Advances in Atomic, Molecular, and Optical Physics, Volume 29

Advances in

ATOMIC,MOLECULAR,AND OPTICAL PHYSICS VOLUME 29

EDITORIAL, BOARD

P. R. BERMAN New York University New York, New York K. DOLDER The University of Newcastle-upon-Tyne Newcastle-upon-Tyne England

M. GAVRILA F.O.M. lnstituut voor Atoom- en Molecuulfysica Amsterdam The Netherlands

M. INOKUTI Argonne National Laboratory Argonne, Illinois S . J. SMITH Joinr Institute for Laboratory Astrophysics Boulder, Colorado

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

Sir David Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN'S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

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

VOLUME 29

@

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Publishers Boston San Diego New York London Sydney Tokyo Toronto '

This book is printed on acid-free paper. @ Copyright 0 1992 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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OF CONGRESS CATALOG CARDNUMBER: 65-18423 LIBRARY ISBN 0-12-003829-3 ISSN 1049-25OX

PRINTED IN THE UNITED STATES OF AMERICA 91929394 9 8 7 6 5 4 3 2 1

Contents

CONTRIBUTORS

ix

Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques Chun C . Lin and L. W. Anderson

I. 11. 111.

IV. V.

Introduction Excitation out of the Ground Level into Nonmetastable Levels Excitation out of the Ground Level into Metastable Levels Excitation out of Metastable Levels Conclusion Acknowledgments References

1 2 12 23 27 30 30

Cross Sections for Direct Multiphoton Ionization of Atoms M. V. Ammosov, N . B. Delone, M . Yu Ivanov, I . I . Bondar, and A . V. Masalov I. 11. 111.

IV. V.

VI.

Introduction Methods of Measuring the Principal Quantities That Characterize Multiphoton Ionization of Atoms The Procedure for Measuring the Quantities Needed to Find the Multiphoton Cross Sections and Its Accuracy The Results of Measuring the Multiphoton Cross Sections of Direct Atom Ionization Analytical Expression for Estimating the Multiphoton Cross Sections of Direct Atom Ionization Conclusion References

34 45 73 90

101 107 108

Collision-Induced Coherences in Optical Physics G . S . Aganval I. 11.

Introduction A General Framework for the Calculation of Nonlinear Optical Phenomena

114 116

Contents

vi

Second-OrderOptical Response and Collision-InducedCoherences Collision-InducedCoherences in Fluoresence and Ionization Spectroscopy V. Collision-InducedCoherences in Third-Order Nonlinear ResponseFour-Wave Mixing VI . Collision-InducedCoherences in Probe Absorption in the Presence of a Pump VII . Relation between Collision-InducedCoherences in Second-Order and Third-Order Responses VIII. Collision-Induced and -Enhanced Resonances in Fifth-Order Nonlinearities IX . Effect of Cross-Relaxationon Collision-Induced Resonances X. Dipole-Dipole Interaction-InducedResonances XI. Collision-Induced Resonances in Spontaneous Processes XI1. Narrowing and Enhancement of Signals Due to VelocityChanging Collisions XIII. Nonlinear Response, Collisions, and Dressed States XIV. Other Sources of Coherences Similar to Collision-InducedCoherences xv. Conclusion Acknowledgments References III. IV.

119 124

129 139 145 147 149 152 155 160 164 166 172 174 174

Muon-Catalyzed Fusion Johann Rafelski and Helga E. Rafelski I. 11.

111. IV. V.

Introduction Nuclear Fusion dr Muon Catalytic Cycle and Reaction Rates Muon Sticking Conclusion Acknowledgments References

177 181

186 199 209 21 1 212

Multiple-ElectronExcitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions

J . H.McGuire I.

Introduction

II. Theory III. Observations and Analysis N. Conclusion Acknowledgments Appendix: Correlation References

217 219 263 312 313 314 315

CONTENTS

vii

Cooperative Effects in Atomic Physics J . P. Connerade

I. 11.

HI. IV. V. VI . VII. VIII .

Ix.

X. XI.

XI.

Introduction Many-Body Effects and the Conservation of Angular Momentum Rydberg Series Non-coulombic Potentials and the Periodic Table Giant Resonances Atomic Giant Resonances in Other Environments-Controlled Collapse and Instabilities of Valence Giant Resonances in Nuclei and in Atomic Clusters Are Giant Resonances in the d and f Sequences Atomic Plasmons? Extending Mean Fields Beyond the Hartree-Fock Scheme Can One Blow Off Complete Shells by Laser Spectroscopy? Interactions Between Giant Resonances and Rydberg SeriesIntershell and Intersubshell Couplings Conclusion References

INDEX

CONTENTS OF PREVIOUS VOLUMES

325 328 330 333 337 34 1 344 348 352 355 358 364 365 369 373

This Page Intentionally Left Blank

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

G. S. Agarwal(113), School of Physics, University of Hyderabad, Hyderabad 500 134, India M. V. Ammosov (33), General Physics Institute, Vavilov str., 38, 117942, Moscow, USSR L. W. Anderson (l), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

I. I. Bondar (33), Uzhgorod State University, Oktyabrskaya str., 54 2924000, Uzhgorod, USSR J. P. Connerade (325), Blackett Laboratory, Imperial College, London SW7 2AZ, UK

N. B. Delone (33), General Physics Institute, Vavilov str., 38, 117942, Moscow, USSR M. Yu.Ivanov (33), General Physics Institute, Vavilov str., 38, 117942, Moscow, USSR Chun C.Lin (l), Department of Physics, University of Wisconsin, Madison, Wisconsin 53706 A. V. Masalov (33), Lebedev Physical Institute, Leninsky prosp. 53, 117924, USSR J. H. McGuire (217), Department of Physics, Tulane University, New Orleans, Louisiana 701 18 Johann Rafelski (177), Department of Physics, University of Arizona, Tucson, Arizona 8572 1 Helga E. Rafelski (177), 5250 N. Foothills Drive, Tucson, Arizona 85718

This Page Intentionally Left Blank

ADVANCES IN ATOMIC, MOLECULAR. AND OFTICAL PHYSICS, VOL. 29

STUDIES OF ELECTRON EXCITATION OF RARE-GAS ATOMS INTO AND OUT OF METASTABLE LEVELS USING OPTICM AND LASER TECHNIQUES CHUN C . LIN and L. W.ANDERSON Department of Physics University of Wisconsin Madison,Wisconsin

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 11. Excitation out of the Ground Level into Nonmetastable Levels. . . . . . . . . 2 111. Excitation out of the Ground Level into Metastable Levels . . . . . . . . . . 12 A. Excitation into the Metastable Levels of Helium . . . . . . . . . . . . . 12 B. Excitation into the Metastable Levels of Neon . . . . . . . . . . . . . . 15 C. Excitation into the Metastable Levels of Argon . . . . . . . . . . . . . 21 D. Excitation into the Metastable Levels of Krypton. . . . . . . . . . . . . 22 E. Excitation into the Metastable Levels of Xenon . . . . . . . . . . . . . 23 IV. Excitation out of Metastable Levels . . . . . . . . . . . . . . . . . . . 23 A. Excitation out of the Metastable Levels of Helium . . . . . . . . . . . . 24 B. Excitation out of the Metastable Levels of Neon . . . . . . . . . . . . . 26 C. Excitation out of the Metastable Levels of Argon . . . . . . . . . . . . 26 D. Excitation out of the Metastable Levels of Krypton. . . . . . . . . . . . 27 V. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

I. Introduction Electron excitation of atoms is one of the most fundamental inelastic atomic collision processes. It provides a powerful means of studying the nature of the electron-atom interaction. The body of cross section data for electron excitation out of the ground level into various excited levels for different atoms that has been accumulated over the past few decades has led to a basic understanding of the excitation processes. Electron excitation is also important for many applications in areas such as gas-discharge lasers, fluorescence lighting technology, and the dynamics of the upper atmosphere. In this chapter, we discuss a new phase of electron excitation: excitation into or out of metastable levels of the rare-gas atoms. Since metastable atoms in laboratory experiments radiate at a very slow rate compared with the total de1 Copyright 0 1992 by Academc Ress. Inc. All nghts of rcpduclion in any form reserved ISBN 0-12-003829-3

2

Lin and Anderson

struction rate, the detection of the metastable atoms is difficult and experiments to measure the electron excitation cross sections into or out of metastable levels require novel experimental techniques. These experiments are, however, very important because they allow one to test whether the physical description of electron excitation derived from the experiments on excitation out of the ground levels is also valid for excitation out of the metastable levels. Furthermore, metastable atoms and molecules play a prominent role in many natural phenomena such as atmospheric processes and gas discharges and plasmas. For instance, in gas discharges containing a rare gas at pressures above about 1 mtorr, the primary mechanism to produce the electron-ion pairs that sustain the discharge is a two-step one involving electron excitation into a metastable level followed by a second electron collision with an atom in a metastable level that results in ionizing the metastable atom. There are usually large numbers of metastable atoms present in a discharge and the processes that lead to the formation or the destruction of these metastables are important. This chapter contains discussions of experiments on the electron excitation of rare-gas atoms into or out of metastable levels. In Section I1 we discuss briefly the electron excitation out of the ground level to nonmetastable levels of rare-gas atoms as background, and then in Section I11 we discuss the electron excitation into metastable levels of rare-gas atoms. There are various methods of measuring total (integrated) cross sections or differential cross sections into metastable levels including methods based on metastable detection, using laser-induced fluorescence (LIF) or absorption, using Auger-electron or channel-electron multipliers, or using electron energy loss measurements. We discuss these various methods but with emphasis on the measurements of total cross sections, especially those measurements using optical detection because of the reliability and the relative simplicity of the measurements based on optical methods. Discussions of other recent measurements of total cross sections using nonoptical methods are also included. In some cases the differential cross section data are used to provide absolute calibration for measurements of total cross sections. Our discussion of differential cross section measurements will focus mainly on those used for calibration of total cross sections. In Section IV we discuss measurements of total cross sections out of metastable levels for rare gases. We do not include measurements of differential cross sections out of metastable levels since they were not used for calibration of total cross sections in the experiments discussed in this chapter.

II. Excitation out of the Ground Level into Nometastable Levels In a typical experiment using the optical method to measure the total excitation cross sections out of the ground level of an atom, an electron beam passes

ELECTRON EXCITATION OF RARE-GAS ATOMS

3

Collision Chamber

Monochromator

Standard Lamp FIG.1. Schematic diagram of the apparatus for measuring optical-emission cmss sections. The rotating mirror allows one to direct separately the radiation from the collision chamber and the radiation from the standard lamp to the monochromator via the same optical path.

through a gas inside a collision chamber and is collected by a Faraday cup as shown in Fig. I . Emission from the atoms excited by the electron beam emerges from a slot in the Faraday cup and passes through a window in the collision chamber. The light emitted from a segment Ax of the electron beam into a known solid angle is directed to a monochromator and the optical signal for a particular i --f j transition is detected by a photomultiplier tube (PMT). Absolute calibration obtained by comparing the PMT current due to the atomic transition with that due to a standard lamp at the same wavelength, gives the number of photons emitted per unit time and per unit length of the electron beam, N ( i + j ) . A detailed description of the experimental apparatus and procedures has been given by Filippelli et al. (1984). The gas density and electron-beam current are kept sufficiently low that one can neglect all the secondary processes that affect the excited-level population such as collisions of excited atoms with electrons or other atoms.' The optical emission cross sections for the i -+j transition, QOpt(i + j), is defined by the equation (I/e)p,Qo,, (i

j) =N i

-+

+

j),

(1)

I In some experiments secondary processes may be a complicated issue and cause considerable errors in the cross sections. See, for example, Filippelli er al. (1984).

Lin and Anderson

4

where I is the electron-beam current, e is the proton charge, and pg is the number density of the atoms in the ground level. Summing the optical-emission cross sections over all the j-levels below a given i-level gives the apparent excitation cross section for the i-level,

which is related to the steady-state number of atoms in the i-level per unit length of the beam, N ( i ) ,

( I / e)p,QaPp(i ) = N(i

(3)

),

where A(i) =

A(i+j),

(4)

j
and A( i +j ) is the Einstein A coefficient for the i +j transition. The notation < i under the summation sign covers all the j-levels that are below the i-level. Equation (3) results from Eqs. (1) and (2) since

j

N ( i + j ) = N(i)A(i + j ) .

(5)

The i-level is populated both by direct excitation from the ground level and by radiative cascade from the atoms that are excited by the electron beam to the various levels above i. To obtain the direct excitation cross section, Q d i r , the cascade contribution from all the higher levels must be subtracted from the apparent cross section, i.e., Qdir(i)

=

Qapp(i>

-

2 ksr

Qopt(k+

i).

(6)

The optical-emission cross sections are the experimentally measured quantities from which the apparent and direct excitation cross sections are deduced. According to Eqs. (1) and (3,the ratio of Qopt(i+ e) to Qopt(i+ k) is equal to the ratio of the corresponding Einstein A coefficients so that Eq. (2) can be rewritten as

If the branching ratio, A(i + e ) / A ( i ) ,is known through spectroscopic measurements or theoretical calculations, then the apparent cross section for level i can be obtained by measuring the optical cross section for only one transition of the type i +j. To determine the direct excitation cross section from Eq. (6) requires optical cross sections for all the k + i emissions, some of which are usually in the infrared or far infrared region and are difficult to measure because of the lower sensitivity of infrared detectors as compared with PMT. However, if one can measure the optical cross section for a transition of the type k + m , which

5

ELECTRON EXCITATION OF RARE-GAS ATOMS

lies in a more favorable part of the spectrum, then one can infer the k -+ i cross section from the relation Qo,,(k * i) = Q,,(k + m)[A(k + i ) / A ( k+ m ) ] .

(8)

Determination of the direct excitation cross sections for the nIS, nlP, nlD, n3S, n3P, and n3D series of He by means of the optical method have been reported by several groups, e.g., St. John et al. (1964), Moustafa Moussa ef al. (1969), van Raan et al. (1974), and Showalter and Kay (1975). Optical cross sections for the nIS * 2’P, nlP + 2’S, nlD ---* 2lP, n3S + 23P, n3P+ 23S, and n3D 4 23P transitions with n 1 3 were measured. All these lines have wavelengths suitable for PMT detection. The transition probabilities are known with a reasonable degree of accuracy so that Eq. (7) gives the apparent excitation cross sections. Likewise, Q. (8) can be used to determine cascade from the higher levels for which the k + m optical cross sections have been measured and for which the remaining cascade is then calculated. To illustrate the main features of the cross section data, in Fig. 2 we show plots of direct excitation cross sections for the 3IS, 3’P, 3ID, 33S, 33P, and 33D levels of He versus the incident electron energy, called the excitation functions, measured by St. John ef al. (1964). Each curve is separately normalized so that all curves have the same maximum height. Levels of the same spin, same L, and different n exhibit a very similar shape for the excitation function except for the different peak values of the cross sections, which decrease with increasing n, and for a slight shift in the position of the peaks due to the slight difference in

0.0

0

50

100

160

200

INCIDENT-ELECTRON ENERGY (eV) FIG. 2. Energy-dependence of the direct cross sections for electron excitation out of the ground level of helium to the 3IS, 3’P, 3ID, 33S, 33P, and 3’D levels measured by St. John er al. (1964). All curves are separately normalized to give the same peak height.

6

Lin and Anderson

the threshold excitation energies. The shape of an excitation function generally falls into one of three categories. A very broad maximum followed by a slow decline is seen for the excitation functions of the nlP levels, which are optically (electric dipole) connected to the ground level. Excitation functions for all the triplet levels exhibit a narrow peak and decrease rapidly at higher energies. The change in the total spin of the target atom in these processes might at first sight seem to require a magnetic spin-dependent interaction between the incident and atomic electrons which is much smaller than the Coulomb interaction. However, instead of a direct spin flip, the mechanism responsible for excitation of the triplet levels is that the incident electron takes up a bound excited orbital and the atomic electron with spin opposite to that of the incident electron is ejected from the atom. This mechanism requires no magnetic spin-dependent interaction but only an exchange between the incident and atomic electron. Such an electronexchange process is efficient if the incident electron is slow enough to remain in the vicinity of the target for an appreciable amount of time. As one increases the electron energy, the duration of interaction decreases, causing a drastic decline in the excitation function. Intermediate between the very broad and very narrow excitation functions is the group of excitation functions for the nlS and n'D levels which correspond to spin-allowed but dipole-forbidden optical transitions to the 1 S ground level. The three kinds of energy dependence of the cross sections are also predicted by theoretical calculations based on the Born-type approximations as illustrated in Fig. 3 where the curves are individually normalized to the same peak height. Furthermore, with the Born-Bethe approximation, 1.0

5

E

w

0.8

0.0

0

50

100

150

200

INCIDENT-ELECTRON ENERGY (eV) FIG. 3. Energy-dependence of the direct cross sections for electron excitation out of the ground level of helium to the 3'5, 3'P, and 3'D levels calculated by the Bornapproximation and to the 33S, 33P, and 33D levels by the Born-Rudge approximation using Hartree-Fock wave functions. All curves are separately normalized to give the same peak height.

ELECTRON EXCITATION OF RARE-GAS ATOMS

7

TABLE 1 PEAKVALUESOF THE DIRECTELECTRON EXCITATION CROSSSECTIONS (in l O - l * cmz) out of the Ground Level into Various Excited Levels of Helium ~

Levels 3's 3'P 31D

~

Qd 0.47 3.2 0.35

Levels 3's 33P 3'D

Q!? 0.89 0.67 0.25

it is shown that at high energies the cross sections for excitation to the n'P levels are proportional to E - I In E where E is the incident electron energy and the cross sections for excitation to niS and n i D are proportional to E - ' (Mott and Massey, 1965; Moiseiwitsch and Smith, 1968). For the triplet excitation both the Born-Ochkur (Ochkur, 1963) and Born-Rudge (Rudge, 1965) approximations predict an E -3-dependence at high energies. The overall agreement between Figs. 2 and 3 is considered satisfactory in view of the experimental uncertainty and the fact that the Born approximation is valid only at high energies. Parenthetically, we should mention that the theory of electron excitation of atoms has progressed much beyond the stage of the Born approximation. In this chapter we use the Born approximation for illustration because of its simplicity. A discussion of the current status of the theory is outside the scope of this chapter. The magnitude of the cross sections also depends on the nature of the transitions. Excitation to the n' P series corresponding to dipole-allowed transitions shows much larger peak values of the cross sections than the n l S and n i D series as exemplified by the results of St. John et al. (1964) shown in Table I. The peak values of the cross sections for the n3S and n3P series are larger than those for the n3D series, but even the former are smaller than the peak values of the cross sections for the n'P series. A simple, intuitive way to understand the behaviors of the excitation cross sections is to think of the excitation process as an absorption-like transition caused by the electromagnetic field associated with the incident electron. If we apply a multipole-type analysis to the electromagnetic field, the dipole component (k = l), which is the strongest one, is responsible for the excitation from the 1's level ( L = 0) to the n'P levels ( L = 1). Excitation to the n'D levels requires the quadrupole component ( k = 2), hence smaller cross sections. To account for the excitation into the n ' S level, one has to introduce the k = 0 component which has no direct optical analog. The essence of this simple picture can be traced to the studies of the stopping power of matter for charged particles by Bohr (1913, 1915) in which he suggested a qualitative similarity between the effect of a moving charged particle on an atom and the effect of an electromagnetic wave in producing the absorption and dispersion (Merzbacher 1984), and to the work of Purcell (1952) on the lifetime of the 2*S levels of hydrogen in

8

Lin and Anderson

an ionized atmosphere. One may wonder whether the implicit use of a projectile path is compatible with the wave nature of the incident electron. However, this simple picture can be placed on a more rigorous foundation by means of the Born approximation which gives the cross section as being proportional to the square of an integral containing the wave functions of the initial and final stages of the projectile-target system and the Coulomb interaction of the incident electron (coordinates r:r, 8 , @) with the atomic electrons (coordinates ?;:ri,ei, Qi).Upon expanding the Coulomb term using the spherical harmonics as

where r , is the greater of r and r i , and r< is the lesser, it is easy to see that the k = 0, 1, and 2 terms are responsible for excitation to the S, P, and D levels, respectively. This picture applies to collisions at high energies in which the impact duration is small compared to the classical periods of the target electrons. For slow collisions the orbital motion of the target electrons may be sufficiently altered by the projectile that modifications are needed for the arguments given in this paragraph. Nevertheless, it often turns out that this simple picture serves as a useful guide for qualitative considerations even for collisions of intermediate energies. The classifications of excitation cross sections and excitation functions of He are based on the quantum numbers of the excited states under study according to the LS coupling. This scheme can be extended to atoms that do not conform to the LS coupling. Consider the Ne atom in the excited 2p5ns configuration. The 2p5 core has angular momenta el = I and s, = 1/2, and the outer ns electrons, e2 = 0 and s2 = 112. The LS coupling gives the 'P, , 3P0,3 P l , and 3P2terms. To describe a non-LS-coupling case, one starts with the LS eigenstates and allows for the deviation from the LS coupling by mixing the various LS eigenstates of the same J since the total angular momentum is rigorously a good quantum number even for the intermediate coupling cases. If we neglect mixing of configurations, the wave functions, $. of the 2p5ns levels can be expanded using the LS eigenfunctions, 4, as = 4 ( 2 p 5 n s , ' P I ) + P4(2p5ns, 3PI), JlZ = P4(2p5ns, ' P I ) - 4 ( 2 p 5 n s , 3PI),

$1

$3

=

$,

=

(10)

4(2p5ns, 'Po), 4(2p5ns, 3P2).

Since 3P0 and 3P2are the only members with the particular value of J within the 2pSns configuration, neither member mixes with any other members. In Paschen's notation the levels corresponding to G I , J13, and 14, for the 2p53s configuration are called I s2, 1s,, 1s3, and 1ss , respectively. For the 2p54s con-

ELECTRON EXCITATION OF RARE-GAS ATOMS

9

figurations the designation of 2s2 through 2s, are used and so on for higher configurations. Since the ns3 and ns5 levels are purely triplet, we expect their direct excitation functions to show a sharp peak near the onset and a steep decline at high energy based on the results of He discussed earlier. The direct excitation functions for ns, and ns4, on account of the mixed singlet and triplet characters of their wave functions, are expected to be a combination of a broad-peak and a narrow-peak component. Unless one of the mixing coefficients is exceptionally small, the large broad peak dominates so that the excitation functions of the nsz and ns4 levels have the shape characteristic of the dipole-allowed excited states, i.e., a very broad peak with a slow decline (E - I In E dependence) at high energies. The 2p5np configuration can be analyzed in the same way. The ten levels of the 2p5np configuration can be considered as superpositions of the ten ISo, 3 S l , IPI, 3P0,3PI, 3 P ~ ID2, . 3D1, 3D2,3D3LS eigenstates with mixing only among members of the same J. Here the 3D3term, being the only member with J = 3, remains unmixed, whereas all the other nine levels have both singlet and triplet characters. Thus, only the level with J = 3 has a sharply peaked excitation function. The other nine levels have excitation functions characteristic of spin-allowed and dipole-forbidden excited levels, i.e., a peak of intermediate width and (l/E)-dependence at high energies. Notice the difference between this and the very broad peak and E-I In E dependence for the J = 1 levels of the 2p5ns configuration because transitions from the ground level (2p6 IS,) of Ne to the J = 1 levels of the 2p53s configuration (Is, and 1s4) are dipole-allowed whereas transitions from the ground level to all levels of the 2p5np configuration are dipole-forbidden. The apparent excitation functions for the four levels of the 2p55sconfiguration (3s in the Paschen notation) and for 9 of the 10 levels of the 2p53pconfiguration (2p in the Paschen notation) measured by Sharpton et al. (1970) are shown in Figs. 4 and 5 , respectively. The apparent excitation functions include the cascade contributions, and the deviation of the shape of the apparent excitation functions from the shape of the direct excitation functions varies from one level to another. Nevertheless, these graphs show qualitatively the three kinds of excitation functions: the very broad ones for the optically allowed levels of the 2p55s configurations (3s, and 3s,); the narrow peaks for the 2p53p, J = 3 (2p9) level, 2p55s, J = 2 (3s5) level, and 2p55s, J = 0 (3s3) level characteristic of spin-forbidden levels; and the intermediate shape for the remaining eight levels of the 2p53p configuration (spin-allowed but dipole-forbidden). The 2p3 curve is not included in Fig. 5 because the apparent excitation cross sections of the 2p3 level have a very large cascade component so that the shape of the apparent excitation function differs significantly from that of the direct excitation function. The 2p5nd configurations can be treated in a similar way (Sharpton et al., 1970) and are not detailed here. The different shapes of the excitation functions for different levels deduced from these analyses are in full agreement with experiments.

Lin and Anderson

10

fTym

"iNm

3c 0

v)

J-1

J-1

0

100

0

200 0

100

200

Electron Energy, eV FIG. 4, Apparent excitation functions for the four levels of the 2ps5s configuration of Ne. All curves are separately normalized to give the same peak height. The levels are labeled by the Paschen notation and the value of J .

U

mm J=2

J=2

40 8o 12Ol6O

40 8o 12Ol6O

40 8o 120160200

Electron Energy, eV FIG. 5 . Apparent excitation functions for nine levels of the 2p53pconfigurationof Ne. All curves are separately normalized to give the same peak height. The levels are labeled by the Paschen notation and the value of J .

A very interesting trend in the magnitude of the excitation cross sections of Ne has been observed experimentally (Sharpton et al., 1970). Within the 2p53p configuration the levels with even values of J generally have larger cross sections than those with odd J , whereas the reverse is true for the 2ps4d configuration. This observation can be explained by the multiple picture described earlier. Excitation from the 2p6 IS, ground level to a level in the 2p53p configuration entails a 2p +-3p transition, or t? = 1 + t? = 1. This can be accomplished by

ELECTRON EXCITATION OF RARE-GAS ATOMS

11

means of the k = 0 and k = 2 components of the multipole interaction. However, the final levels are further characterized by J that ranges from 0 through 3. The k = 0 and k = 2 components just mentioned are effective for transitions from the ground level (J = 0) to a J = 0 level and a J = 2 level, respectively, of the 2p53p configuration, but do not match the parity requirement for transitions from J = 0 to J = 1 or J = 3. While transitions to the o d d 4 levels can take place through higher-order effects (or exchange for J = 3), their cross sections are expected to be much smaller than the cross sections for the even-.! levels. For excitation from the 2p6 configuration to the 2p54d configuration, the p .+d transition (A1 = 1) limits us to the k = 1 and k = 3 components which are compatible with transitions from J = 0 to J = 1 and J = 3, but not with the J = 0 0, 2, 4 transitions. Therefore, in this case we expect to find larger cross sections for the odd4 levels. A rigorous treatment based on the Born approximation and group-theoretical arguments is given by Sharpton et al. (1970) for Ne and by Ballou et al. (1973) for Ar. Fabrikant et al. (1988) and Teubner ef al. (1985) have expressed concern about the reliability of describing the ls3(,PO)and ~ S ~ (levels ~ P as ~ purely ) triplet levels in electron-excitationexperiments. In principle the ,Po (or ,P2) level could be mixed with a singlet level of the same J from a different configuration of the same parity. The usual configuration interaction that arises from the r i k Coulomb interaction term in the Hamiltonian, however, does not produce triplet-singlet mixing because r;' is diagonal in the total spin quantum number. To spoil the pure triplet character of the Is, and Is, levels of Ne, one must resort to mixing of the 2p53s configuration with a higher configuration through interactions of electron spins with orbital motion. Considering the relative magnitude of the spin-orbit-type interaction versus the energy spacings between the 2p53s and the higher configurations, it is clear that such mixings must be very small. This is confirmed experimentally by the gJ factor which is sensitive to the singlet admixture of these nearly pure triplet levels except for the case of J = 0 where there is no Zeeman splitting. The measured value of the gJ factor for the Is, level of Ne is 1.503 (Moore, 1949), which is very close to the theoretical value of 1.501 for the pure ,P2 level. Likewise, the measured gJ factors for the corresponding Is( 3P2)levels of Ar, Kr, or Xe indicate that these levels are also pure LS levels. Of course, one must distinguish the triplet-singlet mixing in the 1s3(3P0)or ls,t3P2) levels from the triplet-singlet mixing in the Is, and Is, levels. In the latter case the mixing is quite appreciable because it occurs between LS eigenstates ()P, and ' P I ) within the same configuration ( 2 ~ ~ 3 sHowever, ). the fact that the 1s2and 1s4 levels are not pure LS levels does not mean that other levels of Ne cannot be pure LS levels. As explained earlier, the Is, and Is, levels have the unique J values within the 2p53s configuration so that they do not mix with any LS eigenstates of the same configuration and therefore can be described as pure LS levels to a high degree of accuracy. In studying electron excitation of the rare-gas atoms, recognition of the pure LS levels among the majority of non-

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LS levels is a key step in the progress of understanding the cross section data from a unified viewpoint in terms of the atomic structure. In addition to measurements of total-excitation cross sections using the optical method, differential cross sections for electron excitation are of interest. Differential cross sections can be measured by detecting the electrons that have lost the exact energy necessary to excite a given atomic level and that are scattered at an angle 8.These experiments provide information on the angular distribution of the scattered electrons that cannot be obtained using the straightforward optical method to measure total cross sections. The differential excitation cross section multiplied by sin 8 and integrated over all angles 8 is equal to the totalexcitation cross section. In order to obtain accurate integrated cross sections, it is necessary to have differential cross sections that are measured over a wide range of angles, and if the differential cross section is strongly peaked in the forward direction, the scattering angle must be measured very accurately. It is usually preferable to measure the total cross sections using the optical method.

III. Excitation out of the Ground Level into Metastable Levels

The measurement of total electron excitation cross sections into metastable levels has been studied for all the rare-gas atoms (He, Ne, Ar, Kr, and Xe) using various techniques. For He the methods utilizing absorption out of the metastable levels, sensitized fluorescence, or time of flight have been used. For Ne, there are extensive measurements including measurements using optical and laser techniques as well as nonoptical methods. Measurements for Ar, Kr, and Xe are somewhat more limited. INTO THE METASTABLE LEVELS OF HELIUM A. EXCITATION

The ground level of He is (Is*) ‘ S o and the two lowest excited levels, (ls2s)2’S and ( l s 2 ~ ) 2 ~are S , both metastable. The LS coupling gives an excellent description for the low-lying levels of He. Since the metastable levels do not radiate, they are difficult to detect. An early optical experiment (Woudenberg and Milatz, 1941) to measure the apparent excitation cross section of the 23S level of He used optical absorption out of the 23S level. They produced the Z3Smetastables by an electron beam in a He gas and used the absorption out of the z3S level to determine the rate of production of He atoms into the 23S level. They carefully studied the effects of pressure and of other experimental factors. They determined the relative values of the apparent cross section and found that QaPp(2)S) has a nmow peak near an electron energy of 25 eV.

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Another optical method for detecting the metastable levels of He is sensitized fluorescence. An excitation transfer collision of a He atom in the 23S level produces a Cd+ ion in an excited (4d95s2)2D,, level by the reaction He(23S)

+ Cd('S) + He(1lS) + Cd+(

'D5/2)

+ e-.

(11)

The excited Cd+( 'D5J ion is detected by its emission of radiation of wavelength 441.6 nm. Bogdanova and Marusin (1975) used the radiation from the Cd+ ions to detect He atoms in the 2 ) s level. The 441.6-nm radiation emitted by the Cd+( 2D5/2) ions is measured first in the absence of He and then in the presence of He. The rate equations for the formation and destruction of He(23S) and Cd+( 2D5/2)levels can be solved in the steady state to give the following expression for the apparent electron excitation cross section for He(23S) atoms,

where Z is the intensity of radiation emitted by Cd+(2D5,2)in the absence of He, AZ is the change in the intensity of radiation emitted when He is present, ( 'DY2)] is the apparent cross section for production of Cd+(2DJ12) by Qapp[Cd+ electron impact, QTis the excitation transfer cross section for the reaction given by Eq. ( l l ) , (QTv) is the thermal average of QT times the relative He-Cd velocity, nHeis the ground-level He density, and ~ ( 2 ~iss )the reciprocal of the steady-state destruction rate of the He(23S) atoms by all mechanisms. Equation (12) is based on the assumptions that AZ is entirely due to the transfer process in Eq. (11) and that the rate of this transfer process is much smaller than [ ~ ( 2 ~ S ) ] -Inl . addition to measurements of A M and Qa,[Cd+ ( *Ds,2)],one must have accurate values of QTand ~ ( 2 ~ins order ) to determine the metastable cross section. Thus, a comprehensive knowledge of the excitation transfer mechanisms is required to obtain reliable results using the method of sensitized fluorescence. Bogdanova and Marusin (1975) used the value of QT as roughly equal to 10-l5 cm2 estimated by Ivanov et al. (1972), and they measured the lifetime of He(23S) from the duration of the afterglow to obtain ~ ( 2 ~ sIn) .a similar manner Bogdanova and Marusin ( 1969) have measured the apparent electron excitation cross section of He(2IS) using the sensitized fluorescence of Ne(3s2). They found a peak value for Qa,(2'S) of 1.6 x lo-'' cm2 at an electron energy of 35 eV, and a value for Qapp(21S)of 1.3 X lo-'* cm at an electron energy of 45 eV. Their results are in serious disagreement with other measurements. An interesting optical experiment using laser-induced fluorescence has been performed by Zetner et al. (1986) to study the electron excitation of the 2IS level of He for energies near threshold. The LIF technique for determining metastable level cross sections was developed by Phillips er al. (1981a, 1985) in an experiment to study the excitation of Ne metastables and is discussed in Section 1II.B dealing with Ne. Zetner et al. (1986) have observed resonances in the 2 ' s excitation cross section.

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In addition to the experiments using optical methods, there are a number of nonoptical experiments that give electron excitation cross sections into the metastable levels. The sum of the apparent cross section for the electron excitation for He into either the 23S or 2IS level, i.e., Qam(z3S) Q,pp(2LS),has been measured by Mason and Newell (1987). They produced an atomic beam of ground level He. A very monochromatic electron beam was incident on the atomic He beam. The He atoms were deflected by the electron-excitation collision. The deflected He atoms in either the P S or 2'5 level were detected by a channel-electron multiplier. The electron-beam current was measured using a Faraday cup. The electron beam was modulated so that the excitedatom-velocity distribution could be determined using time-of-flight techniques. This made it possible to discriminate against a signal from uv photons emitted by He atoms excited to radiating levels. Mason and Newell have measured earn@%) Qam(21S)for electron energies from thresholds up to 140 eV. They obtained absolute cross sections by normalizing their results to the earlier results of Borst (1974) at 20-eV incident electron energy. The peak value of Qam(z3S) Qam(2IS)is 6 X 10-l8 cm2 at an electron energy of 25 eV. The measurements of Mason and Newell agree well with earlier measurements and seem to give reliable values for Qam(2%) Qa,(2'S). Earlier measurements of Qaw(Z3S) Qam(21S)have been carried out both by Borst (1974) using Auger electron detection of the metastables and by Lloyd et al. ( 1 972) using channel-electron-multiplier detection of the metastables. Borst ( 1974) obtained absolute cross sections by calibrating at low energy using the trapped-electron method, whereas the measurements of Lloyd et al. yielded only relative cross sections for electron energies of 19.8-200 eV. Overall the results of these experiments are consistent with the more recent measurements of Mason and Newell (1987). In addition to the experiments discussed, there have been a large number of experiments to measure the near-threshold behavior of the total apparent cross section for the production of He atoms in either the 23Sor 2IS metastable levels. As examples we cite experiments by Brunt et al. (1977), Buckman et al. (1983a), Pichanick and Simpson (1968), and Johnston and Burrow (1983) for investigating the near-thresholdbehavior of Q,(23S) QaPp(2 ' S ) with emphasis on studying the resonances that occur in these cross sections. Because our primary interest is in the overall cross sections as a function of the energy rather than the near-threshold behavior, we do not discuss these experiments further. Several experiments have been carried out to determine separately QaW(2S) or Qapp(2'S).Cermak (1966) has measured separately the relative values of Qapp(21S) and QaW(2'S)from near threshold to 60 eV. The 23S or 2IS levels were detected using Penning ionization of Ar, and the kinetic energy of the electrons ejected in the Penning ionization was measured to detect separately the 23S or 2's levels. Holt and Krotkov (1966) have measured Qapp(z3S) or QaPp(2'S) from 19.0-23.2 eV. They detected the metastables using Auger electron emis-

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sion from a metal surface, and used electric-fieldquenching to eliminate the 2ISlevel atoms from an atomic He beam. They have also used an inhomogeneous magnetic field to permit separate measurements for excitation into the m = 1, 0, - 1 states of the 2% level. Dugan et al. (1967) have measured separately relative values of Qa,,(23S) or QaP,(2'S) using an inhomogeneous magnetic field to separate the 2% and 2's atoms. Their measurements were carried out for electron energies of 25- 135 eV. Their results show a narrow peak in QaPp(2%)near 25 eV and a broad maximum in Qapp(21S)near 100 eV. All these measurements appear to be carefully done and offer useful results. Another nonoptical method used to measure electron excitation cross sections is the inelastic scattering of electrons with adequate energy resolution to determine the level in He that is excited. These experiments yielded direct differential cross sections. One must integrate over all scattering angles 8 the differential cross section times sin 8 to obtain a total cross section. Chamberlain et al. (1970) and Trajmar (1973) have used inelastic electron scattering to obtain differential direct cross sections. The sum of total cross sections for 23S and 2 ' s obtained using Trajmar's differential cross sections lies somewhat below the total apparent cross sections obtained by Mason and Newell (1987). This may be due to cascade effects in the experiments of Mason and Newell (1987) that are not present in the measurements of Trajmar (1973). In summary, for the 23S and 2IS metastable levels of He, there are reliable measurements using nonoptical techniques of Q,,,(23S) Q,,(21S). The early optical measurements of Woudenberg and Milatz (1941) appear to give reasonable relative values of Qapp(23S).The LIF measurements of Zetner et al. (1986) near threshold are reliable and of high quality. There are some other nonoptical measurements of Q,pp(23S)and Qa,p(21S)that appear reasonable. The total direct cross sections obtained by integrating the differential cross sections obtained by inelastic electron scattering also appear reasonable. The totai apparent cross sections obtained by Bogdanova and Marusin (1975) using sensitized fluorescence do not agree well with the results of others and appear to suffer from some systematic uncertainty. It would be desirable to have further experiments using optical methods to obtain separately absolute direct cross sections for the 23S and 2 ' s levels over a wide range of electron energies. There have been only a few measurements of the cross sections for the separate levels and they have yielded only apparent cross sections except for the inelastic-electron-scattering measurements. Optical measurements are especially needed to give both the apparent and direct cross sections.

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B. EXCITATION INTO THE METASTABLE LEVELS OF NEON

The ground level of Ne is 2p6 IS,. The lowest excited configuration is 2p53s which gives rise to four levels designated as Isz, Is,, Is,, and Iss in the Paschen

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notation. As explained in Section 11, the Is, and Is, levels are accurately described by the LS-coupling 3P, and 3P2 terms, respectively. The Is, and Is, levels are superposition of 'PI and 3P1as expressed in Eqs. (10) with Is, being approximately 93% 3P1plus 7% 'PI and the opposite mixture for Is,. The Is, and Is, levels are metastable because a transition from either level to any lower levels is forbidden from consideration of the J quantum number and parity. Both the Is, and Is, levels decay to the ground level by electric-dipole radiation and the lifetime of the Is, level is about 13 times as long as the lifetime of the Is, level. An early optical experiment for measuring the excitation cross section of the Is, level of Ne was carried out by Milatz and Ornstein (1935). In their experiment the density of 1s5 metastable atoms produced by electron impact was determined by monitoring the absorption of a beam of 640.2-nm light by the Is, atoms through the Is, + 2p9 transition. They reported the energy-dependence of the apparent excitation cross section of the Is, level but without an absolute calibration. They observed a rapid decrease of cross section with increasing energy. They used a low Ne pressure (-0.02 torr) but without the sophisticated detection apparatus available today. Hadeishi (1962) used a similar method and operated in the pressure range of 0.4- 1.7 torr where secondary processes may occur. His measured cross sections for the Is, level depend on the pressure and are more than 10 times the cross sections of others. Aside from the difference in absolute magnitude, the energy-dependence of Hadeishi's cross sections is also quite different from that of others. More recently Mityureva and Penkin (1983) also used absorption to monitor the production of Ne atoms in the Is, and Is, levels and measured the apparent cross sections of the Is, and Is, levels in a manner similar to that used by Milatz and Ornstein (1935). They subtracted the cascade cross sections to obtain direct excitation cross sections to the Is, and Is, levels. They found the peak value of the direct cross section for the Is, level is 3.4 X 10-l8cm2at an electron energy of about 23 eV. A new and innovative method to obtain the electron excitation cross sections for the Ne metastable levels using laser-induced fluorescence has been introduced by Phillips et al. (1981a,b, 1985). A dye laser operating at the appropriate wavelength was used to excite the metastable atoms to a higher level that radiates. The LIF was used to detect the metastables. If the rate of excitation out of the metastable level by the laser is much greater than the total decay rate out of the metastable level by all other mechanisms and if the dye-laser intensity and frequency distribution are such that one can saturate the entire Doppler absorption line width out of the metastable level, then the rate of photons produced by LIF is equal to the rate of formation of metastables by direct electron excitation plus cascades. In this case the absolute apparent electron excitation cross section can be obtained from the absolute LIF decay rate, the electron current, and the target density. For Ne the lowest configuration above the 2p53s is the 2p53p configuration.

ELECTRON EXCITATION OF RARE-GAS ATOMS

17

This configuration gives rise to 10 energy levels denoted as the 2p,, . . . ,2p10 levels in the Paschen notation. The wavelengths of many of the optical transitions between the levels in the 2p53p and the 2p53s configurations are in the visible region and are well suited for LIF experiments using cw dye lasers. In order to make concrete the use of LIF for the measurement of the cross section for electron excitation into a metastable level, we discuss the excitation into the Is,(’P2) level of Ne. The relevant processes involved are shown in Fig. 6. The Is, level was populated both by direct electron excitation and by cascade from higher levels. A cw laser operating with a wavelength of 588.2 nm excited atoms in the Is, metastable level up to the 2p2 level. The 2p2 level has a configuration 2ps3p and an electronic angular momentum of J = 1. The 2p2 level decays by radiation to the four 1s levels and the branching ratios for this radiation are known (Sharpton ef al. 1970). The radiation at 659.9-nm wavelength corresponding to the 2p2 4 Is, transition (LIF) was observed. The 2pz level was populated in two different ways. The first method, by which the 2pz level was populated, was the electron excitation of this level by both direct and cascade processes, and the second method was the laser absorption from the Is, metastable level. These two processes for populating the 2pz level are independent and consequently the 2p2 -9 Is, radiation results from the same two processes. The 2p2+ Isz LIF produced by 2p2 atoms excited from the Is, level by the laser was separated from the 2p2 --.* Is, radiation due to electron excitation

Electron Impact Excltatlon

Ground Level FIG. 6. Diagram showing the electron excitation, laser absorption, and laser-induced fluorescence processes involved in the use of the LIF technique for measuring cross sections of electron excitation out of the ground level into a metastable level of neon.

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by modulating the laser beam. A collimated electron beam passed through a stainless-steel chamber filled with about one mtorr of Ne exciting some of the Ne atoms in the ground level into the metastable Is, level. A cw dye-laser beam with a wavelength of 588.2 nm passed through the chamber intersecting the electron beam at right angles. The optical emission emitted from the region of the electron beam crossed by the laser beam was observed along an axis that was perpendicular to both the electron-beam axis and the laser-beam axis. The laser beam was on-off modulated by a rotating mechanical chopper. The optical emission was analyzed using a monochromator and then detected using a PMT. The output current of the PMT was the input for a lock-in detector, the output of which was the y-input of an x y recorder. The x-input to the recorder was the voltage used to accelerate the electron beam. The plot obtained is directly proportional to the electron excitation function. The absolute calibration of the electron excitation cross section posed an important problem. If the intensity and frequency distribution of the dye-laser beam were such that the entire Doppler linewidth of the Is, + 2p2 transition was saturated, then the number of photons emitted per second in LIF would be equal to the rate of electron excitation of atoms into the Is, level. In this situation a measurement of the absolute intensity of the LIF coupled together with the solidangle of the detector would enable one to obtain the calibration of the absolute apparent cross section. This calibration scheme has not yet been carried out because of the high laser intensity required. An alternative method for calibrating the apparent electron excitation cross section is as follows. As explained in Section 11, the Is, is a pure triplet level. The direct electron excitation cross section of a pure triplet level is expected to peak at an energy only a little above threshold and then decrease rapidly with increasing energy. Since most of the levels that cascade into Is, (such as 2p2 and 2p4) have a mixed singlet-triplet character, the excitation cross sections of the majority of these cascading levels decrease with increasing energy rather mildly in comparison with the Is, level. Thus, for the Is, level, the direct electron excitation cross section is expected to be very small in comparison with the cascade cross sections at electron energies of 60 eV or higher. The total apparent electron excitation cross section of the Is, level at energies of 60 eV or higher is therefore very nearly equal to the sum of the cascade cross sections, which are the optical emission cross sections for transitions into Is, from the various higher levels and have been measured absolutely by Sharpton et al. (1970). Thus, the absolute calibration of the apparent cross section for the electron excitation of the metastable 1s, level was obtained by setting the Is, cross section equal to the sum of the cascade cross sections at 90-eV incident electron energy. The direct cross section at energies below 90 eV was obtained by subtracting the sum of the cascade cross sections from the apparent cross section. This calibration procedure of course means that the direct cross section was not obtained for energies of 90 eV or higher.

ELECTRON EXCITATION OF RARE-GAS ATOMS

19

In addition to measurements of the cross sections for electron excitation into the Is, level of Ne, measurements of the cross sections for the electron excitation into the Is, ( ,Po) level of Ne have been made using the same method as for the Is, level (Phillips er af., 1981b). By analyzing the polarization of the LIF as a function of the laser polarization, absolute excitation cross sections for separate Zeeman states (m,-states) of the metastable Is, level have been measured at an impact energy of 18.2 eV (Phelps et af., 1983). At this impact energy the cross sections for exciting the various m,-states are found to be equal to one another within experimental error. In addition to the measurements of the electron excitation cross sections over a broad energy range using LIF, there has been an important extension of the LIF to measurements of the electron excitation cross sections of Ne for nearthreshold energies by Zetner er al. (1986). They have used the LIF method to study resonance structure in cross sections for the electron excitation of the Is, and Is, levels, and they also have measured the cross sections for electron excitation into a single m,-state of the ls, level. The results they obtained show resonance structure in both the Is, and Is, cross sections. There are also several nonoptical measurements of the electron excitation cross sections for the metastable Is, and Is, levels of Ne. Mason and Newell (1987) reported measurements on the total cross section for the sum of the apparent cross sections for the electron excitation into the Is, or Is, levels of Ne for electron energies of 16.2-140 eV, i.e., Qapp(ls3)+ QapP(ls5),using the same method described in Section 1II.A. They normalized their cross sections to those of Teubner et at. (1985) at 26 eV for absolute calibration. Earlier measurements of the sum of the apparent excitation cross sections of the two metastable levels include the work of Dorrestein (1942). He used electron bombardment of a Ne gas to form metastables. The metastables drifted to a detector where Auger electron emission was used to detect them. The difficulties of this method include determining the efficiency for the ejection of secondary electrons from the metal surface by an incident metastable atom and separating the contributions to the total secondary electron emission of the metastable atoms and the ultraviolet photons. He used modulation techniques to overcome these problems. Teubner et af. (1985) also have measured the sum of the apparent excitation cross sections of the Is, and Is, levels using a method somewhat similar to that of Dorrestein (1942) except that they used a time-of-flight technique to eliminate the photon signal. Teubner er al. raised the point of whether the Is, and ls, levels can be accurately described as purely triplet levels. We have discussed this issue in Section II. The works of Teubner er al. (1985) and of Mason and Newel1 (1987) seem carefully done and they obtain cross sections that are reasonably consistent with each other. In addition, relative values of Qapp(ls3)+ Qapp(ls,)have been reported by Theuws et al. (1982). Dunning et af. (1975b) have measured the ratio of the population of the Is,

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and Is, levels produced by electron impact by an optical method. They used a laser to selectively remove either the Is, or Is, atoms from a beam of metastable atoms and detected the metastable atoms by secondary electron emission from a metal surface. Their measurements give the Is,: Is, ratio at electron energies 35, 0.6, 5.1 2 0.4, and 60, 80, and 100 eV as, respectively, 6.9 ? 1.4, 5.8 5.0 2 0.4. Register et al. (1984) have reported differential cross sections for excitation of 16 features in the electron energy-loss spectrum of Ne. By integrating their measured differential cross sections times sin 8 over all scattering angles, they obtain the total cross section for direct excitation. Their integrated results agree reasonably well with those of other researchers. There are a number of papers using nonoptical methods that measure the sum of the electron excitation cross sections for the 1s levels with primary interest in near-threshold resonance structures. Papers reporting such measurements include Pichanick and Simpson (1968), Buckman et al. (1983b), Brunt et al. (1976), Schaper and Scheibner (1969), and Johnston and BUKOW (1981). In summary, optical experiments have been used to measure the individual apparent cross sections into either the Is, or Is, levels, i.e., Qaw(ls3)or Qa,( lsJ, whereas the nonoptical experimentals have been used to measure the sum of cross sections into the ls, or Is, levels, i.e., Qaw(ls3)+ Qa,(ls5). The measurements of Milatz and Ornstein (1935) show a much more rapid decrease in Qa,(ls5) than more recent measurements. The results of Mityureva and Penkin (1983) indicate that above about 30 eV, Q(ls5)is smaller than Q(ls,). This is in sharp disagreement with the results of Phillips et al. (1985) and Dunning et al. (1975b). In particular, Phillips et al. (1985) find that the ratio of the direct excitation cross section for Is, to Is, is equal to the 5 : 1 ratio of the statistical weight of these two levels within experimental uncertainty and that even the ratio of the apparent excitation cross section for Is, and Is, is fairly close to 5 : 1. The results of Phillips et af. (1985) provide individual cross sections measured over a wide energy range and with reasonable values for Qapp(ls3),Qdk(ls3), Qaw(ls5),and Qd,,(ls,). The sum of the cross sections Qapp(ls3)+ Qapp(ls5) from Phillips et al. (1985) can be compared to the results obtained using nonoptical techniques to measure total apparent cross sections. The measurements of Mason and Newell (1987), Teubner et al. (1985), Dorrestein (1942), and Phillips et al. (1981a,b 1985) all place the peak of the sum of the apparent cross sections at an electron energy of about 26 eV and all have a peak value for the cm2. However, the sum of the apparent cross sections of about 3.5 X dependence of Qw(ls3) + Qapp(ls5)on the electron energy at high energies is somewhat different for these papers. The apparent cross sections of Dorrestein and of Phillips et al. agree well and the apparent cross sections of Mason and Newell and of Teubner et al. agree well but the apparent cross sections of Dorrestein and of Phillips et al. decrease from the peak somewhat less steeply at higher

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ELECTRON EXCITATION OF RARE-GAS ATOMS

21

energies than the apparent cross sections of Mason and Newell and of Teubner et al. The differential cross sections of Register et al. (1984) for the Is, and Is, levels times the sin 8 have been integrated over all scattering angles 8 to provide total direct cross sections. The integrated total direct cross sections obtained by Register et al. (1984) agree well with the direct cross sections obtained by Phillips et al. except for electron energies above 60 eV. The disagreement above 60 eV is expected since Phillips et al. took Qdir(1s3)and Qdir(ls5)to be zero at energies above 90 eV.

c. EXCITATION INTO THE METASTABLE LEVELS OF ARGON The ground-level configuration of Ar is 3p6 and the configuration of the lowest excited levels of Ar is 3p54s. The 3P2 and 3P1levels that arise from the 3p54s configuration are virtually pure LS levels and are metastable. Mityureva and Smirnov (1985, 1986) in a pair of papers have measured the apparent cross sections for the production of the ,P2 or ,Po metastable levels. They produced the Ar metastables using electron bombardment in a low-pressure gas. The number density of metastable atoms is detected using absorption out of the metastable level. They have also measured at 22 eV the cascade contribution to the cross section. They found that the peak values of the apparent cross sections for the 3P2 and ,Po levels are, respectively, 3.4 X 10-l7 cm2 and 0.9 x l O - I 7 cm2. The peak values of both cross sections occur at an incident electron energy of about 20 eV. They also found that the peak values of the direct cross sections for the ,P2 and ,Po levels are 4 X 10-l8 cm2 and 8 X lO-I9 cm2,respectively. The sum of the apparent electron excitation cross sections into the meta3P0), has been measured stable 3P2or 3P0levels of Ar, i.e., Qapp(3Pz) + Qapp( using Auger electron emission to detect the Ar metastables by Borst (1974), who obtained absolute cross sections. The sum of the apparent cross sections 3P2) Qapp(3P0) has been measured using a channel electron multiplier Qapp( to detect the metastables by Mason and Newell (1987), Theuws et al. (1982), and Lloyd et al. (1972). Both Theuws et al. (1982) and Lloyd et al. (1972) obtained only relative cross sections. Mason and Newell (1987) calibrated their measurements at 22 eV to the results of Borst (1974) to obtain absolute cross sections. The sums of the apparent cross sections Qa,(-’P2) + Qa,,<3P0) obtained by the various researchers agree reasonably well among each other. The sum of the apparent cross section Qapp( 3Pz) + Qapp(3P0) has a peak value of about 3.55 x lo-’’ cmz at an incident electron energy of about 22 eV. Chutjian and Cartwright ( 1981) have measured the differential direct electron excitation cross sections for the 3P2and 3P0levels. By extrapolating differential cross sections to angles at which the cross sections were not measured and inte-

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grating over all scattering angles 8 the differential cross sections times sin 0, one obtains the total integrated cross sections. The peak values of direct total cross sections thus obtained for the ‘P, and ’Po levels are about 4 X lo-’* cm2 and 7 X l O - I 9 cm2, respectively. Chutjian and Cartwright (1981) have also measured the differential direct cross sections for other higher levels in Ar. If one obtains total direct cross sections by integrating the differential cross sections for these higher levels and then uses these to calculate the sum of the apparent cross sections for the metastables Qapp(3P2) Qapp(3P0),the result agrees well with the results of Borst (1974), Mason and Newell (1987), and others. There are also a number of experiments to measure QapP(’P2)+ Qapp(’Po) near threshold with special reference to resonance structures. Experiments of this type include such papers as Brunt et al. (1976), Buckman er al. (1983b), and Pichanick and Simpson (1968).

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D. EXCITATION INTO THE METASTABLE LEVELS OF KRYPTON The ground configuration of Kr is 4p6 and the lowest excited configuration is 4p55s. This configuration gives rise to 3P2and ’Po metastable levels. Mityureva et al. (1986) have measured the apparent cross sections for the electron excitation into the metastable ’P2 or ’Po levels. They used absorption out of the metastable level to detect the atoms in the metastable level. They also have measured the cascade contribution to the formation of metastables and obtained the direct cross sections. They obtain peak apparent cross sections for the 3P2and ’Po levels of 3.9 x lo-’’ cm2 and 1.0 X lo-‘’ cm2, respectively. The peak values of both apparent cross sections occur at an incident electron energy of about 16- 17 eV. The peak value of the direct cross section for the electron excitation of the ’P2 level is found to be about 1.1 x 10-l7 cm2. The direct electron excitation function for the 3P, level is found to have a peak value at about 16 eV and to fall slowly as the electron energy increases. The slow decrease in the direct cross section as the energy increases makes the data questionable. In addition to the optical measurements of Mityureva et al. (1986), there have been measurements of the sum of the relative apparent electron excitation cross QaPp(’P0)by Mason and Newell (1987) and Theuws et sections Qa,(’P2) al. (1982) using a channel electron multiplier to detect the metastable level atoms. Theuws et al. reported only relative cross sections. Mason and Newell obtained absolute cross sections. They found that the peak value of QaPp(’P2)+ Qapp(3P0)is about 2.84 X 10-17cm2at 17-eV incident electron energy. At energies above the peak value, the cross sections of Theuws et al. decrease more slowly than those of Mason and Newell. The calibration used by Mason and Newell to yield absolute cross sections is obtained as follows. Trajmar et al. (198 1) have measured the differential cross sections for the meta-

+

ELECTRON EXCITATION OF RARE-GAS ATOMS

23

stable 3P2and 3P0levels and other higher levels. The total integrated cross sec3P2)+ Qapp(IPO)are used tions of Trajmar et al. when summed to give Qapp( by Mason and Newell to calibrate their measurements at 15 eV. If we apply this same calibration to the data of Theuws et al. (1982), the summed integrated cross sections of Trajmar et al. are found to lie somewhat above the values of either Theuws et al. or Mason and Newell at higher energies. Again, there have been measurements of Qapp(3P2)+ Qapp(3P0)near threshold to study resonances. Measurements of this type have been reported by various groups including Brunt et al. (1976), Buckman er al. (1983b), and Pichanick and Simpson (1968). E. EXCITATION INTO THE METASTABLE LEVELS OF XENON

The ground configuration of Xe is 5p6, and the lowest excited configuration of Xe is 5p56s, which contains the 3P2and IPOmetastable levels. The apparent electron excitation cross section of the 3P2metastable level has been measured by Penkin and Smirnov (1986) using optical absorption at 881.9 nm (5p56s 3P2+ 5p56p 3D3)to detect the metastable 3P2atoms. They found the peak value of the apparent cross section to be 4.6 X lo-” cmz at 15-eV incident electron energy. Mason and Newell (1987) have measured the sum of the apparent cross sections, Qaw(’P2) Qapp(3PO),using electron bombardment of a low-density Xe gas beam and using a channel electron multiplier to detect the metastables. They normalize their data at 10-eV incident electron energy using the method of Blagoev et al. (1984). They found the peak at 14.7 eV and reported a value for the cm2at 15-eV incident elecsum of the apparent cross sections as 11.69 X tron energy. The apparent cross section falls rapidly with increasing electron energy and is only 4 X 10-l8 cm2at 33-eV incident electron energy. There have been a number of experiments measuring Qapp(3P2) + Qapp(IPO) at near-threshold energies. Examples of experiments of this type include the work of Brunt et al. (1976), Buckman et al. (1983b), and Pichanick and Simpson (1968).

+

IV. Excitation out of Metastable Levels The cross sections for the electron excitation out of metastable levels also are important both for fundamental understanding and for applications. There have been only a few experiments to measure the electron cross sections for the excitation out of metastable levels. The results of the few experiments done to date

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are insufficient to enable one to formulate a simple understanding of the cross sections that is similar to the understanding presented in Section I1 for electron excitation out of the ground level. Gas discharges in rare gases and operating at pressures above a few mtorr involve the electron ionization of atoms in metastable levels as the primary mechanism for the formation of the electron-ion pairs that sustain the discharge. Reactions, such as the electron excitation out of the metastable levels, that determine the metastable concentration in a discharge are essential to understanding gas discharges. A. EXCITATION OUT OF THE METASTABLE LEVELS OF HELIUM

Several experiments reporting electron excitation out of the metastable levels of He have appeared in the literature. Mityureva and Penkin (1975) use two parallel tubes in the same vacuum systems filled with He at pressure of about 0.1 torr. In one tube a gas discharge generated metastables that diffused to the other tube where an electron beam produced excitation out of the metastable levels. They measure the metastable density to be 10" atoms/cm3,but do not specify the ratio of the densities of 23S metastable atoms to 2 ' s metastable atoms. They do not report any electron excitation cross sections. They do, however, observe excitation of the 3'P, 33P, 3ID, and 43D levels but without identifying from which of the metastable levels the excitation originates. They obtained data indicating that the electron excitation cross sections out of the metastable levels are much larger than the corresponding excitation cross sections out of the ground level. Gostev et al. (1980) have also carried out an experiment on the electron excitation out of the metastable He levels. They have an experimental setup with a fast metastable atom beam produced as follows. A He+ ion beam is extracted from an ion source and is passed through a multicapillary array made of carbon. In collisions with the walls of the carbon capillaries the He+ ions are partially converted into fast neutral He atoms some of which are in the 23S or 2IS metastable levels. The density of atoms in the z3S level is measured using absorption by the 23S -+ 23P transition. No density is reported for atoms in the 2IS level, but perhaps they believe it was very small. The fast beam containing atoms in the metastable 23S level is crossed by an electron beam and radiation is detected from excitation to the 33P, 33D, 43D, and 3'P levels. Radiation from these levels is observed both below and above the threshold energy for these levels. They postulate that the below-threshold light is due to dissociative recombination of molecular ions formed along the direction of the fast atomic He beam. They use a model to subtract the light from the postulated dissociative recombination in order to determine the excitation cross sections out of the 23S level. They obtain excitation cross sections for the 33P, 3)D, 43D, and 3'P levels. The cross section for the excitation from the 23S to the 33P level has a peak value of 7 X

ELECTRON EXCITATION OF RARE-GAS ATOMS

25

cm2 at an electron energy of 5 eV. The corresponding results are 1 x 10-l4cm2 and 3 eV for excitation from z3S to 33D, are 7 X cm2 and 4 eV for excitation from 2% to 43D, and are 1 x cm2 and 3 eV for excitation from 23Sto 3'P. Because of the use of the model to subtract the effects of processes other than the direct electron excitation out of the metastable level and because of the possible existence of 2IS level atoms in the beam, the cross sections obtained by Gostev are somewhat suspect. Rall et al. (1989) have carried out measurements of absolute apparent cross sections for the electron excitation out of the z3S metastable level of He into the 33S, 43S, 33P, 33D, 43D, 53D, and 63D levels, and they report absolute direct excitation cross sections from the 2% level to the 33P, 3 3 , and 33Dlevels. Their experiment was carried out as follows. An atomic He beam containing atoms primarily in the I'S, 23S, and 2 ' s levels was formed by effusion through a I-mm-diameter hole in a hollow cathode discharge. The He atomic beam contained mostly He atoms in the 1 IS level but a fraction of about 3 X 10 - 5 of He atoms were in the z3S level. The density of z3S level atoms was about 5 X lo9 atoms/cm3in the collision region. The density of 2 ' s level atoms was about 3 X lo8atoms/cm3in the collision region or 6%of the 23S density. The atomic He beam was crossed by an electron beam. The electron beam was collected in a Faraday cup. The Faraday cup has, in addition to an open end for the electron beam to enter it, three side holes in it so that the He atomic beam can enter it, so that the radiation emitted can exit it, and so that a dye laser beam can pass through it. The three sets of side holes are at angles of 60" to each other. The light is collected and passed through an interference filter to select the emission from only one transition. When all the voltages in the electron gun were kept below 22.7 eV, the energy of the electrons in the electron beam was low enough that excitation out of the ground level was not possible and the radiation emitted was entirely due to excitation out of the metastable levels. Since the z3S population is so much larger than the 2IS level population, the excitation to the seven n3L levels mentioned earlier is almost entirely due to excitation out of the atoms in the z3S level. For electron energies above 22.7 eV the radiation from excitation out of the ground level is so large as to swamp the radiation out of the 23S level. Thus, the measurements were made only for energies less than 22.7 eV. Relative apparent cross sections were measured for excitation out of the 23S level to the seven higher excited levels. The absolute calibration was obtained by first measuring the relative optical efficiencies for the observed emissions from the seven excited levels so that the ratios of the absolute apparent cross sections for the various excitations are all known. Then the absolute cross section for the excitation of the 33P level was obtained by measuring the 3)P + 2's fluoresence, which resulted from exciting the 23Smetastables to the 33Plevel by a laser beam. This is possible because the oscillator strength of the 23S + 23P absorption is accurately known. Direct absolute cross sections have been obtained by

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subtracting the cascade contributions. At 10 eV the apparent cross sections for excitations out of the z3S level to the 33S, 33P, 33D, 43S, 43D, 53D, and 63D 1.5 X levels are 5.6 X 10-l6, 3.0 X 10-I6, 9.4 X 10-I6,1.5 X 10-l6, 0.24 x 10-l6, and 0.11 x cm2, respectively, and the direct excitation to the 33S, 33P, and 3)D levels are 4.8 X 2.1 X 10-l6, and 8.0 x cm2, respectively. The cross sections obtained by Gostev er al. (1980) for the excitation out of the z3S level into the 33P level are of comparable magnitude to those of Rall et al. but the measurements by Gostev er al. for the excitation out of the 23S level into the 33D and 43D levels give much larger cross sections than those by Rall et al. (1989). Mityureva and Penkin (1989) reported measurements of the cross sections out of the metastable levels using stepwise excitation of He. Metastables were produced in a gas discharge tube and diffused into a glass vacuum tube where the electron beam produced excitation. They measured both the 23S and 2IS densities using optical absorption. There were approximately five times as many 2% metastables as 2 ' s metastables. They have measured apparent excitation cross sections out of the metastable levels into the 33P,3IP, 3'D, and 43D levels. They found a peak value of the cross section into the 33P and 43D levels of 2.7 X 10-l4cm2 and 6 X lo-', cm2, respectively. These cross sections are about 70 and 40 times larger than the cross sections reported by Rall et al. (1989). B. EXCITATION OUT OF THE METASTABLE LEVELS OF NEON Mityureva and Penkin (1975) studied the stepwise excitation of Ne using the same apparatus that they used for the stepwise excitation of He reported in the same paper. They observed radiation from the 2p9 (J = 3) level to the Is, ( 3P2) level and from the 2p2(J = 1) level to the Is, level. They detected the metastable atoms using optical absorption. Presumably most of the metastable atoms are in the Is, level though this is not stated by Mityureva and Penkin explicitly. They measured only relative apparent cross sections and found a sharply peaked energy dependence.

c.

EXCITATION OUT OF THE METASTABLE LEVELS OF ARGON

Mityureva (1985) studied the stepwise excitation of the 2p9 ( J = 3) level of Ar. The apparatus included three parallel electron guns. Metastable atoms were produced by the outer two electron guns and diffused into the electron beam produced by the center electron gun. Mityureva estimated that 80% of the metastable atoms were in the I s ~ ( ~ level P ~ )and 20% were in the I S ~ ( ~level. P ~ )The

ELECTRON EXCITATION OF RARE-GAS ATOMS

27

atoms were excited out of the metastable levels into the 2p9 ( J = 3) level of Ar. The excitation of the 2p9level was believed to come primarily from the Is, level. Mityureva determined the peak value of the apparent cross section for excitation out of the Is, level to the 2p9 level to be 9 X 10-l5 cm2 at an energy of 3.5 eV and to fall sharply as the energy increased above 3.5 eV. Further studies of the electron excitation out of the metastable levels of Ar using the stepwise excitation have been reported by Mityureva et al. (1989a,b). They indicated that the density of atoms in the 3p54s 3P2 level exceeded the density of atoms in the 3p54s 3P0 level by more than an order of magnitude (Mityureva et al., 1989b). They reported apparent cross sections from the Ar metastable levels to the 2pl, . . . ,2p9 levels in the 3p54p configuration. The apparent cross sections were very large. For example, they found that the peak value of the apparent cross section into the 2p9 level was 1.8 X 10-I4cm2 at an electron energy of 5 eV.

D. EXCITATION OUT OF THE METASTABLE LEVELSOF KRYPTON Mityureva et al. (1989~)studied the electron excitation out of the metastable levels of Kr. They used a pulsed electron beam both to produce the metastables and to excite the metastables. In the first pulse the electron energy was high enough to produce the metastables. After the radiating atoms produced by this pulse decayed for about 12ps, a second pulse with electron energies too low to produce ground-level excitation but with enough energy to excite out of the metastable levels was produced. The metastable density was measured using optical absorption. It was indicated that the population of the 4p55s 3P2 level exceeded that of the 4ps5s 3P0level by an order of magnitude. By assuming that the majority of stepwise excitation proceeded through the 4p55s 3P2 level, the apparent excitation cross sections for electron excitation out of the 3P2metastable level into the 2pl, . . . ,2p9levels of the 4 p 3 p configuration were obtained. The measured cross sections were again very large. For example, they reported the peak value of the apparent cross section for excitation out of the 3P, metastable level into the 2p9 level as 4.8 x lO-I4 cm2.

Conclusion In this chapter we have described measurements of the total cross sections into and out of the metastable levels of rare-gas atoms. We have tried to present our assessment where we believed that there were any serious problems with the

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measured cross sections. In this section we discuss some general considerations of the various experiments, and we discuss experiments that will help future data needs. The measurements of total electron excitation cross sections into the metastable levels of rare gases using optical absorption detection of the metastables can yield apparent cross sections for individual metastable levels. The use of absorption techniques to measure the metastable density, n, is reliable but requires an optical path, [ such that ( n u A [ )= 1 where uAis the optical absorption cross section. The optical pathlength required for a good signal-to-noise ratio may be very long. Optical absorption can yield absolute apparent cross sections provided both the oscillator strength of the transition and the metastable destruction rate are known. If these are not known, then an alternative method of calibration for optical absorption measurements is necessary to yield absolute cross sections. Laser-induced fluorescence detection is very sensitive and can be used to detect a small number of metastable atoms. Absolute cross sections can in principle be obtained using LIF if the laser power is sufficiently intense that the rate of laser excitation out of the metastable level is much greater than the metastable destruction rate by other processes. In practice, this has not yet been done and absolute cross sections have been obtained only using other calibration methods. Both optical absorption and LIF methods yield apparent cross sections. In order to obtain direct cross sections, one must subtract the cascade cross sections from the apparent cross section. If the direct cross section is very much smaller than the apparent cross section at high energy (as is the case for the metastable 3P2or 3P0 levels in Ne, Ar, Kr, or Xe), then obtaining direct cross sections at high energies is difficult. Obtaining the total absolute direct cross sections by integrating the absolute direct differential cross sections times sin 0 over all scattering angles 8 is possible. However, the measurement of differential cross sections requires good angular resolution and good energy-loss resolution. The measurement of direct cross sections for metastable levels at high energy is one area where the differential cross sections may be especially valuable. The measurement of the sum of the apparent cross sections into all metastable levels using Auger electron or channel electron multiplier detection of the metastable atoms has been studied for all the rare gases. These measurements are interesting and have been carefully executed. However, the measurements usually rely on the assumption that the detection efficiency is the same for different metastable levels. For He metastables Dunning et al. (1975a) have shown that the Auger electron emission efficiencies for He(2'S) or He(23S) striking a stainless steel plate differ by more than 25%. Thus, the assumption of the same efficiencies for other rare gases may not be correct. Another problem in obtaining

ELECTRON EXCITATION OF RARE-GAS ATOMS

29

absolute apparent cross sections is in the calibration. Mason and Newell (1987) carefully give the reference cross section and energy at which they have calibrated their sums of total apparent cross sections. For example, they calibrate their Ne cross sections at 26 eV to the cross sections of Teubner et al. (1985). Teubner et al. in turn calibrated their Ne cross sections by ratioing them to the He cross sections using the ratio of Auger electron detector efficiencies from Dunning et al. (1975a). The absolute He cross sections were then obtained using the total absolute cross sections for the direct excitation of the He(2'S) at 400 eV obtained by Dillon and Lassettre (1975) plus the cascade contributions from the n'P levels at 400 eV. The total absolute cross sections of Dillon and Lassettre were obtained by integrating the differential cross sections measured by electronenergy-loss measurements. Dillon and Lassettre state that the differential cross sections were measured with a 5% error (or 6% if pressure correction is required). The total direct cross section obtained from these differential cross sections may contain additional sources of uncertainty. Teubner et al. (1985) expressed the cascades from the higher n'P levels at 400 eV as a sum, over n, of the products of the branching ratio for the nlP += 2's transition times the direct excitation cross section into the n'P level from the ground level. The correct expression for cascades from the higher n'P levels, however, should be the sum of the products of the branching ratio times the apparent excitation cross section as can be seen by combining Eqs. (6) and (7) derived earlier. In this case the error in the absolute calibration caused by use of the direct excitation cross sections of n'P in place of the apparent excitation cross sections for determining the cascades should be small because the total cascade amounts to only 16% of the total cross section as pointed out by Teubner et al. (1985). Nevertheless, because of the many steps to the absolute cross sections of Mason and Newell, one might be concerned that the final uncertainty could have some systematic components. Experiments measuring the sum of the apparent cross sections using Auger electron or channel electron detection of the metastable rare gas atoms have been carried out for all the rare gas atoms and for a large energy range by several researchers. Measurements by Mason and Newell, Teubner, Theuws, and others seem carefully done and agree well among themselves so that there is a reasonably complete data set for all the rare gases. There are, however, fewer measurements using optical methods to determine the total apparent cross sections for individual metastable levels than there are measurements of the sum of the apparent cross sections to all metastable levels. In some cases, direct cross sections for excitation into a single metastable level have been obtained by subtracting the cascade cross sections. Direct total cross sections have also been obtained by integrating the differential cross sections obtained by electron-energy-loss measurements. The direct cross sections into a metastable level using the two different methods agree fairly well where com-

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parison is possible. However, only for a very few metastable levels have cross sections been determined by the two different methods. Additional measurements are needed for the other levels. The field of measurements of the apparent and direct electron excitation cross sections out of rare-gas metastable levels is only at its initial phase. Reliable data are available only for restricted energy ranges and for a few cases. The major problem in obtaining reliable absolute electron excitation cross sections out of the metastable levels is the low metastable target density and the resulting low rate of optical emission. The cross sections of Mityureva and Penkin (1989) for electron excitation out of the metastable levels of He are very much larger than the cross sections of Rall et al. (1989). Furthermore, Mityureva et al. (1989b,c) have also reported remarkably large cross sections for electron excitation out of the metastable levels of Ar and Kr. Additional measurements of electron excitation cross sections out of the metastable levels of rare gases would be most valuable. This will permit the development of an overall understanding for electron excitation out of metastable levels that is similar to that for ground-level excitation.

Acknowledgments We acknowledge that the preparation of this chapter was supported by a grant (PHY 9005895) from the National Science Foundation.

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Phelps, J. 0.. Phillips, M. H., Anderson, L. W., and Lin, C. C. (1983). J . Phys. B 16,3825. Phillips, M. H., Anderson, L. W., and Lin, C. C. (1981a). Phys. Rev. A 23,2751. Phillips, M. H., Anderson, L. W., Lin, C. C., and Miers, R. E. (1981b). Phvs. Left. 82A,404. Phillips, M. H., Anderson, L. W., and Lin, C. C. (1985). Phys. Rev. A 32, 21 17. Pichanick, F. M. J., and Simpson, J. A. (1968). Phvs. Rev. 168, 64. Purcell, E. M. (1952). Asfrophys. J. 116,457. Rall, D. L. A., Sharpton, F. A., Schulman, M. B., Anderson, L. W., Lawler, I. E., and Lin, C. C. (1989). Phys. Rev. Lett. 62,2253. Register, D. F., Trajmar, S., Steffenson, G., and Cartwright, D. C. (1984). Phys. Rev. A 29, 1793. Rudge, M. R. H. (1965). Pruc. Phys. Suc. (London) 85,607. St. John, R. M., Miller, F. L., and Lin, C. C. (1964). P h w . Rev. 134, A888. Schaper, M., and Scheibner, H. (1969). Beifr. Plasma P h y i k 9,45. Sharpton, F. A., St. John, R. M., Lin, C. C., and Fajen, F. E. (1970). Phys. Rev. A 2, 1305. Showalter, J. G, and Kay, R. B. (1975). Phys. Rev. B 1 1 , 1899. Teubner, P. J. O., Riley, J. L.,Tonkin, M. C., Furst, J. E.. and Buckman, S. J. (1985). J . Ph.vs. B 18,3641.

Theuws, P. G. A,, Beijerinck, H. C. W., and Verster, N. F. (1982). J . Phvs. E 15, 328. Trajmar, S. (1973). Phys. Rev. A 8, 191. Trajmar, S . , Srivastava, S. K., Tanaka, H., Nishimura, H., and Cartwright, D. C. (1981). Phvs. Rev. A 23,2167. van Raan, A. F., Moll, P. G., and van Eck, J. (1974). J . Phys. B 7,950. Woudenberg, J. P. M., and Milatz, J. M. W. (1941). Physica 8, 871. Zetner, P. W., Westerveld, W. B., King, G. C., and McConkey, J. W. (1986). J. Phys. E 19,4205.

ADVANCES IN ATOMIC. MOLECULAR,AND OPTICAL PHYSICS. VOL. 29

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS M . V. AMMOSOV, N . B . DELONE, and M. YU. IVANOV General Physics Institute Moscow, USSR

I . I . BONDAR Uzhgorod State Universiiy Uzhgorod, USSR

A . V. MASAL.0V Lebedev Physical Institute Moscow, USSR

I. Introduction . . . . . . .

.

.

.

. .

.

.

. . .

. .

. .

.

.

.

. . . . . 34

11. Methods of Measuring the Principal Quantities That Characterize Multiphoton

Ionization of Atoms. . . . . . . . . . . . . . . . . . . . . . . A. Principal Relations. . . . . . . . . . . . . . . . . . . . . . . B. Single-Frequency Laser Radiation. . . . . . . . . . . . . . . . . C. Multifrequency Laser Radiation . . . . . . . . . . . . . . . . . D. Measurement of the Power of Nonlinearity of the Ionization Process . . . E. Measurement of the Ionization Probability . . . . . . . . . . . . . F. Measurement of Multiphoton Cross Sections of Direct Atom Ionization . G. The Absolute Method of Measuring Multiphoton Cross Sections . . . . H. The Relative Method of Measuring Multiphoton Cross Sections. . . . . The Procedure for Measuring the Quantities Needed to Find the Multiphoton Cross Sections and Its Accuracy . . . . . . . . . . . . . . . . . . . A. Measuring the Energy of a Laser Pulse. . . . . . . . . . . . . . . B. Measuring the Time Distribution of Intensity in a Laser Pulse . . . . . C. Measuring the Spatial Distribution of Laser Radiation Intensity . . . . . D. Measuring the Number of Ions Produced . . . . . . . . . . . . . . E. Measuring the Density of Neutral Atoms . . . . . . . . . . . . , . F. Calculating the Laser Pulse Intensity, the Effective Pulse Duration, and the Effective Interaction Volume . . . . . . . . . . . . . . . . . . . G. Calculating the Ionization Probability per Unit Time and the Multiphoton Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . H. The General Procedure for Measuring Multiphoton Cross Sections . . . The Results of Measuring the Multiphoton Cross Sections of Direct Atom Ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Expression for Estimating the Multiphoton Cross Sections of Direct Atom Ionization . . . . . . . . . . . . . . . . , . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

111.

IV. V.

VI.

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

.

45 45

46 51 55 61 63

64 65

. . 73 . . 76 . . 77 . . 79 . . 81 . . 83 . . 84 .

.

.

87

. 88

. . 90 . . .

101 107

108

33 English translation copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12003829-3

34

Ammosov, Delone, and Ivanov

I. Introduction This chapter is devoted to cross sections of direct multiphoton ionization of atoms. (We also use the term multiphoton cross sections.) We discuss the methods used to measure these cross sections, the experimental results obtained, and the theoretical relations that enable one to estimate the magnitude of multiphoton cross sections. As in other cases, knowledge of the cross sections of direct multiphoton ionization of atoms enables one to obtain information about observed phenomena: ionization probability, the number of ions produced, the ionization degree of initial neutral medium, and how all these quantities depend upon different parameters characterizing the radiation and the atom medium. The problems related to cross sections of multiphoton atom ionization have been considered in several monographs (Delone and Krainov, 1985; Faisal, 1987) and reviews (Morellec et al., 1982; Chin and Lambropoulos, 1984) and in many original papers. However, recently, two new effects have been discovered-the production of multicharge ions (Suran and Zapesochnyi, 1975) and above-threshold ionization (Agostini er al., 1979)-that have qualitatively changed the principles of multiphoton atom ionization previously summarized in Delone and Krainov (1985). These effects play an essential role both in interpreting the experimental data on multiphoton cross sections obtained earlier and in optimizing the organization of experiments to be made. This is why the authors believe that a discussion of the whole set of data related to multiphoton cross sections of atom ionization will prove to be quite timely. There is also another aspect of the problem that has made the authors write this chapter. There are many areas in the physics of laser radiation interaction with matter where one has to know the probability of multiphoton atom ionization, or, at least, its approximate estimate (Delone, 1989). Rigorous theories of the ionization process require complicated calculations for each atom and each radiation frequency. The approximate relations we present here, on the one hand, are absolutely elementary but, on the other hand, give a satisfactory description of the experimental data. Let us first consider the principal features of the process of multiphoton atom ionization that have been examined ia detail by Delone and Krainov (1985). The process of ionization of atomic particles (atoms, molecules, and ions) under the action of radiation is called nonlinear if the energy ho of radiation quanta is less than the binding energy E , of an electron in a given particle (less than the ionization potential I if the particle is in its ground state):

hw < E n , 1.

