Aduances in
ATOMIC, MOLECULAR, AND OPTICAL. PHYSICS VOLUME 36
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Aduances in
ATOMIC, MOLECULAR, AND OPTICAL. PHYSICS VOLUME 36
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Plank-Institutf i r Quantenoptik Garching bei Miinchen Germany
Editorial Board P. R. BERMAN Uniuersity of Michigan Ann Arbor, Michigan M. GAVRILA F. 0. M. Institute uoor Atoom-en Molecuul’sica Amsterdam, The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois
W. D. PHILIPS National Institute for Standards and Technology Gaithersburg, Maryland
Founding Editor SIRDAVIDBATES
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEWYORK,NEWYORK
Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK INSTITUTFUR QUANTENOPTIK MUNICH, GERMANY
Volume 36
ACADEMIC PRESS
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This book is printed on acid-free paper. @ Copyright 0 1996 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
Academic Press, Inc. 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://w.apnet.com Academic Press Limited 24-28 Oval Road, London NWI 7DX, UK http://w.hbuk.co.uWap/ International Standard Serial Number: 1049-25OX International Standard Book Number: 0-12-003836-6 PRINTED IN THE UNITED STATES OF AMERICA 96 97 9 8 9 9 00 O l Q W 9 8 7 6 5
4
3 2 1
Contents vi i
CONTRIBUTORS
Complete Experiments in Electron-Atom Collisions
Nils Anderson and Klaus Bartschat I. I1 . 111. IV .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 8 24 80 83
Stimulated Rayleigh Resonances and Recoil-Induced Effects J.-Y. Courtois and G. Glynberg I. I1 . 111. IV . V.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stimulated Rayleigh Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recoil-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Recoil-Induced Effects in Atomic and Molecular Physics . . . . . . . . . . Conclusion References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
........................................
88 91 109 121 136 137
Precision Laser Spectroscopy Using Acousto-Optic Modulators W A . van Wijngaarden I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Spectroscopy Using Frequency-Modulated Lasers . . . . . . . . . . . . . . . . . . IV . Hyperfine Structure and Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . V . Starkshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
141 142 148 152 166 179 180
vi
Contents
Highly Parallel Computational Techniques for Electron-Molecule Collisions Carl Winstead and Vincent McKoy
....................................... .......................................... 111. Computational Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Illustrative Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction
I1. Theory
V. Conclusion References
........................................ ........................................
183 186 191 209 217 218
Quantum Field Theory of Atoms and Photons Maciej Lewenstein and Li You I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Hamiltonian of QFTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Properties of BEC in Trapped Alkali Systems . . . . . . . . . . . . . . . . . . . . V . DiagnosticsofBEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Quantum Dynamics of Condensation . . . . . . . . . . . . . . . . . . . . . . . . . VII . Theory of Bosers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Nonlinear Atom Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
222 224 230 234 239 253 267 274 275 275
......................................... ............................
281 287
SUBJECX INDEX
CONTENTS OF VOLUMES IN THIS SERIAL
Contributors Numbers in parentheses indicate the pages o n which the authors’ contributions begin.
NILS ANDERSON (11, Niels Bohr Institute, Orsted Laboratory, Copenhagen, Denmark KLAUS BARTSCHAT (11, Department of Physics, Drake University, Des Moines, Iowa 50311
(871, Institut d’Optique Thkorique el AppliquCe, Orsay, J.-Y. COURTOIS Cedex, France G. GRYNBERG (871, DCpartement de Physique de I’Ecole Normale Supkrieure, Laboratoire Kastler-Brossel, Paris, Cedex 05, France MACIEJ LEWENSTEIN (221), Commissariat B 1’Energie Atomique, DSM/DRECAM/SPAM, Centre d’Etudes de Scalay, Gif-sur-Yvette, France VINCENT McKov (1831, A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125 WILLIAM ARIEVAN WIJNGAARDEN (140, Physics Department, York University, Toronto, Ontario, Canada CARLWINSTEAD(1831, A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125 LI You (2211, Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 20138
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 36
COMPLETE EXPERIMENTS IN ELE CTRON-A TOM COLLISIONS NILS ANDERSEN Niels Bohr Institute 0rsted Laboratory Copenhagen, Denmark
KLAUS BARTSCHAT Department of Physics Drake Unicrersify Des Moines, Iowa
1. Introduction
. ..... ... .... . . . . . . . . .. . . ...... . .. . . . . . .
A.History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Scattering Amplitudes . . . .. . . . . ........ 11. Elastic Scattering. . . . . ... . . . . . . . ... .. A. Light Targets without Spin: He 1’s . . . . . . . . . . . . . . . . . . B. Light Targets with Spin: Na 3 2 S . . . . ....... . .... .. . C. Heavy Targets without Spin: Hg 6’S,. .. . .. . . ..... . . D. Heavy Targets with Spin: Cs 6 2 S , / 2 . ...... . ...... 111. Impact Excitation. . . . . . . . . . . . ..... ..... .. A. Light Targets without Spin in the Initial State: He 1’s + 2’P, 33P . B. Light Targets with Spin: Na 32S + 3 2 P . . . . . . . . . ........ C. Heavy Targets without Spin in the Initial State: Hg 6’S,, + 6’P, . . . . D. Heavy Targets with Spin: Cs 6*S,/, + 62P,/2,,,2 . . . . . . . . . .... E. Higher Angular Momenta: Excitation of He 1’s + 3’ D . . . .... . . IV. Conclusions . . . ... . . . . . . . . .. . . ... .... Acknowledgments . .. . . . . . . . . . ..... . . .. References . . . . . . . . . . . . . . . . .. . .. . . . .... . .
. . .... . . . . . . .. . . . ... . . . . . . . . . . . . . . . ... . .. ..... . . . .. . . .. . . .... . ... . . .... ...... . ...... . ... . . ... . .. .... . . . . .. . .. ...... ....... . ... .... . .. . . . .... .. .
1 1 4 8 8 11
15 20 24 28 34 48 70 71 80 83 83
I. Introduction A. HISTORY
“The most important experimental technique in quantum physics is the scattering experiment.” This statement is the opening sentence of a well-known textbook (Taylor, 1987) and highlights the central role that scattering experiments continued to play in the development of modern physics. A schematic diagram of a generic scattering experiment is shown 1
Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003836-6
2
N. Andersen and K. Bartschat
/
9 0 A
’0
B
FIG. 1 . Schematic diagram of a collision process.
in Fig. 1. The particles A and B are allowed to interact. When emerging from the interaction region, the collision partners generally have changed directions and may be in quantum states different from the initial ones. The aim of theoretical quantum physics is to model as accurately as possible the development of the system in the interaction region, for confrontation of the predictions with actual observables. The experimentalist aims at precisely defining the incident channel, as well as characterizing the collision products as closely as possible. For many important processes in nature, typical observables are averages over key parameters, such as incident directions, scattering angle, velocity, temperature, and so forth. However, the ultimate goal is to establish uniquely the relationship between the ket vectors and which determine the initial and final states of the system. A complete description in the quantum mechanical sense is succinctly expressed in terms of a corresponding set of complex-valued scattering amplitudes. What sets atomic physics apart in this context is that it may perhaps be the area of collision physics that today can present the largest systematic collection of fundamental processes for which this ideal has been achieved. This statement should not hide the fact that, in the words of Sir Harrie Massey, “The techniques required are very elaborate. Indeed, it is probably true that experiments in this field are among the most complicated in atomic physics today. They are very important for deepening our understanding of atomic collisions and it is essential that their complexity should not be allowed to obscure their importance” (Massey, 1983). In a series of influential papers, Bederson (1969a, b) coined the term perfkt scattering experiment to stimulate the development of experiments, or rather combinations of experimental approaches, that would result in a complete
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
3
determination of the quantum mechanical scattering amplitudes. His starting point was a systematic analysis of elastic and inelastic electron-atom collision experiments, using alkali atoms as an example and spin polarization as a key variable. At the time of formulation in 1969, many of these experiments could only be imagined as future goals, since few of the necessary key technologies had been developed to assist experimentalists and theoreticians in their common aim. In the intermediate period, this situation has dramatically improved, partly due to impressive advances in experimental and theoretical methods, but undoubtedly also due to the early, explicit formulation of a quest for a superior, systematic approach. Perfect scattering experiments have now been performed for several fundamental processes in electron-atom and atom-atom collision physics. At first sight, it may perhaps appear paradoxical that determination of scattering amplitudes, often considered somewhat “esoteric” (Callaway and McDowell, 1983) and abstract parameters mapping an event in Hilbert space, may provide us with a more concrete visualization of the collision dynamics. However, as we shall see, this is because the scattering amplitudes or density matrix parameters are ultimately related to directly observable properties, such as a change of spin direction of the scattered electron and, for impact excitation, characteristic properties of the photon radiation pattern that is emitted when the excited state decays. Thus, systematic exploitation of all available information enables a deeper insight into the collision dynamics than determination of a differential cross section-the “classical” observable in a scattering experiment-alone provides. An early example is the discussion by Kohmoto and Fano (1981) on the relationship of attractive and repulsive forces to the sign of the angular momentum transferred to the excited atomic electron in an electron-atom collision event (see Fig. 2). This chapter addresses the advances up to the present in complete electron-atom collision experiments. The aim is to present a series of key examples for fundamental scattering processes, together with the experimental techniques that have been used. The purpose is not a full presentation of all processes studied, nor of all data that have been accumulated; rather, it is to select examples of the most recent theoretical and experimental results that will enable the reader to assess the present level of achievement. We hope that the power of this approach will become evident along the way, in the sense that it provides an efficient framework for a systematic and complete test of the current theoretical understanding. In addition, it may produce specific recipes for ways to select experimental geometries that most efficiently test theoretical predictions, and it may reveal connections between apparently unrelated observables from
4
N. Andersen and K Bartschat
FIG.2. Propensity rule for positive orientation due to attractive forces between projectile and target in a collision process (from Kohmoto and Fano, 1981).
often very different and highly sophisticated experiments, thus providing valuable consistency checks. The presentation is structured in the following way. To begin with, a general analysis of scattering amplitude properties concludes in a recipe for determination of the number of independent parameters necessary to define a complete experiment for a given process. We then proceed to analyze in a systematic way a string of specific cases of elastic and inelastic collisions, with gradually increasing levels of sophistication. Finally, we comment on directions in which future studies could fruitfully be pursued.
B.
SCATTERING AMPLITUDES
As we have pointed the determination of important to derive number depends on
out, a “perfect” or “complete” experiment requires all independent scattering amplitudes. It is, therefore, the number of such independent amplitudes. This the assumptions about the dynamics of the collision
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
5
process, i.e., the individual projectile-target interactions that are taken into account. Throughout this chapter, we will assume that 1. All interactions conserve the total parity of the projectile-target system. 2. All interactions are invariant against time reversal. 3. Atomic hyperfine structure does not affect the collision process.
These assumptions are very well fulfilled for most atomic collisions and certainly for all cases discussed in this chapter. It is, however, important to realize that the resulting simplifications in the theory are not fufjiffed exact& in nature. Wi$ these assumptions in mind, transitions from an initial state IJ,M,,k l m , ) to a final state IJ,M,,z,m,) are described by scattering amplitudes f ( ~ , mM~p, , ; 6 )
=
( ~ , ~ , ; z IA ~ m~ , ~ , ; ~ , m , >(1)
where 9 is the transition operator. Furthermore, J, ( J , ) is the total electronic angular momentum in the initial (final) st_ate ,Of the target and M I ( M , ) its corresponding z component, whereas k, (kf) is the initial (final) momentum of the electron and m, (m,) its spin component. The For simplicity, we will scattering angle 8 is the angle between k, and if. often omit it in the notation, together with an overall normalization factor that is needed in explicit numerical calculations. In practical applications, one must define the scattering amplitudes with respect to a quantization axis for the angular momentum components. A standard choice for numerical calculations is the so-called “collision system,” where the incident beam axis is the quantization (2‘) axis and the y‘ axis is perpendicular to the scattering plane. On the other hand, the algebra often becomes simpler and many observables can be interpreted more easily in the “natural system” where the quantization axis z” coincides with the normal vector to the scattering plane and the X’ axis is defined by the incident beam direction. If not stated otherwise, we will use the natural coordinate frame throughout this chapter. The transformation of the scattering amplitudes from one system to another can be achieved in a straightforward way by transforming the initial and final states through standard rotation matrices and using the fact that the action of the Yoperator must be independent of the particular coordinate system. For details, we refer to Appendix A of Andersen et al. (1996). An important point for the discussion of scattering amplitudes is the fact that the abovementioned assumptions about the symmetry properties
6
N. Andersen and K Bartschat
of the projectile-target interaction lead to conservation laws through the Y operator. These, in turn, will cause interdependences between various scattering amplitudes or simply require some amplitudes to vanish. Consider, for instance, the conservation of the total parity. For our case of interest, electron-atom scattering in a plane, the process must be invariant against reflection in this plane. This reflection operation can be constructed as the parity operation, followed by a 180” rotation around the normal axis of the scattering plane. As shown in Appendix A of Andersen et al. (1996), the result for amplitudes in the natural coordinate frame is
where n, and n, are the parities of the initial and final states, respectively. Equation (2) shows that many amplitudes vanish in the natural frame (namely those where the exponent is an even or odd integer, depending on the product of the parities). This fact is one of the many advantages that can be used when formulating the general theory in this frame. Numerical calculations, on the other hand, are simpler in the collision frame, where one finds phase relationships ( f) between amplitudes with a given set of magnetic quantum numbers and those where the signs of all quantum numbers are reversed. We now discuss the number of independent scattering amplitudes. There are 4(2J, 1)(2Jf 1) possible combinations of magnetic quantum numbers. Due to parity conservation of all interactions determining the outcome of the collision process, the number of independent scattering amplitudes is cut in half, giving a total of 2(2J, + 1)(2Jf + 1) complex amplitudes for each transition between fine structure levels. Subtracting a common arbitrary phase, the total number of independent real parameters is thus 4(2J, + 1)(2Jf + 1) - 1. These are usually parameterized as one absolute differential cross section and 4(2JL+ 1)(2J1 + 1) - 2 dimensionless numbers, namely 2(2J, + 1X2Jf 1) - 1 relative magnitudes and 2(2J, + 1)(2Jf + 1) - 1 relative phases. Without going into details, we note that time reversal invariance of the projectile-target interaction leads, in general, to relationships between scattering amplitudes for inelastic excitation and “superelastic” deexcitation. In the special case of elastic scattering, such relationships may further reduce the number of independent amplitudes. An important example is elastic electron scattering from cesium atoms which will be discussed in Section 1I.D. In addition to the fundamental assumptions of parity conservation and time reversal invariance, it is sometimes also assumed that the total spin S
+
+
+
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
7
and the total orbital angular momentum L of the combined target + projectile system are conserved during the collision. This nonrelativistic approximation is generally good for electron collisions with light targets, such as helium or sodium. It is also the basis for the “fine-structure effect” (Hanne, 1983) in which observables for various fine structure transitions are related by algebraic factors. In the nonrelativistic approximation, the scattering amplitudes depend in a purely algebraic way on the spin quantum numbers, and transitions between fine structure levels are described by standard recoupling techniques. Specifically, the scattering amplitudes (1) can be expressed as
f ( M p p Mimi; 0 )
where S,, S,, M,. , MF,, L , , L , , M L , and M L , are the spins as well as the orbital angular momenta and the ’ corresponding z components of the initial and final target states, and
are nonrelativistic scattering amplitudes that describe transitions between orbital angular momentum states. Equation ( 3 ) expresses the conservation of the total spin S and its component
M,
=
M,,
through the Clebsch-Gordan Together with
f’( M L , , ML,; 0 )
=
+ m l = MLs,+ m,
(5)
coefficients ( j l ,m i ; j z , m 2 I j 3 , m3).
n,n,( - l)’‘,-M’~cfS(
MI>,,M [ , , ;0 )
(2a)
from parity conservation, this approximation reduces the number of independent scattering amplitudes for any transition between members of two
N. Andersen and K Bartschat
8
fine structure multiplets to NsN‘, where Ns is the number of total spin channels and NL
(2L; + 1)(2L/ =
+ 1) + ( - l ) L ’ + L ’ n ; n f 2
(6)
As in the more general case discussed previously, time reversal invariance may further reduce the number of independent amplitudes. This formalism yields, for example, the four independent amplitudes f : f! f; 1, and f l for the transitions 32S,/2 + 32P,/2,3/2in electron impact excitation of sodium, where the superscripts “ t ” and “s” refer to the “triplet” ( S = 1) and “singlet” (S = 0) total spin channels and the subscript ML, = i-1 to the orbital magnetic quantum number of the excited P state. Both ML,(= 0) and the scattering angle have been omitted in the notation. This compares with 24 (!) independent amplitudes for the transitions 62S1/2 + 62P1/2,3 / 2 in electron scattering from cesium atoms, namely 8 for 62S1/2-+ 6’PIl2 and 16 for 62S1,2 -+ 62P3/2.
11. Elastic Scattering We will start with complete experiments in elastic electron-atom scattering, going from the simplest case of a light target without spin and orbital angular momentum (such as helium) to more complicated situations such as light targets with spin (sodium), heavy targets without spin (mercury), and finally, heavy targets with spin (cesium). Except for the cesium target, these are characteristic examples of cases where complete experiments have, indeed, been performed or where only a small amount of information, such as the sign of a phase angle, is still missing. We will only consider initial atomic S states. While experiments for elastic electron scattering from targets with nonzero orbital angular momentum have been performed, these can only be described as the very first steps toward the complete determination of all independent scattering amplitudes. A. LIGHTTARGETS WITHOUT SPIN: He
1’s
From the point of view of a complete experiment, this is the simplest possible case in elastic electron-atom scattering. In the nonrelativistic approximation, there is only one independent scattering amplitude, namely f S = ’ / ’ ( M L ,= 0, M L , = 0; 0 ) = f that needs to be determined for each scattering angle. Figure 3 shows this amplitude as an arrow in the complex
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
9
f
FIG. 3. The scattering amplitude f for elastic e-He scattering.
plane, but given the arbitrary phase, the determination of the magnitude of this amplitude does, indeed, correspond to a complete experiment. This magnitude can be determined via the absolute differential cross section
a=lfl
2
(7)
it.,
f=G for each scattering angle 8. Figure 4 shows the apparatus of Brunger et al. (1991) that has been used for absolute differential cross section measurements for electron-helium scattering. It is a crossed electron-atom beam apparatus where the energy spread of the electron beam is reduced by a monochromator and the scattering angle can be varied by a rotatable electron spectrometer. For details of the experimental procedure we refer to the original publication, but it is important to point out that a major difficulty in this kind of experiment arises from the need of an absolute value of the differential cross section. As pointed out in Section I.B, one such value is required in all complete experiments-if additional parameters are needed, these can be determined on a relative scale. Figure 5 shows experimental data of Brunger et al. (1992) for elastic e-He scattering at energies of 1.5, 10.0, and 20.0 eV. Note how the differential cross section becomes peaked in the forward direction with increasing energy, a typical feature of such collisions. The data are compared with a recent “convergent close-coupling’’ (CCC) calculation of Fursa and Bray (1995). The agreement between experiment and theory is excellent.
10
N. Andersen and K Bartschat
,- Atomic
Electron ODtics
in
"
zoom
YONOCHROMATOR
1s-a
1
Target Region
CEM Detect
5
0
cm
FIG. 4. Schematic diagram of an apparatus for measurements of a differential cross section (from Brunger et al., 1991).
In summary, the complete experiment, in its simplest form, has been achieved for this case. This statement also applies to elastic scattering from other closed-shell targets, such as Be, Ne, or Ar, as long as the electron spin does not enter into the discussion. The situation will change when relativistic effects, such as the spin-orbit interaction between the projectile electron and the target nucleus, need to be taken into account. This will be further discussed in Section 1I.C.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
u OO
'
'
io.
60 90 '120' '150' Scottering Angle (deg) '
'
'
'
11
'180
FIG.5. Differential cross sections for elastic electron scattering from helium atoms at electron energies of 1.5, 10, and 20 eV. The experimental data of Brunger et al. (1992) are compared with the "convergent close-coupling" results of Fursa and Bray (1995).
B. LIGHTTARGETS WITH
SPIN:
Na 3 2 S
The next level of complexity is encountered for a target with spin one-half, as for example atomic hydrogen and the quasi-one-electron targets Li, Na, K, etc. The most detailed picture has been achieved for the Na 3's state, because it is experimentally more convenient than, in particular, H 1's. Due to the possibility of triplet and singlet scattering, two scattering amplitudes, f S = I'ML, = M L , = 0; 6 ) = f ' and f s = ' ( M Lf = M L , = 0; 6 ) = f ' , are required for the description. These are shown as two arrows in Fig. 6. We thus need three parameters, namely one cross section, one relative size, and one angle, to completely determine the amplitude pair. We
FIG. 6. The triplet and singlet amplitudes f' and f" for elastic e-Na scattering
12
N. Andersen and K. Bartschat
obtain the following expression for the cross section for unpolarized beams
where we have defined the relative sizes
If'12
w'
-
4%
and
with 3w'
+ w"1.
While the differential cross section may again be measured with a setup of the type shown in Fig. 4, the triplet and singlet probabilities have been determined in an experiment developed at NIST and sketched in Fig. 7 (McClelland et al., 1989). This is a crossed beam experiment in which both the electron and sodium beams are spin polarized before the interaction. The polarized electron beam is produced by shining circularly polarized Optical Pumping Detector
e
-
Linear Polarizer Laser Diode FIG. 7. Polarized-electron-polarized-atomscattering apparatus used in the NIST experiments for e-Na scattering, GaAs polarized electron source, scattering chambers, sodium oven, and Mott spin analyzer (from McClelland er al., 1989).
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
13
laser light on a GaAs crystal, while the sodium beam is illuminated by circularly polarized light tuned to the resonance transition before reaching the interaction zone. In the geometry shown, it is thus possible to spin polarize both beams perpendicular to the scattering plane, and flip their spin direction individually by inversion of the circular polarization of the photon beams. One may then measure the so-called spin asymmetry parameter
where P and PA are the spin polarization of the electron and atom beams, respectively. A relative phase angle, conveniently chosen as A (see Fig. 6), thus remains to be determined. In Munster, Hegemann et al. (1993) performed the experiment sketched in Fig. 8. Electrons with polarization P perpendicular to the scattering plane collide with unpolarized atoms, and the polarization P' of the scattered electron beam is determined by Mott scattering. The depolarization parameter T is defined as P' T ~ - P= 2 2 ( w'
+
cos A )
(13)
In this way cos A may be determined, and thereby A, except for the sign. This final ambiguity has not yet been resolved experimentally, but it could
180" spectrometer I arge
GaAsP cathode
0
Pockels cell
fi
U
HeNe laser
.__ ...
.6
"0 63
Mott detector FIG. 8. Experimental setup to measure the depolarization factor T = P ' / P for scattering of spin-polarized electrons from unpolarized sodium atoms (from Hegemann et al., 1993).
