Advances in Atomic and Molecular Physics, Volume 13

Advances in ATOMIC AND MOLECULAR PHYSICS V O L U M E 13

CONTRIBUTORS TO THIS VOLUME BENJAMIN BEDERSON PAUL R. BERMAN MANFRED FAUBEL

I. V. HERTEL THOMAS M. MILLER

R. K. NESBET W. B. SOMERVILLE

W. STOLL J . PETER TOENNIES

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D. R. Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST. NORTHERN IRELAND

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

VOLUME 13

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1977

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Contents vii

LIST OF CONTRIBUTORS

Atomic and Molecular Polarizabilities- A Review of Recent Advances Thomus M . Miller und Benjumin Bederson I. Introduction 11. The Calculation of Polarizabilities 111. Experimental Measurements of Polarizabilities

1V. Future Possibilities for Polarizabilities References

i 10 21

47 51

Study of Collisions by Laser Spectroscopy Paul R . Berman I. Introduction 11. Collisions

Ill. IV. V. VI.

Three-Level Systems Transient Systems Experimental Survey-Theoretical Conclusions References

Outlook

51 60 65 90 100 106

110

Collision Experiments with Laser Excited Atoms in Crossed Beams

I . V . Hertelund W . Stoll I. Introduction 11. Basic Theory I l l . Excitation of Atoms by Laser Optical Pumping IV. Theory of Measurements in Scattering Experiments by Laser-Excited Atoms V. Collision Experiments Vl. Atomic Scattering Processes in the Presence of Strong Laser Fields VII. Conclusions References V

113

I I7 129

157 174 21 I 223 224

vi

CONTENTS

Scattering Studies of Rotational and Vibrational Excitation of Molecules Manfred Faubel and J . Peter Toennies 1. 11. 111. IV. V. VI.

Introduction Potential Hypersurfaces Inelastic Scattering Theory Experimental Methods Recent Experimental Results Summary References

229 238 248 257 274 307 308

Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations R . K . Nesber I. 11. 111. IV.

Introduction Theory Methods Applications References

315 318 337 349 378

Microwave Transitions of Interstellar Atoms and Molecules W . B. Somerville 1. 11. 111. IV. V.

VI. VII. VIII. IX.

Introduction Spectroscopic Formulas Atomic Hyperfine Structure Atomic Fine Structure Recombination Lines Structure in Diatomic Molecules Transitions in Diatomic Molecules Rotation in Polyatomic Molecules Inversion in NH References

AUTHOR INDEX SUBJECT INDEX CONTENTS OF PREVIOUS VOLUMES

383 385 387 390 394 397 403 422 428 430 437 45 1 459

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

BENJAMIN BEDERSON, Physics Department, New York University, New York, New York (1) PAUL R. BERMAN, Physics Department, New York University, New York, New York (57) MANFRED FAUBEL, Max-Planck-Institut fur Stromungsforschung, Gottingen, West Germany (229) I. V. HERTEL, Fachbereich Physik der Universitat Kaiserslautern, Kaiserslautern, West Germany (1 13)

THOMAS M. MILLER, Molecular Physics Center, Stanford Research Institute, Menlo Park, California (1) R. K. NESBET, IBM Research Laboratory, San Jose, California (315) W. B. SOMERVILLE, Department of Physics and Astronomy, University College, London, England (383) W. STOLL,* Fachbereich Physik der Universitat Kaiserslautern, Kaiserslautern, West Germany (1 13) J. PETER TOENNIES, Max-Planck-Institut fur Stromungsforschung, Gottingen, West Germany (229)

* Present address: Messer Griesheim GmbH, Diisseldorf. West Germany. vii

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ATOMIC AND MOLECULAR POLARIZABILITIES-A REVIEW OF RECENT ADVANCES THOMAS M. MILLER hfolrcirlur Physics Cerilrr. Sturi/?)ri/ Rcwarclt Irr.sritulc>. Mrlllo Purk. Cfr/i/orrliu

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B E N J A M I N BEDERSON Physics Di,purrnicwr. New York Uiriorrsirj. Ncw York. New York 1. Introduction . . . . . . . . . . . . . . . . . . . . . ..................... A. Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B. Matrix Elements . . . . . . ...................... II. The Calculation of Polarizah ..................... A. Summation of Oscillator Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Effective Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. R-Matrix Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Variational Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Statistical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Experimental Measurements of Polarizab A. Bulk Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Indirect Methods . . . . . . . . . . . . . . . . . . C. Beam Techniques.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Simple Molecules . . . . . . ......................... IV. Future Possibilities for Polari ......................... Appendix . . . . ......................................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I 5 6 10 10 11

12 1X

19

42 46 47 49 51

I. Introduction The electric dipole polarizability of an atom or molecule describes the response in lowest order of the field strength of the electron cloud to an externa! electric field. This property plays an important role in those collision processes where the relevant electric field is that produced between collision partners, one or both of which is a neutral atom or molecule. Indeed, some

'\

7: M . Miller and B. Bederson

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TABLE I MANIFESTATIONS OF THE ELECTRIC DIPOLEPOLARIZABILITY a” Long-range electron- or ion-atom interaction energy Ion mobility in a gas” Relation between a ( v ) and oscillator strengthsc van der Waals constant between systems N, b (Slater-Kirkwood approximation)” “

I/ =

-e2ai2r4

K = 13.876/(ap)”’ cm2/volt sec

15

Dipole-quadrupole” constant C S e

Clausius-Mosotti relationf Dielectric constante Index of refraction” Diamagnetic susceptibility‘

K ( v ) = 1 + 4nna(v) q(v) = I + 2nna(v) zrn‘v e2[uon,a]‘’2/4mc2

Verdet constant’

V ( v )=

Rayleigh scattering cross-section‘ Modified effective range cross-section’

vn d x ( v ) 2mc2 dv U ( V )= (8n/9~‘)(2nil)‘[3a(v)~3r(v’)] 8n2p Z2e2aAk ... o ( k ) = 4nA2 + 3 h2 ~

+ +

Langevin capture cross-section“‘ “0

Casimir-Polder effect“ x is the static polarizability; frequency-dependent polarizabilities are indicated by a(v), which reduce to a for v = 0. Anisotropic effects are not included, except where specifically noted; cgs units are used except where noted. a in units of A3, p is the ion-atom reduced mass in units of the proton mass. Assumes pure polarization interaction - e2a/2r4;classical limit. c,&or is the average oscillator strength over magnetic substates, and refers to a transition between the state whose polarizability is a(vo)and all other states connected to this state by dipole matrix elements. no. , are number of electrons in outer shells of a, 6 . a d , a, are the dipole and quadrupole polarizabilities; ad,aqare the average transition frequencies for dipole and quadrupole transitions (see Stwalley, 1970; Kramer and Herschbach, 1970). For a gas of atoms or of molecules that do not possess permanent electric dipole moments. q ( v ) is the frequency-dependent index of refraction. n is number density. Obtained from Clausius-Mosotti relation, assuming K o 1. q2(v) = K ( v ) . ’ In the approximation that the static polarizability is given by the variational formula

”

@

ATOMIC AND MOLECULAR POLARIZABILITIES

3

low-energy scattering process can be specified accurately by an expression that involves only one parameter-the polarizability. In some circumstances a part of the atom can also be assigned a welldefined polarizability. Thus, the “core polarizability ” refers to the polarizability of the inner shells and is an important parameter in determining quantum defects, nuclear shielding, and polarizabilities of ionized atoms. Clearly, the polarizability is particularly important in interactions where there is little interpenetration of the wavefunctions of the collision partners. The most loosely bound electrons play the largest role in the redistribution of the electron charge in an electric field; the valence electrons generally account for at least 90% of the polarizability of an atom or molecule. The manifestations of the atomic polarizability in collision phenomena are manifold. A testimony to its importance is the frequency with which one encounters an expression involving the polarizability. We have gathered a representative collection and give them in Table I, which also includes some references to frequency-dependent polarizabilities. Some of these expressions have been used as a basis for measuring the polarizability, for example, using dielectric constants or ion mobilities. It is sometimes difficult to track down published polarizability calculations for specific atoms and molecules, since these often appear as intermediate steps in other calculations. One of the best calculations of the polarizability of cesium is that of Norcross (1973), in a paper entitled “ Photoabsorption by Cesium.” Other examples are the papers of Ice and Olson (1975) entitled “Low-Energy Ar’, Kr’, Xe+ + K, Rb, Cs Charge Transfer Total Cross Sections,” and of Garrett (1965), Polarization and Exchange Effects in Slow-Electron Scattering from Lithium and Sodium,” in which atomic polarizabilities were calculated. “

(Table footnotes continued) (see Hirschfelder er 01.. 1954, p. 942 IT.). The correct definition of magnetic susceptibility per atom is

In the very crude approximation that all the energy denominators in the oscillator strength summation for a are equal to the ionization energy, x” = (E/4rnc2)a. j Defined from 0 = V(v)B, where 0 is the angle ofrotation of linearly polarized light through a medium ofdensity n, per unit length, for a longitudinal magnetic field strength B (Faraday effect). a(\,) = $a, ++all; ~ ( v =) a l l - a,. A is the scattering length, p the reduced mass, k the wavenumber of the scattered particle, pu/h (see OMalley et a/., 1962). I ’ is ~ the relative velocity of approach for an ion-neutral pair. V ( r )is interaction energy for two nonpolar molecules, at very large distances [ r 9 c E i / l i ] ,g is a numerical factor, and Ei equals an average over the excitation energies (Casimir and Polder, 1938).

’ ‘

7: M. Miller. a i d B. Bcilerson

4

This article will review recent advances in the experimental and theoretical determination of the polarizabilities of simple atoms and molecules. We will concentrate primarily on static (as opposed to frequency-dependent) polarizabilities, although some discussion of the latter will be presented. Higher-order polarizabilities, which are becoming increasingly accessible to observation because of the availability of high-intensity lasers, will also be briefly discussed. The perturbation of atomic levels by electric fields was first reported by Stark (1913) and independently by LoSurdo (1913). The “Stark effect” was treated theoretically by Epstein (1916), who was later to apply the new quantum theory to the same problem (Epstein, 1926). Polarizability values (related to the quadratic Stark effect ”) are accurately known for the noble gas atoms and for hydrogen, in theory, but the remainder of the periodic table has proven much more difficult to deal with, both theoretically and experimentally. The static electric dipole polarizability of the ground-state hydrogen atom is almost exactly 4.5ai, where a, is the Bohr radius. Across rows of the periodic table, polarizabilities range from hundreds (of a: units) for the alkali metal atoms generally monotonically down to a few for noble gas atoms. Excited atoms have much larger polarizabilities; recent polarizability measurements for atoms in Rydberg orbits have yielded values on the order of 1 0 ’ O a ~In . Fig. 1 we have plotted the polarizabilities of the atoms in the first row of the periodic table. On the same graph we show the polarizability anisotropy, which indicates the importance of the orientation “

FIG.1. Atomic polarizabilities and polarizability anisotropies across the first row of the periodic table. (Results of Werner and Meyer, 1976.)

ATOMIC AND MOLECULAR POLARIZABILITIES

5

of the atom in an external electric field. The anisotropy is largest when the first p electron is added (boron) and becomes smaller for successive P states as the valence shell fills. A. MOMENTS

An atom or molecule in a uniform electric field E has an electric dipole moment p = po

+ a - E + jfl:E2 + 6y iE3 + ...

(1)

We will deal with cases where there is no permanent dipole moment po, only induced moments. Thus this discussion is restricted to the class of all atoms, and homonuclear molecules. For spherically symmetric states p is zero. One is generally concerned only with the polarizability a, a second-rank tensor; however, the kyperpolarizability y (a fourth-rank tensor) is of current interest in relation to laser studies of nonlinear effects in atoms. Higher-order hyperpolarizabilities are not significant in present-day experiments. Practically speaking, for static laboratory fields the induced electric dipole moment is adequately described by the polarizability a. If intense laser radiation fields are also present, then both the polarizability a and the hyperpolarizability y are needed to determine the induced electric dipole moment, although of course for this situation one is referring to a(v), y(v), where v is the frequency of the laser field. For the ground state of the hydrogen atom, the polarizability a, as already mentioned, is 4 . 5 ~ 4 ,while the hyperpolarizability y is exactly 1333.12%; /Hartree. The hyperpolarizability should not be confused with higher-order multipoles of the atomic charge distribution. The full quadrupole moment also contains induced terms, which depend on the electric field strength and on the field gradient. A complete treatment of the dipole and quadrupole expansions and the symmetry properties of the coefficients has been given by Buckingham (1967). All coefficients in these expansions depend upon the frequency of the applied external electric field. We are interested primarily in dc fields and most of our discussion will be for static polarizabilities. The polarizability is not significantly different from its static value for frequencies of the external electric field much less than the frequency corresponding to the lowest excitation energy of the atom. For resonant frequencies or very large frequencies the atom may absorb energy from the external field via excitation or ionization of the atom, and loss of flux to these channels will affect the polarizability. Calculation of the polarizability for complex frequencies yields the total photoionization cross section, proportional to a(io),where i denotes the imaginary part of the angular frequency o. We have not mentioned dipole and quadrupole shielding factors that

T M . Miller and B. Bederson

6

involve different matrix elements than will be emphasized below, but the calculational methods are the same. The dipole shielding factor is the ratio of the change in the electric field at the nucleus due to the redistribution of the atomic charge, to the strength of the external electric field at the nucleus. The dipole shielding factor is determined by the operator P,(cos Oi)/$, where the sum is over the N electrons of the atom, ri and Oi are radial and angular coordinates for the ith electron, and P,(cos Oi) is a Legendre polynomial of order K.The quadrupole shielding factor depends on the operator P,(cos Oi)/r;.Dipole and quadrupole shielding factors are required for the determination of nuclear moments from measurements of atomic hyperfine structure. They correct for the distortion by core electrons of the effectof nuclear moments on the energy of the valence electrons. According to the Hellpann-Feynman theorem (Chang et al., 1968), the dipole shielding factor is unity for a neutral atom and N / Z in general. Application of various theoretical techniques to its calculation provides grounds for comparisons between these techniques.

B. MATRIXELEMENTS It is customary to treat the external electric field as a perturbation. The perturbation hamiltonian is - p E, where p is the dipole operator -e ri . The summation is of the position coordinates ri of the N electrons in the atom. The electronic charge is e. The first-order energy correction (t,b0 I p * E I$o) is zero if the unperturbed state I$o) has definite parity. This is to say that there is no permanent dipole moment. The second-order energy correction W,is quadratic in E,

xi"'

where ri is a unit vector defined by E = En, k denotes a state of the atom of energy & k , and the sum is over all states of the atom except for k = 0. We can compare this energy correction to the negative integral of Eq. (l),

W = -4a:E' (31 in which we have taken po = fl= 0 and have ignored higher-order dipole polarizabilities. We see that the second-order energy correction in perturbation theory may be identified with the polarizability a. The fourth-order energy correction gives 7. If we consider a Cartesian coordinate system (x, y, z), the components of the polarizability tensor are

ATOMIC A N D MOLECULAR POLARIZABILITIES

7

The components of the hyperpolarizability tensor are (Buckingham, 1967) Yijmn

= S(i, j , m, n)

C

kto

where i, j , m,n = x, y, z, and S(i,j, m,n) is a symmetrizing operator implying a summation over all the 24 permutations of i, j, m, n in the expression following S(i,j , m, n). We have already indicated that odd-order energy corrections for spherically symmetric atoms are zero. There are also simplificationsin the a and y tensors for isotropic systems. The a and y tensors themselves must be isotropic; the only second-order isotropic tensors are scalar multiples of the unit tensor 6,; therefore,

aij = aaij (6) where a is a scalar and fully represents the polarizability for S-state atoms. Fourth-order isotropic tensors have nonzero components only when i, j , m, and n occur in pairs. The most general isotropic fourth-order tensor can be written Yijmn

=Bijdmn

+ 4 d i m d j n + d i n a j m ) + v ( a i m 6 j n - dindjm)

(7)

The fourth-order energy correction is w 4=

-2%y34

(8)

which is

where i,j, m,n = x, y, z. Substituting for y i j m the general isotropic formgiven in Eq. (7), we find

W, = -&(A where E Z = Ef

+ 2p)~4

+ E,' + EI. In Eq. (9) we note that

(10) )'xxxx

= yyyy,, = yzzzz =

7: M . Miller urirl B. Bederson

8

A + 2p. so that if we define a scalar 7 = yxxxx = yyyyy = -yzzZz we can write a purely scalar equation,

w4 -- -241'E4 1

(11) For spherically symmetric atoms then, we have the energy given to fourth order by W = Wo - i a E 2 - 1 , E4 241 (12)

The geometric arguments above were given by Grasso et al. (1968). In general, the calculation of atomic polarizabilities will involve nonspherical wavefunctions and the polarizability will depend on the orientations of the atom in the electric field. However, the polarizability tensor components are not necessarily independent. If the electric field direction is chosen to be along the axis ofsymmetry of the atom, the off-diagonal components vanish, and a,, = a,. The dipole operator p is a so-called T operator (Condon and Shortley, 1951) whose components satisfy the same commutation rules with the components J,, J,, and J , of the total angular momentum J as do J,, J,, and J , themselves. The tensor components a,,(m and a,,(rn are then (Bederson and Robinson, 1966) J )

a,,(m,) = A ( J ~+ mf - J ) + B ( J + ~J a&,)

J)

- mf)

+ C(J2 + 35 + 2 + mf) = 2A(J2 - m f ) + 2Bm: + 2C(J2 + 25 + 1 - mf)

where

(13a) (13b)

and p = )PI, and J , J' are the total angular momenta for the t,h0 and j k states, respectively. It can be shown that of the 2(2J + 1) equations represented by Eq. (13), only two are independent. The difference between any two tensor components is a multiple of A - B + C :

+C) - aXx(mJ,) = (m: - m;,)(A - B + C ) - axx(mJ,) = ( J 2 + J - 2mf - m f . ) ( A- B + C )

azz(mJ)- aZZ(mJ,) = 2(m:,

a,,(m,) a,,(m,)

-

mf)(A - B

(14a) (14b) (14c)

ATOMIC A N D MOLECULAR POLARIZARILITIES

9

Therefore, once two independent components are determined, the full polarizability tensor for any mJ can be obtained from Eq. (14). We note from Eq. (13) that the tensor components are independent of the sign of m ,, and that if J = f the polarizability is a scalar. The two independent polarizability quantities are frequently expressed in terms of an isotropic part (the average polarizability) and a nonisotropic part (the anisotropy or the “tensor” polarizability). The average polarizability cl is defined as the average of either a,, or azzover m, . Equivalently, Cr is the average of a,, , a,,,,, and a,, for a given m ,:

or = 4(2a,,

+ a::)

(15)

and B is the same for all m , . The polarizability anisotropy [generally denoted by y(m,), not to be confused with the hyperpolarizability 71 is defined as a,Z(iiiJ) - axx(mJ),a quantity that depends on m,. The “tensor” polarizability a, is similar to the anisotropy except that the m, dependence is removed:

aZ,(m,) = B + c1,(3m$ - J~

- J)/J(2J -

I)

(16)

From Eqs. (13) and (15) it follows that ~ 1 ,=

-+J(2J - l ) ( A - B

+C)

(17)

When the symmetry axis of the atom is exactly parallel to the external electric field, the polarizability is denoted by a ,I ; in the perpendicular case it is aL.In the laboratory the symmetry axis of the atom cannot be oriented exactly along the field direction and instead of a ,I the projection of c1 lI along the field direction is measured, namely, azz (mJ = J). The matrix elements for the polarizability are similar to those for the dimensionless oscillator strengths fok except for the energy factor, where

and

+ 1) = ( J + 1)(2J + 3), q ( J , J - 1) = 4 2 5 - 1) q ( J , J ) = J ( J + I), q(J,J

and me and e are the electronic mass and charge. Comparing Eqs. (13) and (18), the average polarizability may be written

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7: M . Miller and B. Bederson

This equation is useful because frequently values of fOk are available from experimental or theoretical work on atomic transitions, and an estimate of the average polarizability may be calculated from knowledge of these.

11. The Calculation of Polarizabilities Theoretical calculations of polarizabilities have been made by a variety of techniques-semiempirical, perturbation, variational, and statistical-with some intermixing of these concepts. Some calculations are attempts to obtain “exact” results (or rigorous bounds) to an arbitrary degree of accuracy for specific atoms, while others are aimed at finding a rapid technique that can be more easily applied to any atom, sacrificing accuracy. Fortunately, a number of theorists have investigated the relationship between different methods. A. SUMMATION OF OSCILLATOR STRENGTHS

The simplest theoretical technique for the calculation of dipole polarizabilities involves the summation of oscillator strengths, using Eq. (19). The oscillator strengths satisfy a sum rule 1.fOk

=N

where the summation includes oscillator strengths for all possible electronic transitions including ionization. It would not be feasible actually to obtain a l l f o p ,but Eqs. (19) and (20) allow one to place bounds on the polarizability. If& is taken to represent known oscillator strengths (whether experimental or theoretical), and Fk are the remaining unknown oscillator strengths, then Eq. (20) becomes and the contribution of the unknown oscillator strengths Fk to the polarizability is

An upper bound to the unknown contribution Aa is given by taking the smallest possible energy difference &’:in :

ATOMIC A N D MOLECULAR POLARIZABILITIES

11

Then, we can place bounds on CC:

Dalgarno and Kingston (1959) used the oscillator strength sum rule for the alkali metal atoms. Except for lithium, almost all of the contribution to the sum comes from the resonance transition no S-no P, where tio is the principal quantum number of the ground state. Since only the valence electron is important, N z 1. Dalgarno and Kingston used experimental and theoretical estimates of oscillator strengths and estimated the accuracy of the results to be within about 5 % except for cesium (10%). Their results are in accord with the latest experimental measurements of the alkali polarizabilities. Altick (1964) used experimental oscillator strengths to place bounds on the polarizabilities of the alkaline earth atoms, for which there were no other data at the time. Cohen (1967) determined bounds on the polarizabilities of a number of neutral atoms and ions using experimental and theoretical oscillator strengths. Included were the alkali metal atoms, the alkaline earth atoms, and metastable helium states. [Unfortunately, the experimental resonance oscillator strengths used by Altick and Cohen for the alkaline earth atoms were low (Stwalley, 1971).] The alkali atom calculations tend to compare well to experiment, while the alkaline earth results are somewhat low. Hyman (1974) has calculated the polarizabilities of the alkaline earth atoms from experimental oscillator strengths. B. EFFECTIVE QUANTUM NUMBERS

A semiempirical perturbation method has been developed by Adelman and Szabo (1973), which is an improvement over the coulomb approximation of Dalgarno and Pengelly (1966). They utilize an effective (nonintegral) principal quantum number obtained from the ionization potential, and an effective angular momentum quantum number related to the ionization potential of the closest electric-dipole-allowed state to the state under consideration. Explicit core effects are neglected by Adelman and Szabo, but the use of effective quantum numbers compensates for this neglect to some degree. An analytic expression is given for the 2I-pole static electric polarizability for atoms. The expression is evaluated for the dipole and quadrupole polarizabilities of many neutral atoms and ions. The results for S-state monovalent atoms seem remarkably good. The results for divalent atoms and P-state atoms are somewhat less reliable.

12

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C. PERTURBATION THEORY In relation to the early work on the summation of oscillator strengths, it may be said that the goal of perturbation theory is to avoid a summation over states. The average polarizability can be written

where $o is the unperturbed wavefunction and $ the first-order correction. The electronic coordinates are (xi, y i , zi). The first-order correction to the wavefunction satisfies

(Ho -

~ ) l $ l => ei=x1(xi + Yi + zi)1@0)

(251

where H o is the unperturbed hamiltonian and Wothe zeroth-order energy. In the case of spherically symmetric atoms, the average polarizability is simply ~ = 2 e < $ o ICzi1$1> i= 1

(26)

Various approaches have been used to determine Go and $ There is the intuitive idea that a Hartree-Fock procedure should give accurate polarizabilities (Musher, 1967 ; Kreiger and Kaufman, 1976).The most straightforward approach is the so-called coupled Hartree-Fock method of Dalgarno (1959), Allen (1960), and Kaneko (1959). This is a fully self-consistent procedure, which leads to perturbed orbitals that are coupled both by direct and exchange interactions. The solution involves laborious computation (Dalgarno, 1962). However, the results are the best obtainable within the Hartree-Fock approximation, in which the wavefunctions are written as an antisymmetrized product of one-electron spin-orbitals. The perturbed Hartree-Fock wavefunction is variationally determined. The coupled Hartree-Fock method has only been applied to six elements, in the first two rows of the periodic table (also to some ions). There have been efforts to find approximations that uncouple the firstorder Hartree-Fock equations. Comparisons of the various methods have been given by Langhoff et al. (1966), Chang et al. (1968), and Tuan and Davidz (1971).Langhoff and Hurst (1965) have shown that Dalgarno’s original uncoupled Hartree-Fock theory (1962) is equivalent to the application of first-order perturbation theory to the unperturbed Hartree-Fock problem. Other uncoupled approximations retain some additional terms present in the coupled equations but prove easier computationally (Langhoff et al., 1966; Yoshimine and Hurst, 1964; Langhoff and Hurst, 1965). Dalgarno (1962) has shown that his coupled Hartree-Fock method yields polarizabilities correct through first order. The uncoupled schemes all contain error terms of first order. Langhoff et al. (1966)have estimated that the uncoupled

ATOMIC A N D MOLECULAR POLARIZABILITIES

13

Hartree-Fock approximations represent a computational savings between a factor of 4 to 300 over the fully coupled approximation, depending on which uncoupled scheme is used. Tuan et al. (1966), Musher (1967), and Tuan and Davidz (1971) have investigated the possibility of applying double perturbation theory to obtain corrections to the uncoupled Hartree-Fock results, hopefully without a severe increase in computational effort. Musher (1967) has shown that the uncoupled Hartree-Fock approximation corresponds to the leading term in a perturbation expansion, while the coupled Hartree-Fock is correct to one higher order. He shows that an attempt to improve upon the coupled theory with a perturbation correction is fruitless since the self-consistency terms in the coupled theory are caused to vanish, and one ends up with the uncoupled Hartree-Fock approximation. Musher (1967) also shows that the different uncoupled schemes are related as leading terms in a Taylor series expansion of the uncoupled results. He argues that the uncoupling process does not eliminate unwanted ” nonlocal potentials from the problem, but simply creates a series expansion in which the leading term does not contain nonlocal potentials and in which it is hoped that the remaining terms are negligible. Schulman and Musher (1968) performed a perturbation calculation of higher-order terms in the polarizability of the hydrogen molecule. Their results indicated that the corrections followed a geometric progression, and they suggested a “geometric approximation for the polarizability, “

”

(271 where a, and a, are the polarizability and first-order correction to it, obtained with an uncoupled Hartree-Fock approximation with first-order perturbation correction. Amos (1970) showed that the geometric approximation was applicable to all of the uncoupled Hartree-Fock schemes. Tuan (1970) has shown that the geometric approximation is an approximation to the coupled Hartree-Fock result. Tuan and Davidz (1971) have evaluated a number of the polarizability approximations described here, for the helium and beryllium isoelectronic sequences. They find that ageom, evaluated using one particular uncoupled scheme, leads to the closest agreement with the fully coupled result. Tuan and Davidz (1971) allowed for different amounts of self-consistency in their potential, via a parameter 7 that could be chosen between 0 and 1. They used T = 0, J, and 1 in their computations for the helium and beryllium isoelectronic sequences, and found that z = gave the best agreement with the fully coupled results, when the perturbation correction was applied to the uncoupled approximation. The implication is that there is an optimum 7 near f for a given isoelectronic sequence. Stevens and Billingsley (1973) have reported highly refined calculations of ageom = aO(1 - a1 /ao)-

4

14

T M . Miller

and B. Bederson

the polarizabilities of the first-row atoms, including some excited-state polarizabilities. They used an extension of the coupled Hartree-Fock method due to Cohen and Roothaan (1965), which they apply to multiconfigurational self-consistent-fieldwavefunctions. By including higher configurations in their calculations, Stevens and Billingsley were able to investigate the importance of electron correlation on the polarizability. Beryllium showed the greatest lowering of the polarizability below that of the base configuration, as higher configurations were considered. The base configuration yielded a polarizability of 6.75 A3, in agreement with the coupled Hartree-Fack result of Cohen (1965). Mixing of nearby configurations resulted in a polarizability of 5.41 A’. A further interesting result of Stevens and Billingsley is the large anisotropy in the polarizability of boron, which has a 2Pground state. For mL = 0 (external electric field parallel to the atomic symmetry axis), the correlated polarizability is 3.44 A3.For rnL = f 1 (perpendicular case), the correlated polarizability is 2.56 A’, 26% lower. The work of Werner and Meyer (1976), to be discussed below, shows an even larger anisotropy for boron. The work of Stevens and Billingsley is an important contribution to polarizability theory because their method has a wider applicability than do the coupled and uncoupled Hartree-Fock methods, and because an estimate of the accuracy of the calculations can be obtained from the magnitude of the electron correlation effects. Their method can be applied to degenerate as easily as to nondegenerate atoms; excited states may be considered without difficulty, and electron correlation effects may be included. In the work referred to above (Stevens and Billingsley, 1973) the accuracy was estimated to be within 5% for the first-row atoms. However, their result for neon is 10% below the most accurate experimental result. Werner and Meyer (1976) and Reinsch and Meyer (1976) have also applied the method of finite perturbation of Cohen and Roothaan (1965) to determine atomic and molecular polarizabilities. They used two types of wavefunctions, obtained from the pseudo-natural-orbital configuration interaction (PNO-CI) and the coupled-electron-pairapproximation (PNOCEPA) of Meyer (1973). Although the PNO-CEPA wavefunctions are not variationally obtained, they had been found to yield better results in molecular structure calculations.Werner, Reinsch, and Meyer noted two competing electron correlation effects in their calculations: (a) Unoccupied low-lying p orbitals make possible a very effective angular correlation. If one wants to describe a state where two s electrons tend to keep apart from each other by preferring a geometrical configuration on differentsides of the nucleus, one has to admix configurationsto the ground state with the two s electrons excited into two p orbitals. In case there exist

ATOMIC A N D MOLECULAR POLARIZABILITIFS

15

low-lying unoccupied p orbitals in the atom, this means that this type of correlation is favored, in contrast to the case where such orbitals are energetically more expensive. This preference of opposite configurationswill lead to a reduction of the shielding of the nuclear potential and thus a contraction of the valence shell and a reduction of the polarizability. This effect dominates in the left-hand side of rows of the periodic table because there are empty p orbitals available. (b) Electrons tend to avoid each other by one keeping nearer to the nucleus and the other farther away @/out correlation).This is achieved by configurations using orbitals with additional radial nodes and is more effective for a more diffuse distribution since in this case the excitation energies are lower. This effect dominates to the right-hand side of rows of the periodic table. Intershell correlation is mostly of type (a) and is called core polarization. Core polarization is larger in the second row of the periodic table than in the first because the cores are increasing in size. However, core polarization decreases across a row as the valence p orbitals fill up and tend to neutralize each other. Figure 2 shows the relative contributions of electron correlation to the polarizabilities of the first- and second-row atoms, as calculated by Werner and Meyer (1976) and Reinsch and Meyer (1976). Also shown are the corresponding relative changes in (r2)112due to electron correlation, where (r2)l/’ is the mean value of the radial distribution of charge in the valence shell. The calculations ol’Werner and Meyer (1976) cover the atoms lithium through neon, and their results were displayed in Fig. 1. Reinsch and Meyer (1976) extended the calculations through calcium. These polarizabilities compare very well with experiment, where available, and the accuracy is estimated at +2% by the authors. Reinsch and Meyer are extending their calculations to the third row of the periodic table, which will include atoms through krypton. These results will be important since there are presently no reliable theoretical, and no experimental values, across a long row of the periodic table. Some of their preliminary results have been included in a tabulation of average polarizabilities in the Appendix. An interesting result of Reinsch and Meyer’s latest calculations is the finding for gallium that the anisotropy is 70% of the average polarizability. This is the largest atomic polarizability anisotropy yet reported, in the ground state. Werner and Meyer (1976) also point out that ordinary perturbative treatments would have difficulty in including coupling elements between substituted configurations of different pairs, because these effects correspond to three- and four-body interactions, which appear only beyond second-order

16

17: M . Miller and B. Bedrrson

FIG.2. Relative electron correlation contribution to the polarizabilities (upper part) and to the size of the valence shell ? = ( r 2 ) t $ (lower part) for atoms in the first two rows of the periodic table. Dashed lines, valence shell correlation only; solid lines, total including intershell correlation. (From Reinsch and Meyer, 1976.)

perturbation theory. An interesting example is the oxygen atom polarizability where there is a disagreement even in the sign of the polarizability anisotropy, between the results of Werner and Meyer and those of Kelly (1964), who used many-body perturbation theory. Dehmer et al. (1975) and Cummings (1975) have calculated dipole oscillator strength distributions for the first- and second-row atoms, using Hartree-Slater wavefunctions. Their moment S( - 2) is related to the polarizability by u = 4S( - 2)ai and comparisons with more exact treatments of the polarizability were given by these authors. Sternheimer (1962, 1969) has performed extensive atomic structure calculations, numerically computing accurate solutions to the simpler Hartree perturbation equations. In his latest work on the alkali metal atoms, Sternheimer (1969) gave an approximate correction to his earlier polarizabilities, due to shielding of the valence electron by the core from the external electric field. The corrected values are in good agreement with the best experimental results. Sternheimer (1969) remarks that his method is not as accurate, in

ATOMIC A N D MOLECULAR POLARIZABILITIES

17

principle, as the coupled Hartree-Fock method or the BruecknerGoldstone many-body theory. However, the later two methods are considerably more complicated. Kelley (1963, 1964, 1966), Caves and Karplus (1969), Chang et al. (1968), Matsubara et al. (1970), Miller and Kelly (1972), and Doran (1974), have applied the many-body perturbation theory of Brueckner and Goldstone (Goldstone, 1957) to the calculation of atomic polarizabilities. Caves and Karplus (1969) and Chang et al. (1968) have used the diagrammatic method to shed light on the various Hartree-Fock methods of calculating polarizabilities, i.e., which diagrams are excluded from the various approximations. Chang et al. have categorized the contributions of various diagrams in the following way. The intrinsic polarizability is due to the direct perturbation of the individual orbitals by the external electric field. Self-consistency accounts for the further distortion of each electronic orbital through the averaged coulomb and exchange potentials produced by the orbitals. Self-consistency is divided into intra- and intershell consistency, depending on whether the interacting orbitals are in the same electronic shell. Correlation is interpreted as the dynamic dependence of one electron on another. In many-body theory, self-consistency is represented by single-particle excitations,and correlation by two- or more-particle excitations. Conventional Hartree-Fock theory (the coupled theory) includes self-consistency as well as some twobody excitations. The Brueckner-Goldstone many-body calculation is, in principle, exact. If the interelectron potential is chosen to be the Hartree-Fock potential, a number of diagrams vanish. Then, the remaining diagrams may be regarded as improvements over the coupled Hartree-Fock theory. In the case of lithium, treated by Chang et al. (1968), electron correlation proved to be negligible so that their result for the polarizability was close to that of the corresponding coupled Hartree-Fock value. It is useful to display the contributions of the intrinsic, intershell, and intrashell consistency, and correlation diagrams, as Chang et al. have done for lithium. However, in this particular case, the nonintrinsic contributions are on the order of 0.1 %. A similar analysis for atoms such as oxygen or boron would be interesting. We conclude our discussion of perturbation techniques with a remark on the importance of the wavefunctions used in various calculations. Most of the calculations we have referenced utilized analytic functions to represent the single-electron unperturbed Hartree-Fock wavefunctions and the firstorder corrections. Chang et al. (1968) noted that different authors frequently obtain different results using the same method depending primarily on whether their wavefunctions were determined variationally or numerically. If variationally obtained, the properties of the trial functions must be scrutinized. The numerical procedure is more accurate but tedious to set up.

18

T M . Miller and B. Bederson

Because the polarizability is mostly dependent on the valence electrons, the outer part of the radial wavefunctions is most crucial in determining the polarizability. Wavefunctions that differ only slightly can yield greatly different polarizability values. Yoshimine and Hurst (1964) found that careful attention must be paid not only to the selection of the basis functions, but also to the precise determination of the variational parameters. They gave an example in which the solutions to the Hartree-Fock equations were determined precisely enough to give a wavefunction that gave the correct free-atom energy to within 0.0005 hartrees, but the polarizability obtained with this wavefunction was incorrect by a factor of two! Further, it should be noted that the Hartree-Fock wavefunction is not necessarily the best choice for the unperturbed wavefunction, particularly when using an approximate theoretical method. An interesting example is that of Norcross’ (1973) calculation of the polarizability of cesium. He used a semiempirical model potential involving two adjustable parameters, based on the lowest few spectroscopic term values. Norcross’ calculation included the core dipole and quadrupole contributions to the valence-electron hamiltonian (even dynamical effects),a spin-orbit potential with an additional relativistic correction. Norcross’ result is close to that found by Sternheimer, but he shows that individual contributions to the total polarizability are not the same in the two calculations. In Table I1 are given the direct valence-electron polarizability (neglecting the effect of the external electric field on the core) a,; the effect of the dipole moment induced in the core by the valence electron avc; the polarizability of the ionic core a,; and the total polarizability a,, for both the Norcross (1973) and Sternheimer (1969) calculations on cesium. Particularly striking is the difference in the two values for the valence-electron polarizability av. Both values for a, are close to the latest experimental results, (59.6 -t 1.2) x cm3 (Molof et al., 1974a) and (63.3 k 4.6) x cm3 (Hall el al., 1974).

