (upper curve). The wave packet that tunnels through the barrier arrives earher than the one that travels the same distance in air.

transmitted than the anti-aligned component, leading to a rotation not perpendicular to the B-field but toward the B-field. It is debatable to what extent this sort of an alignment should be considered as contributing to an interaction time. Recently, we have studied the energy dependence of the tunneling time, not by tuning the frequency directly, but by angling the dielectric mirror, thus shifting its bandgap. Preliminary results are qualitatively in better agreement with the group delay than with the Larmor time; a more accurate test is currently under way.

D. INTERPRETATION OF THE TUNNELING TIME Although no conclusion about interaction times can be drawn from our experiments, they provide a very clear verification of superluminal tunneling delays. The simplicity of the geometry we used and the absence of extraneous degrees of freedom strongly suggests (within the framework of standard quantum mechanics) that there are no further time scales “hidden” in this particular experiment. At the same time, we would like to understand these superluminal effects in the context of a relativistic theory. We do not believe that any particle travels faster than c at any given time, or that any signal could be transmitted faster than c, and yet it is nontrivial to reconcile these statements with the data. One attempt made recently (Deutch and Low, 1993) relied, like the Larmor time, on attaching an extra parameter such as spin to a tunneling wave packet. [This idea had appeared elsewhere in the literature as well

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(Martin and Landauer, 1992).] The interesting observation was made that if the incident Gaussian wave packet consisted of two different polarizations-one for the first half of the wave packet and an orthogonal (hence distinguishable) one for the second half-then the superluminally transmitted peak would consist entirely of this first polarization, demonstrating the causal “disconnection” of the outgoing peak from the incident one. Further analysis of this example (Steinberg et al., 1994) shows that it involves mixing particle and wave pictures of tunneling in a way that would not be experimentally testable. The addition of the polarization information in question suppresses the interference of multiply reflected waves (in analogy with the quantum eraser discussed earlier) and the transmitted beam is no longer a simple Gaussian; the discontinuity introduces rapid oscillations at later times. It is only the small peak which arrives earliest that has the character described by Deutch and Low. It is not possible in a single experiment to observe both the superluminal transmission and the polarization information. A modified version of this gedanken experiment, due to Rolf Landauer, involves smoothly varying the polarization across the incident pulse. In this case, the transmitted peak appears superluminally, as before, but reproduces the original polarization profile, providing no extra information about where the transmitted particles originate. Even if one believes that no particle can travel faster than light (certainly all serious researchers in this field agree that no signal can, at any rate), it appears that attempts within the framework of quantum mechanics to “label” particles and thus verify this claim are doomed to failure, thanks to the complementarity principle.

IX. Envoi A. PHASE OPERATORS

Let us recall Section 1II.B and difficulty of interpreting the energy-time uncertainty principle. The number-phase uncertainty relation is related to this better known example; energy, after all, is number times hw, while time is phase divided by w. Like time, phase does not correspond to a Hermitian operator in quantum mechanics, and the usual nonrelativistic derivation of uncertainty relations via commutation relations cannot apply.4 The first 4J~stas going from nonrelativistic quantum theory to field theory reduces the position momentum uncertainty relation to the same status as the time-energy relation (see Section IILB), it reduces other uncertainty relations to that of number and phase. For example, polarization is no longer a property of a particle, but a label of different field modes. In a given basis, spin projection along the quantization axis is the difference of the occupation number for the two eigenstates; spin projection in a nonorthogonal basis depends on the phase difference between these two eigenstates.

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attempt at a phase operator was made by Dirac, who reasoned that phase should be the conjugate variable to number, and therefore postulated the existence of a phase operator 9 whose commutator with the number operator N is -i, from which the number-phase uncertainty principle AN A+ 2 1/2 follows automatically (Dirac, 1927). It was later shown (Louisell, 1963) that no such Hermitian operator exists. This is connected with the fact that N has a positive-definite spectrum, and a translation operator for N (which role would play) cannot exist, in precise analogy to the question of a time operator. Nevertheless, there does seem to be some meaningful content to this relation: If we pick a reasonable definition of phase for simple test cases (such as the angular coordinate of the Wigner distribution in electromagnetic phase space), the phase of number states is indeed perfectly undefined, and the phase uncertainty of a coherent state does indeed fall as the reciprocal of its number uncertainty. Other attempts have been made at generalized phase operators, but these are of unclear physical significance and no universal preference has emerged (Barnett and Pegg, 1986; Noh et al., 1993). Mandel’s group has therefore taken a practical approach to the problem (Noh et al., 1991). They have set up a homodyne measurement that is typical of how one might measure phases in classical optics. Classically, the phase can be determined as a function of differences between the photocurrents at various detectors in such a scheme. Mandel’s group defines the operator by taking the expression appropriate for their experiment, and replacing the c-number photocurrents with their operator counterparts. They have shown theoretically that this operator may differ in certain regimes from the other proposals we have discussed, although in the correspondence principle limit they all of course agree. They have experimentally confirmed the validity of this definition, and found results that do not match the other proposed operators. Raymer’s group has performed related experiments (Smithey et al., 1993), in which they are able to measure the total quantum state of an electromagnetic field mode. This differs from the proposal of Aharonov’s discussed earlier. For one thing, the measurement involves studying an entire ensemble, not a single system. For another, it examines the properties of a single-mode field whose occupation is quantized, as opposed to the (firstquantized) wave function of a single particle. It is novel in that it reconstructs the entire density matrix for the state, allowing a calculation of all properties from number statistics to quadrature amplitudes to phase fluctuations. This is done by integrating various “slices” of the Wigner distribution by performing appropriate homodyne measurements, and performing an inverse transformation of the type used in computer-aided tomography to reconstruct the whole function. We do not review these experiments here, but only mention that they can be used to measure directly the number-phase uncertainty product for various states of the field.

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B. NONLINEAR QUANTUM MECHANICS So many quantum mechanical predictions have already been verified that it becomes difficult to see most optics experiments as tests of quantum theory, as opposed to mere ongoing confirmations. The real problem is that, although people are dissatisfied with certain aspects of quantum mechanics (QM), there is very little in the way of convincing extensions or generalizations of QM. One noteworthy exception is Weinberg’s nonlinear quantum mechanics. Linearity, in the sense of the superposition principle, is arguably the heart of quantum mechanics, as of any wave mechanics. Not only does it make calculation easier than in strongly nonlinear theories, but it is really the source of interference effects and particle-wave duality. On the other hand, the most striking problem of quantum mechanics is the discrepancy between linear, unitary (Schrodinger) evolution and the observation in real life of individual events or quantum jumps: the absence of Schrodinger cats in nature. Because information is gained whenever a collapse occurs from some pure linear superposition of states to one particular eigenstate of a given operator, such a process cannot be explained in terms of a linear theory. Perhaps if a small nonlinear component were present in a more complete formulation of quantum theory, this collapse would emerge naturally? Weinberg (1989) proposed an example of such a nonlinear extension of quantum mechanics. In particular, he offered an example of what a nonlinear Hamiltonian might look like for a nuclear spin-3/2 system. The tests of this generalization have relied on optical spectroscopy. For example, Bollinger et al. (1989) have looked for a dependence of the transition frequencies between hyperfine states of ’Be’ on the populations ofthese states. The hyperfine state involves the nuclear spin, and a nonlinear term in the nuclear Hamiltonian would lead to energy shifts based on these populations. A limit of less than 10 pHz was placed on these frequency shifts, implying that less than about lopz6of the binding energy per nucleon could be attributed to a Weinberg nonlinearity. As discussed by Polchinski (1991), if such a nonlinearity does nevertheless exist, it will necessarily lead to great changes in our understanding of QM. It appears that the breakdown of the superposition principle would either lead to the possibility of superluminal commnication or of comrnunication “between different branches of the wave function,” that is, dependence of experimental observables on contrary-to-fact conditions such as what the experimenter would have done in different experimental circumstances! C. ISSUESIN CAUSALITY

It might seem that while optics is a useful tool for investigating complex systems, optics itself is completely understood; this is not the case. In fact,

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some of the most interesting causal paradoxes that have yet to be resolved occur in optics. The most well known of these is Barton’s paradox (Barton and Scharnhorst, 1993), closely related to the Casimir effect [itself a Q E D phenomenon recently given indirect confirmation in a quantum optics experiment (Sukenik et al., 1993)J. In this case the amplitude for light-light scattering, one of the radiative corrections in QED, is modified by the presence of closely spaced parallel plates (an extremely small, but in principle observable, effect). In certain arrangements, it appears that this can lead to propagation of light in vacuum (albeit a vacuum “colored” by the presence of the Casimir plates) faster than c. Similar paradoxes have been pointed out in connection with localization of any particle in a quantum field theory (Hegerfeldt, 1985) and with the Glauber theory of photodetection (Bykov and Tatarskii, 1989).It may be that the high momentum/energy components introduced by localizing a detector to within a wavelength lead to spurious “dark” detections (Ho, 1993), which can be mistaken for causality-violating events.

D. CONCLUSIONS Optics experiments have allowed wide ranging and important tests of QM.

In conjuction with atomic physics, they have been used to investigate such diverse questions as parity nonconservation and nonlinear extensions of quantum mechanics, issues whose proper place is many orders of magnitude above the 1-eV energy scale. Optics is also important in the study of gravitational phenomena, such as gravitational antennas and gravitational lensing by some of the most massive objects in the universe. Thanks to novel optical techniques, we have new abilities for manipulating single atoms and ions, which promise new devices for time keeping (and thus further tests of general relativity), as well as studies of such diverse phenomena as Bose-Einstein condensation, the Casimir effect, the Unruh effect, and atomic interference. The two great revolutions in physics that occured at the beginning of the twentieth century, quantum mechanics and relativity, both arose from optical experiments. At the end of this century, we still find that optical experiments probe deeply our understanding of these two branches of physics. The reason for this is that the photon is at once both a quanta1 and a relativistic particle. The conceptual tension between the fundamentally nonlocal nature of the quantum on the one hand, and the intrinsically local nature of space-time on the other, is most sharply manifested in optical phenomena and provides a strong motivation for pushing optical tests to new frontiers.

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Acknowledgments This work is supported by ONR grant number N00014-90-J-1259.We would like to thank Grant McKinney and Joshua Holden for valuable assistance.

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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL. 34

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS DOMINIQUE DELANDE and ANDREAS BUCHLEITNER Laboratoire de Spectroscopie Hertzienne de I'Ecole Normale Supekieure UniversitC Pierre et Marie Curie Paris. France

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A . Classical and Quantum Chaos . . . . . . . . . . . . . . . . . B . Chaotic Atomic Systems . . . . . . . . . . . . . . . . . . . . C. Ionization: A Complication or a Useful Tool? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Scaling Properties E . Laboratory Experiments versus Numerical Experiments . . . . . . . I1. Time Scales-Energy Scales . . . . . . . . . . . . . . . . . . . . A . Classical Scales . . . . . . . . . . . . . . . . . . . . . . . B . Quantum Scales . . . . . . . . . . . . . . . . . . . . . . . 111. Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . A . Properties of Individual Energy Levels . . . . . . . . . . . . . . B . Widths of Resonances . . . . . . . . . . . . . . . . . . . . . C . Periodic Orbit Spectroscopy . . . . . . . . . . . . . . . . . . D. From Classical to Quantum Mechanics . . . . . . . . . . . . . . IV. Wave Functions: Localization and Scars . . . . . . . . . . . . . . . A . Localization in the Regular Regime . . . . . . . . . . . . . . . B . Localization in the Chaotic Regime . . . . . . . . . . . . . . . C. Localization in the Mixed Regular-Chaotic Regime . . . . . . . . . D . Relationship between Localization and Lifetime . . . . . . . . . . V . Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Time-Independent Systems . . . . . . . . . . . . . . . . . . . B . Time-Dependent Systems . . . . . . . . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 90 91 92 94 94 95

97 97 101 103 108 109 110 111 113 115 115 115 116 121 121

I. Introduction A . CLASSICAL AND QUANTUM CHAOS Atomic systems played a major role in the birth and growth of quantum mechanics. One central idea was to relate the well-known classical motion of the electron of a hydrogen atom-an ellipsis around the nucleus-to 85

Copyright 0 1994 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003834-X

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the experimentally observed quantization of the energy levels. This is the aim of the Bohr and Bohr-Sommerfeld models. These simple semiclassical models were unable to make any reliable prediction on the energy spectrum of the next simplest atom, helium. Because of the great success of quantum mechanics, the problem of correspondence between the classical and the quanta1 dynamics has not received much attention in the last 60 years. The fundamental question is (Gutzwiller, 1990): How can classical mechanics be understood as a limiting case within quantum mechanics? For systems with time-independent one-dimensional dynamics like the harmonic oscillator and the hydrogen atom, the correspondence is well understood. The restriction to such simple cases creates the erroneous impression that the classical behavior of simple systems is entirely comprehensible and easily described. During the last 20 years it has been recognized that this is not true and that a complex behavior can be obtained from simple equations of motion. This usually happens when the motion is chaotic, that is, unpredictable on a long time scale although perfectly deterministic (Henon, 1983). A major problem is that of understanding how the regular or chaotic behavior of the classical system is manifest in its quantum properties, especially in the semiclassical limit. This is the subject of quantum chaos. More precisely, typical questions we would like to answer are as follows (Gutzwiller, 1990): 0

0

0

0

What are the appropriate observables to detect the regular or chaotic behavior of the system in the classical limit? More precisely, how is chaotic or regular behavior expressed in the energy levels and eigenstates of a quantum system? What kind of semiclassical approximations can be used if the system is not separable? What is the long-term behavior of a quantum system?

A relatively nonintuitive result is that chaos may take place in lowdimensional systems. Because atoms are simple microscopic low-dimensional systems that have to be described by quantum mechanics, they are among the best available prototypes for studying quantum chaos. The goal of this chapter is to present and discuss the most important progress in this area. When chaos is present, it is generally impossible to obtain analytic expressions (even approximate ones) for the time evolution of the system. Hence, chaotic systems are intrinsically complex systems lying beyond any traditional analytic description. On the other hand, complex systems are not necessarily chaotic, but may have similar statistical properties. For example,

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recent experimental results on mesoscopic systems (Bouchiat and Montabaux, 1990; Al’tshuler et al., 1991) suggest that the universal fluctuations observed could have the same origin as the spectral fluctuations (see Section 1II.A) of quantum chaotic systems. In that sense, quantum chaos can also be thought of as a first step in the direction of complex systems.

B. CHAOTIC ATOMICSYSTEMS Compared with other microscopic complex systems (nuclei, atomic clusters, semiconductors, etc.), atoms have the great advantage that all the basic components are well understood: These are essentially point particles interacting through a Coulomb static field. Hence, it is possible to write an explicit expression of the Hamiltonian. A definite advantage of atomic systems is that they can be studied theoretically and experimentally. The word experiment must be understood here to mean traditional laboratory experiments, but also numerical experiments. Indeed, currently available computers make it possible to compute numerically properties of complex systems described by simple Hamiltonians. During the last 10 years, the constant interaction between the experimental results and numerical simulations led to major advances in the field of quantum chaos, as is shown later. A crucial result is that classical chaos can only exist in systems where different degrees of freedom are strongly coupled. [this is a consequence of the celebrated KAM theorem (Lichtenberg and Liberman, 1983).] This implies that a small perturbation added to a regular system cannot make it chaotic. In other words, atomic systems where chaos is supposed to play an important role require nonperturbatiue situations, where, in addition to the Coulomb binding force, the electrons are submitted to additional forces of comparable strength. This can be easily realized on the simplest atom, hydrogen, by applying to it an external field. Different situations can be realized depending on the type of field used (magnetic, electric, or both) and on its time dependence. We first consider the case of a hydrogen atom in a static external electric field. Small corrections to the pure Coulomb potential in the hydrogen atom (finite mass of the nucleus, relativistic corrections, spin effects, QED corrections, spontaneous emission, etc.) do not affect the global (regular or chaotic) behavior of the system and can be neglected. Throughout this chapter, we use atomic units, where the mass, the absolute value of the charge of the electron, and the Planck’s constant are equal to 1. In these units, the Hamiltonian is (Delande, 1991):

H

= H,

+ F-r

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where H, is the field-free Hamiltonians:

p and r are, respectively, the momentum and position of the electron, and F is the electric field in atomic units of 5.14 x 109V/cm. Fortunately (unfortunately as far as chaos is concerned!), this Hamiltonian is separable in parabolic coordinates (Harmin, 1982) and the motion is fully regular. The case of the static magnetic field is more interesting. In cylindrical coordinates, and using the symmetric gauge A = t r x B for the vector potential, the Hamiltonian is given by:

H

= H,

Y Y2P2 + -L, +2 8

(3)

where y denotes the magnetic field (along the z axis) in atomic units of 2.35 x 105T, and L, the z component of the angular momentum. Because of the azimuthal symmetry around the magnetic field axis, the paramagnetic term yL,/2, responsible for the usual Zeeman effect, is just a constant. The diamagnetic term, y2p2/8,is directly responsible for the onset of chaos in the system. The competition between the Coulomb potential with spherical symmetry and diamagnetic potential with cylindrical symmetry governs the dynamics. As a crude criterion, chaos is most developed when these two terms have the same order of magnitude. For a quantum state of principal quantum number n, this gives: yn3 N 1

that is, Bn3 3: 2.35 x 105T

(4)

This can be realized in a laboratory experiment with Rydberg states n 1: 40 to 150 (Iu et al., 1991; Holle et al., 1988; van der Veldt et al., 1993). When written in cylindrical coordinates, the Hamiltonian (3) describes a time-independent two-dimensional system belonging to the class of the simplest possible chaotic systems (Lichtenberg and Liberman, 1983). This makes this system an almost ideal prototype for the study of quantum chaos (Friedrich and Wintgen, 1989). A combination of static electric and magnetic fields can also be used. When the two fields are not parallel (for example, perpendicular), the system is fully three dimensional and described by the Hamiltonian:

H

Y 2

= H, + - L ,

Y2P2 +-+ 8

F-r

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Again, the nonperturbative regime is reached when the impact of the three fields (Coulomb field of the nucleus, external magnetic field, and electric field) is comparable. In addition to Eq. (4), this implies: Fn4 N 1 in atomic units,

that is

Fn4 N 1 x 109V/cm in laboratory units

(6)

which is relatively easy to realize in laboratory experiments (Raithel et al., 199 1). Finally, it is possible to replace the external field with the internal field due to another electron of the same atom. This brings back the three-body problem (helium or helium-like atoms) described by the following Hamiltonian ( Z is the charge of the nucleus):

From the chaos point of view, an interesting situation arises when the two electrons have comparable excitations. Strong dynamical correlations between the two electrons are expected, leading to a breakdown of the independent electron picture. The corresponding doubly excited states, lying very close to the double ionization threshold, are autoionizing states with weak optical excitation probabilities from low excited states, which does not make it easy to observe them (Domke et al., 1992; Eichmann et al., 1992; Camus et al., 1989). Different dynamical properties are expected for systems whose Hamiltonian shows an explicit time dependence. The easiest way to introduce such a dependence is to apply an external oscillating electric field on a hydrogen atom. For a monochromatic field of amplitude F and angular frequency o, polarized along the z axis, the Hamiltonian in the length gauges is

H = H, + F Z C O S ~ ~

(8)

Because of the azimuthal symmetry around the electric field axis, the system is effectively two dimensional. In numerical experiments and theoretical investigations, a simplified one-dimensional model of the atom has been widely used (Casati et al., 1988; Galvez et al., 1988; Bayfield et al., 1989; Sauer et al., 1992; Jensen et al., 1989). Chaotic dynamics can be observed only in a nonperturbative regime, that is, when the amplitude and frequency of the external field are comparable

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to the amplitude of the Coulomb field on the n’th Bohr orbit and to the classical Kepler frequency of the electron. This gives:

Fn4 N 1 in atomic units, or wn3 N 1

i

Fn4 N 1 x 109V/cm in laboratory units wn’ 1: 5 x 10i5Hz

(9)

Such a situation can be obtained by applying a strong infrared laser to a weakly excited state. However, the spatio-temporal characteristics of the laser pulse are then extremely important for the dynamics of the atoms, ions, and electrons produced. Such complications usually hide the regular or chaotic aspects (L‘Huillier et al., 1991). Another possibility is to employ highly excited states (n N 60) and a microwave field. This system led to numerous important experimental and theoretical results and is discussed in Section V. For a circular polarization of the external electric field, classical studies have shown that the system is mainly regular and of poor interest as long as chaos is concerned (Zakrzewski et al., 1993).

C . IONIZATION: A COMPLICATION OR A USEFUL TOOL? Chaos or regularity manifest themselves during the long-term behavior of the system. For this reason, it is often studied (in classical mechanics) for bounded systems where the different components may interact during an indefinitely long time. Atoms do not belong to this type of systems, because they can ionize. Hence, they are prototypes for studying the effect of transient chaos, that is, chaos on a jnite time scale (fixed by the lifetimes of the resonances). A complementary point of view is given by scattering experiments, when the dynamics near the scatterer is chaotic. Clearly, the physics is the same, although the language is somewhat different (Smilansky, 1991). On the other hand, ionization provides a simple tool for analyzing the electron dynamics. Experimentally, we can easily detect whether an atom is ionized after interaction with an external field, whereas it is much more complicated to measure its internal state. Ionization is a by-product of the underlying quantum dynamics. Although it is only a very rough measure of the final state of an atom, it can provide much physical insight. The most striking example is the dynamics of the hydrogen atom in a microwave field where most experiments just measure the ionization probability.

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D. SCALING PROPERTIES

One of the main difficulties in the study of the semiclassical limit h + 0 is that the value of the Planck constant is fixed in laboratory experiments. One can get around this difficulty in atomic systems thanks to the existence of scaling laws. Indeed, the Coulomb potential and other interaction potentials discussed earlier are power functions of the position and momentum. Thus, a convenient scaling of all variables and external fields leaves the equations of motion invariant. This scaling transformation is (Delande, 1991):

where I is any positive real number. This means that different initial conditions with different external fields may have exactly the same classical dynamics. A simple example is the field-free motion of the electron, which is an elliptic trajectory whatever its (negative) energy. This is no longer true in quantum mechanics, because of the absolute scale imposed by the Planck constant h. Different scaled situations observed experimentally correspond to the same classical dynamics with different effective values of the Planck constant. Hence, in a real (or numerical) experiment, the semiclassical limit h -P 0 can be studied by tuning simultaneously toward higher excited states the energy and the characteristics of the external fields according to Eq. (10). This possibility is extremely important for understanding the classicalquanta1 correspondence. The best example is so-called “scaled spectroscopy,” where the wavelength of the exciting laser (which determines the energy of the excited state under study) is scanned together with the external static electric and/or magnetic field (see Section 1II.C). The scaled variables are those quantities defined in Eq. (10) that are invariant under the scaling transformation. An example is the scaled energy E of a hydrogen atom in a magnetic field (Delande, 1991; Friedrich and Wintgen, 1989; Hasegawa et al., 1989): E =

EyP2l3

(1 1)

which measures the energy of the electron in units of magnetic field. Because of the scaling law, the classical dynamics, instead of depending both on E

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and y, actually depends only on E , whereas quantum properties depend a priori on both quantities. In the crossed magnetic and electric fields, an additional scaled quantity, measuring the electric field in “units of magnetic field” has to be introduced. For helium-like atoms, the situation is even simpler. It is f = Whatever the (negative) total energy, the entire dynamics is fixed and depends only on the discrete parameter Z . This is in sharp contrast with the quantum properties, which clearly depend on whether the total energy is above or below the first ionization threshold. For time-dependent systems, the situation is somewhat more complicated, because the energy is no longer conserved. However, a standard experiment consists of preparing a Rydberg state with principal quantum number n and applying a microwave field to it. The convenient scaled quantities are the scaled field amplitude and frequency (Casati et al., 1988), defined through [see Eq. (lo)]: F , = Fn4 w,, = con3

In any case, the use of scaled variables is necessary for a correct understanding of the classical-quanta1correspondence. E. LABORATORY EXPERIMENTS VERSUS NUMERICAL EXPERIMENTS From a conceptual point of view, experiments on chaotic atomic systems should be done on excited states of the simplest atoms, hydrogen (in external field) or helium. Because of the high density of Rydberg states, a selective excitation requires the use of lasers. Unfortunately, the desired wavelengths are not always easily obtained with current laser technology. For this reason, many experiments have also been done on Rydberg states of different atomic species. It is reasonable to expect that the perturbation induced by the presence of the atomic core on the Rydberg electron will not affect too much its regular/chaotic properties. However, this does not go beyond a questionable assumption. The atomic core is a small object with a radius of a few Bohr radii; whether any semiclassical approximation can be used to represent its effect is far from obvious. The simplest way is to use an effective potential V,(r) in addition to the Coulomb potential, replacing the zero-field Hamiltonian by:

H - - P2 - - + K1 ( r ) ‘-2 r The core potential is chosen so that it reproduces the observed quantum defects of the different 1 series of Rydberg states. However, different model potentials give roughly the same quantum defects, but different classical dynamics. This shows that the atomic core is a pure quantum object, which

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does not have any straightforward classical counterpart. The main effect of the core potential is to break the Coulomb degeneracy of the energy levels. In most cases, in the strong field regime where all hydrogenic states are supposed to be strongly mixed, this effect is probably weak (see Section 1II.A). However, in the case of the microwave ionization, there are strong experimental and numerical indications that this assumption is not true (Blumel et al., 1991; Arndt et al., 1991; Buchleitner, 1993). From the theoretical point of view, the hydrogen atom is entirely calculable using its dynamical group S0(4,2). The use of group theory allows extremely efficient numerical experiments (Delande, 1991; Friedrich and Wintgen, 1989; Hasegawa et al., 1989). The Hamiltonian of an atom in an external field, when expressed in a basis adapted to its internal dynamical properties, can be represented as a sparse matrix. Using an efficient diagonalization algorithm, one is able to compute very accurately, highly excited energy levels and wave functions. For example, computing the 1000'th excited state of a hydrogen atom in a magnetic field of a few teslas with 8 significant digits is nowadays an easy task! It is even more surprising when it is found to be in exact agreement with the (less accurate) experimental measurement (see Fig. 3 later in this chapter)! This method makes it also possible to compute the dynamics of a hydrogen atom in a time-dependent external field (Buchleitner and Delande, 1993). In addition, use of the complex coordinate method allows for the full inclusion of the effect of ionization and atomic continua (Delande et al., 1991). Recently, it has been successfully extended to the helium atom, taking into account the exact interelectronic repulsion (Wintgen et al., 1992). Another recent advance is the extension to nonhydrogenic atoms, by inclusion of a core potential (Halley et al., 1993). The main advantage of numerical experiments is their flexibility and their ability to give information that is extremely difficult or even impossible to measure (for example, a wave function). On the other hand, they require big and expensive computers. For two-dimensional time-independent or one-dimensional time-dependent problems, the numerical experiments are now possible in realistic conditions. In higher dimensions, this is more difficult-at the border of feasibility-and experimental results are of major importance. From the ensemble of recent experimental and theoretical results, one has to choose subjectively the most interesting ones. From our point of view, the main concern in quantum chaos is to establish an understandable connection between traditional quantum properties-energy levels, eigenstates, wave functions, etc.-and the rather novel point of view of time-dependent dynamics, which is more suitable for comparison with classical dynamics. These different aspects are studied in the following sections. First, we discuss what we think is the skeleton of the classical and quanta1 dynamics, the existence and ordering of the different time scales of the system. We then

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turn to the spectroscopic (in an extended sense) aspects, including wave functions, ionizations, and excitation probabilities. Finally, we discuss the dynamics of classical and quantum regular and irregular systems, with an emphasis on time-dependent systems.

11. Time Scales-Energy

Scales

A. CLASSICAL SCALES Depending on the nature of the classical dynamics, there may be different time scales in the evolution of an atomic system. The first one is the typical period, which can be characterized, for example, by the period To of the shortest periodic orbit. Typically, for a Rydberg atom with principal quantum number around 50, it is of the order of a few picoseconds (Delande, 1991). For a chaoticsystem, neighboringinitial conditions divergeexponentiallywith time. The associated time constant z h a o s is a second time scale, which describes the typical time on which the memory of the initial condition is lost. For a typical strongly chaotic system, this time scale is of the order of the shortest period. An additional complication arrives when the motion is not bounded, that is, when the atom ionizes. The additional time scale Tonis then the lifetime chaos is beaten by ionization; a discussion in of the atom. If terms of regularity or chaos is irrelevant in this case. This is experimentally observed for atoms in ultrastrong, pulsed laser fields (L'Huillier et al., 1991) and not discussed in this chapter. Except in this situation, all experiments in chaotic atomic systems have been performed under conditions where the ionization time is reasonably long, of the order of nanoseconds or microseconds. The most complicated situation is observed when the classical dynamics is in the mixed regular-chaotic regime, depending on the initial conditions. There, one observes classical structures that can trap some trajectories or strongly inhibit the diffusive-like chaotic motion. The exact nature of these classical structures is not extremely clear (Percival, 1991). They seem to be remnants of the invariant tori existing in the regular regime. They are known as vague tori or cantori and apparently have a fractal structure. Contrary to invariant tori, they are not impermeable and a trajectory can escape from such a structure. The associated leaking time is another time constant for these mixed systems. It seems that, in fact, different time scales exist (at various scales). To our knowledge, these phenomena are not well understood even in the classical context. The hydrogen atom in a microwave field provides an example of such a system with mixed dynamics and several

Ton<
CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

95

time scales. This makes the detailed interpretation of the quantum results very difficult.

B. QUANTUM SCALES Additional time scales, depending on h, exist for quantum dynamics. This can be understood from the following example. Consider a time-independent bound quantum system with Hamiltonian H in an arbitrary initial state I $(t = 0)). It obeys the Schrodinger time evolution equation:

The solution of this equation can be expressed using the discrete eigenstates and eigenvalues of the Hamiltonian H : Hl4i) = Eil4i)

(15)

with the following expressions:

where the constant coefficients ci are computed from the initial state by means of ci

=

(4il$(t = 0))

(17)

The autocorrelation function of the quantum system is a diagonal element of the time-evolution operator U ( t ) = exp( - iHt/h): C(t) = (440) I$(tD = ($(O) I U(t)I$(OD

It is a discrete sum of oscillating terms and, consequently, a quasiperiodic function of time. This is extremely dierent from a classical autocorrelation function for a chaotic system, which is decreasing on the characteristic time scale Khaosand does not show any revival at longer times (Lichtenberg and Liberman, 1983). The time-evolution operator is the Fourier transform of the Green's function (Gutzwiller, 1990; Balian and Bloch, 1974):

G + ( E )exp( - iEt/h) dE

(20)

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Dominique Delande and Andreus BuchIeitner

[This expression is valid for t > 0 provided a small imaginary part q is added to the energy so that integrals are well defined; a similar expression can be written for t 0 with G - ( E ) . ] The Fourier transform of the autocorrelation function

-=

exp(iEt/h)C(t) dt

= C2nhlcil26(E - Ei)

(21)

i

is thus a diagonal term of the Green's function:

c ( E ) = ih c $(O) IG'(E)

- G-(E)

I$(O))

(22)

If we now consider the Fourier transform over a finite time interval, instead of over the whole range of time from - 00 to + 0 0 , we obtain a smoothed version of the quantum spectrum:

c,(E) =

,S:

T(E - Ei) sin h exp(iEt/h)C(t) dt = ~ 2 x h I c i l 2 i T(E - Ei) h

(23)

For short IT: the different broadened peaks centered at energy levels Ei overlap, and e,(E) is a globally smooth function as its classical counterpart. In such a situation, it is possible (although nothing proves that it is always the case) that the quantum C,(E) mimics classical chaotic behavior. The important point is that, for large ?: the different peaks do not overlap and the discrete nature of the energy spectrum must appear in C,(E), regardless of the initial state. The typical time needed for resolving individual quantum energy levels is simply related to the mean level spacing AE through

After this break time, the quantum system cannot mimic the classical chaotic behavior, which has a continuous spectrum. Since Tbreak depends on h, one can understand how quantum dynamics tends to classical dynamics as h goes to zero. The mean level spacing is given by the Thomas-Fermi approximation and scales as hf, with f the number of degrees of freedom. For two- (or higher) dimensional systems, Tbreak tends to infinity as h + 0. Hence, on a j x e d interval of time, the quantum dynamics may tend to the classical one as h goes to 0, although the convergence is not uniform with time. This statement has to be understood in its strict mathematical sense: Different values of the fixed interaction time t demand in general different values of A in order to achieve the desired approximation of the classical limit by the real quantum system. The longer the interval of time, the smaller the A needed to obtain the convergence of the mean value of a quantum observable to the value of the corresponding classical quantity. This

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

97

illustrates the noncommuting character of the limits t -, 00 and h + 0. In some sense, after the break time, the quantum system “knows” that the energy spectrum is discrete, it has resolved all individual peaks, and the future evolution cannot bring any essential new information. As a consequence, the system cannot explore a new part of the phase space, so it freezes its evolution, repeating forever the same type of dynamics. Any diffusive-like dynamics is then prohibited by pure quantum effects after the break time. This is the origin of the dynarnical localization (Casati et al., 1988; Fishman et al., 1982) experimentally observed for the dynamics of Rydberg states in a microwave field (see Section V). Since the density of states strongly depends on the number of degrees of freedom, Tbreak may vary over many orders of magnitude in atomic systems. Typical orders of magnitude are nanoseconds to microseconds for atomic Rydberg states studied experimentally. Whether an intermediate time scale exists that is shorter than Tbreak and limits the quantum-classical correspondence is mainly unknown. Finally, the natural lifetimes (due to spontaneous emission) are usually of the order of hundreds of microseconds, much longer than any time scale relevant to regular or chaotic behavior. This is why the system can be considered to be governed by a nondissipative Hamiltonian.

111. Spectroscopy In this section, we discuss the basic spectroscopic properties of chaotic atomic systems, including positions of the different energy levels and related properties like oscillator strengths and lifetimes of resonances. We show that the onset of chaos is characterized by well-defined properties of the quantum spectrum, which have important experimental consequences. A. PROPERTIES OF INDIVIDUALENERGY LEVELS The goal of traditional spectroscopy is to assign quantum numbers to the different energy levels in order to obtain a complete classification of the spectrum. Good quantum numbers are associated with conserved quantities, that is, operators commuting with the Hamiltonian. In a classically ergodic system, the trajectories generically explore uniformly all the available phase space, which discards the existence of conserved quantities (except energy for time-independent systems). The same is valid in quantum mechanics: Quantum chaos is characterized by the destruction of good quantum numbers. This is illustrated in Fig. 1, which shows the evolution of the energy levels

Dominique Delande and Andreas Ruchleitner

98 -40

-42

-44

-46

-48

.-

M a g n e t i c f i e l d [tesla]

FIG.1. Energy levels of a hydrogen atom versus magnetic field for typical Rydberg states (L, = 0, even parity series). At low field (left part of the figure), the classical dynamics is regular and the energy levels (quasi-) cross. The quantum eigenstates are defined by a set of good quantum numbers. In the right part of the figure, the classical dynamics is chaotic, the good quantum numbers are lost, and the energy levels strongly repel each other. The strong fluctuations in the energy levels are a characteristic of chaotic behavior.

of a hydrogen atom as a function of the magnetic field strength. At low magnetic field, when the classical motion is regular, approximate good quantum numbers exist. States with different sets of good quantum numbers do not (or weakly) interact. A given eigenstate can be unambiguously followed in a wide range of field strength, since it crosses (or has very small avoided crossings with) the other energy levels. In this regular regime, it can be shown that, because of the existence of conserved quantities, a given classical trajectory does not explore the entire phase space, but only a subspace known as an invariant torus, almost like a separable system. It is possible to quantize semiclassically such a system using the EBK procedure (Gutzwiller, 1990; Ozorio, 1988). This is a natural extension of the usual WKB quantization rule along each almost separated coordinate and provides an efficient semiclassical quantization scheme for individual energy levels in association with good quantum numbers (see also Section 1V.A for the wave functions in the regular regime). Actually, the energy levels predicted from the EBK quantization should not be distinguishable from the exact quantum result in the left part of Fig. 1. When chaos invades the major part of the classical phase space (right part of Fig. l), this quantization scheme cannot be used The sizes of the avoided crossings increase and individual states progressively lose their identities and the associated good quantum numbers. In the fully chaotic regime, the energy levels and the eigenstates strongly

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

I

I

-

~~

y=3.00

1

I

99

1-

I1

y = 3.01

a 3

w -

-6.0 Energy (cm')

-5.5

-6.0

-5.5

Energy (cm.')

FIG.2. Two numerical simulations of an excitation spectrum of hydrogen atoms in a magnetic field y from the ground state. Although the magnetic field changes by only 3 parts in 1000, the spectra are completely different. This shows the extreme sensitivity of chaotic spectra on small changes of parameters.

Juctuate when the magnetic field is changed. An example is given in Fig. 2 where a relative change of 0.3% in the magnetic field is enough to scramble the spectrum. In that sense, the quantum system shows a high sensitivity on a small perturbation as its chaotic classical equivalent. The energy spectrum of a classically chaotic system displays an extreme intrinsic complication, which means the death of traditional spectroscopy. Such extremely complex spectra have been observed experimentally in several atomic systems in external fields (magnetic field, crossed electric and magnetic fields) and numerically on virtually all chaotic systems. It is probably the simplest and most universal property. Figure 3 shows an experimental spectrum of the lithium atom in a magnetic field, compared with the numerical simulation on hydrogen. Both display the same complicated reproducible details and the agreement is excellent (Iu et a!., 1991).

-30

30

FIG.3. Comparison between the experimental spectrum of the lithium atom in a magnetic field B cz 6.1 13T (L, = 0, even parity series) and the theoretical spectrum of the hydrogen atom computed using the complex coordinate method, shown as mirrors of each other. The agreement is excellent. Aside from an overall intensity scale factor, there is no adjustable parameter.

Dominique Delande and Andreas Buchleitner

100

This also proves that the atomic core is only a small perturbation. This qualitative property has been put on firm ground by a study of the statistical properties of energy levels (Bohigas, 1991). The simplest quantity is the distribution of nearest neighbor spacings (energy difference between two consecutive levels, normalized to the mean level spacing). In the regular regime [see Fig. 4(a)J, consecutive energy levels generally have different sets of good quantum numbers and do not interact. The distribution of spacings is one of uncorrelated levels, that is, a Poisson distribution:

P(s) = e-' (25) which nicely reproduces the numerical results obtained on different systems and the experimental results on Rydberg atoms in a magnetic field (Delande and Gay, 1986; Friedrich and Wintgen, 1989; Hasegawa et al., 1989). In the chaotic regime, the strong level repulsion induces a completely different result [see Fig. 4(b)], with practically no small spacing. A simple model is able to predict the statistical properties of energy levels. It assumes a maximum disorder in the system and therefore models the Hamiltonian by a set of random matrices, which couple any basis state to all of the other ones. Depending on the symmetry properties of the Hamiltonian (especially with respect to time reversal), different ensembles of random matrices have to be considered (Bohigas, 1991). For all atomic systems studied, the Gaussian orthogonal ensemble of real symmetric matrices is convenient. The predicted spacing distribution agrees very well with the numerical results for various atomic systems. It is very close to the so-called Wigner distribution: 7TS

~ ( s )= 2 exp

a

] - LO . '

(- $) b

I p(s)

0.5

0.0

0.0

1.0

2.0

3.0

Spacing s

4.0

FIG.4. Distribution of energy level spacings (normalized to the mean level spacing) for the hydrogen atom in a magnetic field, obtained from numerical diagonalization of the Hamiltonian. (a) In the classically regular regime, the distribution is maximum at 0 and well fitted by a Poisson distribution (dashed line). (b) In the classically chaotic regime, the probability of finding almost degenerate levels is very small (level repulsion). The results are well reproduced by the Wigner distribution (dashed line) and random matrix theory (Delande and Gay, 1986).

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

101

The transition from a Poisson distribution in the classically regular regime to a Wigner distribution in the chaotic regime gives a characterization of quantum chaos, at least for highly excited states. Other statistical properties of the energy levels in the chaotic regime have been studied and are in good agreement with the predictions of random matrix theory (Delande and Gay, 1986). From an experimental point of view, fully resolved clean spectra are needed for extracting spacing distributions. These are difficult to obtain because of the high density of states in the regions of interest and because of possible experimental imperfections (stray fields, etc.), which rapidly spoil the results. Although there is no doubt that the spacing distribution should be Wigner, it has not yet been verified in a real experiment on a chaotic atomic system. Recently, dynamic properties involving different values of an external parameter have been studied and put the sensitivity on external perturbations on a quantitative ground (Zakrzewski et al., 1993b; Zakrzewski and Delande, 1993). Finding universal properties in the local statistical properties of energy levels for chaotic systems is not a real surprise. As discussed in the preceding section, this range of energy (mean level spacing AE) corresponds to a long time behavior (h/AE >> Khaos), where chaos classically fully develops with its universal properties. Universality is also observed in the corresponding quantum dynamics. On the other hand, at shorter times of the order of Khaos, nonuniversal properties exist in the classical behavior. This implies also a deviation from the predictions of random matrix theory on a large energy scale, as has been numerically and experimentally observed (Delande, 1991).

B. WIDTHSOF RESONANCES The model of random matrices describes bound spectra and does not take into account the possible autoionization of the atom. If the typical autoionization time is longer than the break time: Ton

>> Tbreak

(27)

the atom does not ionize significantly during the time needed to develop the detailed quantum dynamics. Hence, the energy spectrum and the eigenstates will be essentially well described by random matrix theory. Autoionization comes into play after, as a small perturbation. Each eigenstate is thus coupled to the continua through some operator, which can be modeled by a random matrix. The resulting width of the eigenstate results from the superposition of all the decay channels. As a first approximation, all decay channels have roughly the same importance and the statistical distribution of the widths depends only on the number of open channels. If this number is small, pure quantum effects are very important. This is especially the case

Dominique Delande and Andreas Buchleitner

102

for the hydrogen atom in a magnetic field. In such a case, the magnetic field confines the transverse motion of the electron: Immediately above this first ionization threshold, the electron can ionize only in the magnetic field direction and is left in the lowest Landau level. There is only one channel and the prediction of random matrix theory for the widths r of the resonances is the so-called Porter-Thomas distribution (Grtmaud et al., 1993):

r

where is the average width, which can be obtained from classical dynamics. This formula implies that the small widths are by far the most probable. This nonintuitive result means that the strong statistical fluctuations in the random matrix model tend to compensate each other, the most probable value of any matrix element being zero. This prediction has been checked using numerical experiments (Grtmaud et al., 1993). The resulting cumulative distribution fP(r)dT is shown in Fig. 5 in comparison with the Porter-Thomas prediction. The experimental observation of extremely narrow resonances in the spectrum of the lithium atom in a magnetic field between the first two Landau ionization thresholds (Iu et al., 1991) confirms the predominance of small widths and its extreme sensitivity (as any completely destructive interference process) to a small change of parameter. The agreement between theory and numerical experiment in Fig. 5 for a nontrivial phenomenon clearly shows the relevance of quantum chaos for the understanding of complex atomic systems.

J

.-

a a

0.4

E,

"

0.2

0.0

0

1

2 3 4 Normalized Width

FIG.5. Cumulative probability of the normalized widths (solid line) compared with the cumulative probability obtained from the Porter-Thomas distribution (dashed line). The agreement is very good, especially for the smallest values. The infinite slope at the origin (square root behavior) is characteristic of only one open channel. The narrow resonances are by far the most probable.

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

103

When condition (27) is not satisfied, the different resonances overlap. The ionization process has to be fully, that is, nonperturbatively, included in the classical and quanta1 dynamics. Such a situation is probably encountered in the experiments on the hydrogen atom in crossed fields as well as in the microwave ionization experiments. Unfortunately, no clean experimental data in this region have been published. Numerical experiments in crossed fields have reached this regime. The random matrix theory here predicts a well-defined signature in the ionization cross section-the so-called Ericson fluctuations (Ericson, 1960)-which are observed in the numerical spectra (Main and Wunner, 1992). This is one more piece of evidence for the relevance of quantum chaos.

C. PERIODIC ORBITSPECTROSCOPY The two previous subsections have demonstrated the universal character of fluctuations. This also implies that a detailed analysis of all energy levels and eigenstates does not make sense: No interesting information can be brought to the physics of chaotic phenomena beyond the statistical aspects. On the other hand, this does not mean that these energy levels do not carry any information; it is just that this information has to be extracted in a different way. More precisely, as the individual specificity of a chaotic system manifests at relatively short times, before universal chaotic features dominate, it has to be found in the long energy range characteristics of the quantum spectra. From the analysis of Section 11, the comparison between quantum and classical properties is more clearly expressed by a Fourier transform of an experimental spectrum, hereby passing from the energy to the more suitable time domain. Indeed, in a typical spectroscopic experiment, a laser is shined on the atoms, and the absorption probability (or related quantities such as the excitation cross section) is measured. The experimental spectrum is close to

C(E) = C I < $ ~ I T I ~ ) I- ~~ ~i )( E

(29)

i

where 1 $i)are the eigenstates with energy Ei, Ig) represents the initial state of the atom, and T is the transition operator, depending on the polarization and properties of the laser used. The finite resolution of the spectra can be accounted for through a simple convolution with the line profile. A straightforward comparison with Eqs. (14) to (23) shows that the Fourier transform of the spectrum is simply a diagonal element of the time-evolution operator [see Eq. (18)] with the initial state:

lw9) = Tlg)

(30)

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Dominique Delande and Andreas Buchleitner

where Tlg) is not a stationary state of the Hamiltonian, but rather a wave packet. Therefore, performing the Fourier transform of the experimental spectrum is tantamount to studying the time dynamics of this wave packet. In other words, this allows for the investigation of wave packet dynamics with cw lasers. In the specific case of excitation of atomic Rydberg states from a low lying state, T l g ) is a small (on the typical scale of Rydberg states) wave packet starting very close to the nucleus. After absorption of a laser photon, the wave packet expands from the nucleus and eventually returns after some time. For such short times, it is reasonable to assume that the wave packet propagates semiclassically along the classical paths. Hence, C(t) should first decrease very rapidly, as the wave packet explodes from the nucleus, then display a bump after the period of a closed orbit, that starts and ends at the nucleus. The amplitude of the bump is directly related to the angular localization of the initial wave packet and thus depends on the initial state and polarization of the laser. Indeed, Fourier transforms of experimental spectra obtained for the hydrogen atom in electric (Harmin, 1982), magnetic (Delande, 1991; Friedrich and Wintgen, 1980) and crossed fields show these bumps at the periods of closed orbits. The full theory of the effect of closed and periodic orbits on quantum spectra is beyond the scope of this chapter, but certainly among the major achievements of the last years (Bogomolny, 1988; Gao and Delos, 1992). It relies on the qualitative arguments given earlier. The starting point is the Feynman path integral formalism, which is strictly equivalent to the usual formulation of quantum mechanics. It expresses the quantum propagator as a sum over all paths (including unphysical trajectories) connecting the initial to the final point, each path contributing with a factor exp(ijL dt/h) where L is the classical Lagrangian. In the semiclassical limit h + 0, the sum over all paths is calculated with the stationary phase approximation, which selects only classical trajectories (along which the action SL dt is precisely stationary). This gives the semiclassical approximation to the propagator, known as the Van Vleck propagator. In the next step, the Green function at energy E , Fourier transform of the propagator, is computed with one further stationary phase approximation. The dominant contributions then come from the classical orbits with energy E. The experimentally measured spectra can be computed using this semiclassical Green’s function. The simplest quantity is the density of states:

d(E) = C 6 ( E - Ei) i

Considering our previous notation [see Eq. (21)], it corresponds to the case where lei[ = 1, and is nothing but the trace of the Green’s function: 1 d(E) = --TrImG+(E) 7T

=

(r(ImG+(E)r)dr

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

105

If the trace is evaluated again by the stationary phase approximation, one obtains the famous trace formula (Gutzwiller, 1990; Balian and Bloch, 1974) (here written for a two-dimensional time-independent system):

d(E) = d,(E)

+ Im

T. exp[i(Sj/h - ajn/2)] I nh

[det(Mj - l)]’’’

(33)

which expresses the density of states as a sum over all periodic orbits j with action Sj(E), period 17;.(E),and their repetitions (same orbit traveled several times). Here M j is the monodromy matrix describing the linearized motion around the periodic orbit; aj is the Morse index (an integer); and d,(E) is the smooth density of states obtained from the Thomas-Fermi model:

Related quantities, such as the density of oscillator strengths observed experimentally, can also be expressed as sums over periodic or closed trajectories. The trace formula and related formulas establish an intuitive link from quantum to classical mechanics. A difficulty is that the characteristics (period, action, etc.) of the periodic orbits smoothly evolve with the energy E. Thus, the positions and intensities of the peaks in the Fourier transform of a spectrum move with energy, which broadens and smears out the peaks. One can get around this problem using scaled spectra (see Section 1.D) obtained from fixed classical dynamics. From Eq. (33), a Fourier transform of the density of states with respect to the effective value of l/h will display 6 peaks at exactly the actions-instead of the periods in the usual spectraof the periodic orbits. This periodic orbit spectroscopy of scaled spectra is actually the best method we know for extracting information from chaotic spectra, where a detailed analysis of the individual lines is not particularly enlighting. An example of periodic orbit spectroscopy is given in Fig. 6 for the spectrum of the helium atom in a magnetic field (van der Veldt et al., 1993). The quantum spectrum contains 3000 energy levels without any apparent order. Its Fourier transform reveals about 50 peaks at the actions of classical periodic orbits. The comparison with the synthetic spectrum obtained from a semiclassical theory inspired by the trace formula (Gao and Delos, 1992) is very impressive. Most of the peaks are reproduced at exactly the right position with a satisfactory intensity. Those splendid spectra have been obtained in a regime where the dynamics is mainly regular. Other spectra have been recorded on the hydrogen atom in the chaotic regime, at much lower resolution (Holle et al., 1988). Figure 7 shows a series of Fourier transforms of scaled spectra,

106

Dominique Delande and Andreas Buchleitner a)

a

R:

6 7

v:

; 1 1 1 7 1 r l R i 11 12131415161718192021 22

9 1011 1213141516

R 3 16 1718 192021 2223 @ -

iai92ozi

I

I

I

I

I

I

I

I

I

I

FIG.6. (a) Fourier transform of an experimental spectrum of the helium atom in a magnetic field y, at constant scaled energy E = -0.7. It displays peaks at the actions of the classical periodic orbits. Comparison with the semiclassical prediction in (b) shows an extremely good agreement. This proves the major role of classical periodic orbits in quantum spectra. (Courtesy of T. van der Veldt.)

obtained at different values of the scaled enery E, in comparison with the actions of the periodic orbits. Each peak can be clearly attributed to a periodic orbit and its evolution with E . The same kind of results have been obtained on scaled spectra of the rubidium atom in crossed electric and magnetic fields (which requires a simultaneous sweep of the laser frequency and the strengths of the two external fields). Again, some peaks are unambiguously related to periodic orbits (Fig. 8) (Raithel et al., 1991). This is an experimental indication that periodic orbits, which are one-dimensional structures, also play a major role in six-dimensional phase space. The experimental data are not of comparable quality for doubly excited states. Numerical experiments on a two-dimensional simplified model of the helium atom, which assumes the two electrons to be localized in opposite directions with respect to the nucleus, have shown equally the importance of periodic orbits. More importantly, the study of Wintgen et aZ. (1992) shows that the most important orbit is the one where the two electrons have the same excitation, but opposite phases (one electron collides with the nucleus while the other one is at its maximum distance and vice versa). This

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

107

FIG.7. Fourier transform of the experimental spectra of the hydrogen atom in a strong magnetic field y, obtained atzonstant scaled energy E. The different spectra, plotted in (a) for various E (here denoted by E), display peaks at the actions of the classical periodic orbits, plotted in (b). The agreement between the experimental quantum Fourier transform and the classical periodic orbits shows their importance for understanding the long range correlations in the chaotic spectra. (Courtesy of K. H. Welge.)

asymmetric stretch is much less unstable than the symmetric one and consequently dominates the spectra (Ezra et al., 1991). This invalidates the traditional approach (Fano, 1983), in which the two electrons are supposed to lie in the symmetric configuration (the so-called “motion” on the Wannier ridge). This important and unexpected result is typical of how the classical mechanics of complex systems can improve our understanding of quantum spectra. Recently, it has been shown on the basis of theoretical and numerical work that the periodic orbits are not the only relevant structures in chaotic spectra. A complete description must also include complex orbits, where time, position, and momentum are made complex (Kus et al., 1993). The relatively small peak at action S = 3.93 in Fig. 6 does not correspond to any real periodic orbit, but to a “ghost” (complex) periodic orbit. The inclusion

108

Dominique Delande and Andreas Buchleitner Fourier Spectra:

Corresponding Orbits: B1,3

w

= -1.57

0

= 3.93

S = 322h

FIG.8. Fourier transform of an experimental scaled spectrum of rubidium atoms in crossed electric and magnetic fields. It displays peaks at the actions of classical closed orbits. [Courtesy of Raithel et al. (1991).]

of these complex orbits, especially in the vicinity of classical bifurcations, should improve the efficiency of periodic orbit expansions. D. FROMCLASSICAL TO QUANTUM MECHANICS

The set of preceding figures (Figs. 6 to 8) shows how some collective (not related to a specific energy level) properties of quantum chaotic spectra can be interpreted in terms of classical orbits. Because periodic orbits are dense in a fully chaotic phase space, it can be argued that they contain all the information. However, we basically do not know how this information is shared and structured in the different orbits. It seems that some hierarchy exists, at least for systems where a systematic coding of the periodic orbits is known: Most of the information contained in the very long orbits can be essentially extracted from shorter periodic orbits (Wintgen et al., 1992). The best way to clarify this issue is to try to compute a quantum spectrum from the knowledge of as many classical periodic orbits as possible. It has been found numerically that this is an extremely difficult problem because of the proliferation of long periodic orbits and because of subtle relations and compensations between them. Resolving individual peaks requires the inclusion in the sum at least all of the orbits of a period shorter than the break time Greak = h/AE. For typical Rydberg states, this means billions of orbits, which is, of course, impossible. Thus the trace formula and related formulas are of no practical use for computing semiclassical approximations to the energy levels. Partially successful attempts to overcome this difficulty have been made. In practice, although the trace formula is supposed to be asymptotically valid for highly excited states, it is able to give satisfactory values for the energies of the lowest states only! For example, inclusion of 13 periodic orbits allows us to compute correct values for the lowest 10 states of the hydrogen atom in a magnetic field (Wintgen, 1988).