14
N. Andersen and K. Bartschat
be removed in a setup where the spin polarizations in the initial channel are orthogonal, say electron spin P in the forward direction and atom spin PA in the scattering plane perpendicular to the beam, and the final electron spin polarization component P ’ perpendicular to the scattering plane is determined. Then a measurement of
determines A uniquely (Andersen and Bartschat, 1993). A determination of the set
thus constitutes a complete experiment. Figure 9 shows theoretical “close-coupling plus optical potential” (CCO; Bray, 1992) and “convergent close-coupling’’ (CCC; Bray, 1994) predictions together with experimental results at 12 eV collision energy for the parameter set ( T , A , V ,cos A). The available experimental data for a; (Gehenn and Reichert, 1972) are unfortunately only relative, but the shape agrees well with the one found theoretically (see Fig. 10). A complete experiment has thus not yet been achieved at this energy, and the situation is similar at 4.1 eV. Nevertheless, we show in Fig. 10 the complete parameter set (uu;w‘;A). After conversion, the choice of sign for the experimental values of A has been guided by the theoretical curve. We notice satisfactory agreement between the theoretical predictions and the experimental results. In the literature, one may often find an alternative choice of amplitudes, f and g, related to “direct” and “exchange” scattering (Kessler, 1985). They are related to the triplet and singlet amplitudes through
and
f” = f + g Thus, if there is no exchange between the projectile and the target electron ( g = 0, or f ‘ = f’),then w‘ = w’= and A = 0, causing A = 0, T = 1, and V = 0. On the other hand, if exchange dominates (f = 0, or f ‘ = -fs), then again w‘ = w s = but A = - T , resulting in A = 0, T = 0, and V = 0.
a
a,
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
0
McClellond e l 01.
0
Scattering Angle (deg)
15
McClelland et 01.
Scattering Angle (deg)
FIG. 9. Experimental results for the parameter set ( T , A , cos A ) for elastic e-Na scattering at 12 eV. The experimental data of the Miinster (Hegemann et al., 1993) and the NIST (McClelland et al., 1992) groups are compared with CCO (Bray, 1992) and CCC (Bray, 1994) calculations. Also plotted are the theoretical predictions for the V parameter (see text).
C. HEAVYTARGETS WITHOUT SPIN:Hg 6’s”
We now move on to heavy targets where relativistic effects, in particular the spin-orbit interaction between the projectile electron and the target nucleus, can no longer be neglected. A classic example of such a target is mercury, with a nuclear charge of Z = 80 and the ground state configuration ( 6 s 2 ) ’ S , . Spin dependent effects in elastic electron scattering from this target have been studied extensively over the past three decades. Detailed discussions of this collision system were given by Kessler (1985, 1991) who used the collision coordinate frame with a “direct” amplitude f and a “spin-flip’’ amplitude g.
16
N. Andersen and K Bartschat
1000
100
10
1
0.1
O.OlL,’ 0.4
“ ‘
30
“
60
‘
”
90
’
”
120
’
“
150
’
I
I
180
. . wt 20
0
true
ghost
-1 .1
Scattering Angle (deg)
Scattering Angle (deg)
FIG. 10. Differential cross section u,, weight parameter w‘, and phase angle A for elastic e-Na scattering at 12 eV. The relative differential cross section measurements of Gehenn and Reichert (1972) have been normalized to the theoretical predictions. There are two possible experimental solutions for A, and we have chosen the “true” and the “ghost” set based on their agreement/disagreement with the theoretical results.
In the formalism outlined in Section I.B, we can use Eq. (2) to determine the number of independent amplitudes in the natural coordinate frame. Since the initial and final atomic states are identical and the target carries no electronic angular momentum, we find the two nonvanishing amplitudes
and
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
17
Note that there are no spin-flipsin the natural frame! However, using the semicl3sical form of the spin-orbit potential as being proportional to i.S: where 1 and are the orbital and spin angular momenta of the continuum electron, one sees immediately that the scattering potentials for spin “up” and spin “down” electrons on a given trajectory_are not identical if this potential has to be taken into account. (Note that 1 is perpendicular to the scattering plane for all classical trajectories.) Hence, this is the mathematical equivalent to the qualitative description of this effect given in Section 3.4.2 of Kessler (1985). The amplitudes f and f 1 can be related to the direct and spin-flip amplitudes by transforming from the collision to the natural flame, as outlined in Appendix A of Andersen et al. (1996). We find
and (Note the similarity of this transformation to the relationship between the triplet/singlet and direct/exchange amplitudes in the previous section.) A schematic picture of the amplitudes as two complex numbers is shown in Fig. 11. Apart from the magnitudes of f and f i, a complete experiment also requires the determination of the phase angle A. The latter replaces the angle y 1 - y 2 , i.e., the phase between the direct and spin-flip amplitudes in Kessler (1990, as the last independent parameter that needs to be determined. To measure the relative magnitude and phase of the two scattering amplitudes, the Munster group developed an experimental setup in which polarized electrons are scattered from unpolarized atoms and the electron
FIG. 11. The “spin up” and “spin down” amplitudes scattering.
fT
and f i for elastic e-Hg
18
N. Andersen and K Bartschat
polarization after the collision is determined. In contrast to e-Na scattering discussed in the previous section, all the information must now come from the preparation and analysis of the electron spin, since the target remains structureless during the whole collision process. This type of experiment allows for the determination of the so-called STU parameters. Because of the necessary extension to inelastic scattering (see Section 1111, Fig. 12 shows a schematic diagram of the “generalized” STU parameters (Bartschat, 1989) that fully describe the change of an arbitrary initial electron polarization through scattering from any ensemble of unpolarized target atoms for both elastic and inelastic scattering. There are seven relative generalized STU parameters with the following physical meaning: The polarization function S, gives the polarization of an initially unpolarized projectile beam after the scattering, while the asymmetry function S, determines the left-right asymmetry in the differential cross section for scattering of spin polarized projectiles. Furthermore, the contraction parameters T,, Ty, T, describe the change of an initial polarization component along the three Cartesian axes, whereas the parameters UKyand U,, determine the rotation of a polarization component in the scattering plane. Together with the absolute differential cross section a; for the scattering of unpolarized electrons, these eight parameters describe the maximum information that can be obtained from preparation and analysis of electron polarization alone.
i A
i
I
I
--
‘-1 I
‘+ SAP,
-7-
T X P X - uxypy
X”
1 +SAP, FIG. 12. Physical meaning of the generalized STU parameters for an initial spin potarization @,which is changed to a final spin polarization @‘through the scattering process (see text).
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
19
Since, in the present case, that is the maximum information about the total collision process, it is not surprising that the generalized STU parameters are not independent for electron scattering from targets without angular momentum. For elastic scattering from such targets, we find
where we have defined
and w1
If
LIZ
=-
2% with
wt
+ w1 = 1
Also, T,
=
1
( 22)
reflects the fact that spin components perpendicular to the scattering plane are conserved. Equations (19) can be inverted to yield the complete parameter set (q,; w ; A ) through wT
=
A
=
s+l ~
2
and ATAN2 ( - U , T )
where ATAN2 is the FORTRAN function for arg(T - iU).This shows the physical meaning of the phase angle A: It describes the rotation of an
20
N. Andersen and K Bnrtschnt
initial electron polarization component in the scattering plane through the collision. The most essential parts of the apparatus used by Berger and Kessler (1986) to measure the STU parameters for elastic electron scattering from mercury and xenon atoms are shown in Fig. 13. Experimental results from their work together with the absolute differential cross section a, of Holtkamp et a f . (1987) for elastic e-Hg scattering at 50 eV are shown in Fig. 14, followed by the set ( w T , A ) in Fig. 15. Note that all relative parameters are in very good agreement with the relativistic calculation of McEachran and Stauffer (1986), while the absolute differential cross section is generally overestimated by the theory. The most important reason for this disagreement is the neglect of “absorption,” i.e., loss of flux into inelastic channels, in the calculation. This problem can be remedied in a simple way by including at least a semiempirical complex absorption potential (Hasenburg et af., 1987).
D. HEAVYTARGETS WITH SPIN:Cs 6’S1/2
We finish this main section with a brief discussion of elastic electron scattering from a heavy target with spin, cesium ( Z = 5 5 ) with the ground state configuration 6’S1,,. This case requires extension of the formalism presented in Section 1I.B to account for relativistic effects. To begin with,
G A S P CATHODE
--
u
U
FIG. 13. Experimental setup for measurement of the change of the electron polarization vector in elastic scattering (from Berger and Kessler, 1986).
21
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
Holtkomp et 01.
- McEachron and Stauffer
0.1
”
’
0
30
’
60
’
’ . 90
“
120
’
150
.
180
0
30
60
90
120
150
180
30
60
90
120
150
180
-0.2 -
- McEachron ond Stauffer 0
30
60
90
120
-0.4 -
150
180
0
Scattering Angle (deg)
Scattering Angle (deg)
FIG. 14. Differential cross section a, and STU parameters for elastic e-Hg scattering at 50 eV. The experimental data of Holtkamp et al. (1987) for au and of Berger and Kessler (1986) for (S,T , U )are compared with results from a relativistic calculation by McEachran and Stauffer (1986).
inspection of Eq. (2) reveals that parity conservation allows for the following eight nonvanishing scattering amplitudes in the natural frame:
f ( + +, + +; 01,
f(+
- 9
+ -;
01,
f ( - +, - +; 01, f(-
- >
-
-; 0)
(24a)
f ( + -,
- + ; e l , f ( - + , + - ; e l , f ( + + , - - ; e > , f(-
fi.
-, + + ; e l (24b)
The amplitudes (24a) are with the abbreviations f for spin values “nonflip” amplitudes, whereas those listed in (24b) correspond to “double-flip” amplitudes, since the spins of both the projectile and the
22
N. Andersen and K Bartschat Hg 6 ' S 0
50eV
".n 7n I"
Berger and Kessler
- McEachran and Stauffer
0.60
0.50
0.40
0.30;'
" '
30
"
60
'
"
90
'
"
120
'
"
'
150
S c a t t e r i n g Angle ( d e g )
'
I
0
180
S c a t t e r i n g Angle ( d e g )
'
FIG. 15. Weight parameter w and phase angle A for elastic e-Hg scattering at 50 eV. The experimental data, obtained from the STU results of Berger and Kessler (1986), are compared with predictions from a relativistic calculation by McEachran and Stauffer (1986).
target electron are flipped, either through exchange or by explicitly spin dependent forces such as the spin-orbit interaction. In a pioneering paper, Burke and Mitchell (1974) analyzed this problem in detail and showed that time reversal invariance reduces the number of independent amplitudes from eight to six. They used a modified natural coordinate system with the quantization axis perpendicular to the scattering plane, the x (their q ) axis along the direction of the momentum This choice of coorditransfer 2, - gi,and the y ( p ) axis along if+ ii. nate system had been discussed earlier (Schumacher and Bethe, 1961, Wilmers, 1972) and yields a transparent system of equations for various spin dependent variables. Specifically, Burke and Mitchell gave equations for the following physical observables (their notation is given in parentheses): 1. The cross section a, (ZJ for scattering of unpolarized beams. 2. The asymmetry function S,(P,); S = S, = S, is a general result for elastic scattering that follows from time reversal invariance. 3. Another asymmetry function ( P , ) that determines a left-right asymmetry in the differential cross section for scattering of unpolarized electrons from polarized target atoms.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
23
4. Asymmetry functions (C,,, Cp!,,C,,, Cr,q7C,J that are generalizations of the exchange asymmetry A discussed in Section 1I.B. 5. Functions ( D i n ,D;!,, Diq> and (D:,f, D;!,, D&) that determine the atomic polarization after scattering of unpolarized electrons from polarized atoms (superscript “1”) or the electron polarization after scattering of polarized electrons from unpolarized atoms (superscript “2”); such parameters are generalizations of the T parameter. 6. Functions ( K ! , f ,K&, Kd,) and (K:,,, Ki,,, Ki,) that determine the atomic polarization after scattering of polarized electrons from unpolarized atoms (superscript “1”) or the electron polarization after scattering of unpolarized electrons from polarized atoms (superscript “2”); these are also generalizations of the T parameter. 7. Functions ( D j Y ,D i p , D;,, D i p ;K;y, K:*, Ki,, K i p ) that are the corresponding extensions of the U parameters for the cases listed in ( 5 ) and (6). Without going into details of the analysis, we point out that even a measurement of the entire set of parameters given by Burke and Mitchell does not yet correspond to a complete experiment. This can be seen immediately by simplifying their equations to the nonrelativistic case. Then all the S and U type parameters disappear, all the T type parameters become identical, C/,4= C , , = 0, and the remaining C type parameters reduce to the exchange asymmetry A of Eq. (12). As shown in Section II.B, however, V type measurements are necessary to obtain complete information about the relative phases of the scattering amplitudes. Given the difficulty of such experiments, we follow Burke and Mitchell and will not provide the generalized equations for such parameters-if such an experiment were to be performed, there should be plenty of time for the experimentalist to derive them while waiting for the signal in the polarization detector! Nevertheless, the e-Cs collision system is a prime example of a case where new information can be obtained by using spin polarized electron and atom beams. In a recent experiment, the Bielefeld group (Leuer et al., (199.5) determined the parameter set ( A “ ” ,A””, Aint),which corresponds to ( - C n n ,P,, P I ) of Burke and Mitchell. The experimental arrangement is shown in Fig. 16, followed by results for elastic e-Cs scattering at a collision energy of 7 eV in Fig. 17. The nonvanishing result for Aint is particularly interesting, since it is the first experimental verification of an interference effect between electron exchange and the spin-orbit interaction. A combination of both effects is required for such a result, as pointed out by Farago (1974) more than two decades ago.
24
N. Andersen and K. Bartschat Monochromator
Laser Diode
n. Polarizer
W
FIG. 16. Polarized electron-polarized atom scattering apparatus used in the Bielefeld experiment for e-Cs scattering (from Leuer et al., 1995).
111. Impact Excitation After elastic scattering, the next level of sophistication is reached when the atom is excited. This step, however, enables us to exploit the rich information contained in the pattern of photons that are emitted in optical decays of the excited state. (We will not deal with excitation of metastable states in the present chapter). For simplicity, the discussion will be focused on S ++ P and J = 0 J = 1 transitions, the main cases of interest for experimental studies to this date. Generalization of the formalism to higher angular momenta is straightforward, as will be illustrated by the example He 1's + 3'D, where interesting new aspects appear. Figure 1Na) shows schematically the basic problem. An atom A is excited from an S state to a P state by impact of an electron, which is deflected by an angle 8, as monitored by a detector in this direction. The excited P state of the atom may be characterized from a study of the radiation pattern Ce., photon direction and/or polarization). This can be achieved with the geometry shown in Fig. 18(b), where the scattered electron is detected in coincidence with the photons emitted in a specific direction, including a possible photon polarization analysis. Two
-
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
c
6
EE
25
0.1 -
.*
0.0-
*-
**
-0.1 -
0 h
*** -~*********=
Q -0.2-0.3 30 40 50 60 70 80 90 1001101201301401
Angle 0 [deg]
-0.31 '
" " " "
" " " " " " '
30 40 50 60 70 80 90 100110120130140150
Angle 0 [deg]
0.041
eV
-0.064 . , .
.
, . , . I . , . . . , . , . . 30 40 50 60 70 80 90 1001101201301401 '0 I
I
I
Angle 0 [deg] Fic. 17. Scattering asymmetries for elastic e-Cs scattering at 7 eV (from Leuer et al., 1995). The nonvanishing values of A'"' indicate the simultaneous importance of electron exchange and spin-orbit interaction.
N. Andersen and IC Bartschat
26
a
b
2"
C
FIG. 18. (a) Schematic diagram of a collisionally induced charge cloud of an atom A cxcited to a P state by an electron scattered at an angle 0. This event may be studied in two ways: (b) Photons emitted in the P + S decay are polarization analyzed (Stokes parameters) in a selected direction and detected in coincidence with the scattered particle. In the time-reversed scheme (c), the atom A is excited by photons coming in from a selected direction, and the number of particles B leading to deexcitation are detected as function of laser polarization. The hvo approaches yield essentially equivalent information.
approaches are here common:
1. Coherence analysis, i.e., a measurement of the Stokes vector ( P , , P,, P 3 ) in one or several suitably selected directions in space. 2. Correlation analysis, in which the angular distribution of the photons in one or two planes containing the collision center is mapped. This is equivalent to a measurement of the two linear light polarizations P , and P, . The correlation approach, therefore, gives less information than (1); nevertheless, it is often used in cases where photon polarization analysis is difficult, such as for resonance excitation of atomic hydrogen or the rare gases.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
27
Figure 19 shows a schematic diagram of the pioneering photon correlation experiment that was built by the Stirling group (Eminyan er al., (1975). The atomic beam (helium) emerges perpendicular to the plane of the diagram. Figure 20 illustrates a later refinement of the experiment, also by the Stirling group (Standage and Kleinpoppen, 19761, which for the first time allowed for a full Strokes parameter analysis of the photons emitted from a planar electron-atom collisional excitation process. An alternative method, which has also been used very successfully, exploits the time-reversed scheme. Figure 18(c) illustrates how this is done. A photon beam from a selected direction excites the atom A. One then maps the P + S deexcitation as function of photon polarization for electrons scattered at an angle 8. For incident energy E, it is equivalent to the information obtained from the reverse S + P excitation experiment at an impact energy of E + A E , where A E is the S-P energy difference. The exploration of the possibilities of this approach was pioneered by Hertel and coworkers (Hertel and Stoll, 1977). In what follows, we shall combine information obtained from all these approaches.
Electron pulses electron analyzer 127O electron monochromator
Q 4 4 Photon pulses
Photon detector
Constant frpction timing discnminator Constant frpqtion timing discnminator
FIG. 19. Experimental setup used by the Stirling group to study electron-photon correlations in electron impact excitation of He 2 ' P (from Eminyan et al., 1975).
28
N. Andersen and K Bartschaf
Analyzer
electrons e beam Scattering plane FIG. 20. Experimental setup used by the Stirling group for electron-photon analysis (from Standage and Kleinpoppen, 1976).
A. LIGHTTARGETSWITHOUT SPIN IN He 1's -+ 2lP, 33P
THE
INITIALSTATE:
We start by discussing the case He 1's + 2lP, a process described by the two scattering amplitudes f s = 1 ' 2 ( M L ,= 1, ML, = 0, 6 ) = f + l and f S = ' I 2 ( M L ,= - 1, ML, = 0; 6 ) = f - shown in Fig. 21. Transitions to the ML, = 0 state is forbidden due to conservation of reflection symmetry in the scattering plane. Hence, this process is determined by three parameters: the differential cross section, a parameter describing the relative size of the two amplitudes, and an angle that fixes their relative phase.
f-1
FIG. 21. The amplitudes f + , and accounting for electron spin.
f-l
for electron impact S
+
P excitation without
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
29
The two independent dimensionless parameters may be chosen as the angular momentum transfer = (O,O, L L ) ,with
(c)
L,
=
lf+1I2
- If-I12
If,
+ If-
It2
112
- lf+1I2 -
lf-1I2
-
=
a;
w+- w _
(25)
and the alignment angle y of the major symmetry axis of the charge cloud in the scattering plane, shown in Fig. 18(a). The alignment angle y , defined mod r , is related to the phase angle 6 (cf. Fig. 21) through
s = -2y*
(26)
7r
All this may be expressed compactly through the density matrix p, which in the IML) helicity basis is given by 1 p=uu2
1+ L, O 0 0 -P,eZiY o
-Ple-2iY
1 - L,
The parameters ( L , , y ) are related to the Stokes vector $ = (P,,P2,PJ for the light emitted in the + z direction in a subsequent P + S decay through P,
2
+ iP2 = P,e2iy = - -U If+llIf-lle-iS
(28a)
and P3 = -L,
(28b)
Determination of the parameter set
thus constitutes a perfect experiment, since the wavefunction of the He2’P state can be written as
1)
=
/=
/ 7
1 + 1) - e2;” -(1
-
L,)
I - 1)
(30)
This wavefunction is normalized as (I) I I)) = uu,consistent with the trace of the density matrix (27).
30
N. Andersen and K Bartschat
In this case, the degree of linear polarization, P! = JPf + P; is a shape parameter that describes the form of the electron charge cloud, with a length/width ratio in the scattering plane given by l / w = (1 +_PI)/ (1 - PI). The total degree of polarization of the emitted light, P = lPl, is unity, i.e., P 2 = L2, P: = 1 (31)
+
An early illustration of how these parameters can be used to visualize the outcome of the collision process is given in Figs. 22 and 23 (Andersen et
al., 1984). The upper part of Fig. 22 shows experimental results at 80 eV 1' ' '
x'*B
4
1
1
1
1
'
1
f' \
\ .
+,
\ \
-
\
8"
',
\
-
,,/'
0,.
-1.
I
1
50
1
I
1
'
I
'
I
8
I
I
I
1 . .
1 \
-
I
t
9
**-.
/
i ................... I
, , .o
I
1
1
1
'
80
1
........... I\*; .................... ....
180
....
I
1:::
]
\
I I I
? - \
,
\tl
0 '
0:
L-7-
, P A
,s*-{
-
-50.
10'
, 7--o-
I
/@
P)
0.5
,/' --
0
0
'1.0
1
A
:/I"
: 0
4
I
,
,
I
...... ................. ..... ,
,
I
,
,
- 280 I
,
,
I
,
l
FIG. 22. The collision-induced angular momentum transfer L , and alignment angle y (bottom) for He l l S + 2’P excitation at an incident electron energy of 80 eV (from Andersen ef al., 1984). The labels (a)-($ between the two figures refer to the situations displayed in Fig. 23. The data are taken from Eminyan et al. (1975) (0)and Hollywood ef al. (1979) (0).
31
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
I
8 It
I
f
h
i
i
Fro. 23. The shape and dynamics of the radiating He 2'P state for electron excitation at an incident electron energy of 80 eV (from Andersen et al., 1984). The scattering angles of interest are O"(af, 25"(b), 35"(c), 50"(d), 65"(e), 70"(f), 90°(g), 110"(h), 130"(i), and lSO"(j). The upper row (a-e) corresponds to counterclockwise rotation of the electron around the atomic core, whereas the lower row (f-j) corresponds to clockwise rotation.
for the orientation parameter L , , and the lower part results for the alignment angle y , here compared with the value predicted by the first Born approximation (FBA), which describes the excitation as creation of a p orbital aligned along the direction of the momentum transfer "k -+ The orientation parameter shows a characteristic dependence on the scattering angle 8: Starting at zero, positive values grow with increasing 8 until an almost circular state is observed, with the alignment angle following the FBA prediction until this point. From then on, the orientation decreases rapidly, changes sign, and approaches its other extremum value. This corresponds again to a circular state, but with opposite sense of rotation (i.e., clockwise) of the excited electron around the atomic core. During this interval, the alignment angle is almost perpendicular to the FBA prediction, with another rapid change near the minimum value of L , . At even larger scattering angles, the orientation and the alignment angle converge back to zero until, at backward scattering angle, the state is again in its initial shape, a p orbital aligned along the beam axis. Figure 23 shows the corresponding behavior of the charge cloud in the collision
zi.
N . Andersen and K Bartschat
32 1000
1
. .
I
.
,
I
0 (1 0-19
.
.
I
.
.
I
.
'
I
.
1 .o
.
cmz/sr) 0
Hall et 01.
0
CartWright e l 01.
1
0.5
1
0.0
?