D. R-MATRIX CALCULATIONS Allison et al. (1972a,b) have described the use of the R-matrix (reactionmatrix) theory of Burke et al. (1971) in the calculation of atomic polarizabilities. The method is especially suitable for the calculation of dynamic polarizabilities since the logarithmic R-matrix boundary conditions are well specified, and loss of flux through the boundary surface is easily calculated (photoionization). The perturbed atomic wavefunction is expanded in terms of R-matrix states, which form a complete and rapidly converging basis set in the region of space where the perturbed atomic wavefunction has nonvanishing probability amplitude. Computer programs to generate the R-matrix

ATOMIC A N D MOLECULAR POLARIZABILITIES

19

TABLE I1 CALCULATED

VALUES

POLARIZABILITY a,,

OF THE VALENCE ELECTRON THE CORE POLARIZABILITY

ELECTRONa,,, THE CORE POLARIZABILITY INDUCED BY THE EXTERNAL ELECTRIC FIELDa,, AND THE TOTALATOMIC POLARIZABILITY U , OBTAINEDBY NORCROSS (1973) AND STERNHEIMER (1969) FOR, cESIU@ INDUCED BY THE VALENCE

Norcross Sternheimer

65.72 71.31

9.24 13.26

2.82 3.14

59.30 61.19

The total polarizability is a , = a, + a, - a,,. The value of the ionic core polarizability a, used in obtaining these results in both cases was taken from another source. The units are cm3 for the polarizabilities.

states for any atom had been written earlier (Burke et al., 1971) in connection with low-energy electron scattering calculations. The R-matrix method has been used by Robb (1973,1974,1975)to calculate dynamic atomic polarizabilities of a number of the first-row atoms and to obtain van der Waals coefficients (as did Allison et al., 1972b). Robb and Reinhardt (1974) have calculated photoionization cross sections.

E. VARIATIONAL CALCULATIONS In a uniform external electric field, the total energy of an atomic system will be decreased. A variational procedure can be used to determine a wavefunction Y that minimizes the total energy in the electric field E. The polarizability can be obtained from the calculated energy change Ad, namely, a = lim(-2 A S / E z ) (28) E-0

Chung and Hurst (1966)have applied the variational procedure to ground and excited states of the helium isoelectronic system using a 72-term wavefunction. In the case of He(2'So) 96 terms were included in the variational wavefunction to obtain convergence. [As an aside, we remark that the accurate calculation of the polarizability of He(2%,) is important as it is used as a standard in many of the experimental determinations of atomic polarizabilities.] Chung (1977) has recently calculated dynamic polarizabilities for He(2lS0) and He(2%,) using a variation-perturbation scheme due to Glover and Weinhold (1976).

7: M. Miller and B. Bederson

20

Schwartz (1961) has pioneered the variational approach in calculating atomic polarizabilities, applying it to H - and He. Sims and Rumble (1973)have used the variational procedure described by Weinhold (1972) to determine rigorous lower bounds to the polarizabilities of the beryllium isoelectronic series. The variational formulation is such that systematic improvement of the bound is possible, to any desired extent. Sims and Rumble (1973) presented both the lower bounds and “recommended” values for the beryllium isoelectronic series, obtained with a 105-term unperturbed wavefunction and a 53-term perturbed wavefunction for beryllium. For C2+ and 04+ an 89-term unperturbed wavefunction was used. They estimate the accuracy of their “recommended” results to be better than 2?6. Sims et al. (1976) have also made calculations for atomic lithium and their result is in close accord with experiment and other accurate calculations. Because lithium has been used as a standard in some recent polarizability measurements (Molof er al., 1974b; Schwartz et al., 1974; Miller and Bederson, 1976) we present in Table I11 a comparison of these results. TABLE 111 A

COMPARISON

OF

RECENT

POLARIZABILITY VALUES FOR

ATOMIC LITHIUM

Source Sims rr a/. (1976) (rigorous lower bound) Sims ct a/. (1976) (variation-perturbation) Molof ef al. (l974a) ( E - H gradient balance experiment) Werner and Meyer (1976) (multiconfigurational perturbation) Adelman and Szabo (1973) (effective quantum numbers)

a

(A3)

23.47 24.27 24.3 0.5 24.38 24.3

F. STATISTICAL CALCULATIONS Dalgarno reviewed various theoretical approaches to polarizabilities in 1962 and noted that calculations based on the Thomas-Fermi (TF) model considerably overestimated the polarizability for neutral atoms. Improvement was seen for increasing atomic mass and increasing excess charge. Bruch and Lehnen (1976) have recently calculated the polarizabilities of neutral inert gas atoms using the T F and Thomas-Fermi-Dirac (TFD) statistical models. They found, as did Roberts and DelleDonne (1975), that the T F atom has injinite polarizability on the basis of a variational calculation. An interpretation of this result is that the TF atom completely screens a

ATOMIC AND MOLECULAR POLARlZABlLlTIES

21

static electric field from its interior. Bruch and Lehnen found the TFD polarizabilities for the inert gas atoms to be many times larger than experimental values although there was considerable improvement for larger atomic number. On the other hand, TFD polarizabilities for the alkaline earth atoms beryllium and calcium turn out to be smaller than accepted values.

111. Experimental Measurements of Polarizabilities In 1971, Teachout and Pack published an extensive compilation of experimental and theoretical values of atomic polarizabilities. It is immediately clear from their tabulation that most of the periodic table has not been touched upon by direct experimental work, although there are several theoretical values available for most elements. Very accurate polarizability values for the noble gas atoms are known from experiments in which the dielectric constant is measured with low-frequency fields. The other elements are more difficult to handle experimentally and beam techniques have been applied in some cases, notably with the alkali metal atoms. Optical techniques in vapor cells, or shock tubes, have been applied in a few other cases. For a number of atoms these methods have been used to determine excited-state polarizabilities, even for Rydberg states. A. BULK EXPERIMENTS

The polarizability of atoms or molecules that are in the gaseous state at normal laboratory temperatures, say 77-400°K, can be measured accurately by determining the dielectric constant K of the gas. The polarizability is then (K - 1)/4m", where N is the number density of the gas. The dielectric constant is best measured with an ac capacitance bridge in which the change in the capacitance of a parallel-plate condenser is determined, with and without the gas present. With this method one obtains the average polarizability. For molecules, the polarizability is further an average over the thermal distribution of rotational and vibrational states. Experiments performed with hydrogen molecules at low temperatures, however, have yielded very accurate results for the polarizability for the ground rotational and vibrational states of H2 (see Kolos and Wolniewicz, 1967). Dielectric-constant experiments with atoms are possible only with the noble gases, and the accuracy is superb, -0.1 2)(Orcutt and Cole, 1967).The other elements are atomic gases only at high temperatures where one runs into a number of experimental difficulties, among them pressure gradients, pressure measurements, excitation, and ionization.

22

T. M . Miller and B. Bederson

Microwave interferometry has been used to determine the index of refraction of molecular gases and noble gases, and hence polarizabilities at low frequencies (Newell and Baird, 1965). Optical measurements of the index of refraction q(o)of gases are useful in determining the static polarizability in cases where the photon energy of the probe beam is small compared to the lowest excitation energy of the gas under study. The (frequency-dependent) - 1]/2~N,where N is the gas number density. polarizability is given by [q(o) Optical experiments in vapor cells possess the same limitations as do the capacitance measurements, with one important exception: by measuring the depolarization of the probe beam in passing through a cell, one can determine the relative value of the polarizability anisotropy. Such measurements for H2 have been important in comparisons between experiments and theory for this “simple” molecule. For atomic systems, only the noble gases would seem viable as candidates for optical refractivity work, insofar as static polarizabilities are concerned. However, Alpher and White (1959) were able to use a shock tube to dissociate N2 and 0,in order to determine the index of refraction of nitrogen and oxygen gases. They performed the experiments for different Mach numbers to obtain different degrees of dissociation (33-53 % for N, ,36-9 1 % for 0,) and used three different wavelengths of light (412.2, 451.5, and 544.6 nm). cm3 for the average polarizAlpher and White found (1.13 f 0.06) x cm3 for oxygen. It is interability of nitrogen, and (0.77 f 0.06) x esting that the ratio of E(N,)/E(N) is 1.55, while Z ( 0 , ) / ~ ( 0 ) = 2.11. That is, the polarizability of N, is less than that of two separated nitrogen atoms, but the polarizability of 0,is greater than that of two separated oxygen atoms. Alpher and White did not see any variation of the polarizability with photon wavelength, within the scatter in their data; the values they report are probably close to the static polarizabilities. In the shock tube method, the major part of the tube is filled with the gas under study. The tail end of the tube is filled with a driving gas, perhaps helium, behind a metal diaphragm that has been scratched along desired rupture lines. When the pressure of the driving gas reaches a high value that can be fairly accurately estimated, say 50 atm, the diaphragm ruptures and compression of the sample gas begins. The sample gas and the driving gas remain well separated by a shock front. Moving ahead of this shock front is the sample gas in thermodynamic equilibrium at a high temperature (say, 5000°K).The temperature, pressure, and percent dissociation in the gas can be calculated accurately. Downstream in the shock tube the gas passes between optical viewing ports and one has typically 0.2 msec for interferometric observation. Marlow and Bershader (1964) have used this technique to measure the polarizability of atomic hydrogen, at 587 nm. In their case, the H, was

ATOMIC AND MOLECULAR POLARIZABILITIES

23

70-85% dissociated (4000-5000"K).In the interferometer a fringe shift of about one fringe was observed, with a precision of 674, due to the introduction of the atomic hydrogen gas. With 27 measurements at different initial pressures, the precision of the experiment was improved to 1.5%. Marlow cm3 for the polarizability of atomic and Bershader found 0.683 x hydrogen at 587 nm. They compared this to the polarizability calculated in cm3 (Podolsky, 1928; Dalperturbation theory for 587 nm, 0.691 x garno and Kingston, 1960), a value that is 3.6% greater than the static polarizability. Marlow (1965) later investigated the temperature dependence of the polarizability of H2, motivated by the possible need for such a correction to the data of Marlow and Bershader (1964).He found that a correction cm3 should be applied to the atomic hydrogen result. of -0.004 x The correction is very small because the fractional dissociation of H2 in the shock tube was large.

B. INDIRECT METHODS In Table I we listed a number of formulas involving the polarizability, any of which could be turned around and used to determine polarizability values, with varying degrees of reliability. The use of the dielectric constant of a gas to deduce the atomic or molecular polarizability has proven to be accurate, for example, and even higher-order corrections can be made if warranted. Other techniques that carry no guarantee as to accuracy we will class as " indirect methods." Ion mobilities have been used to determine polarizabilities (Hackam, 1966), for example. The problem here is that such a result depends on the induced dipole force between colliding ions and atoms being the only important force (McDaniel, 1964). Sometimes this is true; but sometimes it is far from being true, most notably when resonant charge transfer takes place. Indirect results are generally of value only when no direct measurements have been made. The tabulation of Teachout and Pack (1971) does not show any indirect measurements for elements for which no other measurements exist, except for radon. Here, the adsorption rate of radon from a surface has been used to deduce the polarizability of radon (Tuck, 1960). C. BEAMTECHNIQUES I . General Remarks

An atomic beam technique was used by Scheffers and Stark in 1934 to measure the polarizabilities of the alkali metal atoms, not too long after the introduction of atomic beam magnetic deflection methods by Gerlach and

24

i? M . Millcr and B. Bederson

Stern in 1924. It was not until the late 1950s that further atomic beam work on polarizabilities was carried out, inspired by development of the atomic beam resonance method in the 1940s. The beam experiments can be separated into two groups: (a) those in which a transverse inhomogeneous electrostatic field is used to induce a dipole moment in the beam atoms and to exert a deflecting force on them; and (b) those in which a homogeneous electrostatic field is used, and a radiation field (optical, rf, or microwave) serves to induce a transition between two Stark-shifted levels of the atoms. In the first type of experiment the full polarizability for a given atomic level is obtained, while in the second only the polarizability difference for the two relevant levels is found. In the deflection experiment, the tensor components of the polarizability may be determined if the beam is somehow state-selected, since the selection of different m, states permits measurements to be made with the atom oriented in different directions with respect to the applied electric field. In the nondeflecting experiments, the difference polarizability may be essentially equal to the full polarizability of the higher-lying level in many cases, since polarizability values increase rapidly as the principal quantum number n increases (n’ if there is no core penetration). By exciting to different m , states then, the tensor components of the higher-lying level may be specified. Thus, both types of experiments have considerable potential. In the sparce amount of data available, we find complementary results rather than redundancy. 2. Electric Deflection Experiments

Conceptually, the simplest polarizability measurement is made in the manner of Scheffers and Stark (1934). A collimated atomic beam is passed through a region where a transverse inhomogeneous electric field has been established.The electric field E at the beam position induces a time-averaged dipole moment

-

pz = a E = a,, E

(29)

in the atoms, where the z direction is chosen to be the direction of E. about which the polarizability tensor is assumed to be diagonal. The electric field gradient interacts with the induced moment to exert a force F , = pr dEfdz (30) on the atoms, and the beam undergoes a deflection from its original course. If the beam is well collimated, the deflection can be detected downstream from the electric field region of the apparatus. The deflected beam is dispersed by the velocity distribution. (There is also broadening caused by the variation of the electric field gradient across the finite beam width.) It is thus

ATOMIC A N D MOLECULAR POLARIZABILITIES

25

necessary to know the beam velocity distribution in order to infer the polarizability from the deflected beam profile. Scheffers and Stark (1934) used an effusive beam source and hence assumed a Maxwellian velocity distribution. Scheffers and Stark (1936) also measured the polarizability of atomic hydrogen (one of the very few such measurements in the literature, even to this day). They used a water-cooled discharge tube to dissociate H 2 , and a Moo3-coated surface as a detector. Reduction of the MOO, by hydrogen produced a visible image of the beam profile. Their result was less than half that calculated quantum mechanically. But Scheffers (1940) later altered their value of the polarizability of atomic hydrogen to (0.6 k 0.2) x cm3 after finding that the temperature of the hydrogen discharge was closer to 600°K than room temperature. The calculated static polarizability is 0.67 x cm3. Scheffer and Stark's apparatus was miniscule by modern standards-the field electrodes and flight path totaled 14 cm. Their alkali atom polarizability results are of historical interest only (30-50% low), but their contribution to beam technology was considerable. No other deflection experiments are recorded until 1952, when Drechsler and Muller reported measurements on barium and lithium. Their work was followed by Liepack and Drechsler (1956) on the metals nickel, copper, molybdenum, and tantalum. This group used a field electron microscope apparatus. They found that if an atomic beam was passed close to the field-emitting tip, the beam was deflected. They detected the undeflected and deflected beams by letting the metal beam deposit on a glass plate. The greatest uncertainty in the experiment involved the calculation of the electric field strength and the field gradient at the beam position. Their result for lithium has been shown to be at least 28% too small, while the barium figure is 90% too large. Their other results are yet to be tested. Zorn and co-workers at Yale, and later Michigan, have also used an electric deflection technique over many years to measure the polarizabilities of the alkali metal atoms (Zorn and Fontana, 1960; Chamberlain and Zorn, 1960, 1963; Hall and Zorn, 1974), strontium (Hall et al., 1968), and metastable-state argon (Johnson, 1970) and helium (Crosby and Zorn, 1977). This group has improved the electric deflection experiments by (1) using an electrode configuration in which the electric field strength and field gradient can better be determined, (2) using an improved measurement technique (called the " small-shift " method), and (3) using a velocity selector in their latest work. As presently set up (Fig. 3) the experiment consists of a beam source, a rotating-disk velocity-selector, a beam collimating slit, deflection electrodes, and a beam detector at the end of a flight path of about 93 cm. A typical deflection pattern is shown in Fig. 4. The electrodes are of the " two-wire''

26

'I: M . Miller and B. Bederson

FIG.3. Schematic diagram of the electric-detlection apparatus used at Michigan. The dimensions are in centimeters. (From Hall and Zorn, 1974.)

configuration (Ramsey, 1956). The field and the tield gradient can be calculated analytically and it can be shown that the value of E dE/dz does not vary significantly over the beam height. The Michigan group has made electrolytic-tank tests and found that fringing fields effectively add 3 % to the length of their electrodes; The beam atoms with an average polarizability d are deflected by an amount

-

2 = f.(dE dE/dz)L?/mu2 (31) as they pass through the electric field electrodes a distance L The atom mass is m and the velocity is u. The net deflection s at the detector plane is directly proportional to 2.In the small-shift method, the beam detector is located off-center at a point where the undeflected beam profile has a large slope. near the half-intensity point. As the deflectingfield is turned on,the detected intensity changes rapidly by an amount A(2, E ) = (dl/dZ)s

(32)

ATOMIC AND MOLECULAR POLARIZABILITIES I

- 0 7

1

+

6

1

ib

I

ii

I

I

1)

ii

27

FIG.4. Deflected and undeflected beam profiles for velocity-selected (547 m/sec) metastable argon, obtained with the Michigan electric-deflection apparatus. (From Johnson, 1970.)

where I ( Z ) is the detected intensity, dZ/dZ the slope of the beam profile at the detector position Z, and s the net beam deflection. Thus, the measurement of A and I ( Z ) enables one to obtain s, then rji, provided that the electric field strength and field gradient at the beam position are known. These quantities are calculated from the electrode geometry and the applied electric potential. In the older " large-shift " method, the beam deflection is on the order of the beam width, and a more complicated analysis of the experiment is required (see Zorn et al., 1963; and Bederson and Robinson, 1966),because the beam profile is broadened due to the velocity dispersion. (No large-shift measurements have been made with the velocity-selected beam.) The use of a velocity selector makes the experiment straightforward. In the older work, knowledge of the beam velocity distribution was crucial. Hall and Zorn (1974)performed deflection experiments both with and without the velocity selector to study uncertaintities associated with the beam velocity distribution. They deliberately introduced background gas into their electrode chamber in order to distort the velocity distribution, and

?: M . Miller and B. Bederson

28

~+++-;

__---

I

FIG.5. The apparent polarizability of cesium determined with an electric-deflection apparatus, with and without beam velocity selection, as the background gas pressure was varied. The results show that significant error can occur if the data obtained without velocity selection are analyzed under the usual assumption that the velocity distribution is undistorted Maxwellian. (From Hall and Zorn, 1974.)

found results that were consistent with suspicions that the older measurements were erroneously low due to inadequate vacuum. Their results for cesium are shown in Fig. 5. Note that the velocity-selected beam gave a polarizability value that was independent of the background gas pressure. These data place quite stringent restrictions on deflection experiments performed without velocity selection. Greene and Milne (1968) have reanalyzed the data in Hall's Ph.D. thesis (1968) and find less stringent requirements. In their analysis they include scattering due to the background gas in the entire apparatus-not just the electrode chamber-and they find a curve for the non-velocity-selected data that rises more steeply than shown in Fig. 5. While Hall's analysis leaves the impression that the apparent cesium polarizability would be considerably in error even for apparatus background pressures of torr, Greene and Milne find that lo-* torr is an adequate vacuum, for non-velocity-selected experiments in a typical 2 m long apparatus. Nevertheless, velocity selection removes the uncertainty and permits measurements to be made in cases where the velocity distribution in the beam is known to be non-Maxwellian, such as with metastable-state atomic beams. Zorn's group has studied sodium, potassium, rubidium, and cesium with the velocity-selected beam (Hall and Zorn, 1974). They have done unpublished work with a non-velocity-selected strontium beam (Hall et al., 1968) and have unpublished results for the average polarizability of metastablestate argon, (Johnson, 1970). The metastable beam was a combination of 'Po and 'P2 argon, but the polarizabilities probably differ by less than 1%. The atom beam was excited by electron impact. More recently, they have studied metastable-state helium, and are able to separate 2'S, and 2'S,

ATOMIC A N D MOLECULAR POLARIZABILITIES

29

effects using a helium resonance lamp to quench the singlet state via 21S0-21P1 excitation followed by 2’P1-1 ‘ S o radiative decay (Crosby and Zorn, 1977). The results of the Michigan group are in good agreement with theory and other experiments. There are no other measurements available for 2’S0 and 23S, helium, but the variational calculations (Chung and Hurst, 1966; Chung, 1977) for helium are expected to be more accurate than is possible to achieve with present experiments. The experimental test is important, however, because 23S1helium is used as a “standard in other polarizability experiments. Crosby and Zorn (1977) found (108 k 13) and (44.6 k 3) A3 for the He(2’So) and He(23Sl) polarizabilities, respectively. These can be compared to the theoretical values of 118.711 and 46.772 A3, respectively (Chung, 1977). Greene and Milne (1968) operated an electric deflection apparatus at Midwest Research Institute, Kansas City, but their results are unpublished. Their apparatus was similar in principle to the Michigan apparatus except that it did not contain a velocity selector and they incorporated a mass spectrometer detector. Greene and Milne had a tuning fork beam chopper that was used to modulate the atomic and molecular beams and to pulse them for time-of-flight checks on the velocity distributions. Greene and Milne were primarily interested in molecular polarizabilities and dipole moments, but measured the polarizabilities of sodium (25.7 x cm3) as a check on their apparcm3) and potassium (44.5 x atus. These values are within a few percent of accepted values. We will mention their molecular results in Section II1,E. ”

3. N YU Experiments and the E-H Balance Method

At New York University, Bederson and co-workers have used both an electric deflection method and a different technique, the “ E-H gradient balance ” method, to measure atomic and molecular polarizabilities. In doing electric deflection work they have not used the small-shift procedure described above, but have deflected the beams distances that are several times the half-width at half-maximum of the forward beam. Deflection spectra are obtained with the beam detector fixed a distance S away from the axis of the undeflected beam. The potential across the deflection electrodes is swept; at some potential V a maximum is observed in the beam intensity. An example is given in Fig. 6 for rubidium atoms (Molof et al., 1974b). If many such spectra are obtained, a plot of V 2 vs S may be made, as shown in Fig. 7 for rubidium atoms and molecules. A linear plot is expected since the induced dipole moment is proportional to E2 (or V’). Molof et a!. (1974b) have shown that some deviation from the ideal occurs since the beam is of

7: M . Miller and B. Brderson

30

2.0.15 mm

.'. ... . . *

I

*

.

4

t r

1 i

5 5 0 5 0 POTENTIAL APPLIED ACROSS POLE PIECES (kV)

0

FIG. 6. Electric deflection spectra for rubidium atoms. The detector was positioned off the beam axis by the distance Z and the potential across the deflection electrodes was stepped at 50 V per point. (From Molof et a/., 1974b).

-

0 0

0.2 0.4 0.6 BEAM DEFLECTION (mm)

FIG. 7. Typical results obtained from electric deflection spectra for Rb and Rb, at 534°K. The applied potential was that corresponding to a maximum in the beam intensity for a particular detector position off axis. The slope of the resulting line is inversely proportional to the average polarizability. (From Molof et a/., 1974b.)

31

ATOMIC AND MOLECULAR POLARIZABILITIES

finite width, and the data (Fig. 7, for example) show this for small beam deflections. For beam deflections that are larger than the undeflected-beam width, the V 2 vs S lines are linear. The equation for the line for an infinitesimal beam with a Maxwellian velocity distribution is V2 = ( 4 k T / ~ E ) s

(33)

where k is Boltzmann's constant, T the temperature of the beam source, and K a geometrical constant of the apparatus. The constant K is the product of the electric field to potential ratio and square of the field gradient to field ratio for the deflection electrodes, and LI(L, + 2L2)/2.The electrodes are L, in length and the distance from the end of the electrode set to the detector is L 2 . Because of the difficulty in determining the apparatus constant K absolutely, the NYU group has chosen to normalize their electric deflection results to known polarizabilities of the alkali metal atoms (Molof er al., 1974b). The normalization was accomplished by making measurements on atoms, then molecules, in the same beam from a single oven source. An inhomogeneous' magnetic field was used to remove atoms from the beam, acting on the atomic magnetic moment of one Bohr magneton. The molecules, with magnetic moments on order of nuclear magnetons, comprised only about 1% of the full beam. The ratio of the slopes of the V 2 vs S deflection lines (Fig. 7) for the species under study and for the "standard" is equal to the inverse ratio of the respective polarizabilities. We will discuss the results of experiments on molecular polarizabilities in Section 111,E;suffice it to say here that the alkali dimer polarizabilities tend to be 50% greater than those of the corresponding atoms, at the temperatures studied by Molof et al. (typically 600°K). With the normalization procedure, the only uncertainties are in the relative potential measurements, the statistical uncertainty, and the uncertainty in the polarizability of the standard. Schwartz et al. (1974) and Miller and Bederson (1976) have studied the alkaline earth atoms with the electric deflection method. In these experiments the normalization was against the lithium polarizability. Lithium was loaded into the oven source along with the alkaline earth. A mass spectrometer was used to separate the deflections of the two beams. The polarizabilities of the alkaline earth atoms tend to be -60'7, of those of the adjacent alkali metal atoms in the periodic table, for the atoms which have been studied thus far (barium, strontium, calcium). The lighter atoms, magnesium and beryllium, are predicted to have polarizabilities further lower than the corresponding alkali atoms in the periodic table ( f and respectively). A comparison of various results for the alkaline earth atoms is given in Table IV. Both the NYU and the Michigan groups have noted that the statistical uncertainty in their data is mostly due to the small shifts in the beam position for unpredictable reasons. They are dealing with tightly collimated,

-

-

a,

'I: M . Miller and B. Bederson

32

TABLE IV

A COMPARWNOF RECENTALKALINE EARTHPOLARIZABILITY VALUES Atom Be

(A3)

Source

5.297

Sims and Rumble (1973) (rigorous lower bound) Sims and Rumble (1973) (variation-perturbation) Robb (1973) (R-matrix method) Werner and Meyer (1976) (multiconfigurational perturbation) Robb (1975) (R-matrix method) Stwalley (1971) (experimental oscillator strengths) Stewart (1975) (Hartree-Fock approximation) Reinsch and Meyer (1976) (multiconfigurational perturbation) Miller and Bederson (1976) (electric deflection experiment) Stwalley (1976) (experimental oscillator strengths) Reinsch and Meyer (1976) (multiconfigurational perturbation) Hyman (1974) (experimental oscillator strengths) Adelman and Szabo (1973) (effective quantum numbers) Schwartz et al. (1974) (electric deflection experiment) Hyman (1974) (experimental oscillator strengths) Adelrnan and Szabo (1973) (effective quantum numbers) Schwartz et al. (1974) (electric deflection experiment) Hyman (1974) (experimental oscillator strengths) Adelman and Szabo (1973) (effective quantum numbers)

a

5.42 f 0.12 5.652 5.600 Mg

11.1 f0.75 11.11 f 0.44 11.51 10.57

Ca

25.0 f 2.5 25.0 f 1.0 22.8 22.8 24.4

Sr

27.6 f 2.2 28.4 30.5

Ba

39.7 f 3.2 35.8 43.6

ATOMIC A N D MOLECULAR POLARIZABILITIES

33

narrow beams traveling over long distances. In the NYU experiments, the beam-collimating slit on the entrance to the electrode set is 0.005 cm wide; the beam travels over 2 m altogether. The beam deflections must be measured with a precision of about 0.0002cm, and must remain steady for periods of at least 15 min. In practice, it is found that random shifts of this order can occur, due to temperature changes or vibrations or whatever. Bederson and his co-workers at NYU devised a new method of measuring polarizabilities, which they call the E-H gradient balance technique (Salop ef al., 1961). In this method the electrodes of the deflecting field are also the pole pieces of a magnetic circuit. The result is that both an inhomogeneous electric and an inhomogeneous magnetic field may be applied in the interaction region of the apparatus. The object is to balance the electric deflecting force against a magnetic force acting on the magnetic moment of the atom. Since magnetic moments are well known, the balance condition permits one to determine the polarizability. One principal advantage of this method is that the balance condition is velocity independent. Thus, one does not need to know the velocity distribution in the atomic beam, the balanced beam intensity is much greater than that of a velocity-dispersed beam, and the balanced beam width is approximately as narrow as that of the undisturbed beam-which leads to greater precision in the data acquisition. Furthermore, since the value of the effective magnetic moment depends on the mF sublevel of the atom, each sublevel can be brought into balance separately and the tensor components of the polarizability may be obtained. That is, magnetic field permits state-selection of the beam. These experiments are performed on the same apparatus as are the pure electric-deflection experiments described above. The apparatus dimensions are given in Fig. 8, and a cross section of the most recent pole pieces they have used is shown in Fig. 9. The pole pieces are soft iron and are insulated from the magnet yoke by a glass vacuum envelope and from each other by quartz spacers. The position ~ 7 5 . -63 5 . 6 - 7 - 1 2 3 - 1

OVEN

%S

iNTERACTlON REGION

fr?I

DETECTOR u)

d

d

L

.-ElI-

FIG. 8. A diagram of the NYU E-H gradient balance apparatus with dimensions in centimeters. (From Molof et a/., 1974a.)

34

T M . M i l l u and B. Bederson

'DETAIL

FIG.9. A cross-sectional view of the iron pole pieces used in the NYU E-H gradient balance apparatus. The pole pieces are 35.6 cm long and the design parameters were a = 0.159, h = 0.172, and c = 0.066 cm. The atqmic beam passes between the pole pieces at the position shown (at 1.2~1). Quartz spacers are indicated by Q. (From Molof et al., 1974a.)

of the beam in the pole pieces is indicated in Fig. 9 and is the optimum position for the beam according to calculations of the uniformity of E dE/az over the beam height. In the interaction region of the apparatus an inhomogeneous magnetic field H is applied across the beam, which acts on the effective magnetic moment p(m,) of a beam particle in the magnetic substrate mF. The magnitude of the transverse magnetic force is p(mF)d H / d z , where d H / d z is the transverse component of the gradient of the magnitude of the magnetic field. If one also applies an inhomogeneous electric field E across the beam, there is an electric force that for a scalar polarizability a is simply aE aE/dz. The induced electric force is always directed toward the direction of stronger field, while the direction of the magnetic force depends on the sign of p(mF). If p(m,) is negative it is possible to adjust the field strengths so that the electric and magnetic forces are equal and opposite. When the forces are so balanced, U E a q a z = p ( m F )m / a z (34) and since atomic magnetic moments are known, it is possible to obtain a by determining E and H when the balance condition of Eq. (34) prevails. Salop et al. (1961) first used this technique to measure the polarizabilities of the alkali metal atoms. If the electric and magnetic fields are congruent, then the balance equation may be written aE2 = p(rn,)H (35) and the result is independent of the ratio of the field gradient to field strength, C, where c = E-' a q a z = H-' a ~ / d z (36)

ATOMIC A N D MOLECULAR POLARIZABILITIES

35

which has been cancelled on both sides of Eq. (35). It is still necessary to know the magnetic field strength H,and in practice the ratio of the electric field strength to the applied potential, K = E/V

(37)

Thus, the balance equation is a K 2 V 2 = p(m,)H

Salop et al. calculated the geometrical constant K and determined the magnetic field strength at the beam position from knowledge of field strengths at which the effective magnetic moments for certain mF values become zero. Thirteen years later, Molof et al. (1974a) studied the alkali metal atoms on an improved apparatus with better data-acquisition equipment. They were able to determine C and K experimentally for their apparatus and concluded that it is difficult to calculate accurate values of these constants. They depend on the precision with which the pole pieces are machined, their spacing, the accuracy with which the beam position is known; and the importance of the fringing fields. In conventional electric deflection experiments, the product C K 2 must be accurately known. In the E-H balance experiment of Salop ef al. (1961), K 2 was required. Molof et al. (1974a)chose to normalize their measurements to the polarizability of 2’S, metastable helium, a value that has been accurately calculated (Chung and Hurst, 1966). With the normalization, the balance equations for the atom in question and for helium provide the unknown polarizability: (39) where and pHe(1) refer to metastable helium in the m, = 1 state, and V and VHeare the applied electric potentials used to achieve balanced beams without changing the magnetic field strength. Examples of the E-H balance data of Molof el al. (1974a) are shown in Figs. 10 and 11. Figure 10 is a coarse scan of the magnetic sublevels of cesium at an intermediate field strength of 1010 G. As the potential across the pole pieces is increased, the components of the beam corresponding to different magnetic sublevels come into balance. Those with the largest magnetic moment require the largest applied potential for balance. The ( F , m F )= (4,4) sublevel has a magnetic moment of one Bohr magneton, independent of magnetic field strength. When clearly resolved, Molof et al. used this sublevel to obtain their polarizability data since there is no ambiguity in p(mF).In routine data acquisition, Molof et al. scanned only over a range of potentials necessary to capture the top part of the balance spectrum for the sublevel of interest in order to achieve greater precision. Typical data a = aH~[p(m~)/pH~(1)](~He/v)2

7: M. Miller and B. Bederson

36

POTENTIAL APPLIED ACROSS POLE PIECES (kV)

FIG. 10. A coarse scan of the potential applied across the E-H gradient pole pieces of the NYU apparatus at a magnetic field strength of 1010 G. The peaks result from a balancing of electric and magnetic forces acting on cesium atoms in different (F, m F ) sublevels. (From Molof et a/., 1974a.)

10

.

> a

26

-

c m

-aa


0

.x 0. For Im(p) e 1, Re(p) 5 1, Zi(p)

‘V

nl” ex~I-[Reb)]’)

(3.27)

Since -y o x k’. For arbitrary A, the exact equations (3.21)should be used. For I A 1 on the order of several ku, the line shape takes on a more complicated form, with resonances arising when various combinations of the conditions (3.28)-(3.31) are satisfied (Bjorkholm and Liao, 1976; Salomaa and Stenholm, 1976).