(1)

Considering that atomic ionization potentials range from 5 to 25 eV, we see

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

35

that inequality (1) holds for radiation in the ultraviolet and visible regions and for all other regions with greater wavelength. The process of nonlinear ionization is referred to as being multiphoton if the so-called adiabaticity parameter y (Delone and Krainov, 1985) is much greater than unity:

Here w and E are the frequency and strength of the radiation field. In order to eject a single electron from an atom, the latter must absorb an amount of energy that exceeds the electron binding energy (ionization potential); hence, at y >> 1, the atom should absorb the so-called threshold number of radiation quanta, K , given by the relation

where {n} denotes the integer part of x . If the condition (3) is fulfilled, i.e., if the atom absorbs the threshold number K of radiation quanta, the energy E = Khw - I < hw is distributed between the positive ion and the electron and, since the latter is much lighter, practically all of it transforms into electron kinetic energy. In the opposite case (i.e., where y << l ) , the ionization process has the character of a tunneling in an alternating field (Delone and Krainov, 1985). By substituting numerical values into Eq. (2), one finds that for atoms in the ground state with ionization potential I = (0.2 to 1.0) x E, (E, being the atomic energy unit), and for field strengths E << E,, y >> 1 for radiation in the near ultraviolet, visible, and near-infrared frequency ranges. The process of multiphoton ionization is called direct if the energy of any number K' < K of absorbed quanta does not coincide with the energy of any transition in the atomic spectrum from the ground to an excited state allowed by selection rules for multiphoton transitions (Delone and Krainov, 1985). If the E, = 5 X lo6 V cm-I, one can strength of the radiation field is E << neglect the perturbation in the atomic spectrum induced by the radiation field (i.e., the dynamic Stark effect) and use the table data on binding energies and quantum numbers of electron excited states in an atom. In this case, the criterion of direct ionization reads as Ai

=

-

K'hw( >> yi,

(4)

where E;, are the energies of the transitions from the ground state into all excited states allowed by selection rules for multiphoton transitions and y i are the natural widths of these states. The quantities A iare called resonance detunings. When

36

Ammosov,Delone, and lvanov

the strength of the radiation field is E > 10-3Ea= 5 X lo6 V X cm-l, one, as a rule, cannot neglect the perturbation of the spectrum of bound electron states by the radiation field, so the criterion of direct ionization has a more complicated form:

In contrast to Eq. (4),in this formula the resonance detuning A,(E),the energy of the excited state e.(E), and its width r i ( E ) are all functions of the radiation field strength. In certain cases the selection rules that determine the allowed transitions are also different (Delone and Krainov, 1985). The probability of multiphoton ionization per unit time, w , is defined as the ratio of the number of ions Ni produced per unit time to the number of atoms N , upon which the radiation field acts: w = N i / N ,. The total ionization probability per laser pulse is given by the evident relation W = wTK = N i / N a ,where TK is the effective duration of the radiation pulse. (See Section 1I.B.) The principal equation that relates the probability of direct ionization per unit time to the cross section of this process and the radiation intensity reads as w = aKFK.

(6)

Here aKis the effective cross section of ionization, F is the radiation intensity (the number of photons that cross a unit surface per unit time), and index K means that the process involves the absorption of K radiation quanta. From Eq. (6) and the dimensions of the probability ([w]= sec-I) and radiation intensity ( [ f l = photons X cm-2 X sec-l) it follows that the cross section of K-photon ionization has the dimension [ a K ] = cm2KX secK-'. One can see that for one-photon ionization (K = 1) the effective cross section has the standard dimension [ a , ]= cm2. The cross section aKdepends upon the frequency and polarization of radiation, since these two parameters determine the intermediateresonances. (See Eq. (7).) As it is for photoionization, in the case of multiphoton ionization the radiation frequency is a free parameter that can vary continuously. Accordingly, in both cases the tabulation of the cross sections is out of the question, in contrast to single- and multiphoton excitation. The problem is to develop the methods for theoretical description of direct multiphoton ionization and to optimize these methods by comparing the results they give with experimental data. When Eqs. (2)-(5) are satisfied, a quantitative description to the direct process of multiphoton ionization is traditionally given by means of the nonstationary perturbation theory in the first nonvanishing (Kth) order (Delone and Krainov, 1985; Faisal, 1987). In the framework of the nonstationary perturbation theory, the cross section of direct multiphoton ionization is described by a compound matrix element of the Kth order that has the form

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

37

where V, are dipole matrix elements that describe transitions between the ith and the jth bound states (the last transition is between the bound state and a free state E ) , while the expression in each bracket in the denominator is the detuning of the corresponding intermediate resonance. As one can see from (7), with variation of frequency 0 , as one of the detunings becomes smaller and intermediate resonance is approached, the multiphoton cross section becomes greater. The relative role of intermediate resonances can be estimated from the data on atomic spectra. If we assume that the maximum resonance detuning is of the order of the spacing between levels (i.e., around 1 eV = lo4 cm-I), and that the minimum detuning is of the same order as the natural width of a level (i.e., cm-I), we see that the cross section can vary by a factor of 10’. However, in practice, the maximum detuning is several orders of magnitude smaller, at least in the region of excited states, while the minimum detuning is several orders of magnitude higher (owing to such factors as the finite width of the laser radiation spectrum, Doppler effect, field broadening of bound states, and saturation of the ionization process in the resonance), as the range within which the value of the cross section varies between resonances is not so great, though it still amounts to several orders of magnitude. As an example that illustrates a typical dependence of the multiphoton ionization cross section on radiation frequency, in Fig. 1 we show the results of typical perturbation theory calculations. These results show the qualitative features of direct multiphoton ionization: the difference between ionization processes in the field with linear and circular polarization and the range within which the cross section varies in the interresonance gaps.

.I

FIG. 1 . Cross section a2of two-photon ionization of the hydrogen atom in the field of linearity polarized ( - ) and circularly polarized ( - - - ) radiation. n = 2, 3, 4 are the intermediate onephoton resonances with excited states characterized by the indicated main quantum numbers.

38

Ammosov, Delone, and tvanov

The perturbation theory calculations of the cross section require information about the atom spectrum and can only be made by a computer with special programs. Here we shall not consider this aspect of the problem, which has been discussed in detail in Delone and Krainov (1985), Faisal(1987), Morellec et al. (1982), and Chin and Lambropoulos (1984). From the first principles it follows that the perturbation theory in the first nonvanishing (Kth) order is applicable to describing the process of multiphoton atom ionization if the following three conditions are satisfied (Delone and Krainov, 1985): (1) The strength of the external field E should be much less than the atomic strength E, . (2) The adiabaticity parameter y should be much greater than unity. (3) There should be no intermediate resonances with bound electron states, the transitions into which from the initial state are allowed by the selection rules for multiphoton transitions (Delone and Krainov, 1985). These three conditions are in agreement with the well-known facts that the direct process of multiphoton atom ionization results from absorption of a threshold number K of photons given by Eq. (3); that its probability is given by a powerlike relation (6); and that with increase of the field strength (at a fixed radiation frequency) the rnultiphoton ionization gradually transforms into tunneling (as from the case y >> 1 we come to the case y << 1). However, the experiments of the recent years have shown that, in reality, the general scheme of the nonlinear ionization process is more complicated and that the conditions under which the perturbation theory is applicable are different from those presented (Delone and Fedorov, 1989a,b). Two new effects have been discovered experimentally: above-threshold ionization (Agostini et a l ., 1979) and production of multicharge ions (Suran and Zapesochnyi, 1975). The above-threshold ionization is the absorption of more than K photons, where K is the threshold number of photons for direct multiphoton ionization. Its probability is of the same order of magnitude as the probability of absorbing K photons at a field strength E << E,. The principal method of observing and examining this phenomenon experimentally is to record the energy spectra of produced electrons. When a threshold number of photons is absorbed, the kinetic energy of electrons equals &&n = Khw - t ( E ) , where t ( E ) is the atom ionization potential in the radiation field. For above-threshold ionization we have E ~ = (K + S)ho - t ( E ) . A typical energy spectrum of electrons produced by above-threshold multiphoton atom ionization is shown in Fig. 2. The discovery of such an effect means that, in reality, the limits within which the first nonvanishing (Kth) perturbation theory approximation is valid are determined by a much more stringent condition than E << E,. In practice, this means that in the multiphoton region (y >> 1) at E << E,, in addition to the threshold (K-photon) ionization channel, there may also be realized the above-

.

~

~

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

39

'I T

- 0

FIG. 2. Energy spectrum of electrons produced in the six-photon (K = 6) process of ionization of the xenon atom (Agostini et al.. 1979).

+

threshold ((K $)-photon) channels with the same probability. Let us mention that the above-threshold ionization is observed for all atoms (from hydrogen to noble-gas atoms) and for several molecules within a wide range of radiation frequencies, from near infrared to near ultraviolet (Delone and Fedorov, 1989a,b). Here we shall not discuss the theoretical interpretation of the abovethreshold ionization both because this is a very complicated problem and because at present it is impossible to make some of the final conclusions. The current state of this problem is presented, for example, in the reviews by Delone and Fedorov (1989a,b). Another effect that underwent a thorough experimental examination is the formation of multicharge ( A q + ) ions in addition to the singly charged ones ( A + ) in multiphoton atom ionization (at y >> 1) in a field of moderate strength (E << Ea).The production of multicharge ions is observed in ionization of any multielectron atom by laser radiation with the frequency ranging from the nearultraviolet to the near-infrared region (Delone and Fedorov, 1989a). The degree of ionization may be as high as q = 10. As an example, in Fig. 3 we present the experimental data related to formation of single- and double-charge strontium ions. The detailed experimental examination of multicharge ion production that has been made in many studies has enabled one to make two important conclusions about the character of this process. The first conclusion is that the multicharge ions are produced in the process of cascade ionization. The successive stages of

40

Ammosov, Delone, and Ivanov

FIG. 3. Experimental data on the yields of Sr+ and Sr2+ions versus the radiation intensity F in the six-photon (K = 6) process of strontium atom ionization (Aleksakhin er a/., 1979). One may notice that in the region where the saturation of the Sr+ ion yield occurs, the number of Sr2+ions is only several times smaller than the number of Sr+ ions.

the cascade take place within a single laser pulse. The second conclusion is that at each stage of the cascade there are produced ions both in the ground and in a number of excited states. Thus, the simplest case of double-charge ion production corresponds to the following reactions: A A+ (A+)?

+ (KI + S l ) h o + Z A + , (A')?, e + ( K 2 + S2)fiw -+ Z A2+,(A2+)?,e + ( K 3 + S , ) h Z A*+,(A2+):, e . --j

Here K , , K 2 , and K 3 (S,, S1, and S,) are threshold (above-threshold) numbers of photons absorbed, and (As+)* are ions excited into different (ith) states. Accordingly, the ions of any charge multiplicity are produced both as a result of absorption of the threshold number of photons K and as a result of absorption of a greater, above-threshold number of photons K S. At fixed radiation intensity, the probabilities with which the ions are produced in the ground and in the excited states may be comparable in magnitude. Let us point out that the very

+

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

41

fact that ions are produced in excited states indicates that the interaction between the field of laser radiation and atoms that have more than one electron on their external orbit is not of single-electron character. As a concrete example, in Fig. 4 we present the experimental data related to production of double-charge strontium ions. Both of these effects-above-threshold ionization and multicharge ion production-qualitatively break down the simple model of the process of nonlinear ionization which before that was considered to be fundamental. As concerns the problem we are interested in (i.e., whether it is possible to describe the direct process of multiphoton ionization by such quantities as multiphoton cross sections), the influence of these two effects are qualitatively similar. In both cases the process of ion production from a neutral atom is a sum of different channels differing in their final state. For above-threshold ionization, these channels differ in the number of photons absorbed (K, K f 1, K f 2, . . .); the final states of the ion are the same-the ion remains in the ground state; the final states of the free electron differ in the value of its kinetic energy: E ~ =. (K~ ~ S)hw ~ - I ( E ) . When the ions are produced in excited states, the channels differ both in the number of photons absorbed and in the state of the ion and in electron kinetic energy. It is evident that each channel can be characterized by the value of the corresponding multiphoton cross section. As for the total effect of all channels, if the power of nonlinearity K of these channels are different, it is incorrect to introduce an integral multiphoton cross section. (Let us recall that the dimension of a multiphoton cross section depends upon K.) Since, in principle, the multiphoton processes are not of threshold character with respect to the strength of the radiation field, there is, strictly speaking, no certain threshold below which one can assert that a certain ionization channel has been realized. The only thing one can say is that if, at a sufficiently small field strength, a certain channel is dominating, one can approximately characterize the process of multiphoton ionization by a certain multiphoton cross section. The qualitative difference between multiphoton and single-photon processes is that the former depend upon the intensity of radiation while the latter depend upon its energy, i.e., upon the total number of photons that went through the target. It is difficult to find the space-time distribution of the laser radiation intensity on the target with sufficiently high resolution and to take it into account when calculating the multiphoton cross sections. One encounters three main difficulties. First of all, both the spatial and the time distribution of laser radiation are, in principle, nonuniform. Owing to this, they are difficult to measure and one has to introduce a certain normalization. The second difficulty is that the measured quantities enter Eq. (6) in high (Kth) power. This requires them to be measured with high accuracy, though the accuracy with which one can find the cross section is not very high. The third problem arises in those cases where the

+

42

Ammosov, Delone, and Ivanov

f71 WlI

I

2

Energy ( e V )

3

4

(b) FIG. 4. (a) Energy spectrum of electrons produced in the three-photon ( K = 3) process of strontium atom ionization (Agostini and Petite, 1985). Figures near the maxima indicate the number of the channel according to which the Sr+ and Sr2+ions are produced. (b) Schematic diagram of the channels that lead to production of Sr+ and Sr2+ions (Agostini and Petite, 1985). One can see that the efficiency of K-photon absorption with production of the Sr+ ion in the ground state (channel 1) and the efficiency of ( K + 1) photon absorption with production of the (Sr+)* ion in the state ~ P ~ P , , (channel ~ , , , ~ 7) are practically the same.

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

43

laser radiation used is not ideally monochromatic. In the multimode generation regime, the instantaneous intensity of radiation is a random quantity, and the probability of it having a certain value is unknown in the general case. The optimal ways of solving these problems and the accuracies with which one can measure different quantities will be discussed in the corresponding sections of this chapter. The experiments in which one observes and examines the process of multiphoton atom ionization and, in particular, measures the multiphoton cross sections of direct ionization, are usually made according to the traditional scheme (Delone and Krainov, 1985). One uses the radiation of a pulsed laser, which is focused into a vacuum chamber. The pulsed character of the radiation and its focusing are necessary to increase the intensity of radiation in the region where the ionization occurs. The chamber is filled with the gas (or vapor) of examined atoms at a pressure where no collisions between the particles occur within the time the laser pulse acts in the region where the radiation is focused. One can also use an atomic beam in the chamber and focus the laser radiation at the center of the beam. The ions produced in the focusing region are accelerated by a constant or pulsed electric field. After that their masses are analyzed in a timeof-flight mass spectrometer, and the ions are detected by an electron multiplier. In a series of successive laser pulses, one varies the parameters of radiation (its intensity and frequency) and measures the corresponding ion yield. The scheme of a typical experiment and of the apparatus used in it is shown in Fig. 5 . The measuring procedures for different parameters characterizing the laser radiation, the atom target, and the results of their interaction are discussed in detail in the following sections. Finally, let us consider the ranges within which the principal parameters characterizing the atoms and the radiation field in which the direct process of multiphoton ionization may take place and within which the cross section of this process may be measured. As concerns the process of multiphoton ionization, in principle, the conditions of its realization are given by the following two inequalities: y

>>

1 and E

<< E , .

From Eq. (2) for the adiabaticity parameter y one can easily estimate that for classical frequencies of laser radiation, from near infrared to near ultraviolet, at E << E , we always have y >> I , i.e., the ionization process is always of multiphoton character. In the multiphoton limit ( y >> l ) , the direct process can be clearly isolated only if the field strength is not very high, so that the change in the energy of bound states, 8&(E),owing to the dynamic Stark effect (Delone and Krainov, 1985), is much smaller than the difference between the energies of the neighboring states. It is evident that the field strength E at which the condition

44

Ammosov, Delone, and tvanov

FIG. 5. Typical scheme of an experiment where multiphoton cross sections are measured: I , laser source; 2, attenuator made of two polarizers; 3, neutral optical filters; 4, optical wedge; 5, focusing lens and a lens identical to it; 6, windows of the vacuum chamber; 7, vacuum chamber; 8, system of electrodes of the time-flight mass spectrometer; 9, field-free flight gap; 10, ion detector; 11, vacuum gauge; 12, calorimeter; 13, ion-signal detecting and processing system; 14, photodetector; 15, wideband oscilloscope; 16, microscope; 17 camera.

SE(@ << A&,,,*, is realized is the smaller the higher is the main quantum number of the given state n. Since the field is turned on smoothly (the radiation intensity within a pulse has a Gaussian distribution), in those cases where the strength of the laser radiation field is so high that one has S E ( E ) E,,.? , there will always be a moment at which the condition ( 5 ) necessary for realization of the direct process will not be satisfied, and there will appear an intermediate resonance (that may be multiphoton) due to the smooth variation of the transition energy (the dynamic resonance). The data on the dynamic Stark effect available at present (Delone and Krainov, 1985) enable one, in each case, to estimate with good accuracy the maximum field strength allowed. If the conditions just discussed are satisfied, and the direct process of multiphoton ionization takes place, this does not necessarily mean that one can measure its cross section. For this, there exist additional constraints that place the upper and lower limits upon the radiation field strength. On the one hand, the field strength should be high enough so that within a single laser pulse, in the volume where the ionization takes place, there would be produced a certain minimum number of ions (- 1) that can be detected. On the other hand, the field strength should not exceed the value at which the total ionization probability

-

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

45

-

within a pulse, W = wT,, becomes unity (the condition W 1 is equivalent to l/T,). Since the dependence of the ionization probability per the condition w unit time w on radiation intensity F is always very steep (see Eq. (6)),in practice, the measurements can be made only in a very narrow range of intensities in For nanosecond laser pulses (T, the vicinity of E = %= lo-%) this is the values from E lo3 V X cm-' for two-photon processes (K = 2) to E lo5 V X cm-I for six-photon processes (K = 6). We conclude this introduction by emphasizing once again that in the case of multiphoton atom ionization in a field that is not very strong, the direct process is realized. The resonance process of ionization, the conditions of which are expressed by inequalities opposite to (4)and ( 5 ) , requires special selection of the laser radiation frequency. This is why the values of the cross sections of direct multiphoton ionization give one the main information about multiphoton ionization for most of the radiation frequencies. Let us stress once more that in very strong fields, it is impossible to distinguish between direct and resonance ionization processes since the condition &(E) << A&,,"+,no longer holds.

-

v m .

-

-

-

II. Methods of Measuring the Principal Quantities That Characterize Multiphoton Ionization of Atoms A. PRINCIPAL RELATIONS

The experimental observation of the process of K-photon ionization of atoms by laser radiation consists of recording the number of ions produced within a single laser pulse. The relation between the number of ions and the radiation characteristics is based upon Eiq. (6). The ionization probability per unit time w determines the rate with which the ion concentration ni grows at a given point in space as the pulse passes through: ri, = (no - n,)w,

(8)

where no is the initial concentration of neutral atoms. Within the time T the radiation pulse acts, the ion concentration reaches the value

while the total number of ions Ni produced within the whole volume of interaction amounts to Ni = no

I[

1 - exp(

-JT

w d l ) ] du dy dz.

46

Ammosov, Delone, and Ivanov

With account of the dependence of the radiation intensity on time and on the coordinates of the interaction F = F ( f ,x, y, z), we find

Ni = no

\[

1 - exp( - a K

I

F K ( f ,x, y, z) dr

11

dx dy dz.

(11)

This formula includes all particular cases that we shall need in the following calculations. Far from the ion yield saturation, where ni << no and 1 - exp( -x) = x, we obtain the well-known powerlike dependence of the number of ions on radiation intensity:

Ni = noaK

JJ

F K ( f ,x, y . z) dr dx dy dz.

(12)

In experiments on multiphoton ionization, the results of measurements of the space-time intensity distribution F(t, x, y, z) are expressed in terms of effective quantities. For simplicity, let us introduce these quantities in the case of singlefrequency laser radiation. (In this regime the laser generates a single axial mode of the lowest transversal index.)

B . SINGLE-FREQUENCY LASERRADIATION In the case of single-frequency laser radiation, the time and coordinate dependencies in F(r, x, y, z) can be factorized:

F(f,x, y. Z ) = Fo+(f)$(x. y, z),

(13)

where + ( t ) is the pulse-intensity envelope, &a = 1; $(x, y, z) is the spatial intensity distribution created by the lens in the vicinity of the focus, = 1; and F, is the maximum intensity within the interaction volume. The normalization of +(r) and $(x, y, z) to unity and extraction of the maximum value Fo is justified by the nonlinear character of the ionization process, where it is the maximum value of intensity that gives the greatest contribution to the ionization probability. We then obtain N , = no /[l - exp(-(r,F5T,JIK(x, y, z))l dx dy dz,

(14)

and far from saturation we have

N i=

nOakFgTKVK= now(Fo)TKVK,

(15)

where TK = J +(, t ) dr is the effective radiation-pulse duration for the K-photon process and V, = J JIK(x,y. z) dx dy dz is the corresponding effective interaction

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

47

volume. Whereas quantities TK and V K can be measured by recording 4(t)and $(x, y , z), in order to find Fa one also has to know the value of the pulse energy Q: Fo = ( l / h w ) ( Q / S T ) where , T = J 4(t)dt is pulse duration and S = J +(x, y , zo) dr d y is the cross section of the beam in the focal waist. Thus, far from saturation we have

and the power of nonlinearity of the ionization process K can be measured as the slope of lg Ni as a function of Ig Fa or of Ig Q : Ig Ni

=

K Ig F,

+ C = K Ig Q + C’.

(17)

The ionization probability w(Fo) for the value of intensity Fo is calculated according to Eq. ( 1 5):

In order to calculate the ionization cross section when the measurements are made with the absolute method to be discussed shortly, one should use the relation

Ni Ni FgTKVK [Q/(hwST)]K T ~ V ~ ‘

( l l K = - -

(19)

As was already mentioned, Eqs. (17)-(19) can be used in the region without saturation. The boundary of this region can be estimated using one of the following relations:

Ni

S

noVK;

w(F0)TK 6 1;

aKFKTK S 1.

To measure multiphoton cross sections with the relative method, one needs calculation formulae for the intensity-dependence of the number of ions in the region of saturation of the ion signals, where (YKF~TK2 1. In the general case, the corresponding integral (14) cannot be expressed through V K ,and more detailed information about $(x, y, z) is required. According to the qualitative picture of the ion-yield saturation (Arutyunyan ef al., 1970; Cervenan and Isenor, 1975), by the end of the radiation pulse, the ion concentration reaches its maximum value no at those points of the interaction volume where (YKF$TK$~(x, y , z ) 3 1 , and the growth of the total number of ions with intensity is mainly due to the increase of the volume where the ionization is complete. Accordingly, in the saturation region, with increase of Fo the slope of the dependence of Ig Ni on lg Fo decreases from the value K it has far from saturation to

48

Ammosov, Delone, and Ivanov

the value 3/2 in the region of deep saturation (Arutyunyan et al., 1970) (Fig, 6). The value 3/2 describes the increase of the volume with complete ionization in the region of space where the geometric optics is valid and corresponds to conical geometry of the beam (focusing by a spherical lens). When the radiation is focused by a cylindrical lens, the asymptotic slope is 2 (Cervenan and Isenor, 1975). A typical experimental dependence of the ion yield on the laser radiation intensity is shown in Fig. 7. The numerical calculations of the dependence of the ion yield on radiation intensity at different K have been made only for the focused Gaussian beam (Boulassier, 1976). The distribution + ( x , y. z) for such a beam has the form

where A is the radiation wavelength and ro is the beam radius at the focal waist at the level of intensity lle. The radius tois related to the radius R the beam has at the entrance to the lens (also at the level lle) and to the distance L between the lens and the waist: ro = AL/2.rrR. For a Gaussian beam S = and

V , = (rr2riL/KR) X

-

1 . 3 , . . . a(2K - 5 ) 2 . 4 .. . : ( 2 K - 4 ) ’

-

(At K = 2 the factor 1 * 3 . . . (2K - 5)/2 * 4 . . . . (2K - 4)is replaced K - 7/4)) by unity, while at K 2 3 it is approximated by the expression (T( with an accuracy no worse than l%.)The approximation of the actual spatial distribution of intensity by a Gaussian distribution is justified for radiation of 7

Fic. 6. Qualitative behavior of the ion yield as a function of radiation intensity: 1, region without saturation; 2, saturation region; 3, region of deep saturation.

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

49

t'

t:

I

I

I

I I I I I I

2 3 4 6 Intensity, arb. u.

810

FIG. 7. Typical dependence of the ion yield for five-photon ionization of the Na atom (Arslanbekov et al., 1977).

lasers that operate in the regime of a single lowest transversal mode with aberrationless focusing. A spherical lens can focus without aberration only when the focusing angle is small enough. For a focused Gaussian beam, integral (14) can be reduced to the single integral

which can be evaluated according to the approximate formulae of Baravian and Sultan (1985). In the general case, where the spatial distribution of intensity +(x, y, z ) is arbitrary, one is able to describe only the very beginning of the saturation region (Delone et al., 1971). To this end, one expands the exponent in the integral (14) into a series:

Unfortunately, the partial sum of this series can describe the dependence N i ( F o ) only in a narrow range of Fo values. In the region where the series converges fast (aKF6TK<< I), its partial sums differ very little from the first term, while with increase of intensity they deviate sharply from the true dependence N i ( F o )

50

Ammosov, Delone, and tvanov

(Fig. 8). This is why this series is of little use for practical calculations of the ionization cross section from the experimental data. A somewhat better description to the beginning of saturation is given by the two-term approximation

where NT = noa,F~T,V, is the extrapolation of the nonsaturated dependence Ni(Fo)into the saturation region. Figure 8 clearly shows the advantage of this two-term approximation over the partial sums (22). A quantitative analysis of the region where Eq. (23) is applicable shows that it can be used in practical calculations within the region where the difference lg NT - lg N, does not exceed several tenths. Thus, the formula for calculating the cross section acquires the form =

(lg NT - lg N,)/(FgT,(V,/2VK)lg e ) .

(24)

In order to calculate the multiphoton cross section with the intersection point method to be discussed shortly, one has to know the value of aKFgTKat the point on the In Ni versus In Fa diagram where the straight line of the nonsaturated dependence (with the slope K ) extrapolated into the region of high Fointersects the straight line that describes deep saturation with the slope 3/2 extrapolated into the region of small Fo (Fig. 11). The calculated coordinates of the intersection point a , = C X , ( F X ) ~ Tfor , a focused Gaussian beam are presented in

FIG. 8. Calculated dependence of the ion yield on the intensity of a focused Gaussian beam for five-photon ionization. Solid lines: behavior of the partial sums of series (22). (Figures indicate the number of terms in the series.) Dashed line: two-term series (23).

51

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS TABLE I RESULTSOF CALCULATION OF THE INTERSECTIONPOINTCOORDINATES uK

K

aK

2

3

4

5

6

I

2.0

3.5

5.0

6.5

1.9

9.2

Table I. The ionization cross section is easily found from the intensity F 8 at the intersection point measured experimentally:

C. MULTIFREQUENCY LASERRADIATION In the case of multifrequency laser radiation (many axial modes of the lowest transversal index), the pulse envelope +(r ) does not describe the time-dependence of the intensity completely, since the instantaneous value fluctuates during the pulse. These fluctuations can be described by introducing a corresponding factor F(r): F(t, x, y, 2) = F(r)+(t)+(x, z). (26) y7

The normalization of +(r) and +(x, y, 2) remains the same. In order to measure the fluctuations F ( t ) one has to use a complicated apparatus with high resolution time (SlO-"s). This is why such measurements are not made in multiphoton experiments. The pulse envelope +(t ) is measured oscillographically with averaging of F(r). Owing to the fluctuation of intensity, in the case of multifrequency laser radiation, the ion yield changes from pulse to pulse even when the distributions +(r) and $(x, y. z) and the pulse energy Q all remain constant. It thus becomes necessary to average the ion signals. In some experiments one averages the ion signals themselves over a succession of radiation pulses:

but in most cases the averaging is done for logarithms of the signals:

= (In{

no

I[

1 - exp( - a K

FK(t)+"(t)dr $"(x,

y, z )

11 1) dr dy dz

.

52

Ammosov, Deione, and Ivanov

The averaging of logarithms of the ion signals arises from the need to analyze the experimental data on the dependence of the number of ions on intensity in a double logarithmic scale. Far from saturation (the corresponding criterion is presented shortly) one has

(N,) = n 0 4 F K ) T K V K (Ig N , ) = lg(noaKVK)+ (Ig

(28a) F K ( t ) V ( r )d t ) .

(28b)

In the case of multifrequency laser radiation, the mean values that enter the expressions for the ion yield far from saturation are known:

(lg

I

( F K )= gK(FP

FK(t)@W dt) = H ,

(294

+ lg((F)KTK)

(29b)

where (F) is the mean value of intensity at the maximum of its space-time distribution: (F) = Q/(hwST),and g, and H K are respectively the statistical and pseudostatistical factors of radiation; their values depend upon the number of modes generated by the laser and are tabulated by Masalov and Todirashku (1980). Thus, the factors g , and H , enter the expressions that relate the number of ions to the experimentally measured characteristicsof the multifrequency laser radiation:

(Ni) = n@KgK(F),T,VK (Ig N , ) = K Ig ( F ) + H, + lg(noaKTKVK).

(30a) (30b)

One can see that for multifrequency lasers, the boundary of the saturation region (Ni= noV, or w((F))TK= 1) is shifted toward smaller intensities: aKgK(F)KTK = 1. The relations for calculating the ionization probability w((F))= a,(F), and the multiphoton cross section acquire the form (cf. Eqs. (18) and (19)):

The linear dependence of (lg Ni) on lg(F) and on lg Q for multifrequency laser radiation does not change (cf. Fq. (17)):

(Ig Ni) = K lg ( F )

+

C I = K lg Q

+ Cl.

(33)

The analytical expressions for the dependence of the mean values (Ni) and (lg Ni) on the intensity of multifrequency radiation in the region of saturation are unknown. Nevertheless, knowing the statistical factors g, and H , , one is able

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

53

to make estimates sufficient for practical calculations. For instance, by averaging ( 2 3 ) in the beginning of the saturation region we obtain (lg Ni) = Ig NT - a,g.(F)KT,(Vx/2VK) Ig e,

(34)

where Ig NT = H , + I ~ ( c x , ( F ) ~ T , Vand ~ ) , the relation for calculating the multiphoton cross section within the two-term approximation acquires the form (cf. Es.(24))

In order to estimate the dependence of (Ig N , ) on Ig(F) within the whole saturation region (including the asymptotics of high intensities), let us first turn to the case of a large number of generated laser modes (the corresponding criterion will be presented shortly), where the quantity a, J FK(r)$JK(t) dr that enters the expression for (lg N , ) practically does not fluctuate from pulse to pulse and equals its mean value a,g,(F)V'. or aKIOH+')KTK.(In this limit the values ofg, and 10"~are apparently the same.) Under such conditions, there is no need for averaging, and the dependence of (Ig N,) on lg(F) coincides with the dependence of lg N, on Ig F , for single-frequency radiation with the only difference that they are shifted one against the other along the intensity scale by the amount equal to H,/K (Fig. 9). Let us label the dependence of (Ig N , ) on lg(F) in the limit of a large number of modes with the index m. It is clear that for analyzing (Ig one can use all the relations presented previously for single-frequency laser radiation if only one replaces Ig Fo by lg(F) H , / K . If the number of modes generated by the laser is not large enough, the value of a, J FK(r)$JK(t) dr fluctuates from pulse to pulse, and the dependence of

+

FIG. 9. Qualitative behavior of the ion yield as a function of the laser radiation intensity for single-frequency (curve 1 ) and multifrequency (curve 2, dependence (Ig NJ=;curve 3, dependence (lg N,)) lasers.

54

Ammosov, Delone, and lvanov

(Ig Ni)on lg(F) is not only shifted with respect to the case of single-frequency radiation but also changes its form. Let us estimate the distortions in the form of the dependence of (lg Ni) on lg(F) in the first nonvanishing order in the fluctuation of s = Ig(a, J F K ( z ) + K ( fdz). ) To this end, let us use the expansion of Ig N , in powers of s about the value (s) = H, + Ig(aK(F)KTK) and take into account that [dlds * .IS+) = (l/K)d/d(lg(F)) * After averaging, we get

-

a.

(lg Ni) = (Ig Ni),

1 d2(lg Ni),

+((s 2K2 d(lg(F))*

-

MY).

The fluctuations of s are tabulated in Masalov and Todirashku (1980) and are denoted there as c?i = ((s - (s))~);5% becomes smaller with increase of the number of modes. The quantity (d(lg Ni)J/[d(1g(F))] = /3 is the slope of the tangent to the dependence of (lg Ni), on Ig(F); /3 changes with increase of (F) from the value K (its value far from saturation) to 3/2 (in the region of deep saturation). Thus, (lg Ni) = (lg Ni),

6; dp +2K2 d(lg(F))’ ~

(37)

One can see that far from saturation and in the region of deep saturation, where /3 is practically independent of (F), there holds the relation (lg N,) = (lg A’,)=.

In particular, this means that the coordinate of the intersection point is independent of the number of generated laser modes, and, consequently, when the multiphoton cross section is measured according to the intersection-point method and one uses the radiation of a multifrequency laser, there holds the relation (cf. Eq. (25))

In the general case, the correction in Eq. (37) is nonzero and has a negative value, reaching a maximum (in its absolute value) at the “center” of the saturation region. According to rough estimates,

and the maximum deviation of (Ig Ni) from (Ig N J , does not exceed 0.2 X 6;. According to the data on r?; presented by Masalov and Todirashku (1980), these deviations are small compared to the errors with which one measures the ion signals in real experiments (GNiINi= 0.1) within a wide range of mode numbers (>lo) at K = 2 to 6. For this reason, for practical measurements one can neglect the difference between (lg Ni) and (lg Ni), .

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

55

And so, for the rest of the chapter, as main relations for the measured quantities we shall use relations (17)-(19), (24), (25), (31), (32)-(3% and (38).

D. MEASUREMENT OF THE POWER OF NONLINEARITY OF THE

IONIZATION PROCESS

The measurements of the power of nonlinearity K play an important role in the study of multiphoton atom ionization. The agreement between the experimentally observed value of K and its theoretical value {I/hw 1) serves as a basis for identifying the ionization process, indicates that this process is direct, and thereby makes it possible to measure other characteristics of the ionization process. Only that region of intensity values where the observed power of nonlinearity is an integer and agrees with the calculated value can be used for measuring the ionization probability and multiphoton cross section with the absolute method. When one does not know the value of the multiphoton cross section, the power of nonlinearity K is measured by observing the growth of the ion yield with radiation intensity. It is convenient to analyze the dependence of the ion yield on radiation intensity by using a plot with a double logarithmic scale, where this dependence has the form of a straight line with the slope K:

+

For each laser pulse we have a corresponding point on this diagram. By looking at the set of experimental points obtained for different values of intensity one is able to select the range of intensity and ion-signal values where the dependence In Ni on In Fo is a straight line, and find the value of K from the slope of this dependence. The broader the range of the values of radiation intensity and ion signals within which the straight-line dependence is observed, the more accurately K can be measured. The value of K is obtained by approximating the data in this region by a straight line. This is a necessary procedure since the experimental points cluster around a straight line with a certain dispersion. The latter depends on the apparatus error with which one measures Ni and F,,, on the specific procedure used to measure the intensity, and on the mode composition of the laser radiation used in the measurements. The resulting error with which the value of K is measured is determined by the dispersion of experimental points and by the range of intensity values within which the measurements are made and the number of points therein. An evident requirement to the error with which one measures K is the condition 6 K S 0.5. In this case there can be no mistake about what integer number the measured value corresponds to. However, more accurate measurements are also important since they indicate that the direct pro-

56

Ammosov, Delone, and Ivanov

cess of ionization takes place. A favorable circumstance for making high-accuracy measurements is the fact that the measured value of K is independent of the absolutization of the ion-signal and radiation-intensity values. The value of the nonlinearity power K can be found from the experimental values x, = lg F, and y, = lg(Ni), (where index m = 1, 2 , . . . , M labels the pairs of measured values of Ni and F , ) by using the least-square method. The specific formulae used to estimate K within this method may be different and depend on which one of the two errors (Sx or 6 y ) determines the spread of experimental points. If the latter is mainly due to the fluctuations of intensity (K SX >> Sy, the case where single-frequency laser radiation is used, to be discussed shortly), the value of K is estimated by minimizing the sum Ern[x, - ( y , - b)/K]' (by regression of x to y). Then, one has

If however the ion-signal fluctuations are dominating (Sy >> KSx), K is estimated by minimizing the sum Ern( y , - Kx, - b)* (by regression of y to x), and

T K =

XmYm

c

-

(T

xi -

x m ) ( C

(C

Yrn)/M -

=

Kmin.

(41)

Xrn)'lM

Though the values K,, and Kminare close to each other, they do not coincide. The difference between K,,, and Km,"should be taken into account when the error with which K is found is less then K,, - K,. (to be discussed shortly). The error with which K is estimated within the least-square method depends upon the scatter of experimental points about a straight line, and is calculated by using the correlation coefficient of experimental data:

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

57

The value of R characterizes the quality of the experimental data obtained. The smaller the spread of points, the closer R is to unity and the smaller the error of K estimation is:

Moreover, K,, - K d n = K m ( l - R 2 ) . As one can see, at a small number of experimental points, M - 2 < 1 / ( 1 - R 2 ) . the values K,,, and Kmincoincide with each other within the error 6 K . However, in a situation more likely for experiments on multiphoton ionization of atoms by laser radiation, where M - 2 > 1/(1 - R 2 ) , the difference between K,,, and Kminexceeds the error SK, and one has to select a correct expression for K . The correlation coefficient can be predicted basing the data on relative errors with which the radiation intensity and ion signals are measured: 6xllg e = 6F/F and 6y/lg e = 6Ni/Ni,respectively: 6x2

6Y2

t

In the case where the experimental points are evenly distributed throughout the range Ax = x,,, - xmin and Ay = y,,, - ymin, in Eq. (44) one can replace the sums by their approximate values and obtain 1 - R 2 = 12

I (E)’ (z)’I +

L

-1

=

(AY)2

[K26x2+ 6 ~ 2 1 ,

(45)

where we have taken into account that Ax = AylK. Equation (45) enables one not only to make prognoses about the error in the measurement of K in a future experiment, but also to check how realistic are the estimates for 6x and Sy after the measurements have been done. Furthermore, Eq. (45) explicitly contains the contributions of 6x and Sy to the resulting scatter of experimental points. Let us now turn to the estimation of the error in the measured value of K with Eqs. (43) and (45). As was already mentioned, the result of data approximation according to Eqs. (40) and (41) is independent of the units one uses to measure the intensity and the ion signals, i.e., it does not depend upon the error in absolutization of F o and Ni (as well as of n o ) . Thus, Sx and 6y in Eq. (44) are determined only by the errors in Fo and Ni in relative units. The latter include the random errors of the measuring apparatus and the errors that are due to fluctuations of the measured quantities. For 6x we have (Sx/lg e)’ =

($)2

+

(?)*+ if)’+ (F)‘,

(46a)

Ammosov, Delone, and lvanov

58

whereas in 6y2, in addition to the ion-signal-measurement error given by Eq. (21), one should also take into account the errors in n o , V, , and TK: (6y/lg e)2 =

(F)2+ (2)'+ (2)2 + (?)*.

(4%)

If the measurement procedure is such that for each laser pulse one measures the radiation energy Q and the parameters of the space-time distribution of S and T, and, on their basis, calculates Fo (the dependence N,(F,,)), 6 x will contain only the apparatus errors of Q,S, and T. For such a measurement procedure one can expect 6xAg e "- 0.05 to 0.1. If, however, in the process of measurement one controls the radiation energy only (the dependence N,(Q)), 6 x will contain the errors 6S/S and STIT due to technical fluctuations of these quantities. In this case, one can expect Sxllg e = 0.1 to 0.2. The error 6y essentially depends upon what sort of a laser (single-frequency or multifrequency one) is used in the measurements. In the case of a singlefrequency laser, Sy is mainly due to the apparatus error 6N,IN,, to which the technical fluctations 6noln0, 6vK/vK, and 6T,/T, are added. Under such conditions, one can expect 6yllg e = 0.1. For a multifrequency laser 6N,lN, also contains fluctuations associated with nonreproducibility of the fine temporal structure of radiation. Depending upon the number of generated modes N and on the power of nonlinearity of the ionization process, the multifrequency fluctuations 6N,lN, fall within the range of 0.01 (N = lo3, K = 2) to 2 (N = 10, K = 10) (Masalov and Todirashku, 1980) and in realistic experimental situations are dominating. In the case of multifrequency lasers, one should expect 6yllg e = 0.1 to 2. Both the range of variation of the ion signals, Ay = l g ( N y / N y ) , and the associated range of intensities, Ax -- AylK, acceptable for measuring the power of nonlinearity are limited. From below they are limited by the sensitivity of the ion detector; when making estimates we shall suppose that the sensitivity is maximum and N y = 1. From the preceding, the range in question is limited by saturation of the ionization process: N y = noVK= noh31K04( 0 is the angle of radiation focusing into the interaction volume.) For an atom beam no cm and 8 = 1120, we get N Y = 1 O l o ~ r n - ~and , taking h = 0.5 X = lo3 and Ay = In lo3 = 3. When the ionization of atoms occurs in a gas, the atom concentration can be much higher, and the range of measurements can be made wider. In this case, however, one encounters difficulties that are due to the fact that the apparatus that records the ion signals has a limited dynamic range. For this reason, in our further estimations we shall take N-IN= lo3. Let us now present the three resulting estimates:

(1) When one studies the dependence N,(Fo) with a single-freqency laser and 6xllg e = 0.05 to 0.1, 6yl lg e = 0.1, K26xZ> 6y2, and K = K,,,, one has

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

I - R 2 K 2 (0.0005 + 0.002) 6K -- K2 (0.02 t 0.04)lFM.

59 (474 (47b)

The accuracy 6K < 0.5 can be reached with several tens of laser pulses up to K = 10. (2) When one studies the dependence Ni(Q)with a single-frequency laser and 6xllg e = 0.1 to 0.2, 6yllg e -0.1, K 2 6 x 2>> 6y2, and K = K,,, ,one has 1 - RZ = Kz (0.002 + 0.008) 6 K = Kz (0.04 f 0.09).

(47c) (474

Here, for a “reasonable” number of laser pulses, M S 100, the accuracy SK 6 0.5 is guaranteed to be reached within the range up to K = 10. (3) When one uses the radiation of a multifrequency laser, the values of K26x2 and 6y2 can be comparable to each other, and the estimates for 1 - R 2 and 6K strongly depend upon the value of K and the number of modes N. For instance, at K = 5 and N = lo’, and Syllg e = 0.23 < KGxIlg e 0.5, we have

-

I - R 2 0.05 6 K = l.l/-, which ensures the necessary accuracy after several tens of laser pulses. On the other hand, at K = 7, N = 100, Sy/lg e = 0.9, and KGxllg e = 0.7 we have 1 -

R 2 0.26 21

6K = 3.5/<M, which ensures the necessary accuracy at M = 100. We note that the measurement results for K obtained with multifrequency lasers are independent of the magnitude of the statistical factor of radiation. One should note two circumstances that are common to both types of measurements just considered, but are more important for those made with multifrequency lasers. First of all, in cases where the ion-yield fluctuations are high (6Ni/Ni = I), the correct measurement procedure is to average the logarithms of the ion signals rather than the signals themselves. The analysis shows that in such cases the ionsignal distribution is far from normal. For this reason, estimates of the average value are quite sensitive to the loss of signals that happened to be outside the dynamic range of the recording apparatus, and these estimates converge to the true average with increase of the number of measurements slower than l / e . On the contrary, distribution of the ion-signal logarithms is closer to normal, and estimates of the average values are only weakly sensitive to the loss of extreme values and converge to the true value in the “correct” manner. The second circumstance concerns the estimates of the power of nonlinearity based on the experimental data in those cases where K 2 W is comparable to

60

Ammosov, Delone, and lvanov

6y2 (as it is in the case we just considered, where K = 7 and N = 100, so K6x/lg e = 0.7 while 6y/lg e = 0.9). In this case neither Eq. (40) for K,,, nor Eq. (41) for Kmingive correct (nondisplaced) estimates for K. Moreover, K,, - Kminmay be >1 (in our example, K,, - Kmin= 1.8), and then the possibility of reaching high accuracy in K measurements is not used. The account of the comparable contributions of errors 6x and 6y to the spread of experimental points leads to the following formula for K:

This formula is valid if Eq. (44)holds. Without the analysis of the contributions of 6 x and 6y to 1 - R2it is impossible to specify the value of K more precisely inside the interval between Kminand K,,, . To summarize the preceding considerations concerning the measurement of the power of nonlinearity of the ionization proces, we can assert that the existing pulse lasers enable one to make measurements within the range K = 2 to 10 with an error no higher than 6 K G 0.5 for a moderate number of laser pulses, M = 30 to 100. The best accuracy can be reached with single-frequency lasers, when the intensity of radiation is measured (in relative units) for each pulse. Such measurements require rather complicated apparatus, so the intensitymeasurement procedure is justified in those cases where it is necessary to measure K with very high accuracy, or where the high-radiation-energy and lowradiation-energy regimes of the laser are noticeably different (Lompre er al., 1982). On the other hand, if one does not measure the radiation intensity for each laser pulse, the experiment becomes much simpler, while the loss in accuracy is insignificant. By comparing the measurement results obtained with single-frequency and multifrequency lasers, one can see that the former provide higher accuracy. For single-frequency laser radiation, the measurement error for K is mainly determined by the relative error with which one measures the radiation intensity and by the range of observed ion signals. In this case the error 6K is proportional to K2 and at small K (G5) may reach one or two tenths. When one uses multifrequency laser radiation, the fluctuations of the ion yield become greater. This, in its turn, increases the measurement error and makes the data-processing procedure more complicated: it becomes necessary to estimate the contributions of the errors in intensity and in the ion yield to the spread of experimental points, and to select a formula for the nondisplaced estimate of K. Let us note that by increasing the number of measurements M (the number of laser pulses) one cannot increase the accuracy of measurements to any noticeable extent, since owing to

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

61

the square-root-like dependence, the variation of M has little effect on 6 K . The most efficient way to increase the accuracy of measurements made with multifrequency lasers is to increase the number of generated modes: with increase of the number of modes the multifrequency fluctations of ion signals become smaller. And though multifrequency lasers, when used for studying the processes of multiphoton atom ionization, give a worse resulting accuracy than do the singlefrequency lasers, their application is justified by the simplicity of the laserresonator construction and, consequently, by reliability of operation, and, also, by the stability of radiation energy and of the space-time distribution parameters in a series of pulses.

E. MEASUREMENT OF THE IONIZATION PROBABILITY The ionization probability per unit time, w = aKFK, characterizes the rate of K-photon ionization of an atom in a field with fixed intensity. The value of the ionization probability is used in the calculation of ion-concentration dynamics n,(f):& ( f ) = (no - n,)w. Under variation of radiation intensity the ionization probability changes as w' = w(F'/F)K.

(49)

The observation of ion signals in the process of multiphoton atom ionization by pulsed laser radiation enables us to measure the ionization probability. The value obtained will correspond to one of the intensity values that have been realized within the interaction volume during the laser pulse. Since the greatest contribution to the ion yield comes from the intensity at the maximum of the space-time distribution, the experimental measurements are usually referred to that value of intensity. The measurements are based upon the following relation between the integral (over the pulse) ion signal Ni and the characteristics of radiation: Ni = aKFi$oTKVK= nOw(F,)TKVK. For the ionization probability per unit time this gives

If the measurements are made with multifrequency lasers, it is necessary to take into account the statistical factor of radiation, which makes allowance for the time fluctuations of intensity within a pulse:

Ammosov, Delone, and Ivanov

62

In practice one should use the latter formula, since this is what corresponds to the experimental situation, where with observation of ionization in a series of laser pulses the logarithms of the ion signals are averaged. Equation (51) is presented to give a complete picture, and it can only be used in those cases where the number of modes generated by the multifrequency laser is sufficiently large and g, and lWKcoincide within the ion-yield measurement error. It should be emphasized that Eqs. (50)-(52) are correct only far away from saturation, where the ion yield is proportional to the Kth power of radiation intensity. This means that the measurements of the ionization probability should be preceded by the measurements of the ion yield versus radiation intensity dependence. Any point on the straight-line dependence of In Ni on In Fo with the slope K is good for measuring the ionization probability w. It is then evident that in the case of multifrequency lasers the averaging procedure gives (lg Ni), so one should use Eq. (52) and not (51). As was already mentioned, the ionization probability w calculated according to one of the Eqs. (50)-(52) corresponds to the value of intensity at the maximum of the space-time distribution: Fo = Q/(huST). The error with which one can measure w with the method described previously is determined not only by the random error of all the quantities that enter Eqs. (50)-(52) but also by the errors of absolutization of the neutral atom concentration no and ion yield N i values: (!?)2

=

+

(%)2

+

($)2

(T)’ + p$;bs + (?)*

(!3)2 +

(53)

abr

In the case of multifrequency lasers, where the fluctuations 6 N i / N , ,due to the nonreproducibility of the fine temporal structure of pulses, are large, the contribution of multifrequency fluctuations of the ion yield to the resulting error can be suppressed compared to (6NilNi), by averaging over one or two tens of laser pulses. One should keep in mind that the preceding estimate of the error in w expressed through the errors of no, Ni , T, , and V, absolutization is not a complete characteristic of the measured value of w:there is also a “hidden” error due to the error in absolutization of intensity values for which w was measured. This hidden error becomes apparent when it becomes necessary to find the value w ’ of the ionization probability at another value of intensity F’ basing on the measured value w :w ’ = w(FA/F0),. The error in w’ obtained this way now contains the error in intensity 6 F o . Thus, in order to give a complete characteristic to the result of measuring the probability of multiphoton atom ionization, one should know not only w and 6w but also Fo and SFo .

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

63

Owing to the dependence on radiation intensity, the ionization probability w is not a universal characteristic of the process of multiphoton ionization. Such a . this reason, we do universal characteristic is the K-photon cross section a K For not consider here the methods of measuring the multiphoton ionization probability in more detail. In conclusion, let us mention one important case where it is necessary to know the ionization probability: the case where one compares the processes that have different powers of nonlinearity (i.e., that result from absorption of different numbers of photons). Let us recall that, since the dimension of multiphoton cross sections depends upon the power of nonlinearity of the process, one cannot compare these cross sections for those processes that have different Ks. In this case one should compare the probability of the process measured at fixed radiation intensity.

F. MEASUREMENT OF MULTIPHOTON CROSS SECTIONS OF DIRECT ATOMIONIZATION The cross section of K-photon ionization of an atom is the proportionality coefficient between the ionization probability per unit time and the Kth power of radiation intensity: LYK

= W/FK.

(54)

The magnitude of aKdepends upon the frequency of laser radiation and upon its polarization, but does not depend upon the radiation intensity. Measuring these cross sections is one of the most important aims of experimental studies of the process of multiphoton atom ionization in a light field. By comparing the experimental values of aKwith its theoretical values, one is able to check the correctness of the used theoretical models of an atom and of its interaction with radiation. When describing the methods used to measure the ionization cross sections, we shall pay principal attention to the resulting accuracy of these measurements. Let us mention in advance that the modern methods enable one to measure aK with an accuracy of the order of aK = loA'", where 6A = 1. This accuracy may prove to be insufficient to enable one to establish a difference between the rough estimates of aKand the results of detailed calculations. This is why it is important to improve the accuracy of the experimental measurements of aK. There are two substantially different methods of measuring the atom ionization cross sections: the absolute and the relative. They differ in the set of quantities measured in the experiment and, consequently, in the collection of the measuring apparatus, in the methods of data processing, and, in the end, in the resulting

64

Ammosov, Delone, and Ivanov

accuracy of measuring a K .For instance, among other quantities, within the absolute method one has to measure the ion yield Ni and the concentration of neutral atoms no in absolute units, whereas within the relative method it is sufficient to measure Ni in arbitrary relative units, while no does not have to be measured at all. (This is what gives these methods their names.) For this reason, as concerns the accuracy of measurements, the potentialities of the relative method are greater.

G . THEABSOLUTE METHODOF MEASURING MULTIPHOTON CROSS SECTIONS The absolute method of measuring the cross sections of K-photon atom ionization is based upon the interrelation between the ion yield and the radiation intensity far from saturation: CW, =

Ni/n,FgTKVK.

(55)

Equation (55) can be used to measure the ionization cross sections with radiation of single-frequency lasers. When one uses multifrequency lasers, in the formula for the cross section one should take into account the statistical properties of radiation: (YK

= (N,)/(no&'#o)KTKVK)

or

(56)

CXK

= 10'" "l'-'~/(no(Fo)'TKvK).

(57)

Since the absolute method of measuring the cross sections is applicable when the ion yield observed is far from saturation, the measurements of the cross section according to the absolute method should be preceded by the measurements of the ion yield versus radiation intensity dependence. Any point of this dependence from the range where N, F t is fitted for calculating the cross section according to Eqs. (55)-(57). Since the experimental data on the dependence of N, on Fo are analyzed on a diagram with a double logarithmic scale, in the case of measurements made with multifrequency lasers one should calculate aKaccording to Eq.(57). The error in measuring aKwith the absolute method is mainly due to the errors of absolutization of intensity, of the number of ions, and of the neutral atom concentration:

-

In those cases where multifrequency lasers are used, in this relation one should also take into account the error in the value of the statistical factor. The quantity

65

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

66 takes into account the error that is due to technical and random apparatus fluctuations 6Fo and 6 N , : (6Mg e)2 = (6Fo/Fo)2+ (6Ni/Ni)2.These fluctuations manifest themselves in the spread of experimental points on the dependence of lg N ion lg Fa. For this reason, the error 66 can be directly estimated from the spread of experimental points x, = (Ig F,,), and y, = (lg Ni), (rn = 1,2, . . . , M))about the line y = Kx + 6:

The error 66 can be made substantially smaller if a, is calculated, not from a single experimental point from the region of the nonsaturated Ig Ni versus lg Fa dependence, but by using all the experimental values of x, and y, corresponding to the straight-line part of the dependence of Ig N , on lg Fo. In this case, for single-frequency laser radiation we have A

5

Ig

=

[6 Ig e

-

lg(noT~V~)],

(60)

- H,.

(61)

while for multifrequency radiation A = 6 lg e - Ig(n,T,V,)

The value of b is obtained here by averaging over all the experimental values: 6 = (l/M)(Zm y, - K Emx,), and then the error IS^)^ in (59) becomes M = ( I / M ( M - I)) X Z, (y, - Kx, - 6)2. times smaller and amounts to The described procedure of experimental data processing makes the contribution of 66 to 6 A practically zero. Then, the maximum accuracy with which one can measure the multiphoton ionization cross section with the absolute method is limited by the errors in absolutization of radiation intensity, of the number of ions, and of the neutral atom concentration, and the error in intensity is multiplied by the factor K. Taking (6FolFo),, = (IjNi/Ni)abs = (6q,/nO)& = 0.3, we obtain the following estimate for the maximum accuracy of measuring the cross sections: 6A = 0.4343

X

0 . 3 q m =0.13V‘m.

(62)

According to this estimate, the accuracy with which one can measure aK with the absolute method can be within one decimal order up to K = 6 or 7. However, the practical results prove to be somewhat worse. (See Section IV.)

H. THERELATIVE METHODOF MEASURING MULTIPHOTON CROSS SECTIONS The relative method of measuring the cross sections of K-photon ionization of atoms makes use of the dependence of the ion yield upon radiation intensity in the region of saturation, where the product aKFtTK is close to unity, and the ion yield becomes comparable to the number of atoms in the effective volume:

66

Ammosov, Delone, and Ivanov

Ni = n,V, . The fact that in the saturation region the quantity (YKF~TK has definite values is used in the relative method and allows one not to measure the neutral atom concentration, while the values of the ion yield can be measured in arbitrary relative units. This makes the experimental measurements much simpler, and the accuracy with which one can measure the cross sections in this case is limited mainly by the error in absolutization of the radiation intensity value. This is why, as of 1990, the multiphoton cross sections are measured, as a rule, with the relative method. The relative method was introduced into practice by Delone et af. (197 1) and has at least three modifications that differ in the way the experimental data are processed.

1. The Two-Term Series Method In this variant of the relative method one analyzes the experimental data on the lg Ni versus Ig Fodependence in the region of low saturation, where the deviation of experimental points from a straight-line dependence becomes noticeable. In this region, the approximate analytical dependence of lg N, on lg Fo has the simple form of a two-term series Ig Ni = lg N'f - (Y~F:T,(V~~/~VK)I~ e,

(63)

which makes the experimental data processing much simpler. (Here Ig Nf is the straight line extrapolated from the nonsaturated region into the region of saturation. See Fig. 10.) The two-term series method can be used to analyze the experimental data on the lg N , versus lg Fo dependence both in the nonsaturated region and in the region where the saturation begins. The experimental points from the nonsaturated region are approximated by a straight line that is then extrapolated into the 1 h Ni

,Wi"

CROSS SECTIONS OF DIRECT MULTlPHOTON IONIZATION OF ATOMS

67

region of high intensities in order to find the value of lg N T . After that, one selects the experimental point with the smallest possible intensity whose deviation from lg NT is within two or three ion-signal measurement errors. From the deviation of this point from the nonsaturated dependence A = lg NT - lg Ni one calculates the value of the cross section:

With experimental points that deviate from Ig NT by less than two or three errors in 6 N , / N i one gets lower accuracy, whereas the use of points with greater deviation leads to a systematic error in the value of the cross section since such points lie outside the region where the two-term series is valid. When one uses the radiation of a multifrequency laser, the cross section is calculated according to the formula

In this case the statistical factor that corresponds to the ion-signal logarithm averaging procedure is g, and not HK . The error with which one can measure the multiphoton cross section with a two-term series method is mainly determined by the error in absolutization of radiation intensity values:

The term (lg e/A)(6iVi/Ni)is the relative error with which the quantity A = lg NT - Ig N , is measured, in which only the spread SN, is taken into account, while the spread 6NT is considered to be insignificant since the values of lg NT are found from a large number of experimental points. The quantity ) be regarded as the error with which the experimental points (lg e / A ) ( G N i / N i can are approximated by the theoretical curve. According to the approximation recipe proposed, it amounts to (lg e/A)(6Ni/Ni) 1/(2 + 3). One can see that the contribution of this term is not small. The low accuracy of data approximation by a two-term series is one of the weak sides of this method. Its advantages are the following: 2-

(1) It is enough to have a minimum amount of data for the beginning of saturation. (2) Data processing is simple.

68

Ammosov, Delone, and Ivanov

(3) It can be used in the case of an arbitrary spatial distribution of radiation intensity. (4) It can be used in the case where the measurements are made with radiation of multifrequency lasers.

Thus, in order to find the values of multiphoton cross sections with a two-term series one has to measure the following quantities: N , (in relative units), F, TK , and V,/VK. In the case of multifrequency lasers, one should also measure g K . The range of intensity values where the ionization process is to be observed should cover the nonsaturated region and the beginning of the saturation region. The accuracy with which the cross sections are measured is limited by the error of radiation intensity absolutization increased K times, and by the error in approximation of (lg e/A)(GNilNi). 2 . The Intersection-Point Method This method makes use of the experimental data on the ion yield in the region of deep saturation, where the In Ni versus In Fo dependence is a straight line with the slope 3/2. The straight line with slope Kin the nonsaturated region intersects the straight line with slope 3/2 in the region of deep saturation at the point has the value aKthat depends on K and on the shape where the product CUKF~TK of the spatial distribution of radiation intensity at the focus of the lens (Fig. 11). Having found the intersection point and the value of intensity F8 at this point from the experimental data on the ion yield lg N i as a function of lg Fo , one can easily calculate the cross section according to the formula

'

K-

1g Ni

~.-~3/2

Ni-F,K

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

69

If the radiation of a multifrequency laser is used, one should calculate the cross section from the relation

The intersection-pointmethod can be used only in the case where the experimental data on the lg N i versus lg F,,dependence exhibit the slope K in the region without saturation and the slope 3/2 in the deep-saturation region. For this reason, before using this method, one should measure the degrees of nonlinearity of the ionization process both in the region of small intensities and in the region of deep saturation. If from these experimental data one can find intensity ranges where the required slopes are observed, the intersection point x* = Ig F$ can be found by approximating the data within these ranges by straight lines according to the least-square method: x* = b2 - b ,

K - (3/2)'

where b, is the parameter of the line y = Kx + b, far from saturation, b, = ( l / M I )X, ( y , - Kx,), M I is the number of corresponding experimental points, b2 is the parameter of the line y = 3x12 + b, in the region of deep saturation, and b2 = ( 1 / M 2 )C, ( y ; - 3xA/2) for M 2 experimental points. Within this approximation, it is not necessary to use the experimental values of the slopes. It is enough to take the correct integer K and 312. The error with which the intersection point is found can be expressed through errors Sb, and 6b2:

We have presented here the formulae for those errors Sb, and Sb, that come from concrete experimental data. When making prognoses about the error with which ionization cross sections can be measured in an experiment that is being planned, the errors 6b, and Sb, can be estimated by using the following expressions: Sb: = (Syg Sb: = (Sy:,,

+ K2Sx2)/M1,

+ (3/2)'6x2)/M2,

(70d) (70e)

70

Ammosov, Delone, and Ivanov

where Sxllg e = SFo/Fois the expected measurement error for Fo (due to technical and random apparatus fluctuations) in relative units, 6yKllg e = SN,/Ni is the expected ion-signal measurement error in the region without saturation (also in relative units), and 6yW,/lge = 6 N i / N ,is the same error in the region of deep saturation. When the experimental data are such that in each range one has several tens of experimental points, one can expect that the measurement errors 6b, and Sb2 will be at the level of several hundredths, which gives a measurement error for intensity at the intersection point SF%/F8 smaller than the error of Fo absolutization. The resulting accuracy with which the multiphoton cross section can be measured with the intersection-point method amounts to

Thus, KSF$/F%plays the role of the error with which the experimental points are approximated by the theoretical curve and, together with the error of intensity absolutization, limits the accuracy of the cross section measurements. A shortcoming of the intersection-point method is the fact that the values of uKhave been calculated only for the light beam with a Gaussian profile and one does not know the sensitivity of the values of aK to the variation of the spatial distribution of radiation. The advantages of this method are: (1) the relative simplicity of experimental data processing and (2) its applicability in the case where the measurements are made with radiation of multifrequency lasers. Thus, in order to measure the multiphoton cross sections with the intersectionpoint method one should have a laser with a Gaussian spatial distribution and the recording apparatus should be able to measure N , (in relative units), F, , and TK (and also HKin the case of multifrequency lasers). The range of radiation intensities where the ionization process is to be observed should cover both the region without saturation and the deep-saturation region. The accuracy with which the cross section can be measured is limited by the error of radiation intensity absolutization and by the error in intensity at the intersection point increased K times.

3. The Method of Complete Comparison This method is based on comparing the experimental data on the ion yield in the region of saturation with the dependence N , ( F , ) calculated according to Eq. (14) for a concrete spatial distribution of laser radiation intensity. Within this method no preference is given to the data from one or another saturation region. Instead,

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

71

one makes the comparison by using the whole set of experimental points. The method of complete comparison has been introduced into practice by Cervenan and Isenor (1974) and Cervenan et al. (1975). These papers have stimulated the numerical calculations of the Ni( F O )dependence for radiation with a Gaussian intensity profile in the focusing region (Boulassier, 1976). The ideological side of comparison of the calculation results with experimental data and of finding the value of the cross section presents no problem. Since in the calculations, the quantity CYKF~TK is used as an independent variable, a comparison with the experimental data enables one to find the value of intensity Fc for which CYKFETK= 1 . (We have taken unity only to be definite.) The cross section of the ionization process is calculated in this case according to the formula

It is difficult to use the method of complete comparison when one uses the radiation of multifrequency lasers, since the theoretical dependencies N i ( ( F ) ) with averaging over the multifrequency fluctuations are not known. Only when the number of modes generated by the laser is sufficiently high, so that the multifrequency fluctuations are suppressed, the dependence N i ( ( F ) )is the same as it is for single-frequency radiation if only one replaces F t by 1 0 H ~ ( F ) K(In . this limit gK and 10". coincide.) Here the method of complete comparison can be used, and the cross section of the ionization process is calculated according to the formula

It is convenient to compare the experimental data with the calculated dependence on a diagram with a double logarithmic scale: x = lg F0 for the experimental data, x' = (l/K)lg(a,F{Z',) for the calculation data. Fc is found as the value of intensity that coincides with the value x' = 0 given by the theoretical dependence at optimal fitting of data. When the theoretical curve y = f ( x ) is fitted to the experimental points by using the least-square method, one encounters the same problems as when the power of nonlinearity of the ionization process is measured. One should decide which of the two deviations (in y or in x ) should be minimized by comparing the concrete errors 6 x and 6y. When the radiation of single-frequency lasers is used, and the ion-yield fluctuations are relatively small, one should expect that the error Sx, due to the technical and random apparatus fluctuations of radiation intensity, will be dominating: K26x2 > 6yz in the region without saturation and (3/2)26x2> in the deep-saturation region. In this case, in order to find Fc within the least-square method, one should minimize the sum

Ammosov, Delone, and Ivanov

72

Z, (x, - f- ( Y,))~,where f-,( y,) is the dependence inverse to f ( x ) . The error with which the scales along the x axis are compared is given by the relation 6x6 =

(2

lg e ) , = 6 x 2 ( z

+

F)I(K

- (3/2))2.

(74)

Here, when estimating ax,, the experimental points have been divided into two groups: M I is the number of points with x, < xc and M2 is the number of points with x, > x,. It is assumed that for the first group of points the slope of the theoretical curve is close to K, while for the second group of points it is close to 3/2. One can see that at a given number of laser pulses, the best accuracy is reached when the numbers of points in the first and second groups are proportional to the corresponding slopes: M I:M , = K : 312. In this case the measurement error for intensity Fc falls off with increase of the number of pulses as U C M : (6F,/F,) = ( 6 x l f l ) ( K 3/2)/(K - 3/2)lg e. At M I + M 2 = 100, one can expect that the comparison error will be noticeably less than the error of intensity absolutization. If, in the case where the error 6x is dominating, one finds Fc by minimizing the sum Z, (y, - f(x,)),, the resulting value xc will be smaller by the amount =26x2/Ax. (Ax is the range of intensity variation in decimal orders.) Though smaller than 6x, this displacement may nonetheless be greater than the comparison error axc and thus is undesirable. The opposite case, where the dominating error in the spread of experimental data is 6y, will not be considered here. Such a case could be realized if one uses the radiation of a multifrequency laser with a small number of generated modes, when the multifrequency ion-yield fluctuations are high: 6y/lg e = 6Ni/Ni 1. However, as was already mentioned, the method of complete comparison cannot be applied in this case owing to the absence of theoretical dependencies Ni((F)). The resulting accuracy with which one is able to measure the multiphoton cross section by complete comparison is

+

The dominating role here i s played by the error in intensity absolutization. The absence of theoretical calculations that could describe the ion yield versus radiation intensity dependence for an arbitrary spatial intensity distribution and that could take into account the multifrequency fluctuations of the ion signals, is a shortcoming of this method. Its advantages are: ( 1 ) its applicability to the analysis of data in any part of the saturation region and (2) the potentially high accuracy of fitting the calculated curve to the experimental data.

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

73

Thus, in order to measure the multiphoton cross sections with the method of complete comparison, it is necessary to have the radiation of a single-frequency laser (or, at least, of a multifrequency laser with a sufficiently large number of generated modes) with a Gaussian profile of the spatial distribution. The measuring apparatus should enable one to record N,(in relative units), Fo, and TK. The range of intensity variation should include some part of the saturation region. And the accuracy with which the cross sections can be measured is limited by the error in radiation intensity absolutization increased K times. The choice of the method for measuring the multiphoton cross sections is mainly determined by the characteristics of the laser radiation used to observe the ionization process:

(1) by the profile of the spatial distribution of its intensity in the focusing region, (2) by the composition of longitudinal modes, and (3) by the highest intensity at the maximum of the space-time distribution. For instance, it is difficult to use the intersection-point method and the method of complete comparison in those cases where the profile of the spatial distribution of laser radiation intensity is other than Gaussian, since there are no theoretical dependencies of the ion yield on intensity. So in such cases the two-term series method is more preferable. The method of complete comparison cannot be used if one has to work with a multifrequency laser that generates a small number of longitudinal modes since here, also, there are no theoretical dependencies. Finally, the intersection-point method cannot be used in those cases where the intensity of the laser radiation in the focusing region is not high enough to enable one to observe deep saturation of the ion signals.

III. The Procedure for Measuring the Quantities Needed to Find the Multiphoton Cross Sections and Its Accuracy The schematic diagram of the setup for measuring the cross sections of direct multiphoton atom ionization is depicted in Fig. 5 . The radiation produced by a pulse laser is focused into the vacuum chamber filled with the gas of examined atoms under low pressure. As a target, one may also use an atom beam. For producing direct multiphoton atom ionization, the lasers with pulse duration of the order of nanoseconds or shorter are necessary to have high intensity of radiaphotons/ tion: from lo2’ photons/cm2 sec for atoms of alkali metals to cm2 sec for noble-gas atoms (i.e., of the order of lo8 W/cm2 and lOI4 W/cm2, respectively, for the energy of a photon hw = 1.2 eV). The ions produced in the region of laser radiation focusing are extracted by a constant (or pulsed) electric

74

Ammosov, Delone, and Ivanov

field into the flight gap, where they are separated according to their masses and come into the detector. The measured quantities are the laser radiation intensity F in the region of interaction with atoms; the initial atom density no; and the number of ion-produced N,.(See Eq. (II).) The quantities no and N ican be measured directly, but there is no direct method of measuring the intensity F of the laser radiation used for multiphoton ionization. Measuring the laser radiation intensity in absolute units is one of the most important procedures in experiments where multiphoton cross sections are measured. First of all, the error of intensity absolutization gives the largest contribution to the resulting error in the value of the multiphoton cross section since intensity has to be raised to the Kth power. Second, for pulse lasers (and only lasers of this kind can be used for observing multiphoton ionization) the measuring of intensity is a special case, since the space-time distribution of F is essentially nonhomogeneous. Within a laser pulse, at different points of the interaction volume one has different values of intensity. So the measurement of intensity comes to measuring a large number of values corresponding to different moments of time within a pulse and to different points within the interaction volume: F ( t , x , y, z). For measuring the multiphoton cross sections one has to use the radiation of lasers that in the regime of a single lowest transversal TEMm mode generate one or many longitudinal modes (i.e., single- or multifrequency radiation, respectively). Only in this case can one separate the temporal ( t ) and spatial ( x , y, z ) variables, and the intensity distribution F ( t , x , y, z) near the focal plane of the focusing lens can be measured with required accuracy. If the laser generates many transversal modes, the spatial and time variables cannot be separated. As the radiation pulse develops and the resonator quality grows, the modes with lower transversal indices are radiated first, while those with higher indices are radiated later on. When many transversal modes are generated, the intensity distribution along the front of the wave is no longer smooth: it has local regions with very different intensity. For radiation of a single-frequency laser, the space-time intensity distribution in the interaction region can be presented using Eq. (13):

where F , is the value of intensity at the maximum of F(t, x , y, z); Q is the total energy in the pulse; T = J +(r) dt is the pulse duration; S = J $ ( x , y, z,) dx dy is the area of the cross section in the plane of the maximum intensity value F,,; and + ( t ) and $ ( x , y, z ) are dimensionless distributions whose form is smooth and close to Gaussian, and which are normalized to their maximum value. In this case the measured quantities are the energy Q and the distributions + ( t ) and $(x, y. z). Having measured these quantities experimentally, one is then able to

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

75

calculate all the parameters that enter Eqs. (55), (64), (67), and (72) for multiphoton cross sections: T, S, F o ( F 8 , F C ) , T , = $ + " ( f ) dr, and V , = $ $"(x, y, Z) dx dy dz, VXIVK. A single-frequency laser is the most convenient one for measuring the multiphoton cross sections in the respect that the time distribution of its radiation, +(r), has no small-scale oscillations and can be measured reliably by standard methods (as is discussed shortly). In practice, however, one often uses multifrequency lasers (in the regime of the lowest transversal mode generation). There are several reasons why these lasers are used, such as the higher energy in a pulse, the simplicity of the resonator construction, and the higher stability. A shortcoming of these lasers is the existence of small-scale oscillations (of the order of lO-"s) in the time distribution, which are difficult to record without distortion by standard methods. In the case of multifrequency radiation (i.e., generation of many longitudinal modes of the lowest transversal index) we have

Here F( r) describes the fast oscillations inside the pulse envelope +( r). One should keep in mind that, owing to the inertialess nature of direct multiphoton ionization, the atom reacts to the instantaneous value of intensity. This is why in the case of multifrequency lasers, the ion yield fluctuates from pulse to pulse owing to the nonreproducibility of their fine temporal structure. Owing to the high peak values of radiation intensity in the fluctuations, the average ion yield is higher than it is for single-frequency radiation at equal . . pulse energies: (Ni) ( F K ) = g K ( F ) , . In this case the measured quantities are the same (Q,+ ( I ) , and $(x, y, z)), while to the calculated quantities there is added either g K or H K , and F , is replaced by (F). (See Section 1I.C.) The majority of measurements of the multiphoton cross sections have been made with nanosecond radiation, so we shall not consider the problems that arise when one measures multiphoton cross sections with pico- or femtosecond lasers. Thus, in order to find the multiphoton cross sections (Y, one has to measure Q , + ( f ) , $(x, y, z), n o , and N i and then calculate the values of T,S, F, V,, TK, and g K and substitute them into the expressions for a, presented in Sections 1I.G and H. When one observes the process of multiphoton atom ionization, a single laser pulse gives practically no information about the value of multiphoton cross section. In order to measure the latter, one needs a series of pulses both to obtain the data on N i ( F ) dependence and to get reliable data about the average values of the ion signals. The latter is especially important for multifrequency lasers. The need to use a series of laser pulses imposes certain requirements upon the stability of the laser radiation and atom target parameters. The stability of the radiation produced by modern pulse lasers is limited by technical fluctuations:

-

76

Ammosov, Delone, and Ivanov

by the instability of the voltage feeding the pumping lamps, by the fluctuations of the lamp discharge, by the instability of the temperature and of the thermal lens of the active element, and so on. One may also consider the fundamental fluctuations due to the quantum nature of light and the discreteness of atoms, but such fluctuations are not dominating in modem setups. A. MEASURING THE ENERGY OF A LASERPULSE

The measurement of multiphoton cross sections imposes the following requirements upon the device that measures the energy of the laser radiation. First of all, it should measure the energy values ranging from 10-* to several J’s. Second, the absolute gauging of this device must be as accurate as possible. For this reason, of all the ways to measure the laser radiation energy, one selects the calorimeter method. In standard calorimeters, the laser radiation is absorbed by a blackened metal cone whose temperature is recorded by a battery of thermocouples. The stability of these devices is sufficiently high (= 1% for the measurement time one hour), but the systematic error in absolutization of the measured energy value (i.e., the absolutization error) is around 10%. A higher accuracy can be reached with double-chamber calorimeters, in which one measures the temperature difference between the irradiated surface and an identical surface not exposed to radiation. The systematic error in devices of this type is about 3 or 4% (Demtroder, 1982; Rabek, 1982). The calorimeter is installed behind the interaction chamber (see Fig. 5 ) , so in the measurements one should take into account the energy losses due to reflection from the surface of the optical elements, including the chamber windows and lenses, placed between the interaction object and the calorimeter (about 4% at each surface). The energy losses on ionization can be neglected. Let us note that the full power of laser radiation coming into the calorimeter may lead to destruction of the blackened layer upon the receiving element and, consequently, to an uncontrollable increase in the systematic error. The calorimeter can be positioned in another way, where it receives only a small portion of the radiation energy that is directed upon it by a light-dividing wedge. In this case one has to gauge the light divider, and the error in its reflection coefficient will be added to the total measurement error. A weak point of calorimeters is their large inertia-of the order of seconds. No such problem arises with photoelectric detectors of radiation energy, but their response to radiation intensity is not standardized, so in order to use them one has to perform independent absolute gauging of the corresponding devices. Let us now consider the measurement error of radiation energy. The total error with which one measures the energy is made up by apparatus errors (both random and systematic, their typical values were presented previously) and by the

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

77

error associated with technical energy fluctuations from pulse to pulse. When measuring the multiphoton cross sections, a number of measures are taken to stabilize the operation of the laser: the stabilization of the feeding source, thermostabilization of the active elements, sustaining equal time intervals between laser pulses, etc. It is evident that the degree of stabilization should be the greater the higher is the power of nonlinearity of the studied process. Practice shows that for processes with the degree of nonlinearity K S 5 or 6, for which most of the experimental data on multiphoton cross sections have been obtained, it is necessary to stabilize the energy in a series of laser pulses to within 2 or 3%. The variation of the laser radiation parameters from pulse to pulse makes one consider the problem of controlling these parameters in each laser pulse. The measurement of Q , +(r), and +(x, y , z) in each pulse requires complicated and expensive apparatus that should include a computer for accumulating and processing the data. So this control is justified only in those cases where the multiphoton cross sections have to be measured with an accuracy higher than usual. Furthermore, one should take into account the range of variation of the radiation parameters in different regimes of laser operation. Compared with singlefrequency lasers, the multifrequency lasers have better stability of energy and of the spatial characteristics, but the fluctuations of the ion signals due to the multifrequency nature of radiation are high. For this reason, it is very unlikely that a pulse-by-pulse control of radiation parameters in this case would increase the accuracy of measurements. In the case of single-frequency lasers, such a pulseby-pulse control can lead to a noticeable increase of the measurement accuracy and, in addition, would enable one to eliminate those cases where two or more longitudinal modes are generated.

B. MEASURING THE TIMEDISTRIBUTION OF INTENSITY IN A LASERPULSE The standard procedure for measuring the laser pulse duration in the nanosecond region comes to the following. (See Fig. 5 . ) Part of the laser radiation is directed by a light-dividing wedge upon a photodetector. If additional weakening of the signal is needed, one can use neutral optical filters. As a photodetector one uses coaxial photoelements or fast-acting photodiodes that transform the light pulse into an electric signal. The latter is fed to a wideband oscilloscope, and the distribution on its screen is recorded on a film. Such a procedure can give a time resolution of to 1O-Io s. The most fast-acting photodetectors are photodiodes with a time resolution of lo-" s. They can be used together with wideband oscilloscope with a band up to 10 GHz. Wider possibilities are offered by pulse-periodic lasers that enable one to make measurements by using stroboscopic oscilloscopes with the time resolution = l o - " s (Demotroder, 1982; Rabek, 1982).

78

Ammosov, Delone, and Ivanov

By using the apparatus for fast signal processing (such as a transient recorder, a fast transient digitizer, or a boxcar integrater), followed by computer processing of data, enables one to exclude the photographic procedure. For measuring the picosecond pulse durations one uses streak camera with a time resolution below lo-’* sec. The femtosecond durations can be measured by nonlinear-optics methods (Demtroder, 1982; Rabek, 1982). Let us now consider the accuracy with which one can measure +(r). There are two sources of errors: (i) the fluctuations of + ( t ) from pulse to pulse and (ii) the apparatus errors in the +(t)measurements. The apparatus-measurementerror consists of a systematic error and a random error. The first one, 6Tab,,is determined by the time resolution and by the absolutization error. The time resolution of different devices already has been presented. As a rule, the absolutization error in the +(r) measurements is no less than 10% (as it is, for instance, in proper calibrators of oscilloscopes). In order to increase the accuracy of absolutization, different time tags are used. In certain cases one can bring the absolutization error down to several percent. The random apparatus error, 6T,,, is associated with the errors of reading from measuring devices (that are due to the finite width of the line on the oscilloscope screen, of the pointer, and of the scale marks, etc.). Usually, this error is small and does not exceed 1 or 2%. The fluctuations of +(t) are mainly due to the technical fluctuations in the laser-operation regime. They can be made smaller by stabilization. Their typical value is ST, = 1 to 3%. The methods of measuring + ( t ) in the case of single-frequency lasers and in the case of multifrequency lasers are essentially different. Whereas in the former case one should make the detector time resolution as high as possible, in the second case there exists an optimal value of the detector time response 7.On the one hand, the detector should smooth out the small-scale oscillations F( t) (7 > lo-’’ s); on the other hand, it must not distort the envelope +(t) (7 < 10-9 s). Thus, for the limiting error with which one can measure the time resolution, one can write down

+

(6T)2 = (6Tabs)24- (6Trm)2 (6Tn)*.

The preceding analysis clearly shows that the main contribution is given by the systematic error of absolutization, ST,,, , and by the error 6T, due to fluctuations of +(r) from pulse to pulse. The minimum error in +(r) (about several percent) is obtained when one uses single-frequency radiation, minimizes the technical fluctuations of + ( t ) , and takes special measures to gauge the measuring apparatus. The typical values are 6TflIT = 2 to 4%, 6Tab,/T= lo%, and 6TIT = 10 to 15%. By controlling +( r) in each laser pulse, when measuring the multiphoton cross

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

79

sections, one is able to eliminate the contribution 6Tn in the total error 6T. In those cases where a pulse-by-pulse control is impossible, one should measure &t) for a large number M of radiation pulses. The error in the average value will then be given by the formula 6Tn = (6T,,),,,/-.

c. MEASURING THE SPATIAL DISTRIBUTION OF LASERRADIATION INTENSITY Measuring the spatial distribution $(x, y , z) of the laser radiation intensity is a difficult problem since it is necessary to measure a three-dimensional distribution. This problem can be divided into two subproblems. The first one is to measure the distribution $(x, y , zo) in the focal plane of the focusing lens. The second subproblem is to measure the distribution $(x, y , z ) along the z axis of propagation of the laser radiation near the focal plane. The procedure of measuring the distribution of the laser radiation intensity in the focal plane of the lens comes to the following (Fig. 5). A small portion of radiation is directed by an optical wedge 4 upon a lens identical to the one that focuses the main part of the laser beam upon the atom target in the vacuum chamber. The optical paths from the laser to each of the two lenses should be made equal. A microscope with 10- to 100-times magnification reflects the focal plane of the auxiliary lens upon a photographic film or a photographic plate. (The typical values of the area into which the laser radiation is focused are S=3 X to 3 X lo-* cm2.) The intensity of radiation in the auxiliary beam is selected, by using neutral filters, to be within the range where the blackening of the photoemulsion is linear. The intensity distribution $(x, y, zo) is obtained by processing the photographic images with a microphotometer. In more detail, this photometric procedure is presented in Rabek (1982). The error of measuring $(x, y, z,,) is made up by the systematic and the random apparatus errors (6S,,, and as,) and by the error SS, due to fluctuations of $(x, y , z) in a series of laser pulses. The 1 to 2%. The value typical values of the errors are: 6Sab,/S= 10% and 6S,/S of 6Sn/S depends upon the stability of laser operation and upon the mode composition of its radiation. This problem has not been examined in detail in the literature. By controlling $(x, y, zo) at each laser pulse, one is able to eliminate the contribution 6S, in the total error 6s. In those cases where a pulse-by-pulse control is impossible, one should measure $(x, y, zo) for a large number M of laser pulses. The error in the average value will be 6S,,= (tSSIf),Jfl. A typical value of the total error is ~1

(6S/S)2 = (6S,,,)2

+ (6Sn)2 + (6Sm,)2 = 15%.

