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. 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

@

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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

This Page Intentionally Left Blank

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

Dennis P. O’Brien et al.

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

Dennis P. O’Brien et al.

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

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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.

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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

SUBNATURAL LINEWIDTHS

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

+

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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-

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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

SUBNATURAL LINEWIDTHS

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.)

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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.

SUBNATURAL LINEWIDTHS

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.)

SUBNATURAL LINEWIDTHS

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

Dennis P. O’Brien et al.

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

+

SUBNATURAL LINEWIDTHS

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 << Iw2 - O ~ I - ~ , the inverse frequency separation of the two

28

Dennis P. O'Brien et al.

-:7

1(t)

,t

FIG.5. Pulsed excitation of a coherent superposition of excited states 2 and 1. The fluorescence beats at the difference frequency w 2 , .

excited states 11) and )2), creating a coherent superposition evolving in time with the wave function

m1)

io,t - 71 + C20) exP(- i o 2 t - Y2'2/2)12) (68) where o,,o2are the frequencies and y, ,y2 the spontaneous decay rates of the excited states and C,, C,their probability amplitudes. The wave function evolves freely followingthe excitation, and the fluorescent decay to a particular final state 13) is proportional to the modified Golden Rule transition rate I N I( 31 Vlw(t))12,where Vis thedipole interaction Hamiltonian responsible for the coupling to level 13) given in Section II1,A. If y, = y2 = y and V13 = V23 = Vand an equal initial superposition ofll) and 12) is generated so that Cl(0) = 2-'12 = C2(0),the fluorescent intensity Z ( t ) beats at 021 = o2- o, , that is,

Iw(0)

= C,(O exP(-

S(t) = I VI2(1 - cos w2,t)e-Yf

(69)

The time-integrated signal is

and a double-peaked spectrum of width y is obtained symmetrical about w2,= 0 by varying the magnetic field and thus w21. But if one observes selectively the fluorescence following pulsed excitation at t = 0 from time T onward, while fluorescence from t = 0 to time Tis discarded, then only those atoms having survived until Tin the coherent superposition [Eq. (69)] are

29

SUBNATURAL LINEWIDTHS

selected to obtain a narrower signal. They produce the time-biased signal

rm y cos o z l T -

02, sin

02,T

Y2+4 1 For T = 0, we obviously recover the result [Eq. (70)], but for long delay times T = n/y with n large (so that the Lorentzian is slowly varying compared with the oscillating terms), the separation between central maxima becomes Am2, = (2n/n)y.The effective “halwdth” between central maxima of the time-delayed level-crossing signal is therefore about (n/n)natural widths. The oscillatory ringing in Fig. 6 is due to the sharp turn-on of the time integration after delay. If the ringing introduces unwanted complications, it

t

lntensit y

- 4 2 -28-14 0 14 28 42 Magnetic Field FIG. 6. Time-integrated fluorescent intensity for increasing delays T = A1 as a function of wzI(i.e., of the Zeeman magnetic field) observed experimentally by Schenck el al., in pulsed excitation of barium ‘So-lP, transitions. Here = 7 G is the level-crossing half-width magnetic field for zero delay; lifetime = 8.2 nwc. (From Schenk et al., 1973.)

30

Dennis P. O’Brien et al.

can be “apodized” by using a time-biased filter function which rejects the early decay light but turns on and off smoothly (Deech et al., 1974; Figger and Walther, 1974; Series, 1976). C. RAMSEY INTERFERENCE METHOD 1. Radio-Frequency Fields

The method of separated oscillatory fields devised by Ramsey (1949, 1950, 1956, 1980)is shown schematically in Fig. 7.A radio-frequency field is applied over two short regions of length I separated by a relatively long region of length L in which no rf field is present. The principal features of the line shape are indicated for a transition between two stable energy levels for a monoenergetic atomic beam. The overall width corresponds to the transit time 7 through the short regions 1. The narrower interference peaks correspond to the transit time T through the long region L. Because the atom is excited by the two pulses separated by the time interval T, the possibility arises that it could have been excited either by the first pulse V, in region a, or by the second pulse Vb in region b. The total transition probability thus exhibits interference effects between these two regions. To describe this scheme we therefore consider the excitation of a two-level atom with upper and lower levels 12) and 11) by two pulses of duration z and central frequency o1with a detuning A from the atomic The wave function at time t is a superposition ofthe resonance frequency oo. states 11) and 12) given by

IwW)

=

c1w exP(-iw)ll) + Cdt) exp(-io,t)P)

(72)

where we neglect decay for the moment. The equation of motion for the

A

ATOMIC BEAM

-

-

FREQUENCY

FIG.7. Method of separated oscillatory fields.

31

SUBNATURAL LINEWIDTHS

excited-state probability amplitude is C2(t) = - iv2,(t)exp(io,t)

c,(t)

(73)

and similarly for C,(t), where V2,is again the interaction matrix element . thesystem istaken to between levelsll) and12): V,, = ( 2 J e r * E ( l )Initially, be in the ground state: C,(O)= 1, C,(O) = 0. If the perturbation produced by the fields is sufficiently weak we may solve these equations perturbatively by putting C,(t) = C,(O) = 1. The field being spatially confined, the interaction Hamiltonian may be expressed as V = d*E, cos(o,t - (6)F(t), E, being the envelope of the electric field, (6 its relative phase, and F(t) a step function describing its sudden turn on and off. In the rotating-wave approximation V21= - +(21d-EO11)exp(-io,t

+ (6)F(t)

(74)

+

The solution to Eq. (72) for times t > T 22 following the two pulses is ; rr C2(t)= - J V2, exp(i At dt

+

0

-

f

V,, exp(i At

+ i&) dt

(75)

and we have taken the field strength to be the same in both interaction zones. The excitation probability P2(t)= IC2(t)I2of the upper level is then (Knight, 1981) 1 P2(t)= - I V12sin2(Ar/2) 4

A ( T + T)

1

+ d(6

(76)

where the first factor is the usual excitation probability produced by a single pulse of duration T. The second term in Eq. (76) describes the interference between the two excitation amplitudes, d(6 = (6, - & being the phase difference between the two pulses. The inclusion of spontaneous decay in the description of Ramsey fringes can readily be achieved if both levels decay at the same rate y. It is then sufficient to allow the transition frequency to become complex, oo+ w, - iy/2, and the excitation probability becomesP,(t) = IC2(t)12e-Yr. The dependence of the interference term on the phase can be used to isolate the interference term. By measuring the difference in probability for an atom to be in the excited state when the phase &$ is switched from 8(6 = 0 to d(6 = II, one obtains the “interference signal” 1 sin2(A/2) AP=-1VI2 4 cos A(T+ T) 4

4

(77)

32

Dennis P. O’Brien et al.

The phase-isolation technique is particularly important if the atom can decay between pulses, since AP is generated by the interference cross term only, in the cross terms of the modulus squared of Eq. ( 7 3 , and has enormously enhanced fringe contrast compared with Pz . If phase isolation is not used and the delay T > z, the contribution due to the first pulse in Eq. (76) will dominate both that of the other pulse and the cross term, so that fringes will appear as a small riple on a broad single pulse line shape. Note also that great care must be taken to ensure that the two pulses are phase locked. In practice each pulse in a pair in optical Ramsey excitation is derived from a common pulse using beam-splitting techniques. In Fig. 8 the observed Ramsey line shape using phase subtraction is compared with that observed with phase addition (where the fringes are eliminated in favor of the normal broadened line shape). Hughes (1960) suggested that the method of separated oscillatory fields could be used to obtain line widths narrower than the natural width. His specific proposal was that, using a beam of fast hydrogen atoms with a velocity of about lo8cm/sec, the atoms will travel an appreciable distance in sec so that separated oscillatory fields could be realized physito cally. This method was used to measure the Lamb shift in the n = 3 state (Fabjan and Pipkin, 1972; Kramer et al., 1974; Clark et al., 1977) and in the n = 2 state (Lundeen and Pipkin, 1975) of atomic hydrogen. Decay senously reduces the fringe contrast, since the number of atoms which survive to interact in the second cavity and so produce interference is small. For the case of the Lamb 2s + 2p transition, where y = 100 MHz, substantial line narrowing by a factor of 4 of the natural linewidth has been achieved (Lun-

Fie. 8. Typical separated oscillatory field resonance line shapes. (From Lundeen and Pipkin, 1975.)

SUBNATURAL LINEWIDTHS

33

deen and Pipkin, 1975),with a concomitant improvement by a factor of 3 in the precision of the experimental value of the Lamb shift. 2. Optical Fields The extension of the Ramsey interference technique to sub-Dopplerspectroscopy in the optical region, proposed by Baklanov et al. (1976a,b), was originally aimed at reducing the effects of transit-time broadening. Rather than increasing this time by expanding the laser beam diameter (with a consequent loss of intensity), they proposed using two spatially separated beams to obtain structures, in the profile of the resonance, having a width determined as in the case of radio-frequency fields by the time of flight between the two beams. This technique was demonstrated experimentally by Bergquist et al. ( 1977a,b)in the case of the saturation resonances of CH4 and Ne, and by Barger et al. ( 1979)in the case of calcium. Figure 9 illustrates the extension of Ramsey’s idea to Doppler-free two-photon spectroscopy. This time there are at least two interactions in each field region, one with each of the two counterpropagating fields. The total phase factor that a p pears is exp[i(+; 4: - 47 - 491, where each 4 comes from one of the four fields. In saturation spectroscopy a third field region is also used. The reason for this lies in the fact that saturation spectroscopy is a two-step process: first the selection process of a velocity group, second the probing process of this selected group. We must therefore repeat the Ramsey operation twice: once to create a system of fringes on the population holes or peaks and a second time to probe this population change. In a similar approach Salour (1976, 1977, 1978a-c) and Salour and Cohen-Tannoudji ( 1977) obtained interference fringes in the profile of the Doppler-free two-photon resonances by exciting atoms with two time-delayed coherent laser pulses instead of using two spatially separated cw light

+

Atomic Beam

FIG.9. Ramsey’smethod for Doppler-free two-photon spectroscopy.

34

12);p Dennis P. O’Brien et al.

1

T

TIT

Ir>

Ir> 1’)

T+T

lg)

19)

11)

FIG.10. Two quantum-mechanical amplitudes whose sum represents the total amplitude. (From Salour, 1978).

standing waves. Salour (1976) dealt with short-lived atomic states (lifetime = 5 X lo-* sec), so that the transit time through the beam (lo-’ sec) plays no role in the problem. Here the major source of broadening in the observed line shape is due to the spectral linewidth of the laser. The splitting is 1/2T, with T the time delay between the two pulses, and is much smaller than the spectral width l/t of the laser pulse, where, for a Fourier-limited pulse, T is the duration of each pulse. This technique combines the advantages of pulsed dye lasers (power, spectral range) with the high resolution usually associated with a cw dye laser. As before the interference structure can be understood simply if we consider the two-level system of interest. After the first pulse the atomic state is a coherent superposition of the two levels. Atoms in this superposition freely precess and, depending on the point in time at which the delayed pulse arrives, one can see either constructive or destructive interference with the atomic precession. Another interpretation of this interference can be shown as in Fig. 10. This figure represents the sum of two quantum-mechanical amplitudes. The probability of reaching the 12) state is the absolute square ofthis sum, and the cross term appearing in this absolute square represents the interference signal. Figure 1 1a shows the four well-known two-photon resonances of the 3 2S4 2Dtransition of Na, observed in a reference cell with a single pulse. Figure 1 1b shows the same four resonances observed in the sample cell when excited by the two time-delayed coherent pulses with 2T = 17 nsec. An interference structure clearly shows on each resonance. Salour ( 1978c) has verified that the splitting between the fringes is inversely proportional to the effective delay time 2Tbetween the two coherent pulses. Figures 1 lc and d show the same experimental traces except for effective delays of 2T = 25 and 33 nsec, respectively. Note that in these cases (Fig. 1 lc and d) the contrast of the fringes is less than that of Fig. 1 1b, where a shorter delay is used. Obviously if T is large enough one may use this technique to obtain subnatural spectra, given that the signal can still be detected and isolated from the background signal.

SUBNATURAL LINEWIDTHS

I . I

35

I

0

2

L A S E R FREQUENCY ( G H z l

FIG.I 1. The four two-photon resonances of the 32S-42D transition of Na. (FromSalour and Cohen-Tannoudji, 1977.)

Two important requirements must be fulfilled in order to obtain such interference fringes in the profile of the two-photon resonance. First, each pulse must be reflected against a mirror placed near the atomic cell in order to expose the atoms to a pulsed standing wave and in this way to suppress any dephasing due to the motion of the atoms. The probability amplitude for absorbing two counterpropagating photons is proportional to exp[i(wt - kz)]exp[i(wt

+ kz)] = exp(2iwt)

and does not depend on the spatial position z ofthe atoms. Second, the phase difference between the two pulses must be constant during the entire expenment. When the delay is unlocked the interference fringes disappear as w is varied. Phase isolation can again be used to eliminate the diffraction background and thus isolate the interference fringes.

D. TRANSIENT LINENARROWING 1 . General A common feature of the experiments we have discussed so far is to discard the part of the radiation emitted shortly after the preparation of the system and to collect only the delayed and exponentially weakened signal. Meystre et al. (1980) have proposed and analyzed several experiments involving time-delayed measurements in a system of two-level atoms driven by a monochromatic, tunable field. The system is prepared at time t = 0 in,

Dennis P.O'Brien et al.

36

say, the lower excited state 11) and coherently driven by the cw laser of frequency w. Only the radiation emitted after a finite time delay is then detected. The narrow linewidth obtained in this delayed detection scheme is due to the fact that, in the transient regime, the probability for induced transitions is not weighted by a Lorentzian of width y12 = (yl y2)/2 but rather S,, = (yl - y2)/2. Although a general analysis may be camed out fully quantum mechanically in both the small-signal and the strong-signal regimes (Lee et al., 1981; Guzman et al., 1983), for the sake of simplicity we describe here only the semiclassical approach, whose results coincide with the quantum-mechanical ones when spontaneous decay directly between the two unstable states can be neglected. Consider an ensemble of two-level atoms with upper level 12) and lower level 11) decaying at rates yl and y2, respectively, which is prepared at time t = 0 into the lower levelll) and coherently driven by a weak cw field from 11) to the upper level 12) as in Fig. 12. The density matrix equations are (Sargent et al., 1974)

+

P21 =

- W o + Y21)P21 - W(P1, - P22)

where E = Eo cos wt is the driving laser field, wo the energy difference between the two atomic levels, and p the dipole moment of the transition. In the absence of collisional broadening y21 = (y2 y1)/2. For weak fields, and making the rotating-wave approximation, the first-order solution to system

+

E o c o s Vt

*'-\

-9 FIG. 12. Level diagram indicating the excitation from ground state lg) to state [I), the monochromatic incident field driving the 12) to 11) transition, and the decays of levels 12) and 11) to some distant levels 13) and 14).

SUBNATURAL LINEWIDTHS

37

(78) is

t ) for the atom to be in the upper where A = wo- w. The probability level at time t as a function of the detuning is then p22(A, ‘)

= A2 (pEo)2 + 88,

[exp(-y2t)

+ exp(-yl t) - 2 cos(At) exp(-y2, I ) ]

(80)

This expression contains in the denominator S2,,rather than y21as one might have expected. This result was first obtained by Breit (1933), in the context of his theory of optical dispersion. However, its physical implications have been largely ignored, since in general experiments are prepared in such a way that the fluorescencelinewidth is not S2, but rather yZl.This is due to the fact that an integration of pZ2over all times is involved, which corresponds to having atoms entering the interaction region at random times, detectors open for all times, etc. However, Meystre ef al. (1980) have shown that by an appropriate choice of excitation and detection, advantage may be taken of the narrower Lorentzian factor appearing in Eq. (80). They consider an experiment to measure the time-delayed spectrum of the transition, i.e., the number of photons emitted spontaneously in the transition 12) +13) from time t = Ton, with T > 0. This time-delayed line shape is given by w

3

T) = Y2

j-; p22a 0 dt

which on substituting for pZ2(w,t) from Eq. (79) gives

For yI = y2, this is the same result as Eq. (68), provided one replaces the atomic frequency w21by the detuning A between the laser and atomic frequencies. This is because, in the case of level crossing, one prepares the system initially in a coherent superposition which then evolves and decays freely, while in the latter case, the atomic system is coherently driven by the field for all times. In the first situation, resonance is then achieved for exact level crossing (w21= 0), while it occurs here for o = w2, in the case of an optical transition. With this exception, there is no difference between the

38

Dennis P, O'Brien et al. 1

-

-1

a

I (I

0

I

1

1

T-m

FIG.13. Full width at half-maximum ofS(A, T ) ,asa function ofthe delay T, in unitsofy;l, for various values of S,, = (7, - ~ $ 2 .(From Meystre et a/.. 1980.)

level-crossingtechnique and time-delayed detection of an optical transition. Thus the technique discussed here (incoherent preparation, coherent driving, delayed incoherent detection) is intermediate between the level-crossing (coherent preparation, free evolution, and delayed incoherent detection) and Ramsey (incoherent preparation, coherent driving, free evolution, and coherent detection) methods. In the limit T -,0 the familiar Lorentzian line shape of width y2, is recovered, as was also the case before [see Eq. (70)]:

For T = 0, however, the shape of the line is determined by the prefactor in Eq. (82) whenever the exponential in the last term in large parentheses decays rapidly enough to damp the oscillations introduced by the sinusoidal terms. Since it si a, rather than y2, that appears in the prefactor, the line can have a width less than y2, ,but depending on T. In Fig. 13 is plotted the full width at half-maximum of S(A,T) as a function of T, and normalized to the linewidth for no delay (i,e., y 2 , , for y2/yI = 3, 2, and 1.01).We see that for y2 -- yI there is almost no theoretical limit to the narrowing. However, since the signal intensity is exponentially decreasing with T, there is a competition between narrowing of the linewidth and loss of signal intensity.

2. Pulsed Excitation All alternative way to obtain transient line narrowing has been proposed by Knight and Coleman ( 1980). The major advantage of their method is that it is also applicable to ground-state transitions. The basic idea is to drive the transition coherently between a stable state 11) and an unstable state 12) with

39

*--\

SUBNATURAL LINEWIDTHS

PULSE AMPLITUDE

k t

0

3

t

1

-

FIG. 14. Level diagram of a three-level atom, with ground state 11) and excited state 12) decaying to the state 13). The transition 12) 11) is driven by an exponentially decaying laser pulse.

a pulsed radiation field switched on suddenly at t = 0, and decaying exponentially at a rate yL as in Fig. 14. For weak fields and within the semiclassical approximation, the probability of excitation of the state (2) is formally identical to that given in Eq. (80), provided that one replaces y , by yL, the inverse time constant ofthe laser pulse. A further advantage ofthis method is that one can match yL with the decay constant y2. By making yL = y2 in Eq. (80), the probability of exciting the state 12) at time t becomes

In this case most of the absorption at time t is centered within a band whose full width at half-maximum is of the order of 2n/t. Therefore the absorption spectra at times longer than a lifetime have a width narrower than the natural width, and decreasing with time. Thus arbitrarily narrow lines can be produced by an adequate choice of yL and of the delay T.

3. Strong-Signal Regime The strong-signal regime can be analyzed by solving exactly the density matrix Eqs. (78). Again the fully quantum-mechanical results agree with the semiclassical ones provided one neglects spontaneous emission directly from 12) to 11). The general result of this analysis is that under appropriate conditions, the combination of a strong field plus delayed detection can lead to the appearance of a dip at the line center of the time-delayed spectrum. In

Dennis P. O'Brien et al.

40

,?

-8.0

-6.0

-4.0

-2.0

0.0

A-

2.0

4.0

6.0

8.0

10.

FIG. 15. Photocount distribution S(A, T ) ,without the attenuating factor exp(-y,O), for a etal., 1983.) fixed laserintensity,asafunctionofAand T.Timeinunitsofy~'.(FromGuzman

Fig. 15 is plotted the function &'(A, T) for a fixed value of the laser intensity, but omitting the attenuating exponential factor exp(- y2T).The decay constants and Rabi frequency are taken to be, respectively, y2 = 3.0, y , = 1.0, and pEo = 2.0. As expected the spectrum has the well-known power-broadened Lorentzian line shape in the limit T = 0. As T is increased, the linewidth first decreases(transient line narrowing) and thereafter broadens back while a narrow dip appears at line center. This dip becomes deeper and wider as Tis further increased. It is important to realize that the dip appears with a smaller delay, the stronger the field, and that it can have an arbitrary small width (less than d2,). 4 . Phase Switching A generalization of this technique was developed by Shimizu et al. (198 1, 1983), who proposed a method to obtain a coherent transient response in laser-induced fluorescence spectroscopy, while injecting the atoms in the interaction region at random times. This is achieved by suddenly switching the phase of the laser field. In this way a subnatural linewidth can be obtained

SUBNATURAL LINEWIDTHS

41

with the linewidth being determined by the time interval between the switching and observation. Consider again a system of two-level atoms interacting with a near-resonant monochromatic optical field. The atoms enter into the interaction region at random times tl , The equation of motion for the density matrix at time t is still given by Eq. (78), but the upper level population at time t is now given by integrating pZ2over all injection times

The optical field is switched at t = 0 so that

+

t <0

(86a)

+ c.c.,

t>O

(86b)

E = Eoe-i'#'e-iwf c.c., = Eoe-imf

The solution to Eq. (78) in the weak-field limit for such a field gives

)((e-i'#' - l)(e(iA-?izN- e-W) + C.C.

(87)

where A = w - wo. The first term is the steady-state response to the field E,, after the switching. The second term is a transient term with the former half decaying with the decay rate of the off-diagonal element p I 2 .This may be interpreted as the transient response of the coherent signal. Because it contains an oscillatory factor eib the fluoresence intensity observed at time t shows a peak near the line center with a spectral width of 1/2t, which can be narrowed, in principle, to any value, if we wait sufficiently long after switching. The transient response can be isolated by taking the differenceof fluorescence intensities either at two different times or at two different excitation intensities or phases. A resonance of subnatural width has clearly little practical use unless the observed line center coincides with the resonance frequency of the medium. This is not actually the case in Eq. (87), because the coherent term is generally a complex number causing asymmetry in the line shape and a shift in its peak position. However, in most phase-modulation devices the phase of the laser is shifted in an oscillatory manner so that signals for positive and negative phase shifts are obtained in the same operation. A symmetric line shape can be obtained by combining these two signals. In Fig. 16 theoretical spectra obtained from a symmetrized version of Eq. (87) are compared with experimental spectra for the resonance fluorescence spectrum of the Na D , line for various delay times.

42

Dennis P. O’Brien et al.

FIG.16. Experimental spectra and their theoretical fittings ofthe Na D,fluorescence line for (b) y , 2 f = 1.43, ( c ) yL21= 2.86, and (d) yI2f = 4.3. (a) is the natural-width-limited spectrum. (From Shimizu ef a/., 1983.)

E. OTHER COHERENTEFFECTS

I. Polarization Spectroscopy This method of spectroscopy is used when investigating the forward scattering of monochromatic light through a gas. In this case the frequency of the forward-scattered light is not Doppler-shifted with respect to the incident light, and the forward scattered optical paths for scattering from atoms in random positions are all equal (Gawlik and Series, 1979). This implies the possibility that the forward-scattered light from different atoms is mutually coherent and is coherent with the probe beam. A counterpropagating circularly polarized pump beam is used to perturb the atoms so that the scattered light has a different polarization to the probe beam, in which case the forward-scattered light can be separated out. When the frequency of the probe beam is the same as the pumping beam we have a velocity-selective interaction leading to narrow resonances. Gawlik et al. (1982) applied this method to obtain subnatural spectra without a reduction in signal-to-noise ratio so inherent to the other techniques of obtaining subnatural spectra. However, in their experiment, in contrast to others involving polarization spectroscopy, the linearly polarized probe beam perturbs the investigated medium as strongly as the circularly

SUBNATURAL LINEWIDTHS

43

laser frequency

FIG.17. An example of the narrowest subnatural dips obtained, with a width of 2.6 MHz. The crossing angles of the polanzers was f3= 12 mrad; Ip and I, were smaller than 10 mW/cm2. (From Gawlick ef al., 1982.)

+

polarized pump beam. The main feature observed in these experiments, which were on Na, is an additional dip in the standard Doppler-free polarization signal, whose depth, width, and position depend on the intensities ofthe light beams, on the angle between polarizer and analyzer, and on the pressure in the resonance cell. They observed widths as narrow as 2.6 MHz (see Fig. 17), i.e., considerably narrower than the natural width of 10 MHz. Subsequently Gawlik et al. ( 1983) have performed further experiments in order to clarify the mechanisms responsible for the subnatural widths. Two interpretations have been proposed: (1) The resonances might be due to Zeeman coherences induced by the two laser beams which are revealed in the signal by velocity-selective light shifts while scanning the laser frequency (Gawlik et al., 1982).( 2 )Zeeman or optical hyperfine pumping could also be responsible for the observed effects. In particular, nonstationary effects of velocity-selective optical pumping were shown to result in similar line shapes of Doppler-free two-photon resonances in Na (Bjorkholm et al., 1982). In fact the experimental tests and theoretical calculations support the idea that the narrow structures in typical polarization signals are due to nonstationary effects of optical pumping. Whether this still somewhat mysterious narrow resonance can be used as a spectroscopic tool remains to be seen. 2, Trunsient Coherent Rarnun Spectroscopy

With laser excitation sources and modern electronic detection techniques, the spectral resolution of conventional Raman spectra is determined by the linewidth of the vibrational states. In liquids linewidths are usually of the order of 10 cm-I so that the resolution and precision of band positions is several wave numbers. Recently a transient coherent Raman technique was introduced based on a short excitation and a long coherent probe of molecular transitions (Zinth, 1980; Zinth et al., 1982, 1983). This experimental

Dermis P. O'Brien et al.

44

method gives an improvement of spectral resolution beyond the usual limitations of spontaneous Raman spectroscopy. The technique is as follows: During the transient Raman excitation process molecules are driven at the excitation frequency we= w, - w2by two pulses of frequencies o1and w 2 . A Raman transition w, close to o,is therefore coherently excited. At the end of the exciting process the coherent excitation oscillates at the frequency or and decays with the dephasing time T,. A third, delayed probe pulse of frequency ojinteracts with the coherently vibrating molecules and gener-

Raman Shift (crn-l) 2850 2900 2950

b

u

Nl aJ N

2880

2900

2920

Roman Shift (crn-') FIG. 18. Experimental results of transient coherent Raman spectroscopy of C6H,2. (a) Frequency ranger of the various generator liquids used in the experiment. (b)Polarized spontaneous Raman spectrum ofC6HI2recorded with a resolution of 1 cm-I. The frequency positions of the resonances found in the transient spectra are marked with vertical lines. (c) Three transient spectra taken with different generator liquids. New Raman lines are detected and the spectral resolution is improved. [Note the frequency scale in (c) is 3.7 times larger than that in (b).] (From Zinth et al., 1982.)

SUBNATURAL LINEWIDTHS

45

ates a Stokes spectrum of the freely relaxing excitation. The crucial point of this transient excited Raman spectroscopy is the narrow Stokes spectrum produced by the long third pulse. Only molecules vibrating in phase contribute to the coherent Stokes light. Molecules which have suffered collisionsare incoherent and are not observed subsequently. However, molecules which vibrate freely for several dephasing times T2 interact with the delayed third pulse and give a sharp Stokes spectrum. The spectral width of the observed Stokes bands is determined by the spectral width Amp of the third pulse. In their experiments Zinth et al. (1982) worked with Gaussian pulses with duration T~ = 8 psec, which gives Amp = 2 cm-l. In the usual spontaneous Raman spectrum for molecules with homogeneously broadened Raman bands, the spontaneous linewidth Am (FWHM) is determined by the dephasing time T2(Am = 1/nT2= o.32/T2),while for the method used here we have Amp = 2 In 2/nrP.Choosing 7p 2 1 .4T2achieves a higher resolution. The experimental results using this technique are given in Fig. 18 for the case of cyclohexane in the small frequency range 2850 and 2940 cm-l. V. CONCLUSION We have reviewed a number of techniquesdeveloped over the last 20 years or so to achieve a spectroscopic resolution beyond the natural linewidth. Most of these rely on the use of some kind of delayed detection scheme, which we have attempted to describe in a unified picture, so as to draw analogiesand differences between them. The two essential characteristics of these methods are the time delay, which produces the subnatural spectrum, and the coherence generated, which centers the resonance about the atomic transition. This second property is necessary if the method is to be used as a spectroscopic tool. Because of the delayed detection scheme, all of these methods present the major disadvantage of a concomitant loss of signal. Thus it is clear that they will find their most relevant applications in cases where the spectroscopic study of the atom or molecule under consideration is already so advanced that one is interested in its most minute details. A further requirement for the application of such techniques is of course an excellent signal-to-noise ratio. For these reasons, delayed detection techniques do not appear to have as much promise for the future as the other group of methods we have presented, namely those based on the inhibition of spontaneous emission achieved by placing the system under study in a “protected” environment, such as a resonator with only a few modes of the electromagnetic field available for spontaneous decay. It is not completely clear yet how such a procedure could be extended to the optical regime, but as far as microwave

46

Dennis P. O’Brien et al.

transitions are concerned, new exciting results, particularly on Rydberg transitions, should be expected in the near future. We conclude by noting that, in order to keep the size of this review within reasonable limits, we chose to concentrate on those spectroscopictechniques which can roughly be described as “two-level system” techniques. Of course, considerable amounts of research have also been devoted to more complicated level schemes. In the case ofthree-level systems, e.g., several references worth mentioning include Inguscio et al. ( 1980), Omols ( 1979), Hackel and Ezekiel (1 979), and Wong and Garrison ( 1980).

REFERENCES Albrecht, K., and Neuwirth, W. (1967). 2.Phys. 203,420. Alexandrov, E. B. (1964). Opf.Spectrosc. 17, 957. Baklanov, Y. V., Dubetsky, B. Y., and Chebotaev, V. P. ( I976a). Appl. Phys. 9, 17 1. Baklanov, Y. V., Dubetsky, B. Y., and Chebotaev, V. P. (1976b). Appl. Phys. 11,201. Barger, R. L. (1981). Opt. Lett. 6, 145. Barger, R. L., Bergquist, J. C., English, T. C., andGlaze, D. L. (1979). Appl. Phys. Letf.34,850. Barton, G. ( 1970). Proc. R. Soc. London Ser. A 320.25 1. Bergquist, J. C., Lee, S. A., and Hall, J. L. (1977a). Phys. Rev. Letf. 38, 159. Bergquist, J. C., Lee, S. A., and Hall, J. L. (1977b). In “Laser Spectroscopy 111” (J. L. Hall and J. L. Carlsten, eds.), p. 142. Springer-Verlag, Berlin and New York. Berman, P. R. (1982). Phys. Rev. Left. 48, 366. Biraben, F., Cagnac, B., and Grynberg, G. (1 974). Phys. Rev. Lett. 32, 643. Bjorkholm, J. E., Liao, P. F., and Wokaun, A. (1982). Phys. Rev. A 26, 2643. Bjorklund, G. C. (1980). Opt. Lett. 5, 15. Bjorklund, G. C., and Levenson, M. D. (1981). Phys. Rev. A 24, 166. Bloch, D., Raj, R. K., and Ducloy, M. (198 1). Opf.Commun. 37, 183. Bloembergen, N., and Pound, R. V. (1954). Phys. Rev. 95, 8. Bordk, Ch. (1970). C. R. Acad. Sci. (Paris) 271, 371. Bordk, Ch., Carny, G., Decombs, B., and Pottier, L. (1973). C. R. Acad. Sci. (Paris) 277,381. Breit, G. ( 1 933). Rev. Mod. Phys. 5, 9 I . Cagnac, B., Grynberg, G., and Biraben, F. (1973). J. Phys. 34,845. Carlsten, J. L., and Hall, J. L., eds. (1977). “Laser Spectroscopy 111.” Springer-Verlag, Berlin and New Y ork. Carmichael, H. J.. and Walls, D. F. (1975). J. Phys. B8, L77. Carmichael, H. J., and Walls, D. F. (1976a). J. Phys. B 9, L43. Carmichael, H. J., and Walls, D. F. (1976b). J. Phys. B 9, 1 199. Champeau, R.-J., Leuchs, G., and Walther, H. (1978). 2. Phys. A 288, 323. Clark, B. 0..Van Baak, D. A,, and Pipkin, F. M. (1977). Phys. Left.62a, 78. Cohen-Tannoudji, C. ( 1975). In “Laser Spectroscopy” ( S . Haroche, J. C. Pebay-Peyroula, T. W. Hgnsch, and S. E. Harris, eds.), p. 324. Springer-Verlag, Berlin and New York. Cohen-Tannoudji, C. (1977). In “Frontiersin Laser Spectroscopy” (R. Balian, S. Haroche, and S. Liberman, eds.), Vol. I , p. 3. North Holland Publ., Amsterdam. Colegrove, F. D., Franken, P. A., Lewis, R. R., and Sands, R. M. (1959). Phys. Rev. Left.3,420. Copley, G., Kibble, B. P., and Series, G . W. (1968). J. Phys. B 1, 724.

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Copley, G., Kibble, B. P., and Series, G. (1973). Phys. Rev. Lett 31, 189. Corney, A. ( 1977). “Atomic and Laser Spectroscopy.” Clarendon, Oxford. Corney, A.. and Series, G. W. (1964). Proc. Phys. SOC.83,207. Couillaud, B., and Ducasse, A. (1975). In “Laser Spectroscopy” (S. Haroche, J. C. Pebay-Peyroula, T. W. Hansch, and S. E. Hams, eds.), Vol. 43, p. 476. Springer-Verlag, Berlin and New York. Cresser, J. D., Hager, J., Leuchs, G., Rateike, M., and Walther, H. (1982). In “Dissipative Systems in Quantum Optics” (R. Bonifacio, ed.), p. 2 1. Springer-Verlag, Berlin and New York. Deech, J. S., Hannaford, P., and Series, G. W. (1974). J. Phys. B 7, 1131. Dehmelt, H. G., and Toschek, P. E. (1975). Bull. APS 20,61. Demtroder, W. ( I 98 1). “Laser Spectroscopy.” Springer-Verlag, Berlin and New York. Dodd, J. N., Kaul, D., and Wamngton, D. (1964). Proc. Phys. Soc. London 84, 176. Ducas, T. W., Littman, M. G., and Zimmerman, M. L. (1975). Phys. Rev. Lett. 35, 1752. Eberly, J. H., and Wodkiewicz, K. ( 1977). J. Opt. Soc. Am. 67, 1252. Eckstein, J . N., Ferguson, A. I., and Hansch, T. W. (1978). Phys. Rev. Let. 40, 847. Eisenberger, P., Platzman, P. M., and Winick, H. (1976). Phys. Rev. Leu. 36,623. Fabjan, C. W., and Pipkin, F. M. (1972). Phys. Rev. A 6, 556. Figger, H., and Walther, H. ( 1 974). Z. Phys. 267, 1. Fontana, P. R. (1982). “Atomic Radiative Processes.” Academic Press, New York. Franken, P. (1961). Phys. Rev. 121, 508. Gabrielse, G., and Dehmelt, H. (1985). Phys. Rev. Left. 55, 67. Gawlik, W., and Series, G. W. (1979). I n “Laser Spectroscopy IV” (H. Walther and K. W. Rothe, eds.), p. 2 10. Springer-Verlag, Berlin and New York. Gawlik, W., Kowalski, J., Trager, F., and Vollmer, M. (1982). Phys. Rev. Lett. 48, 871. Gawlik, W., Kowalski, J., Trager, F., and Vollmer, M. (1983). In “Laser Spectroscopy VI” (H. P. Weber and W. Liithy, eds.), p. 136. Springer-Verlag, Berlin and New York. Gibbs, H. M., and Venkatesan, T. N. C. (1976). Opt. Commun. 17,87. Grove, R. E., Wu, F. Y., and Ezekiel, S. (1977). Phys. Rev. A 15, 227. Guzman, A., Meystre, P., and Scully, M. 0. (1983). In “Advances in Laser Spectroscopy” (F. T. Arecchi, F. Strumia, and H. Walther, eds.), p. 465. Plenum Press, New York. Hackel, R. P., and Ezekiel, S. (1979). Phys. Rev. Left. 42, 26. Hamillard, D. W., and Floy, G. R. (1968a). Phys. Rev. Left. 21, 724. Hamillard, D. W., and Floy, G. R. (1968b). Phys. Rev. Left. 21, 1468. Hanle, W. (1924). Z . Phys. 30, 93. Hansch, T. W., Levenson, M. D., and Schawlow, A. L. (1971). Phys. Rev. Lett. 26,946. Hlnsch, T. W., Harvey, K. C., Meisel, G.,and Schawlow, A. L. (1974). Opt. Commun. 11,50. Hansch, T. W., Lyons, D. R., Schawlow, A. L., Siegel, A., Wang, Z.-Y., and Yan, G.-Y. (1981). Opt. Commun. 37, 87. Haroche, S., Goy, P., Raimond, J. M., Fabre, C., and Gross, M. ( I 982). Philos. Trans. R. Soc. London Ser. A 307,659. Hams, S. M. (1961). Phys. Rev. 124, 1178. Hartig, W., Rasmussen, W., Schieder, R., and Walther, H. (1976). Z . Phys. A 278, 205. Hassan, S. S., and Bullough, R. K. (1975). J. Phys. E 8, L147. Heitler, W. ( I 954). “The Quantum Theory ofRadiation,” 3rd. Ed. Oxford Univ. Press, London and New York. Heitler, W., and Ma, S. T. (1949). Proc. R . Irish Acad. 52, 109. Hellmuth, T., Leuchs, G., Smith, S. J., and Walther, H. (1980). Proc. Int. ConJ Mulfiphoton Process., 2nd, Budapest. Hellmuth, T., Leuchs,G., Smith, S. J., and Walther, H. (1981). SpringerSer. Opt. Sci. 26, 194. Hogasen, H., Bergheim, K., and Skaavag, K. (1963). Phys. NOD. 1, 153.

Dennis P. O’Brien et al. Holland, R. E., Lynch, F. J., Perlow, G., and Hanna, S. S. (1960). Phys. Rev. Lett. 4, I8 I . Hughes, V. W. ( I 960). I n “Quantum Electronics” (C. H. Townes, ed.), p. 582. Columbia Univ. Press, New York. Inguscio, M., Moretti, A., and Stmmia, F. (1980). Opt. Commun. 35,614. Jaynes, E. T., and Cummings, F. W. (1963). Proc. IEEE 51,89. Kimble, H. J., and Mandel, L. (1976). Phys. Rev. A 13, 2123. Kimble, H. J., Dagenais, M., and Mandel, L. (1977). Phys. Rev. Lett. 39,691. Kimble, H. J., Dagenais, M., and Mandel, L. (1978). Phys. Rev. A 18, 201. Kleppner, D. (1971). In “Atomic Physics and Astrophysics,” (M. Chretien and E. Lipworth, eds.), p. 49. Gordon & Breach, New York. Kleppner, D. (1981). Phys. Rev. Lett. 47,233. Knight, P. L. (1981). Comments Atom. Mol. Phys. 10,241. Knight, P. L., and Coleman, P. E. (1980). J. Phys. B 13,4345. Knight, P. L., Kagan, D., and Radmore, P. M. (1981). Phys. Lett. 82% 288. Kramer, P. B., Lundeen, S.R., Clark, B. O., and Pipkin, F. M. (1974). Phys. Rev. Lett. 32,635. Lange, W., and Mlynek, J. (1978). Phys. Rev. Lett. 40, 1373. Lawler, J. E., Ferguson, A. J., Goldsmith, J. E. M., Jackson, D. J., and Schawlow, A. L. (1979). Phys. Rev. Lett. 42, 1046. Lee Hai-Woong, Meystre, P., and Scully, M. 0. (1981). Phys. Rev. A 24, 1914. Leuchs. G., and Walther, H. (1977). Springer Ser. Opt. Sci. 7 , 299. Leuchs, G., and Walther, H. (1979). Z. Phys. A 293,93. Leuchs, G., Smith, S. J., Khawaja, E. J., and Walther, H. ( 1 979). Opt. Commun. 31, 3 13. Levenson, M. D., and Bloembergen, N. (1974). Phys. Rev. Lett. 32,645. Lundeen, S . R., and Pipkin, F. M. (1975). Phys. Rev. Lett. 34, 1368. Lynch, F. J., Holland, R. E., and Hammermesh, M. (1960). Phys. Rev. 120, 513. Ma, 1. J., Mertens, J., Zu Putlitz, G., and Schiitte, G. (1968a). Z. Phys. 208, 352. Ma, 1. J., Zu Putlitz, G., and Schiitte, G. (1968b). Z. Phys. 208, 252. McFarlane, R. A., Bennett, W. R., and Lamb, W. E., Jr. (1963). Appl. Phys. Lett. 2, 189. Marinero, E. E., and Stuke, M. (1979). Opt. Commun. 30, 349. Meschede, D., Walther, H., and Miiller, G. (1985). Phys. Rev. Lett. 54, 55 I . Metcalf, H., and Phillips, W. (1980). Opt. Lett. 5, 540. Meystre, P. (1974). Ph.D. thesis, Swiss Federal Institute of Technology, Lausanne. Meystre, P., Geneux, E., Quattropani, A., and Faist, A. (1975). Nuovo Cimento 25B,521. Meystre, P., Scully, M. O., and Walther, H. (1980). Opt. Commun. 33, 153. Milonni, P. W., and Knight, P. L. (1973). Opt. Commun. 9, 119. Mitchell, A. C. G., and Zemanskey, M. W. (1934). “Resonance Radiation and Excited Atoms.” Cambridge Univ. Press, London and New York. Moi, L., Goy, P., Gross, M., Raimond, J. M., Fabre, C., and Haroche, S. ( 1 983). Phys. Rev. A 27, 2043. Mollow, B. R. (1969). Phys. Rev. 188, 1969. Mollow, B. R. (1972). Phys. Rev. A 5, 1522. Mollow, B. R. (1975). Phys. Rev. A 5, 1919. Nagourney, W., Janik, G., and Dehmelt, H. (1983). Proc. Nutl. Acud. Sci. U.S.A.80,643. Neuhauser, W . ,Hohenstatt, M., Toschek, P. E., and Dehmelt, H. G. (1978).Phys. Rev. Lett. 41, 233. Neuhauser, W., Hohenstatt, M., Toschek, P. E., and Dehmelt, H. G. ( I 980). Phys. Rev. A 22, 1137. Neuwirth, W. (1966). Z. Phys. 197,473. Oliver, G., Ressayre, E., and Tallet, A. (1971). Lett. Nuovo Cimento 2, 77. Omont, A., Smith, W. W., and Cooper, J. (1972). Astrophys. J. 175, 185. Oniols, G. ( 1 979). Nuovo Cimento 53, I .

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 21

MOLECULAR APPLZCATIONS OF QUANTUM DEFECT THEORY CHRIS H . GREENE Departmeni if Physics and Astronomy Louisiana State Cniver.riry Baton Rouge. Louisiana

Ch. JUNGEN Lahoratoire de Photophysique Molkciilaire du CNRS L'nibtersitP de Paris-Sud Orsay, France 1. Introduction . . . . . . . . . . . . . . . . . . . . 11. Quantum Defect Concepts and Formalism . . . . . .

111.

. . . . . . . . . . . . . . A. Origin of the Rydberg Formula . . . . . . . . . . . . . . . . . B. Multichannel Rearrangement Processes . . . . . . . . . . . . . C. The Eigenchannel Representation. . . . . . . . . . . . . . . . D. Photofragmentation Cross Sections . . . . . . . . . . . . . . . E. Physical Significance of the Eigenchannels . . . . . . . . . . . . Rovibrational Channel Interactions. . . . . . . . . . . . , , . . .

51 54 54 56

58 62 63 66

A. Adaptation of the Quantum Defect Formalism to Molecular

Problems. . . . . . . . . . . . . . . . . . . . . . . . . B. Channel Interactions Involving Highly Excited Bound Levels . C . Channel Interactions Involving Continua . . . . . . . . . . D. Treatment of a Class of Non-Born-Oppenheimer Phenomena. IV. Electronic Interactions at Short Range . . . . . . . . . . . . . A. Theoretical Developments . . . . . . . . . . . . . . . . . B. Electronic Preionization in Molecular Nitrogen. . . . . . . . C. Photodissociation and Dissociative Recombination . . . . . , V. Discussion and Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. . . . . . . . . . .

. . . . . .

.