FORMAL SOLUTION C. LINESHAPE-COLLISIONS: It is relatively easy to arrive at a formal expression for the line shape that includes the effects of collisions. In the impact approximation, the contributions to p from collisions [Eq. (2.111is simply added to the rhs of Eqs. (3.13). Steady-state solutions are still possible and each of Eqs. (3.13) takes the form qij(t’:)ijij(v) + rij(v)Pij(v)

- (. dv’Wj(v’+v)fiij(v’) where

q i j ( r z= ) ;‘i h i j + ( 1 kl2 = tk,

k23

= .fij(v)

+ ikijrz) k13 = k‘ + tk

- hij)(ylij

= k‘,

(3.39) (3.40) (3.41)

qij is given by Eqs. (3.15), Tii and W j by Eqs. (2.3) and (2.2), and the .fii correspond to the rhs of Eqs. (3.13) for fiij, e.g., .fIl(V)

= i%[fi21(V) - filZ(V11

”I?.(’)

= i%[@22(v)

- fill(v)]

+ Jl(V)

(3.42a)

- I’%’fil)(v)

(3.42b)

In perturbation theory, Eqs. (3.39) are most easily solved by taking solutions for fiij(v) of the form

1

Pij(v) = dv‘Gij(v’+v).Yij(v’)

(3.43)

where the propagators Gij(v’+ v ) satisfy the equation [tlij(r:) -

+ rij(v)lGij(v’--*v)

1 ~ V ” U $ ~ ( V ” + V ) G ~ ~ ( V ’ - V=’ ’d(v ) - v’)

(3.44)

STUDY OF COLLISIONS

BY LASER SPECTROSCOPY

75

All the effects of collision are included in Eq. (3.44), which in general will be very difficult to solve. However, the formal solution for the line shape in perturbation theory (in the external fields) now follows directly. Using Eqs. (3.39b(3.43) and (3.13) one sees, for example, that the contribution of chain 1 of Table I, given for the no-collision case by Eqs. (3.16), is now replaced by

p\q(v) = ix

I dv’G,,(v’+v)[~~~~(v‘!- $\‘;(v’)]

p$’;(v)

I d~’G~~(v’+v)[p‘j:)(v’) - fi$’;(v’)]

= ix

3i3i(v) = - i f pi4:(v) = ix’

(3.45)

1 d~’G,,(v’+v)p\~;(v’)

1 d~‘G,,(v‘+v)[p~~:(v’)

- $i3J(v’)]

When this contribution is combined with those corresponding to chains 2 and 3 of Table I and Eq. (3.20) is used, one arrives at the formal line shape 3

p33=

CPi33 i=l

where

(3.46a)

P . R . Berman

76 p:3 = - A

(. dv dv4 dv3 dv2 dv1 d v o G 3 3 ( ~ 4 - + ~ )

x

G 2 3 ( ~ +~4)Gi3(V2+V3)G23(Vi 3

x

[G22(Vo+Vi)A2(Vo)

- G33(v0

-+

+V2) vi)b(vo)]

+ conjugate

(3.46d)

Some simplification of these equations is possible regardless of the collision model utilized. First, if the A,(v) represent thermal distributions as has been assumed, collisions can not change these distributions. In other words,

1 dv’G,,(v’+v)L,(v‘)

=A

~ ( ~ ) / ~ ~

(3.47)

Second, the total number of atoms in a given state is unchanged by collisions, i.e., (. dvG,,(v’+v) = l/y, (3.48) a result that is easily obtained by integrating Eq. (3.44) over v for i = j while making use of Eq. (2.4a). Using these results in Eqs. (3.46) and recalling definitions (3.19), one obtains

(. dv d ~ dv, 2 ~v~[G~~(v~+v)

pi3 = -AN(y3)-’

+ G23(V2-*V)*lG2Z(Vl+V2)[G12(VO+Vl) + G 1 2 b o v1 )*I WO(V0) +

(. dv dv, dvl d v o G 2 3 ( ~ 2 + ~ )

p:3 = -AN(y3)-’ 13(v 1

1 wO(vO)

v2)G 1 2(v0

+ conjugate p:3 =

(3.49a)

(3.49b)

1

AN'(^,)-^ dv dv2 dv, dvo G23(v2--rv)G13(v1

-+v2)

+

x G 2 3 ( v o - - r ~ , ) W O ( ~ O conjugate )

(3.49c)

We now turn our attention to the choice of a collision model, which will give rise to specific propagators. D. LINESHAPE-SPECIFIC COLLISION MODEL In choosing a collision model, the major question to ask oneself is whether or not, to first approximation, the collision interaction is state dependent, since the answer to this question will provide the form for the collision

77

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

kernel, as discussed in Section II,C,2. Another important factor is the activeatom to perturber mass ratio since this also affects the nature of the collision terms. The simplest model and the one we choose here is a model in which collisions are phase changing in their effect on off-diagonal density matrix elements, i.e., iij(v)

1

=

-Y$h(v)ijij(V),

(3.50)

i#j

coll

where y5h is given by Eq. (2.7). As discussed in Section II,C,2, this type of equation will arise for either a large active atom to perturber mass ratio or a strongly state-dependent collision interaction. In the latter case, one should approximate yfih by neglecting the collision contribution from the weakly interacting state. The negative of the rhs of Eq. (3.50) replaces the second and third terms of Eq. (3.39) and using Eqs. (3.39), (3.43), and (3.44), one obtains the off-diagonal propagators Gij(v’+v) = [vij(vZ)

+ Y ~ ~ ( v ) ] 6(v - ’ - v’)

(3.51)

The range of validity of this model for the off-diagonal elements as well as some alternative models will be discussed at the end of this section. For the diagonal terms, there is no problem of interpretation, since collisions are solely of a velocity-changing nature. The only place where velocitychanging collisions enter the calculation is in the G22(v1+v2) factor of Eq. (3.49b). Collisions in levels 1 and 3 do not affect the line shape for reasons discussed above Eqs. (3.49). The line shape will not be critically sensitive to the form of kernel Wz2(v’+v) used in Eq. (3.44) for G,,(v‘+v) and we shall use the phenomenological kernel of Keilson and Storer (1952), WZZ(V’+V)= r , , [ n ( A ~ ) ~ ] - ~exp[’ ~ (v

- uv’)’/(Au)’]

(3.52a)

is the (speed-independent) rate of collisions in state 2, u is a where parameter between 0 and 1, and (Au) = (1 - U ~ ) ’ ’ ~ U

(3.52b)

fi

times the rms velocity change per collision. For u z 1 one has weak is collisions; for u z 0 one has “strong” collisions since, on average, each collision thermalizes the sample. Other kernels, based on physical potentials, have been given by Kolchenko er al. (1972), but Eq. (3.52a) provides a good first approximation with simple analytic properties. Equation (3.44) for G22(v’--+v’)with the kernel W22(v’+v) given by (3.52) and ~ ~ ~ ( = v y2 , ) can be solved by iteration (Keilson and Storer, 1952; Borenstein and Lamb, 1972) and one obtains G 2 2 ( ~ ‘ - -V)+ =

“

qv

- v’)

ri

1

r22n

‘9 ;,(E)

1 exp [n’i2(A~n)]3

v - u”v’

[( -

”

~

Au,,

)]

(3.53)

78

P. R. Berman

where

ri = 7 2 + r 2 2

(3.54)

(Au,,) = (1 - a2")li2u

(3.55)

The first term corresponds to atoms that have not undergone collisions, and the nth term in the summation corresponds to the average effect of n collisions. We now have values for all the propagators needed in Eqs. (3.49). Only phase-changing collisions enter in p53 and p i 3 since the chains leading to these terms (Table 1) do not contain the population f~~~as an intermediate term. The net modification in the line shape formulas owing to phasechanging collisions is the replacement of uij by qij

= qij

+ yfih(v)

(3.56)

In the p i 3 term of Eq. (3.49a), the propagator G,,(v'+v) will enter, but the integrals that arise are not difficult (see the Appendix for the evaluation of this term) provided we neglect the speed dependence ofyfih(v),which we now do.' Consequently, when the propagators (3.51) and (3.53) are substituted into Eqs. (3.49), one obtains the line shape P33

=

where Pi3

=

1P i 3

(3.57a)

2AN ri y3 kuk'u

(3.57b)

[

x Im I , ( i423

__ ii112

"13

k'u' ku ' I k ' + t k l u '

c,

V(k' + ck)]

(3.57~)

It is usually a very good first approximation to neglect the u dependence of rch.especially if the ratio of active-atom to perturber mass is 2 1.

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

79

2AN

P:3

=

- y3(k'u)' I k'

+ ck I u

x Im [ I 3 ( -i q Z 3

k'u ' I k'

+ ck I u ,Y ( k ' + t k ) ) ]

"13

(3.57d)

which is the generalization of Eqs. (3.21) to include collision effects. In the limit of large detunings I A I > ku, corresponding to the no-collision Eqs. (3.24) and (3.25), one finds P33

P33

-

2A(N

-

N')

i613

- 73 (k'+rkluAzZi(Ik'+ek[u)

for A' z - A

(3.58)

2A

- ____ y 3 k'uA2

(3.60)

and for future reference

+

A

+ Sfr 1 = 3 A + A' + Sfi

d = Im q l Z= A

(3.63a)

A' = Im

(3.63b)

~

For near-resonant tuning I A 1 < ku in the Doppler limit

fij4 ku, k'u

(3.64)

Eqs. (3.57) reduce to

(3.65)

P . R . Berman

80

where

tk' Aa = A' - -A k

r c =k' k i + 1k'2 + R3 , r r d

= rb@(k

-

(3.66a)

k' Ab = A' + 7 (A+ A ) k-k

(3.66b)

k A, = -A k

(3.66~)

-

k' +7 (A+ A ) k - k

- k')

+ T,@(k' - k),

+ A,O(k' - k )

(3.66d)

and @(.u) is the step function given in Eq. (3.38). Before analyzing the line shapes (3.58),(3.59),and (3.65). it might be useful to indicate when one would expect this collision model to be applicable. As stated previously, the model is valid for large active-atom to perturber mass ratios or for strongly state-dependent collision interactions. If states 1,2, and 3 belong to different electronic configurations, one could expect the collision interaction to be strongly state dependent. Different electronic configurations for the various levels may be easily realized in the upward cascade of Fig. 2a. On the other hand, in the V " schemes of Figs. 2b and 2c, levels 1 and 3 may be part of the same electronic configuration. In that case one might want to take collisions as velocity changing rather than phase changing in their effect on P l 3 , at least to first approximation. Some consequences of such a model will be mentioned below and, in particular a method for checking the effect of collisions on fiI3 will be given. It should be noted that analytic solutions are obtainable only for very simple collision kernels when one considers the effects of velocity-changing collisions on off-diagonal density matrix elements. If levels 1 and 2, 2 and 3, or all three levels are in the same electronic configuration (e.g., vibration-rotation transitions), another collision model may be necessary that will be much more complicated than the one considered above. The collision model we have chosen should be applicable to many physical systems and has the important advantage of being the simplest type of model that one could imagine. "

E. LINESHAPEANALYSIS Even for the simple collision model chosen above, there are eight collision and z) that must be parameters (rrl,rph,, rph,,Spl, Sp!, Sps, determined. These parameters are related to collision cross sections in the various levels through Eq. (2.7) and to the rate and strength of collisions in

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

81

level 2 through the kernek (3.52). Even though the number of parameters is large, systematic line shape studies can lead to a unique determination of each quantity. To simplify the analysis, we shall take

N = O Physically, this limit corresponds to having only level 1 populated in the absence of applied fields, as would be the case if level 1 were the ground state. The situation in which N' # 0 does not lead to any additional problems (provided one can experimentally determine N and N'), but also does not provide any new collisional information. 1. Large Detunings, IA I b ku

Systematic line shape studies should begin in the large detuning limit with the line shape given by Eqs. (3.58) and (3.59). a. A' 2 -A. Equation (3.58) represents a resonance at

-

A

+ A ' = A + A' + Sp!

=0

having a width that corresponds to the convolution of a Gaussian of width ( I k' + t k I u ) with a Lorentzian of width ( 7 1 3 + rph3).The " resonance condition for each velocity subgroup of atoms that leads to this TQ resonance [corresponding to condition (3.29)in the no-collision case] is

-

A' + A = - ( k '

+ (k)r,

f13

(3.67)

Since L',takes on a wide range of values, Eq. (3.67)can be satisfied by a large range of A', and the resonance is broad, except for the "Doppler-free" case I k' + t k I zi 4 F 1 3 . In any event, studies of the line shape (3.58)as a function of pressure can lead to values for and Sp); . Another interesting feature of the line shape for A' z -A, I A I b ku, is its sensitivity to the collision model. As seen from chain 2 in Table I and Eqs. (3.13f) and (3.67),the line shape in this region is a measure of the effect of collisions on jI3 since it is the TQ contribution that is being probed. If collisions are phase shifting in their effect on j , there is a line broadening and shift given in Eq. (3.58). However, if the collision interaction in levels 1 and 3 were the same, collisions would be velocity changing in nature in their effect on 6 1 3 , Eq. (3.58) is no longer valid, and one should observe a ITUYYOWirrg of the line with increasing pressure (Dicke, 1353; Galatry, 1961; Rautian and Sobelman, 1967).Consequently, the line shape in this region can help to indicate whether collisions are '' phase changing" or "velocity changing" in By the"width" of a Lorentzian. we shall mean its half-width at half-maximum (HWHM). and by the width of a Gaussian. we shall mean its HWHM,'(ln 2)' '.

82

P . R . Berman

their effect on p 1 3 , i.e., whether the collision interaction is the same or different for levels 1 and 3. b. A' % 0. Equation (3.59) with N = 0 corresponds to a broad resonance at A' % O with contributions from both the SW and TQ chains. Velocity-changing collisions in level 2 play no role here since the detuning I d I is so large that the population of level 2, which depends on the absorption of field El through the factor F = f1,/[(fld2

+ (A + ckuz)21

-

fl,/(~)z

(3.68)

is approximately independent of u, . On the other hand, one sees from (3.68) that the contribution of the SW chain will increase with increasing pressure owing to the presence of f,, , and this feature is reflected in the line shape (3.59). Consequently, pressure studies of the amplitude of (3.59) can lead to a value for F:!. Moreover, pressure studies of the width (convolution of a Gaussian of width k'u with a Lorentzian of width f 2 3 ) and shift of the resonance can provide values for re! and S $ . It might be noted that linear spectroscopy on the 1-2 and 2-3 transitions can provide independent measures of r:!, Sp!, re!, and Se!.

2. Near-Resonant Tuning, I A I c ku; Copropagating Waves (t = 1) Apart from Sf! all the phase-shifting parameters can be determined from the large detuning case. In order to obtain values for the rate of velocitychanging collisions r2,and the collision strength parameter a, the field El must be tuned inside the Doppler width of the 1-2 transition. As mentioned in Section III,D, only the SW chain serves to probe the velocity-changing collisions. The SW chain can be isolated in the Doppler limit (3.64) if copropagating waves are used, since the velocity integration leads to a cancellation of the TQ contributions. For c = 1, Eq. (3.65) is

(3.69)

This equation for the SW contribution has a very simple physical interpretation. The first term corresponds to those atoms that have not under-

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

83

gone velocity-changing collisions. For these atoms, the field El excites a velocity ensemble

ku,

=

-A f f l z

(3.70)

and the field E2 will be resonant with this ensemble if

A' = - k'v,

(3.71) f [(k'/k)f12 + f23] In other words, the resonance occurs at A, = A' - ( k ' / k ) A = 0 with a halfwidth at half-maximum (HWHM) of (k'/k)f,, + f z 3 = r,,, which is precisely what one finds in the first term of Eq. (3.69). The nth term in the summation in Eq. (3.69) corresponds to an atom having made n velocity-changing collisions in level 2 before being excited to level 3. Equation (3.70) still describes the excitation by the first field, while Eq. (3.71) is replaced by fz3

= (k'/k)A

6' = -k[f,(u,; A/k

& flz/k) &

f23

(3.72)

where fn(u,; up) represents the distribution of velocities u, after n collisions for atoms initially having velocity up . For the Keilson-Storer kernel, f,(v,; u:) is approximately a Gaussian of width (Au,) [Eq. (3.55)] centered at u, = anup, so that (3.72) represents a resonance centered at A' = d'(k'/k)A with a width that is the convolution of a Gaussian of width k(Au,) with a Lorentzian of width a"(k'/k)flz + f z 3 . This contribution to the line shape may be found in the plasma dispersion function in Eq. (3.69) with a weighting factor (r2*/r\)" giving the probability for n collisions during the lifetime of level 2. The actual line shape (3.69) will depend on the ratio of the parameters appearing in that equation. In particular, the quantity

0) = r 2 2 / 7 2

(3.73)

gives the effective number of velocity-changing collisions occurring in the lifetime of level 2 and consequently determines the number of terms entering the summation in Eq. (3.69). Another important ratio is k(Au,,,)/T,; if this ratio is greater than unity, velocity-changing collisions give rise to a distinct contribution to the line shape, owing to the fact that the velocity changes are large enough to significantly alter the velocity distribution of atoms excited by the field El. If this ratio is less than unity, velocity-changing collision effects are hidden in the homogeneous line width. Line shapes resulting from various valves of ( n ) and k(Au(,,))/r, are discussed below. Recall from Eqs. (3.52) and (3.55) that

(Au,) = (1

- aZn)l'Zu,

(Au) = (Aul) = ( 1 - a2)1'2u

and that the Doppler limit (3.64) is assumed.

84

P . R. Berman

a. ( n ) arbitrary, k(Au(,,))6 y12, ~ 2 3 . In this case, velocity-changing collisions are very weak (a z 1) and have a negligible effect on the line shape. The limiting form of Eq. (3.69)is (3.74) with all velocity-changing collision effects absent. b. ( n ) < 1, k’(Au) > r,,. In this case, only one term in the summation in Eq. (3.69) must be considered. The value of a is sufficiently different from unity to clearly distinguish the effects of velocity-changing collisions,and the limiting form of (3.69) is

The line shape consists of a contribution from atoms that have not undergone velocity-changing collisions, plus a “ shoulder ” arising from atoms that have been partially thermalized as a result of one average collision. Since ( n ) < 1 and r,,< k’(Au), the relative amplitude of the shoulder is small, and a careful line shape analysis will be necessary to verify its existence. c. ( n ) T J k u %. 1, a(”) z 0. In this limit, there have been a sufficient number of collisions to thermalize the sample in level 2 before absorption of field E, . In the sum in (3.69),(Au,,) x u for most n, and (3.69) reduces to

x {r,,(n)d/2(kU)-

exp[ - ( & / k ’ ~ ) ~ ] }

(3.76)

The line shape consists of a broad Gaussian background arising from atoms that have become thermalized in level 2 as a result of collisions. To satisfy the condition ( n ) = r22/y2 % 1 and the Doppler limit

r,,= [r23+ (k’/k)f,,]

6 ku,

one must generally find a system where the decay rate y z is considerably less than y1 or y 3 . d. k(Au) 2 y,,, 7 2 3 ; k(Au(,,))< I-,,. As will be discussed below, it may be possible to satisfy these conditions over a fairly wide range of (n). In the limit that k(Au(,,)) < r,,,velocity-changing collisions are not strong enough to remove atoms from the holes burned in the atomic velocity distribution by the pump field. The line shape will be a Lorentzian, slightly modified by the presence of velocity-changing collisions.

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

85

To determine the range of (n) for which the conditions k(Au) > 712

3

723

(3.77a)

(3.77b) k(Au(n>) < ra and the Doppler limit (3.64) can both be maintained, one must first examine the implications of Eqs. (3.77) on the strength parameter a of the KeilsonStorer kernel. It follows directly from Eqs. (3.55) and (3.64) that condition (3.77b) can be met only if (1 - a2(n))1/2 < Ta/ku 4 1

(3.78)

Thus, condition (3.77b) is achieved only for "weak " collisions a

x I,

x 1

(3.79)

Equations (3.55) and (3.79), in turn, can be used to obtain the relationship (Au,,)/(Au) = [(I - a2")/(l - a2)]1/2x n'/'

(3.80)

The allowed range of (n) follows most easily if one uses Eqs. (3.66a), (3.60), (3.34), and (3.73) to write r, in the form

r, = r(i+ ~ ( n ) )

(3.81)

where (3.82) and T are pressure independent. A lower limit on (n) is determined from Eqs. (3.77b), (3.81), and (3.80) to be

(n) > {[k(A~)Tl(n>int- 1)W-l

(3.83)

+

is the integer part of (1 (n)). An upper limit on (n) is where obtained from the Doppler limit (3.64) and Eq. (3.81) as (n) c ku/TW

(3.84)

For typical values such as W x 5, ku/T z 50, k ( A u ) r x 2, Eqs. (3.83) and (3.84) may be satisfied in the range 0.4 c (n) c 10. Thus at very low pressures ((n) 4 l), k(Au) > r,, so that it is impossible to satisfy Eqs. (3.77). However, in a wide intermediate pressure range, the limiting conditions (3.77) can be satisfied. At still higher pressures ((n> % l), the Doppler limit no longer holds. The line shape is determined from Eq. (3.69). The imaginary part of the plasma dispersion function appearing in Eq. (3.69) will consist of the convolution of Gaussian and Lorentzian terms [see Eq. (3.26)]. If the Lorentzian component is expanded in a series about u, = 0 based on the fact that the

P . R. Berman

86

Gaussian function of width k(Au,) will be sharply peaked compared with the Lorentzian function of width r, ,and if Eqs. (3.79), and (3.77b) are used, one can find the asymptotic form of (3.69) to be P33

-

-

[

4n1I2N exp - ( ; ) 2 ] Y2 kur-0

~

r 2

(3.85)

with m

x2 = ( Y 2 / G )

c (r22/~:)"[k'(A~")12

(3.86)

n= 1

Using Eq. (3.80), one may carry out the summation in Eq. (3.85) and obtain x2 = (~2/r:)[~(~~)12(r22/r~)/[1 - (r22/r;)~2 = [k'(A~)]'( r(1+ B(n>)+ 32 [k(Au)I2 f 1+W(n) -~

(3.88)

Since W ( n ) 2 1 over the range of allowed (n), the net effect of velocitychanging collisions is to add a nearly constant contribution to r,. For example, if W = 8.0 and k(Au)/r, = 2.0, one must have (n) 2 0.2 to satisfy Eq. (3.83). In the range 0.2 < 01)< 1.5, a graph of ravs (n) appears to be linear with the correct slope of 8.0 but with an intercept l . 5 r rather than The possibility of such extrapolation anomalies have been discussed by other authors from somewhat different viewpoints (Alexseev et al., 1973; Kolchenko et al., 1973). Such anomalies have been found experimentally (Bagaev et al., 1972; Meyer et al., 1975; Cahuzac et al., 1976; Mattick et al., 1976), although it is not yet clear that their origin lies in the above process. In summary, studies of near resonant tuning for L = 1 allow for a determination of TZz,a, and Sfi through the use of Eq. (3.69) together with the values of the parameters determined from the large detuning case. The effects of velocity-changing collisions are most discernible in cases where k(Au) > r,. Typical line shapes [Eq. (3.69)] are shown in Figs. 3 and 4 for k' = k, A/ku = 0.5, and collision parameters T , / k u = 0.1, Sqh,/ku = -

r.

STUDY OF COLLISIONS BY LASER SPECTROSCOPY I

I

87

I

A =0.5ku

~0.0 = i a

0

'

A'/ku

FIG.3. The line shape [Eq. (3.69)] p 3 3 (in arbitrary units) vs probe detuning A/ku for k' = k and a pump detuning A/ku = 0.5. The collision parameters are TJku = 0.1, Sqh,/ku = -SPl/ku = 0.02, rz2/yZ = (n) = 1, a = 0.8 (Au = 0.614)corresponding to moderately strong collisions at pressures on the order of several torr.

I

'

A'/ku

FIG.4. The line shape [Eq. (3.69)] p a l under the same conditions as in Fig. 3, except (n) = 5.

P . R . Brrman

88

SP;/ku = 0.02, a = 0.8, and ( n ) = 1 (Fig. 3), or (n) = 5 (Fig. 4). (The different values of ( n ) might reflect different values of ll2.) The collision parameters correspond to moderately strong collisions [(Au) = 0.6~1 at pressures typically on the order of several torr. Each line shape consists of a collision-broadened and shifted Lorentzian component arising from atoms that have not undergone velocity-changing collisions, plus a redistribution component arising from atoms that have become partially thermalized as a result of collisions in level 2. The relative amplitude of the redistribution term to the Lorentzian term increases with ( n ) , as can easily be seen in Figs. 3 and 4.

3. Near-Resonant Tuning: Counterpropagating Waves (6

=

- 1)

Since all the collision parameters can be determined by cases ( 1 ) and (2) above, this situation can provide a consistency check for the values of the variables. The line shape will be more complicated than the copropagating case since chains 2 and 3 of Table I now enter into the calculation. The line shape is given by Eq. (3.65)and can be asymmetric since collisional shifts give rise to somewhat different positions for the resonances. There is also the possibility that the TQ contribution is much narrower than the nocollision part of the SW contribution [e.g.. if r,, 4 r,]. giving rise to a narrow peak superimposed on the Lorentzian contrihution from the SW chain. In Table 11. a summary of the method for obtaining the collision parameters is presented.

TABLE II COLLISION

PARAMETERS DETERMINED USING UPWARD CASCADE LINE SHAPES

Pump" detuning

Copropagating ( + ) or counterpropagating ( - )

IAI % kit

k

6, = 0

[ A 1 % krr

+

A'+A=O

1A1 < kii

+

1A1 < k r r

-

6' = (k' k)8 6' z z " ( k ' k)d 6' z - ( k ' k)h

Resonance posit ion

-

h' z -z

"

(k' k @

Parameters determined

ry:. r;; . s;: ry:. SP: sy:. r12. Consistency check on all parameters

89

STUDY O F COLLISIONS BY LASER SPECTROSCOPY

F. ALTERNATIVE LEVEL 'SCHEMES

I . Altrrrtative Three-Lmel Systems The results for the upward cascade can be translated into other three-level systems as follows: For the inverted V (Fig. 2b). tr+counterpropagating, copropagating, A'+ -(A' - SZp!), (A + A')+ (A - A' + ST!). For the V (Fig. 2c), t = 1 +counterpropagating, t = - I + copropagating, A + -(A - ST!), (A + A')+ -(A - A' - ST!).

t = -1

-,

-

For these level schemes, one is more likely to have the possibility of an equal collision interaction for levels I and 3. 2. Two-Luvrl Systems

If levels 1 and 3 actually coincide, the three-level system reduces to a two-level system. From a theoretical and experimental viewpoint, two-level systems can be more complicated than three-level ones. First, the assumption that field E, drives only the 1-2 transition and E2 only the 2-3 transition is no longer tenable since the 1-2 and 2-3 transitions are now identical. The presence of two fields driving the same transitions can lead to both spatial and temporal beats in the polarization. The basic equations will be of a somewhat different form than those discussed in Section 111.9 but can still be solved easily in perturbation theory. For strong fields, one must use numerical methods to solve the equations (Stenholm and Lamb, 1969; Feldman and Feld. 1970; Holt, 1970), in contrast to the threelevel problem, where analytic solutions were possible. Second, the collision interaction in levels 1 and 3 is now identical so that collisions are "velocity changing" rather than '' phase interrupting" in their effect on p t 3 = p , ,. Consequently, the line shape formula will contain contributions from velocity-changing collisions in both levels I and 2; it is no longer possible to isolate a single level in which to study velocity-changing collisions as was possible in the three-level case.' Since the two-level system is a degenerate inverse V, the case of counterpropagating waves and near-resonant tuning will have contributions only from SW terms. The line shape is analogous to Eq. (3.69).and for detunings

' If one of the states has a much slower natural decay rate than the other. velocity-changing collisions in the more rapidly.decaying state will have little effect on the line shape. In this limit. one can again study velocity-changing collisions in a single level.

90

P. R. Berman

A and A' of the pump and probe, respectively, one finds that the nonlinear probe absorption I, is given by

ifl2( 1 + u") + u"A + A'

zi(

~

k(Au,)

(3.89)

where it has been assumed that k' 5 k. Velocity-changingcollisions in both levels enter the line shape. For copropagating waves the situation is more complex since the TQ processes contribute. The line shape has been discussed by Kolchenko et al. (1973). The TQ processes give rise to narrow resonances of width (yl rl1) and ( y 2 r22) centered at R' = R; in addition, there is the SW contribution. It may be possible to analyze the line shape to obtain the individual rates for velocity-changingcollisions in levels 1 and 2, but due to the presence of both SW and TQ contributions the line shape analysis will be difficult.

+

+

IV. Transient Systems The theory discussed in Section 111 neglected all transient effects and was concerned only with experiments in which the external fields acted for times much greater than the relaxation times associated with the given systems. There is another class of experiments where the transient phenomenon itself is analyzed to obtain information on the relaxation processes in atomic and molecular systems. Clearly, such experiments must be carried out on time scales on the order of the relaxation times associated with the system. Transient experiments are complementary to steady-state ones for collisional studies. Transient and coherent transient methods have been used extensively in NMR work (Abragam, 1961) to obtain values for relaxation and diffusion constants in liquids. Only recently have coherent transient techniques been used in optical spectroscopy, where the short decay times require that the external fields be applied in times on the order of tens of nanoseconds. Techniques for producing transients in the optical domain have involved using short light pulses (Kurnit et al., 1964; Abella et al., 1966; Patel and Slusher, 1968; Bolger et al., 1976; Aartsma and Wiersma, 1976), dc stark fields to switch molecules into or out of resonance with continuously applied

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

91

fields (Brewer and Shoemaker, 1971; Brewer and Shoemaker, 1972; Berman et al., 1975; Loy, 1976), or, most recently, electrooptic frequency switching of a cw laser beam (Brewer and Genack, 1976). In all of these experiments, one effectively applies a pulse or series of pulses to an atomic system and observes the transient response of the system. Transient experiments are well suited for independently determining population and polarization decay rates; in steady-state experiments these two effects were superimposed in the line shape. Moreover, experiments involving two or more pulses (such as photon echo experiments) enable one to determine in a unique manner whether collisions are velocity changing or phase changing in their effect on off-diagonal density matrix elements. Such clear tests were absent in the steady-state domain. On the negative side, transient experiments have usually involved large field strengths and the accompanying problems of power-dependent effects. In addition, the interpretation of collision rates may itself be a function of the excitation pulse structure, as will be discussed below. Several types of coherent transient experiments will be discussed. The theory of collisions in transient experiments has been developed by several authors (Scully et al., 1968; Schmidt et al., 1973; Doktorov and Burshtein, 1973; Heer, 1974; Berman et al., 1975; Stenholm, 1975): we shall point out some of the general conclusions reached by these authors. Only transients in two-level systems will be considered (Fig. 1); transient effects in threelevel systems have attracted recent interest (Shoemaker and Brewer, 1972; Brewer and Hahn, 1973; Loy, 1976; Leite er al., 1976) and may prove fruitful for future collision studies, but they will not be discussed here. In all cases, we shall consider only the eflects ro lowest (nonlinear) order in the applied fields. Strongfield effectswill introduce major complications into the analysis of collision phenomena.

A. FREEINDUCTION DECAY (FID)

In FID, the two-level system is subjected to the pulse shown in Fig. 5a. In the first stage, the applied cw field gives rise to steady-state values for density matrix elements P1l(v), P,,(v), P2,(v). Subsequently the field is abruptly terminated and one observes the transient decay of the system’s polarization. In the absence of collisions, the steady-state value of P,,(v, t = 0) may be obtained by solving Eqs. (3.13) with x‘ = 0. The well-known result (Torrey, 1949; Brewer and Shoemaker, 1972; Hopf et al., 1973) is

P . R. Berman

92

FIG.5. The applied electric field amplitudes E(t), which give rise to the transients discussed in Section IV: (a) gives rise to free induction decay, (b) gives rise to two-pulse nutation, and (c) gives rise to photon echo.

where

+ 2X2Y1z(Y;' + Y

3 (4.2) and the notation is the same as that used in Section 111. We have not yet made the weak-field assumption for reasons that will become obvious. Since the width of the velocity the preparatory field was on for a time ,:;y% distribution created by the field is the power-broadened homogeneous one (Y;2)2

vl2

= (Y12)2

lk*

When the field is removed, P,,(v, t) satisfies the free-atom equation

so that for t

> 0,

which, in the Doppler limit (3.32) reduces to6

The polarization is proportional to -1m(p12) and therefore exhibits a

-

The FID signal also has a rapidly decaying component (decay time l/ku) which is not indicated in Eq.(4.5). I should like to thank Professor M.Feld for a discussion on this point.

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

93

+

simple exponential decay with rate (yI2 y i 2 ) . However, as is evident from Eqs. (4.2) and ( 4 4 , the polarization goes to zero in the weak-field limit. The outlook for collisional studies using FID is not good owing to the necessity for having strong fields to observe detectable signals. Collisional processes such as phase-interrupting effects on pl and velocity-changing effects on p12,pll, or p22 will all tend to increase the bandwidth of excited atoms and consequently lead to a more rapid FID. The time dependence of the different processes might be somewhat different, but it will be difficult to separate the various components in the decay signal. In addition, the switching techniques for FID (Brewer and Shoemaker, 1972) have involved changing the laser or atomic frequency rather than turning off the laser. The new laser frequency can be resonant with another velocity subset of atoms, which will also emit a transient optical nutation signal (Brewer and Shoemaker, 1971; Hopf et al., 1973).The total signal will then consist of both the FID and the optical nutation, which complicates the line shape even more.

ABSORPTION B. TWO-PULSE NUTATION-DELAYED SATURATED The principle of two-pulse or delayed nutation (Berman et al., 1975)and delayed saturated absorption (Hansch et al., 1971; Brochard and Cahuzac, 1976) is really the same. One subjects the sample to the two-pulse sequence shown in Fig. 5b. The first pulse creates changes in the populations of states 1 and 2. In the weak-field limit, these changes are proportional to x2 and manifest themselves as holes burned into the population velocity distributions. The width of the holes will depend on the ratio of ylz or Fl2 to the frequency bandwidth wL associated with the laser pulse (wLis the inverse of pulse duration time). The width will be given by the maximum of wL/kand y12/k;ifwL 4 y 1 2 , the preparation is effectively steady state. Thus, the velocity bandwidth uo excited by the first pulse is given by uo = max(f,,/k, wL/k)

(4.6) The changes in the populations will be denoted by [Ap,,(v, t)], [ApZ2(v, t)]. These population changes evolve in time and are probed by the second pulse, which has the same bandwidth as the first. Either the absorption or optical nutation signal of the second pulse can be monitored. In the case of delayed saturated absorption the pulses are generally chosen to be counterpropagating. If one measures the change in absorption (or amplitude of the optical nutation signal) of the second pulse produced by the presence of the first pulse, assuming both pulses are resonant with the same velocity subset of atoms, then in effect one is measuring the total number of atoms left in the

P . R. Berman

94

hole burned by the first pulse. The delayed absorption signal will be sensitive to all prooesses that have removed atoms from the velocity hole created by the first pulse. Such processes include decays of the atoms to any other levels resulting from spontaneous emission or inelastic collisions, in addition to those velocity-changing collisions that have resulted in velocity diffusion out of the hole. In this way, two-pulse nutation or delayed saturated absorption experiments are well suited for measuring all inelastic relaxation rates and velocity-changingcollision rates for the various populations. An alternative method for studying velocity-changing collisions with delayed saturated absorption is to vary the frequency of the second pulse and measure its absorption spectrum (Hansch et al., 1971). As velocitychanging collisions increase the velocity width of the holes, the absorption spectrum will correspondingly broaden, ultimately reaching the limit of absorption from a thermalized sample. To illustrate the above concepts, we consider the case 7, +Y,, w L + L wLw19rll (4.7) which might correspond to level 1 being the ground or a metastable state. The decay of level 2 is so fast that it may be neglected, and the absorption change signal arises from [APl ,I. The condition wL 9 yl, rl ensures that no relaxation of level 1 occurs during a time equal to the width zL of the pulse. will ensure that the hole width is determined by f,, The condition wL G f 1 2 rather than wL [See Eq. (4.6)]. The lowest-order change in fill is easily calculated in perturbation theory as

where zL is the pulse width. For the Keilson and Storer kernel, the value of [Afil ,(v, t ) ] will be (Keilson and Storer, 1952; Berman, 1974) [APl1(v, t ) ] = (. dv’G(v’+v, t)[APlI(v’, O)]

(4.9)

with G(v’+v, t) =exp[-(y,

+

+ r1,)t]{6(v - v‘)

(n!)-’[n(A~,)~]-~’~ n= 1

x

(rl

exp{- [(v - anv’)/(A~n)I2})

(4.10)

The change in P12 produced by the combined action of the two pulses is given by [see Eq. (3.13d) with x’-0, X+X‘, A+A’] [AfilZ(v, t)] = -WIAPll(v, t)3/[fl2

+ @‘ + ku,)]

(4.1 1)

95

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

where A‘ is the detuning of the second pulse and we have taken k z k. The absorption change signal is proportional to the negative imaginary part of the velocity integral of [Afi,,(v, t)], and using Eqs. (4.9b(4.11)in addition to a simplifying but not necessary assumption that k(Au) B

f,,

one obtains the absorption-changesignal for two pulses separated by time T to be -1m

1dv[Afi,,(v,

T)] = 2 n 1 / 2 ~ 2 ~ exp[’ ~ L N( d / k ~ ) ~ ] ( k u r ~ ~ ) - ~

(4.12)

The signal consists of two parts, which are analogous to the two terms in Eq. (3.69) of the steady-state theory. The first term is the normal saturated absorption signal arising from atoms that have not undergone velocitychanging collisions, and the second is the summation term arising from atoms that have had their velocities changed in collisions. If rllt < 1, the summation term may be neglected and the probe absorption-change signal as a function of A’ will be a Lorentzian of HWHM 2 f 1 , centered at d‘ = cA. For higher pressures or delay times (r, T % l), the summation term dominates the line shape; the probe absorption-change signal will be approximated by a Gaussian of width ku centered at & = 0. Thus, by measuring the width, central frequency, and shape of the absorption-change signal spectrum for different values of rl,z (by varying either the pressure or delay time), one can arrive at some idea of the strength and rate of velocitychanging collisions in level 1. If one chooses not to measure the frequency spectrum of the absorption, but simply the absorption for resonant tuning d = dl = 0, the absorption change signal takes the form

,

-1m

1dv[AP,,(v,

t)] = 2n1~2~Z~’t,N(k~f12)-1H(~) (4.13)

with H ( t ) = exp[-(y,

[i

+ Tll)z] + nl/’f

n= 1

(&)v] k ( W

(4.14)

P. R. Berman

96

For

f12 /k(Au) < 1 as assumed, (4.15)

and one has a simple method for determining rll at various pressures. by Should inelastic collisions be present, one would determine (r, + rine’) /k(Au) is not totally negligible due to pressure or power this methpd. If f I 2 , should keep the second term in Eq. (4.14). In that broadening in f l Zone case, H ( z ) starts at a value of 0.5 at z = 0 and asymptotically approaches ( 7 ~ ~ / ~ f , ~ exp(-y,z) /ku) when rlltb 1. There are some restrictions on two-pulse nutation experiments. It is only possible to easily measure two-pulse nutation signals if all transients between the pulses have died away. These transients decay in a time =Z min[(f12)- l , (oL)‘1, which restricts the time 7 between pulses to (4.16)

On the other hand, to observe velocity-changing collisions in level i, one must have (4.17) (yi + rii)t 5 1 for reasonably large signals. Thus, if Fl2 > oL,then two-pulse nutation experiments are practical only if (yl rl1) < f 1 2 or (yz r22) 4 f 1 2 . This limiting case was discussed above and can provide useful information on velocity-changing collisions in the slowly decaying state. one must If both (yl + rll)and (y2 TZ2)are on the order of f12-, increase the intrapulse transient decay rate by choosing wL b r12. In that case, the width of the hole is given by (4.6) and, consequently, the rate at which velocity-changing collisions deplete the hole will be functions of wL.Under these conditions, two-pulse nutation experiments are best suited for providing a measure of the total hole depletion rate (inelastic plus velocitychanging collision contributions) for a given pulse excitation scheme. These results can be useful in the analysis of the photon echo experiments to be described below.