(77)

Lompre et al. (1982) have proposed a method of measuring S by computer processing of the video image. The laser radiation was recorded by a silicon mosaic vidicon with the diode matrix 225 X 225 elements. The video signal

80

Ammosov, Delone, and lvanov

went through the image amplifiers and to a high-speed digital encoder. From there the information went to the graphic PERICOLOR terminal coupled to a computer. In contrast to the long procedure of photometering, in this setup the value of S is calculated with an error SS/S = 4% in 5 seconds. This method enables one to make a pulse-by-pulse control of the spatial distribution cross section. The spatial distribution $(x, y, z ) along the z axis of the optical system can be measured in two ways. The first way is to record successively the different cross sections of the focal region in a series of pulses. This can be done by both photographing and processing video images. The other way is the method of a mirror wedge, which enables one to record different cross sections of the focusing region in one radiation pulse. The successive reflection from two mirrors produces a series of spatially separated images of the focusing region. By selecting the proper angle between the mirrors and the proper incidence angle of radiation, one can get the desired number of images with necessary spacings between them. A similar method was used by Lompre et al. (1982). There, instead of a mirror wedge, the radiation was reflected by two parallel glass plates, so that, simultaneously, one recorded four different cross sections of the focusing region. Thus, having measured the intensity distribution cross sections at different z , one can obtain the dependence of intensity on the zo coordinate. In those cases where only estimates of the multiphoton cross sections are needed, and there is no need to measure the cross sections with the smallest error possible, the procedure of measuring $(x, y , z ) can be simplified substantially if one makes use of the fact that the intensity distribution in a laser beam is close to Gaussian. As is known, the intensity distribution of a focused Gaussian beam has the form 1 2 yz $(x, y , z ) = 1 ( z A / 2 ~ r 2exp( ) ~ -ri (zA/2~r,)~

+

+

+

where A is the radiation wave length and ro is the beam radius at the focal waist, which is taken to be the distance from the beam axis, over which the intensity lowers e times. As one can see from this expression, the wave length and the waist radius completely fix the intensity distribution of a Gaussian beam. The radius ro is related to the beam radius at the lens entrance R and to the distance L between the lens and the waist:

ro = ALI~TR.

(79)

If the lens diameter is large enough: D > 3R, the intensity distribution in the vicinity of the waist is monotonic and has no local maxima and minima typical of those cases where a plane wave is focused by a lens of finite size (Born and Wolf, 1964). The fundamental difference between the two cases is that in a Gaussian beam the intensity falls off smoothly from the axis to the periphery.

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

81

Thus, for a Gaussian beam, the measurement of the spatial intensity distribution comes to measuring the cross section of the beam in the plane of the focusing lens and the distance between the lens and the waist. In conclusion, let us note that the Gaussian distribution gives a good description to the radiation of single-frequency lasers and of multifrequency lasers with a large number of longitudinal modes in the transversal TEM, mode. However, the intensity distribution at the focus of the lens can be distorted substantially by its aberration. When focusing laser radiation, one can practically always get rid of aberration of slanting beams by thorough adjustment of the system. It is however impossible to eliminate spherical aberration for a single lens. The following inequality gives the condition under which the distortion of the wave front of the beam will be negligible:

CR4 3

-€-A.

f’

4

C = 1.3 for a plano-convex lens (Born and Wolf, 1964).

D. MEASURING THE NUMBER OF IONS

PRODUCED

The ions produced in the region of laser radiation focusing are extracted by an electric field into a detector (Fig. 5 ) . The ion detectors used are either secondary electron multipliers of different types or Faraday cylinders. The operation of a multiplier is based on the effect of an avalanche-like secondary electron emission. In usual multipliers, the proliferation of electrons takes place during the flight through a system of electrodes. In channel multipliers this happens on the walls of a single spiral-shaped channel about 1 mm in diameter and about 100 mm long. In a microchannel multiplier, the electrons are multiplied in a large number of parallel microchannels each about 0.05 mm in diameter and about 5 mm long. The amplification coefficient for the usual multipliers ranges from lo5to lo6; for channel multipliers they are lo6to lo8.In the measurements, the signal from the multiplier goes through a preamplifier matched to the cable and onto the low-impedance input of a wideband oscilloscope or a high-speed ADC. If necessary, the measuring circuit can be supplemented by a wideband amplifier. As a source feeding the multipliers one uses stabilized high-voltage rectifiers. The random error of the measurements done with multipliers is small: around several percent. It is determined by the stability of the source feeding the multipliers, by the fluctuations of the amplification coefficient, by the fluctuations of the dark current, and by the random apparatus error of the oscilloscope. In the relative measurements of the number of ions, N,, the random error determines the total measurement error.

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Ammosov, Delone, and Ivanov

In the absolute measurements of the number of ions produced, the error is determined by the systematic error of absolutization. As a rule, the latter is quite high: 20 to 100%. The reason is that the secondary emission coefficient (and, consequently, the amplification coefficient) depends upon the type of ions, their charge, and their kinetic energy. So it is necessary to make absolute gauging of the multiplier by using a precision source of the studied ions. A shortcoming of the multipliers is the small dynamic range where the amplification of the signal is linear: from two to four orders of magnitude for linear amplification. It can be increased by changing the feeding voltage, but then one has to make additional gauging. Let us note that in the ion-counting regime, where the multiplier detects individual ions (owing to the large amplification coefficient), the measurement error is minimum and is determined only by accidental triggerings of the detector. In order to realize this regime, it is necessary that within a laser pulse there be produced only a small number of ions, i.e., that the ions come to the detector with intervals greater than the proper time of the multiplier (1-30 ns). This regime can be used to measure the multiphoton cross sections with the absolute method. (See Section 1I.G.) The absolute measurements of the number of ions are also made with the Faraday cylinder by measuring the charge of the ions that have accumulated upon the cylinder walls. The error of such measurements is around 25%. In order to isolate the useful signal from the signal of residual gas ions, the ions are separated according to their mass in the flight gap preceding the detector. The scheme of the time-of-flight mass spectrometer with two field-filled gaps and one 50-cm-long field-free flight gap gives the mass resolution m/Am = 100, which is quite sufficient for the experiments discussed. In order to gather all the ions produced into the detector, one has to sustain sufficiently low pressure in the interaction chamber and in the flight gap to avoid collisions of the ions with atoms. The length of the flight gap has to be smaller than the free path length of ions: X < hi = For the ion-atom collision cross section crai = cm2 and X = 100 cm, the density of the gas should torr.) (For pressure this means p S be no S 10" Furthermore, to gather all the ions, one has to use a sufficiently large diaphragm upon the extracting electrode, with diameter d , > e (see Fig. 5), where t? is the effective length of the region where laser radiation interacts with atoms. We mention this trivial requirement because for radiation from the visible frequency region the length e may reach several millimeters. Sogard (1988) has shown that the ion-yield dependence can be substantially distorted if the ions from the periphery of the interaction region are not recorded. This is especially important for processes with a small number of photons, where the length of the effective interaction volume is close to, or may even exceed, the Rayleigh length 2,. (Z, = 27rr3h >> ro is the distance counted off the waist of the Gaussian

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

83

beam, over which the intensity at the beam axis decreases two times.) For a Gaussian beam, the effective length of the interaction region is t!K = VK/SK Z R / f l , where V K is the effective volume of the K-photon process given by Ehq. (87) of Section F and SK = .rrri/K is the effective area of the waist. To gather all the ions produced, one also has to make sure that the extracting potential difference is sufficiently high and that saturation of the ion signal occurs with increase of the extracting voltage. The value of the minimum extracting potential difference that ensures the gathering of all the ions into the detector is determined by Coulomb repulsion of ions (Ammosov et af., 1989):

-

where 5 is the degree of ionization. Let us mention that for [n, the value of Urninmay reach several kilovolts.

3

10'O

E. MEASURING THE DENSITY OF NEUTRAL ATOMS In those cases where laser radiation is focused in a gas of examined atoms, the atom density is measured with vacuum ionization gauges, provided that the density of residual gases is much lower. As a rule, the density used ranges from to lo* to lo'* cm-3. (The corresponding pressure range is torr.) The error is determined by the gauging of the apparatus and is under 10%. When an atomic beam is used, the atom density in the beam can be measured in several ways. In the measurements of multiphoton cross sections one often uses the method of surface ionization or the method of depositing the atoms upon a piezoelectric cell. The device that measures the surface ionization is an emitter made of tungsten in the form of a strip or a filament, upon which the surface ionization of beam atoms takes place. The electrons or ions produced with ionization are collected by a constant electric field upon a collector. The latter is shaped either as a plate (for a strip-shaped emitter) or as a cylinder coaxial to the emitter (when the latter is a filament). For each kind of atoms one selects a certain material for the emitter. For instance, for alkaline and alkaline-earth atoms one uses either pure tungsten (for Li, K, Rb, Cs, and Ba) or oxidized tungsten (for Na, Mg, Ca, and Sr). The measurement error is determined by gauging and amounts to about 20%. In the deposition method, one uses a piezocrystal coupled to a resonance circuit. The atoms are condensed upon the crystal surface, and as the layer of atoms becomes thicker, the resonance frequency of the crystal changes. The measurement error is also determined by gauging and is around 25%. One can also use the ionization method and the method where one observes resonance fluorescence. In the first case the atoms of the beam are ionized by an electron beam or by ultraviolet radiation. The ions produced are recorded by a

84

Ammosov, Delone. and lvanov

detector. The error with which the density is determined is around 30%. In the second case the atoms of the beam are excited by a dye laser (with radiation frequency equal to the resonance frequency of the atoms) and the light they emit is detected by a photoelectron multiplier. The error is of the same order as in the previous case. In both cases it is determined by the apparatus gauging. The most accurate way to measure the atom density is to use the vapor of examined atoms instead of an atom beam. In this case, into the interaction chamber one places an open ampule containing the metal whose vapor is then ionized. The whole chamber is placed into a thermostat and is kept at high temperature, so that the metal vapor fills the chamber. After that, the measurement of density comes to measuring the chamber temperature and to determining from this temperature the saturation vapor pressure. In this case the error is determined by the quality of the thermostat where the chamber is kept and amounts to several percent. F. CALCULATING THE LASERPULSE INTENSITY, THE EFFECTIVE PULSE AND THE EFFECTIVE INTERACTION VOLUME DURATION, Having measured the distributions +( t) and $(x, y , z) and the energy Q in a pulse of laser radiation, one can calculate the value of intensity Fo at the maximum of the space-time distribution: Fo

=

1 Q - [photons/(cm2 x s)],

ho ST

where S = J $(x, y , zo) dr dy and T = J + ( t ) dt. $(x, y, z) and + ( t ) are integrated graphically. This produces an additional error, which is usually small and does not exceed 1 or 2%. When calculating the absolute values of S and T, one should pay attention to the normalization of the $(x, y, z) and + ( t ) distributions. For calculating the multiphoton cross sections (in Eq. (82) for intensity and in any other expression for c y K ; see Section II) both distributions have to be normalized to the maximum value, i.e., the value at the point where the intensity is maximum is taken to be unity. This choice of normalization is related to the nonlinear character of interaction. Some authors have used other normalizations, such as the normalization with respect to the level lle, or l/e2, or 1/2. If the values of S and T have been calculated in one of these normalizations, in order to obtain the maximum intensity value in Eq. (82) one has to use conversion coefficients. They are easily calculated in the case of a Gaussian distribution (with respect to the space and time coordinates):

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

== 1.43 F ( l / e )=z 4.00

F(lIe2)

85

0.83F(1,2).

Considering what has been said previously, when $(x, y, z) is measured along the radius of the laser beam cross section on the focusing lens by using Eqs. (78) and (79) for a Gaussian beam, the value of S is calculated as follows:

S =ma

(84)

where r, is the beam radius at the intensity level lie. The error with which the intensity is measured is made up by the errors in the measured quantities:

($j2 =

($?)2

+

(!E)2+ (!E)2.

As already mentioned, the main contribution to the latter errors comes from the absolutization error. When one uses the radiation of a stable single-frequency laser, measures Q,T, and S for each laser pulse, and takes measures to lower the absolutization error (measures the energy with a precision calorimeter and uses time markers to measure the pulse duration), one is able to reach an error 6FlF =z 10%. For instance, by using the method of Lompre et al. (1982), Normand et al. (1989) have measured the intensity with a 6% accuracy. Such measurements, however, are either very labor-consuming or require very complicated apparatus. As a rule, even when one uses a multifrequency laser, T and S are measured in a separate experiment, and, when measuring the cross section, only the energy is controlled in each laser pulse. The typical values of 6FIF in this case are,25-30%. Finally, let us mention Perry and Landen’s (1988) proposal of a “direct” method of measuring the field strength based on the changes in the ionization potential Z of the atom under the action of an alternating field, by measuring the kinetic energy E , , ~ ” = Khw - Z(E) of the electron produced. However, in reality, it is still difficult to give a strict substantiation to this method owing to the existence of above-threshold absorption (Delone and Fedorov, 1989a,b), and, also, in view of the limits on the conditions under which one is able to distinguish clearly an elementary ionization event from pondermotive effects in the region where the laser radiation is focused (Delone and Fedorov, 1989a,b; Goreslavsky et al., 1989). As it follows from the principal relations presented in Section 11, from the measured quantities 4(t)and $(A y, z) one should also calculate the time T K and the volume V, of interaction that are effective for a K-photon process. Let us recall that the differences between T K and T and between V, and V, and thereby

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Arnrnosov, Delone, and Ivanov

the need to calculate TK and V,, are due to the nonuniformity of the space-time distribution of laser radiation. In principle, these quantities can be measured directly by using multiphoton detectors. For example, it is possible to measure T2 by recording the time distribution of the second harmonic of laser radiation, excited in a nonlinear crystal. However, the measurements with multiphoton detectors are difficult to make correctly (Delone and Masalov, 1980), so this method is practically never used. The value of T K = J 4,(?)dt is found by raising the distribution 4(t)to the Kth power and integrating it. The error with which T K is found is somewhat higher than the error in T since (6TK)ran= K6T,, (and similarly for (8TK),), while the systematic (absolutization)errors are the same: (ST,),, = 6T&,,. If the distribution 4(t)is close to Gaussian, we have T K = T / a , and the total error is 6TK = ST S T G , where STG is associated with the deviation of +(?) from the Gaussian form. The typical values are GTKITK -- 10 to 15%. The value of V, = J JIK(x,y, z ) Cix dy dz is calculated as follows. The distributions I,!@, y, z , ) measured at different values of zi are normalized, raised to the Kth power, and summed up: V , = Az Xi S K I . Here Az is the discrete step in z with which the cross sections of $ ( x , y, z i ) are selected; SKI = J $,(x, y, z , ) ak dy is the effective area of the ith cross section. The error SV, is given by the following expression:

+

Here (SS,/S,) is the error with which the effective area of the cross section is found; (SS,), = KISS, is the error given by the fluctuations of $(x, y , z ) from pulse to pulse; ( S S , ) , = KSS, is the random apparatus measurement error; = 6Sinl is the integration error; (SV,), is the error associated with discrete selection of the cross sections; and e K = V,/S, is the effective interaction length. If the distribution $(x, y . z ) is close to Gaussian, then having measured the beam radius R at the entrance to the lens, the distance L from the lens to the waist, and thereby the waist radius ro (see Eq. (79)), one can calculate V , according to the formula:

1

for K = 2 = [ r ( K - (7/4))]-1’2 for K

(2K - 4)!!

3.

87

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

In this case the error SV, has the form

where 6L is the error in L and ( 6 V K ) ,is the error associated with the departure from the Gaussian form. The typical values of SVK/VK are 15-20%. G . CALCULATING THE IONIZATION PROBABILITY PER UNITTIME AND THE MULTIPHOTON CROSSSECTION The process of multiphoton ionization is characterized by two fundamental parameters: by the K-photon cross section and by the probability per unit time w at a given value of intensity. The relation between these two quantities is given by Eq. (6): w = aKFK. However, the observed quantity is not w , but the total probability W per laser pulse. Let us write down the relations that connect a K , w , and W to the measured quantities we have already discussed. For a singlefrequency laser we have W

= w/TK = ni/(noTK) = Ni/(VKTKno)

ffK =

N,/( iloTKVKFK) ,

(894

for multifrequency radiation we have W

= N,/(noTKFKgK),

LYK

= Ni/(noTKVKFKgK).

(89b)

The error in the probability 6wlw is made up by the errors in the measured quantities:

($)’

=

(F)’+ (2)’+ (z)2 + (2)’.

(90)

As was previously shown, the typical values of the errors are

and, consequently, 6wlw = 40%. The error with which one finds the multiphoton cross section is substantially higher, since it includes K times the intensity-measurement error. Since B a K I a K 3 1, the measurement error can be presented in the form 6A = S(lg aK), where aK = 10A’SAand (SA)2 = (lg e)’

[(T)’+ (?)*+ r2)’+ r?)’+

Kz($)‘].

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Ammosov, Delone, and Ivanov

The typical value is 6FIF c= 30%.For this reason, for example, for three-photon ionization we have 6a3/a3= 100% and 6A = 0.3. H. THEGENERAL PROCEDURE FOR MEASURING MULTIPHOTON CROSSSECTIONS

Before measuring the multiphoton cross section, in a separate experiment one measures the degree of nonlinearity K of the ionization process (Section 1I.D). This is dictated by the need to check whether the ionization mechanism is direct (i.e., that there are no resonances) and to select the range of laser radiation intensity where the measurements should be made. One records the ion-yield dependence on the laser radiation intensity (Figs. 6 and 7). For calculating the cross sections according to the absolute method, one has to make measurements in the nonsaturated region, where the ion yield N i is proportional to F K . The different modifications of the relative method require the measurements to be made both in the nonsaturated region and in the region of saturation (Section 1I.H). The advantage of the relative method is that there is no need to measure the neutral ion concentration no, and one only has to make relative measurements of the ion yield N i and of the effective interaction volume V,. As a shortcoming of the relative method one may consider the need to obtain the experimental data in a wide range of radiation intensities, including the regions where the dependencies N i ( F ) are saturated, which requires high laser radiation intensities. A very important problem for any experiment where the multiphoton cross sections are measured is the selection of the operation regime for the laser. As was already mentioned, it is not recommended to use a laser that generates many transversal modes. The choice between the two possibilities-a laser that generates many longitudinal modes of the lowest transversal index and a laser that generates single-frequencyradiation-determines the experimental methods and, in the end, the resulting accuracy of measurements. When one uses multifrequency lasers (which, compared to single-frequency lasers, have more stable Q,S,and T in a series of successive pulses), the spatial and time distributions of radiation intensity can be measured in a preliminary experiment. Then, the cross section measurements come to measuring the ion yield as it depends upon the laser pulse energy. The method of measuring the cross section in this case is the following. First, one measures the c#~(t)and I,!J(x, y, z) distributions. Then, one measures the power of nonlinearity of the process, i.e., the dependence N i ( Q ) . After that, in the required region of the N i ( Q ) dependence, one generates a series of laser pulses with constant energy, so it is enough to measure the energy in a single pulse of the series. The ion yield, measured for each pulse, is averaged. Let us emphasize that for multifre-

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

89

quency lasers, owing to fluctuations, the peak value of intensity can be two or three times higher than the maximum value of the measured average intensity. For this reason, in this case the dispersion of the ion yield is very large, and for measuring the multiphoton cross sections one must use a larger number of laser pulses than in the case of single-frequency lasers. When measuring the N , ( Q ) dependence, the energy of a laser pulse should be varied in such a way so that the 4(?)and +(x, y , z) distributions would remain the same. There are several ways of doing this: (1) By using neutral filters. (2) By using the polarization method for completely polarized laser radiation. The energy of the laser radiation that has gone through two polarizers is proportional to cos28, where 0 is the angle between the axes of the polarizers. (3) By weakening the radiation by reflecting it from a surface. The laser radiation is let through two identical plane-parallel transparent plates, one of which makes an angle 8' with the optical axis of the system, the other being inclined at the angle n - 8'. By changing the angle of the first plate one can change the transmission coefficient, while the second plate restores the direction of the beam along the system axis. The second method is the most convenient one. The first method does not allow one to vary the energy continuously and, owing to the nonlinear effects on glass filters, it is difficult to use for laser radiation of high intensity. The third method yields to the second one only in the relative complexity of energy-variation calibration. Often, one uses a combination of the first method with the second or third one. One cannot recommend varying the radiation energy in a pulse by changing the operation regime of the pumping lamps for active elements of the generator and/or of the amplifier. As a rule, this changes the space-time distribution of the radiation, which was shown by controlling it in each laser pulse (Lompre et al., 1982). As concerns the measurement of multiphoton cross sections, a shortcoming of multifrequency laser radiation is that one has to measure the statistical factor g, = J F K ( t )d?/(F)K(Delone et al., 1980) (or the pseudostatistical factor H , ) . The direct measurement of F( r) requires apparatus with very high time resolution (no worse than lo-" s). As a rule, one uses the values of g, calculated for Gaussian radiation with a different number N of longitudinal modes (Masalov and Todirashku, 1980). (For N + m, g K + K!.)The number of modes in the laser radiation can be found by measuring its spectral width. In this case, when calculating the cross section, one has to take into account the error associated with the deviation of the envelope 4(t)from the Gaussian form. When one uses single-frequency lasers (which are less stable than the multifrequency ones), the experimental technique is different in that it becomes necessary to measure the radiation energy in each pulse. In addition, one has to

90

Ammosov, Delone, and Ivanov

control the time distribution +(r) in each laser pulse to exclude the cases where two or more modes have been generated. An advantage of using single-frequency radiation is the smaller dispersion of the ion signals at constant radiation energy and no need to measure g K . One can give the following recommendation for measuring multiphoton cross sections with minimum error. First of all, one should have a stable singlefrequency laser. The intensity distribution should be measured in each pulse, which enables one to eliminate the error associated with fluctuations of the energy Q,duration T, and cross section area S. The energy should be measured by a precision two-chamber calorimeter; T should be measured with time tags. This enables one to reduce the main error-the error of intensity value absolutization. S and V, sliould be measured by combining the method of a mirror wedge with computer processing of the video image. By measuring the intensity this way one is able to bring the error down to 6FIF = 10%. (Normand er al. (1989) have reached 6%.) Furthermore, the measurements should be made according to the relative method (see Section ILH), where there is no need to measure the neutral atom concentration and one only has to make a relative measurement of N , and V,. In this case the errors S N , / N , and SVKlV, are substantially lower-around several percent. The values a, have not been measured this way before, evidently, because such measurements require much work. However, this method enables one to expect a minimum error: GaK/a,= KSF/F -- K X 10%.

IV. The Results of Measuring the Multiphoton Cross Sections of Direct Atom Ionization In Table I1 we present practically all the data on absolute values of the cross sections aK measured with laser radiation that have been published up to this time (1990). Most of the data have been obtained by using linearly polarized laser radiation, and only a small portion of them have been obtained with circularly polarized radiation. For all the studied atoms, the measured values of the cross section refer to multiphoton ionization from the ground state; only for He have the values of a, been measured for ionization from the ground state and from the first excited states 2s3S, and 2s1S0. Both the absolute and the relative methods have been used for measuring a,. For frequencies presented in Table 11, the fact that the process of multiphoton ionization was direct has been checked by measuring the dependence of the ion yield Ni on the laser radiation intensity F and thereby determining the power of nonlinearity Kexp= d(lg Ni)/d(lgF ) . In all the cases in Table 11, the measured value of the nonlinearity power was equal, within the experimental error, to the