66 69 16 87 91 91 102 106 I I5 1 I8

I. Introduction Rydberg states of diatomic molecules, in which one electron usually roams far from the nuclei and other electrons, differ dramatically in most 51 Copynghl 0 1985 by Academic Press, Inc All nghls of reproduction in any form reserved

52

Chris H . Greene and Ch. Jungen

respects from valence states. The dominant simplifying element of the valence-state physics is the slow time scale of nuclear vibration and nuclear rotation compared with the time scale of electronic motion. As higher and higher Rydberg electronic states are excited a point is reached at which these time scales become comparable, and in fact the electronic motion eventually becomes the slowest. At this stage the interpretation of molecular spectra cannot be handled by a conventional analysis based on the BornOppenheimer approximation and on the coupling of electronic angular momenta to the internuclear axis. As realized by Mulliken ( 1964), an understanding of this portion of the spectrum would require a radically different method of theoretical analysis. The first element of this analysis was the development of quantum defect theory by Seaton ( 1966)to handle the corresponding multichannel Rydberg spectrum in atoms and in electron-ion scattering. A comprehensive review of this approach has been presented recently by Seaton (1 983). High-resolution spectroscopic studies of H2photoabsorption by Chupka and Berkowitz ( 1969), Herzberg ( 1969), and Herzberg and Jungen ( 1972) provided an experimental impetus which sparked an extension of Seaton’s formulation to molecular Rydberg spectra by Fano (1 970). Fano’s article emphasized two new concepts, namely, the physical significance of eigenchannels of the electron -ion interaction and the necessity for a rotational frame transformation between the eigenchannels at small electronic distances r and the fragmentation channels at large r. Systematic exploitation of these concepts, including their extension to vibrational channel interactions, has permitted a nearly complete accounting of the experimental H2photoabsorption spectrum (Dehmer and Chupka, 1976) up to photon energies of 133,000 cm-’ (Jungen and Raoult, 1981). Mulliken (1976) has referred to Rydberg states as the “stepping stones toward ionization.” He thereby stressed the basic unity which exists between high Rydberg states and the adjacent ionization continua: the short-range physics is common to both while they differ only at long range. Quantum defect theory provides a unified description of discrete and continuous spectra in terms of the same parameters pertaining to the physics at short range. Thereby it accounts for the Rydberg structures as well as for their decay into the continuum. This decay is called preionization (or synonymously autoionization), or predissociation if the continuum is associated with nuclear rather than with electronic motion. Figure 1 displays two molecular spectra which exhibit striking effects due to both preionization and predissociation. Both spectra result from photoexcitation of nitric oxide and they cover the same wavelength region. The upper spectrum is a recording of the ion current resulting after excitation just above the ionization threshold: The small peaks superimposed on the flat continuum are Rydberg structures whose

+

, I

+

4

I

v*= 1

77,000

7 ~ 0 ENERGY / cm”

75,000

I

a

I 1.P vf 0

74,000

FIG.1 Medium-resolution photoionization(a) and photoabsorptionspectrum(b) of NO nearthe first ionization threshold.The spectra are taken with comparablespectral resolution (= 8 and 13an-’,respectively),but they correspondto Merent temperatures(20 and 78 K, respectively). The assignmentsof nplr, u Rydberg bands are those of Miescher and Alberti (1976) and are based on the high-resolutionabsorption spectrum. Sections marked *-* are reproduced in Fig. 21. [(a) From Ono et ul., 1980; (b) From Miescher et al., 1978.1

54

Chris H. Greene and Ch. Jungen

presence reveals that preionization occurs. The lower trace is an absorption spectrum taken under comparable experimental resolution. The strikingly different appearance of the two spectra reveals that a second decay channel, corresponding to production of neutral atomic fragments, must also interfere. Moreover, the different appearance of the Rydberg peaks in the two spectra indicates that predissociation is strong and competes with preionization. The aim of the present article is, first of all, to review the current state of molecular quantum defect theory. Second, emphasis will be placed on describing its applications to typical problems of molecular physics such as the interpretation of the spectra of Fig. 1. Although we do not explicitly discuss electron scattering by neutral molecules in this review, theoretical methods closely related to quantum defect theory have been applied to this class of problems as well. A thorough and fairly recent review of the current status of electron - molecule scattering theory has been presented by Lane ( 1980).

11. Quantum Defect Concepts and Formalism A. ORIGINOF THE RYDBERG FORMULA

The various approaches called “quantum defect theory” are essentially designed to take advantage of one simple fact: When one molecular electron moves sufficiently far from the remaining electrons and nuclei (say at r > 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
r)

tron mass for the system of interest to the electron rest mass. In these expressions z is the charge of the ion, while the phase parameters q and /3 are constants with respect to r : V ( E ) = (zrn/k)In P(E) = K(U

+

+

2k - 1 ~ / 2 arg r(l 1 - izrn/k)

- 1)

(4)

in terms of the "effective quantum number" u = Z ~ / and K the orbital angular momentum quantum number 1. The constant D is given by Seaton (1 966) or by Greene et al. ( 1983), but it will not be needed here. These asymptotic forms of (fl g) will permit a straightforward application of the large-r boundary conditions relevant to scattering or photoionization processes. [In some applications, especially far below threshold, an alternative analytic pair of solutions (fo,go)should be used in place of (f;g).] The bound-level positions follow immediately from requiring the outer field solution [Eq. ( l)] to be finite at r + m. With the large-r forms (3) for E < 0 the constraint obtained for integer I is

+

sin n(u p ) = 0

(5)

equivalent to the empirical formula of Rydberg:

En = - rnz2/2(n- p ) 2

(6) Here the quantum defect p is nearly independent of energy, whereby Eq. (6) compactly parametrizes an infinite number of states.

56

Chris H . Greene and Ch. Jungen

In multichannel situations, of course, both Eqs. ( 5 ) and (6) must be generalized, as discussed below. Likewise Eqs. ( 1)-(3) remain valid for non-Coulombic long-range fields as well if the parameters q, /Iand , D appropriate to that long-range field are used in place of Eq. (4). Applications of this flexibility of quantum defect theory have expanded in recent years (Greene et al., 1979,1983; Watanabe and Greene, 1980), as evidenced, for example, by the treatments of molecular dissociation described in Sections II1,D and IV,A below (Giusti 1980; Colle, 1981; Nakamura, 1983; Mies, 1984; Mies and Julienne, 1984). B. MULTICHANNEL REARRANGEMENT PROCESSES

The quantum-mechanical amplitude for a rearrangement collision such as

e-

+ H,+(i’)

+

e-

+ H2+(i)

(7)

is given by one element of a scattering matrix Sii,. It is defined mathematically through the asymptotic (outgoing wave) form of the stationary-state wave function having an incoming wave component in channel i f only and outgoing scattered waves in all channels i,

The coordinate r in Eq. (8) represents the separation between the outermost electron and the H2+ center of mass, while o denotes all other independent coordinates of the system, including spin and angular coordinates. The factor @Lo) thus represents the complete wave function of H2+in the state i together with the spin and orbital angular momentum wave functions of the outermost electron with appropriate angular momentum coupling factors. The A operator indicates the Fermi antisymmetrization needed, but it satisfies only a formal requirement here since exchange is negligible asymptotically. Lastly f f ( r )represent outgoing/incoming wave solutions of the outer field (electron - ion) radial Schrodinger equation for channel i. Asymptotically these are essentially exp(+ ikir) aside from an additional logarithmic phase factor associated with the Coulomb tail. The normalization v ~ involves the relative velocity of the receding fragments; it ensures that lSii@)12 is the ratio of the outgoing radial probability flux in channel i to the incoming flux in channel i f . Alternative channels i can have different internal energies Ei of the fragments and possibly different reduced masses mi,so that the wave vector ki in f f ( r )at a total energy E is given in atomic units (Bethe and Salpeter, 1957) by

k: = 2rni(E - Ei)

(9)

~

/

~

57

QUANTUM DEFECT THEORY

The dimension of the square scattering matrix is No, the number of open channels [for which kf > 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

QUANTUM DEFECT THEORY

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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,.

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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 <- 1 (not drawn in the figure for clarity) whose intersections with the Lu Fano curve determine the positions of the bound levels. When there is no channel interaction the Lu -Fano curve degenerates into two perpendicular straight lines, each of which represents an unperturbed Rydberg series. Inspection of Eq. (42) indicates that this can arise in two ways, either when the two eigenquantum defects pa coincide or when the frame transformation matrix is diagonal, i.e., the frame transformation angle 8 is zero. As an example Fig. 6 shows the Lu-Fano plot describing

QUANTUM DEFECT THEORY

73

n; (mod I )

00

05 He2 ndA

05

t

I

5P

FIG.6. Lu-Fano plot representingthe u = 0, N = 6, odd-parity triplet levels of He, arising from n d n - and ndA- states. The values n: and n: refer to the u+ = 0, N+ = 5 and 7 limits, respectively.(From Ginter and Ginter, 1983.)

excited levels with J = 6, 1 = 2, and negative parity of molecular helium observed by Ginter and Ginter (1983). (For related work see Ginter and Ginter, 1980; Ginter et al., 1984.) The two interacting Rydberg series have ll- and A- symmetry, respectively, in the Born-Oppenheimer limit and converge toward the N + = 5 and 7 limits of He,+, u+ = 0. Although the frame transformation angles 0 are nearly identical in the H, and He, examples, namely 54.7’ in H2 as compared with 5 1.7” in He,, it can be seen that the channel interaction is much weaker in He,. The reason is that the eigenquantum defects pn and p Adiffer by less than 0.1, as can be read from the plot. This is characteristic of an electron with a high orbital angular momentum 1 which is largely kept away from the core by the centrifugal potential l(l+ 1)/2r2 and thus cannot probe the noncentral core structure efficiently. Quite generally, molecular Rydberg states corresponding to a high I value appear in the spectra as compact “I complexes” (see Dieke, 1929) owing to the small values of all pu,’s, and they are characterized by weak channel interactions. They tend to conform to Hund’s case d even for small values of the principal quantum number, because, even when vI = v2 in Eq. (43), the linear system corresponding to Eq. (42) still yields coefficientsA, far from 1 or 0, again as a consequence of the smallness of the pa’s. In graphical terms, even for small differences vI - v,, the levels will remain on the horizontal or vertical parts of the Lu-Fano curve.

Chris H . Greene and Ch. Jungen

74

3. Rotational Channel Interactions in a Heavy Molecule Labastie, Bordas, Broyer, Martin, and collaborators (Martin et al., 1983; Labastie et al., 1984; Bordas et al., 1984) have recently observed highly excited Rydberg states of the Na, molecule using the method of opticaloptical double resonance, and they have applied quantum defect theory to their highly resolved data. An example of their spectra is shown in the lower part of Fig. 7 corresponding to excitation of 1 = 2 Rydberg states near the v+ = 4 threshold from the intermediate A ‘Zf, v+ = 4, J’ = 22 even-panty level. The final levels thus must have J = 2 1,22, and 23 and odd parity, and therefore involve series associated with odd N + values ranging from 19 to 25. In all, eight different channels are represented in the spectrum of Fig. 7 and involve two three-channel systems and one two-channel system. These excited Rydberg levels are subject to vibrational preionization since they lie above the u+ = 0 ionization threshold-in fact they are detected experimentally by the ionization signal which they produce. However, the resonance widths are smaller than the experimental resolution; so that in a good

24

25

26

I 39,850

27

28

29

30

31

I

32 33 3 4

39,900

35

-rE 3;,950

[Em-’]

FIG.7. Rydberg spectrum of Na, excited from the A Ci:, v = 4, J’ = 22 even-parity level (top: theory; bottom: experiment). The spectrum corresponds to excitation of I = 2 Rydberg states near the v+ = 4 limit and illustrates the fringe effect. The energies corresponding to integer values of v are marked below the experimental spectrum. (After Bordas ef al.. 1984.)

QUANTUM DEFECT THEORY

75

approximation we can disregard the ionization process for the present purposes. In the upper part of Fig. 7 is shown a theoretical spectrum based on the methods outlined in Section 11, in particular the algebraic linear system given in Eqs. ( 17) and ( 1 8). All open channels are omitted and, in fact, the R dependence of p,,(R) has been disregarded altogether, whereby Eq. (40) becomes proportional to S,,”+, .As for H,, Eq. (34) was used to evaluate the rotational 1-uncouplingframe transformation. A fit of the quantum defects p, to the experimental data yielded the valuespd, = 0 . 2 1 , = ~ -0.04, ~ ~ and pda = 0.43. The ionization limits E,+,,,+have also been fitted and can be found in the paper of Martin et al. (1983). The quite good agreement between the experimental and theoretical spectra demonstrates that theory accounts for the rotational channel interactions in Na, as it does in H, and He,. The spectra of Na, illustrate a striking new aspect of channel interactions which is not obvious in the H, example discussed above. The new feature of the spectrum in Fig. 7 is its periodicity, which is related simultaneously to several series. By contrast, the conspicuous level perturbations seen in H, (Fig. 4) arise whenever a N + = 2 level occurs; that is, they are related to the periodicity of just one series. The periodicity of the Na, spectrum manifests itself by simplified, comblike structures, or fringes, which are numbered k = 1,2, . . . . Between the fringes the intensity is distributed over numerous channel components, and no outstanding structure can be discerned. For the present discussion it is sufficient to consider a two-channel situation to which Eqs. (42) and (43) apply. Labastie et al. (1984) have shown that the fringes occur whenever

vI-v2=k, k=l,2,. .. (44) Equation (42) is indeed periodic for integer values of k; one of the coefficients A , of the corresponding linear system is then unity; i.e., a periodic return to the Born-Oppenheimer limit occurs (Hund’s case b), as had been stressed by Fano ( 1975) and Jungen and Raoult ( 1 98 1). A favorable circumstance in Na, is that the dipole intensity for excitation from the intermediate A lZ; state is dominated by the l’l- 2 transition moment since Z- Z is quite weak and A -2is dipole forbidden. As a consequence all intensity is concentrated near energies corresponding to vi = ni - pdn, and different lines coincide owing to Eq. (44). The condition [Eq. (44)] is, however, not yet sufficient to explain the appearance of the fringes. For these to become clearly visible it is also necessary that the interfering Rydberg series remain “in step” over several cycles of nv. Thus, according to Labastie et al., one must also have dv,/dv,

-

1

(45)

76

Chris H. Greene and Ch. Jungen

Since Eq. (43) implies that V , = v,( 1 - 2 ~ AE)-'I2 : (46) where AE is the splitting between the two ionization limits, it follows that, in order to fulfillthe conditions of Eqs. (44) and (45) simultaneously, one must have

2k K v, K ( 2 AE)-'I2 (AE in a.u.) (47) Clearly, then, fringes of the type shown in Fig. 7 are better visible for smaller AE. This explains why they appear in the heavy system Na,, which has a much narrower rotational structure than H, or He,. The fringe pattern becomes progressively narrower as k increases. From Eq. (47) we estimate that the last fringe, k,, , corresponds roughly to AE =r 250/k2,, (AE in cm-') (48) This shows that fringes are not to be expected in vibrational or electronic channel interactions where AE usually is much larger than 250 cm-I. Also, the fringe effect has obviously no counterpart in the continuum, unlike the level perturbations of the type seen in H, which extend into the continuum in the form of preionization profiles (and present also, of course, in Na,). C. CHANNEL INTERACTIONS INVOLVING CONTINUA

The main conceptual feature of quantum defect theory has been the recognition that Rydberg bound states lying below an ionization threshold and the continuum states just above this threshold are governed by the same physics at short distances. This concept has been used widely in other physical problems, of course, such as effective-range theory applications in nuclear and atomic physics. Indeed Sommerfeld ( I 93 1) had long ago exploited this fact in his studies of atomic Rydberg electrons. Now we turn to an explicit demonstration of this unity, whereby the same parameters used to calculate accurate H, bound levels will equally serve to characterize the photoionization cross sections of H,. 1. The Total Photoionization Cross Section

Besides the eigenquantum defects p, and frame-transformation matrix elements U,,given above, the photoabsorption spectrm depends on dipole matrix elements DL? = D E connecting the ground state of H, to the finalstate eigenchannels w,. These are given in terms of Born-Oppenheimer matrix elements d,,(R) in Eq. (4 I). In the absence of an ab inifiocalculation

QUANTUM DEFECT THEORY

77

of these matrix elements, studies to date have chosen d,(R) = dAR) to be a constant, independent of R but slowly varying as a function of energy. The equality of d, and dn is suggested by the united-atom limit (Herzberg and Jungen, 1972). The assumption of R independence should not be taken too seriously,as the H2ground state is confined to a narrow range of internuclear separations and therefore the R dependence is hardly probed by ground-state photoabsorption experiments. It is now relatively straightforward to calculate the photoionization cross section by solving the linear equations of quantum defect theory, Eqs. (37)(38), and then using the general cross-section formula, Eq. (31). At any energy E of interest there are typically both continuum (i E P)and discrete channels (i E Q). The continua provide the path to ionization, that is, to the actual escape to infinity of the electron from the molecular ion. The closed channels provide a mechanism for the formation of short-lived quasibound states which typically autoionize in to sec. In simple atoms, autoionization is often said to result from the electron -electron repulsion terms in the Hamiltonian. In this vein rovibrational autoionization in small molecules must be attributed to a qualitatively different dynamic feature; namely, the R dependence of the quantum defect function,u,,(R) is responsible for “vibrational” autoionization, while its A dependence is responsible for “rotational” autoionization. (Most rovibrational autoionization features involve both physical mechanisms, making it impossible to distinguish the two processes.) This distinction can at least be made in a mathematical sense since the rotational factor of the frame-transformation matrix [Eq. (33)] would induce no autoionization if ,u, were equal to Pn. Similarly, if p,,(R) were independent of R, then the vibrational factor of Eq. (33) would induce no autoionization. When both of these conditions are fulfilled, the matrices Sii,and C,,,of Eq. (40) are proportional to the unit matrix, in which case the absence of channel interactions is obvious. But this weak-coupling limit need not be satisfied, as evidenced for example by H, photoabsorption where channel interactions are indeed so strong as to render any perturbative scheme useless. Only quantum defect theory seems capable of dealing with these multichannel interactions which disturb the positions and intensities of an infinite number of Rydberg and continuum levels at once. Figure 8 displays a quite simple photoionization spectrum measured under high resolution by Dehmer and Chupka (1976). The spectral range shown includes the 8pa and 9pa, u = 2, J = 1 levels ofH, . This spectrum has again been obtained using a cooled sample of para-H, so that it corresponds for the most part to an excited system with J = 1, odd parity. The Rydberg channels involved in this range are represented by Fig. 9: All discrete Rydberg structures at this energy are embedded in the v+ = 0 and 1 , N + = 0 and 2 continua. Between the v = 2 levels with n = 8 and 9 lie numerous “inter-

Chris H. Greene and Ch. Jungen

78

'>

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

+

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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

QUANTUM DEFECT THEORY

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

Chris H . Greene and Ch. Jungen

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 <4;'l(d/dR)&') are of little significance. For Rydberg potential curves described by Eq. (35), however, AU(R) = K 3 ,so that in fact y approaches unity in Hz by n = 5 since A & , is nearly independent of n. The qualitative change in the physics renders the Born Oppenheimer description ineffective for high-lying Rydberg states. As we have seen, experimental manifestations of this non-Born -0ppenheimer behavior include strong rovibronic perturbations and conspicuous intensity modulations in the ionization continuum associated with vibrational autoionization. Yet another conspicuous nonadiabatic process is predissociation, to be discussed in Section 111,D,2. 1. Adiabatic and Nonadiabatic Correctionsto the Discrete Levels

Figure 12 presents energy levels of the Hz C Ill; state as calculated by Jungen and Atabek ( 1977)using the quantum defect procedures described in Section II,C. Shown are deviations between experimental values and theoretical values obtained in QDT and in the strict Born-Oppenheimer approximation. Note how the quantum defect results, derived from the Born Oppenheimer functions pA(R),agree with experiment far better than do the full Born - Oppenheimer energy levels. Among the non-Born Oppenheimer effects included in the quantum defect calculation are the well-known corrections for the H2+ion (Hunter et al., 1974); this ensures that the H2Rydberg levels converge to the proper thresholds. Here this gives the bulk of the correction in Fig. 12, with a comparatively small correction coming from the outer electron. The quantum defect formulation represents these effects in a much simpler fashion than conventional theory (Van Vleck, 1936), which would require separate evaluation of several perturbation integrals as functions of R followed by vibrational averaging.

QUANTUM DEFECT THEORY

89

FIG.12. Deviations of the observed J = 1 levels in the 2pn ‘n; state from those derived ab inirio for H2(circles) and D, (dots) in the Born-Oppenheimer approximation (BO) and using quantum defect theory (QDT). The difference between the two results represents the adiabatic and nonadiabatic corrections. The remaining deviations in QDT correspond to the specific isotope effect. (After Jungen and Atabek, 1977.)

These non-Born - Oppenheimer couplings also cause perturbations between discrete levels belonging to different Born - Oppenheimer potential curves. One reflection of such perturbations is the A doubling as given in Figure 13, for example, which is the splitting between the 3p7r D Ill: and Ill; states. Since T I: interacts with 5; levels while Ill; states experience no such interaction, the A doubling displays the effect of perturbations clearly. Notice how a seemingly random dependence on J and v, corresponding to the erratic occurrence of near degeneracies, again emerges “automatically” in this approach. While agreement with experimental values of Takezawa (1970) is not exact, the general trends are very well represented.

2. R-Matrix Treatment of Predissociation A more extreme failure of the Born- Oppenheimer approximation occurs when an excited molecular electronic state predissociates, thereby converting its internal energy from the electronic degrees of freedom into kinetic energy of the dissociating atoms. Experimentally the strength of this process is measured on an absolute scale by the “reduced” predissociation linewidth

Chris H. Greene and Ch. Jungen

90

0

I

3

5

J

FIG.13. A doubling in the 3pn D TI,state of H, (difference between n: and n; components). Dots: observed values from Takezawa ( 1970). Full lines: calculated values from Jungen and Atabek (1977). The irregular pattern of the splittings illustrates the breakdown of the hypothesis of “pure precession” (Van Vleck, 1929) arising because the vibrational structures of the interacting 3pn and 3pu states are intermingled and, in addition, different (cf. inset).

of rovibrational levels below the first ionization threshold, a dimensionless quantity -

r = r-(dE/du)-i = r / h w (57) Here w is the vibrational spacing and r is the full width at half-maximum. Above the ionization threshold the strength ofdissociation relative to ionization can be measured more directly at each energy by observing the branchThe usual expectation is that this branching ratio remains ing ratio ud/oji. much less than one, since the photoabsorption tends to excite the electrons initially rather than the heavy nuclei. Experiment (Guyon ez al., 1979; Dehmer and Chupka, 1976) and recent theory (Jungen, 1982, 1984) both indicate, however, that this expectation is frequently violated, especially in light molecules and particularly close to resonances.

QUANTUM DEFECT THEORY

91

Predissociation can result from a variety of mechanisms, but it yields to a surprisingly simple treatment when a single electronic channel is present in the body frame. In this case the quantum defect approach gives an essentially exact wave function [Eqs. ( 14) and ( 16)] in terms of the reaction matrix K or the equivalent § and C matrices of Eq. (40). These contain the dynamic information required to describe the processes of vibrational autoionization or vibrational excitation. This type of predissociation can be viewed as a natural extension of the vibrational excitation process treated in Section III,C, e- H2+(u) e- H2+(u’),the only difference being that the final state u’ actually lies in the H H vibrational continuum of the lowest potential curves in the Rydberg series. Realized some time ago by Chang et al. (1973), this concept has been developed into a complete formulation only recently (Jungen, 1982, 1984) through an adaptation of the eigenchannel R-matrix method (Fano and Lee, 1973).The R-matrix method (Burke and Robb, 1975;Nesbet, 1980)obtains a variational solution to the Schrodinger equation at the total energy E of interest, but only within a finite reaction volume R of configuration space. Outside the surface I: which encloses this reaction volume, the solution can be represented by a channel expansion such as Eq. (10) or (14) with the proper form at large distances. By matching the inner variational R-matrix solutions to the outer region solutions on Z, the reaction matrix K is obtained. In this sense the R-matrix method links naturally with quantum defect theory, in that the variational problem is attacked only at small distances where the many-particle system is complicated; the more rapid energy variations associated with the motion at large distances are then represented analytically. The eigenchannel version of R-matrix theory (Fano and Lee, 1973; Lee, 1974a; Greene, 1983; Le Rouzo and Raseev, 1984) solves variationally for the eigenstates of the R matrix, i.e., for states having a constant normal logarithmic derivative (- b) everywhere on the reaction surface I:.The reaction volume R in this problem is the rectangular region r < r,, R < R, shown in Fig. 14. Here r2 is taken just large enough to contain the entire Born - Oppenheimer electronic wave function of each dissociative state. Instead R, is chosen just large enough so that all non-Born-Oppenheimer processes are contained within R < R,. The reaction surface enclosing this volume has two distinct parts: The line ( r = r2,R) connects to the ionization channels in regions 11, while the line (r, R = R,) connects to the dissociation channels in region 111. The wave function in region IV is neglected altogether along with the possibility of three-body fragmentation (dissociative ionization). The region denoted the “reaction zone” in Fig. 14 is characterized by complicated dynamics, but as we saw above the physical solution along the line r = r, is well represented by an R- and A-dependent quantum defect function p,,(R).

+

-

+

+

92

Chris H . Greene and Ch. Jungen

@tion

PRO zone

R (nuclei)

*

FIG.14. Schematic representation of the (r, R) configuration space relevant to the calculation of competing ionization and dissociation processes in HI.The appropriatechannel expansion terms are indicated for different zones. (After Jungen, 1984.)

The aim of the R-matrix calculation is thus to find a set of eigensolutions

w,, of the Hamiltonian with the following properties: wp

=

c

PU+N+(C

R)(RIu+)1Tu+N+,,

(u+N+)€P

r B r2, R < & (58) - Gd(R)sin az,], r r2, R 3 &

X [fu+N+(r) cos mp- gu+N+(r) sin nr,],

w, =

@$Tdg[Fd(R)cos KT,

(59) Here Eq. ( 5 8 ) coincides with the expression [Eq. (19)] for the electronic eigenchannels. Equation (59) is the exact analog on the dissociative part of the reaction surface, with @$ the electronic Born -0ppenheimer wave function in a dissociative channel d. At t g s stage the choice of two independent vibrational continuum solutions (Fd, G,,) in the relevant Born -0ppenheimer potential for the open dissociative channels is somewhat arbitrary. It is useful to choose this base pair through boundary conditions at

R

= Ro,

so that Eq. (59) takes a form familiar from R-matrix theory -aR +tann~,~,=O,

R=Ro, r s r 2

QUANTUM DEFECT THEORY

\

93

(Note that Fd and Gddo not coincide with the usual regular and irregular Born - Oppenheimer solutions Fand G, but are convenient linear combinations of them.) These eigenchannel boundary conditions supplement the behavior along r = rospecifiedby p,,(R), and the usual finitenesscondition at r 00 imposed by algebraic quantum defect procedures. It is convenient to regard the electronic phase parameter zPin Eq. (58) and the vibrational phase parameter zp in Eqs. (59) and (61) as independent variables, disconnected from the constraint ofequality implied by Eqs. (58) and (59),much as in the Lu - Fano plot discussed in Section 111,B,2. In practice the boundary condition [Eq. (6 l)] on the dissociativepart of the reaction surface is imposed first by picking a trial value 7 and thereby deJininga discrete, orthonormal vibrational basis of Hz+, (u+lR)cT).The associated energy levels EFjN+also now depend on 7 , although this dependence is negligible for the low v+ states which do not extend to R = R,,. Figure 15 compares the resulting finiterange spectrum (with Ro = 4) with the infinite-range spectrum in Hz+, showing that the finite-range level density is much lower for high v + as these states reach rapidly up into the vibrational continuum. For this particular (trial) choice of 7 the vibrational integrals are evaluated to give the S7)and C(T) matrices as before, and at the energy E of interest the closed ionization channels i E Q can be eliminated [see Eq. (23)] by solving

-

TB, = tan 7rtphB,

-

L

-0.50-

a

0

(62)

/ '

-

r

W

-0.60-

I

I

I

I

1

1

b

Chris H . Greene and Ch. Jungen

94

The net result of this elimination for each trial choice ofthe vibrational phase 7 is a set of No eigenphase shifts n7, of the open portion of the (ionization channel) reaction matrix, with p = 1 , . , . ,No. Since in general the vibrational surface parameter 7 will not equal any of these eigenvalues 7, (modulo l), it is necessary to search as a function of 7 until this consistency is established. As in the Lu-Fano plot, a convenient graphical representation of this procedure in Fig. 16 plots the electronic eigenphases 7, as functions of the vibrational eigenphase T . The example of Fig. 16 corresponds to two open ionization and two open dissociation channels 2 p n and 3pZ. Although only two electronic 7pare present for each value of T, these two curves 7 f ) display four intersections with the diagonal, giving No Nd = 4 eigensolutions tpas expected. In particular the two arrows show that the rapid variations of 7, lie close to the (unperturbed) dissociative H2Born - Oppenheimer logarithmic derivatives at this energy. In essence these appear as 7 varies across its full range 0 d 7 s 1, when low-lying Rydberg states associated with very high vibrational thresholds sweep through the energy E. A few additional remarks will now show how to extract the final 4 X 4 “grand” reaction matrix W, which refers to ionization and dissociation at once. With the eigenvalues of W given by tan n7,, we require their corresponding eigenvectors. The components of an eigenvector T, in the ioniza-

+

3pP

2pn

I 7P

0.50

0

FIG. 16. Predissociation in H1.Electronic eigenphases functions of the vibrational eigenphase 7 (cf. text).

7p (quantum

defects) plotted as

95

QUANTUM DEFECT THEORY

tion channels i are given by the standard result

The components in the dissociative channels are somewhat more complicated to evaluate in the general case, but in the limit of only one dissociative channel per value of A (i.e., d = nA), it is (see Lee, 1974a)

-

[Although this appears to be ill behaved in the special case T,, f ,there is in vanishes in that limit.] The solutions fact no divergence since (u+IR,)(~J Bi,,,= Bu+N+,p of the homogeneous system [Eq. (62)] are normalized according to

Finally, the grand reaction matrix refemng to ionization and dissociation is

R = T tan ntTT

(68)

In fact, the eigenphase shifts T,, correspond to a constant phase shift in the ionization channels i, but to a constant logarithmic derivative in the dissociation channels nA. A linear transformation [Eq. (3.3) ofGreene (1980)l gives the “usual” reaction matrix in an energy-normalized representation of ionization and dissociation channels:

K = (F’ - FR)(G’ - GR)-’

(69)

Here the diagonal matrices F, F’, G, C ’ take the values Fii,= - Giit= dii,, Gii,= Fji, = 0 in each ionization channel but instead ( F d , Gd)d,, are the values of the energy-normalized regular and irregular Born - Oppenheimer dissociative-state vibrational solutions evaluated at R = R, . Likewise (FL, G;)ddd, are their derivatives with respect to R at R = Ro (details concerning the specification of these vibrational comparison functions can be found in Greene et al., 1983; Mies, 1984). We now discuss the application of these procedures to the 3pn+, v = 3, J = 2 level of H,,to which Fig. 16 actually refers. In this energy range all ionization channels are closed,but to obtain agrand reaction matrix K which varies weakly with the energy the v+ = 3, N + = 1,3channels were artificially opened in this initial stage of the calculation (as in Jungen, 1984).In terms of the resulting 4 X 4 reaction matrix the predissociation can now be calculated in a straightforward manner using the general methods of Section 11. The

Chris H . Greene and Ch. Jungen

96 0.8L

-

\

f 0.6v)

-

W

I

a

z

0.4-

2 0.2W

t FIG.17. Predissociation in H1.Calculated vibrational eigenphasesum plotted as a function of the energy near the 3pn D Il:, u = 3, J = 2 level.

physical boundary conditions at infinity must now be applied; i.e., in the present example the ionization channels v+ = 3, N + = 1 and 3 are at this point treated as closed channels while the dissociation channels 3 Z and 2 ll are open. Figure 17 presents the resulting variation of the eigenphase sum as a function of energy near the position of the 3pU, v = 3, J = 2 level. A dispersion pattern is obtained which arises from the predissociation by the 3pZ continuum. The resonance center and the predissociation width can be determined from it easily: we see that the calculated width is in almost perfect agreement with the measurement of Glass-Maujean et al. (1979). The resonance position is also very well reproduced by the calculation: Ecalc= 119318.3 cm-l as compared with EOb,= 119320.5 cm-I (Takezawa, 1970). Jungen (1984) has applied this approach to the more complicated situation of competition between predissociation and preionization occurring above the v+ = 1 ionization threshold. Some examplesare shown in Fig. 8 and have already been discussed. It is worth mentioning that, unlike most R-matrix calculations, the bulk of the computational effort in the present formulation is not spent in finding variational approximations to eigenfunctions of the Hamiltonian at the desired energy- these are already given to good accuracy in terms of the input function pA(R)through the vibrational frame transformation. Rather the only calculation required here is the determination of appropriate linear combinations of these eigensolutions which obey the consistency requirement [Eqs. (58), (59)] between the electronic phase shifts zp and the vibrational phase parameter z (and which obey the boundedness requirement at r 00). Surprisingly, the information contained in pA(R)alone, together

-

QUANTUM DEFECT THEORY

97

with the preceding discussion, suffices even to describe an inelastic atomic collision process.

IV. Electronic Interactions at Short Range The discussion of the preceding section has been limited to rovibrational channel interactions since it was assumed that the physically relevant fragmentation channels always correspondto the same electronic state of the ion core. We proceed now to the discussion of processes whereby conversion of electronic core energy plays a role in addition to the conversion of rovibrational energy already discussed. In Section IV,A we deal with the formal aspects of the problem. A. THEORETICAL DEVELOPMENTS 1. Rovibronic Channel Interactions

The most straightforwardmethod of including electronic channel interactions starts out from the rovibrational treatment of Section 111. The fragmentation channel labels for ionization are now (i) = (k, u+, N + ) , where k stands for both the electronic state of the ion core and the orbital angular momentum of the distant excited electron, neglecting all spin effects. The complementary eigenchannel labels are (a)= ( p , R,A). The electronic eigenchannel representation /3 may not have any simple physical interpretation; it must be determined from experiment or calculated ab initio for given R and A. The frame transformation matrix U, of Section III,A [Eq. (33)] is now complemented by an electronic factor, giving

u. = ukv+N+,BRA = (~~~)'"'(v'IR)'~"'(N'(A)'~~)

(70) This factorization isjustified by the fact that each factor results from integration over different variables (electronic, vibrational, rotational). Both the eigenchannel representationa of the full system and the relevant levelsof the free core are assumed here to conform to the Born - Oppenheimer approximation. [For diatomic ion cores this is usually a good approximation for the lowest few electronic states, but in polyatomic ions even the electronic ground state may depart significantly and systematically from the BornOppenheimer approximation if an electronic angular momentum is present (Jahn -Teller effect, Renner-Teller effect). The present treatment would

98

Chris H . Creene and Ch. Jungen

require some extension in order to include such situations.] The reaction matrix including electronic interactions now takes the form

Similarly the matrices C,,, and Sii,of Eq. (40) in the preceding section are obtained in generalized form by replacing the tangent function in Eq. (7 1) by cosine and sine functions, respectively. Correspondingly, in the formulation of Eqs. ( 17) and ( 18), the matrices 43, and S, are, e.g., 43. .C

kv+N+, BRA

= (klp)(RA)(U+IR)(kN+)(N+I~)(kJ) cos nppA(R) (72)

and similarly for S,. If nuclear motion is disregarded entirely, Eqs. (70)-(72) define a purely electronic multichannel problem forfixed R and A with frame-transformation elements

u,

,y(RN=

kg

(k

(73)

and a corresponding body-frame reaction matrix

These simplified expressions provide the link with atomic quantum defect theory and they involve the electronic quantities-the reaction matrix or alternatively its eigenphases and vectors- which can in principle be obtained in an electronic ab initio calculation for fixed geometry. For photoabsorption processes the required dipole matrix elements of Eq. (4 1) now have the structure d,,(R). For negative energy the solutions of the generalized eigenvalue problem defined by Eqs. (23)-(25) with U and p from Eqs. (73) and (74) yield the Born- Oppenheimer potential energy curves relevant to a given problem. Unlike in the preceding section, these curves are no longer given by a set of R-dependent Rydberg equations of the type of Eq. (35)with nearly energyor n-independent quantum defects. Rather, strong n dependences will occur owing to the electronic interchannel interactions, and in a potential curve plot of the type shown in Fig. 3a numerous avoidedcrossings between curves associated with different electronic core states may occur. An example is the gerade electronic symmetry ofthe H, molecule [cf. Fig. 1 of Wolniewicz and Dressler (1977)]. We note two further features of the rovibronic treatment just outlined.

99

QUANTUM DEFECT THEORY

First, by their definition the electronic transformation elements include the projections of several pure 1 states on a given eigenstate j?. In other words, 1 mixing, mentioned in Section III,A, is in principle taken into account here. Second, the flexibility in the choice of the vibrational basis used to represent the channel mixing coefficientsABRhcorresponding to Eq. (36) persists in the present extended treatment. One can alternatively use the vibrational wave functions of the different ion states or, if there are strong avoided crossings between potential curves of the neutral molecule, the vibrational wave functions corresponding to the adiabatic curves might be useful. [See the discussion following Eq. (36).] The treatment of rovibronic channel interactions sketched here has been outlined first by Jungen and Atabek (1977) and later by Raseev and Le Rouzo (1983). It has to date not been implemented. Instead another approach, due to Giusti ( 1980), has proved very useful in situations where the electronic channel interactions are relatively weak and lend themselves to a perturbation treatment. In addition, this approach can be extended very easily to treat dissociation processes involving an electron core rearrangement.

2. Two-step Treatment of Electronic Channel Interactions Giusti starts out by considering that the Hamiltonian ofthe system can be conveniently partitioned into two parts, H = Ho V Ho includes the longrange fields and those short-range interactions confined to a single electronic channel. It is useful to further indicate the separate short- and long-range contributions to Ho separately through Ho = T Vo= T VLR ( Vo- VLR), where V L R is the long-range potential (e.g., the Coulomb potential for photoionization of neutral systems). Residual interactions between electronic channels specified, e.g., in an independent-electron model, are described by the short-range operator V. The approach will be most useful when V is small. In the first step of the treatment only Ho is taken into account. The resulting total multichannel wave function t,do)can be written exactly as in Section 11, as

+

+

v(o)(E)=

x

+

+

W;O’(aAy(E)

Y

Here the eigenchannel label y has been used instead of a to indicate that the influence of V has still been omitted. The summation over fragmentation

Chris H . Greene and Ch. Jungen

100

channels i is meant to include alternative rovibrational as well as electronic core states as in Section IV,A, 1. Giusti proceeds now by remarking that each factor ( A cos xpy - gi sin zpy) can be regarded as a new, phase-shifted basis functionf;(r, y ) in the longrange field, which may serve in the second step of the treatment. The linearly independent conjugate basis function g,(r, y), which l a g s j by 90", is cos lccLy +A@) sin w Equation (75) can thus be rewritten as 7) =

Fi(rY

W'O)(E)=

c

y

(76)

w$'(E24,(E)

Y

The main effect of the residual interaction Vis to modify the wave function at large distances ras follows. Each factorj(r, y)A,(E)in Eq. (77) must now be replaced by a sum over degenerate eigenstates(I!of the complete Hamiltonian with an additional phase shift in each channel,

2 U$:)[j(r,Y) cos npLV' - a r , 7) sin npLV)1AII(E)

(78)

(I

Here the npLV)are the additional phase shifts due to V, and the matrix U$V,) transforms the eigenstates of Ho into those of the full Hamiltonian H. After insertion of Eq. (78) into Eq. (77) and using the full expressions forfand g, the following wave function w results (for r 2 ro):

with

§ , =

2 Uiysin n ( p y+ pLV))u::) Y

This wave function has exactly the same form as the function, Eq. (79, of the one-step treatment except that each matrix element C,(I is no longer equal to Uiacos npa but involves the product of two successive unitary matrices, Uiy and UiV,),and the sum of eigenphases nhYand npf) arising from ( Vo - VLR) and V, respectively. The net effect of the short-range interactions contained can be equivalently expressed with the following reaction matrices

QUANTUM DEFECT THEORY

101

If these two reaction matrices are known at the outset, diagonalization yields their eigenvectors and the eigenphases so that the linear system, Eq. (17), can be set up with the matrices S, and C, evaluated with Eq. (80).2 The advantage of the two-step method in comparison with the approach outlined in Section IV,A, 1 is that, by including most of the interactions in Ho , the residual term V can be treated perturbatively. Electronic channel interactions leading to electronic preionization may serve as a first illustration.

3. Interactions between Ionization Channels This application of the two-step approach has been introduced by GiustiSuzor and Lefebvre-Brion ( 1980);we present it here with a slight change of emphasis. In the first step rovibrationalinteractions within a given molecular ionization channel are included, and the mixing of partial waves 1 by the noncentral core field is also taken into account. The fragmentationchannels are labeled ( i ) = (kv+N+)as in Section IV,A, 1 and the eigenchannels of Ho are labeled (7) = (BRA).The transformation matrix U,, has elements U,, = (klp)(RA)(u+lR)(kN+)(N+lA)(~') (82) in exact analogy to Eq. (70). The difference is that the electronic eigenstatesp here are not meant to diagonalizethe full electronic Hamiltonian but rather correspond to the motion of an electron in the field of a core frozen into a given electronic state. Quite a number of ab initio computer programs are available today which furnish numerical solutions of exactly this problem One may ask at this point how the matrices K and K(") of Eq. (81) relevant to a given problem are related to the matrix KCtOUl)describing the same problem in a one-step approach such as outlined in Section IV,A,l [Eq. (71)]. It is in fact simpler to derive a complex matrix S(Ww) = (1 iK(loul))/(l- iK(mw))rather than K('OW)itself. By forming the complex Jost-type matrices (see, e.g., Newton, 1966) with elements j g = C, isi, = U, exp(i inp,) we find that &. = U, exp(2inp,)(UT),. = j&(j-)ii!

+

*

x U

(I

in a one-step treatment. In a two-step approach the matricesj* become with Eq. (80) j&=

2 Uiyexp[fin(py+ p!,"))]Ui:) Y

so that the complex "short-range'' scattering matrix S(low)which combines both steps into one

has elements $:PW)

=

c (xuiyexp(inp,)u:) u

w'

)

exp(2inp;~)(~)~),.exp(inp?,)(v)y,i,

It can be easily verified that this matrix is unitary and therefore that K(mw)is symmetric as it should be.

102

Chris H. Greene and Ch. Jungen

(e.g., Dill and Dehmer, 1974;Dehmer and Dill, 1979;Lucchese and McKoy, 1981; Raseev, 1980; Schneider and Collins, 1981 ; Burke ef al., 1983). In the second step the coupling Vbetween electronic channels associated with different cores is introduced. It arises then from the interelectronic repulsion l/ru. If the interaction is weak the matrix elements of the reaction matrix Kc?, Eq. (81), can be evaluated in good approximation as a matrix elemeny of V over the core region deriving from the LippmannSchwinger equation in first order (see Giusti, 1980, for a comprehensive discussion). Thus KpRAd,R,A, (V) -n(i/F)l Vli/$’) !‘(!I?) d(R - R’)dAAt (83) 88

The electronic interaction matrix V$!(R)is dimensionless, as can be verified from the dimensions of y / ( O ) and V As a quantity resulting from electronic short-range interactions, V$!(R) will generally vary slowly with excitation energy. Its eigenvaluestan npLT(R)and eigenvectors(Pl(~)(~”)are needed for each value of R and A. The labels of the eigenchannels are then (aRA),and the frame transformation for the second step has the form

(PRAlaR’A’)= ( p l ~ ~ )d(R ‘ ~ ”-) R ’ ) d A A ,

(84)

The analog to the matrices 43, of Eq. (72) therefore becomes

Q)k,+N+, nRA = (N+Il\)‘kJ’(U+IR)‘kN”

By comparing Eqs. (72) and (85) we see that the one- and two-step approaches differ in the electronic factor. In Section IV,B below we describe the application of this formulation to electronic preionization in N,. Section IV,C shows how the approach has been extended to dissociativeprocesses in NO and in H, for the case of weak coupling. B. ELECTRONIC PREIONIZATION I N MOLECULAR NITROGEN The preceding formulation has first been used by Lefebvre-Bnon and Giusti-Suzor (1983) for model calculations dealing with the interplay between electronic and vibrational preionization. These authors showed that, as in the case of rovibrational preionization (cf. Section III), a clear distinction between the two processes is often impossible so that one should really speak of “vibronic” preionization. Morin el al. ( 1982) applied the two-step approach to the photoionization of molecular oxygen. They presented an

QUANTUM DEFECT THEORY

103

empirical interpretation of a complicated section in the partial cross sections for vibrational excitation where electronic preionization dominates. Quite recently Raoult et al. (1983) have published a detailed ab initio study of electronic preionization in the Hopfield series of molecular nitrogen [seealso the subsequent paper by Le Rouzo and Raoult (1989, who have presented improved calculations]. We shall now discuss these results in some detail. Numerous Rydberg series are known in N, converging to the X, A, and B states of N2+.The most strikinglybeautiful of these are the so-called Hopfield series (Hopfield, 1930a,b) (Fig. 18) which appear above the N2+A ,nu threshold as regular broad preionization features converging towards the B 22: limit of Nz+. The regularity of the series (i.e., the absence of overlapping vibrational progressions and channel perturbations) no doubt arises because only v+ = 0 is excited in the B ,Stcore owing to the similarity of the potential energy curves of the N, Xand N,+ B states, and because vibrational preionization is not strong. Besides their regularity, the Hopfield series exhibit striking preionization shapes: There is one series which appears with absorption peaks, while another consists of window resonances. In order to account for these observations a multichannel treatment ofthe Rydberg series associated with the B state of N,+ as well as of the continua associated with the lower-lying X and A states must be carried out. In photoabsorption from the X '2: ground state of N,, 'Z: and 'nuelectronic states can be excited. Therefore the channels (PRA)of Section IV,A,3 are superpositions of the following partial waves 1 G 3:

:

A = '2

X 22icore: pa,, fau

(Worley- Jenkins series)

A 21"Iu core: dng

A = 'nu

B 'Z: core: sag,dog

(Hopfield series)

X ,2: core: pn,, fir,

(Worley- Jenkins series)

A

2nucore:

B ,2,, core:

sogB, dog,dS, dirg

(Hopfield series)

The spin variables have for the most part been excluded here: Strictly, the ionization channels should be labeled using a coupling scheme (Q, w ) analogous to ( j ,j ) coupling in atoms, and the triplet states should thus be included in the PRA representation used in the above table. The neglect of these is justified by the relative smallness of spin-orbit coupling (= 80 cm-l in the A state of N2+).Raoult et al. have disregarded molecular rotation and vibration altogether and evaluated the quantum defects pjA(R), transformation elements (kl/3)("^),and dipole amplitudes d,&) separately for each molecular and for the internuclear distance R = Re corsymmetry, A = 5 and 'nu,

:

$1

a

0 0 I-

0

I

a w

1

I-

a _I

w

a I

FIG.18 Relative photoionization cross section for N, X 'El, U" = 0 taken at a temperatureof 78 K. The spectrum shown correspondsto the range between the A 211uand B 2E: thresholds of N,+. (After Dehmer et al., 1984.)

