Advances in Atomic and Molecular Physics, Volume 21

Advntic~es Advntic~esin in ATOMIC AND MOLECULAR PHYSICS V VOLUME 21 CONTRIBUTORS TO THIS VOLUME SHIH-I CHU CHRIS H...

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Advntic~es Advntic~esin in

ATOMIC AND MOLECULAR PHYSICS V

VOLUME 21

CONTRIBUTORS TO THIS VOLUME

SHIH-I CHU CHRIS H. GREENE YUKAP HAHN

CH. JUNGEN M. R. C. McDOWELL PIERRE MEYSTRE R. M. MORE DENNIS P. O’BRIEN HERBERT WALTHER M. ZARCONE

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

Sir David Bates DEPARTMENT O F APPLIED MATHEMATICS A N D 'THEORETICAL PHYSICS T H E Q U E E N ' S UNIVERSITY O F BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT OF PHYSICS N E W YORK UNIVERSITY N E W YORK, NEW YORK

VOLUME 21 I985

@

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Publishers

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COPYRIGHT @ 1985 BY ACADEMIC PRESS. INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTEJD IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOMPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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LIBRARY OF CONGRESS CATALOG C A R D NUMBER: 65-1 8423 ISBN hhhh PRINTED IN THE UNITED STATES OF AMERICA

85 86 87 88

98 7 6 5 4 3 2 I

Contents

ix

CONTRIBUTORS

Subnatural Linewidths in Atomic Spectroscopy Dennis P . O'Brien, Pierre Mey.strr, a n d Herbert Wulther

I. 11.

Ill. IV. V.

introduction Summary of Improvements of Spectroscopic Resolution "Fundamental" Ways to Overcome the Natural Linewidth Time-Biased Coherent Spectroscopy Conclusion References

i 2 10 25 45 47

Molecular Applications of Quantum Defect Theory Chris H . Grrenc unti Ch. Jungtw

I. 11. 111.

1V. V.

Introduction Quantum Defect Concepts and Formalism Rovibrational Channel Interactions Electronic Interactions at Short Range Discussion and Conclusions References

51 54

66 97 1 IS 118

Theory of Dielectronic Recombination Y14licip Hullti

I.

124

11. 111.

128

Introduction Electron-Ion Collision Theory The Dielectronic Recombination Cross Sections 1V. The Dielectronic Recombination Rate Coefficients V . Discussion and Summary Appendix A: Radiative Widths and Coupled Equations Appendix B: Auger Probabilities A , in LS Coupling Appendix C: Radiative Probabilities A, in LS Coupling V

146 157 171 178 180 184

vi

CONTENTS

Appendix D: Scaling Properties of A,, A,, w. and Appendix E: Extrapolation to High Rydberg States References

185 I89 I94

Recent Developments in Semiclassical Floquet Theories for IntenseField Multiphoton Processes Siiiii-I Chrr

I. 11. 111.

1V. V.

VI.

Introduction The Floquet Theory and General Properties of QuasiEnergy States Computational Methods for Multiphoton Excitation of Finite-Level Systems Non-Hermitian Floquet Theory for Multiphoton Ionization and Dissociation Many-Mode Floquet Theory Conclusion References

I97 199

208 226 239 248 249

Scattering in Strong Magnetic Fields

M . R . C . McDowell cittd M . Z m m e I. 11. 111.

1v. V. v1. VII. VIII.

Introduction Center-of-Mass Separation Potential Scattering Ensembles of Landau Levels The Low-Field Limit of the Cross Section Photoionization Photodetachment of Negative Ions Charge Exchange References

255 258 26 1 277 28 I 285 293 297 303

Pressure Ionization, Resonances, and the Continuity of Bound and Free States

R . M . More 1.

11. 111.

IV. V.

Introduction Continuity of Pressure Ionization Resonances Applications Conclusions Appendix A: Properties of the Jost Function Appendix B: Green's Function Appendix C: Electron Density of States

306 318 324 333 346 347 348 349

CONTENTS

Appendix D: Resonance Perturbation Theory Appendix E: Convergence for the &Potential Model References

INDEX CUMULATIVE AUTHOR INDEX: VOLUMES CUMULATIVE SUBJECT INDEX: VOLUMES

1-20 1-20

vii 35 I 352 354

357 369 375

This Page Intentionally Left Blank

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

SHIH-I CHU, Department of Chemistry, University of Kansas, Lawrence, Kansas 66045 ( 197) CHRIS H. GREENE, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 (51) YUKAP HAHN, Department of Physics, University of Connecticut, Storrs, Connecticut 06268 (123) CH. JUNGEN, Laboratoire de Photophysique Moleculaire du CNRS, Universite de Paris-Sud. 91405 Orsay, France (51)

M. R. C. McDOWELL, Department of Mathematics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 OEX, England (255) PIERRE MEYSTRE, Max-Planck Institut fur Quantenoptik, D-8046 Garching, Federal Republic of Germany ( I )

R. M . MORE, Lawrence Livermore National Laboratory, Livermore, California 94550 (30s) DENNIS P. O'BRIEN,* Max-Planck Institut fur Quantenoptik, D-8046 Garching, Federal Republic of Germany ( I ) HERBERT WALTHER, Max-Planck Institut fur Quantenoptik, D-8046 Garching, Federal Republic of Germany ( I ) M. ZARCONE, Istituto di Fisica, 90123 Palermo, Italy (255)

*Present address: C.C.K. Euratom, Physics Division, 21020 Ispra, Italy ix


ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 21

11

SUBNATURAL LINEWIDTHS 1 I IN ATOMIC SPECTROSCOPY DENNIS P. O’BRIEN,,*PIERRE MEYSTRE, and HERBERT WALTHER Max-Planck Institut f i r Quantenoptik Garching. Federal Republic of Germany

........................... . . . . . . . A. Nonlinear Spectroscopy . . . . . . . . . . . . . . . . . . . . B. Quantum Beat Spectroscopy . . . . . . . . . . . . . . . . . . C. Ultimate Spectral Resolution . . . . . . . . . . . . . . . . . . 111. “Fundamental” Ways to Overcome the Natural Linewidth . . . . . . A. Review of Heitler-Ma Theory of Natural Linewidth . . . . . . . B. Purcell Method . . . . . . . . . . . . . . . . . . . . . . . . C. Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . D. HeitlerMethod . . . . . . . . . . . . . . . . . . . . . . . . IV. Time-Biased Coherent Spectroscopy . . . . . . . . . . . . . . . . A. General Remarks . . . . . . . . . . . . . . . . . . . . . . . I. Introduction

11. Summary of Improvements of Spectroscopic Resolution

B. Level-Crossing Spectroscopy . . . . . . . . . . . . . . . . . . C. Ramsey Interference Method. . . . . . . . . . . . . . . . . . D. Transient Line Narrowing . . . . . . . . . . . . . . . . . . . E. Other Coherent Effects. . . . . . . . . . . . . . . . . . . . . V. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 3 7 8 10 10 13

19 23 25 25

26 30 35 42 45 47

I. Introduction It has traditionally been one of the main endeavors of spectroscopists to develop measurement techniques yielding ever higher resolution. The precision of a measurement in high-resolution optical spectroscopy is limited by effects such as Doppler broadening, collision broadening, transit-time broadening, and the natural linewidth caused by spontaneous decay. Doppler broadening can be eliminated with the use of, among others, the two nonlinear techniques: saturated absorption and two-photon spectros-

* Present address: C.C.R. Euratom, Physics Division, 21020 Ispra, Italy. 1

Copyright 0 1985 by Academic Reos, Inc. All rights of reproduction in any form r e ~ e ~ e d .

2

Dennis P. O’Brien et al.

copy (Demtrdder, 1981). Transit-time broadening due to the limited interaction time of the atoms with the light beam can be reduced by using expanded laser beams, the Ramsey technique, and cooled trapped ions. The latter techniques can also be used to eliminate the residual second-order Doppler effect (Dehmelt and Toschek, 1975). It might seem that after the Doppler and other broadening mechanisms had been eliminated, the natural linewidth would remain the ultimate limit to high-resolution spectroscopy (for a review see Rothe and Walther, 1979). Indeed spectroscopists quite often have the problem of investigating structures with a splitting smaller than or comparable to the natural width of a transition. Recently a reduction of the width below the limit imposed by the natural lifetime has been achieved in several experiments, such as with radio-frequency optical double resonance (Ma et al., 1968a,b),level-crossingspectroscopy (Copley et al., 1968), phase switching of a light field (Shimizu et al., 1981, 1983; Berman, 1982), polarization spectroscopy (Gawlik et al., 1982, 1983), and Ramsey spectroscopy (Salour and Cohen-Tannoudji, 1977). The object of this article is to describe some of the recent work on observing subnatural spectra. In Section I1 we briefly review some of the recent improvements in spectroscopic resolution that eliminate conventional broadening mechanisms, and then turn to subnatural spectroscopy. Of all the methods used to penetrate the natural linewidth, probably the most fundamental are the Heitler method of weak excitation in resonance fluorescence and the Purcell method of “switching off ” the vacuum field by modifying the density of modes of a resonator in which the atom decays. In Section 111 we examine these two methods and review the Heitler-Ma theory of natural linewidth and of resonance fluorescence. In Section IV we examine the various time delayed detection techniques of level crossing and Ramsey spectroscopy. Throughout we use units in which Planck‘s constant h = 1, so that frequency and energy have the same dimensions.

11. Summary of Improvements of Spectroscopic Resolution There are essentiallytwo groups of methods available in laser spectroscopy allowing a resolution which may only be limited by the natural width. The first group includes the methods of optical radio-frequency spectroscopy, e.g., the optical double-resonance method, the quantum beat method, and others listed in Table I on the right-hand side. In atomic spectroscopy in particular these methods have played a large role in past years, the excitation originally being achieved with discharge lamps having a spectral distribution

3

SUBNATURAL LINEWIDTHS TABLE I SURVEY ON

THE

METHODS OF HIGH-RESOLUTION SPECTROSCOPY USED FOR THE STUDY OF ELECTRONICALLY EXCITED STATES Broadband excitation Coherent population of two or more states

Narrowband excitation Narrowband absorption Atomic beam

Broadband absorption Saturated absorption Two-photon spectroscopy (stepwise excitation) Fluorescence line narrowing

“Incoherent” population

Time integral observation

Time differential observation

Double resonance method Optical pumping

Level crossing

Quantum beats

Anticrossing

Modulated excitation

comparable to or larger than the Doppler width. With the advent of tunable lasers, and their concomitant high spectral brightness, these methods, and in particular the quantum beat technique, underwent a sort of renaissance. In this last case, it is essential to have short light pulses exciting the atoms or molecules, since the Fourier-limited spectral distribution of the pulse has to be larger than the splitting of the levels under investigation to guarantee a coherent excitation. Laser excitation is therefore more advantageous than classical light sources. The techniques listed on the left-hand side of the table are those only applicable with narrowband excitation as provided by monomode lasers. With the introduction of these methods, especially atomic beam scattering, nonlinear absorption and two-photon spectroscopy began to play an ever increasing role. It is impossible to discuss in this section all methods in high-resolution laser spectroscopy; we limit ourselves to a few examples to show the present state of development of the various methods. In this article mainly nonlinear spectroscopy and the quantum beat method will be discussed. Other methods of Table I will be treated in Section IV in connection with subnatural linewidth spectroscopy. A. NONLINEAR SPECTROSCOPY

Nonlinear spectroscopy effectively started when McFarlane et al. (1 963) and Szoke and Javan (1963) demonstrated the Lamb dip caused by gain


4

saturation at the middle of the tuning curve of a single-mode HeNe laser. Later, with intracavity and external absorption cells, the method was used for high-resolution spectroscopy and for frequency stabilization of lasers. In nonlinear spectroscopy a signal is observed when two counterpropagating laser beams of the same frequency (saturating and probing beam) interact with the same atom in a nonlinear way. The various methods summarized in Table I1 differ in the approach by which this nonlinear interaction is detected, resulting in a different sensitivity and selectivity. A first improvement of the detection sensitivity can be obtained if one of the beams is chopped and the modulation of the other beam traced via phase-sensitive detection (BordC, 1970; HPnsch et al., 197 1). This alternative detection of the signal is especially useful at low pressures, where the total absorption is small. Favorable signal detection may be obtained if both laser beams are chopped with different frequencies, and the fluorescence is detected at the sum frequency, as done in the intermodulated fluorescence technique by Sorem and Schawlow (1972). Since the upper-state population is modulated at the sum frequency as well, the detection of the saturation signal can also be performed using optogalvanic (Lawler et al., 1979) or optoacoustic techniques (Marinero and Stuke, 1979). The latter are two important extensions of the intermodulated fluorescence method.

TABLE 11 DEVELOPMENT OF SATURATION SPECTROSCOPY Detection scheme Coupling of opposite laser beams

Direct signal

Amplitude modulation, one beam

Absorption

McFarlane et a/. BordC (1970); (1963); SzOke Hilnsch et al. and Javan (1971) (1963)

Disperion

BordC et al. (I 973)

Polarization

Optogalvanic detection. Optoacoustic detection. Polarizers are rotated.

Amplitude modulation, two beams

Frequency modulation

Sorem and Schawlow Bjorklund ( 1972); Lawler et (1980) a/. (1979y; Marinero and Stuke ( 1979)b Couillaud and Bjorklund Ducasse (I 975) ( I 980) Hilnsch et a/. Wieman and Hilnsch ( 1976) (1981)e; Hansch et al. ( 1 9 8 1 p

Heterodyne detection Raj et a/. ( 1980)

SUBNATURAL LINEWIDTHS

5

The nonlinear coupling of the counterpropagating beams can also be detected by observing either the dispersion of the medium in an interferometric setup (BordC et al., 1973; Couillaud and Ducasse, 1975), or else the laser-induced birefringenceor dichroism,as is done in the polarization spectroscopy technique introduced by Wieman and Hiinsch ( 1976).Polarization spectroscopy is of particular interest for measurements involving optically thin samples or weak lines, but, since the signal shape depends strongly on the choice of polarization for the saturating laser beam and on the adjustment of the polarizers used to probe the light-induced birefringence and dichroism, asymmetric lines may be obtained. This represents a drawback compared with intermodulated fluorescence. However, polarization spectroscopy has a big advantage compared with the other methods described so far: The broad signal background of the narrow line due to the collisional redistribution of the particle velocities is absent, since the collisions change the light-induced alignment or orientation; therefore, the particles which experienced a collision do not contribute to the signal. A method which combines the advantage of intermodulated fluorescence and polarization spectroscopy has been proposed and used by Hiinsch et al. (1981). The essence of this polarization intermodulated excitation (POLINEX)spectroscopy is that the polarization instead of the amplitude of one or both counterpropagatingbeams is modulated. When the combined absorption depends on the relative polarization of both beams, an intermodulation in the total rate of excitation is observed. The signal detection can therefore be performed either by observing the fluorescence or by using indirect methods, e.g., optogalvanic detection. This new technique has another important advantage compared with intermodulated fluorescence: The POLINEX signal does not have to be detected on a strongly modulated background (modulated at fundamental frequencies),because neither beam alone can produce a modulated signal in an isotropic medium; nonlinear mixing in the detection system therefore cannot produce spurious signals. However, there may be a dependenceofthe signal on external magnetic fields since the polarized excitation can produce a coherent population of Zeeman sublevels affected by the external field. Compensation of the earth’s magnetic field is therefore necessary not only for achieving the highest possible resolution, but also for obtaining the highest possible signal. Another new technique has been recently proposed: the heterodyne detection scheme ofsaturated absorption (Raj ef al., 1980;Bloch ef al., 1981). The method uses resonant near-degenerate four-wave mixing with two close optical frequenciesto perform high-frequency optical heterodyne saturation spectroscopy.The specific features of this method are that the phase delays in the heterodyne signal for the crossovers make it possible to measure the


6

relaxation rates of lower and upper states separately. In this respect the technique resemblesthe phase-shift method in modulated fluorescence.The technique also gives information on line assignment and is generally applicable to any nonlinear spectroscopic scheme, e.g., polarization spectroscopy, where it becomes possible to optimize the signal-to-noise ratio by an adequate choice of the heterodyne frequency. In particular, the influence of amplitude fluctuations of the laser can thus be eliminated when the measurement is performed in a frequency range where the shot-noise limit can be reached. This technique can in addition be applied to Doppler-free two-photon spectroscopy and to Raman spectroscopy. Another important improvement in nonlinear spectroscopyresulted from the introduction of frequency modulation (FM) spectroscopy by Bjorklund ( 1980). The technique allows a sensitive and rapid detection of absorption and even dispersion features with the full spectral resolution characteristic of continuous-wave dye lasers. This technique uses a phase modulation of the probe beam. The optical spectrum of the beam then consists of a strong carrier at frequency w, with two sidebands at frequencies w, f a,,where o, is the modulation frequency. A key concept is that w, is large compared to the width of the spectral feature under investigation, so that it can be probed by a single isolated sideband. Both the absorption and dispersion associated with the spectral feature can be separately measured by monitoring the phase and amplitude of the radio-frequency heterodyne beat signal that occurs when the frequency modulation spectrum is distorted by the effects of the spectral feature on the probing sidebands. Since single-mode dye lasers have little noise at radio frequencies, these beat signals can be detected with a high degree of sensitivity. Furthermore the entire lineshape of the spectral feature can be scanned by tuning either w, or w, (Bjorklund and Levenson, 1981). The methods of nonlinear high-resolution spectroscopy discussed so far all refer to nonlinear absorption in the atomic or molecular ensemble. Another very important nonlinear method which also deserves mention is two-photon spectroscopy. It was pointed out by Vasilenko et al. ( 1970)that two-photon transitions are suitable for eliminating the Doppler width. To understand this effect we assume that the atoms or molecules in a vapor cell are excited by two single-mode laser beams traveling in opposite directions. An atom moving in the cell with a velocity component v, sees the frequencies of the two laser beams Doppler shifted by the amount 1 - v,/c and 1 u,/c, respectively. If the atom performs the two-photon transition by absorbing one photon from each of the two beams, the influence of the linear Doppler effect is cancelled since

+

w( 1

+ UJC) + w( 1 - V J C ) = 2w


7

The essential point is that the Doppler width is compensated for all atoms: The whole ensemble, which is illuminated by the laser beams, therefore contributes to the signal (contrary to nonlinear absorption spectroscopy where only one velocity subgroup is investigated). The two-photon resonance is usually monitored via the subsequent fluorescence. If the polarization of the two laser beams is identical, two photons from the same as well as from different beams may be absorbed with the same probability. Since the two-photon transition with photons from the same beam is not Doppler free, a broad background signal is observed together with the sharp Doppler-free signal. Using different polarizations for either of the two beams (e.g., a+ and a-) eliminates the broadband background if the twophoton transition is only allowed with a+ and a- photons absorbed simultaneously. Two-photon spectroscopy complements nonlinear absorption spectroscopy since it allows dipole-forbidden transitions to be investigated. Since levels having an energy twice as large as the photon energy are measured, the energy range accessible by laser excitation is considerably increased. It was pointed out by Cagnac et al. (1973) that two-photon experiments can be performed even with modest laser powers. The corresponding transition rates may be rather large ifan intermediate state of opposite panty exists which has almost half the energy of the two-photon transition. The first Doppler-free two-photon experiments were performed with pulsed dye lasers (Biraben et al., 1974; Levenson and Bloembergen, 1974), and almost simultaneously with a cw dye laser (Hansch et al., 1974). Meanwhile the method has been used in a variety of investigations (Teets et al., 1977; Eckstein et al., 1978). If two-photon experiments are performed with high power densities it is always necessary to check the influence of the ac Stark effect, since this may well be present and affect the result. B. QUANTUM BEATSPECTROSCOPY The quantum beat method is of essentially the same nature as other methods based on coherent effects in fluorescence, such as the Hanle effect or level crossing, modulated excitation, and also perturbed angular correlation in nuclei. The techniques require that the system be excited into a coherent superposition of substates or neighboring states, and that afterward the evolution of this coherence be monitored as a change either of the polarization properties or of the angular distribution of the re-emitted radiation. The quantum beat method is much easier to understand than the more sophisticated steady-state atomic coherence phenomenon seen in a Hanle or

8


level-crossing experiment. The fact that the experimental demonstration came much later has technical reasons: Until recently it had been very difficult to detect fast modulation signals and also to produce the short and intense pulses needed for efficient excitation of the atomic system. Thus, after the first demonstration of the quantum beat method with classical light sources (Alexandrov, 1964; Dodd et al., 1964), the method was not widely applied until pulsed, tunable lasers became available. Since then a large number of experiments have been performed, especially in connection with the investigation of Rydberg levels. In the standard quantum beat experiments, light pulses are used to excite the coherent superposition of two closely spaced levels. Detection is performed by observing the temporal change of the fluorescence. Quantum beats can also be observed by means of stimulated transitions. In this case, the detection is performed by a second light pulse which measures, for example, the absorption of the system starting from the coherently populated intermediate levels. The quantum beats are obtained by measuring the absorption as a function of the time delay between exciting and probing light pulses. The first quantum beat experiment using this absorption method was performed by Ducas et al. (1975). Furthermore, the quantum beats can be investigated in a corresponding setup observing the laser-induced birefringence and dichroism (Lange and Mlynek, 1978). In comparison with the standard experiments, these new methods have the advantage that the spontaneous lifetimes of the levels investigated can be large, as is the case with Rydberg states. However, the fact that a second resonant probing light pulse must be available complicates the experiment, and a nonresonant probe would have many advantages. In the case of highly excited Rydberg states, field ionization is a very simple way to probe the quantum beats. This was demonstrated for the first time by Leuchs and Walther ( 1979). Owing to the long lifetime of the levels, large delay times could be used, and a rather high resolution was obtained in these experiments. Another way of nonresonant probing of quantum beats is the use of photoionization. Here, either the change in the total current ofphotoelectrons (Hellmuth et al., 1980,198 1) or in the angular distribution can be measured (Strand et al., 1978; Leuchs et al., 1979).

C. ULTIMATE SPECTRAL RESOLUTION With the development of frequency-stabilizedsingle-mode lasers and particularly tunable lasers, and with the introduction of methods eliminating the Doppler width, the natural linewidth sets the limit for the resolution.


9

This limit, however, is only achievable as long as the transition frequency to linewidth ratio is not larger than 1O'O. Long lifetimes as associated with vibrational transitions in molecules or forbidden transitions in atoms allow, in principle, a much higher resolution. However, limitations are set by the transit-time broadening and the influence of the quadratic Doppler effect. The transit-time broadening can be reduced by the use of expanded laser beams. In this way it was possible to resolve the radiative recoil-induced doublets of the hyperfine components of the methane transitions at 3.39pm. This high resolution (5 parts in loLo),derived from an external absorption cell with a 30-cm aperture, nevertheless remained two orders of magnitude larger than the natural linewidth (Hail et al., 1976). As larger cells and associated optics are difficult to realize, alternative schemes had to be found. An attractive solution is the multiple interaction in standing wave fields (Baklanov et al., 1976a,b), the optical analog to the Ramsey technique routinely used in radio-frequency spectroscopy of atomic and molecular beams. In these interference methods, the resolution is limited by the travel time between radiation zones rather than by the transit time through one zone. The first experiments with this technique were reported at the laser spectroscopy conferences in Jackson Lake Lodge (Carlsten and Hall, 1977) and in Rottach-Egern (Rothe and Walther, 1979). The highest resolution to date was obtained in the saturated absorption experiment of the Ca LS0-3P, intercombination line at 657 nm. The radiation beams were spatially separated (Barger, 198l ) by up to 3.5 cm. The linewidth observed was 3 kHz. An important condition for the application of the Ramsey technique is that the phases of the standing-wave fields are stationary, otherwise the fringes will wash away within the averaging time. Therefore, this method also has technical limitations, and it seems difficult at the moment to reduce the linewidth much further than 1 kHz. The new technique of spectroscopy of trapped ions, however, opens up the possibility of overcoming the present limit of ultimate resolution, and several groups are exploring this promising method (Neuhauser et al., 1978, 1980; Wineland and Itano, 1981; Nagourney et al., 1983). Neuhauser et al. (1978, 1980) succeeded in confining a single Ba+ ion in a Paul radio-frequency trap with a localization of about 2000 A via laser sideband cooling to about 10 mK. Recently Wineland and Itano ( 1981) also performed a monoion oscillator experiment using Mg+. They demonstrated a localization of an ion to s 15 pm in their Penning trap. Similar results have been obtained by Nagourney et al. ( 1983)in a radio-frequency trap. It is quite obvious that the new techniques open a new door to ultimate resolution spectroscopy, and it is certain that before long new, exciting results will be obtained in this connection.

10


111. “Fundamental” Ways to Overcome the

Natural Linewidth A. REVIEW OF HEITLER-MA THEORY OF

NATURAL LINEWIDTH The excited levels of an atom have a certain probability of decay by spontaneous emission, and therefore have a finite lifetime. As a result the levels develop a small but finite width r, where r is the total probability per unit time for the state concerned to decay. It is therefore evident that, because of the finite width of the levels, the emitted radiation will not be strictly monochromatic; its frequencies will be spread over a range Am = r. But in order to measure the frequency distribution of the photons with this accuracy the time needed is T >> l/Am = l/r. During this time the level will almost certainly decay by emission, so that perturbation theory is not adequate. We calculate therefore to all orders the total probability for a transition of an atom from some excited level 12) with frequency m2to the ground level Il), which has an infinite lifetime, and is therefore strictly discrete. The first method used to treat spontaneous emission quantum mechanically was devised by Weisskopf and Wigner (Weisskopf, 193 1, 1933; Weisskopf and Wigner, 1930). We use, however, the more general method of Heitler and Ma (Heitler and Ma, 1949; Heitler, 1954; Fontana, 1982), which is the one commonly used in atomic physics. To illustrate the method we apply it to a study of the spontaneous emission from a single two-level atom. The Hamiltonian of a two-level system interacting with the electromagnetic field in the rotating wave approximation is

where wo= m2 - m1 is the resonance frequency of the two levels and oI2= Il)( 21. The dipole approximation is assumed, and the coupling constant is g k = - i(

2aCk/L3)(d €k)e’’’*

(2)

where L3is the volume in which the field is quantized and the polarization vector. Throughout this review we shall absorb the polarization indices in k. We make the following ansatz for the wave function

IT) = W12, (0)) exp (-- im20 +

F bk(f)ll, k ) exp[-

i(m2 + @ f I

(3)

11


where I(0)) and Ik) are, respectively, the vacuum state and the state of the radiation field in which a photon in the mode k is present. The Schrodinger equation determining the amplitude coefficients gives in the interaction picture

where Vf, = (2, (0)l V(1,k). The initial conditions are

b(0) = 1,

(6)

bk(0) = 0

These initial conditions may be incorporated into Eqs. (4)and (5) by modifying them to

ibk = ( V:2)*b eXp[-i(oo - o k ) t ) ]

(8)

These equations are further assumed to hold for all t by requiring that bk(t) = 0 for t < 0. On introducing the Fourier transforms defined by

do B ( o )exp[i(02- o)t]

(9)

Eqs. (7) and (8) reduce to (0- 02)B(W ) =

2 Vf2 &(o) + 1 k

(11) (12)

(0- 01- Wk)Bk(o) = ( Vf,)*B(o)

To solve Eqs. ( 1 1) and (1 2) we introduce the function (Heitler, 1954) [(x) = P( llx) - in 6(x)

(13)

where P denotes the principal part. The solution is then given by B(o) =

1 - O2) - zkI

Vf212c('(0

- wI

- wk)

(14)

Dennis P.O'Brien et al.

12

Inserting Eqs. ( 14) and ( 1 5) into Eqs. (9) and ( 10) gives

I"

exp[i(02 - w)t] b(t) = - do 2ni -o- w2+ S(0)/2

OD

-2ni L/

- 0,- o k ) exp[i(ok + o,- w)t] d o ( Vf2)* ((0 (17) o - o2 i r ( o y 2

+

where

It now follows from the initial conditions and the properties ofthe (function that

The probability distribution of the states Ik) after a long time is therefore given by

where y is the real part of and we have absorbed the level shift into coo.In fact y is given by

I Vf2I2 6(w - a]- o k )

flu)= 2n k

(21)

We now assume that the atom is weakly coupled to the radiation field so that we can make the pole approximation in Eq. (16) and evaluate y at 02. On taking the limit L3 m Eq. (21) for o = o2reduces to

-

= 2n = 2n

I I I dak

p ( o k ) I Vk212 d(O0

dQk [I V?212

P(ok)l%-~

-

dwk (22)

where dokdRk represents the number of field modes in the frequency range to wk dok and dQk is the differential solid angle. The vacuum density of modes per unit volume is

+


13

Since 2x1 Vt21&,-wo)is the probability per unit time for the emission of a photon with frequency oowith fixed polarization and direction of motion, we see that y given by Eq. (22) is the total probability of spontaneous emission from level 12) to level 11).

B. PURCELLMETHOD 1. Background

Equation (22) shows that the spontaneous emission rate depends crucially on the density of modes p(w,) about the atomic frequency 0,.This was already realized many years ago by Purcell, who observed that the spontaneous emission rate for a two-level system is increased if the atom is surrounded by a cavity tuned to the transition frequency oo(Purcell, 1946). Conversely the decay rate decreases when the cavity is mistuned. In the case of an ideal cavity, tuned far off the atomic resonance frequency, no mode is available for the photon, and spontaneous emission cannot occur. In order to eliminate spontaneous emission completely, every propagating mode must be suppressed. A completely enclosed perfectly conducting cavity would accomplish this, but a waveguide below cutoff could also serve the purpose. The waveguide which can be viewed as a cavity with ends removed to infinity is experimentally attractive because atoms can pass through it freely. For an atom which decays in free space by an electric dipole transition, the emission rate y of Eq. (22) becomes 4e203

Y =7 l(r)I2 where ( r ) is the radial matrix element coupling the two levels, and we have used the density of modes in free space

The linewidth is, of course, Am = 2y (26) From the point of view of semiclassical radiation theory and time-dependent perturbation theory, Eq. (24) is a special case of Fermi’s Golden Rule

P,,= 2np(o)lV12

(27) where P,,is the probability per unit time of making a transition from the

14


discrete state 11) to state 12), which lies in a continuum, and p(w)do is the number of states per unit volume in the range ooto oo do.Integrating over the solid angle gives for the density of modes in free space

+

where the factor of 2 accounts for the two possible transverse photon polarizations. If the atom is not in free space, Eq. (25) is only approximately true, and in the limit where the characteristic size of the cavity approaches the radiated wavelength it breaks down completely. An extreme case occurs when the atom is in a cavity tuned to the radiation frequency. Instead of a continuum of modes there is now only a single mode. However, because of the energy dissipation within the cavity, a photon radiated at a well-defined frequency will be “smeared out” over the full spectral width of the cavity response functionf(o). If the cavity is excited at resonance by a photon of frequency w , then the mean spectral energy density at frequency w is Wc(w)= o f ( d / V c

(29)

where Vcis the cavity volume. We may compare this to the energy density in free space when each mode is excited with a single photon Wf(0) = W f ( 4

(30)

Assuming the resonator to have a Lorentzian line shape centered at a,, and of FWHM Amc = 2 r c , we obtain

At resonance, o = oc,this gives

2 1 1 mode pc(o,) = - --= 1 K Vc Amc (cavity volume)(FWHM)

(32)

Thus an atom in a resonant cavity (0, = oo) radiates at a rate

We have taken the characteristic volume of the cavity to be Vc= Ai/4;Q, is the cavity quality factor w/A\w.We conclude that, when an atom is placed in a cavity with a single mode at the transition frequency, it radiates approxi-


15

mately Qc/n2times more rapidly than in free space. This enhancement of the radiation rate was first pointed out by Bloembergen and Pound ( 1954) and is the rationale for using resonant cavities in masers and lasers. Since a tuned cavity enhances the radiation rate, it is not surprising that a mistuned cavity can depress it. Consider for example a cavity whose fundamental frequency is at twice the resonance frequency of the atomic transition. The radiation rate is then

which is substantially decreased compared to the decay rate yf in vacuum. In principle by making Q sufficiently large, yc may be made arbitrarily small. A serious shortcoming of the above proposal is that the cavity will significantly perturb the atom. Aside from direct interactions with the surface, such as those due to dispersion forces, the deep potential well represented by the cavity necessarily truncates the atomic wave function. It is easily shown by uncertainty principle arguments that the resulting shift in energy is large compared to the fine structure. Consequently such a system is not interesting for precision experiments. However, it is still possible to demonstrate experimentally the suppression of spontaneous emission. For instance the radiation lifetime of a beam of atoms will be markedly changed if it passes between conducting plates whose separation is small or comparable to 112. In this case the radiation pattern is two dimensional, with only a single polarization allowed, El (electrical field perpendicular to surface). It can be shown that the density of modes in this case is

where a is the separation of the plates, so that the emission rate becomes y z = y f -P=2 y - = yw pf

f

ca

1

f

4a

When a = 112, y2 = yf/2, which is not a spectacular change. What is perhaps more unexpected is that, as a 0, yz 00; the rate increases rather than decreases! This occurs because the mode configuration for the EL mode is independent of a. However, p depends on the number of modes per unit frequency interval per unit volume, and as a + 0, the volume vanishes. This effect should be observable in an atomic beam experiment with excited atoms or molecules which pass between two conducting plates. Ifthe lifetime --+

--+

16


is comparableto the transit time, then the fraction of excited atoms reaching the detector will change as the plates are brought together.

2. Experiment An experimentalstudy of the radiative properties of atoms in the proximity of conductors was undertaken by Vaidyanathan et al. (198 1). They studied the blackbody radiation transfer of free atoms between conducting planes at a wavelength so long that the conductor was, to a good approximation, ideal. The transfer rate undergoesa discontinuitywith frequency which can be explained in terms of the effect of the elementary mode structure on the spontaneous decay rate (see Fig. 1). The radiative absorption rate is proportional to the mode density do)at the transition frequency. The mode density for the electric field perpendicular to a normal to the planes undergoes a discontinuity at a cutoff frequency o,= c/2d, where d is the plane separation. pL is given by WCW p*(o) = 4n -

0

> o,

= 0,

0

< o,

c3

’

I

1

98

I

I

I

1

1

1

1.02 1.04 1.06 1.08

I

FR E0UEN C Y ( V/Vcl

0

1 I

1.08

(37)

2 3 4 F I E L D (Vlcml 1

I

5 I

I

FIG.1. Blackbody radiative transfer signals in sodium located between parallel conducting plates for 29d + 30p (a) and 28d + 29p (b) as a function of the absorption frequency. The critical frequency is v, 1/2d = 1.48 cm-I, where dis the plate separation.The increase in the transfer rate at v = v, (left-hand side) is due to the “switching” ofthepLmode. (From Vaidyanathan el at., 1981.)

-

1

1.10 1.12 1.14 1.161.18 FREQUENCY ( V / V c l

17


3. Jaynes- Cummings Model To understand further how the spontaneous emission of a photon can be affected by its surroundings, we note that for times less than the transit time from the (fixed) atom to the nearest point of environmental influence (a mirror for example), and back again, the atom decays as if isolated. For longer times, we have the possibility of reabsorption and emission and then the influence of the environment becomes important. The theory of such influence has been discussed by several authors (Barton, 1970; Stehle, 1970; Milonni and Knight, 1973; Wittke, 1975; Kleppner, 1971, 1981; SanchezMondragon et al., 1983). A fully quantum-mechanical theory of spontaneous emission in an ideal cavity is obtained from the Jaynes-Cummings model of a single two-level atom in interaction with a single mode of the electromagnetic field (Jaynes and Cummings, 1963; Stenholm, 1973; Meystre et al., 1975; Von Foerster, 1975; Meystre, 1974). In the rotating-wave approximation, the JaynesCummings Hamiltonian is

H = wo 0 2 2 + oca t c a,

+ do21a c + a t c on)

(38)

where ozl= 12)( 11 etc., and a,, atcare the usual radiation mode operators, In the following we work in the Heisenberg picture. The atom’s radiated power spectrum can be evaluated using the definition of a physical spectrum S(w)introduced by Eberly and Wodkiewicz (1977) (note that stationarity is not achieved here, so that the Wiener-Kitchnine theorem cannot be applied in this case):

I’ IT-‘

S(w) = 27 Re

dz exp[y - i ( o - o , ) ] ~

dt’ exp[2y(T- t’)]D(t’,T)

(39)

+

where the dipole correlation a t , ? ) is proportional to (o2,(t T)alz(t)). Here T is the transit time of the excited atom through the cavity, and y is the half-bandwidth of the spectrometer used to measure the spectrum. The Jaynes-Cummings model being exactly solvable, an exact expression for D(t,r) can be obtained (Sanchez-Mondragon et al., 1983). For an atom initially excited and the field in the vacuum state, one finds

+

+

D(t,z) = ( 2 ~ ~exp[i(oc )-~ A/2)7((vO A/2)2 exp(ivoT) (vo - A/2)2 exp(- ivoz) 2gz cos v0(2t - T))

+

+

(40)

where v o = J(A/2)2+g2 is the vacuum field Rabi frequency and A = oo- 0,. There are two limiting cases of general interest: For the first

18

Dennis P. O'Brien et al.

case we consider a long-time spectrum, with broadband detection, and the atom far from resonance, i.e., A >> y and g > 1/T. Then the spectrum becomes

that is, the spectrum is Lorentzian with a width given by the spectrometer bandwidth y. We see that, if y < A / 2 , where A is the free-space Einstein spontaneous emission coefficient, then we have line narrowing. For the second case we consider again a long-time spectrum, but with the atom near resonance, i.e., y, g > A >> 1/T.Then we have S(0)

+

Y2 c-f

myg"0

- 0,- d2 + y21-'

(42)

In the narrowband detection limit ( y -=c g)f= ,and the spectrum consists of two resolved peaks at 0 = 0, g. This result represents a vacuum-field Rabi splitting (Sanchez-Mondragon ez al., 1983), where 2g plays the role of the Rabi frequency. Figure 2 shows a series of predicted spectra, each one for a different value of the frequency detuning A between cavity and atom. Figure 3 showsthe effect on the emission line shape ofany radiation that may already be in the cavity at the time the atom enters. The various spectra correspond to different values of the initial field strength; vacuum-field Rabi splitting is evident. However, for larger values of the field strength a different type of spectral shape emerges, a triplet of peaks. This triplet is in fact the exact analog of the intense laser line splitting observed in resonance fluorescence (Mollow, 1969). Highly excited atoms, prepared inside a millimeter-wave cavity resonant with a transition connecting two neighboring Rydberg levels, constitute an

+

FIG.2. A set of vacuum spectra, for which T = A-I, for values ofatom-cavity detuning on a logarithmic scale from A = 1/10 (back line) to A = 10 (front line). For small detuning the vacuum Rabi splitting is evident, and for large detuning the spectrum shows pure fluorescence. In between, one sees weak Rayleigh-type scattering at a position near w = w,. (From SanchezMondragon et al., 1983.)


19

-40

FIG.3. A set of spectra, for which T = 20 k', showing the influence of coherent radiation already present in the cavity at t = 0. The parameter2a,which increases by a factorof 10O.l from spectrum to spectrum, is the effective Rabi frequency of the field initially in the cavity. The transition from two-peak vacuum Rabi splitting (small a)to three-peakac Stark splitting (large a)is evident. (From Sanchez-Mondragonel al., 1983.)

almost ideal system for the study of fundamental matter- field coupling effects (Moi et al., 1983; Raimond et al., 1982a,b; Haroche et al., 1982).The very strong electric dipole coupling between nearby Rydberg levels is further enhanced by an amount proportional to the quality factor Q, when the Rydberg states are prepared inside a resonant cavity. Further, the coupling with all the field modes other than the cavity one can be neglected. It is hence possible to realize experimentally the situation where a small sample of two-level atoms is interacting with only one field mode. By increasing the Q of the cavity enough one can in fact reach the situation where the emission threshold corresponds to a single atom at a time in the cavity- the situation discussed theoretically above (Meschede et al., 1985). For the case when the cavity is off resonant for allowed atomic transitions, Rydberg atoms offer a way to investigate inhibited spontaneous emission as we have seen occurs in the case of blackbody absorption. Note that a similar effect has recently been observed in a somewhat different context by Gabrielse and Dehmelt (1 985). These authors observed that the cyclotron decay time of a single electron held in a Penning trap was increased by a factor of four as compared to its free-space value, due to the fact that the trap effectively constitutes a cavity which decouples the cyclotron motion from the free-space radiation field.

