Advances in Atomic and Molecular Physics, Volume 20

Advances in

ATOMIC AND MOLECULAR PHYSICS

VOLUME 20

CONTRIBUTORS TO THIS VOLUME DAVID A. ARMSTRONG

D. R. BATES CHRISTOPHEP dT T CHE R A. W. CASTLEMAN, JR. J.-F. CHEMIN H. FIGGER GORDON R. FREEMAN

J. A. C. GALLAS G. G. HALL

S. HAROCHE G. LEUCHS

T. D. MARK W. E. MEYERHOF

J. M. RAIMOND I. I. SOBEL’MAN A. V. VINOGRADOV

H. WALTHER

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

Sir David Bates DEPARTMENT O F APPLIED MATHEMATICS A N D THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

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

VOLUME 20 1985

@

ACADEMIC PRESS, INC.

(Harcourt Brace Jovanovich, Publishers)

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Contents

ix

CONTRIBUTORS

Ion - Ion Recombination in an Ambient Gas

D.R . Bates I. 11.

111. IV. V. VI.

Introduction Ter-Molecular Recombination Inclusion of Mutual Neutralization Channel Neutral-Neutral Channel Recombination in the Earth’s Troposphere and Lower Stratosphere Appendix References

1

2 21 29

33 37 37

Atomic Charges within Molecules G. G. Hall I. 11.

111. IV. V. VI.

41 42 45 53 57 60 62

Prologue Properties Population Partitioning Point Charges Purport and Prospect References

Experimental Studies on Cluster Ions T. D.Mark and A . W. Castleman, Jr

1. 11.

Ill. IV. V.

66 68 81 102 116

Introduction Experimental Formation of Cluster Ions Dissociation of Cluster Ions Thermochemical Properties V

CONTENTS

vi VI.

Other Properties References

i36 142

Nuclear Reaction Effects on Atomic Inner-Shell Ionization

U’.E. Meyerhof and J.-F. Chemin I. 11.

Ill. IV. V. VI. v11. VIII.

Introduction Survey of Nuclear Reactions Survey of Atomic Inner-Shell Ionization United-Atom Effects of Nuclear Reactions Separated-Atom Effects of Nuclear Reactions Summary Appendix A: Sketch of the Statistical Theory of Nuclear Reactions Auuendix B: K X-Ray Emission in Second-Order Distorted Wave Approximation References

173 176

182 189 208 226 228 23 1 234

Numerical Calculations on Electron-Impact Ionization

Christopher Bottcher I. 11. Ill. 1V.

Introduction The Wave Packet Method Box-Normalized Eigenstates Conclusions References

24 1 246 260 265 265

Electron and Ion Mobilities

Gordon R. Freeman and David A. Armstrong I. 11. 111.

lntroduction Electrons Ions References

267 270 296 320

On the Problem of Extreme UV and X-Ray Lasers I. I . Sobel’man and A . V. Vinogradov 1. 11. 111. IV.

Introduction Preliminary Considerations Inversion Schemes for Multicharged Plasma Conclusion References

327 328 333 342 343

CONTENTS

vii

Radiative Properties of Rydberg States in Resonant Cavities S. Haroche and J. M. Raimond

I. 11. 111. IV. V.

Radiative Properties of Rydberg States in Free Space Brief Survey of Experimental Techniques Single Rydberg Atom in a Resonant Cavity Collective Behavior of N Rydberg Atoms in a Resonant Cavity Conclusion and Perspectives References

350 358 363 383 408 409

Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction- Rydberg Molecules

J. A . C. Gallas, G. Leuchs, H. Walther, and H . Figger I. 11. 111. IV. V. VI. VII. VIII. IX.

Introduction General Properties of Rydberg Atoms Excitation and Detection of Rydberg States Methods of High-Resolution Spectroscopy of Rydberg States Results of High-Resolution Spectroscopy of Rydberg States Interaction of Rydberg Atoms with Blackbody Radiation Radiation Interaction ofRydberg Atoms-aTest System for Simple Quantum Electrodynamic Effects Rydberg States of Molecules Rydberg Molecules References

INDEX CONTENTS OF PREVIOUS VOLUMES

414 416 419 42 1 427 435 440 450 457 460

467 473

This Page Intentionally Left Blank

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

DAVID A. ARMSTRONG, Department of Chemistry, University of Calgary, Calgary, Alberta T2N 1N4, Canada (267) D. R. BATES, Department of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast BT7 lNN, Northern Ireland (1) CHRISTOPHER BOTTCHER, Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (241) A. W. CASTLEMAN,JR., Department of Chemistry, The Pennsylvania State Uni-

versity, University Park, Pennsylvania 16802 (65) J.-F. CHEMIN, Insfitut National de Physique NuclCaire et de la Physique des Particules, Centre d'Etudes NuclCaires de Bordeaux-Gradignan, 33 170 Gradignan, France ( 173) H. FIGGER, Max-Planck-Institut fur Quantenoptik, 8046 Garching, Federal Republic of Germany (4 13) GORDON R. FREEMAN, Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada (267)

J. A. C. GALLAS, Max-Planck-Institut fur Quantenoptik, 8046 Garching, Federal Republic of Germany (4 13) G. G. HALL,* Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, England (41)

S. HAROCHE, Laboratoire de SpectroscopieHertzienne de I'Ecole Normale Sup&ieure, Universitk de Paris VI, 7523 1 Paris Cedex 05, France (347) G. LEUCHS, Sektion Physik der Universitiit Miinchen, 8046 Garching, Federal Republic of Germany (413) T. D. MARK,? Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (65)

* Present address: Division of Molecular Engineering, Kyoto University, Sakyo-ku, Kyoto 606, Japan. t Present address: Institut fur Experimentalphysik,Leopold Franzens Universitilt, A-6020 Innsbruck, Austria. ix

X

CONTRIBUTORS

W. E. MEYERHOF, Department of Physics, Stanford University, Stanford, California 94305 ( 1 73)

J. M. RAIMOND, Laboratoire de Spectroscopie Hertzienne de 1’Ecole Normale SupCrieure, Universitk de Paris VI, 7523 1 Pans Cedex 05, France (347) I. I. SOBEL’MAN, P. N. Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, USSR (327) A. V. VINOGRADOV, P. N. Lebedev Physical Institute, Academy of Sciences ofthe USSR, Moscow, USSR (327) H. WALTHER, Max-Planck-Institut fur Quantenoptik and Sektion Physik der Universitat Miinchen, 8046 Garching, Federal Republic of Germany (41 3)

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 20

ION- ION RECOMBINATION IN AN AMBIENT GAS D.R. BATES Department of Applied Mathematics and Theoretical Physics The Queen’s University of BelJast Belfast, Northern Ireland

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . 11. Ter-Molecular Recombination . . . . . . . . . . . . . . . A. High Gas Densities . . . . . . . . . . . . . . . . . . B. Low Gas Densities. . . . . . . . . . . . . . . . . . . C. IntermediateGas Densities . . . . . . . . . . . . . . . D. Universal Curves . . . . . . . . . . . . . . . . . . . E. High Ion Densities. . . . . . . . . . . . . . . . . . . F. Influence of an Electric Field . . . . . . . . . . . . . . 111. Inclusion of Mutual Neutralization Channel. . . . . . . . . Alkali - Halide Ions . . . . . . . . . . . . . . . . . . . . IV. Neutral-Neutral Channel . . . . . . . . . . . . . . . . . V. Recombination in the Earth’s Troposphere and Lower Stratosphere . . . . . . . . . . . . . . . . . . . . VI. Appendix . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

1

2 2 7 13 16 18 20 21 28 29 33 37 37

I. Introduction Little attention will be given to laboratory work, this being fully covered by Loeb ( 1956), Massey and Gilbody ( 1974), and Armstrong( 1982).Moreover, except during the early years, it has provided little guidance to theorists. It is appropriate to call to mind the age ofthe subject. The earliest research was carried out by Thomson and Rutherford ( 1896), who showed that the conductivity of gases exposed to X rays is due to the presence of ions and recognized that ion-ion recombination must occur. Denoting the rate coefficient for the process by a,they noted that in the absence of a source the ion 1 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003820-X

D.R. Bates

2

density at time t, n(lt),should decay according to 1 --

1 -- at Nlt) n(10) Rutherford ( 1897) verified this prediction experimentally. Ion-ion recombination in an ambient gas comprises ter-molecular recombination A+ B- 2 + AB 2 (2)

+ +

+

and assisted mutual neutralization A+

+ B- + 2- A + B + 2

(3) It is convenient to consider first ter-molecular recombination in the absence of a mutual neutralization channel and then to consider the effect of such a channel. Proton transfer is another channel leading to neutralization. It may be important for some complex ions.

11. Ter-Molecular Recombination The form of the variation of the recombination coefficient a with gas number density n ( 2 ) is well known (Fig. 1). As n ( Z ) is raised, a increases from zero, passes through a maximum, and then decreases monotonically. At 300 K the maximum is typically around 2 X lod cm3/secand is reached near 2L where L is Loschmidt’s number (2.69 X 1019/cm3).The nature of the ions may depend on n ( 2 )because of clustering. The extent of clustering and its effect on a are specific to a particular gas. Partly because of this and partly because of the absence of challenging experimental data (with the masses of the positive and negative ions determined) theorists have concentrated their attention on the idealized case in which the ions are the same at all densities. Ter-molecular recombination represents the simplest possible three-body chemical reaction, the range of the interaction between the reactants being very long compared to that between either reactant and the third body. It would be a reproach to theorists if such a reaction were not thoroughly understood. A. HIGHGASDENSITIES His interest in ionic recombination having been aroused through having worked in Thomson’s laboratory, Langevin (1903a) devised a theory of the

3

ION-ION RECOMBINATION IN AN AMBIENT GAS

0

I

2

3

4

5

6

n (02)

+ +

+

FIG.1. Rate coefficient (Y for 0: 0; O2 [O,] O2at 300 K as function of n ( 0 , ) in units of Loschmidt’s number (Bates and MendaS, 1982a; see Sec. C). +

process. Although he did not specify restrictions on its range of validity his is in fact a high gas density theory. It is transparent in its simplicity. If a pair of oppositely charged ions are at a distance r apart the field between them is - (e/r2)iand hence if Kis the sum of their mobilities, the velocity with which they drift toward one another is

V, = - (Ke/r2); (4) Ions which come together are assumed to recombine. This assumption is valid at high gas densities where the ions do not gain appreciable kinetic energy of relative motion from their mutual attraction. On taking r to be large enough for the local ion concentration to equal the bulk average ion concentration and multiplying through by the surface area 4nr2 it may be seen that the flux obtained corresponds to a recombination coefficient

aL= 4nKe (5) The mobility, and hence aL,is inversely proportional to the gas density. Langevin’s neglect of diffusion was criticized by a number of scientists, notably Townsend (1 9 1 9 , Loeb and Marshall (1929), Harper (1932), JafE (1 940), and Loeb ( 1955, 1956). In the normal case of homogeneous ioniza-

D. R. Bdes

4

tion with ion densities typically lo6- 109/cm3the mean distance between neighboring ions is lov2cm. Commenting that at such great distances the mutual interaction energy of the ions is minute compared to their thermal energies, Harper ( 1932) opinioned that diffusion must therefore be the dominant transport mechanism. He sought to justify his physical intuition using simple mathematics. Fundamental to any comparison to drift is the Einstein relation eD = kTK

(6)

where D is the sum of the diffusion coefficients of the ions and T is the temperature. Unwilling to accept the concept ofthe statistical concentration gradient of an ensemble Harper did not follow the lead of Townsend ( 191 5 ) and apply Fick’s diffusion equation but instead treated diffusion as a Brownian movement as had Loeb and Marshall ( 1929).At time t the mean square separation of a pair of particles, initially distance d apart, is given by (Kennard, 1938) ( r Z )= d 2

+ 6Dt

(7)

Disregarding the distinction between ( r 2 )and r 2 (which causes only minor error) and differentiating with respect to time, Harper deduced a diffusion velocity V D = (3D/r)i

(8) He noted that V, falls off more slowly with increasing r than does V, of Eq. (4) and inferred that diffusion is indeed more rapid than drift at large separations. Proceeding as from Eq. (4) to Eq. ( 5 ) he deduced from Eq. (8) that the rate coefficient for encounters which bring oppositely charged ions to within distance r of each other is

P(r) = 127rrD

(9)

The separation making V, and VD equal is (10)

rH = Ke/3D

= e2/3kT

(= 1.9 X lod cm at 300 K)

( 1 1)

Harper argued that taking r in Eq. (9) to be rH yields the recombination coefficientbecause should the ions be closer than this critical separation they would drift together rather than diffuse apart. He remarked upon the coincidence that the formula obtained on doing so is identical with Eq. ( 5 )although the physical models are quite different. It was realized from the beginning that diffusion speed V,, obtained from Eq. (7) is uncertain and that the reason behind the choice of the critical radius rH is far from compelling.

ION-ION RECOMBINATION IN AN AMBIENT GAS

5

Illustrative of this different values of V, and r,, the combination again leading to Eq. (9,were given by J a E (1940). Nevertheless Harper had changed the perception physicists had of recombination at high ambient gas densities. In recognition of this it has become customary to refer to Eq. ( 5 ) as the Langevin - Harper formula and to use the symbol aLH for the rate coefficient involved. The practice is inappropriate because the change in perception is actually mainly incorrect. While diffusion has to be taken into account it is not rate limiting and is quite unimportant at large separations. This may readily be demonstrated. Fick's transport equation for ions is

n(lr)

F=hr2{F+D$}

where n( I r) is the concentration of ions of one sign at a distance r from an ion of the opposite sign and F is their inward flux toward this ion. Let r, be the separation at which recombination may be presumed to have occurred. All that need be specified concerning this separation (which is not to be confused with rH) is that it satisfies the strong inequality

ro rnu where r,, is the natural unit of length defined by

(13)

r,, = e2/kT (=5.6 X cm at 300 K) (14) Using Einstein's relation [Eq. ( 6 ) ]it is found that the solution for which

.-_

\

'I

L ' U J /

The corresponding recombination coefficient is (Townsend, 19 15)

Because of strong inequality [Eq. ( 1 3)] the exponential in the denominator may be ignored compared to unity so that Eq. (17) reduces to Eq. ( 5 ) of Langevin. Drift in the electric field rather than diffusion is the rate limiting process (Bates, 1975). Consider now as functions of r the contributions to F/n( 100) from drift and

D.R.Bates

6

diffusion, FK(Ir)/n(1") and FD(I r)/n(loo), respectively. Taking the denominator of Eq. (16) to be unity it is seen that

and

As the separation r is decreased from a large value (cf. Fig. 2) the drift contribution &(r)/n( Im, remains exceedingly close to its asymptotic value 4nKe until a very narrow zone just beyond r, is reached when it falls sharply toward zero because this is how n( Ir)/n(1") behaves. In contrast, the diffusion contribution &(r)/n( 103) remains negligible until the very narrow zone mentioned is reached. Through this zone the gradient dn(r)/dradjusts so that there is an increase in the diffusion contribution to match the decrease in the drift. The original argument based on a comparison of the drift speed VK of Eq. (4) and the diffusion speed VD of Eq. (8) is completely without validity because the latter speed is a measure of how fast Brownian movement increases the mean square separation of an isolated pair of particles, whereas what is relevant here is the speed at which the ions come together down the steady-state concentration gradient. An incidental consequence is that the numerical factor on the right of Eq. (9) is wrong (see the Appendix).

I

\

-

I 5 00

I

I 5 04

I

1

5 oe

DISTANCE BETWEEN IONS IN

I 5 12

I

5 16

8

FIG.2. Relative importance ofdrift and diffusion as a function ofdistance between ions at 300 K. The recombination distance r, is taken to be 5 A.

ION-ION RECOMBINATION IN AN AMBIENT GAS

7

A simple derivation of Langevin’s formula [Eq. ( 5 ) ] based on an examination of the contributions of both drift and diffusion to the inward flux has been given (Bates, 1983a). The mobility sum K may be known from laboratory data. Alternatively it may be obtained from the ion-neutral interactions. Putting a tilde over a symbol to indicate that a common practical unit is being used, we have that the contribution to the sum from ion i is

Ki = 1.85, X 104Qi1$(Z)-1&i31/2T-1/2cmZ/V*sec (hard sphere or symmetrical resonance charge transfer) =

1.382X 101a-11z$(Z)-1&;1/2 cm2/V-sec

(polarization)

(20)

where Q i is the diffusion cross section of the ion in the gas in A2, p” is the polarizability of the neutral in A3, $ ( Z ) is the gas density in units of LoSchmidt’snumber and Mi, is the ion-neutral reduced mass on the amu scale (Dalgarno et al., 1958; McDaniel, 1964).

B. Low GASDENSITIES Measurements by Langevin ( 1903b) indicated that the recombination coefficient a is approximately proportional to the square of the gas densityp in the low gas density region. Undue credence was attached to this variation which arose because of failure to allow properly for diffusion to the walls of the apparatus. Langevin (1903b) explained it by taking as the criterion for recombination that two ions having reached a separation of about the mean free path should then make a contact collision with each other. Richardson (1905) came closer to the truth. He correctly assumed that the recombination takes place if the internal energy of a pair of ions (i-e., the sum of their potential energy and their kinetic energy of relative motion) is rendered negative by collisionswith gas molecules. However, convinced ofthe need to reproduce the laboratory data, he wrongly postulated that several collisions are essential. Data showing a to be directly proportional to n ( Z ) in the low density region were obtained by Thirkill ( 1913) working under Thornson’s supervision. Satisfied as to their correctness, Thomson returned to them after he had retired from the Cavendish Chair. He succeeded in devising a neat theory (Thomson, 1924). The recombination mechanism is conceptually straightforward but is complicated because of the different courses a collision may open. One of a pair of oppositely charged ions which have approached each other along an open orbit collides with a gas molecule. This may reduce the

D.R. Bates

8

kinetic energy of relative motion ofthe ions so much that their orbit becomes bound. A subsequent collision may result in a switch back to an open orbit. Whether the ions recombine or separate indefinitely is determined by a sequence of many collisions. Thomson put forward a simplified model. Confining his attention to the case in which the masses of the ions and the gas molecule are equal he made the physical approximation that the kinetic energy of relative motion of the ions is always brought to the mean thermal value, jkT, by a collision. A closed orbit is then entered if, and only if, either ion collides with a gas molecule when within a trapping distance rT of the other such that

On Thomson’s model entry into a closed orbit is regarded as equivalent to recombination. The derived formula for the recombination coefficient is

where M,, is the reduced mass of the ions and Wis the probability that the specified stabilizing collision occurs. The general expression for Wis rather cumbersome (cf. Loeb, 1955). In the low gas density limit it reduces to

Q , 3 and Q23 being the ion-neutral collision cross sections. As the gas density is raised W increases monotonically toward unity. Thomson’s model is remarkably successful in view of the nature of the approximations involved. A version of it even reproduces the classical diffusion theory results of Pitaeviskii (1 962) on

e

+X++ Z + X +

Z

(24)

quite closely (Bates, 1980a).For 40 years after its introduction it was treated as though it might be refined to yield an accurate theory of ion -ion recombination at low gas densities (Loeb and Marshall, 1929;Jaffk, 1940; Natanson, 1960; Brueckner, 1964). However, the model is merely an artifice for carrying out an averaging easily. The averaging must be done properly if accuracy is to be attained. Fortunately the problem presented is not unduly formidable. It may be

ION-ION RECOMBINATION IN AN AMBIENT GAS

9

solved by first calculating n ( E )dE, the number density of ion pairs having internal energy in the interval dEaround the (negative)value E while recombination is proceeding, and then calculating the net rate of downflow of ion pairs past any chosen (negative)energy. The relevant equations may readily be written (Bates and Moffett, 1966) provided it may be assumed that (1) even though the internal energies of the ion pairs are being changed rapidly by collisions, their centers of mass execute thermal motion at the temperature of the ambient gas; and (2) the angular momentum of the ion pairs has the microcanonical distribution (i.e., the distribution in the square of the angular momentum is uniform). While the approximate validity of these assumptions was not in serious doubt it was necessary to check them. A detailed examination of (1) was made by Bates et al. (1971) and of (2) by Bates and MendaS (1975). It was shown by carrying out suitable computations that neither assumption introduces significant error. By denoting the rate coefficient for those collisions between ion pairs and gas molecules in which the internal energy of the ion pair is changed from E to an interval dF around F by K(E,F) dF it may be seen that the energy density distribution must satisfy the integral equation

n(E)

1;

n ( F ) K(F,E)dF

K(E,F)dF =

(25)

where IG 1 is the greatest binding energy an ion pair may have. The boundary conditions for the relevant solution are

n(G)= 0

(26)

and

n(E)=n,(E)

for E 3 O

(27)

n,(E) being the energy distribution in thermodynamic equilibrium. Equating the rate of recombination to the rate of downflow past energy X gives X

n(A+)n(B-)a= n ( Z )

[" [ J E - x JF-G

(n(E)K(E,F)- n(F)K(F,E))dEdF (28)

Naturally a does not depend upon X. It is advantageous to change to the dimensionless variables 3,

-E/kT,

p

-FfkT

10

D.R. Bates

and put

-XfkTGo,

(30)

-GfkT=v

Writing -fl(E) d E = q(A) dA,

-no(E) dE

qO(A)

dA,

P(A)

q(A)/qo(A) (31)

and

-K(E,F) dF

K(A,p) dp

(32)

and using the detailed balancing relation

(33)

VO(J)K(A,P) = tlO(P)K(PJ)

we see that Eqs. (25)-(27) are equivalent to

with and With the aid of Eq. (33) and the classical formula

qo(A) dA - ~ ~exp(A) / dA ~ n ( ~ + ) n ( ~-- ) 2 ( k ~ ) 3 ~ 5 / *

e

~

we find that Eq. (28) reduces to

Given the ion-neutral interaction, K ( A , ~ may ) be determined. The simpler cases have been treated: symmetrical resonance charge transfer (Bates and Moffett, 1966; Flannery, 1980); Langevin, that is, hard sphere core with polarization tail (Bates and Flannery, 1968); hard sphere core (Bates and MendaS, 1975; Flannery, 198 1 a); and polarization (Bates and MendaS, 1982b). Knowing K ( A , ~ the ) integral Eq. (34) may be solved by expressing it as a set of linear equations. The numerical work is eased by the transformation p = 3 tan u

(38) (Bates and MendaS, 1975). The evaluation of the recombination coefficient a from Eq. (38) is then straightforward.

ION-ION RECOMBINATION IN AN AMBIENT GAS

11

To complement Langevin's high gas density Eq. ( 5 ) it is desirable to have comprehensive tabulations giving the low gas density limit. This objective may be met more easily than might have been expected. There are two reasons. First, the recombination coefficient a may be accurately expressed as the sum of two partial recombination coefficients; thus

a = a,+ a2

(39)

where ai is the contribution arising from i - 3 collisions, that is, from collisions between ion i and gas molecules (Bates and Flannery, 1968; Flannery and Yang, 1980).Equation (39) is of key importance in reducing the number of parameters involved in the tabulation. Second, the dependence of a,on the three masses involved, the temperature and (in the case of a single-parameter interaction) that parameter (five entities in all) may be covered by the tabulation of a function F(ai) of one independent variable; Thus it is found that

where Mi, is the i - 3 reduced mass, where

a , = ( M , +M2-Mi)M3/Mi(Ml + M 2 + M 3 ) ,

i = 1 or 2 (41)

and where A , contains a power of the temperature and of the interaction parameter as multiplying factors (Bates and Flannery, 1968). Flannery (1980, 1981a) has given A , and tabulated F(a,) for the symmetrical resonance charge transfer and hard sphere core interaction cases and Bates and MendaS (1982a) have done so for the polarization interaction case. A polarization interaction is generally a better representation than a hard sphere core interaction. The symmetrical resonance charge transfer case arises only where the ions are daughters to the ambient gas molecules. A handy way of presenting the results is in terms of aTi,the Thomson partial recombination coefficient obtained from Eqs. (22) and (23) by excluding from the latter the cross section which does not concern i - 3 collisions. Bates and MendaS ( 1982b) have pointed out that the ratio ai/aTi may be expressed

+

+

ai/aTi = ( ( M , M2 - Mi)/(M1 M2))RT(a,), i = 1 or 2 (42) where &(ai) is simply related to F(ai).Figure 3 illustrates &(ai) for the symmetrical resonance charge transfer, hard sphere core, and polarization interaction cases. Note that in the case of symmetrical resonance charge transfer A+ + A -.A + A +

(43)

D.R. Bates

12

0

cL

-2 0

-1

0

0

1

log a (

FIG.3. The function &(a,) = [(MI+ M 2 ) / M 2 ]/aTl, ~ I where a,is the calculated partial recombination coefficient, aTIis the corresponding Thomson coefficient, and a, = M,M,/ M , ( M , M , M,) is the mass ratio parameter. Curve SR is for the case of symmetrical resonance charge transfer (Flannery, 1980), HS, a hard sphere interaction (Flannery, 198la), and P, a polarization interaction (Bates and Mend&, 1982b).

+ +

or A-+A4A+A-

(44)

we have

Mi = M , from which it follows that the maximum value of ai is unity. As may be seen the symmetrical resonance charge transfer and polarization curves lie very close together. The mobility is the key parameter in the analysis leading to R,(ai) and also determines the recombination coefficient in the high gas density limit. In conjunction with the closeness of the two curves in Fig. 1 this suggests that when treating recombination at any gas density it would be a satisfactory approximation to represent the effect of symmetrical resonance charge transfer collisions by using a polarization interaction with the polarizability assigned the effective value which gives the mobility correctly.

ION-ION RECOMBINATION IN AN AMBIENT GAS

13

For his model Thomson (1 924) took M,=M,=M3

(45)

so that

log a,.= -0.477 (46) By referring to Fig. 1, it may be seen that Thomson overestimated the recombination coefficient but was correct to within a factor of about two, surely a signal achievement. Until now, it has been assumed that the ion-neutral collisionsdo not excite internal energy modes (rotational or vibrational) of either the ion or the neutral. The effect of such inelastic collisions has been investigated by Bates (1 98 1a). He supposed that N collisions would be needed to redistribute the energy of relative motion of a colliding ion and gas molecule equally between the 1 translational degree of freedom and 2-4 rotational (or vibrational) degrees of freedom. His calculations which were camed out by the method of imposed collisions (Section II,C), demonstrated that the effect would be of little consequence even should N be as low as 2, which is a small number of collisions for such a redistribution. ai = 1/3,

C. INTERMEDIATE GASDENSITIES Bates and MendaS ( 1978a) have modified the steady state procedure described in Section II,B so that it is not confined to covering only the low gas density limit. They did this by incorporating orbits into the ion-pair number densities considered. Again, let E denote the internal energy of an ion pair and let r denote the distance between the ions. Bates and MendaS considered ni(E,r)dEdr and nj(E,r)dEdr number densities while recombination is proceeding in intervals dE around E and dr around r with the subscripts i and j indicating pairs in which the ions are moving together and apart, respectively. The steadystate equations for n,(E,r)and nj(E,r)are complicated. They may be solved by a power series expansion in n ( Z ) ,the ambient gas number density. This leads to a similar power series expansion for a,the recombination coefficient. A comparison to the power series expansion for aT,the Thomson recombination coefficient, is instructive. It shows that the a - n ( Z )graph is convex upward like the aT - n ( 2 )graph but that its gradient changes less in the low n ( Z ) region (in agreement with results obtained later by computersimulated experiments). The origins of the curvatures are not the same. According to the analysis of Bates and Mend& (1978a) the curvature arises because as n ( 2 )increases so does the dependence on r of n,(E,r)and nj(E,r)

D.R. Bates

14

and so does the difference (ni(E,r)- nj(&r)) whereas on the Thomson model it arises because the probability that either ion makes a collision within the trapping radius tends toward unity. The modified steady-state procedure gives insight on a facet of mechanism of recombination. However, the power series for a! converges slowly and computational difficulties mount rapidly with each successive term. Consequently, the procedure does not provide a practical means ofbridging the gap between the high gas density region governed by transport and the low gas density region governed by the loss of the internal energy of ion pairs in collisions. One important way of proceeding is by the microscopic theory of Flannery (1982a), which is well summarized in a recent review article (Flannery, 1982b) and will not be discussed here. Another way is through computersimulated experiments (Bates and MendaS, 1978b; Bates, 1980b; Bardsley and Wadehra, 1980). In these, a computer program arranges that a succession of (fictitious) pairs of ions start from some distance R, apart great enough for their relative velocity distribution to be taken to be Maxwellian and that they move in their mutual Coulomb field making random collisions with molecules of the ambient gas. It also arranges that the start of the experiment and a recombination event (should this take place) be recorded in two accumulative stores. Each experiment is terminated at the earlier of recombination and (in the procedure of Bates and MendaS, 1978b) the occurrence of a so-called partial parting or alternatively (in the procedure of Bardsley and Wadehra, 1980) the instant when separation Ro is again reached. The recombination coefficient may be obtained from the entries in the accumulative stores after very many (perhaps several million) experiments. Computer-simulated experiments as just described are very inefficient at low gas density. The steady state method (Section II,B) which has generally been used to cover this region is in some instances inconvenient. Another way of proceeding is by the method of imposed collisions (Bates, 198la). In this variant of the normal computer-simulated experiments a program ensures that one of the pair of ions experiences a random collision in the interval between the start of the experiment and the separation becoming Ro again. By taking into account the probability of the imposed collision in each experiment, it is possible to calculate the derivative of a with respect to n ( Z ) in the low n ( 2 )limit. Figure 1 gives results (Bates and MendaS, 1982a) on 0:

+ 0; + 0,

+

[O,] + 02

(47)

with the ambient gas at 300 K. A polarization ion-neutral interaction is assumed.

ION-ION RECOMBINATION IN AN AMBIENT GAS

15

In the case of a polarization interaction, a at constant n ( 2 )varies as T-3in the low n ( Z ) limit and is independent of T i n the high n ( 2 ) region. This difference does not hinder exact temperature scaling of the a - n ( Z )graph (Bates, 1980~). The scaling law may most easily be established using natural units (nu). The basic nu are e2/kTfor length, M I , for mass, and k T for energy. Derived nu are for frequency and e4/[(kT)3/2Mf$2] for binary rate coefficient. An advantage of changing to nu is that the distribution of the kinetic energy of the ambient gas molecules and the distribution of the internal energy of the ion pairs becomes temperature independent. Choosing any arbitrary distance R , nu let NR,) nu be the rate coefficient for Ro encounters defined as starting when the ions reach Ro of each other and ending when they are infinitely separated and let P(R,) be the probability that an R, encounter leads to recombination. It is evident that a = 4 v o ) ~ ( R onu)

(48)

For the purpose at hand, make the region R < Rofield free and let T(&) nu be the total time that ions are within this region during an R , encounter. Equating the rates at which ions enter and leave the field-free region gives (49) The probability P(R,) which appears in Eq. (48) and the time t(R,) which appears in Eq. (49) control the dependence of the recombination coefficient a on T, the temperature, and on n(Z),the ambient gas number density, which will be kept in absolute ( /cm3) units. Now r(R,) andP(R,) depend on Tand n ( Z )only through the collision frequency v in nu: Thus provided v is held fixed T and n ( Z )do not affect other pertinent physical entities when these are expressed in nu. In the case of a polarization interaction we have that tl(Ro) = 4nRi exp( ~ / R O ) / ~ @ O )

where p is the polarizability of the neutral and Mi3 is the reduced mass indicated by the subscripts (both quantities in absolute units). It is seen from Eq. ( 5 1 ) that v in nu does not vary along a line on which n ( Z ) P l 2is constant. Hence neither does a when expressed in nu nor aT3l2when a is expressed in cm3/secso that the original a - n ( 2 )graph for 300 K may be scaled to temperature T simply by taking the dependent variable to be ( T/3O0p2aand the independent variable to be (300/T)3/2n(Z). At a temper-

16

D. R. Bates

ature of 1200 K, for example, the maximum recombination coefficient is one-eighth that at 300 K and occurs at a gas density eight times as great. Temperature scalingis possible for other single-parameter interactions. In the case of a hard sphere core interaction the independent variable is again ( T/300)3/2a while the dependent variable is ( 3 0 0 / T ) n ( Z )(Bates, 1980b). This scaling law also applies to symmetrical resonance charge transfer.

D. UNIVERSAL CURVES In the low and high ambient gas density limits we have (omitting the symbol indicating the species of ambient gas molecule)

&+An

(52)

-

(53)

and

a

B/n

where, for a given recombination process and a given temperature, A and B are known constants. Noting that A and B are measures of the two physical effects controlling the recombination rate (collisional energy loss and transport) it would seem worth trying whether we can put

a = 4(A,B,n)

(54)

where 4 is some function of the variables indicated (Bates, 1980d). Dimensional considerations and the necessity ofconforming with Eqs. (52) and (53) require that Eq. (54) be of the form

a = (([ 1 +f(r)l/An)+ ([ 1 + g(r)ln/B))-i

(55)

where

r = An2/B and f and g are functions such that f ( r ) and n2g(r)-0 and

n-2f(r) and g(r)

-

as n - 0

0

as n

-

(574

co

(57b)

Regarding recombination process [Eq. (47)] at 300 K, on which the computer-simulated experiments have been done, as the standard case we will affix the letter S as a subscript on all symbols relating to it and will similarly use X for a recombination process about which we require information. We

ION-ION RECOMBINATION IN AN AMBIENT GAS

17

may without loss of generality take the temperature as also being 300 K because the temperature-scaling problem has already been solved. It is clear that scaling parameters A and p exist which make the Aa,.- qn, graph coincide with the as- n, graph (Fig. I ) in both the low and high gas density limits. Their values are 3, = (AsB,/AxB,)'/2

and (59)

tl = (AxBs/AsBx)"2

They make

r, = r, (60) so that insofar as relation (54) is valid the two graphs coincide completely making ( T / 3 0 0 ) 3 / 2 h, (300/T),I2qn, a universal curve. To test this Bates and MendaS (1982a) carried out computer-simulated experiments on a variety of cases. There is no evidence of any significant differences between the recombination coefficients they obtained and those based on the universal curve. If the positive and negative ions have different complexities the former (say)may take part in symmetrical resonance charge transfer collisionswhile the latter may not. Bates and MendaS (1982a) modified their computer simulation program in the simple manner required to model symmetrical resonance charge transfer as indicated earlier (Section 11,B). All that had to be done was to use an effective polarizability (up to about four times the true polarizability)for the positive ion-neutral collisions. The universal curve was found to be a reliable predictor. Sennhauser et al. ( 1980) have made measurements on the mean recombination coefficient CW describing the loss of cluster ions in ammonia NHa(NH,), -tX-(NH,),

+ NH,

+

products

(61)

P.n

( X - probably being C1- or OF).It is given by

wheref: andf;;are the fractions of positive and negative ions havingp and n ammonia molecules clustered and where a!, refers to recombination between the (p,n)cluster ions. They also have given the approximate values of f: ( p = 3 to 6) and f;;( n = 2 to 5 ) and measured the mobilities. Bates and MendaS ( 1982a)used the latter to determine the effective polarizabilitiesand thence (with the help of the universal curve) 5at several gas pressures. The agreement with the measured values is quite satisfactory.

18

D. R. Bates

E. HIGHION DENSITIES In the preceding sections it has been implicitly assumed that the ion density nj = n(A+) = n(B-)

(63) is low enough for collective effects to be ignored. On introducing the Debye- Huckel screening length 1 = (kT/8ne2ni)lI2

(64)

it is plausible to take as a sufficient condition for the assumption to be justified that the dimensionless entity Q = l-Z(e2/kT)2

(65)

be considerably less than unity (which as it happens would ensure that another relevant dimensionless entity ~ z , ( e * / k Tbe) ~also considerably less than unity). At 300 K 0 = 4.3

x

10-1511,

(66)

so that the condition is quite well satisfied even when ni is as great as 1013/cm3.The ion densities encountered were far less than this until the recent past but a need to investigate collective effects has arisen because the ion density may be around 1015/cm3(cf. Nighan, 1982) in modern laser discharges. A superficially attractive approach is to determine a spherically symmetrical effective potential V(r)for an ion by using Poisson’s equation together with either the Boltzmann equilibrium formula (Morgan et al., 1980) or, preferably, the transport equations (Flannery, 198 1 b) to determine the charge distribution. Because of rather similar defects inherent in the two approximations only the former, which is the simpler, will be discussed. It leads (cf. Landau and Lifshitz, 1958) to the Debye-Huckel potential field e U(r)= - exp(- r / l ) r provided aissmall compared to unity. The restriction on aexcludes the high niregion it is desired to cover but this has been disregarded in exploratory work. The practice has been to suppose - eU(r) to be the interaction potential between a pair of oppositely charged ions. This is unsatisfactory because it implies that only one of the pair is screened. Moreover, the screening cloud fixed around the chosen ion has been taken as being without mass and as

ION-ION RECOMBINATION IN AN AMBIENT GAS

19

consisting of a continuum. In consequence, the results of the pioneering computer-simulated experiments on recombination (Morgan et al., 1980) are mainly of historical interest. According to them a decreases at all ambient gas densities as n,is increased above about 1013/cm3.However, a simple argument (Bates, 1981b) shows that there is no such decrease at least in the high ambient gas density limit, the Langevin formula [Eq. (5)] remaining valid. Thus at sufficiently high gas densities oppositely charged ions which are initially nearest neighbors remain nearest neighbors until they recombine and noting that their drift toward each other is mobility controlled Eq. ( 5 ) is easily recovered. This conclusion is supported by results which Morgan et al. ( 1982) have obtained from computer-simulated studies on recombination using a powerful molecular dynamics code which they have developed. The code enabled them to follow the motions of 50 ion pairs in a cell having periodic boundary conditions. These conditions largely eliminate the effect of the number of ion pairs studied being finite. Because of its demand on computer time, the code has been sparingly employed and generally with auxiliary economizing approximations. Morgan et al. (1982) suggest that if ni exceeds 1014/cm3screening reduces a significantly at ambient gas pressures below 1 atm. However, there is an opposing effect: collisional ion -ion recombination X+

+ Y - +X +

(or Y - )

+

XY

+X +

(or U-)

(68)

Bates (1982a) has estimated the addition, ( d c ~ )this ~ ~ ,process makes to a at low ambient gas densities, by scaling a formula for the rate coefficient for collisional electron-ion recombination which Stevefelt et al. (1 975) derived from the results of the classical trajectory analysis camed out by Mansbach and Keck (1967). In the case of

+ + Kr+

Kr+ F-

(or F-)

+

KrF

+ Kr+

(or F-)

(69)

for example, he obtained =3

X 10-7(300/T)4.5(n,/1015) cm3/sec

(70)

in the linear region. The linear region should extend to near the ion density nf at which appreciable overlap between successive ion - ion collisions begins. This is given approximately by 1 ni = (kT/e2)3

2n

The value of nf at 300 K is 1 X 1015/cm3. Much remains to be done on recombination at high ion densities.

D. R. Bates

20

F. INFLUENCE OF AN ELECTRIC FIELD Morgan et al. ( 1982)have drawn attention to the inhibition of ter-molecular recombination by an applied electric field and have used their molecular dynamics code (Section II,E) to obtain illustrative results for the case of Kr+ - F- recombination in helium at 300 K and 3 atm with the ion density at 101’/cm3.Two effects occur. First, a field hinders recombination by raising the kinetic energy of the ions (especially at low ambient gas densities). Morgan et al. (1982) note that when the field to gas number density E / N is 20 Td the drift velocities of Kr+and F- in helium are 1.O X lo’ cm/sec and 1.4 X lo5 cm/sec, respectively, suggesting an effective ion temperature of about 3000 K. Second in the presence of a field E the interaction potential is

-e z r

-

V(r)= -- eE r At 20 Td and 3 atm this potential has a saddle point where its value is - 4 nu at an ionic separation of 0.5 nu. As pointed out by Morgan et al. (1982),

recombination would therefore require that the internal energy of an ion pair be reduced to below - 4 nu by an ion-neutral collision at an ion -ion separation ofless than 0.5 nu, which isclearly a rare event. In harmony with this the molecular dynamics calculations show that a is a factor of about 60 less than its value in the absence of a field (Fig. 4).

10-8

-

0

5

10

15

20

E/N Td

FIG.4. Ter-molecular recombination coefficient for Kr+ - F ions, density lOlS/cm3,in helium at 300 K and 3 atm as function of E/N(after Morgan et a/., 1982). I

1 Td = lo-” V.cm2.

ION-ION RECOMBINATION IN AN AMBIENT GAS

21

The behavior at high gas densities is quite different from that found by Morgan et al. (1982). Let Ei be the applied electric field. Treat one ion as fixed at the origin of coordinates. It may be proved that an oppositely charged ion will drift toward and ultimately reach the fixed ion if, initially, when the separation is large it is upstream and within distance

r, = (4e/E)1/2 (73) of the x axis. On the assumption that the drift is rate limiting we have that a = nraKE (74) Kbeing the sum of the mobilities of the two species ofion. In view ofEq. (73), a of Eq. (74)is obviously independent of E and indeed has the value given by Eq. ( 5 ) of Langevin ( 1903a).

111. Inclusion of Mutual Neutralization Channel Considerableeffort has been devoted to research on mutual neutralization in the absence of an ambient gas (75) (cf. Bransden and Janev, 1983).For present purposes it is sufficient to recall that the process may be regarded as taking place through an avoided crossing between the ionic and covalent potential surfaces (Fig. 5). On the approxiA++B-+A+B

I ’

DISTANCE BETWEEN NUCLEI

FIG.5 . Avoided crossing between ionic and covalent potential surfaces.

D. R. Bates

22

mation due to Landau (1932) and Zener (1932) the probability of mutual neutralization ensuing from a double traversal of the avoided crossing is

P = 2P(1 - P)

(76)

where

p = exP(- v/vx) (77) in which vx is the radial component of the velocity of relative motion at the distance r, between the nuclei at the avoided crossing and q is the Landau Zener parameter defined by q = (n2/he2)(rxAU}2

(78)

AU being the closest separation in energy of the potentials. Provided the thermal energy is much less than the kinetic energy gained in the approach down the ionic potential to the avoided crossing, the rate coefficient a4 may be expressed as

}:{

(

a4 = 16rx 2M:,kr)"2(E3

- E3

with

v,

= (2/M12rx)112

{z})

(79)

(80)

all quantities being in atomic units and E(x)being the exponential integral indicated (Janev and Radulovic, 1977). Some illustrative values of a;are given in Table I. As may be seen, they lie well below the ter-molecular recombination coefficients except at low ambient gas densities. We might therefore be inclined to suppose that mutual neutralization in an ambient gas is not a process of much importance. Computer-simulated experiments (Bates and MendaS, 1978b; Bates, 1980b,e, 1981c, 1983b; Whitten et al., 1983) have shown the supposition to be quite incorrect. It is convenient at this stage to systematize the notation. Subscripts 2, 3, and 2 will be affixed to (Y to indicate the rate coefficients for mutual neutralization, for ter-molecular recombination, and for total ion loss. As already mentioned a$'is the rate coefficient for mutual neutralization in the absence of an ambient gas. Its companion a! is the rate coefficient for ter-molecular recombination in the absence of a mutual neutralization channel. Results in the form of axand ( Y curves ~ at 300 and 2000 K are shown in Figs. 6 and 7. The ions concerned have the same masses and avoided crossing separation r, as have Na+ and C1- but are allotted several values of the Landau -Zener parameter q; the ambient gas consists of mixture [NJ:[HZ]: [HZO] =0.606:0.139:0.255

(81)

ION-ION RECOMBINATION IN AN AMBIENT GAS

23

TABLE I VALUES OF MUTUAL NEUTRALIZATION RATE a$at 300 K for Ion Pairs Having COEFFICIENT Some Reduced Mass"AND SAMEAVOIDED AS Na+ - CICROSSING DISTANCE* ~

~~

~

Landau - Zener parameter q(au) ~

1 -6c 1 -5 1-4

1-33 1-3 1-23

b c

as (cm3/sec)

~~

2.1-10 2.1-9 1 .7-8 3.7-8 5.0-8 2.5-8

1 3 . 9 5 amu. 9.6 A. 1-6' 1 x 10-6.

which is representative of the burnt flame gas of Burdett and Hayhurst ( 1979), whose attempt at measuring ionic recombination in such gas stimu-

lated some of the theoretical work under discussion. The ion-neutral interaction was taken to be ofthe polarization type with H 2 0assigned an effective polarizability. Observe that a2 = ap

- a3

(82) may be much larger than a! (Figs. 6 and 7); that is, that mutual neutralization may be greatly enhanced by the presence of an ambient gas. The explanation is simple. A free ion pair can be converted into a bound ion pair by collisions with the ambient gas. This facilitates mutual neutralization because it allows the avoided crossing to be traversed very many times provided the angular momentum and internal energy of the ion pair satisfy certain conditions. To illuminate the explanation Bates and Morgan (1983) camed out computer-simulated experiments giving P(3,) for several values of 7 where p(3,)d l is the rate at which ion pairs belonging to the steady state distribution and having internal energy in the interval d3, around 3, traverse the avoided crossing. It is evident (Fig. 8) that the contribution to the rate from bound ion pairs may far exceed that from free ion pairs. The ratio ofthe former contribution to the latter isgreatest at low temperature. In the qrange covered, it increases as r,~is decreased. This is a consequence of a corresponding increase in the steady state number density of bound ion pairs. Although the statistical errors in the values of axobtained from the early

24

D.R. Bates

2

I

3

PRESSURE (atm)

FIG.6. Rate coefficients for total ion loss a,(full curves) and ter-molecular recombination a3(- - - ) for ions isometric with Na+ and C1- and having an r, value of 17.8 au in burnt flame gas mixture [Eq. (8 I ) ] at 300 K. Value of Landau-Zener parameter q in au marked on each curve. In the q = 0 case, the at,a 3 ,and a! curves coincide.

computer-simulated experiments (Bates and MendaS, 1978b) made da,/dn(Z)rather poorly determined it was clear that the value ofthe derivative at the origin is high. An analytic proof has been given (Bates, 1979) that shows it is actually infinite. Bates and Morgan (1983) have confirmed this remarkable feature using the method of imposed collisions (Section 11,B). The amount by which mutual neutralization enhances the loss rate A a = a y -a: (83) is of interest in some connections. Using a hard sphere core for the ion-neutral interaction, Bates ( 1980b) has carried out computer-simulated experiments giving A a at several temperatures for 0:

+ 0; + 0,

neutral producfs

(84) He let a4 have a number of values (some untypically high to ease the calculations). As may be seen (Fig. 9), A a passes through a maximum as the gas +

ION-ION RECOMBINATION IN AN AMBIENT GAS

25

2

I

PRESSURE (atm)

FIG.7. Same as Fig. 6, except temperature is 2000 K.

density is raised. This maximum is not as broad as the corresponding termolecular recombination maximum, falls off more slowly with increase in temperature (approximately as T-'I2), and is located at a much lower gas density. It is useful to note that A a vanes approximately as CUB if all other parameters are held fixed. Smith and Adams ( 1 982a) have adduced laboratory evidence for the predicted initial sharp rise in axfrom studies on NO+

+ NO: + He

-.

SF:

+ SF; + He

-

and

neutralproducts

neutral products

(85)

(86) Their measurements give that in the case of Eq. (85) the values of CUB and of apat 10 Torr of helium are 6.5 X and 1.5 X lo-' cm3/sec and that in the case of Eq. (86) the corresponding values are 4.0 X lo-* and 1.0 X

10

0

-10

-20

x

x -30 -40

-50

-60

I

166

0

-5

x

-15

-10

I 0-s

b

00

6b

0 0 0

0 0

OO

-08

Energy (eV)

0

-12

-16

0

0 C

0 0.0

-0.4

%

0

0 - ~

00 1

0

0 Io

-10

10-8

0

0

-8

-6

-4

/ % 0s

0 0

10-8

-04

1

-2

Io-’

0

10-8 04

0

C

10-7

i

E

-0.8

Energy (eV)

-1.2

-

I 0-9 t

, ,

00

, -04

00

O0o* -0.8

1

p, 1

-1.2

-1.6

Energy (eV)

FIG.8. Coefficient p(A) giving the rate oftraversal ofthe avoided crossing. Ions and ambient gas are as in Figs. 6 and 7. (a) 0 = 300 K, (b) 0 = lo00 K, (c) 0 = 2000. Contribution from free ion pairs (); data pointsgive contribution from bound ion pairs with the Landau-Zener parameter q:(0).t ] = lod au, (0).q = lo-’ au, (0). q= au. Upper horizontal scale gives internal energy in nu, lower, in eV (Bates and Morgan, 1983).

27 27

ION-ION RECOMBINATION RECOMBINATION IN IN AN AN AMBIENT AMBIENT GAS GAS ION-ION

,

,

,

I

C

-

1

I

d

-

-

-

-

-

d

FIG.9. Amount A a by which mutual neutralization increases the ion loss rate coefficient. The independent variable at each temperature T is ( T s / T )n ( M ) , where Ts is the standard temperature (300 K)and n ( M ) is in units of Loschmidt’s number; the dependent variable is ( T/Ts)I/*A a with A a in units cm3 sec. The value of a: at 300 K is given on each curve (in as at Tis indicated by the horizontal lines on the leR units cm3/sec) while that of ( T/Ts)1’2 of the vertical scale. (a) 150 K, (b) 300 K, (c) 600 K, (d) 1200 K.

O-’ cm3/sec. cm3/sec.The Theincreases increasesare aremuch muchtoo toogreat greatto tobe beattributed attributedto toter-molecter-molec11O-’ ular recombination which is very slow at 10 Ton of helium as may immediular recombination which is very slow at 10 Ton of helium as may immediately be seen from the results of calculations by Flannery and Yang (1978). ately be seen from the results of calculations by Flannery and Yang (1978). Their relative values are approximately proportional to the relative values of Their relative values are approximately proportional to the relative values of a4 as mentioned in the preceding paragraph and are also dependent on the a4 as mentioned in the preceding paragraph and are also dependent on the masses and and the the ion-neutral ion-neutral collision collisioncross cross sections. sections. masses The laboratory evidence just mentioned must be be viewed viewed with with caution. caution. The laboratory evidence just mentioned must Reference is made to another publication (Smith and Adams, 1982b) for Reference is made to another publication (Smith and Adams, 1982b) for details of the experimental results. The ax pressure curves presented in this details of the experimental results. The axpressure curves presented in this publication are are not not quite quite consistent consistent with with ax axat at 10 10Torr Ton having having the the values values publication

D.R. Bates

28

cited above. More important, the curves are tangential to the pressure axis in the low pressure limit instead of being tangential to the araxis. Returning to Figs. 6 and 7, it is seen that mutual neutralization can severely suppress the rate of ter-molecular recombination. This is because the two processes are in competition and a large number of traversals of the avoided crossing are likely to occur before collisions make the internal energy of an ion pair so negative that the avoided crossing can no longer be reached. Figure 9 shows that

A a 3 a4

(87)

except at high ambient gas densities. It follows from Eqs. (82) and (83) that a2

+ aj 3 a4 + a;

(88)

This inequality signifies that the increase in the mutual neutralization rate due to the gas exceeds the decrease in the ter-molecular recombination rate due to the mutual neutralization channel.

ALKALI- HALIDEIONS

The rate coefficients for reactions between alkali metal and halogen ions are of interest in connection with the direct generation of electric power by MHD (cf. Hayhurst and Sugden, 1963) and in connection with the electrical properties of rocket exhausts (cf. Jensen and Pergament, 197 1). Janev and Radulovit ( 1977) applied the asymptotically correct Landau Herring method (see Janev, 1976) to determine the AU values of many alkali-halide combinations. To calculate a4 from Eq. (79) they then assumed that the A U value involved could be taken to be that for the 2P3,2state of the halogen atom and that the potential surface associated with the 2P,,2 state could be disregarded. Bates (1983b) adopted the same approximation to calculate not only a: but also a 2 ,a;, aj,and axin the burnt flame gas mixture [Eq. (8 l)] at 2000 K. Table I1 contains the results. The ion pairs are arranged in order of decreasing q, which is the most important parameter involved. Minor irregularities in the variations down the columns arise from the dependence ofthe rate coefficients on r,and the masses ofthe ions. It will be seen that mutual neutralization is the major loss process unless q is such that it makes

29

ION-ION RECOMBINATION IN AN AMBIENT GAS TABLE I1 FOR REACTIONS BETWEEN ALKALIAND HALOGEN RATECOEFFICIENTS" IONS AT 2000 K AND IN THE BURNTFLAME GAS MIXTURE^ [(Eq.

Mutual neutralization Ions

Landau-Zener parameter q(au)

Li+ INa+ ILi+ BrNa+ BrLi+ C1Na+ CIK+ IRb+ IK+ Brcs+IRb+ Br-

4.6PC 1.22-3 5.92-4 9. I9-' 6.74-5 5.07-6 3.53-6 3.00-' 4.0Y9 3.57-10 3.0O-ll

Ter-molecular recombination

ff2

1.1-8 1.3-8 1.7-8 4.9-9 4.0-9 4.3-1° 3.9-1° 4.2-11 6.9-13 7.9-14 6.8-15

ff!

2.2-8 2.2-8 3.2-8

1.4-9 l.l-9 1 . P 1.2-9 1.8-9 1.4-9 8.6-1° 4.6-1° 8.6-1° 3.3-10 5.4-1°

1.1-8

1.2-8 2.8-9 3.6-1° ( 1 .O-")

vs

VS

ff3

Total loss

ffz

2.4-11 1.3-" 1.5-Ll 3.5-11

2.2-8 2.2-8 3.2-8

5.O-Il

1.2-8 3.0-9 2.T9 6.2-1° 8.6-1° 3.3-10 5.4-1°

1.9-1° 1.7-1° 2.6-1° 8.6-1° 3.340 5.4-1°

1.1-8

In cm3/sec. Except for as. 4 . W ' = 4.65 X loT3;VS =
0) but this is not possible for polyatomics. In the Mulliken theory the overlap population is itself significant. It mea-

+

ATOMIC CHARGES WITHIN MOLECULES

47

sures the electron density which is attracted to both nuclei and so gives rise to bonding. The total overlap population is then a quantity which should correlate with bond strength. The first attempt to generalize these ideas to polyatomic molecules was made by Chirgwin and Coulson ( 1950)in the context of a K electron theory of conjugated molecules. For the gross atomic population their definition is the natural generalization of the diatomic definition and involves overlap populations from all other atoms. For their bond-order definition they generalized the Huckel theory definition which assumed orthogonality. If the density matrix in terms of the atomic orbitals A, is rs

then the atomic population on A, is

and their bond order between A, and A, is

Mulliken ( 1955) himself made a generalization of this to arbitrary molecules. His formulas allow for sums over all the atomic orbitals associated with each atom in such a way that the result is invariant under an axis rotation. A second attempt to provide a generalization was made by Lowdin (1 950) in connection with his method of orthogonalizing a set of functions in a symmetrical way. The orthogonalized atomic orbitals can be defined as A:

=

2 As(S-1/2)sr S

where S-' is the matrix inverse of the overlap matrix S.The density matrix becomes P=

P:sA:*A:

(16)

rs

where

The diagonal elements are used by McWeeny (195 1) to define the atomic populations and the off-diagonal elements the bond order. Although these definitions are not the same as the Chirgwin-Coulson and Mulliken ones

G. G. Hall

48

[see, e.g., the numerical comparisons for diborane given by Politzer and Cusacks ( 1968)],Davidson ( 1976)has shown that when the overlap integrals between atoms are small they are the same to first order.

B. MODIFIED WEIGHTING A number of authors have attacked the Mulliken definition of the gross atomic population on the ground that an equal division ofthe overlap term is arbitrary. One alternative possibility is to divide it in proportion to the net atomic populations of that orbital. Formulas ofthis type have been suggested by Stout and Politzer ( 1968) and by Christoffersen and Baker (197 1). These achieve the desirable object of ensuring that the net contribution to an atomic population from any molecular orbital is always positive and less than 2. In terms of a diatomic and in the simple notation given above, the Christoffersen -Baker definition of the gross atomic charge becomes qA = u2

+ abS[2a2/(a2+ b2)]

(18) As might be expected, these charges give a more realistic representation of the charge distribution in the sense that, if localized as point charges on the nuclei, the resulting molecular dipole moment is closer to the accurately calculated one. As has been pointed out by Grabenstetter and Whitehead (1972), these formulas are not invariant to a unitary transformation of the occupied molecular orbitals. An alternative form suggested by Ross and Schmit ( 1966) does retain this invariance but has not attracted such support because of its complexity. C. CONSERVING T H E BOND MOMENT The use of the atomic populations themselves results, as B shows, in expressions that become complicated to use and to justify. Lowdin ( 1953) suggested that another criterion for the division might be the conservation of the bond dipole moment. An immediate disadvantage of this idea is that it introduces new integrals, the moments, into the calculation. Attempts to use this idea have been made by Doggett ( 1969)and by Hillier and Wyatt (1969). The results are not greatly changed from the original Mulliken populations though, as Hillier and Wyatt point out, the change is sometimes enough to reverse the charge on an atom. Although the proposal does improve the significance of the calculated atomic charges by ensuring both the conservation of total charge and of total dipole moment, it pays the price of becoming rather more difficult to imple-

ATOMIC CHARGES WITHIN MOLECULES

49

ment. The problem of invariance under rotations is not eased and as a result the charges become basis dependent.

D. DIVIDING SPACE The artificial element in these definitions of atomic charge led Pollak and Rein ( 1967)to suggest a major change in approach. They proposed to define the atomic charge directly by an integration of the electron density over a volume appropriate to the atom. For a diatomic they suggested dividing space by a plane through the midpoint of the bond. At a stroke this gives charges which are invariant to basis and axis transformations. It also gains considerably in intuitive appeal. The use of the midpoint in this definition was seized on by several authors as unnecessarily arbitrary. Bader et al. (197 1) moved the plane to the point along the internuclear axis where the density has a minimum (i.e., to the col point). They pointed out that as a bonus, a radius could then be assigned to the atom so that the internuclear distance is strictly the sum of the radii. Politzer and Hams ( 1970) preferred to define their atomic region by reference to the free atom. Maclagan (1 97 1) offered three definitions. The first was the point where the bare nuclear potentials were equal, i.e., z A / r A = ZB/rBand the second where the forces were equal, i.e., z A / r i = ZBIri. The first gave larger and unrealistic results but the second, which has some justification in terms of the Hellmann -Feynman theorem (see Berlin, 195I), is much more reasonable. A third point obtained by using(2 - 2) for Z for first-row atoms gives qualitatively more acceptable results. While these definitions do solve some of the problems of the Mullikentype approach, they each have their own limitations. Finding a convincing argument for the choice of dividing point will not be too difficult as soon as sufficient evidence for the reasonablenessof the final results is produced! It is much more difficult to see how the definition can be extended to polyatomic molecules since the use of planes to divide space is then no longer sufficient. A recent paper by Grier and Streitwieser (1982) points out that differences in electron distribution following substitution are often localized and so much less sensitive to the definition of the boundary. By an integration differences in the electron populations can be calculated even though the totals remain in some doubt. E. LOCALIZED HYBRIDS

Because part of the problem arises from the need to make the result invariant under rotation of axes at an atom, McWeeny (1 960) suggested that

G. G. Hall

50

there should be a preliminary phase to the calculation in which a “best” hybrid atomic orbital for use in describing the bond is determined. This was developed by Davidson ( 1967) who started by calculating the best least squares fit of the molecular charge density by atomic orbitals. Localized atomic orbitals have also been defined by Ruedenberg et al. ( 1982) not only to describe bonding but to describe its modification along a reaction path. These ideas have been expanded into a full discussion by Roby ( 1974). His work is based on several new ideas. In the first place he rejects the idea of atomic populations implied in all previous work. He associates with an atom all the charge which affects it. When charge is shared by two atoms it is counted in both. This, he claims, gives a more realistic impression of what each atom is contributing to the bonding. Obviously, to conserve total charge, the sum of his atomic occupation numbers has to have the shared parts subtracted so that they are included only once in the molecular total. In terms of the notation in Section III,A, the Roby occupation number is

n A = az

+ 2abS + b2S2

(19)

and the shared occupation number is ,s

= 2abS

+ (a2+ bZ)S2

(20)

so that (21) n~ + n~ - SAB is conserved by the normalization condition. As might be expected these atomic occupation numbers exceed the nuclear charges in all the quoted examples (see Table I). Another aspect of Roby’s work is his systematic use of projection operators. This has several major advantages which are important to his argument. The projection operator is defined in a way that makes it easier to preserve invariant properties of the various kinds that are required. It also makes it possible to generalize from a molecular orbital wave function to any one TABLE I

Rosy OCCUPATION NUMBERS ~~

Liz N, F, LiH

FH

3.870 8.355 9.225 3.562 9.510

3.870 8.355 9.225 1.845 1.396

1.741 2.710 0.447 1.407 0.906

ATOMIC CHARGES WITHIN MOLECULES

51

involving the same set of atomic orbitals. Thus if the natural spin orbitals for the molecules are li) with the occupation number ni then the one-electron density is written

Similarly, the projection operator for the atom A can be defined as

where IAr) are the orthonormal orbitals required to define completely the state of the free atom A. The occupation number of A is then defined as

where Tr is the trace operation applied to the matrix. Since the numbers ni are positive, it is apparent that the atomic occupation numbers must all be positive. IflABt ) is the tth member ofthe set ofatomic orbitals on A followed by those on B and if S,, is the matrix of their overlap integrals then the projection operator into the joint space of A and B is n,

The corresponding occupation number is and from this the shared occupation number is defined as sA,

= nA

+ n B - nAB

(27)

It is readily seen that, if A and B are so remote that all their overlap integrals vanish, S,,becomes the unit matrix and nAB= nA nB so that sM = 0. Roby is also concerned with finding the most appropriate description of the atoms. The matrix S, is in block form with unit matrices in the diagonal positions. The remaining blocks can also be made diagonal. This is done by applying a unitary transformation to the orbitals of each atom separately exactly as in the theory of corresponding orbitals (Amos and Hall, 1961). The significanceof these transformations is that they construct the hybrids in pairs on each center such that their overlap is a maximum. This obviously solves the phase problem at the same time. The Roby theory has been modified by Cruickshank and Avramides (1982). They point out that if the set of atomic orbitals on each atom is extended toward completeness, then each atomic occupation number will tend to the total number of electrons in the system. Their solution is to restrict the atomic orbitals to those which span the Hartree-Fock model of

+

G. G. Hall

52

TABLE I1

ROBYAND MULLIKEN ATOMICCHARGES Mulliken

Roby 4.4

CH, NH3 OH, FH

-0.27 -0.43 -0.46 -0.32

qB

SAE

q.4

4B

Overlap

0.07 0.14 0.23 0.32

1.461 1.341 1.201 1.048

-0.76 -0.92 -0.78 -0.47

0.19 0.31 0.39 0.47

0.760 0.685 0.562 0.461

the atom. This achieves the object of producing values of the occupation numbers that rapidly become independent of basis size though it leaves unresolved the question of what is the appropriate state of the atom. Cruickshank and Avramides also proceed to calculate an atomic charge in the more conventional sense. They reduce the atomic occupation numbers by halfthe shared charge to produce an atomic charge which is conserved. In simple terms, their gross atomic charge is qA = nA - isAB= a2

+ abS - +S2(a2- b2)

(28) which differs from Mulliken bJ terms in the square of the overlap. They also claim that the shared occupation number is a measure of bond strength. For single bonds, sM is approximately (2s)so the operation offinding hybrids on each atom to maximize the overlap is a return to the older valence concept of bonding being determined by the principle of maximum overlap of hybrids. A comparison between the net atomic charges and those of the original Mulliken analysis is shown in Table 11. As is evident, the Roby charges are more moderate than those of Mulliken. Very similar considerations to these arise in the discussion of atomic charge initiated by Rys et al. (1976) and continued by Yanez et al. (1978). They are concerned with the electron density rather than with individual orbitals and so can use a linear combination of contracted spherical Gaussians to fit the density by means of a least squares criterion. The number of these functions is fixed and equal to the number of electron shells but the contractions are determined by an optimized fit to a spherically averaged atomic density. Some provision for the effect of charge transfer and bonding is made by allowing for an adjustable scale factor on the valence shell function. The constraints in this fitting are clearly designed to avoid the difficulties mentioned above of allowing too much freedom to the atomic basis functions while trying to obtain a close fit to the electron density. The

ATOMIC CHARGES WITHIN MOLECULES

53

TABLE 111 COMPARISON OF CHARGES OBTAINED BY FITTING PROCEDURES Density fitting"

Field fittingb

q,4

qH

qA

qH

CH4 NH,

-0.033 -0.658

OH,

-0.614

FH

-0.392

0.008 0.219 0.307 0.392

0.728 -0.252 -0.650 -0.507

-0.182 0.084 0.325 0.507

Yanez et al. (1978). Hall and Smith ( 1 983).

resulting net atomic charges appear to be consistent from molecule to molecule and from one wave function to another. Some examples are shown in Table 111.