TABLE I1 EXPERIMENTAL VALUES OF MULTIPHOTON CROSS SECTIONS oLK OF DIRECTIONIZATION PROCESS ~~~~

'0

Measurement method

K

Atom

2 2 2 2 2 2 2 2 2

K K cs Ca Ca He(2slS0) He(2s'S I ) Xe Xe

3 3 3 3

Na Na* Na

2 2

K

2

3 3

K*

3

Rb Rb*

2 2 2 2 2 2

3 3 3 3 3 3 3

K cs cs* cs

1

I I 2 2 2 2 -

-

1

cs

1 1

Ca Ca

2 2

~~~

w,

cm-'

IgU;

lg U K

gK

References

18.870 18,870 19,940 28,185 29,700 28,797 28,797 51,400 5 1,700

-47.2 ? 0.8 -48.8 ? 0.8 -49.2 ? 0.1 -46.3 t 0.5 -46.0 2 0.5 - 48.6 Z: - 48.8"; -49.4 - 52.0

2 2 2 2 2 2 2 2 2

-47.5 2 0.8 -49.1 ? 0.8 -49.5 2 0.1 -46.6 f 0.5 -46.3 2 0.5 - 48.9 ?: -49.1 T f i -49.7 -52.3

Held et a/., 1972 Delone et a/., 1973 Normand and Morellec, 1980 Alimov et a/., 1986a Alimov er a / ., 1986a Lompre ef a/.. 1980 Lompre et a / ., 1980 McCown et a/., 1982 Bischel et a / . , 1979

14,400 14,400 18,870 14,400

-76.3 +- 0.1 -75.9 t 0.1 -79.6 ? 1 . 1 -78.5 t 0.1

I I 6 I

- 76.3 ?

14,400 16,537 14,400 14,400 14,400 14,400 14,410 14,410 18,790 18.928

-78.0 -78.2 -76.8 -76.5 -76.7 -76.4 -76.0 -77.0 -78.7 - 77.6

1

- 78.0 5

Cervenan ei a/., 1975 Cervenan er a / ., 1975 Delone e t a / . , 1973 Cervenan et a/.. 1975 Cervenan and Isenor, 1975 Cervenan et a / ., I975 Alimov et a / ., 1980 Cervenan ef a/., 1975 Cervenan ei a / . 1975 Cervenan et a/., 1975 Cervenan et a/., 1975 Evans and Thonemann, 1972 Fox e r n / . . 1971, Koganetd.. 1971 Akramova e t a / . . 1984c Bondar and Suran. 1986 Bondar et a / ., I985

t 0.1 k 1.1 ? 0.1 t 0.1 t 0.2

0.2 t 1.5 t 1.5 k 0.5 k

6 1 1

1 1

6 6 6 6

0. I -75.9 + 0.1 -80.4 ? 1 . 1 -78.5 2 0.1 0. I -79.0 ? 1.1 - 76.8 ? 0. I -76.5 2 0.1 -76.7 ? 0.2 -76.4 ? 0.2 -77.8 2 1.5 -77.8 + 1.5 -79.5 ? 0.5 - 78.4

.

(continues)

TABLE I1 (continued)

References

N rg

3 3 3 3 3 3 3 3 3 3 3

Ca Sr Sr Sr Ba Ba Ba He(2s I S,,) He(2s1S,,) He(2s3S,) Xe

2 2 2 2 2 2 2

4 4

K K

1

4

cs

1

1

2 2 1

I

cs

1

Mg Mg Ca Kr

2 2 2 I

16,785 18,790 18,828 17,655 18,790 18,875 16,092 14,415 14,400 14,400 33,860

6 6 6 6 6 6 6 6 6 6 1

9,435 9,435 9,435

- 104.2 f 1.2 -107.8 f 0.2 - 107.0 f 0.2

24 24 24

9,470 18,790 16,790 15,527 33,860

- 108.1 5 0.2 - 104.1 f 0.4

24 24 24 24

-111.1:::

-112.5::; - 115.1 f 0.6

1

- 82.8 -+0U6s -78.8 -77.6':.: - 77.0 -78.6 ? 0.4 -75.3:: - 78.4fO-3 4 5 -79.1:;: - 80.3 - 81.3 f0.2 -0 6 -82.7 f 0.1

Bondar and Suran, 1988 Feldman et a/., 1984 Bondar et a/., 1983 Bondar and Suran, 1988 Akramova et al., 1984b Bondar e t a / . . 1984 Bondar and Suran, 1988 Bakos et a/., 1976 Lompre et al., 1980 Lompre et a/., 1980 Peny and Landen, 1988

- 105.6 2 1.2 - 109.2 2 0.8

Held et a/.. 1972 Delone er a / ., 1973 Delone et a / ., 1973 Arslanbekov er a/., 1975 Normand and Morellec, 1980 Alimov et a/., 1986b Bondar and S u m , 1990 Bondar and S u m . 1988 Perry and Landen, I988

- 108.4 2 0.2 -109.5 f 0.2 -105.5 f 0.6 - 112.5::; - 113.9::; - 1 1 5 . 1 ? 0.6

5 5 5

Na Na Sr Sr Sr Ba Ba

I 2 2 2 2 2 2

9,435 9,435 9,450 9,395 9,435 9,395 9,395

- 138.0 f 1.7 - 136.9 f 0.5 - 140.6'1; - 141.6 - 141.61bA - 138.9 f 1.0 - 138.7'(,:

6 6 6

Ca Ca Ca

2 2 2

9,450 9,395 9,455

-

6 6 6 6 7 7 8

Hg Kr Xe Xe Xe Kr Ar

I 1 2 -

-

14,410 18,870 18,870 17,000 14,410 17,000 17,000

11 I1

Xe Xe

2 2

9,435 9,435

-336 ? 2.0 -338.2

22

He

2

9,435

-731.9;::

5

5 5 5

a

W

1

I 20

- 1 4 0 . 1 2 1.7

1

120 120 120 I20 I20

- 136.9 ? 0.5 - 142.7';; - 143.7 - 143.71;; - 140.0 2 1.0 - 140.81bi

170.2 - 170.3 f 1.2 - 169.4::;

720 720 720

- 173.1'i: -173.2 2 1.2 - 172.311:

- 171.5 f 2.3 - 170.2 f 0.6 - 169 - 173.7 -203.7 f 2.8 -205.5 - 237.5

720 720 I 1 5040 I

-174.4 2 2.3 -173.1 & 0.6 - 169 - 173.7 -207.4 ? 2.8 -205.5 -237.5

Aleksakhin et a / . , 1979 Akramova er al., 1984a.b Bondar et al., 1985 Bondar and Suran, 1986 Chin et al., 1969 Agostini et al., 1970 L'Huillier et al.. 1983a Perry et a / . , 1988 Chin et al., 1969 Perry er al., 1988 Perry et al., 1988

I

-336 f 2.0 -338.2

Arslanbekov and Delone, 1976 L'Huillier er al., 1983b

1

- 731.91::

Lompre et al., 1976

1 1

Delone er a/., 1973 Arslanbekov et 01.. 1975 Aleksakhin et a/., 1979 Feldman er a/., 1984 Bondar et a / ., 1983 Akramova er a/., 1984b Bondar et a / ., 1984

K is the nonlinearity power of the ionization process; o is the laser radiation frequency; ah is the measured value of the cross section; gK is the statistical factor of radiation; a, is the value of the cross section with account of the statistical factor: Ig aK = Ig a; - Ig g K . In the "Measurement method" column, I and 2 refer to the ?bibsolute and relative methods of measuring a,, respectively. All other notations are explained in the text. * indicates the values of a, measured for circularly polarized radiation; all other data have been obtained for linear polarization.

94

Ammosov, Delone, and Ivanov

minimum number of photons that should be absorbed to ionize the atom, i.e., one had Kexp= K. One should note however that the accuracies with which the measured dependencies d(lg N,)ld(lg F) could be approximated by a straight line with the slope Kexp= K were substantially different. This accuracy was determined not only by the type of the laser used and by its stability but also by the type of the ionized atoms. For instance, as a rule, for alkali atoms Kexp= K with high accuracy, and with increase of radiation intensity the powerlike dependence with Kexp= K clearly transformed into the dependence Kexp= 3/2, corresponding to ion-yield saturation. On the contrary, for ionization of alkaline-earth atoms, the dependence Kexp= K , as a rule, is very inaccurate. Sometimes one can qualitatively see that a single powerlike dependence with Kexp= K is not realized (Bondar et al., 1988a). The detailed experimental studies, in which besides ion detection one also measured the energy spectra of produced electrons, have shown that in the case of alkaline-earth atoms, owing to absorption of more than the threshold number K of photons, the single-charge ions are produced not only in the ground state but also in excited states (Delone and Fedorov, 1989a). This process explains the deviation of the observed dependencies d(lg N,)ld(lg F ) from the powerlike ones with Kexp= K. The production of ions in the ground A + and in excited (A+)* states shows that in the standard experiments, where one records only the yield of singlecharge ions, one actually measures a summary effect produced by realization of different channels in which the ions are formed in different final states. Accordingly, all the characteristics extracted from the measured total yield of ions in different final states are also of integral character. This evidently applies both to the ionization probability and to the multiphoton cross section. To a certain extent, the notion of an integral ionization probability as of the probability of production of single-charge ions per unit time (in any final state) is justified, though, as one can see from the principal relations presented in Section I, w is expressed through quantities that depend on K. Numerically, the inaccuracy in w due to the K-dependence of T K is small since the dependence of T K = f ( K ) itself is not very sharp (see Section 1II.F) and the spread in the value of K for processes with production of ions in different final states is also not very large (Delone and Fedorov, 1989a). Both in principle and qualitatively, the situation is worse with cu,, which is determined from the integral yield of single-charge ions produced in different states. From the principal point of view, aKis a function of K and even the dimensions of aKare different at different K. From the quantitative point of view, the strong powerlike dependence of aK = w/FK on F produces a significant inaccuracy in the value of aKdetermined from the integral ion yield. Let us now summarize the situation with the values of the multiphoton cross sections presented in Table 11. On the whole, the data can be clearly divided into

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

95

two groups. The first group is the data on alkali atoms. The experimental results on production of double-charge ions have been obtained only by He et al. ( 1 985) for ionization of alkali atoms by ultraviolet radiation. Double-charge ions have been observed only when the radiation intensity was greater than 200 MW/cmZ (E = 5 X lo5 V/cm). The power of nonlinearity of this process is K = 10. In the visible and near-infrared regions we are interested in, the power of nonlinearity is two to three times higher, so the radiation intensity necessary to produce double-charge ions should be much higher than the value presented previously. However, the measurements of the cross sections of direct alkali atom ionization by radiation from the visible and near-infrared frequency ranges (K =Z 5 ) have been made at E =Z lo5 V/cm. Thus, the values of the multiphoton cross sections presented in Table I1 for alkali atoms have been measured under such conditions (at such frequency w and intensity F of radiation) where both the above-threshold ionization and production of excited ions can be neglected. Thus, the values of multiphoton cross sections for alkali atoms presented in Table I1 are well grounded and, with good accuracy, characterize the threshold ionization channel A

+ Khw-A'

+ e-,

(92)

where the atom absorbs the threshold number K of photons and the ion is produced in the ground state. It is evident that, in principle, these data should be described in the framework of the one-electron approximation by nonstationary perturbation theory in the first nonvanishing (Kth) order. The second group is the data for alkaline-earth atoms. .The experimental data of many studies (see, for example, Suran and Zapesochnyi, 1975; DiMauro et al., 1988; Haugen and Othonos, 1988; and Camus et al., 1989) show that the measurements (whose results are summarized in Table 11) have been made under such conditions (for such w and F) where the ions are produced not only in the ground state (A+) but also in excited (A+)* states. (See, e.g., Figs. 3 and 4.) Thus, the values of multiphoton cross sections for alkaline-earth atoms presented in the table, as a rule, are summary characteristicsfor several ionization channels S of quanta absorbed, in the final state of the ion, differing in the number K and in the electron kinetic energy:

+

Thus, the index K that labels the quantities aKcharacterizes only the threshold (with respect to the number of photons absorbed) channel of ionization. It is evident that any rigorous theoretical description of the ionization process in such cases should be made outside the framework of the one-electron approximation and with account of the structure of concrete channels, of their relative weights, and of the dependence of the latter on radiation intensity. In many cases, the measurements of multiphoton cross sections have been

96

Ammosov,Delone, and Ivanov

made with multifrequency lasers, and in a number of cases with the radiation of the second harmonic generated by such lasers. In all these cases, the calculation of multiphoton cross sections from the experimental results requires that one take into account the statistical factors of radiation, g, or HK . However, the values of these factors are not always known. The situation is the simplest in the case where the measurements are made with the fundamental radiation of multifrequency lasers. Here the statistical factor depends upon the number of generated longitudinal modes N = Av * 2L,. (Av is the width of the laser radiation spectrum in cm-I and L, is the length of the laser resonator.) The value of g , varies from 1 for single-frequencyradiation to K!for radiation with a large number of longitudinal modes. According to Delone et al. (1980), the number of modes for which the asymptotic limit g , = K! is realized depends upon K and varies from N = 10 for K = 2 to N = lo3 for K = 11. The analysis of the experimental conditions under which the results in Table IJ have been obtained shows that in all those cases where multifrequency radiation was used, the value of g , was close to the asymptotic limit K!.For this reason, for the fundamental radiation of multifrequency lasers it was taken that g , = 10". = K !. The situation is more complicated with statistical factors for the second harmonic of multifrequency radiation: here g? depends not only upon the number of modes generated but also upon the efficiency of conversion into the second harmonic. In the limit of low conversion efficiency, where the intensity of the second harmonic is proportional to the fundamental radiation intensity squared, gimcan be expressed through the statistical factors g , and g , of the fundamental radiation: g;to = (FL)/(F2JK= (FZK)/(F;)" = g l K / g f . When the number of modes is large, we have gi- = (2K)!/2K,which is greater than K!.In the limit of high (almost 100%) conversion efficiency, where F, = F,, the statistical factors g? and g , coincide. Thus, for a large number of modes, the statistical factor of the second harmonic of a multifrequency laser, gp, varies between K ! and (2K)!/2K.At K = 2 this is the range from 2 to 6, at K = 3 it is 6 to 90, etc. The absence of any data on the efficiency with which the radiation of a multifrequency laser is converted into the second harmonic, as well as the absence of calculations that would describe the dependence of gi- on this efficiency, increase the uncertainty with which the multiphoton cross sections are found. For this reason, the data in Table I1 on the cross sections are presented in two forms. In the fifth column we present experimental data processing with no account for the statistical factors. These values (denoted as lg ah) can be regarded as multiphoton cross sections only in those cases where the measurements were made with single-frequency lasers. In the case of multifrequency lasers, the quantities lg ah characterize the initial experimental data. In the seventh column we present the data on multiphoton cross sections where the statistical factors, presented in the sixth column of the table, have been taken into account: lg a, = lg a; lg g K .

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

97

As one can see from Table 11, there are several cases where the values of effective cross sections for the same atom under the same conditions (same frequencies and same polarization of radiation) have been measured in several studies. By comparing these data we see that, within the experimental error, they coincide. The exception is the values of the effective cross section of four-photon ionization of the K atom at the frequency of a neodymium laser measured by Held et af. (1972) and by Delone et af. (1973) and of six-photon ionization of the Xe atom at the same frequency measured by Alimov and Delone (1976), by Arslanbekov and Delone (1976), and by L'Huillier et al. (1983a). As of late 1990, there were no additional data that would enable one to tell which result is more correct. It should be noted that the errors presented in Table 11, with which the values of effective cross sections cu, for large K (K = 11, 22) have been found according to the relative method, can hardly be considered reliable. The point is that the studies by Alimov and Delone (1976) and Arslanbekov and Delone (1976) were done a long time ago, and the laser radiation used was not very stable in a series of pulses. Accordingly, the fluctuations in radiation intensity and in the ion yield in a series of pulses were high, but the authors did not always make a correct account of the large deviations of the ion signal from the average value, so the measurement errors presented are probably underestimated. Let us now discuss the results that concern the dependence of effective multiphoton atom ionization cross sections on the radiation polarization. Table I1 presented data on the absolute values of effective cross sections aK measured for circular polarization. Besides this, in a number of studies the authors have measured the ratio of the ion yield N: for circularly polarized radiation to the ion yield N f for linear polarization at fixed F: R = N f / N f . Under the conditions discussed previously, the ratio of the ion yields, R, will coincide with the ratio of multiphoton ionization cross sections for circular and linear polarizations: R = Nf/Nf = ak/ag. It should be noted that since for measuring R one does not have to make absolute measurements of the laser radiation intensity, the value of R can be found with a much higher accuracy than the absolute values of the cross sections. The experimental results for Rcxpobtained for fixed radiation frequencies are presented in Table 111. There also we present the values of R calculated according to the well-known factorial formula (Klarsfeld and Maquet, 1972): R

=

(2K

+

l)!!/K!.

(94)

This relation holds in the one-electron approximation for ionization from an arbitrary initial state with any main (n)and orbital (e)quantum numbers (Krainov and Melikishvili, 1988) in those cases where the power of nonlinearity is not very high (K s 5 ) and the ionization occurs on a frequency for which the compound matrix element Eq. (7) is dominated by the virtual transitions that obey the Bethe rule (n + n + 1; t' + e + 1). The constraint on K comes from

TABLE I11 EXPERIMENTAL A N D THEORETICAL VALUES OF RATIOOF MULTIPHOTON IONIZATION CROSS SECTIONS FOR CIRCULAR A N D LINEARPOLARIZATION OF RADIATION 0,

K

Atom

K

cm-I

R,,,

Rex*

References

2 2 2

cs cs

18,870 14,410 18,870

1.5 1.5 1.5

1.2 -C 0.4 1.28 2 0.19 0.9 -C 0.18

Delone, 1975 Fox e t a l . . 1971; Kogan etal.. 1971 Klewer et a / ., 1977

3 3 3 3 3 3 3 3 3 3 3 3 3

Na Na K Rb cs cs Ca Ca Ca Ca Ca Ca Ca

18,870 14,400 14,400 14,400 14,400 14,410 18,245 18,483 18,504 18,591 18,607 18,790 19,816

2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

0.42 +. 0.08 2.30 2 0.09 2.66 ? 0.11 2.16 2 0.13 2.24 ? 0.11 2.15 2 0.4 0.89 +. 0. I 1 1.01 4 0.14 0.73 2 0.12 0.65 +. 0.10 0.60 2 0.08 I .o 0.03 +. 0.005

3 3

Sr Sr

17,216 17,298

2.5 2.5

2.73 2 0.38 1.47 +. 0.18

Delone, 1975 Cervenan et a / ., 1975 Cervenan et a/., 1975 Cervenan et a/., 1975 Cervenan et a / ., 1975 Foxeta/.. 1971;Koganeta/.. 1971 Bondar et a / ., I988b Bondar et a/., 1988b Bondar et a / . 1988b Bondar et a / . . 1988b Bondar er a/., I988b Akramova et a/. 1984c Bondar et a / ., I984 Bondar and Suran, 1986 Bondar et a / ., 1988b Bondar et a / . 1988b

.

.

.

3 3 3 3 3 3 3 3 3 3 3 3

Sr Sr Sr Sr Sr Ba Ba Ba Ba Ba Ba He(2s3S,,)

18,832 16,642 16,808 16,903 17,106 17,696 I 7.762 17,784 17,894

4 4 4

18,870 14,415

2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

2.5 ? 0.4 2.38 f 0.46 2.85 2 0.35 1.25 ? 0.23 0.79 f 0.11 1.29 f 0.23 1.42 f 0.24 0.91 2 0.18 1.19 ? 0.20 I .o 0.7 2 0.2 2.9 f 1.2

Bondar et a / ., 1983 Bondar e t a / . , 1986, 1987 Bondar e t a / . , 1986, 1987 Bondar e t a / . . 1986, 1987 Bondar ct a / . , 1986, 1987 Bondar er a / . . I988b Bondar et a / ., 1988b Bondar er a / ., 1988b Bondar et a / . . 1988b Akramova er a/. I984c Bondar er a / . , 1984 Bakos ei a / ., 1976

K K

cs

9,435 9,395 9,435

4.4 4.4 4.4

1.67 f 0.51 3.9 2 0.8 4.0

Delone, 1975 Akramova e t a / . . 1983 Delone, 1975

5 5 5 5

Na Sr Ba Ba

9,435 9.435 9,434 9,395

8 8 8 8

0.5 5.5 2 0.090 0.330

6 6

Ca Ca

9.395 9,435

14 14

0.72 f 0.25 1.4 2 0.3

18,790

* 0.1 1.8 2 0.009 f 0.006

.

Delone, 1975 Bondar et a / ., I983 Bondar et a / ., 1984 Akramova et a/.. I984c Akramova er a / ., I984c Bondar et a / ., 1985 Bondar and Suran, 1986

100

Ammosov, Delone, and Ivanov

neglecting the C-dependence of the matrix element that describes the transition into the continuous spectrum. This neglect is justified if the values of C for the final states in the case of linear (e = 1) and in the case of circular (C = K ) polarizations are not very different. With lowering of the contribution to the compound matrix element from the channel where the Bethe rule holds, the value of R also becomes smaller compared to Eq. (94), but still remains above unity: R > 1. The exception is the relatively narrow frequency intervals where there are intermediate resonances allowed in the field with linear polarization and forbidden in the field with circular polarization, so that ak = 0. Thus, one can summarize the situation by saying that under the conditions just specified, the direct multiphoton ionization proceeds more efficiently in the field with circular polarization. The special choice of the frequency that distinguishesthe transitions satisfying the Bethe rule has not been made in any of the studies except Bondar et al. (1988b) and Alimov et al. (1987), where the data for several alkaline-earth atoms have been obtained. Our analysis of the conditions under which the experiments have been made has shown that the necessary criteria are also satisfied by the data obtained by Fox et al. (1971), Cervenan er al. (1975), Delone (1975), Bakos et al. (1976), and Kogan et al. (1971). If of all these data we consider those that can be compared with the factorial relation (94) with sufficient reason, we may conclude that this relation holds for ionization of alkali atoms and breaks down for ionization of alkaline-earth atoms. One should suppose that the discrepancy between the experimental data and Eq. (94) in the case of alkalineearth atoms is associated with the fact that it is impossible to describe the interaction of laser radiation with these atoms in the one-electron approximation, which is also indicated by other data (Delone and Fedorov, 1989a). In a number of studies (Agostini and Lecompte, 1976; Samson, 1982; Bondar et al., 1983, 1984; Akramova et al., 1984c; Bondar and Suran, 1986) the authors have made a detailed examination of how the ion yield N , depends upon the ellipticity parameter 8 of laser radiation (upon the ratio of the polarization ellipsoid half-axes). Under the conditions similar to those that are necessary for the proportionality relation N , aKto hold, the dependencies N,(8) will be proportional to the 8-dependenciesof the ionization cross section a K ,i.e., N,(O) a,(@).In all the studied cases, the dependenciesN,(8) were monotonic curves. With enough justification, the dependencies N,(B) can be compared at present only with the calculations made in the one-electron approximation for the threshold ionization channel (92) (Delone and Krainov, 1985;Smith and Leuchs, 1984). Let us once again turn to the data presented in Table 11. From all these data it follows that at fixed values of the nonlinearity power K,the values of the effective cross section of direct multiphoton atom ionization coincide within the difference between the values of aK at minimum and maximum detuning of the intermediate resonances,

-

-

CROSS SECTIONS OF DIRECT MULTIPHOTON IONIZATION OF ATOMS

101

Let us note in conclusion that despite the relatively large body of experimental data obtained through 1990, there are several reasons why the program of measuring the effective cross sections of direct multiphoton atom ionization cannot be considered complete. In the first place, up to now the detailed experimental studies of multiphoton cross section have been made only for individual groups of atoms (alkalis and alkaline earths). Evidently, one should make similar measurements for other groups of atoms. In the second place, the measurements should be made under such conditions where one knows that neither the above-threshold ionization nor the production of ions in excited states takes place. Accordingly, in addition to detecting the ions, one should also measure the electron-energy spectrum and the luminescence due to relaxation of excited states. The method of recording the energies of produced electrons is well developed and is widely used in the studies of above-threshold ionization (Delone and Fedorov, 1989a, b) and of multicharge ion production (Delone and Fedorov, 1989a). The study of Haugen and Othonos (1989) convinces one that additional information can be obtained by measuring the luminescence. Only when a single ionization channel is dominating can the experimental data be correctly characterized by the value of the multiphoton cross section and the latter can be calculated theoretically in the framework of the corresponding model. When realizing such a program, one should keep in mind that in all the cases one should have constraints upon the intensity of the laser radiation used that arise from the fact that in a strong electric field, there appear dynamic multiphoton resonances where the Stark shift is ~ E ( E2) AE,,"?,. Let us recall that the constraints upon the field strength are actually constraints upon the nonlinearity power of the process that can be recorded.

V. Analytical Expression for Estimating the Multiphoton Cross Sections of Direct Atom Ionization As is known (Delone and Krainov, 1985), in principle, the rigorous calculations of any quantity can be made only for the hydrogen atom, since only there one knows the exact wave function. However, even in this case there arise practical limits in numerical calculations made by a computer. A sufficiently complete account of the results of such calculations made according to the perturbation theory is presented by Karule (1978, 1985, 1988). When one turns to more complicated atoms, even with only one valent electron, there arises a need to use different approximation methods in the framework of the perturbation theory, since neither the exact potential of the frame nor the exact wave functions are known. In practice, the simplest are the methods of the quantum defect and of the model potential (Delone and Krainov, 1985; Faisal,

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1987; Morellec er al., 1982; Manakov et al., 1986). The calculations of the ionization cross sections aK for alkali atoms made with these methods are presented in a number of papers for several values of K and for narrow frequency ranges. However, in order to make such calculations, one has to design special programs for each kind of atom and frequency interval. As was already mentioned, both in the case of hydrogen and in the case of other atoms with a single electron on the external shell (alkali atoms), at sufficiently high field strength, the process of above-threshold ionization begins to play an essential role. For this reason, there always exist constraints upon the field strength more stringent than the requirement E << E,, under which the first nonvanishing (Kth) approximation of the perturbation theory can be used for describing the experimental data. The problem becomes much more complicated as we turn to the atoms that have several equivalent electrons on their external shell. In this case one can no longer ignore the various multielectron effects, which is clearly indicated by different experiments (Delone and Fedorov, 1989a). The theory that makes allowance for multielectron interaction is sufficiently well developed for singlephoton ionization (Samson, 1982; Starace, 1982). As for multiphoton processes, no sufficiently complete theoretical analysis on the subject has yet been made (Wendin et al., 1987). Among the additional difficulties that one encounters in this case, let us mention the need to take into account the various effects produced by a strong light field (Wendin er al., 1987). All this goes to show that for today’s practical needs one is interested in a simple analytical formula that would enable one to estimate the cross sections of multiphoton ionization even if this estimate will have a limited accuracy. One possible way of obtaining a formula such as that is to use the quasi-classical (adiabatic) approximation (Delone and Krainov, 1985). Strictly speaking, this approximation is applicable only to high-lying atomic states, but practice shows that it gives satisfactory quantitative results also for the low-lying states, including the ground state. In our opinion, the fact that when using the quasi-classical approximation for describing the process of multiphoton atom ionization, one loses some of the accuracy, is not of principal importance since the accuracy of the experimental data on multiphoton cross sections is also relatively low. (See Section IV and Table 11.) Besides, in view of the strong powerlike dependence of the probability w on radiation intensity F at sufficiently large K , in practice it is enough to know only the order of magnitude of a K . In the framework of the quasi-classical approximation, the analytical expression for multiphoton cross sections of direct ionization of the hydrogen atom has been obtained by Berson (1981, 1982). Berson has compared his results with the data of the numerical calculations made by Karule (1978, 1985, 1988) and has found that they are in good agreement with each other, including the results for the ground state of the hydrogen atom. For instance, in the case of two-photon

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103

ionization, the difference between the quasi-classical results and the numerical data does not exceed 30%. However, the analytical expressions obtained by Berson (198 1, 1982) are quite complicated: the cross section is expressed through integrals of the complete hypergeometric function whose argument is a combination of the Airy-function and its derivative. For these integrals, approximate analytical expressions have been obtained. Their Comparison with the exact calculation of these integrals by a computer has shown that they coincide within 10 to 30%. As a result, the approximate quasi-classical expression for the ionization cross section of the hydrogen atom from the ground state has the form (y,(w)

=

BKo-(7K+2Y3

(95)

where [a,] = cm2KsecK-1,o is the radiation frequency in eV's, and B, is a coefficient that depends upon the radiation polarization and equals

Bk

1

+

(96)

+

(97)

1.2(35)K10-3'K+'6/(2K I)KU;+1.5

in a field with linear polarization, and

Bi

1.8(41)K10-31K+16/(2Kl)K2K+'.S

in a field with circular polarization. Thus, Eq. (95) enables one to estimate easily the multiphoton cross section of the process of direct ionization of the hydrogen atom with an accuracy no worse than 100%. This is smaller than the standard accuracy of experimental measurements of the multiphoton cross sections. (See Section IV and Table 11.) To see whether it is possible to use Eq. (95) for estimating the multiphoton ionization cross sections in the case of complicated atoms, one may compare its results with experimental data. Such a comparison has been made for linearly polarized radiation with the data presented in Table 11. As it follows from Table 11, the greatest amount of experimental data on the cross sections a, has been obtained for ionization of alkali and alkaline-earth atoms from their ground state by linearly polarized radiation of a ruby laser (w = 1400 cm-I) or of a neodymium laser (o= 9400 cm-I) and of its second harmonic (o = 18,800 cm-I), as well as for several other frequencies close to these, The power of nonlinearity K of the ionization process ranged from 2 to 6. For each value of a, measured experimentally, we have found its ratio j3 to the corresponding theoretical value calculated according to Eq. (95). After that we found the mean value ( p ) and its mean-square error. The values of the coefficients found this way are the following: 3./ = looI ? I ' for alkaline-earth atoms and /3 = 10' for alkali atoms. Let us note that the error in the coefficient j3 is due not only to the approximative character of the formula we used but also to the error in the experimental values of the cross section and to the spread of the data found in different experiments for the same radiation frequency. As for

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noble-gas atoms, for them the coefficients /3 were not found since the experimental data on the ionization cross section of such atoms are very scarce. Thus, the cross section of direct multiphoton ionization of complicated atoms can be estimated according to the following semiphenomenological formula: (YK(W)

= pBKW-‘7K+*’’3,

(98)

where the coefficient /3 depends upon the atom kind (alkali or alkaline earth). Figures 12 through 15 show the calculation results for the aKdependence on K = 2 to 6, obtained according to Eq. (98) for frequencies of the ruby laser, of the neodymium laser and of its second harmonic, and with account of the coefficients p presented previously. Also presented are the experimental results for these and several other nearby frequencies. One can see that, on the whole, there is a satisfactory agreement between the calculations made according to Q. (98) and the experimental data within the indicated errors. Thus, Eq. (98) enables one to estimate the values of the effective cross sections of multiphoton ionization of alkali and alkaline-earth atoms with an accuracy close to the one with which these quantities are measured experimentally. Let us note that from the preceding analysis follows a clear distinction between the cross sections of multiphoton ionization for alkali ( ( p ) = 10) and alkalineearth ((p)= 1) atoms. Earlier, it was usually stated that such a distinction cannot be made within the accuracy of experiments (Delone and Fedorov, 1989a). In reality, as is seen from this analysis, other things being equal, the multiphoton cross sections (and, consequently, the probabilities of multiphoton ionization) in the case of alkali atoms are systematically an order of magnitude higher than they are for alkaline-earth atoms. As of late 1990, there is no unambiguous explanation to this difference. As we have already said, it is very likely that this is a consequence of the fact that, as a rule, in experiments on ionization of alkaline-earth atoms some integral values for several ionization channels differing both in the number of photons absorbed and in the final state of the singlecharge ion were measured. In conclusion, let us note that there are practically no experimental data for other types of atoms. This actually applies to noble-gas atoms too, despite the fact that they have been examined in many studies. The point is that owing to the relatively high ionization potential in these atoms, their ionization in the field of near-infrared and even visible radiation is associated with absorption of a large number of photons. Accordingly, their ionization is observed only at very high field strength, where one can no longer ignore the perturbation of the atom spectrum by the radiation field. The results obtained by Alimov and Delone (1976) and by Agostini and Lecompte (1976) show that under such conditions, it is very difficult to distinguish clearly the direct process of ionization. In order to observe the latter and to measure the multiphoton cross sections for noble-gas atoms, one should use the laser radiation from the ultraviolet frequency range. Only one

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105

-400

-450 3

4

5

6

K

FIG. 12. Effective cross section of direct K-photon ionization aKof alkali atoms by radiation of a neodymium laser (o= 9400 cm-I) versus the nonlinearity power K. The two solid lines show the dependence of aK calculated according to Eq. (98) for two extreme values of the p coefficient for alkali atoms: p, = lo1O + l and = 10' 6. Error bars include the values of aKtogether with their errors measured experimentally for o = 9400 cm-l and several nearby frequencies. Data presented are based on the results of Held er al. (1972). Delone et al. (1973). and Normand and Morellec ( 1980).

-400 - (50

FIG. 13. Effective cross section of direct K-photon ionization ah.of alkali atoms by radiation of a ruby laser (o=I 14,400 cm-I) versus the nonlinearity power K. The notations are the same as in Fig. 12. For comparison, we also present the existing experimental data on ionization of Hg atoms (K = 6) by radiation of a ruby laser. The data presented are based on the results of Fox et al. (1971), Evans and Thonemann (1972). Cervenan er al. (197% Cervenan and lsenor (1974). and Kogan er al. (1971).

Ammosov,Delone, and Ivanov

106

pg(c, [cm2HsecK-’1)

1

2

3

4

K

FIG. 14. Effective cross section of direct K-photon ionization aKof alkali atoms by radiation of the second harmonic of a neodymium laser (w = 18,800 cm-I) versus the nonlinearity power K. The notations are the same as in Fig. 12. The data presented are based on the results of Held et al. (1972) and Delone et al. (1973).

-150

-200 3 4 5 6 7 K FIG. 15. Effective cross section of direct K-photon ionization aKof alkaline-earth atoms by radiation of a neodymium laser (w = 9400 cm-I) versus the power of nonlinearity K. The notations are the same as in Fig. 12. The values of the coefficients for alkaline-earth atoms are PI = 10-oJ+J7 and fi2 = 10-o The data presented are based on the results of Aleksakhin er al. (1979), Bondar ef al. (1983, 1985). Akramova er al. (1984a, b, c), Feldman eta!. (1984), and Bondar and Suran (1986).

’.

such experiment is known (McCown et al., 1982) in which the two-photon cross section of direct ionization of the xenon atom has been measured: a2= cm4 X sec-I. Within the accuracy of the experiment, this two-photon cross section is of the same order of magnitude as the two-photon ionization cross sections for alkali atoms. (See Table 11.) However, as shown by the experimental data we

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107

discussed previously (Table 11), one cannot make any definite conclusions based on the results of a single experiment; for that one needs a large number of measurements. Only the systematic measurements of multiphoton cross sections of direct ionization of noble-gas atoms at different powers of nonlinearity of these processes will enable one to make conclusions about the magnitude of these cross sections and their relation to the cross sections for atoms with other structures of the external electron shell.

VI. Conclusion Summarizing the material we have presented in this chapter, let us accentuate our viewpoint on the state of different directions of study and the value of the results obtained. The methods of measuring the cross sections of direct multiphoton atom ionization are developed well and in detail. This refers to the absolute and the relative methods, to the means of taking into account the nonmonochromaticity of laser radiation, and to the measuring techniques for all the parameters that determine multiphoton cross sections. Unfortunately, there are yet no experimental data on multiphoton cross sections for the hydrogen atom. Such data would have been very valuable in view of the high reliability of the curresponding theoretical calculations. Brewer el al. (1989) have shown that such an experiment can be made at present. The set of experimental data related to multiphoton ionization of alkali atoms clearly shows that under the conditions of experiments where the multiphoton cross sections are measured, the threshold ionization channel dominates, and it is to this channel that the measured values of the cross section refer. Their theoretical interpretation is substantiated in the framework of the one-electron approximation by using the first nonvanishing perturbation theory approximation. The results of such calculations give a good description to the experimental data (Delone and Krainov, 1985; Faisal, 1987; Morellec et al., 1982; Chin and Lambropoulos, 1984). The set of experimental data on multiphoton ionization of alkaline-earth atoms shows that under the conditions of experiments in which these cross sections were measured, the single-charge ions, as a rule, are produced via several channels differing in the number of photons absorbed and in the final state of the ion, which may be either in the ground or in an excited state. This gives one reason to believe that the measured cross sections are summary values over several channels. Of great interest are the experimental data on the cross sections for other atoms with many valence electrons, data that are not available at present. In the case of

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noble-gas atoms, data for processes with a relatively small number of photons can be obtained at present by using coherent ultraviolet radiation. For all atoms and radiation frequencies, there exist constraints upon the field strength at which the direct and resonance ionization processes can be distinguished from each other. These constraints are associated with a large dynamic Stark shift that changes the transition energies. Accordingly, there are constraints on the number of photons the process involves. The practical value is evident for simple analytical expressions that enable one to estimate the cross sections of direct multiphoton ionization of any atom at any frequency of radiation with sufficiently high accuracy for a wide variety of applications. Nevertheless, the existence of such approximate formulae for the cross sections does not mean that there is no need to develop rigorous methods of calculating the multiphoton ionization cross sections for multielectron atoms, that go beyond the one-electron approximation and make allowance for the whole set of multielectron phenomena observed experimentally.

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ADVANCES IN ATOMIC MOLECULAR AND OPTICAL PHYSICS VOL . 29

COLLISION-INDUCED COHERFNC'ES IN OPTICAL PHYSICS G . S. AGARWAL School of Physics University of Hyderabad Hyderabad. India

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. A General Framework for the Calculation of Nonlinear Optical Phenomena . . 111. Second-Order Optical Response and Collision-InducedCoherences . . . . . IV. Collision-Induced Coherences in Fluorescence and Ionization Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Hanle Effect: Collisional Effects . . . . . . . . . . . . . . . . B . Extra Resonance in Fluorescence from Open Two-Level Systems . . . C . Optogalvanic Spectroscopy and Collision-Induced Resonances . . . . V. Collision-Induced Coherences in Third-Order Nonlinear ResponseFour-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . A. Resonance at Optical Frequency. . . . . . . . . . . . . . . . . B . Resonances Induced by an Inelastic Process . . . . . . . . . . . . VI . Collision-Induced Coherences in Probe Absorption in the Presence ofahmp . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Relation between Collision-Induced Coherences in Second-Order and Third-Order Responses . . . . . . . . . . . . . . . . . . . . . . . VIII. Collision-Induced and -Enhanced Resonances in Fifth-Order Nonlinearities . IX . Effect of Cross-Relaxation on Collision-Induced Resonances . . . . . . . X. Dipole-Dipole Interaction-Induced Resonances . . . . . . . . . . . . XI . Collision-InducedResonancesinSpontaneousRocesses . . . . . . . . A . Collision-Induced Radiation from Trapped States . . . . . . . . . . XI1. Narrowing and Enhancement of Signals Due to Velocity-Changing Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI11. Nonlinear Response, Collisions, and Dressed States . . . . . . . . . . XIV. Other Sources of Coherences Similar to Collision-Induced Coherences. . . A . Extra Resonances in the Transient Response . . . . . . . . . . . . B . Extra Resonances Due to the Fluctuations of Pump and Probe: Fluctuation-Induced Extra Resonances . . . . . . . . . . . . . . C . Saturation-Induced Extra Resonances . . . . . . . . . . . . . . XV. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Copyright 0 1992 by Academic F'ress Inc . All rights of reproduction in any form reserved ISBN 0-12-003829-3

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G.S . Agarwal

I. Introduction The optical nonlinear response of a system subjected to external fields has been studied extensively. In most works the relaxation or the damping was assumed to broaden the spectral lines. It was not until Bloembergen and coworkers (1978) discovered that collisions with the atoms of the buffer gas can lead to new coherences in the nonlinear response of the system. These resonances were extensively studied by Bloembergen and coworkers (Prior e f al., 1981; Bogdan et d., 1981a,b; Bloembergen e f af., 1985; Rothberg and Bloembergen, 1984a,b; Zou and Bloembergen, 1986) in a series of four-wave-mixing experiments in sodium and in samarium. Grynberg (198 1a) also predicted collision-inducedcoherences among Zeeman levels that were verified by Bloembergen et al. (1985) and Scholz et af. (1983). Later the present author (Agarwal, 1986) showed that fourwave mixing is not the only way to observe collision-induced coherences. These in fact can also be seen in fluorescence spectroscopy which involves a secondorder nonlinear process rather than third-order process. The collision-induced coherence in fluorescence was verified by Lange (1986). Liao et af. (1979) studied the collisional redistribution in two-photon absorption. They observed twophoton spectra of sodium 3Sl,,-4D,,, transition at various pressures of neon perturber gas. They observed the appearance of the broad resonance near the 3P,,, --f 4D,,, transition frequency. The intensity of the resonance increased with increase in pressure. The collision-induced coherence was also shown to be important in saturated absorption experiments (Grandclement et al., 1987a,b; Gong and Zou, 1988). Collision-induced narrowing of the longitudinal linewidth in four-wave mixing was observed by Lam et al. (1982). Giacobino and Berman (1987) reported collision-induced coherence at optical frequencies. The collision-induced or, rather, -enhanced coherence has also been studied extensively in higher-order (Trebino and Rahn, 1987; Agarwal and Nayak 1984, 1986) nonlinear mixing processes such as six-wave and eight-wave mixing. Resonances similar to those induced by collisions were also shown to occur due to fluctuations (Agarwal and Kunasz, 1983; Agarwal et al., 1987a,b,c; Prior et al., 1985; Agarwal, 1988) of the field and due to saturation or due to higher-order effects in intensities (Agarwal and Nayak 1984, 1986; Friedmann and Wilson-Gordon, 1983, 1984, 1987; Grynberg and Pinard, 1986). Let us start with a classic example. Consider the motion of free electrons in a bichromatic field. The displacement satisfies the equation

where we assume that the input fields are linearly polarized along the x axis. r is the collisional parameter. The response at the frequency (2w, - w2) is easily

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

115

obtained from (1): e3

x'3' =

--

&2

*

IE2e

-i(2wl-w?N

m3c2 D(2ul - w 2 ) ( - iwl)( - 2 i w J

D(w)Ww,)

+ [ - i(o,- 4

+ A}], (2)

1

D(o) = ( - i ~ ) ~i d . The square bracket in response function (2) can be approximated by

Thus, the nonlinear response exhibits the resonance at o1= w2 only if the collisions in plasma are accounted for. It may be added that there is the whole field of collisonal redistribution (Cooper, 1980) of radiation, i e . , the light can be scattered at frequencies that are different than the incident frequencies. The scattering at such frequencies depends on the presence of collisions. Such redistribution of radiation has been extensively studied notably by Cooper and coworkers (Cooper, 1980). As an example, consider the light of frequency we scattered by a two-level atom (Fig. 1) of frequency oo.The spectrum of the scattered light has the form

~

wn

~

'

1

1

-=21,TI

~ I

2

yP+I

T2

FIG. 1. Schematic illustration of a two-level system. Note that 2y is the spontaneous emission rate and r p is the contribution from phase-changing collisions to the decay rate of the off-diagonal element plzof the density matrix.

G. S. Agarwal

116

where T I and T2are the longitudinal- and transverse-relaxation constants. Thus, the resonance w = wo in the spectrum of the scattered light is due to the collisions and is in a sense analog of the collision-induced coherence. Note that in collisional redistribution one deals with the spectrum that involves two time correlations which is in contrast to the physical phenomena mentioned in the previous paragraph, which are determined by macroscopic values like polarization, inversion, and other coherences in the system. The organization of this chapter is as follows: In Section 11 we present a general framework (Kumar and Agarwal, 1986) for the calculation of nonlinear susceptibilities by taking into account spontaneous emission, and elastic and inelastic collisions. We also show how other physical observables such as fluorescence can be calculated. Using these general expressions we demonstrate, in the next few sections, the existence of collision-induced resonance in fluorescence and ionization spectroscopy, saturated absorption spectroscopy, four-wave mixing, and six-wave mixing. We also show the relation between the collision-induced resonances in second-order and third-order response of the system. In Section IX we discuss the effects of cross-relaxation(Kumar and Agarwal, 1987)on collisioninduced coherences. In Section X we show the existence of new resonances due to dipole-dipole interaction (Varada and Agarwal, 1991). In Section XI we exhibit collision-induced resonances in “spontaneous” processes which are described in terms of the two time-correlation functions of the atomic-polarization operators. In Section XI1 we treat the question of narrowing and enhancement of signals due to velocity-changing collisions. In Section XI11 we show how the dephasing-induced phenomena can be understood in terms of the dressed states. In Section XIV we discuss several other mechanisms that lead to coherences similar to collision-induced coherences. The article is concluded with remarks on selective reflection and new directions.

II. A General Framework for the Calculation of Nonlinear Optical Phenomena We present a general formulation (Kumar and Agarwal, 1986) for the calculation of the nonlinear susceptibilities of a quantum mechanical system interacting with external fields. The collisional and radiative decay effects are accounted for. Let Ho be the unperturbed Hamiltonian of the system and H,(r) be the interaction Hamiltonian with external fields. The equation of motion for the density matrix p is I aP _ (5) - -- W,f(f),PI - iLOPI at

h

where Lo is the unperturbed Liouvillean and it includes the effects of collisions

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117

and spontaneous emission. Before the application of the external fields, the system is in equilibrium state p(O)defined by L0p'O' = 0 .

(6)

The structure of the Liouville operator Lo depends on the model of relaxation. Here we take Lo to have structure [ - iLop11, = ( - iwl,Pl,

- r , P , ) ( l - 6,) + 6,

c

(YtkPU

- Yk,P,J.

(7)

Here w,s are the transition frequencies associated with the unperturbed Hamiltonian Ho. The quantity yI, gives the inelastic collision rate from the state 1j ) to the state li). It also includes contribution from spontaneous emission if E, > E l , i.e., if qI> 0. Note that the rate of inelastic collision yl, will be zero for the optical transitions but it could be significant for nearby levels. Note further that y,, can be made nonzero by pumping the level li) from the level l j ) by using a broadband field. The decay of the coherences (off-diagonal elements of the density matrix) is given by Ts. These also include contributions from phasechanging collisions, i.e.,

The model (7) is based on the impact approximation for collisions. The collisions occur instantaneously with respect to various other time scales in the problem. Note that the model (7) assumes transitions within the states of the system. If the population leaks out of the system, then (7) can be generalized to include terms of the form (-iL&Il, = - r : , p , +

PA,.

(9)

Here p I is the rate at which the level li) is being pumped. Such a pumping is essential to maintain the equilibrium if r:, # 0. The pumping terms determine pc0).We shall mostly work with the model (7). The nonlinear-response theory can be developed by expanding p in various powers in H,. We would be considering the response of the system to external electric fields and since we would only be interested in dipolar response, we write H,(t) as

Here da is ath component of the dipole moment operator. Let us introduce the Liouville operator Lf

L,W

= [H,(t), Ilh.

(11)

118

G . S. Agarwal

The quantum mechanical time-dependent perturbation theory then shows that the density matrix pen) to nth order in applied fields is

x L,(w2) .

(&I

- w,,)-'Lf(w,,)pfo).

On combining Eqs. (lo)-( 12), we can calculate the response of the dynamical variable Q to external fields. The nth-order contribution to (Q(t))will be

dQ = [da,

1.

(14)

Here Sym stands for the symmetrization of the nth-rank tensor in front of it in the indices ( a l w l ) (, a 2 w 2 ) ,. . . , (~,Jo,,).Note that if Q is chosen to be the dipolemoment operator, then Slff)gives the usual nonlinear susceptibilities

x$il

an(wl

. . . 0,)

= n S j ~ ,ajw1

. . . wn)(p=dm,

(15)

where n is the density of atoms/molecules. Equation (15) assumes independent particle approximation. In what follows we shall use both S'")and x'"'. The fourwave-mixing and saturated-absorptionexperiments will be described by appropriate third-order nonlinear susceptibilities. For the purpose of calculating the laser-induced fluorescence we need to know, say, the population in the excited state. We shall see that collision-induced coherences can be seen in a variety of experiments involving four-wave mixing, saturated absorption, and laser-induced fluorescence. In order to obtain explicit expressions for the nonlinear response, we need the eigenfunctions of the Liouville operator Lo. From the structure (7) it is clear that Lo(i)(j(= A,lli)(j(,

A,

= 0,) -

iT,],

i # j.

(16)

In addition, Lo has eigenfunctions that depend on the inelastic part of the relaxation matrix. Let us diagonalize the matrix R defined by

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

119

It turns out that "04k

=

iAk4kr

4 k

=

2e

sf'k[e)(el.

(18)

The complete set of eigenfunctions La are given by (16) and (18). These depend on the collisional parameters. Note that the action of a function of Lo on any operator can be obtained from f(Lo)Q

=

x

k#e

Q M ~ ( L o ) I ~ )+ ( ~ IC, Q , k f ( L d ) k ) ( k j

The expression for first-order response is rather simple:

In deriving (20) we have used the property pi? = pi!'&. The first-order response is determined only by the relaxation of the coherences. Note that while the width of the resonance is dependent on the collisional dephasing, the collisions themselves do not lead to any new resonances.

III. Second-Order Optical Response and Collision-Induced Coherences We next consider the calculation of the second-order response. Here the model of relaxation can make substantial difference. We show that the collision-induced coherences appear in the second-order response of appropriate variables. Using (16) and (14) we can show that

120

G. S. Aganval

To simplify (21) we use (17) and (18):

Be,(W)

=

(W

- iAp)-lSqp(S-')pe.

P AP#O

Note that if one had used the decay model (9) (with yii = 0), then one would get the much simpler result (W

- Lo)-qe)(el

=

(W

+ iTee)-llt)(t1,

(23)

which amounts to using Beq(W)

+ See)-'.

= 6,(w

(24)

In view of (24) it is useful to introduce the auxiliary function Cqe(W) = B q d W ) - 6eq(W

+

iree)-l,

(25)

which identically vanishes for the model (9). Using (22) and (25) in (21) we get (Agarwal, 1986)

up =

0,

+

-

w2.

This is the most general expression for the second-order response of a system. Note that the third term (with (YO, ow2) and fourth term in the first square bracket can be combined to yield

s,

=

d$d$Qkj[Ajk

(Afi -

-

Wp

Wp)(Aik

+

i(rjk

-

rji

-

rik)]

- WJCAji - WI)

(27)

121

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

It is obvious from (27) that a resonance in Sjtdlpat w, = A,, can be seen if the decay of coherences is such that

r,, - r,, - rltz 0.

(28)

In addition, Q, terms show that one can have a resonance at w,

= 0 provided that

2r,t # -A,

(29)

where A, is one of the eigenvalues associated with the relaxation matrix defined by (17). The condition (29) depends on both phase changing and inelastic collisions. Thus, for a two-level atom with radiative relaxation and collisional dephasing, the dephasing-induced resonance w, = 0 will have a width 27, i.e., just the radiative width. We have thus shown that the second-order response of appropriately defined physical quantities can have two types of collision-induced resonances: w,k = w, and w, = 0. In the next section we discuss an explicit example of such resonances. The pressure-induced coherence between two ground levels (Bloembergen et al., 1985; Gong and Zou, 1988) can also be seen from (26). Consider a specific situation with two ground states Igl) and (g2)and say one excited state le) (Fig. 2). Let Q represent the operator (gl)(g21.We combine the first two terms in the first square bracket in (26). This leads to

S m

-

=

-

d:&fgl

2(A,,,, -

Pi:\z(Aegl - W)

[

-

-

~ ~ z e ~ ~ g l ~ i ~ ’ , l [u~ pg z ~ il( r e g l

+

rgze

wp)(Ag2e - wl)(Aegl

-

02)

-

]

- w,)

pk:bl(Ag2e

(Ag2e- w N L g 1 -

UP)

2(Ag2gl -

+

rg,gJI

4

(30) (31)

if p(Bq)gl= p60:,,. Expression (31) clearly shows that there is no ground-state Hanle resonance in the absence of collisions and in the absence of population imbalance between is negligible and the two ground levels Isl)and Is2). For Na ground state rg,g2 the Hanle resonance has a width essentially determined by the residual Doppler

FIG. 2. Schematic illustration of a three-level model used in the text for ground-state coherences.

122

G. S. Agarwal

14)
Is> (91

19>
F

G

19>
FIG. 3. Eight diagrams that contribute to the dephasing-dependent second-order response of the variable Q.

width (Bloembergen et al., 1985) which is inversely proportional to pressure in the range 100- 1000 tom of argon. Expression (30) also shows that if one were to approximate the intermediatestate denominators such as (Aeg, - w2)-I by the detuning factor l/A, then all the pressure-induced contributions are missed, e.g., the square bracket in (30) would reduce to (&\, - &\,)/A. Note that the usual polarizability theory of Raman scattering or four-wave mixing will miss the pressure-induced contributions because in such a theory (A, - 0J-I IIA. However, Agarwal and Cooper (1982) showed that one can modify the polarizability theory and describe the dynamics in the projected space consisting of levels Igl) and Ig2) by Bloch-like equations that have additional contributions due to collisional dephasing. We next discuss how the second-order response can be written as a sum of two-sided diagrams. We shall introduce two sets of diagrams: one depending only on the collisional dephasing and the other depending on inelastic collisions. The eight diagrams giving the C-independent terms in (26) are given in Fig. 3. The rules for writing the contribution from a given diagram are as follows: Consider, for example, the diagram E, which consists of the following transitions:

-

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

123

For the transition occurring on the ket side, the state lg) has changed to ( j )so include a matrix element d,. For transition on the bra side, include the matrix element ( - d & ) . The response of Q will involve Tr(Qb)(kl) and hence one should include a matrix element Q,. Next we include two propagators associated with each bra-ket combination after the interaction. Thus, the propagator associated with lj)(gl will be (AjR- w I ) - l whereas the propagator associated with lj)(kl will be (Ajk - w, @*)-I. Note that the frequency of each absorbed (emitted) photon is taken as positive (negative). Thus, the diagram E gives

+

Similarly, the diagram F will be equal to

Note that the combined contribution of E and F is

which is the same as (27). We next consider the contribution of terms such as C in (26). Consider the diagram (Fig. 4) representing the transitions

Here in the last step an inelastic transition (represented by bold lines in the diagram) has taken the system from the state l j ) to ( k ) . From Eq. (22) it is clear

J FIG. 4. A typical second-orderdiagram involving inelastic collisions.

124

G . S. Aganval

that this transition will involve the propagator - Cjk(w, - w2). Thus, the contribution from the preceding diagram J is

A different time ordering of wI and w2 will change the propagator (Ajg- w I ) - l in (35) to (Aa w2)-l. The other diagrams and their contributions can be similarly written down.

+

IV. Collision-Induced Coherences in Fluorescence and Ionization Spectroscopy Very often the atomic structure is studied by fluorescence or by ionization spectroscopy. In such studies the system is weakly excited by an external laser field and one monitors the excited-state populations and coherences by observing fluorescence or ionization signals. Thus, the observed information is contained in the atomic response to second order in fields. We have presented a fairly general expression for the atomic response in the previous section. We next examine the consequences of the general structure (26) of the second-order nonlinear response. We consider specific systems. A. HANLEEFFECT:COLLISIONAL EFFECTS

Consider a typical atomic-cell Hanle experiment involving the transitions between j = 0 to j = 1 levels (Fig. 5). Let the system be interacting with a static magnetic field along the z axis and with a modulated electromagnetic field of frequency we which is, say, polarized along the x axis. The exciting field is taken to have the form &t) = 2E(1

+ M cos

Rt)e-iw(l

+ C.C.

(36)

where M is the modulation index and R is the modulation frequency. The excit-

2s

FIG. 5 . Schematic illustration of a three-level model used for excited-state Hanle resonances.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

125

ing field couples the state lj = 0, m = 0) = 18) and ( j = 1, m = 1) = 11). Ij = 1, m = - 1) = 12). The fluorescence signal can be detected in various directions. Let I, and I - be the signal along the z and x directions with linear polarization x and y respectively. It can be shown that the detected signal not only depends on the populations in the excited states but also on the coherence between two Zeeman sublevels (Saxena and Agarwal, 1982): p l r + p22

I,

* 2Rep12.

(37)

The density matrix elements to second order in the applied fields can be obtained from (26). The d.c. component in fluorescence can be obtained by setting M = 0 , 0 , = w e , w2 = - w e , p! = and by choosing the operator Q in appropriate fashion. For example, to calculate p I 2 ,we take Q = )2)(11, i.e., Q, = S,S,,. According to (27) we shall have a term in pL2with the structure

Note that w12 is equal to the Zeeman splitting between the levels 11) and 12). Thus, a scan of fluorescence as a function of the magnetic field will exhibit a resonance at zero value of the magnetic field as long as r12

#

r13

+

(39)

r23.

Note that (39) requires the presence of phase-changing collisions

r12= 27 + P ( T , ~ ,

L3 = Y + pal39

etc.

(40)

and, thus, r 1 2

-

r 1 3

-

r z 3

= p ( a12 -

(T13

-

(T23)*

(41)

Here p is the pressure of the buffer gas. It is thus clear that the zero magnetic field Hanle resonance is induced by collisions. The modulated component of fluorescence can be similarly obtained, say, by choosing w , = (we + R), w2 = -Up. Equation (27) now shows the presence of the collision-induced resonances whenever wI2 = k R. These collisioninduced resonances can be seen either in the scan of the signal as a function of the modulation frequency (for fixed magnetic field) or as a function of the rnagnetic field (for fixed modulation frequency). In addition, the modulated companent of fluorescence can have an additional resonance coming from the excitedstate population. In order to obtain p I I, we choose Q = 11)(1);then the last term in (26) leads to

The C terms have the structure C,, = (R

+ 2iy-I

and, thus, the resonance at

126

G . S. Agarwal

zero value of fi will be present provided that which on using (40)implies p f 0. Thus, the scan of the modulated component of the ffuorescence will also exhibit a collision-induced resonunce at 0 = 0. The width of this resonance is independent of collisions. Since the system under consideration is rather simple, it is possible to obtain closed-form expressions (Saxena and Agarwal, 1982) for the signals It. The density-matrix equations for the model three-level system in a frame rotating with the frequency ofof the external field can be written as aPll = - 2 y l p l I -

at

ap22 = -2yg22

at

+ ( i ~ x ~ ( t +) pc.c.) ~~

+

(ia2(t)p32

+ C.C.),

Here

and the detuning parameter A is as shown in Fig. 5. For the case of amplitudemodulated field e(t) = .so(1 M cos fit). The signals to second order in the field can be calculated from (44). For the special case aI= a2 = a,y1 = yz = y , and for large detunings A we find that the d.c. part of the fluorescence is given by

+

Clearly the collision-induced coherence is quite significant in the difference (I, - I-). If all the uilare equal to u,then for large pressures we get (I, - I-) = A2

[I

+

+

4s2 p2uzp2u2

1

(47)

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

127

Note that the signal is proportional to A - z unlike the four-wave mixing signal which is proportional to A-6. Thus, for large detunings it may be advantageous to study fluorescence rather than four-wave mixing. The modulated signal can be similarly obtained. Calculations show that

I, =

2MlalZcos Qr A2

2MlaI2 x, + sin QrY, , A2

(48)

where

Y, =

\(2y

+ pu)2 + (2s + Q)'

I

I-

4up.n 4y2 n 2

+

up(Q

+ 2s)

* ((ZY + pu)* + (a + 2s)* - n +

)

-0 .

Collision-induced coherence at Q = +2s, 0 are evident from (49) and (50). Note that p12(Q)can be produced by the following transitions: 13)(31 "2' 11)(31 2' 11)(21, 13)(31 2 13)(21 "%a 11)(21, represented by the diagrams E and F of Fig. 3. The collision-induced coherences have been seen by Lange (1986) using fluorescence from Ba vapor contained in a heated stainless cell. The light from a N z laser pumped dye laser was detuned by 10" to 10I2Hz from the IS,-'P, transition (A = 553.5 nm). The buffer gas was argon. The fluorescence polarized perpendicular to the magnetic field was collected. Figure 6 clearly shows the collisioninduced coherence at zero magnetic field. The amplitude of the Lorentzian in the figure increases linearly with laser power.

B. EXTRARESONANCE IN FLUORESCENCE FROM OPENTWO-LEVEL SYSTEMS We next consider an open two-level system with states denoted by 11) and 12). Let each state be pumped externally at some rate p, and let each state decay to some other levels at the rates rl, and r;, (cf. Eq. 9). Let the two-level system be

G . S . Agarwal

128

120 B (mT) X lO”rad/sto the red. Buffer gas is argon (p-25 hPa). The drawn line is a Lorentzian fit (Lange, 1986). -00

-40

0

40

80

FIG. 6. Hanle resonance in fluoresence. The laser is detuned by 2.5

driven by two fields of frequencies o,and 0,. Let us ignore all other sources of relaxation. The resulting fluorescence intensity can be obtained from (26) by choosing Q = I1)( 1I and by dropping all the terms involving Cs. The component of fluorescence oscillating at the frequency 0 , - o2can show an extra resonance at o,= w2 with a width I‘ll if

r:, - 2r1, z 0 ,

ri2 = wll+ r , ~ ,

(51)

i.e., if r;, # 0. We thus find that the component of fluorescence at w1 - 0, shows extra resonance that is induced due to the decay of the ground state (r;2 + 0).

C. OPTOGALVANIC SPECTROSCOPY AND COLLISION-INDUCED RESONANCES The optogalvanic spectroscopy (Goldsmith and Lawler, 1981) has been used as an important method for studying atomic-molecular transitions in gas discharges and flames. In a discharge the electron density is determined by various collisional and recombination processes. The atoms in the excited states get ionized due to electron impact. In addition, if the discharge interacts with a laser field, then the electron density changes as the laser field pumps population from one

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

129

state to the other state. Thus, the laser field changes the impedance of the discharge. If the field is modulated at a certain frequency R, then the optogalvanic signal can be observed at R, 2R, etc. The observed optogalvanic signal is directly related (Miyazaki et al., 1983) to the populations of the atoms in different states. The changes in the atomic populations can be directly obtained from Eiq. (26). Note that ply) gives the population of the ith state before the application of the field. The pi:) is determined by the collisional and radiative-decay parameters, y and r as well as the pump parameters, p,. For the purpose of treating optogalvanic effects, we have to include both relaxation and pumping mechanisms (7) and (9). Note further that all the collisional dynamics is contained in the matrix C (defined by (25)). For a specific system the optogalvanic effect can be obtained from (26) by choosing Q = le)(e( and by choosing appropriate combinations of w , and w,. For a single field of frequency we modulated at R we can choose 0, = -we, w2 = + w e + R to get a signal oscillating as e-iCli. For two fields with different modulations R, and R2,we can choose W I = wt + R I , w2 = -we + to get the signal oscillating as e-i(ni+nz)r. Note that (26) takes into account all the coherence effects whereas traditionally one obtains optogalvanic signals by using rate equations. Furthermore, Rs need not be small compared with the relaxation coefficients. The optogalvanic signal will exhibit various atomic resonances at Wy = qi if the frequency of the laser is scanned. On the other hand, if we is detuned far from the transition frequencies, then the optogalvanic signal would exhibit a collision-induced resonance at R = 0 or at R, R, = 0 for the two situations mentioned previously. The width of the resonance is determined by the eigenvalues A of the matrix R defined by (17). Finally, note that the interplay between collisional and intense field effects in optogalvanic spectroscopy has been discussed (Tewari and Kumari, 1989).

+

V. Collision-Induced Coherences in Third-Order Nonlinear Response-Four-Wave Mixing In this section we consider the general theory of third-order nonlinear optical phenomena and show the existence of collision-induced coherences. A closed form for the third-order nonlinear response is given by (14). The derivation of the explicit form is similar to the derivation of (26) and we quote the final result

G. S. Agarwal

130

Other contributions C3'and D3)are given by Cs,3v(wl,w2, w3) =

pf)dkndnkdijdji i.n.kJ

I}

+ 5 permutations

(54)

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

131

We have followed Bloembergen et d ' s notation, i.e., papy stands for dGdEkd$dJ. This is the most general form of xI3) for a system undergoing relaxation described by the Eq. (7). All the terms involving C,,(o) are new and these arise as a result of population changes in the system due to collisions and spontaneous emission. If we put Cafl(w)= 0, then the preceding expression reduces to B(3) which is just the result of Bloembergen et al. (1978). If the system has no permanent dipole moment, then the contribution P 3 )vanishes. Thus, the contribution is essentially due to inelastic collisions. In contrast to the contribution B(3),P3)has resonances whenever w, w2 = 0, the width of such Rayleighlike terms being determined from the inelastic collisions. Moreover, such Rayleigh-like terms also have the possibility of two intermediate states resonating with one of the applied frequencies. Consider the two terms in the square bracket in B"'.

+

B E

(Aln -

-

( ' +

4 A,,

Clearly the nonlinear polarization at w, oIn= w , w3 if

+

rjn

#

-

~3

'

A," -

01

1.

(56)

+ w2 + w3 will exhibit a resonance at

rll+ Ti,, .

(57)

Note that the optical transitions to states ( j ) and In) starting from ground state

G. S.Aganval

132

are allowed whereas no optical transition between I j ) and In) is allowed. On using (8) the condition (57) reads

r;" - rt - rfhz o

(58)

provided that the ground state Ii) does not decay. Therefore, the resonance at win = o, o3is induced by the phase-changing collisions. Note that such additional resonances also occur in D3).(See, for example, the squared bracket with a prime in D3).) Various terms in x") can be expressed in terms of double-sided diagrams. These diagrams have been discussed by Fujimoto and Yee (1983) and by Prior (1984) for the case when no inelastic collisions are included and when the spontaneous emission occurs to states outside the states of interest. Kumar and Agarwal (1991) have shown how diagrammatic methods can be generalized to ). are eight basic diagrams as there obtain the inelastic contributions to x ( ~ There are eight ways of putting three interactions on two sides. The contribution from a typical diagram (Fig. 7) can be obtained as follows (Prior, 1984):

+

(1) Use the initial density matrix psa. (2) Four matrix elements are obtained by tracing the left line (ket) upward and the right line (bra) downward. One writes a matrix element for each interaction. One writes a component of the matrix element depending on the polarization of the field involved at the transition. The field at the frequency o, is taken to be involved at the top of the diagram. Thus, for the diagram, we shall write a matrix element d18d",$(

- d$) = dbdzd$d$*.

Here a minus sign is included because one of the interactions is on the bra side.

19)


A

Is>


lg>

a1 C

FIG. 7. Some of the diagrams contributing to ~ ' 3 1 .The collision-induced coherence can be seen by combining tbe contributions of the triplet of diagrams such as these.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

133

(3) The product of three propagators is obtained by advancing in time until one has crossed an interaction vortex. Photons absorbed (emitted) are drawn as lines pointing upward (downward) from left to right. The propagator is given by the inverse of the denominator consisting of the energy difference (complex) of states on both sides of the diagram minus frequencies of absorbed photons plus the frequencies of the emitted photons. The first propagator in the diagram will be (l/Aa - w2). The next propagator will be l/(Akj - w2 + w3) and the last propagator will be l/(Au - wp).Thus, the contribution from the diagram will be

Consider next the two terms obtained by changing the time order of the. absorption of w2 photons (keeping the time order of wIand w3 photons the same). The contributions from B and C are obtained from A by changing the propagators to B = (A, - w,)-'(Akg C = (A,, -

wP)-l(Akg

+ wj)-'(Ak, +

+ U,)-'(A,~-

0 3 01

-

wz)-l

+ as)-'.

The contributions from these three diagrams can be combined to yield A + B + C

- -

d:#d$d%d$J (A,, - wp)(A, +

w3)

{

1

Atg -

+

@I

+

+

w3

m3

(AkJ

- w2) + i r k , - rgl+ O.3 - w2)

1.

rkg)

(59)

The combination of these diagrams shows the existence of the collision-induced coherenceato, w3 - w2 = 0. The collision-induced coherences were first reported experimentally by Prior et af. (1981) and Bogdan et al. (1981a. b). They studied a collision-induced coherence between 3Pl/2 and 3P,,, levels of Na. The four-wave mixing signal at the frequency 20, - w2 was scanned as a function of the frequency of the second laser. A collision-induced resonance occurs when the frequency difference lol - w21 matches the separation between the levels 3P1/2 and 3P,,,. Bogdan et af. (1981a,b) verified (as predicted by theory) (i) proportionality of the width and the integrated intensity of the extra resonance to pressure and (ii) almost no dependence of the peak height on pressure (see Fig. 8). Scholz et al. (1983) verified the collision-induced Hanle resonance in the excited electronic state. They also obtained results in the regime outside impact approximation. This is shown in Fig. 9. The signal is sensitive to the sign of the detuning parameter.

+

1 1

0I -

.

FIG. 8. Experimental scans of the intensity of the 4WM mixing signal at 201 - u 2 ,for three buffer-gas pressures. The lower scan at each pressure was taken with the input beams at o,blocked. Note the nonresonant signal, independent of pressure, and the pressure-induced resonance (Bogdan eral., 1981a).

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

135

M A G N E T I C FIELD / mT FIG. 9. Pressure-induced Hanle resonances in four-wave mixing. Argon partial pressure is p = 1.06 x lo4 Pa. The only parameter changed between (a) and (b) is the sign of A: (a) A = - 1.6 X 10” rad sec-I, (b) A = 1.6 X 10” rad sec-I (Scholz etal., 1983).

Bloembergen et al. (1985) demonstrated the collision-induced Hanle resonance in the ground state of sodium atoms. They reported a very narrow resonance with a width of about 20 KHz. They found (as predicted by theory) (i) linewidth = l/p and (ii) peak height a p 3 over a pressure range of the order of 100- 1000 torr of argon. Rothberg and Bloembergen (1984a) carried out a very systematic analysis of the ground-state coherences (Figs. 10, 11). They reported collisional narrowing of Doppler broadening. At low buffer-gas pressure the width depends on the angle between the pump and probe beams. The line narrows with increase in pressure. At large p the broadening is proportional to Up. They also examined the collision-induced coherence between two hyperfine split levels of groundstate sodium. Zou and Bloembergen (1986) examined the collision-induced (ground-state) coherences in four-wave mixing from samarium atoms. The ground level is a triplet 4f6 6Sz ’F, and the nearly resonant excited state is a singlet 4f6 6 s 6p ’F,,. The linewidth of the Hanle resonance was found to have + A u R D A u where rggV a p; AoRD a Up, and Aw is approxithe form rggj mately independent of pressure.

+

G.S.Agarwal

136

.

Loronlzlan 111

Hollum

6 0 Torr

700 Torr

FIG. 10. Line shape of Zeeman coherence resonances in Na-He mixture for 0 = 3" where 0 is the angle between pump and probe beams (Rothberg and Bloembergen, 1984a).

A. RESONANCE AT OPTICAL FREQUENCY Giacobino and Berman (1987) reported a collision-induced coherence at optical frequency. They considered the excitation of 4D - 3P3,, optical coherence in Na produced via the reaction

Na(3S,,,)

+ He + o1 + 20,

+ He.

(60) The energy-level diagram is shown in Fig. 12 and the experimental results are shown in Fig. 13. The existence of such a coherence can be understood from our general expression (53). Consider the three terms in the second square bracket in (53) and choose the states and frequencies as follows (Fig. 12): li) = 10) = 3s1,2,

lk)

+ Na(4D,

= 12) = 4D,

and I j ) could be either 3P3,, or 3P,/,; w3 =

02, w1 =

3P3/J

In) = 11) = 3P,.

-w , .

(61)

137

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

100

9 K

I

2 z

Y

I=-

k

-zw8

;"i"

€ 0

O

0 0

X

f 2O

r3" 4.

0

I" D

-

J 2 100 -

0\

O 0

5 -

-,

1

50

20

100

200

500

1000

2000

40 = ( 2 )

FIG.12. Schematic illustration of the energy levels used to demonstrate the existence of collision-induced coherence at optical frequencies. INTENSITY

0.

.

t

4/2TI(GHtl

138

G . S. Aganval

Let us assume that the detuning A of w2 from intermediate state IJ) is much bigger than the detuning of o1from 11). The three terms in the second square bracket can be combined to yield

[A,, - (2w2 - wl) - i(r, + rlo- r2~)i - ( 2 ~ 2- ml)I(Ajo - w2)(&1 + w,)(A20 - 2 ~ 2 ) '

(62)

This clearly shows the collision-induced coherence at the optical frequency (4D - 3 P d

B. RESONANCES INDUCED BY AN INELASTIC PROCESS

-

As an explicit example of inelastic contributions consider four-wave mixing in a three-level system such as Ruby (Steel and Rand, 1985; Liu et a f . , 1987) (Fig. 14). Let the fields o1and w2 be acting on the transition 12) 11). The state 11) can also decay to the state 12) via the intermediate state 13). The state 13) in systems such as Ruby is a metastable state. The four-wave mixing signal can be shown to be

The quantity C (which depends on inelastic processes such as spontaneous emission) turns out to be

where we have set the rate y l z , for inelastic transition from ground state to the excited state, zero. Thus, the four-wave-mixing signal will exhibit a collision-induced resonance at o1 = 0 2 .This arises from (o + 2ir12)/[o i(y2] Y ~ ~ The ) ] . signal will also exhibit a resonance at o1 = o2arising

+

+

FIG. 14. Three-level system relevant to transitions in systems such as Ruby. This is also an example of an open two-level atom.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

139

-

24MHZ

PROBE-PUMP DETUNING FREQUENCY FIG. 15. Measured nearly degenerate four-wave-mixing- response as a function i pump-probe . detuning in atomic sodium. The first resonance occurs at 6 = o,- w2 = 0 and the second resonance at 6 = 2(o, - w, ) (Liu ef a[., 1987).

from the leakage of the population from the state 11). The present system is like an open two-level system. If the state 13) is a metastable state, then the fourwave mixing signal would essentially exhibit a very narrow resonance at w , = w z . Such a narrow resonance with a width -9.8 Hz (FWHM) has in fact been seen by Steel and Rand (1985) in Cr:YAlO, . The lifetime of the metastable level is about 33 1 msec. Liu er al. (1987) performed nearly degenerate four-wavemixing experiments on F = 2 ground-state to F = 2 excited-state transitions of the D1 line in sodium. The excited state can decay to F = 1 ground state. In this situation one has an open two-level system (Fig. 14 with y23 = 0) leading to a very narrow resonance at 6 = 0 (Fig. 15) [for further studies see Berman et af. (1988)J.

*

VI. Collision-Induced Coherences in Probe Absorption in the Presence of a Pump We next consider a pump-probe experiment. Consider a quantum mechanical system interacting with a pump field w , and a probe field w 2 . We examine the rate of absorption of energy from the probe in presence of the pump field. We assume that both probe and pump are far detuned from atomic resonances. The energy absorption can be obtained (Griineisen et af., 1988; Mollow, 1972) from the induced polarization P at the frequency 02.Here P(w2)is to be calculated to second order in the pump field and to first order in the probe field: $(t) = $(wz)e-'"2' PJw2) = x:;y*(w2, 0

+ C.C. + . . . , 1

9

- WIbp(w2)Ey(WI)Ed - w A .

(65)

(66)

G. S.Aganval

140

The absorption from the probe field in presence of the pump field is given by the imaginary part of X ( ~ ) ( Ow1 ~ , , - wI).The general structure of ~ ( for ~ a1 multilevel system has already been given in Section V. In this section we consider for simplicity a two-level model. Using (44)we can calculate p(w2) a p13(02) by setting s = 0,

a, = 0,

al(t)= aI

+ azewr,

6 = w,

-

0,.

(67)

Clearly one has =

p13(02)

i(r13

+

+

iAl

+

al(p33(-6)

;a)-'[aZ(p33(0)

-

- plI(0)):

pll(-6))!19

(68)

where (p); stands for mth (nth) order in the field aI(a2) and p(i2) stands for time-dependence of the form p(t) = p(R)ciRr+ C.C.Note that for large detuning from atomic resonance, the first term in (68) will be real and hence will not contribute to the pump-probe absorption experiment. On using p34- 6) = -pII(-6) and the first of Eqs. (44)we find that the terms of interest are

- -2(Ai

~13(wz)

+

6)-liai[2y

+

i S l - 1 [ d ~ 3 i ( 0 ) ) 8- ai(pi3(-6))91.

(69)

Note that PI3 equation shows that

+

+

(pl3(-6))9 = (rl3 iAl i6)-lia2 ( ~ ~ ~ ( 0 )=) ; -iaT(T,, - iAI)-I.

(70)

On combining (69) and (70) we obtain

To this should be added the contribution to first order in a2 and zeroth order in a I :

From (71) we see that a collision-induced coherence at 6 = 0, i.e., at w, = w2, is possible in probe absorption in the presence of the pump since 2 1 2rI3- 2y = - - T2 TI ci p .

(73)

Note that the collision-induced coherence is dispersive and hence the probe gets absorbed (amplified) for 6 > 0 (< 0) if AI > 0. For A, < 0, the probe is amplified for 6 > 0. Grandclement er at. (1987a, b) have observed (Fig. 16) CW oscillation generated by this collision-induced amplification of the probe wave. They used a pump beam slightly detuned from the D, resonance line of sodium contained in a cell in a ring cavity. The buffer gas used was helium at a few torr. The frequency of the oscillating beam was found to differ from the pump beam

COLLISION-INDUCED COHERENCES IN OFTICAL PHYSICS

FIG 16. Laser function of frequency of oscillation is shown (Grandclement et al., 1987a).

-

141

intensity as a

3 -

c)

rd

v

%

H

8,( m i l l g a u s s ) FIG. 17. Absorption-weakening process associated with collision. The drawn line is the experimental result. The crosses represent a Lorentzian fit. (0 = 0.3". T, = 225'12, Ph = 2280 tom, A = 35 GHz below D, resonance, I = 20 mW/cm* in each incident beam) (Gong and Zou, 1988).

by 8-50 MHz. Gong and Zou (1988) have also observed collision-induced state of Na (0, resonance) by using absorption Hanle resonances in the 32S,,2 spectroscopy (Fig. 17). The resonance was observed as the reduction in the net absorption rate. The linewidth of the Hanle signal is inversely proportional to pressure showing a Dicke type of narrowing. The integrated signal is proportional to p 2 and A-4 and is also proportional to the square of the intensity assuming that I , = I * .These features of the absorption signal are consistent with the

G.S. Agarwal

142

structure of ~ ‘ ~ ’ ( 0 ~ 1-, wl, w,) and are related to the Zeeman coherence (30). We in fact demonstrate in Section VII the relationship between the collision-induced coherences in second-order and third-order responses. The full expression for the third-order susceptibility giving the absorption of the probe beam of frequency w2 for the level scheme of Fig. 14 is

where C(0) =

2 w

+ i(yz1+

+ b + i(y2, +iy3

ySl)

I

y31)hJ

+ k)’

(75)

For pump fields on resonance Hillman et al. (1983) observed (Fig. 18) collisioninduced dip at the line center in the absorption curve (pump-probe experiment).

0 c

modulation frequency (tiz) FIG. 18. Attenuation of the modulated component (probe beam) is plotted as a function of modulation frequency. The probe beam experiences decreased absorption at low-modulation frequencies. The width of this hole is 37 Hz for low laser powers. The spectral hole is power-broadened at high laser powers (Hillman er al., 1983).

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

A2 - - - -

I43

-- - -

w2

FIG. 19. Schematic illustration of the energy-level diagram for the two-photon absorption experiment of Liao e t a / . (1979). No + NEON

A

-*-4.0GHz

27r

0 torr

2r FIG. 20. Two-photon excitationspectra of the sodium 3S,,2-4D,i2 transition at various pressures of neon perturber gas. The detuning of the fixed-frequency laser from intermediate-state resonance is below A2/27r = - 4 GHz. The broad resonance is centered near the 3P,,2+ 4D3,, transition frequency. All spectra are taken with the same detection sensitivity. Solid line: experimental curve. Points-theory (Liao er ot., 1979).

The collision-induced dip can be understood from the nonvanishing of C(w) terms. The C(w) terms lead to a resonant structure at o,= w2 with a width that does not depend on the collisional dephasing. Boyd and Mukamel (1984) have evolved diagrammatic methods to understand the collision-induced dip in the absorption curve. The existence of the pressure-induced resonance in two-photon absorption (Figs. 19 and 20) can be understood in terms of the collision-induced transfer of population to the state 3P,,, . The intermediate state is unpopulated in the absence

G . S.Agarwal

144

of the collisions. One can easily calculate the relevant susceptibility using a three-level model (Fig. 19) with ll), 12), and 13) representing the excited, intermediate, and ground state respectively. Let A, and A, be the detunings as indicated in Figure 19. The third-order susceptibility is found to be

+

("-

1 m

3

+

m0-12

+

I).

yz

iAl)

The two-photon absorption rate is obtained from the imaginary part of X(~)(O - w~z, , al).Note that the resonance A, = 0 occurs only in presence of collisions, that is, when rZ3 # y z , rZ3 + T13# rIzJungner er al. (1989) and Khitrova er al. (1988) considered the collisioninduced resonances in a Doppler broadened system. Consider the A configuration (Fig. 2) with two lower levels, 12) and 13). Assume that initially the population is pumped to the state 11). Consider the energy absorption by a probe beam of frequency wI. The two levels 12) and 13) could be, for example, Zeeman levels in level-crossing studies. A third-order perturbation calculation shows that the energy absorption from the probe has the structure

x

[

4

+ r12+

r23 3

+

r13

-

rl

r:3

rZ3

-

rll

I

(w$3

rl2+ rI3 +

(rLZ

+

r13)')

(77) where r,,is the total decay rate of the level 11). In deriving (77), the Doppler limit has been used. Thus, the signal consists of two resonances at ~ 2 3 one : with a width r23 and the other with a width (r13 + rlz). It is expected that rI2 + r13 > rz3.The broad resonance occurs only if rI2 T13- rZ3 - rlI# 0. Thus, the existence of the broad resonance requires the presence of collisions. The ratio of the intensities of the two resonances depends on pressure. The theoretical prediction was verified in experiments (Stahlberg et al., 1985) on the 2P4 ( J = 2)-3& ( J = I) transition in Ne. It is interesting to note that the resonance w13 = 0 with a width rI2 arises from the Doppler averaging of a term such as 1 (78) (o,?+ S,, - O , b ) ( w l 3 - ir,, - O , + kv) and, thus, the broad resonance can arise from the overlap of two Bennett holes. Finally, we mention that several collision-induced nonlinear dispersive effects have been studied by Zou and Gong (1990). They have for example reported collision-induced self-defocusing.

+

+ r,3

+

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

145

VII. Relation between Collision-Induced Coherence in Second-Order and Third-Order Responses We have seen in the previous sections how the pressure-induced resonances can appear in fluorescence spectroscopy and in third-order response which describes processes such as four-wave mixing and saturated absorption. The character of these resonances is the same in both second-order and third-order nonlinear response. One might therefore expect a relation between the two; i.e., one might look upon four-wave mixing and saturated absorption as ways to observe the resonances that are already present in the second-order response. To see this explicitly, we return to the basic Eq. (14) which we write for second- and thirdorder responses: ( - 1)3 s&2013(w1 0 2 1 w3) = -sym Tr[Q(wl + w2 + w3 - iLO)-lda1 3! (79) (w2 + w3 - i ~ ~ ) - ~ d ~-z iLo)-Ici"3p(o)], (w~ t

Suppose we want to calculate the induced polarization in third order. Then typically Q will be li)(jl, with Ii) and lj) connected by dipole transition. Note further that

=

(R - A,j)-l(jlB[i).

Thus, (79) can be written as s!&ap,(wl

9

w2. w3)

+ w2 + w3 - Aj,)-I

= (-1)3 (a1

3!

x sym Tr{14(jl[da1, GI), where G = (w2

+ o3- iLo)-Ici"2(w3- iLO)-'dap(O).

Equation (82) on simplification reduces to

(82)

(83)

146

G. S . Aganval

Note that the quantities that appear in (84) are basically the second-order responses given by (80). Consider now the first (second) term in (84). (See Fig. 21 .) It requires that dipole transitions are allowed between li) and l j ) and between Ij) and lk) (Ii) and Ik)) and, thus, the first (second) term requires the secondorder response between level li) and Ik) (Ik) and Ij)) which are not dipoleconnected. Thus, in the context of Zeeman levels, the second term in (84) would require the second-order response of the coherence between two Zeeman levels. The four-wave-mixing signal at 2w, - w2 can be obtained by choosing w, = w3 = w,, w2 = - w2 after the symmetrization has been carried out. It is thus clear that the third-order response, say four-wave-mixing signal, would exhibit any collision-induced resonances that are exhibited by the second-order response of coherences between two levels not connected by dipole transition as is illustrated in the diagram. For example, in the context of the three-level system discussed in Section 111, the states Ij) and Ik) would be, say, (1) and 12) and the second-order response p$)(wl - w2) exhibits a resonance whenever w, - w2 is equal to Zeeman splitting between 11) and 12). The analysis of this section would in turn imply that the four-wave-mixing signal at 2w, - w2 will exhibit a collision-induced resonance at 0 , - w2 = ? 2s. This in fact is the resonance first predicted by Grynberg (1981a). For a pump-probe absorption experiment we can choose 0 , = - w3. Then the induced polarization on li) t* 1j ) transition will depend on the coherence pik where Ii) and Ik) are, say, two ground levels. Thus, the collision-induced coherence in pik (cf. Eq. (31)) will also be seen in pump-probe absorption experiments as indeed was reported by Gong and Zou (1988).

FIG. 21. Schematic illustration of connection between coherences in second-order and third-order responses.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

147

VIII. Collision-Induced and -Enhanced Resonances in Fifth-Order Nonlinearities It is clear from the discussion in the previous sections that the collision-induced resonances occur quite generally. We have already discussed a host of nonlinear processes that exhibit such resonances. One would expect such resonances in higher-order nonlinear processes as well. For example, six-wave-mixing processes might show these. The higher-order susceptibilitiesbecome very complex and just contain too many terms. However, in specific cases of two-level and three-level systems, the higher-order response can be calculated in closed form. We present a simple example. Consider six-wave mixing in a two-level system (Agarwal and Nayak, 1986). We e5amine the coherent signal produced at 3w1 - 2w2 in the direction, say, 3k1 - 2k2. Calculations using optical Bloch equations show that the induced polarization at 3w, - 2w2 is given by

x

(I- iAl -

T2

T2

x where A, nances at

(-!--

= w,, -

T2

-1

iAl - 2i6)

+ iAl - 2iS

(i

-

i6)(:

-

3i6),

w l , 6 = w1 - w2. This fifth-order response exhibits reso-

It may be recalled that 6 = 0 resonance in four-wave mixing is induced by collisions. Let us examine the character of 6 = 0 resonance in six-wave mixing to see if collisions are needed. Let us assume that A l >> l / T l , 1/T2, 6 so that (85) simplifies to

G . S . Aganval

148

where In the absence of collisions r = 0 and then

The signal shows an inverted resonance at x = 0. It is clear from the structure (87) that for large pressure, the structure of the resonance at 6 = 0 changes. This is shown in Fig. 22. The collisions enhance the resonance w1 = w2 in sixwave mixing. Trebino and Rahn ( 1987) reported collision-enhanced ground-state hyperfine resonances in sodium-seeded hydrogen air flame (Fig. 23). They observed at high intensities additional resonances at & w,/2 in nondegenerate four-wave mixing. Here ohis the ground-state hyperfine splitting. They also performed higherorder wave-mixing experiments in six-wave and eight-wave-mixing geometries. 2 Oh149 and They demonstrated the existence of subharmonics -+ whf21 ? ? wh/5. These subharmonics can be understood in terms of the nonlinear susceptibilities of higher order. For example, the subharmonics ? wh/2( 2 wh/3) can be

X"' NONLINEAR I T Y

-2

-1

0

1

2

X

FIG. 22. Six-wave-mixing signal (87) for different values of the collisional parameter r. The curves (a), (b), and (c) are for r = 0, 1 , and 10, respectively. The maximum values in (b) and (c) are 1.6 X 10' and 1.46 x 104, respectively. The minimum value in curve (a) is unity.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

149

I

6WM Geometry

8WM Geometry

-0.00

-004

0 00 0.04 (ern-')

0 00

wI-wZ

FIG. 23. (a) Spectrum of collision-enhanced resonances in sodium in a flame obtained using a

6WM geometry (3w, - 202, in lowest order). Note the strong Zeernan resonance at zero frequency difference, the sharp four-photon subharmonic resonances at 20.03 cm-I (labeled 2), and the weak two-photon hyperfine resonances at *O.M crn-l (labeled I). Also observe the small dips at 20.02 cm-1 (labeled 3). which are probably due to six-photon resonances at * o , / 3 . The small constant background is due to scattered light and is not a nonlinear-optical effect. (b) Spectrum obtained using an 8WM geometry (40, - 3 0 ~ ) .Observe the subharmonics at 2 112, t 113, 2 1/4, and ?I15 of the hyperfine splitting (labeled 2, 3, 4, and 5, respectively). The subharmonics at tow,/5are due to at least 14WM (Trebino and Rahn, 1987).

understood in terms of (~('9. As indicated in Fig. 23, the explanation of & w,/5 requires at least 14 wave-mixing processes. Trebino (1988) has further shown that dephasing-induced nonlinear optical effects exist in all orders of the pertubation theory. To show this he rewrote susceptibility as a sum of terms all of which are proportional to the combination rU+ r,, - of the dephasing rates.

ck

M. Effect of Cross-Relaxationon Collision-Induced Resonances Our discussion of collision-induced resonances has been based on the assumption that the different lines do not overlap. If the levels are close by and if the pressure

150

G. S. Aganval

w1 13)

1

FIG. 24. Energy diagram of the model system with various relaxation rates. 27 is the radiative relaxation rate of each transition and (+ is the strength of collisional coupling between the two components.

is large, then line-mixing effects become important. In such a case the relaxation model (7) is to be changed. In order to see the effect of cross-relaxation on collision-induced coherences, we have examined a simple four-level model (Kumar and Agarwal, 1987) which is schematically shown in Fig. 24. The figure shows various allowed transitions and the collisional rates. Let 5 be the crossrelaxation parameter. The density-matrix equations for this model are given by u I p l 4 + 5pZ3- W 4 ( p U - p I 1 ) , 4 + Y + uIp23 + (PI4 - W3(p33 - p 2 A pI1= -(u + 2y)pll + w22 - i ( H ; 4 ~4 1p I 4 H L ) , PI4 p23

= -ti(wo + = -[i(oo -

$1 + Y +

(u + 2 7 ) -~ i ~( H ~S 3 ~ 3-2 P23%2),

P22

=

P33

= 2 ~ ~ 2-2up33

~ P II

pa = 2yplI

(90)

+ + up33 -

up4 upU

+ +

i ( H h 2 - P23%2)? i(H14~41 - P14H:I).

Here H ' ( t ) is the atom-field interaction in the dipole approximation. The coherences are coupled to coherences due to the cross-relaxation. These densitymatrix equations are easily solved to obtain the linear and nonlinear susceptibilities. The linear susceptibility x ( I ) ( o , )is found to consist of, in general, two lines X"'(WI)

= -

where

As the cross-relaxation 5 increases beyond s, one of the two lines in the absorp-

15 1

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

tion spectrum becomes narrower and stronger while the other line becomes broader and weaker. Finally, only the narrower line survives. For {/s u/s >> 1 one has

-

Hence, at very high pressure, one observes a single narrow line with width approaching the natural linewidth. This is the well-known line-narrowing phenomena resulting from the mixing of the lines. The third-order susceptibility responsible for four-wave mixing is found to be

x(3)(wI, 0, , - ~

2

)

A+ + A , + ih, +

-6 x[-(

A,

A+ ih+

+

+ A,

c

-

1

+ 2i(y + u) - 6 + A , + ih,

6

A,

+

-

A,

ih,

(6

+ A,

ih-

I (94)

ih-

-6

+ A, +

ih-

+ ih-

1 +

+ 6 + A,

- iX,

A+A-

-

+ A, +

+ A, + ih-

A+

'(6

A-

-6

- ih+

-

6

+

where 6 = w, - w 2, A, = wo - w , , and A, and A , are defined by (92). Let us now examine the effect of cross-relaxation on the collision-induced resonance. Let the fields w, and w2 be detuned far from the atomic resonance, i.e., A , , A2 >> y , (T,etc. Then (94) leads to an approximate expression I X ' Y ~ I w,, 1

-w*))2

a

1

+

2 i ( u - 5) 6 2iy

+

(95)

Therefore, the collision-induced resonance at w, = w2 becomes less prominent

152

G. S. Aganval

as cross-relaxation increases. Note that the width of the resonance is still determined by the radiative relaxation. The numerical results for the behavior of the four-wave mixing signals for a range of relaxation parameters can be found in Kumar and Agarwal (1987). Note that simplified theories of line-mixing effects are given by Hall et al. (1980) for four-wave mixing and by Kothari and Agarwal (1990) for six-wave mixing.

X. Dipole-Dipole Interaction-InducedResonances We have so far considered the resonance induced by collisions with atoms of the buffer gas. In the absence of the buffer gas but' at high densities the dipoledipole interaction becomes important and this can lead to extra resonances (Leite and De Araujo, 1980; Rand and Lam, 1987; Varada and Agarwal, 1991). For example, consider a system of two two-level atoms with frequencies R, and R2. Let this system interact with an external field of frequency w , . We can now study the probability of simultaneous excitation of both atoms by absorbing two photons from the field. If the atoms are noninteracting, then the probability p12of simultaneous excitation is the product of the probabilities of finding each atom in the excited state. Thus, in the absence of direct interaction between two atoms pI2will exhibit resonances at w , = R, ,a2. However, if the dipole-dipole interaction (d-d interaction for brevity) between atoms is significant, then p l Zexhibits an additional resonance (two-photon resonance) at

a, + R2 = 2w,.

(97)

The d-d interaction-inducedresonance (97) can be easily understood in terms of the perturbative expression for the transition probability for two-photon absorption. Let us label the states for the two atoms as shown in Fig. 25. The states Ij, a)and li, p) are connected by d-d interaction. Thus, the states after the d-d interaction part has been diagonalized will have the structure

FIG. 25. Energy-level diagram for two two-level atoms and the two-photon transition between the energy levels of the total system.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

153

where S is a 2 x 2 diagonalization matrix. The matrix element rn for two-photon transition from the state (i, a)to Ij, 0) will be m =

G9

I I%>< -

(WI

+

~ 5 I ~ , Pl ) j ,

(i, a l H , )

q.J

s)(

(loo)

~ I H , \ iP, )

(@I

- 4.p)

A nonzero m will lead to nonzero transition probability. Such a transition probability will show the resonance (97). Note that H, has the form - d, * E - d2 * E where d, and d2 are the dipole moment operators for the two atoms. In the absence of d-d interaction m a

2% (Wl

W,,

- W,rr)(WI

-

q 3 L l

- W,p)

(d,)IJ(d2)@

(101)

and thus the resonance (97) disappears. The dipole-dipole interaction leads to the two-photon resonance (97). This problem can be solved to all orders in the applied field and to all orders in the dipole-dipole interaction V. We have based this calculation (Varada and Agarwal, 1991; Varada, 1990) on the density-matrix equation for a system of two two-level atoms,

where each two-level atom is represented by spin 1/2 operators S , S;. The : steady-state behavior can be studied by first transforming the preceding to the interaction picture. The probability of pairwise excitation is given by p,2 = (S:S;S:S,).

(103)

In Fig. 26 we show p,* as a function of the detuning parameter A = R, + R2- 2~~ for various values of V and the applied field. The figure shows how d-d interaction can give rise to the two-photon resonance which becomes more and more prominent as d-d increases. The spectrum of the emitted fluorescence exhibits the characteristic resonances that arise from dipole-dipole interaction. Consider, for example, two identical atoms with interaction

H

=

hoo(Sf

+ Sg) + V ( S : S , + StS;),

(104)

154

G . S. Agarwal

-2

2

-10

10

FIG. 26. Two-photon resonance induced by d-d interaction. All frequencies are in units of 2y. Detuning parameters are chosen such that A, - A1 = 200, A, + A2 = A. (a) For low fields RI = g2 = 1. (b) For larger field strengths g, = gz = 20. Curves are marked by the d-d interaction V/2y (Varada, 1990; Varada and Agarwal 199 1).

155

COLLISION-INDUCEDCOHERENCES IN OPTICAL PHYSICS

-

la>

FIG. 27. Dipole-dipole interaction-induced excitation of two atoms by the absorption of one photon of frequency w,

-o

~ ~ .

which in terms of the collective operators can be written as H = ~ o s 2+ V(S+S- - 1 - F).

(105)

The eigenstates of (105) are the triplet and singlet states. The triplet states have energies oatV, and - oo. Thus, the spontaneous-emission spectrum will consist of lines at oo f: V. For the case of atoms driven by the coherent field, the spectrum of fluorescence can be calculated from (102). The calculations are rather complex and have to be performed numerically. The actual form (Varada, 1990) depends on the relative magnitudes of the dipole-dipole interaction and the strength of the external field. We can consider other types of situations where new resonances can appear due to d-d interaction. Consider the case of transitions shown in Fig. 27. Clearly, the pairwise excitation of the two atoms to the state Ij, p) is possible even to second order in the field. The dipole-dipole interaction leads to resowpn. This clearly involves matrix elements of the form nance at o, = w,, (pjl Vlyi)(yilH,lai). On the other hand, single-photon excitation involves a matrix element of the form (yilH,lai). Such problems can also be handled in terms of the general expression (26) which gives the fluorescence. In (26) the states V and d is to be replaced by are to be replaced by the eigenstates of Ha d, d l . The probability of pairwise excitation can be obtained by choosing Q = 1j , p)( j , PI. Note that such a problem can also be solved exactly (Rand and Lam, 1987).

+

+

+

XI. Collision-Induced Resonances in Spontaneous Processes In the introduction we discussed how the collisions can lead to redistribution of radiation. The radiation-field spectrum is determined by the two-timecorrelation function of the dipole-moment operators. Such correlation functions can be calculated for any system (Kumar and Agarwal, 1991) by following a procedure similar to that used in the calculation of the nonlinear polarization.

G . S. Aganval

156

For example, such correlation functions to second order in the applied field will show the collision-induced resonances in the radiation emitted by a quantum mechanical system. In order to understand the collision-induced resonances in emission, let us consider the Eq. ( 5 ) for a two-level atom (Fig. 1):

aP -- - - y1 z , ( S + S - p

-

2S-pS'

2

at

1

- -y12(S-S+p - 2S'pS2

+ pS+S-) + ps-S+)

Here y2,(yI2) is the rate of decay of the excited (ground) state to the ground (excited) state and r ( p ) is the collisional dephasing parameter. In the previous sections we calculated the nonlinear response to applied fields, i.e., we evaluated quantities such as (S-(t)), (Pet)). We now need to calculate correlations such as (S+(tl)S-(t2)).Such correlations can for example be calculated by writing Langevin equations equivalent to (106). One can, for example, show that the dipole-moment operator is given by

+ F - ( r ) - -hi [ S - , H,(t)] where F ( - ) ( t )is the noise term such that ( F - ( t ) ) = 0,

( F * ( t ) F z ( t ' ) ) = 20'TEj(t -

t),

etc.

(108)

The diffusion coefficients can be obtained in the standard way and one, for example, finds that 20'-

+ yI2

= T(p)(S+S-)

( 109)

which is equal to TqS'S-) as generally y12= 0. The diffusion coefficient D +has the very interesting feature that it vanishes if the collisional dephasing rate is zero. If r ( p ) # 0, then it depends on the strength of the applied field as (S'S-) depends on the external field. Thus, the fluctuations in the dipole-moment opera-

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

157

tor depend on the collisional parameter P). This can be seen from an explicit calculation of the two-time correlation functions. Assuming that the applied field is monochromatic (frequency o1) and weak, the calculation shows that ((S+(O)S-(7))

- (S-)(S+))e-= d r

A = w0 - w,,

(g( =

l$l.

Hence, the scattered spectrum will exhibit a collision-induced coherence with a width l/Tz. For the multilevel case a similar result is obtained. In place of (107) we have (10

(A) = -(r,, -

iw,)li)(jl

-

ni [IWI, H,(~)I+F,,(~),

i

zj .

(110a)

The correlation function of the noise is found to be (Fi,(f)F,k(t’)) =

2DJNf -

f’),

(1 11)

i # j # k.

(112)

where 2Dt,.,k =

Kk-

r,, - r,k>(pt,),

We again see that the noise correlations such as (1 11) are nonzero only if collisional dephasing exists. The noise correlation is at least of order two in the applied field amplitude. Equation (1 12) shows very generally the existence of pressure-induced contributions to the quantum properties of the system. FROM TRAPPED STATES A. COLLISION-INDUCED RADIATION

We next discuss another example in which collisions play a dominant role. Consider the Raman scattering (Agarwal and Jha, 1979) in presence of strong fields. The model system is shown in Fig. 2. We assume that the inelastic collisions can occur between the levels Is,)= 12) and lg2) = 13). The detailed calculations show that if the fields w1 and w2 are tuned so that A, = A? = A, then in the steady state such a system does not radiate provided that the inelastic collisions between 12) and 13) are absent. In presence of collisions the spectrum of the

158

G. S. Agarwal

(U-tlz

I /y,

FIG. 28. The spontaneous Stokes spectrum in the case of exact resonance and strong fields. The spectrum is identically zero if collisional parameters u , and u2 are set to zero (Aganval and Jha, 1979).

radiation emitted near the frequency w,, consists of spectral peaks at (Fig. 28)

where G I = dl3 * El/fz, G, = d,, * E2/fz.One can interpret the resonances as due to the collision-induced coherence at frequencies given by (1 13). The frequencies (1 13) correspond to the transitions among dressed states of the system. The dressed states and their energies in the special case A, = A2 = 0 are given by

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

159

The density-matrix elements in the dressed state basis obey equations similar to (7): bij

=

-iStjPi,

-

ciJP,J;

=

-2 7J,Pi,+ C ytJPJJ. J

(115)

J

All the relaxation parameters depend on the strength of the applied fields. These are given by

Figure 29 shows the various allowed transitions in the dressed-state picture. The figure also shows the transitions if we let v -+0, i.e., if we do not allow any inelastic collisions between the two ground levels. In such a case all the population is trapped in the dressed state)~)1 as can be seen from the explicit result #6;;(") = [as;, a =

+ as, + ( 1

v,IG114 +

-

2a)6;l),

v2lG2I4

~ ~ ( +1 2 1~ ~1 1 +~~ )v 2 ( I ~ , l +4 2 1 ~ 1 ~+ )~ G ~ ( Y , I G ~ I ~+ ~ ~ 1 ~ ~ 1 ~ '

( 1 17)

FIG. 29. Schematic illustration of the transition among the dressed states in both absence and presence of collisions ( v , , v2 f 0).

G. S. Agarwal

160

Thus, in the absence of collisions, no radiation is possible. This is like no radiation from a system in ground state. Collisions in the presence of strong fields populate the dressed states I+*)and which then radiate as the state 11) is a superposition of the dressed states I&) and

Finally we note that the instantaneous intensity produced by such a system shows an interesting off-on behavior (Agarwal et al., 1988) (quantum jumps) particularly if y, is several orders of magnitude bigger than y 2 v, # 0. The off state corresponds to the dressed state 12) - (G2/G,)13)whereas on states correspond to the dressed states 11) 2 (G,/Go)(2)? (Gl/Go)13).It should be remembered that this off-on behavior occurs only because collisions lead to a nonzero decay rate of the level 12) to 13).

-

xu[.

Narrowing and Enhancement of Signals Due to Velocity-Changing Collisions

We have so far ignored the effect of the velocity-changing collisions (VCC) on the generation of different types of signals. The formulation of Section II can be extended to include such collisions. For simplicity we consider here only the case of two-level optical transitions. The basic density-matrix equation ( 5 ) is modified by the addition of (Berman, 1986)

(1 19)

T,,(v)=

I

W,,(v+ v’) dv‘.

(120)

The quantity r,,(v)gives the rate at which atoms in the state (i) and with velocity v either decay to other states or change their velocity as a result of VCC. W,,(v’ + v) brings population from other velocity subclasses into the subclass v. The quantities T,,(v),W,,( v’ v) ( i # j ) are analogous quantities for the coherence p,,( v). We have made the following simplifying assumptions in addition to those given in Section 11: Changes in velocity occur instantaneously, i.e., the time scale for such changes is much smaller than all the other relevant time scales in the problem.

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

161

The value of v following a jump depends at most on the values v’ before the jump (Markov approximation). The VCC and the phase-interruptingcollisions are assumed to be independent. The dynamic equations for the two-level atom in an external field could now be solved using the model ( 1 19). A reasonable model for the collisional kernel is given by Keilson and Storer (1952). However, the analysis simplifies considerably in the limit of strong collisions. In this model collisions result in rapid thermalization of the velocity distribution of active atoms. The model is especially suited if the mass of the perturber atoms is much bigger than the mass of the active atoms. We thus take