+

+

+

C. PHOTONECHOES

When a two-level system is subjected to the pulse scheme shown in Fig. Sc, it is possible to observe “photon echoes ” (Kurnit et al., 1964; Abella et al., 1966; Pate1 and Slusher, 1968; Brewer and Shoemaker, 1971) that are analogousto spin echoes ( H W , 1950) of NMR.The theory of photon echoes in gases (Scully et al., 1968; Gordon et al., 1969) and the effects of collisions on echoes (Schmidt et al., 1973; Doktorov and Burshtein, 1973; Heer, 1974; Berman et al., 1975) have been discussed by a number of authors. Basically, the theory of photon echoes without collisions is straightforward (Carr and Purcell, 1954). The first pulse creates some value for the

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

97

off-diagonal density matrix element P12(v, t = 0). Between t = 0 and the application of the second pulse at t = r, the atom oscillates freely, with Plz given by PlZ(V.

t ) = expHy12 + i(A + k ~ , ) I t ) P l 2 ( V , 0)

(4.18)

There will be a dephasing of the dipoles due to a distribution of velocities. At t = T a pulse of area II = J x ( t ) dt (Carr and Purcell, 1954) is applied whose net effect is to change P12 into its complex conjugate P12(v, 7 ) = exp(-yl22) exp[i(A + k ~ , ) r ] P 1 2 (O)* ~,

(4.19)

For t > T the atom again oscillates freely, with Pl2 given by Pl,(V,

t > 2) = exp{-[?12

- i(A + kuz)17>

x expHy12 + i(A = exp(-y12t)

+ kuz)l(t - 7)>P12(V, O)*

exp[-iA(t - 27)]

x exp[ - iku,(t - 2r)]P12(v,0)*

(4.20)

One immediately sees that the effect of the second pulse has been to start the dipoles rephasing. At time t = 2r, the dipoles are all in phase again and the echo signal is emitted. The normalized echo amplitude R at t = 22 will be defined here as = Im

1 dvj12(v, t = 2r)/1m (. dvj12(v,o)*

(4.21)

For the no-collision case, (In light of the discussion of Section IV,B, one will have to use a pulse bandwidth wL < y12 in order to observe the echoes.) The use of photon echo experiments for collision studies should now be obvious. Any collisional process that interferes with the dephasingrephasing of the dipoles will diminish the echo amplitude so that studies of G vs the pulse delay r can yield collisional data. Moreover, it will be shown below that 6 can have a dependence on 7 that will determine whether collisions are velocity changing or phase changing in their effect on P12. 1. Phase-Interrupting Collisions

If collisions are phase interrupting in nature, the only effect on the echo amplitude is to replace y I 2 by fyh,and b(z)= exp(-2Qh,

7)

(4.23)

P. R. Berman

98

One finds a simple exponential decay of echo amplitude as a function of pulse separation. By varying the pressure, one can determine rl;; as a function of pressure. If one state dominates the collision interaction, the elastic scattering cross section for that level will be obtained. 2. Velocity-Changing Collisions

If collisions are velocity changing in their effect on P12 there will be two processes that can lead to a decrease of the echo amplitude. First, velocitychanging collisions can remove atoms from the velocity hole burned by the first pulse; such atoms are effectively lost from the system since they will not be resonant with the second pulse. Such collisions act as inelastic ones for the system, and together witk other inelastic processes give rise to a factor is the total inelastic collision exp(-2rQt) in the echo amplitude, where rQ rate. It will be assumed that TQ can be measured in a two-pulse nutation experiment. The second effect of velocity-changing collisions is to interfere with the dephasing-rephasing of the atomic dipoles that led to a perfect cancellation of the Doppler phase factor at time t = 2t. For a rigorous calculation of the echo amplitude, one must solve the transport equation dfiI2(v,t ) P t = -[y12 -

+ i(A + ku,)]P12(v,

t)

(rQ+ T)fi12(v9 t , (4.24)

for the time regions 0 < t < t,t < r < 2t. The quantities W(v'+v) and r a r e the kernel and rate, respectively, for velocity-changing collisions. Without going into a formal solution of Eq. (4.24), one can obtain a good feeling for the time dependence of &(r)by some simple arguments. The Doppler phase factor exp{- [iko,(t - 2r)I) in Eq. (4.20)vanishes at t = 22 in the absence of collisions. Roughly speaking, one should replace this factor at t = 27 by exp[-2ik(AuZ)r], where Ao, is the change in velocity in time 27. This factor will overestimate the effects of collisions since, in reality, the changes do not act over the entire time interval. However, the form of the time dependence will be the same as that in the more rigorous solutions. Thus, we take the echo amplitude to go as (4.25) with the average over collision histories and the assumption that ((Au,)~'" = 0 for integer n. The echo amplitude will have different forms for ( [ k ( A ~ ~ ) t ]4~1) and ([k(A~,)t]~)>> 1.

')

STUDY OF COLLISIONS BY LASER SPECTROSCOPY

a. ([k(Au,)zI2) &(2)

< 1.

-

99

In this limit, Eq. (4.25) becomes

eXp[-2(y1,

rQ)T][1 - 2k2((A~,)Z)~2]

(4.26)

The value of ((Au,)’) will be approximately equal to [the number of colli) one collision = ( A u ) ~ to / ~ within a sions in time 22 = 2 r ~ x] [ ( ( A U ~ ) ~for factor of two for the Keilson-Storer kernel (3.52)]. Therefore, ((Au,)~) 5 rr(Au)2

which, when used in Eq. (4.26) along with the fact that ([k(Au,)~]~) < 1, gives rise to a value for the echo amplitude &(z)

-

exp[-2(y12

+ TQ)?]e ~ p [ - 2 r k ~ ( A u ) ~ t ~ ]

(4.27)

As mentioned above, the effect of velocity-changing collisions was overestimated. More rigorous calculations (Doktorov and Burshtein, 1973; Heer, 1974; Berman et al., 1975) give &(2)

-

],

DISTANCE FROM LASER BEAM AXIS

FIG.9. Spatial profile for excited-state number density (n,)for a typical dye laser. Note the sharp cutoff at around 1.2 mm.

excited-state density profile is much more sharply confined than one might expect from the laser intensity profile. This fact may be very advantageous when performing scattering experiments. As we see, the laser may define a collision volume much more sharply than often possible otherwise in a collision experiment. As we see in Fig. 9 the effect of narrowing the laser bandwidth is of minor importance below 30 MHz. Spatial widening gives a more extended excitation region, which is still relatively well limited.

2. Measurements of Optically Excited Multipole Moments From the discussion in Section III,A,S it is obvious that in an actual experiment one is not sure of finding the stationary multipole moment distribution given in Eqs. (54) and (27). Especially in regions of weaker intensity, long times up to psec may be needed to reach equilibrium. On the other hand, in regions of high power density AvC\ may be typically of the order of the hyperfine splitting. The overlap of the hyperfine levels thus produced changes the characteristics of the pumping mechanism (see Section III,A,3). A similar effect may be caused by radiation trapping in

145

COLLISIONS WITH LASER EXCITED ATOMS

the sodium beam. The reemitted spontaneous radiation has a different polarization and thus disturbs the excitation by n or CJ light. The result, common to all the disturbing effects mentioned, is a poorer degree of anisotropy of the optically pumped state. Thus, the question arises of how to measure the multipole moments, which must be known for a proper interpretation of collision experiments. Fortunately, one often has to know only the multipole moments up to rank 2, i.e., the orientation and alignment optically produced (see Section IV). These (and only these) may be determined by observing the fluorescence light of the atom. This has been outlined in detail by Fano and Macek (1973) in the context of impact excitation and polarization of the emitted light. Fano and Macek give the fluorescence intensity of an aligned and oriented atom as

I x 1 - +h(''A;fec+ $!I(~'A;':cos 2p +

sin 28

(60)

with

while p = 0 or n/2 for z-light detection parallel (III) and rectangular (I,) to a given detector and analyzer direction, and p = +n/4 for left and right circularly polarized light, respectively. The orientation 0;''and the alignment A:" measured in the light detector frame are derived from those in the photon frame by ,:el

=

= Cb"(0, 'Y)Ao"h,

cp(0,0)ogh 1

Adel 2+

- 3c:2:(o,WGh

(611

with the real solid harmonics given in Table I. Equation (61) differs somewhat from those of Fano and Macek since at present we refer to optical atom excitation. 0 is the polar angle of the light detector and Y the polarization analyzer angle with respect to the photon frame. Equations (60) and (61) may be used to determine the multipole moments Yl.(l)/#'(O) and I '(2)/ Y4 '(O), which are connected to Ogh and Agh by Eq. (49a). The linear polarization of the emitted light

allows us to determine Agh and thus 17 '(2)/ Y4 *(O). Circular polarization leads to a determination of Ogh and % '(l)/W '(0) when Agh is known. For n-light excitation Ogh = 0. Particularly simple is the observation of unpolarized light in the plane of the polarization vector of the exciting laser

I. K Hertel and W Stoll

146

c k! z 5

Z w

E2\8=y; I

FLUORESCENCE

Na-BEAM

V

z

W

u In W a

from behind

N? -BEAM

0

2

4e goo 2700 900

270'

goo

POLARIZATION ANGLE

FIG.10. Fluorescence intensity from a sodium 32P,,2,F = 3 atom, excited with linearly polarized laser light, as a function of the polarization angle 0 of the exciting light relative to the fluorescence detector. Maximum (I,,,,.J and minimum (Imin) positions of the E vector of the light are indicated.

light as a function of its polarization direction 0 with respect to the detector. This geometry together with an experimental signal is shown in Fig. 10. Equation (60) simply gives I K (1 - fh(')AP), Eq. (61) gives A?' = Aghf(3 cos' 0 - l),and Eq. (49a) gives Agh

= W(2)/W(0)(2F

+ 1)- "'[F(F + l)u(')(F)]-

When inserting explicit values for d2)(F)and h") for F = 3, F = 2 one obtains' I K 1 0.1732(3 C O S ~0 - l)W(2)/W(O) (62)

+

The ratio of maximum to minimum fluorescence found in the experimental determination (Fig. 10) is Imax/Imin x 1.47, which is a poor example. For stationary pumping [ W(2)/W(O) = -0.9621 ImuJZmin = 1.75, and for the ini= 1.47 would be tial moment of pumping [W(2)/W(O) = -0.6931 Imux/Zmin expected. The experimental fluorescence anisotropy indicates that in the pumping region we either have no stationary conditions, and/or strong radiation trapping, and/or a partial overlap of the hyperfine levels due to saturation. In fact, we observe a decreasing anisotropy for increasing atom beam intensities, which indicates radiation trapping. If the F = 2 level partially overlaps with the F = 3 level or is excited otherwise, then Eq. (62) gives only a first-order approximation.

COLLISIONS WITH LASER EXCITED ATOMS

147

It should be pointed out that these types of determination of W ( 2 ) and W(1) give values averaged over an area seen by the light detector. This spatial averaging has to be done in the same way as in the actual scattering experiment for scattered particles by the particle detector. That is to say, the light detector has to see the same area as the particle detector does later. Under these conditions one may safely use the spatially averaged values W ( k ) for evaluation of the collision experiment [see, e.g., Eq. (12)], where the measured scattering intensity would be given by Ispal. 10

C W(k)spal. av.(t[ok'>*

In Section V we describe an alternative way to measure W(2) directly in the collision experiment. This is possible in particularly favorable cases. Some collision experiments may require knowledge of the higher-order multipole moments induced in the optical-pumping processes. These cannot be measured simply by single fluorescence photon detection. In order to detect higher multipole moments many successive photons have to be observed coherently. That is to say, the detection process has to be in itself a pumping process or a time-dependent observation of the atomic response on a defined change in the excitation conditions. We wish to indicate a few possibilities that, however, to our knowledge have not yet been exploited for the determination of higher multipole moments: (1) One could use the exciting laser by rotating its polarization angle once through 360" in a time short compared to the time for reaching stationarity (see Section III,A,5), but long compared to 7 . The absorbed power, or the fluorescence, changes as a function of @(t), O(t), and would depend on W(k)C$'(@,a).

(2) One could use a second laser for pumping excitation of a higher state out of the first laser-excited state. (3) Alternatively, one could apply a magnetic field. If this is made nonparallel to the photon frame, Zeeman coherence results and the optical multipole moments become nondiagonal and time dependent, each characterized by oscillation frequencies up to 2kw,, where wLis the Larmor frequency in the field. The time dependence has to be observed in times short compared to the stationary pumping times. The time origin could be defined by switching the polarization vector from parallel to H into rectangular position.

One also may observe Hanle signals. Such complex signals have been observed by Ducloy et al. (1973) for a gas laser and have been interpreted as determined by a hexadecapole moment in the 2P4neon levels, CohenTannoudji (1975) has discussed such Hanle signals in simple cases, to mention just two references from a field that is related to the present topic and has been reviewed recently by Decomps et al. (1976).

I . K Herrel and W Stoll

148

No experiments on higher multipole moments directly connected to crossed-beam collision processes have been performed and thus we end the discussion here. 3. Methods for Preparation and Detection of Atoms in the Ground State by Optical Pumping

a. It is interesting to discuss within the previous context an experimental method described by Schieder and Walther (1974), which in principle is capable of determining the multipole moments of the ground state after laser pumping. The situation for ground states is somewhat more convenient because Zeeman coherence does prevail for a long time, since no spontaneous decay destroys it. The experiment of Schieder and Walther is shown in Fig. 11. Two laser beams were produced by splitting the output beam of a A Flucxexence Signal

No- Atomic Beam

cj-

h e r Beam 2

Loser Beam 1

004 Gauss

q v I

I

I

I

I

I

I

-

FIG.I I . Scheme of the experimental setup (the direction of observation is perpendicular to the laser and atomic beam) and fluorescence signal at the second laser beam as a function of the magnetic field. Both laser beams were circularly polarized. From Schieder and Walther (1974).

dye laser. The first intense beam induced the ground-state coherence by optical pumping, whereas the second beam probed the time development of the ground-state coherence after the flight between the two interaction regions. The fluorescent intensity, observed perpendicular to the atomic and laser beams at the second interaction region, was measured as a function of the magnetic field H,.In principle the set up is similar to a Rabi-type apparatus used in atomic beam resonance work. As Schieder and Walther point out, there might be a quite interesting application in collision studies.

COLLISIONS WITH LASER EXCITED ATOMS

149

For this purpose a second atomic beam could be crossed with the first one in the region between the two laser beams. Observing the fluorescent light induced by a second laser beam, phase and hyperfine level changing collisions could be detected. b. An elaborate method for preparing a beam of ground-state sodium atoms in a particular hyperfine level with a simultaneous velocity selection has been demonstrated by Shouda and Stroud (1973). No application to scattering experiments has been reported so far. But the scheme indicates interesting possibilities. In most cases the nuclear spin is decoupled during collision and the dynamics are essentially determined by elastic-scattering amplitudes (see the similar analysis on fine-structure transitions given in Section IV). Valuable information, e.g., on spin exchange amplitudes could be obtained in such experiments, especially when a pure electron spin state is prepared in an experiment using 0' (or 0 - ) light. Then the 1 F = 2, A = 2) state is populated exclusively and one has a pure electron spin state 1 S = i,M , = +&).Such experiments would have their merits in comparison with the usual magnetic selection methods where a loO'?;, polarization is never obtained.

c. As a general tool for detecting scattered atoms, cw dye lasers may have a number of advantages in collision experiments with ground- or excited-state atoms. In particular, when detecting the particles by fluorescence one can simultaneously detect the atoms and measure their velocity. Such a device can in principle be built in a more compact form than the conventional combination of mechanical selector and surface ionization detector. The feasibility of such a Doppler-shifted fluorescence (DSF) detector has been demonstrated by Hertel et al. (1975b), using a cw dye laser to detect a sodium beam. The atom beam to be detected is intersected by the laser detector beam within a length 1 = d/sin 0 (see Fig. 12). A small fraction of the laser beam crosses the sodium beam under 90" to give an unshifted reference frequency. The atomic fluorescence is observed at right angles to both the atom beam and the reference beam. Assuming the atoms to have only one resonance frequency vreS, fluorescence is observed from atoms flying with a velocity v when the exciting laser light has the frequency v, Doppler-shifted by AV = v,, - v = vv cos O/C (634 i.e., v = A AV/COS 0 (63b) Av is determined with the aid of the reference beam.

I . K Hertel and W Stoll

150

Top view schematic

(b)

EL

-%hv

?-

ID4

FIG.12. Schematic diagram of the DSF detector. The sodium beam (Na) is intersected, (a)

by a detector laser beam hv (det) under a small angle Q measuring the Doppler-shifted excitation frequency, and (b) rectangularly by a reference laser beam h ~ ,(ref) , ~ providing a signal at the unshifted resonance frequency.The fluorescencesignal is observed by a photomultiplier PM via a lens L.From Hertel et 01. (1975b). Copyright by the Institute of Physics.

Again, the F = 2 + F = 3 transition is used. Confusion, due to several HFS lines, may then be avoided by optical pumping with high laser intensity, allowing for a sufficiently large number of pumping cycles. Each atom emits several photons when passing through the detection region. During one natural lifetime, each excited atom emits one photon. The total number of photons N emitted by an average atom of velocity u per passage through the detection region of length x is thus given by

This leads to a detection efficiency a l/u, while it is a u with a mechanical selector. Here the passage time is tx = x/u and may be typically 2502 for thermal sodium atoms. A realistic estimate for the overall detection efficiency including the finite collection-and quantum-efficiency of the photomultiplier system may be 35% for each atom entering the detection area. The other property of the DSF detector of interest is its velocity resolu-

COLLISIONS WITH LASER EXCITED ATOMS

151

tion. It is determined by the uncertainty 6(Av) in measuring the Doppler shift Av. From Eq. (63) we obtain for the velocity resolution

6~ = (A/COS 0)~ ( A v )

(65)

Thus the absolute value 6 u for the velocity resolution is independent of u, in contrast to a mechanical selector, which has a constant relative velocity resolution. For sodium we expect a velocity resolution 6u of f 60 msec-' for 0 x 0 and ~ ( A vx) 100 MHz due to saturation. A mechanical selector can easily do better at thermal velocities. However, at suprathermal velocities the DSF detector becomes superior. The results for detecting a velocity selected sodium beam (FWHM x 7%) are shown in Fig. 13 (signals c and b). As the reference point (v = 0, Av = 0)

.

i\JI I

*

. .

I1

/ i:,

F = 2-F.2

J 2500

2000

1500

1000

500

I

1

1

1

I

0

AV [MHz]-

Vmox. XI

FIG. 13. Fluorescence intensity from a mechanically selected sodium beam with most probable velocity u,,,~,. The reference beam signal (a) resolves the HFS splitting of the F = 2 + F transitions. Signals (b) and (c) come from the Doppler-shifted F = 1 + F and F = 2 + F excitations, respectively. From Hertel et al. (1975b). Copyright by the Institute of Physics.

here, the F = 2 + F = 3 transition of reference peak (a) is used. Its relatively weak intensity allows the three different HFS transitions to be distinguished. One may state a fair agreement for the expected and measured profile of -9% FWHM and for the position of the maximum determined by the mechanical selector and measured with the DSF detector. Related experiments investigating atomic Na and Naz velocity distributions in supersonic beams have been carried out by Bergmann et al. (1975, 1976) and more recently for fast sodium atoms by Hammer et al. (1976).

I . 1/: Hertrl and W Stoll

152

4 . Atom Beam Deflection by Laser Optical Pumping When an atom beam having a velocity v is excited by photons intersecting it at right angles, the photon momentum h / l is transferred to the excited atoms. In high-intensity fields most excitation processes will be followed by an induced emission transferring the opposite momentum, and no net momentum transfer to the atom results. However, if spontaneous emission follows (which averaged over 4n carries no momentum away) the atom will be deflected from its original direction with an average transverse velocity uI given by mu, = h/l,where m is the atomic mass. If this happens N times the atom beam deflection angle ( 9 4 1) will be 9 = (vJu)N = 90 N (66) where 9, = h/(mAv) is the elementary deflection. If the atom beams passes through an excitation region of length b,, the number of deflections following spontaneous decay will be N = t e x / T , where t,, is the time the atom spends in the excited state. The probability of finding the atom in the excited state is aM= n , / ( n , n,) (see Section II1,A) and t,, = ( b , / v ) a M .We have for the fraction aM < 4 and $aM = n,/n,, where n,/no is given in Fig. 7. The total deflection angle will be

1

c

c+

1

or 3u2 = k A h, bM ,where k A is an atomic constant. This beam deflection has been discussed and investigated by a number of authors (e.g., Ashkin, 1970a,b; Stroke, 1972a,b; Schieder et al., 1972; Picque and Vialle, 1972) mainly since it offers a possibility for isotope separation. Duren et al. (1975) use a well-collimated and velocity-selected sodium beam (FWHM 20/,), which is excited over an effective length of h,. = 2.5 mm. The atom detector is positioned 630 mm away from the excitation region. The experimental results are shown in Fig. 14 for different sodium beam velocities v N p. As described by Eq. (67) for low velocities, the deflection is highest and one clearly recognizes a splitting of the beam into ground state P = 1 atoms, which cannot be excited (and deflected),and into F = 2 atoms, which have been deflected in the pumping region. Duren and collaborators carefully discuss the broadening of the deflected beam by finite atomic and laser beam profiles. The deconvolution is indicated in Fig. 14. The ratios of the F = 1 peak (undeflected) and the F = 2 peak (deflected) should be 5 : 8 = 0.625. Duren et al. find 0.5 to 0.6, which is probably due to neglect of the beam broadening by spontaneous decay. To illustrate the order of magnitude, we cite from Duren et al. (1975) lo9 degrees cm2/sec2 (681 near its maximum, i. This corresponds to

Svi, = 1.05 x for b,z2.5 mm and

c

bM

COLLISIONS WITH LASER EXCITED ATOMS

153

10

5

h

. o

E:

?

P

k

V,

’0

= 1068 [mlsec]

V,.

= 1237 [mlsecj

cd

v

5

0

0

1.0

2.0

3,O

4.0

5

DETECTOR POSITION (am) FIG.14. Experimental distribution of sodium beam intensity as a function of deflection without laser, (.)with laser. (---) 3’S,,, , F = 1 atoms. (---) 3’S,,,, F = 2 atoms. angle. (0) partially excited to the 3’P,,, , F = 3 state. From Duren et al. (1975).

N % 40 to 75 deflections. In principle the measured deflection allows the determination of the excited-state fraction C o M in the deflected beam, while passing through the excitation region. Bhaskar et al. (1976) make use of this possibility in their measurement of the total cross section for the electron scattering of excited sodium (see Section V,C). As they point out, the broadening of the beam due to to first spontaneous decay in each direction is proportional to (9/3 approximation * and will be given by A Qspont = 9( 1/31~)’’~

(69)

l 6 For quantitative investigations it has to be remembered that spontaneous decay is not necessarily isotropic and Eq. (69) has to be modified.

154

1 . 1/. Hertrl and W. Stoll

.335 DETECTOR POSITION (INCHES)

,290

FIG. 15. Fluorescence beam broadening. Laser off. - - - laser on. . Laser is incident at right angles to direction of motion of detector. From Bhaskar et a/. (1976).

This is illustrated in Fig. 15, taken from Bhaskar et al. (1976). Their detector is moved rectangular to the atom and laser beam.” We conclude by saying that in actual scattering experiments the influence of beam deflection by radiation has to be discussed carefully. It may be used with advantage for the determination of total cross sections as done by Bhaskar et al., but it may also be a disturbing effect for small angles in heavy-particle scattering experiments with high angular resolution.

5. Other Excitation Schemes a. Sodium. The pumping scheme described in Section 111,AJ to excite sodium into the 2P,,2, F = 3 state is restricted to one hyperfine level. Among all the hyperfine levels of the p state this one allows the investigation of the largest number of multipole moments, since k,, = 2F. It lends the highest flexibility to the experiment and in principle all dynamical variables ” The

experimental data are uN,, =

lo5 cm/sec, 9, = 3 x

rad, L = 800 mm (32 in.).

COLLISIONS WITH LASER EXCITED ATOMS

155

may be investigated by this type of target preparation. However, it may be of interest to excite other hyperfine levels or even the 'Pli2 state for comparative experiments. By looking at Fig. 4 it may seem that simultaneous excitation by two laser frequencies from the F = 1 and F = 2 states should allow us to pump all upper hyperfine levels F. This is, however, not possible, as Gerritsen and Nienhuis (1975a) have pointed out: Some ground-level substates form a trap. For example, when pumping with linearly polarized light the induced transition probabilities are x ( F M 10 I FM)',i.e., zero for F = F, M = 0. On the other hand, 1 FM = 0) states are populated by spontaneous decay. Thus after some time all atoms will be found in one of these ground levels even when pumping with two frequencies. For circularly polarized light, the I F = 2, M = 2) state is the trap when either the F = 2 or F = 1 excited hyperfine level is pumped, even from both F = 1 and F = 2 . Gerritsen and Nienhuis suggest using at least three optical pumping frequencies in order to obtain up to 50", of the atoms in the 2P,,zor 'P3 state. One may obtain several closely spaced frequencies in different ways, e.g.. as Hertel and Stamatovic (1975) and Hertel et al. (1977~) have demonstrated, one can make use of spatial hole burning in the active medium to operate a cw dye laser on two or more stable modes simultaneously. Gerritsen and Nienhuis (1975a) have proposed a multidirectional D o p pler pumping. A single-frequency laser beam is to be split in order to intersect the sodium beam under two different angles in the scattering center and thus the Doppler shift may be exploited. Each laser beam may be reflected back after passing through the atom beam. Thus, four frequencies may be matched. However as Gerritsen and Nienhuis (1975b) point out elsewhere, one has to be careful when superposing a beam with its reflection. Standing waves or similar interference patterns may be produced. Carter rt al. (1975a) have described and used a different scheme to excite the 3'P1/2 or 3'P,/, state. They use only one laser beam and reflect it back and forth through an angle to fit transitions from both the F = 1 and P = 2 ground states. At u = 1400 m/sec, we obtain a laser-atom-beam intersection angle of 68" (not 23"as cited by Carter et al.). The trap in the M = 0 states is removed by applying a small magnetic field (5G) roughly perpendicular to the photon frame. As described in Section III,B,2 this induces a Zeeman coherence between different magnetic substates, which subsequently may be pumped due to transitions among other magnetic states. The disadvantages common to all these pumping schemes compared to

156

I.

I/: Hertel

and W Stoll

the F = 3 scheme discussed in Section III,A,l is their complexity. It prevents the knowledge of all components of the multipole moments and the intelligible study of polarization effects will become a formidable task. One thereby renounces the powerful tool of investigating the details of scattering dynamics that otherwise could lead to a much more thorough understanding of particular processes. b. Other Atoms and Other Transitions. As mentioned before, excitedstate differential scattering experiments have been reported so far only for the Na(32P)state excited by Rhodamine 6G cw dye lasers. The current improvement of dye lasers, especially with regard to shorter and longer wavelengths, may make other transitions and other atoms accessible soon (see, e.g., Basting et a/., 1976). Some of the alkaline earth elements, some metal atoms, and some other alkalis should be within range of cw dye lasers now in use. Ga-As diodes could in principle be used, operating in a tunable single-mode cw version (Picque, 1974; Picque et al., 1975). Then C S ( ~ ~ P ) could be excited, which may be of great interest for its large spin-orbit interaction. Frequency doubling of dye lasers could widen the scale of possible applications. Very high powers are needed and one probably would have to use a pulsed laser. Quite generally, the frequency range covered by pulsed lasers is much wider. In fact, total scattering cross sections have been measured for fine-structure-changing transitions in a potassium beam excited to the 42P1,2state by a pulsed laser system, as reported by Anderson et al. (1976). The tunable light was produced by an optical parametric oscillator pumped with a frequency-doubled Nd : YAG 'laser. The experiment is described in Section V,E. An alternative to frequency doubling is the direct two-photon excitation of atoms. Two-photon spectroscopy seems to be very attractive of late (see, e.g.. Cagnac, 1975). It has not yet been exploited in connection with scattering experiments in atom beams. As an example, one may excite atomic cesium in the 92D3,2level by absorption of ruby laser light (Ward and Smith, 1975).Theoretically at least, the application of a chirped laser" has been discussed by Garrison er al. (1976). They discuss the possibility of reaching full inversion for the cesium 92D3,2and 62S1;2 levels. The dynamical Stark shift, changing during the pulse duration, necessitates the modulation of the frequency with speeds of around 10'' MHz/sec. To obtain such a chirping speed one could, for instance, wobble one of the laser mirrors, e.g., by ultrasonic resonance, or change the index of refraction inside the laser (Gerlach, 1973; Hutcheson and Hughes, 1974; Taylor et a/., 1971).

'' That is. a pulsed laser with modulated frequency.

COLLISIONS WITH LASER EXCITED ATOMS

157

Most of these techniques are still speculative as far as scattering experiments are concerned. But ambitious experimentalists are called upon to apply them in a prosperous future.

IV. Theory of Measurements in Scattering Experiments by Laser-Excited Atoms In Section I1 we have given general formulas interpreting, in principle, scattering processes of any type by any polarized target. The scattering intensity has been given [Eqs. (18)-(2011 in terms of multipole moments %*(k) describing the target (in our case the laser-excited atoms) and those describing the collision process. The collision multipole moments would be prepared in the inverse scattering process starting with originally unpolarized atoms. After having outlined in the previous section how to prepare the W ( k ) , some more details of scattering multipole moments have to be discussed now. Also, methods will be described that allow us to make use of the technique to extract specific knowledge on the dynamics of the scattering mechanism. A. THEPERCIVAL-SEATON HYPOTHESIS: ELECTRONSPIN AND/OR

NUCLEAR SPIN

UNCOUPLING

In their theory of the polarization of impact radiation, Percival and Seaton (1957) supposed that the hyperfine interaction plays no essential role during the collision. Then the state parameters do not depend in any significant way upon the nuclear spin, but only upon the electronic angular momentum. To prevent any misinterpretation we note that this statement does not imply that hyperfine-structure transitions do not occur in a collision. On the contrary, they usually do take place and the respective cross sections may readily be computed once the electronic transition amplitudes . f J M J are known. The product states l J M J I M , ) [IM, refer to the nuclear spin and J M J to the electronic angular momentum] have to be coupled in the usual way to form hyperfine states I (IJ)FM,). The squared amplitude of the latter is proportional to the differential cross section for exciting that state. In physical terms, this nuclear spin uncoupling means that during the time rco, of collisional interaction, the electronic and the nuclear angular momenta are completely decoupled and M, does not change. Only before and after the collision does the nuclear spin I precess around J under the influence of the atomic I * J interaction. This Larmor precession takes a time tHFs2 h/AEHFSdetermined by the hyperfine splitting AEHFS.As long as

I . 1/: Hertel and W Stoll

158

tHFs % tcol the assumption of spin uncoupling is certainly justified (we may take the hyperhe interaction during collision to be of the same order of sec, the magnitude as the atomic A E H F s ) . Since tHFs is of the order of assumption is probably good for all collision processes of practical interest. Similarly, one often may assume that the spin-orbit interaction AEFSis weak and consequently that the electron spin is uncoupled from orbital angular momentum. The spin projection quantum number M , is conserved during collision. In this case only the orbital angular momentum excitation amplitudes f L M have to be known to fully describe the scattering process, being determined by Coulomb forces only. Again, the assumption is valid if tFs

h/AEFs

% tcoI.

The spin-uncoupling hypothesis should be a good approximation for most atomic collisions with light atoms (tFS z lo-’’ sec) at not too low energies. For electron collisions where f c o , z 10- l 5 sec, it should be possible to apply it, although no experimental proof has yet been given. One should remember that tcol is not a uniquely defined quantity and depends on the magnitude of interaction energy one is willing to regard as negligible. As Macek (1976) has pointed out, the influence of weak long-range forces cannot a priori be neglected when polarization effects are investigated, even though the averaged differential cross sections may well be obtained without them. Furthermore, when dealing with heavier atoms or very low energies, the Percival-Seaton approximation may break down. Departures of a purely statistical population have been observed in the 6’P3/2: excitation of cesium by electron impact in the electron volt range (Hertel and Ross, 1968); additional cases have been discussed by Fano (1970). Also, for the description of photodetechment from negative alkali ions near the atomic first ’P threshold (Lineberger, 1975) the spin-orbit interaction will have to be taken into account. Finally, we note that tFS + tco,may be written as

,,’

where a is a typical interaction range and v the relative collision velocity. This is just the Massay criterion for nonadiabatic fine-structure transitions, which may not always be valid at thermal energies. We shall discuss this point in more detail later. The purpose of the following treatment is to factor out the unimportant nuclear spin and/or electron spin parts from Eqs. (12) and (19) for the scattering intensity. The procedure has been described by Macek and Hertel (1974). For the present review it seems appropriate for clarity to go into somewhat more detail.

COLLISIONS WITH LASER EXCITED ATOMS

159

We have to evaluate the averaged tensor matrix elements ((i' I # F ) I i)), where i, i' stands for an appropriate representation of the laser-excited atom. We note that by definition (10) $'I is constructed by angular momentum states [here Ijm)= I FM) since we have well-defined initial hyperfine quantum numbers] in the coupling scheme (FF)k, or more precisely in the scheme [ ( I J ) F ( I J ) F ;k]. On the other hand, one may define state multipole operators constructed by the nuclear coordinates in the (II)kn,,coupling scheme and by the electron angular momentum in the (JJ)kel scheme. The product may be of rank k. Thus, may be operators [ d k e l 1 ( J ) @ ~[~n"'(I)]Jlk' decomposed into the product representation as one would d o in coupling three angular momentum states, by application of the respective recoupling coefficients:

1( ( I J ) F ( I J ) F ;k I (JJ)ke,(II)knu;k )

rF1(F)=

kcl knu

X [ T [ k c " ( J ) @ T[knY'(f)]jlk'

where

[See, e.g., Brink and Satchler, 1971. Eqs. (3.23) and (5.4).] For averaging we may write I i ) in any representation, in particular in the l J M J IM,) representation. Noting that the averaged quantities ( T ~ I ' ) ~ and ~ ( ~ k ~ lare ) ,irreducible ,~ tensor operators too (the state multipole moments) we may write

((JM;IM,[T""'(J) @ Ttk""'(~)]jlk'1 J M J I M , ) ) = [(T'kpl'(J))@ (T[knu'( I))]!'

(72)

since J and I are uncoupled during the collision. Then in Eq. (71) the averages ( ) may be carried out on either side correspondingly. Again, as discussed in Section I1 the state multipole moments (T!~I'(J)) and ( T ~ ~ ~ ' (may I ) ) be replaced by virtue of the Wigner-Eckart theorem by multipole moments constructed from angular momenta (T',kell(J))and ( T',nul(I)). Using Eq. (16) we obtain (T!I(F)) =

1[(JI)F(JI)F;k I (II)knu(JJ)ke,;k ] kdnu

I

x (kelMkn,,M' k q ) ( T!$")( Tbul)

(73)

160

I . V Hertel and W Sroll

where d k ) ( j )is given in Eq. (18). The conservation of nuclear spin during the collision implies that the nuclear multipole moment is the same before and after collision. If no nuclear spin analysis is performed after the collision and thus M ,is distributed isotropically, this implies in our time-inverse view that (mI(1)) = 0 unless k,, = 0. Then the summation Eq. (73) reduces to one term: (r:’(F)) = [ ( I J ) F ( I J ) F ;k I ( J J ) k , , ( l l P ; k ] ~ ( ~ “ ) ( J ) i ~ ( O ’ ( 1 )

I

x (k,,@O kq)(T[,ke”(J)>/(700>= f(O)(a - B ) / W ( T:+(kcol))// l/QR)but not so strong as to totally destroy the atom (l/QR9 fFS 2 tCol2 tel) and OR < v (we also recall that typical pumping times are t,, = so that essentially stationary systems are investigated). Energetically these numbers imply that the natural linewidth r < Edipol(the energy of the induced dipole) and certainly Edipol% EFS< Eel. Recall also that the energy resolution AE,,, of present-day scattering experiments is some meV at best. Thus we also have Edipol< AEr,%.To violate these conditions the radiation power would have to be at least four orders of magnitude larger than currently used. Even higher laser intensities would be needed off resonance. So, to a high degree of accuracy, the atomic basis sets for the unperturbed atoms may be used to describe the atom under the influence of field and scattering. The laser field could in principle have a direct influence on a collision process A + B + h v in one of two ways:

’’

(a) By direct interaction with the collision system AB: A

+ B + hv

. +

(AB) + hv -+ (AB)*+ A‘ + B’

(131)

(b) By disturbing the excited atom (dressed atom): A

+ B + hv

.+

A:,,,

+ B + A‘ + B’

(132)

This is more clearly seen by inspecting the total Hamiltonian:

H

=

H,

+ HAF +

HF(AB1

+ HAB

with

Ho

= HA0

+ HBo +

HF,

(133)

Here H A , , HBo, and H,, are the free Hamiltonians for the atom A, particle B, and laser field F, respectively. The interaction terms are H A F for the

212

I . I.: Hrrtel arid W Stoll

atom-field interaction, HF(AB) for the field-collision system interaction (typically a dipole operator for the interparticle coordinates), and HAB for the collisional interaction. When the laser is not in resonance with an atomic transition, HAF may be neglected. Then one usually first solves the scattering problem HI = H,

+ HA,

H2 = H,

+ HAF

(134) and treats the interaction HF(AB)with the field as a perturbation that may induce transitions within the scattering continuum, the so-called free-free transitions. In contrast, for the resonant case HAF is dominant. One possibly may neglect HF(AH)and first solve the atom-field problem (135) The collision HA, may be treated as a small perturbation. In this case, the atom is a time-dependent coherent superposition of ground and excited state, which has been neglected throughout the present work. A. FREE-FREE TRANSITIONS AND SIMILAR PHENOMENA

First we briefly discuss the free-free transitions, going somewhat outside the scope of the present review.25However, we want to show that HF(AB)is of no importance in our experiment. O n the other hand, several very interesting experiments have been reported recently that should not entirely be excluded in an article dealing with scattering and lasers. The most straightforward case is the free-free transition for electron-atom scattering (or inverse bremsstrahlung). There, in the dipole length approximation, H,(,,, = erE, where r is the electronic coordinate and E the electric field. The electron absorbs or emits a photon in the presence of the atom. Theoretical investigations (e.g., Geltman, 1973, Kriiger and Schulz, 1976) predict cross sections for this process. In Born's approximation it is proportional to the elastic cross section:

where i! is the Compton wavelength of the electron, u the radiation density, E the polarization vector of the field, and AK the electronic momentum transferred in the collision. Thus, the free-free cross section increases as AK and fxl/to4. This has been exploited by Andrick and Langhans (1976) to observe the effect experimentally in e + Ar scattering at around 10 eV. The 2s

Recent progress has been reviewed by Gavrila (1977).