QUANTUM DEFECT THEORY

105

responding to equilibrium in the N, ground state. The calculations were done with the ab initio code of Raseev (1980) which is based on the singlecenter frozen-core static exchange approximation. Partial waves up to 1 = 14 had to be included in order to obtain convergence in the core region; however, it was found that outside the core only values l d 3 contribute significantly, and the dimension of the matrix K in Eq. (8 1) is in fact considerably and to N = 5 for ‘Z:. reduced, to N = 6 for ‘nu This first step of the ab initiowork shows that strong s-dmixing occurs in the channels associated with both the A and B states of N2+,whilep-fmixing is negligible in the energy range of interest here. [At much higher energy the occurs (Plummer et well-known shape resonance associated with N2+X al., 1977; Hamnet et al., 1976), which involves strong mixing of the p andf partial waves.] For example, the transformation matrix (kip) of Eq. (82) for so and do electrons associated with the B ,Z states is calculated by Raoult et af. to have elements

: k

P

B2X:, su

B 2 X i , da

B *Xi,“s”a B lX:, “d”a

0.86 0.52

-0.52 0.86

Following these authors “I” is used here to denote the partial wave with largest amplitude I in the p representation. In the second step the residual electronic interaction V s ! ( R )is calculated by evaluating the required two-electron integrals with a computer program written by Le Rouzo (unpublished). None of these interactions exceeds 0.1, and the first-order approximation, Eq. (83), is thus justified a posteriori. The resulting additional phase shifts n,uL? and eigenvectors (pla)then serve to construct the matrices S, and 43, according to Eq. ( 8 5 ) and to set up the linear system of Eq. (1 7). The results thus obtained by Raoult et al. and Le Rouzo and Raoult with their fully ab initio approach include predictions of total and partial cross sections for photoionization, the photoelectron angular distributions, and the polarization of the fluorescence from the N,+ B 2Z: state formed in photoionization. As an example Fig. 19 compares a section of the calculated total photoionization cross section with the corresponding experimental curve recorded by Dehmer et al. ( 1984). Since these authors measured only the relative ionization cross section, it had to be calibrated with the absolute absorption cross section curve of Giirtler et al. ( 1977)assuming that photodissociation is negligible. Figure 19 shows that the observed and calculated curves are in quantitative agreement inasmuch as the continuum back-

Chris H . Greene and Ch. Jungen

106

M

20

t

t

ol-

1

L--1-

680

670

I 1

682

PHOTON WAVELENGTH

684

(%I

FIG.19. Electronic preionization in N,. Calculated (top, Le Rouzo and Raoult, 1985) and observed (bottom, Dehmer et al.. 1984) total ionization cross section near then = 5 membersof the Hopfield Rydberg series.

ground is concerned, and in semiquantitative agreement inasmuch as resonance positions, widths, and shapes are concerned. Thus Raoult et al. have been able to conclude that the Hopfield “absorption” series corresponds to excitation of a quasibound “d”a Rydberg electron while the “emission” series is due to a “ d ’ k and “s”o electron. These resonances appear either because the preionized state has a high absorption intensity (“d”a) or because the preionized state is strongly coupled to an intense continum (“d”n, “s”~). It must be stressed that, in order to be able to interpret the structure of the Hopfield series, a full treatment of the ionization process including electronic preionization has been necessary. C. PHOTODISSOCIATION AND DISSOCIATIVE RECOMBINATION 1. Theoretical Development

The two-step approach provides also a convenient description of dissociation processes which enter into competition with ionization (Giusti, 1980; Giusti-Suzor and Jungen, 1984). Figure 20 illustrates this with the help of a potential energy diagram. Shown are a set of Rydberg potential energy curves converging towards one ionic curve, plus a set of valence state (“nonRydberg”) curves which cross the former at numerous points. In the spirit of

107

QUANTUM DEFECT THEORY

I

I’

90

I

I

-

-n -- - - - -

1.P -

I 7

R (a u.)

FIG. 20. Diabatic potential energy curves of *lIstates in NO. The arrows indicate the avoided crossings occumng in the adiabatic curves as a result of Rydberg-valence-state interactions (cf. the inset). Three energy ranges are marked which correspond to the zones of I: spectral perturbations; 11: predissociation; 111: competition between predissociation and preionization. (After Gallusser and Dressler, 1982.)

quantum defect theory the Rydberg curves are regarded as representing the closed portion of a molecular ionization channel. By analogy each valence state can be viewed as representing a dissociation channel: The vibrational continuum above the dissociation limit represents its open portion while the bound vibrational spectrum corresponds to the closed portion. The situation depicted in Fig. 20 is different from that encountered in Section 111,D,2, since here dissociation proceeds through a valence state external to the Rydberg series, whose electronic structure diflers from that o f the Rydberg

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Chris H . Greene and Ch. Jungen

states. In other words, dissociation in the present case involves electronic rearrangement in the core, while in the situation discussed previously it does not. Electronic couplings between the Rydberg manifold and the valence states are indicated in the inset of Fig. 20 by avoided crossings: These interactions produce spectral perturbations in the discrete part of the spectrum; they lead to electronic predissociation of Rydberg levels situated above the dissociation limit. Above the ionization potential there will be a competition between dissociation and ionization processes. In order to apply the general approach of Section IV,A,2 to this problem we must first define the fragmentation channel labels. The Rydberg manifold corresponds to channels ( i } = (u+N+)if we restrict the treatment to a single electronic core state and a single partial wave 1. The valence states are labeled (dA};the “core” wave function here is the electronic valence state wave function for all electrons, while the radial functionsfand g for relative motion of the fragments are just the regular and irregular vibrational wave functions FLA)(R)and GLA)(R)in each valence state potential. The “core” region or “reaction” zone extends in its electronic dimension to the edge of the ion core, and in its nuclear dimension to the radius& somewhat beyond the outermost curve crossing (see the figure). The first step of the treatment is analogous to the theory of Section 111,A: The rovibrational interactions within the ionization channel are taken into account while the dissociation channel wave functions are still FLA)(R).Thus we have labels (7) = (RAYdA), Next the Rydberg-valence-state interaction Vmust be introduced. The first-order Lippmann - Schwinger equation analogous to Eq. (83) is now an integral over all electron coordinates and the nuclear coordinate, extending throughout the reaction zone. However, since the Rth Rydberg vibrational eigenchannel function is essentially6(R - R ’), we are left with an electronic integral for each value of R , multiplied by the regular valence-state continuum wave function F,(R). Thus the nonzero elements of K(y) become p) 88’ E K (RA,&‘ V) = -n V y ( R ) F y ( R ) (86) Note that, unlike in electronic preionization (Section IV,A), the dimension of the electronic interaction VLA)is (energy)’l2,while Flf)(R)has dimension (energy X length)-’/*, whereby the element K E , is not dimensionless. As a quantity resulting from electronic short-range interactions, V$*)(R)is expected to vary slowly with excitation energy. By contrast, the matrix K(”wil1 vary on the scale of a vibrational energy quantum owing to the factor F,,(R), and it must therefore be evaluated on a sufficiently narrow energy mesh. A complication in practice is the partially continuous character of the basis set (7) = (RAYdA). This difficulty had already been encountered in Section II1,A and the remedy here is similar: Instead of expressing the matrix K(“) in

QUANTUM DEFECT THEORY

109

the representation (7) we must express it in the fragmentationchannel representation ( i ) . We then obtain (see Giusti-Suzor and Jungen, 1984)

K ("+N+, v) "+IN+' Using this reaction matrix one can then proceed with the general method outlined in Section 11. The remaining problem in practice is how to obtain the electronic interaction VLA)(R).One way would be to fit VLA)directly to experimental data. Another way consists of relating VbA) to the size of avoided crossings seen in Fig. 20. From the construction of the first-order expression of the matrix K(v), it can be seen that for small V$*)and low principal quantum numbers n (for which successiveRydberg curves are still well separated) the magnitude of the avoided crossings, 2 V S ) ,will be approximately equal to twice the interaction VLA)multiplied by the discrete Rydberg normalization factor for the nth level. Thus we may put V z ) ( R ) [n - pA(R)]-3'2V(dh)(R) (88) The energies V S )are the quantities entering vibronic perturbation calculations of the type often performed in molecular spectroscopy. A final remark concerns the vibrational wave functions F(R) and G(R). The regular function F is required for the evaluation of the first-order reaction matrix K(") in the second step of the calculation. Unlike the Coulomb functions which are known analytically, F must be calculated numerically. For positive energies this can be done using any standard integration method, while for negative energies the Milne approach can be used (see Greene et al., 1983), which immediately defines then G(R)as well. This approach yields also the negative energy phase parameter which is no longer given simply by A ( V - I) as for a Coulomb field.

2. Application to Competing Dissociation and Ionization Processes in NO Giusti-Suzor and Jungen (1984) have applied this approach to the calculation of predissociation and preionization in the threshold region of nitric oxide. The idea underlying their work was to use the very detailed information available on Rydberg- valence-stateinteractions in the discrete region in order to interpret the decay processes observed at higher energy. Precise data concerning level perturbations in the discrete range stem mainly from the work of Miescher and collaborators (see Miescher and Huber, 1976), who performed extensive analyses of the high-resolution absorption spectra

QUANTUM DEFECT THEORY

111

of several isotopes of NO. The potential energy curve diagram shown in Fig. 20 refers to the Rydberg and valence states of NO having 211 symmetry. Gallusser and Dressler ( 1982)have given a quantitative representation ofthe observed vibronic level positions and intensities in the states shown in the figure, in terms of a set of “deperturbed” potential energy curves and transition moments, and of electronic Rydberg - valence-state interaction energies V:). Their data furnish directly the main parameters required for the implementation of the approach of Section IV,C, I . Figure 1 compares the medium-resolution photoabsorption and photoionization spectra (Miescher et al., 1978;Ono ef al., 1980)at higher energy near the ionization threshold. The assignment of npn, u+ Rydberg peaks derives from the work of Miescher (1976) and is based on the spectra of several isotopes obtained under much higher resolution than the spectrogram shown in the figure. The present recording shows that these peaks dominate the absorption spectrum in this range. (A careful analysis indicates that thepn ionization channel cames nearly half ofthe total absorption cross section in this range.) The high-resolution spectrum shows also that the individual rotational lines are diffuse, with widths that vary strongly from one band to another, but this is not visible in Fig. 1 except for the 5pa, u = 3 peak whose width exceeds the rotational band structure. The ionization cross section is strikingly different: It is essentially flat, with steps occumng near each vibrational threshold u+. Resonances are visible only as fairly weak “wiggles” superimposed onto the staircase structure. Obviously the difference between the two spectra must correspond to photodissociation, i.e., to the cross section for formation of the atomic fragments, mainly, N(2D) O(3P).Note that this difference cannot be taken directly from the figure since the two curves shown are on relative scales which in fact turn out to be very different. The fact that all resonance features appear only weakly in ionization indicates thatpredissociafion plays a dominant role. In the case of the npn, v+ resonances the strong Rydberg state crossings with the B 211 and L 211 valence states furnish the mechanism for predissociation which first comes to mind. Figure 2 1 presents two sections of the spectra of Fig. 1 on an enlarged scale. Drawn also are theoretical curves resulting from the two-step approach

+

..

FIG.2 I . Two sections ofthe photoionization (top) and photoabsorptionspectrum (bottom) of NO (Fig. I ) in the region between the v+ = 0 and 1 (right) and v+ = 1 and 2 (left) ionization thresholds. The observed and calculated spectra are represented by full and broken lines, respectively.Note the difference between the cross-sectionscales in the top and bottom spectra. The calculated spectrum contains only pn resonances, but accounts for the background due to all partial waves II. (After Giusti-Suzor and Jungen, 1984.)

112

Chris H . Greene and Ch. Jungen

outlined in the preceding subsection, implemented with the parameters (in particular interaction energies V , and the R dependence of the quantum defect) taken from Gallusser and Dressler’s work. Molecular rotation has been disregarded in the initial multichannel calculations, its neglect being justified by the basically vibronic origin of preionization and predissociation in NO, and because below n = 10 1 uncoupling in the pn Rydberg series is weak. In order to draw the theoretical contours of Fig. 2 1 Giusti-Suzor and Jungen convoluted each calculated resonance feature with the rotational band structure reflecting the Boltzmann distribution present in the NO sample and the experimental resolution width. Even after this convolution the height of each peak npn, v+ depends sensitively on the level width as well as on the intensity associated with the peak. The ionization signal in turn depends sensitively on the characteristics of the competition between preionization and predissociation. The calculated curves shown in the figure do of course not include resonances other than pn. For these, however, the overall agreement between experiment and theory is quite striking, showing that the interplay between dissociation and ionization processes is on the whole correctly calculated. It is instructive to discuss the competition between the two processes with reference to the Au = 1 selection rule mentioned in Section II1,C. If this rule could be applied directly to the processes rather than only to the channel couplings,one would be tempted to conclude that only Av = 1 resonances (if any) can appear in photoionization, while IAvl> 1 resonances should be absent, i.e., appear only in photodissociation. An early analysis of the photoionization spectrum of NO (Ng et al., 1976), in contradiction with Miescher’s work, had indeed been based on this assumption. It can be seen, however, in Fig. 21 that lAol= 1, 2, and 3 resonances are observed and calculated to have comparable peak heights in the photoionization cross section. Following Giusti-Suzor and Jungen this must be attributed to an indirect process, whereby a bound Rydberg level becomes coupled to the ionization continuum via the valence-state dissociation channel. In other words, the strong electronic interaction induces preionization even where the direct coupling is extremely weak. One might also term this a continuum - continuum interaction which interferes with the discrete continuum interactions. According to the present analysis, the characteristic staircase pattern of the ionization curve of NO- one of the classic examples of this type-is also a consequence of the predominance of electronic predissociation over vibrational preionization. Unlike in Fig. 10, where no step was apparent at the o+ = 4 ionization threshold, the large step at the v+ = 1 threshold in Fig. la reflects the strong influence of predissociation, which affects the resonances far more than the direct continuum background.

QUANTUM DEFECT THEORY

113

3. Dissociative Recombination Giusti-Suzor et al. ( 1983)have applied the same approach to dissociative recombination in low-energy e-H2+ collisions. This is in turn an instance where quantum defect theory has been combinedwith ab initio theory which provided the dynamical parameters for the problem, namely quantum defects, valence-state potential energy curves, and Rydberg- valence-state interaction energies. Unlike photoabsorption in H2, which restricts the excitation to the manifold of singlet ungerade electronic states, dissociative recombination may involve singlet as well as triplet, gerade as well as ungerade, compound electronic states; in addition all partial waves 1 are present in the entrance channel. Giusti-Suzor et al. selected for their calculations those channels which appeared to be the most important for the process studied. Thus, high-1 states were omitted since their small amplitude in the reaction zone implies they are unlikely to provide an efficient reaction path. Ungerade electronicchannels were disregarded since they exhibit no avoided crossings, as shown by Fig. 3a, and the nonadiabatic mechanism discussed in Section III,D,2 is relatively weak. Triplet channels were ruled out on the basis of the same argument. Among the singlet gerade states it is only the manifold of 'Zlpotential curveswhich exhibitsavoided crossingsqualitativelysimilar to those shown in Fig. 20 but with considerably stronger electronic interactions. Among those the sagRydberg channel was omitted, being presumed unimportant, and s-d mixing was also neglected along with molecular rotation. The channels of '2: symmetry which were finally retained have dominant electronic configurations ionization (v+):

(laJ(~daJ

dissociation ( d ) :

(la,)Z

The dynamic parameters were taken from a separate study of Hazi et al. ( 1983).Thus the diabatic ( 1aJ2valence-state curve was taken in part directly

from the ab initio scattering calculations of Takagi and Nakamura (1980) and in part extracted from a deperturbation analysis performed on the very accurate adiabatic (noncrossing) '2: curves of Wolniewicz and Dressler ( 1977, 1979).This deperturbationcalculation yielded also the R dependence of the quantum defect, p..c'T(R).Finally, the electronic interaction Vd including its R dependence was obtained by the Stieltjes moment method. The calculationswere carried out in terms of the resulting parameters in a way exactly analogousto the NO example,except that, instead ofthe incoming wave boundary condition of Section II,D, the outgoing wave boundary

Chris H . Greene und Ch. Jungen

114

conditions discussed in Section II,B were employed (and of course the dipole transition moments were not required). Figure 22a presents an example of the results obtained by Giusti-Suzor et al. We notice, first, that the background cross section decreases nearly in proportion to E-’ as is expected since the background parts of the K and S matrices are nearly energy independent on this scale so that only the factor E-’ from the general cross-section formula remains. Second, all Rydberg resonances which are supenmposed on the background are window resonances. The reason is related to the weakness of the R dependence of the quantum defect in this example, as can be appreciated as follows: Electron capture constitutes the first step of the dissociative process and hence plays a role analogous to photoexcitation in all previous examples. Near threshold the population of the Rydberg levels n“d”a, v+ > 0 by electron capture is governed by their vibrational coupling to the entrance channel “ d ” q u+ = 0, and this turns out to be quite weak owing to a weak R dependence of p near equilibrium. The situation is then

L

-

0. 0 5 ENERGY ( s V

1

0.1

0.2

0.5

FIG. 22. Cross sections for dissociative recombination of electrons with Hz+ions (after Giusti-Suzor el a/., 1983). (a) Solid curve: calculated cross section for the u+ = 0 state of H2+. Dips are due to ndu, u Rydberg states, with the values of n and u indicated for the most prominent dips. Dashed curve: result obtained when all closed channels are omitted. (b) Cross section for e-H,+ recombination with a mixture of ions in the u+ = 0, I , and 2 states: solid curve, calculation; dashed curve, experiment (Auerbach et a/., 1977).

QUANTUM DEFECT THEORY

115

analogous to that in which a closed Rydberg channel has a nearly vanishing optical transition moment and is embedded in a strong continuum: It is well known that window resonances appear under these circumstances (Fano, 196 1). In turn, the large widths of the resonances seen in Fig. 22a are due mainly to predissociation. In an experiment, H2+levels with v+ > 0 are also present, and for the comparison of theory with the observations an appropriate convolution of the calculated cross sections must be made which takes account also of the experimental resolution. Figure 22b presents such a convoluted cross-section curve, designed for the comparison with the merged-beam experiment of Auerbach et af.(1 977) (also shown). It can be seen that the two curves are quite similar, although neither the resonance positions nor the absolute cross-section values do quite coincide.

V. Discussion and Conclusions There appears to be some misconception about multichannel quantum defect theory being primarily “empirical” since it is based on parameters which supposedly cannot be obtained directly by ab initio computational schemes. Much in the previous sections has been designed to show that this is not the case. First of all, electron-ion scattering calculations (such as, e.g., those of Takagi and Nakamura, 1980, 1983) yield phase shifts and possibly transition moments for fixed molecular symmetry and geometry which are exactly of the type used in molecular quantum defect theory. The most recent ab initio study of doubly excited states in H, has been performed by Raseev (1985). It is true that no ab initio calculations appear to have been camed out as yet at negative energy which would yield directly quantum defects, although this is now routinely done in atomic calculations (Lee, 1974a; Greene, 1983; O’Mahony and Greene, 1985). Instead quantum defect curves have been extracted from potential energy curves calculated by quantum chemical methods as discussed in Sections 111 and IV,C. In electronic multichannel situations this procedure would become less straightforward (cf. Section IV,A,l), but there appears to be no conceptual problem. Obviously ab initio methods of the R matrix type, where the electron structure calculation is restricted to a finite volume encompassing the reaction zone, complement the quantum defect theory of the outer zone in the most natural way. But fixed-geometry ab initio results of any type can be combined with rovibronic multichannel calculations as the examples discussed here show. Reliable ab initio computations no doubt will play an ever-increasing role in the future in the analysis ofexperimental

116

Chris H . Greene and Ch. Jungen

data relating to molecular decay processes. As an example, the analysis of the Hopfield series of N2reviewed in Section IV,B requires knowledge of, in all, 47 dynamical parameters all of which have been calculated ab initio: It is difficult to envisage how these could have been obtained empirically from experiment given the fact that molecular rotational structure is virtually washed out at these energies by the fast decay processes. Applications of quantum defect theory to polyatomic molecules have been scarce thus far. Dagata et al. (1981) have published a Lu-Fano plot pertaining to the sand d series in CH,I convergingtowards the two spin states of CH,I+. Their plot has been derived from a high-resolution absorption spectrum and resembles very closely the correspondingplot for the Xe atom, as anticipated by the single-channel analysis of Wang et al. ( 1977). Apparently vibrational motion does not completely alter the channel interactions in this case, as a remarkable similarity between the ionic fine-structure effects in the atom and in the polyatomic molecule emerged. Fano and Lu (1 984) have sketched the application of quantum defect theory to the Rydberg molecule H, . In a different context Golubkov and Ivanov (1984b) have applied quantum defect methods to the problem of slow collisions between an atom and a diatomic molecule. By taking into account simultaneously the ion core and the perturber atom, they obtained an expression for the potential surfaces of the quasimoleculeXf A. In a series of papers (Golubkov and Ivanov, 1981,1984a; Ivanov and Golubkov, 1984)a parallel reformulation of the essentials of quantum defect theory has been developed in terms of integral equations. Using a Lippman - Schwinger equation starting from the Coulomb Green’s function, integral equations for the transition matrix have been derived. The connection with the approach of Sections I1 and 111 has not been explicitly determined, and few applications of this approach have been published. This review has concentrated almost excusively on the use ofthe quantum defect method for studying photoabsorption by neutral molecules or electron scattering by positive ions. Closely related work has likewise been accomplished in parallel for the problem of electron scattering by neutrals and also for the photodetachment of molecular negative ions. For example, it was for electron- neutral scattering that the vibrational frame transformation was introduced by Chang and Fano ( 1972).An extensive review by Lane ( 1980)points out many such applications. Chang ( 1984)has recently considered the extension of frame transformation theory to polyatomic molecules. An important difference in the physics of negative ions is the absence of a long-range Coulomb potential between the outermost electron and the core. This fact alone need not invalidate the use of quantum defect theory as described in Section 11, but it implies that the outermost electron moves far more slowly in the vicinity of r = r, than it would in the presence of a charged core. Accordingly the fixed-nuclei approximation may be inappropriate

+

QUANTUM DEFECT THEORY

117

even within r C r,. Some difficultieswith the frame-transformationmethod have in fact been noticed (Momson et al., 1984;Jerjian and Henry, 1985),in numerical studies of near-threshold vibrational excitation of H2by electron impact. Nevertheless the problem of resonant vibrational excitation (e.g., as in electron scattering by N2)seems to be well understood followingthe work of Herzenberg (1968) and of Schneider et al. (1979). Electron scattering by neutral polar molecules has received much attention in recent years (see particularly Norcross and Collins, 1982, and references therein). One conceptual problem is posed by an electron in the field of a polar molecule which was absent when the Coulomb attraction dominates. That is, the r2dipolar potential experienced by a distant electron in the body frame becomes converted into an r4polarization potential once the electron escapes so far that the molecule is left in a definite rotational level. Analytical studies close in spirit to quantum defect theory (Clark, 1977, 1979, 1984; Engelking, 1982)and also some numerical studies (Collins and Norcross, 1978)have sorted out some of the effects of this change of frames, but a comprehensive theory still seems to be lacking. Another area of electron - neutral scattering theory which has been explored in depth is dissociativeattachment (e.g., e- HC1+ H C1-). This is one of the simplest processes in which “electronic” kinetic energy can be converted into nuclear motion, and as such it has fundamental chemical significance. Here much progress has been made by Gauyacq (1983), who treats separately the short- and long-range physics in a formulation closely related to the frame-transformationmethod. Two opposite limiting cases, of energy-independent short-range scattering phase shifts d,(R) and of very strong energy dependences (such as near a resonance, see Bardsley, 1967b; Gauyacq and Herzenberg, 1982)have been treated. Recent progress using an R-matrix-type formulation has also been reported by Launay and Le Dourneuf( 1984).Nonetheless,further work seems desirableto encompass both of the limiting regimes and all intermediate situations as well within a single theoretical framework. The remarkable upshot of all this is the widespread applicability of the quantum defect approach, which has now reached far beyond the original intentions of Seaton (1966) or of Fano ( 1970).It has been proven capable of describing spectral features in problems approaching chemical complexity, and should thus serve as an appropriate interface between experiment and ab initio theory.

+

+

ACKNOWLEDGMENTS We thank Annick Giusti-Suzor for suggestionsand comments on this manuscript.One ofus (C. H.Greene) was supported in part by the National Science Foundation and in part by an Alfred P.Sloan Foundation Fellowship.

118

Chris H . Greene and Ch. Jungen REFERENCES

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL . 21

THEORY OF DIELECTRONIC RECOMBINA TION YUkAP HAHN Department of Physics University of Connecticul Storrs Connecticut

.

1. Introduction . . . . . . . . . . . . . . . 11. Electron-Ion Collision Theory. . . . . .

............ . . . . . . . . . . . . . A . Reaction Channels . . . . . . . . . . . . . . . . . . . . . .

B. Scattering Amplitudes and Cross Sections . . . . . . . . . . . . C. Cascade Effect . . . . . . . . . . . . . . . . . . . . . . . . 111. The Dielectronic Recombination Cross Sections . . . . . . . . . . . A . LiSequence . . . . . . . . . . . . . . . . . . . . . . . . . B. Sill+, S3+, Vz+. and CaZ+ . . . . . . . . . . . . . . . . . . . C. Mg+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Ca+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Other Systems. C+. CI6+.and ArZ+ . . . . . . . . . . . . . . . 1V . The Dielectronic Recombination Rate Coefficients. . . . . . . . . . A . H Sequence. N = I . . . . . . . . . . . . . . . . . . . . . . B . He Sequency. N = 2 . . . . . . . . . . . . . . . . . . . . . . C. Li Sequence. N = 3 . . . . . . . . . . . . . . . . . . . . . . D . Be Sequence. N = 4 . . . . . . . . . . . . . . . . . . . . . . E. Ne Sequence. N = 10 . . . . . . . . . . . . . . . . . . . . . F . Na Sequence. N = I I . . . . . . . . . . . . . . . . . . . . . G . Mg Sequence. N = 12 . . . . . . . . . . . . . . . . . . . . . H . Other Sequences. N = 18. 19 . . . . . . . . . . . . . . . . . . V . Discussion and Summary . . . . . . . . . . . . . . . . . . . . . A . Configuration Interaction and Intermediate Coupling . . . . . . . B. Overlapping Resonances and Interferences. . . . . . . . . . . . C. External Field Effect . . . . . . . . . . . . . . . . . . . . . . D. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Radiative Widths and Coupled Equations . . . . . . . . A . Inner-Shell Electron Transition . . . . . . . . . . . . . . . B . Outer-Shell Electron Transition . . . . . . . . . . . . . . . Appendix B: Auger Probab es A, in LS Coupling . . . . . . . . . A . 2e Systems . . . . . . . . . . . . . . . . . . . . . . . B. 3e Systems . . . . . . . . . . . . . . . . . . . . . . . . C. 4e Systems . . . . . . . . . . . . . . . . . . . . . . . . D. 5e Systems . . . . . . . . . . . . . . . . . . . . . . . . E . Active-Electron vs. Core-Electron Couplings . . . . . . . . . . .

124 128 132 138 144 146 146 149 151 153 156 157 158 159 161 165 167 168 169 170 171 172 173 175 176 178 178 179 180 180 181 182 183 183

123 Copyright 0 1985 by Academic Pms. Inc. All rights of reproduction In any form reserved.

124

Yukap Hahn Appendix C: Radiative ProbabilitiesA, in LS Coupling. . . . . . . . A. 2e Systems . . . . . . . . . . . . . . . . . . . . . . . . B. 3e Systems . . . . . . . . . . . . . . . . . . . . . . . . C. 4e Systems . . . . . . . . . . . . . . . . . . . . . . . . Appendix D: Scaling Properties of A,, A,, a,and aDR. . . . . . . . A. The Z Scaling. . . . . . . . . . . . . . . . . . . . . . B. The nb Scaling . . . . . . . . . . . . . . . . . . . . . . C. The I Dependence. . . . . . . . . . . . . . . . . . . . . Appendix E: Extrapolation to High Rydberg States. . . . . . . . . . A. Extrapolation from Low-n States . . . . . . . . . . . . . . B. Extrapolation by Quantum Defect Theory . . . . . . . . . . C. Dipole Approximation. . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

I84 I84 I85 185 185 186 188 189 189 I90 19 I 192 I94

I. Introduction Dielectronic recombination (DR) is by now a well-known process in astrophysical and laboratory fusion plasmas. Together with the radiative decay of collisionally excited states, DR and direct radiative recombination (RR) are important modes by which high-temperature plasmas cool radiatively. Precise determination of the behavior of high-temperature, low-density plasmas requires an accurate estimate of rate coefficients for these atomic processes for many different ions. In tokamak plasmas, metal atoms (Ni, Cr, Ti, Fe, Mo, . . .) from container walls and current limiters are sputtered into the plasma, where they are stripped to high degrees of ionization. Residual gases (C, N, 0, . . .) also contribute to the impurity population. In addition, some noble gas atoms are introduced into the plasma for diagnostic purposes. These atoms will be stripped of electrons, collisionally excited and/or ionized, and recombine, often with emission of characteristic radiation. The electron temperature and density profiles are determined by measuring Doppler shifts, laser light scattering, and line emission intensities, while the distribution of ionic states may be studied by constructing a set of rate equations which require as input various atomic reaction rates. Of main interest here is the capture process of continuum electrons by ions, which may proceed either by a direct radiative recombination or, more frequently, by DR and its higher-order processes. Since DR was first described by Massey and Bates (1942), much progress has been made both experimentally and theoretically in understanding DR and its implications in various physical phenomena. In particular, availability of high-temperature tokamak plasmas and highly stripped ion beams for

THEORY OF DIELECTRONIC RECOMBINATION

125

collision experiments have allowed systematic studies of these capture and ionization reactions. Paralleling these developments, improved theoretical procedures have been formulatedto estimate the reaction cross sections and rate coefficients. It is the purpose of this article to review recent theoretical progress, with emphasis on the DR process. For a general historical background and astrophysical applications, two lengthy theoretical reviews are available (Seaton and Storey, 1976; Dubuv and Volont5, 1980). Fusion-related work has been summarized by Post et al. (1977), DeMichelis and Mattioli (1981), Drawin (1983), and Griem (1964, 1974). Here we present the theory of electron capture (and ionization) from a general scattering theory point of view, with a coherent summary of the more recent developments. To put the entire field ofelectron -ion collisionsinto proper perspective, we shall review not only the DR process in detail but also some of the related higher-order electron -ion collisions which proceed via resonant intermediate states. As will become clear, many of these processes are mutually related, invariablythrough the resonance phenomena, often with drastic enhancement in their cross sections (Hahn, 1983, 1984). Electrons and ions in a plasma undergo various collision processes through the electron -electron, electron - ion, and ion -ion interactions. The electron -electron interaction is usually very rapid and produces a statistical distribution of electronsin energy. Since the electron -electron interaction is very rapid, depletion of an electron population by inelastic collision and capture will be replenished immediately. The ion - ion interaction is also important, resulting in excitation, ionization, and charge exchange, thus affecting the ionic distribution and transport properties. We direct our discusion here exclusively to the electron - ion interaction, leading to target excitation and ionization, as well as to the capture of incoming electrons. Thus, a set of rate equations may be set up for a specific plasma model (McWhirter, 1965) to describe the behavior of a plasma at electron temperature T,. In the collisional radiative model, for example, it is assumed that the population density nr(Z,s) of an ion with the degree of ionization Z and at a state s varies in time due to collision processes as

Yukap Hahn

126

where p is the collisional ionization rate, X the collisional excitation rate, A, the radiative decay rate, and a the radiative capture rate. Evidently, more elaborate models require other reaction rates as well. The rates are defined (in units of cm3/sec) as thermal averages of the velocity-weighted reaction cross sections. Thus, their evaluation requires accurate cross-section estimates, either experimentally or theoretically. Since ions of high charge states are not readily available experimentally, most of the available data needed for Eq. (1) are obtained theoretically. We note that in general these rate coefficients should depend on the electron and ion densities, n, and n,, respectively. Furthermore, the definition of the rates should reflect the way they are used in Eq. (1); that is, often the (infinite set of) coupled equations [Eq. (l)] are truncated to include only a few low-lying states explicitly. In such cases, the rates should not include those states which appear explicitly in Eq. (1). This adjustment has never been made in practice and may seriously affect the analyses unless a proper set of rates is used. The experimental situation has changed drastically during the past few years, and at present there are several direct measurements of the coefficient aDR (Breton et al., 1978a,b; Brooks et al., 1978;Isler et al., 1982;Bitter et al., 1979)and the cross sections (Belic et al., 1983;Mitchell et al., 1983; Dittner et al., 1983; Williams, 1984; Tanis et al., 1982, 1984). They have provided valuable checks on the theoretical calculations and stimulated much theoretical discussion. For the latest reviews, we refer to Datz ( 1984),Briand et al. (1984b), Dunn et al. (1984a,b), and Dittner et al. (1984a). The electron -ion collision processes of interest are the following: (a) Collisional excitation (X): (,-

+ A'+

_ _ . -

'>-I

+ (Az+)*

Direct cxcitation (DE)

Reson a nt cxcita t io ti ( R E ) )**

(..Il:-ll+

._..._

(..f(:-ll+

)*

+

;I

Dielcctronic recornhination (DR)

(b) Radiative capture (a): e-

+ ,4:+

*

(,'{t:-lH

)*

+ j,

Radiative recornhination (RR)

Dielectronic recornhination (DR) (.A(:-

11'

)**

...___

L'-'

+(.Ti.+)*

Resonant excitation (RE)

THEORY OF DIELECTRONIC RECOMBINATION

(c) Collisional ionization + ,1’+

(p): *

[ > r

127

+

+

e-‘

(,@+I)+)*

Direct ionization (DI)

Auger ionization (AI) (I-

+ (,.I:+)** Resonant excitation double Auger ionization (REDA)

(d) Photoionization (7): 3,

’

+ .,ll>II+

\

1 3

e-

+ A?+

Photoelectric effect (PE)

Photo-Auger ionization (PAI) = (DR)-’

(;p-ll+)**

___.__

‘.

y’

+ (A(;-’)+)*

Raman scattering (Rams)

Note that the photo-Auger ionization in (d) is the inverse of DR described in (b). A pictorial representation of the processes (a)-(d) is given in Fig. 1, which will be useful in formulating a “diagrammatic” method of describing the various processes, to be discussed in the next section; there are many higher-order processes to consider, including the cascade effect, and it is convenient to construct a procedure from which scattering amplitudes may be readily written down. Since theoretical approaches followed by other groups have been reviewed already, we concentrate in this chapter upon the formulation we have developed and applied in our work on the rate and cross-sectioncalculation for the past several years. Furthermore, rather than listing the details of the results obtained by often lengthy computations, we shall try to emphasize the basic theory and various approximations introduced, thus clearly defining the limitations of the approach and suggesting further improvements. Vast amounts of theoretical data have been accumulated on the Auger and radiative transition probabilities, fluorescenceyields, cross sections, and rate coefficients, which are potentially useful in plasma diagnostics and modeling. They will be summarized in a future publication. Those readers who are mainly interested in a summary of the recent experimental and theoretical development may skip Section I1 and proceed to Sections I11 and IV. To read these sections, one needs Eq. (60)for ZDRand Eq. (64) for aDR, both of which contain the excitation-capture probability V, and the fluorescence yield a.

128

Yukap Hahn

FIG.I . Pictorial view of the second-order processes, all of which proceed through the resonant intermediate states d; (a) electron collisional excitation, (b) dielectronic recombination, (c) collisional Auger ionization, and (d) photo-Auger ionization. Note that the process (d) is exactly the inverse of (b).

11. Electron - Ion Collision Theory There have been many theoretical approaches formulated to treat the electron-ion collision processes, with or without the effect of radiation fields. However, a systematic formulation in its full generality was only recently explored (Hahn ef al., 1982a; Hahn and LaGattuta, 1982). We present here the basic derivation of this approach (Gau and Hahn, 1980a) and show various ways by which approximations may be introduced. By systematically developing the theory and stating explicitly the approximations introduced to reduce the theory to a manageable level of complexity, it is possible to assess the region of validity of a given formulation. More importantly, it will facilitate improvements of the theory in the future; different phenomena, such as resonances and cascades, can be brought out more clearly with different expressions deduced from the same general theory. For those intermediate resonance states which involve high Rydberg state (HRS) electrons, the QDT can be applied; it often nicely complements

THEORY OF DIELECTRONIC RECOMBINATION

129

the present approach. Due to the complexity of the many-electron system involved here and also to the often dominant higher-order effects, a simple diagrammatic procedure may be helpful and will be introduced in this Section (Hahn, 1983, 1984). A diagrammatic procedure was proposed recently to facilitate handling of higher-order processes in electron - ion collisions. The main features of the proposed diagrams are: (1) the explicit energy versus time scale describing sequential development and (2) the incorporation of intermediate bound (and resonant) state representation. This allows one to readily construct the corresponding scattering amplitudes, with or without the full cascade effect, and with allowance for the full width in case of unstable states. Thus, the diagrams corresponding to the processes in Fig. 1 are given, together with the scattering amplitudes, in Fig. 2. The notation is as follows: ( 1 ) The vertical scale is for energies, the horizontal scale for time. To be consistent with Fig. 1, the time increases from the left to right. This will result in the time-reversed amplitudes, but with the correct structure. (2) The lenses represent bound states of the target ion. The squares represent various intermediate bound states, many of which are unstable resonance states. These have full widths r arising from coupling to all the available open channels, both Auger and radiative. Obviously, therefore, these diagrams are not simply related to a perturbative series. (3) Thin lines denote occupied electron states; holes created in an intermediate state are expressed as empty spaces between thin lines. The thick line at the bottom of the diagram denotes the nuclear core. This immediately suggests that the present diagrammatic technique can also be extended to describe ion - atom collision processes, in which complicated inner-shell excitation and ionization are possible. (4) The electron -electron interactions are denoted by the vertical dotted lines and I/. Radiation, either emitted or absorbed, is indicated by the wiggly lines, with the electron -photon vertices given by the operator D.

The corresponding matrix elements are self-explanatory. A generalized cascade theory of photocapture was presented earlier in a mathematically consistent way (Gau and Hahn, 1980), using a time-independent scattering theory and by taking properly into account the effect of the radiation field. The theoretical basis for other processes can also be given as special cases of the general amplitude. Here we concentrate on the capture process. The total Hamiltonian for N 1 electrons plus radiation field is given by

H,,

+ = He + Hd + D = H + D

(2) When the projectile electron is singled out as r, in the symmetric H e , we

I

u < i l D If>

f

FIG.2. Diagrammatic representation of the processes described in Figure I , together with their first-order contributions; (a) direct collisional excitation and resonant excitation, (b) radiative recombination and DR, (c) direct collisional ionization and Al, (d)photoelectric effect and PAL In addition, (e) cascade effect on the DR of(b), in which the branching ratio to (e') is to be incorporated, and ( f ) resonant excitation double Auger ionization contribution to (c). The vertical scale is the energy, showing vacancies created during the collision by the absence of electron lines. To make the presence of vacancies transparent, the energy scale is made uniform in (d)-(f). The two lens shapes at the edges denote the bound-state ions, while the squares describe intermediate states with full widths.

THEORY OF DIELECTRONIC RECOMBINATION

131

where

and

The radiation field is described by

with and the interaction between the electrons and radiation field is

for electric multipole radiation, and similarly for the magnetic multipoles. The radiative transition probability is then given by UJ = (2m)lMfi12P/

(6)

where

and the cross section by

when the incident wave is normalized as exp(ipi*ro)(or its Coulomb counterpart to be specified in detail later). The state vectors for the entire system described by H , are the direct product of state vectors for atomic states and photon states. We take the initial state with zero photons, Ii,,,O). Since we do not consider processes in which the electron number changes, it is sufficient that only the radiation field is quantized. In M f i ,the final state IJk) is a

Yukap Hahn

132

+

solution of H = He H , while ~Y,,) = Iimt ,0) = I",) with the correct physical boundary conditions.

is a solution of H,,,

A. REACTION CHANNELS

We partition the space of solutions of Hm,Y,= EY, into subspaces. The wave functions in each subspace describe states containing a particular number of photons, nta. Projection operators can be defined, I,,, = R(O)+ + R(2)+ . . . (8) where R(0)= p + Q, p2 = p = Pt, R(1) R R2 = R = Rt R(2)+ R(3)+ . . S,

Q2

= Q = Qt

.

and where the orthogonality relationship among the projection operators is given by PQ = PR = Q R = 0 and also R(l)R(j)= R(J)6., The component R(O)with zero photons is divided into two parts, P and Q, where P contains those states with only one electron in the continuum, while the Q space covers all the other states with zero, two, three, etc., continuum electrons (which includes the multiple ionization channels as well as the closed channels). In the following, the subspace S , with more than one photon, reached from R(O)by cascade through R(I),will be neglected for simplicity but will be incorporated later. (This will modify RF,R to R F f R = RF,R - RF, GfF, R ) . Thus, the total wave function Y, may be written as Y, = Y,

= PY,+ QY, + R Y ,

(9)

Substitution of Eq. (9) into

( E - H,,,)Y, = 0 F,Y, (10) where H,,, is given by Eq. (2), generates a complete set of equations PF, P Y , = -PF,Q 'P, - PF, R'P, QF,QY, = - QF, P Y , - QF, R Y , RF, R Y , = -RF, P'P, - RF,QY,

( 1 1)

In the following, we assume that the initial boundary condition is such that an incident wave is present only in one of the channels of P Y , . In the photon-initiated processes, a similar formalism can be developed. There are several different ways by which Eq. (1 1) can be studied.

THEORY OF DIELECTRONIC RECOMBINATION

I33

1. Elimination of the R Channels

We start with this option, although elimination of the P or Q channels is also possible. We define the Green's function G R= (RF,R)-' for the ( N I ) electrons and one photon. Then

+

R Y t = - GRFtP Y , - GRF,QYl

(12)

and thus

PFfPYt = - PFPQY, QFf QY, = - QFf P Y , where Ff = F, - F1GP F, .Incidentally, the operator F f has the orthogonality property that

RFf

=F f R =0

(15)

which can be used to generate a set of states orthogonal to the R space without ever explicitly constructing the P and Q operators. Such a procedure was used earlier in the treatment of exchange and rearrangement collisions (Hahn, 1970,1971) in a mathematically consistent way. By setting Gf = Gf - in S(RF,R), where Gf is a Green's function with standing-wave boundary conditions for the electronic part of the states, we have

FP = (F, - F,

cf F t ) + inF, S(RF, R)F,

(16)

The F, Gf Ft part gives rise to a radiative shift of the states in the P and Q spaces, while the F, SF, term provides the radiative damping (broadening). If we denote the various channel projections as P

9

r

(where the index r should not be confused with the electron coordinates), then Eqs. ( 13) and ( 14) may be expanded into a set of coupled equations

and

134

Yukap Hahn

In Eqs. ( 18)and ( 19),we neglected the radiative shift and used the definition

where the r' sum is over all the open channels in R space such that the energy E f l , including the photon energy, is equal to E. (The corresponding electronic states may or may not be open.) In general, we have

r

r'

R(E: - H e - H&)RY& = 0 Thus the radiative vertex functions are defined by (p:

= DR,Y&

(22)

Note that Eq. (2 1) is a set of (infinite) coupled equations for the individual states RrY&, and only those states with E = E: are included in Eq. (20). Furthermore, due to the possible presence of an SIcomponent and also a resonant part in RYf ,the E: states will in general have finite widths. Therefore we may correctly replace S(E - E R )by

S(E - E : )

+

A:

rR(r)/2n ( E - E:)2 rR(r)2/4

+

The explicit expression for the raidative channel potential for specific cases is analyzed in Appendix A. It is seen that the potential can be a constant independent of the space variables in the case of an inner-shell transition, and a correct vertex function is derived here in the case of outer-shell transitions. The connection between Eqs. ( 18)and ( 1 9) and quantum defect theory is also explored in Appendix A. When the coupling to the R component is strong, as in processes involving highly ionized target ions, its effect on PY, and Q Y , can be very large and perhaps lessens the importance ofthe coupling among the channels within the Por Q spaces, as well as the coupling between P and Q through PHeQ and QH, P. Since the set of equations in Eq. ( 1 1) is completely symmetric in P, Q, and R, it is also possible to formally eliminate the Q or Pcomponent from the set and obtain a new set of coupled equations for the remaining two components. We forego such analysis here, except to note that the combination Q R after the elimination ofthe Pspace may be useful in studying in detail the cascade transitions of resonance states.

+

THEORY OF DIELECTRONIC RECOMBINATION

135

2. Elimination of the P and R Channels The set of coupled equations, Eqs. (1 3) and ( I 4), correctly takes into account the radiation channels represented by RY, . However, the coupling between the incoming channel component P Y , and the intermediate resonance channels QY, can be very strong, especially for light ionic targets at low degrees of ionization. To exhibit explicitly the effect of this coupling on resonance structures, it is convenient to formally eliminate the P and R components in the QY, equations. Thus, in analogy to Section II,A, 1, we have from Eq. (1 I),

P Y , = P Y p - GPF,QY, - GPF,RY, (24)

RY,= - G f F ,QY,- GPF,P Y , which in turn gives

P Y , =P Y p~ GpFfQY, (25)

R Y , = - GPpF,P Y , - GPpFrQY, where

G P= (PFP)-', Gf

F =E -H

= F,

+D

= (RF,R)-'

G p = [P(F,- F,GfF,)P]-' = (PFfP)-'

(26)

PFP P Y p = ~0 etc. for other quantities. Substitution ofEq. (26) into the Q Y , equation in Eq. (1 1) then gives

QFf'QY,

= QBf'

(27)

where

F f P = F, - F,GFFf

- F,GPpFP

B;RP= F , ( P Y p ~ GPpRF,P Y p )

(28)

It should be stressed that the treatment of Eq. (27), with the resulting quadratures Eq. (25), is completely equivalent to solving the set of equations ( I 1); i.e., once QY, is determined from Eq. (27), the rest of the functions P Y , and R Y , are obtained immediately from Eq. (25). The operator FRPis convenient to study, because the Q space contains by definition all those channels with zero, two, three, etc., electrons in the continuum, and FRPcontainsnot only the coupling to both the P and R spaces but also their mutual interac-

Yukap Hahn

136

tions. Incidentally, we note that from Eqs. (24)-(26) and from the definitions of all the Green's functions in these equations, it is straightforward to show that RFfPP= PFRP = F;RPR = FFPP = 0 t (29) RBFP= PB;Rp= 0 The orthogonality properties, Eq. (29), were first discussed by Hahn (1970, 197 1 ) and are important in practice when the exchange and rearrangement effects are to be incorporated. Since we are not interested in collisional ionization, the Q space defined as Q = Q(oe)

+ Q(W + QOe) + .