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

109

TABLE 1

ENERGYLEVELS OF ‘Se SERIES OF HELIUM ATOM ~

~

~

~~~~~~~

State

Periodic Orbit

Cycle Expansion

Exact quantum

lsls 2s2s 3s3s 4s4s

- 3.097 -0.804 -0.362 -0.205 -0.131

- 2.932 -0.777 -0.353 -0.199 -0.129

- 2.904

SSSS

-0.778 -0.354 -0.201 -0.129

Note. Some energy levels of the I S p series of the helium atom, compared to the simple semiclassical quantization using only the assymetric stretch periodic orbit and the more refined “cycle expansion,” which includes a set of unstable periodic orbits. The agreement is remarkable, which proves the efficiency of semiclassical methods for this chaotic system. (Courtesy of D. Wintgen.)

A very interesting example in this context is the helium atom, for which the use of cycle expansion (an improvement of the periodic orbit expansion) gives a remarkable value for the energy of the ground state (see Table I). This shows how modern semiclassical mechanics (which is aware of chaos and of the properties of periodic orbits) can solve a simple problem having been discussed for more than 70 years (Wintgen et al., 1992). Yet, we are far from the goal of computing individual, highly excited states of a chaotic system from the knowledge of its classical dynamics. Using periodic orbits, we can at most compute low-resolution spectra.

IV. Wave Functions: Localization and Scars In this section, we discuss the general properties of the wave functions of a classically chaotic system, with emphasis on the localization properties. As for individual energy levels, the wave functions are extremely sensitive to any change of a parameter or any perturbation. In a first approximation, random matrix theory predicts that the wave functions are random waves, uniformly distributed over the entire phase space. This is the quantum equivalent of an ergodic classical trajectory. However, because of quantum interference effects, differences must exist, especially at long times. Any eigenstate of a quantum Hamiltonian is invariant under time evolution and not uniformly distributed over the phase space. Hence, quantum ergodicity is possibly observable in the wave functions only on a scale larger than the typical oscillations, that is, the de Broglie wavelength. The most ergodic wave function one can imagine is then composed of random oscillations at

110

Dominique Delande and Andreas Buchleitner

the de Broglie wavelength superimposed on a smooth uniform background. Beside the previous general description of chaotic wave functions, specific structures might exist that are related to peculiarities of the system, exactly as periodic orbits are responsible for specific structures in the energy spectrum. The simplest manifestation of such a specificity is a preferential localization (enhancement) of the wave function in some region of phase space. Unfortunately, there is a large amount of confusion in the current literature regarding different types of localization, which have different physical origin. Among them, the following ones have been discussed and (numerically) observed or predicted: 0

0

0

Localization in the vicinity of invariant tori in a classically regular system (see Section 1V.A) Localization on residual structures (such as cantori) in systems with mixed regular-chaotic dynamics (see Section 1V.C) Localization in the vicinity of unstable orbits in globally chaotic systems. Such a localization has been baptized a “scar” (Heller, 1984) of the unstable periodic orbit (see Section 1V.B).

In atomic systems, direct experimental observation of wave functions is almost impossible. Hence, we have only indirect proofs of totally or partially localized wave functions. Such proofs rely on measurements of matrix elements involving highly excited chaotic states, which are, of course, affected by the localization. Because of the lack of experimental data in atomic systems, we will mainly present results of numerical experiments.

A. LOCALIZATION IN THE REGULARREGIME For a regular system, the EBK quantization predicts a localization of the regular wave functions on the invariant tori of the classical motion (Ozorio, 1988; Gutwiller, 1990). This has been confirmed numerically on several quantum regular systems. The simplest one is the hydrogen atom in a static electric field. Using its separability in parabolic coordinates and semiclassical quantization along these coordinates, we can compute absorption spectra (and consequently wave functions) in excellent agreement (Harmin, 1982) with the experimental observations. Anothcr example has been recently discovered in the dynamics of the helium atom: There are eigenstates localized on classical invariant tori for the “frozen planet” configuration (Richter et al., 1992). Similar phenomena have been found in the hydrogen atom in a microwave field. The wave functions of the Floquet states evolve at the period of the microwave field. In the simplified one-dimensional model, there is a resonance between the electronic Kepler motion and the microwave external

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field when their two frequencies are equal. The nonlinear terms create the following standard phenomenon: There is a stable periodic trajectory where the electron motion is locked on the external frequency (with a well-defined phase) surrounded by invariant tori and an unstable periodic trajectory with opposite phase embedded in a stochastic separatrix layer (Buchleitner, 1993). Figure 9 shows the time evolution of a quantum wave function associated with an invariant torus. It is clearly an atomic wave packet oscillating back and forth at the frequency of the external microwave. Contrary to normal atomic wave packets, which spread rapidly because of dispersion, this wave packet is nondispersive and reproduces itself identically after one period. This is a typical nonlinear stabilization phenomenon, which might be of interest in the context of wave packet studies.

B. LOCALIZATION IN THE CHAOTIC REGIME The localization of chaotic wave functions in the vicinity of unstable periodic orbits, although it might appear a little surprising at first sight, is after all extremely natural if one comes back to the equations describing the quantum dynamics. For times of the order of the shortest periods, the

radial distance (lo4a.u.) FIG.9. Wave functions of Floquet eigenstates of a one-dimensio?al hydrogen atom in a microwave field. The states evolve in time with the periodicity of the external microwave field. We show the wave packets at phases 0, 4 2 , and s (from left to right). Top: When the classical motion is regular, the quantum wave packet is localized in phase space, locked on the external frequency, and nondispersioe. Bottom: When the classical motion is chaotic, the quantum wave packet may be partially localized on complex phase space structures, with a quasirandom background. Note the phase difference of a s between the regular and chaotic wave packets.

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semiclassical Van Vleck propagator can be used, which means that a localized wave packet essentially evolves classically. Let us consider an initial wave packet J$(O)) lying close to the classical orbit with an initial momentum close to the classical momentum on the orbit, for example, a coherent state. Its localization is limited by the Heisenberg inequalities, but in the semiclassical limit k + 0, an accurate localization is possible. For short times, it will evolve as a dispersing wave packet approximately following the classical periodic orbit. On the other hand, its time evolution can also be described by the quantum equations (14) to (23). This has two consequences: Because the modulus of the overlap between the wave packet 1 $(t)) and an eigenstate 14i) is constant with time, the latter cannot be localized only on one point or one part of the periodic orbit. It is either scarred by the whole orbit or not scarred at all. The autocorrelation function C(t) = ($(O) 1$(t)) displays peaks at multiples of the period T,because the wave packet comes back at its initial point. Ofcourse, because of the dispersion of the wave packet, the returning peaks get smaller and smaller and finally disappear. However, the low-resolution Fourier transform of C(t) will display peaks at the inverse period 2zk/T From Eq. (23), these peaks are unresolved superpositions of the contributions of the various energy levels. Consequently, some of the eigenstates must have enhanced values of Icil = l(4il$(0)) 1. In other words, some wave functions of eigenstates must have an enhanced amplitude on the initial position, that is, a scar of the periodic orbit. Moreover, these arguments make it possible to predict quantitatively where the peaks are located and thus where scars are expected to be. The resulting condition is known as the scar quantization condition (Delande, 1991; Gutzwiller, 1990): S(E) =

i

(

p*dr=2zh n + -

(35)

where S(E) is the action of the periodic orbit, a its Morse index, and n a non-negative integer. This formula, although it is strongly reminiscent of the WKB quantization condition, is of a different nature. It is only an approximate indication of where scarred states have the highest probability of being. Indeed, nothing proves that the different unresolved states lying under the peaks of the low-resolution Fourier transform are democratically scarred. All we know is that collectively they carry a well-defined scar strength. The detailed repartition depends on the long-term behavior of the wave packet and cannot be estimated with this approach. The existence of scars makes the difference between real wave functions and random waves. Scars appear because they are the only possible intense manifestations of chaotic fluctuations compatible with the dynamics.

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The scarring phenomenon is nothing but intrinsic fluctuations plus semiclassical propagation. It also follows that scars are stronger if the orbit is short and weakly unstable. An example of a scar, obtained in a numerical experiment for the hydrogen atom in a magnetic field, is shown in Fig. 10. An indirect experimental proof of the existence of partial localization of chaotic wave functions has been obtained on the rubidium atom in crossed electric and magnetic fields. It uses an inhomogeneous electric field to induce deflection of an atomic beam proportionally to the static electric atomic dipole moment. If the wave functions were purely random waves, the electric dipole moment would be close to an average fixed value (classical average over the accessible phase space). On the contrary, experimental results, displayed in Fig. 11, indicate the existence of large electric dipole moments and large variations with the different parameters (Raithel et al., 1993). Although it is not obvious that we should associate the large dipole moment with a specific classical structure (saddle fixed point, periodic orbits, etc.), this nevertheless proves that localization phenomena exist in this real chaotic system.

c. LOCALIZATION IN THE MIXEDREGULAR-CHAOTIC REGIME Localization phenomena are much more difficult to analyze in mixed systems. Indeed, contributions come from the remaining invariant tori

FIG.10. Wave function of the hydrogen atom in a magnetic field. We show the 474th excited state (L, = 0, odd parity series) at scaled energy E = -0.2, scarred by the classical almost circular unstable periodic orbit shown in superimposition.The dotted line is the line of classical turning points.

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B.0.27 fa73

8=0.35T t i 1.70

8=0.51 c: 1.12

I

e:0.97 t=OS6

1

I

-

iL

’..

B.121

c.o.37 I

( 1

Od

Scaled @ W e Moment

I

‘0

FIG.1 1 . Experimental measurements showing the dipole-deflection pattern as a function of a scaled electric field. The variations of the average electric dipole moment are probably related to localization of the eigenstates on classical structures (Courtesy of Raithel et al., 1993.)

together with various structures of the phase space. The latter cannot be reduced to isolated periodic orbits, as in a fully chaotic system. The main reason is the existence of lots of different time scales, which strongly prevents the quantum system from approaching ergodicity. Additional information is needed to interpret the quantum spectra properly, such as sizes and geometrical shapes of the unstable regions surrounding unstable orbits, geometries of separatrices between different types of motion, existence and geometries of partial barriers, etc. A typical example is the one-dimensional hydrogen atom in a microwave field. The phase space is mixed and structured by the different resonances between the atomic and the external frequencies. Figure 9 shows an atomic state, evolving periodically with the microwave frequency, localized near the unstable periodic orbit, in phase opposition with the stable resonance, and the surrounding phase space. It is the counterpart to the quantum state localized on the principal resonance discussed earlier. One clearly sees the wave packet oscillating back and forth emerging from a rather erratic background. Analysis of phase space plots of such a state shows that it contains essential contributions from the stochastic layer surrounding the unstable periodic orbit. Hence, analyzing such a system in terms of the scar of an unstable periodic orbit (Jensen et al., 1989; Sauer et al., 1992) is an

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adventurous extrapolation and certainly an oversimplified view of localization in mixed systems. Note also that strictly analogous localized wave packets, periodic in time with no dispersion, have also been observed in numerical simulations of three-dimensional atoms (Buchleitner, 1993).

D. RELATIONSHIP BETWEEN LOCALIZATION AND LIFETIME An important question is one of whether the localization of atomic wave functions on bound phase space structures implies some stability versus ionization. In general ionization takes place in an outer region of the atom, where the binding Coulomb field is relatively weak, while significant phase space structures are localized in the inner part, where all interactions have the same order of magnitude. Thus, any relation between localization and the ionization rate indicates the existence of a long-range correlation in phase space. This certainly happens in regular systems where the relevant structures, invariant tori, extend throughout phase space, as has been demonstrated for the hydrogen atom in a static electric field (Harmin, 1982) and for the frozen planet configuration of the helium atom (Richter et al., 1992). For a strongly chaotic system, long-range correlations are not expected, and scarring by an unstable periodic orbit should not increase (or decrease) significantly the lifetime of an atomic state. Strong numerical evidence of this decorrelation exists for the case of atoms in a magnetic field above the ionization threshold (see Section 1II.B). This explains why residual regularities as scars are not relevant for the ionization rate and the success of the random matrix predictions. Mixed regular-chaotic systems are much more complicated. Residual long-range correlations exist, but not in a systematic way. Hence, scarring might favor low ionization rates, as suggested by Jensen et al. (1989), but there is no systematic one-to-one correspondence. It is certainly oversimplified and essentially wrong to associate a reduced ionization rate with an individual scar of an unstable periodic orbit.

V. Dynamics A. TIME-INDEPENDENT SYSTEMS Although the study of the temporal dynamics of a quantum system provides us with the clues for understanding quantum chaos (see Section 11), very few

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direct measurements have been done in the chaotic regime. Actually, as shown earlier, Fourier transforms of spectra obtained with cw lasers allow for the same information in an easier way. Furthermore, there is no equivalent of scaled spectroscopy in the time domain. Any wave packet excited in a real experiment by a short laser pulse has a certain dispersion in energy and will consequently spread quite rapidly, which is avoided in scaled spectroscopy. Consequently, it is not very likely that new interesting physical information can be obained from such time-resolved experiments on time-independent systems.

B. TIME-DEPENDENT SYSTEMS For explicitly time-dependent Hamiltonian systems, the situation is completely different. Experimental results are badly needed, especially for the long-time dynamics, which is difficult to study in numerical experiments. The present status of technology allows us to excite and probe atomic systems on very short time scales, of the order of 1ps or less. For excited Rydberg states, this is shorter than the Kepler period of the electron. A new generation of experiments in atomic physics is born. Most probably, fascinating unexpected results on quantum chaotic dynamics will be discovered in the near future. The hydrogen atom in a microwave field is a first step in this direction, because it contains an explicit time dependence, but can also be reduced to a static problem by use of the Floquet theorem as a consequence of its time periodicity. The exchange of energy between the different degrees of freedom, which is usually difficult to measure in a microscopic multidimensional system, is equivalent here to the number of photons exchanged between the atom and the external field, a much more accessible quantity. A rough estimation of the energy exchange is the ionization probability of the atom, by far the easiest quantity to measure in an experiment. It contains some information on the dynamics, but in a strongly convoluted form. In view of the success of spectroscopy of chaotic time-dependent systems, it should be interesting to perform the spectroscopy of the Floquet states of the microwave problem, eventually followed by a Fourier transform. Although in principle feasible, Floquet spectroscopy has not yet been performed. In a typical experiment, the atoms are prepared in an excited Rydberg state, then exposed to the microwave for a finite well-defined interaction time and finally analyzed. The analysis can simply be the ionization probability or, in a more refined experiment, the internal state of the atom is analyzed by (static) field ionization. The shape of the ionization signal versus the static field strength is sensitive to the values of the principal

CLASSICAL AND QUANTUM CHAOS IN ATOMIC SYSTEMS

6

10

14 10 22 Uicroware Electric Fidd Amplitude (V/cm)

117

28

FIG.12. Experimental ionization probability of hydrogen atoms after interaction with a microwave field versus the field amplitude. The curve has a typical threshold behavior, a signature of a classical phenomenon. (Courtesy of P. Koch.)

quantum number populated after the interaction with the microwave. Figure 12 shows a typical experimental result: The ionization probability plotted versus the microwave field strength displays a threshold behavior (Sauer et al., 1992). The field strength at which 10% of the atoms is ionized, defines the 1OOA ionization threshold, which, in turn, is a smooth function of the microwave frequency. This behavior is in sharp contrast to the standard photoeffect where the threshold behavior is observed on the frequency dependence of the ionization probability, independently of the field amplitude. The latter is a signature of a quantum effect, whereas the former is rather characteristic of a classical phenomenon. The key observation is that the gross features of the ionization process depend only on the scaled variables of Eq. (12): Different ionization curves for different initial states, but at the same scaled frequency and plotted versus the scaled amplitude, do approximately coincide. As explained in Section I.D, this is a clear signature of a classical phenomenon. It does not discard the possibility of finding a quantum interpretation, but it proves that a classical one, without specific quantum interferences, has to exist. Conversely, any deviations from this scaling behavior are the signature of a quantum effect. Figure 13 shows the scaled 10% ionization threshold versus the scaled frequency, as observed experimentally (Bayfield et al., 1989). In the lowfrequency regime, the threshold tends to the static threshold, as it should. At larger frequencies, it decreases up to a scaled frequency wo 3: 1 where it starts to increase again. For the classical system, the phase space is mixed regular-chaotic, because of various resonances between the atomic and external frequencies. The classical ionization threshold is associated with the disappearance of the last invariant torus separating the initial state of the

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118

I

1

0.5

1.0

2.0

1.5

2.5

J 3.0

WO

FIG.13. Scaled 10% ionization threshold versus the scaled frequency, from the experimental results of Bayfield et al. (1989). Above w,, = 1, the experimental results (symbols, dashed line) significantly deviate from the classical result (dotted line). This supports the theory of dynamic localization, whose prediction is indicated by the solid line. (Courtesy of I. Guarneri.)

atom from the continuum. Above this value, the classical dynamics is essentially a diffusive-like motion leading to relatively fast ionization. The comparison between the classical threshold and the quantum experimental results shows a globally excellent agreement up to coo N 1; above, the experimental result lies systematically higher, which signifies that the quantum system is more stable than the classical one. Strong experimental and numerical proofs show that this phenomenon is of quantum origin: Starting from higher values of the principal quantum number no, at the same value of coo, leads to a quantum threshold a little lower. At fixed coo, the threshold tends to the classical prediction with increasing no, as it should do in the semiclassical limit (Buchleitner, 1993). When broadband noise is added to the microwave field, the threshold decreases and tends to the classical one (Arndt et al., 1991). The noise amplitude needed to induce such an effect is very small, not sufficient to affect strongly the classical dynamics. The correct interpretation is that noise destroys the subtle phase coherence effects in the quantum system. Numerical experiments show that the ionization probability is very sensitive to small changes (of the order of of the scaled frequency at a fixed value of no. This has been related to the sensitivity of individual Floquet states on small changes of parameters (Casati et al., 1988; Buchleitner, 1993), a quantum consequence of classical chaos, as shown in Section 1II.A.

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loglo interaction time t (s) FIG.14. Scaled 10% ionization threshold of a one-dimensional hydrogen atom (initial principal quantum number no = 23) in a microwave field versus the interaction time t (numerical experiment). The different curves correspond to different values of the scaled frequency (from top to bottom, wo = 2.7, 3.0, 2.0, 1.43, 1.3, 0.6, and 0.8). This shows a weak dependence over more than 4 orders of magnitude in the interaction time, apparently of the algebraic type. This result strongly supports the theory of dynamical localization.

Numerical experiments where the interaction time with the microwave is changed over several orders of magnitude show a very weak dependence of the 10% threshold with the interaction time (see Fig. 14) without displaying a systematic dependence on wo (Buchleitner, 1993). This is inconsistent with the diffusive-like chaotic excitation and ionization predicted by classical dynamics. It agrees with an experimental observation on rubidium atoms (Arndt et al., 1991). We currently interpret these results to be an effect of dynamical localization due to the relatively long interaction time between the atom and the microwave (few hundreds periods): The quantum time dynamics initially follows the diffusive classical one, but “freezes” after the break time (Casati et al., 1988; Fishman e al., 1982; Bliimel et al., 1991). Of course, this is only a qualitative argument: Ionization does not stop after the break time, but it is strongly reduced. A direct experimental observation on rubidium Rydberg states, analyzed by static field ionization, effectively shows this freezing of the atomic evolution after some break time (see Fig. 15) and supports the theory of dynamic localization (Bliimel et af., 1991). By means of simplified classical

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ramp field IVlcml

ho"

30 05

-

[loki c p

'35 6

18 3

0

ramp field N k m I

[lolcA{

-E

05

N 0

5 0

366

18 3 ramp field IVlcmI

0

FIG.15. Field ionization signal of rubidium Rydberg states. (a) Initial state 84p,,,. (b) The same initial state after interaction with a microwave field during 10011s.(c) Same as (b), but interaction time equal to 1OOOns. The absence of evolution between (b) and (c) strongly supports the theory of dynamic localization (Bliimel et al., 1991).

time dynamics and a simplified one-dimensional model of the atom, quantitative predictions on the effect of dynamical localization on the threshold behavior (Casati et at., 1988) have been obtained, which agree quite well with experimental results (see Fig. 13). Numerical experiments (Buchleitner and Delande, 1993) also show that the same phenomenon exists for a three-dimensional atom as well as for a simplified one-dimensional model. In the former situation, because of the additional degree of freedom, the mean level spacing between Floquet eigenstates is much smaller and the simple expression for the break time [Eq. (24)] predicts a much longer time for the manifestation of dynamical localization. The reason is probably that Eq. (24) is valid for a strongly chaotic system, but not for mixed systems. In the real experiments, the influence of remaining structures in phase space on the break time is not known.

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VI. Conclusions We believe that the experimental, theoretical, and numerical results discussed in this review clearly show how the concepts of classical chaos in lowdimensional Hamiltonian systems provide one with new and interesting physical insights into the dynamics of complex quantum systems. The general behavior of a chaotic quantum system may have different complementary aspects depending on which physical quantities are probed: 0

On a narrow energy interval-roughly at the level of individual eigenstates-the quantum structures display strong apparently random fluctuations and a high sensitivity to any small change of an external parameter. This is the quantum counterpart of the classical sensitivity to initial conditions on long time scales.

0

On a large energy scale, where spectral properties are averaged over several states, the individual features of the studied system become manifest, the best example being the importance of periodic orbits and scars.

The two preceding aspects are well separated in the semiclassical limit, that is, for highly excited states. A major unsolved problem is to understand what happens for weakly excited states, far from the semiclassical limit. There, the standard characterization of quantum chaos by the statistical properties of the energy levels breaks down. This is highly relevant for experimental systems such as molecular physics, where the complexity comes not from the strong excitation of a few degrees of freedom, but from the simultaneous excitation of many different degrees of freedom. A second very difficult problem is the quantum behavior of systems with mixed regular-chaotic dynamics-by far the most common ones. Theoretical guides are lacking here. It is likely that progress will be achieved in this direction in the near future.

References Al’tshuler, B. L., Lee, P. A., and Webb, R. A. (1991). Mesoscopic Phenomena in Solids, North-Holland, Amsterdam. Arndt, M., Buchleitner, A., Mantegna, R. N., and Walther, H. (1991). Phys. Rev. Lett. 67,2435. Balian, R., and Bloch, C. (1974). Ann. Phys. 85, 514. Bafield, J. E., Casati, G., Guarneri, I., and Sokol, D. W. (1989). Phys. Rev. Lett. 63, 364. Blumel, R., Buchleitner, A., Graham, R., Sirko, L., Smilansky, U., and Walther, H. (1991). Phys. Rev. A 44, 4521. Blumel, R., and Smilansky, U. (1987). Z. Phys. D6, 83.

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Bogomolny, E. G. (1988). J E T P Lett. 47, 526. Bohigas, 0. (1991). In: Chaos and Quantum Physics (Giannoni M.-J., Voros, A,, and ZinnJustin, J., eds.), Les Houches Summer School, Session LII, North-Holland, Amsterdam. Bouchiat, H., and Montabaux, G. (1990). J. Phys. (Paris) 50, 2695. Buchleitner, A. (1993). Ph.D Thesis, Universitb Pierre et Marie Curie, Paris. Buchleitner, A., and Delande, D. (1993). Phys. Rev. Lett. 70, 33. Camus, P., Gallagher, T. F., Lecomte, J. M., Pillet, P., Pruvost, L., and Boulmer, J. (1989). Phys. Rev. Lett. 62, 2365. Casati, G., Guarneri, I., and Shepelyansky, D. L. (1988). IEEE J. Quantum Electron. 24, 420, and references therein. Casati, G., Guarneri, I., and Shepelyansky, D. L., (1990). Physica A 163,205. Delande, D. (1991). In: Chaos and Quantum Physics (Giannoni, M.-J., Voros, A,, and Zinn-Justin, J., eds.), Les Houches Summer School, Session LII, North-Holland, Amsterdam. Delande, D., Bommier, A., and Gay, J. C. (1991). Phys. Rev. Lett. 66, 141. Delande, D., and Gay, J. C. (1986). Phys. Rev. Lett. 57, 2006. Domke, M., Remmers, G., and Kaindl, G. (1992). Phys. Rev. Lett. 69, 1171. Eichmann, U., Lange, V., and Sandner, W. (1992). Phys. Rev. Lett. 68,21. Ericson, T. (1960). Phys. Rev. Lett. 5, 430. Ezra, G. S., Richter, K., Tanner, G., and Wintgen, D. (1991). J. Phys. B 24, L413. Fano, U. (1963). Rep. Prog. Phys. 46, 97. Fishman, S., Grempel, D. R., and Prange, R. E. (1982). Phys. Rev. Lett. 49, 509. Friedrich, H., and Wintgen, D. (1989). Phys. Rep. 183, 37. Galvez, E. J., Sauer, B. E., Moorman, L., Koch, P. M., and Richards, D. (1988). Phys. Rev. Lett. 61, 2011. Gao, J., and Delos, J. B. (1992). Phys. Rev. A 46, 1455. Gremaud, B., Delande, D., and Gay, J. C. (1993). Phys. Rev. Lett. 70, 615. Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics. Springer, Berlin. Halley, M. H., Delande, D., and Taylor, K. T. (1993). J. Phys. B 26, 1775. Harmin, D. A. (1982). Phys. Rev. A 26, 2656. Hasegawa, H., Robnik, M., and Wunner, G. (1989). Prog. Theor. Phys. 98, 198. Heller, E. J. (1984). Phys. Rev. Lett. 53, 1515. Henon, M. (1983). In: Chaotic Behaviour of Deterministic Systems (Iooss, G., Helleman, R., and Stora, R., eds.), Les Houches Summer School 1981, North-Holland, Amsterdam. Holle, A., Main, J., Wiebusch, G., Rottke, H., and Welge, K. H. (1988). Phys. Rev. Lett. 61, 161. Iu, C. H., Welch, G. R., Kash, M. M., Kleppner, D., Delande, D., and Gay, J. C. (1991). Phys. Rev. Lett. 66, 145. Jensen, R. V., Sanders, M. M., Saraceno, M., and Sundaram, B. (1989). Phys. Rev. Lett. 63, 277 1. KUS, M., Haake, F., and Delande, D. (1993). Phys. Rev. Lett. 71,2167. L'Huillier, A., Schafer, K. J., and Kulander, K. C. (1991). J. Phys. B 24, 3315. Lichtenberg, A. J., and Lieberman, M. A. (1983). Regular and Stochastic Motion. Springer, New York. Main, J., and Wunner, G. (1992). Phys. Rev. Lett. 69, 586. Ozorio de Almeida, A. M. (1988). Hamiltonian Systems: Chaos and Quantization. Cambridge University Press, Cambridge. Percival, I. (1991). Recent Developments in Classical Mechanics (Giannoni, M.-J., Voros, A., and ZinnJustin, J., eds.), Les Houches Summer School, Session LII, North-Holland, Amsterdam. Raithel, G., Fauth, M., and Walther, H. (1991). Phys. Rev. A 44, 1898. Raithel, G., Fauth, M., and Walther, H. (1993). Phys. Rev. A 47,438. Richter, K., Briggs, J. S., Wintgen, D., and Solov'ev, E. A. (1992). J. Phys. B 25, 3929. Sauer, B. E., Bellerman, M. R. W., and Koch, P. M. (1992). Phys. Rev. Lett. 68, 1633, and references therein. Smilansky, U., (1991). In: Chaos and Quantum Physics (Giannoni, M.-J., Voros, A,, and

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Zinn-Justin, J., eds.), Les Houches Summer School, Session LII, North-Holland, Amsterdam. Van der Veldt, T., Vassen, W., and Hogervorst, W. (1993). Europhys. Lett. 21, 903. Wintgen, D. (1988). Phys. Rev. Lett. 61, 1803. Wintgen, D., Richter, K., and Tanner, G. (1992). Chaos 2, 19. Zakrzewski, J., and Delande, D. (1993). Phys. Rev. E 47,1650. Zakrzewski, J., Delande, D., Gay, J. C., and Rzcazewski, K. (1993a). Phys. Rev. A 47,R2468. Zakrzewski, J., Delande, D., and Kus, M. (1993b). Phys. Rev. E47, 1665.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 34

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS THAD WALKER and PAUL FENG Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin I. Introduction . . . . . . . . . . . . . . . . . . 11. Collisions in Optical Traps: General Considerations . A. Optical Trapping . . . . . . . . . . . .. . B. Interpretation of Ultracold-Collision Experiments in the Presence of Light . . . . . . . . . . . . 111. Collisions of Ground State Atoms . . . . . . . . . A. Hyperfine-State-Changing Collisions . . . . . . B. Thermalization in Magnetic Traps . . . . . . . C. Frequency Shifts of Atomic Clocks . . . . . . . D. Open Questions for Ground-State Collisions . . . IV. Collisions Involving Singly Excited States . . . . . A. Energy-Transfer Mechanisms . . . . . . . . . B. Collision Dynamics . . . . . . . . . . . . . C. Practical Implications of Trap-Loss Collisions for Laser Cooling and Trapping . . . . . . . . D. Absolute Measurements-Small Detunings . , . E. Frequency-Dependent Experiments . . . . . . . F. Open Questions for Singly Excited State Collisions V. Collisions Involving Doubly Excited States . . . . . A. Collision Dynamics . . . . . . . . . . . . . . B. Summary of Experiments . . . . . . . . . . . C. Open Questions for Doubly Excited States . . . . Acknowledments . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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I. Introduction A major motivation for the development of optical trapping and cooling techniques (Foot, 1991) is the desire to study atoms free from the usual sources of spectral line broadening and shifts that arise from atomic motions and collisions. With temperatures of 1pK now attainable, great progress has been made toward the elimination of motional effects. However, it was recognized very early in the development of optical traps that several novel collisional processes would be important at low temperatures. These collisions are distinguished from the usual atomic collisions in several ways. First, the low energies imply that the collisions can be highly quantum mechanical in nature, involving only a few partial waves. Second, the 125

Copyright 0 1994 hy Academic Press, Inc. A11 rights of reproduction in any form reserved

ISBN 0-12-003834-X

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collisions are unusually sensitive to long-range interatomic forces. Third, the long collision times mean that absorption and emission processes can significantly affect the collision dynamics. Finally, the possibility of spontaneous emission during the collisions requires new ways of thinking about these collisions, both theoretically and experimentally. To date, a variety of experiments have been performed that investigate collisions of laser-cooled atoms. In this chapter we review these experiments with an emphasis on the issues that have and have not been addressed. For more information on the theoretical aspects of ultracold collisions that will not be discussed in detail here, see the recent review by Julienne et al. (1992). As illustrated in Fig. 1, the experiments naturally fall into three classes, depending on which states the atoms are in when they reach the point of closest approach. Each of these classes is discussed separately. The first class of ultracold collisions, where both atoms are in their ground electronic states (henceforth, referred to as ground state collisions),

Naz++ e-

3s + 3P3,2 3 s + 3P1/2

A

1,i

3s +3s

Interatomic Separation R FIG.1. Basic collision processes for laser-cooled Na atoms. Ground-state collisions occur on the 3s + 3s curves. Singly excited states collisions involve (1) excitation by the lasers at large R, (2) acceleration with possible spontaneous decay on the excited state potential, and energy transfer via (3a) radiation or (3b) change of fine-structure state at small R. Doubly excited state collisions involve (i) excitation by the lasers at large R, (ii) acceleration with possible spontaneous decay on the singly excited state potential, (iii) excitation to a doubly excited state at intermediate R, and (iv) possible spontaneous decay or motion to small R where (v) associative ionization may occur.

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has the most features in common with higher temperatures collisions. Here the collisions are conventional in the sense that the atomic motion is governed by conservative interatomic forces. Depending on the temperature, the collisions can be dominated by only the first few partial waves, and they are extremely sensitive to details of the interaction potentials. Thus, for example, it may be possible to affect the collision processes with weak electric or magnetic fields. In various contexts both elastic and inelastic processes are observed. The current interest in these ground state collisions stems from (1) the important role of the elastic collisions in applications of optical traps to other areas of study such as frequency standards and the possibility of attaining a weakly interacting Bose condensate, and (2) the limitations that inelastic collisions place on the number of atoms that can be trapped. These collisions are discussed in Section 111. The second class of collisions, to which we will refer as singly excited state collisions, involves excitation of one of the colliding atoms by the light fields, while the other remains in the ground state. These collisions are of interest because they display many of the features mentioned in the opening paragraph. The excited state interaction potentials arise from the resonant dipole-dipole interaction, and accordingly are very long range ( R - 3, where R is the interatomic separation); this interaction can significantly affect the ultracold atoms at very large distances, of the order of I/27~,where I is the wavelength of the atomic resonance transition. The long-range potentials are modified by the presence of strong, near-resonant light fields. Spontaneous emission plays an essential role in the collision dynamics because the collision times are in general longer than excited state lifetimes. In the absence of acceleration of the atoms by the excited state potential, spontaneous emission would terminate the excited state collision. However, the potentials are strong enough in some cases to accelerate the atoms to relatively high energy (1 K or so) and, consequently, to small interatomic separation, without interruption of the process by spontaneous emission. A subsequent change in the molecular state or radiation of a red-shifted photon at small interatomic separation can transfer some of the internal energy of the atoms to kinetic energy, resulting in ejection of the atoms from the trap (trap loss). Although most of the experiments done to date have focused on the roles of spontaneous emission and the nature of the energy transfer process, the collisions have also been used to perform a novel type of rotationless free-bound molecular spectroscopy. These collisions are discussed in Section IV. The third class of collisions, doubly excited state collisions, is distinguished by both atoms being excited by the light fields. These collisions are also sensitive to the long-range potentials, interaction with external light fields, and spontaneous emission. Since two excitations of the atoms are involved, the rates are small-however, direct detection of final-state ions is highly efficient. Because of the two-photon nature of the process, the collision

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dynamics are correspondingly more complex than for the singly excited states. For example, motion occurs on both singly excited and doubly excited potentials. Because the doubly excited potentials are relatively weak, with an R-dependence of R - 5 , they cannot accelerate the atoms rapidly enough to reach small R without spontaneous emission occurring. Thus, in order for the atoms to reach small interatomic separations in a doubly excited state, substantial acceleration of the atoms in a singly excited state must occur first, followed by absorption of another photon to a doubly excited state. As in the singly excited case, most of the experiments have focused on understanding the collision process, and observation of vibrational structure in the frequency dependence of the collision rates is a promising new technique for molecular spectroscopy. These collisions are discussed in Section V. Other phenomena occur in optical traps that involve interactions between the trapped atoms. In particular, collective behaviors have been observed that arise from long-range forces between the atoms (Walker et al., 1990; Sesko et al., 1991; Hemmerich and Hansch, 1993). We distinguish these phenomena from collisions because they involve other than pairwise interactions, the relative motions of the atoms are likely not important, and they may depend sensitively on the details of the trapping method used. Before detailing specific experiments (in Sections 111, IV, and V), in the next section we present some of the basic features of optical traps that are necessary for understanding ultracold collisions. In this context, it is natural to discuss the collisions from a generic point of view in order to see how conclusions about the collisions can be deduced from the measurements.

11. Collisions in Optical Traps: General Considerations In this section we present an overview of how collisions arise in optical traps. Although a detailed description of the techniques of optical trapping and cooling is beyond the scope of this review [see Foot (1991) for more information], we summarize in Section 1I.A the features of optical traps that are important for understanding the collisions. We then describe in Section 1I.B the basic picture that has emerged to describe a generic collision between laser-cooled atoms, and that is used to interpret the experiments.

A. OPTICAL TRAPPING We now describe an apparatus that is typical of those used for collisions experiments so that we can discuss many of the techniques that have been used to make these measurements. None of the experiments conforms

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exactly to the one described here; later in this section we describe two of the major variations and note significant differences when the experiments are discussed in the following sections. Figure 2 is a schematic diagram of a magneto-optical trap (MOT) apparatus that is suitable for measurements of collisions of trapped sodium atoms. An atomic beam of Na enters an ultrahigh vacuum chamber (- 10-"Torr), providing a source of atoms for trapping. Three mutually orthogonal counter-propagating pairs of laser beams intersect the atomic beam inside the vacuum chamber, producing an l-cm3 volume of space where trapping and cooling occur. The laser light consists of two narrowband frequencies, one tuned typically 10 to 20 MHz below the F = 2 + F' = 3' transition in Na (Fig. 3), and the other tuned to the F = 1 + F' = 2' transition. The F = 1 -P F' = 2' light serves to optically pump the atoms into the F = 2 state, whereas the F = 2 + F' = 3' light produces the cooling and trapping forces on the atoms. Cooling of the atoms to microkelvin temperatures is produced by several mechanisms, including Doppler cooling and polarization-gradient cooling (Foot, 1991). A pair of anti-Helmholtz magnetic field coils (not shown) produces a quadrupolar magnetic field that is zero at the center of the intersecting laser beams. The trapping-laser photons are spin polarized in such a way as to provide a restoring force that

-

-

FIG.2. Schematic of a typical optical trapping apparatus. Important features not shown are the ultrahigh vacuum chamber inside which the laser and atomic beams intersect and the magnetic field coils that produce a quadrupolar field at the position of the trapped atom cloud.

Thad Walker and Paul Feng

130 0

3'

-60 MHz

2'

-94 MHz -1 10 MHz

0'

1'

1 32P,,, ,

FIG.3. Hyperfine structure of the 3S,/, and 3P,,, states of Na. In the traps used for most collision experiments, the population of the 3S,,,(F = 1) state is small compared to the F = 2 state.

confines the atoms. As a result of the cooling and confining forces, slow atoms from the atomic beam are captured by the trap. Having obtained a sample of trapped atoms, it is often necessary to measure its density and temperature. To measure the density, either absorption or emission measurements are performed, combined with measurements of the spatial distribution of the cloud of trapped atoms. For absorption, a weak probe laser is sent through the cloud. By means of known atomic matrix elements, an integrated column density is determined. For emission, a calibrated lens and photodiode/photomultiplier system is used. The total number of atoms can be calculated from the photon detection efficiency, the lifetime of the excited state of the atom, and the fraction of atoms in the excited state. Knowledge of the excited state fraction is usually obtained from calculations based on the relevant atomic matrix elements and the intensity and frequency of the trapping laser. For either absorption or emission measurements, it is also necessary to obtain the spatial distribution of the trapped atoms. The spatial distribution is deduced by analyzing camera images of the fluorescence from the cloud of trapped atoms. If the number of trapped atoms is small ( - lo4) the distribution is a Gaussian with a width that depends on the temperature of the atoms and the strength of the confining force. If the number exceeds that value,

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS 13 1

radiation trapping causes the cloud to expand as discussed later. Typical values of the peak density are lo9 to 1011cm-3,with typical experimental uncertainties in the density of 30%. Temperature measurements can be made by a variety of techniques, the most popular probably being releasing the atoms and measuring time-offlight signals. Typical values for MOTS are in the range of 20 to 300pK. Little emphasis has been placed on the effects of temperature in most of the experiments to date. The reasons for this are explained in Section 1V.B. Collisions are typically detected in one of two ways. The first is to observe the final state produced. For Na collisions involving doubly excited states, associative ionization can occur, so detection of the Na: ions that are produced is possible using an electron multiplier. This can be done with excellent efficiency, so with knowledge of the density distribution n(r) of the atoms in the trap and the ion count rate I , collisional rates can be measured absolutely. The ion count rate is given by r

J

I = /3 n2(r)d3r Here /3 is the desired rate coefficient. Note that, in contrast to higher temperature experiments, the convention for collisions between laser-cooled atoms is that /3 is defined with respect to the total atom density, not the density of excited state atoms. This is a result of a fundamental difference between low- and high-temperature experiments, as explained in Section 1I.B. Many experiments on ground state and singly excited state collisions rely on indirect detection. When the result of a collision is the transfer of internal energy to kinetic energy of the atoms, the atoms leave the trap if the resulting velocity of the atoms exceeds the capture velocity of the trap. This process is sometimes referred to as trap loss. Thus the collision rate can be measured by observing the time dependence of the number of atoms in the trap. This is typically done by loading a number of atoms into the trap from the atomic beam, then shuttering the atomic beam and observing the subsequent decrease in the number of atoms with time, due to trap loss. The rate equation governing the density distribution (now time dependent) of the trapped atoms is

3 must In addition to 8, the collisional rate coefficient of interest, the term l be included to take into account collisions between the cold atoms and the residual vapor in the vacuum system. If the number of atoms is small enough that the shape of the density distribution is constant in time, a

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similar equation is obeyed by the total number of trapped atoms (Bardou et al., 1992), and Eq. (2) has a simple transient solution. When more than lo4 to lo5 atoms are contained in the trap, long-range repulsive forces between the atoms, which arise from radiation trapping, begin to limit the atom density. As the number of atoms increases further, the density distribution approaches that of a uniform-density sphere or ellipsoid. Because the intratrap collisions depend on the square of the density, this has important implications for collisions measurements. Figure 4 shows how the density distribution affects the time dependence of the number of trapped atoms. For large numbers of atoms, the uniform density distribution produces a constant, uniform collision rate, resulting in exponential decay of the number of trapped atoms at a rate that depends both on collisions between the trapped atoms and collisions with hot background atoms in the vacuum chamber. As the number of atoms decreases, radiation

--

-

%

Gaussian Density Dist. Background Gas Collisions

0.1 =

--

FIG.4. Time dependence of the fluorescence from a cloud of optically trapped Cs atoms, illustrating the effects of collisions and radiation trapping. For large numbers of atoms (small time), the decay is dominated by intratrap collisions and has an exponential dependence on time since the density distribution approximates a uniform sphere whose size changes as the number of atoms changes. As the number decreases further, radiation trapping is no longer important and the density distribution becomes independent of the number of atoms. Intratrap collisions in this regime produce nonexponential decay. After a very long time, the decay becomes dominated by collisions with vacuum residuals so the time dependence again becomes exponential.

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trapping eventually becomes weak enough that the density begins to drop while the shape remains constant, usually a Gaussian. In this regime the trap loss is dominated by intratrap collisions, so the decay is nonexponential. Finally, when the number of atoms becomes very small, the density is likewise sufficiently small that the intratrap collisions are no longer impoEtant, so the collisions with hot background gas atoms cause the decay to once again become exponential in time. The large difference in the slopes of the small time and large time exponential decays in Fig. 4 is the result of the intratrap collisions. We should, of course, study the dependence of /? on various parameters. In principle, this can be done by changing the intensity and frequency of the trapping laser. The frequency is difficult to vary since the trap only operates over a narrow range of detunings slightly to the red of the atomic resonance. This is unfortunate because as we will see, a large amount of information is contained in the frequency dependence of /?. A useful technique that bypasses this problem is to introduce an additional laser to the trap, called a catalysis or probe laser. As long as the frequency of this laser is not too close to the atomic resonance frequency, it does not affect the trapping and cooling processes, but it can produce an increase in the collision rate. Varying the frequency or intensity of this laser allows study of the excited state collision rates in detail. Two important variations on this apparatus include the use of vapor-cell loading and dipole-force trapping. In the vapor-cell trap, there is no atomicbeam source of atoms. Instead, a background vapor of the desired atoms is maintained in the vacuum cell, and slow atoms from the low-velocity tail of the Maxwell-Boltzmann distribution are directly loaded into the trap. This leads to a higher pressure in the vacuum system, but it eliminates the complications of an atomic beam. Dipole-force trapping depends not on the momentum imparted to atoms by scattered photons, but rather on the fact that polarizable particles such as atoms are attracted to points of electric field maxima; in a dipole-force trap, a highly focused laser beam provides this maximum. The laser is not tuned to the atomic resonance and in fact may be greatly detuned. Because the dipole force is weak, these traps typically confine far fewer atoms than the MOT with higher temperatures as well, but densities can be higher because radiation trapping is not present. B. INTERPRETATION OF ULTRACOLD-COLLISION EXPERIMENTS IN THE PRESENCE OF LIGHT

Experiments on collisions of optically trapped atoms must be interpreted differently than higher temperature experiments due to interactions of photons with the colliding, low-temperature atoms. For example, at high temperatures the rate at which the associative ionization process occurs in

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a sample can be written Kn*’, where n* is the excited state atom density, and the rate coefficient K is independent of the laser intensity. For low temperatures, it is more appropriate to write the collision rate as pn’, where n is the total atom density. This is necessary because of the important role of spontaneous emission in the collision process (Julienne, 1992). To understand this role qualitatively, consider two cold trapped atoms moving toward each other from distances of the order of A. As the atoms approach each other they absorb photons from the lasers, but then they spontaneously emit before moving very far-only tens of Angstroms. Thus the atoms are continuously cycled from ground to excited states at these large distances. This contrasts with high-temperature experiments where the atoms move much more than a wavelength in a natural lifetime. As the atoms come closer still, they begin to be affected by long-range interactions. By far the most important of these is the resonant dipole-dipole interaction that acts when one of the atoms is in the excited state while the other is in the ground state. Both repulsive and attractive curves can be excited, but as the atoms reach separations where the dipole-dipole interaction is comparable to R / t , where z is the excited state lifetime, the attractive curves are preferentially excited due to the lasers being tuned to the low-frequency side of the atomic resonance. For atoms with degenerate states, the optical pumping process will be affected by the presence of the other atom, but this has not been studied to date. At some point the dipole-dipole interaction decouples the atoms from the light, and the evolution of the collision from this point forward is what distinguishes the various classes of collisions. Most of the atoms that are in the excited state when the interaction with the light is decoupled by the dipole-dipole interaction radiate back to the ground state. Because the light is now decoupled from the colliding atoms, ground state collisions proceed without reexcitation of the atoms. Because this is the case regardless of how many times the atoms were excited before reaching the region where the interaction with the light is decoupled by the resonant dipole-dipole interaction, the ground state collision should be independent of intensity. This is in contrast to a conventional collision experiment in which the ground state flux of atoms would be reduced by appreciable excitation of the atomic excited states; that is, at high temperatures the ground state collision rate would be given by p(l -fe)’n2, where f, is the fraction of atoms in the excited state. In contrast, at ultracold temperatures the effect of spontaneous emission at large distances makes the collision rate essentially independent of the saturation of the free atoms and hence it is given by fin2. The lack of intensity dependence of the ground state collisions should be a good approximation, but cannot be completely true. For example, the interactions of the colliding atoms with the light at large separations could effect both the collision trajectory and the relative velocity. In

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS 135

addition, altered optical pumping may occur at large relative distances. However, it is expected that these processes will induce only a slight intensity dependence. For singly excited state collisions, the observable effects require the atoms to be in the excited states at small interatomic separations. In order for this to occur, the light must excite atoms that are close enough to survive in the excited state to small distances without radiation occurring. Because of spontaneous emission, we expect the main effect of the light at distances larger than where the dipole-dipole interaction decouples the atoms from the light will be to provide a slight attraction of the atoms toward each other, due to preferential excitation to attractive potential curves. So the principal question concerns the production of excited states in the region where the dipole-dipole interaction decouples the atoms from the light. If the atoms are left in the excited state after they are decoupled from the light, they can, subject to spontaneous emission, be accelerated to small distances where energy transfer processes can occur, leading to ejection of the atoms from the trap. Again, as for the ground state collisions, the atomic excited state fraction is irrelevant so the collision rate depends on the total atom density, not the excited state density. In the absence of saturation effects, the excitation rate is linear in intensity, so the singly excited state collisions are also linear in intensity. The interpretation of doubly excited state collision experiments is similar in most ways to the singly excited state collisions. Again the coupling of the free atoms to the light at very large distances is thought to play a minor role, while the production of excited states in the region of decoupling of the atoms from the light is very important. The two-photon nature of the process means that the rates are quadratic in intensity. Interpretation of measurements of B is not possible without knowledge of whether ground state or excited state collisions are contributing. For collision processes where the final-state products such as ions are directly detected, this is not usually a problem. However, for trap-loss measurements it is essential to distinguish which collision processes are producing the trap loss. Here the intensity dependence is vital. As discussed earlier, singly excited state collisions are expected to be linear in intensity, while ground state collisions should be independent of intensity. Because of this feature, experiments that use additional lasers significantly detuned from the atomic transition frequencies in order to probe the frequency dependence study only excited state collisions. However, for studies where only the trapping laser intensity dependence is measured, care must be taken to isolate the excited state and ground state collisions. Figure 5 shows data from Wallace et al. (1992) of @ as a function of the intensity of the trapping laser. At small intensities the rate is a rapidly decreasing function of intensity, whereas for large intensities the rate increases roughly linearly. This is interpreted as follows. At the very lowest intensities the trap is shallow enough so that

Thad Walker and Paul Feng

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1 0-l1 n

cn

Q

1 0-l2

0

5

10 15 I, (mW/cm2)

20

FIG.5. Measurement of the trap-loss rate coefficient B as a function of the intensity of the trapping laser for a Rb MOT. Solid lines are curves to guide the eye. The initial rapid decrease of B with increasing intensity arises from suppression of ground state collisional loss as the trap becomes sufficiently deep to capture atoms that have undergone a collisional change of hyperfine level. The linear increase at higher intensity is interpreted as trap loss from singly excited state collisions. [From Wallace et al. (1992), by permission.]

ground state collisions that change the hyperfine level of the colliding atoms cause the atoms to leave the trap. As the laser intensity increases, these atoms are captured by the trap with increasing probability, so p decreases. The linear increase in /3 with further increase in intensity represents the increased contribution from excited state collisions. In this way excited state and ground state processes are distinguished from each other.

111. Collisions of Ground State Atoms A number of different collision processes can occur for laser-cooled atoms colliding in their ground states. The effects that have been observed are hyperfine-state-changing collisions (Sesko et al., 1989; Wallace et al., 1992; Walker et al., 1989), thermalization in a magnetic trap by elastic scattering (Monroe et al., 1993), and frequency shifts in a laser-cooled atomic clock

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(Gibble and Chu, 1993; Verhaar et al., 1993). Because the experiments are very different for each case, they are discussed separately in the following subsections. Other related work concerns ultracold collisions of magnetically trapped hydrogen atoms (Masuhara et al., 1988), hydrogen atoms with surfaces (Doyle et al., 1991), He-He scattering (Mester et al., 1993), and frequency shifts in hydrogen masers (Walsworth et al., 1992). The techniques used in these experiments are substantially different from the laser-cooling experiments and are not discussed in this paper.

A. HYPERFINE-STATE-CHANGING COLLISIONS An important contribution to trap loss in many cases is the hyperfinestate-changing process, which for alkali atoms A with nuclear spin I and total angular momentum F is

(3) with an increase in the kinetic energy of each of the colliding atoms of hVhfs/2, where Vhfs is the atomic ground state hyperfine splitting. Another possibility is that both atoms can change hyperfine state, that is,

Currently there are no means to distinguish the two processes. As explained in Section II.B, both of these processes should be insensitive to laser intensity. The first unambiguous observation of trap loss from ground state collisions was made for cesium by Sesko et al. (1989) in a MOT similar to that described in Section 1I.A. When the trapping lasers were carefully aligned, an intensity dependence similar to Fig. 5 was observed, with /l decreasing rapidly from 8 x 10,' cm3/s at 2 mW/cm2 intensity to a minimum of 3 x 10-12cm3/s at 4mW/cm2, followed by a linear increase at higher intensities. As explained in Section II.B, this rapid decrease is interpreted as resulting from the increasing ability of the trap to capture fast atoms as the intensity increases. This interpretation is further supported by the observation that if the trapping lasers were misaligned slightly (presumably reducing the trap depth), then was observed to be completely independent of intensity, as would be expected if the trap were unable to capture the fast atoms that result from the collision. Similar measurements were made by Wallace et al. (1992) for Rb, where the hyperfine-state-changing collisions were seen to be captured at different

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intensities for the two naturally occurring isotopes 85Rb and "Rb (Fig. 5). The intensity at which /3 begins to decrease is larger for 87Rb than "Rb by a factor of 1.5, due to the larger hyperfine splitting of 87Rb (6.83GHz) compared to 85Rb (3.04GHz). Because the capture velocity of the trap is the factor that determines trap depth (Monroe et al., 1990), the observed value of 1.5 is consistent with the ratio of the velocities of the exiting atoms [(6.83/3.04)'/2 = 1.51 if the capture velocity is assumed to be linear with intensity, as is reasonable for the small intensities used.

B. THERMALIZATION IN MAGNETIC TRAPS

A second process that can occur for ground state atoms is elastic scattering. This is a very unimportant process in a MOT because the energy transfer rates due to elastic collisions are overwhelmed by the cooling and heating processes induced by interactions with the light field. However, when laser-cooled atoms are loaded into a magnetic trap, the elastic collisions are the only mechanism for thermal equilibration of the magnetically trapped atoms. The thermalization rates are particularly important for attainment of Bose-Einstein condensation by means of evaporative cooling. In addition, in the regime where only s-wave scattering is important, strong magnetic-field-dependent resonances in the scattering rates are predicted (Tiesinga et al., 1992a). Monroe et al. (1993) have measured the elastic scattering rates in an anisotropic magnetic trap via observation of thermalization of the atoms. Cesium atoms were trapped in a vapor-cell MOT (Monroe et al., 1990), then cooled by polarization-gradient cooling (Foot, 1991), optically pumped into the weak-field-seeking F = 4, mF = 4 state, and loaded into the magnetic trap. The V ( p .B) force, combined with gravity, produced an anisotropic confining potential, so oscillations of the atoms along the x, y , and z directions occurred at different frequencies. Because of asymmetries when loading the atoms from the MOT into the magnetic trap, the temperatures measured in different directions were also unequal. The method used for measurement of the temperatures was to quickly turn off the magnetic field and illuminate the atoms with a resonant laser beam. Analysis of the resulting image gave the density profile of the atoms, which was converted to a temperature using the well-known trapping potential due to the magnetic field. As collisions occurred, the temperature anisotropy between the various directions was reduced and from this the elastic scattering cross section was deduced. The effect was enhanced by heating of the motion along one of the directions via a parametric drive of a frequency twice the value of the natural frequency of oscillation along that direction. Data were taken at high and low densities, allowing a separation of intratrap collisions

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from glancing collisions with hot background gas that also heat the sample. A Monte Carlo simulation of the heating process due to elastic collisions was performed in order to extract the collision cross section. The measured cross sections as a function of temperature are shown in Fig. 6. At the temperatures given, the elastic cross section should be mostly given by s-wave and d-wave contributions (p-wave is forbidden due to Bose symmetry for spin-polarized atoms). Since the temperature dependence is flat compared to a T-' dependence if an s-wave resonance were dominant, or a T 3 contribution from d-waves, the most straightforward conclusion is that the s-wave contribution dominates the scatttering cross section and the scattering is nonresonant as T + O . However, Verhaar et al. (1993) have suggested that the data can also be explained by a d-wave contribution that increases with temperature, compensating for a decreasing s-wave contribution. In principle, the weak-field-seeking states that are trapped in a dc magnetic trap are susceptible to spin-flip collisions that can change the hyperfine or Zeeman state of the atoms, thereby ejecting them from the trap. Cornell et al. (1991) have demonstrated an ac magnetic trap that works for

a

0

--c

--

--

-

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strong-field-seeking states, so the atoms can be confined in their lowest energy state, making spin-flip collisions endothermic and, therefore, suppressed at low temperatures. C. FREQUENCY SHIFTS OF ATOMIC CLOCKS One reason for using laser-cooled atoms for improved frequency standards is the reduced systematic effect of frequency shifts due to collisions. The collisional frequency shift can still be very important, however, since the inherent linewidth is so much smaller, especially in a fountain geometry where very large transit times can be achieved (Kasevich et al., 1989). Gibble and Chu (1993) measured the frequency shift in a prototype fountain-based Cs frequency standard. Atoms from a vapor-loaded MOT were launched upward by frequency shifting the upward-going laser beams with respect to the downward-going ones. The atoms were cooled by polarization-gradient cooling (Foot, 1991) in the moving standing wave to a few microkelvin,and optically pumped into the lower ( F = 3) Cs hyperfine state. The atoms entered a microwave cavity tuned to the 9.192-GHz Cs hyperfine resonance frequency. The field in the cavity applied a x/2-pulse to the atoms that traversed the cavity. After the atoms reversed direction due to gravity, they experienced another n/2-pulse. Then the number of atoms in the F = 4 state was detected by laser-induced fluorescence. In this way, Ramsey fringes were generated for the F = 3, m = 0 + F = 4,m = 0 magnetic-field-independent transition. Collision-induced frequency shifts were measured by varying the density of the cold atoms in the fountain. The time-dependent density was determined by absorption of a resonant laser coupled with assumptions about initial velocity and spatial distributions of the atoms. The frequency-shift data are shown in Fig. 7. Two cases were measured: collisions with a random distribution among the F = 3, m, states, and collisions with only the mF = 0 states occupied. The results are in order-of-magnitude agreement with the theory of Tiesinga et al. (1992b), which is consistent with the extreme sensitivity of the calculations to the detailed shapes of the potential curves. In addition, the differential fractional frequency shift between collisions with F = 3, m, = 0 and collisions with F = 4,m, = 0 was measured to be -0.1 k 0.2 (Verhaar et al., 1993). Despite significant experimental uncertainties, the combination of the frequency-shift measurements and the elastic scattering measurement provides important experimental constraints on the theoretical analysis of Cs ground state collisions (Verhaar et al., 1993). The theory is sensitive to the positions of the last bound states for the triplet and singlet potentials. The results obtained to date suggest that for both potentials the first unbound state is very close to the dissociation limit, having an effective vibrational

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FIG.7. Frequency shift of a Cs fountain-based atomic clock as a function of the density of atoms, for a random distribution of atoms among the F = 3, mF states (solid line), and for the F = 3, mF = 0 state (dashed line). [From Gibble and Chu (1993),by permission.]

quantum number in the range of -0.05 to -0.1 (modulo 1). This gives a large negative triplet scattering length of - 200 to - 1000 Bohr radii, or cross sections in the range of lo5 to lo7A. Further frequency-shift measurements will more tightly constrain these results.

D. OPENQUESTIONS FOR GROUND-STATE COLLISIONS With the increasing interest in the use of laser-cooled atoms for a variety of experiments including atomic frequency standards, atom interferometry, and studies of quantum collective effects such as Bose condensation, the need to understand the effects of ultracold ground state collisions will become ever more important. Because the theoretical calculations are so sensitive to the energies of the last bound states of the potentials, it is vital to make many different types of measurements in order to constrain the theory unambiguously. For example, measurements of the magnetic-field-induced resonances predicted by Tiesinga et al. (1992a) for the total cross section would strongly limit the range of parameters for the calculations. Similarly, larger numbers of accurate frequency-shift measurements would tightly constrain the positions of the magnetic-field resonances. A similar situation exists for ultracold collisions of non-laser-cooled species. For example, significant discrepancies between theory and experiment for frequency shifts of both room temperature and cryogenic hydrogen

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masers have been observed (Walsworth et al., 1992; Hayden et al., 1993; Walsworth, 1993). However, recent preliminary results for collisional broadening are within a factor of 2 to 3 of theory (Walsworth, 1993). Again, the challenge is to obtain improved accuracies and increased numbers of complementary measurements in order to obtain a thorough understanding of the collisional processes. The rates of ground state hyperfine-changingcollisions should be strongly reduced if the colliding atoms are spin polarized. This effect has not yet been studied, but should be possible using spin-polarized traps such as the spin-polarized vortex trap of Walker et al. (1992) or a spin-polarized dipole trap (Gould et al., 1988; Miller et al., 1993a).

IV. Collisions Involving Singly Excited States In this section we focus on experiments investigating the properties of singly excited state collisions of laser-cooled atoms. By this we mean collisions for which the collision dynamics and energy transfer occur principally in the S + P states (Fig. 1) that are produced by exciting the colliding ground state atoms with a single photon. Singly excited state collisions are of current interest due to the sensitivity of the collision dynamics to interactions with light, both in the form of external driving fields from the lasers and in the form of vacuum fluctuations that produce spontaneous emission. The case in which the energy transfer occurs in doubly excited states is presented in Section V. We begin our discussion of the singly excited state collisions by describing the energy transfer mechanics that are known to apply (Section 1V.A). Then the collision dynamics are discussed (Section 1V.B) and we point out some of the basic physical issues involved in understanding these collisions. Since these collisions are known to be the dominant source of trap loss for the traps that have been built to date, we then give (Section 1V.C) a brief analysis of some of the practical implications these collisions have for trapping and cooling of atoms. We then discuss the experiments that have been performed. We have somewhat arbitrarily divided these into absolute measurements (Section 1V.D)and those that measure the frequency dependence of the collision rates (Section 1V.E). We conclude by discussing some of the open questions concerning these collisions (Section 1V.F). A. ENERGY-TRANSFER MECHANISMS As explained in Section II.A, the effects of singly excited state collisions are usually observed by measuring the time dependence of the number of atoms

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in the trap. In order for the atoms to leave the trap due to collisions, energy transfer must occur, changing internal energy to relative kinetic energy of the colliding atoms. Since final-state products are not directly measured, the trap-loss rate coefficient fl is in general a result of two principal mechanisms that produce trap loss for alkali atoms, shown in Fig. 1 and discussed in more detail in Julienne et al. (1992). The first energy-transfer mechanism is radiative redistribution, also known as radiative escape, which results in trap loss when the energy difference between a photon from the laser beam absorbed by the colliding atom pair and a spontaneously emitted photon leaves the atoms with kinetic energies that are greater than the trap depth. The Franck-Condon principle dictates that, in a semiclassical picture, the absorption occurs at large interatomic separations due to the small detuning of the laser from resonance, whereas the emission must occur at small distances in order to cause trap loss. The necessity of transporting the initially slow atoms to small interatomic distances within a radiative lifetime brings the dynamics associated with spontaneous emission into play, because spontaneous emission at large distances does not lead to trap loss, whereas spontaneous emission at short distance does. A special type of radiative redistribution is radiative association in which the spontaneous emission results in a bound state of the molecule. The second energy-transfer mechanism only arises for excitation of the atoms by light tuned near the P,,, state. If the atoms reach small interatomic separations without spontaneous decay, a fine-structure-changing transition can occur:

and the outgoing atoms have a kinetic energy equal to one-half of the fine-structure splitting, which is much greater than the trap depth for all of the alkalis except possibly lithium. Again, it is necessary to transport the atoms to small interatomic separations for this process to occur, so spontaneous emission at large distances is important. In principle, a third possible mechanism is excited state hyperfine-changing collisions: (F = I

+

k) +

(

A2PJ -+ A2S1,2 F

9+

= I --

A2PJ

(6)

with a resulting kinetic energy of hvhfs/2 for the outgoing atoms. This mechanism will only cause trap loss when the ground state mechanism does also (i.e., when the resulting velocity exceeds the capture velocity of the trap) and is suppressed by spontaneous emission as compared to the analogous ground state process. Thus its contributions to trap loss should always be small compared to other processes.

144

Thad Walker and Paul Feng

Another excited state energy-transfer mechanism is possible for trapped metastable He. This is Penning ionization through the ls2p3P state:

+ H@P,) -,He(%,) + He+ + e - ,

(7)

and is the only singly excited state process for which the final states have been directly observed.

B. COLLISION DYNAMICS The common link between each of these mechanisms is in the collision dynamics: production of the excited atom pair at large interatomic distances followed by transport of the atoms to small separations where these energy transfer processes can occur. A number of issues arise in this context, with the principal theoretical uncertainty being how to treat the interaction of the colliding atoms with the light, both from external laser fields and spontaneous emission. These issues are reviewed in detail in Julienne et al. (1992). A simple semiclassical description of the excited state collision process can be understood with reference to Fig. 1. A colliding atom pair, initially with both atoms in the ground state, absorbs a photon from the laser to a strongly attractive excited state potential curve. The atoms are then accelerated toward each other. If they spontaneously emit a photon before reaching small distances where energy transfer can occur via one of the mechanisms described in Section IV.A, the excited state portion of the collision is terminated. The collision proceeds on the ground state curve, and the atoms remain in the trap after the collision. On the other hand, if the atoms reach the energy transfer region at small interatomic separations, one of the above-mentioned energy transfer processes can eject the atoms from the trap. This is the basic picture of how these collisions proceed. The particular interest in excited state collisions stems from the poorly understood nature of each of the different parts of the process. The proper treatment of the role of the external fields is uncertain, especially with regard to the possibility of nonresonant absorption of the light. The subsequent motion on the excited state potentials also is not understood, both due to nonadiabatic effects of hyperfine structure as well as the roles of vastly different ground state and excited state trajectories. The effects of interruption of the collision process by spontaneous emission are novel. Finally, the energy-transfer process is subject to quantum interference effects that are not averaged out due to the very small spread in initial velocities of the colliding atoms (Dulieu et al., 1994). For excited state collisions as viewed by semiclassical models (Julienne and Vigue, 1991), the necessity of strong acceleration of the atoms on the excited state potential curves means that the probability of the excited atom

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

145

pair coming to small interatomic separations without radiating is fairly insensitive to the initial atomic velocity. For this reason, trap-loss rates should be insensitive to temperature. Preliminary experimental results suggest that this is the case (Gould and Wallace, 1993). However, Boesten et al. (1993) have predicted that a strong quantum interference occurs when treating the problem in an entirely quantum mechanical manner, causing a reduction of the collision rate with decreasing temperature, with a predicted T 2 dependence. C. PRACTICAL IMPLICATIONS OF TRAP-LOSS COLLISIONS FOR LASER COOLING TRAPP~G

AND

The majority of excited state collision experiments have studied the effects of collisions induced by the lasers that are used to trap the atoms. As we have emphasized, there are many poorly understood issues concerning these collisions that motivate current research efforts. However, we should point out that there are important practical implications to studies of trap-loss collisions since they can limit the number of atoms that can be trapped or the storage times for those atoms. To see how this works in a particular case, we consider a vapor-cell-loaded trap (Monroe et al., 1990) for alkali atoms. The rate equation governing the number of trapped atoms is determined by integrating Eq. (2) over the volume of the trap, with an additional term for the continuous loading of atoms into the trap from the background vapor:

dN _ - IA - kAN - p dt

s

n2(r,t) d3r

Here N is the number of trapped atoms, IA is the loading rate of atoms from the background alkali vapor of pressure A, and k A is the rate at which atoms are ejected from the trap from collisions with the background vapor. In general, collisions from the vacuum residuals occur as well but we assume they can be made negligible. As explained in Section II.A, radiation trapping makes the density become approximately a constant uniform value, n,, when N is large. In this approximation, the steady-state number of atoms is N =

1A kA

+ pn,

(9)

The maximum number of atoms, N,,, = l/k, is attained when kA >> Bn,. Under this condition N,,, is independent of B. Thus the excited state collisions do not limit the number of trapped atoms. However, the condition kA >> fin, does limit the lifetime of individual atoms in the trap to be l / k A without sacrificing the total number of atoms trapped. This becomes

-

146

Thad Walker and Paul Feng

an important consideration for experiments such as collisions in magnetic traps that are loaded from laser traps for which it is desirable to have as long a lifetime as possible without sacrificing the number of trapped atoms. In other situations the trap-loss collisions limit the attainable number of atoms. This would be the case for atomic-beam-loaded traps where the atomic beam has no overlap with the optical trapping region. In this case the atoms can be loaded from the atomic beam without the beam itself limiting the lifetime. The number of trapped atoms would then be a balance between the loading rate and the collisional loss rate. A second situation where the number of atoms is limited by p is for a dual optical trap scheme where atoms from a vapor cell trap would be transported via radiation forces through a low-conductance tube to a second chamber without any vapor. In principle, the loading rate into the second trap could be a significant fraction of IA without the k A loss mechanism. Here again the number of atoms is limited by the trap-loss collisions. D. ABSOLUTE MEASUREMENTS-SMALL DETUNINGS In this section we describe the singly excited state measurements that have been made where the collisions were induced by the laser that traps the atoms, and in which the effects of the collisions were studied by varying the laser intensity. In all cases discussed in this section the lasers used were tuned within a few linewidths of the atomic resonances. The experiments are sensitive both to collision dynamics and energy-transfer probabilities.

1. Cesium The first identification of singly excited state collisions was made by Sesko et al. (1989) with a Cs MOT very similar to the apparatus described in Section 1I.A. This experiment was mentioned before in the context of ground state hyperfine-changing collisions (Section 1II.A). The collisional rate coefficient /? was deduced from trap-loss transients in the limit of small numbers of atoms. The density of the atoms was determined from absorption measurements coupled with observation of the Gaussian spatial distribution of the atoms using a CCD camera. Identification of the excited state collisions was made by studying the intensity dependence of p, obtaining results similar to Fig. 5. Comparison of the result, /?= 1.0 x 10- l 1 cm3/s at an intensity of 10mW/cm2, to a model prediction of Gallagher and Pritchard (1989) showed that the results were about a factor of 5 higher than the model. Subsequently, refinements of the model by Julienne and Vigue (1991) and Band and Julienne (1992) found excellent agreement with this measurement. However, Dulieu et al. (1994) have shown that this agreement was fortuitous.

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

147

2. Rubidium The agreement of the models and the experiment for Cs was shown not to hold for Rb in experiments reported by Hoffmann et al. (1992) and Wallace et al. (1992). The Hoffman et al. result was that the 85Rb trap-loss rate, 3.6 f 1.5 x 10-'2cm3/s at 10mW/cm2, is a factor of 5 smaller than had been predicted by the models that agreed with the Cs results (Julienne and Vigue, 1991). Wallace et al. confirmed the value for 85Rb (3.4 1.5 x 10-'Zcm3/s) and in addition measured a still smaller value for 87Rb, 1.0 x 10-'Zcm3/s. These results led to the realization of the principal problems of comparing the models to experiment: sensitivity of the energy-transfer calculations to the details of the inner parts of the potential curves, and the importance of hyperfine interactions for the collision dynamics. The Hoffman et al. (1992) experiment used a vapor-cell-loaded MOT to study trap-loss collisions. In this type of apparatus a large number of atoms are trapped so that the density distribution is not a Gaussian, in contrast to the experiments where small numbers of atoms are used. The change in the density distribution arises from repulsion of the atoms due to radiation trapping (Walker et al., 1990; Sesko et al., 1991). In the limit of very large numbers of atoms, the distribution becomes nearly independent of position except at the edge of the trapped atom cloud. In this case the rate equation (8) becomes dN _ - IA - kAN - pfn, N dt

where f = fnzd3r/n,fnd3raccounts for the deviation of the distribution from a uniform sphere, and is deduced from the spatial distribution measurements. As shown by Hoffmann et al. (1994), f is a very slowly varying function of N, so the transient solution of Eq. (10) is well approximated by an exponential with time constant. y-

' = ( k A + fifn,)- '

(11)

The excited-state collisions were isolated from other types of collisions by addition of a catalysis laser to the trap, which allowed measurement of the resulting change in p. This technique and the other conclusions of this experiment are described in more detail in Section 1V.E. The density distribution was determined from camera images and absorption measurements. The Wallace et al. (1992) experiment was done with a MOT of a design similar to that described in Section 1I.A. Rubidium atoms were loaded into the trap from a Rb atomic beam that was slowed using frequency-chirped light from a diode laser (Foot, 1991), and transients similar to Fig. 4 were observed, considering only the times for which the number of atoms was

148

Thad Walker and Paul Feng

small enough that radiation trapping was not affecting the density distribution. A CCD camera was used to measure the Gaussian spatial distribution of the trapped atoms. This, coupled with an absorption measurement with the trapping lasers on, gave the density of the atoms. With the trapping lasers on, the excited state fraction must be measured or calculated. The excited state fraction was measured in a novel way using photoionization (Dineen et al., 1992). The 413-nm light from a krypton-ion laser photoionizes excited state atoms but not ground state atoms. Introduction of the 413-nm light produced an additional loss term for the atoms that was deduced from the transient decay. When the trapping lasers are very intense, the excited state fraction is 50% and the photoionization rate is maximum. At lower intensities, the excited state fraction was therefore found by comparing the photoionization rate to the maximum saturated rate and dividing by 2. This enabled a measurement of the density of the trapped atoms. The Wallace et al. (1992) results are shown in Fig. 5 for the trap-loss rate as a function of intensity for the two isotopes 85Rb and 87Rb.As discussed in Section III.A, the rapid decrease with intensity at low-intensity signals the suppression of ground-state hyperfine-changing collisions. The further increase of the trap-loss rate with intensity, signaling excited state collisions, was observed to be highly isotope dependent, with the ”Rb rate being a factor of 3.3 k 0.3 larger than for 87Rb. This large isotope effect was completely unanticipated by the models. Since the hyperfine structure is the feature that is most different for the two isotopes, it is the most likely origin of the isotopic effect, although mass effects can also be important for the fine-structure energy-transfer process due to sensitive phase-dependent interferences (Dulieu et al., 1993).

3. Sodium While the observation of trap-loss collisions was first reported by Raab et aE. (1987) in the original paper on the MOT, the first investigation of the collisions was that of Prentiss et al. (1988). Collisional loss of trapped Na was measured from the time dependence of the number of trapped atoms, and the dependence of the loss on the intensity of the trapping lasers was studied for a MOT similar to that described in Section 1I.A (“type I”) and for one (“type 11”) where the lower ground hyperfine state was principally occupied by the atoms instead of the upper as for type I. (The transitions used for the type I1 trap are F = 1 --t F‘ = 0 and F = 2 -+ F’ = 2’). Surprisingly, no intensity dependence was observed for either type of MOT; the cause of this remains unclear. Marcassa et al. (1993) used a vapor-loaded trap to measure the excited state collisions in the presence of radiation trapping in a manner similar to Hoffmann et al. (1992). The transients are exponentials, but the time

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

149

constant depends both on collisions with background vapor and trapped-atom collisions [see Eq. (1l)]. To separate the two contributions, the trap-loss rate was measured using both type I and type I1 traps. The type I1 trap was much lower in density and thus its trapped-atom collision rate could be ignored. Assuming that k is the same for the two types of traps, the type I1 transient determines kA, allowing Pfn, to be deduced from the transient of the type I trap. The densities were determined from fluorescence measurements, a calculation of the excited state fraction, and the spatial distribution of the trapped atoms. The measured intensity dependence of is shown in Fig. 8. The intensities used are much higher than for the other experiments, and an optical Bloch equation analysis was presented that predicted saturation of the trap-loss rate. It is not evident from Fig. 8 whether or not this prediction is supported by the data, although the theory (without hyperfine structure or dynamics taken into account) agrees with the experiment to within a factor of 2. 4. Lithium

Lithium is significantly different from the other alkalis in that the finestructure splitting is only 0.48K. Thus it is possible to suppress the fine-structure-changing mechanism for trap loss, leaving radiative redistribution as the only means for producing trap loss. Kawanake et al. (1993) A A

10 9

aec-1

0 7 6

5 4

I

3 2 1

0 0

100 200 Intensity (mWcm-0)

300

FIG.8. Intensity dependence of the trap-loss rate coefficient for Na atoms in a MOT, compared to results of model calculations. [From Marcassa et al. (1993), by permission.]

Thad Walker and Paul Feng

150

made such measurements using a four-beam Li MOT and the fluorescence decay method in the limit of small numbers of atoms. Densities were measured using fluorescence, calculated excited state fractions, and CCD camera measurements of the spatial distribution. The trap depth was varied by turning off the trapping lasers for a variable amount of time. As the duty ratio was varied, the value of /3 changed as shown in Fig. 9. As the duty cycle increased, the trap was increasingly able to capture the fast atoms produced by the fine-structure-changing loss mechanisms, so /3 decreased. Once all of the atoms produced by fine-structure-changing collisions were caught by the trap, fl again increased due to excited-state collisions where radiative redistribution causes the trap loss. This interpretation is supported by a measurement of the trap depth made by applying a variable-length pulse of light to the atoms with the trapping lasers off, then on. The observed values of /3 were 4.5 x 10- l 3 cm3/s at 100% duty cycle and the intensities of the two lasers used for trapping and optical pumping were 7.5 and 2.5 mW/cmz, respectively. Due to the small hyperfine structure of Li, significant populations of both hyperfine levels existed in the trap. It is not known how much contribution to the trap loss is attributable to atoms in

lo-" t 00 0

0

lo-"

0 0 Omo 0 0 0 0 ~

lo-"

1

0

20

.

1

.

1

60 80 Duty ratio('/.)

40

.

1

.

100

FIG.9. Collisional loss rate coefficient for Li atoms as a function of the fraction of time the trapping laser is on. The rapid decrease with duty ratio is similar to Fig. 5, but the process being suppressed by the increased trap depth is the fine-structure-changing process. The remaining excited state collisional loss is from radiative redistribution.[From Kawanake et al. (1993), by permission.]

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

15 1

each of the two hyperfine levels. In any case the measured rates are much larger than had been predicted (Julienne and Vigue, 1991). Ritchie et al. (1994) have also observed suppression of the fine-structure-changing collision mechanism by studying the trap-intensity dependence of B, similar to Fig. 5. Using a three-dimensional model of the trap depth, order of magnitude agreement of the radiative escape rates calculated by the optical Bloch equation method (Band and Julienne, 1992) with experiment was obtained. 5. Metastable Helium

-

The trap-loss rate coefficients for the alkali atoms are typically 10- cm3/s. For metastable 4He(3Sl), the corresponding process was measured by Bardou et al. (1992) to be 10-7cm3/s. In their apparatus, a cryogenic (15 K) supersonic beam of metastable He was transversely cooled by a 2-D optical molasses region in order to optimize capture into the trap. The beam was slowed by means of Zeeman-tuned slowing (Foot, 1991). The atoms were then captured into a MOT at the exit of the tapered solenoid for the magnetic field. Typically 1043S, atoms were trapped using this apparatus, as determined from a calibrated CCD camera. The temperature was about 1 mK. Collisions were detected by observing ions produced by Penning collisions between the trapped atoms and either background atoms or other trapped atoms. Due to trap loss arising from both collisions with background atoms and trapped atoms, the ion current varied as a function of time in a manner similar to Eq. (2). The deduced value of the collisional rate coefficient is /3 = 7 x lo8cm3/s at a laser intensity of x 27 mW/cm2 and a detuning of about 5MHz. To verify that this results from excited state collisions, the change in the ion current was monitored when the trapping beams were rapidly (< 1,us) shut off, as shown in Fig. 10. The excited state contribution was then studied as a function of density, with a resulting linear dependence indicative of excited state collisions. Absolute measurements of ion collection efficiencies indicated that the observed rates were consistent with trap loss arising solely from Penning ionization (7), although in principle trap-loss processes analogous to those present for the alkalis (Section 1V.A) could be operating as well. The extremely large rate coefficient results from the long excited state lifetime and the small mass of He, both of which favor excitation of the He atoms at extremely large distances. Applying the semiclassical model of Gallagher and Pritchard (1989), one estimates that the He atoms can be excited at distances exceeding 2000 8, and still survive without spontaneous emission to small interatomic separations where the Penning ionization occurs. Since this distance is comparable to or longer than the distance at which the light is decoupled from the atoms by the dipole-dipole interaction,

-

152

Thad Walker and Paul Feng

1

0

I

I

100 200 time (ps)

I

300

trap density n ( 0 ) (xlO' at/cm3) FIG.10. Method for determining the origin of Penning ionization for trapped metastable He. The ion signals abruptly change when the trapping light is turned off, as shown in (a), indicating that 4,," - 4orrarises from excited-state collisions with either other trapped atoms or background atoms. No density dependence is observed for whereas the ratio - ~ o f f ) / ~ o fdepends r linearly on density, as shown in (b), indicating that excited state intratrap Penning collisions are producing the excited state part of the ion current. [From Bardou et al. (1992), by permission.]

it is likely that the considerations of Section 1V.B d o not apply for this experiment, and it is appropriate to interpret the experiment in terms of the excited state and ground state densities. In this case, Bn2 should be replaced by P*nn*, where n* is the excited state density. When analyzed in this way, the value for b* is 2.8 x 10-'cm3/s.

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLEDATOMS

153

E. FREQUENCY-DEPENDENT EXPERIMENTS From the comparison of model calculations to the results of the substantial number of absolute, single-frequency measurements for all the alkalis that have been trapped, it is clear that it is difficult for current models to reproduce the experimental results. A combination of at least two factors is involved: (1) difficulty in calculating energy-transfer probabilities (Dulieu, 1993), and (2) neglect of effects arising from hyperfine structure (Walker and Pritchard, 1994). The purpose of measuring frequency dependences of the collision rates is to concentrate on the dynamics, since the energy-transfer probabilities are relatively insensitive to frequency. We will also discuss how in certain cases the effects of hyperfine structure may be minimized as well. In addition, observation of discrete structure in the trap-loss spectra shows that in certain regimes these collisions can be considered a novel means for obtaining spectroscopic information about the molecules.

1. Small Detuning Measurements- Collision Dynamics Sesko et al. (1989) first pointed out the importance of performing frequency-dependent measurements in order to learn about the dynamics of the collisions. The idea is based on the correlation between the frequency and the interatomic distance at which the light interacts with the atoms. At small detunings from the atomic resonances, the atoms are excited at large separations, while at large detunings the atoms are excited when they are closer together. Because of spontaneous emission, the probability of survival of the atoms to very small separations where energy transfer can occur depends critically on how far apart the atoms are when they interact with the light, and hence depends on the frequency of the light used to excite the atoms. Since varying the frequencies of the trapping lasers over a range greater than a few linewidths is not possible, it is advantageous to add an independent catalysis laser to produce increased collisional loss without affecting the trapping processes. Because ground state collision rates are independent of intensity, the catalysis laser technique inherently studies only excited state collisions. The trap-loss rates are measured using the techniques described previously, but it is the change in the trap-loss rate produced by the catalysis laser that is of interest. Care must be taken that the laser does not affect the trapping and cooling forces-in practice this restricts the catalysis laser frequency to be detuned five or more atomic linewidths from any atomic resonances, depending on the intensity of the laser. For comparison purposes, measurements of /Ias a function of frequency (trap-loss spectra) are shown in Fig. 11 for Cs and the two Rb isotopes.

Thad Walker and Paul Feng

154

1.E

0 0

1.4

o =Rb

h

VJ

im

FJ

-

0.E

87Rb

A

0

d

'0

0

Z. 0.6 a.

0

AOA

0.3 0.0 a.

5.0

3 4.0

cs

a a

im

.a I

* a

.

* !

.a

a

3.0

h z 2.0 a

1.0

0.0 -1.0

*a

f ' I

-0.8

.

:

a**

I

m

-

1

-0.6 -0.4 Detuning (GHz)

I

-0.2

I

0.0

FIG.11. Frequency dependence of the trap-loss rates for Cs, "Rb, and "Rb, for a catalysis laser tuned near the P,,, states of the various species. Detunings are measured from the highest frequency hyperfine transitions for each atom. Symbols on the axes show the positions of the lowest frequency excited state hyperfine transitions. The correlation of the peaks with the hyperfine structure suggests the importance of hyperfine structure in determining the dynamics of the collisions. [Adapted from Sesko et al. (1989), Walker et al. (1989), and Feng et al. (1993), by permission.]

The Cs spectrum was measured by Sesko et al. (1989). The catalysis laser intensity was 10 to 25 mW/cm*. The experimental techniques for determining /3 were described in Sections 1I.A and 1V.D. The data showed the general structure expected: As the detuning goes from the atomic resonance to increasingly red detunings, the trap-loss rate first increases, reaches a maximum, and decreases. This is as expected since for small detunings the atoms are excited at large interatomic separations and de-excite by sponta-

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

155

neous emission before reaching the energy-transfer region. As the detuning is increased in magnitude, the survival probability increases so the trap-loss rate increases. Eventually the collision rate decreases due to fewer atoms being available at small separations for excitation. Subsequent catalysis laser measurements on "Rb by Hoffmann et al. (1992) and Feng et al. (1993) for 87Rb showed that the spectra cannot be interpreted so simply. In these experiments the catalysis laser was used with a vapor-cell MOT, and /? was determined from observations of the change produced in the loading time constant when the catalysis laser was introduced. To place the collision rates on an absolute scale, the density distribution was determined from absorption measurements and analysis of a CCD camera image. Details of the methods used are given in Hoffmann et al. (1994). The most difficult part of the experiments was the determination of the density distribution. Therefore, it was desirable to avoid having to determine the density distribution for each data point if possible. Once the linearity of with intensity was established, the density distribution was held fixed as the detuning was changed by varying the intensity of the catalysis laser in order to keep the number of atoms N constant. According to the simple models of excited state collisions (Gallagher and Pritchard, 1989), the maxima of the Cs and Rb spectra should be at very nearly the same detuning. Figure 11 shows that this is not the case. This fact clearly shows the essential role that the excited state hyperfine interaction plays in determining these spectra. Note that for each of the three species studied, the maximum occurs near the position of the lowest allowed atomic hyperfine transition. (The positions of these transitions are marked on the graphs.) Several other interesting features are evident from the Rb data. First, the 85Rb and "Rb spectra are the same within experimental scatter at large negative detunings, below the atomic hyperfine structure. Second, for 87Rb two maxima are observed, both near the atomic hyperfine transitions. Third, the significant difference in the trap-loss rates for 85Rb and "Rb first observed by Wallace et al. (1992) at small detunings is limited to the region between the two highest frequency atomic hyperfine transitions. The consequences and possible explanations of these results are considered in more detail in Hoffmann et al. (1994) and Walker and Pritchard (1994). Briefly, strong mixing of the excited state potentials by the hyperfine interaction means adiabatic and nonadiabatic effects are important for determining the collision dynamics. In addition, both repulsive and attractive potential curves are important. The similarity of the 85Rband 87Rbspectra in the detuning region below the atomic hyperfine structure suggests that hyperfine effects are less significant for excitation in this range of detunings. Presumably this is because the complicated mixing of the atomic hyperfine levels occurs at larger distances than those at which the atoms are being excited for these large detunings, so the potential curves are similar for the two isotopes.

Thad Walker and Paul Feng

156