-0.5
i
0.1
d '
.iO. '$0'
'sb'
1;O'
'140' 'lS6''o0
1 .o
1 .o
0.5
0.5
0.0
0.0
-0.5
-0.5
-1
.o
0
30
60
90
120
150
Scattering Angle (deg)
186'"O
30
60
0
Beijers et 01.
A
Eminyon e l 01.
90
120
V
30
60
90
150
180
Khokoo e l 01.
120
150
180
Scattering Angle (deg)
FIG.24. Differential cross section mu and Stokes vector ( P , , P , , P3)for He 1's -+ 2'P excitation at an incident electron energy of 50 eV. The experimental data of Hall et al. (19731, Cartwright et al. (1992), McAdams et al. (1980), Beijers et al. (19871, Eminyan et al. (1974), and Khakoo et al. (1986) are compared with CCC calculations of Fursa and Bray (1995). Except for the direct P3 measurement reported by Khakoo et al. (1986), the magnitude of P3 was obtained by assuming full coherence of the radiation, and the sign was guessed by using the Khakoo et al. data and the theoretical results as a guide.
plane, with the upper strip corresponding to L, > 0, and the lower one to L, < 0. The behavior may be conveniently summarized by plotting the corresponding movement of the Stokes vector on the PoincarC sphere (Andersen and Hertel, 19861, which, however, is outside the scope of the present chapter. The current state of the art is shown in Fig. 24, which summarizes experimental and theoretical values of the parameters (a,, P I ,P z , P 3 ) at an impact energy of 50 eV. These values are converted in Fig. 25 to the standard set (u,;L ,; y , Pl), of which the first three are independent,
33
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS 1000
100
10
0
Halletal.
0
Cartwriqht et 01.
1
0.1
0
30
30
60
90
120
150
186"'
60
90
120
150
186900
0
30
Scattering Angle (deg)
60
90
120
150
180
Scattering Angle (deg)
FIG.25. Differential cross section uu and coherence parameters (LL, f', y ) for He 1's + 2' P excitation at an incident electron energy of 30 eV. The data were transformed from the measurements shown in Fig. 24.
whereas the fourth determines the shape of the charge cloud. Due to experimental difficulties associated with the circular polarization measurement of the 2'P + l l S line at 58.4 A, the magnitude of P3 was, in most experiments, obtained via lP,l = i.e., assuming completely polarized light. Then the sign was guessed by using the theoretical results and the direct measurement of P3 by Khakoo et al. (1986) as a guideline. It is important to point out that the indirect approaches forfeit the opportunity for an experimental consistency check by using the independent measurement of all three components of the Stokes vector. The importance of such a consistency check will be further discussed in Section III.C, where it is seen to be much more critical than in the present case. Furthermore, a complete experiment requires the experimental determination of the sign
dm,
34
N. Andersen and K Bartschat
of P3. This has, indeed, been achieved for 50 eV and some other energies by the Windsor (Khakoo et al., 1986) and Perth groups (Williams, 1986). We note good agreement between the CCC theory of Fursa and Bray (1995) and experiment at all scattering angles, a situation that was far from being satisfactory during the early exploration stages of this process. In summary, He 1’s -+ 2lP excitation serves today as a standard example of a perfect experiment on inelastic scattering. Similar, but less extensive, results have been obtained at other energies, and for higher members of the He n’P series. We end this section by briefly summarizing the situation for excitation of the He 1’s + n3P series, using results for 33P at 30 eV as an example. This process is interesting since, for helium, excitation of a triplet state can only occur by electron exchange. T h t set of Stokes parameters measured for the 33P + 23S photons (3889 A) is not a unit vector due to the influence of the fine structure coupling, which will effectively depolarize the electron charge cloud between collision and decay. This effect, however, can be remedied by introducing so-called “reduced” Stokes parameters (Andersen et al., 1979), labeled by a bar. Appendix B of Andersen et al., (1988) gives the following equations:
E.,
and 41 - 27
F --P3 The reduced Stokes vector is again a unit vector. The results for the coherence parameters are displayed in Fig. 26. Again, we see a satisfactory agreement between theory and the set of complete experimental observables.
B. LIGHTTARGETSWITH SPIN:Na 3,s
+
32P
3
For Na32S -+ 32P excitation, the projectile and target spins of s = double the number of scattering amplitudes from two to four (recall that fo = 01, since we have the possibility of triplet ( t ) and singlet (s) scattering, bringing us to the situation of Fig. 27.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
35
1
0.1
0.01 1 .o
0.8 0.6 0.4
0.2 0.0
0
30
60
90
120
150
Scattering Angle (deg)
18
Scattering Angle (deg)
FIG. 26. Same as Fig. 25 for He 1's + 3 3 P excitation at an incident electron energy of 30 eV. The experimental data are obtained from measurements of Donnelly et al. (1988).
Neglecting an overall phase, we thus need determination of seven independent parameters for a complete experiment. In addition to the differ-
36
N. Andersen and K Bartschat
FIG.27. Schematic diagram of triplet (f)and singlet (s) scattering amplitudes in the natural frame for 'S j 2 P transitions by electron impact. Note that A + + 6' = A - + 6'.
ential cross section o;, six dimensionless parameters must be defined, three to characterize the relative lengths of the four vectors, and three to define their relative phase angles. As a start, we parameterize the density matrices in analogy to Eq. (27), as (see also Hertel et al., 1987)
and
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
37
where
++
- 4- and as = I,!+- I,!-. In the case where unpolarized with 6' = beams are used, the total density matrix becomes the weighted sum of the two matrices p s and p', i.e.,
where
N. Andemen and K Bartschat
38
The parameters w' and w s are related to the parameter r by Hertel et al. (1987) through W'
=
WS =
1 - ws 1- 3 w ~ 3ws W'
=-
=
(+'/us used
(38)
However, we prefer the use of w'.' to r for reasons of mathematical symmetry and simplicity. At this point we have thus introduced a total of six parameters, namely gU,w', L', , Ls, , y', and ys,leaving one parameter, a relative phase, still to be chosen. Inspection of Fig. 27 suggests, for example, the angle A + . Note that the fourth angle, A - is then fixed through the relation A + - A - = 8' - 8 s
= 2 ( y S- y ' )
(39)
The first equality sign follows from inspection of Fig. 27, the second one from Eq. (26), applied individually to the singlet and triplet components. We thus use the following complete set:
These parameters 1 . Allow for a complete description of the scattering process 2. Are a natural generalization of the parameters used for unpolarized beams 3. Can be interpreted in simple physical pictures 4. Are accessible in "partial" (i.e., noncomplete) experiments The reduced Stokes vector @ of an unpolarized beam experiment is given by the weighted sum of the singlet and triplet (unit) Stokes vectors @','as fi = 3w'@' + w s i j , s
(41)
from which the set of parameters ( L L ,y , P,) for an unpolarized beam experiment may be evaluated. In particular,
L,
=
3W'LL
+ w";
(42)
Since, in general, L\ # L: and y' # ys,this causes the (reduced) degree of polarization P to be smaller than unity, i.e., P I 1 for an unpolarized electron beam. We now want to express the STU and V parameters discussed in Section I1 in terms of the complete set of parameters (40). Provided that electron exchange is the only spin dependent effect of importance for the excitation, and that the fine structure energy splitting is negligible com-
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
39
pared with the initial and final energies of the projectile, the STU parameters get vastly simplified (see Appendix C of Andersen et al., 1996). The seven polarization, asymmetry, contraction, and rotation parameters for each fine structure level reduce to the following set of only four independent parameters (the superscripts denote the J value of the excited target state):
sP -= s 'P/ 2 =
- 2s;/2
sA -= s'/2 = A T
~ 1 / = 2
u
~ 1 1 2=
(43a)
-2sy T;/Z = ~ z 1
(43b) / = 2 ~ 3 1 2= ~ 3 1 = 2 ~ 3 1 2 X z Y
112 = - 2 ~ 3 1 2 = - 2 u Y ; / 2 XY
(434
UYX
XY
(43c)
The results are
s,
s,
=
1
=
+ WSL1
-W'L\
-[
a+p+ cos A + -
24 1
T = -[a+p+ 2 a; U V
1
=
-[
COSA-] - 2w'L:
COSA++ a - @ -COSA-]
+2
~ '
(44b) (44c)
a+p+ sin A + -
2% 1
=
(44a)
-[a+@+
2 a;
sin A + + a - p - sin A-1
(44e)
The amplitude sizes a+ - and p, may be eliminated from Eqs. (44) by using
P+
-a* =
2 flu
iw'wS(1 f LL)(l f L?)
(45)
We now investigate to what extent the perfect scattering experiment has been achieved to date. We begin with the key experiment performed by the NIST group (McClelland et al., 1989) and shown in Fig. 7, but with the beam overlap modified so that scattering may take place also from the excited state. Spin polarized electrons with polarization vector perpendicular to the scattering plane were scattered superelastically from spin polarized sodium atoms in the 32P state. This state was produced by pumping with circularly polarized laser light. By reversing the directions of the two polarizations individually, the experiment allows for the determination of L: , L; , and w' (for details, see Hertel et al., 1987).
40
N. Andersen and K Bartschat
This experiment does not determine the alignment angles y f and ys. However, the off-diagonal elements of Eq. (36) show that ple2iy = 3wrp;e2i~r+ wsp;e2i~’
(46)
As illustrated in cig. 28, $is complex equation corresponds to addition of the two vectors Pt and P/, multiplied by+weighting factors 3w‘ and ws, respectively, to form the resulting vector PI. Hence, elementary geometry (the true can be applied to obtain two pairs of solutions, ( y l , ySItrue solution) and (y ‘ , ys)ghost (the other possibility):
y”y*-
* 2
where the angles y, and )I are defined in the figure. Provided experimental data are available for the parameter set ( P I ,y , w‘, L: , L:) at a given collision energy and scattering angle, two sets of possible angles ( y f , y s ) can be determined. It was pointed out by Hertel et af. (1987) that the foregoing ambiguity is mathematically identical to the one found for S + D excitation processes (Andersen et al., 1983). This will be further discussed in Section 1II.E. The ambiguity could have been resolved with the following modified version of the NIST experiment. If the laser light propagates in the collision plane perpendicular to the electron beam direction, with the electron spin polarization still parallel or antiparallel to the atomic spin polarization,
FIG.28. Vector diagram corresponding to Eq. (46). For a discussion, see text.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
41
one can determine an asymmetry parameter B analogous to the A parameter of Eq. (12). The result is (Andersen and Bartschat, 1993) B = -
w‘(1 - P ; ) - w”1
1 - P,
-
P;)
(48)
Thus, a consistency check between the results of Eqs. (47) and (48) will eliminate the ghost solution. Such an experiment, unfortunately, has not been performed yet. The ambiguity for the alignment angles can, however, be removed with help from state of the art theory. While data for all the parameters necessary for determination of the two sets of angles ( y ‘ , y s ) have indeed been obtained for electron-sodium (de)-excitation, the energy and scattering angle combinations investigated by the Adelaide (Teubner and Scholten, 1992) and the NIST (McClelland et al., 1985; Scholten et al., 1991) groups, unfortunately, do not overlap at all. Consequently, we replace the missing experimental data for PI and y with theoretical results of Bray’s calculation (Bray, 1992) at a total energy of 4.1 eV to “invert” the NIST data at this energy. This approach seems justified in light of the excellent agreement between Bray’s theory and experiment for all available collision energies. The results of the inversion are shown in Fig. 29 for the angles y r and y s . The error bars on the experimental points were obtained by first changing the theoretical results for the set of input parameters
(w‘,L> ,L y ) (recall that = 41 - (,!,‘is)’ ) by a small amount and then looking at the effect on the inverted theoretical results for the two pairs of ( y ‘ , 7’). This gives partial derivatives of y f and y s with respect to the three input values and thus allows for the calculation of error bars. (This is a somewhat pessimistic estimate of the error bars, since we assume an independent determination of the parameters w‘,L: , L: .) Since the true and the ghost solutions evaluated from theory are in very good agreement with the experimental values, we select the “true” experimental data as those that follow the true theoretical solution. Note that the two sets of solutions can cross each other, and that it is impossible to stay on the “true” experimental curve by assuming, for example, a smooth angle and energy dependence of the phase angles. Next, we note from inspection of Eqs. (44) that information on the still missing phase difference between a singlet and triplet amplitude can only be obtained from STU parameters. The T parameter was measured by Hegemann et al. (1991, 1993) at 4.0 and 12.1 eV total collision energy, whereas the NIST experiments were performed at 4.1 and 10.0 eV, respectively. Again, due to the unfortunate lack of data from different experiments at the same energy, we demonstrate an inversion method for
42
N. Andersen and K Bartschat
"1 h
cn 0
z
ul
.?
theory
- ahost
30
o - 30 - 60
30
60
90
120
150
I
180
Scottering Angle (deg)
60 A
cn
- theory
____
I
ghost experiment
30-
FIG. 29. Alignment angles y' and y s calculated from the NET (McClelland et al., 1989) data for ( w f , L: , L:) and theoretical results for (P,, y ) from scattering amplitudes of Bray (1992) for electron impact excitation of the 3*P state of sodium at an incident electron energy of 4.1 eV; 0, two sets of inverted experimental data as well as true (-) and ghost (- - -) theoretical solutions (from Andersen and Bartschat, 1993).
the A angles by using theoretical data (Bray, 1992) for the parameter T at 4.1 eV. (If experimental results for the U or I/ parameter were available, evaluation of the missing phase angle is straightforward from Eqs. (44).) The idea of the second inversion procedure is illustrated in Fig. 30. Equation (444 for the T parameter corresponds to a nonlinear equation for A' and A-. In addition, the difference between these two angles is related to the difference between the alignment angles y' and y s through Eq.(39). Consequently, solutions for A + and A - can be found by searching for crossings between the lines determined by A cos A + + B cos A - = C
(49)
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS AcosA+
43
+ BCOSA- = C
180
120
60 h
[r
s
o
I
a - 60
-1 20
-1 80
-
FIG.30. Determination of the singlet-triplet phase angles A’ and A - from experimentally observable parameters (see text).
where the constants A , B, and C are evaluated from Eq. ( 4 4 ~ )and the in Fig. 30. Due to the lines defined by Eq. (39) labeled (7’ - yt)true,shost ambiguity in the sign of the arguments in the cosines, and if the ambiguity in the pair ( y t , y s ) is taken into account, one will usually find four solutions, only one of which is correct. This is illustrated in Fig. 30 for inversion of the theoretical data at a scattering angle of 40” and a total collision energy of 4.1 eV (Andersen and Bartschat, 1993). Note that the problem can be reduced to searches in the first quadrant, since the “ghosts” in the second and fourth quadrants may be found via intersections with the dashed lines in Fig. 30, which are mirror images of the difference lines in those quadrants seen in the first quadrant. Since the slopes of the dashed mirror lines are reversed compared with the original difference lines, the actual crossing points in the second and fourth quadrants can easily be reconstructed, while the only remaining crossing, in the third quadrant, is related to the one in the first quadrant through a simultaneous sign change in A\+ and A-. The results for A’ and A - as a function of the scattering angle for a collision energy of 4.1 eV are shown in Fig. 31. For simplicity, only one ghost solution (where A \ + > 0) is shown. Again, the theoretical results help
44
N. Andersen and K Bartschat
-::I -1 800
,
, l ,
30
,
60
7,
theory , , , ghost experiment
90
,
120
,
1
150
180
60 90 120 150 Scottering Angle (deg)
180
Scottering Angle (deg)
180 120 CTI
60
0)
s LI
o - 60 -1 20
-l8O0
experiment
30
FIG. 31. Singlet-triplet phase angles A' and A - (Andersen and Bartschat, 1993) calculated from data for (w',L: ,LS, ) of McClelland et al. (19891, the corresponding alignment angles ( y ' , y s ) presented in Fig. 29, and theoretical T parameter results from scattering amplitudes of Bray (1992) for electron impact excitation of the 3*P state of sodium at an incident electron energy of 4.1 eV; 0 , two sets of inverted experimental data as well as the true (-) and one ghost (- - -) solution (see text).
to identify, in most cases unambiguously, the true solution among the possibilities obtained from an inversion of experimental data alone. Figures 32 and 33 show the full set of results obtained in this way at collision energies of 4.1 and 10 eV, respectively. We notice a very satisfactory overall agreement between the theoretical predictions and the complete set of experimental parameters (40). Nevertheless, the importance of such detailed benchmark measurements is demonstrated very clearly by looking at the parameter L: at 10 eV. Only the most sophisticated CCC theory of Bray (1994) can reproduce the experimental results for this
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
45
Na(32P) 4.1 eV
O(deg)
FIG. 32. Survey of alignment and orientation parameters for excitation of the Na 3*P state by spin polarized electrons at an incident electron energy of 4.1 eV. The differential cross sections are given in units of &sr. The experimental data of the NIST and Miinster groups have been transformed to the parameter set (40); they are compared with CCC (-; Bray, 1994) and CCO (- -; Bray, 1992) results and a 10-state close-coupling calculation of the JILA group ( ......; Zhou et al., 1995).
46
N. Andersen and K Bartschat
FIG. 33. Same as Fig. 32 for an incident electron energy of 10 eV. The dotted line shows results from a second order distorted-wave calculation by Madison et al. (1992). The differential cross section data are from Srivastava and VuskoviE (1980).
observable; in contrast, the parameter L\ is much less sensitive to the quality of the theoretical model. Although a complete set of parameters at the present time could only be extracted from available experimental data after two guesses guided by theory, the discussion showed that a complete experiment is within reach
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
47
for this case, in particular if the experimental programs of the participating groups aim at common choices of angles and energies. Indeed, the somewhat complicated inversion procedure could be avoided and the perfect scattering experiment directly be achieved, for example, as follows: A measurement of the in-plane asymmetry parameter B could resolve the ambiguity in the two ( y ' , 7') pairs. Furthermore, since the equations for the pairs ( S , , U ) and ( T , V ) can be recast in a form similar to Eq. (41), determination of one of the pairs gives a geometrical ambiguity in the ( A + , A - ) pair, as discussed for ( y ' , y'). This ambiguity can be removed by measurement of one more of the remaining S T W parameters. If ( 7 , P I ) are known from unpolarized beam experiments and (L: , L: ,w ' ) from a NIST type experiment7 any three of the five parameters B,S,, T , U , and I/ will suffice to achieve a perfect scattering experiment. The scattering amplitude information contained in the atomic density matrix (i.e., the Stokes parameters) and the reduced density matrix of the scattered electrons (i.e.7 the STU parameters) is illustrated in Fig. 34. From a Stokes parameter analysis, one obtains information about the relative phase between the two fil and fi, amplitudes and the relative and f-,amplitudes, as well as the relative phase between the two sizes of all four amplitudes. However, the Stokes parameters do not depend on the relative phases between any triplet and singlet amplitude. The STU parameters, on the other hand, can be used to determine the relative phase A + between the two fil and fil amplitudes and the relative phase A - between the two fi, and f: amplitudes, provided that the relative sizes of all four amplitudes are known from a Stokes parameter measurement.
fsl
a
Stokes
b
STU
f?, FIG.34. (a) This diagram shows which relative amplitude sizes and phases can be evaluated from a Stokes parameter analysis. (b) This diagram shows which relative amplitude sizes and phases enter into the equations (44) for the STU parameters.
48
N. Andersen and K Bartschat
The preceding discussion also demonstrates how the inversion procedures may serve as consistency checks among separate experimental data sets. Consistent experimental data should always allow for inversion within experimental uncertainties. We end the discussion of Na32P excitation by pointing out one additional consistency test. Equation (44a) points to an interesting link between Stokes parameters and STU parameters. Nickich et al. (1990) performed a measurement in a similar geometry to the NIST setup. The essential difference was the choice of a target density high enough to ensure unpolarized target atoms due to the depolarization effect of radiation trapping. For each fine structure level, the asymmetry function
was measured for the superelastic transition 32P -+ 32S. Because of time reversal invariance of the interaction, however, the S, measurement for the deexcitation process is equivalent to determination of the polarization function S, for excitation. One can thus use Eq. (44a) to predict the results of this experiment from the data set (L:, L:,w') of the NIST experiment (Hertel et al., 1987; Balashov and Grum-Grzhimailo, 1991). The data of Nickich et al. (1990) and the results predicted from the NIST experiment (Scholten et al., 1991) are shown together with theoretical results calculated from Bray's amplitudes (Bray, 1992) in Fig. 35. The consistency between the two independent experimental data sets and their agreement with the theoretical prediction underlines the power of the present formalism. C. HEAVYTARGETS WITHOUT SPININ THE INITIALSTATE: Hg 6lS, + 63P1 For the Hg6lSO+ 63P, excitation process, the argument in Section 1.B yields six independent scattering amplitudes for a J = 0 + J = 1 transition, thereby requiring the determination of one absolute differential cross section, five relative magnitudes, and five relative phases. For heavy atoms, explicitly spin independent forces, such as the spin-orbit interaction or other relativistic effects, must be taken into account-in the T operator for the collision and/or in the description of the target states (e.g., by an intermediate coupling scheme). The large number of independent parameters reflects the additional degrees of freedom that the problem presents. Figure 36, for instance, shows that for electron spin polarization in the scattering plane the charge cloud symmetry axis is no longer restricted to
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
SD 0'4
49
r------
0.2
0.0
-0.2
-n
Y .
A I
0
20
40
60
80
100
120
140
160
180
Scattering Angle (deg) FIG. 35. Spin polarization function S, for electron impact excitation of the ( ~ P ) ~ P , / , , , / , states of sodium at an incident electron energy of 10 eV; data of Nickich et al. (1990) for and Sp(2P,/2) ( 0 )from the Sp(2PI/2(0)and S,(2P3,,) ( W k prediction of S,(2P,/2 (0) NIST data (Scholten et af., 1991) for (w',L i , L$ k the theoretical curves for S,(*P,/,) (-1 and S,(2P3,2X- - -1 were calculated from scattering amplitudes of Bray (1992).
Z
X
FIG. 36. Example of a tilted and twisted charge cloud due to an in-plane spin polarized incident electron beam (from Raeker er al., 1993). The figure is for Hg 63P, excitation with impact energy 8 eV, initial spin polarization Py = 1, and scattering angle 0 = 30".