..

p (30) for different numbers of electrons in the continuum (and with no photons) will be approximated by the first term. That is, Q = Q(&)is a closed-channel space with no continuum electrons. The extrapolation to energies above the ionization thresholds in multichannelquantum defect theory (MQDT) corresponds to the transition Q(&)+ Q(2e).Incidentally, Q(") is different from Qqof Eq. (1 7) in that Q? specifically isolates a given electronic configuration, while Q ( @can involve in general an infinite number of such configurations. Now, we analyze the structure of Eq. (29) and FRp,with the approximation of Eq. (30).The operator of QFWQ generates a set of states as Q(Oe),

QFF'Q@d

Q(W

=(E

- gy)Q@d

(31)

which correspond to resonance (autoionization)states when 6; are degenerate with the states in the P space. By setting

G;R

= Cf

- in S(RF,R )

and similarly for the other Green's functions, we then have (Q@dlFfPIQ@d)= E - 62 2 :

E - E B - AZ - A:

+q

4 / 2

+ ir,(4/2

(32)

where

E - E , Q = (Q@dIFtIQ@d) and where the A's are the real-part contributions to FRPand

(33)

THEORY OF DIELECTRONIC RECOMBINATION

137

and

In Eqs. (34) and (35), the PpYpand RrYRfunctions represent specific configurations ( p and r) in the respective spaces as defined by

PF, PyP

=0

RF, RYf

=0

where, as in Eq. (17),

PYf

=

z

.

.

PpYf

P

That is, in general PpYf,for example, is obtained by a set of coupled equations in Eq. (36). In the distorted-wave Born approximation, Eq. (36) is replaced for each Pp or R, component by an equivalent optical potential U, yielding the equations

Pp(PF,P - UpP)PpYf 0 Rr(RF,R - U f ) R , Y f

0

(38)

Finally, we note that in general both the QQd and R,Yf states may have sizable widths. Most of the formalism takes into account only the spread in QQd, and neglects the effect of width in the R space; this can cause appreciable error and should be investigated further. (The cascade effect, to be discussed in Section II,C, is an example.) Note that Eq. (32), with Eqs. (33)-(37), is an approximation in which GpR and GpP in FfPare replaced by Gf andGf ,respectively. Improvements ofthis in which corrections to GP and Gf are included can be derived trivially from GpR and GpP; this will result in modified forms for A$ and I-, due to the coupling of R to P and vice versa. This effect may, of course, be partially incorporated through the cascade effect of Section 1I.C.

3. Elimination of the R and Q Channels In addition to the above cases, we can also try to eliminate the R and Q channels from Eq. ( 1 1). Then we obtain for the P Y , function

PFQRPYt= 0

(39)

138

Yukap Hahn

with and

G F = [Q(F,- F,GfF,)Q]e ( Q F f Q)-' G ~=Q[R(F,- F,GPF,)R]-'

(RFP R)-'

(41)

On the other hand, from Eqs. (13) and (14),we may also obtain QY, = - G F Q F f P Y t

(42)

PFmPY, =0

(43)

and thus with FQR= Ff - FfGPFf

(44) Note that we used the same notation for FQRin Eqs. (43)and (44)as in Eqs. (39)and (40);they are completely equivalent. However, in Eq. (40)both the Q and R channels appear symmetrically in two separate terms, while in Eq. (44)the effect of the R channels is mainly through the operator Ff . Depending on the strength of the couplings (Vand D) in a particular situation, one form may be more convenient than the other. The usual coupled-channel method treats Eqs. (39)and (43)very accurately with D = 0 and F f = F; a large number of channels are often included in P Y t ,and G Qis treated by a set of closed channels by means of pseudostates. For low-energy scattering by light ionic targets, G ~isQ always neglected. However, as the scattering energy increases and/or the target becomes heavier and more highly stripped, the effect of G f can no longer be neglected.

B. SCATTERING AMPLITUDES AND CROSSSECTIONS The DR and RR amplitudes corresponding to the processes (Fig. 2b) can be expressed rigorously as

Mfi = ( ~ ~ I D t ~ ~ i ) (45) where Yt,iis given by Eq. (9)with outgoing wave boundary conditions, and ",-is defined for the final state (f)by ( E - H ) Y j = O e FYf (46) where H = H,, - D, Y,-contains at least one photon, i.e., Y, = R Y p The

THEORY OF DIELECTRONIC RECOMBINATION

139

amplitudes for the other processes in Fig. 2 can also readily be constructed; this will be briefly discussed later. We first analyze the initial-state wave function Yhi.From Eqs. (27) and (32), QYt = SQRPQBP~ ==

x l Q O d ) [ E - E f - A d + ir(d)/2]-'(Q@dIQBfP) (47) d

where

Ad=Ai+Af;

u 4 = T*(4+ T r ( 4 = (QFfpQ)-l

~ Q R P

E

Gro

From Eq. (13) we have

PYt,,= PY? - GPPF;RQYt,i

- GPPFfgQppQBtr (48) where G P = (PFPP)-'. Similarly, from Eq. (24),we can express the RYt,, component of Yt,,in terms of PYtY,,, and QYt,,as = PY?

(49) RYt,,= - GPFtQYLi- GPFtPYt,, Finally, Yfapproximately satisfies Eq. (46), without the Q Y and P Y components, because Yf= RYfand RP = RQ = 0. Thus, Eq. (45) for M j becomes Mfi = ( R ~ f l D l P ~ t + , i () R y ~ l D l ( - ) g ~ ~ Q B t r )

+

3 Mfi" M y (50) where the term ( R Y f ID(RYhi)is neglected, as R = R(')is assumed. (However, for a more general R including R(2),R(3),etc., this term can be important.) In the distorted-wave Born approximation (DWBA), we may set

PYhi Pi$:, -QBfr

QVP$r,

*

Pi(F - Up)PiYr= 0 R Y f = Rf*f

and obtain

MfiRR = (Rf$flDIPi$r) MfiDR= (Rf$f(D9QRpVI Pi$:)

(51)

The first term in Eq. (5 1) is the direct radiative recombination (RR) amplitude, while the second is for the DR process. With the inclusion of ~ Q R Pas

Yukap Hahn

140

defined in Eq. (47), we have

MfiDR

I:(R,@-flDIQod)[E - Ef - Ad + ir(d)/2]-' d

(52) where we have dropped the t subscript in Fsince both the Pand Q spaces are without photons. Refinements of Eq. (52) which incorporate the cascade effect and allow for the energy spread of the states in the R space will be examined in Section II,C. The capture cross section is given by Eq. (7), with pfand Ji as defined there. Thus, neglecting all the higher-order terms, d o , == ?! &IMJR + M D R 2 r - h Ji /iI X

(Q@dIFIPi%)

+ dugR+ dujfR

(53) where dujfRdenotes the interference between MJRand MfiDR,First consider the RR term; for a transition cl, nl, we have in the electric-dipole, nonrelativistic approximation (and in the uncoupled representation) = duJR

-

&,(93(31

1 duRR= dnk 4

( N + ~ ) j ( R ~ Y f ) ( ~ * r o ) ( P i Y 27 ) d(54) rol

where

with Rc,jro)

-'J

=Pc

sin (pp0

z In +ln(2pcro)- 2 + ole+ 6, Pc

and m

R#f

'(A

=

*

r0

R ~ , ( r o ) * ~ ( ~ o ) * ~ ~ (0rR;/ ) ) ,dr, = 1

Obviously, lc = 1 1. Summing over the magnetic quantum number rn and over the emitted photon polarization, we have

THEORY OF DIELECTRONIC RECOMBINATION

141

where (Yt = Ry = 13.6 eV) RlfiE

q,= e2/ h c Detailed discussion of uRRwas given by Hahn and Rule ( 1977) and by Lee and Pratt (1975, 1976). Next, consider the DR cross section in Eq. (53). From Eqs. (6) and (47),

X [ E - E(d)

+ ir(d)/2]-'(Q@dlFlpi@f)12

(56)

+

where E ( d ) = E f A(d). In general, we have the resonance states d with very small widths T ( d )as compared with distances between the resonances. In such cases, an isolated resonance approximation (IRA) may be introduced by taking only the diagonal terms in the sum in Eq. (56). Typically, for ions with low degrees of ionization, ZI 6 5 , we have ra>> r, if the states in Q@d do not contain high Rydberg orbitals. On the other hand, for Z , 3. 10, those components of r, which involve inner-shell radiative transitions can sometimes become very large and thus ra4 r,. Then, there is a chance that the overlap between the resonance states d can be important. In the IRA, we have

After the dQk integration and summing over the polarization and magnetic quantum numbers, we finally obtain

In Eq. ( 5 8 ) we introduced the following quantities Va(i

-

d ) = (gd/2gi)A,(d

+

i),

gd and gi = statistical factors

142

Yukap Hahn

and

T~ = ao/vo= 2.42 X lo-’’

sec

The cross-terms d’ # d i n Eq. (55) can be important and will be discussed further in Section V. Obviously, the capture cross section is appreciable only if the target ion is charged ( Z , a I), and a multichannel quantum defect theory can be applied to examine this problem when high Rydberg states are involved. The Auger and radiative widths are defined by

r,(d) = C A , ( ~ + i’) i’

=~

2 7 4 ( v4 l i y i i t / h

i‘

= CA,(d-.f) Y

(59)

= C2xl(flDld)l2iy/fi

P

Explicit formulas for A, and A, in LScoupling are given in Appendix Band C for 2 - 5-electron systems. In general, there are double infinities of states d which must be summed over in Eqs. (56) and (57), and the i’ andf’ sums in Eq. (59) each contain an infinite number of states. Since the oDR(i + d) are sharply peaked around E = Ed and there are many such peaks within a small energy interval, it is convenient to average them over an energy bin of size Ae,. So long as this bin size is chosen to be much smaller (at least a factor of- 3 - 5 or more) than the actual experimental beam width, such an averaging procedure will contain exactly the same information as the original oDR. Otherwise the choice ofAe, is arbitrary. Thus we define

For actual comparison with experiments, ZDRmust also be folded over the beam profile B(ec),as

I I

ogR(e,)= with

ZDR(ei)B(e: - ec)dei

B(e,) de, = 1

THEORY OF DIELECTRONIC RECOMBINATION

143

Often B assumes a Gaussian shape, in which case the peak value of ZDR is reduced by roughly a factor of 0.8 For plasma diagnostics and modeling purposes, the cross section Eq. ( 5 8 ) is integrated over a Maxwell distribution of the projectile electron velocity and the DR rate is obtained, as

(=a).

aDR( j

+d ) =

(62)

where

and e, 3 fm, vf

= pf/2me. That

is,

which was first derived by Bates and Massey (1942) many years ago. Obare closely related, since they contain essentially the viously, ifDRand aDR same physical information. From Eqs. (60) and (64) we have

Except for the temperature- and ec-dependent factors in the square brackets in Eq. (65), aDR and ZDR are interchangeable. This fact is important in and ZDR.In the followcomparing the theory with experimental data on aDR will be applied to ing Sections I11 and IV, Eq. (60) for ZDRand (64) for aDR various ionic targets. The necessary formulas for A, and A, are summarized in Appendix B and C. Modification of Eqs. (60) and (64), which incorporates the cascade effect, is given in Section I1,C. The theoretical framework presented above for the RR and DR processes is quite general and can readily be applied to other resonant processes of some interest, such as Auger ionization (AI) and photo-Auger ionization (PAI), as described in Fig. 2. Thus, in the IRA, the A1 amplitude is given by

M$' and hence

-

= (fIFGrF(i )

ZA1(i d ) - VF(i

-

d ) [1

- o(d)]

(66)

(67) wherevr is the collisional excitation probability (without capture) and 1 -

Yukap Hahn

144

o is the Auger yield. Equations (66) and (67) are to be compared with Eqs. (5 I ) and (60). Similarly, the PA1 is described by

(fIFGrDli)

(68) which is the inverse of DR, Eq. (51). If we define the frequency-weighted cross section as MJAI =

r

then yPM

-

1 - o(d) ]

(70) where xpA1is the photoexcitation probability. In the electric dipole approximation and in the Bethe approximation, xpA1is related to V, of Eq. (64), while V : is simply an analytic continuation of V, into the positive energy oDR, continuum. Therefore, theoretical and experimental studies of aDR, oA1,and ypA1are mutually complementary. Much experimental and theoretical work has been done on the A1 process (Dolder and Pert, 1976; Hahn, 1977; Crandall, 1982) and also on the PA1 (LaGattuta and Hahn, 1982b; Hahn, 1978), in which the interrelationship between the various reactions is emphasized. XPA"

C. CASCADE EFFECT When the states f in Eqs. (57) and (58), which are reached from the intermediate states d by radiative emission, are stable against Auger emission, the capture process is essentially complete. In some other cases, the states f which contribute to the sum may not be Auger stable, and the fluorescence yield given in Eq. (58) may overestimate the overall capture probability. (There is a possibility that some statesf may decay radiatively several times before an Auger emission becomes allowed. Such a mode must also be incorporated.) Formulation of the cascade effect requires inclusion of the Sspace in Eqs. (8) and (9),which has thus far been completely neglected. We recall that the space S contains all those channels with two or more photons. In order to systematically include these additional channels, we change the notation slightly and write R(O)= P Q = P(O) Q(O) Q(O)', where Q(O)' contains the channels with more than one continuum electron. The superscript (0) indicates that the number of photons is zero. Analogously, we set in general R(r)= pCr) + Q ( r ) + Q(r)' (71)

+

+

+

145

THEORY OF DIELECTRONIC RECOMBINATION

and In the following, we will neglect the 9‘‘)’ components without loss ofgenerality. They are of course important in the theory of collisional ionization, including single- and double-Auger contributions. The next step in the derivation of the cascade amplitude is to note that the = Gro as introduced in Eq. (47) is a crucial part of the Green’s function !i’& DR amplitude MjRin Eq. (50). In particular, we have from Eq. (28) Gro = [QW(Ft- FtGP$)Ff?(” - I;1GPb”Fy)Q(O)]-l

- Gro(Q(0);p@),Q ( 1 ) ) =

(73)

which was obtained by eliminating the Po) and Q ( ’ )channels. The effect of these two channel spaces appears explicitly in Gro through the two Green’s functions in FFP.In fact, it is straightforward algebra to show that Gro = Q@)(l/Ft)Q“J) Q(o)GtQ(o) (74)

+

+

within the limited subspace of P(O) Q(O) Q(l? Therefore, as additional channel spaces are incorporated into Gro of Eq. (73), extra terms will appear in the denominator, which is in turn expanded as Gro + Gro + GrlF,Gro + * * * More rigorously, in analogy to Eq. (74), we have Gt = 1/F, = Gro GrlDGro Gr2DGrlDGro

+

+

+

*

*

*

+ G{

(75)

where

Grr E Grr(QW;p(O, QV+l))

= Q(r)Gr,QW

and G( in Eq. (75) includes many other terms such as

2 O

G: =

x m

(p(r)Gtp(r’)+ p(r)G,Q(r‘)+ Q(r)GtQ(r’>r)

r O r’-0

+ pWG, Q(r‘)’ + QWG, QV’Y + .

.

.)

Note that in Grr only the nearest-neighbor subspaces P(‘) and Q(r+l)are explicitly coupled; this is, of course, an approximation. It is now a simple matter to derive the cascade-corrected w(d). For each Grr we introduce the representation

Grr= zlQ(r)@dt,))[E - I?$,; dtn

+

- Ad(,) ir(dc‘))/2]-’(Qcr)@dt,,l

In the final-state sum overfin Eq. (58), we include only those states which

146

Yukap Hahn

are Auger stable. The new cascade-corrected o(d), to be inserted in oDR of Eq. (58), ZDRof Eq. (60), and aDR of Eq. (64), is given by

where

o(d(d

+ dV+l))

A,(dV) + d(r+U)/r(dW)

and all the dcr)in Eq. (76) are by definition Auger unstable.

111. The Dielectronic Recombination Cross Sections During the past several years, theoretical estimates of aDR and oDRhave been obtained by a number of groups, and, as a result, values for several isoelectronic sequences have become available. In the following, we summarize these data, most of which were obtained in the nonrelativistic HartreeFock approximation, with LS coupling and in the isolated resonance approximation (IRA), as shown in Eq. (57). Possible refinements are discussed in Section V. Discussion of oDRand aDR separately in the following material is purely can be obtained from oDR by Eq. (65) and vice versa, for convenience, as aDR in comparing with the available experimental data. We start with the oDR, which have been calculated only recently, mainly because of the availability of experimental data during the past year and a half. A. LI SEQUENCE

Specifically the reaction of interest here is the Ant = 0 process for the ionic targets B2+, C3+,and 0 5+ (McLaughlin and Hahn, 1982b, 1983a,b). Sill', S13+,and Fe23+with An, # 0 will also be discussed. Thus, for example,

e-

p,l,

+ C3+

--+

i-1~22~

+

(C2+)**+ (C2+)* y d-ls22pnl

f-lsl2snl

and similarly for B2+and 0 5+. Here n, denotes the principal quantum number of the target electron (2s ---* 2p). The data are summarized in Fig. 3, and the formulas for A, and A, are given in Appendixes B and C.

147

THEORY O F DIELECTRONIC RECOMBINATION I

1

1

1

I

I

-

FIG.3. DR cross section ZDR, for C3+,2s 2p, averaged over Ae, = 0.0 I Ry. As compared with Figure 5 , the large-n contribution near the threshold e, = 0.6 Ry dominates. The peaks in the region 0.01 -0.03 Ry persist, with some uncertainty. They correspond to 2p4d ' F a n d )P, and 2p4f 3D and 3G of C2+.The dotted bar is for 28 c n 67; the hatched, for 68 c n. (From McLaughlin and Hahn, 1982b.)

For the An, = 0 excitation -capture process considered here, with large n, the resonance states decay predominantly by radiation emission ( 2 p + 2s) with the corresponding A: being independent of n. The resonance condition, e, = (e2, - e2,) en,3 0, gives the minimum value of n = no, for which e, = 0. Depending on the size of the excitation energy (e2p- ez,),no can be as small as 2 (C3+,05+,B2+) or as large as 13 ( Fe23+).Of course, all n with n 3 no are allowed. As discussed fully in Appendix D, A , and A , for the Ant = 0 mode scale with the effective charge Z as

+

-

A: - Z , A, Z (77) in the limit of Z - CQ. However, for n =S 100, we found that A , >> A , (or r, B r,)and thus w - n-3 ez 1. From Eq. (60), we have, with Ae, 2,

-

which is independent of A , and also nearly independent of n, except for the where L = 2 L 1. To compare with experiment, statistical factor g, = k?,

+

148

Yukap Hahn

we define a partial sum S=

C ZDR(i-

d = 2pnf)

(79)

n-nm

where n , is the minimum n value contained in the highest bin and nFis the cutoff value determined by a beam-analyzing field F; i.e., nF Z3/4F-1/4. (In general, there should be an f dependence of n,in such a way that higher field strengths are required to ionize an electron w t h higher 1. That is, for a given F, nF will be smaller if f is larger. For details, see Section V.) Therefore,

-

-

d = 2pn, f ) n ,

-Z 1 I 4

(80) For capture into states of larger f (I b 7), A, can become comparable to A, so that aDR is no longer independent ofA,. For light ions with Z d 15, explicit calculation shows that, with a simple choice of Aec 2 I , S behaves as - 2' for low 2 and Z-II4 for high 2. Experimental results (with merged beams) of Dittner et af.(1 983) are compared with the theory in Table I and Fig. 4. An attempt was made to examine uDRfor heavier ions, in particular Fe23+.The 2 p -+ 2s (An, = 0 ) initial excitation accompanied by the capture of the incoming electron into a high Rydberg state (n = nb 3 12) was studied earlier (Jacobs et a f . , 1976) with and without the effect of external field. The outstanding feature of this ion, as compared with the light ions, is that the S = ZDR(i

-

-

TABLE I SUMS OVER THE DIELECTRONIC RECOMBINATION CROSS SECTIONS, FOR THE TARGET IONS B2+,

c3+, AND 0 "'

z = z, fl=(-

Z3")

nm

SNIY S/&hy (C 3+) S/Ssd, (C +) S/SE& 3+) S;icU/Skald(C

'

2 22 6 1.8 0.56 0.67 0.44

- 0.5

3 30 8 3.2 1.00 1.00 1.00 1.0

5 44 13 4.9 1.53 1.67 1.8 -2

* S is defined by Eq. (79) (Mchughlin and Hahn, 1983b). Sums are compared with the experimental data of Dittner ef al. ( 1 983). The scaling of S in Z is taken to be S Z . More recent data (Dittner et al., 1984b)indicate that the field effect is large, but theirs ratios are unchanged.

-

THEORY OF DIELECTRONIC RECOMBINATION

149

3r-----l

t

N

E V

a

n

b

FIG.4. Experimental DR cross sections for B2+(0)and C3+(0)by Dittner ef al. (1983). The :oretical result is obtained from ZDR by folding over a beam profile of width 3 eV. The iximum n values included are n,(B2+) = 22 and n,(C3+) = 30. The theoretical curve ( - - -) C3+is multiplied by a factor of 1.5; this discrepancy may be due to a field enhancement, is not adjusted. Pradhan ( 1984)obtained resultssimilar to this. More ile the B2+curve (-) :ent data (Dittner ef ul., 1984b)show similar cross sections, but enhanced in magnitude by a :tor of 3 to 4, indicating a strong field effect.

mtribution from the low-n states is very large and becomes small rapidly as increases. This is in strong contrast with the light-ion case where ZDRas a nction of n is nearly constant. As will be discussed further in the next ction, this feature is not altered when external electric field F is added in the teraction region.

B. SI I * + , S3+, Vz+, and CaZ+ The Li-like Si, S, and V ions (McLaughlin et al., 1985a)were also studied bth experimentally and theoretically. In particular, isonuclear sequences :re successfully compared. The reaction of interest here is the one that involves An, # 0 transitions

Yirkap Hahn

150

and for which an experimental study was carried out (Tanis et al., 1982, 1984; Clark et al., 1985). Explicitly

where we have I q + = Sill+, S3+, V19,20*21+; Ca16J7J8+ and T = He, Ne, and Ar. The cross section for the Sill+ ion is given in Fig. 5 , as calculated by McLaughlin and Hahn ( 198 I), which may be compared with Fig. 4. The relationship between the above reaction (resonant transfer excitation, RTE) and DR comes from the fact that the electrons in the target atom T assume the role of projectile electrons in the DR process, exciting the 1s electrons in the incoming ion I as they are captured into 1q-l states (Brandt, 1983). The binding effect of the electrons in Tis included by foldingthe DR cross section over the momentum distribution of the electrons in T. Obviously there are I

I

I

I

1

3

A

N

E u

R I

-

'0 Y

a

n

Ib 1

C

ec ( Ry 1 FIG.5. The energy-averaged DR cross section ZDRfor the Sill+ target, where the I selectron is excited during the initial stage. The bin size is Aec = I Ry. The peaks at low e, correspond to the intermediate states with low principal quantum numbers n = 2. aDR for SI3+isalso given by McLaughlin and Hahn (1981).

THEORY OF DIELECTRONIC RECOMBINATION

151

some uncertainties as to the effective number of electrons participating in RTE, as well as the possibility of more than one electron participating simultaneously in a reaction involving single ion ZQ+. Comparison of such analyses with experimental data indicated that indeed RTE and DR are related (Brandt, 1983).Agreement in the peak values is obtained overall to within a factor of two. Conclusive evidence was obtained more recently by a study of RTE, not in an isoelectronic sequence, but in the isonuclear sequence for Vq+ (and Caz+) with q = 19,20,21 corresponding to the Be-like, Li-like, and He-like ions. Insofar as the theoretical result for the DR cross sections used in this study, the B-like and Li-like data were obtained earlier. The Be sequence, with Is electron excitation, is more complicated, however, due to the presence of more than one Auger channels which contribute to r,.Explicitly,

+ ls22s2 p:1: + ls22s2p pJ

p:l: pyl:

7

F== ls2s22pnl-

1s22s2nl+ yK

\ ls2s2n’l’+ y

(81)

+ ls22snl + ls22pnl

The cross section consequently decreasessignificantly as q decreases; i.e., the number of electrons in the target increases. Table I1 contains the DR cross section ratios for V9+with Ae, = 30 Ry, which was calculated without configuration mixing. To eliminate some of the experimental uncertainties, the following procedure was adopted. Invariably, the relevant DR process here involves initial excitation of one of the 1s electrons into either the n = 2 shell or n 3 3. These two processes are fairly well separated in energy and thus grouped into peaks A and B. With three different ions, we can therefore obtain six peak ratios which are presumably less sensitive to some of the expenmental uncertainties. As shown in Table 11,the agreement between the theory and experiment is excellent. Even stronger evidence was obtained in the CaQ+case. C. MG+ The Mg+ ion has been the most studied target for the DR process during the past year and a half, mainly because the cross section was large and excellent experimental data became available (Belic et al., 1983). The reaction of interest is (3s + 3p) e-

+ Mg+

-

(Mgo+)**

+

(Mgo+)* y

(83)

The theoretical study was carried out by LaGattuta and Hahn (1982a,

152

Yukap Hahn TABLE I1 THEORETICAL CALCULATION"OF THE PEAK VALUE RATIOSFOR THE TARGET IONS V19+.20+.21+ Ratio

Theory (DR)

Experiment (RTE)b

~~

~~

~

McLaughlan ef a/. (1985a). * Compared with the experiment of Tanis ef a/. (1984). The two peaks A and B correspond to the transitions ls22s+p,lc-r ls2s2p2, ls2s22p and ls2s2pn/, 1 s2s2n/withn 2 3. They are separated by a large difference in excitation energies. Presumably, some of the uncertainties in the analysis of the resonant-transfer-excitationprocess in terms of the basic dielectronicrecombination mode are eliminated in these ratios.

-

1983b). The excitation energy Ed - Ei = 0.323 Ry and Ar(3p 3s) = 2.80 X lo8 sec-'. All the other Ar(3pnl .--* 3pn'l') are small. Individually they do not scale in n like n-3, especially when n' is close to n, but the sum Z,,,<, Ar(nf + n'l') scales nearly as n-3. As in the B2+and C3+cases, the n dependence of ZDR(i = 3s + d = 3pnl) is nearly flat for each f until n reaches n, = 200, beyond which it starts to decay as r3. Therefore the overall behavior of ZDR for n d n, is similar to Eq. (78); only one Auger channel contributes to r, so that V,/T = VJr, = gd/2gi,independent of the A,, Of course this is an approximation, and the cross section decreases slowly as n increases. The actual calculation incorporates this slow change by explicitly evaluating A, and A,, including all the n-dependent radiative transitions. In the experiment of Belic et af.(1983), however, the n contribution was cut off at nF = 64 by the beam-analyzing field, which was placed outside the interaction region. This reduces the net total cross section by a factor of four from the full value with n 00. The theoretical cross section, with n d n,and folded over the experimental beam profile, was compared with the experimental data. Agreement was very poor; the theoretical prediction gave a cross section which was smaller than the experimental peak by a factor of six, although the full cross section without the n, cutoff gives a closer agreement. Several later calculations +

THEORY OF DIELECTRONIC RECOMBINATION

e,

+

I53

(eV)

FIG.6. DR cross section for thee- Mg+system is compared with the experimental data of Belic et al. ( 1983).The dashed curve is obtained without the external field effect, while the solid curve incorporates the Stark mixing by the Lorentz field of 24 V/cm in the interaction region 3p,,J reduces this peak by (LaGattuta and Hahn, 1983a). The fine structure effect (3p,,, about 20%. +

(Pradhan, 1984; Pindzola et al., 1984; Geltman, 1985) also confirm this discrepancy (Fig. 6). One possible explanation of this discrepancy was suggested ( LaGattuta and Hahn, 1983a) in terms of the effect of an external electric field. In the crossed-beam experiment, a magnetic field of 200 gauss was imposed along the electron beam direction to focus the beam. This translated in the ion rest frame to a Lorentz electric field ofabout 24 V/cm. A preliminary calculation indicated that the DR cross section may be enhanced by the presence of this field in the interaction region by as much as a factor of 8- 10, which is sufficient to bring the theory into agreement with the experimental data. However, there are postcollision perturbations which may further complicate the situation, and further research is required to clarify this problem. The effect of fields on the DR process will be discussed again in Section V.

The process of interest here is again a An, = 0 transition, but now involves the outer 4s electron in Ca+,

e-

pclc

+ Ca+ i-4s

(Cao+)** 4 (Caw)* d -4pnl f-4snl. ...

+y

(84)

Yukup Huhn

154

Explicit referenceto the core electrons ofCa+is omitted for simplicity.Other modes of collisional excitation-capture are also important, in which the 4s electron may be excited to the 3d state or to other higher states, or excitation of the core electrons (3p, 3s, etc.) to higher states as well. Except for the 4s 3d excitation, these modes require higher threshold energies than the 4s -,4p case above, and thus we omit them from our consideration here by limiting the range of incident electron energy to e, < 0.25 Ry. The 4s 3d mode is also omitted for the simple reason that its principal decay mode after the excitation-capture is by Auger emission;thus its contribution to the DR cross section is small. Nevertheless, the fact that the 4p- 3dAuger channel is open has a drastic effect on the DR cross section for the mode (4s 4p); that is, the intermediate states d = 4pnl have for n > 5 two Auger channels available to which they can decay

-

-

-

I-, ( d )= A , (d

-

i = i; = 4sp, 1, )

+ A , (d

---*

is = 3dpf 1;)

We have found that

A,(d- ii) a A , ( d + i), = IOA,(d+i),

1Z 1

I= 1

(85)

thus the contribution for the 1 = 1 states to oDRis greatly suppressed (by a factor of lo), while all the other cases are reduced by a factor of 2 to 3 due to the presence of this additional channel is. For n B 8, we found that both A,(d --., i) and r, scale as n-3 for each 1. On the other hand, r, will have a piece A,(d = 4pnl -fi = 4snl) and a small part A,(d = A= 3dnl), both of which are essentially independent of n. The remaining part of r, approximately scales as r3. Thus for n S n, = 300, where A,(d +f:) = r,, we have r, z+ Tr and

+

+

Here V, = (&/&)A, is used with the statistical factors gd = (2L 1)(2S 1) and gi = 2. As expected, ZDRgiven above is nearly independent of n;that is, ZDRis constant for all n G n,. As n > n,, A, = r, z+ r, so that w -,1 and

This part of the contribution is generally small. We also note that this is the region where A, produces large enough widths to cause overlaps among Rydberg resonance states. The energy-averaged cross section ZDR for a Gaussian beam profile of width 0.3 eV is given in Fig. 7. The general structure is quite similar to that

155

THEORY OF DIELECTRONIC RECOMBINATION I

I

1

1

I

e,

I

(eV)

+

FIG.7. DR cross section for the e- Ca+ system, n 80, is compared with the experimental data of Williams (1984). The solid curve includes the contribution from n n, = 80, but without the external field effect, F = 0. The dashed curve is obtained by incorporating the Stark by an mixing due to a stray field of 0.5 volts/cm (Nasser and Hahn, 1984) and multiplying oDR arbitrary factor of 2. Therefore, there still exists a discrepancy of a factor of -2-3.

obtained for the Mg+ and B2+ cases. The 1 dependence is given in Fig. 8, where ZDRis summed for n d 80 = n,for each 1. The drastic reduction in the 1 = 1 contribution caused by a sudden drop in A,(d + ii = i) due to cancellation in the matrix element is shown. The contribution from 1 Z= 8 is found to be negligible. Now we compare our result with the recent experiment (Williams, 1984) camed out by a crossed-beam technique. Using an analyzing-field cutoff at nF = 80 and Aec = 0.3 eV, we obtain (Nasser and Hahn, 1984) the cross-section peak value of

-

nF

S,

=

2 ZDR(i nl

d = 4pnl) = 2.3 X

cm2

d = 4pnl) = 1.8 X lo-’’

cm2

while m

S=

2 ZDR(i

+

nl

Note the drastic reduction in Swith the n,cutof€, a factor of 8; this is in sharp contrast to the e Mg+ system, where the reduction with n, = 64 was a factor of 4. Typically, the folding of Sover a beam profile of width 0.3 eV reduces the

+

156

Yukap Hahn

i

b

I

\ , ' '\

0 0

i I

I

2

I

I

I

4

I

6

I

I

I

a

.f

+

FIG.8. The 1 dependenceof Z, ZDR(nl)for n < nF= 80, e- Ca+,Aec = 0.01 Ry, is given for the process (i) = 4s pcIc+ ( d )= 4 pnl, An, = 0. The large dip at I = 1 is due to an accidental cancellation in the matrix element for A,(d i ) and also large A:(d i' = 3d p : / i ) .

+

+

+

+

above values by a factor of roughly 0.7 f 0.1; we have (SF)= 1.6 X lo-'* cm2, which is to be compared with the experimental value (aDR)= 1.8 X 1O-I' cm2, which is a factor of 12 higher than the theoretical prediction. As will be discussed in Section V a stray electric field in the interaction region of about 0.3 - 3 V/cm can partially account for this discrepancy (Nasser el af., 1985).

-

E. OTHERSYSTEMS, C+,C L ~ +AND , AR=+ In addition to the systems discussed above, some work has been done on C+(2s22p3 2s2p2,An, = 0) by Mitchell et al. (1983) using a merged-beam apparatus. The theoretical cross section (LaGattuta and Hahn, 1983a) was roughly a factor of 10 smaller than the experimental data; presumably, a stray field of a few volts per centimeter may account for this discrepancy. The e- C16+was also investigated very recently by the Oak Ridge group (Dittner et af., 1984) and theoretically by Pradhan (1984). The process involves the 3s + 3p, An, = 0 transition, and agreement with experiment is good. Apparently, higher charge states (2, > 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

THEORY OF DIELECTRONIC RECOMBINATION

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.

THEORY OF DIELECTRONIC RECOMBINATION

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.

-

+

-

+

-

THEORY OF DIELECTRONIC RECOMBINATION

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

THEORY OF DIELECTRONIC RECOMBINATION

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).

-

+

THEORY OF DIELECTRONIC RECOMBINATION

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

THEORY OF DIELECTRONIC RECOMBINATION

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

THEORY OF DIELECTRONIC RECOMBINATION

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.

THEORY OF DIELECTRONIC RECOMBINATION

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

THEORY OF DIELECTRONIC RECOMBINATION

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)

THEORY OF DIELECTRONIC RECOMBINATION

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))

THEORY OF DIELECTRONIC RECOMBINATION

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

THEORY OF DIELECTRONIC RECOMBINATION

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 )

THEORY OF DIELECTRONIC RECOMBINATION

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, << A,. Incidentally, it is interesting to compare the behavior of A, in Eq. ( 127) with the corresponding A, for inelastic scattering A:(&--* sc)

- [l2- Z - 2

( 129)

187

THEORY OF DIELECTRONIC RECOMBINATION

Equation (129) shows that the collisional excitation to a continuum c" (i.e., ionization) is weaker by a factor of Z-2 as compared to an excitation to a bound orbital b, and this is the main reason why for high-2 targets collisional excitations dominate over ionization. Of course, the sum over the continuum states (with the density - Z2)restores the Zodependence and the sum rule. The scaling properties Eqs. ( 127)and (128) are usually badly broken when the degree of ionization 2,for the target ion is small (Z,G 3). However, for Z, >z Zc/2, where Zc is the nuclear core charge, a judicious choice of the effective charge Z can make the scaling law effective. For example, Z = (Zc 2,)/2 works well for A,. The dependence of the fluorescence yield o on Z is more complicated as w = r,/rwith r = r, r,.Thus

+

+

Zo

W -

- Z4 in turn gives

r, r, K r,

for Tr >>

and

for

and w << 1

W =

1

aDR Z-3

-

for o -- 1 (high-2 ions)

- Z1

for o << 1 (low-Zions)

(131)

where we assumed that the temperature factor kBTeischosen to scale as Z2; the factor exp(-e,/k,T,) is then effectively 2 independent since e, Z2 according to the resonance condition e, es= e, eb. The energy-bin-averaged cross section ZDR behaves in the two extreme limits as (with Ae, Z2)

+

+

-

-

ZDR- Z-4 -Zo

for o = 1 for o << 1

2. An, = 0 Case

By definition, the sublevels of different 1:s for given n, are nondegenerate in this case due to non-Coulombic screening by the core electrons, so that the scaling in 2 is broken. Therefore, it is in general difficult to give a simple behavior as in Case 1. But, noting that often

-

Ae = enolo - enal, 2

we have e, = Ae

+ eb- Z if nb >> 1 so that eb= 0. Then v/, - Z3I4and A,

- Z'I2

(1 33)

A,

- 2'

(1 34)

and

Yukap Hahn

188

These are quite distinct from Eqs. (127) and (128). Deviations from Eqs. ( 133) and ( 134) occur when nb becomes small, so that eb is no longer negligible. Explicit numerical calculations indicate that the behavior of Eq. (1 33) for A, is approached for high Z, but at low Z (610) A, ZPb,where &, = 2 for low lb and Pb = 6 for high lb b 6, and Pb 4 for Z t 15 and for all 1,. If we assume the simple Z dependence of Eqs. ( 133) and ( 134), however, then

-

o-Zo

for o=l

- Z1J2 for and, with the choice k,T, - Z, we have Similarly, with Aec

o e~ 1

-

( 1 35)

- Z,

B. THEnb SCALING Initial capture of the continuum electrons by the ionic target, accompanied by Ana = 0 excitation (a + s),often results in an intermediate state din which one of the electrons is placed in a high Rydberg state (nbb = n!). Evaluation of the corresponding A, and A, is difficult, and there are basically two methods which are available in practice. One method is to employ the quantum defect theory and evaluate the A’s at energies eb b 0 above the threshold ( n = 00). An extrapolation across this threshold to the desired n then gives the desired probabilities (see Appendix E). Another approach is to extrapolate from the lower n. First, explicit calculation is camed out for the A’s using the distorted wave functions, up to the values n = 21, where the n scaling usually sets in. It is then a simple matter to extrapolate to higher n once the appropriate scaling properties are known. For all the orbitals involved in A, and A,, we assume that the shape of the wave functions in the overlap regions of interest is unchanged, except for the bound-state normalization n-3/2.The 1 dependence of this assumption is obvious; for large 1, the wave function is pulled in more and consequently

THEORY OF DIELECTRONIC RECOMBINATION

--

189

affected more by the screening, unless n is higher. Thus, we have as nb + m

sc)

1/n:

A,(b -+ t )

l/ni

A,(ab and

(1 38)

(1 39) When the b-state electron is not directly involved in the transitions, however, we have instead

-

sbc) + n:

-1

(140)

A:(ab -,tb) + nj

-1

(141)

A:(&

and similarly

-

:

Hence, at very large n = n b , A and A : are expected to dominate and o(n,, A / ( A A :). Examples are found in Sections III,A,C, and D.

nb + m)

+

C. THE1 DEPENDENCE The 1dependence of A, and A,, thus of aDR and aDR, is more complicated and often shows irregular fluctuations due to accidental cancellations in the radial matrix elements. It was found generally (Gau and Hahn, 1978) that the 1dependence is approximately Gaussian in form at low 1, exponential for intermediate 1, and a steep I-' for 1 d n. The usual inverse power dependence of the form I-" with u = 3,4, etc., tends to badly overestimate the quantity at large 1. We have found that the main contribution to aEFcomes from 1 d 8 for low-2 ions and 1 % 15 for high-2 cases.

Appendix E: Extrapolation to High Rydberg States For all Anf = 0 transitions in the DR process, and also for some Anf # 0 transitions involving M- and higher-shell electrons, the initial excitation energy is small so that one of the electrons in the intermediate states d must be in a high Rydberg state (HRS), in order to satisfy the resonance condition E, = Ed. Often important contributions come from the states with n d n, = 300 and 1 d 12, depending on the relative magnitudes of I-, and r, in o.We describe below three different methods which have been used to estimate the contribution from such HRS. We use the Li-like target systems of Section II1,A as examples (LaGattuta and Hahn, 1981b).

I90

Yiikup Huhn

A. EXTRAPOLATION FROM LOW-n STATES Typically, states with n b 21 and f Q 7 begin to scale in n, as described in Appendix D. We parametrize the widths as

- a + b/n3 r,(d= nala, nf) -f + g/n3 T,(d = nofa, nf)

where the constants u, b, 1; and g are determined by fitting the explicitly calculated data at lower n 6 21. The constants a andfcome from transitions of the types in Eqs. (140) and (141), while b and g are for the types in Eqs. ( 138) and ( 139). The fluorescence yield w is then parametrized as

+

In general, (b g)/u is very large, so that w will increase like n3 until n reaches the value n n, = [(b g)/(a +f)l1l3. For n 3 n,, w starts to level off and approaches a constant a/(a +f).For light ions and low degrees of ionization, n,can be as large as 500, e.g., Ca+, while for heavier ions and high degrees of ionization n, can be as small as 10, e.g., FeZ3+.When Eq. ( 143) is put into the cross-section formula Eq. (60) and allowing for the behavior V, n-', we have

+

i=

-

ZDR(i

-

d)

- constant,

for n

- n-3,

for n 2 n,

6 n,

(1 44)

It is important to note that a direct extrapolation of ZDRand aDR to large n can easily lead to erroneous results, often grossly underestimating high-n contribution. In the evaluation of total DR rates, the sum over all the allowed intermediate states d has to be performed. Insofar as the HRS are concerned, the exponential factor in Eq. (64) is not sentitive to (nf), so that we may set

With d = nalanlin the Li-like ions for example, we limit our sum over d to a sum over nf, and obtain

191

THEORY OF DIELECTRONIC RECOMBINATION

where Aa(d- i) mula

- b,/n3 is used. For n,

3

15, we have a convenient for-

where

,[, (

FA(xm)= - -In 3 2

X&--X,+l (xm+1)2

) + T f- if n itan-l(

2xm fi- 1

)]

When x, << 1, FA(xm) = 1.21 - xmis a good approximation.

B. EXTRAPOLATION BY QUANTUM DEFECTTHEORY The quantum defect theory (QDT) as developed by Hamm (1955) and Seaton ( 1983) is based on the analyticity of the pure Coulombic amplitude and wave functions in the energy variable. The spectrum generated by the long-range Coulomb potential is countably infinite below the ionization threshold, with an accumulation point at the threshold energy, and a continuum starting at this same point. With proper normalization of the bound and continuum wave functions, it was shown to be possible to continue the various amplitudes across the threshold. Thus, if all the continuum functions involved in A, and A , are normalized as in Eq. (54), with a factor ( 2 / ~ p f ) ' /then ~,

In Eq. ( 148),n-3 is often replaced by ( r ~ * ) - for ~ low-lcases, where n* = n - p with the quantum defect p defined in terms of the phase shift 6 asp = @a.In practice, several values of ef are chosen which lie close to the threshold, and the corresponding continuum wave functions are calculated; A, and A, are

Yukap Hahn

I92

evaluated using these functions. They are in general smooth functions of e:, so that continuation below the threshold is straightforward. There are some exceptions, however, as in the Ca+ case. Due to strong screening, extrapolation to very low n states is not reliable; the method described earlier in Section A nicely complements the QDT method, and provides a useful check.

C. DIPOLEAPPROXIMATION

The third alternative in the evaluation of A, for HRS is to simplify the radial matrix element by (b = nl) a

dr, ~:va(~*)v&lh It was shown (Gau and Hahn, 1978)that Eq. ( 149)should be a good approximation for 13 3. Now the integral involving the r2 potential can be converted to a standard dipole integral, as

where K, = (e,/%))Lf2/Z =p , / Z

g(nl;K,lc)

7

l-

z2 n

dr r3~1b(Wc

The function g(nl;K,I,) was tabulated by Burgess (1964a) and can be expressed in terms of hypergeometric functions. Equations ( 149)and ( 150) are especially useful for high n and high 1. The behavior of Rkis illustrated in Fig. 15; the approximation in Eq. (150) seems valid for 1 3:4.

193

THEORY OF DIELECTRONIC RECOMBINATION

10

1c

0

FIG.15. Validity of the Bethe approximation in the evaluation ofA, for high Rydberg states is examined (Gau and Hahn, 1978). The transition potential V(r) for the system e- Mo3*+ (2s2 p,l, 2s2pnl) is plotted and compared with the asymptotic form V o r 2 For . r b 0.3a0, Vis asymptotic. The integral 1 6 v., ypc,cVdr’ is plotted for the cases 1 = 0 and 4.

+

+

+

ACKNOWLEDGMENTS The theoretical work summarized in this review was carried out in close collaboration with Dr. J. Gau, Dr. K. LaGattuta, Dr. D. McLaughlin, andwithI. Nasser, M. Dube, N. Shkolnik, J. Retter, and R. Luddy. Numerous discussions with many experimental and theoretical colleaguescontributed to the progresswe have made during the past several years. In particular, we benefited greatly from the discussionsand correspondence with Drs. M. Bitter, C. Bottcher, R. Cowan, D. Crandall, C. Crume, S. Datz, P. Dittner, G. Dunn, D. Griffin, R. Isler, R. McFarland, P. Miller, M. Pindzola, R. Hulse,D. Post, and J. Tanis. The research reported here has been supported by a DOE grant AC0276ET53035.