Another region where the curves would be expected to be both similar and relatively simple is for states where hyperfine interactions are large, so the collision dynamics of interest occur at interatomic separations that are larger than the distance at which hyperfine mixing occurs. Figure 12 shows calculated adiabatic potential curves for the states of "Rb that demonstrate this. Accounting for the effects of the hyperfine interaction on the collision dynamics should be relatively straightforward for excitation by light tuned to the red of the F' = 1' state, due to the lack of interaction of repulsive and attractive curves in this detuning region. states of 85Rb Peters et al. (1994) have measured the spectra for the and 87Rb(Fig. 13), using the same technique as described above for the P3/2 states. As expected, the shapes of the trap-loss spectra for the two isotopes are nearly identical. In addition, the shape expected from the model of Gallagher and Pritchard (1989) very crudely modified to account for hyperfine effects is seen to agree fairly well with the data. The detuning dependence supports the treatment of multiple orbits of the atoms by the model. 2. Large Detuning Measurements- Free-Bound Spectroscopy When the colliding atoms are excited at small interatomic separations (i.e., large detunings), they oscillate many times in the potential well before

4

3

2 C, / hR3 (GHz)

1

0

FIG.12. Calculated potential curves that correlate to the 5S,,,(F' = 2 ) + 5P,,,(F') states of "Rb at large interatomic separations. Due to the large hyperfine splitting, the potential curves that emanate from the F = 1' state are mostly attractive and follow a nearly K 3dependence. The lack of distortion of the curves implies that the hyperfine effects on the dynamics are small, and meaningful comparison to theory should be possible. [From Peters et al. (1994), by permission.]

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

1. -I

-1200

85Rb 87Rb I

I

I

- 1 0 ~~

157

-

I

-600 -400 Detuning (MHz)

-800

I

-200

0

FIG.13. Spectra of "Rb and "Rb for light tuned close to the P,,,!F' = 1') state. The hyperfine dynamics are less important here than for the P3/2state as evidenced by the very similar spectra obtained for the two isotopes. The solid line is a fit to the simple model of Gallagher and Pritchard (1989), and the dashed line is the same model without the effects of multiple orbits. The inset shows the experimental values for BA'16, which is predicted to be a constant [From Peters et a!. (1994), by permission.]

radiating. Thus vibrational structure should be observable in the trap-loss spectrum. This free-bound spectroscopy can have extremely high resolution due to the low temperatures, which means there is little complication due to rotational structure. Miller et al. (1993a) have observed striking vibrational structure in the Rb trap-loss spectrum for detunings from about -60 to -llOOcm-l. These observations were made with atoms trapped in a far-off-resonance dipole trap using light detuned in the range just mentioned (Miller et al., (1993b). Heating of the atoms from spontaneous emission occurs very slowly in this trap due to the large detunings. The Rb atoms were first produced in a MOT, then loaded from the MOT into the dipole trap. Only about 2700 atoms were trapped, but the density was high, being on the order of 1 0 " ~ m - ~ After . 100ms in the dipole trap, the trap was turned off and laser-induced fluorescence was used to detect the number of atoms. The reduction in the number of atoms was attributed to collisions. The spectrum obtained by repeating such measurements using different detunings is shown in Fig. 14. Two regular vibrational series are observed to make up about 65% of the observed lines, with level spacings that agree with theory to a few percent. In addition, the variations in the intensities of

Thad Walker and Paul Feng

158

Q

OO-?-

_-

---

-

1 0 -: I ;[,:I\

__----j (

jl

:I

_----.

,I

,I

iI

;I

;I

iI

;I

;I ; I ; I

;I

the lines correlate well with a simple model of photoassociation suggested by Thorsheim et al. (1989) and Weiner (1989). The rate coefficient at the strongest peaks in the spectrum is in the range of 1.4 x lo-" to 1.7 x 10-'ocm3/s. This large value is not surprising since the intensity of the light is 1 MW/cmZ. This more than compensates for the extremely large detunings that can sample only a small fraction of the interacting pairs.

-

F.

OPEN

QUESTIONS FOR SINGLY EXCITED STATE COLLISIONS

In this section we summarize what has been learned experimentally concerning singly excited state collisions, and point out some of the aspects of the collisions that have not yet been studied. A linear intensity dependence of fi has been observed in the experiments on Cs (Sesko et al., 1989; Walker et al., 1989), Rb (Wallace et al., 1992; Hoffman et al., 1992), Na (Marcassa et al., 1993), and Li (Kawanake et al., 1993), both for small and large detunings. In the Na case, with intensities of up to 300mW/cmz, the data are also consistent with a slight saturation

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

159

predicted from an optical Bloch equation model. For the extremely high intensity molecular spectroscopy work (Miller et al., 1993a), the intensity dependence was not studied. Clear observation of saturation effects might help distinguish between the semiclassical and optical Bloch equation models. For trap-loss collisions induced by the trapping lasers in MOTS, absolute measurements of trap-loss rates have been made for all the alkalis that have been studied (Wallace et al., 1992; Sesko et al. 1989; Walker et al., 1989; Marcassa et al., 1993; Kawanaka et al., 1993; Ritchie et al., 1994), as well as metastable He (Bardou et al. 1992). As a result of these experiments, the models are now understood to be incapable of quantitatively accounting for these results, due both to the extreme sensitivity of the calculated energy-transfer probabilities to the details of the potentials curves (Dulieu et al., 1994), and to the untreated dependence of the dynamics on hyperfine interaction effects (Walker and Pritchard, 1994). Spectroscopic-quality knowledge of the potential curves is necessary in order to have confidence in the energy-transfer calculations. Studies of the collision dynamics using frequency dependences have been made for Cs (Sesko et al., 1989; Walker et al., 1989) and Rb (Hoffmann et al., 1992, 1994; Feng et al., 1993; Peters et al., 1994). Measurements for the P3,2states revealed the importance of hyperfine interactions on the collision dynamics. These effects are difficult to treat quantitatively by the models. However, the lack of isotopic effect in the measurements for PI/, states of Rb (Peters et al., 1994) suggests that detailed treatment of the hyperfine effects by the models is not as important in this case, so progress on understanding other novel aspects of the collision processes should be possible. Agreement between a simple model of multiple orbits of the colliding atoms and the Rb data suggests that further progress should be possible for experiments that are carried out in regime where hyperfine dynamics are not important. At this time most of the basic predictions of the models have not been tested. The ability to perform free-bound vibrational spectroscopy using traploss collisions (Miller et al., 1993a) opens up many new possibilities for detailed studies of the molecular structure of excited states of the alkali dimers. It should be possible to map the various states in great detail to within a few gigahertz of the dissociation limit. The technique is inherently of extremely high resolution, and does not suffer from complications due to the high density of rotational states in these molecules. Another interesting possibility is to perform spectroscopy of higher excited states by studying the change in the trap-loss rate as a second laser excites the higher energy states. An important issue that has not yet been studied in depth is that of a possible temperature dependence. As mentioned earlier, Boesten et a1 (1993) have predicted a T 2 dependence of arising from a quantum interference

160

Thad Walker and Paul Feng

effect, in contrast to semiclassical models that predict a much weaker temperature dependence (Julienne and Vigue, 1991). There is another effect that would produce a reduced collision rate as the temperature is decreased. Recent experiments report observations of confinement of individual atoms in wavelength-sized potential wells formed by the standing waves of the trapping light (Westbrook et al., 1990; Grynberg et al., 1993). Under these conditions, the long residence times of the atoms in these potential wells would produce a simple steric reduction in the collision rates. For excitation by light tuned within a few linewidths of the free atom resonance, the production of excited states is an unresolved theoretical issue, since survival probabilities are small due to excitation being at large distances. The Gallagher-Pritchard model (1989) asserts that nonresonant excitation occurs at small distances due to radiative broadening. A competing method due to Band and Julienne (1992) uses optical Bloch equations to study the production of excited states. No experiments done to date have clearly addressed this question. Differing predictions for intensity dependences may be one method for distinguishing these theories. Other excellent tests are studies of the trap-loss spectra at small detunings if done in a regime where hyperfine dynamics can be treated simply (Peters et al., 1994). For excitation by light that is tuned many linewidths from the atomic resonances, all models predict that the excitation is produced in a relatively small range of interatomic separations where the light is resonant with the molecular potentials curves. Another interesting question is a possible dependence of p on the atomic spin polarization. Using a spin-polarized trap (Walker et al., 1992) and a circularly polarized catalysis laser, the trap-loss rates may be significantly reduced, depending on the degree of coupling of the electronic angular momentum to rotation. This issue has not been yet studied. As a particular practical consequence, if collisional loss of metastable He could be reduced in this way, the number of He atoms that can be trapped might be significantly increased. Finally, the “dark spot” trap recently demonstrated by Ketterle et al. (1993) should have interesting collisional properties. In this trap, radiation-trapping density limitations are reduced by putting atoms in a region of space where there is no optical pumping light, so the atoms accumulate in the lowest ground state hyperfine level. Since ground state hyperfine-changing collisions are not possible, and there is little light scattering, one might expect collisional loss to be small. Collisional loss can still occur, however, due to molecular absorption of the trapping light. Though this light is detuned a few gigahertz from the nearest atomic resonance, and thus the trap-loss rate coefficient is expected to be fairly small, the increased density might compensate, so excited state trap loss could still be quite important.

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

161

V. Collisions Involving Doubly Excited States In this section we describe the experiments done on collisions where the energy-transfer occurs in the doubly excited states, i.e., the P3/2 + P3/2 manifold of Fig. 1. The experiments done to date have studied the associative ionization process for Na, which is usually written Na3’P3,,

+ Na32P3,2 + Na; + e -

(12)

This process is energetically forbidden for the other alkalis. Many of the issues concerning the dynamics of these collisions are similar to the singly excited case discussed in Section IV.B, except at least two photons are involved so the phenomena are more varied. In keeping with the observation that it is inappropriate to consider excited state ultracold collisions as resulting from the collision of two excited state atoms, it is better to think of the process as a collision of two ground state atoms in which two photons are absorbed, resulting in the molecular ion, as depicted in Fig. 1: Na32S,/2 + Na32S112+ 2y -+ Na:

+ e-

(13)

Consistent with this interpretation, the term photoassociative ionization has come to be used for this process. A. COLLISION DYNAMICS As mentioned earlier, the dynamics of doubly excited state collisions are similar in principle to singly excited state dynamics. Here again some of the important issues are production of the singly excited states, rapid acceleration on the singly excited state potentials, and the possibility of spontaneous emission during the acceleration process. However, additional features arise for doubly excited state collisions since the process involves absorption of two photons. These features are discussed by Julienne et al. (1992) and Gallagher (1991). The basic picture is illustrated in Fig. 1. Due to spontaneous emission, we consider the collisions as beginning in the ground states. Absorption to the singly excited state potentials results in acceleration of the atoms toward each other. Absorption of a second photon to a doubly excited state then occurs, followed by associative ionization if the atoms reach small interatomic separations without radiating. In addition to the issue of spontaneous emission while on the singly excited state curves, the amount of kinetic energy accrued by the atoms before the second photon is absorbed is also of prime importance, since the weakness of the doubly excited state potentials means little acceleration occurs in the doubly excited states. Thus

162

Thad Walker and Paul Feng

the acceleration of the atoms on the singly excited state curves governs the time required to reach small interatomic separations where associative ionization occurs. This depends of course on the frequency of the second photon that is absorbed, which in turn suggests the importance of studying two-color frequency dependences of the collision rates in order to map out the collision dynamics. €3. SUMMARY OF EXPEFUMENTS Photoassociative ionization was the first measured collisional process observed between trapped atoms, by Gould et al. (1988). The trap used was a near-resonant dipole trap formed by two focused (beamwaists N 100pm) 40-mW laser beams. The detuning of the trapping lasers was about -700MHz from the Na3Sl12(F= 2) -+3P3/2(F'= 3') transition. The trapping lasers were alternated in 3-ps intervals to avoid dipole heating from standing waves; both beams were off for 3 p s to allow cooling by conventional optical molasses that was loaded from a Zeeman-cooled atomic beam. The density distribution and the temperature were deduced from CCD camera images, with an effective density of 10'0cm-3 and a temperature of about 0.75 mK. After a number of trapping and cooling cycles, a pulsed electric field was used to remove the ions from the trap. An electron multiplier detected the ions. Time-of-flight analysis of the resulting ion signal positively identified the ions as Na; as opposed to Na'. By changing the number of atoms loaded into the trap, the signal was observed to be proportional to the square of the density. The measured rate coefficients are on the order of 10- l 1 cm3/s. A second associative ionization experiment by Lett et al. (1991) dramatically demonstrated the appropriateness of the different interpretation of ultracold collisions as compared to high-temperature collisions, as exemplified by Eq. (13) and as discussed in Section 1I.B. Using essentially the same apparatus as in the first experiment, ions produced from the photoassociative ionization reaction were continuously swept into the electron multiplier, and a time-dependent ion signal was observed as shown in Fig. 15. While the number and density of the atoms were essentially the same for the cooling and trapping phases of the experiment, and the excited state fraction changed only by about a factor of 2 between the two phases, the ion signal was larger by about a factor of 100 for the trapping phase as compared to the cooling phase. The lack of correlation between the collision rate and the excited state fraction is a clear demonstration of the importance of interpreting the collision rates as depending on the total atom density, not the excited state density. The frequency dependence of the collision rate was studied by changing the frequency of the trapping laser. The observed ion signal is shown in Fig.

-

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS 14

163

1

- Molasses Phase

I

Trapping Phase

2 ,

0 0

S

I0

IS

20

I

1

2s

30

Time (ps)

FIG.15. Time-dependent ion and fluorescence signals observed for molasses and trapping phases of a near-resonant dipole trap. The fluorescence (and thus the excited state fraction) differs for the two phases by only a factor of 2, whereas the ion signal varies by a factor of 100. This supports the interpretation of ultracold collision depending on the total atom density and the detailed properties of the light, not just on the atomic excited-state fraction. [Adapted from Lett et al. (1991).]

16. Due to changing densities and temperatures as the detuning was changed, the ion signal only roughly corresponds to a measurement of /3. As the frequency is changed to more negative detunings, a simple model would suggest that the ion signal should decrease rapidly since the two-photon process becomes increasingly nonresonant. In contrast to this model, the ion signals are roughly independent of detuning up to -4GHz, except for reproducible structure in the spectrum. An explanation for this spectrum was presented by Julienne and Heather (1991). They suggested that the trapping laser excites the colliding atoms into a 0-, potential curve that correlates to 3s + 3P3/2atomic states at large interatomic separation. The 0-, state is one of the so-called weakly bound long-range molecular states (Stwalley et al., 1978) that results from decoupling of the atomic spin-orbit interaction by the resonant dipole-dipole interaction. The inner turning point of the 0; state is at about 30 A, giving a good Franck-Condon overlap for resonant excitation to a 'El state that correlates to the 3P3/2+ 3P3,, atomic states and from which associative ionization can occur. Therefore, the process is doubly resonant: excitation into the 0-, state, followed by excitation of the : X' state that produces associative ionization. The structure seen in Fig. 16 is interpreted as resulting from the vibrational structure of the 0-, state. Wagshul et al. (1993) observed effects of ground state hyperfine structure

Thad Walker and Paul Feng

164

-a

n

4-

w

3-

: Y

0

2-

E

m iij 1-

-c 0

on photoassociative ionization using a MOT loaded from a Zeeman-slowed atomic beam. The trap was alternated in time (50% duty cycle) with a separate, tunable laser beam and the ion rates were measured only when the tunable laser was on. Strong ion signals were observed when the laser was tuned greater than about 0.8 GHz to the blue of the trapping laser frequency, as shown in Fig. 17. The signals were proportional to the square of the intensity, verifying that the ionization was produced by a two-photon process. By turning off the optical pumping beam (see Fig. 3) for a variable time before the trapping beam, the population in the F = 1 ground state could be changed due to optical pumping. The ion signal was found to have a maximum as the optical pumping time was changed, indicating that the process being observed arises from collisions of F = 1 and F = 2 atoms. The spectrum can be understood with reference to the simplified potential curves in Fig. 17. For blue detunings of less than approximately one-half the ground state hyperfine splitting, excitation can occur to the 2 + P3,zsingly

165

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

10

-

n

I

'

I

'

I

'

I

' ; I

'

I

l

~

l

'

l

'

l

'

-

8

v1

.LI

C

'

=e

-

6

(J

-

W

U

a

2

-

N

w

0

0.8 1.2 6,- (CHZ)

0.4

1.6

2.0

FIG.17. Observed photoassociative ionization rates for light tuned to the blue of the Na trapping transition, and simplified potential curves that explain the signal. For atoms colliding along the 1 2 ground state, excitation to the 2 P,/, state is resonant. The atoms are accelerated until the laser can again resonantly excite them to the P3/2 + P,,, state where associative ionization can occur. The frequency threshold for the process is shown by the dashed arrow. [From Wagshul et al. (1993).]

+

+

excited state potentials but the photon energy is insufficient to excite to the P3/2 + P3/2 states. When the detuning is greater than one-half the hyperfine splitting, the atoms can be accelerated along the 2 P3/2 potential and then be resonantly excited to the P3/2 P3/2 states where associative ionization occurs. The observed threshold for this process is within 10MHz of that predicted from this simple picture. For blue detunings greater than the ground state hyperfine splitting, the process is again nonresonant and the rates are reduced. A model calculation reproduced the spectrum fairly well, assuming as explained earlier that the associative ionization occurs principally through the 0-, singly excited state potential. Bagnato et al. (1993) studied the photoassociative ionization process in a similar way, but with different laser frequencies and initial states for the colliding atoms. The trap used was vapor-loaded MOT, with trapping and optical pumping frequencies generated from a dye laser beam by an electro-optic modulator. Depending on whether the carrier was tuned for trapping or optical pumping, variable populations in the F = 1 and F = 2 states could be obtained. The ion signal was measured as a function of the frequency of a tunable probe laser, and peaks blue of the trapping and

+

+

Thad Walker and Paul Feng

166

optical pumping frequencies were observed, which can be understood by considering mechanisms similar to that described in the previous paragraph. Measurements were also made with the sidebands turned off with a fast rf switch; depending on whether the carrier was providing trapping or optical pumping, the atoms were quickly optically pumped into the F = 1 or F = 2 states. The resulting spectra are shown in Fig. 18, featuring three major peaks in the spectra with diagrams of proposed explanations roughly placed above the associated peaks. The rightmost peak, occurring at a frequency roughly twice the ground state hyperfine splitting to the blue of the trapping frquency, is linear in the probe intensity, suggesting that one probe and one trapping photon is involved. The collision mechanism is explained as absorption from the trapping laser from the 1 + 1 ground state to a 1 + P3/2 state. After acceleration on this curve, excitation to the P3,z+ P3,z states occurs from the probe laser. The central peak is quadratic in probe intensity, suggesting a two-photon, single-frequency process. The proposed explanation involves

r :

2+P I+P

Z+P

2+2 I +2 I+l

I

2CP

----- I +P

2+2 I *2

2+2

I +z

I+l

1+1

R

I

iIII

R

i k I II

(b) -4000

-2000

0

Zoon (wP-wI)

4000

6000

BOO0

(MHz)

FIG.18. Two-color photoassociative ionization rates as a function of detuning of a probe laser upwith respect to the trapping frequency wl, with proposed mechanisms displayed above the three main spectral features. (a) Spectrum measured in the presence of the probe laser and the optical pumping laser of frequency w2, with the colliding atoms in the state 1 1. (b) Spectrum in the presence of the probe laser and the trapping laser, with the colliding atoms in the state 2 + 2. [From Bagnato et al. (1993), by permission.]

+

MEASUREMENTS OF COLLISIONS BETWEEN LASER-COOLED ATOMS

167

excitation from the 1 + 1 ground state to a singly excited state that is a mixture of 1 + P3/2 and 2 + P3/2 states. The mixing is produced by the resonant dipole-dipole interaction. After acceleration on the singly excited curve, further excitation to the P312 + P3/2 state results in associative ionization. Similar asymptotically forbidden processes are discussed for singly excited state trap loss by Walker and Pritchard (1994). For the leftmost peak, which is linear in probe intensity, the atoms begin in the 2' + 2 state and are excited by a red-detuned probe, then excited to the P3/2 + P3/2 state. In principle, this mechanism is exactly analogous to trap loss, where the energy-transfer process is the excitation to the doubly excited state followed by associative ionization. It is remarkable therefore that the spectrum is so different from the Rb and Cs trap-loss spectra (Section 1V.E). This may be indicative of the unusual properties of the long-range 0-, state mentioned earlier. Lett et al. (1993) have studied the production of Na: ions in a MOT for red detunings as far as - 100GHz from resonance. They observe discrete structure in the ion signals, as shown in Fig. 19. Two series of vibrational bands exist for detunings below -5GHz. Note that these bands are essentially rotation free which greatly simplifies interpretation of the spectra. Instead of associative ionization being the collision mechanism, direct photoionization out of an attractive singly excited state to Na: was proposed. The evidence for this is the lack of modulation in the strengths of the signals expected in associative ionization due to bound states in the doubly excited potential. The observed resonances are consistent with the expected vibrational splittings of either the 1, or the 0: singly excited states. However, using the hyperfine structure calculations of Williams and Jul-

-100

-80

-40 Frequency (GHr)

-60

-20

0

FIG.19. N a i production rates as a function of frequency. Two regular vibrational bands are observed, which are identified as arising from direct photoionization of 1, singly excited state molecules. [From Lett et al. (1993).]

168

Thad Walker and Paul Feng

ienne (1994), the observed substructures of the individual resonances are consistent with that expected for the 1, state. Related work has been done on associative ionization in atomic beams that studies this process at higher temperature than in traps (Thorsheim et al., 1990). Under these conditions the collisions are not as sensitive to spontaneous emission, but more detailed studies of the molecular physics can be made since the internuclear and photon-polarization axes are well defined. C. OPENQUESTIONS FOR DOUBLY EXCITED STATES

As stated before, the issues concerning the dynamics of ultracold doubly excited state collisions are similar in principle to trap-loss collisions (Section 1V.B). As is the case for most of the trap-loss measurements, the interpretation of the associative ionization experiments done to date is clouded by the poorly understood effects of hyperfine structure. For trap loss, excited state hyperfine structure seems most important, while the ground state hyperfine structure is more important for associative ionization (Wagshul et al. 1993; Bagnato et al. 1993). In addition, the interpretation of these experiments as being dominated by the unusual properties of the long-range bound 0-, state (Julienne and Heather, 1991) makes comparison to simple models not yet possible. Use of a different doubly excited state collision process that is more general, such as the direct photoionization process proposed by Lett et al. (1993),may allow for more direct tests of the models of the collision dynamics. The use of ultracold collisions as a new form of essentially rotationless molecular vibrational spectroscopy promises to improve greatly our understanding of alkali dimers, as discussed in Section 1V.F. Doubly excited state spectroscopy, while similar in principle to the singly excited state case, has the additional capability of being able to probe states that do not readily produce trap loss. Thus, for example, it may be possible to use associative ionization to observe the very weak minima that arise in the interatomic potentials due to hyperfine structure (Walker and Pritchard, 1994). This demonstrates some of the potential of ultracold collisions to give new insights into molecular physics near the dissociation limit.

Acknowledgments This work is supported by the National Science Foundation and the David and Lucile Packard Foundation. T. Walker is an Alfred P. Sloan Research Fellow.

MEASUREMENTS O F COLLISIONS BETWEEN LASER-COOLED ATOMS 169

References Bagnato, V., Marcassa, L., Tsao, C., Wang, Y., and Weiner, J. (1993). Phys. Rev. Lett. 70, 3225. Band, Y., and Julienne, P. (1992). Phys. Rev. A 46, 330. Bardou, F., Emile, O., Courty, J., Westbrook, C., and Aspect A. (1992). Europhys. Lett. 20,681. Boesten, H ., Verhaar, B., and Tiesinga, E. (1993). Phys. Rev. A 48, 1428. Cornell, E., Monroe, C., and Wieman C. (1991). Phys. Rev. Lett. 67, 2439. Dineen, T., Wallace, C., Tan, K., and Gould P. (1992). Opt. Lett. 17, 1706. Doyle, J., Sandberg J., Yu,I., Cesar, C., Kleppner, D., and Greytak, T. (1991). Phys. Rev. Leu. 67, 603. Dulieu, O., Julienne, P., and Weiner J. (1994). Phys. Rev. A 49, 607. Feng, P., Hoffmann, D., and Walker T. (1993). Phys. Rev. A 47, R3495. Foot, C. J. (1991). Contemp. Phys. 32, 369, gives a basic introduction to laser cooling and trapping. Further details appear in special issues of Laser Physics, to be published, J. Opt. Soc. Am. B 6, 2020 (1989), and Laser Manipulation of Atoms and Ions, Proc. Int. School of Physics “Enrico Fermi”, (E. Arimondo, W. Phillips, and F. Strumia Eds.), North Holland, Amsterdam, 1992. Gallagher, A. (1991). Phys. Rev. A 44, 4249. Gallagher, A., and Pritchard, D. (1989). Phys. Rev. Lett. 63, 957. Gibble, K., and Chu, S. (1993). Phys. Rev. Lett. 70, 1771. Gould, P., Lett, P., Julienne, P., Phillips, W., Thorsheim, H., and Weiner, J. (1988). Phys. Rev. Lett. 60, 788; Gould, P., Lett, P., Julienne, P., Phillips, W., Thorsheim, H., and Weiner J. (1988). AIP Con$ Proc. 172,295. Gould, P., and Wallace, C. (1993). Private communication. Grynberg, G., Lounis, B., Verkerk, P., Courtois, J.-Y., and Salomon, C. (1993). Phys. Rev. Lett. 70, 2249. Hayden, M., Hiirlimann, M., and Hardy, W. (1993). IEEE Trans. Inst. Meas. 42, 314. Hemmerich, A., and Hansch, T. (1993). Europhys. Lett. 21, 445. Hoffmann, D., Feng, P., and Walker, T. (1994). J. Opt. SOC.Am. B 11, 712. Hoffmann, D., Feng, P., Williamson, R., and Walker, T. (1992). Phys. Rev. Lett. 69, 753. Julienne, P., and Heather, R. (1991). Phys. Rev. Lett. 67, 2135. Julienne, P., Smith, A., and Burnett, K. (1992). In Advances in Atomic. Molecular. and Optical Physics 30,141. (B. Bederson and H. Walther, Eds.), Academic Press, San Diego. Julienne, P., and Vigue, J. (1991). Phys. Rev. A 44, 4464. Kasevich, M., Riis, E., Chu, S., and DeVoe, R. (1989). Phys. Rev. Lett. 63, 612. Kawanake, J., Shimizu, K., Tanaka, H., and Shimizu, F. (1993). Phys. Rev. A 48, R883. Ketterle, W., Davis, K., Joffe, M., Martin, A., and Pritchard, D. (1993). Phys. Rev. Lett. 70, 2253. Lett, P., Jessen, P., Phillips, W., Rolston, S., Westbrook, C., and Gould, P. (1991). Phys. Rev. Lett. 67, 2139. Lett, P, Helmerson, K., Phillips, W., Ratliff, L., Rolston, S., and Wagshul, M. (1993). Phys. Rev. Lett. 71, 2200. Marcassa, L., Bagnato, V., Wang, Y.,Tsao, C., Weiner, J., Dulieu, O., Band, Y.,and Julienne, P. (1993). Phys. Rev. A 47, R4563. Masuhara, N., Doyle, J., Sandberg, J., Kleppner, D., Greytak, T., Hess, H., and Kochanski, G. (1988). Phys. Rev. Lett. 61, 935. Mester, J., Meyer, E., Reynolds, M., Huber, T., Zhao, Z., Freedman, B., Kim, J., and Silvera, 1. (1993). Phys. Rev. Lett. 71, 1343. Miller, J., Cline, R., and Heinzen, D. (1993a). Phys. Rev. Lett. 71, 2204. Miller, J., Cline, R., and Heinzen, D. (1993b). Phys. Rev. A. 47, R4567. Monroe, C., Cornell, E., Sackett, C., Myatt, C., and Wieman, C. (1993). Phys. Rev. Lett. 70,414.

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Monroe, C., Swann, W., Robinson, H., and Wieman, C. (1990). Phys. Rev. Lett. 65, 1571. Peters, M., Hoffmann, D., Tobiason, J., and Walker, T. (1994). Phys. Rev. A, to be published. Prentiss, M., Cable, A,, Bjorkholm, J., Chu, S., Raab, E., and Pritchard, D. (1988). Opt. Lett. 13, 452. Raab, E., Prentiss, M., Cable, A., Chu, S., and Pritchard, D. (1987). Phys. Rev. Lelt. 59, 2631. Ritchie, N., Abraham, E., Xiao, Y., Bradley, C., Hulet, R., and Julienne, P. (1994). To be published. Sesko, D., Walker, T., Monroe, C., Gallagher, A,, and Wieman, C. (1989). Phys. Rev. Lett. 63, 961. Sesko, D., Walker, T., and Wieman, C. (1991). J. Opt. SOC.Am. B 8, 946. Stwalley, W., Uang, Y., and Pichler, G. (1978). Phys. Rev. Lett. 41, 1164. Tiesinga, E., Moerdijk, A., Verhaar, B., and Stoof, H. (1992a). Phys. Rev. A 46, R1167. Tiesinga, E., Verhaar, B., Stoof, H., and van Bragt, D. (1992b). Phys. Rev. A 45, R2671. Thorsheim, H., Wang, Y., and Weiner, J. (1990). Phys. Rev. A 41, 2873. Thorsheim, H., Weiner, J., and Julienne, P. (1987). Phys. Rev. Lett. 58, 2420. Verhaar, B., Gibble, K., and Chu, S. (1993). Phys. Rev. A 48, R3429. Wagshul, M., Helmerson, K., Lett, P., Rolston, S., Phillips, W., Heather, R., and Julienne, P. (1993). Phys. Rev. Left. 70, 2074. Walker, T., Feng, P., Hoffmann, D., and Williamson, R. (1992). Phys. Rev. Left. 69, 2168. Walker, T., and Pritchard, D. (1994). Luser Physics, To be published. Walker, T., Sesko, D., Monroe, C., and Wieman, C. (1989). A I P Con$ Proc. 205, 593. Walker, T., Sesko, D., and Wieman, C. (1990). Phys. Rev. Lett. 64,408. Wallace, C., Dineen, T., Tan, K., Grove, T., and Gould, P. (1992). Phys. Rev. Letr. 69, 897. Walsworth, R. (1993). Private communication. Walsworth, R., Silvera, I., Mattison, E., and Vesset, R. (1992). Phys. Rev. A 46, 2495. Weiner, J. (1989). J. Opt. Soc. Am. B 6 , 2270. Westbrook, C., Watts, R., Tanner, C., Rolston, S., Phillips, W., Lett, P., and Gould, P. (1990). Phys. Rev. Lett. 65, 33. Williams, C., and Julienne, P. (1994). To be published.

ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 34

THE MEASUREMENT AND ANALYSIS OF ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS J. E. L A W L E R Department of Physics University of Wisconsin Madison. Wisconsin

and

D.A. DOUGHTY Corporaie Research and Development Center General Electric Company Schenectady. New York

I. Introduction

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

11. Theory of the Stark Effect

A. Stark Effect in Atomic Systems . . . . . . . . . . . B. Stark Effect in Molecular Systems . . . . . . . . . . 111. Electric Field Mapping Based on the Stark Effect in Atoms . . IV. Electric Field Mapping Based on the Stark Effect in Molecules V. Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

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

171 173

173 177 179 196 204 205

I. Introduction Interest in glow discharge plasmas has remained high for many decades because of their widespread application as a source of incoherent and coherent light, in plasma processing of materials, in pulsed power devices, and in other technologies. Plasma etching of semiconductors and various plasma deposition processes emerged as major applications during the 1980s. The technological significance of plasma processing is described in Plasma Processing of Materials (Proud and Gottscho, 1991). More fundamental work on glow discharges also advanced greatly during the 1980s. For example, substantial progress was made through the use of laser diagnostics to study glow discharges and as a result of the dramatically increased computing power that became available in the 1980s to model glow discharges. Many of the laser diagnostics are described in Radiative Processes in Discharge Plasmas (Proud and Luessen, 1986). Kinetic theory 171

Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003834-X

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J. E. Lawler and D. A. Doughty

models, in particular, became far more sophisticated and realistic during the 1980s (see Kushner and Graves, 1991, and subsequent articles). This article is a review of recent work that used optical diagnostics to study electric fields in glow discharge plasmas. Alternative methods for measuring electric fields in plasmas include electron beam deflection (Warren, 1955) and electrostatic probes (Hershkowitz and Cho, 1988). Optical techniques have important advantages over these methods: They can be used at higher pressures and discharge current densities than electron beam deflection; and they are noninvasive, unlike electrostatic probes. In addition, optical techniques are usually easier to apply in a highly pure system than either of the alternative methods. One of the early, important studies of the Stark effect in a nonhydrogenic system was that of Foster (1924, 1928) in which emission spectroscopy was used to study the Stark effect in a Lo Surdo discharge tube. Although emission spectroscopy is still used to observe the Stark effect, much of the recent activity in this research area involves applying laser spectroscopy to the study of electric fields in glow discharge plasmas. Laser techniques generally offer superior spatial and ,temporal resolution. Laser techniques, like emission spectroscopy, can be nonperturbing when a short pulse duration laser is used. This last advantage requires slightly more explanation. Although a few nanosecond duration laser pulse does perturb the discharge, usually on a microsecond time scale, the unperturbed electric field is measured during the laser pulse by the presence or absence of a Stark shifted spectral line at the laser frequency. Much of the more fundamental interest in glow discharges today is motivated by the common goals of diagnosing, understanding, modeling, and ultimately predicting nonhydrodynamic phenomena in glow discharges. Fluid or hydrodynamic approximations have traditionally been used in modeling the central region of glow discharges. These approximations are less suited to modeling boundary regions such as the cathode fall and negative glow regions of dc discharges. The breakdown of fluid approximations is due to the proximity of the boundary and the resulting large gradients in electric field, electron density, average electron velocity, and average electron energy. For example, a fluid approximation might employ drift and diffusion terms to describe transport. The concept of diffusive transport has an underlying requirement that the fractional change in particle density be small in a mean free path, (mfp). If this requirement is not met in some region, then the concept of diffusion may have rather limited usefulness in describing the transport in that region. A kinetic theory approach or a direct solution of Boltzmann’s equation is often necessary to describe transport realistically in such regions. Hydrodynamic approximations fail for similar reasons in the very low-pressure rf and microwave discharges that are widely used in semiconductor processing. The local field approximation is probably the most important approxi-

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

173

mation used in gaseous electronics. It is a fluid approximation that is often used in the central region of moderate or high-pressure glow discharges. In this approximation the local reduced field or E / N (ratio of electric field to gas density) is assumed to determine the phase space distribution function of the charged particles. This assumption implies that the important discharge parameters including mobilities, diffusion coefficients, excitation coefficents, and ionization coefficients can be parameterized in terms of E,”. Although this approximation breaks down in the nonhydrodynamic regions described in the preceeding paragraph, it is still of great interest to map the electric fields in these nonhydrodynamic regions. Space-charge densities and ion current densities can be deduced from field gradients and fields. Experimental electric field maps serve as input data for discharge simulations that are not self-consistent. Self-consistent simulations, which involve the simultaneous solution of tranport equations and Poisson’s equation to predict electric fields, can be tested with experimental electric field maps. The next section of this article is a review of some of the fundamentals of the Stark effect in atoms and in diatomic molecules. The third section describes recent glow discharge studies using the Stark effect in atoms to map electric fields. The fourth section describes recent glow discharge studies using the Stark effect in diatomic molecules to map electric fields.

11. Theory of the Stark Effect

A. STARKEFFECTIN ATOMICSYSTEMS The basic theory of the Stark effect in atoms is reviewed in this section. Figure 1 shows a partial Grotrian diagram of a typical atomic system that could be used to observe the Stark effect. Levels c and d are separated by a small energy gap, W, - W,, at zero electric field. These two levels are connected by a far-infrared or microwave transition that is allowed by electric dipole and has a substantial dipole matrix element. It is important to recall that the angular momentum along the field axis, mh, is quantized for all values of the electric field. Thus, only states with the same m can mix such as the ljem > and Ij,m > states were, j , and j , are total angular momentum quantum numbers. Often it is possible to analyze the Stark effect by including the mixing of states in level c and d with each other and neglecting the mixing of these states with states from other levels. We refer to this system as an isolated two-state system. An experimentalist might observe a shift in level c by measuring the

174

J. E. Lawler and D. A. Do ughty C

Icb

Ica

I , a

FIG.1. Partial Grotrian diagram of a typical isolated two-state system used to observe the Stark effect. Levels c and d are separated by a small energy gap at zero electric field, and are connected by an electric dipole-allowed transition.

change in the wavelength of the c -,b emission. In an isolated two-state system the calculation of the energy W,‘ (m) of a perturbed state from level c only requires the diagonalization of a 2 x 2 matrix. The result is

w2 (4= ( 4+ K)/2 - [(W, - w 2 / 4 - I(jcmIzlj,m)

)ze2E2]”2

(1)

where e is a unit charge. The familiar result from second-order perturbation theory is recovered by keeping the first nontrivial term of a Taylor series expansion of the radical in the limit I Wr(rn)- W,l = I A K(m)l<< W, - W,. This yields =

- I(jcmlzljdm>12eZE2/(W,

-

(2)

The matrix element in this equation is the same matrix element that determines the strength of the d + c infrared or microwave transition. It is often convenient to express the matrix elements in terms of the dimensionless oscillator strength of the d + c transition, which is usually tabulated as a gf value (the product of a level degeneracy and an oscillator strength). [The tabulation of gf or log (gf) values is advantageous in that we no

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

175

longer need to distinguish between emission and absorption oscillator strengths.] Atomic oscillator strengths have been tabulated by Wiese et al. (1966, 1969), Martin et al. (1988), Fuhr et al. (1988), and others. We find, after using the Wigner-Eckart theorem, 10’crnlzljdrn>12

= I(jc1rnolidm)12

3gfA:rnec2/[8n2(2jd+ ‘)(K - K)]

(3) where Ac is the Compton wavelength of the electron (2.43 x lO-”cm), m,c2 is the rest energy of the electron (511 keV), and the first term on the right side of the equation is a Clebsch-Gordan coefficient. If the individual rn states are resolved in an observation of the c-+ b emission, then the expression for W,‘(rn)is directly applicable. If the individual rn states are not resolved, then it is necessary to compute a weighted AK. The weighting in this average over rn is determined by the angle between the observation direction and the electric field (quantization axis) and by the polarization of the observed emission. Although detailed formulas are not given here, we note that the Wigner-Eckart theorem enables one to calculate efficiently the weighting factors for a particular experimental arrangement. It is worth noting that in cases where the gf value is unknown, we can measure the relative value of the electric field as a function of position in the glow discharge. If such a relative map extends from electrode to electrode, then it can be normalized using a voltage measurement. This constitutes a measurement of the unknown gf value. An experimentalist might also or instead observe the Stark mixing of levels c and d by measuring the ratio of the “forbidden” c a emission to the intensity of the “allowed” c -+ b emission. This intensity ratio, averaged over all observation directions and polarization, can be written in terms of Einstein A coefficients as -+

This intensity ratio saturates if the electric field is large enough to make I(jcrnlzljdm)12e2E2z (Wd - Wc)2. The Stark shift AWc(rn) changes from quadratic to linear in E as the intensity ratio saturates in the isolated two-state system. The line intensity ratio method has been used both with “natural” plasma excitation and with laser excitation. (Although it is conceivable that energetic, highly directional electrons in a glow discharge could produce nonisotropic excitation, it is extremely unusual.) A detailed analysis of line intensity ratio measurements with isotropic or natural excitation requires specification of the direction and solid angle observed with respect to the electric field, the polarization observed again referenced to the electric field, and the observed spectral bandpass. The analysis of a laser excitation or laser-induced fluorescence (LIF) experiment of this type

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J. E, Lawler and D. A. Doughty

requires that we specify additional parameters including the laser polarization with respect to the electric field, the laser bandwidth, the laser pulse duration, and the laser pulse energy. The LIF experiments in low collision rate glow discharges can produce atomic coherences, which result in Stark quantum beats. A more detailed analysis can be found in Bethe and Salpeter (1977), which is required for a complete interpretation of the angular and polarization-dependent effects in a particular experiment. The simple Stark effect described in the preceding paragraphs is useful if level c and a' are near each other and are widely separated for other levels. Typically, such calculations are used on low excited levels of atomic systems. The calculation of the Stark effect in highly excited levels, which are Rydberg in character, often requires a more sophisticated calculation. Rydberg states mix with many other nearby Rydberg states due to the Stark effect. Interest in Rydberg atoms increased dramatically during the 1970s when good pulsed dye lasers became widely available. Once it became possible to produce copious quantities of Rydberg atoms, many excellent studies of their spectroscopic and collisional properties were performed. For example, Zimmerman et al. (1979) performed extensive calculations of Stark effects in alkali atoms and compared their calculations to experimental observations. Their basic approach for calculating Rydberg atom Stark effects is a variation of that used by Foster (1928). It was adapted, with some further modifications, when Rydberg atom Stark spectroscopy was used to study glow discharges (Doughty and Lawler, 1984; Doughty et al., 1984). A Hamiltonian matrix diagonalization is used in a calculation of the Stark effect in Rydberg atoms. The basis set is made of eigenfunctions of the zero-field Hamiltonian. The optimum choice of principal quantum number n is determined by requiring that the Stark effect be large enough to make accurate measurements, but small enough to avoid mixing states of different n. This simplifies the matrix to an (n-m) x (n-m) diagonalization where mh is the component of orbital angular momentum along the field axis. The diagonal matrix elements, Wnl,are given by the empirical Rydberg formula

where yon is the ionization energy, R, is the Rydberg constant, and 6, is the quantum defect. Quantum defects can be deduced from tables of experimental atomic energy levels (Moore, 1971; Sugar and Corliss, 1985). The few nonzero off-diagonal matrix elements connect states that differ in orbital angular momentum quantum number 1 by one. Hydrogen matrix elements given by

(nlm(eEzln(1- 1)m) =

(i) ['"' -il:)y2 -

eEa,n

m2

177

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

where E = Ei is the electric field, e is the unit charge, and a, is the Bohr radius, are used with some corrections. The corrections are necessary because of the departures from a perfect (6, = 0) hydrogenic system. These corrections to the radial matrix elements have been tabulated as a function of quantum defect by Edmonds et al. (1979). The linear Stark effect of atomic hydrogen is recovered from this calculation in the 6, = 0 limit. The size of the Stark effect and thus the sensitivity of the electric field diagnostic is rapidly increasing function of n. Figure 2 is a theoretical Stark map for the n = 11, rn = 0 singlet states of He. Pure rn = 0 states are excited by polarizing the laser along the electric field and exciting the Rydberg states from the 2lS metastable level of He. Information on the field strength is available from the position and relative strengths of the Stark resonances. The relative strength is determined by the amount of the original P state mixed in each state. The relative strength of “forbidden” and “allowed Stark resonances is very field dependent at low fields.

B. STARK EFFECTIN MOLECULAR SYSTEMS Similar Stark mixing of different electronic states occurs in atoms and in molecules.The Stark effect from electronic state mixing in molecules has not been used as much in glow discharge studies as the Stark effect from the

1.0

0.6

.-n

0.2

4-

-s -

I

3

I

0.1

-

0.05

0.02 -

0

Electric Field (Vlcm)

0

400

800

Electric Field (Vlcm)

FIG.2. (a) Theoretical Stark map for the n = 11 singlet rn = 0 levels of He. (b) Theoretical relative intensities of selected Stark components as a function of electric field. [From Den Hartog et al. (1988), Fig. 2, p. 2474, by permission.]

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J. E. Lawler and D. A. Doughty

“permanent” electric dipole moment of heteronuclear diatomic molecules. The Stark effect from the permanent electric dipole moment is in general easier to interpret and easier to measure at low fields than Stark effects from mixing different electronic states. A diatomic molecule in an electronic level with a nonzero electronic angular momentum along the internuclear axis, such as a ll or A level has nearly degenerate levels with opposite parity. Consider a ‘Il electronic level. The electronic angular momentum along the internuclear axis A couples to the molecular rotational angular momentum N to form the total angular momentum 3. Each rotational level specified by quantum number j in the ‘II electronic level has a 2 j + 1-fold degeneracy due to different projections of the angular momentum mh on the z axis defined by the electric field. There is also a twofold degeneracy due to the electronic wavefunction having a plus or minus symmetry with respect to reflection to a plane containing the internuclear axis and normal to R. The overall parity of the rotational level is determined by whether the quantum number associated with angular momentum 3 is even or odd and by whether the electronic wave function is plus or minus. Nearly degenerate states of opposite parity with the same rotational angular momentum 5 result from the reflection symmetry of the electronic wave function. These states are in general not perfectly degenerate because of a small amount of A doubling (Herzberg, 1950). Any molecule in a pure state in field free space has a definite parity and thus cannot have a “permanent” electric dipole moment. However, in general, a substantial electric dipole matrix element connects these nearly degenerate levels of opposite parity in heteronuclear diatomic molecules. This matrix element is the “permanent” electric dipole moment of our heteronuclear diatomic molecule in the ‘II electronic state. The Stark effect from the permanent dipole moment of a heteronuclear diatomic molecule is usually analyzed as an isolated two-state system when it is used to measure electric fields in glow discharges. The calculation of this Stark effect in heteronuclear diatomic molecules is very similar to the calculation outlined for the isolated two-state atomic system discussed at the beginning of this section. The rotation of the molecule adds a slight complication, which can be treated using the semiclassical vector model to project the permanent dipole moment p onto the z axis. Figure 3 illustrates this vector model calculation. The atomic quantity I (jcmlzlj,m)12eZ is replaced by p z m z A 2 / [ j 2 ( j 1)2] from the vector model, and (W, - W,) is replaced by zero-field splitting due to A doubling. Electric field measurements in glow discharges using molecular Stark effects are usually made by observing the ratio of the intensity of forbidden lines to that of allowed lines in the same band of an electronic transition. The Einstein A coefficients in the earlier expression for an intensity ratio of atomic lines must now be replaced by Honl-London factors. The calculation of these factors is described by Hougen (1970). The Stark effect in heteronuclear diatomic molecules has one of the

+

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

179

7 3 B FIG.3. Vector model for computing the Stark effect in a heteronuclear diatomic molecule. The permanent dipole moment fi is aligned with A along the internuclear axis. The magnitude of r( projected onto 1 is p A / J . The magnitude of p projected onto the z axis is p A j ..?/J* in a vector model calculation.

important advantages of the Stark effect in Rydberg levels of an atom. The zero-field splitting due to A doubling usually has a strong j dependence [it scales a s j ( j + 1) in a I l l level], thus a natural mechanism exists for varying the sensitivity of the Stark effect diagnostic.

111. Electric Field Mapping Based on the Stark Effect in Atoms Nakajima et al. (1983) reported a very interesting experiment in which laser optogalvanic spectroscopy was used to map the electric field in the cathode region of commercial hollow cathode lamps. Such lamps are widely used in atomic absorption spectrophotometers. They observed transitions from the 4ps5p to 4ps8d configuration of Kr. The relatively small quadratic Stark shifts were measured using a cw single-frequency dye laser. Some of these measurements are shown in Fig. 4. The good (< l-mm) spatial resolution

J. E. Lawler and D. A. Doughty

180

4 EO v 17384.70

17384.60

17384.50

FIG.4. Frequency of the Kr 2p6-7d; transition ( 0 )and corresponding electric field (0)as a function of position in a commercial 4-mm-diameter “see-through” hollow cathode lamp operating with a 4.73 mA/cm2 (316.8-V) dc discharge in 6 Torr of Kr. The arrow indicates the location of the cathode surface. The Stark shift is measured using laser optogalvanic spectroscopy. [From Nakajima et al. (1993), Fig. 3, p. C7-499, by permission.]

clearly demonstrates one of the key advantages of electric field diagnostics based on laser spectroscopy. Doughty and Lawler (1984) and Doughty et al. (1984) reported the development of techniques for mapping electricfields in glow dischargesbased on laser optogalvanicdetection of transitions to Rydberg levels.Laser excitation of Rydberg levels of most atoms,even He, is readily achieved in a glow discharge. One need not use the laser to excite from the ground state because copious quantities of excited metastable atoms are found in a typical glow discharge.The very large linear Stark effects that occur in true Rydberg levels gave electric field measurements of high accuracy and precision. Although transitions to high Rydberg levels (n 11) do not fluorescein a typical glow discharge,they can be readily detected using optogalvanic effects.Atoms in Rydberg levels are usually collisionally quenched by processes such as associative ionization:

-

He’

+ He -,He; + e-

(7)

This collisional ionization process perturbs the local ionization balance in the glow discharge and thus produces an optogalvanic effect. A natural

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

181

amplification of the optogalvanic effect occurs when excess electrons are released near the cathode. Such excess electrons are accelerated through the cathode fall region and produce a small avalanche of additional ionization. The dramatic improvement in sensitivity achieved by exciting true Rydberg levels, which produce linear Stark effects, is illustrated by the spectra of Fig. 5. Electric fields of less than 1 kV/cm resulted in quadratic Stark shifts of << lcm-' in the experiment by Nakajima et al. (1983) and were essentially undetectable in their experiment. Electric fields of lkV/cm resulted in well-developed linear Stark patterns spread over many inverse centimeters in the experiment by Doughty and Lawler (1984). Doughty and Lawler (1984) also demonstrated that the line intensity ratio method of observing the Stark effect could be used with laser optogalvanic spectro-scopy. Two criteria must be satisfied to use the line intensity ratio method. The first criterion is that the laser intensity must be reduced below the saturation intensity so that the number of Rydberg atoms produced is proportional to the strength of the particular Stark component. The second criterion is that the optogalvanic effect must be equally efficient on all Stark components. Doughty and Lawler (1984) argued that the rapid mixing of the populations of levels with different 1 and with the same n = 11 in their experiment should ensure that the second criterion was met. Cross sections for this type of mixing of n = 11 He Rydberg atoms are 1.85 x cm2 (Gallagher et al., 1977). Doughty et al. (1984) further extended the diagnostic based on the Stark effect in Rydberg atoms with the use of multistep excitation of the Rydberg atoms from intersecting laser beams. This enabled them to make pinpoint measurements of the electric fields in the glow discharge. Such techniques are valuable if one wishes to make a measurement that is not an average along the laser path through the discharge. A significant effort was started at the University of Wisconsin to use the electric field diagnostic based on optogalvanic detection of Rydberg atoms and other diagnostics based on laser spectroscopy to map thoroughly the cathode fall and negative glow regions of a clean He discharge. These experiments were designed for comparison to kinetic theory simulations. Helium was chosen because electron impact cross sections on He are better known than cross sections for other gases, and because He produces minimal sputtering of the cathode. Heavier inert gases sputter the cathode and produce a more complicated mixture of inert gas and metal atoms in the discharge. Great care is necessary to produce a clean He discharge because the high ionization potential of He results in any impurities (even at the part-permillion level) being preferentially ionized and possibly dominating the ionization balance. Den Hartog et al. (1988) at Wisconsin used a glass and metal discharge system, a very clean vacuum pump station, ultrahigh purity He, and a cataphoresis cleaning system to further purify the He.

J. E. Lawler and D. A, Doughty

182

i

Electric Field = 800 V/Cm Distance From Cathode = 0.10 cm

n OD

r 0

s

p Y

-c* 4 c. Po

Q)

CI

C

- 2

0

4

0 -4 Displacement (cm-’)

-8

8

Electric Field = 6 10 VlCm n

26

Distance From Cathode = 0.25 c

I

C

a

e

m “4

-

>1 c

m

C

Q

C

- 2

0 4

0

-4

-8

Displacement (cm-‘) FIG.5. (a) Stark spectrum of the He 2’s-n = 11 Rydberg manifold observed 0.10 cm from the aluminum cathode of a 0.51 mA/cm2 (426-V) obstructed dc discharge in 2.0 Torr of He. Calculated positions and intensities of the 10 Stark components are indicated by vertical lines with cross bars. Open circles are experimental (integrated) intensities. (b) Stark spectrum observed 0.25 cm from the cathode. The Stark patterns are measured using laser optogalvanic spectroscopy. [From Doughty and Lawler (1984), Fig. 2, p. 612, by permission.]

Meaningful comparison between experiment and kinetic theory is more easily achieved if the experiment is essentially one dimensional. Den Hartog et al. (1988) used a nearly one-dimensional discharge between plane parallel

183

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

aluminum electrodes. Tests to verify the uniformity of the discharge were performed. A cathode-to-anode spacing of 6.2 mm and a He pressure of 3.5 Torr were chosen. A range of current densities from 0.190 mA/cm2, which corresponds to a cathode fall of 173 V, to 1.50 mA/cmZ,which corresponds to a cathode fall of 600 V, was used in the experiments. Figure 6 is a schematic of the experiment apparatus. A N, laser pumped dye laser system, tunable from 205 to 700 nm, was used. The dye laser bandwidth was 0.3 cm-' without an etalon and 0.01 cm-' with an etalon. Optogalvanic, fluorescence, and absorption detection were used in the experiments as shown in Fig. 6. The discharge was on a precision translation stage in order to make spatial maps without disturbing the laser alignment. Spatial resolution down to 0.1 mm was achieved through careful control of the laser beam width and depth of focus in the experiments. Figure 7 shows a series of Stark spectra recorded in a 3.50-Torr He discharge with the laser at different distances from the cathode. Note that the field magnitude and resulting Stark effect increase rapidly as the laser is moved from the edge of the negative glow at 0.382 cm toward the cathode. Figure 8 shows an electric field measured by Den Hartog et al. (1988) as a

Dual Chart

Etalon

Recorder

Photodiode

\ Photodiode

-

v

/ r

Power

i

n

U KDP

" I

,

Crystal (used only in Rydberg atom excltatlon)

FIG.6. Schematic of the experimental apparatus used at the University of Wisconsin for extensive studies of the cathode fall and negative glow regions of a clean He discharge. Three detection methods-optogalvanic, fluorescence, and absorption-are shown. [From Den Hartog et al. (19881, Fig. 1, p. 2472, by permission.]

J. E. Lawler and D. A. Doughty

184

Calibration Fringes

0.025

A 4

z

0.076

Q

I

v)

0

I

z 4

>

4 U

0.127

Q 0

I= P

0

0.178

0.229

0.280

I -8

I -4

I

I

0

4

0.33 1 0.382

LASER FREQUENCY OFFSET (cm-1) FIG.7. Optogalvanic Stark spectra of the He 2’s-n = 11 Rydberg manifold in a 1.50 mA/cm’ (600-V) dc discharge between plane parallel aluminum electrodes in 3.50 Torr of He. The distance from the cathode in centimeters at which each spectrum is taken is given at the right [From Den Hartog et al. (1988), Fig. 3, p. 2475, by permission.]

function of distance from the cathode for five discharges in He at 3.50 Torr. The solid lines are linear least-square fits to the data. The field measurements are known to be accurate to 1% on average by comparing a voltage Vef from the spatially integrated field to digital multimeter readings Vnm. These voltages are compared in Table I. These field maps can be directly compared to numerical simulations. They can also be further analyzed to yield the ratio of ion current to electron

-

185

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

I

I

I

I

I

1

1

Helium Dlscharge Pressure: 3.5 Tom

1

-

Electrode Separatlon: 0.62 ern

0

0.1

0.2

0.3

I

I

I

0.4

0.5

0.6

DISTANCE FROM CATHODE (em)

FIG.8. Electric field as a function of distance from the cathode for dc discharges at five current densities between plane parallel aluminum electrodes in 3.50 Torr of He. The lines are linear least-square fits to the measurements (solid circles) using optogalvanic detection of Rydberg atoms. The anode is at 0.62 cm. [From Den Hartog et al. (1988), Fig. 4, p. 2476, by permission.]

current or current balance at the cathode (Doughty et al., 1987). The ion current density at the cathode, . I : , is the product of the ion charge density p + and the average ion velocity at the cathode (u,)O,. Essentially all of the space-charge density in the cathode fall region is due to ions, thus Gauss's law determines p + from the field gradient. Table I OF THE ELECTRIC FIELD AND GASDENSITY MEASUREMENTS TO SU2rlPrlARY OF THE ANALYSIS DETERMINE THE CURRENTBALANCE AT THE CATHODE SURFACE

0.190 0.897 0.382 0.173 0.171 11.2 2.08 1.66

0.519 1.426 0.301 0.211 0.215 10.8 4.20 2.82

7.12 0.148 3.52

9.28 0.390 3.02

0.846 1.870 0.282 0.261 0.264 10.3 5.88 3.89 23.3 10.9 0.641 3.13

1.18 2.395 0.300 0.356 0.359 9.48 7.07 5.71 22.1 13.2 0.933 3.78

1.50 3.017 0.396 0.600 0.597 8.01 6.74 8.92 21.1 16.5 1.11 2.85

Note. The dimensionless ratio J"+/(Jo-JO,) is the ion current density divided by the electron current density at the cathode. It is essentially the same as the number of ions produced in the cathode fall region per (net) electron emitted from the cathode.

186

J. E. Lawler and D. A . Doughty

The average ion velocity at the cathode surface is the equilibrium drift velocity for the E / N (ratio of electric field to gas density) at the cathode surface. Ions, unlike electrons, are a hydrodynamic equilbrium throughout most of the cathode fall region. The short ion equilibration distance is due to the symmetric charge-exchange reaction: He+(fast) + He(s1ow) + He(fast) + He+(slow)

(8)

This reaction restarts the ions from a thermal (slow) speed after each collision; thus, an equilibration distance comparable to the mfp for charge exchange is expected. The mfp, (No)- where 0 is the charge-exchangecross section must be compared to the thickness d, of the cathode fall region. Typically Nod,>> 1, which means that the ions are equilibrated throughout the cathode fall except in a narrow region at the cathode fall negative glow boundary. The preceding arguments are very intuitive but not strictly rigorous because of the production of new ions from electron impact ionization throughout the cathode fall and because of the large field gradients found in the cathode fall. Both of these complications were included in a derivation of the equilibration distance of ions in the cathode fall by Lawler (1985). Lawler used Green’s function solutions to Boltzmann’s equation for the ions. The Green’s function solutions are exact for Boltzmann’s equation with an electric field that varies linearly with position and with a symmetric charge-exchangecollision term. The production of new ions was included by calculating equilibration distances in two limiting cases. It was generally known at the time that the electron impact ionization rate per unit volume peaked near where the cathode fall field extrapolated to zero. Lawler defined the equilibration distance to be the distance required for the average ion velocity to reach 90% of the equilibrium drift velocity, (Y,)+

= [2eE/(m,1roN)]’~*

(9)

where m, is the ion mass. Lawler found an equilibration distance of 1.7 mfp by assuming all of the ions started near where the electric field extrapolated to zero. He found an equilibration distance of 5.7 mfp by assuming a spatially uniform electron impact ionization rate per unit volume. The physically correct case is between these two limits; thus, the ions are certain to be equilibrated at locations more than 5.7 mfp from the cathode fall negative glow boundary. The calculation by Lawler (1985) was an integral part of analyzing the field maps. The calculation is also of interest because it is a rigouous, kinetic theory calculation that is analytical. The field gradient map yields directly p + , the ion charge density form Gauss’s law. The field maps and the gas density N are needed to determine the average ion velocity (v,)O, at the cathode. Substantial gas heating and a corresponding gas density reduction occurs in an abnormal cathode fall. Den Hartog et al. (1988) carefully measured the gas kinetic temperature

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

187

using a narrowband laser to probe a He transition whose linewidth is dominated by Doppler broadening. The complete analysis of the electric field maps of Fig. 8 by Den Hartog et al. (1988) is summarized in Table 1. The average ion velocity at the cathode for the two lowest current discharges was determined from the E / N at the cathode using precise drift velocities for He+ in He measured by Helm (1977). Theoretical charge-exchange cross sections (Sinha et al., 1979) were used to determine the average ion velocity for the three higher current discharges. This was necessary because the E / N at the cathode surface in these discharges exceeded the highest E / N used in Helm’s (1977) measurements. Den Hartog et al. (1988) included, at least approximately, the weak energy dependence of the charge-exchange cross section CT in their analysis. The average kinetic energy, (m+u,2/2)0,, of the ions at the cathode is proportional to the reduced field, Eo/N, at the cathode (10)

(rn,v$/2)0+ = eEof ( 2 N a )

The hyperbola defined by Eq. (10) was plotted on a graph of CT as a function of ion energy. The intersection determined the average ion energy at the cathode and the effective charge-exchangecross section CT for calculating the average ion velocity [ 2 e E o / ( m + m N ) ] ” Zat the cathode. All of these parameters are included in Table I. The difference between discharge current density J , and the ion current density J: at the cathode determines the electron current density at the cathode. The current balance is defined as J? /(J, - J?). The current balance, which averages 3.3, is surprisingly small. It implies a rather large electron emission coefficient of 0.3 for these cold cathode discharges.This electron emission coefficient, however, is actually a composite coefficient for ion, metastable atom, and UV photon bombardment of the A1 cathode. The metastable atomic flux to cathode, which was also measured, was consistent with the large composite electron emission coefficient. A comparison of the empirical current balance to results from Monte Carlo simulations yielded interesting insights on the source of ions reaching the cathode. Den Hartog et al. (1988) concluded that most of the ions reaching the cathode were produced between the surface of the cathode and the position where the electric field extrapolated to zero. Ganguly and Garscadden (1985a) applied optogalvanic spectroscopy to measure Rydberg transition linewidths and series terminations in a He positive column discharge. They used a 6-mm-diameter tube filled with 1.25 Torr of He and operated at a dc current of 1.2 mA. The He n’Po series was excited with a frequency-doubled dye laser pumped by the second harmonic of a Nd:YAG laser. The laser bandwidth was 0.3 cm- Five sample spectra from their experiment are shown in Fig. 9. The laser beam was perpendicular to the positive column axis. Spectrum C was recorded with the laser

’.

J. E. Lawler and D. A. Doughty

188

x4

U

A

A

~

1950

31925

31900

31875

x4

cm'l

FIG.^. Optogalvanic spectra of the He Rydberg series excited from the 2's level in a 6-mm-diameter, 1.2-mA dc positive column discharge in 1.25 Torr of He. The laser intersects (at 90") the column axis in C.The spectra labeled B and D are taken with the laser 1.5 mm off axis. The spectra labeled A and E are taken 2.25 mm off axis. [From Ganguly and Garscadden (1985), Fig. 2, p. 541, by permission.]

189

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

intersecting the column axis, spectra B and D were recorded with the laser beam aligned to give a closest approach to the axis of 1.5 mm, and spectra A and E were recorded with a closest approach distance of 2.25 mm. The observed linewidths increase and the principal quantum number of the Rydberg series termination decreases as the laser beam is moved further from the column axis. Series termination occurs when Stark line broadening becomes comparable to the line separation. An analysis of their spectra produced the field map and metastable density map as a function of column radius shown in Fig. 10. Ganguly and Garscadden achieved an impressive sensitivity of & lV/cm at fields of 10V/cm. Ganguly and Garscadden (1985b) demonstrated a technique for measuring the direction of the electric field at off-axis positions in a positive column. This experiment was a further refinement of their positive column experiment described in the preceding paragraphs. The discharge tube and laser system were similar. The laser beam propagated through the positive column discharge and was perpendicular to the column axis. The linear laser polarization was varied until, at a particular radial location, the laser polarization was orthogonal to the positive column field. (It is important to recall that the electric field in the positive column has both an axial and radial component at off-axis locations.) An orthogonal polarization sup100

-

-

19

-

-

- 17 - 5 - 15 > - w

I2 0 0 -

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400

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3 300 m

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0 A

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c w

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100-

2

11

I

I

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1

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RADIAL POSlflON

FIG.10. Experimental radial profiles of the electric field (+) and the relative metastable density (*) in a 6-mm-diameter, 1.2-mA dc positive column discharge in 1.25 Torr of He. [From Ganguly and Garscadden (1985), Fig. 3, p. 541, by permission.]

190

J. E. Lawler and D. A. Doughty

pressed the Am = 0 transition from the Z3S metastable level to the n3S level. This condition was easily observed because the larger quantum defect of the n3S level resulted in the 23S+n3S forbidden transitions being well resolved from the 23S+n3Po allowed transition. Figure 11 shows some of the spectra recorded by Ganguly and Garscadden (1985b).

WAVE NUMBER (cm'l) FIG.11. Optogalvanic spectra of the He Rydberg series excited from the 2's level in an 8-mm-diameter, 0.8-mA dc positive column discharge in 1.2 Torr of He. The laser intersects (at 90") the column axis in B, and is polarized parallel to the axis. The spectra labeled A and C are taken with the laser 2 mm off axis with the laser polarized at +50% and -40% with respect to the axis. [From Ganguly and Garscadden (1985b), Fig. I, p. 2544, by permission.]

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

191

Ganguly (1986) used optogalvanic spectroscopy to measure Rydberg series terminations in the negative glow region of a He discharge. The negative glow studied by Ganguly extended from the end of a hollow cathode to the positive column. He was able to establish a lower limit on the Holtzmark field and thus the ion density in the negative glow but was unable to clearly separate the microscopic Holtzmark field from the macroscopic space-charge field in the negative glow plasma. Shoemaker et al. (1988) exploited a technique for measuring electric field gradients that was suggested by Doughty and Lawler (1984). The spectral width of “outer” components in a well-developed linear Stark pattern from Rydberg states is determined by the local field gradient in the laser beam. The theoretical Stark map in Section I1 of this paper clearly shows that the energy of the highest and lowest states in a linear pattern varies rapidly with electric field, while the energy of the middle state is almost independent of the field. The width of the outer components is thus determined by the product of the spatial width of the laser beam and the local field gradient. Lee et al. (1987) used Doppler-free saturation spectroscopy to observe quadratic Stark shifts in transitions between low excited levels of He. Their highly abnormal or high-voltage (- 5-kV) cold cathode glow discharge is similar to those used for generating high-intensity electron beams. This type of electron beam source is used for laser excitation and materials processing. Typically a pressure of 1 Torr of He is used in this type of glow discharge. Lee et al. (1987) chose the 21P0-+31D,transition of He1 at 668 nm for their work. Although the Stark effect is not very large for the 3’D, level, the 668-nm transition is at a convenient wavelength for single-mode cw laser experiments. The observed homogeneous width of the line was 400 MHz; this width is due to the short radiative lifetime of the 2’P0 level and due to resonance collisional broadening. Lee et al. reported being able to measure Stark shifts as small as 30 MHz. The calculation of the quadratic Stark shift is straightforward using the formula presented in Section I1 of this paper. The 2lPo level is not significantly perturbed, while the 3’D level is perturbed only by the nearby 3’P0 level. Well-known He1 oscillator strengths can be used to derive the quadratic Stark coefficient (Wiese et al., 1966). Figure 12 includes electric field maps produced by Lee et al. in two different discharges. A flat 3.2-cm-diameter Mo cathode was used. They found a nearly linear decrease in the strength of the electric field with distance from the cathode over most of the cathode fall region. The field strength dropped more quickly near the negative glow. Calculated results from a self-consistent fluid model are indicated by the solid lines in Fig. 12. Zeller et al. (1993) used optogalvanic detection of Rydberg atoms to map electric fields in He and He-0, discharges. They studied a range of pressures from 0.2 to 3 Torr, a range of voltages from 1 to 5 kV, and several cathode materials including aluminum covered with an oxide layer, clean aluminum, and magnesium. The substantial range of parameter space explored in their N

J. E. Lawler and D. A. Doughty

192

12

-

4-

32-

'-

b I

0

1

2 3 4 1 DISTANCE FROM CATHODE lmm)

=

'4'

FIG.12. (a) Electric field as a function of distance from the molybdenum cathode in a 4.35 mA/cmZ (5.11-kV) dc discharge between plane parallel electrodes in 1.3 Torr of He. Measurements (open circles) and model results (solid line) are shown. Saturated fluorescence spectrum (insert) of the He 21Py-3'D, transition with reference frequency markers. (b) Electric field as a function of distance from the cathode in a 8.70 mA/cmZ (4.97-kV) dc discharge in 2.3 Torr of He. [From Lee et al. (1987), Fig. 2, p. 410, by permission.]

193

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

study enabled them to derive empirical scaling laws for cold cathode discharges in He. Figure 13 from Zeller et a!. (1993) includes field maps for discharges with cathodes made of aluminum covered with an oxide layer, made of clean aluminum, and made of magnesium. These measurements beautifully illustrate the higher electron emission coefficients achieved with an oxide layer on the aluminum cathode and the resulting higher electric field and ion density in the cathode fall region. All three discharges ran at 1 kV in 1 Torr of He. The discharge current density was 4.4 mA/cm2 in the discharge with the aluminum cathode covered with an oxide layer, and was 1.1 mA/cm2 in the discharge with the clean aluminum cathode. Zeller et al. (1993) also explored the breakdown of the simple one-dimensional description of the cathode fall region. Barbeau and Jolly (1991) mapped the electric field in a hydrogen discharge using traditional emmision spectroscopy to observe the Stark effect. Their discharge occurred between 30-mm-diameter iron electrodes separated by 30 mm with a gas pressure of 0.4to 1.2 Torr. Current densities were in the range of 1 mA/cm2. They used the profile of the Balmer-a line to assess Doppler broadening because it is relatively insensitive to the Stark effect. Hydrogen atoms were known to have quite large kinetic energies near

"

" * 3

8

0 00

I

2 3 4 5 POSITION (millimeters)

1

6

7

FIG.13. Electric field as a function of distance from the cathode in three 1.0-kV dc discharges at 1 Torr of He or He and a trace of 0,. The Al,O,-coated cathode (open circles) operated at 4.41 mA/cm2, while the aluminum cathode (solid squares) operated at 1.10 mA/cmZ.Optogalvanic detection of Rydberg atoms is used to map the field. [From Zeller et al. (1993), Fig. 3, by permission.]

J. E. Lawler and D. A. Doughty

194

the cathode (Benesch and Li, 1984), in part because energetic ions incident on the cathode backscatter as neutral atoms. The observed Balmer-ct profile was convoluted with a theoretical Stark pattern to yield the calculated line profile for Balmer-6 shown in Fig. 14. Barbeau and Jolly (1991) reported achieving sensitivity to a few tens of volts per centimeter and spatial resolutions of 0.5 mm. This sensitivity and spatial resolution are good for an emission experiment. Their spatially integrated electric field measurements agreed with the discharge voltage within 4%. Various techniques, especially those of laser spectroscopy, described in the earlier paragraphs, have been used extensively for fundamental studies of

.0.4

-0.4

-0.2

0

0.2

0.4

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Delta Lambda (nm) FIG.14. Emission spectra of Balmer-d observed 3 and 10 mm from the cathode of a 0.85 mA/cmZ (900-V) dc discharge between plane parallel iron electrodes in 0.6 Torr of H,. Measured (+) and fitted (solid line) profiles are shown. [From Barbeau and Jolly (1991), Fig. 2,p. 238, by permission.]

195

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

continuous, low-power glow discharges. Stark effect observations on transient, high-power glow discharges are more difficult. Emission spectroscopy has been used more extensively in transient, high-power discharges. Allario and Wainfan (1969) reported one of the early experiments of this type. They studied the formation of the cathode fall during breakdown in He and He-H, mixtures. Maron et al. (1986, 1987) used time-resolved emission spectroscopy on high-power, magnetically insulated ion beam diodes to study the spatial and temporal dependence of the electric field in the diode gap. These pulsed power devices have numerous applications including: generating intense pulsed ion beams, switching high currents at high voltage, and generating intense pulsed microwave radiation. Optical diagnostics are essential in these high-power diodes because of the extreme conditions. Electric fields of 1.0 MV/cm, magnetic fields of 5 kG,current densities of 50 A/cmZ, and plasma densities of 10’’ cm-3 are found in these devices. Figure 15 shows a schematic of their experiment. Many of these parameters evolve on a time scale of tens of nanoseconds. Maron et al. used a spectrograph equipped with a detection system that provided adequate spectral resolution and a temporal resolution of 8 ns. The detection system consisted of an array of seven separate fiber bundles, which transmitted light from the exit focal plane of their spectrograph to individual photomultipliers with oscilloscopes. Maron et al. observed the Stark shift of an emission line near 450 nm from the 4d configuration of doubly ionized Al. Figure 16 is a plot of

-

N

CATHODE

‘INE

N

TO SPECTROMETER

ANODE ( +V 1

agnetic Field

TO Fibsr BUndkS and PM Tubes

peclrometer

FIG.15. (a) Method used to map electric fields in a high power magnetically insulated diode. (b) Schematic of the planar magnetically insulated diode and optical apparatus. Mirror M is translated. [From Maron et al. (1986), Fig. 1, p. 699, by permission.]

J. E. Lawler and D.A. Doughty

196

1

I

1

1

X (cm) FIG.16. Experimental electric field as a function of distance from the anode of a magnetically (8.0-kG) insulated diode with a gap voltage of 360 kV. The field is mapped 35 ns after the beginning of the discharge. [From Maron et al. (1986), Fig. 3, p. 700, by permission.]

some of the electric field measurements by Maron et al. (1987). The direct comparison of measurements by Maron et al. (1987) to detailed numerical simulations has shed considerable light on the operation of these diodes, particularly with respect to plasma propagation speeds and expansions in the diode gap.

IV. Electric Field Mapping Based on the Stark Effect in Molecules Moore et al. (1984) at Bell Labs were the first to use the Stark effect in heteronuclear diatomic molecules, such as BC1, to map electric fields in glow discharge plasmas. Mixtures of BCl, and Ar are used as a feed gas for plasma etching of Si, Al, and 111-V compounds. Copious quantities of the BCl radical are produced in the plasma. The A'IItX'C' (0,O)band of BCl is ideal for electric field measurements using Stark effects. The Bell Labs group headed by R. A. Gottscho subsequently used the diagnostic in extensive studies of capacitively coupled rf glow discharges used for etching semiconductors. ' band. The analysis Figure 17 is a schematic representation of a 'nt' X of the LIF experiment by Moore et al. (1984) is essentially the analysis of the Stark effect in the isolated two-state atomic system described in Section 11. The 'n rotational levels are split in a plus parity and a minus parity level

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

te -f

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te

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t e

J

FIG.17. Schematic energy-level diagram for a 'n-'C+band. Electric dipole-allowed transitions are shown as solid lines and forbidden transitions are shown as dashed lines. [From Moore et al. (1984), Fig. 1, p. 538, by permission.]

for a given value ofj at zero electric field due to A doubling. Transitions in a 'nt'X+band, which are forbidden at zero electric field, are represented by the dashed lines in Fig. 17. Dipole-allowed transitions are represented by

J. E. Lawler and D. A. Doughty

198

solid lines. It is important to note that a Ill rotational level at zero electric field is coupled to 'C+ rotational levels either through R and P branch transitions or through a Q branch transition depending on the parity of the 'll rotational level. Moore et al. found it convenient to excite the R(8) transition with their laser and observe fluorescence of the P( 10) transitions. It was impractical to observe fluorescence of the R(8) transition because of laser light scattering from windows and other surfaces. The Q(9) transition does not fluoresce at zero electric field. The relative intensity of the Q(9) verses the P(10) fluorescence is proportional to the square of the electric field for low values of the field. Figure 18 shows observed spectra for two values of the field. The Q(9) fluorescence is nearly invisible in Fig. 18(a), but

,

I

ANODE

a

-

-

271.8

272.0

272.2

WAVELENGTH (nm) FIG.18. Laser-induced fluorescence spectra from excitation of the R(8) line of the (0,O)band of the BCI A'n-X'Z' system in a parallel-plate rf discharge (50 kHz, 1.2 W/cm2) in 0.15 Torr of BCI,. (a) 1 mm from the powered electrode with V,,, (anode) and (b) same location with Vmin(cathode). [From Moore et al. (1984), Fig. 3, p. 540, by permission.]

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

199

is nearly as strong as the P(10)fluorescence in Fig. 18(b). A detailed analysis of relative Q/P intensity ratios is involved, but relatively straightforward. If we assume the molecules fluoresce before they collide, then we need to specify the laser polarization, laser bandwidth, observation direction and solid angle, and observed polarization. Absolute calibration requires knowledge of the dipole moment of the molecule. If the dipole moment is unknown, then the spatially integrated relative field can be normalized to a voltage measurement. Gottscho and Gaebe (1986) used the electric field diagnostic and an optogalvanic method to probe negative ion densities in capacitively coupled rf discharges. They found that C1- is the dominant negative ion in C1, discharges, and that BCl; is the dominant negative ion in BCl, discharges. They observed quite dramatic effects of the negative ions on the sheath structures including the formation of double layers. Their experiments revealed a strong rf dependence in the negative ion formation and destruction mechanism. Gottscho (1987) further extended these studies by exploring a larger range of radio frequencies and powers and by including a more electropositive feed gas (a dilute mixture of BCI, in Ar). The rf range studied, from dc to 10 MHz, covered the transition from a resistive to highly capacitive discharge. The extensive spatial and temporal field maps produced by Gottscho (1987) have been extraordinarily valuable for providing insight and for direct comparison to rf discharge simulations. Derouard and Sadeghi (1986) chose the NaK molecule for electric field measurements using the Stark effect. The B'lIcX'Z' band system of NaK is accessible to cw dye lasers. Alkali atoms and molecules can be introduced at low concentrations into a glow discharge and their density controlled by an appropiate cold spot. Derouard and Sadeghi demonstrated the potential use of the NaK molecule in a cell experiment. They reported a measurement of the product of the permanent electric dipole moment and radiative lifetime and a measurement of the ratio of the A doubling constant to permanent electric dipole moment. They observed electric fields are as small as 5 V/cm in their cell experiment. Figure 19 shows sample spectra from their experiment. Derouard and Alexander (1986) presented a detailed theoretical analysis of the effects of coherence in pulsed and cw laser-induced fluorescence experiments for measuring electric fields. They used a standard density matrix approach to calculate the quantum beats, which can be observed in the fluorescence under a variety of conditions. Variations in the quantum beats due to differences in laser bandwidths, pulse durations, and laser and detection system polarization were explored in their calculations. Derouard et al. (1989) reported an extensive set of measurements on the B'IIcX'Z=+ band system of NaK. Radiative lifetime and collisional quenching cross-section measurements for the B'lI vibrational levels were

200

J. E. Lawler and D. A. Doughty

100 60 40

20

10

5

OVkm

FIG.19. Laser-induced fluorescence spectra from excitation of the P(5) line of the u' = 5, u" = 0 band of the NaK B'lI-X'Z+ system in a cell with various static electric fields. The laser is polarized perpendicular to the field, and fluorescence polarized parallel to the field is observed at 90" to both the field and laser beams. [From Derouard and Sadeghi (1986), Fig. 1, p. 240, by permission.]

reported. Time-average and time-resolved laser-induced fluorescence measurement of Stark mixing enabled them to extract an accurate value for the permanent electric dipole moment of B'II level of NaK. They also directly observed Stark quantum beats. One of the most impressive experiments on a glow discharge using the Stark effexct in NaK was reported by Debontride et al. (1989). They used a discharge in 98% Ar and 2% K with a trace of Na at a total pressure of 0.4 Torr occurring between plane parallel 3-cm-diameter stainless steel electrodes separated by 3.5 cm. A range of steady-state current densities from 14 to 70 pA/cm* were studied. The steady-state glow discharge was perturbed by illuminating the cathode with a 30-11s duration laser pulse, which caused a dramatic increase in electron emission due to photoelectric effect. Debontride et al. monitored the discharge current, the spatially resolved spontaneous emission, and the spatially resolved field in the cathode fall region of the glow discharge. All three quantities were temporally resolved throughout the 100-ps-long perturbation. The 100-ps recovery time is much longer than the ion transit time through the cathode fall region and yet shorter than the ambipolar decay time of the negative glow. Figure 20 shows the temporal dependence of the current perturbation. Figure 21 is a series of spatial maps of the electric field and optical emission, each at a different time after the perturbing laser pulse. Debontride et al. (1989) used a hybrid model to simulate this experiment. The high-energy electrons produced in the cathode fall region were modeled using a single beam description. This model was based on a nonequilibrium fluid approximation and incorporated particle and energy balance equations. The low-energy electrons in the negative glow were also modeled using a fluid approximation that incorporated particle and momentum balance equations. Their model was self-consistent because it involved

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

20 1

+J

C (u

L L

3

u m 0

a

FIG.20. Photoemission optogalvanic signals and pulse shape of the laser illuminating the cathode as a function of time for three different (unperturbed) discharges. The 14 pA/cmZ (lOO-pA, 154-V), 28 pA/cmZ (200-pA), and 71 pA/cm2 (500-pA, 224-V) discharges occur between plane parallel stainless steel electrodes in 0.4 Ton of Ar with 2%K. [From Debontride et nl. (1989),Fig. 2, p. 5209, by permission.]

solving Poisson’s equation to find the instantaneous electric field in the cathode fall and negative glow regions. Good qualitative agreement was achieved between the simulations and the experiment. This agreement provides convincing evidence that the dominant physical mechanisms in this transient discharge perturbation are understood. Alberta et al. (1993) used the Stark effect in NaK to study a rf discharge in 0.4 Torr Ar with 2% K and a trace of Na. Driving frequencies from 35 kHz to 4 MHz and capacitively coupled voltages from 155 to 410 V were used in this investigation. The rf discharge occurred between asymmetric electrodes separated by 3 cm. The diameter of the larger electrode was 3 cm and that of the smaller electrode was 1.5 cm. Alberta et al. mapped the spatial and temporal dependence of the electric field in the discharge and mapped the direction of the field in their asymmetric rf discharge. The direction of the field was measured by analyzing the polarization of the laser-induced fluorescence from NaK. Figure 22 is a schematic of their experiment; note the rotating birefringent halfwave plate in front of the monochromator. Alberta et al. (1993) observed the extent of “fringing” fields within about 2 mm of the edge of both the small and large electrodes. Their 35-kHz (low-frequency) discharges generated a substantial self-bias, as expected. This asymmetry in the electric field in the two electrode sheaths is shown in Fig. 23. The ion currents to both electrodes were nearly equal and thus the higher average current density on the small electrode requires a larger voltage when the small electrode is the instantaneous cathode. Their 4-MHz

J. E. Lawler and D. A . Doughty

202

x 3570-*--..

0-0

0

'

2

5 ps 4

m

0 2 4 6 8 POSITION (mm)

6 8

FIG.21. Electric field (solid lines) and Ar 419.8-nm emission from the 5p[~]'-4s[$]~ transition (dashed lines) as a function of distance from the cathode for different times after the laser perturbs the electron emission from the cathode. The unperturbed 14 pA/cm2 discharge occurs between plane parallel stainless steel electrodes in 0.4 Torr of Ar with 2%K. Laserinduced fluorescence on NaK is used to map the field. [From Debontride et al. (1989), Fig. 5, p. 521l), by permission.]

(high-frequency) discharge also generated a self-bias. One of the most interesting results from this investigation was that different discharge sustaining mechanisms can occur in different parts of the same rf discharge because of the unequal area electrodes. The larger electric field near the small electrode resulted in the y-mechanism being very significant. The y-mechanism refers to the emission of electrons from the electrode by ion bombardment and the subsequent acceleration of the electrons in the

203

ELECTRIC FIELDS IN GLOW DISCHARGE PLASMAS

0.1pF

---_---___ Monochromator _____ ____ 6

Translation stages

COMPUTER

’

optical transients analyser

FIG.22. Schematic of the experiment for mapping the magnitude and direction of electric fields in a rf discharge between asymmetric electrodes in 0.4 Torr of Ar with 2%K. [From Alberta et al. (1993), Fig. 1, p. 107, by permission.]

sheath. These high-energy electrons are often described as “beam” electrons, although the degree of anisotropy in their velocity distribution is pressure dependent. The higher energy electrons produced a purple-colored discharge near the small electrode. The smaller electric field near the large electrode resulted in the tc-mechanism being very significant. The a-mechanism refers to electron heating by sheath expansion. These heated electrons are often 2

2

small electrode

small electrode

4 - 1 ;k 0

2

4

8

0

2

4

6

position from electrode (mm) 2

electrode

electrode

6T/t6

7T/l6 12T/16

T/18 0

2

4

6

0

2

4

6

FIG.23. Electric field as a function of distance from the electrodes of a 35-kHzrf discharge between asymmetric plane parallel electrodes in 0.4 Torr of Ar with 2% K. The rf voltage amplitude is 410V.Five or six different rf phases are included on each plot. Laser induced fluorescence on NaK is used to map the field. [From Alberta et al. (1993), Fig. 4, p. 110, by permission.]

204

J. E. Lawler and D. A. Doughty

described as “wave-riding” electrons. An orange discharge occurred near the large electrode, which was due to the emission produced by lower energy electrons that excite the alkali resonance levels. The great detail in the experimentalmaps produced by Alberta et al. (1993) has not yet been reproduced in numerical discharge models, but advanced modeling efforts are under way.

V. Conclusion Optical techniques for measuring electric fields in glow discharge plasmas are now highly developed from work during the past decade. Optical emission and laser spectroscopy provide a noninvasive means of studying the electric field structure of gas discharges under a variety of conditions. Laser spectroscopies in particular (laser-inducedfluorescenceand optogalvanic spectroscopy)offer nanosecond time resolution, submillimeter spatial resolution, and precise spectral selectivity. Laser techniques for measuring electric fields were first applied in somewhat idealized research discharge systems. The laser techniques were subsequently improved and refined so that they became useful in less idealized discharge systems. The versatility of these methods can be seen by the wide parameter range in which they have been used. Electric fields from a few volts per centimeter to > lo6 V/cm have been measured in discharges with current densities ranging from pA/cm2 to >A/cm2. Fields have been measured in discharges operating from dc to 13 MHz, as well as in a pulsed mode. Both atomic and molecular species exhibit Stark effects so that virtually any glow discharge can be studied by means of the methods described in this review. In addition to electric field measurements, optical techniques are widely used to measure other important plasma parameters. Relative measurements of the density of crucial species in the discharge, such as metastable atoms and reactive radicals, are performed by a variety of techniques. Optical absorption techniques are particularly useful for making absolute density determinations. Atomic or molecular translational temperatures or nonthermal distributions are measured using Doppler-shifted laser-induced fluorescence, for example. Such measurements are useful in discerning the directionality of ions in the cathode sheath. Rotational and vibrational temperatures or nonthermal distributions for molecular species which emit in the discharge are readily measured by optical emission spectroscopy or by laser techniques for nonradiating species. With such a battery of optical diagnostics it is possible to quantify accurately the important parameters of a particular discharge. Such a data set, especially when combined with a state of the art plasma model, leads to a deep and highly quantitative understanding of the physical and chemical processes in a gas discharge system.

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205

References Alberta, M. P., Debontride, H., Derouard, H., and Sadeghi, N. (1993). J. Phys. 111 France 3, 105- 124. Allario, F., and Wainfan, N. (1969). J. Appl. Phys. 40, 675-681. Barbeau, C., and Jolly, J. (1991). Appl. Phys. Lett. 58, 237-239. Benesch, W., and Li E. (1984). Opt. Lett. 9, 338-340. Bethe, H. A., and Salpeter, E. E. (1977). Quantum Mechanics of One- and 7bo-Electron Atoms. Plenum, New York. Debontride, H., Derouard, J., Edel, P., Romestain, R., Sadeghi, N., and Boeuf, J. P. (1989). Phys. Rev. A 40,5208-5919. Den Hartog, E. A., Doughty, D. A., and Lawler, J. E. (1988). Phys. Rev. A 38,2471-2491. Derouard, J., and Alexander, M. H. (1986). J. Chem. Phys. 85, 134-145. Derouard, J., Debontride, H., Nguyen, T. D., and Sadeghi, N. (1989). J. Chem. Phys. 90, 5936-5943. Derouard, J., and Sadeghi, N. (1986). Opt Commun. 57, 239-243. Doughty, D. A., Den Hartog, E. A., and Lawler, J. E. (1987). Phys. Reo. Lett. 58, 2668-2671. Doughty, D. K., and Lawler, J. E. (1984). Appl. Phys. Lett. 45,611-613. Doughty, D. K., Salih, S., and Lawler, J. E. (1984). Phys. Lett. A 103, 41-44. Edmonds, A. R., Picart, J., Tran Minh, N., and Puller, R. (1979). J. Phys. B: Atom. Molec. Phys. 12,2781-2787. Foster, 3. S. (1924). Phys. Rev. 23, 667-684. Foster, J. S. (1928). Proc. Roy. SOC. A. (London) 117, 137-163. Fuhr, J. R., Martin, G. A., and Wiese, W. L. (1988). J. Phys. Chem. Ref: Data 17, Suppl. 4. Gallagher, T. F., Edelstein, S. A,, and Hill, R. M. (1977). Phys. Rev. A 15, 1945-1951. Ganguly, B. N. (1986). J. Appl. Phys. 60, 571-576. Ganguly, B. N., and Garscadden, A. (1985a). Appl. Phys. Lett. 46, 540-542. Ganguly, B. N., and Garscadden, A. (1985b). Phys. Rev. A 32,2544-2545. Gottscho, R. A. (1987). Phys. Reo. 36, 2233-2242. Gottscho, R. A., and Gaebe, C. E. (1987; IEEE Trans. Plasma Sci. 14,92-102. Helm, H. (1977). J. Phys. B: Atom. Molec. Phys. 10, 3683-3697. Hershkowitz, N., and Cho, M. H., (1988). J. Vac. Sci. Technol. A, Vac. Sut$ Films 6, 2054-2059. Herzberg, G. (1950). Molecular Spectra and Molecular Structure 1. Spectra of Diatomic Molecules. Van Nostrand Reinhold, New York. Hougen, J. T. (1970). The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules. National Bureau of Standards (US.) Monograph 115, US. GPO, Washington, DC. Kushner, M. J., and Graves, D. B. (1991). IEEE Trans. Plasma Sci. 19, 63. Lawler, J. E. (1985). Phys. Rev A 32, 2977-2980. Lee, S. A,, Andersen, L.-U. A,, Rocca, J. J., Marconi, M., and Reesor, N. D. (1987). Appl. Phys. Lett. 51, 409-411. Maron, Y., Coleman, M. D., Hammer, D. A. and Peng, H.4. (1986). Phys. Reo. Lett. 57,699-702. Maron, Y., Coleman, M. D., Hammer, D. A., and Peng, H.-S (1987). Phys. Rev. A S , 2818-2832. Martin, G. A,, Fuhr, J. R., and Wiese, W. L. (1988). J. Phys. Chem. Ref: Data 17, Suppl. 3. Moore, C. A,, Davis, G. P., and Gottscho, R. A. (1984). Phys. Rev. Lett. 52, 538-541. Moore, C. E. (1971). Atomic Energy Levels. National Standards Ref. Data Series National Bureau of Standards (US.) No. 35, U.S. GPO, Washington, DC. Nakajima, T., Uchitomi, N., Adachi, S., Maeda, S., and Hirose, C. (1993). J. Phys. (Paris) C7, 497-54. Proud, J., and Gottscho R. A. (1991). Plasma Processing of Materials: Scientific Opportunities and Technological Challenges. National Academy Press, Washington, DC.

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Proud, J. M., and Luessen L. H. ed. (1986). Radiative Processes in Discharge Plasma. NATO Advanced Study Institute, Series B. Vol. 149, Plenum, New York. Shoemaker, J. R., Ganguly, B. N., and Garscadden, A. (1988). Appl. Phys. Lett. 52,2019-2021. Sinha, S., Lin, S. L., and Bardsley, J. N. (1979). J. Phys. E: Atom. Molec. Phys. 12, 1613-1622. Sugar, J., and Corliss C. (1985). J. Phys. Chem. Re$ Data 14, Suppl. 2. Warren, R. (1955). Phys. Reo. 98, 1650-1664. Wiese, W. L., Smith, M. W., and Glennon B. M. (1966). “Atomic Transition Probabilities VoI. I Hydrogen Through Neon.” Natl. Stand. Ref. Data Ser., Natl. Bur. Stand. (U.S.) Circ. No. 4, U.S. G P O Washington, DC. Wiese, W. L., Smith, M. W., and Miles B. M. (1969). “Atomic Transition Probabilities Vol. I1 Sodium Through Calcium.” Natl. Stand. Ref. Data Ser., National Bureau of Standards (U.S.) No. 22, US.GPO, Washington, DC. Zeller, Ph., Moriya, S., Yu,Z., and Collins, G. J. (1993). “Sheath Electric Field and Impedance of a Cold Cathode Abnormal Glow Discharge,” to be published. Zimmerman, M. L., Littman, M. G., Kash, M. M., Kleppner, D. (1979). Phys. Rev. 20, 2251-2275.

ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 34

POLARIZATION AND ORIENTATION PHENOMENA IN PHOTOIONIZA TION OF MOLECULES N. A. CHEREPKOV State Academy of Aerospace Instrumentation St. Petersburg. Russia 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Spin Polarization of Photoelectrons Ejected from Unoriented Molecules . . .

Ill. IV.

V. VI.

A. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . B. Comparison with Experiment . . . . . . . . . . . . . . . . . . . C. On the Influence of Rotational and Vibrational Motion . . . . . . . . D. Chiral Molecules . . . . . . . . . . . . . . . . . . . . . . . . . Photoionization of Oriented Molecules . . . . . . . . . . . . . . . . . A. Spin Polarization of Photoelectrons . . . . . . . . . . . . . . . . . B. Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . Circular and Linear Dichroism in the Angular Distribution of Photoelectrons A. Qualitative Consideration . . . . . . . . . . . . . . . . . . . . . B. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . C. Experimental Observations . . . . . . . . . . . . . . . . . . . . . Optical Activity of Oriented Molecules . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 209 209 212 216 221 222 222 225 228 228 233 236 243 245 246

I. Introduction The technique of photoionization of molecules is commonly used for investigations of molecular structure and properties. Until recently, however, measurements and calculations were usually restricted to the total and partial photoionization cross sections and the angular asymmetry parameter p. At the same time, great progress occurred relative to investigations of atoms due to the inclusion of consideration of photoelectron spin (Fano, 1969; Cherepkov, 1983; Kessler, 1985; Heinzmann, 1985). Measurements of the angular distribution of atomic photoelectrons with defined spin polarization, combined with the partial photoionization cross-section data, enable one to perform so-called “complete” or “perfect” quantum mechanical experiments and to extract from these data all five theoretical values, namely, three dipole matrix elements and two phase shift differences, which are necessary to describe the photoionization of an atom in the electric207

Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003834-X

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dipole approximation (Heinzmann, 1980; Heckenkamp et al., 1986; Schafers et al., 1990). Measurements of the spin polarization of molecular photoelectrons also began more than 10 years ago, but progress here is less pronounced (Heinzmann, 1985; Lefebvre-Brion et al., 1989). The general theory of the spin polarization of molecular photoelectrons for rotating molecules appeared together with the first measurements (Cherepkov, 1981a), but until now numerical calculations have been scarce (Lefebvre-Brion et al., 1985; Reseev et al., 1987).The first steps are now being made toward investigation of the influence of vibrational and rotational motion on the photoelectron spin polarization, both in experiments (Bowering et al., 1992; Huth-Fehre et al., 1990a,b; Drescher et al., 1993) and in theory (Raseev and Cherepkov, 1990; Buchner et al., 1992). The problem of the complete quantum mechanical experiment in molecules is much more complicated than in atoms due to the fact that the orbital angular momentum in molecules is not a good quantum number. Nevertheless, the partial wave expansion for continuous spectrum wave functions is still used; therefore, the number of partial waves contributing to the process is, in principle, infinite, as is the number of allowed dipole transitions. In practice, the partial wave summation can be restricted to only a few terms, but at any rate the number of theoretical parameters will be larger than five. On the other hand, from the measurements of the angular distribution and spin polarization of molecular photoelectrons one can extract, as in atoms, only 5 parameters (up to 10 parameters in the case of chiral molecules). More than five parameters can be extracted from investigation of either molecules fixed in space (for example, adsorbed at a surface or in a liquid crystal), or aligned and oriented molecules, such as molecules in a gas phase excited by polarized light and having nonstatistical distribution over the states with different projections of the total angular momentum M,. In the following, both fixed-in-space and aligned or oriented molecules are referred to as oriented for brevity. The angular distribution of photoelectrons ejected from oriented molecules, even without spin analysis, in principle, is characterized by an infinite number of parameters (Dill, 1976) and therefore can give more information than the investigation of the spin polarization of photoelectrons ejected from unoriented molecules. Since this angular distribution is quite complicated, it is worthwhile to simplify it by considering the difference between photoelectron currents ejected at each angle either by left and right circularly polarized light, or by linearly polarized light of two mutually perpendicular polarizations. The values obtained in this way (after Ritchie, 1975), are called the circular and linear dichroism in the angular distribution of photoelectrons (CDAD and LDAD), respectively. In the very first “complete”experiment for aligned NO molecules in the excited A2C+ state (Reid et al., 1992; Leahy et al., 1992),

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209

the CDAD and the angular distribution measurements were used to extract 12 theoretical quantities characterizing the photoionization process in some approximation. These quantities were compared with corresponding theoretical values. Measurements of CDAD and LDAD can be used not only as a sensitive test of existing theories, but also, for example, as a method to determine the molecular orientation. Finally, oriented molecules reveal the optical activity, that is, they rotate the polarization plane and exhibit circular and linear dichroism (Cherepkov and Kuznetsov, 1991b). All of these effects, discovered and investigated mainly during the last decade, are reviewed in this article. They already appear in the electric dipole approximation; therefore, higher multipoles are not discussed here. Our consideration is restricted to one-photon processes with the ejection of only one electron.

11. Spin Polarization of Photoelectrons Ejected from Unoriented Molecules A. GENERAL THEORY If molecules are unoriented, the angular distribution of photoelectrons is defined by two vectors. The first of them, the polarization vector e in the case of linearly polarized light, or the unit vector q in the direction of the photon beam in the cases of circularly polarized and unpolarized light, describes the photon beam. The other one is the unit vector K, which defines the direction of the photoelectron ejection, K = p/p where p is the photoelectron momentum. The angular distribution depends on one angle between these two vectors [see Eq. (22)], and, as in atoms, is described by one parameter, the angular asymmetry parameter B (Tully et al., 1968; Buckingham et al., 1970). To obtain more information on the molecular structure, one can also measure the spin polarization of photoelectrons ejected at a definite angle. Then the process is characterized by three vectors, the third vector being the unit vector in the direction of the photoelectron spin s. If molecules are unoriented, the angular distribution of the photoelectrons with deJned spin polarization (ADSP) again has the same form as in atoms and is characterized by three polarization parameters (Cherepkov, 198la):

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where ai(w) is the partial photoionization cross section of the i subshell, w is the photon energy, P , is the second Legendre polynomial, and m characterizes the light polarization. For circularly polarized light m = f 1, for linearly polarized light m = 0 and the vector q must be replaced by the polarization vector e. For unpolarized light m = 1 and linear terms in rn must be omitted. The polarization parameters A', y', v]', like the angular asymmetry parameter fi. are expressed through the dipole matrix elements and phase shifts, and contain the dynamic information on the process. All dependence on the angles (the geometrical part) is explicitly given in Eq. (1) and is independent of the photon energy or molecular structure. For chiral molecules ADSP contains five additional terms, which are discussed later. The degree of spin polarization of photoelectrons ejected at a definite angle, P(u, s,q), is defined as usual (Blum, 1981):

The term proportional to A' in Eq. (1) is the only one that survives after integration over electron ejection angles. It gives the degree of spin polarization of the total electron flux. This polarization is different from zero for circularly polarized light only and is parallel to the direction of the photon beam. The next term proportional to y' also contributes for circularly polarized light only. It describes the spin polarization in the reaction plane, defined by vectors q and K (see Fig. 1). The component of the spin polarization in the reaction plane perpendicular to q is completely defined by the parameter y'. The last term in Eq. (l), proportional to the parameter v]', gives the contribution for any polarized or unpoiarized light. It describes the spin polarization perpendicular to the reaction plane (the latter for linearly polarized light is defined by the vectors e and K). General expressions for the spin-polarization parameters A', yi, and v]' were first derived by Cherepkov (198la) for linear molecules in the Hund's case (b), and later for arbitrary polyatomic molecules (Cherepkov, 1983) by means of the partial wave expansion for the continuous spectrum wave functions. Chandra (1989b) proposed the incorporation of the symmetry properties characteristic for nonlinear molecules by using the expansion over generalized harmonics introduced earlier (Van der Lage and Bethe, 1947; Burke et al., 1972). The relation between these two formulations is discussed by Cherepkov and Kuznetsov (1991a). Chandra (1989b) and Raseev and Cherepkov (1990) have also derived the expressions for the spin-polarization parameters using the angular momentum transfer formulation introduced by Buckingham et al. (1970) and Fano and Dill (1972). All of the alternative formulations mentioned previously are connected to each other by algebraic transformations and therefore are equivalent. As an example, we present the spin-polarization parameter A for the particular case of the ionization of a x subshell of a diatomic molecule with

POLARIZATION AND ORIENTATION PHENOMENA

21 1

FIG.1. Definitions of vectors in photoionization by circularly polarized and unpolarized light. 2113/2 final ionic state. Retaining in the partial wave expansion of the continuous spectrum wave functions only terms with I = 0, 1, and 2, and assuming that the Hund‘s case (b) is applied, we find (Cherepkov, 1981a):

1 A3’2 = -(ld2212 2B

- I d O O l Z - 1 4 0 1 2 - ld2oI2)

(3)

where

B = Id,o12

+ Idl0l2 + ldl,12 +

ld2012

+ ld2,I2 + ldz212

(4)

and d,, denotes the dipole matrix element, with I and M the orbital angular momentum of a photoelectron and its projection on the molecular axis, respectively. The spin polarization of molecular photoelectrons in the electric-dipole approximation appears to be a result of spin-orbit (or the spin-axis) interaction, which manifests itself in the multiple splitting of molecular levels, and in the difference between continuous spectrum wave functions corresponding to different projections of the electron spin on the molecular axis. Although the spin-orbit interaction is always small, of the order of ( Z Z ) where ~ c1 is the fine-structure constant and Z is the nucleus charge, there is an essential difference between these two effects. If the fine-structure

212

N. A. Cherepkov

splitting is resolved, that is, the photoelectrons corresponding to different fine-structure components of the molecular ion state are separated, the degree of spin polarization of photoelectrons is of the order of unity. In this case the smallness of the order of (az)’ manifests itself in the magnitude of the fine-structure splitting, whereas the spin polarization appears to be due to the dipole selection rules and therefore does not contain any small parameter. If the fine-structure splitting is not resolved, the spin polarization appears to be a result of the direct influence of the spin-orbit interaction on the molecular wave functions, and it is small-of the order of (az)’. These conclusions follow immediately from the fact that if one neglects the spin-orbit interaction, the polarization parameters A’, yi, and q’ for two fine-structure components appear to have opposite signs and magnitudes that are inversely proportional to the statistical weights of these states, so that after averaging over these states the net results, as in atoms (Cherepkov, 1983), is exactly equal to zero. The possible nonzero values of the degree of polarization are connected with the slightly different magnitudes of the dipole matrix elements corresponding to different fine-structure components. In general, the spin polarization of molecular photoelectrons is expected to be lower than in atoms due to the averaging over all possible orientations of the molecular axis with respect to the photon polarization, and due to a greater number of allowed transitions in molecules. For linear molecules in Hund’s case (b) it was shown (Cherepkov, 1981a) that parameter A’ is restricted by the condition IA’I < 0.5, so that the greatest attainable degree of polarization of the total photoelectron flux from linear molecules could not exceed 50%, whereas in the case of atoms it can reach 100%. But there is at least one exception from this rule when photoelectrons had to be described in Hund’s case (e), which is discussed in the Section 1I.C.

B. COMPARISON WITH EXPERIMENT The first numerical calculations of the spin-polarization parameters have been performed for HI molecules in the region between the ’IIllzand ’113/’ ionization thresholds where the spin-orbit autoionization takes place (Lefebvre-Brion et al., 1985). Calculations in an extended photon energy region for the znp outer shell and d inner shell photoionization of HBr and HI molecules were published later. (Raseev et al., 1987). These authors used the self-consistent field approximation for the ground state of the molecule, and the frozen core static exchange approximation for the continuous spectrum wave functions. The spin-orbit autoionization has been studied by the multichannel quantum defect method. Figure 2 shows the results of calculations for the outer znp subshell of

POLARIZATION AND ORIENTATION PHENOMENA

0

20

40

60

0

20

213

40

PHOTOELECTRON ENERGY

FIG.2. (a). Angular asymmetry parameter j and spin-polarization parameters A, 7, and 5 = q/2 as a function of photoelectron energy for X2lTIi2(n4p-') of HBr. Experimental points for j are from Carlson et nl. (1984b).(b). The same for HI(n5p-I). [From Raseev et nl. (1987).]

HBr and HI molecules (Raseev et al., 1987). Since these molecules do not differ much from the corresponding isoelectronic rare gas atoms Kr and Xe, it is natural to compare the parameters presented in Fig. 2 with the analogous results for the outer np subshells of rare gas atoms (Huang et al., 1981;Cherepkov, 1983). This comparison shows that the angular asymmetry parameter /? in molecules has approximately the same values and the same energy dependence as in the corresponding isoelectronic atoms, whereas the polarization parameters, having the same energy dependence, are approximately two times lower in magnitude compared to the corresponding atomic values for the ,PI,, final ionic state. It supports the general conclusion that the spin polarization of molecular photoelectrons should be lower as compared to that of atomic photoelectrons. On the other side, the atomic-like behavior of the parameters shown in Fig. 2 is explained by the fact that the znp molecular orbital is a nonbonding one. In this respect it would be interesting to perform calculations of polarization parameters for molecules where the molecular character of orbitals is pronounced and where, for example, the molecular shape resonances occur. The experimental investigations of the spin polarization of molecular photoelectrons have already been performed for many molecules. Heinzmann et al. (1980) measured the spin polarization of the total photoelectron flux for CO, and N,O molecules near the ionization threshold in the spinorbit autoionization region using circularly polarized synchrotron radiation.

N. A. Cherepkov

214

The polarization parameter q i has been determined at several photon energies for Br2, I,, CH3Br, and CH31 with unpolarized light (Schonhense et al., 1984). The most comprehensive measurements up to now have been performed for HI (Bowering et al., 1991, 1992) and HBr (Lefebvre-Brion et al., 1989). Figures 3 and 4 show the partial photoionization cross section and the polarization parameters A , y, and ( = q / 2 measured by Bowering et al. (1991) for two spin-orbit components, 2111/2and 2113/2, of HI. All three parameters have opposite signs for the two spin-orbit components and approximately the same magnitude (except for the parameter t), in accord with theoretical predictions (Cherepkov, 1981a). Indeed, the zl13,2and 2r11,2states have equal statistical weights; therefore, the corresponding polarization parameters should have the same magnitude and opposite signs at equal photoelectrons energies, provided the spin-orbit interaction is neglected. Deviations from this equality are connected with the influence of the spin-orbit interaction on both initial and final-state wave functions. We should mention that rather large deviations from this equality for the parameter have also been observed in other molecules (Schonhense et al.,

BOB

.

I

1

I

I

1

90

80

70

60

80 60 n

P

z4.40

v

b

20 0 120

110

100

FIG.3. Partial photoionization cross section for the final ionic states HI' 2113,2(u = 0) and

2n,,2(v = 0) at a spectral resolution of 0.17 nm, normalized to the absolute data of Carlson et

al. (1984a), shown by solid circles. The relevant ionization limits are indicated by vertical lines. [From Bowering et al. (1991).]

215

POLARIZATION AND ORIENTATION PHENOMENA

120

110

100

90

a0

70

60

A(nm) FIG.4. Wavelength dependence for the spin-polarization parameters A, 5 = q/2, and y tor HI' 2113,2( u = 0) (solid circles) and (u = 0) (open circles) final ionic states (spectral resolution of 0.5nm). The curves represent the nonrelativistic ab initio calculations of Raseev et al. (1987). [From Bowering et al. (1991).]

1984). In general, reasonable agreement exists with the nonrelativistic calculations of Rasev et al., (1987). In the region between the 2111,2 and 'E+ limits, the electronic autoionization resonances of Rydberg series converge to the HI' A 2 V ionic state; therefore, parameter A was measured here with a much higher point density. In this region a strong differenceexists between the theoretical predictions and the experiment because the autoionization resonances were not taken into account in calculations. The data for the spin polarization parameters can be used to estimate the strength ofdifferent channels.Since parameter A of Eq. (3) contains only squares of dipolematrix elements,it can be expressedthrough the partial cross sectionsC T ~ : A

=

1 2

5 -(aa - CT,)/O

(5)

where CT = 0, + CT, + oa is the total cross section, the plus sign refers to the 2113jtfinal state, and the minus sign to the 2111,2 one (Cherepkov, 1981a;

N, A. Cherepkov

216

Raseev et al., 1987). The limiting values A = f0.5 occur when only one partial cross section is different from zero, either CJ, or oh,and A = 0 when CJ, = oh.The observed positive values of A for the 2113,2 state show that the ED contributions are larger than the E 6 contributions. From the two measured quantities, CJ and A, one could not extract three partial cross sections oU,on,and oh,but one can define two linear combinations, which are given here for the 2113/2final ionic state (Bowering et al., 1992):

a(6, T I ) = oh+ 0,/2 = (0.5 + A ) .CJ a(o,n) = ou + an/2= (0.5 - A ) . o

(6)

In Fig. 5 these linear combinations are shown for the HIf2113/2(u = 0) final ionic state in the region of electronic autoionization. The resonance is clearly seen in only one of these linear combinations, which means that there is a strong enhancement of the oh partial contribution, or in other words, that the corresponding resonance decays mainly into the E& continuum.

c. O N THE INFLUENCE OF ROTATIONALAND VIBRATIONAL MOTION It is rather evident that vibrational motion cannot directly influence the spin polarization of photoelectrons. Therefore, although the expressions for the spin-polarization parameters have been obtained in the general form, which allows us to include the vibrational excitations, they have not been considered explicitly in the numerical calculations reported up to now.

E

W

b"

10

11 12 13 EPhoton (ev>

FIG.5. Energy dependence for sums of partial contributions u1 to the cross section for HI' X21'I,,, (u = 0) final ionic states (open circles, us + u,Jz solid circles, ue + uJ2). [From Bowering et al. (1992).]

POLARIZATION AND ORIENTATION PHENOMENA

217

On the other hand, in experiments different vibrational excitations have been resolved and studied. Bowering et al. (1992) measured vibrationally resolved photoelectron spectra for H I + X 2 n (v = 0, 1, 2, and 3) final ionic states and defined the partial cross sections and the spin-polarization parameters in the region of electronic autoionization between the 2111/2 and 'Z' limits. They found that for a given photon energy the results for different vibrational excitations differ substantially. This is connected mainly with the fact that the same electronic resonances appear at different photon energies depending on the vibrational excitation of the final state. Therefore it is necessary to resolve the vibrational structure when the spin polarization measurements are performed in the region of autoionization resonances. More detailed investigations of electronic autoionization, both theoretical and experimental, have been performed by Lefebvre-Brion et al. (1989) for the HBr molecule between the and 'C+ limits (see Fig. 6), where well-defined vibrational progressions of the 5pn and 4dn states have been observed in the cross section. The measured values of the polarization parameter A in this region reveal the rapid variation, although the experimental points are not sufficiently dense to outline completely the wavelength dependence of A in full detail. The theoretical investigation shows that A has an oscillatory behavior in this region, which, in principle, agrees with the experimental findings (as shown in Fig. 6), but a detailed comparison between theory and experiment at this stage is impossible. In the region of the spin-orbit autoionization between the 21-In3/2and 2111/2limits of HI where the resonances are rather narrow, even the rotational structure can be resolved. Such good resolution can now be obtained with the new VUV laser light sources based on the nonlinear frequency-mixing technique (Huth-Fehre et al., 1990a). The resonances in this region are connected with the Rydberg series converging to the 21-1112 (v = 0) threshold and autoionizing into the 2113/2(u = 0) continuum via spin-orbit interaction. Since only one spin-orbit channel is opened here, one need not analyze photoelectrons in energy in order to observe the spin polarization of photoelectrons. So, in the first experiments with the laser VUV source, the photoionization cross section (Huth et al., 1988), the spin polarization of the total photoelectron yield (Huth-Fehre et al., 1990a,b), and the angular asymmetry parameter fi (Mank et al., 1990) were measured. The laser source delivered r 10" photons/s with a bandwidth of 0.4 cm-' in the range from 83,500to 88,500cm-', with a repetition rate of 11 Hz and a pulse length of 5ns. The rotational structure was clearly visible, but all attempts to assign rotational progressions gave ambiguous and inconclusive results (Huth et al., 1988). On the other hand, the observed spectrum strongly depends on the temperature of the molecular beam (Hart and Hepburn, 1989). Therefore, subsequent measurements have been performed on a rotationally cooled sample with the temperature T,,, 13 K, at which only two ground state rotational levels are considerably populated. Careful

-

218

N. A. Cherepkov

J

90

88

Wavelength 0.2

9L

92

I

’

i

.

96

(nml “

’

I

”

’

’

b

AA-0.L nm

0 A

2

es

90

.

.

.

i

95 Wavelength (nml

.

.

.

.

J

?a

FIG.6. (a) Calculated photoionization cross section (top) and the angular asymmetry parameter fi (bottom) for the X217 (v = 0) final ionic state of HBr. Vibrational progressions of autoionizing resonances are indicated. (b) Calculated spin-polarization parameter A, convoluted to the experimental resolution I = 0.4nm, in comparison with the experimental data for Znr,2 (solid circles) and 2113,z(open circles, multiplied by -1) final ionic states. [From Lefebvre-Brion et al. (1989).]

analysis of the spectra observed at low temperatures by Mank et al. (1991) shows that there is a transition from Hund’s coupling case (c) to case (e) with an increase in the principal quantum number n of the Rydberg states. Mank et al. also showed that the introduction of rotational autoionization and perturbation in the multichannel quantum defect theory treatment is essential in explaining the difference in shape between the successive Rydberg levels, which is typical for molecules.

POLARIZATION AND ORIENTATION PHENOMENA

219

Even at a low rotational temperature, however, the observed spectra are too complicated and could not be identified unambiguously at the moment. As an example, Fig. 7 shows the observed photoelectron yield and the polarization parameter A in the region of ( 5 p ~ ) ~ n IRydberg A states with n = 6 (Huth-Fehre et al., 1990b). The results of the ab initio calculation (Lefebvre-Brion et al., 1985), which neglects any rotational effects are also presented. Figure 8(a) shows an expanded view of the central part of Fig. 7(b) for spin-polarization parameter A. Here the rotational structure is

85500

86000

86500

WAVENUMBER (cm-’1 FIG.7. (a) Measured photoelectron yield from a jet-cooled HI sample. (b) The corresponding electron spin-polarization parameter A. All experimental data are plotted as vertical error bars, indicating their experimental uncertainties. The result of ab inifio calculation (Lefebvre-Brion et al. (1985) of A is shown as a dotted line. [From Huth-Fehre et al. (1990b).]

N. A. Cherepkov

220

0.2

I

I

I

z

ow k 9 a

d e O z E u,

-0.1

66100

86000

WAVE NUMBER (crn-l) 0.41

b

A

0.0

-0.4

86115

86165 86215 Pholon Energy(cm-1)

86265

FIG.8. (a) An expanded view of the central part of Fig. 7(b). [From Huth-Fehre et 01. (1990b).] (b) Calculated spin-polarization parameter A with rotation taken into account. [From Biichner et al. (1992).]

clearly seen, but due to an overlap between rotational branches of different electronic states the spectrum remains unidentified.

POLARIZATION AND ORlENTATION PHENOMENA

22 1

Theoreticalinvestigationsof the polarization phenomena when the rotational structure is resolved have also begun. Raseev and Cherepkov (1990) derived the expressions for the spin-polarization parameters for the transition between a definite initial rotational level of the neutral molecule and a definite final rotational level of the ion. The first ab initio calculation using this formalism was performed by Buchner et al. (1992). Figure 8(b) shows spin-polarization parameter A calculated by them in the photon energy region that corresponds to the experimentalresults presented in Fig. 8(a).The theoreticalcurve was obtained by averagingover initial rotational levels at the temperature of the experiment, and was convoluted with the experimental resolution. From the comparison between Figs. 8(a) and 8(b) we see that there is no direct correspondence between theory and experiment.Therefore,further theoretical investigationsare required. Recently, new data appeared for the situations where Hund's case (e) is applied (Herzberg, 1950), for example, in the region of autoionization resonances in hydrogen halides between the 2113,2and thresholds (Lefebvre-Brion, 1990; Haber et al., 1991; Mank et al., 1991). For rather high principal quantum numbers n of the Rydberg state, the electron velocity becomes so small, and its radial distance from the nuclei so large, that it cannot follow the molecular rotation and becomes uncoupled from the molecular axis. Then this Rydberg electron, as well as the photoelectron after the autoionization of this Rydberg state, is characterized by the quantum numbers 1 and j = 1 _+ 1/2, as in atoms. Equations for the spin- polarization parameters in this coupling case have been derived by Raseev and Cherepkov (1990) [the resonanceform, Eqs. (lo), (17)-(19)]. The analysis of the equation for parameter A shows that in Hund's case (e) the degree of spin polarization can exceed 50%, as was observed experimentally by Drescher et al. (1993),and for a given initial rotational state it strongly depends on the rotational state of the residual ion. These conclusions are very important for the identification of autoionization resonances.

D. CHIRAL MOLECULES If molecules are chiral, that is, if they have neither center of inversion nor reflection planes and rotation-reflection axes, the expression for ADSP becomes more complicated and contains, in addition to the terms given by Eq. (l), five new terms (Cherepkov, 1983):

222

N. A. Cherepkov

with the same notation as in Eq. (1). The first term here gives the CDAD predicted originally by Ritchie (1976). The second term gives the component of the transverse polarization of photoelectrons, which has an opposite sign for right and left circularly polarized light. The difference between the transverse spin polarizations given by the last term in Eq. (1) and by the second term in Eq. (7) for several electron ejection angles is demonstrated in Fig. 9. The last three terms in Eq. (7) characterize the longitudinal polarization of photoelectrons, which exists for any polarized or unpolarized light, whereas in the case of achiral molecules the longitudinal polarization of photoelectrons appears for circularly polarized light only. Thus, ADSP for chiral molecules is defined by 10 independent parameters. All terms in Eq. (7) already appear in the electric-dipole approximation, whereas the well-known phenomena of optical rotation and circular dichroism in the photoabsorption by chiral molecules are caused by the electric dipole-magnetic dipole interference terms, which are a times smaller. CDAD given by the first term in Eq. (7) is not connected with the spin-orbit interaction, and to observe it one need not resolve the fine-structure splitting. Four other terms in Eq. (7) describe the spin polarization of photoelectrons, and therefore one has to resolve the fine-structure splitting for their observation. All five parameters in Eq. (7) are proportional to differences of pairs of dipole matrix elements having opposite signs for all projections of orbital momenta and spins. For achiral molecules these differences are identically equal to zero, whereas for chiral molecules they are different from zero. A realistic estimation of these differences without calculations is hardly possible. They should be proportional to an asymmetry factor q discussed by Rich et al., (1982), which depends on the degree of dissymmetry in the structure of the molecule and can be of the order of lo-’ (Campbell and Farago, 1985; Blum et al., 1990). Unfortunately, until now there have been no measurements of the spin polarization of photoelectrons ejected from chiral molecules.

111. Photoionization of Oriented Molecules A. SPINPOLARIZATION OF PHOTOELECTRONS Situations can occur in which the initial state of a molecule does not possess spherical symmetry and can be characterized by at least one vector. This is the case when rotating molecules are aligned or oriented, for example, through excitation by linearly or circularly polarized light, respectively, which produces a nonequal population of states with different projections M on a given direction n of the total angular momentum J, where J # 0.

223

POLARIZATION AND ORIENTATION PHENOMENA

t'

a

+

-L

s

5

FIG.9. Schematic presentation of the transverse spin polarization of photoelectrons ejected by circularly polarized light from (a) achiral and (b) chiral molecules.

Molecules are said to be aligned if states with projections M and - M are equally populated, and oriented otherwise (Blum, 1981). The alternative possibility is to freeze the molecular rotation. For example, molecules adsorbed on a surface, or molecules in liquid crystals, are not rotating. Molecules can be oriented by an external field (Kaesdorf et al., 1985; Parker and Bernstein, 1989). It is also possible to select the photoionization processes of fixed-in-space molecules in a gas phase by detecting in coincidence the fragment ions and photoelectrons, provided the molecular ion decays rather fast after ionization. Fixed-in-space molecules, depending on their structure, can be characterized by one, two, or even three vectors. In the following both fixed-in-space and aligned or oriented molecules will be referred to as oriented for brevity. In many respects fixed-in-space and aligned or oriented molecules behave identically, that is, geometrical parts describing particular processes are the same for these two cases, whereas the dynamic parts (the corresponding parameters) are different. The photoionization process for oriented molecules is described naturally in the molecular frame defined by the direction of molecular axis for fixed-in-space molecules or by the direction of molecular alignment or orientation otherwise. Then ADSP depends on three vectors and can be presented as an expansion in spherical functions (Cherepkov and Kuznetsov, 1987a, b):

*

A?MrxS.

'

YTM~

(a)

'

&($)

'

%&$)

(8)

where l,, is the highest orbital angular momentum retained in the partial wave expansion of the continuous spectrum wave function, and the vector q in the case of linearly polarized light (m = 0) should be replaced by the polarization vector e. The coefficients of this expansion, Ak,"M,xS., contain the dynamic information on the process and are expressed through the dipole matrix elements and phase shifts. The corresponding equations are given by Cherepkov and Kuznetsov (1987a,b) using the partial wave expansion for

224

N. A. Cherepkov

the continuous spectrum wave functions, and by Chandra (1989b, 1991) using the expansion over the generalized harmonics (see also Cherepkov and Kuznetsov, 1991a). The coefficients are normalized by the condition Ag!oo = 1. Consider now some general conclusions, which can be drawn from Eq. (8) in the simplest case of fixed-in-space linear molecules. In Hund's cases (a) and (b), which in the absence of rotation coincide, projections of the orbital angular momentum and spin on the molecular axis are conserved separately. Therefore, the photoelectron spin s is always directed parallel to the molecular axis irrespective of the direction of the photon beam, its polarization, or the direction of photoelectron ejection. Formally this statement follows from the dipole selection rules, which lead to the condition 5 = 0 in Eq. (8) (Cherepkov and Kuznetsov, 1987a), and as a result, the photoelectron spin s can appear in ADSP only through scalar product (sn), where n is the unit vector in the direction of the molecular axis. This conclusion is valid also for the total photoionization cross section obtained from Eq. (8) by integration over electron ejection angles

from which we can find the degree of polarization of the total photoelectron yield

The parameter &goo here plays the same role as the angular asymmetry parameter p does in the angular distribution of photoelectrons, while the parameter which defines the degree of polarization, is proportional to the parameter A' introduced in Eq. (l), = -$-Ai. The parameter as well as A', is expressed through only square moduli of the dipole matrix elements. Therefore, experimental determination of the parameters A' and &goo can be used as a method for extracting the contributions of og, o,, and og.All other parameters also contain the contribution of interference terms between different continua multiplied by sine or cosine of the corresponding phase shift difference. This makes the procedure of extraction of theoretical quantities from the measured data much more difficult. In the approximation used in Eqs. (3) and (4), the parameter &goo is

Note that although in the case of unoriented molecules belonging to Hund's case (a) all three components of the photoelectron polarization vector are

POLARIZATION AND ORIENTATION PHENOMENA

225

different from zero [see Eq. (l)], for fixed-in-space molecules only one component survives that is directed along the molecular axis. But the degree of spin polarization for fixed-in-space molecules can reach 100% (Cherepkov, 1981b), whereas for unoriented molecules in Hund's cases (a) and (b) it could not exceed 50%. Both of these facts are easily explained by the rotation of molecules, which leads, on one hand, to averaging over all orientations of the spin quantization axis in the laboratory frame connected with the photon beam, thus enabling all three components of spin in this frame to be different from zero. On the other hand, this averaging makes the degree of polarization for unoriented molecules lower. In Hund's case (c) the projection of spin onto the molecular axis is not conserved; therefore, all three components of the spin-polarization vector are different from zero. This case is discussed in more detail by Cherepkov and Kuznetsov (1987a).

B. ANGULARDISTRIBUTIONS Although investigations of ADSP can give the most complete information, up to now nobody measured the spin polarization of photoelectrons ejected from oriented molecules because of the complexity of such experiments. The spin-polarization measurements are connected with the loss of at least three orders of magnitude in intensity (Kessler, 1985). To make the experiment simpler, one has to exclude the spin-polarization measurement and to investigate the angular distribution of photoelectrons. The corresponding analytical expression can be obtained by summation over spin projections in Eq. (8):

where AyMToo. Such an expression was obtained for the first time by Dill (1976) and analyzed for the case of fixed-in-space molecules by Dill et al. (1976) and Davenport (1976). The number of terms in Eq. (12), in principle, is still infinite so that the measurements of Eq. (12) give quite exhaustive information on molecules. In practice, these measurements can be restricted by the difficulty of extracting many parameters from one measured curve. Therefore, it is worthwhile to reduce the number of terms describing the process. This can be done by considering a particular geometry when many terms disappear, for example, by setting two of three vectors n, K, and q to be parallel or perpendicular, or by investigating the difference between photo-

226

N. A. Cherepkov

electron currents ejected at a definite angle by the light of two orthogonal polarizations. The first of these two possibilities was realized by Golovin et al. (1990,1992) who investigated 0, molecules in a gas phase by coincidence technique. In this experiment the molecular oxygen was ionized by unpolarized He1 resonance radiation, and photoelectrons were analyzed in energy in order to select only those that originate from the following reaction:

0, + hv(21.2eV) .+ O:(B2C;)

+ e-

The ions Oi(B2C;) are unstable and completely decay, mainly into the channel O(3P) O'(4S). An estimation shows that the decay time is short enough compared to the period of rotation at room temperature. Therefore, the direction of the atomic ion motion coincides with the molecular orientation at the moment of absorption of a photon. In the experiment the atomic ions were detected in coincidence with the photoelectrons. Axes of the photoelectron energy analyzer and of the mass analyzer were perpendicular to the direction of the light beam, and the angle between the axes was varied from 45" to 135". So, of three vectors characterizing the process, n, q, and K, two vectors, n and q, were fixed by the condition n Iq, while the third vector K was varied in the plane perpendicular to the vector q. If we retain in the expansion of the photoelectron wave function in partial waves only the principal term with 1 = 1, the sum over T in Eq. (12) will be restricted by two terms with T = 0 and 2. Then for the geometry of the experiment the general Eq. (12) is greatly simplified:

+

where cp is the angle between K and n, C and B are the parameters connected with the parameters and oc and cr, are the cross sections corresponding to the pa and pn channels, respectively. The results of measurements are presented in Fig. 10, where the vertical bars show the statistical error with a confidence coefficient of 0.68. The solid line is the approximation of the experimental data by Eq. (13) with B = 0.145 0.059. From the expression for the parameter B we can extract the ratio of cross sections for this photon energy, oa/on = 0.67 k 0.08. In the simplest geometry used by Golovin et al. (1990) and with the restriction T = 0 and 2, it was possible to extract only one parameter. Changing the geometry of the experiment, one can extract up to four parameters retaining the condition T = 0 and 2. Improving the statistics, it would be possible to extract the contributions of terms with T > 2, which give the next terms in the angular dependence. An advantage of this method is the fact that molecules under investigation are free and not perturbed by any external field, contrary to the case of molecules adsorbed on a surface.

221

POLARIZATION AND ORIENTATION PHENOMENA

I"" 4.2

t

T

1.4

1.0

0.9 0.8 I

60"

f20°

I

1

240'

180'

I

300'

/

'p

FIG.10. Dependence of the coincidence signal on the angle cp between directions of the atomic ion beam and the photoelectron beam (see text for details). Solid curve: approximation of the experimental data by Eq. (13) with B = 0.145. [From Golovin et al. (1990).]

On the other hand, the use of the coincidence technique is time consuming. Similar ideas have been used by Yagishita et al. (1992) who measured the total K shell photoionization cross sections of linear molecules for the cases when the polarization vector e of linearly polarized radiation was either parallel or perpendicular to the molecular axis. The direction of the molecular axis was also defined by detection of atomic ions after dissociation of the molecule. The corresponding analytical expression can be easily obtained by integration of Eq. (12) over the electron ejection angles. It has the form of the usual angular distribution [see, e.g., Eq. (22)] where 8 is the angle between the vectors e and n, and /? = -$A;:. For the case of K shell ionization the parameter fl is (Dehmer and Dill, 1978)

Substituting it into Eq. (22), we find that for e (1 n and e In only pa and pn channels contribute, respectively,

f,,=-aa; 3 471

I --.-3

an

l-411

2

In this way o, and 6, partial cross sections as a function of photon energy have been measured directly by Shigemasa et al. (1992) for the K shell of

N. A. Cherepkov

228

h

In

U

.F!

c

7.

n

L

m

B

H

t U

w

3 0 U V

410

415

420

425 430 435 PHOTON ENERGY (eV)

440

445

FIG.11. Symmetry resolved u,, (dots) and u, (solid line) photoabsorption spectra above the K shell threshold in N,. The dashed curves represent the theoretical calculations by Dehmer and Dill (1976) with the n-component normalized to the experimental 6, at 445eV. The dot-dashed curves represent the theoretical calculations by Rescigno and Langhoff (1977) with the n-component normalized to the experimental u, at 425eV. [From Shigemasa et al. (1992).]

N, molecules, by Kosugi et al. (1992) for the K shell ionization of 0, molecules, by Yagishita et al. (1992) for the N and O K shells of NO molecules, and by Shigemasa et al. (1993) for the C and O K shells of CO molecules. As an example, Fig. 11 shows the experimental results for N, molecules in comparison with calculations. Although the calculations give a qualitative description of the experiment, there are considerable quantitative differences. This example demonstrates the efficiency of experiments with oriented molecules for the most severe test of theoretical methods.

IV. Circular and Linear Dichroism in the Angular Distribution of Photoelectrons A. QUALITATIVE CONSIDERATION Consider now the other way to simplify Eq. (12) by defining the difference between photoelectron currents ejected at a definite angle by left and right circularly polarized light, or by linearly polarized light of two orthogonal polarizations (called CDAD and LDAD, respectively). CDAD for fixed-inspace molecules in the electric dipole approximation was predicted at first

POLARIZATION AND ORIENTATION PHENOMENA

229

by Cherepkov (1982), and was shown to exist also for the case of aligned molecules by Dubs et al. (1986). LDAD was considered at first by Cherepkov and Schonhense (1993). We start with the qualitative description of the origin of CDAD following Cherepkov (1992) [see also Schonhense (1990)l. Suppose that the light beam is propagating along the x axis of the molecular frame defined by the molecular axis or by the molecular alignment or orientation. Then the dipole operator for left and right circularly polarized light is given by

Suppose for simplicity that the initial state of an oriented molecule is characterized by a wave function that can be expressed through the spherical function with a given 1 (summation over 1 does not alter the answer but makes the equation more complicated):

The photoelectron wave function $p-($), which contains in the asymptotic region the superposition of a plane wave propagating in the direction of the electron momentum p and a converging spherical wave, is presented as a partial wave expansion:

With these assumptions the dipole matrix element after integration over spherical angles of r can be expressed through the reduced matrix elements as follows:

The angular distribution of photoelectrons is proportional to the square modulus of this matrix element. The key point is that in (18) there are terms of the same sign for two circular polarizations, and terms of the opposite sign. As a result, the angular distributions for two circular polarizations will also contain a group of terms having the same sign and a group of terms having opposite signs. The latter will contribute to CDAD, and the former

230

N. A . Cherepkov

will cancel. In the case of unoriented molecules, the laboratory frame can be always taken to be coincident with the photon frame in which the dipole operator for right and left circularly polarized light is proportional to only one spherical function, Yl l(r) or Yl - l(r), and CDAD does not appear. If the initial states with different m are degenerate and equally populated, the angular distribution has to be averaged over rn:

Using Eq. (18) and the condition that the dependence on m is given solely by the spherical function xm(f),one can show that the terms that have different signs for two polarizations disappear from (19). And vice versa, if there is any unequivalence between states with different projections m, these terms will give nonzero contribution and will cause the appearance of CDAD. Unequivalence between states with different rn can be connected either with unequal population of states with different m (alignment and orientation of molecules), or with the dependence of a radial part of a wave function in our laboratory frame on the projection m, like it takes place when molecules are fixed in space and states with different rn are not degenerate. The same arguments hold for polarized (aligned or oriented) atoms that also reveal CDAD (Dubs et al., 1986; Cherepkov and Kuznetsov, 1989). From the consideration presented, we see that CDAD directly follows from the dipole selection rules given by the 3j symbols in (18), and therefore is expected to be of the order of unity (Cherepkov, 1982), that is, of the same order of magnitude as the differential cross section for the same angle. The general equation for CDAD can be written through the parameters introduced in Eq. (12) as follows: I&DAD(K, Q) = I', I(K,Q) - 15 1(K,Q) =

FIG.12. Illustration of the origin of CDAD. Absorption of light is different depending on whether the screw senses of the light and the experimental arrangement are the (a) same or (b) opposite. [From Cherepkov (1992).]

POLARIZATION AND ORIENTATION PHENOMENA

23 1

An analysis of this equation for some particular cases is given by Cherepkov and Kuznetsov (1987a). CDAD can be considered as a kind of optical activity. It is well known that the circular dichroism (CD) of unoriented chiral molecules is caused by a dissymmetry in their structure (Barron, 1982), due to which they can be presented as a helix. The absorption of light is different depending on whether the screw senses of light and of molecules are the same or the opposite. This leads to the appearance of C D in photoabsorption by chiral molecules. CDAD from oriented molecules appears due to dissymmetry in the geometry of the experiment. Three vectors describing the experiment, q, K, and n, can form a basis for a left or right coordinate system provided they are noncoplanar. Suppose that these vectors are mutually perpendicular. Then they can be arranged as shown in Fig. 12 to form a part of a left-handed or right-handed helix. Again, absorption of light will be different depending on whether the screw senses of light and of the experimental arrangement are the same or the opposite. As a result, CDAD will appear. Consider now LDAD. Here it is convenient to start the derivation in the photon frame with the z’ axis directed along the photon beam. Then the dipole operators for two orthogonal linear polarizations along the x’ and y‘ axes are given by

Again we have both terms with the same sign and terms of opposite signs for two polarizations. The latter lead to the appearance of LDAD. But contrary to CDAD, LDAD is different from zero even for unoriented molecules. To demonstrate this, let us consider the usual angular distribution of photoelectrons for the case of unoriented molecules:

where 9 is the angle between K and e. This equation is written in the coordinates frame with the z axis directed along the polarization vector e. Transforming it to the photon frame with the z’ axis directed along the photon beam, and x’ and y’ axes directed along the polarization vectors e, and ey, respectively, we find

I~_(K) = -

3a.(w) /3 . sin28’ . cos 2 cp’ 8a

where 9’ and cp’ are the spherical angles of the vector K in the primed coordinate system. The contribution of Eq. (23) disappears when

232

N. A. Cherepkov

cp’ = (2n + 1)n/4, for n = 0, 1, 2, 3. Then the nonzero value of LDAD is solely a consequence of orientation of molecules. The general expression for LDAD in the molecular frame is (Cherepkov and Schonhense, 1993) W

CDAD(K,

9) = CJK,9) - C x ( ~9), =

f i . ai(m) C C T,MT

. YTMT(KI)

*[D2-Kz(SZ)

AFMT

1

(- 1)“

K

+

D2-K-z(n)]

(24)

where D’,,.(O) is the Wigner rotation matrix (Varschalovich et al., 1988) with the Euler angles SZ = {u, fl, y } , which define the rotation from the molecular frame to the photon frame. From the general definitions of CDAD (20) and LDAD (24) one can come to the following conclusions: 1. Neither CDAD nor LDAD depends essentially on the spin-orbit interaction, and if the fine-structure splitting of an initial or final state is not resolved, both CDAD and LDAD remain essentially unchanged, like the angular asymmetry parameter 8. This is in contrast to the spin polarization of photoelectrons, which becomes small, of the order of (aZ)’, if the fine-structure splitting is not resolved. Therefore, CDAD and LDAD can be studied in light molecules where the photoelectron spin-polarization measurements are actually impossible due to very small spin-orbit splitting. 2. CDAD and LDAD are defined by different parameters ApMT,with k = 1 and k = 2, respectively. Therefore, CDAD and LDAD measurements supplement each other, giving additional information for extracting the dipole matrix elements and phase shifts from the measured quantities. 3. In principle, LDAD measurements can give more information than CDAD ones because the general expression for LDAD contains more terms. In linear molecules CDAD is defined by the parameters that contain the interference terms between degenerate photoelectron continua differing by & 1 in the projection m of the orbital angular momentum. For example, in photoionization of a 0 orbital, CDAD will be due solely to the interference between the a and n continua (Dubs et al., 1985). LDAD is described by the parameters that contain the interference terms between continua differing by Am = 0, f 1, and +2. 4. CDAD and LDAD have characteristic zeroes from which one can define the direction of molecular orientation, and from the magnitude of the effect one can find the degree of orientation. By choosing the proper geometry for an experiment, one can exclude the contribution of some terms from LDAD and CDAD so that the number of parameters to be extracted from a particular measurement can be essentially reduced. By changing the geometry, one can select the parameters to be defined, and from a series of measurements one can extract quite a few parameters.

POLARIZATION AND ORIENTATION PHENOMENA

233

B. NUMERICAL EXAMPLES

To demonstrate the general properties of CDAD and LDAD, consider a model case of a fixed-in-space heteronuclear diatomic molecule for which we accept the following simplifying assumptions: 1. An initial o orbital is ionized to only so, pa, and pn continua. 2. The dipole matrix elements for transitions to the pa and pn continua are equal and are two times larger for the sa continuum. 3. The phase shifts are d,, - d,, = n/4,6,, - d,, = n/18.

Under these assumptions only terms with T < 2 remain in Eq. (12). For linear molecules the following conditions are fulfilled (Cherepkov and Kuznetsov, 1987a):

ApM,= (- l)kAk,TMr and

K

+M , =0

so that the total number of parameters in (12) is reduced to 10 (and only 5 of them are independent in this model case). To reduce further the number of parameters, consider the particular geometry shown in Fig. 13. If we introduce the spherical angles 9, cp and S, p,, of the vectors K and q,

tz

FIG.13. Specification of the angles for CDAD and LDAD consideration.

234

N. A. Cherepkov

FIG.14. Numerical results for the model case of a fixed-in-space heteronuclear diatomic molecule (see text for details). (a) The angular distributions (26) 1; (solid curve) and 1: (dashed curve). (b) CDAD (25) as a function of an angle 9. [From Cherepkov (1992).]

respectively, in the molecular frame, then this geometry will correspond to Sq = p = 54.7" [so that P2(cos9,) = 01, cpq = 0, 0 d 9 < n, and cp = 90" or 270". Under these conditions two terms contribute to CDAD:

whereas the angular distribution for the same geometry is defined by five parameters:

Figure 14 shows the results of our model calculation for this particular case. Evidently, it is much easier to extract two parameters from the CDAD curve in Fig. 14(b) than five parameters from the angular distribution curves in Fig. 14(a). Consider now LDAD for the geometry shown in Fig. 13 when the primed frame is obtained from the unprimed one by rotation defined by the Euler

POLARIZATION AND ORIENTATION PHENOMENA

235

FIG.15. LDAD for the model case of a fixed-in-space heteronuclear diatomic molecule (see text for details). (a) The angular distribution If, (28) (solid curve) and If, (dashed curve). (b) LDAD (27) as a function of the angle 9.[From Cherepkov (1992).]

angles c1 = 0, /? = 54.7", and y = 45". In this case only two terms contribute to LDAD. I ~ ~ ~ , , ( K , Q )a= L 4 $4. [ A : ; ~ + J ~ - A : ;cos91sin9 ~ (27) 271 whereas the angular distribution of photoelectrons ejected by linearly polarized light is defined by five parameters:

Results for the model calculations of LDAD are given in Fig. 15. It is interesting to note that the terms in (26) and (28), which contribute neither to CDAD nor to LDAD, coincide. A numerical example of CDAD from a nonlinear CCI, molecule has been studied by Chandra (1989a). Consider now a set of experiments from which one can extract all 10 independent parameters AFM,with T < 2, which are different from zero in the case of heteronuclear diatomic molecules. This set can include (Cherepkov and Schonhense, 1993) the following: 1. Measurement of CDAD described by Eq. (25), which gives two parameters: A : ; and A:; 2. Measurement of LDAD in the geometry described earlier [Eq. (27)], which gives two other parameters: A:; and A:; l .

'.

236

N. A. Cherepkov

3. Measurement of LDAD in the geometry of the experiment when p = 54.7" and y = 0 (see Fig. 13). The corresponding analytical expression is

from which one can extract two parameters, A;: and A::, and a linear combination of the parameters A;: and A;;,. 4. Measurement of the angular distribution of photoelectrons ejected either by circularly or by linearly polarized light in the particular geometry as is given by Eqs. (26) or (28), respectively, from which one can extract two parameters, A?: and A;:, and a new linear combination of the parameters A;: and A:;,. It is essential that no more than three parameters are extracted from each measurement proposed above. Evidently, many other combinations of experiments can give the same information. C. EXPERIMENTAL OBSERVATIONS The first calculations of CDAD for the fixed-in-space CO molecules were performed by Dubs et al. (1985). Later it was realized that aligned molecules must reveal the same behavior (Dubs et al., 1986), and the first experimental evidence of the existence of CDAD was obtained for aligned NO molecules (Appling et al., 1986,1987) using the so-called resonantly enhanced multiphoton ionization (REMPI) process. In these experiments NO molecules were excited to the AZC state by absorption of one or two linearly polarized photons from the pump laser beam and then ionized by the counterpropagating probe laser beam. The laser beams crossed the molecular beam, which entered the interaction region at right angles to both the propagation direction of the laser beams and the detector axis of the electron spectrometer. The angular dependence of CDAD was obtained by rotating the linear polarization vector e of the pump laser beam, which gives the direction of molecular alignment. The REMPI technique allows for population of a single rotational level of the resonant intermediate state, so that the subsequent photoionization of this state gives a rather simple photoelectron spectrum. Using highresolution photoelectron spectroscopy, one can separate photoelectrons corresponding to the transition from a given rovibrational excited state to a given rovibrational state of molecular ion. The angular distributions of these photoelectrons, as well as CDAD and LDAD, provide the data necessary for the complete quantum mechanical experiment on the excited +

POLARIZATION AND ORIENTATION PHENOMENA

-IT

0

2

Theta

.,

237

3 2

FIG.16. CDAD intensity as a function of the angle between K and n for (2 + 1) REMPI of the A2Xf (u = 0) level of NO via (a) the R,, + Sll(l/2) and (b) the S21(1/2) rotational branch least-squares fit to Eq. (30); ---, theoretical calculalines; experimental data points; -, tions of Dubs et a/. (1986). [From Appling et a/. (1986).]

state (Dixit and McKoy, 1985;Reid et al., 1991). From calculationsfor different moleculesit follows that for a given intermediaterovibrational state the angular distributions strongly depend on the vibrational and rotational quantum numbers of the final ionic state (Wang et al., 1992, and references therein). Figure 16 shows the results of (2 + 1') REMPI measurements of NO in the reaction

NO(XZlll,z,Y" = 0, J" = 1/2) + 2y(450-452 nm, linearly polarized) NO*(A2Z+, u = 0, J ) + y'(266 nm, circularly polarized) -+ NO+(X'C+,u' = 0) + e-. Here the two-photon excitation process is used to prepare an aligned distribution of JM, substates, and the one-photon ionization process gives the CDAD effect. CDAD for the experimental arrangement is described by the following equation:

9) + a,P:(cos 9) (30) where Pi are associated Legendre polynomials and 9 is the angle between ICDAD

= a,P:(cos

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N. A. Cherepkov

the linear polarization and electron propagation directions. The contribution of P i is clearly visible in Fig. 16(b) since CDAD is not symmetric about + x/4. The observed CDAD curves are normalized to the photoelectron intensity at 9 = 0. Although the theory (Dubs et al., 1986) overestimates the contribution of Pi, the overall agreement between the theoretical and experimental results is reasonable. Since the parameters aL are proportional to the corresponding state multipoles of the alignment for the resonant state, the CDAD measurements can be used to study the alignment of molecules. Winniczek et al. (1989) have used the CDAD measurements to probe the alignment of ground state NO fragments produced by the UV photodissociation of methyl nitrite, CH,ONO. Using the (1 + 1’) REMPI process, the first “complete” experiment has been reported recently (Leahy et al., 1991, 1992; Reid et al., 1992) in which aligned NO molecules in the excited A%+ (u = 0, N = 22) state have been ionized by circularly polarized light. Due to high resolution of the electron spectrometer, rotationally resolved angular distributions have been observed. The new data for CDAD and angular distributions with circularly polarized light have been combined with the previous data on the angular distributions of photoelectrons measured with linearly polarized light (Allendorf et al., 1989). As a result, as many as 12 parameters have been obtained: seven dipole matrix elements corresponding to the transitions to the 1I continuum states with 1 = 0, 1, 2, 3, and I = 0, 1, and five phase shift differences. The agreement between the parameters extracted from the experiment and calculated by Rudolph and McKoy (1989) is rather good. Evidently, further investigations in this direction are very promising. CDAD for fixed-in-space molecules were first observed by Westphal et al. (1989, 1991). They investigated both diatomic (CO and NO) and polyatomic (benzene, CH,I) molecules oriented by adsorption on a Pd( 111) or graphite(0001) surface. Adsorption on a single-crystal surface yields a perfectly oriented ensemble of molecules with a rather high target density, about 2 orders of magnitude larger than in the gas phase, resulting in large photoelectron intensities. As a source of circularly polarized light, they used synchrotron radiation. A disadvantage of these experiments is connected with the fact that photoelectrons are ejected into a hemisphere outside the solid, which is further restricted by the systematic effect of refraction of photoelectrons due to the electrostatic surface barrier. Therefore the grazing emission parallel to the surface is equivalent to the emission angle for the case of free molecules, which can be substantially smaller than 90”. Investigation of the influence of the substrate on the angular distribution of photoelectrons ejected from adsorbed molecules has been performed rercently by Budau et al. (1993). They have shown that the backscattering from the surface changes only slightly the angular distribution of photoelectrons ejected from the lone pair 40 orbital of CO located at the oxygen

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239

F FIG.17. Typical photoelectron spectra of NO on Pd(ll1) at t40"C.The +and -denote the spectra for right and left circularly polarized light, respectively. The binding energy scale refers to the Ferrny energy, EF,of the substrate crystal. The dashed line denotes the Pd spectrum without NO. [From Westphal et at. (1991).]

end, whereas for the 5a and In orbitals, the angular distribution patterns change completely. Therefore, the comparison of experimental data with calculations performed for free molecules should be preferentially made for intermediate subshells, such as the 40 subshell of CO. CDAD from the outer 40, SO, and lx shells of CO and N O molecules adsorbed on a Pd(ll1) surface was studied in detail by Westphal et al. (1989,1991). The geometry of the experiment corresponds to Fig. 13 with = 130" and the angle 9 varied. For experimental reasons, the measured CDAD effect was normalized by defining the ratio ACDADof the difference to the sum of intensities for two light polarizations at each angle. The observed values of ACDADwere usually large, up to 80%, and were in reasonably good agreement with the theoretical calculations performed for an isolated CO or NO molecule with Hartree-Fock orbitals for both the bound levels and the photoelectron [see Lucchese et al. (1982) for further details of calculations]. Figure 17 shows typical photoelectron spectra of NO adsorbed on Pd(l11) that were obtained with circularly polarized light. The spectra show the structures corresponding to the 40 and Sa/lx orbitals (the latter could not be resolved), as well as the Pd4d bands close to the

N. A . Cherepkov

240

80 60

5

40

2

0

20

0

a -20 -40

ei n FIG.18. Normalized CDAD intensities for the 4u levels of (a) CO and (b) NO molecules, calculated for various tilt angles (indicated in the figure) and measured experimentally. [From Westphal et nl. (1991).]

Fermi level. The substrate structures are not discussed here [see Schonhense et al. (1991) for a discussion]. The strong dependence of the photoelectron intensities originating from the molecular orbitals on the photon polarization is clearly seen. Examples of the observed CDAD spectra are shown in Fig. 18. They all have the characteristic feature of going through zero at 9 = 0, which follows directly from the general equation (24). Written in the vector form (Cherepkov and Kuznetsov, 1987a), this equation is proportional to the product [q(wn)], which turns out to be zero when all three vectors are coplanar. This fact can be used to determine the direction of molecular orientation if it is unknown. In the case of adsorbed molecules, another question arises regarding if molecules are standing upright or if they tilt and rotate around the surface normal. This question can be solved only by comparison

POLARIZATION AND ORIENTATION PHENOMENA

0"

20"

40"

60"

24 1

80"

8 FIG.19. Normalized CDAD intensities for the totally symmetric initial orbital 3a,, of benzene at hv = 30.7eV for a monolayer [(a), 300"K] and two higher coverages [(b) and (c), 140"KIon Pd(ll1). [From Westphal et al. (1991).]

between the CDAD intensities calculated for different tilt angles and the experiment. Figure 18 shows CDAD intensities calculated for CO and NO4a photoemission at hv = 30.7eV for various angles of tilt. For each angle of tilt, the result corresponds to an orientational average around the surface normal. Since the 40 level is only weakly influenced by the substrate, calculations for the 40 levels of free molecules should be reliable here. It is important that the CDAD intensities change significantly with increasing angle of tilt and even change the sign at some angle. Comparison of these calculations with experimental points shows that CO molecules are adsorbed normal to the surface, whereas NO molecules are tilted by an angle of 35" to 45" with respect to the surface normal. These results are consistent with earlier angle-resolved photoemission studies (Miyazaki et a!., 1987) and demonstrate the potential of CDAD measurements as a probe of adsorbate orientation. LDAD measurements can be also used as a probe of molecular orientation (Cherepkov and Schonhense, 1993). For a particular case of linear molecules and for the geometry shown in Fig. 13, from the general equation (24) it follows that LDAD is identically equal to zero for arbitrary 9 when u = y = 0, f l = 0 (or n), and cp = n/4 (or 37r/4). Therefore, to establish molecular orientation, one should look for zero of LDAD in the geometry when K lies in the plane containing q and forming an angle +n/4 with the

N. A. Cherepkov

242

Benzene 3ala

0"

20"

40"

60"

80"

0 FIG.20. Variation of normalized CDAD intensity with the photoemission angle for different photon energies for the 3a,, orbital of a benzene monolayer on graphite (0oO1). [From Westphal et at. (1991).]

X ' O Z plane of the photon frame, by varying the angle between q and n. Then a zero of LDAD will correspond to n 11 q or n 11 (-q). As an example of a polyatomic molecule, Fig. 19 shows the normalized CDAD intensity for benzene adsorbed on Pd(ll1) as a function of the collection angle for the geometry considered earlier (Westphal et al., 1991). For a monolayer coverage when benzene molecules are lying flat on the surface, there is a pronounced minimum; for higher coverages the curves are almost structureless. This large difference in behavior of CDAD intensities indicates a different adsorption geometry. It is likely that for coverages above a monolayer not all molecules are lying flat on the surface, but some are strongly tilted or even standing upright. Figure 20 demonstrates the strong dependence of CDAD on photon energy for the 3 4 , orbital of

POLARIZATION AND ORIENTATION PHENOMENA

243

benzene physisorbed on graphite(0001). This pronounced variation can be explained by a sharp variation of the dipole matrix elements and/or phase shift differences as a function of the photoelectron energy. More recent investigations show that even photoelectrons ejected from a K shell of fixed-in-space molecules reveal CDAD. Bansmann et al. (1992) observed CDAD for the carbon K shell of CO molecules adsorbed on a Pd(ll1) surface and found that the normalized CDAD intensity is as large as for the case of the outer 4a and 5c~/lzshells of CO. The carbon K shell in CO is essentially an atomic-like s-orbital, but CDAD cannot occur in photoemission from a filled atomic subshell due to the lack of alignment or orientation (Cherepkov and Kuznetsov, 1989). Consequently, the nonzero CDAD effect from the molecular K shell can be explained solely by the difference between radial parts of PO and p.n continuum partial waves, which coincide in the case of atoms. The parameters describing CDAD are proportional to the sine of the phase shift difference 6,, - dPG;therefore, from a large magnitude of the observed CDAD effect it follows that this phase shift difference could not be small. Numerical calculations are desirable for a better understanding of the observed results. On the other side, as pointed out by Bansmann et al. (1992), the high magnitude of the normalized CDAD intensity exceeding 50% is well suited to analyze the circular polarization of the x-ray radiation throughout a wide photon energy region. CDAD for fixed-in-space CO molecules was recently studied theoretically by Raseev (1992) in the region of electronic autoionization between 17 and 18.4eV. CDAD shows a sharp variation in each resonance, which can be used to distinguish between resonances of different symmetries.

V. Optical Activity of Oriented Molecules The usual optical activity of chiral molecules appears to be due to the absence of a center of inversion, reflection planes, and rotation-reflection axes in individual molecules (Barron, 1982). The optical activity reveals itself in rotation of the polarization plane of linearly polarized light in transparent regions, or in circular and linear dichroism (CD and LD, respectively) in regions of absorption. CD is a difference in absorption of left and right circularly polarized light, and LD is a difference in absorption of linearly polarized light of two orthogonal polarizations. The optical activity of chiral molecules is described by electric dipole-magnetic dipole interference terms, which are a times smaller than the electric dipole terms, where LY is the finestructure constant. It became clear quite awhile ago that oriented molecules that lack a center

N. A. Cherepkov

244

of inversion but possess reflection planes or a rotation-reflection axis (so that they are superposable on their mirror images) can show optical activity for certain directions of the light beam (Barron, 1972). It is important that optical activity of oriented molecules already appears in pure electric dipole approximation (Chiu, 1970; Cherepkov and Kuznetsov, 1987a) and therefore does not contain any small parameter. The analytical expression for CD in the case of fixed-in-space molecules can be easily obtained by integration of Eq. (20) over the electron ejection angles &(q) =

3. oi(w)[ - A::. (nq) + $.

ReAAA. (qn,)

- $-ImAAA.(qn,)]

(31)

where n, n, and n,, are the unit vectors in the directions of the z, x, and y axes of the molecular frame, respectively. It is implied here that the vector n is directed along the molecular axis of the highest symmetry, and if there is a plane of symmetry parallel to the molecular axis, it coincides with the yz plane of the molecular frame. It can be shown that if the yz plane is a plane of symmetry, then (Cherepkov and Kuznetsov, 1987a) = 0,

Im&A = 0

(32)

and only one term remains in (31). If both the yz and x z planes of molecular frame are the planes of symmetry, then all terms in (31) turn out to be equal to zero. Therefore, CD in the electric-dipole approximation manifests itself only if oriented molecules have rather low symmetry. Rotating molecules, like polarized atoms, reveal CD only if they are oriented (Cherepkov and Kuznetsov, 1989), while aligned molecules do not reveal CD. The analytical expression for LD in the case of fixed-in-space molecules follows from Eq. (24) if we integrate it over the electron ejection angles &(q)

= fi*CTi(W)

CAig(-I)"

[DTKz(SZ)

D5.-2(n)]

(33)

K

LD is different from zero for fixed-in-space molecules of all symmetries. In particular, for the case of diatomic molecules, only one term with IC = 0 remains in (33): 3 &(q) = -oi(w) . A;: * sin2P cos2y (34)

$

and the angles fi and y are defined in Fig. 13. The parameter A;: for one particular case is given by Eq. (11). Rotating molecules, again by analogy with polarized atoms, reveal LD if they have nonzero components of an alignment tensor (Cherepkov, 1992). The optical rotation by fixed-in-space molecules in the electric dipole

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245

approximation was predicted at first by Chiu (1970) and discussed in more detail by Cherepkov and Kuznetsov (1991b). It exists for all molecules starting from diatomics. The rotatory power of the nematic liquid crystal of p-azoxyanisole observed earlier by Williams (1969) can be explained in the framework of this theory. Contrary to the case of unoriented chiral molecules (Barron, 1982; Healy, 1974), CD and the optical rotation in the electric dipole approximation are not connected by the KramersKronig relation. Qualitatively one can say that for the appearance of the optical activity, either the molecule or the experimental arrangement must not coincide with its mirror reflection. In the case of fixed-in-space molecules the experimental arrangement had to be characterized by three noncoplanar vectors, forming the basis for the right or left coordinate system. In the case of LD the photon beam is characterized by two polarization vectors e, and ey. If molecules are fixed-in-space, it adds at least one vector, the direction of molecular axis n, and if these three vectors are noncoplanar, the LD appears. The symmetry of molecules is not essential here, and LD is nonzero even in the simplest case of linear molecules. Circularly polarized light is characterized by only one vector, the vector of photon spin s,, which is parallel or antiparallel to q depending on the light polarization. Therefore, the fixed-in-space molecule had to be characterized by at least two vectors for the appearance of CD. This is why CD takes place for fixed-in-space molecules of relatively low symmetry. For unoriented molecules the experimental arrangement can be characterized by no more than two vectors, ex and ey. Therefore, the optical activity can appear only due to the properties of the molecule itself. As is well known, chiral molecules could not be superposed on their mirror images, which leads to the optical activity of chiral molecules.

VI. Conclusions The new experiments discussed in this review open very broad possibilities for investigations of molecular structure and spectra. Of course, these experiments are much more involved compared to the usual photoionization cross section and angular asymmetry parameter measurements with unoriented molecules, but the qualitatively new information that can be obtained from them justifies these efforts. The most exciting aspect is the possibility of performing the complete experiment with molecules in some approximation and to extract more than a dozen of the parameters necessary for theoretical description of the process. These measurements give the new information for more stringent tests of the existing theories of

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molecular photoionization, which are far from being perfect at the moment. Of special interest are the measurements of CDAD and LDAD from fixed-in-space or aligned and oriented molecules. They enable one to get rid of many terms, giving the contribution to the usual angular distributions with a given polarization of light, and in this way to specify the type of information to be extracted from the particular measurement. Changing the geometry of the experiment or the kind of measurement (CDAD, LDAD, CD, or LD), one can extract quite a few parameters from a set of measurements provided only a few parameters are extracted from each measurement. CDAD measurements have already been shown to have some practical applications. In particular, they can be used to probe molecular orientations of any kind, for example, those produced by scattering processes, molecular reactions or external fields, as well as by adsorption on a surface. Another possible application of CDAD is the measurement of the degree of circular polarization of x-ray radiation. CDAD measurements have been proposed for use in the search for Cooper minima in molecules (Rudolph et al., 1990). Without any doubt, future research on LDAD, CD, and LD will reveal other applications. Therefore, further investigations, both theoretical and experimental, are desired in order to realize all of the potential possibilities of these methods.

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ADVANCES IN ATOMIC, MOLECULAR, A N D OPTICAL PHYSICS, VOL. 34

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION IN PROJECTILE ELECTRON EXCITATION AND LOSS E . C. MONTENEGRO Departamento de Fisica Pontifcia Universidade Catdica do Rio de Janeiro Rio de Janeiro, Brazil

W. E. MEYERHOF Department of Physics Stanford University Stanford. California

and

J. H. McGUIRE Department of Physics Tulane University New Orleans, Louisiana

. . . . . . . . . . . . . . . . . . . . . . . . . . A. PWBA Formalism . . . . . . . . . B. Closure Method . . . . . . , . . . C. Improvements of Closure Method . . D. Molecular Form Factors . . . . . . E. SCA Formalism . . . . . . . . . . F. Impulse Approximation . . . . . . . G. Higher Order Processes . . . . . . . 111. Comparison with Experiment . . . . . . A. Projectile-ElectronExcitation . . . . I. Introduction

11. Theory

.

.

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B. Projectile-ElectronLoss Cross Section . C. Projectile-ElectronLoss and Capture . . D. Separation of Screening and Antiscreening E. Electron-Loss Probability . . . . . . . F. Electron Spectroscopy . . . . . . . . IV. Conclusion . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . References . . . . . . . . . . . . . . .

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Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any Corm reserved. ISBN 0-12433834-X

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E. C, Montenegro et al.

I. Introduction In an atomic collision, an electron orbiting one atomic nucleus may influence what happens to an electron on the other atomic center. For example, the Coulomb repulsion between the two electrons may cause one or more of the electrons to change state. Another possibility is that the electric field of the negative electron weakens the field of its positively charged atomic nucleus, which in turn decreases the rate at which transitions take place in the second atomic center. There are also other, more subtle ways in which electrons on two colliding centers may affect the transition rates. Projectile electron loss is a process in which an ion interacts with an atomic or molecular target and at least one electron is ejected from the projectile as a result of this interaction. For the large majority of atomic and molecular targets, the loss process occurs by the coulomb interaction between the (target, 2,) nucleus and the (projectile, 2,) electron, with the target nuclear field being attenuated by the screening of the target electrons, as illustrated in Fig. 1. In the projectile frame, projectile electron loss can be viewed as the ionization of the projectile by the target. If the target nuclear charge is small compared with the projectile nuclear charge and if the projectile velocity is large compared with the Bohr velocity of the active projectile electron, first-order theories such as the plane wave Born approximation (PWBA), can be used to describe the loss process. However, the analogy with the classical ionization problem (Bethe, 1930), in which the ionization is due to a simple, bare, charged particle, is not sufficiently general. In most cases of interest, the target is neutral and screening of the target nucleus plays an

FIG.1. Diagrammatic representation of first-order projectile electron loss by a target with nuclear charge Z,. As a result of the nucleus-electron interaction (represented by a dashed line), the projectile active electron makes a transition from an initial state [I+*) to a final continuum state I$f). The projectile nuclear charge and velocity are Z , and u, respectively. This process is called the screening efect.

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

25 1

important role in the calculation of the electron-loss cross sections (Bates and Griffing, 1953). A neutral target cannot be simulated by a bare one for the study of projectile-electron loss. In fact, the effectiveness of the screening is not the same for all impact parameters for which projectile-electron loss can proceed (Toburen et al., 1981). This effect, which must be included in any realistic theoretical approach, makes the description of the loss process significantly different from the ionization by a bare particle. Another important difference between the ionization caused by a bare particle and by a neutral atom is the possibility that the target electrons also act as ionizing agents. It turns out that in this case, the projectile-electron loss is due solely to the electron-electron interaction, with the target nucleus acting as spectator. The simplest theoretical approach to account for electron loss is the free-collision model or classical impulse approximation (Bohr, 1948; Dmitriev and Nikolaev, 1963; Bates and Walker, 1966, 1967; Walters, 1975; Dewangan and Walters, 1978; Meron and Johnson, 1990; Riesselmann et al., 1991), in which the active projectile electron is supposed to be free, colliding with the target atom with a relative velocity equal to the projectile velocity. In the impulse approximation, the projectile electron is described by a velocity distribution, which reflects, to some extent, the bound nature of this electron. The projectile is ionized if the impulse given by the target atom to the projectile electron during the collision transfers an energy greater than the projectile ionization energy. Recently, Lee et al. (1992) used a different approach to calculate the electron-electron contribution to the loss. Viewed from the projectile frame, the target electron can be considered to interact with the projectile as a free particle if the collision time is short compared to the orbiting time of the active projectile electron. Under these conditions, a quantum mechanical impulse-like approximation can be used to calculate the electron-electron interaction part of the loss, connecting this process to the ionization of ions by free-electron impact. The full quanta1 treatment of the electron-electron interaction was first carried out by (Bates and Griffing 1954, 1955). In this case, the free-electron hypothesis is not used, and both the projectile and target electrons are allowed to change states, in their respective centers, as a result of the electron-electron interaction. Figure 2 gives an illustration of this mechanism, which has been called antiscreening by several authors (McGuire et a/., 1981; Anholt, 1986; Montenegro and Meyerhof, 1991a) in contrast to the effect on the nucleus-electroninteraction, illustrated in Fig. 1, and called the screening efect. Stolterfoht (1989, 1991) discusses various possible twoelectron interactions and notes that the antiscreening interaction can be classified as a two-center scattering correlation. On the other hand, the screening effect is classified as a single-electron interaction. The mutual change of states by the two interacting electrons in the

E. C. Montenegro et al.

252

TARGET

PROJ E1 C Tl LE

4

t

FIG.2. Diagrammatic representation of a first-order projectile electron loss mechanism due to the electron-electron interaction (representedby the dashed line). While the active projectile electron goes from a state I$J to a state I$,) in the continuum, the active target electron performs a simultaneous transition, which brings the target from a state I@,J to a state I@.). This process is called the antiscreening effect.

antiscreening mode introduces additional difficulties in theoretical calculations, because two electronic transitions must be calculated and, in most cases, continuum states are the more likely final states. Furthermore, in most measurements of electron loss, the final state of the target is not observed and a sum over all possible final states must be evaluated. This is a particularly difficult task if the sum is carried out state by state for targets with more than two electrons. The most widely used procedure to circumvent this difficulty is via closure (Lodge, 1969; Gillespie, 1977; McGuire et al., 1981; Anholt, 1986; Hartley and Walters, 1987; Montenegro and Meyerhof, 1991a, 1992). An approximation based on closure has two important merits besides avoiding a state-by-state sum. First, the completeness of the atomic states is maintained, which assures that at asymptotically high energies this kind of approximation is exact (Hartley and Walters, 1987). Second, the calculations use form factors based only on the target ground state wave functions, avoiding the need for excited state wave functions (including the continuum) and simplifying considerably the calculations. In its crudest form, the closure method neglects the energy differences between the various electronic transitions in the target, which can be reached via the electron-electron interaction, assigning an average value to the energy

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

253

transferred in all possible transitions of the target. Although this assumption substantially simplifies the calculations, it results in a strong overestimation of the electron-electron contribution to the loss in the intermediate- to low-velocity regime. Most of the recent work within the PWBA framework has been directed toward removing this deficiency (Anholt, 1986; Hartley and Walters, 1987; Montenegro and Meyerhof, 1991a). The theoretical approaches just discussed are designed to study singleelectron loss. For collisions involving highly charged ions, however, multiple-electron transitions are not only possible but can have a sufficiently high probability of distorting an interpretation of the experimental measurements by means of a single-channel analysis. Under these circumstances, the independent-particle model (IPM) together with an impact-parameter analysis has been widely used to obtain workable results (see McGuire, 1992, for a recent review of multiple-electron processes). Double-electron loss or simultaneous capture and loss are examples of processes involving electron loss where the IPM can be advantageously used. An impactparameter analysis (semiclassical theory) of electron loss was not available until recently (Montenegro and Meyerhof, 1991b, 1992). Based on timedependent perturbation, this theory gives for the total cross sections the same results as those obtained within the PWBA and shows new features of the electron loss process, such as the long-range character of the electron-electron interaction (antiscreening mode). The dominance of the electron-electron interaction at large impact parameters can be related to the observation of low-energy electrons in collisions, in which both the projectile and the target are ionized, because of the small momentum transfer associated with distant collisions (DuBois and Manson, 1990; Manson and DuBois, 1992; Heil et al., 1991, 1992). An analysis of the ejected electron spectra through the second-order Born approximation (JakubassaAmundsen, 1992) supports this interpretation based on impact-parameter analysis, indicating the importance of the electron-electron interaction when the energy and the emission angle of the ejected electron decrease. There is a second-order process associated with antiscreening that plays an important role in experiments where the recoil ions or the ejected electrons are simultaneously measured with the projectile in electron loss collisions (Montenegro et al., 1992a; Heil et al., 1992). This process, which has been called two-center double ionization (Montenegro et al., 1992a), incoherent projectile-target ionization (Jakubassa-Amundsen, 1992) or uncorrelated double inelastic scattering (Wang et al., 1992), is a projectile-electron loss process due to the screened target nucleus, occurring simultaneously with target ionization by the screened projectile nucleus. It results in the same final electron states as antiscreening. A knowledge of the differences between the dynamic signatures of these two processes is important in order to understand the way in which they compete for the simultaneous ionization of the projectile and the target above the antiscreening threshold.

254

E. C. Montenegro et al.

11. Theory

A. PWBA FORMALISM Figure 3 is a sketch of the coordinates of the electrons that participate in a collision resulting in projectile one-electron loss for the two modes illustrated in Figs. 1 and 2. The projectile with charge Z , e and velocity v carries an electron (only the active electron is considered) with coordinates p with respect to the projectile nucleus. Due to the nucleus-electron and elec-

FIG.3. Sketch of projectile (2,) electron loss due to a target with nuclear charge 2,. The active-electron coordinates of the projectile and target are represented by p and gj, respectively. The nucleus-electron and electron-electron interaction are represented by dashed lines. As a result of these interactions, the projectile goes from a state to a state I$,), while the target either stays in the ground state I@,) or undergoes a transition to the state IDn) with a transition energy A& [From Montenegro and Meyerhof (1991a).]

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

255

tron-electron interactions, indicated by dashed lines in Fig. 3, the projectile goes from a state to a state I$,), while the target (nuclear charge Z2e) undergoes a transition from a state IQo) to a state Ian>. The j'th target electron has a coordinate kj with respect to the target nucleus. The projectile-nucleus coordinate with respect to the target nucleus is R. The cross section for the loss process can be written within the PWBA as (McDowell and Coleman, 1970):

where

40 =

E , - Es 7

and

E n - Eo

AEn

q"=hu= hV For most cases of interest, the maximum momentum transfer qmaxcan be set equal to infinity. Using the Bethe integral,

to calculate the Fourier transform appearing in Eq. (2), and discarding the cases where the projectile does not change state, one obtains for osf,on:

where

and vo = e2/h is the Bohr velocity of H. In the case of projectile loss, the projectile final state is in the continuum and, if the ejected electron is not observed, a sum over all possible final

E. C.Montenegro et al,

256

continuum electron states must be performed. Denoting the final projectile by l $ k , ) and normalizing this wave function by state <$kp

I $kb)

= (2n)3s(kp-

(10)

kp)

the cross section for projectile loss can be written Os*On=

c

contSk,)

Jr

=(2t)3

dkp Osk,.On

(1 1)

with k,,, = 00 for most of cases of interest. The summation in this equation denotes the sum over continuum states of the ejected projectile electrons. Another important situation to be considered is when the target final state is not observed and one must sum over all possible target final states. Equation (8) then gives:

when the first sum includes, besides all bound target states, the continuum target states in the same way as indicated by Eq. (1 l), with the integration being carried out over the wave numbers of the ejected target electron. One sees that projectile electron loss without the observation of the final target states involves two electrons in the continuum, and the computational work increases considerably. There are very few calculations for projectile-electron loss in which the various target final states are explicitly considered. The simplest system for which the PWBA theory can be used is the H + H collision. Bates and Griffing (1955) calculated projectile-electron loss cross sections for cases where the target is left in various specific final states, including the continuum. A similar calculation was carried out by Boyd et al. (1957) using He' as the projectile. These calculations involve a one-electron target (or projectile). Extension to collision systems with two electrons (target or projectile or both) was first performed by Bell et al. (1969,1970) and Bell and Kingston (1971, 1974, 1976). To obtain the total cross section for electron loss for H H, Bates and Griffing (1955) used a state-by-state calculation for Eq. (12). Bell et aE. (1969,1970) and Bell and Kingston (1971) also used the state-by-state approach to determine the electron loss cross sections for H and He' projectiles on He. To our knowledge, there is no reported calculation based on the state-by-state approach for multielectron targets other than He. For a general atomic or molecular target, the state-by-state method is very cumbersome to perform. All such calculations involving many-electron targets within the PWBA framework are performed through the use of closure.

+

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

257

B. CLOSURE METHOD

In its simplest form, the closure method assigns the same average energy AE,, or equivalently, using Eqs. (4) and (6), the same average momentum transfer ijmin, to all possible transitions of the target. Under these conditions, Eq. (12) can be written as:

because the integration limits do not depend on n any more. The sum can now be calculated in a closed form, with the result expressed in terms of the target ground state wave function. If the antisymmetry of the target ground state wave function is ignored and this wave function is expressed by one-electron orbitals I (PA(t)), so that IQO)

=

n a

(14)

Ma)

it can be shown (Montenegro and Meyerhof, 1992) that for any sum of one-body operators d ( t j ) , the use of the closure relation (Messiah, 1961) gives

where

and 2

Sa(q) = 2

2

- ~al < + a 1 e + 1 4 a ) l

(19)

The quantities S, and Sa can be considered effective square target charges for the screening and antiscreening processes, respectively. As stated before, the closure approximation has three merits: (1) it strongly simplifies the calculations compared with the state-by-state

E. C. Montenegro et al.

258

method; (2) the only state of the target needed in the calculation is the ground state; and (3) it gives the correct asymptotic limit as a consequence of the explicit use of the completeness of the target wave functionuhe lack of a definite criterion to choose the average energy transfer AE, is the starting point of difficulties associated with this method. Apparently, the dynamic behavior of the electron-electron interaction is sensitive to the differences in the energy transferred. Figure 4 compares, for H + H collisions, the exact PWBA result (Bates and Griffing, 1955) obtained through the state-by-state method withthe closure approximation calculated with the simple choice of setting AE, equal to the ionization energy of the hydrogen atom, I,. It can be seen that at intermediate to low velocities, the closure approximation strongly overestimates the cross section. At higher velocities, the two curves coalesce, indicating that, in this regime, the differences in the energies of the target excited states are not as important as the completeness of the target wave function. An interesting application of the closure approximation at high velocities was carried out by Gillespie (1977) who calculated the electron detachment

10

ENERGY ( k e V )

FIG.4. Results from projectile loss calculations in H + H collisions. Curve (a) is the exact PWBA calculation (Bates and Griffing, 1955); curve (b) is the closure approximation with AEn = I,. [Adapted from Montenegro and Meyerhof (1991a).]

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

259

cross section in collisions of H - with H and He. Taking advantage of the fact that the H - ion has only one bound state, the total detachment cross section is calculated using the closure relation to evaluate the target and the projectile form factors (double closure).

c. IMPROVEMENTS OF CLOSURE METHOD 1. General Considerations The cross section for electron loss given by Eq. (12) can be separated into two parts, which correspond to the major channels through which the electron loss can proceed. Separating the n = 0 term from the others, we have

and

Equation (20) corresponds 'to the elastic channel (from the target point of view). In this channel, the target electrons do not change state throughout the collision. The projectile loss is due to the nucleus-electron and electron-electron interactions, with the nucleus effectively screened by the target electron potential (McGuire et al., 1981). If Eq. (14) is used to describe the target groud state, the last term in Eq. (20) in S,(q), is given by Eq. (18). In general, there is no difficulty in computing this screening contribution to the electron loss from Eq. (20). In contrast with the screening mode, Eq. (21) gives the contribution from the inelastic channels (from the target point of view). Here, the nucleus assumes a passive role, providing the momentum distribution of the target electrons. The projectile electron loss proceeds solely via the electronelectron interaction, which causes both the projectile and the target to change states. This is the antiscreening mode (McGuire et al., 1981). If an average value qminis used for the minimum momentum transfer, the sum rule Eq. (15) can be used to evaluate Eq. (21), resulting in the expression:

As shown in the previous sections this simplification does not give good results at intermediate velocities and efforts have been made to remedy this defect. Essentially three methods have been used to this end: free-collision methods, correction-factor methods, and the sum rule method.

E. C. Montenegro et al.

260

2. Free-Collision Methods The free-collision methods use momentum conservation arguments to find reliable values for gmin.These methods are based on the assumption that the major contribution for the sum appearing in Eq. (21) comes from the target continuum states. With this premise, Eq. (21) can be approximated by

where a one-electron target is considered for simplicity. If the wave functions of the two electrons in the continuum are approximated by plane waves, i.e., +f eik;P and +kt oc eikk, one has, from Eq. (9),

-

-

-

-

The major contribution to the integrals in Eq. (24) occurs for 1q + k,l Iq - k,l small. In other words, q k, and q - k,, which essentially express momentum conservation in a binary encounter. This result, together with the assumption that only continuum target states contribute to the loss, gives AE, hk:/2m hk,2/2m. Consequently, using Eqs. (4) and (6), qmin can be expressed in terms of the final wave number of the projectile electron and the sum rule can be used. This kind of reasoning has been used by Lodge (1969) and Day (1981) (see also Hartley and Walters, 1987) to study electron loss collisions within the closure approximation. If the free-electron model and momentum conservation arguments are taken one step further, the maximum momentum transfer to the projectile electron in a binary collision with the target electrons is given by rnulh. Following this argument, Dmitriev and Nikolaev (1963) used this value for the maximum momentum transfer in Eq. (22), instead of infinity. The same procedure has been employed by other authors (Victor, 1969; Dewangan and Walters, 1978; Hartley and Walters, 1987) to improve the simple models described earlier, which consider only changes in the value of ijmin.

-

-

3. Correction Factor Methods

The correction factor methods are also based on the assumption that the main contribution to electron loss comes from the continuum target states. However, instead of making different choices for the integration limits in Eq. (22), an ad hoc correction factor is introduced in the integrand of Eqs. (22) or (23) to improve the behavior of the cross section at intermediate velocities, keeping the correct asymptotic behavior at high velocities as given by the closure approximation. Anholt (1986) used the analogy of the antiscreening process with a

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

26 1

free-electron impact ionization of the projectile to obtain a correction factor for the closure approximation. If the binding energy of the active target electron is neglected, the antiscreening contribution to loss can be simulated, in the projectile frame, by the action of a free electron. The latter is able to ionize the active projectile electron with binding energy I , only if 1

-mu2 2 ( E / - E~),,,~" = I, 2

(25)

This picture is essentially the same as the free-electron model. Anholt assumes that the closure method fails to give good results at intermediate velocities, because this method does not take into account properly the electron-electron interaction near the projectile ionization threshold by a beam of electrons. The near-threshold effect is included in the theory by replacing S,(q) in Eq. (22) with S,(q)",(u)/aB(u), where a,(u) is the freeelectron impact ionization cross section of the projectile (with threshold effects included), and O,(U) is the same cross section for a proton (Born cross section). The ratio C J ~ ( U ) / O , ( U ) goes to zero at the threshold velocity (u,,, = )-/, and goes to unity if u>> u,,, because in this case CJ,(U) + o,(u), assuring the correct behavior given by the closure approximation at high velocities. Hartley and Walters (1987) also assume that the main contribution to the electron loss comes from the continuum target states. For a one-electron target this assumption is equivalent to considering only the antiscreening mode as described by Eq. (23). However, this equation does not take into account the completeness of the target final states and, hence, the correct high-energy behavior is not assured. The target state completeness is inserted in the calculations by modifying Eq. (23) as follows:

Because the sum over the final continuum target states in Eq. (26) is not permuted with the integration over the momentum transfer as in the closure approximation, the energy transfer to each final target state is properly evaluated, at least for the continuum states. As a consequence, the cross section is not overestimated at intermediate velocities, as it is within the closure approximation. The term between brackets in Eq. (26) apparently is not effective at intermediate velocities; it is included to assure the correct behavior of the cross section at high velocities. In this regime, it is valid

262

E. C. Montenegro et al.

to permute the first sum with the integral in Eq. (26) and, as a result of this approximation, only the sum over all target states remains, assuring the completeness. 4. Sum Rule Method

The sum rule method was introduced by Montenegro and Meyerhof (1991a) to retain the advantages of the closure method, without introducing any ad hoc factors or making any additional hypotheses regarding the larger contribution that the continuum states make compared with that from the bound, excited target states. Splitting the region of integration into two parts, Eq. (21) can be rewritten as:

where qo and q,, are defined in Eqs. ( 5 ) and (6), respectively. The first sum corresponds to the closure approximation. Because the lower integration limit does not depend on q,,, the sum can be permuted with the integral. The second sum can be viewed as a correction to the closure approximation. Montenegro and Meyerhof (1991a) considered the functional defined by the second integral in Eq. (27) and developed this functional in the vicinity of qo up to first order in q,,. The Bethe sum rule for energy loss (Bethe and Jakiw, 1968a) is then used to calculate the sum

c qnl<@nlFeiqt'

n#O

and to obtain a correlated cross section

with

and

IQO>

I

(28)

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

263

The lower integration limit qo + 6 increases rapidly if the projectile velocity decreases and approaches the threshold, resulting in a fast decrease of the antiscreening contribution in this region. Figure 5 shows some features of the screening-antiscreening calculations for the H H system. From a numerical point of view, the difference between the Anholt (dotted curve c) and Montenegro and Meyerhof (dashed curve f) calculations for the total loss cross section is very small, except near the onset for the antiscreening mode, where Anholt’s method gives a sharp threshold, but the sum rule method gives a smoother onset. This smooth onset can be clearly seen from the dot-dot-dashed curve e, where the antiscreening contribution is shown separately. The projectile energy corresponding to the threshold velocity ( u , = ~ is 25 keV but the sum rule method (as well as the exact calculations of Bates and Griffing) gives a residual contribution even for projectile velocities below this value, as a consequence of the momentum distribution of the active target electron. It is also interesting to note that for this particular system, at high projectile energies, the antiscreening contribution to the loss is significantly larger

+

Jm)

E N E RG Y ( keV 1 FIG.5. Results from projectile ionization in H + H collisions. Curves (a) and (b) are the same as in Fig. 4. Curve (c) is from Anholt (1968). Curves (d), (e), and (f) are the screening, antiscreening and total contributions for the loss, respectively, following Montenegro and Meyerhof (1991a). [Adapted from Montenegro and Meyerhof (1991a).]

E. C. Montenegro et GI.

264

than that of the screening contribution (thin solid curve d), indicating the importance of the electron-electron interaction with respect to the simple screened nucleus-electron interaction.