COLLISIONS WITH LASER EXCITED ATOMS

213

Scattered electron energy lev1

FIG.43. Energy loss spectrum of e--Ar scattering, upper line with, lower line without laser beam. Counts per data point are plotted against energy of the scattered electrons. Incident energy is 11.55 eV. The arrows indicate an energy gain (or loss) of 117 meV, corresponding to free-free transitions. From Andrick and Langhans (1976).Copyright by the Institute of Physics.

collision angle was 160" (large AK). They used a C 0 2 laser at 10.6 pm wavelength and an intensity of about 6 x lo4 W crn-'. High-energy resolution and extreme background suppression allows us directly t o see the effect on either side of the elastic peak profile (Fig. 43). Andrick and Langhans (1976) give a preliminary experimental value for y2 = 4 x which is in rough agreement with the theoretical value [Eq. (136)] of y 2 x 1.4 x We immediately see that for the scattering from laser-excited atoms this effect is generally completely negligible. Since y 2 sc l/04, in the visible one would need at least lo7 W/cm2 to observe an influence of less than lop3of the cross section. It is exciting to follow the free-free transition through an electronscattering resonance in the initial or final elastic wave. Andrick and Langhans (1978) have done this for the Ar- resonance in elastic scattering. Figure 44 shows the resonance structure (without laser) in the elastic peak as a function of the incident energy. It has a well-known fine-structure splitting (Weingartshofer et al., 1974). Figure 44 also displays the free-free channel where the resonance is doubled due t o its influence in both the incoming and outgoing channels.26 Otherwise the free-free cross section still remains essentially proportional to the elastic scattering without laser.

*' In Fig. 44 the inelastic free-free channel is observed. Thus the resonance contribution from this outgoing channel is shifted 117 meV to higher energies.

I . K Her-trl and W Stoll

214

x lo3

.-

INCID. ELECTRON

ERGY

-

FIG. 44. Resonances in differential electron scattering from argon at 160". -, ordinary elastic scattering; 0 , resonance in the free-free channel. From Andrick and Langhans (1978).

The situation changes completely when the laser line fits the difference between two resonances. Then the free-free process effects a transition between two quasi-bound negative ionic states and a marked increase in the free-free cross section may be expected. Langendam et al. (1976) have recently reported the first experimental observation of this resonant " freefree absorption by an electron in the field of a neon atom. The Ne levels and Ne- resonances involved are shown in Fig. 45. Langendam et al. detect the process by spontaneous emission of either the 5882 or the 6030 A line of the neutral neon, since the upper 3p level may be populated only due to the 18.95 Ne- resonance. The latter is excited by a freefree transition from the 16.8 eV resonance (Fig. 46). N, laser-pumped Rhodamine 6G and Cumarine 47 dye lasers are used, respectively, to induce the free-free transitions. The laser power is lo4 W in 20 nsec.,' Related processes may occur in heavy-particle collisions. These laserinduced transitions take place between the potential energy curves of the colliding atoms A and B and may be seen to some extent inverse to the "

*' Most recently even multiphoton transitionshave been observed in the free-free e-scattering by Weingartshofer et a/. (1977).

215

COLLISIONS WITH LASER EXCITED ATOMS

I

NeNe

llSo ELECTRONS

7, /-

,=/”’

GROUND LEVEL

FIG.45. The Ne and Ne- levels involved in the investigated transitions (Ne level energies from Schulz, 1973). From Langendam et al. (1976). Copyright by the Institute of Physics.

c2 3

ji I-

z

I 1

I.: I

9

2l

>

3

i,i

0:

I l l

>

0

25

2

t-

5 zI

t Ln

5

I I

F1

I

0:

I

M 15 W

0

53

5891 I

O! 0 5890 5900 5910 5! LASER WAVELENGTH (8)

20

0

ELECTRON ENERGY (eV:

FIG.46. Intensities of Ne I emission at 5882 and 6030 A, induced by optical transitions from an Ne- level at 16.8 eV (see Fig. 45). (a) For a fixed electron energy of 16.8 0.5 eV, as a function of dye laser wavelength. The largest peak occurs at 5897 A; a smaller one lies at 5891 A, i.e., at 2 meV separation. Dots and crosses refer to separate data runs. (b) For a fixed laser wavelength of 5897 A, as a function of electron energy. The 3P,(3p) excitation function (solid curve), obtained as the total 6030 A emission rate without gated counting, serves as a calibration of the energy scale and an indication of the energy width of the beam. From Langendam et al. (1976). Copyright by the Institute of Physics.

I . V; Hertel and W Stoll

216

photon emission in the "spectra of colliding atoms" (Gallagher, 1975). A typical process is A + B + (n)hw--* A B*

+

where ( n ) b (n = 1, 2 , 3 , . . .) is not in resonance with the excitation energy of B* for infinite separation R , . The energy difference of the AB and AB* interatomic potentials becomes resonant to n h o at the smaller internuclear distances formed during collision. Recent theoretical treatments, e.g., by Kroll and Watson (1976) and by Lau (1976a), underline that high laser intensities are needed for these transitions. A similar mechanism for a laserinduced atom-atom transition is A* + B

+ h v + A + B*

where the difference energy between A* and B* excitation at large internuclear distance R , nearly equals hv. Since in the molecular picture the transition is resonant for a large range of internuclear distances, it should have much higher cross sections than the previous case. Such processes have recently been reported by Harris et al. (1976) and Falcone et al. (1977) in the Sr + Ca system at a laser intensity of 5 x lo5 W/cm2. Laser-induced energy transfer to both the calcium 4p2 ' S and 5d'D states has been observed detecting the fluorescence from the excited calcium states:

+ + Ca(4s2 'S) + hv(4711 A) = Sr(5s2 'S) + Ca(5d'D)

~ r ( 5 p ' ~ O+)ca(4s2 's)+ hv(4977 A) = sr(5s2 's) ca(4p2 's) Sr(5p'P')

The term scheme is shown in Fig. 47. The experimental fluorescence signal as a function of the wavelength of the transfer laser light shows a maximum at around the R = 03 wavelength A = 4976.8 A, with a FWHM of 14 cmAgain, the theoretically predicted cross section depends linearly on the laser x 9 (W/cm') cm2 = 5 x intensity (McGinn, 1969) and is oc = 9 x 10- " an2.The experimental value of 9 x lo-'* cm2 is in good agreement. We conclude that these effects may be completely discarded in the experiments otherwise discussed in the present work.

'.

B. COHERENT SUPERPOSITION OF GROUND AND LASER-EXCITED STATES IN APPROXIMATELY RESONANT ATOM EXCITATION Neglecting the interactions HF(AB) that have been discussed in the previous section, we may now concentrate on the distortions an atom experiences when it is excited with a strong, nearly resonant laser radiation. The problem is closely related to saturation or power broadening, which has been discussed in Section III,A,2. In fact, the behavior of the atomic ensemble in the

COLLISIONS WITH LASER EXCITED ATOMS

217

Co I

Sr 1

FIG.47. Energy level diagram for laser-induced transfer from strontium 5p'P0 to calcium 4p2 ' S . From Falkone et a/. (1977).

laser field [H, = H, + HAF,Eq. (1391 may be fully understood in terms of optical Bloch equations. As we have shown, the latter also yield the power broadening. The full atomic density matrix B contains in general nondiagonal terms connecting ground and excited states. These nondiagonal terms have been neglected throughout the paper and only the diagonal excited state part (re has been used. A proper treatment would have to contain the total a(t)as well as the scattering density matrix p(t). The experiment then essentially measures a time-averaged Tr B * p. Several serious problems in defining the basis sets prevent us at present from doing so. Nevertheless, we are concerned about the neglect. A simple but appropriate view is in terms of wave functions. As well known, in the presence of the laser field, the atom is a coherent, timedependent superposition of ground and excited state:28

I + b,,(t)(2),

YAY= a&) 1)

v = 1, 2

(137) where for simplicity we just treat the atom as a two-level system with a stationary unperturbed ground state 11) and an excited state (2). The circularly pumped sodium atom may be taken to represent such a system to a good approximation: ( 1 ) = I F = 2, M = 2) and 12) = ( F = 3, M = 3). The collisional interaction may then be treated in the usual perturbative approximation. However, a fully time dependent treatment has to be applied. 2 8 The two solutions 'PA,(?) and YA,(t)correspond to different initial conditions: ~ ~ (= 01, ) b,(O) = 0, and a,(O) = 0, b,(O)= 1.

218

I . I.: H w r l and W Stoll

The influence of coherent superposition of states on the collision may not be discarded by arguing that during the collision the resonance condition for the photon does not hold. The scattering process is in any case a small perturbation to the atomic ensemble as such, whose time dependence is dominated by the near-resonant laser field ( HAB) < ( HAF). Thus the timedependent wave functions Y A , ( t(there ) are two orthogonal ones) have to serve as a basis set in a proper quantum mechanical treatment of the scattering problem. We will see, nevertheless, that the phenomena arising from coherent superposition will hardly be observable experimentally unless the fields are very large. Hahn and Hertel (1972) were the first to attack the problem and have discussed the experimental implications for the scattering by laser-excited sodium 32P. Gersten and Mittleman (1976) have used an essentially identical treatment and applied it to derive an elaborate formula for electron scattering by laser-excited hydrogen. No discussion is given by the latter authors on the experimental phenomena to be expected; in particular, no numerical estimates of the importance of coherent-state superposition on scattering by laser-excited atoms. Thus we essentially follow Hahn and Hertel (1972), who used a fully quantum mechanical formulation of atom, field, and scattering process. They neglected spontaneous decay. The field quantization is an elegant but unimportant formalism in this case and identical results are obtained in a semiclassical treatment of the laser field (Hahn, 1972). The rotating wave approximation (see, e.g., Paul, 1963; Brunner er al., 1964; or Paul, 1969) is used to obtain the time-dependent atomic wave functions Y &). It is a good approximation for near-resonant excitation (laser frequency w % w2 - wl, the resonance frequency), possibly with a The coupling of the atomic levels by the field small detuning Aw e w 2 - q. is given by K = D,, X [ A , 1/2h = *OR, where D12 is the atomic dipole matrix element, d',, the (real) field amplitude, and OKthe Rabi frequency (see Section III,A,3). The rotating-wave approximation gives a periodic behavior of the timedependent amplitudes a$), b,(t) with essentially

I b,(r) l2

K sin2 2 Wr

where W 2 = K 2 + (Ac0/2)~< w2

The squared amplitudes I a(t)12, I b(r)l2 give the probability of finding the atom in the ground or excited state, respectively. This is illustrated in Fig. 48. K 2 / W 2gives the maximum, and

n,/n, = * K 2 / W 2

(140)

COLLISIONS WITH LASER EXCITED ATOMS

219

FIG.48. Timedependence of the probabilities l u > ( t ) l ’ , Ib,(t)l’ to find an atom i n the excited and ground state, respectively. Curve 1 represents lu2(t)1’ = Ibl(t)l’, while curve 2 is lu,(t)12 = I b2(t)I2.K Z / W Zgives the maximum probability of finding the atom in the excited state, having been originally in the ground state. From Hahn and Hertel (1972). Copyright by the Institute of Physics.

is the time-averaged fraction of upper state p~pulation.~’For resonant excitation, K = W and n , / n , = i,as it must be. The averaged ground state population is

1 - K2/2 W2)

(141) The rotating wave approximation allows an interpretation of these oscillations as splitting of the (nondegenerate) atomic levels, arising from the modulation as side bands. A schematic is given in Fig. 49. These split levels again exist only in a coherent superposition. They cannot be observed in the total fluorescence. When w is tuned through resonance, the splitting just determines the saturated linewidth. They may, however, be seen when the fluorescence light is frequency analyzed as three distinctive components: the Rayleigh scattering w, the anti-Stokes fluorescence w 2W (which is near w2 - w1 for larger detuning), and the Stokes shifted fluorescence w - 2 W. Experiments have resolved these three components (Shouda et al., 1974; Wu et al., 1975). In addition, Carlston and Szoke (1976a,b) and Carlston et ul. (1977) report on collision induced fluorescence under off-resonant conditions (Aw# 0). The rotating-wave approximation even allows us to estimate the relative population ” of these virtual levels. Hahn and Hertel derive a time-dependent close-coupling system [without exchange, which is included by Gersten and Mittleman (1975)l for the collision problem, investigating transitions from yAl-+yAland v~~ --* yA2. The resulting scattering amplitudes would be explicitly time dependent. The (Ilg/H0)(

+

“

”) It should be noted that Eq. (140). which neglects spontaneous emission, ditTers numerically somewhat from the stationary solutions of the optical Bloch equations [Eq. (95)] even in the limit of r = I/T + 0. The numerical consequences of the present discussion therefore have to be modified slightly when spontaneous radiation decay is included.

I . I/: Herrel and W Stoll

220

FIG.49. Schematic illustration of the atomic “level splitting” for a two-state system. The level splitting 2W is small compared to the level distance m2 - wI. A positive detuning AUJis assumed,where w 2 - (ul 2 w;criis thefield frequency. From Hahn and Hertel(l972). Copyright by the Institute of Physics.

experiment time-integrates the scattered wave function. When this is done, a set of delta functions arises describing different energies klm after collision, when the initial energy was homo. The close-coupling system could be solved stationary. Thus, the coherent superposition of states leads to inelastic transitions with a typical energy loss or gain. All transitions among the four levels indicated in Fig. 49 are possible. In addition, multiphoton transitions could occur for very high fields. The close-coupling equations given by Hahn and Hertel allow in principle the following final energies: tic,,,, = hCOmo

+ h[(mo- m)w - 21W]

(142)

where mo refers to the initial and m to the final number of photons in the laser field, and the level-splitting transfer is given by 1 = 0, L 1. For not too high intensities, mo - m = 0, k 1 and, since W < w, we may distinguish the following process: quasielastic

hlm = h~~,,,,,

superelastic

hl,,, = h ~ ~ ,+, ,ho ~

inelastic

1 +0 klm hr-;o,, - ho I

or

+2Wh (143)

=

where ho is nearly the atomic excitation energy. Hahn and Hertel(l972)have given expressions of the differential scattering cross sections for the various processes in first Born approximation.

22 1

COLLISIONS WITH LASER EXCITED ATOMS

TABLE IV DIFFERENTIAL CROSSSECTIONS FOR PARTICLE SCATTERING I N RADIATION FIELDS, CALCULATED I N THE FIRSTBORN APPROXIMATION^

Transition*

‘lrn

1’ 4 1‘; 1“ -+ 1’’ 2’ 4 2’; 2” -+ 2“

2“ -+ 2’; I ’

4

1”

2’ -+ 2”; 1” -+ 1’ 2’ -+ 1”; 2” 4 1‘

2” -+ 1” 2’-+ 1’

I ’ -+ 2”; 1” -+ 2’ 1‘ -+ 2‘

l”42”

T21

EOm0

-w

+2 w

$(I

I2

- & ) 3 ~ ~ 2 1 ~ 2

hi,,,is the energy after collision and T j are Born amplitudes. The designation of the various transitions is to be understood in terms of Fig. 49.

Their results are given in Table IV, where T,, and T22denote the Born elastic scattering amplitudes for pure ground and excited state scattering, and T,, the inelastic and T,, the superelastic Born amplitudes, respectively. Characteristic are the interference terms in the quasielastic scattering, which reflect the coherent superposition of ground and excited state. The question is whether this interference could actually be observed in the experiment. Recall that W is essentially the Rabi frequency (for resonant excitation). It is typically lo* Hz for the intensities used to excite the atoms. This corresponds to an energy splitting of 2 peV, which is hardly observable with any conventional scattering techniques resolving 2 meV at best.

I.

222

I/:

Hortc.1 and W Stoll

Thus one has to sum over the unresolved energy losses. Then one obtains the following cross sections:

superelastic

inelastic

da K2 dQ- 2W2

--

do

dR

~

[ T 1 2 \ 2 h

(144)

ko 2

kolno+1 ko

processes

If we recall from Eqs. (140) and (141) that the average population of the ground state is 1 - K 2 / 2W 2 and of the excited state is K 2 / 2W 2 ,these cross sections are just the ones expected if nothing were known about the level splitting and the consequent interference terms. Thus, neglecting coherence terms used throughout the paper is correct in a Born's approximation. We believe, however, that this outcome is typical for any set of scattering amplitudes that do not vary significantly over an energy range of some peV. It should be possible to obtain such a theorem in a more rigorous way, using a density matrix formulation for the laser excitation as well as for the scattering, including the nondiagonal terms. Hahn and Hertel (1972) and Gersten and Mittleman (1976) have neglected spontaneous decay. while in a new paper of Mittleman (1976) it is included. This is advisable, especially under the circumstances relevant to the present paper, where the spontaneous lifetime 7 is of the same order of magnitude as R R .The difficulty with spontaneous decay is that the atomic wave functions become nonnormalized unless a fully quantized treatment of the field arid spontaneous decay is used. On the other hand, the optically excited density matrix is essentially stationary and one could obtain stationary solutions (including the nondiagonal terms) from the optical Bloch equations. Again, a density matrix formalism should be used for the whole problem. However, it is p!ausible that interference terms as given in Table IV should rather have the tendency to wash out. Equally, instead of a pure splitting by W , a broadening of the lines is to be expected when T = n/R,. In summary, we may safely assume that neglecting coherence between ground and excited states does not affect the experimental analysis given in the present review. Possible observation of the laser-induced splittings in collision processes could be made in connection with hyperfine-structure transitions that have the same energy separations. Also, for very high laser fields, six orders of magnitude above the ones presently used, the splitting may become resolvable with scattering techniques.

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223

One may also observe multiphoton transitions. As Hahn and Hertel (1972) show, these may become important when the collision time tco,is comparable to n/QR.For these high powers the free-free transition probabilities discussed in Section V1.A may also be of comparable magnitude.

VII. Conclusions It will not have escaped the attention of the reader that the review presented here was not meant t o give a comprehensive survey on all scattering experiments involving lasers. Whole areas, such as excitation of molecules, high rydberg states, or gas cell experiments, have been left out deliberately in order to keep the size of the article within reasonable limits. It has been the intention of the authors to provide a solid basis for planning, preparing, and performing future scattering experiments with laserexcited atoms in crossed beams. Particular attention has been paid to the possibilities of detailed investigations on scattering dynamics, which are typical of the target preparation by laser optical pumping. It is our firm belief that these possibilities open up a novel field in atomic collision physics, whose first successful performance we have just witnessed. We hope we have illustrated this by the considerations of how to test particular theories and by the experimental examples presented. We also have shown how to pose new critical questions on atomic interaction dynahow to obtain answers without the necessity of mics and-possibly-even going into detailed computations. Typical applications are the study of the range of validity of the Born or Glauber approximation, the importance of spin coupling or uncoupling in a collision, or the critical discussion of angular momentum quantum numbers to be conserved in a collision as predicted, e.g., by the so-called elastic” theory in heavy-particle scattering. Thus the new methods may provide fresh insights into old problems. In addition, we have illustrated how a quantitative evaluation of the measurements has to be performed. The language of multipole moments used throughout the review has proved to be a valuable guide thereby. Even though it looks complex at first sight, it gives us a very direct perceptual view of what happens to an atom during collisions and allows intuitive interpretations of many experimental findings. ”

ACKNOWLEDGMENTS We would. of course. like to stimulate further work in this flourishing field. And it is a particular pleasure to us to thank all experimental and theoretical groups actively engaged in these problems for supporting us with their experimental material. numerical data, and helpful com men ts.

224

1. VI Hrrtsl and W Stoll

R. Diiren, especially, has been a patient instructor in the field of elastic heavy-particle collision and C. McDowell and D. Moores have continuously supported us with their theoretical results on scattering amplitudes. One of us (I.V.H.)wishes to take the opportunity to give his special thanks to all those who are or were members of our group in Kaiserslautern during the past years. They have in one way or another contributed to part of the work reported here: S. Azarkadeh, L. Hahn, H. Hermann, H. Hofmann, W. Miiller, W. Reiland, K. Rost. A. Stamatovic, W. Sticht, and last but not least my co-author W. Stoll. Without their endeavors we would not have been able to contribute to this fascinating topic in atomic physics. H. Hermann, H. Hofmann, and W. Reiland were particularly helpful in reading the manuscript and suggesting improvements. Finally, we wish to acknowledge gratefully the continuous financial support from both our university and the Deutsche Forschungsgemeinschaft.

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SCA TTERING STUDIES OF RO PA TION AL AND VIBRATIONAL EXCITA TION OF MOLECULES M A N F R E D F A U B E L and J . P E T E R T O E N N I E S Max-Planck-lnstitut ,fur Stromungsforschung, Gottingen, West Germany 1. Introduction A. Scope of

ew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Methods for Studying Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . C. Simple Models of Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 230 23 I 234

A. Introductory Remarks . . . . . . . . . . . . . . .

C. Potential Models

........

111. Inelastic Scattering Theory

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

A. The Quantum Mechanical '* Close-Coupling" Method . . . . . . . . . . . . . . B. Approximate Methods.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Some Computational Results . . . IV. Experimental Met A. Basic Techniques .............. B. Design Considera ent . . . . . . . . . . . V. Recent Experimen A. Survey of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Scattering from H, Molecules ..................... C. Scattering from Heavier Linear Molecules . . . . . . . . . . . . . . . . . . . . . . . D. Scattering from the Alkali Halides . . . . . . . . . . . . . . . . .... VI. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

248 248 252

257 259 274 280 296 303 307 308

I. Introduction This review deals with the following inelastic scattering process involving an atom (or ion) A and a molecule M: A

+ M(ji, m j i ,ni)

E; 9.9

A

+ M(jf9

nf)

+ AEinel

(1)

where j and m denote the rotational angular momentum quantum numbers, 229

230

M . Faubel and J . P . Toennies

n the vibrational quantum number, 9 and cp the polar and azimuthal scattering angles, respectively, and AEine,the internal energy change, which for single-quantum transitions ranges from eV for a j = 0- 1 transition in CsF to 0.5 eV for a vibrational transition in H 2 . The collision energies E dealt with will lie in the range between 0.1 and 10 eV. For gases of atoms and molecules with closed outer electronic shells the energy transfer processes (1) determine the rate at which equilibrium between the external (translational) and the internal molecular degrees of freedom is established. These processes are therefore of fundamental importance for understanding the bulk kinetic properties of gases. Examples are the transport coefficients, relaxation times in sound waves, nozzle expansions, and shock waves. In chemically reactive systems, usually involving open-shell atoms or molecules, the reaction rates will generally depend on the internal states of the molecules. Thus the rate of chemical processes may also be determined and influenced by processes (1).Inelastic collisions also compete with radiative energy transfer and thus affect linewidths and line shifts, and in some cases they determine the intensities of spectral lines. In addition to their fundamental importance these processes have recently become of interest in connection with laser technology and laser isotope separation and in interpreting the spectral lines of interstellar molecules observed with giant radiotelescopes. In all three cases further advances are limited by a lack of a more detailed knowledge of quantum state to state inelastic rate constants. A.

SCOPEOF THISREVIEW

About ten years ago virtually all our knowledge on energy transfer came from bulk measurements of the properties mentioned above. Theoretical models were primitive: rotational excitation was treated using semiclassical perturbation theory, while theories of vibrational relaxation were all based on a one-dimensional collinear collision model, in which the interaction potential .was estimated from the addition of atom-atom potentials. Since that time a large number of experimental results on the detailed scattering cross section for Eq. (1) have become available. Simultaneously exact quantum mechanical techniques have been developed to the extent that the measured cross sections can be compared directly with a priori theory. Both developments have had a great impact on our understanding of the elementary collision process. This review starts with an overview of the interrelationship between the various experimental approaches and the newly developed theory. Some simple models that provide insight into many of the phenomena of inelastic

SCATTERING STUDIES OF EXCITATION MOLECULES

23 1

scattering are dealt with at the end of this section. The following two sections are devoted to a more detailed discussion of the theoretical approach. The emphasis is on the capabilities and limitations especially as they pertain to the experiments to be discussed later. In the fourth section we survey the experimental methods and discuss the factors influencing the resolving power and intensity in an energy loss experiment. In the final section a number of experiments involving collisions between various scattering partners and H,, N,, CO, and CO,, N,O, and the alkali halides are discussed in detail. In each case an attempt is made to show how the transition probabilities depend on energy and scattering partners. Wherever possible the results are interpreted in terms of the most recent theory. Shorter reviews describing some of the experiments on the system Li +-H,, carried out in our laboratory have been presented by both of us (Faubel, 1973; Toennies, 1974b). Rotationally and vibrationally inelastic scattering has also been dealt with in sections of recent reviews by Doering (1973), Koski (1975), Pauly (1973), and Toennies (1974a). This review is designed to complement a recent shorter review by Toennies (1976), which contains all the recent literature up to the winter of 1975, while leaving out most of the experimental details. B. METHODS FOR STUDYING ENERGY TRANSFER Figure 1 summarizes the theoretical approach to the problem of energy transfer and also shows the contributions from various types of experiments. In the first step of the theoretical pathway, quantum chemical methods are used to calculate a potential hypersurface. The potential hypersurface is a function of the three Jacobi coordinates: R, the distance between the center of mass of BC and A ; r, the bond distance of BC; and the angle y between R and r (see the insert in Fig. 1). In the second step scattering theory is used to calculate the differential cross sections da"'(E; 9, cp)/do for elastic, and rotationally and vibrationally inelastic scattering. The scattering angles 9 and cp are defined in the insert of Fig. 1. E,, is the collision energy in the center of mass system. In a first approximation, the R dependence of V ( R ,r, y ) determines largely the angular distribution of the elastic (or total) scattering. The y dependence determines the torque exerted on the molecule and thereby the probability of rotational excitation, while the r dependence determines the collisioninduced forces along the bond direction and the probability of vibrational excitation. The integral inelastic cross section oi+'(E) is obtained from doi"( 9 ) / d o by integration over all angles and therefore only depends on the collision energy. If a local Maxwellian velocity distribution is assumed, a

M . Faubel and J . P . Toennies

232

Theoretical pathway

-Expts.

Quantum chemistry

s

t IpTl-'

-

bulk relaxation

FIG.1. Schematic diagram showing on the left the important steps in the theoretical pathway leading from the quantum chemical calculation of potentials to bulk relaxation times. The arrows indicate the direction in which the calculation can be carried out. The experimental contributions are indicated at the right. The insert shows the coordinates used to describe the potential and the differential cross section.

rate constant for the bulk behavior can be obtained by integration of C J ~ + ~ ( E ) over the energy distribution in the gas (Shuler et al., 1969): 31.7

k"'(T) =

:i

dEa"'(E) exp( -

&)

(2)

where p is the reduced mass and kB the Boltzmann constant. In the final step the rate constant is related to the experimentally observed relaxation time z, which is usually defined by x(t)/x(t = 0)= e-'Ir

(31 where X is some bulk property of the gas. Equation (3) is only correct for the limiting case of a two-level system or for the pure vibrational relaxation of harmonic oscillations (Clarke and McChesney, 1964, p. 116). The two-level case is a good approximation for the slow vibrational relaxation of hydrogenic diatomic molecules (e.g., Hz , HCI) where the much faster rotational relaxation can be neglected. Furthermore if low temperatures are assumed, then

( P z ) - l z ki'f(T)pi(T) (4) where P is the pressure and pi the density of molecules in state i. More complicated expressions corresponding to a multimode relaxation result if

SCATTERING STUDIES OF EXCITATION MOLECULES

233

higher temperatures and more states are involved (Oppenheim et al., 1967). As mentioned in the introduction, significant advances have been made in the last few years in calculating realistic cross sections. This has made it possible to carry out all the steps in the theoretical pathway for a number of simple systems such as the vibrational relaxation of He-H2 (Rabitz and Zarur, 1975; Verter and Rabitz, 1976) and the rotational relaxation of H2-H2 (Rabitz and Lam, 1975). The “bottleneck” in extending such calculations to other systems is the lack of data on the potential hypersurfaces. Thus experiments are needed to acquire information on hypersurfaces and to test approximate theoretical models for calculating potentials. As indicated in Fig. 1, essentially three different types of experiments are available: bulk relaxation, laser-excited . spectroscopy, and beam-scattering experiments. Bulk studies of energy transfer are reviewed in Gordon et al. (1968), Gordon (1973), and Amme (1975). Laser experiments have been dealt with by Weitz and Flynn (1974), Ormonde (1975), Smith (1976a), in several articles in the book edited by Levine and Jortner (1976), and briefly in Toennies (1976). In bulk relaxation experiments ( P t ) - ’ is measured directly either in a sound wave (Jonkman et al., 1968a,b,c),gas expansion (Gallagher and Fenn. 1974), or in a shock wave (Dove and Teitelbaum, 1974). The method is limited to relatively slow processes, such as vibrational relaxation or rotational relaxation of hydrogenic systems at low temperatures. In their theoretical calculation ofvibrational-rotational relaxation in He-H, , Rabitz and Zarur (1975) had to include up to 66 rate constants at 500°K. Thus information on a single state-to-state rate can only be obtained at temperatures low compared to the energy level (Jonkmann et a/., 1968a,b,c). The large number of spectroscopic experiments usually involving the use of lasers can be grouped into two categories: those in which the net transfer rate into or out of a given state is measured and those in which the rate between two sets of states is observed. With but a few exceptions expenments of both categories are limited to infrared active molecules and vibrational excitation. Experiments of the first category include the spectrophone (Read, 1967-1968). laser Raman excitation (Audibert ut a/.. 1976). laser infrared fluorescence decay (Ormonde, 1975; Weitz and Flynn. 1974). Examples of experiments of the second category include microwave and infrared double resonance studies (Oka, 1973; Freund er al., 1973; Hinchen, 1975; Hinchen and Hobbs, 1976)and the infrared luminescence experiments (no laser involved) of Ding and Polanyi (1975). Recently a number of investigations have been performed in which the rate for single-collision state-to-state transitions are observed. These include the infrared double-resonance lamb dip fluorescence experiments of Meyer and Rhodes

234

M . Faubel and J . P. Toennies

(1974) and Bischel and Rhodes (1975), the infrared laser-induced fluorescence experiments of Lang et al. (1976), and the visible laser fluorescence spectroscopy of electronically excited states (Ennen and Ottinger, 1974; Steinfeld, 1972). Despite the considerable detail available from these experiments there is evidence that the information content of state-to-state rate constants and state-to-state integral cross sections as a function of energy is limited (Procaccia and Levine, 1975,1976). Thus there is now some question whether these rate constants will prove useful in studying the details of potential hypersurfaces or the dynamics of collisions. In the scattering experiments to be discussed in this review the differential cross sections are measured directly. Since the results are not integrated over the angular distribution they contain considerably more information on the potential and collision dynamics. The major drawback of scattering experiments compared to other experiments is that they are much more difficult to perform.

C. SIMPLE MODELSOF INELASTICSCATTERING A convenient expansion of the interaction potential is

and

Here re is the equilibrium internuclear distance and x = r - re. As we shall see in the next section, the spherically symmetric term uo is usually larger than the higher-order terms (see Fig. 5). Furthermore the potential varies only slightly with r over the range of mean vibrational amplitudes at room temperature (see Fig. 4).' Thus in first approximation V ( R , r, y) 'v uo(R, re), only elastic scattering can occur. The elastic scattering from a spherically symmetric potential is now very well understood. For recent reviews see Buck (1975), Pauly (1975) Toennies (1974a), and Bernstein (1966). Figure 2 shows in a composite diagram the potential Vo(R)(upper left), some typical trajectories seen from above (left center), and the resulting classical and quantum mechanical angular distributions (right side). As shown in the upper left corner, the potential well depth is designated by E and its location by R,. Both the classical and quantum mechanical differential cross sections are characterized by two angular re-

' Typical values of mean vibrational amplitudes at T = 298°K are between 0.09A (H2) and 0.03 A (N,) (Cyvin, 1968).

SCATTERING STUDIES OF EXCITATION MOLECULES

235

FIG.2. Schematic diagram showing the spherical symmetric part of a typical atom molecule potential (upper left), classical trajectories for various impact parameters (left center), and the classical and quantum mechanical differential cross section (right). The differential cross section is characterized by two regions separated by the rainbow angle 9,. Most trajectories leading to smaller angles probe only the attractive potential, while the large-angle trajectories probe mostly the repulsive potential. The quantum interferences pattern can be attributed to an interference between trajectories of type 1 and 2 leading to the slow undulations, and to an interference between these trajectories and those of type 3 leading to the fast undulations.

gions. At small angles the cross section is largest and decreases rapidly with increasing angle. The trajectories arriving at small angles have large impact parameters b and probe only the long-range potential. The magnitude and angular dependence of the cross section depends therefore on the strength and range of the attractive potential. With increasing angle the differential cross section increases to an intermediate maximum, which is called the “ rainbow.” The angular location of the classical rainbow 9, is given to a good approximation by 9, ‘V AEIE (6) where A = 1.92 f 0.15 for a wide range of different potential models (Hundhausen and Pauly, 1965). The quantum interference structure seen in Fig. 2 can be attributed to an interference between trajectories of type 1 and 2 leading to the slow undulations, and to an interference between these trajectories and those of type 3 leading to the fast undulations. All trajectories arriving at angles greater than 9, have small impact parameters and penetrate into the repulsive part of the potential. Thus at large scattering angles

236

M . Faubel and J . P. Toennies

the differential cross section depends almost entirely on the strength and shape of the repulsive potential. Reasonably good agreement between the classical and angle-averaged quantum differential cross sections as indicated by Fig. 2 is found for heavy particles (reduced mass greater than 20 amu) and also for lighter particles at higher energies provided that R,/1 >, 20, where 1= h/p is the de Broglie wavelength. Neglecting the angular anisotropy seems to be an acceptable first-order approximation, as indicated by the observed angular distributions for atomand ion-molecule scattering. [See Kuppermann et al. (1973), Bickes et al. (1975), and Section V.] Since the inelastic processes are not resolved, the observed cross section is the sum over all inelastic and the elastic differential cross section, and is called the total cross section. The total cross section shows in all cases a primary rainbow, which is partially smeared out by the averaging over different orientations (Barg et al., 1976).The fast undulations and secondary rainbows (see Fig. 2) are usually not observed since they are quenched by inelastic processes (Cross, 1970). For some scattering partners additional features attributed to the potential anisotropy are observed in the differential cross section (Giese and Gentry, 1974; Buck et al., 1975a). For a potential that is anisotropic and depends on r there will be a finite probability for the occurrence of an inelastic transition in the course of the trajectory. If we assume that the classical trajectory is not disturbed by the transition, then the transition probability can be estimated from first-order time-dependent perturbation theory,

where the integral is over the classical trajectory with t = 0 at the classical turning point (point of closest approach), V ( b ,t ) is usually set equal to the difference between the full potential and the symmetric part uo , and b is the impact parameter. According to Eq. (7) the transition probability is proportional to the square of the magnitude of V'. In addition, the transition probability also depends on the exponential term, which oscillates as a function of time. This oscillatory term is modulated by the preexponential factor, which will be appreciable only during the collision. Thus according to Eq. (7) the transition probability will be large if tCOll/T

51

(8 1

where z = h/AE is the effective period of the excited motion. This resonant condition is also referred to as the Massey criterion (Massey and Burhop, 1952, p. 441). Collisions in which tcoll/t % 1 are frequently referred to as adiabatic, and collisions with tcol,/z 5 1 are called diabatic or nonadiabatic.