I94

Yttkap Hahn REFERENCES

Bates, D. R., and Massey, H. S. W. ( 1943). Phil. Trans. Roy. SOC.A239, 269. Belic, D. S., Dunn, G. H., Morgan, T. J., Muller, D. W., andTimmer, C. (1983). Phys. Rev. Lett. 50,339-342. Bell, R. H., and Seaton, M. J. (1985). J. Phys. B. Bely-Dubau. F., Dubau, J., Faucher, P., Gabriel, A. H., Loulergue, M., Steenman-Clark, L., Volonte, S., Antonucci, E., and Rapley, C. G. (1982a). Mon. Not. R. Astron. Soc. 201, 1159-69. Bely-Dubau, F., Faucher, P., Steenman-Clark, L., Bitter, M., von Goeler, S., Hill, K. W., Camhy-Val, C., and Dubau, J. (1982b). Phys. Rev. A 26, 3459-2469. Bitter, M., Hill, K. W., SanthofF, N. R., Efthimion, P. C., Meservey, E., Roney, W., von Goeler, S., Horton, R., Goldman, M., and Stodiek, W. (1979). Phys. Rev. Lett. 43, 129- 132. Breton, C., DeMichelis, C., Finkenthal, M., and Mattioli, M. (1978a). Phys. Rev. Lett. 41, 110-3. Breton, C., DeMichelis, C., Finkenthal, M.. and Mattioli, M. (1978b). J. Quant. Spectrosc. Radial. Transfer 19, 367-379. Briand, J. P., Charles, P., Arianer, J., Laurent, H., Goldstein, C., Dubau, J., Loulergue, M., and Bely-Dubau, F. ( I984a). Phys. Rev. Lett. 52,6 I7 -620. Briand, J. P., Charles, P., Arianer, J., Laurent, H., Goldstein, C., Dubau, J., Loulergue, N., and Bely-Dubau, F. ( 1984b).In “Electronic and Atomic Collisions” (J. Eichler ef al., eds.), pp. 795 - 800. Elsevier, Amsterdam. Brooks, R. L., Datla, R. U., and Griem, H. R. (1978). Phys. Rev. Left. 41, 107- 109. Burgess, A. (1964a). Mem. R . Astron. SOC.69, I . Burgess, A. (1964b). Astrophys. J. 139, 776-780. Burgess, A. (1965). Astrophys. J. 141, 1588- 1590. Burgess, A., and Tworkowski, A. S . (1976). Astrophys. J. 205, L105-LI07. Cheng, K. T., Kim, Y.K., and Desclaux, J. P. (1979). At. Data Nucl. Data Tables 24, I I I . Clark, M., Brandt, D., Swenson, J. K., andShafroth, S. M. (1985). Phys. Rev. Lett. 54,544-546. Cowan, R. D. (198 I). “The Theory ofAtomic Structure and Spectra.” Univ. ofCalifornia Press, Berkeley. Crandall, D. H. (1982). ORNL Report ORNL/TM-8453. Damburg, R. J., and Kolosov, V. V. (1979). J. Phys. B 12,2637. Datz, S. ( I 984). In “Electronic and Atomic Collisions” (J. Eichler, J. Hertel, and V. Stolterfort, eds.), pp. 795 -800. Elsevier, Amsterdam. DeMichelis, C., and Mattioli, M. (1981). Nucl. Fusion 21, 677-754. Dittner, P. F., Datz, S., Miller, P. D., Moak, C. D., Stelson, P. H., Bottcher, C., Dress, W. B., Alton, G . D., and Neskovic, N. (1983). Phys. Rev. Leu. 51, 31 -34. Dittner, P. F., Datz, S., Miller, P. D., Moak, C. D., Stelson, P. H., Bottcher, C., Dress, W. B., Alton, G. D., and Neskovic, N. ( I984a). In “Electronic and Atomic Collisions” ( J . Eichler et al., eds.), pp. 819-825. Elsevier, Amsterdam. Dittner, P. F., Datz, S., Miller, P. D., Moak, C. D., Stelson, P. H., Bottcher, C., Dress, W. B., Alton, G. D., and Neskovic, N. (1984b). In preparation. Dolder, K. T., and Pert, B. ( 1976). Rep. Prog. Phys. 39,693-749. Drawin, H. W. (1983). At. Mol. Process. Control. Thermonucl. Fusion, pp. 19-30. Dubau, J., and Volonte, S. (1980). Rep. Prog. Phys. 43, 199-251. Dubau, J., Gabriel, A. H., Loulergue, M., Steenman-Clark, L., and Volonte, S . (1981). Mon. Not. E. Astron. SOC.195, 705. Dube, M., Rasoanaivo, R., and Hahn, Y. (1985). J. Quant. Spectrosc. Radiar. Transfer 33, 13-26.

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Dunn,G. H., Belic, D. S., Morgan, T. J., Muller, D. W., and Timmer, C. ( 1 984a). In “Electronic and Atomic Collisions” (J. Eichler et al.. eds.), pp. 809-817. Elsevier, Amsterdam. Dunn, G. H., Belic, D. S., Djuric, N., and Mueller, D. W. (1984b). Int. ConJ At. Phys. Seattle. Fujimoto, T . , and Kato, T. (1981). Astrophys. J. 246, 994- 1003. Fujimoto, T., Kato, T., and Nakamura, Y. (1982). Inst. PIasmaPhys. Nagoya U . IPPJ-AM-23. Gau, J., and Hahn, Y.(1978). Phys. Lett. 68A, 197-200. Gau, J., and Hahn, Y. (1980). J. Quant. Spectrosc. Radiat. Trander 23,121 - 130. Gau, J., Hahn, Y., and Retter, J. A. (1980). J. Quant. Spectrosc. Radiat. Transfer23, 13 I - 146; 147- 156. Geltman, S. (1985). J. Phys. B 18, 1425- 1442. Griem, H. R. (1964). “Plasma Spectroscopy.” McGraw-Hill, New York. Griem, H. A. (1974). “Spectral Line Broadening by Plasma.” Academic Press, New York. Hahn,Y.(1970).Phys. Rev. C1, 12-16. Hahn, Y. (1971). Ann. Phys. 67, 389-405. Hahn, Y. (1977). Phys. Rev. Lett. 39,82-84. Hahn, Y. (1978). Phys. Lett. A 67,345-348. Hahn, Y. (1980). Phys. Rev. A 22,2896-2898. Hahn, Y. (1983). Comments At. Mol. Phys. 13, 103- 114. Hahn, Y. (1984). In “Electronic and Atomic Collisions” (J. Eichler et al., eds.), pp. 801 -807. Elsevier, Amsterdam. Hahn, Y., and LaGattuta, K. (1982). Phys. Rev. A 26, 1378- 1384. Hahn, Y., and Rule, D. W. (1977). J. Phys. B 10,2689-2698. Hahn, Y., Gau, J. N., Luddy, R., Dube, M., and Shkolnik, N. (1980a). J. Quant. Spectrosc. Radiat. Transfer 24, 505 - 5 15. Hahn, Y., Gau, J. N., Luddy, R., and Retter, J. A. (1980b). J. Quant. Spectrosc. Radiat. Transfer 23,65-72. Hahn, Y., LaGattuta, K., and McLaughlin, D. (1982). Phys. Rev. A26, 1385-1391. Hahn, Y., LaGattuta, K., McLaughlin, D., and Nasser, 1. (1985). Phys. Rep., in press. Ham, F. S. (1955). Solid State Phys. 1, 127. Hickman, A. P. (1984). J. Phys. B 17, LlOl-LlO6. Huber, W. A., and Bottcher, C. (1980). J. Phys. B 13, L399-L402. Hulet, R. G., and Kleppner, D. (1983). Phys. Rev. Lett. 51, 1430-1433. Isler, R. C., Crume, E. C., and Arnarius, D. E. (1982). Phys. Rev. A 26, 2105-21 16. Jacobs, V. L., Davis, J., and Kepple, P. C. (1976). Phys. Rev. Lett. 37, 1390- 1393. Jacobs, V. L., Davis, J., Kepple, P. C., and Blaha, M. (1977). Astrophys. J. 211,605-616. Kachru, R., Tran, N. H., van Linden, van den Heuvell, H. B., and Gallagher, T. F. (1 984). Phys. Rev. A 30,667-669. Mar, H. (1981). J . Phys. B 14,4165-4170. LaGattuta, K. (1984). Phvs. Rev. A 30, 3072-3077. LaGattuta, K., and Hahn, Y. (1981a). Phys. Rev. A 24,785-792. LaGattuta, K., and Hahn, Y. (1981b). Phys. Lett. A 84,468-472. LaGattuta, K., andHahn, Y.(1982a).J. Phys. B 15,2101-2107. LaGattuta, K., and Hahn, Y. 1982b). Phys. Rev. A 25,411-416. LaGattuta, K., and Hahn, Y. 1983a). Phys. Rev. Lett. 50,668-671. LaGattuta, K., and Hahn, Y. 1983b). Phys. Rev. Lett. 51, 558-561. LaGattuta, K., and Hahn, Y. 1 9 8 3 ~ )Phys. . Rev. A 27, 1675- 1677. LaGattuta, K., and Hahn, Y. 1984). Phys. Rev. A 3 0 , 316-324. LaGattuta, K., and Hahn, Y. 1985a). Phys. Rev. A 31, 1415- 1418. LaGattuta, K., and Hahn, Y. 1985b). Phys. Rev. A (forthcoming). Lee, C. M., and Pratt, R. H. ( 975). Phys. Rev. A 12, 1825-1829.

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Lee, C. M., and Pratt, R. H. (1976). Phys. Rev. A 14,990-995. McGuire, E. J. (1975). In “Atomic Inner-Shell Processes” (B. Craseman, ed.), Vol. I , pp. 290- 309. McLaughlin, D., and Hahn, Y. (1981). Phys. Lett. 88A, 394-397. McLaughlin, D., and Hahn, Y. (1982a). J. Quant. Spectrosc. Radiat. Transfer 28,343-353. McLaughlin, D., and Hahn, Y. (1982b). Phys. Rev. A 27, 1389- 1392. McLaughlin, D., and Hahn, Y. (1983a). Phys. Rev. A 28,493-494. McLaughlin, D., and Hahn, Y. (1983b). J. Phys. E 16, L739-L743. McLaughlin, D., and Hahn, Y. (1984). Phys. Rev. A 29,7 12-720. McLaughlin, D. J., Nasser, I., and Hahn, Y. (1985a). Phys. Rev. A 31, 1926- 1928. McLaughlin, D. J., LaGattuta, K., and Hahn, Y. (1985b).J. Quant. Spectrosc.Radiat. TransJer McWhirter, R. W. P. (1965). In “Plasma Diagnostic Techniques” (R. H. Huddlestone and S . L. Leonard, eds.), Chap. 5, pp. 210-264. Academic Press, New York. Massey, H. S. W., and Bates, D. R. (1942). Rep. Prog. Phys. 9.62-74. Merts, A. L., Cowan, R. D., and Magee, N. H. (1976). LASL Report No. LA-6220-MS. Mitchell, B. A., Ng, C. T., Forand, J. L., Levac, D. P., Mitchell, R. E., Sen, A., Miko, D. B., and McGowan, J. Wm. (1983). Phys. Rev. Lett. 50, 335-338. Nasser, I., and Hahn, Y. (1983). J. Quant. Spectrosc. Radiat. Transfir 29, 1-8. Nasser, l., and Hahn, Y. (1984). Phys. Rev. A 30,1558- 1560. Nasser, I., LaGattuta, K., and Hahn, Y. (1985). Phys. Rev. A (forthcoming). Peterkop, R. (1983). J. Phys. E 16, L587-L593. Pillet, P., Smith, W. W., Kachru, R., Tran, N. H., and Gallagher, T. ( 1983). Phys. Rev. Lett. 50, 1042- 1045. Pindzola, M. S . , Griffin, D. C., and Bottcher, C. (1984). Post, D. E., Jensen, R. V.,Tartar, C. B., Grasberger, W.H., and Lokke, W. A. (1977). At. Data Nucl. Data Tables 20, 397-439. Pradhan, A. (1984). Phys. Rev. A 30,2141 -2144. Rau, A. R. P. (1983). J. Phys. E 16, L699-L705. Richards, D. (1984). J. Phys. E 17, 1221 - 1233. Rolfes, R. G., Smith, D. B., and MacAdam, K. B. (1983). J. Phys. B 16, L535-L538. Roszman, L. (1979). Phys. Rev. A 20,673-676. Roszman, L., and Weiss, A. (1983). J. Quant. Spectrosc. Radiat. Transjer 30, 67-71. Seaton, M. J. (1983). Rep. Prog. Phys. 46, 167-257. Seaton, M. J. (1984). J. Phys. E 17, L531 -L533. Seaton, M. J., and Storey, P. J. (1976). In “Atomic Processes and Applications” (P. G. Purke and B. L. Moiseiwitsch, eds.), pp. 133- 197. North Holland Publ., Amsterdam. Silverstone, H. J. (19711). Phys. Rev. A 18, 1853. Stebbing, R. F., and Dunning, F. B. (1982). “Rydberg States of Atoms and Molecules.” Cambridge Univ. Press, London and New York. Tanis, J. A., Bernstein, E. M., Graham, W. G., Clark, M., Shafroth, S. M., Johnson, B. M., Jones, K. W., and Meron, M. (1982). Phys. Rev. Lett. 49, 1325- 1328. Tanis, J. A., Shafroth, S. M., Willis, J. E., Clark, M.,Swanson, J., and Strait, E. N. (1984). Phys. Rev. Lett. 53, 2551 -2554. Temkin, A. (1982). Phys. Rev. Lett. 49, 365-368. Temkin, A., and Hahn, Y. (1974). Phys. Rev. A 9,708-724. Wannier, H. G. (1953). Phys. Rev. 90, 817-825. Williams, J. F. (1984). Phys. Rev. A 29,2936-2938. Wilson, R. (1962). J. Quant. Spectrosc. Radiat. Transfer 2,477-490. Younger, S . (1983). J. Quant. Spectrosc. Radiat. Transfer 29,67-74.

II

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 21

RECENT DEVELOPMENTS IN SEMICLASSICAL FLOQUET THEORIES FOR INTENSE-FIELD MULTIPHOTON PROCESSES SHIH-I CHU Departmenr of Chemistry University of Kansas Lawrence, Kansas

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 11. The Floquet Theory and General Properties of Quasi-energy States . A. The Floquet Formalism . . . . . . . . . . . . . . . . . . . B. Shirley’s Time-Independent Floquet Hamiltonian . . . . . . . C. The Time Evolution Operator and Transition Probabilities . . . D. Properties of Quasi-energy States. . . . . . . . . . . . . . . 111. Computational Methods for Multiphoton Excitation of Finite-Level Systems.. . . . . . . . . . . . . . . . . . . . . . . . . . . A. Peturbation Methods . . . . . . . . . . . . . . . . . . . . B. Extensions of the Floquet Hamiltonian Method . . . . . . . . C. Time-Propagator Methods. . . . . . . . . . . . . . . . . . D. Other Methods . . . . . . . . . . . . . . . . . . . . . . . IV. Non-Hermitian Floquet Theory for Multiphoton Ionization and Dissociation. . . . . . . , . . . . . . . , . , . . . . , . . . A. Non-Hermitian Floquet Matrix Formalism . . . . . . . . . . B. Other Multiphoton Ionization and Dissociation Methods. . . . V. Many-Mode Floquet Theory . . . . . . . . . . . . . . . . . . A. GeneralFormalism. . . . . . . . . . . . . . . . . . . . . B. Generalized Rotating-Wave Approximation . . , . . . . . . . C. SU(N) Dynamical Symmetry and Quantum Coherence . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

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197 199 199 201 202 204

. . .

208 209 209 2 17 224

. . . . . . . . .

226 227 237 239 240 242 247 248 249

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I. Introduction There is currently much interest in the study of the structure, spectroscopy, and dynamics of atoms and molecules in intense externally applied 197 Copyright 0 1985 by Academic Pms,Inc. All rights of reproduction in any form reserved

198

Shih-I Chid

electromagnetic fields. In particular, the development of intense and powerful tunable lasers has revolutionized optical spectroscopy and led to the discovery of distinctly new nonlinear molecular phenomena. Foremost among these are collisionless multiphoton ionization (MPI) of atoms and multiphoton excitation (MPE) and dissociation (MPD) of polyatomic molecules. Recent reviews on MPI/MPD subjects are extensive (see, for example, Lambropoulos, 1976; Letokhov and Moore, 1976; Bloembergen and Yablonovitch, 1978; Shulz et al., 1979; Cantrell et al., 1980; Morellec et al., 1982;Chin and Lambropoulos, 1984).While considerable experimental and theoretical work has now been directed toward understanding these phenomena, much remains to be explained. The main obstacles stem from the complexity of the molecular level structure and the lack of suitable a priori theoretical and numerical methods. The theory of multiphoton processes can be formulated in a fully quantum-mechanical or semiclassical formalism. In the former approach, both the system and the field are treated quantum mechanically, while in the semiclassical approach, the system is described by a time-dependent Schrodinger equation in which the effect of the radiation field is represented by an effective Hamiltonian consistent with Maxwell’s equations. The semiclassical approach usually leads to Hamiltonians periodic in time through an assumed sinusoidal time variation of the electromagnetic field. In addition to being considerably more tractable, the semiclassical theory has been shown to lead to equivalent results of the fully quantized theory in strong fields (Shirley, 1963, 1965; Peg, 1973; Stenholm, 1973). Indeed, much recent multiphoton research is couched in terms of semiclassical theory, particularly the Floquet theoretical approach. The solutions of the differential equations of a quantized system interacting with a sinusoidal monochromatic field can be written as a Fourier series. As far back as the period 1 883 - 1900, Floquet ( 1883)studied the solutions of such linear differential equations with periodic coefficients, and Poincark ( 1 892,1893,1899) considered the construction ofthe solutions. The Floquet theorem was later used by Autler and Townes (1955) to obtain the wave function for the two-level system in terms of infinite continued fractions. However, the application of Floquet theory to quantum system began to grow only until after the mid-1960s (Shirley, 1965; Ritus, 1966;Zel’dovich, 1966). In a well-known paper, Shirley (1965) reformulated the problem of the interaction of a quantum system with a strong oscillating field by relating the solution of the Schrodinger equation to a time-independent infinite Floquet matrix. While this is literally a semiclassical theory nowhere involving explicit field quantization, Shirley has shown that his Floquet states can be interpreted physically as quantum field states. Indeed his Floquet quasienergy diagram is identical to the dressed-atom picture derived by CohenTannoudji and Haroche (1969).

SEMICLASSICAL FLOQUET THEORIES

199

Most of the subjects to be discussed in this article are concerned with the extension of Shirley’sapproach in publications after 1976. A comprehensive survey of different Floquet approaches for two-level systems published before 1976 is given by Dion and Hirschfelder ( 1976), which also includes a description of the nine Floquet theorems. Some of the earlier applications of Floquet theory to excited atomic and molecular states in strong fields can be found in the review by Bayfield (1 979). An alternative semiclassical approach involving the development of the method of transformations to rotating coordinate frames, particularly suitable for magnetic resonance problems, has been recently reviewed by Series (1978). Section I1 is devoted to a survey of Shirley’s Floquet Hamiltonian method and general properties of quasi-energy states (dressed states) in monochromatic sinusoidal fields. Recent new developments in Floquet theories and quasi-energy methods are described in Sections 111- V. Section I11 is devoted to new computational methods for bound - bound multiphoton excitation (MPE) of finite-level systems, with particular emphasis on rovibrational MPE in simple molecules. The non-Hermitian Floquet theory, which generalizes the conventional (finite-level)dressed-atom or quasi-energy picture to include the complete set of bound and continuum states, allowing nonperturbative treatment of frequency- and intensity-dependent multiphoton ionization (MPI) and dissociation (MPD) processes, is described in Section IV. Exact extension of Shirley’s(one-mode) Floquet formalism to a generalized many-mode Floquet theory (MMFT) is discussed in Section V. The MMFT makes possible reformulation of the time-dependent problem of finite-level systems exposed to polychromatic fields (such as multimode laser fields or several monochromatic fields with different frequencies) as an equivalent time-independent infinite-dimensional eigenvalue problem. The last section is a summary of the advantages of the Floquet approaches in multiphoton researches.

11. The Floquet Theory and General Properties of Quasi-energy States A. THEFLOQUET FORMALISM Consider a quantum system whose Hamiltonian is periodic in time with the period 7 : H(r, t 7 ) = H(r, t ) , where 7 = 2n/w and w is the field frequency. The time-dependent Schrodinger equation for the system isgiven by

+

%(r, t)Y(r, t ) = 0

(1)

200

Shih-I Chu

where %(r, t ) = H(r, t ) - i h a p t

(2)

and

+

H(r, t ) = Ho(r) V(r, t )

(3)

with V(r, t

+ T) = V(r, t)

(4)

The unperturbed Hamiltonian Ho(r) has a complete orthonormal set of eigenfunctions

Ho(r)a(r)= E%(r), (P(r)la(r)) = S , (5) The wave function Y , called the quasi-energy state (QES), can be written, according to the Floquet theorem (Floquet, 1883; Poincart, 1892; Moulton, 1930), in the following form

Y(r, t) = exp(- i d / h )@(r, t)

(6)

and @(r, t

+ T) = @(r, t )

(7) where E is a real parameter, which is unique up to multiples of 2nn/r, called the Floquet characteristic exponent or the quasi-energy (Ritus, 1966; Zel'dovich, 1966, 1973). Substituting Eq. (6) into Eq. (I), one obtains the eigenvalue equation for the quasi-energy (8) %(r, t P k ( r , t ) = ~ k @ k ( r ,t ) subjected to the periodicity condition, Eq. (7). Here the subscript k corresponds to the kth Floquet mode. For the Hermitian operator %(r, t), Sambe (1973) introduced the composite Hilbert space R 0 T. The spatial part is spanned by square-integrable (L2) functions in configuration space, with the inner product defined by

(a(r)lP(r))=

a*p dr = S

The temporal part is spanned by the complete orthonormal set of functions eimor, where m = 0, 1, +2, . . . , and

20 1

SEMICLASSICAL FTOQUET THEORIES

The eigenvectors of % satisfy the orthonormality condition

and form a complete set in R 0 T:

It follows from Eqs. (6) and (7) that the two vectors @,(r, t) and Q1,(r, t) = Ok(r,t) exp(inot), n = 0, f 1 , +2, . . . , belonging to the quasi-energiesE k and Ek n h o lead to one and the same state Yk(r,t) and, therefore, are physically indistinguishable. However, they are different vectors of the Hilbert space R 0 T. They thus satisfy the orthogonality relation, Eq. (lo), and appear as independent terms in the completeness condition, Eq. ( 1 1). It is evident that one can always reduce any quasienergies E k to a point in a zone such as e0 - h o/2 S E S e0 h o/2, specified by a real number e0. Therefore, physically different quasi-energy states can be partially characterized by their reduced quasi-energies, which lie in the same zone. The choice of zone is, however, arbitrary. Further, if e2 = E, n h o, the vectors (r, t) and (r, 2) = mC2(r,t) exp(- i n o t ) are linearly independent and belong to the same quasi-energy E , . The degeneracy of quasienergy levels plays an important role in (single- and multiphoton) resonance transitions in the quantum system (Shirley, 1965).

+

+

+

B. SHIRLEY’S TIME-INDEPENDENT FLOQUET HAMILTONIAN

Following Shirley ( 1969, we introduce the Floquet-state nomenclature

Ian) = Ia)ln),where la) are atomic or molecular eigenstates of Ho(r)and In) are the Fourier vectors, such that ( t i n ) = exp(inot). Expanding @(r, t) and H(r, t), which are both periodic in time, in terms of the Floquet-state basis,

and m

(a(r)lH(r,t)lP(r))

H$einof

= “--m

and substituting these expansions into the Schrodinger equation, one obtains an infinite set of recursion relations for .@ ’; They can be rewritten in

Shih-I Chu

202

the form of a matrix eigenvalue equation

(anlA,lyk)(Dl:8'=

(14)

where I?, is the time-independent Floquet Hamiltonian (Shirley, 1965) which has the matrix elements

(anlA,lprn) = H$?]

+ n h wscrasnm

(15) It follows from Eq. ( 14) that the quasi-energies are eigenvalues of the secular equation detlAF - €11 =0 A typical Floquet matrix structure is shown in Fig. 1.

(16)

+

The eigenvector associated with the eigenvalue E,,(=E? n h o)isgiven The ) . components of the eigenvectors satisfy by lean), with Cog; = ( a n l ~ ~ the following periodicity relation:

( a , n + PIEg.m+p)

= ( a ,nIEpm)

(17)

C. THETIMEEVOLUTION OPERATOR AND TRANSITION PROBABILITIES The time evolution operator is a linear operator U(t, to)which transforms an initial state " ( t o ) at time to into state Y(t)at time t :

W) =

w ,t 0 ) W o )

(18)

U(t, t o ) satisfies the initial condition U(to,to)= 1 and is a solution of the Schrodinger equation

ih dU(t, to)/dt = H(t)U(t,to)

(19)

For a Hermitian Hamiltonian H ( t ) , U(t, to) is unitary and can be constructed according to the following relation: c,t,

U(t, t o ) = Y(t)Y(to)-1 (20) c-, where Y ( t )is a solution (column vector) of Eq. (1). The matrix element Uaa(t,to) = (PIU(t, to)la)is then the amplitude that a system initially in quantum state a at time to evolves to state p by time t according to the time-dependent Schrodinger equation [Eq. ( l)] with Hamiltonian H(t). Using the Floquet Hamiltonian method, Shirley (1965) showed that

~ ~ t~o ) = ( I] t (fin1 , exp[-iA,(t n

- to)/h11a0) exp(inot)

(2 1)

203

SEMICLASSICAL FLOQUET THEORIES ... n = 2

n = l

n=O n=-1 n = - 2

n‘= 2 n’= 1 n‘ = 0

1 HFI =

n’ = - 1

n’ = - 2 Where

v‘ = 0

A =

v‘ = 1

and

v = l

v = o

~ v ’ = o

,

0 =

v’ = 1

FIG.1. Structure of the Floquet Hamiltonian for multiphoton excitation ofdiatomic molecules. The Hamiltonian is composed ofthe diagonal Floquet blocks, oftype A , and off-diagonal blocks of type B. E$y) are the unperturbed vibration-rotational energies, and b”,,”,,,= -f(uJlp.Eolv’j’ > Fd, withp theelectricdipole moment operator, E,-,thepeakamplitude, and 6 the phase of the monochromatic field.

In practice, the computation involves the diagonalization of a truncated Floquet matrix, Eq. ( 14), yielding N-independent eigenvector solutions and N Floquet eigenvalues (quasi-energies)coo(for an N-level system; a ranges from 1 to N). In terms of the quasi-energy eigenvectors and eigenvalues, Upact,to)can be rewritten as Up&, t o ) =

c n

(my/) exP[-

q

l

- to)/h. l(cy,laO) e x p ( i n 4

y

(22) Equation (2 1) shows that Ua,(t, to)can also be interpreted as the amplitude that a system initially in the Floquet state la0) at time to evolves to the

204

Shih-I Chu

by time t according to the time-independent Floquet Floquet state I@) Hamiltonian A,, summed over n with weighting factors einwr.The latter interpretation enables one to solve problems involving Hamiltonians periodic in time by methods applicable to time-independent Hamiltonians. The transition probability to go from the initial quantum state a and a coherent field state to the final quantum state p, summed over all final field states, is given by (Shirley, 1965)

Pa+& to) = I Ua,
(pkl exp[- ifi,(t - to)/h11a0) X exp(imot,)(aml exp[- ifi,(t

- to)/h](pk)

(23) The quantity of experimental interest, however, is the transition probability averaged over initial times to(or equivalently averaged over the initial phases of the field seen by the system), keeping the elapsed time t - to fixed. This yields Pa+g(t - to) =

I(P4 exp[-ifi,(t - t O ) / ~ I I ~ O ) l Z

(24)

Finally, if one also averages over t - t o , one obtains the long-time average transition probability

D. PROPERTIES OF QUASIENERGY STATES

1. Variational Principle The variational form of the quasi-energy eigenvalue equation, Eq. (8), is given by (Sambe, 1973)

W@I= 0,

€[@I

= ((@lwq)/((@l@))

(26) where @(r, t) and its variation 6@(r, t) are both in the composite Hilbert space R 0 T. The eigenfunctions Qk(r, t) of Eq. (8) are given by the stationary solutions of the variational equation [Eq. (26)], and the corresponding eigenvalues Ek are given by the stationary values E [ @ ~ ] of the functional

€[@I* 2. Hellmann -Feynman Theorem Similar to the stationary states in a conservative system, an analogous Hellmann - Feynman theorem can also be derived for quasi-energy solu-

SEMICLASSICAL FLOQUET THEORIES

205

tions in a periodically time-dependent system. If the Hamiltonian H(t, A) depends on a time-independent parameter A and the periodic relation H(t 7,A ) = H(t, A ) holds for any A, then (Sambe, 1973)

+

d W ) / d A = ((@ la SPAI@))I( (@ I@ ))

(27)

a)= ((@lwq)/((@l@))

(28) where (€(A), @(r, t, A)) are the solution of Eq. (8). Similarly, one can easily show that the quasi-energy-statesolutions also satisfy a theorem analogous to the hypervirial theorem (Hirschfelder, 1960) for stationary states.

3. Mean Energy It is well known that, for an explicitly time-dependent Hamiltonian, the energy is not a conserved quantity. However, if the Hamiltonian is periodic in time, it is possible to define the notion of “mean energy” (Fainshtein et al., 1978)

H, =( ( W ) I H ( t ) l W ) ) ) = E + ((@,(t)lih a/atl@,(t)))

(29) where E is the quasi-energy and the quasi-energy-statewave functions Y,(t) and @,(t)are defined in Eq. (6). Using the Fourier-series expansion m

m

one obtains

Thus H , may be interpreted as the energy accumulated in each harmonic of @,(r, t ) and averaged over the weights of these harmonics. Milfeld and Wyatt (1983) also showed that the Floquet-state “mean energy” plays a major role in the static part ofthe molecular -state energy fluctuations.Using the Hellmann- Feynman theorem, the general connection between the mean energy and the quasi-energy is shown to be (Fainshtein et al., 1978)

H , =E -w a E / a o

(31)

4. Symmetry of Quasi-energy States

The spatial symmetry of the Hamiltonian %, Eq.(2), in general is lower than the symmetry of the molecular Hamiltonian H&), Eq. (3). However, as a consequence of the periodic time dependence of 7f, the symmetry is

Shih-I Chi1

206

increased on going to the composite Hilbert space R 0 T. Let the symbol T ( A t )denote the operator for a displacement in time by At. The Hamiltonian then commutes with the operator f f ( z ( 2 )= ?(7/2)f, where f is the operator for inversion of the coordinates of the electrons and the nuclei. Since the eigenvalues of the operator f?( r/2) are -t 1, the periodic function @(r, t ) , Eq. (7), can be chosen to be even or odd with respect to its action (Bordo and Kiselev, 1980). This symmetry property of the periodic functions @(r, t ) can be used, for example, in the study of the emission of a molecule. Other related symmetry properties of a two-level system have also been discussed by Braun and Miroshnichenko ( 1980). Salzman (1974a) showed that the time evolution operator satisfies the relations

U(t

+ z, z) = U(t,0)

(324

U(t 4-z, 0) = U(t, O ) U ( 7 , 0 )

(32b)

so that

U(nr, 0) = U(7,O)"

(33)

Thus the time evolution operator over one period, U(z, 0), provides essentially all the information we ever need about the long-time behavior of the system with a periodic Hamiltonian. In the truncated basis set, U(z, 0) may be diagonalized by some unitary transformation S, S+ U(q 0 ) S = e-iD

(34)

where D is a diagonal matrix. Thus U(7 , O ) = Se+"+

whereupon

U(nz, 0) = U(7,O)"= Se+*S+

(35)

Additional symmetry properties of the periodic functions @(t)are noted by Milfeld and Wyatt (1983) and others: @ * (z/2)

=

a(7 / 2 )

@*(O) = @(O)

(364 (36b)

and @*(f/2 - t ) = @(2/2

+2)

(36c)

These relations reduce the numerical efforts of solving the time-dependent coupled equations, Eq. ( 19), to only the first half of the optical cycle.

SEMICLASSICAL FLOQUET THEORIES

207

o./o

FIG.2. (a) Quasi-energies (measured in unit ofw) and (b) time-average multiphoton transition probabilities P,,#as functions of wowith b/w = +.w,, is the unperturbed level spacingofa two-level system (such as a spin? system subjected to a static magnetic field), w the frequency of the driving monochromatic radiation field, and b the atom-field dipole coupling strength. Note the correlation between the avoided crossings of the quasi-energy levels and the multiphoton resonance peaks.

5. Quasi-energy Plot

Much information can be learned from the plot of the quasi-energy eigenvalues (or the characteristic exponents) of the Floquet Hamiltonian, Eq. (14). The main feature of the quasi-energy plot is illustrated in Fig. 2 for the simple case of a two-state system of energy spacing ti w o subjected to the dipole interaction V ( t )= 2b cos ot with an intense monochromatic field of frequency w (Shirley, 1965). In Fig. 2a, six ofthe quasi-energies E are plotted as a function of the level spacing w o ,using w as a scaling parameter. The corresponding time-average transition probability P, Eq. (25), is shown in Fig. 2b. One notices that resonance occurs (near wo= w , 30, 5 0 , 7 w ) whenever the quasi-energy curves for two eigenvalues approach each other closely but do not cross (avoided or anticrossing). The minimum separation of two branches of the (quasi-energy) hyperbola is correlated with the width

208

Shih-Z Chu

o.d

0

1

I

I

1

I

2

3

4

5

4

6

7

FIG. 3. Time-average transition probabilities Fa,, as a function of wo with (a) b/w = 3 , (b) b/w = 4, and (c) b/w = 1.0. Note the power-broadening and ac-Stark-shift effects as b increases. Notations same as Fig. 2.

of the resonance peak, whereas the location of the minimum separation determines the resonance position. At this rather high field strength, b = w/3,all four resonances can be clearly seen to be substantially shifted from their respective unperturbed positions (the Bloch - Siegert shifts, Bloch and Siegert, 1940). The one-photon peak is further strongly broadened, although the five- and seven-photon peaks remain discrete and are much narrower in width. As one increases the coupling strength b, each resonance in Fig. 2b further broadens and shifts toward smaller values of oo(Fig. 3) until it becomes lost in the ever-widening background.

111. Computational Methods for Multiphoton Excitation of Finite-Level Systems Exact analytical solution of the time-dependent Schrbdinger equation with a temporally periodic Hamiltonian is possible only in exceptional cases.

SEMICLASSICAL FL,OQUET THEORIES

209

Thus it is in general necessary to develop approximate methods for the treatment of multiphoton excitation (MPE) of atoms and molecules. A. PERTURBATION METHODS Several variants of the perturbation theory have been proposed or employed to calculate the quasi-energy spectrum and transition probabilities (Shirley, 1965; Ritus, 1966;Young et al., 1966; Sen Gupta, 1970;Langhoff et al., 1970; Sambe, 1973; Zon and Katsnel’son, 1974; Manakov et al., 1976; Barone et al., 1977; Sokolov, 1978; Manakov and Fainshtein, 198 1 ;Aravind and Hirschfelder, 1984). A detailed review of various Floquet perturbation methods for two-level systems has been given by Dion and Hirschfelder (1976). A more recent discussion of the application of Floquet perturbation methods to atomic systems can be found in the review article by Bayfield ( 1979). The perturbation methods have been quite successful in explaining the main features of multiphoton phenomena at low intensities. At high intensities, however, higher-order perturbation terms begin to contribute significantly and eventually dominate the lower-order terms, and a typical problem of convergence of the perturbation series manifests itself. In these cases, nonperturbative methods are required to solve the problems. The following two subsections describe some of these nonperturbative Floquet methods recently developed. OF B. EXTENSIONS

THE

FLOQUET HAMILTONIAN METHOD

Shirley’sFloquet matrix method described in Section II,B, which is equivalent to summing the entire perturbation expansion to all orders (Faisal, 1976), can be extended to the nonperturbative study of rovibrational multiphoton excitation of a molecule within a single electronic state. In the electric dipole approximation, the interaction potential energy between the quantum system and the classical field is given by V(r, t ) = -p(r)

E, cos(ot

+ 8)

(37) wherep is the dipole moment operator of the molecule, E,, and dare the field amplitude and phase, respectively, assumed to be independent of time, and r represents the internuclear coordinates. The Floquet matrix possesses a block tridiagonal form as shown in Fig. 1. The determination of the vibration-rotational quasi-energies and quasi-energy states (QES) thus reduces to the solution of a time-independent Floquet matrix eigenproblem. Figure 1 shows that fiF has a periodic structure with only the number of 0 ’ s in the diagonal elements varying from block to block. This structure endows the quasi-energy eigenvalues and eigenvectors of fiF with periodic properties.

Shih-I Chu

2 10

Using a simplified version of this Floquet matrix, Moloney and Faisal ( 1979)have computed rovibrational multiphoton excitation probabilities in

CO. Chu et al. (1982) have extended the Floquet matrix method to include the effect of an additional external static electric field. The quasi-energy and MPE spectra of the HF molecule are studied as functions of field strengths and frequency. Nonlinear effects such as power broadening, dynamical Stark effect, Autler - Townes multiplet splitting, hole burning, and S-hump behaviors, etc., are observed and discussed in terms of quasi-energy diagrams (Fig. 4). Many of the salient features in the spectral line shapes may be qualitatively understood in terms of an analytical three- or four-level model (Tietz and Chu, 1983a). The addition of a dc electric field removes the

m 0 LT

a z

0 ! I

v, z 6

c l x

0

60

FREOUENCY ( c m - ' )

FIG.4. (a) Quasi-energy plots and (b) time-average MPE transition probabilities P,,, for the HF molecule subjected to both laser (E,c = 1.0 TW/cm2) and dc electric fields (Edc= a.u.) simultaneously. (- * -), one-photon peaks; ( - - -), two-photon peaks; and (-), three-photon peaks. Nonlinear effects such as power broadening, dynamical Stark shift, Autler-Townes multiplet splitting, hole burning, and S-hump behaviors, etc., are observed and can be correlated with the quasi-energy diagram. The introduction of an external dc field strongly enhances the MPE probabilities and results in a much richer spectrum. (From Chu et a/.. 1982.)

SEMICLASSICAL FLOQUET THEORIES

21 1

restriction of the rotational dipole selection rule and causes significant intermixing of the bare molecular vibrator states. Due to the greater number of strongly coupled nearby states in the dc field, nonlinear-optical effects such as those mentioned above appear at a much lower ac field strength than they would in the absence of the dc field. The introduction of an external dc field, therefore, strongly enhances the MPE probabilities and results in a much richer spectrum, in accord with experimental observation (Duperrex and van den Bergh, 1980).Thomas ( 1983)has recently carried out analytical and numerical Floquet treatments of model two-level systems through their interaction with intense laser and static electric fields. He also found the enhancement of the MPE spectrum whose resonances will be more power broadened and will appear at higher frequencies than they would in the absence of the static field. Multiphoton resonance lineshape characteristics in N-level systems were considered by Hermann and Swain (1 977) and by Wang et al. (1985). Shirley’s Floquet Hamiltonian method, powerful as it may be, is subjected to some practical limitations. In dealing with molecular multiphoton problems, the Floquet matrix is truncated to N X N in dimension where N = NFN,Nj, where NF is the number of Floquet photon blocks, N, the number of vibrational levels included in one Floquet block, and N, the number of rotational states included in one vibrational level (Fig. 1). As N increases rapidly with the size of the molecule and with the order of multiphoton processes, and as the computational expense grows as N 3 , the full Floquet analysis can become prohibitively costly even for simple diatomic molecules. One is thus led to seek new approximation techniques capable of providing accurate results yet involving much smaller Floquet matrix manipulations. The situation here is analogous to the field of molecular collision theory, where a variety of angular momentum decoupling methods (for recent reviews, see Bernstein, 1979)have been developed in the last decade to alleviate the problem of a large number of coupled equations. In the following subsections, some such approximate Floquet treatments recently developed will be described.

I. Nonadiabatic Theory for Resonant Multiphoton Excitation In the process of IR multiphoton absorption (MPA), the laser frequency is of the same order of magnitude as the molecular vibrational frequency and considerably exceeds the rotational frequency. More explicitly, since the Hamiltonian contains fast (vibrational motion and electric dipole interaction) and slow (rotational motion) parts, one anticipates the wave function will have the adiabatic form of the product of a rapidly oscillating function times a function slowly varying in time. One can determine the rapidly

212

Shih-I Chu

oscillating part by considering the Schrddinger equation without the rotational Hamiltonian. This gives rise to an effective quasi-vibrational energy (QVE) which depends parametrically on the orientation of the molecule. The slowly varying parts of the wave function can then be solved by using the effectiveQVE as the potential energy for the rotational motion. This adiabatic picture has been used previously (Braun and Petelin, 1974; Makarov and Fedorov, 1976)to obtain approximate analytical solutions for the rotational spectra of diatomic molecules in some special cases. It provides an adequate zeroth-order description and a convenient basis for the development of an economical and feasible approach for MPA studies. However, the adiabatic approximation tends to break down nearby resonant transition regions, and nonadiabatic angular couplings among nearly degenerate adiabatic QVE states are required for a proper MPA analysis (Chu et a/., 1983b; Ho and Chu, 1983). The Schrddinger equation for a heteronuclear diatomic molecule interacting with a coherent monochromatic field E = Eo cos(wt 6) has the following form ( h = 1) i dY(r, t ) / d t = H(r, t)Y(r,t )

+

where

+

H(r, t ) = HLo)(r) HRR(?)- p(r)Eocos 8 cos(wt

+ 6)

(38) Here Hlo)(r)is the diatom vibrational Hamiltonian and HRR(?)is the rigidrotator Hamiltonian. The vibrational - rotational interaction has been ignored for simplicity. p(r) is the molecular electric dipole moment, and 8 is the angle between the direction of the field (Eo//z) and the axis of the molecule. As the interaction energy is independent of the azimuthal angle 4, the rotational magnetic quantum number miis a constant of motion. In the adiabatic approximation, one first solves for the QVE states with the rotational motion frozen. The resulting Schrddinger equation (with 8 fixed) is

+

i m(r,t, 8)ldt = [Hto)(r)- p(r)Eo cos 8 cos(ot 6)]O(r, t, e) (39) Coflesponding to Eq. (39), an equivalent time-independent Hamiltonian &") may be written in analogy with the semiclassical Floquet Hamiltonian of Shirley. The resulting matrix block structure [with the dimension only (&Nu)x (&Nu)]is Similar to that given in Fig. 1. The solution O& (athe molecular index, k the photon index) can be written in terms of an orthonormalized vibrational-field basis lu,n), where u runs over all unperturbed vibrational states and n is the Fourier index steps from - a~ to a ~ : a&, t; 8) = exp(iA&t)FJr, t; 8) (40) where

+

SEMICLASSICAL FLOQUET THEORIES

213

and is the quasi-vibrational energy wave function correspondingto the (orientation-dependent)QVE [email protected] QVE spectrum (A&) possesses a well-defined band structure (indexed by P) characterized by the laser frequency a.All P bands (P= 0, f 1 , k 2, . . .) have the same QVE structure and differ from one to the other only by a phase factor. The QVEs within each P band consist of the subset (wherea = 0, 1, 2, . . .) and are nearly degenerate and strongly coupled by the slow rotational motion. The interband rotational couplings are, however, orders of magnitude smaller (due to the much wider separation of QVEs) than those of intraband couplings and can be totally ignored. Thus, to an excellent approximation, the IR MPA process of interest can be properly studied within any P band. For convenience let us choose the P = 0 band below and introduce the following shorthand notation: Fp

Fp,-p,

Ap

Ap,-p,

Funs

(vnIAp,-p)~

Approximate solution of the total Hamiltonian in Eq. (38) can now be obtained by appropriate superposition of the QVEs Fp:

Y$y)(r,t ) = exp(- i~gt)(27r)-”~ exp(irnj+)

zap(O)F,(r, t; 0)

(42)

P

where p runs over all QVEs in the P- 0 band, and the total quasivibrational - rotational energies cp are to be determined. Substituting Eq. (42) into Eq. (l), averaging over a small time interval, say, one period (7 = 2n/o), and expanding xppin terms of orthonormalized associated Legeiidre polynomials Py, ~ p p ( O= )

CI

bp,pjPjml (COS

0)

(43)

j-lq

one obtains the following eigenvalue equations

where gii = ( P ~ ~ / ~ A ~ ~ P F ) S + PH

2 2 (P,”’IFy,,p(l?,Fvn,dIP~J)) u

(45)

n

The structure of the total quasi-energy“G” Hamiltonian is of the dimension only of (NUN,)X (NUN,). Finally, after making a certain unitary transformation, one arrives at time-dependent and long-time-averaged (state-to-state) MPA transition probabilities. For example, the latter is given by

F$!v3, = C m

F C ~ ( v Y ’rn, - nl~.l;”))

(vj, - n l e ~ m ~ ) ) 1 2

n

(46)

I

SEMICLASSICAL FLOQUET THEORIES

215

where

The utility ofthe nonadiabatic method is demonstrated by a detailed study of the sequential MPA spectra for the CO molecule, including state-to-state multiquantum transitions (Fig. 5 ) and transitions from initially thermally distributed states as a whole (Ho and Chu, 1983). Excellent agreement of the MPA spectra obtained by the nonadiabatic approach and the exact Floquet matrix method was observed in all of the fine details (Chu et al., 1983b). Since the nonadiabatic approach is usually computationally order@) of magnitude faster than the exact Floquet analysis, it may provide a practical nonperturbative method for generating high-resolution resonant MPA spectra of small polyatomic molecules.

2. Most Probable Path Approach The most probable path approach was introduced recently by Tietz and Chu ( 1983b)in an ab initio study ofhigh-order multiphoton excitation ofthe SO2molecule. The feasibility of multiphoton excitation (MPE) and dissociation (MPD) of triatomic molecules is a subject or experimental controversy. Some groups have reported experimental observation of collisionless MPD in triatomic molecules (SOz,0 3 OCS) , at a 1 GW/cm2 laser field strength range (Bailkowski and Guillory, 1979; Proch and Schroder, 1979), while others have disputed these claims (Wolk et al., 1980; Simpson and Bloembergen, 1981). It is therefore desirable to carry out comprehensive theoretical studies of the MPE/MPD dynamics of these sparse intermediate case molecules. A brute-force attempt to calculate polyatomic MPE would quickly become impossible due to the large size of the Floquet matrix needed for convergence. For typical ten-photon calculations for SO2, for example, a matrix on the order of 5000 X 5000 would have to be diagonalized at each frequency and field strength. In any exact Floquet calculation, however, the majority of the molecular-field states are unimportant due to extreme detuning or very small coupling matrix elements. The most probable path approach (MPPA) is a practical strategy introduced to determine which ~~

FIG.5. Time-averaged MPE transition probabilities for CO molecules initially prepared at ~ 3, m,) with (a) m, = 0, (b) rn, = I , (c) m, = 2, and (d) m, = 3, respectively, at stateslu = 0 , = 50 GW/cm2 of laser field strength. Note the change of the spectra as the rotational magnetic quantum number m, (which is a constant of motion) vanes. Line patterns same as Fig. 4. (From Ho and Chu, 1983.)