C. RESONANCE FLUORESCENCE Resonance fluorescence, in particular the problem of theoretically and experimentally determining the spectrum of the fluorescent light radiated by


20

a two-level atom driven by an intense monochromatic field, has been the subject of numerous studies. For sufficiently strong fields, it is found that the spectrum of the scattered light splits into three peaks consisting of a central peak, centered at the driving field frequency with a width r / 2 (r-l= Einstein A coefficient)and having a height three times that of two symmetrically placed sidebands, each of width 3r/4 and displaced from the central peak by the Rabi frequency. In addition there appears a delta function (coherent) contribution also positioned at the driving frequency. In the limit of strong fields, the energy camed by this last contribution is negligible compared to the three-peak contribution (Mollow, 1969, 1972, 1975; Carmichael and Walls, 1975, 1976a,b; Wodkiewicz and Eberly, 1976; Kimble and Mandel, 1976; Swain, 1975; Hassan and Bullough, 1975; Oliver et al., 1971; CohenTannoudji, 1975, 1977; Schuda et al., 1974; Wu et al., 1975; Hartig et al., 1976; Grove et al., 1977). Here we concentrate on the weak-field limit of resonance fluorescence, which yields a “subnatural” fluorescence spectrum. Although not per se a spectroscopic tool, since the spectrum is always centered at the exciting laser frequency, this example illustrates again that the natural linewidth is by no means a fundamental limit. Since the resonance fluorescence spectrum depends upon the intensity distribution of the incident radiation in the region of the atomic transition, we shall for the present assume a general form for the distribution of incident photons. In a given mode k there are nk photons with a specified direction and polarization. Denoting as before the ground state of the atom by 11) and the excited state by 12) (assuming no degeneracy), we confine ourselves in the weak-field limit to intermediate states where the atom is excited and one light quantum of frequency ck, = wkois absorbed, wkobeingalmost equal to the resonance frequency woofthe atom. In the final state the atom is again in the ground state 11) with another light quantum a,., emitted. The relevant states of interest are thus

11, R ) ,

12, R - lko),

11, R - 1ko + lk,)

(43)

where the incident distribution of photons is given by the state IR)

The Schrddinger equation for the amplitudes of the various states of interest in the rotating-wave approximation reduces to

it, = 2

v k o b k o exp[i(o,

- wk)t]

ko

ibko= V t o bexp[- i(oo- W k ) f ]

+ b(t)

(45)

21


ibkokl

= Vtokl

bko

exP[-

i(okl

- wO)tl

(47)

where we have used an obvious notation for the amplitudes and V k 0 = ( 1 , RI V12,R - lko)y etc. As for our treatment of spontaneous emission we Fourier transform these equations to obtain (0- oa)B(w) =

- ob)Bko(0)

=

2 V k o B k o ( ~ )+ Lo

(48)

V&B(o) +

(49)

z

VkoklBkokI(~)

ki

(o- ac)Bkokl(o) = Vzokl Bk,(o) where for convenience we have introduced the frequencies

+

+ 0,-okay

+ 0,- +

(50)

(51) The solutions of Eq. (4) are obtained exactly as for the case of spontaneous emission and are 0, wry

@b

=0

2

0 1

oko

okl

where we have introduced the functions

and

For o = q,,y is just the damping constant that arose in Section III,A for the spontaneous emission of level 2, and is independent of k o . The real and imaginary parts of y are thus the total transition probability for emission from the excited state and the level shift of the excited state, respectively. If we neglect the small y in Eq. (56), we see that the imaginary part of r for o = onis a contribution to the self-energy of the ground state due to the

22


absorption of photons from the incident beam. Here we are not interested in level shifts and shall therefore put the imaginary parts of y and r to zero. So we have r

=

I vkolz (w - o 0 ) Z

+ y2/4

The frequency distribution of the emitted radiation for t by calculating

-

00

is obtained

so that the probability distribution for emission is

Since r is very small, the probability is strongly peaked at Wk, = w k o . This means that for a given incident photon only photons of the same frequency have an appreciable probability of being emitted. Further discussion depends on the form of the incident intensity distribution. We shall confine ourselves to the examples of a broadband excitation about the natural line breadth and of a monochromatic excitation (sharp compared to the natural linewidth y). Using Eq. (61) the probability for the emission of a photon of frequency Wk, after excitation by broadband radiation is

This is the same shape as the spontaneously emitted line. Similarly, by summing over k, in Eq. (61), we obtain the probability that a photon of frequency has been absorbed. The result is

For broadband excitation, the shape of the resonance fluorescence line behaves therefore as if two independent processes, an absorption and a subsequent emission, took place.


23

For the case of excitation by monochromatic radiation the integration in Eq. (6 1 ) yields

The intensity distribution is essentially determined by two factors. First, ) , emitted line has the same since the intensity is proportional to ~ o ( ~ k l the shape as the incident excitation and is thus much narrower than the natural line. Second, the denominator in Eq. (64) is practically constant where I, is nonzero. This factor therefore determines the intensity. Since we do not obtain the spontaneously emitted line in this case, we see that here the resonance fluorescence has to be considered as a single coherent process.

D. HEITLERMETHOD In the theory of Heitler the atom absorbs only one photon. After emission of the secondary photon the atom returns to its ground state without interacting anymore with the primary radiation field. The absorption and emission of a photon is a single coherent process. This is a good approximation so long as the incident radiation field is weak. As we have just discussed, in weak-field resonance fluorescence calculations, excitation with nearly monochromatic light produces an emitted spectrum that has the same shape as the primary one, in contrast to the case of strong-field excitation (induced transition rate comparable to, or greater than, the spontaneous emission rate), where the spectral distribution consists of three peaks. The weak-field spectrum has been observed by several groups (Wu et al., 1975; Hartig et al., 1976; Gibbs and Venkatesan, 1976; Eisenberger et al., 1976), and the fluoresence linewidth was shown to be indeed less than the natural width. In related experiments, the second-order correlation of the field was also measured in this same weak-field limit (Cresser et al., 1982)as well as in the strong-field regime, in which case photon antibunching is observed(Kimbleetaf., 1977, 1978;Cresseretal., 1982).Wediscuss herethe experiment of Gibbs and Venkatesan (1976), using the 80-MHz-wide 2852 A transition of 24Mgatoms in an atomic beam irradiated by a frequencydoubled cw dye laser. The atomic system consisted of two levels with the atomic resonance, incident, and scattered frequencies being wo, w 1, 02, respectively. In contrast to the treatment of the preceding section, the lower level had a width rl/27r determined by optical excitation and translational motion through the light beam. This was about 1 MHz. The excited state width was 80 MHz. Because of the allowed decay of the lower level the theoretical spectrum is a generalization of the Heitler spectrum (Omont ec

24

Dennis P. O'Brien et al.

where

A = 47r2 (11P112)2( llF21q2Pll r 2

(66)

and pll is the lower state density. The matrix elements are the dipole moments, including the polarization effects of the incident and scattered photons. The scattered intensity is

( o 4 W a c 4 ) W 1 m2)

(67)

. \!

a

I

,...*r

* .

"...

40 M H r

I

I

D I F F ERE NC E

I

FR EO U ENCY

FIG.4. (a) Fabry - Perot transmission with finesse, drift, and laser jitter averaged over a few minutes. (b) Narrower than natural linewidth fluorescence. The fluorescence was observed through the Fabry-Perot as the transmission frequency was scanned up and down about 500 times over 30 min. About 20 MHz is estimated to arise from residual Doppler broadening; the remainder is from the laser and Fabry-Perot largely from Fabry-Perot drift. (c) Large-angle fluorescence versus magnetic field scan of the "Mg resonance frequency ( I .4 MHz/G). No Fabry-Perot was used, so that the width results mostly from natural breadth and residual Doppler broadening of the atomic beam excited at right angles. The solid curve in (c) is a Lorentzian curve superimposed on the data; the solid curves in (a) and (b) are hand drawn through the data points. (From Gibbs and Venkatesan, 1976.)


25

for a monochromatic incident intensity I , at frequency w , .The first term in Eq. (65) expresses energy conservation, i.e., that the scattering is elastic; the scattered frequency differs from the incident one by no more than the width of the lower state. If the incident field has a finite frequency width, Eq. (67) must be averaged over those frequencies, and consequently the fluorescence exhibits this same width. In their experiment, Gibbs and Venkatesan observed a fluorescence width less than half the natural width, as illustrated in Fig. 4.

IV. Time-Biased Coherent Spectroscopy A. GENERAL REMARKS A common feature of the various line-narrowing techniques that will be discussed subsequently is to observe the radiation from naturally decaying states a certain time interval after they are populated, so that the observation is limited to the set which has survived in the excited state for a longer time than the average. Related ideas have been used in Mdssbauer spectroscopy (Lynch et al., 1960; Wu et af., 1960; Holland et al., 1960; Hams, 1961; Neuwirth, 1966; Albrecht and Neuwirth, 1967; Hamillard and Floy, 1968a,b;Hogasen et al., 1963). In atomic spectroscopy the first experiments of this type were performed by Ma et af. (1968a,b) and by Copley et af. (1968). The experiment of Ma et al. (1968a,b) was a double resonance experiment on the long-lived 3P,levels of Sr and Cd. The atoms were investigated in an atomic beam, the regions where the atoms are excited and observed being separated so that a time delay was naturally produced by their time of flight. The linewidth observed was a factor of 1.4 below the natural width. At about the same time Copley et af.(1968) performed a level-crosslevel of sodium, using ing experiment on the hyperfine structure of the 3 2P3,2 pulsed excitation. The light source was a sodium vapor lamp chopped by a Kerr cell. The fluorescence light was observed 2.5 lifetimes after the excitation. Because the fluorescencedecays exponentially, the signal-to-noiseratio decreases with increasing delay time. Due to the weak initial intensity the achieved accuracy was not better than in former experiments. With the development of pulsed dye lasers, light sources tunable from the near ultraviolet to the far infrared became available. The bandwidth ofthese lasers can be reduced to 0.07 A or less using frequency selective devices inside the cavity. Thus specific atomic and molecular fine structure levels can be excited with an intensity sufficient to saturate the levels. If a delayed obser-

26


vation is performed, after eight lifetimes, a fraction of 3 X of the atoms is still in the excited state. This is about the same portion of excited atoms as obtained with classical light sources immediately after the excitation pulse. Figger and Walther ( 1974)performed a similar experiment to that of Copley et al. ( 1968)on 23Na.However, they used a nitrogen pumped dye laser for the excitation of the Na atoms. The fluorescent light was observed in time intervals which were initiated up to seven lifetimes after excitation. Therefore the signal was only determined by atoms having survived in the excited state up to the initiating time. The minimum linewidth observed was six times smaller than the natural width. Neighboring crossing signals which overlap in a level-crossing experiment using the time integral observation of the fluorescent light were resolved, and the hyperfine constants of the 3 2P3,2 level were improved. This experiment used an “on-line” electronic apodization technique to eliminate the spectral sidebands inherent in this kind of delayed detection technique. We return to this point in the next section. Another experiment worth mentioning is that of Champeau et al. (1978), who combined the techniques of two-step laser excitation and delayed detection to measure the fine structure constant of the 3 2D level of lithium. An important question is to determine if anything can be gained by discarding data, as is done in such time-delayed spectroscopic techniques. In general, there is indeed nothing to be gained by discarding data, as long as the information is fully understood. But spectroscopists never have complete information about their signal shapes, and therefore selective (and biased) deletion of the data can be of great help and may even be necessary. Metcalf and Phillips ( 1980)have shown that, despite the loss of signal associated with time-delayed detection, it may still prove very useful in a number ofapplications.

B. LEVEL-CROSSING SPECTROSCOPY In the Hanle experiment (Hanle, 1924) the depolarization of scattered light by a magnetic field is proportional to the atomic lifetime of the excited state. A variation of the “Hanle effect” is the level-crossing experiment of Colgrove et al. (1 959) (see also Franken, 196 1). They measured the intensity of the polarized scattered radiation emitted in a Zeeman transition as a function of an applied static field and observed a sharp resonance when the magnetic field produced a degeneracy in the atomic frequencies. During excitation a coherent mixture&) = C,(l) C212) oftwo excited states 1 and 2 is produced. When the two states are Zeeman substates differing in their magnetic quantum numbers by +2, this coherent mixture is produced by excitation with light which is linearly polarized perpendicularly

+


27

to the external field. A single photon produces a coherent superposition of the two states, whereby excitation is started from the same lower level. At small magnetic fields where the two coherently excited states have equal energies E , and E2 within the limit of the natural width, the condition IE, - E21 < l/z is fulfilled. In this case the phase relation for the wave function under the continuous excitation process will be conserved when the atoms or molecules decay to the ground state, so that the angular distribution of the reemitted radiation corresponds to that of the absorbed radiation. However, when IE, - E21 > l/z, the change of the wave function is so rapid that the phase relation produced during excitation is destroyed for the ensemble and the fluorescence is isotropically emitted. Under the condition that the fluorescence is observed perpendicularly to both the external magnetic field and the excitation direction, the signal shape is an inverted Lorentzian where the minimum is observed for E , = E2.The halfwidth of the Lorentzian is determined by the natural bandwidth. The change in angular distribution of the fluorescent radiation observed in the region around El = E2 is independent of whether the “level-crossing” El = E2 occurs close to zero magnetic field (the so-called Hanle signal, Mitchell and Zemansky 1934; Corney, 1977), or at higher fields. The effect can therefore be used to determine the crossings which belong to Zeeman substates of different fine structure or hyperfine structure states. In addition, it should be mentioned that the effect can also be observed when an external electric field or a combination of magnetic and electric fields is applied. When level-crossing signals at higher fields are observed, the method, in principle, compares the magnetic or electric splitting with that at zero field. It therefore provides the parameters of the zero-field splitting in a ratio to the field-induced energy change, which must be determined by other methods, e.g., the optical radio-frequency double resonance method, in order to obtain the zero splitting absolutely. Using the resonance fluorescence of atomic vapors, this method has been extensively applied to the measurement of the hyperfine interaction constants of many excited states. The precision of these measurements depends on the overlapping natural widths of the excited energy levels concerned. The width of the level-crossing curves and the precision of the measurement are thus determined by the natural widths of the excited states. In order to narrow the level-crossingcurves, Copley et al. ( 1968)suggested observingthe fluorescent light not from all the atoms of the vapor, but from a sample biased toward those which have survived as excited atoms for a longer time than the mean lifetime (Copley et al., 1973; Corney and Series, 1964). To describe the delayed level-crossing method we consider an atomic system depicted in Fig. 5 (Knight, 1981). The atom is excited by a short pulse of duration At r,), it experiences a purely Coulombic attraction. At any energy two independent solutions exist to the radial Coulomb Schrodinger equation, one of which is well behaved and regular at r = 0 while the other diverges and is accordingly irregular at r = 0. The radial wave function of the outermost electron must then reduce to a linear superposition of regular and irregular Coulomb wave functions (f;g),

r > r, y ( r ) = f ( r ) cos I I ~- g(r) sin np, (1) The constant p is called the quantum defect and is the usual scattering phase shift 6divided by II.In scattering theory one is usually interested in the phase shift at positive energies (E > 0) only, but in quantum defect theory the expression [Eq. ( 1 )] is meant to apply equally at E < 0, wherep then provides information about the bound-state spectrum. The underlying physical element which makes the quantum defect approach useful is the near energy independence of any property of the system (such as the quantum defect p ) determined at small radial distances. This

QUANTUM DEFECT THEORY

55

follows immediately from the strong attraction prevalent at small distances; that is, the local kinetic energy at r 4 ro is large even at and below the ionization threshold because of the deep Coulombic well. At large distances instead the wave function changes dramatically from an oscillatory character just above E = 0 to an exponential decay just below E = 0. Fortunately this energy dependence stemming from large radii is given in closed analytical form in terms of well-known, tabulated properties of Coulomb wave functions (Seaton, 1966). For example, the asymptotic form of the energynormalized regular and irregular solutions is

+ (zrn/k)In r + q] - ( 2 r n / ~ k ) ' /cos[kr ~ + (zrn/k)In r + q]

f(~, r) g(E,

r)

at positive energies E

f(~, r)

( 2 r n / ~ k ) sin[kr '/~

= k2/2rn

> 0, and

(rn/nK)1/2[sin/3 (D-lrrrnlKeKr ) - cos p (DrzmlKe-w 11 (3)

- (rn/n~)'/*[~os p (D-lr-zm/KeKr ) + sin p (Dr*mlKe-Kr)] at negative energies E = - ~ ~ / 2 0 in Eq. (9)]; this is also the number of linearly independent solutions Yi. of the Schrodinger equation at the total energy E of interest obeying regularity boundary conditions at the origin. But while these No solutions and especially their asymptotic forms completely characterize the scattering information, they depend strongly on the energy for two reasons: 1. The scattering matrix has pole-type structures in the vicinity of autoionizing and predissociating resonances. 2. Even away from resonances the eigenphase shifts of S ( E ) are highly energy dependent, particularly close to fragmentation thresholds, because of the phase shift contributed by the long-range field [see Eq. (4)].

The main result of Seaton’s quantum defect theory (1966, 1983) is the parametrization of the complicated, energy-dependent S matrix in terms of a smooth, short-range reaction matrix K and standard parametersB(E) and ME) characteristic of the long-range potential. In many applications K can even be taken independent oftheenergy,and for this reason it lends itselfto a direct physical interpretation far more readily than does the scattering matrix. The reaction-matrix representation of the N independent solutions for an N-channel problem utilizes (A,gi),two independent solutions to the outerfield Schrodinger equation in channel i. Specially, the independent solutions outside a “reaction surface” at r = ro must be the generalization of Eq. (1):

Ti@)= A

N

Qi(w)[f;(r) Sii, - gi(r)K,,.],

r > ro

(10)

i- 1

Note that the independent solutions of Eq. (10) are quite different from the S-matrix representation, Eq. (8). For one thing they are real, with Kii.symmetric. For another, there are more independent solutions (a total ofN) than the number of open channels (No).This reflects a key (and initially surprising) feature of the reaction-matrix representation: The boundary conditions at r + m have not yet been applied, whereby the N, = N - Noclosed channel components ( i = No 1, . . . , N) are exponentially divergent. The N X Nsmooth reaction matrix K contains the scattering information in slightly disguised form. It characterizes the large-r form of Nindependent solutions having an unacceptable divergence. In constructing physically acceptable solutions we utilize the fact that the form of this divergence is specified precisely by Eqs. (3) and (10). This allows us to find a new set of solutions Y j ( E ) ,withj = 1, . . . ,No, which are linear combinations of the Y,(E) in Eq. (10) that remain well behaved at r - m. This procedure is usually referred to as the “elimination” of the closed channels from the wave

+

58

Chris H . Greene and Ch. Jungen

function (see footnote 1 for details). The result is an No X No open-channel reaction matrix

K ( E ) = Kw - K"[K"

+ tan B(E)]-lKCo

( 1 1)

in terms of the parts of the original smooth reaction matrix referring to open and closed channels at a given energy E,

The B(E) in Eq. ( 1 1) is a diagonal N, X N, matrix whose elements are the negative energy phase parameters [Eq. (4)] in each closed channel, which diverge at each ionization threshold. This simple result gives in algebraic form all of the complicated energy dependences which are associated with closed-channel resonances. From the point of view of ab initio calculations, this is of considerable practical importance since the calculation can be confined to within a small volume of configuration space and to a coarse energy mesh, typically E 2 1 eV, yet it is capable of representing sharp resonance features in a scattering (or photoabsorption) experiment on an energy scale of cm-' units or less. It is also of practical importance for semiempirical attempts to account for experimental observations, which can thus be accomplished by fitting to (nearly) energy-independent quantities such as K. It may be useful to give the correspondence between our notations and the matrices introduced by Seaton (1 983). Thus our No X No reaction matrix K ( E ) in Eq. (1 1) is denoted R by Seaton, who called it the reactance matrix. The short-range N X N reaction matrix K is denoted 3" by Seaton. Our N X N frame transformation matrix U is denoted X by Seaton, and for our "short-range scattering matrix" (see footnote 2, p. 101), Seaton uses the symbol x. Lastly, Seaton's No X No matrix T coincides with our open-channel eigenvector matrix given in Eq. (20).

C. THEEIGENCHANNEL REPRESENTATION Fano ( 1970, 1975) has singled out the eigenvalues and eigenvectors of the reaction matrix as having special physical significance. These, denoted tan qua and U,, respectively, are found by diagonalizing K:

K = U tan(np)UT (13) where a superscript T has been used to denote the transpose of U. The eigenchannel wave functions yl,(E),a! = 1, . . . ,N, have a common phase shift npa, often called an "eigenquantum defect," in each of the fragmenta-

59


tion channels i. [In this article Greek indices (a,. . .) always refer to the eigenchannels, while italic indices (i, j , . . .) refer to the fragmentation channels.] The independent eigenchannel solutions have the following form outside the reaction volume:

In matrix notation these eigenchannels are related to the solutions ‘Pi@) in the reaction-matrix representation by

W=wCOS R/l

(15)

Here w and Y are square matrices whose rows represent the different fragmentation channel components i, and whose columns represent separate independent solutions of the Schrodinger equation at r > ro. The elimination of divergent closed-channel components is quite analogous to the procedure leading to Eq. ( 1 1) above. Any allowed solution must be a linear combination of the eigenchannels,

with the coefficientsA,(E) to be determined by large-r boundary conditions. The superposition [Eq. ( I6)] must decay exponentially in every closed channel (i E Q), while each open-channel component ( i E P)will be required to have a common eigenphase shift RZ. These two conditions when combined with Eqs. (3) and (4) imply the linear equations [see Lu (1971)l N

where

{U,

uia W P i

Fia(E) =

+~ a ) ,

sin n(- z +pa),

+

iEQ iEP

sin picia cos PisM, - sin w7Cia cos nr§ia,

+

iEQ

(18)

iEP The second set of equations, for example, with 43, = Viacos npa, will be used below. This homogeneous system of equations has a nontrivial solution only if det(Fia) = 0, which can be satisfied by No values of the “collision eigenphase shift” n.s,,(E) and No solution vectors A,@) of Eq. (17), with p = 1, . . . , N o . The resulting “collision eigenchannel” solutions have the following form in the fragmentation zone ( r > ro),

Chris H. Greene and Ch. Jungen

60

where the orthogonal matrix T(E) is given in terms of the normalized A, by N

Tip(E)=

a- 1

UIa cos n(-

Tp

+ PaMw

and their normalization is now specified according to

The matrix T is unitary as can be seen as follows. By forming the complex superpositions of the collision eigenchannel solutions y/,(E),

we find after some manipulation a wave function with the asymptotic form of Eq. (8), with the elements of the scattering matrix given by

From this expression we see that the matrix T is in fact the matrix of eigenvectors of the scattering matrix and therefore is unitary, and in fact if a real normalization is adopted, it is a real orthogonal matrix (i-e., T-l = TT).' In applications to date using the eigenchannel formulation of quantum defect theory, the linear system of Eqs. ( 17) and (1 8) is solved by iteration. That is, a search is executed for values of T at which det{Fia) = 0. This The orthogonality of T can also be verified directly from the expressions of Eq. (20). We present the considerations here for the reader interested in details because they permit at the same time a verification ofthe form ofthe open-channel reaction matrix K(E)given in Eq. ( 1 1). First, we reexpand the channel coefficientsA,, in terms of new coefficients

I: U,,,cos np,,A,, N

Z, =

a- I

The linear system [Eqs. (17), (18)] then takes the form (tan

[!AT}

a,,.

+ Kii,)Z,, = 0,

for each i E Q, P

where the reaction matrix K is related to U andp by Eq. (13). Second, we partition the system into two parts referring to closed and open channels, respectively,

rani L

Km tan(- ;T:

+ Km] [E:]

The "closed" portion of the linear system is now used to express the N, coefficientsZ c in terms of the No coefficients Z o according to Zc = -(tan

B + Km)-lKmZo


61

iterative calculation is needed because the “eigenvalue” z enters F, nonlinearly through a trigonometric function. One drawback of this iterative calculation is that the linear algebra cannot be solved “automatically,” using standard computer programs. It is possible neverthelessto put this system of equations into the standard form of ageneralized eigenvalue problem for tan atpwhich is convenient for numerical applications; this has not been used in the previous work, e.g., of Jungen and Raoult (198 1):

TA = tan at AA

(23)

with

and

A,

=

{O’U ,

iEQ iEP

cos ap,, This eigenproblem (Wilkinson, 1965) has No nontrivial solutions also since the rank of the A matrix is No. Standard routines are available for its efficient solution. Equations (23)-(25) apply also when all channels are closed, except that the homogeneous system, Eq. (23), then possesses a nontrivial solution only at certain discrete energy levelsEn.These must be determined in general by a numerical search. The concept of “energy-normalized” wave functions is no longer appropriate in this purely discrete regime, where the wave functions should be normalized to unity. In terms of an unnormalized solution vector A, of Eq. (23), the normalization integral is (Lee and Lu, 1973; Greene, 1980): N: = a-’

c [A, uj,

COS(P,

+ ap,)]

i,a

(26) d X - AanU,, sin(P, ap,,) IdE 1E-E“ The main energy dependence contributing to the derivative in Eq. (26)

+

(I’

Insertion of this expression into the “open” portion ofthe system leads immediately to the form of Eq. ( 1 I ) for the No X No open-channel reaction matrix K(E),which is symmetric. Finally, we verify from the linear system, Eqs. (17), (18), using the expression, Eq. (20), for T and the definition of the coefficients Zi, that

Z & = Ti, cos mp The orthogonality of T now follows immediately from the orthogonality of Zo.

62


comes from the long-range phase parameter pi defined by Eq. (4)for a long-range attractive Coulomb field. To a good approximation then, particularly for a high Rydberg level, this result simplifies to

D. PHOTOFRAGMENTATION CROSSSECTIONS

-

Application of the finitenessboundary condition at r 03 has reduced the number of physically relevant stationary state solutions wPat any energy E to No, the number of open channels. Various photoabsorption experiments performed using photons at a definite energy access specific linear combinations of these independent solutions. Scattering experiments can be used to study different linear combinations. Here we summarize the formulas which describe photoabsorption from a single bound state having total angular momentum Jo. In the electric dipole approximation the information required to calculate a photoabsorption cross section consists of matrix elements of the molecular dipole operator i * r , with E^ the photon polarization vector. We assume the incident light to be linearly polarized. Other possibilities are easily treated by the same methods. The initial and final stationary states, yo and w, will be assumed to have definite angular momenta Jo and J, respectively, which is permissible since J commutes with the field-free molecular Hamiltonian. Consequently the dynamics is isolated in Nreduced dipole matrix elements = (aJII r(l)llJo),using notation and conventions ofSobel’man for each J, D‘,“) (1972). Because yois typically limited to a small spatial region where ty, is a slowly varying function ofenergy, these matrix elements are likewise smooth in E. (Photoabsorption by an initial Rydberg state as occurs, for example, in a multiphoton experiment, represents a notable exception.) A smaller subset of No matrix elements D$J)is obtained using the eigenvector A, from Eq. (23h N

DLJ)(E)=

I]DLJ)Aw(E) a- I

(28)

Of course these “collision eigenchannel” matrix elements can vary rapidly with energy, since they incorporate resonance effects. The incoming wave boundary condition, described thoroughly by Starace ( 1982), must be satisfied when photofragments are observed in one channel i. In particular the real wP must be superposed to eliminate all outgoing


63

spherical wave components in channels i’ # i. This is accomplished by the complex superposition

which is thus connected to the initial state by a reduced dipole matrix element

The partial cross section (in a.u.) for photofragmentation into channel i is then an incoherent summation over all final-state angular momenta:

with a the fine-structure constant and o the photon energy in a.u. The total cross section is a sum over all of the No a, at each energy, and reduces to the simpler form involving the real DbJ),

Note that these expressions are applicable to either photoionization or photodissociation, or to both competing processes at once. In addition the dipole matrix elements of Eq. (30) serve as input into theoretical formulations of other observable properties such as the photofragment angular distributions (Dill and Fano, 1972; Fano and Dill, 1972), fragment spin polarization (Lee, 1974b), or the alignment and orientation of photofragments (Klar, 1979, 1980; Greene and Zare, 1982). The calculation of various observables in the photoionization of rare gas atoms has been reviewed by Johnson et al. ( 1980). E. PHYSICAL SIGNIFICANCE OF THE EIGENCHANNELS Strictly speaking, the reaction matrix representation, Eq. (lo), and the eigenchannel representation, Eq. ( 14), of independent solutions are equivalent, and related linearly by Eq. (1 5 ) . Yet the eigenchannels often have a simple physical significance,whose systematic exploitation can be extremely helpful. In particular, if the short-range Hamiltonian is approximately diagonal in some simple, standard representation, then the short-range reaction matrix ought to likewise be nearly diagonal in that representation. This follows from the (heuristic) quantum defect view of a scattering process

64


which is depicted schematically in Fig. 2. Consider a specific transition which results after an electron collides with the ionic core, causing a transition from a rovibrational state i of H2+ to a final state i’. Because r, was chosen such that this transition cannot take place at radii r > r,, it occurs as follows: First, an electron comes in toward the H2+ ion along the long-range potential Vi(r) which convergesat r 00 to a corresponding level Ei of H2+.It remains on this potential until it reaches r = r,, the edge ofthe reaction zone. Within the reaction zone the motion is complicated and not described by any local potential, but the net result of this collision with the core at r < r, is that the electron can be scattered and re-emerge from the core in a diflerenf long-range potential Vit(r).It can then move outward to the detectors at infinity along this potential. Ifwithin the core there is no interaction between wave functions with the channel structures i and i’ (i.e., if the Hamiltonian matrix element Hiit= 0), then clearly there should be no scattering from i to i’ within the core (i.e., Kii.= 0). Thus diagonality of the short-range Hamiltonian in a standard representation implies that K should also be diagonal in that representation. This simple concept can be developed considerably further. Thus, since the scattering eigenchannels (a)are identical to the eigenstates of the shortrange Hamiltonian, the same orthogonal transformation Uiareduces both K and H to diagonal form. Put another way, the matrix element Viashould be

0 FIG.2. Potential and kinetic energies of an electron outside an ion core. The total energy E corresponds to a situation where the electron is bound with respect to the three ionization limits E , , E2, and E,.


65

interpreted as the projection of a fragmentation-channel wave function Ii) onto an eigenstate la) of the collision (or of the short-range Hamiltonian). For example, in electron scattering by a core having a spin-orbit splitting, the fragmentation channels i must be characterized in j j coupling, since the asymptotic electron wave vector kiis different for the different core statesj,. On the other hand, the predominance of the exchange interaction in the short-range dynamics implies that the eigenchannels (a)are LS-coupled solutions. A factor of the frame transformation matrix U, should thus be given by the standard recoupling coefficient (jjlLS).Examples of this “finestructure frame transformation” have been most extensive in the contexts of atomic photoionization and negative ion photodetachment, as in the studies by Lee and Lu ( 1973), Lee ( 1979, Johnson et al. (1 980), and Rau and Fano (1971). Perhaps even more striking applications of the frame transformation viewpoint are the treatments of vibrational and rotational interactions in diatomic molecules (Fano, 1970; Atabek et al., 1974; Jungen and Atabek, 1977; Dill and Jungen, 1980; Jungen and Dill, 1980). Here the fragmentation channel labels (i) are just the ionic rotational and vibrational quantum numbers (u+, N + ) , since together they identify the energy of a distant electron. The crucial, nontrivial idea which permits a surprisinglysimple formulation of complex rovibrational interactions among Rydberg levels has been are (at r < ro)precisely the realization that the short-range eigenchannels (a) the wave functions obtained in the Born - Oppenheimer approximation. This is ensured by the small electron -nucleus mass ratio (Born and Oppenheimer, 1927) or by greatly different time scales in other contexts (Chase, 1956). Moreover, the short-range electronic phase shift p(R) is generally a slowly varying function of the energy, whereby the electronic time delay is negligible on the vibrational time scale. In this event the electron experiences the instantaneous field of the nuclei, which can then be regarded as frozen in this while the electron moves within the core. The eigenchannels labels (a} case are then (R,A), A representing the projection of the electronic angular momentum onto the internuclear axis I?, with R representing the internuclear separation. This deceptively simple statement should not be confused with the conventional Born - Oppenheimer approximation, which assumes that the nuclei are frozen in space for all electronic radii r, and which clearly fails to describe molecular Rydberg states properly. The eigenquantum defects in this case are Mulliken’s p,,(R) (1969), and they can be calculated from the (known) Rydberg-state Born - Oppenheimer potential curves U,(R): The reaction matrix K (e.g., a 20 X 20 matrix for the H, npA, J # 0, states if 10 vibrational channels are included) would be essentially impossible to obtain without this simplification. Besides this “practical” significance ofthe eigenchannel formulation, it also shows that the eigenchannel parame-

66


tersFa and U , and the state vectors have a clear physical significance- in this case coinciding with the fixed-nuclei parameters and wave functions.

111. Rovibrational Channel Interactions A. ADAPTATIONOF THE QUANTUM DEFECTFORMALISM TO MOLECULAR PROBLEMS

The photoabsorption spectra of diatomic molecules in the vicinity of numerous rovibrational thresholds can be quite complicated. In a zerothorder picture of the spectrum, one might imagine a simple Rydberg series of levels converging to each ionic threshold from below, with a smooth adjoining continuum above each threshold. While simple, this picture is not even qualitatively correct. The actual spectra are dominated by severe perturbations of level positions and intensities, and the photoionization continua display broad and narrow autoionization profiles interwoven in a complex fashion. We discuss in this section the adaptation of quantum defect theory which has accounted for these complex spectral features to a remarkable extent in H, and Na,. As indicated above in Section II,D, the fragmentation channel labels for this problem are ( i ) = (u+, N + ) ,and the complementary eigenchannel labels are (a)= (R, A). Each element of the frame transformation matrix Viaof Section I1 consequently factors into a product of two projections

uia uu+~+, R A = (V+lR)(N+)(N+IA)('J)

(33) The factor (v+IR)(") is the vibrational wave function corresponding to a of the molecular ion in its ground electronic state. (Interspecific level Eu+N+ actions among different electronic channels are discussed in Section IV below.) The rotational factor of the frame transformation in Eq. (33), denoted (N+lA)("),transforms a state from the Hund's coupling case relevant within the reaction zone at small r (typically case b) to the Hund's coupling case relevant at large r (case d). (Different Hund's coupling cases are described by Herzberg, 1950.)If the ground ionic state is a Z+ state, as is true for H2+, this factor is given by (N+IA)(lJ) = (-

[2/( 1

dAo)]"2(N+oll- A, JA)

(34) This result and its generalization to non-X+ core states are derived by Chang and Fano (1972). In Eqs. (33) and (34) lis the orbital angular momentum of the outermost electron. As it stands, Eq. (33) presumes that spin effects are negligible and that only one l has appreciable amplitude in the fragmentation 1)J+A-N'

67


zone. This is an excellent approximation in the case of the npa and npn Rydberg levels of H,, but for other symmetries and for many other molecules an expansion of the outer wave function in I is often needed. (Even in H,, for example, s-d interactions may be significant.) When this more complicated situation occurs, the full-frame transformation matrix, Eq. (33), must of course be modified to reflect the fact that the orbital momentum of the outer electron can be changed when it collides with the core. Moreover, this full-frame transformation matrix is no longer a purely geometrical quantity, but contains dynamical information which may require a separate calculation. Likewise ionization thresholds may exhibit splittings due to spin -orbit coupling, as in the rare gases, or to spin -rotation coupling (e.g., in a Z ionic core). This results in additional channel interactions not considered in this review. The eigenquantum defectspa are related to the usual Born-Oppenheimer potential energy curves of the neutral molecular Rydberg states, U,,(R), and the ground-state ionic Born - Oppenheimer potential U+(R)by U,,(R) = U+(R)- (2[n - P*(N12)-1 (35) All energies in Eq. (35) are given in atomic units. Some nA potential curves of H2are given in Figure 3a to show the range of energies and internuclear radii of interest. Despite the quite different shapes of these potential curves, they are accurately described by Eq. (35) withp,,(R) independent ofn. As Fig. 3b shows, a residual n dependence is in fact present, particularly at IargeR for the lowest Z state, but for our present discussion we will neglect this bodyframe energy dependence. Through Eqs. (33) and (35) the short-range quantum defect parameters are thus known, whereby the treatment of Sections II,C and II,D can now be ' -1 H'+ H(ls) e

n.2.

R (a

u)

-05 0

I

I

2

I

I

4

1

4

6

I

R h u )

FIG.3. Potential energy (a) and quantum defect (b) curves for the lowest ungerade singlet Rydberg states of molecular hydrogen.

68


used to find the energy levels and the photoabsorption spectrum of the molecule. The actual implementation of this procedure involves an additional complication not anticipated in Section 11, associated with the fact that the eigenchannel index (a)= (R, A} is partly continuous. Consequently the system, Eq. ( I7), is apparently underdetermined, as it involves an infinite number of unknowns A,, but only a finite number (N) of equations. This might be resolved in a variety of ways, but the most straightforward approach is to represent the A R A themselves as a finite superposition of rovibrational channel functions with unknown Bi =

In practice any orthonormal vibrational basis (RI v) can be used in the superposition [Eq. (36)] to express the problem in terms of unknown coefficients BUN+, By a suitable choice of the vibrational basis it is possible to bring the system Eq. (17) [or Eq. (37) below] into nearly diagonal form locally, in a given energy range. This circumstance has been exploited in the applications to low Rydberg states which are described in Section III,D,l below (see Jungen and Atabek, 1977). With any such choice this modifies the system [Eq. (17)] to the form N

2 Fii@)B,(E)

=0

(37)

i'- I

with Fij.(E)=

+

sin PiCiit cos pi§ii., -sin n7Cii, cos nrSi,,,

+

i€Q

i EP

(38)

Equation (28) is also modified and takes the form

with

(39)

With Eq. (36), the matrix elements C,, of Eq. (38) take the form

and an analogous expression applies for §,i, with sin np. The reduced dipole matrix elements DLJ) used in Eq. (39) have been

69


further expressed in terms of purely body-frame matrix elements dA(R)in Eqs. ( 1 7)-( 19) of Dill (1972). [See also Eqs. (18)-(23) of Jungen and Dill ( 1980).] The resulting expression is

DL’)

= (2J

+ 1)1’2d~(R)(~,IR)(’o’(AlJO)(L’)

(41) where J, and (u,lR) are the total angular momentum and the vibrational wave function of the initial state, respectively. The superscript 1 in the matrix element (AIJo)(lfirepresents the multipolarity of the incident electric dipole photon and not the orbital angular momentum of the escaping photoelectron. B. CHANNEL INTERACTIONS INVOLVING HIGHLYEXCITED BOUNDLEVELS 1. Rotational Perturbations in H2

The observation by Herzberg, in 1969, of high Rydberg levels of H, assohas ciated with series converging to alternative rotational levels of Hz+, historically been the first documented example of rovibrational channel interactions (Herzberg, 1969; Herzberg and Jungen, 1972). Figure 4 displays E (crn-I) 124,300

I ‘

22

1124,200

21 16

20 unidentified 1.0

0.5

PO

0

np0,v.o np2,v.O

FIG.4. Rotational perturbations between highly excited Rydberg levels in H,. The spectrogram is adapted from Herzberg and Jungen (1972) and representsabsorption to J = 1, odd-parity final-state levels. To its right are shown the (strong) np series converging to the v + = 0, N + = 0 level of H,+ and the perturbing(weak) seriesconverging to the v+ = 0, N+ = 2 level. The stick spectrum to the left of the spectrogram has been calculated using quantum defect theory. Far left: observed (circles) and calculated (full lines) quantum defects of the individual levels evaluated with respect to the v+ = 0, N + = 0 limit.