IV. Partitioning A. BISECTION

Even a cursory inspection of the experimental charge densities of molecules obtained from X-ray crystallography or by electron scattering experiments shows the strong peaks of charge associated with the heavy atoms and the thin threads of charge that hold them together. The natural consequence is a definition of the atom by bisecting the bonds, using planes, at their col points. For simple enough molecules these planes divide up space into atomic regions and the atomic charge can be defined as the integral over this region. Various authors have suggested the extension of this process of bisecting bonds to calculated electron densities so that direct comparisons could be made between theory and experiment. The idea works well for diatomic and linear molecules since the planes remain parallel and the definition is unambiguous. It can be extended in a plausible way to planar molecules but it becomes increasingly difficult, for more complicated molecules, to allocate all the volumes cut offby the planes

54

G. G. Hall

in a nonarbitrary way. An alternative analysis of experimental densities is given by Hirshfeld ( 1977). B. LOGES

The Lewis ( 1916) theory of atomic structures saw the electrons within an atom as localized in inner shells or in specific bonding directions. The prequantum electronic theory of valence (see, e.g., Remick, 1943) built this up into an interpretive scheme which correlated a mass of empirical observations and had considerable predictive power. Daudel ( 1953)and his collaborators (for a recent review and references see Aslangul et al., 1972) have set themselves the task of finding the rigorous quantum analog ofthis theory. They propose the partitioning ofthe space of the molecule into loges such that the probability of a fixed number of electrons being separated is greatest. Thus for an atom such as Li there is a definite radius of a sphere such that the probability of two electrons being inside it and one outside is a maximum. The loge boundaries are generally either spheres (for inner shell loges) or planes. The loge partitioning has considerable chemical appeal with its visual interpretation of the two-electron bond and the lone pair. It suffers from the disadvantage that it requires the N particle probability functions and that it uses an information theory criterion which is difficult to manipulate. A simpler approach using pair distributions has been discussed by Bader and Stephens (1975). C. TOPOLOGICAL ATOMS A more fundamental attack on the problem of partitioning the electron density in a theoretically sound way has been initiated by Bader and his collaborators (Bader and Beddall, 1972;Bader et al., 1973).They considered the orthogonal trajectories ofthe electron density, i.e., the collection of paths each ofwhich represents the motion of a point starting off, usually at infinity, and climbing up the density by the steepest route possible, which will cut the level surface ofp orthogonally, i.e., in the direction of Vp. Almost all of these trajectories terminate at one or other of the nuclei. The trajectories which terminate at a nucleus fill up the space assigned as belonging to that atom. In the terminology of Thom (1975), the nucleus is an attractor and the atom its basin. The behavior ofthe electron density at a nucleus is dictated by the fact that the potential becomes infinite there. To satisfy the Schrodinger equation

ATOMIC CHARGES WITHIN MOLECULES

55

exactly this divergence has to be eliminated by a canceling term. The source of this is the kinetic energy operator. If the wave function has a discontinuity in slope at the nucleus then it has a divergent kinetic energy contribution of exactly the right kind. For an isolated atom the wave function will be symmetrical so the electron density has a conical cusp at the nucleus (Kato, 1957).For an atom inside a molecule the electric field at the nucleus due to the rest of the molecule has to be included since this has the effect of pulling the cone to one side. In the Hartree-Fock theory, where the effect of the other electrons is averaged, the electric field at the nucleus is the force acting on the nucleus. By the Hellmann-Feynman theorem this will vanish when the nucleus is in equilibrium and consequently the electron at the nucleus will experience no other force there and so will have a symmetrical conical singularity. In nonequilibrium configurations and particularly for H atoms the cone may be so pulled by the field that the nucleus ceases to be a local maximum. In general, orthogonal trajectories start and finish at critical points, i.e., points where V p = 0, or at infinity. Although the nucleus is, strictly speaking, a singularity rather than a critical point (Lea,V p is not defined at the nucleus) it behaves, topologically, exactly like a maximum. In the neighborhood of a critical point the leading term in the local approximation to p is quadratic, i.e.,

p-po=r+Hr where H i s the Hessian matrix with elements

(29)

H~ = gplax, axj By classifying the forms of H the different species of critical point are identified (Collard and Hall, 1977),H has three eigenvalues, and all are real since it is symmetrical. Each can be positive or negative. The number of nonzero eigenvalues is the rank and the difference in the number of positive over negative eigenvalues is the signature of H. The maximum has rank 3 and signature - 3 which is abbreviated to (3,- 3). A minimum would have (3,3) since all its eigenvalues are positive. There are two kinds of col: (3,- 1) and (3,l). The (3,- 1) col is the one which lies between nuclei. There are two trajectories which originate there and one goes to each of the bound nuclei. It is this which Runtz et al. (1977)define as the bond. The negative eigenvalues at the col define a surface which is the boundary surface between the two atoms. The (3,l)col is found inside a molecular ring. Its positive eigenvalues define a finite surface, or cap, whose edges are the bonds while its negative eigenvalue is an axis. A minimum occurs inside a cage where these ring axes originate. This analysis of the electron density has the advantage of being completely

56

G. G. Hall

independent of axes and of basis sets provided these are sufficient to describe the critical points. Not only does it define a partition of space into atomic volume it also defines bond paths and ring caps whose shapes contain important information about the bonding. For a planar molecule containing a ring of identical atoms the situation raises new complications. At the center of a three-membered ring, p must have three-fold symmetry and consequently H will be degenerate. If the ring is large enough p may become rather flat there with zero eigenvalues. This would turn the point into a rank 1 maximum ( 1 ,- 1). In the cap the significant term in the approximation to p will be the cubic one. The form of the trajectories at the center is that ofFig. 3 in Collard and Hall ( 1977).Similarly, for a four-membered ring the major term is quartic with four hills and valleys meeting at the center. There are some disadvantages in this solution to the partitioning problem. In the first place there are some features that do not exactly match chemical ideas. Thus, for example, two approaching He atoms produce a col and a bond between them just as two H atoms would in forming H, and despite their inability to form a chemical bond. To distinguish these situations topology alone is insufficient and quantitative criteria have to be added. Another difficulty has been that since the atomic boundaries are often complicated surfaces it is necessary to develop new numerical techniques to manipulate them. Even the atomic charges, defined by an integration over the atomic volumes, are far from trivial to calculate. A suggestion to overcome this has been made by Hall and Smith ( 1984) who revert to an earlier device of taking planes through the col points and pointing out that there is very little density in the outer disputed regions. Their calculations involve a fitting ofthe molecular density by Gaussians so the outer parts have even less charge than in the true density. Some examples are shown in Table 111. A major argument in favor of this partitioning is that it fits naturally into molecular quantum mechanics. Bader has shown (for a recent review see Bader and Nguyen-Dang, 1981) that a virial theorem can be proved for each atomic region and that various quantum mechanical results true for the molecule can be extended to apply to the regions because of the special properties of the dividing surfaces. He also shows that many molecular properties, in particular the total energy, can be expressed as additive functionals of the atomic regions. Stutchbuq and Cooper ( 1 983) have demonstrated the practical utility of this Bader definition by discussing the atomic charges and dipoles of a series of alcohols and amines. In particular, they point out that the role of methyl groups in neutral saturated molecules is minimal whereas in ions they play an important role as electron sources or sinks.

ATOMIC CHARGES WITHIN MOLECULES

57

V. Point Charges A. EMPIRICAL

The idea of using point charges and point dipoles to represent the electron distribution over a molecule is a classical idea (cf. Polya and Szego, 1951) and has been applied by many people without using quantum theory. Julg ( 1971) has reviewed these attempts, which rely principally on dipole moments, and shown their utility in relation to NMR and ESCA observations. He also demonstrated the existence of a minimum in the electrostatic potential around H20inside the lone pairs. Reproducingthis minimum is a severe test for a simple model because Earnshaw’s theorem ensures that the potential cannot have a minimum at a point in space free of charge. A systematic attempt to build point models of simple molecules has been made by Kollman (1978). He uses the electronegativity of the atoms to partition charges between atomic centers. Lone pairs are represented by point charges located at the van der Waals radius, or perhaps half of it, from the nucleus. The results are used in calculations of intermolecular forces. Noel1 and Morokuma ( 1976) are concerned by the hydration of ions and complexes. They develop point charge models with the aim of reproducing intermolecular forces derived from calculations. These calculations show that the electrostatic terms are the controlling ones in the angular dependence of the forces. B. LOCALIZATION The idea of using the Mulliken populations to give a point charge model has appealed to many authors. Replacing the electronic density of an atom by a delta function implies that the atomic orbitals are well localized and close to spherically symmetric. In some molecules this may be true but it is not always so. A molecule containing 0 or N atoms frequently has significant lone pairs. For these atoms, at least, the point charges must be supplemented by point dipoles to achieve a plausible representation of the electron density. The possibility of using moments at a single convenient central point to represent the molecular charge density has been investigated by Rein ( 1973). He is interested in calculating molecular interaction energies and demonstrates that the expression for these in terms of molecular moments becomes divergent even when they are well separated. By comparison, using an ex-

58

G. G. Hall

pansion on every atom produces much more satisfactory and compact results. This work suggests that one criterion for a suitable set of point charges and dipoles must be the accuracy of the resulting electric potential at typical intermolecular distances. A very similar approach to the problem has been made by Scrocco and his colleagues(for a recent review see Scrocco and Tomasi, 1978).They prefer to expand around the charge centers of the localized molecular orbitals. This avoids the need for any arbitrary division of an overlap density. It also abbreviates the expansion by locating the atomic charge at the center so that the dipolar term vanishes. In a recent paper (Etchebest et al., 1982)a modification ofthis procedure is suggested. The K shell electrons are absorbed into the nucleus. The other localized orbital distributions are expanded about their centers of charge up to the octuple. These additional moments have the effects of reducing the error in the electrostatic potential to about one-quarter of its value obtained using the Mulliken point charges and of allowing the representation to be used up to 2.5 (A) of any nucleus. On the other hand, Huzinaga and Narita (1980) have preferred to return to the Mulliken populations. Since p is a quadratic form in the atomic orbitals they take the charge and dipole of each orbital product and allow charges to be transferred by adding dipoles to preserve the moment. This gives a model consistingof point charges and point dipoles which are determined to ensure a good local representation. C. SHRINKING GAUSSIANS The process of deriving point charge models received a considerable boost when Hall ( 1973;also Tait and Hall, 1973)showed that spherical Gaussians could be shrunk into delta functions without changing any of the spherical moments. Since this preserves an infinite number of moments it is a major advance over a fitting which retains only a small number of them The derivation ofthe model begins with the electron density expressed as a quadratic form in Gaussians multiplied by powers of the coordinates. Now the product of two Gaussians is itself a Gaussian but its center is moved to a point on the line joining the original centers. The remaining factors can easily be transformed to this new center. By this technique the double summation is turned into a linear summation. For an FSGO wave function, i.e., one in which only spherically symmetric Gaussians are used but their locations are optimized, all the products will themselves be spherical Gaussians and the summation takes the form

ATOMIC CHARGES WITHIN MOLECULES

59

The point charge model of this is then

A factor (x - x,) can be introduced into an orbital expressed in terms of Gaussians by applying the operator (d/ax). When higher Gaussians are used (see Martin and Hall, 1981) the single summation can be written as

where Ds(V)is a function of the operators (dldx), (d/ay),and (a/&) so that the correct power factors are produced. The point model then replaces this by

Thus, for example, a p function will introduce the point dipole VS. By using the identity

it is readily shown that "

"

for any function F which satisfies V 2 F= 0 (37) This includes all the spherical moments. The first moment which is not of this kind has F = r2. This point charge model provides approximate electrostatic potentials which agree closely with the true ones outside a radius of about 6 bohr from the nuclei. Its disadvantage is that the number of points can become large. N Gaussians in the original basis can lead to as many as N ( N + 1)/2 points when the products are rewritten as single Gaussians. This has led a number of authors to suggest ways of combining many of the smaller terms into larger units to achieve a more compact result without too much loss of accuracy. Shipman (1975) retains the charges corresponding to the squared terms and adds to them the overlap charges weighted by the exponents of the Gaussians. This has the effect of retaining both the total charge and the total dipole moment. Huzinaga and Narita (1980) have rediscovered the same allocation and extended it to higher Gaussians. For the special case ofa Frost FSGO wave function, where the number of Gaussians is exactly half the

G. G. Hall

number of electrons, Amos and Yoffe (1975) show that an even simpler model is possible in which two electrons are placed at the center of each Gaussian. Stone ( 198 1) has given formulas for relating moments at one point to moments at a different point and comments on the advantages of a choice which remains compact but ensures that the neglected higher order moments do not diverge. He advocates that squared terms, which are naturally centered on the atoms, should be retained in full but that the overlap terms should be redeveloped around a center, probably the centroid, and terminated. An alternative procedure for abbreviating the point charge expressions has been developed by Hall and Smith (1984). They use a variation principle, developed by Hall and Martin (1980) and Hall (1983), to optimize the fitting of the electron density by a linear combination of Gaussians on the atoms. The shrinking ofthese into delta functions is then possible without introducing new centers. In effect this redistributes the charges in an optimal way. When the exponents of the Gaussians are also optimized, some become small corresponding to diffuse functions spreading over several nuclei. They suggest that these functions should be partitioned spatially at the col points using planes and that charges and dipoles within each atomic region should be centered on the nuclei. This theory seeks to combine the utility of the point charge expression with the conceptual clarity of the topological partitioning.

VI. Purport and Prospect In strict terms, atoms do not exist in molecules as peas do in a pod and the atomic charge is an artificial concept which can be validly defined in different ways. Earlier sections have outlined three alternative ways of looking at the problem. The key issue on which they differ is their understanding of localization. Clearly the atomic charge must presuppose some definition of a localized electron density around a particular nucleus. In the population approach this is defined using atomic orbitals since these do concentrate in the appropriate neighborhood and are natural to the form of the wave function. The partitioning approach provides a definite volume around each nucleus to constitute the atomic region so the localization is strictly nonoverlapping and precise. The point charge approach cames localization to its ultimate form by using distributions such as delta functions and their derivatives. Distributions themselves are not very meaningful but they produce simple expressions for molecular moments and for the electrostatic potentials. The three approaches can also be contrasted in what they regard as the

ATOMIC CHARGES WITHIN MOLECULES

61

purpose of the atomic charge. In the population approach the atomic charge occurs as a major term in the total energy. It quantifies the extent of ionic bonding in the molecule and leaves the overlap population to relate to covalent bonding. On the other hand, in partitioning it is the electron density which is dominant and its differential topology which motivates the definitions. Virial theorems show that a partitioning of the energy into atomic contributions does result but these contributions depend on the shape of the boundary surfaces and on various functions defined over them. The direct comparison possible, when exactly the same atomic orbitals are used to define the atom in any molecule, has been lost. The point charge approach is not intended to explain molecular binding but rather to simplify the calculation of the intermolecular forces. Each of the approaches has its own technical problems which are slowly being solved. The problem of providing definitions which incorporate the desired invariance properties has been a long-standing one in population studies. The systematic use of density matrices and projection operators has led to a solution though there are problems remaining in the specification of the atomic states and their orbitals. Calculations of atomic chargesand other properties in the partitioning approach are relatively few because of the difficulties of specifyingand using the bounding surfaces. The use of strongly localized functions to fit the electron density and the recognition that some contributions are too small to merit accurate calculation have reduced the magnitude of this problem. Any use of point charges is subject to obvious mathematical problems since the self-energy of a point charge is divergent and some other physical quantities cannot be expressed since distributions cannot be multiplied. The device of carrying out the approximations using Gaussians and shrinking these into delta functions as the final step has decoupled the two operations and permits conventional techniques to be used most of the time. Although it might seem that at present there are three essentially different definitions of the atomic charge, each with its own advantages and disadvantages, some convergence in the approaches can be discerned. There is the recognition within the population approach that localization in the Hilbert space of atomic orbitals is not enough and that spatial localization is also necessary. In the partitioning approach the converse applies since spatial localization alone brings computational problems which are considerably eased by the use of localization in the Hilbert space. Even the point-charge approach benefits by an intermediate stage in which localized functions play an important role. Finally, the use of a variation principle to determine fitting seems to open up the possibility of an eventual unified definition. The goal of defining an atomic charge, within a molecule, which can be calculated readily and used in discussions of molecular bonding and reactivities has not yet been reached. As the different definitions are refined in

62

G. G. Hall

response to objections and become numerically closer, hope rises that it will eventually be achieved. This process of deriving a concept, the atomic charge, from the wave functions and mechanics of the electrons is an important example of scientific synthesis, the deduction of a higher level concept from lower level ones. The difficulties of this process have been discussed by Hall (1 959) and, more recently, by Primas ( 1981). A chemical concept, which does not enter into the Schrodinger equation, has been shown to have a real existence, though with limitations because the definition is not yet precise, within the solutions describing the electronic motion inside the molecule.

REFERENCES Amos, A. T., and Hall, G. G. (1961). Proc. R. Soc. London Ser. A 263,483. Amos, A. T., and Yoffe, J. A. (1975). Theor. Chim. Acta 40,221. Aslangul, C., Constanciel, R., Daudel, R., and Kottis, P. (1972). Adv. Quantum Chem. 6, 94. Bader, R. F. W., and Beddall, P. M. (1972). J. Chem. Phys. 56,3320. Bader, R. F. W., and Nguyen-Dang, T. T. (1981). Adv. Quantum Chem. 14,63. Bader, R. F. W., and Stevens, M. E. (1975). J. Am. Chem. Soc. 97,7391. Bader, R. F. W., Beddall, P. M., and Cade, P. E. (1971). J. Am. Chem. SOC.93, 3095. Bader, R. F. W., Beddall, P. M., and Peslak, J., Jr. (1973). J. Chem. Phys. 58, 557. Bader, R. F. W., Keavenly, I., and Cade, P. E. (1967). J. Chem. Phys. 47, 3381. Berlin, T. (1951). J. Chem. Phys. 19,208. Bonham, R. A., Lee, J. S., Kennedy, R., and St. John, W. (1978). Adv. Quantum Chem. 11, 1. Chirgwin, B. H., and Coulson, C. A. (1950). Proc. R. Soc. London Ser. A 201, 196. Christoffersen, R. E., and Baker, K. A. (1971). Chem. Phys. Lett. 8,4. Collard, K., and Hall, G. G. (1977). In;. J. Quantum Chem. 12, 623. Coppens, P., and Stevens, E. D. (1977). Adv. Quantum Chem. 10, 1. Cruickshank, D. W. J., and Avramides, E. J. (1982). Philos. Trans. R. SOC.Ser. A 304, 533. Daudel, R. (1953). C. R. Acad. Sci.Paris 237,601. Davidson, E. R. (1967). J. Chem. Phys. 46, 3319. Davidson, E. R. (1976). “Reduced Density Matrices in Quantum Chemistry.” Academic Press, New York. Doggett, G. (1969). J. Chem. SOC.A , 229. Epstein, S. T. (1974). “The Variation Method in Quantum Chemistry.” Academic Press, New York. Etchebest, C., Lavery, R., and Pullman A. (1982). Theor. Chim. Acfa 62, 17. Feynman, R. P. (1939). Phys. Rev. 56, 340. Freed, K. F., and Levy, M. (1982). J. Chem. Phys. 77, 396. . Grabenstetter, J. E., and Whitehead, M. A. (1972). Theor. Chim. Acta 26, 390. Crier, D. L. and Streitwieser, A. (1982). J. Am. Chem. SOC.104, 3556. Hall, G. G. (1959). Rep. Prog. Phys. 22, 1. Hall, G. G. (1973). Philos. Mag. 6, 249. Hall, G. G. (1973). Chem. Phys. Lett. 20, 501. Hall, G. G. (1983). Theor. Chim. Acta 63, 357. Hall, G. G., and Martin, D. (1980). Isr. J. Chem. 19, 255.

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Hall, G. G., and Smith, C. M. (1984). Inter. J. Quanf. Chem. 25,881. Hellmann, H. ( 1937). “Einfiihrung in die Quantenchemie.” Deuticke, Leipzig. Hillier, I. H., and Wyatt, J. F. (1969). Int. J. Quantum Chem. 3, 67. Hirshfeld, F. L. (1977). Theor. Chim. Acta 44, 129. Hohenberg, P., and Kohn, W. (1964). Phys. Rev. 136B, 864. Huzinaga, S., and Narita, S. (1980). Isr. J. Chem. 19, 242. Julg, A. (1 975). Top. Curr. Chem. 58, 1. Kato, T. (1957). Commun. Pure Appl. Math. 10, 15 I. Kollman, P. (1978). J.Am. Chem. SOC. 100,2974. Lewis, G. N. ( 1 9 16). J. Am. Chem. SOC. 38,762. Liiwdin, P. 0. (1950). J. Chem. Phys. 18, 365. Lowdin, P. 0. (1953). J. Chem. Phys. 21, 374. Maclagan, R. G. A. R. (1971). Chem. Phys. Left.8, 1 14. McWeeny, R. (1951).J. Chem. Phys. 19, 1614. McWeeny, R. ( 1 95 1). J. Chem. Phys. 20,920. McWeeny, R. (1960). Rev. Mod. Phys. 32, 335. Martin, D., and Hall, G. G. (198 1). Theor. Chim. Acta 59,28 1. Mulliken, R. S. (1935)J. Chem. Phys. 3, 573. Mulliken, R. S. (1955). J. Chem. Phys. 23, 1833. Noell, J. O., and Morokuma, K. ( 1 976). J. Phys. Chem. 80,2675. Politzer, P. (198 1). In “Chemical Applications of Atomic and Molecular Electrostatic Potentials,” p. 7. Plenum, New York. Politzer, P., and Cusacks, L. C. (1968). Chem. Phys. Lett. 2, 1. Politzer, P., and Harris, R. R. (1970). J.Am. Chem. SOC.92,6. Politzer, P., and Truhlar, D. G. (1981). “Chemical Applications of Atomic and Molecular Electrostatic Potentials.” Plenum, New York. Pollak, M., and Rein, R. (1966). J. Theor. Eiol. 11, 490. Polya, G., and Szego, G. (195 1). “Isopenmetric Inequalities in Mathematical Physics.” University Press, Princeton. Primas, H. (198 I). “Chemistry, Quantum Mechanics and Reductionism.” Springer-Verlag, Berlin and New York. Remick, A. E. ( I 943). “Electronic Interpretations of Organic Chemistry.” Wiley, New York. Rein, R. (1973). Adv. Quantum Chem. 7,335. Roby, K. R. (1974). Mol. Phys. 27,81; 28, 1441. Ros, P., and Schmit, G. C. A. (1 966). Theor. Chim. Acta 4, I. Ruedenberg, K., Schmidt, M. W., Gilbert, M. M., and Elbert, S. T. (1982). Chem. Phys. 71,41, 64, 65. Runtz, G. R., Bader, R. F. W., and Messer, R. R. (1977). Can.J. Chem. 55, 3040. Rys, J., King, H. F., and Coppens, P. (1976). Chem. Phys. Lett. 41,383. Scrocco, E., and Tomasi, J. (1973). Top. Curr. Chern. 42.95. Scrocco, E., and Tomasi, J. (1978). Adv. Quantum Chem. 11, 115. Shipman, L. L. (1975). Chem. Phys. Leff.31, 361. Stone, A. J. (1981). Chem. Phys. Lett. 83,233. Stout, E. W., and Politzer, P. (1968). Theor. Chim. Acta 12, 379. Stutchbury, N. C. J., and Cooper, D. L. (1983). J. Chem. Phys. 79,4967. Tait, A. D., and Hall, G. G. (1973). Theor. Chim. Acta 31, 31 1. Thorn, R. ( 1975). “Structural Stability and Morphogenesis.” Benjamin, Reading, Massachusetts. Wahl, A. C. (1964). J. Chem. Phys. 41,2600. Warshel, A. (198 1). Acc. Chem. Res. 14, 284. Yanez, M., Stewart, R. F., and Pople, J. A. (1978). Acta Crysfallogr.Sect. A 34,641.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS.VOL. 20

EXPERIMENTAL STUDIES ON CLUSTER IONS T. D . MARK* and A . W. CASTLEMAN, JR . Department of Chemistry The Pennsylvania State University University Park. Pennsylvania

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . A . Molecular Beam Ionization Technique . . . . . . . . . . . . . B. High-pressure and Drift Cell Techniques . . . . . . . . . . . . C. Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Formation of Cluster Ions . . . . . . . . . . . . . . . . . . . . . A . Ionization of Neutral Clusters . . . . . . . . . . . . . . . . . B. Association Reactions . . . . . . . . . . . . . . . . . . . . . IV . Dissociation of Cluster Ions . . . . . . . . . . . . . . . . . . . . A . Unimolecular (Metastable) Dissociations . . . . . . . . . . . . B . Collision-Induced Dissociations. . . . . . . . . . . . . . . . . C. Photodissociation . . . . . . . . . . . . . . . . . . . . . . . V . Thermochemical Properties . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B. Equilibrium Measurements. . . . . . . . . . . . . . . . . . . C. Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Bonding to Positive Ions . . . . . . . . . . . . . . . . . . . . E . Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . F . Some Structural Considerations. . . . . . . . . . . . . . . . . G . Systems Containing an Organic Constituent . . . . . . . . . . . H . Bonding to Negative Ions . . . . . . . . . . . . . . . . . . . VI . Other Properties . . . . . . . . . . . . . . . . . . . . . . . . . A . Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . B. Recombination . . . . . . . . . . . . . . . . . . . . . . . . C. Transport Properties . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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* Visiting Professorofchemistry (1983)in the Department ofchemistry. The Pennsylvania State University. Permanent address: Institut f i r Experimentalphysik.Leopold Franzens Universiat. A-6020 Innsbruck. Austria. 65 Copyright 0 1985 by Academic Press. Inc. All rights of reproductionin any form reSeNed.

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I. Introduction The last decade has been marked by a rapidly growing increase of interest in the properties of both neutral and ionic clusters (e.g., see reviews by Kebarle, 1968, 1972a,b, 1974, 1977;Zettlemoyer, 1969; Milne and Greene, 1968, 1969; Narcisi and Roth, 1970; Ewing, 1972, 1975, 1976; Wegener, 1974; Klemperer, 1974, 1977; Knof, 1974; Good, 1975; Blaney and Ewing, 1976; Christophorou, 1976; Sinfelt, 1977; Muetterties, 1977; Smirnov, 1977; Smalley et al., 1977; Hoare, 1979; Ferguson et al., 1979; Stein, 1979; Castleman, 1973a, 1974, 1979a- d, 1982a,b,d- f; Castleman and Keesee, 1983a,b, 1984; Castleman et al., 1982a,b; Muetterties et al., 1979; Castleman and Mark, 1980; Geoffroy, 1980; Gingerich, 1980a,b; Smith and Adams, 1980, 1983b; Hamilton, 1980; Levy, 1980, 1981;van der Avoird et al., 1980;Perenboom et al., 1981;Beswick and Jortner, 1981;Hagena, 1981; Gspann, 1982; Sattler, 1982; Ferguson, 1982; Mark, 1982a,b; Mots, 1982; Bohme, 1982; Lewis and Green, 1982; Ng, 1983; Martin, 1983; Dehmer, 1983; Buttet and Borel, 1983; Friedel, 1983; Geraedts et al., 1983a). This interest is due to the many faceted aspects ofcluster research which pertain to important questions of both a fundamental, as well as an applied nature. Cluster ions comprise a nonrigid assembly of components having properties between those of large gas phase molecules and the bulk condensed state and many of the studies have provided data which serve to bridge the gap between atomic and molecular physics on one hand, and condensed matter physics on the other. Additionally, it is well recognized that research on the properties of cluster ions is valuable in obtaining a more complete understanding of the forces between ions and neutral atoms or molecules, where, for example, data on energetics provide a direct measure of the depth of the potential well of interaction between the ion and the collection of neutral molecules. Work on increasingly higher degrees of aggregation has had a direct bearing on the field of interphase physics which is concerned with elucidating the molecular details of the collective effects responsible for phase transformations (nucleation phenomena), the development of surfaces, and ultimately solvation phenomena and the formation of the condensed state. A natural extension ofwork on large gas phase ion clusters is the attempt of various investigatorsto connect their properties to those of ions in liquids or solids. Such research enables a further understanding of bonding and reactivity of complexes known to exist in the solution phase that have gas phase analogies. For instance, exciting new results and experiments involving both ions and neutral clusters suggest that relatively few molecules must be coclustered for either the ion or the neutral system to begin to display properties normally associated with the condensed phase (Castleman, 1979a,

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1982a-d,f, 1983; Castleman et al., 1978a,b, 1981c, 1982a,b; Kay et al., 1981; Castleman and Keesee, 1981a, 1983a,b, 1984; Lee et al., 1980a,b; Keesee et al., 1979a, 1980; Holland and Castleman, 1982a,b).Furthermore, studies of the attachment of electrons to neutral clusters provide a technique for investigating the solvated electron which is a specific example of the general class of systems termed polarons (Emin, 1982). Systems composed of metal atoms are beginning to draw the attention of theoreticians interested in theories of nonmetallic- metallic transitions and the onset of metallic conductivity. In other areas of physical chemistry cluster research is being utilized to unravel important problems pertaining to energy transfer and energy redistribution within molecules; the results are expected to have an important bearing on the further development of statistical theories of reaction rates and unimolecular decomposition. Finally, the surface-cluster analogy has drawn the particular attention of inorganic chemists who are beginning to lay the foundation for interpreting the molecular details of catalytic processes (Messmer, 1979; Shustorovich and Baetzold, 1983; Muetterties, 1977; Muetterties et al., 1979; Block and Schmidt, 1975; Beauchamp, 1983; Somorjai, 1981) and to lay a physical basis for understanding catalysis. Furthermore, both ion and fast atom bombardment techniques are utilized in surface chemistry and solid state physics to investigate surfaces and reactivity of surface sites. Data on gas phase cluster ions are useful in interpreting the results. Also, small clusters are currently being used as prototypes of surface sites in theoretical efforts to model electronic states and properties of the surface state (Gatos, 1981). In terms of applied fields, examples of the importance of cluster research are legion. The existence and role of cluster ions in the upper atmosphere have been recognized for many years and have drawn the attention of numerous researchers (Ferguson et al., 1979; Castleman and Keesee, 198lb; Keesee and Castleman, 1982; Arnold and Joos, 1979; Arnold and Fabian, 1980; Arnold et al., 1982; Bohringer and Arnold, 1981; Arijs, 1983; Arijs et al., 1982, 1983a-d; Mohnen, 1971a,b; Turco et al., 1982). Recognition of their potential role in the lower atmosphere is becoming increasinglyevident as the result of recent observations of positive and negative cluster ions below the troposphere by Arnold ( 1983) and suggestions due to Ferguson (1977), Castleman (1 980), Castleman and Keesee ( 1981b), Keesee and Castleman (1 982), and Arnold ( 1980). A general view of the earth’s environment, consisting of a weakly ionizing plasma surrounding the globe is given by Smith and Adams ( 1980).Cluster ions are known to exist throughout most of the earth’s atmosphere (Ferguson et al., 1979) and are implicated as being important in maintaining the charge balance as well as potentially effecting certain nucleation processes leading to aerosol particle formation (Castleman and Keesee, 1981b; Keesee and Castleman, 1982). Additionally, the

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formation of interstellar molecules has been suggested to be initiated via radiative stabilization involving ion clusters (Smith and Adams, 198la; Villinger et al., 1983).Other applied areas to which cluster ion research contributes include combustion science, where ion clusters are also known to form during combustion processes, and they have been suggested as playing a role in soot formation, radiation chemistry, biochemistry, electrochemistry, analytical detection of trace species, and the transport of fission products in nuclear reactors, and even mechanisms of corrosion in high-temperature systems (see for example Franck, 1978; Pitzer, 1983). Others have suggested the use of cluster ions for fuel injection into fusion reactors (Henkes, 1964) and cluster ions are also present in other high-temperature systems such as those designed as sources of energy employing magnets, hydrodynamics, and others too numerous to mention. In a general sense a cluster (or, as they are sometimes referred to, a microcluster) is defined as an aggregate of atoms, ions, and molecules so small that an appreciable number of the constituents are present on its surface at any give time, i.e., setting an upper limit in the region of lo4- los constituents/ microcluster (Hoare, 1979). In the present article, only a selected class of clusters, i.e., the ionic clusters, will be treated. They can be categorized into two groups, one consisting of cluster ions formed by the attachment of neutrals, L, to preexisting ions, I+, (I+ * L,, n = 1,2, . . .), and one consisting of ionized van der Waals molecules (A:, AB:; n = 1, 2, . . .). There are three types of experimental approaches to produce and study these cluster ions depending on different degrees of physical isolation, i.e., clusters moving freely in space (gas phase studies), clusters supported upon a substrate surface, and clusters existing in solids and liquids. In part because excellent reviews are available (Ozin, 1977, 1980; Hoare, 1979; Moskovits, 1979; Muetterties et al., 1979; Miller and Andrew, 1980; Hamilton, 1980; Perenboom et al., 1981; Lewis and Green, 1982) and in part to keep the present review article to manageable proportions, herein attention is given to the more limited subject of gas phase cluster ions. As a prelude to a discussion of the known properties of cluster ions, some of the main types of experiments now available for the production and analysis of cluster ions will be selectively reviewed first. Following this will be a discussion of their production, loss, and properties in the gas phase.

11. Experimental Generally, a cluster ion experimental system consists of a suitable cluster ion source coupled to a conventional apparatus designed for the study of ion

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properties. Since these apparatus have been characterized previously, or are available commercially, only brief mention of them will be made where necessary. Therefore, attention is given to a description of the pertinent details of the cluster ion source, with the inclusion of a discussion of those cases where cluster ion production is part of the experimental setup itself. A. MOLECULAR BEAMIONIZATIONTECHNIQUE Although the existence of condensation in high-velocity expanding jets has been known since at least 1942 (Oswatitsch, 1942), it is only recently (after some pioneering studies by Becker et al., 1956, 1962; Becker and Henkes, 1956;Bentley, 1961;Henkes, 1961,1962) that this fact has been put to use with the advance of mass spectrometry (e.g., Gspann and Korting, 1973; Beuhler and Friedman, 1980; Sattler et al., 1980; Stace and Shukla, 1980;Stephanetal., 1980a, 1983a;DietzetaL 1980;HelmetaZ., 1981;Echt ef al., 1984) and the advent of an array of supersonic nozzle designs, i.e., ranging from the use of a simple hole (Klemperer, 1974; Helm et al., 1979; Dehmer and Poliakoff, 1981) to the use of more sophisticated designs employing differential pumping, seeding, various nozzle shapes, multiexpansion, laser vaporization, and/or pulsed operation (e.g., Becker and Henkes, 1956; Hagena, 1963, 1974, 1981; Hagena and Obert, 1972; Larsen et al., 1974;Gentry and Giese, 1977, 1978;Behlen et al., 1979;DeBoer et al., 1979; Liverman ef al., 1979, and earlier references therein; Otis and Johnson, 1980; Beuhler and Friedman, 1980; Levy, 1980; Brutschy and Haberland, 1980; Bowles ef al., 1981; Saenger, 1981; Dietz et al., 1981; Cross and Valentini, 1982; Kay et al., 1982; Reyes-Flotte et al., 1982; Kappes et al., 1982a; Kappes and Schumacher, 1983; Delacretaz et al., 1983a; Peterson et al., 1983, 1984a). In principle, careful control ofthe nozzle and stagnation chamber parameters (ie., pressure, gas mixture, temperature) can lead to the production of quite high concentrations of bound neutral clusters (van der Waals molecules) from the dimer upward to the n = lo8 range (see also reviews on supersonic nozzle beams by Anderson et al., 1966; Milne and Green, 1968, 1969; Andres, 1969; Anderson, 1974; Hagena, 1974, 1981). This neutral cluster production has been studied (and/or used) with a variety of experimental techniques, i.e., ionization detector with retarding Jield (Bauchert and Hagena, 1965; Hagena and Henkes, 1965; Falter et al., 1970; Hagena and Obert, 1972;Hagena and von Wedel, 1974;Obert, 1977,1979), electron diflaction (Audit, 1969; Stein and Armstrong, 1973; Raoult and Farges, 1973; Farges ef al., 1973, 1977, 1981, 1983; Yokozeki and Stein, 1978; DeBoer et al., 1979; DeBoer and Stein, 1981; Bartell et al., 1983; Heenan et al., 1983; Buttet and Borel, 1983, and references therein), light scattering

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(Stein and Wegener, 1967; Stein, 1969; Klingelhofer and Moser, 1972; Wegener and Wu, 1976; Wu et al., 1978a,b; Konig et al., 1982; Godfried and Silvera, 1983), electric and magnetic deflection (with mass spectrometry) (Gordon et al., 1971; Dyke et al., 1972a,b; Dyke and Muentner, 1972; Novick et al., 1972; Klemperer, 1974, 1977, and references therein; Knight etal., 1978;Odutolaetal., 1979;Bartonetal., 1979;Castlemanetal., 1981a, 1983c; Kremens et al., 1984; Sievert et al., 1983; Sievert and Castleman, 1984; Keesee et al., 1984; Knight, 1983), beam scattering with or without mass spectrometry (Becker et al., 1962; Burghoff and Gspann, 1967; Van Deurson et al., 1975; Herschbach, 1976; Van Deursen and Reuss, 1977; Buck and Meyer, 1983),single and multiphoton ionization mass spectrometry(Robbins et al., 1967;Fosterer al., 1969;Feldman et al., 1977;Herrmann et al., 1977, 1978a,b, 1979; Mathur et al., 1978, 1979; Rothe et al., 1978; Cook and Taylor, 1979a,b, 1980; Cook et al., 1980; Leutwyler et al., 1980, 1981, 1982a,b;Hopkinsetal., 1981, 1983;WaltersandBlais, 1981;Dehmer andpoliakoff, 1981;Dietzetal., 1981;Fungetal., 1981;Duncanetal., 1981; Ono and Ng, I982a,b; Dehmer, 1982; Dehmer and Pratt, 1982a,b; Pratt and Dehmer 1982a-c, 1983; Delacretaz et al., 1982a,b, 1983a,b; Eisel and Demtroder, 1982; Martin et al., 1982; Kappes et al., 1982a; Michalopoulos et al., 1982;Powers et al., 1982, 1983; Shinohara andNishi, 1982;references summarized by Ng, 1983;Kappes and Schumacher, 1983; Rademann et al., 1983; Bronner et al., 1983; Haberland et al., 1983a; Choo et al., 1983; Rohlfing et a)., 1983; Stanley et al., 1983a,b;Castleman et al., 1983d;Shinohara, 1983;McGeoch and Schlier, 1983; Peterson et al., 1983, 1984a,b; Dao et al., 1984; Echt et al., 1984), photoelectron spectroscopy and electronelectron spectroscopy (Mathis and Vroom, 1975; Dehmer and Dehmer, 1977, 1978a,b;Carnovale et al., 1979, 1980; Kubota et al., 1981; Yencha et al., 1981; Poliakoff et al., 1981, 1982; Tomoda et al., 1982, 1983; Tomada and Kimura, I983), electron impact ionization mass spectrometry (sometimes combined with additional diagnostic tools, such as lasers, velocity selectors, etc.) [Greene and Milne, 1963, 1966; Leckenby et al., 1964; Cuthbert et al., 1965; Leckenby and Robbins, 1966; Milne and Greene, 1967a,b; Buchheit and Henkes, 1968; Knof and Maiwald, 1968; Golomb and Good, 1968; Tay and Dawson, 1970; Tay et al., 1970; Golomb et al., 1970, 1972; Henkes and Isenberg, 1970; Harbour, 1971; Milne et al., 1972; Fricke et al., 1972; Yealland et al.. 1972; Gspann and Korting, 1973; Dorfield and Hudson, 1973a,b;Lin, 1973;Larson et al., 1974; Gspann and Krieg, 1974; Calo, 1974; Van Deursen and Reuss, 1973, 1975, 1977; Henkes and Mikosch, 1974; Van Deursen et al., 1975; Henkes et al., 1977; Lee and Fenn, 1978; Stephan et al., 1978, 1980a,b, 1981, 1982a-c, 1983a-c,e, 1984; Gough et al., 1978; Herrmann et al., 1978b, 1982; Helm et al., 1979, 1980a,b, 1981; Market al., 1980a, 1982, 1983;Castleman et al., 1980a,b, 198lb,e; Dun and

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Stevenson, 1980; Stace and Shukla, 1980, 1982a-c; Gspann and Vollmar, l980,198l;CookandTaylor,1980;Cabaudetal., 1980;Buelowetal., 1981; Futrell et al., 1981a,b, 1982; Kay et al., 1981; Geraedts et al., 1981, 1982, 1983b; Casassa et al., 1981; Lisy et al., 1981; Yamashita et al., 1981; Hoareauetal., 1981;Gspann, 1981;Sattleretal., 1981;Echtetal., l981,1982a,b; Stephan and Mark, 1982a-c, 1983; Gough and Miller, 1982; Vernon et al., 1982,1983; StaceandMoore, l982,1983;Soleretal., 1982(seealsoHagena, 1983); Saito et al., 1982, 1983; Dreyfuss and Wachman, 1982; Kay and Castleman, 1983;Haberland et al., 1983a; Labastie et al., 1983;Muller et al., 1983; Ding and Hesslich, 1983a,b; Stace, 1983a,b;Bomse et al., 19831, and electron attachment to van der Waalspolymers (Gspann and KOrting, 1973; Klots and Compton, 1977, 1978a,b; Klots, 1979; Bowen et al., 1979, 1983; Armbruster et al., 1981; Haberland et al., 1983b; Stephan et al., 1983d). The extent of condensation in a supersonic expansion depends upon the time scale of expansion (Hagena and Obert, 1972)and the degree of cooling in the expansion, which scales with the stagnation pressure,po,nozzle diameter, D, and nozzle-stagnation temperature, To.Obert (1979) has reported a scaling law for hydrogen clusters p o . DI.5 . T-2.4 = constant (1) The practical limit to production of neutral clusters is the requirement of sufficient pumping capacity to handle the gas load discharged through the nozzle. Besides the use of larger and larger pumps (combined with the differential pumping stages), there have been introduced recently several other techniques, i.e., pulsed nozzles and laser vaporization within the nozzle (see references above). Translational temperatures down to 0.00 15 K have been predicted (Toennies and Winkelmann, 1977) and almost achieved (Levy, 1980).The ultimate temperatures achieved in a given supersonic expansion usually follow the order TmS < T,, < Tvibr, due to the fact that the density of the gas drops as the expansion proceeds and vibrational (and rotational) cooling is less facile than translational, requiring far more collisions. Because condensation is a much slower process than rotational or even vibrational relaxation, extensive cooling of the internal degrees of freedom can be achieved before condensation takes place, i.e., rotational distributions corresponding to less than 0.2 K and vibrational temperature of down to 20 K have been observed ( e g , Levy, 1981, and references therein). Since condensation involves the production of clusters via statistical processes, it is generally difficult to obtain narrow size cluster distributions. However, the production of a relatively narrow distribution has been achieved with a special source design ( e g , Bowles et al., 1981). It is sometimes desirable to minimize the production of clusters heavier than a particular one under study. A pressure- temperature relationship given by Van Deursen and Reuss ( 1977)

72

T. D.Mark and A . W. Castleman, Jr.

can be used to select conditions that serve to cut offthe production of heavier clusters, but usually this can only be applied to separate dimers and trimers from heavier clusters (Ng, 1983; see also Dehmer and Pratt, 1982b). Finally, it should be mentioned that thus far parameterized studies of factors influencing cluster distributions have been largely confined to those with species which are gaseous or have substantial vapor pressures at room temperatures. However, Schumacher and co-workers (e.g., Herrmann et al., 1978a,b; Kappes et al., 1982a) have investigated the influence of the nature of the carrier gas mass on alkali clusters, and other highly refractory compounds have been recently clustered (e.g., Preuss et al., 1979, and references therein; Dietz ef al., 198 1; Michaloupoulos et al., 1982; Powers et al., 1982, 1983; Hopkins et al., 1983; Rohlfing ef al., 1983). In principle, a cluster ion can be produced from any of these neutral cluster beams with an ionizing agent of sufficient energy and intensity. Although different ionization (electron, single and multiple photon, Penning ionization, heavy particle impact) and detection techniques have been used, the basic features of virtually all molecular beam ionization mass spectrometers are similar. Two examples are described in what follows to provide the reader with some information about the actual design ofsuch an apparatus. Figure 1 shows a detailed cross section of the molecular beam production section, ionization region, and ion optics in front of a modified commercial double-focusing mass spectrometer used in our Innsbruck laboratory (Stephan et al., 1982a-c, 1983a). The gas (mixture) under study (gaseous atoms and/or molecules) is expanded from a high-pressure (up to 10 bar) variable temperature (down to 100 K) stagnation reservoir through an aperture nozzle (10- to 50-pm diameter) located 4 cm from the electron beam which intersects the molecular supersonic jet at 90”.The molecular beam isdefined by the nozzle and a collimator (5-mm diameter) in front of the ionization chamber. A 500 liter/sec turbo pump is used to evacuate the open ion source chamber and to handle the gas load imposed by the supersonic jet (giving a background pressure between and Torr, depending on the stagnation pressure and gas mixture). The ion source is isolated from the mass spectrometer by the mass spectrometer entrance slit ( S , ) , serving as a differential pumping aperture. Ions produced by electron impact ionization are extracted at right angles (“off line detection”) from the ionization region by an electric field penetrating into the ionization chamber (penetrating field extraction, see Stephan et al., 1980~).Special deflection plates (deflection mass spectrometry, see Stephan et af., 1980c) in the ion optics between the ion source (commercial, Nier type) and the mass spectrometer entrance slit serve to sweep the extracted ion beam across the mass spectrometer entrance slit, thus allowing ions formed by cluster ionization to be distinguished from

73

EXPERIMENTAL STUDIES ON CLUSTER IONS

ELECTRON BEAM

MASS

ROMETER

F

FIG. I . Schematic view of supersonic molecular beam electron impact ionization mass spectrometer after Stephan et al. (1982a-c, 1983a). Nozzle N is a m dia. pinhole located 4 cm from the electron beam crossing region. S is the stagnation chamber and A an aperture. Electrode L, , plate P and chamber C are at common source potential, while electrode L, produces a field penetration extraction field. Beam centering plates L3, L, locate the extracted ion beam on a defining aperture D, whereas the earth slit, L,, constitutes the end of the accelerating region. Deflection plates L6,, and L, and L, centerthe beam on the mass spectrometer entrance slit, S, , or sweep the beam across S, . Beam flag F is used to interrupt the beam diagnostically.

background ions with the same m/z (Helm ef al., 1979, 1981; Stephan ef al., 1982b,c, 1983a;see also Stace and Shukla, 1980). This simple experimental set-up allows the production of atomic and molecular clusters up to n = 50 with sufficient intensity for (also high-resolution) mass spectrometric studies, i.e., measurement of appearance energies, study of metastable decay, etc. Figure 2 shows an overview of another, rather novel, technique involving pulsed laser vaporization within a pulsed supersonic nozzle and (subsequent) mass-selective resonant two-photon laser ionization (Dietz et al., 1980, I98 1;Michalopoulos ef al., 1982; Powers ez al., 1982, 1983; Hopkins et al., 1983). Generation of ultracold metal clusters (e.g., for Mo, T,,, < 6 K, Trot- 5 K, TYib - 325 K; Hopkins et al., 1983) is achieved by the followingtechnique. The vaporization laser (Nd :YAG second harmonic) hits a rotating target rod within the throat ofa pulsed supersonic He expansion. As the vapor cools in the high-density He gas, clustering of metal atoms within the exit channel of the expansion begins to occur and further clustering and cooling occur as the gas freely expands into the vacuum. The length of the exit channel controls the extent of cluster production, i.e., producing up to n = 30 clusters of Cu, Fe, Ni, W, and Mo at 8 atm He stagnation pressure, 1-mm orifice, an extension channel of 2.5-mm length and 3-mm diameter, 10-Hz pulsed operation, and typical base pressures with pulsed beam-on of

T. D.Murk and A . W. Castleman, Jr.

74 V4fQRRATlON LASER

\

PROBE LASER

MOLECULAR BEAM

FIG.2. Schematic view of supersonic molecular beam two-photon ionization mass spectrometer after Dietz ef al. (1980), Powers ef al. ( 1 982), Hopkins ef af.( I 983). ( I ) Main chamber containing pulsed nozzle with laser vaporization; (11) probe chamber, (111) detection chamber containing time of flight mass spectrometer; (A) repeller plate; (B) draw-out grid; (C) flight tube grid; (D)deflection plates; (E) flight tube; (F) support rods; (G) copper cryoshield; (H) stainless steel liquid nitrogen Dewar.

8X 1 X lod, and 4 X lo-* Torr in chambers 1-111, respectively. The clusters are detected by a time-of-flight mass spectrometer (a vertical cross section is shown in Fig. 2) 1 18 cm downstream from the nozzle. The clusters are interrogated using two-color ionization by scanning a Nd :YAG pulsed dye laser and using a second color of sufficient energy (provided by a Nd :YAG harmonic or excimer laser) to ionize the excited clusters. Again, ions are extracted at right angles into the mass spectrometer and deflection plates have to be used to counteract the translational velocity of the molecular beam. In ending this discourse on supersonic expansions (the most commonly used technique for generating neutral clusters), it is interesting to note two other techniques for both neutral and ionized cluster production, i.e., the expansion of ionized gas (Dole et al., 1968; Mack et al., 1970; Searcy and Fenn, 1974;Armbruster et al., 1981;Udseth et al., 1981; Beuhler and Friedman, 1982a,b; Yamashita and Fenn, 1983; Haberland et a/., 1983b) and inert gas (“cold bath”) condensation (“quench flow”) of evaporated metals, etc. (Pfund, 1930a,b;earlier references in Perenboom et al., 1981; Duthler et

EXPERIMENTAL STUDIES ON CLUSTER IONS

75

al., 197 1; Mann and Broida, 1973; Eversole and Broida, 1974; West et a/., 1975; Nishida and Kimoto, 1975;Granqvist and Buhrmann, 1976;Kimoto and Nishida, 1977; Knight et al., 1978; Schulze and Abe, 1980; Kimoto et al., 1980; Sattler et al., 1980a,b, 1981, 1982; Bondybey and English, 19811983; Bondybey, 1982; Sattler, 1982; Muhlbach et al., 1982a,b; Pfau et al., 1982a,b; Riley et al., 1982;Gole et al., 1982; Heaven et al., 1982a;Martin et al., 1983; Bondybey et al., 1983). The latter technique has been particularly successful in producing refractory metal clusters (also in combination with laser evaporation). As pointed out by Riley et al. (1982),generallythese latter sources will not result in clusters as cold internally as do supersonic expansions. B. HIGH-PRESSURE AND DRIFTCELLTECHNIQUES Although early observations of ion clusters were obtained in ion sources operated in the neighborhood of lo4 Torr, equilibrium conditions were generally not attainable with the few collisions taking place and thermodynamic parameters usually could not be measured with confidence. Early attempts by Field ( 196l), Melton and Rudolf (1 960), and Wexler and Marshall ( 1 964) were made using essentially conventional mass spectrometric ion sources but with small ion exit slits and improved pumping. These new methods enabled reactions requiring a third body for stabilization to be observed, but it was generally impossible to ensure that complete thermalization ofthe ions and the attainment ofion equilibrium had occurred. Major developments were made by Kebarle and co-workers during the 1960s and early 1970s that have led to the investigation of bonding and reaction for a number of clustering sequences (see sections devoted to thermochemistry and association reactions). Kebarle and his group have used several different types of high-pressure mass spectrometers, including a high-pressure alpha particle instrument capable of operating at pressures up to several hundred Torr, a 100-keV proton beam mass spectrometer, and a 4-keV pulsed electron beam apparatus (Kebarle, 1972b, 1975). Important data have been derived from these techniques by Kebarle and by many others who have adapted similar devices. Related high-pressure instruments operate as drift cells and employ internal electrodes that function to move the ion clusters from the point of production to a suitable sampling orifice. There, the ions effuse into vacuum and are introduced into a mass spectrometer for identification and quantitative measurement of requisite intensities. In the limit of zero field, the two apparatuses are identical, but as operated in actual practice the latter do potentially impart some field energy to the ions being investigated. The drift

76

T. D. Mark and A . W. Castleman, Jr

field techniques generally do have the advantage of providing relatively high ion intensities at pressures of several Torr, but it is important that measurements be made as a function of field energy ( E / N ) to establish that true equilibrium is being achieved (see for instance Castleman et al., 1978a). The general high-pressure technique is adaptable to two configurations, one in which the ions are injected at a constant flux and the other where the beam is pulsed and the evolution of the cluster reactions is followed as a function of time. With suitable corrections for loss at the boundaries of the apparatus and electrodes, the ion ratio at long times can be taken as proportional to the equilibrium ratios (Kebarle, 1972). Deriving exact correction methods to account for the diffusion and loss of ions and separating these from the equations pertaining to cluster evolution as a result of sequential clustering reactions has introduced some approximations and thereby uncertainties. However, within experimental error, studies by the two techniques have led to basically the same thermodynamic parameters. The high-pressure pulsed electron beam apparatus used in the laboratories of Kebarle is shown in Fig. 3. It is composed of an electron filament, suitable electrodes for electrostatic focusing of the beam, deflection electrodes, and magnetic and electrostatic shielding for the electron beam region. Appropriate carrier and clustering gases are introduced into the ion source through an entrance port. The electron gun is constructed with conventional cathode ray focusing geometry and is provided with a deflection plate for the positioning of the beam. Suitable electrodes enable the ions to be focused and directed into the mass spectrometer for identification. Ion source temperatures of - 20 to 600 "Ccan be achieved with the heating coils and cooling channels located in the copper mantle. During operation, the electron beam enters the ion source through a slit of X 4 mm and has an intensity of about lo-' A when operated continuously. The ions exit the ion source through a narrow slit constructed of stainless steel razor blades that has an approximate dimension of X 3 mm. In operation, a pulsing sequence is initiated in which the electron beam is on for a short period of time. After an appropriate delay time, an ion gate is opened for another short period (generally a few microseconds) and the ions, after having resided a specific number of microseconds in the ion source, are collected over a specified differential time duration. (In other versions of operation, the apparatus does not utilize an ion gate but the current is collected continuously.) The ion source is field free and the ions created in the electron beam diffuse throughout the cell and through the ion exit slit. In pulsed operation, the total ion intensity at first increases, then reaches a maximum, and thereafter gradually decreases to zero. Reaction conditions must be selected such that the ion concentration changes due to reactions are

+

EXPERIMENTAL STUDIES ON CLUSTER IONS

77

FIG. 3. Electron beam, high-pressure ion source after Kebarle (1972b). (1) Electron filament; (2)-(6) electrodes for electrostatic focusing of electron beam; (7),(8) deflection electrodes; (9) magnetic and electrostatic shielding of electron beam; (10) ion source with copper heating mantle; ( 1 I ) electrostatic wire mesh screen with high pumping conductance for shielding of ions and efficient pump out of neutrals; (1 2) electron entrance slit; (1 3) electron trap; (14) copper lid holding ion exit slit flange; ( 1 5 ) gas in- and outlet; (16)-( 18)ion source supports and insulation; (19)-(26) ion beam acceleration and focusing; (27) mass spectrometer tube to 90" magnetic sector; (28) 84x1. pumping lead leading to 6-in. baffle and diffusion pump; (29)gas line heaters.

much faster than the decrease of total ion concentration caused by diffusion of the ions to the walls (Kebarle, 1972). The data composed of normalized ion intensities are analyzed as if they represented concentration changes for a homogeneous reaction system in which, at time zero, a unit concentration of primary ions is created. The ions are then considered to react in a pseudo

78

T. D. Mark and A . W. Castleman, Jr.

first-order fashion with the total ion concentration remaining constant. The treatment assumes equal diffusion coefficients for all ions, and also the absence of reactions which are second order in ion concentration. Modifications of this apparatus have been made which enable the production of positive metal-like ions using thermionic source techniques (see for instance Searles and Kebarle, 1969). Field and co-workers (e.g., Beggs and Field, 1971a), Conway and coworkers (e.g., Conway and Janik, 1970), Pack and Phelps (1966, 1971), Puckett and Teague ( 197la), Keller and Beyer ( 197la), Rakshit and Warneck (1979), Castleman and co-workers (Castleman et al., 1978a; Keesee et al., 1980), Meot-Ner and co-workers (e.g., 1978, and Meot-Ner and Sieck, 1983),and Wince1and co- workers (e.g., Wlodek et al., 1980),among others, have used adaptations of the high-pressure technique sometimes in conjunction with the drift cell method. A typical ion source and mass spectrometer arrangement is shown in Fig. 4. In the case of positive ions, a thermionic emission source is employed with the application of a positive voltage to a heated filament on which the source material is dispersed. In the case of negative ions, the thermionic source is replaced with a platinum filament on which a barium zirconium material is dispersed; the heated filament then becomes a copious source of electrons. In this case a negative voltage is applied to the filament. The ions are focused by means ofa repeller assembly, the potential of which is set a few volts below that of the filament. Ion energies are further controlled by means of two electrodes placed before and just after the reaction cell. The potential of the “top gate” is adjusted to ensure field-free conditions in the reaction cell where clustering reactions and the final equilibrium process take place. The temperature of the high thermal conductivity (gold-plated copper block) reaction cell is established by a combination of electrical heating and cooling jacket surrounding the high-pressure vessel. Appropriate thermocouples are located throughout the system and a temperature within 0.5 “C of the same value is established on all of them. The high-pressure vessel is mounted inside a vacuum chamber (---); see Fig. 4. A 10-in. diffusion pump plus liquid nitrogen cold trap is used to maintain pressures less than 1 X 10-6 Torr in this chamber. Ions and ion clusters diffuse through a 50- to 75-pm hole in a platinum disk into the vacuum chamber. There they pass through suitable ion optics for focusing the ionic species into the mass spectrometer for quantitative determination of intensities using pulsecounting techniques. A crucial feature of this apparatus is its ability to operate at moderate pressures (typically 5 -25 Torr) and, more importantly, to ensure the acquisition of equilibrium. Measurements of apparent equilibrium constants are made as a function of both nature and pressure of the carrier gas and of the electric field parameters in the ion source (with particu-

r-

I I

I I

I

ION COUNTING EOUIPMENT

CMNNELTRON MULTIPLIER

OUADRUPOLE

I

I

I

I

I I I

79

EXPERIMENTAL STUDIES ON CLUSTER IONS

------------1

ocl 50

I

I I

I

I I

OUTER JACKET

I

I I I I I I

SCALE 1

1u

u

CAPACITANCE MANOMETER

FIG.4. Schematic ofthe ion clusteringapparatus, after Castleman ef a/.(1978a). The figure isdrawn approximately to scale, with internal supportsomitted forclanty. ---,A region ofhigh vacuum. For further discussion refer to the text.

80

T. 11. Miirk uncl A . M '. Castleman. J r

lar attention to the top gate potential). A plateau region in the plot of apparent equilibrium constant versus E / N (where E is the electric field and N is number density) is sought to ensure that field penetration from elsewhere in the ion source is not imparting significant energy to the ions above thermal and thereby influencing the equilibrium measurements.