216

Shih-I Chu

molecular-field states are, in fact, important at each step of the multiphoton processes. The procedure is derived from algorithms which utilize artificial intelligenceto prune the number of choices at each node (photon order) of a decision tree (Winston, 1979).Similar to some minimax game-playing programs, the MPPA examines the possible paths to take at each photon order iteration with the static evaluation function given by Nth-order perturbation theory (this is a breadth-first search). If all paths were followed exhaustively, the problem would be beyond practical solution. In game theory, one answer is to ignore paths which start with very poor moves. The MPPA likewise uses a breadth-limiting heuristic technique and discards any paths for which the Nth-order coupling term is small (with respect to other Nth-order terms). The MPPA begins by calculating all possible second-order perturbative terms. The Nplargest couplings (where Npis the number of paths to keep at each step) are chosen as the most probable paths through second order. The initial state (of course) and the intermediate states of the chosen paths are marked as important and are used in the final calculations. At each iterative step, the method calculates all possible (N 1)st-order couplings (paths) using only the NpNth-order paths saved in the last iteration. The (N 1)storder couplings are then examined and the largest Npare saved for further traversal. Nth-order states which have now become intermediate to a large (N 1)st-orderpath are “important” and are marked for later use. By iterating long enough, one can traverse the entire Floquet molecular-field basis space, saving only those states which are important to the various ith-order processes. The reduction of the basis set is quite substantial and leads to many orders of magnitude savings in computer time, yet maintains good accuracy in most cases. Using the MPPA, Tietz and Chu ( 1983b)found that collisionlessMPD of SO2will not be achievable at a laser intensity under 20 GW/cm2 (Fig. 6), in agreement with the recent experimental results of Simpson and Bloembergen ( 1983). The latter experiment, however, has extended the laser power further up to 300 GW/cm2and found that appreciable MPD yields begin to occur, and that the process is controlled by the laser intensity and not the laser fluence. A further MPPA study (Tietz and Chu, 1985) showed that the MPE of SO2is primarily a one-ladder pumping phenomenon dominated by the power-broadening effect and that MPD is likely to occur (though the yield is predicted to be small, P < at a laser intensity above 100 GW/ cm2. Several other quantum-mechanical MPE studies of triatomic molecules have appeared recently. Quack and Sutcliffe (1984) have studied the possibility of mode-selective IR-MPE of 0 3using , a quasi-resonant approximation, in which they neglect interactions with states that are off resonance by more than 0 / 2 . Milfeld and Wyatt (1985) have studied the MPE of OCS, using the Magnus approximation in the Floquet Hamiltonian.

+

+

+

217

SEMICLASSICAL FLOQUET THEORIES

1.0

n

0.1

0.01

0.001

0.1

1.0

10.0

100.0

I (GW/cm2) FIG.6. Dependence oftheaverage number ofphotons absorbed(5) by SO, on the excitation laser intensity. The different traces correspond to the indicated excitation frequencies(cm- I ) of the laser. A typical IR MPD of SO, requires about forty 9.3ym photons. The results shown in this figure indicate that MPD of SO2cannot be achieved for I < 20 GW/cm 2. (From Tietz and Chu, 1983b.)

It appears that detailed ab initio studies of MPE dynamics in small polyatomic molecules are now becoming feasible. Such a theoretical investigation would provide useful new physical insight and complementary information to the experimental results. The feasibility of experimental observation of IR MPE of small molecules (OCS, NO2,SO,, DN,) has been recently reviewed by Bloembergen et al. (1 984).

C. TIME-PROPAGATOR METHODS Instead of using Shirley's Floquet Hamiltonian method, an alternative way for studying multiphoton processes is to solve for the time-evolution operator U(t,to)directly by considering the Schrodinger equation, Eq. ( 19), with H(t 7)= H(t)and U(to,to)= 1. This may be achieved by solving Eq. (19) either numerically or perturbatively. A detailed comparison of several numerical methods for solving Eq. (19) was given by Smith et al. (1982).

+

1. Numerical Integration Method To solve for U(T, 0), Burrows and Salzman (1977) and Salzman (1977) first transformed U(t, to) into the interaction representation, U,(t, to),and

218

Shih-IChu

then use standard finite-differencetechniques to numerically integrate the Schrodinger equation for U,(t, to)from t = 0 to t = 7. U(n7,O)may then be computed for arbitrarily large integer n by means of matrix multiplication, Eq. (35). The transition probability from the initial state ti) to the final state If) is obtained by

Pfi(nZrn7)= I(flU(2rn7,O)”li)12 at intervals of 2m7, where n and m are integers. Using this procedure, Salzman ( 1977) calculated one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom. Leasure and Wyatt ( 1980) have considered explicit construction of the Floquet characteristic exponent by numerically integrating the coupled first-order differential equations for the propagator elements Uji(t,to):

where Eo is the field amplitude, E!” is the stationary energy of the ith molecular state, and P,k are the dipole coupling matrix elements. The Floquet amplitudes are given by

+

F(t)= @(t)eMut, @(t T) = @(t) (49) wherep, the characteristic exponent matrix, is a real-valued N X Ndiagonal matrix. Propagating F(t)across the first optical cycle [0, 71, 7 = 27r//o, with U(7,0),and rearranging the propagation equation F(t) = U(7, O)F(O),yields the matrix eigenvalue equation @t(O)U(z,O)@(O) = exp(2i7rp) (50) Solution of this equation gives the characteristic exponents firn (eigenvalues of p), and the periodic function @ at t = 0. For any time t = n7, use of F(n7) = U(n7, O)F(O) gives U(n7,O) = @(0)eiwflTWt(O) from which the transition probability may be obtained,

(51)

Pji(nr)= (17,~(n7,0)l2 Leasure and Wyatt (1980) applied this method to nonrotating HF, with particular emphasis upon the determination of dynamic Stark shifts and power-broadening parameters.

2. The Methods of Meath, Moloney, and Thomas In a series of papers, Moloney and Meath have considered matching power-series expansion techniques (Moloney and Meath, 1975, 1976) for

SEMICLASSICAL FLOQUET THEORIES

219

the solution of U(7,O).In particular, they emphasized the importance of carrying out the phase and temporal average transition probabilities in multiphoton transitions. They have also considered the effect of static fields on the multiphoton spectra of some model finite-level systems (Moloney and Meath 1978a,b) and the nature and behavior of the Floquet characteristic exponent with particular emphasis on the frequency-sweep experiment (Moloney and Meath, 1978~). In the Schrodinger representation, the time-dependent wave function for an N-level system, interacting with a sinusoidal electric field, can be written (in the notation of Moloney and Meath) as Y(r,t ) = a7(r)a(t),

at(t)a(t) = 1 (52) The state amplitudes uj(t) = (q5j(r)lY(r, t)) satisfy the coupled first-order system of equations ( h = 1 ), d i - a(t) = H(t)a(t) (53) dt where H(t) = E(O)- Eop C O S ( ~ ~S) (54)

+

The column vector a, the row vector @(r), and the square matricesp and E(O) are defined by (a)j = uj(t), (a7@)), = 4j(r), (p), = p, and (EcO)), = Ej S,, respectively. Here 4j and EY) are the wave function and energy of the jth stationary state of the unperturbed system, E,, o,and 6 are the amplitude, frequency, and phase of the sinusoidal field, and p, = (4ilpl$j) are the transition dipole matrix elements. Transforming to the dimensionless variable 0 = ot S, the solution of Eq. (53) can be written in terms of the time evolution operator U(0,O)

+

4 0 ) = u(0, O)a,(J) (55) where a,(S) = U-'(S, O)a(O),a(0) is the initial (t = 0) value of the state amplitude, and U(0, 0) satisfies

Making use of the periodic properties, Eq. (33), one obtains

a(O

+ n2n) = U(0,O)[U(2n, O)]"ao(S),

n = ~ 1, , 2 , .

..;

o ~ e a 2 ~

(57)

From Eq. (35), one obtains

a(O

+ n27r) = U(O,O)SAnS-la,(S)

(58)

where A is a diagonal matrix containing the eigenvalues of U(2n, 0) and S is

Shih-I Chu

220

the unitary matrix diagonizing U(2n, 0). Since U(8,O) is unitary, these eigenvalues may be written as

1, = exp(iAii2n) where Aii is a characteristic exponent defined by

(59)

+

Aii = tan-'[Irn(Aii)/Re(lii)]/2n q/2 (60) and q = 0, k 1, f2, . . . determines the branch of the multivalued inverse tangent functions. Substitution of Eq. (60) into Eq. (58) gives a(8

+ n2n) = U(8,O)S exp(iAn2n)S-'ao(d)

(61)

for 0 c 8 c 2n. For arbitrary 8, a(8) can be written as a( 8) = L(6) exp(iA8)S-la0(6)

(62) where the Lyapunov matrix (Pullman, 1976; Arnold, 1978) L(@= U(8,O)S exp(- iA8) is periodic in 8 with period 2n. The time evolution operator U(8,O) over 0 s 8 G 2n can be constructed by using matching power series techniques (Moloney and Meath, 1975, 1976) or the Riemann product integral method to be discussed below. It is convenient to introduce the Laplace time-averaged value of the transition probability to state Ik) defined as 1

r-

7r

o

P&(w,6)= ' J dt exp(-t/zr)lak(ot

+ 6)12

where 7, has units of time and may be viewed as representing a mean relaxation time, i.e., the time over which the system interacts with the applied field. If this interaction process is interrupted through, say, the collisions between the members in an ensemble of N-level systems, then for finite t r , P&(w, 6) represent the collisional damped phase-dependent transition probability. Since the initial phase of the field experienced by the system is not well defined, the quantity of physical interest is the phase-averaged transition probability

(Note that the phase-averaging procedure in the Moloney - Meath approach is equivalent to the average over the initial time to in Shirley's method.) Finally, if relaxation effects are negligible, the quantity of experimental interest is the time-averaged and phase-averaged transition probability

&(o) = lim 7,--

1

Tr

dt P&(w)

SEMICLASSICAL FLOQUET THEORIES

22 1

This last quantity is equivalent to Shirley’s long time-average transition probability, Eq. (25). A related algorithm recently considered by Thomas and Meath (1983) and Thomas (1983) is based on the Riemann product integral (Dollard and Friedman, 1977) representation of U(8,O) in conjuction with Frazer’s method of mean coefficients (Frazer et al., 1960). This method allows one to evaluate U(8,O) over 0 s 8 s 2a as

U(8,O) = f lim

do‘ H(8’))

g 0 0

where 8,= 2a/np+ 8,-, for k = 1, 2, . . . , np and with O o = O and 8np= 2a. The time-ordering operator farranges the product in chronological order from right to left since the terms in the product do not generally commute. Thus to compute U(8,O)through Eq. (66) one effects a discretization of the interval 0 S 8 S 2a into a sufficientnumber npof subintervals, evaluates the exponential matrix over each subinterval, and applies the group property of the evolution operator at adjacent subintervals. Essentially the method, by averaging over - i/oH(8) in each subinterval, replaces a time-dependent system Eq. (52) by a time-independent system and so is similar in spirit to the stroboscopic method of Minorsky (1962) and applied by Askar (1974) to the study of transitions induced by resonant fields in N-level systems. One major advantage of the Riemann product integral representation algorithm is that it can be applied to both nonsinusoidal and sinusoidal fields. Using this method, Thomas and Meath (1982) have calculated P k ( o )for nonrotating multiphoton vibrational excitations of the rare gas dimers HeXe, NeAr, and NeXe. They have also camed out a model calculation for the interaction of a two-level system with a Gaussian pulse (Thomas and Meath, 1983).

3. The Magnus Approximation The Magnus formulation (Magnus, 1954; Pechukas and Light, 1965) determines the time evolution operator in an exponential form, U(t,to) = exp[A(t, to)], via the differential equation

where C,X = [A, XI and A(to, to) = 0. The formal solution to Eq. (67) is aA/at = CA[exp(CA) - l ] - l H ( f ) / i h

(68)

222

Shih-I Chu

Using an iteration procedure and expanding, m

Equation (68) can be solved, and the results yield

The truncation of the series at A , is the nth-order Magnus approximation. Each A , is anti-Hermitian, thereby preserving the unitary nature ofexp(A) at any order. For periodic time-dependent Hamiltonians, the symmetry properties for U(t, f?), Eqs. (32)-(36), can be invoked to simplify the time propagation calculations. Using the first-order Magnus approximation, Walker and Preston ( 1977) have carried out an instructive comparison of the quantum and classical behavior in the MPE of nonrotating HF Morse oscillator. Their results indicated good agreement between the classical and quantum predictions for averaged quantities such as the energy absorption averaged over laser pulse times. However, the classical trajectory method fails to describe multiphoton resonance phenomena completely. The time development of the driven Morse oscillator is shown in Fig. 7. The system is initially (7 = 0) a nearly minimum uncertainty (Gaussian) wave packet and starts off in its ground state. Tick marks locate the eigenstates of the oscillator. Time is measured in units of optical cycles, and the strip shown here covers the time span between 9.0 and 10.0 optical cycles. The probability density is plotted above a line drawn through the potential at the average energy of the wave packet. The histogram at the left of each plot shows the relative populations ofeach of the unperturbed eigenstates. Arrows point to the average position and the average quantum number of the wave packet. Note that, as the anharmonic oscillator absorbs energy and moves up the vibrational ladder, it does not remain a coherent, minimum uncertainty wave packet but spreads and eventually breaks up. However, its average position executes a classical sloshing motion in time. Schek ef al. (1979, 1981) have applied the first-, second-, and third-order Magnus approximation to the study of multiphoton excitation dynamics of a truncated nonrotating Morse oscillator. They also examined the validity of the rotating-wave approximation (RWA). They found that the RWA breaks down for high-order MPE of a sparse many-level molecular level structure. This finding differs from a recent model study of the MPE of SF, (Clary,

SEMICLASSICAL FLOQUET THEORIES

223

FIG.7. The time development of the quantum HF Morse oscillator in a periodic driving field. The oscillatorstarts off in its ground state at time zero, and the time development from 9.0 to 10.0optical cycles is shown here. Note the quantum anharmonic oscillatordoes not remain a coherent, minimum-uncertainty wave packet, but that its average position executes a classical sloshing motion in time. (From Walker and Preston, 1977.)

1983). Using the first-order Magnus approximation and considering only the v3 mode excitation, Clary found that the RWA and the Floquet -Magnus calculations are in good agreement for two- and three-photon transitions. Leasure el al. (198 1) have carried out one- and two-photon excitation calculations for the rotating Morse oscillator diatomic molecules LiH, CO, IBr, and HF, using the first-, second-, and third-order Magnus approximation. Average photon absorption spectra are studied with respect to power broadening, dynamical Stark shifts, and line shapes. In some cases, effective two-state perturbative results agree well with the numerical results. The I4CO and I3COenrichment ratio is found to decline with increasing laser intensity and initial rotational temperature. Milfeld and Wyatt ( 1983)have reexamined the Magnus propagator. They gave formal expressions for the Magnus A operator up to fifth order. Also, they have derived the third-order interaction picture, and the third- and fourth-order Schrodinger picture matrix elements using the Magnus approximation for the semiclassical Hamiltonian. Using the second-order interaction-picture Magnus approximation, Wyatt et al. ( 1983) have recently

Shih-I Chu

224

carried out a computational study of the mode-selective MPE of a HCnonHeiles model anharmonic-oscillator system,

H = -f(#/dx2 + 8/dy2) + f(x2+ y2)+ A(xy2 - x 3 / 3 ) Two types of quasi-periodic states are known to exist in the Htnon - Heiles system (Hose and Taylor, 1982): Q’,the normal mode, and Q”,a local (bond) mode; highly mode-mixed states are designated as N (nonquasi-penodic). It is found that efficient (mode-selective)multiphoton excitation up a ladder comprising Q1states may be obtained by adjusting w and Eo. This result is analogous to the finding in the ab initio most probable path approach study of the MPE of SO2(Tietz and Chu, 1983b), where they found that efficientMPE pumping up the symmetric-stretch vibrational ladder is achievable and that the couplings to other ladder states (particularly the bending-mode states) are not as important as expected. D. OTHERMETHODS

I. Recursive Residue Generation Method Based on an extension of the recursive approach developed by the Cambridge solid-state physics group (Haydock, 1980)for the study of local state densities in disordered solids, Nauts and Wyatt (1983) have recently proposed a recursive residue generation method (RRGM) for the calculation of time-dependent transition probabilitiesin multistate quantum systems. The residues of Green’s functions, and hence transition amplitudes, can be computed directly from three sets of eigenvalues,without explicitly constructing eigenvectors.In order to generatethese sets of eigenvalues,the time-independent Hamiltonian H (such as the Floquet Hamiltonian) is converted to a Jacobi (tridiagonal) matrix J by the Lanczos recursion method (Lanczos, 1950).Let a, and b,, I (n = 0, 1,2, . . . ,N - 1) denote, respectively,diagonal and off-diagonal elements in row (n 1) of J;this self-energy and nearest-neighbor coupling energy define a link in the one-dimensional chain used to portray J. In the recursion scheme, each additionalstep “forges a new link in the chain.”Given the recursion vectorsin - 1) and In) in fast storage, the next vector, self-energy, and coupling energy are formed by the threeterm explicit recurrence relation (Haydock, 1980)

+

In

+ l)b,+, = Hln) - In)a, -In

- l)b,

(70)

where a, = (nlHn), b, = (nlHln - 1) = (n - 1 Hln), and where b,+l is chosen to normalizeln 1). The starter is1 l)b, = H ( 0 ) - lo)&. Since each recursion step generates a more distant environment of the transition of interest, the method usually converges for N, .Q: N. Similar to the most-

+

SEMICLASSICAL FLOQUET THEORIES

225

probable-path approach, most of the physics of the i +f transition is found to be concentrated in a relatively small subspace of the full Hilbert space. Nauts and Wyatt (1983,1984) have applied the RRGM method, in conjunction with the Floquet theory and the Magnus approximation for the time propagator, to the study of multiphoton excitation of overtones in a model anharmonic Morse oscillator coupled with a multimode harmonic bath, involving more than 3000 states. 2. Rotating Frame Transformation Method Whaley and Light ( 1984) have recently introduced a rotating-frametransformation (RFT) method, similar in spirit to the method developed by Energy

L u

f

----------10.-1)

10.-2,

10,-3>

10.-4>

10,-5>

FIG.8. Energy-level scheme for the fully quantized molecule-fieldstates of an anharmonic molecular system with a single laser frequency w,showing the effect of the rotating-frame transformation. Molecular-field states are denoted by la,- n), where a is the molecular state index and n is the photon number. Thick solid lines refer to the levels within the energy range fu/2 about the initial state l0,O). One-photon couplings between this subset of states are indicated as dotted lines. (From Whaley and Light, 1984.)

226

Shih-I Chu

Pegg and Series in magnetic resonance studies (Series, 1978),for multiphoton excitation of molecules. When one starts with the equations of motion in the interaction representation, the system is transformed with a unitary rotating-frame transformation to yield a separation of time-independent from time-dependent terms in the Hamiltonian. The transformation is chosen such that the former contain essentially all of the information relevant to the most nearly resonant multiphoton transitions from the ground state to all other states. Neglect of the time-dependent terms gives an equation of motion with only time-independent interactions which can be solved for the distribution of molecule-state populations as a function of time. The conventional tridiagonal rotating-wave approximation is seen to result from a particularly simple choice of the RFT. Although the Floquet theory is not employed in the RFT method, the time-independent RFT matrix is a submatrix of the infinite-dimensional Floquet matrix. In particular, the RFT submatrix is a selection of the molecule-field states closest in energy to the initial state within * 0 / 2 , subject to the restriction of retaining only one molecule-field state per molecular state (Fig. 8). The construction of such a finite matrix thereby destroys the periodicity of the Floquet solutions. The RFT method has been applied to the study of multiphoton dissociation (simulated by the addition of a phenomenological decay width to the energy of the highest state) of a model diatomic molecule (Whaley and Light, 1984). The R m method can be extended to interactions with several laser fields.

IV. Non-Hermitian Floquet Theory for Multiphoton Ionization and Dissociation The Floquet matrix methods described in Sections I1 and 111 involving the time-independent Hermitian Floquet Hamiltonian provide nonperturbative ab initio techniques for the treatment of bound-bound multiphoton transitions. These methods cannot, however, be applied to bound - free transitions such as multiphoton excitation (MPI) or dissociation (MPD) processes. A major recent extension of the Floquet theory is to generalizethe conventional (finite-level)dressed-atom or quasi-energy picture to include the complete set of bound and continuum states of atoms and molecules. This has the effect of giving each of the dressed or quasi-energy levels an intensity-dependent imaginary part (width) in addition to the usual field induced shift. Proper interpretation of the frequency and intensity depen-

SEMICLASSICAL FLOQUET THEORIES

227

dence of the complex quasi-energies give rise to MPI or MPD rates. In this section we discuss the recent development in the non-Hermitian Floquet theory and some other related approaches. A. NON-HERMITIAN FLOQUET MATRIXFORMALISM

The non-Hermitian Floquet matrix formalism was first developed by Chu and Reinhardt (1977, 1978) and Chu (1978a) in studies of MPI of the H atom in intense monochromatic laser fields. The theory was later extended to a complex quasi-vibrational energy formalism for MPD of molecules (Chu, 198 I ; Chu et al., 1983a). The theories employ the use of two major conceptualizations, namely, complex coordinate (or dilatation) transformation (Aguilar and Combes, 197 I ;Balslev and Combes, 197 I; Simon, 1973; Reinhardt, 1982; Junker, 1982; Ho, 1983) and square-integrable (L2)continuum discretization (Yamani and Reinhardt, 1975). 1. Multiphoton Ionization in Linearly Polarized Fields

+

Corresponding to the time-dependent Hamiltonian H = H(atomic) F r cos wt describing the interaction of an atom with a monochromatic field, an equivalent time-independent Hamiltonian I?Fmay be written in analogy with the semiclassical Floquet Hamiltonian of Shirley. The resulting block structure is shown in Fig. 9, where th_e VJ,, ? are dipole coupling elements and the angular momentum blocks S,P, D . . . represent the projection of the atomic electronic Hamiltonian onto states of total L = 0, 1,2, . . . , etc. Thus, for example, in the case of the H atom, the S block consists of the Is, 2s, 3s, . . . , ns, . . . bound states and the entire ks Coulomb continuum. The Hamiltonian of Fig. 9 has no discrete spectrum, and the existence of Floquet solutions of the time-dependent Schrodinger equation is not established (Young et al., 1966; Zel’dovich, 1973). However, writing the time-evolution operator as

‘I

exp(- i H F t l h )= -

2xi

dz

exp(- iztl h ) Z-fiF

(71)

gives the usual result (Goldberger and Watson, 1964) that the time depennear the real axis but on higher dence is dominated by poles of ( z Riemann sheets, and that the complex energies of the poles are related to positions and widths of the shifted and broadened dressed states. These complex pole positions may be found directly from the analytically continued Floquet Hamiltonian ,fiF( O), obtained by the dilatation transformation r + eier. This transformation effects an analytical continuation of

Shih-I Chu

228

\

\

WHERE

A =

\

\

AND 8.

\

\

FIG.9. Structure of the non-HermitianRoquet Hamiltonian for MPI of H atom in linearly polarized monochromatic fields. The Hamiltonian is composed of Roquet blocks, of type A , which are in turn composed of angular momentum blocks S, P,D, . . . ;both types of blocks are coupled by the dipole coupling elements V,,,*, . (From Chu and Reinhardt, 1977.)

SEMICLASSICAL FLOQUET THEORIES

229

(z - AF)-Iinto the lower half-plane on appropriate higher Riemann sheets, allowing the dressed states to be determined by * * solution * of a non-Hermitian eigenproblem. In practice the atomic blocks S, P,D, . . .were discretized by use of a finite subset of the complete discrete Laguerre basis r1+1e-Ar'2 L:1+2(lr),n = 0, 1, 2 . , . , which gives a Pollaczek (Yamani and Reinhardt, 1975)quadrature representation of the bound and continuum contributions to the spectral resolution of the hydrogenic Hamiltonian. To the

I

I

0.30

0.35

I 0.40 w (0.u.)

I 0.4 5

I 0.50

FIG.10. Intensity-dependentgeneralized cross section, a,, for ionization ofthe bare 1 sstate of the H atom in the frequency region dominated by resonant two-photon ionization -, F,, = 5 X lo-' a.u.; - -, F,, = 0.01 a.u.; - . -, F,, = 0.025 a.u.

Shih-I Chu

230

extent that enough Floquet blocks (as well as the basis size and the number of angular momentum blocks) are included to obtain convergence, a completely nonperturbative result is obtained in that all orders of perturbation theory are included, and all relevant processes involving different photon numbers are simultaneously and self-consistentlyincluded. Using this procedure, Chu and Reinhardt ( 1977)obtained converged intensity-dependent “generalized” cross sections (Lambropoulos, 1976), aN = (ionization rate)/(intensity)N (72) for an H atom subjected to intense monochromatic single-mode linearly polarized radiation. Figure 10 shows the flux dependence of resonant twophoton ionization across sections a, in the neighborhood of 1s 2p and 1s -B 3p one-photon resonances. At low fields(F< 10-4 a. u.) a, is independent of field strength, except very close to resonance, and reduces to the perturbation results (Chan and Tang, 1969).As the fields approach 1 or 2% of atomic field strengths, the intermediate 2p state broadens substantially and the strong near-resonant enhancement close to o = 0.375 (a.u.) is remains essentially field independent. quenched; however, off resonance CT, More recently, Maquet ef al. (1983) have exploited in great detail the relationship between the complex poles of the resolvent of the Floquet Hamiltonian and continued-fraction perturbation theory (Swain, 1975; Gontier et al., 1976; Yeh and Stehle, 1977; Mower, 1980). Connection is also made between various truncations of the matrix Floquet Hamiltonian and diagrammatic representations of the infinite-order perturbation summations implicit in extraction of eigenvaluesof the Hamiltonian. Detailed discussion of time evolution in two-, three-, and four-photon ionization processes are given in the paper by Holt et al. ( 1983)where they make connection of the non-Hermitian Floquet theory with the standard two-level model (Allen and Eberly, 1975; Beers and Armstrong, 1975). Experimental study of resonant MPI of atomic hydrogen has become feasible only very recently (Kelleher et al., 1985).

-

2. Multiphoton Ionization in Circularly Polarized Fields The extension of the non-Hermitian Floquet matrix method to the study of MPI of atoms induced by intense circularly polarized radiation was considered by Chu ( 1978a)and more recently by Tip ( 1983). The semiclassical time-dependent Hamiltonian in the electric dipole approximation is H(r, t ) = Hatomic(r) - eEo(xcos ot

+

+ qy sin ot)

(73)

where q = 1(- 1) corresponds to the left (right) circularly polarized light, and the solution of the Schrodinger equation has the form given in Eq. (6), Y(r, t ) = exp(- k t / h ) @ ( r ,f ) . The time dependence of the periodic function

SEMICLASSICAL FLOQUET THEORIES

23 1

@(r, t), Eq. (7), can be removed by transforming to a system of coordinates that rotates with the frequency of the field (Bunkin and Prokharov, 1964)

@kt) = exp(iqotL,/A.M,(r)

(74)

where 4,(r) satisfies the time-independent eigenvalue equation &r>4,cr>= If4tomic(r)-

WL- eEoxIA(r) = E4,W

(75)

Here L, is the z component of the orbital angular momentum operator, Q is the quasi-energy operator, and E is the quasi-energy eigenvalue. Salzman ( 1974b)showed that the time-evolution operator for a spherically symmetric system driven by a circularly polarized field can be solved in closed form: U(t) = A(t)eiQf/*, where A(t) = exp(iqotL,/h) is a unitary factor which is periodic in time and may be ignored in calculatingtransition probabilities. Similar to the Floquet Hamiltonian in the linearly polarized field, the quasi-energy operator Q has no discrete spectrum. The complex poles of the resolvent (E - Q)-'are located near the real axis, but on higher Riemann sheets. These complex quasi-energy states may be determined by the analytical continuation of the quasi-energy operator obtained by the complex coordinate transformation (Chu, 1978a): Q(r) 4 &reto) = [Hammic(reio) - qoL, - e ~ , x e ~ ~ ] (76) A typical structure of the quasi-energy operator in matrix representation is shown in [email protected]. It consists of diagonal angular momentum blocks L, (where L = S, p, d, . . . ,etc., and rn are the magnetic quantum numbers) and off-diagonal dipole coupling blocks of types Xand Y [typeXcouples L, to (L 1),-12 whereas type Ycouples L, to (L 1),+,]. The quasi-energy spectrum of Q(reie) may be discretized by the orthonormal Lagueme basis described before. Using this procedure, Chu ( 1978a)calculated intensity-dependent generalized two-photon ionization cross sections u, for an H atom in a strong circularly polarized field. The spectral line profiles of the resonant u2versus frequency are similar to that shown in Fig. 10. The quenching of the resonant profile in very strong fields is seen to be due to the large avoided crossings of quasi-energy levels. Similar observations were noted by Preobrazhenskii and Rapoport ( 1980),who studied the same problem using the method of complex coordinates and perturbation theory.

+

+

3. Photoionization in Intense Magnetic Fields The method of complex quasi-energy formalism may be extended to the problem of photoionization in intense magnetic fields. The Hamiltonian for

232

Shih-I Chu

FIG.I 1. Structure ofthe non-Hermitianquasi-energy Hamiltonian for MPI ofan H atom in circularly polarized monochromatic fields. The Hamiltonian consists of diagonal angular momentum blocks L , (whereL = S,P,D, . . , ,etc., and m are the magnetic quantum numbers) and off-diagonal dipole coupling blocks of types X and Y.

a hydrogenlike atom in the presence of both a uniform magnetic field B 11 z and a coherent monochromatic field of frequency o is given by

H(r, t) = H,(r)

+ V(r, t )

where H , is the Zeeman Hamiltonian,

+

Hz(r) = HnmrniC(r) ~

i+,+ p ~ Z ( +x ~y 2 )

and V(r, t ) = -eE,z cos o f = - eEo(1 - qn/w)(x cos of

+ qy sin at)

(LP)

(CP) where R = eB/2pc is the cyclotron frequency. Analogous to the B = 0 cases, the time-dependent Hamiltonian H(r, t) can be transformed to a time-independent Floquet Hamiltonian I& or the quasi-energy operator Q depending upon the light polarization. In the case of a circularly polarized (CP)field, for example, the quasi-energy operator becomes

Q = Hamrnic(r) + (Q - qo)L, + ipiP(x2 + y2) - eE,( I - q ~ / o ) x(77) Note that this Hamiltonian is similar in form to that of atoms in crossed

SEMICLASSICAL FLOQUET THEORIES

233

electric and magnetic fields: Assuming B 11 z and E 11 x , the Stark-Zeeman Hamiltonian H,, is (Chu, 1978b) Hsz

+ Qi,+ ipQ2(x2+ y 2 )- eEox

= Hat0~&r)

(78) Thus the time-dependent magnetophotoionization problem is essentially equivalent to a static crossed electric and magnetic field problem. For magnetic field strength B above a certain critical value B,= rn$e3c/h3= 2.35 X lo9 G, the structure of the hydrogen atom consists of bound states (below the first N = 0 Landau threshold) and series ofautoionizing states (below each excited Landau threshold N 3 1) (Edmonds, 1971; Chu, 1978b). These autoionizing resonances arise from the coupling of the electronic motions in the transverse and longitudinal directions. The determination of these autoionizing states can be facilitated by the use of the method of complex coordinates (Fig. 12). Recently Bhattacharya and Chu ( 1 983) have developed a complex-coordinate coupled-Landau-channel (CCCLC)method for the study of autoionizing resonances in intense magnetic fields of astrophysical interest. They have calculated the positions and widths of several autoionizing resonances of the H atom below the first two excited Landau thresholds for magnetic fields in the range of lo9to 1OI2 G. Friedrich and Chu (1983) have studied the same problem using the conventional close-coupling-scatteringmethod. More recently, Bhattacharya and Chu ( 1985) have extended the complex quasienergy method along with the Dilototion tronsformotion HZ

-

Hz(Q)

CT [ HZ(ff)]

cr ( HZ) Quasi- Londou

x x....

Bound stotes

Resononces (hidden)

Bound stotes

Resononces (exposed)

FIG. 12. Effect of the dilatation (or complex coordinate) transformation, r 4reia,on the spectrum,a(H,), ofan atomiczeeman HamiltonianH,. The bound states [theCoulomb series below the first ( N = 0) quasi-landau threshold] are invariant to the transformation while the continua rotate about their respective quasi-landau thresholds, exposing complex resonance Coulomb series(abovethe N = 0 Landau threshold)in appropriate strips ofthe complex energy plane. These exposed autoionizing resonances may be determinedby the use ofthecomplex-coordinate coupled-landau-channel method.

234

Shih-I Chu

CCCLC method for the study of the (bound to quasi-bound) resonant photoabsorption spectrum of hydrogen atom in intense magnetic fields. 4. Complex Quasi- Vibrational Energy Method for

Multiphoton Dissociation Unimolecular MPD reactions of polyatomic molecules induced by intense pulsed IR lasers have received much attention since their discovery in the early 1970s (Isenor et al., 1973). Many excellent review articles of experimental progress are available which also include discussions of various theoretical approaches (see, for example, Letokhov and Moore, 1976; Bloembergen and Yablonovitch, 1978; Shulz et al., 1979; Cantrell et al., 1980; Quack, 1982). Although the current theoretical models are able to explain qualitatively the distinctive features of MPD phenomena, a priori study has so far been hampered by the extreme complexity of the molecular level structure involved. A diatomic molecule is the extreme case of a small polyatomic molecule which has only one vibrational mode and no quasi-continuum. It is the ideal place to examine the effects of coherent MPE. Experimentally, MPD of diatomic molecules from ground vibrational states has not been observed at the field strengths currently available (
where rand R stand for electronic and nuclear coordinates, respectively, and the semiclassical time-dependent Hamiltonian 7f in the electric dipole ap-

SEMICLASSICAL FLOQUET THEORIES

235

proximation, is given by

N(r, R, t ) = H(r, R)

+ p(r, R)

E, cos ot

(80)

Here,

+

H(r, R) = fR &(r, R) is the field-free total Hamiltonian (with fRand fie, being the nuclear kinetic energy operator and the electronic Hamiltonian, respectively),p(r, R) is the dipole moment operator, Eo is the electric field amplitude, and o is the EM frequency. One first seeks solutions of the field-free Schrodinger equation,

M r , R)tk(r, R) = E',O'&(r, R) (81) In accordance with the usual assumption of weak coupling of nuclear with electronic motion, one can consider R as a slowly varying parameter, and the solution of Eq. (8 1) can be written as tk(r, R) = @k(R)V/k(r, R) 82) where mk(R)and yk(r,R) are the nuclear and electronic wave functions, respectively. In the adiabatic representation, the electronic wave function yk(r, R) is an eigenfunction of H&, R) with eigenvalue U,(R): i.e., m r , R)Wk(r, R) = Uk(R)Wk(r, R) (83) We shall assume (yk)forms a complete orthonormal set. Without loss of generality, let us consider the vibronic transitions between two electronic states. Since %(r, R, t ) is periodic in time, the dynamic problem Eq. (79)can be transformed into an equivalent static Floquet eigenvalue problem fiF(R)6c(R) = d c ( R ) (84) In Eq. (84), the time-independent Floquet Hamiltonian fiFis an infinite matrix with rows identified by the pair of indices an, and columns by pm: ( f i F ) a n $ m = fR

+ UJR) + n h 018,

Jnm

+ [+~czp(R) E01 8 n n , m - n * l ( 1

- dczp)

(85)

where

p,(W

=

dr W 3 , R)P(r, R)y/B(r, R)

is the electronic transition dipole moment. Here we use Greek letters corresponding to electronic states, while italic letters will denote photon Fourier components. We order the components so that a runs over electronic states

Shih-I Chu

236

before each change in n. In the case of two electronic states, for example, the Floquet Hamiltonian I?Fhas the structure shown in Fig. 13. We see HFhas a periodic structure with only the number of w's in the diagonal elements varying from block to block. The structure endows the eigenvalues and eigenvectors of fiFwith periodic properties. The Floquet matrix so constructed is a real, symmetric matrix and contains no discrete spectrum in the real energy axis. Similar to the MPI of atoms described earlier, the complex poles of the resolvent [z - fiF(R)]-' may be found directly from the analyticallycontinued Floquet Hamiltonian RF(Reie)obtained by applying the dilatation transformation to the dissociative nuclear coordinateR, R + R exp i0.In practice, the diagonal electronic blocks NJR exp i0) are expanded and discretized by use of appropriate L2 radial (such as harmonic oscillator type) vibrational bases. The desired com-

A-2&11

A- 4 0 1

0

5

B=

0

0

FIG.13. Structure ofthe Floquet Hamiltonian for molecular multiphoton dissociation from the ground ( i = 1) to an excited electronic state ( i = 2). The Hamiltonian is composed of diagonal Floquet blocks, of type A, which in turn are composed of two electronic blocks characterized by the kinetic energy operator fRand the potential energy curves U,(R), i = 1,2, where R is the internuclear separation. The rovibrational structure within each electronic block can be discretized by using appropriate L2basis. The rovibronic states are coupled by the electric dipole interaction. (From Chu, 198 I.)

237

SEMICLASSICAL FXOQUET THEORIES

plex quasivibrationalenergy (CQVE)states can be identified by the stationary point of the 6 trajectories of the eigenvalues of fiF(Reie).Once the CQVEs are found, the intensity-and time-dependent MPD rates can also be determined. The CQVE method has been extended to the study ofphotodissociation ( lsag- 2 ~ 0 , )of Hz+ in intense laser fields (Chu, 198 1) and to two-photon dissociation of vibrationally excited Hz+(1 sap vi = 6 - 12) (Chu et al., 1983a). An independent calculation of (weak-field)two-photon dissociation cross sections a, was also performed using the inhomogeneous differential equation (IDE) approach of Dalgarno and Lewis (1955). The two sets of data were found to be in very good agreement. The cross sections 0, for the low-lying vibrational states are negligibly small but increase rapidly with increasing vibrational quantum number q, suggesting that experimentally accessiblepowerful lasers can be used to probe the highly excited vibrational states of the ground electronic state of a homonuclear diatomic molecule. The Hz+ cross sections are largest close to the two-photon dissociation thresholds and decrease monotonically with increasing photon energy. The pattern is different for HD+, where the dominant features are multiphoton resonant structures (Laughlin and Chu, 1985).

B. OTHERMULTIPHOTON IONIZATION AND DISSOCIATION METHODS Instead of analytically continuing the Floquet Hamiltonian into higher Riemann sheets via the method of complex coordinates, HF(r)+ HF(rele), as discussed in Section IV,A, Faisal and Moloney ( 198 1 ) have developed an alternative way of constructing the non-Hermitian Schrddinger equation. They derive a non-Hermitian Hamiltonian from a given Hermitian one by first identifyingthe part of the vector space associated with the continuum of interest and then projecting it out from the total vector space under the outgoing boundary conditions. The motion in the discrete space is thus modified (due to the elimination of the continuum) and shows decaying behavior as a function of time. In order to impose the outgoing boundary condition, they used the well-known projection operator technique to isolate the Green’s operator associated with the continuum part of interest. The Hilbert space of a Hermitian Hamiltonian can be divided into two parts (Feshbach, 1962; Cohen-Tannoudji, 1968) with the corresponding projectors P and Q,such that Q projects onto the continuum of interest. P and Q satisfy the usual properties of projection operators (P Q = 1, P = Pz, Q = @, PQ = Q P = 0). Starting from the Schrddinger equation in the whole vector space, Hly/) = lily/), with the state vectors I y / ) and the energy

+

238

Shih-I Chu

spectrum E, they arrive at a reduced Schrodinger equation, fi(pW) = E(pW), in terms of the non-Hermitian Hamiltonian fi: 1 QHP Q(E-H)Q , - iaPHQ S [ Q ( E- H ) Q ] Q H P

where P signifies taking the Cauchy principal value. The 6 part shows the required outgoing nature of the Green’s operator through the factor - in. The continuous range of its spectrum allows the S functional distribution in Eq. (86) to be meaningfully defined as an integration operation. By comparing the properties of the original Hermitian and the reduced non-Hermitian Hamiltonians, it can be shown that this non-Hermitian formulation is consistent with the theorem of Fock and Krylov (1947) for the description of truly decaying states. Faisal and Moloney (198 1) further discussed a self-consistent nonunitary algorithm to obtain expressions for the time-dependent transition probabilities. They then applied the method to the study of the time dependence of two- and three-photon (resonant and nonresonant) ionization of an H atom in intense monochromatic fields. An interesting effect, that of field-induced quantum beats in MPI between I-degenerate states, was noted. Figure 14 shows such an effect for the three-photon ionization of H( 1s)via the intermediate l-degenerate 3s and 3d states. Figure 14a shows a sequence of quantum beats in the total ionization probability Pio,(t), while Fig. 14b exhibits a similar effect for the survival probability P,,(r) of the initial ground state. The beats occur with a period T = 2a/R, where R = ejs - E~~is the difference of the perturbed quasi-energies of the initially degenerate ec) and e\Od) states. Leforestier and Wyatt (1982, 1983) have recently considered two semiclassical methods for the time-dependent studies of laser-induced dissociation (LID) processes. The first method (Leforestier and Wyatt, 1982) is essentially a time-dependent version of Wigner- Eisenbud R-matrix (Lane and Thomas, 1958)theory. The method employs the Floquet amplitudes to construct a set of field-dependent L2basis functions (dressed states) within a “reaction shell.” Then, by standard R-matrix procedures, the internal wave function is matched to scattering wave functions for the multiple coupled dissociation coninua in the external region. They applied this method to one photon dissociation from a truncated square well which supports two bound states. Their second method (Leforestier and Wyatt, 1983)involves the use of L2 discretization of the dissociation channels. To accomplish this, they use an optical potential method allowing the continuum channels to be represented by a set of pseudobound states coupled to the actual bound states of the molecule. The evolution of the system is thus amenable to the standard

SEMICLASSICAL FLOQUET THEORIES

239

0.5 1.o Ti FIG. 14. Time evolution of the doubly degenerate resonant multiphoton ionization of the hydrogen atom. Shown are (a) the total growth probability of ionization, and (b) the survival probability of the initial ground-state atom. The field photon energy h w = 0.22222 a.u., corresponds to a simultaneous two-photon resonance to thedegenerate 1s- 3sand Is- 3dtransitions. The time scale parameter y-' = 77 I .4 ps is related to the width of the dressed state corresponding to the unperturbed initial state. The field strength is Eo = lo-' a.u. Note the field-induced quantum beats in (a) and (b). (From Faisal and Moloney, 1981.)

0

problem of laser-induced transitions between bound states. The long-time behavior is computed by using Floquet analysis. The method is applied to elucidate the multiphoton dissociation dynamics of a model nonrotating six-bound-state Morse oscillator. Whaley and Light (1 982) have developed a quantum-mechanical method for MPD of diatomic molecules, using a quantized representation ofthe laser field and a quantum description of the molecule. In this method they use the time-independent R-matrix formulation to obtain transition matrix elements between the ground vibrational state and a scattering state matched asymptotically to the outgoing component of a dissociative state of the diatom. Floquet theory is not employed in this method. For a trial calculation on HF, they found dissociation at experimentally detectable levels in 1 ns) for field strengths of loLsW/cm2, depending on (Pdiss 3 the particular form used for the dipole function. The intensity dependence of the dissociation rate was found to vary considerably with different functional forms of p ( r ) .

V. Many-Mode Floquet Theory The Floquet approaches described in previous sections require the semiclassical Hamiltonians to be explicitly periodic in time and are therefore

240

Shih-I Chu

applicable to problems involving strictly monochromatic radiation fields. Recently there have been extensive studies of atomic and molecular processes involving the use of two lasers with different frequencies. Examples are: multiphoton double-resonance experiments (Jacques and Glorieux, 1980; Galbraith et al., 1982), collisions in two laser fields (de Vries ef al. 1980; George, 1982), MPD of polyatomic molecules by two IR lasers (Ambartzumian and Letokhov, 1977; Alimpiev et al., 1983; Koren, 1983), and multiple quantum transitions in double-frequency pulsed NMR experiments (Zur efal.. 1983), etc. In addition, a broad class of various phenomena pertaining to nonlinear optics is based on experiments performed with multimode laser fields (for a recent review, see Delone et al., 1980). Exact treatments of these multifrequency problems are beyond the scope of conventional Floquet theories. Most previous semiclassical approaches are based on approximations of one kind or another. The most commonly used approach is the so-called “generalized rotating-wave approximation” (GRWA)- an extension of the RWA for the single monochromatic field case (Ramsey, 1955; GuccioneGush and Gush, 1974; Tsukada, 1974; Bonch-Bmevich et al., 1979; Goreslavskii and Krainov, 1979; Kancheva ef d.,198 1; Zur ef al., 1983; Zur and Vega, 1983). This approximation takes into account only graphs in which the absorption of a photon of a particular type is accompanied by the subsequent emission of a photon (not necessarily ofthe same type) and vice versa. The GRWA method is expected to give reasonable results only in some special cases, for example, a two-level system subjected to bichromatic fields with frequencies a,and a2close to the level separation oo.For problems involving arbitrary detunings, the GRWA in general cannot be justified and fails to predict many nonlinear features and multiphoton transitions in strong fields. Recently an exact extension of Shirley’s (one-mode)Floquet formalism to a generalized many-mode Floquet theory (MMFIJ has been found (Ho et al., 1983; Chu and Ho, 1984; Ho and Chu, 1984a). This makes it possible to treat the time-dependent problem of any finite-level system exposed to polychromatic fields as an equivalent time-independent infinite-dimensional eigenvalue problem.