D. MOLECULAR FORMFACTORS The relative importance of the antiscreening mode with respect to screening increases if the atomic number of the target decreases. As can be seen from Eqs. (18) and' (19), the screening effective squared charge S,(q) increases essentially as Zg, whereas the antiscreening factor increases as Z , at high projectile velocities. Consequently, the relative importance of the electron-electron interaction in the loss processes is enhanced for low-Z targets essentially as l/Z2. In this respect, molecular hydrogen is a particularly attractive target due to the difficulty of obtaining atomic hydrogen targets. The electron loss process in molecular hydrogen targets was studied within the PWBA framework by Meyerhof et al. (1991). An interference due to the two atomic centers in the molecule is the origin of the main differences between atomic and molecular targets. Such an effect affects significantly the screening and antiscreening factors [Eqs. (18) and (19)], which are the basic quantities appearing in all methods using the closure approximation, as discussed previously. The factors S,(q) and S,(q) are given in the molecular case by the following expressions: cos(q * 111I 1 - F,(q) I2 1

(32)

P

where

Here, 1 is the vector connecting the two nuclei of the molecule, dR, is the solid angle subtended by this vector, and m0 is the H, molecular ground state wave function. The results of calculations are sensitive to the various possibilities of the molecular orientation and depend on the molecular wave function as well as the averaging procedure used. The important point in this study is the investigation of the differences between a molecular target and a molecule composed of two hydrogen atoms. Three possibilities were considered, resulting in significantly different results for the various projectiles analyzed. Figure 6 shows the results for the ratio of electron loss cross sections for Ho projectiles by H, and Ho targets. The Weinbaum, Stewart, and Hub-

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

0

0.1

0.2

265

0.3

PROJECTILE ENERGY (MeV/u) FIG.6. Ratio r I 2 of Ho projectile-electron loss cross sections for H, and Ho targets. The short-dashed, solid, and dashed-dotted curves represent the Weinbaum, Stewart, and Hubbell-Copper model results, respectively. Experimental data are from McClure (1968) (squares) and from Wittkower et al. (1967) (circles and triangles). [From Meyerhof et al. (1991).]

bell-Cooper (Meyerhof et al., 1991) models correspond to different alternatives to calculate the molecular form factor given by Eq. (34). The differences among these models are due to the region of small momentum transfers, which plays an important role in the cross-section calculations for intermediate to high projectile velocities. E. SCA FORMALISM

1. General Considerations The impact parameter approach was first introduced to study projectile excitation or loss with the aim of improving the results of the first-order Born approximation through the close-coupling approximation. These extensive calculations (Flannery and Levy, 1969; Levy, 1969; Flannery, 1969a,b; Bell et al., 1973, 1974; McLaughlin and Bell, 1983) were carried out for H collisions with H and He targets to study the effects of distortion,

266

E. C. Montenegro et al.

exchange, or coupling to the ground state or other excited states, when the collision partners are singly or mutually excited. Due to computational difficulties, projectile electron loss was not considered until recently. Schiwietz and Grande (1992) calculated the total ionization cross section for H + H collisions using a screened Bohr-like potential. Fritsch and Lin (1993) calculated the screening contribution for electron loss in C 5 + + He collisions with emphasis on the dependence of the loss probability on the impact parameter. Another important scenario for which the impact-parameter formulation can be extremely useful is in conjunction with the IPM. This allows a study of more complex collisions systems where multielectronic processes are likely to occur (McGuire, 1992). In such cases, second-order theories present severe computational difficulties and often can be used only in some particular situations. The general behavior of the colliding systems and the cross section or probability dependence on the various collision parameters often can be obtained through the use of a simpler analysis based on first-order theories. Furthermore, because of the equivalence between the first-order semiclassical approximation (SCA) and the PWBA discussed in the previous section, we limit this review mainly to the first-order SCA. There are essentially no early examples of the use of SCA for projectile excitation or loss. Vegh and Sarkadi (1983) studied 1s -+ 2p target excitation using a factorization of the screened projectile potential to calculate the excitation cross section, within the framework used for bare projectiles (Bang and Hansteen, 1959). Only recently, a more comprehensive treatment of projectile loss within the SCA has been obtained by Montenegro and Meyerhof (1991b, 1992), Ricz et al. (1993), Kabachnik (1993), and McGuire and Montenegro (1993). The development of SCA theory brings out new features of the projectile loss process. The basic points of this theory are discussed in the next sections. 2. Screening and Antiscreening within the SCA Figure 7 shows a sketch of the coordinates useful for projectile ionization by a target with nuclear charge 2,. The notation in this figure is the same as in Fig. 3. In the following discussion, the projectile is assumed to follow a straight-line trajectory with velocity v and impact parameter b. If the active projectile electron performs transitions from the state IJ/,)to a state while the target atom goes from the state I@,) to a state I@,), the semiclassical amplitude is given by (Montenegro and Meyerhof, 1992):

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

267

FIG.7. Same as Fig. 3 with b being the impact parameter and R(t) being the time-dependent internuclear separation. [From Montenegro and Meyerhof (1991b).]

+

+

where R(t) = b vt, w = wo w,, wo = uq, and wn = uq,. The first term in the integrand of Eq. (35) corresponds to the interaction of the active projectile electron with the target nucleus. The remaining sum corresponds to the interaction of this electron with all target electrons. In cases where the final state of the target is not observed, the probability of projectile excitation or loss is given by

and the corresponding cross section is: Pm

usf = 2n J

dbbPsfu,b)

(37)

0

In the case of projectile loss, the sum over the continuum states of the projectile is performed according to Eq. (1 1). Equation (36) can be separated into two parts, corresponding to the cases where the target electrons remain in their initial states (screening) or are excited, through an electron-electron interaction with the projectile electron, to any other final state (antiscreening). Then, we have

E. C. Montenegro et al.

268

where PScreen and Pantican be written

and

(40) In the absence of target electrons, the usual SCA probability for a target nucleus of unit charge is easily recovered from Eq. (39) (Bang and Hansteen, 1959): pSCA(V, b) = IaSCA(u, b)I*

(41)

where

3. Screening

The screening probability can be written in terms of the SCA amplitudes for bare particles by direct comparison of Eqs. (39) and (42), using the relation

rj = b - Sj

(43)

(see Fig. 8). Now, Eq. (39) can be rewritten Pscreen(v,

b) = ZACA(~, b) - (@o

I C ~SCA(V, rj) IQo) j

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

269

sketched in Fig. 8. The difference between these two terms corresponds to the transition amplitude for the screened target nucleus. The screening effect can be recognized more directly if Eq. (39) is rewritten as

The potential between square brackets in Eq. (46) is the screened target potential as seen by the projectile and can be represented by a screening function 4(R + p) defined as

Equation (46) now reads:

indicating more clearly the action of the screened target potential on the projectile electron. Vkgh and Sarkadi (1983) used Eq. (48) as the starting point to calculate

FIG.8. Sketch of the projectile (2,) electron loss in the projectile reference frame caused by the target (2,) electronic cloud with local density [&<)Iz. The impact parameter for the collision is b, while the effective impact parameter of the cloud is rl. [From Montenegro and Meyerhof (1992).]

E. C. Montenegro et al.

270

target inner shell excitation by screened projectiles. The screening function + p) was approximated by the zeroth-order term in the Taylor series expansion (p << R), so that @(R + p) 'v $(R), simplifying the calculations. Montenegro and Meyerhof (1991b) showed that through the use of a Bohr type potential with

4(R

+(R

+ p) = Z2e-'IRS.P/

(49)

Eq. (48) can be written

where d , = Ib + pII and K O is the modified Bessel function of order zero. From this equation, the SCA probability for a bare target (Hansteen, 1989; Madison and Merzbacher, 1975) is immediately recovered by making a = 0. Equation (50) also suggests that the screening probability can be obtained in an approximate way from previously calculated SCA probabilities for bare particles. Montenegro and Meyerhof (1991b) proposed a scaling rule in the form:

Pscreen(u,

O) =

Z2aSCA(u,

O) - ( @ O 1

aSCA(u, j

e

j)

I@O>

iz

(52)

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION r

271

I

I

-

30-MeV C5++ He 1s 2p ( m = o )

P)

0

08

,

--a.

--- - - - _ _ _

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

I m p a c t Parameter

b (a.u.)

FIG.9. Impact-parameter dependence of the probability P for the electric dipole excitation of the projectile in 2.5 MeV/uC5+ + He collisions. Dashed line: calculation with bare target nucleus; thick solid line: calculation of Ricz et al. (1993); dotted line: factorization approximation (Vtgh and Sarkadi, 1983); thin solid line: scaling approximation (Montenegro and Meyerhof, 1991b). [Adapted from Ricz et al. (1993).]

small energy transfers (excitation), the screening effect is expected to be highly enhanced. On the other hand, if the energy decreases and processes involving higher energy transfer are considered, as in projectile loss, the screening effect should be small. This is indeed the case for the 0.83 MeV/u C5 He electron loss probability (below the antiscreening threshold), studied by Fritsch and Lin (1993) within the close-coupling approximation, and illustrated in Fig. 10. A very small screening effect is present here. +

+

4. Antiscreening

The probability of loss via electron-electron interaction is given by Eq. (40). Here, the difficulty found in the PWBA approach in performing the summation over target final states is also present. Montenegro and Meyerhof (1992) calculated Eq. (40) using the closure approximation with an average value for the energy transfer, hw, and an average velocity parameterized by

v = v(l + 6 / w o ) - 1

(53)

where wo = vq, [Eq. ( S ) ] . The parameters ii, and V are determined using the formal equivalence between the SCA and the PWBA and setting the minimum momentum transfer obtained in the PWBA equal to the minimum

212

E. C. Montenegro et al.

0.010

1

.

1

1

1

.

1

.

1

1

n

-

v

Q

n

0.002-

1

,

I m p a c t Parameter b (a.u.1 FIG.10. Calculated total electron loss probability for 0.83 MeV/uC5+ impinging on HeZ+ ions (solid line) and on a screened He nucleus (dashed line). [From Fritsch and Lin (1993).]

momentum transfer obtained by the sum rule method described in Section II.C4. The parameter 61 . is then given by CT, = u6

(54)

where 6 is given by Eq. (30). If this procedure is followed, the antiscreening probability can be conveniently expressed in terms of the semiclassical amplitudes for bare particles, in a way similar to Eq. (44) for the screening mode (Montenegro and Meyerhof, 1992):

where aSCA is given by Eq. (42). Using expansion (14), Pantican be rewritten as

where r = b - 5. Unless one deals with a one-electron target, Eqs. (55) and (56) are not identical term by term, but only as a whole (see the appendix of Montenegro and Meyerhof, 1992). We now discuss briefly Eqs. (55) and (56). The first sum in Eq. (56)

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

273

corresponds to a classical view of the electron-electron interaction due to an electron distribution moving with the projectile velocity. We can realize this if we recall from Eq. (41) that the SCA probability for a bare particle with charge Z , is z:pSCA(O, r, = z; laSCA(fi, 1’ (57)

F

Also, 14A(5)12 = l@(5)12 is the target electronic density at a point with coordinate 5. Then we have, using Eq. (45):

Figure 8 displays the collision in the projectile frame. The impact parameter for the collision is b, while the effective impact parameter of the chargedensity element 10((5l2 )d35 is rl. Because the “effective” ionizing charge is not a point particle but is distributed in the vicinity of the target nucleus Z,, antiscreening has a broader impact parameter dependence when compared

Q 00

2

4

6

8

10

12

b (units of a,) FIG.11. Probability of electron loss of C 5 +on He at 2.0MeV/u as a function of the impact parameter b. The probabilities for a bare and screened target nucleus are denoted by Z2Psc, and Ps,,c,n,respectively. The probability for the antiscreening process is called Pan,i.[From Montenegro and Meyerhoff (1992).]

E. C. Montenegro el al.

274

with screening. This is illustrated in Fig. 11 where the bare target (SCA), the screening, and the antiscreening distributions for electron loss in a 2.0 MeV/u C5 on He collision are displayed. The screening probability distribution is confined to impact parameters smaller than those associated with a point particle. Antiscreening, on the other hand, is the dominant mode at large impact parameters, i.e., for distant collisions. We consider now an interesting relation that can be obtained for the second term of Eq. (55). If the closure is used in its simplest form, i.e., 0 = D, this term is the same as that appearing in Eq. (44) for the screening mode (McGuire and Montenegro, 1993; Sulik and Stolterfoht, 1993). If one defines the screening amplitude, uscreen(u,b), in such a way that I ascreen(ui, b) I’ = Pscrcen(0, b), then Eq. (44) gives: +

(@o

I C asca(u, rj) I@o)

= ZzaSCA(u, b) - ascreen(U3

b)

(59)

i

The relation shows that Panti(u, b) is related to the degree of target screening. In particular, if target screening is negligible, which corresponds to the second term in Eq. (46) being negligible or a + 0 in Eq. (50), the right-hand side of Eq. (59) is zero. In that case, only the first sum in Eq. (55) is effective. F. IMPULSE APPROXIMATION

The preceding sections discuss theoretical approaches that are based on the PWBA or the SCA, both of which are well suited to treat interactions of first order in the momentum transfer. It is of interest to consider theoretical approaches that include higher order effects, without being too complex. The impulse approximation is one such approach. The impulse approximation has been used to describe a variety of interactions ranging from nuclear collisions to various atomic collisions (Goldberger and Watson, 1964a,b, c). This approximation is particularly well suited to collisions in which the particles undergoing a transition (i.e., the projectile electrons in the case of electron excitation or loss) can be considered as “quasi-free,” uncorrelated particles that interact strongly with the perturbing collision partner (namely, the entire target, in the present case). If one starts from a many-body theory, many-body operators are replaced by two-body operators in the impulse approximation (McDowell and Coleman, 1970b). This simplifies the many-body problem considerably. On the other hand, the two-body operators are the full two-body scattering T matrices, so that there is a significant improvement over the first Born approximation in which only an effective interaction potential V,,, appears. The active projectile electrons are regarded as uncorrelated, effectively free electrons, and the cross section is one for the transition of one or more

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

275

quasi-free projectile electrons weighted by a momentum distribution that is determined by the projectile electron wave functions. The impulse approximation is valid if the collision time is much shorter than the orbiting time h/Zp of the projectile electrons, where I , is their binding energy (Mott and Massey, 1965). A less restrictive criterion is that I, must be much smaller than the projectile kinetic energy (Goldberger and Watson, 1964a). The term impulse approximation can be confusing because it is used differently by different authors. In the standard treatments, for example, Goldberger and Watson (1964a, b, c) or McDowell and Coleman (1970a,b), the effective target is the physical target, but in the present context it is the projectile. Also, the effective projectile is an unstructured projectile, which means that in the present context the impulse approximation applies only to an unstructured target, such as an electron, a proton, or a rigid (i.e., unexcited) atom. On the other hand, in projectile loss by a neutral target, the target electron(s) can be excited, so that the usual impulse approximation needs to be modified. Since one-center electron correlation is sometimes important and N electron effects can be of interest, we first consider electron loss by a projectile with N correlated electrons interacting with the target via a potential Kff, namely, quasi-elastic scattering (Goldberger and Watson, 1964a),which includes many-body effects on one center (the projectile in our case) but does not address higher order projectile-target scattering that are included in the impulse approximation. Then we consider simpler systems where there is only one projectile electron and define the impulse approximation for projectile-electron loss via an interaction V,,, with a structureless target. In the simplest version (Goldberger and Watson, 1964c), the cross section in the impulse approximation is described by the cross section for a projectile electron interacting freely with I/eff, averaged over the bound state momentum spectrum of the active projectile electron. Finally, we discuss one attempt to go beyond the usual impulse approximation and include the structure of the target in the projectile loss process. I . Quasielastic Scattering

The full many-electron problem is difficult to solve exactly. If the electron-electroncorrelation between the projectile electrons is retained, one obtains a widely used approximation that is in some ways intermediate between the exact many-electron solution and the impulse approximation. This is called quasi-elastic scattering (Goldberger and Watson, 1964a). Consider a projectile with N identical correlated electrons interacting with a target whose detailed internal structure is rigid. Quasi-elastic scattering occurs when a small excitation energy is imparted to the projectile by the target but electron correlation in the projectile is retained. For a projectile with N identical, correlated electrons, the differential cross section

216

E. C.Montenegro et al,

for projectile electron loss (with a sum of elastic plus inelastic scattering of the projectile) may be expressed in the quasi-elastic scattering approximation as (Goldberger and Watson, 1964a)

where q is the momentum transfer of the target to the projectile, F(q) is the projectile one-electron form factor corresponding to Eq. (9), but with f = s, v is the volume of the projectile, C(q) is a multiple-electron projectile correlation function and dqwo-body/dR is the differential cross section for scattering of the target from a single free projectile electron into a solid angle do. Some physics common to the ideas of this review is evident in Eq. (60). Incoherent scattering from the N projectile electrons occurs for q >> 1/ rinteraction where the two terms in square brackets go to zero and da/dR becomes linear in N , the number of projectile electrons. At the other limit, where q becomes small, elastic scattering of the target dominates, which is coherent so that the cross section varies as N 2 [since F(0) = 1 and C(0) = 01. The transition from a coherent to an incoherent cross section for a projectile with N electrons is also discussed by McGuire (1992). In the impulse approximation, C(q)/v is omitted, that is, multiple-electron projectile correlation is neglected, and the projectile electrons are treated independently, but the two-body cross section is folded with density distributions of the active projectile electron.

2. The Impulse Approximation Now let us proceed to the impulse approximation (IA) including the two-center electron-electroninteraction. We keep the target rigid, which is equivalent to the screeninginteraction in the loss process. We consider only a single projectile electron (the extension to N projectile electrons is briefly discussed later). In terms ofthe interaction, v, = I/ + Z,eZ/IR - gjl, where I/is given by Eq. (3) and the second term is the interaction of the target electrons with the projectile nucleus, which was ignored in Eq. (3), the exact transition operator T may be expressed as (McDowell and Coleman, 1970b; Wang et al., 1992)

5:

m

T=&

C

(G:Q"

n=O

where higher order scattering (e.g., with the projectile nucleus) is fully included. Here GZ is the Green's function describing the unperturbed propagation of the projectile and target electrons. We now neglect the electron-electron interactions within the target (or approximate them by a mean field potential). To obtain the impulse

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

277

approximation for a system in which the interaction Kff is a sum of interactions, we neglect in T the ZZ,e2/IR - gjl interaction of the target electrons with the projectile nucleui and replace V, by (Wang et al., 1992) Z2e2

e2

Kff = - I R + ( , J + F I R + p - 5 , 1 If Zle2/hv<< 1, it is not expected that the interaction of the projectile charge Z , with the target electrons is important in the transition of the projectile electron. This last idea is the essence of the impulse approximation. Only the target perturbs the projectile electrons in this approximation. The transition operator is now given by

c W

T,, = Kff

(90+Kff)”

n=O

where gof is the one-electron free-particle Green’s function. The n = 0 term corresponds to the PWBA approxirnation discussed in Section 1I.A. In the impulse approximation, the cross section for the transition of a projectile electron caused by the interaction V,,, is sometimes further approximated in term of the two-body cross section from the two-body matrix qAdefined by Eq. (63), namely,

da,, = sd3Pltb ,(PI 1’

dotwo-bod dR y

dR where P is the projectile-electron momentum in the projectile frame and dctwo-body/dR is the exact cross section for scattering of an unbound projectile electron with the target. If the target were a single charged particle, then dqwo-body/dR would be evaluated from the off-shell two-body coulomb T matrix for scattering of the projectile electron by the charged (target) particle (Wang et al., 1991). This expression is commonly used (Goldberger and Watson, 1964~).Equation (64) is intuitively understandable because it describes a two-body cross section folded with the momentum spectrum of the active bound projectile electron. Wang et al. (1992) point out that Eq. (64) is valid only if the projectile emits a fast electron in the backward direction. Otherwise the momentum space density distribution I $,(P) l2 of‘the ground state wave function should be replaced (McDowell and Coleman, 1970b) by the inelastic projecile form factor ($fle-iP’f’I$,). It is only when the final state is approximated by an outgoing plane wave that Eq. (64) ia valid (Wang et al., 1991). If there is more than one projectile electron, ($fle-ip‘Pl$ts) is replaced by N

($fle-lF*P~l$s)and one may recover the N dependence of Eq. (60) with j= 1

C = 0. An excellent discussion of the relation of the impulse approximation to second-order Born theories is given by Jakubassa-Amundsen (1992) who

E. C. Montenegro et al.

278

also describes how to include the antiscreening term (inelastic target scattering) in the impulse approximation. Both Wang et al. (1992) and Jakubassa-Amundsen (1992) give emphasis to cross section differentials in the energy and angle of the ejected electron in projectile electron loss (Section 1II.F). A useful impact-parameter formulation of the impulse approximation has been developed by Briggs (1977).

3. Related Models Based on the work of Brandt (1983), Zouros et al. (1989,1993) have used Eq. (64) as the basis for an intuitively appealing way to express the four-body cross section (projectile nucleus and electron, colliding with a target nucleus and electron) in terms of the three-body cross section (projectile nucleus and electron, colliding with a target electron). They use the latter cross section at a given relative velocity v between the target electron and the projectile, and weight it by the Compton profile of the target electron. If po is the momentum (in atomic units) of the target electron in the target frame, then its momentum in the projectile frame is p = po v. In many cases u >> po so that p = (u2 2vp0, + p:)’” x (v2 2~p,,)’/~. Then, in analogy with Eq. (64),

+

+

ofour-body

=

SIh

+

dpo,Jo( Po,) othree-body(u2

+ 2Up0,)

(65)

Here, Jo(poz) = ffdpo,dpo,)#o(po))2is the Compton profile, which is the probability of finding a target electron with a momentum component poz. The wave function 40(po) is the ground state momentum wave function of the target electron. There does not appear to be any clear theoretical relation between Eqs. (65) and (64). In particular, the physical assumptions are different: For loss, Eq. (64) assumes that the projectile electron is weakly bound (JakubassaAmundsen, 1992), but Eq. (65) assumes that the target electron is weakly bound. Nevertheless, the model is physically reasonable because it folds a three-body cross section for the transition of a projectile electron over the momentum distribution of the target electron in the spirit of Eq. (64) (Montenegro and Zouros, 1994). It has been successfully applied to the antiscreening parts of projectile excitation (Zouros et al., 1989) (Section 1II.A) and loss (Lee et al., 1992) (Section 1II.B). In particular, it shows very clearly that the antiscreening threshold, Eq. (25), is spread out by the electron momentum distribution in the target, in agreement with observation (see Fig. 13, Section 1II.A). G. HIGHERORDERPROCESSES

The relative importance of the antiscreening effect with respect to screening in projectile loss increases as the target atomic number decreases (Section

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

279

1I.D). On the other hand, collisions involving highly charged ions and light targets are very likely to produce copious ionization of the target. Under these conditions, one could argue that there should be no target electrons left to produce an effective screening of the target nucleus or to participate in the electron-electron-induced projectile ionization. In this section we discuss why first-order theories described in the previous sections ignore the final ionization state of the target, but are able to give good agreement with experiment for a broad range of collisions systems. Basically, we assume that the processes involving the target electron can be described within the IPM (McGuire, 1992) and that there is no relaxation of the target core within the collision time (Wang et al., .1992). Calling P,,,(b) the target ionization probability per target electron by a one-electron projectile, we have for a two-electron target the following possibilities for the screening and antiscreening probabilities P, and PA, respectively:

is the projectile-electron loss probability by the screening In Eq. (66), PScreen mode computed for a neutral two-electron target such as H, or He, and in is the antiscreening probability per target electron. Eq. (67) Panti Equation (66) considers all the possibilities for the final target states, so Ps = PSCIeen (Wang et al., 1992). This result is valid for measurements of the total projectile-electron loss cross section. However, if the recoil ion is simultaneously measured, the experiment selects only events resulting, for example, in He' or H i ions as the final state of the target. In that case, only the middle term of Eq. (66) remains. This term corresponds to a simultaneous ionization of the target and the projectile by two-center double ionization (DI) through the mutual nucleus-electron interaction and is illustrated in Fig. 12(a). In fact, DI can be associated with second-order terms of the Born expansion of the transition operator (Wang et al., 1992; Jakubassa-Amunsden, 1992). Antiscreening [illustrated in Fig. 12(b), for comparison] results in the same final states as DI. These two processes should, in principle, be added coherently in the determination of the total cross section. We return to this point later. The antiscreening process, on the other hand, needs at least one target electron to be available to participate actively in the loss process. As a consequence, target ionization can affect significantly the net antiscreening probability, as Eq. (67) shows. Targel ionization and antiscreening, however, have a completely different dynamic behavior. The antiscreening process involves small momentum transfer and large impact parameters (Montenegro and Meyerhof, 1992) and its contribution increases at small

280

E. C. Montenegro et al. a

0 ' FIG.12. Sketch of the contributions to simultaneous projectile (2,) target (Z,) ionization: (a) by double ionization and (b) by antiscreening. [From Montenegro et al. (1993b).]

emission angles (Jakubassa-Amundsen, 1992). Target ionization, as well as the DI process, involve high momentum transfers and small impact parameters (Montenegro and Meyerhof, 1991b; Jakubassa-Amundsen, 1992). Consequently, even if Pionis large (near unity), the impact-parameter range and the continuum electron energies to which this process contributes most have a small overlap with the corresponding impact dependence of Panti. Hence, one can assume PA x 2Panti.The same argument can be used to neglect interference terms arising in the cross-section calculations where DI and antiscreening must be considered, and a sum of probabilities can be used instead of a sum of amplitudes as a reliable approximation (see also Section 1II.D) (Wang et al., 1992; Montenegro et al., 1992a).

111. Comparison with Experiment We now discuss some of the experiments that have been done in the last few years to examine specifically the contributions of the electron-electron

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

28 1

interaction to projectile electron excitation and loss. These experiments have dealt mainly with cross sections, both singly and in coincidence between the projectile and the target or between the projectile and the emitted electrons. Very few impact-parameter probability measurements have been made; some are still in progress. On the other hand, electron spectra have been measured, both singly and in coincidence with the projectile. The effect of the two-center electron-electron interaction is strongest in the case of H, and He targets (where the nuclear effects are weakest), so we limit our discussion to such targets. Since for the present review the most relevant interactions are screening, antiscreening, and DI, the emphasis is on experimental results that characterize and isolate these interactions.

A. PROJECTILE-ELECTRON EXCITATION Zouros et al. (1989) [see also Richard (1990)l were able to avoid the action of the screening process and to isolate the antiscreening interaction by examining the excitation of a transition in a three-electron projectile (05+, F6+),which is forbidden for a screened coulomb interaction, but is allowed for an antiscreening interaction: the spin-flip transition ls22szS to ls2s2p4P. Using high-resolution 0" Auger spectroscopy, Zouros et al. (1989) determined the excitation cross section of this transition in H, and He targets. Figure 13 shows their results for F6+. For the H, target, there is a clear rise in the cross section above the antiscreening threshold, Eq. (25), indicated by an arrow on the abscissa. This rise is smeared out by the momentum distribution of the target electrons. The dashed and dash-dotted curves give the expected rise for each target, computed with an impulse-like approximation (Section II.F.3). At lower energies, there is an increasing background, which might be due to a transfer-loss process. This two-step process could also lead from the ls22s2S to the ls2s2p4P state through the intermediate formation of the ls22s2p state by capture of a target electron, with subsequent loss of a 1s electron by action of the (screened) target nucleus, in the same collision. Because capture would be involved, the cross section for this process would decrease rapidly with energy, distinguishing it from the antiscreening process, which increases with increasing energy near the threshold. Zouros et al. (1989) mention that an estimate of the transfer-loss cross section is 2 orders of magnitude smaller than the experimental cross section and point to another process that could also form the low-energy background: an excitation of the projectile electron, by the screened target, to the ls2s2p2P state with subsequent exchange with a target electron to the 4P state. A calculation of this cross section has not been made.

E. C. Montenegro et al.

282 12-

n

,

1

1

1

1

1

1

1

1

1

1

1

1

1

10 -

1

1

1

1

1

1

0 F6; He

N

E 0

0

v r

0 -

8 -

+

6 -

0

eeE (x1.78)

4 -

4

2 -

* 0

W

6 Q

I=

d

00

eeE (r 1.73) --*.?Y' J, , I , 8 12 16 2 0 24 2 8 32 36 40 0 I

I

4

**

a!! I

I

I

I

I

ENERGY (MeV) FIG.13. Production cross section for the ls2s2p4P excited state in F6++ H, PROJECTILE

(O), He (0) collisions, as a function of projectile energy. The arrow on the abscissa indicates the location of the antiscreeningthreshold, Eq. (25). The dashed and dash-dotted curves give the antiscreening cross sections, called eeE on the figure, for H, and He, respectively, computed by an impulse-like approximation, Eq. (65). [From Zouros et af. (1989).]

B. PROJECTILE-ELECTRON LOSS CROSS SECTION Projectile-electron loss has been studied with many one-electron projectiles (H', He+, Liz+,B4+,C 5 +,07+ ) colliding with Ho, H,, and He targets. With many-electron projectiles, multiple-electron loss can occur, a subject beyond the scope of this review. We select a few experiments specifically designed to determine the screening and the antiscreening cross sections. Hulskotter et al. (1991) have measured one-electron loss cross sections for various projectiles between Liz+ and U90+ in H, and He targets at energies from 1 to 400 MeV/u. The charge change of the projectiles on traversal of a gas cell was determined. For the lower energies, with Li2+ to 0 ' ' projectiles, a 6-cm-long windowless gas cell was used at the Stanford F N Tandem Van de Graaf accelerator. For the high-energy work, with AuS2+ and U86,90+ projectiles, a 2-m-long gas cell was constructed at the Lawrence Berkeley Laboratory Bevatron accelerator. This cell was closed by flapper valves, which opened only during the beam pulse from the accelerator, thus keeping the pressure in the cell nearly constant. Figures 14 and 15 give sample results for 07+ and AuS2+ one-electron loss cross sections. In each case, the curves were calculated with the PWBA expressions using the Anholt prescription for an antiscreening cross section (Section II.C.3). The dashed curves show the continuation of the screening

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

0

1

2

283

3

PROJECTILE ENERGY ( M e V l n ) FIG.14. Electron loss cross section for 07+projectiles colliding with (a) H, and (b) He targets, as a function of the projectile energy. The arrows give the location of the antiscreening threshold, Eq. (25). The solid curves give the loss cross sections computed with the PWBA formulation of Section II.C.3. (Anholt method). The dashed curves give the continuation of the calculated screening cross sections above the antiscreening threshold. (c) Ratio of H, to He target cross sections. The solid curve is the ratio expected on the basis of the PWBA. The dashed curve is the ratio of the screening cross sections. [From Hiilskotter et al. (1991).]

cross section above the antiscreening threshold, Eq. (25). The solid curves are the sum of the screening and antiscreening contributions. Experiment and theory are in very good absolute agreement, but the separation into two contributions cannot be based on this experiment alone. In taking the ratio of the H, to the He cross sections, an approximate threshold can be discerned [Fig. 14(c)] and a clearer picture emerges. At the lower energies, the cross-section ratio (H, molecule/He atom) is nearly equal to the ratio expected for the unscreened nuclei: ( 2 x 1)/( 1 x 4) = 0.5. At higher energies, the ratio is higher, consistent with an additional contribution due to the target electrons, which is relatively higher for H, than He. The solid curve is the theoretically expected ratio. Figure 15(c) shows the same ratio for Au5,+ projectiles.

284

E. C. Montenegro et al. U 8

n

1 1 I 1 1 1 1 l

I

I 1 1 1 1 1 1 1

1

I 1 1 1 1

a

4

t v

-0 ‘ 0 5 12 Lu

m m m

0 a

8

0

4

0

0

0.7

4

a 0.6 u r 0.5

I”

0.4

10

lo2

10’

PROJECTILE ENERGY (MeVlnucleon)

FIG.15. Same as Fig. 14, but for AuszCprojectiles. [From Hulskotter et al. (1991).]

Loss measurements have been made by Lee et al. (1992) with C 2 + and 04+ projectiles. These authors fit their data, as well as that of Fig. 14, by using a PWBA formulation for screening and an impulse-like approximation for antiscreening (Section II.F.3). C. PROJECTILE-ELECTRON Loss AND CAPTURE If the projectile energy decreases and its atomic number increases, capture of a target electron by the projectile is expected to increase to such an extent that loss and capture may occur in the same collision. Because such a process would not change the charge of the projectile, if loss is determined by charge change, a correction will have to be applied to the observed yield of charge changed projectile ions. In principle, two electrons are available for capture from a target such as

ROLE OF TWO-CENTER ELEC'TRON-ELECTRON INTERACTION

285

H, or He, but actually this is true only for the screening process. In the case of the antiscreening process, one target electron needs to be in an excited or ionized state to be able to contribute the momentum transfer necessary for projectile loss. Hence the observed loss cross section is (Montenegro et al., 1992b) gloss

= 2~

:s 1:

Pscreen(1

+ 2~

N gscreenC1

- PJ2bdb

Panti(1 - Pc)bdb

- Pc(0)12

+ OantiCl - pc(O)I

(68)

are the screening and antiscreening loss probHere, Psc,,,,(b) and Panti@) abilities, calculated as described in Sections II.E.3 and II.E.4, respectively. The probability P,(b) is the capture probability for one target electron, and 1 - Pc is the probability that a target electron is not captured, The second expression is approximately correct if the target is much lighter than the projectile, in which case Pc(b) extends to much larger impact parameters than either Psc,,,,(b) or Panti(b).One sees that if P,(O) becomes appreciable can become considerably smaller than if capture with respect to unity, o'loss is ignored. Figure 16 shows one-electron loss cross sections for C3+ + H,, He measured by the charge change method (Montenegro et al., 1992b). At low energies, the measured cross sections are considerably smaller than the PWBA cross sections predicted by the formulation of Section II.C.4 where capture is not taken into account (solid curve in Fig. 16; the dashed curve is LT~,,,,,,).Because Pc(0) is poorly known for this case, Eq. (68) was used to deduce Pc(0) semiempirically and then compare it with theoretical models. The comparison, shown by solid curves in Fig. 17, follows from a unitarization procedure proposed by Sidorovitch et al. (1985) together with an OBK expression for the single-electron capture probability, whereas the dashed line is based on suggestion of Anholt et al. (1987) that all singleparticle probabilities should be set to 0.999 if the model calculation exceeds unity. The former procedure appears to be the better one. Similar results are obtained for O5+ projectiles.

D. SEPARATION OF SCREENING AND ANTISCREENING By measuring the final target charge site in coincidence with projectiles that have undergone electron loss, it is possible to separate experimentally the screening and antiscreening contributions (Montenegro et al., 1992a, 1993b). Consider the collision of the one-electron projectile He+ with H,. In principle, any collision producing He++,but leaving H, in its ground state,

E. C. Montenegro el al.

286

h

P

E

h

I]

E

0

0

2

4

6

8 1 0 1 2

E (MeV) FIG. 16. Projectile one-electron loss cross section for C3+ on (a) H, and (b) He. Experiment: solid circles from Montenegro et al. (1992b), open squares from Anholt et al. (1988), open triangles from Goffe et al. (1979). Theory: dashed curve is the screening and full curve is the screening plus antiscreening cross section calculated according to the PWBA formulation of Section II.C.4. [From Montenegro et at. (1992b).]

must be due to the screening process, whereas if H: is produced, the antiscreening process must have been effective. However, if one detects only H, or H: in coincidence with H e + + ions, background processes can exist. In the collision

He+ + H, +He+ +

+ H,(+e-)

(69)

the background process is target excitation (TE). This process leaves the target recoils neutral and, hence, cannot be distinguished from the screening process. TE is part of the antiscreening interaction, but turns out to add only a minor fraction to the measured cross section. In the collision He+ + H,

--t

H e + + + H:(+2e-)

(70)

the background process is two-center DI: The screened target nucleus can ionize the projectile in the same collision in which the screened projectile nucleus ionizes the target. This process leaves the projectile and target ions and the electrons in exactly the same final states as the antiscreening

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

287

U

h

0

v

a" 0.4

_ _ _ - -- - \ \

0.2

0.0 0

1

2

'.

3

4

E (MeV) FIG.17. One-electron capture probability at zero impact parameter for C3* ions on (a) H, and (b) He. The semiempirical points are obtained from Eq. (68),using the data shown in Fig. 16. For an explanation of the curves, see Section 1II.C of the text. [Adapted from Montenegro et al. (1992b).]

process. As discussed in Section II.G, the two processes in principle should be added coherently, but do not need to be. DI decreases rapidly with increasing projectile energy, so that above 2 MeV it contributes only a small amount to the total cross section for collision (70). Another minor correction has to be applied to the calculated cross section, though: Because only the target ionization part of the antiscreening is determined experimentally, the target excitation part has to be subtracted from the threshold total antiscreening cross section. [This is just the part that has to be added to the cross section for collision (69).] The simplest DI calculation considers the two ionizations as independent events (see also Wang et al., 1992). If only one electron on the H, target is ionized, the probability of ionization by the (screened) He' projectile nucleus is equal to 2Pi0,(b)[1 - Pi,,,(b)], where Pion@)is the probability of ionizing a single target electron. If the probability of projectile loss by the (screened) target nuclei is PScreen, the DI cross section is

-

The probability PS,,,,.is calculated as described in Section II.E.3. The probability Pioncannot be calculated by perturbation theory because it is

E. C. Montenegro et al.

288

close to unity. It has been estimated by Montenegro et al. (1993b) by means of a prescription of Sidorovitch et al. (1985). This prescription can be checked indirectly by calculating the ordinary target ionization cross section nion= 472

J ,Pion(1

- Pion) b db

and comparing this with a cross section measurement using the collision He+ + H,

+ He+

+

H;(+e-).

(73)

Experimentally, the cross section for collision (70) is determined from a coincidence measurement between H: target recoils and charge analyzed H e + + projectiles. The cross section for collision (69) is obtained from a

12

8

4

12 8 4

0 0.0

1.0

2.0

3.0

4.0

FIG.18. (a) Cross section for electron loss of Het on H, not followed by target ionization, as a function of the projectile energy. The theoretical curves, computed with the PWBA formulation of Section II.C.4, have the following meaning: S, screening; TE, target excitation part of antiscreening; BToT= S f TE. (b) Cross section for electron loss of He' on H,, in coincidence with H l recoils, Eq. (70). The theoretical curves have the following meaning: DI, double ionization; A, antiscreening; TE, target excitation part of antiscreening; C,,, = DI + A - TE. [From Montenegro et al. (1992a, 1993b).]

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

289

He+ singles measurement, from which the cross section for collision (70) is subtracted. When that is done, the result shown in Fig. 18(a) is obtained. This cross section is dominated by screening (S), with a minor contribution from TE, resulting in the total cross section for (69), B T o T = S + TE. The cross section for collision (7oj is shown in Fig. 18(b). Here, the cross section is analyzed in terms of DI, which dominates at smaller energies, and antiscreening (A) from which TE has been subtracted, which dominates at larger energies. The overall cross section is CT o T = DI A - TE. Figures 18(a) and (b) show the successful separation of the screening and antiscreening contributions, essentially on a purely experimental basis. The shift from the DI to the A process, when both the projectile and the target are ionized, is illustrated in impact-parameter space for the collision Hef + He + He' + + He +(+ 2e-) in Figs. 19(a) to (c) (Montenegro et al., 1993b). At low energies, the DI contribution, which follows essentially the product of the target and projectile screening probabilities, dominates. As +

+

0.08 0.06 0.04

0.02 0 h

P v

n

P

0.04

3.0MeV 0.12

C

0

0s 1.0 125

2.0 2 5 3.0

b(ad FIG.19. Probability distribution, multipliad by impact parameter, for electron loss of He' colliding with a He target, accompanied by target excitation and ionization, as a function impact parameter. The impact parameter is in atomic units. Three different projectile energies are illustrated in (a), (b), and (c), respectively. The evolution of the double- ionization (DI) and antiscreening (A) processes can be seen as the projectile energy increases, explaining the trend of the DI and A cross sections shown in Fig. 18(b) for a H, target. [From Montenegro et al. (1993b).]

290

E. C. Montenegro et al.

the energy increases, the wider A distribution takes over the cross section, as expected.

E. ELECTRON-LOSS PROBABILITY As discussed in Section II.E.4, the electron loss probability as a function of impact parameter is different for the screening and antiscreening process. In particular, the antiscreening probability is wider, since it represents the interaction of an excited target-electron cloud with the projectile, whereas the screening interaction represents the interaction of the screened target nucleus with the projectile. One can obtain the impact-parameter distribution for projectile-electron loss from the scattering angle distribution of those projectiles that have undergone electron loss. This type of measurement has been made by Montenegro et al. (1993a). A beam of Liz+ (l.lMeV/u) or C5+ (0.9 - 3.1 MeV/u) was passed through a pair of collimators (0.3mm in diameter, separated by 2m) into a windowless gas cell containing H, or He. The emerging beam was charge analyzed by a pair of electrostatic deflection plates and then impinged on the surface of a microchannel plate. A two-dimensional impact pattern of that part of the beam that had undergone electron loss was obtained. By integrating the concentric rings of the pattern, a scattering angle distribution was determined. The integrated distribution, representing the total loss cross section, was normalized to the direct cross-section measurements of Hiilskotter et al. (1991). Typical scattering angles in such a direct measurement can be determined down to a few milliradians, which is large enough so that impact parameter (b) can be computed directly from the scattering angle. A major effect, however, of the physical width of the beam and of the resolution of the microchannel-plate surface develops on the derived impact-parameter distribution: an artificial peak can appear in the distribution. For this reason, we refer to the measured probability distribution later as the apparent distribution, Papp(b)* A comparison between experiment and theory is made in Fig. 20 for 1.1MeV/u Li2+ + H, electron loss. Figure 20(a) gives bPapp(b). The units of b are in Liz+ Bohr radii. Curves S and A are the calculated curves for screening and antiscreening, respectively, folded with the resolution function. Agreement between theory and experiment is very good. Nevertheless, the separation into two contributions cannot be made on the basis of this experiment alone. If we divide the experimental distribution by the calculated (resolution-folded) distribution for screening only, we obtain the result shown in Fig. 20(b). At small b values, the ratio is close to unity, showing that screening is dominant there. At larger b values, the ratio exceeds unity, showing that antiscreening exceeds screening at large impact

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

29 1

0.1 0

0.00

n

0.06

v

n '

0.04

n

0.02 0

b 4 -

-

0 0

3 -

1 -

0 0.1

-

0 I

I

I

I

IIIII

1

I

I 1 1 1 1 1 1

10

b(a1) FIG.20. (a) Apparent electron loss probability, multiplied by impact parameter, as a function of the impact parameter, for 1.1 MeV/u Liz' + H, collisions. The impact parameter is in units of the Liz+ Bohr radius (al). The curves marked S and A are the screening and antiscreening probabilities, respectively, computed with the SCA and folded with the overall beam and detector resolution. The solid curve is the sum. (b) Ratio of the measured loss probability to that computed for screening alone. The solid curve is the ratio expected on the basis of the SCA. [From Montenegro et al. (1993a).]

parameters. The solid curve is the theoretically expected ratio. The results for 0.9 to 3.1 MeV/uC5+ + H, electron loss are similar, but with a He target, discrepancies with the theory are found in all cases. This breakdown of the theory may be caused by the use of atomic wave functions for the projectile and the target. With a He target and such relatively light projectiles, a molecular distortion of the wave function is likely at low energies. At present, an experiment is under way (Froschauer et al., 1993) to determine the impact-parameter distribution for electron loss in C 5 + + He by recoil-ion time-of-flight spectroscopy (Ullrich et al., 1991). In this method, the time of flight of recoil ions (He' or H e + + )emitted approximately at 90" to the beam is used to determine the scattering angle of the projectile to a precision of a few mkroradians. Those projectile ions that have undergone electron loss are selected by electrostatic deflection. At the time of writing, the results are not yet complete, but one hopes to observe

292

E. C. Montenegro et al.

a separation between the antiscreening and DI events similar to that shown in Fig. 19 for He' + He.

F. ELECTRON SPECTROSCOPY Electron loss appears as a definite feature in electon spectra from ion-atom collisions with projectiles carrying one or more electrons. One might hope that these spectra reveal the loss mechanisms discussed previously-screening, antiscreening, and two-center DI-but perhaps also others not observed in cross section or impact-parameter studies. It is instructive to compare electron spectra for bare and one-electron projectiles colliding with the same atom. One of the simplest examples is given by H + + He and Ho + He. Figures 21(a) and (b) show spectra for 0.5and 1.0-MeV impact energy, respectively, at an emission angle of 30" (lab.) (Heil et al., 1992). The dashed lines (B) give experimental spectra for the bare H + projectile and are typical of target ionization spectra by bare

lo2

10'

E(eW

FIG.21. Doubly differential electron emission cross sections from H + He collisions, at a laboratory angle of 30", as a function of the electron energy, at a projectile energy of (a) 0.5 MeV and (b) 1.0MeV. All the curves are experimental and have the following meaning: B, singles electron spectrum for H + projectile; E, singles electron spectrum for Ho projectile; and P, electron spectrum from Holoss events. [Adapted from Heil et al. (1992).]

ROLE OF TWO-CENTER ELECTRON-ELECTRONINTERACTION

293

projectiles (Rudd and Macek, 1972). They consist of a smooth continuum peaking at very low energies and end in a binary-encounter peak due to collisions between the projectile and target electrons, in which the target nucleus does not absorb any appreciable momentum. Such electrons are typically called “loosely bound” (Drepper and Briggs, 1976). For each Ho spectrum, two data curves are given. One, marked E, is the total singles electron spectrum. The other, marked P, is the spectrum of electrons in coincidence with H projectiles emerging from the scattering cell. Hence, P is the spectrum of target and projectile electrons emerging during the loss process. Heil et al. (1992) have normalized the curve P to E at the loss peak, but actually these doubly differential cross sections differ by the cross section for electron ejection from the target, as the work of DuBois and Manson (1990) shows. The most prominent difference between the H + + He and Ho + He spectra is the just-mentioned loss peak, which occurs at an electron velocity very close to the projectile velocity. This peak represents the electrons emitted from the projectile by action of the (screened) target nucleus. It contains both the broad continuum, and the binary-encounter peak, but transformed from the projectile to the laboratory frame of reference and slightly shifted because of the projectile-electron binding energy (Drepper and Briggs, 1976; Jakubassa-Amunsden, 1993). In the total electron spectrum E, the loss peak sits on a broad continuum, which is similar to the spectrum B of target electrons. There is one difference between the two spectra at low electron energies: the doubly differential cross section for E is less than for B. The reason is that these electrons originate from large impact parameter, low momentum transfer collisions where projectile screening is most effective (Heil et al., 1992). From the point of view of this review, it is most interesting to examine the loss spectrum P. In this spectrum also, the loss peak sits on a broad spectrum whose shape is quite similar to B, except for the absence-or at least strong reduction-of the binary encounter peak. The broad continuum has been ascribed to target electrons emitted together with the projectile loss electrons in the same collision (DuBois and Manson, 1990). This becomes apparent in a theoretical analysis of the spectra. Manson and DuBois (1992) have analyzed spectrum E of Fig. 21(a), using only the first-order PWBA described in Sections 1I.B and 1I.C. The results are shown in Figs. 22(a) and (b). The dotted curve in Fig. 22(a) is the spectrum of electrons emitted from the target (He) by action of the screened nucleus of the projectile (HO). The dash-dotted curve is the spectrum of electrons emitted from the target in the antiscreening process. At low electron energies, this process dominates the spectrum for reasons mentioned at the end of Section 1I.G: The antiscreening process produces low-energy electrons because of the low momentum transfer in these collisions. The low-energy electrons !from the projectile are shifted into the +

E. C. Montenegro et al.

294 1 O-lg

I

I

1

1

I

I

0.5MeV

h L

I

H-He 1

v)

> u \

E

h

L

v)

,

0

I

10

I

1

20

I

,

30

I

I

40

I

50

ENERGY ( R y ) FIG.22. (a) Analysis of curve E in Fig. 21(a) using the first-order PWBA. Dotted and dash-dotted curves represent the electron spectra from the target, emitted by the projectile screening and antiscreening processes, respectively. The short-dashed and long-dashed curves represent the electron spectra from the projectile, emitted by the target screening and target antiscreening processes, respectively. The solid curve is the sum. For details, see Section 1II.F. (b) Comparison of the sum curve from (a) (solid curve) with the experimental doubly diaerential cross section (dashed curve), curve E of Fig. 21(a). [From Manson and DuBois (1992).]

loss peak, but those from the target appear as low-energy electrons in the laboratory system. The target spectrum is discussed in detail by Manson and DuBois (1992). The binary encounter peak is not examined in this reference, but a summarizing discussion can be found in Jakubassa-Amundsen (1993). The short-dashed, peaking curve in Fig. 22(a) is the electron spectrum emitted from the projectile (Ho) by action of the screened nucleus of the target (He), transformed to the laboratory system. The long-dashed curve is the projectile electron spectrum produced in the antiscreening process. The shift to lower energies is due to the lower momentum transfer in these collisions, as mentioned earlier (Manson and DuBois, 1992). Figure 22(b) compares the sum of the calculated spectra (solid curve) with experiment [dashed curve, identical to curve E of Fig. 21(a).] Agreement is good, except in the region below the loss peak. Various authors have

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

295

examined the loss spectrum (P) of Fig. 21(a) more carefully. They find that at forward electron emission angles, the region around the loss peak can be fitted quite well with first-order Born approximation (Heil et al., 1992; Jakubassa-Amundsen, 1992, 1993), but at lower energies this theory fails to match the experimental cross section, as noted by Manson and DuBois (1992). At backward angles, where measurements by Kover et al. (1988) exist between 95" and 170" for the similar collision system of He' + He, it is not possible to fit even the loss-peak region by first-order Born approximation. A second-order approximation must be invoked, which is essentially the two-center DI process (Jakubassa-Amundsen, 1992, 1993; Wang, et al., 1992; Kuzel et al., 1992). The latter authors point out that the electron-electron interaction (antiscreening) cannot provide the momentum transfer needed to propel projectile electrons into the backward direction. A central (small impact parameter) scattering on the target nucleus is required in order to account for the backward cross section, exactly as occurs in DI. It is satisfying to realize that in this way a consistent picture of the loss process is beginning to emerge.

IV. Conclusion Figure 23 summarizes in an approximate manner the loss mechanisms discussed in this chapter by means of Feynman-type diagrams in which distance is displayed horizontally and time progresses vertically upward. The projectile (P) and target (T) nuclei (N) are represented by thick lines and the electrons (e) by thin lines. The momentum transfer q involving nuclei is indicated by a thick dashed line, and that involving only electrons is shown by a thin dashed line. The processes summarized under (1) are of first order and those under (2) of second order in q. The simplest loss process is that induced by a bare target, in which the momentum transfer is produced by a single coulombic interaction between the target nucleus and the projectile electron. The plane wave Born approximation (PWBA) and the semiclassical approximation (SCA) are well suited to compute cross sections and impact-parameter dependent probabilities, respectively. The equivalence of these two theories for cross-section calculations has been demonstrated (Bethe and Jakiw, 1968b). There are many experimental tests of total cross sections, but the use even of the simplest targets (H, or He) in the study of electron spectra from ionization by bare projectiles produces discrepancies with the theory (Rudd and Macek, 1972; Madison and Merzbacher, 1975). The simplest two-center electron-electron interaction appears in (oneelectron) projectile loss by screening;, in which the (single) target electron

E. C.Montenegro et al.

296

LOSS

MECHANISMS

Bare target

Screening

Target excitation \An

Target ionization

tiscreeni ng

(b)

Double ionization

Transfer- loss

FIG.23. Summary of loss processes: (a) first-order processes and (b) second-order processes. The thick solid lines represent nuclei (N), the thin solid lines, electrons (e). On the left side of each diagram the target (T) is represented, on the right side, the projectile (P). The thick horizontal dotted lines are momentum transfers (4)in which the nuclei are involved; the thin dotted lines are momentum transfers in which only electrons are involved. Only the presently ascertained second-order loss processes are shown. The transfer-loss process does not result in a net projectile loss.

remains in its ground state throughout the collision. Again, the PWBA and the SCA have been fruitfully applied. The calculations show that one should not think of screening as a simple potential shielding effect, because it is q dependent. This q dependence has been observed in electron spectroscopy (Heil et al., 1992). In recent years, much attention has been given to antiscreening, the enhancement of projectile loss or excitation by momentum transfer from an excited or ionized target electron. This process had already been predicted

ROLE OF TWO-CENTER ELECTRON-ELECTRON INTERACTION

297

in the 1950s by Bates and Griffing on the basis of the PWBA. They pointed out that, in this theory, a direct momentum transfer to the projectile electron from the target electron requires that the latter be excited or ionized. The target nucleus acts only as a spectator. This fact makes the impulse approximation (IA) an attractive alternative to the PWBA, because it provides implicitly the sum over unobserved target states, which is needed for most comparisons with experiment. Experimental total one-electron loss cross sections agree with the PWBA or IA approaches, especially if in the former the sum over target states is evaluated with improved closure methods. As soon as loss cross-section measurements become differential, for example, in target charge or in emitted-electron energy and angle, the need for processes beyond first order becomes apparent. We indicate two such processes in Fig. 23, Section (2). The main distinguishing feature of these processes in relation to the first-order processes is the role played by the projectile nucleus in the loss process, in ionizing (scattering) or capturing the target electron. Through either mechanism, projectile loss is affected: Ionization enhances loss, capture decreases the observed loss. The theory of second-order loss processes is a subject of current interest (Jakubassa-Amundsen, 1992, 1993; Wang et al., 1992). A simplified approach uses the independent-particle model (Montenegro et al., 1992a, 1993b). Further advances in this field are to be expected. From the experimental point of view, electron spectroscopy and recent studies in recoil momentum spectroscopy (Froschauer et al., 1993; Cocke, 1993; Dorner et al., 1994; Wu et al., 1994) promise to reveal new details about the role of second-order processes. Such theoretical and experimental investigations will keep the field of two-center electron-electron interactions an active area of research.

Acknowledgments This work was supported in part by CNPq (Brazil), National Science Foundation Grants PHY-9019293 and INT-9101057 (Stanford University), and by the Division of Chemical Sciences, Office of Energy Research, U.S. Department of Energy (Tulane University).

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Goldberger, M. L., and Watson, K. M. (1964b). Collision Theory, Chap. 11, Sec. 2, Eq. (200), Wiley, New York. Goldberger, M. L., and Watson, K. M. (1964~).Collision Theory, Chap. 11, Sec. 2, Eq. (60b), Wiley, New York. We neglected a factor of u(P), which is approximately unity in our applications. Hansteen, J. M. (1989). In Spectroscopy and Collisions of Few-Electron Ions (Ivanescu, M., Florescu, V., and Zoran, V., eds.), p. 173, Academic, New York. Hartley, H. M., and Waiters, H. R. (1987). J. Phys. B 20, 1983. Heil, O.,DuBois, R. D., Maier, R., Kuzel, M., and Groeneveld, K.-0. (1991). Z. Phys. D 21,225. Heil, O., DuBois, R. D., Maier, R., Kuzel, M., and Groeneveld, K.-0. (1992). Phys. Rev. A 45, 2850. Hiilskotter, H.-P., Feinberg, B., Meyerhof, W.E., Belkacem, A., Alonso, J. R., Blumenfeld, L., Dillard, E. D., Could, H., Guardala, N., Krebs, G. F., McMahan, M. A,, Rhoades-Brown, M. E., Rude, B. S., Schweppe, J., Spooner, D. W., Street, K., Thieberger, P., and Wegner, H. E. (1991). Phys. Rev. A 44, 1712. Jakubassa-Amundsen, D. H. (1992). Z. Phys. D 22, 701. In this work, the DI process is called incoherent projectile-target ionization and the antiscreening process (less target excitation) is called coherent projectile-target ionization. The screening interaction is called elastic and process resulting in target excitation, inelastic. Jakubassa-Amundsen, D. H. (1993). Proc. Fifrh Workshop on Fast Ion-Atom Collision Processes, Debrecen, Hungary, July 17-19, 1993. Nucl. Instrum. Meth. B, 86, 82. Kabachnik, N. M. (1993). J. Phys. B26, 3803. Kover, A., Szabo, Gy, Gulyas, L., Tokesi, K., Berknyi, D., Heil, O., and Groeneveld, K . - 0 . (1988). J. Phys. B 21, 3231. Kuzel, M., Heil, O., Maier, R., Lucas, M. W., Jakubassa-Amundsen, D. H., Farmery, B. W., and Groeneveld, K.-0. (1992). J. Phys. B 25, 1839. Lee, D. H., Zouros, T. M. J., Sanders, J. M., Richard, P.,Anthony, J. M., Wang, Y. D., and McGuire, J. H. (1992). Phys. Rev. A 46, 1374. Levy 11, H. (1969). Phys. Rev. 184,97. Lodge, J. G . (1969). J. Phys. B 2, 322. Madison, D. and Merzbacher, E. (1975). In Atomic Inner-Shell Processes (Craseman, B., ed.), p. 2, Academic, New York. Manson, S . T., and DuBois, R. D. (1992). Phys. Rev. A 46, 6773. McClure, G . W. (1968). Phys. Rev. 166, 22. McDowell, M. R. C., and Coleman, J. P. (1970a). Introduction to the Theory of Ion-Atom Collisions, Chap. 7, Sec. 2, North-Holland, Amsterdam. McDowell, M. R. C., and Coleman, J. P. (1970b). Introduction to the Theory of Ion-Atom Collisions, Chap. 6, Sec. 8, North-Holland, Amsterdam. McGuire, J. H. (1992). Adu. At. Mol. Opt. Phys. 29, 217. McGuire, J. H., and Montenegro, E. C. (1993). Int. Conf Phys. Electron. At. Collisions, 18rh, Book of Abstracts (Anderson, T., Fastrup, B., Folkman, F., Knudsen, H., eds.), p. 656, University of Aarhus, Denmark. McGuire, J. H., Stolterfoht, N., and Simony, P. R. (1981). Phys. Rev. A 24, 97. McLaughlin, B. M. and Bell, K. L. (1983). J. I’hys. B 16, 3797. Meron, M., and Johnson, B. (1990). Phys. Rev. A 41, 1365. Messiah, A. (1961). Quantum Mechanics. North Holland, Amsterdam. Meyerhof, W. E., Hiilskotter, H.-P., Dai, Qiang. McGuire, J. H., and Wang, Y. D. (1991). Phys. Rev. A 43, 5907. Montenegro, E. C., Belkacem, A., Spooner, D. W., Meyerhof, W. E., and Shah, M. B. (1993a). Phys. Rev. A 47, 1045. Montenegro, E. C., Melo, W. S., Meyerhof, W. II., and de Pinho, A. G. (1992a). Phys. Rev. Lett. 69, 3033.

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Montenegro, E. C., Melo, W. S., Meyerhof, W. E., and de Pinho, A. G. (1993b). Phys. Rev. A 48, 4259. Montenegro, E. C., and Meyerhof, W. E. (1991a). Phys. Rev. A 43, 2289. Montenegro, E. C., and Meyerhof, W. E. (1991b). Phys. Rev. A 44, 7229. Montenegro, E. C., and Meyerhof, W. E. (1992). Phys. Rev. A 46, 5506. Montenegro, E. C., Sigaud, G. M., and Meyerhof, W. E. (1992b). Phys. Rev. A 45, 1575. Montenegro, E. C., and Zouros, T. J. M. (1994). Phys. Rev. A (in press). In this reference, it is shown the relationship between Eq. (65) and PWBA [Eq. (21)]. Mott, N. F. and Massey, H. S. W. (1965). The Theory of Atomic Collisions, Chap. XII, Sec. 5.2, Oxford University Press, Oxford. Richard, P. (1990). In X-Ray and Inner-Shell Processes, AIP Conf. Proc. No. 215 (Carlson, T. A., Krause, M. O., and Manson, S. T., eds.), p. 315, American Institute of Physics, New York. Ricz, S., Sulik, B., Stolterfoht, N., and Kadar, J. (1993). Phys. Rev. A 43, 1930. Riesselmann, K., Anderson, L. W., Durand, L., and Anderson, C. J. (1991). Phys. Rev. A 43, 5934. Rudd, M. E., and Macek, J. H. (1972). Case Studies At. Phys. 3, 47. Schiwietz, G., and Grande, P. L. (1992). Nucl. Instrum. Meth. B 69, 10. Sidorovitch, V. A,, Nicolaev, V. S., and McGuire, J. H. (1985). Phys. Rev. A 31, 2193. Stolterfoht, N. (1989). In Spectroscopy and Collisions of Few-Electron Ions (Ivascu, M., Florescu, V., and Zoran, V., eds.), p. 342, World Scientific, Singapore. Stolterfoht, N. (1991). Nucl. Instrum. Meth. B 53,477. Sulik, B., and Stolterfoht, N. (1993). Personal Communication. Toburen, L. H., Stolterfoht, N., Ziem, P., and Schneider, D. (1981). Phys. Rev. A 24, 1741. Ullrich, J., Dorner, R., Lencinas, S., Jagutzki, O., Schmidt-Bocking, H., and Buck, U. (1991). Nucl. Instrum. Meth. B 61, 415. Vegh, L., and Sarkadi, L. (1993). J. Phys. B 16, L727. Victor, G. A. (1969). Phys. Rev. 184,43. Walters, H. R. J. (1975). J. Phys. 8 4 , L54. Wang, J., Reinhold, C. O., and Burgdorfer, J. (1991). Phys. Rev. A 44, 7243. Wang, J., Reinhold, C. O., and Burgdorfer, J. (1992). Phys. Rev. A 45,4507. In this work, the screening process is called singly elastic. The DI process is called uncorrelated double inelastic and the antiscreening process, correlated double inelastic. Wittkower, A. B., Levy, G., and Gilbody, H. B. (1967). Proc. Phys. SOC.London B 91, 306. Wu, W., Wong, K. L., Ali, K., Chen, C. Y.,Cocke, C. L., Frohne, V., Grese, J. P., Raphaelian, M., Walch, B., Dorner, R., Mergel, V., Schmidt-Bocking, H., and Meyerhof, W. E. (1994). Phys. Rev. Lett. 72, 3170. Zouros, T. M. J., Lee, D. H. and Richard, P. (1989). Phys. Rev. Lett. 62, 2261. Zouros, T. M. J., Lee, D. H., Sanders, J. M., and Richard, P. (1993). Nucl. Instrum. Meth. B 79, 166.

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ADVANCES IN ATOMIC MOLECULAR. AND OPTICAL PHYSICS. VOL. 34

INDIRECT P R 0CESSES IN ELECTRON I-MPACT IONIZATION OF POSITIVE IONS D. L. MOORES Department of Physics and Astronomy University College London London. United Kingdom

and K. J, REED High Temperature Physics Division Lawrence Livermore National Laboraiory Livermore. California 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . B. Excitation-Autoionization (EA) . . . . . . . . . . . . . . . . . . C . Resonant-Excitation (Capture) Process . . . . . . . . . . . . . . . D. Ionization-Autoionization (IA) . . . . . . . . . . . . . . . . . . I1 Basic Ideas: The Independent Processes Model . . . . . . . . . . . . A . Excitation-Autoionization . . . . . . . . . . . . . . . . . . . . . B. Resonant Capture Processes . . . . . . . . . . . . . . . . . . . . C. Multiple Ionization . . . . . . . . . . . . . . . . . . . . . . . . I11. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Method of Close-Coupled Target States . . . . . . . . . . . . . . IV. Comparison of Theoretical and Experimental Data . . . . . . . . . . A . Helium-like Ions . . . . . . . . . . . . . . . . . . . . . . . . . B. Li-like Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Ions with 2s2, 2pq Configurations . . . . . . . . . . . . . . . . . D . Na-like Ions . . . . . . . . . . . . . . . . . . . . . . . . . . E. Mg-like Ions . . . . . . . . . . . . . . . . . . . . . . . . . . F. Heavier Monovalent Ions . . . . . . . . . . . . . . . . . . . . . G. Metal Ions with 3pq and 3dq Ground States . . . . . . . . . . . . H . Heavy Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Multiple Ionization . . . . . . . . . . . . . . . . . . . . . . . J . Very Highly Charged Ions . . . . . . . . . . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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301 302 303 303 303 305 305 309 310 311 311 324 324 328 336 337 342 344 353 362 379 398 421 422

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I Introduction Although a precise theory would not make any distinction between the two. it is both customary and practical to distinguish between direct and indirect 3101

Copyright 0 1994 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003834-X

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contributions to electron impact ionization of positive atomic ions. In direct ionization, bound electrons are expelled from an outer or inner shell by direct impact with the incident electron. The term indirect ionization is usually reserved for processes that occur via intermediate autoionizing states. In this review we examine the various indirect ways by which electrons can be ejected from an atomic system. The most important of these processes is excitation-autoionization (EA) in which the incident electron excites the target ion (usually but not always from an inner shell) into a short-lived state with energy in excess of that required to ionize the system, which subsequently decays by autoionization (an Auger transition). Alternatively,the incident electron may be captured by the target ion into a short-lived state of a compound ion, which then decays by two successive autoionizing transitions. This sequence of processes is called resonant-excitation double autoionization (REDA) although the words capture or recombination might be more appropriate than excitation in this context. If the compound state decays directly by simultaneous ejection of two electrons, the process is called resonant-excitation auto-double-ionization (READI). These three processes all lead to single ionization of the original target ion. Multiple ionization events (which by analogy could be assigned the acronyms EMA, REMA, REAM1 where M stands for multiple) can also result if the state into which excitation or capture occurs lies above the energy required to eject more than one electron. The principle indirect mechanism for multiple ionization is however ionization-autoionization (IA) in which direct inner shell ionization (primary ionization) by the incident electron is followed by single or multiple autoionization, either by successive, cascading transitions or by simultaneous ejection of two or more electrons. In all processes involving intermediate autoionizing states, competition between autoionization and radiative decay occurs. An autoionizing state is also able to make a radiative transition to a pure bound state, the final step in the dielectronic recombination process. This radiative stabilization will reduce the overall ionization cross section by an amount depending on the relative probabilities for autoionization and radiative decay. In the limit that the radiative decay probability is much the larger of the two, this process dominates and the indirect contribution to ionization is very small. If X'+ denotes a z-times ionized ion, *represents an excited state, and ** a doubly excited state, the principal mechanisms for ionization can be summarized by the following:

A. DIRECTIONIZATION xz+ +

-,

X(z+n)+

+ (n -t 1)e

(1)

The ionizedion X ( z + n ) +can be left in either the ground or an excited state.

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B. EXCITATION-AUTOIONIZATION (EA)

Then, either

or (Xz+)* X z ++e+hv --$

C. RESONANT-EXCITATION (CAPTURE) PROCESSES REDA. (x(Z-u+1**

+

(xz+)*+ e,X"+l'+

+e+e

(3)

READ1

D.IONIZATION-AUTOIONIZATION (114) Xz+

+e

-+

(X('+')+)** t e +e

-+

P+")+ + (n + 1)e

(5)

In processes (3) and (4) any of the autoionizing states may also undergo radiative decay as in (2) but for clarity this has not been shown here. In process ( 5 ) steps involving intermediate stages of ionization (cascades) have likewise been omitted. A schematic cross section for single ionization, showing contributions from A, B, and C is plotted in Fig. 1. Structure due to EA was already a recognized feature of the ionization cross sections of neutral atoms (Fox et al., 1953) when Goldberg et al. (1965) suggested that it might also contribute significantly to elecrtron impact ionization of ions. They estimated that the inclusion of the process would increase the ionization rates of highly ionized Fe ions by factors of 2 to 3, leading to corresponding changes in ionization balance curves for Fe in the solar corona. Bely (1967) included excitations from further inner shells and found even larger enhancements. Jordan (1969) included EA in her calculation of the ionization equilibrium of elements between carbon and nickel in low-density plasmas, finding that in some cases it reduced the tendency of dielectronic recombination to raise the estimated temperature at which a species has its maximum abundance. The first experimental demonstration of indirect ionization in ions was by

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plus direct inner shell

READ1 resonances L-

m

c

+

5

b c

0 .c U

a J

m

direct outer shell

,/

... ..

v)

m

e

U

/'

plus excitation autoionization

Electron energy FIG.1. Schematic cross section for electron impact single ionization of a positive ion showing contributions from direct outer and inner shell ionization, excitation-autoionization,REDA, and READI. [From A. Miiller (1991), Fig. 2.13.1

Peart and Dolder (1968) who revealed the dramatic effect of EA on the ionization cross section of Ba' using an intersecting-beams apparatus. Measurements on other alkali-earth ions soon followed. Subsequent experimental work showed that in some complex ions EA could totally dominate direct ionization. Following the suggestion by LaGattuta and Hahn (1981) that REDA should make an important contribution to indirect ionization, experiments were performed in an attempt to detect it. The measurements of Gregory et al. (1987) in the Na-like ion FeI5+ revealed complicated fluctuations in the cross section that could be attributed to REDA, but failed to confirm the large predicted enhancement. Calculations by Tayal and Henry (1989) gave only a small. REDA contribution, but subsequent work by Chen et aE. (1990) including more contributing states brought theory and experiment into good agreement. Meanwhile, REDA was identified in heavy metal ions by Miiller et al. (1988a). Whenever EA is important it is also likely that it will be accompanied by a corresponding REDA contribution. REDA is doubtless present in much of the early experimental data in unresolved form. The expression auto-double-ionization was coined by Henry and Msezane (1982) to distinguish processes that occur only through simultaneous double

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

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Auger emission. A good example is to be found in Li-like ions where doubly excited configurations ls2s2p21 of the He-like system formed by electron capture are lower in energy than the lowest autoionizing configurations ls2sz and ls2s2p of the Li-lieke system. Single autoionization of ls2s2p21 can only occur to bound states of the Li-like ion. (An energy-level diagram is shown later in Fig. 14.) Although the effect is very small, high-precision experimental studies have been able to produce clear evidence for it in several Li-like ions (Muller et al., 1988b; Hofmann et al., 1990). Multiple ionization resulting from inner shell ejection followed by Auger transitions was, like EA, a well-known mechanism for ionization of neutral atoms (Fox, 1960) before it was first demonstrated in Ar ions by Muller and Frodl (1980). Numerous examples of indirect multiple ionization in rare gas and heavy metal ions have since been recorded, with ejection of up to four electrons. In some ions, multiple ionization is initiated by inner shell excitation or electron capture rather than ionization. Since the first unambiguous experimental demonstration of an indirect ionization process in a positive ion, enormous progress has been made and indirect ionization in all its various forms has been observed and measured in a wide range of ions from Li' to U86+. The early crossed-beam work was restricted to ions of low charge but the development of ion sources such as the electron beam ion source (EBIS) (Donets, 1976), the Penning ion gauge (PIG) source (Mallory and Crandall, 1976), and the electron cyclotron resonance (ECR) ion sources (Meyer, 1985) has enabled the measurements to be extended to more highly charged ions. More recently the availability of heavy ion storage rings such as the TSR at Heidelberg and the ESR at Gesellschaft fur Schwerionenforschiing in Darmstadt, and the construction of Electron Beam Ion Traps (EBIT:) (Levine et al., 1988) have enabled very highly stripped ions to be studied. Work with energy-resolved electrons with a spread of around lOOev (Peart et al., 1989a) and the development of the fast scan technique (Muller et al., X988a) has enabled the structures to be examined in much finer detail than before. At the same time, theoretical work has kept step with the experimental advances to the extent that comparisons may now be made between calculated and measured data of a comparable degree of accuracy, and thus meaningful interpretations can be made of the behavior of the cross sections.

11. Basic Ideas: The Independent Processes Model A. EXCITATION-AUTOIONIZATION We assume here that the interference between direct and indirect ionization can be neglected and the two processes are treated as independent. In many cases this assumption is an excellent one and we shall refer to it as the

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D. L. Moores and K J. Reed

independent processes model. The ionization cross section for an incident electron energy E , above the first autoionization threshold may then be written

+ 1Q;"(E0)Bi

Q'"'(Eo) = Qio"(Eo)

(6)

i

where the sum is over all energetically accessible autoionizing states i; Q'"'(E,) is the total cross section; Q'""(E,) the direct ionization cross section; and QP(Eo) the excitation cross section of autoionizing state i, regarded for this purpose as if it were a pure bound state. The branching ratio B,is

where A:j is the probability of autoionization to state j of the ionized ion and Aik is the probability of radiative decay of state i to a bound state k of the target ion. The size of the EA contribution thus depends on the relative values of the excitation and the ionization cross sections and also on the relative values of the autoionization and radiative decay probabilities. For ions of low charge z the autoionization probabilities tend to be much larger than the radiative decay probabilities, i.e., A'<< A" and B + 1 (low charge)

(8)

Thus for ions of low charge we have Q'"'(E,) = Qi""(Eo)

+ 1 QI"(Eo)

(9)

i

The EA contribution is then obtained merely by adding the excitation cross sections for all contributing states to the direct ionization cross section. Knowledge of A" and A' is not required. Systems for which we might expect EA to be particularly significant will be those with filled or almost filled subvalent shells (of p, d, or even f electrons) immediately below a smaller number of valence electrons. In these systems the probability of excitation of an inner electron is relatively higher owing to the larger occupation number of the subvalent shell. A good example is the excitation of the 2p shell in Na-like ions: 2p63s -, 2p53snl Na-like ions have been the subject of much attention, both theoretically and experimentally. Along an isoelectronic sequence the direct ionization cross section scales as z4 while the excitation cross section scales as zz, so we expect the size of

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

307

the indirect contribution to increase relative to the direct contribution with increasing charge on the ion. This is illustrated later in Figs. 16, 17, and 18 in the Na sequence Mg', A12+,Si3'.. The same effect is observed along an isonuclear sequence, although the changing atomic structure as valence electrons are successively removed also affects the shape of the cross sections. This is illustrated in Fig. .SO for Xe ions 4d"5s25p6-q. For the lower values of 4, EA is comparable. with direct ionization but by Xe6+ it is a factor of 10 larger. Another effect must be considered.,however, in highly charged ions. As z increases, the radiative probability A' for a given state increases as z4 but the autoionization probability remains roughly constant. Thus Bidecreases, and as a result EA is reduced, along an isoelectronic sequence; the excited states prefer to stabilize rather than autoionize. In the limit, A' >> A" and B + 0 (high charge)

(10)

For sufficiently large z, radiative decay can destroy the EA contribution completely and only direct ionization will be important: Q'"'(E,)

:=

Qion(Eo)

We should emphasize, however, that (8) and (10) just mark general trends. For all ionic charges there will be some individual transitions for which neither of the inequalities may be assumed, and EA contributions can persist even for very highly stripped ions such as U8'+ (Reed et al., 1991). Also, not all radiative decays lead to stabilization, since an autoionizing state may make a radiative transition to another autoionizing state with a branching ratio close to unity, which promptly autoionizes. This should be taken into account when making calculations. The detailed behavior of the ionization cross section depends on the structure of the target ion. At each autoionization threshold an additional term will be added to the sum (6). Since the excitation cross section of a positive ion is finite at its threshold, this will lead to an abrupt jump in the total cross section. As more and more states are excited, more and more jumps will occur, their height decreasing as the excitation cross sections decrease. They eventually merge into a semicontinuum converging on an inner shell ionization threshold. If the lowest configuration attainable by excitation of an inner shell electron lies well above the outer shell ionization threshold, as in Na-like ions, the cross section will display a series of fairly well-defined jumps well above threshold, starting in the region of or above the peak in the direct ionization. The schematic cross section shown in Fig. 1 is an example of this type of behavior, which corresponds to category (i) distinguished by Burgess et al. (1977) and Burgess and Chidichimo (1983). If the autionization threshold lies well below the outer shell ionization

D. L. Moores and K J. Reed

308

threshold, as in B-like ions [category (ii) of Burgess et al.] the EA contribution will manifest itself as a large number of small jumps, usually not resolved, immediately above the outer shell ionization threshold and up to the inner shell threshold. As z increases the principal quantum number no of the first autoionizing state increases (Moores, 1979), ranging from a value of 4 in N2+ to 11 in Fe21+.For such systems one would not expect to observe structure in the cross section, the EA giving instead a smooth overall enhancement detectable as a faster rise from threshold than that due to direct ionization alone (Fig. 2). The cross sections for many ions, such as Ti2+,combine the features of both categories. In other configurations, such as the 2p6 ground state of Ne-like ions, the EA contribution is small and very little structure is seen. As z increases along either an isoelectronic or an isonuclear sequence, the binding energy of a given autoionizing state will increase faster than the level separations in the ( z + 1)-times ionized ion, leading to a progressive switch from autoionizing to bound state at some value of z. This brings about sudden variations in the size of the EA contributions at points along an isoelectronic sequence, since when a state becomes bound it no longer appears in the sum (6). In such circumstances, estimates of the EA contribution by isoelectronic extrapolation or interpolation (“scaling”) should only be attempted with extreme caution after a careful study of the energy level structure of the systems of interest. In the example of B-like ions mentioned earlier, the 2s2pnl states “move down” relative to the 2s2p states with increasing z, giving a corresponding increase in no. An example in an isonuclear sequence occurs in Sb4+and Biq+ (Muller et al., 1985~)in which the ndQ(n+ l)s2(n l)p3-q configurations (n = 4,5) switch from autoioniz-

+

48

68 78 88 98 108 Incident Energy (ev) FIG.2. Electron impact ionization cross section of N 2 + in 10-’8cmz. Dashed curve: Coulomb-Born approximation, omitting EA; full curve: including EA. [From D. L. Moores (1979). Experimental points due to Aitken et al. (1971).]

58

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

309

ing to bound as q increases from 1 to 2. This leads to unusual behavior in the ionization cross sections of these ions.

The resonant capture processes can be regarded as a special case of EA in which the intermediate state is a resonance in the scattering of an electron by the target. The effects of REDA appear as peaked structures in the total cross section below each autoionization threshold. Each state into which capture occurs is a member of a Rydberg series of autoionizing states, which converges onto a series limit, which is itself an autoionizing state. The large possible number of states may mean that to observe the structure high experimental resolution is required and it can be difficult to distinguish between resonant capture and the onset of higher EA processes. REDA may be included in a theoretical treatment by using some approximation that includes resonances such as close coupling to calculate the excitation cross sections appearing in (6). If allowance is to be made for the fact that the resonances may also decay radiatively it may be easier to include the resonant capture process by adding to (6) terms of the following form:

where QrP(Eo), the capture cross section, is obtained on application of the principle of detailed balance:

where AE is an energy bin width and E , the energy released in the inverse process, 1, is the ionization energy of hydrogen in the same units as AE and Eo, and gk and gi are the statistical weights of the intermediate state k and the initial state of the ion, respectively. The branching ratio for REDA is given by

If this procedure is followed, an approximation that does not allow for resonances should be used to obtain the excitation cross sections in (6). In the case of READI, a different formula must be used. Instead of (12) one must add

3 10

D. L. Moores and K. J. Reed

where QFP(E,) is the capture cross section into the intermediate state f and depends on the single autoionization rate of state f via an expression of the form of (13). The capture cross sections depend on the energy bin width AE, which is to some extent arbitrary. It is often allowed for by convoluting each resonance with a Gaussian of width commensurate with the typical experimental energy resolution. The branching ratio in (15 ) is

The auto-double-ionization probability A:! depends critically on the electron-electron correlation and is best calculated by means of many-body perturbation theory (Pindzola and Griffin, 1987). The calculated value is very sensitive to the basis set chosen and to the continuum phase selected in the matrix elements.

C. MULTIPLE IONIZATION The ejection of two of more electrons may take place directly or via the indirect inner shell IA process, in which primary ionization creates a hole state which decays either by a single Auger transition or by a cascade of such transitions to give net multiple ionization. Multiple ionization may also occur via multiple primary ionization, multiple Auger decay of a hole state, or any other energetically allowed combination of these processes. It may also occur via inner shell excitation or even resonant capture followed by multiple Auger transitions. Direct multiple ionization is in general of higher order in the interaction than the corresponding indirect process, and the only theory available suitable for low-impact energies is the classical binary encounter approximation of Gryzinski (1965). Fortunately, the direct cross sections decrease with increasing ionic charge and also with the number of electrons ejected, and for complex ions ionized more than once multiple ionization is usually dominated by the indirect processes, which, because they generally involve one-electron transitions, are easier to deal with theoretically. If the only important contributors are direct ionization and IA, the total double ionization cross section is given by

Qvz(E,) = Q i ~ ~ d+i C r QP(E0)Bi i

where Q$"idir is the direct double ionization cross section, QY(E,) is the single ionization cross section corresponding to hole state i, and the sum is over all hole states that can be created which contribute to double ionization. The branching ratio is Bi:

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

311

where r is the sum of all possible decay probabilities of state i. The threshold for double ionization corresponds to the minimum energy required to eject two electrons, the onset of direct double ionization. Below the IA threshold, only direct double ionization can occur. The onset of the IA process is marked by a sudden change of slope in the cross section. A good example of this behavior is shown later in Fig. 59 for Ar ions. From Eq. (17) we see that if the direct cross section is small and the branching ratio unity, the total cross section is just determined by the sum of the cross sections for single ionization from the inner shells. If excitation followed by double autoionization (EDA) is also important, a term

C Q,”‘(Eo)BP i

has to be added, where Qjx(E,) is the cross section for excitation of an inner shell electron to state j and BF is the appropriate branching ratio. This process will contribute below the IA threshold, where it will augment the double ionization. Because net multiple ionization can result from many possible combinations of primary ionization (single or multiple) and decay pathways, the general expression for the cross sec:tion equivalent to (17) will be correspondingly more complicated if it is to include all contributions. The notation C T ~ , ~ +is, , also used to define the cross section for n-fold ionization of ionization stage q of a species.

111. ‘Theory A. METHOD OF CLOSE-COUPLED TARGET STATES 1. Fundamentals In this method, introduced by Jakubowicz and Moores (1981), no distinction is made between direct and indirect ionization. EA is included by inserting into the matrix element foir the ionization amplitudes an ejected electron wave function, which is a solution of the close-coupling equations describing the motion of the electron in the field of the ionized ion. The autoionizing levels appear as the closed and bound channel resonances, which arise naturally in the solutions of the close-coupling equations. Radiative decay may be taken into account by modifying the matrix

312

D. L. Moores and K. J. Reed

elements describing ionization according to the radiation damping theory of Davies and Seaton (1969). We consider the electron impact ionization of a complex positive ion with nuclear charge 2 and (N + 1) electrons. Let I , be the energy in Rydbergs required to ionize the (N + 1)-electronion and leave the N-electron ion in state a. If kg is the incident electron energy, conservation of energy requires that

ki - I,

+ kh

= 4.2

= E,

(19)

where qh and k,Z are the ejected and scattered electron energies, respectively. Neglecting the possibility that the incident electron may be captured nonresonantly and two bound electrons ejected, the ionization cross section may be expressed in the form (in units of nag)

Q,(ki) = {oEu'2

c.(ki, 4:) dqh

(20)

where cr,(k$, qh) is the single-differential cross section o,(G 4:) =

d(Gl4:)

+ 4 ( k k 4:)

- d!"(k?i, 4:)

(21)

and

and

where crd, oe, and d"'are, respectively, the direct, exchange, and interference contributions to the single-differential cross section; and f,(q,, k,) is the direct and q,(q,, k,) the exchange ionization amplitude. In Eqs. (22) to (24), oois the statistical weight of the initial state. The sum over m symbolizes summation over all initial and final azimuthal quantum numbers. (The term direct is used here to differentiate direct from exchange ionization rather than to distinguish direct ionization from indirect ionization as in Section I.) The exchange amplitude is assumed to be given by dqa, 'a) =f(ka, S 3

(25)

which is the correct relationship if exact wave functions could have been used. With these assumptions, the exchange single-differential cross section is given by

&(ki, 4:)

=

4%k 3

(26)

313

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

Jakubowicz and Moores (1981) introduce an approximation for calculating the amplitudes and hence cross sections in which the incident and scattered electrons are represented by distorted waves calculated in an effective central potential, and the initial state of the ( N + 1)-electron ion and its final continuum state are described by close-couplingwave functions. The direct ionization amplitude is expressed in the form

f,(qa,ka) =

- (2n)-

e (y'/(out, qa) I V(r 1 . . * r N + 1~ ko, ka) IT0S

512 i A

L n O>

(27)

where So, Lo, noare the total spin, total orbital angular momentum, and total parity quantum numbers of the initial state, respectively, and A(qa, k,) is a phase factor. The continuum function Yf(out, qa) is of the form (Burke and Seaton, 1971)

x e:
(28)

+

where represents the ( N 1)-electron quantum numbers = SLHM,M,; z1 = 2 - N ; a(T,ya) is an angular coefficient and q,(z, k) is the Coulomb phase: q,(z, k ) = argr(l

+ 1 - iz/k)

(29)

The sum over y is over all possible channels y associated with state CI of the N-electron ion. The lya,my. are the quantum numbers of the ejected electron and q: is the channel energy. The functions YsoLono and YFm(out,& are ( N + 1)-electron wave functions of the standard close-couplingform (Burke and Seaton, 1971)

i= 1

p= 1

The first summation in Eq. (30) is over n, free channels and the second over nb bound channels. These functions are solutions of the set of coupled integrodifferential equations describing the interaction of an electron with is a solution an N-electron ion, which is diagonal in SLI1. The term with boundary conditions corresponding to all channels closed. The functions YTE(out,4:) correspond to at least one channel open. The free channel functions @$(out,q;), which form part of Y;z(out, q:), are of the form

where d is an antisymmetrizing operator and Yi(rN:1)is a function of all coordinates of the (N + 1)-electron system except the radial coordinate rN+ 1. The asymptotic form of Fi,(out, r ) for open channels is

314

D. L. Moores and K. J. Reed

where S is the scattering matrix and

+?

= q;

exp & iCi

lit

(33)

The notation out in the preceding equations signifies that the boundary conditions correspond to an outgoing wave only in channel y, if y is open. For closed channels i, Fi,(0Ut, r)

- 0.

(35)

r-0

where +(k,r) is a distorted wave expressible in the form of a partial wave expansion. The radial components of +(k, r), G,(k,r), satisfy a radial equation of the form

l(1 + 1)

[$+kz-

rz

1

+ V ( r ) G,(k, r) = 0

with U(r) being some spherically symmetric potential satisfying U(r)

-

r-tm

2(Z - N - 1) r

(39)

with boundary conditions

G,(k, r)

N

r-m

k-”’ sin[C

+

r,(k)]

(41)

where z,(k) is the non-coulomb part of the phase. When the amplitude is expressed in the form of Eq. (27), the similarity between this formulation of the theory and that of photoionization becomes apparent since the form of the matrix element in Eq. (27) is similar to that of the transition matrix element for photoionization but with the dipole

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

315

operator replaced by the more general multipolar expression V(r ...rN+ 1, k,, ka). Unlike its counterpart in the case of photoionization, however, this operator is energy dependent, though it is slowly varying as a function of energy. Substituting Eqs. (25) through (41) into Eqs. (19) through (24) and performing all possible angular integrations, sums over quantum numbers and other algebraic operations necessary to simplify the resulting expressions, the single-differential cross sections (SDCS) direct or exchange can be expressed in the form

where qr(OUt, 6) = (yta(out, 4:) I Qr yo)

(43)

In Eqs. (42) and (43), we have put 2e = 4:. The quantity I v,‘”,) incorporates the electron-ion interaction and the initial state wave function and it varies slowly, that is to say, nonresonantly, as a function of E. For simplicity we assume that all angular factors and coefficients arising from the reduction of Eqs. (25) through (41) have been included in Resonant behavior arises as a result of bound or closed channels included in the final-state wave functions appearing in Eq. (43).

vf.

2. inclusion of Radiative Decay Modifications to Eqs. (42) and (43) to allow for the possibility of radiative decay of the resonance states may be obtained from the time-dependent theory developed to treat dielectronic recombination by Davies and Seaton (1969) and Bell and Seaton (1985). Iff,(&,t ) is the amplitude for the state in which the system composed of an ionized ion plus electron plus radiation field is in state a with energy E at time t with no photon present, Davies and Seaton show that

L(E, 00)

=

{ l - 2nzD(~)[1+ L ( E ) ] - ~ D * ( E ) ) ~ , ( E , O )

(44)

where D(E)= < y a ( & ) IdIY,)

(45)

is the matrix element associated with a radioactive decay from state a to a bound state p, and L(&)= --in

where the limit

c

-+

s

D*(&’)D(&’) de’ E’

-E-ic

0 is taken after evaluating the integral with 5 > 0. The

D. L. Moores and K. J. Reed

316

matrix elements (43) are then replaced by qr(Out,

E)

=

[l - 2n2D(1 + L)-'D*] (YFb(OUt, q:)(yrY o )

(47)

3. Two Simple Cases

To illustrate the method, we now consider two simple cases-one open plus one bound channel, and a Rydberg series of resonances due to one closed and one open channel.

a. One Open Plus One Bound Channel. In this case (30) has the form 1 r

Y(E)= Y -F(E)

+ C(E)@

(48)

Following the methods of Bell and Seaton (1985) we find Y(out, E ) = 2ie-'"

(E

- a*)-'[(& - 8') 'Po)+ Y ( ' ) ]

(49)

where q is a slowly varying phase shift, YcO) is an unmodified continuum function, Y ( ' )a bound channel function modified by interaction with the continuum and

where A" is the autoionization probability. One then obtains from (43), neglecting radiative decay, qr(out,

E)

= -2ieiq(& -

where

a)-'[(& - cO)P + Q]

(51)

P = (P') v Yo)

(52)

Q = (Y(')lV Yo)

(53)

If the possibility of radiative decay is taken into account, one obtains an where expression similar to (51) but with %" replaced by 9'

9"

= EO

i 2

- - (A" + A')

(54)

where A' is the radiative transition probability. Introducing the variables X =

2(E

A"

- 80)

+ A'

and q =

2Q P(A" A')

+

we find

I qr(out, E ) 12 =

+qy + x2

4P(X

1

(55)

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

317

This expression is the well-known Fano profile (Fano, 1961) and describes the behavior of the SDCS in the vicinity of a resonance.

b. One Open Plus One Closed Channel. To treat this case we use the many-channel quantum defect theory [reviewed by Seaton (1983)l. Bell and Seaton (1985) show that under certain assumptions the scattering matrix may be expressed in the form s11 = x11

- x12cxz:!

- g(v)e-2"'v1-1 x 2 1

(57)

where channel 1 is the open channel and channel 2 the closed channel. The matrix x is the scattering matrix at energies for which both channels are open and its elements vary slowly with energy. The effective quantum number v in the closed channel is obtained from

where 8,is the energy of the ion core and

+

(59) where A' is the radiative transition probability for a transition within the core, assumed to be the principal decay mode of the resonances. Following a method similar to that of Dubau and Seaton (1984) for photoionization, substitution of the asymptotic forrn of the continuum function into the matrix element (43) yields g(v) = exp[7rv3A'/(z

T(0ut) = T,(out) -

i(l - e-4na)1/2exp(in6 - ina - A) T2(0Ut) exp[ -274 f l f A)] - exp[ -2ni(v a)]

+

(60)

where

A=

v3Ar 2(z 1)2

+

The scattering matrix and hence the matrix element T(out) can be seen from (60) to have poles at

v = n - [:a + i(/3 + A)]

(64) where n is an integer. The squared modulus of (60) gives the resonance profile. When A -,0, we obtain I ~ ( o u t )= l ~II'~~(x where P and q are real and

+ 4)2/(1 + x2)

(65)

D. L. Moores and K. J. Reed

318

x = tan n(v

+ cl)/tanh nj

(66)

which is of the same form as that obtained by Dubau and Seaton (1984) for photoionization. For very narrow resonances, (65) reduces to a Fano profile with

i:

x = ( E - EO) -A’

These results show how resonant structure is produced in the SDCS as a result of including bound or open channels in the final-state wave function for the target continuum state.

4. Application to Ne-like Se

A good example of an application of the theory is the calculation by Moores and Reed (1989) of ionization of the 2p53s state of Ne-like Se. The configurations 2s22p5,2s2p6, and 2s22p43sof the F-like ion were included in the expansion (30). Some results are shown in Fig. 3 in which the SDCS for an incident energy of 240Ry is plotted as a function of ejected electron energy between 37.9 and 39.1Ry. The peaks are a result of 2s22p43s3d 3F.

(1930)

(2500)

(1150)

Incident energy = 240 Ry

4010)

y\ I 37.9

I

38.1

I

I

I

1

38.3 38.5 38.7 Ejected electron energy (Ry)

3P

I

38.9

I 39.1

FIG.3. Resonances in the single-differential cross section of SeZ4+.[From Moores and Reed (1989), Fig. 3.1

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

resonances, i.e., the transitions Se24+(2s22p53s)+ e

-.

319

Se24+(2s22p43s3dSL) +e