C. OTHERS In addition to the above-mentioned techniques for the production of cluster ions in the gas phase, there is a number of methods where the production process involves the transfer of energy to the surface of a sample and the subsequent emission of neutral and/or ionized (singly and multiply) clusters. Within the space available in this article, it is not possible to give a detailed description of these techniques; however, the reader is referred to some recent papers and reviews on these methods, e.g., evuporatron (direct or Knudsen) (Drowart and Goldfinger, 1967, and earlier references therein; Margrave, 1968; Gingerich, 197 1, 1980a,b; Gutbier, 196 1; Lin and Kant, 1969; Donovan and Strachan, 197 1 ;Gingerich and Finkbeiner, 197 1; Emelyanov ct ul., 197 1 ; Marr and Wherrett, 1972; Evans et al., 1972; Brundle et a / , 1972; Schaaf and Gregory, 1972; Wagner and Grimley, 1972; Piacente and Gingerich, 1972; Neckel and Sodeck, 1972; Cocke and Gingerich. 1972, 1974: Hildenbrand, 1972; Guido and Balducchi, 1972; Boschi and Schmidt, 1973; Stearns and Kohl, 1973a,b, and references therein; Kordis and Gingerich, 1973a,b, 1977; Cabaud et al., 1973, and references therein; Dehmer et uI , 1973; Gingerich et ul., 1974; Potts et a/., 1974; Wagner and Grimley, 1974; Berkowitz, 1975; Ihle and Wu, 1975; Richardson and Weinberger, 1975; Cocke et a/., 1975: Streets and Berkowitz, 1976a.b; Wu, 1976, 1979, 1983; Potts and Price, 1977; Drowart ef a/., 1977; Smets ef a/., 1977: Smoes r" u I , 1977; Zmbov et a/., 1977; Grimley ct a / , 1978; Kingcade ef a1 , 1978; Biefeld, 1978; Potts and Lyns, 1978; Gupta and Gingerich, 1978a,b, 1979; Neubert, 1978; Gingerich and Cocke, 1979; Wu el ul., 1979; Potts and Lee, 1979; Hilpert, 1979; MacNaughton et a/., 1980; Berkowitz et a/., 1980; Pitterniann and Weil, 1980, and references therein; Hilpert and Gingerich, 1983; Rosinger ef a/ , 1983),fie/cI &sorption (also including the electrohydrodynamic and electrospraying techniques) (Beckey, 1960; Schmidt, 1964, and earlier references therein: Dole et al., 1968; Mahoney et al., 1969: Anway, 1969; Mack ef al., 1970; Huberman, 1970; Evans and Hendricks, 1972: Colby and Evans, 1973; Simons ef ul., 1974, and references therein; Heinen et ul., 1974; Rollgen and Schulten, 1975a,b; Schulten and Rollgen, 1975; Block and Schmidt, 1975; Iribarne and Thomson, 1976; Stimpson et a/., 1978; Rollgen and Ott, 1978; Rollgen e/ a/., 1978, 1980; Sudraud et ul.,

EXPERIMENTAL STUDIES ON CLUSTER IONS

81

1979; Thomson and Iribarne, 1979; Culbertson et al., 1980; Waugh, 1980; Dixon et al., 1981a,b; Block, 1982a,b; Moller and Helm, 1983; Helm and Moller, 1983a,b; Yamashita and Fenn, 1983), laser evaporation (see also above, and Berkowitz and Chupka, 1964; Bykovskii et al., 1971;Fiirstenau et al., 1979; Fiirstenau and Hillenkamp, 198I), laser-inducedjield evaporation(Kellogg, 1981, 1982;Jentschetal., 1981, 1982;Block, 1982a;Drachsel et al., 1982; Tsong et al., 1983), spark evaporation (Mattauch et al., 1943; Sasaki et al., 1959; Dornenburg et al., 1961; Franzen and Hintenberger, 1961; Hintenberger et al., 1963; Cornides and Morvay, 1983; Becker and Dietze, 1983),Penning ion source evaporation (Kaiser et al., 197 l), sputtering (Honig, 1958, and earlier references therein; Krohn, 1962; Hortig and Miiller, 1969; Clampitt and Jefferies, 1970; Joyes, 1971; Tantsyrev and Nikolaev, 1971, 1972; Staudenmaier, 1972; Leleyter and Joyes, 1973, 1975; Herzog et al., 1973; Richards and Kelly, 1973; Honda et al., 1978, 1979; Szymonski et al., 1978; Taylor and Rabalais, 1978; Wittmaack, 1979; Lancaster et al., 1979a,b;Jonkman andMich1, 1979, 1981;Leleyter et al., 1981; Orth et a/., 1981a,b, 1982a,b; Devienne et al., 1981; Pedrys et al., 1981; Campanaef al., 1981, 1983; Barlaketal., 1981, 1982;Leleyter, 1982;Standing et al., 1982;Devienne and Roustan, 1982,and referencestherein; Winograd et al., 1982; Kimock et al., 1983; Ens et al., 1983; Joyes and Leleyter, 1983; Stulik et al., 1983; Michl, 1983), and electron-induced desorption (Clampitt and Gowland, 1969; Floyd and Prince, 1972). Furthermore, cluster ions have been observed and studied in various types of discharges and ion traps (e.g., Narcisi and Roth, 1970; Loeb, 1973, and references therein; Colby and Evans, 1974; Schwarz, 1978; Mathur and Hasted, 1979; Kaufman el d., 1980)and in the atmosphere and ionosphere (e.g., see Ferguson and Fehsenfeld, 1969; Narcisi and Roth, 1970; Sechrist and Geller, 1972; Loeb, 1973; Meyerett et al., 1980; Carlon, 1980, 1983; Ferguson and Arnold, 1981; Arijs, 1983).

111. Formation of Cluster Ions A. IONIZATIONOF NEUTRALCLUSTERS

1. Ionization Processes und Mechanisms

If the energy of an electron, photon, or beam of ions (or fast neutrals) colliding with a gas phase cluster target is greater than a critical value (ionization energy or appearance energy), some of the neutral clusters, depending

T. D.A4iirk mil ‘4. W. Casileman, Jr.

82

on the corresponding ionization cross section, will be ionized resulting in the production of an ion (and sometimes a neutral) and the ejection of one (or more) electrons (e).As the energy of the electrons (or photons) is increased, the abundance and variety of the resulting ions will increase (e.g., see Rosenstock rt al., 1977; Mark, 1982c,d, 1984a,b). The ionization process (and concomitant changes in vibrational and rotational excitation) is governed by the Franck-Condon principle (and the selection rules) and may proceed via different reaction channels, i.e., for the simple case of a dimer .4,

+ h v ( e ) - .A: + e(+e) --A$+ + 2r(+ e )

single ionization double ionization

e(+e)

autoionization

+A’(+(>)

predissociation

-,4:(+e)-’4:+ --A

-AA:*+e(+e)-,il++A

-

-

/I+

.4:+ ‘4;

+e(+~)

+ 2e(+e) + e + hv’(+e)

+ A + e(+ e)

fragmentation autoionization radiative transition dissociative ionization ion pair forination

Moreover, in addition to being produced by ion pair formation (lo), negative ions may be produced by two other generalized processes: .-I2

+ e - .4;*

-

/I-

+ /I

resonance capture

(1 1)

dissociative resonance capture

(12)

The A;* ion in reaction ( I I ) is unstable and stabilization must occur by radiation emission before it decomposes by autodetachment. Furthermore the endoergic charge transfer process

+A;

(13) can be used (with X a fast alkali beam) as a relatively gentle electron attachment to weakly bound clusters (e.g., Bowen et al., 1983). Reactions (2), ( 3 ) , (9), (lo), and (12) can be classified as direct ionization processes, whereas reactions (4)- (8) can be categorized as two-step processes. Photon ionization cross sections of rare gas and alkali dimers have been found to show only autoionization lines (due to molecular states; see X + A,--rX+

EXPERIMENTAL STUDIES ON CLUSTER IONS

83

also Carlson ef al., 1980, 1981) near the first ionization threshold and virtually no contribution from direct ionization due to a poor Franck-Condon overlap between the neutral and the lowest vibrational states of the ion (e.g., Ng el al., 1976, 1977a,b;Dehmer and Poliakoff, 1981; Dehmer, 1982; Leutwyler et al., 1981, 1982a; Pratt and Dehmer, 1982a,c; Dehmer and Pratt, 1982a,b; Martin et al., 1982). Observed autoionization lines corresponding to the optical allowed atomic Rydberg series have been attributed to associative ionization reactions occurring in the ionization region via interaction of excited monomers and background gas monomers (e.g., Dehmer, 1982, and references therein). Conversely, Ng et af. (1977~)have reported for nitric oxide dimers that the dominant ionization mechanism is direct ionization such that only one of the NO molecules in the dimer is excited to different vibrational states of NO+ (XIZ+)forming an NO-NO+ (lZ+u’) complex (for other examples see also Linn et al., 198la,b; Erickson and Ng, 1981). Moreover, molecular dimers may be ionized either directly or via a vibrational autoionization process in one of the moieties as described by Anderson et af. (1980) for (H2), H,-H,

+ hu -,H,-H:(n,o)

-,H,-HI(u’)

+e+

H:

+H +e

(14)

(see also Tiedemann et af., 1979;Ono eta/., 1980;Linn and Ng, 1981) or via an autoionization process involving association ionization of loosely bound molecules, sometimes termed chemiionization and studied in detail by Gress et af. (1980) and Ono et af. (198 la,b), i.e., determining the relative reaction probabilities of various product channels of the chemiionization processes as a function of n

+ CS + e, -,S: + 2CS + e, -+C,S: + S + e, -,CS: + C + e,

CS:( V,n) - CS, + CS:

-

(CS,): + e,

Similar to the last step in reaction (14) a number of cases have been reported where ionization of a cluster involves an internal ion molecule half reaction, e.g., producing such ions as H30+from (H,O), (Milne et af., 1970; Ng el al., 1977d), H,O+(H,O), from (H,O), (Hermann et af., 1982), OH-(H,O), from (H,O), and NO-(N,O), from (N,O), (Mots and Compton, 1978a,b),N H t from (NH3)2 (Ceyer et af.,1979a; Stephan et al., 1982c; Choo et al., 1983),NHt(NH,), from (NH,), (Stanley et af., 1983a; Echt et al., 1984),C,H: or C,Hf from (C2H4),(Ceyer et af., 1979b),Of,,,from 0; (a 4111,,u) - (O,), (Linn et af., 1981b), C,Of from (CO,), (Stephan et al.,

84

T. D.Mark and A . W. Castleman,Jr.

1982b), CO$.O, from (CO,), (Stephan et al., 1983d), CHf(CH,), from (CH,), (Ding and Hesslich, 1983b). In some cases (depending on reaction energetics and dynamics) this leads to a situation where the protonated cluster ions are by far the dominant ions appearing in a mass spectrum, whereas the unprotonated cluster ions are either not detectable or order of magnitudes smaller (Fricke et a!., 1972;Lin, 1973; Van Deursen and Reuss, 1973; Ng et al., 1977; Ceyer et al., 1979a; Tiedemann et al., 1979; Anderson et al., 1980; Castleman et al., 1981e; Stephan et al., 1982c; Hermann et al., 1982; Dreyfuss and Wachman, 1982; Shinohara, 1983).Obviously, such an internal ion molecule reaction can be viewed in terms of an initial rearrangement within the cluster ion and a subsequent unimolecular decomposition. It is interesting in this conjunction to point out recent fragmentation studies by Ono and Ng (1 982a,b), Stace and Shukla (1982c), Stace and Moore (1982), and Stace (1983a,b). Moreover, Biermann and Morton (1 982) have studied the production of protonbound dimers via unimolecular decomposition of molecules following photoionization. Lifetimes with respect to unimolecular decomposition have been estimated from measured and calculated reaction rate constants for dimerization and trimerization reactions involving protonated species (e.g., see Olmstead et al., 1977; Davidson et al., 1977; Neilson et al., 1978; D. Smith et al., 1982; Adams et al., 1982, and references therein). The observation of multiply charged cluster ions (e.g., Domenburg et a!., 1961; Dole et al., 1968; Henkes and Isenberg, 1970; Gspann and Korting, 1973; Simons et al., 1974; Henkes et al., 1977; Stimpson et al., 1978; Sudraud et al., 1979;Mark et al., 1980a; Helm et al., 1980a, 1981;Dixon et al., 1981b; Kellogg, 1981, 1982; Sattler el al., 1981; Stephan el al., 1982a; Jentsch et al., 1981, 1982; Echt et al., 1982a;Dreyfuss and Wachman, 1982; Ding and Hesslich, 1983a; Becker and Dietze, 1983; Drachsel et al., 1982; Cornides and Morvay, 1983; Tsong et al., 1983) raises an interesting point about their production and stability. According to experimental evidence and calculations presented by Helm et al. (1 98 1) and Stephan el al. (1 982a), doubly ionized heteroatomic rare gas dimers are produced by direct ionization from the ground state to either weakly bound attractive potential curves lying below the dissociation limits of the respective atomic ions (e.g., see potential energy curves of NeXe2+in Fig. 5) or to quasi-bound molecular levels (e.g., ArXe2+,Neb2+).The existence of stable mononuclear trimers (like Ni:+, Au:+, and W;+) has been explained by Jentsch et al. (1982) by a simple empirical rule relating neutral bond energy to the Coulomb energy (see also observation of PtHe$+by Tsong et al., 1983). Henkes and Isenberg ( 1970)reported the existence of very large, multiply charged N, clusters (also produced by multielectron ionization) and explained their stability by the fact that multiply charged clusters (with the exception of very small clusters

EXPERIMENTAL STUDIES ON CLUSTER IONS 6

h

2 z e -5 .-m 5

v

-

B

1

1

1

1

1

1

85

I

4-

3-

21-

01

2

3

4

5

6

(A) FIG.5. Qualitative potential energy curves for low lying states of NeXez+ after Helm et al. (1981). (a) Xe2+(1So)+ Ne(’S,); (b) Xe2+(’D,) + Ne(lS,); ( c ) Xe+(2Pl/2)+ Ne+(,P); (d) Xe2+(3P,) Ne(lS,); (e) Xe2+(’P,) + Ne(’S,); ( f ) Xe+(2P,,,) Ne+(,P); (g) Xe2+(’P,) Ne( IS,). Internuclear separation

+

+

+

as discussed above) are only stable if their size exceeds a certain critical value, where the repulsive Coulomb energy (minimized by distributing the charges uniformly on the surface) is smaller than the binding energy (see also an earlier prediction by Henkes, 1962, and later studies on critical sizes by Gspann and Korting, 1973; Henkes et al., 1977; Sattler et al., 1981, 1983; Echt et al., 1982a; and Ding and Hesslich, 1983a, on Pb, NaI, Kr, Xe, H,, N,, CO, and N,O clusters). See also the occurrence of multiply charged small clusters of polystyrene (Dole et al., 1968) and doubly charged Fee G,, and (Mg.G,) clusters, where G is glycerol (Simons et al., 1974; Stimpson et al., 1978). As a possible production mechanism for triply charged dimers of W, Mo, Re, and Ir observed by field evaporation by Kellogg (1981, 1982) postionization of singly charged dimers has been proposed. Another excitation (ionization) mechanism, used recently quite successfully for detailed information on the ionization process (i.e., fragmentation), on autoionizing Rydberg states (threshold spectroscopy) and intermediate states (excitation spectroscopy)is multiphoton excitation (ionization) (Feldman et al., 1977; Herrmann eta]., 1977, 1978a,b, 1979; Rothe et al., 1978; Mathur et al., 1978a,b; Leutwyler et al., 1980, 1981, 1982a,b; Fung et al., 1981;Hopkinsetal., 1981, 1983;Michalopoulosetal., 1982;Shinoharaand Nishi, 1982; van Raan et al., 1982, and references therein; Martin et a!., 1982; Eisel and Demtroder, 1982; Riley et al., 1982; Delacretaz et a!., 1982a,b; Powers et al., 1982, 1983; Bernheim et al., 1983; Rademann et al., 1983;Stanley et al., 1983;Shinohara, 1983;Choo et al., 1983;McGeoch and Schlier, 1983). Depending on the type of information desired, either one or

86

T. D. Mark and A . W. Castleman, Jr.

two color, resonant or nonresonant, two photon or multiphoton, mass selective photoionization has been employed. Henmann et al. (1977, 1978b), Rothe et al. (1978), and Leutwyler et al. (1980) have discussed in detail the corresponding ionization mechanisms and photon lunetics (including the respective power dependencies).

2. Ionization Cross Sections and Appearance Energies Ionization of neutral clusters as a function of the energy of the ionizing agent is described in quantitative terms by the ionization cross section function, i.e., partial, total (charge production), counting (ion production), and effective (total divided by number of units in the cluster) cross sections (for a definition see Mark, 1982d, 1984a,b). Although these cross sections (in particular partial cross sections) are an indispensible prerequisite for the quantitative detection of clusters (see also Section III,A,3), unfortunately, little is known about their absolute magnitude and/or fragmentation yields (partial cross section ratios). This is mainly due to the facts that (1) it is not possible to produce targets of neutral clusters of known density and definite cluster size and (2) fragment ions ( e g , from atomic clusters) appear in the mass spectrometer at mass to charge ratios of smaller clusters. In the following section we will discuss first total and counting ionization cross sections, then partial ionization cross sections and ratios, and finally threshold studies leading to the determination of appearance energies and molecular levels.

a. Total and counting ionization cross sections. For the determination of these cross sections no mass analysis of the product ions is necessary, thus facilitating the experimental task. However, no total cross sections for a specific cluster size have been measured to date, only cross sections for certain cluster size distributions (usually not very well characterized and only measured after ionization!). Studies performed for large clusters include absolute and relative effective electron impact ionization cross sections for CO, clusters (Hagena and Henkes, 1965; Falter et al., 1970), absolute effective electron impact ionization cross sections for H, (Henkes and Mikosch, 1974, see Fig. 6), and absolute electron impact counting ionization cross sections for H, (Tay et al., 1970). In all of these experiments a shift of the maximum to higher electron energies with larger cluster distributions has been observed (e.g., see Fig. 6). Moreover, the effective ionization cross section appears to decrease, indicating that, at least for large clusters, no simple additivity rule ( e g , see for normal molecules Otvos and Stevenson, 1956; Lampe et al., 1957; Stevenson and Schissler, 1961; Batabyal et al., 1965; Hamson et al., 1966; Pottie, 1966; Beran and Kevan, 1969; Grosse

EXPERlMENTAL STUDIES ON CLUSTER IONS

87

Electron energy (eV)

-

FIG.6. Absolute effective electron ionization cross sections as a function of electron energy for various H2 cluster size distributions (H,)" e ions (with n the mean cluster size); (a) n = l,(b)n = 340,(c)n = 1700,(d)n = 8750,(e)n = 48,00O,(f)n= 85,00O(afterHenkesand Mikosch, 1974). Also shown for comparison is the total ionization cross section function for H, measured by Rapp and Englander-Golden (1965).

+

and Bothe, 1970;Alberti et af.,1974;Bartmess and Georgiadis, 1983)can be used.

b. Partial ionization cross sections. Relative partial ionization cross section functions (i.e., giving the general shape of the ionization efficiency) have been measured for a number of cluster systems for both electron impact ionization (Leckenby and Robbins, 1966 [Arf and (CO,)f]; Milne et al., 1970 [(H,O),,H+]; Gspann and Korting, 1973, (N2)g((n=11838) and (H,)S+(n= 65132) with z = 1 to 4; Herrmann et al., 1978b (Na;); Helm et al., 1979 (Art, Krf,Xef, ArKr', and KrXe+);Castleman et al., 1981e, and Hermann el al., 1982 [(H,O),H+)]) and photoionization (e.g., Robbins et al., 1967 (Nat); Ng et af.,1977 (XeKr+, XeAr+, and KrAr+); Herrmann ef af.,1978a,b (Na:, K:); Jones and Taylor, 1978 [(CO,):]; Ono ef al., 1980, 1981a,b [(CS,):, OCS:]; Walters and Blais, 1981 [(H2S):]; Dehmer and Pratt, 1982b (Ar:); Ono and Ng, 1982a,b [(C,H,):]; Peterson et al., 1984 (Na,+)I. Because of the reasons mentioned above, no absolute partial ionization cross sections have been measured. Moreover, some of the relative functions may be falsified by contributions from higher clusters (e.g., see studies of Ar, photon ionization cross sections as a function of stagnation gas pressure by Dehmer and Pratt, 1982b). To calibrate mass spectrometric detection, the additivity rule mentioned above has sometimes been applied, i.e., the usual

88

T. D.Mark and A . W. Castleman. Jr.

working rule has been that, for example, the neutral dimer-to-monomer ratio is half the ratio of the dimer-to-monomer ion current measured by the mass spectrometer (e.g., see Leckenby and Robbins, 1966; Milne and Greene, 1967a-c; Yealland et al., 1972; Dorfeld and Hudson, 1973a; Lee and Fenn, 1978; Pittermann and Weil, 1980; Cook and Taylor, 1980).This working rule tacitly assumes that there is no fragmentation of the clusters

-200 -150 -100 - 5 0

0

RELATIVE FREQUENCY (ern-')

-200 -150 -100 -50

0

50

RELATIVE FREQUENCY ( cm-'

FIG.7. (a) One-color resonant two-photon ionization spectrum ofa benzene cluster beam monitoring the photoion signal in the benzene dimer channel after Hopkins et al. (1 98 1). The frequency scale is relative to the monomer origin at 38086. I/cm. Note by comparison to (b) that most spectral features in this scan result from fragmentation of the trimer into the dimer signal channel. (b) Two-color resonant two-photon ionization scan (after Hopkins cf al., 198 1 ) ofthe absorption bands ofthe benzene dimer, trimer, and tetramer in the region of the lB,,(nnl*) + 'A, origin of benzene monomer. The intensity for each spectrum has been normalized here to a constant level. Actually, the two-color observed peak photoion signals were in the ratio 1 :0.05 :0.03 for dimer :trimer :tetramer, respectively. The zero of the relative frequency scale is set to the position ofthe forbidden origin ofthe benzene monomer at 38086. I/cm. The red shifts ofthe most prominent features from the monomer origin are - 40, - I 15, and - I49/cm for the dimer. trimer, and tetramer, respectively. The strong features in the trimer spectrum centered 22 and 40/cm to the blue ofthe origin are due to progression activity in the van der Waals modes of the cluster.

EXPERIMENTAL STUDIES ON CLUSTER IONS

89

during ionization, which has recently been shown to be incorrect (see below). Helm et al. ( 1979) and Mark (1982c,d) have recently proposed for rare gas dimers a modified additivity rule theoretically taking into account the dissociative channels (see also Drowart and Goldfinger, 1967; Stafford, 1971; and Gingerich, 1980b). c. Partial ionization cross-section ratios. Partial ionization cross sections and their ratios will play an important role, if the ionization process leads to more than one ion, e.g., either multiply charged ions and/or fragment ions. Cross section ratios (or measured ion current ratios) have been reported for various multiply charged cluster ions produced by electron impact (Gspann and Korting, 1973; Helm et al., 1981; Sattler et al., 1981; Stephan et al., 1982a; Echt et al., 1982a; Dreyfuss and Wachman, 1982) and range from 0.1% (ArXe2+/ArXe+,Helm et al., 1981)to 100%[(CO,):+/(CO,):, Echt et al., 1982al.Conversely, there is only very little information on cross section ratios between fragment ions and their respective cluster parent ion (i.e., fragmentation ratios). Determination of this ratio necessitates the unambiguous identification and quantitative analysis of fragment and parent ions. The task is also complicated by the fact (see Sections III,A,3 and IV,B) that part of the fragmentation may take place after the ions have exited the source region. Moreover, fragmentation of some molecular clusters has been easily verified (i.e., via dissociative channels leading to an ion with different mass than the neutral clusters). While fragmentation of atomic and/or molecular clusters has been found to be negligible or ruled out for some systems (e.g., Milne and Greene, 1967a; Yealland, et al., 1972; Walters and Blais, 1981; Lisy et al., 1981; Kappes et al., 1982a; Dreyfuss and Wachmann, 1982), it was recently possible via improved experimental detection techniques to demonstrate beyond a doubt that in the case of others (at least for small clusters) considerable fragmentation is occurring upon electron or photon impact' sometimes already several tenths of an electron volt above threshold [e.g., Henkes, 1962; Milne et al., 1972; Wagner and Grimley, 1972, 1974; Van Deursen and Reuss, 1975, 1977; Smets et al., 1977; Herrmann et al., 1978b; Jones and Taylor, 1978; Lee and Fenn, 1978 (see also Helm et al., 1979; Gentry, 1982); Grimley et al., 1978; Lisy et al., 1981; Duncan el al., 1981; Hopkins et al., 1981 (e.g., see Fig. 7); Vernon et al., 1982, 1983; I As Mark (1 982d, 1984a,b)has recently pointed out, there is, in contrast with general belief, no large difference in fragmentation yields between electron and photon ionization threshold spectra.This should be true also for clustersand warrants further investigation. In this conjunction it is interestingto point out that Hoareau et al. (198 1) were able to detect Nazclusters up to n = 20 despite the fact that Herrmann ef al. (1978b) could only observe Na,+clustersup to n = 3 by electron ionization. See also results by Cook and Taylor (1980), Riley ef al. (1982). and Haberland ef al. (1 983a).

90

T. D. Mark and A . W. Castleman, Jr.

Poliakoffet al., 1982; Dehmer and Pratt, 1982b;Dehmer, 1982; Riley et al., 1982;Ono and Ng, 1982a,b;Delacretaz etal., 1982a,b;Geraedts et al., 1982, 1983; Gough and Miller, 1982; Haberland et al., 1983a; Buck and Meyer, 1983; Ding, 1983; Peterson et al., 19841. This question, however, is still a matter of dispute (see also discussion by Echt et al., 1982b; Hermann et al., 1982;Haberland etal., 1983a; and Section III,A,3), although it appears at the moment that at least for most ofthe small clusters appreciable fragmentation does occur (e.g., for large clusters see studies by Gspann and Korting, 1973; Gspann and Vollmar, 1980, 1981; Echt et al., 1982b; Sattler, 1983a,b). Moreover, there exist three recent sophisticated studies reporting the determination of absolute cross section ratios (Gough and Miller, 1982;Geraedts et al., 1982; Buck and Meyer, 1983), e.g., yielding at an electron energy O f 100 eV a[CO+/(CO)2]/a[(CO)z/(CO),] = 0.8 5/0.25 ; a[SF f / ( sF6 )2 ]/ a[sFf'SF~j/(SF6)2] 100, a [ S F ~ / ( S F 6 ) , ] / a [ S F f ( S F 6 ) z / ( S F 6 ) ,a] 150, a[SFf / ( s F 6 ) 3]/a[ SFfSF6/ (SF, ), ] 3 15; and a(Ar+/Ar,)/a(Ar f/Ar, ) = 0.70/0.30, a(Ar+/Ar,)/o(Arz/Ar,) = 0.6 1/0.38 [with a(Ar:/Ar,) 01(preliminary results), respectively. In addition, Wagner and Grimley (1 974) and Grimley et al. ( 1978)have reported electron impact fragmentation ratios for Bin and (LiF), , Hermann el al. ( 1982) apparent fragmentation ratios for (H,O), (n = 3 to 10)as a function of electron energy and stagnation conditions, and Ono and Ng (1982a,b) have recently reported fragmentation ratios for (C2H2)3 photoionization and a clastogram for the (C,H,), photoionization in the wavelength region between 1200 and 600 A.

-

d. Threshold studies. The importance of low-energy ionization studies of clusters produced in expanding jets has been recognized first by Lee and co-workers and Dehmer and co-workers with emphasis toward determination of adiabatic and vertical ionization (appearance) energies and elucidation of the respective potential energy curves and electronic levels. This subject has been recently reviewed in great detail for single photon ionization by Ng (1983) and for electron impact ionization by Mark (1982a,b), both including literature up to the end of 1981 (see also compilation by Levin and Lias, 1982). Ng (1983) has not included work in the area of multiphoton ionization. Moreover, both authors have not included the measurement of appearance energies by mass spectrometry of high-temperature systems (e.g., see reviews by Drowart and Goldfinger, 1967; Berkowitz, 197 1, 1979; Gingerich, 1980a,b;and other recent studies by Lin and Kant, 1969; Donovan and Strachan, 1971;Gingerich and Finkbeiner, 1971;Gingerich, 197 1; Emelyanov et al., 1971; Marr and Wherrett, 1972; Schaaf and Gregory, 1972; Wagner and Grimley, 1972; Piacente and Gingerich, 1972; Neckel and Sodeck, 1972; Cocke and Gingerich, 1972, 1974; Hildenbrand, 1972; Stearns and Kohl, 1972a,b, and references therein; Kordis and Gingerich,

EXPERIMENTAL STUDIES ON CLUSTER IONS

91

1973a,b, 1977; Cabaud et al., 1973, and references therein; Gingerich et al., 1974;WagnerandGrimley, 1974;Cockeetal., 1975;Wu, 1976,1979,1983; Drowart et al., 1977; Smets et al., 1977; Smoes et al., 1977; Zmbov et al., 1977; Grimley et al., 1978; Kingcade et al., 1978; Biefeld, 1978; Gupta and Gingerich, 1978, 1979a,b;Neubert, 1978; Gingerich and Cocke, 1979; Wu et al., 1979; Hilpert, 1979; Pittermann and Weil, 1980, and references therein; Hilpert and Gingerich, 1983; Rosinger el al., 1983). In the following we will as an example briefly review the new literature on single-impact ionization and in addition the studies employing multiphotoionization employing supersonic expansions (for older appearance energies see also Rosenstock et al., 1977). In contrast with the above-mentioned studies on the absolute magnitude of the ionization cross section functions, in threshold ionization studies it is only necessary to obtain accurate energy dependencies of the relative ionization cross section functions. Clearly, photoionization and in particular molecular beam laser multiphoton ionization gives more accurate threshold energies than electron impact studies. On the other hand, electron ionization is simple (and can be quite accurate if appropriate electron monochromators are used, in particular for the determination of the double-ionization threshold not accessible by direct photon ionization) and it has been demonstrated that there is good agreement between thermochemical values (binding energies, proton affinities) derived from electron impact threshold energies and those deduced in high-pressure mass spectrometry or drift cell studies (see Section V). Newer studies (not included in the review of Mark, 1982a,b) include rare gases, CO, ,ArCOZ,NH, ,N, ,Pb, NayI, Hg, clusters (Harbour, 1971; Cabaud et al., 1980; Hoareau eta)., 1981; Stephan et al., 1982a-c, 1983a,d, 1984; Mark et al., 1982a, 1983; Saito et al., 1982). A similar good agreement has been noted for photoionization threshold energies (Ng, 1983). It should be mentioned, however, that these comparisons are only meaningful if the obtained appearance energies correspond to the adiabatic threshold (Anderson et al., 1980; Dehmer and Poliakoff, 1981) and the AH values correspond to the same temperature, i.e., correcting for heat capacity changes from translational, vibrational, and rotational degrees of freedom (e.g., see Conway and Janik, 1970; Teng and Conway, 1973; Castleman et al., 1982a). Single photoionization thresholds have been determined for rare gases, NO, 0, , HF, HCl, HBr, HI, COYN2, H2, CS2, OCS, COz,N,O, SOz, HzO, NH,, C2H2,C2H4,and acetone clusters (as reviewed by Ng, 1983) and in addition for rare gases, alkali, oxidized alkali, ArCO, ,H2S,Na,Cl, Cu, CuBr, and acetic acid clusters (Hudson, 1965; Robbins et al., 1967; Foster et al., 1969;Herrmann et al., 1978a,b;Cook and Taylor, 1979b;Waltersand Blais, 1981;Pratt and Dehmer, 1982a,c, 1983;Dehmer and Pratt, 1982a,b;Powers

T. D. Mark and A . W. Castleman, Jr.

92

et al., 1983; Martin et al., 1983; Castleman et a!., 1983d; Peterson et a/., 1983, 1984; Dao et al., 1984). For some cases it was possible to measure a whole sequence of ionization energies, e.g., from the monomer up to Fe25 (Rohlfing et al., 1983), to (H2S), (Walters and Blais, 1981) to Na,, and K, (Herrmann et al., 1978a,b), and to Na, (Robbins et al., 1967; Castleman et al., 1983d; Peterson et al., 1984a) and recently for N%5(Kappes and Schumacher, 1983).The results on Nanhave been compared to theoretical calculations, and are in fair agreement (see discussion by Peterson et al., 1984a) with the classical description of the work function of a spherical metal drop (see also the electron ionization results of Saito et al., 1982a,b, for Pb, up to n = 7 and of Neubert, 1978, for Ten up to n = 7), e.g., see Fig. 8. But the agreement can be improved by a different choice of radius (see Kappes and Schumacher, 1983). 2.5

I

1

-2

-3

I

I

I

I

I

4

5

6

7

8

1

--

a 3 4.0-

4.5

-

5.0

1

n (otomr/clurter)

FIG.8. Plot of work function versus number of atoms in a Na cluster using W(R)= W ,

+

3 e2/R,where W represents the work function, R the radius of equivalent sphere, and e the

elementary charge. The conversion between number and equivalent spherical size was made using (a) R = 1.86 A for the nearest neighbor distance, (b) R = 1.54 A based on the covalent radius (Demitras et al., 1972), and (c) R = 2.1 I A based on the bulk density. (0)Experimental results of Peterson et al. (1984a). The polycrystalline work function ( W, = 2.75 eV) was taken from Whitefield and Brady (1971); note that the results for different crystal faces range from 2.65 to 3.10 eV.

EXPERIMENTAL STUDIES ON CLUSTER IONS

93

Even more accurate determination of ionization energies (and/or dissociation energies) is possible by sequential two-photon ionization and/or optical - optical double-resonancespectroscopy of Liz, Na, ,Na, ,K2,NaK, Cu,, fluorobenzene,benzene-Ar (Herrmann et a/., 1978a,b, 1979; Mathur etal., 1978b;Leutwyleretal., 1980, 1981, 1982a;Fungetal., 1981;Eiseland Demtroder, 1982; Martin et al., 1982;Powers et al., 1983; Rademann et al., 1983;Delacretaz et al., 1983b;Bernheim et al., 1983;McGeoch and Schlier, 1983). In the case of two-color two-photon ionization, the whole FranckCondon accessible range of the intermediate rovibronic statesis mapped out by the subsequent ionization steps, thus assisting the determination of the adiabatic ionization energies. This may explain the slight but significant differences between single- and two-photon ionization results (i.e., see also recent three-photon ionization results by Delacretaz et al., 1983b). According to Martin eta/. (1982), however, the true ionization potential can only be deduced by extrapolation of Rydberg states, because ionization threshold methods lead to slightly different results dependent upon the experimental condition (see also Eisel and Demtroder, 1982; Leutwyler et a/., 1982a; Delacretaz et al., 1983b). Complementary information to that gained from the methods discussed above can be obtained by photoelectron spectroscopic studies. The photoelectron spectrum gives direct information on the vertical appearance energy, and relative vertical transition strengths (if the excitation is nonresonant, i.e., see Ref. 3 in Poliakoff et al., 1982)and indirect information on the structure from Franck - Condon envelopes. Studies include photoelectron spectra of Ar,, Kr,,. Xe,, (H,O),, methanol dimers, formic, acetic, and trifluoroaceticacid dimers in supersonic expansions (Dehmer and Dehmer, 1977, 1978a,b; Carnovale et al., 1979, 1980; Tomoda et al., 1982, 1983; Tomoda and Kimara, 1983), and of high-temperature vapors (e.g., Berkowitz, 1971, 1975, 1979; Evans et al., 1972; Brundle et al., 1972; Dehmer et al., 1973; Boschi and Schmidt, 1973; Potts et al., 1974; Richardson and Weinberger, 1975; Streets and Berkowitz, 1976a,b; Potts and Price, 1977; Potts and Lyus, 1978; Potts and Lee, 1979; MacNaughton et al., 1980; Berkowitz et al., 1980). Since the methods of neutral cluster production (see above) generally result in a distribution of cluster sizes and the spectral features of these different clusters are frequently overlapping, conventional photoelectron spectroscopyis not unambiguous. Poliakoff et al. (198 1,1982)have recently shown for the case of Xe, that mass analysis of the ion cluster in coincidence with the photoelectron spectrum removes these problems. Moreover, this photoelectron- photoion coincidence technique has been helpful in assessing fragmentation of the cluster under study (Poliakoff et al., 1982). (For comparison there are photoemission spectroscopic studies of the electronic

94

T. D. Mark and A . W. Castleman, Jr.

structure of supported clusters, e.g., see Kubota et al., 1981; Yencha et al., 1981 ; Lee et al., 1981; Mason, 1983, and references therein.) Finally, it is interesting to point out that there exist some studies on cross sections for electron attachment to clusters (H,O, N,O, NO2, CO,, SO,, H,S, C1,) and their respective electron affinities (Dillard, 1973; Mots and Compton, 1977, 1978a,b; Mots, 1979; Bowen et al., 1979, 1983; Stephan et al., 1983d; Quitevas and Herschbach, 1983).

3. Mass Spectra of Cluster Distributions A large number of cluster studies has been concerned with either the distribution of neutral clusters produced in supersonic expansions, inert gas condensation, etc. (see above), and detected by ionization mass spectrometry, or the distribution of ionic clusters as, for instance, ejected from surfaces by ion bombardment (e.g., see references given in Section 11, A and C). For both cases observed anomalies (“magic numbers”) were interpreted and/or correlated with particularly stable or unstable cluster geometries (structures) often under the assumption that these anomalies arose in the primary cluster production process. In light of some recent findings, i.e., fragmentation and metastability (e.g., see Sections III,A,2 and IV,B), such interpretations should be accepted with some caution and treated on a very individual basis, i.e., some of the measured distributions (especially for very large clusters) may reflect the actual distribution before the detection process, whereas in other cases the measured distribution is strongly distorted by secondary processes introduced by the detection process (e.g., see discussions by Echt et al., 1982b; Sattler, 1983a,b; Haberland, 1983). In the following we will discuss, out of the vast amount of systems studied, well-documented examples of particular interest, including an atomic, molecular, and metallic cluster species.

a. Ar, cluster distributions. Argon cluster mass spectra have been recorded by a number of groups (Greene and Milne, 1963; Milne and Greene, 1967a; Worsnop et al., 1981; Dreyfuss and Wachman, 1982; Ding and Hesslich, 1983a). In each case neutral clusters are produced in a supersonic nozzle expansion. The detection has been achieved by electron impact ionization (with electrons at least several electron volts above threshold) with subsequent mass spectrometric analysis. Intensity variations have been observed in the mass spectra as a function of cluster size at various expansion conditions. Similar irregularities have been reported for water clusters (see below) and other rare gas clusters (Echt et al., 1981; Ding and Hesslich, 1983a), although it should be pointed out that there are distinct differences, i.e., the first magic numbers [corresponding to complete shell icosahedra (Hoare,

EXPERIMENTAL STUDIES ON CLUSTER IONS

95

1979)], 13 and 5 5 , observed in Xe, are not observed in Ar or Kr expansions. Conversely, the magic number 19 appears to be common to all three rare gases and has also been observed in most studies. Milne and Greene ( 1967a) ascribe these anomalies, due to kinetic processes during “nucleation,” to lower stability of the neutrals, or to differing behavior of the ions, whereas Ding and Hesslich (1983a) (following the approach of Echt et al., 1981, 1982b)concluded that the observed structure in the mass spectra mirrors the abundance of the neutral cluster distribution. (Recent studies also suggest some contribution to magic numbers following ionization; Ding, 1983.) However, it seems that from the mass spectra alone (and also from changes in the expansion conditions and in the electron energy, unless all the way down to the appearance energy) it is not possible to establish the origin of these variations. Recent studies (see also Sections III,A,2 and IV,B) on the direct fragmentation of Ar, and Ar, upon electron impact (Buck and Meyer, 1983),on the direct fragmentation of Ar, clusters with n > 5 upon electron, photon, or Penning ionization (Haberland ef al., 1983a), on the metastable decay of Ar ;(Stephan and Mark, 1982b, 1983),and on the metastable decay of Ar:(Stace and Moore, 1983; Mark et al., 1984) cast serious doubt concerning the interpretation of these intensity distributions in terms of structure and stability of the neutral clusters.

b. (H,O), cluster distributions. Again, several groups reported measurements of the relative intensities of clusters of water molecules detected by mass spectrometry in supersonic expansions of water vapor (Lin, 1973; Holland and Castleman, 1980b; Castleman et al., 1981e; Hermann et al., 1982; Dreyfuss and Wachmann, 1982). There are intensity variations, the most prominent of which occurs at (H20)21H+.Due to the fact that this irregularity at n = 2 1 is observed at the same cluster size n = 2 1 whether ( Hz0)21H+ are produced by electron impact ionization on neutral clusters present in the expansion of water vapor (see above references) and in atmospheric air (Carlonand Harden, 1980;Carlon, I98 1) in studiesof the growth of water clusters formed during the coexpansion of water vapor and protons in a supersonicexpansion(Searcyand Fenn, 1974;Burke, 1978;Beuhler and Friedman, 1982b),or during studies of the secondaryion mass spectra of ice (Lancaster et al., 1979a), Hermann et al. (1982) (see also discussion in Searcy, 1975; Kassner and Hagen, 1976;Searcy and Fenn, 1976)concluded that this discontinuity observed in the neutral expansion studies originates after the ionization process. These conclusionsare supported by recent findings on fragmentation (Vernon et al., 1982), metastable decay (Stace and Moore, 1983) and the stability of (H20)!1H+ via dodecahedra1 clathrate structure where an “excess” proton remains in the cage structure and an unbonded H,O is trapped in the center (Holland and Castleman, 1980b). It

96

T. D. Mark and A . W. Castleman, Jr.

is interesting to note in this conjunction the reported existence of stable ion structures in hydrogen bonded ion clusters (Stace and Moore, 1982), preferential solvation of protons in mixed clusters as detected by monitoring the competitive decomposition processes via metastable peak intensities (Stace and Shukla, 1982b)and the fact that Kay et al. (198 1) have recently reported some indirect evidence of solvated ion pair formation for aqueous nitric acid clusters by electron impact mass spectrometry [see also Herschbach, 1976, for solvation in the scattering of HI from (NH3),,].

c. (CsI), cluster distributions. This is another interesting case where it has been recently demonstrated that measured cluster ion distributions have been modified prior to detection by metastable decay: Secondary cluster ions ejected from alkali- halide surfacesby energetic Xe+ion bombardment have beenobservedwithSIMS(Campanaetal., 198l;Barlaketal., 1981,1982).It was found that the ion intensity showed distinct anomalies, i.e., the most prominent of which occurs at (CsI),,Cs+.These anomalies were correlated to particularly symmetrical cubic-like structures. Similarly, Sattler et al. (see Martin, 1983) related anomalies observed in mass spectra of alkali- halide clusters produced by supersaturation of the vapor in cold He gas to a multiring structure of the neutral clusters. However, very recent studies by Standing et al. (1982) and Ens et al. (1983) (see also Campana et al., 1983) have clearly demonstrated that the anomalies observed by SIMS studies are due to unimolecular decay after the ion formation, i.e., the yield of cluster ions decreases smoothly with n when observed at very short effective times after ion formation (e.g., Fig. 9). Because secondary ion clusters (CsI),Cs+ with n > 7 produced by 8 keV Cs+bombardment were found to be mainly metastable, with lifetimes NH, > H,O > SO, > CO, > CO 3 HCI > N, 3 CH,. Additional considerations (Peter-

EXPERIMENTAL STUDIES ON CLUSTER IONS

-

\

h

5 30E

1 m xU

-

-$20 I

7

125

-

10-

1

0.1

1

1

1

I,2

2.3

3.4

I

4.5

1

5.6

t. -AHy

n.n+ I

+

FIG.12. -AH,,,+, versus cluster reaction, n,n 1, for (a) DME, (b) NH,, (c) H,O, (d) SO,, ( e ) CO,, (f) CO, (g) HCI, (h) N,, and (i) CH, clustered to Na+, after Castleman et al. (1983a).

son, 1982; Castleman et al., 1983b) suggest that DME forms a bidentate bond with Na+. Comparing the stronger bond for ammonia, compared to water in light of the electrostatic properties given in Table I, reveals the importance of the ion-induced dipole interaction in governing bond strength. Although ammonia has the smaller dipole moment, its much larger polarizability enhances its bonding at small cluster sizes. The role of the quadrupole moment is also seen by comparing the relative bonding of SOzto NH, . Both have approximately comparable values of dipole moments and polarizabilities, but the quadrupole moment of SOz leads to a repulsive interaction compared to the attractive one of NH,, consistent with the smaller bonding strength of SOz.The role of polarizability is further seen by comparing CO and HCl, where the latter has a larger dipole moment but both have significant polarizabilities. Molecules without permanent dipoles have comparatively low bond energies as seen for Nz and CH,; the relatively

TABLE I PROPERTIES OF SMALL MOLECULES"

Dipole moment

Dz

Polarkability %,a y , a,

Quadrupole moment Qzz,

QW9

8,

IP(a) (eV)

Proton affinity (kcal/mol)

Ligand

(Debye)

(A3)

( 10-26esu/cm2)

H20

1.85 (a)

1.452, 1.651, 1.226 (b)

-0.13, -2.5, 2.63 (c)

12.6

SO2

1.63 (f)

2.653,4.273, 4.173 (f)

1.3,4.0, -5.3 (g)

12.34

166.7 (x) 169.3 (d) 133- 159 (h)

CO

0.1098(j)

2.601, 1.624, 1.624 (k)

-2.5, 1.25, 1.25 (I)

14.013

139.0 (d)

N2

0 (a)

2.39, 1.46, 1.46 (m)

- 1.2,0.6,0.6 (n)

15.769

113.769 (d)

AH,,,(exp) (kcal/mol) for attachment to Na+

BE, (kcal/mol) 0

-24 (e)

H

- 18.9 (i)

- 15.8

- 12.6"

-11.6

- 8.0 (i)

-7.7

/ \

H

O\ /O S C

II

0

N

II

N 0

co*

0 (a)

4.05, 1.95, 1.95 (0)

-4.32,2.16, 2.16 (p)

13.769

126.8 (d)

- 15.9"

-11.1

II

C

II

0

CH,

0 (a)

2.56 (9)

0

12.6

128.2 (d)

NH,

1.47 (r)

2.388,2.1,2.1 (s)

-2.32, 1.16, 1.16(t)

10.2

202.3 (d)

-29.1 (u)

HCI

1.084 (a)

2.727, 2.477, 2.477 (v)

3.8, - 1.9, - 1.9 (1)

135 (w)

- 12.2 (i)

12.74

- 7.2"

-2.75

H

" Table from Peterson (1982). Original source ofeach entry indicated by letter: (a)Handbook ofchemistry and Physics ( 1 974); (b) Liebmann and Moskowitz (197 1);(c) Verhoeven and Dymanus (1970); (d) Kebarle (1977); (e) Dddic and Kebarle (1970); (f) Patel er al. (1979); (8) Pochan er al. (1969); (h) Smith and Munson (1978); (i) Perry ef al. (1980); ( j ) Muenter (1975); (k) Hirschfelder ef al. (1964); (1) Stogryn and Stogryn (1966); (m) Trapy ef al. (1974);(n) Momson and Hay (1979); (0)Koide and Kihara (1974);(p)Barton er al. (1979);(9)Maryott and Buckley ( 1953);(r) McClellan (1962);(s) Koch ef al. (1962);(t)Kukolich (1970);(u) Castleman ef al. (1978a);(v) Gianturco and Guidott (1978);(w) Polley and Munson (1978);(x) Collyer and McMahon (1983).

EXPERIMENTAL STUDIES ON CLUSTER IONS

127

stronger bonding of CO, compared to N, and CH4is accountable in terms of the large polarizability of CO,. Another interesting trend is seen by comparing the binding of various ligands to K+and Li+(seeCastleman et al., 1978a;Woodin and Beauchamp, 1978). Interestingly, the bond strengths correlate inversely with differences in ionization potential between the respective ligand and the neutral metal. This approach is able to predict (Castleman el al., 1978a) the ordering of ion - ligand bond energies for many chemically similar systems, but fails in a few cases, for example, the bonding order of Li+ to methylamines where NH, < MeNH, < Me,NH, Me,N; but Me,N < Me,NH. The anomolous position of Me,N in this series most likely arises from repulsive interactions between Li+ and the methyl groups of the ligand, a factor not accounted for in the simple correlation with ionization potential differences (Woodin and Beauchamp, 1978). As pointed out by Kebarle (1977) alkali ion-molecule interactions can be considered in terms of a Lewis acid-base interaction, since there is so little electron transfer in these systems. However, the Lewis acid-base concept is not completely general. Rather convincing evidence that the bonding of molecules to certain ions does involve partial transfer of an electron comes from a number of experimental measurements which show that the bonding strengths exceed those expected on the basis of simple electrostatic considerations. In the case of metal ions, the first data suggestive of this fact were derived from measurements by Tang and Castleman (1972) for the bonding of water to Pb+, an ion having an electronic configuration 6sz 6p'. Despite the relatively large size of the ion (1.5 A, comparable to that of Rb+, 1.48 A) the bonding of the first water molecule has a AHvalue of 22.4 kcal/mol. This is closer to that of water bonded to sodium than to comparably sized closed shell ions. More recently, Castleman and co-workers (Tang and Castleman, 1974;Tang et al., 1976;Holland and Castleman, 1980a, 1982a)have shown similar behavior for water bound to other metal ions including Bi+ (6s26p2 valence shell), Sr+ (5s configuration) and for both water and ammonia bound to Ag+ and Cu+. In the case of each of these ions, the values were much larger than expected on the basis of electrostastic considerations. For Ag+ a AH value of 33.3 k 2.2 kcal/mol was obtained, while for Cu+ the value exceeded approximately 40 kcal/mol. Even the addition of the second ammonia ligand to Ag+ has a AH of 36.9 f 0.8 kcal/mol, also pointing to a bond energy for the first complex well in excess of 40 kcal/mol. Among the more interesting results are the stabilities of the cluster ions Na, 02,and (CO);, all of which are much greater than expected for ion-induced dipole interactions in the case of the first two, and ion - dipole interaction for the last species. It is suggested that bonding must arise due to the sharing of an electron by the two molecules (Conway, 1969, 1979 and

I28

T. D. Mark and A . W. Castleman, Jr.

Kebarle, 1977).Fehsenfeld et al. ( 1975)have suggested that the interaction of water with NO+ and NO: produces protonated nitrous and nitric acid, respectively. But the bond energies are within range of the usual values for small ions bound to water, i.e., 18.5 and approximately 21.2 kcal/mol (French el al., 1973; Fehsenfeld et al., 1975). Compared to studies of the binding of ligands of a single species to ions, relatively few multiligand complexes involving simple inorganic cations have been studied in the gas phase. In the case where both ligands are inorganic, data are available for the pairs H,O/NH, bound to H+ (Payzant et al., 1973), H,O/H,S to H+(Hiraokaand Kebarle, 1977a),and H 2 0 / C 0 2to Na+ (Peterson et al., 1984).Other data are available for a few systems where one or more ligands are organic (Hiraoka et al., 1972; see also Sunner et al., 1981 ;Yamdagni and Kebarle, 1976).Until now little attention has been paid to these systems, but it is slowly becoming recognized that the results are of value in understanding solvation (Peterson et al., 1984b; also see a related discussion in Arnett er al., 1972).

E. SOLVATION Kebarle ( 1972b, 1974), Kebarle et al. ( 1969,1977),and Kebarle and H o g (1965) were apparently the first to draw attention to the usefulness ofdata at successively larger degrees of clustering for gaining insight into the solvation of ions in the liquid phase. More recently, Lee et al. ( I 980a) and Holland and Castleman ( 1982b) have considered the ratio of the solvation energy in the liquid phase to the summed gas phase enthalpy values up to a given cluster size and shown a rather remarkable convergence with size for a wide variety of different cations clustered with both water and ammonia (see Fig. 13). Rather unexpectedly, at cluster sizes beyond five, the differencesin the ratios between a variety of both simple and complex ions are within experimental error. These findings lend confirmation to some of the very simple Born concepts (1920), albeit that they require corrections for surface tension effects (see Lee et al., 1980a; Holland and Castleman, 1982b).Klots ( 1981) has considered the role of the surface potential in similar relationships. Convergence of these ratios to approximately the same value is indicative of the fact that beyond the first solvation shell the majority of the contribution to solvation is from electrostatic interactions between the central ion cavity and the surrounding medium. However, in some cases (e.g., Sr+)for which there are no single-ion solvation results to compare, the AHvalues far exceed the heat of condensation of water to degrees of clustering far greater than expected (Tang et al., 1976).Although few data are available for ligands other than water, an ultimate limiting value of A H for very large cluster sizes,

7

I

1

1

I

I

a

6

5-

5 - p 5 A

4

s B om

0

=,

$12

3

4-

0

3-

2-

2

---_

I

I 2

I

I

3

4

n

I

I

5

6

I-

Y

OO

I

2

3

4

5

6

n

FIG.13. (a) Ratio of Randles’total enthalpy () of solvationto the partial gas phase enthalpy of hydration (25°C)for positive ionic cluster size n, after Lee et al. (1980a).Thomson’s drop (---). (b) Ratio of total enthalpy of solvationto the partial gas phase enthalpy of ammoniation(- 33°C) for Coulter (---), Li+ (A), Na+ (0),K+(V),Rb+ (0). positive ionic cluster size n, after Lee et al. (1980a). Senozan (mean) (---),

130

T. D. Mark and A . W. Castleman, Jr.

leveling off at a A H value corresponding to the heat of vaporization of the liquid is always indicated; and the approach is always evident at surprisingly small cluster sizes (e.g., see treatment of alkali metal solvation in ammonia, Lee et af., 1980a). Further evidence for the relationship ofcluster data to ion solvation comes from observations of discontinuities in otherwise smooth trends in A H with size. See, for instance, Kebarle et al. (1967c), Searles and Kebarle (1968), Davidson and Kebarle (1976a), Yamdagni and Kebarle (1972), Castleman et al. (1978, 1983d), and Holland and Castleman (1982a). For further discussions of the importance of ion clustering in understanding solvation, the reader is referred to Schuster ef al. (1 977), Schuster (1976), Abraham and Liszi ( 1978),Veillard ( 1977, 1978),Smith efal. ( 1982),and Clementi (1 976). The relationship to proton conductivity is discussed by Weidemann and Zundel ( 1970). CONSIDERATIONS F. SOMESTRUCTURAL Several experiments have indicated the existence of some particularly stable structures ofcluster ions, for instance, the eigenstmcture H,O+( H20), (Eigen and DeMaeyer, 1958; Fang et al., 1973)based on both experimental findings (Hermann et al., 1982) and theoretical computations (Newton, 1977). Yet, the thermodynamic measurements on the bonding of water to H,O+(DePazefal., 1968, 1969;Arifovefal., 1971a, 1973a;Bierbaumetal., 1976; Beggs and Field, 197la,b; Bennett and Field, 1972a; Cunningham ef al., 1972; Good et al., 1970a,b; Kebarle e?al., 1967; Lau ef al., 1982; MeotNer and Field, 1977; Puckett and Teague, 197la) fail to reveal a particularly dramatic change in bond energies for the addition of additional water molecules to the eigenstructure. Indirect evidence has also been reported (Searcy and Fenn, 1974;Holland and Castleman, 1980b;Castleman et al., 198la,c, 1983e;Castleman, 1979a, 1980; Hermann et al., 1982; Dreyfuss and Wachman, 1982; Stace and Moore, 1983; Kassner ef al., 1980; see also Kassner and Hagen, 1976; Lancaster el d.,1979a)for the unusual stability of H,0+(H20)20,But, there are no bond energy measurements for clusters of this size to support the inferences from the various experimental studies. More recently, Stanley et al. (1983a) and Echt et af. (1984) have gained some evidence for a special structure involving ammonia, namely NH:( NH,), , via multiphoton ionization studies of ammonia clusters. The species has an analog to the eigenstructure in the water system. The reported (Tang and Castleman, 1975;Arshadi and Futrell, 1974;Payzant et al., 1973; Searles and Kebarle, 1968; Stephan et al., 1982c, 1983a;Wincel, 1972;Hogg

EXPERIMENTAL STUDIES ON CLUSTER IONS

131

et al., 1966;Arifov ef al., 1973;Ceyer et al., 1979a;Fehsenfeld and Ferguson,

1973; Long and Franklin, 1973; Puckett and Teague, 1971; Luczynski and Herman, 1979) bond energy measurements do not provide conclusive evidence of a solvation shell at this cluster size. AN ORGANIC CONSTITUENT G. SYSTEMS CONTAINING

Despite the general thrust of this article to trends in small inorganic systems, there are several cases where those from organic ones can be generalized and are germane to the subject at hand. The influence of ligand size resulting in steric hindrance in subsequent ligand attachment was found in studies of the clustering of dimethoxyethane about the monovalent sodium ion (Peterson, 1982; Castleman et al., 1983a). Other systems where at least one component, either the ion or ligand, is inorganic and the other organic include Hiraoka and Kebarle ( 1975a,c, 1976, 1977b,c),Kebarle ( 1977), Lau and Kebarle (1981), Davidson et al. (1979b), Yamdagni and Kebarle (1973a), Gimsrud and Kebarle (1973), Lau et al. (1981), Kebarle et al. (1967a), Meot-Ner (1978a,b, 1983), Sunner ef al. (1981), Holland and Castleman ( 1982a),Peterson et al. ( 1984b),Castleman ef al. ( 1983e), Upschulte et al. (1983), Wlodek et al. (1983), Beggs and Field (197 lb), Meot-Ner and Sieck (1983), Meot-Ner and Field (1974a), Meot-Ner et al. (1978), Bennett and Field ( 1972b,c),Field and Beggs ( 197l), Davidson and Kebarle ( 1976ac), Field ( 1969),Speller ef al. ( 1982a,b),Davidson ef al. ( 1978, 1979a),Staley and Beauchamp (1979, Cordennan and Beauchamp (1976), and Reents and Freiser ( 1981). More recently, findings of rather large bond energies for the clustering of molecules of metal ions have been derived from the measurements of Staley and co-workers (Jones and Staley, 1982a-c; Kappes and Staley, 1982a,b; Kappes et al. 1982b; Uppal and Stelay, 1982a-c). They have investigated the attachment ofa number ofligands to Al+, Mg+, Mn+, Ag+, Cu+,Cr+, Ni+, and Fe+, all showing rather large relative binding energies to these ions of open electronic configuration. In terms ofatmospheric science, evidence ofthe importance ofthe binding of certain organic molecules to ions has come from the work of Bohringer and Arnold ( 1981) in studies made of water and methyl cyanide coclustered to protons. The results show that the hydration equilibrium constants for heteromolecular clusters containing ethyl cyanide molecules are generally lower than those for the corresponding proton hydrate ions; the values decrease with increasing number of methyl cyanide ligands contained in the complex. The data suggest that interaction between water and methyl cyanide, and among the methyl cyanide molecules, is weaker than that between

132

T. D. Mark and A . W. Castleman, Jr

water molecules. An exception in the general trends was found for H+(CH,CN),H,O, where it is the most stable ion containing four ligands in the system. The authors explain the high stability of this ion by a structural rearrangement in which H,O+ is expected to form the core ion and the methyl cyanide molecules are believed to symmetrically attach to the three hydrogen atoms sharing the positive charge. Among the ions with six ligands, a similarly enhanced stability was also found for the ion H+(CH,CH), ( H,O),. Evidently, when the charged site can switch upon ligand addition, rather different bonding effects are displayed. In the context of solvation effects, a rather interesting observation has come from the work of SenSharma and Kebarle (see Kebarle, 1977)regarding the bonding of water to the ion C2H,0Ht. Although not really conclusive, they have presented data which suggest that the binding energies of this system pass through a minimum, where at cluster size six the enthalpy value exceeds that for the fourth water molecule addition. This is taken as an indication ofthe deprotonation ofthe base C2H50H.Although C,H,OH isa relatively strong base in the absence of water, it is weak compared to water. These findings may be in accordance with the rather interesting observations of Stace and co-workers concerning the crossover in ion stabilities noted for mixed clusters (Stace and Shukla, 1982b). For certain organic systems, Davidson et al. (1979b) have also made a comparison of proton transfer free energies between the gaseous and solution phase. They concluded that the large attenuation due to the substituent effect in solution must result from an effect of the substituent on solvation that partially cancels the effect on the isolated molecule. For instance, an electron-withdrawing substituent-like CN, which increases the intrinsic acidity of a material like phenol, is expected to unfavorably affect the solvation of cyanophenoxide ion and thus reduce the acidity increase of cyanophenol in aqueous solution. Cluster ion research has a great deal to contribute to the field of solvation involving organic constituents, but further discussion is beyond the scope of this article.