A. GENERAL FORMALISM For simplicity let us consider now the nonlinear response and multiphoton excitation dynamics of a two-level (or spin-f) system driven by two intense linearly polarized monochromatic fields. The Schrddinger equation,

24 1

SEMICLASSICAL FLOQUET THEORIES

within the electric dipole approximation, can be written as in Eqs. ( I ) and = 1, %(r, t ) = H(r, t ) - i apt %(r, t)Y(r, t ) = 0,

(a,

and

+

i = 1, 2 Vi(r, t ) = - A E, cos(wit $i), (88) Ho is the unperturbed Hamiltonian with orthornormal eigenstates (la),I/?)) and eigenvalues (E,,EB),p is the electric dipole moment operator, E i , wi, and $iare, respectively, the (peak)amplitude, the frequency, and the initial phase of the ith monochromatic field (i = 1, 2). For the Hermitian operator %(r, t), one can now introduce the composite Hilbert space R 0 T , 0 T2. The spatial part R is spanned by (la), IP)), and, in the two-mode temporal part, T, is spanned by exp(inlwlt) and T, by exp(in202t),where n,,n2 = 0, 1, f 2, . . . . In the two-mode Floquet theory (Ho and Chu, 1984a),the time-evolution operator in its matrix form is shown to be

*

Uylyl(t,to) = (71 I =

x nl

w ,to)lrz), (ylnln21

71 Y

72

= a or

P

exp[-ifiF(t - tO)11y2°0)

nl

X exp[i(n,w,

+ n202)t]

(89) Here fiFis the time-independent two-mode Floquet Hamiltonian defined in the generalized Floquet state basis, lyn1n2)

= Ir) @In,) @In,)

and satisfies the infinite-dimensional eigenvalue equation,

g

(71 4n21fiFIY2kl k2)(y2k&219

= A(y1n1 n2V)

(90)

Y1 I 1

where (Ylnln2lfiFly2klk2)

=!!~~~kL'n2-k21

+ ( n l a l + n 2 0 2 ) d y t y l dnIkl dnzkl with

(91)

Shih-I Chu

242

In Fig. 15 the structure of the matrix BFisdisplayed by ordering components so that y runs over the unperturbed states, a andp, before each change in n, , and n, , in turn, runs over before n, changes. The quasi-energy eigenvalues (Ayflln2) and their corresponding eigenvectors ( IAyflln2)) of fiFhave the following periodic properties forms, namely,

(95) (71 ,n,+ ql,n2 + q21&.nl+ql,nz+q2) = ( ~ l n 1n21Ay2nln2) Using Eq. (89) and the periodic relations Eqs. (94-99, the transition probability averaged over the random initial time to can be written as

Performing the long-time average over t - to in Eq. (96) gives

Furthermore, it is easy to show that Eq. (97) can be cast into an alternative form, with w othe unperturbed energy spacing of the two-level system, -

pa+, = f[l - 4 ( d ~ y m / ~ w , ) z l (98) which is in essence the extension of Shirley's (1 965) expression for the case of a two-level system in a monochromatic field. B. GENERALIZED ROTATING-WAVE APPROXIMATION Let us now compare the exact two-mode Floquet theory (TMFT) with the GRWA, the latter being equivalent to a two-rotating-field treatment in this case. We shall use the notation [nI , k n,] to denote n photons of the first field ( w I ) are absorbed, and nz photons of the second field ( w2)are absorbed (+) or emitted (-). The allowed transition types, according to the TMFT, are [n 2m 1, kn],with n, m = 0, 1, 2, . . . , and the number of MPE pathways for each type is Np= (2n 2m l)!/[n!(n 2m l)!]. On the other hand, only the [n 1, - n] type is allowed in the GRWA and Np= 1.

,

+ +

+

+ +

+ +

243

SEMICLASSICAL FLOQUET THEORlES

H, =

***

A =

C

X

0

c

=

0

0

F

]

C-ZW,l

.=mi

FIG. 15. Linearly polarized bichromatic Floquet Hamiltonian for the two-level system, constructed in a symmetric pattern. w , and w2 are the two radiation frequencies, and V W , i = I , 2, are the electric dipole coupling matrix elements for the ith field. Note that the diagonal block A possesses an identical Floquet structure to that ofthe one-laser problem. (From Ho and Chu. 1984a.)

The GRWA bichromatic Hamiltonian is shown in Fig. 16 for comparison. Note that it possesses a periodic structure with only the number of (oI - WJ’S in the diagonal elements varying from block to block. Figure 17 shows an example of the comparison of the T M R and GRWA results for the nonlinear response of a two-state system in an intense linearly

E, - 2w, + 2w2 b:

HFWA=

0

b2

0

0

0

0

0

0

0

0

0

E,-w,+w2

b,

0

b:

E0-2w,+w2

0 0

b2

b:

0

0

E.

bi

0

4

b:

0

b:

E,-w,

0

0

0

0

E,+wl-w2

b:

0

b:

0

0

0 i

b,

E,-3w,+2w2

0

0

0

0

0

b,

Eo-

0

0

b:

0

w2

0

b2

0

0

E,+2w,-2w2

b,

b:

E, + w , - ~ w ,

FIG.16. GRWA bichromaticHoquet Hamiltonian for the two-level system. wI and w 2are the two radiation frequenciesand b,, i = 1,2, is the dipole coupling matrix element for the ith field.

245

SEMICLASSICAL FLOQUET THEORIES a a

0.c

As 0 -1 All 3 -3 As-1

0

A02-2

-O*‘P

As-2

1

1-1

As-3

2

boo a

0.7

0.6 0.5 0.4

0.3 0.2 0.1

o.a

I

1

-5

-3

I -1

I

I

1

3

I 5

X

FIG. 17. Bichromatic quasi-energies A and time-average transition probabilities Fa+ as functions of the dimensionless resonance parameter x = (20, - o1- 02)/(w, - W J . Solid curves for exact MMFT calculations and dotted curves for GRWAs. Parameters used are wI= 1.0, w2 = 0.799, Ib,l= 0.05, and lbll = 0.075 (arbitrary units).

polarized bichromatic field. Figure 1 7a depicts the bichromatic quasienergies I , as a function of the dimensionless resonance parameter x = ( 2 0 , - w 1- w2)/(w1 - w 2 )at w , = 1.0, w 2= 0.799, V$ = 0.05, V”$ = 0.075, and 4I= 42= 0 (arbitrary units) for both the TMFT (solid curves) and the GRWA (dotted curves) calculations. The corresponding long-time

Shih-I Chu

246

average transition probabilities Fa+Bareshown in Fig. 17b. The parameterx indicates the order and the (field-free)position of resonant MPE transitions. Positive x implies that photons of the higher frequency (0in this case) are absorbed while those of lower frequency ( 0 2are ) emitted, and vice versa. One sees that for x < 0, the GRWA predicts larger shift and width for each resonance than the exact TMFT, whereas the reverse is true for x > 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-

SEMICLASSICAL FLOQUET THEORIES

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.

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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.

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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.

SEMICLASSICAL FLOQUET THEORIES

25 1

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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 << 1 or B e 109~. We will define a strong magnetic field as one in which the magnetic interaction becomes dominant ( h a c> 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)

SCATTERING IN STRONG MAGNETIC FIELDS

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

SCATTERING IN STRONG MAGNETIC FIELDS

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)

SCATTERING IN STRONG MAGNETIC FIELDS

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

SCATTERING IN STRONG MAGNETIC FIELDS

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

SCATTERING IN STRONG MAGNETIC FIELDS

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)

SCATTERING IN STRONG MAGNETIC FIELDS

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

<1

(86)

This gives the elastic amplitudes

and for inelastic transitions

The corresponding expressionsfor the cross sections are, in each rn subspace, closely approximated by

and

knt I Vnn(kn T kn,)12kf, y ti4 kn (a2 kf,)(P2 kit)

K p2

gnn’ (+)=---

with

+

+

SCATTERING IN STRONG MAGNETIC FIELDS

27 1

It is clear that the first Born singularity is removed except in the unlikely event that p2 = 0. Ohsaki has applied these results to scattering by a screened Coulomb potential

at magnetic field strength parameters 0.10 and 0.20. At y = 0.2 the first Landau level lies at 2.72 eV and the second at 8.166 eV. His results, shown in Figs. 1 and 2, show that both forward and backward elastic and inelastic processes give cross sections which are finite everywhere. The corrections to the FBA are small away from threshold in this model. Note that the forward and backward amplitudes are equal at a threshold in this model.

1 OZt

1

o2

0

2.72 5

10

15

INCIDENT ENERY (eV)

FIG. 1. First Born approximation (FBA) and partially summed Born series approximation (HMBA) for electron scattering from the100k) state by the screened Coulomb potential (2.54) in a magnetic field ofstrength y = 0.1. Here (+) indicates forward and (-) backward scattering. (From Ohsaki, 1983a.)

M. R.C. McDowell and M. Zarcone

272

-

L

1 o2

0

10 15 5 INCIDENT ENERY (eV)

FIG.2. First Born approximation and partially summed Born series approximation for electron scattering from the 10, 0, k ) -B II , 0, k ) transition. (From Ohsaki, 1983a.)

D. COUPLED-EQUATIONS FORMULATION The cross sections given in the previous section for the FBA and the HMBA cases are independent of the sign of the potential; that is, repulsive and attractive potentials give the same result. This was expected in the FBA, while in the HMBA it is due to the straightforward extension of the field-free Green function formalism. In this formalism the solution of the scattering equation is determined by the boundary conditions at positive and negative infinites along the field direction, while the behavior at the origin (the center of the scattering potential) is not used. We note further that the FBA matrix element diverges logarithmically for a Coulomb potential with weak screening for transitions with zero momentum transfer (k = k' - k-0). In this case for the elastic process lnrnk) lnmk) we have (Ventura 1973) +

SCATTERING IN STRONG MAGNETIC FIELDS

273

) r'(v)lr(v) is the logarithmic derivative of the r function. In where ~ ( v = order to overcome such difficulties, a higher-order treatment is required. Onda (1978), in the energy range far from Landau resonances, and Ohsaki ( 1983a,b), for near-resonant collisions, have developed a coupled-equations formalism for potential scattering in a strong magnetic field. The scattering function may be expanded as yYt(r)=

C

Qnm

(P)gnm( z )

(94)

n. m

Substituting into Eq. (42), and projecting out on the Qnm(p),we obtain the standard coupled equations

with Vnm.n*m#(Z)= 2

I

Q,*,(P)@n~m~(P)

(96)

If V(r)has axial symmetry, m is conserved and the coupled equations reduce to

n,n'=O, 1 , 2 , . * . for each rn subspace separately. Let us further assume that (nl V(r)ln')is symmetric with respect to inversion through the origin, and drop the label rn, writing (98) For such potentials parity is a good quantum number, and we can form even and odd solutions, decoupled from each other Vnn4z) = V n n 4 - Z )

. approxiThe coupled equations must be solved in the half-space ( 0 , ~ )We mate by truncating the expansion to N states and, to obtain N linearly independent solutions, require that at z = 0 g $ ( O ) = d g t b ( O ) / d z = d&fl

( 1OOa)

g$$O) = dggJ(o)/dz= 0

( 1OOb)

and we distinguish by g @ ) ( z ) the solutions coupling different pairs of chan-

274

M . R. C. McDowell and M. Zarcone

nels. We then form the general solution N

A @ )=

[l,S:b(z)

+ Jpg2JI

(101)

so that for an incident wave in the nth channel the asymptotic solution is A(z)

+ Ana(+)leik~'

( 102a)

e i k + A n a (-)e-ikaz

( 102b)

= [Sfla

z++-

=

na

z--m

It is convenient to choose asymptotic forms g:&)

-

I-+-

-

z-8-m

Fa,COS(ka z + Ja/9) FasCOS(~, z - Saa)

(103a) (103b)

and ( 103c)

(103d) Using Eqs. ( 100)-( 103),the scattering amplitudes may be expressed in terms of the phase shifts S,,, qaa and the mixing coefficients Fa,, Sa,(als= 1, . . . ,N), i.e., 4N2quantities. Thus we need to solve the Nsecond-order coupled equations 2Ntimes (Neven, Nodd). Knowing these parameters, the cross sections are given by

Ohsaki ( 1983b) then considers a two-state approximation explicitly. We have

($+ kf)gg."(z)

=

+

Viigy)(Z) V i j g p ( Z ) i, j = 1, 2

(105)

For elastic scattering we can write the even and odd solutions as G2g~~-N(')cos(k1zfSy)

$?=G,gy/+

( 106a)

and $ ( OI)

+ H2g10,)- No)sin(k, z f q )

= H ,gK)

(106b)

and similarly for inelastic scattering. Imposing the boundary conditions of

275

SCATTERING IN STRONG MAGNETIC FIELDS

Eq. ( 102),the partial cross sections for elastic transmission and reflection are then (711(+)

= (n/y)[*( 1

+ cos[2(6 - q)] + 1 - cos 26 - cos 2q]

- cos[2(6 - V)l) while the total elastic cross section (rn fixed) is all(-) = (n/r)f(l

(107)

(n/y)(2 - cos 26 - cos 211)

(108) where 6 and q are the even and odd phase shifts relative to the complex numbers N @ )and W O ) , respectively. Ohsaki has studied the two-state approximation for both attractive and repulsive screened Coulomb potentials Gel=

V(r)= k re-'

(1 09)

at y = 0.05 (= 2.35 X lo8 G ) ,in which case the lowest Landau level 10,0, k) is at 1.36 eV and the first excited level I 1, 0, k) at 4.08 eV. He finds that 6 is smoothly varying through this threshold but q changes rapidly by about n/2 in the energy interval from 3.8 to 4.0 eV (Fig. 3) and interprets this as indicating a Feshbach resonance of width r - 0.14 eV lying just below the I 1 , 0, k) threshold.

I

3.5

1

3.7

I/

1 h I 1 I 3.9 4.1 4.3 INCIDENT ENERGY (eV)

I

FIG.3. Phase shifts for the even- and odd-panty cases in elastic scattering in a two-state approximation, for elastic scattering of electrons by the attractive screened Coulomb potential in Eq. (109). (From Ohsaki, 1983a.)

276

M . R.C.McDowell and M. Zarcone

Ohsaki (1983a) also shows that within the two-state approximation for axially symmetric potentials satisfying

that a

k,*lkn,

knt a 1

( 1 11)

so that the inelastic cross section, in the absence ofa zero energy resonance, is zero at threshold. Clark ( 1983)gives a simple argument which confirms this result. In the absence of the potential he defines even and odd solutions $y)(z)

= N cos kiz

$y’(z)

= N sin

kiz

where N = ( nki)-1/2,so that the functions $ i are normalized on the energy scale. With the interaction on, then at the boundary z = zo of the interaction region the open-channel (k: > 0) boundary conditions may be written in R-matrix form as

both channels being open. In the asymptotic region (z > zo) the even and odd solutions may be written

my)= kT1/ZAf)sin(k,z + 6J, @y)= k71/2Ay’sin(kiz + ai),

z > zo z > zo

(1 14a)

and

a?)=kT1lZAy)sin(-kiz + di), @y)= k;1/2Af‘) sin(kiz - qi),

z < -zo z < -zo

(1 14b)

Consider the even solutions at z = zo. Equation ( 1 1 3) yields (dropping the superscript)

+ 6,) = A ,k IIZR k cos(k, zo + 6,) + A,kf/2R12k~/2 cos(k,z, + 6,) A , sin(k2zo+ 6,) = A k:lZRzlk cos(kl zo + 6,) + A2k:/ZR2,k:/2 cos(k,z, + 6,)

A sin(kl zo

11’

(1 15)

277

SCATTERING IN STRONG MAGNETIC FIELDS

which have a unique solution if

+

+

det(Sijsin(kizo Si) - kj/zRjik,!/2cos(kizo Si)(Z 0,

j , i = 1, 2 (1 16)

-

We are interested in the i = 2 threshold, with k, constant and k2 O+. Let A, = 1 be the main component of the eigenvector in this channel, then Eq. (1 15) gives sin(k,zo As k,

-

+ 8,) = A , k~/2kf/2Rz, cos(k, zo + 6,)+ k2RZ2cos(k,zo + 6,) (1 17)

0, defining

+ 8,)

(1 18)

k2(Rzz- zo)= -k,a,

(1 19)

k, R,, then

S,

-

= tan(k, zo

say, and

near the threshold. Again the condition, Eq. (1 16) is equivalent to there behaves as k, so being no zero-energy resonance. The reaction rate OlZ(k2) a

kdkl

IV. Ensembles of Landau Levels We have seen in Section 111 that it is possible to formulate the problem of the scattering of a charged particle by a potential in a magnetic field and to define cross sections for scattering from one Landau level to another. McDowell (1982) raised the question of what experimental situation was being considered. The simplest situation would be one in which a beam of neutral atoms was photoionized while passing through a region in which there was a constant magnetic field. We do not know the angular distribution of photoelectrons for this process, but if we photoionize from a I 1SO) state and the potential seen by the ejected electron is axially symmetric, we may suppose that m continues to be a good quantum number. We must therefore eject into an m = 0 state. However, because of the residual Coulomb potential seen by the electron, it is clear that this cannot be an unperturbed Landau level. An alternative would be to photodetach from a beam of negative ions.

278

M. R. C.McDowell and M. Zarcone

B=o

I

B+o

I

B=o

m detect or

FIG.4. Schematic description of a scattering experiment in which a beam of electrons, representable by a plane wave, enters a region (11) in which there is a uniform magnetic field along the initial direction of motion.

Then the ejected electron sees a short-range potential which may be treated perturbatively. Sufficiently close to the photoionization threshold and at sufficiently large fields only the I 1, 0, k) Landau level at energy 47 Ry above the threshold is accessible. One is then in a position to consider a scattering a.u.; experiment. At laboratory fields (a4X lo4 G) y is on the order of so one does not have this situation in practice, and one has to take channel couplings into account (see Section V). The standard scattering experiment is illustrated in Fig. 4.A beam of charged particles, represented by a plane wave, is incident from a field-free region on a region in which there is a constant static magnetic field. We can have two different geometries. A. MAGNETIC FIELDPERPENDICULAR TO THE Z DIRECTION

In this case the orbit of the electron in the field is a portion of that which would be followed ifthe field were present in the whole plane; then classically the electron is always deflected by the magnetic field and the electron beam will never penetrate deeply into region 11. A quantum-mechanical description of this event can be obtained matching the energy-dependent wave functions for region I and region I1 at the boundary. This problem was treated long ago by Uhlenbeck and Young (1930) for the case where the motion of the electron in region I is along the z axis. A generalization of this work to the case when the electron velocity in region I has a component along the magnetic field has been given by Johnson ef al. ( 1983) both classically and quantum mechanically.

B. MAGNETIC FIELDPARALLEL TO THE z AXIS In this case the electron beam can penetrate inside region 11, and it is necessary in order to calculate reaction cross sections to know the occupation numbers of the Landau states before any collision.

SCATTERING IN STRONG MAGNETIC FIELDS

279

Faisal( 1982) and Ohsaki (1 983a) examined this problem independently and gave apparently different results. The discrepancy was resolved by McDowell et al. ( 1983). They treated two distinct cases; in the first, simpler case the field is suddenly switched on. In the second case the transition from the zero-field region to the region with a field is adiabatic. In the sudden case we expand

where Qnmk(r)is the Landau state Inrnk,). Since the Landau wave functions are orthogonal,

and d,, is the Fourier transform with respect to k of the Landau function. Writing

+

k2 = k t ( 0 ) ki(0)

(1 23)

in terms of the field-free components of k,

which is a Landau function with spatial variable Q = yp2 replaced by y-’k:(O). Notice that all states are occupied, and k,(O), ki(0)are not separately conserved, However, the only states with ldnm12>> 0 are those for which

y-Ik:(O)

= 2n

+ Irnl+

1

(125)

so the occupancy of the j t h Landau level decreases rapidly with increasingj up to j,,

= k:(0)/2y

( 126)

In the “sudden” case there is no return to a plane wave when the field is switched off. The sudden description will be valid when the rise time of the field is short compared with the cyclotron orbit period T = 2a/o,. Ifthe incident particle has velocity V,along the field,it will move a distance xs = ny-’k2(0)in time

280

M. R . C.McDowell and M . Zarcone

T; so for validity of the sudden approximation the rise distancex,, of the field must satisfy

< ny-'k,(O)

(1 27) In any laboratory situation xois a macroscopic quantity, say, xo* 10" a.u., and, since typically k 1 a.u., we require y < lo-" for Eq. (3.5) to hold, or fields of less than 25 G. At typical kilogauss fields it will only be valid for relativistic electrons. The sudden expansion is an exact expansion of the plane wave and contains all the Landau eigenstates including those not degenerate in energy with the field-freeplane wave. It gives the correct result when the field is suddenly switched on. If the field then later goes adiabatically to zero it does not go over to a plane wave. The adiabaticexpansion is in terms of degenerate Landau states only (corresponding to that part of the energy of the plane wave which is perpendicular to the field), with different m(n).Thus the perpendicular and parallel components of k are xo

-

k:(O) = 2(2n

+ rn + Irn(+ 1)y

Ry

(1 28)

and

k,(O) = k = Ikl where in region I ( 130) k = (k,(O), k,(O)) In the adiabatic regime we need to restrict the space occupied by the field to a region of dimension L perpendicular to the field. This is not a real restriction, since for a Landau orbital Inrnk) the mean-square radius perpendicular to the field is

and the cyclotron radius is

Rc = W ) / 2 Y so all we require is Rc GE L Making this restriction, a linear combination of Landau states

(1 33)

with mmax= ki(0)/4y - f , is obtained which goes into a plane wave when Y40; lim X(r) = Y ( r ) Y-0

SCATTERING IN STRONG MAGNETIC FIELDS

28 1

provided

where & is the azimuthal angle of the vector k. Moreover, since the squared moduli of the coefficients are independent of rn, the occupation numbers of the Landau states are also independent of rn, and are proportional to l/ZV,, where N , is the number of Landau states degenerate in energy present in an area L2 percpendicular to the field. Thus

IC m (n)I

= ( 1/L2)n//r

(1 36) The problem that remains is to show that not all negative integer values of m occur, or the total density of states would diverge. Such a restriction is in fact thrust upon us by Eq. (1 34), and a detailed analysis (McDowell et al., 1983) shows that the maximum positive value of rn allowed is

rnt = k:(0)/4y

(137)

and the maximum negative value is

mi=yL2-2mt with L the maximum dimension perpendicular to the field. The completeness relation is then

(1 38)

s = c 2 C~(rn)C,(rn’)(nrnkJnrn’k) m ml

-

-

so that S = 1 is recovered both as y 0 and as L 0 faster than y - ’ . Alternatively, we can look on Eq. (1 39) as saying that the adiabatic representation is valid if (1/2y)k1(0) -=cL ; so for a finite apparatus dimension xowe require (1/2y)k,(O) -=c xo (140) which is the reverse of the sudden approximation inequality, Eq. (127). Clearly for laboratory fields y < k 1, and xo= lo*, this is always satisfied.

-

V. The Low-Field Limit of the Cross Section One cannot proceed directly from the Landau wave functions to the field-free case because of the change of symmetry. Nevertheless, it is impor-

M . R . C. McDowell and M . Zarcone

282

-

tant to show that, for example, the elastic cross section goes to the field-free case in the limit y 0. The problem has been addressed by Ohsaki ( 1983a) and by Bivona and Ferrante ( 1984). We give a different approach. The cross section for scattering by a potential V(r) from some initial plane-wave state xiwill be the limit as y + 0 of

oil = FifJ;A

(141)

where

Pi,=

( 2 ~ /’)ITi# h 6(E,- Ei)

( 142)

is the transition probability per unit time and Ji, is the incident flux. We have for a particular final state 2,

Tv=

z C:,(ni)Cmr(n,)(nfm/k/I TInimiki)

m.mr

In C i(lm~-lmrl)e-i(mf-mrM(nfm,k,lTln,rniki) -(143)

L’ Y mpnj For a potential with axial symmetry rn, = rnf= m so that

Putting C#I = &, - 4k, and taking the square of the modulus,

Averaging over the azimuthal angle,

The incident current density is

Again averaging over azimuth,

n

hk,

Ji,, = - L-’ -

Y P By completeness (Ventura, 1973)

C (nirnkiln,mki) m

z (nirnkiln,mki)

= L-’

m

K

Y

(149)

SCATTERING IN STRONG MAGNETIC FIELDS

283

so J.inc = L-3 hkiJp for normalization in a box of side L. Then

where the sum is over all final states with energy

E = E f = kj

+ 4y(n,+

+ + 1))

+(/mi m

Ry

(152)

Writing

and noting that for a one-dimensional system the density of final states per unit energy interval is

p ( E f )= ~ L / 2 n h ' k f

( 1 54)

we have

Now, for a fixed energy perpendicular to the field, y + 0 implies n -,03 for all finite m.Consequently we can replace the sum over n,by

r where p(n,) is the density of states in the plane perpendicular to the field

Now

k,,

=k

sin 8,

k,, dk, d+

= k2 cos 8 d R

where 0 = cos-'(ki k,). We can then write

2 n/

with

-& /

k2 cos 8 dR

284

M. R. C.McDowell and M. Zarcone

We specialize to elastic scattering, so that in the field kikf= k2 cos 8,

8 = 0 or K

( 160)

and

To proceed further we choose to examine the Coulomb potential, since in that case alone we can put, for the field-free case,

I Tir12 = I V&2

( 162)

and we can replace ( 16 1 ) by the first Born approximation

We wish to prove that for then

du?)(y ) - UOP lim -r+O df2 4k2sin2812 the Rutherford differential cross section. We suppose there is initially a well-collimated beam so that kL is initially zero, and

Using Ventura's result for V,,,,, where Ha, b, x) is a confluent hypergeometric function (CHF)of the second kind and

where q is the z component of the momentum transfer

4 = kf- ki

SCATTERING IN STRONG MAGNETIC FIELDS

285

The CHF may be expressed in terms of the incomplete y function (Erdelyi et al., 1953, p. 133) as Now as y

-

w( 1, I - nf, X)=

0, nf+

m,

xwr(- 5 , ~ )

( 170)

so we can use the expansion

-+

w(1, 1 - nf,X)

(1

1 n,+X)

(171)

++

( 1 - (1 1 n,+X)2 n,

+...)

Retaining only the leading term, lim w( 1, 1 - n,, X)= 4y/(q2

n,+m

+ ki,)

(172)

Inserting Eq. ( 172) into Eq. ( 167) and noting that q2

+ kt,=

(kf- k)2+ k2 sin2 8

(1 73) we have Eq. (1 65) as required. The leading-order correction to Eq. ( 172) is

- -4ykj (1 + n,) - (1 n,+ X)2 - (42 kZ,,)2

+

+

and hence

cF)(y ) = a?'( 0 )+ O(y ) in agreement with Bivona and Ferrante (1984). The argument is far from rigorous in view of what we have said about the FBA near a Landau threshold, since as y 0 these are infinitely dense in any interval.

-

VI. Photoionization We have already remarked on the comparative simplicity of the conceptual problem when we consider photoionization or photodetachment rather than the full scattering problem. Wunner et al. (1 982) gave an expression for the photoionization cross section of a one-electron (hydrogenic) system in a magnetic field, and showed that it satisfied simple scaling laws. We again use box normalization with L3 = V, so that 0" = vw/c

(1 76)

286

M. R. C.McDowell and M. Zarcone

with the transition probability w given by wfi

= (2nlh2)l(fIH,li)12Pf(k/)

( 177)

Here

and

Pf(kf)= Lmlhkf

( 179)

The vector potential A. of the static magnetic field is given by

A.

= f BX

r

( 180)

and A,, the vector potential of the radiation field, by

in the usual second quantized notation (cf. Sakurai, 1967). We write B,=VXA,

( 182)

and p B = eh/2m ( 1 83) is the Bohr magneton. Given infinite nuclear mass the wave functions scale as

&Z, B, r) = Z3l2$(1, Z-2B, Zr)

( 184)

for an ion of charge Z while the bound-state energies scale as

E(Z, B) = Z2E( l’Z+B)

(185) as noted by Surmelian and O’Connel(l974). The Z 3 factor produced by the wave functions is cancelled by the change in V. The term in H I scales as 2, provided

-

Z-’k( 1)

( 186)

L(Z) = Z-’L( 1 )

(187)

k(Z) and the density factor has and

kf(Z,B ) = Zkf(lyZ-2B)

(188)

SCATTERING IN STRONG MAGNETIC FIELDS

287

so p(Z, B) = Z-'p( 1, Z-2B)

(1 89)

Consequently

a,(Z, B, Ej, E f , k) = Z-'aV(1, Z-'B, Z-2Ei, Z-'Ef, Z - ' k ) (190) where

Ei+ hkc=Ef (191) This implies that we must know the cross section for hydrogen at energy and momentum values off the energy shell to determine a,. This difficulty disappears when the dipole approximation eik-r + 1

(1 92) is made and H , is now free of scaling. This gives the dipole cross-section scaling as

ald)(Z,B, E j , E f , k) = Z-201d)(l,Z 2 B , Z - 2 E j , Z-2Ef, Z-2k)

(193) If we average over the angle of orientation of k with B,we obtain the simple scaling between two hydrogenic systems of charges 2,,Z 2 at corresponding as fields Zy2Bo,Zr2Boand frequencies v 1 = ZTv,,v2 = Z$vo

Schmitt et al. (1 98 1) calculated the photoionization cross section for atomic hydrogen at very high fields ( y > lo4) by using a one-state approximation (Canuto and Kelly, 1972): where an,(p,) is a Landau function and

and g".")(Z)has an asymptotic Coulomb phase

gt:j(Z)

Ias, efikZexp(+ k-I In k ~ )

(1 98)

Parallel work by Kara and McDowell(l98 1) is vitiated by their failure to take account of the Coulomb phase. In addition, for the photoionization problem

288

M. R. C.McDowell and M. Zarcone

we must ensure that there is no outgoing wave as Z + --oo. Schmitt et al. obtain the two linearly independent solutions with

gO(0) = 1,

dg(')(O)/dZ= 0

( 199a)

g(")(O)= 0,

dg(")(O)/dZ= 1

( 199b)

and combine these to satisfy the boundary conditions. They calculated the cross section in the energy regime

and only. Thus the final state must have n = 0 and mfs 0. However, mi = 0; so to an excellent approximation only mf= 0 need be considered. Schmitt et al. found that a,(O = n/2) is a slowly varying function of energy, at y = 100: a, = 0.6~7,

(202)

(where a, is the Thomson cross section), at photon energies up to 10 Ry and then decreased rapidly, in agreement with the Born approximation. Further the cross section was a maximum for photons incident perpendicular to the field and zero parallel and antiparallel, for the transition considered due to the sin2 8 factor associated with the linear polarization. They found that at large photon energies the cross section decreased as v-9/2 in contrast to the field-free results of v-'l2. Greene (1983) has generalized their treatment for the case where more than one channel ( hw 2 h w,) must be considered. He treats the final state of the atom in the field by supposing that a finite region around the nucleus can be treated as the region of strong interaction, and the solution obtained in it matched to the asymptotic Coulomb solutions on its boundary. The inner region is taken to be a cylinder with ends at Z = & Zo and radius p,. No specificchoice ofp, is made since it merely needs to be large enough to accommodate the finite set of Landau functions used in any real calculation. The choice of Z, is made so that for the potentials may be represented by their (Coulomb) asymptotic form. An R-matrix approach is adopted in which one calculates the normal derivative of the wave function on the boundary Z of the interaction region R. For a single channel one has Gy/Sn

+ by = 0 on Z

(203)

SCATTERING IN STRONG MAGNETIC FIELDS

289

If the wave function is then expanded in a set of basis functions yk ( k = 1, . . . , n )

the boundary value b can be expressed in terms of the C k using the usual variational expressions for the energy and including surface terms. One obtains b[cl = with =

T7

cdA,cl

In [-vyk

*

vYl

/

7

cksklcl

+ 2yk(E - v)yll d7

(205)

(206)

and

If we then require b to be stationary with respect to small variations in the parameters

k = 1, dbfdck=O, one obtains the eigenvalue problem

. . . ,n

Ac = bSc

(208) (209a)

or (A- bS)c=O (209b) The solution for each open channel consists of eigenvalues and associated eigenvectorsCBwhich are orthonormal with weight function S on Z. Greene points out that Eqs. (203) and (204) give an approximate value

correct to order &, whereas Eq. (205) is correct to order Sty2; thus the difference is a useful measure of convergence. In the external region the flh solution can be written in terms of regular and irregular one-dimensional Coulomb functionsfn(2) and g n ( 2 )as WB =

@nm(P)[~nBfn(z) n

- JnBgn(z)I

(21 1)

290

M . R. C. McDowell and M . Zarcone

where I d , Jd are constants. Using Eq. (203) -by/, = n

anm(p)[znBjL(z) - J n B d 1 ( ~ ) 1

(2 12)

so a system of 2Nlinear equations for the 2Nconstants may be obtained by projecting Eqs. (2 1 1 ) and (2 12) onto each Qnm in turn. The Kmatrix is then

K = JI-I (2 13) This may be put in the more usual many-channel quantum-defect form by diagonalizing to obtain U-IKU = tan np (214) As an example Greene considers one- and two-state approximations to calculate the photoionization cross section of atomic hydrogen in a field of strength 4.7 X lo9G ( y = 2.0) for m = 0 only. He diagonalizes the Hamiltonian in the basis YN =

an& Pm)(2/Zo)”z cos[n(j - +)z/zol n = 0 , 1; j = 1 , . . . , 10

(2 15)

and

since only the n = 0, 1 Landau states enter significantly into the groundstate wave function. A binding energy of 1.00238 a.u. was obtained for 6 = 2.5 which compares with the accurate variational value of 1.0222 (Wunner and Ruder, 1983). For the final state the expansion used was yN=Qn0(p)sin(+jZ), n = 0 , l ; j = 1, . . . , 15 Greene writes the two independent solutions obtained as

(2 17)

2

and obtains the R-matrix elements as

The results obtained with this variational treatment agree closely with those obtained from a direct solution of the two coupled equations. At this field strength the first Landau level (n = 0) lies 1 a.u. above the ionization limit, and the first excited Landau level is at 3 a.u. At energies up to 2.4 a.u. coupling was found to be unimportant and a one-channel solution

SCATTERING IN STRONG MAGNETIC FIELDS

29 1

adequate. The bound states may be obtained once the quantum defect is known by the usual condition p v = n (n = 0, 1,2, . . .). It was then possible to obtain the discrete oscillator strengths as well as those in the continuum. His results for the discrete oscillator strengths are in good agreement with those of Wunner (unpublished), although the oscillator strength sum rule is not satisfied exactly. One notable feature is that about 75%of the oscillator strength out of the ground state is to other bound states, compared with 56.5%in the field-free case. At higher energies (hv > 2.4 a.u.) there are significant contributions from the excited (n = 1, m = 0) state, the modulus squared of the off-diagonal S matrix element increasing to 0.16 at E = 3.2 a.u. Writing the solutions in the two channels in terms of the eigenphases [Eq. (2 14)]

+

@a

,

'

w~(I)pn urn cos ~a

= 8.n

(220)

one finds that p is slowly varying in this energy range but p2increases rapidly by almost 7r/2 as E increases from 2.6 to 2.9 a.u., as in the model potential study by Ohsaki (1983b). In addition to the bound states, a single Rydberg series of autoionizing states is embedded in the continuum. These produce a series of resonances in the cross section (or equivalently the differential oscillator strength distribution) starting at E = 2.75 and converging on the n = 2 threshold (see Fig. 5). A more detailed study of these autoionizing levels has been reported by Friedrich and Chiu (1983) for the /mi=0, 1, and 2 subspaces. They also restrict themselves to field strength y > 1 since for this range the negative energy solutions for each subspace (m,K ) form a single nondegenerate hydrogenic sequence,

En = - I/(n

+ a,,)

(22 1)

Rather than diagonalize H in a basis and obtain autoionizing levels by a variant of the stability method, Friedrich and Chiu solve the coupled equations [Eq. (95)] for 0 E 2y Ry with boundary conditions

and

wherefo(2) and h,(Z) are the regular and irregular one-dimensional Coulomb functions.

M. R.C.McDowell and M. Zarcone

292

1

1.6-

1.2

1

a

-

0.8 -

0.4 \.

I

1

I

I

-

AAL.

E(a.u.1

FIG.5. Total photoionization oscillator strengths of H at y = 2.0 near the first excited I10k) Landau threshold. (a) is the length and (b) the velocity result. (From Greene, 1983.)

The autoionizing resonances are signalled by a rapid increase by a In S(E), with resonance position at the local maximum of the gradient and width

r = a[dS(E)/dE]-'

(224)

They note (Seaton, 1955) that lim (&E)la]= - lim Sn

&+O

n--

for each (m, R). For each autoionizing state found in (0,2 y ) one can define an effective quantum number n, - nL2 = E, - 2y

(226)

SCATTERING IN STRONG MAGNETIC FIELDS

293

TABLE I

ENERGIES OF M

-0.64 -0.194 -0.094

= 0,n=

+, HYDROGEN

1.25 2.27 3.26

0.17 0.03 0.01

a Energies (relative to 100k) at 2y Ry) of the three lowest autoionizing states (m= 0, n = +) of hydrogen in a magnetic field of strength y = 0.4. (By permission of the American Institute of Physics from Friedrich and Chiu (1983). Phys. Rev. A 28, 1423).

+

and these are shown in Table I for m = 0, n = at y = 0.4. Friedrich and Chiu note that successive - n, differ by close to unity, indicating that they belong to a single Rydberg series. As y decreases, successivehigher resonant statesof a given subspaceappear to pass below the ionization threshold, and appear among the bound statesas a “pseudoresonance.” For the (0, +) subspace at y = 0.2, this lies at E, -0.03 Ry with a width of 0.20 Ry. For small Im(and y -,0, the autoionizing widths can be large (>O. 1) and lifetimes correspondingly short (lo+ s). The widths decrease with increasing y and Irnl and for given (m,n) with increasing n,.

VII. Photodetachment of Negative Ions The only experimental information currently available on continuum states of atomic systems in a magnetic field (other than the well-known studies by Garton and Tomkins (1969) and others on photoabsorption) arises from ion-trap experiments on photodetachment of negative ions of Sby Blumberg and his collaborators (Blumberg et al., 1978, 1979) and by Larson and Stoneman on SeH- (Larson and Stoneman, 1982). The ions are stored in a Penning ion trap, illuminated by monochromatic light and counted by measuring the current to the trap electrodes while driving the remaining ions by an electric field. The fraction of the ions surviving is measured as a function of frequency of the radiation at a fixed (but variable) magnetic field, of the order of lo4 G (1 T). Typical results for !?(Fig. 6) and SeH- (Fig. 7) are shown below. There isa pronounced oscillation as a function of frequency. Blumberg et al. (1978,

294

M. R. C.McDowell and M. Zarcone

- 32.0

0.0 RELATIVE

32.0 64.0 FREQUENCY ( G H r )

96.0

FIG.6. Photodetachment of S - by light of u polarization at E = 1.07 X lo4G. (From Larson and Stoneman, 1982.)

1979)and Larsen and Stoneman (1982) argue that this phenomenon can be understood with the following argument. Since the potential seen by a slow electron on the field of a neutral atom is composed ofa short-range term [and a long-range (r-4)polarization interaction] in the absence of the field, it will not be much affected in the case of weak fields when fioi < Ebb, the attachment energy. They argue that the final state is changed from a plane-

PHOTON ENERGY (cm-1)

FIG. 7. Photodetachment of SeH- for E = 1.3 X lo4 G. From Larson and Stoneman ( 1982).

SCATTERING IN STRONG MAGNETIC FIELDS

295

wave state to a Landau state, to first order. The boundp electron (in the ions they consider) is promoted to an s- or d-wave continuum. The transition rate is = (2n/h>al(flvli)12p(~/)

(227)

with

E / = Ei

+ hv

(228)

Near threshold, the d continuum is likely to be less important, because of the angular momentum barrier, and we need only the s wave. They suppose that, for all r,

If)

a

k-*l2(sin kr

+ So cos kr)

(229) and hence I(flvli)12 is independent of kfsince Vis short range. The effect of the magnetic field is to transform p(Ef)from a three-dimensional density of states proportional to kfto a one-dimensional density of states proportional . the cross section is also proportional to kT1and singular at each to k ~ 'Thus successive Landau threshold. In the experiment sharp resonances are not seen, as they are broadened due to the ion motion due to motional Stark level of S- and shifts and Doppler broadening. In addition the ground 2p3/2 the ground 3p2level of the S atom are split into a number of Zeeman sublevels, and each of these leads to a different threshold for the nth Landau level, which must be averaged over with appropriate weights. A calculation based on these ideas by Blumberg et al. (1979) appeared to give a good description of the experiment, but it included three adjustable parameters (Fig. 6) to allow for the unknown velocity distribution in the ion trap. Larson and Stoneman ( 1982) considered the effect of the final-state interaction. They write, for the case of detachment to the lowest Landau state, look/),

so that d2g/dZ2- (2m/h2)U(Z)g(Z)= Eg(Z)

(23 1)

with

and

r2 = p 2

+ Z2

(233)

M . R. C.McDowell and M. Zarcone

296

The potential is approximated by a spherical well of range r, -sz R, and depth V, which leads to a one-dimensional well of depth

so that the effective depth is reduced by the ratio of the time the electron spends in the well to the time outside it. Since any one-dimensional potential supports at least one bound state, there should be new observable negative ion states in a refined experiment. In the present case, since in the absence of the field V, 1 eV, r, la,, they conclude that V, = 1 GHz, and finalstate interaction effects are much too small to be observable at a field of one tesla. The above explanation of the structure has been disputed by Clarke (1983), in view of his, and independently, Ohsaki’s, result that the cross section for electron -atom elastic scattering is nonsingular at a Landau threshold. Clarke investigates a model analogous to that for the quasi-Landau resonances observed in photoabsorption. It was first shown by Edmonds (1970) that these correspond to states in which the electron is localized in the z = 0 plane. In Fano’s language (Fano, 1983) they are localized “on the potential ridge.” The potential behaves at large r as

-

-

V(r)= - l/r

+ i y 2 r 2 sin2 0

(235)

so that in the z = 0 or t9 = a/2 plane the repulsive term is maximal. The quasi-Landau states which lie on this ridge are supposed to decay by falling off it, though the mechanism is not understood. However, in photodetachment the long-range Coulomb term is absent and is replaced by a short-range interaction, whose effect can be represented by a scattering length. Nevertheless Clark argues that resonances similar to quasi-Landau resonances will still occur at energies E , = (2n

+ 1 -p)ho,

Ry

(236)

wherep is the quantum defect ofthe particular system in a given rn subspace. For S- he takes V(r)= -a/2r4

+ (1/8)y2r2 sin2 0,

r > 10a,

(237)

with a! the static dipole polarizability of sulphur, about 10.43~:(Miller and so the asymptotic Bederson, 1977). At 10.7 kG (1.07 T), y is 4.55 X potential has a zero for 0 = a/2 at about 125U0.Clark obtains the scattering length a, for the 2p3,2state of S- from Rau and Fano (197 1). He investigates the consequences of supposing that the observed resonances are wave functions confined to the z = 0 plane.

SCATTERING IN STRONG MAGNETIC FIELDS

For the m

= 0 subspace he

297

solves

by noting that V(p) has a zero at 125 a.u. and matching there at zero scattering energy to the boundary condition

4 d&dP

Ip-p0

=b

(239)

The boundary value b is deduced from the known scattering length, and found to be 0.9 1 a.u. The solutions can then be obtained in terms of known solutions for a zero-energy electron in a polarization potential. For the m = 0 series Clark calculates a quantum defect ofO. 1. This implies that the actual resonances are not at their Landau thresholds but shifted to lower energies by 0.1 y Ry. Since in the case studied y < 1, the effect is not detectable in the experiment. We take the point of view that no completely convincing theory of the observed structures in photodetachment in a magnetic field exists, but that it has been established that the cross sections are finite at the Landau thresholds, and that (see above) one would in appropriate cases expect series of Feshbach resonances converging on a threshold from below, which in the circumstances of ion-trap experimentscould yield an observed absorption of the type seen.

VIII. Charge Exchange A different approach is required to the problem of ion-atom collisions in a magnetic field since the collision energy E is generally much greater than h w,. Thus many Landau states are accessible. Clearly at very low energies and very high fields ( ho,> E) this is not the case, but there has been no study of this regime up to the present. The two cases considered have been moderately strong fields 0 d y d 0.5. While Bivona et al. (1984) looked at symmetric resonance charge exchange at low energies [4 d u d 18 X lo6 cm s-' (1.8, - 3 d u d 8.2, -2 a.u.)] up to 0.17 keV amu-l in the laboratory frame, Grozdanov and McDowell(l984) looked at the asymmetric case at energies from 25 to 75 keV amu-I. A quantal approach must be used in the first case, and, since at this stage one is interested in establishing the general features of any effect of the field rather than in making precise numerical estimates, a two-state approach (McDowell and Coleman, 1970) is appropriate. Bivona ef al. found that, within the two-state model, not only

298

M. R. C. McDowell and M. Zarcone

did the magnetic field modify the bound-state wave functions, but, more importantly, it introduced a new phase factor in the matrix elements. This acts in a similar way to the well-known momentum-transfer factor (Bates and McCarroll, 1958) in damping the transition probability oscillations and sharply reducing the cross section. Specifically, ignoring the cyclotron motion of the center of mass of the two ions (H+, H+, say), and adopting straight-linetrajectories for the heavy-particle motion, the electronic Hamiltonian becomes

compared with the field-free case in which A = 0. The gauge must be common to wave functions on both centers, and it is convenient to define them with respect to a gauge with its center at the center of mass of the nuclei, r = 0, so that

A(r,)

= A(r) =4

+ S(B X R)

r ) + VrL

with

fA = +(BX R)

r

in the usual notation, and similarly forf,. This gauge transformation is unitary S(A) = exp(- iefA/hc)

(243) With these changes the usual result is obtained, that the transition probability is given by

P(v, B, p) = sin2 q

S=

I

cp$$A

(244)

dr

and go = 0 when A = 0, but in general go = (e/2hc)(B X p) r

when both the initial velocity and the field are taken along the z axis. Electron-transfer factors have been omitted.

299

SCATTERING IN STRONG MAGNETIC FIELDS

0.01

0.02

r

0.03

0.04

+

FIG.8. Cross section for resonant symmetric charge exchange Rb+ Rb Rb magnetic field of strength y at u = 7.2 X lo6 cm s-'. (From Bivona ef al., 1984.) .-)

+ Rb+ in a

Bivona ef al. give results for Rb+ on Rb, treated as a single-electron problem, but with a model potential V(r)= r - l [ - 1.0

+ 1.494 exp(-0.6606r)]

(249) so that all the matrix elements can be reduced to a quadrature. They find (Fig. 8) that the resonant symmetric charge-transfer cross section is a rapidly decreasing function of y in (0,0.5) for fixed velocity. In the low-field limit

P = p0 + 0 7 2 + o(y3)

(250)

with (Y

= -4 sin2(2qoq,)

(251)

where qo is the zero-field phase shift and

the superscripts referring to the zero-field and perturbed parts. For large impact parameters, qo and q, have the same sign and lqol < n/2; so P < Po. For small impact parameters (sin2 q) = 0.5 with or without the field, so the prediction that the cross section behaves as

Q = u - by2 should hold in general for symmetric resonance at low velocities.

(253)

300

M. R. C. McDowell and M. Zarcone

At higher velocities the electron-transfer factors will play an important role so that in effect go is replaced by g=go

+v

r = (172 X p + v) r

(254) where y is the field strength in atomic units. Typically for v > 1, capture takes place at relatively small impact parameters; so, unless 14ypI > 1, the effects of the field will be less significant than in the case considered by Bivona el al. The two-state atomic-orbital results obtained by Bivona et al. have been confirmed, in principle, by Wille (1984). He uses a molecular-orbital treatment and solves the stationary two-center one-electron problem in the magnetic field. With the z axis along the internuclear axis and the y axis perpendicular to that and to the field, then the electronic Hamiltonian is

- + y 2 [ p 2 - (x2- Z 2 )sin20 - X 2 sin 281

where 8 is the angle between the field and the z axis.