50
N. Andersen and K Bartschat
*+
f+l f+l
+
f-1
J
FIG.37. Schematic diagram of scattering amplitudes in the natural frame for J =
1 transitions by electron impact. Note that A + + 6
=
=
0
+
A-+ 6 T.
this plane; instead, it may tilt away from the plane and even twist (Raeker et al., 1993). This will necessarily lead to considerable complications in the
algebra. Omitting J = 1, Ji = M i= 0, and 8, we parameterize the six scattering amplitudes f{Mf, mf,m i ) in Fig. 37 as (Andersen and Bartschat, 1994):
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
51
The quantities (51a-d) represent no-$@ amplitudes that leave the projectile spin unchanged, whereas (51e, f) describe the cases where the electron spin is j-l@ped. The up/down arrows correspond to the initial spin projection in the natural frame. As for e-Na excitation, we want to express the photon and electron polarization properties in terms of the amplitude parameters, as well as the density matrix for the excited state. The analysis is done most systematically in the reduced density matrix formalism, with an expansion of the density matrix elements in terms of state multipoles. In the following, we shall not repeat the derivations, which are given elsewhere (Andersen et al., 1996), but just state the results. Since most information is contained in the emitted light, we start by addressing the photon pattern. 1. Generalized Stokes Parameters The emitted radiation can be analyzed systematically in terms of the "generalized Stokes parameters" introduced by Andersen and Bartschat (1994). They are defined in such a way that all four possible combinations of photon polarization analyzer and initial electron polarizations enter on an equal footing. We use the following notation: The quantity 1: p x , y , ( p ) is the light intensity transmitted by a linear polarization analyzer oriented at an angle p for incident electron polarizations in the x , y , or z direction, with the light being observed in the direction denoted by A. Similar definitions are made for the intensities transmitted by circular polarization analyzers. This gives, for example, 1: pJ45")
=
t [ 1;
p,
+ cIp2);Pz]
(52a)
and the total intensity for unpolarized incident electrons can be constructed as 1,'
=
a[ 1Y,pJ450)+ I Y P J 4 5 O ) + IU+P,(1350)+ ZYPJ1350)]
(53)
We define the generalized Stokes parameters Q$m measured with a photon detector in the direction Ti and electron spin polarization along the direction rh by taking the three other independent linear combinations of the four intensities. The second subscript j = 1,2,3 refers to the sign combinations + - -, - - +, and -, whereas the first subscript i = 1,2,3 refers to the photon polarizer settings (O", 90"), (45",135"),
+
+
+ +
N. Andersen and K Bartschat
52
and a-,a+),as for the standard Stokes vector. See Fig. 38 for sign conventions. For example, we get Zy+pJOO) QYZ
=
l1 -
Z:pJOo)
+ ZY_p(Oo) - Zy+p,(900)- ZY_p19Oo) + ZY,(O") + Z'+p,(900)+ ZYPJ9O0)
(54a)
The standard three component Stokes vector is thus replaced by a 3 matrix. All denominators in Eqs. (541456) are equal to the intensity
X
3
I,' of
Eq. (53). The first three components (54a, 55a, 56a) form the usual Stokes vector (P1,P,,P,), as measured with an unpolarized beam. The other columns contain additional information, which we shall now explore. For this purpose, we first address the density matrix. It no longer describes the electronic charge could, but instead the excited state ( J = 1) distribution.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
53
FIG. 38. Coordinate frame for definition of generalized Stokes parameters (Andersen and Bartschat, 1994). The linear polarizer settings in the directions it = x , y , z are shown for polarizer angles p = o”, 45”, 9o”, and 135”, following the notation of Blum (1981). The incident electron beam is characterized by spin polarization components * P I , +P’, or * P z , as indicated.
Since this state radiates like a set of classical oscillators completely 1 analogous to a P, state, we shall maintain our previous notation by replacing the term “charge cloud density” by “oscillator density.” We also maintain the parameter name L , though, strictly speaking, .TI would be more appropriate. We now parameterize the density matrix for the classical oscillator density by further generalizing the method used previously for sodium. For excitation with an unpolarized electron beam, the density matrix for heavy atoms is decomposed into a pair of matrices, one having positive reflection symmetry with respect to the scattering plane and the other having negative reflection symmetry. The importance of excitation of states with negative reflection symmetry and the corresponding spin-flip processes is measured by the height parameter h (Andersen et al., 1988). We consider the two cases: 1. Electron beam polarization perpendicular to the scattering plane, i.e., along the z direction of the natural coordinate system. The natural extension of the earlier decomposition is a pair of density matrices, one for spin-up electron impact excitation and one for spin-down excitation, where
54
N. Andersen and K Bartschat
“up” and “down7’ correspond to the initial spin component orientation with respect to the scattering plane. Hence, 1+Ll
-P,?e2iY =
wtpT
0 0 O
1 - L,+
0
0
0
+ w”p“ 0
1+L;T
-p,+Te-2iyT\
0 0 -p; t e 2 i ~ t 0
1
0 l-L,+’
1 + L,+1
0
0
0 0
-p;
1 e 2 i ~I
,
!
-p,+~e-ziY’ 0 1 - L y
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
55
(58i)
2y'
=
2yJ
=
&- 4+*
*--
*,&
+
7T=
-6'
7T=
-6J +7T
- 7T
(59f)
(59g)
Consequently, the decomposition of the density matrix (57) is described in terms of the cross section uu and seven dimensionless independent parameters: (uu; W T , L y , L,f l , h' ,h' ; y
1,
y
1)
(60)
This set leaves three relative phases still unknown. Before searching for additional information about these phases using electron polarization in the scattering plane (to be discussed in (2) following), we first map the results of the radiation pattern analysis with electron polarization P, by showing the relationship between the parameter set (60) and the general-
56
N. Andersen and K Bartschat
ized Stokes parameters. For an electron beam polarized along the z axis, the generalized Stokes vector matrix for observation in the $2 direction is given in terms of the density matrix parameters by (Andersen et al., 1996):
ti
(IQ;;)=-
(1-h)P,
w'(l-h')P,'
(I-h)P,
wt(l-hr)PJ -w'(l-hi)f)
-wl(l-hl)ft
f[w'(I -3h')-wJ(l-3h')]
(1 - h)P,
w ' ( 1 - ~ ' ) P J-
+ [ . ' ( I- 3 / 1 9 - w J ( 1 - 3 h ' ) ]
w l ( ~-
$[w'(l -3hr)-wJ(1-3h1)]
hJ)fj
with the (normalized) light intensity
I =I,'
=
;(I
- h)
and (1 - h ) P i = ~ ' ( 1 h')Pi'
+~
i
' ( 1 hL)Pi"
=
1,2,3 ( 6 1 ~ )
We now assume that the height of the charge cloud for unpolarized electron impact, h, is known from a standard P4 measurement (Andersen et al., 19881, i.e., a P , measurement with photon detector along the y direction in the scattering plane. One may then use the sum and the difference of the elements in the+first two columns (i.e., six paLameters) of Eq. (60) to obtain w r ( l - h T ) P T / ( l - h ) and w L ( l - h l ) P i / ( l - h), where 9 = (P,' ,P 2 , P3f ) and @ = (P,' ,P$ ,P: ). Since the degrees of = I? ' I = 1 for the two initial spin projections, one polarization P + canfirstextract c T w T ( l - h T ) / ( l - h)and c i = w"1 - h c ) / ( l - h ) from the sum of the squares of the individual components, and subsequently L:', L T L , y ' , and y ' from the Stokes vectors S T and 3'. (Since c T c c = 1 theoretically, any measured deviation from this relationship should be remedied by renormalizing all elements of the generalized Stokes matrix by a common factor.) The last column determines [w (1 - 3h ) - w (1 - 3h )]/(1 - h), which, when combined with c and c allows for determination of w h ? , and h I.Thus, knowing h , the seven dimensionless parameters of (60) can be determined from QY. The independent determination of h cannot be replaced by Eq. (59~);this equation is not independent from the others and can only serve as a consistency check. Switching now to the matrices measured in the two remaining directions y and x , equations for the nonvanishing elements in the first row of the and (QY) are written here as a column for convenience matrices (Andersen et al., 1996): 9
''
+
',
',
(Qr)
- [ w f ( l - 3h')
+wJ(l - 3h1)]+ [w'(l
-[w'(l -3h')[w'(1
+h') -~
w'(1 - 3 h l ) ] ' ( + 1 h')]
-
- h')Pg + w l ( l - hL)P,']
[ w ' ( l -h')P,'
+ [w'(l
- w'(1 - hL)P,']
- h')P: - w J ( l
-hl)P,']
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
57
with the normalized light intensity
I =I,' and
(z]:[ -
-[(l
4
+ h ) + (1 - h ) P , ]
- 3h') +wl(l - " I ) ]
-[w'(l
=
3
=
-[w'(l - 3h')-
[w'(I
wl(l
~
3h1)]
+ h ' ) - w'(l + h i ) ]
+ [w'(l +
- h')P?
[w'(I -h')P,f
+w'(l
- h')P,']
- w'(l - h l ) P { ]
- [w'(l - h')PJ - w'(l - h')P,']
with the normalized light intensity
r=z,.=
3 -4[ ( I + h )
-
Except for the P4 measurement, we see from inspection that no additional information can be obtained from generalized Stokes parameters observed in other directions (such as x or y ) with electron beam polarization vector perpendicular to the scattering plane. On the other hand, such additional measurements with photon detectors in the scattering plane can provide valuable consistency checks. As found previously, three more relative independent phases are needed to determine the scattering amplitudes uniquely. In analogy to the sodium case, we define
A-E
++- $+ +-- $-
(64b)
A"
4" - I)~
(64c)
A+=
(64a)
Inspection of Fig. 37, however, shows that only two of these are independent, since A + - A - = 6 ' - S i = 2( Y ' - Y ' ) (65) in analogy to Eq. (39). Therefore, it remains to fix the phase of the spin-flip amplitudes fJ ' relative to the nonflip amplitudes. As will become clear, a convenient choice for the remaining phase angle is
'
611
4,-
(66)
$0
A complete set of independent parameters is then given by ( u Uw ;',
L I T L, , f L h, ' , h l ; y ' , y ' , A + , A", 6
')
(67)
i.e., one absolute cross section, five relative sizes, and five relative phases. Information about the remaining three phase angles may be sought for in
58
N. Andersen and K Bartschal
experiments with in-plane spin polarization, a possibility that we will now explore. 2. Electron beam polarization in the scattering plane. No additional information is obtained with such an electron polarization if the photons are observed in the z direction (Andersen and Bartschat, 1994). However, in the directions x and y one obtains eight nontrivial components, namely IQ,"i" and Z Q P with = A X , xy, yx, and yy. To clarify the algebraic structure of the general expressions for these components, we introduce the following abbreviations: A,
=
a+Po
(68a)
A,
=
P+ao
(68b)
A,
=
a-po
(68c)
A,
=
p-ao = 4+- *o
(684
w,
st l cpo = s T L =
w, =
*+-
w, =
cp-- $bo = S T J
( 69a) -Ao-A+
(69b)
- 8'
(69c)
w4=$b--40=ssJ -Ao-Ab'-SJ
(69d)
where the phases are defined in Fig. 37. With this notation, we find
(;g) :(
+
+ + + A3 cos w 3 - A, cos
-Al sin w , A, sin o2 A, sin w 3 - A , sin w, - A , cos 0,+ A, cos w , A , cos wg - A, cos w4
=
(;g)=?(
3 A , cos W , - A, cos W , - A , sin w1 + A , sin 0,- A, sin 3
(%)=5(
w,
0 ,
+ A, sin w,
)
+
- A l cos w 1 - A, cos w 2 + A 3 cos 0, A, cos W , A , sin w1 + A , sin w , - A , sin w , - A , sin w,
(;g)=-( ++ 3 A , sin w , 2 A , cos w ,
A, sin w , + A , sin w 3 + A , sin w4 A, cos w , + A3 cos w3 + A, cos w,
)
)
(70a)
(70b)
)
(70c)
(704
Inspection shows that a measurement of the generalized Stokes parameters in the y and x directions with in-plane electron beam polarizations P,, and P, will provide the four relative magnitudes and the four relative phase angles defined in Eqs. (69). If analysis in the z direction with polarization P, has been completed, all relative magnitudes are already
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
59
known, and the four A parameters may serve as consistency checks. For example, the parameters L I and L; can be derived as
Additional information may be obtained from the w angles. They determine the relative phase angles within the amplitude triplets (f,!,, f J l ,fd ) and 2 is straightforward, as well as the derivation of formulas for the corresponding triple, quadruple, etc. coincidence measurements, or partially incoherent steps on the way. Since such developments are still challenges for the future, we refrain from further discussion.
+
+
FIG.50. Decomposition of the D state electron density in the scattering plane (from Andersen et al., 1983). Negative and positive contributions are indicated as hatched and crosshatched areas, respectively.
EXPERIMENTS IN IN ELECTRON-ATOM ELECTRON-ATOM COLLISIONS COLLISIONS EXPERIMENTS
79 79
Turning now now to to the the experimental experimental situation, situation, results results from from photon photon cascade cascade Turning coincidence analysis are presently not available in the literature, but work coincidence analysis are presently not available in the literature, but work is in progress (Wang and Williams, 1996). In the meantime, we shall is in progress (Wang and Williams, 1996). In the meantime, we shall demonstrate inversion of experimental data sets for ( P , , P 2 , P 3 , P,), using demonstrate inversion of experimental data sets for (P,, P 2 , P 3 , P,),using 28 and and resolving resolving the the again the the geometrical geometrical technique technique introduced introduced in in Fig. Fig. 28 again sign ambiguity for the pair ($+, $-)-and thereby for T--by comparison sign ambiguity for the pair (T+,?-)-and thereby for 7-by comparison 1's+ +3'D 3'D with theoretical theoretical predictions. predictions. For For this this purpose purpose we we select select the the He He1's with excitation process at 40 eV, for which experimental results with good excitation process at 40 eV, for which experimental results with good quality are are available available (McLaughlin (McLaughlin etet al., al., 1994; 1994; Mikosza Mikosza etet al., al.,1994). 1994).The The quality theory selected selected for for comparison comparison isis again again the the CCC CCCcalculation calculation of of Fursa Fursa and and theory Bray (1995). (1995). Figure Figure 51 51shows shows results results for for the the four four Stokes Stokesparameters, parameters, which which Bray
FIG.51. Stokes parameters ( P , , P , , P 3 , P4) for electron impact excitation of He 1's + 3'D transition at an incident electron energy of 40 eV. The experimental data of Mikosza et al. (1994) and McLaughlin er al. (1994) are compared with CCC calculations of Fursa and Bray (1995).
80
N. Andersen and I2 Bartschat
are used to produce the set (91) displayed in Fig. 52. Before discussing the inversion procedure, we point out that the observed radiation is nearly incoherent P = 0) at scattering angles of 100" and 120", despite a completely coherent excitation in this case. Figure 53 displays the results of our inversion procedure, starting with the unique determination of i', and i; from the available L , and poo data. As in the case of Hg63P, excitation, the individual i: do not have to vanish for scattering in the forward and backward directions. Also note that the two individual channels correspond to nearly complete circular polarization for scattering angles between 80" and 120", whereas the average angular momentum transfer is almost zero. The nearly circular nature of the radiation in the individual channels also explains the difficulty in solving for the alignment angles j ~ *in this angular range. Nevertheless, Fig. 53 shows how a reliable theory can be used to distinguish between the true and the ghost solution. In fact, the remaining problems in the ( P I ,y ) * (T', T-) inversion can, in part, be traced back to the relatively large magnitude of PI (cf. Fig. 52). (Mathematically, this results in one side of the triangle in Fig. 28 being longer than the sum of the other two, in which case we set JI = x = 0 and thereby get j ~ + =?-= y.). Based on the good agreement between theory and experiment for the parameter set (i:,i; , y ) and the fact that the theoretical data are internally consistent (even if they do not describe nature perfectly), one might suspect that the experimental data for PI are slightly overestimated. If this were indeed true, the agreement between experiment and theory might further improve in Fig. 52.
IV. Conclusions The series of examples presented in this chapter shows that the field of quantum mechanically complete experiments has developed to considerable maturity within the area of electron-atom collisions since the time of formulation of a "perfect scattering experiment" more than 25 years ago. Today, a collection of simple elastic and inelastic scattering processes in fundamental systems serve as benchmarks for current state of the art scattering theory. Several more systems are close to completion, perhaps in many cases closer than was actually realized at the time of data accumulation. We found that the analysis of incomplete data sets can often be completed with the assistance of inversion procedures and .guidance from theoretical predictions. The examples presented in this chapter span the range from the simplest case, completely described by a single amplitude,
81
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS 10
u
(1 0 - ~ ~ ~ ~ 2 / ~ ~ )
He 3'D
40eV
1 .o
1
0.0 0.1 -1
.o - ccc
,2.O0'
0.01 1.0
,
,
1
,
.
I
,
,
1
,
,
1
'
'
I
'
30
' .
" "
60
90
'
"
120
'
"
'
150
'
180
90
'
:PI
"
.
60
0
.
1.0
,
I
,
,
1
,
,
I
.
.
I
. .
I
.
1 .o
'
: Po0 0.8 -
-
0.6 -
- 0.6
0.4
-
-
0.8
0.4
0
Scottering Angle (deg)
30
60
90
120
150
180
Scattering Angle (deg)
FIG. 52. Differential cross section a, and coherence parameters ( L , , P,, y , poo, P ) for electron impact excitation of He 1's + 3'D transition at an incident electron energy of 40 eV. The experimental data of Mikosza ef al. (1994) and McLaughlin et al. (1994) are compared with CCC calculations of Fursa and Bray (1995).
82
N. Andersen and K Bartschat
He 3 ’ 0
40eV
2.0
1 .o
0.0
-1
.o
-2.0
0
30
60
90
120
150
180
0
30
60
90
120
150
180
90 60 30
0 -30
- 60 -90
Scattering Angle (deg)
FIG.53. Sublevel resolved coherence parameters (i:,y’ ) and (i;, T-) for electron impact excitation of He 1’s + 3’D transition at an incident electron energy of 40 eV. The experimental points have been calculated from the data of Mikosza et al. (1994) (“down” triangles) and McLaughlin et al. (1994) (‘‘up” triangles). They are compared with results obtained from CCC scattering amplitudes of Fursa and Bray (1995). The “true” (?+, y - ) pair (full symbols) was guessed by using the theoretical results as a guide.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
83
to a present maximum of six scattering amplitudes. All these cases were conveniently discussed within a common framework by parameterizing the change of the scattered electron polarization by means of generalized STU parameters and, for excitation, by a complete evaluation of generalized Stokes parameters for full characterization of the radiation pattern. Systematic mapping of the various dimensionless parameters provides us with a much closer insight into the detailed collision dynamics than the single differential cross section parameter would permit. On the other hand, one should not underestimate the need for accurate differential cross section measurements. It is the only absolute observable, and a truly complete experiment cannot be achieved with relative measurements alone. As is evident from our analysis, several of the existing cases deserve further experimental refinement in terms of a larger angular range or smaller error bars. It is also clear, however, that for cases of significantly greater complexity than those presented here, future progress will require further development of sophisticated coincidence setups and the ability to handle very long data accumulation times under stable conditions. For higher angular momenta and targets with spin, the ideal of completeness may quickly become an unrealistic goal within the foreseeable future. More progress is expected, in particular, from scattering and (de)excitation studies involving optically prepared states. We hope that the systematic framework presented here will serve as a helpful guide for the future exploration of this fascinating field.
Acknowledgments We thank Igor Bray and Al Stauffer for communicating data in electronic form, and John Broad for producing some of the figures. This work was supported, in part, by the Danish Natural Science Research Council (NA) and the United States National Science Foundation (KB).
References Andersen, N., and Bartschat, K. (1993). Comments At. Mol. Phys. 29, 157. Andersen, N., and Bartschat, J. (1994). J. Phys. B 27, 3189; Corrigeudurn (1996). ibid 29, 1149. Andersen, N., and Hertel, I. V. (1986). Comments At. Mol. Phys. 19, 1. Andersen, N., Andersen, T., Cocke, C. L., and Pedersen, E. H. (1979). J . Phys. B 12, 2541. Andersen, N., Andersen, T., Dahler, J. S. Nielsen, S. E., Nienhuis, G., and Refsgaard, K. (1983). J . Phys. B 16, 817. Andersen, N., Hertel, I. V., and Kleinpoppen, H. (1984). J. Phys. B 17, L901.