H. BONDINGTO NEGATIVE IONS

Fewer data are available for the clustering of molecules to anions; systems composed of inorganic anions and ligands include those of Payzant and Kebarle ( 1972),Keesee and Castleman ( 1980b),Lee et al. ( 1980b),Keesee et al. (1979a, 1980c), Peterson et al. (1984b), Wlodek et al. (1980, 1983), Robbiani and Franklin (1979), Fehsenfeld and Ferguson (1974), Conway

EXPERIMENTAL STUDIES ON CLUSTER IONS

133

and Nesbitt (1968), Davidson et al. (1977), Spears and Ferguson (1973), Fehsenfeld et af. (1975), Kebarle et al. (1968, 1972), Castleman (1982a), Paulson and Henchman (1982), Pack and Phelps (1966, 197l), DePaz et al. ( 1970), Arshadi et al. ( 1970), Hiller and Vestal ( 1980, 198l), Lifshitz et al. ( 1978), Cosby el al. (1 978), Moseley et al. (1976), Wu and Tiernan (198 l), Arshadi and Kebarle (1 970), Larson and McMahon (1983), Yamdagni and Kebarle ( 1974b), Yamdagni et al. (1973), and Payzant et al. ( 1971, 1972). For some systems there is sufficient information to observe trends in factors governing cluster stability. As in the case of the positive ions, the most extensiveset of measurements is available for hydration. Also, considerable data are available for the clustering of HCl, SO,, MeCN, MeOH, and HCOOH to C1-. The same general tendency of the AH values to approach the AH of vaporization of the individual ligands at large cluster sizes is seen for these as with H20. Interestingly, however, data for MeCN bound to C1-, and water to I-, show that it is possible for a given bond energy to fall below the heat of vaporization at intermediate cluster sizes. This is understandable in the case of systems where the ion - ligand bond is comparatively weak and more than one binding site is available between ligands in the condensed phase. A rather interesting trend (Keesee et af.,1980c)in bond energies is seen for the case of the first ligand to a variety of ions where the trends for systems where data are available including SO,, water, and C02 show the inequality to be as follows: OH-> F - , O - > O ~ > N O ; > C l - > N O y > C O y > S O ~ = S O ~ = B r - > I -

(27)

The fact that the bromide ion is nearly equivalent to SO 3 is based on experimental hydration results (Arshadi and Kebarle, 1970) and is also supported by electrostatic calculations (Spears, 1972b, 1977;Keesee, 1979).The above inequality appears to parallel the order of gas phase basicity of the negative ions where the strongest bases exhibit the largest bond dissociation energies. Such a correlation has been suggested on the basis of similar studies of gas phase hydrogen-bonded complexes of the type A-. HR (Yamdagni and Kebarle, 197 1). In that work, the descending order of the heterolytic bond dissociation energies was found to correspond to the increasing order of the bond dissociation energies of A- - HR. On the basis of considerations in terms of acid -base concepts, the noted trend of negative ions appears to be a general one (Keesee et al., 1980c). An interesting comparison is that ofthe relative bond dissociation energies for a given ion with different ligands such as SO,, H20,and CO,. Hydrogen, sulfur, and carbon are the centers which are attracted to a negative charge. Water has a slightly larger dipole moment than sulfur dioxide but the quad-

134

T. D. Mark and A . W. Castleman,Jr.

rupole moment of water is repulsive when the dipole is attractive to a negative ion. Alternately, the quadrupole moment of SO, is attractive to a negative ion and carbon dioxide has no dipole but does have a significant quadrupole. Considering only charge - dipole or charge - multipole interactions, the bond strength for a ligand attached to a given ion would be expected to be in the order of H,O = SO,, but both being greater than CO, . For a weakly basic or large ion like I-, the order SO, > H,O > CO, is observed. However, as the ions become smaller or more basic, SO, bonds relatively more strongly than water. With 0: the enthalpy change for addition of CO, becomes comparable to that of H,O. Finally, for a small ion like 0-, the order SO, > CO, > HzO actually occurs. Interestingly, the mean polarizabilities for SO,, CO,, and HzOfollow the same order as their relative bind energies to small ions. Polarization energies are known to be relatively more important for smaller ions due to the ability of the neutral to closely approach the ion, and thereby become more influenced by the ionic electric field with the attendant result of a larger induced dipole. Qualitatively, consideration of the polarizabilities partially explains the increased bonding strength of SO, and CO, over that of water in clustering to smaller ions. In some cases, the bonds are strong enough to be considered as actual chemical ones instead of merely weak electrostatic effects. In other words, the bond may be of a covalent nature which is equivalent to stating that significant charge transfer occurs between the original ion and the clustering neutral. An example is OH-.CO, which is more properly considered as HCO,. Likewise, C1-.S02 has a rather strong bond energy for the first cluster addition, but a very weak one for the second addition, whereupon the first cluster addition may be thought of as forming a “molecule” over which the negative charge becomes relatively widely dispersed (Keesee and Castleman, 1980a,c). In terms of charge transfer, the electron affinity of the clustering neutral is a relevant factor to consider in assessing the relative bonding trends. SO, has an electron affinity of - 1 .O eV (Janousek and Brauman, 1979) and forms a stable gas phase negative ion, whereas the negative ion of water has not been observed (Caledonia, 1975).CO, has been detected in high-energy processes in which the linear neutral can be bent to form the ion although it is short lived and autodetaches (Paulson, 1970). Nevertheless, (CO,); is apparently stable (Hots and Compton, 1977; Stephan et al., 1983d; Rossi and Jordan, 1979).Collective effects are evidently important in the formation and stabilization of certain negative ion complexes. This point is further realized by recent observations of Haberland et al. (1983b), who have reported observations of the solvation of electrons by as few as 1 1 or 12 molecules. An examination of the bonding of neutral molecules onto ions has partic-

EXPERIMENTAL STUDIES ON CLUSTER IONS

135

ular value in elucidating the properties which govern stability. An interesting result is found by comparing the association of ammonia, water, sulfur dioxide, and carbon dioxide to both Na+ and C1-. The ion - dipole interaction is the most important electrostatically attractive force between an ion and a neutral molecule. Both sodium and chloride ions have closed electronic configurations and spherical symmetry, although the ionic radius of Na+ is considerably smaller. Consequently, with other factors being equal, a neutral molecule would be expected to bind more strongly to the smaller Na+. However, bonding and charge transfer are also important. Based on experimentally determined stepwise heats of association, it is evident (Castleman and Keesee, 1983b, 1984)that only sulfur dioxide binds more strongly to C1- than to Na+ for the first association step. Ammonia exhibits the largest difference between Na+ and C1-, with its binding being much weaker to the negative ion. The relative stability of the ion-molecule complexes for ammonia, water, and SO2are in reverse order for the two ions. In terms ofmagnitude, after the first ligand addition, the enthalpy change for the clustering of SO2onto C1- is much smaller than for Na+. Many of these trends are in agreement with expectations from electrostatic considerations. For instance water, sulfur dioxide, and ammonia have similar dipole moments. However, when the dipoles are aligned in the electrostatic field of the ion, the small quadrupole moments ofwater and the considerably larger one of ammonia are repulsive to a negative charge and attractive to a positive one. In the case of sulfur dioxide, the situation is reversed. Thus, the ion quadrupole interaction is consistent with the different ordering of the first bond energies for water, ammonia, and sulfur dioxide between the positive Na+ and the negative C1-. A number of authors have utilized thermochemical data to interpret trends and expectations concerning ions in liquids. The interested reader is referred to a few relevant references (see Schuster et a!., 1977;Kebarle, 1974; Castleman, 1979a, 1982a; Castleman et al., 1982a; Taft ef al., 1978; Watts and McGee, 1976; Murrell and Boucher, 1982). In the case of large clusters, a relationship has been observed for negative ions similar to that for positive ones, where the ratio of solvation enthalpy to the summed gas phase values of the cluster enthalpies is almost independent of systems beyond five or six molecule additions (Lee ef al., 1980a). The finding suggests that only five or six bond energies of a given ligand to an ion are needed to predict the solvation energy in the liquid phase. The importance of accounting for interaction among the ligands and solvent molecules was discussed in a recent article by Arnett ef al. (1972). Castleman ( 1979a) has considered this effect for the ion OH- solvated in water and clustered with CO, to form HCO,. Through a complete Born-

136

T. D. Mark and A . W. Castleman, Jr.

Haber cycle analysis, a direct analogy to the bond energies between OH- and CO, in the gas phase and interactions of OH- and CO, in the aqueous phase is made. A few Systems of cations bound to ligands have been investigated where one of the constituents composing the cluster is inorganic and the other is organic. The investigations include Larson and McMahon (1 983), Arshadi et al. ( 1970),Yamdagni et al. (1973e), Yamdagni and Kebarle ( 1971,1972), Wlodek et al. (1980, 1983a,b), Keesee et al. (1980c), Kebarle (1977), and Luczynski et al. (1978). As mentioned earlier, Kebarle (Davidson et al., 1979b; Yamdagni and Kebarle, 197 1) have reported a relationship between the basicity of A- and the hydrogen bond in the monohydrate A-.H,O complex which showed that the strength of the hydrogen bond increases with the gas phase basicity of A-. Similarly, it was found that the hydrogen bond in BH+.H20increases with the gas phase acidity of BH+. These relationships for the bonding of a water molecule in terms of the basicity of A- or the acidity of BH+ provided an explanation on a one-molecule solvation basis for the attenuation mechanisms observed in the solution phase. Thus, it was suggested that a substituent that increases the acidity of AH decreases the basicity of A- and therefore decreases the hydrogen bonding in A-.H,O. As with the cation clusters there is much to be learned about solutions of organic constituents.

VI. Other Properties It is outside the scope of this article to give a detailed review of some of the other gas phase properties of cluster ions, i.e., including reactivity, recombination, mobility, and diffusion. Therefore, this section will be only illustrative rather than exhaustive. Moreover, as indicated below, some of these properties have not been studied and much more work needs to be done before a general overview of the subject is warranted. A. REACTIVITY Association reactions, and their backward analog and declustering reactions (collision-induced dissociation), have been discussed already in Sections III,B and IV,B. Ligand switching reactions (see also Albritton, 1978; Dotan et al., 1978), in general (Bohme, 1982), proceed rapidly ( k 3 cm3/sec) when exothermic. Depending on the energetics of solvation, the successiveexchange may come to a halt at some intermediate stage of mixed

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137

solvation (e.g., see Smith er al., 1981). Here we will only consider reactive (cluster)ion - molecule reactions, including charge transfer, proton transfer, and ion - atom interchange processes. According to a recent review by Ferguson ( 1982), reactivity of cluster ions is an area in which very little research has been done. Studies have been reported by Good et al. (1970a,b), Puckett and Teague ( 197 1 a,b), Fehsenfeld et al. ( 197 1 a,b, 1976, 1978), Fehsenfeld and Ferguson ( 197 1,1973,1974),Young and Falconer ( 1972), Howard er al. (1974), Nestler et al. (1977), Smith etal. (1978b, 1980, 1981), Sieck (1978), Rakshit and Warneck (1979, 1980a,b, 1981), Bohme er al. (1979, 1982), Viggiano et al. ( 1 980), McCrumb et al. (1980), McCrumb and Warneck (1981), BohmeandMackay(l981),Hierl eral. (1981),Adamseral. (1982), Mackay et al. ( 1982), Fahey et al. ( 1982), Rowe et al. ( 1982), Kleingeld and Nibbering (1983), and Henchman er al. (1983). For information on the reactivity of Fe(C0): species, see Foster and Beauchamp (1975). The effect of clustering on the reactivity of ions is important for an understanding of ion chemistry in plasmas, as well as solvation chemistry in general (see McIver, 1980). Smith er al. (1978b) found that, for example, association of N, to N r does not seriously modify the reactivity of the N t except that it reduces somewhat the energy available for reaction. The most common mechanism apparently is direct charge transfer, usually followed by fragmentation with the nitrogen -nitrogen bonds in the reacting ions remaining intact [see also reactions of (CO,): observed by Rakshit and Warneck, 1979, 1980a, 198 11. Similarly, for ligand exchange (see above), some (nonresonant) charge transfer processes of OF(H,O), ,and reactions of hydrated hydronium ions, increased hydration of the reactant ion does not significantly decrease the reaction constant (Fehsenfeld et al., 1978; Viggiano er al., 1980; Fahey et al., 1982; Bohme, 1982). See also isotope exchange reactions studied by Smith et al. (1980) and Adams et al. (1982). In contrast, for some (resonant) charge transfer processes (Fahey el al., 1982) and in several ion-atom interchange and/or nucleophilic displacement reactions (Fehsenfeld and Ferguson, 1974; Fehsenfeld er al., 1976; Bohme and Mackay, 198 1; Hierl er al., 198 1; Mackay er al., 1982; Ferguson, 1982; Henchman et al., 1983), increased solvation of the reactant ion resulted in a significant decrease in reactivity (see also results obtained by Howard et al., 1974, for associative detachment of negative ions, in which the reactivities are also reduced by clusteringwith H20). Similarly, Bohme er al. ( 1979) found that stepwise hydration leads to a decrease of the reaction rate constant for proton transfer from H30+,e.g., to H,S with a concomitant change in AGO (see also Bohme, 1982; Bohme et al., 1982, for solvated proton transfer reactions). In case of nucleophilic displacement reactions (e.g., see Fig. 14) these striking results may be interpreted, according to Bohme and Mackay (198 I), and Henchman et al. (1983), in terms of the

T. D. Mark and A . W. Castleman, Jr.

138

10-0

I

1

1

1

I

1

I

1

\ n

I

I

I

n

FIG. 14. Reaction rate constants for gas phase nucleophilic displacement reactions of solvated anions with methyl chloride and methyl bromide as a function of the extent of solvation, after Bohme (1982).

qualitative model developed by Olmstead and Brauman ( 1977), and Pellerite and Brauman (1980). On the other hand, an enormous reactivity enhancement has been found by Rowe et al. (1982) for a new class of ion-catalyzed reactions between neutrals occumng in cluster ions in which the central ion does not form chemical bonds, e.g., the rate constant for the homogeneous gas phase reaction of N205with NO is smaller than cm2/sec,whereas the rate for this reaction on alkali ion cluster is increased at least seven orders of magnitude in the case of Na+ as central ion, and in excess of nine orders of magnitude in the case of Li+ (see also Kappes and Staley, 198lc). Other cluster ion reactions have been reviewed recently by Dillard ( I 973),

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Good (1 979, Smirnov (1 977), Smith and Adams (1980), Ferguson (1982), and Bohme (1982) (see also compilation of ion-neutral reaction rates by Albritton, 1978). See Cumming and Kebarle (1978) for a discussion of entropy changes in proton transfer reactions involving negative ions.

B. RECOMBINATION Although a large number of results is available concerning electron-ion and ion-ion recombination (Bardsley and Biondi, 1970; Mahan, 1973; Moseley et al., 1975b; Flannery, 1976; Smith and Adams, 1980; Brouillard and McGowan, 1983), until recently few reliable data were available on cluster ions. Smirnov ( 1977) and recently Smith and Adams (1980, 1982, 1983b) have reviewed ion cluster recombination data of environmental interest up to 1983. It was found by Smith and his colleagues (Smith et al., 1976, 1978a, 198lb; Smith and Church, 1977) that the binary rate coefficient for mutual neutralization is remarkably independent of the complexity of the recombining ions (- 6 X lo-* cm3/sec), more variation occurring with temperature than with size of the cluster ion. This is in contrast with electron - ion dissociative recombination (as studied for cluster ions by Kasner et al., 1961; Kasner and Biondi, 1965, 1968; Weller and Biondi, 1968; Gerard0 and Gusinow, 1971; Sauer and Mulac, 1971; Wilson and Armstrong, 1971;Plumber al., 1972; Leuet al., 1973a,b;Huanget al., 1976, 1978; Whitaker et al., 1981a,b; MacDonald et al., 1983) where, e.g., the results showed a marked increase in n for the cluster series H30+(H20) cm3/secat n = 6). Further data are required especially for (reaching large clusters. It is interesting to add here some recent studies on electron - Hirecombination yielding cross section functions for the electron-induced dissociation of H t and H; (Peart and Dolder, 1974; Mathur ef al., 1978, 1979). Moreover, E. E. Ferguson (personal communication) and, independently, Coffey and Mohnen ( 1971) have suggested that in some cases the product of the recombination of positive and negative cluster ions upon interaction might lead to zwitterions which would be electrically neutral overall, but have separated charges contained within. This proposal has been extended recently by Arnold and Fabian (1980) to include the growth of multiple ion clusters into a particle. Recent experimental results on clustering between nitric acid and water (Castleman et al., 198la; Kay et al.. 1981) indicate that ion pair formation does occur in a manner analogous to that anticipated by the above authors.

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Studies of the mobilities and diffusion coefficients of ions in gases are an important source of information on ion-neutral interactions, e.g., interaction potentials, collision frequencies, etc. Atomic and simple molecular ions have been extensively studied and results were reviewed some time ago by McDaniel and Mason (1973) and Loeb (1973). However, there is much less work on cluster ions, because the controlled production and detection of cluster ions require somewhat different experimental conditions than those usually available in drift tubes used for those studies. In addition, reactions occurring in the drift tube (e.g., clustering, declustering, switching) may obscure the true nature of the observed phenomenon [e.g., see for the case of (N,)f: Keller et al., 1965; Varney, 1968; Moseley et al., 1969; Varney et al., 1973; Sejkora and Mark, 1983; Sejkora et al., 1983; Mark and Mark, 19841. Recent cluster ion studies (available data up to 1973 have been reviewed by McDaniel and Mason, 1973, and by Loeb, 1973) on mobilities and diffusion coefficients include investigations by Young et al. (1970), Young and Falconer ( 1972), Bricard et al. (1 972, 1977), Elford and Milloy ( 1974), Huertas et al. (1974), Golden and Frain (1975), Burke and Frain ( 1975),Dee ( 1976),Bierbaum et al. ( 1976),Helm ( 1976a- c), Dotan et al. ( 1976, 1977a), Nestler and Warneck (1977), Nestler et al. (1977), Alger et al. (1978), Helm and Elford (1978), Rakshit and Warneck (1979), Hodges and Vanderhoff (1980), Takebe et al. (1981), Jowko and Armstrong (1982), Headley et al. (1982a), Sejkora et al. (1983), Girstmair et al. (1983), Schlager et al. ( 1983), Bohringer and Arnold (1983a), and Mark ( 1984c). As to be expected, mobilities (e.g., see Fig. 15) and diffusion coefficients decrease with the size of the cluster (see also Kilpatrick, 1971; Karasek et al., 1971), however, it should be pointed out that charge exchange reactions can reverse this trend [e.g., see results on (N,):for n = 1 and 2 by Varney, 1968; Moseley et al., 1969;Sejkoraetal., 1983;Girstmairetal., 1983)orresultson(He):forn= 1 to 4 by Helm (1976a). According to Schlager et al. (1983) and others, the observed zero-field mobilities of cluster ions are considerably smaller than the calculated Langevin mobilities. According to these authors this indicates that for cluster ions the polarization effects are relatively unimportant compared to elastic scattering processes (for a theoretical treatment see also Hagen et al., 1975). Moreover, according to Meyerett et al. (1980) and Keesee and Castleman ( 1983), not only the mass but also the effective radius of the ion is also expected to be a factor in determining mobility of large clusters. This could have implications on the common procedure (Bricard et al., 1977; Meyerett et al., 1980) of determining ion mass from mobility

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E /N (Td)

FIG.15. ReducedionmobilitiesofH,0+andH,0+-H20,(n = 0, 1,2)inHe(297K),after Dotan ef al. (1977a). 0.25 Torr (A&, 0.45 Tom (0,O).

spectra in the lower stratosphere(seealso ion mobility spectra measurements and size conversion by Hagen et al., 1975; Iribarne and Thomson, 1976; Northby and Akinci, 1978; Akinci and Northby, 1979; Akinci et al., 1980; Suck et al., 1983). Comprehensivereviews of the experimentaland theoretical work up to 1983 will be given in the forthcoming books by McDaniel and Mason ( 1984) and Lindinger et al. ( 1984).

ACKNOWLEDGMENTS This work was partially supported by the Fonds zur F6rderung der wissenschaftlichen Forschung (Austria). Moreover, the preparation of this articlewas made possibleby a sabbatical leave of T.D.M. as Visiting Professor in the Department ofchemistry, The Pennsylvania State University, with the support of the National Science Foundation, Grant ATM-82040 10. A.W.C. gratefully acknowledges the U.S. Department of Energy, Grant DE-AC02-82ER60055, the National ScienceFoundation, Grant ATM-8204010, and the Department of the Army, Grant DAAG29-82-K-0160,whose financial support enabled this article to be written. One of the authors (A.W.C) is especially indebted to Dr. Robert G. Keesee for helpful discussions during the course of writing this article, and both authors offer a special work of thanks to Ms. Barbara Itinger for the organizationofthe referencesand typing ofthis manuscript. T.D.M. gratefully acknowledgeshelpful discussions with Dr. Olof Echt, Univemitat Konstanz.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 20

NUCLEAR REACTION EFFECTS ON ATOMIC INNER-SHELL IONIZATION W. E. MEYERHOF Department of Physics Stanford University Stanford, Calforn ia

J.-F. CHEMIN Institut National de Physique Nuclkaire et de la Physique des Particules Centre d’Etudes Nuclkaires de Bordeaux-Gradignan Gradignan. France I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11. Survey of Nuclear Reactions . . . . . . . . . . . . . . . . . . A. Light Projectiles . . . . . . . . . . . . . . . . . . . . . . B. Heavy Projectiles . . . . . . . . . . . . . . . . . . . . . 111. Survey of Atomic Inner-Shell Ionization. . . . . . . . . . . . . A. Low-Z Projectiles . . . . . . . . . . . . . . . . . . . . . B. High-Z Projectiles . . . . . . . . . . . . . . . . . . . . . IV. United-Atom Effects of Nuclear Reactions. . . . . . . . . . . . A. Experiments with Low-Z Projectiles . . . . . . . . . . . . . B. Experiments with High-Z Projectiles. . . . . . . . . . . . . C. Positron Emission . . . . . . . . . . . . . . . . . . . . . V. Separated-Atom Effects of Nuclear Reactions . . . . . . . . . . A. Experiments with Low-Z Projectiles . . . . . . . . . . . . . B. Experiments with High-Z Projectiles. . . . . . . . . . . . . V1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Appendix A: Sketch of the Statistical Theory of Nuclear Reactions. VIII. Appendix B: K X-Ray Emission in Second-Order Distorted Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction The present article deals with one small area of overlap between atomic and nuclear physics, the influence of nuclear reactions on the ionization of 173 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

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atomic inner shells. Inner-shell ionization by ions typically involves MeV bombarding energies (1 MeV/amu corresponds to a velocity of 1.4 X lo9 cm/sec), for which a time description of the projectile motion is valid. Hence, the influence of a nuclear reaction on the atomic processes usually expresses itself as a time delay during the atomic collision at “zero” internuclear separation R . Here, the electrons of the collision partners overlap, i.e., they are in the united-atom (UA)’ configuration. As in most time-dependent perturbations, one would expect the largest effects on the electronic processes to occur if the nuclear delay times are comparable in magnitude with the relevant electronic times in the UA state. Typically, the former lie between and sec (Lane, 1960; Nolan and Sharpey-Schafer, 1979; Massa and Vannini, 1982). There are two important UA electronic times, orbiting times (transition energies) and innershell vacancy lifetimes (level widths) (Merzbacher, 1982a,b). Transition energies in the UA Kshell can exceed 1 MeV (Reinhardt and Greiner, 1977),so that orbiting times down to sec are available in the UA configuration [(energy in electron volts) (time in seconds) = 0.66 X The K-shell vacancy lifetimes can be as short as lo-’* sec (Anholt and Rasmussen, 1974; Soff and Greiner, 1981). Hence, the relevant electronic and nuclear time ranges overlap and conditions should exist under which nuclear reactions appreciably affect inner-shell ionization. This was recognized about 20 years ago (Gugelot, 1962; Ciocchetti et al., 1963; Ciocchetti and Molinari, 1965), but experimental proof had to await improvements in techniques and in theoretical understanding (Chemin, 1978; Blair et al., 1978). It is somewhat too early to assess the full impact of this field of research on atomic and nuclear physics, but certain benefits can be appreciated now, which provide the motivation for further investigations. In atomic physics, half-collision processes have been of interest, such as ionization during alpha and beta decay (Freedman, 1974; Walen and Briancon, 1979, because they impose severe tests on atomic ionization theories, which have not as yet been fully met (Anholt and Amundsen, 1982). In principle, UA ionization effects occurring in nuclear reactions provide information about the incoming half-collision ionization probability (Gugelot, 1962), which is complementary to the outgoing probability obtained in radioactive decay. As is discussed in Sections I11 and V, the separated-atom (SA) ionization probability depends on the interference, i.e., phase relationship, between the incoming and outgoing ionization amplitudes. This phase relationship is altered by the time delay introduced by a nuclear reaction at R = 0 (Ciocchetti et al., 1963). The abbreviations used throughout this article are CE, compound elastic; c.m., center of mass; CN, compound nucleus; DI, deep inelastic;DWBA, distorted wave Born approximation; ER, evaporation residue;IC, internal conversion; MO, molecular orbital; IAR, isobaric analog resonance; SA, separated atom; SCA, semiclassical approximation; UA, united atom.

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If the time delay is known, e.g., because of the presence of a nuclear resonance of known width, the absolute phase of the incoming ionization amplitude at R = 0 can be determined. Target electron capture by a projectile should also be sensitiveto a nuclear time delay at R = 0 (Horsdal-Pedersen ef al., 1982). Calculations of the expected effects have been made (Amundsen and Jakubassa-Amundsen, 1984).The effect has been detected (Scheurer ef al., 1984). It is possible that with the help of time delay effects one may be able to distinguish expenmentally whether capture is a first- or second-order process (Shakeshaft and Spruch, 1979; Briggs et al., 1982). For nuclear physics, the sensitivity of inner-shell ionization to nuclear time delay provides a tool for nuclear reaction time determinations down to sec. With rare exceptions (Phillips, 1964),these shortest times have not as yet been reached by other methods (Massa and Vannini, 1982;Nolan and Sharpey-Schafer, 1979). Lifetimes of unbounded nuclear states can also be determined. If these states are separated, the lifetimes can be obtained directly by nuclear determinations of level widths. But if the states are overlapping, nuclear methods are indirect (Ericson and Mayer-Kuckuk, 1966; Brody ef al., 198l), whereas inner-shell ionization allows a direct comparison of nuclear widths and inner-shell vacancy widths (von Brentano and Kleber, 1980; McVoy efal., 198I). On a more refined level, even correlations between overlapping nuclear levels (Brody et al., 1981) might be determined by inner-shell ionization studies (McVoy and Weidenmiiller, 1982; Koenig ef al., 1983; Reinhardt et al., 1983). In the last few years, theoretical studies of the time development of nuclear reactions with heavy ions by means of the Hartree-Fock approximation have become more prevalent (Davies et al., 1981; Stocker et al., 1982). One may speculate that details of the time development will be revealed in innershell ionization phenomena. A similar refinement is taking place in the study of crystal blocking patterns (Massa and Vannini, 1982). A fundamental link between atomic and nuclear physics has been established through the study of atomic positron emission from inner-shell vacancy decay in GeV heavy-ion collisions such as U U and U Cm (for recent reviews see Muller ef al., 1982b; Schwalm, 1984; Backe and Kozhuharov, 1984). The observation and interpretation of line structure in the positron spectra point to unusually long reaction times, implying unusual reaction mechanisms (Muller et al., 1982b; Rhoades-Brown et al., 1983; Hess ef al., 1984). To provide a self-sufficient review of nuclear reaction effects on innershell ionization, Sections I1 and I11 survey the general features of nuclear reactions and of atomic collisions, respectively. In both cases, different models apply to light and heavy projectiles. In Section IV, experiments

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involving U A ionization effects are discussed. Here, the essential “atomic clock” is the inner-shell vacancy lifetime. In Section V, experiments involving SA ionization effects are reviewed. In that case, the UA electronic orbiting time provides the atomic clock. A conceptual summary is given in Section VI. Some theoretical details are presented in Appendices.

11. Survey of Nuclear Reactions For the present purpose, it is useful to emphasize the time-sequence point of view of nuclear reactions. Nevertheless, the uncertainty principle sets limitations on the concept oftime intervals, ofwhich one must remain aware (Weisskopf, 1957; Feshbach, 1958, 1960; Yoshida, 1974). As mentioned, different nuclear models are useful for light and heavy projectiles. Typical projectile energies lie between - 1 and 10 MeV/amu in the ionization experiments reviewed in this article. As a measure of nuclear collision time, one can take the quantity (Gobbi and Bromley, 1979)

t, = R,/v, (1) where R, is the sum of the radii of the projectile and target nuclei and u1is the projectile velocity. Typical values of R, are of the order of I -2 X lo-’* cm and v1 lies between 2 - 5 X lo9cm/sec for the bombarding energies under discussion, giving t, = 2 - 10 X 10-22 sec. The reason for having different models for light and heavy projectiles becomes apparent if one compares the projectile de Broglie wavelength LI with R,. For MeV nucleons, L1 is of the order of 10-l2cm, i.e., the same as R,. Hence, the wave nature of the projectile is important in nucleon-induced reactions. For MeV/amu heavy projectiles, say, with mass number - 100,Al is of the order 1 O-I4crn, i.e., much smaller than R,. Hence, a classical view of heavy-ion induced reactions is approximately valid. Further details of these two cases are now discussed.

-

A. LIGHTPROJECTILES

Consider a target nucleus which is bombarded by a proton of several MeV energy. In analogy with the scattering of light, the scattering between the two nuclei can be treated by introducing an imaginary nuclear potential called optical potential. The parameters of this potential are weakly dependent on the energy of the projectile, on the mass, and on the atomic number of the target nucleus. The imaginary part of the potential accounts for the flux of

INNER-SHELL IONIZATION

177

the incident projectile wave which is absorbed by the target nucleus. The optical model has been very successful in reproducing the gross features of reaction cross sections induced by light projectiles (Hodgson, 1971). Nevertheless, some reaction cross sections display highly irregular behavior as a function of energy, which fails to be reproduced by the preceding scattering potential. The compound nucleus (CN) model of nuclear reactions explains this behavior by assuming that at some energies the composite system can be formed in one of its many excited states. Then, the dependence of the cross section on the projectile energy E will display sharp resonances. At low-excitation energy of the CN, the resonances are isolated and the lifetime of the CN is defined by

(2)

T, = h / r

where is the width of the resonance. At high-excitation energy, the compound states become broad and the density of levels is high (Gilbert and Cameron, 1965). The width of each resonance r may become larger than the mean level spacing between resonances D, so that the cross section at any particular energy is influenced by several resonances, depending on the spread ofthe projectile energy AI.If the mean energy of the beam is changed by AE, with AE G AI, a new set of compound levels is excited, but some of the previously excited levels still play a role in the decay of the CN. This leads to correlations in the reaction cross sections measured at E and at E AE, which induce fluctuations in the excitation function (Ericson, 1960).These fluctuations are characterized by a correlation coefficient C ( A E )and a coherence width r,,where

+

+

C ( A E )= r:/(r: A E ~ )

(3)

If one assumes that all the widths r of the excited compound levels are the same, r = r,, a mean lifetime of the CN can be measured in experimental situations where AZ S r

T, = tt/r,

(4)

A brief outline of the statistical aspects of nuclear reactions is given in Appendix A. In reality, this picture appears to be oversimplified. Time evolution functions based on S-matrix theory show that correlations between the resonance parameters, and correlations between the scattering amplitudes through the different decay channels, lead to nonexponential decay of the compound system (Yoshida, 1974; Yoshida and Yazaki, 1977; Luboshitz, 1977). This can be understood in the following model which is summarized in Fig. 1, based on another point of view of nuclear reaction mechanisms. The projectile wave function is first scattered by the optical potential

W. E. Meyerhof and J.-F. Chemin

I78

INDEPENDENT PARTICLE STAGE

COMPOUND SYSTEM STAGE

FINAL STAGE

Direct interaction Incident

Multiple collisions

D

CN a

I CN decay

Shape elastic scattering

CE scattering

FIG. I. Sequence of stages of a nuclear reaction initiated by 1 - 10 MeV/amu protons or neutrons. D = doorway state, CN = compound nucleus (adapted from Weisskopf, 1957).

(shape elastic scattering). The projectile which enters the target nucleus undergoes a sequence of collisions with the target nucleons, forming a chain of stages of increasing complexity with n particles and m holes. This includes doorway states and preequilibrium states. At the end, a fully equilibrated CN is reached. At each stage in the sequence, the composite system can decay in a particular channel, leading to a continuous distribution of decay times which generally is not exponential. By measuring mean decay times of the CN in the region of overlapping resonances, one can gain useful information about the meaning of interaction times in nuclear physics (Massa and Vannini, 1982) and about statistical properties of nuclear levels at high-excitation energy (see Section IV,A). B. HEAVYPROJECTILES

-

In 1 - 10 MeV/amu collisions between a complex projectile nucleus (2,, A , ) and a complex target nucleus ( Z 2 , A 2 )a, nearly classical description of the collision sequences is valid, as mentioned above. In contrast with few nucleon-induced reactions, the Coulomb and centrifugal (angular momentum) barriers play a decisive role (Lefort, 1980; Norenberg, 1980). Figure 2 sketches the main evolutionary paths which can occur if two complex nuclei collide (Lefort, 1976a,b, 1980;Gobbi, 1981 ). After contact, a nuclear composite is formed in which the radial component of the incident

179

INNER-SHELL IONIZATION

\

FIG.2. Stages in the evolution of 1 - 10 MeV/amu heavy-ion reactions. CN = compound nucleus, ER = evaporation product, F = fission composite, DI = deep inelastic composite, n = nucleon emission, y = gamma emission (adapted from Lefort, 1976a).

kinetic energy is dissipated, whereas most of the tangential component is available to set the system into rotation. Details of the radial energy dissipation mechanism and, particularly, its time scale are still a matter of controversy (Schroder and Huizenga, 1977; Riedel et af., 1979; Moretto and Schmitt, 198 1; Lefort, 1981; Gobbi, 1982; Huizenga er af.,1982). The predominant evolutionary pathway of the nuclear composite depends mainly on the Coulomb bamer and, to some extent, on the energy and the impact parameter of the collision. One can crudely characterize the nuclear barrier by the product Z,Z,, since nuclear radii vary only with the cube root of the mass. If Z , Z , 7 2000, at not-too-largeimpact parameters the system evolves to a CN. In heavy-ion reactions, the CN generally is formed at a higher excitation energy than in nucleon-induced reactions because of the higher bombarding energies involved. Hence, one would expect the CN to be quite short-lived (Fig. 3 and Appendix A). But, in heavy-ion reactions, even at relatively small impact parameters, large angular momenta can be imparted to the nuclear composite, lengthening the CN lifetime. Therefore, lifetimes up to sec should occur even in heavy-ion reactions. If Z,Z, 7 1000, nucleon evaporation from the CN nucleus is typical, producing so-called evaporation residues (ER) with Z and A not far from those of the CN. The ER are highly excited and decay to their ground states

-

180

W. E. Meyerhof and J.-F. Chemin

EXCITATION ENERGY (MeV )

FIG.3. Representative lifetimes ofa compound nucleusA = 180, with nuclear spinsJ = 10 and 60, respectively ()(from Lefort, 1976a). Reaction time in the deep inelastic nuclear reaction 8.2 MeV/amu a% 166Er(---)(from Riedel et al., 1979). The times are plotted as a function of the nuclear excitation energy.

+

by gamma emission. If Z,Z, 5: 1000, the CN is more likely to fission, approximately symmetrically. The fission products, also, are formed with high excitation and decay by nucleon and gamma emission. Fission ofthe CN can also occur after one, two, or more nucleon evaporations, in which case it is called second, etc., chance fission (Grant, 1976). Nucleon evaporation decreases the excitation of the CN and lengthens the lifetime against fission sufficiently, so that it can be determined by the crystal blocking method, for example (Gibson, 1975; Andersen et al., 1976b; Massa and Vannini, 1982). If the product Z , Z 2has the values near 2000 - 2500, the fission saddlepoint can be reached without going through the CN stage. This mode of evolution of the nuclear composite has been called “fast fission” (Grkgoire et al., 1980,1982; Ng6, 1982). If Z,Z, 5: 2000, the Coulomb and centrifugal barriers prevent the complete amalgamation of the nuclear composite and the system tends to evolve via a deep inelastic reaction. During the cohesion of the nuclei, an essentially random exchange of nucleons takes place, so that the product nuclei have Z and A distributions around Z , , A , and 2, , A 2 .Hence, one speaks of “projectile-like” and “target-like’’ products. The dissipation of a large portion of the incident kinetic energy, mentioned above, causes the product nuclei to be

INNER-SHELL IONIZATION

181

highly excited and to decay by nucleon and gamma emission, or even to fission. Model studies of deep inelastic reactions and comparison with experimental results indicate that the total excitation energy, called -Q here, is a function of the (nuclear) impact parameter b of the collision (Schroder and Huizenga, I977),

-Q=fQ(b)

(5) Maximum excitation, i.e., maximum dissipation of the initial kinetic energy, takes place at small impact parameters. As the impact parameter increases, -Q decreases until it reaches values close to zero at the impact parameter where the nuclei graze each other. By introducing the concepts of radial and tangential friction, which dissipate the respective kinetic energies, the trajectory of the center of the projectile nucleus with respect to the target nucleus can be calculated. One can then define a sticking or reaction time T as the excess time of the collision with respect to a pure Rutherford trajectory. One finds that T is also a function of b, with increasing values of T occurring as b decreases from its grazing value Deformation of the nuclei can have a considerable effect on this relationship (Schmidt et al., 1978). By combining Eqs. ( 5 ) and (6), one can establish a parametric relationship between -Q and T which can be compared to experiment (Section V,B). For example, Schroder et al. (1978) found

T = Y exp(l -Q IIP) (7) where pand y are constants. Figure 3 gives a theoretical result by Riedel et al. ( 1979). The reaction time T also enters in model calculations of the variances 4 and 05 of the 2 and A distribution of the reaction products. Simple diffusion models give the result typical of a random-walk situation (Gobbi and Norenberg, 1980; Moretto and Schmitt, 1981). Any model of deep inelastic nuclear collisions must obtain consistency among relations ( 5 )-(8) and, of course, with experiment. This has been accomplished for many, but not for all, deep inelastic reactions. Microscopic, time-dependent Hartree - Fock calculations confirm many features of heavy-ion collisions discussed above and indicate the expected time scales (Davies et al., 1981; Stocker et al., 1982). Other quantum me-

182

W. E. Meyerhof and J.-F. Chemin

chanical ab initio calculations have also provided insight (Maruhn et al., 1980).

111. Survey of Atomic Inner-Shell Ionization Since to date only K X-rays have been used in the experiments we shall discuss, this survey is limited to a discussion of K-vacancy production in atomic collisions. Typical projectile energies lie between 1 and 10 MeV/ amu. For such collisions, the semiclassical approximation (SCA) has been very popular, in which the nuclear motion is treated classically, but the electronic processes are calculated quantum mechanically (Bang and Hansteen, 1959; Briggs, 1976). The relation of the SCA to a full quantum mechanical calculation has been examined (Briggs and Taulbjerg, 1978). Although the SCA gives equivalent results to the full quantum mechanical calculations in most atomic collisions at MeV/amu energies, if a nuclear reaction occurs in the collision, the SCA can give misleading results (Blair et al., 1978; Blair, 1980; Anholt, 1984). This is particularly so for nucleon-induced reactions, where a full quantum mechanical calculation such as the distorted-wave Born approximation (DWBA) must be used (Feagin and Kocbach, 1981;Blair and Anholt, 1982; McVoy and Weidenmuller, 1982). On the other hand, it appears that for heavy-ion reactions, where a classical description of the nuclear motion is usually assumed to be valid, the quantum mechanical treatment reduces to the SCA (Heinz et al., 1983). In inner-shell processes, the motion of the electrons can be treated as independent in most situations (Briggs, 1976).As the projectile atom passes through the target atom, the time-varying electric field acts on the “active” ( K )electron and can cause an electronic transition from the Kshell to empty bound or continuum states. In the deexcitation of the K-shell vacancy, X rays or Auger electrons can be emitted; in super-heavy systems, positron emission can occur (Reinhardt and Greiner, 1977). If the deexcitation takes place during the collision, i.e., while the projectile and target electron clouds overlap, “noncharacteristic” or “molecular-orbital” X rays can be emitted. If the vacancy deexcites after the collision, “characteristic” or “separatedatom” X rays can be produced. A useful definition of the relevant atomic collision time, analogous to Eq. (I), is the expression l K = aK/vI

where aK is the Bohr radius of the UA system Z , i- Z,.

(9)

INNER-SHELL IONIZATION

183

Since the full quantum mechanical treatment of an atomic collision is quite complex, the SCA is used in the present survey of atomic collision models. Results of the DWBA are presented in Appendix B. One can write for the complete Schrodinger equation,

i h m l a t = H,,@(R,r,t) (10) where R is the internuclear separation, r the coordinate ofthe active electron, 0 the combined atomic and nuclear wavefunction and Hmtthe corresponding Hamiltonian. Then, the SCA assumes that the nuclear wavefunction can be factored out of @ and that the nuclear motion can be represented by the classical trajectory R(t) (Brigs, 1976). The resulting equation for the electronic wavefunction v/[R(t),r,t] is ihdy/lat = Hey/ [R(t),r,t]

(1 1)

Here, He is the electronic Hamiltonian

He = -(h2/2m)V,

+ V[R(t),r]

(12)

where I/ is the potential acting on the active electron. Two main models are in use. In the “adiabatic” or “molecular” model one solves Eq. (1 1) at each instant of time t, (13) E[W )lw = H,y/[R(t Lrl where E is the energy of the system if it would remain infinitely long at the internuclear separation R(t). In Eq. ( 13), y/ is then a normal diatomic molecular wavefunction. One assumes that in a collision the actual wavefunction w will evolve as (Schiff, 1968),

where the sum includes continuum states. In contrast with this molecular model, in the atomic model one assumes that the wavefunction of the active target electron vl(r,t) is only weakly perturbed by the projectile. Then, the Hamiltonian He is written H e = Ho(r)

+ Vl[R(t),rl

(15)

where H, is the Hamiltonian for the electron in the target atom,

EY = ~ o y / ( r )

(16)

u1 is the perturbing potential due to the projectile atom, and r is defined with respect to the target nucleus. For an unscreened projectile nucleus, u1 = -Z,eZ/IR(t) - rJ

(17)

184

W. E. Mej)erhof and J.-F. Chemin

In Eq. (16), E is the energy of the electron in the target atom. In this model, one assumes that in a collision the actual wavefunction will evolve as (Schiff, 1968)

Experimentally, one finds that the molecular model is a good approximation for atomic collisions with high-Z projectiles moving with velocities u, less than Bohr velocity v, of the higher 2 partner. The atomic model is appropriate for low-Z projectiles with u, < u, and for all collisions with u1 > u, (Meyerhof and Taulbjerg, 1977; Mokler and Folkman, 1978). Further details of these models are discussed below.

A. Low-Z PROJECTILES

For light projectiles, the condition Z, dt

(19)

where o is the transition frequency 0=(E,

+ E)/h

(20)

and EK is the K-electron binding energy. In the experiments to be described, the quantity under consideration is the

185

INNER-SHELL IONIZATION

ionization probability PK of the target K shell

PK = 2 ~-la”(w)l2dE The factor 2 has been introduced to account for the two K-shell electrons. To solve Eq. ( 19),several methods are used. One consists of expanding the interaction potential V , , Eq. (17), into its multiple components [Kocbach, 1976; Appendix B, Eq. (BS)]. This method has the advantage of allowing a clear separation between the properties of the atomic system and dynamics of the collision. For details of calculations we refer to several review papers describing atomic ionization by light particles (Meyerhof and Taulbjerg, 1977; Kocbach ef al., 1980; Andersen et al., 1982; Merzbacher, 1983). In connection with nuclear reaction effects on ionization, two peculiar effects of the atomic model may be noted, although for simplicity they are neglected in the remainder of this article. First, it is convenient to use the atomic wavefunction v/ of the active electron appropriate to the charge 2,. But, in collisions in which a projectile enters the nucleus, the wavefunction should be that ofacharge 2, Z,. This change of charge alone can produce an additional term in the ionization amplitude, Eq. (1 9), sometimes called adiabatic “shake off” or “shake up” (Kocbach, 1978). The correction affects only the monopole term (Andersen ef al., 1976a).This term alone constitutes the entire transition amplitude if the time-dependent perturbation acts suddenly, i.e., in a shorter time compared to the reciprocal of the transition frequency (Merzbacher, 1983), as is the case of ionization accompanying beta decay. Second, the motion of the active electron should be referred to the center of mass of the colliding system, but it is often more convenient in the evaluation of Eq. ( 19)to refer it to a coordinate system centered on the target nucleus. In such a noninertial frame of reference, non-Newtonian forces arise. This results in an additional term V, in V , in Eq. (15), which is the potential due to the force of the acceleration of the target nucleus when struck by the projectile (Ciocchetti and Molinari, 1965; Rose1 et al., 1982). The term VRcan be written as (Amundsen, 1978)

+

VR = (rn/M,)r

. VR V,(R)

(22) where rn is the electron mass, M, the mass ofthe target nucleus, and V,(R) is the potential interaction between the target nucleus and the projectile including the Coulomb and short-range nuclear potentials. The form of VR suggests that this term should play an important role only in close collisions and that in this case only the dipole part of the matrix element should be affected. The resulting net effect of VR on PK is the following. For slow and close collisions, the “recoil effect” reduces the ionization probability com-

186

W. E. Meyerhof and J.-F. Chemin

pared to a nonrecoil calculation (Chemin et al., 1978). For fast or distant collisions, the recoil effect enhances the ionization probability. If a nuclear reaction takes place in the collision, one important quantity is the ionization probability in the incoming (or outgoing) part of the collision, which we call Pi,, pi, = 2

:1

del-(i/h)

eim'

(y+lv,lyo)dt 12

(23)

Using the multipole expansion of Vl, Eq. (23) can be written as a sum of the different multipole ionization amplitudes bl(E,e) excited in the collision

For K-shell ionization, 1is just the angular quantum number of the electron in the final state. A comparison between Pi, (or Po,,) and PK in collisions induced at the same energy and zero impact parameter would be very interesting. Unfortunately, experimental situations in which Pi, and PK could be compared occur very rarely. The best comparison comes from Po,, measurements associated with alpha decay of polonium isotopes and PK measurements at the same energy and nearly zero impact parameter in helium scattering on lead (Lund et al., 1977). Depending on the projectile velocity, the ratio (PK/Pout) vanes from 2 at low velocity to 3 at higher velocity. Information concerning the ratio PK/Pincan be found by comparing the values of the ionization probability in (p,y)or (p,n)nuclear reactions with the values of PK in ( p , p )scattering (Chemin, 1978;Duinker, 1981;Anholt, 1984)(see Section IV,A).

B. HIGH-ZProjectiles Theoretically, the solution for the expansion coefficients A,(t) in Eq. (14), which determine the transition probabilities relevant in the present context, proceeds in two steps. First, one has to solve Eq. ( 13)for the molecular orbital (MO) energies E,,(R);second, one has to solve Eq. (14). For high-2 atomic systems, it is important that the Dirac equation be used, instead of Eq. ( 13). This problem was first solved for one-electron systems by Muller et al. (1 973) (see also Miiller and Greiner, 1976) and for many-electron systemsby Fricke et al. ( 1975).Figures 4 and 5 show the energy levels of the lowest MO for the one-electron systems 531 , 9 A ~and 92U 92U, respectively. One special feature of the latter system is the fact that, as R 0, IE(R)I continues to increase, whereas for lower 2 systems the binding energiesbecome constant. The increase ofJE(R)lisdue to the very large (relativistic) contraction of the

+

+

-

187

INNER-SHELL IONIZATION

/-=-

I

Iw

1

I

+

FIG.4. Binding energies of one-electron molecular orbitals for ,rlI ,&u as a function of the internuclear separation R. The Isu-ionization amplitude A ( t ) is indicated, as well as (a) compound nucleus, (b) molecular orbital, and (c) separated atom X-raytransitions (adapted from Muller and Greiner, 1976).

electron wavefunction for high-Z systems as R 40 (Reinhardt and Greiner, 1977). For lower 2 systems, the wavefunctions stabilize as R + 0, a feature already noted in the early work of Bates (Bates and Carson, 1956). The contraction of the wavefunction for high-Z systems affects not only E(R), but also the ionization probability of the MO, which can become many orders of magnitude larger than extrapolations from nonrelativistic scaling laws would predict (Betz et al., 1976).

I

4

+

FIG. 5 . Binding energies of one-electron molecular orbitals for U U as a function of the internuclear separation R. For R 7 35 fm,the Isu molecular orbital dives into the negative-en(from Muller and Greiner, ergy Dirac sea. Finite nuclear sizes (- . -), point nuclei (-) 1976; Betz, 1980).

188

W. E. Meyerhof’and J.-F. Chernin

The ionization probability is computed from IAn(w)I2[Eq. (14)] integrated over all electron kinetic energies E in the continuum

In first-order perturbation theory, A,(t) is given by (Schiff, 1968; Briggs, 1976) A,(t) = -

where

1.

dt (w,lalatjwo) exp[ i

[

w,(t’)dt’

3

(26)

+ l~o“)llJ/t2.

(27) The subscript zero indicates the initial state. For high-Z systems, the abovementioned contraction of yoas R + 0 profoundly affects the magnitude of A , ( a ) through the size of the overlap integral ( y,ld/dtlwo) (Betz et al., 1976). The operator dlat can be written (Briggs, 1976) wco = (E

alat = ka/aR - i l k , (28) where 19is the angle between R and the incident projectile velocity and L , is the component of the orbital momentum perpendicular to the collision plane. The operators a/aR and L , and the wavefunction w have to be evaluated in the frame rotating with the molecular axis R.They give rise to radial and rotational coupling matrix elements, respectively. In collisions involving nuclear reactions, the rotational coupling can usually be neglected. The molecular model automatically contains the adiabatic shake-offterm mentioned in Section III,A because the wavefunction is always appropriate to the total nuclear charge. The model also contains the recoil term, because the wavefunction is referred to as the center of charge, which, usually, is very close to the center of mass (Amundsen, 1978; Jakubassa and Amundsen, 1979). Figure 4 indicates the ionization amplitude, Eq. (26), as well as three types of X rays which are important in the subsequent discussion: (a) CN X rays emitted during nuclear sticking (b) MO X rays emitted on the way into and on the way out ofthe collision (c) SA X rays emitted after the collision

-

Typical atomic collision times t K [Eq. (9)] are shorter than lo-’’ sec, but K-vacancy lifetimes are longer than sec (Anholt and Rasmussen, 1974;Mokler and Folkmann, 1978; Soff and Greiner, 1981). For this reason, it is usually possible to separate the theoretical treatment of CN and MO X-ray emission from that of SA X-ray emission (Anholt, 1982).

-

INNER-SHELL IONIZATION

189

IV. United-Atom Effects of Nuclear Reactions There are two different ways to investigate united-atom (UA) effects of nuclear reactions: one is by searching for compound nucleus (CN), i.e., UA, X rays; the other is by searching for CN positrons, also called spontaneous positrons (Reinhardt and Greiner, 1977). These terms are further explained below. The motivation for this work, for atomic physics, is the possibility of systematically studying the inner-shell ionization probability on the way into a collision (Kocbach, 1978) and, eventually, perhaps of detecting characteristic X rays from super-heavy systems(Rafelski, 1979).For nuclear physics, the interest lies in obtaining information about the nuclear level density, such as its spin dependence, from the CN lifetime (Ericson, 1960). Also, there is an indication from the CN positron studies that unusually long-lived nuclear compound modes might be formed in heavy-ion reactions (Muller et al., 1982a,b; Rhoades-Brown et al., 1983; Hess et al., 1984). Since different atomic and nuclear collision models apply to low-2 (light) and high-2 (heavy) projectiles, these two cases are discussed separately. In each case, a theoretical introduction is followed by a description of the experimental results. WITH A. EXPERIMENTS

L O W - z PROJECTILES

1. Theoretical Background

Consider a proton or alpha particle bombarding a target nucleus and forming a CN with a cross section 0,.If Pinis the K-shell ionization probability in the way into the collision, the product ocPi,is the cross section for making a compound atom with a K-shell vacancy. This statement is valid only if the nuclear and atomic processes can be considered as independent. Then, the excited compound atom can decay by the filling of the K-vacancy with a probability per time unit A, or by disintegration of the CN with a probability per time unit Ac (= l/Tc). The relative probability for K X-ray emission in the UA configuration is given W, &/(Ac + A,) (29) Here, W, is the fluorescence yield of the UA. Hence, the cross section for CN X-ray emission is (Gugelot, 1962)

w,Tc /( 1 + AKTc (30) A, is related to the lifetime of the K-shell vacancy in the UA T, and to the 0cx

= 0d’in

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W. E. Meyerhof and J.-F. Chemin

natural width of a K-shell vacancy in the UA rKby K-shell vacancy lifetimes range between and lo-'* sec, depending on the atomic number of the UA (Mokler and Folkmann, 1978; Anholt and Rasmussen, 1974)so that CN lifetimes can be determined in this time range by using relation (30), if all quantities but T, are known. The derivation of Eq. (30) assumes a single exponential decay law of the CN, which is very questionable in the case of CN excited in the region of overlapping levels. Furthermore, the assumption that atomic and nuclear excitations can be treated independently might not be correct in a quantum mechanical treatment which is required if the wavelength of the projectile is comparable to the nuclear radius (Section 11). Several authors have considered this problem (McVoy et al., 1981; von Brentano and Kleber, 1980; Anholt, 1982). A general treatment of K X-ray emission accompanying a nuclear reaction in the distorted wave Born approximation (DWBA) is outlined in Appendix B, following Anholt (1982). By combining relations (Bl) and (B9), one obtains the cross section for emission of a photon of frequency w, in the UA configuration of the system, coincident with a particle emitted into a solid angle dQ,

where bd(E,E)is the ionization amplitude, E the kinetic energy given to the electron, h ~ is ,the UA level separation, rxis the partial radiative decay width of the K vacancy in the UA, and f(8,E) is the nuclear scattering amplitude for emission of a particle at an angle 8, for a projectile of c.m. energy E. After integration of Eq. (32) over the photon frequencies ox,one obtains the cross section da,ldO for emission of a CN X ray coincident with a scattered particle. Different cases must be considered. First, if one assumes thatf(8,E) is the nuclear scattering amplitude for exciting an isolated resonance, f(8,E) consists of a direct partfD(8E) and a resonant part which is described by a Breit - Wigner function,

f ( e , E ) =fD(e,E) +fm (e,E) f E s - ( ~ R - E - ir/2)-1

(33) (34)

The resonance width r is related to the mean lifetime T, by Eq. (2). We assume thatfD(8,E) remains constant over the energy interval of interest, of

191

INNER-SHELL IONIZATION

the order of r. Inserting Eqs. (33) and (34) into Eq. (32) and integrating over the photon frequency, one obtains (for two K electrons),

If r >> E , the nuclear amplitude can be taken out of the integral, giving the result where we have used Eq. (24) and the relation If Eq. (36) is integrated over dQ, after replacing r, by its value W'I',, one obtains the Gugelot formula Eq. (30) with oC= ac(E- EK).Second, one can assume that the CN is excited to the region of overlapping resonances by a beam for which the energy resolution AZ is larger than the width r of CN states. The observed CN X-ray cross section now is an average over the many excited resonances. We have seen (Section I1,A) that the mean decay time of the CN in the statistical model is related to the coherence width of the fluctuations rc[Eq. (4)].In this model, the amplitudesf, at two different energies E and E AE are related by (Ericson, 1960)

+

The same manipulation of Eq. (32) as used previously, but averaging over the beam energy resolution, leads to an average cross section (daJdQ) given by Eq. (36) with rcin place of r. 2. Experiments with CN X Rays

Two groups have been successful in detecting CN X rays from proton-induced reactions. The systems which have been analyzed are p Io6Cdat 10 and 12 MeV (Chemin, 1978; Chemin et al., 1979) andp 112Snat 10and 12 MeV (Rohl et al., 1979, 1981). In both cases, the experimental techniques were similar. The X rays were detected in Si(Li) detectors in coincidence with inelastically scattered protons detected at backward angles. After subtraction of random coincidences, a net X-ray coincidence spectrum is obtained. Such a spectrum is shown in Fig. 6 forp Io6Cdcollisionsat 12 MeV. A peak due to the In K, X ray, the UA X ray, is clearly visible in the spectrum. The detection of X rays at the energy of the UA X ray does not unambigously signal the emission of CN X rays. The CN formed in the reaction can decay by neutron emission, leading to an excited residual nucleus, with the same nuclear charge as the CN. The residual nucleus can decay by

+

+

+

W. E. Meyerhof and J.-F. Chemin

192

500

I

1

I

1

400 J

w

2

$ Y In

2

300

200

3

0 V

I00

19

21

23 X-RAY

25

ENERGY

27

29

(keV)

FIG.6. X-ray spectrum from 12 MeV proton bombardment of IwCd in coincidence with inelastically scattered protons ( 5 . 5 - 10 MeV energy window). Accidental coincidences have been subtracted. The In K, line is due to CN formation; Cd KO/ lines are due to decay by IC of the residual Cd nuclei (from Chemin er a/.. 1979).

internal conversion (IC), producing X rays whose energy is identical to those of the CN (Deconnink and Longequeue, 1973; Meyerhof et al., 1979; Karwowski et al., 1979). This problem can be avoided only if very neutron-deficient CN are formed, in which case neutron emission is very much reduced due to the large binding energy of neutrons in these nuclei. Furthermore, the requirement of a coincidence between an X ray and a charged reaction product will suppress the neutron effects. The requirement of using neutron-deficient CN severely limits the number of systems in which CN X rays can be observed. The area under the UA X-ray line (Fig. 6 ) is used to determine the mean lifetime T, of the CN with the help of Eq. (30). The corresponding results are given in Table I. The large uncertainties associated with the lifetimes are due to the fact that the UA K, X-ray line lies at the foot of an intense target X-ray peak and on a continuous background. The target X rays and background are both caused by the deexcitation of residual nuclear states formed after proton evaporation from the CN, either by IC or by gamma rays. The latter produce a Compton continuum in the Si(Li) detector. These background processes also put severe limitations on the detection of UA X rays in the heavy-ion-induced reactions, discussed in Section IV,B. In interpreting the CN X-ray cross sections and extracting the CN lifetimes by means of Eq. (30), Pi, must be known. The classical ratio PK/Pi,,= 2 has been used to determine the T, values given in Table I. Detailed estimates of Pi, are reported by Rohl et al. (1 98 1). The lifetime values in Table I can be compared to the theoretical values given by the statistical model of nuclear reactions. For strongly overlapping

193

INNER-SHELL IONIZATION TABLE I

COMPOUND NUCLEUS LIFETIMES MEASURED IN PROTON-INDUCED REACTIONS

Reaction 10 -MeV p 12 -MeVp 10 -MeV p 12 -MeVp

+ Io6Cd + Io6Cd + 112Sn + I12Sn

Compound nucleus and excitation energy

T, (measured)

lo71n*(14 MeV) 1071n*(16MeV) I1,Sb*(13 MeV) lYjb*(15 MeV)

(6.5 f 4)10-17 (5.0 k 2.5)10-17 (4.0 f 3.8)10-17 (3.4 f 2.0)10-17

T, (calculated) Reference

(=) 11 X 5X

Chemin et al. (1979) Chemin et al. (1979) Rohl et al. (1 979) Rdhl et al. (1979)

CN levels which occur in the nuclear reactions investigated to date, the statistical theory, briefly sketched in Appendix A, predicts that the width TJ of compound states of spin J at an excitation energy in the CN, E *, is given by (Ericson, 1963; Vonach and Huizenga, 1965),

rru

rJ(E*)= ( ~ ~ ) - ' D J ( E * ) v, ,s

P(E:,K)wv)dE,

(39)

where the sum extends over all the decaying channels from the state Jof the CN, including the different types of evaporated particles v, with different nuclear angular momenta 1, spins S,and kinetic energies Ev ranging from zero to em, leaving the residual nuclei in different states of excitation energy E: and spins K. The different decay channels are characterized by the transmission coefficients T,(Ev)which can be calculated from an optical model potential. The term p(E:,K) is the density of levels of spin K at the excitation energy E: of the residual nucleus. D J(E*)is the spacing of the CN levels of spin J a t the excitation energy E* such that theDJ(E*) = p(E*,J)-'. p(E*,J ) has approximately an exponential dependence on E* and J(Gi1bert and Cameron, 1963,

p(E*,J) = C exp 2(aE*)" exp[-(J+ f)2/2c93

(40) where 0 is the spin parameter related to the moment of inertia of the CN, a is the level density parameter, and Cdepends on E*, J, a, and c.The influence of the spin of the CN on the width TJ is illustrated in Fig. 3. Experimental results must be compared to the average width r corresponding to the different spins of the excited levels

where 0 is the cross section for the formation of a CN with a spin J. The theoretical values of T, = h/T are reported in Table I. The agreement with the experimental values is rather good.

W. E. Meyerhof and J.-F. Chemin

194

3. Other UA Efects Besides leading to emission of CN X rays, ionization of the inner shells of the target atom on the way into a nuclear collision can also lead to effects on the nuclear cross section. In the reaction 4He(a,a)4He,in the neighborhood of 184 keV, several structures have been observed and attributed to collisions in which the projectile produces an inner-shell vacancy on the way into the collision and a nuclear reaction (Benn et al., 1968).The cross section for this process duuA/dQ can be evaluated with Eq. (35), except for the lifetime factor Eq. (29), which, here, is replaced by unity,

EK- €)I2

(42)

If, in Eq. (34), - 4Ap(r)

IKi,1s22p6,C!&p,yt>

IKf, 1s22p5,CJA,,rt, >

- xy (K/,R) 4fW

]

(B2)

respectively. The term CzA, specifies the number of electrons in the continuum state, y; specifies the number of photons emitted, K is the c.m. momentum, x*(K,R) are projectile (ejectile) wavefunctions, and 4(r) denotes the electronic wavefunctions. The perturbing Hamiltonian V coupling these states is the sum of the Coulomb interaction V, (Eq. 17) and the radiative Hamiltonian H , . Introducing the Coulomb Green function G$, the second-order DWBA amplitude is given by (Messiah, 1970) TcAp

= (K/Ki)"2(2nhu)-'

(~74/IvI4tp) G$( 4ApI vIxt4i)

(B3)

The wavefunctions x* are given by a partial wave expansion (Blair and Anholt, 1982),

x* = 4n(KR)-'

C(i)'

Yg( K ) Y,,,,(R) e*i61F,(KR)

( B4)

m

where F,(KR) is the radial wavefunction of the nuclear Hamiltonian and S,is the sum of Coulomb and nuclear phase shifts for the Ith partial wave. The Coulomb potential is expanded in spherical harmonics of r and R,such that

W. E. Meyerhof and J.-F. Chemin

232

the ionization matrix element is given by (Kocbach, 1976)

(4ApIvcI4i) = [ 4 ~ / ( 2 1 +1)1"2 Vp(R)GA(R)

(B5) where G,(R) is a radial electronic matrix element. Omitting for simplicity the effects of polarization and angular anisotropy of emitted photons, which do not play any role, the radiative amplitude is written (hv)-'

( 4flqhbAp ) =

(B6) = rX/27r,where r x i sthe total KX-ray

with the normalization condition transition rate. Substituting Eqs. (B4)- ( B 6 )in Eq. (B3), T,,, could be evaluated after a cumbersome integration over the radial coordinate R. Blair and Anholt ( 1 982) suggest breaking this integral into two parts, for R smaller and larger than a critical radius R,. The radius R , is chosen such that

R, c R , R,, the Green function and projectile radial wavefunctions are replaced by their asymptotic values (Messiah, 1970). For R < R, , the decay processes correspond to UA or CN X-ray emission. The region R > R , accounts for SA X-ray emission. The matrix element T,,, can be evaluated using the relations connecting the momenta in the different states of the system (M, = reduced mass), (Kf

- K 2 ) / 2 M p= E ,

+ E,

for R < R , (K,?- K 2 ) / 2 M p= h(w - us;") for R > R , (K,? - K 2 ) / 2 M p= h(w - myA)

(B8)

To account for the decay of the 1s state, O, is taken to be complex hw,

= Ex - i r K / 2

where Ex is the transition energy of the K X-ray line and rK is the partial decay width of the 1s state for the considered transition. These quantities are evaluated in the UA configuration or the SA configuration depending on the value of R.

INNER-SHELL IONIZATION

TY$, corresponding to photon emission in the internal region R given by T$$

= (-

233

< R,

is

1)'d6,, 6: DUA(w,-

X [f(0,E- EK-E)-~(O,E-EK-E--,+

(B9)

where b,*is the ionization amplitude

b,*= L d R G,(R) exp(iijR) and

4R = (Ki - K)R + (q - qi)ln 2KR

q =Z,Zzez/hv

(Bll) The term f ( 0 , ~is) the nuclear amplitude for spinless particles defined by

f(0,~) = (2iKi)-l

7(21+ 1)P,(cos

B)(ezidt- 1)

The physical interpretation of Eq. (B9) is simple. The first term in the brackets is connected with a nuclear scattering energy (E - EK - E ) and corresponds to the creation of a vacancy in the 1s state which decays during the CN formation. The second term corresponds to the fraction of the 1s vacancies created in the collision which have decayed before the nuclear scattering occurs. In general, the latter term can be neglected, since the vacancy lifetime h / r Kis much longer than the atomic collision time [Eq. (9)]. From Eq. (B9), Eq. (32) can be obtained. Emission of target X rays is obtained in the same formalism by evaluating the matrix element Ts& for the region R > R,, Neglecting the emission of SA X rays in the incoming part of the collision, for the reason noted above, Tsf,, is given by TZf6,= DSA(0x - 0KSA )- 1 X [Diob,f(E) + (-l)'d,ob,*f(E-EK-~)l

(B13)

Evaluating the corresponding cross section, Eq. (57) is obtained.

ACKNOWLEDGMENT This work was supported in part by US. National Science Foundation Grants PHY-80I5348 and INT-8 1-06105.