70


a section of the spectrum obtained by Henberg. By working with cold para-H, ,and owing to the fact that the H, ground-state rotational levels have a comparatively wide spacing, he obtained a gas sample in which virtually all molecules were in the (Jo=) J” = 0 (even-parity) level. Dipole absorption thus yields the excited system of J = 1 states with odd parity. The outer electron in H, has virtually pure 1 = 1 character and therefore only the N + = 0 and N + = 2 rotational levels of H,+ can be formed in photoionization under these circumstances. Correspondingly, the Rydberg series seen in Fig. 4 are associated with the u+ = 0, N + = 0 and 2 thresholds. It is however evident from Fig. 4 that strong perturbations occur which affect both level positions and intensities in the two series. These perturbations stand out in the level pattern drawn at the right of the figure where an attempt is made to associate each observed line with either the N + = 0 or the N + = 2 threshold. To the left of the experimental spectrum in Fig. 4 is shown a theoretical stick spectrum obtained by solving the linear system [Eq. (37)] with the matrix elements [Eq. (40)] and the analytic frame transformation [Eq. (33)]. The transition matrix elements have been evaluated using the united-atom approximation, by setting d,(R) = dn(R)= d. The quantum defect curves were initially extracted from the potential curves shown in Fig. 3a by use of the Rydberg Eq. (35) with n = 3 and 4, but a slight adjustment oftheir values (= 5%) was subsequently found necessary and reveals a slight energy dependence of the p’s. The good agreement between observed and calculated spectra underlines the correctness of the concepts outlined above in Sections II,E and III,A. Although the spectral range shown in Fig. 4 exhibits only lines associated with the interacting levels of the u+ = 0, N+ = 0 and 2 Rydberg series, the influence of vibrational interactions with channels u+ > 0 is not negligible: In order to attain the quality of the theoretical calculation shown (residual discrepancies of less than one wave number unit, see Jungen and Raoult, 1981 ), it has been necessary to include 10 vibrational channels in the calculation. 2. Lu - Fano Plols

From a more qualitative point of view we can nevertheless regard Fig. 4 as representing a two-channel, two-limit system. It is instructive to see how the level perturbations in the two Rydberg series affect the empirical quantum defect as given by Eq. (6). This is shown on the left of Fig. 4: Here the quantum defect pN+-ocalculated for each successive line with the known ionization limit E(u+ = 0, N + = 0) is plotted. (Historically the inverse procedure has been followed since the multichannel treatment of the rotational interactions was used to determine the ionization potential of H, .) The plot

71


does not distinguish between levels associated with the N + = 0 and 2 limits since strictly this distinction is meaningless. The overall effect of this procedure is that each time a level belonging in a first approximation to theN+ = 2 series occurs, the quantum defect P,+-~ rises by unity. This arises because one level “too many” has been counted in the N + = 0 Rydberg series so that the quantum defect must also increase in order to keep v = n - p constant. The interesting fact displayed in Fig. 4 is that in this way the level perturbations can be made to stand out very clearly. Since the “extra” N+ = 2 level interacts with the nearby unperturbed N + = 0 levels and pushes them away from their unperturbed positions, the increase of the quantum defect is not just a step function but exhibits a characteristic dispersion pattern. The stronger the interaction, the broader is this pattern. It is repeated each time a zeroth-order N + = 2 level occurs. Further, it can easily be appreciated that the dispersion patterns, if plotted with respect to the effective quantum number v,+-~ = [2E(v+= 0, N+ = 2) - 2E]-’12,will occur in periodic intervals as do the zeroth-order N + = 2 levels. The dispersion curves obtained in this way can finally be telescoped into a single unit square by plotting the itself. The result of these points versus v,,,+-~ (modulo 1) instead of v,+, operations is shown in Fig. 5, where it can be seen that all observed points now come to lie on (or close to) a single cuwe. This curve then represents in condensed form all of the perturbations arising from the interaction between the two channels. This type of diagram has been introduced by Lu and Fano (1 970) and has

- 0.5

0

0.5

v2(mod I )

FIG.5. Lu-Fano plot representing the perturbations shown in Fig. 4.

72


both practical and theoretical significance. The use of Lu-Fano plots assists the experimentalist in the spectral analysis of perturbed Rydberg series; it helps to detect channel interactions and to assess their strength on an absolute scale without any calculation effort. Numerous Rydberg levels can be represented graphically in compact form. Theoretically, the Lu - Fano curve represents the solution of the linear system, Eq. ( 17), for a bound two-channel, two-limit system, namely,

[

+

det cos 8 sin a(v, p , ) -sin8sina(v2+pl) cos 0 = U , , = U22,

+

sin 8 sin n(vl p2) cos8sin n ( v 2 + p 2 )] = o

(42)

sin 8 = UI2= - U2,

The effective quantum numbers v, and v2( v ~ + -and ~ v,+,~ in the example of Fig. 5 ) are regarded here as variables, disconnected from the channel electron energies E , and c2in terms of which they were originally defined through

E = E i + e i = E i - 1/2~: (43) Read in this way, Eq. (42) and the equivalent graphical curve depend just on the dynamical parameters pa and U,, which are nearly independent of the energy, while Eq. (43) includes the strong energy dependences. In other words, the Lu-Fano plot can be viewed as an elegant graphical way of removing the boundary conditions on the wave function at infinity inasmuch as they govern the detailed level positions, and hence to make stand out the short-range dynamics common to all the levels. The detailed properties of Lu- Fano plots have been discussed by Lu and Fano ( 1 970) and by Giusti-Suzor and Fano (1 984). Here we mention only that the channel interaction is reflected by the departure of Fig. 5 from a step function, and that, as is clear from Eq. (42), the intersection points of the curve with the subsidiary diagonal straight line v1 (modulo 1 ) = v2 (modulo 1) give directly the values of pa, a = 1, 2. At low energy (small v values), observed levels will lie near this diagonal if the core levels are sufficiently closely spaced so that their differences can be neglected in Eq. (43) and hence v, = v2.These levelsconform to the Born -0ppenheimer approximation. At higher energy Eq. (43) is represented in Fig. 5 by lines with negative slope

0.8 0.4 -

H~fm*=lJcolc. I-

0

mou. 50 1

h

0.04

nfn=2I+nflr J co /c

122300

127,200

127,100 E (ern-')

127,000

12 6,900

FIG.8. Preionization and predissociation near the u+ = I , N+ = 2 ionization ' threshold ( 1 26773.6 cm-l) in H, ( J = 1, J" = 0). Top: observed photoionization spectrum of Dehmer and Chupka (1976). Bottom: calculated partial vibrational photoionization spectra and photodissociation spectrum (see text for details).

lopers" which are associated with higher ionization limits and correspond to n < 8. Note that all the assignments given in Figs. 8 and 9 have no strict meaning owing to the strong channel interactions which take place between all the channels; they are useful just for "bookkeeping" purposes since of course the number of levels occurring in the whole range is not changed by the couplings. The quantum defect calculation (Raoult and Jungen, 1981; see also Takagi and Nakamura, 1981) is based on exactly the same parameters and d used in the calculation described in Section III,B, 1. The calculation yielded all the partial cross sections point by point on a conveniently chosen energy mesh. These theoretical partial cross sections are shown in the lower part of Fig. 8 for each v+ final state, summed over the N + = 0 and 2 rotational contributions. Since Dehmer and Chupka (1 976) did not discriminate between the different photoelectron groups, they obtained the total photoionization cross section. More precisely, their spectrum gives the photoionization efficiency, that is, the ratio of the total ionization to the transmitted light, but this is very nearly proportional to the ionization cross section except near the very strongest resonances. For the exact comparison between experiment and theory, the theoretical partial cross sections should be summed up and convoluted with the experimental resolution width of 0.016 A. The result of this procedure at the wavelengths corresponding to the peak maxima is indicated in the experimental spectrum by horizontal arrows. Quite good agreement is found on


79

Fic. 9. Schematic illustration of vibrational-rotational preionization and predissociation in H, (J = 1, odd panty). Ionization and dissociation continua are indicated by vertical and oblique hatching, respectively. For each given u+ of the H,+ion there are two continua corresponding to rotational quantum numbers N+ = 0 and 2 of the ion ( J = 1). Selected Rydberg levels are indicated below the vibrational ionization limit with which they are associated.

the whole. (The theoretical spectrum is normalized to coincide with the experimental one at the 8p0, u = 2 peak.) Similarly, it can be seen that the continuum background is quite accurately reproduced; one must remember that the vibronic perturbations which affect the Rydberg levels are quite strong. These perturbations are not so obvious in Fig. 8, but their presence has been established empirically by Herzberg and Jungen ( 1972) by the use of Lu-Fano plots, and they are also well borne out in the channel coefficients [Eq. (37)]resulting from the calculations. The vertical arrows in the top spectrum of Fig. 8 mark the peak positions measured by Herzberg and Jungen in the absorption spectrum. Their wavelength determination is more accurate than can be attained in the photoionization experiment: It can be seen that the theoretical peak energiesgenerally agree better with these values.

80


Turning now to the discussion of the resonance widths, we remark that, both in the experiment and in the calculation, the broadest features correspond to the lowest vibrational quantum numbers. This means that the ionization process is favored when only a single quantum of vibrational energy has to be exchanged. Quantitatively this can be understood with reference to Eq. (40):Assuming a linear R dependence of sin np,(R) and cos np,,(R), and harmonic-oscillator vibrational wave functions, one obtains coupling matrices C, ,+, and S , , ,,,+,whichobey the selection rule Av+ = k 1 familiar from optical infrared spectra. In reality, vibrational anharmonicity as well as the interactions among the closed channels break this selection rule, but its approximate validity is still clearly borne out in the simpler portions of the H2photoionization spectrum, such as that shown in Fig. 8, where the resonances are relatively well isolated. The Av+ = k 1 selection rule as applied to photoionization spectra has first been discussed by Berry ( 1966) and Bardsley ( 1967a). Berry termed it a “propensity rule” because in a first approximation the strongest channel couplings (e.g., the off-diagonal elements of a reaction matrix) indicate the preferred decay paths: Indeed, Fig. 8 shows that the v+ = 1 ionization channel is calculated to carry the bulk of the ion signal arising from the resonance features. By contrast, the flat parts of the total cross section are shared by the ionization channels according to the Franck - Condon factors for excitation from the ground state of H,; they correspond more nearly to a “direct” photoionization process. An experimental test of these predictions has been made by Dehmer and Chupka (1 977) and more recently by Ito et al. ( 198 1 ) for a few selected resonances. The agreement between experiment and theory is again satisfactory. An example is furnished by the 6pn, v = 6, J = 1 (odd-parity) resonance at 752.866 A, which is reproduced in Figs. 10 and 1 1 . The observed (Dehmer and Chupka, 1977) and calculated (Jungen and Raoult, 198 1 ) final vibrational-state distributions in photoionization are

Theoretical (%) Experimental (%)

v+ = 4

v+ = 3

82 82

16 15

v+

=2

2 3

Close inspection of Fig. 8 reveals some details which have not been accounted for in the discussion so far. For one thing, in the excitation energy range corresponding to the figure, the H2molecule is subject not only to preionization but also to predissociation. The latter process occurs within the p-type molecular Rydberg channel because the Rydberg electron might transfer some of its energy to the core vibrational motion, enabling the molecule to dissociate into atoms. This is the exact opposite of vibrational

81


preionization, where the electron picks up energy from the core. Energetically this is possible since the dissociation limit lies lower than the ionization limit (cf. Fig. 3). Figure 8 includes at the bottom the theoretical photodissociation cross section, calculated using an extension of quantum defect theory which will be discussed in Section III,D,2 below. Notice that only resonances corresponding to high vibrational quantum numbers and low principal quantum numbers are noticeably affected by predissociation; that is, these are the levels whose “vibrational energy” most closely approaches that of the vibrational continuum state leading to dissociation. Thus we see that predissociation in the present situation conforms to a rule similar to the Au+ = 1 rule for preionization: In both cases the preferred decay paths are those which correspond to exchange of the minimum amount ofenergy. Note the analogy with the “energy-gap law” familiar from the theory of nonradiative transitions in larger molecules (see, e.g., Robinson and Frosch, 1963). The correctness of the quantum defect treatment of the competition between ionization and dissociation processes is demonstrated by the 4pu, u = 6 and the overlapping 3pn, u = 8 and 6pu, u = 3 peaks: These resonances absorb very strongly but are calculated and observed to appear only weakly in photoionization. (Note the very different scales of the dissociation and ionization cross sections in Fig. 8.) Figure 10 displays a more complex section of the photoionization curve of

1

-- -5 -

I

r6 p 6n

I

I

I

I

7

6PU v . 6 1

vr

1.0-

v,

$

.-0

9pn

- v=5 : I

;P!j .’

I

#

FIG.10. Preionization near the u+ = 4, N + = 0 and 2 thresholds in H, ( J = 1, J" = 0). The observed and calculated total oscillator strengths are shown as functions of photon wavelength. The experimental points from Dehmerand Chupka (1976) have been shifted by -0.068 A so as to bring the observed and calculated 9pq u = 5 peaks into coincidence. The calculated spectrum is broadened to a resolution of 0.0 16 A to correspond to the experimental measurements. (After Jungen and Raoult, 1981.)

82


H, corresponding to excitation wavelengths near 754 A. At this energy the u+ = 0 to 4 channels are all open; the u+ = 4, N + = 0 and 2 limits fall into the range shown and are indicated, but they are not directly visible in the spectrum as a threshold discontinuity since the averaged below-threshold resonance structure matches smoothly onto the continuum cross section (Gailitis, 1963). However, a new type of feature appears just below the u+ = 4, N + = 2 threshold. An interloper with low n and high u (6~17,u = 6) falls among the dense manifold of the high n/low u series (np2, u+ = 4). Interaction between these closed channels leads to a transfer of intensity from the strong 6pa, u = 6 line to the very weak np2, u+ = 4 lines. The remarkable result is the formation of a “complex resonance” whose width exceeds by far the widths ofits individual components. Dehmer and Chupka( 1976) in their experiment were just able to resolve some of the fine structure ofthe complex resonance (cf. the figure), but in an earlier spectrum (Chupka and Berkowitz, 1969) the resonance is not resolved and has an apparently perfect Lorentzian shape with a width of about 15 cm-’. As a result one might mistakenly take this global width as the one governing the preionization decay rate. It is interesting to examine the calculated partial cross sections near the 6pa, u+ = 6 complex resonance. They are shown in Fig. 1 1 . Obviously this is an instance where the Av+ = 1 selection rule, as applied to such processes, is bound to break down: As a consequence of the strong perturbations affecting the Rydberg levels, no definite vibrational quantum number u+ can be ascribed to any preionized level. We see indeed that the complex resonance is calculated to appear as such in the u+ = 4, 3, and 2 ionization channels, although with diminishing intensity. It virtually disappears, however, in the u+ = 1 channel where only the narrow central peak (labeled 6pu, u = 6) subsists. Complex resonances have been found not only in H, (Jungen and Dill, 1980; Jungen and Raoult, 198 1) but also in N, (Dehmer et al., 1984) and in atoms (Connerade, 1978; Gounand et al., 1983). We think that they are probably a more typically molecular phenomenon because in molecules, owing to the vibrational and rotational degrees of freedom, dense manifolds of levels are always present. We may suspect that many unresolved resonance features observed in molecular spectra at high energy are actually of this type. Giusti-Suzor and Lefebvre-Brion ( 1984) and independently Cooke and Cromer ( 1985) have used quantum defect theory to study analytically the simplest case of a complex resonance, involving two closed and one open ionization channel. They showed how the global or “effective” width of the resonance depends on the coupling between the closed channels. The significance of complex resonances, however, seems to lie more in the role they play in the competition between alternative decay processes. Indeed, the level scheme underlying the complex resonance, corresponding to an iso-

83


0.0.4

0.00 0.08

0.04

0.00

-->

0.08

I

0)

5

0)

0.0.4

c aI c L In

;0.00

I

,

I

I

I

I

1

I

I

I

I

I

I

1

1

1

1

m

z .u

1

1

1

1

1

1

1

1

1

-71

+-

0.8

n p 2 v =I nPO

u)

0

0.4

I 0.0

08

04

00

[I

-

max.x

I

I 753

'

75 5

754 Wavelength (

I

)

FIG.1 1. Calculated partial vibrational photoionization oscillator distributionsfor the spectral range shown in Fig. 10. Near the u+ = 4, N + = 2 limit the data have been averaged over the dense Rydberg structure arising from the np2, u = 4 (n > 36) Rydberg series (broken line). (After Jungen and Raoult, 1981.)

84

Chris H. Greene and Ch. Jungen

lated level facing a dense set of states, both being embedded in a set of continua, has been the building block for the description of the unimolecular decay of metastable molecules (see, e.g., Mies and Krauss, 1966; Rice et al., 1968; Lahmani et al., 1974). In that theory a time-dependent language is usually adopted and attention is focused, for example, on nonexponential decay patterns. The analog in the present context of complex resonances is their departure from Lorentzian shape: Note how in the example shown in Fig. 1 I the effective width is different by orders of magnitude depending on whether the u+ = 4 or the u+ = 1 ionization channel is monitored. 2. Photoelectron Angular Distributions The angular distribution of ejected photoelectrons provides an additional experimental handle on the short-range molecular dynamics of interest. For processes excited by an electric dipole photon linearly polarized along an axis i, the fraction of photoelectrons observed in a particular channel i along an axis k is proportional to 1 /3iP2(cosO), in terms of the second Legendre k. The quantity ofinterest here is polynomial P2(x);its argument is cos 0 = i* the so-called “asymmetry parameter” p. The importance of measuring p stems largely from the fact that it depends not only on the relative amplitude but also on the relativephaseofthe alternative partial waves lcomprising the final continuum state. The dynamic information needed to calculate pi is contained in the reduced dipole matrix elements D:!-)of Eq. (30). We will refer to this element here with the notation DIN-)to indicate its dependence on the orbital angular momentum 1 of the escaping photoelectron explicitly. This amplitude connects the initial state having angular momentum Joto an incoming-wave, energy-normalized final state of angular momentum J, where J = Jo,Jok I . These amplitudes correspond to an angular momentum coupling,

+

Jo +j, = J

= Nt

+1

(49)

The angular momentum contributed to the photoionization process by the incident electric dipole photon is designated by j ? = 1; that of the final ionic fragment state i is N t . In writing Eq. (49) the spin angular momentum s = f of the photoelectron and the spin Si of the molecular ion have not been written explicitly. This amounts to the assumption that these spins remain coupled throughout the ionization process to the value of the spin Soin the initial state. [More precisely, it presumes the relation So= S, s to hold in addition to Eq. (49).] The implications and limitations of this frequent assumption are discussed by Watanabe ef al. ( 1984). In small molecules it is

+


85

an excellent approximation except in very limited energy ranges, such as the range between two spin-rotation thresholds of the molecular ion. As mentioned in Section II,D, the reduced dipole amplitudes D Y - ) determine the asymmetry parameter Pi. This relationship takes the general form

as can be extracted from Jacobs ( 1972), Lee ( 1974b),or from references cited in these works. The symbol o( ) denotes a complicated geometrical quantity, involving Wigner 3j and 6j coefficients.Notice that the expression, Eq. (50), represents a coherent summation over 1 and J, which complicates the analysis and requires considerable attention to ensure that the phases of the amplitudes DIN-) are correct. In two papers, Fano and Dill ( 1 972) and Dill and Fano ( 1972) recognized that the coherence originates in the quantum-mechanical incompatibility of the operators l2 and J 2with the observable 0. Dill and Fano use an alternative set of amplitudes S,( j,) characterized not by J 2but by j:, where j, is the angular momentum transferred between unobserved photofagments. Since j: commutes with 8, amplitudes having differentj, then contribute incoherently to P. More explicitly this defines j, by

---

j , = N t - JO = j Y - 1

(51)

The second equality follows from Eq. (49), and implies that at most three partial waves 1 =j,, 1 =j , k 1 can be present for a given angular momentum transferj,. Parity conservation also implies that only even 1or else only odd 1 can contribute coherently in any given ionization channel i. The net result of the angular momentum recoupling is that the expression for /3 in terms of the Sl(jt)is now incoherent:

Here each j, is classified as parity-favoredor parity-unfavored according to whether j , - 1 is odd (k1) or even (0), respectively. For parity-unfavored transfers j,,

86


while for parity-favored transfers

In Eqs. (53) the S&) are abbreviations for S,,,(j,). Whereas parity-unfavored contributions to p are fixed at the value - 1, parity-favored contribu6 2. Finally, the amplitions can lie anywhere in the full range - 1 G pfav(jt) tudes S,(j,) have been expressed in terms of reduced dipole amplitudes by Dill ( 1973),

The preceding results were utilized by Dill (1972) to study the angular distribution of photoelectrons ejected from H, (earlier work by Buckingham et al., 1970, had introducedj, as a dummy summation index, but without the physical interpretation stressed by Dill and Fano). The dominance of the 1 = 1 partial wave in the asymptotic wave function of H, photoelectrons was pointed out above in Sec. III,B,I. If all other partial waves 1 # 1 are neglected, the asymmetry parameter pi appropriate to each ionic rotational channel N t can be evaluated explicitly. In particular, when the target molecule has J, = 0, only the two values IVt = 0, 2 are permitted by angular momentum conservation, with asymmetry

p ( N t = 0 ) = 2.0;

P ( N t = 2) = 0.2

(55)

independently of both the energy and the vibrational state of the ion. Table I compares the above predictions to some ofthe more recent experiTABLE I EXPERIMENTAL ASYMMETRYPARAMETERS IN H2PHOTOIONIZATION

I(nrn) P(N+ = 0, u+ 58.4

1.918"

73.6

I .903"

= 0)

Ruf et al. (1983). Pollard et al. ( 1982).

f l N + = 2, U + = 0 )

0.81 f0.17' 0.87 f 0. 19b 0.54 f 0.16' 0.08 f 0. I 5 b

87


mental results. While p values close to 2 are observed in the N + = 0 channel as expected, a sizable discrepancy is present in the N+ = 2 channel where p = 0.2 is not observed at either photon energy. This discrepancy is of considerable interest since it has immediate implications about the amplitudes of 1 # 1 partial waves in the final-state wave function. In view ofthe excellent agreement between theoretical (1 = 1 only) and experimental integrated cross sections, the deviation ofp(N+ = 2, u+ = 0) from has been a surprise. Indeed, a wide range of incompatible p values has been published for this channel (e.g., at 73.6 nm in Table I). (See Ruf et nl., 1983, references therein for some discussion of the past and present experimental situation.) Theoretical efforts to understand the unexpected departure of p from 4 have been undertaken by Itikawa ( 1979) and by Ritchie ( 1982), which consider an admixture of 1 = 3. In essence the origin for the difficulty of determiningp for this channel, both theoretically and experimentally, stems from the small value of the N + = 2 partial cross section. The dominant ( I = 1) oi experiences a substantial cancellation which reduces it to an order of magnitude smaller than the N + = 0 cross section; this can be seen from the (purely 1 = 1) calculations of Raoult et al. ( I 980). This opens the door to a possible influence byfwaves despite a complete absence of (observed) 1 = 3 states in the known discrete spectra of H,. The very recent results of Hara (1985) and of Hara and Ogata (1985) are in essential agreement with the measurements of Ruf, Bregel, and Hotop (1983). At 73.6 nm, for instance, these authors calculate p(u+ = 0, N + = 0) = 1.891 and p(u+ = 0, N+ = 2) = 0.643 (see Table I). Finally, there have been several calculations (Raoult et al., 1980; Raoult and Jungen, 1981;Jungen and Raoult, 1981) of the asymmetry parameter in the vicinity of autoionizing resonances in H,. Just as this spectral range shows rapid variations in the total and partial cross sections, the rotationally unresolved p oscillates in a complicated fashion also. To date no experimental work has been done in this range to test the sensitive predictions of quantum defect theory. The effects of electronic preionization on the p parameter have been studied in a fully ab initio approach by Raoult et al. (1983). This pilot calculation accounts successfully for p in the X- and A-state channels of N,+. This work is reviewed in Section IV,B.

+

D. TREATMENT OF A CLASSOF NON-BORN - OPPENHEIMER PHENOMENA In valence states of diatomic molecules the nuclei and the outer electrons are confined within the same volume of space, typically within a radius of a few Bohr radii. In addition the molecule is bound, implying that the forces

88


on the nuclei and on the electrons are of comparable magnitude. This standard argument immediately suggests that the time scale of nuclear motion is orders of magnitude slower than the (valence) electronic time scale, whereby the Born-Oppenheimer concept of defining an electronic energy U(R) at each internuclear separation makes good physical sense. A rough estimate of the validity of the Born - Oppenheimer separation of electronic and nuclear variables is given by the ratio of vibrational level spacings to the spacing between successive potential curves of the same symmetry [see Dressler (1 983)1, = A U(R)/A&ib.

(56)

When y is large compared to unity (e.g., y = 25 for the H2ground state with v = 0) the Born-Oppenheimer approximation is adequate, and nonadiabatic effects embodied in electronic matrix elements such as 1) are less sensitive to external field perturbation. The behavior of individual resonance states which are involved in the DR

+

THEORY OF DIELECTRONIC RECOMBINATION

157

process was investigated recently by Briand et al. (1984a,b) using the EBIS ion source for Ar13+.14+JS+.Although the cross sections inferred from this experiment are consistent with the theoretical estimate, this promising approach requires further improvements in accuracy.

IV. The Dielectronic Recombination Rate Coefficients Calculation ofthe DR rate coefficientaDR requires a theoretical procedure similar to that employed for uDRin Section 111. As noted in Eq. (65), the essential structure of aDR and the energy-averaged ZDRare the same; both require the excitation -capture probability V, and the fluorescence yield o(d). The degree of sophistication in the theoretical procedure one adopts and the number of intermediate states to be included are dictated largely by eventual usage of the rate coefficient. (1) For diagnostic purposes, one often studies a small group of satellite lines emitted by a plasma. The number of intermediate states involved are then small, and the necessary calculation of the energy levels A, and A, can be camed out carefully with high precision, incorporating the configuration interaction (CI) and intermediate coupling (IC) effects as well as other refinements. (2) For modeling purposes, however, a complete set of aDR is needed for each ionic species with core charge Z,, which is present inside a plasma, and for all possible degrees of ionization ZI.To reduce the calculation to a manageable level, drastic approximations are introduced even when treating a small subset of isoelectronic sequences as benchmark calculations. Eventually, a simple empirical formula has to be found that can generate all the rates needed. Complete calculation of aDR for a given isoelectronic sequence is often lengthy,due to the multistep nature of the DR process. Free-electroncapture to a double infinity of intermediate (resonant) states has to be incorporated. Further complicationsarise when these resonance states decay to final states which are themselves unstable against further Auger emission (i.e., the cascade effect). As a result, only a limited number of ions have been treated theoretically, and various semiempirical formulas are employed in practical applications. Burgess ( 1964b; Burgess and Tworkowski, 1976) proposed a phenomenological formula for ions of Z,% 20, where, at low temperature, the An, = 0 process is dominant. Merts et al. (1976) later modified the formula to incorporate the transitions which are important for Z , 3 15 and ZI b 10.An improved formula was recently proposed by Hahn ( 1980)based on a limited set of benchmark calculations. To examine the effectivenessof these formulas, it is necessary to obtain further benchmark cases. Thus far,

Yukap Hahn

158

the following isoelectronic sequences have been treated: N = 1, 2, 3,4, 10, I 1, and partially N = 12, 18, 19, and 5. Here Nis the number of electrons in the target ion before capture, N = 2, - 2,. We summarize below these results, with a discussion of features which are specific to each sequence. Generally, treatment of each sequence takes anywhere from six months (N = 1,2) to three years (N = I I , 12, etc.). The general procedure adopted here is similar but not identical to that adopted for ZDR.All the radial matrix elements for both radiative and Auger transitions are evaluated in the distorted-wave approximation using orbitals obtained in the nonrelativistic, single-configuration Hartree - Fock approximation and in LScoupling. The continuum function is generated using the direct local and exchange nonlocal potentials, which are constructed from the bound HF orbitals. Since there are in general a double infinity of resonance states to consider, we first evaluate the entire aDR in the angular-momentum-averaged (AMA) approximation. This initial step is much simpler than the calculation done in LS or LSJ coupling; typically 300 500 states are examined. It is well known, however, and is borne out by our extensive calculations, that the AMA approximation generally overestimates the aDR by roughly 50%. For some individual cases, this factor could be as large as 2, and some other cases even smaller than 1, but the dominant contributions are often increased in the AMA. The total sum of the aDR’sover many intermediate states is relatively stable against different coupling and configuration-mixing effects. Intermediate coupling generally gives values which are lower than those obtained in L S coupling, but this is not without exceptions (as in the Anl = 0 case). Out of an exhaustive and complete set of AMA calculations, we select a subset of dominant states which contribute at least 70% of the total AMA result, although this subset usually contains roughly a quarter or less of the total number of states. Therefore, the amount of computation involved is much less and manageable when this subset is treated in L S coupling. The total af,P in L S coupling is then estimated using a simple scaling formula

-

A. H SEQUENCE,N = 1

This is the simplest of all systems and naturally much work has been done (Dubau et al., 1981; Fujimoto and Kato, 198I ; Burgess and Tworkowski, 1976). A summary of the work was compiled by the Nagoya group (1982) earlier. The process involves only the Ana # 0 transition of the Is electron in


159

where no and nb 3 2. The final states reached from the intermediate states d by radiation emission may further decay by either radiation or electron emission, or both. In cases when electron emission is allowed, a cascade modification of the fluorescence yield o ( d ) for the d state should be incorporated. Agreement among the various calculations is reasonable, but none of the results is complete in the sense that not all the important intermediate states are considered, although Dubau's result is presumably the most detailed. The most dominant transitions involve the intermediate states d = 2snl and 2pn1, where nb=5 n 3 2. Details of the theoretical quantities are given by Dubau et al. (1981). As an important side remark, we should note here that the problem of electron capture to HRS accompanied by a core-electron excitation in this two-electron system is of fundamental importance in the general three-body scattering problem with Coulomb fields. In particular, collisional ionization in the threshold region has been the subject of intense research in recent years to determine the dominant energy dependence of the cross section. The in the case of the hydrogen target classical Wannier law ( 1 953) of o has been supported by a semiclassical treatment (Peterkop, 1983; Rau, 1983; Klar, 198 1). However, a purely quantum-mechanical treatment with full screening and polarization effects is more difficult (Temkin and Hahn, 1974; Temkin, 1982). Analytic continuation of the amplitude from below the ionization threshold requires careful handling of an infinity of overlapping Rydberg resonances. Similarly, oDRmay be continued to energies above the threshold and obtain oionK. The formulas for A, and A, for this two-electron problem (in LScoupling) are summarized in Appendixes B and C, respectively.

-

B. HE SEQUENCE,N = 2 This is the first sequence that is not hydrogenic and still simple enough to be treated in detail. The main difference between this sequence and the H sequence discussed above lies in the presence of an additional electron, which not only screens the nuclear charge but also affects the overall statistics. Several extensive theoretical studies are now available (Bely- Dubau er

160

Yukap Hahn

al., 1982a,b; Dubau et al., 198 1 ;Younger, 1983; Nasser and Hahn, 1983) in addition to a review article by Dubau and Volontk ( 1980).Within a spread of approximately f2096, all the theoretical work is in agreement for most of the ions with Z,'s, except for lighter ions where some discrepancies persist. On the experimental side, Bitter et al. (1979) deduced the DR rate for Fe24+(An,#0) from the PPPL tokamak data, which are again in good agreement with the theoretical calculations, Table 111. The process of interest is pel,

+ 1s'

lsnalanblb * ls2n,/,, ls2nb/b +

+

+

lsn:,l:,nblb,lsnalanLIL y'

(90)

where the second set of final states (f') could be Auger unstable. Important intermediate states d for Ox, Ar, Fe, and Mo were explicitly listed by Nasser and Hahn (1983), where the dominant r,, r,, o are also tabulated. One feature of o to be noted there is that, for low-lying resonance states, the fluorescence yield w is small for light ions and approaches 1 as we go to heavier ions. This is obviously due to the scaling behavior of A, - Z4(Appendix D) while A, Zo 1 . Furthermore, for nala= 2 p and nb large, o invariably approaches 1 for all 2 because of the n scaling; A, - nb3, while A, n j 1 . In general, as in the H sequence, I-, is dominated by the 2 p --., 1 s

- -

- -

TABLE I11 DIELECTRONIC RECOMBINATION RATECALCULATION" FOR THE He-LIKE TARGET IONS~ Ion

Mo Fe Ar A1 0

C

k,T,(keV)

4.00 8.00 2.00 4.00 0.925 I .85 0.46 0.I57 0.314 0.080

N-H

1.65 1.99 4.64 5.88 7.12 9.63 7.7 8.31 14.5 7.5

Y

B

BD

4.3

3.6

3.8

5.6 6.9 9.0 10.

Nasser and Hahn ( 1 983). Compared with the results of Younger (1983)and of Bely-Dubau ef al. (I 98 I ), and also with the experiment of Bitter ef al. ( 1979).Overall agreement is reasonable considering the numerous approximations introduced.


161

transition whenever the 2 p state is initially populated. The three-electron formulas for A, and A,(in LS coupling) are listed in Appendixes B and C.

C. LI SEQUENCE,N = 3 This sequence is much more complicated to treat than the H and He sequencesdiscussed above, because of the presence of additional excitation modes. For the first time, we have both the An = 0 as well as the A n # 0 excitations. That is,

p,l,

+ ls22s

-P

ls2sn,l,nblb,

An, # 0,

n, and nb 3 2

--*

ls2n,l,nblb,

An, # 0,

n, and nb 3 3 (91)

+

1s22pnJby

An,

=0

The first process above requires higher threshold energiesthan the other two, which involve a 2s electron excitation. We will discuss below the Li sequence in detail, as it can serve as a model for all the rest of the sequenceswhich have more complicated electronic structure. The discussion follows closely the work by McLaughlin and Hahn (1982a, 1984). 1. 2s, An,

= 0 Excitation

In the An, = 0 mode, we have the 2s electron excited to 2 p as the continuum electron is captured to a high Rydberg state (HRS), nbf b = nl. Since a large number of these states ( n 5 500, 1 4 15) can be involved, special care is required to estimate their contribution (Appendix E). The scaling property ofA, and A, in n is very important in determiningthe behavior ofaDRat large n and 1. Extrapolation of the individual A,, A,, and V . is needed for the evaluation of aDR; often aDR is nearly constant as a function on n, which implies that o n3 while V, r 3 .A direct extrapolation of aDR can introduce serious errors. Besides, as the energy separation AE,between the 1 s22s and ls22p states is small, more accurate determination of the continuum energy e, than that calculated by the single configuration resonance condition has to be made (see Appendix D). From the table of Cheng et al. ( 1979), we have AE,d = 0.889,2.51,4.35, and 12.5 Ry, respectively, for 05+, AP, These + . values in turn determine the lowest allowed values of FS3+andM o ~ ~ nb (for which e, = 0); iib = 6 , 10, 12, and 12 for the above ions. The large nb contribution is dominant, and the maximum values of n6 to be included are n, = 300,200, 150, and 80 for the ions considered above. The cutoffs at high nb are determined by comparing the magnitude of r, and A,(2p --.* 2s), which is independent of nb, as explained in Appendix E. The high-lcontri-

-

-

Yukap Hahn

162

butions are also found to be important; for 05+, 1 S 10 are important while for the heavier ions, 1 d 15 should be included. Almost always, this mode of excitation dominates at low temperature. Figures 9 and 10 illustrate the contribution for 05+ and FeZ3+. 2. 2s, An, # 0 Excitation Overall, this mode of excitation-capture dominates the total DR rate mainly because the 2s electron is relatively easy to excite and there are many more intermediate resonance states to which the 2s electron can go, as compared to the An, = 0 transitions ( 2 s + 2p). The 1s electrons are purely speculators, and the system behaves essentiallylike a two-electron version of the H sequence. We have

+

ls22s i-pclc+ ls2n,l,nblb +f y where n, b 3 and nb 3 n,. The procedure followed here is the same as in the He sequence; first, a complete set of intermediate states d is treated in the angular-momentum-averaged (AMA) scheme. A dominant subset of d is then selected which contribute at least 70% or more of the total AMA result, and the DR rates are recalculated for this subset of states in LScoupling. The total LS result is then obtained by a simple scaling, using Eq. (88). The

I

1s. An90

I

+

FIG.9. DR rate for the e- 05+ system calculated by McLaughlin and Hahn (1984)in LS coupling and cascadecorrected.Contributions from the different excitation modes are shown as system (McLaughlin eta/., 1985b);the overall functions of kJ,. The circle is for the e- 04+ features in these two systems are the same.

+

THEORY OF DIELECTRONIC RECOMBINATION I

0.01 I 0

I

I

1

2

4

6

k,T,

(keV)

163

T

1

a

+

FIG.10. Same as in Figure 9, for the system e- Fez)+.Note the large contribution from the 2s, Anl # 0 mode, while the 2s, Anl = 0 mode is less important except at very low temperature; the opposite is the case with OJ+in Fig. 9.

temperature k,T, in aDR was chosen such that the Maxwellian factor exp(-e,/k,T,) is approximately the same for all ions of the Li sequence. Since e, scales as Z 2 ,where Z is an effectivecharge roughly determined from the simple formula Z = ( Z , 2,)/2, the value for k , T, = 1 keV was chosen for Fe23+and k,T, = 5.17, 33.3, and 200.8 Ry for 05+, A P , and MoJ9+, respectively. The dominant contributions from the nu = 3 state excitations are given in Table IV. The rates were calculated explicitly for the states n, = 3,4 with nb = 3 - 7, and the contribution ofthe high-n,tail (nb 3 7) was obtained by fitting T,(d)and T,(d)as functions of nb. Such an extrapolation is given in Table IV. The cascade effect is and scaling result for the total aDR also shown, which reduces the rate considerably for Ar*'+and FeZ3+.This will be discussed further later. We note that, unlike An, = 0 case, the HRS contribution here is small. The fluorescence yield o ( d ) is usually large for low nu and nb, and quickly approaches the value unity (or some other constant if r,, B rr).Both rrand r,,scale as nb3 and n;3n;3, so that aDR nb3 and n;3n;3.

+

-

3. Is Excitation, Ana # 0

The rate coefficient for the Is transition is generally small at low temperatures due to the large 1s excitation energy, which also results in a large e, in the exponential factor exp(- e,/k, Te).But as much as 40%of the total DR rate can come from the Is excitation mode at high temperature. For heavier

164

Yukap Hahn TABLE IV DIELECTRIC RECOMBINATION RATECOEFFICIENTS aDR FOR THE Li-LIKE TARGET IONS' d

OJ+

Ar

Fe13+

~ 0 3 9 +

3snp 3snd 3snf 3dnd 3dnf 3dng Sum Total Cascade corrected

0.598 2.88 0.786 0.442 0.867 0.462 6.04 14.9

2.17 7.85 5.35 2.69 10.7 4.36 33. I 76.4

3.7 1 10.2 3.13 3.20 8.73 4.4 I 34.0 74.6

2.89 7.04 I .62 3.62 4.97 I .45 21.6 48.6

14.9

68.9

66.2

46.4

'Given for the dominant transitions2s, An, # 0 at the electron temperatures scaled as Z2.The scaled kBT, are used.

ions (i-e.,Fe, Mo, . . . ) over 90%of the total 1s contribution comes from the 1s + 2 p transitions

-

+ ec(s,d)

ls2s2pnbp

+eC(p,f)

lS2S2pnbd

ls22s

+

-

(92)

On the other hand, for lighter ions (2, Q 1 9 , the 1s 2 p mode contributes only about 30% of the total 1s rate. The LS-coupled formulas for A, and A, are listed in Appendix B and C (for four-electron systems). There is no large nb tail contribution for this process because of the presence of an additional Auger channel ls2dpnbib ls2nb/b e!li. The probability A: for that process dominates the Auger width raand is a constant independent on nb. Thus the fluorescence yield o ( d )at large nbapproaches a small but constant value A,(2p 1 s)/A:. aDR scales as V, nF3at large nb. The calculated 1s contribution is given in Figures 9 and 10 for 05+andFeZ3+.This is consistent with the 1 s excitation contribution in the He and Be sequences, when the effects of additional spectator electrons are taken into account. Note the relatively large contribution of the 2s, Ana = 0 and 2s, Ana # 0 for 05+, as compared to the Fe23+case. As discussed in Section 111, the 1s excitation has been studied experimentally through the resonant-electron transfer-excitation in S i- Ar, Si -t- Ar, and Si He, etc., by Tanis et al. (1982, 1984). Due to the extremely small cross sections, the Ana # 0 process has never been directly measured in a laboratory collision experiment.