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Andersen, N., Gallagher, J. W., and Hertel, I. V. (1988). Phys. Rep. 165, 1. Andersen, N., Bartschat, K., and Hanne, G. F. (1995). J . Phys. B 28, L29. Andersen, N., Bartschat, K., Broad, J. T., and Hertel, I. V. (1996). Phys. Rep. (in press). Balashov, V. V., and Grum-Grzhimailo, A. N. (1991). Z. Phys. D: A4 Mol. Clusters 23, 127. Bartschat, K. (1989). Phys. Rep. 180, 1. Bartschat, K. (1993). J. Phys. B 26, 3595. Bartschat, K., and Madison, D. H. (1988). J. Phys. B 21, 2621. Bederson, B. (1969a). Comments At. Mol. Phys. 1, 41. Bederson, B. (1969b). Comments At. Mol. Phys. 1, 65. Beijers, J. P., Madison, D. H., van Eck, J., and Heideman, H. G. M (1987). J . Phys. B 20, 167. Berger, O., and Kessler, J. (1986). J . Phys. B 19, 3539. Blum, K. (1981). “Density Matrix Theory and Applications.” Plenum, New York. Borgmann, H., Goeke, J., Hanne, G. F., Kessler, J., and Wolcke, A. (1987). J . Phys. B 20, 1619. Bray, I. (1992). Phys. Reu. Left. 69, 1908. Bray, I. (1994). Phys. Reu. A 49, 1066. Brunger, M. J., and Buckman, S. J., Newman, D. S., and Alle, D. T. (1991). J. Phys. B 24, 1435. Brunger, M. J. Buckman, S. J., Allen, L. J., McCarthy, I. E., and Ratnavelu K. (1992). J. Phys. B 25, 1823. Burke, P. G., and Mitchell, J. F. B. (1974). J. Phys. B 7, 214. Callaway, J., and McDowell, M. R. C (1983). Comments At. Mol. Phys. 13, 19. Cartwright, D. C., Csanak, G., Trajmar, S., and Register, D. F. (1992). Phys. Reu. A 45, 1602. Donnelly, B. P., Neill, P. A., and Crowe, A. (1988). J . Phys. B 21, W21. Diimmler, M., Bartsch, M., GeeSmaM, H., Hanne, G. F., and Kessler, J. (19901, J . Phys. B 23, 3407. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M. C., and Kleinpoppen, H. (1974). J . Phys. B 7, 1519. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M. C., and Kleinpoppen, H. (1975). J. Phys. B 8, 2058. Farago, P. S. (1974). J . Phys. B 7, L28. Fursa, D. V., and Bray, I. (1995). Phys. Reu. A 52, 1279. Gehenn, W., and Reichert, E. (1972). Z. Phys. 254, 28. Goeke, J., Hanne, G. F., and Kessler, J. (1989). J. Phys. B 22, 1075. Hall, R. I., Joyez, G., Mazeau, J., Reinhardt, J., and Scherrnann, C. (1973). J. Phys. (Orsay, Fr.) 34, 827. Hanne, G. F. (1983). Phys. Rep. 95, 95. Hasenburg, K., Bartschat, K., McEachran, R. P., and Stauffer, A. D. (1987). J. Phys. B 20, 5165. Hegemann, T., Oberste-Vorth, M., Vogts, R., and Hanne, G. F. (1991). Phys. Reu. Lett. 66, 2968. Hegemann, T., Schroll, S., and Hanne, G. F. (1993). J. Phys. B 26, 4607. Hertel, I. V., and Stoll, W. (1977). Adu. At. Mol. Phys. 13, 113. Hertel, I. V., Kelley, M. H., and McClelland, J. J. (1987). Z. Phys. D: At.. Mol. Clusters 6, 163. Hollywood, M. T., Crowe, A., and Williams, J. F. (1979). J. Phys. B 12, 819. Holtkamp, G., Jost, K., Peitzmann, F. J., and Kessler, J. (1987). J . Phys. B 20, 4543. Kessler, J. (1985). “Polarized Electrons,” Springer-Verlag, Berlin and New York. Kessler, J. (1991). Adu. At. Mol. Phys. 27, 81. Khakoo, M. A., Becker, K., Forand, J. L., and McConkey, J. W. (1986). J. Phys. B 19, L209. Klose, M. (1995). Ph.D. Thesis, Universitat Munster, FRG.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS Kohmoto, M., and Fano, U. (1981). J. Phys. B 14, L447. Leuer, B., Baum, G., Grau, L., Niemeyer, R., Raith, W., and Tondera, M. (1995). Z. Phys. D : At. Mol. Clusters 33, 39. Madison, D. H., Bartschat, K., and McEachran, R. P. (1992). J . Phys. B 25, 5199. Massey, H. S. W. (1983). I n “Fundamental Processes in Energetic Atomic Collisions” (H. Lutz, J. S. Briggs, and H. Kleinpoppen, eds.) Plenum, New York. McAdams, R., Hollywood, M. T., Crowe, A,, and Williams, J. F., (1980). J . Phys. B 13, 3691. McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1985). Phys. Reu. Lett. 55, 688. McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1989). Phys. Rev. A 40, 2321. McClelland J. J. Lorentz, S. R., Scholten, R. E., Kelley, M. H., and Celotta, R. J. (1992). Phys. Reu. A 46, 6079. McEachran, R. P., and Stauffer, A. D. (1986). J. Phys. B 19, 3523. McLaughlin, D. T., Donnelly, B. P., and Crowe, A. (1994). Z. Phys. D : A t . Mol. Clusters 29, 259. Mikosza, A. G., Hippler, R., Wang, J. B., and Williams, J. F. (1994). Z . Phys. D: At. Mol. Clusters 30, 129. Miiller, H., and Kessler, J. (1994). J. Phys. B 27, 5933; corrigendum: ibid. B 28, 911 (1995). Nickich, V., Hegemann, T., Bartsch, M., and Hanne, G. F. (1990). Z. Phys. D: A t . Mol. Clusters 16, 261. Raeker, A,, Blum, K., and Bartschat, K. (1993). J . Phys. B 26, 1491. Scholten, R. E., Lorentz, S. R., McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1991). J. Phys. B 24, Lf553. Schumacher, C. R., and Bethe, H. A. (1961). Phys. Reu. 121, 1534. Scott, N. S., Burke, P. G., and Bartschat, K. (1983). J. Phys. B 16, L361. Scott, N. S., Bartschat, K., Burke, P. G., Nagy, O., and Eissner, W. B. (1984). J . Phys. B 17, 3775. Sohn, M., and Hanne, G. F. (1992). J. Phys. B 25, 4627. Srivastava, R., Zuo, T., McEachran, R. P., and Stauffer, A. D. (1992). J . Phys. E 25, 2409. Srivastava, S. K., and VuskoviE, L. (1980). J. Phys. B 13, 2633. Standage, M. C., and Kleinpoppen, H. (1976). Phys. Rev. Lett. 36, 577. Taylor, J. R. (1987). “Scattering Theory,” Krieger Publishing, Malabar. Teubner, P. J. O., and Scholten, R. E. (1992). 1. Phys. E 25, L301. Thumm, U., and Norcross, D. W. (1992). Phys. Reu. A 45, 6349. Thumm, U., and Bartschat, K., and Norcross, D. W. (1993). J . Phys. E 26, 1587. Wang, J. B., and Williams, J. F. (1996). Aust. J . Phys. 49, 335. Williams, J. F. (1986). Aust. J . Phys. 39, 621. Wilmers, M. (1972). Ph.D. Thesis, Universitat Mainz, FRG. Zhou, H.-L., Whitten, B. L., Trail, W. K., Morrison, M. A., MacAdam, K., Bartschat, K., and Norcross, D. W. (1995). Phys. Reu. A 52, 1152.
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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL. 36
STIMULATED RAYLEIGH RESONANCES A N D RECOILINDUCED EFFECTS J.-Y COURTOIS Institut d’Optique Thioorique el Appliqu6e Orsay. France
G. GRWBERG Laboratoire Kastler-Brossel Dkpartement de Physique de I’Ecole Normale Supkneure Pans. France
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Stimulated Rayleigh Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Stationary Two-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Other Examples of Stimulated Rayleigh Resonances in Atomic Physics D . Examples in Molecular Physics .......................... E . Stimulated Rayleigh Resonances in Solid State Materials . . . . . . . . . . . . I11. Recoil-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Recoil-Induced Resonance as a Stimulated Rayleigh Resonance . . . . . B. Experimental Observation of Recoil-Induced Resonances C . The Recoil-Induced Resonance as Raman Processes between Different Energy-Momentum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Atomic Bunching in the Transient Regime E . Coherent Atomic Recoil Laser (CARL) IV . Other Recoil-Induced Effects in Atomic and Molecular Physics . . . . . . . . . . A . Recoil Effects in Saturated Absorption Spectroscopy B. Recoil Doublet of Optical Ramsey Fringes . . . . . . . . . . . . . . . . . . . . C . The Ramsey-Bordt Matter Wave Interferometer D . The Experiment of Kasevich and Chu ...................... E . Recent Advances in Atom Interferometry Based on the Photon Recoil . . . . F. Recoil-Induced Inversionless Lasing of Cold Atoms . . . . . . . . . . . . . . . G. Atomic Recoil and Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . H . Other Related Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Copyright Q 1996 by Academic Press. Inc. All rights of reproduction in any form resewed . ISBN 0-12-003836-6
88
.I.-Y. Courtois and G. Grynberg
I. Introduction Light scattering occurs as a consequence of fluctuations in the optical properties of a material medium. It is indeed well known that a completeZy homogeneous material can scatter light only in the forward direction (see, for example, Fabelinskii, 1968). A light scattering process is said to be spontaneous if the fluctuations that cause the light scattering are excited by thermal or by quantum mechanical zero-point effects. By contrast, a light scattering process is said to be stimulated if the fluctuations are induced by the presence of a light field. One of the simplest technique for investigating stimulated scattering mechanisms is pump-probe transmission spectroscopy, which is illustrated in Fig. l(a). This figure shows a material system interacting with two externally applied laser beams, namely an intense pump beam of frequency w and a weak probe beam of frequency wp = w 6. Under the most general circumstances, the probe transmission spectrum obtained by recording the intensity of the probe beam after transmission through the scattering medium as a function of the pump-probe frequency detuning 6 has the form shown in Fig l(b), in which Raman, Brillouin, and Rayleigh features are present. Raman scattering typically results from the interaction of light with the vibrational modes of the molecules constituting the material system. Brillouin scattering is the scattering of light from sound waves, that is, from propagating pressure (hence density) waves. Rayleigh scattering is the scattering of light from any other nonpropagating modulation of material observables. It is also known as quasi-elastic scattering because there is almost no frequency shift between the incident and scattered beams. The reason that these scattering processes can lead to stimulated amplification or absorption of the probe beam (see Fig. lb) is that the interference of the pump and probe fields contains a frequency component at the difference frequency S. The observables of the material system can be driven by this light interference, which therefore acts as a source for the generation of macroscopic observable modulations. Thus, the beating of the pump wave with the modulation of the material observable tends to reinforce the probe wave, whereas the beating of the pump wave with the probe field tends to reinforce the observable modulation. Under proper circumstances, the positive or negative feedback described by these two interactions can lead to exponential growth or decay, respectively, of the amplitude of the probe wave. More precisely, the interaction of the pump field with the observable modulation yields a macroscopic polarization of the material system having the same characteristics (frequency, polarization, wavevector) as the probe beam. However, the phase of this polariza-
+
STIMULATED RAYLEIGH RESONANCES
89
scattering medium
T
F 0P
=0+6
Brillouin
anti-Stokes I
0
6 FIG. 1. Investigation of stimulated scattering by pump-probe spectroscopy. (a) A pump beam of frequency w and a probe beam of frequency wp are sent into a nonlinear medium. Because of the excitation of an observable in the nonlinear medium, their propagations are coupled. The probe beam intensity is recorded versus S = up - w at the exit of the nonlinear medium. (b) Typical probe beam transmission spectrum. The stimulated Rayleigh resonance is centered at 6 = 0 and usually displays a dispersive shape. The lateral resonances correspond to stimulated Raman and Brillouin scattering.
tion generally differs from that of the probe field because the modulation of the material observable is phase-shifted with respect to the pump-probe excitation wave (typically, this phase shift results from the finite response time of the material system). In other words, the polarization of the medium exhibits a nonzero component being 7~/2 phase-shifted with respect to the probe beam. The work of the probe field onto this component leads to the modification of its intensity as displayed by the probe transmission spectrum. The fact that the energy transfer between the pump and probe beams via the material system exhibits resonances, or 2 component of the material polarequivalently that the ~ / phase-shifted
90
J.-Y Courtois and G. Grynberg
ization presents resonant enhancements, is therefore related to resonant variations of the phase shift in the medium response and/or of the modulation amplitude of the material observable. Stimulated Rayleigh scattering was discovered in dense molecular media at the end of the 1960s (Mash et al., 1965; Bloembergen and Lallemand, 1966; Chiao et al., 1966; Cho et al., 1967; Fabelinskii, 1968). It was then identified in stationary two-level systems (Mollow, 1972) and in optical crystals (Giinter and Huignard, 1988), where it is better known as the photorefractive effect. During the 1980s, stimulated Rayleigh processes induced by optical pumping were discovered in various dilute atomic systems (Grynberg et al., 1990). Experimental interest in this subject has been renewed because of the discovery of many original stimulated Rayleigh processes in laser-cooled atomic vapors (Courtois and Grynberg, 1992, 1993; Lounis et al., 1992; Hemmerich et al., 1994; Courtois et al., 1994). One of the most striking examples of these new mechanisms are the so-called “recoil-induced resonances,” which were first predicted and interpreted by Guo et al. (1992) in terms of stimulated Raman processes between energy-momentum states differing because of the momentum exchange between the pump and the probe fields during photon redistribution processes. Soon afterward, these resonances were observed experimentally and reinterpreted by Courtois et al. (1994) in terms of stimulated Rayleigh scattering involving atomic spatial bunching. Beside its intrinsic interest in the framework of nonlinear optics, the Rayleigh-Raman duality of recoil-induced resonances provides a clear illustration of the possible ambiguity in the identification of a photon recoil-induced effect. Such an ambiguity is often found in atomic and molecular physics because many effects that are easily explained in terms of momentum exchange between atoms and photons can be alternatively interpreted without involving explicitly the photon recoil. The organization of this chapter is as follows. We present in Section I1 the basic ideas about stimulated Rayleigh scattering by considering more particularly the situation where it arises from a relaxation process going on in the material system, and we describe a few experimental observations made in atomic and molecular physics. We then consider the case of nonstationary two-level atoms, and we derive the shape and characteristics of the recoil-induced resonances (Section 111). In particular, we show that these resonances can be interpreted either as originating from a stimulated Rayleigh effect or as a stimulated Raman phenomena between atomic energy-momentum states having different momenta. Finally, to make a clear distinction between the physical phenomena that pertain directly to recoil-induced processes (i.e., that actually permit the measurement of the photon recoil) and those for which the introduction of the
STIMULATED RAYLEIGH RESONANCES
91
recoil constitutes a mere physical convenience, we review in Section IV some indisputable manifestations of the photon recoil in atomic and molecular physics.
11. Stimulated Rayleigh Resonances This section is devoted to the presentation of the basic physical ideas about stimulated Rayleigh resonances (Section 1I.A) and of some experimental illustrations (Sections 1I.B-El. For the sake of clarity in the presentation, we restrict ourselves to those manifestations of stimulated Rayleigh scattering that are observable on pump-probe transmission spectra. However, the concepts presented hereafter can be readily extended for describing resonant variations of the nonlinear refractive index, or fourwave mixing (and phase conjugation) spectra. A. INTRODUCTION As previously mentioned, stimulated Rayleigh resonances displayed by pump-probe transmission spectra originate from the diffraction of the pump wave onto nonpropagating observable modulations that are phaseshifted with respect to the pump-probe interference pattern. It appears from this scheme that the interpretation of a stimulated Rayleigh resonance consists essentially in identifying (1) the modulated observable V , (2) its driving mechanism, (3) the physical origin of its phase shift with respect to the excitation, (4)the diffraction mechanism of the pump beam onto the observable modulation. In order to make a satisfactory compromise between generality in the theory and clarity in the presentation, we will only consider in this section the most commonly encountered situation, where the phase shift of the observable modulation originates from relaxation mechanisms taking place in the material system. This restriction will be relaxed in Section 111. We consider a set of atoms or molecules interacting with two incident beams E (pump) and Ep (probe) with respective frequencies w and wp =
0
+ 6:
E = R e [ E e x p { - i ( w t - kar)}] Ep = Re[ Ep exp{ --i( wpt For the sake of simplicity, the fields are presently considered as scalar quantities. Effects associated with the vectorial character (polarization) of the field will be discussed later on. We also assume that each beam
92
J.-Y Courtois and G. Grynberg
separately does not undergo linear absorption during propagation, although this assumption can be easily relaxed in the case of a weakly absorbing medium. The quantity 77 = Ep/Z,equal to the ratio of the probe to the pump fields, is assumed to be small (1771 }I}
(2)
In a nonlinear medium, the electrical susceptibility x is a function of one or several atomic observables V that are assumed to depend on the field intensity.’ For the present discussion, we assume that there is just one relevant observable, the time evolution of which is characterized by a singZe relaxation rate y. In other words, V evolves according to dV
- + y ( V - V,,) dt
=
0
(3)
where V,, is the equilibrium value of V for a time independent field intensity. Because the influence of the probe beam reduces to a small spatiotemporal modulation of the total field intensity, it is possible to deduce the probe-induced modification of the medium from the intensity variation of the steady state observable V,, in the vicinity of its value V,, obtained in the absence of the probe beam (the intensity then merely coincides with the pump intensity Z = l8I2/2). If the medium had an instantaneous response time to the pump-probe excitation, the effect of the probe beam would be readily obtained through the expansion of V,, up to first order in the small parameter Q = gP/g:
-i[ 6 t
-
(kp - k) * r]})
(4)
Equation (4) somehow characterizes the “instructions” given by the probe to the material system to adapt to its presence. How the medium actually complies with these instructions, though, depends on its response time.
’ In the general case, these observables may also depend on the field polarization.
94
J.-Y. Courtois and G. Grynberg
Using Eq. (3), we thus find that the actual steady state value for V in the presence of the probe reads: Y
y-iS
-
exp{ -i[ S t - (k, - k) r]))
(5)
The functional dependence of the macroscopic polarization on the observable V being given, it is possible to derive from (5) the expression of the susceptibility x of the medium up to first order in 7.This yields
Y
dV dl
y-iS
exp{-i[ S t
-
(k, - k)
- 1-11}
and hence the total polarization P is
P
= E,
Re Eo
( x ( V,) [ 8exp{ - i ( w t - k . r)} + 8, exp( -i(
+-Re 2
d x dV0 ( d V dl
[
w,t -
k,
*
r))]}
Y y-iS
- - 2YP8*-
xexp{-i[St-(k,-k).r]]
1
+ c . c . Pexp{-i(wf-k.r)}}
(7)
Because we are interested in the probe propagation through the material system, we only consider the component P, of the polarization that radiates a field in the probe direction. By setting
P,
=
Re[ 9, exp{ -i( wPt - k, * r)}]
with k, = ~ ~ < w , / and c )x ‘ the real part of x(V,), and by using the slowly varying envelope approximation, one obtains, as the propagation equation for the probe field,
For a nonabsorbing medium, i.e., a medium for which the imaginary part x(V,) is equal to 0, Eqs. (7) and (8) yield
x” of
STIMULATED RAYLEIGH RESONANCES
95
with g=
(
1 d-x 2dV
YS
OP
+i
dV d l Z c ~ ~ ) [ - y 2 + S 2 y 2 + S 2
The real part of g, which describes the modification (absorption or amplification) of the probe amplitude,* is thus a dispersive function of 6 for a nonabsorbing medium? In particular, the asymptotic value of g for S / y + + m is 0. This is because the translational motion of the light interference pattern becomes so fast that the medium is almost uniformly excited and the contrast of the material grating becomes vanishingly small. Equation (10) shows that the gain also vanishes as 6 goes toward 0. This is because in that case the pump-probe modulation is so slow that the medium can adapt quasi-instantaneously to the excitation wave; hence, the interference pattern and the material grating exactly coincide. The extra rr/2 phase shift undergone by the pump beam during diffraction on the grating leads to a transmitted probe being rr/2 phase-shifted with respect to the diffracted pump. Therefore, no constructive nor destructive interference takes place between these two waves. This shows that to observe gain for S = 0, it is necessary that another physical mechanism induces a spontaneous phase shift between the stationary light pattern and the material grating. An example of such a behavior can be found in photorefractive crystals such as BaTiO, (Giinter and Huignard, 1988). Finally, although we do not discuss the stimulated Rayleigh structures displayed by four-wave mixing spectra, we note that in such a situation, both the 0 and the rr/2 phase-shifted components of the material polarization contribute to the signal. Therefore, the physical contents of the Rayleigh resonances in probe transmission and in four-wave mixing generation are slightly different. B. STATIONARY TWO-LEVEL ATOMS The identification of a stimulated Rayleigh resonance on the probe transmission spectrum of an ensemble of stationary two-level atoms can be
* The gain coefficient for the intensity is 2 Re(g).
+
In the case of a weakly absorbing medium ( x''
Amplifier
/by X
FIG. 12. Stark shift measurement apparatus. Details are given in the text (Li and van Wijngaarden, 1996, by permission).
PRECISION LASER SPECTROSCOPY
169
shifts of transitions in barium (Li and van Wijngaarden, 1995a) and calcium (Li and van Wijngaarden, 1996). The generation of an atomic beam and the fluorescent detection has been discussed in Section 1V.B. Ytterbium atoms were excited by a laser in a field-free region and in a uniform electric field. The electric field was generated using two highly polished stainless steel disks having a diameter of 7.62 cm. The spacing was determined to be 1.0163 0.0003 cm using precision machinist blocks, the size of which was specified to within 2.5 X cm. Plate voltages of up to 50 kV were continuously monitored using a precision voltage divider that reduced the voltage by a factor of 5000 with an accuracy of 0.01% (Julie Labs KV-50/01). The reduced voltage was measured by a voltmeter with an uncertainty of less than 0.002%. The electric field shifts the transition by an amount
Av
=
KE2
( 18)
where the Stark shift rate
Here, m, has been set to zero since the laser was linearly polarized along the quantization axis, which was specified by the electric field. Hence, only ~ ) state was populated. Equation (19) the m = 0 sublevel of the ( 6 . ~ 63P, holds for the even isotopes of ytterbium that do not have a nuclear spin. For 171,173Yb, the hyperfine interaction must also be considered. For simplicity, only the transition in I7'Yb to the F = level of the ( 6 ~ 6 p ) ~ P , state was studied, which has a Stark shift rate given by
,
The tensor polarizability of the 'PI state could then be determined by subtracting (20) from (19). The laser frequency was tuned across the transition while fluorescence produced by the radiative decay of the excited state to the ground state was detected by two photomultipliers (PM1 and PM2). The signals were processed by separate lock-in amplifiers. Spectra similar to that shown in Fig. 6 were obtained, and the frequency was calibrated as has been described in Section 1V.B. The results of nearly 500 wavelength scans taken at various electric fields are shown in Fig. 13. A least squares fit of a straight line y = kE2+ y o to the data yielded K = - 15.419 k 0.048 kHz/(kV/cm)2. The frequency shift at zero field y o was 5.33 MHz. This offset arises from a small difference of the intersection angle of the laser and atomic beams in the field and field-free regions. The tensor polarizability was found to be a2 = 5.81 f 0.13 kHz/(kV/cm)*. The result is in good agreement with
170
W.A. van WQngaarden
E2 ( kV
/ cm ) 2
FIG. 13. Frequency shift versus electric field squared for the ytterbium ( 6 ~ ) ’ ~ s ~ ( 6 ~ 63P1 ~ ) transition (Li and van Wijngaarden, 1995b, by permission). --f
5.99 f 0.34 kHz/(kV/cm)2 found by an optical double resonance experiment (Rinkleff, 1980) and 6.04 f 0.21 kHz/(kV/cm)2 obtained using quantum beat spectroscopy (Kulina and Rinkleff, 1982).
C. CESIUM6P3/2
+
nS,/, (n
=
10-13) TRANSITIONS
An example of a Stark shift measured as illustrated in Fig. 4(b) is an
experiment that studied the 6P3/2 (10-13)S,/2 transitions in cesium (van Wijngaarden et al., 1994). The apparatus was similar to that shown in Fig. 12. An oven generated two cesium atomic beams propagating in opposite directions. One atomic beam passed through a field-free region, whereas the other beam traveled through a uniform electric field. Atoms were excited from the 6S,/2 ground state to the 6P3/2 state by a diode laser that generated a few milliwatts of light at 852 nm. A ring dye laser then excited the 6P3/: + nS,/, ( n = 10-13) transition. Part of the dye laser was frequency shifted by an acousto-optic modulator. The unshifted laser beam at frequency Y excited the atoms in the field-free region, --j
PRECISION LASER SPECTROSCOPY
171
whereas the laser beam having frequency v - vAo excited the atoms passing through the electric field. Fluorescence from the field and field-free regions was recorded as the dye laser frequency v was scanned across the resonance. The frequency interval separating the fluorescent peak observed in the two regions was given by A = -hvAo - K E 2
(21)
where the shift rate is
K = -1{ 2
- a0(6P3/2)
- a2(6p3/2)}
(22)
A was plotted versus the electric field squared, as is shown in Fig. 14. A line was fit to the data and the field such that atoms in the field-free and field regions were simultaneously in resonance, i.e., A = 0 was found. The polarizabilities a0(nS,/,) were found using the small contributions of ao(6P312) = 407 and a2 = -65.1 kHz/(kV/cm)* calculated by Zhou and Norcross (1989). The results listed in Table V agree with those found
FIG. 14. Frequency separation A of fluorescent peaks observed in field and field-free regions versus electric field squared for excitation of the cesium 13S,,, state using an acousto-optic modulation frequency of 350 MHz (van Wijngaarden et a/., 1994, by permission).