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Miiller, U. ( 1984). GSI Report 84-7. Gesellschaft fur Schwerionenforschung, Darmstadt. Miiller, B., and Greiner, W. (1976). Z. Nuturforsch. 31a, 1. Muller, B., and Oberacker, V. (1980). Phys. Rev. C 22, 1909. Miiller, B., Rafelski, J., and Greiner, W. ( I 972a). Z. Phys. 257,62. Miiller, B., Rafelski, J., and Greiner, W. (l972b). Z. Phys. 257, 183. Miiller, B., Rafelski, J., and Greiner, W. (1973). Phys. Left. 47B, 5. Miiller, U., Reinhardt, J., Soff, G., Miiller, B., and Greiner, W. (1980). Z. Phys. A 297, 357. Miiller, U., Reinhardt, J., de Reus, T., Schliiter, P., Soff, G., Greiner, W., and Miiller, B. (1982a). In “X-Ray and Atomic Inner-Shell Physics-1982’’ (B. Crasemann, ed.), p. 206. American Institute of Physics, New York. Miiller, U., Aboul-El-Naga, N., Reinhardt, J., de Reus, T., Schliiter, P., Seiwert, M., Soff, G., Wietschorke, K. H., Miiller, B., and Greiner, W. (1982b). Lect. Notes Phys. 168, p. 388. Muller, U., Reinhardt, J., de Reus, T., Schliiter, P., Soff, G., Wietschorke, K. H., Miiller, B., and Greiner, W. (1983a). In “Quantum Electrodynamics of Strong Fields” (W. Greiner, ed.), p. 195. Plenum, New York. Miiller, U., Soff, G.,de Reus, T., Reinhardt, J., Miiller, B., and Greiner, W. (1983b). Z. Phys. A. 313, 263. Miiller, U., Soff, G., Reinhardt, J., de Reus, T., Miiller, B., and Greiner, W. ( 1984). Gesellschaft fur Schwerionenforschung, Darmstadt, preprint GS 1-84-10, Phys. Rev. C, in press. Natowitz, J. B., Namboodiri, M. N., Kesiraj, P., Eggers, R., Alder, L., Gonthier, P., Cerruti, C., and Allemann, R. (1978). Phys. Rev. Lett. 40, 75 I. NgB, Ch. (1982). Lecr. Notes Phys. 168, 185. Norenberg, W. (1980). In “Heavy Ion Collisions’’ (R. Bock, ed.), Vol. 2, p. I. North-Holland Publ., Amsterdam. Nolan, P. J., and Sharpey-Schafer, J. F. (1979). Repf. Prog. Phys. 42, 1. Phillips, G. C. (1964). Rev. Mod. Phys. 36, 1085. Rafelski, J. ( 1979). Personal communication. Rafelski, J., Muller, B., and Greiner, W. (1977). Z. Phys. A 285, 49. Rafelski, J., Fulcher, L. P., and Klein, A., (1978). Phys. Rep. 38C, 227. Reinhardt, J., and Greiner, W. (1977). Rep. Prog. Phys. 40,219. Reinhardt, J., Miiller, B., Greiner, W., and Soff, G. (1979). Z Phys. A 292, 21 1. Reinhardt, J., Soff, G., Miiller, B., and Greiner, W. (1980). Prog. Part. Nucl. Phys. 4, 547. Reinhardt, J., Miiller, B., and Greiner, W. (1981a). Phys. Rev. A 24, 103. Reinhardt, J., Miiller, U., Muller, B., and Greiner, W. (1981b). Z. Phys. 303, 173. Reinhardt, J., Miiller, B., Greiner, W., and Miiller, U. (1983). Phys. Rev. A 28, 2558. Rhoades-Brown, M. J., Oberacker, V. E., Seiwert, M., and Greiner, W. (1983). Z. Phys. A 310, 287. Riedel. C., and Norenberg, W. (1979). Z. Phys. A 290, 385. Riedel, C., Wolschin, G., and Norenberg, W. (1979). Z. Phys. A 290,47. Rohl, S., Hoppenau, S., and Dost, M. (1979). Phys. Rev. Left.43, 1300. Rohl, S., Hoppenau, S., and Dost, M. (1981). Nucl. Phys. A 369, 301. Rosel, R., Trautmann, D., and Baur, G. (1982). Nucl. Insfrum. Methods 192,43. Saboya, B., Chemin, J. F., Andriamonje, S., Scheurer, J. N., Roturier, J., Thibaud, J. P., and Bruandet, J. F. (1983). Unpublished results. Schadel, M., Kratz, J. V., Ahrens, H., Briichle, W., Franz, G., G a d e r , H., Warnecke, I., Wirth, G., Herrmann, G., Trautmann, N., and Weiss, M. (1978). Phys. Rev. Left.41, 469. Scheurer. J. N., Baker, 0. K., and Meyerhof, W. E. (1984). Submitted. Schiff, L. 1. (1968). “Quantum Mechanics”, Sect. 35. McGraw-Hill, New York. Schliiter, P.,de Reus, T., Reinhardt, J., Miiller, B., and Soff, G. (1983). Z. Phys. A 314, 297.

INNER-SHELL IONIZATION

239

Schmidt, R., Toneev, V. D., and Wolschin, G. (1978). Nucl. Phys. A311.247. Schroder, W. U., and Huizenga, J. R. (1977). Annu. Rev. Nucl. Sci. 27,465. Schroder, W. U., Birkelund, J. R., Huizenga, J. R., Wolf, K. L., and Viola, V. E., Jr. (1978). Phys. Rep. 45, 30 1 . Schwalm, D. (1984). I n “Electronic and Atomic Collisions” (J. Eichler, 1. V. Hertel, and N. Stolterfoht, Eds.), p. 295. North-Holland Publishing Co., Amsterdam. Schweppe, J., Gruppe, A., Bethge, K., Bokemeyer, H., Cowan, T., Folger, H., Greenberg, J. S., Grein, H., Ito, S., Schule, R., Schwalm, D., Stiebing, K. A., Trautmann, N., Vincent, P., and Waldschmidt, M. (1983). Phys. Rev. Lett. 51,2261. Shakeshaft, R., and Spruch, L. (1979). Rev. Mod. Phys. 51,369. Skapa, H. ( I 983). Ph.D. thesis. Faculty of Science and Mathematics, University of Heidelberg. Soff, G., and Greiner, W. (1981). J. Phys. B 15, L681. Soff, G., Reinhardt, J., Miiller, B., and Greiner, W. (1977). Phys. Rev. Lett. 38,592. Soff,G., Reinhardt, J., Miiller, B., and Greiner, W. ( 1979). Phys. Rev. Letf.43, 1981. Soff,G., Reinhardt, J., Miiller, B., and Greiner, W. ( 1980). Z. Phys. A 294, 137. Stocker, H., Cusson, R. J., Lustig, H. J., Hahn, J., Mahrun, J., and Greiner, W. ( 1982). Z. Phys. A 306,235. Stoller, Ch., Chemin, J. F., Anholt, R., Meyerhof, W. E., and Wolfli, W. (1983). Z. Phys. A 310, 9. Stoller, Ch., Morenzoni, E., Nessi, M., Wblfli, W., Meyerhof, W. E., Molitoris, J. D., Grosse, E., and Michel, Ch. (1984). Phys. Rev. Lett. 53, 1329. Thompson, W. J., Wilkerson, J. F., Clegg, T. B., Feagin, J. M., Ludwig, E. J., and Merzbacher, E. ( 1 980). Phys. Rev. Lett. 45,703. Tomoda, T. (1982). Phys. Rev. A 26, 174. Tomoda. T., and Weidenmiiller, H. A. (1982). Phys. Rev. A 26, 162. Towle, J . H., and Owens, R. 0. (1967). Nucl. Phys. A 100,257. Trautvetter, H. P., Greenberg, J. S., and Vincent, P. (1976). Phys. Rev. Left. 37, 202. Von Brentano, P., and Kleber, M. (1980). Phys. Lett. 92B, 5 . Vonach, H., and Huizenga, J. R. (1965). Phys. Rev. 132B, 1372. Walen, R. J . , and Bnancon, C. (1975). In “Atomic Inner-Shell Processes” (B. Crasemann, ed.), Vol. 1, p. 233. Academic Press, New York. Wang, W. L., and Shakin, C. M. (1970). Phys. Lett. 32B, 421. Weisskopf, V. F. (1957). Rev. Mod. Phys. 39, 174. Wigner, E. P. (1955). Phys. Rev. 98, 145. Wolschin, G. (1977). Nucleonika 22, 1165. Wolschin, G . (1983). Unpublished results. Yoshida, S. (1974). Annu. Rev. Nucl. Sci. 24, 1 . Yoshida, S., and Yazaki, K. (1977). Nucl. Phys. A 255, 173.

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II

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 20

NUMERICAL CALCULATIONS ON ELECTRON-IMPACT IONIZA TION CHRISTOPHER BOTTCHER Physics Division Oak Ridge National Laboratory Oak Ridge, Tennessee I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A. ScopeofWork . . . . . . . . . . . . . . . . . . . . . . . . B. Experimental Situation. . . . . . . . . . . . . . . . . . . . . C. Available Theories. . . . . . . . . . . . . . . . . . . . . . . 11. The Wave Packet Method. . . . . . . . . . . . . . . . . . . . . A. Time-Dependent Formalism . . . . . . . . . . . . . . . . . B. Numerical Methods . . . . . . . . . . . . . . . . . . . . . . C. Selected Results . . . . . . . . . . . . . . . . . . . . . . . . 111. Box-Normalized Eigenstates . . . . . . . . . . . . . . . . . . . . A. R-Matrix Formalism for Ionization . . . . . . . . . . . . . . B. A Pilot Calculation . . . . . . . . . . . . . . . . . . . . . . IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

241 241 242 242 246 246 250 253 260 260 263 265 265

I. Introduction A. SCOPEOF WORK

The purpose of this article is to describe recent calculations on the electron impact ionization of single-electron systems at energies near threshold, e.g., e + H(1s) -,e + e + H+

(1)

No attempt will be made to cover the whole field of electron impact ionization, but a selective account of other developments will help to give some perspective. Because of the difficulties in conventional approaches, we have concentrated on the time-dependent wave packet method described in Section 11. 24 I Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003820-X

242

Christopher Bottcher B. EXPERIMENTAL SITUATION

The total cross section for electron impact ionization of atomic hydrogen was measured early in the modern era of atomic collision physics by Boyd and Boksenberg,whose results have been little altered by subsequent investigators (e.g., Fite and Brackman, 1958; McGowan and Clarke, 1968). Recent work on total cross sections has emphasized the threshold region (Baum et al., 1981). Interesting variants of the straightforward experiment are the double photodetachment from H- measurements of Donahue et al. (1982) and the polarized electron-polarized atom measurements of Alguard et al. (1977). Impressive as these total cross section results are, it is the general view that increased understanding of mechanisms can only come from differential and coincidence experiments, which examine the energies and directions of the collision products. The most comprehensive differential measurements are those of Erhardt and his colleagues (Shubert et al., 1979) with helium targets, mostly at high impact energies. They have done enough at low energies ( 35 eV) to gain some insight into threshold behavior. The most subtle and original studies are those of CvejanoviC and Read ( 1974) on the angular and energetic distribution ofoutgoing electrons within an electron volt of threshold. Beyond the realm of “fundamental” experiments a vast body of data has built up on more complex targets, e.g., ions of interest in high-temperature plasmas (Crandall, 1983). Ionization of complex systems is frequently dominated by “indirect” mechanisms, e.g., the excitation of an inner-shell electron to form an autoionizing state (Pindzola et al., 1983). At high impact energies the interpretation of the ionization process simplifies,and becomes in practice a different subject (Inokuti, 1971; Bell and Kingston, 1974).

-

C. AVAILABLETHEORIES The essence of any calculation of Eq. (1) is to find a wave function which satisfies Schrodinger’s equation

( H - E )Y =0

(2) with appropriate boundary conditions. For an infinitely massive proton the nonrelativistic Hamiltonian is (atomic units will be used henceforth, unless otherwise stated)

H=

5‘”? + 0 3 + ($- )-; 1

-

(3)

243

ELECTRON-IMPACT IONIZATION

We denote the positions of the outgoing electrons by (rl ,r2) and their momenta by (k, ,k,). The first serious attacks on ionization at low energies (Peterkop, 1963; Rudge and Seaton, 1965) attempted to extend the Born approximation to three-body processes. They wrote the scattering amplitude as f(k,,k,)

=

I

dfl

d?, WH- E)@(k,,k,)

(4)

where 0 represents two ingoing Coulomb waves, @@I

,kz) = X(CI,-.klb l M

C 2 ,-kzIrz

1

(5)

c.

andX(C,klr)is a Coulomb wave associated with nuclear charge The essence of the problem is that 0 must be better than a simple product for Eq. (4) to represent a finite amplitude. The authors cited derived the famous screening condition,

If Y were exact, the amplitude would satisfy

(7)

f(kz $1 ) = (- 1)'f(ki , k ~ )

depending on the total spin S. With approximate wave functions, however,f may not behave properly under interchange of momenta. Applications of Eq. (4) have been limited by the complications of satisfyingEq. (6). But an even more serious problem is that of finding an approximation to Y meaningful in the ionization region. The next step, in an ideal world, is the generalization of Eq. (4) into a variational principle,

6f= -6

I

dr, dr, Yr ( H - E)Yf

where Yi and "',are solutions of Eq. (2) with ingoing waves in the initial and final channels, respectively. The initial function must contain outgoing terms

I

d3k,d3kzf(k,,k,) exp(ik, rl

+ ik,

rz)

+

+

X 6(E - k: - k;)

(9)

which can be equivalently expressed in hyperspherical coordinates (Morse

Christopher Bottcher

244

and Feshbach, 1957).In coordinate space we write

r, = p cos a,

r2 = p sin a

(10)

and in momentum space

k, = K cos p,

-

k, = K sin j?

Then Eq. (9)becomes (asymptotically, as p

(1 1)

00)

where F is related tof by some normalization factors. Thus, the variational trial for Yi should contain terms like Eq. (1 2),probably with F expanded in hyperspherical harmonics

A complete variational calculation of breakup has been camed to convergence for a model problem with short-range forces by Bottcher et al. ( 1974). They followed up the formal work of Lieber et al. (1972).Even in the short-range calculation, one point is very clear: the wave function cannot be uniformly represented by an expansion in hyperspherical harmonics. To obtain convergence a hybrid basis set had to be used, describing the wave function where pa is finite, but p 00, A variational calculation on the Coulomb problem is not yet possible for this reason. Coulomb distortion is omitted in Eqs. (9)and (12):the exponential in Eq. (12)should contain a complicated function G ( ~ , p , ato) account for this distortion. We can guess forms of G valid for a n/4 or a 0 but no uniform approximation is available. Progress has recently been made by Altick (1982)and Peterkop (1982). This last-mentioned point is at the heart of current controversy about the correct threshold law for Eq. (1). Wannier (1953)made three predictions about ionization near threshold based on a classical model.

-

-

-

as E + 0 (the ionization threshold), (1) The total cross section -AE where a -- 0.127. (2) The differential probability that the outgoing electrons have energies (E,E- E ) , per interval &, is independent of E . (3) Electrons are ejected in a range of relative angles to each other A 8 around 8 = n, where A 8 = BE1l4.

ELECTRON-IMPACT IONIZATION

245

This theory, which has been placed on a quanta1 foundation by Rau (197 l), Peterkop (197 1, 1982) and Peterkop and Liepinsh (198 l), rests on the assumption that all orbits for double escape can be traced back to a region around p = R , (Y = n/4,and 8 = a. The reaction zone radius R is rather ill defined but certainly exceeds E-I. This assumption is undoubtedly valid for ( ~ ( p m ) > some S(E ) .Thus the behavior described under headings (2) and (3) above should be expected for a > S.However, if 6remains finite ( >0) as E + 0, the possibility arises of another term in the scattering amplitude. Such a term has been derived by Temkin (1974, 1982)in his dipole theory. The predicted signatures of Temkin’s mechanism are as follows:

-

(1) The total cross section follows a modified linear law (bear in mind that this gives an amplitude added to the Wannier amplitude)

-

(2) The differential probability has an envelope (within which it may oscillate in a complicated way) rising as E 0. (3) For E E/2 in our plots. Figure 6c presents the corresponding probabilities for L = 1 and E, = 27 eV. The rapid falling off of P for E > 0 is due to the centrifugal barrier which makes it difficult for the incoming electron to transfer much energy to the ejected electron. In contrast with the L = 0 case, most ofthe ionization is at 8 = n/2, in accordance with recent work by Mar and Schlecht ( 1976)and Greene and Rau (1983) in Wannier theory for higher partial waves. When PLK(E,8)is integrated over E and 8 it is possible to extract the breakup probability by fitting the integrated probability to a function pL=ALE+ BL

(48)

The effect of the initial energy spread and contamination by excited states near a = 0 are entirely contained in BL so that the first term gives the cross section (Bottcher, 1982a).We found A , = 0.157, A I = 0.0238, A, = 0.0225 which leads via Eqs. (31) and (32) to a cross section (0.683E)lra~near threshold; Callaway and Oda ( 1979) found (0.665E)ng. When higher par-

257

ELECTRON-IMPACT IONIZATION

tial waves are added (Burke and Taylor, 1965) either estimate is within experimental uncertainty. More insight can be obtained from polar plots of EP(E,8) versus 8 for prescribed values of E and E . Figure 7 shows such plots for S waves in e H( 1s) ionization at three impact energies. The full lines correspond to E = E / 2 and the dashed to E = E/6. The angular correlations are striking in all cases, though more so when the ejected electrons have equal energies. The individual plots illustrate how the process changes as E increases. (a) E = -0.25: most of the wave packet is subthreshold so the probability is a single lobe around 6 = x; (b) E = 0.1: the enhancement around 6 = x and hole around 6 = 0 are still dominant features, but a more isotropic process is emerging, presumably binary encounter “knockout.” Some evidence may be discerned (near 8 = x / 2 ) ofinterference between the two mechanisms-a

+

a,&oo 1800

I

,

0.2

0.3

00

I

0

I

0.1

FIG.7. Polar plots of ejected electron distributionsfor selected ratios ofthe energies of the and electronswhose energies bear a ratio outgoing electrons: electron of equal energy (-) equal to 5.85 (---). The mean energies above threshold are (a) E = -0.25, (b) E = 0.01, (c) E = 0.5. In (b) we show the result when the incoming projectile is a positron (labeled ef).

Christopher Bottcher

258 I

I

0 I

I

I

I

0.2 0.4 0.6 I

I

I

I

FIG.8. Similar to Fig. 7 except that the target differs from H(Is) as follows: (a) He+(Is) and (b) H(2s). In each case the mean energy above threshold is 0.2 times the initial binding energy.

phenomenon which experimentalists might look for; (c) E = 0.5: the Wannier characteristics are disappearing and knockout coming to dominate. We have investigated a number of other processes related to Eq.(I), in particular replacing H( Is) by He+(1s) and H(2s). Figure 8a is a polar plot of EP(E,8) for the target He+(1s) with L = 0 and E = 0.4. The full and dashed lines have the same significance as in Fig. 7, i.e., pertaining to equal and unequal ejection energies, respectively. The angular correlation is almost as large as for H( 1s). This is probably true for ionic targets in general, at least in the low partial waves. Figure 8b is a similar plot for H(2s) with E = 0.025 and L = 0. Though a hole is evident at 8 = a, knockout appears to be dominant. This may be characteristic of excited targets in general, and would support the application of classical formulas in these cases (Percival and Richards, 1975). A fascinating variant of Eq. (1) is to replace the incident electron by a positron. In Fig. 7b we have superimposed a plot of E P ( q 8 ) for e+ H( 1s) corresponding to the same parameters as the e H( 1s) collision, viz. L = 0, E = 0.1, E = E/2. The outgoing particles are predominantly at 8 = 0, as we might expect, but a small isotropic component is discernable. Figure 9 shows the variation of EP(q8) with E / Eat several 8. As 8 0 the peak around E = E/2 becomes sharper, indicating that the ejected pair e+e- are predominantly in bound states. The area under the peak corresponds to a partial cross section for positronium formation in the *Schannel of O.OO5aai in agreement with the accurate variational calculations of Humberston ( 1979).Most of the observed cross section is in the ' P channel which we have not yet investigated.

+

+

-

ELECTRON-IMPACT IONIZATION

259

3

4

-

ul \y^ a

0.3

W

0.1

0.03 472'

1 1

I

I

I

I

I

I

I

/E

FIG.9. Similar to the plots in Fig. 6 but for a positron projectile(cf. Fig. 7). The mean energy the total probabilabove threshold is E = 0.1 and the angle 0 (in degrees)is indicated (-); ity (---).

We conclude with some remarks on the characteristics of density plots in the three-dimensional case, of which an example is given in Fig. 10. The two frames (a),(b) correspond to 8 = 172 and 8";other parameters are as for Fig. 7b. The lack of density around 8 = 0, a = n/4 is evident. The irregular contours are not numerical artifacts but correspond to complicated normal modes introduced by the variation of the potential in space. Similar phenomena are found in wave optics when the refractive index varies with position (Born and Wolf, 1965). While a formal connection has not been worked out these oscillations may correspond to one-electron states of high angular momentum which arise naturally in Wannier theory (Fano, 1976).

Christopher Bottcher

260

2 4

6 8 40 42 W I

I

1

I

2

4

6

8

I

I

4 0 4 2

2

4

6

8

4 0 4 2

FIG. 10. Contour plots of density at the end of time integration for a three-dimensional model (S-wave, E = -0.1). The axes define the hyperspherical planes (cf. Fig. 2) for given angles,(a)O = 172",(b)8".TheOr,,0r2scalesareindicatedtothebottomandleftofeachframe.

111. Box-Normalized Eigenstates A. R-Matrix Formalism for Ionization

The wave packet method has obvious limitations if results are needed at precise energies, e.g., close to threshold. It is not difficult to set up a formalism for solving the stationary Schrodinger equation in the ionization regime (Bottcher, 1982b),

( H - E)V/ = 0 The boundary condition is written

(49)

where s is a point on the boundary and Y, are channel eigenfunctions. The wave function is uniquely defined if an incoming wave is only present in one channel, say p = 0. If D is the operator of differentiation normal to s,a set of R-matrix states is defined by the eigenvalue problem

( H - En)& = 0,

D$,,= 0

(51)

26 1

ELECTRON-IMPACT IONIZATION

Then by familiar manipulations, the solution of Eq. (49) is given by v/ = iRAQt,u

+ RQY,

(52)

where Q is a delta function on the boundary and A is defined in terms of the channel wave numbers k,,

The R-matrix kernal is given by

Equivalently, v/ satisfies

( E - H - iAQ)v= QYo (55) and so could be determined in principle without matrix diagonalization. The obstacle to implementing Eqs. (49)- (55) is that accurate and uniform expansions of Y,,are not available for the Coulomb problem. However, some progress can be made without this knowledge. Suppose we are interested in double photodetachment which involves evaluating a matrix element of the dipole operator X between a bound state 4, and a continuum state v,,(E),

The label p contains other information on the final state, e.g., the energies of the ejected electrons, and can be identified with the index in Eq. (50). Since 4, is confined to a limited region of space, Eq. (56) should not be too sensitive to the asymptotic region or approximations to the channel eigenfunctions. Thus we take the states ( 5 1) and calculate partial and total widths r w = I ( YpI4n ) 12,

r n = Cr w

(57)

P

using approximate channel eigenfunctions. Then it may be shown (Bottcher, 1983) that

where xn

= ( 4oIXI4n )

(59)

+

The left hand side of Eq. (58) is averaged over a band of energies(E,E A E ) while on the right hand side the sum is over all n such that Enlies in this band.

262

Christopher Bottcher

The relation is valid if AE is large enough to contain many eigenvalues. Intuitively IxnI2is the probability ofexciting the “resonance” n,and T,/Tn is the probability that n decays into the channel p. Summing over p gwes an equally appealing result,

If photodetachment can be calculated, it ought to be possible to extract information on the final states. Indeed we can also show that the expectation value of Z with respect to y/,(E), say

under the same conditions as Eq. (58), where

The integral in Eq. (61) extends over the finite region within the boundary. Summing over p we find

Suppose Z is a function only of a and 8, then by virtue of Eq. (26) we can equate r

where p,, is the ejection probability associated with y,,.The probability near threshold is probably independent of E and p so we may take

By calculating Eq. (64) for a range of moment functions Z the general features of P, can be extracted from Eq. (66).

263

ELECTRON-IMPACT IONIZATION

B. A PILOTCALCULATION We have made one application of the formalism in Section III,A, to the two-dimensional Hamiltonian described above,

The “box” was defined by rl ,r, d 93a0 and 44 finite elements were used in each dimension. It appears that a box of at least this size is needed to extract meaningful information on the threshold region. If the R-matrix states [Eq. ( 5 1)] are represented as vectors in the space of finite elements, we have a very large generalized eigenvalue problem which is not amenable to solution as it stands. Our approach was to replace the finite elements by linear combinations which diagonalize Ho, (68)

Ho w,, = E,, w,,

These functions can also double as channel eigenfunctions. The eigenvalue problem is now expressed in the space ofproducts wPwyand we are interested in eigenvalues near E = 0. Thus, we define a zeroth order subspace P in which the diagonal elements of H are close to E. Then by matrix partitioning the full problem can be reduced to diagonalizing

The resolvent in Q space can be constructed by iteration since the diagonal elements of E - H , are larger than the off-diagonal. Having obtained a set of 128 eigenvaluesin the range - 1.6 to 1.6 eV we examined the moment functions

+

Z, = sin 2a, 2, = cos22a which have average values in Wannier theory

(70)

1

G2)W

=j

Figure 1 1 shows a histogram of the expectation values of 2, versus En for En > 0 and states symmetric under interchange of rl and r2 .The expectation Using Eq. (64) we calculated values have a small variation about (2,)w. (Z)*”for both operators and eigenvectors with En > 0, 0, < O are rather different.

+,-

TABLE I

Eigenvector E>O,+ E > 0, E < 0, E < 0, -

+

(Z, 0.946 0.908 0.873 0.833

(Z2).,J+ 1.115 1.280 I .360 1.490

ELECTRON-IMPACT IONIZATION

265

IV. Conclusions Extensions of the methods described above will probably lead to a complete solution of the ionization problem for simple targets. The great sticking point remains the asymptotic boundary condition. Possibilities in the realm of multielectron targets are almost unlimited.

ACKNOWLEDGMENT This research was sponsored by the U.S.Department of Energy, Division of Basic Energy Sciences under Contract W-7405-ENG-26 with Union Carbide Corporation.

REFERENCES Altick, P. L. (1982). Phys. Rev. A 25, 128. Alguard, M. J., Hughes, V. W., Lubell, M. S., and Wainwright, P. F. (1977). Phys. Rev. Lett. 39, 334. Baum, G., Kisker, E., Raith, W., Schroder, W., Sillmen, U., and Zenses, D. ( 1 98 1). J. Phys. B 14,4377. Bell, K. L., and Kingston, A. E. (1974). Adv. At. Mol. Phys. 10,53. Born, M., and Wolf, E. (1965). “Principles of Optics.” Pergamon, Oxford. Bottcher, C. (198 la). J. Phys. B 14, L349. Bottcher, C. (1981h). Int. J. Quantum Chem.: Quantum Chem. Symp. 15,671. Bottcher, C. (1982a). J. Phys. B 15, L463. Bottcher, C. ( I 982b). In “Recent Developments in Electron-Atom and Electron-Molecule Collision Processes” (W. Eissner, ed.), pp. 39-43. Daresbury Laboratory Report DL/SCI/R18, Warrington, U.K. Bottcher, C. ( I 983). In “Invited Papers of the XI11 ICPEAC” (J. Eichler, ed.). North-Holland, Amsterdam. Bottcher, C., Bryce, W. M., and Mandl, F. (1974). J. Phys. B 7,769. Burke, P. G., and Taylor, A. J. (1965). Proc. R. Soc. London Ser. A 287, 105. Byron, F. W., Joachain, C. J., and Piraux, B. (1982). J. Phys. B 15, L293. Callaway, J., and Oda, D. H. (1 979). Phys. Left. 72A, 207. Coulter, P. W., and Garrett, W. R. (1978). Phys. Rev. A 18, 1902. Crandall, D. H. (1983). Phys. Scripta T3, 249. Crothers, D. S. F. (1982). J. Phys. B 15, 2061. CvejanoviC, S.,and Read, F. H. (1974). J. Phys. B 7, 1841. Donahue. J. B., Gram, P. A. M., Hynes, M. V., Hamm, R. W., Frost, C. A., Bryant, H. C., Butterfield, K. B., Clark, D. A., and Smith, W. W. (1982). Phys. Rev. Lett. 48, 1538. Fano, U. (1976). J. Phys. B 17, L401. Fite, W. L., and Brackman, R. T. (1958). Phys. Rev. 113, A1 141.

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Christopher Bottcher

Goldberger, M. L., and Watson, K. M. (1962). “Collision Theory.” Wiley, New York. Greene, C. H., and Rau, A. R. P. (1982). Phys. Rev. Lett. 48, 533. Greene, C. H., and Rau, A. R. P. (1983). J. Phys. E 14,99. Humberston, J. W. (1979). Adv. At. Mol. Phys. 15, 101. Inokuti, M. (1971). Rev. Mod. Phys. 43,297. mar, H., and Schlecht, W. (1976). J. Phys. B 9, 1699. Kulander, K. C. ( I 98 1). NucL Phys. A 353,34 1. Levin, F. S. (1981). Nucl. Phys. A 353, I. Lieber, M., Rosenberg, L., and Spruch, L. (1972). Phys. Rev. D 5, 1330. McCullough, E. A., and Wyatt, R. E. (1971). J. Chem. Phys. 54, 3578. McGowan, J. W., and Clarke, E. M. (1968). Phys. Rev. 167,43. Morse, P. M., and Feshbach, H. (1957). “Methods ofTheoretical Physics,” Vol. 11, pp. 17191728. McGraw-Hill, New York. Percival, I. C., and Richards, 1. (1975). Adv. At. Mol. Phys. 11, 1. Peterkop, R. (1963). Sov. Phys. JETP 16,442. Peterkop, R. (1 97 I ). J. Phys. E 4, 5 13. Peterkop, R. (1982). J. Phys. B 14, L75 I . Peterkop, R., and Liepinsh, A. (1981). J. Phys. E 14,4125. Peterkop, R., and Rabik, L. (1972). J. Phys. B 5, 1823. Pindzola, M. S., Griffin, D. C., and Bottcher, C. (1983). Phys. Rev. A 27,2331. Rau, A. R. P. (1971). Phys. Rev. A 4 , 207. Richtmeyer, R. D., and Morton, K. W. (1967). “Difference Methods for Initial Value Problems.” Wiley (Interscience), New York. Rudge, M. R. H., and Seaton, M. J. (1965). Proc. R. SOC.London Ser. A 283,262. Strang, G., and Fix, G. J. (1973). “An Analysis of the Finite Element Method.” Prentice Hall, New York. Schubert, E., Schuck, A., Jung, K., and Geltman, S. (1979). J. Phys. B 12,967. Temkin, A. (1974). J. Phys. B 7, L450. Temkin, A. (1982). Phys. Rev. 49, 365. Temkin, A., and Hahn, Y . (1974). Phys. Rev. A 9, 708. Wannier, G. (1953). Phys. Rev. 90,917. Zienkiewicz, 0.C. (197 1). “The Finite Element Method in Engineering Science.” MdjrawHill, New York.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 20

ELECTRON AND ION MOBILITIES GORDON R. FREEMAN Department of Chemistry University of Alberta Edmonton. Alberta, Canada

DAVID A . ARMSTRONG Department of Chemistry University of Calgary Calgary, Alberta, Canada

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Low-Density Gases (n/n, < 0.1). . . . . . . . . . . . . . . . B. Dense Gases and Low-Density Liquids (0.1 < n/n, < 2.0) . . . . 111. Ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Low-Density Gases (n/n, < 0.01) . . . . . . . . . . . . . . . B. Dense Gases and Low-Density Liquids (0.01 < n/n, < 2.0) . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

267 270 270 28 1 296 297 3 18 320

I. Introduction Empirical information about certain aspects of the transport of electrons and ions in fluids has thrust uncomfortably ahead of the related theory. The theories of electron and monatomic ion scattering by monatomic and diatomic gases at low number densities n are relatively advanced. Scattering by polyatomic molecules and in the multibody regime are now exciting challenges. Several examples illustrate the rich field of questions in urgent need of theoretical investigation. Two newly recognized phenomena have been interpreted only qualitatively or semiquantitatively. One is a molecular sphericity effect on electron scattering at energies 1.5 X l 0-30m3 exhibit a scattering cross section minimum for electrons at an energy < 1.O eV, the Ramsauer-Townsend (R-T) minimum (Massey et al., 1969).The R- T minimum results in a mobility (drift velocity/field strength) maximum as the electric field strength is increased and the electrons become heated. The R-T effect has a quantum mechanical interpretation. However, a similar mobility maximum occurs for simple ions in many gases as the field is increased (McDaniel and Mason, 1973). The interpretation is based on the energy-sensitive balance of the effects of the long range attraction and short range repulsion between the ion and molecule (Wannier, 1970; Bates, 1982a). It is curious that the electron and ion mobility maxima occur at similar drift velocities and energies (vd lo3 m/sec, E - 1 eV; Huang and Freeman, 1978a, see p. 1360; Takata, 1975; Thackston et al., 1980b). Furthermore the threshold drift velocity vp,above which the steady state mean energy of the charged particles is significantly greater than the thermal energy of the gas, is - lo2 m/sec for monatomic cations (Gatland et al., 1978) and anions (Thackston et al., 1980a) as well as electrons (Huang and Freeman, 1978a, 1981) in monatomic gases. The value of vF for both electrons and heavy ions is - 40% of the average speed of the monatomic gas molecules. For transport purposes the electron may be considered as a light anion. While that analogy is considered to be self-evident by some, it is thought by others to be misleading. At very low number densities, where the electron magnetic moments do not interact with each other, the analogy should be pursued to discover its limits. In particular, the phenomenological R-T scattering minimum is worthy of a new, more general examination by theoreticians. On another front, the mobilities of polyatomic ions in low-density gases do not conform to present treatments (Parent and Bowers, 1981 ). The clustering of neutral molecules about ions (Thomson et al., 1973; Sennhauser and Armstrong, 1978;Jbwko and Armstrong, 1982a)may be considered as a dynamic complication of the polyatomic ion problem. Quantitative interpretation of the mobilities of clustered ions has as a prerequisite an adequate model of the mobilities of polyatomic molecular ions. The important contribution of rotational transitions to the scattering of thermal energy ions by polar molecules has recently been emphasized (Takayanagi, 1980;Celli et d.,1980).The process is probably also significant for thermal ion scattering by nonspherical nonpolar molecules (Gee et al.,

-

ELECTRON AND ION MOBILITIES

269

1982).The importance of rotational transitions in low-energy electron scattering is illustrated by the depth and elegance of theoretical treatments ofthe process for diatomic molecules (Massey et al., 1969; Takayanagi and Itikawa, 1970; Golden et af., 1971; Golden, 1978; Burke, 1979). A different theoretical approach with different approximations might be required for more complex molecules. The molecular sphericity effect (Gee and Freeman, 1983a) might be a good point of initial attack. Excellent reviews of scattering by monatomic and diatomic molecules have been given previously in this series (Takayanagi and Itikawa, 1970; Golden, 1978; Burke, 1979) and in books by Huxley and Crompton ( 1974) and Mason and McDaniel ( 1985). Electron and ion transport in a selection of these gases at low densities is used in the present article as a comparative basis for that in polyatomic molecular gases. After increasing the number of atoms per molecule the next logical step is to increase the number of molecules per unit volume. Effects of increasing the fluid density through the gas, critical (n,) and into the liquid phase are described. There is a great need for deeper theoretical interpretations of electron and ion behavior at all densities. For example, the density-normalized mobility of electrons, going from the normal gas to the liquid, can either increase or decrease by several orders of magnitude depending on the molecular structure. Experimental techniques (McDaniel and Mason, 1973; Elford, 1972; Elford and Milloy, 1974a; Huxley and Crompton, 1974; Helm, 1978) and the conditions for steady state drift (Lin et af., 1977; Kumar, 1981) have been critically examined elsewhere. Although most measurements of mobilities have been made with drift tubes, the microwave conductivity method (Warman, 1982) and ion cyclotron resonance method (Huntress, 1971; Ridge and Beauchamp, 1976) are being increasingly used. An impressivebody of information on mobilities and momentum transfer integrals for the simpler ion-gas systems is summarized in Atomic Data and Nuclear Tables (Ellis et af., 1976b, 1978). Electron and ion mobilities in the “normal” liquid phase, which has a density more than double that of the critical fluid, > 2.0 n,, is beyond the scope of the present article. Suffice it to say that electron mobilities in dielectric liquids are strongly influenced by the molecular properties. If the molecules are spherelike, nonpolar, and do not capture the electrons, electrons tend to reside in a conduction band and have very high mobilities, up to 1 m2/V sec (Huang and Freeman, 1978a). If the molecules are distinctly nonspherical or highly polar, electrons tend to reside in localized states and have low mobilities, down to those of ions, lo-’ m2/V * sec (Schmidt, 1972). Mobilities may be found at all values between these extremes depending on the molecular properties and on the temperature and density of

-

-

-

270

Gordon R . Freeman and David A. Armstrong

the liquid (Dodelet et al., 1973). The study of electron transport in liquids serves as a probe of the liquid state and leads to the question of the nature of the electron itself.

11. Electrons A. LOW-DENSITY GASES(n/n, < 0.1)

1. Monatomic Molecules

The density-normalized mobilities np, of electrons, and their dependencies on the density-normalized applied electric field strength E/n, in helium (Pack and Phelps, 1961) and argon (Huang and Freeman, 1981) are compared in Fig. 1. These two examples were chosen because helium has a positive scattering length (6.3 X lo-" m; O'Malley, 1963) and is (of all molecules) the one that is most nearly a rigid sphere, while argon has a negative scattering length (-9.0 X lo-" m; O'Malley, 1963) and has a much cozier relationship with electrons. The polarizabilities of helium and argon are respectively, 2.1 and 16.4 ( m3;Landolt-Bornstein, 195 1). The former is not large enough for the charge-induced dipole attraction to dominate the short range repulsion even at the lowest energy and largest distance at which scattering has been detected. The latter polarizability provides a charge-induced dipole attraction that dominates the short range repulsion at low energies. The density of argon represented in Fig. I , 95 X 102smolecule/m3 or 35

1 10-3

10-2

10-1

1

Eln (Td)

FIG.1. Density-normalized mobilities ripe of electrons in low-density gaseous helium and argon, as functions of electric field strength. (0),He, 77 K, 4.0 X molecule/m3 (Pack and Phelps, I96 1 ); (O),Ar, 12 1 K, 95 X I 025molecule/m3 (Huangand Freeman, I98 I); Td = lo-*' V . m2/molecule.

ELECTRON AND ION MOBILITIES

27 1

Amagats, is at the upper limit of the low-density regime (Huang and Freeman, 1981). The mobility in helium is independent of field strength up to about 3 mTd (mTd = V m2/molecule), then decreases monotonically with increasingfield (Fig. 1). The decrease of mobility is due to heating ofthe electrons by fields 2 3 mTd, combined with the slight increase of the momentum transfer cross section with increasing energy (Fig. 2; Frost and Phelps, 1964).Electronsgain energy by acceleration in the field and lose it by collisions with molecules. At low field strengths the average amount of energy gained between collisionsis less than that normally exchanged during a collision, so the electrons remain near thermal equilibrium with the medium. At higher field strengths the electrons gain more energy between collisions than they normally exchange, so they increase in energy and velocity until a new steady state is reached..If the electron collision frequency changes as a result of the increased velocity, the mobility changes. It can either increase or decrease, depending on the velocity dependence of the momentum transfer cross section. In contrast with helium, the mobility in argon increases at fields > 1.4 mTd, passes through a maximum at 5 mTd, then decreases at higher fields (Fig. 1). This behavior reflects electron heating by fields 2 1.4 mTd, combined with the R-T minimum in the momentum transfer cross section at -0.23 eV (Fig. 2; Milloy et al., 1977; Haddad and OMalley, 1982). The mean energy of the electrons increases much more rapidly with field strength in argon than in helium (Warren and Parker, 1962). The shape of the electron energy distribution in argon at fields above the heating threshold is

-

v(l&m/sec) 5.9 1

i((11

0.01

I

'

18.8 8

I ""I

I

0.1

I

I

I

""r

59

1

E(W

FIG.2. Electron momentum transfer cross sections urn,= of low-density helium and argon and Phelps, 1964);Ar gases, as functions of electron energye and velocity 0.He (-)(Frost (---)(Frost and Phelps, 1964), (- - -) (Milloy el al., 1977), (-) (Huang and Freeman, 1981).

272

Gordon R. Freeman and David A. Armstrong 10

H

E

0 c \

0

c

0

001

01

E

10

(ev)

FIG.3 . Isotropic energy distribution function of electrons in argon gas at 293 K, at the V . m2/molecule)(Makabe and Mori, 1982). R - T indicated field strengths E/n in mTd indicates the location oftheRamsauer-Townsend minimum. Reproduced with min (---) the kind permission of the authors and publisher.

strongly influenced by the minimum in the scattering cross section at 0.23 eV (Fig. 3; Makabe and Mori, 1982).

2. Diatomic Nonpolar Molecules The diatomic molecules most frequently used in electron scattering studies are hydrogen and nitrogen (Pack and Phelps, 1961; Frost and Phelps, 1962; Warren and Parker, 1962; Lowke, 1963; Frommhold, 1968; Nelson and Davis, 1969; Robertson, 1971). Interest continues today (Taniguchi et al., 1978; Reid and Hunter, 1979; Braglia et al., 1981, 1982; Wada and Freeman, 198 1; Pitchford and Phelps, 1982). The value of np, for thermal electrons at 77 K in hydrogen is fourfold smaller than that in nitrogen (Pack and Phelps, 196 I), so at - 10-2 eV the momentum transfer cross section of hydrogen ( - 8 X m2; Englehardt and Phelps, 1963) is fourfold larger than that of nitrogen ( 2 X m2; Englehardt et al., 1964). The energy dependence of the cross section of hydrogen is much smaller than that of nitrogen at E < 0.1 eV. The comparison is somewhat analogous to that between the cross sections of helium and argon at a 10-fold higher energy (Fig. 2). The question is whether an R - T minimum exists in the cross section of nitrogen (1.7 X mz at 15 meV; Wada and Freeman, 198 1). The relative scattering behaviors of hydrogen and nitrogen imply that the cross section of hydrogen does not possess an R - T minimum, but that of nitrogen might. Among the molecules that have isotropic polarizabilities, the scattering length of helium is large and positive, whereas that of argon is large and negative (Table I; O'Malley, 1963). The scattering length of neon is small and positive (O'Malley and Crompton, 1980). Interpolation of the values of a and A in Table I indicates that if the isotropic polarizability were

-

273

ELECTRON AND ION MOBILITIES TABLE I PHYSICAL PROPERTIES He TJKY

5.19

n,(1OZ6 molecules/m3)" 104.3 E(10-3' m3)b a,,- ai( m3)b

2.1

8 , ( i O - 3 eV). a,( m2)d A( 10-l'm)p

0.0 5 6.3

0

Ne

Ar

Xe

H2

N2

CH4

CzH,

CzH,

190.6 305.5 282.4 44.4 150.9 289.7 33.3 126.2 60.8 40.6 46.7 50.5 92.7 67.1 144.4 80.5 40.1 8 18 26 45 40 4.0 16.4 0 0 0 3 7 0 8 21 0.0 0.0 0.0 7.5 0.25 0.65 0.08 0.08 0.3 6 -116 8 2 -20 -20 -11 1.1 -9.0 -34.4

Critical temperature and density; Reid ef al. (1977). Polarizability mean and anisotropy; Landolt-Bornstein (195 1); Bridge and Buckingham ( 1 966). Rotational constant; Herzberg (1950, 1967). Electron momentum transfer cross section at 0.01 eV; Frost and Phelps (1962, 1964);OMalley and Crompton (1980);Gee and Freeman (1981b). Scattering length; O'Malley (1963); O'Malley and Crompton (1980).

6X m3 the scattering length would be small and negative. When the polarizability is anisotropic a larger mean polarizability is required to produce an R - T minimum. For example, hydrogen has a mean polarizability of 8 X m3 (Bridge and Buckingham, 1966) but no R-T minimum, m3. The mean polarizability presumably due to the anisotropy of 3 X nitrogen is double that of hydrogen, but the electron momentum transfer cross section of the former at lo-* eV is only one-quarter that of the latter (Table I). Nitrogen either possesses an R - T minimum or is on the verge of doing so. The mean polarizability of nitrogen is larger than that of argon, but the anisotropy of the former would cause its R-T minimum to be much shallower and at a lower energy than that of the latter (Freeman ef al., 1979). The anisotropy effect is not understood beyond an unsatisfying qualitative rationalization. The detailed interpretation is evidently more complex than we would first suspect. Whether nitrogen possesses an R-T minimum can be resolved by refined measurements of np, at temperatures below 100 K and fieldsbelow 10 mTd (Wada and Freeman, 198I ). The values of ripe at 293 - 300 K agree from one laboratory to another within 3%(Fig. 4). However, at 78 _+ 1 K the values of Wada and Freeman (1981) are 10- 15% lower than those of Lowke (1963) and Pack and Phelps ( 1961) at fields < 10 mTd. The threshold drift velocities for electron heating seem unrealisticallylow in the earlier data for 78 k 1 K, but new measurements are required to settle the question. The spacing between rotational levels in nitrogen is sufficientlysmall (0 2 is 68, = 1.5 meV, Table I; Herzberg, 1950), and the anisotropy of polarizability is suffi-

-

274

Gordon R. Freeman and David A. Armstrong

a

FIG.4. Density-normalized mobilities n,ue of electrons in nitrogen as functions of electric field strength and temperature. Pack and Phelps (1961): (O),77 K; (+), 300 K. Lowke (1963): ( 0 ) 78 K; (o), 293 K. Wada and Freeman (1981): 79 K (A), 157 K; 295 K.

(o),

(o),

ciently large that inelastic energy exchanges between thermal electrons and the molecules are significant at 77 K (Englehardt et al., 1964). Inelastic scattering is the dominant process that removes from the electron the energy gained from the field, even at the lowest fields. The threshold drift velocity upfor electron heating in nitrogen should therefore be greater than the speed of sound c,, which is related to the average speed of the molecules (Gee and Freeman, 1979; Paranjape, 1980; Wada and Freeman, 1981). When the energy is removed from the electrons mainly by elastic collisions up is a few tenths of the average speed of the molecules and is referred to co by analogy with a similar phenomenon in semiconductors (Shockley, 195 1). The ratio up/coequals 0.5 in argon at 12 1 K, characteristic of energy loss by elastic collisions (Huang and Freeman, 1981). In nitrogen at 77 K the ratio should be significantly greater than one. The data of Pack and Phelps (1961) for nitrogen at 77 K correspond to u p / c o= 5 k 2; Wada and Freeman (1981) reported 10 & 2; Lowke's data (1963) correspond to a ratio < 2.1. The disagreement between these values reflects the difficulty of the measurements, but the true value at 77 K is probably 5 - 10. By contrast, the spacing between rotational levels in hydrogen is so large (0 + 2 is 6Bo = 45 meV, Table I ) that rotational excitation by thermal electron swarms is negligible at 77 K. At this temperature the value of up is 3 10 m/sec (Pack and Phelps, 196 1 ) and co = 669 m/sec, so u,"/co = 0.5 in agreement with the value for argon and energy loss by elastic collisions.

-

3. Polyatomic Nonpolar Molecules (Sphericity Eflect) The electron momentum transfer cross sections, om,,,of isotropically polarizable polyatomic molecules possess an R-T minimum at E = 0.25 k

ELECTRON AND ION MOBILITIES

275

0.05 eV, near that of argon. Examples are methane (CH,) (Ramsauer and Kollath, 1930;Cottrell and Walker, 1967; Pollock, 1968;Gee and Freeman, 1979; Sohn et al., 1983),silane (SiH,) (Cottrell and Walker, 1967; Pollock, 1968), and neopentane [C(CH3),] (McCorkle et al., 1978; Freeman et al., 1979). Upon replacing one of the H atoms of methane by a CH3group, making ethane, the molecule nearly doubles in volume but the value of nb, in the gas increases (Fig. 5). The average value a,,,,, = ( v ) / ( v/a,,,) of the larger molecule is smaller than that of the smaller molecule for thermal electrons at 200-300 K (Gee and Freeman, 1979, 1980a). The reason is that ethane possessesan R - T minimum at 0.12 eV, which is nearer the range of energies of the thermal electrons than is the R-T minimum for methane at 0.25 eV. This lowers the value of a,,,, for ethane at energies S O . 12 eV, covering the thermal region in question. The shift of energy of the R - T minimum, going from methane to ethane, is mainly the result of the change of the degree of sphericity of the molecule, rather than the change of molecular size. The spherelike methane (CH,) and neopentane [C(CH,),] are very different in size, but their R-T minima are at similar energies, 0.25 and 0.2 1 eV, respectively. By contrast, the distinctly nonspherelike ethane ( H3CCH3) and n-pentane (H3CCH2CH2CH2CH3) have R - T minima at 0.12 and 0.13 eV, respectively (Gee and Freeman, 1980a; Freeman et al., 1979). Results for many hydrocarbons are summarized in Table 11. The effect of molecular shape on am,,at E < 0.3 eV is illustrated by the cross sections of the three isomers of pentane, CSHIZ,in Fig. 6. There is a simple relationship between the external shape of the molecule and the energy dependence of its am.,,even for such complex molecules. This should

-

2 I00

300

200

T (K)

FIG. 5 . Density-normalized mobilities ripe of electrons in CH, (0)and C,H, (A) gas; n = 7 X loz5molecule/m3; Gee and Freeman (1979, 1980a).

TABLE I1

ELECTROF; SCATTERING PARAMETERS OF HYDROCARBOX GASES

Aknnes 0.0

0

0

I .0 7

90-160 294

-

0.13

23.2

0.14

1.2

15

325

0.0

52

0.21

1.7

5

300- 340

0.13

1.0

29

300

so.1

A

0.Y

15.7

0.05 0.132 ( 8 k T / 1 r M , ) ~ /the ~ ] , Maxwell model (constant collision frequency), the Lorentz model (m > M). In addition, important studies by Wannier (195 1, 1952) had provided insight into higher field behavior. These were exploited later in the 1970s, when further attacks on the solution of the Boltzmann equation were made. In a valuable review, which also discusses the space-time development of swarms, representations of the collision integral and other important aspects of theory, Kumar et al. (1980) have classified and examined in detail approaches to the solution of the Boltzmann equation. “The method of most general applicability is that of moment equations,” and papers dealing with the required basis functions and other mathematical procedures have appeared (Kumar, 1980a,b). A lucid account of that method has also been given by Mason and McDaniel(l985). Viehland and Mason (1975a, 1978) devised a procedure for using basis functions, in which the ions of mass rn had a temperature Ti different from that of the neutral gas and defined as Ti = rn ( of ) /3k. According to this “two-temperature theory” the mobility and drift velocity is given at all E/N by

The term a, which is normally less than 0.1 and zero in first approximation, includes all of the higher order kinetic theory approximations. It depends in a complicated way on E/n, T, the ion and molecule masses m and M , and the ion -neutral molecule potential. The effective temperature Teff is defined by the relations

jkT,,

+M )

= +k(mT+M T i ) / ( m

(254

=+kT++Mv$(l +p)

(25b)

The term /3 is similar to cy and is also zero in first approximation. The effective temperature characterizes the mean ion eneigy in the center ofmass frame of the ions and neutrals and appears in the W.’)collision integral. Clearly when +Mvi 1 at high field strengths. Thus the Viehland- Mason theory is the first rigorous kinetic theory, which is valid at arbitrary field strengths and has no restriction on the form of the ion -neutral potential function or on the m/Mratio, with the above exception.

b. Comparison of’ theory and experiment and the determination o j ion neutral potentiul ,fimctions ,from mobilities. During the past 7 years the two-temperature theory has been used extensively to test potential functions by comparing calculated mobilities with experimental results (see, e.g., Gatland et al., 1977a,b; Lamm et al., 1981; Gatland, 1981; Viehland et al., 1981b). Also the inverse process of finding potentials from mobilities over a range of field strengths has been undertaken (see, e.g., Gatland et al., 1977b; Gatland, 1981 ;Viehland et al., 1981b). Both procedures are nontrivial computational exercises, and careful tests have been made to establish an acceptable level of accuracy (Viehland et al., 1976; Viehland, 1983).Given that the best gas phase mobility data are now accurate to k 290,Viehland et a/. ( 1976) suggest that it should be possible to determine ion- neutral potentials to within about 5% over a wide range of separation distances. The effect of small changes in potential function on mobility is aptly illustrated in Fig. 20a,b, where some data reported by Gatland ( 1981) for Li+ ions in argon are reproduced. The circles in Fig. 20a are the experimental mobilities of Akridge et al. (1975) for standard argon density (2.687 X 1019 molecule/cm3) and temperature T = 300 K, while the curves are mobilities calculated from the five potential functions shown in Fig. 20b with the aid of the Viehland- Mason theory. The Gordon- Waldman (GW), Kim Gordon (KG),self-consistentfield (SCF), and configuration interaction (CI) potential functions are seen from Fig. 20a to give reasonable and progressively better fits to the experimental mobilities, but there is still a significant difference. The best fit is given by the solid line, which was calculated from a potential determined directly from the mobility data. The fact that it differs significantly from the CI potential only from 6 to 8 a.u. internuclear separation indicates the sensitivity of the mobility to the form of the potential and demonstrates the utility of mobility data for the refinement of such functions. As T,, and the kinetic energy of relative motion of the ion - neutral pairs change, different regions of the potential energy function are sampled. This causes the mobility to exhibit a certain form of dependence on T,, (or E/n). For example, at low T,, the long-range r4 polarization attraction potential

30 1

ELECTRON AND ION MOBILITIES

20

50

200

I00

E/n (Td)

0'01

-

2.5

k"

I

~

'

~

1

'

~

"

b Li Ar

7.5

5.0

I 0

R(a.u.)

FIG.20. (a)Reduced mobilitiesofLi+ionsinargongasat300 Kderivedfrom the potentials in (b) and experimental data. (b) The Kim-Gordon (KG), Gordon-Waldman (GW) and Olson and Lin SCF and CI theoretical potentials for Li+-Ar, together with the potential determined directly (DD) from the experimental data. Both R and V are in atomic units (0.5292 A and 27.2 1 eV, respectively). The crossing point 0 and minimum v, in the DD curve are indicated. From Gatland (1981) by kind permission of the author and publisher.

dominates and the mobility tends toward pWl,

0

PWl=

n m r

where the constant 0 is independent of T and E/n and a is the molecular polarizability (McDaniel and Mason, 1973; Mason and McDaniel, 1985).

302

Gordon R. Freeman and David A . Armstrong

At high TeB,short-range repulsion dominates. Generalized mobility curves, such as the idealized example in Fig. 21, take the form of p/ppl plotted against T,*,= kTe& where E is the depth of the potential well (Gatland et al., 1977a;Takebe, 1983).The maximum in Fig. 2 1 can be shown to occur in the region where the short-range repulsive forces most effectively cancel the longer range attractive forces. This usually corresponds to ion - neutral interaction distances in the range of 0 to rmin Fig. 20b. Its height aboveppl is a measure of the extent to which this cancellation occurs. For example a “softer repulsion” yields a higher and broader maximum, while addition of shorter range (e.g., 1O-a) attractive forces augments the polarization forces and reduces the maximum. Mobility ratiosp/pWlfor the alkali ions Li+ through Rb+in He, Ne, and Ar at 300 K are shown as a function of E/n in Fig. 22. These data from Gatland et al. ( 1977a)illustrate the types of mobility dependence on E/n (or TeB)seen in real systems. The circles are ratios ofexperimental mobilities topwl,while the curves are for calculated mobilities based on Waldman- Gordon potentials. Considering their relative simplicity, the latter give quite good overall agreement with experiment. The values of E in the required potential functions have been shown to vary over quite a wide range (Takebe, 1983).If the E / n scale in Fig. 2 1 were converted to T,*,, the curves for different ion neutral combinations would therefore correspond to different parts of the generalized curve in Fig. 21. Thus for Li+-Ar, with E = 0.55 eV (Takebe, 1983), the data cover roughly the same T:ffrange as the generalized curve, while for K+- He with E = 0.023 eV they are pushed to the right-hand part of the generalized curve. This effect of E and the factors discussed above can

I

10.‘

I

10.8

10

kT,fi/E

FIG.2 I . Generalized curve of ion mobility as a function of effective temperature. The arrows indicate regions dominated by the potential minimum rm and crossing point u. The curves in Fig. 22 correspond to fragments of the generalized curve starting from different T,, as indicated by the short vertical lines (where well depth e2 < c, ). From Gatland et al. ( 1 977a) by kind permission of the authors and publisher.

303

ELECTRON AND ION MOBILITIES '

I

I 10'

10'

10'

10'

10'

10'

10'

10'

(Td)

FIG.22. Calculated and experimental ion mobilities at 300 K as a function of the ratio of the electric field strength to the gas number density. The circles represent experimental measurements, their diameters the experimental uncertainty. Solid curves are calculated from Waldman-Gordon electron-gas model potentials. From Gatland ef al. (1977a) by kind permission of the authors and publisher.

account qualitatively for the different forms of curve seen in Fig. 22. The fall-off in p/pWlat high E/n for all systems is due to the dominance of the repulsive interaction. As a means of improving the accuracy of the theory for m/M > 1 at high fields, Lin et al. ( 1979c)built anisotropy into the ion temperature used in the basis functions for the moment equations, defining temperatures perpendicular to, T L ,and parallel to, TI,,the field direction by the equations

The theory, thus derived, is now called the "three-temperature theory." The numerical results were rapidly convergent and gave much better agreement (0.5%at third approximation) for m/Mas large as 4 in the high E/n region. At the same time, the agreement achieved in the two-temperature theory for lower m/M was unspoiled. However, a heavy price in increased computational complexity is paid in the three-temperature theory and it will probably only be used where absolutely necessary.

c. Special effects. This section is concerned with special types of interaction or potential. i. Systems with valence interaction: H+-He and others. The hydrogen ion - helium gas system exhibits a number of special effects and warrants

304

Gordon R. Freeman and David A. Armstrong

separate consideration. The solid line and points showing the temperature dependence of the low field mobility of H+ in He in Fig. 23 were both calculated from the accurate Kolos-Peek potential for that system with the aid of the Chapman-Enskog theory. The line was obtained by Lin et al. ( 1979a) using a classical cross-section formulation, while the points represent Dickinson and Lee’s quantum mechanical calculation ( 1978). The quanta1deviations seen below 50 K will be discussed later. Here we may note that above that temperature the two sets of calculations agree within their uncertainty limits with each other and with the experimental measurement of Orient (1 97 1) at 300 K, which is shown by the lower triangle. This is particularly gratifying, for the accuracy of the Kolos- Peek potential is considered to be such that in principle the mobility is now calculable to a greater accuracy than that to which it can be measured (Lin et al., 1979a). Other measurements of H+ and D+ mobility in He have been compared by Howorka et al. (1979). There is general agreement within the limits of the experimental uncertainties. The upper triangle in Fig. 23 is the mass-scaled mobility of D+ from Orient ( 1972). The molecular ion HHe+ is a homolog of H, with a binding energy of 2 eV and rm = 0.8 A.Indeed at sufficiently high pressures and low E/n it is stable in helium and its mobility has been determined (Snuggs et al., 1978).Thus to observe the true mobility of H+ in He Orient worked at helium pressures of ~ E Q ( ~ ) ( E )where , Q(’)(E) is the momentum transfer cross section and E the mean kinetic energy of relative motion of the ion and gas molecule. They also made the first prediction of a runaway effect for a specific systemH+(and D+) in He. Figure 25 presents their plot of EQ(I)(E)vs E for that system. The curve never rises above the first maximum, where the value is 1.1 X lo-” eV cm2. This is equivalent to the ion in a field strength of 1 10 Td. Thus runaway should be seen when the mass-scaled field strength, [(m M)/m]l/Z(E/n), significantly exceeds that. However, these authors pointed out that even below 1 10 Td a significant high-energy tail should develop on the ion distribution. Experimental verification of a runaway effect for H+, and indeed also D+, in He was first provided by the results of Howorka et al. (1979). Contrary to the usual finding of a narrowing in the amval time histogram with increasing field, they observed a broadening at E/n = 70 Td and an “early toe.” The latter effect is attributable to a high-energy tail as discussed above, while the

+

-

+

308

Gordon R. Freemun und David A. Armstrong

10

Id

1

cleVl

FIG.25. Collisional momentum loss as a function of relative collision energy, showing the conditions for runaway; calculated for the actual H+-He potential (); calculated for the (8-4) potential model (---). The energy limits for orbiting collisions are marked for both potentials. From Lin ef a/. (1979a) by kind permission of the authors and publisher.

broadening is an indication of the effect of runaway on the longitudinal diffusion coefficient D,, (see below). Their mass-scaled mobilities for D+ and H+ have been plotted against the mass-scaled field strength in Fig. 26 as filled and open triangles. The upturn in mobility at 80 Td is most unusual for ions in helium and is thought to be a further indication of runaway. This figure was actually taken from the paper of Lin and co-workers (1979a) and shows also their curves for the first, fourth, and fifth orders of approximation for mobilities calculated from the Kolos- Peek potential by means of the Viehland-Mason theory. In the region below 40 Td, where steady drift occurs, convergence of the fourth and fifth approximations is seen. However, above 60 Td convergence ceases. The increasing divergence there was attributed to runaway, since the higher order moments solutions are more sensitive to the high-energy tail. Finally Moruzzi and Kondo (1980) have shown that the apparent mobility depends separately on E and n, and there appears to be no doubt ofthe runaway phenomenon for H f a n d D+ in He. A corresponding phenomenon for electrons in gasses has of course been known for some time. Acceleration need not continue indefinitely, since it will usually be terminated by inelastic collisions.

309

ELECTRON AND ION MOBILITIES

-% F

1L

> N \

-k P 9 -f +

13

E

r.