W J ~+ ’ v)

=

r,(v)M(v),

(121)

where M(v)is the Maxwellian velocity distribution given by

~ ( v = ) (~‘Gv,~)-le-v+~fh.

(122)

The four-wave-mixing signal can be (Singh and Agarwal 1988b, 1990; Gorlicki et af., 1988; Lam et at., 1982) calculated depending on the geometry. The results in the Doppler limit and in the limit of no saturation are found to be:

( I ) Phase-ConjugationGeometry: Xgi(WI9

0 1 9

-%)

+2 A1

=

-

wI,

f i [ 2 r + ~ 7(r22 ~ ~ - rll)y - i6

6

=

(wI

-

(rI,+ rZ2)w1

~ 2 ) .

In this configuration, the signal shows two resonances at 6 = 0, and 2Al. For no buffer gas both resonances have a halfwidth 2y. The resonance at 2Al broadens with increase in the buffer-gas pressure. The resonance 6 = 0, however, narrows and gets enhanced with increase in pressure. Its width now is determined by the velocity-changing rate rZ2 of the ground state. Generally one expects Tzz << rll,y. Lam et al. (1982) observed this behavior (Fig. 30) in four-wavemixing experiments on the D2 line in sodium. The experiments used neon as the buffer gas.

162

G . S . Agarwal

b. I‘ 82 MHz

wl2n

+ w/2n C

t wi2n tf

FIG. 30. Comparison between theory and experiment of the NDFWM spectral response for various neon buffer-gas pressures. The amplitude scaling varies with each figure. The separation between the double peaks is 82 MHz. (a) Theory, A, = -41 MHz, 0 torr. (b) Theory, A, = -41 MHz, 2 tom. (c) Theory, A, = 0, LO torr. (d) Experiment, A, = -41 MHz, 2 tom. (e) Experiment, A, = -41 MHz, 2 tom neon. (f) Experiment, A, = 0, 33 torr neon (Lam ef a / . , 1982).

( 2 ) Forward Four-Wave Mixing: Xj,3’(Wl, W I ,

-4

+ 2 v G [ 2 ~ I l ~ Z +Z y r zy* -- i8

rll)

-

u-1,

+

r*Z)isl

The resonance at 2 6 , is now missing as it is washed out by the Doppler averaging. The behavior of the resonance S = 0 is similar to that in the phase-conjugation geometry.

163

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

The intensity-dependent collisional enhancement of the four-wave-mixing signals was also observed by Steel and McFarlane (1983) who also developed a rate-equation model to explain their observations. Singh and Agarwal (1990) have solved the basic equations for the two-level atoms to all orders in the pump field for the forward four-wave-mixing geometry. The result is shown in Fig. 3 1. At low powers the signal exhibits a very narrow resonance at 6 = 0 which is in accordance with the result of third-order perturbation theory. For moderate pump powers the signal is enhanced. With further increase in pump power, AC Stark split resonances appear. Finally Fig. 32 shows how the four-wave-mixing signals change with increasing pump powers. The saturation leads to a hole at the line center as long as rl,# rZ2, i.e., as long as the VCC makes the two-level system an open system. Rothberg and Bloembergen ( 1984a) studied in detail the collision-induced coherences in the ground state of Na. They demonstrated the collisional narrowing of the Zeeman coherences. The collisional narrowing occurs because during A WI ,~the Na atom can undergo many collisions leading to averaging out of its velocity. The collisions in this system are velocity-changing collisions. They extracted VCC cross sections from their data.

'I'

I

- 0.08

-0.04

I. \

0

0.04

0.08

6 FIG. 31. FWM signals S as a function of the pump-probe detuning 6, for various values of pump amplitude g, and for moderate values of P(=2y), when VCC's parameter is small; rzz(=0 . 1 ~ ) Curves . a, b, c, and d correspond to gl = O.ly, y, 3y, and 5 y , respectively. The magnitude of the signal for the curve as shown in the figure is lo3times the actual value. Because in we have made the choice of r,, = 3rz2and r12 = rz2; all general, for optical transitions, r,,> rzz parameters scaled in terms of Doppler width yo 5 lOOy (Singh and Aganval, 1990).

G . S.Agarwal

164

S

0

-0.03

0.03

6 FIG. 32. FWM signals S as a function of 6 for various values of pump amplitude gl for large = 0 . 1 ~ )Curves . a, b, c, collisional dephasing parameter P(=8y) and small VCC parameter r22( . 5y, Sy, and lOy, respectively. The magnitude d, and e correspond to the values of g, = 0 . 1 ~ 3y, of the signal shown in the figure for curve a is lo4times the actual value. These results are for the rlz = rz2. Note that the saturation of transition in the presence of VCCs leads case when I-1, = 3rzz, to a dip at the line center provided that rll# r22 # 0 (Singh and Agarwal, 1990).

XIII. Nonlinear Response, Collisions, and Dressed States In order to understand the collision-inducedcoherences, a dressed-state description is especially useful (Grynberg, 1981b). Consider first a two-level atom interacting with a monochromatic field of frequency oI. The Hamiltonian in the rotating wave approximation and in a frame rotating with the frequency w1 can be written as E H = h A S hG(S+ S-), A = WO - w,, G = . (125) h

+

+

-a.

~

The dressed states are the eigenstates of the Hamiltonian (125) given by

with energies ? R/2, R =

VA2 + 4G2. For field strengths G << A, these states

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

165

can be approximated by

One can now show that the phase-changing collisions and the radiative relaxation lead to the transfer of population from one dressed state to the other. These rates denoted by pa@are found to be pI2= 2 p 4 ( ~ 4+

Note that IpI

w),

pzl = 2p4(y

+ w).

- CIA and, thus, to second order in the applied field G2 pI2- 217-, - 2( + A2

(128)

y).

p21

Thus, we have population distribution among the dressed states. In steady state the dressed state I&) has a population pe (T/y)(G2/A2). Note that this population is proportional to the dephasing parameter r. The dephasing-induced additional population in the dressed state is responsible for the additional structure at the frequency wo in the spectrum. This is because the frequency of the photon emitted in the transition + I&) is

-

+

;[

-

(-31

-01

+A

= w,

+ a = w, + VLY + 4GZ

= w1

+

(

~

-0 wl)

=

~

0

.

Grynberg and Berman (1989, 1990), and Berman and Grynberg (1989) have adopted a dressed-state description to describe the interaction of a two-level atom with two fields of frequencies, w1 and wz. In a frame rotating with the frequency we = (wI+ w2)/2, the two fields appear as a static field modulated at the frequency (0, - up).The Hamiltonian H ( z ) now acquires modulation. Assuming slow time-dependence of H ( z ) , one can introduce adiabatic states of H(z). Thus, the states I&.), and the transfer rates (129) will be slowly varying functions of time t as the Rabi frequepcy G become time-dependent. The modulated component of the fluorescence can be obtained from the rate equations for the populations of the dressed states. Clearly the modulated fluorescence will show a resonance at wI - w2 = 0 provided that r # 0. Thus, the adiabatic dressed-state description explains nicely the origin of the resonance w1 - w2 = 0 in (49) and (50). The four-wave mixing in the two level atoms also shows the collision-induced extra resonance at w1 = wz with a halfwidth 2 y .

166

G . S. Aganval

This arises from the modulated population in the excited state at frequency wl - w2. This collision-induced modulation of the excited state is also responsible for the collision-inducedresonance in pump-probe experiments in two-level systems. Let us then consider very briefly the pressure-induced resonances in different phenomena in the three-level systems with two excited states, /el), le2),and the ground state, Ig). In order to explain all kinds of pressure-induced resonances, we only need to show that the excited-state coherence peIP2 has a modulation at w1 - o2 and that the amplitude of modulation exhibits a resonance at lo1- w21 = [we,czland that it is proportional to pressure. Let us for simplicity consider that the field wl (a2)acts on the transition lei) ( g ) ( ( e 2 ) (g)). We next make a canonical transformation to eliminate the fast time-dependence. The dressed states to lowest order in the field can be written down as

- -

Note that in the original frame I&) -+ e-;ulf , IJIz) + e-iwz' . We can now obtain the coherence (JI1I P ( $ ~ ) between the dressed states (JIl) and )J12) to second order in the applied fields. Such a coherence has a free time-dependence e-i(ol-wz)'. A simple calculation shows that

Thus, the coherence between the two excited dressed states is proportional to pressure. Note further that the resonance denominator in (131) corresponds to the energy separation between the dressed states /GI>and (J12).

XIV. Other Sources of Coherences Similar to Collision-Induced Coherences In the last part of this chapter I describe briefly the resonances that can arise from other sources of dephasing. These other sources of dephasing lead to resonances that are similar to the collision-induced resonances. It has been found that:

167

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

(a) The transient nonlinear response (Kumar and Agarwal, 1989) exhibits resonances similar to collision-induced resonances. (b) The fluctuations of the pump and probe (Agarwal and Kunasz 1983; Agarwal et al., 1987a,b,c; Prior et al., 1985) lead to new resonances even in the absence of any collisional relaxation. (c) The saturation effects (Agarwal and Nayak 1984, 1986; Friedmann and Wilson-Gordon 1983, 1984, 1987) lead to new resonances even if the system has radiative relaxation. The details of the results can be found in the published works cited under (a)-(c) above. Here I discuss one illustrative example of each of the preceding three cases. A. EXTRA RESONANCES IN THE TRANSIENT RESPONSE Consider the transient four-wave-mixing signal produced by a system interacting either with cw fields or with long pulses. Using the third-order perturbation theory, Kumar and Agarwal (1989) have proved that the transient susceptibility X [ ~ ) ( W ~w2, , w3, t) can be obtained from the steady-state susceptibility ~ ( ~ ) (w2, w w3) ~ , as follows:

XWJ~, w2, w 3 , r)

= m(t){x%~,, w2, w,))

(132)

where the operation m( t ) stands for

+ terms obtained by A, * A2 and AI * A3

/.

The full expression for the time-dependent nonlinear response can be obtained by substituting the usual 48 terms of x(31in (132). The existence of the new resonances can be seen by considering a typical term in xI3).For example, a term 1

X(") = ( u p

+

- L)(wi

- &,)<wi

w2

-

(134)

A,,)

in the steady-state response leads to the following terms in the transient response: e d w P- A,,,

x(t) = (Up

+ +

-

)I

A,,,)(w3 - wrIA+ ir,,,,)(w2 + w3

+ c,,)

- on,

eilwl + w > - A1,)f

(wI

+

.

..;

O2

-

r,,

Ak$)(w3

-

= r,,, +

a n t

+

irnrk)(W2

r,,;- r,,

-

wk,

+

irk,,)

(135)

168

G . S. Agarwal

where . . . denotes two other terms that we have not written down explicitly but these are obtainable from (133). In Eqs. (134) and (135) we have not written for brevity, the dipole matrix elements and we have used the notation AtJ = w,] - ir,,,and where r,,are the decay parameters associated with the 0th element of the density matrix. The relaxation model is same as in the work of Bloembergen and coworkers. The expression already shows the existence of the extra resonances in transient response even if the collisional dephasing is absent. The first term in (1 35) can lead to a resonance between two unpopulated states w2 wj + w,, = 0. The second term can lead to resonances of the form w1 + w2 = 0 if k and i are same. Thus, our result (132) for the transient fourwave-mixing (FWM) response establishes the existence of the different types of extra resonances in transient experiments. As a simple illustration of the preceding, consider the transient FWM from a two-level system. Consider an experiment in which the generation of the FWM signal is monitored at different times. The system under consideration is shown in Fig. 33. The generated signal is at the frequency 2w, - w2. The signal is scanned as a function of w1 - w2. Each of the fields is detuned far from atomic resonance. The results for various times of observation are shown in Fig. 33. This figure shows the well-defined extra resonance at w1 = w2 for different times of observation. The width of the resonance

+

1

S 0.5

0

-I

0

-2

2

I

%g,,

FIG. 33. The transient FWM signals for a two-level optical transition as a function of fll?ryll, = o1- 02. The curves a, b, and c correspond respectively to the times of observation given by y 3 , t = 0.05, 0.025, and 0.1. All curves are normalized with respect to the peak height of the curve (a). The pump is far detuned from resonance A = 5ooOy3,. The steady-state signal, which on this scale coincides with x axis, is of the order 11.394 X lo-'. The dashed curve gives the signal (150s) for the case when the fields are switched on over a scale P - I with /3 = 250yll and y,, r = 0.1. For all the curves the scale on the x axis is magnified by a factor 20 (Kumarand Aganval, 1989).

fl

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COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

1-5-

1-

s-

-.

-' - - ''-'

..-.__'

,5:---.,

'.-/

'-, (b)

1

1

I

-

_-..

- _ - I

L

The terms that are ignored in (136) are of higher order in A-I. The usual steadystate extra resonance produced by dephasing collisions is of the order of lIA6. Thus, the transient FWM signal is quite significant. The approximate result (136) explains the numerical results of Fig. 33 very well. The Hanle resonance can also be obtained in transient four-wave-mixing experiments as shown in Fig. 34.

B. EXTRARESONANCES DUETO THE FLUCTUATIONS OF PUMP AND PROBE: FLUCTUATION-INDUCED EXTRARESONANCES We start with a simple problem that demonstrates how the fluctuations of the pump can create new resonances. Consider the resonance fluorescence produced by a two-level optical transition driven by a weak coherent field of frequency w, . The spectrum of resonance fluorescence is 6(w - w,) for a monochromatic field.

170

G . S . Agarwal

However, for a source with width yc, the spectrum S(w) for large detuning A has the form

Thus, the fluorescence peak at the natural frequency wo is induced by the fluctuations. We also point out that the spectrum differs from (137) if the dephasing due to collisions is there. The spectrum (Eq. 4) continues to have a coherent component S(w - w I ) even if collisional width Tp# 0 and a fluorescence component with a width ( y r p )showing that the details of the spectrum depend on the model of dephasing. Fluctuation-induced resonance can also occur in fluorescence in a way similar to the collision-induced resonances. For example, the fluorescence produced by j = 1 to j = 0 transition shows a fluctuation-induced Hanle resonance similar to the one in (31). The modulated fluorescence also shows a fluctuation-induced resonance at modulation frequency equal to Zeeman splitting (Saxena and Agarwal, 1982). These resonances have been recently seen in experiments by Vemuri et al. (1991). In order to understand in a simple way the fluctuation-induced resonance in fluorescence, consider the contribution S,. (Eq. 38) in a case when the exciting field is fluctuating. We need to average (38) over the spectrum of the field: Yc

+

x (wjz + w1 - ir,2)-l(ul,- wI - irl,)-I. Here we assume a Lorentzian spectrum for the exciting field. The integral (138) leads to

x [ o ,~ G , - i(rl,+ y c ) ~ - ~ [ -+~G~ 2 3- i(rZ3 + yc)1-i. This shows the existence of the resonance at wI2 = 0 due to either the collisional dephasing, pump fluctuations, or both. It may be noted that the integral in the limit of large yc has the same form a; the integral (77) in the Doppler limit. We next turn our attention to the question of the existence of fluctuationinduced resonances (FIER) in nonlinearly generated coherent radiation (Agarwal and Kunasz, 1983; Agarwal et al., 1987a,b,c; Prior et al., 1985; Agarwal, 1988; Kofman et al., 1988). We have considered two different models of fluctuations of the pump and probe so that the existence of FIERs is established under rather general conditions. We present an illustrative example. Consider the case when the input fields have stabilized amplitudes but with phases cp,(t) such that 9:s ( t ) are Gaussian, Markovian, and delta-correlated:

171

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

(+i(r)c$j(r’)) = 2yij6(r - r’). For such models exact results can be obtained, i.e., even the saturation effects can be included. The fluctuations of the fields lead to a correlation between different atoms. Hence, one has to solve for the ensemble average of single-atom and two-atom density matrices. Moreover, since there is considerable redistribution of the emitted radiation, we have to separate out the contribution responsible for FWM. This can be done from an analysis of the spectrum of the emitted radiation. The details of the procedure can be found in Agarwal and Kunasz (1983), Agarwal et af. (1987a,b,c), Prior et al. (1985), and Agarwal (1988), which discuss the existence of FIER in a number of FWM situations. Here we consider the case of Hanle resonances. Consider the degenerate FWM in the phase-conjugation geometry in a threelevel systemj = 0 to j = 1 transition. The geometry is shown in Fig. 35. This figure shows the fluctuation-induced Hanle resonance in phase-conjugation ge-

a - Pump

11,+1> = Il> 11,-1> = 12>

Probe

=-

Conjugate

-

10,0> = 13> (a)

10-10

I -~ ~~

~

7c=0

I

(x 100)

=+

Pump

172

G . S. Agarwal

ometry. Here yc is a measure of the pump fluctuations. We have assumed that the pump atom detuning is much bigger than Doppler broadening which in turn is much bigger than natural width. We add that we have shown that the correlated fluctuations of the pump and probe lead to new resonances in a variety of nonlinear phenomena just like the phenomena associated with other sources of dephasing. The detailed character of the new resonances, however, depends on the particular dephasing mechanism. For the observation of FIER it is best to use sources whose fluctuations can be controlled. In this figure g/A = 1/5OOO and hence the signal will be many orders of magnitude bigger if g is increased by, say, a factor of 100. C. SATURATION-INDUCED EXTRARESONANCES The previous sections' discussion of collision-induced resonances was mostly based on third-order suceptibilities. A description based on third-order xs becomes inadequate as one reduces detuning and increases field strengths. The nonlinear response in such situations has been examined numerically by Agarwal and Nayak (1984), Levine er al. (1987), and Chencinski er al. (1990). The nonlinear response is found to exhibit extra resonances even for the radiative relaxation of the atom. A simple illustration of this is shown in Fig. 36. We find that the four-wave-mixing signal produced by a two-level system leads to a resonance at w1 = o2in the absence of collisions but when the saturation effects are important. The existence of such a resonance can be understood in terms of the nonlinear response functions involving dressed states (Kumar and Agarwal, 1991).

XV. Conclusion In this chapter I have demonstrated how collisions and other sources of dephasing lead to new physical phenomena in a variety of experimental situations such as fluorescence and ionization spectroscopy, nonlinear spectroscopy, and Raman spectroscopy. Such dephasing-induced phenomena not only provide useful spectroscopic and atomic information but also could find applications in the production of laser-maser systems. I have already mentioned one such situation following Q. (73). I mention another situation here: consider the excited-state coherence induced by collisions and by excitation by a bichromatic field with frequencies w1 and 02. The excited-state coherence has a time-dependence e-i(ul-wz)'.We can use pSle, to drive a two-photon maser transition between the levels ]el)and le.). This would provide us with another example of a new class of laser-maser systems based on the idea of correlated emission (Scully, 1985).

COLLISION-INDUCED COHERENCES IN OPTICAL PHYSICS

ur*

173

-

FIG. 36. Four-wave-mixing signal produced by a two-level system as a function of wI - 0 , . Curve a (inset): A, = 200, Ti = T, = I , (2 . El)/fi = gI = 4, (2 . E,)/fi = gl = 15; and for A, = 0, TI = 0.5, TI = 1.0. Curve b: g~ = I , g, = 15. Curve c: g, = 4, g, = 15 (Agarwal and Nayak, 1984).

This work has not discussed the effect of wall collisions on the optical properties. The wall collisions, when they are significant, i.e., when k ~>> , r~ >> y lead to interesting new spatial coherence efects. These make the linear and nonlinear response nonlocal (Woerdmann and Schurmann, 1980; Singh and Agarwal, 1986). The response functions acquire new resonant terms that are significant near the surface, i.e., over a length -vlh/y. Such terms lead to the generation of Doppler free signals in reflection (Woerdmann and Schurmann, 1980; Singh and Agarwal, 1986). Note further that this chapter is based on the usage of the impact approximation for collisions. The impact approximation for collisions will start breaking down at large detunings or at very large pump powers. In such a case one has to use a non-Markovian description rather than Eq. (7). We do not pursue this subject here and refer to the published literature

174

G. S. Agarwal

(Singh and Agarwal, 1988a; Devoe and Brewer, 1983; Agarwal, 1985; Wodkiewicz and Eberly, 1985; Schenzle e t a / . , 1984; Hanamura, 1983). Dephasing-induced resonances could also occur in entirely different types of situations. For example, one could consider the quantum electrodynamic interaction of atoms with a radiation field in cavities. The transmission spectra in the limit of very weak incident fields and in the limit of strong atom-cavity field coupling are known to exhibit vacuum field Rabi splittings (Agarwal, 1984). The dephasing in such a situation leads to additional vacuum field Rabi splittings (Agarwal, 1991) in a manner similar to the collision-inducedcoherences.

Acknowledgments The author thanks Prof. H. Walther for his hospitality at the Max Planck Institut fur Quanten Optik, where a major part of this chapter was written. REFERENCES Aganval, G. S. (1984). Phys. Rev. Lett. 53, 1732. Agarwal, G. S. (1985). Opt. Acta. 32, 981. Aganval, G. S. (1986). Opt. Commun. 57, 129. Aganval, G. S. (1988). Phys. Rev. A37,4741. Agarwal, G . S. (1991). Phys. Rev. A43, 2595. Agarwal, G. S., and Cooper, J. (1982). Phvs. Rev. A26, 2761. Aganval, G. S., and Jha, S. S. (1979). J . Phys. B12,2655. Agarwal, G . S., and Kunasz, C. V. (1983). Phys. Rev. A27,996. Aganval, G . S., and Nayak, N. (1984). J. Opr. Soc. Am. B1, 164. Aganval, G. S., and Nayak, N. (1986). Phys. Rev. A33,396. Agarwal, G . S., Kunasz. C. V.,and Cooper, J. (1987a). Phys. Rev. A36, 143. Aganval, G. S., Kunasz, C. V.,and Cooper, J. (1987b). Phys. Rev. A36,5439. Agarwal, G . S., Kunasz, C. V.,and Cooper, J. (1987~).Phys. Rev. A36, 5654. Aganval, G. S . , Lawande, S. V., and D’Souza, R. (1988). Phys. Rev. A37,444. Berman, P. R. (1986). J. Opt. Soc. Am. B3, 564, 572. Berman, P. R., Steel, D. G., Khitrova. G . . and Liu, J. (1988). Phys. Rev. A38,252 (1988). Berman, P. R., and Grynberg, G. (1989). Phys. Rev. A39,570. Bloembergen, N., Lotem, H., and Lynch, R. T. Jr. (1978). Indian J . Pure and Applied Phys. 16, 151. Bloembergen, N., Zou, Y. H., and Rothberg, L. J. (1985). Phys. Rev. Len. 54, 186. Bogdan, A., Downer, M., and Bloembergen, N. (1981a). Phys. Rev. AM, 623. Bogdan, A,, Downer, M.,and Bloembergen, N. (198 1b). Opt. Lett. 6, 348. Boyd, R. W., and Mukamel, S. (1984). Phys. Rev. A29, 1973. Chencinski, W. M., Schreiber, W. M., Levine, A., and Prior, Y. (1990). Phys. Rev. A42, 2839. Cooper, J. (1980). In “Laser Physics” (D. F. Walls and J. D. Harvey, eds.), Academic, Sydney. Devoe, R. G., and Brewer, R. G. (1983). Phys. Rev. Lett. 50, 1269. Ducloy, M., and Nienhuis, G. (1988). Phys. Rev. A38,5197. Friedmann, H . , and Wilson-Gordon, A. D. (1983). Opr. CeN8, 617.

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Singh, S., and Agarwal, G. S . (1986). Opt. Commun. 59, 107. Singh, S., and Agarwal, G. S. (1988a). J . Opt. SOC. Am. B5, 254. Singh, S., and Agarwal, G. S. (1988b). J . Opt. SOC. Am. B5,2515. Singh, S., and Aganval, G. S. (1990). Phys. Rev. A42,3070. Stahlberg, B . , Lindberg, M., and Jungner, P.(1985). J . Phys. B18,627. Steel, D. G . , and McFarlane, R. A. (1983). Ph.ys. Rev. A27, 1217. Steel, D. G . , and Rand, S. C. (1985). Phys. Rev. Lett. 55,2285. Tewari, S . P., and Kumari Krishna, M. (1989). J. Phys. B22, 1 1 15. Trebino, R. (1988). Phys. Rev. A38, 2921. Trebino, R., and Rahn, L. A. (1987). Opt. Len. 12,912. Varada, G. V. (1990). Thesis submitted to the University of Hyderabad, Hyderabad, India. Varada, G. V., and Agarwal, G. S. (1991). To be published. Vemuri, G., Cooper, J., Smith, S., and Agarwal, G. S. (1991), to be published. Wodkiewicz, K., and Eberly, J. H. (1985). Phys. Rev. A32, 992. Woerdmann, J. P., and Schurmann, M. F. H. (1980). Contemp. Phys. 21,463. Zou, Y . H., and Bloembergen, N. (1986). Phys. Rev. AM, 2968. Zou, Y. H . , and Gong, Q. (1990) in “Resonances” Editors M. D. Levenson e t a / . . (World Scientific Publishers) p. 455.

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ADVANCES IN ATOMIC. MOLECULAR AND OPTICAL PHYSICS. VOL. 29

Muon-Catalyzed Fusion JOHA" RAFELSKI Department of Physics University of Arizona Tucson. Arizona

HELGA E . RAFELSKI Insiitut fiir Theoretische Physik J . W. Goethe Vniversitat Frankjiuri. Germany

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Nuclear Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . A . pdFusion . . . . . . . . . . . . . . . . . . . . . . . . . . . B . &Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . C . dt Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Other Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . E . d-r Nuclear-Coulomb Interference . . . . . . . . . . . . . . . . . 111. dt Muon Catalytic Cycle and Reaction Rates . . . . . . . . . . . . . . . A.Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . B . The q,, Problem . . . . . . . . . . . . . . . . . . . . . . . . . C . Properties of Muomolecules . . . . . . . . . . . . . . . . . . . . D . Properties of Electromuomolecular Complexes . . . . . . . . . . . . E . Resonance Condition for Muomolecule Formation . . . . . . . . . . . F. Resonant Muomolecule Formation . . . . . . . . . . . . . . . . . G . Fusion Following Resonant Formation of the (1 1) State . . . . . . . . . H . In-Flight Fusion . . . . . . . . . . . . . . . . . . . . . . . . . I . Pseudoresonant and Complex Near-Threshold Direct-Fusion Processes . . . IV. Muon Sticking . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Initial Sticking Probability . . . . . . . . . . . . . . . . . . . . . B . Regeneration . . . . . . . . . . . . . . . . . . . . . . . . . . C . Recapture . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Muonic X-rays after Fusion . . . . . . . . . . . . . . . . . . . . E . Experimental Values for the Sticking Probability . . . . . . . . . . . . F. Density-Dependence of Sticking . . . . . . . . . . . . . . . . . . V. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 181

181 182 183 183

184 186 186 187 189 190 192 195 196 196 197 199 199 201 204 204 206 208 209 211 212

.

I Introduction The natural existence of a heavy electron. the muon. bridges the enormous energy gap between the atomic and nuclear domains and facilitates spontaneous

177

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Copyright 0 1991by Academic Ress Inc . Ail rights of reproduction in any form reserved. ISBN 0-12-003829-3

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J . Rafelski and H . Rafelski

nuclear-fusion reactions of hydrogen isotopes. Because of this interconnection of atomic, molecular, and nuclear phenomena, the chain of atomic and molecular processes, into which a single muon engages in a target consisting of a mixture of hydrogen isotopes, is very complex. The cycle of reactions, in which a single muon repeatedly initiates nuclear fusions during its lifetime, is termed muoncatalyzed fusion or MuCF. At the origin of the diverse effects is the muonic hydrogen atom, a small neutral object, much like a neutron, capable of entering into chains of complex resonant reactions at thermal energies. Consequently, the entire wealth of neutron-like physics repeats itself here, with the added complication that the neutral object is now polarizable. In the most interesting fusion cycle, involving the deuteron ( d ) and the triton ( t ) (nuclei), a single muon has been shown to catalyze more than 150 fusion reactions (Jones et a f . , 1986) before being lost in some undesirable process. Loss processes include the natural weak decay of the muon, which occurs in a time T -- 2.2 psec and the binding of the muon to an element with 2 > 1, including the capture (sticking) of the muon in the actual fusion reaction by the emerging helium nucleus (Jackson, 1957). Every d-t fusion releases 17.6 MeV and hence the maximal direct energy yield per muon is presently 2.6 GeV, stemming from about 150 fusions. Note that in the experiments with catalyzed dt fusion the so-called scientific breakeven point has been exceeded. That means that the amount of fusion energy released by a single muon during its catalytic cycle exceeds the minimal energy required to make the muon, which is 2rn,cZ. However, major improvements in MuCF are needed before this scientific break-even could ever lead to economically useful applications. For one, the 150 fusions were reached at cryogenic temperatures in liquid-hydrogen targets, and it is clear that in a reactor environment we could not yet reach this fusion yield. Furthermore, we must consider the energy cost per muon. As of 1990, the practical path to the high-intensity muon beams, required in MuCF, employs hadronic interactions, similar to those induced by cosmic rays in the upper atmosphere and which generate a natural flux of approximately 150 muons m-2s- I . The negatively charged muons, which are important here, result from decays of negative pions. Isospin considerations suggest neutron-neutron reactions as being the most efficient way of producing r s , and the preferred projectile beam and target should therefore be rich in neutrons, such as deuterium or tritium (Petrov, 1980; Takahashi et a l . , 1980; Jandel er a l . , 1988). However, Shin and Rafelski (1990a) suggest that a rather high energy (approximately5 GeV) stored proton beam colliding with an internal target could provide even a more energy-efficient source of negative pions and hence muons. Still, we have to find one order of magnitude in the reduction of the energy cost of the muon production and/or in the MuCF fusion yield before the MuCF approach to fusion energy can be seriously considered as a viable fusion alter-

MUON-CATALYZED FUSlON

179

native. As we shall illustrate, understanding the many reactions that muoncatalyzed fusion involves cannot be viewed as even approximately complete. Thus, the missing order of magnitude may indeed be found in the diverse atomic and molecular phenomena encountered in MuCF. The first theoretical studies of MuCF were performed by Frank (1947), Sakharov (1948), and Zel'dovich (1954), anticipating the first observation of muon-catalyzed pd fusion by Alvarez er al. (1957) in a hydrogen bubble chamber. The following theoretical study by Jackson (1957) has, however, led to the belief that the muon-cycling rate was much too small and the muon loss much too high for MuCF to be considered a viable path to fusion-energy production. Since the physics of the MuCF cycle was believed to be well understood at that time, the interest in the subject was dormant for the following 10 years until experiments with dd-catalyzed fusion (Dzhelepov er. al., 1966) revealed a strong and unexpected temperature-dependence in the cycling rate. This observation was explained by Vesman (1967) in terms of the resonant formation of ddp molecules, requiring the existence of extremely weakly bound muomolecular states capable of resonating with the ordinary electromolecules. Detailed calculations have shown that a weakly bound state with rotational and vibrational quantum numbers (JY) = ( I 1) is indeed present in the ddp molecular ion. This molecular state is also present in the (drp)+-ion (Gershtein and Ponomarev, 1977) and is bound exceedingly weakly (on the muonic scale) by -0.6 eV relative to the asymptotic state d + ( t p ) , sThat . this state is somewhat remarkable may be seen by noting that this is just 0.02% of the muonic Rydberg unit. In comparison, the difference in binding energies of the tp and d p K-shells, due to the reduced mass effect, is 48 eV. In Table 1 all quantities of relevance to the MuCF d-t cycle are given to the precision required in determining the energies of the quantum states with absolute precision of 1 meV. A considerable amount of effort has been devoted to precisely determining the energies and wave functions of the almost unbound (1 1) state, as the resonant formation rate, in particular of the dtp (1 I ) state, is extremely sensitive to the binding energy and it is believed that much of the nuclear activity arises from within the bound muomolecules. In the resonant process d p or tp collides with a d in an electromolecule and forms a muomolecule, while the electromolecule absorbs the reaction energy and is excited to a specific rotational-vibrational state. The energy absorbed by the electromolecule must match the energy released in forming the muomolecule, leading to a resonance condition. The resulting rnuomolecular ion effectively becomes a heavy hydrogen center within the excited electromolecule. The theoretical prediction (Gershtein and Ponomarev, 1977) that the muomolecular formation rates would exceed lo8 s-' led to the first experiments in dr fusion (Bystritskii er. al., 1981; Jones el. al., 1983; Breunlich el. al., 1987) in which 150 fusions per muon were ultimately reported.

J . Rafelski and H . Rafelski

180

TABLE I Atomic constants Electron mass Rydberg constant Fine structure constant Electron Bohr radius Schrodinger-Coulombmuo-atomic energies QP,, %I*

EL, Muonic constants Muon mass Muon Bohr radius Lifetime of muon Muon decay rate T J ~ Hydrogen properties Proton mass Deuteron mass Triton mass (Atomic) liquid-hydrogendensity (LHD) Nuclear properties Neutron mass a-particle mass Light helium mass (m, + md - m. - mJc2

m. RY a ai

5 10,999.1 eV/c2 13.605 698 eVfc2 U137.035 989 (6) 52,917.7 fm

185.8408 Ry 195.7416 Ry 199.2726 Ry

2,528.494 eV 2,663.201 eV 2,711.242 eV

mP

206.7683 m, 255.928 fm 2.197 psec 4.5.52 x los /sec

a; 7r

A,

mP md

m, Po

m. ma m3He Qd,

1836.153 m, 3670.484 rn, 5496.922 m, 4.24 x 1W2 cm-’ 1838.684 m, 7294.300 m, 5495.886 m, 34.422 m.

All muomolecules, in particular p p p , d p p , t p p , and ttp, can be formed by Auger emission of an electron from the target molecule. For example, the reaction for p d p molecule formation is

The most loosely bound P-muomolecular state is the (J,v) = (10) state, with a binding energy of O(100 eV), requiring the emission of an energetic electron from the host molecule. The nonresonant molecular formation rates are essentially constant at thermal energies, i.e., they change little within the thermal kinetic energy range of the muonic atom in the hydrogen target and are of the order of lo6 s- I . Electromagnetic muomolecule formation rates, accompanied by emission of a photon, are insignificant in the energy range of 30-100 eV in comparison to the Auger processes. The achievable number of fusions per muon, Y,is obtained from the ratio of the cycling rate A, to the muon loss rate Al. The latter also contains, aside from the rate of muon decay, the rate at which the muon catalyst is poisoned. This is the product of cycling rate A, and the probability of muon poisoning per cycle W,,

181

MUON-CATALYZED FUSION

of which the main cycle sticking wp' is the dominant contribution: A y = L = A, h,lA,

1

+ w? +

1

1

<-<-. SW, W, w $

(2)

Typical best values today are Y = 150, A, -- lo8 s-I. Muon sticking and the muon-cycling rate are thus the key obstacles limiting the number of fusions per muon and possible practical applications of MuCF. Contrary to the popular belief, it is not the speed of the nuclear reaction that imposes the limit on the number of fusions possible per muon. The key quantity, which can be studied directly in an experiment simply by watching the frequency of occurrence of the fusion reaction neutron, is the muon-cycling rate A,. Each muon catalytic cycle contains a large number of atomic and molecular processes, which with some small branching can delay and/or neutralize the catalytic function of the muon. It is therefore necessary to understand the cycle history, including alternatives that occur with a small probability of the mder of O( 1,OOO), in order to understand theoretically the measured rates. The processes involved will be discussed in Section 111. Unlike A,, w, is determined indirectly in the experiments, allowing for much divergence among different research groups. We shall return to this important issue in Section IV. In the next section we shall first briefly survey the diverse nuclear-fusion reactions that can be catalyzed by muons.

II. Nuclear Fusion A.

F FUSION

In MuCF exothermic fusion reactions can occur between several combinations of hydrogen isotopes. A list of reactions of interest is given in Table 11. The p d reaction was the MuCF process first experimentally observed by Alvarez (1957), but certain important aspects of this reaction (Bystritskii e?. af., 1976; Aniol et. al., 1988, 1989; Petitjean et. a l . , 1990) are still under experimental study as of 1990: the (normalized to standard LHD density) reaction rate is found to depend on target conditions (e.g., temperature), even though the molecule-formation process is expected to be completely nonresonant. This unexpected behavior may involve a new path to MuCF: the pseudoresonant process, which we shall address in Section 111.1. The pd nuclear reaction proceeds primarily by the emission of a 5.4-MeV y . Since this is an electromagnetically mediated process, fusion occurs rather slowly, the dominant rate occurring from the hyperine state F = 112 of the (dp)-atom, at a rate of 3.5 x lo5 s-'

J . Rafelski and H . Rafelski

182

TABLE I1 FUSIONREACTIONS Reaction p + d

Branching ratio

- 84% - 16%

+

S-wave d + d p + t d + t t + t

+

Reaction products

- 52% - 48%

-

4

+

P-wave 42% 58%

'He (5.4 keV)

+y

(5.48 MeV)

'He(0.20 MeV)

+p

(5.29 MeV)

(1.01 MeV)

+p

(3.02MeV)

3He(0.82 MeV)

+n +y

(2.45 MeV)

t

OHe (52 keV) 4He(3.56 MeV) 4He + n n

+n

+

(19.76MeV) (14.03 MeV) (11.33 MeV)

(Bogdanova et. al., 1988), which is about as fast as the natural muon-decay rate. The F = 312 rate is three times less likely and the muon-conversion rate is five times less likely than the F = 1/2 fusion rate (Hartmann et. al., 1990). B. dd FUSION

dd fusion is considerably faster, but fusion in the dd molecule is unusual in that it is theoretically expected to occur from a nonzero angular momentum state of the molecule ddp. This is experimentally proven by the observation of the branching ratio into the two different isospin symmetric channels: the -'He p branch is 1.4 times greater at room temperature than the t p branch (Balin et. al., 1984a). The ddw molecular (Jv) = (1 1) state is formed resonantly, and due to the exchange symmetry of the two deuterons a transition to a J = 0 state can only occur if accompanied by a spin flip of one of the deuterons, a process much slower than the nuclear-fusion rate: the rate of this transition is 37.3 ? 1.5 x lo6 s-I (Balin et. al., 1990), while the fusion rate from (Jv) = ( 1 1 ) state is hfd1 = 0.44 X lo9 S K I .The magnitude of this rate has been computed d reactions measured at low extracting the P-wave reaction strength from d energies with polarized beams (Ad'yasevich el. al., 198 1 ; Bogdanova et. a f . , 1982a, 1986; Zmeskal et. al., 1990). Formation of the (11) state occurs at a rate of about 3 x lo6 s-I at room temperature (Jones et. al., 1986; Breunlich et. al., 1989). At very low temperature the resonance is too far away and the resonant rate decreases to the point that other fusion mechanisms can dominate, and the near symmetric S-wave branching ratio is found at T < 50 K (Balin et. al., 1990).

+

+

+

MUON-CATALYZED FUSION

183

C. dr FUSION The dr MuCF cycle is by far the most interesting case theoretically, experimentally, and from the point of view of practical applications among all hydrogenhydrogen cycles, and is the one emphasized in this chapter. The main reasons for this interest are: 1. The dtp system possesses a (Ju) = ( 1 I ) molecular state that is believed to be bound by just 0.66 eV. See Section 1II.C. 2. The dt nuclear system has a J" = (3/2)+ nuclear resonance just 50 keV above the d f threshold, with a width of 70 keV (Hale et. al., 1987). Consequently, the d-t nuclear reaction is 100 times faster than any other involving hydrogen isotopes. 3. The Q-value for the d r --* a n reaction is 17.60 MeV, making it one of the most energetic of all hydrogen-hydrogen reactions. This is beneficial not only because of the high energy yield per fusion, but also the high recoil velocity of the a-particle reduces the probability that the muon will become bound to the a-particle (Section 111). We shall return to discuss the dr MuCF cycle and fusion at length in the remainder of this chapter.

+

+

+

D. OTHERFUSIONREACTIONS Two more hydrogen-hydrogen reactions are also listed in Table 11: p t and tf. These have been studied experimentally by Hartmann er al. (1988) and Breunlich (1987). In addition to reactions between hydrogen isotopes, catalyzed reactions involving elements with Z > 1 have also been theoretically considered by Kravtscov et al. (1984) and Harley et al. (1990a). A particular practical advantage of such systems is a possible absence of neutrons in the final state as well as the avoidance of tritium. The key drawback is that once the muon is transferred to deeply bound states in the 2 > 1 nucleus, it is lost from the cycle of fusions. Clearly, this obstacle, if looked upon superficially, precludes even one fusion per muon, as it can be expected that the muon is primarily transferred to the more bound 2 > 1 system. On the other hand, we can search for situations in which the branching ratio in, say, a p p collision with a Z-nucleus is tilted toward the nuclear reaction either as was discussed in detail by Harley et al. (1990a) in molecular processes or (as it appears to be equally of interest) in direct-fusion reactions. Due to the screening effect of the muon-wave function and the small reduced mass of the proton, the Coulomb penetration factor in direct reactions remains of acceptable size for practically all reactions of interest. It remains to be seen if a single catalyzed fusion involving nuclei, other than hydrogen, will be observed.

184

J . Rafelski and H . Rafelski

E. d-t NUCLEAR-COULOMB INTERFERENCE For the dtp system the nuclear-resonance parameters (Hale et al., 1987) are of the same order of magnitude as the binding energy of the muon to the compound 5He system, 11 keV. The kinetic energy of the muon, should it become attached to the fusion product, the (Y particle, is 86 keV, just the width of the intermediate nuclear state. This suggests that the energy-dependence of the nuclear interaction may interfere with the muonic system. We now try to assess the likelihood that a nonperturbative treatment will lead to a significant revision of the conventional perhubative reaction picture. When considering this question it is helpful to realize that the muomolecular state dtp, which decays in the fusion reaction either into a two- or three-body final state, d d

+ t + p+

+ t + p+

a

+

p (ap)J

+ n, + n,

(3) (4)

is actually itself a resonance, with a meV width, as determined by perturbative calculations of the rate of fusion within the S-states in the molecule. The dominant channel is the three-body final state, Eq. (3), involving a free muon, since the Q-value of the reaction, Q = 17.59 MeV, leads to the a-particle velocity v, = 5.98ac, which is significantly higher than the muonic velocity in the muomolecule. We note that in the reaction in Eq. (4), va,+ = 5 . 8 2 ~ . The muomolecules are quite stable in the sense that their natural oscillation frequency is 100,OOOtimes faster than the (perturbative)fusion rate. This implies in a perturbative treatment of the influence of strong interactions on the muomolecular structure that only a negligible influence can be found on the branching ratio between the two reactions (3) and (4) (Hale et al., 1989; Struensee et al., 1988). Of course, this is a circular result inherent in the perturbative approach and cannot be taken as an independent proof of the conjecture. Danos et al. (1989) claim that when the validity of the perturbative treatment of the nuclear effects is not presupposed, a different picture emerges. Why is it necessary to proceed beyond the perturbative description? The reaction (4), in which the muon emerges bound to the a-particle, arises from a small fraction of the total amplitude (Section 1V.A). When computing the sticking amplitude, the question is what the muon is doing while the nuclei are close together. It may be entirely improper to use classical physics arguments in order to discount the influence of the nuclear interaction on the behavior of the muon. Classically, the muon can be in only one place at a time, and certainly it cannot penetrate the Coulomb barrier. The nuclear interaction has little opportunity to greatly influence the semiclassical part of the system. This argument applies only to the semiclassical components of the molecular amplitude, but not to the typically small, truly quantum components, here those driving the nuclear-

MUON-CATALYZEDFUSION

185

fusion reaction. Thus, considering the inverse process of neutron scattering with the (ap)+-ion, we see an elastic resonance of such a width at the appropriate energy. This consideration in our opinion shows that (a) The coupling of the muomolecular d t p wave function to n a p channel must involve a rapidly changing phase, as a function of the eigenstate energy. (b) The effective off-diagonal coupling term in the Hamiltonian must be O( 100eV). Danos et al. (1989) further suggest that in order to obtain the proper eigenstate of the system reaching into both d t p and a p n configuration space domains, one has to consider how the state has been prepared. Details of this effect have not been studied numerically and remain unresolved. Harley et al. (199Ob) discuss how the fusing state is formed by the Auger transition from the J = 1 doorway electromuo resonance and show that the fusing-state amplitude is populated as a function of energy within the perturbative width of the stationary state with a weight corresponding to the strength of the Auger transition rate. This approximation is similar in spirit to the one suggested by Danos et al. (1989), but since the problem is solved in the adiabatic approximation, the muon amplitude in fusion is fixed irrespective of what the nuclei do. Thus, sticking could not be considered in this work. Beyond this discussion, which leaves the issue of principle yet unsolved, we see the following phenomena that may influence the molecular wave function, and hence the /L sticking probability (branching ratio of reactions in Eqs. (3) and (4)): 1. Influence on the d t p three-body wave function, so that Jldlw(r,,,Rdl + 0 ) is changed in shape. This effect was studied initially by the method of Green’s function (Rafelski and Miiller, 1985). As in more refined studies (Struensee et al., 1988; Hale et al., 1989; Kamimura, 1989), an allowance is made in the relative nuclear-wave function for a contribution from the irregular part of (Coulomb) wave function, which may be very large at resonance. For the parameters of the nuclear resonance now accepted, a slight increase in screening will result as first noted by Rafelski and Muller (1985). This effect is accounted for in the results presented in Section 1V.A. 2 . Energy-dependence of the nuclear-reaction amplitude leading to a dif ferent reaction strength for the various muonic 5He components. For a singleenergy-component wave function (such as in the adiabatic approximation (Section IV.A), in which the muon amplitude becomes exactly the 5He-ls state) this effect disappears in the sticking probability, which is a branching ratio. Thus energy detuning of the reaction can only occur if there is a true sharing of energy in the initial state between the muon and the nuclear motion. A somewhat differ-

J . Rafelski and H . Rafelski

186

ent position was taken by Danos et al. (1986), who considered the detuning of energy in the outgoing channel; a continuum muon has an average energy of 5 keV, whereas the kinetic energy of sticking muon is about 86 keV. The difference just detunes the resonant amplitude known to have a width of about 50 keV (Hale et al., 1987). However, as one can see clearly from, for example, work using Wigner function to study the final channel at the time of fusion (Shin and Rafelski, 1990b), in the adiabatic approximation the energy of the muon component is determined from the beginning. As the muon emerges from the Coulomb potential of the produced a-particle it can acquire, with the small probability characteristic of sticking, a higher translational kinetic energy at the expense of the reduced potential energy. Thus sticking probes those components in the initial Wigner-phase space distribution that are deeply bound (geometrically close to the fusing nuclei). Thus, we learn that sticking comes from the part of the three-body wave function that is most affected by nuclear-Coulomb competition. However, the detuning effect is operative for the nonadiabatic components of the wave function. There is always an amplitude of the initial threebody state that corresponds to, say, 100-keV continuum muons and hence to a nuclear relative energy detuned by the same amount. Because these nonadiabatic components reduce sticking, the additional effect of detuning reduces the nonadiabatic components and hence the nuclear force again has the effect of increasing the muon sticking ever so slightly! However, this significant effect is not accounted for in results presented in Section 1V.A. From this discussion we see that the perturbative treatment of nuclear interaction is potentially not fully accounting for the nuclear interaction’s impact on the muomolecule and the reaction branching ratio (sticking). We shall use the perturbative approach for purely pragmatic reasons in the remainder of this chapter, pending further clarification of the issues involved.

III. dt Muon Catalytic Cycle and Reaction Rates A. OVERVIEW

Present theoretical and experimental results suggest that only the dt catalytic cycle (Fig. I) can be repeated as often as a few hundred times during the lifetime of one muon (2.2 ps). The main steps of the MuCF-dt cycle occurring in a D-T mixture (see Fig. 1) are summarized here: 1. Muons are stopped within 10- lo s. 2. Capture occurs primarily by Auger capture proportional to relative concentrations C, or C, (Soff and Rafelski, 1990).

MUON-CATALYZED FUSION

187

FIG. 1. Key d-t-F fusion-cycleprocesses.

3. (a) If the muon is captured by a deuteron, it undergoes transfer to the heavier tritium isotope. Transfer competes with the cascade processes. From the state transfer occurs more slowly, at a rate of 3 X lo8 s-I (Men'shikov and Ponomarev, 1984), and transfer must compete with ddp formation or directfusion reactions. (b) If the muon is captured by a triton, the muon cascades down to the (?,u)~~ state within lo-" s. 4. The (fp)lsatom (with thermal or epithermal energy) collides with a D2or DT molecule and formes the excited (dtp)I1muomolecular-ion or there is a direct nuclear-fusion reaction. 5 . In the molecule, Auger transition with AJ = +- 1 follows, primarily to the (Jv) = (01) state. This occurs within lo-" s. 6 . Ultimately the cycle of atomic and molecular processes is terminated by nuclear-fusion reaction d ?-+ a(3.5 MeV) + n(14.1 MeV), which takes place within lo-" s or by muon decay. After fusion, the muon is either bound to the product a-particle or is set free, and the sequence of the processes repeats. 7. Further atomic processes during the period of slow down of the comoving a-p in D-T the target matter are affecting the rate of muon loss to the fusion product.

+

B. THEqIsPROBLEM The population of the deuterium ground state plays an important role in muoncycle dynamics, and a significant fraction of the average dz-MuCF cycle time is taken up by the time the muon spends in this state. Once in the ground state, the transfer process dplS I -+ ?PIS+ d is known to occur relatively slowly, at a

+

188

J . Rafelski and H. Rafelski

rate of 2.8 X lo8 s-I (Men’shikov and Ponomarev, 1984). In view of this remark it is clear that we have to better understand the transfer rate from excited states of d p , which should be comparable to that of the de-excitation rate, so a significant fraction of transfers should occur from the (dp),,,l states before the muon reaches the ground state. The fraction of muons reaching the (pd),,ground state is denoted q I s .These muons are significantly delayed in comparison to muons that avoid this state. To see this we consider the normalized fusion cycling rate A,, (ignoring here changes in muonic atom F-spin in collisions, viz. F = 0 f* 1 for t p and F = 112 c, 312 for d p ) which can be written as:

The first term describes the average time the muon must wait once it reaches the 1s state of deuterium with the probability Cdqls:the two C,terms describe the reactions with tritium, which are believed to be limited to the transfer process with the rate A d r , but which could also include direct-fusion reactions in flight Adp+r+fus. The CDproportional term describes all reactions the d p will have with another deuteron, such as ddp formation or direct fusion. This term is believed to be much smaller than the tritium-directed reactions and is negligible unless the tritium concentration CT is quite small. The last term in Eq. (5) describes the time a rp, made with probability C, CD(l - q I s )+ C,f, waits for a reaction, which in the conventional wisdom is dominated by the muomolecularformation rate, but which in principle comprises all possible cycle reactions, in particular also a direct nuclear reaction in which the neutral atom rp can engage. In the probability to reach the rp we encounter the direct capture, = C,, as well as the transfer during the cascade in the deuterium, = 1 - qIs;in the last term there is the feeding from the 1s-state, which has a probability f as some of the Is-& will form a ddp or undergo a direct-fusion reaction instead of a transfer reaction: f = CThd,/(CThdt + C&p+r+fus CDAdd,,). Clearly, if f = 1, we simply have a unit probability to pass through the rp atom in the MuCF cycle. From this analysis emerges the insight that a significant fraction of the average dt MuCF cycling time is taken up by the transfer of the muon from the d to t . The fraction qIsof muons reaching the ground state has been first theoretically computed, and was predicted to be strongly density-dependent (Men’shikov and Ponomarev, 1985), with the result that in a high-tritium-density environment the value of qIsshould be exceedingly small. Subsequent experiments (Jones er al., 1986) culminating in a comprehensive analysis by Anderson (1989) have shown that these results are completely incompatible with experiment. An empirical explanation was proposed by Miiller er al. (1989b), which had as a main feature the suppression of transfer whenever the energy gain in the transfer between the n-shell of the d- and r-p, 48/n2 eV, was less than the molecular dissociation

+

+

MUON-CATALYZED FUSION

189

energy. While this model is agreeing well with all data, it remains controversial as its assumption turned out to be difficult to prove. C. PROPERTIES OF MUOMOLECULES From the beginning of muomolecular physics in the 1960s, it has been correctly recognized that due to the changed energy and distance scales in muonic physic, the adiabatic molecular approximation has a rather limited range of validity. Since then, much effort has been invested in the understanding of the quantumchemistry structures required in computations of the diverse rates of molecular reactions. Generally speaking, there are three methods to approach the problem of few bodies interacting via a l l r long-range Coulomb potential: (a) An adiabatic expansion method, in which the motion of a subset of (two) bodies in an effective potential is slow, permitting us to freeze their position for the purpose of computing the property of the fast component (Vinitskii et al., 1982); (b) A variational approach, in which a suitable and very large set of basis functions is used to diagonalize the Hamiltonian (Szalewicz et al., 1987); and (c) A finite element method, in which the Hamiltonian is solved, assuming in each small volume interval a rather simple form of the wave function, and varying the local Parameters. The difference between (b) and (c) is that the former method assumes a form to be valid in the entire multidimensional space, and hence a great flexibility (large number of parameters) is needed to reach an accurate solution. The latter method needs little local flexibility due to its local variational nature, but has the need for many volume cells. Nevertheless, method (c) holds without doubt an important methodological advantage over the method (b) as the wave function is certain to converge with a similar precision as the eigenvalue does. It is the critical deficiency of the method (a) that the energy may have converged within the numerical precision, but the wave function, in particular in the Coulombtunneling-forbidden region, is entirely wrong. This mistake is difficult to notice in method (a) as at all times the total approximant is a smooth, well-behaved desired function. Since one of the key objectives of muomolecular-structurecalculations is the understanding of the muon amplitude in the (Coulomb-forbidden) point of nuclear contact, method (a) has turned out to be impractical, while method (b) requires an enormous numerical effort. Table 111 shows the muomolecular spectra obtained in high-accuracy variational calculations of muomolecules. In the ddp and dtp molecules, the

J . Rafelski and H . Rafelski

190

TABLE I11 SCHRODINGER-COULOMB MUOMOLECULAR SPECTRUM IN eV ~~

00 01 10 11 20 30

253.152 107.265 -

-

221.549

213.840

97.498

99.126

-

325.073 35.844 226.681 1.974 86.45

-

-

-

-

-

319.139 34.834 232.471 0.660 102.654

-

362.909 83.771 289.141 45.205 172.65 48.70

Source: Alexander and Monkhorst, 1988; Breunlich er al., 1989.

(J,v) = (1 1) state is bound by 1,974 meV and 660 meV relative to the asymptotic states d + ( d ~and ) d~ ~ ( t p ) , srespectively. ,

+

D. PROPERTIES OF ELECTROMUOMOLECULAR COMPLEXES When a neutral muonic atom approaches one of the nuclei of the electromolecule, it can simply attach to it. The muomolecular (1 1) binding energy is smaller than the dissociation energy of the DX electromolecule (X = D or T), which is of the order of 4.5eV. Consequently, in the process of the formation of this (large) molecule, the biding energy for the (1 1) states must be absorbed by the vibrational and rotational states of the host electromolecules. The total binding energy of the (dtp),,molecule including relativistic and finitesize corrections is estimated to be 596 meV. (See Section 1II.E.) The vibrational levels of the D, molecule are spearated by about 300 meV (Table IV). Thus, in the resonance process the electromolecule needs to be excited to its second vibrational state. This is a forbidden transition in the harmonic approximation. After the formation of the ( 1 1) state, the following processes may occur:

+

1. The molecular complex may back-decay into rp DX (Lane, 1983); 2. The muomolecule can de-excite by Auger transitions (Bogdanova er al., 1982b; Scrinzi and Szalewicz, 1989); 3. The electromolecule can de-excite in collisions with other molecules (Ostrovskii and Ustimov, 1980).

These three processes determine the full width of the muoelectromolecular complex resonance, which is believed to be typically several meV (Petrov, 1985; Cohen et al., 1989; Petrov and Petrov, 1989). However, this important number is still being debated. Clearly, the molecular-formation rate effective in fusion is reduced by the back-decay, and the effectiveness of this phenomenon depends on the total width of the state arising in particular from the last two processes,

191

MUON-CATALYZED FUSION TABLE IV

ROTATIONAL AND VIBRATIONAL QUANTA IN meV OF THE [(dpr)d2e] AND [(dpt)r2e] MUOMOLECULES AND THE DISSOCIATION ENERGY OF THE GROUND STATE E,WITH AND WITHOUT rp Jv Em[(dpr)d2e] = 4587.8 &[dd2e] = 4556.3 A& = -31.5

&[(dpr)t2~] = 4608.0 &[dr2e] = 4572.8 A& = -35.2

0

1 2 3 4 5 6 7 8 9 10 0 1

2 3 4 5

6 7 8 9 10

0

1

L

3

4

0.0 5.2 15.2 30.2 50.2 75.0 104.5 138.8 177.6 221 .o 268.6

311.7 316.7 326.7 341.7 361.7 386.5 416.0 450.3 489.1 532.5 580.1

613.1 617.9 627.7 642.2 661.5 685.6 714.3 747.5 785.2 827.2 873.4

904.3 909.0 918.4 932.5 951.3 974.6 1002.4 1034.6 1071.2 1111.8 1156.6

1185.4 1190.0 1199.1 1212.8 1231.0 1253.6 1280.5 1311.7 1347.1 1386.5 1429.9

0.0 4.1 11.8 23.2 38.5 57.6 80.3 106.7 136.7 170.2 207.2

273.1 277.0 284.7 296.1 311.4 330.5 353.2 379.6 409.6 443.1 480. I

538.4 542.1 549.6 561.0 575.7 594.2 616.4 642.1 671.3 703.9 739.9

795.8 799.5 806.7 817.7 832.1 850.2 871.8 896.8 925.2 957.0 992.0

1045.6 1049.1 1056.2 1066.9 1081.O 1098.5 11 19.5 1143.8 1171.5 1202.4 1236.5

Note: A& is the associate zero-point energy shift of the molecular ground state. Source: Scrinzi and Szalewicz, 1989; Faifman et al., 1986; and authors' own estimates.

which ultimately lead to fusion: subsequent to their occupancy the electromuomolecular resonance becomes a bound state. We note that fusion occurs predominantly following the Auger de-excitation of the (1 1) state. The first Auger transition from the initial drp (1 1) state proceeds to the (OO), (Ol), or (20) states of dtp: We thus see that a second and indeed sometimes a third transition must occur before tdp reaches a ( J = 0) state, from which fusion can occur. The latter process requires that an electron be recaptured in the intervening period. The Auger transition rates for the [(tdp)e], [(rdp)de]+,and [(tdp)d2e] complexes are given in Table V (Scrinzi and Szalewicz, 1989). For the configuration [(tdp)de]+ the results with a final electronic state in a charge 2, = 2 have been given. We see that, independent of the detail of the Auger cascade, a fusing state (00)or (01) is reached by about 91% of all muons within lO-"s. However, about

192

J . Rufelski and H . Rufelski TABLE V RATESOF AUGERTRANSITION FOR DIFFERENT INITIAL ELECTROMOLECULE CONFIGURATIONS Transition

x 10"

Rates

sec - I

Source: Scrinzi and Szalewicz, 1989.

9% of all muons reach the (20) state, which dominantly undergoes a second Auger transition to the (10) state. Only if an electron has been recaptured during the elapsed time of both Auger transitions, a third Auger transition to the J = 0 fusing state can occur. Estimates of such recapture rates for the typical D-T environment (Cohen and Leon, 1986) suggest that this is not a bottleneck of the reaction.

E. RESONANCECONDITION FOR MUOMOLECULE FORMATION In order to compute the resonant formation of the muomolecules, it is necessary to understand the binding energy of the (1 1) state of the drp molecule and the associated resonant-excitation spectrum in the electromolecule in the process:

-

w + (ddee)Kl.,=o [(drp)lldee3Kf.Y=~ w + (dtee)K,.v=O [(drp),,teelKf,Y=2 -+

(7)

(8)

to a precision within the total width of the electromuomolecularcomplex, which is believed to be of the order of 0 (1-3 meV), depending on the density. In the preceding we assumed that since the energy of the transition is 0 (600 meV) the second vibrational state will always be excited (Table IV). The muomolecule is the new Coulomb center of the electromolecule and hence the electromolecular excitation energies also contain, aside of the usual terms associated with the changes in vibrational and rotational quanta, further energy differences arising from: 1. The change of the zero point energy of the molecule due to a greater reduced mass involving the muomolecular center of the electromolecule, which shifts the electromolecular states of the newly formed system downward by some 30 meV

MUON-CATALYZEDFUSION

193

(Table IV) in comparison to the usual electromolecular energies, reducing by so much the excitation energy in the complex electromolecule. This quantity, AEm, has to be added to all energies in Table IV in order to identify the resonance condition. 2. The effect of the increase in the size of the electromolecular center, due to the rather large size of the weakly bound (1 1) muomolecule, has turned out to be insignificant 0 (0.3 meV) (Scrinzi and Szalewicz, 1989). On the other hand, there are numerous and significant corrections to the Coulombic Schrodinger energy of the threshold state (1 1): 1. The l/r-Coulomb potential is modified, due to the presence of virtual electronpositron pairs, by the vacuum polarization potential. This slightly, O(0. l%), strengthens the magnitude of all Coulombic interaction potentials at the distance scale of 200 fm, and there is a corresponding correction in all binding energies, also of the energy of muonic atoms. This translates into a small, but significant correction O( 17 meV) in the muomolecule (Aissing et af.,1990; Bakalow, 1988; Swe Myint er af., 1989), which reduces the molecular Coulomb binding. 2. The finite nuclear size reduces the muomolecular binding by O( 13 meV) (Swe Myint et af., 1989). 3. There are a number of small corrections, such as the 1,s coupling and higher relativistic phenomena of order (v/c)*,which total O( 1 meV), even though the atomic corrections are O(50 meV) (Aissing et af., 1990; Bakalow 1988; Swe Myint et af., 1989). However, we must question this perturbative estimate: taking the Dirac-muon wave function, we would have introduced the relativistic small components (which lead, e.g., to a I-s coupling) into the wave function 1) amplitudes at all times. In the of the molecule, which hence contains (1 J = 1 muomolecule this could unsettle the balance between the angular momentum sharing of muonic and nuclear motion. 4. Aside from the Coulomb interaction, there is the magnetic interaction, due to the magnetic moments of t (and d), with that of the muon, which we now discuss.

*

The hyperfine splitting in the (rp)lSF = 0 and F = 1 states is 237.3 meV. The F = 0 state has an added binding of 177.9 meV; the binding of the F = 1 state is reduced by 59.3 meV. The F = 0 (ground) state is believed to have the highest molecular formation rate in collisions with D, molecules, and the F = 1 in collisions with DT molecules (Leon, 1984). Hyperfine quenching of the F = 1 state occurs by charge exchange in the reaction (tp)rS=I t + t (rp):;o at a rate of 1.3 X 109s-' at liquid hydrogen density (LHD) (Bracci et af., 1989). The magnetic interaction is short-ranged, centered around the d and t, and of significantly different strength: it is O(200 meV) in the tp atom, but only

+

+

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J . Rafelski and H . Rafelski

O(50 meV) in the d p atom. This interaction thus attracts a greater portion of the molecular wave function toward tritium. In that regard it acts similarly to the reduced mass effect, which in the first place makes the (1 1) molecule highly asymmetric, as shown in Table VI. In order to compute the energy shift in first order, the highly asymmetric dtp Coulombic wave function is used and a reduction of hyperfine binding relative to the tp F = 0 state is found to be 35.2 meV (Aissing et af., 1990). However, this must be taken as the upper limit of this effect, as the impact of the interaction is to increase the asymmetry of the molecule. In the limit that most of the wave function would be near ot the tritium, the molecular hyperfine energy shift is the same as in the atom and hence there would be no correction to the muomolecular energy. In order to resolve the problem of the hyperfine shift, one needs to include the magnetic interaction (as well as the vacuum-polarization potential and relativistic Dirac form of the muon wave function) in the ab initio calculation of the structure of the drpl, state. We note that the strong hyperfine attractive binding in the t p atom arises in the F = 0 state. There is also a corresponding reduced binding in the F = 1 case, which is one quarter as strong, AEf,=’ = 8.8 meV, which now leads to an increase of the molecular binding, since it corresponds to a reduction of a positive shift in the muonic atom. Taking the perturbative hyperfine shift, we find that the energy of the ( 1 I ) state is reduced from the Coulombic value by 64 meV: E71=0 = -596 t 2 & 20 meV. The first uncertainty is arising from the nuclear formfactor data for tritium and the polarization of the nuclei by the muon; the second (being our estimate of the systematic uncertainty) is arising from the perturbative treatment of the different interactions, which change the muomolecular Coulomb energy by lo%, but which have an absolute value, often rivalling that of the weak (1 1) binding. Our estimate reflects on our belief that the wave function could be TABLE VI DISTANCES FOR EXPECTATION VALUESOF THE INTERPARTICLE THE THREELOWERSTATES OF dty

(r‘d (d,)

(rd (1)

0-d (rz,,)

2.648 7.697 1.950 5.013 2.041 5.463

4.974 28.451 2.639 10.923 3.791 20.802

2.918 9.305 2.079 5.748 2.198 6.368

9.09 128 2.22 8.91 8.36 124

Note: The unit length is the muon Bohr radius a,* = 265.5 fm. Source: Haywood et a/. (1989) and Scrinzi el al. (1988).

MUON-CATALYZED FUSION

195

greatly affected, leading in turn to a change in the perturbative evaluation of the other interactions. The situation is somewhat more clear in the F,, = 1 case; in particular, the important hyperfine effects are four times weaker. Thus, we take = -640 ? 2 2 10meV. It is perhaps worth mentioning that in the case of the ddp molecule the situation with the hyperfine effect (and other perturbative corrections) is much simpler. Since the effect is much smaller in absolute terms, O(50 meV), it is a much smaller fraction of the binding energy, 0(2%), and the system is symmetric between both molecular centers. Thus, no major impact on the molecular wave function can be expected.

F. RESONANTMUOMOLECULE FORMATION The resonant molecular formation is mediated by the multipole interaction between the d-t composing the muonic molecule and the potential due to the second deuteron and the electrons. The molecular-formation rate is given by

where the initial and final electromuomolecularstates are '?, = IIDX]K,,,(tp),s) and qf = )[(dtp):lx2e]$,J.E is the energy of the initial state relative to the d ( t ~ )threshold. ~ , EE:Kf is the position of the vr = 2 rotational resonances with respect to this threshold. It is the difference in energy between the electromolecular initial and electromuomolecularfinal states, reduced by the ( 1 1) state binding energy. The temperature-dependence of the molecular formation rate enters in PK,, the probability that the target D2 molecule is in the K,th rotational state, and f.,,, is usually assumed to be Maxwellian, although should there be fast resonant reaction mechanisms, there would be no opportunity for the QL. atom to become thermally equilibrated following its formation. Below-threshold resonances (with Eres< 0) are inaccessible if the resonances have negligible width. However, the resonant states are broadened by internal and external de-excitations of the (1 1) state (see page 190). The delta function in Eq. (9) then becomes a Lorenzian, and below-threshold resonances are able to contribute to the molecular-formation rate. The molecular-formation rate in the limit T + 0 does not vanish, but goes to a constant. This behavior has been observed for cp + D, (Jones, ef al., 1983), which could indicate the presence of a below-threshold resonance contributing to the measured rate of lo%-'. A fit to the experimental data is consistent with the existence of a ( K p , ) = (00)+ (Kfvf) = ( 1 2 ) resonance 26.0 meV below threshold, assuming the ( I 1) binding energy of 605 meV. This value differs from the best theoretical values to date, which we reviewed in the preceding section, but is within our estimated systematic error range (see page 194).

+

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J. Rafelski and H. Rafelski

The observed density-dependence of the overall cycling rate at T < 130 K suggests that there exists a nonlinear density-dependence in the molecularformation rate, involving the target molecule (dxee) at high tritium concentrations (Jones et al., 1986). If the low-temperature dtp formation rate is dominated by below-threshold resonances, this density-squared-dependence can be attributed at least in part to the collisional broadening of the resonances due to three-body collisions (Men’shikov and Ponomarev, 1986; Petrov et al., 1988). However, this mechanism is unable to account for all of the observed density dependence. We return to these questions in Section 111.1. G. FUSIONFOLLOWING RESONANT FORMATION OF THE (1 1) STATE In the dd and dt systems, near room temperature, it is the firm majority view, which we cannot share that the formation rate of the (1 1) molecular state dominates under all conditions all other fusion processes, and fusion is therefore always mediated by this molecular state. In the dt system, the (11)-state will within 0.84 x 10-l2 s make a dipole Auger transition to the (Ol), (20), or (00)molecular states, with branching ratios respectively (follows from Table V). Fusion of the of 0.89, 0.11, and 2 X d and t occurs almost exclusively from the J = 0 states, and it proceeds also practically exclusively via a 5He (3/2) + nuclear resonance located 50 keV above t threshold. The fusion rate has been computed by a number of authors the d using either an optical potential or an R-matrix formalism (Struensee et al., 1988; Kamimura, 1989; Harley et al., 1990~).These calculations give the fusion rate from the (00) and (01) states to be of the order of 1.3 X lo1*s-’ and 1.4 x 10l2s-I, respectively. The fusion rate directly from the (1 1) state is estimated to be of the order of lo8 s-I (Hu, 1987), although this is somewhat uncertain due to the lack of low-energy data for dr-P-wave reactions.

+

H. IN-FLIGHT FUSION

+

Fusion can occur above the d (rp)],threshold, when the d and t tunnel in collision through the Coulomb barrier screened by the muon. The cross section for the direct-fusion reactions is expected to obey the “ l h ” law at small collision energies due to the short range of the effective Coulomb (d-r) interaction in the presence of the muon, so that the fusion rate tends toward a finite value at very low temperatures. The resonant muomolecular-formation mechanism, however, tends toward zero with decreasing collision energy (Vesman, 1967) in the absence of near-subthreshold resonances, and the direct-reaction processes can therefore be expected to dominate at sufficiently low temperatures. This in-

MUON-CATALYZED FUSION

197

flight fusion was one of the first considered for dd MuCF (Zel’dovich, 1954). This early estimate of the direct-reaction fusion rate has been performed in the WKB (Wentzel-Kramers-Brillouin) approximation for which a fusion rate of 240 s-I at LHD was obtained in the limit vd.,,, -+ 0. More complex numerical calculations of Harley et a/. (19904 based on the R-matrix description of the dt system give fusion rates of 2 X lo5s-I in the same limit, in excellent agreement with work of Bogdanova (1982b) based on optical potentials. The R-matrix approach (Harley et al., 1990~)leads further to the elastic d-tp cross section in agreement with potential calculations (Adamczak and Melezhik, 1988). Nonetheless, we do not consider the subject to be completely settled, as all these approaches treat the asymptotic-polarization potential incorrectly and neglect completely the possible impact of continuum resonances induced by the dynamic (not static) presence of the muon. In order to resolve these issues, one will need to do a calculation using continuum three-body states, a task of formidable proportions. Interest in in-flight fusion was in part stimulated by the suggestion that a tepid plasma might be a suitable environment for MuCF to take place (Men’shikov et al., 1987; Men’shikov, 1988). Work here was interrupted following the realization that temperatures in excess of 30 eV were needed in order to have a reduction, rather than enhancement, of the stopping power, and that T > 100 eV is needed before a decrease in the stopping power of the plasma by a factor of 2-3 is likely to significantly enhance the regeneration of the muon (see Section 1V.B) (Jandel et al., 1989). Since at these temperatures hydrogen molecules are unstable, limiting the rate of resonant-molekule formation, one needs a new fusion channel. Rapid direct-nuclear-fusion reactions catalyzed by muons accompanied perhaps by a small intrinsic sticking would be ideally suited for MuCF in tepid plasmas. In addition, there is evidence for the existence of Feshbach resonances within a few eV of the t + ( d ~ threshold ) ~ ~ (i.e.. about 50 eV above the d (tp),,threshold), for which the fusion rate is estimated to be loL2 s-I at resonance (Froelich et af.,1989). In addition, the intrinsic sticking fraction from the resonant states is believed to be smaller than that for the bound J = 0 states.

+

1. PSEUDORESONANT AND COMPLEX NEAR-THRESHOLD DIRECT-FUSION PROCESSES In addition to fusion from muomolecular states and in-flight fusion, it is possible for the hydrogen nuclei to make a transition to the nonresonant continuum that ( t ~ threshold ) , ~ (up to - 17.59 MeV). The exists both above and below the d continuum exists as a consequence of the coupling of the dtp channel to the anp continua, and the stationary states have been constructed with the help of the Rmatrix method in the adiabatic approximation by Harley and Rafelski (1990a). This global stationary continuum penetrates deep into the d-t region near to the

+

J . Rafelski and H . Rafelski

198

three-body Coulomb eigen energies, with resonantly enhanced amplitude, and generally has an amplitude about smaller away from these eigen energies (Harley et ul., 1990b). Furthermore, just below the d (rp),*threshold, the amplitude diverges weakly as the energy of the dtp system approaches the d (tp),sform below. To be able to involve this subthreshold continuum in the nuclear-reaction processes, there must be other bodies to pick up the (small) surplus energy, much like the resonant formation of the (1 1) state. Naturally, the energy sharing can occur with the host electromolecule. Transition rates to these subthreshold states depend sensitively on the size of the transition energy; ideally, transition energies of less than a few meV are desirable. Transitions involving the excitation of the first vibrational and rotational state in the host molecule result in transition rates of the order of lo3s-' (Harley e l ul., 1990b). For threebody collisions, in which two D2molecules interact, placing one of the D,s off mass-shell with respect to the t p and enabling the tp to make the transition to the subthreshold state around one of the deuterons, the fusion rate (Harley and Rafelski, 1990a) is

+

+

(A,) --

(pIp0)*

1K T X

-

A,,,

X,

E

lo7 s - ' ( T < 5 K).

At cryogenic temperatures this fusion rate becomes competitive with the (1 1) molecular-formation rate. This could have an interesting impact on sticking, if the sticking in the pseudoresonant fusion channel is significantly different from sticking in the J = 0 states. This can be expected on the basis of the observation (Hu, 1987) that fusion from the (1 1) state is greatly reduced. However, this result has just been challenged (Haywood et al., 1991). In addition, this mechanism can compete with the transfer reaction d p t + d + tp, which has a rate of 2 X lo8 s-I. It has been suggested that the existence of a pseudoresonant channel has the potential to clarify a number of experimental and theoretical discrepancies in the dt cycle (Harley and Rafelski 1990a,b). It is worth mentioning here that there are other hints of a direct-fusion channel in both dd and pd fusion. It has been reported that the ratio of dd 3Hen to dd + tp reactions drops slightly below unity at T = 50 K, as compared to the room-temperature value of 1.39 (Balin et al., 1990). This suggests that there is a direct-fusion channel able to compete with the (ddp),,formation rate at low temperatures, allowing the deuterons to fuse from an S-wave rather than from the nuclear P-wave in the (1 1) state. Nevertheless, in this case it is not possible to rule out conventional direct-fusion channels, such as the Auger formation of the (ddp)state with J = 0. However, a second observation is difficult to understand without the pseudoresonant direct-fusion mechanism: the pd fusion rate becomes enhanced by a factor of 2 as temperature is reduced to 25 K (Aniol et al., 1989). This is very surprising as ( p d p ) molecular formation should be completely nonresonant and the Auger formation of ( p d p ) cannot have any temperature-dependence. Only if transition energies of meV are governing the fusion process, such as in pseudoresonant fusion, can there be any influence from

+

-

MUON-CATALYZED FUSION

199

temperature on this rate. If, as predicted here, competitive direct-fusion channels can be shown to exist and even sometimes to dominate the molecular resonant rates, this would provide an important new avenue of research in the field of muon-catalyzed fusion, and would require us to reevaluate the practicability of this approach to nuclear fusion energy.

IV. Muon Sticking A. INITIALSTICKING PROBABILITY The probability of the initial sticking is the reaction-branching ratio 0: =

r(dtp-+ n

r(dtp-+ n a p)

+ +

+ (~p)

+ T(dtp-+ n + ap)’

(11)

which is small, but significant in the context of possible MuCF applications. The total initial sticking probability 0:is actually the sum of the probabilities with which the muon sticks in any bound state directly after fusion:

Assuming that the nuclear interaction can be accounted for as a perturbation, the branching ratio of the reaction is

where In/)are the final states with a stuck muon and Ic) those with a free continuum muon, and Vnu, is the nuclear interaction. All bra and ket states in Eq. (13) are solutions of the Coulomb problem noting that H p ’ ( t ) changes at t = 0, i.e., when the nuclear reaction occurs. It is possible to evaluate the sticking fraction, Eq. (13), exactly, allowing for: Change at t = 0 of the Coulomb Hamiltonian; The time-dependence of the Coulomb potential due to the finite velocity of the a-particle in the rest frame of the muon fusion amplitude (Rafelski et al., 1987, 1991). Up to nonperturbative effects of the nuclear interaction (see section I1 E) sticking to the (n1)th state and final-state interactions of the a-particle thus is:

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J . Rafelski and H . Rafelski

We next obtain sticking in the adiabatic approximation. The three-body wave function given in the Born-Oppenheimer adiabatic approximation is $(r* R) =

$TC(G

R)x(R),

(15)

where the two-center (TC) amplitude of the muon refers to a solution for a frozen position of the hydrogens at separation R. The relative nuclear-motion amplitude x ( R ) describes nuclear vibrations under the influence of the mutual Coulomb repulsion of the nuclei, kept together by the molecular potential V,, = E(R), where E(R) is the Coulomb eigen energy to which at a given separation R the amplitude JrTc is an eigenstate. Inserting Eq. (15) into Eq. (14) we obtain the expression first obtained by Jackson (1957) and since reused many times (Bracci and Fiorentini, 1981):

Here JrTC(r;0) is the muon wave function computed for the combined nuclei, i.e., the muonic SHe-ls eigenstate. Note that in these early discussions of the sticking process, it was customary to forget that the muon is described in terms of the time-dependent a-comoving state. This omission led to claims for the so-called sudden approximation, i.e., that the sticking occurs instantly after fusion, in which case it is not necessary to consider the time evolution of the muon amplitude. However, this claim is actually wrong as implied by results of Shin and Rafelski (1990a), who obtained the initial phase space distribution of the muons after fusion and determined its further time evolution, showing that the asymptotic conditions are reached only when the a-particle traveled several A. Therefore, it is essential to understand why sticking can follow from Eq. (14), despite the incorrectness in principle of the sudden approximation. As mentioned previously, this is a consequence of using the proper initial- and final-state time-dependent Coulombic wave function (Rafelski et al., 1987). The result, Eq. (14), is rather simple, and understanding the miracles that lead to its validity permits us to account for possible interferences arising during the buildup of the final sticking amplitude. In particular, recognizing that the sticking probability evolves during an appreciable distance along with the traveling a-particle, it must be kept in mind that there can be other interactions, such as Coulomb scattering from other matter acting on the muon during the time its postfusion amplitude develops. To be able to describe this phenomenon, Shin and Rafelski (1990~)developed a Monte Car10 Wignerfunction approach to the evolution of the fusion amplitude. How and if these considerations affect sticking is yet to be fully determined. The sticking adiabatic probability is a: 1.2% as shown in the first column of Table VII. More sophisticated three-body nonadiabatic wave functions (Ceperly and Alder, 1985; Hu, 1987; Haywood et al., 1988, 1989) lead to a some-

-

20 1

MUON-CATALYZED FUSION TABLE VII PARTIAL STICKING PROBABILITIES on,(%) FOR THE (01) STATE FOR THE REACTION(dtp.) + (p4He) + n nl

&(BOP

1s 2s 2P 3 4

0.9030 0.1287 0.0321 0.0509 0.0220

41(01)*

5

-

All others

-

0.6840 0.0981 0.0240 0.0386 0.0167 0.0086 0.0181

1.1645

0.8881

04

=

Total

wb(OlP 0.71 0.10 0.02 0.04 0.02 0.01 0.02 0.92

sticking fractions in Born-Oppenheimer approximation (Jackson, 1957). bCoulombsticking fractions from Haywood era/. (1988, 1989). cSticking fractions including perturbative nuclear interaction modifications of the wave function (Jeziorski et al.. 1990; Kamimura, 1989).

what smaller value of @ = 0.89%, which is slightly dependent on the initial molecular state. The nonadiabatic results presented in Table VII are for the (nl) = (01) state from which most of the nuclear reactions from within the molecular system occur. If for some unexpected reason the fusion reaction occurs from the (11) state, sticking may be different (Hu, 1987). In the last column of Table VII the initial sticking fractions into the nl states are presented as obtained allowing for the R-matrix perturbative nuclear impact on the muon fusion = 0.92% is viewed by the majority of MuCF reamplitude. As of 1990, searchers as the best value for the total initial sticking probability and is questioned only in the context of possible nonperturbative nuclear phenomena. (See Section I1.E.) At this point we note that the energy-detuning effect discussed in Section 1I.E is not allowed for Table VII. Following the argument of Danos et al. (1989) the reduction of sticking from the Born-Oppenheimer value to the nonadiabatic value is reduced by as much as 30%, which would imply that the theoretical sticking value is indeed again closer to 1%-the detuning effect restores the (partial) validity of the BO-sticking calculation.

B . REGENERATION Once the muon sticks to the a-particle, it is not entirely lost from the cycle of reactions. At the initial velocity of about v, = 5.82ac, it carries about 86 keV kinetic energy, which is insignificantly greater than the energy _ _ needed, 11 keV,

J. Rafelski and H.Rafelski

202

to strip it from the a-particle. Even more importantly, it takes many atomic collisions before the ap+-ion loses its energy, approximately 3.5 MeV. To relate the initial sticking, W: = w,(t = O), to the final sticking, W, after regeneration, it is customary to introduce R,the reactivation probability, which can be densitydependent. The final sticking is W,

= ~ : ( l-

R).

(17)

The diverse muon-stripping processes compete with the rate of energy loss of the (ap) -ion in the hydrogen medium, which depends upon the stopping power S(v) and density p of the hydrogen: +

dE dt

- = -pvS(v). The time required to bring the (ap)+-ionto rest in liquid hydrogen is of the order of trtop= 4 X lo-" at LHD, so muon stripping, if it occurs, does not have any impact on the cycling rate of the muon. In the simplest first estimate of the significance of regeneration let us assume that all muons captured in excited states rapidly fall to the Is state, which is thus populated with the initial probability w!. Allowing only for one-step processes, we find

where h,,(v) = usu(v)pvis the rate of muon stripping from the ground state, with the stripping cross section uslr( v) (sum of ionization and transfer cross sections). Using the energy rather than the slowdown time of the a-particle as the integration parameter and combining Eq. (18) and Eq. (19), the stripping fraction for the ground state is

where Eo is the initial and E, the final energy of the (ap)+-ion. The stripping fraction is therefore essentially the ratio of the muon-stripping cross section to the energy-loss-weighted electron-ionization cross section (stopping power S). In this simple calculation, RC,(for E, = 0) is found to be about 0.3. In a complete treatment of the muon-reactivation process, not only the 1s state, but also all excited states must be included in the population equations, each of which has different initial amplitudes and different stripping cross sections. The excited-state populations are coupled by radiative, Auger, and Coulomb-induced de-excitation and Stark mixing (quenching) processes. The reactivation probability R of the muon after fusion can be calculated including all the preceding processes by solving the following coupled differential equations numerically:

203

MUON-CATALYZEDFUSlON

dp, = dt

2 ACZPi,

with

i

S(r) is the population of the (ap)+-ionstates as a function of time t; P, is the number of stripped muons; The reactivation probability R is

The multiple-step processes involving radiative transitions are the source of a weak density-dependence in the reactivation probability. See Table VIII. (See also Rafelski et al., 1989; Rafelski and Muller, 1990). One finds in the complete analysis that Eq. (20) is correct in the limit of zero density, in which case the transitions toward the (ap) ground state dominate all intermediate excitation processes. There are a priori two paths to enhance regeneration. One possibility is to +

TABLE VlII

REACTIVATION COEFFICIENTS OF THE (ap)' REACTIONR AS A FUNCTION OF DENSITY WITH QUENCHING CROSS SECTION ffb+ = 0.0037 a$-' AND WITHOUT QUENCHING (Q = 0) Density

R

0.0 0.2 0.4

,2782 ,2908 ,3093 ,3251 ,3376 .3478 ,3561 ,3633 ,3696 .3752 .3802

0.6 0.8 1 .o

1.2 1.4 1.6 1.8 2.0

Source: Rafelski er a / . , 1989.

R

(Q = 0) ,2782 ,3181 ,3441 .3653 ,3815 .3938 ,4036 ,4115 .4183 ,4239 ,4290

204

J . Rafelski and H . Rafelski

identify external conditions in which the stopping power is reduced. While the stopping power S in the first approximation is only dependent on the particle density, it varies greatly with temperature above the molecular dissociation energy (Men’shikov and Ponomarev, 1987). A study of this effect by Jiindel et al. (1989) has shown that a significant enhancement of muon regeneration could occur in a dense plasma with temperatures above 100 eV. Harley et al. (1991) have found an even more pronounced effect for degenerate dense plasma (S > 100 LHD). Another possibility is to reaccelerate the ap+-ion, which has been considered in some detail by Kulsrud (1989), and it requires an essentially matter-free space, which is, however, inconsistent with the assumption that ap+-ionsare generated by fusion processes in a hydrogen target. C. RECAPTURE In addition to stripping, there is a possibility of capturing continuum muons. The final-state Coulomb interaction of the negative muon with the outgoing a-particle results in an enhancement of the muon phase-space distribution in the vicinity of the a-particle (Miiller et al., 1989a; Shin and Rafelski, 1990a, b). Some of these so called convoy muons can recombine with the a-particle by external Auger or radiative capture. The fraction of muons escaping with a velocity somewhat greater than that of the a-particle are decelerated more rapidly due to their smaller mass, enabling the a-particle to catch up with these muons and capture them into a loosely bound state. Environmental conditions, such as target density or the presence of magnetic fields, may influence the fraction of muons available for capture. Computations (Miiller et al., 1989a) show that the fraction of convoy muons can be as great as 0.8%, compared with less than 0.1% when the final-state a-p interaction is neglected. Convoy muons contribute in principle an additional term to the sticking probability, which could be strongly density-dependent due to the competition between the stripping of captured muons from outer shells and the density-independent radiative transitions to the tightly bound inner shells. Clearly, the convoy effect adds to the sticking fraction presented in Section 1V.A. Therefore, if the convoy effect is indeed a cause for the density-dependence of sticking, the intrinsic sticking must be much smaller than expected as of 1990, the value of which is already smaller by a factor of two than the current experiments. (See Section 1V.E.) It should be emphasized at this point that as of 1990, the theoretical calculations show that muons convoy the fusion a-particle. Any suggestions relating to large muon capture rates are still conjectures.

D. MUONICX-RAYSAFTER FUSION Should the muon be bound at any time to the a-particle, there will be some muonic x-ray transitions. These arise either from the initial population of the excited states or from the excitation of the ground state during the a-particle

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MUON-CATALYZED FUSION

slowdown process. In either case the radiative transitions occur in competition with the other density-driven Coulombic processes and, hence, their observed intensity provides key supplementary information about sticking. The K-series x-ray yield per muon fusion, is obtained from the population probability Pj(t) of the (ap)+states, given the transition rates Arad:

A detailed investigation (Rafelski et al., 1989) of the dependence of the K X yield on the diverse phenomena controlling the history of the (ap)+-ion has shown that significant differences are only brought about through the choice of the Stark mixing in the L-shell. Turning off the mixing reduces the K , x-ray yields by 30%. On the other hand, significant modifications of the stopping power could impact the KX-yields. In particular, a reduction of the stopping power, which leads to greater reactivation (and hence smaller final sticking), increases the yield of muonic x-rays emitted after fusion. A qualitative expression for P, (Rafelski et al., 1989) shows that the K, x-ray yield due to excitations into the L-shell is

where the ratio of the K-L excitation cross section to the stopping power enters. Consequently, a (density-dependent) reduction of the stopping power enhances the KXm yield. This observation precludes any ad hoc manipulation of the stopping power of the (ap)+-ion with the goal of reducing the final sticking. This leads to a very similar proportional enhancement of the theoretical X-ray yield. Taking the present canonical sticking value of 0.92% to renormalize prior theoretical results on x-ray yields in accordance with Eq. (25), we arrive at the theoretical values shown in Table IX. It seems that the most sophisticated theoTABLE IX OF X-RAYSPER 100 d-f FUSIONS: XY, NUMBER

Experiment: Hartmann er a / . (1990) Nagamine (1990) Theory: Cohen (1988) Markushin (1988) Takahashi (1988) Rafelski et al. (1989) Stodden et al. (1990)

4,- 1 0.19 0.05 0.20 k 0.06 4, = 1.2 0.26 0.26 0.25 0.31 0.31

*

X

100 FOR DENSITIES 4, = p/po

qJ = 0.1 0.31 0.31

-

0.39 0.36

Note: Theoretical results are renormalized to 00 = 0.915%

4,=1 0.08

S

4,-1 0.02 ? 0.013

-

-

4, = 1.2 0.12 0.12 0.18 0.07 0.082

4, = 1.2 0.019 0.018 0.024 0.012 0.021

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J . Rafelski and H . Rafelski

retical results for XY,, of Rafelski et al. (1989) and Stodden et al. (1990) are a factor of 1.5 too large compared with the experimental values (Hartmann et al., 1990; Nagamine, 1990). As this difference is most pronounced (up to errors in measurement) in the absolute yield, it may be taken as a further suggestion that the initial sticking is indeed smaller than expected. (See Section 1V.E.) E. EXPERIMENTAL VALUES FOR THE STICKING PROBABILITY After the experiment by Jones et al. (1986) many laboratories, including LAMPF, LNPI (Leningrad Nuclear Physics Institute), PSI (Paul Scherrer Institute, formerly SIN), Triumph, KEK (Japan), and RAL, pursued measurements of the total initial sticking fraction in either dd or dt fusion. Most of the USLAMPF &-fusion sticking fraction results were obtained considering the cycling rate of muons and measuring the rate of muon loss. Similar experiments have been performed with a somewhat different analysis of kinetics of the neutronemission and cycle dynamics at the PSI. Furthermore, the PSI experiment has been measuring the x-ray emissions from the MuCF cycle (Hartmann et al., 1990) from which the sticking probability can be extracted as well through a theoretical analysis. In a tritium-rich environment this is a very difficult experiment, as the natural triton decay generates a background just in the energy window of interest. Therefore, the PSI experiment was carried out at a very small tritium concentration, C, = 4 X This issue was taken up at the KEK laboratory at C, = 0.3, where a strong magnetic field was applied to limit the range of the @-electrons,and an intense pulsed muon beam allowed an enhancement of the signal-to-noise ratio. A summary of the experimental measurements of &' is given in Table X. While there is only a slight density-dependence arising from the regeneration phenomenon in the theoretical calculations using pure three-body Coulomb theory, there is a pronounced trend (within the error bars) for a decrease of sticking with density in the LAMPF data, which are just barely in agreement with the PSI neutron-based sticking data. However, the density-dependence in the LAMPF data could also be related to the fact that the high-density points were obtained by changing the temperature near or in the liquid-DT phase. (See Section 1V.E.) The interesting point is that these results are nearly half as small as the theoretical expectations. There is no doubt, in qualitative terms, about the experimental sticking being significantly smaller than the theoretical expectation, considering the enormous neutron yield reported per muon, which can be as high as 150 neutrons (Jones, 1986). Interestingly, the PSI and KEK x-raybased sticking is in much better agreement with the LAMPF neutron-based data than is the neutron-based PSI data. The result shown in Table X was extracted using the theoretical yield of K, x-rays per stuck muon: K J o , = 0.53. This agreement between neutron and x-ray data is significant, as the sticking probability measured by observing neutron yield suffers from the need to apply

207

MUON-CATALYZED FUSJON TABLE X THE( u p ) +STICKING FRACTION FOR DIFFERENT d-r TARGET DENSITIES 4 = p/pu

4 = 1.2 Experiment (from neutron detection): Jones efal. ( 1 986) Breunlich er al. (1987) Nagamine (1987) Bossy er al. Experiment (from X-raydetection K J o , = 0.53): Hartman et at. (1990) Nagamine (1990) Theory using o? = 0.915: (I-R) 04 (R: see Table VIII)

*

4

= 0.1

0.07 0.35 0.45 2 0.05 0.42 2 0.07 0.42 ? 0.14

1.1 2 0.5 0.50 f 0.10

0.36 0.09 0.38 f 0.11

*

-

0.59

0.65

-

corrections that account for the muon-loss probability per catalytic cycle from any cause. In particular, in a D-T environment we expect a certain small number of dd and dt fusions, which all have a significantly greater intrinsic sticking, simply because these reactions are less energetic, and the double charge reaction product is less energetic as compared with the dr case. Another important phenomenon is the capture or transfer of the muon to helium impurities, arising primarily from tritium decay. In either of these phenomena there can be a complex temperature- and density-dependence, and many of the involved branching ratios are still ill understood. Thus, most of the corrections are based on information and systematic behavior derived from differing experimental conditions and therefore there can be some difference about the magnitude of the individual effects and corrections to be applied. In particular, the improperly understood value and density-dependence of qIr(Section 1II.B) is a major source of some of the uncertainty, as once the muon reaches the ground state of dp, there is an appreciable probability that it will enter the dd-fusion cycle. All the available experimental data (as of 1990) for o,from LAMPF, PSI, and KEK are given in Fig. 2 along with theoretical curves. A weak theoretical-density-dependence arises from the regeneration factor. All experiments so far have yielded a sticking significantly below expectations, though we anticipated finding a greater-than-expected value (allowing for a yet unknown loss mechanism, etc.). This stimulated a new series of experiments that are presently in progress. LAMPF, in collaboration with RAL, has developed an apparatus in which coincidence measurements of a neutron in conjunction with an a + +or an (ap)+can be performed. Aside from the directness of the measurement, the other advantage of this project is that only a minor correction is required to account for stripping of the muon from an (ap) during the passage through the gas target and mylar window, as the muonic ion loses only a fraction of its energy. (Consider Eq. (20) for Ef= Eo.) LNPI has previously developed a direct method to measure sticking in the d-d reaction in a +

J . Rafelski and H . Rafelski

208

Final sticking fraction

1.0

I

I

1

4

I

I

t

I

1

1

,

-

-- q-3-e-:

0.8 -

-

1(

h

K

3"

-

- --

o.6gJ T

f-

1 1:

0.4

x

0.2 -

-

0 PSI 4

0.0

LAMPF

I

'

KEK Theory

'

" ' '

'

I

I

'

wire-ionization chamber. This group joined forces with PSI to carry out a direct measurement of sticking in a much-refined wire-ionization chamber. At Los Alamos, the intention is to study high-density, low-temperature sticking with the help of a novel target. A Berkeley-Livermore-PSI group is considering hightemperature, high-pressure targets.

F. DENSITY-DEPENDENCE OF STICKING There is a clear disagreement between predicted values of w, and the experimentally observed values at high density with which unusually high fusion yields (more than 150 neutrons per muon (Jones, 1986)) are associated. The source of this discrepancy could lie with any of the possibly misunderstood aspects of the catalyzed fusion cycle, which is density-dependent. But any density-dependence (beyond the trivial linear dependence of all two-body rates) requires either the competition between a two-body rate and a density-independent rate, typically a radiative transition or the competitive presence of three-body processes. A third alternative is to understand the density-dependence as a temperature-dependence in the range of 10 to 40K (see Section III.I), since the high-density points were obtained changing temperature and density at the same time. If the density-

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MUON-CATALYZED FUSION

dependence of sticking is to be explained, its understanding must be related to the five known density-dependent processes in the MuCF cycle: 1. The probability to reach the ground state of dp (the so-called q,sprobability, Section 1II.B). Here radiative K,, K, transitions in dp compete with transfer rates dp+ tp. 2. The muon-recapture probability (the convoy effect). (See Section 1V.C.) 3. The probability of muon regeneration, where a small density-dependence arises from the competition of 2p- Is transition with excitation into higher orbits. (See Section 1V.B.) 4. The nonlinear density-dependence in electromuomolecule resonance formation. (See Section 1II.F.) 5 . The possibility of a small sticking side cycle of MuCF which dominates at high density and/or at low temperature. (See Section 111.1.) Other effects have been explored, e.g., a density-dependent stopping power (Rafelski er al., 1989). While a density-dependent reduction of the stopping power could explain the density-dependence of the effective sticking, the associated modification of the x-ray spectrum from transitions within the (Hep) -ion increases the already present discrepancy between theoretical predictions and experimental data. (See Section 1V.D.) +

V. Conclusion Of all the hydrogen-hydrogen MuCF reactions that have been studied since 1980, the d-t reaction has attracted by far the most theoretical and experimental attention. However, after 15 years of modern MuCF studies, central questions are still being debated, and there is today a clear perception that we are far from understanding muon-catalyzed processes, let alone deciding if MuCF will ever lead to any significant practical applications. One of the reasons for the recent surge of interest in MuCF has been the recognition of the important role diverse resonant reaction may play in speeding up the catalytic cycle. Until this was understood, the key bottleneck in the chain of reactions was the slowness of the in-flight (Auger) formation of the muomolecule which was believed to be a necessary step prior to a fusion reaction. Resonant molecular formation occurs when a neutral tp muo-atom enters a hydrogen molecule and binds extremely weakly to a deuteron, the muomolecular binding energy being picked up by the (second) vibrational band of the host electromolecule. The rate of formation of such resonances is highly temperaturedependent. After the initial experiments established temperature-dependence of the dtp cycle rate, much effort in the world has been devoted to establishing a reliable

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value for the sticking. Most of LAMPF dt-fusion sticking fraction results were obtained by considering the cycling rate of muons and measuring the rate of muon loss. A different analysis of the kinetics of the neutron emission and cycle dynamics was performed at PSI, where the x-ray muo-atomic transitions from the MuCF cycle have also been measured. X-ray emissions have also been studied at KEK, in the presence of an intense magnetic field. The punch line of these experiments is the very low sticking of at LHD, with values of 0.35% (LAMPF) and 0.45% (PSI and KEK)-the results are within error bars of each other. The conventional theoretical high-density value is about 0.6%,which is about 1.5 times the experimental value. Furthermore, in theory there is only a slight density-dependence arising from the stripping, while in the experiment there is a pronounced trend (within the error bars) for an increase of sticking with decreasing density in the LAMPF data, a result that may be seen to disagree with the PSI data. A further disagreement at densities below LHD between LAMPF and PSI may be traced to differences in corrections applied to the raw sticking, which are required in the D-T environment due to a small number of dd and rt fusions which however have a significantly greater intrinsic sticking than dt fusion. From the point of view of applications, what matters is the uncorrected muon loss per cycle (excluding muon decay) W,, and on this quantity the groups nearly agree: muon loss per cycle is significantly below theoretical expectations. Should the muon become bound to the a-particle, there will be some muonic x-ray transitions (Section 1V.D). The theoretical x-ray yield is found to be 1.5 times greater than in the experiment, while the sticking, as we saw, is also a factor 1.5 times larger than in the experiment. Thus, there are two discrepancies between experiment and theory of profound importance and together they suggest that the initial sticking is smaller than expected, at least at high densities. But how small is the sticking really? All the surprising findings have stimulated a new series of experiments in progress as of 1990 that are to measure the initial sticking (LAMPF and RAL) and perform a direct measurement of sticking in a wire ionization chamber (LNPI and RAL). Extreme target conditions are also to be studied. The discussion of recent progress in understanding o,was one of the points emphasized in this survey. We have shown that the issue arises at least in part from a possible misunderstanding of the reaction chain leading to fusion, and in part from the theoretical oversimplification of the complex reaction processes. In our opinion the present data suggest the existence of a small sticking side cycle of d-t MuCF which becomes particularly active at high density and/or at low temperature. However, there are a number of other important results connected primarily with dt fusion that are not yet satisfactorily understood and for which a deeper understanding is required if the practicability of MuCF is ever to be properly assessed: 1. The observed nonlinear density dependence in the dtp molecular formation rate.

MUON-CATALYZED FUSION

21 1

2. The observed high probability qIsof the muon reaching the long-lived and inactive state (dp)],. 3. The low-temperature behavior of the cycling rate and, in particular, the molecular formation rate in D2 and DT molecules. 4. The observed temperature and molecular-hydrogen composition-dependence of the pdp fusion. Experiments as of 1990 arrive at the molecular rates during the process of complex analysis of the kinetics of the reaction-rate data and fusion yield obtained at varying conditions of density, isotope composition, molecular composition, and temperature with data taken as a function of time. The difficultiesjust noted may simply imply that our global understanding of the MuCF cycle of reactions is indeed incomplete, in the sense that parts of the MuCF cycle proceed via different chains of reactions than we allow for in the analysis. After an intensive search for such alternative paths we have reason to believe (Harley and Rafelski, 1990a, b) that direct in-flight reactions compete with the conventional molecular reaction picture. Many of the MuCF puzzles and foremost the sticking problem may have simply arisen from the ignorance of a possible important direct-reaction channel. A similar study could also help clarify peculiarities noted in the d and p d cycles. Should this explanation be correct, we must revisit in detail two areas of MuCF that have been rather quickly dismissed: 2 > 1 fusion and the tepid plasma environment (Section 1V.E). Even if dr-MuCF-based cost effective power production can be achieved, the dt reaction poses severe health and technological problems and requires a significant inventory of tritium in order to stop muons in a DT target. Ironically, the aneutronic and nontritium pd-reaction is suppressed in its rate by the relatively high Q-value, which suppresses the muon-conversion probability. Disregarding these concerns, which are similar to those voiced in other approaches to fusion, we note that in principle the required progress appears to be small in comparision to more conventional fusion approaches, such as plasma fusion or inertial confinement fusion. However, as highlighted in this chapter, it is generally believed that MuCF is already close to the limits posed by a combination of practical approaches with fundamental laws of physics, and hence further improvement is exceedingly difficult. But in principle a much greater number of fusions per muon will be possible than the present record of about 150, once all the atomic and molecular processes in the MuCF cycle are properly understood.

Acknowledgments Between 1982 and 1990, most of the progress in MuCF research has arisen directly from a coherent program of research sponsored by the Advanced Energy Projects Program of the U.S. Dept. of Energy under the leadership of its former

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+