for SL = ,P, ,D, ,F. Interference effects between neighboring resonance profiles and the background are apparent, but they are not large. No experimental measurements have yet been made of SDCS for electron-ion ionization. A given resonance will appear in ad when q2 attains the threshold value and in be when k2 attains its threshold value. Thus, as the incident energy increases, it first appears in be (because k2 > q2), when k 2 is first able to reach the threshold energy at q2 = 10. At a resonance in either ador d it is found that the direct-exchange interference term is small. The course of the SDCS becomes increasingly complicated as patterns of resonances appear in both cdand be and overlap with each other as the energy changes. As the Rydberg series of resonances approaches the series limit, the resonances become closely packed and eventually only their average magnitude is meaningful. Techniques based on MQDT for averaging over the resonances have been described by Moores (1989). If radiative decay is neglected, the averaged cross section immediately below each threshold for excitation of the ionized ion does not join smoothly to the above-threshold cross section. This discontinuity, the so-called Gailitis jump (Gailitis, 1963) is a result of the loss of flux into newly open channels. When radiative decay is included the discontinuity disappears, the averaged resonant contribution tending to zero at threshold. The effect of a resonance on the total cross section is illustrated in Fig. 4 where we show the SDCS as a function of the slow (ejected) electron energy for three incident energies El, E,, and E , straddling a resonance. If E, is the resonance energy and E~ is the fast (scattered) electron energy, Eq. (17) gives

Suppose that

The cross-section curves as we pass through this resonance have the forms shown in Fig. 4(a), (b), and (c). The total cross section Q [Fig. 4(d)] is obtained by integrating a(tJ over E, and it is seen that there will be an abrupt change in Q(E,) as E increaser;from El to E , due to the appearance of the resonance from the left. At E , the resonance has not yet appeared; at E , it is beginning to appear; and by E , the whole profile lies within the energy range of integration. Owing to the width of the resonance state, the jump in the cross section will not be in the form of a discontinuity in slope, the onset being smoothed out by the resonance profile. Only in the limit of a delta-function profile will a discontinuity of slope be obtained.

Incident Energy E l

Ejected Electron Energy

I

Incident Energy Ep

Ejected Electron Energy

Incident Energy E3

Ejected Electron Energy

V

I

I

I

Incident Electron Energy

FIG.4. Schematic picture of the behavior of the single-direrential cross section at three incident energies straddling a resonance, together with the corresponding total cross section.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

321

5. The Independent Processes Model Regained

a. One Open Plus One Bound Channel. We now examine the circumstances under which the theory discussed in the present section reduces to (6). For the case of one open plus one bound channel, the SDCS is given by b(E)

4CI(&- EO)P+ Q12 - E ~ ) ' + $(A" +

=(E

We assume the resonance to be sufficiently narrow that P and Q may be considered constant over the profile. We also have y(1)= ( g X ) - 112

Y@

(70)

where y2 = A". Integrating Eq. (69) over the profile we find that the cross term is equal to zero and

Bearing in mind the definitions of YcO) and @ we see that the first term in Eq. (71), when averaged over the resonance, is proportional to the differential ionization cross section don(&), and the squared matrix element l<~lV~0N2

is proportional to the excitation cross section. The total cross section obtained by integrating (71) over the whole ejected electron energy range is given by, since the off-resonant contribution of the second term is negligible,

Qt(Eo)=

+ ~A QA"+ Ae

IT(&)de = Qi""(Eo)

x ( E E )

(72)

in agreement with (6).

b. One Open Plus One Closed Channel. A similar result is obtained in the case of one open and one closed channel. At a pole in the S-matrix

eP = b2 - (z + 1)2/2(n - [a

+ i(B + A)]I2

(73)

Assuming the resonances to be narrow compared with their separations, so that /3 + A << ct, this becomes Ep

i 2

= 8 , - -(Aa

+ A')

(74)

where b, = 8, --

(z

+

1)2

2(n - a)2

(75)

D. L. Moores and K, J. Reed

322

+

A" = 48(z 1)' 2(n - a)3 A' =

4A(z + 1)2 2(n - a)3

Expression (60) may be integrated over a resonance using the techniques based on contour integration discussed in detail by Bell and Seaton (1985) and Dubau and Seaton (1984), which are similar to those used to obtain Gailitis averages. We find that the cross term vanishes and that (vo = n - a):

where G(vo) = exp(4nA) - 1 x 4nA = 4nArv:/2(z

+ 1)'

(78)

We have 1x121'

x 4x8

= 4nA"v:/2(~

+ 1)2

(79)

Thus,

Note that I TI1' is proportional to the cross section into the unmodified open channel; in other words, the direct ionization cross section, while I T2I2gives the cross section into a single closed channel resonance (since the resonances do not overlap), or the excitation cross section. Again, we recover an expression of the form (6). 6. Discussion

We have shown that Eq. (6) gives a good representation of the total cross section if the individual contributions to direct and indirect ionization are interpreted as averages over resonances, provided these are narrow compared with their separations. For the two simple cases considered in detail, the cross terms giving the interference between direct and indirect ionization actually vanish when the averaging procedure is carried out. The closecoupled target state (CCTS) method must be used if the structure in the differential cross section is required or if the resonances are sufficiently broad to invalidate the assumptions leading to (6). An advantage of the method is that it does give some of the best direct cross sections available since an elaborate ejected electron wave function including full term dependence is used. It also provides a good method for treating direct ionization with simultaneous excitation of the target. If the total cross sections are

INDIRECT PROCESSES IN EILECTRON IMPACT IONIZATION

323

required, it is usually sufficient to use (6). Indeed it is often to be preferred because it is simpler and because excitation cross sections of the highest accuracy can be employed. The majority of theoretical studies to date have used this method. In the CCTS method, the excitation cross sections are necessarily restricted to a distorted-wave-type approximation and REDA can only be included by means of (12) and (13). Use of excitation cross sections calculated by a method that includes the effect of scattering resonances allows REDA to be automatically incorporated. If Eqs. (6), (7), (12), and (13) are used (the independent processes approximation), the theoretical problem reduces to the calculation of large numbers of cross sections for inner shell excited states, autoionization, and radiative transition probabilities, and the success of any calculation will depend on how well this can be done. In calculations of excitation cross sections of positive ions it is important to make adequate allowance for the interaction between the continuum electrons and the target ion, and to include the Coulomb interaction, exchange, distortion, and correlation. Moreover, an accurate representation of the target is absolutely essential. Methods that have been used to calculate the excitation cross sections contributing to the indirect ionization of positive ions are, in increasing order of sophistication, the Coulomb-Born, distorted wave, and close-coupled approximations. Many different variations of these methods exist. In the Coulomb-Born and distorted-wave methods, a wide range of choices for the target orbitals is available, and in the distorted-wave method the effective potentials for the continuum electrons may also be chosen in a number of ways. In ionization of ions with incomplete shell structures, both initial and final configurations will admit a large number of terms. In the configuration average (CA) distorted-wave approximation of Pindzola et al. (1986a), a single cross section for transitions between two configurations is determined, thereby making enormous savings in labor over a level-to-level calculation that would calculate a separate cross section for each pair of initial and final levels. The calculations have been extended to allow for the energy-level spread within each clsnfiguration by an average statistical model (ASM) in which the CA cros:s section is statistically partitioned over all levels of the final configuration. If distorted-wave methods prove to be of insufficient accuracy, then we find it necessary to use a close-coupling approximation, which has reached its apotheosis in the most recent form of the R-matrix method (Berrington et al., 1987). When applying the close-coupling method to excitation of inner shell excited states, it is usually necessary to drop an infinite number of intervening excited states, and hence open channels, from the expansion, but this has not proved to be a major source of error in practice. Once it is assumed that direct and indirect ionization are independent

324

D. L. Moores and K. J. Reed

processes so that Eqs. (6) or (9) may be used, then for the purposes of comparison between theory and experiment we can justify the extraction of the direct and indirect contributions from the experimental data. Comparisons can then be made with each part separately. A common procedure is to fit the experimental data, at energies below the autoionization threshold, to a reliable estimate of the pure direct ionization cross section, such as that given by the Lotz formula (Lotz, 1967, 1968) or the distorted-wave calculations of Younger (1980, 1981a,b) scaled if necessary to match experiment. The direct part is the substracted from the total measured cross section to give the pure indirect contribution, which may be compared with the results of excitation calculations. When branching ratios are required, the radiative and autoionization transition probabilities may be calculated by the methods of Chen (1989, Badnell(1986), and Badnell and Pindzola (1989).

IV. Comparison of Experimental and Theoretical Data A. HELIUM-LIKE IONS

Li+ Ionization a. ?he Ground State. Because of the simple structure of the ground state of He-like ions, a single closed 1s' shell, indirect ionization resulting from single excitation cannot occur, since all singly excited states lie, by definition, below the first ionization energy. Extremely accurate experimental work with Li', however, has revealed two other mechanisms by means of which it may take place. The first is double excitation followed by autoionization (DEA):

Li+ + e

-+

Li'(2121')

+ e + Li2+ + e + e

(81)

and the second is simultaneous double excitation plus capture followed by auto-double-ionization: Li+ + e -+ Li(2121'21") -+ Liz+ + e

+e

(82)

Both processes proceed via correlated multiple-electron transitions, which in positive ions have very small cross sections compared to those for direct ionization. In Li+ they give rise to structures with magnitudes of about 0.3% of the total cross section at that energy, so that to see them it is necessary to perform experiments of extremely high precision. Figure 5 shows results obtained for ionization of Li' by Miiller et al. (1989a) using their energy scanning technique, which enables relative cross sections to be measured to

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

325

Electron energy CeVl FIG.5. Electron impact ionization cross section of Lit between 130 and 160eV after a fit to the direct ionization has been subtracted out. [From Muller et al. (1989).]

an uncertainty of less than 0.01%. The points plotted in Fig. 5 are the remainder of the experimental data after a fit to the direct ionization in this energy range has been subtracted out. Dips are seen in the cross section at about 137 and 140eV, with obvious interference with the background: prominent step-like features above 146eV and possible additional resonant structure at higher energies. To bring the size of the structures into perspective, note that the direct ionization cross section varies smoothly from about 2.6 to 3.3 x 10-'*cm2 in this energy range. The dips are attributable to simultaneous double excitation plus capture into the triply excited states 2s22p 'P and 2s2p2 'D of neutral Li [Eq. (82)], and the steps to double excitation of terms of the configuration 2121' [Eq. (Sl)]. The higher energy features may be due to excitation of 2131'31" states. Structure due to triply excited s,tates has also been observed in higher resolution studies of the ionization. of the isoelectronic species neutral He (Qutmentr et al., 1971). Spectacular behavior of the electron detachment cross section of H - reported by Peart and Dolder (1973) (Fig. 6) was attributed to the same phenomenon but recent theoretical work by Robicheaux et al. (1994) has shed doubts on the existence of triply excited resonance states of H'- and hence on the interpretation of the experimental data. The calculation of the cross sections for these higher order processes

D. L. Moores and K. J. Reed

326 r

xQ-"4

-

CI

-k "

3-

.c En

.

s b 2-

L -

1

Is

Intrrnctton energy (ev)

I 20

FIG.6. Electron detachment cross section of H - , measured by Peart and Dolder (1973). The inset shows the region of 17.2-eV plotted on an enlarged scale. Solid curves, crosses, and open circles refer to measurements with different ion beam energies.

presents a stiff challenge to the theoreticians, requiring techniques as refined as the experimental ones. An investigation of the structure observed near 137eV was made by Pindzola and Griffin (1990) who calculated the cross section for triple excitation [process (82)] of the 2s22p 'P state of Li' using high-order perturbation theory. A fourth-order calculation was found to give fair agreement with experiment in regard to the magnitude, and it was shown that by including sufficient higher order diagrams to allow for window resonance structure, fair agreement with regard to shape could also be obtained for a reasonable choice of an empirical parameter. The cross section for the DEA process may be written

Qtat(E0)= C Q'""(E0) X

A:: + -QQdx'"(E,J r x

where Q>(Eo) is the double excitation cross section and r is the sum of all decay probabilities of doubly excited state x. If the branching ratio is taken to be unity, the cross section (in Fig. 5 ) above 145eV is then just equal to C,Q;". The QF are best calculated in a close-coupling approximation in which the ground state and both singly and doubly excited states 2s2, 2s2p, and 2p2 are included in the eigenstate expansion. The latter states have to be treated as pure bound, owing to the difficulty of taking into account their autoionizing nature in the calculation.

327

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

L

12

-

0 -

145

150

155

160

Energy (eV) FIG.7. Double excitation of Li'. Comparison between measured data of Fig. 5 (full circles) and 37-state R-matrix close-coupling calcula.tion.[From Griffin et a/. (1992), Fig. 3.1

Griffin et al. (1992) found it necessary to perform a 37-state R-matrix calculation before agreement between theory and experiment both in magnitudes of the cross section and general form of the step structure was obtained (Fig. 7). The calculation used pseudo-orbitals to circumvent difficulties arising from the strong term-dependence of Hartree-Fock orbitals for singly and doubly excited states encounted in previous work (Pindzola and Griffin, 1990).

b. Ionizationfrom Excited States. In ionization out of the excited states ls21 of He-like ions, the excitation of doubly excited states 2~21'only involves a single electron jump S~Dis relatively more probable than in ionization of the ground state. Of particular interest is the ionization of metastable states via an indirect tramsition, for example, 06+(ls2s3S)+ e + 06+(2s2p3P)+ e + 0 7 + ( l s )+ e

+e

(84)

The cross section for 06+(ls2s3S) has been estimated by Rachafi et al. (1989) including several doubly excited states and radiative decay. Their

328

D. L. Moores and K. J. Reed Q (

lo2ccm2)

58.m

4m.m

31.11

28.N

1r.n

8.88 8.W

8.61

1.21

1 .w

2.41

3.88 E (kev)

FIG. 8. Calculated electron impact ionization cross sections of ls2s’S state of 0 6 +Solid . curve, direct ionization only; dotted curve, direct plus EA; dashed curve, direct plus EA plus radiative decay. [From Rachafi et al. (1989).]

results, which look similar to the structure due to Is excitation in Li-like ions, are shown in Fig. 8.

B. LI-LIKEIONS Owing to a conjunction of advantageous factors, light alkali-like ions have been intensively studied both experimentally and theoretically. In colliding-beam experiments, the absence of metastable ions in the parent beams means that measured cross sections are well defined. The ground configuration of these ions, a single valence electron outside closed shells, leads to a particularly large contribution from indirect ionization, and their relatively simple structure makes accurate calculations a possibility.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

329

Investigations of Li-like ions have provided excellent examples of EA, REDA, and READI. The EA processes e

+ X"+(ls22s)+ e + X2+(ls2s21SL)+ X("+"+(ls2)+ e + e

(85)

have now been observed in all ions of an entire row of the periodic table from Be+ to Ne7+. Crossed-bea.m measurements for C3+, N4+, 05+ (Crandall et al., 1979); Be' (Falk and Dunn, 1983); 05+, B2+ (Crandall et al., 1986), and Ne7+, 05+ , and N4+ (Defrance et al., 1990) all reveal structure near the autoionization threshold, though in these experiments the separate states are not resolved. The height of the step is observed to increase with ionic charge as predicted by theory. Some results are shown in Fig. 9, together with calculated results from Henry (1979) and Crandall et al. (1986) in which Q'""(E,) of Eq. (9) was taken from the distortedwave calculations of Younger (1980), normalized to experiment below the autoionization threshold, and the excitation cross sections were estimated from a six-state close-coupling calculation performed at two above-threshold energies. Four t e r m s - 1 ~ 2 ~ ~ 'S, 1~2s(~S)2p 4P, ls2s('S)2p 'P, and 1~2s(~S)2p~P-were included in the sum in (9), and branching ratios were taken to be unity. Except in Be+, satisfactory agreement with experiment was obtained below the threshold for excitation of n > 2. Above it, neglect of 2s31 states is apparent. Calculations by Jakubowicz and Moores (1981) for C, N, 0,and Ne using the CCTS method described earlier gave similar agreement although these omit the 4P autoionizating states. Resonance profiles in the differential cross section, yet to be investigated experimentally, were also calculated in this work; some results for Ne7+ are shown in Fig. 10. Measurements with improved experimental accuracy for O5 (Rinn et al., 1987) and Bz+ to F6+(Muller et al., 1988b, 1989b; Hofmann et al., 1990) have not only resolved the excitation of the four individual states but have also shown a great deal of more detailed structure. The cross sections look similar in detail for all the ions; a typical set of results is shown in Fig. 11 for C3+. The relative heights of the steps do not agree well with Henry's calculations, although the sum is albout the same as the calculated value. The experiments reveal cross sections rich in structure, most of which is reproduced well by the 17-state R-matrix calculations of Tayal and Henry (1990, 1991), who included 5 bound and 12 autoionizing states in the close-coupled expansion, but also by the independent processes calculations of Reed and Chen (1992), which employed relativistic distorted-wave and multiconfiguration Dirac-Fock methods to evaluate the cross sections from Eqs. (6), (12) and (13), assuming unit branching ratios for radiative decay. Although the independent processes method has the drawback of not allowing for channel coupling and interference resulting therefrom, it has the advantage of enabling many more (uncoupled) channels to be included in the problem. In some cases this can have important effects on the cross +

D. L. Moores and K J. Reed

330 2.6

I

I

1

I

I

I

2.0

I

I

I

I

400

300 ENERGY IeV) 15

1.3

Ftc. 9. (a) Electron impact ionization cross section of C 3 + .Data points from Crandall et al. (1979). The direct ionization calculation of Younger (1980, 1981) has been multiplied by 0.81 to fit the experimental data between 255 and 290eV. The upper solid curve is the sum of Henry's close coupling excitation cross-section calculations and the scaled direct cross section. (b) Electron impact ionization cross section of N4+.Normalization of the direct cross section was not required.

sections. Theory and experiment are compared in Fig. 12 for 0 5 +Both . experiment and theory show contributions due to REDA between 550 and 570eV in the form of closely spaced Rydberg series ls2s21nl' with n 2 5 converging on the autoionizing states 1~2.~21 included in the theory. The strongly peaked, conspicuous structures seen in the experimental data above 580eV are due to REDA via Rydberg series of levels ls2s31n'l'

331

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

10-11

I

I

41.92

1

I

I

/;

I

1

I

I

41.97 49.21

I

1

;

I

I

1

,

,

4926 49.69

1 49.74

k Z (RydlFIG.10. Resonances in the single-differential electron impact ionization cross section of Ne”. [From Jakubowicz and Moores (1981).]

2.6

0

c

0 c

2.5

U

:

b

.-c 0 c 0 Q

2.4

v)

2.3 1 s2s(JS)2p 2P.

240

260

280

300

320

340

360

Electron energy CeVl FIG.11. Electron impact ionization cross sections ofC3+measured by Miiller et al. (1988b). READ1 peaks are clearly visible between 240 and 260eV. The solid line is the sum of the direct ionization cross section (Younger, 1980), multiplied by 0.85, and the EA contributions calculated by Henry.

D. L. Moores and K. J. Reed

332

+ 05+ N n

1.05

I -Distorted-wave

1

Close-coupling

........ ..... Experiment .I

0.90

540

560

580

600

620

640

660

Electron energy (eV) FIG.12. Electron impact ionization cross section of 0’+. Comparison of distorted-wave independent process calculations by Reed and Chen (1992) (solid curve) with 17-state R-matrix close-coupling calculations by Tayal and Henry (1991) (dashed curve) and with crossed-beam measurements by Hofmann et a1 (1990). The DW direct ionization cross sections have been normalized to the direct ionization cross sections used by Tayal and Henry.

converging on the ls2s31 EA thresholds. At higher energies, REDA series ls2s41nl convergingto ls2s41EA thresholds are also seen. The overall agreement is excellent, most of the detailed features appearing in all three sets of data. In the smooth region between the ls2s21 thresholds and the ls2s31nl’ resonances, the close-coupling and distorted-wave results are in near perfect agreement.The ls2s3sz resonance at 584 eV is almost exactly the same height in both calculations. Just after the ls2s3sz resonance there is a noticeable dip in the close-coupling results, which does not appear in the distorted-wave results, and the largest 1~2~3131’ peak that immediately follows is higher in the close-coupled results. These differences are due to interference between the REDA and EA channels neglected in the independent processes approximation but included in the close-coupled calculation and which is clearly observable in the experimental results. The three peaks that immediately follow the largest feature are distinct in the distorted-wave results and the experimental data but not as clear in the close-coupling results. The 1~2~3141 feature at about 615eV is taller in the close-coupling results. In the same region the experimental results lie between the distorted wave and the close coupling. Figure 12 shows that in the region of the ls2s2pnl resonances the close-coupling results are noticeably higher than the distorted-wave results, but the latter are closer to experiment. The same is true for all ions of the

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

333

sequence studied. To explain this difference, we recall that the REDA process

occurs in two steps, the first of which involves the formation of a doubly excited Be-like state, which decays to an autoionizing state of the Li-like ion ls2sn’l’. However the Be-like intermediate state can also decay to an excited bound state of the Li-like ion, thus

This process does not contribute to’ the cross section for ionization of the Li-like ion. The pathways represented by Eq. (87) are referred to as loss channels because they decrease the calculated REDA contribution. If loss channels are not included in the calculation, the REDA contribution will be overestimated. In their close-coupling calculations Tayal and Henry included bound states ls’nl with n < 3. In their paper it is suggested that bound states with n = 4 may be necessary to correct the overestimation. However, in the distorted-wave calculations, the Coster-Kronig transitions ls2s2pnl+ ls2s2 + e (that is, the first Auger decay in the REDA process) become energetically possible only when n is greater than 4 for these ions. Reed and Chen found that the most important effects are due to bound states with 10 3 n 3 5. Figure 13 shows how the loss channels affect the REDA cross sections in 0’’. In the region of the ls2s2pnl resonances, the

..------_ Without loss channels

-

With loss channels

580 600 620 640 660 Electron energy (eV) FIG.13. The effect of loss channels o n the REDA cross section for 0 5 + . 540

560

3 34

D. L. Moores and K. J. Reed

ls2s41 ls2s31

ls2s31n"l" --

ls2s2pnl'

+REDA

-

REDA ls2s2

b

+

EA

ls2s2pZ

b

READI

lS222P

b

READI

1s2

ls22s

Be-like

Li-like

FIG.14. Schematic energy-level diagram for Li-like ions.

He- li ke

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

335

REDA cross section is too large by a factor of 3 if the loss channels are neglected. The relativistic distorted-wave results become less reliable as the ionic charge decreases. Also, interference and other effects of coupling become increasingly important. Nevertheless, the distorted-wave results are still in reasonably good agreement with both experiment and the close coupling for N4+ and C3+. The REDA resonances become less prominent as 2 decreases. In F6+ the largest REDA feature is nearly equal in magnitude to the EA cross section, but in C3+ this peak is only one-fifth its magnitude. Reed and Chen calculated the branching ratios in Eqs. (6), (7), (12), (13), and (14) to estimate the effects of radiative decay in these ions. The effect on the REDA contribution was found to be small, of the order of 1%, but the EA contribution was more affected. Even then, in F6+, the case most affected, the EA contribution was decreased by 20%, an amount equal to 3% of the direct ionization. Pindzola and Griffin (1987) have discussed the possibility of observing READ1 in electron impact ionization of Li-like ions as a result of the processes e + X'+(ls22s)

+

X'"-')+(ls2s2p21)

-+

X('+')+(ls2)+ 2e

(88)

Because the Be-like states ls2s2p21 created by resonant capture have energies less than those of the ls2s" and ls2s2p Li-like configurations (see Fig. 14), they can only contribute to single ionization by undergoing auto double ionization. As mentioned earlier, double electron jumps rely on strong electron correlation. Pindzola and Griffin (1987) used many-body perturbation theory to calculate upper and lower bounds to the contribution to single ionization via this process from the ls2s22p'P term for Be+ to Ne". In all cases the maximum contribution was calculated to be less than 1% of the direct ionization background. Nevertheless, using the fast energy scanning technique it proved possible to detect the effect first in C3+ and subsequently in B2+,N4+, and 0 5 +(Hofmann et al., 1990). In C3+, small peaks are observed between 240 and 255 eV (Fig. 1l), which are shown on an expanded scale in Fig. 15. The results look similar in the other ions. Excitations to three terms-ls2s22p 'P, ls2s2p2 'P, and 3D-were identified, the energies being in excellent agreement with Auger spectroscopy data and with calculations by Safronova and Lisina (1979). Hofmann et al. made Gaussian fits to the data after subtracting the direct ionization and were thus able to estimate the resonance strength S=

1

n(E) dE

(89)

for the three transitions. Comparison with the calculations of Pindzola and Griffin shows that the experimental data for Z = 5 to 8 lie within the predicted theoretical range.

336

D. L. Moores and K. J. Reed

n

E 2.37

.g 2.35 t u Q)

v)

240

245

250

255

Electron energy CeVl FIG.15. Electron impact ionization cross section of C 3 + .Enlarged part of Fig. 11 for the region around 250 eV.

C. IONS WITH 2s2, 2pq CONFIGURATIONS These are not ideal systems for studying indirect effects in ions of low charge. The contributions are relatively small, and in experimental work beams of these ions are often infested with an uncertain fraction of metastables. Experimental data that exist such as those for Si6+ and Si" (Zeijlmans van Emmichoven et al., 1993) show little sign of structure in the cross sections. Calculations for Ar9+ in a CCTS approximation using the R-matrix method for the close-coupling wave functions (Laghdas et al., 1994) reveal beautifully intricate resonance structure in the single-differentialcross section due to 2s22p4nland 2s2p5nl states, but this has a small effect on the total cross section. The most promising candidates for exhibiting a noticeable EA effect are the Be-like and B-like ions, owing to their low occupation number in the outer shell. In Be-like ions one might expect to find the EA process c

+ Xz+(ls22s2)-+e + Xz+(ls2sznl)-+

X'"+ ')+(1s22s)+ 2e

but it has not been observed even for nl = 2p, the strongest transition. Also, it is possible that high-precision experimental work could reveal higher order transitions analagous to those observed in Li-like ions. In B-like ions, excitation of the 2s electron via

+ 2e e + Xz+(2s22p)-+ e + xz+(2s2p(SL)nl)-,X(=+l)+(2s2)

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

337

can occur. Calculations for N2+(Fig. 2) show that the summed effect of these excitations, occurring for large n, should cause the cross section to rise more steeply from threshold than expected on the basis of direct ionization alone. There is some indication that this may be the case in the experimental data but the situation is obscured by the presence of metastable ions. A similar effect should occur in C-like ions but as z increases the total cross section becomes increasingly dominated by direct ionization. Calculations of indirect effects in highly stripped, h.eavy Be- and B-like ions are discussed in Section IV.J.3 of this article.

D. NA-LIKE IONS The principal mechanism for indirect. ionization of Na-like ions is from EA transitions of the type X’+(2p63s) + e -, X+(2p53s(SL)nlS’L‘)+ e X(Z+1)+ (2p6) + 2e ~

(90)

Bely (1968) predicted large enhancements, increasing in size relative to direct ionization with z, due to these processes in ions from Mg+ to Fe’”. The substantial jumps calculated for Mg’. were not confirmed by the experiment of Martin et al. (1968). Any structure present was smaller than the experimental uncertainty of 7%. On the other hand, the difference in slope above 50eV between the experimental curve and the sum of the 3s and 3p direct ionization cross sections calculated by Moores and Nussbaumer (1970) suggested that a small effect could be present. Moores and Nussbaumer also pointed out that the scaling procedure employed by Bely could easily give considerable overestimates of excitation cross sections for lower members of the sequence. Starting from Eq. (9) they carried out detailed calculations in which the excitation cross sections for n = 3,4, 5 and I = 0, 1, 2 were calculated in a no-exchange Coulomb-Born approximation, and the sum for n >, 6 was obtained by extrapolating differential cross sections for 2p ionization to negative energies. The computed EA enhancement of 20%-though considerably smaller than Bely’s estimate-was still much larger than the experimental upper limit of 7%. The difference was attributed to the use of the no-exchange Coulomb-Born approximation in the calculations. Partly to resolve this discrepancy, further experimental (Crandall et al., 1982) and theoretical work (Griffin et al., 1982a) on the isoelectronic ions Mg’, A12+,and Sii3+was carried out. Experiment showed clear signs of structure at the predicted energies in the cross sections for all three ions, although individual transitions could not be resolved (see Fig. 16, 17, and 18). The measured direct ionization cross section for Mg’ was about

D. L. Moores and R J. Reed

338

5-

t L-

-

5

3-

2 6

-

?3

$

2-

u

1-

70

L

I

I

20

30

LO

, 50

60

'0

Interactm energy lev1

FIG.16. Electron impact ionization cross section of Mg'. Full circles, Peart et al. (1991b); crosses, Martin et al. (1968); triangles, Crandall et a!. (1982). Curve M is the calculation of Moores and Nussbaumer (1970) and curve Y is the distorted-wave direct ionization cross section from Younger (1981b), scaled by 0.9.

\4 12 - (0 -

1

16

I

1

I

1

I

I

I

I

-

N

E

{

1

8 -

-

Q

S6: 4 -

-

e+AI2+

2 -

0 r

r

-

li

t t I

-2

20

--

30

I 40

I

I

50

60

I

I 80

70 ENERGY (OW

-

tftt I 90

I

iw

I

I

(20

FIG.17. Electron impact ionization cross section of AI2+. Experimental points from Crandall et al. (1982). The lower solid curve is the distorted-wave direct ionization cross section of Younger (1981b), scaled by 0.65. The upper curve shows the results of adding the distorted wave excitation cross sections of Griffin et al. (1982a).

INDIRECT PROCESSES IN EL.ECTRON IMPACT IONIZATION

-

N

i2

-I

40

-

I

l

~

l

I

I

I

I

I

I

(

I

I

339

I

0 -

E

u

m

Q

v

0

b"

6 4

X$

$

2

I

1

e+si3+

I

0 l

-2 40

l

l

J

80

1

i

l

l

120

l

l

'

l

l

160

l

l

'

l

200

l

'

l

240

ENERGY ( eV 1 FIG.18. Same as Fig. 17 but for Si3+.No scaling of the direct cross section was reqi required. uired.

10% lower than that of Martin el al. (1968). After removing the direct ionization contribution by subtracting out distorted-wave cross sections derived from the work of Younger (1981b) (multiplied by a scale factor where necessary to fit experiment), the contributions to EA at an energy just below the 2p ionization threshold were found to be 18% in Mg+, 40% in A12+, and 65% in Si3+. Griffin et al. (1982a) used a distorted-wave exchange method to calculate excitation cross sections (90) for n = 3, 4 and 1 = 0, 1, 2. Unit branching ratios were assumed. The calculations overestimated the total EA contribution by about a factor of 2 in all three ions (Figs. 17 and 18). The largest individual contribution was found to be due to the monopole transition 2p63s-,2p53s3p, a fact also noted by Moores and Nussbaumer (1970). The large jumps due to this transition were not observed in the experimental data-indeed, if they were omitted, quite satisfactory agreement between theory and experiment was apparent for A12+ and Si3+.The reason for these discrepancies was unclear. Griffin et al. were able to rule out radiative decay as being of major importance and suggested that the disagreement was a result of the inadequacy of the distorted-wave approximation for these transitions. Henry and Msezane (1982) carried out a two-state close-coupled calculation for 2p -P 31 transitions in A12+ and Si3+,estimating the Mg' values by extrapolation. The inclusion of channel coupling did improve agreement with experiment, although the 2p --* 3p jump was still overestimated. They also investigated REDA by means of a model three-state calculation for

340

D. L. Moores and K. J. Reed

AIZ+,finding a series of resonances due to e

+ A12+(2p63s)

+

Al+(2p53s3pnl)-+ A13+ (2p6) + 2e

(91)

lying between the 2p53sz and 2p53s3p excitation thresholds in the range of 73 to 80eV with a particularly large enhancement attributed to an n = 3 resonance close to threshold. When averaged over a 2-eV Gaussian to simulate the experimental energy spread, the shape of the experimental curve between the thresholds was reproduced quite well-evidence for REDA in Na-like ions. Even in ions of such relatively simple structure, an enormous number of intermediate states 2p53snl and 2p53snln'l' can contribute to the net ionization. Many of them will be crowded together, producing no resolvable structure, observable merely by change of slope in the cross section. As high-resolution studies make it possible to reveal increasingly complicated structure, ever more elaborate and sophisticated calculations must be performed to interpret it. Miiller et al. (1990) have used the fast energy scanning technique to search for fine details in a relative measurement of the Mg' cross section between 46 and 73eV, with an energy resolution of 300 meV. Peart et a/. (1991b) have measured absolute cross sections from threshold to 70eV for the same ion using energy-resolved electrons with a spread of 170meV and energy defined to within 40meV. Tayal (1991) has performed 19-state R-matrix calculations of the excitation cross sections of 16 autoionizing states arising from the 2ps3s31,1 = 0, 1, 2 configurations. The results of Peart et al. are compared with those of Muller et al. in Fig. 19 and with those of Tayal in Fig. 20. To compare these three sets of data, various adjustments have been made. The indirect ionization contribution was extracted from the experimental data by subtracting out the appropriately scaled direct cross section of Younger (1981b). Peart et al. (1991b) normalized the relative measurements of Miiller et al. (1990) to fit their absolute data below 50eV. They also substracted 600meV from the energy scale assumed by Muller et al. (1990) to match the position of the first resonance, and reduced Tayal's (1991) calculated energies by 2.5 eV. In the following discussion, we refer to the energy scale adopted by Peart et al. (1991b). The agreement between the three sets of data is quite good, but there are some daerences. All three show considerable structure, and in many cases the positions and heights of peaks and dips correspond. It is quite difficult to distinguish between individual REDA peaks, the net effect of a number of REDA peaks, and EA steps. One thing on which all agree is the existence of the large peak at 50.9 eV immediately above the 2p53sz excitation threshold. This is an obvious feature of all three sets of results, and is a result of the resonance 2 ~ ~ 3 s 3 p ( ~ S ) The 3 s . structure above this is due to several overlapping Rydberg series of resonances that are difficult to disentangle. The net effect

34 1

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION 4.1

I

d.r

E , " C.t

-9

C

.-

c u YI w v)

0

3

4.4

4.2

FIG.19. Electron impact ionization cross section of Mg+. Measurements by Peart et al. (1991b) with energy-resolved electrons (full circles) compared with those using the energy-scan technique of Miiller et a!. (1990) (full curve).

T

t CO

I

50

60

70

Interaction energy (eV 1

FIG.20. Electron impact ionization cross section of Mg+. Comparison of the data of Peart et al. (1991) with the R-matrix close-coupled calculations of Tayal (1991) (curve T).

D. L. Moores and K. J. Reed

342

seems to have the form of a broad feature centred at 53eV. A sharp peak seen at 52eV by Muller et al. (1990) does not appear in the data of Peart et al. (1991b). The steps at 54.78 and 57.21 eV are associated with excitation of 2p53s('P)3p 'S and 2p53!@P)3p 'S, respectively. The two sets of experimental data display differing behavior above 58eV. The sum of the inner shell excitation cross sections to all 16 autoionizing states considered by Tayal is in better agreement with Peart et al. at 70eV (above the REDA threshold), but the calculations include only contributions from nl = 3s, 3p, 3d. The differences in the data may be explained in a number of ways. First of all, different techniques were used in the two experiments. On the theory side, the 2p53snl autoionizing states are treated as pure bound states, their width not taken into account; direct and indirect processes are treated as independent, interference between them is ignored, and radiative decay is neglected. Calculations are performed in LS coupling. Not all states 2p53s(SL)3p(S'L') can autoionize in LS coupling; some can radiate, and quartet terms might do neither. Some 2p53s3p states could be sufficiently long lived to survive the passage from collision region to detector and not be detected as ionized ions. It is known that there is a problem-especially in the more highly charged systems-with metastable ions 2p53s3p in the parent beam decaying before they reach the collision region and producing a background signal that becomes stronger with z and makes it difficult to measure cross sections beyond Si3 . This could also explain the overestimate of the 3p contribution by theory, which assumes that all possible 2p53s3p terms contribute to EA. Although a minor problem in Mg+, it could be more serious in more highly charged ions in the sequence. +

E. MG-LIKEIONS The Mg sequence only differs from the Na sequence in having two 3s valence electrons in the ground configuration instead of one, so that similar types of transitions contribute to indirect ionization. Crossed-beam measurements for S4+, C15+,and Ar6+ by Howald et al. (1966) display large EA features due to the transitions X'+(2p63s2)

+ e + X'+(2p53sZnlSL)+ e + X(z+')+(2p63s)+ 2e

(92)

The cross-section curves look similar for all three ions but with the ratio of the indirect to direct cross sections at the peak increasing from 0.49 in S4+ and 0.69 in CI5+ to 1.1 in Ar6+. As a representative example, results for Ar6+ are shown in Fig. 21. The excitation cross sections for these processes have been calculated in the distorted-wave approximation by Pindzola et al. (1986a) and a 21-state R-matrix method by Tayal and Henry (1986) including the ground 2p63s2'S state plus six autoionizing configurations (2s22p53s2)3p,4s, 3d

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

343

C

.c -0

3-

2v) v)

2 u

I -

, '

*7!00 ' Electron

I

' ' " " ' 1000 2000

Energy (eV)

FIG.21. Electron impact ionization cross section of Ar6+. Experimental data: Howald et al. (1986). Theory: dashed curve, distorted-wave direct cross section calculated from Younger's parameters; dotted curve, Lotz formula; full curve, sum of direct cross section and excitation cross sections calculated by Tayal and Henry (1986).

and (2s2p63s2)3p,4s, 3d in the close-coupling expansion. The R-matrix calculations confirmed the predicted rise in the cross section at about the same energies, but with peaks slightly lower and displaced with respect to experiment (Fig. 21). The relative contributions from the forbiden 2p -+ 3p and allowed 2p +(4s + 3d) excitations were found to decrease with increasing z at low energies although the forbidden transitions fall off more rapidly with increasing energy. Results for the 2p -+ 3p forbidden transitions were in good agreement with the distorted-wave calculations, but the inclusion of channel coupling and configuration interaction in the R-matrix calculation gave significantly smaller cross sections for 2p + 3d. At energies for which some channels were closed, Tayal and Henry saw evidence of REDA to which they attributed the unusual shape of the cross section in the low energy region but no attempt was made at a detailed study. In Mg-like ions the transitions involved result from captures of the form

+

(2s22p63s2) e -+ 2s22p53s2nl(SL)n'l'STLT and

of which there are a very large number.

(93)

344

D. L. Moores and K. J. Reed

A common feature of both sets of calculations was the much faster decrease of the cross sections at high energy compared with experiment. This may be attributed partly to the neglect of excitation of higher nl states in the calculations, but principally to the fact that in the experiment an unknown but significant fraction of the ions coming from the ECR source are in metastable 2p63s3pstates, which live longer than the transit time from source to scattering region. It is clear from the experimental data that the observed onset of ionization occurs below the ground state threshold. The slower decrease of the experimental cross section at high energies is due to direct inner shell ionization of the metastable state, which leaves the ionized ion in the 2p53s3p configuration. Some of the terms of this configuration autoionize and so contribute to double ionization by the IA mechanism but a large fraction do not, surviving the passage to the detector still only once ionized. On the other hand, direct inner shell ionization of the ground state yields ions in the 2p53sz configuration, which autoionizes immediately to 2p6, thus contributing to double rather than single ionization. The same process for the metastable state gives a significant contribution to single ionization and this accounts for the high-energy differences between experiment and theory. Pindzola et a1. (1986a) were able to resolve the discrepancy assuming that 2/3 of the 2p53s3p terms would contribute. Other evidence for the metastable presence is found in the slight mismatch in the observed and calculated autoionization structure, but the experimental data are not sufficiently detailed to make any definite identifications. Tayal and Henry also did a 15-state R-matrix calculation for Al+. In this ion, 2p + 3p excitation dominates other transitions over all energies considered, but the total EA contribution was less than 10% of the direct ionization. Reasonable agreement was obtained between the R-matrix calculations, augmented by the direct ionization cross section taken from a parametrization by Younger, and the experimental data of Belic et al. (1986) and Montague and Harrison (1983) after an estimated 9% metastable contribution had been subtracted. In Si2+ (DjuriC et al., 1993) the contribution is a little larger with a well-defined step, giving the cross section a similar appearance to that for Si3+ (Fig. 18). Agreement between experiment and CA distorted-wave theory is only fair but the issue is clouded by the presence of metastable ions in the ECR source beam.

F. HEAVIER MONOVALENT IONS It is scarcely surprising that indirect ionization was first observed in Ba' since some of its most dramatic manifestations are found in ionization of heavy alkali-like ions, as a glance at Fig. 22 confirms. In this section we discuss the experimental and theoretical data for Ca', Sr+, Ba' [ground

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

x

345

i Electron energy (eV)

FIG.22. Measured cross sections for electron impact ionization of alkali-earth ions showing the effect of EA. [From Peart and Dolder (1975).]

configurations np6(n + l)s] for which the excitations due to An = 1 transitions np6(n

+ 1)s -,np5(n + 1)snd

(95)

give a particularly large EA contribution, and for Ti3+, Zr3' Hf3', and Ta3+,which have ground configuration np6nd and for which the excitations np6nd -+ np5nd2

(96)

are likewise very important. 1. Alkali-Earth Ions The early measurements for Ca', Sr', and Ba' (Peart and Dolder, 1968, 1975; Feeney et al., 1972), which first confirmed the existence of EA, have been repeated for low collision energies with energy-resolved electrons with the aim of examining the structure more closely (Peart et al. 1989a,b, c; Figs. 23,24, and 25). In both Sr' and Ba', the ionization cross section rises from threshold with slight undulatory behavior before making a steep rise at about 21.3eV in Sr', 15eV in Ba'. After that the curve flattens, though it shows many jumps and dips, more pronounced in Ba'. The structure is undoubtedly due to EA and REDA but it is very difficult to identify the individual intermediate states responsible from among the large number present. Some of the autoionizing states may be identified by comparison with theory, ejected electron spectra of the neutral atom, or resonances in the photoionization spectra of the ions. In Sr' the onset of strong ionization

346

D.L. Moores and K. J. Reed I

I

FIG.23. Electron impact ionization cross section of Sr' measured with energy-resolved electrons. [From Peart et al. (1989b).]

is identifiable with the transition 4p65s 44 ~ ~ 4 d ( ~ P2P )5s

(97)

)6s 5p66s+ 5 ~ ~ 5 d ( ~ P2P

(98)

and in Ba' with but the large jump is not due to this transition alone. The form of the measured cross section is similar in Ca'. The onset of autoionization is due to 3p64s + 3 ~ ~ 3 d ( ~ P'P) 4 s

(99)

Electron mergy (eVl

FIG.24. Electron impact ionization cross section of Ba' measured with energy-resolved electrons. [From Peart et al. (1989c).]

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

347

ll

111

t"

1.6 l . i

.z 1.011u1*'+"

0.8 -

b

7 )#;L

PBG

xL

f t t1

0.6

4

1 lit+$

Calculations have been carried out for this system by Burke et al. (1983) who used the R-matrix method, coupling together the ground state 3p64s and four autoionizing states 3p54s2'P, 3~'3d(~P)4s 2*4Pand 3p53d('P)4s 'P; by Griffin et al. (1984a), a unitarized distorted-wave calculation including contributions from all nine terms of 3p53d4s and allowing for radiative decay; and by Pindzola et al. (1987a) who performed a close-coupling calculations, still including only three terms of 3p53d4s, but using a 3d orbital in the 3p53d('P)4sZP term different from that in the 's4P terms (term dependence). 3~'3d(~P)4s The three calculations yielded results differing considerably from experiment and from each other as Fig. 26 shows. The two close-coupling calculations both obtained a cross section dominated by a large feature, not seen in the experiment, identified as due to REDA via the intermediate states 3p53d24s1*3F. Badnell et al. (1991) subsequently carried out a 13-state calculation in which all nine terms of the 3p53d4s configuration were included using the R-matrix method as coded for the Opacity Project to compute the excitation cross sections. A scaled Lotz formula was used to estimate the small direct ionization contribution and radiative decay taken into account using the branching ratios of Griffin et al. (1984a). In the R-matrix calculation, orbital term dependence was simulated by the use of a 68-term configuration interaction expansion. The results (Fig. 27) show far better agreement with experiment; in particular, the spurious broad resonant feature is absent. The difference is due to the inclusion of all nine terms of 3p53d4s rather than a restricted set of them. In the calculations in which only a few of these terms

D. L. Moores and K. J. Reed

348

01 10

I

20

Lo

30

Electron energy lev)

50

FIG.26. Electron impact ionization cross section of Cat-comparison of theory and experiment. Curve E, experimental data of Peart et al. (1989b) normalized to earlier data (Peart and Dolder, 1975; full circles). Curves CCB, CCP, and DW are, respectively, the close-coupled calculations of Burke et al. (1983), Pindzola et al. (1987), and the Distorted-wave calculations of Griffin et al. (1984a).

v) v)

0

cf

0.01

20

'

25

'

I

30

'

'

35

*

40

'

I

45

Energy (eV) FIG. 27. Electron impact ionization cross section of Ca'. Calculations by Badnell et al. (1991), full curve, excitation cross sections, 13-state close-coupled calculation. Dashed curve, four-state calculation (both convoluted with a 0.2-eV FWHM Gaussian) added to a direct contribution, estimated from the Lotz formula scaled to experiment. Dotted curve, experimental data of Peart et al. (1989b).

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

349

are included, important 20-electron bound channels associated with parent 19-electron states excluded from the expansion are omitted, and in addition the number of possible Auger decay channels is limited. When this restriction is lifted, improved agreement with experiment is obtained.

2. Zn+ and Gat Strong EA contributions also dominate the low energy ionization cross sections of Zn+(3d1'4s) and Ga+(3d"4sz) whose structure resembles that of the alkali-earth ions in having valence s electrons outside a closed shell. Measurements with energy-resolved electrons for Ga' (Peart and Underwood, 1990) and Zn' (Peart et al., 1991a) reveal complicated structure due to excitations of the type

The measurements for Zn+ when normalized to the absolute data of Rogers et al. (1982) give excellent agreement in shape, but delineate more of the details owing to the improved energy resolution. The smooth undulations in the cross section rising from threshold are due to the summed effect of excitation of many overlapping terms of the configurations 3d94snl(Fig. 28).

m1111 I

n

3 1 9 ~ 3d)bsf ~ 1 ~

3d9 4 p 2 2 -

-=

"-

4-

__--------

3; .--p 2.. --

r\s> I I '

, '

"

1-

O

M

I

350

D. L. Moores and K. J. Reed

FIG.29. Electron impact ionization cross section of Ga' measured with energy-resolved electrons by Peart and Underwood (1990) (black circles). Also shown are data of Rogers et al. (1982) (solid curve) and estimates of the direct ionization from the Lotz formula (L) and the scaled plane wave Born approximation (PWB) by McGuire (1977).

In Ga', steplike features due to excitation of 3d94s24pare observed (Fig. 29). Good agreement with distorted-wave calculations by Pindola et al. (1982) is found between 21.6 and 22.9 eV except for a large overestimate by theory of the excitation cross section for the optically allowed transition to the 'PI state at 22.8eV. 3. Ions with Ground Conjiguration np6nd Measurements of the single-ionization cross sections of Ti3+ (Fig. 30), Zr3+ (Fig. 31), Hf3+ and Ta3+ by Falk et al. (1981, 1983) reveal that EA completely dominates, exceeding the direct contribution by large factors, despite the fact that out of 45 possible states of the configuration np5ndz only 6 can autoionize in Ti3+ (n = 3), 8 in Zr3+ (n = 4), and 34 in Hf3+ (n = 5). First attempts at the difficult task of calculating the excitation cross sections for these transitions (Falk et al., 1981), Griffin et al. (1982a,b) by a simple distorted-wave method retaining only dipole interactions reproduced the shape of the measured cross section for Ti3+ and Zr3+, though not for Hf3+, but the calculated magnitude was about 2.5 times larger than experiment. (The direct ionization adds only a small amount to the cross section.) Estimates of branching ratios showed that radiative stabilization could not account for the differences. More elaborate distorted-wave calculations (Bottcher et al., 1983) led to improved agreement with experiment mostly as a result of including exchange; the inclusion of higher multipole interactions tended to be

351

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION 70

‘

60

h

N

s

-.P

50 40

c

J

I

I

: p“‘

g 3 0 v)

- 0 0

.

8

{

0 0

FIG.30. al. (1983). ionization.

160

-

140 N 120 a0 ‘0 loo-

I

I

.: 1:-. ‘ . I

I

Y

-

5 80.c V t*n 60-

1

1

1

I

.*

h

6

1

I

I

I

I

Zr 3+

I

-

c

I

I

-

*.

.r

I

-

* .

I

Electron Energy (eV) FIG.31. Electron impact ionization cross section of Zr3+. Experimental data from Falk et d.(1983). Theories, direct ionization only: dashed curve, Lotz formula; solid curve, scaled plane wave Born approximation of McGuire (1977).

352

D. L. Moores and K. J. Reed

counteracted by the effect of unitarization. The calculations still overestimated the experimental results in Ti3+ and Zr3+, but lay below them in Hf3+. The remaining discrepancy was attributed by the authors, to the neglect of channel coupling, inclusion of which would redirect flux from allowed to forbidden channels, reducing the cross sections in Ti3+ and Zr3+. This would not apply to Hf3+ where the distorted-wave results were already dominated by dipole transitions. In Hf3+ the low calculated values could be attributed to neglect of REDA in the theory, excitations other than np + nd being ruled out as giving too small an effect. More extensive calculations have been performed in Ti3+. Burke et al. (1984) carried out an R-matrix calculation including the ground state 3p63d plus nine terms of the 3p53d2 configuration in the expansion. Improved agreement over the distorted-wave results was obtained, but the calculated cross section was still too high. Gorczyca et al. (1994) have subsequently carried out a detailed study of this ion using the R-matrix code developed for the Opacity Project (Berrington et al., 1987). Their work showed coupling between the 19 terms of the 3p53d2configuration and six singly excited 3p6nl states to be unimportant, 26-state, 21-state. and even 4-state calculations giving a very similar EA contribution which also agreed with the 10-state results of Burke et al. (1984), apart from an energy shift. However, inclusion of the singly-excited states is necessary to provide an alternative decay path for resonances. To obtain good agreement with experiment it proved necessary to include target configuration mixing with correlation configurations formed by double electron promotions 3p2-3d2. Comparison of unified R-matrix and distorted-wave results was used to diagnose potential problems with pseudo-resonances, which were eliminated by removing 3p43d4 and 3p33d5 configurations from the R-matrix wave function. The results of a 14-state calculation are compared with experiment in Fig. 32. The calculation also includes radiative decay, although the branching ratios for the 3p53dnl states were all close to unity, apart from 3p5[3d2(3P)]'D and 3p53d('D)4s 'D, for which they were 0.854 and 0.329, respectively. It is anticipated that a calculation of at least a comparable degree of complexity would be required to achieve a similar level of accord between theory and experiment in Zr3+ and Hf3+.Similar effects have been recorded in the ions Ni9+ and Fe7+ isoelectronic with Ti3+ though to date only distorted-wave calculations have been performed. In Ni9+ the excitations 3p --t 5f, 6p, 6f and 2p + 3d are important and in Fe7+,3p -,51, 3s --t 41. In both ions high-n transitions give large contributions. G. METALIONSWITH 3p4 AND 3d4 GROUND STATES

Owing to their importance in magnetic fusion research, Fe and Ni ions have been closely studied both experimentally and theoretically. Detailed, critical

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

353

r J 40

45

50

55

60

Energy (eV) FIG.32. Electron impact ionization cross section of Ti3+.Solid circles, experimental data of Falk et al. (1981, 1983); solid curve, 14-state calculation with target CI, omitting correlating states, convoluted with a 2-eV FWHM Gaussian, by Gorczyca et al. (1994).

reviews of the data for the Fe isonuclear sequence have been made by Pindzola et al. (1987) and for the Ni sequence by Pindzola et al. (1990). Other ions for which measurements have been made are Ti+,Ti2+(Diserens et al., 1988; Mueller et al., 1985); Cr’ (Man et al., 1987); and Cu2+,Cu3+ (Gregory and Howald, 1986). A number of general conclusions can be made about these ions. Calculations are complicated owing to the large-often enormous-number of states that contribute to EA. For some ions agreement between experiment and CA distorted-wave calculations has been found to be very good, for others less so, though comparison is rendered difficult in some cases by metastable ion contamination. In general, some of the terms SL of an inner shell excited configuration will be autoionizing, others will not. As z increases, the principal quantum number of the first autoionizing configuration will increase. Excitation and ionization of 2p and 3s shells also contribute. In some ions such as Fe2+,states formed by direct 3p ionization will be autoionizing, leading to net double ionization via the IA mechanism. The relative importance of excitation compared to direct ionization will decrease as z increases. All of these factors lead to an indirect ionization contribution that varies erratically in magnitude and shape as the atomic structure changes. Pindzola et al. (1987b) in their conclusion warn that “Cross sections and branching ratios are not amenable to scaling

354

D. L. Moores and K. J. Reed

along an isonuclear sequence.” The same may also be said for an isoelectronic sequence. 1. Ions with Configuration 3p63dg(q > 1)

In these ions the EA process e

+ X’+(3p63dg) -+ e + X”+(3p’3d4nlSL)-+ e +

X(’+’)+(3p63dq-’)

+e (loo)

gives large contributions to the total. We first examine the subsequences Ti’, Fe5+,Ni” (ground configurations 3d24s,3d3) and Ti2+,Fe6+,Ni8+ (3d’). Results of crossed-beam measurements for TiZ+by Diserens et al. (1988) and by Mueller et al. (1985) are shown in Fig. 33. The small differences between them are attributible to a fraction of metastable 3d4s ions thought to be present in the experiment of Mueller et al. but not of Diserens et al.

Electron energy lev)

FIG.33. Electron impact ionization cross section of TiZ+.Circles, squares, and solid lines, experimental data of Diserens et al. (1988); triangles, experimental data of Mueller et al. (1985); long dashes, Lotz formula; short dashes, formula of Burgess and Chidichimo (1983).

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

355

Both experiments reveal a large contribution due to indirect ionization from the EA process e

+ Ti2+(3p63d2) -P

e + TiZ+(3p53d2nlSL)-+ e + Ti3+(3p63d)+ e

(101)

The onset at about 32eV is due to the An = 03p -+ 3d transition for which there are 386 terms SL lying between 32 and 48 eV. Partly overlapping with these are jumps due to n = 4 transitions (a total of 674 terms). Add to this the probability that there are contributions from REDA transitions e

+ Ti2+(3p63d2)+ e + Ti+(3p53d2nln'l') -+ e + Ti3+(3p63d)+ e

(102)

and it becomes clear that the structure seen-though having the appearance of several discretejumps-is in reality the integrated effect of a huge number of transitions. In fact, the formula of Burgess and Chidichimo (1983), in which the total EA effect is approximated by lowering the inner shell continuum threshold, reproduces the data quite well below 50eV. In Fig. 34 we show the experimental data of Gregory et al. (1986) for Fe6+ compared with the ASM calculations of Pindzola et al. (1986b), who included excitations 3p -+nl; n = 4,5; 1 = 1, 2, 3; and 3s 44s, 5s. All the 3p53d3states, 241256 of the 3p53d24pand 2041386 of the 3p53d24d terms are pure bound and so do not contribute to ionization. The 3p -+ 4f makes the largest contribution to EA but no transition really dominates the others. Agreement with experiment is good up to about 200eV but not above. Results of a similar calculation for Nia+ (Griffin and Pindzola, 1988) are compared with experimental data of Wang et al. (1988) in Fig. 35. In this ion, the 3p53d24p states are bound; EA from 3p -+ 5p makes the largest contribution, and 3p --t 6p, 6f, 5f make up most of the rest. Theory and

1

DL

'

1

'

1

'

1

'

E

0

7

--

-

12.0 -

0

-

Electron Energy (eV) FIG.34. Electron impact ionization cross section of Fe6+, ground state 3p63d2. Theory: ASM/DM calculations of Pindzola et al. (1986b). Dashed curve, direct ionization only; solid curve, total cross section. Experimental data from Gregory et al. (1986).

D. L. MooresandK. J. Reed

356 0

in

3 - 7

-€0 D

2 c

O c)

v

.-3 _0

M" I n n

m

ps

3

0 0 L

0

300

600

900

1200

1500

Electron Energy (eV) FIG.35. Electron impact ionization cross section of Nis+, ground state 3p63d2. Theory: ASM/DW calculations of Griffin and Pindzola (1988). Dashed curve, direct ionization only; solid curve, total cross section. Experimental data from Wang et a!. (1988).

experiment agree well up to about 500eV. At higher energies, the theoretical results fall off more rapidly with energy than experiment, which shows some structural features reminiscent of thresholds. The reasons for the discrepancy are not understood, but it is a common phenomenon in many Ni and Fe ions. The irregular shape of the measured Ti+ cross section (Fig. 36) is a sure sign of a large indirect contribution. Also shown in Fig. 36 are two theory curves, which allow for direct ionization only: the scaled plane wave Born calculation of McGuire, which lies more than a factor of 2 below experiment, and the Lotz formula, which gives a cross section larger than the experimental one despite the lack of allowance for any EA contribution. One is tempted to conclude that the Lotz formula overestimates the direct ionization in this case. A similar overestimate, by the DW approximation rather than the Lotz formula, appears to exist in other singly charged ions Fe+ and Ni+ (see later). The ground state of Ti+ is 3d24s, whereas for the other two ions-Fe5+ and Ni7+-it is 3d3. The sharp rise seen in the Ti+ cross section is due to the An = 0 transition

e

+ Ti+(3p63d24s)-+ e + Ti+(3p53d34s)+ e +

+

Ti2+(3p63d2) e

(103)

Similar rises due to excitation of n = 3 electrons are found in the other two ions.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION I

I

I

1

1

1

1

1

1

I

I

I

I

I

I I l l

357 1

Electron energy ieVI

FIG.36. Electron impact ionization cross section of Ti'. Experimental data (solid curve, triangles) from Diserens et al. (1988); long dashes, Lotz formula; short dashes, scaled plane wave Born calculation by McGuire (1977).

In Fe5+ the 3p53d4 states are all bound and so are 57/213 of the 3d34s. The excitation e

+ Fe5+(3p63d3)

e

+ Fe5+(3p53d34p)+ e + Fe6+(3p63d2)+ e

(104)

in which 419/613 terms are autoionizing gives the largest contribution but 3p + 4d, 4f, 5p, Sd, 5f and 3s -+ 3d, 4s also play a role. Agreement between the ASM calculations of Pindzola et al. (1986b) and the experiment of Gregory et al. (1986) is very good (Fig. 37). They also calculated the cross section for total ionization of the metastable state 3d24s of Fe5+ in order to estimate its contribution to the total measured cross section. The result (Fig. 38) is in poor agreement with experiment-the ionization threshold occurs at about 30eV lower than the observed onset and the low-energy cross section is a factor of 2 too high and a different shape. This provides evidence that metastables, if initially present in the beam, have decayed despite their long lifetime before the collision region. Analogous calculations for Fe6+ gave a similar result and the same conclusion. As in Ti+ the ionization cross section from 3p63d24sis dominated by the 3p + 3d excitation and it is seen that the general shapes of the measured cross section for Ti+ in Fig. 36 and the calculated one for metastable Fe5+ in Fig. 38 are very similar.

D. L. Moores and K. J. Reed

358

20.0

-

-

-

_ __._-- - - - - - - - _ _ _ _ _ _

10.0-

5.0 -

-

0.0

I

,

I

,

I

.

Electron Energy (eV) FIG.37. As in Fig. 34, but for Fe5+ ground state.

In Ni", 3p -+ 4s, 3s -+ 3d transitions do not lead to autoionization and the largest contributions arise from 3p + 4f, 5p, 5f and 3s -,4s. The 3p + 4p contribution is reduced because only 44/613 of the terms are autoionizing. Agreement between theory and experiment is good below about 500eV (Fig. 39), but at higher energies a similar type of discrepancy to that seen in Ni8+ is found.

t**

#

....... .,..',.._.............._.

,

I

.:.-." , : , :

/'

0

100

200

300

400

Electron Energy (eV) FIG.38. As in Fig. 37, but for the 3p63d24sstate.

500

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

2,

359

I

For ions with configuration 3p63d4(Ni6+)and 3p63d5(Ni”) the picture is very similar, with large contributions from EA and good agreement between the distorted-wave calculations of Griffin and Pindzola (1988) and the experiment of Wang et al. (1988) at low energy but with the experimental points lying above the experimental curves at high energies. In Ni3+(3p63d7)the CA DW calculations by Griffin and Pindzola (1988) included 3p + 3d excitation and direct ionization out of the 3d and 3p shells. Only about half of the 3p ionization leads to single ionization since 54 out of 110 levels of the 3p53d7 configuration lie above the ionization limit of Ni4+ and thus contribute to double ionization. This was taken into account by statistical partitioning. The results agree reasonably well with the experimental data of Gregory and Howald (1986), the maximum discrepancy being about 30% in the region of the peak (Fig. 40). Burke et al. (1987) have carried out an R-matrix calculation including 6 LS terms of the 3d7 configuration out of a possible 19, whereas Griffin and Pindzola (1988) and Pindzola et al. (1990) have applied the distorted-wave method to the same system. When the direct ionization cross sections (of 3p and 3d) calculated by the latter are added to the excitation cross sections calculated by the former authors, reasonable agreement is found between the two sets of calculations. The R-matrix results (Fig. 41) show a wavy structure at low energy due to REDA; this is not included in the DW theory.

D. L. Moores and K. J. Reed

360

9

0 n-

“E0 m

9

‘ 9 8 v .-6

5a N2

!A in

Kj

(3

9 0

200

400

600

800

:OOO

Electron Energy (eV) FIG.40. Electron impact ionization cross section of Ni3+. Theory as in Fig. 35. Experimental data: Gregory and Howald (1986).

N

E * 0T0 - 0 40.0 5

-

mu

Y

nn

t

n I

Electron Energy (eV) FIG.41. Electron impact ionization cross section of Ni3+ below 100eV. Full curve, total ionization cross section in the CA approximation. Broken curve, direct ionization only, Griffin and Pindzola (1988). Chain curve, R-matrix, excitation cross section from Burke et al. (1987) added to the C A direct cross section. Experimental points from Gregory and Howald (1986).

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

0

100

200

300

400

36 1

500

kctron Energy (eV) FIG.42. Electron impact ionization cross section of Fe'. Theory as in Fig. 34. Experimental data from Montague et al. (1984).

In similar calculations for Ni2+,the total indirect contribution was found to amount to less than 10% of the total. No experimental data are available for this ion. In Fe" (Fig. 42) and Ni', calculated direct ionization cross sections alone are higher than experiment (Montague et al., 1984; Montague and Harrison, 1985). Further theoretical work is required on these singly charged ions. 2. Ions with Conjigurations 2s22p63s23pq Just as in the case of 2pq ions, EA is not of major importance in these ions until the outer shell is nearly empty. There are two main types of possible transition, involving excitation of 3s and 2s, 2p electrons, respectively. The transitions 3s23p4-,3s3pqnl+ 3s23p4-

+e

(105)

will only be important for small q and for low ionic charge and should manifest themselves in a sharper rise from threshold than expected from direct ionization alone. There is some evidence for this in, for example, the measured data for Ar3+ (Gregory et al., 1983) (Fig. 43) while the effect is very strong in the Al-like ion Si+ (DjuriC et al., 1993). As the charge on the ions increases, L shell EA processes such as 2s22p63s23pq+ 2s22p53s23pqnl+ 2s22p63s23pq-' + e

(106)

D. L. Moores and K. J. Reed

362 1

I I

l l

I

I

I

I

I

I l l (

20 -

e t Ar3'

-

Ar4' + 2e

1

!3

lo0

200

I 500

I

I

I

I

I

lo00

ELECTRON ENERGY (eV) FIG.43. Electron impact ionization cross section of Ar3 +.Experiment: solid circles, Gregory et al. (1983); open circles, Miiller et al. (1980). Theory: solid line, direct ionization cross section calculated by the DW method of Younger (1981b); dashes, Lotz formula.

become important. Along a given isonuclear sequence, the magnitude of these contributions increases as q decreases and the ion is changed from an Ar-like closed shell configuration to an Al-like 3s23pconfiguration, which has more of the character of a Mg-like ion. The ion is also becoming more highly charged. Thus, a jump is just about detectable in Ar3+ (Gregory et al. 1983) (Fig. 43). Calculations for the Fe sequence (Pindzola et al., 1986b) predict a contribution equal in magnitude to a few percent of the direct cross section in Fe8+(3p6) rising to more than 90% in Fe13+(3p). This increase is confirmed by the measurements of Gregory et al. (1987). Pindzola et al. (1986b) have shown that the best agreement between theory and experiment is obtained if very large contributions from ionization of excited states 3p4-'3d are assumed to be present in the experimental data. This affects the fine details of the EA structure but not its overall relative magnitude, which is similar in the two cross sections. A similar situation if found in Nil" and in Nil4+. No account of radiative stabilization, which could be important in ions of such a high charge state, was made in the calculations.

H. HEAVYIONS Both direct and indirect ionization out of nd and nf subshells of heavy ions in low stages of ionization can be profoundly affected by the existence of

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

363

so-called “giant” resonances. The giant resonance phenomenon, which has also been recorded in the photoionization of atoms and ions, in molecules, clusters, and solids, can be interpreted as a collective effect resulting from cooperative behavior due to short-range correlations beween electrons (Connerade, 1991). Ideally, an accurate theoretical description should be based on a many-body approach such as many-body perturbation theory or the random phase approximation, but such calculations are difficult and much effort is required to produce good quantitative results. It is also possible to account for giant resonances in atomic systems within the framework of an independent-particle models. The formation of giant resonances is then explained by the same mechanism that produces the sudden changes in binding energy of d or f electrons, which precede the onset of the transition, lanthanide, and actinide series of elements in the periodic table (Fermi, 1928; Mayer, 1941; Griffin et al., 1969). These changes are due to the appearance of a double-well structure in the effective potential experienced by d- and f electrons. In a Hartree-Fock approach, the radial wave functions P)?(r) are calculated in an effective potential

(107)

where 2 is the nuclear charge, 2YLS(nl;r)/r and 2XLS(nl;r)/r are the direct and exchange potentials, and is a Lagrange multiplier (Froese-Fischer, 1977). In general, v,qS(r) and hence P)?(r) will be different for different LS terms, this being referred to as term dependence. For d and f electrons, the effective potential can have the form of a well-pronounced inner potential well separated from a shallower outer one by an intervening potential barrier. A strong barrier can have the effect of keeping an electron in the outer well; weakening of the barrier can cause its wave function to “collapse” into the inner well. Examples are shown in Fig. 44 where effective potentials for the 4f electron in Ba (6s4f) and La (6s5d4f), together with the corresponding 4f wave functions, are plotted as a function of distance from the nucleus. In Ba the barrier is stronger and the 4f electron resides in the outer well; whereas in La the weaker barrier causes the 4f wave function to collapse completely into the inner well, with a corresponding abrupt decrease in effective principle quantum number. Collapse of an excited or continuum electron may result in a greater degree of overlap with an inner orbital and give rise to enhanced excitation or ionization cross sections at certain energies, or giant resonances. Alternatively, increased destructive interference could occur, leading to reductions in cross sections.

D. L. Moores and K. J. Reed I'

T

I

1

I

I

I

I

I

Ba

I 6841

La I 685d41-

- --

RADIUS (bohr units) FIG.44. Effective potentials yffand radial wave functions P for the 4f electron in neutral Ba (solid curves) and neutral La (dashed curves) calculated in a Hartree plus statistical exchange approximation (Griffin et al., 1969). The radial functions are plotted on a linear scale, the zero of which lies at the corresponding eigenvalue, labeled caT.

The relative heights of the potential barrier and depths of the potential wells, (and consequently the location of the orbital wave functions) exhibit sensitive term dependence. In the important cases of the p5d and d9f configurations large positive dipole exchange interactions in the P effective potentials produce a strong double-well structure in ions of low charge. For other terms of these configurations, the exchange interactions are much smaller, resulting in less pronounced structures. This is illustrated in Fig. 45 for Sb3+ where it is seen that the barrier in Kff('P) pushes the 4f('P) further into the outer well compared to the same orbital for all other terms (labeled CA). Thus approximations that do not allow for term dependence, such as configuration average methods, may fail to provide an accurate description of, for example, 4d excitation or ionization when it is dominated by the 'P partial wave. Attempts to compensate by the device of using a Rydberg series configuration interaction may work in some cases but frequently require large basis sets in order to achieve comparable accuracy with term-dependent Hartree-Fock calculations. For sufficiently highly charged ions the effects of the strong coulomb potential depress the potential barriers to the extent that resonance effects

365

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disappear, even though term dependence of the effective potentials may persist. In Xe6+, 4d 4f calculated excitation cross sections still depend critically on the final term value (Griffin et al., 1984b), and in a study of 4d ionization of palladium-like ions (Younger, 1986b) term dependence was found to be significant up to an ionic charge of 10. The picture of well/barrier structure of the potential, its term dependence, and the consequent location of the wave function in the inner or outer well has provided a satisfactory explanation of the large resonant feature observed in the 4d photoionization cross section of neutral Ba. This is interpreted as being due to a giant resonance in the continuum rather than, as has been suggested, a result of 4f excitation followed by autoionization. In fact, precise calculations show that the 4d94f state is not autoionizing. In Ba potential barriers force the excited nf orbitals to reside in the outer well. Only cf continuum functions have appreciable overlap with 4d; all the oscillator strength is stolen by a fairly narrow band of the continuum, giving a giant resonance. The shape of the cross section and the differences in relative ion yields in the multiple photoionization of Ce, Gd, and Dy atoms (Zimmerman, 1989) may also be explained in terms of wave function collapse as the nuclear charge increases from Ce to Dy.

366

D. L. Moores and K J. Reed

1. Excitation-Autoionization Pindzola et al (1983a) have carried out theoretical studies of potential barrier effects on EA due to transitions 4d"5s2

-+ 4d95s2nf

(108)

(n = 4,5) in the Cd isoelectronic sequence and Griffin et al. (1984b) have made a similar investigation of the transitions 4d 05s25p6- 4 -+ 4d95s25p6-4nl

(109)

in the Xe4' isonuclear sequence for 1 d q d 6 and nl = 5p, 4f, 5d, and 4f. The ion Xe6+ belongs to both sequences. In both sets of calculations, a distorted-wave method was used to calculate the relevant direct ionization and excitation cross sections. For each ion, the dominant contribution to EA comes from 4f excitations. Radial wave functions corresponding to both CA and term-specific Hartree-Fock potentials were calculated; in the case of 4f, only the 4f 'P results differed significantly from the CA, which were consequently used for all terms other than 'P. The results may be interpreted by referring to the variation of Kff(CA) and Kff('P) and hence P4f(CA) and P4f(1P) as a function of increasing nuclear charge, in the case of the Cd isoelectronic sequence, or of increasing effective charge q in the case of the Xe ions. The trends are similar in the two cases. For low charge (In' and Xe'), Kfr(CA) and Kff('P) may differ significantly in shape and in barrier height, but both CA and 'P 4f wave functions reside in the outer well where they closely resemble each other (Fig. 46). Overlap with 4d is small. As the charge increases (Xe2+ and Sb3+),the barrier in Kff(CA) is virtually destroyed whereas in Kff('P) it is merely reduced; P4f(CA)but not P4f(1P)has partially collapsed into the inner well with increased overlap with 4d. By Xe6+ the barrier in Kff(CA) has gone while in Kff('P) it is small and negative (Fig. 47). P4f(CA)has collapsed into the inner well and P4,('P) is on the verge of doing the same. Overlap between P4f(CA) and 4d is large. The details of the EA contribution may then be explained by noting that the excitations to 4f 'P are dipole-allowed and depend on the size of the overlap between 4d and P4f('P); excitations to the other 4f levels are dipole-forbidden and depend mainly on the exchange contributions, which in turn depend on the overlap between 4d and the outgoing continuum wave, as well as the overlap between the 4f (CA) orbital and the incoming continuum wave. Cross sections for nondipole transitions fall off rapidly with electron energy whereas those for dipole transitions fall off much more slowly.

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2. In' and Sb3+ In these two ions all 20 levels of 4d95s24f are autoionizing and contribute to EA. In In' the small overlap between the 4d and 4f orbitals gives very small excitation cross sections and the total EA contribution is correspondingly small. In Sb3+ the overlap between 4d and 4f 'P is still small, giving small cross sections for dipole-allowed transitions, but 4f(CA) has a large amplitude in the inner well region. The result is a large EA contribution dominated by the nondipole transitions to 4f and also to 5f (5f 'P also being small), which exhibit a rapid fall-off with energy. The experimental results of Gregory and Howald (1986) (Fig. 48) are consistent with this, but comparison with the calculations of Pindzola et al. (1983a) (Fig. 49) shows evidence of contributions from higher Rydberg series members and of REDA, neither of which were included in the calculations.

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E l e c t r o n Energy (eV) FIG.49. Electron impact ionization cross section of Sb3+ near threshold. Solid points, measurements of Gregory and Howald (1986). Dashed curve, Lotz formula for 5s direct ionization; solid curve, sum of dashed curve and EA contribution from 4d1OSsZ+ 4d9Ssznf,n= 4,s calculated by Pindzola et al. (1983a).

the same reasons as those given for In+. As the ionic charge increases, P,,(CA) moves further into the inner well, while P&P) also shifts and begins to reside partially in the inner and partially in the outer well, as can be seen in Fig. 51(a), (b), and (c) and Fig. 47. The contribution from dipole-allowed 4f ‘P excitations is growing relative to that from nondipole 4f excitations. Thus we might expect, after an initial sharp rise associated with the EA thresholds, a rapid decrease from the maximum in Xe2+ and Xe3+ but a slower one in Xe4+ to Xe6+. This trend is clearly followed in Fig. 50. Results of theory and experiment from Griffin et al. (1984b) are compared in Figs. 52(a) through (e). Agreement is only fair, except for Xe6+. The overall magnitudes of the EA contributions tally quite well and the slower fall-off from the maximum with increasing q is also reflected in the calculations.The jump due to 4d + 4f in Xe” is considerably overestimated by theory. But the major source of discrepancy is in the large enhancements in the experimental cross sections below the various thresholds for EA. This must be mainly due to the effects of REDA. In Xe6+,for which this process is estimated to contribute very little, the same kind of discrepancy is not seen and theory and experiment are in excellent agreement. In X3+ both theory and experiment are compatible with a E - 3 fall-off between 100 and 150eV, characteristic of nondipole transitions. Some of the differences may be due to a “spectator” approximation made in the calculations.

D. L. Moores and K. J. Reed

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INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

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4. Ionization of Heavy Ions of Low Charge Younger (1986a, 1987a,b) has made a detailed study of the effects of giant resonances on the direct ionization of positive ions. Strictly speaking, ionization via a giant resonance is not an indirect process in the sense of our definition given in Section I, since intermediate autoionizing states are not involved. However, the formation or potential formation of giant resonances can have profound implications for indirect ionization also, so discussion of this work is included here. The first evidence for giant resonance behavior in an ionization cross section came from Younger's (1986a) distorted-wave calculations for 4d ejection from Cs'. Hartree-Fock potentials were used to calculate the continuum orbitals: a nonlocal term dependent potential for the 4d9kf' P ejected wave and an approximation to the frozen-core potential of the initial state for the incident and scattered waves. The calculated cross section showed a strong resonance feature close to lOOeV incident electron energy. Though the effective potential for the ejected electron wave exhibited a marked double-well structure, this was not the cause of the resonance, which was shown convincingly to be due to a shape resonance resulting from the well and barrier structure of the potential experienced by the scattered electron. At the resonance energy the f-wave scattered electron phase shift was found to increase sharply through 71, indicating the formation of a true shape resonance. In 4d ionization of Cs' 4d"5s25p6

-,4d95s25p6+ e

(110)

and in similar transitions in many other heavy ions, the final state lies above the double ionization threshold and decays by autoionization with near unit branching ratio. Thus the net result is ejection of two electrons via the IA mechanism and calculations should be compared with experimental data on double ionization. Comparison between Younger's calculations and the double-ionization experiment of Hertling et al. (1982) in Fig. 53 show excellent agreement. There is some evidence for an onset of double ionization below the IA threshold, perhaps due to direct ionization or excitation-double autoionization not included in the theory. Thanks to Younger's extensive calculations for a large number of systems, it has been possible to study the effects of giant resonance phenomena along isoelectronic and isonuclear sequences and for singly charged ions along a sequence in the periodic table. a. Periodic Table Sequence and Singly Charged Ions. Figure 54 gives the results of systematic calculations of distorted-wave exchange cross sections by Younger (1987b) of the 4d state of all singly charged ions from Ag' ( Z = 47) to Ce' (2 = 58). The solid curves represent calculations with

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

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FIG.53. Electron impact ionization of 4d shell of Cs'. Solid circles, measurements of double ionization cross section by Hertling et a/. (1982). Theory of Younger (1987a): solid curve, DWBE with term-dependent ejected f waves and ground state correlation; long dashes, DWBE with semiclassical exchange potential and no ground state correlation; short dashes, same as long-dashed curve but without scattering exchange; dot-dash curve, same as short-dashed curve but scattered waves calculated in a Z = 1 coulomb potential; dash-double-dot curve, same as dot-dash curve with incident waves also calculated in a 2 = 1 coulomb potential. The ordinate is the incident energy divided by the 4d ionization energy.

term-dependent ejected electron f waves; the long-dashed curves represent a non-term-dependent semi-classical exchange potential for the f waves. The short-dashed curves are the same as the long-dashed but neglect scattering exchange. The dot-dash curve is the same as the solid one but also allows for some ground state correlation. Low energy enhancements, as indicated by the shift in the peak to lower energy are seen to become significant by 2 = 49 (In'). By I' and persisting through the rare-earth ions, large resonance structures appear due to the shape resonance in the 1 = 3 scattered electron wave, showing the phenomenon to be a general one, not restricted to Cs'. The results of the

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INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

377

calculations are seen to be highly sensitive to the form of the incident, scattered, and ejected waves, especially for the three heaviest ions studied, so it is not surprising that agreement with experimental data is patchy. In some cases such as Xe+, the use of term-dependent ejected f waves overcompensates for the inadequacy of simpler approximations, and the measured peak cross section (Achenbach et al., 1983) is larger than that calculated. In I + both theory and experiment predict a sharp low-energy rise but the rapid fall-off in the cross section at higher energy is not reproduced by any of the experimental curves. In Ba' theory predicts a sharp resonance, whereas the experimental data for double ionization (Hirayama et al., 1987) suggest a much broader feature with large contributions below the calculated peak. About a third of this is due to 5s ionization and the rest to REDA and possibly READ1 following 4d excitation. Peart et al. (1993) used energy-resolved electrons to resolve structure presumed to be due to these processes between 80 and 120eV. For Z < 55 the term-independent, semiclassical exchange (SCE) cross sections exceed the term-dependent Hartree-Fock, sometimes by a large amount, but for Z > 55 the reverse is true. Younger (1987b) attributes this to the increased overlap of the term-dependent f-waves with 4d and increasing cancellation in the SCE matrix elements as a result of shifts in potential barriers as 2 increases. Also for 2 < 55, scattering exchange has a very small effect on resonant cross sections. The calculations for Ba+(4d"5s25p66s), La+(4d1°5s25p65d2),and Ce+(4d105s25p65d24f)predict that as the atomic weight increases the resonance becomes sharper and shifts to lower energies. This is due to the decreasing height of the potential barrier. But with every increment of unity in the atomic weight, an extra valence electron is added. Just as the resonance is about to disappear, the added electron becomes a 4f (Ce') whose presence increases the height of the barrier again, saving the resonance, which persists into the rare-earth elements. An extra 4f electron may be effectively "added" by excitation of an existing valence electron in a lower orbital. Thus a comparison of the 4d ionization cross sections for the ground state 5d2 and excited states 5d4f and 4f2 of La+ (Fig. 55) shows a broadening and shift of the resonance to higher energy as the barrier height increases with the addition of extra 4f electrons. A similar effect is responsible for the enhancement of the cross section of Ba2+ from an excited 5p54f configuration compared with that from the ground state 5p6 (Younger, 1987b).

b. Isoelectronic and Isonuclear Sequences. Younger (1987a) has also studied the systematic behavior of giant resonance effects along the Xe isoelectronic sequence Xe, Cs+, Ba2+,La3+ (Fig. 56). As the nuclear charge increases, so does the depth of the inner potential well, and the shape resonance in the scattered electron channel becomes narrower and shifts to lower energy. In Ba2+ it is incomplete, lying only partly in the continuum.

D. L. Moores and K J. Reed

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By La3+ it has disappeared from the continuum, since now there are no 4f valence electrons to maintain the barrier height. A similar trend is found in the iodine sequence I, Xe', Cs2+, Ba3+ and also along the Ba and Xe isonuclear sequences (Younger, 1987a; Pindzola et al., 1984b). More will be said about these data in the section dealing with multiple ionization. Younger (1987b) also investigated ionization of the 4f'O subshell in a number of ions, finding resonant behavior only in Tm', where it arises in the incident and scattered channels at different energies, giving a complex, double structure. A large resonance was also found in the 4f ionization of Eu+4f76s occurring in the scattered channel. In some ions (e.g., La2+)the hole state created by 4d ejection lies above the triple-ionization threshold, so that primary 4d ionization can give net double or triple ionization. One might expect a resonance in the doubleionization cross section to show up in the triple-ionization cross section too. In such a case some assumption about the branching ratio for double or triple ionization is required before comparison with experiment can be made.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION 25

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I. MULTIPLE IONIZATION 1. Rare-Gas Ions Since the pioneering studies by Muller and Frodl (1980) in Ar, extensive experimental investigations have been made of indirect contributions to electron impact multiple ionization of Ar ions (Muller et al., 1985a,b; Pindzola et al., 1984; Tinschert et al., 1989); Kr ions (Pindzola et al., 1984b; Tinschert et al., 1987); Xe ions (Achenbach et al., 1983; Pindzola et al., 1983b, 1984b; Muller et al., 1984; Howald et al., 1986; Gregory and Crandall, 1983); and triply charged Ar, Kr, and Xe ions (Gregory et al., 1983). Data are now available for double ionization of Ar' to Ar7+, Kr+ to Kr4+, Xe' to Xe4+,Xe6'; triple ionization of Ar+, Ar", Kr+, KrZ+ and Xe' to Xe3+,Xe6+ and for quadruple ionization of Ar', Xe+, Xe2+.In Kr and Xe ions in particular, multiple ionization cross sections are sufficiently large that the process may not be neglected by comparison with single ionization in modeling the ionization balance of high-temperature plasmas

D. L. Moores and K. J. Reed

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FIG.57. Electron impact double-ionization cross section cq,q+2 of Ar ions, measured by Tinschert et a!. (1989), Miiller et al. (1985a, b): circles, q = 1; squares, q = 2; triangles, q = 3; diamonds, q = 4; inverted triangles, q = 5; stars, q = 6; solid circles, q = 7.

a. Double Zonization. The results for Ar and Kr presented in Figs. 57 and 58, respectively, show a common trend. The onset of double ionization occurs at the threshold for direct ejection of two electrons. At an energy corresponding to the ionization of one inner shell electron, 2p for Ar and 3d for Kr, the cross section shows a change of slope due to the onset of the IA process:

e + Arq+(2p63~23p(6-q)) -,e +e

+ Ar(q+')+( 2 ~ ~ 3 ~ ' 3 p () ~+-eq ) + A1-(~+')+(2p~ + 2e

(111)

in Ar ions, and e + KrQ+(3d"4~'4p(~-~)) + e + Kr (q+1)+(3d94~24p(6-q)) +e

+ Kr(qf2)" ( 3 d " 4 ~ ~ 4 p ( ~ -+~2e) )

(112)

for Kr ions. Below this second threshold any measured cross section must

INDIRECT PROCESSES IN ELECTRON IMPACT IONiZATION

381

Electr3n energy l e v ) FIG.58. Electron impact double-ionization cross section of Kr ions u ~ , ~measured + by Tinschert et al. (1987), and 3d single-ionization cross sections calculated from the classical binary encounter approximation of Gryzinski (1965) (dashed curves); circles and curve A, q = 1; squares and curve B, q = 2; triangles and curve C, q = 3; diamonds and curve D, q = 4.

be due to a combination of direct double ionization and the indirect process of inner shell excitation followed by double autoionization (EDA) or auto-double-ionization (EADI) or perhaps even resonant capture triple autoionization. Given the absence of reliable theoretical estimates for these processes, their relative contributions remain a matter of speculation. In some cases, such as for Ar3+, it could be concluded that the entire cross section below the 2p ionization threshold is due to direct double ionization since the data can apparently be fitted very well (see Fig. 59) to the sum of the direct double-ionization cross section as given by the classical binary encounter theory (suitably scaled) of Gryzinski (1969) and the 2p single ionization cross section given by the Lotz formula. In others, such as Kr' and Kr2+ (Fig. 60), such a fitting procedure is not possible, indicating that some other process such as EDA could also be important. The combination of the uncertain accuracy of the Gryzinski cross section with an arbitrary choice of scaling factor and the neglect of indirect processes make the validity of the fitting procedure questionable and, until further theoretical work is done to clarify the issue, it should be regarded merely as a convenient parameterization of the data. The measured cross section below the IA threshold is large in Ar' and Kr+, its maximum value exceeding that reached above threshold. As the

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FIG.59. Electron impact double-ionization cross section of A r 2 + and Ar". Experimental data from Tinschert et al. (1989). Short-dashed curve, direct double ionization, classical binary encounter theory of Gryzinski, multiplied by 0.12 (0.19). Long-dashed curves, Lotz formula for L shell single ionization multiplied by Auger branching ratios. The solid curve is the sum of the two dashed curves.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

383

L

200 500 1000 Electrsn energy ( e V ) FIG.60. Electron impact double-ionization cross section of Kr'. Experimental data from . Solid curve, binary encounter theory of Gryzinski for direct double Tinschert et ~ l (1987). ionization multiplied by 0.2; dashed curve, same theory for 3d single ionization, multiplied by 0.8. 50

100

ionic charge increases, however, the below-threshold contribution decreases in relative size, becoming small by Ar4+ and Kr4+. For Na-like Ar", 2p ionization of the ground state does not lead to an autoionizing level and thus results in single ionization only; the small measured cross section is attributed to metastable ions in the ECR source. Xe ions are of interest because indirect double ionization results from primary 4d ejection: e

+ Xe4+(4d'05s25p(6-q))+ e + Xe(q+')+(4d95s25p(6-4)) +e + e + Xe'4+2'+(4d105s25p(4-q)) + 2e

(113)

and as we saw in the previous section 4d ionization can be strongly affected by giant resonance effects and calculated cross sections are sensitive to the potentials used. Comparison with experimental data for double ionization of Xe ions gives a sensitive test of theory and provides a good means for shedding extra light on the giant resonance mechanism. The calculations of direct single 4d ionization along the Xe isonuclear sequence by Pindzola et al. (1984) thus yield information on the doubleionization cross sections provided we assume that the hole states have negligible radiative decay probabilities and thus unit branching ratios.

D. L. Moores and K. J. Reed

384

Contributions to double ionizations will also arise from EDA and from direct double ionization, both processes having a threshold energy below that for IA. Results for Xe+ to Xe4+ are shown in Fig. 61. The most conspicuous feature of the measured cross sections (Achenbach et al., 1983; Miiller et al., 1984) is the large resonant-like contribution at low energies. In Xe' this feature closely resembles the peak in the 4d photoionization cross section of neutral Xe. A similar resonance is observed in the double-ionization cross section of I + . Along the Xe isonuclear sequence, the relative magnitude of the resonance decreases with increasing ionic charge. 400 80

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INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

385

The double ionization cross section for Xe' is quite small below the IA threshold in contrast with the behavior of the corresponding cross section in Ar+ and Kr'. This may be attributed to the resonant nature of the 4d ionization giving a cross section well in excess of that for direct double ionization, while at the same time displacement of the 4d95s25p5nfcollision strength into the continuum leaves only a small EDA contribution. In Xe2+ and to a lesser extent Xe3+ a significant EDA contribution is required to explain the measured cross section below the IA threshold, because in these ions direct double ionization is predicted to be very small. By Xe4+ IA dominates although the complication arises that only about 1/3 of the hole states are above the double-ionization threshold and thus able to contribute. Pindzola et al. (1984) have given a detailed theoretical explanation of the observed data within the framework of Hartree-Fock distorted-wave theory. This work clearly brings out the importance of allowing for term dependence in the potentials. Calculations using TDHF potentials are consistently in better agreement with experiment than those using CA potentials. In Xe+ term-dependent effects give a lower cross section than CA, whereas in the more highly charged ions the reverse is true. Pindzola et a/. (1984) have explained this behavior on the basis of the structure of the potentials. In Xe+ the low energy kf'P continuum orbital is forced to reside in the outer region by a large potential barrier, whereas in the CA case a reduced, negative barrier gives a kf continuum orbital residing in the inner well where overlap with 4d is large. As the continuum energy increases, the kf'P orbital collapses into the inner well, increasing its overlap with 4d. The net result is a resonance in both cases but the CA calculations yield a larger low energy cross section, which is more sharply peaked and with a maximum closer to threshold. Similar results were obtained by Younger (1987b). With increasing ionic charge, the potential barriers are weakened but persist in the TDHF potential. The kf CA orbital collapses into the inner wells in such a way that cancellation with 4d occurs, whereas the corresponding TDHF orbital still resides in a region where overlap with 4d is large, leading to a larger cross section. At the same time, the weakening of the double-well potential structure reduces the size of the resonance, which shifts to lower energy and ultimately vanishes from the continuum. The superior agreement of TDHF calculations with experiment provides support for these hypotheses. In Xe6+ ( 4 ~ ~ 4 p ~ 4 d ' ~the 5 s IA ~ ) process , is initiated by 4s or 4p inner shell ionization. Distorted-wave calculations for direct 4s, 4p ejection however predict a cross section a factor of 2 smaller and with a higher threshold than the measured data (Howald et al., 1986) (Fig. 62). Direct double ionization and 4p EDA have been proposed to account for the discrepancy; the observed threshold is compatible with either process.

386

D. L. Moores and K. J. Reed

FIG.62. Electron impact double ionization cross section of Xe6+. Experimental points by Howald et al. (1986). The dashed curve is a DW calculation of direct 4s + 4p ionization.

b. Triple and Quadruple Ionization. Triple ionization of Ar and Kr ions is dominated by single inner shell ionization (2s and 2p in Ar, 3s and 3p in Kr), followed by double Auger decay. Since the onset of the process is observed to occur at the minimum energy needed to remove three electrons, it is clear that direct triple ionization does contribute, but its contribution decreases with increasing initial ionic charge. It is also possible that direct double ionization followed by single Auger decay could also occur. The triple ionization cross section for KrZ+is shown in Fig. 63. The onset of the dominant process, single 3p ionization plus double autoionization, is clearly seen by the change in slope at 242eV. The curves show the results of an attempt to fit the data to scaled Lotz-type formulas for direct triple ionization and inner shell 3p ionization. The cross sections for triple and quadruple ionization of Xe ions shown in Fig. 64 are very large compared with analogous cross sections in Ar ions. In Xe ions multiple ejection can result from a large number of possible indirect processes. Because of the number of inner shells involved and the lack of pronounced energy separations between levels, it is difficult to distinguish between the various contributions in such complex ions. As the incident electron energy increases, the ratio of multiple- to single-ionization cross sections increases, although the actual size of the ratios decreases with the number of ejected electrons. This is illustrated in Fig. 65 for Xe ions. A similar effect is found in Kr ions, although here the ratios are smaller. This behavior is consistent with the mechanism of inner

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

20

387

7

t

200

103

s 00

1000

Zlec tron energy ( e V ) FIG.63. Electron impact triple ionization of Kr2+. Experimental data of Tinschert et

at.

(1987): Dot-dash curve, Lotz formula for 3s and 3p single ionization multiplied by 0.65; dashed curve, a fit to the experimental data between the direct triple-ionization threshold and the 3p single-ionization threshold. The solid curve is the sum of the other two.

shell ionization because it reflects the relative ease of ejection of the inner shell compared with more loosely bound electrons. c. Triple Ionization of X e 6 + . Howald et al. (1986) found the onset of triple ionization of Xe6+ (Fig. 66) to occur at 374eV, the approximate threshold energy for ejection of the three most loosely bound electrons

e

+ Xe6+(4d105s2)-, Xe9+(4d9)+ 4e

which is a multiple-electron process. Estimates showed that the absolute minimum energy for any single-electron primary process (3d excitation plus capture followed by quadruple autoionization of XeJ+) leading to production of Xe9+ was 614eV. The thresholds for 3d excitation-triple autoionization and ionization-double autoionization were, respectively, 670 and 760eV. Thus the large measured cross section between 400 and 600eV could only be due to a process in which primary ionization involved a direct multiple ionization, for example, direct double ionization followed by Auger decay:

e

+ Xe6+(4p64d"5s2) + Xes+(4p54d95s2)+ 3e + Xe9+(4p64d9)+ 4e

A correlated direct two-electron process such as this has seldom been

D. L. Moores and K. J. Reed

388

1

I

0 c

c 0