172
W.A. uan Wjngaarden TABLE V SCALARPOLARIZABILITIES OF THE CESIUM (10-13)S,,2 STATES
n
Fredriksson and Svanberg (1977)
van Wijngaarden et al. (1994)
Theory
10 11 12 13
123 & 6 322 + 16 720 45 1650 + 170
119.06 f 0.28 309.70 f 0.26 713.48 f 0.58 1491.20 f 1.22
118 309 709 1490
*
by Fredriksson and Svanberg (1977) but are much more accurate. The latter group used a lamp that excited atoms in an atomic beam to the 6P3/2 state. The 6P3/2 + (10-13)S,/2 transitions were excited by a dye laser having a linewidth of about 75 MHz. Data was taken at fked dye laser frequency as follows. The electric field applied across the atomic beam was increased from 0 to a maximum of 7 kV/cm while fluorescence produced from the radiative decay of the excited nS,,, state was monitored by a photomultiplier. Flourescent peaks occurred whenever the dye laser excited one of the hyperfine levels of the 6P3/2 state to the Starkshifted n S , / , state. The Stark shift rate was then found using the hyperfine splittings of the 6P3/2 state along with the field strengths corresponding to the peak maxima. The experimental accuracy was limited by uncertainties in the electric field determination and by the accuracy of the hyperfine data of the 6P3/2 state then available. The data listed in Table V agree closely with results computed using the method developed by Bates and Damgaard (1949). These results were found using experimentally measured energies and assuming a Coulomb potential to describe the interaction of the valence electron and the nucleus plus the inner core electrons. This approximation has been used to compute polarizabilities in several alkali atoms (Gruzdev et al., 1991; van Wijngaarden and Li, 1994~1,and good agreement with the experimental data has been obtained for all but the lowest P states. This is not surprising since the Coulomb approximation best describes excited states that have minimal penetration of the inner electron core and have a small spin-orbit interaction.
D. CESIUMD LINES Several groups have studied the Stark shifts of the 6S,,, + 6P1/2,3/2 transitions in cesium. Hunter et al. (1988) used two glass cells loaded with
173
PRECISION LASER SPECTROSCOPY
cesium atoms. One cell was also filled with 6 Torr of nitrogen gas that pressure shifted the cesium resonance by 40 MHz. The second cell was made by gluing two metal plates onto a pyrex cylinder. These plates served as the electrodes that generated an electric field. The fluorescence signal observed by a photodiode in the pressurized cell was used to lock the laser to the transition frequency. A n acousto-optic modulator shifted part of the laser beam by 40 MHz. The frequency-shifted laser beam was incident on the second cell, across which an electric field was applied. The modulation frequency needed for the laser to excite the Stark-shifted resonance was measured at various electric field strengths. Data were taken at laser powers of about 1 pW to minimize ac Stark shifts, optical pumping, and saturation effects. The result for the D , line Stark shift is listed in Table VI. The 6S,,2 + 6P312 transition was excited to the various hyperfine levels of the The experiexcited state to determine the tensor polarizability ~t2(6P3/2). ment was done using a ring dye laser and repeated with a diode laser. The diode laser experienced slightly less frequency jitter and therefore produced data having a smaller statistical variation. The experimental accuracy was limited by the determination of the electric field. The cells were plagued with systematic uncertainties including leakage currents and needed to be coated with surfa-sil to minimize field inhomogeneities. Unfortunately, the coatings deteriorated noticeably after several months. 6P3l2 Tanner and Wieman (1988a) studied the Stark shift of the 6S,/2 transition using apparatus similar to that shown in Fig. 10. An atomic beam passes through two plates that were separated by 0.3950 k 0.0002 cm. One plate had a transparent conductive coating that permitted the laser to be transmitted, whereas the second plate was a gold-coated mirror. Electric fields were generated by applying voltages of up to 18 kV to the plates. Voltages were determined using a high voltage divider and a digital voltmeter. The divider drifted slightly with temperature, limiting the fracThe diode laser tional uncertainty of the voltage calibration to 6 X frequency was locked to the cesium resonance using a saturation signal observed in a cell. Part of the laser was frequency shifted by an acoustooptic modulator and excited the atoms passing between the field plates. The Stark shift was determined by measuring the modulation frequency needed to keep the atoms in resonance. The result listed in Table VI is somewhat lower than that found by Hunter et al. but has a five times smaller uncertainty. The accuracy was limited by the determination of the electric field. A very accurate measurement of the Stark shift of the D , line was done by Hunter et al. (1992) using the apparatus illustrated in Fig. 15. Two diode lasers excited the 6S,,, + 6P,,, transition at 894 nm. One laser was --f
CL
TABLE VI
4 P
PRECISION STARK SHIFTSUMMARY
Transition l-+u
Atom Ba
(6s)' 'So
-+
(6s6p)'PI
Polarizability a&)
-
2a2(u) - ao(l)
4 4 )
( 4 ~'SO) ~
cs
6S1/2 + 7s1/2 6P,/2 6S1/2
-+
( 4 ~ 4 p ) ~ ~ I a,(u)
+
6S,/2
+
6P3/2
-
2 a 2 ( ~)
a&) - ( Y O U ) a&) - a&) a,(u) - a,(l) a,(u) - (Y"(1) a,(u) - a,(l) a,(u) - ao(l) a,(u) - a,(l) a,(u) ff,(U)
6p3/2
12s1/2
a&) - ao(l) a,(u) - ao(l) a,(u) - an(/)
Wp
aJu)
-
ao(l)
~4P1/2
a,(u)
-
ao(1)
2p,/2
a&) - ao(l) a&) - ao(I) a,(u) - a,W a&) - a,(l)
+
6p3/2
+
6p3/2
6P3/2
K
4
Li
2SI/2
2s1/2
s
-+
-
'OSl/2
'lSl/2
+
* 2p3/2
Ci,(U)
Reference
57.06 k 0.12 k 0.10 - 10.79 f 0.29
Li and van Wijngaarden (1995a) Kreutztrager and von Oppen (1973) Hese et al. (1977)
24.628 k 0.082
Li and van Wijngaarden (1996)
- 10.72
4 u ) Ca
Value (kHz/(kV/cm)2 Y
+
1420.6 4.8 241.2 + 2.4 230.44 k 0.03 230.5 314.2 f 3.2 308.6 f 0.6 308.0 2.0 -64.7 -65.3 k 0.4 -65.1 118,720 280 309,360 f 260 713,140 f 580 1,490,900 f 1200
+
78.800
0.010
+
-9.243 0.004 -9.234 & 0.082 - 9.272 -9.281 If: 0.100 0.408 0.011
+
Watts et al. (1983) Hunter et al. (1988) Hunter et al. (1992) Zhou and Norcross (1989) Hunter et al. (1988) Tanner and Wieman (1988a) Zhou and Norcross (1989) Hunter et al. (1988) Tanner and Wieman (1988a) Zhou and Norcross (1989) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) Miller e r a / . (1994) Hunter ef al. (1991) Windholz et af. (1992) Pipin and Bishop (1993) Windholz et al. (1992) Windholz et al. (1992)
3 & C
3
i!a
TABLE VI (continued)
Atom
Transition I+u
Value (kHz/(kV/cd2Y
Polarizability
0.399
(YO(1)
a
Note that 1 kHz/(kV/cmI2 = 4.0189ai/h and at = 1.4818 X
Reference Pipin and Bishop (1993)
48.99 f 0.11 49.28 f 0.15 -21.97 k 0.10 40.56 f 0.14
Windholz and Neureiter (1985) Windholz and Musso (1989) Windholz and Musso (1989) Ekstrom et al. (1995)
122.306 f 0.016
Miller et al. (1994)
-91.8 k 0.4 13.29 f 0.06
Neureiter et al. (1986) Neureiter et al. (1986)
30.838 f 0.096 5.81 k 0.13 5.99 f 0.34 6.04 f 0.21
Li and van Wijngaarden (199%) Li and van Wijngaarden (1995b) Rinkleff (1980) Kulina and Rinkleff (1982)
cm3, where a, is the Bohr radius and h is Planck's constant.
176
W A. van Wijngaarden
Atomic Beam
- - - - _ _- - - 1
-------
FIG. 15. Stark shift measurement apparatus for studying the alkali D lines (Hunter el al., 1992, by permission).
locked to the D, line using the saturated absorption signal observed by a photodiode (PD) in a cell while the second laser excited an atomic beam as it passed through an electric field. Considerable care was spent designing the field electrodes to permit an accurate determination of the electric field. Two h/10 optical quartz flats were coated with indium tin oxide, which has a transmission coefficient of over 80% at 894 nm. The fluorescence was therefore transmitted through the electrodes and detected by a photomultiplier (PM). The plate separation distance of about 2.8 mm was precisely determined using four 80% reflecting aluminium pads 2 mm in diameter that were placed at the corners of a 1.3-cm square centered on each electrode. The aluminium pads on the two electrodes were aligned to form four separate Fabry-Perot etalons. The electrode spacing could then be monitored throughout the experiment by measuring the free spectral range of the four etalons using a ring titanium sapphire (Coherent 899-21) laser and a wavemeter. This permitted the electrode spacing to be determined with a fractional uncertainty of 40 ppm. Voltages of up to k7.5 kV were applied to the plates. The voltage was determined using a high voltage divider chain accurate to 60 ppm that was constructed using
PRECISION LASER SPECTROSCOPY
177
precision wire wound resistors. The reduced voltage was measured with a voltmeter to a precision of 30 ppm. The experimental procedure was as follows. Part of the second diode laser was frequency shifted 60 MHz by an acousto-optic (A01 modulator. The frequency-shifted laser beam was then used to lock the diode laser to the Stark-shifted transition found by passing the atomic beam through the electric field. Part of each of the two diode laser beams was focused onto a fast photodiode (FPD). The acousto-optic modulator shifts the beat note to a higher frequency that is relatively insensitive to noise effects, which in general have lower frequency components. The beat frequency was measured by a counter as a function of the electric field to determine the Stark shift. Data were taken at various laser powers and voltages to check for systematic effects. The results given in Table VI are substantially more accurate than data found in their earlier experiment (Hunter et al., 1988). Tanner and Wieman (1988a) also found results that were lower than those obtained by Hunter et al. (1988) in their study of the cesium D, line Stark shift. Hunter et al. (1992) attributed this discrepancy to an underestimate of the electric field uncertainty in their initial work. The improved experiment used an atomic beam instead of a cell for observing the Stark-shifted transition and therefore did not suffer from the various problems discussed earlier. Hunter et al. have also used diode lasers to study the Stark shifts of D ,lines in lithium (Hunter et al. 1990, potassium, and rubidium (Miller et al., 1994). Several theoretical estimates of the D line Stark shifts have been made. The most accurate is that of Zhou and Norcross (19891, which is listed in Table VI. They used a semiempirical potential composed of a Thomas-Fermi potential plus a term describing the polarization of the inner electron core. The various potential parameters were adjusted to obtain optimum agreement of computed and measured excited state energies (Weber and Sansonetti, 1987). This potential was then used to solve the Dirac equation for the single valence electron. The close agreement between the calculated values and the measured results obtained for the Stark shifts of the D lines and the tensor polarizability of the 6P3,2 state is impressive. E. PRECISION STARKSHIFTSUMMARY A summary of Stark shifts measured with uncertainties of less than 0.5% is given in Table VI. Many of these results have been found using experimental techniques that have already been presented and are therefore not further discussed. The group led by Windholz has studied a number of transitions in lithium (Windholz et al., 19921, sodium (Windholz and
178
W. A. van Wjngaarden
Neureiter, 198.9, and samarium (Neureiter et al., 1986). They used a ring dye laser to excite an atomic beam in a field-free region and in a uniform electric field. Fields of up to 300 kV/cm were generated by applying high voltages to stainless steel plates. Silver-coated optical glass flats could not be used as electrodes since the coatings were destroyed by occasional sparks at these high voltages. The laser frequency was scanned across the resonance, and the fluorescence was detected by photomultipliers. The change in laser frequency was monitored by passing part of the laser beam through a Fabry-Perot interferometer as is shown in Fig. 1. The accuracy of their results was limited to a few tenths of a percent by uncertainty in the frequency marker positions generated using the etalon. For the case of the lithium D, line, their data are in good agreement with the more accurate results found by Hunter et al. (1991). Both of the experimental determinations are a few percent lower than a theoretical estimate (Pipin and Bishop, 1993). The latter used a combined configuration interaction Hylleraas method to calculate wavefunctions and matrix elements. An experiment by Ekstrom et al. (1995) used the recently developed technique of atom interferometry (Adams et al., 1994) to determine the ground state polarizability of sodium. Their apparatus consists of a threegrating Mach-Zender interferometer. The transmission gratings have a 200-nm period and generated two atomic beams separated by 55 pm. One beam passed through an electric field created by applying a voltage across two metal foils. This generated a relative phase shift between the two beams given by
where u is the velocity of the atoms, a. is the ground state polarizability of sodium, E is the electric field, and L is the length of the electric field region. The two atomic beams were then recombined, and the resulting interference pattern was studied using a hot wire detector that was mounted on a translation stage. Phase shifts of up to 60 rad were observed using fields of several kV/cm. The scalar polarizability a 0 ( 3 S , / , ) was determined to be 40.56 k 0.14 kHz/(kV/cm)’. The uncertainty is due to statistical and systematic effects. The latter was dominated by geometrical effects such as fringing electric fields that affect the interaction length L. These were studied using electrode foils of 7 and 10 cm in length and using guard electrodes to minimize the fringing fields. Another complication was modeling the velocity distribution of the atoms in the atomic beam to estimate the average velocity u. The final result for ffo(3S1/2) is substantially better
PRECISION LASER SPECTROSCOPY
179
than the value of 41.0 f 2.9 kHz/(kV/cm)2 (Hall and Zorn, 1974), which was determined by measuring the deflection of an atomic beam in an inhomogeneous electric field. The result obtained by the atom interferometric experiment can be combined with the measured Stark shifts of the sodium D lines (Windholz and Neureiter, 1985; Windholz and Musso, 1989) to obtain values of 89.55 f 0.18 and 89.84 k 0.19 kHz/(kV/cm)’ for the scalar polarizabilities of the 3 P , , , and 3P3/’ states, respectively.
VI. Concluding Remarks The technique of precisely measuring frequency shifts using acoustooptically modulated laser beams has been demonstrated in a number of experiments. The apparatus is relatively straightforward consisting of an atomic beam, a frequency-modulated laser beam, and a photomultiplier. Large fluorescence signals having little noise can be generated using either dye or diode lasers. The method has a number of advantages when compared with other techniques. It does not require large and very uniform magnetic fields, as are needed in level crossing and in optical double resonance experiments. The data analysis is also less complicated since only a Lorentzian or Gaussian function is fitted to the observed spectral line. Short temporal resolution using relatively expensive transient digitizers is also not required, as is the case in quantum beat spectroscopy. Most significantly, the frequency calibration is much simpler than using a Fabry-Perot etalon. Interferometers are plagued by numerous problems including vibrations and sensitivities to pressure and temperature fluctuations, necessitating the use of frequency-stabilized lasers locked to an atomic transition to stabilize the cavity. In contrast, computer-controlled frequency synthesizers can quickly and conveniently generate a much wider range of frequencies with an accuracy of one part per million. Over the past decade, experiments using acousto-optically modulated lasers have yielded isotope, hyperfine, and Stark shifts representing frequency intervals ranging from a few megahertz to several gigahertz. Data of unprecedented accuracy with uncertainties as low as several parts in lo5 have been obtained. These results pose a stringent test for theories of multielectron atoms. The measured Stark shifts can also be used in conjunction with data obtained using novel new methods such as atom interferometry to determine scalar polarizabilities of excited states that heretofore could not be determined. Hence, the method of using
180
W. A. van Wjngaarden
frequency-modulated lasers is very versatile, and optical modulators are likely to play an increasingly important role in precision laser spectroscopy.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada and York University.
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 36
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES FOR ELECTRON-MOLE CULE COLLISIONS CARL WNSTEAD and KINCENT McKOY A . A . Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, California
I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Electron-Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Parallel Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1.Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Computational Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Designing a Parallel Program . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Parallel SMC Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Illustrative Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Performance.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. CrossSections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Conclusion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183 183 185 186 191 191 196 209 209 212 214 217 218
I. Introduction A. ELECTRON-MOLECULE COLLISIONS
Electron-molecule collisions at low impact energies have long been of fundamental interest because of the variety of phenomena exhibited in such collisions and the possibility for gaining insights therefrom into molecular spectra, particularly the electronic structure of the ground and low lying excited states (Schulz, 1973; Lane, 1980; Hall and Read, 1984; Allan, 1989). At the same time, the rates and outcomes of such collisions have been of great practical interest to those seeking to understand partially ionized gases, including the atmospheres of Earth and of other planets, gas lasers, and the edge regions of fusion plasmas. Recently, the increasing importance of low temperature plasmas in a variety of materials 183
Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003836-6
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processing tasks has been a driving force behind the study of low energy electron-molecule collisions (National Research Council, 1991). These nonequilibrium plasmas, widely used in the manufacture of semiconductor microelectronics, rely on collisional energy transfer between hot electrons and cold molecules to generate reactive species, such as free radicals and ions, that can effect desired chemical and physical changes at an exposed surface. Though hot in human terms, the electrons in these plasmas, with effective temperatures of thousands and tens of thousands of Kelvin, fall in the low energy regime with respect to collisions because they have kinetic energies comparable with those of the molecular valence electrons (1 eV = 12,000 K). Though most of our current knowledge of electron-molecule collisions derives from experiment, accurate measurements of collision cross sections are in fact quite difficult, especially for inelastic processes, and few groups worldwide have undertaken this challenging work. Demand for cross section data already exceeds supply, and the list of “critical” but absent data grows longer daily. Moreover, the species of interest include not only stable molecules but those radicals and ions whose populations within the plasma may be significant, and experiments will be all the more difficult for such transient species. The lack of reliable cross section data threatens to impair the accuracy of numerical models of low temperature plasmas and thus ultimately to impede the development of computer-aided design and optimization tools for plasma reactors (National Research Council, 1991). What can theoretical studies contribute? The calculation of electron cross sections at high impact energies (several hundred electron volts or more) is relatively straightforward: Since the first Born approximation (Schiff, 1968) applies, the problem effectively reduces to that of calculating a matrix element between bound electronic states of the target molecule. The well-developed methods of modern quantum chemistry may be brought to bear in calculating accurate approximations to such bound states. At low energies (roughly speaking, below 100 eV), matters are far less simple. A proper accounting for the identity of electrons requires antisymmetrization of the projectile electron with the N electrons of the target; that is, a single wavefunction for the ( N 1)-electron system must be determined, subject to appropriate (scattering) boundary conditions. Even if, as is often the case, nuclear motion may be neglected, the resulting problem in continuum electronic structure is formidable. Moreover, it must typically be solved for many different collision energies and for all possible combinations of directions of incidence and departure. Yet much progress has nonetheless been made toward the development of widely applicable methods for carrying out such calculations.
+
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Most recent theoretical studies of electron-molecule collisions have relied on variational approximations to the scattering amplitude or to some closely related quantity, thereby avoiding direct numerical solution of Schrodinger’s equation. Using one such variational method, we have been able to carry out calculations of elastic and inelastic cross sections for a variety of molecules of interest in plasma processing, including such large molecules (in terms of electron count) as SiF,, AlC,H,, and C,F,. However, studies of many-electron systems remain numerically intensive despite the choice of an efficient theoretical approach, and our work has depended on exploiting the prodigious advances in computational power that have resulted from the development of massively parallel processors.
B. PARALLEL COMPUTATION The advent in recent years of commercial parallel computers has opened new possibilities in computational physics and chemistry. As machines based on large numbers of powerful microprocessors have begun to supplant conventional vector supercomputers, not only has the absolute performance of the largest machines increased at a sharply accelerated rate, but the ratio of performance to price has improved as well. As a result, many calculations that seemed impossibly vast 10 or even 5 years ago would now be more or less routine. However, applications of massively parallel computational methodology to problems in atomic and molecular physics have been rather limited, especially in comparison with other areas of physics (e.g., fluid dynamics). This relative lack of progress no doubt stems from a number of causes, including perhaps the perception that parallel machines are not suited to problems that do not have an obvious spatial decomposition, as well as the perception that any effort invested in the arduous hand parallelization of a complex program is soon to be vitiated by the advent of automatically parallelizing compilers. The chaotic competition between architectures and paradigms, accompanied by the frequent appearance and disappearance of vendors, certainly justifies a degree of hesitancy and skepticism also. Nevertheless, we would argue, the widespread application of massively parallel machines to atomic and molecular problems is not only feasible and appropriate, but overdue. While it is undeniably easier to conceive strategies for parallelizing physical problems that involve low dimensional spaces and local interactions, we will show by example that an obvious physical decomposition of the problem is not prerequisite to an effective computational decomposition. Moreover, automatic parallelization, while the topic of vigorous research, is enormously difficult, and the design and
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implementation of complex programs that scale efficiently to hundreds of processors seems likely to remain a task for humans into the foreseeable future. Granted, we must still choose from an array of parallel programming methods and target architectures, with little assurance that today’s favored archetype will be in favor-or even in existence-tomorrow. However, in the past couple of years there has been some narrowing (by bankruptcy) in the range of hardware choices, accompanied by the emergence of de fact0 standards for at least the message passing model of parallel software. The latter development affords some assurance that investments in program development can be recouped on a variety of architectures, ranging from networks of workstations to conventional shared memory vector supercomputers. In this chapter, we will describe our methodology for implementing the calculation of low energy electron-molecule scattering cross sections on massively parallel computers. Our emphasis here will be on the adaptation of a problem in molecular quantum mechanics to parallel computation and on the performance achievable, and our hope is that some of the techniques described here will be of use to others who are contemplating the parallelization of similar computations. The present work updates and extends a previous account (Winstead and McKoy, 1995) along similar lines; for a discussion of various computational methods for electron collisions and a survey of recent results obtained by those methods, see Winstead and McKoy (1996). In Section 11, we summarize the theory behind our numerical method, showing how we arrive at working equations for the scattering amplitude beginning from Schrodinger’s equation. Section 111 discusses the parallel implementation of our method and associated issues. An illustrative example is presented in Section IV, and concluding remarks are offered in Section V.
11. Theory The approach we use to solve the electron-molecule scattering problem is referred to as the Schwinger multichannel (SMC) method (Takatsuka and McKoy, 1981, 1984). The SMC method is a straightforward extension of the original variational method of Schwinger (1947) to multichannel problems in many-electron systems. A recapitulation here of its main points will aid in understanding the associated computational issues that are our main subject. Consider a molecule possessing N electrons and it4 nuclei. In the collision problems of interest, the nuclei can almost always be considered
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fixed, since the duration of the collision is short on the time scale of nuclear motion (Chase, 1956). The purely electronic Hamiltonian for the molecule is then
where atomic units (ti = e = me = 1) are employed, as they wjll be is the coordinate vector of electron i, and Z j and R j are throughout, the charge and coordinates of nucleus J . An electron colliding with this molecule interacts with it through a potential V,+
v,), a new effect is possible. A laser that is blue-detuned for low motional levels will necessarily become red-detuned for high motional levels. As a result, the atom will be heated from below and cooled from above. For a single atom, such an interplay of heating and cooling leads typically to a final distribution centered close to the state for which the actual detuning is close to zero. In Cirac et al. (1995) it was shown that such a mechanism leads, when N is sufficiently large, to a perfect condensation of bosons in a single trap level, not necessarily the lowest one. Moreover, for a given value of laser detuning, there are in general several neighboring states into which the condensation may occur (see Fig. 12). The system thus exhibits multistability and hysteresis effects when the detuning is adiabatically and cyclically
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
a
259
1
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changed (see Fig. 13). The stationary states of the system display some algebraic similarities to Bose-Einstein distributions (BEDS), and they have been termed generalized BEDS.
D.