231

SCATTERING STUDIES OF EXCITATION MOLECULES

6

4

Relative collision energy [eV]

FIG.3. The energy-dependent collision times tc,,ll are compared with the characteristic periods of rotational and vibrational motions of some typical diatomic molecules. The collision times were estimated from r , , , = 6 A/g, where g = (2E/p)”* is the relative velocity for the indicated reduced mass p. The rotational period is estimated from t l M= h/B and the vibrational period from rvib= v - ’ . The horizontal lines start at the energy of the first excited state. If the collision time is less than the period of internal motion then the excitation is probable.

Figure 3 compares collision times with characteristic periods tor the rotational and vibrational motions of molecules, for a wide range of energies. The collision times shown as the diagonal solid lines were calculated from tEo,,= 6 A/g, where g was calculated from the collision energy using the indicated reduced mass p. The rotational period was calculated from ,z, = h/B, where B is the rotational energy constant, and the vibrational period from t,ib = l/v, where v is the vibrational frequency. The horizontal lines for the rotational and vibrational periods start at the energy required to excite the first excited state from the ground state. According to the Massey criterion, the transition is probable if the appropriate diagonal solid lines lie below the horizontal dotted line. Figure 3 thus indicates that rotational excitation is highly probable at low energies and that vibrational excitation is only probable at collision energies greater than about 1 eV.

238

M . Fuirbel arid J . P . Toennirs

The Massey criterion has implications for the experiments: The ratio of periods can be written as t c o , , / t.v 1 AEIE (91 where 1 = pbg/h is the orbital angular momentum quantum number. Since I is proportional to the impact parameter, it depends on the scattering angle. In general, for not too large scattering angles 1 will be fairly large (250) so that a large transition probability implies a small value of AE/E. This is an unfortunate circumstance, since it makes experiments difficult. For most scattering experiments both a large transition probability and a large value of AE/E are important prerequisites for detection of quantum transitions. Conservation of angular momentum and energy places an additional angle-dependent constraint on the transition probabilities. In most of the experiments discussed in this review, the initial rotational angular momentum quantum number ji will be small. If j i =0 then the total angular momentum Ji before the collision is equal to the orbital angular momentum l i . Since the anisotropic terms in the potential are usually small the direction off = (p/h)R x g will not change significantly. Moreover, the initial and final orbital angular momenta are usually parallel, and hence j, d

li

This constraint suggests that rotational excitation will not be probable in near-central collisions with small impact parameters. Vibrational excitation does not require angular momentum but, because of the large level spacing, may be limited by the available kinetic energy. The available kinetic energy may be estimated from the square of the velocity component along the direction of R upon the impact of two hard spheres (Shuler et al., 1969): (11) Ecenters = Ecd1 - b2/R2) By substituting the classical deflection for hard spheres b = R, cos 9/2, we get AEvi,, d EcenlerS = &M sin2 912 (12) implying that near threshold, vibrational excitation will be most probable for large scattering angles. This result is consistent with classical (Barg et ul., 1976) and quantum mechanical (McGuire, 1975) calculations of vibrationally inelastic scattering using a realistic potential (see also Fig. 9).

11. Potential Hypersurfaces A. INTRODUCTORYREMARKS Despite very significant advances in the last few years very little is known about atom-molecule potentials, especially about their anisotropy and

SCATTERING STUDIES OF EXCITATION MOLECULES

239

dependence on the molecular bond length. The major source of information is quantum chemical calculations. Recently scattering measurements of integral total cross sections with polarized molecular beams (Reuss, 1975) and the careful analysis of pressure-induced infrared spectroscopy (Le Roy and Van Kranendonk, 1974) and line-broadening data (Shafer and Gordon, 1973) have also yielded results on the anisotropy of the H,-rare gas potentials. For related work on HCl-Ar see Dunker and Gordon (1976). Several fundamentally different types of interactions are expected, depending on the electronic states of the partners. These are summarized in Table I. When one of the partners has nonzero orbital angular momentum, the interaction can remove the degeneracy. Several hypersurfaces, with an energy spacing that usually increases with decreasing R, will result. At some point the energies of two electronic states with different vibrational levels can cross and an electronically nonadiabatic transition corresponding to a local breakdown of the Born-Oppenheimer approximation can lead to vibrational excitation. The probability can be large since no change in translational energy is involved. Nikitin (1975) and Smith (1976a) have presented evidence that these so called vibronic transitions can lead t o a considerable increase in the transition probabilities. Potentially reactive systems also are expected t o have enhanced vibrational transition probabilities (Smith, 1976a). Beam experiments on H+-H,, discussed in Section V,B, show the expected enhanced vibrational excitation. The effect of different electronic states on rotational excitation is not known at the present time. Since most of the theoretical work has been on closed-shell systems the remaining discussion is restricted to them. In all calculations the Born-Oppenheimer approximation, the validity of which has been tested by spectroscopic experiments, has been used.

CHEMICAL CALCULATIONS OF B. AB INITIO QUANTUM POTENTIAL HYPERSURFACES~ In discussing potential calculations it is useful to distinguish between three different regions of R : (i) R c R , . At small intermolecular distances the self-consistent field (SCF) method is now generally considered to be a reliable approximation for describing the repulsive potentials3 This is related to the fact that the SCF approximation is capable of accounting for exchange and classical electrostatic effects, which predominate in this region.

* The standard methods for calculating potential hypersurfaces are described in the reviews of Certain and Bruch (1972) and Balint-Kurti (1975b). Recently, however. Reinsch and Meyer (1976) have found that intraatomic correlation effects not included in the SCF calculation make a 30'2, contribution t o the repulsive potential of Ne-Ne at R , .

TABLE I

POSSIBLE ELECTRONIC STATES IN ATOM (ION~DIATOM INTERACTIONS States during collision

(Smith, 1976b)

is b

Examples

Electronic states of partners Linear geometry

Bent geometry

(Cwv, y =0)

(Cs, y # 90, # 0)

Atom (ion)

Molecule

'S

'I:+

'I:

'A'

zS

'Z

21:

ZA'

'Z

'S

+

+

+

,A' +

2n

2A, + ZA" ZA'

+

ZA"

Q-

Readon possible

Vibronic transitions possible

He-I:,

no

no

H-Hz

Yes

no

Cl-HCl

Yes

%' 3

W-NO

a0

2

Partners

I

4

s.

24 1

SCATTERING STUDIES OF EXCITATION MOLECULES

(ii) R x R, . In this intermediate attractive region, electron correlation effects compete with overlap repulsion. For neutral systems this region can only be treated in an ab initio calculation by using the very time consuming configuration interaction (CI) wave function expansion. An exception is the interaction of a ' S ion with a neutral molecule. Here since correlation is small compared to classical electrostatic and induction effects, the potential is quite accurately predicted by a SCF-type calculation (Schaefer and Lester, 1975). (iii) R > R , . This attractive asymptotic region is dominated by interatomic correlation. Perturbation theory has been successfully used to obtain the anisotropy and small r dependence (Meyer, 1976; Pack, 1976). The greatest difficulties in the calculation of potentials for neutral systems are presented by region (ii). Since perturbation theory does not function well in this region (Hirschfelder, 1967; Claverie, 1971) the potential has to be obtained from the difference of large energies: V(R,r, Y) = &A-BC(R, r,

Y) - [ & A ( R = m) + & B C ( R =

111

(13)

where EA-BC is the total electronic and nuclear energy for the A-BC combined system and and cBCare the energies of the separated partners. Small errors in any of these quantities can lead to a sizable error in V (Kutzelnigg, 1976). Limits are set on the number of configurations included in a CI calculation and on the accuracy by the computing time tcomp,which increases with the number of electrons roughly as (Pople et al., 1976)

(14) where n, is the number of electrons and N the number of atomic basis functions. For this reason precise ab initio calculations have been restricted to systems with four or less electrons. In addition, a large number of geometries must be calculated. Unfortunately there is no simple rule for the number of geometries, since it depends on the scattering problem and required accuracy. A rough upper limit can be estimated by assuming that about 20 points are needed per coordinate. The total number of normal coordinates (equal to the number of Jacobi coordinates) for N nuclei is 3N - 6, and therefore the total number of geometries G is given by tcomp% n:N4

G =2 p - 6

(15) Thus for an atom-diatom system about lo4 geometries are needed. In practice this number is reduced by symmetry. In the largest CI calculation carried out up to now, Kutzelnigg et al. (1973) calculated about 300 geometries for Li+-H2. However, only two angles y were considered, which was only enough to determine the P , term in the potential [see Eq.. (sa)]. Table I1 summarizes recent ab initio calculations of potential hypersur-

TABLE I1 SOME

RECENTLYCALCLJLATED AB INlTlO POTENTIAL HYPERSURFACES FOR A-BC

Number of electrons

2.

3

System H'-H,

H- H,

Type of surface'

Method

Region probed

Number of points

vibr.

SCF-MO-CI

att.

vihr.

CI

85

vibr.

c1

isosceles triangle minimum

rigid

VB

rep.

14

vibr.

CI

saddle point

40

+ extrap.

249

69

3

He-Hl

collineai

SCF

minimum

80

4

He-H,

vibr.

SCF

rep.

32

vihr.

IEPA-PNO

att.

vibr.

SCF

rep.

200

rigid

VB

att.

26

+ rep.

37

4

Li+-H,

rigid vibr. vibr.

SCF SCF CI

att. + rep. att. + rep. att. + rep.

70 120 300

4

Li+-HD H,-H,

rigid vibr.

SCF CI

att. att.

+ rep.

4

70 220

5

Li-H,

vihr.

SCF

+ rep. att. + rep.

12

Ne-H,

vibr.

c1

rep.

13

Li-HF

rigid

SCF

att.

+ rep.

40

14

Li '-N2

vibr.

SCF

att.

+ rep.

134

14

Lit-CO

vibr.

SCF

att.

+ rep.

111

20

HF-HF

rigid

SCF

att.

+ rep.

394

50

Ill

AND

AB-CD

Potential modelb

SYSTEMS

Reference Csizmadia rl al. (1970) Bauschlicher rf al. (1973) Carney and Porter (1974) Norbeck ef al. (1975) Lin (1973), Yates and Lester (1974) Brown and Hayes (1971). Sat hyamurt hy er (11. (1976) Gordon and Secrest (1970) Tsapline and Kutzelnigg ( I 973) Wilson ef al. (1974) Geurts er ul. (1975) Lester (1970a) Lester (1971a) Kutzelnigg ec al. (1973) Lester (1971b) Silver and Stevens (1973) Karo ef ul. (1973) Birks rf a/. (1975) Lester and Krauss (1970), Lester (1970b) Staemmler (1975) Staemmler (1976) Yarkony ef ul. (1974)

vibr., vibrotor; rigid, rigid rotor.

* The model is given in the reference of the next column unless a reference is listed in this column. Giese and Gentry (1974).

' Kuntz (1972).

Models are in Gordon and Secrest (1970) and in Alexander and Berard (1974). Fremerey and Kendall (1974). Alexander er a/. (1977). 242

243

SCATTERING STUDIES OF EXCITATION MOLECULES

faces that have enough detail to be suitable for calculating inelastic cross sections. One of the most extensively studied neutral systems is He-H2. Here the uncertainties in the potential are nevertheless still quite large. Table I11 shows recent a b initio results for the well depth c and well position R, for the collinear (Cmu)and perpendicular (C2,,)geometries. Even though R, is usually easy to predict, the values differ by as much as 4%. The differences in E are largest for C, and amount to 27%! However, it now appears as if the results in the last column are close to the correct value (Hariharan and Kutzelnigg, 1976).

C . POTENTIAL MODELS For use in a scattering calculation it is desirable to have the potential in an analytic form with as few parameters as possible. Unfortunately in most cases it has been difficult to find a simple analytic expression to fit a large number of potential points covering many different geometries. The most frequently used formula for the interaction with a homonuclear molecule is an extension of a model first suggested by Krauss and Mies (1965). For the interaction with an ion, the terms in Eq. (5b) are given by

u2 = (a, + b , x ) exp(a,R

+ Blx) + (a2 + b 2 x ) exp(cc2R + P 2 x )

+ (cl + d , x + e , x 2 ) R P 3+ (c2 + d 2 x + e2x2)R-4

(16) with x = r - r e . For a heteronuclear molecule, additional terms of odd TABLE 111 COMPARISON BETWEEN

INITIO WELLPARAMETERS FOR COLLINEAR (Cz,.. y = O ) = 90) FOR He-H, PERPENDICULAR (C,,,,

AB

Reference

Method"

R,(A)

Tsapline and Kutzelnigg (1973) Geurts rt al. (1975) Hariharan and Kutzelnigg (1976)

IEPA-PNO VB IEPA-PNO 1-3 CEPA Full CEPA Analysis of experimental data

3.3 1 3.48 3.32 3.35 3.44 3.48

Shafer and Gordon (1973)

E

(mev)

R,

1.84 1.20 1.65 1.47 1.21 1.30

3.33 3.40 3.30 3.33 3.39 3.33

i:

AND

(mev) 1.20 1.09 1.29 1.16 0.98 1.38

The following abbreviations are used: IEPA, independent electron pair approximation; PNO, paired natural orbitals (includes interpair correlations only): CEPA, coupled electron pair approximation; 1-3 CEPA, includes the singlet-triplet interpair coupling but no intra- or inter-intra-coupling; full CEPA. includes all coupling.

244

M . Faubel and J . P . Toennies

parity ( A = 1,3, . . .) are needed. This model, which contains altogether 42 parameters, was used by Lester to fit 120 SCF calculated points for Li+-H2. The errors range from 20 to 30% for large values of R, to well under 1% at small values of R. For H+-H2 another model, based on the addition of two-particle potentials, has proven to be more useful (Giese and Gentry, 1974). The standard error of the difference between the calculated points and the analytic model was about 0.1 eV, although for some points errors as large as 0.4 eV were found. For vibrational excitation and collision-induced dissociation of high-lying vibrational states a similar additive model with an additional term to take account of three-particle forces has been used for He-H, (Wilson et al., 1974). However, as the range of geometries increases, the search for a simple precise model becomes more difficult. Thus Blais and Truhlar (1976) in a semiempiricalstudy of the Ar-H, potential hypersurface for collision-induced dissociation no longer attempt to find a simple closedform expression. Instead they offer a computer subroutine to prospective users to enable them to generate their own potential data. For rotational and low-level vibrational excitation, two other methods appear promising. Billing and Hunding (1976) have developed a procedure for finding the most efficient mathematical expression to fit a given set of potential points. In another approach, Tang and Toennies (1978) have used a simple Buckingham potential with corrections to take account of both overlap and the asymptotically divergent character of the long-rangedispersion series (Tang and Toennies, 1977) to calculate the anisotropy. The problems associated with fitting a calculated potential are well illustrated by the experience of Alexander and Berard (1974). They found that the ab initio points calculated for He-H2 (Gordon and Secrest, 1970) could be nearly reproduced by five different fits (see also Secrest, 1974). They calculated vibrational transition probabilities and the results for the five different models differed by as much as a factor of 2. It is difficult to visualize an atom-molecule potential hypersurface that extends into four dimensions. Figure 4 shows three different projections of the potential hypersurface for Li+-H2 based on the potential model in Eq. (17). The upper right and lower left curves show the radial R dependence of the potential for fixed r with re = 0.74 A, the equilibrium distance, and r = 1.0 A for the two angles y = 90 and 0", respectively. The lower right section shows the equipotential curves as a function of y. Compared to He-H2 (see Table 111) E is much greater, while R, is smaller. The well depth at y = 90" with E = 250 meV, R , = 2.0 A is much greater than at y = 0", where E = 40 meV and R, = 2.5 A. The large anisotropy is due largely to the classical charge quadrupole and charge-induced dipole terms, which account for more than 90% of the potential for R > 2.5 A (Lester, 1971a). Figure 5 shows the R dependence of the u,(R, r = re) terms of the cal-

SCATTERING STUDIES OF EXCITATION MOLECULES

.-Potential

245

energy lev]

4 0

a)

i

R [A]

FIG.4. Composite diagram of the SCF calculated rigid-rotor potential hypersurface of Li+-H, (Lester, 1971). In (a) and (c) V ( R ) is shown for y = 90 and 0". respectively, for two different values of the internuclear distances: r = r e = 0.74 A (-)and r = 1.0 A (..). In (b) the equipotential tines for r = re are shown as a function of y.

culated Lif-H2 potential. Both the SCF energies of Lester (1Y70a) ana tne somewhat greater values obtained from the calculation of Kutzelnigg et al. (1973) are shown. For Li+-H2 the u2 term is the predominant anisotropic term. However, for near-central collisions,the u4 term is of increasing importance. u2 and u4 are directly proportional to the matrix elements that govern the rotational excitation [see Eq. (2211. Figure 6 illustrates the r dependence of the Li+-H2 potential hypersurface by showing at the bottom the equipotential lines of the uo term in the R, r plane. At the top, the radial dependence of the vibrational coupling matrix elements for this term are shown. The vibrational wave functions are shown at the bottom right. Note that the uo(R, r ) , which is the only term remaining after spherical averaging, only contains the pure vibrational coupling. The uo term is used in the well-known breathing-sphere approximation (Tarr et al., 1976). For Li+-H2, the u2 and u4 terms also lead to appreciable vibrational matrix elements corresponding to combined rotational and vibrational coupling (Schaefer and Lester, 1975). Thus the breathing-sphere approximation

246

M . Faubel and J . P . Toennies

FIG. 5. The R dependence of the various Legendre terms in the Lit-H, potential for of Lester (1971) lie somewhat above the small effect on dispersion. The r 2 and r 4 terms are directly proportional to the potential matrix elements for rotational excitation. Note that the positive ordinate is a log scale and the negative ordinate a linear scale.

r = r , = 0.74 A are compared. The SCF results (---) C1 results of Kutzelnigg et a / . (1973) (-) showing the

does not work well for Li+-H, (Schaefer and Lester, 1973) and fails for many other systems. The equipotential lines in Fig. 6a show that in the attractive region of large distances ( R > 1.9 A) the molecule experiences a force tending to pull the atoms of the molecule apart. In the repulsive region at smaller distances the direction of the force is reversed and tends to push the atoms together. At R = 1.9 the net force on the molecule is zero and the vibrational matrix elements plotted in Fig. 6b also pass through zero. The off-diagonal vibrational coupling matrix elements are largest with respect to the diagonal ones at small internuclear distances. This suggests that vibrational excitation will be most probable at large scattering angles. D. APPROXIMATE METHODS Because of the difficulties in calculating ab initio potential surfaces for systems with more than about four electrons, there has been considerable interest in more approximate methods. The simple addition of atom-atom potentials to obtain an atom-molecule hypersurface has recently been tested and found to be only of limited accuracy (Barg et a!., 1976). Recently Gordon and Kim (1972, 1974) have revived and modified an electron gas

SCATTERING STUDIES OF EXCITATION MOLECULES

247

iction

i i

-0.2

1.0

1.5

FIG.6. In (a) the equipotential lines of the uo(R, r) term in the Legendre expansion of the Li+-H, potential are shown. The area enclosed by the dashed lines shows the region for which the potential was calculated. The vibrational wave functions shown at the right project out a part of the potential to yield the R dependence of the vibrational matrix elements shown in (b).

model originally applied to the calculation of intermolecular potentials by Lenz (1932) and Jensen (1932). The big advantage of this method is that the computing time is short and independent of the number of electrons. Comparison of potential parameters for rare gas atom- and alkali atom-rare gas atom interactions shows the model to be accurate to within about 15% (Gordon and Kim, 1972). Several improvements have been suggested (Kim and Gordon, 1974; Rae, 1975).The calculation of potential hypersurfaces for Li+-H, and He-H,CO has been tested by extensive comparisons with SCF energies (Green et al., 1975). Qualitative discrepancies were found, which could lead to significant errors in calculated cross section, especially for vibrational excitation. Nevertheless the method is now used in a routine way to generate anisotropic potentials for atom-molecule interactions. For references to recent work, see Toennies (1976) and Parker et al. (1976).

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111. Inelastic Scattering Theory A. THEQUANTUM MECHANICAL " CLOSE-COUPLING " METHOD

The quantum mechanical calculation of inelastic cross sections for the scattering of a 'S atom from a 'Z molecule is now well understood. The other cases listed in Table I each require a different, more complicated formulation, which is only available for reactive scattering without vibronic transitions (Kuppermann, 1975; Kuppermann and Schatz, 1975a,b) and for P atom (Rebentrost and Lester, 1976) and a I7 molecule the scattering of a ' (Klar, 1973). Two formulations of the ' S atom-lz molecule problem have been presented. In the formulation of Arthurs and Dalgarno (1960) the orbital and rotational angular momenta are defined with respect to a spacefixed coordinate system with z axis parallel to the incident momentum. In the other formulation, due to Curtiss and co-workers (Curtiss and Adler, 1952; Curtiss, 1953, 1968; Gioumousis and Curtiss, 1961; Hunter and Curtiss, 1973; see also Chang and Fano, 1972) the rotational angular momenta are defined in a rotating coordinate system with z axis parallel to the R vector. This is also called the R helicity representation. In a modification, called P helicity, the rotational angular momentum is defined with respect to the direction of linear momentum P. Both these representations have the advantage that the orbital angular momentum L = R x P has a vanishing projection. These body-fixed representations are useful starting points for a hierarchy of efficient approximations. However, since all of the exact calculations have used the Arthurs and Dalgarno space-fixed formulations we briefly review its essential steps here. For more details see Lester (1971c, 1975). The nuclear Schrodinger equation in the center of mass system is - -Vi

1

+ V(R, r) - En,] Ynj(R, r) = 0

- Hint(r)

where the terms in brackets are, from the left, the relative kinetic energy operator, the Hamiltonian for internal rotational and vibrational motion, the potential energy surface, and the total energy (translational and internal) of the system with the molecule initially in the state nj. p is the reduced mass of the entire atom-molecule system. The potential V(R, r) couples the relative motion to the internal motion of the molecule. The wave function Vnj(R, r) is expanded in a basis containing the wave functions of the molecule in a total angular momentum representation:

249

SCATTERING STUDIES OF BXCITATION MOLECULES

Here J is the total angular momentum of the entire system, M its projection along a space b e d axis, 1 the orbital angular momentum before collision, and the primes denote possible quantum numbers after collision. The function %$’(R, r) contains the entire angular momentum of the system in the J M representation: g$M(RIr) =

C (jlmjmll jlJM)Yjmj(r)Kml(R)

(19)

mjmr

where the Y(x) are the angular momentum wave functions and the I...) are the Clebsch-Gordon coefficients. The Xnj(r) are the vibrational wave functions for the rotating oscillator and the Uijl,,,j,l,(R)are the radial functions of relative motion. Substituting Eq. (18) into Eq. (17) and carrying out the usual manipulation leads to the following set of coupled equations for the Jth partial wave: ( . - a

where k2 is proportional to the kinetic energy in the n j channel:

with E , , ~ the internal energy of the molecule. For the Legendre expansion of the potential, Eq. (5a), the matrix elements in Eq. (20) are given by

where the c A , which contain the angular momentum coupling, are the Percival-Seaton coefficients. They have been tabulated (Percival and Seaton, 1957) or can be calculated with the aid of Wigner 3j and 6j symbols. The integral on the right is the vibrational matrix element (see Fig. 6b). Note that the u1 determine the strength of rotational coupling (see Fig. 5). The dimension of the set of coupled equations (20) depends on the number of molecular states n, j incliided in the basis set Eq. (18). The set of coupled equations has to be solved numerically with the condition that the U ( R )vanish at the origin. The S-matrix is obtained from the usual asymptotic conditions at R 00. The scattering amplitude is cal---+

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250

culated from the S-matrix using the following efficient expression (Jacob and Wick, 1959; Goldberger and Watson, 1964): fnjmj-n,j*mj,($)=

(-)’‘’‘(24j)-’

1 (25 + 1) dLjmy(9) J

x [(21’

+ 1)(21 + 1)]1’2[S:ar - daat]

(23) where the d&,,j,(9) is the Wigner rotation matrix and the abbreviation a = njl has been used. The differential cross sections can be calculated directly from the scattering amplitude, and for the njrnj+n’j’mJ transitions are given by

It should be noted that m is defined with respect to the incident relative velocity. A more general expression for the 9 and q dependence and arbitrary orientation of mi, as well as of m j , ,is easily obtained from the Arthurs and Dalgarno expression for the scattering amplitude (Alexander et al., 1977; Fitz, 1977). The differential cross section for degeneracy-averaged nj + n’j’ transitions of unpolarized beams results by simply averaging Eq. (24) over initial mi and summing over final mi:

For comparison with measured angular distributions, without discrimination of inelastic processes, Eq. (25) can be simply summed over all final states nj to yield the total differential cross sections. Finally, integration over the entire solid angle yields the integral cross section:

Each term in the sum over J is called a partial cross section. The formulation presented here is exact except for the need to truncate the basis set in the expansion of Eq. (18). In this connection it is convenient to use the term channel; where a single channel refers to a set of quantum numbers njl for a given J . Since at the present time no satisfactory convergence criterion is known, the number of channels included in a calculation has to be determined by trial and error. In most calculations of rotational excitation, convergence has been achieved by including all energetically accessible (open) channels of interest and a few adjacent channels (Lester

SCATTERING STUDIES OF EXCITATION MOLECULES

25 1

and Bernstein, 1967; Eastes and Secrest, 1972). This rule may break down if the coupling is strong as in Li+-H2. Usually it is not necessary to include energetically inaccessible (closed) channels unless these lie adjacent to the ones of interest. However at low energies closed channels may lead to new cross section features such as Feschbach resonances (von Seggern and Toennies, 1969; Micha, 1973; Eastes and Marcus, 1973). The computing time for a close coupling calculation of the S-matrix for one partial wave depends on the number of channels as tcompK NF,where m lies between 2 and 3 depending on N , and the algorithm used (Gordon, 1971; McGuire and Kouri, 1974). The number of channels for pure rotational excitation is (Fremerey, 1975) Nc =

Ymax+++ 1

&,ax

for heteronuclear diatoms, ji = 0 for homonuclear diatoms, ji= 0

ll2, 2)’,

2)(jmax+ l),

for homonuclear diatoms

(27)

ji # 0

where j,,, is the largest rotational state included in the basis set. For rovibrational excitation, N, has to be multiplied by the number of vibrational states nmax if the same size rotational basis set is used for each vibrational state. Thus the computing time increases with j,,, at least as j:,,! An approximate expression for estimating the total computing time for all partial waves at a fixed energy, while including all open channels, is

where q = 1 for a heteronuclear and q = 2 or 4 [see Eq. (27)] for a homonuclear diatom. From the condition that all open channels be included we get E N Bjiax (B is the rotational constant), E N hvn,,,. Furthermore with J,,, N kR,,,, Eq. (28a) can be written as tcompa ( ~ B V ) - ~ ( ~ ) ~ / ~ R , , , E ~ ~ + ~

Equation (28b) indicates that the choice of energy has the greatest effect on the computing time. Moreover, for systems with a large value of B and a short range of interaction, the computing time will be shortest. This is one reason why most calculations have been for H2 or hydrogenic systems. The fastest system for calculations is probably He-para H2withji = 0. At 1.2 eV collision energy, seven rotational states j = 0, 2, . .., 12 are energetically accessible and there are 49 channels. To solve the coupled equations requires somewhere between 8 and 60 hours on a UNIVAC 1108 (McGuire and Kouri, 1974). The largest calculation reported to date has included 118 channels (Schaefer and Lester, 1977). Whereas over 30 close-coupling computational studies of rotational exci-

252

M.Faubel and J . P . Toennies

tation have been reported (for a recent tabulation see Toennies, 1976),only a few calculations on vibrational excitation have been carried out. So far as we are aware, no fully converged three-dimensional calculations of vibrational excitation with full account of rotations has been performed under conditions of interest for a scattering experiment, where the transition probability is greater than say lo-’.

B. APPROXIMATE METHODS In view of the limited applicability of the close-coupling method, a large number of approximate methods have been developed in recent years. In all the approximate methods a reduction in computing time is achieved at the expense of accuracy. The accuracy depends both on the system as well as on the collision energy in a way that is usually difficult to assess with certainty at the outset. Most approximate methods are reasonably accurate (10-3073 for predicting integral inelastic cross sections, which, although of interest for interpreting bulk studies, are difficult to measure in beam experiments. Usually the differential cross sections are less accurate since these depend not only on the absolute magnitude but also on the phase of the scattering amplitude. A list of some of the more important quantum mechanical approximations that lead to a reduction in the computing time for calculating Smatrices is given by Toennies (1976). For a more detailed discussion of these approximations see also Secrest (1973), Baht-Kurti (1975a), Miller (1979, and Rabitz (1976). There are also a large number of semiclassical approximations in which classical mechanics is used to simulate the relative translational motion. State to state transition probabilities are then obtained either by a quantum mechanical superposition of classical trajectories (Marcus, 1972; Miller, 1975)or by perturbation theory (for a review see Balint-Kurti 1975a) or, as in the case of vibrational excitation, by analogy to the forced-oscillator problem (for a review see Duff and Truhlar, 1975b). In the last case of a forced harmonic oscillator the distribution of quantum transition probabilities is given by a Poisson distribution (Kerner, 1958): 1 Po,, = - (BEYexp( - A&) .f !

(29)

where A&= AECl,,,/hv. In the DECENT model (Giese and Gentry, 1974) AEclassis calculated for three-dimensional classical trajectories leading to a given scattering angle. For each trajectory the transition probability is then calculated using Eq. (29). The differential cross sections calculated in this

SCATTERING STUDIES OF EXCITATION MOLECULES

253

way have been found to be in excellent agreement with scattering experiments for H+-H2 and Li+-H2 (see Section V,B). In our opinion the most convenient and reliable approximation for comparison with presently available scattering experiments is the straightforward use of classical mechanics. Figures 7 and 8 show the type of agreement

FIG.7. The rotationally inelastic (j= O-+ 2) integral cross section is shown as a tunction 01 energy for Li+-para-H, . The quantum mechanical results are in reasonable agreement with the pure classical mechanical results. The small bump at about 0.1 eV and the larger quantum cross section at lower energies are attributed to the temporary excitation of closed channels (Lester and Schaefer, 1973). 0,Lester and Schaefer (1973); A, Schaefer and Lester (1975); 0,McGuire (1974); 0 ,La Budde and Bernstein (1973).

that can be achieved with exact quantum mechanical differential cross sections for Li+-H2, j = 0-2, and other transitions. These and other comparisons suggest that classical mechanics is accurate so long as the transition probability is sufficiently large (P 2 l-lO%) and provided that the quantum interference effects, discussed in connection with elastic scattering in Section I,C, are not too important. These conditions are often fulfilled in scattering experiments since only transitions with a large probability are observed. Moreover, the angular resolution is frequently insufficient for resolving interference effects. As opposed to quantum

M . Fairhel and J . P . Tornnies

254

10.~

0

10

20

30

40

Center of mass scattering angle

50

60

J [.I

FIG.8. Comparison of elastic and inelastic differential cross sections for Li+-para-H, ( j = 0) at E,, = 0.6 eV calculated using classical mechanics (-) and quantum mechanics (-- -). Note that the inelastic cross sections 0 - 2 and 0+4 have been shifted downward by two orders of magnitude with respect to the elastic 0+0 cross section. There is good agreement at angles greater than the classical rainbow at 9 = 35".At smaller angles quantum interferences

lead to a shift of the primary rainbow to about 30" and to fast undulations.

mechanical approximations, there are no inherent limitations imposed by basis set considerations. Thus the classical method is applicable to any system and the numerical error can be estimated simply from the statistics, which depend only on the available computing time (Barg et al., 1976).

SCATTERING STUDIES OF EXCITATION MOLECULES

255

COMPUTATIONAL RESULTS C. SOME Figure 7 shows the results of several independent calculations of the energy dependence of integral inelastic j = 0-2 cross section for Li+-H2. The inelastic cross section rises steeply at the threshold of 44 meV and increases nearly linearly in the range investigated. The small differences in the two curves based on close-coupling calculations are largely due to the larger basis set ( j = 0, 2, 4, 6, 8) used in the calculation of Lester and Schaefer (1973). Figure 7 shows that there is reasonable agreement with the classical Monte Carlo calculations of La Budde and Bernstein (1971, 1973). Although a similar energy dependence has been observed in most other systems, an entirely different behavior, consisting of a sharp increase to a maximum at threshold followed by a gradual falloff has been found in He-HCI (Green and Monchick, 1975), He-HCN (Green and Thaddeus, 1974), and He-CO (Green, 1976). Figure 8 shows a comparison of close-coupling differential cross sections for the 0-0, 0-2, and 0-+4 transitions of Li+-para-H2 at 0.6 eV (Schaefer and Lester, 1975) with the results of a calculation using classical mechanics (Barg et al., 1976). At angles beyond the rainbow at 35" the agreement between the classical and quantum results is good ( z10%). At smaller angles large discrepancies (50-100%) are found. The differences are attributed to two different quantum interference effects which leads to (1) several widely spaced rainbows and a shift in the location of the primary rainbow on which is superimposed (2) closely spaced undulations (see Fig. 2). Figure 9 shows the results of a similar quantum mechanical closecoupling calculation for Li+-para-H2 at E,,,,, = 1.2 eV. At this energy the primary rainbow angle has been reduced to about its previous value and is at lo". Whereas the differential cross sections for rotational excitation all decrease with increasing angle, the small differential cross section for vibrational excitation shows, aside from a small forward peak, a broad maximum in the backward direction. All these features are roughly consistent with the simple considerations of Section 1,C [Eqs. (6), (9), and (12)]. Computational results of this type provide useful insight into what to expect in a scattering experiment. Theoretical studies can also serve to explore the dependence of various cross section features on potential parameters. Unfortunately few such studies have been reported to date, presumably because of the large amount of computer time needed for a systematic study. In an early close-coupling study of rotational excitation in He-N, , Erlewein et al. (1968) used a simple Lennard-Jones type potential:

M . Faubel and J . P . Toennies

256

0

20 LO 60 80 100 120 140 160 180 Center of mass scattering angle9

["I

FIG. 9. Quantum mechanically (close-coupling) calculated differential cross sections for rotationally and vibrationally inelastic scattering of Li+-para-H, at ECM= 1.2 eV (Schaefer and Lester, 1975).The basis set used for the pure rotational transitions is n = 0, j = 0.2,4,6,8, 10. 12. The expansion for vibrational transitions is n = 0, j = 0, 2, 4, 6, 8, 10; n = I, j = 0, 2, 4; ti = 2, j = 0, 2, and is probably not sufficient for complete convergence. For this reason slight corrections have been applied to the original results to remove spurious undulations (Schaefer, 1976).

where q2, 12 and q2,6 are anisotropy parameters. At E,,,,, = 4.3 meV they found from extensive numerical calculations CTo-'2

(q2. 1 2 - l.8qZ. 6)' + 0*21q2.1 2 4 2 , 6 (31) This result suggests that the repulsive and attractive anisotropies can to a large extent cancel each other. This cancellation effect has since been confirmed (Dickinson and Richards, 1974). In a related study Schaefer and Lester (1975) found only small differences in the rotationally inelastic cross sections calculated for the similar SCF and CI potential hypersurfaces shown in Fig. 5. A number of quantum mechanical and semiclassical studies have shown that vibrational transition probabilities are very sensitive to both the R and r dependence of the potential (Mies, 1965; Hayes et al., 1971; Alexander and Berard, 1974; Schinke and Toennies, 1975; Duff and Truhlar, 1975a). In

SCATTERING STUDIES OF EXCITATION MOLECULES

257

all cases the molecule was held at a fixed angle y during the collision. Since rotational coupling can have a large influence on vibrational transition probabilities (Lester and Schaefer, 1973), the conclusions are expected to be modified at least somewhat by rotational excitation. In the course of their extensive studies of Li+-H2 Schaefer and Lester (1975) did observe that the partial cross sections for vibrational excitation were much more sensitive to the small differences between the SCF and CI potentials than the rotationally inelastic cross section. More studies of this type are needed to help guide scattering experiments.