Electron energy ( e V ) FIG.64. Cross sections for threefold and fourfold ionization of Xe ions. Solid squares, u , , ~ ; open circles, C T ~ , open ~ ; triangles, uz,s;opensquares, C T ~ ,solid ~ ; circles, u ~ ,[From ~ . Miiller et al. (1984).]

recorded as occurring with such strength in such a high stage of ionization. In the experimental data, the onset of IA leading to net triple ionization after two Auger decays

+

e + Xe6+(3d'04s24p64d105s2)+ Xe7f(3d94s24p64d'05s2) 2e + Xes+(3d'04s4p64d'05s)+ 3e + Xe9+(3d104s24p64d9) + 4e is clearly seen from the change in slope near 760eV.

389

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION (a1

t

I

0 81

I

03

..

* 0 .

q.2

I

E l e c t r o n energy l e v )

E l e c t r o n energy led1

LI

9: 2 q= 3 q=l

-

'

0

p:

00

1.

/- .

100

.200

500

Electron energy I e V I

FIG.65. Ratios of multiple-ionization to single-ionization cross sections in Xe ions. (a) with 4 = 1, 2, 3, 4, (b) uq.q+3/oq,q+l with 4 = 1, 2, 3, and (4oq,q+4/op.q+, with q = 1, 2. [From Miiller et a/. (1984).

2. Single and Double lonization of Bi and Sb Ions In a study of single and multiple ionization of Bi4+ (q = 1, 2, 3) some unexpected results were found (Miiller et al., 198%) (Fig. 67). For energies greater than 200eV, the single-ionization cross section for Biz+ was found to be larger than the corresponding cross section for Bi+ despite the higher ionization energy of the doubly charged ion. The double-ionization cross section for Bi', 61.3, is exceptionally large, with a maximum of

D. L. Moores and K. J. Reed

390

0.3

0.2

0.1

0.0

200

500

800

1100

1400

Electron Energy (eV) FIG.66. Electron impact triple-ionization cross section of Xe6+ measured by Howald et al. (1986). The dotted curve is the calculated IA cross section.

7.5 x 10-"cm2. The cross section is a factor of 3 smaller than 01.3 but ~is ,only ~ slightly smaller than 02-4. The ground state coal collapses into the inner well, increasing its overlap with 4d. The net result is a resonance in both cases but the CA calculations yield a larger low energy cross section, which is more sharply peaked and with a maximum closer to threshold. Similar results were obtained by Younger (1987)autoionizing to bound. Thus, 5d ionization from the 5d"6s26p ground state of Biz+ leads to single, not double, ionization. These facts account for the larger single ionization cross section in Biz+, since it contains contributions from 6s, 6p, and 5d shells, whereas the cross section for Bi' only contains contributions from 6s and 6p shells. The double-ionization cross section for Bi' is large because it results from ionization of the 5d shell, which has an occupancy of 10; for Biz+ it is small because 5d ejection results in single ionization, leading to a loss of flux in the double-ionization channel. Similar results were observed in Sb ions, which have the homologous configurations 4d'O5s25pq. o

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

3

1

1

I

I l r l r l

1

1

I

I

1

391

1 1 1 -

-

--

- 1

10

20

I

1

I

l

50 100 200 500 ELECTRON ENERGY ( e V )

I

1000

10 8 - 6 -€u f L

9 1

b 2

0

1

50

---___

1

1

1

100 200 500 ELECTRON ENERGY (eV 1

1

1

.

1000

FIG.67. Electron impact single- and double-ionization cross section for Bi ions. [From Miiller et al. (1985c).] (a) Single ionization. Closed circles, experimental C T ~ , open ~ ; circles, experimental solid curve, direct ionization cross section u1,2,Lotz formula including 6s and 6p ejection; dashed curve, direct-ionization cross section u ~ ,Lotz ~ , formula including 6p, 6s, and 5d ejection. (b) Double ionization. Closed circles, experimental ul.3; open circles, ~ ; experimental dashed curve, direct double-ionization cross experimental u ~ , triangles, , classical binary encounter approximation; dot-dash curve, single 5d section u , , ~ Gryzinski ionization of Bi", Lotz formula; solid curve, total calculated u l , 3 including indirect 5d IA contribution plus direct double ionization.

3. Single and Multiple Ionization of Heavy Metal Ions (54 2 Z 2 58)

Many fine structural details and examples of new indirect multiple events have been uncovered by studies of ionization of heavy metal ions with nuclear charge in the range of 54 to 58 using the fast energy scan technique.

D. L. Moores and K. J. Reed

392

14 n 12

% 0

-

!2

10

b U

8

t m-

0

+ 0

6

0) v) v) v)

4

0

L

0

2 0 100

1000

Electron energy CeVI FIG.68. Electron impact ionization cross sections of La3+ and Cs+ ions. Squares, experimental data for La3+;circles, experimental data for Cs+ (Miiller et al., 1988b). Also shown is Younger's (1987a) calculated cross section for 4d ionization of Cs+. The Cs' data have been multiplied by 2 for ease of display.

Miiller et al. (1988b) have reported measurements of single ionization of La3+,single and double ionization of Cs', Ba2+,and La2+,and double and triple ionization of Ce' and Ce2+. Miiller et al. (1989) measured cross sections for single ionization of La', La2+,and La3+, double ionization of La+ and La2+,and triple ionization of La2+. Single- and double-ionization measurements for Ba2+ and' Ba3+ by Tinschert et al. (1991) used two alternative ion sources; an ECR source producing a beam with a predicted high metastable ion content and a PIG source delivering mainly ground state ions, thus yielding direct information on ionization from both ground and metastable ions simultaneously. The cross section for single ionization of the Xe-like ion La3+ (Fig. 68) exhibits a peaked structure with a shape resembling that of the double

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

393

ionization cross section of the isoelectronic ion Cs+, for which it was attributed to a giant resonance in the 4d ionization. However, a similar explanation cannot apply to La3+,because the peak occurs at 105eV, well below the 4d ionization threshold of about 140eV. Besides that, 4d ejection would lead to net double ionization. Distorted-wave calculations (Fig. 69), including direct ionization of the 5p6 ground state plus EA due to excitation of 4d95s25p64f,predict a rapid rise in the cross section due to the latter process, but the rise does not occur until 95 eV. Below this energy, the large measured cross section must be due to a process neglected in the calculations. The estimated 5p54f metastable content in the experimental data (5%) is insufficient to account for it. The explanation consistent with the observed data is that below 95 eV the cross section can only be due to REDA processes, with capture into a 4d95s25p64fnlconfiguration and subsequent double autoionization. Energy scan measurements (Fig. 70) have been able to reveal some of the fine details of the structure, which overlaps the region where 4d94fz levels are located. Although peaks as narrow as 0.4eV show up, it is doubtful if these features can be identified with individual states owing to the complexity of the

r

I

I

I03 I29 Electron Energy ( e V )

I 5

FIG.69. Electron impact ionization cross section of LaJf.The solid line is a DW calculation of the sum of direct single ionization (dashed line) plus EA contributionsfor the 5sz5p6 ground configuration.The dot-dash curve is a similar calculationfor the 5s25p54fexcited configuration. [From Miiller et al. (1989b).]

D. L. Moores and K J. Reed

394

n

V

120

s

0 i Y

2 100 .C

0

c V

al v) v) v)

80

0

I

0

ao

70

90

100

110

120

Electron energy CeVl FIG.70. Electron impact ionization cross section of La3+;energy scan measurement. Open circles are data from Fig. 69. The arrows indicate the ranges for contributions from EA via the 4d94f configuration of La3+ and from REDA processes via the 4d94f2 doubly excited configuration of Lazf. [From Miiller et a/. (1988b).]

spectrum. They are more likely to be blends of many states. Figure 71 shows the results of scan measurements of similar details in single and double ionization of Cs+, Ba2+, and La2+ and for double and 15 1 .o

z

.5

0 c

U

n La"-

1.

74

76

?6

80

85

90

95

La-

100

Electron energy CeVl FIG.71. Scan measurements of single (upper row) and double (lower row) ionization of Cs+ [(a), (b)], Ba2+ [(c), (d)], and La2+ [(e), (f)] ions after subtraction of approximated nonresonant contributions.

395

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

:i:_i n

39.5

N

E

t

c)

-

c

I

0

U

u

0

v,

1.6

II)

2

0

1.4 ls5 1.3

'-2

1

b ) La2'

1

;\/ I.

11

lS1 1.0

t:! 90

1 1 8 8 1 1

3

4

Q9 ;b ,

La'+

I

-

0

1 8

92

a 1 1

6

3

1 I *

94

a

8

!a

1 1

1"

96

1 '

1 1 ' 1 ' '

""1""!

98

100

Electron energy CeVl FIG.72. Comparison of energy scan measurements of (a) single and (b) double ionization of La2+ ions. [From Miiller et al. (1989b).]

triple ionization of Cef and Ce2+. In each case a wealth of resonance structure is revealed. After an approximation to the background ionization has been subtracted, it is found that many of the individual features line up energetically when the data for ejection of different numbers of electrons from the same target ion are compared. The peaks must therefore be due to the same original intermediate state, formed by capture, decaying by alternative pathways (which could also include radiative decays) that lead to different ion yields. La2+ provides a good example. Data for single and double ionization are compared in Fig, 72. The feature at about 95eV is due to resonant capture

D. L. Moores and K. J. Reed

396

to a La' configuration 4d95s25p65d3.This state cannot sequentially triply autoionize, so to produce La4+ it must eject three electrons by some sequence that includes one multiple-electron emission, such as autoionization followed by auto-double-ionization (or the reverse) or auto-tripleionization. Two sequential single autoionizations of this state should lead to net single ionization. This is confirmed by the appearance of the same state in both single- and double-ionization cross sections. The data for Ce ions (Fig. 73) show peaks at identical energies in double and triple ionization of Ce' and Ce2'- These are due to intermediate states formed by 4p excitation plus capture, branching into net double or triple ionization by ejection of three and four electrons, respectively, thus providing examples of RETA and REQA. In Ba' (Fig. 74) intermediate states that decay either by emission of four or five electrons (quintuple autoionization) with a relative probability of 4: 5 giving net triple or quadruple ionization have also been observed. The strong resonances seen between 80 and 105eV in single ionization of La3+ are not seen also in double ionization of the same ion because the double ionization threshold is too high, at 110 to 118eV. .0 n

"E

1.o

.6

0

2

-

b U

.4

.5

.2

g 0.

0. .20 .15

f:

.4

.10

.2

.05

L

0.

190

200

210

0.

180

200

220

Electron energy CeVI FIG.73. Energy scan measurements of double (upper set) and triple (lower set) ionization of Ce' and Ce2' ions after subtraction of approximated nonresonant contributions from the original data. [From Miiller et al. (198Sb).]

397

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

n

N

E 0 0 N I

-

0 U

K 0 .c 0 Q) v) v)

v)

9 L

0

*

.*

4l

770

0

780

*.\

-.

..

*..-t-nw.wb:

.

790

800

810

820

830

Electron energy CeVl FIG. 74. Comparison of energy scan measurements of (a) net triple and (b) net quadruple ionization of Bat ions after subtraction of approximated direct contribution from the measured total cross section. [From Hofmann et al. (1990).]

In the systematic studies of La and Ba ions (Miiller et al., 1989b; Tinschert et al., 1991), a wide variety of indirect processes involving excitation or ionization of different subshells followed by Auger decays have been identified in addition to the resonant capture multiple autoionization events described earlier. The data have been analyzed by comparison to the results of energy-level calculations with the Dirac-Fock atomic structure code of

398

D. L. Moores and K. J. Reed

Grant et al. (1980) and with distorted-wave calculations of direct-ionization cross sections. Interpretation presents a problem owing to the huge number of terms arising from these open-shell configurations. For example, the 5p55d2 configuration of La2+ straddles the 5s25p6of La3+ so that 5p ionization of La+ 5p65d2is divided between single and double ionization. In triple ionization of La2+ the onset is compatible with the predicted threshold for direct triple ionization (128 to 136eV) and also with the threshold for 4d ionization followed by auto-double-ionization (IADI) (125 to 133eV). At higher energy (230 to 248eV), 4p ionization followed by two sequential autoionizations can also give net triple ionization. An excited hole state formed by inner shell ionization or excitation plus capture may branch into a number of ionization stages via combinations of sequential single or multiple-electron transitions. Add to this the possibility that radiative decay may occur at any stage and the situation becomes very complicated indeed.

J. VERYHIGHLY CHARGED IONS As discussed in Section 11, in lighter ions the Auger channel dominates the decay of the intermediate states, and the importance of EA increases relative to direct ionization as the charge state of the ion increases. However, in heavier ions the branching ratio for decay of some of the intermediate states is much smaller than unity, and those intermediate states that decay radiatively do not contribute to ionization. The radiative channel diminishes the contribution of indirect processes to the ionization cross section, the magnitude of the diminution increasing with increasing Z . Also, relativistic effects, which are negligible in lighter ions, can be very important in heavy ions. Relativity influences the atomic structure and the energy levels of highly charged heavy ions and can critically affect the cross sections for electron impact ionization and excitation. In addition, the cross sections and branching ratios for particular intermediate states can be strongly affected by configuration interaction in some highly charged ions. This can have striking effects in ions where level crossings occur and can result in abrupt changes in the EA cross sections for the affected intermediate states. 1. Na-like Ions

a. Excitation-Autoionization. Griffin et al. (1987) have made a theoretical investigation of the effects of EA on the electron impact ionization of several Na-like ions with 22 < 2 < 28. For these ions the radiative rates are comparable to the Auger rates. The target wave functions were calculated

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

399

by Hartree-Fock methods with relativistic modifications, including massvelocity and Darwin corrections. A CA distorted-wave approximation was used to determine the excitation cross sections from the initial configuration to all final autoionizing configurations included in the calculation. It was found that configuration interaction among the final autoionizing states has a relatively small effect on the overall inner shell excitation cross sections for these ions. However, configuration interaction changes some of the autoionizing rates by as much as 3 orders of magnitude with a consequent pronounced effect on the branching ratios. The total ionization cross section was obtained under the independent processes assumption. When radiative decay of the intermediate states is taken into account, the total ionization cross section including EA is about five times the direct cross section for all of these ions. The results for Fe’” were in fairly good agreement with calculated results for EA reported earlier by LaGattuta and Hahn (1981) although the latter authors used averaged branching ratios in their calculation. The good agreement, however, appears to be a fortuitous result of the fact that the average branching ratios used by LaGattuta and Hahn are too small for the 2p53s3d configurations and too large for the others. In a later calculation for Fe15+ using detailed branching ratios, Chen et al. (1990) found that the E A cross sections from the C A calculations were larger than the detailed level calculations by as much as a factor of 3 in the near threshold region. This discrepancy is due to the fact that not all states from the same configuration are energetically accessible in that region, and their inclusion in the averaging process results is gross overtimation of the cross sections. Crossed-beam measurements of ionization cross sections for Ar’ + between 18 and 1186eV (Zhang et al., 1992) show a marked E A effect and are in good agreement with cross sections calculated in the DW approximation and including E A contributions, using the methods of Griffin et al. (1987). The first fully relativistic calculations of E A contributions to ionization of a highly charged heavy ion were carried out for Na-like Au (Reed et at., 1990a). In this investigation separate relativistic calculations were carried out for the direct ionization cross section, the cross sections for excitation to the intermediate states, and for the Auger and radiative rates for decay of the intermediate states. The following types of process were included in the calculations:

+

+

2s22p63s e + 2s22p6

2e

2s22p63s+ e -,2s22p53s31+ e -,2s22p6+ 2e

+ 2s22p63s+ e + 2s22p53p2+- e

+ 2s22p6+ 2e

2s22p63s e -,2s2p63s31+ e -,2s22p6 2e 4

The direct ionization cross sections were calculated using a full partial wave

D. L. Moores and K. J. Reed

400

approximation in which the bound, incident, scattered, and ejected electrons are computed in Dirac-Fock potentials (Moores and Pindzola, 1990). A relativistic DW method in which relativistic configuration interaction wave functions for the target ion were generated using a Dirac-Fock atomic structure code (Hagelstein and Jung, 1987) was used to compute the cross sections for exciting the 2s and 2p electrons to the n = 3 subshells. All of the 2s22p631, 2s22p5313Y and 2 ~ 2 ~ ~ 3 1 3(1,l' 1 ' = 0, 1, 2) configurations were included in the target structure calculation. The branching ratio for each intermediate state was calculated from Eq. (7), the individual Auger and radiative rates being calculated using the MCDF model (Grant et a]., 1980) and Chen (1985). The energy levels and wave functions for the excited states were calculated explicitly in intermediate coupling, including configuration interaction within the same complex using the MCDF model in the average-level (AL) scheme (Chen, 1985). Excitation cross sections and branching ratios for 80 intermediate states were computed and some of these are shown in Table I. The results for Na-like Au are shown in Fig. 75. The dashed curve is the direct ionization cross section, a smooth function of incident electron energy peaking at about 2.5 times the 3s ionization energy. The dotted curve shows the sum of the direct ionization and the cross sections for excitation of the n = 2 inner shell electrons. The steep rise in this curve near 10.20keV is

I ,... ..........

I ."""""'~"~~ '.....I............................................ I

~

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

Excitation-autoionization without radiative decay Y

......

r 6

\ Excitation-autoionization

.-c0

with radiative decay

v) v)

v)

0

'Direct ionization

10.0

11.0

12.0

13.0

14.0

I 15.0

Energy (keV) FIG.75. Cross sections for electron impact ionization of Au6'+. Dashed curve, direct ionization; dotted curve, sum of direct plus EA but without radiative decay; solid curve, total ionization cross section including radiative decay.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

401

TABLE I ENERGYLEVELS, EXCITATION CROSS SECTIONS ( U ) , AND BRANCHING RATIOS(B) FOR AUTOIONIZINGINTERMEDIATE STATES OF A d 8 +

Level 1 6 8 12 15 18 25 33 34 43 44 100 114 120 122 129 138 141 150 154 162 166 175

Configuration

Energy (eV)

cr(cm2) at 15 keV

B

00.00 9592.13 9608.12 10101.17 10128.82 10200.16 10308.07 10426.50 10439.78 10454.56 10492.48 11330.86 11467.65 11532.63 11880.37 12062.11 12242.07 12336.31 12524.60 12587.58 12731.27 12846.10 12901.02

1.89( - 23) 1.89(-23) 1.15(-23) 1.65( -23) 2.64( - 22) 1.10(-23) 1.71(-23) 1.44(-23) 5.68( - 22) 2.67( - 22) 4.18( - 24) 4.41( -24) 1.31( -22) 7.23( - 23) 1.25( -23) 1.06(-22) 8.82(-24) 4.82( -24) 5.02( -26) 3.23( - 24) 1.23(- 23) 4.68( -24)

4.45( - 2) 4.80( - 1) 1.00 4.77( - 4) 8.96( - 1) L3S( - 1) 1.00 5.49( - 5) 9.72( - 3) 3.66( - 3) 3.54( - 2) 4.65( - 2) 8.52( - 1) 9.36( - 1) 3.46( - 1) 6.60( - 2) 1.96(- 2) 3.03( - 1) 8.98( - 1) 5.17( -4) 1.00 4.84( - 1)

Note. Numbers in parentheses are powers of 10.

largely due to the opening of the channel for exciting the (2p,~z3s3p31z),/, state (level 18 in Table I). In the notation used here the first orbital in the configuration is the inner shell having a vacancy, while the following two orbitals give the outer shell occupations; the subscript outside the parentheses gives the total angular momentum of the state. Thus the above configuration means a state with a 2p3/, vacancy, electrons in the 3s and 3p3,, orbitals, and a total angular momentum of J = 1/2 for the state. The cross section for exciting this state is 3.59 x lo-’’ cm’, which is nearly four times the direct ionization cross section at this energy. The next large increase occurs at 10.45keV and is mostly due to the (2p3/23~3d5/2)3/2 state (level 43), which has a cross section of 6.23 x lo-’’ cm’ near threshold, and the large increase near 10.50keV is due to the (2p3/23~3d5/2)1/2 state (level 44) with a cross section of 2.93 x 10-”cmZ at this energy. The solid curve in Fig. 76 is the total ionization cross section with radiative decay taken into account. Although the effects of CI on the Auger

D. L. Moores and K. J. Reed

402

I

I

I

I I

7.0

0.6 0.5

1 I

I

4.0

6.0

..

.

l

-1

1 I

I

10.0 Energy (keV)

8.0

I

12.0

14.0

1

FIG.76. Cross sections for electron impact ionization of Xe43+. Dotted curve, sum of ionization cross sections for n = 2 inner shell electrons; dashed curve, total direct ionization cross section; solid curve, computed total ionization cross section; open circles, measurement by Donets (1983); solid circles, measurements by Schneider et al. (1990).

and radiative rates of Na-like doubly excited states are very important for the ions with 22 < Z < 28 (Griffin et al., 1987) they are much less significant for Au6'+, except for states arising from the 2p53s3d and 2p53pz configurations. The excitation of the 2p53pz states from the Na-like ground state is forbidden in a single-configurationapproximation. These states contributed slightly more than 5% of the total ionization cross section in Au6'+. Radiative decay diminishes the EA contribution considerably in Au6' +. For example, the (2p3,z3s3d5,z)31zstate (level 43) has the largest excitation cross section. However, this state can decay to the ground state by a strong electric dipole allowed 2p-3d transition. It has a branching ratio of only 9.72 x and because of this the contribution of this intermediate state to the total ionization cross section is small. On the other hand the (2p31z3s3p31,),l,(level 18), which also has a large excitation cross section, has much weaker 2p-3s El transitions. This state decays principally via Auger emission with a branching ratio of 0.9. Hence this excitation makes a significant contribution to the total ionization cross section. In Table I one can see the large range in the magnitudes of the excitation cross sections and branching ratios of the intermediate states. Most of these intermediate states make very little contribution to the ionization cross section because either their excitation cross sections or branching ratios are very small. Although radiative decay reduces the overall EA contribution by a factor of about 4, even after this has been taken into account EA still

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

403

enhances the ionization cross section by more than a factor of 4. This enhancement is due to a few levels with large excitation cross sections and also large Auger decay probabilities. Detailed calculations of cross sections and branching ratios for each intermediate state are necessary in order to observe the large contributions of these few intermediate states. In a subsequent calculation, Clark and Abdullah (1991) showed that the computed EA enhancement is considerably smaller if configuration averaging is substituted for detailed calculations. Similar calculations for Na-like Xe were carried out later by Reed et al. (1990b) using the same distorted-wave methods. These calculations included intermediate states produced by excitation to the n = 4 subshells as well as to the n = 3 subshells. Direct ionization from the 2s and 2p subshells was also included. Above 10 keV the direct 2p ionization cross section exceeds the 3s cross section; the 2s contribution is smaller, but not negligible. In Fig. 76 the theoretical results for Xe43+ are compared to the cross sections obtained in two different experiments. The measurements of Donets (open circles) were made using an electron-beam ion method (Donets, 1983). The solid circles represent data obtained in measurements using the EBIT (Schneider et al., 1990). The results for Na-like Xe were similar to the results for Na-like Au, but there were some interesting differences in the contributions of different intermediate states, some of which are discussed later. The variation of EA effects along the Na isoelectronic sequence was studied in an investigation that included nine Na-like ions with 26 < 2 < 92 (Reed et al., 1991).Direct ionization, excitation cross sections, and branching ratioswerecalculatedforFe'5+,Se23+,Mo31+,A g46+,Xe43+,Ba45+,E~52+, and U*'+ at energies where the effects of direct ionization of the 2p and 3s electrons would not be significant and where contributions from excitation of inner shell electrons to n = 4 states of the ions are also unimportant. While the relative magnitude of the contributions from particular intermediate states varies from one ion to another, a few states make prominent contributions in every ion, notably the (2p1/23s3p1/2)1/2, the ( 2 ~ 3 s ' ) ~and /~, (2p3,23s3d5,2),,2states. This work included a detailed examination of how the contributions of certain intermediate states vary with atomic number. In Fig. 77 electron impact excitation cross sections calculated at an incident energy just above the excitation threshold are shown as functions of atomic number for three intermediate states. They generally decrease rapidly with increasing atomic number, apart from two notable exceptions. For the (2pl/23s3p1/2)1/2 state, the cross section for 2 = 54 is less than the cross section for 2 = 56; and for the (2p,,,3~3p,,~),,~ state the cross section for 2 = 26 is about one-half the cross section for Z = 34. In contrast, the cross section for (2pl/23s3p1,2)1/2in FeI5+ (Z = 26) is nearly five times larger than it is in Se2,+ (2 = 34). The 3s direct ionization cross sections for incident energies of 1.02, 1.10, and 1.75 in thresholds units all decrease smoothly and rapidly and drop

D. L. Moores and K. J. Reed

404 I

I

I

I

1

I 50

I

I

I

I

60

70

80

90

I

I

I

20

30

40

z

I

I

I

I

loo

FIG.77. Electron impact excitation cross sections for some intermediate states in Na-like ions.

nearly 3 orders of magnitude between Fe’” and U a l + . Figure 78 shows the ratios of the total EA contributions (product of excitation cross section and branching ratio) to the direct ionization cross sections for the three intermediate states just discussed. The ratio for the (2p3/23s3p32z)122 state starts out very low at Z = 26 and rapidly increases with increasing atomic number. For the EA contribution for this single intermediate state is 2.7 times the direct ionization cross section, and in Us’ the corresponding figure is 3.5. The ratio for the (2S3S2)1/, state is slightly higher than it is for the (2p3223s3p3/2),/,state in Fe’” but increases much more gradually with increasing Z . For Z 2 79, its value is about 0.5. For the (2pl,,3s3pl,,)l~2 state, the largest ratio is for Fe15+ for which the EA contribution is equal to the direct ionization contribution and far exceeds the contribution of any other intermediate state. For the rest of the ions the contribution of this state varies between 70% and 100% of the direct ionization contribution, except for Xe43+ ( Z = 54), for which the ratio drops suddenly to 0.06 after reaching 0.95 at Z = 47. It is back up to 0.96 at Z = 56. The reason for this decrease at 2 = 54 can be seen in TabIe 11, where the branching ratios for the (2p3/z3s3p31z)llz, ( 2 ~ 3 s ~ ) and ~ ~ ,(2p11,3s3pl,z)ll~ , states are collected. The branching ratio for the (2p1/23s3p1,2)1/2 state is 0.9293 at Z = 47, and +

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

405

4.0

3.5 3.0 2.5

a" -r

0: 2.0

m 1.5 1.o

0.5

0 20

60 70 80 90 100 Z FIG.78. Ratios of EA cross sections (BiQi) to direct ionization cross sections (Qd) for three intermediate states in Na-like ions.

30

40

50

Branching ratios

26 34 42 47 54 56 63 79 92

0.9383 0.9866 0.9870 0.9816 0.9718 0.9682 0.9499 0.9364 0.9164

0.9807 0.9902 0.9856 0.9796 0.9681 0.9641 0.9476 0.8962 0.8379

0.9985 0.9929 0.9569 0.9293 0.0846 0.9583 0.8564 0.8523 0.8379

406

D. L. Moores and K. J. Reed

0.9583 at 2 = 56, but for 2 = 54 it is 0.0846. The dramatic drop occurs because the (2p1123s3pl12)llz eigenvector has very large (2p1/23~3d5/2)1/2 and (2pl/23s3d312)1/2 coefficients.Very steep level crossings occur for these states at 2 = 54, and hence strong configuration interaction among these states is manifested in Xe43+. Both the (2p1/23~3d512)l12 and (2pl/23~3d312)1,2 states have very large radiative decay rates and the strong mixing with these two states at Z = 54 results in the very small Auger branching ratio for the (2plj23s3p,12)lj2state. In all the other ions considered the (2pl~23s3pl/2)l~2 eigenvector has small mixing coefficients with these other states. Another interesting example of the effect of configuration interaction is noted for the (2p1/23S2)1/2intermediate state. Reed et al. (1990b) point out that the excitation cross section for this state is relatively large in Xe43+, The large cross section in Xe43+ but is relatively small in Fe15+ and is due to mixing of this state with the (2p3/23~3d5/2)1,2 state. This (2p3/23~3d5/2)1/2 level has one of the largest excitation cross sections among all the intermediate states. In Fig. 79 the excitation cross section for the (2p1/23S2)1/2 state is plotted for all the ions. The magnitude of the (2p3,23s3dS/2)1/2 coefficientin the (2p1/23S2)1/2eigenvector is also shown. It is very small in Fe”+ (0.0335) and increases slowly with increasing Z. At 2 = 54 it rises steeply to 0.2005 and the (2p1/23s2)1i2cross section also increases sharply. The magnitude of the (2p3/23~3d5/2)1/2 coefficient is even 4.0

3.5

1.o

0.5

z FIG.79. The (2p,i,3s3d,,,),i, component of (2p,,T3s2),i, eigenvector and electron impact excitation cross section for (2p,i23s2),i2level in Na-like ions.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

407

larger (0.4016) at Z = 56. But at 2 = 56 the (2p1/23S2)l/2excitation is less than its value at Z = 54. This decrease in the cross section occurs because the dominant (2p1123s2)1/2 coefficient in the eigenvector and the (2p3/23~3d5/2)1/2 coefficient have opposite signs for Z = 56. For Xe4,+ and all of the lighter ions these two coefficients have the same sign. They have opposite signs in A U ~ ~but + , in this case the magnitude of the (2p3/23~3ds/2)1/2 coefficient is sufficiently small that the effect is not very pronounced. Similar irregularities in the Auger rates of doubly excited 3131' states of Na-like ions have been noted previously and were explained as due to the effects of level crossings among various states (Chen, 1989). In Fig. 80 the direct ionization cross sections and the total ionization cross sections with and without radiative decay are shown for Se2,+ and U81+.The EA cross section for U a l + is dominated by a large increase near 8 keV, corresponding to the (2p,i23s3p,,2)1,2 excitation. The contributions of the other intermediate states are small by comparison. In contrast to this, the Se2, + EA cross section exhibits a number of step-like increases. This illustrates a general trend. As Z increases along the Na-like sequence, the EA contribution becomes increasingly concentrated in fewer and fewer transitions. In Fe'" the 10 largest EA cross sections constitute about 70% of the total EA contribution; in Ba4'+, 80% but in U6'+ the 10 largest constitute 97% of the total. The (2p1/23s3pli2)1/2 state, which makes the single largest contribution in Fe'", constitutes 25% of the total EA contribution in that ion. In contrast to this, the largest single contribution in U6'+,which comes from the (2p3/23s3p3/2)1/2 state, is more than 50% of the total EA contribution in this ion. It can be seen in Figs. 80(a) and (b) that the effect of radiative decay is noticeably greater in Ual+than in Se23+.This illustrates another trend in the sequence; the effect of radiative decay of the doubly excited states generally increases with Z. The effect of the Breit interaction was also studied in this work and it was found that for a few intermediate states significant changes in the branching ratios resulted when it was included, some of them changing by several orders of magnitude. But the overall effect on the total EA cross sections was not significant. Table I11 illustrates these effects for Se23+and Xe43+. Also, the effects of radiative decays among the n = 3 states of the Na-like ions were examined.Since radiative rates increase rapidly as Z4, one would expect the n = 3 to n = 3 radiative decays to be most pronounced in the heavier elements. Table IV shows how the branching ratios and the EA cross section for the (2p,/,3~3d)~/~ state are affected by including them. The effect is nearly negligible for Z = 34 and only slightly larger for Z = 56. But for Z = 92 the branching ratio and the EA cross section decrease by more than an order of magnitude for this state. The overall effect on the total EA contribution is negligible in Se2,+ and only about 3% in Ba4'+. In U" where the effect is most pronounced, these radiative decays reduce the EA cross section by slightly more than 10%. +

D. L. Moores and K. J. Reed I

I

I

a ~e"+

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

10 765 -

~

..>

<

........, 1

4-

3!-"-

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2-

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15.0

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17.0

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19.0

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Energy (keV) FIG.80. Cross sections for electron impact ionization of (a) Sez3+ and (b) Us"; Dashed curve, direct ionization; dotted curve, sum of direct plus EA but without radiative decay; solid curve, total cross section including radiative decay.

b. REDA in Highly Charged Na-like Ions. LaGattuta and Hahn's CA model predicted a factor of 5 enhancement of the ionization cross section of FeI5+ for electron impact energies between 750 and 780eV due to REDA.

409

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION TABLE I11 EFFECTOF INCLUDING BREITINTERACTION IN CALCULATION OF AUGER RATES ~

(a) Effect on Intermediate States Branching ratio 2

Level

Configuration

(without Breit interaction)

(with Breit interaction)

Total cross section (cm2) ~~

~~

(without Breit interaction)

(with Breit interaction)

4.493( - 20) 3.290( - 21)

4.596( - 20) 3.415(-21)

34 54

Note. Numbers in parentheses are powers of 10.

TABLE IV RADIATIVEDECAYS AMONGn = 3 LEVELS EFFECTOF INCLUDING

Z

34 56 92

Branching ratio

Cross section (cm2)

(without n = 3-3 decays) (with n = 3-3 decays)

(with n = 3-3 decays) (with n = 3-3 decays)

1.0000 1.oooO 1.0000

0.9988 0.9727 0.0013

1.261( - 21) 1.275(-22) 9.826( - 24)

1.261(- 21) 1.204(- 22) 1.285(- 25)

(b) Effect on Total Excitation-Autoionization Cross Sections 2

Branching ratio (without n = 3-3 decays)

34 56 92

4.385( -20) 3.448( - 21) 2.642( - 22)

Note. Numbers in parentheses are powers of 10.

Cross section (cm2) (with n

=

3-3 decays)

4.382( -20) 3.341( - 21) 2.303( - 22)

D. L. Moores and K. J. Reed

410

This enhancement was clearly absent from the experimental data obtained by Gregory et al. (1987). Later, a close-coupling excitation calculation by Tayal and Henry (1989) indicated that the REDA contribution to the cross section was small. However, Tayal and Henry included only contributions from the 2p53s31nl’Rydberg series. They also neglected radiative decays in the first step of this two-step process, and not all possible Auger channels were included in their calculations of the branching ratios for the first step. Subsequently,a comprehensive calculation of the REDA cross sections for Fe’” using a detailed Dirac-Fock model was carried out (Chen et al., 1990) in which more than 10,000 intermediate autoionizing states were included. The REDA process for Na-like ions can be represented schematically by e

+ ls22183s+ -+

ls22173snlnl’+ ls22173sn”l”+ e ls22s22p6+ e

+e

Explicit calculations were carried out for resonant states from 2s22p53s41nl ( n = 4-7), 2s22p53s31nl’(n = 7-12), 2s22p63s31nl’(n = 4-10), 2s2p63s4141’, and 2s22p53s5151“configurations. For the first three cases, extrapolation to n = 30 was accomplished using an n - 3 scaling law for Auger transitions. The total ionization cross section were obtained in the independent processes approximation for incident electron energies from 600 to 1000eV, the inner shell excitation cross sections and the detailed Auger and radiative rates being calculated using the same methods as those described above for A d 8 + . In the calculations of the two-step double Auger branching ratios, all possible Auger channels and radiative decays leading to bound states were taken into account. The REDA cross sections are reduced by about 50% by radiative transitions.To facilitate comparison with the experiment of Gregory et al., each REDA resonance was convoluted with a 2-eV Gaussian. This comparison is shown in Fig. 81. The overall agreement between the theoretical and experimental results is rather good. However, some strong narrow peaks such as those at 760,780,858,and 865 eV might have been missed by the experiment. Experiments with higher energy resolution and good statistics are currently being performed at the Heidelberg Storage Ring TSR, and may make possible a better comparison with the REDA peaks that appear in theoretical results. Figure 82 shows a comparison of the three sets of calculations of the REDA contributions in Fe’”. In the 750- to 800-eV region the cross sections from the CA approximation with a 20-eV bin width are about three times larger than the cross sections from the detailed MCDF calculation and would have been more than an order of magnitude larger if the same bin width of 2eV had been used. This demonstrates the importance of performing detailed level-to-level calculations in the treatment of the REDA process. The cross sections from the close-coupling calculations are a factor of 2 smaller than the results of the detailed calculations.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

41 1

4.0

1.o

0.0

'

600

t

700

I

1

I

800

900

lo00

Energy (ev) FIG.81. Electron impact ionization cross section for Fe'". The dashed curve is the direct ionization cross section. The lower solid curve is the sum of direct and EA cross sections, whereas the upper solid curve is the total calculated cross section with the REDA contribution (Chen et al., 1990). The solid circles with error bars are the experimental data of Gregory et a/. (1987).

Chen et al. (1990) showed that the most important contributions to the REDA cross sections in this energy region arise from the 2 ~ ~ 3 ~ 4 1 and 41' 2s2p63s31nl'intermediate states. These states were omitted in the close-coupling calculations, resulting in the under estimation of the cross section in this region. Studies of REDA contributions to ionization of Na-like KrZ5+and Xe43+ have also been reported (Chen and Reed, 1993a) and also show numerous narrow resonances due to the effect. In KrZ5+the REDA contribution is as much as 30% of the total ionization cross section including EA, and in Xe43+ it is as much as 25%. Recent high-resolution measurements of the ionization cross section of Xe43+ using the Electron Beam Ion Trap also show the REDA resonances and are in good qualitative agreement with the calculated results (Schneider et al., 1993). 2. Highly Charged Heavy Li-Like Ions

Relativistic distorted-wave calculations of electron impact ionization cross sectionsincluding contributions due to direct 2s ionization EA [Eq. (85)l and REDA [Eq. (86)] have also been reported for Li-like ArI5+,FeZ3+,Kr33+,and Xe5'+ (Chen and Reed, 1992).The same Dirac-Fock methods used for Na-like

D. L. Moores and K. J. Reed

412 4.5

4.0

1.o

0.5 0.0I

600

700

060

900

1000

Energy (eV) FIG.82. Theoretical electron impact ionization cross sections for Fe'". Solid curve, MCDF results, Chen ef al. (1990); dotted curve, close-coupled results, Tayal and Henry (1989); dashed curve, results of configuration average approximation, LaGattuta and Hahn (198 1).

ions were used to obtain the Auger and radiative rates for the Li-like ions. Relativistic multiconfiguration target wave functions, which included ls221and ls21nl' (n = 2,3) configurations, were used in the distorted-wave calculation of the 1s - nl (n = 2,3) excitation cross sections needed to obtain the EA contribution. For the REDA ionizing process [Eq. (86)] explicit calculations were carried out for resonant states from ls2s31nl' (n = 3-6, 1 < 3), ls2s2pnl configurations. The onset of 2s - 2pnl (no < n < 18, 1 < 3), and 1~2~4141' Coster-Kronig transitions occurs at no = 9 and 10 for ArI5+ and Fe23+, respectively. Extrapolation to n = 50 was accomplished for the first two cases using the n - 3 scaling law for the Auger transition rates. In the calculations of the Auger matrix elements, the contributions from both Coulomb and generalized Breit interactions were included for the 1~2121'autoionizing states; for the others the Breit interaction was neglected. In Table V, energy levels, excitation cross sections, and branching ratios for autoionizing states that contribute significantly to the ionization are listed. Among these important states, the 1 ~ 2 (J ~= 5/2) 2 state ~ ~deserves ~ ~ some attention. This state is metastable against Auger decay via the Coulomb interaction as well as against radiative decay via electric dipole

413

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

TABLE V ENERGY LEVELS,EXCITATION CROSS SECTIONS AND BRANCHING RATIOS FOR SOME IMPORTANT AUTOIONIZINGSTATES IN LI-LIKEIONS Autoionizing state

2J+ 1

Energy (eV)

uex(cm2)

B"

2 6 2 2 4 2

ArI5+ 3078.33 3089.11 3111.50 3124.18 3124.79 3617.55

3.22( - 22) 3.62( - 22) 5.25( - 22) 2.10( - 22) 2.92( - 22) 6.30( - 23)

0.967 0.812 0.232 0.786 0.910 0.973

2 6 2 2 4 2

Fe23+ 6599.8 6626.7 6651.6 6675.9 6678.8 7779.9

7.59( - 23) 7.48( - 23) 1.03( - 22) 6.83( - 23) 5.10( - 23) 1.52( - 23)

0.893 0.500 0.145 0.290 0.940 0.913

2 6 2 4 2

Kr33+ 12929.3 13010.1 13075.6 13083.9 15277.6

2.23( - 23) 2.05( - 23) 2.48( - 23) 1.50( - 23) 4.47( - 24)

0.755 0.212 0.0448

2 2 6 4 2

XeS1 30054.6 30185.7 30459.1 30583.1 35636.9

5.05( - 24) 3.27( - 24) 3.75( - 24) 3.68( - 24) 1.01( - 24)

0.487 0.0716 0.0487 0.132 0.578

0.848 0.805

+

~~

Note. Numbers in parentheses are powers of 10. The cross sections were calculated at electron energies 3651, BOOO, 15,400, and 35,840eV for ArI5+, FeZ3+,Kr33+, and Xe'", respectively. The energies of the autoionizing states are referenced to the ground state of the Li-like ions.

emission. An accurate branching ratio is only obtained when the Breit interaction is included in the calculation of both Auger decay rates and radiative decay rates, which occur by magnetic quadrupole x-ray transitions. From Table V, one can see that the branching ratio becomes progressively smaller as atomic number Z increases from 18 to 54. In the REDA calculations, all possible Auger and radiative transitions were used in the determination of the branching ratios. Each REDA resonance was convoluted with a Gaussian of 20eV in width. The REDA cross sections are reduced by factors of 1.75, 2, 3, and 5 for Ar'", FeZ3+, Kr33+,and XeS1+,respectively, due to the inclusion of radiative transitions. The direct ionization, EA, and total ionization cross sections including

414

D. L. Moores and K. J. Reed

REDA contributions are displayed in Figs. 83(a) through (d). Due to the large difference between the 1s and 2s binding energies, the indirect processes occur between three and four times the ionization threshold instead of near the ionization threshold, as is the case in Na-like ions. The contributions of EA to the total decrease from 8% to 2% as Z increases from 18 to 54. For low-Z ions the ls2s2p states are the most important intermediate states, whereas at high 2 the ls2sz becomes the most prominent due to the increasing dominance of the radiative transitions in the decay of the ls2s2p states (see Table V). For the REDA ionization process the ls2s2pnl Rydberg series contributed very little to the ionization cross sections for ArI5+ and Fe23+.Hence, the contribution from this Rydberg series was neglected for Kr33+and Xe51+. The resonance structures seen in Figs. 83(a) through (d) arise from the ls2s31nl' intermediate states. The first three groups of peaks denoted by A, B, and C are from n = 3,4, and 5, respectively. Near the region of the KMM resonances, the indirect processes contribute 15%, 11%, and 8% for Fe23+, Kr33+,and Xe5'+, respectively. The gradual decrease in the enhancement is due to the increasing dominance of radiative decay with increasing atomic number. The 1~2~4141' states were found to contribute only 1.2% to the ionization cross section. Hence, the contribution from the higher n members of the ls2s41nl' Rydberg series was neglected in these calculations. Despite the strong radiative decay in these ions, the indirect processes still contribute about 15% to 20% to the ionization cross section at electron energies near the 1~2~3131' resonances. These contributions are larger than the enhancement found for lower Z ions (Reed and Chen, 1991; Tayal and Henry, 1991). From this it is clear that the effect of indirect processes, especially REDA, on the electron impact ionization of the Li-like ions persists to very highly charged members of this isoelectric sequence. The effects of relativity on REDA in Li-like ions have been studied by comparing the cross section with and without relativistic effects (Chen and Reed, 1993b). The comparisons are made in Figs. 84(a), (b), and (c). Relativistic effects shift the resonance positions to higher energies-by 60 eV in Fe23+and 1200eV in Xe5'+-and also enhance the cross sections -by 20% in FeZ3+to a factor of 3 in Xe5'+. The relativistic enhancement is principally due to increases in Auger rates, which result from several factors, including changes in the energies, shifts in wave functions, spin-orbit coupling and the inclusion of magnetic and retardation interactions in the two-electron operator. Other calculations for highly charged Li-like ions, by Badnell and Pindzola (1993), are discussed in the next section together with their results for Be-like and B-like ions.

I

2.2

2.1

-

2.0

-

1

1

I

I

I

1

e + Ar j5+

N

B g

19-

P

1.8-

?J

-5

u) n

E 1.7

-

1.6

-

1.5

2.8

3.0

3.2 3.4 Electron energy (KeV)

3.6

See legend p. 416.

3.8

4.0

-

D. L. Moores and K. J. Reed

416 (c)

1.06

I

1.04

I

-I

I

e + Kr3)*

1.02

0.90

--.... 1.64

-

c9

-

.

-------..-..

-

7

I

1.60 L 28

--.

I

30

I

I

1

1

32 Electron energy (keV)

34

36

FIG.83. Electron impact ionization cross sections for (a) Ar'", (b) FeZ3+,(c) KrJ3+,and (d) Xe'". The dotted, dashed, and solid curves show the direct, direct plus EA, and total crass sections, respectively. The peaks A, B, and C indicate the KMM, KMN, and KMO resonances, respectively.

417

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION I

I

1

I

e

- - - - - --

+ Fez3+

Relativistic Non relativistic

-

FIG.84. See legend p.418.

D. L. Moores and K. J. Reed

418

1.1

I

I

I

I

-

1.0 -

0.9 -

e + xe51+

-

Relativistic Non relativistic

-

_____--

0.8 -

0.7

-

-

0.6

-

-

0.5

-

-

-

0.4

-

J!

0

I

L

$ ,;

I

FIG.84. REDA cross sections for (a) Fez’+, (b) Kr33t, and (c) Xe” ’. The solid curves indicate the results of the relativistic calculation with the Breit interaction. The dashed curves show the results of the nonrelativistic LS coupling calculation.

3. Other Isoelectronic Sequences a. Be- and B-like Ions. Badnell and Pindzola (1993) have studied EA and REDA in electron impact ionization of Li-, Be-, and B-like ions of Fe, Kr, and Xe. Cross sections were calculated in the independent processes model [Eqs. (6), (7), (12)-( 14)] in which the direct ionization was calculated by the CA distorted-wave approximation, excitation cross sections by a level-to-level distorted-wave approximation with configuration interaction and intermediate coupling, using semirelativistic wave functions. All Auger and radiative rates required to evaluate the branching ratios (7) and (14) and the capture cross section (13) were calculated by the code AUTOSTRUCTURE (Badnell, 1986). The resonances were convoluted with a 1.5 Rydberg Gaussian. The results obtained look similar in form to Figs. 83(a) through (d). They found that in the absence of radiation damping there is only a slow variation between the Li-, Be-, and B-like ions of all

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

419

three species. However, when radiative damping is included, results within an isonuclear sequence can change dramatically from one ion to the next. In all cases the dominant group of REDA resonances are the KMM transitions. Radiation damping reduces the KMM cross sections by a factor that increases significantly with Z in Li-like ions the factor is 3 in Fe, 5.6 in Kr, and 24 in Xe. The corresponding reductions in B- and Be-like ions are less but still large. For Li-like ions most of the radiation damping is on the second autoionizing step; this damping is reduced in B-like ions and further reduced in Be-like ions. Hence the resonances persist more strongly at high-charge states in Be-like ions than in Li-like or B-like ions. The absolute values of the REDA cross sections for the Be-like ions are much greater than for the ions in the other two sequences. However, the direct ionization cross section increases by a factor of about 2 from Li-like to Be-like ions and by another factor of about 2 from Be-like to B-like ions. The Be-like EA cross sections are greater than their Li-like and B-like counterparts by a factor of 2 to 3. This means that relative to the slowly varying "background" of direct ionization plus EA, the Be-like resonance contributions are only a little larger than the Li-like, but both are relatively substantially larger than the B-like contribution. This fact may make the Li- and Be-like sequences preferable for a high-resolution experimental search for these resonances to the B-like sequence or the more complex second-row ionic sequences. Large differences between the calculations of Chen and Reed (1992) and Badnell and Pindzola (1993) may be tracked down to differences in the Auger yield of the l s 2 ~ 2 p ~ state P ~ , ~used in the double Auger yield (branching ratio) employed in the two studies. When the same value is used (Chen and Reed, 1993b), the difference is only 20% and this can be put down to the different approximations used to calculate the collision cross sections.

b. Mg-Like Ions. Indirect contributions to ionization of Mg-like Kr ions have been investigated by Chen and Reed (1993a) using the independent process and distorted-wave approximations. In these ions the EA and REDA processes are described by Eqs. (92), (93), and (94). Figures 85(a) and (b) show the effect of radiative decay in Mg-like and Na-like Kr, respectively. In the first step of the REDA process radiative decay produces only a 10% reduction in the cross section, but in the second step it is more important. The REDA cross sections for Kr24+ are reduced by 30% in contrast to a factor of 2 reduction for Kr25+.The presence of two 3s electrons in the Mg-like autoionizing states increases the Auger width and drastically reduces the radiative damping for Kr24+. In this ion the enhancement due to indirect processes is about 2.5 times the direct ionization cross section.

420

D. L. Moores and K J. Reed 3.5

[

3.0

-

I

I

I

I

I

..

....

I -

m. a .

*.

a . a .

ta

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..

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-

---......... No radiative ---- Radiative in 2nd step -

E

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Radiative in both steps

v

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-

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1.7

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FIG.85. The REDA cross sections for the (a) 2p53s24141'intermediate states of KrZ4+and (b) 2 ~ ~ 3 ~ 4 1 intermediate 41' states of KrZ5+.The dotted curves indicate the results without radiative damping. The dashed curves show the values with radiative decay in the second step. The solid curves show the values with radiative decay in both steps.

INDIRECT PROCESSES IN ELECTRON IMPACT IONIZATION

42 1

V. Conclusions The extensive theoretical and experimental studies made during the last 30 years have confirmed that so universal is the phenomenon of indirect ionization that there is scarcely a positive ion for which the electron impact ionization cross section does not show one or another of its manifestations. EA is particularly important for complex ions with well-stocked subvalent shells, often dominating direct ionization in such systems. Particularly good examples are provided by the heavier alkaline earth ions (Fig. 22) and ions such as Ti3+ and Zr3+ (Figs. 30 and 31), in which the effect is especially spectacular. REDA contributions are also by no means negligible in most complex ions, as comparison between experimental data and the results of calculations that do not take the process into account reveal. In many complex ions the indirect contribution results from the summed total effect of excitation of an enormous number of intermediate states and is observed as an overall enhancement of the cross section, irregular in shape, rather than a series of well-defined identifiable features. In general, unless interference between resonances in differential cross sections is the subject of study, the independent processes model ( 6 ) is adequate. The theoretical work then uses separate calculations for the direct ionization cross sections, excitation cross sections, and branching ratios. In some ions a distorted-wave approximation can be sufficiently accurate to calculate the excitation cross sections. In many others, however, nothing short of a many-state R-matrix close-coupled calculation is capable of giving agreement with the measured EA contribution. This is demonstrated by the work described in this review on Li', Li-like ions, Ti3+,and Ca'. In Ca+ the inclusion of all nine terms of the 3p53d4s configuration in the closecoupled expansion is of crucial importance (Badnell et d.,1991), whereas in Li' the calculation of accurate cross sections for double excitation requires the use of pseudo-orbitals in the configuration interaction expansion of the bound states (Griffin et al., 1992).The close-coupled approximation also has the advantage of automatically incorporating the REDA effect via closedand bound-channel resonances. In heavy ions of low charge, the existence of a double well and barrier structure in the potentials experienced by the electrons in an independent particle model and its dependence on the final term of the ionized ion can provide an explanation of some of the observed behavior of the ionization cross sections, such as the so-called giant resonances in the 4d ionization of Cs . Some observed phenomena, for example, the double excitation plus capture in He-like ions, READ1 in Li-like ions, the triple ionization of Xe6+ -any process involving simultaneous ejection of more than one electron in a single step-are intrinsically multiple-electron transitions, which can only +

422

D. L. Moores and K. J. Reed

be described theoretically through correlated many-electron interactions. Accurate calculations are difficult and are best approached through manybody theory. More theoretical work is required in this area. Direct multiple ionization also falls in this category. In highly charged ions, a radiative decay diminishes the contribution of indirect ionization, the effect becoming more pronounced with increasing charge on the ion. Relativistic effects too become important, influencing the atomic structure and affecting the behavior of branching ratios. Calculations have revealed the extent to which the detailed ion structure determines the total EA and REDA contributions in very highly charged ions and have also shown that despite the influence of radiative decay, a significant amount of indirect ionization can persist even in the most highly charged systems. The most important contribution to indirect multiple ionization of positive ions has been found to be from the IA mechanism, which gives a larger contribution than the higher order direct process except in singly charged ions at lower energies. If a unit branching ratio is assumed, the IA contribution is given by the direct single ionization cross section out of an inner shell, which can be calculated with reasonable reliability. In some ions there is also a large REDA contribution to this process, which in combination with the direct ionization, for which a reliable theory is yet to be found, complicates the interpretation of the data. The recent discovery of resonances that may branch either into net n-fold or ( n + 1)-fold ionization also provides a challenge to quantitative theoretical interpretation. Finally, we conclude with a repetition of the warning given in Section 1V.G. Indirect ionization is a very common phenomenon and is apt to be encountered in some measure in the electron impact ionization of all positive ions. Because of the dependence on ionic charge, the simultaneous contributions from alternative mechanisms, and the possibility of radiative decay, the cross sections often vary irregularly and erratically from one ion to the next, even though their ionic structure might look superficially similar. Thus, any extrapolation from one system to another, by so-called “scaling”, even of separate contributions to the total, should be approached with extreme circumspection.

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ADVANCES IN ATOMIC, MOLECULAR, A N D OPTICAL PHYSICS, VOL. 34

DISSOCIATIVE RECOMBINATION: CROSSING AND TUNNELING MODES DAVID R. BATES Department of Applied Mathematics and Theoretical Physics The Queen’s University of Belfast Belfast, Northern Ireland

. . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Diatomic Hydrogen Ions . . . . . . . . . C. Monohydride Ions . . . . . . . . . . . . D. Valence-Bound Nonhydride Ions . . . . . .

I. Introduction

11. Upper Limit to Rate Coefficient . . . . 111. Crossing Dissociative Recombination . . A. Theory . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

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

427

. 433 . 434

.

434 440 443 445

E. Homonuclear Ions in Which One Partner Has a Closed Outer Valence Shell; and Dimer and Trimer Ions . . . . . . . . . . . . . . . . . . IV. Tunneling Dissociative Recombination . . . . . . . . . . . . . . A. Theory . . . . . . . . . . . . . . . . . . . . . . . . . B. Ions on Which Calculations Are Done . . . . . . . . . . . . . C. Valence-Decrease Class of Ions . . . . . . . . . . . . . . . . D. Proton-Bridge Bond Ions and Related Clusters . . . . . . . . . . V. Signature of Polyatomic Ion Dissociative Recombination . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

455 461 46 1 467 47 1 477 479 48 1

.

I. Introduction The earliest mention of dissociative recombination was made by Kaplan (1931). In an attempt to explain the strong emission of the 557.7 nm ‘S -r3P line of 01 from the nightglow and the aurora he supposed that

0: + e-+ 2O(’S)

(1)

may well be very rapid because of the “remarkable coincidence” that the recombination energy equals the energy needed to dissociate the oxygen molecule into two ‘S oxygen atoms. However, because of the poor quality of the data he had, he overestimated the energy available for the excitation of the free oxygen atoms by 1.4 eV; and, in any case, close electronic energy balance in such a process does not, as was commonly believed at the time, have a “resonance effect.” 421

Copyright 0 1994 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003834-X

428

D. R. Bates

In his Bakerian Lecture to the Royal Society of London, Appleton (1937) described the exploration of the ionosphere with the aid of radio waves. He reported that during much of the day the electron density at the peak of the E-layer scarcely departs from its equilibrium value as the Sun's zenith angle changes, from which he deduced that o! > lo-' cm3 s-', o! being the recombination coefficient. Setting out to interpret the inequality and other results of Appleton in terms of the collision physics, Massey (1937) listed the types of processes that might be occurring. Among the recombination processes, he mentioned dissociative recombination but dismissed it as being too slow to be significant on the grounds that it involves interaction between electronic and nuclear motion within the molecule. He was strongly influenced by an earlier Bakerian Lecture by Chapman (1931) in which it was asserted that the observed geomagnetic variations require that the ratio A of the number density of negative ions to the number density of electrons is very large in the E-layer. Massey thought this was the key to understanding recombination in the ionosphere. After a lengthy set of quanta1 calculations (Bates and Massey, 1943) the conclusion was reached (Bates and Massey 1946) that 1 is actually so small that any notion that negative ions affect the recombination rate must be abandoned. Having examined and dismissed every other possibility they could envisage, Bates and Massey (1947) tentatively proposed that dissociative recombination may, after all, be rapid. In support of this they recalled that in their research on the photoionization of caesium vapor Mohler and Boeckner (1930) found that ionization occurs when the energy is insufficient to produce it directly due, they thought, to associative ionization processes such as Cs(n2P) + Cs -,Cs:

+e

(2)

For II = 9 Mohler and Boeckner estimated that the cross section is as great to as cm2, which would imply that the rate coefficient for the inverse process of dissociative recombination is not small. Bates and Massey noted that similar associative ionization had been observed by Arnot and McEwan (1938, 1939) in mercury vapor and in helium. Yet the theoretical case for dissociative recombination being a very slow process seemed to remain overwhelming. It was correctly presumed to be close kin to dissociative attachment for which Massey (1938, 1950) had given a straightforward formulation. According to this formulation, the rate of transitions from the initial state i to the final statefis 4n2 -I h

K,12

(3)

where Vir is the matrix element of the kinetic energy of relative motion of the pair of atoms. Denoting the reduced mass of the nuclei by M , the nuclear coordinates by R, the electron coordinates relative to the nuclei by r, the

DISSOCIATIVE RECOMBINATION

429

initial and final nuclear and electronic wave functions by t,hi(R) and $/(R) and by q5i(r,R) and 4f(r,R), respectively,

Since the electronic wave functions are orthogonal and vary much more slowly with R than the nuclear wave functions, this reduces to

The large magnitude of M when expressed in atomic units makes I V, 1’ very small. A stirring change in the position came when Biondi and Brown (1949a, b) began their classic application of microwave techniques to the measurement of recombination between electrons and positive ions in helium, neon, argon, hydrogen, nitrogen, and oxygen afterglows. They reported that at 300 K the recombination coefficients in the inert gas afterglows (taken in order) and 3.3 x are 1.7 x lo-*, 2.1 x cm3 s-’ and that those in the other gases (again taken in order) are 2.5 x 1.4 x and 2.8 x cm3 s-’. They assumed that the ions in the former group are monatomic and that the ions in the latter group are diatomic with larger recombination coefficients because the spacing of the vibration and rotation levels is comparable with the mean electron energy, Applying a classical theory of radiative recombination involving an effective nuclear charge on the ion (Eddington, 1926), they obtained values of o!for the inert gases about ten times the measured values, unaware of quanta1 calculations (Bates et al., 1939) on 0’ radiative recombination that had given a recombination coefficient of only about 3 x 10-”crn3 s - l at 300 K and had shown that the use of an effective nuclear charge is unjustified. The paper on the helium afterglow appeared first. Discussing it Bates (1950a) reasoned that a recombination coefficient of 1.7 x 10-%m3 s-’ could not be due to atomic ions; that because of 2He

+ He’

4

He;

+ He

(6)

the ions are diatomic and that the loss process is dissociative recombination. As it happens, the helium results were flawed owing to the effect of ambipolar diffusion but this mattered little overall, the neon and argon results being sound. (c.f, Bardsley and Biondi, 1970). Complex ions, H: and N:, were responsible for the measured high recombination coefficients in the hydrogen and nitrogen afterglow. Convinced that the laboratory researches of Biondi and Brown had proved that dissociative recombination is indeed very rapid, Bates (1950b)

D. R. Bates returned with determination to the problem of understanding how this could be. After he had again verified that formula (5) offers no hope, a sketch of how the potential energy curves of the ion and molecule might lie (Fig. 1) showed that coupling of the nuclear and electronic motions is not involved and that the way dissociative recombination may occur is simple. For the sake of being definite, suppose that the ion AB' is in the zeroth vibrational level of its ground state. If the potential energy curve is crossed near to its minimum by a repulsive potential energy curve belonging to an electronic state of AB (now called the resonant state), a double-electron radiationless transition may occur whereby the ion plus free electron enters the resonant state. Because of their mutual repulsion the atoms begin to

LSeparation FIG.1. Schematic potential energy curves for crossing mode. The ionic curve includes the energy of the free electron.

DISSOCIATIVE RECOMBINATION

431

separate. By preventing the inverse process of autoionization this stabilizes the recombination AB+(X) + e P AB(Res) -,A

+B

(7)

The explanation for dissociative recombination being far more rapid than radiative recombination is that radiationless transitions are far more rapid than radiative transitions as observations by Shenstone (1948) and others on the widths of spectral lines broadened by autoionization had shown. Bates posited that the recombination coefficient could be some 10-'cm3 s-'. This theory met with general acceptance. Tests were, of course, needed. By making measurements in various argon-helium mixtures, Biondi (1951, 1963) checked to determine whether the recombination coefficient was high or low depending on whether the ions are molecular or atomic. The signature of the mechanism envisaged is the generation of suprathermal atoms. Using a Fabry-Perot etalon, Connor and Biondi (1965) and Frommhold and Biondi (1969) saw the signature on the profiles of lines emitted from neon and argon afterglows. The favorable crossings proposed were conjecture in 1950 but, as subsequent research has shown, the conjecture was reasonable. For example, in the case of O:(X, 2) = 0) dissociative recombination, an investigation by Guberman (1983a) has shown that favorable crossings are made by four potentials of 0 , that belong to states that can be reached without spin change and whose configuration does not differ from the configuration of the ion plus free electron by more than two orbitals. It was not until almost 30 years after the physical model of dissociative recombination was described that an ab initio computation on a moderate complex ion was performed: that on 0: by Guberman (1987). A brief descriptive name for the mode of dissociative recombination on which attention has, so far, been focused is the crossing mode. In contrast with the almost immediate recognition of the crossing mode that followed the crucial laboratory measurements of Biondi and Brown the recognition of the second mode of dissociative recombination was long delayed. The prime reason for this is that scientists in the field, knowing that a favorable crossing may lead to rapid dissociative recombination, were irrationally convinced that rapid dissociative recombination necessitates a favorable crossing of potentials. Subsidiary reasons are that most thought that the sophisticated mathematical treatment that had been developed by, in particular, Giusti (1980) must surely be complete; and that sufficient attention was not paid to reading the guiding signals provided by experiment. In retrospect, theorists should have been alerted to the true position by the very large recombination coefficients observed for the hydrate cluster ions H,O+(H,O), by Leu et al. (1973a), for the polyhydrogen ion H: by

432

D. R Bates

Leu et al. (1973b) and for the ammonium cluster ions NH:(NH,) by Huang et al. (1976). They were not. Instead they erroneously turned (Smirnov 1977, Bottcher 1978) to the indirect process of Bardsley (1968b) which, like the original process, entails a crossing (see Section 1II.A). Another opportunity was provided by the measurements of McGowan et al. (1979) showing that the recombination coefficients of H 3 0 + and many like ions (see also Adams and Smith, 1988a, b) are large. It was missed. Considering the channel H30+

+ e+H,O + H

(8)

Bates (1989) noted that the energy available is not enough for the products to be electronically excited, that there is a unique final potential and that this is also the case for six similar saturated polyatomic hydride ions that are known to have large recombination coefficients. Reasoning that the chance of favorable crossings in all seven instances is so remote that it must be dismissed, he suggested channels of the types H 3 0 + + e + OH + 2H

(9)

H 3 0 + + e + OH + H,

(10)

or

a facet of which seemed to make the problem for these less acute. However a seminal series of investigations by Adams et al. (1989, 1991) and Herd et al. (1990) on the products of dissociative recombination showed that while channels of types (9) and (10) are important so are channels of type (8). Bates (1991b) sought to rationalize this well established laboratory result by the conjecture that as a rule the repulsive potential in such cases lies just outside the inner wall of the attractive potential. Consideration of hydronium and ammonium cluster ions led to the first advance. The kernel of each series is a proton-bridge ion; thus the kernel of the hydronium cluster ions H,O+(H,O), may be written H z O - H + - H 2 0 (Schuster, 1976; Janoschek, 1976). Taking this system as an example, Bates (1992a) noted that the energy made available to the neutral products of dissociative recombination in a low-temperature plasma is 4.84 eV . Hence electronic excitation is excluded and the only possible channel is H,O . H + H,O

+ e + H(1s) + 2H,O

(1 1)

in which the free electron enters a 1s orbital around the H + ion by a radiationless transition. It is evident that the transition is of the singleelectron type. Calculations (Bates, 1992a) showed that the rate of singleelectron transition may be as high as around 1015to 10l6/s, which is several

DISSOCIATIVE RECOMBINATION

433

powers of 10 greater than the rate typical of a double-electron transition. The radiationless transition involved in

dissociative recombination is also of the single-electron type. Ab initio computations by Kraemer and Hazi (1989) have shown that the potentials do not cross and that a substantial potential barrier must be passed through in order to reach the equi-energy configuration of the neutral system from the initial configuration of the ionized system. Yet the measured recombination coefficient (Section 1V.B) at 300 K is around 2 x IO-'cm3 s-'. Bates (1992b) proposed a multistep mechanism (Section 1V.A) that enables the necessary tunneling to take place as readily as required via vibrational levels of Rydberg states. Support for the proposal came from complementary calculations on H:and D: dissociative recombination (Bates et al., 1993) where the circumstances are much as those for HCO' dissociative recombination. During the course of the investigation on process (12) an elementary aspect of single-electron transitions that had, hitherto, been overlooked was recognized. This is that a single-electron transition precludes the crossing of the potentials of the states concerned: thus, if they cross as in Fig. 1, then on the left of the crossing the energy of the system would be raised by the active electron switching from a free state to a bound state, which is not possible. Shortly afterward it was perceived (Bates, 1993a) that singleelectron transitions govern not only the recombination of the three ions just mentioned but also the recombination of proton-bridge cluster ions, H 3 0 + , and many other polyatomic hydride ions. In all cases in which the recombination coefficient has been measured it is high. A brief descriptive name for this second very common mode of dissociative recombination is the tunneling mode. It is idle but intriguing to speculate on how ideas on dissociative recombination might have developed if, in 1950, a group of ions such as H 3 0 + had been among those whose recombination coefficients had been measured in the laboratory and if it were at that time recognized that there could be no question of the relevant potentials crossing.

11. Upper Limit to Rate Coefficient Many of the measured dissociative recombination coefficients are so large that when considering them it is revealing to make comparison with the upper limit a(l) to the recombination coefficient for free electrons of

D. R. Bates

434

azimuthal quantum number I that may be derived. Using the maximum cross section theorem of quantum mechanics (see Mott and Massey, 1965) it may be seen that for a Maxwellian energy distribution

a(l) = h2(2n/m3kT)112(21 + 1) = 5 x 10-7(300/T)1/2(21+ 1) cm3 s-’ (13) Only partial waves having fairly small I need be taken into account because the overlap between the higher partial waves and the molecular wave functions is not appreciable. Occasionally formula (13) may, with advantage, be recast as

a(l) = 5 x 10-’(300/T)’/~q,J(l, m) cm3 s p l

(14)

where m is the magnetic quantum number for some chosen axis and f(l, m), which is generally unity but for certain m may be zero, is the fraction of (I, m) electrons open to dissociative recombination. An example is instructive. Discussing O:(X21-In,) dissociative recombination along the exit channel and taking the line between the nuclei as axis, Guberman and GiustiSuzor (1991) report that only pn free electrons contribute significantly. In a particular case there may be electron spin restrictions but this cannot usefully be incorporated into the comprehensive formula for the upper limit to the recombination coefficient.

111. Crossing Dissociative Recombination

A. THEORY Because of the simplicity of the physics it is worth retracing the early derivations (Bates, 1950b; Bates and Dalgarno, 1962) of the formula for the rate coefficient aD for the original process envisaged where the subscript D is affixed because it is now commonly called the direct process. The close kinship with dielectronic recombination will be made evident. As a preliminary, consider a neutral system X having a doubly excited level p energy E above its ionization threshold. Suppose that the system is in a plasma at temperature Tand that X(p) autoionizes at rate A(au) and that its neutrality is stabilized at a rate A(st) by energy in excess of E being removed. Letting r denote the ratio of the statistical weight of X(p) to that of X + in its ground level we have that, in the steady state A&)

}

A(au) rh3 exp( -c/kT) A(au) 2(2nn~kT)~/’

+

DISSOCIATIVE RECOMBINATION

435

where the term in braces is the factor by which the steady-state number density of X(p) is below its equilibrium number density in the absence of a stabilization process. The rate coefficient for (dielectronic) recombination through level p is therefore

This formula may readily be recast to make it applicable to the crossing mode of dissociative recombination. The factor

in formula (16) is the fraction of the X(p) generated that is stabilized. In dissociative recombination the atoms move apart so quickly that the stabilization rate is normally much greater than the autoionization rate. Consequently, it is a sufficient approximation to take S, which is here called the survival factor, outside an integral that will shortly appear and formally write it as

where &(st) and d ( a u ) are mean values of the stabilization and autoionization rates. The A(au) appearing in the numerator of the modification of formula (16) must naturally be treated more precisely. It is such that A(au) de is the autoionization rate into the interval E to e + de. The total autoionization rate when the nuclei are held a fixed distance R apart is 1

d ( R ) = - r(R)

A

2nh me4

= -1 V ( R )1’

where T ( R ) is the resonance width and V(R) is the matrix element of the interaction causing autoionization. Letting the vibrational wave function of the ion be +,(R) and the energy-normalized radial wave of the separating fragments of dissociation be cp,(R) it may be seen that the necessary Franck-Condon factor is included by taking

D. R. Bates

436

and hence, referring to formula (16), that the dissociative recombination coefficient is given by

If q,(R) is represented by the Winans-Stuckelberg delta function

cp&(R)= 6(R&- R)/{U",)J''Z

(22)

where U'(R,) is the derivative of the repulsive potential at the separation RE at which the potential is energy E above its asymptote, formula (21) reduces to

for a low-temperature plasma, R, being the separation of the crossing. Note that in determining R , the potential energy curve of the ion must be raised so that it includes the energy of the free electron. Using resonance theory Bardsley (1968b) obtained the same formula but with the survival factor given by

[

S = exp - b ( R ) dr]

N

exp[ - d(aur)/d(sr)]

(24)

Giusti (1980) obtained

from molecular multichannel quanta1 defect theory. Expressions (17), (24), and (25) are not valid unless S is close to unity but in this circumstance the first two terms in expansions of them agree. They are of services less for calculating S than for checking whether or not it is a good approximation to assume complete survival. Approximation is inherent in the factorization needed to obtain an explicit expression for the survival factor. However, MQDT (see later) automatically takes the effect properly into accounnt (Giusti, 1980). From the parent to Eq. (23) it is obvious that the Winans-Stuckelberg delta function approximation to the cross section Q(u, E ) for the dissociative recombination of an ion in vibrational level u with an electron of energy E vanishes if the crossing separation R,(E) coincides with the separation for any of the nodes of the vibrational wave function. As E is increased, RX(&) decreases from R,(O) toward R,(co) at a rate that depends on the gradient of the repulsive potential. Should RX(&)pass through the separation for a

DISSOCIATIVE RECOMBINATION

437

node in the vibrational wave function the delta function approximation to Q(u,E)would also pass through a node. Guberman (1986) has carried out model calculations that show that at each of these nodes the true Q(v,E ) versus E has a deep minimum. Concerned by an apparent difference between the observed and predicted dependence of the recombination coefficient on the temperature, Bardsley (1968b) introduced indirect dissociative recombination. This involves a sequence of two radiationless transitions, the first (due to a breakdown in the Born-Oppenheimer approximation) into a Rydberg state in an excited vibrational level, the second into the same repulsive state that is reached in the direct process. Because of a propensity rule proved by Berry (1966), the vibrational quantum number in the first transition changes by unity so the process may be written

AB' (X,u)

+ e % AB(Ry, u + 1) % AB(Res) + A + B

(26)

Bardsley (1968b) derived an expression for the rate coefficient tlI using the Breit-Wigner resonant scattering cross-section formula. This may be recovered by the application of the second-order perturbation theory expression (see, for example, Bates, 1961) for the rate of transitions from an initial state a to a final state d through intermediate states s when first-order transitions are forbidden:

where V and E are the matrix element and energy indicated by the subscript and p is the energy density of final states. Writing me4/h2 = s

(28)

the corresponding cross section for an electron of energy E is

or

nh2r rasrsd d ( E ) = Cs 4m.5 ( E - E,)' i-ir2 where r is the total width. Multiplication by the velocity and integration over a Maxwellian distribution gives a rate coefficient of

D. R. Bates

438

which if the width is small compared to kTreduces to

where Eo is the energy of the inital system. The primes on the n and a signals (for reference in Section 1V.A) denote that one intermediate state is included. Note that the treatment is only valid if first-order transitions are forbidden. If they are not, perturbation theory (c.f., Bates, 1961) shows interference between them and second-order transitions. The direct and indirect cross sections and rate coefficients are therefore not additive. The phenomenon of interference causing structure in the dissociative recombination cross section uersus electron energy curve was discovered by McGowan et al. (1979) from merged beam measurements on H i dissociative recombination. We will return to this later. O’Malley (1971, 1981, 1989) has pointed out that the direct process is also affected by Rydberg states. At a crossing between the repulsive potential and the potential of one of the Rydberg states, transition from the resonant state to an excited vibrational level of the Rydberg state may occur. The coupling is electronic whereas that in the indirect process of Bardsley (1968b) is vibronic. Clearly Berry’s Au = 1 propensity rule does not apply and low Rydberg states can be involved. A neat treatment by O’Malley (1989) gave that the ratio of the dissociative recombination cross section as modified by the electronic coupling to a level of energy E(n,u) to the unmodified cross section is

( E - E(n, u)}’ { E - E(n,u)}’ +

(33)

E being the total energy and r the resonance width. According to this the effect is to produce a pure dip or window in cross section versus energy curve. OMalley (1989) refers to the dissociative recombination as being interrupted. Dissociative recombination may now be written AB+(v) + e

AB(n, u

1_ AB(Res)

+ 1)

AB(n’, u’)

-

A

+B (34)

in which AB(n, v + 1) is involved through vibronic coupling and AB(n’, u’) is involved through electronic coupling. The exciting discovery of structure by McGowan et al. (1979) in a merged ion-electron beam experiment showed the need to develop a unified theory

DISSOCIATIVE RECOMBINATION

439

that incorporates the original direct process, the indirect process of Bardsley (1968b), and, later, after OMalley (1981) had pointed them out, the interruptions. Theorists responded with powerful sophisticated mathematical formulations of the scattering problem. No attempt is made here to describe these adequately. The outlines sketched are primarily to allow reference to certain facets of the formulations to be made when giving accounts of particular investigations. One approach (the CI method) is based on the theory of configuration interaction used by Bardsley (1968a) in his discussion of molecular states that may decay by autoionization or predissociation. Giusti-Suzor et al. (1983) extended this to allow vibrationally excited Rydberg states that may decay in this manner and doubly excited electronic states to be treated simultaneously.To render the set of linear algebraic equations that arise and are tractable, Hickman (1987) introduced an independent resonance approximation (IRA) whereby the effect of the many Rydberg states contributing to dissociative recombination may be assessed. He found that large Au transitions due to electronic coupling are important; and that, dependent on the phase of the complex matrix element involved, the interference with the direct process instead of being destructive [as Giusti et al. (1983) found to be the norm] may be constructive; that is, instead of leading to a decrease in the cross section, interference may lead to an increase. The generally adopted approach began with the work of Lee (1977) who used multichannel quantum defect theory (MQDT) to unify the treatment of dissociative recombination. Some inconsistencies were overcome by Giusti (1980) in an influential paper. Naturally MQDT, like the CI method, is an approximation and, as such, has been subjected to refinements. The convergence of successive refinements of a theoretical treatment is usually, but not necessarily, uniform. As Giusti (1980) recalled atomic quantum defect theory (Seaton, 1958, 1966; Greene and Jungen, 1985) proceeds by extrapolating from a Rydberg series to electrons scattered by the ion with the quantum defect of the former becoming the phase shift (divided by n) of the radial wave function of the latter. A generalized quantum defect theory of Greene et al. (1979) permits a unified treatment of the electron-molecular ion and the atom-atom continua of dissociative recombination. Based on this, MQDT yields an S matrix that includes all open channels. A basic element of MQDT is the introduction of an innermost reactive zone containing all short-range strong interactions that determine the phase shift for each long-range radial wave function and the mixing coefficients for the long-range wave functions. The vibronic interaction between the ionization channels is included through mixing coefficients that, knowing the quantum defect for the Rydberg series, may readily be calculated. Electronic coupling is more troublesome but may also readily be calculated to first order. Guberman

D. R. Bates

440

and Giusti-Suzor (1991) have described how the next order of perturbation may be obtained. Hickmann (1989) proved that for a single resonance IRA and MQDT give identical results provided that in the latter full account is taken of electronic coupling.

B. DIATOMIC HYDROGEN IONS Figure 2 shows the relevant potential energy curves for H i dissociative recombination. The lowest doubly excited state (2pa")' 'ZJ has a repulsive potential energy curve that crosses the H i ground state potential sufficiently close to its minimum to dominate even the dissociative recombination of H:(u = 0) with thermal energy electrons. It goes to H + + H - in the adiabatic limit. The H - has been observed by Peart and Dolder (1975) in an experiment involving inclined beams of electrons and H: ions having a wide distribution of vibrational levels. Neutral products of dissociation lying above H(2sp) are not accessible unless the ion is vibrationally excited or the electrons are supra thermal in energy. The rapidity of H:

+ H, - + H+ H:

(35)

(Theard and Huntress, 1974) renders H: much less prevalent than H: in natural environments, as was first pointed out by Martin et al. (1961), and prevents the study of HZ recombination in afterglows. Clearly, however, the dissociative recombination of HZ is of the utmost theoretical interest in its own right because of its relative simplicity. It is, therefore, fortunate that the merged ion-electron beam measurements pioneered by McGowan have been continued by Mitchell and his associates in Western Ontario. In the research on H: the ions were formed in a source by electron or photon impact on H, and therefore were initially distributed through all 19 available vibrational levels. These have long radiative lifetimes (Hus et a/., 1988). However, the ions were stored in an rf trap containing helium or neon with which they could react by H:(v)

+ He -+ HeH+ + H

(36)

H:(v)

+ Ne -+ NeH+ + H

(37)

or Processes (36) and (37) are exothermic only for u > 2 and u > 1, respectively, so that it is possible, in principle, to produce H:(u = 0, 1,2) or H:(o = 0, 1) as discussed by Sen et al. (1987). There is evidence (Yousif and Mitchell, 1989) that the internal energy of the ions introduced into the beam depends on the potential of the electrode used to draw the ions out of the trap, a low

DISSOCIATIVE RECOMBINATION

1 1.0

I

1.4

I

ia

I

I

I

I

2.2

2.6

30

34

I

38

R(ao)FIG.2. Potentials of doubly excited autoionizing H, states and also H l ground state and (dashed lines) 'Z: and ,nuexcited states. The H, states based on H i 'Z, are labeled (1) and (2) to indicate whether they are the first or second root of the symmetry, those based on H i 2rI, are the lowest root of the symmetry and bear no reminder of this (Guberman 1983b).

extraction potential favoring ions in low internal energy states. The widely cited dissociative recombination cross-section curve due to Van der Donk et al. (1991b), which is shown in Figure 3, was obtained with a low extraction potential. It is claimed that the curve refers to H:(u = 0) on the grounds of the good match between their observed and their MQDT calculated windows. On increasing the extraction potential, the central window was found to disappear due, it is claimed, to the H:(u = 1) population being raised and washing out the window. However, according to the later MQDT calculations of Schneider et al. (1991), the cross-section

442

D. R. Bates

FIG.3. Comparison between the calculations of Schneider et al. (1991) on Hg(u = 0) dissociative recombination and merged beam measurements by Van der Donk et al. (1991b). The dashed curve is the theoretical cross section convoluted with a 2.5-meV triangular apparatus function (Schneider et al., 1991).

curve for H:(u = 1) has a window at closely the same energy as that of the window that disappears. A possible explanation is that at the low extraction potential the H:(u = 1) gains enough energy to be lost preferentially by process (37) and that at the high extraction potential excitation to H:(u 2 2) occurs. The uncertainty weakens the rigor of the experimental check on theory that is described shortly. When assessing the uncertainty it may not be irrelevant to bear in mind the merged beam measurements of Noren et al. (1989) on N: dissociative recombination. The ions extracted by a low potential were believed by Norden et al. to be N:(u = 0) but Mitchell (1993) now believes they were actually mainly N:(u = 1) because the measured recombination coefficient otherwise seems inexplicable. In a singular development Forck et al. (1993) have measured the dissociative recombination cross-section curves for HD' using an ion storage ring in which the beam has a residence time of about 5 s. Their curve certainly refers to HD+(u = 0) since, as they point out, calculations by Colbourn and Bunker (1976) have shown that the radiative lifetimes of vibrationally excited ions are 34 ms or less. The energy range from 0.3 to 40 eV was covered. Below 1 eV there is a fairly good fit to an inverse energy line drawn through data points of Hus et al. (1988) scaled to HD' and

DISSOCIATIVE RECOMBINATION

443

stated to refer to a H:(u = 0,l) mixture. We return to the high-energy results of Forck et al. in Section 1II.E. Following on early research by Bottcher (1976) many calculations on H i dissociative recombination have been made. Attention is restricted here to the more recent of these. Using IRA and taking full account of the electronic coupling, Hickmann (1987, 1989) obtained cross sections that are too large. Guberman and Giusti-Suzor (1991) have reasoned that this is due to interference between resonances not being allowed for properly in the IRA approach. Important MQDT calculations with accurate information available on quantum defects, electronic coupling, and the dissociative potential energy curve (see Schneider et al., 1992; Van der Donk et al., 1992) have been carried out by Schneider et al. (1991) who evaluated the v = 0 and u = 1 cross-section curves, with second-order effects of the electronic interaction (Gubermann and Giusti-Suzor, 1991) included. They started with the molecular data that Urbain et al. (1991) had found appropriate for the inverse associative ionization process very slightly adjusted so that they get, in the v = 0 case, the same number of resonances between 20 and 80 meV as Van der Donk et al. (1991b) observed (see earlier discussion for observational procedure in this regard). Fig. 3 compares the calculated curve with the measured curve shifted 5 meV towards lower energies (as is permissible from the experimental viewpoint). A conclusion regarding the success attained cannot be reached until the precise ratio of [H:(v = l)] to [H:(u = O ) ] in the ion beam has been indubitably determined. The calculated v = 1 cross section of Schneider et al. (1991) is around 10- l4 cmz up to about 40 meV , and between this energy and about 80 meV is around cmz owing to the combination of several wide windows. An extensive tabulation of computed recombination coefficients has been published (Schneider et al. 1993). Takagi (1993a, b) has shown that the resonances are markedly affected by the H: rotation. The difference between the cross section energy curves that he obtained for the ortho and para forms of the ion by averaging over the rotational levels appropriate to 145 K is great enough to be detected by a merged beam experiment. Takagi et al. (1993) have also investigated the consequence of including higher order effects of the interaction. They found that it is considerable. This research emphasizes the need for the H:(v = 0) cross-section energy curve to be measured with a good energy resolution in a manner that does not depend in any way on theory.

C . MONOHYDRIDE IONS Using the FALP method, Geoghegan et al. (1991) have established that at 300 K a(KrH+) and ol(XeH+)are less than 2 x lo-' cm3 s-' and 4 x lo-'

D. R. Bates

444

cm3s - l , respectively. Another member of the group, HeH+, is discussed in Section 1V.B. Measurements on the dissociative recombination coefficients of other monohydride ions have been made only by the merged beam method exploited by the group in Western Ontario. They give that at 120 K a(CH+) is 2.2 x cm3 s - ' (Mitchell and McGowan, 1978), cr(NH+) is 6.8 x cm3 s - ' (McGowan et al., 1979), and a(OHf) is 5.9 x lop8cm3 s-' (Mu1 et al., 1983) a calibration error, reported by Mitchell (1990), that reduces merged beam cross sections published between 1977 and 1985 by a factor of 2 being taken into account and other unspecified changes by Mitchell being accepted (as done elsewhere in the article). Because of its interest in connection with interstellar cloud chemistry, the dissociative recombination of CH + has attracted considerable theoretical attention (c. f., Compton and Bardsley, 1984). The 'I3 state of CH that goes adiabatically to H(1s) + C('D) is the resonant state. Takagi et al. (1991) found that its potential crosses the potential of the ion close to the left classical turning point of the first vibrational level. They applied the MQDT treatment of Giusti (1980) modified to take into account both the pn and dn partial waves of the free electron. Figure 4 shows the CH+(u = 0, 1) cross section curves they obtain. The windows are not as wide as the H:(u = 0) windows (Fig. 3) calculated by Schneider et al. (1991). At 120K the corresponding dissociative recombination coefficient is 1.1 x cm3 s whereas that for CH+(u = 1) is 2.1 x l o T 7cm3 s-'. The values for the direct process are about the same as these.

',

10-13

v':O-+

Diss

\..

Y

10-2

Incident energy (eV1

1

.

lo-'

Incident energy (eV1

FIG.4. Calculated CH+(v = 0 and 1) dissociative recombination cross sections. The dashed curves refer to the direct process, whereas the full curves include the effects of the Rydberg states (Takagi et al., 1991).

DISSOCIATIVE RECOMBINATION

445

D. VALENCE-BOUND NONHYDRIDE IONS The recombination coefficients given in this and later sections relate to 300 K and to ions in the zeroth vibrational level unless otherwise specified.

I . Nitrogen Guberman (1991) has carried out a MQDT computation on the dissociative recombination of N: including the second-order terms in the electronic matrix element (Section 1JI.A). Figure 5 shows the striking structure he found in the cross section versus energy curve. The value of ct(N:) that he obtained from his most refined calculations, that is, those taking the Rydberg states into account, is 1.6 x 10-7cm3s-', which is significantly smaller than his value 2.7 x 10- cm3 s - ' for the simple direct process. Several determinations of ol(N:) have been made from the decay of "$1 in stationary afterglows using a mass spectrometer for ion identification. cm3 s - l and Kasner and Biondi (1965) and Kasner (1967) got 2.9 x 2.7 x 10-7cm3s-' respectively, whereas Mehr and Biondi (1969) got 1.8 x cm3 s - The last is generally cited as the preferred value in the

'.

1o - ~

I

1 0-2 10-1 ELECTRON ENERGY(eV)

100

FIG.5. Calculated NZ(u = 0) dissociative recombination cross sections. The dashed curve refers to the direct process, whereas the full curves include the effects of the Rydberg states (Guberman, 1991).

446

D. R. Bates

literature presumably on the grounds that it is the most recent of the three. However, Mehr and Biondi (1969) comment that the decay of [Nz] in their experiment does not follow a recombination curve well and express some puzzlement regarding the reason. In his section of a review (Bardsley and Biondi, 1970) Biondi refers to the three papers and, evidently attaching the same weight to each result, recommends the value 2.5 x lo-’ cm3 s-’. An analysis that Bates and Mitchell (1991) carried out on afterglow measurements by Zipf (1980) gave that a(N:) is 2.6 x l o p 7 cm3 s-’. A value of 2.6 x cm3 s - ’ was also obtained from a flowing afterglow Langmuir probe (FALP) investigation by Canosa et al. (1991), whereas 2.0 x cm3 sC1 was obtained from a similar investigation by Geoghegan et al. (1991). Thus, strong experimental evidence implies that a(N:) is close to 2.5 x lo-’ cm3 s - l , in good agreement with the calculated value for the simple direct process. The MQDT refinement used yields a value that seems too low. Guberman (1991) calculated that branching ratios P () for the channels: N:(u

= 0)

+ e + 2N(’D) +

N(’D)

-+

N(’P)

a

+ N(4S) + N(4S)

b

(38)

C

are r(a) = 0.12,

r(b) = 0.88,

r(c) = 0.

(39)

These results depend on his finding that only the 2311, state of the neutral molecule has a repulsive potential that makes a favorable crossing with the ground state potential of the ion and has a large width r for the electron capture. A crossing is avoided between this repulsive potential and the C3n, potential that has channel (b) as its asymptote (Fig. 6). A Landau-Zener calculation by Gubermann gave that the separating atoms arising from the dissociative recombination have an 0.88 chance of traversing the avoided crossing and thus moving along channel (b). Virtually all the 0.12 fraction that remain on the C3n, potential traverse another avoided crossing onto a potential that has channel (a) as its asymptote. Guberman’s calculated chance of traversing the first avoided crossing is in good agreement with the chance that Helm and Cosby (1989) obtained from measurements on product branching in predissociation. Using this as the critical parameter determining the products of the dissociative recombination, his values for the branching ratios, which are important in connection with the escape of N15 from the Martian atmosphere (Fox and Dalgarno, 1983; Fox, 1989), should be reliable. According to Guberman a higher value of r(a) may be appropriate for u > 0. Queffelec et al. (1985) have made spectroscopic measurements on the N(4S), N(’D) and N(’P) atoms generated by N:(u = 0) dissociative recom-

447

DISSOCIATIVE RECOMBINATION

+'D

:+'P

\

IfD

4 8 2.2 2.6 3.0 3.4 3.8 4.2

Internuclear Distance (Bohr) FIG.6. Potential of ground N: state and of N, states relevant to dissociative recombination (Guberman, 1991).

bination. They found that the quantum yield f('D) of N('D) atoms per recombination exceeds 1.85, which would imply that the branching ratio r(a) exceeds 0.85, which is in serious conflict with the value given in Eq. (39). It is difficult to credit that Guberman's calculations on this, supported as they are by the measurements of Helm and Cosby, could be so seriously in error. The discrepancy has not yet been explained. 2. Nitric Oxide By using resonant scattering theory to relate spectroscopic data on excited states of molecules to the cross section for dissociative recombination, Bardsley (1983) calculated that a(NO+)is 4.3 x lo-' cm3 s - l . A MQDT computation has been done by Sun and Nakamura (1990). Their dissociative recombination cross section curves (Fig. 7) are not affected as much by the inclusion of the Rydberg states as are the corresponding N: curves of Guberman (1991) or the corresponding 0: curve of

D. R. Bates

448

1 1

.-C0

l

c

0

$1

Electron Energy (eV) FIG.7. Calculated NO' (D = 0, 1, 2) dissociative recombination cross sections via the B2n resonant state. The scale on the right is for u = 0, that on the left for the other levels. The effects of the Rydberg states are included (Sun and Nakamura, 1990).

Guberman and Giusti-Suzor (1991). Sun and Nakamura calculated that a(NO+) is 1.6 x lo-' cm3 s-' with the value for the direct process about the same. Ion trap measurements by Walls and Dunn (1974) and stationary afterglow measurements by Dulaney et al. (1987) both gave that cl(NO+) is cm3 s-'. FALP measurements by Alge et al. (1983) and by 4.2 x cm3 s- which is in excellent agreement Spanel et al. (1993) gave 4.0 x with the value just reported. The laboratory results are consistent with an analysis by Torr et al. (1977) of ionospheric data obtained with the aid of an Atmosphere Explorer satellite. They are in close accord with Bardsley's result. Note that Bardsley's result refers to the simple direct process. Commenting on their rather poor success, Sun and Nakamura (1990) remark that the three crucial quantities-potential energy curve, electronic coupling, and quantum defect function-are not easy to estimate purely theoretically.

',

DISSOCIATIVE RECOMBINATION

449

An experimental study by Kley et al. (1977) has given a branching ratio of 0.76 for N(2D) formation. Sun and Nakamura do not discuss the products of dissociation.

3. Oxygen The position regarding 0,' disociative recombination is unusual. It has been the subject of much computing effort based on elegant mathematical physics but this has been restricted to the channels generating metastable atoms. Consequently, dependable experimental checks such as those that normally accompany the development of a theory have not been possible. The values of ~(0:) obtained from the stationary afterglow measurecm3 s - ' and 1.95 x lo-' ments of Kasner and Biondi (1968) are 2.2 x cm3 s-', respectively. A FALP measurement (Alge et al. 1983) also gave 1.95 x lo-' cm3 s-'. The ion trap measurements of Walls and Dunn (1974) gave 1.9 x cm3s- for 01(0:whereas ), FALP measurements by Spanel cm3 s-'. Torr et al. (1976) found that et al. (1993) gave 2.0 x Atmosphere Explorer data in the range from 1000 to 2000 K could be fitted satisfactorily with a(0,' IT) = 1.6 x

(T/300)-0.55cm3 s - l

(40)

It is clear that a ( 0 : ) is known sufficiently accurately to provide a useful check on theory. The channels leading to one or more metastable atoms are as follows:

O:(X)

+ e + 02('X;)

+ O('D) + 5.0 eV + O2('AU)+ 20('D) + 3.0 eV O,('C:) O('D) + O('S) + 0.8 eV -+

+ O('P)

(41) (42) (43)

Figure 8 shows the potentials involved. The first-order repulsive potentials shown are based on a [3s, 2p, Id] contracted Gaussian basis set and first-order configuration interaction wave functions (Guberman, 1979), and the second-order repulsive potentials are based on larger scale calculations with [6s, 3p, 2d, lfl contracted Gaussian basis sets and configuration wave functions that include all single and double excitations to the virtual space and contain some 150,000 terms (Guberman, 1987,1988). This brief account of them is given in order to illustrate the extreme thoroughness of the theoretical investigation. The bound state potential curve is the Rydberg-Klein-Rees curve of Krupenie (1972) at the experimental ionization potential above the ground state of the molecule and shifted by 0.0180 Bohr to larger R to compensate for the difference between the calculated and observed equilibrium separations. Confining his attention to the simple direct process, Guberman (1987, 1988) evaluated the recombination coefficient for channel (43) and then the

D. R. Bates

450

-.6

0

-.62

d

2 +

-.?

1.8

2 2.2 2.4 2.6 INTERNUCLEAR DISTANCE (BOHR)

2.0

FIG.8. Potential energy curve of ground state of 0;(with wave functions and positions of lower vibrational levels) and first- and second-order (see text) potentials of 32;, 'A,, and, '2: resonant 0, states (Guberman, 1989).