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Jackson, J. D. (1957). Phys. Rev. 106,330-339. Jandel, M., Danos, M., and Rafelski, J. (1988). Phvs. Rev. C37,403-406. Jandel, M., Froelich, P., Larson, G., and Stodden, C. D. (1989). Phys. Rev. A40, 2799-2802. Jeziorski, B., Moszynski, R., Szalewicz, K., Scrinzi, A., Zhao, X., Kolos, W., and Velenik, A. (1990). University of Delaware, Newark. Unpublished. Jones, S. E., Anderson, A. N., Caffrey, A. J., Walter, J. B., Watts, K. D., Bradbury, J. N., Gram, P. A. M., Leon, M., Maltrud, H. R., and Paciotti, M. A. (1983). Phys. Rev. Lett. 51, 17571760. Jones, S. E., Anderson, A. N., Caffrey, A. J., Van Siclen, C. D. W., Watts, K. D., Bradbury, J. N., Cohen, J. S., Gram, P. A. M., Leon, M., Maltrud, H. R., and Paciotti, M. A. (1986). Phys. Rev. Lett. 56, 588-591. Jones, S. E. (1986). Nature321, 127-133. Kamimura, M. (1989). In “AIP Conf. h c . 181: Muon-Catalyzed Fusion” (S. E. Jones, J. Rafelski, and H. J. Monkhorst, eds.), pp. 330-343. American Institute of Physics, New York. Kravtscov, A. V., Popov, N. P., and Solyakin, G . E. (1984). JETP Lett. 40, 124- 126. Kulsrud, R. M. (1989). In “AIP Conf. Proc. 181: Muon-Catalyzed Fusion” (S. E. Jones, J. Rafelski, and H. J. Monkhorst, eds.), pp. 367-380. American Institute of Physics, New York. Lane, A. M. (1983). Phys. Lett. 98A, 337-339. Leon (1984). Phys. Rev. Lett. 52, 605. Markushin, V.E. (1988). Muon Catalyzed Fusion 2,395-420. Men’shikov, L. I., and Ponomarev, L. 1. (1984). JETP Lett. 39,663-667. Men’shikov, L. I., and Ponomarev, L. I. (1985). JETP Lett. 42, 13- 16. Men’shikov, L. I., and Ponomarev, L. I. (1986). Phys. Lett. B167, 141-144. Men’shikov, L. I., and Ponomarev, L. I. (1987). JETPLetr. 46, 312-315. Men’shikov, L. I. (1988). Muon Catalytic Processes in a Dense Low-Temperature Plasma. Insritut Atomnoj Energii im. I.B. Kurchatova March 1988. Preprint IAE-458912, Moscow, USSR. Miiller, B., Rafelski, H. E., and Rafelski, J. (1989a). Phvs. Rev. A40, 2839-2842. Miiller, B., Rafelski, J., Jandel, M., and Jones, S. E. (1989b). In “AIP Conf. Proc. 181: MuonCatalyzed Fusion” (S. E. Jones, J. Rafelski, and H. J. Monkhorst, eds.), pp. 105-110. American Institute of Physics, New York. Miiller, B., Rafelski, J., Jandel, M . , and Jones, S. E. (1989b). 105-1 10. Nagarnine, K. (1987). Muon Catalyzed Fusion 1, 137- 150. Nagamine, K., Matsuzaki, T., Ishida, K., Watanabe, Y., Sakamoto, S., Miyake, Y., Nishiyama, K., Torikai, E., Kurihara, H., Kudo, H., Tanase, M., Kato, M., Kurosawa, K., Sugai, M., Fujie, M., and Umezawa, H. (1989). In “Proceedings of an International Symposium on Muon Catalyzed Fusion pCF-89” (J. D. Davies, ed.), pp. 27-32. Rutherford Appleton Laboratory, Oxon. Nagamine, K. (1990). X-ray observation of a-sticking phenomena in muon catalyzed fusion for liquid and high T concentration D-T mixture. FCF ‘90 International Conference on Muon Catalyzed Fusion, Vienna. Ostrovskii, V. N., and Ustimov, V. I. (1980). Sov. Phys. ZETP 52,620-625. Petitjean, C., Lou, K., Ackerbauer, P., Breunlich, W. H., Fuchs, M., Jeitler, M., Kammel, P., Marton, J., Nagele, N., Werner, J., Zmeskal, J., Bossy, H., Crowe, K. M., Sherman, R. H., Baumann, P., Daniel, H., von Egidy, T., Gruenewald, S., Hartmann, F. J., Lipowsy, R., Moser, E.,and Schott, W. (1990). The p p d fusion cycle. pCF ’90 International Conference on Muon Catalyzed Fusion, Vienna. Petrov, Y. V. (1980). Nature 285,466-468. Petrov, Y. V. (1985). Phys. Lett. 163B, 28-30. Petrov, V. Y., and Petrov, Y. V. (1989). In “AIP Conf. Proc. 181: Muon-Catalyzed Fusion” (S. E. Jones, J. Rafelski, and H. J. Monkhorst, eds.), pp. 124-144. American Institute of Physics, New York.

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Petrov, Y. V., Petrov, V. Y., and Shlyakhter, A. I. (1988). Muon CatalvzedFusion 2, 261-272. Rafelski, J. (1980). Hydrogenic mesomolecules and muon catalyzed fusion. Exotic Atoms '79, Vol. 4 (K.Crowe et al., eds.), pp. 177-205. Plenum, New York. Rafelski, H. E., Miiller, B., Rafelski, J., Trautmann, D., Viollier, R. D., and Danos, M. (1987). Muon Catalyzed Fusion I, 315-332. Rafelski, H. E., Muller, B., Rafelski, J., Trautmann, D., and Viollier, R. D. (1989). Progr. Part. and Nucl. Phys. 22,219-338. Rafelski, H. E., and Muller, B. (1990). Comp. J . Phys. Comm. 59, 521-525. Rafelski, J., and Miiller, B. (1985). Phys. Lerr. 1648, 223-227. Rafelski, H. E., Harley, D., Shin, G . R.. and Rafelski, J. (1991). J. Phys. B: Arom. Mol. Opt. Phys. 24, 1469-1517. Sakharov, A. D. (1948). Report of the Physics Insritute. Academy of Sciences, Unpublished, referred to by Zel'dovich (1954). Scrinzi, A., Szalewicz, K., and Monkhorst. H. I. (1988). Phvs. Rev. A37, 2270-2276. Scrinzi, A., and Szalewicz, K.(1989). Phvs. Rev. 39, 2855-2861. Shin, G. R., and Rafelski, J. (1990a). Nucl. Instr. Methods A287, 565-569. Shin, G. R.. and Rafelski, J. (1990b). J. Phvs. G: Nucl. Part. Phvs. 16, L187-Ll95. Shin, G. R., and Rafelski, J. (1990~).Phvs.Rev. A43, 601-602. Soff, G., and Rafelski, 1. (1990). Z. Phvs. D14, 187- 190. Stodden, C. D., Monkhorst, H. J., Szalewicz, K., and Winter, T. G . (1990). Phys. Rev. A41, 128I - 1292. Struensee, M. C., Hale, G. M., Pack, R. T., and Cohen, J. S. (1988). Phvs. Rev. A37, 340-348. Swe Myint, K., Akaishi, Y.,Tanaka, H., Kamimura, M., and Narumi, H. (1989). 2. Phvs. A334 423-428. Szalewicz, K.,Monkhorst, H. J., Kolos, W.. Scrinzi, A. (1987). Phvs. Rev. A36, 5494-5499. Takahashi, H., Kouts. H. J. C., Grand, P., Powell, J. R . , and Steinberg, M. (1980). AtomkernenergieKerntechnik 36, 195- 199. Takahashi, H. (1988). Muon Caralvzed Fusion 2,453-458. Vesman, E . A. (1967). JETP Lett. 5,91-93. Vinitskii, S. I., Ponomarev, L. I . , and Faifman, M. P. (1982). Sov. Phys. JETP 55, 578-581. Zel'dovich, Y. B. (1954). Dokl. Akad. Nauk S. S. R. 95, 493-498. Zmeskal, J . , Kammel, P., Scrinzi, A, Breunlich, W.H., Cargnelli, M.. Marton, I . , Nagele, N., and Werner, J. (1990). Phys. Rev. A42, 1165-1177.

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ADVANCES IN ATOMIC MOLECULAR AND OPTICAL PHYSICS VOL . 29

MULTIPLE-ELECTRON EXCITATION. IONIZATfON. AND TRANSFER IN HIGH-VELOCITY ATOMIC AND MOLECULAR COLLISIONS J . H. McGUIRE Department of Physics Kansas State University Manhattan. Kansas

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

A . Formulation . . . . . . . . . . . . . . . . . . . . . . . B . Independent-Electron Approximation . . . . . . . . . . . . . C . Multiple-Electron Effects (Electron Correlation) . . . . . . . . . D . ExpansioninZp/v . . . . . . . . . . . . . . . . . . . . . E . Many-Body Perturbation Expansions in Both Zp/vand u . . . . . . F. Special Features of Electron Capture, Nonorthogonality,and Special Kinematics in Many-Body Scattering . . . . . . . . . . . . . G . Projectile Electrons and a Classical Atomic Limit . . . . . . . . H . Molecules and the Transformation Between a(B) andf(Q) . . . . . I . Methods of Computation for Few- and Many-Electron Transitions . . I11. Observations and Analysis . . . . . . . . . . . . . . . . . . . A. Simple Analysis . . . . . . . . . . . . . . . . . . . . . . B . Multiple Ionization. . . . . . . . . . . . . . . . . . . . . C. Multiple Excitation . . . . . . . . . . . . . . . . . . . . D. Multiple-Electron Capture . . . . . . . . . . . . . . . . . . E. Ionization and Excitation . . . . . . . . . . . . . . . . . . F. Transfer and Ionization . . . . . . . . . . . . . . . . . . . G . Transfer and Excitation . . . . . . . . . . . . . . . . . . . H . Projectile Electrons . . . . . . . . . . . . . . . . . . . . I . Molecules . . . . . . . . . . . . . . . . . . . . . . . . IV. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . Appendix: Correlation . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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217 219 219 226 230 232 237 239 244 247 250 263 264 270 286 288 291 292 297 299 302 312 313

314 315

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I Introduction The many-body and many-electron problem is common in various areas of physics as well as in chemistry and biology. Basic understanding of phenomena ranging from the nature of matter at the creation of time to the properties of useful 217

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Copyright Q 1992 by Academic Press Inc. A11 rights of reproduction in any form reserved. ISBN 0-12-003829-3

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materials in the human environment is limited by the boundaries of our knowledge of the many-body problem. There is an advantage in studying the manybody problem in atomic physics since the two-body and parts of the three-body problem are understood. Furthermore, both the mystery of the meanings of quantum mechanics and the mystery of the transition from microscopic time-reversible atomic processes to the dynamics of macroscopic time-irreversible aggregates of atomic particles is inherent in the many-body problems of atomic interactions. Thus, by studying the many-body problem in atomic physics we are able to develop effective tools to discover insights that provide both meaning and utility in our lives. The few-body problem in atomic physics is not completely understood. Although our knowledge is growing (Stolterfoht, 1989, 1990; Reading and Ford, 1987a,b; Briggs and Macek, 1991; McGuire, 1987; McGuire and Straton, 1990a,b), understanding is limited in a number of key areas. One limit of understanding, for example, is encountered in collisions of atomic hydrogen with charged particles at moderately high velocities (Madison, 1990; Walters, 1988; Williams, 1986). While there is good agreement between theory and experiment for total and differential cross sections, relative phase information observed in electron impact excitation to the first excitation state of atomic hydrogen differs even in sign at large scattering angles from present calculations done on some of the largest computers now available. Another limitation in the Coulomb fewbody problem is the asymptotic wavefunction for three charged particles needed for calculations of ionization of atoms, ions, and molecules by charged particles (Brauner et al., 1989). Despite these limitations, some reasonable understanding of multielectron processes such as multiple-electron ionization is still possible under certain conditions. For example, if electron correlation is small, then it may be possible to apply the independent-electron approximation to describe total cross sections, even when a sum over many final states is included (McGuire and Macdonald, 1975; McGuire and Weaver, 1977; Sidorovitch and Nikolaev, 1983). As more details about the final states are specified, it is often necessary to include more and sometimes new details (e.g., many-electron effects) in the analysis of atomic and molecular reactions. While looking closely at atomic interactions tends to require new understanding of many-particle effects for individual atoms, this many-body understanding is also useful in dealing with aggregates of atoms. The many-electron effects needed to understand specific detailed cross sections in many-electron atoms and molecules may also be required to understand the transition from microscopic atomic properties to properties of macroscopic systems of atoms. Specifically, we may get a clearer notion of correlation. In studies of both microscopic and macroscopic effects (Goldberger and Watson, 1964; Balescu, 1975) the notion of correlation is an important concept. In a general sense correlation is the interdependency of individual members of a group or, perhaps more simply, how individuals affect one another. This concept of correlation is used in both atomic physics and statistical mechanics. (Cf.Appendix.) In these contexts correlation

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

2 19

usually refers to electron correlation which is caused by the Coulomb interaction (or correlation potential) acting between electrons, although other kinds of correlation (e.g., time [memory] or spin) are possible. While a physical definition of correlation could be based on the correlation interaction between electrons, the more common definition is mathematical (Balescu, 1975; Huang, 1987; Van Kampen, 1981; Eadie et al., 197 1 ; Gardiner, 1985). Mathematically correlation is that which is not described by a product of terms. For example, a wavefunction is correlated if "(r,, rz)# $(r,)+(rJ.The physical and mathematical definitions of correlation differ a little. In this chapter the mathematical definition of correlation is used and some, but not much, distinction is made between correlation and multielectron effects. The focus of this chapter is multielectron transitions in high-velocity atomic (and molecular) transitions, i.e., multiple ionization, electron transfer, excitation, and combinations thereof. Multiple-electron transitions are intrinsically many-body (many-electron)phenomena in contrast to single-electron transitions which may or may not depend significantly on multielectron effects. At high collision velocities the description of both single and multiple transitions is relatively simple both conceptually and mathematically. The emphasis here is on multielectron effects in single atoms and molecules, although brief attention is given to an example of a transition to a classical limit of quantum mechanics in a few-electron system that is observable. A little consideration is given to some statistical methods. Auger transitions and dielectronic recombination are largely ignored. Unless otherwise specified the projectile is considered to be a fastcharged particle that is much heavier than an electron, and the target is a neutral atom that is not strongly correlated. This means that the projectile moves faster than the target electrons and that the independent-electron approximation can be sensibly defined. (Cf.Fig. 11 .) Where possible different processes are analyzed in a unified way. For example, some connection is made among collisions of atoms (and molecules) with photons, electrons, positrons, protons, antiprotons, fully stripped ions, partially stripped ions, and neutral atoms.

II. Theory A. FORMULATION The Hamiltonian for scattering of an atom of nuclear charge Z, with N electrons by a particle of charge Z, and mass M using atomic units is (McGuire, 1987)

= K

+ V + Ho,

(2)

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McCuire

where M,is the reduced mass,

K = --V2 2M,'

(3)

and

-cv,+ N

- N)

zP(zT

j= I

R

where long-range Coulomb effects vanish when 2, = N and

Here R is the internuclear coordinate and rJ the coordinate to the jth electron. The (rt - rj - I Coulomb (electron-correlation) interaction gives rise to spatial electron-electron correlation in the unperturbed (static) atomic Hamiltonian H,. It is now assumed that the internuclear motion may be separated from the electron motion and that the internuclear motion may be treated classically, so that the internuclear trajectory R(t) is well defined, e.g., R(t) = B vt where B is the impact parameter of the projectile. The resulting equation (McGuire and Weaver, 1986) for the electron motion is now time-dependent, namely,

+

The Hamiltonian for the electrons, H e 1 ,defined as H,1 = Ho

+

i

V, = H,

+V-

-

zF'(zT

R

N)

(7)

is now explicitly time-dependent since the V, depend on R which is explicitly time-dependent. Consequently, the evolution operator U is not simply given by exp[ - iH,, ( t - t o ) ] ,but rather by a more complicated expression containing time ordering, namely, EQ.(10) which follows. It is useful to work in the intermediate representation, where one takes full advantage of the fact that the eigenfunctions of H, are known (or nearly known). In the intermediate representation, the evolution operator U ( t , t o ) is governed by

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

22 1

V,(t) is not a sum of single-electron operators because H,, in Eq. ( 5 ) is not a sum of single-electron terms due to the Irk - r,l- electron-electron interaction. Equation (8) may be formally solved for U ( t , f a ) (with T as the time-ordering operator):

namely, Dyson’s equation (Goldberger and Watson, 1964). The probability amplitude for scattering from the asymptotic initial state I#~to the asymptotic final state 4fis given by a =

(4fllCli) = (+flu( +

- m)l6i).

~7

( 1 1)

The probability, P(B),for a transition from 4, to r$f is given by the absolute square of a, and the corresponding cross section by

mif =

I

dB P(B) =

I

ds luI2.

(12)

This result holds for an arbitrary number N of electrons, including the fundamental case of N = 1. Some special considerations are as follows.

1 . Mulfielectron Eflects and Electron Correlation Electron correlation is a multielectron effect. That is, electron correlation arises because there is more than one electron in our system and the electrons affect one another. Correlation (discussed in the Appendix) is rigorously defined as the difference between exact and uncorrelated quantities. A quantity is uncorrelated if that quantity is determined from a sum of single-electron Hamiltonians, even though that quantity obeys the Pauli exclusion principle and appropriate symmetry requirements such as rotational symmetry (conserving angular momentum) and parity. For example, the total wavefunction, the probability amplitude, and the probability are uncorrelated if (ignoring symmetries) q ( r , , r z ) = $(rl)+(r2),a(1, 2) = a(l)a(2) and P(1, 2) = P(l)P(2), respectively. Note, however, that u( 1, 2) f a(l)a(2) due to the integration over impact parameters in Eq. (12). As the Hartree-Fock approximation is usually defined to be the uncorrelated static wavefunction, we use the time-dependent Hartree-Fock (TDHF)

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McGuire

approximation (Devi and Garcia, 1984; Stich et al., 1985;Gramlich et al., 1986) to define the uncorrelated dynamic wavefunction. In practice this requires defining a new basis set of wavefunctions using the variational principle every time the system changes significantly during the collision (which can be difficult). Also, for highly correlated systems the uncorrelated limit is sometimes not easily defined (e.g., there may be no sensible single-electron configuration). In this chapter we shall primarily consider systems where correlation is not very strong so that the uncorrelated (independent-electron) limit is well defined.

2. Scattering, Relaxation, and Asymptotic Regions In a scattering event the boundary between the scattering region and the asymptotic region is not easily localized. In this chapter we shall assume that the interaction usually occurs within about one Bohr radius of the target nucleus. At high velocities the collision time is usually shorter than the relaxation time required for the final state to decay, usually by either X-ray decay or Auger emission. So the excitation and de-excitation process can be decoupled in most cases we consider. The correlation that occurs outside the scattering and relaxation regions (in both space and time) is referred to as atomic, static, or asymptotic correlation. The Auger effect corresponds to a de-excitation process in which the electron-electron interaction causes a metastable doubly excited state to decay with one electron going to a lower state and the other to a higher state. This is sometimes referred to as dynamic correlation (Stolterfoht, 1989). Correlation occurring during the collision is called scattering correlation.

3 . Waves and Particles

Quantum amplitudes may be formulated in either the wave picture using the scattering amplitude, f,or in the particle picture using the probability amplitude, a. As shown by McCarroll and Salin (1966) and demonstrated explicitly in Section ILH., the amplitudes f and a are related by a Fourier transform, namely,

f(Q) =

I

dB eiQsa(B)

where Q is the momentum transfer, p is the reduced mass, v the velocity of the projectile, and B is the impact parameter of the projectile. Heref = 0 if there is no scattering (e.g., if the target does not interact with the projectile) so that U is replaced by U - I in Eq. (1 1). In this chapter both f and a are used according to convenience of presentation. The duality of these quantum mechanically equivalent representations presents a difficulty, however, when correlation is

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

223

considered. Uncorrelated probability amplitudes do not usually correspond to uncorrelated scattering amplitudes since a product of a’s does not transform by Eq. (13) to a product off’s. In studying correlation we shall generally use the particle picture so that uncorrelated means that the electron probability amplitude is expressed in product form. We shall consider in Sections 1I.G and III.1.2.d examples where the scattering amplitude is in product form. Since this amplitude arises from the electron-electron interaction, we are reluctant to say that electron correlation is not present in this case. A resolution of this dilemma is suggested by Balescu (1975), who suggests the use of Wigner distributions. In most experiments the wavepacket of the projectile is much larger than the size of the target. Under these conditions it is always possible to determine the scattering angle accurately, but it is not always possible to determine the impact parameter accurately. The conditions under which there is a well-defined one-toone relationship between the impact parameter and the scattering angle of the projectile (namely 2v/ZpZT< 1, where v is the collision velocity) were established by Bohr in 1948. Sometimes these conditions are not met in the examples we consider, and then the impact parameter, B, is considered as a mathematical rather than a physical parameter.

4. Interaction Representation In dealing with probability amplitudes we generally use the interaction representation, the projectile is localized so the trajectory is well defined (often well approximated by a straight line), and the influence of the target electrons on this trajectory is neglected (i.e., the trajectory is determined by elastic scattering from the nucleus) as discussed by McGuire and Weaver (1986). In principle one could work in another representation such as the Heisenberg representation. In practice this is difficult because of the coupling between the projectile and the target expressed (Magnus, 1954) as an infinite series of commutator terms in an exponential. Without these terms the exp{iwt} term (e.g., Eq. (40)) is missing in the first-order amplitude in an expansion in Z , and the resulting cross sections become infinite as with the Magnus results of Eichler (1977). 5 . Long-Range Coulomb Terms

The long-range nature of the Coulomb interaction causes some difficulties. The total two-body Coulomb cross section is infinite. The kernel of the LippmannSchwinger equation is not compact so that there is no guarantee of uniqueness of the solution and corresponding numerical difficulties arise (Joachain, 1975; Merkuriev, 1976; Chen and Chen, 1972). The on-shell limit of the off-energy-

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McGuire

shell Coulomb T matrix does not exist due to a phase divergence (Gau and Macek, 1975; Roberts, 1985). Fortunately, the long-range Coulomb terms are almost shielded in nature at large distances. For scattering from neutral atoms the long-range problems may be avoided by formulating scattering (McGuire and Weaver, 1986) in terms of the difference between the instantaneous Coulomb interaction and the average of this Coulomb interaction over the initial- (or final-) state electronic densities of the target. This difference of terms (similar to Eq. (4))eliminates the long-range Coulomb tail in interactions of a charged projectile with electrons of a target atom. In the case of neutral targets, the Coulomb interaction between the projectile and the target nucleus is replaced by the nonCoulombic static potential between the projectile and the entire atomic target with a frozen density distribution of target electrons. Thus unnecessary asymptotic Coulomb terms are removed, e.g., in the case of p + H + H + p where asymptotic Coulomb terms are not required. This also provides a starting point for density functional theory mentioned in Section II.1 where the electron density is no longer frozen. For scattering of charged particles with ions in this formulation, the electronic interactions remain non-Coulombic (i.e., short range) and scattering from the nucleus involves (if necessary) only two-body Coulomb terms that are well understood. This formulation also removes to first order the difficulty arising if nonorthogonal initial and final states are used. The longstanding problem of the asymptotic wavefunction for three charged particles is not, however, solved by this formulation. In many cases, however, there is an advantage in retaining long-range Coulomb terms since mathematical techniques have been developed that are useful in analytic evaluation of matrix elements. A useful formulation of scattering amplitudes in terms of Coulomb wavefunctions, for example, has been developed by Belkic et al. (1979).

6. Exclusive and Inclusive Cross Sections In a rigorous quantum mechanical calculation a theorist may evaluate a probability amplitude a, for a transition from a known
MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

225

particular transition. For example, single K-shell ionization data usually include any event that occurs together with ionization of a K-shell electron. Quantities that include a sum over all possible final states of electrons are called inclusive. As a basic example of inclusive cross sections, consider what happens to the full wave function, q (McGuire and Macdonald, 1975). As the system evolves, the initial state Pidevelops into a linear combination of final states, i.e.,

where ..Ir, represents a particular state of ionization and a, represents the probability amplitude for evolving from Pi to 'Ps.Choosing {PIS} as an orthonormal set we have

corresponding to conservation of probability (unitarity). Let us now represent the wave function of the total system as partially separable

W Pr l; , r2, . . . , r N - l ;{ d ) =

$K(rp)+R,

(16)

then we may consider the evolution of each part according to *i

2 *f

=

2

=

aisqs

(5)

C

aik$Kk

(k)

2 t4

airqRr.

(17)

Using the orthonormality of the {$Kk} and {qR,} we see that the probability amplitude ai, for a transition from a particular initial state to a particular final state is a product of two factors corresponding to (18)

ai, =

so that the probability of producing such a state is also a simple product, namely, lais/*= /aikl2lair)2 = P~P,.

(19)

The corresponding cross section for producing excitation to state k for the participating K electron and state r for the remaining target is ub =

I

PkP, dB,

(20)

where B is the impact parameter of the projectile. If we sum over all final states of the remaining target then using unitarity C P,=l (r)

(21)

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McGuire

we see that c{ rl g & r = I p &( 4c p r d B= I P k d B = g k ,

(22)

where crkcorresponds to a standard single-electroncalculation for exciting the K electron. The K electron is often referred to as the active electron and the remaining R electrons as passive electrons. In practice, neither the experimentalist nor the theorist usually keeps account of everything that happens in a collision involving N electrons, especially when N is large. Usually one chooses to examine just one (or sometimes a few) active electrons and ignores the remaining passive or spectator electrons. Typically calculations for single-electron probabilities or cross sections, which simply omit the passive electrons, are compared to data where the active electron undergoes the specified transition regardless of what the other electrons do. And to the exent that the separability condition of Eq. (16) holds, this comparison is correct. B. INDEPENDENT-ELECTRON APPROXIMATION The independent-electron approximation (or uncorrelated limit) serves as a useful starting point for understanding many-electron systems when multielectron effects are weak. This independent-electronapproximation (Hansteen and Mosebekk, 1972; McGuire and Weaver, 1977; Hazi, 1981; Sidorovitch and Nikolaev, 1983) is exact under the following conditions: (i) The projectile is a point charge. (ii) The internuclear motion is separated from the electronic motion and treated as elastic classical scattering. (iii) The Irk - rj(- I electron-electron interactions are approximated by single-electron potentials. Condition (i) is fully satisfied in collisions of atoms with electrons or bare ions. Condition (ii) is well satisfied for incident heavy ions and may be satisfied for high-velocity incident electrons whose de Broglie wavelength is small compared to atomic distances. Constraints due to condition (iii), which corresponds to ignoring effects of electron-electron interactions during the collision, are considered in further detail in Section 1I.C. We note that the independent-electron approximation does not depend on an expansion in the projectile charge, Z , , but does depend on the weakness of the electron-electron interaction. Electron-electron correlation potentials appear in Eq. (1) where Ha may be written as a sum of operators H’, which reduce to single-particle operators when correlations are ignored. Specifically, from Eq. (3,

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227

Because k (k>11

is not a single-electron operator, the operators Hi are not single-electron operators. However, if the electron-electron interactions (which give rise to cork. (k>Jl)

are approximated by an average potential, i.e.,

(k>j>j)

then the H:, terms are indeed single-electron operators and correlation disappears. Now if electron-electron correlations are ignored, then [Hb, H i ] = 0 since the Hi0 are single-particle operators. Recalling from Eq. (4)that V is a sum of single-particle potentials V, (plus a possible two-body internuclear term that we omit here) and noting that Ha is a sum of commuting single-particle terms, Vl(r) = e-HofVeiHo' =

2 exp[ - i 2 Hir] VI exp

[7 i

Hhr]

That is, without electron-electron correlations, V / ( t )is a sum of single-particle operators V{(r),and as a consequence the evolution operator reduces to a product of evolution operators, i.e.,

The electrons evolve independently during the collision. Finaliy, if Pauli exclusion (discussed shortly) is ignored, then the probability

McGuire

228

amplitude is a product of independent-electron amplitudes, namely,

alf = ~ 1 9=)(4fIWl)=

JJ (+JIu,I+;)

=

J

JJ a;',

(29)

J

where represents the full-electron wave function and = llJ4, is the asymptotic wave function. Consequently, the transition probability laifIz is a product of independent single-electron probabilities in the independent distinguishableelectron (or time-dependent Hartree) approximation, thus simplifying considerably the many-body problem. A binomial distribution of single-electron probabilities results (Section 11.1) if the electrons (or a subgroup of the electrons) have the same transition probability.

1 . Pauli Exclusion Electron symmetries are included in our definition of an uncorrelated wavefunction (Section 1I.A). However, other authors (e.g., Balescu, 1975; Becker, 1988) regard the effects of electron symmetries as corrections to the uncorrelated limit and refer to these corrections as Pauli correlation. Let us include the effect of Pauli exclusion (following Reading and Ford, 1980; Ford et al., 1979; Becker et al., 1980) in a two-electron system in which ( a ,b) + la, 6). In the preceding independent distinguishable electron approximation (without Pauli exclusion), the probability for a two-electron transition quite simply is

P(a9 PI

= =

l(4(Q)4(P)I +(a)Jl(b)l* l(4(4 I W))I21(4(P) I Jr(b))12

(30)

= P(a)P(P).

The inclusive probability, P a , for producing vacancy a in the ground state of A can be found using closure, namely

P, =

z2

u#a

P(QV(P)=

@

z

*#a

P(a) = 1 - ) ( 4 ( a )I

(31)

If more than one state on A is initially occupied, then one could approximately impose exclusion by using Pa

= 1 -

C

occupied a

[($(a) I $(a))Iz.

(32)

In the independent-electron approximation with Pauli exclusion the probability for la, b) + la,p) is P(a7

P)

=

I(4(aW(P)I

1

( J l ( a ) W ) - Jl(b)Jl(a)))l2.

(33)

It is then straightforward (but a little tedious) to show (Reading and Ford, 1980; Ford, 1981; Becker, 1988) that the inclusive probability is

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

Pa

=

2 C p ( a , PI

a#@ @#a

= 1

-

c

occupied k

= 1

229

- I(4(a) I +(a))12 - I((4(a I +(b))12 (34)

I(4W I +(a))12.

This result is similar to the independent distinguishable electron approximation of Eq. (32). Now consider the inclusive probabilities for producing vacancies in both a and b. For distinguishable electrons, Pub = Pup,. For indistinguishable electrons (again after some algebra) it may be shown (Ford, 1981)

Pab =

papb

-

I2 k+a.b

($a

I

$k)(4b

I +,>I2

papb.

(35)

The term subtracted accounts for Pauli exclusion in occupied states.

2. Coupled Channels: Reduction to Independent-ElectronApproximation Consider a system comprised of two electrons, 1 and 2, each of which can be in two states, j and k. This is a four-channel problem. Let represent the full wavefunction and Q> the asymptotic wavefunction. Then the possible transition probability amplitudes are defined by

*

In the independent-electron approximation, 11' = $I* = i,h2 and @ =
McGuire

230

where V = V , + V,, the {I&} are orthogonal and w = (E, - Ek).Using (37) and V;k = ($$lVYIc#4)then one has, after a little algebra, ih: = Via:

+

Vfke-’”‘a[.

(40)

This is the usual equation for the amplitude for each single electron. Thus, in the independent-electron approximation, the coupled channel equation for two electrons in two identical states may be solved in terms of two coupled channel equations, one for each electron (McGuire, 1988). Lin (in Hall et al., 1983) has extended this analysis to nonorthogonal basis sets for application to electron capture. It may be possible to generalize this result to more channels and more electrons. A binomial distribution of single-electron probabilities results (Section 11.1) of the electrons (or subgroup of the electrons) have the same transition probability.

C. MULTIPLE-ELECTRON EFFECTS (ELECTRON CORRELATION) Let us now consider effects of electron-electron Coulomb interactions. These multiple-electron effects are missing in the independent-electron approximation and, for the most part, they may be regarded as electron correlation. Since the probability amplitude is usually expressed in terms of a scattering operator acting between asymptotic states, +i and &, it is convenient to consider separately multielectron effects on the scattering operator and on the asymptotic wavefunctions. As has been noted in Section II.A.2, this division is somewhat arbitrary and does not explicitly address relaxation of excited final states (e.g., Auger and photon decay). For simplicity we consider here a two-electron system. Generalization to a system of N electrons is straightforward.

1, Asymptotic States The unperturbed Hamiltonian for a two-electron system from Eq. (3) may be rewritten (McGuire, 1987) as

+

[rhl - u , ( r , ) - u2(r2)1.

Here the mean field potentials, u , ( r l )and u 2 ( r 2 ) have , been introduced. (Cf.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

23 1

Eq. (26).)A common example of such a potential is u I ( r I )= s/rI

(42)

where s is a screening factor (for example s = 5/16 in variational calculations of the ground state). The correlation potential, u, is defined by v(r) =

- ul(rl) - u2(r2)l.

(43)

Thus, H, = HA

-+ HZ + u ,

where HA and HZ are one-electron operators. The asymptotic states + I and +f are eigenstates of H, . These asymptotic states are not known exactly. However, a number of methods are available to evaluate these asymptotic, static wavefunctions approximately, especially for the ground state and also for singly and doubly excited states of two-electron systems. Continuum wavefunctions are, however, not well known, especially for two strongly correlated continuum electrons (although some useful analytic expressions have been developed (Brauner et al., 1989)). One common method used to evaluate multielectron static wavefunctions is the configuration interaction (CI)method, where added to the independent-electron wavefunction are additional terms called configuration interaction terms corresponding to u # 0 in Eq. (43). Usually and are approximate. Sometimes +, and are regarded as eibecause in some cases correlagenfunctions of different approximate H, and Hf, tion and screening are physically different initially and finally. In these cases and +r are nonorthogonal. This nonorthogonality can result in nonvanishing matrix elements for certain (e.g.. multiple) transitions. Since there is a physical cause for these nonzero matrix elements, they are retained (unlike spurious nonorthogonality terms that are often discarded). A simple example is the shake matrix arising from a change of screening during the collision (e.g., s changes in Eq. (42)). If the correlation potential u is now ignored, the resulting probability amplitude for shake transitions of N- 1 electrons following an interaction of one electron with the projectile is +f

+f

+l

N

The ratio of double- to single-ionization cross sections is independent of the charge and the velocity of the projectile (Mittleman, 1966). Amplitudes of this type may (or may not) dominate for multiple-electron transitions at high collision velocities under certain conditions. (C’ Sections III.A.2-4.) Since this amplitude is expressed as a product of single-electron terms (since u = 0), it is usually not considered to be electron correlation, although it is a many-electron effect.

232

McGuire

2 . Scattering Operator The interaction potential, corresponding to Eq. (9), is the interaction representation, v , ( t ) = e-iU,rV(t)eiHd = e-ilHbf+H6f+vf)

x

[v,(t)+

+

~,(t)]ei(~b= ~ +V:~ ~ ~V:+ ~ ~ )

(45)

N

because [Hh,u ] # 0, it follows that Vj is a many-electron operator. This is not an artifact of the intermediate representation. Rather, it is one way to express the influence of the correlation interaction, u, occurring during the collision. Formulation in other representations (e.g., the Heisenberg representation) is usually ' Section II.A.4.) C more difficult. ( It is sometimes useful to write the many-electron scattering potential, Vj (t),as V j ( t ) = VjO(t)

+ Vj"(t),

(46)

where Vjo is the limit of Vj as the correlation potentials lri - rj[ go to zero, corresponding to the independent-electron approximation. The many-body operator for scattering correlation Vjc is the difference between the full operator of Eq. (45) and the independent-electron operator with u = 0 in Eq. (45). This scattering correlation potential may be used to express the amplitude for scattering correlation to first order, namely, U - Uo = - i

L:

VF(t) dt

(47)

where the independent evolution operator, U , , is defined by Eq. (28). This amplitude for scattering correlation has been further considered by McGuire (1987). D. EXPANSION IN Zp/v

At high collision velocities, v, it is sometimes useful to expand the probability amplitude of Eq. (11) in powers of the scattering interaction V, (Briggs and Macek, 1991; Madison and Merzbacher, 1975; Inokuti, 1971). Since the full solution to the scattering problem usually is not known, expanding in V, is sometimes a useful method of approximation. Physical interpretation of the terms in this expansion is also sometimes possible. The expansion in V, corresponds to an expansion in Zplv where Zp is the charge of the projectile. In most cases this series converges when Zplv << 1. An interesting exception to the uniformity of convergence occurs in the case of electron capture where the leading-order

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

233

term is classically forbidden by conservation of energy and momentum so that the second-order quantum term in Zplv dominates. (Cf.Section 1I.F.) From Eq.(10, 11) using V(r,, - t o ) we have

a = (+f I T exp{i

v,(t) dt)4i)

=6if+al+a2+a3+....

(48)

In this expansion the correlation potential, u, is included to all orders in each term of the expansion. If T = 1, there is no time ordering and it does not matter in what sequence the interactions between the projectile and the target electrons occur (although equivalent regions are included n! times in the exp{ - iV,(r) dr} form). 1. Firsr-Order Term in Zplv

In the high-velocity limit the first-order term of the expansion in Zplv usually dominates. Since the first-order term is relatively simple, this is a convenient limit in which to test theory. From Eq. (48) the first-order term is a1 = - i

lim io-=

/:lo

(4r I V,(rI)l+i) dfI*

(49)

There is no time ordering in the first-order term since the interaction V, occurs only once. If the members of the unperturbed basis set (4) are real (as in the case of single and multiple excitation with bound states), then a , is purely imaginary.

234

McGuire

In the absence of multielectron effects, a , is identically zero for multipleelectron transitions. Mathematically this corresponds to the orthogonality of the single-electron wavefunctions for those electrons that do not interact with the projectile in the independent-electron approximation. Physically multipleelectron transitions cannot occur with only one interaction with the projectile unless the electrons interact among themselves. The first-order term in the high-v limit for charged projectiles is sometimes related to scattering by photons. This occurs because photons with energies below I keV carry little momentum and because for charged particles the minimum momentum transfer, Qmin= AE/2v (except electron capture), is also usually small at high v where cross sections are strongly peaked about values of Q near em,.. However, in photoionization the energy imparted to the ejected electron is essentially equal to the energy of the incident photon minus the binding energy, while in ionization by charged particles at high velocities electrons are ejected with a distribution of electron energies that is largely independent of the energy of the projectile (except for binary encounter events). Thus, when Qmi,<< 1, scattering by charged particles may in some cases be related to scattering by photons. (C’ Section III.B.2.c and Inokuti, 1971.) For two-electron transitions between doubly excited states described by CI wavefunctions, the first-order transition amplitude, a, , has been reduced (McGuire and Straton, 1990b) to closed analytic form. (C’ Section 11.1. I). Extension of their result to N-electron excitation is straightforward.

2. Second-Order Term in Z,lv The second-order term corresponds to two interactions between the projectile and the target electrons. From Eq. (48) the second-order term is given by

If there are no multielectron effects, then for a two-electron transition a’ reduces to a product of two first-order terms in the independent-electron approximation. This follows from Eq. (28) and (29) for a two-electron transition to second order in V , . If 4, and 4, arc real, then a’ is real in the independentelectron approximation. Since a , is purely imaginary, there is in this case no interference between the first-order term and the independent-electron approximation portion of the second-order term (Macek, 1989; McGuire and Straton, 1990b). a. Time Ordering and Energy Nonconservafion. Time-ordering effects are possible starting with the second-order term (McGuire and Straton, 1990b). In the context of this review where the projectile wavepacket is much larger than the

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

235

dimensions of the target, time ordering corresponds to effects of the sequence of interactions with the projectile within the coherence length of the projectile wavepacket. This is a quantum effect. The effect of time ordering may be analyzed by setting T = l (T - l ) . Here T = l is the limit in which time ordering effects vanish (Ryufuku and Watanabe, 1979). Hence we define T - I to be the operator that carries the effects of time ordering. Keeping in mind and using the step function 8, the intermediate state propagator from the e'lHof factors between V(r,) and V ( r , )in the second-order term in Eq. (48) with to-* m is

+

The principle value term P(l/R - E ) vanishes in the limit as T - t 1. Thus, the - ilrS(n - E ) term in Eq. (51) corresponds to T = 1 and the P(lli2 - E ) term corresponds to T - I , which carries the effects of time ordering. We note that in the second-order term T - 1 may be replaced by 28(t, - t l ) = 2(8 - g) where = 1/2 so that T - 1 corresponds to a time variation of the integrand in Eq. (51) from its average value. In this sense T - 1 may be regarded as a time correlation (or memory). Using this in Eq. (51) it may be shown that all matrix elements for both single and double excitation are real so that the second-order term may be represented by Zb(ic2 - C) where 1;, arises from T = 1 in Eq. (50) and c2 arises from the operator T - 1 which gives the effects of time ordering. The time-ordering term c2 also represents effects of energy nonconservation in intermediate states during the collision. This may be seen in Eq. (51) where T - 1 is associated with P( lli2 - E ) which restricts E # R so that intermediate energy is not conserved. The energy nonconserving contributions are purely quantum mechanical and may be ascribed to virtual intermediate states. Energyconserving intermediate states, corresponding to the -lrS(n - &) terms in Eq. (5l), also contribute to the double-excitation probabilities and cross sections for the C2 terms. The energy-conserving terms may be directly related to on-shell physically observable processes, and are also present in classical calculations. It may be shown that c2 and C2 obey a dispersion relation so that in the secondorder amplitude the energy-nonconserving (time-ordered) contribution may be expressed as a integral over energy-conserving (non-time-ordered) contributions and vice versa, namely

F*(&) =

+ -i P

c2(i2)dn

(53)

236

McGuire

Such a dispersion relation also holds in the vicinity of a Thomas singularity in electron capture (Weaver and McGuire, 1984).

3. Higher-Order Terms in Z,lv Calculation and analysis of higher-order terms becomes more difficult as the order of the term is increased. From Fq. (48) the nth-order term may be expressed as

(54) In the independent-electronapproximation for a basis set of real {4}the terms of odd order in Z, are all imaginary and the terms even in Z, are real if time ordering is ignored. In this approximation there are only even-order terms in Z , in the observable transition probabilities and cross sections. These observables are invariant under change of sign of the charge (e.g., replacing proton projectiles by antiprotons). In the independent-electron approximation the lowest-order nonzero term for an N-electron transition is a product of N first-order transitions. That is, Nthorder perturbation theory is approximated by a product of N first-order transitions. This is a significant reduction in complexity. This result is modified when the projectile carries electrons (C'. Section 1I.G.) In higher- (n > 2) order terms time ordering and intermediate energy nonconservation are not in one-to-one correspondence. The energy nonconservation term introduces a relative phase factor of i every time the principle value term in Eq. (51) occurs. The nth-order term contains n - 1 propagators, any one of which may, or may not, be off the energy shell. Higher-order terms in Z, are generally required when the condition that Z,lv < 1 is not met. However, for a group of independent or uncorrelated electrons with similar principal quantum numbers, if the number of electrons is large, then products of first-order amplitudes are, nevertheless, sometimes sufficient. (Cf.Eq. (88).) If multielectron effects are weak, then the first N - 1 amplitudes for an Nelectron transition may be small since these terms go to zero as multielectron effects vanish. In this limit the leading-order term in Z,lv is the Nth-order independent-electron approximation term. As the strength of electron correlation is increased or the collision velocity is increased, the first N - 1 terms become relatively more important. 4 . Z; Inte$erence

At high collision velocities the perturbation series is often convergent. The first two terms are then adequate for two-electron transitions, namely

MULTIPLE-ELECTRONEXCITATION, IONIZATION,AND TRANSFER

a

= a,

+ a2 = CZ, + C’Zb

237 (55)

where C and C’are complex coefficients independent of Z,. Transition probabilities and cross sections vary as laI2.Thus, if both C and C’ are nonzero, then a Z; term representing interference between first- and second-order amplitudes is possible. In this case cross sections for particles of opposite charge are different. For a two-electron transition C = 0 if there are no multielectron (usually electron correlation) effects. (CJ Section 1I.D. 1.) Consequently, in two-electron transitions multielectron effects are required for Z; interference terms to be present at high v. If r#Jj and r#Jf are real, then C is purely imaginary and we may set C = - ic, where c1is real. In the second-order coefficient there are both real and imaginary terms, i.e., C’ = F2 - icz where c2 and Fz are real. It may be shown (Section II.D.2) that c2 arises from time-ordering effects (i.e., T # 1 in Eq. (48)). Eq. (55) may now be expressed as

a = -ic,Zp

- (F2 -

ic2)Z;

=

-i(cl - c2Zp)Zp- TzZ;.

(56)

We have seen that a nonzero c, carries effects of spatial (i.e., rsL)electronelectron interactions and a nonzero c2 carries time ordering. The term F2, which has no time ordering, may carry some electron correlations and does not include the lowest-order independent-electron approximation, where a reduces to a simple product of first-order probability amplitudes. Cross sections, IT, and scattering probabilities, laI2, may be expressed by IT =

=

la12

d~ --.

J (c:z;

c:zg - 2c,,z;

+ c:zg + c22z8) dB + c:zg + C:z;! + O(ZSp) -

2c,c2z;

(57)

where B is the impact parameter of the projectile. The difference in double excitation by particles of opposite charge (cf. Section III.B.2) is given by

w( - ) -

IT(

+ ) = 4c,,(zPp.

(58)

This is nonzero only if effects due to both spatial correlation (in c,) and time ordering (in cz) are present. This result may not apply to capture or ionization since r#Ji and & are assumed to be real. Also, coupling to a continuum background is ignored. (C’ Section 1II.C.1.)

E. MANY-BODY PERTURBATION EXPANSIONS IN BOTHZ,/v

AND u

In this section we consider expansions in powers of both the scattering potential, V,, (i.e., Z,/v) and the correlation potential, u . Let us begin with the first-order probability amplitude in Z,/v from the previous section and further expand this term in powers of u.

238

McGuire

Through first order in u we have,

+

t(V?[&, v'l - [Hb, ufIV?)}l4,)

where VF = e-fHhefHbr and where 4, and 4fare eigenstates of Hb and HL which differ if there is a change in screening. For a two-electron transition the first term (shake) is nonzero only if Hoand H i differ. The second term (ground state correlation) may be regarded as representing the effect of u to first order on 4,. The third term (TS 1, defined shortly) may be regarded as either final-state correlation or an electron-electron interaction following interaction with the projectile. The last term is the lowest-order approximation for scattering correlation. (Cf. Section 1I.A.) Exchange terms (not explicit in Eq. (59)) arise from the antisymmetry of and &. The independent-electron approximation term (not explicit) is a term second order in V, and zeroeth order in u. All such terms obtained by expansions in V, and u many be represented by diagrams (or combinations of diagrams) from many-body perturbation theory (MBFT)(Carter and Kelly, 1981; Ishihara and Hino, 1989) partially illustrated in Fig. 1. In these diagrams excited particles propagating forward in time are +I

(C)

FIG. 1 . Many-body perturbation theory diagrams (Carter and Kelly, 1981). The ground state (or vacuum), which is not shown, propagates upward until electron-electron interaction (---) or interaction with the projectile (-) occurs. Only diagrams first order in correlation are shown: (a) ground state correlation, (b) TSl (two step with one electron-projectile interaction) or final-state correlation, (c) shakeoff.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

239

denoted by upward arrows and holes (equivalents to particles propagating backward in time) are denoted by downward arrows. MBPT techniques are well developed and have been applied in various areas of physics. There are welldefined rules for writing amplitudes corresponding to each diagram (Fetter and Walecka, 1971). These diagrams sometimes can provide images to guide our understanding of possible physical mechanisms for multiple- (as well as single-) electron transitions. Some commonly used terms corresponding to physical mechanisms include:

1. Ground-state correlation. This refers to the effect of the electron-electron interaction, u , in the ground state. 2. TSI (two step-one) or final-state correlation. Here one electron collides with another after the first electron is hit once by the projectile. This is a two-step process with one interaction with the projectile. Sometimes this is alternatively regarded as the effect of electron correlation in the final state following the collision. The interpretation depends on where the boundary between the collision and the asymptotic region is chosen. (Cf.Section 1I.A.) 3. Shake. This corresponds to rearrangement in the final state of the target after collision with the projectile. This rearrangement is due to a particle-hole interaction which may occur because one electron isn’t present in its initial state. The wavefunction collapses because of a lack of screening. The MBPT shake amplitude is first order in u while the overlap term in Eq. (44)usually contains all higher-order terms in u. 4. Exchange terms. These are Pauli correlation effects. (C’ Section 1I.B. 1.) 5 . TS2 (not shown in the figure). This ia a two-step process with two interactions with the projectile. It corresponds to the independent-electronapproximation. 6. Scattering correlation. This is included in all diagrams in which electronelectron interactions are included in the scattering region. (The boundary of the scattering region is not usually well defined (Section 1I.A.2).) The approximate scattering correlation term in Eq. (47) corresponds (approximately) to a difference of second-order diagrams. F. SPECIAL FEATURES OF ELECTRON CAPTURE, NONORTHOGONALITY, AND SPECIAL KINEMATICS IN MANY-BODY SCATTERING Atomic scattering is categorized by groups of final states, namely, elastic scattering, excitation to bound states, ionization, and electron capture. These categories are not truly distinct (e.g., high Rydberg excited states mix into the continuum in the presence of external electromagnetic fields). For multiple- (and single-) electron transitions we are primarily concerned with excitation, ionization, and electron capture. In some ways electron capture is different from excitation and ionization. Un-

McGuire

240

like excitation and ionization, the quantum amplitude for electron capture converges to the second-order term (Shakeshaft and Spruch; Alston, 1989) in Zplv at high v because the first-order term is restricted (but not strictly forbidden) by conservation of overall energy and momentum (McGuire et al., 1989). Except at very high velocities (often relativistic) second-order effects in total cross sections are rather small, however. In electron capture the perturbing asymptotic potentials differ initially and finally. For excitation and ionization the minimum momentum transfer is given by = AE/v which becomes small at high v while for capture (which includes an electron translation factor) Qmin= v12 AE/2v which is large at high v. Since scattering usually occurs in regions where Q * B = 1, capture tends to occur at small impact parameters, B, while excitation and ionization occur at large B at high v. In electron capture the final and initial asymptotic scattering potentials differ. Perturbation theories (Taulbjerg, 1990) exact through second-order include the strong potential Born approximation (first order in the weaker asymptotic potential) (Alston and Macek, 1982; Alston, 1989; Taulbjerg e? at., 1990) and the continuum distorted wave approximation (higher order in both asymptotic potentials) (Belkic et al., 1979). Scaling of scattering probabilities and cross sections in Z,, Z,, and v (cf. Section 1II.A) tend to be quite different for capture than for ionization and excitation at high v. Nonorthogonality of asymptotic wave functions arises mathematically in electron capture due to translation factors. Because of the translation factor overlap between the final state and the initial state is nonzero corresponding to a finite probability for a transition even when the target does not interact with the projectile. This nonphysical finite transition probability can be removed in a variety of ways (McGuire, 1985). One may use Gramm Schmidt orthogonalization, e.g., replace ($3 by 14f) - ( 4 i ) (1~&). i Or one may use Bates (1958) correct derivation of first-order perturbation amplitudes where the instantaneous scattering potential, V, is replaced by the difference of V and its averaged value, (4ilVl$i). Equivalently one may use the correct first-order distorted wave result derived by Bassel and Gerjuoy (1960). A Green’s function approach using a single asymptotic Hamiltonian is also available (Halpern and Thomas, 1979). All of these approximate methods may be shown to be mathematically equivalent to first order in V at small momentum transfer, Q.Nonorthogonality may also arise due to a change in screening in asymptotic wavefunctions. (Cf. Sections 1I.C.1 and III.A.2.) This type of nonorthogonality is physically meaningful. Generally, effects of nonorthogonality are retained when they are physically meaningful and eliminated when they arise from a mathematical mistake. Special kinematic conditions arise that lead to effects observable in differential cross sections for electron capture (Briggs and Taulbjerg, 1979). Because of the restriction that two particles leave the scattering region together in particle trans-

em,.

+

-

MULTIPLE-ELECTRON EXCITATION, IONIZATiON, AND TRANSFER

I (2,3) +

24 1

(1,2) +3

-+

7

FIG. 2. Two-step particle transfer diagram (McGuire er nl., 1989). The intermediate mass M' may equal M I ,Mz,M3.The mass of the upper (lower) particle on the diagram is taken as Mf(Mf) in text and M,(Rf) may equal either MI(M2) or M,(MI).

fer, electron capture in a one-step process is classically forbidden by conservation of energy and momentum at high collision velocities. Consequently, the simplest classical mechanisms for particle transfer is the two-step process proposed by Thomas in 1927. The kinematic diagram corresponding both to the classical model of Thomas and to second Born terms is shown in Fig 2, where particle 1, 2, or 3 scatters twice. Here, since particles 1 and 2 go off together with the same velocity, the entire collision is coplanar. If all the masses and the incident velocity v are known, then there are six unknowns, v', v f , and v3 as defined by Fig. 2. Conservation of momentum gives two equations of constraints for each collision. Conservation of overall energy gives a fifth constraint. And conservation energy in the intermediate state gives a sixth constraint. With six equations of constraint, all six unknowns may be completely determined. For p + it is easily verified that in the notation of example, for p + + H + H Fig. 2, (Y = (me/M,)sin(60"), p = 60", and y = 120', where M' = M f = M , = me, an electron mass, and M , = M3 = M p , a proton mass. The standard Thomas peak in p H at a = 0.027' has been observed (Vogt, et al., 1986). In general, conservation of overall momentum and energy gives three equations of constraint in the four unknowns vf and v3, namely,

+

+

M,v = ( M , + Mf)vf + M3v3, $M,v: = &(Mf+ Mf)vj f M , v $ ,

+

where Mf (fir)is the mass of the upper (lower) particle in the final bound-state system shown in Fig. 2, in which M' is the mass of the intermediate particle

242

McGuire

( M , , M 2 , or M 3 ) . From Eqs. (60) and (61) it is easily shown that the velocity of the recoil particle is constrained by the condition

+

where r = ( M I M2 momenta defined by

+ M , ) M 2 / M l M 3 .Here K

is a dimensionless ratio of

Equation (62) specifies the values of v, allowed by overall energy and momentum conservation independent of the intermediate states of the system. In the simplest two-step process, illustrated in Fig. 3, we impose conservation of energy in the intermediate states, namely,

+

Em,, = $ M I v 2 = &%f’vf2 &M,v:.

(64)

After some algebra, we obtain for M‘ = M I , M 2 , or M3 the general condition for intermediate energy conservation, namely,

K = 1.

(65)

This corresponds to v3 = M ’ v / M 3. The classical Thomas peak occurs when both Eqs. (62) and (65) are satisfied. In two-step processes for transfer, the intermediate particle can in general be particle 1, 2, or 3. For the case where where the intermediate particle is particle 2, i.e., the transferred particle, the loci for the overall energy and momentum conservation are shown in Fig. 3 at various values of r (McGuire et al., 1989). Also shown is the intermediate-energy-conservingThomas peak at K = 1. Even when the classical Thomas peak is kinematically forbidden, the tail of the broadening may extend into the physical region if the energy uncertainty in the intermediate states AE is large enough. Two-step binary encounter Thomas effects are also present in differential cross sections for ionization (Zhang, er al., 1990) although they are less pronounced because the first-order terms in Z J v are not small. One-step binary encounter effects are prominent in some differential cross sections for ionization. Another special kinematic peak occurs when an electron goes off together with a positive

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

243

1.0 .0.8

-

O6

-A

jo'

<> o.2 ';r 0.0 I1 h v) -a2 0.4

0

0

-0.6 -0.8 -0.4

-1.0 L,

%

Jq

I

4

16

64 256 10244112 16448

FIG. 3. Recoil angle versus recoil velocity for two-step particle transfer (McGuire ef al., 1989). Overall energy- and momentum-conserving loci at 2 cos y = rK - ( I I K ) for various values of r = (MI + M 2 + M3)IM1M3,and intermediate-energy-conservation locus at K = 1 when the intermediate particle is the captured particle, i.e., M' = M,.These loci are invariant under the interchange of M I and M 3 . In the equivalent figures for M' # M , , the only change is a shift by M'IM2 of the locus of intermediate energy conservation. The uncertainty principle permits particle transfer within intermediate energies of +. AE of the Thomas peak along the locus of overall momentum and energy conservation.

(or possibly neutral) projectile in an unbound state. For positive projectiles this electron-capture-to-the-continuum peak (which we somewhat arbitrarily include in the category of ionization) arises from the long-range behavior of the Coulomb interaction (a counterexample to the argument of Section II.A.5 that long-range Coulomb are usually negligible). These are examples of special kinematics in the many-body problem that are relatively simple conceptually and mathematically, and are observable experimentally.

244

McGuire

G. PROJECTILE ELECTRONS AND A CLASSICAL ATOMIC LIMIT Thus far we have considered collisions of structureless projectiles with a multielectron target. Now let us consider collisions in which the projectile also carries electrons. Now in addition to the projectile nucleus the projectile electrons may influence the scattering process (Briggs and Taulbjerg, 1978; Inokuti, 1979; Matsuzawa, 1980; Anholt, 1986; Meyerhof and Huelskoetter, 1990). If the projectile electrons interact with the target electrons (possibly changing the state of both the target and a projectile electron with each electron-electron interaction), there may be two-center multielectron effects. Effects of this type have been variously labeled as two-center scattering correlation (Stolterfoht, 1989), antiscreening (McGuire et al., 1981) inelastic scattering (of the projectile) (Bates and Griffing, 1955), and the free collision model of Bohr (1948) where the projectile electrons scatter incoherently with other particles on the projectile. I . First Born Approximation

Let us consider scattering of a projectile of nuclear charge Zp carrying N electrons from an effective one-electron target of nuclear charge Z, . In the first Born approximation the scattering amplitude for an inelastic transition from +I to $f in which the state of the target electron is changed is

AQ>

+

where = +p+, functions and

=

J d~ eiQ*

(+pf+LT

IV,~J+pi$ti)

(66)

corresponding to a product of projectile and target wave-

Due to orthogonality of the target states, the first two terms in VIMare zero. The next term contributes only when the projectile remains in its initial state. The last term is the sum of electron-electron interactions between the N projectile electrons and the target electron. Fourier transforming the potential terms using J eiQRIIR- r( dR = 4n/Q2eiQ",we convert the sum to a product form, namely,

The differential cross section may now be written as

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

245

If a sum over final projectile states is taken, one has

where 2Zff,the square of the effective charge of the Z , = 1 projectile, is expressed as

+ St",.

Z M Q ) = ( Z P - F)*

(71)

Here N

F =

(+PI

IC eQ ' J I 4,) and I- 1

S,,

c

=

N

(+,f

ffl

I

c

J=I

+pl)12

etQq

(72)

are the atomic form factor and the incoherent scattering function, respectively. S,, arises entirely from the electron-electron interaction. As expected, Z2ff reat small Q and to 2* N at large Q as illustrated in Fig. 4. duces to (2, For transition of N electrons on a single target center, a minimum of N interactions with the projectile nucleus is required if there are no electron-electron interactions. For two centers both carrying electrons, a minimum of N / 2 electron-electron interactions is required. In general, a nucleus can excite at most one electron per interaction and an electron can excite at most two per interaction (itself and its interaction partner).

w2

+

1

I

I

I

I

NUCLEUS AND N INCOHERENT ELECTRONS

w

,

I

2, MOMENTUM TRANSFER,

Q

FIG. 4. Effective charge squared of a projectile of nuclear charge Zp carrying N electrons as a function of the momentum transfer Q (Cj.Eq.(71); McGuire eta!., 1981.) At small Q (large B) Zp is fully screened, and at large Q (small B) the charges act incoherently, corresponding to the free collision model of Bohr (1948).

McCuire In the wave picture in the first Born approximation the scattering amplitude for a two-electron transition Eq. (62) is a product of three terms: an internuclear term times a projectile term times a target term. In a mathematical sense this product form might be uncorrelated. In a physical sense, however, it is the correlation (electron-electron) interaction that is causing the two-electron transition. In the impact parameter representation (cf. Section II.A.3) the probability amplitude is not a product of single-particle terms and the process in this picture is not uncorrelated. This technical dilemma is discussed further by Balescu (1975), who suggests the use of Wigner distributions.

2. The Measurement Problem and a Classical Limit A long-standing problem in physics is the measurement problem: where and how does the reduction of quantum mechanics to classical physics occur? The classical limit of quantum mechanics is sometimes associated with the disappearance of interference terms as in the Schrodinger cat paradox. A reduction from interfering quantum amplitudes to an incoherent classical limit on the atomic level is contained in Z& illustrated in Fig. 4.Here at small momentum transfer Q , the projectile charge is fully screened and & = (Z, - Mz,which is proportional to N 2 . For large Q the amplitudes become incoherent as the exp{iQ . (rk - r,)} interference terms in S,,, disappear and Z& = 2; + N corresponding to incoherent scattering by the particles in the projectile and linear in N . This is a limit that could have been found from classical methods without quantum mechanics. While the measurement problem in its full form includes a variety of classical limits (h = 0, n = m, distinguishable states, onset of entropy, decoupling of a detector from observation, etc.), the change of Zzfffrom coherent screening to incoherent scattering of individual particles is one of the simplest examples of the reduction of quantum mechanics to classical physics. We now make a general observation on the nature of quantum mechanics. Physicists typically try to understand macroscopic phenomena in terms of microscopic particles (which sometimes have never been directly observed, e.g., an electron or a proton). Thus, it is natural for the physicist to ask what the classical limit of quantum mechanics is. Lindhardt (1988) has suggested a somewhat more direct question: how may one describe the process in which an experimentalist creates conditions to observe quantum coherence in the midst of often noisy statistical classical macroscopic phenomena? Weaver ( 1990) points out that the quantum mechanics provides an apparently necessary bridge from a classically prepared macroscopic system to classical detection. That an experimentalist is able to coax (often reversible) coherent quantum processes from a sea of fluctuating (usually irreversible) macroscopic events seems worthy of tribute.

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247

H. MOLECULES A N D THE TRANSFORMATION BETWEEN a (B) A N D ~ ( Q ) Molecules are collections of atoms bound in geometric patterns determined largely by the interactions of valence electrons. Thus, scattering from molecules may be formulated in terms of scattering from atoms influenced by both geometry and molecular wavefunctions. Sometimes (but only sometimes as discussed in Section 111.1) a molecule may be regarded as a collection of atoms held at fixed interatomic distances. To establish an expression for the molecular T matrix (hereafter denoted as T M )of scattering from a diatomic molecule, we shall treat the diatomic molecule as a system containing two effective atomic scatterers centered on each nucleus. Using the well-known expansion (Messiah, 1965) to treat multicenter scattering and ignoring rescattering (or memory) between the effective centers, one can express TM as (Wang et al., 1989)

TM

=

(Ti + T2) + (TICoT2 + T2GOTI) + . . .

Ti

+ Tlr

(73)

where Ti ( j = 1, 2) stands for the individual T matrix for each effective center. An effective atomic center in a molecule is not quite the same as an isolated free atom. It will be shown that the T matrix for an isolated atom and an atomic center in a molecule differ by an overall constant. For high-velocity collisions we ignore vibration and rotation for the molecule, which is effectively frozen during the relatively quick collision. For scattering from two centers illustrated in Fig. 5, it is convenient to place

Phase difference

=(&- K,).p^ c.; FIG. 5 . Illustration of superposition of scattered waves for a diatomic molecule (Wang er a / . . 1989). On the left is the incident plane wave with wave vector K,; on the right are the scattered waves from center 1 and center 2 with wave vector Kf. Here Q = K, - K,. Illustration of ImpactI

parameter picture for a diatomic molecule. p is the molecular axis vector that defines orientation of the molecule (Wang e t a / . . 1989). CM denotes center of mass of atom I and atom 2. Zpis the charge of the projectile, B is the impact parameter. The total probability amplitude. a, is related to a( I ) and a(2).the amplitudes from center 1 and center 2, by Eq. (77).

248

McGuire

the origin of all coordinates at the center of mass of the system. Then the atomic centers are displaced from the center of mass by (1 - p)p and - pp for center 1 and center 2, respectively, where p = M , / ( M , M 2 ) . It is well known (Messiah, 1965) that displacing the origin of the coordinates by a distance d corresponds to

+

(74)

T,(d) = e-'QdT,(0),

with Q = Kf - K,. Consequently, we have

TM

TI

+ T2 == Tl(0)e-t(l-+)QP + T,(O)elfiQ~ t,e-dI-+H)~+

(75)

t2pQP

with respect to the center of mass of the molecule. Here T,(O) = t, is the T matrix for center j where the projectile interacts with the effective atomic center with no displacement. Thus, the relative phase between T I and T2 due to the internucelar displacement p is Q * p , independent of the choice of origin of the coordinate system. As a consequence the quantum waves scattering from the diatomic molecule produce an interference pattern characteristic of scattering from two centers. Now we find the corresponding probability amplitude for scattering by a particle with a straight-line trajectory R = B + vt passing through the center of mass of the molecule at c = 0 and B = 0. The probability amplitude a(B) is in general related to the T matrix by a Fourier transformation (corresponding to Eq. (13)), namely,

'I

a(B) = V

e'QlBTd2Ql,

(76)

where Ql is perpendicular to the projectile velocity v taken to define the z axis of the system. Thus, taking and using Eqs. (75) and (76) we have a general expression for the probability amplitude, namely,

'I

aM(B)= U

B(t,e-dl-+)(QI PI+Q;P

~ I Q L

e - d l - +)Q,P, U

I

e*QlIB-(I

)+r,el+cQlPL+Q,P,))

- w b ~ l l t , d2Q

#Q,

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

249

where S = Q,p,, and ei(l-P)Q+'* is an overall phase dependent on the choice of origin of the coordinate system. This overall phase may be ignored in the evaluation of physical observables such as Then the net probability for scattering from a diatomic molecule is given by

PM

=

=

1~11'

+

1~21'

+ 2Re(a,a2ei6)).

(78)

This is of the form of a vector sum, i.e., uM = a, + u2where PM = Note that B - (1 - p)pL = 0 and B ppl = 0 correspond to trajectories passing through centers 1 and 2 , respectively. Also the phase difference 6 is independent of the choice of origin of the coordinates, i.e., independent of p. In the impact parameter treatment illustrated in Fig. 6, the probability amplitude uM is the sum of an amplitude a, from center 1 at position (1 - p)p relative to the center of mass, plus an amplitude u2 from center 2 at - p p . The amplitudes u1and u2 have a phase difference 6 = Qzpzarising from the separation of the two centers. For heavy projectiles, i.e., Mp>> 1, such as we consider in this chapter, one has to order (l/Mp), Ql = Qmin= Kf - K , , the minimum momentum transfer. For electron capture, using overall energy conservation, one can obtain Qmin= Kf - Ki z v12 - AEl2v = Q l , where AE = Ef - Ei and for excitation and ionization Qmin= AE/2v. An alternative derivation of the probability amplitude has been obtained by Shingal and Lin (1989) in the impact parameter representation by combining a translation factor, exp{iQ * p } with the difference in the transit time of the projectile. Because of the phase difference, 6, probabilities and cross sections for scattering from molecules depend on the orientation of the molecule (Cf. Fig. 5 ) .

+

p h s e difference = PzQ,

FIG. 6. Illustration of impact-parameter picture for a diatomic molecule. p is the molecular axis vector that defines orientation of the molecule (Wang et d., 1989). CM denotes center of mass of atom 1 and atom 2. Z, is the charge of the projectile. B is the impact parameter. The total probability amplitude, a, is realted to ~ ( 1 )and a(2), the amplitudes from center I and center 2, by Eq. (77).

McGuire

I. METHODS OF COMPUTATION FOR FEW-A N D MANY-ELECTRON TRANSITIONS I . Few- (Mostly Two-)Electron Transitions Cross sections for transitions of a few electrons are generally more difficult to evaluate than for single transitions and less difficult than for many-electron transitions. More work has been done for few-electron transitions than for manyelectron transitions. However, we give only a brief overview of calculations for transitions of few electrons since these are relatively well documented in the present literature. Three methods for dealing with the more difficult manyelectron transitions are given here with more detail.

a. First and Second Order. Calculations based on perturbation expansions in the scattering potential V, and the correlation potential, u , are often relatively simple both mathematically and conceptually. Even so, as of early 1991 no calculation exact through second order has yet been done (although coupled channel and forced impulse calculations described shortly below should be noted). Calculations exact through first order in ZJv and all others in v have been completed for double excitation by McGuire and Straton ( 1990b). Calculations including ionization are more difficult because correlated continuum wavefunctions are not easily determined and electron capture is difficult because of translation factors. Double-excitation calculations through second order in ZJv have been done (McGuire and Straton, 1990b) using closure in the intermediate states and confining intermediate states to real {$}. Mechanisms corresponding to specific many-body perturbation terms have been considered by Sorensen (Andersen er al., 1987) and by Ishihara and Hino (1989). Some interesting models including some second- and third-order contributions have been proposed by Vegh (1988). Calculations in the independent distinguishable electron approximation have been done for double ionization and for double capture. And some progress has been made in the case of simultaneous electron transfer and excitation where a combination of amplitudes involving both V, and u have been evaluated. Further discussion of these calculations may be found throughout Section 111. b. Classical Calculations. Olson (1987) has extensively applied fully classical calculations to few electron transitions. In these calculations the electron that classically orbits the target nucleus interacts with an incoming projectile. The orbit of the electron can then be evaluated from coupled equations and followed in time. These calculations are detailed in their description and do not depend on use of perturbation theory in V , . Including effects of the electron correlation potential, u, is more difficult since the atom becomes classically unstable when

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

25 1

u is introduced. However, Olson has incorporated some correlation by simply keeping the electrons on opposite sides of the atom (Zajfman and Maor, 1986). Other classical calculations (Peach et al., 1985; Reinhold and Falcon, 1986; Horbatsch, 1986) include some multielectron effects using model potentials.

c. Time-Dependent Hartree-Fock. The independent-electron approximation including Pauli exclusion is defined (Section 11.A. 1) by the time-dependentHartreeFock approximation. Devi and Garcia (1984, 1990). Gramlich et al. (1986), Stich et al., (1985), and Bottcher et al. (1990) have done a number of calculations using TDHF methods. The trajectory of the projectile is considered to be classical and known. The TDHF wavefunction is found self-consistently at each time step in the collision by optimizing the effective central one-electron potentials so that the effect of the remaining correlation potential u on the energy is minimized. Thus, a new set of TDHF basis functions is determined at each time step of the collision. The basis functions used to determine the TDHF wavefunctions include atomic expansions, molecular orbital expansions, pseudostate expansions, and three-center expansions. Translation factors and continuum states are included. d. Forced Impulse Method. In the forced impulse method developed by Reading and Ford (1987a,b), the system evolves without correlation (i.e., u = 0) for short time steps (as in TDHF). The new set of states is then used as a basis set together with the full Hamiltonian with V, fixed and u # 0 to define a correlated wavefunction at the new time. This correlated wavefunction then evolves without correlation to define the next set of basis states which are again used to evaluate a new wavefunction with correlation at the new time step. Thus, dynamic correlation is ignored only for short time steps. As the duration of the time step is reduced to zero, the result becomes exact. Reading and Ford have used two time steps so that their results are valid through second order in Z,/v. These results may be extended to include more time steps. e. Coupled Channel Calculations. In the method of coupled channels the full scattering wavefunction is expanded in a particular set of basis states. The full Hamiltonian including both V, and u is then used to define a set of coupled differential equations for the coefficients of the expansion of the exact wavefunction. Choice of the set of basis states can be important since the complete expansion must in practice be truncated. The numerical difficulty of these calculations increases rapidly as the number of basis states is increased. Nonetheless, the coupled channel method is the most complete practical method now available. Coupled channel calculations for few electron transitions have been carried out

252

McGuire

(e.g., Shingal et al., 1986; Fritsch and Lin, 1983, 1986, 1990; Reading et al., 1984; Bottcher, 1982; and Kimura et al., 1985). A review is given by Fritsch and Lin (1991).

f. Other Velocity Regimes. For the most part in this chapter we omit both low collision velocities and relativistic velocities where related work has been done (Barat, 1988; Rocin et al., 1989; Barany, 1990). At intermediate to low collision velocities where the molecular orbital methods may apply, scattering correlation becomes more important and has been examined. Also, the question of successive versus simultaneous interactions during a collision has been considered. 2 . Many-Electron Transitions The theory of many-electron transition amplitudes usually requires more detail than the theory of few-electron transitions. If we recall (Section I) that it is not at present possible to fully describe single excitation for impact by electrons (or other charged particles) of atomic hydrogen to the first excited state, then the difficulty of dealing with many-electron transitions becomes apparent. For manyelectron effects optical potential methods (Dreizler, et al., 1990) and statistical models are sometimes effective. The statistical methods include density functional techniques (a generalization of the use of the static potential mentioned in Section II.A.5) (Kohland and Dreizler, 1990), information theory (Aberg et al., 1983; Blomberg et al., 1986), Fokker-Planck (Section II.1.2.c) statistical energy deposition (Russek, 1963; Section II.1.2.b), and fluid dynamic methods (Horbatsch and Dreizler, 1986). Although we concentrate on few-electron transitions in this chapter, three different statistical methods are developed for multiple-electron transitions. a . Binomial distributions, (t)Pn(l - P)N-n.If events 1 and 2 are independent (i.e., uncorrelated), then the probability that both 1 and 2 occur is

P

=

P , x P2

(79)

if Pi ( j = 1, 2) are independent probabilities for event 1 and 2. Let us consider a model system of two distinguishable electrons each of which has two states, $i and &. Let P, be the probability that the jth ( j = 1, 2) electron has made a transition and Qj be the probability for no transition. Since total probability is conserved, then after a collision each electron is either in the initial state +i or the final state &, that is, P,

+ Qj = 1 .

(80)

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

-

After the collision there are four possible states: neither makes a transition

P?

=

only #1 makes a transition

P;O

=

only #2 makes a transition

P!'

=

both make a transition

P;'

(1

-

P,)(l

-

Pz)

-

253

( 1 - P)2

P1=Pz=P

PI(] - P2)

P(I - P )

P,=P2=P

P2(1 - P I )

P(I - P )

PI=P2=P

=

PIP2

sum = 1

sum = 1

sum = 1

The right-hand column gives the uncorrelated probabilities when P I ---* P2 = P . All columns sum to 1, reflecting the fact that the total probability for all events is one. It is at this point straightforward to find all uncorrelated probabilities for a system of two distinguishable electrons with three possible states: ground state, ionization, and capture where P , is the ionization probability and P , is the capture probability. In the preceding example with P I = P2 = P , we can write the uncorrelated probability that n = 0, 1, 2, of the two electrons made a transition as

P = (?)P.(l

- P)2-n

where ( 3 is the binomial coefficient, i.e., (2) = 1 . Q) = 2 and (z) = 1. Such binomial distributions are commonly encountered (Eadie er al., 1971) in statistical physics. A simple and useful generalization of Eq. (81) may be obtained by expanding ( P Q). where Q = 1 - P , namely,

+

1 = (P

+ Q). = =

n=o

i:

"=O

(:)P8QN-.

(;)P.(l

-

P)N-n

where the binomial coefficient (f) is the number of ways of arranging n of N electrons. The quantity

254

McCuire

is called the probability function, corresponding to the transition of exactly n of N electrons each one of which has a single-electron transition probability, P . In the notation of Section II.A.6 this is an exclusive probability. Using the binomial distribution it is straightforward to show that the average number of electrons ionized, (n), is given by ( n ) = N P , and that cr2 = (nZ) - (n)' = (n(n - 1))' - (n2) - (n) = NP(1 - P ) . A simple sum rule is easily developed from Eq. (83). Let us consider an atom with J shells. The exclusive probability for exactly n, electrons undergoing a transition in shell 1, n2 electrons in shell 2, . . . , and n, in shell J is given by

Let us assume that P , = P N s<< 1 in some shell s. Then the inclusive probability for removing one electron from the sth shell, summing over all other shells, is P =

,rl=O

(;;)P1'(1

-

P1)NI-"' . . . (?)PS

x ...

where we have ignored terms of order Pf and smaller and used Eq. (83) to do the sums. The result is that the inclusive probability is equal to N,P,, i.e., just what one would expect if one had only N , independent electrons with equal probability P , . In this picture it is the inclusive probability (not the exclusive probability) that is conceptually simple. This extends in a useful way the notion that passive electrons are frozen. If the passive electrons are independent, a sum over all final states (corresponding to many experimental situations) gives a probability of 1 that each passive electron did or did not make a transition. If P is not small compared to 1, then one may not ignore terms of order P 2 and it may be easily shown that

where the middle term is the binomial inclusive probability. Ben Itzhak ef al. (1988) made a useful discovery that justifies in some cases the use of perturbation theory in computing total cross section, even if P is not

MULTIPLE-ELECTRON EXCITATION. IONIZATION, AND TRANSFER

255

small at small impact parameters, B . Consider a P ( B ) that is montonically decreasing in B. Then the maximum of P f ( B )occurs at B,,, given by

= (:)[n

dP dB

- N P ] P n - l ( l - P)N-n-l] -

whence

If nlN << 1 is small, then P is small for the B that contributes most to the total cross section, then perturbation theory may be applied. In other words the factors restrict P to small values if N >> n. of [ 1 Finally we note that it is not necessary to use the same value of P for all electrons in a given shell. Different electrons may have different values of P (e.g., due to different binding energies). In this case a multinomial distribution (Eadie et al., 1971; Aberg, 1987) may be used, corresponding to

where Zy=, Pi = 1. In this distribution the final state is specified more completely than in the binomial distribution of Eq. (83). Both binomial and multinomial distributions are applicable in the independent distinguishable electron approximation (Section I1.B.; Section I1I.B. 1). Binomial distributions apply if the single-electron transition probabilities, P, are the same (or nearly the same) for a group (or shell) of N electrons. Overall energy conservation is not applied. For (heavy or very fast light) projectiles the energy loss of the projectile is usually negligible in contrast to the statistical energy deposition model considered next. b. Statistical Energy Deposition Model. In the statistical energy deposition (SED) model developed by Russek (1963), it is assumed that energy, ET, obtained by an atomic (or molecular) target during a collision is statistically distributed to all of the N electrons in the system. One simply counts the number of ways that ET can be distributed among the N equivalent electrons. If any electron

256

McGuire

has an energy greater than the ionization energy, E,,, then it is ionized. P; is the probability function for ionizing n of the N electrons. The following mathematical development is largely due to Tulkki ( 1987). According to Russek (1963) P;(m) =

(3

j=O

K,,(j)Q&-"(M- rn - j)lK,,(m)

(90)

gives the probability that when m units of energy are statistically distributed among N electrons, n and only n electrons will receive r units of energy each allowing their ionization. In Eq. (90)

where [vlh] is the integer part of ulh. The task is to determine P;(ETIEion)= lim P',")(m)

(93)

m--

such that x = mlr = ET/Eio,stays finite. Tulkki observed that the summation over j can be carried out exactly. Substituting Eqs. (91) and (92) in Eq. (90) and observing from Eq. (91) that j S m - rn - ri gives

where

c (I+4- l ) ( m

m - m - ri

L',h3(m,r; i) =

-

j=O

rn

+

- j - ir N - n - 1 m - m - j - ir (95)

Using the abbreviations a = n - 1 b = m - rn - ir N c = N - n - 1 +a.

+

- n - l + a

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

257

Eq. (95) can be written as b-c

b + l

(97)

c - a

j-0

(3[(mz)’r’;

This sum rule is given by Vilenkin (1971). Thus Eq. (94) reduces to

~l,i(~

n ) ( m - rn

i=O

- ir N - 1

+N

-

1

PN(m) =

(98)

which is exact. Now let m and r go to infinity:

[ ( m - m)/r]+ [x - n](x

-

n - 1 < i,,,

x

- n)

for all i except for i = x - n which gives a negligible contribution if it occurs. Using the formula

+w

v fixed

in Eq. (98) yields the Russek formula

x = ETIEi,

where k must be chosen such that n

+k

S x< n

+k+

1.

(102)

Equation (101) implies that the energy has been shared statistically between the N - n electrons that remain and the n electrons that are ejected. Thus, it corresponds to a distribution where the ions are left in excited states. Since decay of excited states often occurs by autoionization, the SED distribution is not usually observed experimentally. Hence, justification of this simple method is incomplete (Aberg, el al., 1984). The binomial P;;(B) and the SED PE;(AE,) are compared in Figs. 7 and 8. The impact parameter B and energy transfer AEm may in some cases be related

N

FIG.7. The probability P i for removing n of N electrons from a target for N = 8. The binomial distribution corresponding to the independent distinguishable electron approximation is shown on the left for various values of n as a function of the impact parameter.

0

2M

400 600 AE (eV) FIG.8. The probability Pg for removing i of N electrons for N = 8 in the statistical energy deposition model (Mueller ef al.. 1985) is shown on the right as a function of the energy transfer AE to the system. See Fig. 15 for a comparison of these models to observations.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

259

classically (Cocke, 1979). In any case the binomial and SED P,! are not the same. For example, the maximum values of Pg differ for 0 # n # N . For N = 6, one has Max P; n

IEA

SED

1 2 3

40% 33% 31%

70% 59% 43.5%

The SED distribution tend to more successful at low collision velocities, while ' C the binomial distributions are often successful at high collision velocities. ( Section 1II.B.1 .) In chemistry the basic concept of SED has been modified by using quantum transition probabilities (e.g., harmonic oscillator probabilities for vibrational transitions in molecules). This modifies the Pg probability functions. This procedure (Rosenstock et al., 1952) goes by the name of quasi-equilibrium theory. (Cf.Sec. 111.1.1.) c . Fokker-Planck Method. The Fokker-Planck method (Balescu, 1975) is a statistical method, widely used in nuclear physics, for analyzing distributions of events as a function of time. This method may also be applied to atomic collisions. In an ion-atom collision the atomic charge can change during a collision according to a probability distribution, W ( N , t), which changes during the time of the collision, 0 < t < T. Initially the number of electrons N = No is well defined, corresponding to W ( N , t < 0 ) = 6 ( N - N o ) . During the collision the width of the distribution, r, increases. When the collision is over and the systems have readjusted, W ( N , t > T ) no longer changes. This is the basic picture that directly corresponds to observed data in some cases.

(i) Fokker-Planck Equation. The equation governing the probability distribution W ( N , t ) is the Fokker-Planck equation, dW(N' ') = [ V ( N , t)W(N, r)] dt dv

61

+[D(N, i)W(N, t)]. dN'

(103)

This is a diffusion equation where V is a velocity coefficient and D is a diffusion coefficie5t. This equation is the same as the Schrodinger equation i dqtdr = H ( - V, V z ) 9 with time rotated by 90" in the complex plane. Equation (103) corresponds to Eq. (92) of Balescu who gives a clear derivation and explanation, with W = W , , N = y, V = A, and D = 112B. The basic assumption is that the process is a Markov process: a statistical process where in each step infor-

260

McGuire

mation regarding previous steps is lost (i.e., the memory time is short). The first term on the right-hand side of Eq. (103) corresponds to a frictional slowing corresponding, as is later illustrated, to the fact that the system is losing electrons. The second term corresponds to diffusion, or the time rate of change of the width in Fig. 9. Note that if V and D go to zero, then dWldt = 0 and W remains frozen in time. Mean values of physical observables may be determined from W, e.g.,

r

(N") =

J cw N"W(N, r)

(moments of N )

,-

(E) =

J cw E,W(N,

r)

( E is energy, for example).

We may also note the following mathematical properties:

I

W ( N , r) 3 0

CW W(N;t )

= 1

(nonnegative probability distribution)

(105)

(conservation of total probability).

Finally, V and D are related to the first two moments (corresponding to a linear second-order partial differential equation) via

d v=(N) dt

( 106)

(ii) Application to Atomic Collisions. The statistical distribution function W may be applied to atomic collisions by relating V and D to atomic collision quantities. There are various ways in which this may be done. For example, one may introduce transition rates and Fermi-Dirac distributions for colliding atomic systems, as has been done in nuclear collisions. An alternate method, used here,

OctcT

0 N NO FIG. 9. The Fokker-Planck distribution function W ( N , t ) before, during, and after a collision occurring during at time T.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

26 1

is to consider one shell of an atom that is losing electrons (e.g., via direct Coulomb ionization) with a single-electron probability, p, that is the same for all electrons within that atomic shell. With this picture in mind, one may express V and D in terms of the singleelectron transition probabilities, P ( B , t) by using the definitions of V and D from Eqs. (106) and (107). Noting that n N = No where n = number of vacancies, N = number of electrons, and No = total number of states in that atomic shell, it is easily shown that

+

d

(N) dt d dt

=

-d

(n) and dt d dt

- {(N2) - (N)2} = - {(n’)

-

(n)2}.

If we use, for example, a binomial distribution for the probability of producing n vacancies from N electrons, namely P f = (:)pn(l - p)N--n, it is straightforward to show that

where p = P(B, r) is a single-electronprobability for an atomic transition. Then

D

=

d +-{(n’ dt

-

dP (n)2)}= N-(1 dt

- 2p).

Now V and D are known functions of N and t. They may be used in the FokkerPlanck equation, Eq. (103), to find W(N, t). Note that after the collision, V and D go to zero, so that W(N, t > T) does not change. Next let us consider some limitations of this method. The Fokker-Planck equation assumes that a Markov process is valid, i.e., the system forgets its history before the collision. This implies that the collisions are well separated in space and time (e.g., Eq. (73) holds). From the atomic point of view, each shell is treated independently, each B is treated independently, and p(r) may be computed accurately. Also, electrons only leave and do not return. (This can be corrected by a more complicated model with particles moving back and forth, as has been done in nuclear physics.) Since the Einstein relation is not used, W does not go over to a Maxwellian distribution. (C.’ Balescu, 1975. This may not always be a limitation.) It is also required that No >> 1, so that Jp dN W(N, t) = J; dN W(N, t) or Jg0 CW W(N, r) << 1. Note that 1 = Z, (f) pn(l - p ) N - n = J; dN W(N, t ) corresponds to conservation of total probability.

262

McCuire

(iii) Simplest Model. If V and D are constant in Eq. (103), then an analytic solution is known to the Fokker-Planck equation. Specifically, consider otherwise

0

otherwise.

The Fokker-Planck Equation then is dW dr

- = -Vo-

dW

dN

+ Do-.d2W dN2

The solution is for 0 < r < T

This is a Gaussian distribution in N . As t + 0, W ( N , r ) 4 6(No - N), and for t > T, W ( n , t > T ) = W ( N , T ) . The behavior is illustrated in Fig. 10. This corresponds to taking p ( t ) and dpldt as step functions is time. Also, Vo and Do are independent of N . Thus take

Note that if Po << 1/2, then - Vo = Do = a l N , , whence

(1 19)

This distribution depends only on one parameter, a = NoPo, which is equal to (No - N ) = ( n ) , the average number of vacancies in the spectrum. This model for Po << 1/2 makes a specific prediction between the position of the peak of the Gaussian distribution and the width of the Gaussian distribution, namely that (since - V, = D o ) ,

(No - N ) = ( N 2 ) - ( N ) 2 = a = (16 In 2 ) P

( 120)

consistent with a binomial distribution when p << 1/2. A more sophisticated model may be able to explain deviations from a binomial distribution.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

263

t t FIG. 10. The Fokker-Planck single-particle probability p ( t ) as a function of time, r. T is the duration of the collision.

Note that this model gives some insight into the meaning of V and D in the Fokker-Planck equation. The “diffusion coefficient,” D, is the simply the time rate of change of the halfwidth (with a factor of [ 16 In 2]’/2),and the “frictional velocity,” V, tells us how fast the peak of the distribution moves from 0 < t < T. Since observed data are often reduced using best fits to Gaussian distribution, and since Fokker-Planck is based on Gaussian distribution, the Fokker-Planck method is well suited for analysis of experimental data, although explicit understanding in terms of microscopic atomic properties is now lacking. Information theory (Aberg et al., 1987; Blomberg et al., 1986; Aberg, 1987) is a statistical method, related to the Fokker-Planck method in its origins, for extracting the maximum information from experimental data with noise by using the smallest number of parameters. Useful interrelations between information theory, multinomial distribution, binomial distributions and other statistical methods are documented by Eadie et al. (1971).

III. Observations and Analysis The first Nobel Prize in physics was awarded in 1901 to Roentgen “for the discovery of x-rays.” By 1913 various observations of X-rays (or “Roentgen rays”) were beginning to resolve conflicts between J. J. Thompson’s model (19 13) of an atom as continuous tubes of electrical charge and Bohr’s new quantum model (1913) of atoms and molecules. In 1921 Wentzel wrote about “multiple ionization by cathode ray particles” and by 1926 the Auger effect (Rosseland, 1923; Auger, 1926) had been observed, and some of the basic quantum properties of atoms had been established. The status of atomic and molecular collisions in 1969 was summarized in a five-volume set of books by Massey et al. Some of the history of fast ion-atom collisions has been given by Merzbacher (1984) and Briggs (1988). By the time of the Atlanta conferences on atomic inner-shell processes (Fink er al., 1973; Crasemann, 1975) new work on multielectron transitions had begun. In this chapter the time frame from about

264

McGuire

1970 to early 1991 is partially covered with emphasis on total cross sections for few- and many-electron transitions in collisions of neutral atoms (and molecules) with fast (and usually massive) projectiles.

A. SIMPLE ANALYSIS In this section we begin with some very simple models for multielectron transitions. These beginning models serve to form a simple (and incomplete) conceptual basis of understanding for the analysis of collisions in which multipleelectron transitions occur. These initial models become somewhat more complete as specific and recent observations are considered later in this section. Ratios of cross sections for multiple to single transitions are often used here. The effects of the single collision tend to cancel so that the nature of the higherorder transition mechanism is often clearer in the ratio than in the cross section itself. 1. Applications of Binomial Distributions

For collisions of weakly correlated atomic targets with particles of charge Z, at high incident velocities, v, the independent distinguishable electron approximation often provides a sensible starting point for the analysis of total cross sections. (Cf.Section I1.B.) In this independent distinguishable electron approximation the probability for the transition of n of N electrons with the same single-electron transition probability, P, is given by a binomial distribution (cf. Section II.I.2.a), namely,

P;;= (;)P(l

-

P)N-n.

The idea of multiplying single-electron probabilities for two-electron transitions was introduced in 1965 in classical calculations of Gryzinski. Use of binomial distributions was independently considered by various people (Burch, Hansen and McGuire, Hansteen and Mosebekk, Macek) and first published by Hansteen and Mosebekk in 1972. A variety of methods are available for obtaining the single-electron probabilities used in the binomial distribution. Coupled channel calculations provide one of the most complete methods. In a number of cases calculations based on perturbation theory (Briggs and Macek, 1991; Madison and Merzbacher, 1975; McDowel1 and Coleman, 1970) are sufficient as are classical methods. (Cf.Section II.I.1.) For ionization the tabulated first-order (in Zdv)calculations of Hansteen, el al. (1975) are especially useful. In this case the single-electron probability

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

265

P ( Z p , Z, , v ) = (ZdZ,)zP(Zp= 1, Z, = 1, v/Z,).This scaling law may also be applied to first-order probabilities for excitation, tabulated for P(Zp = 1,2, = 1, v ) by Van den Bos and De Heer (1967). Such first-order scaling tends not to be so accurate for electron transfer although qualitative estimates can be simply obtained using the first-order Brinkmann-Kramer approximation. (Cf.Section 111.1.1.) Examples of the application of binomial distributions in the analysis of data are given in Section 1II.B-D.

2 . Analytic Expressions for Two-Electron Excitation and Ionization at High Velocities and Large Impact Parameters At moderately high v and when correlation is not too strong, the independentelectron approximation usually provides a reasonable approximation since the first-order term in Z J v is often relatively small. (Cf.Section 1I.D.) In the independent-electron approximation the probability amplitude is a product of singleelectron amplitudes. (C' Section 1I.B.) At large impact parameters, B , the single-transition amplitudes for excitation or ionization are proportional to dipole . +i) and the probability amplitude for excitation or matrix elements, Dif = (& 111 ionization of two electrons is (McGuire, 1987) = a,a2 = - +(Zs/v2)D\fDir2e -(Qb+QbB/B2

(122)

where Qo = Qminis the minimum momentum transfer of the projectile. This term varies as (ZP/v)*. At very high velocities the first-order term in ZJv dominates. A qualitative estimate of this term is sometimes given by the simple shake contribution to the full first-order term of Section 1I.C. 1. From Eq. (44) using the large B approximation for a,, we have aSwc=

al (ur/u,)= - ir(Z,Jv)D;fe-q@/B (631

4;).

(123)

Some simple estimates of (4f 1 4i)are given in the following subsection 3 and somewhat more reasonable estimates are discussed in Section 1II.B.2. If the collision is not too fast, then there may be sufficient time during the scattering event for correlation interactions between electrons to influence the transitions. The approximate expression for the lowest-order scattering correlation term (Cf.Section II.C.2) may also be expressed in closed form at large B (McGuire, 1987), namely, as =

-2i

5 Q"t&(qoB) v3

-

(90B)KdqB)l

( 124)

where K is a Bessel function of the second kind. Since this term varies as Zplv3, it tends to be small at large v and when electron correlation is weak.

266

McGuire

I

Mechanisms for Multiple Vocancv Production

I V'VOREIT

FIG. 11. Overview at many-electron transition mechanism. Here u is the electron-electron correlation interaction and V is the interaction with the projectile nucleus. If u -+ 0, then there is no correlation and the independent-electron approximation is valid. If the collision velocity v is large, asymptotic correlation (e.g., shake) dominates and the process is first order in V. If correlation, u, is strong, then scattering correlation may also be significant and all correlation terms must be considered. A 0 and MO denote the atomic orbital and molecular orbital regimes. At low v the collision time may be long enough to permit a statistical energy distribution (SED)or a Wannier-like process. In this chapter we consider high collision velocities (v/vmb,,3 1) for systems that are not too strongly correlated (u/V 2 1).

These simple analytic expressions (which may be easily generalized to N electron transitions) give a simple qualitative picture of mechanisms for multiple excitation and ionization by charged particles illustrated in Fig. 11. At very high velocities the small terms first order in Zdv dominate. At moderately high velocities and when multielectron effects are small, the independent-electron approximation dominates. As the collision velocity is further decreased and correlation becomes stronger, correlation occurring during the scattering event becomes significant. At low v the molecular orbital (MO) picture (Madison and Merzbacher, 1975) and perhaps the statistical energy deposition model (cf. Section II.1.2b.) comes into play. Near-threshold scattering correlation may overlap with Wannier threshold effects arising from the electron-electron interaction. Some of the limitations of this simple picture will become evident shortly.

3. Simple Shake Factors Before considering realistic generalized shake probabilities that employ correlated initial-state asymptotic wavefunctions, we begin with the simple shake probability amplitude defined (Aberg, 1967, 1976; Carlson, 1967) by us,,&== (4, I 4,). (Cf.Section 1I.C.1.) In this simple shake amplitude (c#J~ I 4i)is nonzero because Hi and H, are different due to differences in electron screening. We note that the

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

267

simple shake probability is uncorrelated because a = alashake where a, is the probability amplitude for scattering the first electron. Furthermore, the simple shake probability is usually independent of the projectile, i.e., shake represents a postcollision decay. In the simple shake picture the probability that the shake electron does not remain in its ground state is given using closure, Eq. (31), by 1 - l(4i1 +;)I*. Here 4iand 4: differ due to a change in the screening of the nucleus by electrons. For a target of bare nuclear charge Z and screening parameter s using ground state wavefunctions, we obtain with Z' = Z - s:

= I -

+

3 -s 2 1 - 3(s/Z) 3(S/Z)2 - 1 - 3s/Z + ( 1 5 / 2 ) ( ~ / 2 ) ~ 4 Z2'

For excitation and ionization the simple shake probability has the same dependence on Z as the first order perturbation probability for direct excitation or ionization by the projectile, i.e., Z-2. The total shake probability is the probability that the electron goes to a state other than the ground state of the target. It is also straightforward to compute probabilities for shake into specific states. The simple shakeup probability (Jacobs and Burke, 1972) for excitation to the 2s state is given by Prup

=

K42. I 4L)I2

We note the excitation is about 40% of the total shake probability for helium in this simple model. Most of the shake electrons usually are ionized electrons corresponding to shakeoff. The simple shakeoff probability in helium has been computed by McGuire and Heil (1987). In general, Aberg (1990) suggests that as a rule of thumb shakeup dominates over shakeoff for outer-shell (valence) electrons whereas for inner-shell electrons shakeoff dominates.

268

McGuire

In addition to shakeup and shakeoff it may be possible to have shakeover where the electron collapses onto the projectile after the collision. The simple shakeover probability has been evaluated by McGuire et al., (1988), namely,

This simple shakeover probability is dependent on the properties of the projectile ( Z , and v), unlike shakeoff or shakeup. To our knowledge there is no experimental evidence for shakeover. However, using Eq. (127) for transfer ionization, it is easily shown that the ratio of total cross sections for transfer ionization to single-electron transfer are predicted to increase with collision energy if shakeover dominates over direct capture. (Cf.Section III.F.3.) The simple shake probabilities are expressed in terms of the change in screening, s. In a variational calculation is easily shown that s = 5/16 to first order in a 1/Z expansion. For K-shell electrons s = 0.3 is commonly used as a fit of hydrogenic wavefunctions to more accurate numerical wavefunctions. Another estimate of the screening constant based on Hartree-Fock-Slater (HFS) wavefunctions has been discussed by Mukoyama (1986) where the screening constant is expressed in terms of the expectation value of the radius, i.e., s = ( r H ) / ( rHere ) . rHis the expectation value of r for a hydrogen atom, namely &[3n2- E(l + l)] and ( r ) is the mean value for a HFS wavefunction. Values of ( r ) and the corresponding screen constant, s, for helium are listed in Table I. Simple calculations sometimes useful for qualitative understanding have been done for other system by Aberg (1967) and Carlson et al. (1968). 4 . Observed High-Energy Limits of Ratios for Double to Single Transitions in He

In the limit of high collision velocities (or energies) the cross section for a double-electron transition might be proportional to the cross section for a singleelectron transition. This may occur if the transition probability for correlation is stronger than the probability for the brief interaction with the projectile. In the high-velocity limit it is useful to study the ratio of double- to single-electron transition cross section since the single collisions’ effects tend to cancel in the ratio. Observed values of various rates in He are tabulated in Table 11 and discussed throughout Section 111 (as noted in Table 11). Observations in targets other than He are less complete (Barnett, 1990). The ratios for He vary, in some cases by an order of magnitude. Theory suggests that some of these ratios are interrelated, as discussed later in this section, but experimental confirmation of these predictions is incomplete. Of central importance is the value of the high-energy photoionization ratio, which is not well determined. (Cf.Fig. 22.)

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER TABLE I AVERAGE RADIIAND SCREENING CONSTANTS FOR GROUND STATEAND EXCITED STATES OF HELIUM USING A HARTREE-FOCK-SLATER PROGRAM

0.940 0.764 4.35 0.764 4.64 0.755 10.58 0.756 11.08

0.405 0.038 0.621 0.038 0.922 0.013 0.724 0.016 0.872

TABLE I1 EXPERIMENTAL HIGHENERGY RATIOSOF CROSS SECTIONS FOR DOUBLETO SINGLE-ELECTRON TRANSITIONS IN HELIUM Process

Ratio

Reference

Photoionization (total u)

4

Sec. 111. 8.2

Ionization by p ' , e' (total u)

3 x 10-3(220%)

Sec. III.B.2

Ionization by N+' (total u)

1 x 10-2(?)

Sec. III.B.2.h

Ionization by e (differential u,small Q )

Same as photons

Sec. III.B.2.e

Ionization by p + (differential u,binary encounter, large Q)

2 x W Z(230%)

Sec. III.B.2.e

Ionization by neutral atoms (total u)

1 x 10-2(?)

Eq. (139)

Photoexcitation to n (total a)

5%(?)

Sec. III.C.2