-E

12

0

50

100

Km + M)/mI” (E/N (Td)

FIG.26. Mass-scaled mobilities of H+ and D+ in He at 300 K as a function of mass-scaled field strength, showing the onset of runaway. The curves are calculations of Lin et a/. (1979a): H+ (-), D+ (---), the numbers refer to the order of approximation. Smoothed values (&A) from the measurements of Howorka ef al. (1979). From Lin ef a/. (1979a) by kind permission of the authors and publisher.

d. Ion difiuion -generalized Einstein relations. The fact that ions drifting in a gas under the influence of a field may also undergo diffusion is of practical as well as fundamental importance, since the arrival time histogram is thereby broadened and must be carefully deconvoluted to extract the drift time and mobility (McDaniel and Mason, 1973).Diffusion occurs both parallel and transverse to the field direction with diffusion coefficientswhich are, respectively, D,,and D, . At low fields, where the energy distribution of the ions is Maxwellian, these two are equal and related to the mobility by Eq. (30), -qD - kT P

Weinert and Mason ( 1980) have pointed out that although it is most commonly referred to as the Einstein Relation, Eq. (30) was in fact derived independently and prior to Einstein’s work on Brownian motion by both Nernst and Townsend. Strictly speaking it is therefore more properly called the Nernst - Townsend - Einstein relation (Mason and McDaniel, 1985). In principle both D,,and D, can be determined experimentally, but the measurements are more difficult and the present accuracy ( 2 10%)consid-

-

310

Gordon R. Freeman and David A. Armstrong

erably less than for mobility. For this season there is great practical value in having relations similar to Eq. (30), which are valid at all field strengths. The first such generalized Einstein relation (GER) was derived by Wannier ( 1952) on the basis of a simplified model,

1

--k7',,[1+ 4DIl d dln(E/n) lnK P This form was later confirmed by Robson (1972) using nonequilibrium thermodynamics. Since those publications extensive theoretical work has been done in this area (Viehland et al., 1974; Viehland and Mason, 1978; Skullerud 1976; Robson, 1976; Waldman and Mason, 198la; Waldman et al., 1982), largely because the derivation of GER provides insight into the parallel and transverse components of the ion energy. Furthermore, where D,, and D, have been determined experimentally their comparison to values calculated from experimental mobility with the aid of GER provides an important test of the reliability of theory and experiment. Experimental values ofp and D, in several systems have been used in tests with GER based on Eq. (31) and ion temperatures calculated from the two-temperature theory (see, e.g., Viehland and Mason, 1975b; Thackston et al., 1980a; Holleman ef al., 1982, and references therein). The results in general agree within the uncertainties ofthe determinations ofD,. The most rigorous theoretical treatment is that of Waldman and Mason (198 la), which employs the formalism of the three-temperature theory to develop GER from the Boltzman equation without restrictions of m/Mor E/n. It has been tested against the calculations of Viehland and Mason and Skullerud, and experimental D,,and D , for the alkali ion-inert gas systems. The agreement for the three theoretical approaches was comparable. However, in principle the three-temperature theory is more adaptable as well as rigorous. Finally we may note that GER have been modified to take into account the case of resonance charge transfer (Waldman et al., 1982). 2. Polyatomic Ions and Gases

Polyatomic systems introduce three new major complications: (1) The ion - neutral interactions may no longer be spherically symmetric, (2) energy contained in internal degrees of freedom ofthe ion or molecule must now be included in the energy balance equations, and (3) inelastic collision processes must be included in the kinetic theory treatment. The Wang Chang, Uhlenbeck and de Boer equation is a generalization of the Boltzmann equation for

ELECTRON AND ION MOBILITIES

31 1

particles with internal degrees of freedom and can be made to take into account the inelastic collisions. With this as a starting point and some modifications of the techniques used earlier for the Bolzmann equation, Lin et al. (1 979b) and Viehland et al. (1 98 la) have, respectively, derived solutions for mobilities of structurelss and polyatomic ions in polyatomic gases. The form of the master equation for mobility in first approximation is the same as Eq. (24) with a! = 0, except that Te5is replaced by a new quantity Tk,&. The latter can be found from #kTk,eE= [ t k T + fMv:1[ + (M/m)t(Tk,e5)l-' = jk(mT MTk)/(m 4- M )

+

(32a)

(32b) Here Tk is the ion translational temperature, while Tk,&is the effective translational temperature in the center-of-massframe of the ion and neutral molecule. The quantity t(Tk,,5), which does not appear in the corresponding equations in Section III,A, 1,a for atomic systems, is a dimensionless ratio characterizing the energy loss due to inelastic collisional exchanges. It is defined as = ( APz ) / ( 2P

AP )

(33) Where p is the momentum in the ion-molecule center-of-masssystem and A indicates the change in a collision. c(Tk,eE)

a. Polyatomic and clustered ions in monatomic gases. Detailed balance shows that when steady state conditions are reached in these systems the internal temperature of the ions, T,, must be equal to Tk,,(Viehland ef al., 1981a). Also ((Tk,e5) is zero so that Eq. (32a) reduces to Eq. (34a), an expression of the same form as that for atomic systems, [#kT+ 3Mv$1 (344 With these similarities in the kinetic theory expressions, the major differences from purely atomic systems lie in the possible asymmetry of the interaction potentials and the fact that they are now averaged over the rotational and vibrational levels of the ion. The effects of these factors appear to be unspectacular, because the field dependences of mobilities of molecular ions, for example, COfand N20+ in He, Ne, and Ar (Ellis et al., 1976b, 1978),exhibit similar trends to those seen in Fig. 22 for K+or Rb+. However, it remains to establish experimentally whether changes in Tk,,Eproducedby altering E/n are equivalent to those induced by changing the gas temperature T, and experimental data (above and Dotan et al., 1976)warrant further analysis. Most of the studies of larger ions, like S F f , SF;, and S02Fywere contkTk,e5=

312

Gordon R. Freeman and David A . Arrnstrong

ducted only at low field (Patterson, 1972). However, it was analysis of this kind of data which emphasized the need to consider more realistic potentials than the hard sphere repulsive polarization attraction and other simple models (Patterson, 1972; Mason et al., 1972; McDaniel and Mason, 1973). Usually the mobilities can be fitted with potential functions of the form

with n ranging from 8 to 12. This also appears to hold for clustered monatomic ions like C1-*(H,O), in argon, krypton, and xenon (Jbwko and Armstrong, 1982a). A core model, consisting of a (12-4) central potential displaced from the origin, can also be used to reproduce the mobilities of polyatomic ions (Mason et al., 1972). Both this and the n-6-4 potential may drastically alter the dependence of mobility on Tea,completely suppressing the maximum seen in Fig. 2 1. This resembles what is seen experimentally when temperature is varied, but n - 6 -4 potentials do not give realistic agreement (Parent and Bowers, 1981). A special type of interaction can occur in the case ofdiatomic inert gas ions in their parent gases (Helm and Elford, 1978). The process involves the transfer of the monatomic ion from one neutral atom to another, e.g., He:+ He

-

He

+ He:

(35)

It has a relatively high cross section and causes the mobilities of the dimer ions to be lower than for ions of similar mass with normal interaction potentials (Mason and McDaniel, 1985).Helm and Elford point out that the phenomenon of fragment ion exchange may apply to more complex systems as well.

b. Monatoinic ions in nonpolar polyatomic gases. The introduction of internal degrees of freedom into the buffer gas molecules has no obvious effect on the field dependencies of mobilities measured at a given temperature, which frequently resemble those in monatomic gases (see Ellis et al., 1976b, 1978). Also we find that p approaches pp, at low field. However, experiments by Viehland and Fahey (1983) have provided evidence for the parameter c ~ T ~which , ~ ~occurs ) , in Eq. (32a). The open squares in Fig. 27, which is taken from their paper, represent zero field mobilities of C1- measured in nitrogen gas at several temperatures and plotted against these temperatures as the values of Tk,eaon the abscissa scale. The open circles and triangles show mobilities all measured at 300 K, but with different E/n. They are plotted against values of Tk,,,calculated on the assumption that Zl). Because Z2> 2, it is possible to have T2> T Iand provide significantly higher effective temperature of excitation in pumped (lasing) plasma. One of the problems arising in resonantly pumped laser schemes is to find resonances with 'precision about the width of corresponding transitions. In neutral atoms this precision is about 10-5 - 10-6and such resonances are very rare. In high-temperature laser-produced plasma due to large Doppler broadening, resonances should be accurate to within The use of resonant line pairs for pumping short wavelength lasers has been suggested by Vinogradov et al. (1975b), Norton and Peacock (1975), and Bhagavatula ( 1976). Experimental evidence on selective photoexcitation has been reported by Bhagavatula (1976). The La radiation of Cs+and ( l ~ ) ~ls)(2p) -( radiation of C4+,respectively, are the souces of pumping n = 2 to n = 4 transitions of MglI+ and Mg", resulting in overpopulation of the level n=4. In the earlier works the pumping with helium-like resonances was considered. As a rule in this case the resonances are not precise and require either opacity broadening or large Doppler shift to pump effectively. In the case of L-shell- K-shell resonances there are many precise resonances for which neither opacity broadening nor Doppler shift are required, as has been shown by Hagelstein ( 1981). For example, some pairs of resonantly coupled transitions are presented below (Hagelstein, 1982). Pumped transitions

Pumping transition

Ne8+, ( ls)(4p), 1 = 1.1000 nm hw,, = 55 eV A = 1.0239 nm Ne9+, (ls)-(3p), 1 = 1.0240 nm hw32= 189 eV 07+, (ls)-(3p), 1 = 1.6006 nm 1 = 1.6007 nm 121 eV

Ni20C, ( 2 ~ ) ~ - ( 2 p ) ~ ( 3 dI )= , 1.1000 nm Ga22+, ( 2 ~ ) ~ - ( 2 ~ ) , ( 3 s A) ,= 1.0239 nm

VIE,

(2~)~-(2~)(2p)',1 = 1.6007 nm

When lasing at relatively long wavelengths may be of interest, the resonantly coupled pair Li2+,(ls)-(3p), I = 11.3905 nm, and V14+, ( 2 ~ ) ~ - ( 2 ~ ) ( 2 p ) ~ , I = 11.393 nm, the lasing transition being (3p)-(2s), I = 73 nm. The effective temperature of excitation, T, in the discussed inversion scheme is determined by the number of photons per mode in pumping

342

I . I. Subel’man and A . V. Vinugraduv

radiaton,

v = [exp(+)

-

11’

According to estimation it is necessary to provide v = 0.01 -0.1 to obtain sufficiently high gain. The resonantly pumped photoexcitation schemes in Ne-like and H-like ions, where lasing transition energies h w are 55,15 1, and 189 eV, have been considered by Hagelstein (198 1). The results show that gain g - 50/cm may be attainable if for the resonance line radiation field v exceeds 0.015 photon/mode. The calculations of v are very difficult and should include the treatment of radiation transfer in resonance spectral lines broadened by opacity and Doppler effect in expanding plasma. The main advantage of resonant photopumping is that this method can ensure high effective temperature of excitation in pumped plasma having relatively low temperature and density. As a consequence the limitations imposed on geometry of lasing plasma by opacity and refraction are not so strong.

IV. Conclusion The various laser schemes based on different pumping mechanisms including collisional pumping, recombination pumping, charge exchange, and resonant photoexcitation pumping were treated theoretically and experimentally during the last few years. However, no real lasers at 1 < 100nm are available at present, not to mention X-ray lasers. The main reasons are that because of lack of high-quality resonators the gain -length product gL needed in extreme-UV and soft X rays are 1O2- 1O3 times larger than in the visible spectra, the needed pumping power very rapidly increases with decrease of 1,and very strong limitations on plasma dimensions are imposed by plasma opacity and refraction. The first experimental works in search of inversion and lasing at short wavelengths were based on relatively simple theoretical schemes. The populations of ion levels were calculated taking into account elementary processes of excitation and deexcitation in plasma with given temperature, electron, and ion densities. As a rule plasma was considered to be optically thin. The importance of detailed hydrodynamics simulation, including energy absorption, plasma heating and expansion, kinetics of ionization and

ON THE PROBLEM OF EXTREME UV AND X-RAY LASERS

343

excitation, the choice of pumping laser with optimal wavelength, and so on, was not fully understood. As a result experiments were carried out under conditions which differed greatly from that assumed in theoretical predictions. It is clear now that for successful attempts toward lasing at short wavelengths the feasibility of the whole chosen scheme of experiment and design should be considered including high gain, opacity, and refraction requirements as well as available pumping lasers, targets, plasma, and level popultions diagnostics and so on. Considerable progress in understanding computer codes for the design and simulation of laser schemes, diagnostics, and experimental technique was achieved during the last few years. The more powerful high-quality lasers may be used now in experiments on providing plasma with necessary plasma parameters. As a rule conventional cylindrical optics for focusing in line is not as good as optics for point focusing. The quality of cylindrical focusing can be essentially improved now with the help of new techniques based on the wave front conjugation (Plotkin and Ragozin, 1981). Much work has been done on spectroscopy of multicharged ions. For example, when experiments in search of lasing on (3s)-(3p) transitions of Ne-like ions began, their spectra were not very well known. The ion with the highest charge Z for which (3s)-(3p) transitions were identified was A13+ (Artru and Kaufman, 1975). The estimations of populations were based on theoretical values of energy levels. The accuracy of energy level calculations permitted making crude estimations of level populations and inversion but was not high enough for transition identification. During the last few years the transitions (3s)-(3p) were identified for Ne-like ions Si4+,P5+,S6+, C17+, using triggered spark and beam-foil technique and also specAre+,and K9+, tra of laser-produced plasma (Brillet and Artru, 1976; Eidelsberg and Artru, 1977; Garnir et al., 1978; Kononov et al., 1983; Kramida et al., 1983a,b). All of these achievements give a more solid ground for theoretical predictions, design of laser systems, and experiments directed to further progress in the field of shortwave lasers.

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Anderson, D., McCullen, J., Scully, M. O., and Seeley, J. E. (1976). Opt. Commun. 17, 226. Artru, M.-C., and Kaufman, V. (1975). JOSA 65,594. Attwood, D.T., andHenke, B. L.,eds. (1981). Proc. ConJLow EnergyX-RayDiagn., 75th, Am. Inst. Phys., New York. Baldwin, G. G., Solem, J. C., and Goldansky, V. I. (1981). Rev. Mod. Phys. 53,687. Bhagavatula, V. A. (1976). J. Appl. Phys. 47,4535. Bhagavatula, V. A., and Yakobi, B. (1978). Opt. Commun. 24,331. Bohn, W. L. (1974).Appl. Phys. Lett. 24, 15. Boiko, V. A,, Brunetkin, B. A., Bunkin, F. V. ef al. (1983). Kvantovaya Electron. 10, 901 (in Russian). Bokor, J., Babsbaum, P. A., and Freeman, R. R. (1983). Invited Paper, CLEA-83. Baltimore. Brillet, W.-U. L., and Artru, M.-C. (1976). Phys Scripfa 14,285. Colombant, D. J., Whitney, K. G., Tidman, D. A., Windsor, N. K., and Davis, J. (1975). J. Phys. Fluids 18, 12. Dahlback, G.D., Dukart, R., Fortner, R., Dietrich, D., and Steward, S. (1982).Appl. Phys. B28, 152. Danilychev, V. A., Zvorykin, V. D., Kholin, I. V., and Chugunov, A. Y. (1982). Kvantovaya Electron. 9, 92 (in Russian). Dewhurst, R. J., Jacoby, D., Pert, G. J., and Ramsden, S. A. (1976). Phys. Rev. Lett. 37, 1265. Dixon, R. H., and Elton, R. C. (1977). Phys. Rev. Lett. 38, 1072. Dixon, R. H., Seeley, J. F., and Elton, R. C. (1982). Proc. Top. Meet. Laser Tech. EUV Speclrosc., Boulder, March 8- 10. Eidelsberg, M., and Artru, M.-C. (1977). Phys. Scripfa 16, 109. Elton, R. C. (1982). Opt. Eng. 21, 307. Feldman, V., Bhatia, A. K., and Suchwer, S. (1983). J. Appl. Phys. 54,2188. Gaponov, S. V., and Salashchenko, N. N. (1982). Izvest. Akad. Nauk SSSR 46, 1543 (in Russian). Garnir, H. P., Baudinet-Robinet, Y., and Dumont, P. D. (1978). Phys. Scripta 17,463. Gudzenko, L. I., and Shelepin, L. A. (1964). Sov. Phys. JEPT 18,998. Haelbich, R., Segmiiller, A., and Spiller, E. (1978). DESY SR-78/2 I . Hagelstein, P. L. (198 I). Unpublished Ph.D. thesis, Lawrence Livermore Lab. Holstein, T. (1947). Phys. Rev. 72, 1212. Holstein, T. (I95 I). Phys. Rev. 83, 1159. Iljukhin, A. A., Peregudov, G. V., Ragozin, E. N., Sobel’man, 1. I., and Chirkov, V. A. (1977a). Pis’ma JETP 25,569. Iljukhin, A. A., Peregudov, G. V., Ragozin, E. N., and Chirkov, V. A. (1977b). Kvantovaya Electron. 4, 9 19 (in Russian). Irons, F. E., and Peacock, N. J. (1974). J. Phys. B 7, 1109. Jacoby, D., Pert, G. J., Ramsden, S. A,, Shorrock, L., and Tallents, G. J. (I98 la). “Lasers and Applications,” Vol. 26, p. 228. Springer-Verlag, Berlin and New York. Jacoby, D., Pert, G. J., Ramsden, S. A,, Shorrock, L. D., and Tallents, G. J. (1981b). Opt. Commun. 37, 193. Jaegle, P. ( I 98 1). Invited paper, Gen. Con/: Eur. Phys. SOC.,5th, Istanbul, 7- 11 Sept. Jaegle, P., Carillon, A., Dhez, P., Surean, A., and Cukier, M. (1971). Phys. Left A 36, 167. Key, M. H., Lewis, C. L. S., and Lamb, M. J. (1979). Opt. Commun. 28, 331. Kononov, E. Y., Koshelev, K. N., Levykin, Y. A,, Sidel’nikov, Y. V., andChunlov, S. S. (1976). Kvantovaya Electron. 3, 576 (in Russian). Kononov, E. Ya., Kramida, A. E., Podobedova, L. I., Ragozin, E. N., andchirkov, V. A. (1983). Phys. Scripla 28, 330.

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Kramida, A. E., Podobedova L. I., Ragozin, E. N., and Chirkov, V. A. (1983a). Lebedev Physical Institute, Preprint n"236 (in Russian). Kramida, A. E. et al. ( I 983b). Spectroscopy Institute, Preprint n" 16 (in Russian). Latush, E. L., and Sem, M. P. (1973). JETP 64,2017 (in Russian). Norton, B. A,, and Peacock, N. J. (1975). J. Phys. B S , 989. Palumbo, L. J., and Elton, R. C. (1977). J. Opt. SOC.Am. 67,480. Pixton, P. M., and Fowles, G. R. ( I 969). Phys. Lett. A 29, 654. Plotkin, M. E., and Ragozin, E. N. (1981). J. Tech. Phys. 51, 361 (in Russian). Rejntjes, J., Eckardt, R. C., She, G. Y., Karangelen, N. E., Andrews, R. A., and Elton, R. C. (1 976). Phys. Rev. Lett. 37, 1540. Rejntjes, J., She, C. Y., Eckardt, R. C., Karangelen, N. E., Andrews, R. A., and Elton, R. C. (1977). Appl. Phys. Lett. 30,480. Silfvast, W. T., and Green, J. ( 1 977). Proc. CLEA, Washington D.C. Tkach, R., Mahr, H., Tany, C. L., and Hartman, P. L. (1 980). Phys. Rev. Left.45, 542. Vinogradov, A. V., and Shlyaptsev, V. N. (1980). Kvantovaya Electron. 7, 1319 (in Russian). Vinogradov, A. V., and Shlyaptsev, V. N. (1983a). Kvantovaya Electron. 10,5 16 (in Russian). Vinogradov, A. V., and Shlyaptsev, V. N. (1983b). Kvantovuya Electron. 10,2325 (in Russian). Vinogradov, A. V., and Sobelman, I. I. (1972). JETP63,2113 (in Russian). Vinogradov, A. V., Skobelev, I. Y., Sobelman, I. I., and Yukov, E. A. (1975a). Kvantovaya Electron. 2, 2 189 (in Russian). Vinogradov, A. V., Sobelman, I. I., and Yukov, E. A. (1975b). Kvantovaya Electron. 2,105 (in Russian). Vinogradov, A. V., Sobelman, I. I., and Yukov, E. A. (1977). Kvantovaya Electron. 4,63. Vinogradov, A. V., Sobelman, I. I., and Yukov, E. A. (1978). J. Phys. (Paris) 39, C4-61. Waynant, R. W., and Elton, R. C. (1976). Proc. IEEE66, 1059.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS.VOL. 20

RADIATIVE PROPERTIES OF RYDBERG STATES IN RESONANT CAVITIES S . HAROCHE and J . M . RAIMOND Laboratoire de Spectroscopie Hertzienne de I'Ecole Normale Supkrieure UniversilO de Paris VI Paris. France

I . Radiative Properties of Rydberg States in Free Space . . . . . . . . . A . Electric Dipole Transitions in Alkali Rydberg Atoms: Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . B. Observation of Spontaneous Emission Processes in Rydberg States . C. Interaction of Rydberg Atoms with Blackbody Radiation . . . . . D . Collective Radiative Properties of Rydberg Atoms in Free Space . . E. Spectroscopy of Rydberg Atoms . . . . . . . . . . . . . . . . I1. Brief Survey of Experimental Techniques . . . . . . . . . . . . . . A . Atom Preparation . . . . . . . . . . . . . . . . . . . . . . . B . The Resonant Cavity . . . . . . . . . . . . . . . . . . . . . C. Realization of a Quasi "Two-Level'' Atom System . . . . . . . . D . Detection of Rydberg Atom Evolution. . . . . . . . . . . . . . E . External Radiation Acting on the Atoms. . . . . . . . . . . . . 111. Single Rydberg Atom in a Resonant Cavity . . . . . . . . . . . . . A . The Theoretical Framework . . . . . . . . . . . . . . . . . . B. Spontaneous Emission of a Single Atom in a Resonant Cavity at T=OK . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Single Atom Interacting with Blackbody Radiation in a Resonant Cavity at T # 0 K: Rabi Nutation in a Chaotic Field . . . . . . . D. Single Atom in a Coherent Cavity Field . . . . . . . . . . . . . IV . Collective Behavior of N Rydberg Atoms in a Resonant Cavity . . . . A . Collective Emission in the Cavity in the SchrlMinger Picture . . . . B. Collective Emission of N Atoms in the Cavity in the Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Observation of Collective Emission of Rydberg Atoms in a Resonant Millimeter-Wave Cavity: Quantitative Check of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Collective Absorption in a Resonant Cavity . . . . . . . . . . . V . Conclusion and Perspectives. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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341 Copyright 0 1985 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12003820-X

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Rydberg atoms are excited atomic systems in which an electron has been promoted to a level with a very large principal quantum number n. These systems have been the object of much theoretical and experimental investigations in past years. Their radiative properties are very interesting for several reasons: first, the size of these atoms and hence the electric dipole matrix elements between neighbor levels- proportional to n2-are typically ( n 30) three orders of magnitude larger than the corresponding quantities for ground or low excited atomic or molecular states. As a result, the intrinsic coupling of these atoms to the radiation field is unusually strong. Second, transitions between Rydberg levels fall in the millimeter-wave range. Quantum effects involving ultimately the interaction of a single photon with a single atom are observable, which is quite novel in this frequency domain. Third, these atoms have relatively long spontaneous emission lifetimes, which means that in spite of their very strong coupling to microwaves, they can live for a very long time in excited states and basically behave as metastable species on which very precise spectroscopic investigations can be made. This is the first article on Rydberg atoms in the present volume. The article, by Gallas, Leuchs, Walther, and Figger (GLWF), is concerned pnmarily with general methods and recent advances in production, detection, and spectroscopic techniques, whereas the present article is concerned pnmanly with the interaction of Rydberg atoms with radiation fields inside resonant cavities. Among all the radiation - Rydberg atom experiments, those performed in a cavity are especially interesting. It is indeed possible to prepare the atoms in a relatively large (cm size) low-order resonant cavity in which they can stay for a time long enough for the coherent coupling with the cavity mode to become important. Very efficient and sensitive field ionization techniques can be used to study the evolution resulting from this coupling. This simple experimental situation corresponds to a well-known quantum optics model developed in a large number of theoretical papers over the last 20 years (Jaynes and Cummings, 1963; Tavis and Cummings, 1969; Bonifacio and Preparata, 1970; Scharf, 1970; Allen and Eberly, 1975; Knight and Miloni, 1980). It is the quasi-ideal realization of an ensemble of two-level atoms interacting with a single radiation mode. A whole class of effects had been predicted with this model, but has been never observed so far, because it was very difficult to realize the corresponding experiments with “ordinary” atomic transitions. The spontaneous emission of a single atom in the cavity is expected to be drastically modified as compared to its free space properties. In some cases, it is enhanced by the presence of the cavity walls (Purcell, 1946;Goy et al., 1983). It can in other instances- if the cavity finesse is very

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high- be replaced by a regime of reversible exchange of a photon between the atom and the field. If the cavity is off-resonant for the atomic transition, spontaneous emission can even be altogether suppressed (Kleppner, 1981). All these effectscan be understood in terms of modifications of the boundary conditions for the vacuum field around the atom due to the presence of the cavity. If the field in the resonator is initially excited (by thermal or coherent sources), various kinds of Rabi nutation phenomena have been predicted (Cummings, 1965; von Foerster, 1975; Eberly et al., 1980;Knight and Radmore, 1982; 1983). IfNatoms are initially prepared in the resonator, they essentially behave as a single quantum object equivalent to a spin N/2. The collective spontaneous emission of such a system, starting from a state of maximum atomic energy, is closely related to the well-known problem of superradiance (Dicke, 1954; Aganval, 1970; Bonifacio et al., 1971;Gross and Haroche, 1982).In fact, the system is then formally equivalent to a pendulum initially in its unstable equilibrium position and triggered away from this state by quantum or thermal fluctuations. This system is known to exhibit macroscopic fluctuations throughout its evolution, which reflect the randomness of its initial triggering. The experimental study of Rydberg atoms in a cavity provides -a very complete and precise check of the predictions of this model (Raimond et al., 1982b; Kaluzny et al., 1983, 1985; Kaluzny, 1984). The collective absorption properties of a sample of Natoms initially in the lower state of the transition resonant with acavity mode at temperature Tare also interesting. The collective system is then analogous to a pendulum exhibiting Brownian motion near its stable equilibrium position. Here again, Rydberg atoms in cavities provide an elegant way to analyze experimentally this new kind of Brownian motion (Raimond et al., 1982a). Over the last few years, several articles have been devoted to various aspects of Rydberg atom radiative properties in free space or in cavities (Haroche, 198 1, 1984;Gallagher, 1983;Fabre, 1982;Moi et al., 1983;Fabre and Haroche, 1983; in addition to GLWF). Since new results have very recently been obtained, we intend in this review to present a description of the latest experiments dealing with Rydberg atoms in cavities, together with a comprehensive theoretical framework which gives a simple interpretation of these results. In Section I, we briefly recall the main radiative properties of Rydberg atoms in free space (no cavity around the atom) and rapidly describe the experimental studies in which these properties have been investigated. In Section 11, we give a brief survey of the experimental techniques developed to study the coherent coupling of Rydberg atoms to a cavity mode. Section 111is devoted to the analysis of single atom effects and Section IV to collective radiative behavior in the cavity.

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I. Radiative Properties of Rydberg States in Free Space In this section, we consider details of the radiative properties of alkali Rydberg atoms in free space and analyze the general consequences of their very strong coupling to the low frequency part of the electromagnetic spectrum, which makes them very attractive to study matter - field coupling effects at an unusual scale (see GLWF for a more general discussion). We recall first some orders of magnitude about isolated Rydberg atoms and their coupling to the vacuum field (Section I, A) and briefly describe some experiments in which spontaneous emission from Rydberg states has been detected (Section 1,B). We then discuss the effects of blackbody radiation on these atoms which turn out to be quite dramatic at room temperature (Section 1,C). The collective radiative properties of these atoms is then discussed (Section 1,D) and we conclude the section with a brief analysis of experiments in which Rydberg atoms interact with an applied coherent field (Section 1,E). A. ELECTRIC DIPOLETRANSITIONS IN ALKALIRYDBERG ATOMS:ORDERSOF MAGNITUDE

Rydberg states currently studied in laboratory environment have principal quantum numbers n ranging from about 10 to 100,whereas radioastronomy observations deal with n values in the range of a few hundred (Hoglund and Mezger, 1965;Dalgarno, 1983).For such states, in a classical picture, the outer electron revolves far away from the core made of the nucleus and the other electrons (at a distance of the order of n2uowhere a, is the Bohr radius). In first approximation, this core behaves as a point-like charge. The Rydberg spectrum is thus quasi-hydrogenic (the energy of level n being -R/n2, where R is the Rydberg constant taking the reduced mass of the electron into effect). In fact, the finite core size slightly perturbs this simple model in a way which -for alkali atoms at least -is very simple to describe with the help of the so-called quantum defect parameters (Seaton, 1958).The energy En,of a level of angular momentum 1 simply reads (see Table 1, GLWF)

En/= -R/(n -

(1)

where 6,is a number-at most of the order of a few units- which depends to a very good approximation only upon 1(and of course upon the species). The term 6, describes in fact the dephasing of the wave function of the Rydberg electron which is continuously scattered by the nonhydrogenic

RADIATIVE PROPERTIES OF RYDBERG STATES

35 1

atomic core. Apart from the spectrum modification described by Eq. (1) and the corresponding change of the wave functions, the radiative properties of all alkali Rydberg states are very similar to the hydrogen ones and the orders of magnitude of the important radiative parameters can be easily estimated with the help of the hydrogen model (Fabre, 1982). The typical binding energy of an nl state is Ed -R/n2 (about - 10 THz for n = 30). The energy difference between nearby levels are of the order of R/n3(about 100GHz for n = 30). The electric dipole matrix elements corresponding to such transitions are proportional to the atom size, ofthe order of qa,nz, where q is the electron charge. For a value n = 30, they are thus about three orders of magnitude larger than typical electric dipoles between ordinary atomic states corresponding to low quantum numbers. There are exact analytical expressions of these matrix elements for hydrogen (Bethe and Salpeter, 1957).For alkalis, we have to take into account the quantum defect dephasings and the reduced electric dipole matrix element between nearby levels nl and n’l’ can be expressed as (Fabre, 1982)’

-

dnr,n.r. = ( nll Iqrl ln’l’)

= zquon2g(n*- n’*)

(2) where g(n* - n’*) is a dimensionless function of n* - n’* (n* = n - 6,and n’* = n‘ - drpare the so-called effective quantum numbers of levels nl and n’l’). This universal function (independent of the species) has been calculated and tabulated by various authors (Picart et al., 1979; Davydkin and Zon, 1981). It starts from 1 for n* = n‘* and exhibits damped oscillations versus n* - n’*, with quasi-periodic cancellations resulting from destructive interference between the corresponding dephased wave functions. Equation (2) is valid only if n* - n’* n are also easy to obtain (Fabre, 1982). In particular the dipole matrix elements to low-excited states with n’ 1 are shown to be of the order of a,. The wave function of the final n’l’ level is indeed confined to a very small region of the order of a. .The dipole matrix element is thus proportional to the amplitude of the nl wave function around r = 0, which scales as i r 3 l 2 . For n 30, such matrix elements are of the order of 10-2 a.u., which explains why relatively strong light intensities are required to prepare these Rydberg states from ground state with a good efficiency. These electric dipole matrix elements are directly related to the partial

-

-

I The reduced quantities d,l,n.l.and the matrix elements ( rn,,,)*of GLWF [Eq. (I)] are related by ( rnn,)z= (rnax([J’)/(21+ 1)) d;,,,.,

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spontaneous emission rates between the corresponding levels [see Eq. (l), GLWF]:

is the transition pulsation). For transitions between nearby levels, and provided g remains of the order of unity, di/,,,,, scales as n4and coil,n,p as nT9. rg&,,, then varies as n-5, with a typical order of magnitude 100/sec for n - 30. Although this appears as a small absolute rate, it is indeed quite large compared to the emission rates of “ordinary” microwave transitions in low 10-9/sec range). lying states atoms or molecules (ris then in the For transitions toward n‘ 1 states, d;l,n,l,scales as n-3, whereas conl,njl,is with a typical value of the nearly n independent. rgf’n.r, then scales as order of 105/secfor n 30. The total spontaneous emission rate for an nl level is

-

-

r?= q n’

(4)

’

For a low angular momentum state (l En,,,),the induced rate described by Eq. (7) merely adds to the spontaneous emission rate, the total (spontaneous induced) emission probability being proportional to (n+ 1). Typical rates of transition at room temperature and for w 100 GHz are I‘kpkyY 104/sec.Such rates largely dominate the partial spontaneous emission rates on the same transition (6- 100 >> 1 for T = 300 K and o 10” Hz). In fact, blackbody transfer rates on An = 1 or An = 2 transitions are about 10- 20%the total spontaneous emission rates of nl level with 1 20) and can be interpreted as the mean vibration energy of a free electron in the blackbody field. We find then the value

-

-

of the order of 2.4 kHz at room temperature. For low lying states (n - l), the blackbody-induced shift is very small ( 0.03 Hz at 300 K) (Barton, 1972; Knight, 1972). As a result, the optical transitions connecting low-lying levels to high Rydberg states have a temperature-dependent frequency. This dependence has been recently observed in an experiment using very monochromatic dye lasers (Hollberg and Hall, 1983). (See Section VI, GLWF for a further discussion of Rydberg atoms interacting with blackbody radiation.)

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D. COLLECTIVE RADIATIVE PROPERTIES OF RYDBERG ATOMS IN FREESPACE

We have so far discussed single atom effects, in which isolated atoms interact with the vacuum or thermal field. In fact, if a large enough number

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of atoms is initially prepared in a Rydberg level, the field radiated by the atoms reacts back on them and collective radiative effects become observable. This is of course a common feature to all atomic systems. In the case of Rydberg atoms, however, the very strong coupling to the radiation field makes the threshold of these collective effects, known as superradiance, very low. To observe them, it is sufficient to initially prepare a sample of atoms in a given nl state ( n 30) and to monitor the population transfers toward lower n - 1 levels. The rate of this transfer undergoes a dramatic increase when the number N of atoms is larger than a threshold number (typically of the order of lo5) (Gross et al., 1979). The reciprocal of the characteristic emission time TSRofthe superradiant emission between two level nl and n'l' is proportional to N and to the corresponding spontaneous emission rate (Gross and Haroche, 1982),

-

T;; = NpI-Er-,,,,, (9) wherep is a geometric factor (essentiallyproportional to the solid angle of the emission) depending on the ratio of the sample size to the transition wavelength. For sample dimensions of the order of the wavelength, typically p - lo-'. Equation (9) shows that Ti; rates of the order of 106/sec are obtained on An = 1 transitions around n - 30 for N - lo5 atoms. T;; is then of the order of or larger than all the other atomic decay rates (I-JF:?k?!',Y * * ) and the collective effects dominate the system evolution. These effects can occur on cascading transitions. The details of the emission dynamics depend on the initial phase of the process, during which strong interatomic correlations build up. This phase is dominated-at room temperature -by blackbody-induced emission so that superradiance in Rydberg levels can be described as a transient amplification of the blackbody background by the atoms. This effect has been observed on a large number of millimeter-wave and infrared transitions. Its detailed analysis is complicated by various diffraction and propagation effects of the field along the sample. We describe in section IV a similar collective radiation effect occumng in a resonant cavity, which is much simpler to analyze theoretically and to study experimentally.

E. SPECTROSCOPY OF RYDBERG ATOMS We have considered so far the coupling of Rydberg atoms to incoherent or self-radiated fields. Let us conclude this review with a brief description of spectroscopic experiments in which Rydberg atoms have been coupled to a coherent monochromatic field. These experiments were primarily aimed at the precise measurement of energy intervals in order to test the quantum

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defect model. Some experiments were carried out with tunable lasers exciting transitions from low-lying states to Rydberg ones (Fabre and Haroche, 1983, and references therein). Others make use of a double resonance scheme: a laser excites an initial Rydberg level and a millimeter-wave or radiofrequency source induces a transition toward a nearby state (Fabre et al., 1978;Goy et al., 1982).Figure 2 shows a typical example ofspectra obtained in such double resonance experiments. These experiments have allowed a very precise determination of quantum defects, in which the very slight variation of these quantities with n has been put in evidence. A very striking feature of those Rydberg - Rydberg millimeter-wave transitions is the very low power necessary to saturate them. The saturation is obtained when the Rabi frequency dfl,,fl,,,&/h, where & is the field amplitude, becomes of the order of the linewidth (determined either by spontaneous emission broadening or by Doppler effects, transit time, or stray electric fields Stark broadening). As dfll,fl.l, is very large, the saturation power of an nS+ nP transition in alkalis, for instance, is only in the 10-lo to 10-l2 W/cm2 range ( n - 30)! Two-photon nS + ( n 1)s transitions are also very easily induced in alkalis, since the relay level nP stands almost exactly in between the S states: saturation power is typically as low as W/cm2 for n 30 (Goy et al., 1982). These spectroscopic investigations of Rydberg states spectra are interesting since the measurement of a small number of parameters allow us to predict, with good accuracy (using the quantum defect formula), the frequency of any transition between highly excited states: they provide a frequency reference grid in the far infrared (Goy et al., 1982).

+

-

213.025

FIG.2. Millimeter-wavespectrum ofthe 28S,/, 28P,/, transition in cesium obtained by a double-resonance scheme (from Goy ef al., 1982). A laser prepares atoms in the 28S,,, level. A tunable microwave is applied on these atoms, which resonantly transfers population to the 28P,,, level. This transfer is monitored by the field-ionizationtechnique (see Section 11,D).The hyperfine splittings of 28S,,* and 28P,,, levels are resolved (the lines are identified by the numbered arrows in the insert). Horizontal scale in gigaHertz. +

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11. Brief Survey of Experimental Techniques The experiments we describe in this section deal with the interaction of a sample of Rydberg atoms with a single mode of an electromagnetic field in a resonant cavity. The aim of the experiments is to test various simple effects of quantum optics, predicted in a large variety oftheoretical papers over the last 20 years, but which have been very difficult or impossible to observe so far due to experimental difficulties. The advent of techniques permitting us to prepare and study the evolution of Rydberg atoms in millimeter-wave cavities has opened the way to these studies and permitted us to check in details these theories for the first time. To allow for a precise quantitative comparison between experiments and theory, we have to realize very clean experimental conditions in which a sample of N quasi-ideal “two-level” atoms is prepared at agiven time in a well-defined position in a high Qresonant cavity and we should be able to study the evolution of this sample at subsequent time. We briefly introduce in this section the experimental methods we have developed to meet these requirements. Since these methods have been described already in previous publications (Goy et al., 1983; Raimond et al., 1982a,b),we restrict ourselves here to a very general description.

A. ATOMPREPARATION

The sketch of the set-up is shown in Fig. 3. Rydberg states of alkali atoms

(nS,,, states of sodium in these experiments) are prepared by pulsed laser excitation of a thermal atomic beam. The stepwise excitation scheme is 3S1/2-3P1/2-nS1/2,involving 5896 and 4100 A laser wavelengths. The pulse duration is about 5 nsec, much shorter than all characteristic evolution time of the system, so that we can consider the peparation to be instantaneous. The laser power and atomic beam density can be varied within large ranges, so that the mean number Nof atoms excited at each laser pulse can be as small as 1 or as large as 106.2

-

Laser excitation with a small number of optical photons allows us of course to prepare Rydberg states with low land m quantum numbers (Sstates in our experiments).It has recently been shown that the use of optical excitation in conjunction with a succession of microwave transitions in the adiabatic rapid passage regime permits us to prepare selectivelyand efficiently high m states (“circular” states with m = I = n - 1) (Hulet and Kleppner, 1983).These states have interesting radiative properties (see Section I) which certainly will be investigated experimentally in the near future.

RADIATIVE PROPERTIES OF RYDBERG STATES

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$s Frequency multiplier

I

I

359

Signal

-

Alanic ban

-

Field ionizatim plater

____

FIG.3. Scheme of the experimental set-up for Rydberg atom cavity experiments showing the atomic and laser beams and the Fabry- Perot cavity. The coils whose section is shown in the figure provide the axial magnetic field needed to isolate a two-level atom system. The electrode provides the inhomogeneouselectric field required for the time resolution of the system evolution detection. Atomic detection is performed downstream of the atomic beam by applying an electric field between the field ionization plates and detecting the resulting electrons in the electron multiplier (EM). If needed, a coherent driving field is generated by the frequency multipliedX-band klystron; incoherent blackbody radiation can also be coupled into the cavity with the help of the electrically heated wire W.

B. THE RESONANT CAVITY The atomic excitation occurs within a millimeter-wave cavity which, in all our experiments, is an “open” semi-confocal Fabry-Perot resonator. In a typical cavity, the intermirror length L is 1 cm (mirror radius r = 2L 2 cm). The cavity sustains a Gaussian mode with awaist w, 2 mm. The effective mode volume is Y = 7rLw34 0.1 cm3.The typical quality factor Q is in the range lo4- lo6 and the cavity finessef= QA/2Lis of the order of lo2- lo4. The average time spent by atoms flying across the cavity waist is At - 2w0/v, 3 psec (vth is the thermal drift velocity lo3 m/sec). The above cavity parameters are only indicative. Other types of millimeter-wave cavities can of course be considered. The exact calculation of the cavity parameters will be discussed in the next section, and further details about the cavities used in each experiment given later.

-

-

-

-

-

-

c. REALIZATION OF A QUASI“TWO-LEVEL” ATOMSYSTEM In most experiments, we excite an nS,,,level and tune the cavity into transition: Each of the levels implied in the resonance with an nS,,,+ n’P,,,

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S. Haroche and J. M. Raimond

*

transition has a two-fold degeneracy3 (magnetic sublevels m, = 3): the cavity sustains in this case the two a+ and a- circularly polarized transitions, corresponding to photons propagating along the cavity axis 02. By applying a magnetic field B , parallel to this axis, we induce a Zeeman effect which removes the degeneracy of the a+ and a- modes. Tuning the cavity towards the a+ (or the a-) Zeeman component amounts to selecting a single mode, the coupling of the field with the off-resonant transition being basically suppressed. This Zeeman tuning is achieved with the help of small Helmholtz coils, coaxial with the cavity, which deliver magnetic fields ofthe order of 50 G. From now on, we will call le) and Ig) the two Zeeman sublevels selected by this process. Of course, spontaneous and blackbody-induced decay to lower state -or blackbody absorption processes to upper levels-are in fact also perturbing the atoms and the two-level model is only an approximation, valid for times shorter than the typical transfer times due to all these incoherent processes (of the order of a few microseconds in our experiments). D. DETECTION OF RYDBERG ATOMEVOLUTION The coherent coupling of Rydberg atoms with the field mode resonant on the le) + 18) transition induces a time evolution of the probability of finding the atoms in the corresponding levels. The most sensitive way of experimentally probing this evolution is to make use of the selective field ionization technique (Ducas et al., 1975). The method consists of applying a homogeneous electric field to the atoms after they have left the cavity. This field F , ( t ) is produced between two parallel condenser plates (see Fig. 3). We make use of the fact that the threshold for field ionization is different for levels le ) and 18) which have different binding energies: basically, the electric field F , (t) is applied as a saw-toothed pulse, which permits us to time-resolve the ionization of levels le) and Ig). The timing of the experiment is sketched on Fig. 4:at time t = 0, the laser pulses prepare the initial Rydberg level le) or 1s). During time At, the atoms interact with the field mode in the cavity. They then leave the cavity and enter inside the condenser. At time t, the ramp of electric field F , ( t )is started: it reaches at times t,and t,, respectively, the threshold for ionizing levels le) and 1s).The electrons resulting from ionization are accelerated towards a grid placed on one of the plates, cross it and reach a high-gain electron multiplier. The time-resolved electron peaks around time t, and t, have an area proportional to the number of atoms in We neglect here the hyperfine structure which, in Rydberg states, is generally too small to couple the electronic and nuclear spins during the time At.

RADIATIVE PROPERTIES OF RYDBERG STATES

1 0

I

I

I I

t

At

t,

I

]

i~ I

I

Ie

tg

36 1

F2(t)

*

TIME FIG.4. Time sequence of operationsto study the coherent evolution of Rydberg atoms in a the field applied on the “freezing” cavity.F , (f) is the field ionization detection field ramp, F2(f) electrode (see text).

the corresponding level at the detection time. During the time interval t , - At outside the cavity, the atoms evolve according to known processes (spontaneous and blackbody radiation-induced ones) so that it is possible to make the necessary corrections and to deduce the state of the system at the time At when the atoms leave the cavity mode. The method thus allows us to probe the state of the atomic system at the end of the atom-single mode interaction process (“final” state of the evolution). Directly probing the atoms inside the cavity to get information about the system before time At could be attempted by arranging the condenser plates inside or around the cavity. This is difficult, however, due to contradictory requirements of high finesse (for the cavity) and high field homogeneity (for the condenser): the presence of the condenser spoils the cavity Q and the cavity damages the field homogeneity. There is, however, another elegant technique to sample the system evolution during time At, with the condenser platesoutside the cavity as shown in Fig. 4.This is the “freezing electrode” technique (Raimond et al., 1982b) sketched in Fig. 4.At a given time t < At, a nonuniform electric field F2(t)is applied inside the cavity with the help of a small electrode (the electrode is entirely outside the mode waist, so that the Q of the cavity is not spoiled: see Fig. 3). This nonuniform field is too small to ionize the atoms: it Stark shifts (Zimmerman et af., 1980)the Rydberg levels out of resonance and interrupts the coherent coupling of the atomic system with the cavity mode without modifying the coupling to the continuum oftransverse modes (the continuum width is much larger than the Stark shift). From time t on, the coherent evolution is “frozen” and the ionizing field F , ( t )applied at time t , basically probes the system as it was inside the cavity at time t (here again, the evolution between times t and tl is incoherent and can be computed from the knowledge of the free atom spontaneous and blackbody radiation-induced transfer rates, since the effect of the Stark field on these incoherent processes is negligible). The system evolution is reconstructed by varying t and resuming the sequence.

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The electron multiplier (EM) gain calibration allows us to determine the absolute value ofthe number ofatoms in each level at time t (N,andN,). The experiment is run by a computer which defines boxcar gates around times t, and t,, measures the electron signals, normalizes according to the EM gain, drives the “freezing electrode,” corrects fort - t , evolution, makes statistics, stores and displays the data, etc. A very large counting dynamics is achieved by changing the EM high voltage. It is thus possible to study the evolution of very small systems (N - 1) or rather large ones (N los). This latter possibility will be taken advantage of in Section IV. In Section 111, we restrict ourselves to experiments in which the number of atoms excited at each laser pulse is very small, so that collective effects are negligible (see below).

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E. EXTERNALRADIATION ACTINGON THE ATOMS We now turn to the description of the external sources of radiation acting on the Rydberg atoms in the cavity. The cheapest one is no doubt the incoherent blackbody radiation, which is always present in the cavity mode if T > 0 K. It comes partly from the finite temperature cavity walls, but also from the “outside world” through diffraction coupling via the mirror edges (mainly at low frequencies, where the mode waist on the mirrors is not very small compared to the cavity transverse dimensions). We will see that this radiation plays a very important role in several interesting Rydberg atom-cavity experiments (we already noticed in Section I the extreme sensitivity of Rydberg states to the blackbody field). It is possible to reduce the blackbody background by cooling the apparatus to liquid helium temperature. It is also possible to increase it above room temperature, by means of a small electrically heated tungsten wire, slightly coupled to the mode (Fig. 3) (Raimond et al., 1982a) or by feeding radiation from a black microwave source in the mode through coupling holes pierced in the mirrors. By these techniques we are able to control the cavity mode temperature over the 4.2-900 K range. When it is necessary to apply a monochromatic field on the atoms, we use the harmonics of an X-band klystron locked on a quartz oscillator, produced in a high-frequency Schottky diode (Goy, 1982). This system provides continuously tunable radiation, between 40 and 300 GHz, with powers of the order of to W. If higher powers are required (two photon Rabi precession experiments for instance), we use backwave oscillator tubes, in the same frequency range, whose output power is in the 10 mW - 1 W range. These sources are frequency stabilized by harmonic mixing with the X-band klystron (Goy, 1982).

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363

RADIATIVE PROPERTIES OF RYDBERG STATES

111. Single Rydberg Atom in a Resonant Cavity We describe in this section the simplest atom -cavity coupling situation, where a single two-level atom is coupled to the cavity mode. (Experimentally this situation can be realized by decreasing the pumping laser intensity down to a level such that one atom at most is prepared in the cavity per laser shot.) This simple case will provide us with some important examples of coherent coupling between the atomic system and the field, even in the absence of any external coherent source of radiation. The analysis of the atom-field interaction in terms of irreversible transition rate made in Section I is valid only in the case of atomic coupling to a continuum of field modes, defined by a mode density distribution. In quantum optics, a wide range of effects results basically from the coherent interaction of the atomic system with a single radiation mode (defined by a resonant cavity). Reversible processes such as optical nutation are expected in this case. They play an important role for the understanding of laser and maser action in particular. Expressed in angular frequency units, the intrinsic coupling constant between the atom and the field mode is

R = d / h Jhw,/2~,,Y where ?I is the effective volume of the cavity and d the electric dipole matrix element of the transition (frequency coo). The term R can be considered as the Rabi nutation pulsation associated to the electric field of a single photon in the cavity. For “ordinary” atoms at optical frequencies,the cavity is necessarily of an “open” Fabry -Perot type and transverse modes, with wave vectors not pointing along the cavity axis, are significant. The term Sl is then a very small quantity as compared to the spontaneous rate r,describing the coupling to the continuum of transverse modes. For a typical optical transition, we have rT 10’- 108/secwhereas Y lo-’ in an order of magnitude estimate (for d ho, S.I. units) yields R - lo4- 105/sec.As a result, the “coherent” coupling of a single atom to a single mode of the field is negligible, unless there is a large average number i of photons in the mode such that R fi>rT,i.e., E>

-

-

-

106- 108.

In the case of Rydberg atoms, we consider-as described in Section II,B -millimeter-wave cavities resonant with a transition connecting two nearby lying states Inl) and ln’l’ ). These cavities can in principle be either closed and operating on their fundamental mode, or open (Fabry-Perot type) and operating on a higher order mode. As discussed in Section II,B, we have for practical reasons chosen this latter configuration. In either case, they

364

S. Haroche and J. M. Raimond

have a high finesse only at long wavelengths and do not appreciably modify the mode structure at optical frequencies. In spite of the small oovalues, the R parameters of such systems remain rather large, due to the dnr,ntr. huge values. We have typically R - 105/secfor d,r,,.l.ho,and ?I = m3. This coupling has now to be compared to the total spontaneous emission rate r$which -at least for low 1values -mostly corresponds to spontaneous emission at short wavelengths not affected by the cavity. The condition R > r$can be met and we clearly see that coherent atom-cavity coupling effects are expected to be observable for very small photon numbers ( n = 0 ultimately). In other words, as soon as the cavity is tuned to the frequency of a large dipole-electric allowed Rydberg transition, a situation in which a quasi two-level atomic system is coupled to a single cavity mode is automatically realized. A large variety of interesting effects have been (or could be) studied with this simple system using the experimental technique described in Section 11: enhancement of the spontaneous emission rate in the cavity (Purcell, 1946), Rabi oscillations either in the self-radiated field or in a blackbody chaotic field (Cummings, 1965; von Foerster, 1975; Knight and Radmore, 1983), and Rabi nutation collapse and revivals in an external coherent field (Cummings, 1965; Faist et al., 1972; Eberly et al., 1980; Knight and Radmore, 1982). We present in this section a simple theoretical model accounting for all these effects and we describe the first results of experiments in which they have been put in evidence. A. THETHEORETICAL FRAMEWORK

We have shown in Section I1 that it is possible to prepare a quasi-ideal two-level Rydberg atom coupled to a single field mode and to monitor its time evolution. We now proceed to analyze this system theoretically. We start by introducing the framework necessary for the description ofthe cavity mode and its relaxation (due to the finite cavity quality factor) and we present the equations describing the atom cavity system evolution, which will be solved under various conditions in the next section.

+

I . The Single-Mode Field The cavity-selected single-modefield is described by its creation and annihilation operators, a+ and a. These operators obey the standard commutation rule [a,a+] = 1

( 1 1)

RADIATIVE PROPERTIES OF RYDBERG STATES

365

The action of a and a+ on the state In) (photon occupation number n) is given by aln) = di In - 1 )

+I)

a+ln) =

(12)

The electric field operator in the mode is defined by

+

E(r) = [aef(r) a+e*f(r)] (13) In this expression, is a complex vector describing the polarization of the mode, assumed to be uniform in the cavity [e = (ex ie,,)/fi for o+ polarization, (ex - ie,,)/ fifor o- one, where ex and e,, are unitary vectors perpendicular to the cavities axis];f(r) is the relative field amplitude at point r in the standing wave pattern of the mode [f(r) is equal to unity at points where the field is maximum]; o,the cavity selected frequency, is assumed to be close to the atomic frequency 0,; Y is the cavity mode effective volume. It should be emphasized that, contrary to the case of most quantum electrodynamics calculations, where the field is quantized in an arbitrary virtual box of large dimensions, Y here has a well-defined value,

+

If we consider a cofocal, or semicofocal cavity sustaining a Gaussian mode, whose waist w, is w, =

[L: intermirror distance, L = (q 3, = 2nc/o],we get

(15)

+ 1/4) A12 with q integer, at resonance,

The volume of the cavity is thus of the order of 5A3 for a typical cavity in the tenth order. The fact that Y is not very large compared to A3 shows that the one photon electric field in the cavity may be rather large, providing an important atom -mode coupling.

2. The Cavity Mode Relaxation The damping time for the electromagnetic energy stored in the cavity (T,, = Q/o) ranges in our experiments from 10 nsec to 1 psec. It is of the order of the atom-cavity interaction time At, and of the typical atomic evolution times. It is thus clear that a realistic description of the system evolution must include these field relaxation processes.

366

S. Haroche and J. M . Raimond

Giving a complete description of the field relaxation theory would lead us far beyond the scope of this paper. We will thus only briefly recall the results derived in various references(Louise11and Marburger, 1967; Agarwal, 1970; Cohen-Tannoudji, 1976). The mode relaxation can be described in the Schrodinger picture, where the system density matrix is a function of time, as well as in the Heisenberg picture, where the system state is constant, and where the operators themselves are functions of time. Both pictures will be used in this paper. To describe the field relaxation in detail, a model for the “reservoir” to which the field is coupled is necessary. Fortunately, the form of the relaxation equations do not depend on the precise nature of this reservoir, which is due to the fact that the coupling ofthe field mode to the reservoir is always a linear combination of a and a+. Only the exact value of the Q factor depends on the reservoir model. Since we can always measure the Q from independent experiments, we do not need to derive it from first principles and we will not worry any more about the exact description of the reservoir. The rate equation describing the relaxation of the field mode, represented by its density matrix h,can be shown to be, in the Schrodinger picture,

dh/dt

(l/ih)

[HF,&] -k

AFh

(17)

where

is the field hamiltonian and AFis a relaxation Liouvillian operator defined as

In this equation, 6is the average number of blackbody photons per mode at temperature T given by Eq. (8) and Q( T) is a phenomenological temperature-dependent cavity quality factor. The first two terms describe obviously energy losses from the field to the reservoir, the two last ones absorption from reservoir blackbody energy. It can be checked very easily that this rate equation admits the steady state solution hnn, = dnn,(1 - p ) p”, when developed along the In ) states basis, which describes a field at thermal equilibrium [ p = exp - ( h co/k,T)]. In the Heisenberg picture, the action of the reservoir can be described by a

RADIATIVE PROPERTIES OF RYDBERG STATES

367

random “force” (Langevin force) Fa, the evolution of a being given by -d-a =-

1

0

+

[-a,H,] - --a Fa dt ih 29 The second term of this equation describes the relaxation of a (i.e., of the field) to its equilibrium value with a time constant 2Q/0. The last term describes the fluctuations of the field coupled to a large reservoir. Fa,resulting from the superpositionof a large number of random processes, has a very short correlation time t, of the order of the reciprocal of the reservoir frequency width, i.e., of the reciprocal of the mode frequency, and Gaussian statisticalproperties. From the field equilibrium state, we get the correlation properties of Fa, (Fa)= O ( FXt) Fa(t + 7)) = Da+o6r,(t) ( Fa(t) Ft;(t+ 7)) = Daa+67c(~) (F 3 t

(21)

+ 4 1 ) ) =Da+aerc(t)

where drCis a quasi-6 function having a time integral of 1, and being nonvanishing only for 191< t,, and O J t ) is defined by

& ( r )= 0

3. Atom

for

t

>0

+ Field Evolution

Let us now couple the two-level atom system to the field described in the previous section. It is convenient to introduce at this stage the spin 4 like operators for the two-level atom system D+, D- and D3 defined as

S. Haroche and J. M . Raimond

368

which obey the standard commutation rules

[D+,D-]= 2D3 [D3,D*]= kD’

The atomic energy is

and the atom -field interaction is written as

[we assume here that the atom is at an antinode position in the field, where f(r) = 1 and we neglect atomic motion through the cavity mode during A f : we also perform the rotating wave approximation] (Allen and Eberly, 1975). If we assume now that the atomic system has no relaxation of its own, we immediately get for the combined atom field density matrix the following rate equations in the Schrodinger picture

+

in which the commutator describes the “coherent” evolution processes and the AF term the “irreversible” decay ones. To get this equation, it has been implicitely assumed that the field relaxation mechanisms are not affected by the coupling to the atom. This is always true, since R is much smaller than T,’

.

In the Heisenberg picture, on the other hand, we get after a straightforward computation of the [a,HF HA,] and [D+,HA HM] commutators,

+

daldt

= -i o a

dD-/dt

+

- ( o / 2 Q )a = io,D-

+ i n D - + F,

- 2iRaD3

(30)

Atomic relaxation processes could be of course included in the theory, but we would then have to take into account the other levels of the Rydberg atom to which le) and Ig) are coupled by spontaneous and blackbody-induced radiative transfers. Since we wish to restrict ourselves to two-level atom effects, we will neglect these processes. This is justified by the fact that in our experiments, the atomic relaxation rates are of the order of 105/sec, smaller than the reciprocal of the atomic transit time through the cavity which sets an upper limit to the duration of each experiment.

369

RADIATIVE PROPERTIES OF RYDBERG STATES

B.

EMISSION O F A SINGLE ATOMIN RESONANT CAVITYAT T = 0 K

SPONTANEOUS

A

As a first very simple example, we discuss in this subsection the evolution of a single Rydberg atom initially prepared in the upper level le) of the transition resonant with the cavity mode, at T = 0 K (spontaneous emission in the cavity). Using the Schrodinger picture, we first theoretically describe the two regimes ofevolution, which depend on the respective values ofR and o/Q. We show that a “dressed-atom’’ description of the atom field system leads to a very clear physical interpretation. We then describe an experiment in which the regime corresponding to w/Q >> Q has been observed, which can be described as an effect of spontaneous emission rate enhancement due to the cavity.

+

1. The Two Regimes of Evolution

The system evolution is then described by Eq. (29) with E= 0. Only three states are relevant for the system description: the initial state 11 ) = le,O ) corresponding to an excited atom in an empty cavity, and the states 12 ) = Jg,1 ) and 13 ) = Jg,O), respectively, describing the atom in its lower level with one or zero photons in the mode. Of course, in the limiting case where w / Q 40, only the two states11 ) and 12 ) are coupled and the system evolution reduces to a reversible Rabi oscillation between these two levels at the exchange rate Q. The third state 13 ) is reached via field relaxation processes (photon decay in the mirrors: o / Q # 0). Developing the rate Eq. (29) along this three-state basis, we get a linear first-order differential equations system with the eigenfrequencies, w x0=--

o

o

x*=-- 2 Q f @

2Q

(’-

16t:Q2)

Thus, two regimes of evolution have to be distinguished, as follows.

a. Oscillatory regime. If the condition, o / Q < 4R is fulfilled, x* have an imaginary part of the order of 2Q. The population of states J 1 ), ) 2 ) ,13 ) contain oscillatory terms at this frequency, these oscilla-

370

S, Haroche and J. M . Raimond

tions being damped at the rate w/2Q (see Fig. 5a). We basically observe the quantum mechanical oscillations between the coupled discrete states leO ) and lgl ). These oscillations have a very clear physical meaning: condition (32) expresses that the photon emitted by the excited atom is stored in the cavity long enough for the atom to reabsorb it. Ofcourse the excitation of the cavity eventually decays away with a rate w/2Q. This regime is thus a sew Rabi nutation in the single-photon field emitted by the atom.

b. Overdamped regime. If, on the other hand, the cavity damping rate is high, w/Q > 4R

(33) x, are real eigenvalues. The populations of states 11 ) and 12 ) exhibit an irreversible damping. The time constant of this damping is essentially determined by the lowest eigenvalue x+: as soon as Ix+l > 4R), the decay is quasi-exponential with a rate,

Figure 5b shows the irreversible decay of the initial state population in this case. It is instructive to compare r,, to the free-space spontaneous emission rate on the same transition

FIG.5. Single two-level atom in a resonant cavity at T = 0 K: the two emission regimes (a) oscillation of the probability P,(t) of finding the atom excited at time I in case of weak cavity damping (w/Q = 0.2R), (b) irreversible decay ofP,(f) in case ofstrong cavity damping(w/Q = 5R).The time unit is the same for both curves (a) and (b) (equal to 2/R).

RADIATIVE PROPERTIES OF RYDBERG STATES

37 1

we find where

The spontaneous emission rate inside the cavity is thus greater than the one in free space by a ratio qcavof the order of magnitude ofthe cavity quality factor (if Y A3). This effect of spontaneous emission enhancement in a resonator has already been pointed out by Purcell in 1946, in the context of radiofrequency magnetic resonance. However, it was then a mere theoretical curiosity impossible to observe experimentally. We describe in Section III,B,3 an experiment in which we have taken advantage of the large coupling of Rydberg atoms to a millimeter-wave field and put in evidence this effect. The original argument developed by Purcell is somewhat different from the one just presented. Let us outline it here very briefly. We have described above the atomic spontaneous emission as an indirect relaxation process: the atom is not directly coupled to a continuum, as in the case of free-space spontaneous emission, but indirectly connected to the field reservoir via its coupling to the discrete cavity mode. This is not the only possible picture of the physical situation. It would be equally justified to describe the cavity mode as resulting from a superposition of modes with continuous frequenIn this alternative point of view, the atom cies, over a frequency range o/Q. would appear to be directly coupled to a radiation continuum of width o / Q . Condition (33) expresses in fact that this continuum width w/Q is much larger than the atom-to-field coupling matrix element Q. In this case, it is possible to analyze the irreversibleatomic damping by a Fermi Golden Rule argument. The spectral volumic density of a Lorentz-shape distribution of width Q/obeing (2/n)( Q / w )the rate of radiative transfer from level le) to Ig)is, according to the Fermi Golden Rule:

-

which is the same value as the one given in Eq. (35). In this picture, the spontaneous emission enhancement appears as due to the enhancement of the field mode spectral volumic density, from the free space value No(w)= w2/3a2c3 to the cavity mode value (2/n)(Q/w) (l/V). Another very simple physical interpretation of the spontaneous emission enhancement can be given in terms of electric images. The effect of the cavity mirrors on the atom evolution can be modelized by replacing them by the Q

3 72

S. Haroche and J. M. Raimond

-

images of the atoms in these mirrors (V A3). As the cavity is resonant with the atomic transition, all the dipoles of these images are in phase with the atomic one. We have thus Q aligned antennae in phase. A given antenna in this array radiates Q times faster than an isolated one. In other words, the atomic energy is dissipated in this system Q times faster than in free space: we find again the order of magnitude of the qcavenhancement factor. If V >> A3 (high-order mode cavity), there is an extra geometrical factor describingthe solid angle ofcoherence emission of the array ofantennae to take into account, which explains that the enhancement factor is then smaller than Q by a factor of the order of A3/V.