-

+

-

+

-


165

4. Cascade Eflect

The decay of the intermediate states d radiatively contributes to the DR process, but when the final statesfreached are not stable against further Auger emission, then the fluorescence yield w(d) for the original state d has to be modified to take this effect into account. The cascade effect always reduces w and thus aDR. We have from Eq. (76)

where the states labeledf are stable against Auger emission but d’ are Auger unstable. For example

+

i, = ls22s e,,

+

i2 = ls22p eili

* ls23s4d-,

-

+

ls22p4d= f, ls22p3s =fi

(94)

ls23s4p =f4

The statesf 3 and f4 are Auger unstable, so thatA,(d -,f 3 = d;) and A,(d + = d;) are to be reduced by the factors w(d;) and w(d;), respectively. The cascade effect is summarized in Table IV for the different ions of interest here, mainly for the 2s excitation, Ana # 0. The 1sexcitation contribution is hardly affected by this because the dominant decay mode for the ls2s2pnb1, states is such that A,(2p -,1s)and A, > A,(d--, ls2s2pnili). The effects of CI and IC are important here, especially in the intermediate is often very much affected by this, and final states. Each individual aDR(d) but the total aDR (sum over d) is in general less sensitive to both CI and IC.

f4

D. BE SEQUENCE,N = 4 This sequence is very much like the Li sequence discussed above in that all three excitation modes are possible. An earlier treatment of this sequence (Gau et al., 1980; Hahn ef al., 1980a) grossly underestimated the contribution from the 2s excitation, Ana = 0, especially for lighter ions. This has been corrected in later publications (LaGattuta and Hahn, 1983c; McLaughlin ef al., 1985b). The theoretical method used is exactly the same as for the Li Ar14+, sequence, and we simply summarize in Table V the results for 04+, FS2+,and Mo3*+.The general features of aDR are very much like those of the Li sequence, except for the presence of an additional electron in the 2s shell initially. This can sometimes affect the excitation probability V, by a factor

166

Yukap Hahn TABLE V DIELECTRONIC RECOMBINATION RATECOEFFICIENTS aDR FOR IONIC TARGETS OF THE Be SEQUENCE WITH N = 4p 04+~ B T , ( R Y )

Transition 2s, A n # 0 2s,An=O IS, A n # 0 Total

2. I 2.2(- 12) 3.3(-11) 1.7(-20) 3.5(- 11)

4.3 1.9(- 12) 1.6(-11) 1.9(- 16) 1.8(- 11)

8.5 1.1(- 12) 7.1(-12) 5.4(- 15) 8.2(- 12)

17 4.8(- 13) 3.1(-12) 1.8(- 14) 3.6(- 12)

ArI4 k, T, (Ry) ~~

Transition 2s, A n # 0 2s, A n = 0 Is, A n # 0 Total

17 1 3 - 11) 1.1(- 11) 4.3(- 16) 2.6(- 1 1 )

32 1.3(- 1 1 ) 4.2(- 12) 2.9(- 14) 1.7(- 1 I )

65 7.2(- 12) IS(- 12) 1.6(- 13) 8.9(- 12)

I29 3.4(- 12) 5 3 - 13) 2.3(- 13) 4.2(- 12)

Fez2+ k, T, (Ry) ~

Transition 2s, A n # 0 2s, A n = 0 I s, A n # 0 Total

37 1.3(- 1 1 ) 1.0(-11) 9.8(- 16) 2.3(- 1 1 )

74 9.7(- 12) 3.9(-12) 4.8(- 14) 1.4(- 1 1 )

147 4.9(- 12) 1.4(-12) 2. I(- 13) 6.5(- 12)

~

~~~

294 2.1(- 12) 5.1(-13) 2.6(- 13) 2.9(- 12)

Mo3*+ k, T, (Ry)

Transition 2s, A n # 0 2s, A n = 0 Is, A n # 0 Total

I03 1.0(- 11) 9.0(- 12) 6.1(-17) 1.9(- 1 1 )

206 6.5(- 12) 3.4(- 12) 3.4(- 14) 9.9(- 12)

412 3.1(- 12) 1.2(- 12) 1.2(- 13) 4.4(- 12)

824 1.3(- 12) 4.3(- 13) 1.4(- 13) 1.9(- 12)

a From McLaughlin e? d o1985a. aDR in units of cm3/sec. The values for the 2s, An, = 0 contribution was grossly underestimated previously (Hahn ef a/., 1980a) and has been corrected here using an improved theoretical procedure (Appendix E) for treating the high Rydberg states.

of 2, but may also reduce the radiative width r, by a similar amount, especially for the dominant transitions. The net result is that (Y may not change much from that for the Li sequence. This point is important in generating an empirical rate formula applicable for all sequences. The treatment thus far (McLaughlin and Hahn, 1984) does not include the effects of configurating interaction nor intermediate coupling. As seen in the work by LaGattuta


167

( 1984a)and Roszman and Weiss (1983), the CI effect on the total aDR is not as drastic as that on some of the individual states. Nevertheless, this point needs further study. Calculation of A, and A, in the case of 1s electron excitation, when 2s electrons also participate, requires a formula involving five active electrons; it is listed in Appendixes B and C.

E. NE SEQUENCE,N = 10 This is one of the first sequences that have been treated in detail theoretically. We refer to the work by Hahn et al. (1 980b) and Gau et al. (1980) for a complete summary of this sequence. Some work was also published by Roszman ( 1979)on M o ~ ~ in + which , a small subset of low-lyingexcited states d is included without the cascade effect; additional states, which were newhile the cascade effect reduces it; thus the compensatglected, increase aDR ing errors roughly cancel with each other. We list in Table VI some sample results for various cases. This sequence is relatively simple, as compared to the Li and Be sequences, in that there are no contributions from the A n , = 0 transitions, and the dominant mode is An, # 0, in which 2 p electrons are excited predominantly to n , = 3 levels. The n,, nb 3 4 contribution is drastically reduced by the presence of additional Auger channels (i’) which contribute to the r,, and also by the cascade effect. The 2s, A n , # 0 contribution is small, of the order of 3 - 4% of the total aDR.

-

TABLE VI DIELECTRONIC RECOMBINATION RATE COEFFICIENTS FOR THE Ne-LIKE TARGET IONS AT SCALED TEMPERATURES’ d

I,

Ar8+

FeI6+

Mo3*+

2p53dnd 2p53pnd 2p53dnf 2s2p 63dnf Sum Total k J , (keW y=2/3

3 2 4 5

14.5 7.8 22.9 0.14 45.3 78 0.38 92

31.3 12.3 33.8 2.5 79.9 140

52 18.5 33 4.2 108 195 3.1 195

1 .o

131

(From Hahn ef al., 1980b.) Units are in lo-” cm3sec. aDR is fitted by ZY.

-

168

Yukap Hahn F. NA SEQUENCE,N

=

11

A complete study of this sequence was carried out recently ( LaGattuta and Hahn, 1981a, 1984a). The calculation is divided into five parts; (1) 2p, An, # 0 excitation, (2) 2s, An, # 0 excitation, (3) 3s,An, = 0 excitation, (4) 3s, An, # 0 excitation, and finally ( 5 ) 1s excitation, A n , # 0. Again, three ions are singled out for discussion; Mo3I+,Fe15+,and Ar7+.The results are summarized in Table VII and Fig. 1 1. As noted in the Li sequence, the An, # 0 contribution is very large for cases with Z, 3 15 and at high temperature, while the A n , = 0 contribution dominates for ions with Z, 4 15 and at low temperature. Obviously, this is a general trend, which is valid for all the ions considered. The earlier work by Burgess and later by Jacobs and others on light ions included the A n , = 0 contribution alone; this is well justified for Z , 6 15. But, as noted first by Merts et a/. (1976), the An, # 0 transitions are very large for large 2,. Therefore, entire calculation of aDR for heavier ions has to be carried out differently from that for lighter ions.

1

I

I

TOTAL

-"

10

-

-7

5

N I 0

-

a: n -8

/

-

/

1 -

3s.An~O

/

I 0

I

10

I

20

30

40

50

vs nuclear core charge Z , at scaled temperatures; FIG. I I . DR rate coefficient aDR k,T, 22,where Z = ( Z , Z,)/2 for the degree ofionization Z,, N = 1 1. That is, k,T, = 25.9 Ry for Ar7+, 69.6 Ry for Fe15+,and 221 Ry for Mo3’+.The filled circle is the rate for Si”+at k,T, = 12 Ry computed by Jacobs ef al. (1977).

-

+


169

TABLE VII DIELECTRONIC RECOMBINATION RATE OF THE COEFFICIENTS (IIDR FOR IONIC TARGETS Na SEQUENCE' Excitation class

2P 2s 3s, Ant f 0 3s, Ant = 0 Total k,T,(keV)

4kB Tt f kB T, 2kBT,

Ar7+

FelS+

M031+

0.35 0.017 0.01 3 0.35 0.73 0.352

1.11 0.055 0.088 0.24 1.49 0.946

1.21 0.060 0.24 0.2 1 1.72 3.00

2.78

3.38 2.57 0.70

4.52 3.18 0.8 1

1.44 0.3 1

N = 1 1 electrons; coefficients in units of lo-" cm3/sec. Contributions from the different excitation modes are also indicated.

G . MG SEQUENCE,N = 12 As in the Be and Li sequences, the Mg sequence with N = 12 electrons in the target ion was treated and compared with the Na sequence data; thus far, only the 3s electron excitations, An, = 0 and An, # 0, were considered (Dube et al., 1985). We sumarize the results here and make a comparison with that part of the Na sequence involving the 3s electron excitation. The method used here is the same as that in all the other work done by the University of Connecticut group. The calculation was camed out in the LdSd scheme, where a and b denote the quantum numbers of the two active excited electrons in the intermediate state, given by 3sn,l,,nblb. That is, the a and b electrons are coupled first to form L+,Sab,then this pair is coupled to the spectator electron, which in this case is in a 3s state. However, when (n,I,) is in the same shell as 3s = n, I, ,it is more appropriate to couple the 3s and (naIa) first; we may recouple the earlier states II,Ib[Labscrb], I, [LSI) to II,!, [La,S,], / b [LS]).Several sample calculations of CYDRwere performed using the L,S,, coupling scheme, and the results were compared with those using the Labs a b coupling scheme. Individual aDR for each L and S are changed drastically, but, as before, little or no change in the DR rate coefficients were found when summed over La, and S,,.Sample data are given in Figure 12.

170

Yukap Hahn

I

1

I

1

2

1

I

3

(E, 1 FIG. 12. DR rate coefficient aDR vs electron temperature T, for the 3s excitations with An, = 0 and An, # 0 for ions of the Mg sequence with twelve electrons (N = 12), as given by Dube ef al. (1985). The circle is for the same excitation modes for the system FelJ+ e-, as given by LaGattuta and Hahn (1984a). The 2p excitation mode, which is neglected here, is Mo ( E , = 206 Ry); - - -, Ar (32.3 Ry);---, expected to be large at higher temperature. -, Fe (73.5 Ry); and . * * , Si (1 1.8 Ry). k,Te

+

The study of the Na sequence indicated that the contribution of the 2 p electron excitation is very large, especially at higher Zc and at higher temperature. Work is in progress to complete this part of the problem. H. OTHERSEQUENCES, N = 18, 19

In connection with the Ca+ case discussed earlier, bDRwas estimated for the case of 4s, Ana = 0 (4s -,4p) excitation. The usual conversion factor in Eq. (65) provides a check on aDR for given ZDR.As expected, the rate is very large at low temperatures, such as might be found in astrophysical environments and near the container walls in tokamaks. On the other hand, this particular mode with Ana = 0 involves many high nb states; they will eventually be affected by fields which may be present in the electron-ion


171

interaction region and by the density effect (Wilson, 1962). This problem will be discussed in Section V. To extend the existing empirical formula for aDR to N = 18, it is necessary to obtain benchmark calculations ofaDRfor at last two or three isoelectronic sequencesin this region. Work is in progress to meet this need for N = 18 and 19. Since these systems are much more complicated than the ions discussed thus far, careful study of the spectral structure of the intermediate states is important before reliable A, and A, can be evaluated. The versatile structure code of R. D. Cowan ( 1981, personal communication) is used to improve on the results obtained by our simple MATRIX code.

V. Discussion and Summary The theory of DR developed in Section I1 was general enough to include a variety of higher-order processes and also enable us to identify various simplifying approximations employed in actual calculations, so that systematic improvements can be made as situations dictate. The explicit calculation of aDRand aDR summarized in Sections 111 and IV is based on the simplest procedure developed in the isolated resonance approximation and within the single-configuration,nonrelativistic Hartree - Fock framework. The active-electron LScoupling scheme ( L , Sabscheme)was adopted throughout. Obviously, the validity of these approximations should be checked for each specific case. Improved understanding of the theoretical procedure is being achieved by the continuing efforts of many people. We summarize some of the progress made during the past two years, and indicate those critical problems which require further study. Among the topics to be considered are: (1) the effect of configuration mixing and intermediate coupling. Also included here is a discussion of the active-electroncoupling scheme vs. a more conventional core-electron coupling. (2) The question of the effect of overlapping resonances is of interest, since the enhancement in DR and A1 cross sections is brought about by the presence of many intermediate resonance states. The interference between direct radiative recombination and DR amplitudesshould also be examined. (3) Finally, the effect of external fields on aDR and aDR is only beginning to be appreciated,especially for those cases in which high-Rydberg-state captures are involved. The enhancement due to a small external electric field (- 10 V/cm) can be as high as a factor of 10 in some cross sections and rates. The effect is more pronounced for small Z , ( Z , d 15) and for large nb.

172

Yukap Hahn A. CONFIGURATION INTERACTION AND

INTERMEDIATE COUPLING

The configuration-interaction (CI) effect on the individual reaction cross sections is known to be large when configurations of the same symmetry are nearly degenerate in energy. Thus, 2s2 2p2,3s2 3p2 3s3d 3d2,etc., are the typical examples. More recently, the problem of CI in the context of DR cross sections and rates was examined by Roszman and Weiss (1983), Mchughlin and Hahn (1984), and by LaGattuta (1984). In the latter two reports, intermediate-state mixing in the Li and Be sequences was considered; e.g., configuration mixing of ls22s2 ls22p2, ls23s5p ls23p4s, ls23s4p ls23d4p, ls23s3d ls23p2,etc. The change in the individual rates is large, but the total sum ofaDRis hardly affected. Table VIII illustrates this point in the case of Li sequence. A similar result was also found by Roszman and Weiss, and by LaGattuta. We note here that, unless w is changed drastically by the mixing, CI has an effect similar to a unitary transformation of the states involved, insofar as the total aDR is concerned, and thus should not change the total aDR, The effect of intermediate coupling (IC) on DR rates and cross sections has been studied recently by LaGattuta ( 1984)and by Pindzola et al. (1984). This effect is expected to be large when the strength of the spin-orbit interaction (-Z) is comparable to the electron -electron interaction, only the latter

+

+

+

+

+

+

+

+

TABLE VIII

EFFECTOF CONFIGURATION

INTERACTION FOR THE SYSTEM ~

Configuration

w(d)

3s4p 1P 3p4s IP 3s3d ID 3p2 ID 3s4d ID 3d4s ID 3s4d ' D 3d4s ' D 3s4p 'P 3d4p 'P

0.069 0.083 0.074 0.040 0.105 0.166 0.403 0.773 0.135 0.447

(Y &R

0.70(- 14) 1.39(- 14) 2.01(- 13) 4.65(- 15) 6.46(- 14) 8.38(- 14) l.l1(-13) 2.36(- 14) 4.46(- 14) 2.48(- 15)

e- -k

~~~

Mixing coefficient; first configuration

w(d)C1

0.8446 -0.5354 0.7477 -0.6640 0.7938 -0.6082 0.7992 -0.601 I 0.9904 -0.1383

0.042 0.363 0.049 0.070 0.106 0.6 I3 0.878 0.472 0.426 0.1 10

EF 1.56(1.33(9.32(7.80(1.17(3.65(3.99(I .67(1 . 1 1(3.44(-

14) 14) 14) 14) 13) 14) 14) 13) 14) 14)

a I keV temperature. Two configurations are mixed at a time. The mixing coefficients for the first configurationare given explicitly;the mixing coefficients for the second configuration are obtained by reversing the numbers with one of them changing sign. The numbers in in units of cm3/sec. parentheses are powers of 10 for aDR

173


FIG. 13. Effects of different coupling sequences and configuration interaction are studied for the e- Ar"+ system, in which the 2s2 pclc-D 2s2pnl transition is involved (LaGattuta, 1984). -, 2s2p(L,S,) coupling plus CI; ---, L,,,S, coupling without CI; - - -, 2pnl (L,S,) coupling; ... ,result of L,S, coupling with CI given by Seaton and Storey (1976).

+

+

preserving total L and S as good quantum numbers. Both interactions bewhen electrons in high Rydberg states are involved. Additional have as r 3 relativistic effects can also be significant for 2, b 15. The calculations performed thus far do not include these effects in any systematic way; only a limited number of test calculations are available. Finally, the effect of the order of coupling in a many-electron system was studied recently (LaGattuta, 1985;Dube et af.,1985).For example, consider the DR transition

ls22s2

+ eClc2 l s 2 2 s 2 p n l 2 ~

+

2 y

s

~ (95)

Here, in V,, the active electrons are 2 s , pc f c 42pnf.Therefore, the formula for V, is simplest if we couple these active electrons first as I 1 s22s, 2pnf(Ld Sab),LS): active electron coupling. On the other hand, we may also couple the core states first, as1 ls22s2p(L, So,),n f ( L S ) ) ,the core-electron coupling. Explicit calculations show that the difference between these two coupling schemes is more pronounced for heavier ions, and generally the core-electron coupling gives a result which is closer to the intermediate coupling values. This is illustrated in Fig. 13.

B. OVERLAPPING RESONANCES AND INTERFERENCES The formula Eq. (57), in the IRA, is valid only when the distances between resonances are much larger than the widths T(d).Typically for low-lying T ( d )are of the order of loLssec-l, i.e., 1 X states with small Z,,

-

-

Yukap Hahn

174

Ry. This is to be compared with the distances between resonances ofthe order lo-' Ry. On the other hand, when high Rydberg states with n b 10 are involved, we have

r(ni)- ~ , ( n i ) 0.iln3

(96)

RY

which is to be compared with the spacing between resonances

AE,

= n-2

- ( n + i)-2

- 2/n-3

RY

(97)

Typically, A, involving core electrons is lo-* Ry, which becomes comparable to A, at n, = 200. Therefore, for n b n,, A, B A, and w = 1 . On the other hand, in this region of n, oDR and aDR-rr3, for n B n, (98) overlap between resonances starts to set in at n values which are 2 to 3 times larger than n,, as shown by the comparison between Eqs. (95) and (96). Therefore, the question of overlapping resonances is not so important in the calculation of aDR. The problem of overlapping resonances is still of interest, however, because there are many exceptions to the above typical cases, where different 1 states for a given n are involved. Accidental degeneracy in the resonance levels and mixing of the levels by external fields are also possible. An extremely simple model was considered recently by Hickman ( 1984) and by LaGattuta and Hahn (1985a). It is a simple modification of the multichannel quantum defect approach (Seaton, 1983), in which the energy is made complex to simulate the effect of a radiative width for the core electron. For a realistic set of parameters, it was found that the effect of overlap can be large, but, as mentioned above, the net contribution to the total rate is small in a realistic situation; this is consistant with the earlier findings by Bell and Seaton ( 1 985). Figure 14 illustrates the points discussed above. Incidentally, the modification made by Hickman on the energy parameter (real to complex) is disputed by a recent note of Seaton (1984). For inner-shell radiative transitions, our formulation gives a prescription similar to that of Hickman, while the radiative channel contribution corresponding to the outer-shell electron is quite different, as discussed in Section I1 and Appendix A. This point has never been discussed systematicallybefore, and some additional numerical as well as formal study is desirable. The interference contributions between M y and MzRin Eq. (50) and between the terms with d # d' in the sum in Eq. (56) are in general expected to be small. In both cases, the (principal value) resonant factors with different phases integrate to approximately zero when energy averaged. A more careful study is needed, however.


t

175

1M 1

n FIG.14. Estimated dependence of":a vs n and the effect of overlapping resonances as ~ ~ ~ . set of measured by the parameter M for ions of Z = 3, where M = ( U ) , ~ / ( U )A~ realistic parameters is used in the model. Note that u is decaying as n-3 when M starts to deviate from unity.

C. EXTERNAL FIELDEFFECT Atomic collision processes take place inside tokamaks and in stellar environments, often in stray electric and/or magnetic fields. In laboratory experiments involving electrons and ions, external fields may be imposed in the interaction region to collimate the electron beam or simply to influence the process itself. The effect of such fields on the DR rates and cross sections is not very well understood, however. In particular, when the intermediate states dreached by the initial capture involves very high Rydberg states, then the presence of an external field can seriously affect the outcome; the weak Stark mixing of different 1 levels of a given n state can be dramatic, as these states are nearly degenerate. A recent experiment by Belic et al. ( 1983) has shown that the experimental result obtained in a crossed-beam apparatus is about 5 6 times larger than the theoretical prediction (LaGattuta and Hahn, 1982a). However, in the experimental setup, a Lorentz electric field of about 24 V/cm was placed in the interaction region to collimate the electron beam. This field was shown to have a drastic effect on aDR at large n(b20). The theoretical prediction ( LaGattuta and Hahn, 1983b) is given in Fig. 6.

-

176

Yukap Hahn

The large enhancement is caused by strong 1mixing, which can be described by the Stark representation in the case ofcomplete mixing. In the absence of I mixing, only those I states with 1 d 8 contribute to oDR and aDR, because the capture probability V , is very small for 1 b 8. On the other hand, in the Stark representation, all 1's contribute, with large-1 states weighted by the 3 - j symbol, which varies as 1-I. In the cases of Mg+ and Ca+, n d 65 and n d 80, respectively, participate in the DR capture process; contributions from the large n states are cut off by the analyzing field of 36 V/cm and 12 V/cm, respectively. Since oDR and aDR are effectively constants for n =sn, = 200, as given by Eq. (78), the effect of a field appears largely through an increase in the statistical factor g d in V,. A rough estimate confirms this picture. After the recombined ion (Mg("or Cao+)leaves the interaction region, it travels through a zone where the electric field changes. This may shift the (n,n,,m ) population of the HRS electron (Richards, 1984; Pillet et al., 1983, 1984; Rolfes and McAdam, 1983; Hulet and Kleppner, 1983) in the Mg and Ca beams to higher n, and m states (where n, is the electric quantum number). Electrons in high n, and m states are difficult to field ionize (Damburg and Koslov, 1979;Silverstone, 1978). In the case of Ca+,the theoretical DR cross section (Nasser and Hahn, 1984) is about 7 to 10 times smaller than the experimental peak (Williams, 1984).The stray field in the interaction region is estimated to be less than 0.3 V/cm. We have found that, for this small field, oDRwill still be increased by a factor of - 3 -4, thus explaining away a large portion of the existing discrepancy (Hahn et al., 1985)' as summarized in Table IX. [However, see the discussion by Dunn et al. (1984a,b).] In general, the field effect on HRS is a very complex and delicate subject to study. However, often the effect on cross sections and rates is very large. The behavior of HRS under static and time-varying external fields is being studied by a variety of experimental techniques. Both I- and rn-changing collisions (Stebbing and Dunning, 1982) may be relevant in understanding the field effect on DR. Magnetic fields of tesla strength can also be important (Huber and Bottcher, 1980). D. SUMMARY We have summarized here the DR theory as it is applied to the evaluation and aDR. Much progress has been made during the past few years both of oDR experimentally and theoretically. Some of the major problems have been identified and are being studied in detail. New areas of research on HRS electrons and on field effects are yet to be explored. As stressed repeatedly, the aDR calculation is very cumbersome, even in the simplest of approximations. Obviously, a more streamlined approach is


177

TABLE IX VARIOUS PARAMETERS RELEVANT TO THE SYSTEMS e- Mg+AND e- CA+

+

+

Mg+ A, ec(max)

nF AeC 0:; 0:;

(peak) (peak)

Electric field

2.80(+8) sec-l 4.4 eV 64 0.3 eV 2.0(- 18) cm2 I .2(- 17) cm2

24 V/cm 1.6(- 17) cm2 Enhancement 8 factor

.RY:.F

+

Ca+ 1.60(+ 8) sec-l

3.1 eV

80 0.3 eV I .6(- 18) cm2 1.8(- 17) cm2 0.5 V/cm 0.7(- 17) cm2 4

+

* For e- Mg+,3s + 3p, An, = 0; for e- Ca+, 4s + 4p, An, = 0. These parameters are compared, and the cross-section peaks are summarized. Large discrepanciesexist between the experiments and the theory which does not include the field effect. (See also Figs. 6 and 7.)

needed to reduce further the computational problems. A variational approach and a sum-rule-type procedure are being considered. Finally, it is of potential importance to realize that, as pointed out in Section I1 with Eqs. (67) and (70),Auger ionization and photo-Auger ionization are closely related to DR. In view of the difficultiesinvolved in carrying out the DR experiments, the study of A1 and PAI, as well as the RTE experiments, can be as profitable as treating the DR process directly. Over the past several years, a vast amount of theoretical data on aDR have been accumulated.The related A,, A,, and o(d)are themselvesuseful. These data will be summarizedin a future report (PhysicsReport articleby Hahn et al., 1985). Eventually, a semiempirical formula for the DR rates, of the form given by Burgess ( 1965),Merts et al. ( 1976), or Hahn (1 980), is needed to generate a complete set of aDR which are to be incorporated into the rate equations of the type in Eq. ( 1). The existing formula already gives the DR rate to within a factor of two. Further refinements are needed for the analysis of current experimental data; an accuracy in the formula of about f 30% is desirable. The benchmark calculationsdescribed in Section IV are used to improve the empirical formula. An efficient computer code is available (R. Hulse, personal communication) which treats the An, = 0 and An, # 0 contributions separately.

Yukap Hahn

178

Appendix A: Radiative Widths and Coupled Equations In this appendix, we further analyze the term in Eq. ( 19) which contains the effect of the R-space channels and derive an explicit form of the vertex function. In particular, we show that the simple complex energy model (Hickman, 1984) is a special case of a more general formulation presented earlier (Hahn ef al., 1982a,b).The coupling ofthe Qspace to the R space with one photon gives rise to a potential of the form

where the vertex function and the resonance form factor are given by qf = QDRaR,

For definiteness, we consider a case in which only two electrons are active; e.g., the e- Mg+ system discussed in Section II1,C. The Q space = (3p, nl; 0) and the R space = ( 3 p , n'l'; k ) or (3s, nl; k). Two distinct cases will be considered based on the two different R-space configurations given above, and the differentialequation will be derived with the potential shown in Eq. (99) for electron 2 (in the state nl). The exchange effect is neglected since we are mainly interested here in the HRS.

+

A. INNER-SHELL ELECTRON TRANSITION

When Eq. ( 19)describesthe motion ofan outer-shell electron, while one of the inner-shell electrons makes a radiative transition, then we may write the potential as U ; = -iaI3p,n1;0)(3p,n1;0ID1 13s,nl;k)h;

-

X (3synl;klD,13p,nl;0)(3p,nl;01 =-(i/2)A;r1(3p

3s)Q,(d= 3p,nl)

where

r , ( 3 p + 3s) = 2~1(3p,nk01D,13s,nl;k)1~ = 2n1(3plD113s)I2 Qd(d= 3p,nl) = I3p,nl;0)( 3p,nk0 I

R, = 13synl;k)(3s,nl;k(

(101)


179

That is, aside from the projection operator Qd(d= 3p,nl), Us’ is a constant independent of the coordinate of the nl electron. The presence of Af in Eq. (101) is important, because U f = 0 when E # EF. The form of Eq. (101) seems to be valid when resonances are isolated; when resonances in the R space overlap, A: of Eq. (100)may have to be modified, but Eqs. (13), (14), ( 1 8), and ( 19) are still valid. Resonances in the Q space associated with high nl states are still treated correctly by Eq. (19) with UF of Eq. (101). The precise relationship between the present formulation and multichannel quantum defect theory of Seaton (1983) is not entirely clear and will require further examination (Seaton, 1984). In some cases, fine-structure splitting of the core electron (3p electron in the above example) can cause an additional spread of resonance levels as a result of an increase in the Auger width (Kachru et al., 1984). Generally this will result in the reduction of the DR cross section (Dunn et al., 1984b), and also bring about further overlap among the resonance levels.

B. OUTER-SHELL ELECTRON TRANSITION If we take the other form of the R-space function, Uf becomes instead

UF = - ial 3p,nt0)( 3p,nk0 ID213p,n’l‘;k)Af X

(3p,n’l‘;klD213p,nl;0)(3pynkO~

= -(i/2)Qd(d= 3p,nl)AFTr(nl,n’l’) (102) Unlike the case of Eq. (10 l), UF depends on both n and n’ through T@, n’l’), in addition to Qd. Therefore, Ur must be an integral operator in the Q space. More explicitly, in the Q space with Q = &Qd, we have

U R= -iwQD2AR(2,2’;k)D2Q

(103)

where

In Eq. (104),the sum is over all the states which can be reached from the Q space by one photon emission. Each term in this sum has the property that asymptotically it decays exponentially with r2 or r ; . In general, both Eqs. (101) and (103) are present in Eq. (19). Obviously, Eq. (101 ) must be important for high-n states, while Eq. (103) is the dominant term for low-n states. A useful sum rule may be developed using Eq. ( 19) once the projection operator AR is explicitly constructed.

Yukap Hahn

180

Appendix B: Auger Probabilities A, in LS Coupling In the present LS-coupling scheme, all the closed subshells which do not participate in the transition are omitted, and the open-subshell electrons which act as spectators are counted as a single coupled particle. The remaining active electrons are explicitly coupled first, followed by the coupling to other inactive electrons. Alternate coupling sequencesare also explored. The discussion will be divided according to the number of particles that are to be coupled.

A. SYSTEMS

This is the simplest of all systems and includes the cases with N = 3, 1 1, and I9 in which the 2s, 3s,and 4s electrons, respectively,are excited. Here, N is the number of electrons in a target ion. Then ( 105) A, ( n , I , , n b l b n s ec I , ) = ( 1 / ~ & )fa f b fs fc 1 2 ( L a b s o b where N& = 1 if n , I, # nb Ib and N& = 2 if n, 1, = nb / b and where f = 21 1. In Eq. (105) +

+

x ('s

lb

la

kt

]

Lab

where the R k are the usual radial integrals Rk(ls1cIcalb) =

rk< drl r: dr2 r : v s ( r 1 ) v c ( r 2 ) k+l W a ( r I ) V / b ( r 2 ) r> (107)

The bound-state functions v,, vb,and vSare normalized to unity, while the continuum function ycis energy normalized; that is, it behaves asymptotically as (see Eq. (54))


181

B. SYSTEMS This applies to cases with N = 4 , 12, and 20, in which the 2s, 3s, and 4s electrons, respectively, are excited. Also included are the cases with N = 2, 10, and 18, in which the 1 s, 2p, and 3 p electrons, respectively, are excited. There are two distinct cases:

sL)

Casez. Id) = I(ntlf)mStLt,(nala)(nblb)sabL&,

+ti)

= I(n,I,)""s:L;,

pel,, S'L')

We have

where li

(la,lb,lc,lt) GZl;Z= fractional parentage coefficient If (n,l,)m+lis a closed subshell, then S: = L: = 0 and m above A , simplifies to a form

+ 1 = 2(, and the

Averaging over the final set of quantum numbers LsL&s&, we have

and for a = b, -

L 6+ 1)

= 4(4Ia

where S,

+ La, = even.

Lzw

&,L,I(Laa,S,)*,

a =b

Yukap Hahn

182

When averaged, A , becomes, for a # b =

17.12. 4 2Lab sabEabI(Lab

3

Sab)’

and for a = b

C. S SYSTEMS This includes the cases with N = 3, 1 1, and 19, where the Is, 2p, and 3p electrons, respectively, are excited. There are four distinct cases. The intermediate state d is given by s s d la)(nb I, l b ) [ L a b s a b l , Ls) Id) = ~ ~ ) ~ ( n d ~ d ) [ L s d (Na Only the closed subshell with rn 1 = 2fs is considered here for simplicity.

+

Case 1. Id)

+

li)

= (ns/s)m+’(ndld)

+ e,l,,

with a # d, b # d


Case 4. Id) --* li)

+ eclc,

= (n,l,)m(ndld)(nrl,)

183

with t f d

D. SYSTEMS This covers the cases for N = 4, 12,and 20,in which the Is, 2p,and 3p electrons, respectively, are excited and in which the 2s2,3s2, and 4s2electrons, for example, are “participating.” More specifically, we consider the DR process

+

-

-

i = ls22s2 pclc d = ls2s22pnl ‘i

=

+

ls22s2p pili

Thus, we define

Id) Ii’)

= l{Lefe[Leel,

lalb[Labl)Lx,

Lt[L1)

Ltlt[LrrI)Ly,L-[Ll) If we specialize to a case with Lee= See= 0 and L,, = S,, = 0, then L, = La, and S, = See.We further assume that the t orbital is closed in the i’ state, then +

= I(Lela[Lael,

More complete expression involves six 9 - j symbols inside the square.

vs. CORE-ELECTRON COUPLINGS E. ACTIVE-ELECTRON As an example, we consider the 3esystems as described by Eq. (109).With

rn = 1 and 1, = 0, we may write the d state as

Id’) = ((nr lt)(nala)satLat, (nblb)Sblb), sL) Then, in terms of A , of Eq. ( 109),

184

Yukap Hahn

Averaging the A , of Eq. (1 16) over LSL,,S,, we obtain, for a # b for example,

This result agrees with that given by McGuire (1 975).

Appendix C: Radiative Probabilities A, in LS Coupling As with the A , formulas given in Appendix B, we catalogue the A , formulas according to the number of active particles which participate in the coupling. A. S SYSTEMS

-

There are three possible cases:

If)

= (nala)(nJs),

with a # b, a # s, b # s

A,( Lab s a b L s ) = A)':

(1 18)

Case I . Id) = (nala)(nblb)

where A i0) is the one-electron radiative probability given by

with RD(nblb ---* nsls)=

1, Case 2. Id) = (n,la)(n,16)-.,If)

where L,,

+ S,

=

I

r2 dr vnsl,rvnblJr)

= larger of

1, and 1,

= (nala)2,

with a # b

even.

-

Case 3. Id) = (n,la)2-If) = (@,,/,)(nblb), with a # b A,(L,S,LS) = 2 A!')(Ia / b )


185

B. S SYSTEMS

C. S SYSTEMS

Appendix D: Scaling Properties of A,, A,, w, and aDR The DR cross section crDR given by Eq. ( 5 8 ) and the rate coefficient aDR given by Eq. (64) are composed mainly of V. and o ( d ) , which are in turn given in terms ofA, and A,. Therefore, it is useful to analyze first the behavior of A, and A, as functions of the effective charge Z , the principal quantum number n, and the orbital quantum number 1 of one of the states involved.

Yukap Hahn

186

Such information is useful in treating the contribution of HRS, interpolating the values at intermediate 2 and even to test the accuracy of numerical calculations. The scaling properties to be discussed below are based on the Coulombic nature of the wave functions involved. Therefore, any scale breaking is caused by the screening effect and other non-Coulombic interactions. As will become clear, the scaling properties are quite different for the two cases An, # 0 and Ana = 0. A. THEZ Scaling

We divide the discussion into two parts, first for the An, f 0 transitions, and then for the An, = 0 transitions, whose transition energies are non-Coulombic. 1. An,, f Case

A,(ab --* sc) contains three bound orbitals a, b, and s, and one continuum orbital c. Assuming that they are purely Coulombic, the differential equations they satisfy scale in Z. Therefore, the explicit Z dependence appears only in the normalization factors, as

wherep,

- Zand r - Z-I. Hence, with r i i - Z and dr: dr: - Z-6, we have A , - Il2 - 1Z3/2Z3/2ZZ3/2Zl/2. 2 - 6 1 2 - ZO

(127)

On the other hand, A,(b +.f) involves two bound orbitals, so that

-

Ar(b +.f)

(eb - ef)3112

- z 6 ( z 3 / 2 z - l z 3 / 2 . 2-312 - 2 4

( 128)

-

with e, ef- Z 2 .These results are standard, and show that as Z increasesA, can become comparable in magnitude to A,, although for small Z , usually A, 0. The GRWA also fails to predict multiphoton resonances other than the [n 1, - n] and [-n, n I ] types (not shown in the graph). It is interesting to note that all resonances are shifted towards the center x = 0. And in these rather high field strengths, the two one-photon ( x = k 1) peaks have merged into a single broad band. Useful analytical expressions for the shift and width of each multiphoton resonance can be obtained by extending the nearly degenerate perturbation theory of Salwen (1955) to the two-mode Floquet Hamiltonian, Fig. 15, or the GRWA Hamiltonian, Fig. 16. In the case of [n 1 , -n] transitions ( w 1 > 0 2 )for , example, the TMFT transition probability at any instant of time t , evaluated to the lowest order, can be written as

,

+

+

+

(u,/2I2

=[

OO

- (n

+ l)w, + nw2 +

S,]2

sin 2( q,t)

(99)

where

S, = 2b2

1

+f3(n)

0 0

-0

)+2R2b2( w,+w, 1

+

0 0 - 0 2

)

( 100)

u, a IR"b2"+II

and

4 4 = [ w o- ( n

+ l ) w , + n u 2 + S,12 + ( ~ , / 2 ) ~

(101) (102)

Here 6, is the bichromatic Bloch-Siegert shift, u, the corresponding resonance width, b = V$j exp(-i41), R = ( V t j / V $ j )exp[i(+, - 42)]rand O(n) = ( 1 - Sflo).The resonance width can be evaluated more explicitly, e.g.,

=41Rb3(

( 0 0 - WA2

+

(00

+02)(00

-0 1 )

(l03a)

and

( 103b)

SEMICLASSICAL FLOQUET THEORIES

247

for the three- and five-photon resonances, respectively. The corresponding resonance shift and width in the GRWA limit becomes, for n # 0, ( 104a)

and ( 104b)

Comparison of Eqs. ( 104a)and ( 104b)with Eqs. ( 100) and ( 103) shows that the influence of the antirotating factors become important when the detunings of the fields, oo- o and coo - w 2 ,are large, and, therefore, cannot be ignored in the calculations. The MMFT has recently been applied to the study of the time evolution of spin4 systems in multiquantum NMR conditions (Zur et al., 1983) driven by intense bichromatic linearly polarized radio-frequency fields (Ho and Chu, 1984) and by bichromatic circularly polarized fields (Chu and Ho, 1984).

,

C. SU(N) DYNAMICAL SYMMETRY AND QUANTUM COHERENCE It has long been known that, for two-level systems, the description of magnetic and optical resonance phenomena can be greatly simplified by the use of the Bloch spin or pseudospin vector (Feynman et al., 1957;Allen and Eberly, 1975). However, extension of the vector description to more complex systems has not been achieved until recently. Hioe and Eberly (1 98 1) found that the dynamical evolution ofN(2 3)-level systems can be expressed in terms of the generalized rotation of an ( N 2- 1)-dimensional real coherence vector S whose property can be analyzed by appealing to the S U ( N ) group symmetry. For example, the time evolution of three-level systems can be described by a coherent vector of constant length rotating in an eight-dimensional space (Elgin, 1980; Hioe and Eberly, 1982). Furthermore, the existence of a number of unexpected nonlinear constants of motion that govern the density matrix of an N-level system was noticed. In particular, for a three-level system under the two-photon resonance condition, the time evolution of the eight-dimensional coherent vector S can be analyzed in terms of the time evolution of three independent vectors of dimensions three, four, and one, rotating in three disjoint subspaces ofthose dimensions, provided that the rotating-wave approximation (RWA) is valid. The three nonlinear constants of motion in this case correspond to the squares of the lengths of these three subvectors. The dynamical symmetry underlying in

248

Shih-I Chu

the three-level system is a reminiscence of the Gell-Mann SU(3) symmetry in particle physics. Thus the subspaces of three, four, and one dimension of S are analogous to the subspaces of pions (n+,no, ll-), kaons ( K + ,ko, K+,KO),and eon (qo), respectively. In practice, however, if the laser-atom interactions occur away from the two-photon resonance, or if the RWA is not valid, or ifdecays are taken into account, then the dynamical subspaces (8 = 3 0 4 0 1) discussed by Hioe and Eberly will be no longer completely independent. The Gell-Mann SU(3) symmetry of the system will then be broken. The study of the SU(N) dynamical evolution of the coherent vector S and the symmetry-breaking effects embodied in N-level systems subjected to an arbitrary number of monochromatic fields can be greatly facilitated by the use of the MMFT (Ho and Chu, 1985). Thus the ( N z - 1)-dimensional coherent vector S ( t ) can be obtained directly from the relation j = 1, 2, . . . , N 2- 1 (106) Sj(t)= Tr[i(t)s,), where s, are appropriate SU(N) generators, and the density matrix ? ( t ) is determined by

Here i ( t o )is the density matrix at the initial time to (initial conditions) and the time-evolution operator 0(t,to) can be determined by the method of MMFT described in Section V,A and expressed in terms of a few quasi-energy eigenvaluesand eigenvectors. Furthermore, the generalized Van Vleck (GVV) nearly degenerate perturbation theory (Kirtman, 1968; Certain and Hirschfelder, 1970; Aravind and Hirschfelder, 1984) can be extended to analytical treatment of the time-independent many-mode Floquet Hamiltonian. In the case of three-level systems near two-photon resonance, for example, the GVV treatment reduces the infinite-dimensional Floquet Hamiltonian to a three-by-three effective Hamiltonian, from which useful analytical properties of the eight-dimensional coherent vector can be readily obtained . Using the MMm-GVV method, Ho and Chu (1985) have recently exploited pictorially the geometry, dynamical evolution, and symmetry-breaking effects in two- and multiphoton excitations of three-level systems under the influence of intense bichromatic fields.