~~~

=

2

X

10-2 (280%)

Excitation by e * , pi (n = 2)

?

Sec. III.C.2

Capture

?

Velocity-dependent Sec. III.D.2

Transfer ionizationltransfer

2-4 x

Sec. 1II.F

269

270

McGuire

B. MULTIPLE IONIZATION In this section we begin consideration of observation and analysis of multielectron transitions. We first consider ionization where data is relatively plentiful.

I. Ionization of Many Electrons In the early 1970s a number of observations of X-ray spectra were done (Crasemann, 1975) in high resolution revealing structure not evident under lower resolution. This structure, illustrated in Fig. 12, corresponds to various stages of ionization. Reading from the left in Fig. 12 corresponds to ionization of 1K electron and OL electrons, then 1K and lL, then lK, 2L, IK, 3L, lK, 4L, lK, 5L. This represents single K-shell ionization in coincidence with various stages of L shell ionization. In this case since PK<< 1 and PK(B)PL(B) = PK(B)PL(0) since PL(B) = PL(0)for the range of B where fK(B) is nonnegligible,

A fit to a binomial distribution of a single P L ( 0 )is shown in Fig. 13 where P L ( 0 ) is a fitting parameter. Note that under lower energy resolution the peak would appear as one peak with o, = 2, (+K,"L,in accord with the sum rule of Section II.A.6. Similarly, under higher resolution, ionization in the M shell could be resolved. Even for proton impact the M shell is highly perturbed at most collision energies. A useful and simple method for fitting cross sections to a single ionization probability P = Poe-B'Rowhere Po and Ro are fitting parameters has been developed by DuBois and Manson (1987). Good agreement with observation is obtained. In relativistic collisions, Anholt et al. (1987) have also found good fits to observed data for both heavy and light particles. More recent data for both heavy and light projectiles have been collected for various targets by Mueller et at. (1987, 1985) using binomial distribution. Olson (1988) has done a number of calculations using classical trajectory Monte Car10 (CTMC) methods to compute single-electron probabilities and applying the independent-electronapproximation. These calculations generally are

Energy (Ireit!

Enerqy

(keVI

Energy (Lev)

TitODOMeV)

u

340

Waielenglh

(il

Wavelenqlh

Ewqy

Energy IhWl

WaWlenqlh

IkrV)

5.90

Ii)

Energy lkeVl

6.00

6.10

L

V

Wouelenplh

lb

2 45

Wavelenqlh

Wnvclenqfh (A1

IAl

FIG. 12. Experimental yield versus target-binding energy corresponding to various degrees of multiple target ionization (Kauffmaner al., 1973). Here there is always one K shell vacancy and the number of L shell vacancies increases to the right.

t ul

In I

Y-O 3 I

303

z

E w02

c

4 W

O0

KLQ K L ~ KL' KL'

K a

t

04

KL. x i 1

00

X-RAYS 30 MeV

Ct+O

30 MeV

04t

'

KL'

KT

{

2°.4t

Ka

X-RAYS 30 MeV Mn'o

1

i

KL'

FIG. 13. Comparison of measured relative satellite intensities (solid bars) from Fig. 12 to the best least-squares fit assuming a binomial distribution in P,(O) for K X-ray spectra of 30-MeV oxygen on various targets (Kauffman et al.. 1973).

212

McGuire

in fairly good agreement with observation. Olson and Cocke have both noted improvement of theory with observation if increased binding energies with increasing stages of target ionization are included in the independent-electron approximation. The increased binding energy might occur because the ionized electrons do not leave the target quickly so that there is some readjustment of energies even at high collision vehicles. Also, the ionization potential per electron is not independent of the number of electrons removed. Olson’s calculations are in good agreement with observations of Berg et al. (1988) for multiple ionization of noble gasses by 120 Meviamu U+wions. Low-velocity projectile data from Giessen (Mueller et af., 1983) shown in Fig. 14. is compared with the statistical energy deposition model. (Cf.Section 11.1.2.) As was noted near the end of Section II.I.2.b, there are in principle some observable differences between the independent-electron approximation where interactions with the projectile are independent and statistical energy deposition, where energy from the projectile is statistically distributed among the target electrons (and decay of the final states is ignored (Aberg et al., 1984)). In practice, however, both distributions tend to provide approximate fits as illustrated in the 100

0

100

e

50

0

1-

c

100

0 0

5

n o

5

K)

AE / < I s >

FIG. 14. Comparisons of multiple-ionization data with the statistical energy deposition model (Mueller et al., 1985). Here the energy deposited in the system, A&, is given in units of the ionization energy I , .

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

273

FESIDUAL CHARGE STATE FIG. 15. Comparisonsof multiple-ionizationdata to binomial distribution (IEA) and to statistical energy deposition (SED model) (Cocke, 1979).

data and analysis of Cocke (1979) shown in Fig. 15. Generally, however, the independent-electron approximation is usually used at moderately high collision velocities while the statistical energy deposition model is used for low-velocity collisions. Multiple ionization by electron impact (Hazi, 1981) has been reviewed by Schartner (1990). Collisions of highly charged ions at low velocities have been reviewed by Barany (1990).

2. Double ionization Two-electron transitions are the simplest of the multielectron transitions that are the subject of this chapter. Because helium and molecular hydrogen are two

274

McGuire

of the simplest targets in which to study two-electron transitions, these targets have been more thoroughly considered than more complicated systems. a. Independent-Electron Approximation. Application of the independent distinguishable electron approximation to analysis of data for single and double ionization of helium is shown in Figs. 16 and 17 (Sidorovitch et al., 1985) which does not include the relatively accurate data of Shah and Gilbody (1985). At the lower velocities shown, first-order perturbation theory used for the singleelectron probability, P, is inadequate in both single and double ionization. The independent-electron approximation agrees with single ionization within about 40% at higher velocities and misses the data for double ionization by a little

SINGLE IONIZATION 10-1

=

10-16

10-l~

100

1000 E ( KeV/arnu) FIG. 16. Cross sections for the single-electron ionization of He by the H+,He2+,and Liz+nuclei versus nuclei energy (Sidorovitch et al., 1985). The various calcuations are all based on perturbation theory.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

275

more than twice the error in single ionization as one might expect since double ionization varies as P2 compared with single ionization which varies as P (for small P). Unless one uses different binding energies per electron for single and double ionization it is difficult to fit both data sets better than shown in Figs. 16 and 17. For highly charged ions ( Z , up to 44) incident at 1.4 MeV/amu on helium, the Giessen group observed a saturation in the increase in both single and double ionization with increasing Z , shown in Fig. 18. Salin (1987) has interpreted this saturation with the independent-electron approximation using a multiple expansion defined on one center that limits the single-electron probability to values less than one. A simple cutoff of first Born results gives similar results (Ben-Itzhak and McGuire, 1988). Thus, while the overall trends of the

I

I

DOUBLE I ONlZATlON

E (KeV/amu)

FIG. 17. Cross sections for the two-electron ionization of He atoms by the H + , He*, and Li3+ nuclei versus nuclei energy (Sidorovitch er al., 1985). Theoretical curves are all based on perturbation theory.

276

McGuire

r'7

-r--IEA - wder V 1st

in

Incident ton Chorpc

, Zp

FIG. 18. Ratio of single to double-ionization cross sections in helium versus projectile charge, Zp (McGuire et al., 1987). The curve labeled IEA is a calculation of the independent-electronapproximation. The curve labeled 1st order in V is the observed high-energy limit of the ratio.

data are generally qualitatively reasonable (except for the very high velocity data for proton impact explained in terms of shake-like models in Sections III.A.3 and 1II.B.b.-e), it is difficult to obtain accuracy better than a factor of two using the independent-electron approximation in the case of helium. b. Simple Shake. At sufficiently high collision velocities it is expected that effects first order in Z,,/v will dominate. (C' Sections II.D.l and III.A.3.) The simplest picture in this regime is the simple shake picture in which double ionization follows single ionization via final state rearrangement independent of the properties of the projectile. In this simple picture the ratio of double to single ionization is expected to be independent of Z , or v. Ratios of double to single ionization were first considered by the Aarhus group in 1982. Figure 19 shows that this ratio does go to a constant value independent of both v and 2, at high v as qualitatively predicted by the simple shake model. The observed ratio for 40 MeV electrons (80GeV/amu) from Giessen (Salzborn of 2.67 X and Mueller, 1986) nicely confirms this trend. Comparable results for H, observed by Edwards er al. (1988) are shown in Fig. 20.

MULTIPLE-ELECTRONEXCITATION, IONIZATION, AND TRANSFER

r

0.5

\

!.

I

mi mm

2

5

I

1

10

277

20 I

He

'J 0.5

E, (MeV) 5

2

I

10

20

FIG. 19. Ratio of single- to double-ionization cross section in helium versus collision velocity (Haugen et al., 1982). The curve labeled IEA is a calculation of the independent-electron approximation. The curve labeled 1st order is the observed high-energy limit of the ratio. The top energy scale is for electrons and the bottom scale is for protons.

7

6 - I H'on H2 i e- on H2 5-

,Q 4 -

0

3-

-

2?

, € €

I-

0

1

I

I

,

,

,

McGuire

278

c. Relation between Charged Particles and Photons. In the simple shake theory (Mittleman, 1966; Amusia et al., 1975; Sampson, 1990) the ratio of double to single ionization is the same for impact by charged particles and photons. As pointed out by Eckhardt and Schartner (1983) this prediction is not in agreement with observed data. The ratio of total cross sections for double to single ionization for charged particles differs by an order of magnitude from the ratio for photons. This difference can be largely explained by the fact that photons impart all of their energy to the ionized electron while charged particles impart a distribution of mostly relatively small energies. From Eq. (68) without the inelastic projectile electron terms (or from McDowell and Coleman, 1970, Chapter 7) the scattering amplitudes may at high v be expressed as

The amplitude for photon impact may be expressed in terms of

f = (4f1e-E 220

(+[

*

IE

*

PI 44

PI

+i)

-

(+f

IE rl *

(photons)

( 130)

+i)

where K is the momentum of the photon (K = w / c ) . Here E is the polarization vector of the photon and p = iV. Below about 1 keV the value of Kmi. is small enough to use the dipole approximation. Since the matrix elements are the same up to overall factors, the cross sections for ionization (or dipole-allowed excitation) by charged particles and photons are related at small momentum transfer as shown by Byron and Joachain (1967), namely

where E is the energy of the ejected electron, and E is the energy of the incident particles. For charged particles, there is a distribution, p ( E ) , of ejected electrons while for photon impact E = E ~ - ~I , i.e., ~ each , ~electron ~ energy E corresponds to a single photon energy. Consequently, if we take a ratio of double- to single-ionization cross sections

where R = cr++/c+ for both Z and y. Thus, the ratio for ionization-charged

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

PHOTON ENERGY, E

279

lev)

FIG. 21. Energy distribution, p t , of the final state (left-hand side) and ratio, R,, of single to double photoionization (right-hand side) in helium as a function of photon energy, E (McGuire, 1984). Here E = I E where I(&) is the binding (ejected-electron) energy and A is the difference in the single- and double-ionization threshold energies. Note that I p$ d~ = 1, and that R, is equal to the area under the product of the curves illustrated here. The data (Schmidt et al., 1976; Wright and Van der Wiel, 1967; Holland er al.. 1979; Carlson, 1967) are taken from McGuire (1984). A useful datum at 600 eV due to Carlson (1967) is included in the next figure.

+

particles is equal to the ratio for photons weighted by density of ejected electrons and integrated over the ejected-electron energies. This relation illustrated in Fig. 21 was first suggested by Hvelplund (1982). (Cf.McGuire, 1984.) Consequently, the ratios of total cross sections for total double to single ionization can be quite different for photons and charged particles, but are nonetheless related at high particle velocities. Numerical calculations (McGuire, 1984) are in qualitative agreement with the observed results in Table 2.

d . Aberg's Generalized Shake Probability. A conceptually useful generalized shakeoff probability has been introduced by Aberg (1976) who treats the first outgoing electron as a plane wave and ignores antisymmetrization of the wave functons, i.e., the exchange terms are ignored. Following Aberg, let state & b e an uncorrelated antisymmetrized wave function with an outgoing plane wave electron of momentum k and a residual electron wave function Cbf. (For double ionization one might simply use a plane wave of momentum k2 for 4f.) Thus,

280

McGuire

For a two-electron system the scattering amplitude is given by

+ - 4TZp -{F(Q

v5Q

I

I

1

dr, +r(r,)eiQrl drz e-ik.r2+i(r,,r 2 )

-

k)

+ exchange term}.

If we ignore the exchange term, then the matrix element for a shake process to a particular final state cbf is expressed in terms of generalized shake probability amplitudes, Ff(Q - k). It is easily shown that f di(Q - k) (+f I + i ) + (+f )eiQrll &(k) where +i(p)is a momentum space wave function and + i ( r i ,r z ) = +1(rl)+i(r2). Also it is easily shown that the exchange term (i.e., the second term) may be ignored when Q - k = 0 for large Q and large k. Taking the ratio of cross sections for double to single ionization or excitation, and ignoring the exchange term one may obtain Aberg's generalized shake result, namely,

-

-

P A , = N ( Q - k) =

PAbe,g(Q

I

I

dr,ei(Q-k).ri dr2 +f(r2)+i(rL,r2)

- k).

(135)

where N(Q - k) is a normalization factor given explicitly by Aberg. Here Q is the momentum transfer of the projectile, k the electron momentum. This probability is a function of the momentum Q - k, transferred to the residual ion. At a fixed Q - k this probability is independent of how the state f was formed. Thus, in this approximation shake processes for photons and charged particles, capture, and Compton scattering may be interrelated. e. Shakeoff Observation. Aberg's generalized shake probability suggests that shakeoff may be different for photons than for fast ejected electrons in binary encounter collisions with charged particles since Q - k is different. For photons Q = 0 so that the residual momentum transfer of the target is k, which can be large. For binary encounters with charged particles Q = k so that Q - k = 0. The experimental results of Kamber er al. (1988) for binary encounters with charged particles combined with photon results are shown in Fig. 22 with data from Schmidt et al. (1976), Wright and Van der Weil (1976), and Holland et al. (1979). In the electron velocity range shown, k = 15. A discontinuity from the photon results to the charged particle results is evident indicating that the shake

28 1

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

L

6.0

d,k(Carter

8,Kelly

50 -

&?

v

40-

+m

5

30-

m

I

20 -

i 4 ’

1.0 -

I

0 01

, , ,b, , ,

,

01

p’ data

(Q=k)

,I

1.0

1

0

I

I ! *

10.0

PHOTON ENERGY (keV) FIG. 22. Ratio of double to single ionization in helium in equivalent photon energy. Data and theory are taken from Kamber et al. (1988). Note the discrepancy between Q = 0 data for e - and y and Q = k data for p + . The Q = 0 data corresponds to the data in Fig. 21 except that the open triangles in Fig. 21 are omitted and a useful observation of Carlson (1967) has been included at 600 eV. The high-energy limit for photoionization is not well determined experimentally.

limit does depend on Q - k. The physical picture is that shake is different if the collision is a close encounter (large Q) or a distant encounter (small Q). A larger change in screening could occur for collisions at close encounters than for distant encounters. This suggests that there is not just one simple shake limit, but rather a range of high Zdv limits for the ratio of double to single ionization. (C’ Section II.ZA.4.) However, more data would be helpful including more complete data for photoionization.

f. MBPT Calculations. The limit of high ejected-electron velocities for highvelocity collisions may be one of the simplest limits in which multielectron effects dominate. It is clear from the previous discussion that this limit is not well understood as of 1991. The range of ratios with Q is not clear, and the interrelation (if any) of various processes is also not entirely clear. (CJ Section III.A.4.) Many-body perturbation theory calculations done by Ishihara and Hino (1989) indicate that the shake concept may not be sufficient to describe this limit. In addition to contributions from the many-body perturbation theory first-order shake diagram, there are also contributions for photoionization from other firstorder diagrams (Cf.Section 1I.E) as shown in Fig. 23. Similar calculations for impact of charged particles have not yet been done. The observations of Kamber et al. (1988) as well as Aberg’s calculation (discussed in Section III.B.2d) sug-

282

E (eV) 23. MBPT calculation for ratio of double to single photoionization versus photon energy, E (Ishihara and Hino, 1989). The total result includes amplitudes for ground state correlation (GSC), two step-l (TSI). and shakeoff (SO) illustrated in Fig. 1. Calculations using each MBPT amplitude alone are also shown. No one amplitude dominates at high energies. FIG.

gest that the limit of high ejected-electron velocities and high collision velocities depends on the momentum transfer, but neither complete calculations nor reliable qualitative understanding is available. g. Zp?Terms. The ratio of double- to single-total-ionization cross sections compiled by Haugen (Hangen et a/., 1982) in Fig. 19 presented at the first European Conference on Atomic Physics in 1981 posed an interesting problem. In the velocity regime considered single-ionization cross sections are well described by the first Born approximation which varies as Zp?and gives no significant difference between cross sections for ionization by protons or electrons. At such high velocities apparently well within the region of validity of perturbation theory it was difficult to understand the clear experimental factor of two in the difference between double ionization by protons and electrons. Data confirming this effect in Hz taken by Edwards et al. (1988) is shown in Fig. 20. In 1982 McGuire suggested that the observed difference between protons and electrons was due to Z; interference arising from the square of a sum of firstand second-order amplitudes in the absence of multielectron effects. (C’. Section 11.D.4.) The second-order amplitude includes effects of the independent-electron

283

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

approximation. It was counterargued that the observed difference could be due to trajectory effects caused by the large difference in the masses of the protons and electron. An experiment of Edwards et al. (1986) showed that protons and deuterons gave the same ratios. This data provided some evidence against the argument for a mass effect. The issue of charge versus mass was resolved by an experiment by Andersen er al. (1986, 1987) using protons and antiprotons in LEAR at CERN. Their data, shown in Fig. 24, show that for double ionization the antiproton data lie above the proton data by a factor of about two, confirming a clear 2, dependence in double but not single ionization. 10-16

I?I'-lL84

0

PROTON DATA

o

ANTIPROTON DATA

-

10-l~

N

1420-

0

I

I

I

1

2

3

I

4

I

5

I

6

I

7

8

E (MeV) FIG.24. Observed cross sections for single and double ionization of helium by protons and antiprotons (Andersen ef af.. 1986, 1987).

284

McGuire

As of early 1991, theorists tend to agree that the difference between protons and antiprotons (and electrons) is due to correlation. However, there is much disagreement as to the physical mechanism for correlation. McGuire's suggestion of interference of quantum amplitudes is challenged by Becker (1985; McGuire and Burgdoerfer, 1987) who argues that the interference may be restricted by selection rules. Olson (1987) suggests that the effects may be explained classically without quantum interference. Vegh (1988) suggests that polarization of the charge cloud of the target may be especially important in double (' Section 11.1.1) ionization. The relatively complete forced impulse method C calculations of Reading and Ford (1987a,b) indicate that correlation must occur between at least two collisions with the projectile. Sorensen suggests (Andersen er al., 1987) that correlation can occur after one collision with the projectile when one electron hits another. Meanwhile data for positron impact by Charlton er al. (1988) shown in Fig. 25 nicely completes observations for total cross sections. Threshold effects at lower

0.1

0.2

0.5

1

2

5

10

20 "80,000

E [MeV arnu-'1

FIG. 25. Observed ratios of double to single ionization of helium by proton ( p + ) ,antiprotons ( p - ) , electrons ( e - ) , and positrons ( e + )of the same incident velocity in MeV/amu. Data taken from Charlton er al. (1988). The 80-GeVlamu electron point is from Salzborn and Mueller (1986). The differences between negative and positive projectives above one MeV/amu are thought to be caused by correlation, but the physical mechanism for this correlation is not clear.

MULTIPLE-ELECTRONEXCITATION, IONIZATION, AND TRANSFER 100

i

a x

n

V

A

+

285

ELECTRON PROTON NITROGEN HELIUM

10

z 1

1

0

10

20

30

40

50

ENERGY (MeV/amu)

FIG. 26. Comparison of He double-to-single-ionization cross section ratios (Heber et al.,1990). The solid curve shows the predictions of a semiempirical calculation of Knudsen er al. (1984) and the dotted curve shows the predictionsof Reading and Ford. The high energy limit for N'' projectiles appears to differ from the lower Z projectiles.

velocities are evident for the projectiles with a small mass. The Z 3 effects are related to a difference in stopping powers observed for positive and negative pions by Barkas (Barkas et al., 1956; Basbas, 1982; Andersen et al., 1989). Clear conceptual understanding, however, of collision mechanisms for the 2; terms remains a challenge.

h. High v Limit for Large Z,. Heber et al. ( 1990) observed the ratio of total cross sections for double to single ionization by heavy ions with Z, = 7 at values of Zp/v comparable to protons at 5-10 MeV. Heber et al. observed a ratio that differs from the ratio observed for projectiles with Z , = 1 (C'. Table LI) and with the empirical fit in Zplv obtained by the Aarhus group (Knudsen et al., 1984) using electrons and some ions with Z , 2 1. Heber et d ' s results shown in Fig. 26 are in possible conflict with present theory which agrees with the fit of the Aarhus group. (Also see Andersen et al., 1990.) The high u limit is not well established for large Z, . 3. Differential Cross Sections It is evident from the preceding discussion that observations of multielectron transitions are likely to be useful in developing understanding of few- and manybody effects. In principle differential cross sections (Rudd et al., 1985) can yield more detailed information. Unfortunately, relatively few differential studies of multielectron transitions at high collision velocities are available. One differential cross section that has been observed is the ratio of double to

McCuire single ionization of helium by protons as a function of the scattering angle. The observed data of Giese and Horsdal (1988) show a peak at 8 = 28, where 8, is the scattering angle for a single binary encounter. Vegh (1989) first proposed a three-step model including a double binary encounter between an electron and the projectile. A more complete classical calculation by Olson et al. (1989a) (including a double binary encounter events) gives qualitative agreement with the data. Single- and double-ionization cross sections triply differential in the angle and energy emission of the electron and in the recoil angle of the projectile have been observed by Skogvall and Schweitz (1990). For double ionization differences are found with coupled channel calculations indicating a breakdown of the independent-electron approximation. Another set of cross sections differential in the recoil angles and energies of the ejected electrons have been observed by Schmidt-Bocking and analyzed by Olson (Olson et al., 1989a). These observations show that a large amount of recoil momentum can be carried by both electrons and the recoiling nucleus. The topic of recoil distribution has been reviewed by Cocke and Olson (1990). C. MULTIPLE EXCITATION Understanding multiple excitation uses many of the ideas and methods applied in the previous subsection to multiple ionization. However, multiple excitation is easier to understand than multiple ionization where the asymptotic Coulomb wavefunctions are not known. Furthermore, one may search for effects of selection rules more directly in multiple excitation. On the other hand, observations of multiple excitation tend to be more difficult than for multiple ionizations so that only a little data for double excitation is available as of 1991. 1. Fano Profiles

It is common to measure double excitation in helium by observing resonances in either photon or electron emission. The resonances are imbedded in a continuum background as illustrated in the case of Auger electron emission in Fig. 27. Analysis of these resonant (Fano) profiles has been discussed by various authors (Fano, 1961; Mehlhorn, 1978, 1985; Shore and Menzel, 1968; Stolterfoht, 1971; Bordenave-Montesquieu and Benoit-Cattin, 1971; Prost et al., 1977; Bordenave-Montesquieu et al., 1982; Arcuni and Schneider, 1987; Gundnov et al. 1989). At a given energy and angle of the scattered electron the autoionization resonance and the continuum background must be added coherently. Fano (1961) briefly notes after Eq. (110) in his paper that one may unitarity to separate the resonance peak from the continuum background, i.e., using J l'I'E)('PE[ = J (14)(41 IJIE)(+EI) = 1. Here a sum is taken over the energy and direction of the ejected particles. Under this condition (i.e., summing over

+

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

33

35

287

37

e- ENERGY (eV)

FIG. 27. The fits of Fano profiles to the doubly differential emission cross sections for (a) electrons and (b) protons (Giese et al., 1990). The dashed line is the total fit, the dotted line includes contributions of only 2 p resonances, and the dashed-dotted line includes only the 2s2p resonances in the main peak.

both energies and angles of the ejected particles) the resonance is decoupled from the background of continuum electrons if the continuum background is constant. This decoupling does not hold, however, if the background varies with energy and the q parameter (Fano, 1961) is not larger than one.

2. Double Excitation Cross sections for double excitation of helium into the first excited states have been reported by Pedersen and Hvelplund (1989) and by Giese et al. (1990), although other results were available earlier (e.g., Rudd, 1956; Bruch et al. 1984; Stolterfoht, 1985). The more recent results are shown in Fig. 28 together with various calculations. Calculations of Fritsch and Lin (1990) are based on the method of coupled channels (Section 11.1.1). Preliminary results (not shown) of Reading and Ford using the forced impulse method (Section 11.1.1) are close to those of Fritsch and Lin for the (2p2) ID transition. McGuire and Straton (1990b) have done calculations complete through second order in ZJv (Section 1I.D). The theoretical differences are relatively large. Differences between Zp =

288

McGuire

0.1 0.0

'

'

"

"

'

'

'

"

I.o

-.-. ._ '

'

, . - ,

'

'

'

2.0

Projectile Energy (MeV/amu) FIG. 28. Cross section for double excitation of helium to 2s2p('P) (McGuire and Straton, 1990b). Experimental data due to Giese et al. (1990) and Pedersen and Hvelplund (1989). FL is the result of a coupled channel calculation of Fritsch and Lin (1990). Interference with the singleionization continuum background is not included in the theory.

+

1 and Z , = - 1 appear to be smaller than the factor of two found for double ionization. This means that the effects of space and time correlation that give rise to Z; effects (C' Section II.D.4) may be relatively small for double excitation to the first excited level. Observations of excitation of both electrons in A+16 in collisions with He at high collision velocities have also been made (Chetioui, 1990), but no theoretical calculations have been done as of early 1991. Double excitation of helium by photon impact has been considered by Lindle et al. (1985) who quote a constant high energy calculated ratio of double to single excitation into all states of n = 2 of about 4.8% in rough agreement with observation. The connection of ratios of double to single exctiation by various projectiles (e.g., charged particle and photons) has not yet been considered. (See Table 11.) D. MULTIPLE-ELECTRON CAPTURE Multiple capture, like multiple excitation, may be regarded as transitions to discrete bound states. However, as noted in Section KF, capture differs at high velocity from excitation both kinematically (e.g., the Thomas process) and in its scaling laws with projectile charge and velocity.

I . Multielectron Capture Capture of as many as six electrons in a single fast collision has been discussed by Schlachter et al. (1990). The observed data are well fit by a binomial distri-

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

289

0.3 0.2 0.1 0.0

0

1

2

3

4

5

6

7

0

1

2

3

4

5

8

7

0.3 0.2 0.1 0.0

L- and M-&W electrons ~aptwed FIG. 29. Typical fit of data (solid) to binomial distribution (hatched) of probabilities given by the independent-electron approximation (Schlachter er al., 1990) for Ca+” Ar at 47 MeV.

+

bution using a single capture probability, P, as shown in Fig. 29, corresponding to the independent-electron approximation (Section 1I.B). Similarity to the binomial analysis for ionization in Fig. III.B.2 may be noted. Binomial distributions have been previously used for multiple capture by Anholt et al. (1987) including capture in relativistic heavy-ion collisions.

2 . Double Capture Data for total cross sections for single- and double-electron capture for He++ He from Sidorovitch et al. (1985) are shown in Figs. 30 and 3 1 together with calculations in the independent-electron approximation (Theisen and McGuire, 1979) using the Bates-Born approximation. (C’ Section 1I.F.) At high collision velocities both single and double capture go primarily to the ground state. As in the case of single and double ionization (Figs. 1II.B. 5 and 6) the error for double capture is about twice that for single capture, and the independent-electron approximation appears to be somewhat incomplete. More recent calculations using a continuum distorted wave approximation in an independent-electron model using correlated asymptotic wave functions by Crothers and Dunseath (1990) give some improvement in the comparison of theory to observation. Coupled channel calculations at intermediate energies by Jain et al. (1989) are in fair agreement with data by Zouros er al. (1987) for formation of 2 t 2 t ’ double excited projectile states for He++ + He. As the collision velocity decreases, an

+

290

McCuire

SINGLE CAPTURE

10-22

-

10-23

too

1000 E (KeV/amu)

FIG. 30. Cross sections for the single-electron capture to the ground state in the charge transfer of the H + , He2, and Li3+ nuclei in helium versus nuclei energy (Sidorovitch et al., (1985). The theoretical curves are based on the first-order Bates-Born approximation.

increasing fraction of multiple capture goes into excited states. A brief overview of multiple capture at lower velocities, including an examination of simultaneous vs. successive transitions, is given by Barat (1988). In the high-velocity limit it has been suggested that multiple capture may be dominated by shakeover (Section III.A.3; Miraglia and Gravielle, 1987; McGuire et af., 1988). Gravielle and Miraglia (1990) have developed a two-step model for double capture including shakeover. No experimental evidence for shakeover is available as of early 1991.

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

29 1

E (KeWarnu) FIG. 31. Cross sections far the two-electron capture to the ground state in the charge transfer of the H', He2, and Li3+nuclei in helium versus nuclei energy (Sidorovitch er al., 1985). The theoretical curves are based on the first-order Bates-Born approximation.

E.

IONIZATION AND

EXCITATION

Relatively little work has been done on ionization and excitation at high veIocities. A few observations for ionization and excitation in He and Ne in collisions with various ions have been made by Bmch et al. (1979) Comparison of ionization plus excitation in H, by protons and by electrons has been made by Edwards (1990). The only calculation was done by Kocbach (Bruch et al., 1985) using the independent electron approximation with first-order single-electron

McGuire

292

0

u

V

9

m

H

on H e e on H e

+

" 3

1

2

3

4

5

6

7

8

9

Velocity [v/v,]

FIG. 32. Cross sections for excitation plus ionization of helium by impact of protons and electrons (Bruch et al. 1990).

probabilities for excitation and ionization. Qualitative agreement between these calculations and data for C4+ + He at velocities from 167 to 417 keV/amu. Data of Bruch et al. (1990) comparing ionization plus excitation of helium in collisions with protons and electrons, shown in Fig. 32, is consistent with observations of Pedersen and Folkmann (1990). A review is given by Hippler (1988).

F. TRANSFER AND IONIZATION At high collision velocities most studies have been of single and single-electron transfer (Salzborn, 1990). An overview of transfer ionization in helium is shown in Fig. 33 developed by Mueller (McGuire et al., 1987; Hvelplund el al., 1980). In this figure the ratio of transfer ionization to single transfer is plotted over a wide range of projectile energies for projectiles of charge Z , ranging from 1 to 3. At the higher velocities shown this ratio is consistent (HorsdalPedersen and Larsen, 1985; Kristensen and Horsdal, 1990) with a combination of independent-electron approximation plus shakeoff, i .e. the probability for Pumrfer. (Cf. Section transfer ionization is approximated by (Pion+ Pshkeofl) III.F.3.) The value of PshkeoE is approximated by the high-energy photon limit R, = 3.5% since the transferred electron leaves the collision region with a ve-

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

r-

293

II

Energy

(KeVIamu)

Fic. 33. Ratio of ionization plus capture to single-capturecross section in helium versus velocity (McGuire er nl., 1987). Note approximate similarity of limiting value of R at high energy with the limit for photoionization shown in Fig. 21.

locity equal to the velocity of the projectile, which is large. The data for proton impact is anomalous in that the observed ratio shows no peak at intermediate velocities (i.e., about a few hundred keV/amu). Also, studies of transfer ionization in collisions with helium with U+q and I + *about several hundred keV/amu have been reported by Datz et al. (1990), who suggest that electron correlation may be significant due to the large projectile charge which promotes correlated double capture. At lower collision velocities transfer ionization is indistinguishable from double-electron transfer followed by autoionization. At these lower velocities double transfer rates are expected to be greater than direct ionization plus single capture.

1. Independent-Electron Approximation Comparison of a calculation (Sidorovitch et al., 1985) using the independentelectron approximation with data is shown in Fig. 34. Since the single-electron probabilities are computed using first-order perturbation theory which overestimates the true result at low velocities and capture to excited states is ignored (also a source of significant error at lower velocities), the theory is not reliable at the lower velocities. At the higher velocities the agreement of these calculations with observation is consistent with the error in single capture and single ionization. (C’ Sections 1II.B and D.)

McGuire

294

10-16

10-23 10- 24 10- 25

E (KeV/amu) FIG. 34. Cross sections for the ionization of helium with simultaneous electron capture to the ground state of nuclei in collisions of He atoms with the H+,He2+,and Li’ nuclei versus nuclei energy (Sidorovitch er nl., 1985).

2. Thomas Ridge Since electron capture is classically forbidden in a one-step process (cf. Section ILF), classically allowed two-step processes may play an important role in differential cross sections that include electron transfer. To illustrate the nature of second-order ridges, the distribution of recoil particles, daldv, , given by a second Born calculation (Briggs and Taulbjerg, 1979; Ishihara and McGuire, 1988; McGuire et al., 1989) for the four-body reaction p+ He -+ H He++ + e - is shown in Fig. 35. Here M , = M, and

+

+

MULTIPLE-ELECTRON EXCITATION, IONIZATION. AND TRANSFER

295

M' = M 2 = M 3 = mc so that r = 1 in Fig. 3 and v3 is the velocity of the ionized (recoil) electron. In this case the He nuclear charge Z , localizes the two electrons but plays no direct role in the transfer process. The presence of this fourth particle, however, broadens the constraint of overall three-body energy and momentum conservation. A sharp ridge corresponding to overall energy and momentum conservation, and a broad ridge cresting v3 = v are both evident for 200-MeV proton impact shown in Fig. 35. The broad ridge corresponds to a pole in the Green's function. The crest of this ridge occurs when the intermediate energy equals the total energy. The classical Thomas peak is found at the intersection of these two ridges. The widths of both ridges are consistent with the uncertainty principle. The broad energy-conserving ridge has a width P E = Z,V' x hla,,, corresponding to the discussion in Section 1I.F. with v' = v = v3. The sharp ridge has a width given by the momentum distribution of the target electron which is proportional to Z , . This width goes to zero as ZT+ 0. In the ridge shown in Fig. 35 one target electron collides with another in the second step of a two-step collision (somewhat like the TSl process in double ionization of Section IILB). Because of the interactions between the electrons, this ridge corresponds to a multi-electron effect. Observation of this Thomas singularity was reported by Palinkas et al. (1989, 1990) and their data are shown in Fig. 36 together with the original approximate second Born calculations of Briggs and Taulbjerg (1979) as well as more recent

Recoil Angle, 7

+ He + H + He+ + e - for 200-MeV p+ (Ishihara and McGuire, 1988). Here vois the Bohr velocity. The sharp ridge, which permits energy nonconservation in intermediate states, intersects the broad energy-conserving ridge at u, = v and y = 90". (Cf. Fig. 34.) At lower projectile energies both ridges are broader and weaker. The broad peak is indistinct at energies near or below a few MeV where experiments are feasible. FIG.35. Differential cross sections duldv, in the velocity of the recoil electron for p+ +

296

McGuire

t

102*

t . 60

90

120

Emission angle [deg.] FIG. 36. Observations in one-MeV P + He by Palinkas el al. (1989, 1990) of the Thomas peak due to electron-electron rescattering.The solid line is the theory of Briggs and Taulbjerg (1979) and the dotted line of Ishihara and McGuire (1988) corresponding to a cut at u = u,, in the ridge in Fig. 35 (for a 1-MeV proton).

calculations of Ishihara and McGuire (1988). Here v3 is fixed at vj = v and the emission angle y is varied. Data suggesting this effect were reported by Horsdal et al. in 1986. At the collision velocities observed the second Born calculations shown in Fig. 35 are inaccurate (and show little structure). A different mechanism for the observed peak has been proposed by Gayet and Salin (1990) who use a diffraction mechanism without electron correlation. Their calculations give qualitative agreement with the observation of Horsdal et al. shown in Fig. 37. 3. High-Velocity Limit In the limit of high collision velocity, one might use Ptmnsre, PsMeofffor the probability for transfer ionization. Then the ratio of total cross sections for transis independent of the fer ionization to single transfer, proportional to Pshakeoff, collision velocity, v (Sections III.A.3 and 4).The data of Kamber et al. (1988) (Section ILI.B.3) then suggest that this ratio depends on Q - k, the momentum transfer to the residual ion, consistent with the generalized shakeoff probability

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

0.15

-

0.10

-

0.05

-

297

+-

B ++

B

v

\

+

x

0.00

1

2

Scattering angle (mrad)

A

FIG. 37. Ratio of transfer ionization to single transfer. The theory, diffraction model of Gayer and Salin (1990). is one of various possible explanations. Data are from Horsdal er al. (1986).

of Aberg (Eq. (135)). For transfer ionization IQ - kl lies halfway between the value of k for photoionization and zero for ionization via a binary encounter between the projectile and a target electron. Assuming that the ratio varies monotonically with Q - k, one may expect that the ratio for transfer ionization to be between the ratio for photoionization (large )Q - kl) and the ratio for ionization via binary encounters with charged projectiles (small lQ - kl). Data for this ratio (Kristensen and Horsdal, 1990, listed in Table 11) is consistent with this prediction, but the data also appear to increase with projectile velocities. An alternative possibility for this high v limit of the ratio of transfer ionization to single transfer is suggested using the simple shakeover probability of Eq. (127). In the preceding discussion it was assumed that at very high v, the transfer ionPsh&eoB. If shakeover (collapse of the electronic ization probability is PmSfer wavefunction onto the moving projectile) occurs, then the dominant transfer ionPsh&eover. Using this one has a ratio of ization probability is given by Pionnation transfer ionization to single capture that increases linearly with v at in the high v limit. As noted already, observations (Kristensen and Horsdal, 1990) are inconclusive.

G . TRANSFER AND EXCITATION The two-electron transition in which a target electron is transferred and a projectile electron is excited has received considerable attention experimentally stimulated by an observation in 1982 by Tanis et al., who discovered a resonance in

298

McGuire

the total cross section. The observed resonance was interpreted as an inverse Auger process (or dielectronic recombination) in which the projectile electron is excited by interaction with a captured target electron. This process occurs when the kinetic energy of the projectile electron matches the transition energy, i.e., a resonant condition in the collision velocity. This process has been referred to as resonant transfer and excitation (RTE). In some cases RTE dominates the total cross section for transfer excitation. In 1983 Pepmiller et al. observed a nonresonant process for transfer and excitation occur due to independent interactions of the two electrons with the nuclear charges Z , and 2, of the projectile and target, respectively. Thus, RTE is a process with electron correlation and NTE is an uncorrelated process. Theoretically Shakeshaft and Spruch ( 1979) had briefly suggested the possibility of RTE in 1979. The first calculation of RTE was done by Brandt (1983) using quasi-free electrons in the impulse approximation (Goldberger and Watson, 1964, Chapter 11). In Brandt’s approximation the RTE cross section is equal to the cross section for dielectronic recombination weighted with a velocity distribution for the target electron, namely,

where J is the probability for finding a target with a velocity component v, along the beam axis, i.e., J is the Compton profile (Richard, 1990). In 1984 Feagin er al. formulated an amplitude for transfer excitation that includes both RTE and NTE, namely, =

acornelated

+

aunconelatcd

(137)

-

where auncomtated - aexcitation atrangrer, corresponding to NTE, and acornlared is an amplitude for RTE proportional to the electron interaction that includes Brandt’s RTE result. However, the relative phase between the RTE and NTE amplitudes may be difficult to obtain correctly. Coupled channel calculations for He+ H at intermediate velocities (Fritsch and Lin, 1988) are in good agreement with observations. The experimental technique of zero-degree Auger spectroscopy has been developed by Stolterfoht and coworkers (Itoh er al., 1985; Swenson et al., 1986; Stolterfoht et al. 1987). The Fano profiles of angular distributions of Auger electrons and photons have been considered by Bhala (1990) and Kunikeev and Senashenko (1989). Hahn (1990) has discussed the intercorrelation of various resonant processes and developed a formulation that includes contributions from a third electron. Evidence for this two-center correlation is unclear (Schulz et al., 1989; DePaola et al., 1990). A large number of observations and several reviews of transfer excitation are now available (Tanis, 1990; Graham, 1990; Schuch et al., Mokler, 1989). One comparison between theory and experiment is shown in Fig. 38.

+

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

299

FIG. 38. Cross secton o$ for transfer excitation (TE) in S"+ + He collisions (Tanis, 1990). The dashed curve is a calculated RTE cross section multiplied by 0.85. The solid curve is a calculated NTE cross section normalized to the data.

H. PROJECTILE ELECTRONS Previously in this section we have considered projectiles that do not carry electrons into the collision (although they may carry electrons out). Now we consider the effect of projectile electrons. There are two simple limits. If a projectile electron tightly bound, then the electron simply screens the nuclear charge by one atomic unit. If the electron is loosely bound, then the projectile electron and the projectile nucleus may scatter incoherently from the target corresponding to the free collision model of Bohr in 1948 (also Dewangen and Walters, 1979; Andersen et al., 1988; Richard, 1990). In the context of the first Born approximation the projectile may be considered as a point particle with an effective charge, Z,, , given from Eq. (7 1) as

ZfC = (ZP

-

o2- S,",

(138)

where F is the atomic form factor and Sin,is the incoherent scattering function defined by Eq. (72). F and Sinearise from the interaction between a projectile and a target electron, i.e., F and S,,, are mutltielectron terms. Zeffgoes to the

McGuire

300

coherent limit of full screening at small Q and goes to the free collision limit at large Q. In Fig. 4,Z,, is illustrated. Anholt (1986) has shown that Sin=may be zero (or nearly zero) at velocities below the threshold for ionization by free electrons so that there may be a threshold for the effects of the projectile electron-target electron interaction carried by Sin,. The full screening limit is well established experimentally. For example, 07+ and N7+are known to yield the same scattering cross sections in collision with systems of loosely bound electrons. A clear example of the limit of incoherent scattering by the electrons and the nucleus is shown by the data of Wang et al. (1986) in Fig. 39. Here the cross section for collisions of a high Rydberg hydrogen atom with argon is equal to the sum of cross sections by impact with a free proton and by a free electron. The threshold for multielectron effects has been observed by Zouros et al. (1989) and Huelskoetter el al. (1989) shown in Figs. 40 and 41. Zouros et al. (1990) analyzed their results using an impulse approximation similar to that introduced by Brandt ( 1983) for resonant transfer and excitation. (Cf.Section 1II.G.) Liu and Starace (1990) have used the first Born approximation described pre-

--

1

2

I

I

5 I

I

1

20

10 I

I

I l l 1

-

50 -

-

:&++$A aC."".

0

E

20

's

-.-.-.-

I

*.a-

-.*.

-.I.

c

P

*'., A.-.-. .

10-

z : u l ul

e

u

'a.

I'

..J'

-

..*.#

.8'*

-

'. .'

...'

5-

-

'. '.

f

'.\

*.*'

-

E .'

..a

,8*.

21

2

I

I

I

5

I l l l l

70

1

20

I

1

!

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

1 22r-l , ,

PROJECTILE ENERGY (MeV)

II

tI

, , , , , , , , , , , , , , ,,

,

,

,

30 1

~

o F6*+He F6**t-$

8-

J

1

0

' 0

4

8

12 16 20 24 28

32 36 40

PROJECTILE ENERGY (MeV) FIG. 40. Cross sections for the production of ls2s2p4P states by Is + 2p projectile excitation in collisions of (a) (Y and (b) of F6+ (1~2s)projectiles with He and H2 targets versus projectile energy (Zouros er al., 1989). Calculation: electron-electron excitation (eeE) cross sections using ls2s2p4P theoretical electron impact excitation cross sections folded by the Compton profile of the target. Dashed lines, calculated eeE for H1 targets; dash-dotted lines, calculated for eeE for He targets. Arrows (at 16.3 and 25.0 MeV) indicate the projectile energies corresponding to the threshold for 1s + 2p electron-impact excitation in 05+and F 6 + ,respectively.

viously to analyze data of Duncan and Menendez (1977) for doubly differential detachment cross sections for collisions of fast H - with argon. A distinct peak for excitation of the n = 2 level of H is apparent in Fig. 42. The large peak at small angles is almost entirely due to excitation of H (n = 2) in the Sincterm of Q. (138). The remaining discrepancy between theory and experiment may be due to large experimental angular resolution. Wang et al. (1990) have recently used this first Born approximation to predict the ordering of ratios of single to double ionization of helium by projectiles with and without electrons. The analysis of Sec. III.B.2.c was extended to include

McGuire

302 2..oe+s

(a)

I

B Rojecrile a ~ g WeVN y Fic. 41. Electron-loss cross section for Ot7 + HI(Huelskoetter efa / ., 1989). Effects of electronelectron interactions are included in the calculations with the solid line (C'. Eq. 71) but not included with the dashed line where Zen = Zp.

projectile electrons by replacing 2; in Eq. (13 I) by 2iff.Using R to denote the ratio of double to single total-ionization cross sections Wang et al. (1990) predict RH+ = RHe++< RHe+< RHeo< Rphoton and R H + < R,Q < R H e o .

(139)

Partial confirmation of this prediction is given by data of DuBois. (CJ Wang et al., 1990; DuBois and Kover, 1989; and Table 11.)

I. MOLECULES Since molecules are groups of atoms, some molecular properties are similar to atomic properties, but other properties are not describable in atomic terms. For example, since H, has an internuclear separation of 1.4 Bohr radii, one might expect the cross section for H2 to differ from the cross section for two atoms of hydrogen.

I . Application of Atomic Mechanisms Understanding of atomic collisions may to some extent be applied to molecular collisions. The fragmentation of a molecule, for example, is conceptually somewhat similar to ionization of an atom. Data for fragmentation of methane, CH4, into various fragments, CH,' ,

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

?SO

200

240

290

320

360

303

A00

DETACHES Z L C R C N 3 E X Y (4)

160

200

2A0

290

320

360

1

10

60 45

30 15

0 1

DETACHED E:CT;(ON ENmGY (sV)

FIG. 42. Differential cross section for 0.5-MeVH - + Ar 4 H+ + e + Ar* (Liu and Starace, 1990). The contribution from n > 1 excited states comes from electron-electron ineractions.

CH: , CH: , CH+, and C +, was collected and analyzed by Malhi et al. (1987). The ratios of cross sections for CH:/CH; in collisions with electrons and protons are shown in Fig. 43 as a function of the collision velocity, v. Except for the lowest value(s) of n the ratios of CH,+/CH,+ are independent of both v and the sign of the projectile charge. This is consistent with the shake-type mechanism we used previously. (C’. Section III.B.2.) An alternative picture that may be applicable is a statistical energy distribution model (cf Section 11.1.2) in

304 100.0

-

t*

I 90.0

0 \

tm

I

-

16.0 14.0 12.0 10.0

-

c

0 .c d

2.5

0

-

-

-

*

-- +

-

+ R3

+

-+

80.0

0 70.0

-

- -

+

-

-*

*

-RP\ +

-

+

+

* * +

+

+++

1 .

\

+

0

1.0 0.5

+ +

+

+

+

+

I

I

2

3

+

*+++*

+

I

4

5

v (10' c m / w c ) FIG. 43. Ratios of fragmentation cross sections for methane, CH,, collision versus velocity (Malhi et al.. 1987). Note similarity to fragmentation of atoms (i.e., ionization) shown in Fig. 19. which energy deposited by the projectile is statistically distributed after the collision producing various degrees of fragmentation. In chemistry this statistical model is referred to as quasi-equilibrium theory (Rosenstock et al., 1952). The shake and the statistical mechanisms are not necessarily different. These data are consistent with a combination of shake and independent-particle mechanisms. This picture is reinforced by observations of the Z, dependence of the fragmentation ratios shown in Fig. 44. Here the ratios for lower-order fragmentation are independent of Z, while again the C+/CH,+data are different. The CH+ data could be intermediary. Another example of a similarity of collision mechanisms is found in the case of particle transfer. In Section 1I.F mechanisms of transfer via two binary collisions were described. Confirmation of one Thomas mechanism in electron transfer was first obtained in 1983 by Horsdal-Pedersen et al. This Thomas mechanism was first observed, however, in capture of atoms from molecules by Cook

MULTIPLE-ELECTRONEXCITATION, IONIZATION, AND TRANSFER

-+ u ; +* I

0

0

90.0

0

85.0 80.0. 75.0 70.0.-

+* I

20.0

-

15.0

-

0

305

P

1

~

n

0

O

D

A A

-lR2 A

a

A

A

10.0

-

0

.c!f? +'

a,'

16.0 12.0 8.0 4.0 - 0 0.0

--

o o o o

o o o o o 0

16.0 12.0 0.0 4.0 -

P

I

20.0

0

O.O..?

# 0 0

. 0 0 0

' 2

1

1

4

1

'

I

6

Charge

I

I

8

'

1

I

1012

Zp

FIG. 44. Ratios of fragmentation cross sections for methane versus projectile charge, Z, (Malhi et al., 1987). Note similarity to fragmentation of atoms (i.e., ionization) shown in Fig. 18.

et al. in 1975 and further observed by Breinig and Sellin (1981). Another predicted Thomas mechanism, corresponding to the ridge in Fig. 35, has been observed in electron transfer but has not been observed in atom transfer in molecular collisions.

2 . Two-Center Scattering Two-center scattering introduces at least one more particle into scattering at high velocities. In Section 1I.H scattering from a diatomic molecule was approximated as scattering from two atomic centers. Here we consider this problem with an extra center in a little more detail. The molecular T matrix was expressed in Eqs. (73 and 75) as TM -- TI + T2 = f,e-i(I-fiK)v+ t2eifiQ.P

(141)

306

McGuire

where TI and T2 have a relative phase difference of exp{iQ * p } where Q is the momentum transfer of the projectile, p is the displacement vector between the two atomic centers, and pfi is the distance of center 2 from the molecular center at mass. To illustrate application of two-center effects and specific evaluation of probability amplitudes, we evaluate 1s- 1s cross sections for electron capture from diatomic molecules at high collision velocities following Wang ed al. (1989). First, we consider homonuclear molecules, in particular H, . Then we consider heteronuclear molecules. In this section we shall apply the simple BrinkmanKramers approximation, i.e.,rj = V,. a. Homonuclear Molecules. For homonuclear molecules, since by symmetry, ti = t, = t for gerade state, rl = - t, = t for ungerade state, and EL. = 112, we have from Eq. (141)

TM = t(e-iQ.(p/2)

+

e-iQ.(p/Z) )3

(142)

where ( - ) stands for gerade (ungerade). The wave function of a diatomic molecule can generally be obtained by linear combination of atomic orbitals (LCAO) or other methods. (Cf. Sections I1I.B-C.) However, the complete LCAO wave function may not be necessary when computations involve high-velocity collisions in which only inner-shell orbitals are important. Therefore, we begin by representing the molecular wave function by a simple product of 1s hydrogenic wave functions centered at each scatterer. The spherical symmetry of 1s wave functions on each center allows the simplest algebra without losing the basic physics of the process. Similarly, the ionic wave function can be expressed as a sum of 1s hydrogen wave functions centered on each scatterer. Thus we have for the simplest symmetrized molecular wave function aM and ionic wave function ai

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

307

where NM and N,' are normalization constants for the molecular and ionic wave functions; 4, 4, u , and u are hydrogenic wave functions. Since the two atoms are identical, we have 3, and 4 ( u and u ) with the same form of function. The effective charge carried by molecular 3, and Q, (ionic u and LJ) has been written as ZM(Z,)in Eq. (145). We point out that AM(A,) is an overlap integral of the molecular (or ionic) wave function. In the preceding development we have introduced two T matrices: the molecular matrix TM and the effective atomic matrix t. A further relationship holds for t and its atomic limit tA which describes scattering from a free atom. Following a similar procedure to Tuan and Gerjuoy (1960), it is straightforward to show, ignoring an exchange term (not justified for excitation and ionization in H2),

W, ITMI*J

= N&"(A,M X

*

rA(elQ IP)~)

* elQ XIM)

( 146)

IP'~)),

with the initial wave function 'Po = erKxR'PM and the final wave function qb= elK,R'@Iui,in which QM and Ql are defined in Eq. (143) and (144), ul, is the 1s bound state hydrogenic electronic wave function on the projectile. Here A I M and xIM are overlap integrals between ionic and molecular wave functions given (Wang et al., 1989) by AIM

=

uTs(r)3,dr) dr

= 8 ( z , z ~ ) ~ / ~f / (ZzM ~)~,

(147)

+ Z,(PV + 4Z,)e-Z+'] where v = Z,Z - Z& . In Eq. (146) we have replaced the BK amplitude I,, of Tuan and Gerjuoy (1960) by the more general matrix t A . Comparing Eqs. (146) and (141) we have t = NMN,?(A,M

* X,M)

NMN,N,'ltA E NjtA.

(149)

The effective atomic matrix t differs from its atomic limit tA by an overall constant N: . Since T M = t ( e - ' Q p ' 2 ? erQ@z) and t = N j f A , it is primarily N: that determines the difference between the molecular and twice the atomic total cross sections. By the preceding definition, N j is a constant combining the molecular wave function (NM), the ionic wave function (N,),and their overlap (N,'l), which depends on the form of the wave functions. The preceding expression for the overlap factor N&, derived for wave functions given by Eqs. (143) and (144), may be generalized.

308

McGuire

The reflection symmetry provides properties of a homonuclear diatomic molecule may be used to determine N: under two interesting limits, i.e., the separated atom limit ( p + m) and the united atom limit ( p + 0), without specific calculations. For the limit as p m, the total normalization constant for gerade N: and ungerade N; are the same. Namely, when p + 03, we have NM = N: = N; = (1/2)’”,N& = 1, and N: = N; = 1/2. The limit as p +S 0 is considered following Eq. (162). As an example, we now use the simple BK approximation amplitude tlK

-

in Qs. (146) and (142), where 2, is the charge of the projectile and ZT is the effective charge of the nuclei in the molecule, also denoted as Z, , to obtain T&K

= NZI t BAK ( e - i Q ( p / Z )

(151)

eiQW2)).

By Q. (77), the corresponding probability amplitude afCIKis found as

B,

=

B

2

p1/2.

These integrals can be done analytically, namely,

where Kz is the Bessel function of the third kind and second order and

x+

=

(B,I(G +

Qt)’/* -+ B p , COS(&-

X

(Z+

+ QT)’/2,

1

4p)

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

309

where +B and +p are the azimuthal directions of B and p . Thus,

X

[x2_K2(x-) e - iQlp:/2

?

x: K , (x

+

)eiQ+:i2].

The probability as a function of impact parameter is given by

P$K(B) =

The total single-capture cross section ukKis

PLK(B)dZB.

crhK = 2

(159)

The factor of 2 comes from the indistinguishability of the two electrons. When substituting Eq. (159) into (158), uIKcan be written as

where u1and u2are due to individual centers and u12 is due to the interference. The integrals for ul,2in Eq. (159) can be computed analytically to give the BK cross section, i.e.,

while vI2may be computed numerically. For the limit as p -+ m, the interference term uI2approaches zero for both gerade and ungerade. Meanwhile, in Eq. (160), the individual terms u1and vz summed over gerade and ungerade give (TI

=

+.-

+,-

=

gz =

2(4)*gA

=

(N:)CTA

+ (N;)'~A

(162)

4CA.

Thus, the total cross section uM in Eq. (164) yields 2UA9 twice the atomic cross section. For two electrons in the limit as p + 0, again N,? + 1/2, but cos(a,p,) -+ 1 and the ungerade contribution vanishes, i.e., N ; + 0, ZM+ 22, so that uiKp=o = 2uuA.Here uAdenotes the cross sections for an isolated atom,

McGuire

310

and vUAis the 1s-1s cross section per electron for the atom with the united nuclear charge. If there are four electrons, two go to the 1s level and two to the 2p level in the united atom limit. b. Application to H, . The preceding development provides a method to calculate capture from homonuclear diatomic molecules using a simple, but generalizable, molecular wave function. For H, a somewhat better wave function is the Weinbaum wave function, also used by Tuan and Gerjuoy (1960) and other authors. Specifically, we choose @M

=

+ $Is(~)~Is(~) + C[$ls(1)$lr(2) + 4lS(l)+lS(2)1}

NM{$ls(1)41s(2)

(163)

and ionic wave function = Ni’[uls(l)

@i

+-

~1s(1)1,

( 164)

where the normalization constant N,‘ is given by Eq. (145) and NM is given by

+

+

CAM]}"'. NM = 1/{2[(1 C2)(1 + A&) (165) The preceding Weinbaum wave function differs from the previous sample LCAO wave function in Eq. (149) by a bonding term characterized by c # 0. The overlap constant Ni&in this case is found to be slightly different from that in Eq. (165), namely, N& =

(AiM

5

XiM)(l

? C).

(166)

The form of Eqs. (142) and (149) is not changed. If we take the bonding term c + 0, then N& reduces to the simpler form in Eq. (149). For H, at equilibrium, weusep = 1.4, 2, = 1.193, c = 0.256, andZ, = 1.4.

c. Heteronuclear Diatomic Molecules. For heteronuclear diatomic molecules we again choose a simple LCAO wave function with only s wave contributions. Now we use I and 2 to denote the electrons, and A and B to denote the two different atomic nuclei. The molecular wave function is given by NM[auMA(l) + buME(1)l[auMA(2) + buMM(2)1, where NM = [(a4 + b4) + 2a2b2(1 + 2AL) 4ab(a2 + b2)AM]-’”. (167)

4M

Here A,

=

=

+

J ~ & A ( l ) ~ w ( 1 ) and, d r l for example,

The ionic wave function is given by

4i

+

=

+

Ni[a’~,(2) + b’ui,(2)1,

( 170)

with N, = [(a’)2 (b’)2 2a’b‘Ai]1’2and A, = J u ~ A u , B dr. Using these wave

3 11

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

functions, it is straightforward to show that to an overall phase

TM

TA

+

TBeiQ'p,

(171)

-

where it is convenient to choose the origin at the center of mass of the molecule. The mathematical evaluation of probability amplitudes is now similar to the homonuclear case, which is recovered as a b and a' + b', corresponding to c = 1 in Eq. (163). Although the algebra is quite similar for the homonuclear and heteronuclear cases, the identification of final states and counting of electrons is a little different. If A # B in molecule AB, then A + B is a different final state than AB+.Thus TAdiffers from TB. At the same time gerade and ungerade labels no longer apply since there is reflection symmetry. We also note that for most molecules, there are four electrons in the ground state, twice as many as in H, . Hence, except for two-electron systems such as HZ,c&zm2(cA cB) where c A , B is a 1s atomic cross section per electron for center A and B .

+

d . Other Examples. This two-center analysis may be applied, if an exchange is included, to ionization and excitation as well as to the case of electron transfer considered here. In each of these cases this theory predicts that the cross section depends on the orientation of the molecule as illustrated in Fig. 45. This depenIO4

H++H,

3 0

3

10'

-

I

5 MeV

.-

0

40

80

120

160

ORIENTATION ANGLE, 8, (degree) FIG. 45. Capture cross section from H2as a function of the orientation angle O,, of the H2 molecular axis relative to the beam direction (Wang e t a / . . 1989).

McGuire

312

dence on molecular orientation may be observed in Coulomb recoil distributions from collisions of charged particles with molecules. Some data were taken in 1963 by Dunn and Kieffer corresponding to single ionization of H, in collisions with electrons and were analyzed by Zare (1967). Coupled channel calculations for electron impact on H, and other molecules are common (Collins and Schneider, 1988; McConkey et al., 1988; Shimamura and Takayanaki, 1984), but few calculations for impact by heavy projectiles have been done. A few observations of Coulomb explosions in collisions with charged heavy particles are in progress as of early 1991. Data dealing effects of multielectron transitions in Coulomb explosions may be available soon. For multiple-electron transitions one may apply the independent distinguishable electron approximation. One must ensure, however, that the two amplitudes for B p/2 and B - p/2 are properly separated (especially at large p ) when applied to Eq. (29). This can be done by distinguishing which center has a vacancy in the final state. For symmetric molecules such as H,, proper separation may result from proper application of Pauli exclusion. For ionization and excitation of molecules by projectiles carrying electrons the scattering amplitude may be evaluated in the first Born approximation by simply replacing 2, by Z,, as discussed in Section 1I.G. I . The resulting T matrix (or scattering amplitude) is then given by

+

T = 4r/Q2Zf'(0) cos(Q

*

P)fatornie(p)

(172)

corresponding to a product of internuclear scattering times the effective projectile charge times the two-center interference times an atomic scattering amplitude. ( C ' discussion at the end of Section 1I.G. 1.) Data testing the validity of this simple form is not available as of 199 1.

IV. Conclusion Multielectron transitions are many-body phenomena. When more than one electron undergoes a transition in an atomic or a molecular collision, the event contains information about the dynamics of many-body systems. The electrons in these dynamic events may act independently or they may influence one another. In this review both independent and correlated systems of electrons in highvelocity collisions have been considered, leading to a few general observations as follows. Our understanding of many-electron effects is incomplete, even in fast collisions involving only a few electrons, i.e., relatively easy examples of manyelectron phenomena. Some understanding has been achieved in systems in which electron correlation is weak so that an analysis using the independent-electron

MULTIPLE-ELECTRON EXCITATION, IONIZATION, AND TRANSFER

3 13

approximation is sensible. However, incorporating exclusion effects of identical electrons in the independent-electronapproximation is incomplete. Correlation arising from the Coulomb interaction between electrons is often complicated and hard to understand. Some analysis of the role and nature of correlation in two-electron transitions has been possible. Both perturbative methods and coupled channel methods have been useful in the analysis of observed data. However, the nature of specific physical mechanisms responsible for dynamic correlations remains mostly unclear, although experiments in the last several years using antiprotons and positrons have led to some clarification of our understanding. Nevertheless, even the simplest cases-for example, ratios of double- to single-electron transitions in helium in the high energy limit (summarized in Table II)-remain only partially understood at present. Various approaches have been used to help understand multielectron effects in atomic and molecular collisions. Kinematic restrictions imposed by conservation of energy and momentum have been helpful in interpreting some many-body effects in terms of combinations of two-body interactions. Effects of mutual interaction of electrons on two centers in relative motion have been observed and analyzed with some success within the confines of first-order perturbation theory. Two-center effects in the analysis of molecular dynamics have been predicted and may be soon tested by experiment. But both theory and experiment becomes more difficult as the number of centers, the number of electrons, and the number of states involved increase. Eventually as the complexity of the systems grows, our understanding may be limited to statistical descriptions, an intellectual position that is common with our understanding both in basic quantum mechanics and in classical chaos. In summary, understanding the transition from how two bodies interact to how many bodies interact provides a conceptual basis for understanding various aspects of our environment. The human environment is sufficiently rich in both complexity and possibility to challenge the human mind as well as the human spirit. The truths that emerge from this challenge can provide intellectual satisfaction and at the same time provide meaning and alternative in our lives.

Acknowledgments I gratefully acknowledge discussion with 0. L. Weaver, J. Straton, J. P. Giese, C. D. Lin, C. L. Cocke, I. Ben-Itzhak, P. Richard, R. Shingal, T. Ishihara, N. Stolterfoht, W. Meyerhof, T. Aberg and Y.D. Wang as well as some guidance over many years from D. R. Bates. This work has been partially supported by the Division of Chemical Sciences, Office of Basic Research, U. S. Dept. of Energy.

McGuire

314

Appendix: Correlation In a dictionary correlated means interdependent; that is, correlated particles influence one another. Electron-electron correlation occurs because electrons interact with each other. In a technical sense there are different ways to specify this kind of correlation. It is possible to specify that electron correlation is any effect arising from the Coulomb interaction between electrons. It is also possible to define correlation as a deviation from a product of electron probability densities. This latter definition of correlation is sensible because the probability for any reaction may be written as a product of individual single-electron probabilities if the electron-electron interaction is ignored. This definition is also sensible since it corresponds to the definition of correlation used in stochastics and statistical mechanics. To the extent that understanding macroscopic properties in terms of microscopic atomic properties is interesting, it is sensible to choose the broader stochastic definition of correlation (Balescu, 1975; Huang, 1987) since statistics are ultimately useful in dealing with large numbers of atoms. Correlation is often introduced in stochastics and statistical mechanics (Balescu, 1975) as a generalization of the notion of statistical standard deviation of a distribution,

&=

((x -

(173)

(x))2).

If c2is not zero, then the distribution is not localized at a single value ( x ) , but rather the distribution is spread out. This leads to the idea that correlated functions are also spread out, i.e., not confined to a single x, (denoting a single particle), but connect x,and x,. Consider the one-particle distribution function given by f i ( x , ) = p(x,)

=

4*4.

(174)

The two-particle distribution function is uncorrelated iff2(xl,x2) = f l ( x , )fl (xz). In general, however, f2 is correlated and may be written as f2(XIr

x2)

= fl(xl)fl(x2)

+

gz(x1,

(175)

x2).

g, is called the two-particle correlation function. For three particles one may generally write h(XI

9

x29

x3)

=

fl(xl)fl(x2)fl(x~)

+ fl(xl)g2(x29

+ fi(X3)g2(XIr

x3)

+ fl(X2)g2(x1,

x2)

+ g3b-1,x 2 r

x3)

(176)

x3)

and so forth for f4, fs, . . . fN, . . . . This is called a cluster expansion of the distribution function fN in terms of the jth-order correlation functions g, . Correlation has been defined and studied in both space and time. Here we

MULTIPLE-ELECTRONEXCITATION, IONIZATION, AND TRANSFER

3 15

shall consider spatial correlation. Examples of time correlation include timeordering effects and memory. Most studies have been confine to two-particle or pair correlation since experimental studies of higher-order correlation are often difficult. We do note, however, that the N-particle distribution function can be related to the N 1 particle distribution function via the two-particle correlation potential u(xi - x j ) . This is referred to as BBGKY hierachy. The generalized correlation coefficient T is defined as follows. If P is a oneparticle operator, then for a two-electron system

+

75 =

1I (1fi

&(XI,

x * ) p ( ~ I ) p ( rdrl z ) dr*

( r V W dr) -

(1f,

(177)

(r)P(r)dr)*

such that (i) T = 0 if the electrons are uncorrelated and (ii) [TI < 1. Most importantly, T is independent of choice of basis functions. This generalized correlation coefficient has been applied by Christensen-Dalsgaard ( 1988) to atomic structure, but as of early 1991 application has not yet been made to dynamic correlation in atoms and molecules. Finally, let us note that detailed effects of correlation tend to disappear at both large and small space (and time, etc.) scales. On the large scale, correlation effects may average out. For example, fluctuating phase terms may average to a relatively small value on a sufficiently large scale. On a small scale the independent particle limit may dominate if the separation between the particles is large compared to the size of the region of interaction. For example, if the separation between electrons is much larger than the typical interaction distance of a probe (e.g., projectile), correlation effects may be small for relatively well separated electrons. REFERENCES Aberg, T.(1967). Phys Rev. 156, 35. Aberg, T. (1976). In “Photoionizationand Other Probes of Many Electron Interactions” (F. Wuillemier, ed.), p. 49. Plenum, New York. Aberg, T. (1987). Nucl. Instr. Merh. A262, 1 . Aberg, T. (1990). Private communication and Atlanta Proceedings (1973) (R. W. Fink er nl., eds.), US At Eng. Corn. No. Conf. 720404, p. 1509. Aberg, T., Blomberg, A . , Tulkki, J . , and Goszinski 0. (1983). Phvs. Rev. Left. 52, 1207. Aberg, T., Blomberg, A , , and MacAdam, K. B. (1987). J . Phvs. B20, 4795. Alston, S. (1989). Phys. Rev. A. 40,4907. Alston, S . , and Macek, J. (1982). Phys. Rev. (SPB). Amusia, M. Y . , Drukarev, E. G., Gorshkov, V. G., and Kazachkov, M. D. (1975). J . Phys. BS, 1248.

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ADVANCES IN ATOMIC. MOLECULAR, AND OFTICAL PHYSICS. VOL. 29

COOPERATNE EFFECTS IN ATOMIC PHYSICS J . P. CONNERADE Blackett Laboratory Imperial College London, Great Brirain

I. 11. 111. IV. V. VI. VII. VIII. IX. X. XI. XII.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body Effects and the Conservation of Angular Momentum . . . . . Rydberg Series . . . . . . . . . . . . . . . . . . . . . . . . . . Non-coulornbic Potentials and the Periodic Table . . . . . . . . . . . Giant Resonances . . . . . . . . . . . . . . . . . . . . . . . . . Atomic Giant Resonances in Other Environments-Controlled Collapse and Instabilities of Valence . . . . . . . . . . . . . . . . . . . . . . . Giant Resonances in Nuclei and in Atomic Clusters . . . . . . . . . . Are Giant Resonances in the d and f Sequences Atomic Plasmons? . . . . Extending Mean Fields beyond the Hartree-Fock Scheme. . . . . . . . Can One Blow Off Complete Shells by Laser Spectroscopy?. . . . . . . Interactions between Giant Resonances and Rydberg Series-Intershell and Intersubshell Couplings. . . . . . . . . . . . . . . . . . . . . . . Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction In low-energy physics and, indeed, in physics at achievable energies, the main unsolved problem is the many-body problem. The three-body problem was uncovered by Newton in classical mechanics. In quantum mechanics, exact solutions for the few-body problems are not known, although the existence of closed shells in central fields is extremely helpful in developing excellent approximations. In relativistic quantum field theory, there is no such thing even as a system of just two particles. The atom provides the ideal few-body quantum system: not only can it be isolated as a “free” atom, but it actually consists of a selectable number of bodies (electrons) that interact with each other through the best known law of force, the inverse square law. The only aspect of the problem that then remains unsolved is precisely the one of interest. However, in spite of all these good reasons, many-body effects do not figure prominently in the dictionary of most atomic physicists. In fact, the subject is

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dominated by the independent particle approximation (IPA), which takes a diametrically opposite point of view, namely that the electrons are dynamically independent. The predictions of the IPA are uncannily accurate over a wide range of energies and phenomena. Indeed, they seem so good that physicists have come to treat atoms as systems whose many-body characteristics have thankfully disappeared from view with the help of some excellent approximations. An important question to ask is why the independent particle approximation works so well for atoms, and whether conditions can be found under which significant breakdown might be expected. At first sight, almost every aspect of the independent particle model, apart from some QED corrections, seems well worked out. The chances of discovering dynamic many-body effects therefore seem rather small. The first ray of hope in this quest is glimpsed when we ask the question: Is there a single atomic mean field that can be used to represent not only ground state atoms, but also excitation and ionisation processes? It soon transpires that there is not, that there cannot be, and indeed that many different choices of potential are appropriate according to circumstances. The issue of how best to represent many-body phenomena in the calculation of excitations is arguably the most important one remaining in atomic physics. One may also ask the question: Should a mean field or an atomic potential be used in the first place? It would be surprising, in a many-electron atom, if electrons could be regarded as even remotely “independent.” Of course, they are not truly so (despite the name) even within the IPA since they are coupled through an averaging over all the motions enshrined in the central field approximation. What one really means is that they are dynamically independent, and the search for cooperative effects can be interpreted as a search for the breakdown in this aspect of the description. There are, however, more subtle points as to the nature of the breakdown. For example, the price one pays, in the self-consistent field theory of the atom, for reducing a many-electron problem to what appears to be a one-electron description is that each nonequivalent particle “sees” a different effective potential, due to averaging over the motions of all the others. Furthermore, when we come to excited states, each configuration (configuration average approximation) and, in some cases, each term (LS-dependent Hartree-Fock approximation) of the atom may require a different potential, at which point (regarding physical validity as opposed to computational convenience) the whole idea of using potentials to represent the atom becomes rather contorted. How could one excite the system from one state to another without completely altering its mean field? What, then, is the use of a potential? The point at which one decides that a description in terms of average potentials should be given up is to some extent a matter of taste, since quantum mechanics, being a linear theory, allows even quite complicated subsidiary excitations to be represented through the superposition of different configurations. As a rough rule, the more the fully self-consistent potential of the excited state differs from

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that of the ground state, the more complicated and contorted the IPA interpretation becomes. When dealing with photoionisation, a further problem is to decide which system the potential should be calculated for, since one may choose the atom (called V N ) ,the ion ( V N - I ) ,or indeed some kind of interpolation between the two. Alternatively, one may prefer to adopt Wigner’s scattering theory, a formulation that has the advantage that it does not depend explicitly from the outset on the choice of wavefunctions but merely on the existence of a Schrodinger-type equation. Precisely how one extends the IPA can all too easily turn into a technical discussion of computational methods, which would obscure the real issue: Are there new e$ecrs due to the presence of many electrons in the system that one might justifiably call cooperative effects? Our first task is to explore what they might be in order to ensure that appropriate experiments can be performed. However, the question posed here does not originate from experiment, but from fundamental principles. An independent particle model, augmented by low-order perturbation theory, successful though this combination may be, cannot fully represent the dynamic response of a manybody system. The experimental (and theoretical) challenge is then: Where are the missing parts of the description or where (in energy) do cooperative effects arise in the spectral response of many-electron atoms? This is a more specific question than merely asking where theory might break down or becomes inaccurate. Rather, one seeks to identify behaviour involving the correlated motion of many electrons and then relate it to an observable breakdown in the independent electron description. One thinks at first of double excitations, which are the most obvious example of a dynamic correlation effect. In this context, the question would become: can the simultaneous excitation of many electrons result in a specific spectral feature (an atomic plasmon)? The uncertainty as to whether an effective single electron potential may or may not adequately represent many-body effects is a familiar feature of manybody theory: one speaks of quasi particles, which resemble particles in all respects except that they are dressed, i.e., that they experience a potential different from that of free particles, and also quite often of the other particles in the system. Usually, this provides a very satisfactory description, which atomic physicists use and accept under the name independent (or “single”) electron theory. For example, an inner shell vacancy or hole in a nearly complete shell is just such a quasi particle. When this description breaks down, one speaks of a many-body or cooperative effect. From this standpoint, the Hartree-Fock model is merely the first term in a series (the linked cluster expansion, see, e.g., Brout and Carruthers, 1963) which, if convergent and pursued to all orders, should represent the many-body system correctly. Unfortunately, the solution to the many-body problem with an inverse square

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law of force is not known, even in classical mechanics. It was observed a long time ago by Landau and Lifshitz (1977) at a fundamental level that a problem cannot be solved in quantum mechanics unless its solution in classical mechanics is also known. The dynamics of the few-electron problem in quantum mechanics are unsolved. A further complication is that, for systems with N > 2 particles, classical motion becomes a nonlinear problem, so that irregular or chaotic orbits may appear simply as a result of the increase in the number of degrees of freedom. Since free atoms with many electrons are the ideal quantum analogue of such a classical system, an interesting (and unsolved) question arises: Do correlations involving many electrons trigger something similar to chaotic behaviour in the spectral response of many-electron systems? Such behaviour, if observed, would clearly signal a new form of breakdown of the independent electron approximation and would also qualify as a cooperative effect. This chapter addresses the issues just raised and attempts to illustrate by experimental examples many-body effects that signal pronounced departures from the independent particle approximation. There are certain aspects that the present chapter will not address. Beyond a brief discussion, cooperative motion involving only two electrons (as reviewed, for example, by Berry, 1986) will not be covered. Attempts to expand the N body problem in terms of a two-electron basis (Sinanoglu, 1960, 1961) do not seem so far to have been particularly fruitful. It seems fair to comment that helium is not the hydrogen of correlations and that the ideas needed to understand and to classify doubly excited states in He (Herrick, 1978) translate even less into more complex spectra than do the properties of hydrogen for the development of the independent particle picture. Likewise, the double excitation, double escape, and Wannier ridge problems (Wannier, 1953; Fano, 1976) are interesting as a manifestation of the breakdown of the independent particle model but are not central to the subject of multielectron correlations. We shall concentrate on the case where an atom contains several subshells. In other words, we are interested in the atom as a many-body system rather than in the long-range correlations of the three-body problem.

II. Many-Body Effects and the Conservation of Angular Momentum A historical approach to the subject of cooperative effects in light atoms (specifically, in two-electron systems) has been given by Berry (1986), who has attempted to project out from the best available correlated wavefunctions for systems with two outer electrons, probability distributions that would somehow convey the nature and symmetry of their correlations.