IV. Experimental Methods A. BASICTECHNIQUES The measurement of quantum-resolved inelastic scattering cross sections is an exceedingly difficult experimental problem. Two different techniques are used to identify an inelastic event. In the energy change EC method, the small changes in the velocities of one of the scattered particles resulting from the energy lost to or gained from the internal degrees of freedom are used to detect the occurrence of a transition. In the state selection SS method, the changes in the quantum state of the molecule are observed directly. In both methods the state of the molecule must be defined before the collision in order to assign the observed changes to a specific transition. It is convenient to rewrite Eq. ( 1 ) as

where g is the relative velocity (gi = vA, - vBC,i) and c1 = (j,m j , n ) . Thus altogether two relative velocities and six quantum numbers are needed to completely specify the inelastic collision in the center of mass system. Some of these parameters, with the exception of mji and mj, can be eliminated by conservation of energy: 1

2 ~

+i Eidji, n i ) = hd + Eint(.jf7nf)

(33) Figure 10 shows schematically the operation of an idealized apparatus. The beam of BC molecules leaving its source will usually have an initial distribution (indicated by { }) of laboratory velocities and quantum states. Filters are used to select out a narrow range of velocities and a single quantum state. Only a velocity filter is needed to prepare the beam of structureless particles A. The location of the detector defines the direction of 2

M . Faubel and J . P. Tornnirs

258

BC

T filter

A

{GI

FIG. 10. Schematic diagram of an ideal apparatus for measuring inelastic quantum transitions in scattering of a structureless particle A from a diatomic molecule BC. Each of the filters picks out a narrow range of laboratory velocities or a quantum state out of an initial distribution indicated by brackets { }. The detector at laboratory angles 0 and @ must analyze for either the velocity of one of the collision partners or for the quantum state of the molecule in order to identify the transition. In the former case the method is referred to as an energy change EC experiment, in the latter case as a state selecting SS experiment.

the final velocity by specifying the LAB scattering angles 0 and a. In the SS method the detector, either alone or in combination with a filter, is capable of analyzing the scattered particles for a specific set of final quantum numbers a f . In the EC method the detector, usually in combination with a velocity analyzer, is made sensitive to the magnitude of the final velocity. Since the initial state is specified, conservation of energy Eq. (33) may be used to identify the final quantum states from the measured velocity change. These two experimental methods complement each other. State selection experiments usually utilize inhomogeneous electric or magnetic fields (e.g., Stern-Gerlach fields or electrostatic quadrupoles). However, a number of different spectroscopic techniques in combination with lasers appear promising for the future. The SS method has the advantage that the momentum selection of initial states is not critical, that it can detect Am transitions as well as transitions for which the velocity changes are too small to be resolved in an EC experiment. In the past, the SS method has only been successfully applied to rotational transitions of alkalihalides. The EC method is more universally applicable, although only transitions with AE 2 10 meV have been resolved up to now. The EC experiments are made difficult by the fact that the resolution requirement of a large fractional change in collision energy A E L A B / E L A B implies, as discussed in Section l,C, a small transition probability. Nevertheless because of its universality most experiments have used the EC method. For this reason and since the state selection experi-

SCA'ITERING STUDIES OF EXCITATION MOLECULES

259

ments have been discussed in detail elsewhere (Pauly and Toennies, 1968) the remainder of this section deals with energy change experiments.

B. DESIGN CONSIDERATIONS FOR AN ENERGYCHANGE EXPERIMENT The design of an energy change scattering experiment involves a careful optimization of the scattered intensity at a given resolution sufficient to resolve the quantum transitions of interest. The number of inelastically scattered primary beam particles arriving per second in a given solid angle element of the detector dad,,at angles 0 and CD is given by4:

dNi+ =n,n2g

(34)

where n, and n2 are the beam densities, g = v 1 - v2 the relative velocity before collision, A V the volume of the region crossed by the two beams, and I dw/dR I a Jacobian relating the solid angles in the CM system to those in the LAB system. In order to discuss the resolving power of an energy change experiment, Eq. (34) has to be elaborated to take into account the effect of the angular velocity and spatial distributions of the incident beams on the velocity spreads of the scattered beam. A more general expression for the number of particles with average final velocity U1 in the range AuI scattered into the angular direction 0,6,is

The following notation is used throughout to distinguish between quantities in the laboratory (LAB) system and center of mass (CM) system: Capital Greek letters are used to denote LAB scattering angles 0, @, and solid angle R. Small Greek letters denote CM scattering angles 9, cp, and (I). The velocities in the LAB system are given by vl, v 2 , and those with respect to the center of mass are given by -m, 1112 u, =-

m1 + m 2

g,

u2 = -~

m1+

m2

where g is the relative velocity. To simplify the notation, final angles and velocities are indicated by a prime and initial angles and velocities are unprimed.

260

M . Faubel and J . P . Toennies

where the vector velocities have been expressed in terms of polar coordinates; q(ul, 0;) is the transmission of the velocity analyzer set at the final velocity u; ; r denotes the spatial Cartesian cooidinates of the scattering volume; 9(0,@) and q ( 0 ,@) are the CM angles corresponding to the LAB angles; a( )/a( ) is the Jacobian for transforming angles and velocities from the LAB to CM system; u; is the velocity of particle 1 with respect to the center of mass after a transition i -+f; and 6(u, - u ; ) is a delta function that is needed to make the integrand single valued in the velocity space. Figure 11 shows the vector velocities in the laboratory system.

tx

FIG.11. Definition of velocity coordinates in the LAB frame. 0 is in the y-z plane, while the angle @ is used to denote rotation about the coordinate center, but perpendicular to the y-z plane.

Equation (35) can be rewritten in a more useful form as (Duren et al., 1977)

x G(9,~p,8 ; 0, 6,u,, 52, F)

(36) where C( ) is a distribution function describing the weighting of the differential cross section in the three-dimensional CM space resulting from a given choice of apparatus distributions. The integration of Eq. (36) leads to a distribution of scattered intensity over small intervals Au; at different D',for a given apparatus characterized by incident beam distributions at the scattering region described by nl(v,, rl) and n2(v2,r l ) and a detector with solid angle Aflde,. This approach, while being exact, does not lend itself to providing simple yardsticks for designing an experiment. For this reason we will use a simpler, more approximate first-order treatment later on. In the remainder of this section, we first present expressions for the final

SCATTERING STUDIES OF EXCITATION MOLECULES

26 1

LAB velocity and the equations governing the transformation from the LAB to the CM systems, for an idealized apparatus. Next we derive simple approximate formulas for calculating the half-widths of apparatus distributions needed to resolve a given transition. These formulas are used to derive an expression for the scattered beam intensity for a given ratio of primary beam to secondary beam masses. Finally, methods used to analyze and present the results are discussed. 1. Velocity Changes in an Idealized Energy Change Experiment

If both incident beams have delta function distributions and the detector is infinitesimally small, then conservation of energy and momentum yields the following expression for the final LAB velocity u; for a given loss AE:

f

{

[(Pl

+ P 2 ) * a2- m1 m+2m 2

I

x 2Pl ’ (P1 + P 2 ) -

P1 2

m1+ m l m2

+ 2m2 A E ] } 1 ’ 2 )

(37)

where $2= (0, @) is the direction of the detector. This expression simplifies for the important special beam geometry called “ perpendicular plane ” frequently used in inelastic scattering experiments (Greene et al., 1969). In this geometry shown in Fig. 11 the scattered particles are detected in a plane perpendicular to the secondary beam direction and passing through the primary beam direction (pl I p2, I p2). The result is v; =

u1

(cos 0 f

1 + (m2/m1)

[ k)’(g) I-

- sin2 @ ] ’ ” }

(38)

where E&,, = [m, / ( m l + m2)]ElLAB is the center of mass collision energy for a stationary target. Equations (37) and (38) are fundamental to the discussion of the transformation of CM cross-section quantities to and from the LAB system and for the discussion of the resolving power.

2. The Transformation; Center of Mass to Laboratory System The equations governing the transformation of angles and velocities from the LAB to CM system are conveniently derived and illustrated with the aid of a Newton diagram. An example is shown in Fig. 12. Newton diagrams are

M . Fauhel and J . P . Toennies

262

+

”P

inelastic spheres

Fic. 12. A Newton diagram in velocity space is used to illustrate the relationship between the velocities and scattering angles in the LAB and CM systems for the special case of scattering of Lit (1) from H, (2). v I and v, are the velocities of the primary and secondary beams in the LAB before scattering. The initial relative velocity is gi = v I - vz = uI - u z , where uI and u2 are the velocities with respect to the center of mass (denoted by cm). Note that u1 = [ m , / ( m , + m,)]gand uz = - [ n i z / ( ~ ~+i l mz)]g. As a result ofscattering g is rotated through the CM scattering angle 9. l g f l (and I U ’ ~ ~is) related to A E by conservation of energy: tpg: = tpg’ - AE. Thus for agiven LAB angle, 0, scattering events with different 9 and A E are observed.

discussed in detail in many reviews of molecular beam scattering (see, for example, Pauly and Toennies, 1965; Fluendy and Lawley, 1973; Toennies, 1974a) and therefore will not be treated in detail here. Figure 12 shows that the locus of all final LAB velocities of particle 1 for a given A E lies on a sphere with center at the velocity of the center of mass (denoted by cm). It also illustrates how the LAB angle 0 and final velocity u; are related to the CM angle 9 and CM velocity u ; of particle 1. Table IV summarizes the formulas for transforming the cross-section quantities for two important special cases: (a) stationary target, and (b) moving target in the perpendicular plane arrangement. Table IVb indicates that the latter results are directly related to those for a stationary target. For more general transformation equations see Helbing (1968),Warnock and Bernstein (1968), and Pauly and Toennies (1968). 3. The Resolution of a Nonidealized Energy Change Apparatus

The effects of finite changes of v2 and detector angle A@ on the final velocity are shown in Fig. 13, the Newton diagram for a parallel plane arrangement. The change Av, leads to a shift in the location of the center of mass from cm to cm’. In addition, the radius of the sphere of final velocities is contracted. As a result of both effects the locus of final velocities in the CM system is changed from the solid-line circle to the dashed-line circle. The finite detector acceptance angle A@ leads to an additional smearing of Au;. In similar fashion, altogether eight distributions (neglecting the finite scatter-

TABLE IV FORMULAS FOR TRANSFORMING ENERGIES, ANGLES,AND DIFFERENTIAL CROSSSECTIONS-FORAN IDEALIZEDAPPARATUS-FROM THE LAB TO CM SYSTEMS FOR DISCRETE RESOLVEDENERGY LOSSES A€ (a) Stationary target (c,

9 O = cx

=0)

(b) "Perpendicular plane" arrangement (vI I v,, R I vz)o

+0

where

,

where

For u; see Eq. (38) a

3 2

sin-=---

Superscript zero denotes CM quantities calculated in (a).

sin 9O/2 A

(VIII)

M . Fuuhel

264

uitti

J . P. Toennicx

FIG. 13. For in-plane scattering and rn I /tnz = 3.5. thc effcct of a given change in v 2 , denoted by A v 2 , and the effect of a finite detector aperture A@,,, on the final LAB velocity is shown in a Newton diagram. In addition to leading to a spread in V; the changes also lead to a spread in the CM cncrgy and to a shift and spread i n scattering angle.

ing volume) with variables a,,@,, i l l , u Z x . u 2 y . u Z z ra d e , , and adel shown in Fig. 1 1 are expected to contribute to the overall smearing of 0 ; . In order to deal with all distributions, we introduce a multidimensional with components x i , which contains all the apparatus vector x = (v,, v 2 , variables. Expanding about the most probable values xio ,the change in the final LAB velocity Ad, in the direction Ode,,Ode,is given in first order by a Taylor expansion :

a)

Ad, =

1 c i ( x 0 ) AX; i

(39)

where ci = ( ( % ~ i / ( ? ~To ~ )obtain [ ~ = ~analytic ~. expressions for the distribution of Ad, we assume that the CM cross sections are constant over the range covered by the x i and furthermore that the A s i have statistically independent Gauss distributions. As a consequence the distribution in D; is also given by a Gauss distribution (Pfanzagl, 1966; Schwartz, 1965) with variance

~ ’ ( A u ; )= C c’s:(Ax~) i

(404

or in terms of half-widths (fwhm), denoted by Axi,

The spread in velocity is related to the spread in energy by mu; In order to resolve an inelastic transition we simply require

= cinrlAE.

SCATTERING STUDIES OF EXCITATION MOLECULES

265

In general the Taylor coefficients ci can be obtained by numerical differentiation of Eq. (37). For perpendicular plane scattering, this differentiation has been carried out analytically. The results are summarized in Table V. Examination of these coefficients indicates that c4 is zero, which is a special feature of the perpendicular plane geometry (Greene et al., 1969). Furthermore, c2 and c, are identical. In many experiments u 1 + u2 and in such cases c3 and c 8 , which are both proportional to t i 2 , can be neglected. Another simplification results in the usual situation in which the distribution of v2 is cylindrically symmetric with respect to the .Y axis (Fig. 11). For this special case we can replace the coordinates Au2y and AuZzby Ao,, (in the y-z plane) and c5 and c6 by c,. = a,.. Thus the contribution to the final LAB velocity from the target velocity spread reduces to AU;(AV,)

= C T AV2,

(41b)

With these approximations the resolution condition simplifies to

where the velocity-independent Taylor coefficients ainel= cine,/mlu,. a , = al = c,, a@= a2 = c2 /v,, and uT = c,. have been used (see Table V). Since transition probabilities are usually largest for small AE/E, we can neglect AE/E&,, and the four parameters are only functions of m , / m 2 and the LAB scattering angle 0.Figure 14 shows all four parameters as a function of 0 with m , / m 2 as parameter. For m , / m 2 > 1, both small- and largeangle CM scattering regions appear at small angles in the LAB system. Figure 14 can be used to estimate the apparatus half-widths needed to resolve a given inelastic transition. From Fig. 14 it is immediately apparent that with the exception of a, all coefficients are smallest at small angles and small mass ratios m l / m 2 . Thus for ml/m2 > 1, the best resolution is achieved at small angles. Figure 15 shows the energy resolution achieved in some recent inelasticscattering experiments. U p to the present time, the best resolution attained in an energy change experiment corresponds to a AE/E of about 5 x lo-'. The best resolution of about has been reached in state selection experiments. 4 . Scattered Beam Iriterisities in an Energy Chaiige Experimenf

With the aid of the results of the previous section we c;n estimate the maximum available scattered intensity from Eq. (34) for a given energy resolution AEillel.As in the previous section, the transmission of the velocity analyzer will not be considered. To estimate the densities in the incident

TABLE V

TAYLOR COEFFICIENTS NEEDEDFOR ESTIMATING THE RESOLUTION OF PLANE GEOMETRY USING

Apparatus component

Coordinate Ax: (see Fig. 11)

A N ENERGYLoss EXPERIMENT WITH PERPENDICULAR EQ. (37)"

Definition of velocityindependent coefficient 0,

Taylor coefficient c t c . d

,

1 Primary beam

Ar

2 Primary beam

A 0I

= 0,

-

s

3 Primary beam 4

Secondary beam

=

0

0

At3ZX

o3 . c2

?

s-

5 Secondary beam

= 0s

8 Detector

= (Ig . c2

a

The velocity spread Ar', is related to the energy spread AE by Ar; The coordinates are shown in Fig. 11 The following abbreviation is used throughout: (

11'2

=

= qneIAE.

g)

[(2]'(1-

- sin2

with:

ctneI=

* m l u l [m2 ml( ~

~

)I2

1

-mlcl ~

01'"

The plus sign applies for forward CM scattering (3 5 n/2) and the minus sign applies for backward CM scattering (3 2 4 2 ) .

267

SCATTERING STUDIES OF EXCITATION MOLECULES

1

1

1

I

I

I

1

I

I

1

1

1

1

I

4

1

I

la^,

'

1

I " '

I

I

I

0

0'

'io'

'$0' '90' '1iO'

I

I

I

I

-

'ko' '180

e FIG.14. The most important velocity-independent expansion coefficients determining the final LAB velocity smearing in an energy change experiment are plotted as a function of the LAB scattering angle with m,/rnZ as parameter. For m,/m,> 1 both small angle (-) and large angle (- - - ) CM scattering appear in a restricted range of LAB angles. (Upper scale for dashed curves.) 8

Collision energy ECM[eV]

FIG. 15. The energy loss resolved in energy change (0) and state selection ( 0 )inelasticscattering experiments is plotted as a function of the collision energy. The diagonal lines connect regions of equal resolution AEIE.

268

M . Fauhrl and J . P. Toennies

beams we introduce the brightness B. which is the number of particles per second per steradian. For a nozzle beam source the brightness is equal to about one-half the pumping capacity in the nozzle exhaust chamber. Typical pumping capacities are between 1 and 10 torr liter/sec, corresponding to a brightness of 1.5 x lOI9 to 1.5 x lo2' particles/sec sr. The density n at a distance L from the source is given by n = B/vL2 and the scattering volume by A V = L: A2QlL2A@,, where A 0 2 is the angular spread of the secondary beam along the direction parallel to vl. With the aid of these two relations, Eq. (34) may be written as

The angular half-widths Acl, appearing in this expression are restricted by the energy level resolution to their maximum values

where we have assumed that only the five distributions appearing in Eq. (43) contribute and that the intensity contribution from each increases linearly with A q . For the special case ofa perpendicular plane geometry the analytical expression for the ci given in Table V can be used to derive analytical expressions for the ma,. Further simplification is achieved by assuming u2 G ul, expressing u1 in terms of the collision energy E,, and v 2 in terms of the most probable velocity in a nozzle beam, y

2hB T2

(45)

where 1.' is the ratio of specific heats. Equation (43) then becomes

where

The function F contains the entire information on the dependence of the intensity on the mass ratio and CM scattering angle. Figure 16 shows a plot of F for various values of ml /m2 and for a range of scattering angles. Before discussing Fig. 16 we note that Eq. (46) can be further simplified by setting

SCATTERING STUDIES OF EXCITATION MOLECULES

269

FIG. 16. The function F(m,/m,, 9) governing the scattered beam intensity for a given energy resolution is plotted as a function of 9 for different values of m ,/in,.

T, = 300°K and L2 = 3 cm and by expressing B , and B2 in torr liter/sec, AE and E in eV and m2 in atomic mass units (amu), with the result

Thus the signal in a given energy change experiment falls off rapidly with decreasing AEine,and also with increasing m , / m z . Moreover, the signal depends rather strongly on the scattering angle and is expected to be smallest near 90". This is an unfortunate circumstance since, as pointed out in Section V and found in a number of experiments, the inelastic transitions are most probable near 90". In other systems the inelastic cross sections are largest in the vicinity of the rainbow angle and, if this angle is not too small, this is a favorable region t o explore. Although Fig. 16 suggests that back-

270

M . Faubel and J . P . Toennies

ward scattering is also favorable when ml/rn2 is large, one must keep in mind that for small m, /% , 9 3: 0,so that the backward scattered particles are scattered back into the primary beam source. Finally we consider an example taken from our own study of Li+-H2 scattering: ml/m2 = 3.5, B , = 1 torr liter/sec, ECM= 0.5 eV, AEi,,, = 0.04 eV ( j = 0+2), B , = lo9 ions/sec/sr ( z3 x lo-" torr liter/sec). From Eq. (48) we get ions/sec N -=3

*/2

AZ/sr

In the actual experiment the signal was about 3 ions/sec at the rainbow, where do/dw 3: 20 A2/sr, which is consistent with the fact that the resolution was somewhat better than 0.04 eV. Since the detector noise in ion scattering experiments is still smaller than 3 ions/sec, these measurements could be performed successfully. In a neutral beam particle study B , would be greater by 11 orders of magnitude, but the detector noise will also be much greater.

5 . Evaluation of Energy Loss Spectra The immediate result of an energy loss measurement is the velocity spectrum, the energy spectrum, or the time of flight spectrum of scattered particles for a given LAB scattering angle, collision energy, and the distribution of initial states of the molecule BC. To evaluate the measured spectrum for elastic and inelastic transition probabilities, the areas of the partially resolved peaks corresponding to energetically different transitions have to be determined. First we assume that the peak shapes and positions are all known. The normalized peak shapes (unit area) are denoted by hk(x - xk) for the kth peak centered in the channel xk of the spectrum. The determination of the relative areas can then be carried out by building up a synthetic spectrum:

The adjustable amplitudes ak that fit the measured spectrum are directly proportional to the area of the kth transition. From the values of ak the LAB transition probability of the kth transition is obtained directly as P L Z ( E , 0 )= a k / C ak k

(50)

Multiplication of the individual transition probabilities with the measured total differential cross section leads to the differential inelastic LAB cross sections

SCATTERING STUDIES OF EXCITATION MOLECULES

27 1

The LAB differential cross sections can be converted into the C M system using the equations summarized in Table IV. Note, however, that for any given LAB scattering angle the CM differential cross sections correspond to different CM scattering angles, which depend on the relative energy loss AE,,,,/E of the transition. Thus strictly speaking, CM transition probabilities at a gven CM angle can only be obtained by interpolation. This means that the measurements in the LAB system must be so closely spaced in scattering angle that they reveal all the desired angular structure. The actual fitting of the amplitudes a, of Eq. (49) to the measured spectrum usually employs the least-squares method. In this procedure the expression s = Wi(Ni- f ; ) Z

1i

is minimized with respect to the ak,where N i is the content of the ith channel of the measured spectrum,f;. the value of the fit function defined in Eq. (49), and wi a weighting factor taking into account the accuracy of the measured data, usually taken as wi = N ; '. The minimum condition Eq. (52a) leads to a set of equations

aspa,

=o

(52b)

which are linear in the ak and can be solved directly. The positions xk of the respective peaks entering Eqs. (49) and (52) are calculated directly from the kinematical relations Eq. (37). A real problem, however, is the determination of the exact line shape from Eq. (35). The linearized analytic approximation (Section IV,B,3) of the convolution integral results in Gaussian line shapes. In our own work on Li+-H, the halfwidths of these Gauss functions as calculated by Eq. (40b) from the experimental apparatus distributions were found to agree within 10%with the actual half-widths of the measured T O F spectra (see Section V,B). Only the tails of the observed line shapes were not well described by a Gaussian function. The analytic approximation together with the explicit expressions of Table V for the expansion coefficients ci also reveals an increase of the linewidth with increasing energy change AEi'f. This effect is often so small ( 5 lo%), however, that it can be neglected. More sophisticated nonanalytical approaches to line shapes are described in van den Bergh et al. (1973), Eastes et al. (1977), and Duren et al. (1977). In a number of studies a purely empirical approach has been used. Gaussian line shapes with constant half-width were simply estimated from the measured spectra. Procedures of this type have been used by Petty and Moran (1970), Udseth et al. (1973), David et al. (1973), and Rudolph and Toennies (1976). For a typical example of a spectrum fit see Fig. 34. The four Gaussian shaped peaks in Fig. 34 represent the functions a, h, of Eq. (49).Their sum gives the fitting function .f of this equation.

212

M . Faubel and J . P. Toennies

Other properties of an energy loss spectrum can be derived directly from the state to state transition probabilities Pi+f. A quantity of considerable interest in connection with semiclassical theories discussed in Section III,B is the mean energy transfer defined by (LIE), =

f

AEi’fPi‘‘

(534

In many cases the resolution is not sufficient to determine Pi‘I for individual transitions. (AE), can, however, still be determined with good accuracy by direct integration of an unresolved energy loss spectrum. Similarly, higher statistical moments of order n of the probability function Pi’f(AEi+f) can be calculated :

(AE“),

=

CI (AEi+f)”Pi‘f

(53b)

They can be used to determine the variance (n = 2) and higher-order characteristics (Abramowitz and Stegun, 1965, p. 928) of the distribution of an unresolved energy loss spectrum. In determining these high-order moments, corrections for the resolving power of the apparatus become increasingly important. 6. Determination of Experimental Errors

To describe the accuracy of experimental differential inelastic cross sections three errors must be specified: the errors of the transition probabilities or cross sections, the intervals over the collision energy E,, and the scattering angle gcm5averaged over by the experiment. While the error in the transition probabilities can be estimated in a straightforward way from the energy loss spectra, the averaging intervals in E,, and ,9, are determined in a complicated way by the function G defined in Eq. (36) and expressing the distribution of the CM vector (E, 9) as a function of the apparatus distribution vector x introduced in Section IV,BJ. Without dealing explicitly with the convolution integral involved in Eqs. (35) and (36), G can be calculated approximately to second order in the following way. By the aid of the “covariance” matrix 6: a multidimensional distribution h ( x ) is described by the generalized Gauss function h(x) = k exp[-$(x

-X

~ ) ~ C :-- xo)] ~(X

(54)

where (x - x ~is the ) ~transpose of the column matrix (x - xo). When x has only one dimension, Eq. (54) reduces to the normal Gauss function with a: = c I 1= a*,the variance of the distribution. In general (see, e.g., Brandt, Assuming there is no cp dependence of the cross section.

SCATTERING STUDIES OF EXCITATION MOLECULES

273

1964) the matrix elements of c are defined as the second statistical moments of a function f(x):

i"

c..= f'(x)xixj dx

'J

For i # j , the cijdescribe the correlation between the different coordinates x i of the vector x. For statistically independent distributions (such as the apparatus distributions introduced in Section IV,B,3) the cij vanish for i # j and c is a diagonal matrix with 2

Cll

= 01

(533)

the variances in the coordinates x, . If the vector x is linearly transformed into a vector

Y = 4 x - xo) + Yo

(564 the corresponding covariance matrix c, transforms into another covariance matrix (56b) describing now the distribution of y when y and c, are substituted into the generalized Gauss function Eq. (54). The linear transformation from the apparatus distribution vector x to the CM vector y = (E, 9) may be obtained numerically from a linear expansion both of Ec,= i~~ with g = v1 - vz and of g g' = gg' cos 9 (Faubel, 1976). With this transformation the desired distribution G(E, 9) of Eq. (36) is easily determined to second order as a generalized Gauss function [Eq. (5411 with the covariance matrix c,, calculated by the matrix product [Eq. (56b)l from the covariance matrix c, of the independent apparatus distributions defined in Eq. (55b). The diagonal elements cEEand cggof c, are directly the variances of the distribution of E and of 9, respectively, and provide a description of the averaging over the collision energy and the scattering angle. To visualize the averaging of an experiment over expected cross-section structures in the Ecm, 9,, plane it is useful to examine the error ellipses defined by the half-widths of the two-dimensional Gauss function (Faubel, 1976). Figure 17 shows such error ellipses for the elastic (AE = 0) transition of a rotationally resolved Li+-H, scattering experiment at E , , = 0.6 eV. described in detail in Section V.B. The projection of the ellipses on the energy axis and therefore the averaging interval of the CM energy is independent of the scattering angle, as expected. I t is interesting to note that the resolution of the CM energy (2: 43 meV) in this experiment is considerably lower than the energy level resolution of 25 to 30 meV prec, = ac,d

-

M . Fauhrl arid J . P . Tornnies

274

'*CM

0

0.03 rod

.ao

0 -0

e

= 10"

0-0

* AECM

20 rneV

FIG. 17. Error elfipses for the experimental averaging of cross sections over Ecmand ,YC,,. The ellipses are the half-value contours of the two-dimensional Gauss distributions used to describe in second order the experimental weighting of cross sections. The ellipses have been calculated with the apparatus distributions of the .Lit -H, scattering experiment described in detail in Section V.B for E,, = 0.6 eV and for different 'LAB scattering angles. The error ellipses for the j = 0-2 inelastic transitions are similar to those shown for elastic scattering.

dicted by Eq. (40b) and actually observed in the T O F spectra shown in Fig. 20. With increasing scattering angle, both the half-widths and the correlation between the scattering angle and the energy change. As a result the ellipses in the E,,, 9,, plane c h a n g their shape and inclination. The latter two properties also change with the inelasticity AE/E of the considered inelastic transition. For the Aj = 2 transitions observed in this experiment, however, this change was found to be only very small.

V. Recent ExpePimen&atalResults A.

S U R V E Y OF

EXPERIMENTS

Tables VI-VIII survey all of the inelastic scattering experiments at collision energies up to 20 eV that were known as of September 1976. Of these many experiments, a few studies with the best resolution for which a comparison with theory is possible are selected and discussed in detail in this section.

SURVEY OF NEUTRAL SYSTEMS STUDIED BY

Collision partners HD-Ne D2-K HCI-Ar HCI-HCI CsC1-Ar Csl-Ar Csl-Xe CsF-Ar CsCI- Ar CsBr-Ar Csl-Ar KBr-Ne, Ar. N, , CO. C 0 2 KBr-MeOH, H,O, NH, KBr-CH,NO, , (CH,),O. C,H,OH, C,H,, CH,OH CO2-K C0,-Ar, Kr C0,-Ar C0,-Kr CS,-Ne. Ar. Kr. Xe N 2 0 - A r . Kr

Energy transferred”,

+R

-R +R +R

THE

Collision energy (eV)

CM angle (deg)

0.03 0.02 0.05 0.05

50 108 50 50

1

I, +R. V

1

1

1

1

1 +V.R

I

I

i

0.06-0.17

0.35-1.7

i

0.05 ( +1.8 eV in vibrations)

i +R +R +R +R, V +R +R

ENERGY CHANGEMETHOD

0.1 0.08 0.07-0.19 0.16 0.08-0.32 0.08

i

10- 120

Cross section

Experimental technique‘

-

Noz.+TOF 2 vel. sel. Noz. + T O F Noz. + T O F

0.1 A’jsr -

1 )

0.3-350 A’jsr

I

2 vel. sel. Noz. + T O F

40-170

References d e

,f .f 9 9 9

h h

h i j, k

45-80

I 6-28 0- 180 5-180 0- 180 30-180 0-180

1 vel. sel.

I m. n

2 vel. sel.

0

Noz.+TOF

P q

-

r 10 A 2

S

P

Individual quantum transitions could only be resolved in the first two experiments (Buck rr a/.. 1977; Blythe a/., 1964). -, deexcitation; +, excitation; R, energy transfer was attributed to rotational, and V. to vibrational degrees of freedom. The following abbreviations are used: vel. sel.. velocity selector; Noz.. nozzle beam; and TOF, time of flight. The first-named device was used for velocity selection, the second for velocity analysis. Crim er al. (1973). Buck rr al. (1977). ’Beck and Forster (1970). ” Blythe rr al. (1964). Chou rr al. (1973). Farrar ef al. (1973). Donohue cr nl. (1973). Giese and Chow (1975). Loesch (1976). Crim er d.(1974). Armstrong rr a/. (1975). ‘ Farrar (1974). King ( 1974). King rr al. (1973). ” Donohue rr al. (1972). Blais C I ol. (1976b). ‘ Loesch and Herschbach (1972). a

‘

’

‘

TABLE VII SURVEY OF SOMERECENT HIGH-RESOLUTION LOW-ENERGY ( 5 10 eV) Loss SCATTERING EXPERIMENTS USINGTHE ENERGY CHANGE METHOD Collision partners

Initial state ji

Observed transitions'.

Collision energy (eV)

CM angle (den)

Typical cross sections

Experimental technique'

Ref.

3 -7

Hz-Ht HD-H' D,-H+ Hz-H + Hz-H'

I I

0.1

H,-Na' HD-Na'

I

D,-Na+ D,-Ar' NZ-H' CO-H' HF-H' HC1-H'

1

' An = 0. 1, 2. 3, j = (0, 2 )

I T = 300 K 0 0, 1 0, 1 0. 1 0 0, 1

H,-Li+

1

0, 1

, T = 100 T = 300 T=77 T=77 T=300 7 ~ 3 0 0

An

= 1. 2. An = 0, 1 ;

3, 4 Aj =(O. 2). 4. _ _ .20 . An = 0, 1; Aj ={O. 2). 4. _ _ 20 .. An = 0, 1, 2, 4 Aj = 0, 2 An = 0, 1, 2. 3 Aj = (0, 2) Aj = (0, 2j4. 6. 8. 10 An = 0. 1. 2, 3 Aj = (0. 2)

I. ( A € ) = 0 . 1 - 4 . 0 I

= 1.2 (AE) c o (AE) c 0.1 (A€) = 0.35-0.57 (A€) = 0.20-0.45

An

I

' 4-21

I

3-1000 3.7 3.7 6.6-10 0.6 3.6 0.5-3.6 3.6-8.8 2.4-5.0 8-29 3-17 11-17 10-30 10-30 10-30 10-30

1

6-22

'

I

0 20-60 20-60 0-35 14-32 150-175 10-40 125- 170

1

180

I

I

0

I

i

'Isr 0.01-1.5 -

10 AZ/sr -

0.1 A'jsr ~

~

5, 11

A'

-

EDF EDF EDF EDF TOF TOF EDF TOF TOF

d d d e

f .f

9 17

c

t

3-

a

a 3

P

I-

*a Y

2

I

3

TOF

.I

3.

EDF EDF EDF EDF EDF EDF EDF EDF

I

TOF

k 1 1 in 11 It It

n

2

CH,-H+ CD4-H+ N,-Li+ CO-Li O;-Ar C0'-Ar Hi-Ne H,O-Li * C02-H+

T=77 T=77 T,,, r 15 T,,, 2 8.5 ? ? ? T=300 T=300

(AE) = 1.6 (AE) = 1.6 An = 0. 1, 2; A j 2 {O-50) An = 0. 1, 2; A j 5 {O-SO) An = 0, 1, 2, ..., 12 A n = 0 , 1 . 2 ,..., 12 (AE) 2 0.7 (AE) 2 3 000 -+ nOO, OnO, OOn ( M = 1, 2, 3)

C0,-Li+

T,,,

N,O-Li+

T,,,= 30

OO0-+010,020,030 A j = {0-101

+

2

30

I

8-20

8-20 4.2. 7.1 4.2, 7.1 7.5 1-15 16 7.2, 12.6 15-50

10 10 3749 37-49 0-45 0-42 0-11.5 5-20 0-12

2.8, 6.9

10-40

2.8, 6.9

10-40

-

-

EDF EDF TOF TOF EDF EDF EDF EDF EDF

-

TOF

0.5 F\'lsr 0.5 A2jsr -

-

-

TOF

(AE) is the first moment of the energy loss distribution in eV. If listed, specific quantum transitions could not be identified. The states in curly brackets { ) are thought to be excited, but were not resolved. ' Only the analyzing field is listed; EDF, electric deflection field. Udseth et a/. (1973). " Udseth et a/. (1974). Moore and Doering (1969). Gentry et a/. (1975). Bottner ef a/. (1976). Rudolph and Toennies (1976). q Cosby and Moran (1970). Schmidt et a/. (1976). Petty and Moran (1972). van den Bergh er a/. (1973). ' Moran et a/. (1971). Faubel (1976). ' Petty and Moran (1970). j Faubel et a/. (1975). " Siskind er a/. (1976). ' David et a/. (1973). " Krutein and Linder (1977). Dimpfl and Mahan ( 1974). Eastes el af. (1977). Moran and Cosby (1969).

'

'

278

M . Faiihel and J. P. Toennies

Table VI lists the energy change experiments on neutral particle collisions. It is interesting t o note that in only two cases (first and second entry) has it been possible to resolve quantum transitions, testifying to the difficulties of such experiments. Several of the more important of these experiments are discussed in detail in Sections V,B-D. Table VII lists ion beam experiments. The use of ion beams has several advantages over neutral beams:

(1) Beam energies above 1 eV at which vibrational excitation becomes probable (see Fig. 3) are easier to achieve. (2) Ions can be easily energy selected using electrostatic deflection fields. (3) Ion detectors have an efficiency close to 100% and a low background ( 5 lo-' ions/sec; Faubel, 1976). (4) The detector is fast enough (21 10 nsec) to make high-resolution time of flight experiments possible. Because of the higher energy available to ion beams, vibrational transitions have been resolved in a large number of systems, a number of which are discussed in Sections V,B and V,C. On the other hand, rotational transitions are more difficult to resolve because of the difficulties in achieving low beam energies. Rotationally inelastic studies of Li+-H2 and H+-H, are discussed in the next section. Finally, Table VIII summarizes neutral beam experiments utilizing the state selection method. The first two experiments are the only ones in which the primary molecular beam was state selected and initially in one defined state. In all the other experiments the molecular beam had a distribution over a number of initial states. The third entry refers to a particularly interesting study of inelastic differential cross sections combining nozzle beam cooling with electron impact stimulated fluorescence for state analysis of the beam before and after scattering. Unfortunately, despite extensive rotational cooling in the nozzle beam expansion to T , , z 9"K, six rotational levels were still appreciably populated. Since about the same number of levels are found in the scattered beam the analysis requires a fit of at least 14 different elastic and inelastic cross sections connecting the initial and final state. Similar problems were encountered in the next set of experiments on HCl and HF. Here the rotational spectrum of the spontaneous infrared emission from the I I = 1 state was used to determine the rotational populations before and after scattering. Because of the requirement of having a significant n = 1 population, the beams were heated to T = 1900°K and as a consequence contained about 16 rotational states. Only total cross sections could be measured in this apparatus. In the scattering of H F from HCI (fifth entry) there is evidence for resonant rotational energy transfer. In the last experiment indirect information on energy transfer is obtained using a molecular

TABLE VIII SURVEY OF STATE-SELECTED INELASTIC-SCATTERING EXPERIMENTS

Colhsion partners" TIF-He. Ne, Ar. Kr, Xe, H,, O , , N,. H,O, N,O. CH,. SF,. CF,CI, , NH, CsF-Ne, Ar, N,. CO, CO,, CHF, , C,H6 N,-Ar HCI-He, Ar, Kr. HF, HCl, HBr, HI. H,S. C,H, HF-Ar, HCl LiF-Ar, Xe, N , , 0,. NO, CO, HF, HCI, CO, , H,O, SF6 NH3, NO3 CHFC1,

Observed final states

Initial statesh j . m = 1. 0:

j . m = 1, 0;

2,O; or 3.0

Collision energy (ev)

CM angle (d&

0.02

(1-4)

4-600

5-600

2, 0; or 3 , O

j. rn = 1, 0; 1, 1;

.j, m = 1, 0;

0.02

(1-4)

2.0; 3,O j = [0, .... 51

2, 0; 3. 0; 3, 1 j = 0, .. ., 5

0.05

9-21

j = [0, _ _161, _ .n = 1

Cross Section

,j = 0, _ _16; _ ,n = 1

j = [0, . . . , 161, n = 1

j

Thermal at 1100°K

.j = 1, rn = 0, f1 = 0. 1, 2, 3

= 0, . _ . ,16; n = 1

0.2-1.3

(0-180)

0.2-1.3

0-180

0.09

o,,, = 20

A'

-

~

-

0.02 A'jsr

Experimental technique

Referenee

electrostatic quadrupole

d

electrostatic quadrupole electron fluorescence spectra infrared

e..%g

emission spectra infrared emission spectra electric resonance spectroscopy

/I,

i

j

k

3

* With only a few exceptions (Borkenhagen et a/., 1976; Mariella et a!., 1974) the target beam state distributions are at T = 300°K or somewhat below. Transitions can occur from all the states in brackets [ ] to all the observed final states. The parentheses indicate the range of scattering angles analyzed. Toennies (1965). Scott and Mincer (1971). " Borkenhagen er a/. (1976). ' Scott er a / . (1973). Borkenhagen et a/. (1978). ' Ding and Polanyi (1975). Borkenhagen (1977). ' Mariella er a / . (1974).