VI. Conclusion In this article we have reviewed the recent developments in semiclassical Floquet theories and their applications to multiphoton excitation, ioniza-


249

tion, and dissociation processes in intense laser fields. Many other subjects have to be left out of this review due to limited space. In particular, the extension of Floquet theory to the study of laser-induced collisions is a new endeavor of fruitful research. Current interests in this direction include inelastic collisions (Vetchinkin et al., 1976; Chu, 1980; Mohan, 1982), chemical reactions (Mohan et al., 1983), and charge-exchange reactions in slow ion-atom collisions (Ho et al., 1984), and free-free transitions (Gavrila and Kaminski; 1984) in laser fields. The utilities and advantages of the Floquet matrix formalism described in this article may be summarized as follows: (1) It is a nonperturbative approach applicable to multiphoton processes involving arbitrary high field strengths. (2) It provides a simple physical picture for the intensity-and time-dependent multiphoton phenomena in terms of avoided crossings of a few number of real or complex quasi-energy levels. (3) Simplicity in numerical computations- mainly an eigenvalue problem. (4) In the case of complex quasi-energy formalism, it takes into account self-consistently all the intermediate level shifts and broadenings and multiply coupled continua. Furthermore, only square-integrablefunctions are required, and no asymptotic boundary conditions need to be enforced in MPI/MPD calculations.

ACKNOWLEDGMENTS This work was supported in part by the Department of Energy, Division of Chemical Sciences,and by the Alfred P. Sloan Foundation. Acknowledgment is also made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work. The author thanks his colleaguesand collaborators,particularly Professor William Reinhardt, Dr. T. S. Ho, and Dr. J. V. Tietz, together with whom many recent results presented in this article were obtained. He is also indebted to Dr. C. Laughlin for reading the manuscript and making several useful comments.

REFERENCES Aguilar, J., and Combes, J. M. (1971). Commun. Math. Phys. 22,265. Alimpiev, S. S., Zikrin, B. O., Sartakov, B. G., and Khokhlov, E. M. (1983). Sov. Phys. JETP 56, 943.

Allen, L., and Eberly, J. H. (1975). “Optical Resonance and Two-Level Atoms.” Wiley, New York. Ambartzumian, R. V., and Letokhov, V. S. (1977). I n “Chemical and Biochemical Applications of Lasers” (C. B. Moore, ed.), Vol. 3, pp. 167-314. Academic Press, New York. Aravind, P. K., and Hirschfelder, J. 0. (1984). J. Phys. Chem. 88,4788.

250

Shih-I Chu

Arnold, V. I., (1978). “Ordinary Differential Equations.” MIT Press, Cambridge, MA. Askar, A. (1974). Phys. Rev. A 10,2395. Autler, S. H., and Townes, C. H. (1955). Phys. Rev. 100,703. Bailkowski, S . E., and Guillory, W. A. (1979). Chem. Phys. Lett. 60,429. Balslev, E., and Combes, J. M. (1971). Commun. Math. Phys. 22,280. Barone, S. R., Narcowich, M. A., and Narcowich, F. J. (1977). Phys. Rev. A 15, 1 109. Bhattacharya, S. K., and Chu, S. I. (1983). J. Phys. B 16, L471. Bhattacharya, S. K., and Chu, S. I. (1985). J. Phys. B 18, L275. Bayfield, J. E. (1979). Phys. Rep. 51C, 317. Beers, B. L., and Armstrong, L. A. (1975). Phys. Rev. A 12,2447. Bernstein, R. B., ed. (1979). “Atom-Molecule Collision Theory.” Plenum, New York. Bloch, F., and Siegert, A. (1940). Phys. Rev. 57, 522. Bloembergen, N., and Yablonovitch, E. (1978). Phys. Today 31,23. Bloembergen, N., Burak, I., Mazur, E., and Simpson, T. B. (1984). Isr. J. Chem. 24, 179. Bonch-Bruevich, A. M., Vartanyan, T. A., and Chigir, N. A. (1979). Sov. Phys. JETP 50, 901. Bordo, V. G., and Kiselev, A. A. (1980). Opf.Specfrosc.49, 563. Braun, P. A., and Petelin, A. N. (1974). Sov. Phys. JETP39,775. Braun, P. A., and Miroshnichenko, G. P. (1980). Opt. Specfrosc.49, 561. Bunkin, F. V., and Prokharov, A. M. (1964). Sov. Phys. JETP 19,739. Burrows, M. D., and Salzman, W. R. (1977). Phys. Rev. A 15, 1636. Cantrell, C. D., Letokhov, V. S., and Makarov, A. A. (1980). In “Coherent Nonlinear Optics” (M. S. Feld and V. S. Letokhov, eds.), pp. 165-269. Springer-Verlag, Berlin and New York. Carrington, A., and Buttenshaw, J. (1981). Mol. Phys. 44,267. Certain, P. R., and Hirschfelder, J. 0. (1970). J. Chem. Phys. 52,5977. Chan, F. T., and Tang, C. L. (1969). Phys. Rev. 185.42. Chin, S. L., and Lambropoulos, P., eds. ( 1984). “Multiphoton Ionization of Atoms.’’ Academic Press, New York. Chu, S. I. (1978a). Chem. Phys. Leu. 54, 367. Chu, S. 1. (1978b). Chem. Phys. Left. 58,462. Chu, S. 1. (1979). Chem. Phys. Left.64, 178. Chu, S. I. (1980). Chem. Phys. Leu. 70,205. Chu, S. I. (1981).J. Chem. Phys. 75,2215. Chu, S . I., and Ho, T. S. (1984). Isr. J. Chem. 24,237. Chu, S. I., and Reinhardt, W. P. (1977). Phys. Rev. Left 39, 1195. Chu, S. I., and Reinhardt, W. P. (1978). In “Multiphoton Processes” (J. H. Eberly and P. Lambropoulos, eds.), pp. 171 - 178. Wiley (Interscience), New York. Chu, S. I., Tietz, J. V., and Datta, K. K. (1982). J. Chem. Phys. 77, 2968. Chu, S. I., Laughlin, C., and Datta, K. K. (1983a). Chem. Phys. Lett 98,476. Chu, S. I., Ho, T. S., and Tietz, J. V. (1 983b). Chem. Phys. Letf.99,422. Clary, D. C. (1983). J. Phys. Chem. 87, 735. Cohen-Tannoudji, C. (1968). “Cargese Lectures in Physics.” Gordon & Breach, New York. Cohen-Tannoudji, C., and Haroche, S. (1969). J. Phys. (Paris) 30, 153. Dalgarno, A,, and Lewis, J. T. (1955). Proc. R. SOC.Ser. A 233, 70. Delone, N. B., Kovarskii, V. A., Masalov, A. V., and Perel’man, N. F. ( I 980). Sov. Phys. Wsp. 23,472. De Vries, P. L., Chang, C., George, T. F., Laskowski, B., and Stallcop, J. R. ( 1 980). Phys. Rev. A 22, 545. Dion, D. R., and Hirschfelder, J. 0. (1976). Adv. Chem. Phys. 35,265.


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Dollard, J. D., and Friedman, C. N. (1977). J. Math. Phys. 18, 1598. Duperrex, R., and van den Bergh, H. (1 980). J. Chem. Phys. 73,585. Edmonds, A. R. (1971). J. Phys. CON.31, C4. Elgin, J. (1980). Phys. Lett. 80A, 140. Fainshtein, A. G., Manakov, N. L., and Rapoport, L. P. (1978). J. Phys. B 11,2561. Faisal, F. H. M. (1976). Nuovo Clim. B 33,775. Faisal, F. H. M., and Moloney, J. V. (1981). J. Phys. B 14,3603. Feshbach, H. ( I 962). Ann. Phys. N. Y. 19,287. Feynman, R. P., Vernon, F. L., and Hellwarth, R. W. (1957). J. Appl. Phys. 28,49. Hoquet, G. (1883). Ann. Ecole Norm. Sup. 12,47. Fock, V. A., and Krylov, S. N. (1947). Z. Eksp. Teor. Fiz 17,93. Frazer, R. A,, Duncan, W. J., and Collar, A. R. (1960). “Elementary Matrices.’’ Cambridge Univ. Press, London and New York. Friedrich, H., and Chu, M. (1983). Phys. Rev. A 28, 1423. Galbraith, H. W., Martin, D., and Steinfeld, J. I. (1982). Phys. Rev. A 26, 1528. Gavrila, M., and Kaminski, J. Z. ( 1 984). Phys. Rev. Lett. 52,6 13. George, T. F. (1982). J. Phys. Chem. 86, 10. Goldberger, M. L., and Watson, K. M. (1964). “Collision Theory.” Wiley, New York. Gontier, Y., Rahman, N. K., and Trahin, M. (1976). Phys. Rev. A 14,2109. Goreslavskii, S. P., and Krainov, V. P. (1979). Sov. Phys. JETP 49, 13. Guccione-Gush, R., and Gush, H. P. (1974). Phys. Rev. A 10, 1474. Haydock, R. (1 980). Solid State Phys. 3 5 2 1 5. Hermann, J., and Swain, S. (1977). J . Phys. B. 10, 1815. Hioe, F. T., and Eberly, J. H. (1981). Phys. Rev. Lett. 47, 838. Hioe, F. T., and Eberly, J. H. (1982). Phys. Rev. A 25, 2168. Hirschfelder, J. 0. (1960). J. Chem. Phys. 33, 1462. Ho, T. S., and Chu, S. I. (1983). J. Chem. Phys. 79,4708. Ho, T. S., and Chu, S. 1. (1984). J. Phys. B 17,2101. Ho, T. S., and Chu, S.1. (1985). Phys. Rev. A 31,659. Ho, T. S., Chu, S. I., and Tietz, J. V. (1983). Chem. Phys. Lett. 96,464. Ho, T. S., Chu, S. I., and Laughlin, C. (1984). J. Chem. Phys. 81,788. Ho, Y. K. (1983). Phys. Rep. 99, 1. Holt, C. R., Raymer, M. G., and Reinhardt, W.P. (1983). Phys. Rev. A 27,2971. Hose, G., and Taylor, H. S. (1982). J. Chem. Phys. 76, 5356. Isenor, N. R., Merchant, V., Hallsworth, R. S., and Richardson, M. C. (1973). Can. J. Phys. 51, 1281.

Jacques, A., and Glorieux, P. (1982). Opt. Commun. 40, 201. Junker, B. R. (1982). Adv. A f . Mol. Phys. 18,207. Kancheva, L., Pushkarov, D., and Rashev, S. (1981). J. Phys. B 14,573. Kelleher, D. E., Ligare, M., and Brewer, L. R. (1985). Phys. Rev. A 31,2747. Kirtman, B. (1968). J. Chem. Phys. 49, 3890. Koren, G. (1983). J. Appl. Phys. 54,2827. Lambropoulos, P. (1976). Adv. At. Mol. Phys. 12, 87. Lanczos, C. (1950). J. Res. Natl. Bur. Sfand.45,255. Lane, A. M., and Thomas R. G. (1958). Rev. Mod. Phys. 30,257. Langhoff, P. W., Epstein, S.T., and Karplus, M. (1970). Rev. Mod. Phys. 44,602. Laughlin, C., and Chu, S. I. (1985). To be published. Leasure, S. C., and Wyatt, R. E. (1980). Opt. Eng. 19,46. Leasure, S . C., Milfeld, K.F., and Wyatt, R. E. (1981). J. Chem. Phys. 74, 6197. Leforestier, C., and Wyatt, R. E. (1982). Phys. Rev. A 25, 1250.

252

Shih-I Chu

Leforestier, C., and Wyatt, R. E. (1983). J. Chem. Phys. 78,2334. Letokhov, V. S., and Moore, B. C. (1976). Sov. J. Quantum Electron. 6, 129, 259, Magnus, W. (1954). Commun. Pure Appl. Math. 7,649. Makarov, V. P., and Fedorov, M. V. ( 1 976). Sov. Phys. JETP 43,6 15. Manakov, N. L., and Fainshtein, A. G. (1981). Teor. Mat. Fiz.48, 375. Manakov, N. L., Ovsyannikov, V. D., and Rapoport, L. P. (1976). Sov. Phys. JETP43, 885. Maquet, A., Chu, S. I., and Reinhardt, W. P. (1983). Phys. Rev. A 27, 2946. Milfeld, K. F., and Wyatt, R. E. (1983). Phys. Rev. A 27,72. Milfeld, K . F., and Wyatt, R. E. (1985). J. Chem. Phys. To be published. Minorsky, N. ( 1962). “Nonlinear Oscillations.” Van Nostrand-Reinhold, Princeton, N.J. Mohan, M. (1982). J. Chem. Phys. 75, 1772. Mohan, M., Milfeld, K. F., and Wyatt, R. E. (1983). Chem. Phys. Leff.99,411. Moloney, J. V., and Faisal, F. H. M. (1979). J. Phys. B 12,2829. Moloney, J. V., and Meath, W. J. (1975). Mol. Phys. 30, 171. Moloney, J. V., and Meath, W. J. (1976). Mol. Phys. 31, 1537. Moloney, J. V., and Meath, W. J. (l978a). J. Phys. B 11, 2641. Moloney, J. V., and Meath, W. J. (1978b). Mol. Phys. 35, 1163. Moloney, J. V., and Meath, W. J. (1978~).Phys. Rev. A 17, 1550. Morellec, J., Normand, D., and Patti, G. (1982). Adv. At. Mol. Phys. 18, 97. Moulton, F. R. ( 1930). “Differential Equations.” Macmillan, New York. Mower, L. ( 1 980). Phys. Rev. A 22,882. Nauts, A., and Wyatt, R. E. (1983). Phys. Rev. Left.51, 2238. Nauts, A., and Wyatt, R. E. (1984). Phys. Rev. A 30,872. Okuniewicz, J. M. (1974). J. Math. Phys. 15, 1587. Pechukas, P., and Light, J. C. (1965). J. Chem. Phys. 44,3897. Pegg, D. T. ( I 973). J. Phys. B 6,246. Poincark, J. H. ( I 892, 1893, 1899). “Les Methodes Nouvelles de la Mechanique Celeste,” Vols. I, 11, IV. Paris. Preobrazhenskii, M. A., and Rapoport, L. P. (1980). Sov. Phys. JETPSI, 468. Proch, D., and Schroder, H. (1979). Chem. Phys. Lett. 61,426. Pullman, N. J. (1976). “Matrix Theory and Its Applications.” Dekker, New York. Quack, M.(1982). Adv. Chem. Phys. 50,395. Quack, M.,and Sutcliffe, E. (1984). Chem. Phys. Lett. 105, 147. Rabi, 1. I. (1937). Phys. Rev. 51,652. Ramsey, N. F. (1955). Phys. Rev. 100, 1191. Reinhardt, W. P. (1982). Annu. Rev. Phys. Chem. 33,223. Ritus, V. 1. (1966).Sov. Phys. JETP51, 1544. Salwen, H.(1955). Phys. Rev.99, 1274. Salzman, W. R. (1974a). Phys. Rev. A 10,461. Salzman, W. R. (1974b). Chem. Phys. Lett. 25, 302. Salzman, W. R. (1977). Phys. Rev. A 16, 1552. Sambe, H. (1973). Phys. Rev. A 7,2203. Schek, I., Sage, M. L., and Jortner, J. (1979). Chem. Phys. Left.63,230. Schek, I., Jortner, J., and Sage, M. L. ( 1 98 I). Chem. Phys. 59, 1 I. Sen Gupta, N. D. (1970). J. Phys. A 3,618. Series, G . W. (1978). Phys. Rep. 43C, 1. Shirley, J. H. (1963). Ph.D. thesis, California Institute of Technology. Shirley, J. H. (1965). Phys. Rev. 138, B979. Shulz, P. A., Sudbo, A. S., Krajnovitch, D. J., Kwok, H. S., Shen, Y. R., and Lee, Y. T. (1979). Annu. Rev. Phys. Chem. 30, 379.


253

Simon, B. (1973). Ann. Math. 97,247. Simpson, T. B., and Bloembergen, N. (1981). Opt. Commun. 37,256. Simpson, T. B., and Bloembergen, N. (1983). Chem. Phys. Lett. 100,325. Smith, L. N., Augustin, S. D., and Rabitz, H. (1982). J. Comp. Phys. 45, 417 Sokolov, V. V. (1978). Teor. Mat. Fiz.35, 339. Stenholm, S. (1973). J. Phys. B6, 1650. Swain, S. (1975). J. Phys. A 8, 1477. Thomas, G. F. (1983). J. Chem. Phys. 79,4912. Thomas, G. F., and Meath, W. J. (1982). Mol. Phys. 46,743. Thomas, G. F., and Meath, W. J. (1983). J. Phys. B 16,951. Tietz, J. V., and Chu, S. 1. (1983a). In “Spectral Line Shapes” (K. Burnett, ed.),Vol. 2, pp. 863-877. De Gruyter, Berlin. Tietz. J. V., and Chu, S.I. (1983b). Chem. Phys. Leu. 101,446. Tietz, J. V., and Chu, S. I. (1985). To be published. Tip, A. (1983). J. Phys. A 16, 3237. Tsukada, N. (1974). Phys. Lett. 47A, 265. Vetchinkin, S. I., Bakhrakh, V. L., and Umanskii, I. M. (1976). Opt. Spectrosc. 40,28. Walker, R. B., and Preston, R. K. (1977). J. Chem. Phys. 67,2017. Wang, K., Ho,T. S., and Chu, S. I. (1985). J. Phys. B (in press). Whaley, K. B., and Light, J. C. (1982). J. Chem. Phys. 77, 1818. Whaley, K. B., and Light, J. C. (1984). Phys. Rev. A 29, 1188. Winston, P. H. (1979). “Artificial Intelligence.” Addison-Wesley, Reading, MA. Wolk, G. L., Weston, R. E., and Flynn, G. W. (1980). J. Chem. Phys. 73, 1649. Wyatt, R. E., Hose,G., and Taylor, H. S.(1983). Phys. Rev. A 28,815. Yamani, H. A., and Reinhardt, W.P. (1975). Phys. Rev. A 11, 1144. Yeh, S., and Stehle, P. (1977). Phys. Rev. A IS, 213. Young, R. H., Deal, W. J., and Kestner, N. R. (1966). Mol. Phys. 17,369. Zel’dovich, Ya. B. (1966). Sov. Phys. JETP51, 1492. Zel’dovich, Ya. B. (1973). Sov. Phys. Usp. 16, 427. Zon, B. A., and Katsnel’son, B. G. (1974). Sov. Phys. JETP38,470. Zur, Y., and Vega, S. ( 1983). J. Chem. Phys. 79,548. Zur, Y., Levitt, M. H., and Vega, S. (1983). J. Chem. Phys. 78, 5293.


ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 21

SCATTERING IN STRONG MAGNETIC FIELDS M. R. C.MCDOWELL Department of Mathematics Royal Holloway and Bedford New College University of London Egham. Surrey. England

M. ZARCONE Institute of Physics University of Palermo Palermo, Italy

.......................... ................... 111. Potential Scattering . . . . . . . . . . . . . . . . . . . . . . . A. First Born Approximation . . . . . . . . . . . . . . . . . . I. Introduction

11. Center-of-Mass Separation .

IV.

V. VI.

VII. VIII.

B. The Bremsstrahlung Problem . . . . . . . . . . . . . . . C. The Born Series . . . . . . . . . . . . . . . . . . . . . . . D. Coupled-Equations Formulation . . . . . . . . . . . . . Ensembles of Landau Levels . . . . . . . . . . . . . . . . . A. Magnetic Field Perpendicular to the Z Direction . . . . . . B. Magnetic Field Parallel to the Z Axis . . . . . . . . . . . The Low-Field Limit of the Cross Section . . . . . . . . . . . Photoionization. . . . . . . . . . . . . . . . . . . . . . . . . Photodetachment of Negative Ions . . . . . . . . . . . . . . Charge Exchange . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

..

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

255 258 26 1 266 267 268 272 277 278 278 28 1 285 293 297 303

I. Introduction The study of atomic collisions in the presence of strong fields has received increasing attention in recent years. The presence of an external field changes, significantly,the conditions of the scattering processes; the electromagnetic field, exchanging energy and momentum with the projectile and the target, can play the role of a third body, opening new channels and 255 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form resewed.

256

M. R. C. McDowell and M. Zarcone

allowing the observation of collision parameters which would not otherwise by observable. [A rather complete and updated list of contributors on these topics may be found in the Proceedings of the International Colloquium on Atomic and Molecular Physics Close to Ionisation Threshold in High Fields (Connerade et al., 1982).] In this article we shall consider collision processes in the presence of a strong magnetic field. Usually under laboratory conditions the energy changes caused by a magnetic interaction are small compared with the characteristic energies of the system; so that the scatteringprocesses are not affected by the presence of the field while its interaction with the target atom can be treated perturbatively. However, in experiments with highly excited atoms, in solid state physics and in astrophysics, situations are encountered where perturbation theory is not applicable to the target, and the effect of the field on the collision process is not negligible. Magnetic fields are of much interest in astrophysics. This interest dates from the discovery by Hale in 1908 of magnetic fields in sunspots from the Zeeman splitting of their spectral lines. More recently the discovery by Kemp et al. ( 1970) of circularly polarized continuum radiation from a white dwarf and its interpretation as being due to a magnetic field of lo7G has led to renewed interest in the study of atomic properties in strong fields. Since then the existenceof large magnetic fields in pulsars, thought to be up to 10l2 G at the surface, has also been demonstrated (see Rudeman, 1972). The situation in a laboratorycontext is very different. In fact, the strongest magnetic field used in the laboratory is about 1 O6 G. Most classical Zeemaneffect studies have been performed routinely at fields typically in the range 2 X lo4- 5 X lo4G . Several laboratories can produce fields of 20 X lo4 G over a useful distance (- 5 cm). Higher magnetic fields, up to 1.6 X lo6 G , have been produced transiently by discharging a large capacitancethrough a single turn coil by, e.g., Furth et al. (1957). Considerably higher fields have been produced by implosion techniques. A field of between lo4and lo5G is produced and then the field is rapidly compressed in microseconds by an explosive device. Fowler et al. ( 1960) reported a field of 1.4 X lo7G lasting for 2 ps. This technique has the disadvantage of being self-destructive. Owing to the intense stress in the walls of the containing vessel and the thermal heating caused by eddy currents during pulses, an upper limit exists for the production of steady fields in the laboratory. This limit depends somewhat on the available materials but is typically about lo6 G. These intensities are not very high; in fact, in atomic units, the unit of the magnetic field is equal to 4.96 X lo9 G . However, in some particular cases the effect of high fields may be mimicked at low fields. For instance, in solids (1) the mass of an electron in

SCATTERING IN STRONG MAGNETIC FIELDS

257

motion is represented by the effective mass m *, which may be several orders of magnitude smaller than the mass of the electron in free space m, and (2) the dielectric constant is not equal to 1, as in the case of free space, but may have a value in the range 10 to 50 (see, e.g., Praddaude, 1972). Both of these facts contribute significantly to the change in the ratio ofthe magneticenergy to the Coulomb energy (denoted by y ) from the case where the atom exists in free space. We have

where o,= eB/m * is the cyclotron gyrofrequency and %'* = m*e4/2 h2D2

(2)

is the effective Rydberg, with D the dielectric constant. Now, if we suppose D = 50 and m* = 0. lm, then y is a factor of 2.5 X los greater than for the case where D = 1 and m* = m. In other words, if a magnetic field of strength lo4G (a fairly weak field) was applied to the solid the effects observed could be those of a field of 2.5 X lo9 G (a strong field) in free space. In free space D = 1, the energy corresponding to ho, is 1 Ry when y = 1; i.e., B = 2.48 X lo9 G. From now on, a weak magnetic field will be referred to as one in which the ionization energy 1, of the target dominates the Landau energy h a , of the magnetic field so that the magnetic potential may be treated as perturbation and both the projectile and the target as unperturbed system. The region of field strengths in which this occurs for atoms in their ground state is y ZJ. For the case of scattering of a charged particle by a Coulomb potential, the unperturbed system is given by the projectile embedded in the magnetic field, while the Coulomb potential is considered perturbatively. In this case y > 1 and B > lo9 G. Moreover, a magnetic field also affects the structure of the atomic target. 1 it causes the ordinary Zeeman splitting, for y 6 1 the quadratic For y Zeeman term also becomes important, and for y > 1 the magnetic field completely dominates the Coulomb field except in the field direction, and we move into what is known as the quasi-landau regime where the motion of the atomic electrons is close to that of free electrons in a magnetic field, and the atom takes a characteristic cigar shape along the magnetic field.These effects on the atomic levels are described more fully by Garstang (1977) and by Clark ef al. (1984).

258

M. R. C.McDowell and M. Zarcone

11. Center-of-Mass Separation The formal separation of the center-of-mass motion in a scattering problem in the absence of an external field is straightforward. This separation is possible if the total Hamiltonian can be expressed as a sum of two parts: one relative to a fixed center of mass and another describing the uniform centerof-mass motion. In such a case the two-body scattering problem reduces to the study of the interaction of a single particle with reduced mass p = m, m,/M (and M = m, m2)interacting with a central potential. The situation for collisions in the presence of a constant and uniform magnetic field is more complex. In this case, if the total system of interacting particles has a net charge Z,e different from zero, the center of mass moves with a cyclotron frequency wG = Z,eB/Mc in the magnetic field, and the center of mass system is not an inertial system. In other words, if the net charge is zero, the magnetic action on the whole system is zero (no external force, inertial system), but it is not zero on the single particles. If the net charge is not zero, then a Lorentz force Z , e( I /c)VG X B acts on the closed system composed of the interacting particles. This external Lorentz force makes the center-ofmass frame a noninertial frame, and the total momentum of the center of mass is not conserved. In this case it is not particularly convenient to study the process in the center-of-mass frame. We will consider only the case when the net charge is zero or near to zero so that the cyclotron motion of the center of mass is very slow and is negligible compared with the relative motion of the colliding particles. The general question of the separability of the center-of-mass motion for an N-particle system in a uniform magnetic field has been examined in detail by Avrom et al. (1978) following Carter (1968), who solved, exactly, the problem for a neutral two-body system. For two particles, with positron vectors r, and r2 in the laboratory frame, masses rn, and rn2, charges Z , e and Z2e interacting through a potential V(r, - r2), the Hamiltonian in the laboratory frame is given by

+

where Hi is the kinetic momentum of the ith particle

I l i = - ih V, - (2,e/c)A(ri) (4) We choose the z axis of our coordinate system in the direction of the magnetic field B with B = V,A(r) (5) Following Carter, we denote the center-of-mass coordinate by R = (X, Y,Z)


259

and the relative coordinates by r = (x,y,z). They are related to position vectors ri by

r = r2 - rl

(6a)

and

R =a r l

+ pr2

where (64 a = m , / ( m ,+ m2), P = m2/(m, + m2) Requiring that A(r) be linear in r, we obtain the Hamiltonian in terms of the coordinates R and r as H = - - Vh2 f-2p

h2

2m

V$

1 + V(r) + ?[DA2(R) + 2EA(R) 2c

A(r)

+ FA2(r)]

where we have assumed

V, A(r) = 0

(8)

to ensure that A and V, commute. In Eq. (7) the constants are

+

D = e2(Z:/ml Z $ / m 2 ) E = e 2 ( a Z $ / m2 pZ:/m,)

+

F = eZ(P22:/m, a 2 Z $ / m 2 ) G = e(Z,/m2 - Z,/m,) Z = e ( Z , Z2)/M

(9)

+

J = e ( Z , m 2 / m+ , z2m1/m2)/M

Choosing the Landau gauge

A,,(r) = B,,

A,(r) = A,(r)

=0

(10)

Carter showed that

n,, = -ih a/au) and

(1 1)

260

M . R.C. McDoweN and M. Zarcone

commute with Hand are constants of motion, but in general llxand Il,are not both constants of motion. Their commutator

[n,, n Y l = (Ph/ic)e(Z, + 2,) (13) is zero only if the net charge of the system 2, = 2, 2, is zero. For a total charge of the system equal to zero, ll, and l l y are, then, simultaneously constants of motion, and the most general simultaneous eigenstate of these operators and the Hamiltonian H i s

+

w(X,Y,r) = exp[i(Px + eB,,W] exp(iP, Y )4(r) (14) where Px and P, are the eigenvalues of l l x and n,, respectively. Px and P, are then the X and Y components of the center-of-mass momentum P. The Schr6dinger equation Hy/=Eyl

(15)

reduces to

How = E w where

1 - - e r (v X B) C

e2BZ +-2Mc,

(XZ

+ YZ)

where a is a vector potential for the effective magnetic field b = gB = 0,X a

(18)

with g = (m2- m1)/(m2 + m,)

(19)

s o t h a t g = 1 ifm, >> m,. In Eq. ( 12) v = P/M is the velocity of the center-of-mass system, and the term involving v is the potential for the electric field

E = (l/c)(v X B)

(20)

which is present for an observer moving with velocity v relative to the laboratory frame. In the center-of-mass system, taking P = 0, Hois given by


26 1

and one obtains an effective Hamiltonian formally identical to that for a particle of reduced mass ,u moving in a reduced magnetic field and in a harmonic-oscillator potential. OConnell ( 1979), choosing the gauge

in the Hamiltonian [Eq. (2 l)], obtained

H,

=

fi2 e -Vf + -gBL,

2P

2P

e2 +B2(x2+ y z )+ V(r) 8,u

(23)

where L, is the z component of the angular momentum. The Hamiltonian [Eq. (23)] is formally identical to the usual Hamiltonian for the case of m2 infinite, with m, replaced by the reduced mass p and B by the reduced magnetic field b = gB.

111. Potential Scattering The first attempts to treat potential scattering in a magnetic field arose in connection with the problem of Bremsstrahlung (“free - free transitions”) of an electron in the field (Canuto and Chiu, 1970). The first Born approximation matrix element for a Coulomb-potentialinduced transition between two Landau states (see below) was first evaluated by Klepikov (1952) in another connection. Tonnewald (1959) made a JWKB approach to a special case, and Goldman (1 964) gives a variant of his result. Canuto et al. (1969) attempted to tackle the Bremsstrahlung problem for very high fields using a relativistic treatment, but simplifying by replacing the Green’s function by the field-free Green’s function. Canuto and Chiu ( 1970) merely give a closed-form result for the first Born matrix element, using the expression for the integral involved given by Klepikov. The problem was also addressed by Vitramo and Jauho ( 1975),but restricted to elastic transitions in the lowest state. We return, briefly, to the Bremsstrahlung problem below. The first detailed treatment ofthe nonrelativistic scattering problem is due to Ventura (1973), and his treatment using a Green’s function approach has been extended in several ways by Brandi et al. ( 1978) and by Ferrante et al. ( 1980).

262

M. R. C.McDowell and M, Zarcone

We consider a particle of reduced mass and charge e in a static uniform magnetic field, whose direction is taken to be the z axis,

B = B, The Hamiltonian for this problem is Ifo = ( 1/2m)l12,

(24)

ll = - ih V, - (e/c)A(r)

(25) where r is the position vector of the particle with respect to an arbitrary origin. It is convenient to adopt the Landau gauge

A=fBXr We adopt cyclindrical polar coordinates and write

(26)

Ho'Y%(p, 4, Z ) = E n ~ ~ ' c o 4, ) ( Zp), (27) The solutions are the well-known Landau functions and may be written Y$,&(r) = L;1/2eikl@nm(p, 4)

(28)

=L; lI2eikr@nm (p)

where we have restricted the motion along the field to the interval (-Lz, Lz), so that the wave functions are products ofa plane-wave factor for the motion along z and a two-dimensional harmonic-oscillator function in the plane perpendicular to the field. Here @nnt(P) = eidlnm(P)

(29)

where LB,(x)are associated Laguerre polynomials defined as

and fs=y p 2

(32)

y = eB/2ch

(33)

while gives the field strength. Spin may be neglected, as its only effect is to make all levels (except n = 0) doubly degenerate. The energy eigenvalues are

End

= k2

+ 4y[n + f ( m + Iml+ l)]

Ry

(34)


263

The z component of the angular momentum is also a constant of motion

i,y i % k

(35)

= mh y i % k

In the literature, the Landau wave functions, Eq. (28), and energies, Eq. (34), are often given in terms of the quantum numbers Nand s such that EN,k =

+

k 2 4y(N

+ 4)

Ry

(36)

and the component of angular momentum along the field is mh, where

m=N-s (37) The quantum number pairs (N, s) and (n,m) are linearly dependent, with N =n

+ +(lml+rn)

(38)

The principal quantum number N determines the energy of the motion perpendicular to the field and the radius of the electron's classical orbit RN=

+

+ + l)]

y-'IZ(N+ +)'I2 =~ - ' / ~ [ n +(m (ml

(39)

while the degenerate quantum number s specifies the distance of the center of the classical orbit from the z axis,

R, = y-'12(s

+

+)'IZ = y-'12[n

+ +(lml- m + l)]

(40)

Now suppose there is also a potential present, so that

H

= H,

+ V(r)

(41) Provided we can treat the potential as a perturbation, we can now develop a suitable version of scattering theory in which the unperturbed states Inmk) are the Landau states and the perturbation is considered as causing transitions between them. Most discussions to date have supposed that we can consider that at infinity the system is in a single Landau state, say, In,,rn,b) and scatters to some final Landau state. Clearly one needs to be able to calculate the probability of such a transition, but it does not necessarily lead to expectation values of observables. Any real ensemble of systems will be in an ensemble of Landau states. The only case to have been considered in detail is the case when outside the field the system may be considered as an incident plane wave exp(i k r). The nature of this ensemble will be considered in the next section. In this section we restrict ourselves to the simple case of transitions out of a single Landau state. If we write

+

( H , v- k)Y = EY, € > 0 (42) for the Schrodinger equation, that is, we consider a complex energy

264 W=E

M. R. C. McDowell and M. Zarcone

+ k, then we can formally write down an integral equation y+= yco) - -V Y + Ho - w

(43)

H,Y‘a = Ey(O) (44) which is analogous to the usual Lippman -Schwinger equation. We stress that Eq. (43)is purely formal and has not been established on a wave-packet basis, unlike the field-freecase. Nevertheless, Eq. (43)is the starting point for Ventura (1973),Brandi et al. (1978),and Ferrante et al. (1980). If we denote (Ho - W)-l by the Green’s operator G $ , the corresponding Green’s function satisfies (Ho - W)Go(r, r’) = -d(r, r’)

(45)

which has the solution

for some g,,(k). Substituting Eq. (46)into Eq. (43,

+

gn,(k) = [(E - Enm k ) - h2k2/2p]-’

(47)

We then have, using Eq. (34)for En,,

introducing the quantity

+

k;, = (2p/h2)(E- Enm) k 2

(49)

The integration in dk gives the outgoing wave solution (Ventura, 1973)

For any open channel (k;, > 0) the wave equation for the scattering process in the integral representation is given by Yr(r) = ‘€‘y)(r)

+

I

G:(r, r’; E)V(r’)YF(r’) dr’

(51)

265


Using Eq. ( 5 1) and the asymptotic form of Yt, relative to transitions from a single Landau state, and writing kf for k,, lim Yt(r) = lim L ; 1 / 2 e i b Z @ n m ( p ) 1ZI-m

IM--

+

@n’m’(p)L;We-ikpA

fi(-

kfH-

a]

n’m‘

is the transmitted, A,(- kf)the

in which 8(x) is the step function and A&/) reflected, scattering amplitude,

A&,)

i/ L ti2 h 2kf

I

= - - dr exp(-

ikfz)@,,,, (p) V ( r ) Y t(r)

( 5 3)

which we write in terms of the usual T matrix as

A / i ( k f ) = - ( i / L z / h ’ k / ) (fITIi) At least formally one can make the usual Born series expansion

(J IT 1) i ) = T,

= (f

and letting

1 V li) + ( f I VG: V li)

-

A Eab = ‘n,,

then explicitly

Tfi= v,

m. k.

+ c VAEia avai + a

-

+ -*

‘fiVabvbi

(55)

(56)

rnb kb

A Ei, A Eib

(54)

+

...

(57)

where the summations imply an integral over the corresponding continuum. The incident flux in the direction of the field is given by

A uniform-density beam is represented by a statistical mixture of degenerate electron states of different values of m(n)with the same weight, so

Then the cross section for a transition from a particular initial statelnmk) to a final state In’m’k’) is given by the total scattered current in the state

266

M . R. C.McDowell and M . Zarcone

(n'rn'k') divided by the incident flux. From Eq. (52) we have k' k' dp L, [email protected] = -IAntm.12 Lz Using Eqs. (54), (59), and (60), we have

Zmt =

(60)

A. FIRSTBORNAPPROXIMATION

The first Born matrix element has been evaluated by various authors. Following Ventura ( 1973) we consider a screened Coulomb potential

V(r)= Ar-'e-P', /3 > 0 Then (noting that because of axial symmetry rn = rn'

(62) = rn),

where we have used

(64)

(Gradshteyn and Ryzhik, 1980, No. 3.96 1.2) and KOis the modified Bessel function of imaginary argument and zero order, while 0'

= y - ' 0 = p2

(65)

and k is the momentum transfer

k = kf- k,

k=l k I The integral over 0 can be evaluated in closed form and yields a confluent hypergeometric function of the second kind, so that finally

with

X = (4y)-'( /I2 + k2) The result for the pure Coulomb potential follows in the limit /3 + 0. Para-

267


metric differentiation with respect to /?gives the result for the potential N

V(r) =

2

ajrje-br

(69)

j-- I

which is general enough for most purposes. Use of the more general expression

Rev>-+,

Rex>O,

a>O

allows reduction to a single quadrature for inverse power potentials r-“, n being an integer. The expression for (nrnkl Vln’rn’k’) is only weakly dependent on k’. Consequently the first Born approximation to the cross section is

This diverges as k’-’ when k’ +0, and this occurs at the threshold for excitation for In’m’k’); n’ > n. It now appears (Ohsaki, 1983a) that these singularities do not appear in practice. That is, the Born series does not converge to its first term. This is not unexpected, since the potential is effectively one-dimensional,and appears to be connected with the fact that all attractive one-dimensional potentials support at least one bound state (Clark, 1983).

B. THEBrernsstruhlung Problem Using the results for the Born matrix element, Lauer et al. (1983) have readdressed the problem of electron - ion Bremsstrahlung in a very strong magnetic field of order lo7 to lo9 T, the range of interest in neutron star studies. They assume a static Coulomb potential and a uniform static magnetic field

B = (0,0, B) (72) in the z direction, and consider, in the nonrelativisitic approximation, the radiation emitted, in the dipole approximation when an incoming electron in the lowest Landau level 10, - m , k,) is decelerated to a Landau state 10,m’, k i ) , emitting a photon of energy

h v = k2 - ki2= hw

(73)

M . R. C.McDowell and M . Zarcone

268

They give an expression, reduced to a quadrature, for the cross section averaged over m, summed over m’, which to lowest order in x ,

x = 4 AZ(ki

+ k;)

when the only (n, m, m’) combinations allowed in the triple sum over n, m, and m’ are (Virtamo and Janho, 1975) (0, m, m), (1 , m, m l), (1 , m, m - 1) reduces to

+

d2u - mE5 mc2 d R d( h w / E ) - 2nlk2k:l

hw -

X [Ie- I’Q ? - G ~ ( P+ ) Ie+ I’Q:+GI( P)

(74)

+ lezlZQiLQf+G,(~)l G J P ) = (1 + P)ePJvP) - 1

[n w c - w ( 2 + g$ hW2 Q,+=w nw,+w cos + 2mc2 [ ( Q,-=w

1--

cos 8 )

cos2 8

1--

8)

cosz 8

where E,(x)is the exponential integral. Also

I‘

u = azra

(76) where 8is the polar angle of the emitted photon and the polarization vector is e = (e+, e-, ez). They give detailed tables, but in view of what we have to say below about the convergence of the Born series, the expression in Eq. (74) and the results must be treated with reserve, except far from Landau thresholds.