COOLING OF A
GAS WITH ACCIDENTAL DEGENERACY
All of the results discussed so far rely on the fact that off-diagonal elements of the density matrix vanish. As mentioned, this is true provided
260
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the trap potentials are sufficiently anharmonic. In the opposite limit, when the potential is close to perfectly harmonic, the effects of accidental degeneracy dominate the dynamics of the system. Let us enumerate by rSi the eigenstates of a single-atom Hamiltonian in the rotationally symmetric harmonic trap of frequency o,where rSi is a natural number in one dimension, is a pair of natural numbers in 2D, a triple in 3D, etc. When we consider an ensemble of N atoms, the states of such an ideal gas can be written in the Fock representation as In, where the n,- denote the occupation numbers of the corresponding S t h
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
261
c
4
FIG. 11. Mean dispersion of the delay time between emission of two successive photons and mean fluorescence intensity (inserts) as functions of the detuning for N = 10 and u = 1Oy (top), for N = 50 and v = 1OOy (bottom), for bosons (‘‘0”) and fermions (“x”). The results were obtained from numerical simulations of the master equation.
eigenstate. For noninteracting atoms, there are two kinds of degeneracies in such a system. First, there is a degeneracy of energy levels due to rotational invariance; that is, for the states for which the sum of rid's with a fixed sum of the components of 6 ,which are themselves fixed. Obviously, such degeneracies are not present in 1D. We shall not discuss them here, since we shall focus on the case of one-dimensional gas. Second, there exists an accidental degeneracy, due to the particular symmetry of the harmonic potential. This degeneracy occurs even in the case of 1D: for instance, for the states 10,2,0,. . . ) and ll,O, 1,0,. .. >.Here, the state with two atoms in the first energy level has an energy 2 X w , which is equal to
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M. Lewenstein and Li You
* z a
6/r FIG. 12. Phase diagram for the bosonic system as a function of laser detuning and the dimensionless parameter d, = N,/(2ug). In each closed area, we indicate the possible phases by the level that can be occupied ( y = r).
the energy of the state with one atom in the ground level and another atom in the second excited level (1 X Ow + 1 X 20). As we said, both kinds of degeneracies are lifted up if one considers anisotropic trap with anharmonic energy levels. If one then assumes that the resulting energy level shifts are larger than cooling rates, one can evoke standard secular arguments to reduce the ME to a diagonal form in the basis of the bare ideal gas states (Cirac et al., 1994b,c). In the opposite case, i.e., when the effects of accidental degeneracy dominate the dynamics of the system, it turns out that 1. There exist nN(l) so-called vacuum states loL,,) that are annihilated m gig,, + by the “jump” operator A = C:= d
,,
A
lo/,,>
=
0
(64)
where the index 1 = 0, or 1 = 2,3,. .., N energy of the corresponding states,
- 1
indicates the bare
m
c wnat,an lo,,,)
=
0 1 IO/,J,
(65)
n=O
whereas s = 1, ..., n N ( l ) . Each of the vacuum states is a linear combination of the accidentally degenerate energy eigenstates. The number of the accidentally degenerate states of the energy wl in the N atom system p N ( l )is given by a solution of the partition problem
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
IcX0.5 a
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'
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' f
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FIG. 13. Occupation probability 7ik of the ground ( k = 0, solid line) and excited ( k = 1, dashed line) harmonic levels as a function of the laser detuning 6/r for N = (a) 100, (c) 1000, and (e) 2000. Fluorescence intensity (in arbitrary units) as a function of the laser detuning 6/r for N= (b) 100, (d) 1000, and (f) 2000. The results were obtained by solving the master equation with Monte Carlo simulations. Only (e) and (f) show hysteresis behavior. The parameters are ve = 1050r, v8 = 1OOOr.
j
of the number theory (Hardy and Ramanu'an, 19181, and is extravagantly large (c.f. p,(l) = O(exp(.rr 21/3 )) for 1 I N ) . The number of the vacua is given by n,(l) = p,(l) - p N ( l - 1).The very existence of multiple vacuum states is thus a direct consequence and, at the same time, a signature of the accidental degeneracy. 2. The vacuum states are orthonormal, (O,, I O,,, $,) = S,,,Sssr. 3. The Fock-Hilbert space of the system splits into an infinite number of Fock subspaces corresponding to each of the vacuum states. The Fock states in the (1, s)th subspace are constructed as
with k = O,1,. . . . They are also mutually orthonormal, and they are eigenstates of the energy operator with the corresponding eigenvalue w ( f + k ) . They are also highly degenerate (for k + 1 = k' + 1').
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The dynamics exhibits in the Lamb-Dicke limit two time scales. On a faster time scale of the order of T - ~ it, is nonergodic, i.e., it does not mix the different 1 subspaces. After a short time, all coherences between the Ikl,,) and lki,,s,) for k # k’ vanish. Within each 1 subspace, the system approaches the thermal equilibrium characterized by the density matrix diagonal in k, with undamped off-diagonal elements for s # s’, and some temperature (related to the temperature of the “heat bath,” i.e., the system that provides energy dissipation). The dynamics, however, cannot be reduced to a Poisson jump process (i.e., a sequence of random jumps between the various I kc s ) states with the transition probabilities governed by the detailed balance conditions characteristic for the thermal equilibrium). The reason is that coherences corresponding to 1 # 1‘, k = k ’ , as well as to 1 = l ’ , but s # S ’ do not vanish. The ergodicity is restored on a much longer time scale, of the order 77-4, when the dynamics begins to mix various 1 subspaces. The coherences for 1 # 1’ die out, and eventually the system approaches an equilibrium state described by the “canonical” distribution with respect to the energy, with arbitrary coherences between the states with the same k, 1, but s # s’. The latter coherences are then damped on an even longer time scale.
E. SYMPATHETIC COOLING Another route to cold samples of particles is sympathetic cooling (Wineland et al., 1978, 1985; Phillips et al., 1985; Drullinger et al., 1980; Larson et al., 1986; Gabrielse et al., 1989, 1990; Lewenstein et al., 1995, and references therein). With this technique, a gas of particles (,‘A”) is cooled via its interactions with another gas (,‘B”) which is already at a low temperature. Typically, one can assume that the number of particles in B is very large and/or that they are kept cold by another mechanism (such as laser cooling or evaporate cooling). Then B can be regarded as a thermal bath, and therefore the final temperature of A will be very close to that of B. Here, as in the case of evaporative cooling, the required thermalization occurs due to particle-particle collisions. To our knowledge, the idea of sympathetic cooling of neutral particles, and in particular atoms, has not been exploited in the literature. In Lewenstein et al. (19951, such a possibility was discussed concentrating on the following physical situation: the gas of alkali atoms B is confined in a large and rather loose trap, such as magneto-optical trap (MOT). qpically, for alkali atoms such traps have frequencies of the order of 10-100 Hz and sizes of a few micrometers. The gas B is cooled by some mechanism (laser cooling, evaporative cooling, etc.) to a temperature T,. The temperature T, might still be relatively high for the B atoms, which are additionally
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assumed to be relatively heavy. The gas A is composed of other alkali atoms, assumed to be stored in a tight trap such as a far-off resonance dipole trap (FORT) (Miller et al., 1993) located inside the MOT. Tight traps may have frequencies in the range of few kilohertz and sizes of = 0.1 p m . A atoms have smaller mass, but not necessarily much smaller than that of B atoms. In Lewenstein et al. (19951, the ME method has been applied to a model describing the quantum dynamics of sympathetic cooling. We considered a gas of particles A trapped in an harmonic potential and interacting with other particles B that can be regarded as a bath at a given temperature. The interactions between the particles are due to atom-atom collisions, which we have modeled using the standard shape independent potential approximation,
where a:., a,.,, bt(Z>, b ( z ’ ) denote creation and annihilation of A and B atoms in the harmonic oscillator, or plane wave states, respectively, whereas
whereas +&?’) is the wavefunction corresponding to the Zth level of the harmonic oscillator, and C is a constant depending on the &dimensional scattering length a%.For example, in three dimensions,
with p the reduced mass. Again, the methods borrowed from quantum optics were used to derive an ME for the reduced density operator of the system A. The ME describes cooling through transitions between different trap levels. The rates at which these transitions occur depend on the specific properties of the atomic collisions, as well as on the characteristics of the trap and the temperature of the atoms of the bath. For processes involving transitions from n” to Z and 6’to 6 that fulfill
c
I=x,y..
(It,
.
-4)= a ,
( m , - m ’I ) I=x,y
.. .
=
-a
(70)
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the rates are
x n ( Z ) [ n ( Z ’ )+ l ] S [ ~ ( k )- ~ ( k ’+) ahv]
(71)
where a is defined through (701, so that a > 0 ( a < 0) corresponds to processes decreasing (increasing) the energy. In principle, the rates (71) contain all the information concerning the cooling process. The main result (Lewenstein et al., 1995) consists of analyzing these rates and deriving accurate analytic formulae for them. The results and the techniques developed (stationary phase for other asymptotic methods) can be generalized to study other problems, such as an analogous problem of evaporative cooling of atoms in a loose MOT trap with a tight FORT in the center.
F. EVAPORATIVE COOLING As we already have mentioned, perhaps one of the most difficult problems in quantum dynamics concerns the formation of a condensate in the final (thermalization) stage of evaporative cooling. Here again, the relevant atom-atom interactions are elastic collisions. Several authors have attempted to estimate the time scale of the formation of the condensate in a homogeneous system and came to contradictory conclusions (Levich and Yakhot, 1977; 1978; Snoke and Wolfe, 1989; Eckern, 1984; see also Kagan et al., 1992). As carefully discussed in Kagan et af. (1992), most of the authors used quantum kinetic equations and applied them in the so-called coherent region in which a coherent condensate has been already formed, and in which kinetic equations are no longer valid, strictly speaking. Stoof pointed out that the evolution is characterized by the two time scales: the time scale of the slow growth of the condensate, preceeded by the much faster nucleation of the coherent population of the zero momentum state (Stoof, 1991, 1992). Kagan et af. performed more appropriate analysis, dividing the evolution into three regimes: coherent regime (in which the dynamics is essentially governed by the time dependent GPG equation), kinetic region in a linear regime (which concerns the hottest atoms, which do not exhibit degeneracy effects and can be well described by a quasiequilibrium distribution), and kinetic region in a nonlinear regime (which concerns colder atoms, which exhibit degeneracy). The paper by Kagan et al. does describe the basic physics of the condensate formation, but it has
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two drawbacks: (i) it does not address the condensation in trapped systems; (ii) it lacks uniformity in the description of various stages of dynamics. For these reasons, several authors have attempted to study the dynamics of evaporatively cooled gas with the help of the ME technique (C. W. Gardiner and P. Zoller, 1996; M. Wilkens and M. Lewenstein, unpublished, 1995). These can be done either by modeling interactions of atoms with some external heat bath of a given temperature, or by treating the system as closed. The latter approach corresponds more closely to the situation realized in experiments. Here, the ME can be derived by eliminating some of the degrees of freedom: either by using some kind of coarse graining procedure, or by eliminating the hot atoms (i.e., those that, as pointed out in Kagan et af. (1992), evolve according to linearized Boltzmann equations, and whose state can be conveniently described as a quasi-equilibrium distribution with time dependent parameters). This problem, however, has not yet been solved, and it remains one of the most challenging problems of the quantum critical dynamics. In Quadt et af. (1995), the authors derived an ME for the lowest mode in the trap, eliminating all the others. The equilibrium state in such a case is a grand canonical ensemble with a Hamiltonian accounting for atom-atom collisions. The collisions cause nonclassical properties of the equilibrium state (sub-Poissonian statistics of the fluctuating number of condensed particles). From what we have said, one should expect that such a phenomenological treatment is definitely physically sound in the limit of tight traps when the behavior of the lowest state can indeed be separated from other states of the trap. It cannot, however, be valid if condensation occurs via the nonlinear kinetic region discussed in Kagan et af. (1992) for a homogeneous system (i.e., an “infinite” trap). The quantum dynamics of gases of cold atoms can be very well treated with the help of the ME technique, and it exhibits, in our opinion, an enormous richness of interesting physical and mathematical phenomena, such as multistable, exotic stationary states, multistage dynamics, and so forth. There are still, however, basic questions to be answered, and further studies are required to gain more understanding of the new physics, including the development of other statistical physics tools, such as diffusion equations, hydrodynamic limits, and so forth.
VII. Theory of Bosers The idea of a coherent source of atomic matter waves has been developed by several groups independently: Holland and Burnett at Oxford, BordC at UniversitC Paris-Nord, Cirac, Lewenstein and Zoller at J I M , Gardiner in
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New Zealand, Spreeuw and Wilkens in Konstanz, and others. For the first time, it was thoroughly discussed and several models were proposed at the Workshop on QFTAP at JILA in 1994 by Burnett, Cirac, Gardiner, Holland, Lewenstein, and Zoller. Since then several proposals have been formulated (Wiseman and Collet, 1995; Holland et al., 1995; Olshan’ii et al., 1995; Spreeuw, 1995; G u z m h et al., 1995). One should stress that most of the proposals are quite speculative, and for the moment they seem to be quite far from experimental realization. In this section, we shall discuss in some detail a prototypical model of a boser formulated by Cirac, Lewenstein, and Zoller, and presented at the JILA Workshop. Before doing that, however, it is worth mentioning that the same ideas have been developed in the context of condensation of excitons. In particular, a model of a boser that generates a coherent population of nonequilibrium excitons has been proposed (Imamoglu and Ram, 1996). Also, the first experimental evidence of spontaneous buildup of the coherent exciton polariton population in a microcavity has been reported (Pau et al., 1996). A. A PROTOTYPICAL BOSER Boser theory is formulated in analogy to laser theory. A boser is an open system into which atoms can be pumped and from which they can be taken away. The pumping is assumed to be incoherent and consists of putting atoms (in some quantum states) into a black box, called a boser. Atoms are lost also in an incoherent way-in a continuously working boser, atomic losses occur analogously to photon losses from a cavity; in a pulsed boser, the output is realized by letting the atoms leave the boser periodically in analogy to Q-switched lasers. Inside a black box, there is an atomic cavity -usually it is some kind of an atomic trap with well-defined mode structures. Bosing consists in mode selection, and it is caused essentially by quantum statistics. As soon as the atoms start to occupy one trap level (i.e., one atomic cavity mode), other atoms that enter the system will tend to do the same. Of course, this effect must be mediated by some kind of interactions. Such interactions in fact favor transitions to some state-the boser models are based on cooling schemes and contain dynamical mechanisms that help to populate the ground state of the trap. In contrast to the standard laser theory (in which atom-photon coupling has a coherent character and may, for instance, cause Rabi oscillations), the mode selection mechanisms in bosers are frequently incoherent. Until now, two kinds of mechanisms were considered: (i) those based on elastic atom-atom collisions, analogous to the ones used in evaporative cooling (Holland et al., 1996); (ii) those based on laser cooling and spontaneous transitions
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(Wiseman and Collet, 1995; Olshan’ii et al., 1995; Spreeuw, 1995; GuzmQn et al., 1996). The prototype model of a boser consists of two atomic modes (two trap levels): the bosing level (ground state of the trap), described by the annililation and creation operators b,, bi; and the pumping mode, described by b,, b!. The boser dynamics is most conveniently described using the ME technique. The ME for a density matrix p of the two-level cw boser model is
i,
=
-iv[ b:b,’ p ]
The first term here describes free evolution (with zero frequency for the ground state, and some trap frequency v for the pumping mode). The second and third terms describe the statistically enhanced transitions from level 1 to 0 at the rate r+, and from 0 to 1 at the rate r- , respectively. Of course, we expect that bosing will only be possible if the mode selection occurs, i.e., r+>r- , so that the system prefers transitions to the ground state. The fourth term in Eq. (72) describes incoherent pumping of atoms into the level 1 at the rate K , . Finally, the last term describes incoherent atom losses from the trap at the rate K,. Further analysis of this model follows the standard lines of quantum optics. If the rate I?+ is also larger than the pumping rate K , , the pumping level can be eliminated adiabatically. Introducing Glauber’s Prepresentation for the ground state operators (Glauber, 1963a, b) (i.e., a diagonal representation in coherent states 1 a )o)
it is then possible to derive a semiclassical equation for the mean value of the ground state field amplitude,
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which has exactly the same form as the semiclassical equation describing a single-mode laser. The cooperation number C is here defined by
(74) whereas atom saturation number is defined by
r+(75) r+-rAs we see, bosing is possible, provided r+> K ] > r _ and C > 1. In such a no =
K1
case, just as in a laser, the ground state amplitude attains a nonzero stationary value, la12 = n,,,(C - 1). Using standard methods of the quantum noise theory (Gardiner 19911, it is easy to show the foregoing model describes indeed an analog of a laser. To this aim, first we demonstrate that the distribution of the number of atoms in the ground state undergoes a transition from a thermal-like to a Poissonian-like form at the boser threshold. Second, we show that above the threshold the boser spectrum is very narrow, since it essentially arises from the phase difSusion process. In fact, the boser bandwidth far above the threshold is ybos= K O ( r - / K I ) and is much less than the trap loss rate (i.e., the analog of the atomic cavity width) K". We stress that proving those two coherence properties is essential for any model of the boser.
B. BOSERMODELS Various boser models employ different mode selection mechanisms. The model of Holland et al. (1996) assumes that atoms (in the ground electronic state) are pumped into a magnetic trap, where they undergo elastic collisions, just as in the case of evaporative cooling. Due to the complexity of the problem, the authors in fact limit the number of accessible levels to three: a ground level, an intermediate pumping level (to which the atoms are incoherently pumped), and a high energy level from which evaporation-like losses might occur. First, the high energy level is eliminated adiabatically, resulting in a two-level model very much analogous to the one just discussed. The model works in the limit when the collisional redistribution rate (i.e., the rate at which two atoms from the pumping level collide, sending one to the high energy level and the other to the ground state) is fast. Unfortunately, in the same limit collisions between the atoms in the ground state of the trap become faster, which might seriously reduce the coherence properties of the boser.
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In Wiseman and Collet (19951, a boser model employing dark state cooling has bee proposed (Aspect et al., 1988). Ideally, dark states are atomic states (of atoms at rest) that are not coupled to the laser field. In the dark state, cooling atoms may undergo laser excitation from nondark states and spontaneously emit into the dark state. If they enter the dark state, however, they remain there forever. In Wiseman and Collet (19951, a many-body ME for dark state cooling is analyzed. Again all atomic field operators other than for the ground state are eliminated, and the ME for the ground state operators is considered. Two other models in which pumping into the bosing mode is provided, as in the previous case, by spontaneous emission concern two opposite limits. Olshan’ii et al. (1995) considered a general case of pumping atoms into a ground state level of a large trap from a single (electronically) excited level. In an experimental realization, this could correspond to pumping atoms from other ground electronic states in the hyperfine manifold to the final ground state via Raman transitions. If in such a process the effects of photon reabsorption and optical thickness are disregarded, the system does exhibit a boser transition, and it may accumulate atoms in the ground state of the trap (or, strictly speaking, the state from which losses are minimal). Interestingly, in this model spontaneous emission does not select any level-condensation occurs due to quantum statistical enhancement of the emission to the level from which the loss is the smallest. The same model has been analyzed taking into account reabsorption effects. It turns out that due to the large absorption cross section of atoms ( = A’), reabsorption effects may totally prevent the boser action in large traps. Using a noisy laser might help to reduce absorption cross sections, and to restore the possibility of bosing. Spreeuw (1995) considered a similar model but in the Lamb-Dicke limit, Their system consists of metastable Ar atoms trapped in a single minimum of a so-called dark optical lattice created by crossed laser beams. Such a trap is in fact a blue-detuned FORT. The trap is loaded with laser-cooled atoms, which undergo the transfer into the trapped states through spontaneous emissions. The mode selection takes place here because the ground state of the trap turns out to have the lowest loss rate (due to the scattering of trap-laser photons) and the highest pumping efficiency. Finally, in G u z m h et al. (19961, a one-dimensional atomic cavity is considered in which atoms are ‘‘localized’’in a quantum ground state with respect to the transverse motion (with the help of a FORT) and can occupy a discrete set of motional (standing wave) states aligned with the cavity axis (Zhang et al., 1995). In contrast to other schemes, the nonlinear effects of quantum statistics are provided here by atom-atom interactions
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due to near-resonant dipole-dipole forces. Even though the lasers used for cooling and trapping the atoms are far from resonance, and atoms are thus hardly excited, they are subjected to effective two-body dipole-dipole forces. For practical calculation, the authors reduce the model to a three-level case: with the highest pumping level and two active cavity levels.
C. BOSONACCUMULATION REGIME
Obviously, the quantum statistical effects are necessary to achieve the boser action. Three of the models we have discussed involve spontaneous emission. It is therefore challenging to study the role of quantum statistical effects in spontaneous emission more carefully, taking this account reabsorption effects and dipole-dipole interactions. In Cirac and Lewenstein (1996), we have considered the behavior of an atomic Bose-Einstein condensate in the presence of an atom in an excited electronic level. We analyzed the boson accumulation regime (BAR), defined by the relation No >> qev,N - No, where N is the total number of atoms in the ground electronic state, No is the number of atoms in the condensate, and ale"is the number of levels to which the excited atom can effectively decay. In this regime, quantum statistical effects related to the boson nature of the atoms predominate in the process of spontaneous emission. Simple arguments (based on rate equations) suggest that the proportion of atoms in the condensate decreases after the spontaneous decay. Using a more appropriate approach based on an ME description, we have demonstrated that, in general, these simple arguments may lead to erroneous conclusions. Under certain conditions, the ME approach predicts that the proportion of atoms in the condensate increases. We have given an interpretation of this phenomenon in terms of quantum interferences between processes that include reabsorption of the emitted photons and the dipole-dipole interactions between the atoms. Namely, in the BAR limit, the processes that give the leading contributions are presented in Figs. 14 and 15. The process in Fig. 14 is dominant and describes spontaneous emission accompanied by a direct transition to the condensate; processes (a) and (b) in Fig. 15 contribute in the first order of the ratio alev/N,or ( N - N o ) / N , both decrease the condensate fraction, but both start and end in the same state. Their amplitudes necessarily interfere, and amazingly their interference is destructive! Finally, process (c) in Fig. 15 contributes also in the first order of the ratio alev/N, or ( N - N o ) / N , and increases the condensate fraction.
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273
10’
FIG. 14. The model consists of one (internal) excited level and a + 1 ground levels. Initially, there is one atom excited and N atoms in the ground state, distributed among the a + 1 levels.
The surprising physical effect considered in Cirac and Lewenstein (1996) may help to pump atoms into a condensate and to compensate for atomic losses in the atomic trap. Mathematically, the main achievement of this paper is the demonstration that in the BAR it is possible to construct a systematic expansion of the solutions of the ME in a series of parameters alev/N, or ( N - N , ) / N . To estimate these parameters, we may assume a
b
C
FIG. 15. (a) Process in which the excited atom decays directly into the kth ground level. (b) Process in which the excited atom decays into the condensed level; the emitted photon is absorbed by an atom in the same level, and it subsequently decays into the kth level. (c) Process in which the excited atom decays into the condensed level; the emitted photon is absorbed by an atom in the kth ground level, and it decays back into the condensed level.