25

20

- 15 5

N

0

0

F

10

5

1

1

0.2

1

1

0.4

1

1

0.6

1

1

0.8

1

1

1

1.0

v la.u.1

FIG.9. Total (two-state)resonant symmetriccharge exchange for H+, H at y = 0,0.5, 1 .O at energies up to 25 keV. The dashed lines (- - -) show the effect of imposing a straight-line for a Coulomb trajectory (by permission of Dr. U. Willie). trajectory; the solid (-)

SCATTERING IN STRONG MAGNETIC FIELDS

30 1

Wille considers 8 = 0 only and diagonalizesH in a Hylleraas basis, for the Z, = Z2 = 1 case, to obtain a set of "dressed" basis states, (YJ. He can then calculate the cross section for resonant charge exchange using the dressed lsa

and 2pa energies. It is found that, at a fixed impact energy, the cross section is a decreasing function of y in the range 0 6 y C 1.0 in accordance with the LCAO result, but the magnitude of the decreaseis smaller. Wille's results for y = 0,0.5, 1.0 are shown, for energies up to 25 keV, in Fig. 9. The other case considered (Grozdanov and McDowell, 1984) has been

+

He2+ H + He+

+ H+

(256)

at keV amu-' energies. They used the classical trajectory Monte Carlo (CTMC) method of Abrines and Percival(1966), which for this reaction at these energiesgives results both for the total cross section and for capture into individual n levels of He+ in excellent agreement with the recent atomicbasis calculations of Bransden et al. (1 983), which include all orbitals with n d 3 about each center. The classical equations of motion of the three-body system may be written (257)

U=O

m2m3 m2 m3

+

io=Q C

("- ,",'

v

+ w ) x €3

(259)

where in our case the projectile is particle 1,the electron particle 3. We work to order ym,/m,. Determining the constant vector u, the velocity of the center of mass, from the initial conditions, neglecting the deflection of the heavy particles due to interparticle interactions, and retaining only leading terms in an expansion in powers of (me/mp),these reduce to a single vector equation for the electron motion

If y is not too large (y < 1 .O) the ground-state wave function of H is not significantly perturbed by the field, and its momentum distribution remains unaltered to a good approximation. Grozdanov and McDowell assumed that the initial microcanonical ensemble of initial two-body orbits could be

M. R. C.McDowell and M. Zarcone

302

used, and solved Eqs. (257)-(259) with various values of y on this assumption, for the case where €3 II

"

(261)

They found that the capture cross section was enhanced by the field for all y in (0, 0.20) at all three energies investigated (see Table 11). The maximum effect was 16Yo.In the field-free case 90%of the capturesat, e.g., 50 keV amu-' are not into the ground state. The excited states are significantly perturbed by the field which at y = 0.1 corresponds to 0.4 for He+. They tend to become needle shaped; that is, the electron-densitydistribution in the final state is forced to lie along the field direction. The ejected electron is also confined by the field; so its match with the final-state distribution is closer, and the capture probability enhanced. Detailed examination shows that a much higher proportion of the captured electrons have an initial binding energy less than 0.2 Ry than in the field-free case, and that while the capture probability Pc(y, p) at field y and impact parameter p is always larger than p,(O, p) when y # 0, the major enhancement is at large impact parameters.

TABLE I1

TOTAL CAPTURE AND IONIZATION CROSS SECTIONS FOR HeZ+-H COLLISIONS E (keV amu-I) 25

Y

50

74

C Ca

Qi

Qc

Qi

Qc

Qi

0.10

9.76 (0.2 I ) 10.00 (0.22) 10.39

0.15

10.25

0.20

10.16 (0.21)

0.58 (0.08) 0.63 (0.08) 0.67 (0.08) 0.67 (0.08) 0.69 (0.08)

4.96 (0.16) 5.50 (0.18) 5.66 (0.19) 5.73 (0.19) 5.46 (0.18)

2.96 (0.14) 2.80 (0.16) 2.18 (0.16) 2.81 (0.16) 2.76 (0.16)

2.01 (0.10) 2.16 (0.13) 2.23 (0.14) 2.36 (0.14) 2.19 (0.13)

4.58 (0.14) 4.83 (0.19) 4.65 (0.19) 4.50 (0.18) 4.67 (0.18)

0.00

0.05

(0.22) (0.22)

Total capture, a&), and ionization, oi(y), cross sections in units of

nu8 cmz for He2+,H collisions with a magnetic field of strength y along the initial relative velocity. The figures in brackets are the standard errors.

SCATTERING IN STRONG MAGNETIC FIELDS

303

REFERENCES

(London), Ser. A 88, 873. Abrines, R., and Percival, I. C. (1966). Proc. Phys. SOC. Avrom, J. E., Herbst, I. W., and Simon, B. (1978). Ann. Phys. 114,43 1-45 1. Bates, D. R., and McCarroll, R. W. (1958). Proc. R. SOC. Ser. A 245, 175. Bivona, S., Spagnolo, B., and Ferrante, G. (1984). J. Phys. B 17, 1093. Bivona, S., Spagnolo, B., and Ferrante, G. (1984). J. Phys. B. 17, 1093. Blumberg, W. A. M., and Jopson, R. M. (1978). Phys. Rev. Lett. 40, 1320. Blumberg, W. A. M., Itano, W. M., and Larson, D. J. (1979). Phys. Rev. A 19, 139. Brandi, H. S., Koiller, B., Lins de Barros, H. G. P., and Miranda, L. C. M. (1978). Phys. Rev. A 18, 1415. Brunsden, B. H., Noble, C. J., and Chanellev, J. (1983). J. Phys. B 16,4191. Canuto, V., and Chiu, H. Y. (1970). Phys. Rev. A 2, 5 18. Canuto, V., and Kelly, D. C. (1972). Astrophys. Space Sci. 17, 227. Canuto, V., Chiu, H. Y., and Fassio-Canuto, L. (1969). Phys. Rev. 185, 1607. Carter, B. P. (1968). J. Math. Phys. 10,788. Clark, C. W. (1983). Phys. Rev. A 28, 83. Clark, C. W., Lu, K. T., and Starace, A. F. (1984). Prog. At. Spectrosc. C. Connerade, J. P., Gray, J. C., and Liberman, S. (1982). Colloq. CJ. Phys. 2, 11. Edmonds, A. R. (1970). J. Phys. CoNoq. C-4 31,7 1. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). “Higher Transcendental Functions,” Vol. 2. Mcgraw-Hill, New York. Faisal, F. H. M. (1982). J. Phys. B 15, L739. Fano, U. (1980). Phys. Rev. A 22,2260. Friedrich, H., and Chiu, M. (1983). Phys. Rev. A 28, 1423. Ferrante, G., Nuzzo, S., Zarcone, M., and Bivona, S. (1980). J. Phys. B 13, 731. Fowler, C. M., Garn, W. B., and Caird, R. S. (1960). J. Appl. Phys. 31, 588. Furth, H. P., Levine, M. A., and Waniek, R. W. (1957). Rev. Sci. Instrum. 28,949. Garstang, R. H. (1977). Rep. Prog. Phys. 40, 105. Carton, W. R. S., andTomkins, F. S. (1969).Astrophys. J. 158, 839. Goldman, R. (1964). Phys. Rev. A 133,647. Gradshteyn, I. S., and Ryzhik, I. M. (1980).“TableofIntegrals, Seriesand Products.” Academic Press, New York. Greene, C. H. (1983). Phys. Rev. A 28,2209. Grosdanov, T. P., and McDowell, M. R. C. (1984). J. Phys. B 18,921. Johnson, B. R., Hirschfelder, J. O., and Yong, Kuo-Ho (1983). Rev. Mod. Phys. 55, 109. Kara, S. M., and McDowell, M. R. C. (1981). J. Phys. B 14, 1719. Kemp, J. C., Swedlund, J. B., Landstreet, J. D., and Angel, J. R. P. (1970). Astrophys. J. 161, L77. Klepikov, N. P. (1952). Zh. Eksperim. Tor. Fiz. 26, 19. Larson, D. J., and Stoneman, R. (1982). J. Phys. Colloq. C-2 43,285. Lauer, J., Herold, H., Ruder, H., and Wunner, G. (1983). J. Phys. B 16, 3673. McDowell, M. R. C. (1982). In “Recent Developments in Electron-Atoms and Electron Molecule Collision Processes,” p. 93. SERC, Daresbury. McDowell, M. R. C., and Coleman, J. P. (1970). “The Theory of Ion-Atom Collisions.” North-Holland Publ., Amsterdam. McDowell, M. R. C., Zarcone, M., and Faisal, F. H. M. (1983). J. Phys. B 16,4005. Miller, T. M., and Bederson, B. (1977). Adv. At. Mol. Phys. 13, 1. OConnell, R. F. (1979). Phys. Left. 70A, 389-90. Ohsaki, A. (1983a). J. Phys. SOC.Jpn. 52,431.

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Ohsaki, A. (1983b).J. Phys. SOC.Jpn. 52,448. Onda, K.(1978).J.Phys. SOC.Jpn. 45,216. Praddaude, H.C. (1972).P h p . Rev. A 6, 1321.

Rau,A.R.O.,andFano,U.(1971).Phys. Rev.A 4 , 1751.

Ruderman, M.(1972).Annu. Rev. Astron. Astrophys. 10,427. Sakurai, J. J. ( 1967).“Advanced Quantum Mechanics.” Addison-Wesley, Reading, MA. Schmitt, W., Herold, H., Ruder, H., and Wunner, G. (1981).Astrun. Astrophys. 94, 194. Seaton, M.J. (1955).C. R. Acad. Sci. 240, 1317. Surmelian, K.,and OConnell, R. P. (1974).Astrophys. J. 190,741. Tonnewald, L. M. (1959).Phys. Rev. 113, 1396. Uhlenbeck, G.E.,and Young, L. A. (1930).Phys. Rev. 36, 1721. Ventura, J. (1973).Phys. Rev.A 8,3021. Virtamo, J., and Janho, P. (1975).Nuovo Cim. 26B, 537. Wille, U. (1985).Proc. NATO AS1 Fundam. Process. At. Collisions Phys. Wunner, G., Ruder, H., Schmitt, W., Herold, H., and McDowell, M. R. C. (1982).Mon. Nut. Roy. Astron. SOC.198, 769. Wunner, G., and Ruder, H. (1983).Private communication to C. H. Greene.

ll

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. 21

PRESSURE IONIZA TION. RESONANCES. AND THE CONTINUITY OF BOUND AND FREE STATES R . M . MORE* Lawrence Livermore National Laboratory Livermore. California

1. Introduction . . . . . . . . . . . . . . . . . . A . Self-Consistent-Field (Average-Atom) Model .

......... .......... B. Limitations of the Average-Atom Model . . . . . . . . . . . . . C. Pressure Ionization-Qualitative . . . . . . . . . . . . . . . . D . Continuum Wave Functions . . . . . . . . . . . . . . . . . . E. Continuum Density of States . . . . . . . . . . . . . . . . . . II . Continuity of Pressure Ionization . . . . . . . . . . . . . . . . . A . Proof of the Continuity Theorem . . . . . . . . . . . . . . . . B. Some Unsatisfactory Calculations . . . . . . . . . . . . . . . . C. The Spectral Window . . . . . . . . . . . . . . . . . . . . . 111. Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Resonance Wave Functions . . . . . . . . . . . . . . . . . . B . Properties of Resonance Wave Functions . . . . . . . . . . . . C. Resonance Perturbation Theory . . . . . . . . . . . . . . . . D . Convergence of the Expansions . . . . . . . . . . . . . . . . . E . Estimate of Barrier Height and Resonance Energy . . . . . . . . IV. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Thermal Occupation of Resonance States . . . . . . . . . . . . B. Electron Density . . . . . . . . . . . . . . . . . . . . . . . C. Ionization State . . . . . . . . . . . . . . . . . . . . . . . . D . Pressure Formulas . . . . . . . . . . . . . . . . . . . . . . . E. Pressure I- Pressure-Tensor Method . . . . . . . . . . . . . . F. Pressure I1 -Vinal Theorem . . . . . . . . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Properties of the Jost Function . . . . . . . . . . . . . Appendix B: Green's Function . . . . . . . . . . . . . . . . . . . Appendix C: Electron Density of States . . . . . . . . . . . . . . . Appendix D Resonance Perturbation Theory . . . . . . . . . . . . Appendix E:Convergence for thed-Potential Model . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

306 307 309 311 313 317 318 319 322 322 324 324 325 328 329 330 333 334 337 338 339 340 343 346 347 348 349 351 352 354

* Work performed under the auspices of the U S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W.7405.ENG.48 . 305 Copyright 0 1985 by Academic Press. Inc . All rights of reproduction in any form reserved.

306

R. M . More

I. Introduction Early theoretical study of matter at extreme conditions ofhigh density and high temperature was aimed at understanding pressure balance and energy flow in stellar interiors (Cox, 1965; Cox and Giuli, 1968). In recent years there has been a growth of interest in the atomic properties of hot dense plasmas, motivated by laboratory experiments involving high-power pulsed lasers, electron- and ion-beam irradiation of solid targets, and other techniques providing high energy concentration (Caldirola and Knoepfel, 1971; Brueckner, 1976; More, I98 1; Rozsnyai, 1982). Although this research is closely connected to immediate practical applications, there is also a need for improved fundamental understanding of partially ionized matter at high temperature and density, conditions where the plasma environment modifies atomic structure. A typical phenomenon encountered in high-density atomic plasmas is the process by which the combined effects of continuum lowering, plasma screening,and interaction with neighbor ions move electron bound states up through zero energy into the continuum. This process is known as pressure

ionization.

Pressure ionization has obvious spectroscopic consequences; the destruction of excited (unoccupied) levels with high quantum numbers modifies the plasma emission or absorption opacity. At very high densities, pressure begins to release electrons from occupied core levels. This increases the average ionization state and changes plasma energy, pressure, and transport coefficients (More, 1981, 1982, 1983). Exactly what happens when a bound state crosses zero energy and an electron which was previously bound becomes free? Many published calculations find abrupt discontinuities in plasma properties. However, with few exceptions, physical observables are continuous functions of the conditions of observation. Either on the basis of a formal principle of continuity in nature, or more simply as a manifestation of common sense, we expect that observable plasma properties change continuously through the transition associated with pressure ionization. In this article we shall examine these questions in the context of the spherical-cell self-consistent-field model for atoms in hot dense plasmas; in this case the continuity property is established by rigorous mathematical theorems reviewed in Section 11. As bound states move into the continuum, they evolve into low-energy scattering resonances; these shape resonances occur only at high densities where there is strong screening of the atomic potential. Electrons which occupy resonance states are neither strictly localized nor totally free. For short times the resonance electrons are trapped by the centrifugal barrier; over longer times they escape to large distance.

PRESSURE IONIZATION

307

The resonance states can be given a rigorous theoretical description in terms of the Jost-function formulation of quantum scattering theory; this theory is reviewed in Section 111. In Section IV we apply this method to derive new formulas for the thermal occupation, density, and pressure of electrons in resonance states. Throughout this survey, we will find close connections between many high-density atomic phenomena. Pressure ionization, resonances, and the continuity principle are linked because the scattering resonances interpolate smoothly between localized bound states and freely moving continuum states. Many plasma properties, including equations of state, electrical conductivity, bremsstrahlung emission, and x-ray opacities, are strongly affected by resonances. Even the meaning of the plasma ionization state is clarified by understanding the physical character of resonance states.

A. SELF-CONSISTENT-FIELD (AVERAGE-ATOM) MODEL The self-consistent-field method provides a general framework for calculating properties ofmatter at high density and temperature. It is well suited to the treatment of heavy atoms having many thermally excited bound electrons. In addition, the SCF method incorporates the main density effects: pressure ionization, short-range ion screening, and the possible degeneracy of free electrons. In self-consistent-field calculations, the atom is represented as a point nucleus embedded in a spherical cavity in a continuous background positive charge. The cavity radius R, is determined by the plasma density ( p = 3M,/ 4aR2, where M,, = atomic mass) (see Fig. 1). The self-consistent electrostatic potential V(r)is determined by V 2 V = -4a[Zeb(r)

+ p+(r) - en(r)]

(1) where p+(r) is the charge density of the continuous external background charge and n(r) is the electron number density. In the work of Liberman (1979, 1982), p+ is constant for r > 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:

.

:<

ni = 6.76 x 1022 cm-3

T=3.16x104K

...= . .

=%

* -

0.1

1

0-O

1

I

I

1

d

0

IlAE "0

In the second form the nl bound-state contribution drops continuously to zero when Ed + 0. This implies the continuity of thermodynamic properties, although it is not a rigorous proof of analyticity; that was developed for the degenerate case by Kohn and Majumdar (1 965).Petschek (1 971) and Petschek and Cohen (1972)examine the partition function for several shortrange model potentials in order to explicitly illustrate the continuity property.

E

322

R. M. More B. SOMEUNSATISFACTORY CALCULATIONS

Is any of this controversial? Are there any calculations which violate the continuity principle? In the older literature, we often find a quantum treatment of bound electrons combined with a statistical model for the free electrons. This method ignores the phase-shift contribution to the continuum density of states in Eq. ( 17). When a state n, 1 is pressure ionized, the number of free electrons per atom suddenly jumps from Z* to Z* 2(21+ l)f(O), where f(~) is the Fermi distribution function. In the degenerate rangef(0) = 1, and the pressure suddenly jumps by a factor [ 1 2(21+ l)/Z*]5/3. Using this hydrid method, Zink ( 1968) calculates the equation of state for hot plasmas; one example reproduced in Fig. 4a shows a sudden pressure jump of almost two orders of magnitude at density - 15 g/cm for iron at kT = 7.7 eV. There is, of course, no indication of such a dramatic discontinuity in the experimental shock Hugoniot of iron. More recently, Kirzhnitz et al. ( 1976) predict a sequence of discontinuous (first-order) phase transitions at densities corresponding to pressure ionization. Alternatively, the plasma pressure is a discontinuous function of nuclear charge. Again the difficulty appears to originate in the continuum density of states (Fig. 4b). In an effort to address this problem, Lee and Thorsos ( 1978) introduce a spherical-cell model calculation of resonance states at positive energies, and calculate a resonance pressure in addition to the ideal-gas pressure. Only one resonance is allowed for each I in their calculation, and this restriction probably causes the sharp discontinuities they find in pressure as a function of Z (Fig. 4c). In a recent review article, Bushman and Fortov (1 983) quote the previous results without critical comment. To be fair, we must note that the continuity principle is rigorously established only for plasma properties calculated within the spherical-cellcentralfield model. (The calculations considered here, however, are all one-atom central-field models.) It has not been proven that an arbitrary many-atom model could not have a discontinuous transition.

+

+

C. THESPECTRAL WINDOW The next example is also very instructive. The group led by G. E. Norman (Kobzev et al., 1977) has proposed to explain plasma emission experiments

PRESSURE IONIZATION

323

in terms of a “spectral window” just below the lowered continuum in highdensity plasmas (Fig. 4d). If there were low-opacity regions of this type, they would have dramatic implications for the dynamical behavior of dense plasmas having high energy densities. Norman’s proposal is based on a well-known calculation of Weisheit and Shore ( 1974), who examined the radiative properties of hydrogen atoms in a dense plasma modeled by imposing Debye screening on the nuclear potential. For densities where the upper state n’l’ is close to the lowered continuum, Weisheit and Shore find that the oscillator strength of individual absorption lines nl n’l’ is reduced relative to the free-atom oscillator strength. A recent reexamination of the same model by Hohne and Zimmerman (1982) verifies the calculations of Weisheit and Shore. However, Hohne and Zimmerman add the observation that while the oscillator strength for the transition,f(nl-+n ’l’) falls to zero as E,,,/,+0, the level spacing dE,,,/dn’ also tends to zero in such a way that the ratio

-

f(n1- n‘I‘)/(dE,,.,./dn‘)

(23)

remains approximately constant. This statement implies that the oscillator strength per unit frequency interval remains approximately constant. There is little change in the observed opacity and, in particular, no prediction of a transparent “spectral window.” It is easy to understand the result of Hohne and Zimmerman with the use of the WKB approximation for the wave function of the upper state w,,,,,

In Eq. (24), q(r) is the local wave vector, q(r) = [k2- u(r) - ~ ~ ( r ) ] ’ / ~ . The WKB normalization factor A is given by (Landau and Lifshitz, 1958)

Because the other factors in the expression for the oscillator strength

fW-

n’l’)

a l(wfl/l~l~fl,,,)12

(26)

approach constant values in the limit E,,,/,-,0, it is clear that the ratio fl(dE,,/,/dn’) approaches a finite (nonzero) constant.

324

R. M. More

111. Resonances Resonances supported by a one-electron potential barrier are called shape resonances.2In this section we summarizethe quantum theory ofresonances includinga discussion of the normalizationof resonance wave functions, the representation of the Green’s function and Smatrix as a sum over resonance states, and the perturbation theory of resonances. The theory is based on rigorous and general results of scattering theory which apply to essentially any potential having a finite range &. We conclude the section with a practical method to predict where resonances occur; the result is that prominent shape resonances occur only in very dense plasmas ( p > 1 g/cm3 or electron density n, > 1023/cm3). In the picture of scatteringtheory, a resonance affectsall of the continuum eigenfunctions within an energy range zkr,,!of Ed. However, there is an alternative way of thinking, in which we concentrate upon the resemblance of the resonance to a bound state. The resonance is located at a definite (quantized)complex energy, Ed = Ed iTd.In this second viewpoint, each resonance is a unique generalized eigenstate and proves to be characterized by a single wave function.

+

A. RESONANCE WAVEFUNCTIONS

Using the scattering theory summarized in Eqs. (8)-( 16), a resonance wave function qnrcan be defined by Equation (27) specifies q,,/ in terms of the Jost solution f;(Q,,,; r) and a normalization constant Again, f‘ is an abbreviation for dfldk. The resonance wave vector Q,,, is determined by f;(Q,,/)= 0. Resonance states following this definition are studied by Humblet (1952), Zel’dovich (196 I), Berggren (1968), More (1971), More and Gejuoy (1973), and other authors quoted below. The wave function bp /(r)obeys the Schr6dinger equation with the comAt the origin, qn,/approaches zero plex energy eigenvalueJ!!, = ( h2/2rn)Qi,r. because A(Q,,/)= 0. At large radii, the behavior is less satisfactory; It is important to distinguish shape resonances from the resonances associated with autoionizing levels of multiply excited ions which are very important at lower plasma densities, but which are not described by the average-atom model.

PRESSURE IONIZATION

325

1m(Qn,/)> 0, and we have I q n , / ( r ) l ~lNn,/l ex~IWQn,/)rl+O0

(29)

Because of this property, we must be careful to avoid writing meaningless formulas. However, with appropriate assumptions on the potential [such as the assumption that V(r)= 0 for r > &I, the most important formulas involving qn,/are mathematically well behaved. Can we explain or justify the definition? First, Eqs. (27) and (28) are identical to Eq. ( 1 1) for the bound-state wave function w , , ~This . implies that any expression involving wn,/will automatically continue into a formal generalization involving the resonance wave function In some cases, of course, the direct generalization is mathematically divergent or unsatisfactory. [One example is the bound-state normalization equation, which is modified for resonances, as shown in Eq. (3 1) below.] A better way to understand the definition of qn,/is to examine the radial one-electron Green’s function G/(r, r’, k). This operator is defined by the formal expression GI = I/(€ - ITI), where HI is the Hamiltonian for the radial Schr6dinger equation and c = h 2k2/2m.G, is a single-valuedfunction of wave vector k (but not of the energy E). G!is written in terms of the Jost functions in Appendix B. The Green’s function G, has simple poles at k + - Q,,/;the residue of G/at the resonance is easily shown to be (see Appendix B)

give a straightEquation (30)shows that the resonance wave functions q~,,/(r) forward generalization of bound-state wave functions (More, 1971;GarciaCalderon and Peierls, 1976). OF RESONANCE WAVEFUNCTIONS B. PROPERTIES

Equations (27) and (28) apply to any finite-range potential, as does Eq. (30).It can be shown that when the resonance lifetimes are sufficiently long, the wave functions q J r ) automatically become real, normalized, orthogonal functions on a fixed interval 0 < r < R (More, 1971). For large radii (kR > I), we have (see Appendix A)

326

R. M . More

These equations apply to any finite-range potential. Equation (3 1) shows that resonance states cannot be normalized on the infinite range (0 < r < 03) because the right-hand side of Eq. (3 1) diverges as R 03. However, as the bamer becomes high or as the resonance approaches the zero-energy threshhold, it can be shown that the normalization factor Nn,, approaches zero: N i , , a Im(Q,,/). This implies that the right-hand side of Eq. (31) approaches unity as the lifetime becomes long. In this limit, p,,/ is automatically normalized to unity on thefixed range 0 < r < R. It can also be shown that pn,[becomes real as Im(E,,,) + 0. Equation (32) shows that two resonance states (n # - m ) become orthogonal in the same limit (i.e., the limit of long lifetimes). For a bamer of finite height, Eq. (3 1) can be written

-

where Z,,/ = Z,,,(R) is determined by Eq. (3 1). This quantity Zn,,can be interpreted as a renormalization factor; i.e., it is given by ZnSl=1/(1 dH,,,/de),where H,,/ is an energy-dependent effective Hamiltonian for the finite subsystem defined as the region 0 < r < R with transmitting boundary conditions at r = R. This intuitive interpretation of the normalization (or renormalization) of the resonance states is developed in detail by More and Gejuoy (1 973). It is possible to construct from pn,,(r)a special initial state for a time-dependent decay calculation. This initial state begins localized behind the barrier and decays in an exponential fashion (with the usual short-time and longtime nonexponential corrections). The state constructed from pn,,is pure in the sense that it has no other exponential decay mixed with it, even when other resonances are nearby in energy. The construction applies to an arbitrary short-range potential with a bamer (More, 1971). Equation (30) shows that the residues of the Green’s function at its poles are products of the resonance wave functions. For both r, r’ < R, (= range of the potential), it follows that the Green’s function is given by a series expansion (More, 1971; More and Gejuoy, 1973; Bang et al., 1978, 1980; Berggren, 1982)

This expansion, based on the Mittag- Leffler theorem, does not converge beyond the range of the potential. The surprising form of Eq. (33) is essential; the denominator as written locates the resonance poles of GIin the unphysical sheet of the energy plane,

PRESSURE IONIZATION

327

i.e., in the lower half k plane (Fig. 3). Thus Eq. (33) explicitly gives the analytic continuation of the Green’s function onto the second sheet of the energy plane. Another useful result, proven by Regge ( 1 958), represents the Jost function as an infinite product of simple functions of k

Equation (34) is derived from the Hadamard product representation of an arbitrary entire function in terms of its zeroes (Newton, 1960). It is valid for potentials of finite range R, [e.g., a self-consistent-field potential with the ion-sphere boundary condition V(r)= 0 for r > &I. Equation (34) also gives a representation of the partial-wave S matrix S,(k)=f;(k)/J;(-k). . Both Eqs. (33) and (34) are written for a potential with no bound states of angular momentum 1. In this case each resonance contributes a pair ofterms, which we denote with positive and negative index n (recall Q-n,,= Q:,). If there are bound states, they contribute similar terms, but are paired with virtual states occurring on the positive imaginary Q axis. From the asymptotic expansion of the Green’s function one can derive an equation which expresses the completeness of the set of resonance functions (More and Gerjuoy, 1973; Romo, 1975; Bang et al., 1978; BangandGareev, 198 1 ; Berggren, 1982; Garcia-Calderon, 1982)

Along with this equation, many of the authors quoted demonstrate another summation theorem which proves that the resonance functions form an overcomplete set, and cannot be mutually orthogonalized. In a loose manner of speaking, the set of resonance functions includes twice the proper number of functions [i.e., it includes states qn,,(r)and q-,Jr) which scarcely differ for high barriers]. Bang and Gareev (1981) employ a continuation with respect to the strength of the potential to show that the factor f in Eq. (35) is literally associated with double counting in the limit where the potential becomes that of a very deep well (in that limit, the wave function of a deeply bound state and its virtual or antibound pair become identical). Garcia-Calderon ( 1982) gives a numerical test of Eq. (35) for the &potential model described in Appendix E; the seriesconvergesto the expected answer. Bang et al. (1978) use Eq. (35) to recast Eq. (33) in several interesting alternative forms. It is not possible to alter the normalization of qn,,in order to remove the factor f in Eq. (35) without sacrificing the other desirable properties expressed in Eqs. (30), (31), (33), and (36). The completeness relation Eq. (35)

328

R. M . More

and possible alternative expansion theorems are examined in great depth in recent works by Romo (1979, Bang et al. (1980), and Berggren (1982).

C. RESONANCE PERTURBATION THEORY Another beautiful property of the resonance wave functions follows from the Green’s function representation. This is a perturbation theory, which we can write as

agn,/= ( G ~ / I ~ V I V ~+/ )O(JV2

(36)

SVis a (small) perturbation potential. Equation (36) is unlike the usual textbook treatment of resonances because it does not attempt to calculate the decay process by perturbation theory. Instead, we assume that the resonances of one potential have been exactly determined, and are now changingthe potential by SV.The question is then: how do the resonances move? Equation (36) gives the change in the complete complex energy; is%.,it unites expressions for the changes in both real and imaginary parts of Ed. It is rather surprising that Eq. (36) is correct, in view of the fact that the wave functions pn,/cannot be normalized. The result is limited to perturbations W ( r )which are nonzero only within the range of the original potential. thus the diagonal matrix element of the perturbation, (+n/lSvlqn/) =

[

p i / ( ~ ~ dr (r)

(37)

is made convergent by SV(r) and not by the wave function itself. The perturbation theory can be physically understood in terms of the energy dependence of the barrier penetration probability. When the perturbation SV(r) changes the real part of by SE, then the tunneling energy changes and this implies a corresponding change in the tunnel decay rate. The first-order changes SE and R are intimately coupled together, and this coupling is expressed by the appearance of the complex square pil of the wave function in Eq. (37) rather than the absolute value IpnrlZ. The first-order resonance perturbation theory has been rediscovered a number of times following Humblet ( 1952)(see Zel’dovich, 1961;Berggren, 1968;Baz’ et al., 1969).Appendix D gives a brief sketch of the derivation of the second-order perturbation theory from the representation (33) of the Green’s function, More and Gejuoy ( 1973) give the relationship of these results to the more conventional perturbation theory of Kapur-Peierls resonances.

PRESSURE IONIZATION

329

A cruder approximate treatment of resonances could be developed by imposing some simple boundary condition [e.g., ~ ( r=) 01 either at r = &or at the peak of the barrier potential. A real-energy wave function defined by this boundary condition would presumably be a close approximation to the resonance wave function, and with it we could expect to predict changes in En,with reasonable accuracy. However, the crude wave function would not be able to correctly link together the perturbations in Ed and r,, and in this sense the resonance wave functions q,,, are much more powerful.

D. CONVERGENCE OF THE EXPANSIONS

In the last few years, there have been several careful studies of resonance series expansions within the context of theoretical nuclear physics. The central issue is whether these expansions converge well enough to provide a useful representation of continuum states. Bang et al. (1978) report a numerical study for 1 = 0 potentials of 6 function and square-well form. They examine convergenceof expansions of the Green’s function, continuum wave functions, and S matrix. Bang ef al. find the series converging to an accuracy circa 1 part in lo4 for 100 terms. In an appendix, they give an interesting argument showing that coincident complex poles cannot occur for the rectangular well or other similar potentials. They also discussthe generalization of the theory to include a Coulomb potential in the large-r region. Bang et al. (1980) give a systematic analysis of expansionswith respect to nine basis sets, including the resonance wave functionsqnr,and various other sets defined by Kapur - Peierls and Weinberg boundary conditions. They find the resonance expansion [i.e., Eq. (33) above] converges very well, especiallywhen convergenceis acceleratedby a subtraction technique which extracts the part of the Green’s function which has simple energy dependence. Their test is particularly demanding because they consider nuclear potentials of square-well and Saxon- Woods type, which do not have a barrier for I = 0. Berggren ( 1982) also examines the numerical convergence for the Saxon - Woods potential with 1 s coupling. In these cases, it is necessary to sum many terms of the series in order to achieve good convergence. However, for potentials which have strong barriers, there is a region of energies in which a few resonances of small width totally dominate the numerical results. In such cases, we find that the expansions of Eqs. (33) and (34) achieve excellent numerical accuracy with only one or two terms.

-

-

330

R. M. More E. ESTIMATE OF BARRIER HEIGHTAND RESONANCE ENERGY

Where do resonance levels occur in practice? We now develop simple approximate formulas suitable for finding the most prominent resonance of a given angular momentum for dense atomic plasmas described by the self-consistent-field model. The formulas predict the density at which the n, n - 1 level ( e g , 2p, 3d, 4f;. . .) is pressure ionized. The results are good enough to guide the startup of more accurate search techniques. The formulas are obtained by continuing a bound-state eigenvalue formula into the positive-energy range. The bound state is predicted to give rise to a resonance if it moves into the energy interval (Fig. 5a) 0 d En, V,,, where V,, is the height of the centrifugal potential barrier. The barrier height is approximately given by the centrifugal potential evaluated at r = &, the ion-sphere radius (see Fig. 5b)

with A the atomic mass; p the density in g/cm3. Because the maximum potential occurs at somewhat smaller radii, the actual barrier height is slightly larger than Eq. (38); for a number of examples we find, Eq. ( 3 8 ) is accurate to about 10%. The energy level En,can be estimated with the help of a WKB screening model (More, 1982). The general formula is

The quantities on the right side of this equation are defined and evaluated in the following paragraphs. Because the most prominent resonances occur for the highest densities, we consider only the series 2p, 3d, 4J . . . ,for which n = I 1 . For these states it is reasonable to neglect the quantum defect An/. The plasma ionization state Z* is required. For the purpose of developing a useful approximate formula, we take Z* from an approximate fit to the equilibrium Thomas- Fermi ionization state (see Table IV for details). Although this formula omits shell-closureeffects which are quite prominent in low-density ionization calculations, it will be used at high densities, where it is reasonably accurate. Using this ionization state, and assuming the ion core (containing the tightly bound electrons) has radius -sz Ro, the electrostatic potential is approximately given by the ion-sphere formula based on a uniform free-elec-

+

33 1

PRESSURE IONIZATION

-

-401 -60 20

I-

01

-I

25

30

35

40

Density (g/cm3)

Radius r (10%ml

Density (g/cm3)

FIG.5. (a) Analysis of 3dresonances for plasma consistingofargon at constant temperature * * ), I = 2 centrifugal barrier height from Eq. (38). (- - -), 3deigenvalue predicted by Eq. (39), using the Thomas- Fermi ionization state given by Table IV. This model predicts a resonance state (0 < EJd< VmU)for densities of 24 to 32 g/cm J. The points marked x represent 3 d eigenvalues obtained from the relativistic SCF program of D. Liberman, predicting occurrence of a resonance beginning at p = 30 g/cmJ. (b) Effective potential -eV(r) ( h 2 / 2m)[l(l+l)/r2] for the radial Schrddinger equation for argon at T = 1 keV, p = 30 g/cm 3. The electron density is = 7 X 1OZ4/cm3. The potentials are taken from Thomas- Fermi calculations and show the conditions leading to a 3dresonance state. It is evident that the maximum bamer potential is nearly equal to h21(1+ 1)/2mR$. ( - - -), I = 0; ( - - -), I = 1; (- - -), I = 2; (-), 1 = 3. (c) Density- temperature regions in which argon is predicted to exhibit pressure ionization (resonance levels) according to the simple model of Eqs. (38)-(4 I). The calculations refer to equilibrium (LTE) plasmas. 1 keV. (

+

332

R. M. More TABLE IV IONIZATION STATE^ THOMAS-FERMI (a) T - 0 x=dPlzA)s

pin g/cm3

X

f= 1+ x z+=fZ

+ m x

az+-z* /3 ---ap ~ W X

a = 14.3139

p = 0.6624

(b) Any temperature a , = 0.003 323

TO TF= 1 To A = a , T p a,T$ B=-exp(bo+b,TF+b2TZ) c c1 TF cz Q I- A R B Q - ( R C + QF)IIC x aQP

+

+

+

-

Z+=

a2

-

a3 9.26 i 48 X i O-' a, = 3.10165

b, = - 1.7630 b , = 1.43175 b2 0.3 1546 cI = -0.366667 6

ZX ~2

1 + x + m x

= 0.9718

= 0.983333

An approximate fit to Eq. (5 I).

tron distribution

From the effective charge Z(r) we compute the inner-screening charge Q(r) as

Q(r) = Z(r) - rZ'(r) = Z*[ 1 - (r/Ro)2] This is the effective charge which governs the electric field, E = - Q(r)e/r2. The average orbit radius r,, is determined by r,, = aon2/Q(r,,).This equation is easily solved by Newton's rule, except at densitieswhere a , n 2 / Z *becomes too close to &. The interesting solutions occur before this situation arises. Now the energy level is calculated from

PRESSURE IONIZATION

333

Combining these approximations, we obtain a chain of algebraic equations which predict the conditions required for resonance states. The formulas are simple enough to be used in searchesover wide ranges of density and temperature (see Fig. 5c); and the results are good enough to initiate more accurate quantum calculations. For example, Fig. 5a compares the present formulas to fully relativistic quantum self-consistent-field calculations (D. A. Liberman, unpublished); these calculations find resonances in a density range overlapping that of the analytic model. For the case considered in Fig. 5a (argon plasma at T = 1 keV, p 30 g/cm3),an estimate of the plasma line broadening (Griem, 1964; H. Griem, personal communication) shows that it is likely to exceed the barrier potential V-. This would not be the case for plasmas containing more highly charged ions. The most surprising qualitative feature of these results is the prediction that resonances of small angular momentum are more prominent. This is true despite the fact that the barrier height rises with angular momentum 1. The reason is that, for the lowest resonance of each series, the principal quantum number n = 1 1 is smaller for smaller 1. For this reason, the density is higher when the state is pressure ionized, and this increase in bamer height overcomes the decrease due to selecting a smaller angular momentum. As a numerical approximation,

-

+

V-

+ 1)’

= (1.7 eV)1Z2/(1

(41)

gives a reasonable estimate of the barrier heights [see Eqs. (7) and (38)]. This then indicates the range of energies over which the resonance can exist. In typical cases, this range of energies is associated with less than a factor two in densities.

IV. Applications The theory of resonances provides the mathematical tools to describe the physics of pressure ionization-at least within the context of the sphericalcell model. In this section we will develop some representative applications: the counting of electrons in resonance states and their contribution to the plasma pressure. In each case, a formula for the continuum quantity (number or pressure of continuum electrons) is transformed to isolate and exhibit the contribution of the n, 1 resonance. The objectivesare to clarify how the continuity principle operates, to set the stage for perturbative improvements of the self-con-

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R. M. More

sistent-field approximation, and to develop an alternative representation of resonance contributions that is computationally preferable to the direct use of a very fine energy scale to resolve narrow resonances. A. THERMAL OCCUPATION OF RESONANCE STATES Considering the resonance levels to be characterized by a complex energy

En,= En,+ ir,,,we immediately encounter an interesting question about

the equilibrium thermal occupation probability of the resonant state. The Boltzmann factor exp(- E,,/kT) must be generalized in some way; one might conjecture_various forms such as Re[exp(- &,JkT)], exp(- Re[&/ kT]), or exp(-I&,J/kT). All of these forms agree to zero order in the width r,,,but what about the corrections to this limit? In the average-atom model, the Boltzmann factor is replaced by a Fermi function f(~) = [ 1 exp(c - p)/kT]-', and an equivalent question arises. Naturally one desires not only a formula but also a clear statement of what the formula means. This question can be given a rigorous and exact answer, starting from the electronic density of states given by Eq. ( 17). To evaluateg(E), we employ the infinite-product representation of the Jost function [see Eqs. (1 2) and (34) above]:

+

If there are bound states of angular momentum 1, this product must be augmented by additional factors. A series expansion for S,(k) is obtained by taking the imaginary part of the logarithm of this product representation. The derivative with respect to k is 1 dk n k - Qn/ However, the low-energy phase shift obeys dS,/dk a k2' as k 0. For 1 3 1, this vanishes as k-0. The case I = 0 is special; dSo/dk approaches a constant = - a (a is the scattering length) as k 40. In order for Eq. (42) to reproduce this behavior, the resonance zeros Qnl must obey

-ds, - -R,

4- Im

-

(43)

TIm(&)=a-Ro,

1=0

PRESSURE IONIZATION

335

Using these sum rules and combining together the terms which arise from and Q+, we obtain zeroes at Q,,,

where S,,is unity if 1 = 0 and zero otherwise. From the density of statesg(E),the total number of continuum electrons is N, =

[

g(E)f(e) de = No

2 " + -d i(21 + 1)

The ideal-gas continuum contribution is No =

1

go(4f(4

Inserting the series for dS,/dk, we find N, = NA

+ 3SN,,,

In this equation, Nh is a modified free-electron contribution containing the scattering-length correction for 1 = 0 in Eq. (44),and SNn,is the resonance population given by SN,,,,= 2(21+ 1) Re[F(&,)]

(45)

where the resonance thermal occupation function or generalized Fermi Dirac distribution is a Cauchy integral of the real-energy Fermi distribution

Equations (44)and (45)answer the question posed at the beginning of the section. They give the average thermal occupation of the n, 1resonance state in an explicit form; the result is exact even for broad and/or overlapping resonances and applies to any potential that obeys the ion-sphere boundary condition [ V(r)= 0 for r > 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.

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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
S ij

d

j

=~

+ ~ ( 2+) uc

( 1 )

(79)

This is a generalized virial theorem applying to an arbitrary region surrounding one nucleus. Applied to the ion-sphere surface in the case of a neutral cell, the electric field and therefore Tijvanishes on the boundary of the cell, giving the final result

+ +

(80) 3P,V= K(1) 4 2 ) Uc As mentioned at the beginning of the section, this form is a reasonable generalization of the virial theorem to a finite region (in this case a neutral region) imbedded in a larger volume. The pressure which appears on the left side of the equation is indeed the rr component of the pressure tensor, as in Eq. (65). However, the result is not equivalent to the Liberman or Pettifor pressure formulas, which differ by a term proportional to the gradient of n(r). This difference originates in their assumption of a virial theorem for which the right-hand side is U 2 4 , ) . Of course, when applied to a crystalline proximity cell surrounding one atom, the difference vanishes as V n has no perpendicular component (by symmetry).

+

346

R. M. More

In the spherical-cell model this gradient term is nonzero, and we conclude with a preference for the tensor pressure defined by Eq. (65). The issue can only be finally resolved by a calculation of pressure as a thermodynamic derivative. Although we are reluctant to leave these interesting unresolved questions, the original objective has been accomplished: Eq. (69) shows how to analyze the pressure into additive resonance contributions. These contributions are formally additive and independent, even when resonances are nearby in energy. The entire analysis is independent of specific details of the potential aside from the assumption of a finite range &.

V. Conclusions This article has reviewed a variety of techniques and questions concerning the continuous evolution of bound eigenstates into free states under the influence of increasing pressure. As a general principle, a continuously evolving system has continuously changing properties. Although the universality of the principle can be questioned -surely there are some interesting exceptions -nevertheless it is clear that the continuity principle gives useful general guidance. We have also reviewed a powerful formal method for treating resonances like discrete eigenstates. Although the resonance wave functions cannot be normalized, they can be employed as a basis for the expansion of properties of the continuum electrons and give closed-form expressions of great generality. It is hoped that the numerical application of these methods can be explored in future work. We have emphasized two serious difficulties or limitations of the spherical-cell model: One category of difficulties is revealed when we attempt to push the theory to reach the complete thermodynamic consistency already achieved by the Thomas-Fermi class of theories; the other category is a fundamental physical problem, stemming from the neglect of fluctuations in the plasma environment. Simple estimates show that Stark broadening is likely to exceed the barrier height for most cases of interest in the spectroscopy of hot plasmas, except possibly in future experiments at very high nuclear charge (Z* > 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 << R, the phase shifts are significant in magnitude only for small angular momenta 1 G kR,, and so the important or interesting second term is given correctly by this simple derivation. A more elaborate analysis by Anderson and McMillan ( 1967) shows that one obtains Eq. (1 7) by carefully forming the density of states from the one-electron Green’s function 1 g(E) = - R

T 2(21+ 1)

Im[G,(r, r ; E ) ] dr

(97)

The issue of asymptotic forms for large angular momentum arises again; Anderson and McMillan convincingly settle the question by manipulation of the exact radial eigenfunction. The third derivation is very simple but rather formal. Using the three-dimensional unperturbed Hamiltonian fi, = - ( h2/2m)V2, a Green’s function &,(E) and scattering T matrix ?(AT)are defined by

&E)

=

1/(E- fro)

F(E)= P +

Ve,(E)P(E)

(98) (99)

where P = V(r)is the central potential regarded as an operator. These operators obey the differential equations

- dGo/dE = Ga

-dF/dE

=

FGiF

(100)

(101)

Using these equations and the invariance of the trace, one can transform the density of states obtained from the complete Green’s function 6 =

e, +

PRESSURE IONIZATION

35 1

GOTGOto find

+ 7l Im Tr (-&In T )

1 g ( E ) = - Im[Tr G ( E ) ]= go(E)

n

(102)

This is easily evaluated in the partial-wave representation, in which the operator f is diagonal a

.

TI= - ersl(k) sin S,(k)

k

Again, one obtains the expected result, Eq. (17).

Appendix D: Resonance Perturbation Theory In this appendix we review the second-order resonance perturbation theory. The starting point is an unperturbed system which has resonances of finite width (e.g., a potential having a finite barrier). The perturbation is an extra potential U(r),such as that arising in the Stark effect, which shifts the resonances and alters (in general) their widths. The unperturbed problem is described by a Green’s function Go of the type described in Appendix B. The perturbation theory is derived from the formal Dyson equation G = Go

+ GoUG

(103) where Go and G are unperturbed and exact Green’s functions, and U is the perturbing potential, assumed to vanish for r > 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

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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

CUMULATIVE SUBJECT INDEX

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

CUMULATIVE SUBJECT INDEX

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