C. THEBORNSERIES The Lippmann-Schwinger equation is, by Eq. (5 1) Y:,Jr)

= YfLk(r)

Iterating by putting series Y:mk=(l

+

I

for Y& :

+ G:V+

G:(r, r’, E)V(r’)Yzmk(r’)dr’

(77)

under the integral, we obtain the Born

G:V+ G:VG$V+

.*.)‘Pioh

(78)


269

Consider the second term, with q = (n,m):

I

G i ( r ,r ’ ; E)V(r’)Yi:mOk,(r’) dr’ =

/

G$(r, r ’ ; E)V(r’)Yr)(r’)dr‘

(79)

which we write as

with

and for an axially symmetric potential, m = m‘. Provided V0,Jk)has no poles other than the cyclotron resonance (and this occurs provided we are off resonance even for a structured target), we can complete the integral over intermediate wave numbers to obtain (Ohsaki, 1983a)

-

Let IzJ so that the first Born approximation to the amplitude for the In, k ) -In’, k ’ ) transition is CQ

for fixed m. With the same assumption the full amplitude is

270

M. R. C.McDowell and M . Zarcone

with

This may be formally summed if we allow only one inelastic transition and N - 1 elastic transitions in the Nth order. Ohsaki calls this the higher-order modified Born Approximation (HMBA) and obtains

for elastic transitions, and

for the inelastic transition. Both expressions may be summed as geometric series provided I(h2kn/p)vnn(0)I < 1,

I(h2knt/p)vn,nl(0)I

R, and vanishes for r < R, (the ion-sphereboundary condition). Dharma-Wardana and Perrot ( 1982) calculate p+(r) from an ion pair-correlation function g(r) obtained by the hypernetted-chain approximation (a fluid-structure theory). Other approximations are examined by Berggren and Froman ( 1969), Davis and Blaha ( 1982), Cauble et al. ( 1 984), and Perrot ( 1 982). In terms of this potential, the nonrelativistic one-electron Schrodinger equation is h2

-2m V2vs- eW)vs= w s ( r )

(2)

308

R . M. More

FIG. 1. Schematic representation of the spherical-cell model. A nucleus of charge Ze is located at the center of a cavity of radius Ro in a positive charge background p+(r). The cavity radius is fixed by the matter density. The electron distribution is calculated from a self-consistent average potential V(r)and the chemical potential is chosen so that the cell is electrically neutral. If the exterior positive charge densityp+(r) exactly equals the exterior electron density en@),then the potential vanishes for all r 3 R,.

The one-electron quantum number s specifies the energy, angular momentum I, and z component of orbital and spin angular momentum. Most authors add exchange and correlation corrections to the potential V(r),but these are not essential to our discussion and will be omitted for simplicity. Some other corrections and caveats are discussed below. We assume that V(r)-+ 0 for r m. If the electron chemical potential is chosen properly, the entire system is electrically neutral and this boundary condition will be satisfied. A stronger boundary condition, V(r)= 0 for all r > R,, could also be imposed. This would require that the cell itself be neutral and also that the external positive charge density be identically equal to the external electron density [i.e., p+(r) = en(r)].This stronger boundary condition is also sometimes called the ion-sphere model.

-

PRESSURE IONIZATION

309

The Schrodinger equation has two classes of solutions which have immediate physical interpretation. These are: (1) bound states with

which correspond to a purely imaginary wave vector k = - iy and negative energy E , = -( h2/2m)y2,and (2) continuum states

with wave vector k and positive energy E = ( h2/2m)k2. The scatteringphase shift S,(k) is determined by integrating the Schrodinger equation out from r=O. Resonances appear in this description as a rapid rise in 6,(k),occurring over an energy range of width r,, near energy En/.This behavior of S,(k) reflects the existence of a special solution of Schradinger's equation having a quantized complex energy En/=En, irn,.There is also a pole of the Green's function at energy En,.The precise definition of En/is considered in Section 111. In thermal equilibrium,the real-energy (physical)eigenstatesare assumed to be occupied according to the Fermi- Dirac distribution,

+

f ( E J = [1

+ exP(E, - p)/kTI-'

(3) where p is the chemical potential and kT is the temperature in energy units. The electron number density n(r) is obtained by summing over a complete set of bound and free states,

This electron density must be consistent with that assumed in forming the original potential V(r). The SCF method is often called the average-atom (AA) model, because Eq. (3) assigns a fractional occupationto each one-electron state according to the Fermi - Dirac statistics of noninteracting electrons. Equations (3) and (4) give the average occupation and charge density of eigenstates v, with real energies. Formulas which give the corresponding information for resonance states are developed in Section IV.

B. LIMITATIONS OF THE AVERAGE-ATOM MODEL One limitation of the AA is its smoothed representation of the exterior environment. The potential V(r) is understood to be averaged over the

R.M. More

3 10

positions of plasma ions and electrons (with the central nucleus held fixed at r = 0). The debate between ion-sphere, Debye-Huckel, or hypernettedchain approximations can easily miss an important point, namely, at large radius, the average potential V(r)approaches zero but its fluctuations do not. Asymptotically, the fluctuations produced by the random spatial arrangement of distant ions become constant (i.e., independent of distance). A simple estimate of the rms potential fluctuation at large distances follows from the ion-sphere model. Consider a field point r far away from the central ion. For each configuration of the plasma ions, this point falls somewhere within the proximity cell of the nearest ion. We form SV&,,= ( V 2 ( r ) )- ( V(r))2as an average over positions within this distant ion sphere and find SV,,

=

= 0.8 Ze

Ro

In a qualitative sense, the effect of these fluctuations is to produce ion Stark broadening. The fluctuations are large perturbations for any state (bound or free) having energy

k s l e SV,,

(6)

In a real plasma the environment near the atom has a lower symmetry than assumed in the spherical-cell picture, giving rise to electric microfields which fluctuate on time scales associated with electron and ion velocities. These microfields are ignored by the basic self-consistent-field model, although there is no doubt about their important consequences (e.g., line broadening). Despite these related difficulties, we continue with the average-atom model to see precisely what it predicts for electron states near the continuum boundary. Another important limitation on average-atom calculations is associated with the density and temperature dependence of average-atom eigenvalues. These eigenvalues change strongly with density and must not be taken too literally. In a conventional spectroscopic experiment, one resolves and identifies the lines of a specific ion (e.g., heliumlike silicon). These lines depend much less strongly on density than do the average-atom K-shell eigenvalues, which do not describe the energies of heliumlike ions but rather an average of K-shell eigenvalues of all ions in the plasma. is )caused by Much of the density dependence of the eigenvalue E , ( ~ , T changes in the plasma ionization. This is illustrated by approximate eigenvalues for a niobium plasma given in Table I. (The numbers are given to more figures than physically significant.) The eigenvalues are obtained from the nonrelativistic WKB approxima-

31 1

PRESSURE IONIZATION

TABLE I DENSITY-DEPENDENT AVERAGE-ATOM EIGENVALUES (Z=41) Case 1

Case 2

Case 3

200 eV 0.0 1 g/cm

200 eV 0.1 g/cm3

276.13 eV 0.1 g/cm

26.87 -19.615 keV - 3.787 - 3.605

23.29 - 19.326 keV - 3.523 - 3.335

~

T P

Z* IS

2s 2P 3s 3P 3d

- 1.445 - 1.382 - 1.284

- 1.247 - 1.179 - 1.070

26.87

- 19.684 keV - 3.835 - 3.658 - 1.452 - 1.391 - 1.293

tion for a TFD potential (More, 1982).Cases 1 and 2 refer to equal temperatures, and Cases 1 and 3 have the same ionization state Z*. Comparing eigenvalues at the same temperature (Cases 1,2), one sees changes of 10% per decade of density. If we instead compare eigenvalues for the same charge state Z* (Cases 1,3),however, the changes are only about 2% per decade. The 2% changes are probably too small to be significant. Another class of limitations on the average-atom model is associated with the use of Eq. (3) for the average occupation of one-electron states (More, 1981, 1983; Green, 1964; Grimaldi and Grimaldi-Lecourt, 1982). Finally, there are questions about the definition and consistency of the SCF model. Many points of detail are glossed over by the simplified statement of Eqs. (1)-(4). Is the cell itself exactly neutral, or only the general region around it? What feature of the model distinguishesan argon atom in a hydrogen plasma from an argon atom in an argon plasma (at equal pressure)? How should we separate properties of the compressed atom from those ofthe background plasma? Liberman (1 979) distinguishes two theories (Models A and T) according to the scheme adopted for effecting this separation. In Section IV we find this distinction is important for thermodynamic purposes. It remains an open question which method is more powerful and/or consistent.

-

IONIZATION-QUALITATIVE C. PRESSURE The simplest qualitative description of pressure ionization asserts that the nth shell moves into the continuum at a density where the internuclear separation is approximately equal to the average orbit radius, r, = R ,

(7)

312

R.M . More

where r,, = uon2/Q,,is the orbit radius, a,= hz/me2,Q,is an effectivecharge, and R , = (3/4nn,)’/’ is the ion-sphere radius. From the energy viewpoint, a state is pressure ionized when the continuum lowering equals the binding energy. This statement translates into the same numerical estimate (see More, 1982, for further discussion of this point). Another qualitative criterion is that an electron becomes unbound when the Holtzmark nearest-neighbor field Fo = Ze/R%equals the nuclear attraction for the state in question (Burgess and Lee, 1982). This equation is algebraically equivalent to Eq. (7). The qualitative criteria are not sufficient to satisfy our curiosity about pressure ionization; we would like a more detailed understanding. For example, there must be a significant difference in the behavior of subshells for a given principal quantum number; although s-wave bound states are lowest in energy, their wave functions extend to large radii and are most easily affected by compression. Numerical predictions from Eq. (7) are not very accurate, although they scale correctly with density, atomic number, and principal quantum number n. Better qualitative formulas are developed in Section 111. At this point it is useful to illustrate the discussion with an example of pressure ionization. Figure 2 shows a theoretical electronic density of states g(E) for aluminum at density 2.7 g/cm3 and temperature 50 eV, conditions which can be attained in laser-heated targets. The calculation (D. Liberman, unpublished) shows prominent resonance peaks arising from pressure-ionized 3p and 3d states. Shape resonances such as those illustrated in Fig. 2 will make significant contributions to calculated thermodynamic properties. The resonant scattering has a strong effect on plasma electrical conductivity as evaluated by the Ziman formula (Lee, 1977; Lee and More, 1984). Resonance effects are visible in calculations of Bremsstrahlung emission in the high-energy “tip” region (Feng et al., 1982; Lamoureux et ul., 1982; Feng and Pratt, 1982). Shape resonances probably enhance the recombination rate in a plasma, because a free electron easily enters a broad resonance and then has good opportunity for radiative recombination through the analog of a line transition. Likewise, resonance states probably increase the rate of electron -ion heat exchange in a nonequilibrium plasma with T, # Ti. Another speculation is that a resonance level might appear spectroscopically as a line past the lowered continuum. This would be very interesting, but seems unlikely in view of Eq. ( 5 ) (see also Section 111). In thinking about resonance states, we encounter many questions of a deeper theoretical character. What is the modified Fermi function that determines the occupation of the resonance? How can we calculate the Stark

PRESSURE IONIZATION

313

E (RY)

FIG.2. Theoretical electronicdensity of states for aluminum calculated by the self-consistent field model (D. Liberman provided these data). Shading has been applied to enhance the contrast of 3p and 3d resonances from the ell2 background continuum density of states. The density is 2.7 g/cm3and the temperature is 50 eV.

splitting of resonance levels? What about the interaction of resonance electrons with other bound electrons (i.e.¶what are the Slater F and G integrals involving resonance electrons)?To address these questions we require welldefined wave functions and a perturbation theory for the resonance states. These tools can be developed using well-known results from quantum scattering theory.

D. CONTINUUM WAVEFUNCTIONS We begin with definitionsof the Jost functionl;(k) and Jost solutionf;(k; I ) of the radial Schr6dinger equation. These are convenient functions which obey standardized boundary conditions(Newton, 1960).The Jost solution is defined by the differential equation

d2

--f

dr2

I

+ [u(r)+ q(r) - k2]f;(k;r) = 0

and a boundary condition imposed at large radii:

f;(k; r)

i 1exp(- ikr)

(9)

314

R.M . More

As written, Eqs. (8) and (9) define a solution even for complex wave vectors corresponding to complex energies. The reduced Coulomb and centrifugal potentials are

~ ( r=) -(2rne/h2)V(r); u,(r) = l(l+ I)/+ It is important to notice two points about the definition. First, at large radii the boundary condition fixes both the value and derivative off;(k; r); so it completely determines f ; .Second, no boundary condition is imposed at small radii; the Jost solution does not necessarily approach zero as r 0. Equation (8) is singular at r = 0 because the nuclear potential v(r)and the centrifugal potential vl(r) both diverge. Expansion around the singularity shows there are two independent solutions: one is well behaved near the origin, proportional to rl+' for small r ; the other grows like l/rl. The coefficient of the growing solution is extracted by the limit

This equation defines the Jostfunctionf;(k).For a finite-range potential,f;(k) is analytic for all k; i.e., it is an entire function (Newton, 1960). Iff;(K) is zero, thenf;(K; r) is a solution ofthe Schrodinger equation which obeys the proper boundary condition at small radii. Iff;(K,,)=O with Im(K,,) < 0, then it can be shown that Re(K,,) = 0, and we have a bound state of the usual type. The bound-state wave function w,,(r) is proportional to the Jost solution,

This wave function is automatically normalized (f'= djdk). For positive real energies, &(k) is never zero; so f;(k; r) does not behave properly at r 0. We construct a satisfactory continuum eigenfunction by combining two Jost solutions which have the same energy; these are sohtions associated with wave vectors kand -k. The trick is to exactly cancel the singular part off;(k; r) against the singular part off;(-k; r). In carrying out this calculation (see Appendix A), it emerges that the scattering phase shift is simply the phase of the Jost function

f;(k)= lf;(k)leid/(k) real k only

(12)

The continuum solutions of Eq. (8) then obey the boundary conditions

y(k;0) = 0 w,(k; r) 7 A,(k) sin[kr - ln/2

+ d,(k)]

PRESSURE IONIZATION

315

and are related to the Jost function by

The normalization of wI(kr) is often taken to be A,@) = 4 q i

(15) With the Jost-function terminology, it is possible to give a precise definition of the resonance energies (Humblet, 1952; Zel’dovich, 1961; Berggren, 1968; More, 1971; More and Gerjuoy, 1973). Resonances are associated with zeroes of the Jost function in the upper half k plane,

f;(Q,> = 0,

Wen,)> 0

(16) These zeros occur in pairs, symmetric about the imaginary axis, and can Im k

FIG.3. Schematic representation of the k plane showing the various solutions of the equationf,(k) = 0. Solutions having Im(k) < 0 must have zero real part and correspond to negativeenergy bound states. Resonancesoccur in pairs reflected with respect to the imaginary axis, with Im(k) > 0. Virtual or antibound levels occur on the positive imaginary axis. It is customary to refer to the lower-half k plane as the physical sheet of the energy surface, and the upper half-plane as the unphysical sheet.

316

R. M . More TABLE I1 DEFINITIONS OF CONTINUUM WAVE FUNCTIONS

Function

Definition

Energy

b.c. at origin

w,(k, r) h(k, r) bOn/(r) G,(r, r‘, k)

Eqs. ( I3), (15 ) Eqs. (81, (9) Eqs. (27), (28) Eqs. (88)-(90)

Real Complex Only Complex

Vanishes May diverge Vanishes Vanishes

Asymptotic form sin(kr - 1x12 i’ exp(- ikr) a exp(- iQ,r) a exp(ikr,) a

+ 6,)

be labeled so that Q-,,/ = - QZ (Fig. 3). The complex resonance energy is Enl= ( h2/2m)Qfl. As shown in Appendix B, the one-electron Green’s function has poles at wave vectors k = -Q,,,. Because the partial-wave S matrix is S,(k) =f;(k)/ A(- k), the zeros of the Jost function cause poles ofthe Smatrix at&- k) = 0; i.e., k = - Q,,,.Therefore, Eq. ( 16) agrees exactly with the usual definition of resonance energies (Siegert, 1939). The relation to the more complicated Kapur- Peierls resonance theory is workedout by More and Gerjuoy ( 1973). At this point we have mentioned three of the four distinct representations of states in the continuum

w/(k,r)

Real-energy scattering wave functions

f;(k,r)

Jost solution, defined for complex or real energies

bpnl(r) Gl(r,r’;k)

Resonance eigenfunction, defined for E = En, Radial Green’s function

To help the reader sort through the properties of these functions, we collect their definitions and interrelations in Tables I1 and 111.

TABLE 111 FORMULAS CONNECTING CONTINUUM SOLUTIONS

PRESSURE IONIZATION

317

E. CONTINUUM DENSITY OF STATES We next consider the density of states of an electron gas in the presence ofa spherically symmetric potential of short range. This quantity is given by

where go(€)= ( l/2n2)(2rn/h2)3/2V&is the unperturbed density of states of the uniform free-electron system. ( V is the volume of a large spherical volume surrounding the compressed atom.) Using a terminology introduced by Liberman, Eq. (17) is the Model T density of states; i.e., it includes all the effects of the central ion [compare to Eq. (20) below, which includes only the changes in the range 0 < r < R , ] . Anderson and McMillan (1967) give an interesting account of issues and problems which arise in the proof of Eq. ( 17) (see Appendix C for details). Equation ( 17) is closely related to a classic theorem of Levinson ( 1949)

(18) n1= ( 1 /n)[4(4 - 4(9l which relates the scattering phase shift to the number nlof bound states of angular momentum 1. It is also related to the Friedel sum rule 2=

3 (21 + 1) d1(EF)

1-0

This is the condition of perfect screening of an impurity of charge 2 in a metallic electron gas (eF is the Fermi energy; the formula assumes a free-electron band structure) (Friedel, 1967). Both the Friedel formula and the more general density-of-states formula are discussed in many textbooks of solidstate physics (e.g., Kittel, 1963). Equation (19) can be extended to models of interacting electrons (Langer and Ambegaokar, 196 1) and metals with nonspherical Fermi surfaces (Friedel, 1969). Equation ( 17) for the density of statesg(e) corresponds to a specific model ofthe dense plasma, in which all changes produced by the central nucleus are ascribed to that atom. In the terminology of Liberman ( 1 979), this is the Model T approach. Most of the formal investigations of plasma continuity properties work within this model. An alternative class of theories is based upon the spatial location of electrons (Liberman's Model A). In this approach, one chooses to separate out the part of the problem contained within the spherical cell. The corresponding density of states is

318

R.M. More

where iys is an eigenfunction normalized in all space. This approach has many convenient properties, and may ultimately be as satisfactory as the Model T theories. As far as we know, the issue of continuity has not been investigated through rigorous analytic study within this approach.

11. Continuity of Pressure Ionization Continuity theorems for a noninteracting electron gas under the influence of an attractive spherically symmetric potential have been developed by Butler (1962), Kohn and Majumdar (1969, Peierls (1979), Rogers et al. ( 1971), Petschek ( 197I), Petschek and Cohen ( 1972), D'yachkov and Kobzev (198 l), and Hohne and Zimmerman (1982). Most ofthese resultscan be paraphrased by selecting an observable quantity Q(1)which varies with a parameter 1characterizing the central potential. The goal is to prove that Q(1)is continuous or even analytic as a function of 1. If the parameter 1 represents the density dependence of the potential caused by screening,the theorem will show that Q is a continuous function of density. Alternatively, the theorem will show that Q varies continuously as a function of atomic number if the parameter 1measures the charge 2 of the central nucleus. We temporarily assume the potential is stronger for larger 1, so the n, 1 bound state exists only for 1> ?In1. If the quantity Q(1) is an additive function of the atomic eigenstates, then

Here, QF and Qb are contributions of the continuum and other bound states (excluding the state n, I). Because Qn/appears abruptly at 1= An/, Qn,(l) is necessarily nonanalytic there. Mathematically, the singularity of Qn,(A)is exactly cancelled by a singularity of QF(A) so the total Q(1)is analytic through AnI. The physical reason for the singularity in Q,(A) is, of course, the low-energy scattering resonance which exists for 1< An/.' As 1 ?Inl the resonance approaches zero energy, its lifetime becomes long, and its contribution to QF becomes identical to that of the bound state.

-

I For I = 0, there are no resonances, but instead the so-called virtual states located on the imaginary k axis. These states have generally similar properties and consequences for our present purposes.

PRESSURE IONIZATION

319

A. PROOFOF THE CONTINUITY THEOREM

The first studies of this continuity property were motivated by experiments on positron annihilation in metals. In the interpretation of these experiment, the question arose whether positronium (an electron - positron bound state) could form despite the strong screening caused by the metal’s conduction electrons. Ifthere were a critical electron density above which no positronium bound state could exist, that would imply a strong density dependence of the annihilation rate. These issues are briefly summarized in the interesting book of Peierls ( 1979),who sketches out the continuity proof for the case Q(A)= n(O),the electron density at the center of attraction. The theorem implies that there is no discontinuous boundary between low- and high-density behavior. In this case, the continuity theorem is proven as follows. First, the observable Q is formulated as an integral over bound and free wave functions. The wave functions are expressed in terms of the Jost solutionsf;(k; r) [see Eqs. (4), (1 l), ( 14),and ( 15)].The next step is to transform the integrand into an analytic or meromorphic function of k. For Q = n(0)this is easy; the nonanalytic expression ll;(k)l* is replaced by f;(k)f;(-k), which is numerically equal for real k but which is analytic for general k. Once this transformation is) is accomplished, a straightforward contour deformation proves that &I analytic as a function of A. Using the same approach, D’yachkov and Kobzev (198 1) examine Q = Ky the monochromatic Bremsstruhfungopacity. This is written as an integral over continuum wave functions and expressed in terms of Jost functions. As AV(r) increases to produce a bound state, the photoelectric cross section which appears abruptly at I = Inlis exactly compensated by a reduced Bremsstrahlung absorption, so that the total monochromatic opacity is analytic as a function of density. In this calculation, it is necessary to require that the bound-state population be in thermal equilibrium with the continuum (the average-atom model tacitly makes an equivalent asumption). Rogers et uf.(197 1) examine the correction to the partition function of a gas of Maxwellian electrons produced by an ion’s potential V(r).They begin with the correction to the free-electron partition function

This is integrated by parts, using Levinson’s theorem, to give

R.M . More

320 I o8

1

I

I

1

I

I

a

I' 0

1 o6

I o5

k 10'

r"

T -

I

W

a: lo3

2

fn W

r

p

2

10

I o1

1 00

Y

/

I

---

TF-L

t FIG.4. Examples of discontinuous predictions from spherical-cellcalculations in the literature. These predictions all violate the continuity theorem@)discussed in the text. (a) Pressure of compressed iron at a constant temperature of 7.7 eV obtained by Zink (1968). The pressure jumpsabruptly by almost two orders ofmagnitude atp = 15 g/cm3. (b) Schematic prediction of pressure-volume and volume vs. atomic number from work of Kirzhnitz ef al. (1976). The prediction is that phase transitions are associated with the pressure ionization of bound electrons. (c) Numerical calculations of pressure vs. atomic number from work of Lee and Thorsos ( 1978). Especially at low temperatures there are sharp discontinuities in the pressure as a function of atomic number. The continuity theorem of the text argues that such discontinuities cannot result from the correct solution of a spherical-cell model. (d) Figures from Kobzev ef al. (1977), which schematically indicate a proposed transparent region in the spectrum of highdensity plasmas. The upper curve is a representation ofthe oscillator strengthf,,. vs. final-state energy En. and the lower curve is a schematic absorption cross section vs. photon frequency. The text gives an argument against this prediction.

32 1

PRESSURE IONIZATION b

"t

100

1'

ni = 3.1 6 x

TC

cm-3

1

b

\

5

a

1:

.

:
R,]. Approximate formulas bearing a qualitative similarity to Eqs. (45) and (46) have been obtained previously (see, for example, the treatment of the Anderson model for local moments given in Chapter 18 of Kittel, 1963). However Eqs. (45) and (46) give the exact result for the one-electron nonrelativistic potential scattering theory. One unique feature of this result is the inside the integral F @ ) . square root of €/En, Figure 6 displays the real and imaginary parts of F ( E ) for several choices of the chemical potential p (r= 0.05 eV, kT = 10 eV). For small width r,,, the real part of f'(k,,,) is very close to the Fermi function f(E,,). The imagi-

R.M. More

336 0.200

1

I

I

I

1.o

1

I

I

0.175 0.8

--

1

1

b

\

\

0.150

\ \

0.6 -

0.125

o.look

‘YRe

0.075

\ \

F

/ReF \

\

0.4

0.2 0.025

0

0

10

20

30

40

I

50

I

I

10

15

20

Energy E (eV)

Energy E (eV)

1.2

I 5

1

1

1

I

I

Energy E (eV)

FIG.6. Numerical calculations of the resonance thermal occupation function F ( E )defined in Eq. (46) of the text. In each case, the temperature is fixed at 10 eV, the resonance width r = 0.05 eV, and the real part ofthe resonance energy variesover an indicated range. (a) Chemical potential p = - 10 eV. (b) Chemical potential p = 10 eV. (c) Chemical potential p = +40 eV. In this case there is a small discrepancy between Re F(E) and the usual Fermi function at low energies; this discrepancy would disappear if the resonance width r were not held constant but rather allowed to approach zero as E + 0.

+

nary part of F, which plays an interesting role in some of the equations to follow, falls off slowly with energy; i.e., it decreases as 1/E2 rather than as exp(- E/kT). Equations (45) and (46) actually run deeper than may appear at first sight. Consider these equations in the context of the traditional argument which connects level populations with the principle of detailed balance. That argu-

25

PRESSURE IONIZATION

337

ment shows that the rates of forward and reverse transitions are related, in a general way independent of the detailed mechanism of the process, so that their ratio reproduces the equilibrium Boltzmann or Fermi - Dirac distribution. However, if the lifetime of a quantum state is sufficientlyshort that we cannot neglect its energy width, the usual detailed balance argument must be modified insofar as the equilibrium population is changed from the Boltzmann form. Equations (45) and (46) give this change for one specific case, and thereby appear to give a rigorous basis for a fundamental extension of the principle of detailed balance. B. ELECTRON DENSITY We can proceed a step further to express the local electron number density n(r) as a sum of resonance contributions. For this purpose, we observe

Equation (47) determines the electron density in terms of the radial Green’s function G,(r, r‘, E) defined in Appendix B. The equation omits the contribution of possible bound states. Equation (47) can be simplified by use of the resonance series expansion for the Green’s function, Eq. (33). Again gatheringtogether the terms with and - n, we obtain

+

where F(&) is the generalized Fermi function defined above. Equation (48) gives the density of continuum electrons in terms of the resonance wave functions and their thermal average occupations. In order to obtain a total number of electrons, we can form the integral J4nr2n(r) dr. The difference between this result and Eq. (45) is evidently a question of definition; Eq. (45) counts electrons throughout all space, the Model T formulation, while the integral of Eq. (48) counts only electrons within the spherical cell and corresponds to the Model A theory. In the case of a narrow resonance confined by a high potential barrier, Eq. (3 1) shows that the two forms become equal. If a perturbation SV(r) is applied to the electrons, the first-order change of total energy is obtained from Eqs. (36) and (48) in the interesting form

rm

SE = J 4ar2n(r) SV(r) dr = 0

3 2(21+ 1 ) Re[F(&,,) S&,]

Both this result and Eq. (48) show that the imaginary part ofF(&) has some physical significance.

338

R. M . More

C. IONIZATION STATE The traditional definition of the atomic ionization state Q is a simple matter: one merely adds up the populations of the various bound states, subtracts from the nuclear charge, and averages (if necessary). In the average-atom model this prescription is

While this definition is perfectly reasonable for low-density plasmas, we will argue that it is a bit naive and at high densities would be misleading. Naturally, the discussion will center on the effects of resonance states. First, let us recall an analogous situation arising in the Thomas-Fermi theory. In that case there are two distinct definitions of the ionization state Z*:

Z& =

6

d3r

d$f(

&-

eV(r)),

p 2 < 2rneV(r)

(50)

Equation (50) is a classical approximation to the quantum formula, Eq. (49), because it counts all electrons having positive energy relative to the ionsphere zero of energy. The second formula determines the free-electron density from the boundary density n(&). [The analytic fit given in Table IV refers to this second definition, ZA,(p, T).] By analogy to Eq. ( 5 1) we can introduce a second definition of the quantum charge state,

Assuming the quantum-cell model is precisely defined, e.g., by the constraint that the cell itself be neutral, this definition uniquely fixes Q,,, as a function ofmatterdensityp = AMp/(4aR;/3) and temperature. There is no reason to expect Q(l)to equal Q(2,. In the Thomas-Fermi case, Eqs. (50) and (51) give similar charge states for hot, low-density plasmas but disagree significantly at high densities. One often finds cases where the integral 2:)exceeds the boundary-density definition Z;, by a factor of two (in the Thomas-Fermi model it is obvious that Z & > 26, in any case). The difference between 2:) and Z & is associated with electrons of positive energy, counted in Eq. (50), which remain at small radii r < R,, and therefore do not contribute to 28,. These electrons occupy a part of phase space

PRESSURE IONIZATION

339

which is the classical equivalent of the resonance; i.e., they have large angular momentum and are confined by the centrifugal potential. It seems clear that electrons in quantum resonance states would be counted as free in the definition of Qrl, [Eq. (49)] but would not contribute to Q(2), just as in the Thomas-Fermi theory. Weak external perturbations do not readily affect electrons in resonance states, and it appears that those electrons cannot make alarge contribution to the electrical conductivity or other transport coefficients. Similarly, electrons in narrow resonance states probably participate in radiative processes as if they were bound. On this reasoning, we see that both 28, and Q,)may overestimate the number of electrons per atom that are free to participate in plasma conduction and screening. When there is a significant difference between Ql,and Q(z,, and when the resonances are narrow, it appears that the smaller charge Q(z,probably would give a better account of the physics of conduction and transport processes. These issues are very familiar in solid-state physics (Kittel, 1963; Anderson and McMillan, 1967). Metals such as iron and nickel, which have partially filled 3d shells, are poor electrical conductors. The narrow d bands are often described as resonances imbedded in a continuous free-electron s-p band. Although the d states have energies within the partially occupied conduction band, their contribution to electrical conduction is very small; indeed, the conductivity is further decreased by scattering of freely moving s-p camers into the resonance states.

D. PRESSURE FORMULAS In the remainder of this section, we examine several formulas for the pressure of continuum electrons with the objective of resolving it into additive resonance contributions Sp,,,. There are at least three categories of pressure formula: formulas derived thermodynamically, i.e., the volume derivative of the free energy; formulas derived from a kinetic viewpoint, i.e., pressure as a momentum flux or stress tensor; and formulas in which the pressure is related to the kinetic and potential energy densities via the virial theorem. In a completely consistent theoretical model, these three approaches would give the same result (More, 1979, 198 1); this is the case for the Thomas-Fermi theory and certain of its improvements. At present, we are unable to formulate the quantum SCF cell model with the same complete consistency. Possibly the difficulties stem from minor aspects of the theory such as the question whether self-consistency is im-

340

R. M . More

posed everywhere or only within the central ion sphere (0 < r < &),the question how the kinetic energy and entropy of the central atom are to be distinguished from the background plasma, and certain questions concerning the electron-electron interaction. It is not clear that these issues have any large quantitative effect on the final predictions of the model. Nevertheless, without a complete resolution of these difficulties, one remains dissatisfied with the theory. E. PRESSURE I-PRESSURE-TENSORMETHOD The discussion begins with the development of a pressure formula based on the quantum-mechanical stress tensor or momentum flux (Pauli, 1958). This tensor is written in Cartesian coordinates as

wheref, is the Fermi function and {w,) is a complete orthonormal family of eigenfunctions. It is readily verified that P,,reduces to the usual isotropic scalar pressure of a free-electron gas (=pd,,) when evaluated with plane-wave eigenfunctions. Nevertheless the reader may desire more convincing proof that Eq. (53) correctly gives the pressure when applied within self-consistent-fieldtheory. One line of reasoning is based upon the theorem

where E(r) = -grad Vis the electrostatic field. The proof of this theorem uses Eqs. (2)and (4),and the result is valid only if V(r)is self-consistent;i.e., Eq. (54)would fail for models which make approximations to Eqs. (l), (2), or (4). Equation (54) appliesthroughout the compressedatom, and may be intuitively understood as a condition for hydrostatic equilibrium of the electron fluid. It is worth comment that Eqs. ( 5 3 )and (54)can be extended to a self-consistent-field theory including a local-density exchange correction in a straightforward fashion. The pressure tensor acquires an additional contribution (for example, at T = 0)

-(e2/47r)(372 2n)*I3,(

r) 8 ,

and Eq. (54)is unchanged [in particular, the electric field is still the gradient of the electrostatic potential V(r)only]. The Schrddingerequation, Eq. (2),is altered by the addition of an exchange potential contribution.

34 1

PRESSURE IONIZATION

In spherical polar coordinates, Eq. (53) becomes

""I

aw - C O S ~ --1 wf s wf r

dr

a8

r2 sin 8

These quantities can be simplified by using the partial-wave eigenfunctions

WWtk r)Ylm(& (PIX0

(56) and then employing the spherical-harmonic addition theorem to simplify the sums over m. With w s =(

cos y = cos 8 cos 8'

+ sin 8 sin 8' cos(y,- q')

the addition theorem reads

This readily gives useful formulas such as

and similarlyfor the other derivatives.The important physical assumption is that the occupation probabilityf, depends only on the energy E: = h2k2/2m, independent of m (isotropic thermal occupations). With these formulas, it immediately follows that the pressure tensor is diagonal, i.e., (6 1) Pe.p = Pr.e = Pr.q= 0 The radial stress is given by

P,=Ec2(21+21)f(~X)[(d~)l---(--)] waz w (62) 2m

kl

4nr

ar r

r ar2

342

R . M . More

The two perpendicular components are equal to each other,

Both P, and Pee are functions only of radius. For the special model of an exactly neutral cell with a compensating exterior background [i.e., p+(r) = en(r) for all r > R,],the divergence of P is exactly zero for all r > Ro. This condition implies

showing that the pressure components asymptotically become equal. In the limit ( r 4m), the effect of the distant atom is negligible, and the diagonal pressure tensor simply reduces to the ideal-gas pressure of the distant electron gas. Because we imagine the real plasma to consist of an assembly of closely packed ion spheres, it appears reasonable to calculate the pressure by evaluating P,, at the ion-sphere radius R,,but there is no formal justification for this assumption. The expression for Prrcan be reduced on the assumption that V(r)= 0 to the form

This gives the continuum pressure in terms of real-energy continuum radial wave functions (possible bound-state contributions are ignored for simplicity). The integrand in Eq. (65) can readily be expressed in terms of the Green’s function G/(r,r’, k ) by use of the result

wl(k, r)w,(k, r’) = - ( h 2 k / n m ) Im[G,(r, r’; k ) ]

(66)

which is verified in Appendix B. We again invoke the resonance expansion given in Eq. (33) to find

Those readers who are mathematical purists will duly note that the series is

PRESSURE IONIZATION

343

being used at its radius of convergence if r = R,. From these expressions, it is not difficult to gather together terms corresponding to Qnl and Q-,,, with the final result

In this equation, the additional resonance energy integral F ( ’ )is defined as

We have not performed a numerical test of Eq. (69). However, it is evident from examples shown in Fig. 4 that the resonance pressure has a substantial quantitative effect on the equation of state in the shock-wave compression region.

F. PRESSURE I1-VIRIAL THEOREM In this last section, we will show that the pressure given by the preceding formula is in agreement with a slightly modified or extended version of the virial theorem. Several authors have developed well-known pressure formulas on the basis of the virial theorem (Liberman, 1971; Pettifor, 1976; Ross, 1969) and unfortunately we find that these results do not agree with Eq. (65). Thus it is necessary to address a question of consistency of the theory. The question arises before the introduction of resonance wave functions; i.e., the question concerns the calculation of pressure in the self-consistentfield theory of the spherical-cell model. As we will see, it is not very surprising to find a disagreement with a naive application of the virial theorem in the simple form p

= (2K

+ U ) / 3v

(71) because that equation is not necessarily valid when applied to part of a large system. In general, there are electrostatic and boundary-gradient corrections to Eq. (7 I). Similar corrections occur in the quantum-corrected ThomasFermi statistical model (More, 1979),where the tensor pressure formula can be proven to give the same result as the thermodynamic pressure -dF/dV. In order to generalize the virial theorem to apply to an arbitrary region i2 imbedded in a larger volume, we first examine the trace ofthe pressure tensor

344 to establish

R. M. More

In

Tr P d3r= K,,,

+ 4,)

In this equation, the two quantities Kt1)and K(,, are given by

are both candidates to describe the kinetic and K(2) The quantities energy contained in the volume R, and Eq. (72) is already at the heart of the issue. If the region R had special additional properties, so that we could impose either Dirichlet or Neumann boundary conditions on the wave would be equal and no further discussion would functions,then K(,!and K(2! be required. That simpler situation occurs, for example, when R is a crystalline proximity cell surrounding one atom in a simple lattice, the case originally contemplated by Liberman (1 97 1) and Pettifor (1 976). Next, we require some properties of the Maxwell electrostatic stresstensor T,j,which may be defined as T,j = ( 1/4n)(f E 'Sij - Ei Ej) For this quantity, we have

(75)

and

Equation (77) can be understood to provide a definition of the Coulomb energy U,contained in the region R, but the reader is warned that the formal expression diverges close to the nucleus unless a finite nuclear radius is introduced. (This divergence exactly cancels a corresponding contribution to A below, so that the final result remains valid.) Now to develop a virial theorem, consider the integral A=

In r

E(r)p(r)d3r

(78)

The integration is taken only over the finite region R (in the case of practical

PRESSURE IONIZATION

345

interest, this is the ion-sphere volume). Inserting Eq. (76) for the divergence of Tij and integrating by parts we find I

I

S is the surface surrounding the volume and dAj is the outward vector surface area. Also, the charge density p(r) appearing in Eq. (78) can be separated into positive and negative parts, giving rise to a separation

+

A = A(+) A(-)

r A(-) = - e

n(r) r E(r) d3r In The integral A(-) is then transformed by a similar parts integration to give A(-) = -

In + Tr P d3r

x,Pij &Ij

At this point the reader can verify that the small-r divergence of A,,) exactly cancels the contribution from the nuclear region in Uc. Omittmg these terms, the result is

c xi 30). The resonance physics is less compromised by Stark broadening in the high-density regime of shock-wave physics, and promises to play an important role in the calculation of electronic properties of hot dense matter.

PRESSURE IONIZATION

347

Appendix A: Properties of the Jost Function In this appendix we summarize some useful formulas which will assist the reader who wishes to check the equations given in the text. More detail is available in the review by Newton (1960). From the large-r boundary condition, Eq. (9), it is easily seen that

J;(-k; r) = (- l)y-f(k*; r) (81) The two Jost solutions J;(k; r) and J;(-k; r) obey the same differential equation, and therefore their Wronskian W [ u ,u] = u dv/dr - u du/dr is a constant, independent of radius r. The constant can be evaluated at large r, using Eq. (9): W[J;(k;r),J;(-k; r)] = (- 1)"c

(82) Next, observe that with ion-sphere boundary conditions, the potential u(r) is zero for all r > R,. In this case, the solution is simply a spherical Hankel function,

1;( k; r) = - ikrh j2)(kr)

(83)

where hj2)(x)=j , ( x ) - in,(x). At small radii, the Jost functionf;(k) has been defined so that

In combination with Eq. (81), this implies

J;(-k) =f,*(k*) A power-series expansion ofJ;(k;r) at small rshows that every term in the series (up to and including terms proportional to r') is generated from the first term by a recursion relation, and thus each of these terms is proportional toJ;(k). For this reason, iff;(@ = 0, all these terms vanish. We have used the symbol Qnlto denote the special values of k for whichJ;(Q,,) = 0. In this case, a careful study of the power series [in conjunction with Eq. (82)] shows

The proof that the bound-state wave function determined by Eq. ( 1 1) is normalized can be constructed by combining Eqs. (4.18), (4.20), and (4.21 ') of Newton's article. Equations (84) and (86), together with the symmetry properties (8 1) and

348

R.M. More

(85), are sufficient to show that Eq. (14) obeys the small-r boundary condition of Eq. (1 3). The large-r boundary condition is established from Eqs. (9) and ( 12). The normalization Eq. (3I ) of the text is proven by forming the integral from q > 0 to R of

The integral is first differentiated with respect to k, and then k is set equal to Q,,. Finally, q is allowed to approach zero. Equations (84) and (86) are key properties in the evaluation of the result, which can be transformed into Eq. (31).