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that the excited atoms have a mean energy of the order of one recoil, so that one can estimate ale" = ( ~ , / h v )where ~ , eR is the recoil energy and v is the trap frequency. For the current experiments dealing with BEC, ~ , / ( h v )2 30, which would require a number of condensate atoms N s 25,000. According to Anderson et al. (1995) and Bradley et al. (19951, this seems to be within the reach of experiments in the near future. On the other hand, it seems that the BAR has been achieved, or is close to being achieved, in the MIT experiment (Davis et al., 1995~).Another interesting situation to test predictions of Cirac and Lewenstein (1996), however, will ) 3; in this case, the BAR condition will be be the one in which ~ , / ( h v = fulfilled with a small number of particles. To realize such a situation, the trap frequency must be of the order of 1 kHz (or larger). This is the typical case for a dipole (FORT) trap. We expect that studies of the ME equation in the BAR will soon bring further fascinating results.
VIII. Nonlinear Atom Optics As we have already mentioned in Section IV, nonlinear excitations of BEC may lead to various matter wave analogs of nonlinear optics (Edwards et al., 1996). Such phenomena, however, deal with atoms in their ground electronic state and are thus mediated by elastic collisions described by Zg [see (1411. It was, however, suggested by Meystre and his collaborators, and Zhang and Walls, that similar phenomena could also occur in the area of atom optics, and these were termed nonlinear atom optics (NAO). Atom optics is developing vary rapidly. It mainly concerns manipulations of the matter waves in analogy with light waves (Mlynek et al., 1992), and often it involves manipulations of the matter waves using laser light. In such situations (even if the lasers used are far from resonance, and atoms are hardly excited), the resonance dipole-dipole interactions between the excited-ground state atoms play an essential role, and they involve quantum statistical effects, if the temperatures of the atomic beam are sufficiently low and densities sufficiently high. The relevant interactions for kinds of phenomena involve 3&(161, but in the first place the interactions are due to exchange of photons, described by%, (17). Eliminating photon degrees of freedom from the theory, one arrives at an effective (nonlinear) potential for atoms, and one treats the atom by a Hartree-type approximation. This approach is completely analogous to elimination of atomic degrees of freedom in order to derive equations of nonlinear optics. In this sense, Maxwell-Bloch equations provide a unified view of nonlinear optics
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and nonlinear atom optics (Castin and Molmer, 1995). Zhang (1993) has studied, in this manner, atom correlations induced by dipole-dipole forces. Effective atom-atom potential has been frequently approximated by a local potential, in analogy to Eq. (16). In view of the long range character of the dipole-dipole forces such approximations might be questionable (Castin and Mdmer, 1995). Nevertheless, using such an approximation, Lenz et al. (1993,1994; Schernthanner et al., 1994) and Zhang et al. (1994; Zhang and Walls, 1993, 1994) described nonlinear quantum statistical effects that lead to creation and propagation of atomic Thirring solitons. The basic theoretical tool of their analysis is a set of two coupled nonlinear Schodinger equations (for ground and excited state atoms). The theory of NAO was further developed in the works devoted to the study of spontaneous emission effects on atomic solitons. In particular, a self-consistent Born-Markov-Hartree-Fock ME for nonlinear atom optics has been derived (Lenz and Meystre, 1994; Schernthanner et al., 1995). The two-body dipole-dipole interaction potential between atoms in a nonlinear optical cavity (Zhang et al., 1995) is an essential ingredient of the boser model (Guzmh et al., 19961, discussed in the previous section.
IX. Conclusions We hope that the readers of this chapter will appreciate its main message: In the advent of the recent experiments on Bose-Einstein condensation, quantum optics and atomic and molecular optics have entered a new phase! Theoretical quantum optics is merging with many-body theory, condensed matter physics, and statistical physics, and it is becoming a theory of equal, if not higher, complexity. Quantum optics does not only use the methods of quantum field theory and many-body physics-it also starts to contribute its own methods to open new paths in quantum field theory.
Acknowledgments This chapter would not have been written if we did not have the honor and pleasure of collaborating and exchanging ideas with Ignacio Cirac, Jinx Cooper, and Peter Zoller over the last few years. We acknowledge also fruitful and enlightening discussions with, and the help in the preparation of this chapter of, Vanderlei Bagnato, Keith Burnett, Yvan Castin, Claude Cohen-Tannoudji, Eric Cornell, Alex Dalgarno, Jean Dalibard, John Doyle, Ralph Dum, Mark Edwards,
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Mariusz Gajda, Crispin Gardiner, Roy Glauber, Rick Heller, Murray Holland, Atac Imamoglu, Juha Javanainen, Wolfgang Ketterle, Pierre Meystre, David Politzer, Gora Shlyapnikov, Christoph Salomon, Boudewijn Verhaar, Jook Walraven, Carl Wieman, and Martin Wilkens.
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Index A
Absolute differential cross section, 16, 20 Acousto-optic modulator, 148-149 Alkali systems, Bose-Einstein condensates, 234-238 Atom counting, 239 Atomic bunching, in the transient regime, 119-120 Atom interferometry, 121, 130-133 Atom optics, 274 Automatic parallelization, 192-193
B Barium, Stark shift, 174 Bogoluibov-Hartree (BH) theory, 235,238 Bose-Einstein condensates (BEC), 222, 225-230 light scattering coherent weak light, 239-245 incoherent light, 250-253 short intense pulses, 245-250 in trapped alkali systems, 234-238 excitations, 238 Ginsburg- Pitaevski-Gross equation, 235-236 negative scattering length, 236-238 Bose-Einstein distribution (BED), 224, 258-259 Boser, 254, 267-273 Bosing, 268, 269 Boson accumulation regime, 271-273 Brillouin scattering, 88
impact excitation, 70-71 Stark shifts, 170-175 Close-coupling plus optical potential (CCO), 14 Coherent atomic recoil laser (CARL), 110, 120 Coherent light scattering, 239-250 Computers parallel SMC method, 191-217 architecture, 191-192, 212-213 load balance, 195 performance, 212-213 programming models, 192-193,209-212 scalability, 194-195 Convergent close-coupling (CCC) calculations, 9, 14 Cooling with accidental degeneracy, 259-264 evaporative cooling, 229 Bose-Einstein condensates, 234-238 quantum dynamics, 266-267 laser cooling recoil-induced effects, 135-136 stimulated Rayleigh scattering, 104 sideband cooling, ideal gas, 256-258 sympathetic cooling, 264-266 Crossover, 122 Cross sections, trimethylaluminum, 214-21 7
D
Data parallelism, 194 Degeneracy, accidental, cooling a gas with, 259-264 Differential cross section, 9, 10, 19, 216-217
C
Calcium, Stark shift, 174 Cesium electron-atom scattering, 8, 20-24, 25 hyperfine structure and isotope shifts, 163-166
E Elastic scattering, 8 heavy targets, 15-24, 25 light targets, 8-15 281
282
INDEX
Electron-atom collisions, 1-3, 80, 83 elastic scattering, 8 cesium, 8, 20-24, 25 helium, 8-11 mercury, 15-20 sodium, 8, 11-15 generalized Stokes parameters, 51-66 generalized STU parameters, 66-70 higher angular momenta, helium, 71-80 impact excitation, 24, 26-28 cesium, 70-71 helium, 28-34 sodium, 34-48 scattering amplitudes, 3-8 Electron-molecule collisions, 183-186 application, 209-217 highly parallel computation, 191-209 theory, 186-190 Evaporative cooling, 229 Bose-Einstein condensates, 234-238 quantum master equation, 266-267
F Fabry-Perot etalon, laser atomic beam spectroscopy, 142, 145, 151 Far-off resonance dipole trap (FORT), 264, 266 First Born approximation, 31 Free electron laser, 110 Frequency-modulated lasers, 148-152
G Generalized diffraction theory, 245 Generalized Stokes parameters, 51-66 Generalized STU parameters, 66-70 Ginsburg-Pitaevski-Gross (GPG) equation, 235-236
H Helium Bose-Einstein condensate, 228 electron-atom scattering, 8-11 impact excitation, 28-34, 71-80 Hydrogen, Bose-Einstein condensate, 228
Hyperfine structure, 152-153 cesium, 163-166 sodium, 158-163 ytterbium, 153-158
I
Impact excitation, 24, 26-28 heavy targets, 48-71 light targets, 28-48 Incoherent light scattering, 250-253 Integral elastic electron cross section, 214-217 Interferometry atomic interferometry, 121, 133 Fabry-Perot etalon, 145-146, 151 Ramsey-BordE matter wave interferometer, 128-130 Inversionless lasing, of cold atoms, 133-135 Isotope shifts, 152-153 cesium, 163-166 sodium, 158-163 ytterbium, 142-143,144,147-148,153-158
L Laser atomic beam spectroscopy, 142-152 Laser cooling recoil-induced effects, 135-136 stimulated Rayleigh scattering, 104 Lasers coherent atomic recoil laser (CARL), 110, 120 free electron laser, 110 frequency-modulated lasers, 148-152 inversionless lasing of cold atoms, 133-135 light scattering, 245-250 magneto-optical traps, 229, 264 recoil-induced inversionless lasing of cold atoms, 133-135 Laser spectroscopy, 142-148 hyperfine structure and isotope shifts, 152-166 optical modulators, 148-149 Stark shifts, 166-179 Light scattering, 88, 236-237, 239 Brillouin scattering, 88 coherent weak light scattering, 239-245
283 incoherent light, 250-253 Raman scattering, 88 Rayleigh scattering, 88 recoil-induced resonance, 109-137 short intense pulses, 245-250 stimulated Rayleigh scattering, 90-109 stimulated scattering, 88 Lithium, Stark shift, 174
M Magneto-optical traps, Bose-Einstein condensate, 229, 264 Many-body theory, 223, 224, 253, 275 Massively parallel processors (MPPs), 191-192,217 electron-molecule collisions, 185, 191-195 Master equation (ME), 253-259 many-body theory, 224 nonlinear optics, 274 sideband cooling of ideal gas, 256-258 sympathetic cooling, 265 Mercury electron-atom scattering, 15-20 impact excitation, 48-70 Message passing, 193
N Network of workstations (NOWs), 191 Nonlinear atom optics (NAO), 273-274 Nonlinear optics, 236-238, 273-274 Nonlinear Schrodinger equation (NLSE), atoms with negative scattering length, 236-237
0
Optical lattices, stimulated Rayleigh scattering, 104-107 Optical modulators, 148-149 Optical potential, weak light scattering, 242 Optical Ramsey fringes, recoil doublet, 125-128 Optical spectroscopy, 142-148
P Pairing theory, 235, 238 Parallel SMC method architecture, 191-192, 212-213 electron-molecule collisions, 185, 191-217 electron scattering computations, 196 input/output, 208-209 integral transformation, 204-206 one- and two-electron integrals, 197-199 parallel integral evaluation, 202-203 program outline, 207-208 quadrature and scaling, 199-202 scaling of transformation step, 206-207 load balance, 195 programming models, 192-194,209-212 scalability, 194-195 Perfect scattering experiment, 2-3, 80 Photorefractive crystals, stimulated Rayleigh resonance, 169 Photorefractive effect, 90 Polaritons, 242 Potassium, Stark shift, 174 Precision laser spectroscopy, 141-142, 152-180 Pump-probe spectroscopy, 88, 89
Q Quantum dynamics boser, 254, 267-274 condensation in cold atoms, 253 generalized Bose-Einstein distribution, 258-259 master equation, 253-259 nonlinear atom optics, 273-274 sideband cooling of ideal gas, 256-258 Quantum field theory of atoms and photons, 223-224 Bose-Einstein condensates, 234-275 Hamiltonian of, 230-234 master equation, 253-259 many-body theory, 224 nonlinear optics, 274 sideband cooling, 256-258 sympathetic cooling, 265 Quantum master equation, 253
284
INDEX
Quantum optics, 223 light scattering, 88, 236-237, 239 coherent weak light, 239-245 incoherent light, 250-253 recoil-induced resonance, 109-137 short intense pulses, 245-250 stimulated Rayleigh scattering, 90-109 many-body theory, 223, 224,253,275 nonlinear atomic optics, 273-274 Quasi-elastic scattering, 88
R Raman scattering, 88 Ramsey-Bordt matter wave interferometer, recoil-induced effects, 128-130 Ramsey fringes, recoil doublet, 125-128 Rayleigh scattering, 88 Recoil doublet, optical Ramsey fringes, 125-128 Recoil-induced effects, 121-122 atom interferometry based on atom recoil, 133 inversionless lasing of cold atoms, 133-135 Kasevich-Chu experiment, 130-132 laser cooling, 135-136 Ramsey-BordC matter wave interferometer, 128-130 recoil doublet of optical Ramsey fringes, 125-128 in saturated absorption spectroscopy, 122-125 Recoil-induced resonance, 90,109-1 10, 137 atomic bunching in the transient regime, 119-120 coherent atomic recoil laser, 120 experimental observation, 115-117 as Raman process between different energy-momentum states, 117-119 as stimulated Rayleigh resonance, 110-1 15 Rotating wave approximation, 223-224 Rubidium, Stark shift, 175
S
Samarium, Stark shift, 175 Saturated absorption spectroscopy, recoil effects, 122-125
Scalability, highly parallel computational techniques, 194-195 Scattering, 88 Scattering amplitudes, 3-8 Scattering equation, weak light scattering, 240-241,243 Scattering experiments, 1-3, 80, 83 elastic scattering, 8 heavy targets, 15-24, 25 light targets, 8-15 highly parallel computation, 196-209 impact excitation, 24, 26-28 heavy targets, 48-71 higher angular moments, 71-80, 81, 82 light targets, 28-48 parallel SMC method, 196-209 input/output, 208-209 integral transformation, 204-206 one- and two-electron integrals, 197-199 parallel integral evaluation, 202-203 program outline, 207-208 quadrature and scaling, 199-202 scaling of transformation step, 206-207 pump-probe spectroscopy, 88, 89 scattering amplitudes, 3-8 Schwinger multichannel (SMC) method, 186, 190, 191 parallel SMC, 196-209 Sideband cooling, ideal gas, 256-258 Single-program multiple-data architecture, 194 Sodium electron-atom scattering, 8, 11-15, 71 hyperfine structure and isotope shifts, 158-163 impact excitation, 34-48 Stark shift, 175 Spectroscopy frequency-modulated lasers, 148-152 laser atomic bean spectroscopy, 142-148 optical spectroscopy, 142-148 precision laser spectroscopy, 141-142, 152-180 pump-probe spectroscopy, 88, 89 saturated absorption spectroscopy, 122- 125 Stark shift, 166-168 barium, 174 calcium, 174 cesium, 170-175
INDEX lithium, 174 potassium, 174 precision data, 173, 177-179 rubidium, 175 samarium, 175 sodium, 175 ytterbium, 168-170 Stimulated Rayleigh scattering, 88,90,91-95, 136-137 laser-cooled atoms, 104 in molecular physics, 107-109 optical lattices, 104-107 recoil-induced resonance as, 110-1 15 in solid-state materials, 109-1 15 stationary two-level atoms, 95-102 sub-Doppler radiative cooling, 102-104 Stimulated Rayleigh wing scattering, 107 Strong light scattering, 245-250 Sub-Doppler nonlinear spectroscopy, 124 Sub-Doppler radiative cooling, 102-104 Symmetric multiprocessor machines (SMPs), 191 Sympathetic cooling, 264-266
285 T
Time orbiting potential, BEC, 229 Trapped alkali systems, Bose-Einstein condensates, 234-238 Trimethylaluminum cross sections, 214-217 electron scattering, 185, 209-212 Two-beam coupling, 109 Two-wave mixing, 109
W Weak light scattering, 239-245
Y Ytterbium isotope shifts, 142-143,144,147-148, 153-158 Stark shift, 168-170
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Contents of Volumes in This Serial Volume 1
Volume 3
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall andA. T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Taknyanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P. Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fen
The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood
Volume 2
Volume 4
The Calculation of van der Waals Interactions, A . Dalgamo and W. D. Dauison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A . R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner
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 . Perciual Born Expansions, A. R. Holt and B. L. Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke
287
288
CONTENTS OF VOLUMES IN THIS SERIAL
Relativistic Inner Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W. Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosphere, A. Dalgamo The Direct Study of Ionization in Space, R. L. F. Boyd
Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, F . C. Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A . Ben-Reuuen The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAs’”p~, C. D. H. Chisholm, A . Dalgamo, and F. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 6 Dissociative Recombination, J. N. Barhley and M. A . Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman
The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu 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. Griuet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A . J . Greenfield
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C . Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C . Y. Chen and AugustineC. Chen
CONTENTSOF VOLUMES IN THIS SERIAL Photoionization with Molecular Beams, R. B. Cairns, Halrtead Harrison, and R. I. Schoen The Auger Effect, E. H. S. Burhop and W. N. Asaad Volume 9
Correlation in Excited States of Atoms, A. W . Weiss The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi O h The Differential Cross Section of LowEnergy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10
Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K. L . Bell and A. E. Kingston Photoelectron Spectroscopy, W. C . Price Dye Lasers in Atomic Spectroscopy, W . Lunge, J . Luther, and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, WesZey T . Huntress, Jr. Volume 11
The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Perciual and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton
289
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. Lmine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M . F. Golde and B. A . Thrush
Volume 12
Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R. -J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J . C. Lehmann, and J . vigue' Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, WIhelm Raith Ion Chemistry in the D Region, George C. Reid
Volume 13
Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, PaulR. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. V. Hertel and w. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet
290
CONTENTS OF VOLUMES IN THIS SERIAL
Microwave Transitions of Interstellar Atoms and Molecules, W . B. Somewille
Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weisold and Ian E . McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Chikf Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobasheu Rydberg Atoms, S. A . Edelstein and T. F. Gallagher U V and X-Ray Spectroscopyin 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. Griflth Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisionsat Low Energies, J. B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H . Bransden
Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H . B. Giibody 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. Celotta and D. T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopyof Laser-Produced 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. Fortson and L. W e t s
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 . Payne, C. H . Chen, G. S. Hurst, and G . W . Foltz
CONTENTS OF VOLUMES IN THIS SERIAL Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L . Dufion and A . E. Kingston
Volume 18
Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talber? 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 Ouasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A . Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W. Norcross and L. A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W. F. Drake
Volume 19
Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B . H. Bransden and R. K. Janeu Interactions of Simple Ion-Atom Systems, J . T. Park High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. BIum and H. Meinpoppen
291
The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. JenZ The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Fuubel Spin Polarization of Atomic and Molecular Photoelectrons, N . A. Cherepkou
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 InnerShell Ionization, W. E . Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and Dauid A . Armstrong On the Problem of Extreme UV and X-Ray Lasers, I. I. Sobel’man and A. V. Vinogradou Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J . A . C. Gallas, G. Leuchs, H. Walther, and H . Figger
Volume 21
Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn
292
CONTENTS OF VOLUMES IN THIS SERIAL
Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M. R . C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More
Volume 22
Positronium-Its Formation and Interaction with Simple Systems, J. W. Humberston Experimental Aspects of Positron and Positronium Physics, T . C. Grifith Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H . B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R . Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain
Volume 23
Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R . Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Hany 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-Amoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, F. J . Wuilleumier, D. L. Ederer, and J.L. Picque'
Volume 24
The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N. G. A d a m Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J . Smith and G . Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J . Knize, Z. Wu, and W . Happer Correlations in Electron-Atom Scattering, A. Crowe
Volume 25
Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements. Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He+-He Collisions at KeV Energies, R. F. Stebbings Atomic Excitation in Dense Plasmas, Jon C . Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G . Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H . G . Reid Electron Impact Excitation, R. J. W. Henry and A. E. Kingston
CONTENTS OF VOLUMES IN THIS SERIAL Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C . Allison High Energy Charge Transfer, B . H . Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W . R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F. Drake and S. P. Goldman Dissociation Dynamics of Molecules, T. Uzer
Polyatomic
Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstelllar Molecules, John H . Black
Volume 26
Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L . Moiseiwitsch The Low-Energy, Heavy Particle Collisions -A Close-Coupling Treatment, Mine0 Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, FranGoise Masnou-Sweeuws, and Annick Giusti-Suzor On the p Decay of I8’Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Lany Spruch
Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko
293
Volume 27
Negative Ions: Structure and Spectra, Dauid R . Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler Electron-Atom Scattering, I. E. McCarthy and E. Weigold Electron-Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengvel and M. I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28
The Theory of Fast Ion-Atom Collisions, J . S. Briggs and J . H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W . Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E . A. Hinds Volume 29
Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L . W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M. V. Ammosou, N. B. Delone, M . Yu. Iuanou, I. I. Bondar, and A. V. Masalov Collision-Induced Coherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Johann Rafelski and Helga E . Rafelski Cooperative Effects in Atomic Physics, J. P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J . H. McGuire
294
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and 1. C. Nickel The Dissociative Ionization of Simple, Molecules by Fast Ions, Colin J . Latimer Theory of Collisions between Laser Cooled Atoms, P. S. Julienne, A. M. Smith, and K . Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in Ion-Atom Collisions, Derrick S. F. Crothers and Louis J . Dub6
Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. w. F. Drake Spectroscopyof Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H . Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Duren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas. Michkle Lamoureux
Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H . Bransden and C. J . Noble Electron-Atom Scattering Theory and Calculations, P. G. Burke Terrestrial and Extraterrestrial H; , Alexander Dalgamo Indirect Ionization of Positive Atomic Ions, K. Dolder
Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W. F. Drake Electron-Ion and Ion-Ion Recombination Processes, M. R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H . B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I. P. Grant The Chemistry of Stellar Environments, D. A. Howe, J . M. C. Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J. W . Humberston How Perfect are Complete Atomic Collision Experiments?, H . Kleinpoppen and H . Hamdy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S. F. Crothers Electron Capture to the Continuum, B. L . Moiseiwitsch How Opaque Is a Star? M. J . Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing AfterglowLanemuir Technique, David Smith and Pairik Spang1 Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J . F. Williams and J . B. Wang
Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A. R. Filippelli, Chun C. Lin, L. W. Andersen, and J . W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmr and J . W . McConkey
CONTENTS OF VOLUMES IN THIS SERIAL Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R . W . Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Bany I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M . A. Dillon, Isao Shimamura Electron Collisions with N,, 0, and 0: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R . Celiberto, and M . Cacciafore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews, and Compendia of Data on Atomic Collisions, E . W . McDaniel and E . J. Mansky Volume 34
Atom Interferometry, C. S. Adams, 0. Carnal, and J . Mlynek Optical Tests of Quantum Mechanics, R. Y. Chiao, P. G. Kwiat, and A. M . Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleifner Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J . E . Lawler and D. A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N . A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E . C. Montenegro, W . E. Meyerhof, and J . H . McGuire
295
Indirect Processes in Electron Impact Ionization of Positive Ions, D . L . Moores and K . J . Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates
Volume 35
Laser Manipulation of Atoms, K . Sengstock and W . Ertmer Advances in Ultracold Collisions: Experiment and Theory, J . Weiner Ionization Dynamics in Strong Laser Fields, L . F. DiMauro and P. Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U.Buck Femtosecond Spectroscopy of Molecules and Clusters, T . Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. T . Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, w. R. Johnson, D. R. Planfe, and J . Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everitt and F . C. De Lucia
Volume 36
Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen, and Klaus Bartschat Stimulated Rayleigh Resonances and Recoil-Induced Effects, J.-Y. Courtois and G. Gtynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W . A. van Wijngaarden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincenf McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewensfein and Li You
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