2. The Emission Process Revisited in the Dressed-Atom Picture Instead of analyzing the system evolution in the “uncoupled” basis of states le,n) and Ig,n), it is very convenient to study it on the basis of the eigenstates of the total atom field hamiltonian H A H, H,. This is the so-called “dressed-atom’’ picture, which has been developed in a very large number of atom radiation problems (Haroche, 197 1 ; Cohen-Tannoudji and Reynaud, 1977, 1978). We will see that this picture allows us to retrieve very simply the results of Section III,B, 1. The formalism introduced here will also be useful in Sections II1,C and III,D. The energy diagram of the hamiltonian HA H , H , consists of a ground state Ig,O ) and an ensemble of two-state manifolds, separated by the photon energy h w . The two “dressed” eigenstates in the nth manifolds are

+ +

+

+ +

+

I+ n ) = cos q,Jen) sin q,Jgn + 1 ) I- n ) = -sin qfllen)+ cos q,lgn + 1 ) with qnbeing defined by tan 2q,

=

2R

m

(at resonance qn = n/4)

0,- w

(39)

The splitting in the nth manifold is -

o,(n) = 14R2(n

+ 1 ) + (ao- o)211/2

(40)

and at resonances reduces to -

o,(n) = 2~ &Ti

(41) Obviously, those states are worth considering only if w - wo > 1. This phenomenon puts a strong limitation on the temperature in a single atom self-Rabi nutation experiment, which is of course much more easily met at high frequencies. The discussion carried above in the Schrodinger picture can of course be resumed in the Heisenberg formalism (Sachdev, 1984).

D. SINGLEATOMIN

A

COHERENT CAVITYFIELD

The last example we will consider is the case of a single atom in a coherent field present in the cavity. It is well known that such a field has a Poisson distribution of thephoton number and, as a result, the Rabi nutation washing out effect encountered in the last section is also bound to occur. This effect has been studied by various authors (Faist et al., 1972; Eberly et al., 1980; Knight and Radmore, 1982). We briefly discuss it here, along with a curious effect of “revivals” of the Rabi nutation signal at very long times which has been predicted but never observed so far in experiments. The Rydberg atom systems described in this paper certainly open promising opportunities of observing these effects in the future. I . Rabi Nutation in a Coherent Field Sustained by an Infinite Q Cavity

Consider an atom, initially prepared in state Ig) inside a cavity whose damping time is so long that field relaxation processes can be completely neglected. We will consider that this cavity has been somehow filled at time t = 0 by a coherent field in a Glauber state (Glauber, 1965),

Here no is the mean number of photons in the coherent field. As relaxation processes are neglected, the derivation ofP,(t)is very similar to the one presented in preceding paragraphs, and will not be reproduced

S. Haroche and J. M. Rairnond

380 here. We get

The difference in the sign of the modulated terms in Eqs. (48) and ( 5 1) comes from different assumed initial conditions (atom, respectively, in Ig) and le) at t = 0). The physical interpretation of this equation is obvious. As in Eq. (48), the signal appears to be the sum of Rabi oscillations in the field of n photons, weighted by the probability of having n photons in the coherent field. The dispersion of the photon number in a coherent state being of the order of the Rabi oscillations are expected to be washed out after a time,

6,

(Faist et al., 1972; Eberly et al., 1980; Knight and Radmore, 1982). This effect can be described more precisely by developing the function appearing in Eq. (52), around no:we can write

P,(t) can then be written as

L

x exp For short enough times, at/ can be developed as

P,(t) can be written finally as

(-

I) +

noeim/a

cc

(53)

is small and the e i n t / mterm in P,(t)

RADIATIVE PROPERTIES OF RYDBERG STATES

38 1

The modulation depth of the Rabi oscillations obeys thus a Gaussian law, and becomes negligible after a time tcouapse,given by

a value close to the one given intuitively in Eq. (53). The number of observable oscillations before the collapse is of the order of Solution (54) is valid only at short times

G.

dno+ 1 t > 1. Moreover, as the field is supposed to be large, and for times short enough (t < l/R), the dispersion of the Rabi frequencies around R can be neglected, and we get at resonance,

P J t ) = f[ 1 - cos ( 2 n m ) t l

(57)

A typical signal exhibiting this Rabi oscillation in a coherent field with

no = 650,000 photons is shown in Fig. 8. Note that the washing out effect observed on this figure is not due to the fundamental effects described above, but to experimental limitations (finite transit time in the cavity, dispersion of the atoms around the antinode position, etc.). Note also that there is more than one atom at a time in the cavity ( N - 1000) but that the radiative coupling effects between atoms are still negligible so that the signal is essen-

0

0.5

1

t(psec)

FIG. 8. Rabi nutation signal, recorder on the 34s- 34P,,, transition in sodium. The probabilityP,(r) of finding the atom excited at time r is plotted as a function of time (the atom is initially in state Ig)). The field in the cavity corresponds to a mean number of photons no = 650,000.

383

RADIATIVE PROPERTIES OF RYDBERG STATES

tially described here by a single atom model. The main novelty ofthis kind of Rabi oscillation is the extremely low field power required to observe them (ordinary Rabi precessions in magnetic resonance require n, lOI5- 10'' photons!). We see, however, that a lot of improvement remains to be done to observe the single photon oscillation effects discussed above (we recall that the Rabi pulsation varies as the single photon precession in the same cavity would thus occur in the 100-psec time scale). This Rabi oscillation phenomenon could have interesting applications as an absolute microwave power meter, for very low fields where conventional power meters are not very accurate: determining the Rabi frequency amounts to determine directly the field amplitude (the dipole matrix elements between Rydberg states are known to a good accuracy). Rabi nutation phenomena are not limited of course to photon dipolar transitions. Two photon nS + (n 1)s transitions are quite easily observed in sodium, since the transition takes benefit of the np level, close to the middle of the two S levels. Rabi oscillations are also observable on such transitions (Kaluzny, 1984).

-

6:

+

IV. Collective Behavior of N Rydberg Atoms in a Resonant Cavity We describe in this section the evolution ofNtwo-level atoms coupled to a single mode of the electromagnetic field in a cavity, and compare the predictions of the theory with experiments recently performed with Rydberg atoms. The fact that an ensemble of identical atoms excited in a resonator behave as a collective system is well known. It can be easily explained as resulting from the radiative coupling between the atoms via their common interaction with the spatially coherent field in the cavity. This coupling is usually described in terms of a collective damping rate ( T T ) - l = NT", proportional to the atom number N. In most usual cases, the observation of these effects requires very large atom numbers for T i 1to be larger than other relaxation rates (threshold condition for collective behavior, superradiance, maser action, etc.). The novelty of Rydberg atom physics is to provide situations where these effects occur for very small N values, in emission as well as in absorption experiments. In fact, with the set-up described in Section 11, these phenomena are very easy to observe since the pulsed laser excitation is strong enough to prepare a number of atoms of the order of lo4- lo6, far above the threshold of collective behavior (the single atom

384

S. Haroche and J. M. Raimond

signals described above are obtained by strong attenuation of the pumping laser intensity). We will discuss mainly here the simplest geometrical situation, where all the atoms are prepared in a region corresponding to a constant field amplitude in the cavity ( e g , an antinode position). For this configuration (very easy to realize with Rydberg atoms excited in a millimeter-wave cavity) the atomic sample is analogous to an angular momentum J = N/2, with N 1 equidistant nondegenerate levels. The various collective effects then appear as resulting from the simple resonant coupling of this angular momentum with the damped harmonic oscillator representing the field mode in the cavity (see Fig. 9). In this respect, the phenomena studied in this section generalize to a spin J > 1/2 the results obtained in the previous section for a single atom in the cavity ( J = 1/2): we will analyze the overdamped and oscillatory regimes of spontaneous emission of a large angular momentum in the cavity as well as the absorption of blackbody or coherent radiation by this system and describe various experiments in which these effects have been put in evidence with Rydberg atoms. We start by a brief qualitativediscussion of the two regimes of spontaneous emission in the Schrodinger picture (Section IV,A), which will allow us to make aclear connection with the results of the previous section. In the case of large N's, however, the Heisenberg picture proves more economical and we adopt it to give in Section IV,B a quantitative analysis of collective emission in the cavity. A description of recent experiments in which the predictions of the theory have been tested in details is presented in Section IV,C. We consider at last in Section IV,D the effect of collective absorption by an ensemble of Rydberg atoms in a cavity filled with a thermal or a coherent field.

+

FIG. 9. Diagram symbolizing the collective system under study: an atomic system A, analogous to an angular moment J = N / 2 ( N 1 energy levels) is coupled by the interaction HAFto the field mode F, analogous to a harmonic oscillator. This latter is in turn damped by a reservoir at temperature T (damping rate w / Q ) .

+

RADIATIVE PROPERTIES OF RYDBERG STATES

A. COLLECTIVE EMISSION I N THE CAVITY IN SCHR~DINGER PICTURE

385

THE

Let us consider a sample of N two-level atoms, motionless at points r, , r2 * * * rN in the cavity. We call D; and D? the spin-like operators for atom i. The atomic hamiltonian H A and the coupling H , is written as

H,

= -hR

+

[aDTf(ri) a+Drf(Fi)] i

(59)

For the sake of simplicity, let us also consider that all atoms are initially prepared in a small volume compared to A3, close to an antinode position in the standing wave pattern of the field, Equation (59) then simplifies with all f ( r i ) = I and we are naturally lead to introduce the collective operator,

D+=ZDT i

and similarly

D3=xDDj i

These operators obey the standard commutation rules of angular momentum. HA, now simply becomes

+

HAF= - h R(aD+ a+D-)

(62) and is clearly invariant by atom permutation. If the atomic system is initially prepared in a state also invariant by such permutation (e.g., in the level le, e, . . . ,e) corresponding to all atoms excited at t = 0), it will obviously remain at all times in a symmetrical state. These are known as the Dicke states (Dicke, 1954) of maximum cooperation number and are all generated by the repeated action of D- on le, e, . . . ,e).The N -I- 1 states obtained in this way are isomorphous to the eigenstates of an angular momentum J = N / 2 . This connection is made clear by writing them as IJM) = J ( J + M ) ! / N !( J - M ) ! (D-)J-MIe,e,

. . . ,e)

(63) The IJM) level is nondegenerate, has an atomic energy hw,M, and is the fully symmetrical Natom state in which J Matoms are excited in level l e ) and J - M are in level Ig). It is instructive to calculate the correlation between the dipoles of atoms i and j in the level IJM) (this quantity being obviously independent of the

+

386

S. Haroche and J. M. Raimond

couple i,j ) . Using the well-known expressions for the angular momentum operators matrix elements, we get

(JMIDTD,-JJM) =

J 2- M 2 N(N- I)

which shows that maximum correlations between the dipoles exist around the half de-excited level ( M 0): The spontaneous emission of the atomic system in the resonant cavity appears as a cascade emission down the ladder of the IJ M ) states, initially prepared in the level M = J. As this cascade proceeds, the field-mode energy ladder tends to fill up with photons, this process being in competition with the field relaxation mechanism which empties the cavity at the rate w/Q. As in the single-atom case, two regimes of emission have to be distinguished.

-

1. Regime of Collective Oscillation in a High Q Cavity

Let us first consider the case of an infinitely long cavity damping time, at T = 0 K. We expect that, in this situation, self-Rabi oscillations phenomena arise, as in the single-atom spontaneous emission: the field radiated by the atoms is stored in the cavity a long time, and is eventually reabsorbed by the atoms. The situation has been studied by Tavis and Cummings (1969), Bonifacio and Preparata (1970),and Scharf ( 1970). The evolution of the atom field system is described, without any relaxation term (since atomic relaxation is neglected), by

+

the initial density matrix being of course PA+F(O)

=I J J )

( JJI@IO ) ( 01.

Since the atom -field interaction H , couples only those states with the same atom field energy, we can restrict ourselves to the N + I states IJM,n ), with n = J - M. To simplify the notation, we write

+

IJ,M,n = J - M ) = IM)

(66) the ( N + 1)IM)states correspond all to the same eigenvalue of H A HF hamiltonian. The problem thus reduces to the diagonalization of the interaction term H , in this restricted basis. The matrix elements of H , are

+

( M + lIH,IM)=-hQ(J-M)JJ+M+ and H , is a symmetrical tridiagonal matrix.

1

(67)

RADIATIVE PROPERTIES OF RYDBERG STATES

387

The eigenstatesof this matrix constitute a dressed atom manifold, generalizing to the N atom case the two-state manifold ) introduced in Section III,B,2 to describe the evolution ofa single atom in the cavity at T = 0 K. To stress this correspondance, we will call IM, n = 0 ) the dressed states of this manifold. Of course, they are now N 1 levels in the manifold, which are merely linear combinations of the IJM n = J - M) levels. Knowing the dressed energies E z K e a c h level and developing the initial state IJJ, 0 ) = IJ) along the IM,O ) basis, we obtain immediately the wave vector of the combined atom field system at time t ,

1 s

+

Jm)

+

the average atomic energy ( H A), which is equal to (N/2)hcoo - ( HF), since the total energy is conserved, may be written from Eq. (68), ( H A ) = O O ( w(t)lD31u/(t))

This quantity exhibits oscillationsat the Bohr frequencies between dressed atoms levels. Analyzing the very complex Rabi oscillation obtained is possible only for a very low atom number, where H , can be explicity (or numerically) diagonalized, or asymptotically for large N values. For instance, for N = 2, there are three levels of energies f hi2 and 0, for N = 3 the four eigenvalues are f hi2 For greater N, up to 500, computer calculations remain feasible. The signals obtained clearly exhibit oscillatory atom - field energy exchange, with complicated beating patterns for low atom numbers, due to the summation of different frequencies components (see Fig. 10). The exact diagonalization of H?, takes longer and longer computer time as Nincrease (we have performed it up to N = 500). To get results which can be generalized for arbitrary N values, it is useful to analyze the asymptotic properties for large Ws of the HAFeigenvalues and eigenstates. Such an analysis-rather mathematical and difficult-was carried out by Scharf (1970). The main result of this study is that the dressed state energy scale is made of levels remaining nearly equidistant over a large energy range, with a spacing between nearby levels of the order of hi2 fi.As a result, the average atomic energy is predicted to evolve quasi-periodically,with a pseudo-period of the order of 1/Q fi.To illustrate this feature, we present, in Fig. 1 1 the result of a numerical calculation of ( HA(f)) for N = 500. The pseudo-periodic character of the system evolution is obvious. Let us remark that the increase of the oscillation frequency as fihas a simple explanation. Since

w.

388

S. Haroche and J. M. Raimond

FIG. 10. Evolution of the average atomic energy ( H A ( f ) in ) a perfect cavity (Q = ”) for A‘= 5 . The atoms are initially excited and the cavity empty ( T = 0 K). The time unit is I/Q. Note the occurrence of several frequency components in the evolution, leading to a beating pattern.

the average number of photons in the cavity is of the order of N , the average field amplitude is proportional to n a n d the Rabi nutation in this field must have a frequency of the order of f2 Another important feature of this phenomenon is the existence of important fluctuations (Bonifacio and Preparata, 1970) in the number of excited atoms at time t: a large distribution of Mlevels is populated, whose width is

m.

t

-500hI

*

, 0.2IR

1

I

1

TIME

FIG. 1 I . Evolution of the average atomic energy ( H A ( f )in) a perfect cavity for N = 500 (atoms initially excited and cavity empty). The time unit is 0.2/R (five times smaller than in Fig. 10).

389

RADIATIVE PROPERTIES OF RYDBERG STATES

also an oscillating function. A consequence of these fluctuations can be seen in Fig. 1 1 : the mean atomic energy always remains strictly larger than the ground state energy - ( N / 2 )h w,, . If the cavity is not perfect, the ocillations are expected to be damped with a time constant of the order of Q/w. The above description is of course no longer valid, and the system should be described by the whole atom field density matrix. The resulting equations are much too complicated to have analytical solutions, and only computer simulations are practical, provided though that N is not too high. We shall show (Section IV,B) that the Heisenberg picture is much more appropriate to describe this regime.

+

2. Overdamped Regime of Collective Emission in a “Moderate Q” Cavity If the cavity damping becomes of the order of or larger than S2 fi,the oscillations disappear and the emission becomes monotonous. In fact, in the limiting case where Q fi > E, the solution is

[This solution had been derived in other papers (Agarwal, 1970; Bonifacio et al., 197 1; De Giorgio and Ghielmetti, 197 1; Haake et al., 1979) in the T = 0 K case.] The term p M ( t )appears as a distribution exhibiting an exponential

390

S. Haroche and J. M. Raimond

behavior as a function of M at short times ( t < NT,,), then evolves into a bell-shaped distribution covering a wide range of M values and eventually ends up as a narrow distribution around M = - N/2 (see Fig. 12). (This behavior can also be obtained by numerical computation of Eq. (70); Bonifacio et al., 1971.) The characteristic evolution time of the system is TR= ( N P V ) - and l is thus N times shorter than the corresponding evolution of a single atom in the same cavity. This N-fold increase of the system evolution rate, typical of superradiance, is due to the build-up of strong dipole - dipole correlations for M around zero [see Eq. (64)]. The average delay of the emission process, defined as the time when the mean energy is zero, is

The results obtained in the Schrodinger picture are difficult to generalize to more complex situations (arbitrary value of the damping parameter, the case where the atoms are located in nonequivalent position in the field, a description of atomic motion through the mode, etc.). The difficulty of this

t

FIG. 12. Atomic evolution in a strongly damped cavity: the p M ( f distribution ) for N = 200, obtained by numerical integration of Eq. (70) for increasing values oft (from Bonifacio ef a/., 1971). Curves (a), (b), (c), (d), (e) correspond, respectively, to t / T , = 2.65,3.97,5.3,6.62,and 7.95. Note the evolution ofthe distribution which is narrow at small times [curve (a)], considerably broadens at intermediate times to cover practically the whole range of M values [(b), (c), (d)] and becomes narrow again at the end of the evolution (e).

RADIATIVE PROPERTIES OF RYDBERG STATES

39 1

approach comes from the fact that the density matrix of the system contains much more information that is required to analyze a simple experimentand thus leads to very complicated equations. We show now that the Heisenberg approach permits us to get a very simple solution for the problem and can be easily generalized to describe the various complications mentioned above. B. COLLECTIVE EMISSION OF N ATOMS IN THE CAVITY IN THE HEISENBERG PICTURE

From this point of view, the collectiveatomic polarization on one side and the electric field in the cavity on the other are described as coupled operators evolving in time. The atomic operator is in fact a vector (the so-called Bloch vector) (Abragam, 1961) representing the angular momentum J = N / 2 already introduced in the Schrodinger point of view. This vector is initially in an unstable position (correspondingto all atoms being excited) and is triggered away from it by a random torque describing the combined effects of vacuum and thermal fluctuations. This “picture” amounts to describingthe system evolution in term of the amplification of the Brownian motion of this Bloch vector. The Heisenberg evolution equations for the field and atomic operators, taking into account the field damping described by Eq. (20) are

(73)

(we adopt here the interaction representation and define ii = ae-jot and D+ = e-jotD+ and we assume resonant condition w o= o;Faand Fa have obviously the same statistical properties). At time t = 0, we consider that the field is in thermal equilibrium at temperature T; ii(0)obeys a Gaussian statistic with a mean value for the second moment,

(ii(0)ii+(O))

=

n+

1

(74)

On the other hand, the atomic system is fully excited and we have ( k ( 0 ) b+(O)) =0

(75)

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S. Haroche and J. M. Raimond

1. Linear Regime at the Beginning of the Emission

We consider first the beginning of the emission process, when the fluctuations of ii and Fa are important. In this regime, we have

and with the notation change

D+=nd

and

D-=nd+

(77)

which implies

[d,d+] = 1

(78)

Equation (73) becomes

These linearized equations describe the coupling of an inverted oscillator (creation and annihilation operators b+ and b) with the field oscillator. We have introduced the collective Rabi frequency

R,=Rfi

(80)

The general solution of this set of differential equations is obvious. Introducing the eigenfrequencies,

RADIATIVE PROPERTIES OF RYDBERG STATES

393

and

Equation (82) shows that 6 ( t )depends linearly upon the initial condition for the atomic polarization and for the field and upon the Langevin force between 0 and t. Since all those quantities have Gaussian statistics, 6(t)is also a Gaussian quantity, completely determined by its second moment. After some straightforward calculations based on the correlation properties of pa [Eq. (2 111, we get

which is valid in the linear regime, as long as (6+(t)h(t))> 1 >> eLt in Eq. (85),

( d + ( t )d ( t ) ) =

*m + l ) ( + &) (A- -

e2A+r

A+)2

(89)

Comparing Eqs. (88) and (89) immediately leads to the identification,

-4

( +2 3&

- 222 (ii+ 1) 1

=

n+ 1

It is thus equivalent to have a random noise acting on an initially fully

RADIATIVE PROPERTIES OF RYDBERG STATES

395

inverted system, or an initial tipping angle with randon Gaussian statistics, without any noise. It is thus justified to compute a statistical ensemble of classical trajectories with this much simpler random condition. Only one initial value O(0) and p(0) has to be chosen for each trajectory. The angle bp(0) is obviously equiprobablebetween zero and 2n and e(0) obeys to a Gaussian statistic with the following probability distribution,

around an average tipping angle g , , = 2 w N The initial derivative of 8 is on the other hand equal to zero, since it is proportional to &(O),

Computingthe trajectories with these initial conditions amounts to making use of the so-called tipping angle model of the superradiance theories (Bonifacio et al., 1971; De Giorgio, 1971; Bonifacio and Lugiato, 1975), which has been derived here under quite general conditions ( T = 0 K, arbitrary values to Q fiand o/Q). The procedure just outlined to reconstruct the statistics of classical trajectories is not the only one possible. Instead of assuming a small initial polarization with no electric field present in the cavity, it is equivalently possible to start at t = 0 from 8 = 0, but to apply a small random initial field on the atoms, which amounts to giving to 8 an initial nonvanishing velocity. It would also be possible to combine the 8 f 0 and (dO/dt)# 0 initial condition and to describe the evolution triggeringas a combination of polarization and field fluctuations. All of these procedures give of course the same result for the system statistics, after a time t such that eA+f>> 1 . Since the emitted field is too small to be observed before that time, the choice of a “triggering” condition is only a matter of computation convenience. Which of these equivalent “mathematical” choices gives a better description of the “physical reality” can be discussed. The reader can refer (if he is interested in this question)to the detailed discussion given by Dalibard et al. (1983) about the respective roles of atomic polarization fluctuation and vacuum field fluctuation in spontaneous emission. It is worth noting at this stage that the classical trajectories introduced in this discussion have a deep physical meaning, directly related to the description of a single realization of a collectiveemission experiment. Let us assume

396

S. Haroche and J. M. Raimond

that we perform a measurement of the field (or atomic polarization) at a time when enough photons have been emitted for the randon force Fa to have become negligible. This_occurs in the linear phase we are studying here provided that ii Kcondition). An ensemble of classical trajectories for this pendulum motion is obtained by solving -for all times- the nonlinear equations with the random initial conditions defined by Eqs. (91)-(93). For SZ >> w/Q, these trajectories exhibit an oscillatory behavior, which corresponds to the oscillations in the Dicke state ladder of the Schro-

RADIATIVE PROPERTIES OF RYDBERG STATES

397

dinger point of view (see Sect. IV,A, 1). For R fi 20,000,correspondingto R, TR 1.

-

404

S. Haroche and J. M. Raimond

and 0‘=72-8 Let us turn now to a more interesting situation, where novel effects not observed on “ordinary” atomic systems do occur: the collective absorption of an incoherent blackbody field by Rydberg atoms in a cavity. Let us consider the evolution of an ensemble of N two-level atoms, initially prepared in the lower state 18) in interaction with a single radiation mode damped by a reservoir at temperature T. The equilibrium reached by this system -after a transient regime discussed below -is obviously the state of thermal equilibrium of an angular momentum N / 2 at temperature T, the probability of having p excited atoms at equilibrium is

An interesting limiting case is obtained for N >>E the atomic system which remains close to its ground energy can be approximated by a harmonic oscillator. The thermal equilibrium of this system is then the exact replica of the field one, p,=(l

-P)PP

(102)

and the collective system of N atoms has a mean thermal energy equal to k,T. This is clearly at variance with the result we would get by considering the atoms as being coupled independently from each other to the mode. In this case indeed, each atom would get a mean energy equal to hw/2 (if k,T >> hw) and the atomic sample would have a total thermal energy N( h 0/2), much larger than k , T. The fact that the sample absorbs much less than a gas of independent particles is certainly the striking feature of this collective absorption phenomenon. It can be related to the well-known properties of Bose - Einstein statistics: as long as their radiative properties are concerned, the atoms behave as indiscernable particles, with only one quantum state per degree of excitation. The absorption of energy thus occurs without increase of entropy and the atomic sample has basically zero heat capacity. As a result, it gets in thermal equilibrium with the field by absorbing a very small amount of heat. To study the transient evolution ofthe atomic sample toward equilibrium, it is convenient to make use of the Heisenberg point of view. This approach will enable us to give a simple interpretation of the collective absorption phenomenon in terms of a Brownian motion of the Bloch vector around its stable equilibrium position.

RADIATIVE PROPERTIES OF RYDBERG STATES

405

Defining in a similar way as in Section IV,B the operators

and

D+

c+=J=N and assuming small departure from the equilibrium position D3----

(104)

2

we can linearize the Heisenberg equations (73) and we obtain a set of equations describing the coupling of an atomic oscillator (creation and annihilation operators c+ and c) with the damped field mode

di; -- iQNa dt -dii =--

dt

0

2Q

a + ilRNZ + Fa

The solution of this system can be derived exactly in the same way as in Section IV,B, 1, the only difference being that the “atomic-oscillator” is not inverted. As a result, the atomic energy will not diverge exponentially and the evolution will remain restricted to the linear regime with the fluctuating thermal field Fa always being important. After straightforward calculation, we get

with

[

!- 2 + \l(02/4Q2)- 4Q; * =2 2Q-

2

1

(108)

2(t) being Gaussian, all the moments of i; can be obtained from this formula. It is easy to check that for t + 03, ( Z+(t) Z ( t ) ) + Z,which expresses the thermal equilibrium condition discussed above. The transient regime de-

406

S. Haroche and J. M. Raimond

a,.

pends on the respective values of I+and I-, i.e., of o/Qand Here again, we encounter the two oscillatory and overdamped cases. In the oscillatory regime [(o/Q) > SZ I, are real and the mean atomic energy evolves irreversibly toward equilibrium, according to an exponential law,

m],

( ? + ( t )Z ( t ) )

= E[ 1

- e+ITn]

(1 11)

with the same time constant

as in the emission case. Equation ( 1 1 1) leads to a very simple physical interpretation of the collective absorption phenomenon: for times short compared to T R , it reduces to

- iit

(?+(t)? ( t ) ) -= EN r = V t TR

( 1 12)

and we can then define an absorption rate per atom W = ZP', precisely equal to the rate we would get for an isolated atom. We thus see that the atomic sample starts to absorb as if the atoms were independent. After a time of the order of T R , the phase correlations between dipoles belonging to different atoms arise. This tends to damp the absorption process. In other words, the single atom absorption rate W competes with the collective damping rate TRI to induce in each atom a relative excitation WTR = $N, Le., an average number of excited atoms precisely equal to i. The overdamped regime of collective blackbody field absorption has been experimentally observed recently (Raimond er al., 1982a).Sodium atoms in the 30S1/2 state absorb blackbody radiation in a cavity tuned to the 3OSlI2+ 3OPlI2transition at 134 GHz. The cavity mode temperature can be varied (see Sect. I1,E) between 300 and 900 K. We measure, by comparing the ionization signals for resonant and nonresonant cavities the mean num-

RADIATIVE PROPERTIES OF RYDBERG STATES

407

ber AN of atoms excited in the atomic sample, as a function of N, at the end of the atom cavity interaction time At. Since the experiment is performed in zero magnetic field, the system is in fact made of two independent two-level atom systems, each interacting with a mode of a+ or a- polarization in the cavity. The above thermodynamical arguments apply to each of these independent systems and the variation with N of AN are immediately deduced from Eq. (1 1 l),

The experimental results are shown in Fig. 17 for two temperatures (300and 900 K). The dashed curves give the theoretical variation of A N calculated from Eq. (1 13). Using a cooled cavity environment, we have also studied the statistical properties of the number of excited atoms: Fig. 18 shows the experimental histogram representing P ( A N ) (probability of having AN excited atoms), obtained in a T = 85 K cavity mode. The solid line shows the theoretical distribution (here also, the a+ - a- degeneracy is not removed, and we get the square of convolution of the Bose- Einstein distribution). It should be emphasized that the collective Rydberg atom system acts in these experiments as an absolute radiation thermometer, since the field temperature is not modified by the interaction with the atom. The temperature is measured by a mere particle counting, independent of the atomic or cavity parameters.

FIG. 17. Experimental observation of the collective absorption of blackbody radiation by Rydberg atoms in a resonant cavity: plot of the mean number ofexcited atoms ANat time At as a function of N for two radiation temperatures [T = 300 K (X), T' = 900 K (A)]. Note the saturation for large Ns correspondingto a limit 2E(E = 47 and 130 for Tand T', respectively). (from Raimond el al., 1982a). Theoretical values of AN given by Eq. ( 1 13) (---)

-

S. Haroche and J. M . Raimond

408

t

AN FIG. 18. Absorption of thermal radiation by Rydberg atoms in cavity (30S,,* .+ 30P,,2 transition in Na at I34 GHz): histogram of the number AN of atoms having absorbed the thermal field for N = 1900. (The histogram is constructed with 300 pulses). The atoms are in zero magnetic field, so that they interact with two independent modes of the field. Theoretical distribution ofthe blackbody photon number in two modes at temperature T = 85 K (-). Horizontal scale: 5 atoms per box.

Let us note in conclusion that the above effects-which are not a priori restricted to very excited atoms- are, however, completely unobservable with ordinary atomic systems in microwave cavities. In these systems, the time constant TRis indeed very long compared to At, unless Nbecomes very large ( N lo1$). Since the number of excited atoms at equilibrium is always E, the Bloch vector is tilted by a very small angle E/N, which is clearly too small to be observed. In other words, only Rydberg atoms provide Bloch vector “light” enough for the Brownian motion ofthe atomic polarization to be observable.

-

V. Conclusion and Perspectives We have shown in this article that the system made of NRydberg atoms in a resonant millimeter-wave cavity is ideal to test fundamental quantum optics effects involving single or collective atomic systems. Most of the effects predicted by theory have been already observed. The single atom-single photon Rabi nutation effect and the phenomena of collapse and revival of Rabi oscillations in a coherent field made of a few photons are among the main effects which remain to be observed and will require significant improvement of the present techniques (cavity Q should be increased and the atomic velocities should be reduced). We have studied here the resonant case where the atom and the field mode have the same frequency. Another interesting situation arises when the cav-

RADIATIVE PROPERTIES OF RYDBERG STATES

409

ity is completely off-resonant for the atomic transitions. Spontaneous and blackbody-induced effects should then be suppressed (Kleppner, 198 1). An experiment exhibiting the inhibition of blackbody field absorption in such an off-resonant cavity has been recently published (Vaidyanathan et al., 1981) (see GLWF). Observation of spontaneous emission inhibitions will require the preparation of Rydberg atoms in states which can radiate only at long wavelengths (the microwave cavity cannot affect optical transitions). The methods recently developed (Hulet and Kleppner, 1983) to prepare Rydberg atoms in high m states will certainly be very useful in this context. Beside these fundamental aspects, the experiments described in this review suggest a few interesting practical applications of Rydberg atoms. The experiments in which the fluctuation of emission and absorption in the cavity have been studied show that it is indeed possible to use Rydberg atoms as counters of millimeter-wave photons, able to analyze not only the average intensity, but all the statistical properties of the radiation at millimeter and submillimeter wavelengths. A quantitative anal sis of our measurements leads to a detectivity of the order of lo-” W/ Hz for our Rydberg atom beams in the cavity, which is quite comparable to the best available solid state detectors in the 100-GHz range. The interest of Rydberg states lies in the possibility of extending this range to short wavelengths (detectivity is essentially frequency independent). Good detectors in this domain have a noise determined by the background temperature. Rydberg atom collective absorption properties (in the high Q limit) could also be used to reduce the cavity temperature and thus to decrease the detection noise of such systems.

+

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. 20

R YDBERG ATOMS: HIGH-RESOL UTION SPECTROSCOPY AND RADIATION INTERACTIONRYDBERG MOLECULES J . A . C. GALLAS. G. LEUCHS. H . WALTHER. H . FIGGER'

and

'Max-Planck-Instirutf i r Quantenoptik and 2Sektion Physik der Universitat Minchen Garching Federal Republic of Germany

.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . General Properties of Rydberg Atoms . . . . . . . . . . . . . . . Excitation and Detection of Rydberg States . . . . . . . . . . . . Methods of High-Resolution Spectroscopy of Rydberg States . . . . A . AtomicBeam . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Photon Absorption . . . . . . . . . . . . . . . . . . . C. Double Resonance . . . . . . . . . . . . . . . . . . . . . . D. Quantum Beat and Level Crossing Techniques . . . . . . . . . V . Results of High-Resolution Spectroscopy of Rydberg States . . . . . A . Alkaline Atoms-Fine Structure . . . . . . . . . . . . . . . B. Alkaline Atoms-Hyperfine Structure . . . . . . . . . . . . . C. Alkaline Earth Atoms . . . . . . . . . . . . . . . . . . . . VI . Interaction of Rydberg Atoms with Blackbody Radiation . . . . . . VII . Radiation Interaction of Rydberg Atoms-A Test System for Simple Quantum Electrodynamic Effects . . . . . . . . . . . . . . . . . A . Single Atom in Resonant Cavity- Modification of Spontaneous Emission Rates . . . . . . . . . . . . . . . . . . . . . . . B. Single Atom in Resonant Cavity-Disappearance and Revival of Optical Nutation . . . . . . . . . . . . . . . . . . . . . . . C. N Atoms in Resonant Cavity-Collective Behavior . . . . . . . D. N Atoms in Resonant Cavity-Collective Absorption of Blackbody Photons . . . . . . . . . . . . . . . . . . . . . VIII . Rydberg States of Molecules . . . . . . . . . . . . . . . . . . . A . Rydberg States of Diatomic Molecules . . . . . . . . . . . . . B. Rydberg States of Large Molecules. . . . . . . . . . . . . . . IX . Rydberg Molecules . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. I1. I11. IV .

414 416 419 421 421 423 423 425 421 421 431 433 435 440 441 446 446 449 450 452 456 451 460

413 Copyright 0 1985 by Academic &ss. Inc. All rights of reproduction in any form m~ed . ISBN 0-12-003620-X

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I. Introduction This article is one of two in the present volume which discuss recent research on Rydberg atoms and molecules. It is concerned with general properties of Rydberg atoms, methods of production and detection, and spectroscopic techniques and results, mainly for alkali and alkali-earth metals, and for the interaction of such atoms with blackbody radiation. The spectroscopy on Rydberg molecules is discussed in the last two sections. The contribution by Haroche and Raimond (H.R.) is devoted to one branch of Rydberg atom research, the radiative properties of such atoms in resonant cavities. When a valence electron of an atom is excited in an orbit with high principal quantum number and therefore far from the ionic core, the energy levels of the atom can simply be described by the Rydberg formula. This is the reason why atoms in these highly excited states are often called Rydberg atoms. Similary the highly excited valence electron of a molecule will sense the molecular ion essentially as a positive point charge. Consequently its energy levels can be described by a Rydberg formula also. Differences between Rydberg states of atoms and molecules arise from rotational vibrational excitations of the molecular ion. This leads to drastic effects when the frequency of the orbiting electron becomes comparable to the rotational vibrational frequencies. As a result one finds a breakdown of the BornOppenheimer approximation and observes autoionization if part of the rotational vibrational energy is transferred to the Rydberg electron. For a certain class of molecules even low-lying excited states show typical properties of Rydberg states. In the literature they are therefore referred to as Rydberg molecules. These molecular aspects are addressed in Sections VIII and IX, whereas the following discussion, as well as Sections II-VII, concentrate on atomic Rydberg states. The energy changes among highly excited states of atoms are small compared to the large changes between the lower levels. Since smooth changes are characteristic of classical systems (in which energy changes are continuous), Rydberg atoms can be expected to show classical properties. In particular, according to Bohr’s correspondence principle, the frequency of electromagnetic radiation emitted for transitions between neighboring states approaches the frequency at which the electron rotates around the ionic core. This suggests that many properties of these atoms can be understood in simple classical terms. Nevertheless, some very surprising properties of Rydberg atoms have recently been found, which has led to a steady increase in the number of experiments being performed on these atoms. The interest in Ryberg states is manifold:

RYDBERG ATOMS

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( 1 ) The outer electron is a very good probe for the interatomic potential; quantum defects due to the penetration and polarization of the electron core are therefore being investigated, as well as fine- and hyperfine-structure splitting. (2) Radiation effects are different from those for atoms in low-lying states owing to the large matrix elements for transitions to neighboring levels; radiation-induced effects therefore overcome spontaneous emission. The Rydberg atoms in high n states become sensitiveto blackbody radiation, and maser emission with only a small absolute number of radiators can also be observed. Observation and study of these effects allows testing of fundamental theories on light -matter interaction, which is not possible with ordinary atoms. (3) Collisional interaction becomes very important owing to the size of the atoms; their influence shows strong dependence on the main quantum numbers. It is thus found, for example, for the collisional angular momentum mixing in the low n region that the cross section increases in proportion to the geometric size of the atoms, i.e., n4,As the size of the Rydberg orbit increases further, the electron distribution becomes very diffuse and the cross section goes through a maximum and decreases. This is the case for collisions with neutral particles, with which the Rydberg electron interacts only weakly via the induced polarization. For charged particles which interact with the Rydberg electron via the long-ranging Coulomb interaction, the cross section keeps increasing. (4)The binding energy of the electron to the ionic core is very small; the Ryberg atom is therefore strongly influenced by external electric and magnetic fields.

This article is mainly concerned with high-resolution spectroscopy and interaction of Rydberg atoms with radiation. The other fields, e.g., collisions and interaction in external electric and magnetic fields, were covered some time ago and will therefore not be included here (see Edelstein and Gallagher, 1978; Stebbings, 1979; Feneuille and Jacquinot, 198 l ; Kleppner, 1982; Kleppner et al., 1983). The first observation of Rydberg states dates back to the end of the last century when Livering and Dewar (1 879) described the observations of long series in alkali spectra. Ten years later Rydberg (1889) proposed the famous formula En = - R/(n - 8)2,where the quantum defect 6 is approximately constant in a given series, so that the effective quantum number n* = n - S increases by integers. At the beginning of this century Bevan ( 19 12) observed 31 members of the principal series of the cesium spectrum, while Wood ( 19 16) photographed 57 members in sodium. Observation in emission was

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impossible since the electrons in highly excited states are so weakly bound that the atoms are ionized by collisions before they can radiate. Emission from isolated Rydberg atoms was first observed in interstellar space: many emission lines corresponding to transitions between neighboring levels with high quantum numbers are detected in radio astronomy. In space the Rydberg atoms are created by the recombination of electrons and protons to hydrogen; the atoms radiate as they cascade to lower and lower states. The first line discovered in radio astronomy is at 5.4 GHz and corresponds to a transition between the states n = 1 10 and n = 109 of hydrogen. It was observed in the Orion nebula (Kardashev, 1960;Hoglund and Mezger, 1965). The population of highly excited states under collision-free conditions became possible with the advent of tunable laser sources. Individual states can now be populated and their properties studied in detail in the laboratory.

11. General Properties of Rydberg Atoms The properties of the Rydberg atoms are very much hydrogen-like. Their energy is given by the Rydberg formula (Table I). R is the Rydberg constant and 8, the phenomenological quantum defect of the states of angular momentum 1. For states of low 1, where the orbits of the classical BohrSommerfeld theory are ellipses of high eccentricity, the penetration and polarization of the electron core by the valence electron lead to large quantum defects and strong departures from the hydrogenic behavior. As 1 increases, the orbits become more circular and the atom becomes more hydrogenic: 8,changes with F5. In Table I the scaling laws for further properties of Rydberg atoms are compiled: the radius of the charge distribution of the TABLE I SCALING LAWSFOR PROPERTIES OF RYDBERG ATOMS” Energy: Radius: Lifetimes: Fine-structure interval: a

En,= R / ( n - S,)*= R/n*2 n* effective quantum number, S,quantum defect (r) nlC2 7 - n*3 (low angular momentum states) 7 n*5 (high angular momentum states)

-

-

-

For details see text.

n*-3

417

RYDBERG ATOMS

valence electron scales as n*2, and for n* = 50 the linear dimension of the atom is already comparable with the wavelength of light in the visible region and competes with the size of the large biomolecules. The electric polarizability for the quadratic Stark effect increases as n*7, and the diamagnetic interaction as n*4. This allows one to perform experiments at field strengths high enough to make the interaction energy in the external electric or magnetic field comparable with or larger than the Coulomb energy of the atom. For practical reasons the corresponding field strengths for ground state atoms cannot be reached in the laboratory. The study of highly excited atoms in external electric and magnetic fields is therefore interesting in itself. For reviews see Feneuille and Jaquinot (1 98 l), Kleppner ( 1982), Kleppner et ul. (1983), and Delande and Gay (1983). The sensitivity of Rydberg atoms to external electric fields also means that the atoms readily ionize in rather weak fields. This opens the possibility of very effective detection, as will be discussed later. The large Rydberg atom orbitals are characterized by natural lifetimes much longer than those of less excited atoms. In the case of hydrogen Rydberg states, the dependence of the lifetime on n can be obtained by fully quantum mechanical radiation rate calculations involving hydrogenic coulombic wave functions. For Rydberg states of other species the lifetimes (and the other radiative parameters) do not scale exactly as a power of n but rather as a power of n*. The n* scaling law can be determined by using calculations of the Bates and Damgaard ( 1 949) type. The lifetimes scale either as n*3 (when I is small) or as n*5 (when 1 = n). In the following we will give a simple explanation of this scaling law. (In this discussion we do not discriminate between n and n*.)The rate of spontaneous emission of radiation for a transition from a state n to n‘ is given by the Einstein A coefficient: A,,,,

=

167r3v3e2(rn,t)2/3~ohP

(1)

where o is the transition frequency and (r,,,) the matrix element of the electric dipole operator between the initial n and the final state n’. For the case n’ AEfh, where AE is the energy distance between the two excited states. The subsequent evolution of the atoms is observed time resolved via the fluorescence or by probing the atoms with an additional light pulse after a variable delay time. The signal as a function of time (fluorescence)or of the delay time (probing pulse) exhibits a periodic variation, the frequency of which gives the energy separation of the states under consideration. The reason for this oscillating behavior in time is that the phases of the wave functions of the excited states evolve differently in time and their changes are related to the frequency separation of the states. In the density matrix formalism the signal beats result from the time variation of the non-zero off-diagonal elements of the density matrix. In order to observe the quantum beats, the detection process has to be sensitive to the off-diagonalelements of the density matrix. This means that the detection process should not discriminate between the states excited by the light pulse or, in other words, the excitation and detection channels via the neighboring states should be indistinguishable by analogy with Young's double-slit experiment. Otherwise quantum beats will not be observed. In the case of higher lying Rydberg states it is difficult to observe quantum beats in fluorescence or in absorption to higher bound states. However, when the second step is a bound-free transition into the continuum, quantum beats are easier to observe. The quantum interferences are also observable when field ionization is used for detection (see Fig. 2). The first experiments of this kind were performed by Leuchs and Walther (1977 and 1979) on the n2D fine-structure states of sodium ( n = 2 1 to 30). Later the application of the method was extended to levels up to n = 40 by Jeys et al. (1980). Detection of quantum beats by field ionization is possible since the different magnetic sublevels m,ionize at different fields (Rausch von Traubenberg et al., 1930;Gallagher et al., 1977;Jacquinot et al., 1977). With the strength of the ionizing electric field pulse properly adjusted, field ionization allows one to detect the spatial anisotropy of the excited states. The other require-

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426

spectrum of laser pulse

atomlC level

scheme

external field applied after d a b k deby tine

FIG. 2. Scheme for quantum beat experiments using field ionization for detection. The excitation-ionization channels via the excited states have to be indistinguishable as in Young’s interference experiment. The quantum interference signals are observed in the total electron current measured as a function of the time delay between the excitation and field ionization.

ment, the indistinguishability of the ionization via the different levels, can be met by using short rise times for the time-delayed field ionization pulse, so that at least one of the adiabatically avoided crossings existing between the different Stark substates is practically crossed diabatically. The dependence of the quantum beat signal on the rise time of the field ionization pulse was nicely demonstrated by Jeys et al. (1980). The frequency resolution is determined by the maximum variation of the time delay between the exciting light pulse and the electric field pulse used for ionization. A quantum beat signal obtained by this method is shown in Fig. 3. The maximum time delay in this measurement was 5 p e c .

0.8

SlS

Delay Time [psecl Fourier Spectrum

b

Frequency IMHzl FIG.3. Quantum beat signal ofthe fine-structure splitting ofthe 23 *Dstate ofsodium. The

upper trace shows the experimental recording. In the lower trace the slowly varying background which is caused by geometric effects in the electron detection is subtracted. The fine-structure splitting frequency is obtained by a Fourier analysis of the quantum beat signal. For details see Leuchs and Walther (1979).

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The level crossing technique is essentially a time-integrated observation of quantum beats. Whenever the quantum beat frequency is too high, the integration will average out the quantum beat oscillations. Only if the beat period is longer than the lifetime of the excited states-or whatever determines the effective integration time-can a net effect be seen in the time-integrated signal. In a level crossing experiment usually a magnetic field is applied, causing Zeeman splitting. The difference in the phase variations of the two Zeeman levels goes through zero whenever two levels cross, giving an extreme value in the time-integrated signal. So far the level crossing technique has been applied to the spectroscopy of Rydberg states only in combination with the detection of the fluorescence.

V. Results of High-Resolution Spectroscopy of Rydberg States A. ALKALINE ATOMS-FINESTRUCTURE As discussed in Section I1 an empirical hydrogen-like respresentation of alkaline atoms in widespread use assumes the valence electron to move around an inner core with electrons in filled shells. In this representation the effective quantum number n* is used to account for the incomplete shielding ofthe nuclear electric field by the core electrons. In the hydrogenic model the fine and hyperfine splitting of the energy levels of the valence electron should show an n*-3 dependence. This simple model does not consider more sophisticated effects which may influence the core, e.g., polarization effects (singlecore electron excitation) and correlation effects (multiple excitation). This means that deviations from the hydrogenic theory should be expected whenever these effects have an influence. In fact, departures of the experimental fine and hyperfine-structuresplittings from the empirical hydrogenic description are normally regarded as indications of these effects. The accurate measurements of such departures can thus serve to uncover details of higher order many-body effects, which are of course also of general interest and may occur for other atoms. In this respect, the methods of high-resolution spectroscopy described above present a sensitive and unique way of achieving this goal. In recent years many investigations of the fine-structure splitting of the alkaline atoms have been published. Owing to the modern techniques of laser spectroscopy applied in most of the measurements a rather high accuracy has been achieved, so that to date a rather good survey of the effects

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involved can be given. Table I11 summarizes recent papers on the fine structure of alkaline atoms. The observed values of the fine-structure splittings have been fitted to the empirical formula Ah = A/n*3

+ B / P S+ C/n*7

(3) In spite of the success of this empirical treatment, the physical interpretation of the parameters A , B, and C in Eq. (3) is not well identified. Recently, Pendril(l983) revised the discussion on the validity of omitting even powers of n*. By an analytical evaluation of the n dependence of the energy eigenvalues of Rydberg states based on the quasi-hydrogenic character of the single-particle eigenfunctions (Chang, 1978; Chang and Larijani, 1980) a somewhat different formula was obtained Afs = N,(n) (a

+ b/n2 + c/n4 +

*

*

)

I

with Nl(n)= n-3 JJ [(n' - p2)/n2]

(4)

P-0

Equation (4) was used to analyze quantitatively experimental data (Chang and Larijani, 1980). More than one term is especially needed in order to reproduce the low n behavior. The need for terms involving other than an i P 3dependence clearly indicates the breakdown of the simple hydrogenic picture. The precise ab initio calculation ofthe fine structures is very difficult and has been undertaken for only a few Rydberg levels of helium (Chang and Poe, 1976) and of more complex alkalis by using high-order perturbation theory (Foley and Sternheimer, 1975; Sternheimer et al., 1976), many-body formalism (Holmgren et al., 1976; Lindgren and Morrison, 1982), pure relativistic central field approach (Luc-Koenig, 1976). The experimental fine-structure intervals, except for Li, are usually larger than the screened hydrogenic values and can even be inverted, as is the case for n 2Dterms of Na, K, and for the n 'F terms of Rb and Cs. These inversions are particularly interesting because they violate predictions of the simple hydrogenic model, clearly indicating the presence of other effects. In the following the fine structure of the n 'D terms is discussed in more detail. In Fig. 4 we compare the fine structure of the Na n 'D states to that of hydrogen and other alkalis. The references are given in Table 111. The values for the n 'D splittings are multiplied by n*3 and plotted as a function of n. In the case of a pure l/n*3 dependence horizontal lines are expected. The deviation for Na, K, and Rb is obvious. In the case of Na and K the finestructure splitting is inverted for all n states, whereas for Rb only the n = 4 fine structure has a negative sign. The 1 dependence is very well fulfilled

429

RYDBERG ATOMS TABLE I11 SURVEY OF THE FINE-STRUCTURE MEASUREMENTS PERFORMED ON ALKALINEATOMS Element Li

Na

Series P D

F,G P D

K

Rb

cs

F G.H S D

S P D F S P D F

n

Reference

2 7Sn510 4 3 45n57 7dnSll 1 6 5 n d 19 23 5 n S 4 1 95nd16 45nd9 15, 16, 17 21 d n S 3 1 32 S n 5 40 1 I, 13, 14 13SnS17 9dnS46 8SnS19 15Sn520 5,6 7SnS46 9 5 n d 116 28 5 n S 60 7 5 n S 124 45nS9 6 7 7SnSll 12 5 n d 35 27 5 n d 44 7511513 23 S n S 42 15, 17, 19 11 S n 5 4 8 32 9 5 n 5 12 1 0 5 n 5 17 24,29, 30

Mingguang ef al. (1982) Cooke ef al. (1977a) Fredriksson ef al. (1978) Champeau ef al. (1978) Wangler ef al. ( 1981) Cooke ef al. (1977a) Cooke ef al. (1977b) Fabre ef al. (1978); 1980) Fabre ef al. (1975) Fredriksson and Svanberg (1976) Gallagher et al. (1977b) Leuchs and Walther (1979) Jeysefal. (1981) Gallagher et al. (1977a) Gallagher ef al. (1976) Lorenzen et al. (1981) Harper and Levenson (1976) Gallagher and Cooke (1978) Nilsson and Svanberg ( 1 979) Lorenzen et al. (1981) Stoicheffand Weinkrger (1979) Liberman and Pinard (1979) Stoicheff and Weinberger (1 979) Johansson (1961) Farley and Gupta (1977) Eriksson and Wenhker (1970) Lorenzen et a/. (1980) Goy ef al. ( 1982) Lorenzen and Niemax (1979) Goy et al. (1982) Curry et al. (1 976) Lorenzen et al. (1980) Goy ef al. (1982) Eriksson and Wenhker (1970) Fredriksson et al. (1980) Goy et al. (1982)

for the higher n states. In the case of Li the fine-structure values are almost the same as for the hydrogen (deviations are less than 0.5%) and a perfect l/n*3 dependence is found for all n states (Wangler el al., 1981).

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430

Hx

5

25

45

65

n

FIG.4. Fine-structuresplitting of the n 2Dstates of the alkaline atoms and of hydrogen.The values for the n *D splittings are multiplied by n*'. For references see Table 111.

Theoretical calculations for the 2Dfine-structure splitting have only been performed for the Rb 4 2Dstate and for the Na n 2Dsequence, n = 3 to n = 16. In the case of the Rb 4 2D state Lee et al. (1976) could verify the negative sign of the splitting by calculating the exchange core polarization contribution. However, the value is more than 50%smaller than the experimental result. Holmgrenbet af. (1976) and Sternheimer et al. (1978) performed manybody calcuations for the Na 3 2Dto 6 2Dstates including polarization effects to all orders. The negative sign of the splitting is found. However, the values are systematically 25% too small, which the authors attributed to correlation effects not taken into account in their calculations. A better agreement with the experimental values is reached by Luc-Koenig (1976), who calculated the fine-structure splitting of the Na 2Dstates for principal quantum numbers from n = 3 to n = 16 with a relativistic central field approximation; the inverted splitting obseped in the experiment could be reproduced to within = lo%, which is a very good agreement. Pyper and Marketos (198 1) presented a calculation where the fine-structure inversions in highly excited D and F states are explained as first-order relativistic corrections to the Hartree - Fock energy. Recently refined many-body perturbation calculations also gave a good agreement with the experiment showing that correlation effects are rather small (Lindgren and Martensson, 1982). To summarize the theoretical results, it seems obvious that the main reason for the inversion of the fine-structure splitting of the n 2Dlevels is the polarization of the inner core by the outer electron. An almost quantitative

RYDBERG ATOMS

43 1

description is obtained when relativistic effects are included. It is important that for large n the fine-structure splitting of the alkaline 2D states scales as l/n*3 no matter whether the splitting is inverted or not. B. ALKALINE ATOMS-HYPERFINE STRUCTURE The hyperfine structure arises basically from the interaction of the magnetic dipole moment of the nucleus with the magnetic moment of the angular motion and the spin of the valence electron. In a simple hydrogen-like picture ofthe alkaline atoms the magnetic dipole interaction constant can be calculated from the semiempirical formula (Kopfermann, 1958;Belin et al., 1976a,b)

for s electrons and from

for non-s electrons. In these formulas g ; is the nuclear g factor referring to Bohr magnetonspu,,Z, is the effective nuclear charge at a large distance from the nucleus (i.e., Z, = 1 for a neutral atom; Zi = Z for s electrons, and it is generally set to Zi = Z - 4 for p electrons and Zi = Z - 1 I for d electrons), Fr(j,Zi)is a relativistic correction factor; 6 takes into account the finite charge distribution and E the distribution of the nuclear magnetic moment throughout the nuclear volume. The matrix element ( r 3 is) given by ( - 3

)- 4 n3

*34r

Zi

++)(r + 1)

(7)

where n* = z,(hCRy/&)1/2,Eb being the experimental binding energy and a,, the Bohr radius. The derivative dn*/dn is obtained as the slope of a fit function n* = n*(n). From these formulas one sees that the a factors are proportional to Eb3/2,and that the hydrogenic model therefore predicts straight lines in a plot of log a versus log Eb. Hyperfine splittings in alkali atoms are quite small (e.g., a few kilohertz at n = 20 in Cs n 2Dstates) and relatively difficult to measure. Since the study of the hyperfine structure of heavy alkalis is not so difficult as for light alkalis, the number of states already investigated increasesas one goes down in group I of the periodic table of elements. Although experimental results for highly excited states are not yet available, a number of states have already been investigated. The experimental data on K, Rb, and Cs were summarized in

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the review papers of Belin et al. in 1975, 1976a, and 1976b, respectively. In the following emphasis will be given to the splitting of the D states. Such data have been obtained, e.g., by Farley et al. (1977) and Deech et al. (1977). Recently, Fredriksson et al. (1980) reported an investigation ofthe hyperfine structure of the 5 2D3/z,5/2 133Cs.As for many other alkali metal 2D states (Gupta et al., 1972; Hogervorst and Svanberg, 1975; Belin et al., 1976) the hyperfine splitting of the 5 2D5/2was found to be inverted, while that for the 5 2D,12level was normal. From a theoretical point of view the 5 ZDstates are quite interesting because they represent the lowest D states in cesium where relativistic effects can be expected to be important. The inversion of alkali metal atom 'D5/2 states was explained by Lindgren and co-workers (Lindgren et al., 1976a,b). By using many-body perturbation theory very good agreement with experiment was obtained, especially for the 4 2D5/2,3/2states of rubidium (Lindgren, 1976b). While detailed inclusion of correlation effects is necessary for obtaining accurate theoretical values, the 'D5/2 state inversions are basically due to polarization effects induced in the core owing to the presence of the valence electron. Very recent results include measurements of the hyperfine structures of Cs I5,16 'D3/2 (Nakayama et al., 198I), n SlI2and n PlI2for 23 5 n 5 28 (Goy et al., 1982)and the results ofNa n 'P3/2 5 d n d 9 of Zhan-Kui et al. (1982). With the exception of the D5/2 states, all experimental results showed the expected linear behavior in a log - log plot of the magnetic dipole interaction constant versus binding energy as discussed above. This means that the hydrogenic model is able to describe the hyperfine structures, with the exception of the 'D5/2 states. To describe the inversions of the 'D5/2 states, it is necessary to use a more refined model where several radial parameters are used to describe the hyperfine interactions. For D states the following expressions are obtained (Lindgren and Rosen, 1974):

In these formulas ( r P 3 ) , ,( r - 3 ) d , and ( r-3)care the orbital, spin-dipole, and contact radial parameters. By taking ( r 3 = ) ,( r-3)sd= ( r 3and ) ( r-3)c= 0 the above formulas reduce to the hydrogen-like ones. By using, for example, many-body perturbation techniques it is possible to calculate individual values for the different radial parameters, considering polarization effects and correlation effects. For alkali D states core polarization effects are very strong, yielding a large ( r-3)c. In this way it is possible to explain the magnetic hyperfine structures of the anomalous D states (Lindgren et al., 1976a;

RYDBERG ATOMS

433

Belin et al., 1976; Svanberg, 1977) since this term appears with a negative coefficient in the formulas above. In special cases where the calculations have been performed to higher order, an agreement with experiment within a few percent is obtained (Lindgren et al., 1976b).

C. ALKALINE EARTHATOMS As seen in the previous section, high-resolution laser spectroscopy techniques have allowed one to obtain a rather good picture of phenomena occurring in alkaline atoms. In contrast, the Rydberg spectra of the alkaline earth atoms are much more complicated than those of the alkalis since the presence of two electrons in the open shell endows their Rydberg states with additional properties. Besides the Coulomb interaction between the electrons, the additional spin -spin contribution causes the appearance of Rydberg series with predominant singlet or triplet signatures. For bound-state Rydberg series only one of the valence electrons is excited and the configurations are of the type msnl (m = 4, 5, and 6 for Ca, Sr, and Ba, respectively). Furthermore, as is to be expected, the presence of doubly excited configurations will strongly perturb the series with one excited electron. As a consequence, one should expect to have singlet - triplet and/or configuration mixing more or less localized in energy. An example is the interaction between the levels of the 6snd series for n 5 40 with levels of the 5d6d and 5d7d configurations in Ba (Rinneberg, 1984; Aymar, 1984). All the interactions between the various Rydberg series are described in the general framework of the multichannel quantum defect theory (MQDT) (Seaton, 1983; Fano, 1983). The basic idea of MQDT lies in the observation that interactions between the electron and the ionic core can be separated into long-range and shortrange effects. Far from the ionic core the effects are described by Coulomb radial eigenfunctions. Near the core, where interactions are stronger, one uses eigenfunctions which are characterized by two sets of parameters: quantum defects pa and matrix elements Uh,which describe the change of coupling between the electron and the core as their distance changes. One then speaks of “collision channels” i and of “close-coupling channels” a. The former correspond to the asymptotic Coulomb eigenfunctions, while the latter describe the state of the inner region. The problem then consists of connecting the collision channels i and the close-coupling channels a,i.e., of determining the “mixing coefficients.” These mixing coefficients depend on the two sets of parameters pa and Viaand are expected to vary only weakly with the energy. The values of these parameters may be either calculated ab initio or derived from a numerical analysis of perturbed Rydberg series

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through the so-called Lu-Fano plots. Up to now only this latter empirical fitting approach has been used to analyze heavy alkaline earth spectra and to obtain the two sets of parameters. The parameter pa and Uiaallow eigenfunctions for each individual Rydberg level to be constructed. The eigenfunctions obtained in this way cannot be uniquely determined if only the level energies are known experimentally. Additional experimental data such as hyperfine structure, isotope shifts, radiative properties, and interactions of individual Rydberg levels with external fields are therefore very much needed in order to determine the eigenfunctions unambiguously and use them to predict new properties (Aymar, 1984). In 1980, using a relatively broad-band laser (50 MHz), Barbier and Champeau reported a systematic study of hyperfine structures as well as isotope shifts of 6snd 3D,Rydberg states of Ybl. Simultaneously, Liao et al. (1980) reported a complete determination of the hyperfine structure of 2 3P and of 3 3Dstates of 3He in which significant singlet-triplet mixing was observed. They identified the mixing as arising from the hyperfine interaction of the 1s open-shell electron and the interaction as being basically independent of the outer electron. These results are very clear indications that the hyperfine structure of excited state atoms with two valence electrons is very sensitiveto state mixing. At present, the hyperfine structure and isotope shifts of Ca, Sr, and Ba [as well as the Group IIB (Zn, Cd, Hg) and even the Group I11 elements (Belfrage et al., 1983)]are being extensively studied, mainly by very active groups in Amsterdam, Berlin, Goteborg, Gel, and Lund. In all cases the hyperfine structure of single excited Rydberg states, i.e., of configurations like msnl is dominated by the strong Fermi contact interaction between the nuclear magnetic dipole moment and the innermost ms valence electron. As for alkalis, the contribution of the nl Rydberg electron decreases as n*-3 and, in virtually all cases can be neglected. It is therefore possible to resolve the hyperfine structure of the msnl Rydberg states, even for high n, while much higher precision is needed to measure hyperfine structures of Rydberg states of alkali-type systems. As mentioned before, the correlation energy between the two valence electrons is very important in the study of the hyperfine structure of two-electron atoms. Indeed, singlet and/or triplet character is important for the magnetic properties of Rydberg states and hyperfine measurements are very direct and particularly sensitive ways of determining singlet - triplet mixing. In Table IV we present detailed references on hyperfine data already available for two-electron systems. As the excitation of the outer electron increases, the hyperfine coupling of the rns electron remains essentially constant while singlet - triplet separations and fine-structure splittings of msnl Rydberg states decrease proportionally to n*-3. Eventually (when the

435

RYDBERG ATOMS TABLE IV RESULTS ON HYPERFINE STRUCTURES OF ALKALINE EARTHRYDBERGATOMS Element

Series

n

Reference

Ca

4sns ISo 4snd IDz 5sns 5 snd 5 ~ 6 p '.'PI 5 snd lJDz 5d7d ID, ISo 6sns IS, 'SI 'Sl 6s6p 'P, 6s7p 'PI 6snd ID,

u p to 21 7 5 nS42 Up to- 180

Beigang ef a/. (1982a) Beigang ef al. (1983) Beigang and Timmermann (1982a,b) Beigang and Timmermann (1983) Eliel ef al. (1983) Beigang ef al. ( 1981) Rinneberg and Neukammer (1982a) Rinneberg ef al. (1983) Beigang and Timmermann (1982a) Neukammer and Rinneberg (1982a) Hogervorst and Eliel(1983) Neukammer and Rinneberger (1982b) Eliel ef al. (1983) Rinneberg and Neukammer (1982b; 1983) Grafstrdm ef al. (1982) Neukammer and Rinneberg (1982~) Eliel and Hogervorst (1983a) Eliel and Hogervorst (1983b)

Sr

Ba

'DZ

'*'D2 6snf

U p t o - 70 11 5 n 5 5 0 21 5 n S 5 0 14 5 n S 23 15-18,20

10 5 n S 50 12 S n 5 24 11 S n 5 2 7

u p to 20 45n-30

Fermi contact interaction is about of the same order as the fine-structure splittings), strong mixing between msnl fine-structure components (corresponding to a recoupling of the several components of the total angular momentum F ) provokes not only shifts but also intensity variations in the observed hyperfine spectra. For high enough n, even hyperfine-induced n mixing can occur (Beigang and Zimmermann). As is obvious, besides singly excited Rydberg states, both valence electrons might occupy higher orbits. Such doubly excited states may strongly perturb msnl Rydberg series by configuration interaction, displacing the Rydberg levels and causing intensity variations.