Appendix B: Green’s Function In this appendix, we recall the definition of the one-electron Green’s function G,(r, r’, E) for the radial Schradinger equation and express G in terms of Jost functions. The formulas apply to scattering by an arbitrary short-range potential. The result shows how to continue GIto complex energies e. The effective Hamiltonian for the one-dimensional radial motion is

+---h2 d2 h2 I ( / + 1 ) eV(r) + -2m dr2

2m

r2

(87)

Formally, the Green’s function is the resolvent I/(€ - QI),or (E - fil)Gl= S(r - r’)

(88) The Green’s function is symmetric: Gl(r,r’, e) = Gl(r’,r, E). The boundary conditions which complete the definition of G are

0, I -,0 (89) Gl(r,r‘; e) a eik, r -,03 (90) The second boundary condition is the outgoing-wave boundary condition. This boundary condition brings k a &into the problem, and so the Green’s function is single valued when considered as a function of k (rather than energy e). Using previously established properties of the Jost functions, it is easy to verify that Eqs. (87)-(90) are satisfied by G,(r, r’; E)

--+

Equation (88) is verified by integrating from r = r’ - q to r = r’

+ q; the

PRESSURE IONIZATION

349

left-hand side of the integrated equation is then proportional to the Wronskian W[f;(k; &A(- k; r)],which was given in Eq. (82). Equation (91) is valid for any r, r', or k and in particular is valid in either half of the k plane. However, it does not present such a simple or immediate view of the k dependence of G/ as does Eq. (33) of the text. For k = - Qn, dk, we have

+

f;(-k) =f;(Qn/ -Skf;(Qn/) With this, the residue of GIin the k plane is clearly

and the residue in the energy (E) plane becomes Eq. (30) of the text. Working from Eq. (88) for real values of k, the symmetry properties in Eqs. (81) and (85) can be combined to show 2m 1 Im[G,(r, r'; k)] = - --Wf;(klfl(k; r)l Im[f;(W(k; 0 1 h2k If;(k)12 With the continuum wave function of Eqs. ( 14) and (1 5 ) we also have

so that h 2k w,(k; r ) y ( k ;r') = - -Im G/(r,r'; k) am

(92)

This equation applies for real wave vectors only. It may be analytically continued to give the usual representation ofthe Green's function in terms of continuum wave functions,

This formula is valid only in the upper half-plane and must be corrected if there are bound states.

Appendix C: Electron Density of States We will sketch three derivations of Eq. ( 17). The simplest works directly from the asymptotic form of the radial continuum eigenfunction w,(k; r) a sin[kr - ln/2

+ d,(k)]

(94)

R. M . More

350

valid for kr >> 1. If we apply the boundary condition that y ( r ) must vanish on the surface of a large sphere of radius R >> R,, then the number of eigenstates of angular momentum 1 having energy G ( h2/2rn)k2is N,, determined by RN, = kR - 1x12

+ 6,(k)

(95)

The total number of states having energy d E is then N=R

T (21+ l ) ( k R - I s ) + f 7 (21 + 1) S,(k)

(96)

When differentiated with respect to energy, this formula gives Eq. (17), except for the technical point that the first term (the free-electron contribution) does not sum to the correct vacuum density of statesgo(€).The reason is very simple: For very large angular momenta we cannot use the asymptotic form of the wave function, Eq. (94). Because Ro Ro exactly as did the zeroorder potential V(r)which is used in the definition of Go. Both perturbed and unperturbed Green’s functions are expanded by convergent series of the form of Eq. (33). Taking the residue of Eq. (103) at k = - Qnr,we obtain

where Qnlis the resonance zero associated with the exact Green’s function G, and Q$)is that associated with Go. The matrix element of the perturbing potential is defined by

Equation (104) is now expanded in powers of the perturbation U. The

352

R . M. More

resonance wave vector and matrix element are expanded as Qnl= Q$)

+ 9'')+ Q2' + nl

From the first two orders of the expansion of Eq. (104), we find

This is the essential result. In terms of the energy Ed,Eqs. (106) and (107) take the form

Equation ( 108)reduces to the usual second-order perturbation theory in the case of long-lived resonances (More, 197 1).

Appendix E: Convergence for the &Potential Model This simple model problem has served as a testbed for a number of authors (More, 1971; Romo, 1975; Bang et al., 1978; Garcia-Calderon, 1982). We reproduce some useful formulas, together with comments on the numerical calculations in order to facilitate further exploration of resonance-state expansions. The model considers 1 = 0 scattering from a repulsive &function potential shell,

u(r) = V, 6 ( r - b) (h2/2m= 1 in the sequel). The Schrodinger equation forf(k; r) is

-(d2/dr2)f+ u(r)f= k2f

PRESSURE IONIZATION

353

The Jost solution is then

The Jost function is

f ( k )= 1

VO (1 - e-2ikb) +2ik

The scattering phase shift d(k) is the phase off(k), defined for real values of k only. The derivative of d(k) with respect to k is important to forming the free-particle density of states [see Eq. ( 17) of the main text]

+

d4k) - b + kab Vsin2 kb dk k2 2kV0 sin kb cos kb + Va sin2 kb By taking the limit of this equation as k 0, we extract the scatteringlength -=

+

a [see Eq. (43) above]:

+

a = Vob2/(1 bVo)

Finally, the exact Green’s function is simply expressed by sin kr‘ f(- k ; r) k f(-k) All these formulas may be tested against appropriate expansion formulas of the text, i.e., Eqs. (44), (43), and (33). The resonances are located by numerically solvingf(Q) = 0. This equation is easily solved by Newton’s method, provided a good first estimate of Q, is available. Satisfactory estimates are given by

G(r, r’; k ) = -

~

A more general method for finding Q,,can be constructedby exploitingthe nearly uniform spacingof resonances at large n. The first few Q, can easily be obtained by starting Newton’s rule with guesses inferred from the maxima in 6’(k). The equationf(Q,,) = 0 can be expressed as e-2iQn = 1

+ 2iQJ Vo

354

R. M. More

Using this equation we find (- Qn) 1 Nf,= f ---

2b (1

iJ’(Qn)

b 41

1

+

+ V0/2iQn)21 +

1 l/b( Vo 2iQJ

+

sin Qnr l/b(Vo 2iQn)

+

Using FORTRAN complex arithmetic, it is a simple matter to establish the rapid convergence of expansions in terms of the resonance functions. For example,

These series converge well when - 100 terms are used. In the energy range near the first resonance (e.g., 0 < k Z< 10 for Vo= 10, b = l), a one- or two-term series does quite well.

ACKNOWLEDGMENTS We are very grateful to Drs.J. Cooper, J. Green, H. Griem, D. A. Liberman, and A. McMahan for helpful discussions on the form and content of this article.

REFERENCES Anderson, P. W., and McMillan, W. (1967). In “Theory of Magnetism in Transition Metals” (W. Marshall, ed.), p. 50. Academic Press, New York. Bang, J., and Gareev, F. A, (198 1). Lett. Nuovo Cirnento 32,420. Bang, J., Gareev, F. A., Gizzatkulov, M. H., and Goncharov, S. A. (1978). Nucl. Phys. A 309, 381.

Bang, J., Ershov, S.N., Gareev, F. A., and Kazacha, G. S. (1980). Nucl. Phys. A 339.89. Baz’, A. I., Zel’dovich, Ya. B., and Perelomov, A. M. ( I 969). “Scattering, Reactions and Decay in Nonrelativistic Quantum Mechanics” (Israel Program for Scientific Translations, Jerusalem; translated from Rasseyanie, Reaktsii i Raspady v Nerelyativistskoi Kvantovoi Mekhanike, Izdatel’stvo Nauka, Glavnaya Redaktsiya, Fiziko-Matematicheskoi Literatury, Moskva, 1966).

PRESSURE IONIZATION

355

Berggren, K. F., and Froman, A. (1969). Ark. Fys. Kungl. Svenska Vetenskapsakad. 39,355. Berggren, T. (1968). Nucl. Phys. A 109,265. Berggren, T. (1982). Nucl. Phys. A 389,261. Brueckner, K. (1976). In “Laser-Induced Fusion and X-ray Laser Studies” (S. F. Jacobs et al., eds.). Addison-Wesley, Reading, MA. Burgess, D., and Lee, R. L. (1982). J. Phys. Colloq. C2 43,413. Bushman, A. V., and Fortov, V. E. (1983). Sov. Phys. Usp. 26,465. (Usp. Fiz. Nauk 140, 177, 1983). Butler, D. (1962). Proc. Phys. SOC.80,741. Caldirola, P., and Knoepfel, H., (1971). “Physics of High Energy Density.” Proceedings of International School of Physics “Enrico Fermi”, Course. 48, Italian Physical Society. Academic Press, New York. Cauble, R., Blaha, M., and Davis, J. (1984). Phys. Rev. A 29, 3280. Cox, A. N. (1965). In “Stellar Structure” (L. H. Aller and D. B. McLaughlin, eds.), Ch. 3. Univ. of Chicago Press, Chicago. Cox, J . P., and Giuli, T. R. ( 1 968). “Principles of Stellar Structure,” Vol. 1. Gordon & Breach, New York. Davis, J., and Blaha, M. ( I 982). I n “Physics of Electronic and AtomicCollisions” (S. Datz, ed.), p. 81 1. North-Holland Publ., Amsterdam. Dharma-Wardana, M. C., and Perrot, F. (1982). Phys. Rev. A 26,2096. Dyachkov, L. G., and Kobzev, G. A. (1981). J. Phys. E 14, M89. Feng, 1. J., and Pratt, R. H., Jr. (1982). J. Quantum Spectrosc. Radiat. Transfer 27, 341. Feng, I. J., Lamoureux, M., Pratt, R. H., and Tseng, H. K. (1982). J. Quantum Spectrosc. Radial. Transfer 27, 227. Friedel, J. (1967). In “Theory of Magnetism in Transition Metals’’ (W. Marshall, ed.), p. 283. Academic Press, New York. Friedel, J. (1969). In “Physics of Metals” (J. Ziman, ed.), Vol. I . Cambridge Univ. Press, London and New York. Garcia-Calderon, G. (1982). Leu. Nuovo Cimento 33,253. Garcia-Calderon, G., and Peierls, R. (1976). Nucl. Phys. A 265, 443. Green, J. M. ( I 964). J. Quantum Spectrosc. Radial. Transfer 19,639. Griem, H. R. (1964). “Plasma Spectroscopy.” McGraw-Hill, New York. Grimaldi, F., and Grimaldi-Lecourt, A. (1982). J. Quantum Spectrosc. Radial. Transfer 27, 373. Hohne, F. E., and Zimmerman, R. (1982). J. Phys. B 15,2551. Humblet, J. (1952). Mem. Snc. R. Soc. Sci. Liege 12, 9. Kirzhnitz, D. A., Lozovik, Yu. E., and Shpatakovskaya, G. V. (1976). Sov. Phys. Usp. 18,649. Kittel, C. (1963). “Quantum Theory of Solids.” Wiley, New York. Kobzev, G. A., Kurilenkov, Ju. K., and Norman, G. E. (1977). Teplofiz. Vis. Temp. 15, 193. Kohn. W., and Majumdar, C. (1965). Phys. Rev. A 138, 1617. Lamoureux, M., Feng, 1. J., Pratt, R. H., and Tseng, H. K. (1982). J. Quanium Spectrosc. Radiat. TransJer 27,227. Landau, L. D., and Lifshitz, 1. M. (1958). “Quantum Mechanics.” Pergamon, Oxford. Langer, J., and Ambegaokar, V. (1961). Phys. Rev. 21, 1090. Lee, C. M., and Thorsos, E. I. (1978). Phys. Rev. A 17, 2073. Lee, Y. T., and More, R. M. ( 1984). Phys. Fluids 27, 1273. Lee, P. H. (1977). Ph.D. dissertation, University of Pittsburgh. Levinson, N. (1949). Kgl. Danske Videnskab. Selskab.. Matt.-Fis.Medd. 25 (9). Liberman, D. A. ( I97 I). Phys. Rev. E 3,208 I , Liberman, D. A. (1 979). Phys. Rev. B 20, 498 I . Liberman, D. A. (1982). J . Quantum Specirosc. Radial. Transfer 27, 335.

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More,R. M.(1971).Phys. Rev. A4, 1782. More, R. M. (1979). Phys. Rev. A 19, 1234. More, R. M. ( I 98 1). Unpublished report UCRL-8499 1, “Atomic Physics in Inertial Confinement Fusion.” Lawrence Livermore National Laboratory, Livermore, CA. More, R. M. (1982). J. Quantum Spectrosc. Radial. Transfir 27,345. More, R. M. ( 1983).“Atomic and Molecular Physics ofControlled Thermonuclear Fusion” (C. Joachain and D. Post, eds.), p. 399. Plenum, New York. More, R. M., and Gejuoy, E. (1973). Phys. Rev. A 7, 1288. Newton, R. G. ( 1960). J. Math. Phys. 1, 3 19. Pauli, W. (1958). Handb. Phys. 5, 1. Peierls, R. ( 1 979). “Surprises in Theoretical Physics,” p. 137. Princeton Univ. Press,Princeton, NJ. Perrot, F. (1982). Phys. Rev. A 26, 1035. Petschek, A. (1971).Phys. Left. 34A,411. Petxhek, A,, and Cohen, H. (1972). Phys. Rev. A 5, 383. Pettifor, D. (1976). Commun. Phys. 1, 141. Regge, T. ( 1 958). Nuovo Cimento 8,67 I. Rogers, F. J., Graboske, H. C., and DeWitt, H. E. (1971). Phys. Lett. 34A, 127. Romo, W. J. (1975). Nucl. Phys. A 237,275. Ross, M. ( 1969). Phys. Rev. 179,6 12. Rozsnyai, B. F., ed. (1982). “Radiative Properties of Hot Dense Matter” [special issue of J. Quantum Spectrosc. Radial. Transfir 27, (3)). Siegert, A. J. F. (1939). Phys. Rev. 56,750. Weisheit, J. C., and Shore, B. W. (1974). Astrophys. J. 194, 519. Zel’dovich, B. Ya. (1961). Sov. Phys. JETP12, 542. Zink, W. (1968). Phys. Rev. 176,279.

A

AA, see Average-atom model Active-electron LS coupling scheme, 17 I AMA, see Angular-momentum-averaged approximation Angular-momentum-averaged approximation, 158, 162 Ar2+,DR cross section, 156- 157 Artificial intelligence, in most probable path approach, 2 16 “Asymmetry parameter” p, 84-87 Atomic and molecular processes using two lasers of different frequencies, 240 Auger ionization, 127 amplitude for, I43 Auger probabilitiesA, in LS coupling, 180-184 active-electron vs. core-electron couplings, 183-184 2e systems, I80 3e systems, 18 1 - 182 4e systems, 182- 183 5e systems, 183 Autler-Townes multiplet splittings, 210 Autoionizing states, effective quantum numbers defined for, 292-293 Average-atom model, 309,see also Selfconsistent-field method density dependence of eigenvalues, 310-31 I limitations of, 309-31 I

B Bamer height, equation for, 330 Be sequence, DR rate coefficient for, 165- 167

Bethe approximation, validity in evaluation ofA, for HRS, 193 Blackbody radiative transfer in sodium between conducting plates, 16 Bloch-Siegert shifts, 208 biochromatic, 246 Bloch spin or pseudospin vector, 247 Born approximation, see First Born approximation Born series, in potential scattering, 268-272 Bremsstrahlung “free-free transitions,” of electron in magnetic field, 26 1 opacity, 319 problem of electron-ion in strong magnetic field, 267-268 C

C+, DR cross section, 156 Calcium ion DR cross section, I53- 156 crossed-beam technique experiment, 155

Capture cross section enhanced by magnetic field, 302 Carbon monoxide, multiphoton absorption spectra for, 2 I4- 2 I5 Cascade effect, 144- 146 cascade amplitude, 145 CCCLC, see Complex-coordinate coupled Landau channel Center-of-mass separation, in a scattering problem, 258-261 in absence of external field, 258 in presence of magnetic field, 258 -26 1 Channel interactions involving highly excited bound levels, 69- 76

357

358

SUBJECT INDEX

Lu-Fano plots, 70-73 rotational channel interactions in a heavy molecule, 74-76 rotational perturbations in H,, 69-70 Channel interactions involving continua, 76 - 87 photoelectron angular distributions, 84-87 total photoionization cross section, 76- 84 Charge exchange, 297 - 303 CHF, 285 CI, see Configuration interaction C P , DR cross section, 156 Classical trajectory Monte Carlo method, 301, 302 Closed-channel resonances, 58 Coherent effects in fluorescence, 7 Collisional radiative model, 125 “Collision eigenchannel” solutions, 59 Complex-coordinate coupled Landau channel method, 233,234 for autoionizing resonances in intense magnetic fields, 233 Complex coordinate transformation, 227 effect on spectrum of atomic Zeeman Hamiltonian, 233 Complex quasi-energy formalism, 23 I , 233 Complex quasivibrational energy states, 237 Complex resonances, 82 -84 calculated partial cross sections illustrated, 83 in competition between alternative decay processes, 82 in H, and N,, 82 quantum defect theory applied to, 82 width dependence on coupling between closed channels, 82 Computer code, for some DR rates, I77 Configuration interaction, 157, 165, 172 effect on DR rates, table of, 172 Configuration interaction and intermediate coupling, in DR theory, 172- 173 effect of order of coupling in a manyelectron system, I73 Continuity in positron annihilation in metals, 3 I9 spectral window, 322-323 unsatisfactory calculations, 322 discontinuity from spherical-cell calculations, 320

Continuity theorems for electron gas, 3 18-32 I Continuous-wave dye lasers, 6 Continuum density of states, 3 17-3 18 equation for, 3 17 Friedel sum rule, 3 I7 model T density, 3 17 theory based on spatial location of electrons, 3 I7 Continuum wave functions, 3 13- 3 I6 definitions of, 3 I6 interrelations between, 3 16 Jost function and solution, 3 I3 - 3 I6 Convergence for the &potential model, 352-354 Coulomb Green’s function, 116 Coulomb Schrodinger equation, 54 Coupled-equations formulation, in potential scattering, 272 - 277 CQVE, see Complex quasivibrational energy states Cross section for resonant symmetric charge exchange, 299 Cross section for transitions between Landau levels, 265-266 Cross section, low-field limit of, 28 1-285 CTMC, see Classical trajectory Monte Carlo method Cyclohexane, experimental results of transient coherent Raman spectroscopy, 44

D Dielectronic recombination theory of, 123- 196 Dielectronic recombination cross sections, 146- 157 forC3+, 147 C+, C P , Ar2+, 156- 157 Ca+, 153- I56 experiment and theory compared, for Bz+ and C3+, 149 for Fe23+,a heavier ion, 148- 149 isonuclear sequences compared experimentally and theoretically, 149, 151, 152 Li sequence, I46 - I49 Mg+, 151-153

359

SUBJECT INDEX nonrelativistic Hartree- Fock approximation, with LS coupling, 146 S13+,Sill+, V2+,Ca2+, 149- 151 Dielectronic recombination rate coefficients, 157-171 Be sequence, N = 4, I65 - 167 for Be sequence target ions, table of, 166 for e- FeZ3+system, calculated, 163 fore- OJ+system, calculated, 162 H sequence, N = I , 158-159 He sequence, N = 2, 159- 161 for isoelectronic sequences, 157 - 158 Li sequence, N = 3, 161 - 165 for Li-like target ions, table of, 164 in LS coupling, 158 Mgsequence, N = 12, 169-170 Na sequence, N = I I , 168 for Na sequence target ions, table of, 169 Ne sequence, N = 10, 167 for Ne-like target ions, table of, 167 sequencesN= 18, 19, 170-171 Direct radiative recombination, 124 Dissociative recombination, 106- 1 I5 Distorted-wave Born approximation, I39 Doppler-free two-photon spectroscopy, 33, 43 DR, see Dielectronic recombination DR, astrophysical applications, I25 DR, fusion-related work, 124- 125 DWBA, see Distorted-wave Born approximation Dynamical Stark effect, 210

+ +

E Earth’s magnetic field, compensation for, 5 Eigenchannel representation, 58 -62 Eigenchannels identical to eigenstates, 64 -65 Eigenchannels, physical significance of, 63-66 Eigenphase shifts, energy dependence of, 57 Eigenquantum defects, 65, 67 definition of, 58 Einstein spontaneous emission coefficient for free space, I8 Electron density. 337 as sum of resonance contributions, equation for. 337 Electron density of states, 349-35 I

Electronic interactions at short range, 97- I I5 electronic preionization in molecular nitrogen, 102- 106 Hopfield series, 103, 104, 106 interactions between ionization channels, 101- 102 computer program calculations of, 101- 102 photodissociation and dissociative recombination, 106- I15 rovibronic channel interactions, 97 -99 theory, 97 - 102 two-step treatment of electronic channel interactions, 99 - 10I Electronic preionization in molecular nitrogen, 102- 106 Electron-ion collision processes, 126 - 128 collisional excitation, 126 collisional ionization, 127 photoionization, 127 radiative capture, I26 representation of, 129- I30 Electron -ion collision theory, 128- 146 cascade effect, I44 - 146 reaction channels, 132- 138 scattering amplitudes and cross sections, 138- 144 Electron-molecule scattering theory, 54 Electron-neutral scattering, 116- 117 Electron outside an ion core, its potential and kinetic energies, 64 “Energy-gap law”, 8 I “Energy-normalized’’ wave functions, 6 1 Excitation-capture probability V,, I57 External field effect, in DR theory, 175- I76

F Fermi-Dirac distribution, 335, 337 Fermi function, 335-336, 337 Fermi’s Golden Rule, 13 Field-free Green function formalism, 272 “Fine-structure frame transformation,” 65 First Born approximation, in potential scattering, 266-267 Floquet characteristic exponent, or quasi-energy, 200 Floquet formalism, 199- 201

360

SUBJECT INDEX

Floquet Hamiltonian, for two electronic states, structure of, 236 Floquet Hamiltonian method, extensions of, 209-2 17 Floquet matrix structure, 203 Floquet methods, nonperturbative electric dipole approximation, 209 external dc field introduced, 2 10-2 I 1 most probable path approach, 2 I5 - 2 I7 nonadiabatic theory for resonant multiphoton excitation, 2 11 -2 15 Floquet perturbation methods, 209 Floquet quasi-energy diagram, 198 Floquet-state nomenclature, 20 1 Floquet theorem, 200 Floquet theory and general properties of quasi-energy states, 199- 208 Floquet formalism, 199- 20 1 quasi-energy states, properties of, 204 - 208 Shirley’s time-independent Floquet Hamiltonian, 201 -202 time evolution operator and transition probabilities, 202 -204 Floquet theory, application to quantum system, 198 Floquet theory for study of laser-induced collisions, 249 Fluorescence linewidth, less than natural linewidth, 23-25 Fluorescence yield w(d), 157, 163, 164, 165, I72 Flux dependence of resonant two-photon ionization cross sections, 229 Fragmentation channels, 58 - 59 Fragment spin polarization, 63 Frame transformation angles, 73 Frame transformation, rotational, 52 Frequency modulation spectroscopy,6 Frequency sweep experiment, in behavior of Floquet exponent, 2 19 Frequency-weighted cross section, defined, 144 Friedel sum rule, 3 17 Full-frame transformation matrix. 67 G Generalized rotating-wave approximation, 240,242-247

bichromatic Floquet Hamiltonian for two-level systems, 244 Generalized Van Vleck theory, 248 MMFT-GVV method, 248 Green’s function, 3 16, 325, 327, 337, 342, 348-349, 353 Green’s function for the (N 1) electrons and one photon, I33 GRWA, see Generalized rotating-wave approximation GVV, see Generalized Van Vleck theory

+

H Hamiltonian Floquet, for two electronic states, 236 for N 1 electrons plus radiation field, 129, 131 periodic in time and time-independent, solutions of, 204 semiclassical and time-dependent, in the electric dipole approximation, 230 “Hanle effect,” 26 Heitler-Ma theory of natural linewidth, 10- I3 Heitler method in overcoming natural linewidth, 23 -25 strong-field spectrum, 23 weak-field spectrum, 23 Henon - Heiles anharmonic-oscillator system, 224 He sequence, DR rate coefficient for, 159- 161 Heterodyne detection scheme, 5 -6 in Doppler-free two-photon spectroscopy, 6 in nonlinear spectroscopy, 6 in Raman spectroscopy,6 H, experimental study of resonant multiphoton ionization, 230 HF Morse oscillator, quantum and classical MPE behavior compared, 222 HF, multiphoton processes studied by numerical integration, 2 I8 Higher-order modified Born approximation, 270 High-resolution spectroscopy for electronically excited states, 3 laser spectroscopy, 2, 3 High Rydberg state, 159- 193, passim

+

36 1

SUBJECT INDEX High Rydberg states, extrapolation to, 189193 dipole approximation, 192- 193 extrapolation from low-n states, 190- I9 1 extrapolation by quantum defect theory, I91 - 192 Hilbert space, composite, in Floquet formalism for quasi-energy states, 200-201, 204 HMBA, see Higher-order modified Born approximation Hole burning, 2 10 Hopfield series of molecular nitrogen, 103, 104, 106 HRS, see High Rydberg state H sequence,DR rate coefficient for, 158- I59 Hund‘s coupling cases, 66,75 1

IC, see Intermediate coupling effects IDE, see Inhomogeneous differential equation approach Inhomogeneous differential equation approach, 237 “Interference signal,” in Ramsey method, 31 -32 Intermediate coupling effects, 157, 165 on DR rates and cross sections, I72 Intermodulated fluorescence, 5 Ion-atom collisions in a magnetic field, 297 bound-state wave functions modified, 298 Hamiltonian for this process, 298 new phase factor in matrix elements, 298 Ionization state, 338 - 339 definition of, 338 effectsof resonance states, 338 Ion-sphere model, in SCF, 308 Ion-trap experiments on photodetachment of S-and SeH-, 293-295 IRA, see Isolated resonance approximation Isolated resonance approximation, 14 1

J Jaynes-Cummings model, for spontaneous emission, 17 Jaynes-Cummings Hamiltonian, 17 Jost function, properties of, 347- 348

Jost function and solution, 3 13 - 3 16, 3 19, 327, 334,353 JWKB approach, 261

K Kapur-Peierls theory, 316, 328, 329 1

Lamb dip, 3-4 Landau functions, 262, 263 Landau gauge, 259,262 Landau levels, ensembles of, 277-281 magnetic field parallel to z axis, 278-281 magnetic field perpendicular to z axis, 278 Laplace time-averaged value of transition probability, 220 Laser-induced dissociation, 238 Lasers in optical spectroscopy, 198 Level-crossingspectroscopy, 26 - 30 delayed level-crossingmethod, 27 - 30 hyperfine interaction constants of excited states measured, 27 pulsed excitation of barium, 29 pulsed excitation of coherent superposition of excited states, 28 time-integrated fluorescent intensity for increasing delays, 29 Li sequence, for DR cross sections, 146- 149 reaction AN, = 0 process for ionic targets B*+,C3+,andO3+, 146- 147,148, 149 Li sequence, N = 3, DR rate coefficient for, 161-165 Is, An, # 0 excitation, 163- 164 2s, An, = 0 excitation, I6 1- I62 2s, An, ;P 0 excitation, 162- 163 cascade effect, 164, 165 excitations An = 0 and An # 0, 161 LID, see Laser-induced dissociation Linearly polarized bichromatic Floquet Hamiltonian for two-level system, 243 Lippmann-Schwinger equation, 102, 116, 264,268 Lu-Fano plots, 70-73 for detection of channel interactions, 72 graphical way to remove boundary conditions on wave function at infinity, 72 of He,, with weak channel interaction, 73

362

SUBJECT INDEX

of perturbations, 7 I for spectral analysis of perturbed Rydberg series, 72

M Magnetic field capture cross section enhanced by, 302 effect on collision processes, 286 effect on structure of target, 257 laboratory production and limitations, 256 strong field defined, 257 weak field defined, 257 Magnetic fields in astrophysics, 256 Magnetic fields, scattering in, 255 - 304 Magnus propagator, 223 Many-mode Floquet theory, 239-248 generalized rotating-wave approximation, 242-247 SU(N) dynamical symmetry and quantum coherence, 247-248 Matter-field coupling effects, I9 Methane, saturation resonances of, 33 Methyl iodide, Lu - Fano plot of, I I6 Mg+, DR cross section, 15 I - I53 Mg sequence, DR rate coefficient for, 169-170 MMFT, see Many-mode Floquet theory Model A theory, 337 Model T density of states, 3 17, 337 Mode-selective IR-MPE of 03, 216 Molecular applications of quantum defect theory, 51-121 Mossbauer spectroscopy, 25 Most probable path approach, 2 15 -2 16 MPA, see Multiphoton absorption MPD, see Multiphoton dissociation MPE, see Multiphoton excitation MPI, see Multiphoton ionization MPPA, see Most probable path approach MQDT, see Multichannel quantum defect theory Multichannel quantum defect theory, I36 Multichannel rearrangement processes, 56-58 Multiphoton absorption, 2 I I , 2 12, 2 14- 2 15 Multiphoton dissociation, 198, 2 15 Multiphoton excitation, 198, 2 15, 242, 246

Multiphoton excitation of finite-level systems, computational methods for, 208-226 Floquet Hamiltonian method, extensions Of, 209-217 perturbation methods, 209 recursive residue generation, 224-225 rotating frame transformation, 225 -226 time-propagator methods, 2 17 -224 Multiphoton ionization, collisionless, 198 Multiphoton processes fully quantum-mechanical or semiclassical formalism, 198 Multiple interaction in standing wave fields, 9 Ca1So-3Plintercombination line at 657 nm, 9 N

Natural linewidth, “fundamental” ways to overcome it, 10-25 Heitler-Ma theory, 10- 13 Heitler method, 23 - 25 Purcell method, I3 - I9 Resonance fluorescence, 19 - 23 Neon, saturation resonances of, 33 Neon sequence, DR rate coefficient for, 167 Nitric oxide, photoexcitation spectra, 52 - 54 Nitrogen, electronic preionization in, 102- 106 Nitrogen pumped dye laser, 26 Non-Born - Oppenheimer phenomena, 87-97 adiabatic and nonadiabatic corrections to the discrete levels, 88 - 89 quantum defect calculations, 88 -89 R-matrix treatment of predissociation, 89 - 97 Non-Hermitian Floquet Hamiltonian, structure of, for MPI of H atom, 228 Non-Hermitian Floquet matrix formalism, 227-237 complex quasi-vibrational energy method for multiphoton dissociation, 234-237 multiphoton ionization in circularly polarized fields, 230-231 multiphoton ionization in linearly polarized fields, 227 - 230

SUBJECT INDEX photoionization in intense magnetic fields, 231 -234 Non-Hermitian Floquet theory for multiphoton ionization and dissociation, 226-239 matrix formalism, 227 - 237 Nonlinear molecular phenomena, I98 collisionless multiphoton ionization, 198 multiphoton dissociation, 198 multiphoton excitation, 198 Nonlinear spectroscopy, 3 -7 detection methods, 4 0 “On-line” electronic apodization technique, to eliminate spectral sidebands, 26 Optoacoustic techniques, in saturation spectroscopy, 4 Optogalvanic techniques, in saturation spectroscopy, 4, 5 Overlapping resonances and interferences, in DR theory, I73 - 175 P PAI, see Photo-Auger ionization Paul radio-frequency trap, 9 Penning trap, 9, 19, 293 Perturbation methods, for multiphoton excitation, 209 Phase-isolation technique, 32 Photoabsorption from a single bound state, 62 Photo-Auger ionization, 143- 144 Photodetachment of negative ions, 293 - 297 of SeH-, graphic results, 294 of S-, graphic results, 294 Photodissociation and dissociative recombination, 106- I 15 application to competing dissociation and ionization processes in NO, 109- 1 12 dissociative recombination, 1 1 3 - 1 I5 theory, 106- 109 diabatic potential energy curves of states in NO, 107 Photoelectron angular distributions. 84-87

363

angular momentum recoupling, 85 “asymmetry parameter” /?,84-87 in short-range molecular dynamics, 84 Photofragment angular distributions, 63 Photofragmentation cross sections, 62 -63 Photofragments, alignment and orientation of, 63 Photoionization, 285 -293 cross section of one-electron system in magnetic field, 285 efficiency, ratio of total ionization to transmitted light, 78 oscillator strengths of H, 292 spectra, Av+ = k I selection rule applied to, 80 theory and experiment, 80 spectrum of cooled para-H,, 78 theory versus experiment, 78 - 79 Plasma ionization state, 330 Plasma model, rate equations for, 125 Plasmas, hot, dense, 306 equation of state for, 322 pressure ionization in, 306 - 307 Polarization intermodulated excitation spectroscopy, 5 Polarization spectroscopy, $ 4 2 -43 narrowest subnatural dips obtained, 43 POLINEX, see Polarization intermodulated excitation spectroscopy Potential scattering, in magnetic field, 26 I -277 Born series, 268-272 Brernsstruhlung problem, 267- 268 coupled-equations formulation, 272 - 277 first Born approximation, 266-267 Power broadening, 2 10 PPPL tokamak data, 160 Predissociation for H, in excitation energy range, 78, 80-81 R-matrix treatment of, 89-97 Preionization and predissociation spectra, 53 Preionization in H, displayed, 8 1 Pressure formulas, 339- 346 condition for hydrostatic equilibrium of the electron fluid, 340 for pressure of continuum electrons, 339 pressure resolved into additive resonance contributions, 346

364

SUBJECT INDEX

pressure-tensor method, 340- 343 viral theorem, 343-346 Pressure ionization bound state crosses zero energy into continuum, 306 continuity of, 318-323 continuity theorems, proof of, 319-321 described by theory of resonances, 333 in electronic density of states, 3 12, 3 I3 linked to resonances and continuity principle, 307 in plasmas, 306 - 307 qualitative, 3 1 I - 3 13 Pulsed dye lasers, of narrow bandwidth, 25 Purcell method, in overcoming natural linewidth, I3 - I9 for atom in free space, 1 3 - 14 for atom not in free space, 14- 15

Q QES, see Quasi-energy state Quadratic Doppler effect, as a limit to resolution, 9 Quantum beat method field ionization as probe, 8 laser-induced birefringence, 8 laser-induced dichroism, 8 photoionization as probe, 8 Quantum beats, field-induced, in multiphoton ionization, 238, 239 Quantum beat spectroscopy, 7-8 Quantum defect, 54, 296 Quantum defect and potential energy curves, for lowest ungerade singlet Rydberg states of Hlr 67 Quantum defect calculation, 78 Quantum defect concepts and formalism, 54 - 66 adaptation to molecular problems, 66 -69 eigenchannel representation, 58-62 eigenchannels, physical significance of, 63-66 multichannel rearrangement processes, 56-58 photofragmentation cross sections, 62-63 Rydberg formula, its origin, 54-56 Quantum defect theory similarities of Rydberg bound states and continuum states, 76

Quantum-mechanical method for multiphoton dissociation of diatomic molecules, 239 Quasi-energy, eigenvalue equation for, 200 Quasi-energy states definition of, 200 Hellmann -Feynman theorem, 204-205 mean energy, 205 plot of eigenvalues of Floquet Hamiltonian, 207-208 properties of, 204 - 208 symmetry of, 205 - 206 variational principle, 204 Quasi-Landau regime, 257 Quasi-vibrational energy, 2 I2 QVE, see Quasi-vibrational energy

R Radiative probabilities A, in LS coupling, 184-185 2e systems, I84 3e systems, 185 4e systems, 185 Radiative width and coupled equations, in DR theory, 180- 184 inner-shell electron transition, 178- 179 outer-shell electron transition, 179 Ramsey fringes, 3 I Ramsey interference method, 30- 35 Doppler-free two-photon spectroscopy, 33-34 optical fields, 30 - 33 radio-frequency fields, 33 - 35 Reaction channels, 132- 138 elimination of the P and R channels, 135-137 elimination of R and Q channels, I37 - 138 elimination of the R channels, 133- I34 Reaction-matrix representation, 57 Rearrangement collision, quantum mechanical amplitude for, 56 Recursive residue generation method, 224-225 Renormalization factor, for resonance states, 326 Resonance fluorescence, 19- 23 Resonance fluorescence spectra of Na D, line, theoretical and experimental, 42 Resonance perturbation theory, 35 1-352

SUBJECT INDEX Resonance wave functions, 324- 325 completeness of the set, 327 - 328 properties of, 325 - 328 Resonance wave vector, 324 Resonances, 306-307, 324-333, see also Complex resonances bamer height, estimate of, 330-333 convergence of the expansions, 329 Green’s function, continuum wave functions, and S matrix examined, 329 electron density, 337 perturbation theory, 328-329 first-order, 328 second-order, 328, 35 1 quantum theory of, 324 resonance energy, estimate of, 330-333 scattering theory for, 324 thermal occupation of resonance states, 334-337 Resonant transfer excitation, 150- I5 1 RFT, see Rotating frame transformation Riemann product integral representation, 22 I R-matrix treatment of predissociation, 89-97 eigenchannel version of R-matrix theory, 91 -92 electronic eigenphase as function of vibrational eigenphase, for H,, 94 infinite-range and finite-range vibrational spectrum of H2+,93 Rotating frame transformation method, 225-226 energy-level scheme showing effect of RFT, 225 Hamiltonian time-dependent and time-independent terms separated, 226 Rotating-wave approximation, 222,223, 247-248 Rotational channel interactions in a heavy molecule, 74-76 Rotational perturbations between highly excited Rydberg levels in H,, 69 Rotational perturbations in H,, 69 -70 Rovibrational autoionization, 77 Rovibrational channel interactions, 66-97 adaptation of quantum defect formalism to molecular problems, 66 - 69

365

channel interactions involving continua, 76-87 channel interactions involving highly excited bound levels, 69 - 76 first documentation, 69 treatment of a class of non-BornOppenheimer phenomena, 87-97 RR, see Direct radiative recombination RRGM, see Recursive residue generation method RTE, see Resonant transfer excitation RWA, see Rotating-wave approximation Rydberg channels illustrated, in vibrational- rotational preionization and predissociation in H,, 79 Rydberg formula, origin of, 54- 56 Rydberg levels, by quantum beat method, 8 Rydberg spectrum of Na,, 74 periodicity, related to several series, 75 Rydberg-state Born - Oppenheimer potential curves, 65 S Saturated absorption, to eliminate Doppler broadening, 1 Saturation spectroscopy, development of, 4 Scaling properties of A,, A,, w, and aDR, 185-189 I dependence, 189 nc scaling, 188 - 189 z scaling, I86 - I88 Scattering amplitudes and cross sections, 138- 144 Auger and radiative widths, 142 DR cross section, I4 1 initial-state wave function, 139 Scattering experiment, beam of electrons enters region of uniform magnetic field, 278 Scattering in strong magnetic fields, 255 - 304 center-of-mass separation, 258-26 1 charge exchange, 297 -303 ensembles of Landau levels, 277-281 low-field limit of the cross section, 281 -285 photodetachment of negative ions, 293-297 photoionization, 285 -293 potential scattering, 26 1 - 277