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The origin of the subject can be sought in an early controversy between Bohr and Langmuir: when setting up the aufbau principle, Bohr (1913) took the bold step of supposing that the angular momenta of individual electrons could be quantised. It was remarked by Langmuir (1921) that this assumption was unlikely to be quite correct since, in many cases, electron-electron interaction would be significant, so that it might be more reasonable to quantise, e.g., the angular momentum of a pair of electrons. In his own words: Bohr assumes that the angular momentum of every electron round the centre of its orbit is h. However, it is an attractive hypothesis to assume that, in the case of coupled electrons, the quantum theory is concerned not with the angular momentum possessed by one electron but rather with the angular momentum which, by being transferred from electron to electron, circulates in each of two directions about the nucleus.

Various models for helium were developed using the old quantum theory. (See, e.g., Kramers, 1923.) However, Bohr’s early point of view soon predominated even for many-electron systems. At the outset neither Bohr nor Langmuir could have suspected the reason why Bohr’s approach was successful, since it really emerged from the new wave mechanics: by a happy coincidence, the existence of closed shells, the Pauli principle, and the mathematical simplicity that the biaxial theorem introduces in sums of spherical harmonics all conspire to save the central field approximation, within which the conservation of angular momentum is the natural approximation. In this respect, the quantum many-body problem is a good deal simpler than any classical analogue. However, Langmuir’s (1921) approach was indeed a sensible one: when we examine how many-body effects can arise even as a breakdown of the independent particle model, the most likely cause is a breakdown of the angular momentum labeling for individual electrons. Formally, this is how configuration mixing and two electron jumps are introduced in the standard atomic theory. Recently, there has been a great revival of interest in the study of the dynamic aspects of two-electron correlations. It has borne fruit: much progress has been made in the search for new quantum numbers to describe the two-electron system (e.g., Lin, 1974; Herrick, 1978; Berry, 1986). The essential conclusion of the work of Berry (1986) and others is that the electronic states of atoms with two outer electrons in a closed subshell exhibit collective rotations and vibrations rather than predominantly independent particle behaviour. Thus, the low-lying excited states of helium and the alkaline earths resemble collective rotor-vibrators. They are not the quantum analogue of solar systems. The purest independent particle system, hydrogen, possesses +fold degeneracy of its levels because of symmetry under the O(4) rotation group. The independent particle symmetry of the two-electron problem would therefore be

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O(4) x 0(4),but this is broken by correlations. When electron-electron repulsion is included, the appropriate symmetry turns out to be O(4) with an additional invariant (A, - A2I2,where A, and A, are the Runge-Lens vectors of the two electrons, i.e., vectors proportional to the Kepler ellipticities of their orbits, and pointing in the directions of their semimajor axes. The invariance of /A, - A$ signifies simply that the ellipses are “locked,” i.e., that the orbits precess together (Herrick and Sinanoglu, 1975). This is an example of cooperative motion. It was also noted (Nikitin and Ostrovsky, 1976, 1978) that the principal quantum numbers n , and n2 must be the same for correlations to be strong, because, if n, is very different from n 2 , one simply reverts to the independent particle model. Thus, electrons from the same shell are the most highly correlated. It is also found that valence and inner valence electrons are more highly correlated than inner electrons. This arises because they are held out from the deep core region and can never achieve a high kinetic energy. Consequently, their motion is much affected by electron-electron repulsion within the subshell. There has been much interest in manufacturing double Rydberg states with n, n2 that would exist as large, highly correlated states of the atom. This leads to the double-excitation/double-escapeproblem and manifestations of the longrange correlations of the three-body system which, as stated in the introduction, are not part of the main thrust of this chapter. How much of what has been said in this section concerning cooperative motion can be transferred to the many-electron problem is an open question. There can be little doubt that progress in extending the considerable increase in our understanding of two-particle collective motions to the full N-electron atom would constitute the most significant breakthrough in atomic physics since the work of Hartree. There might also be important implications for theoretical chemistry: the electronic shells of some atoms (as remarked by Berry, 1986) might possess an internal geometry that would then only need to be rotated in order to form directional bonds.

-

III. Rydberg Series We are primarily concerned with excitations in many-electron atoms, and we must first decide what the spectral signature of a clean independent electron system is likely to be. The one-electron atom has the signature En = E , - R/n2, where E, and R depend only on fundamental constants. When we move to a many-electron system, the following correlations are required: (i) There is no longer a single ionisation potential E,, i.e., several series limits occur for excitation from the ground state. (ii) These series limits no longer depend only on the fundamental constants, but become properties specific to the atom (dressing

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

33 1

of the quasi particle called the vacancy), whereas only small numerical corrections to R are required. (iii) Finally we introduce a quantum defect that appears as a new quantity in the Rydberg formula and that is approximately constant for all the excited states nl in which 1 is constant. These Rydberg states form the discrete part of a single-excitationchannel. The quasiconstancy of the quantum defect for a given angular momentum I signals that, to a good approximation over most of the space occupied by the excited electron, angular momentum is conserved. It is intuitively obvious, and is established by a beautiful mathematical theory (Seaton, 1966a), that the region of conservation of angular momentum lies outside a core, and that the region of breakdown in which the electron is scattered by many-body interactions lies inside the many-electron core, where the potential is non-coulombic. The origin of the quantum defect p is well understood, and the mathematical formalism for it has been reviewed so often in the literature (see, e.g., the elegant and pedagogical review by Starace, 1976) that there is no point in repeating it here. Rather, it may be useful to extract some simple physical arguments that are relevant to our theme. The near-constancy of p for a given 1 can be traced back (as remarked by Mulliken, 1964) to the recapitulation of atomic orbitals, i.e., to the fact that the nodes of the wavefunctions, as n increases, tend to repeat at nearly the same radial distance from the nucleus. This distance is modified in the presence of a core of finite radius r, , when a matching condition of the coulombic (large r) and core (small r ) wavefunctions is applied at r,, and thus the quantum defect p can also be interpreted as resulting from a phase shift, generalised to bound states of high n. It is then a fundamental result of quantum defect theory that this phase shift v p joins smoothly to the phase shift 6 of the adjoining continuum states. Taken together, the Rydberg series and its adjoining continuum form a whole, described as an excitation channel. It will emerge later in this chapter that continuity across the threshold (first noted by Sugiura, 1927, for hydrogen, and extended by Gailitis, 1963, and Seaton, 1966b, to other atoms) is a fundamental property, valid even when the constancy of the quantum defect breaks down: if a channel is perturbed or distorted by many-body interactions, oscillator strength may actually be transferred across the threshold, from the continuous to the discrete part of an excitation channel or vice versa. This principle has been verified experimentally even for a situation where a resonance straddles the threshold (Connerade el al., 1988). Because of (i) and (iii) together, it is also possible to find situations in which two channels will overlap significantly in energy. Elementary perturbation theory tells us that the coupling between two channels increases. dramatically if the difference in energy between the channels is small. If the electrons move nearly independently of each other, we expect that the two quantum defects p , and p2 remain nearly constant. (See the dashed lines in Fig. 1.) However, if the electrons are not independent, the individual quantum defects cease to be constant

332

Connerude 1

-

h

71

.

Z O

v CI

s

0

0.5 1.0 h/z(mod 1) FIG. I . A two-dimensional plot of quantum defects representing the interaction between a doubly excited and a core-excited channel of Yb I. In the absence of any coupling between the channels, the quantum defects would be constant and follow the dashed lines. Correlations are responsible for the avoided crossing.

and, in cases of moderately strong coupling involving just two channels, can result in the simple graph shown in Fig. 1. It is a major achievement of multichannel quantum defect theory that patterns of the kind shown in Fig. 1 and indeed much more complex ones (see, e.g., Armstrong et ul., 1977) are interpretable in detail using but few adjustable parameters. When p ceases to be approximately constant as a result of interchannel coupling, we are dealing with a breakdown of the independent electron approximation, i.e., the first step beyond the standard atomic model referred to previously. Quantum defect theory confines the many-body effects within a core region, but deals with excited states that lie well outside r, . Thus, the excited electron does not have much opportunity to “see” rearrangements involving many electrons. It is therefore a natural question to ask whether the many-body effects that are beginning to emerge in perturbed Rydberg series might become much more dramatic if somehow the excited state were more compact or loculised. In the language of quantum mechanics, one looks for situations in which the spatial overlap between the initial and the final-state wavefunctions is as large as possible. The hope is that such cases will favour the observation of dynamic rearrangements within the atom. Unfortunately, the near-constancy of quantum defect, which served as a “background” against which many-body effects could be recognised, then disappears from the problem. The constancy of quantum defects is definitely a feature of extended Rydberg orbitals and does not apply to compact states. Thus, we lose an important reference point when we try to make many-body effects large. The familiar Rydberg series becomes unrecognisable because the concept of a quantum defect is not truly applicable. In other words, the regularity of the spectrum no longer exists.

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

333

There are other ways in which Rydberg series may become absent from the spectra of many-electron atoms. For example, if a dramatically large resonance appears in a Rydberg channel, then the sum rule for oscillator strengths implies that oscillator strength must be redistributed from elsewhere in the spectrum: thus, the presence of a strong resonance in the continuum would imply that the intensity of the discrete transitions would be much reduced. Another form of disappearance arises through lifetime effects involving the series limit or parent ion state in cases where it lies above the double-ionisation threshold: if the lifetime is shortened by Auger decay, then once the lifetime broadening becomes comparable to the energy interval between two successive series members, all higher members of the Rydberg series are wiped out. Interestingly, it is also possible for autoionising states to be “stabilised” by interactions, so that they look much narrower (longer lifetimes) than other Rydberg members within the same channel, i.e., that they do not conform to the regular law for lifetime breadths A r = rAugcr rAu,olonlsa,,on/n*3 expected within a series of resonances. A spectacular example of this stabilisation effect is to be found in the spectrum of CaI, where interference between two different doubly excited channels leads to a strong perturbation, and since each of the excitation channels itself arises by breakdown of the 1 characterisationof the outer electrons of Ca, the mixture of the two channels implies that the one electron labels are even further corrupted. Likewise, intensities may be enhanced among upper members at the expense of lower ones, an effect that is ubiquitous in perturbed series. All the preceding effects, which lead to the spoiling and eventual disappearance of Rydberg series, are many-body effects. In a very general sense, we can regard the “quality” of Rydberg series and the accuracy with which their number is predicted by the independent electron model as the best markers for the significance of many-body effects within a given spectral range: the longer and more regular they are, the less the situation is interesting as far as extracting manybody effects is concerned. We thus arrive at the idea that non-Rydberg specfroscopy is the study of many-body effects (Connerade, 1978a, 1984).

+

IV. Non-coulombic Potentials and the Periodic Table As we have seen, the conservation of angular momentum outside the core and the occurrence of closed subshells within the core underpin the aufbau principle and are fundamental to our understanding of the Periodic Table. Since we are searching for breakdown in the independent electron scheme, we should begin by asking just how good this understanding is, and whether it really rests completely on the aufbau principle, or whether non-coulombic effects due to the presence of many electrons in the core must also be taken into account. Elementary textbooks present the explanation of the Periodic Table of the

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Elements as one of the early triumphs of quantum mechanics-a perception so widespread that one often forgets to qualify this statement, which is only strictly applicable to the first three periods. The aujbuuprinzip is usually attributed to Bohr, who actually divided the electrons equally among the subshells. However, it was really Stoner (1924) who first put the appropriate number of electrons in each subshell and arrived at the correct structure. The Bohr-Stoner principle was of course worked out before modern quantum theory, and so the notions of an independent particle model, with a static mean central field and the use of spherical harmonics to represent closed shells, were incorporated later. Even so, the argument neglects important features of the radial equation and only works properly for the first few rows, where the ordering of the levels in the many-electron system is not too markedly different (apart from the lifting of 1degeneracy) from that of hydrogen. Beyond argon, matters become more complicated, and the qualitative, semiclassical notion of screening that has to be introduced as an ad hoc supplement of the aufbau principle (see., e.g., Herzberg, 1935) is quite unsatisfactory, as was first noted by Fermi (1928). The elements of a proper understanding were advanced by Goeppert-Mayer (1941) following Fermi’s (1928) suggestions, but the numerical methods then available did not suffice to follow the explanation through convincingly, and it was only with the progress made possible by the advent of fast computers that a quantitative explanation became possible (Griffin eral., 1969, 1971). The crucial point is that the 1-dependence of the radial Schrodinger equation manifests itself in subtle ways that can be understood by quantum mechanics but not by semiclassical models based on progressive screening. Interestingly, it is a manifestation of elementary quantum mechanics (the filling properties of shortrange asymmetric potential wells) that governs the sudden d and f subshell contraction, and it is the non-coulombic nature of the radial potential (i.e.. a manybody effect which can partly be “absorbed” into a static mean field) that controls the filling of the long periods. The basic physics of the lanthanide contraction is associated with the emergence of a double-well effective radial potential due to the combined effect of a non-Coulombic core and the positive, repulsive centrifugal term in the radial Schrodinger equation. This term has the form l(1 l)h2/2rnr2.If we consider, first, the case of hydrogen, it is clear that the potential at large r must be dominated by the attractive coulombic part which falls off only as llr, whereas the potential at small r is dominated by centrifugal repulsion. Thus, the high 1 states in hydrogen live in a potential well with a more or less hard center depending on the value of I, which is why the f electrons are held well away from the nucleus. All atoms resemble hydrogen at large r (i.e., outside the core radius r,-this is the basis of the quantum defect theory discussed in Section III), and this accounts for the outer well of the 1 = 3 double-well potential in the lanthanides.

+

335

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

As the electron penetrates further inward (inside r o ) ,the screening due to the core electrons becomes less effective, and there is a very rapid increase in absolute magnitude of the attractive Coulombic term, so that the effective potential becomes binding again. At very small values of r, the repulsive centrifugal term must eventually win again, since it depends on l/r2, and this accounts for the inner well of the double-well potential. Asymmetric short-range wells have very different properties from long-range wells: the former may be completely empty of states and will suddenly acquire bound states, one by one, as the binding strength of the well is progressively increased. The number of bound states is always finite. This is a purely wave mechanical effect, not a consequence of semiclassical screening theory. Longrange wells, on the other hand, always contain an injinite number of Rydberg states. The atomic double-well potential contains both, separated by a potential barrier. As the atomic number is increased, a state may thus suddenly drop from the outer into the inner well, where filling is energetically very favourable. The Mayer-Fermi theory of the lanthanide contraction explains (i) why the filling of the long periods occurs out of the expected order for quite high 1 values (1 = 2 and 3), (ii) why the effect is sudden and depends critically on atomic number, (iii) why this filling, when it occurs, happens deep inside the core rather than in the outer reaches of the atom, and (iv) why there is s-d competition, and so on. Since the Mayer-Fermi model is covered by earlier reviews (Connerade, 1978a, 1984; Griffin et al., 1987), we concentrate here on aspects that are relevant to possible breakdown of the quasi-particle approximation, i.e., to the characterisation of vacancies. As an example of the problems that can arise, consider the ionisation of atoms in the long d periods of the periodic table. From available information on the ground states of the atoms and ions, we can construct a table of configurations (Table I). It is a corollary of the aufbau principle that, if one electron is removed from TABLE I TABLEOF GROUND CONFIGURATIONS OF THE ATOMSA N D IONS IN THE FIRSTLONGPERIODS OR TRANSITION ELEMENTS Ca sc Ti V Cr Mn Fe

co Ni cu

zn

. . . 4s’ 3d4’ 3d24s2 3d34s2 3d54s 3d54sZ 3db4s2 3d84s2 3d84s’ 3dIU4s 3dI04s2

. . .4s 3d4s 3d24s 3d4. . . 3d5. . . 3d54s 3d64s 3d8. . . 3d9. . . 3dI0 3d1°4s

Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

. . . 5s’ 4d5s2 4d25s’ 4d45s 4dS5s 4d55s2 4d75s 4d85s 4 d i 0 .. , 4dL05s 4dIU5s2

. . . 5s . . . 5s’ 4d5s 4d4. . . 4dS.. . 4d55s 4d7. . . 4ds . . . 4d9. . . 4diu 4d%

Ba La Hf Ta W Re 0s Ir Pt Au Hg

. . . 6s’ 5d6s2 5d26s2 5d36s2 5d46s2 5d55s2 5d66s2 5d76s* 5d96s 5dI06s 5dlu6s2

. . . 6s . . . 5d2 5d6s2 5d16s 5d46s 5d56s 5d66s 5dY 5d10 5dI06s

Connerade

336

the atom to produce an ion, the ordering on the nl levels should remain the same in the ion as it is in the atom. Table I, which of course is based on experimental information, shows clearly that the ordering for atoms and for ions is not always the same, i.e., that the periodic table for ions is not, in all cases, simply obtained from the one for atoms. There are four elements (V, Co, Ni, and La) from which it is impossible to remove one electron leaving the ion in its ground configuration. Quite clearly, if we accept the nl independent electron labels as meaningful, excitation of one electron cannot occur without a rearrangement of the system. In fact, it is pretty clear that the individual nl labels are not very meaningful in these cases and that the independent electron description is not satisfactory. Situations similar to the one just described can be created by inner-shell excitation: with an inner shell excited, barium resembles lanthanum as far as its outer electrons are concerned. Consequently, they seek to arrange themselves in a configuration involving 5d electrons, such as 5d6s or 5d2rather than remaining 6s2 as in the ground state. In other words, a simple, one-electron excitation scheme becomes impossible. Suppose we look at the resulting ion state (series limit) rather than at the full Rydberg series, for example, by using photoion spectroscopy rather than photoabsorption. We then find that the situation is much as shown in Fig. 2: each channel excited results in a single line, and there are essentially three possible situations. (a) The incident photon may only possess enough energy to excite one electron. This is optical spectroscopy, and is unlikely to yield much information on many-body excitation, since the energy is insufficient to drive Electron Affinities

Valence

Subvalence

~

Core

Photon Energy FIG. 2. (Schematic) The difference between spectra due to core excitation, subvalence excitation, and the excitation of optical electrons as it emerges from graphical solutions of the many-body Dyson equation. (See Connerade, 1984, and references therein for further details.)

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

337

cooperative modes. Only one line is observed, corresponding to the excitation of one particle. (b) The incident photon may possess enough energy to excite the deepest shells. This is X-ray spectroscopy and is, again, unlikely to provide much information on many-body effects. The reason here is that the initial state wavefunction is much smaller than that of the final state, so that the electron escapes very fast, leaving behind a well-characterised vacancy from which it is screened by all the intervening shells. Experimentally, one sees a single line, surrounded by a few satellites, which are a measure of the small breakdown of the quasi-particle approximation. (c) The incident electron may possess sufficient energy to excite a subvalence electron, but not enough to excite a deep shell. In this range, the spatial overlap between the vacancy and the excited-state wavefunction is greatest. Rearrangements can occur while the electron is escaping, and instead of a single line one then finds a jumble of features into which the quasiparticle line has fragmented. The preceding description is backed up both by experiments and by calculations, which all points to the fact that the vacuum ultraviolet and soft x-ray ranges of the electromagnetic spectrum are ideally matched to the study of manybody effects in atoms.

V. Giant Resonances The soft x-ray spectra of atoms, molecules, and solids corresponding to the transition elements and rare earths contain remarkably strong, isolated features that can be traced back right through the sequences (i.e., horizontally across the Periodic Table) past the onset of the long sequence to the corresponding rare gases, in which they appear as broad resonances in the continuum. These features possess many of the properties that have been listed previously as desirable or favourable to many-body excitation. By analogy with collective oscillations in atomic nuclei, they have been baptised “giant resonances,” and their study has been regarded as an ideal testing ground for many-body theories of the atom. The giant resonances can be understood (at least qualitatively) as arising from the presence of the short-range inner well in the effective potential of the excited electron, i.e., as a consequence of the non-Coulombic core. It is clear that, when a short-range well suddenly acquires a new bound state, it must come from somewhere. Quantum mechanics shows that it comes down from the continuum, where it exists as a resonance just before the short-range well becomes sufficiently binding to hold a new discrete state.

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Since the wavefunction of the excited electron can become resonantly localised in the inner region, the electron takes longer to escape than if it were promoted to a pure continuum state in a Coulombic potential, and it therefore can witness rearrangements within the atomic core. In fact, the spatial overlap between the initial- and final-state wavefunctions can be so large that the oscillator strength sum rule is exhausted within the broad feature, which therefore deserves the name giant resonance. When all the oscillator strength is gathered up into this resonance, no Rydberg series is observed in the same channel, i.e., the discrete part of the spectrum is quenched. When the sum rule is exhausted, this means that, for the 4d + ef resonance in La, for example, integrating over the feature yields an oscillator strength of 10, corresponding to the presence of 10 electrons in the subshell. In this sense, the whole shell is involved in the excitation. It is also the case that calculations of these features in Xe, Ba, and La become extremely sensitive to the mode of approximation used. In the many-body perturbation theory (MBF’T) applied to atoms (Kelly, 1964, 1966), an approach in which the many-body perturbation series is treated order by order and all the different interactions, represented by their Feynman graphs, are included for each order and evaluated on a complete independent particle Hartree-Fock basis of states, only a finite number of orders can be included for practical reasons. In the random phase approximation with exchange (or RPAE), a certain class of diagrams (the forward “bubbles” and their exchange counterparts) can be summed to all orders algebraically, but all diagrams cannot be included. It is also possible to perform calculations with suitably tailored Hartree-Fock potentials tuned specifically to obtain agreement in this excitation channel. All these many-body approaches lead to results significantly different from each other and very different from the configuration average approximation. The many-body theories are in broad agreement with experiment (see Fig. 3) in the critical example of Ba as well as in other nearby element such as Xe and La. Unfortunately, the situation is not simple and it is not the case that a glance at Fig. 3 will immediately allow one to distinguish which is the best theory, because there are other variables (partial cross sections, angular distributions of photoelectrons, photoion production rates, etc.) as well as many other atoms that exhibit many-body effects. Also, the experimental data in Fig. 3 were not obtained for free atoms, but for a solid in which structure due to reorganisation of the 6s2 electrons (see Table I) is not important. When an inner-shell vacancy is created in Ba I, many complex mixing effects involving the outer electrons occur and these are not included in the theory as of early 199 1 . There is, however, a purely experimental approach that can be used to estimate the importance of correlated motions in giant resonances, and this consists in studying the relative production rates of different stages of ionisation as one tunes

339

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

/ 100

110

120

130

140

150

w (eV) FIG. 3. Comparison between different many-body calculations (each model has its own acronym) and experimental data for the 4d + Ef giant resonance of Ba I. Note that the experimental data used by the theorists are in fact for polycrystalline Ba, which is preferred because the complex doubly excited structures that interfere with the giant resonance cannot be readily included in the theoretical model and are absent from data for the solid. The origin of the second bump on the experimental curve is unclear. It might be due to backscattering of the photoelectron in the solid. (a) The comparison published by Kelly er u / . (1982). (b) The comparison published by Amusia et a/. ( 1990a).

through a giant resonance. Figure 4 shows some examples taken from studies by Zimmermann ( 1989) in Berlin. What emerges from these studies is that the 4d --f Ef giant resonances behave differently at both ends of the 4f sequence. Near the beginning of the sequence (around Ce), ionisation is dominated by the production of triply charged ions,

Dv

I

It

h

$

n L KJ

v

b

30 40

o(4F)

n E

30

20

o(4F) o(4D) o(5P) 45s)

o

+ 0

v

b

30 10

0

90

100

110

120

130

energy [eV]

140

150

80

100

120

140

160

180

energy [eV]

FIG. 4. Photoion and partial yield cross sections for (a) Ce and (b) Dy after Zimmermann (1989).Note that, in (a) multiple ionisation dominates, whereas single ionisation dominates in (b), and that this is accompanied by a qualitative change in the shapes of the profiles from the typical form for a giant resonance to a Beutler-Fano autoionising profile. It is also interesting that all the partial cross sections are enhanced in the vicinity of the giant resonance.

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

34 1

but as the atomic number increases, the level of doubly charged ions becomes more important (at Dy), until, by the time one reaches Dy, the production of singly charged ions is the dominant process, as is to be expected in photoionisation if the independent-particle model is valid. An interesting and very suggestive change is apparent in these spectra, which had also been noticed by Radtke (1979) in earlier photoabsorption experiments. The characteristic asymmetric profile of the giant resonances (tall asymmetric features with a low energy minimum tied to the 4d thresholds and a long tail toward high energies), which can be parametrised by a simple model based on scattering theory (Connerade, 1988) occur around Ce but gradually turn beyond Dy into a typical autoionising profile (Fano, 1961) with a pronounced minimum adjacent to the resonance energy, characteristic of quasi-discrete state decaying into a photoionisation continuum. Thus, the experiment signals clearly that the cooperative effects are most significant toward Xe, and that, as the inner well becomes more binding with increasing atomic number and the Ef resonantly localised continuum state gradually turns into a 4f resonance, its cooperative nature dissipates and it becomes a more normal singly excited feature. In some ways, this picture is quite satisfying, but when one carefully examines the data for Ce (Fig. 4), one may wonder why, if multiple ionisation is to be the marker for cooperative behaviour, quadruple ionisation is not more significant. The answer seems to be that the outermost three electrons of Ce (with ionisation potentials of 5.5387, 10.85, and 20.198 eV, respectively) are fairly easy to shake off, that the fourth, at 36.758 eV, is significantly more difficult, but that shaking off any more than four requires one to break a closed shell open and is energetically inaccessible at primary photon energies of around 120 eV.

VI. Atomic Giant Resonances in Other Environments-Controlled Collapse and Instabilities of Valence As we have repeatedly stressed, one of the important criteria for giant resonances is that the spatial overlap between the initial and final states should be very large, so that the transition takes place well within the atom. This criterion was arrived at by our desire to favour collective processes as much as possible. There is, however, another consequence of the strong localisation of the excited states in giant resonances, which is the survival of the spectral features in a variety of different environments. If we consider a given atom A, then a molecule involving A or a solid composed of atoms A or containing an impurity A or indeed a surface with atoms A upon it can all be expected to

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exhibit a strong feature at the energy of the giant resonance, the spectral profile of which may exhibit distortions reflecting the change in the environment of the atom. Potentially, this is a powerful probe of different phases of matter, and is important also as a bridge between disciplines. Readers interested in this aspect of giant resonances will find the subject well covered in the proceedings of a summer school devoted to this subject (Connerade et al., 1987). In molecules, if we imagine atom A surrounded by a symmetric array of identical, electronegative halogen atoms (a halogen “cage”) that tend to draw electrons away from the nucleus (Dehmer, 1972), then some control over the balance between the inner and outer wells of the double-well potential can be achieved by altering the molecular environment (e.g., using different halogens). This can lead to changes in both width and energy of the giant resonances, and can even lead to sharper resonances than in A, as occurs for 4d + Ef excitation in LaF, (Connerade et al., 1980). Returning to the description of the Mayer-Fermi model of the lanthanide contraction, we may ponder whether the binding strength of the inner well could be altered in order to control the transfer of the electron from the outer to the inner well, and indeed whether an electron could remain poised, with part of its wavefunction in the inner and part in the outer well. This problem is referred to as conrrolfed collapse (Connerade, 1978b,c). In principle, many different external fields can be used for this purpose: laser excitation of outer electrons has been suggested and would enable the finest control, but laser ionisation (Lucatorto et al., 1981) and even the properties of the chemical environment (Maiste et al.. 1980a,b) have been experimented with to good effect. There is a single naturally occurring instance in which the nf electron possesses a bimodal wavefunction with a comparable probability of being found in either potential well. This situation has been analysed by Connerade and Mansfield (1975). (See also Section IX in this chapter.) Under these circumstances, all the oscillator strength is not concentrated into a single feature, and a Rydberg series re-emerges, but with a grossly distorted trend of the quantum defect, because recapitulation of the atomic orbitals breaks down. Connerade (1982) has developed quantum defect theory for a double-well potential by exploiting effective range theory, and has shown that the function coth(1 - p ) rather ~ than p itself should be plotted against energy so that the threshold relation between the phase shift in the continuum and the quantum defect in the bound states can be generalised to take Levinson’s theorem into account for the short-range well. Difficult situations arise when the atomic orbital is on the verge of collapse or is only just collapsed. In such situations, term sensitivity can be large, and it is not even certain whether the self-consistent field method is entirely reliable or whether two viable solutions might not exist in some cases, as argued by Band and Fomichev (1980). If so, one needs to decide whether they both have physical validity (which is quite conceivable in connection with intermediate valence) or

343

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

whether they are artifacts of the numerical approach. This rather speculative question is discussed by Connerade and Karnatak (1989). Yet another kind of molecular giant resonance can be envisaged that does not rely simply on excitation in an atomic subshell, but involves excitation of electrons from valence shells. Thus, the molecules NZ,CO, NO, and 0, have been considered from this point of view by Lefebvre-Brion (1987), who plots the relationship between the width r and the resonance energies, and finds a behaviour similar (but not identical) to the universal curve for atoms (Connerade, 1984). As she emphasizes, the molecular nature of the resonances emerges from the fact that no resonance at all occurs in the united atom cases. It seems unlikely (see also the discussion in Section VII) that such excitations could lead to pronounced collective modes when the fields are not spherical. There have also been reports of giant resonances in large molecules (see, e.g., Brint er al., 1985) and the general question of whether such features might generally occur in even larger systems of appropriate symmetry (such as cyclopropane, cyclohexane, neopentane, or uranium hexafluoride) has been considered by Robin (1985, 1986). It is likely that each case should be examined separately, since the conditions relating to stability can vary widely and the mechanism involved can be purely molecular in some cases. In solids, giant resonances play an important role as they are often to be found in X-ray spectra and their existence underpins such useful practical tools as resonant photoemission. Even in such apparently unrelated areas as the photonstimulated desorption of hydrogen from surfaces (important to both hydrogen storage and catalysis), quasi-atomic giant resonances are to be found. Also in the condensed phase, an interesting situation exists for d f excitation in the lanthanides, where instabilities of valence may occur that can be probed directly by the excitation of giant resonances, since they relate to the number off electrons that are localised, which may depend on the environment. This is usually discussed in the context of Anderson localisation, but quasiatomic models have also been proposed in which the short-range inner part of the double-well potential, which survives in the solid, plays a crucial role. Experiments on clusters have been attempted, in which the instability of valence is tracked from the atom to the condensed matter limit by varying the size of the clusters. Readers interested in this subject may wish to read Connerade and Karnatak (1989). Finally, one should remark that many of the ideas involved in giant resonances are closely related to models used in the physics of high-density plasmas to describe the electron spectrum near the continuum. The problem here is to treat the energy levels of a disordered system. In one category of models, the central field approximation is used, with the ion represented as a point charge at the center of a spherical cavity in an otherwise uniform positive-charge distribution. A Thomas-Fermi model (More, 1982) or a model based on the Dirac equation

-

344

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followed by suitable averaging (Liberman, 1979) can be used. One finds narrow resonances near the base of the continuum (especially for large angular momentum t; the resonances at higher energy are broad and eventually become indistinguishable from the continuum. More elaborate models treat a small cluster of nearby atoms immersed in the plasma background by multiple scattering theory. The subject is reviewed by More (1982).

MI. Giant Resonances in Nuclei and in Atomic Clusters At this point, it is useful to consider the subject of giant resonances more generally, nor merely confining the discussion to electronic shells of atoms in diverse environments, since collective oscillations occur in a wide variety of situations, some of them indeed more favourable than single atoms to the excitation of collective modes. Historically, the earliest examples of giant resonances were found in nuclei, where the first use was made of dynamic collective models for the interpretation of resonances of sperical shells. The resonances were observed by Baldwin and Klaiber (1947, 1948) in photoabsorption experiments to study induced radioactivity using gamma rays of around 17 MeV, when cross sections about two orders of magnitude higher than expected were discovered. Giant resonances in nuclei are attributed to collective oscillations of the protons against the neutrons. Several modes are found, the most important ones being the giant electric dipole deformation, the giant quadrupole surface oscillation, and the giant monopole breathing mode. One also distinguishes between the protons and neutrons vibrating together (isoscalar mode) or in opposition (isovector mode). Absorption experiments usually detect giant dipole resonances which are the purest case of a giant resonance and will also be the situation of interest here. The subject of giant resonances in nuclei has a long history which was reviewed by Berman and Fultz (1 975). The nuclear dipole resonances exhibit single (for spherical nuclei) or double (for spheroidal nuclei) peaks depending on the forms of the shells and are seen even for very small nuclei such as 4He. It is also interesting that single-particle (or, in the language of many-body theory, excizonic) structures occur, giving sharper peaks that are superposed on the broader collective resonances and may interact with it. A closely similar situation for atoms has been considered by Connerade and Lane (1987) within the framework of K matrix theoryoriginally developed for the interpretation of nuclear resonances. The analogies between the fields are therefore very clear. The damping mechanisms of the collective resonances in nuclear physics have also been studied: they decay mainly through Landau damping and coupling to nuclear shape oscillations (Bertsch et al., 1983).

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

345

Thus, giant resonances in atomic nuclei occur at energies of around 20 MeV and in electronic shells of atoms, at energies of around 100 eV. The former are held together by a very short range potential, leading to a shell structure and resonances well accounted for in the shell model of Nilsson (1955), while the latter exist in a hybrid, double-valley potential which is short-range in the inner well but long-range at large radii. In the former case, different modes have been detected corresponding to departures from sphericity of the oscillating shell. This has not been found in atoms, for reasons that will emerge more clearly below. A completely new class of giant resonances has been discovered in atomic clusters. They occur in nearly spherical metallic clusters, typically those formed by alkali atoms, which are found to be particularly stable around the magic numbers 8, 18,20, 34,58,92, etc. This sequence of numbers is well accounted for (see, e.g., de Heer et al., 1987) within a spherical jellium model (Chou et al., 1984) or by using self-consistent pseudopotential local density functional calculations (Martins er al., 1985). an approach closely related to the very successful many-body methods developed by Zangwill(l987) for the calculation of atomic giant resonances. One can also start from a simple model such as the Woods-Saxon potential (Woods and Saxon, 1954), which Knight et al. (1984) used in their early work on shell closure in spherical clusters. Thus, stable closed shells of delocalised electrons have been shown to exist in certain clusters, reminiscent of the shells in nuclei and in atoms, but on a yet much larger scale. Just as the closed shells in nuclei or in atoms may exhibit a collective mode, so they do in near-spherical clusters (Hecht, 1979; Kreibig and Genzel, 1985; de Heer et al., 1987), and observations of the resulting giant resonances have been extended to free clusters (Selby et al., 1989) which may even be mass selected (Brbchignac et al., 1990). Experimentally, the giant resonances found by laser spectroscopy in mass selected alkali clusters are among the most impressive examples of collective oscillations yet discovered. Indeed, this is a very clean situation, not far short of the ideal as far as characterising collective resonances is concerned. The cross section rises to an extremely high value within a rather narrow resonance (see Fig. 5 ) which dominates the response, and detailed spectroscopy shows no sign of excitonic (single-particle) structures within the range of the resonance. Since the giant resonance is narrow, interference between different modes of oscillation can sometimes be observed, and is traceable to ellipsoidal modes when there is a slight distortion from spherical symmetry, a situation also encountered in nuclear physics. The work of Selby et al. (1989) shows that the closed-shell clusters exhibit a single plasmon resonance, while, for open-shell clusters, the resonance splits up into several peaks. This fact, added to the interference between the different spheroidal modes observed by Brbchignac et al. (1990) and to the general increase in spectral complexity with departures from the spheroidal model (which can be studied within a simple harmonic oscillator model by introducing at least

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FIG. 5. Giant collective resonances in the photoevaporation cross sections of the clusters (a) K; and (b) KL after Brbchignac er al. (1990). Note that, although the resonance frequency is not much affected by size, the cross section is enormously increased and is essentially the area presented by the cluster.

two oscillation frequencies, R,, and Q, along or perpendicular to an axis of symmetry; see the treatment by Clemenger, 1985) shows that the strong, isolated collective resonance is a property of spherical systems, a fact that will be relevant to the discussion of recent results in strong laser fields. (See Section X.) Clearly, this situation is very similar to the one in atoms. It is incidentally of some interest (but does not yet seem to have been investigated) that there exists a theoretical upper limit on the size of a convex spherical cluster composed of absolutely equivalent atoms. It is a general result of group theoretical geometry (see, e.g., Coxeter, 1961) that not more than 20 objects can be arranged on a sphere as vertices of a regular solid with all faces identical (the regular dodecahedron that is the largest of the five platonic solids). If one relaxes the requirement that all facets be identical, then an Archimedian polyhedron of 60 vertices (the 60-hedron or soccer ball) is the largest regular geometry possible (Coxeter, 1973). Above this number of atoms, the giant resonance might break up into modes indicative of the cluster acquiring a nonspherical shape, which would be an elegant way to demonstrate this upper limit experimentally if a

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

347

sufficiently good mass selection were possible. For very large clusters, there is of course the possibility of superstructures, e.g., a 60 sphere of 60 atom clusters, etc. A simple model for the collective resonances is readily constructed in terms of the Mie solution for an oscillating conducting sphere driven by an electromagnetic field (Mie, 1908). For free particles arranged in a sphere, the photoabsorption cross section has a simple damped resonance form,

where the resonance frequency w,, , also referred to as the classical Mie frequency or plasmon frequency, is the frequency of propagation of a surface plasma wave on the sphere of N free electrons, and the resonance arises from an oscillation of the spherical cloud against a fixed background of positive charges. In the case of clusters, the valence electrons oscillate against the background of the positive ions. The theory is based on using the Drude dielectric function, namely

where E is the complex dielectric constant and relies on the assumption that all of the available dipole oscillator strength is exhausted by the single resonance at w". This is known as the surface-plasmon pole approximation (Eckart, 1985). For conducting spherical clusters, the Mie theory is extended by setting w; = N e 2 / m , a , where (Y is a static polarizability (equal to the particle radius cubed in the bulk limit, to be discussed shortly) that needs to be measured or calculated in order to determine wo. (Near-spherical geometries can be taken into account in the manner of Clemenger (1985), and there are often two nearly degenerate frequencies corresponding to oscillations of a prolate or oblate spheroid.) The damping constant r is then the only free parameter, and has been found both to be small and not to vary much with cluster size. For a classical monovalent conducting sphere, (Y = Nr,3 where r, is the WignerSeitz radius. An estimate for clusters of Na yields a wavelength A, = 3650A for r, = 4 a.u. A basic assumption of the Mie (1908) model is that (a) the driving field should be spatially homogeneous over the sphere, i .e., that 2rkrs A

-<< 1

(3)

where k is the absorption coefficient, and (b) the time taken for the sphere to acquire a given polarisation should be very short compared to the driving field, i.e., that r,n 1 << c

w

(4)

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where n is the refractive index. One thus requires that the wavelength of the radiation be much greater than the size of the sphere. For clusters and the number just given, the condition is clearly well satisfied. For nuclei, with frequencies in the tens of MeV, one has wavelengths of around 100 x 10-l5 m which are also much larger than nuclear shell sizes. The interesting point is that for atomic giant resonances in the 50-100A range, ht2n becomes not much greater than the dimension of the object so that the approximations involved in the Mie ( 1908) theory are not strictly applicable. By analogy with nuclear physics the many-body formalism known as the RPA (random phase approximation) has been extended to clusters (Brack, 1989). However, the example of atomic physics suggests that electron-exchange terms are an important correction and will need to be included, much as for the p + d and d + f resonances. In summary, giant resonances in nuclei occur around 20 MeV and in clusters, around 2 eV. Narrow, isolated resonances are characteristic of near-spherical systems. In both nuclei and clusters, the resonances are well characterised (weakly damped) and, although collective in character, can be described using a pseudopotential that is of short range (typically Woods-Saxon shape). The profiles are well represented in the classical Mie formalism.

VIII. Are Giant Resonances in the d and f Sequences Atomic Plasmons? This is an appropriate point, with the examples of the previous section in mind, to discuss a question hinted at in the introduction: are the giant resonances, as exemplified by 4d + Ef excitation in the elements Xe and beyond, or again in elements near the onset of the d transition periods appropriately described as cooperative oscillations or plasmons? Various answers (mostly in the negative) have been given to this question, although some papers (see, for example, Nuroh et al., 1982) have emphasised this description. Some authors have expressed the view that if a dressed particle or pseudopotential description is appropriate, then it is unnecessary to introduce the manybody formalism in atomic spectroscopy. This debate also occurs in nuclear physics (Berman and Fultz, 1975) where (as previously discussed) a Woods-Saxon potential can be used, although the collective nature of the resonances is not seriously in question. A more interesting issue is whether the giant resonances in atoms are identifiable as a collective oscillation, i.e., a definite dynamic mode analogous to the sustained vibrations observed in clusters or in nuclei. It is of course part and parcel of collective models to calculate the dielectric response of the atom by treating the polarizability as a sum over dispersive oscillators. (See, e.g., Brandt and Lundqvist, 1967.) From the standpoint of the

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

349

RPAE many-body theory, Amusia et al. (1 976) have considered the possible occurrence of plasmons and concluded that atomic giant resonances are not plasmons by analysing the pole structure of the particle-hole effective interaction r ( w ) where w is the particle-hole interaction energy, This of course does not negate their strongly collective character as evidenced by so many calculations and by the observed photoion spectra (Zimmermann, 1989, and Section V of this chapter). The question of dynamic integrity of the plasmon oscillation in atomic shells can be settled more directly by appealing to experiment. One is at first tempted to argue that the resonance is identifiable as a collective mode simply because so much oscillator strength is concentrated into a single feature (cf. the surfaceplasmon pole approximation in the previous section) that, integrating over the resonance, one generally finds all the oscillator strength available to the N electrons of the subshell in the whole channel. This and the consequent absence of Rydberg features establish that the feature involves all N electrons, but is not enough to show that their motion is a cooperative mode. In a reply to Nuroh er al. (1982), who suggested that the giant resonance could be considered as a harmonic oscillator representing the collective motions of the whole 4d subshell centered at hwo 105 eV, Connerade (1983) gives a simple argument against plasmons by extracting the typical lifetime from observed profiles and comparing it with the typical period of the presumed collective oscillations (which is readily calculated from the observed resonance energies). One finds that the lifetime is always shorter than the period, in other words, that the oscillation is overdamped and never has time to build up. The reason one can be fairly confident that this conclusion applies generally to most atoms rather than just to one or two cases is shown in Fig. 6 which illustrates a general rule (Connerade, 1984) connecting the lifetimes of giant resonances with their resonance energies: it emerges from this rule (which is merely a statement of the uncertainty relation for virtual states above a threshold) that the only narrow resonances that might exist are those that occur just above the thresholds, i.e., that relatively few atoms (for example, Mn) are in this situation. Consequently, we can regard it as rather well established by observation that the collective mode driven by weak electromagnetic fields in atomic subshells is overdamped and cannot oscillate. This has the interesting consequence that higher harmonics of the plasmon frequency will not be observed either. Otherwise, one would have expected them to appear. We may take the argument yet a little further, again by comparison with the situation in atomic clusters, where the giant resonances are due to oscillations sustained over many periods. As we have seen, the trend in atoms is that the oscillations are more collective the broader they are (i.e., toward Xe), but become more and more excitonic (i.e., quasi-particle) in character and narrow down as the potential becomes more binding. Typical autoionisation lifetimes

-

Connerade

350 C h

.P;.

Gadolinium 4d --t f (Data for Atomic Vapour)

I

120 130 140 150 160 170 180 190 2 0

"5% Experiment 05,

a

Theory 4d

-

f

10 30 50 70 90 110 Energy Above T h r , 4 d d [c\ )

FIG. 6. Typical profile for a giant resonance. Note the characteristic asymmetry with the long tail to high energies and the minimum close to threshold, where discrete structure occurs. (b) The connection between width and energy above threshold for giant resonances. (After Connerade, 1984)

are of the order of lo-" s, whereas the lifetimes of the giant resonances in Xe and Ba are typically three orders of magnitude shorter. There is, however: one remarkable exception in the case of the 3p + 3d resonance in Mn (see Fig. 7) for which the lifetime is not shorter than the period of the electromagnetic field, a case to which we return in Section XI. The reason why the lifetimes of the atomic giant resonances are generally so short seems to be connected with the forms of the potentials. In nuclear physics and also in the physics of spherical clusters, the effective potentials in which the particles move are short-range potentials for all displacements. Indeed, the Woods-Saxon potential (see the previous section) is used in the interpretation of giant resonances both in clusters and in nuclear physics. For small disturbances about the equilibrium position in such potentials, the restoring force is strong enough to sustain an oscillation. In the atomic situation, if a single particle escapes to a large enough radius, it finds itself no longer in the region of the short-range inner well, but in the outer reaches of the atoms, where the potential becomes Coulombic

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

35 1

(long-range), and the probability of a reflection is much smaller. It is no accident that the exception just noted in Mn occurs for a resonance that lies just below threshold and is best interpreted as a bound state autoionising into an underlying continuum. In other words, because the collective resonance occurs well above the barrier between the two wells in the double-well potential, there is little chance of a sustained oscillation. Furthermore, as the energy of the resonance is raised (toward Xe, where collective effects dominate), it stands yet further above

00 7+

'

0

60-

0.X

X

4

0

y55-

0

50 45 40 35 30 25 20 15 10 -\

0-

4

50

60 65 55 Photon Energy (eV)

FIG. 7. Theoretical and experimental profiles for the 3p + 3d giant resonance in Mn (after Garvin et a / ., 1973). The solid curves are theoretical results, the lowest being for the 4s partial cross section. The dashed curves are also theoretical and the experimental data are shown as points. (For further information, see Garvin et a/., 1973.)

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the barrier. If it is lowered, the lifetime lengthens, but the collective effects also generally decrease. (Again, Mn may be an exception. See Section XI.) In a nutshell, it emerges as a property of the long-range Coulomb potential that interpretations based on effective mean fields are quite successful, and that plasmons in atomic shells possess no sustained oscillatory mode. The potential can, however, become dramatically different for different states of the atom, signalling the importance of many-body effects.

IX. Extending Mean Fields beyond the Hartree-Fock Scheme The discussion in the previous section suggests that one might also try to absorb many-body effects entirely within a potential and the question arises whether the Hartree-Fock potential is truly optimal for calculating atomic excitations (as opposed to, e.g., ground state properties). In order to test such an optimised mean field model, the most critical situation (in the light of the concluding remarks of the previous section) must be that in which the eigenfunctions extend over both the short- and long-range parts of the effective atomic potential, i.e., where they become bimodal. There is just one such naturally occurring instance in the Periodic Table, viz. the nf states of Ba', which were discussed in the context of quantum defect theory. A mean field theory of the atom beyond Hartree-Fock has been developed by Dietz and collaborators in Bonn. It is based on a derivation from the Lagrangian of quantum field theory in which all available freedom to optimise the mean field is exploited, i.e., the most general Schrodinger equation is obtained. It is interesting that the resulting differential equation is essentially different from the Dirac-Fock equation, and that they are not perturbatively connected. Bearing in mind that the first term in the linked cluster expansion is just the Hartree-Fock term, this is a genuine many-body theory. It has been the subject of a detailed review (Connerade and Dietz, 1987), to which we refer the interested reader, and merely present here some conclusions relative to double-well potentials. The bound nf states of Ba+ have been calculated within this model. Since the problem is a rather large one, it could not be tackled ab initio but was treated using a parameter g that expresses all the existing freedom in the choice of a mean field to optimise the Lagrangian of quantum field theory (QFT). In principle, it is clear that g should be determined ab initio. However, within the QFT, then it must also be true that, once g is optimised with respect to one observable, the wavefunction is automatically optimised for the calculation of another. Thus, g was optimised on the transition energy in Ba+, a procedure that could be performed exactly for each state (see Fig. 8) and another quantity-the spinorbit interaction-was then available for comparison with experiment.

-I

I

np series

112

2.5

2.0

1.5 1 .o 0.1 0.2 0.3 Binding Energy (eV)

(a) (b) FIG. 8. Quantum defect plots for the anomalous 5d + nf series of Ba'. (a) Compared with a g-Hariree plot. (SeeSection IX: and Connerade and Dietz, 1987, for details.) (b) As straightened out by a modificationof quantum defect theory for a double-weII ptential. (See Section VI and Connerade, 1982.)

3 h

I

E,

2,250 M .I

c.’ c,

.d

n

(I)

.* 200

-fl

9 .-

a

(I)

150

1oc

5c

9f

100/n*~ FIG. 9. Spin-orbit splittings calculated for the series in Fig. 8 under different models (DiracHartree-Fock, mutticonfigurationat Dirac-Hartree-Fock, and optimised g-Hartree) compared to experiment. (See Connerade and Dietz, 1987.)

:11 7f

4f

5f

6f

8f 9f

FIG. 10. Theoretical relative intensities for the series in Fig. 8. Note the anomaly that is not due to the presence of any perturber from outside the channel. (See Connerade and Dietz, 1987.)

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

355

That this procedure is highly successful is illustrated in Fig. 9, from which we conclude that all the correlations present in nf states of Ba+ can indeed be absorbed into a mean field. Calculations of intensities in the 5d + nf spectrum have also been performed that exhibit interesting anomalies (see Fig. 10) but as of early 1991 there exist no quantitative experimental data for comparison. The g-Hartree method has not yet been extended to calculations of photoionisation.

X. Can One Blow Off Complete Shells by Laser Spectroscopy? The discussion of plasmons suggests an intriguing possibility, first mooted by Boyer and Rhodes (1985), namely that, by driving cooperative motion of the outer electrons very fast using powerful oscillatory electromagnetic fields, one might excite inner shells (Szoke and Rhodes, 1986) or blow complete shells off the atom in a single step in a multiphoton excitation scheme. Such experiments must be performed using very short pulse lasers of very high power (see, e.g., Luk et al., 1985), so that as many as 100 photons are absorbed by the atom and a significant fraction of the energy might be deposited in the cooperative mode. In principle, the experiment is a very simple one, although the requirement of producing extremely intense electromagnetic fields requires the construction of extremely powerful laser sources that, when focused down, provide intensities of loi5or loi6Wcm-2. A description of the system in use in the University of Illinois at Chicago Circle is given by Boyer ef al. (1990). The intense beam of the laser is focused down into as small a volume as possible inside a gas jet, and fluorescence, electron emission, and ionisation spectra are all accessible. There are many interesting aspects of experimentationon atoms and molecules in extremely high electromagnetic fields, but we confine our discussion to the question of whether or not a collective mode analogous to the atomic plasmon which was shown in Section VIII to be overdamped at low intensities of the electromagnetic field can be driven at very high intensities. Experiments in which about 100 photons from the laser beam are absorbed have been performed in Xe, and a high abundance of six times (and more) ionised atoms have been seen (Luk et al., 1985). However, as argued by Lambropoulos (1985), one cannot be sure that the ionisation was produced by dumping the laser energy into a collective mode, because the possibility of a sequential ionisation process occurring early within the 5-ps laser pulse also exists, and because it is impossible to “turn on” the electromagnetic field instantaneously. Burnett ( 1989) has considered theoretically how collective motions of the two outer electrons in Mg could be driven by a laser field. He emphasises that the Rabi frequency must be large as compared with the detunings from the pertinent

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atomic transition widths, a condition that may or may not be readily satisfied, depending on the particular atom. However, he also notes that a collective oscillation generated in ultraintense fields will generally have an extremely short lifetime (- 10-l4 s due to photoionisation). Since there is no time resolution within the laser pulse in the original experiment of Luk et al. (1985), it is impossible to decide experimentally whether collective modes were excited in their experiment. However, Codling and Frasinski (1990) have performed experiments on the molecules HI and N, in laser fields of about lOI5 Wcm-2, as a result of which they conclude that a sequential process is more likely. The principle of their method is as follows: if several electrons are ejected collectively on a time scale of 10-l5 s or less, the interatomic separation within the molecule does not have time to change, and therefore the resulting molecular ion will fragment into products of high energy as a result of a Coulomb “explosion.” On the other hand, if electrons are ejected sequentially as the molecule dissociates, the kinetic energy released in each fragmentation is considerably smaller. Fragmentation processes take place on a time scale of about 20 X s, so that a good enough time resolution is achieved even though the laser pulse lasts for 5 ps. The experiment of Codling and Frasinski (1990) has opened up the important area of molecular fragmentation in strong laser fields. However, it may not be entirely relevant to the problem of collective oscillations in atoms, for at least two reasons: (1) As we have seen in Section VII for many different examples, it is actually important that the undisturbed shell be nearly spherical in order to drive a sustained collective mode with electromagnetic radiation; the molecular situation is quite different, especially for the outermost electrons. (2) Since a molecule can fragment on a time scale of 10-15 s or so and the time scale of the collective oscillation might be of the same order, the molecular field is changing fast with time and it is hard to see how a collective oscillation of well-defined frequency could be sustained. An additional point, made by Boyer et al. (1990), is that their experiments were probably performed at about 10 times the intensity available to Codling and Frasinski (1990). The question of the correct time scale to consider for the collective oscillations is a difficult one, but the facts appear as follows. At low field strengths, the collective mode cannot oscillate for reasons given in Section VIII, and therefore for single-photon excitation an atom is subjected to a collectively enhanced photoionisation process (with likely simultaneous ejection of several electrons) s). Note, however, that the elecon a very short time scale (shorter than trons ejected in this case are produced by shakeoff of the outer shells: we do not see several electrons blown out from the subshell originally excited, which is a subvalence shell in the experiments of Zimmermann (1989). Thus, if we are to drive a single collective mode in an electromagnetic field, thefield itselfmust be strong enough to lengthen the 1Sfetime of the collective resonance. Looked at

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

357

spectrally, this implies that the field must sfabilise the collective motion. That this indeed might occur can be surmised from the presence of sharp structure embedded in the photoionisation continuum (the so-called AT1 peaks) which imply the very stabilisation required for the absorption of many photons to occur. If similar sharp structure can be induced in a giant resonance (a problem that may be related to the interaction between giant resonances and discrete structure for atoms in low fields-see Section XI), then the time may become long enough for a collective resonance to be driven by the oscillating field. Experiments so far have been inconclusive, and we have therefore suggested an alternative approach: instead of studying molecules, an intercomparison between multiphoton ionisation spectra of I - and neutral I should be performed in strong laser fields. Such an experiment has the merit that the difference between two spectra is much more accessible to study than comparisons between experiment and theoretical expectations for atoms in strong fields. The negative ion Ipossesses a closed 5p6 outer shell and is therefore isoelectronic with Xe. However, this shell is much more loosely bound than in Xe and could be blown off with about 170 eV as against about 250 eV required to remove the six outer electrons of Xe. The importance of electron-electron correlations in the mophoton spectrum of I - was investigated theoretically by Lhuillier and Wendin (1988) because of the fact that lasers in the 12-25-eV photon range are not yet available to probe the two-photon ionisation of Xe directly. The ionisation spectrum of neutral iodine in a strong laser field was already studied by Luk e f al. (1985). However, no experiments on multiphoton spectroscopy of I- comparable to the work of Luk et al. (1985) appear to have been attempted. The singleelectron photodetachment threshold of I- lies at about 3 eV, which means that only one or two photons need to be absorbed by I- in order to produce neutral I. Consequently, in an experiment on I-, if a sequential process of ionisation is initiated, then neutral I will be produced in a laser field of almost undiminished strength, and will then experience virtually the same multiphoton ionisation processes as a beam of neutral I in the same laser field. On the other hand, if all six outer electrons that form the spherical outer subshell are driven collectively to simultaneous ionisation, then the resulting spectrum of multiply charged ions will show a quite different distribution from the spectrum of neutral I. The negative ion I- is known to possess a strong collective resonance in the 5p + Ed continuum (Amusia ef al., 1990) which lies at too low an energy for more than the double process detachment plus ionisation to I' to be enhanced by singlephoton excitation. However, in a multiphoton process, resonant enhancement, for example, by three- or five-photon excitation to energies near the peak of the collective resonance will readily be achieved. This comparatively straightforward experiment would distinguish clearly whether collective oscillations can or cannot be driven by strong laser fields, and would not be open to the objections raised in the case of molecules since the

358

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system is spherical and does possess an alternative dissociation channel. It would also seem that weakening the forces that bind the closed shell to the center of force and providing a potential within which at least individual particles do not “see” an asymptotically coulombic field should provide the most favourable conditions for the excitation of a collective mode. This situation also relates to the observation (Section 11) that collective effects are greater for electrons of lower kinetic energy. Of course, the laser intensity and turn-on times will remain crucial factors. Since stabilisation must be created by the laser field, the lifetime widths of the laser-induced structures in the photodetachment continuum and in particular their interaction with the giant resonance must be considered.

XI. Interactions between Giant Resonances and Rydberg Series-Intershell and Intersubshell Couplings It was remarked in Section VII in connection with the nuclear giant resonances that excitonic structures or single-particle excitations tended to be superimposed on the giant resonances and could interact with them. In atomic physics, the situation where the single-particle spectrum and the giant resonances overlap in energy gives rise to interactions between fairly sharp Rydberg series of autoionising states (typical lifetimes of about 10- l 3 s) and giant resonances that are normally an order of magnitude broader. The interaction between a very broad resonance and Rydberg series of sharp resonances is a characteristic problem of atomic spectroscopy in the subvalence energy range that is very well described by the same techniques of scattering theory as are employed in nuclear physics. (See Connerade and Lane, 1989.) In the specific situation where the broad feature is a giant resonance, the oscillator strength sum rule is exhausted within its width, which implies that it rises to the unitary limit at its peak. It follows as a theorem in this situation (Connerade and Lane, 1987) that any additional structure in the K matrix can only give rise to transmission resonances, a fact that has indeed been observed in clusters (Br6chignac, 1990). It is a clear manifestation of the many-body nature of giant resonances that there is a mutual influence between the strong and weak channels, i.e., that one can extract the influence of an outer shell on the photoionisation of an inner one and also the effect of inner shells on the photoionisation of outer ones. This is most easily revealed in theoretical treatments, (See in particular Amusia et al., 1974.) One pair of giant resonances that has received a lot of attention in this connection occurs in Mn and Cr. The case of Mn was originally investigated by Con-

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

359

nerade et al. (1976), who ascribed the resonance to a discrete transition from the 3p6 shell into the half-filled 3d5 shell broadened by interaction with the 3d5 + Ef continuum. The observations were improved and extended by Bruhn et al. (1978), and the asymmetric profiles were analysed by Davis and Feldkamp (1978) and Amusia et al. (1978) within the RPAE. Partial cross sections and asymmetry parameters for the ejected photoelectrons were reported by Krause et al. (1984). A more refined analysis was presented by Amusia et al. (1981), who introduced spin-polarised functions and obtained much better agreement between theory and experiment by introducing two different thresholds, one for nl t and one for nl.1 of the half-filled subshells. Further experiments and an MBPT analysis were then reported by Garvin et a l . , 1973. This has been followed by Amusia et al. (1990) extending the model yet further, which concludes that the influence of the 3p + 3d giant resonance on other channels extends much more widely in energy than the width of the giant resonance itself and can be detected at photon energies of up to 90 eV. The paper just cited (Amusia et al., 1990) uses a sophisticated RPAE model to reach this conclusion, but the underlying cause of the effect may actually be quite simple. The width of this resonance is about 2 eV and its resonance energy is about 51 eV. Consequently, its lifetime is roughly 3 X s against a period of oscillation of electromagnetic radiation of 8 x 10-I’~.This simple fact does not appear to have been pointed out before: the giant resonance in Mn comes closer to a viable atomic plasmon than the other cases that have been considered, because its lifetime is actually greater than the period of oscillation. This may well explain why its influence on the remainder of the atomic response is so pronounced: it is impossible for oscillations that are overdamped to develop harmonics. Collective oscillations should, if sustained, exhibit harmonics, since they are essentially nonlinear. The second harmonic of the giant resonance in Mn would lie around 100 eV. Very beautiful examples of extended Rydberg series interacting with a giant resonance occur within the 3p + 3d spectrum of Cr I, originally observed by Mansfield (1977), and the critical dependence of the profile shapes on the energy degeneracy between the different channels has been emphasised by Cooper et al. (1989) from experimental studies of Cr I and Mn 11, which are isoelectronic but nevertheless totally different in their spectral structure near the giant resonances. (See Fig. 11.) This example illustrates a general principle that seems to obtain in all known examples: whenever a giant resonance overlaps in energy with other excitation or ionisation channels, the latter are always enhanced at the expense of the giant resonance, and the giant resonance always tends to lose its identity and become fragmented by the interaction. Another excellent example occurs in the 4d + Ef resonance of Ba I, where the tendency of Ba to resemble La in the presence of an inner-shell vacancy (see Section IV) corrupts the one-electron excitation

Connerade

360

3p Photoabsorption I

,

I

55

60

1

1.O

0.0

E 0

45

50

I

I

I

I

I

45

50

55

60

65

70

65

70

.-c

P

g

1.0

.-> -2

0.0

2 Q)

4-

Q

42

43

44

45

46

47

48

Energy (eV) FIG. 1 I . The 3p + 3d giant resonances in Cr, Mn, and Mn 11, which are isoelectronic with Cr and nevertheless possess very different forms of giant resonance because of interchannel interactions. (After Cooper er al.. 1989.)

scheme and results in a highly developed manifold of double excitations within the giant resonance. We thus uncover another aspect of cooperative effects in atoms. At the same time as they quench the independent electron spectrum (disappearance of Rydberg series within the dipole-allowed channels of the independent electron scheme), many-body effects can dramatically enhance double and multiple excitations, leading to an unexpected profusion of Rydberg series in the range where cooperative effects are expected to occur. Thus, although no actual cooperative oscillations occur, we witness a destruction of the outermost subshell structure of the atom which is due to a complete breakdown of the independent particle model.

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

36 1

Perhaps the best example of this behaviour is to be found in Ba I, where, according to the independent electron scheme, excitation from the 5p subshell should proceed as: 5p66s2'&+ 5p56s2(*P,,2,3,,)ns, nd converging on just two ionisation limits. This excitation scheme rests on the assumption that the 6s2 subshell is well behaved, i.e., that it is a passive spectator to the transition, so that the 5p5 electrons together form a single quasi particle defining just two states split by the spin-orbit interaction. However, because of the breakdown of independent electron characterisation discussed in Section IV, the quasi-particle approximation actually breaks down and, instead of just two limits, one observes a multitude of series converging on at least 14 states of the parent ion, which can be described as a thorough mixture of the configurations 5p56s2,5p56s5d, and 5p35d2. Formally, the transitions arise as two or even multielectron excitations, but the mixture in the photoabsorption spectrum of the atom is so complete that the transitions can only be approximately ordered into sequences, and the spacings only begin to resemble those of a Rydberg series in the immediate vicinity of the series limit. Situations where large spectral densities of interacting levels are observed occur frequently in nuclear physics and have been analysed in terms of the random matrix theory (Porter, 1965). Interestingly, it is this subject that led to the concepts of spectral rigidity and to the Wigner level distribution used as of early 1991 in discussions of chaos in high Rydberg states. However, the situation in both nuclear physics and the present example is different in two important respects. First, we are not under semiclassical conditions (very high values of the principal quantum number n). Second, we deal with a many-electron problem, i.e., irregular motion in the present example would occur simply because the number of dynamic variables is large. The first suggestion that the techniques of nuclear spectroscopy should be applied to the analysis of complex atomic spectra appears to have been made by Rosenzweig and Porter (1960) in an important paper that attempted to extract the form of the level distributions from existing knowledge of the optical spectra of the lanthanides. The paper of Rosenzweig and Porter (1960) was important because of the novelty of the idea and because it laid the conceptual foundation for a statistical approach to the study of atomic spectra. However, one must doubt the significance of the conclusions they were able to reach (i) because they worked with data from an energy range where independent electron models may still be valid, despite the complexity of the spectra and (ii) because the data were both incomplete and unreliable; specifically, the tabulations available to them had been com-

I I

627

Wavelength (A)

I 590

FIG. 12. A portion of the 5p spectrum of Ba, showing the high density of discrete transitions and the high spectral rigidity that results (upper plot: the accumulated density of states).

363

COOPERATIVE EFFECTS IN ATOMIC PHYSICS

0.9

0.5

0

i

2

3

Relative spacings x

FIG.13. Statistical distribution of the level spacings for the sample of Fig. 12, showing the tendency toward a Wigner distribution of states, characteristic of a chaotic system.

piled from different experiments and contained levels of many different J values, which had to be “binned” by J value since levels of different J do not interact. In such complex spectra, the procedure was fraught with uncertainty. The 5p excitation spectrum of Ba I seems to provide a much more satisfactory example (Connerade et al., 1990). First and foremost, the data provide a manifold of high spectral density, whose spectral rigidity is high as can be seen from Fig. 12, all the levels having the same J value, since they are all reached in photoabsorption from the IS, ground state. Second, the distribution of levels does indeed signal a trend toward the Wigner law (see Fig. 13) which favours equal spacings between levels. Thus, there is clear evidence that cooperative effects in atoms can lead to a breakdown of the shell structure in many-electron atoms, and that this breakdown is associated with the emergence of irregular or chaotic motion, as hinted at in the Introduction (Section I). Furthermore, the homologous sequence Ca, Sr, and Ba has been studied, and, for the corresponding spectra of the lighter alkaline earths, although similar energy degeneracies exist, one finds that fairly regular, interacting Rydberg series can still be recognised, becoming more complex (more departures from the independent electron scheme) in Sr than in Ca. Thus, the breakdown of the external shell structure due to cooperative excitations in Ba is clearly associated with the larger number of electrons than are present in the other alkaline earths.

364

Connerade

XII. Conclusion We have presented the main ideas and the experimental background associated with the study of cooperative effects in atomic physics. The field is an active area of research. Many of our conclusions are therefore provisional, and the development of new experimental techniques, in particular intense lasers and also mass-selected beams of clusters, is expected to have a major impact on the subject over the next few years.

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Index

A Above-threshold Ionization, 34, 38, 41, 95,101 Adiabaticity parameter, 35, 38, 39,43 Antimatter, 283, 284 Antiscreening, 244, 299 a-Particle. See Helium. Apparent electron excitation cross section, See Electron excitation. Atomic physics, 325 Aufbau principle, 329, 334 Autoionisation. 333

B Bethe rule, 97, 100 Binomial probabilities, 252, 264, 270 Branching ratio, 4

C Cascades, 4. 9, 26,29 Classical limit, 246, 250 Closed shells, 329 Clusters, 344 Collective processes, 341, 349, 356 Collisional redistribution, 114- 116 Collision-induced coherences, I 14- 174 cross-relaxation, 149- 152 dipole-dipole interaction, 152- 155 dressed states, 165

369

four-wave mixing, 129- 139 Hanle effect, 124- 127 open two-level system, 127 in probe absorption, 139- 144 relationships in second- and third-order responses, 145 resonances in fifth-order nonlinearities, 147149 responses in spontaneous processes, 155160 signal changes, 160- 163 Collision-induced resonances. See CollisionInduced Coherences Computational methods, 250 Controlled collapse, 34 1 Cooperative effects, 325 Correlated emission, 172 Correlation, 314 dynamic, 222 general, 218, 221, 226, 230, 251,266, 312, 314 scattering, 222,265 static, 222, 266 two-center, 244,299, 312 Correlations, 328 Coulomb interference, 184 Coulomb potentials, 223 Cross relaxation, 149- I52

370

Index D

Differential electron excitation cross section. See Electron excitation. Dipole-dipole interaction, 116, 152, 155 Direct electron excitation cross sections. See Electron excitation. Double excitations, 327 Dressed states, 116, 158- 160, 164- 166 Dynamic stark effect, 35,43

E Eight-wave mixing, 114, 148. See also Wave mixing. Einstein A Coefficient-A, 4 Electron Beam Current-I, 3 , 4 Electron capture double, 289 with excitation, 297 with ionization, 292 multiple, 288 single, 239 special features, 239 Electron excitation, 5, 8, 10 apparent cross section-Q,, , 4 , 9, 28 differential cross section, 2, 12, 28 direct cross sections, 4, 6, 28, and rare-gas atoms, 12-27 Energy balance in MuCF, 178, 208 Energy conservation and non-conservation, 234,240,294 Exclusive probabilities, 224, 254 Excitation with capture, 297 double, 287 with ionization, 291 multiple, 286 single, 265

F Faraday cup, 3,25 Fifth-order nonlinearities, 147 Fluctuation-induced resonances, 169- 171 Fluctuations, 114, 156, 167, 169, 170-172 Fluorescence and ionization spectroscopy, 1 16, 124, 172 Fokker-Planck method, 259 Four-wave mixing, 129-139, 152, 161- 165, 168, 172. See also Wave mixing. G Giant resonances, 337 g-Hartree, 352-354

H Hanle effect, 124 Hanle resonance, 121, 125, 133, 135, 169 Hartree-Fock, 326 Helium, 180- 183 muonic, 184, 187 nuclear masses, I80 High electromagnetic fields, 355 Hydrogen, 178-211 helium impurity, 207 nuclear masses, 180 plasma, 197, 204 stopping power, 197, 202 target for MuCF, 178 I Inclusive probabilities, 224, 254 Independent electron approximation, 226, 229, 252,270,274,288, 293, 312 Independent particle approximation. 326 Inelastic collisions, 117, 121, 131, 138, 157, 159 Innershell, 327, 336 Intershell and intersubshell coupling, 358 Ionization with capture, 292 double, 268,273 with excitation, 291 multiple, 270 probability, 36,45,47,52,61-63, 87.94 single, 250 lonization process, nonlinearity power of, 47, 55.60, 103

K Kinematics, 240, 295 L Lanthanide contraction, 334 Laser-induced fluorescence (LIF), 2, 28, 118 ofHe, 13, 15 of Ne, 16-19 Line mixing, 150, 152 LS coupling, 8, 1 I

M Manybody effects, 325 Many-body perturbation theory (MBPT), 237, 281, 338. See also Perturbation theory. Many-body problem, 217,312,266 Many-electron problem formulation, 219, 232

37 1

INDEX general, 217, 312, 266 two-center, 244, 299, 312 Mayer-Fermi Model, 335 Mean Fields, 352 Mie Sphere, 347 Molecular Scattering, 247, 302 MuCF. See Muon-catalyzedfusion. Multicharge Ions, 34, 39, 101 Multielectron excitations, 361 Multifrequency laser radiation, 51, 60, 65 statistical factor of, 52, 59, 89, 96 Multiphoton cross section, 34- 1 I 1 polarization dependence of, 97, 100 Multiphoton excitations, 357 Multiphoton ionization, 34- 11 I direct, 34-36, 44 measuring methods, 45-72 measuring procedures, 73-89 measuring results, 90- 101 resonance, 45 probability of, 36 Muomolecular, 179, 189- 196, 200 Auger transitions, 187, 191- 192 Born-Oppenheimer approximation, 200 electro-muomolecularcomplexes, 190- 192 formation rate, 192, 195 geometry (radii), 194 nuclear effects, 185, 193 QED corrections, 193- 194 resonant formation, 192, 195 spectra, 189-191 Muon, 178-181, 208 Muon-catalyzed fusion (MUCF), 177-21 1 catalytic cycle, 186- 198 cycling rate. 179-181, 186-188 density dependence, 196, 203, 206-209 energy balance, 178, 208 fusion, 179- 187, 196- 199 muon loss, 180-181, 187 muon sticking, 186, 199-209 Q-value. 181-184 regeneration, 201 -204 temperature dependence, 195, 198, 209 transfer dCL---)tp,187 x-rays after fusion, 204-206

N Nonlinear susceptibilities, 116-1 18, 148 Non-Rydberg spectroscopy, 333 Nuclear dr fusion, 183- 186 branching ratio. 199 parameters, 183, 196

reaction products, 182- 184 R-matrix approach, 196 resonance detuning, 185 Nuclei, 345

0 One-electron approximation, 95 Open two-level systems, 127, 139 Optical emission cross section-Q., , 3-5 Optogalvanic spectroscopy, 128

P Perturbation theory, 232, 244, 250. See also Many-body perturbation theory. Phase conjugation geometry, 161, 171 Photoionization, 268, 278, 281 Plasmons, 348 Pressure-induced coherence, 121 Pressure-induced resonance, 166 Probabilities, 221, 224,252-254.264 Projectile electrons, 244, 299

Q Quantum jumps, I60 Quasiparticle, 335

R Raman scattering, 122, 157 Random phase approximation, 339 Ratios of cross sections, 268, 277, 280, 285, 288,291, 296, 301 Runge-Lens vector, 330 S Saturated absorption, 114- 118, 145 Saturation-inducedextra resonances, 172 Scaling laws, 264 Shake, 265-268, 279-280, 288, 296 Aberg’s hodel, 279 simple, 276 Single frequency laser radiation, 46, 53, 65 Six-wave mixing, 114- 116, 147, 152. See atso Wave mixing. Spatial coherence effects, 173 Stabilisation, 333, 357 Stark shift, 101, 108 Statistical energy deposition, 255, 272 Subharmonics, 148

T Thomas singularities, 240, 294 Time-dependent Hartree-Fock (TDHF). 25 1

372

Index

Time ordering, 234 Transfer. See Electron capture. Transient response, 167 Transition probability, 5 Trapped states, 157 Two-photon absorption, 143, 152 Two-photon maser transition, 172 Two-photon resonance, 152- 154 Two-sided diagrams, 122, I32 Two-time correlation, 116, 157

W Wall collisions, 174 Wave mixing eight-wave, 114, 148 four-wave, 129-139, 152, 161-165, 168, 172 six-wave, 114- 116, 147, 152. Wigner distribution, 363 Woods-Saxon potential, 348

U

X X-Ray spectroscopy, 337

V

Zi terms, 236,281

Ultraviolet, 337 Vacuum field Rabi splittings, 174 Velocity-changing collisions, 116, 160, 163

Y-2

Contents of Previous Volumes

Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G . G. Hall and A. T. Amos Electron Affinities of Atoms and Molecules, B . L. Moiseiwirsch

The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer

Atomic Rearrangement Collisions, B. H . Bransden Mass Spectrometry of Free Radicals, S. N. The Production of Rotational and Vibrational Foner Transitions in Encounters between Molecules, K. Takayanagi Volume 3 The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H.PAU~Y The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart and J . P. Toennies High-Intensity and High-Energy Molecular Radiofrequency Spectroscopy of Stored Ions I: Storage, H.C. Dehmelt Beams, J. B. Anderson, R. P. Andres, and J . B. Fenn Optical Pumping Methods in Atomic Spectroscopy, B. Budick

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D. Davison

Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C . Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder

Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S.Carton

Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood 373

374

Contents of Previous Volumes

Volume 4

The Calculation of Atomic Transition Probabilities, R. J. S. Crosslev

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

Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations skss‘ppq,C. D . H . Chisholm, A . Dalgarno. and F. R. Innes

Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson. F. C. Fehsenfeld, and A. L. Schmeltekopf Experiments with Merging Beams, Roy H. Ne,ynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. C. Dehmelr

Relativistic 2-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Volume 6 Dissociative Recombination, J . N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikaru Itikawa The Diffusion of Atoms and Molecules, E. A . Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Volume 7 Physics of the Hydrogen Master, C. Audoin. J . P. Schermann, and P. Criver Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C. Browne Localized Molecular Orbitals, Hare/ Weinstein, Ruben Pauncz. and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States o f Molecules-Quasi-Stationary Electronic States, Thomas F. 0’Malle.y

The Spectra of Molecular Solids, 0. Schnepp

Selection Rules within Atomic Shells, B . R. Judd

The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven

Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak. H. S. Tavlor. and Robert Yaris

CONTENTS OF PREVIOUS VOLUMES A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J . Greenfield

315

A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr.

Volume 11 Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Hatstead Harrison. and R. I . Schoen

The Theory of Collisions between Charged Particles and Highly Excited Atoms, I . C. f e r cival and D. Richards Electron Impact Excitation of Positive Ions, M. J . Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen

The Auger Effect, E. H. S. Burhop and W. N. Asaad

Stark Broadening, Hans R. Griem

Volume 9

Chemiluminescence in Gases, M. F. Golde and B . A. Thrush

Correlation in Excited States of Atoms, A . W. Weiss

Volume 12

The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge

Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev

Collision-Induced Transitions between Rotational Levels, Takeshi Oka

Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R . J . Champeau

The Differential Cross Section of Low-Energy Electron-Atom Collisions, D. Andrick

Topics on Multiphoton Processes in Atoms, f . Lambropoulos

Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English

Optical Pumping of Molecules, M . Brover, G. Goudedard, J . C. Lehmann. and J . Vigui

Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElrov

Highly Ionized Ions, Ivan A. Sellin

Volume 10

Ion Chemistry in the D Region, George C. Reid

Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong. Jr. and Serge Feneuille

Volume 13

The First Born Approximation, K. L. Bell and A. E. Kingston

Time-of-Flight Scattering Spectroscopy, Wilhelm Raith

Photoelectron Spectroscopy, W. C. Price

Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson

Dye Lasers in Atomic Spectroscopy, W. Lunge, J . Luther. and A . Steudel

Study of Collisions by Laser Spectroscopy, Paul R. Berman

Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett

Collision Experiments with Laser-Excited Atoms in Crossed Beams, I . K Hertel and W. Stoll

376

Contents of Previous Volumes

Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies

Ion-Atom Charge Transfer Collisions at Low Energies, J. B. Hasted Aspects of Recombination, D. R. Bates

Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet

The Theory of Fast Heavy Particle Collisions, B. H. Bransden

Microwave Transitions of Interstellar Atoms and Molecules, W.B. Somerville

Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster. Michael J . Jamieson, and Ronald F. Srewart (e, 2e) Collisions, Erich Weigold and Ian E . McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in IonAtom Collisions, S. V. Bobashev Rydberg Atoms, S. A. Edelstein and T. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree

Volume 15 Negative Ions, H. S. W. Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J . W. Humberston Experimental Aspects of Positron Collisions in Gases, T. C. Grifith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernsrein

Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H . B. Gilbody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W. 0.Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G. Burke

Volume 16 Atomic Hartree-Fock Theory, M. Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R. J . Celorta andD. T.Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Praduced Plasmas, M. H. Key and R. J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L.Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N. Forrson andL. Wilets

Volume 17 Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M. F. H. Schuurmans, Q.H . F. Vrehen, D. Polder, and H . M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G.

CONTENTS OF PREVIOUS VOLUMES Payne. C. H . Chen, G. S. Hurst, and G. W. Foltz

Inner-Shell Vacancy Production in Ion-Atom Collisions, C . D. Lin and Patrick Richard Atomic Processes in the Sun, P. L. Dufton and A. E. Kingston

Volume 18

377

The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov

Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J . Morellec. D. Normand. and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen

Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T. D. Mark and A. W. Castleman. Jr. Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W.E. Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Eottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong

Model Potentials in Atomic Structure, A. Hibbert

On the Problem of Extreme UV and X-Ray Lasers, I. I. Sobel'man and A. V. Vinogradov

Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W. Norcross and L. A. Collins

Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M . Raimond

Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W.F. Drake

Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J . A. C. Gallas, G. Leuchs, H . Walther, and H. Figger

Volume 19 Volume 21 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H . Bransden andR. K. Janev Interactions of Simple Ion-Atom Systems, J . T. Park

Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O'Erien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H . Greene and Ch. Jungen

High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M. llano, and R. S. Van Dyck, Jr.

Theory of Dielectronic Recombination, Yukap Hahn

Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. Blum and H. Kleinpoppen

Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Rocesses, Shih-I Chu

378

Contents of Previous Volumes

Scattering in Strong Magnetic Fields, M. R. C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M , More

Volume 24 The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Smith and N. G. Adams Near-Threshold Electron-Molecule Scattering, Michael A . Morrison

Volume 22 Positronium-Its Formation and Interaction with Simple Systems, J. W. Humbersron Experimental Aspects of Positron and Positronium Physics, T. C. Grijith Doubly Excited States, Including New Classification Schemes, C. D.Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody

Angular Correlation in Multiphoton Ionization of Atoms, S. J . Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J. Knize, 2. Wu. and W. Happer Correlations in Electron-Atom Scattering, A. Crowe

Volume 25

Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Peart

Alexander Dalgarno: Life and Personality, David R. Bates and George A . Victor

Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn

Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane

Relativistic Heavy-Ion- Atom Collisions, R. Anholr and Harvey Gould

Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy

Continued-Fraction Methods in Atomic Physics, S. Swain

Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson

Volume 23

Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson

Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal

Differential Scattering in He-He and He+-He Collisions at KeV Energies, R. F. Srebbings

Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J . Bauche, C. Bauche-Arnoult. and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J . Wuilleumier. D. L. Ederer, and J . L. Picqui

Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A . Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H . G. Reid

CONTENTS OF PREVIOUS VOLUMES Electron Impact Excitation, R. J . W. Henry and A. E . Kingston Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. P. Dewangan Relativistic Random-Phase W. R. Johnson

Approximation,

Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F. Drake and S.P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstellar Molecules, John H.Black

379

Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in ElectronAtom Collisions, Joachim Kessler Electron-Atom Scattering, I . E. McCarthv and E. Weigold Electron-Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengyel and M . I . Havsak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule

Volume 28 The Theory of Fast Ion-Atom Collisions, J . S. Briggs and J . H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh

Volume 26

Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy

Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein

Cavity Quantum Electrodynamics, E. A. Hinds

Electron Capture at Relativistic Energies, B. L. Moiseiwirsch The Low-Energy, Heavy Particle Collisions-A Close-Coupling Treatment, Mineo Kirnura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Franqoise Masnou-Sweeuws, and Annick Giusri-Suzor

Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M. V. Ammosov. N. B. Delone. M. Yu. Ivanov, I . I. Bondar, and A. V. Masalov Collision-Induced Coherences in Optical Physics, G. S. Aganval

On the /3 Decay of I8’Re: An lnterface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch

Muon-Catalyzed Fusion, Johann Rufelski and Helga E. Rafelski

Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko

Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J. H. McGuire

Cooperative Effects in Atomic Physics, J . P. Connerade

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