VI. Interactions of Rydberg Atoms with Blackbody Radiation The effect of blackbody radiation on Rydberg atoms is mainly to induce transitions to nearby states. As a result the population will evolve as a function of time after pulsed laser excitation. Changes in the populations typically appear on a microsecond time scale. The most straightforward way

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of proving that blackbody radiation affectsthe Rydberg atoms is to vary the temperature of the environment. Experimentally, this requires a careful design of the interaction chamber so that no radiation leaks into the chamber. In the first experimental verification of blackbody radiation effects Gallagher and Cooke (1979) therefore did not vary the temperature but measured the lifetime of high-lying p states of Na and found values three times smaller than theoretically predicted. When, however, room-temperature blackbody-induced population transfer to nearby states is included, good agreement with the experimental data was achieved. Haroche et al. (1979) observed that the decay of the 25s sodium state occurs with an important population transfer to nearby states, such as the 25p level, which cannot be accounted for by ordinary spontaneous emission (since the 25p level lies above the 25s one). This transfer ofpopulation can be quite consistently explained as a 20% blackbody radiation-induced mixing between the 25s and 25p states. In a slightly different approach Beiting et al. ( 1979)used field ionization of Rydberg atoms with a linearly rising electric field pulse. As described in Section 111, different Rydberg states ionize at different electric field strengths. With such a set-up, the population distribution among the Rydberg states can be measured in a single shot. Beiting et al. (1 979) measured this population distribution for various delays between the pulsed laser excitation and the ionizing electric field pulse. It was observed that the population initially prepared in one Rydberg state is transferred to higher lying states, the more the longer the delay. Using a similar set-up, Rempe ( 1981) measured the population transfer from the initially laser-excited 23'F, state of strontium to nearby d and g states (Fig. 5). In these experiments the temperature of the environment was again not varied. However, it was carefully checked that the population transfer is due to the blackbody radiation. The first direct observation of the temperature dependence of the population transfer was performed by Koch et al. (1980) by heating the environment, and in an experiment by Figger et al. (1980) where sodium atoms of a thermal beam in a cooled environment were excited to the 22d state using two continuous wave (cw)dye lasers. In the latter case the interaction region was inside a copper box cooled to 14 K (Fig. 6). The sodium Rydberg atoms could be exposed to the radiation of a blackbody source by opening a flap at the side of the box. After interacting with the blackbody radiation for about 50 psec the atoms entered a dc electric field which acted like an optical edge filter: all atoms in Rydberg states higher than the initially laser-excited 22d state are ionized and detected. Figure 7 shows the ionization signal for different temperatures ofthe blackbody source. The signal rises linearly with temperature, in good agreement with the Rayleigh - Jeans limit of Planck's

RYDBERG ATOMS

437

2 T=lps

T: 2 p

[ I l r r l " r -

0

500

1000 Vlcm

FIG.5 . Blackbody-induced transitions between Rydberg states. The 23 IF3 level of strontium is excited by the laser radiation. The detection of the Rydberg states is performed by an electric field increasing linearly in time. The field ramp starts 1,2,6, and 12psec after the pulsed laser excitation. The field ionization signal at smaller field strengths results from Rydberg levels populated by blackbody radiation of the apparatus at 300 K.

J . A . C.Gallas et al.

438

fl

No-atomic beam

FIG.6. Experimental set-up for demonstrating the interaction of blackbody radiation with Rydbergatoms. The flap at the right side ofthe cooled box ( 14 K) allows the thermal radiation of the infrared source to enter the box. The 22d state was excited by cw laser radiation. The Rydberg atoms in the 22p level were detected by field ionization.

radiation formula. The effect of the blackbody source mounted outside the box is reduced since the atoms only see the source at a small solid angle. Aside from population transfer, blackbody radiation also affects Rydberg atoms in a more subtle way. The spectral energy density distribution of the blackbody radiation at 300 K has its maximum at about 2 X 1013 Hz. A typical electric dipole transition starting from the ground state of an atom has 1014- 10LSHzandatransition between twoRydbergstates hasabout 10" Hz. It is thus apparent that for a ground-state atom the blackbody radiation appears as a slowly varying field, whereas for a Rydberg atom it appears to be rapidly varying leading to an ac Stark shift of the Rydberg levels. The black-

] 1000 ion counts per 310 K

260 K

I

hap open

seconj

LOO K

500 K

600 K

I

flap closed

FIG.7. By opening and closing the flap (see Fig. 6) the ion signal is changed. The background (not shown in the figure) is about five times the signal induced by the infrared source at 310 K and is due to the 14 K blackbody radiation emitted by the walls of the box.

RYDBERG ATOMS

439

body-induced ac Stark shift Awnfor a Rydberg atom in the state n can be expressed as (Townes and Schawlow, 1955)

where E2, is the squared electric field of the blackbody radiation in a band width dw, at a frequency a b . Accurate evaluation of the shift was performed by Farley and Wing (198 1). A rather good estimate is obtained when it is assumed that the frequencies of the strong transitions are much lower than the frequency ofthe blackbody radiation, i.e., wnnt 15 at 300 K. All Rydberg states experience roughly the same energy shift of about 2.4 kHz at 300 K. Since the shift of all Rydberg states by blackbody radiation is the same, it can be detected only as a change of the optical transition frequency which connects to the ground state. Consequently, the line width and the stability of the dye laser used for excitation has to be and the spectral resolution in the experiment has to be correspondingly high. This challenge was accepted by Hollberg and Hall (1983). They used Doppler-free two-photon absorption to excite the 5s-36s transition in rubidium atoms. With the Ramsey method of separate fields, the spectral width of the signal was decreased to 40 kHz. The line center could be determined with an accuracy of 150 Hz. The atoms were exposed to the radiation of a blackbody source, and a chopper periodically blocked this radiation. Thus, the experiment was insensitive to long-term drifts. When the temperature of the blackbody source was raised to about 500 K, a shift in the line position of 1.4 kHz was observed. This is 10 times larger than the uncertainty. A first study of the temperature dependence also shows agreement with the predicted T2dependence. Gallagher et al. ( 1981) measured the blackbody radiation-induced level

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shift in two-electron atoms where a state of a doubly excited configuration has an energy so close to a singly excited Rydberg state that the transition frequency is in the microwave region. Since the doubly excited state is not shifted by the blackbody radiation, the shift of the Rydberg state is measurable at the comparatively low microwave frequency. As pointed out by Gallagher et al. ( 1 98 l), the measurement of blackbody-induced level shifts may in turn also be used to determine the absolute temperature of the environment. See Section I,C (H.R.) for further discussion ofthis interesting topic.

VII. Radiation Interaction of Rydberg Atoms -A Test System for Simple Quantum Electrodynamic Effects The invention of the maser has generated a great deal ofinterest in theoretical models describing the interaction of two-level atoms with a single mode of an electromagnetic field (e.g., Jaynes and Cummings, 1963; Allen and Eberly, 1975;Knight and Milonni, 1980).Although the first models treated a purely academic problem, modified versions were stimulated. These then led to an understanding of a major part of the experimentally observed phenomena, including the even larger variety of effects found after the laser was invented. In the experiments on atoms in low-lying states it is always necessary that large numbers of atoms and photons are present. This is due to the fact that the matrix elements describing the interaction between radiation and atoms are small. A small number of photons in an experiment has the consequence that the atom- field evolution time indeed usually becomes much longer than other characteristic times such as the atomic relaxation, the atom-field interaction time, and the cavity mode damping time. The theories involving single electromagnetic modes and small photon occupation numbers are therefore not very realistic. They, however, predict some interesting and basic effects. Among them are the following: ( 1 ) Modification of the spontaneous emission rate of a single atom in a resonant cavity. (2) Oscillatory energy exchange between a single atom and the cavity mode. (3) Disappearance and quantum revival of optical nutation induced on a single atom by a resonant field.

441

RYDBERG ATOMS

Rydberg atoms are very suitable for observing these effects for several reasons: They have a very strong coupling to the radiation field, as already mentioned; the transitions to neighboring levels are in the region of millimeter waves, which allows one to build cavities with low-order modes that are reasonably large to ensure rather long interaction times; finally, the Rydberg atoms have long spontaneous emission times, and therefore only the interaction with the selected cavity mode is important and the coupling of the atoms to other cavity modes can be neglected. In the following, several phenomena observable with Rydberg atoms are discussed in more detail. A. SINGLE ATOM IN RESONANT CAVITY -MODIFICATION OF SPONTANEOUS EMISSION RATES

The energy levels of the combined two-level atom and field system can be described in the dressed atom picture (Haroche, 1971; Haroche et al., 1982). The lowest energy of the system is represented by [go) describing the atom in its ground state (g) with no photon in the cavity. The higher energy levels are separated by the energy of a photon. The states I n ) are a superposition of the states le,n), (e stands for excited atomic state and n for the photon number) and Ig,n 1 ) of the system without interaction between the cavity field and the atom:

*

+

I

* + >]/a

n ) = [le,n> lg,n 1 (13) The energy separation between the levels I n ) and I - n ) is 2h Q where is the coupling strength between the field and the atoms. There is a small change proportional to h when the field strength is increased. The energy levels of the dressed atom taking the coupling with the field into account are shown in Fig. 8. In a realistic description of the interaction the dissipative processes also have to be considered. Since Rydberg atoms have lifetimes longer than the atom - field interaction time, their relaxation can generally be neglected. However, the relaxation of the cavity field is important: the harmonic oscillator representing the field is coupled to a thermal reservoir at temperature T representing, for example, the cavity walls. The scheme shown in Fig. 9 gives the corresponding “coupling constants.” The thermal equilibrium of the field mode is obtained in the characteristic time Q/o,where Q is the quality factor of the cavity and w the frequency of the resonant mode. The behavior of an atom entering an empty cavity (i.e., at T = 0 K) in the excited state le) depends on the relative size ofR and w/Q. If Q > o/Q(small damping of the cavity), the probability of finding the atom in the state le)

+

m,

J. A . C. Gallas et al.

442 2hnW

{, I

I I I I I

FIG. 8. Energy levels of a single two-level atom in the dressed atom description with resonant coupling to a cavity mode.

undergoes a damped oscillation. This regime can be considered as a self-induced Rabi nutation in the field ofthe single photon emitted and reabsorbed by the atom. If Q< w/Q, the probability decreases exponentially at a rate ravity = 4 R2Q/m. There is a cavity-enhanced decay rate which is related to the spontaneous rate in free space rspont in the following way:

r,,,,

(14) QA3 rspont/J' where V is the volume of the cavity and A the wavelength of the radiation. This relation was predicted long ago by Purcell (1946). Physically, the cavity enhances the strength of the vacuum fluctuations at the resonance frequency; as a consequence the transition rate is increased. (T,,,/TSpon, is obtained when the number of oscillator modes per unit frequency interval in a resonant cavity is divided by the corresponding value in free space.) The opposite effect, the decrease of the decay rate, is obtained when the cavity is detuned. If the transition frequency of the atom lies below the fundamental frequency of the cavity, spontaneous emission is significantly inhibited. In an ideal case no mode is available for the photon and therefore spontaneous emission cannot occur (Kleppner, 1981). To change the decay rate of an atom, in principle no resonator has to be present; any conducting surface near the radiator affects the mode density and, therefore, the radiation rate. Parallel-conducting planes can somewhat = (3/4n2>*

FIG.9. Schematic description of the atom-single field mode system. The coupling is described by the one-photon Rabi frequency R and the characteristic damping time Q/w.

RY DBERG ATOMS

443

alter the emission rate but can only reduce the rate by a factor of 2 because of the existence of TEM modes, which are independent of the separation. The effect of conducting surfaces on the radiation rate has been studied theoretically in a number of investigations (Purcell, 1946; Milonni et af., 1973; Wittke, 1975; Kleppner, 1981). To demonstrate experimentally the modification of the spontaneous decay rate, it is not necessary to go to single-atom densities in both cases. The experiments where the spontaneous emission is inhibited can also be performed with higher densities. However, in the opposite case, when the increase of the spontaneous rate is observed, a large number of excited atoms increases the field strength in the cavity and the induced transitions disturb the experiment. The first experimental work on the inhibited spontaneous emission was done by Drexhage (1 974). The fluorescence of a thin dye film near a mirror was investigated. Drexhage observed an alteration in the fluorescence lifetime arising from the interference of the molecular radiation with its surface image. An experiment with Rydberg atoms was recently performed by Vaidyanathan, Spencer, and Kleppner ( 1981). They observed a wavelength-dependent cutoff in the absorption of blackbody radiation by Rydberg atoms arising from a discontinuity in the density of modes between parallel-conducting plates. Absorption at a wavelength ofjcm by atoms between planes+ cm apart was measured at a temperature of 180 K. The discontinuity in the absorption rate occurred when the absorption wavelength was varied across the cutoff of the parallel-plate modes. The experiment was performed with Na atoms and the transition employed was 29d + 30p. For the tuning ofthe atomic resonance across the cutoff frequency a small electric field was applied to the parallel plates. Inhibited spontaneous emission was observed clearly for the first time by Gabrielse et al. (1984) and by Gabrielse and Dehmelt (1984). In these nice experiments on a single electron stored in a Penning trap they observed that the cyclotron excitation shows a lifetime which is up to 10 times larger than that calculated for a cyclotron orbit in free space. The electrodes of the trap form a cavity which decouples the cyclotron motion from the vacuum radiation field leading to a longer lifetime. The first observation of enhanced atomic spontaneous emission in a resonant cavity was published by Goy, Raimond, Gross, and Haroche (1983). Their experiment was performed with Rydberg atoms of Na excited in the 23s state in a niobium superconducting cavity resonant at 340 GHz. Cavity tuning-dependent shortening of the lifetime was observed taking advantage of the very strong electric dipole of these atoms and of the high Q value of the superconducting resonator. This cooling, necessary for superconducting operation, also had the advantage of totally suppressing the blackbody field

444

J. A . C. Gullus et ul.

effects ( n = 0) required to test purely spontaneous emission effects in the cavity (see Haroche and Raimonde, this volume, Section III,B,3.) It was shown that the partial spontaneous emission probability on the 2 3 s - 22P transition in Na is increased from its free space value rspont = 150 sec-l up to r,,, = 8 X lo4 sec-'. This enhanced rate is still 35 times smaller than the damping rate m/Q = 2.8 X lo6sec-' ofthe field in the cavity. This means that the photon emitted in the mode is absorbed in the mirrors much faster than the atoms decay. With a 10-fold increase in Q, the values of r,,, and o/Qwould be of the same size, so that the emitted photon would be stored in the cavity long enough for the atom to reabsorb it. This would approach the regme of quantum mechanical oscillations between a two-level atom and a single electromagnetic field mode mentioned at the beginning of this section. The self-induced single-photon Rabi nutation is much more difficult to observe than the collective Rabi oscillation (which will be described later) because it occurs at a rate f i t i m e s smaller (Nis the number of atoms in the cavity) and thus requires the atom to be kept in the cavity for much longer times. Experiments to observe this single-atom - single-photon interaction are presently under way at the Ecole Normale SupCrieure in Pans and in our laboratory. The set-up used in our laboratory (Meschede, 1984) is shown in Fig. 10.

FIG. 10. Experimental set-up for observation of single-atom -single-photon interaction. The Rydberg atoms are prepared in a specific velocity subgroup by a modulated stepwise excitation either with two laser beams or with modulated laser and microwave fields. (The figure shows the position oftwo laser beams.) The length ofthe cylindrical cavity is about 20 mm. The microwave cavity and the surrounding parts are cooled to 2 K with liquid helium (figure according to Meschede. 1984).

RYDBERG ATOMS

445

The superconducting niobium cavity has a Q value of 8 X lo8.In this experiment the atoms are velocity selected during the stepwise excitation into the Rydberg states by using the modulated radiation of two lasers. Instead of the second laser a modulated microwave field can also be used in order to prepare a selected velocity subgroup. The latter method has the advantage that the first excitation step can already be performed into a Rydberg state with high n and long radiative lifetime. This reduces the losses due to spontaneous decay between the two excitation regions. With the set-up shown in Fig. 10 maser oscillation with single atoms was recently observed (for the first experiments the velocity selection of the atoms described above was not operational). The transition 63p3/,-6 ld3/2at about 2 1.5 GHz between 85RbRydberg states was used. The atoms ofa beam were excited by the frequency-doubledlight of a cw dye laser. The cavity was operated in the TE,?, mode. Its high Q value of 8 X lo8allowed us to observe maser oscillation with an average number ofatoms in the cavity ofonly 0.06. An increase in the density of the atoms caused power broadening of the transition and ultimately for about one atom also caused a dynamic Stark shift (Fig. 11). The experiments clearly show that the set-up should also be suitable for observing the predicted quantum revival (see next section). For this observation the interaction time of the atoms with the cavity field has to be well defined and therefore a velocity selection of the atoms in the thermal beam is necessary. Experiments along these lines are at present under way.

g!

e

c

Covity frequency IMHz)

FIG.1 I . Maser operation with single atoms. Shown is the signal of the atoms in the 63p,/, level when the cavity is tuned through the resonance for the 63p3,,- 6 Idllz transition of *'Rb. The flux of the beam is changed for the three curves. The three curves from top to bottom correspond to only 0.06,0.30, and 1.6 atoms being simultaneously in the cavity.

J. A . C. Gallas et al.

446

B. SINGLEATOM I N RESONANT CAVITY -DISAPPEARANCE AND REVIVAL OF OPTICAL NUTATION At temperatures T>O K the cavity also contains thermal photons. The effects described above therefore become more complicated since the atom evolves through an oscillatory or irreversibly damped transient regime toward a final state distribution corresponding to the thermal equilibrium. The transient behavior is again dependent on whether there is weak (aB w / Q ) or strong damping (Q 20,000), oscillations in the atomic population evolution becomes clearly observable. This collective self-nutation regime has also been discussed in the context of superradiance theories (Bonifacio et al., 1975;McGillivray and Feld, 1976; Haake et al., 1979;Polder et al., 1979). It is then generally referred to as the “ringing” regime of superfluorescent emission. In the case of free-space superradiance this phenomenon has not yet been clearly observed since the simple Rabi nutation is then masked by multimode diffraction and propagation effects. D. N ATOMSIN RESONANT CAVITY -COLLECTIVE ABSORPTION OF BLACKBODY PHOTONS

In the previous section the case where the N atoms were initially in the excited Dicke state IJ, J ) was discussed. In the following, the Natoms are now assumed to enter the cavity in the lowest state IJ, -J ) ; furthermore, it is assumed that the cavity field is in thermal equilibrium at a temperature TZOK. The thermal photons represent a Bose-Einstein distribution with an average photon number C # 0. As the time evolves, the atoms gain energy at the expense of the mode which is then supplemented by the thermal reservoir. The time constant for reaching thermal equilibrium depends on the values of N , C, and w/Q. Since the atomic energy diagram consists of nondegenerate equidistant levels with the same spacing as the field levels, the atoms will obviously reach an equilibrium described by a Boltzmann law quite similar to the Bose- Einstein distribution of the photon number in the field mode. (The only differencebetween the two distributions is that the number of levels for the atomic system is finite; this changes the normalization of the Boltzmann distribution.) As a consequence, the number of absorbed photons AN is limited, no matter what value N has, and is equal to the average blackbody photon number per mode (as soon as N > E):

+

AN = C = [exp(hw/kT) - 11-*

(16)

which is close to kT/hw in the Rayleigh-Jeans limit. The energy absorbed by Natoms in the cavity is not identical with the sum of energy that would be absorbed by N independent atoms. In this process the atomic sample evolves in a collective mode and behaves as a single quantum system exhibiting basic effects of Bose - Einstein statistics and Brownian motion (Raimond et af.,1982a). A detailed study of the pulse-topulse random variations of AN around A E should allow one to probe the fluctuations of the cavity mode and to reconstruct their Bose-Einstein

450

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distribution. There is a connection between the absorption and emission of N atoms in a cavity: The atomic indiscernibility, which is responsible for superradiance when the system is initially excited, leads to a kind of “subabsorption” (Raimond et al., 1982a) when it starts from its lower state. The experimental demonstration of the effects just described was first performed by Raimond et al. (1982a); see Haroche and Raimond, this volume, Section IV,C, for a detailed discussion.

VIII. Rydberg States of Molecules Compared with the great amount and large variety of work performed on Rydberg states of atoms there are very few studies on highly excited states of molecules. This is the result of the great complexity of the absorption and emission spectra caused by the vibrational and rotational structure in addition to the closely spaced Rydberg levels. Selective excitation of the highly excited levels is almost impossible owing to the thermal population of the rotational and vibrational levels of the electronic ground state. This is the reason why in many investigations the molecules are cooled to low temperatures or why molecular beams with nozzle expansion are used, which cools down the internal degrees of freedom. The new methods oflaser spectroscopy are also ofgreat help in simplifying the complex spectra, so that more and more studies have been published in recent years. The Rydberg levels of diatomic molecules show some very interesting peculiarities. A molecular Rydberg level can lie above the lowest level of the corresponding molecular ion and can autoionize if part of the vibrationalrotational energy can be transferred to the Rydberg electron. This coupling between electronic and nuclear motions corresponds to a breakdown of the Born - Oppenheimer approximation. The study of the autoionizing resonances, their line profiles, and positions is of fundamental interest for the investigation of the interaction between bound states and the continuum. The predominant coupling scheme ofangular momenta in the lower lying states of many important diatomic molecules such as H2, the alkali dimers, and NO is Hund’s case (b): the orbital angular momentum L is strongly coupled to the internuclear axis with A as projection. A couples with N to K and K with S to the total angular momentum J. If an outer electron is excited to orbitals with higher main quantum numbers n, the coupling of L to the nuclear axis gets weaker and weaker. At very high n values, L is completely decoupled since the outer electron now sees an isotropic Coulomb force.

RY DBERG ATOMS

45 1

This corresponds to Hund’s case (d): L and the molecular rotation vector N couple to J (spin orbit coupling is not regarded in this case). This decoupling with rising n was studied in detail in the case of H, by Herzberg and Jungen ( 1972)(see also Takezawa, 1970). This work shows nicely the main features of the highly excited diatomic molecules and will be discussed here as a characteristic example. H2 has the great advantage compared with the other diatomic molecules that its rotational constant is very large, leading to clear, high-resolution absorption spectra in the ultraviolet (835 - 765 A). The spectra ofpara-H, taken at a temperature of 80 K show that all molecules were in the lowest rotational level. Each of the vibrational levels of Hz was found to be the limit of two Rydberg series. At low n (Hund‘s case b), these are the npa and npn series. At high n values, according to Hund‘s cased, the ldecoupling takes place and as a result the two series converge to the appropriate N = 0 and N = 2 levels of the Hf ion and are designated as npO and np2, respectively (the para-character limits N to even values). For intermediate n values there are strong perturbations between these two series. They arise from the coupling of the orbital motion of the Rydberg electron with the rotation of the Hf core. In addition to these Av = 0 perturbations, there are also strong interactions with the vibrations of the core. They occur when a Rydberg level with u 1 is close to one with u ( Av = 1 perturbations). After first attempts using perturbation theory (Herzberg and Jungen, 1972), the multichannel quantum defect theory (MQDT) initiated by Seaton, and elaborated by Fano, Lu, and others (see, for example, Fano, 1970, and the reviews by Seaton, 1983 and Fano, 1983) gave the complete interpretation of the very complicated spectra of H, . This theory was also applied to discuss the more recent results on Na, by Martin et al. (1983) obtained by laser spectroscopy. The main features of MQDT for atoms have already been mentioned in Section V,C. In the following the theory will be briefly outlined with respect to molecules. In the frame of MQDT excited states, autoionization and ionization are described as scattering processes of an electron at the positive core whereby energy and angular momentum are exchanged. The above mentioned mixing of the spectral series are described in terms of channel mixing, where the channels are determined by the energy of the core, the angular momenta of core and electron, and their respective coupling. As mentioned in Section V,C, the main idea of MQDT is that interactions between the electron and the ionic core can be separated into long- and short-rangeeffects. When the electron is far from the core there is a Coulomb interaction and the ion -electron system is described by “collision channels” i which are analytically known. The effect of short-range non-Coulomb interaction close to and inside the core is characterized by “close coupling channels” a. Effective quantum numbers ui are introduced for each ioniza-

+

452

J.A . C. Gallas et al.

tion limit Ii of the different series (i = 1 . . . M ) by E = Ii - R/ = 02, where E is the energy of the state and R the mass-corrected Rydberg constant. The discrete level is therefore described by a set of uivalues that satisfy the equation for the energy. The equation

det[ lJia sin n (ui +pa)] = 0 ensures correct asymptotic behavior of the wave functions for r + 03; pa are called the quantum defects and describe the close coupling channels a, and Viaare the elements of an orthogonal matrix, which transform the collision channels i into the close-coupling channels a. U, and pa are treated as parameters in a fit of the above equation to the experimental data. The graphical representations of ui versus uj are called Lu - Fano plots. In the case of H, , the two channels npO and np2 are taken into account, with ionization limits I,, I,. The effective quantum numbers are uo and 0,. For this case, the energy equations are of the form

E = I, - 13.6 eV/ua = I, - 13.6 eV/uT All the discrete levels observed by Herzberg and Jungen can be fitted using this two-channel model as has been demonstrated in a Lu-Fano plot of the fitted and experimental values. In particular the 1 uncoupling and also the autoionization which is observed in H, for levels above the N = 0 or u = 0 limit is fully accounted for (Fano, 1975; Jungen and Raoult, 198 1; Dehmer et al., 1984). A. RYDBERG STATESOF DIATOMIC MOLECULES 1. Na,

In the case of the Na, molecule conventional absorption and emission spectroscopy did allow just a few lower lying states to be indentified, but not Rydberg states. The investigation of the two-step labeling methods of laser spectroscopy was a prerequisite for tackling this complicated task. As will be discussed below, the application of these methods simplifies the spectra of the highly excited states. In addition the angular momenta of the states are obtained unambiguously. The optical - optical double resonance method combined with polarization spectroscopy, which gives Doppler-free lines, was successfully applied in studies of Na,, Cs,, and Li, . The principle of the experimental set-up used in these experiments is shown in Fig. 13. A polarized narrow-band laser is tuned into resonance with a known molecular transition. The laser light populates the degenerate angular momentum

453

RYDBERG ATOMS BROADBAND PROBE LA%ER AXJ~OOA

POLARIZER

PUMP LASER AA=O.OlA

SPECTOGRAPH

FIG.13. Experimentalarrangement of a two-step polarization labeling experiment as used by Teets ef al. (1976) and Carlson ef al. (1981).

sublevels of the upper state unequally, so that an orientation or alignment is obtained and, simultaneously, the sublevels of the lower state are nonuniformly depleted. This produces optical anisotropy in both levels belonging to the pumped transition. Ifa second linearly polarized probe laser is tuned into resonance with another molecular transition starting from one of the levels pumped with the first laser, the induced optical anisotropy causes a change in the direction of polarization of the second laser. This change can be detected using an analyzer in the second beam. The spectrum which is obtained when the second laser is tuned is greatly simplified since only transitions having a common level with the pumped transition can be observed. Highly excited states close to the ionization limit can be studied if the probe laser further excites the molecules starting from the upper level of the pumped transition. When this upper level is known, the selection rules help to determine the angular momenta of the highly excited states. The labeling method does not require a narrow-band laser to resolve the Rydberg states as is shown in the investigation of Carlson et al. ( 198 l), who used a pulsed broad-band (300-A) probe laser. In their work 24 new electronic states with excitation energies between 28,000 and 39,000 cm-I were found, for which A as well as the rotational and vibrational constants were determined. The states were identified as low-lying members of Rydberg series with n = 3 to 14. By extrapolation the vibrational and rotational constants of the ground state of Nai are derived. Alkali dimers such as Na,, K,, and NaK in supersonic beams can be resonantly ionized by two steps via selected rovibronic intermediate states with the light of two tunable dye lasers (Herrmann et al., 1978). In all these cases it was found that autoionization is important when the excitation is up to 1000 cm-’ above the ionization threshold. In this case part of the rovibronic energy of the inner core is transferred to the outer Rydberg electron, i.e., there is strong coupling between electronic and vibrational energy resulting in a breakdown of the Born- Oppenheimer approximation.

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454

The vibrational constants of the Nat ,Z; ion core were investigated in more detail by Leutwyler et al. (1982). They obtained good agreement with the value of Carlson et al. (198 I), which was determined by polarization spectroscopy. The Rydberg series observed by Leutwyler et al. converge to five different ion core vibrational states above the ionization limit which were closely spaced. Another result of the work was an estimate of the ionization potential of Na, which was derived from an extrapolation to

v=o.

The first assignment of very high Rydberg levels of Na, (30 < n < 75) was performed by Martin et al. ( 1983).The spectra were taken with a supersonic beam. The pulsed pump laser populated a well-defined intermediate rovibrational level u'J' of theA'Z:or B'H, state of Na,. The probe laser is tuned to autoionizing Rydberg states, and so Nazions could be detected (Fig. 14). Because of the 3p character of the intermediate level, Rydberg series noted as ns '2: and nd 'A:, nd 'n, and nd are excited. From extrapolation to n m, the quantum defectsas well as the molecular constants ofNagand the

-

60 39,970 I

70

39,980 I

80

90 I f 0 39,990 1

m

40,000

'

ndN+=23 nd N+=21 E [crn-'1

FIG. 14. Rydberg spectrum of Na, taken by Martin et al. (1983) using the optical-optical double-resonance method and autoionization for detection. The signal is shown in the lower part of the figure. The lines start from the intermediate level u' = 4, J' = 22 of the A state. There aretwoRydbergseriesv = 4 , n d N + = 2 l , a n d u = 4 , n d N + = 23withnupto75.Thisspectrum is compared with a calculated one being obtained by means of MQDT as shown in the upper part.

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ionization potential of Na, were determined. Similar to the studies in H, the uncoupling of the angular momentum 1 from the internuclear axis can be observed. Then 1 is more strongly coupled to the rotation (Hund's case d), according to J = N 1. Also the mixing of states at lower n was observed as in H, . Multichannel quantum defect theory was again successfully applied (Fig. 14).

+

2. Li, The experimental methods which have been used to study Rydberg states of Na, have also been successfully applied to the spectrum of Li, . The latter is of special interest since Li, is the least complex stable diatomic molecule after H,. Bernheim et al. (1979) began with the optical-optical double resonance labeling to investigate higher Rydberg states of Li, . They excited Li, in a heat pipe with two pulsed dye lasers. For the first excitation step one laser was tuned to the transition A T : +XIZ; and the second laser was scanned from 530-370 nm exciting gerade Rydberg states. The excitation was monitored by observing the radiation produced by collisional transfer from the gerade to ungerade states, which could then decay to the ground states by single-photon transitions. Bernheim et al. (1 979) identified 3 1 new electronic states belonging to the Rydberg series ns IZ;, nd lZ;, nd 'n, having principal quantum numbers up to n = 15. The molecular constants of the Rydberg states were determined. By extrapolating (n + 00) the data of the ground state of Liz were also obtained as well as the ionization potential of Liz. Ab initio calculations by Konowalow and Rosenkrantz (1979) are in good agreement with these experimental results. Extensive high-precision experiments on Li, were performed by Demtroder et al. (1983). For most of the measurements the optical-optical double resonance method was used. The first laser populated the levels ofthe B'II,,state, the second laser is scanned through the spectral region ofinterest. The excited Rydberg levels decay by autoionization and the molecular ions are extracted by a small electric field and are monitored as a function of the wavelength of the second laser. The adiabatic ionization potential could be derived from the onset of the ionization continuum. Furthermore, the dissociation energy of Lif was determined, being appreciably larger than that of the neutral ground state. Because of the high complexity of the spectra resulting from perturbations when 1uncouples from the figure axis with increasing n,it is also necessary to apply Doppler-freelabeling methods, such as two-step polarization spectroscopy. This polarization double-resonance spectroscopy allows a distinction to be made between P, Q, and R lines since the double resonance signals differ in sign and profile for the different transitions. The level width of upper

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Rydberg levels can vary greatly, owing to predissociation, if repulsive potential curves cross the Rydberg potentials. This can be measured by Dopplerfree two-photon spectroscopy.

3. NO Perhaps the most complete measurements on Rydberg states of heteronuclear diatomic molecules have been performed on NO by emission and absorption spectroscopy in the vacuum ultraviolet. Most of the work was done by Miescher and co-workers (1 966) and by Jungen ( 1970). To study Rydberg- Rydberg transitions in NO another interesting version of the optical -optical double-resonance method is used in work by Cheung et al. (1983). NO gas in a cell is excited by two pulsed dye lasers, the pump laser exciting theA2X+level starting from the ground state. The probe laser is tuned to populated higher states. The excitation is monitored by resonance ionization of the highly excited level and by detection of the ions. The paper demonstrates, again, advantages of the methods of laser spectroscopy. Two new Rydberg transitions were analyzed and strong multistate interactions among Rydberg and valence states just below the dissociation limit were observed. 4. H,

Finally, Rydberg states of H, were also studied by using laser techniques. Rottke and Welge ( 1984)generated the 1067-A radiation necessary to excite the intermediate state B*C:(u = 0; J = 0,1,2) by frequency tripling an XeCl excimer laser-pumped dye laser in a noble gas cell. A second dye laser pumped by the same excimer laser is used for the second step to reach the Rydberg states. Various Rydberg series with n up to 75 converging to different rotational states of H: were observed. The excitation of the Rydberg states was monitored for high n by field ionization and for low n by dissociation followed by one photon ionization of the product H*. Furthermore, photoionization of H, was observed in these experiments.

B. RYDBERG STATESOF LARGEMOLECULES Absolute lifetimes of Rydberg states of gas phase benzene, perdeuterobenzene, toluene, and perdeuterotoluene were measured by Wiesenfeld et al. ( 1 983). The Rydberg states were excited with 0.19 psec dye laser pulses. The probe beam, which was split off from the pump beam, was delayed by a

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variable time and was used to ionize the excited state. The time evolution of the excited state was determined by the decay of the ion current as a function of increasing time delay. In this way absolute lifetimes from 70 to 170 fsec were measured. This is the shortest molecular process ever measured directly. The decay mechanism is unclear. However, it is probable that internal conversion to high vibrational levels of the ground state is the reason for the fast relaxation.

IX. Rydberg Molecules For the molecules discussed in Section VIII, Rydberg series are only observed at higher excitation energies. There exist, however, molecules for which all the transitions observed must be assigned to states showing the properties of Rydberg states, i.e., their energy is described by a Rydberg formula. The ground state of these molecules is usually dissociative. He, and ArH are well-known examples of this class of molecules, which may also be called Rydberg molecules. More than 50 bound excited states of He, have been identified during the last 50 years. In recent years much work has been done by Ginter et al. ( 1980) to analyze and classify the spectra. In particular, they applied MQDT to analyze the triplet levels of the configuration ( loJ2( lo,,) npA, n = 6- 17. Extensive channel mixing leads here to a breakdown of conventional band models for the levels with higher n. Metastable a(2so) 3X: He, molecules produced in a dc discharge in a flowing He stream were excited by pulsed dye lasers by Miller et al. ( 1979). Laser-induced fluorescence spectra for the (npn) %,tseries for n = 4 - 9 and the series n = 5 - 15 already known from emission spectra were observed. In addition, relaxation and fluorescence yield measurements of these states could be performed. Spectra of excited Rydberg states of ArH and ArD were found by Johns (1970) near 7670 A in an electric discharge through mixtures of argon and hydrogen or deuterium. So far no spectra of other rare gas monohydrides could be observed. Also polyatomic Rydberg molecules are known, as, e.g., H,, D, ,and their isotopic mixtures or NH, and ND,. These polyatomic molecules have in common that they can be thought of as being built up from their parent molecules H,, D,, NH, , and ND, by adding a proton, which is possible because of their high proton affinity. These ions are then neutralized by capturing an electron in a Rydberg orbital (Dabrowski and Herzberg, 1980). The most extensively and successfully studied polyatomic Rydberg mole-

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cules are H3and D3. Herzberg et al. (Herzberg, 1979; Herzberg et al., 1980, 1981, 1982) observed their optical spectra in the light of a hollow cathode discharge through H, and D,, respectively. Several parallel and perpendicular electronic band systems between 5600 and 7 100 A and in the infrared near 3600 and 3950 cm-I were found to belong to a symmetric top molecule. The molecular constants were derived using model band calculations. In particular, the rotational constants were found to be close to those of Hfand Df, respectively, which were predicted theoretically by Carney and Porter (1976) and measured by Oka (1980) and Shy et al. (1980). In this way the model of a Rydberg molecule with a triangular Hfcore with D,, symmetry and one electron in a higher orbital was established. The bands in the visible (at 5600 and 7 100 A) were interpreted as transitions between different electronic states with the principal quantum number n = 3 in the upper and n = 2 in the lower state. Predissociation in the lower 2s state leads to a line width of the lines in the two bands of 3 to 8 A. Owing to their large width and their high density, many lines within the bands are blends. However, the very complicated perpendicular band of D, at 7 100 A, for example, could be resolved into single lines by a special laser labeling method (Figger et al., 1983). For this purpose a line of the D, emission spectrum of the hollow cathode discharge was selected using a monochromator. Then the beam of a dye laser is directed through the discharge and tuned. The intensity of the observed emission line changes when the laser wavelength coincides with a transition starting from the upper level of the selected transition. In this way simple spectra are obtained which are easy to assign. In particular they are free from the D, background lines. A large amount of transitions between n = 3 and n = 2 could be measured by this method. With a color center laser operating in a spectral region between 2.3 and 3.3 pm, lines have been found which seem to correspond to transitions to Rydberg states with n = 4. Such transitions could not be found spectroscopically by the conventional methods. Before the discovery of the spectra of triatomic hydrogen, bound states with lifetimes in the microsecond region were found by neutralizing Hfand Dfionic beams by transfer of an electron when the ions were passed through different target gases. The molecules H, and D, were detected by a mass spectrometer (Devienne, 1968;Nagasaki el al., 1972).It is quite obvious that this production of triatomic hydrogen molecules by charge exchange is also suitable for direct observation of the spectra. For this purpose intense beams of the corresponding triatomic ions are necessary. If the source is combined with a mass selector, then the different isotopic mixtures of the triatomic hydrogen molecules can be produced and analyzed. Such experiments were recently performed by Figger et al. (1984) and will be described in the following.

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In the apparatus a hollow cathode combined with a beam extraction system was used as ion source. The hydrogen gas pressure in the source was set at a few Torr. The acceleration voltage for the ion beam was between 4 and 20 kV. The ion beam was passed through a homogeneous magnetic field for mass selection from which beams of D I and H: could be separated. Beams of D2H+and H2D+could also be obtained with a suitable mixture of H, and D, in the hollow cathode. Typically, an extraction voltage of 12 kV yielded a 12-pA ion beam of the selected species. For neutralization, the ion beam was passed through an alkali vapor, which provides more effective resonant charge transfer than H, or Ar, used in former experiments. About lo-, Torr of alkali vapor in a 1-cm-long cell was sufficient to neutralize about 80% of the ions. The neutralized ions appeared, at least partly, in highly excited electronic levels. They cascade to lower levels through emission of photons or simply predissociate. Spectra of D,H, H,D, D,, and H, can be observed through a side window of the cell or after the neutralized molecular beam passed the cell (see Fig. 15).This experimental set-up offers a convenient method of measuring the lifetimes of single rotational states by observing the decrease of the intensity of the corresponding spectral line emitted along the beam after passing through the cell. The latter can be measured by moving the cell toward the ion source having the detector fixed in space. The procedure is similar to that in beam foil experiments. The lifetimes are then obtained by dividing the l/e decay distance by the velocity of the molecules in the beam, which is assumed to depart little from the ion velocity before the charge exchange cell. Several of the lifetimes measured in this way for D, agree with theoretical values calculated from electric dipole transition moments given by King and Morokuma (1979). In the cases of H,D and D,H the measured lifetimes are much shorter than the calculated radiative ones, showing that other decay mechanisms such as predissociation can also have an influence on the n = 3 states, which does not seem to be the case for the D3 molecule. Because of the great proton affinity of NH, and ND,, the ammonium radicals NH4 and ND, are also expected to be Rydberg molecules with one outer electron orbiting around the stable NHaand NDaion cores. For NH, and ND, two bands called Schuster and Schuler bands have been identified in the light emitted by a discharge through NH, and ND, (Schuler et al., 1955; Schuster, 1872; Herzberg, 1981). The Schuster band was tentatively assigned to the forbidden transition 3d *F2 3s ZAl(Herzberg, 198I), where 3s 2A, is the ground state of the ammonium radical. The assignment of the Schuler band was discussed in the literature some time ago. Calculations of the electronic energy level system by Broclavik et al. (1982) and Havrilak and King (1983) suggest that it has to be assigned to the 3p ,F2-,3s ,Al

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5600

5700

a

1

50

25

0

17910

17780

17620

-

17160 cm-’

FIG. 15. (a) Emission spectrum of D,H obtained with the beam apparatus (Figger ef a/., 1984). (b) Computer-simulated spectrum for the 3p ,B, 2s ,A, band of D,H. The numbers are asymmetric-top quantum numbers JK-K+.It can be seen that each peak ofthe experimental spectrum could be explained in the computer simulation.

transition analogously to the Na-D lines. This is also supported by two recent experiments: Gellene et al. ( 1982)found a fair degree ofstability ofthe ground state of ND, (T 3 sec) in a neutralized ion beam experiment. This leads to rather clear band systems which end on the ground state, as is observed for ND, . Whittaker et al. ( 1984)observed many lines ofthe Schuler band by laser frequency modulation spectroscopy. Here ND, is formed in a photochemical reaction of ND, . They also favor the assignment mentioned above. REFERENCES Allen, L., and Eberly, J. H. (1975). “Optical Resonance and Two Level Atoms.” Wiley, New York. Aymar, M. (1984). J. Opt. Soc. B 1,239; Phys. Rep., in press.

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A

Acetic acid cluster ions of, 93 binding, 133 dissociation, 1 15 Acetone, cluster ions of, 9 I Aluminum, cluster ions of, 13I Alkali-halide ions, rate coefficients for reactions between, 28-29, 31 -33 Alkali Rydberg atoms electric dipole transitions in, 350-352 high-resolution spectroscopy of, 427 -43 1 Alkaline earth Rydberg atoms, highresolution spectroscopy of, 433-435 Alkaline Rydberg atoms, high-resolution spectroscopy of, 43 1 -433 Ammonia cluster ions of, 9 1 association reactions, 100 bonding to positive ions, 128 structure, 130- 131 thermodynamic properties of, I26 Argon cluster ions of, 87, 89, 90, 9 1, 93 dissociation, 104, 107 mass spectroscopy, 94-95 electron mobilities in, 270-272 ion mobilities in, 30 I physical properties of, 273 Atmosphere cluster ion formation in, 8 1, 13 I Atomic charges within molecules, 41 -63 conservation ofbond moment and, 48-49 dividing space for, 49 in localized hybrids, 49- 53 loge partitioning and, 54 partition studies, 53-56 467

point charges, 57-60 population and, 45 - 53 shrinking Gaussians and, 58-60 space division and, 49 in topological atoms, 54-56 Atomic inner-shell ionization from high-Z projectiles, 186- I88 from low-Z projectiles, 184- 186 survey of, 182- 188 Atomic populations, 45 -53 modified weighting in, 48 overlap populations and, 45 -48

B Barium, Rydberg atoms, high-resolution spectroscopy of, 435 Beam scattering, in studies of cluster ions, 70 Benzene, cluster ions of, 93 Bicarbonate, cluster ions of, 1I2 Blackbody radiation, Rydberg atom interaction with, 354-355, 435-440 Bloch vector, 39 I Born-Oppenheimer definition of molecular energy, 42-43,45 Burnt flame gas, ion-ion recombination in, 22-23,29 C

Calcium, Rydberg atoms, high-resolution spectroscopy of, 435 Carbon dioxide cluster ions of, 86, 87, 89, 9 1 bonding, 128, 134 photodissociation, 109- 110 stability, 127 thermodynamic properties of, 126

468

INDEX

Carbon disulfide, cluster ions of, 9 I Carbon monoxide cluster ions of, 9 1 association reactions, 100, 102 thermodynamic properties of, I26 Cesium cluster ions, photodissociation, 108 Rydberg atoms, high-resolution spectroscopy of, 429 Cesium halides, gaseous, ion-ion recombination in, 29, 3 1 Cesium iodide, cluster ions of, mass spectroscopy, 96, 97 Chirgwin-Coulson theory of atomic populations, 47 -48 Christofferson-Baker definition of gross atomic charge, 48 Chromium, cluster ions of, binding, 13 1 Chromium carbonate, cluster ions of, dissociation, 1 14 Cluster ions, 65 - 172 association reactions of, 96- 102 bonding to negative ions of, I32 - I36 bonding to positive ions, 123- 128 dissociation of, 102- I I5 collision-activated, 102, 104- 106 photodissociation, 106- 1 1 5 unimolecular, 103- 104 entropies of, 120- 122 equilibrium measurements on, I 18- 120 experimental methods for, 68 - 8 1 formation of, 8 I - 102 high pressure and drift cell techniques for, 75-81 ionization cross sections of, 86-94 ionization processes for, 8 1 -96 molecular beam ionization technique for, 69-15 multiply charged, 84 reactivity of, I36 - I39 recombination of, I39 solvation of, 128 - 130 structural aspects of, 130- 13 1 in systems with organic constituents, 131-132 transport properties of, 140- 14 I Cobalt carbonate, cluster ions of, dissociation, 114 Compound elastic scattering, separatedatom effects of nuclear reactions and, 208-226

Compound nucleus (CN) model of nuclear reactions, 177- 178, 189- 195 Copper cluster ions of, 9 I , 93 binding, I3 1 Cosmic rays, as cause of atmospheric ionization. 35

D Dimethoxyethane, clustering of, I3 1 Discharges, cluster ions in, 8 1 Distorted wave Born approximation (DWBA) K X-ray emission in, 23 1-233 for nucleon-induced atomic reactions, 182, 183, 190 Drift cell technique, for cluster ions, 75-81

E Earth, ion-ion recombination in atmosphere of, 33-37 Einstein relation, 4, 5 Einstein-Smoluchowski relation, 37 Electric and magnetic deflection, in studies of cluster ions, 70 Electric power, ion-ion recombination and generation of, 28 Electron-impact ionization available theories on, 242-245 box-normalized eigenstates for, 260-264 numerical calculations on, 241 -266 selected results on, 253-260 R-matrix formalism for, 260-262 wave packet method for, 246- 250 Electron mobilities, 267 - 325 in dense gases and low-density liquids, 28 I -296 in low-density gases, 270-281 Electrostatic potential, of molecules, 44-45 Electron attachment to van der Waals polymers, in studies of cluster ions, 7 I Electron density, of molecules, 43-44 Electron diffraction, in studies of cluster ions, 69 Electron-electron spectroscopy, of cluster ions, 70

469

INDEX Electron impact ionization mass spectrometry, of cluster ions, 70 Electron-induced desorption, cluster ion formation by, 8 1 Energy, of molecules, 42-43 Equilibrium measurements, on cluster ions, 118- 120 Ethane, physical properties of, 273 Ethyl alcohol, cluster ion studies on, 132 Ethyl cyanide, binding to ions, 13 1 Ethyl ion, association reactions of, 101 Ethylene cluster ions of, 9 I physical properties of, 273 Evaporation, cluster ion formation by, 80

F Fast fission, concept of, 180 Fick’s diffusion equation, 4 Fick’s transport equation, 5 Field desorption, cluster ion formation by, 80 Fluorobenzene, cluster ions of, 93 Formic acid, cluster ions of, 93 Franck-Condon principle, 82 G Gas(es) density, ter-molecular recombination and, 2-7 ion mobilities in, 297-320 ion-ion recombination in, 1-40 electron field effects, 20-2 1 intermediate-density gases, 13- 16 low-density gases, 7 - 13 neutral-neutral channels and, 29- 33 low-density electron mobilities in, 270-28 1 ion mobilities in, 297 - 3 18 H Hartree-Fock calculations, on heavy-ion collisions, 18 I Hartree-Fock theory, 55 Hauser-Feshbach formula, 229

Heisenberg picture, collective emission of N atoms in cavity in, 39 1 - 398 Helium cluster ions of, 104 photo dissociation, 107 electron mobilities in, 270, 27 1 , 282 -283 ion mobilities in, 309, 3 12 physical properties of, 273 Hellmann-Feynmann theorem, 43,45,49,55 Huckel theory of atomic populations, 47 Hydrobromic acid, cluster ions of, 9 1 Hydrochloric acid cluster ions of, 9 1 binding, 133 thermodynamic properties of, 126 Hydrofluoric acid, cluster ions of, 9 1 Hydrogen cluster ions of, 83, 86 - 87, 102 dissociation, 105 physical properties of, 273 Rydberg states of, 456 Hydrogen peroxide, cluster ions of, 93 Hydrogen sulfide cluster ions of, 9 1 bonding to positive ions, 128 Hydroiodic acid, cluster ions of, 9 1

I Iodine, cluster ions of, 9 1 Ion-ion recombination of alkali-halide ions, 28-29, 31 -33 in an ambient gas, 1-40 electric field effects, 20-21 universal curves in, 16- 19 in Earth’s troposphere and lower stratosphere, 33-37 Ion mobilities, 267- 365 in dense gases and low-density liquids, 318-320 in low-density gases, 297 - 3 18 Ion traps, cluster ion formation in, 8 1 Ionization, electron-impact type, numerical calculations on, 241 - 266 Ionosphere, cluster ion formation in, 8 1 Ionization detector, with retarding field, in studies of cluster ions, 69 Ionization mass spectrometry, in studies of cluster ions, 70 Iron, cluster ions of, binding, 13 1

4 70

INDEX

Iron carbonate cluster ions of, 137 dissociation, 1 14- I I5

K Knudsen cell technique, in ion cluster studies, I16 Krypton cluster ions of, 87, 93 photodissociation, 107

L Landau-Zener parameter, 23, 2 6 , 3 0 derivation of, 22 Langevin-Harper formula, 5, 7 Langevin’s high gas density equation, 3, 1 I , 19 Laser evaporation, cluster ion formation by,

a1

Laser-induced field evaporation, cluster ion formation by, 8 I Lead, cluster ions of, 9 1 Light scattering, in studies of cluster ions, 69-70 Lithium cluster ions of, 93 binding to positive ions, 127 Rydberg atoms, high-resolution spectroscopy of, 429 Rydberg states of, 455-456 Lithium halides, gaseous, ion-ion recombination in, 29, 3 1 Localized hybrids, atomic charges in, 49- 53 Loge partitioning, in atomic charge studies, 54 Loschmidt’s number, 2, 27

M Magnesium, cluster ions of, 13 I Manganese, cluster ions of, 13 I dissociation, 106 Manganese carbonate, cluster ions of, dissociation, 114 Mass selective photoionization, cluster ion formation by, 86

Mass spectroscopy, of cluster ions, 94 -96 Mercury, cluster ions of, 9 1 Methane electron mobility in, 292 physical properties of, 273 thermodynamic properties of, 126 Methanol, cluster ions of, 93 Methyl alcohol, cluster ions of, binding, 133 Methyl cyanide, binding to ions, 13 1, 133 Molecular beam ionization technique, in studies of cluster ions, 69-75 Molecules atomic charges within, 41 -63 electron density of, 43 - 44 electrostatic potential of, 44-45 energy of, 42-43 properties of, 42 -45 Rydberg states of, 450-457 Mulliken atomic charges, 52 Mulliken theory of atomic populations, 45-48 Multiphoton excitation, cluster ion formation by, 85 - 86 N Neon cluster ions of, 84-85 photodissociation, 107 ion mobilities in, 307 physical properties of, 273 Nernst-Townsend-Einstein relation, 309 Neutral-neutral channel, in ter-molecular recombination, 29-33 Neutralization channels, inclusion of, in ion-ion recombination in gases, 2 I -29 Nickel, cluster ions of, binding, 13 I Nickel carbonate, cluster ions of, dissociation, 114 Nitrogen cluster ions of, 9 I , 102 association reactions, 100, 102 dissociation, 104 stability, I27 electron mobility in, 288-292 ion-ion recombination in, in atmosphere, 33 ion mobilities in, 3 13 physical properties of, 273 thermodynamic properties of, 126

47 1

INDEX Nitrogen dioxide cluster ions of, 9 1 dissociation, 1 13 Nitrous oxide cluster ions of, 9 I photodissociation, 108, 109 stability, 128 Rydberg states of, 456 Nuclear reactions compound nucleus model of, 177 effects on atomic inner-shell ionization, 173-239 from heavy projectiles, 178- 182 from light projectiles, 176- 178 positron emission from, 201 -208 separated-atom effects of, 208 -226 from high-Z projectiles, 2 19 -226 sequence of stages of, 178 statistical theory of, 228-230 survey of, I76 - I82 united atom effects of, 189- 208 experiments with high-Z projectiles, 195-201 experiments with low-Z projectiles, 189-195

0 OCS, cluster ions of, 9 1 Oxygen cluster ions of, 9 1, 102 photodissociation, 108- I 10, 1 I 1 - 1 12 ion-ion recombination in, 3 in atmosphere, 33

P Penning ion source evaporation, cluster ion formation by, 8 1 Photodissociation, of cluster ions, 106- 1 15 Photoelectron spectroscopy, of cluster ions, 70 Polar molecules, electron scattering in, 279-281 Positrons, emission from nuclear reactions, 201-208 Potassium cluster ions of, 93 binding to positive ions, 127

Rydberg atoms, high-resolution spectroscopy of, 429 Potassium halides, gaseous, ion-ion recombination in, 29, 31 Pitaeviskii theory of classical diffusion, 8 Point charges of atoms, 57-60

R Rare gases cluster ions of, 9 1 bond energies, 1 I7 - 118 Recombination coefficient, 9 derived formula for, 8 Refractory metals, cluster ions of, 75 Roby atomic charges, 52 Roby occupation numbers, 50 Rubidium, Rydberg atoms, high-resolution spectroscopy of, 429 Rubidium halides, gaseous, ion-ion recombination in, 29, 3 I Rydberg atoms excitation and detection of, 360-362, 4 19-42 1 in free space collective radiative properties Of, 355-356 interaction with black body radiation, 354-355,435-440 preparation of, 358 radiation interaction of, 440-450 spectroscopy of, 356-357 by atomic beam, 421 -423 by double resonance, 423 -425 high-resolution type, 4 13-466 by quantum beat and level crossing, 425-427 by two-photon absorption, 423 Rydberg molecules, 457 -460 Rydberg states in resonant cavities experimental techniques for, 358 - 363 radiative properties of, 347 -4 1 1 collective behavior, 383 -408 single atoms, 363- 383

s Sackur-Tetrode equation, I2 1 Schrbdinger equation, 62, 183, 242 integration of, 250

47 2

INDEX

Schrodinger picture, collective emission of Rydberg atoms in cavities in, 385 - 39 1 Semiclassical approximation (SCA), for atomic collisions, 182, I83 S l F I technique, 102, 1 17 Silver, cluster ions of, binding, 127, 131 Silver chloride, cluster ions of, dissociation, 103 Sodium cluster ions of, 91, 93 bonding, 124-125, 135 Rydberg atoms, high-resolution spectroscopy of, 429 Rydberg states of, 452-455 Sodium halides, gaseous, ion-ion recombination in, 29, 31 Spectroscopy, of Rydberg atoms, 356-357, 4 13-466 Sphericity effect, in polyatomic nonpolar molecules, 274-279 Stratosphere, ion-ion recombination in, 33 - 37 Strontium, Rydberg atoms, high-resolution spectroscopy of, 435 Sulfur dioxide cluster ions of, 9 1 binding, 133, 134 dissociation, 114 thermodynamic properties of, 126

T Ter-molecular recombination in gases, 2 - 2 1 neutral-neutral channel in, 29- 33 Thomson model for ion recombination in gases, 7, 8, 33 Thomson partial recombination coefficient, 11, 13 Topological atoms, atomic charges in, 54- 56 Transport properties, of cluster ions, 140- 141 Trifluoroacetic acid, cluster ions of, 93 Troposphere, ion-ion recombination in, 3337

U UV and X ray lasers, 327-345 gain coefficient scaling in, 330-331 in version schemes for multicharged plasma, 333 - 342 computer simulation of, 38 opacity limitations in, 33 1 -332 plasma refraction in, 332 -333 resonator or superradiance in, 328-330 V Vanadium carbonate, cluster ions of, dissociation, 114 W Wannier theory, 263, 264 Water cluster ions of, 87, 9 1 binding, 134 mass spectroscopy, 95 -96 thermodynamic properties of, 126 Wave functions, of molecules, 41 -42

X X rays CN-type, from proton-induced reactions, 191-193 effect on ion-ion recombination in gases, 1 K-type, emission in second-order distorted wave approximation, 231 -233 separated atom type, from nuclear reactions, 208-226 types of, from atomic collisions, 188 Xenon cluster ions of, 93 photodissociation, 107 electron mobility in, 284 -287 physical properties of, 273

Contents of Previous Volumes Volume 1

Volume 3

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies. H . Pauly and J. P. Toennies High-Intensity and High-Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J. B. Fenn AUTHORINDEX-SUBJECTINDEX

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. WOK Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, F. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood AUTHORINDEX-SUBJECT INDEX

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. Mum, 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. deHeer Mass Spectrometry of Free Radicals, S. N. Foner AUTHORINDEX-SUBJECTINDEX 473

Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H . S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G.Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0.Mohr

474

CONTENTS OF PREVIOUS VOLUMES

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 . Dalgarno The Direct Study ofIonization in Space, R. L. F. Boyd AUTHORINDEX-SUBJECTINDEX

Volume 5 Flowing Afterglow Measurements of Ion-Neutral 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. Dehmelr The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A . Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations ~ ' s ' ~ pC. q ,D. H. Chisholm, A. Dalgarno, and F. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle AUTHORINDEX-SUBJECTINDEX

Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A. S. Kauf man The Rotational Excitation of Molecules by Slow Electrons, Kaziio Takayanagi and Yukikazyu Itikawa The Diffusion of Atoms and Molecules, E. A . Mason and R. T. Marrero Theory and Application of Sturmain Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A . E. Kingston AUTHORINDEX-SUBJECTINDEX

Volume 7 Physics of the Hydrogen Master, C. Audion, J. P. Schermann, and P. Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Hare1 Weinstein. Ruben Paunez. and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules-QuasiStationary Electronic States, Thomas F. 0 '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, A. J. Greenfield AUTHORINDEX-SUBJECTINDEX

Volume 6

Volume 8

Dissociative Recombination, J. N. Bardsley and M . A. Biondi

Interstellar Molecules: Their Formation and Destruction, D. McNally

CONTENTS OF PREVIOUS VOLUMES

Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I. Schoen The Auger Effect, E. H. S. Burhop and W. N. Asaad AUTHORINDEX-SUBJECTINDEX

Volume 9 Correlation in Excited States of Atoms, A. W. Weiss The Calculation of Electron- Atom Excitation Cross Sections, M. R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of LowEnergy Electron - Atom Collisions, D. Andrick Molecular Beam Electronic Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McEIroy AUTHORINDEX-SUBJECTINDEX

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. Lange, J. Luther, and A. Steudel Recent Progress in the Classification of

47 5

the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr. SUBJECT INDEX

Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and w.D. Robb Role of Energy in Reactive Molecular Scattering: An Information - Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Greim Chemiluminescence in Gases, M. F. Folde and B. A. Thrush AUTHORINDEX-SUBJECT INDEX

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. Gouedard, J. C. Lehman, and J. Vigue Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid AUTHORINDEX-SUBJECTINDEX

476

CONTENTS OF PREVIOUS VOLUMES

Volume 13

Atomic and Molecular Polarizabilities - A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. V. Hertel and W. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfied Faubel and J. Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K . Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B. Sommerville AUTHORINDEX-SUBJECTINDEX Volume 14

Resonances in Electron, Atom, and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transition in One- and TwoElectron Atoms, RichardMarrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobashev Rydberg Atoms, S. A . Edelstein and T. F. Gallagher

UV and X-Ray Spectroscopy in Astrophysics, A . K . Dupree AUTHORINDEX-SUBJECT INDEX 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. Humherston Experimental Aspects of Positron Collisions in Gases, T. C. Grijith Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion- Atom Charge Transfer Collisions at Low Energies, J. B. Hasted Aspects of Recombination, D. R . Bates The Theory of Fast Heavy-Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody Inner-Shell Ionization, E. H. S. Bicrhop 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 AUTHORINDEX-SUBJECTINDEX 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

CONTENTS OF PREVIOUS VOLUMES

Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of 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. Wilets INDEX 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 Spectroscopyin Atomic and MolecularPhysics, M. G. Payne, C. H. Chen, G. S. Hurst. and G. W. Foltz Inner-Shell Vacancy Production in IonAtom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L. Dufton and A. E. Kingston INDEX Volume 18

Theory of Electron- Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand, and G. Petite

477

Classical and Semiclassical Methods in Ineleastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Andersen 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 Few-Electron Atomic Systems, G. W. F. Drake INDEX

Volume 19

Electron Capture in Collisionsof Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K. Janev 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. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. Jenajc The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov INDEX

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