366

SUBJECT INDEX

SCF, see Self-consistent-field method Schrodinger equation for heteronuclear diatomic molecule interacting with coherent monochromatic field, 2 12 for molecular system in electromagnetic field, 234 for system with periodic Hamiltonian, 199 - 200 Seaton’s quantum defect theory, 57 Self-consistent-field method, 307 - 309 electrostatic potential in, 307 matter at high density and temperature, 307 spherical-cell model, 306, 308 Semiclassical Floquet theories for intensefield multiphoton processes, recent developments in, 197-253 Separated oscillatory field resonance line shapes, 32 Separated oscillatory fields method of Ramsey, 30 SF,, model study of multiphoton excitation, 222 - 223 Shape resonances, 324 Shirley’s time-independent Roquet Hamiltonian, 201 -202 S-hump behaviors, 210 Sill+, DR cross section calculated, 150 SO,, multiphoton excitation of, 215 dependence of MPD on laser intensity, 216, 217 most probable path approach, 224 Sodium D,line, resonance fluorescence spectra of, 42 four two-photon resonances of the 32S-42D transition, 34-35 highly excited Rydberg states of, 74 Sodium sequence, N = 1 I , 168 - 169 DR rate coefficient for, 168 plot of DR rate coefficient versus nuclear core charge, I68 “Spectral window,” 322 - 323 Spectroscopic resolution, improvements of, 2-9 nonlinear spectroscopy, 3 - 7 quantum beat spectroscopy, 7-8 ultimate spectral resolution, 8 -9

Spherical-cell model, 308 limitations of, 346 Spontaneous emission, its elimination, I3 Square-integrable (L2)continuum discretization, 227, 234 Standard recoupling coefficient ( j / L S ) ,65 Stark mixing, 175 - 176 Stark-Zeeman Hamiltonian, 233 Stokes spectrum, narrowed by transient coherent Raman spectroscopy, 45 Sub-Doppler spectroscopy, 33 Subnatural linewidths, atomic, I -49 SU(N) dynamical symmetry and quantum coherence, 247 - 248 Gell-Mann SU(3) symmetry in particle physics, analogous to, 248

T Thermal occupation of resonance states, 334-337 resonance population, equation for, 335 resonance thermal occupation function, 335, see also Fermi - Dirac distribution calculations of, 336 Thomas-Fermi ionization state, 330, 338, 339 Thomas-Fermi theory, 339, 343 Time-biased coherent spectroscopy, 25 -45 level-crossing spectroscopy, 26 - 30 Ramsey interference method, 30- 35 Transient line narrowing, 35-42 Time development of quantum H F Morse oscillator in periodic driving field, 223 Time evolution operator and transition probabilities, 202 -204, 227 long-time average transition probability, 204 Time-propagator methods, for multiphoton excitation, 2 I7 - 224 Magnus approximation, 22 1 - 224 time evolution operator in exponential form, 22 1 Meath, Moloney. and Thomas methods, 218-221 numerical integration method, 2 17 - 2 18 TMFT, see Two-mode Roquet theory Tokamak plasmas, 125

367

SUBJECT INDEX Total capture and ionization cross sections for He2+-H collisions, 302 Total photoionization cross section, 76-84 Transient coherent Raman spectroscopy, 43-45 experimental results for cyclohexane, 44 Transient line narrowing, 35 -42 phase switching, 40-42 pulsed excitation, 38 -39 strong-signal regime, 39-40 photocount distribution S for fixed laser intensity, 40 Transit-time broadening, as a limit to resolution, 9 Trapped ion spectroscopy, 9 Two-mode Floquet Hamiltonian, 24 I Two-mode Floquet theory, 24 I , 242- 246 generalized rotating-wave approximation compared to, 245-246 Two-photon spectroscopy, I , 6 - 7 for dipole-forbidden transitions, 7

Doppler-free experiments, 7 to eliminate Doppler broadening, 1

U Ultimate spectral resolution, 8 - 9 Unimolecular decay of metastable molecules, 84 Unimolecular multiphoton dissociation reactions of polyatomic molecules, 234 V VI9+, DR cross section, theory and experiment compared, 15 I , 1 52 Vacuum-field Rabi splitting, 18, 19 “Vibronic” preionization, 102 W Window resonances, 114- I 1 5 WKB approximation, 3 10, 323, 330


CUMULATIVE AUTHOR INDEX: VOLUMES 1- 20 A

Amos, A. T., seeHall, G. G., 1 Amusia, M. Ya., Collective effects in photoionization of atoms, 17 Andersen, N., Direct excitation on atomic collisions: studies of quasi-one-electron systems, 18 Anderson, J. B., High-intensity and high-energy molecular beams, I Andres, P., see Anderson, J. B., 1 Andrick, D., The differential cross section of low-energy electron -atom collisions, 9

Armstrong, David A., see Freeman, Gordon R., 20 Armstrong, Lloyd, Jr., Relativistic effects in the many-electron atom, 10 Asaad, W. N., seeBurhop, E. H. S., 8 Audion, C., Physics of the hydrogen maser, 7 B Bardsley,J. N., Dissociative recombination, 6 Bates, D. R. Electronic eigenenergies of the hydrogen molecular ion, 4 Use of classical mechanics in the treatment of collisions between massive systems, 6 Aspects of recombination, 15 Ion-ion recombination in an ambient gas, 20 Bauche, J., Recent progress in the theory of atomic isotope shift, 12 Beder, F. Chanoch, Quantum mechanics in gas crystal -surface van der Waals scattering, 3

Bederson, Benjamin, see Miller, Thomas M., 13 Bell, K. L., The first Born approximation, 10 Ben-Reuven, A., The meaning of collision broadening of spectral lines: the classical oscillator analog, 5 Berman, Paul R., Study of collisions by laser spectroscopy, 13 Bernstein, R. B. Role of energy in reactive molecular scattering: an information- theoretic approach, 11 Reactive scattering: recent advances in theory and experiment, 15 Biondi, M. A., see Bardsley, J. N., 5 Blum, K., Spin-dependent phenomena in inelastic electron -atom collisions, 19 Bobashev, S. V., Quasi-molecular interference effects in ion-atom collisions, 14 Bottcher, Christopher, Numerical calculations on electron-impact ionization, 20 Boyd, R. L. F., The direct study of ionization in space, 4 Bransden, B. H. Atomic rearrangement collisions, 1 The theory of fast heavy-particle collisions, 15 Electron capture in collisions of hydrogen atoms with fully stripped ions, 19 Browne, J. C., Molecular wave functions: calculation and use in atomic and molecular processes, 7 Broyer, M., Optical pumping of molecules, 12 Buckingham, R. A,, Applications of quantum theory to the viscosity of dilute gases, 4 369

370

CUMULATIVE AUTHOR INDEX

Budick, B., Optical pumping methods in atomic spectroscopy, 3 Burgess, A., Classical theory of atomic scattering, 4 Burhop, E. H. S. H. S. W. Massey-A sixtieth birthday tribute, 4 The Auger effect, 8 Inner-shell ionization, 15 Burke, P. G. The R-matrix theory of atomic process, 11 Theory of low-energy electron- molecule collisions, 15 C

Cairns, R. B., Photoionization with molecular beams, 8 Castleman, A. W., Jr., see Mark, T. D., 20 Celotta, R. J., Sources of polarized electrons, 16 Champeau, R.-J., see Bauche, J., 12 Chemin, J.-F., seeMeyerhof, W. E., 20 Chen, Augustine C., seeChen, Joseph C. Y., 8 Chen, C. H., seePayne, M. G., 17 Chen, Joseph C. Y., Nonrelativistic off-shell two-body Coulomb amplitudes, 8 Cherepkov, N. A., Spin polarization of atomic and molecular photoelectrons, 19 Child, M. S., Semiclassical effects in heavy-particle collisions, 14 Chisholm, C. D. H., Tables of one- and two-particle coefficients of fractional parentage for configurations s h ’ p p q , 5 Cohen, Maurice, see Weinstein, Harel, 7 Cohen, H., Atomic Hartree-Fock theory, 16 Collins, L. A,, see Norcross, D. W., 18 Crossley, R. J. S., The calculation of atomic transition probabilities, 5 Crothers, D. S. F., Nonadiabatic charge transfer, 17 Csanak, Gy., Green’s function technique in atomic and molecular physics, 7 D Dalgarno, A. The calculation of van der Waals interactions, 2 Collisions in the ionosphere, 4

see Chisholm, C. D. H., 5 Atomic physics from atmospheric and astrophysical studies, 15 Davison, W. D., see Dalgarno, A., 2 deHeer, F. J., Experimental studies of excitation in collisions between atomic and ionic systems, 2 Dehmelt, H. G . Radiofrequency spectroscopy of stored ions I: storage, 3 Radiofrequency spectroscopy of stored ions 11: spectroscopy, 5 Dickinson, A. S., Classical and semiclassical methods in inelastic heavy-particle collisions, 18 Doyle, Holly Thomis, Relativistic Zdependent corrections to atomic energy levels, 5 Drake, G. W. F., Quantum electrodynamic effects in few-electron atomic systems, 18 Dufton, P. L., Atomic processes in the sun, 17 Dupree, A. K., UV and X-ray spectroscopy in astrophysics, 14 Duren, R., Experiments and model calculations to determine interatomic potentials, 16 E Edelstein, S. A., Rydberg atoms, 14 English, Thomas C., see Zorn, Jens C., 9

F Faubel, Manfred Scattering studies of rotation and vibrational excitation of molecules, 13 Vibrational and rotational excitation in molecular collisions, 19 Fawcett, B. C., Recent progress in the classification of the spectra of highly ionized atoms, 10 Fehsenfeld, F. C., see Ferguson, E. E., 5 Feneuille, Serge see Armstrong, Lloyd, Jr., 10 Atomic Rydberg states, 17 Fenn, J. B., see Anderson, J. B., 1 Ferguson, E. E., Flowing afterglow measurements of ion-neutral reactions. 5 Figger, H.. seeGallas, J. A. C., 20

37 1

CUMULATIVE AUTHOR INDEX Folde, M. F., Chemiluminescence in gases, 11 Foltz, G. W., see Payne, M. G., 17 Foner, S. N., Mass spectrometry of free radicals, 2 Fortson, E. N., Parity nonconservation in atoms: status of theory and experiment, 16 Fraser, P. A., Positrons and positronium in gases, 4 Freeman, Gordon R., Electron and ion mobilities. 20 G Gal, E., see R. A. Buckingham, 4 Gallagher, T. F.,see Edelstein, S. A., 14 Callas, J. A. C., Rydberg atoms: high resolution spectroscopy and radiation interaction- Rydberg molecules, 20 Carton, W. R. S., Spectroscopy in the vacuum ultraviolet, 2 Gerratt, J., General theory of spin-coupled wave functions for atoms and molecules, 7 Gibbs, H. M., seeSchuurmans, M. F. H., 17 Gilbody, H. B., Atomic collision processes in controlled thermonuclear fusion research, 15 Golden, D. E., Resonances in electron, atom, and molecule scattering, 14 Gouedard, G., see Broyer, M., Optical pumping of molecules, 12 Greenfield, A. J., A review of pseudopotentials with emphasis on their application to liquid metals, 7 Greim, Hans R., Stark broadening, I1 Griffith, T. C., Experimental aspects of positron collisions in gases, 15 Grivet, P., see Audion, C., 7 H Hall, G. G. Molecular orbital theory of the spin properties of conjugated molecules, 1 Atomic charges within molecules, 20 Hansteen, Johannes M., Inner shell ionization by incident nuclei, 11 Haroche, S., Radiative properties of Rydberg states in resonant cavities, 20

Harrison, Halstead, see Cairns, R. B., 8 Hasted, J. B. Recent measurements on charge transfer, 4 Ion -atom charge transfer collisions at low energies, 15 Heddle, D. W.0. Measurements of electron excitation functions, 4 Excitation of atoms by electron impact, 15 Hertel, 1. V., Collision experiments with laser-excited atoms in crossed beams, 13 Holt, A. R., Born expansions, 4 Humberston, J. W., Theoretical aspects of positron collisions in gases, 15 Huntress, Wesley T., Jr., A review of Jovian ionospheric chemistry, 10 Hurst, G. S.,seePayne, M. G., 17 Hutcheon, R. J., seeKey, M. H., 16 I Innes, F. R., see Chisholm, C. D. H., 5 Itano, Wayne M., see Wineland, D. J., 19 Itikawa, Yukikazyu, see Takayanagi, Kazuo, 6

J Jacquinot, Pierre, see Feneuille, Serge, 17 Jamieson, Michael, J., see Webster, Brian C., 14 Janev, R. K. Nonadiabatic transitions between ionic and covalent states, 12 see Bransden, B. H., 19 Jenc, F., The reduced potential curve method for diatomic molecules and its applications, 19 Judd, B. R., Selection rules within atomic shells, 7 Junker, B. R., Recent computational developments in the use of complex scaling in resonance phenomena, 18 K Kaufman, A. S., Analysis of the velocity field in plasma from the Doppler broadening of spectral emission lines, 6 Kauppila, Walter E., see Stein, Talbert S., 18

372

CUMULATIVE AUTHOR INDEX

Keck, James C., Monte Carlo trajectory calculations of atomic and molecular excitation in thermal systems, 8 Keesing, R. G. W., see Heddle, D. W. O., 4 Key, M. H., Spectroscopy of laser-produced plasmas, 16 Kingston, A. E. see Bates, D. R., 6 see Bell, K. L., 10 see Dufion, P. L., 17 Weinpoppen, H. Coherence and correlation in atomic collisions, 15 see Blum, K., 19

Mohr, C. B. O., Relativistic inner shell ionization, 4 Mohr, Peter J., see Marms, Richard, 14 Moiseiwitsch, B. L. Electron affinities of atoms and molecules, 1 see Holt, A. R., 4 Relativistic effects in atomic collisions theory, 16 Morellec, J., Nonresonant multiphoton ionization of atoms, 18 Mum, R. J., see Mason, E. A., 2

L

Nesbet, R. K., Low-energy electron scattering by complex atoms: theory and calculations, 13 Neyaber, Roy H., Experiments with merging beams, 5 Nielsen, S . E., see Andersen, N., 18 Norcross, D. W., Recent developments in the theory of electron scattering by highly polar molecules, 18 Normand, D., see Morellec, J., 18

Lambropoulos, P., Topics on multiphoton processes in atoms, 12 Lange, W., Dye lasers in atomic spectroscopy, 10 Lehman, J. C., see Broyer, M., 12 Leuchs, G., seeGallas, J. A. C., 20 Levine, R. D., see Bernstein, R. B., 11 Lin, C. D., Inner-shell vacancy production in ion-atom collisions, 17 Luther, J.. see Lange, W., 10

M Mark, T. D., Experimental studies on cluster ions, 20 Marrero, R. T., see Mason, E. A., 6 Marms, Richard, Forbidden transition in one- and two-electron atoms, 14 Mason, E. A. Thermal diffusion in gases, 2 The diffusion of atoms and molecules, 6 Massey, H. S . W., Negative ions, 15 McCarthy, Ian E., see Weigold, Erich, 14 McEachran, R. P., seeCohen, M., 16 McElroy, Michael B., Atomic and molecular processes in the Martian atmosphere, 9 McNally, D., Interstellar molecules: their formation and destruction, 8 Meyerhof, W. E., Nuclear reaction effects on atomic inner-shell ionization, 20 Miller, Thomas M., Atomic and molecular polarizabilities-a review of recent advances, 13

N

0

Oka. Takeshi, Collision-induced transitions between rotational levels, 9 O’Malley, Thomas F., Diabatic states of molecules-quasi-stationary electronic states, 7 P Park, J. T., Interactions of simple ion-atom systems, 19 Pauly, H., The study of intermolecular potentials with molecular beams at thermal energies, 1 Paunez, Ruben, see Weinstein, Harel, 7 Payne, M. G., Applications of resonance ionization spectroscopy in atomic and molecular physics, 17 Percival, 1. C. see Burgess, A., 4 The theory of collisions between charged particles and highly excited atoms, 11

373

CUMULATIVE AUTHOR INDEX Peterkop, R., The theory ofelectron-atom collisions, 2 Petite, G., see Morellec, J., 18 Pierce, D. T., seecelotta, R. J., 16 Pipkin, Francis M., Atomic physics tests of the basic concepts in quantum mechanics, 14 Polder, D., see Schuurmans, M. F. H., 17 Price, W. C., Photoelectron spectroscopy,9

R Raimond, J. M., see Haroche, S., 20 Raith, Wilhelm, Time-of-flight scattering spectroscopy, 12 Reid, George C., Ion chemistry in the D region, 12 Reid, R. H. G., see Bates, D. R., 4 Richard, Patrick, see Lin, C. D., 17 Richards, D. see Percival, I. C., 11 see Dickinson, A. S., 18 Robb, W. D., see Burke, P. G., 11 Rosenberg, Leonard, Theory of electronatom scattering in a radiation field, 18 Rotenberg, Manuel, Theory and application of Sturmain functions, 6 Rudge, M. R. H., The calculation of electron-atom excitation cross sections, 9

S Samson, James A. R., The measurement of the photoionization cross sections of the atomic gases, 2 Schermann, J. P., see Audion, C., 7 Schmeltekopf, A. L., seeFerguson, E. E., 5 Schnepp, 0.. The spectra of molecular solids, 5 Schoen, R. I., seecairns, R. B., 8 Schuurmans, M. F. H., Supertluorescence,17 Seaton, M. J. Atomic collision processes in gaseous nebulae, 4 Electron impact excitation of positive ions, 11 Sellin, Ivan A,, Highly ionized ions, 12

Smith, Francis J., see Mason, E. A., 2 Sobel’man, I. I., On the problem of extreme UV and X-ray lasers, 20 Sommerville, W. B., Microwave transitions of interstellar atoms and molecules, 13 Stebbings, R. F. Some new experimental methods in collision physics, 4 Collisions of highly excited atoms, 15 Stein, Talbert S. Positron -gas scattering experiments, 18 Stewart, A. L., The quanta1 calculation of photoionization cross sections, 3 Stewart, Ronald F., see Webster, Brian C., 14 Steudel, A., see Lange, W., 9 Stickney, Robert E., Atomic and molecular scattering from solid surfaces, 3 Stoll, W., see Hertel, I. V., 13 Swain, S., Theory of atomic processes in strong resonant electromagnetic fields, 16

T Takayanagi, K. The production of rotational and vibrational transitions in encounters between molecules, 1 The rotational excitation of molecules by slow electrons, 6 Taylor, H. S., see Csanak, Gy., 7 Thompson, D. G., The vibrational excitation of molecules by electron impact, 19 Thrush, B. A., see Folde, M. F., 11 Toennies, J. P. see Pauly, H., 1 see Faubel, Manfred, 13 V

Van Dyck, R. S., Jr., see Wineland, D. J., 19 Veldre, V., see Peterkop, R., 2 Vigue, J., see Broyer, M., 12 Vinogradov, A. V., see Sobel’man, I. I., 20 Vrehen, Q. H. F., see Schuurmans, M. F. H., 17

374

CUMULATIVE AUTHOR INDEX W

Walther, H., seeGallas, J. A. C., 20 Webster, Brian C., The accurate calculation of atomic properties by numerical methods, 14 Weinstein, Harel, Localized molecular orbitals, 7 Weiss, A. W., Correlation in excited states of atoms, 9 Wilets, L., see Fortson, E. N., 16 Wineland, D. J., High-resolution spectroscopy of stored ions, 19 Wise, Henry, Reactive collisions between gas and surface atoms, 3

Wolf, H. C., Energy transfer in organic molecular crystals: a survey of experiments, 3 Wood, Bernard J., see Wise, Henry, 3 Y Yaris, Robert, see Csanak, Gy., 7

1

Zorn, Jens C.. Molecular beam electronic resonance spectroscopy, 9

CUMULATIVE SUBJECT INDEX: VOLUMES I - 20 A

Affinities, electron, of atoms and molecules, 1,61 Atmosphere, Martian, processes in, 9, 323 Atmosphere, terrestrial D region, processes in, 12, 375 Atomic collision processes in controlled thermonuclear fusion research, 15, 293 Atomic collisions, see Collisions Atomic energy levels Hartree-Fock and related, 16, 1 isotope shift, theory of, 12, 39 relativistic Z-dependent corrections to, 5, 337 Atomic transition probabilities calculation of, 5,237; 16,l; 17,370; 18.3 I3 forbidden, in one- and two-electron systems, 14, I8 I measurements, 3, 83; 12, 244 Atomic physics from atmospheric and astrophysical studies, 15, 37 Atomicpolarizabilities, 11,189; 13, 1; 14,102 Atomic properties. accurate computation of, 14,87 Atomic structure, model potentials, 18, 309 Auger effect, 8, 163 Autler-Townes effect. optical, 16, 190

B Basic concepts in quantum mechanics, atomic physics tests, 14, 28 I Beams, accelerator, characteristics of ions produced in, 12,217 Beams, molecular crossed, chemiluminescence, 15, 182 crossed, collision experiments with laser-excited atoms in, 13, 1 I3

electron resonance spectroscopy, 9,243 high intensity and high energy, 1, 345 merging, collision experiments with, 5, 57 modulated mass spectrometry, 2, 4 I7 photoionization measurements, 8, I3 I polarized atomic spectroscopic measurements with, 5, I24 pulsed supersonic nozzle jet and resonance ionization spectroscopy, 17,262 thermal energy for study of intermolecular potentials, 1, 195 Bell’s inequality, 14, 306 Biorthogonal sets and resonance phenomena, 18,207 Born expansions, 1,90; 4, 143 Bremsstrahlung, 18, 6 C

Charge transfer, see collisions Chemiluminesence in crossed beams, 15, 182 in gases, 11, 361 Classical-oscillator analog, collision broadening, 5 2 0 1 Cluster ions, experimental studies, 20, 65 Coefficients of fractional parentage (cfp), tables of, 5, 297 Coherence and correlation in atomic collisions, 15, 423 Coherent emission by excited atoms without dipole moment, 17, 167 Collective effects in photoionization of atoms, 17, 1 Collision broadening of spectral lines, theory, 5, 201 Collision experiments: electron impact coincidence (e, 2 4 measurements, 14, 127 375

376

CUMULATIVE SUBJECT INDEX

differential cross sections, low energy, 9, 207

excitation, coherent and incoherent, 15, 437

excitation functions, 4, 267; 15, 381 interference effects in ionization, 15, 432 with laser excited atoms in crossed beams, 13, 113

positive ion, excitation, 11, 92 resonances, with atoms and molecules, 4, 175; 14, 1; 19,234

spin-dependent phenomenon, in atom inelastic, 19, 187 Stokes parameter analysis, 19,225 threshold behavior, 15, 391 Collision experiments: heavy particles charge rearrangement, 5, 72 charge transfer fast, 2, 364; 5, 62; IS, 303 slow, 4, 237; 5, 62; 15,205 differential cross section, 17, 336 coherence and correlation, 15,423 double K-electron transfer, 17, 343 electron capture, see Charge transfer excitation and capture in fast, 2, 327 fine structure changing transitions, 13,200 impact parameter dependence for K- K charge transfer, 17, 342 inner shell vacancy production, 17,275 intermolecular potentials, determination of, 1, 195 ion-ion in crossed beams, 15, 317 ion-neutral, ion-ion, and neutralneutral reactions in merging beams, 5, 57

ion - neutral reactions in flowing afterglows, 5, 1 with laser-excited atoms in crossed beams, 13, 113

by laser spectroscopy, 13, 57 merging beams, use of, 5,57 methods, 1967, new in, 4,299 multiply charged ions and H( Is), 15, 300 perfect direct excitation experiment, 18, 279

reactive between gas and surface atoms, 3, 29 I reactive between neutrals, 15, 167 rotational and vibrational excitation of molecules, 13,229; 19, 345

+

+

between simple (H+ H, H+ He, He+ H) systems, 19,67 solid surfaces, atomic and molecular scattering from, 3, 143 spin exchange with polarized atomic beam, 5, 124 Collisions, in naturally occurring environments and applications controlled thermonuclear fusion research,

+

15,293

D region of terrestrial atmosphere, 12, 375 gaseous nebulae, 4,33 1 general atmospheric and astrophysical, many types of process, 15,37 Jovian ionosphere, 10,295 interstellar medium, 8, 1 ionosphere, 4,38 1;6,48 Martian atmosphere, 9, 323 recombination in troposphere and lower stratosphere, 20, 33 Sun, 17,355 Collisions, positron and positronium experiment 4,63; 15, 135; 18,53 theory 1, 141; 4.63; 15, 101 Collision theory: charge transfer classical mechanics, treatment by, 6, 297; 16, 312; 19, 50

into continuum, 15, 286 diabatic states, 7, 232 first Born approximation, 1, 104 fully stripped ions, electron capture from H( Is), 19, 1 general surveys, 1, 102; 15,274; 17,55, 3 10

phase integral method, J7,63 psuedostate expansions, 19, 35 radiative capture, 15, 367; 16, 315 relativistic effects, 16, 307 second Born approximation, 1, 11I ; 15, 280

using Sturmian functions, 6, 255 transitions between ionic and covalent states, 12, I translation factor, 15, 274; 16, 307; 17, 83 Collision theory: electron capture; see Collision theory: charge transfer Collision theory: electron impact asymptotic states, 18, 14 Born expansions, 4, 143 Born, first, range of validity, 10, 53

377

CUMULATIVE SUBJECT INDEX classical mechanics excitation and ionization, 2, 306; 8,64; general survey on use of, 4, 109 and Rydberg atoms, 11, 1 close coupling method, 2, 27 1, 287; 4, 196; 6,242 effective range, 2, 3 12; 4,206 fast, 9, 49 general surveys, 2,263; 9,47; 13, 3 I5 Green’s function technique, 7, 32 1 with highly polar molecules, 18, 341 inner shell ionization, 4, 22 1 ; 15, 329 ionization of H(Is), numerical, 20, 241 low-frequency approximations, 18, 37 optical potential, 2, 283 with positive ions, 11, 83; 17, 381 psuedo-state expansions, 9, 1 18 in radiation field, 18, 1 relativistic effects, 4, 22 I ; 16, 28 1 resonances, 2, 285; 4, 173;9,99; 13, 324; 18,207 R-matrix method, 11, 143; 13,338; 15, 492; 20,266 rotational excitation, 6, 105 with Rydberg atoms, 11, 1 second-order approximations, 9, 1 I8 slow, with atoms, 9,93; 13, 315 slow, with molecules, 15, 471 Sturmian functions, use of, 6, 233 threshold behavior, 9, 1 1 1; 13, 330 time-dependent wave-packet method for ionization, 20,24 I vibrational excitation, 15,495; 19, 309 Collision theory: heavy particles; see also Collision theory: charge transfer Born and close coupling approximations, 15,267 classical atomic scattering, 4, 109 classical and semiclassicalmethods, 18, 165 close coupling method: rotational vibrational transitions, 13, 248 elastic atom-atom, 7, 79; 14, 233 excitation of quasi one-electron (e.g., Be+ Ne) type, 18,265 fast, classical treatment, 6, 269 fine structure, proton excitation, 17,403 gas crystal surface, van der Waals scattering, 3, 205 impulse approximation, 1,93 information theoretic approach, 11,2 15

+

inner shell ionization, 4, 221; 11, 299; 15, 335 inner shell vacancy production, 17,275 ionization, 15, 286 Monte Carlo trajectory calculations of atomic and molecular excitation in thermal systems, 8, 39 nuclear reactions and inner shell ionization, 20, I73 perturbed stationary state method, 1, 119; 17, 83; 19, 16 quasi-molecular interference effects in ion-atom, 14, 341 role ofenergy in reactive molecular, 11,2 15 rotational excitation by atoms, 18, 170 rotational transitions in encounters between molecules, 1, 149; 9, 127 rotational and vibrational excitation by ions and neutrals, 13,229 between Rydberg atoms and charged particles, 11, 1 between Rydberg atoms and neutrals, 14, 385; 15,77 semiclassical effects, 14, 225 solid surfaces, scattering from, 3, 187 in strong resonance electromagnetic fields, 16, 159 vibrational transitions in encounters between molecules, 1, 1 19 Complex scaling, use of in resonance phenomena, 18,208 Correlation in atoms, 9, I ; 14, 92 Correspondence principle, and optical properties of excited atoms, I I, 18 Coulomb amplitudes, nonrelativistic off-shell, 8, 7 1 Crystals gas-surface van der Waals scattering, 3, 205 organic molecular, energy transfer in, 3, 119

D Diabatic states of molecules, 7, 223 Diagnostics, spectroscopic,of laser-produced plasmas, 16,25 1 Diffusion of atoms and molecules, 6, 155 Diffusion, thermal, 2, 33

378


Dissociative attachment, 7, 24 I ; 15, I3 D region, ion chemistry of, 12, 375

E Effective range theory in scattering, 2, 3 12; 4,206 Electrical discharge and breakdown phenomenon, 15,28 Electron capture, see Collisions Electron cooling in ionosphere, 4, 390 Electrons fast, slowing down, 4, 38 I Electron-ion recombination, see Recombination Energy transfer in organic molecular crystals, 3, 1 19 Excitation by collision, see Collisions F Faddeev equations, applications of Sturmians to, 6, 245 Field correlation effects in multiphoton processes, 12, 109 Flowing afterglow studies, 5, I Forbidden lines in gaseous nebulae, 4,356 Forbidden transitions in one- and twoelectron atoms, 14, 181 Fractional parentage, coefficients of (cfp), tables, 5, 297 G Gamow-Siegent states, 18,210 Green’s function technique, 7, 287 Groups and selection rules in atomic shells, 7,252

Hyperfine structure atomic, 3, 88 molecular, 9,289 H i electronic eigenenergies, 4, I3

I I,, laser investigation ofexcited states, 12,201 Inner shell ionization, 4,22 I; 11,299; 15,329 influence of nuclear reactions on, 20, I73 Inner shell, radiationless reorganization of, 8, 163 Inner shell thresholds, collective effects near, 17, 32 Inner shell vacancies, collectivization of, 17, 40 Inner shell vacancy, fluorescence, yield, 8, 186 Inner shell vacancy production in ion-atom collisions, 17, 275 Interstellar atoms and molecules, microwave transitions, 13, 383 molecules, formation and destruction, 8, 1 Ion cooling in ionosphere, 4, 394 Ionic reactions, 15.23 Ion -ion recombination, see Recombination Ionization by collision, see Collisions Ionization in space, direct study of, 4, 4 I 1 Ionosphere, collisions in, 4, 381; 12, 375 Ions, cluster, experimental studies, 20,65 Ions, highly ionized classification of spectra, 10, 223 general studies on, 12,2 I5 Ions, storage of, 3, 53; 5, 109; 19, 135 Isotope shift, theory of atomic, 12, 39 J

Jovian ionospheric chemistry, 10,295 H L Hanle effect, 12, 169 Hartree- Fock equations, time dependent, numerical solution, 14, 1 I 1 Hartree-Fock theory for atoms, 16, 1 Hund’s rule and spin-coupled wave functions, 7, I80 Hydrogen maser, physics of, 7, I Hydrogen solid, spectra of, 5, 187

Lamb shift, 18, 399 Laser-excited atoms, theory of measurements in scattering experiments by, 13, 157 Laser field atomic processes in, 13,21 I ; 16, 159 electron-atom scattering in, 18, 1

379

CUMULATIVE SUBJECT INDEX Laser-produced plasmas, spectroscopy of, 16,201 Lasers,dye in atomic spectroscopy, 10, 173 properties, 10, 174 tunable, 10, 197 Lasers,extreme UV and X-ray, 20,327 Lepton spectroscopy, 19, 149 Level-crossingexperiments, 3, 83 Line broadening in laser-produced plasmas, 16,225 Liquid metals, pseudo-potentials, application to, 7, 363 Localized molecular orbitals, 7,97 Long-range forces between atoms, 2, 1

M Magnetic depolanzation, 12, 169 Maser, hydrogen, physics of, 7, 1 Massey, tributes to, 4, 1; 15, xv Mass spectroscopy of free radicals, 2, 385 high precision, 19, 135 Metals, liquid, application of pseudopotentials to, 7, 363 Microwave transitions of interstellar atoms and molecules, 13, 383; 15, 56 Mobilities, electron and ion, in gases and low-density liquids, 20, 267 Model potentials in atomic structure, 18,309 Molecular dissociation, 8, 52 Molecular orbital theory of spin properties of conjugated molecules, 1, 1 Molecular parameters, measurement by optical pumping, 12, 165 Molecular polarizabilities, 13, I Molecules, atomic charges within, 20.4 1 Molecules, diatomic, reduced potential curve, 19,265 Monte Carlo trajectory calculations, 4, 128; 8, 39; 20, 14 Multiphoton ionization of atoms antiresonances, 18, 140 nonresonant, 18,97 Multiphoton processes in atoms, 12, 87 Mutual neutralization in ambient gas, 20,21 Mutual neutralization, merged beam measurement, 5, 83

N Nebulae, gaseous, atomic collision processes in, 4, 33 1 Negative ions detachment energies, 1, 61 in D region of atmosphere, 12, 399 general survey, 15, 1 spectroscopy, 19, I76 Neutral current interaction and parity nonconservation, 16, 32 1 Non-adiabatic transitions between ionic and covalent states, 12, I Nonresonant multiphoton ionization of atoms, 18,98 Nuclear reactions and atomic inner-shell ionization, 20, I73 Nuclear spin-induced decay, 14,2 1 1 Null matrix elements and selection rules in atomic shells, 7, 25 1 0

Off-shelltwo-body Coulomb amplitudes, 8, 71 Operators, effective, for relativistic effects in many-electron atoms, 10, 2 I Optical double resonance, 3, 74 Optical pumping of molecules, 12, 165 methods in atomic spectroscopy, 3,73 Optical rotation experiments and panty nonconservation, 16, 338 Oscillator strength and branching ratios, from astrophysicalmeasurements, 15,59

P Panty nonconservation in atoms, 16, 3 19 Perturbation methods, Green’s function technique, 7, 339 Perturbation theories and threshold laws, 4, 126 Perturbation theory many body, 17,4 of multiphoton processes, 12, 89 Phase interference phenomena in collisions, 18,341 Photodetachment and photodissociation, 10, 194; 15, 18

380


Photoionization in atomic gases, measurements, 2, I77 of atoms, quantal calculations, 3, 1 collective effects, 17, 1 using dye lasers, 10, 194 and electron impact ionization, I I, 184 inner shell, 4,233 molecular beam measurements, 8, I3 I photoelectron spectroscopy, 10, I3 I spin polarization of atomic and molecular photoelectrons, 19, 395 Plasmas, analysis of velocity field in using Doppler effect, 6, 59 Plasmas, laser-produced, spectroscopy of, 16,201 Plasmas, multicharged, inversion schemes for, 20, 333 Polarization and angular momentum effects in multiphoton processes in atoms, 12, I33 Polarizabilities, atomic and molecular, 11, 189; 13, 1; 14, 102 Polarized electrons, sources of, 16, 101 Positronium, formation and scattering of, 1, 141; 15, 135; 18,53 Positrons and positronium in gases, 4 6 3 Potentials, intermolecular model calculations and experimental determination, 16, 55 molecular beam studies, 1, 195 quantal calculation of, 13, 229 for simple ion-atom systems, 19, 67 Probe, electron and ion, systems, 4 4 17 Pseudo-potentials,applications to liquid metals, 7, 363

Q Quantum-beat experiments with dye lasers, 10, 185 Quantum defect theory, 4, 370; 11, 100; 18, I04 Quantum electrodynamic effects in few-electron atoms, 18,399 radiative interaction of Rydberg atoms, 20,440 Quantum mechanics, basic concepts, atomic physics tests of, 14, 28 I

Quasi-molecular interference effects in ion-atom collisions, 14, 341 Quasi-stationary electronic states of molecules, 7, 223

R R-matrix theory of atomic processes, 11, 143; 13, 338; 15,492; 20, 266 Radiationless reorganization of atomic inner shells, 8, I63 Radicals, free, mass spectrometry of, 2, 385 Random phase approximation with exchange, 17,4 Recombination, aspects of electron -ion and ion-ion, 15,235 Recombination, dissociative experiments with merging beams, 5, 101 general survey, 6, 1 in ionosphere, 6,48; 15, 38 quasi-stationary state representation, 7, 236 Recombination ion-ion in ambient gas, 20, 1 ions, complex, 15, 238; 20, 139 Recombination, radiative in gaseous nebulae, 4, 332 in nightglow, IS, 235 Reduced potential curve method for diatomic molecules, 19,265 Relativistic addition to Schradinger equation for molecular wave functions, 1, 3 effects in atomic collision theory, 16, 28 I effects in many-electron atoms, 10, I inner shell ionization, 4, 221 magnetic dipole transitions, 15, 67 model potentials, 18, 332 Z-dependent corrections to atomic energy levels, 5, 337 Resonance, double, 3,83 Resonance fluorescence, 16, I7 I Resonance ionization spectroscopy in atomic and molecular physics, 17, 229 Resonance multiphoton processes, 12, 1 14 Resonance phenomena, use of complex scaling in study of, 18,207 Resonance in photoionization continuum, 3, 16

38 I

CUMULATIVE SUBJECT INDEX Resonances in electron scattering by atoms and molecules, 4, 173; 9, 99; 14, 1; 18, 207 Resonant electromagnetic fields, theory of atomic processes in strong, 16, 159 Rotational transitions, in encounters with ions and neutrals, 13,229; 18, 170 molecules, 1, 149; 9, 127 slow electrons, 6, 105 Rydberg atoms collisions with charged particles, 11, 1 collisions with neutrals, 14, 385; 15, 77 high-resolution spectroscopy and radiative interaction, 20,4 I3 preparation and detection, 17, 101 quantum electrodynamic effects, 20,440 radiative properties in free space and resonant cavities, 20, 347 spectroscopy and field ionization, 14, 368; 17, 119 Rydberg molecules, 20, 452

S Scattering, see Collisions Selection rules within atomic shells, 7, 25 I Semiclassicaleffects in heavy-particle collisions, 14, 225 Spectra, see Vibrational spectra atomic in vacuum UV, 2, 121 Auger, 8,208 highly ionized atoms, classification of, 10, 223 molecular solids, 5, I55 molecular in vacuum UV, 2, 141 recombination, in gaseous nebulae, 4, 332 solar, 10, 262 solid hydrogen, 5, I87 Spectral line broadening collision, 5, 201 Stark, 11, 331 Spectroscopy dye lasers in, 10, 173 high resolution of stored ions, 19, 135 hyperfine, 7,29 of laser-produced plasmas, 16, 201 laser, study of collisions by, 13, 57 molecular beam, electron resonance, 9,243 neutron, 12,297

optical pumping methods, $ 7 3 photoelectron, 10, 131; 12, 317 positron, 12,320 radiofrequency of stored ions, 5, 109 resonance ionization in atomic and molecular physics, 17,229 time-of-flight scattering, 12, 28 1 transmission, 12,314 UV and X-ray in astrophysics, 14, 393 in vacuum UV, 2,93 Spin-coupled wave functions of atoms and molecules, 7, 141 Spin-dependent phenomena in inelastic electron-atom collisions, 19, I87 Spin exchange and polarized atomic beam, 5, 124 Spin polarization of photoelectrons, 19, 395 Spin properties of conjugated molecules, 1, 1 Spontaneous radiative dissociation, 15,62 Stark broadening, 11,33 1 Stark interference experiments, 16, 357 Strong resonant electromagnetic fields, theory of atomic processes in, 16, 159 Sturmian functions, theory and application of, 6,233; 19, 89 Sun, atomic processes in, 17, 355 Superfluorescence, 17, 167 Surfaces atomic and molecular Scattering from, 3, 143 gas-crystal van der Waals scattering, quantum theory, 3,205 reactive collisions on, 3,291

T Thermal diffusion in gases, 2, 33 Thermonuclear fusion research, atomic collision processes in controlled, 15,293 Time-dependent Hartree -Fock method, 14, 109 Transition probabilities, atomic calculation of, 5, 237; 16, I; 18, 313 Transition probabilities, measurements, 3, 83; 12,244 Transitions, see rotational and vibrational forbidden in one- and two-electron atoms, 14, 181

382

CUMULATIVE SUBJECT lNDEX

microwave, of interstellar atoms and molecules, 13, 383 nonadiabatic between ionic and covalent states, 12, 1 Troposphere and lower stratosphere, recombination in, 20, 33 Two-photon decay, 14, 199

Vibrational spectra intramolecular, 5, 176 lattice, 5, I55 Viscosity of dilute gases, quantum theory, 4, 37

V

Wave functions atomic charges within molecules as partial alternative, 20,4 1 atomic, Hartree-Fock theory, 16, 1 of conjugated molecules, 1, 1 correlation in excited states of atoms, 9,1 frozen core approximations for atoms, 16,

Van der Waals gas-crystal surface scattering, quantum theory, 3,205 interactions, calculation of, 2, 1 Vibrational transitions in encounters with electrons, 15, 495; 19, 309 ions and neutrals, 13, 229 molecules, I, 149; 9, 127

W

16

localized molecular, 7, 97 variational, 5, 257

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