recombination coefficients for channels (41) and (42). These were the earliest ab initio computations of their kind. Later Guberman and Giusti-Suzor (1991) applied MQDT to channel (43), handling the electronic matrix element in the same way as in the N: case (discussed earlier) and getting similar striking structure in the cross-section curve (Fig. 9), the effect of which is to reduce the rate coefficient. The possibility of checking the results of the computations is provided by the emission of the forbidden green and red lines of 01, 557.7 and 630.0 nm, from the F-region of the ionosphere. This emission is believed to be due to 0; dissociative recombination. By making ground-level observations on the profile of the 557.7-nm line with the aid of a Fabry-Perot interferometer, Hernandez (1971) confirmed that the nascent O('S) atoms have suprathermal energy. Similar observations, but with the interferometer mounted on the Dynamics Explorer satellite and thus avoiding the relatively strong 557.7-nm chemiluminescent emission from the 100-km region, enabled Yee and Kileen (1986) to infer that some of the nascent O('S) atoms have more energy than could be provided by the disociative recombination of 0: ions in the zeroth vibrational level. This is in qualitative accord with the results

DISSOCIATIVE RECOMBINATION

10-16 &

,

45 1

. , . . . ,,,

ELECTRON ENERCY(cV)

FIG.9. Calculated 0;(u = 0, 1, and 2) dissociative recombination cross sections via the 'Z: resonant state. The dashed curves refer to the direct process; the full curves include the effects of the Rydberg states (Guberman and Giusti-Suzor, 1991).

D. R. Bates

452

of Guberman (1987) according to which a('Zgf,u), the channel 'Z: recombination coefficient for ions in vibrational level u, increases sharply as u is increased from zero due to the lie of the potentials concerned (Fig. 8). Quantitative checking requires altitude discrimination and is not straightforward because the 0,' ions are generated by O+

+ 0, + 0 + o:(u)

(44)

in an unknown initial vibrational distribution and experience vibrational deactivation by

O,'(u') + 0 + o:(u< u')

+0

(45)

on which the information available is meagre, and by other processes. Fortunately, the measurements of Bohringer et al. (1983) on

0:(4 + O,(O)

+

O,(v)

+ o:(o)

(46)

provide a useful lower limit to the deactivation rate. Letting a(Z, u) denote the O,'(u) dissociative recombination for all channels and letting a('S,u) and cc('D,u) be the rate coefficients for generating O('S) and O('D) atoms, the ratios F,(u)

a(%, u)/a(C,u)

(47)

= a('D, u)/cc(Z,u )

(48)

3

and F,(u)

are material. In his studies of F-region 557.7 and 630.0 nm emission to be described, Bates (1990,1992~)assumed that a@, u) does not vary much with u and in accord with result (40) took it to be 1.6 x (300/77°.55cm3 sK1 except that, influenced by the rate coefficients computed by Guberman (1987, 1988), he took a(Z,2) to be 1.05 x l o w 7(300/T)0.55cm3 s-'. Values of F,(u) and F,(u) are given in Tables I and 11. The very low value of F,(O) is central to the main difficulty encountered.

TABLE I VALUESAT 750 K OF Fdo) OF EQ. (47) 0

0

1

2

3

4

FLU) F,(4

a 0.0025 b 0.0020

0.053 0.032

0.244 0.129

0.052 0.028

0.052 0.027

Note. The values in row a are based on Guberman (1987), those in row b on Guberman and Giusti-Suzor (1991).

453

DISSOCIATIVE RECOMBINATION

VALUES AT

TABLE I1 750 K OF F,(u) OF EQ.(48) ~~

u

0

1

2

3

4

FDh)

1.498

1.212

1.180

(0.111)

(0.109)

Note Based on Guberman (1988) for channels 1411 and [42] and on Guberman and Giusti-Suzor (1991) for channel [43], which enters to a slight extent because of cascading.

Measurements (Takahashi et al., 1990 Sobral et al., 1992) made during a nocturnal rocket flight through the equatorial F-region provided simultaneous data on the electron number density and on f ( ' S ) and f('D) quantum yields of O('S) and O('D) atoms generated per 0: ion recombination, which, of course, depends on the vibrational distribution. The first paper gave only the ratiof('S)/f('D) but the second paper gave f ( 'S) and f( 'D) separately. In his 1990 analysis, Bates had available only the f('S)/f('D) data and the results of the ab initio computations (Guberman, 1987, 1988) on the simple direct process. He judged that the rocket measurements furnish telling evidence that F,(O) is not as minute as in Table I. Noting that the minuteness of the value in Table I is rooted in the very small Franck-Condon factor (see Fig. 8) he suggested that the true F,(O) may be larger because of the indirect process-in the limit of a vanishingly small Franck-Condon factor this would surely be inevitable. The suggestion was encouraged by Hickman (1987, 1989) having reached such a conclusion from his IRA calculations on H: dissociative recombination. Bates (1992~)returned to the problem after the rocket measurements on f( 'S) and f('D) were published separately and Guberman and Giusti-Suzor (1991) had carried out a MQDT treatment of the ' C: channel. Making an allowance for the effect that the suprathermal energy of the nascent O('D) atoms has on the altitude distribution of the emission, he obtained satisfactory agreement with the measured f ( 'D) through the altitude range studied (190 to 320 km). This success reduced the probability that there could be errors due to the attitude of the photometers during the rocket flight not being known accurately and, hence strengthens the evidence that F,(O) is underestimated by the ab initio computations. At low altitudes, where vibrational deactivation of 0 2 + ( v ) is most effective, there is an apparent shortfall in the 557.7-nm emission due to dissociative recombination. Comparison of the two rows of Table I shows that allowing for the effect of the Rydberg states by MQDT increases the discrepancy just mentioned. Two points are to be noted. First, similar allowance brings the computed a(N:) to below the seemingly well-determined experimental value. Second-

454

D. R Bates

ly, the pattern of MQDT computations (Guberman and Giusti-Suzor, 1991; Guberman, 1991; Schneider et al., 1991) appears to be that the Rydberg states introduce conspicuous windows in the cross-section curves and thereby reduce the recombination coefficients. If, as seems likely, this should happen for the and 'A, channels of 0: dissociative recombination the satisfactory agreement on f( 'D) would be lost. It is worth focusing attention on the N: case in view of the laboratory evidence that incorporating the Rydberg states by MQDT reduces the accuracy of the computed recombination coefficients. On the experimental side, a high-precision determination of ct(Ni) would be expedient. 4. Other Ions

Merged beam experiments have given that a(C:) is 3 x cm3 s - l (McGowan et al., 1979; Mu1 and McGowan, 1980) and that a(C0') is 1 x lop7 cm3 s - ' (Mitchell and Huss, 1985), which is in satisfactory agreement with the FALP value (Geoghegan et al., 1991) of 1.6 x cm3 s-'. Afterglow measurements by Mentzoni and Donohoe (1969) had given a much larger value for a(C0') but they were criticized by Whitaker et al. (1981a) as being flawed by clustering. The value of a(C0;) is needed in connection with the Martian ionosphere and there was an ill-founded notion that it might be as high as 1x cm3 sK1 (c.f., Hunten, 1968). To facilitate modeling, Weller and Biondi (1967) carried out an afterglow investigation. They found that ct(C0:) is 3.8 x cm3 s-'. In good agreement with this Geoghegan et af (1991) obtained 3.1 x cm3 s-' by the FALP method. Emission from the CO generated is of interest in connection with the Martian (Fox and Dalgarno, 1979) and Venusian (Fox and Dalgarno, 1981) upper atmospheres. It has been observed in the laboratory. Measurements by Wauchop and Broida (1972) have given that 55% of the CO is in the a31J state, 0.8% in the a3A state and 0.04% in the a' 3C state; while measurements by Gutcheck and Zipf (1973) have given that 5% is in the A'II state. Vallee et al. (1986) have found that the CO(A'II) vibrational distribution is linked to the CO: vibrational distribution. A FALP measurement has given that a ( S 0 : ) is 2.5 x lo-' cm3 s - ' (Smith, 1988). As would be expected, the recombination coefficients for the triatomic ions are comparable in magnitude with those for the diatomic ions under consideration. The median of the recombination coefficients of the valcm3 s - l so, on referring ence-bound nonhydride family is about 3 x to formula (13) and assuming that a p free electron is involved, we see that there is a median chance of 1 in 5 of recombination taking place in a collision between a p electron and an ion. The range of values is quite narrow.

455

DISSOCIATIVE RECOMBINATION

E. HOMONUCLEAR IONSIN WHICH O N E PARTNER HASA VALENCESHELL, AND DIMER AND TRIMER IONS

CLOSED OUTER

Partly because of its importance for inert gas alkali halide lasers Biondi and his associates carried out cardinal laboratory research on the dissociative recombination of inert gas dimer ions (Philbrick et al., 1969; Frommhold and Biondi, 1969; Shiu et al., 1977; Shiu and Biondi, 1977, 1978). Table I11 gives the values of the recombination coefficients they determined by their stationary afterglow method. An estimate of the upper limit to cl(He:) is included for the sake of completeness. It is almost certainly a generous upper limit because, on the one hand, there is no relevant crossing of potentials (Mulliken, 1964) and because, on the other hand, the tunneling mode is inhibited by a large potential barrier, the distance between the minimum of the He: potential and the equi-energy point on the He-He repulsive potential being approximately 1.1 Bohr. Measurements have been made on the dissociative recombination of only one other ion, Hg:, of the type under discussion. Unfortunately, the results obtained (Table 111) disagree badly. Mass identification of the recombining ion was carried out in neither of the two investigations but this is scarcely germane to the problem. Jog and Biondi (1981) used a helium-xenonmercury afterglow, the chemistry of which is not simple. Both they and Majetich et al. (1991) found a satisfactory match between the decay of the electron number density and the decay of the intensity of the emission from the Hg atoms generated. By a novel method Majetich et al. (1991) ensured that Hg: was the only species of ion in their experimental cell. They excited Hg(6’S0) in ambient nitrogen to Hg(6’PP,) by a laser. This led to Hgz through the sequence of quenching Hg(6’P1)

DIATOMIC IONS Ion

+ N,(u = 0)

IN WHICH

-+

Hg(6’P0)

TABLE 111 ONE PARTNER HAS A

Recombination Coefficient (300 K; to-’ cm3 sC1) <0.1

1.7 9.1 16

i‘.’ 44

+ N,(u = 1)

CLOSED OUTER VALENCE SHELL

Reference Bardsley and Biondi (1970) Philbrick et al. (1969) Shiu and Biondi (1978) Shiu and Biondi (1977) Shiu et al. (1977) Jog and Biondi (1981) Majetich et al. (1991)

(49)

456

D. R. Bates

and associative ionization

Their study leaves little room for doubt. Since their binding is due to weak one-electron exchange forces, the inert gas dimer ions (and Hg:) have low dissociation energies (around 1 eV). Rydberg states of the atoms are energetically accessible in dissociative recombination. Many are populated and emission by allowed radiative transitions occurs. As was first demonstrated by Connor and Biondi (1965) the profiles of the lines provide information on the kinetic energy of the nascent atoms. Frommhold and Biondi (1969) have observed neon lines with a “multishouldered” structure, indicating several different dissociation kinetic energies. They suggested that several states of Ne,’ are involved. Barrios et al. (1992) have introduced a time-of-flight method to determine the kinetic energy of metastable atoms formed. They found evidence for 5 s metastable atoms being formed in Kr: dissociative recombination. Spectroscopic observations, in particular by the Pittsburgh group (c.f., Shiu et al., 1977; Shiu and Biondi, 1977, 1978), show that many states are populated by the dissociative recombination of the inert gas dimer ions. When the afterglow electrons have thermal energy most of the emission in the spectral range studied came from the lower p states. No f states lie low enough in energy to be accessible. Emission from accessible s and d states is not in the spectral range studied by the Pittsburgh group. Measurements by Malinovsky et al. (1990) on a neon afterglow give that the ratio of the rates of production of 3d and 3p atoms is 0.35 approximately. In their research on Hg,’ dissociative recombination Jog and Biondi (1981) found strong emission from s,p, and d states. Any from f states would have been too far in the infrared for them to record. These states would be accessible if, as the spectral lines detected by Jog and Biondi appear to prove, the dissociative energy of Hg: has the value of 0.66 eV calculated by Michels et al. (1979) using relativistic self-consistent field density functional methods rather than the 0.9 eV value obtained experimentally by Arnot and Milligan (1936). The main problem is to understand how the recombination coefficients of the heavier ions can be as large as they are. Formula (13) shows that free electrons of azimuthal quantum number 2 or even higher must contribute much. This does not happen in 0,’ dissociative recombination (Guberman and Giusti-Suzor, 1991). From accurate potential energy curves calculated by Michels et al. (1978) using density functional methods, Bates (1991a) found that the entity U’(R,) that is in the denominator of formula (23) for the recombination coefficient is around one-tenth the value typical of valence bound ions. However partly offsetting this and the gain from many states being involved, the capture width r is inversely proportional to the cube of the effective principal

457

DISSOCIATIVE RECOMBINATION

quantum number (Bardsley, 1968b). Another feature must enter. As is shown later, the further feature stems from the matrix element of the interacion, e2/r12,between the pair of electrons concerned which has a special characteristic for an ion bound by exchange forces. A linear combination of the same atomic orbitals (which will be chosen to be real) may be adopted for the wave function of the bonding electron of the ion and for the wave function of this electron when in an antibonding orbital of the molecule. Hence these wave functions may be written

where rAZand rB2 are the position vectors of the electron relative to nuclei A and B a distance R apart and s is the overlap integral. Suppose, for simplicity, that the magnetic quantum number rn is zero and we denote the wave function of an electron 2 that in the separated atoms limit is in a Rydberg state of azimuthal quantum number 1 by Y(i,rl) and denote the I’ partial wave function of the free electron by @(I’,r1), rl being the position vector relative to the midpoint of AB. On expanding e2/rI2in the standard manner and carrying out the integration over the space of electron 2 on the assumption that x is a very much more compact function than Y or @, we find that the first term, vi, of the e2/r12matrix element is given by

a

(53)

the letter a signaling that the product of the wave functions involved is antisymmetrical. Using the approximation (Mulliken, 1939) that the dipole length of the “charge transfer transition” undergone by electron 2 is Q = R/2(1 - s2)’l2

(54)

we can see that the matrix element uii of the second term of the expansion of e2/r12is twice as large. The two terms have the same sign and are thus additive. Calculations (Bates, 1992a) that have been done on a singleelectron matrix element rather similar to that in Eq. (52) indicate that it leads to a capture width considerably greater than the capture width characteristic of double-electron transitions. The dissociative recombination of ions bound by exchange forces may hence be said to be governed by quasi-single-electron transitions, the qualification quasi being necessary because a crossing of potentials occurs instead of being precluded. There is, now, no difficulty in understanding the remarkedly large magnitudes of some of the recombination coefficients measured. A possible explanation for u(Ne:) being relatively small is that

458

D. R. Bates

in this instance x is not sufficiently compact compared with Y and 0 to make the analysis leading to formula (53) valid. Measurements by Macdonald et al. (1983) have given that cr(Ne:) is 1.1 x cm3 s-’. The increase by a factor of 6.5 over cr(Ne:) may merely be due to the combination of U’(R,) being smaller and to there being more accessible Rydberg states and more repulsive potentials. The alkali dimer ions are another group in which the binding is by weak one-electron exchange forces. Their dissociation energies are low; that for Cs,’, for instance, is only 0.61 eV (Huber and Herzberg, 1979). The dissociation recombination coefficients are expected to be large. A measurement of cr(Cs:) would be of interest and may be feasible by the method introduced by Majetich et al. (1991). The system would lend itself to the computational check that is needed. Returning to H: it is, of course, also bound by exchange forces. However, they are strong enough to give a dissociation energy of 2.65 eV (Huber and Herzberg, 1979) so that product atoms in states higher than 2sp cannot be generated in the dissociative recombination of H:(u = 0) with thermal energy electrons. In their test storage ring research on HD+(u = 0) dissociative recombination, Forck et al. (1993) discovered that the cross section passes through a maximum of about 7 x lo-’’ cm2 at 8.6 eV of energy of relative motion (Fig. 10). They attributed this to the HD+ attractive potential when raised in position by 8.6 eV being crossed near its minimum by the repulsive potential (see Fig. 2) of the HD (2pa,, nl) molecule so that (switching to the common isomer)

H:(u = 0) + e -+H(1s)

+ H(n1)

(55)

may occur. Zhdanov and Chibisov (1978) have given an elegant treatment of this process. They simplified the problem by exploiting the relative compactness of the lsag and 2pa, wave functions (much as was done earlier). On expanding e2/r12 in order to facilitate the evaluation of the transition matrix element, they did not notice the rather unusual circumstance that the leading term gives a contribution, analogous to ui of Eq. (53). Instead they retained only the term analagous to uii of Eq. (54). They obtained an analytical formula from which they calculated that the maximum cross sections for the production for n = 3 and n = 4 H atoms are cm2 and 6 x cm’, and that the fall-off is as n P 3 so that 1.3 x the total disociative recombination cross section at 8.8 eV is some 2.4 x cm’. This is greater than Forck et al. (1993) measured, but the survival factor for the dissociative recombination of energetic electrons is much less than the survival factor for the dissociative recombination of thermal electrons. The difference arises because with thermal electrons a slight movement of the nuclei suffices to stabilize the recombination, whereas with energetic electrons a substantial movement is needed owing to the possibility of autoionization that leaves the ions in high vibrational

459

DISSOCIATIVE RECOMBINATION n 01

lo+

E0 U

c

0

ul ul

0 L

10-l'

0

0.3

1

10

40

c.m. energy [ev] FIG.10. Measured HDf(u = 0) dissociative recombination cross section using the ion ~ through the merged electron ion storage ring method. The dashed line on the left is an 1 / line beam data points of Hus et al. (1988) scaled to HD+.The calculated cross sections of Derkits et al. (1979) (multiplied by 3) for the ( 2 p a , ) ( 3 ~ ) ' . ~resonant 2~ states and initial vibrational levels u = 0, 1, and 2 are also shown (Forck et al., 1993).

levels u: thus H2(2pa,, nl) + H;(lsa,,u)

+e

(56)

Derkits et al. (1979) calculated that for the H(3s) channel the factor S by which the cross section of Zhdanov and Chibisov is reduced owing to autoionization is around 4 x and dismissed dissociative recombination through H2(2pa,, nl) as of rather minor importance. Taking the leading term, corresponding to ui of Eq. (53) into account, such dissociative recombination is, despite the effect of autoionization, promising as the explanation of the maximum at 8.6 eV. A very large number of channels must contribute. Forck et al. (1993) suggested that the subsidiary maximum is due to Hz(2pn,, nl) dissociative recombination. The ui term here vanishes but the uii term is comparable in magnitude with the corresponding term for Hz(2pa,nl) and the position of the subsidiary maximum is in accord with the suggestion. The subsidiary maximum entails an addition of only about 1 x lo-'' cmz to the main cross-section curve. A computation on both maxima is desirable. Using the merged beam technique Yousif et al. (1994a) observed the same broad high-energy maximum. In their experiment part of the energy came from the ions being vibrationally excited.

460

D. R. Bates

One of the most surprising, and at the time perplexing, discoveries that the Pittsburgh group made in their investigations using stationary afterglows is that, if a monomer ion AB' forms a dimer ion AB.ABf the recombination coefficient is increased by a large factor with the recombination coefficients of the monomer and dimer unrelated. The classic set of papers pertaining to the discovery, the ionic species concerned being as indicated is, as follows: N:, Kasner and Biondi (1965), Whitaker et al. (1981b), Cao and Johnsen (1991); NO', Weller and Biondi (1968); O z , Kasner and Biondi (1968), Dulaney et al. (1988); and CO+, Whitaker et al. (198 la). Table IV compares the recombination coefficients. Those for the monomer ions are the measured values cited in Section 1II.C and those for the dimer ions are taken from the most recent of the relevant Pittsburgh papers. Table IV also compares the measured dissociation energies of the monomer ions (Huber and Herzberg, 1979) with the measured dissociative energies of the dimer ions (Ng, 1983). The latter are small. The binding of the dimer ions is due to exchange forces (Conway and Yang, 1965). Pointing to evidence (Smith and Lee, 1978; de Castro et al. 1981; Jarrold et al., 1984) for the existence of suitable repulsive potentials, Bates (1991a) reasoned that the recombination process is similar to that for the inert gas ions and predicted that emission from Rydberg states must ensue. The prediction was quickly confirmed by Cao and Johnsen (1991) who identified Second Positive band emission from N2(C211,v = 0,l) generated by N,*N: recombination. In view of this it is safe to accept the correctness of Bates's conclusion. In going from the CO'.CO dimer to the CO+-(CO)2 trimer, the recombination coefficient increases from (13 f 3) x lo-' cm' s - l to (19 f 4) x lo-' cm3 s - l (Whitaker et al. 1981a), which is much less than the relative increase in going from the Ne: dimer to the Ne: trimer (discussed earlier). When a recombination coefficient is as large as that for CO+-CO any increase is restricted by formula (13) and the greatest azimuthal quantum number of accessible Rydberg states. The phrase superdissociative recornbination is appropriate for the recombination of ions bound by exchange forces because of the remarkedly high values the rate coefficients may attain coupled with there being novel features (including quasi-single-electron transitions). TABLE IV RECOMBINATION COEFFICIENTS a(300 K; cm3s- I ) AND DISSOCIATION ENERGIES D (eV) OF MONOMER AND DIMER IONS

a ~

N; N;.N,

D

a ~

4.2 2.6 8.7 NO+ 26 0.9 N O + . N O 17

a

D ~~

10.8 0.6

D

a

~

0; Ol.0,

2.0 42

D

~

6.7 0.4

CO'

2

C O + . C O 13

8.3 1.0

DISSOCIATIVE RECOMBINATION

46 1

IV. Tunneling Dissociative Recombination A. THEORY

If the potential energy curves do not cross but instead are as sketched in Fig. 11, a potential barrier must be passed through in going from the configuration of the ion in, say, its zeroth vibrational level, to the equienergy configuration of the neutral molecule (that is, in Fig. 11, to point P on the repulsive curve). Such passage is not easily effected and the tunneling mode of dissociative recombination would be slow compared with the crossing mode unless a compensating element entered. The element that compensates for the tunneling Franck-Condon factor being much smaller is

x

P 0) C

UI

t

Neutral

Separation FIG.11. Schematic potential energy curves for the tunneling mode. The ionic curve includes the energy of the free electron. The equi-energy point (P) on the neutral curve is marked.

D. R Bates

462

that in the tunneling mode a single-electron radiationless transition replaces the double-electron radiationless transition that (triple-electron transitions being neglected) controls the crossing mode. As pointed out in Section I, a single-electron radiationless transition and an associated crossing of the potentials of the ion and the molecule cannot coexist. I . Single-Electron Transitions

The descriptive phrase single-electron transition is to be taken as meaning that only a single electron actually switches orbitals; that the distortions experienced by other orbitals leaves them nonorthogonal to the original orbitals. In the important valence decrease class a free electron 1, wave function q5(rAl)enters a bound orbital x(rAl),which had been empty, around nucleus A of atom A in a polyatomic ion; and the valence bond orbitals of the (2,3) electron pair that had bound A to atom B (usually hydrogen) contract into atomic orbitals around nuclei A and B so that the bond is dissolved and dissociation occurs. Let the atomic orbitals be $ ( r A i ) and $(rBi), i = 1,2, and for simplicity take all wave functions to be real and represent the valence bond orbitals by the lcao approximation

in which s is, again, the overlap integral (which is positive for a valence bond orbital). The matrix element responsible for the radiationless transition is

{rea)

Integration over the space of electrons 2 and 3 gives I/=

e2(1

+ s)

+

4(rAl)x(rAl) dz,

(59)

rB 1

wheref(r,,) and g(rBl) tend to unity as rAl and rBl tend to infinity. They need not be specified here beyond noting that if s is zero and if $,,(rA2)and $b(rB3)are spherically symmetrical then

(60)

In view of the form of matrix element (59) it may be said that a singleelectron operator is involved. The other important class concerns cluster ions that contain a proton-bridge bond (see Janoschek, 1976). Dissociative recombination takes

DISSOCIATIVE RECOMBINATION

463

place through the free electron entering the Is orbital around the proton (c.f., Bates, 1992a). This nullification of the bond is accompanied by the orbitals $&rB2) and $c(rcz) of neighboring atoms, B and C being distorted. Representing the distorted orbitals by the same symbols with primes and denoting the initial and final wave functions of the active electron as before the matrix element concerned is here

The overlap integrals between tjb(rB2)and $b(rBl) and between IClC(rc3)and $r(rc3) may be taken to be unity. On integrating over the space of electrons 2 and 3 formula (61) reduces to

J'(rBl) and g'(rcl) being similar tof(r,,) and g(rB1) of Eq. (59). This replaces an incorrect expression for V given elsewhere (Bates, 1992a). The g(rBl)/rBl term in Eq. (59) and both terms in Eq. (62) enable changes in the azimuthal quantum number 1 to take place. Single-electron radiationless transitions may have a rate of some IOl5 to 10l6 s-'. The earliest indication of a major gain relative to double-electron transitions is to be found in Landau-Zener calculations by Butler and Dalgarno (1980) on charge-transfer collisions. The enhancement they found in a single-electron transition matrix element compared with a double-electron transition matrix element depends markedly on the crossing distance between the ionic and homopolar potentials being by a factor of about 5 at Sa, and by a factor of about 300 at 10a,. If such a high transition rate as that just indicated occured in the crossing mode autoionization would prevent it being reflected in a corresponding high recombination rate. In the tunneling mode, the survival factor is never significantly less than unity. 2. Role of Intermediate States

A high radiationless transition rate d is not in itself sufficient to account for the values that dissociative recombination coefficients commonly attain when there is no crossing of potentials. For instance, taking d to be 10l6 s - ' and calculating cr(HC0') by straightforward use of formula (21), which is equivalent to treating quanta1 tunneling in the standard manner, gives a value of only 2 x lo-" cm3 s - l , whereas experiment (Section 1V.B) has shown the ol(HC0') is about 2 x lo-' cm3 s-'. Just as the application of second-order perturbation theory yields formulas (30) and (32) for the cross section and rate coefficient of the indirect

464

D. R. Bates

process of Bardsley (1968b) so the application of third-order perturbation theory yields a cross section

and rate coefficient

in which there are now two intermediate states, s and t (the number of such states being indicated by the number of primes on D and IX) and X is the atomic unit of energy (Bates, 1992b). There is a Au = 1 change and the Rydberg states are such that E, and E , are nearly equal. Similarly fourthorder perturbation theory yields the cross section

and rate coefficient

(44)

Throughout, the subscripts s, t, and u indicate the vibrational levels of Rydberg states of the molecule. Normally @’ is much smaller than the square of the difference in the energies of the intermediate states and r,, is the greatest partial width. Comparing the rate coefficients given by formulas (32) and (64) at a temperature high enough for the exponentials to be taken as unity, it may be seen that the effect of introducing an extra intermediate state t is to

DISSOCIATIVE RECOMBINATION

replace the predissociation width

465

rSdby

with

where r t d is the predissociation width of the further intermediate state and the term ar2has been neglected in the denominator. Computations show that except near a node of the overlap integral of the vibrational wave function of the ion and the wave function describing the separating fragments of dissociation rtd

>> rsd

(69)

For typical vibrational energies it is likely that I E, - E, 1 is 0.1 eV or less. It may hence be seen from Eq. (62) that F exceeds unity unless r,, is less than 3 x eV, which (c.f. Bates 1992b) would be an improbably low value. Because of strong inequality (63) the recombination coefficient is therefore increased. Similarly comparing the rate coefficients given by formulas (64) and (66) we can see that the effect of introducing an extra intermediate state u is to replace the predissociation width rtdby = F,rud

(70)

where

and r u d is the predissociation width of the still further intermediate state. Computations show that >> rrd

(72)

except near a node. An increase in the calculated rate coefficient is again likely. As additional intermediate states are included, the increase continues until the separation between the lower Rydberg states that become involved is too great to permit the necessary close energy match. Quantitative prediction for even the simplest polyatomic ion would be a formidable task. Our limited objective is to show that the inclusion of an intermediate state avoids the very low rate coefficients that otherwise had seemed inevitable. The formulation does not allow for interference between the terms arising from the successive orders of perturbation theory. Even the temperature variation is not easy to predict because of the summations entailed in, for example, formula (66) and the occurrence of nodes in the predissociation overlap integral. Keep in mind that an increase

466

D. R. Bates

in the energy of the free electron causes an increase in the separation of the potentials of Fig. 11, which would tend to decrease the recombination cross section. The case of H: will be considered again later, but it is appropriate to draw attention here to one aspect of the ion storage ring measurements of Larsson et al. (1993) on the dissociative recombination cross section uersus energy curve of the cold ion. This aspect is that the measurements provide graphic evidence (Fig. 12) of the occurence of both dissociative recombination modes. It is clear from the potential energy curves shown in Fig. 13 that although the crossing mode is inappreciable for thermal energy electrons as predicted by Michels and Hobbs (1984), it should become important as the energy of the electrons is raised because as this is done the attractive potential of the ion is raised in position so that it is crossed near its minimum by the potential of the resonant state of the molecule. This is responsible for the maximum at 9.5 eV. Larsson et al. also observed an increase in the cross section toward low energies that can only be due to the tunnelling mode. At low energy of relative motion E (in electron-volts) the cross section is 5.3 x lo-'' cm'. Other studies in the high-energy region (Peart et al., 1979; Yousif et al., 1994b) have confirmed the expectation that H- is a product.

h

"€

0

v

Relative energy (eV) FIG.12. Measured H :(u = 0) dissociative recombination cross section using ion storage ring method. The low-energy part of the curve is a result of the tunneling mode, whereas the peak at 9.5 eV is due to the crossing mode (Larsson et al., 1993).

467

DISSOCIATIVE RECOMBINATION

A

? W

-1.4-

C

w

-15-

-I .6

-I

-

70

x- Bohr FIG.13. Potential of ground H: state and of some relevant H, states. [after Michels and Hobbs (1984).]

B. IONSON WHICHCALCULATIONS AREDONE Several measurements on the rate of

HC' = 0 + e+H(ls)

+ CO

(73)

have been made. The values of a(HCO+) obtained are given in Table V. Their mean is 2.0 x lo-' cm3 s- '. The agreement between them is not close. The ion concerned is a member of the valence decrease class (Section 1V.A) in the sense that two of the valence shell electrons of the C atom in CO form a lone pair (c.f., McWeeny, 1979). A single-electron transition is involved in TABLE V MEASURED HCO' DISSOCIATIVE RECOMBINATION COEFFICIENT AT 300 K u(HCO+, u = 0) (10- 'cm3 s - ')

Method

Reference

2.0 1.1 2.4 2.8 2.2 1.5

Stationary afterglow FALP Stationary afterglow Infrared monitoring of ion FALP FALP

Leu et al. (1973a) Adams et al. (1984) Ganguli et al. (1988) Amano (1990) Rowe et al. (1992) Smith and Spanel (1993a)

468

D. R Bates

process (73) and ab initio computations by Kraemer and Hazi (1989) confirm the absence of a crossing of potentials. Calculations on the tunneling mode (Bates 1992b) show that the measured a(HCO+) is understandable if there are good energy matches (see Section 1V.A) up to about u = 3, which seems reasonable. Owing to its formation from H: by process (35) H i is an important ion in natural environments; for example, in the ionospheres of the Jovian planets (see Dalgarno, 1988) and in interstellar clouds (see van Dishoeck and Black, 1988). Some of the more recent measured values of a(H:, u = 0) are given in Table VI. Smith and Spanel (1993a, b), having demonstrated that their low tc is repeatable, dismissed the high values obtained prior to 1993 on several grounds: In the case of the stationary afterglow and other FALP investigations, they claimed that the ions were vibrationally excited while, in the case of the infrared monitoring investigation, they objected that [el was not determined but was assumed to equal [H:(v = O)] and proposed (but did not prove) the hypothesis that the H + ions, which were certainly formed and were lost only slowly, were so abundant that they made [el >> [H:(u = O)] and thence led to tc(H3+, u = 0) being badly over estimated. Even the lowest of the a(H;,u = 0) given in Table VI is far too high to be attributed to the crossing mode treated by Michels and Hobbs (1984). However, if there are good energy matches up to about u = 3, the results are understandable for the tunneling mode (Bates et al., 1993). Smith and Spanel (1993b) believe that the very low recombination coefficient ( 10cm3 s-') deduced from earlier (Adams and Smith, 1988b) FALP investigations is wrong. Smith and Spanel (1993a, b) leveled no criticism against the storage ring measurements of Larsson et al. (1993) who reckon that any systematic error in their results is only 10%. Their cross-section curve (discussed earlier) corresponds to a rate coefficient of 1.7, x 10-7(300/T)0.94 cm3 s-', in remarkedly good agreement with the first three results in Table VI. Incorrect N

TABLE VI MEASURED Hl DISSOCIATION RECOMBINATION COEFFICIENT AT 300 K Reference 1.5 1.6 1.1" 1.7 0.1-0.2 "At 650 K.

Stationary afterglow Infrared monitoring of ion FALP Ion storage ring FALP

Macdonald et al. (1984) Amano (1990) Canosa et al. (1992) Larsson et al. (1993) Smith and Spanel (1993a)

DISSOCIATIVE RECOMBINATION

469

lower values for this rate coefficient have been given in the literature. They are based on the value of the cross section at 2 kT, which is an inappropriate choice when the energy variation is as reported by Larsson et al. (1993). (The appropriate choice is the value of slightly more than 3kT.) The afterglow and storage ring results, as well as being in good agreement, have a persuasive internal consistency. Thus Amano (1990) found that the plot of [H:(u = O)]-' against time is accurately linear and that the derived o! is independent of the H, pressure between 0.1 and 1 Torr; and he advanced his very low [H:(u = l)]/[H:(u = O)] ratio as evidence that the rate coefficient for H: vibrational quenching by H, is high, which would ensure that H:(u = 0) was indeed the ion to which the results of Macdonald et al. (1984) and Canosa et al. (1992) refer. Acceptance of them would leave the c1 of Smith and Spanel the puzzle. It is more difficult to envisage a defect in an afterglow experiment that would make the measured o! too low than one that would make the measured o! too high; but set against this is the storage ring evidence. The merged beam measurements of Van der Donk et al. (1991a) show that the D: dissociative recombination cross sections are comparable with the corresponding H: cross sections (Fig. 14). This is a striking result since doubling the mass normally reduces the rate of quantum tunneling through a substantial barrier like that involved by an extremely large factor. When allowance is made for the intermediate states, this is offset by the low vibrational frequency, which facilitates energy matching. In the case of D: sufficiently close energy matching should occur down to a rather lower effective principal quantum number than for H: with the associated vibrational quantum number u significantly higher, which is important (Bates et al., 1993b). Information on HeH dissociative recombination is needed by astrophysicists (see Roberge and Dalgarno, 1982; Moorhead et al., 1988). The position regarding it has become very interesting. Because of the absence of a crossing of potentials (Michels, 1989), the process had confidently been expected to be slow for thermal energy electrons in harmony with which a FALP measurement of Adams and Smith (1988b) gave that a(HeH+)is less than 1 x lo-'' cm3 s-'. However, using a merged beam, Yousif and Mitchell (1989) found the cross section to be around lo-'' cm2. Only the tunneling mode could be responsible. It is fortunate to have a possible example of this mode in such a simple system. Exploiting the circumstance Sarpal et al. (1993) carried out a set of calculations in which they treated the intermediate states by the R-matrix method. They reproduced the results of Yousif and Mitchell satisfactorily. Meanwhile Tanabe et al. (1993) had made measurements in a storage ring. These likewise gave that dissociative recombination occurs with thermal energy electrons (although the absolute cross section was not determined) and also showed a peak at about 20 eV (Fig. 15), with a cross section of about cm2 [if the low-energy cross +

D. R Bates

470

10-2

I I l v l '

10

I

1

I l l

t

I

I

I 1 1 1 1

? 02

1

o3

Center-of-MassEnergy (mev) FIG.14. Measured H; (closed triangles, low-extraction results; closed circles, high extraction results) and D: (open circles, low extraction results; open triangles, rf source results) dissociative recombination cross sections using the merged electron-ion beam method. Attention may be focused on the low-extraction results-the other results illustrate how much the cross sections may be affected by the extraction energy and source conditions (Van der Donk et al., 1991a).

section may be assumed to be cm2 as measured by Yousif and Mitchell (1989)l. Tanabe et af. attributed the 20-eV peak to the crossing mode invoking two-electron excited states, which form Rydberg states converging to the first, and more highly excited, states of the ion much as in Fig. 10. The apparent harmony was soon disturbed. By means of merged beam measurements, Yousif et af. (1993) also found a high-energy peak, but with a cross section of only around lo-'' cm2. As in the earlier merged beam

47 1

DISSOCIATIVE RECOMBINATION

1000

I

I

#

1

DR-A

n v)

.LI 4

C

=!

9, 100

DR-B

DR-B

(d

Y

A

C

.-0 2 c1

VY

10

v) v)

8

I

60

40

20

0

20

40

60

FIG.15. Measured relative HeH+(u = 0) dissociative recombination cross section using ion storage ring method. The right and left halves of the spectrum correspond to V ,> V;. and F> V , respectively, where V , and V;. are electron and ion velocities, respectively. The low-energy peak is due either to the tunneling mode or to metastable ions recombining by the crossing mode (see text); the high-energy peak is due to the crossing mode (Tanabe et al., 1993).

research (Yousif and Mitchell, 1989) they obtained large cross sections at thermal energies. However, they adduced evidence that this was due to some of the HeH' ions being in the metastable a3X+ state and disappearing by the crossing mode of dissociative recombination: First they found excellent agreement between the positions of the vibrational energy levels of this state and the positions of their observed resonances; second, on adding oxygen to the ion source in the expectation of quenching the metastable ions, they found that the thermal energy recombination disappeared (which would bring their results into qualitative agreement with those of Adams and Smith, 1988b) but, significantly, that the 20 eV recombination remained. Sarpal et al. (1993) dispute the validity of the evidence citing preliminary unpublished results by Larsson that suggest that oxygen has no effect on the cross section. CLASS OF IONS c . VALENCE-DECREASE The tunneling mode of dissociative recombination is remarkable for its prevalence amongst polyatomic ions (Bates, 1993a). Many commonly occuring ions belong to the valence-decrease class. The possibility of dissociation along several channels often arises. Progress would not have been possible without the guidance on the branching ratios that was furnished by

D. R. Bates

472

measurements on some of the dissociation products that were carried out primarily by Adams, Rowe, Smith, and their associates. 1. Dissociation Products In view of a recent review by Adams (1992) only an outline need be given. Letf(Z/A) denote the numbers of atoms or molecules of species A generated per recombination of ion of species I and let r(-) be the branching ratio of the channel identified in parentheses. By incorporating spectroscopic techniques into the FALP apparatus, determinations have been made off(Z/O) by Rowe et al. (1988), of f(l/H) by Rowe et al. (1988) and Adams et al. (1991), and off(Z/OH) by Adams et al. (1989) and Herd et al. (1990). It was stated that f(Z/H) and f(Z/OH) are correct to about 25%. Error bars for f(Z/O) are not given but presumably are not less. The information obtained combined with the obvious necessity for the branching ratios to sum to unity has enabled many branching ratios to be found. Taking the total recombination coefficient a(Z) from Adams and Smith (1988a, b) or Mitchell (1990) this gives cr(Z)r(-), the channel-specific dissociative recombination coefficient. The values that will be cited here, or be implied in describing a channel as “rapid (by which we mean having a rate coefficient of around cm3 s-’) were derived as just indicated, either by the experimental group or by Bates (1991b). Note that at the time the general misbelief prevailed that dissociative recombination necessitates a crossing of potentials. Consequently, erroneous attempts at rationalization were introduced. Spectroscopic studies (Adams et al., 1989, 1991; Herd et al., 1990) have also given data on the extent of the vibrational excitation of the product (see Section V) and, in a few instances, on electronic excitation (Adams et al., 1989; Tsuji et al., 1991; Sonnenfroh et a/., 1993). 2. Channels Attention is first directed to dissociative recombination channels that are known to be rapid, entail valency decrease and are of the tunnelling mode. Next, some ions for which the rapid dissociation recombination channels have not been identified experimentally are considered. Returning to the ions that possess the channels to which attention was first directed, a question is raised regarding their other rapid dissociative recombination channels. Finally (although it is really out of place in this section since only the crossing mode enters) an account is given of the dissociative recombination of a valence unsaturated polyatomic ion. The rapid dissociative recombination process H:O

+ e + H,O(’A,) + H(1s) + 6.4 eV

(74)

does not release enough energy for the products to be electronically excited. It clearly involves a single-electron transition; thus, the free electron must

473

DISSOCIATIVE RECOMBINATION

enter an orbital 'p, around 0' (see Table VII) and thereby reduce the number of valences from three to two so that a H atom is freed. As has been shown (Section I) a single-electron transition precludes a crossing. The following similar processes are also rapid: HzFf

+ e - t HF + H + 9.5 eV

(75)

+ H + 6.1 eV H4N' + e + N H , + H + 4 . 8 eV N = N'H + e + N , + H + 8.5 eV 0 = C = O'H + e + C O , + H + 7.9 eV 0 = C = S + H + e - + O C S+ H +6.8 eV H3S+ + e + H , S

(76) (77) (78) (79) (80)

as probably are CH,0+H2 C,H,O+H,

+e +e

-t

-t

+ H + 5.8 eV C,H,OH + H + 5.4 eV CH,OH

(8 1) (82)

and

+

CH3N+H3 e -t CH,NH,

+ H + 4.4 eV

(83)

A characteristic common to all of these processes is that the heavy atom that formally carries the charge of the ion necessarily loses a valency on neutralization. Even if the heavy atom formally carrying the charge is carbon, a loss of valency may occur as it does for process (74) and the analogous processes HC' = S

+e

-t

CS

+ H + 5.4 eV

(84)

and HC' = N H

+ e - + C N H+ H + 7.3 eV

(85)

Other instances are to be expected. Having noted that ab initio computations by Wang and Freed (1989) have shown that the l A , state of the methylene radical is only 0.52 eV above the TABLE VII PAIRED AND UNPAIRED ORBITALS Species C+

N+,C O', S', N F', 0, S F

Orbitals in Valence Shell

Valency

D. R. Bates

474

ground ,B, state, Bates (1991b) suggested that in CH: recombination the free electron enters any of the orbitals q1 to (p3 (Table VII), and that this orbital and the similar orbital of opposite spin form a pair and thereby reduce the number of valences so that a H atom is released: CH:

+ e -,CH,('A,) + H + 4.8 eV

(86)

Alternatively, or additionally, the crossing mode may occur. An experimental determination of the state of the nascent CH, is needed. According to the recalibrated (see Section 1II.C) merged beam measurements of Mu1 et al. (1983) a(CH:) is 3.5 x lo-' cm3 s-'. The carbohydride ion that has received the most attention, CH:, is often written CH: 'H, to indicate that two of its H atoms are fairly close together, their distance apart being slightly less than 1A compared with the H-H distance of 1.8 A in methane, so they may be regarded as forming a H, molecule. The binding is by a three-center two-electron bond formed from the (p, orbital of the C atom (Table VII) and the 1s orbitals of the H atoms (Dyczmons et al., 1970). If the free electron enters any of the orbitals q1to (p, a H atom is ejected as in process (86) and the CH,('Al) relic and H,(lC,) would not have enough energy to separate and so would form methane in its singlet ground state: CH:.H,

+e+

CH,

+ H + 7.9 eV

(87)

The measured value off(CH:IH) is 1.18 but because of the error bars (see earlier discussion) this cannot be said to necessitate a channel in which two atoms are released. Replacing one of the H atoms that remain bound in channel (86) by CH, can scarcely affect the rate coefficient greatly so it is likely that the large value of cr(C,H:), 7.4 x lo-' cm3 s-', is at least in part due to CH3C+H, + e --r CH,CH('A,)

+ H + 3.4 eV

(88)

As for CH:, there is another possibility: CH: and CH3-CH2+may instead, or may also, recombine by the crossing mode. It is relevant to note that an example of slow dissociative recombination along an exothermic channel of the valence decrease type is not known; and neither is an example of slow dissociative recombination of a polyatomic ion where only the crossing mode could operate. Abouelaziz et al. (1993) have made measurements on the recombination coefficients of some cyclic and polycyclic hydrocarbon ions, finding that a(C3H:), a(C,H:), a(C,H,f), a(C,H:), and a ( C , , H ~ )are 7, 9, 10, 7, and 3 x lo-, cm3 s - l , respectively. They did not investigate the products. Taking the case of the benzene ion for the sake of definitiveness unless it is an exception to the apparent general rule a rapid dissociative channel is as for process (86) with the C + being converted to divalent C and a H atom being released. The high value of a(C,H6+) compared to a(CH:) may be

475

DISSOCIATIVE RECOMBINATION

because the charge on the benzene ion is so mobile that a C + is always presented to an approaching electron. The channel in which an H atom is ejected is not the only major dissociative recombination channel for most of the ions that have been considered. For instance, dissociative recombination of H 3 0 occurs along +

H30++ e+OH

+ H, + 5.8 eV

(89)

and along H30'+e+OH+2H+1.3eV

(90)

(in which, as always, the exothermicities refer to ground state products). The branching ratios r(74), r(89), and r(90) are approximately equal. Recall that several sets of products may ensue from the photodissociation of a polyatomic molecule by line absorption (c.f., Slanger and Black, 1982; Biesner et al., 1989). One possibility is that channels (89) and (90) arise from the crossing mode. However, since the products in these channels may be reached adiabatically from the products in channel (74), a rather simpler alternative is that the same radiationless transition leads to all three channels, which would require that the multidimensional potential energy surface on which the products move does not have barriers to the channels concerned. Further research is needed on this. Using the trapped ion technique Heppner et al. (1976) obtained on H 3 0 + . ~ for energy of relative cross section represented by 4.6 x lo-'* E - ~ cm2 motion E between 0.038 and 0.110 eV. In the case of NHf, by contrast with OH:, the channels releasing two H atoms (either free or bound together as H,) are unimportant (Bates, 1991b). Reliable prediction of the products of dissociative recombination is not within easy reach. As well as proceeding by channel (79), dissociative recombination of HCO: proceeds along

0 = C = O'H

+e+CO

+OH

+ 6.8 eV

(91)

with a branching ratio of 0.34. The crossing mode must be responsible (it being manifest that a single-electron transition cannot break a double bond). Herd et al. (1990) have detected OH(A2Zf 4 X'II) emission. Dissociative recombination of N 2 0 H + is rapid, a(N,OH +) being 4.2 x lo-' cm3 s-', Calculations by Rice et al. (1986) have shown that "+OH is the most stable form of the ion. Product measurements (see Adams, 1992) give that the total branching ratio of the channels leading to OH is 0.31, presumably mainly or entirely

N

= N'OH + e + N , + OH + 10.4 eV

(92)

by the crossing mode, with OH(A3Z+ + X2n)emission observed (Foley et al., 1993). They also give a branching ratio of 0.84 for the channels leading

D. R. Bates

476

to H, which implies that the bond between two neutral atoms 0 and H is broken by the dissociative recombination. This is not anomalous, being no different in kind from OH being electronically excited. The observation of NH(A311 + X3C) emission coupled with proof that it stems from recombination (Foley et al., 1993) is different because it is hard to understand how an H and N atom on the left of Eq. (92) could come together. The N 2 0 H + was formed by proton transfer from H: to N,O and it is likely that some HN = N + = 0 was also formed and that its recombination (by the crossing mode) was responsible for the emission. Ions such as H 2 0 + that are not valence saturated cannot undergo dissociative recombination through valency decrease. However, the dissociation fragments have several repulsive potentials that facilitate the crossing mode and the norm is for the recombination to be rapid. Merged beam measurements (Mu1 et al., 1983; Mitchell, 1990) have given that a(H,O+) is 3.5 x cm3 s-'. The open dissociative recombination channels are H,O+

+ e+ OH + H + 7.5 eV + H, + 0 + 7.6 eV + 2 H + 0 + 3.0 eV

(93) (94) (95)

in which the exothermicities,as always, are for ground state products. Rowe et al. (1988) have determined the branching ratios, finding that r(93) is 0.55 and 495) is greater than 0.24 so that at least two channels are important. Futhermore they have found that some of the OH in channel (93) is in the AZX+state and estimated that the branching ratio for this is 0.05 to 0.15. In envisaging the process, it is not helpful to base the wave functions on a valence bond model.

3. Recombination Coeficients We saw in Section II1.D that the crossing mode recombination coefficents of the nonhydride valence-bound ions investigated are around 3 x cm3 s - ' and in Section 1II.C that the corresponding coefficients for monohydride ions may be much less, presumably because none of the relatively few repulsive potentials of the neutral system make a favorable crossing with the potentials of the ion. When account is taken of special factors, such as channel degeneracy, the recombination coefficients of the ions that have been considered in this section are curiously similar in magnitude to those of Section 1II.D. For cm3 s- but there are four equivalent example, a(H,N+) is 1.4 x channels so that the recombination coefficient in each is 3.5 x lo-' cm3 s-'. The rapidity of single-electron transitions relative to double-electron transitions appears to compensate quite closely for the effective Franck-Condon

'

DISSOCIATIVE RECOMBINATION

477

factor being smaller in the tunneling mode than in the crossing mode. This may not be by chance in that the distribution of the measured dissociative recombination coefficients of polyatomic ions shows the influence of an upper limit or limits. In a tabulation by Adams and Smith (1988a, b) the recombinations of 16 species are given. They range from 1.1 x lop7cm3 s - l to 1.4 x lov6cm3 s-'. Of the 12 in which the ionized atom is trivalent as many as five are in the narrow range of 8.8 x to 1.1 x lop6cm3 s - l ; whereas, of the four in which the ionized atom is tetravalent, two have the value 1.4 x lop6cm3 s-' (see Section 11). D. PROTON-BRIDGE BONDIONSAND CLUSTERS

Stimulated by the surprising results of mass spectrometric determinations in the D region of the ionosphere (see Narcisi 1967) the Pittsburgh group used the microwave stationary afterglow method to measure the recombination coefficients of the hydronium clusters H:O(H,O), (Leu et al., 1973; Huang et al., 1978; Johnsen, 1993) and of the ammonium clusters NH;(NH,), (Huang et al., 1976). The kernels of these clusters are the proton-bridge ions H 2 0 . H +.H,O and NH, . H + .NH, (see Schuster, 1976). Their measured recombination coefficients are 2.2 x lop6cm3 s - l and 2.8 x cm3 s - l , respectively. The energy released by the recombination ensures that the only open channels are those giving ground state products H 2 0 - H + - H 2 0+ e + 2H,O NH,.H+.NH,

+e+

2NH,

+ H(1s) + 5.0 eV + H(1s) + 3.6 eV

(96) (97)

As is, patent single-electron transitions are involved. It was for those processes that the phenomenon of single-electron transition dissociative recombination was first recognized (Bates, 1992a). Because of this feature, which at the time was thought unique, coupled with exceptionally large magnitudes of the recombination coefficients the description superdissociatiue recombination was judged appropriate (as it was for the recombination of dimer ions-see Section 1II.E). It has since been realized that dissociative recombination through a single-electron transition is common (Section 1V.C). An explanation must therefore be sought for processes (96) and (97) being exceptionally rapid. Indeed, they are so rapid that it may be seen from formula (14) of Section I1 that free electrons of azimuthal quantum number up to at least 2 must have a significant probability of recombining. Admittedly cl(H,O+ *H,O), for example, at 2.5 x 10-6cm3s-' is only rather over twice a(H,O+) at 1.0 x cm3 s - A factor of just over 2 might seem no great matter were it not for the evidence given at the end of Section 1V.C that the distribution of the measured recombination coeffi-

'.

D. R. Bates

478

cients indicates an upper limit. Again account should be taken of the channel degeneracy. Hence the comparison should properly be with so a factor of 7.5 enters. +G~(H,O+) One possibility is that matrix element (58) is less than matrix element (61) owing to the greater distortion of the wave functions of the spectator electrons. Another, much more attractive, possibility is that the properties of the proton bond (see Janoschek, 1976) are responsible, these being dramatically different from the properties of the valence bond. This has yet to be explored. The measurements of Leu et al. (1973) have shown that the recombination coefficient of H,O+(H,O), increases as n is increased, whereas the measurements of Huang et al. (1976) show that the recombination coefficient of NH: (NH,), have little, if any, dependence on cluster size. Bates (1992a) has suggested that the difference arises from the near equality of the ionization potentials of H and 0 causing high charge mobility in hydronium clusters so that the proton bond moves to be as close as possible to an incident electron. A similar effect has been suggested in connection with C,H 2 recombination (Section 1V.C). Huang et al. (1976, 1978) had reported that rf microwave heating of the electron gas produced unexpectedly little reduction in the recombination coefficient. Johnsen (1993) has resolved the puzzle. He showed that the heating had much less effect on the electron temperature than was thought because proper account was not taken of the high rate coefficient for inelastic collisions between electrons and polar molecules. Geoghegan et al. (1991) have carried out FALP measurements on {(CH,),CO},H+, the largest proton-bridge ion kernel to have been studied. There is, obviously, no question of the recombination coefficient being enhanced by charge mobility (see earlier discussion). Its value is 1.4 x l o p 6cm3 s-'. The simplest proton-bridge ion is H i , which is best written H, * H + * H, (Yamaguchi et al, 1987; Okumura et al, 1988). Stationary afterglow measurements by Macdonald et al. (1984) have given that the recombination coefficient is 1.8 x l o p 6cm3 s- Several channels are open:

'.

H:

+ e -+

+ H + 9.0eV

(98)

+H2+3H+4.5eV

(99)

+ H, + 3.5eV

( 100)

2H,

+H, -+

5H + 0.0eV

(101)

The exothermicity in channel (100) relates to H, in the 2s2A1, which is rapidly predissociated (Dabrowski and Herzberg, 1980). Miderski and Gellene (1988) have observed that the dissociative recombination leads to the emission of the 3s2A; + 2p2A; system of H, with the lines Doppler

DISSOCIATIVE RECOMBINATION

479

broadened. The resemblance to Rydberg emission due to the recombination of dimer ions such as N, . N: (Section 111. E) is only superficial in that H: is not bound by one electron exchange forces. It is possible that a single-electron transition is involved. For n 2 2 the hydrogen ion clusters H:(H,), have H i as the kernel (Okumura et al., 1988). They are thus not proton-bridge ions. They have very small dissociation energies (Okumura et al., 1988) but are not closely similar to the dimer ions considered in Section 1II.D. It seems likely that H: is the hydrogen ion cluster with the greatest recombination coefficient.

V. Signature of Polyatomic Ion Dissociative Recombination The signature of diatomic ion dissociative recombination is suprathermal atoms. Naturally suprathermal fragments form part of the signature of polyatomic ion dissociative recombination. The remainder of the signature is that molecular fragments are generally vibrationally excited and may be rotationally excited. Duley and Williams (1992) have discussed the possibility of detecting infrared emission from such fragments in dark interstellar clouds. Spectroscopic measurements (see Adams, 1992) have shown that some of the OH generated by the dissociative recombination of, for example, 0 2 H + , HCO i, NzOH+,and H,O+ carries vibrational energy. Considering the simple case of a linear triatomic ion, presumed to be in the ground vibrational level, ABC'

+ e + A + BC + E

(102)

Bates (1993b) pointed out that the molecular product is liable to be vibrationally excited because of two effects. Taking the usually lesser effect first, the vibrational wave function of the ion is a linear combination (with random phase) of the vibrational wave functions of the u = 0 levels of the two normal vibrational modes. At the instant the bond to A is broken, the molecule BC carries a relic of this wave function. The relic differs significantly from the u = 0 vibrational wave function of BC, even though the B-C bond length and force constant are virtually the same as in the ion. Hence the molecule cannot be formed entirely in its u = 0 vibrational level. It is easy to show (c.f., Bates, 1993b) that the odd vibrational levels are not populated by the effect. Illustrative calculations on the relative populations P(u) of the vibrational levels u of the CO generated in HCO+ dissociative recombination along channel (74) were done. These show that P(O), P(2), P(4), P(6), and P ( 8 ) would be 0.78, 0.12, 0.06, 0.03, and 0 .01 if only the influence of the normal modes of the ion had to be taken into account.

480

D. R Bates

The influence of the repulsion between the separating fragments is more important. In many instances it may, provided C of process (102) is not an H atom, be treated as an impulse acting between the ejected atom A and the adjacent atom B of the molecule. This causes the momentum of B relative to C to change by

where M is the mass of the species indicated by the subscript. It is convenient to rewrite Eq. (103) as

in which the masses are now on the amu scale and E is in electron-volts. The momentum wave function representation is the most appropriate to the problem and the simple harmonic oscillator approximation is adequate. Using standard spectroscopic notation (Herzberg, 1950), except that the subscript e is omitted from the symbol for vibrational frequency in wave numbers, the parameter a of the SHO approximation is given by a = 2npvosc/fl

(105)

a = 8.31 x 10-3pWau

(106)

or

in which the reduced mass p of BC is in amu and w is in cm-'. It may be shown that the vibrational distribution of BC is determined by

A = 6/a'/2

(107)

and a tabulation for the range of interest has been given (Bates, 1993b). If

I exceeds 1.18au more than half of the BC molecules are vibrationally excited. In the case of HCO' dissociative recombination 1 is 1.56au and the tabulation gives that the relative vibrational populations P(O), P( l), P(2), P(3), and P(4) are 0.30, 0.36, 0.22, 0.09, and 0.03, respectively. Even more extensive vibrational excitation takes place if a heavy atom is released as in CO: dissociative recombination. When the products are in their ground electronic states (see Section 1II.D) ;1 is 4.47 and virtually none of the nascent CO is in the zeroth vibrational level. This qualitative conclusion is correct even though it is not a good approximation to treat the repulsion between the products as an impulse when both products are fairly massive. If C is an H atom formula, (104) is invalid because the neglected weak impulse between A and C is then of dominant importance. If E' is the kinetic energy of relative motion due to this weak impulse the momentum change is

DISSOCIATIVE RECOMBINATION

48 1

Prediction of E would require ab initio computations. The values needed to account for the experimental data are plausibly small. Measurements by Herd et al. (1990) on the OH released in OH: dissociative recombination along channels (89) and (90) show that a fraction 0.29 is vibrationally excited. Of this, 0.18 arises from the relic of the normal modes. A shortfall of 0.11 remains. For this to be attributed to the impulse on the H atom, E need only be 0.09 eV (Bates, 1993b). Franck-Condon considerations may also be relevant in the present context. They dictate, for instance, that much of the H, formed in channel (89) is vibrationally excited. If the ion that experiences dissociative recombination is bent, the impulses exert an impulsive couple, which can cause a high degree of rotational excitation. To illustrate the effect, calculations on the dissociative recombination of OCOH’ along channel (79) have been done (Bates, 1993b). The rotational quantum number J of the CO, formed is predicted to be 62 for a J = 0 ion.

References Abouelaziz, H., Comet, J. C., Pasquerault, D., and Rowe, B. R. (1993). J . Chem. Phys. 99, 237. Adams, N. G. (1992). Adu. Gas Phase Ion Chem. 1, 271. Adams, N. G., Herd, C. R., Geoghegan, M., Smith, D., Canosa, A., Comet, J. C., Rowe, B. R., Queffelec, J. L., and Morlais, M. (1991). J. Chem. Phys. 94, 4852. Adams, N. G., Herd, C. R., and Smith, D. (1989). J. Chem. Phys. 91,963. Adams, N. G., and Smith, D. (1988a). Chem. Phys. Lett. 144, 11. Adams, N. G., and Smith, D. (1988b). In Rate Coejicients in Astrochemistry (Millar, T. J., and Williams D. A,, Eds.), p. 173, Kluwer Academic, Dordrecht. Adams, N. G., Smith, D., and Alge, E. (1984). 1.Chern. Phys. 81, 1778. Alge, E., Adams, N. G., and Smith, D. (1983). J. Phys. B: At. Mol. Phys. 16, 1433. Amano, T. (1990). J. Chem. Phys. 92, 6492. Appleton, E. V. (1937). Proc. Roy. SOC.London A 162, 451. Arnot, F. L., and McEwan, M. B. (1938). Proc. Roy. SOC.London A 165, 133. Arnot, F. L., and McEwan, M. B. (1939). Proc. Roy. SOC.London A 171, 106. Amot, F. L., and Milligan, J. C. (1936). Proc Roy. Soc. A 153, 359. Bardsley, J. N. (1968a). J . Phys. B: A t . Mol. Phys. 1, 349. Bardsley, J. N. (1968b). J . Phys. B: A t . Mol. Phys. 1, 365. Bardsley, J. N. (1983). Planet. Space Sci. 31, 667. Bardsley, J. N., and Biondi, M. A. (1970). Adv. Atomic Molec. Phys. 6, 1. Barrios, A. Sheldon, J. W., Hardy, K. A., and Peterson, J. R. (1992). Phys. Rev. Lett. 69, 1348. Bates, D. R. (1950a). Phys. Rev. 77, 718. Bates, D. R. (1950b). Phys. Rev. 78, 492. Bates, D. R. (1961). Quantum Theory (Bates, D. R., Ed.), p. 251, Academic Press, New York. Bates, D. R. (1989). Asrrophys. J. 344,531.

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Index A

Absolute gravimeter, 16 Achiral molecules, 222 Adiabatic approximation, 4 Aharonov-Bohm effect, 5, 18, 67 Alkaline earth ions, electron-impact ionization, 345-349 Aluminum ions, electron-impact ionization, 337,340 Ammonium cluster ion, 477 Amplitude splitting, 12 Angular correlation, 55-56 Angular distribution, 207-209, 225-228 dichroism, 208,209, 228-243 Antifringes, 49 Antimony ions, electron-impact ionization, 368, 389-390 Antiscreening, 251, 253, 264, 285-290, 296-297 electron loss probability, 290 semiclassical approach, 271-274, 278-279 Apparent distribution, 290 Argon ion double ionization, 379-384 electron-impact ionization, 361- 362, 379-394, 399,412,414-418 REDA, 412,414-418 triple ionization, 386 Associative ionization, ultracold collisions, 133-134, 161, 162, 168 Astrophysics, dissociative recombination, 469 Atom gyroscope, 4 Atomic beam, 2, 3, 11-14 Atomic clocks, frequency shifts, 140-141 Atomic coherence, see Coherence Atomic core, 92-93 Atomic flux, 4

487

Atomic fountain, interferometry, 2, 28 Atomic intensity, 8, 10 Atomic polarizability, 19 Atomic quantum defect theory, 439 Atomic systems chaotic, 87-90 electric field mapping, 179-196 quantum mechanics, 86 scaling properties, 91-92 spectroscopy, 97- 109 Stark effect, 173-177 time scales, 94 Atomic wave, phase evolution, 4-6 Atomic wave function, 3, 18, 31 Atom interferometry gravity, 14-16, 28 history, 2 properties of atoms, 19 quantum mechanics, 18- 19 rotation, 17-18 Atoms, laser-cooled, collisions between, 125-168 Auto-double-ionization probability, 310 Autoionization, 101, 217-221 electron-impact ionization, 366-367 excitation-autoionization, 302, 303, 305-309, 366-367 ionization-autoionization, 303-305

B Balmer-a profile, 193-194 Bandwidth, 70 Barium ions double ionization, 392, 394 electron-impact ionization, 345-346, 377-378 giant resonancz, 365

Barton’s paradox, 79 Beam electrons, 203 Beam splitters, 11-14 Bell’s inequalities, 51-56 Benzene ions, dissociative recombination, 474 Berry’s phase, 42-43, 44 Beryllium-like ions electron-impact ionization, 335-336 REDA, 418-419 Bismuth ions, 389-391 Blackbody radiation, 36 Bohm theory, 63-67 Bohr model, 86 Bohr-Sommerfeld model, 86 Boron-like ions electron-impact ionization, 336 REDA, 418-419 Boson, 36-37 Bragg diffraction, 1 Breit interaction, 407, 409

C Calcium, Ramsey type interferometry, 24, 25 Calcium ions, electron-impact ionization, 345-349,421 Cantori, 94 Capture, resonant capture, 309-310 Capture cross section, 310 Carbohydride ions, 474 Casimir effect, 39, 79 Catalysis laser, 133 Causality, 78-79 Cavity quantum electrodynaniics, 39 CDAD, 208, 209, 228-243, 246 Cerium ions, electron-impact ionization, 374, 376, 396 Cesium atomic clock, frequency shift, 140-141 laser-cooled atoms detuning measurements, 154-155 ground state collisions, 137, 138 singly excited state collisions, 146, 158, 159 Cesium ions, electron-impact ionization, 374, 375, 394 Chaos, 86-121 atomic systems, 87-90 classical, 87, 94-95 localization, 111-113, 115 quantum chaos, 86-87,97 spectroscopy, 105

time scales, 94-97 transient chaos, 90 wave functions, localization, 109-1 15 Chiral molecules, 208, 221-222 circular dichroism, 231 optical activity, 243 CI method, 439 Circular dichroism, 208, 209, 228-243, 246 Classical impulse approximation, 251 Clauser, J. F., 37 Close-coupled target states, 311-324 Closure method, 256, 257-259 Coherence, 6-8, 10,49-51 Collapse effects, 45 Collisions laser-cooled atoms, 125-168 doubly excited states, 127-128, 161-168 dynamics, 144145,153-156 frequency shifts, 140 ground state atoms, 127, 134, 136-142 optical trapping, 128-133 singly excited states, 127, 142-160 Complementarity, 47-51 Complex systems, 87 Compton effect, 36 Compton profile, 278 Configuration interaction, 439 Copper ions, electron-impact ionization, 353-354 Correction factor method, 260-262 Counterfactual approach, 60 COW experiment, 15 Crossing dissociative recombination, 434-460 diatomic hydrogen ions, 440-443 monohydride ions, 443-444 nitric oxide, 447-449 nitrogen, 445-447 oxygen, 449-454 theory, 434-440 Crossing mode, 431 Cryptography, 58-59 cw-laser-induced fluorescence, 199 Cyclic hydrocarbons, dissociative recombination, 474

D Dark spot trap, 160 de Broglie guided-wave theory, 61-63 Delayed-choice experiments, 19 Detuning, singly excited state collisions, 146-158

Index Dielectric mirror, tunneling time, 72-75 Dielectrics, 71 Diffraction, 1-2, 12 Diffraction gratings, 12 Dimers, dissociative recombination, 455 Dipole-dipole interaction, ultracold collisions, 134 Dipole-force trapping, 133, 162 Direct ionization, 302 Dispersion cancellation, 69-70 Dispersion relaxation, 5 Dissociative recombination, 427-434 crossing dissociative recombination, 434-460 indirect dissociative recombination, 437 polyatomic ion dissociative recombination, 479-481 superdissociative recombination, 460, 477 tunneling dissociative recombination, 461-479 Distorted-wave method, 323 Doppler-free saturation spectroscopy, electric field mapping, 191, 204 Doppler shift, 15- 16 Double closure, 259 Double ionization, 380-386, 389-391 argon ions, 379-384 barium ions, 392, 394 bismuth ions, 389-390 cesium ions, 394 krypton ions, 379-381 lanthanum ions, 393-396 two-center double ionization, 253 xenon ions, 379, 383-385 Double-slit interferometer, 7-8 gravity effects, 14 spontaneous emission, 9-10 Young’s double-slit interferometer, 14, 19-21 Doubly excited state collisions laser-cooled atoms, 127-128, 161-168 collision dynamics, 161-162 Down-conversion, 39-40, 46 energy correlations, 44 parametric, suppression, 51 vacuum-induced coherence, 49-51 Dynamic localization, 97, 119 Dynamics quantum chaos, 115-120 ultracold collisions, 127, 128 doubly excited states, 161-162, 168 singly excited states, 144-145, 153-156

489 E

EBIS, 305 EBIT, 305 ECR, 305 Eigenstates, 4, 6 Einstein, Albert, 36 Einstein-Podolsky-Rosen paradox, 51-52, 69 Elastic scattering, magnetic traps, 138-140 Electric field gradient, interferometry, 29-30 Electric field mapping, 179-204 Electromagnetic field, quantization, 39 Electron beam ion source, 305 Electron cyclotron resonance, 305 Electron-electron interactions, 250-301 Electronic coupling, 438, 439 Electron-impact ionization 2s22p63s2pqions, 361-362 2s2, 2p4 ions, 336-337 3p6 + 3dq metal ions, 354-361 alkaline earth ions, 345-349 aluminum ions, 337, 340 antimony ions, 368, 389-390 argon ions, 361-362, 377-378,393-394, 399,412,414-418 barium ions, 345-346, 377-378 beryllium-like ions, 335-336 bismuth ions, 389-391 boron-like ions, 336 calcium ions, 345-349, 421 cerium ions, 374, 376, 394 cesium ions, 374, 375, 396 copper ions, 353-354 excitation-autoionization, 303, 305-309, 366-367, 398-408 gallium ions, 349-350 gold ions, 399-403 hafnium ions, 350-353 heavier monovalent ions, 344-345 heavy metal ions, 362-379, 391-398 helium-like ions, 324-328 indium ions, 366-368 iron ions, 353-354, 412,414-418 krypton ions, 379-381, 386,412,414-418 lanthanum ions, 377-378, 393-396, 398 lithium ions, 324-328, 421 lithium-like ions, 328-335,411-418 magnesium ions, 337 magnesium-like ions, 342-344,419 multiple ionization, 302, 305, 310-311, 379-398,422

Index

490 Electron-impact ionization (continued) nickel ions, 353-361 np6nd ions, 350-353 rare gas ions, 379-389 silicon ions, 339 sodium-like ions, 337-342, 398-41 1 strontium ions, 345-346 tantalum ions, 350-353 titanium ions, 350-357 uranium ions, 407 very highly charged ions, 398-420 xenon ions, 365, 366, 368-373, 377, 379, 383-385,403,406,412, 414-418 zinc ions, 349-350 zirconium ions, 350-353 Electron interferometry, 1, 2, 18 Electron loss probability, 271, 290-292 Electron spectroscopy, 292-295, 297 Emission spectroscopy electric field mapping, 193-194, 195 Stark effect, 172, 195, 204 Empty waves, 68-69 Energy-time uncertainty principle, 43-45 Energy transfer, singly excited state collisions, 142-144 Entangled states, 18-19, 44, 48 EPR paradox, see Einstein-PodolskyRosen paradox EPR states, dispersion cancellation, 69-70 Ericson fluctuations, 103 Excitation-autoionization, 302, 303, 305-309, 366-367 sodium-like ions, 306, 398-408 uranium ions, 407

F Falling corner cube technique, 28 Floquet spectroscopy, 116 Floquet state, 110, 116 Fluorescence collapse effects, 45 cw-laser-induced, 199 laser-induced, 175-176 optically trapped atoms, 130, 132 parametric, 39, 69-70 Franck-Condon principle, 143 Fraunhofer limit, 12 Free-bound spectroscopy, 156- 158, 159 Free-collision model, 251, 260 Frequency shifts, atomic clocks, 140-141

G

Gailitis jump, 319 Gallagher-Pritchard model, 160 Gallium ions, electron-impact ionization, 349-350 Geodetic precession, 17 Giant resonance, 363, 365, 374 Glow discharge plasma, 171-204 Gluon, 37 Gold ions, electron-impact ionization, 399-403 Grainger, P., 37, 42 Gravimeters, 16, 28 Gravitational phase shift, 14, 15 Gravity, atom interferometry, 14- 16, 28 Greenberger-Home-Zeilinger apparatus, 59-61 Ground state collisions, laser-cooled atoms, 127, 134, 136-142 Guided wave theory, 61-63

H Hafnium, electron-impact ionization, 350-353 Hanbury-Brown-Twiss experiment, 37 Hardy experiment, 61 Heavy metal ions, electron-impact ionization, 362-379, 391-398 Helium energy levels, 109 laser-cooled atoms, singly excited state collisions, 144, 151-152 localization, 110 Helium-like ions, electron-impact ionization, 324-328 Hong-Ou-Mandel interferometer, 46-47, 48, 70, 71 Hydride ions, dissociative recombination, 443-444 Hydrocarbons, dissociative recombination, 455 Hydrogen atom dynamics, 93 energy levels, 97-98 excitation spectra, 99 resonances, 102 wave function, 110, 113-117 Hydrogen atoms, dissociative recombination, 440-443 Hydrogen masers, 141-142 Hydronium cluster ions, 432, 477, 479

Index Hyperfine-state-changing collisions, 137- 138, 142, 143

I

IA, 422 Idler photon, 39 Impulse approximation, 274-278, 297 Incoherent projectile-target ionization, 253 Incoherent sum, 10 Independent particle model, 253, 363 Independent processes model, 306, 321-322 excitation-autoionization, 305-309 multiple ionization, 310-311 resonant capture, 309-310 Independent resonance approximation, 439 Indirect dissociative recombination, 437 Indirect ionization, 303-305, 422 Indium ions, electron-impact ionization, 366-368 Inert gas dimers, dissociative recombination, 455 Intensity ratio, 175 Interference, nonclassical, 42, 43, 46-47 Interference pattern, 10, 20 Interferometers atom interferometers beam splitters, 11-14 magnetic field, 30 optical Ramsey interferometer, 22-26 Raman transitions, 13, 16, 26-29 static electric field, 29-30 three-grating interferometer, 21-22 Young’s double-slit interferometer, 19-21 beam splitters, 11- 14 double-slit interferometers, 7-8 applications, 11-14 spontaneous emission, 9-10 Young’s, 19-21 history, 1-2 Hong-Ou-Mandel interferometer, 46-47, 48, 70, 71 longitudinal Stern-Gerlach interferometer, 30 Mach-Zender interferometer, 54, 68 neutron interferometer, 17 optical interferometer, 17 optical Ramsey interferometer, 22-26 Raman transition interferometer, 13, 16 ring laser interferometer, 17 sensitivity, 4 single-crystal interferometer, 3 1

49 1

Stern-Gerlach interferometer, 30 three-grating interferometer, 21 -22 Young’s double-slit interferometer, 19-21 Interferometry atom interferometry, 2, 18-19 electron interferometry, 1, 2, 18 magnetic field, 30 matter-wave interferometry, 14-19, 28, 30-31 neutron interferometry, 1, 2, 17, 18 quantum mechanics, 18- 19 static fields, 29-30 Invariant torus, 94, 98, 110, 111 Ionization, 90, 94, 101 associative ionization, 133-134, 161, 162, 168 direct ionization, 302 double ionization, 253, 380-386 argon ions, 379-384 barium ions, 392, 394 bismuth ions, 389-390 cesium ions, 394 krypton ions, 379-381 lanthanum ions, 393-396 two-center double ionization, 253 xenon ions, 379, 383-385 electron-impact ionization 2s22p63s23pq.ions, 361-362 2s2, 2pq ions, 336-342, 398-41 1 3p63dq.metal ions, 354-361 alkaline earth ions, 345-349 aluminum ions, 337, 340 antimony ions, 368, 389-390 argon ions, 361-362, 377-378, 393-394, 399,412,414-418 barium ions, 345-346, 377-378 beryllium-like ions, 335-336 bismuth ions, 389-391 boron-like ions, 336 calcium ions, 345-349, 421 copper ions, 353-354 excitation-autoionization, 303, 305-309, 366-367, 398-408 gallium ions, 349-350 gold ions, 399-403 hafnium ions, 350-353 heavier monovalent ions, 344-345 heavy metal ions, 362-379,391-398 helium-like ions, 324-328 indium ions, 366-368 iron ions, 353-354, 412,414-418 krypton ions, 379-381, 386, 412,414-418

492

Index

Ionization (continued) lanthanum ions, 377-378, 393-396, 398 lithium ions, 324-328, 421 lithium-like ions, 328-335, 41 1-418 magnesium ions, 337 magnesium-like ions, 342-344, 419 multiple ionization, 379-398 nickel ions, 353-361 np6nd ions, 350-353 rare gas ions, 379-389 silicon ions, 339 sodium-like ions, 337-342, 398-41 1 stzpntium ions, 345-346 tantalum ions, 350-353 titanium ions, 350-357 uranium ions, 407 very highly charged ions, 398-420 xenon ions, 365,366, 368-373, 377, 379, 383-385, 386,403,406, 412,414-418 zinc ions, 349-350 zirconium ions, 350-353 excitation-autoionization, 302, 303, 305-309, 366-367 Floquet spectroscopy, 116- 119 incoherent projectile-target ionization, 253 indirect ionization, 303-305, 422 ionization-autoionization, 303-305 localization and, 115 microwave, 93 multiple ionizations, 310-311 Penning ionization, 144 photoassociative ionization, 161 quadruple ionization, 386 resonantly enhanced multiphoton ionization, 236-238 threshold ionization, 117- 118 triple ionization, 386-389 argon ions, 386 cerium ions, 396 krypton ions, 386 lanthanum ions, 398 xenon ions, 387-389 Ionosphere, 448,450, 454 IRA, 439 Iron ions electron-impact ionization, 353-354, 412, 414-418 REDA, 408, 410,412,414-420

K KAM theorem, 87 KDP crystal, see Potassium dihydrogen phosphate crystal

Kronig-Penney model, 73 Krypton ions double ionization, 379-381 electron-impact ionization, 379-381, 386, 412,414-418 REDA, 412,414-420 triple ionization, 386

L Lanthanum ions, electron-impact ionization, 377-378, 393-396, 398 Larmor time, 74 Laser-cooled atoms collisions between, 125-168 doubly excited atoms, 127-128, 161-168 frequency shifts, 140- 141 ground state atoms, 127, 134, 136-142 optical trapping, 128-133 singly excited states, 127, 142-160 thermalization, 138- 140 Laser-induced fluorescence, electric field mapping, 175-176 Laser optogalvanic spectroscopy, 179-191, 199,204 Lasers, 133 LDAD, 208, 209, 228-243, 246 Lense-Thirring precession, 17 LIF experiments, 175-176, 196-204 Light conjugate photon pairs, down-conversion, 39-40 de Broglie model, 62 propagation time measurements, 71-72 quantum hypothesis, 36 quantum nondemolition, 41-42 quantum properties, 38-42 squeezed states, 40-41 vacuum fluctuations, 38-39 Linear dichroism, 208, 209, 228-243, 246 Lithium, 331 excitation spectra, 99 laser-cooled atoms, singly excited state collisions, 149-151 resonances, 102 Lithium ions, electron-impact ionization, 324-328,421 Lithium-like ions autoionization, 305 electron-impact ionization, 328-335, 411-418 REDA, 412-418 Local field approximation, 172-173

Index Local hidden variable theory, 52, 59 Localization dynamic, 97, 119 wave functions, 109-115 chaotic regime, 111-113 lifetime and, 115 mixed regular-chaotic regime, 113-115 regular regime, 110- 111 Longitudinal velocity, 6 Loss channels, 333

M Mach-Zender interferometer, 54, 68 Magnesium ions, electron-impact ionization, 337 Magnesium-like ions, electron-impact ionization, 342-344, 419 Magnetic electric field, chaotic atomic systems, 88, 89 Magnetic field gradient, interferometry, 30 Magneto-optical trap, 129-131, 133, 137, 146, 147, 148, 150, 159, 163 thermalization, 138- 140 Many-body theory, 274 Masers, 142 Matter-wave interferometry gravity effects, 14-16,28 outlook, 30-31 quantum mechanics, 18- 19 rotation, 17-18 Measurement theory, 48 Mechanics classical, 92 semiclassical, 86 Mesoscopic systems, 87 Mixed regular-chaotic regime, localization, 113-115 Molecular systems electric field mapping, 196-204 Stark effect, 177- 179 Molecular vibrational spectroscopy, 168 Molecules achiral, 222 chiral, 208, 221-222, 243 oriented, photoionization, 222-228 quantum mechanical parameters, 208 Monohydride ions, crossing dissociative recombination, 443-444 MOT, see Magneto-optical trap MQDT, 439 Multichannel quantum defect theory, 439

493

Multiple ionization, 302, 305, 310-311, 422 antimony ions, 389-391 argon ions, 379-384, 386 barium ions, 392, 394 bismuth ions, 389-391 cerium ions, 396 cesium ions, 394 double ionization, 253, 380-386, 389-391 electron-impact ionization, 379-398 heavy metal ions, 391-398 krypton ions, 379-381 lanthanum ions, 393-394, 395-396 quadruple ionization, 389 rare gas ions, 379-389 triple ionization, 386-389 xenon ions, 383-389

N Neutron interferometry, 1, 2, 17, 18 Nickel ions, electron-impact ionization, 353-361 Nitric oxide, dissociative recombination, 447-449 Nitrogen, dissociative recombination, 445-447 Nonclassical interference, 42-47 Nonlinear quantum mechanics, 78 0 Optical interferometer, 17 Optical pumping, ultracold collisions, 134, 135 Optical Ramsey excitation, 3 Optical Ramsey interferometer, 22-26 Optical resonators, 39 Optical traps dark spot trap, 160 dipole-force, 133, 162 ground state collisions, 127-136 magneto-optical, 129-131, 133, 137, 146, 147, 148, 150, 159, 163 vapor-cell, 133, 145, 147, 148, 165 Optics, 79 Oriented molecules, 208, 222-228, 244 Oxygen, dissociative recombination, 449-454

P Pancharatnam’s phase, 42 Paradoxes Barton’s, 79 EPR, 51-52,69 quantum field theory, 79

Index

494

Parametric down-conversion, 5 1 Parametric fluorescence, 39, 69-70 Partial wave expansion, 208 Particle physics, 36-37 Particle-wave duality, 36 Path integral approach, 4 Path integral wave function, 5 Penning ionization, 144, 305 Periodic orbit spectroscopy, 103- 108 Phase evolution, atomic wave, 4-6 Phase operators, 76-77 Phase uncertainty, 4 Photoassociative association, 161, 162, 163, 165 Photoelectric effect, quantum mechanics, 37 Photoelectrons photoionization, 222-228 spin polarization, 209-222 Photoionization, 207, 222-228 Photon recoil, 3, 13-14 Photons, 36, 37, 39-40 PIG, 305 Pilot wave, 62 Planck, Max, 36 Plane wave Born approximation, 250, 254-256, 264,295,297 Plane wave limit, 5 Plasmas, glow discharge, electric fields in, 171-204 Polarization Bell's inequalities, 52-53 Einstein-Podolsky-Rosen paradox, 51-52 spin polarization, 209-222 Polyatomic ion dissociative recombination, 479-481 Polycyclic hydrocarbons, dissociative recombination, 474 Porter-Thomas distribution, 102 Potassium dihydrogen phosphate crystal, 39 Probe laser, 133 Projectile electron excitation, 281 Projectile electron loss, 250, 251, 282-285 Projectile nucleus, 286 Propagation time, single-photon experiments, 71-72 Proton-bridge bond, 462 Proton-bridge ions, 478-479 Pump photon, 39 PWBA, 250, 254-256, 264,295,297

Q QED, see Cavity quantum electrodynamics

QND, see Quantum nondemolition Quadruple ionization, 386 Quantum information content, 56-58 mechanics, 207, 208 Quantum chaos, 86-87, 97 dynamics, 115-120 Quantum chaotic dynamics, 116 Quantum cryptography, 58-59 Quantum defect theory, 439 Quantum eraser, 47-49 Quantum ergodicity, 109 Quantum field theory, 79 Quantum hypothesis, 36 Quantum interferometry, 18 Quantum jumps, 45 Quantum mechanics atomic systems, 86 chaotic, 87-90 atom interferometry, 18-19 Bell's inequalities, 51-56 Bohm's deterministic model, 63 Einstein- Podolsky- Rosen paradox, 51-52,69 electron interferometry, 1, 2 energy-time, 43-45 history, 36-37 momentum-position, 43 neutron interferometry, 18 nonlinear, 78 spectra, 108- 109 uncertainty principle, 43 wave function, 3, 18 Quantum nondemolition, 41 -42 Quantum optics, history, 37 Quantum potential, 63, 64 Quantum propagation, 71 Quantum spectra, 108-109 Quantum teleportation, 57-58 Quantum &no effect, 45 Quasi-elastic scattering, 275-276 Quasimomentum, 74

R Radiation damping, 418-419 Radiative decay, 315-316, 407 Radiative escape, 143 Radiative redistribution, 143 Raman transition interferometry, 13, 16 Ramsey fringes, 3, 22 Ramsey technique, 2, 3, 22 Random matrix theory, 101, 102, 103

Index Rare gas ions, electron-impact ionization, 379- 389 Rarity-Tapster experiment, 53, 54, 59 READI, see Resonant-excitation autodouble-ionization Recoil, 11, 13-14 Recoil momentum spectroscopy, 297 Recombination dissociative crossing dissociative recombination, 434-460 polyatomic ion dissociative recombination, 479-48 1 tunneling dissociative recombination, 461-479 history, 427-433 radiative, 429, 43 1 superdissociative, 460, 477 Recombination coefficient, 433-434,476-477 REDA, see Resonant-excitation double autoionization REMPI process, 236-238 Resonances, width, 101-103 Resonant capture, 303, 309-310 Resonant-excitation auto-double-ionization, 302,303, 304, 309-310 Resonant-excitation double autoionization, 302,303, 309 beryllium-like ions, 418-419 boron-like ions, 418-419 lithium-like ions, 412-418 magnesium-like ions, 419-420 sodium-like ions, 408, 410-41 1 Resonantly enhanced multiphoton ionization process, 236-238 Ring laser interferometer, 17 Rotation, atom interferometry, 17-18 Rotational motion, spin polarization of photoelectrons, 216, 221 Rubidium laser-cooled atoms detuning measurements, 155-156 ground state collisions, 138 singly excited state collisions, 147-148, 158-159 Rydberg states, 119-120 Rydberg states, 92 rubidium, 119-120 Stark effects. 176 S

Sagnac effect, 17,26

495

SCA, 265-274, 295 antiscreening, 271 -274,278-279 screening, 266-271 Scalar interference, 3 Scaled spectroscopy, 116 Scaling, 91-92,422 Scar quantization condition, 112 Scars, 112-1 13 Scattering atomic interferometry, 19 experiments, 90 quasi-elastic scattering, 275-276 uncorrelated double inelastic scattering, 253 Schrodinger cat state, 19 Schrodinger equation, 4, 12 Screening, 251, 264, 285-290 electron loss probability, 290 semiclassical approach, 266-271 Semiclassical approximation, 265-274, 295 antiscreening, 271-274,278-279 screening, 266-271 Semiclassical mechanics, 86 Signal photon, 39 Silicon ions, electron-impact ionization, 339 Silver ions, electron-impact ionization, 374, 376 Single-crystal interferometer, 3 1 Single-electron transition, 462-463 Single-photon experiments propagation time measurements, 71-72 tunneling time, 69-76 Single-photon interference, 42-43 Singly excited state collisions laser-cooled atoms, 127-129, 131, 142-160 cesium, 146, 158, 159 collision dynamics, 144- 145 detuning, 146-158 energy transfer, 142-144 free-bound spectroscopy, 156-158, 159 helium, metastable, 144, 151-152 lithium, 149-151 rubidium, 147-148, 158 sodium, 148-149, 158-159 trap loss collisions, 145-146 Sodium, laser-cooled atoms, 148- 149, 158-159, 161 Sodium-like ions electron-impact ionization, 337-342, 398-41 1 excitation-autoionization, 306, 398-408 REDA, 408,410-411 Space flight experiments, dissociative recombination, 450, 452, 453

Index

496

Spectroscopy, 97-109 Doppler-free saturation spectroscopy, 191, 204 electron spectroscopy, 292-295, 297 emission spectroscopy, 172, 193, 195, 204 Floquet spectroscopy, 116 free-bound spectroscopy, 156- 158, 159 laser optogalvanic spectroscopy, 179-191, 199,204 molecular vibrational spectroscopy, 168 recoil momentum spectroscopy, 297 scaled spectroscopy, 116 Spin-orbit autoionization, 217, 221 Spinor interference, 3 Spin polarization, 207, 222 Spontaneous emission, 8-11, 39, 127 Spontaneous relaxation, 2 Squeezed states, light, 40-41 Stark effect, 172, 195, 204 atomic systems, 173-177, 179-196 molecular systems, 177-179, 196-204 State operator, 6 Static electric field chaotic atomic systems, 87-89 interferometry, 29-30 Static potential, 5, 6 Stern-Gerlach interferometer, 30 Strontium ions, electron-impact ionization, 345-346 Sum rule method, 262-264 Superconducting gravimeter, 16 Superdissociative recombination, 460, 477 Superluminality, 72

T Tantalum ions, electron-impact ionization, 350-353 Target nucleus, 286 Target states, close-coupled, 311-324 Taylor’s experiment, 37 Teleportation, 57-58 Thermalization, in magnetic traps, 138-140 Thomas-Fermi model, 105 Three-body problem, 89 Three-grating interferometer, 21-22 Time scales, 94-97 Titanium ions, electron-impact ionization, 350-357 Transmission grating, 12 Trap loss, 127, 131 doubly excited state collisions, 168 hyperfine-state-changing collisions,

137-138, 142, 143 singly excited state collisions, 143, 145-146, 147, 151, 159 Traps, optical, see Optical traps Triple ionization, 386-389, 396, 398 Tunneling, 72 Tunneling dissociative recombination, 46 1-479 Tunneling mode, 433 Tunneling times, 75-76 dielectric motors, 72-75 single photon, 69-76 wave theory, 64-67 Two-center double ionization, 253 Two-center electron-electron interaction, 250-301 Two-photon interference, 43, 46-47 Two-photon light source, 39-40 U

Ultracold collisions associative ionization, 133-134, 161, 162, 168 laser-cooled atoms, 125-168 doubly excited states, 127-128, 161-168 frequency shifts, 140-141 ground state atoms, 127, 134, 136-142 optical trapping, 128-133 singly excited states, 127, 142-160 non-laser-cooled atoms, 141-142 theory, 126 thermalization, 138- 140 Uncertainty principle, 43-45 Uncorrelated double inelastic scattering, 253 Unoriented molecules angular distribution, 209 spin polarization, photoelectrons ejected from, 209-222, 245 Unperturbed wave plane, 5 Uranium ions, excitation-autoionization, 407

V Vacuum fluctuations, 38-39, 69 Vacuum-induced coherence, 49-5 1 Vague tori, 94 Van Vleck propagator, 104, 112 Vapor-cell trap, 133, 145, 147, 148, 165 Velocity distribution, 7 Vibrational motion, spin polarization of photoelectrons, 216-220 Vibronic coupling, 438, 439

Index W

Wannier ridge, 107 Watched pot effect, 45 Wave front splitting, 12 Wave function, 3, 18, 31, 61-69 de Broglie guided wave theory, 61-63 localization, 109-1 15 chaotic regime, 11 1 - 113 mixed regular-chaotic region, 113- 115 regular regime, 110- 11 1 Wave packet, 6, 8, 38, 66-67, 72 localization, 112 nondispersive, 11 1, 11 5 Wave-particle duality, 36 Wave-riding electrons, 203 Wave theory, 1, 37-38 Which path experiments, 19,48 Wien filter, 5 W i p e r distribution, 69

497 X

Xenon ions electron-impact ionization, 365, 366, 368-373, 377, 379,403, 406, 412, 414-418 multiple ionization, 379, 383-389 REDA, 412, 414-420

Y Young’s double-slit interferometer, 14, 19-21

Z Zinc ions, electron-impact ionization, 349-350 Zirconium ions, electron-impact ionization, 350-353

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Contents of Volumes in This Serial

Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos Electron Affinities of Atoms and Molecules, B. L Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P. Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fen

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson

The Theory of Electron-Atom Collisions, R. Peterkop and !I Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F . J . de Heer Mass Spectrometry of Free Radicals, S. N. Foner

Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A. L Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood

Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop

Contents of Volumes in This Serial Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G. Reid

Relativistic 2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal

Volume 6

Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I . C. Percival Born Expansions, A. R. Holt and B. L Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, J . B. Hasted Measurements of Electron Excitation Functions, D. W 0. Heddle and R. G. W Keesing

Dissociative Recombination, J . N . Bardsley and M . A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu ltikawa The Diffusion of Atoms and Molecules, E. A. Mason and 7: R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Some New Experimental Methods in Collision Physics, R. F. Stebbings

Volume 7

Atomic Collision Processes in Gaseous Nebulae, M . J . Seaton

Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. Griuet

Collisions in the Ionosphere, A. Dalgarno

Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen

The Direct Study of Ionization in Space, R. t F. Boyd

Volume 5 Flowing Afterglow Measurements of lonNeutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A. L Schmeltekopf Experiments with Merging Beams, Roy H . Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuuen The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAsS’’pq,C. D. H. Chisholm, A. Dalgarno, and F. R. Innes

General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H . S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J . Greenfield

Volume 8 Interstellar Molecules: Their Formation and Destruction, D . McNally Monte Carlo Trajectory Calculations of

Contents of Volumes in This Serial Atomic and Molecular Excitation in Thermal Systems, James C. Keck

The R-Matrix Theory of Atomic Process, P. G. Burke and W D. Robb

Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Z Chen and Augustine C. Chen

Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D.Leuine

Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I . Schoen

Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen

The Auger Effect, E. H. S. Burhop and W N. Asaad

Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M . F. Golde and B. A. Thrush

Volume 9 Correlation in Excited States of Atoms, A. W Weiss The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron-Atom Collisions, D.Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy

Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuiile The First Born Approximation, K. L Bell and A. E. Kingston Photoelectron Spectroscopy, W C. Price Dye Lasers in Atomic Spectroscopy, W Lange, J. Luther, and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett

A Review of Jovian Ionospheric Chemistry, Wesley ?: Huntress, J r . Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I . C. Perciual and D. Richards Electron impact Excitation of Positive Ions, M . J. Seaton

Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K . Janeu Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M . Broyer, G. Goudedard, J . C. Lehmann, and J . ViguC Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, W i f helm Raith Ion Chemistry in the D Region, George C. Reid

Volume 13 Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, 1. K Hertel and 19: Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W B. Someruille

Contents of Volumes in This Serial Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J . Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M . Pipkin Quasi-Molecular Interference Effects in lonAtom Collisions, S. V. Bobasheu Rydberg Atoms, S. A. Edelstein and 7'. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K.Dupree

Volume 15 Negative Ions, H . S. W Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J . W Humberston Experimental Aspects of Positron Collisions in Gases, 7'. C. Griffith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, J . B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H . B. Gilbody Inner-Shell Ionization, E. H . S.Burhop Excitation of Atoms by Electron Impact, D. W 0. Heddle

Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G. Burke

Volume 16 Atomic Hartree-Fock Theory, M. Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Duren Sources of Polarized Electrons, R. J . Celotta and D. 7'.Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M. H. K e y and R. J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N . Fortson and L Wilets

Volume 17 Collective Effects in Photoionization Atoms, M. Ya. Amusia

of

Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M. F. H. Schuurmans, Q. H. F. Vrehen, D. Polder, and H . M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M . G. Payne, C. H. Chen, G. S. Hurst, and G. W Foltz Inner-Shell Vacancy Production in IonAtom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L Dufton and A. E. Kingston

Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg

Contents of Volumes in This Serial Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W Norcross and L A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W F. Drake Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K . Janev Interactions of Simple Ion-Atom Systems, J . 7: Park High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M . Itano, and R. S. Van Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. Blum and H. K leinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov

Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, Z D. Mark and A. W Castleman, Jr. Nuclear Reaction Effects on Atomic InnerShell Ionization, W E. Meyerhofand J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, 1. I. Sobel'man and A. I.: Vinogradou Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M. Raimond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J . A. C . Callas, G. Leuchs, H. Walther, and H . Figger Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O'Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M. R. C. McDowell and M . Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M . More Volume 22 Positronium-Its Formation and Interaction with Simple Systems, J . W Humberston

Volume 20

Experimental Aspects of Positron and Positronium Physics, 7: C. Griffith

Ion-Ion Recombination in an Ambient Gas, D. R. Bates

Doubly Excited States, Including New Classification Schemes, C. D. Lin

Contents of Volumes in This Serial Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K . Dolder and B. Peart Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods Physics, S. Swain

in

Atomic

Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M . Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J . Bauche, C. Bauche-Arnoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J. Willeumier, D. L Ederer, and J . L Picqut! Volume 24 The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Smith and N . G. Adams Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J . Smith and G. Leuchs

Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M . Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He+He Collisions at KeV Energies, R. F . Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M . Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H . G. Reid Electron Impact Excitation, R. J . W Henry and A. E. Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W R. Johnson

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

Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W F. Drake and S. P. Goldman

Correlations in Electron- Atom Scattering, A. Crowe

Dissociation Dynamics of Polyatomic Molecules, Z Uzer

Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor

Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H. Black

Contents of Volumes in This Serial Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L Moiseiwifsch The Low-Energy, Heavy Particle CollisionsA Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, K Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Francoise Masnou-Sweeuws, and Annick Giusti-Suzor On the j3 Decay of '"Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko

Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in Electron- Atom Collisions, Joachim Kessler Electron-Atom Scattering, I . E. McCarthy and E. Weigold Electron-Atom Ionization, I . E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, K I . Lengyel and M. I . Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule

Volume 28 The Theory of Fast Ion-Atom Collisions, J . S. Briggs and J. H . Macek

Cavity Quantum Electrodynamics, E. A. Hinds

Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M . K Ammosov, N . B. Delone, M. Yu.Ivanov, I. I. Bondar, and A. K Masalov Collision-Induced Coherences in Optical Physics, G. S. Agarwal Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J . P . Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions. J . H. McGuire

Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J . C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J . Latimer Theory of Collisions Between Laser Cooled Atoms, P. S. Julienne, A. M. Smith, and K . Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in IonAtom Collisions, Derrick s. F. Crothers and Louis J. Dub6

Volume 31

Some Recent Developments in the Fundamental Theory of Light, Peter W Milonni and Surendra Singh

Energies and Asymptotic Analysis for Helium Rydberg States, G. W F. Drake

Squeezed States of the Radiation Field, Khalid Zaheer and M . Suhail Zubairy

Spectroscopy of Trapped Ions, R. C. Thompson

Contents of Volumes in This Serial Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Duren and Jacques Robert

Studies of Electron Attachment at Thermal Energies Using the Flowing AfterglowLangmuir Technique, David Smith and Patrik Span61 Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain

Atomic Physics and Non-Maxwellian Plasmas, MichPle Lamoureux

Atoms in Cavities and Traps, H. Walther

Volume 32

Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J . F. Williams and J . B. Wang

Photoionisation of Atomic Oxygen and Atomic Nitrogen, K . L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransdenqand C. J . Noble Electron-Atom Scattering Theory and Calculations, P . G. Burke Terrestrial and Extraterrestrial H3+, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K . Dotder Quantum Defect Theory and Analysis of High-Precision Helium Term Energes, G. W F . Drake

Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A. R. Filippelli, Chun C. Lin, L. W Andersen, and J . W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, s. Trajrnar and J. W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R. W Crompton

Electron-Ion and Ion-Ion Recombination Processes, M . R. Flannery

Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody

Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. B. Gilbody

The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry 1. Schneider

Relativistic Electronic Structure of Atoms and Molecules, 1. P. Grant

Analytic Representation of Cross-Section Data, Mitio lnokuti, Mineo Kimura, M . A. Dillon, Isao Shimamura

The Chemistry of Stellar Environments, D. A. Howe, J . M . C . Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J . W Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Hamdy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D. S. F. Crothers Electron Capture to the Continuum, B. L. Moiseiwitsch How Opaque Is a Star?, M . J . Seaton

Electron Collisions with N2, 0 2 , and 0: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto, and M . Cacciatore Guide for Users of Data Resources, Jean W Galfagher Guide to Bibliographies, Books, Reviews and Compendia of Data on Atomic Collisions, E. W McDaniel and E. J . Mansky

Contents of Volumes in This Serial Volume 34

Atom Interferometry, C . S . Adams, 0. Carnal, and J . Mlynek Optical Tests of Quantum Mechanics, R. Y. Chiao, P. G. Kwiat, and A . M . Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J . E. Lawler and D.A . Doughty

Polarization and Orientation Phenomena in Photoionization of Molecules, N. A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E . C . Montenegro, W E . Meyerhof; and J . H . McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D . L. Moores and K . J . Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates

ISBN 0-12-00383q-X