M . Faiihel and J. P . Toennies

280

beam resonance apparatus to measure the vibrational state distribution of one rotational state ( j , m ) = (1, 0) of LiF with and without scattering. Since the beam was appreciable excited (56:/0 in n = 0, 31% in n = 1, 10'9,1 in n = 2, and 3",, in n = 3) the results in this case could only be interpreted in terms of relative vibrational temperature changes. The interpretation is difficult since at least five different types of scattering processes contribute to the observed effect.

B. SCATTERING FROM H2 MOLECULES 1. The Rotational and Vibrational States of H2

The hydrogen molecule is both the simplest of all molecules and has the = 43.9 meV, AEj="3 = 72.8 meV, largest energy level spacing (AEj"" and BE"="' = 0.516 eV). The dissociation energy is 4.48 eV and the first excited electronic level is at 11.4 eV. The large level spacing not only makes collisions with H2 easier to resolve in energy change experiments but also attractive from the theoretical point of view (see Section 111). Hydrogen has two forms: para-hydrogen (p-H,) with even j states, and ortho-hydrogen (0-H,) with odd j states. Transitions between the two forms are forbidden in nonreactive collisions by parity conservation rules, and therefore only even numbered Aj are observed in inelastic collisions. According to the nuclear spin weights (21 I), natural hydrogen (n-H,) occurs with an o : p abundance ratio of 3 : 1. Normal H2 can be converted to p-H2 by means of a catalyst and scattering experiments are possible with p-H, . Small amounts of 0-H2 can also be prepared in the pure form but the procedure is more complicated and not suitable for scattering experiments. At liquid nitrogen temperatures (77°K) H2 is nearly completely ( 5 99%) in its ground states (n = 0; and j = 0 or j = 1, resp.). At 300°K the j = 0 , 2 , 4 states are populated in the ratios 0.53,0.46,and 0.01, respectively,while thej = 1 , 3 , 5 states are in the ratios 0.89, 0.1 1, and 0.001.

+

2. Inelastic Scattering of Li+-H2 This four-electron system has been more extensively studied experimentally than any other system. At E,, 5 1 1 eV the only energetically accessible processes are rotational and vibrational excitation and dissociation. Since Li' is detected, rn, is greater than m2 (m,/m2 = 3.5) making it possible to observe CM backward scattering. Since = 2/9ELAR, the beam intensity, which is limited by space charge effects and therefore is not so much of a problem. These advantages are offset depends on ELAB, by the small signal resulting from the very high angular resolution (see Fig. 16).

SCATTERING STUDIES OF EXCITATION MOLECULES Unscattered ions

28 1

/

7

Inelastically scattered ions

Elastically scattered ans

source

Electrostatic sector field

FIG 18 Schematic perspective diagram of the time of flight apparatus used to study inelastic scattering of Li+-H, The ion beam source is at the bottom left The ion beam energy IS selected by a 127" electrostatic sector field The beam is then electrically chopped and I S scattered from a skimmed nozzle beam of H, The flight time to one or more secondary electron multipliers is measured

Figure 18 shows a perspective schematic diagram of the time of flight apparatus (van den Bergh et al., 1973; Bottner et al., 1976; Faubel, 1976). Isotopically pure 'Li' ions are produced at 1000°K in a commercially available surface ionization source. The ions are accelerated and energy selected by a 127" electrostatic sector field energy analyzer (radius 5 cm, A E / E = 0.4%). The ion beam is then pulsed at lo4 sec-' by a deflection plate arrangement located between the sector field and target beam. Normally the ions are deflected to one side. By switching off the voltage for a short time ( 2 1 psec) ions can pass into the scattering region. Because of the finite flight time in the deflection plates the ion pulse is considerably shorter ( 2 100 nsec). The H2 target is a skimmed nozzle beam with an angular spread of 3 to 4' and an effective width of 2 mm. At a nozzle stagnation chamber temperature of 300°K the velocity component along the primary beam direction is only 1.7 x cm/psec. This corresponds to an effective target temperature of only 1°K. The nozzle stagnation chamber can be cooled t o 900K to produce a target containing either only j = 0 (p-H,) or j = 0 and 1 states (n-H,). The ions are detected by a venetian blind multiplier or by an array of five channeltrons (continuous secondary electron multipliers with a semiconductive layer). The absolute flight time of each ion arriving at one of the detec-

M . Fatlbel and J . P . Toennies

282

tors is measured by a digital crystal clock circuit and stored in a minicomputer. In a series of experiments each with improved resolution and decreased noise, both rotational and vibrational excitation have been resolved in the energy range 2.77 (0.6) -S EL, (ECM)< 39.6 (8.8) eV. References to the published work are given in Table VII. Here we discuss only some of the latest results, some of which have not been published previously. For a comparison with close-coupling calculations, experiments at the lowest possible energies are desirable. Figure 19 shows an example of a

LA8 scattering angle B

FIG. 19. Intensity of Li ions as a function of laboratory scattering angle measured with and WithoBt scattering gas. The ion beam chopper was turned ON.Both the primary rainbow at 6" and the maximum possible LAB angle at 17" can be clearly seen in the signal measured with scattering gas. Lif +n-H2, ELAB= 2.77 eV. +

measurement of the total intensity (without TOF analysis) of the unscattered and scattered Li' beam as a function of the laboratory angle at EL, = 2.77 eV. Note that the unscattered beam intensity has fallen off by a factor lo-' at OLAB N 5". A narrow, well-collimated beam of this quality is needed since the scattered beam signal amounts to only lo-' of the unscattered beam signal. Indeed, the sharp rise of the unscattered signal at

SCATTERING STUDIES OF EXCITATION MOLECULES

283

0 5 4" produces so much noise that scattering measurements at smaller angles are not possible. The primary rainbow is clearly visible in the scattered signal at 6".For the Li+-H2 mass combination, kinematics limits the scattering in the LAB to 0 d 17" and as seen in Fig. 19 no scattering is observed beyond this angle. The background signal is only about 4 x lo-' ionslsec.

Time of flight spectra at 15 LAB angles

FIG.20. Time of flight spectra measured at 15 LAB scattering angles for Li+-p-H2( j = 0 ) at EL, = 2.77 eV are plotted according to their angular location (Faubel, 1976). The peak heights have been adjusted to correspond to the measured total differential cross sections measured without time of Right resolution. The total Right time of the left elastic peak is r 100 psec. The TOF channel width is 0.16 psec. The total integration time for one spectrum is 100 hours.

Figure 20 shows a series of TOF spectra measured at closely spaced angles with a five-channeltronarray ofdetectors each separated by about A@ 2 0.3". The peak heights have been adjusted to correspond to the total differential cross section (see Fig. 21). At each angle two peaks are visible. The left peak at shorter times is due to elastic scattering ( j = 0-0) and the right peak is due to inelastic scattering ( j = 0-2). Aj = 4 transitions are much less probable by an order of magnitude. In addition to the rainbow at 6" the fast undulations in the total cross section are visible as a rise and fall in both peaks. A careful examination of the relative heights of the maxima shows that the relative heights of the elastic and inelastic peaks also change with angle. For example, at 7.0 and 7.7"elastic scattering is more probable, while at 7.3 and 8.0" inelastic scattering is more probable. Using the fitting procedures discussed in Section IV,B,S experimental transition probabilities and differential inelastic cross sections have been determined from the time of flight and the total differential cross section data. The final results for the total, the j = O - r O elastic and the j = 0+2

M . Faubel and J . P . Toennies

284

,001

LAB scattering angle 8

FIG. 21. Experimental differential cross sections measured at E,,, = 2.77 eV (E,, = 0.6 eV) for total, elastic (,j = 0-0) and inelastic ( j = 0 4 2 ) scattering of Li+-H, are compared with theoretical results for the nominal experimental energy of E,, = 0.6 eV in (a) and for a smaller energy of&, = 0.5eV in (b). The absolute values of the experimental results have been adjusted to match the absolute theoretical total cross sections. Excellent agreement both in the shape and relative magnitudes of elastic and inelastic cross sections is only found at E,, = 0.5 eV. No satisfactory explanation for the apparent shift in energy has been found. 0, experimental; -, theoretical.

inelastic differential cross sections at E,, = 0.6 eV are displayed in Fig. 21. They are compared with the theoretical cross sections of Lester and Schaefer at two CM energies. For the comparison the theoretical results were transformed into the LAB system and folded with a Gaussian angular distribution with A@ = 0.40" to take into account the apparatus smearing. The scaling factor from the measured experimental intensities to the absolute value of the total differential cross sections was adjusted to match the theoretical value. The agreement between the experimental total differentialcross section and the theoretical result for E,, = 0.6 eV (Fig. 21a) is rather poor. The theoretical rainbow structure appears at 15 to 20% smaller scattering

28 5

SCATTERING STUDIES OF EXCITATION MOLECULES

angles than the experimentally observed structure. However, excellent agreement is found for 0 2 5.5" with calculations for ECM= 0.5 eV, both with regard to the relative sizes of the elastic and inelastic cross sections and the fast undulations. The remaining discrepancy at 0 5 5.5" is partly due to experimental errors (see Fig. 19) and may also be due in part to numerical errors in the calculation (Schaefer, 1976). The good agreement with the ECM= 0.5 eV theory suggests that the experimental energy calibration may be in error. A recent exhaustive analysis of the absolute peak positions in all available TOF spectra indicates, however, that the actual collision energy in the scattering region agrees with the nominal energy to within better than 7%, indicating that E,, = 0.6 k 0.04 eV (Faubel, 1976). The shift in the rainbow could also be explained by an error in the calculated potential well depth E. However an adjustment of E would probably not affect the ratio of the elastic to inelastic cross sections, which would then still be in disagreement. To understand the remaining small disagreement it would be interesting to carry out a computational search for slightly modified potentials that fit the measurements. Many more results are available at higher energies. Figure 22 shows an example of a TOF spectrum measured at ELAB= 13.76 eV (E& = 3.06 eV) with n-H, . The large first peak at the left is due t o small-angle scattering in the CM system (as in Fig. 20). The smaller slow peak near channel 400 is due

', '

.lo+

50 '

'

-

Flight time [psec] 55 80 ' ' I " Li*- n - H2

85

forward scattering (\)t#23-20'1 /J=1-]'.7

400

C

r u

300 10-3

f

;

ul

200 backward scattering IbcM=160.1 j=l-j= 7 5 31

t+n

100

C 3

8 c 0

0

100

-

2

200 3 00 400 Channel numberllch =40 nsecl

FIG.22. Time of flight spectrum measured at ELAB= 13.76 (EcM = 3.1 eV) and 0 = 5" for Li '-H2. The large peak at the left is due to CM forward scattering, while the small peak at the right is due to CM backward scattering. The small peak at channel 30 (47.5 psec) is due to "blastthrough" of the unscattered primary ion beam. The CM angles given in the figure are for elastic scattering. The measuring time was 65 hours.

286

M . Faubel and J . P . Toennies

AE inol

FIG.23. The relativeCM transition probabilitiesmeasured for Li+-H2 at three different LAB angles at ECM= 3.1 eV are plotted as a function of AE,,,,.With increasing LAB angle, transitions with larger AE,,,,become more probable.

to large-angle scattering in the CM system. Rotational transitions with AJ > 2 are clearly resolved in the fast peak. Although not resolved, the elastic peak is smaller than the Aj = 2,4, and 6 peaks. Similar spectra have been taken with p H 2 (Faubel et al., 1975). The measured transition probabilities for both p-H2 ( j i = 0 )and o-H2 ( j i = 1) transitions (from n - H , experiments) are plotted as a function of AEine,in Fig. 23. Although the transition probabilities are different, the smooth curve connecting all the data indicates no significant symmetry effect. The curves show that the most probable transition shifts to larger AEine,with increasing scattering angle. The transition In a similar way, the probability for a given AEine,also increases with OLAB. transition probabilities measured at ECM= 0.6 eV for the j = 0-2, 0-4, 1-3, and 2-4 transitions were also found to lie on a smooth curve when plotted against AE,,,,. In spite of this evidence for a functional dependence on AEinelpoor agreement was found with the Raff-Winter (1968) -PolanyiWoodall (1972) semiempirical correlation for rate constants nor with linear surprisal (Bernstein and Levine, 1975). The results presented in Fig. 23 rule out extensive vibrational excitation in forward scattering. Extensive vibrational excitation would lead to a deviation in the probability distribution for the j = 0-8 transition, which is not observed. As expected from Eq. (12) and Fig. 3, vibrational excitation was, however, observed in backward scattering at EL.AR 2 16.6 eV (David et al.,

SCATTERING STUDIES OF EXCITATION MOLECULES

No. of

Lit-p

- H2

287

Not of

traj.

ions

2000

1500

1000

1250

0

t

t t tAn,=l 4 20-j

6

t ttAn= 4 20-j'

0

FIG.24. Comparison of experimental (0,Faubel, 1976) and calculated (-, Barg et al., 1976) time of flight spectra for Li+-H2 at ECM= 3.6 eV showing vibrational inelastic maxima. The calcuhted transition probabilities were assigned Gaussian profiles to take account of apparatus smearing and adjusted so as to match the "elastic" peak. The dotted line shows the sum of the Gaussian spectra. The Lester SCF potential was used in the calculations. In the experiment the total measuring time was 24 hours and the channel width is 200 nsec. QCM = 170".

1973). Recent experimental results for scattering from p-H, (Faubel, 1976) are shown in Fig. 24. These and other new spectra indicate that the n = 0- 1 transition (summed over rotational transitions) has a probability of about 10.5 k 2% and is nearly canstant at CM angles between 155 and 170". Although rotational transitions could not be resolved, the small width of the vibrationally elastic peak indicates that the A j = 2 transition contributes less than 30% at 170". Figure 24 also shows the results of a classical Monte Carlo trajectory calculation (Barg et al., 1976).The measured vibrational transition probability agrees with theory within 20%. The DECENT model has been applied to this problem and also provides good agreement with the experimental results (Gentry and Giese, 1975). Comparable agreement has also been achieved with a semiconverged quantum rnechanical scattering calculation with 118 channels (Schaefer and Lester, 1977). 3. Inelastic Scattering of H+-H2

With only two electrons, the H+-H2 system provides the simplest example of a chemical reaction and a molecular interaction exhibiting curve crossing. Thus even at energies below the dissociation threshold (4.48 eV)

M . Faubel and J . P . Toennies

288

three additional endothermic processes not present with Li +-H2, are possible (Krenos et al., 1974):

H’

+ D2+{

D+ + HD, AEo = 0.04 (0) eV

H

+ Dl,

inelastic scattering

(57a)

reaction and charge transfer

(57b)

AEo = 1.85 (1.83) eV charge transfer

(574

where D2 is used to distinguish nuclei and the energies AEo in parentheses are for Hf-H2. Extensive ab initio calculations of the hypersurface have been carried out (see Table 11). Two different projections of the potential hypersurface are shown in Fig. 25. Figure 25a shows the Legendre terms in the anisotropic rigid rotor potential (McGuire, 1976) derived from the analytic potential fit of Giese and Gentry (1974). Compared to Li+-H2 (Fig. 5) E is much greater and R , much smaller. This is expected for a reactive system with a chemically stable intermediate state, but the difference is especially large in this case because there are no electrons on the proton to begin with. The u2 term is about the same size as in Li+-H2 but is negative at intermediate distances. At distances greater than 3 A the u2 term is dominated by the classical charge-induced dipole and charge-induced quadrupole terms. The u2 term is slightly positive because of the positive sign of the quadrupole moment (McGuire, 1976). At R < 1 A the potential shows a very strong anisotropy, which can be simply attributed to the close approach of the proton to the H2 as well as to the strong chemical bond, which favors the triangular H: structure. The reactive part of the potential hypersurface for collinear orientation is shown in Fig. 25b (Krenos et al., 1971).Note that at the minimum the H2 bond is “diluted ” by the presence of the proton and expands from re = 0.74 to 0.81 A. The expansion is even greater for the triangular geometry. The first quantum-resolved inelastic experiments were reported by Moore and Doering (1969), and a number of other studies have followed. We start our discussion, however, by first reviewing some recent TOF experiments on rotational transitions at low energies (Rudolph and Toennies, 1976) since they provide the most direct comparisons with theory. The apparatus used is very similax to that shown in Fig. 18.

SCATTERING STUDIES OF EXCITATION MOLECULES I

I

I

I

I

289

6.0

I

I

1.o

I

I

2.0

I

R [A1

3.0

I

I

H*+H2JH+ H;

3.0r2

[A1 2.0-

1.0-

-

'

1.o

I

I

I

ot,

t=O

2 .o

3.0 r1

[A1

FIG.25. Two projections of the potential hypersurface of the lowest electronic state of H : . In (a) the Legendre terms of the anisotropic rigid rotor potential derived by McGuire (1976) from the Giese and Gentry (1974) analytical potential model are plotted as a function of R. In (b) the equipotential lines of the reactive potential hypersurface for collinear approach calculated by the diatomics-in-molecules method (Krenos et al., 1971)are shown. r , and r2 are the respective interatomic distances. Note that the Legendre terms in (a) are a function of the Jacobi coordinate R = rl + ir2 (for the collinear arrangement).

290

M . Fauhel and J . P . Toennirs

LABangle

in

degrees

FIG.26. The total scattered H + ion intensity, which is nearly proportional to the total differential cross section, is plotted as a function of laboratory scattering angle at ELAR = 5.58 eV ( E & = 3.7 eV). The measunng time was between 100 and 1000 sec/angle. The primary rainbow and several fast undulations are clearly resolved. The arrows indicate the angles at which the time of Bight spectra in Fig. 27 were measured.

A plasma ion source with mass a spectrometer was used as a source of H f ions. Figure 26 shows the scattered H + intensity at E,,, = 5.58 eV (E&, = 3.7 eV) as a function of LAB angle. As with Li+-H, the scattered signal is nearly directly proportional to the differential cross section. The rainbow in this case is at 36". The numerous undulations at smaller angles are probably due to fast undulations, which because of the different mass ratio, smaller reduced mass, and R , value appear to be more widely spaced than in Li+-H2. Figure 27 shows some representative TQF spectra taken with p-H2 and n-H, at the angles marked by arrows in Fig. 26. The spectra show several striking differences to Li+-H2 spectra at the same energy (see Figs. 22 and 24). For one the elastic cross section is always considerably greater than the inelastic cross section. Furthermore, altogether many more transitions with large Aj (up to 20) have been observed (Rudolph and Toennies, 1976). Other striking effects are the observation of a minimum in the distribution of peak heights, especially noticeable for Aj = 6 at 0 = 15" and the rather sudden transition from rotational to vibrational excitation at the rainbow angle 0 = 36".Finally we note some apparently spurious maxima between ,j' = 14 and 16 at 0 = 15". These could be due to para-ortho transitions accompanying reactive scattering and provide a possible means of distinguishing between charge transfer and reactive processes. The experimental transition probabilities shown in Fig. 28 confirm the observations made above. The distribution of transition probabilities is seen

SCATTERING STUDIES OF EXCITATION MOLECULES

29 1

delay time in Usec

FIG.27. Time of flight spectra for H+-p-H, (Q = 10 and 15") and H+-n-H, (0= 36") are shown at three different laboratory scattering angles. The collision energy is Em = 3.7 eV. The number of ions normalized to the maximum, registered over the entire measuring time (25-60 hours) in each time of flight channel (40 nsec), is plotted as a function of the flight time (flight length = 1.25 m). Typically N,,, varies between 10,OOO and 500. The vertical lines show the absolute calculated locations of the arrival times corresponding to the indicated final rotational states. The center of mass scattering angles for elastic scattering corresponding to the three laboratory angles are in increasing order S,, = 15.0, 22.4, and 53.1".

to be extremely sensitive to the scattering angle, which is especially apparent in going from 20 to 21.6". The results of measurements for p-H2 and o-Hz transitions are shown at 20". As found in Li+-H2 at about the same energy, there are no essential differences between the two species. This justifies a comparison of the o-H2 data with calculations for p-H2 transitions. The results of two quantum mechanical scattering calculations for the Giese and Gentry potential fit are also shown in Fig. 28. In each case approximations were necessary because of the large number of rotational transitions. Thus Choi and Tang (1976) in their close-coupling calculations

292

M.Faubel and J . P. Toennies I

I

I

I

4 6 8

"'1. 0

I

10 12

I

I

I

16

16

18

7 , s

1 1 , 1;

lr

0.5

1.0

1.5

I l l

I

I

20

lf

1;,

2.0

AE (j -j') Lev1

, 2.5

3J

FIG.28. Experimental rotational transition probabilities (Rudolph and Toennies, 1976) are plotted against AE,,,,for p H , ( 0 )and o-H, ( 0 )transitions. The results are compared with close coupling calculations (A) with a limited basis set (Choi and Tang, 1976) and an approximate calculation ( 0 )using the j;conserving method (McGuire, 1976). Vibrational channels not observed experimentally at these angles were not included in either of the calculations.

had to restrict the basis set to j = 0, . . ., 12 but found evidence that their results for small A j converged in the angular range studied. Since McGuire (1976) used the j,-conserving approximation he was able to include j = 0, ...,30 in his basis set and get results for the higher transitions. Both calculations agree qualitatively with the experimental trends. The small transition probability A j N 4and 6at lOand 15" has been attributed by McGuire (1976)

SCATTERING STUDIES OF EXCITATION MOLECULES

293

to trajectories passing through the region near R z 1.3 A, so that the contribution from the v2 term nearly averages to zero. Finally the deviations of the j,-conserving results and experiment at large Aj have been attributed either to competition from reactive transitions, which as noted earlier may have been observed in the experiments, and/or vibrationally inelastic transitions, or to errors in the potential hypersurface in the region of R 5 1 A (McGuire et al., 1976). Additional insight into the collision dynamics is provided by the classical calculations of Giese and Gentry (1974), which predicted a large maximum in the first moment of the energy loss distribution [Eq. (53)] at an angle of about one-half the rainbow angle. This prediction of an inelastic " rainbow " has been confirmed by experiments (Rudolph and Toennies, 1976) and quantum theories (Choi and Tang, 1976; McGuire, 1976). The classical calculations suggest that the large quantum transitions producing the inelastic rainbow are due to " hard " collisions from the repulsive core (trajectories of type 3 in Fig. 2). The large observed differences between the rotational energy loss for Li+-H2 and H+-H2 are due to a number of effects and are by no means fully understood (Rudolph and Toennies, 1976). Timeresolved classical trajectory calculations (Barg and Toennies, 1977; Barg rt al., 1976) provide some insight. They indicate that Li+-H2 is highly rotationally excited during the collision, enabling the molecule to orient itself in such a way that the potential energy of interaction is maximized. The inelasticity maximum described above is only observed in H + - H 2 . It is consistent with an impulsive collision in which reorientation cannot occur, and thus the interaction is not so strong. Vibrational excitation was first resolved by Udseth et a/. (1971, 1973) at E:,,, = 10 eV at scattering angles up to the rainbow angle. In their apparatus, mass-selected ions from a nearly monoenergetic novel ion source were scattered in a liquid-N,-cooled scattering chamber and energy analyzed by an electrostatic deflection field. Their results have recently been improved upon by Schmidt et al. (1976),who used two tandem 127"electrostatic sector fields for selection and analysis. A sophisticated ion optical system enables them to obtain a large-beam intensity of 2 x 10- l o A with AE = 30 meV at ELAB= 10 eV and AO1 2: 0.5-1". The target was a nozzle beam similar to that used in Li+-H2 (Fig. 18). The overall angular resolution was better than about 2" and the overall energy resolution was about 45-55 meV. Figure 29 shows a sample energy loss spectrum measured at E:,,, = 10 eV and @LAB = 13". This angle is less than the rainbow at about @LAB 2: 15" (&.. = 23") (Udseth et al., 1973). A large amount of vibrational excitation with transitions up to n,= 4 are observed. The low intensity between 150 ImeV) indicate that very maxima and the small widths of the peaks ( little rotational excitation is present. This observation is of interest in con-

M . Faubel and J . P . Toennies

294

.. . I

.. ..

n'=l

2 '

d.2

Energy of scattered ions [eV]

Re.29. Energy loss spectra measured for H'-H,

with an apparatus using tandem 127" electrostatic sector fields as selector and analyzer (Schmidt et al., 1976). Transitions up to An = 0+4 can be observed. The measuring time was of the order of 1 hour, ELAe= 15 eV, = 10 eV, 0 = 13".

em

nection with the DECENT model calculations of Giese and Gentry (1974), which predict that each vibrational transition is accompanied by a " tail" of large quantum jump rotational transitions similar to those measured at low energies (see Fig. 27). The new data indicate that the contribution from rotational transitions is smaller than predicted. Figure 30 shows a comparison of the experimental results of Udseth et al. (1973) and Schmidt et al. (1976kwithout corrections for rotational excitation-with the semiclassical DECENT model calculations. The overall agreement between the two experiments is quite good. The comparison suggests that the theory tends to underestimate somewhat the transition probabilities. This still holds even if the (probably too large) corrections for rotational excitation are applied. Competition from the other processes in Eq. (57) could affect the experimental transition probabilities especially at large angles. However, on the basis of the measured energy dependence of the integral cross section (Ochs and Teloy, 1974) it appears that reactions (57b) and (57d) are improbable at these energies. Only charge transfer processes are probable and at E,, = 10 eV have integral cross sections of 0.7 A2,which is to be compared with the theoretical integral cross sections for vibrational excitations of 6 A2 for n = 0- 1, 2.5 A2 for 0-2, and 1 A2 for all higher transitions (Collins et al., 1974). The large differences in the vibrational excitation probabilities observed for Li+-H2 and H+-H2 can be understood in terms of the different types of interactions. In H+-H2, strong valence forces are largely responsible for vibrational excitation. These forces have a long range and are nearly propor-

SCA'ITERING STUDIES OF EXCITATION MOLECULES

0.02

**

0.

10"

295

/ 20'

30'

Scattering angle &-M

FIG. 30. Vibrational transition probabitities for H+-n-H2 measured at EL, = 15 eV (Efm= 10 eV) by two groups, Udseth et 01. (1973) ( 0 )and Schmidt et al. (1976) are compared with DECENT model calculations of Giese and Gentry (1974) (-). The experimental results have not been corrected for rotational excitation.

(o),

tional to the radial forces producing the deflection (Giese and Gentry, 1974), thus explaining the predominance of vibrational excitation in the rainbow region. In Li+-H2, strong attractive valence forces are absent and "harder " repulsive interactions are needed to excite the vibration. 4 . Inelastic Scattering of HD-Ne As noted in connection with Table VI, only a few experiments with quantum state resolution have been successfullycarried out with neutral collision partners. The major source of experimental difficulty is the discrimination of the scattered beam signal against the background in the mass spectrometer detector. Nevertheless, rotational structure could be resolved in a very recent preliminary study of HD-Ne scattering (Buck et al. 1977). In this experiment the scattered HD beam is detected in the plane of the incident beams, which are generated by nozzle beam sources. As a result of extensive liquid-nitrogen cooling and two differential pumping stages in the detector, the pressure in the ioniiation region was about lo-" torr. At these low pressures most of the background is H, and for

296

M.Faubel and J . P . Toennies

Flight time [paeC]

FIG. 31. Measured time of flight spectrum lor inelastic scattering of HD from Ne at E,, = 30 meV and 0 = 50". A pseudorandom-chopping technique was used to discriminate the 1 % signal against the noise of the large background signal.

this reason HD, which has a low background signal, was detected. Figure 3 1 shows one of the first measured TOF spectra obtained with this apparatus (Buck et al., 1977). The j = 0- 1 maximum is clearly resolved from the elastic peak. As indicated, the background intensity was about 3 x lo6 counts/sec and the inelastic signal was only about 2 x of the background. In order to suppress the background noise a pswdorandomchopping TOF technique was used. The results shown in Fig. 3 1 are so recent that they have not yet been fully interpreted. Since the interaction potential is referred to the molecular center of mass, the interaction potential for HD-Ne will contain a sizeable PI-term coming from the known displacement of the mass, center away from the geometric center (van Montfort et al., 1972; Kreek and LeRoy, 1975). Thus only a small contribution of between 20 and 30% is expected from the physically interesting P,-term predicted by an ab initio calculation. C. SCATTERING FROM HEAVIER LINEARMOLECULES 1. Molecular Properties of N, ,CO, N 2 0 , and C 0 2

Because of the wide energy level spacing of their rotational and vibrational energy levels, H, and the hydrogen-halides are not representative of most collision partners. The study of rotational excitation of more typical molecules such as N2poses formidable experimental problems having to do

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297

with the narrow rotational level spacing ( B = 2.491 x eV). This means that the cooling in a nozzle beam will not suffice to ensure that all the molecules are in the lowest rotational state. Moreover, the resolution requirements of the apparatus must be increased by nearly two orders of magnitude. Vibrational excitation is easier to resolve since the vibrational spacing of many molecules is greater than about 100 meV. N, and CO have especially large spacings of 0.29 and 0.27 eV, respectively. These two molecules are ideally suited for a comparative study since their masses are identical and most molecular properties are nearly identical. The asymmetry of CO leads to a small dipole moment (0.13 D) not possessed by N, and a 50% larger quadrupole moment (-2.5 x esu) than N, (- 1.5 x esu). The linear molecules CO, and N,O also have equivalent masses and similar molecular properties. They have the following vibrational modes: bending (010) with AE(C0,) = 83 meV and AE(N20)= 73 meV; symmetric stretch (100) with AE(C0,) = 172 meV and AE(N,O) = 159 meV; and asymmetric stretch (001) with AE(C0,) = 291 meV and AE(N,O) = 276 meV. Of these, the (010) and (001) modes of CO, are infrared active as are all the modes of NzO. Being asymmetric N,O has a small dipole moment (0.17 D). Its quadrupole moment (Q = -3.7 x esu). CO, N,, esu) is smaller than that of CO, (Q = -4.3 x CO,, and N,O have fairly large dissociation energies of 11.1, 9.8, 5.5 ( C O , - + O+ CO), and 1.7 eV (N,O-+N, + 0), respectively. They also have fairly large electronic excitation energies of 8.1, 8.6, 5.7, and 4.0 eV, respectively.

2. Inelastic Scattering of Li+-N2/CO Using an apparatus very similar to that shown in Fig. 18, Bottner et al. (1976) have recently reported TOF spectra for Li+-N2 and CO. The target was in each case a nozzle beam with a rotational temperature of about 10°K with a most probable ji= 1 (CO) and 2 (N,). TOF measurements were made at two energies, E,, = 4.23 and 7.07 eV, over a wide range of angles up to 9 N 60°,well beyond the rainbow. Angular distributions without TOF analysis were also measured between 3 and 140".Both systems had nearly identical rainbow angles corresponding to similar well depths of about E = 0.33 k 0.03 eV. At larger angles the angular distributions showed no differences. The TOF distributions, some of which are shown in Fig. 32, show distinct differences, which increase with energy and scattering angle. At the smallest angle and energy the TOF distribution shows a maximum, which is considerably broadened compared to the peak measured in scattering from

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-----

' I

87

88

89

90

91

92

9367

68

69

70

71

72

73

Flight time [psecl 01

bl

CJ

FIG.32. Measured time of flight spectra for Li -N, (0.) and Li ' -CO (0) are compared at two different collision energies and two different angles (Bottner er d..1976). The arrows indicate the expected location of maxima for pure vibrational transitions. Also shown at the left for purposes of comparison is a TOF spectrum for elastic scattering from Ar at E,,,, = 5.23 eV ( E m= 4.45 eV). The shift between observed and expected peak locations is attributed to rotational transitions. +

Li+-Ar. However, most of the energy loss is less than that expected for a vibrational excitation and is attributed to unresolved rotational excitation. After transformation to the CM system, the distributions for CO and N, both show a peaking at j' z 20, with CO having about a 50",,wider distribution than N, . At the same energy but increased scattering angle, an additional peak appears in the CO spectrum but not in the N, spectrum (Fig. 32b); the new CO peak is shifted with respect to the expected location for pure vibrational excitation by about the same amount as observed in the vibrationally elastic scattering. Finally, in Fig. 32c several additional peaks are found in both spectra and the differences between the two systems become more striking. In all cases the additional "vibrational" peaks are shifted by roughly the same amount as the vibrationally elastic peak. Whereas N, has a large peak for n = 0- I , corresponding to a 30 to 35'1;,, probability in the CM systcm, CO shows a strong n = 0+2 peak but only an apparently weak n = 0- 1 peak. In the C M system both are about 20"4 probable. Several attempts have been made to explain the observed differences. Bottner et al. (1976) were able to show that the differences cannot be explained by an impulsive collision nor are they consistent with differences in the matrix elements appearing in first-order perturbation theory. Although important features of the potential hypersurface have been calculated (see

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Table 11) a comparison with close-coupling or even j,-conserving calculations is clearly out of the question because of the large number of rotational states. Bottner et al. (1976) therefore attempted to explain the differences in terms of the vibrational matrix elements by leaving out the rotationvibration coupling, as is done in the 10s approximation. This model also did not explain the observations in an entirely satisfactory manner, which could be due in part to uncertainties in the potential hypersurface. Work is presently in progress using classical trajectories on an improved potential hypersurface (Thomas, 1976). 3. Inelastic Scattering .from Linear Triatomic Molecules

A large number of experimental studies have been reported on the systems

COz, N,O, and CS, . Resolution of quantum transitions was only possible in the ion scattering experiments at collision energies greater than 3 eV, which lead to almost pure vibrational excitation. The available work can be conveniently grouped into three categories, which appear to be related to three different types of interactions depending on the scattering atom or ion and collision energies. Differences in the energy transfer behavior have also been observed for the different molecules, but these effects are smaller. a. Rotational excitation in thermal energy scattering of atoms. Table VI lists five experimental studies using the energy change method at collision energies between 0.07 and 0.32 eV. With the exception of the first study of this series on K-C02 (Beck and Forster, 1970) these experiments used a crossed nozzle beam apparatus with TOF analysis. Evidence for excitation of vibrations was only observed for Kr-C02 (010) at 9 = 120" and E,, = 3.77 kcal/mole (Farrar, 1974). In all other experiments, structureless Gaussian-shaped velocity distributions, which were shifted and broadened compared to elastic-scattering measurements, were observed. The data were interpreted in terms of either the most probable energy transfer (Blais et al., 1976b) or in terms of effective differential cross sections corresponding to rotational transitions from the most probable initial j state to various finalj states. Figure 33 shows an example of data of this type for Ar-C0, at E,, = 0.19 eV (Loesch, 1976). At this and the other (lower) energies the rotational excitation cross sections especially for small Aj are strongly peaked in the forward direction. Since the rainbow angle is estimated to be at about 15" CM, most of the observed scattering probes the repulsive part of the potential. With increasing angle there is a marked trend to increased energy losses. The most probable fractional energy loss AE/E is about 30% and is peaked at about 9 5 60-90. The experimental results have been shown to be in reasonable agreement with classical calculations for a semiempirical rigid rotor potential (Loesch, 1976) and for a theoretical potential

M . Faiibel and J . P. Toennies

300

I

I

I

1

I

., ,.. ..

a

1

' 0

30'

I

I

I

80' 90' 120' Chi scattering angle

I

150.

4

I

180'

FIG.33. Measured differential cross sections assigned to different (experimentally unresolved) rotational quantum transitions are plotted as a function of scattering angles (Loesch, 1976). The collision energy is E , , = 0.19 eV. The energy changes accompanying the indicated transitions are as follows: .j = 6+ 14. A E = 0.008 eV; ,j = 6+32. AE = 0.048 eV; ,j = 6-56. AE = 0.156 eV; and j = 6 4 6 0 . AE = 0.176 eV ( B = 5.9 x eV, j i = 6).

obtained from an electron gas model calculation (Parker et al., 1976). The peaking in the rotational energy transfer has been explained by a '' compromise" between restrictions imposed by the interaction potential, which favors large Aj transitions at large angles, and angular momentum considerations, which favor large impact parameters and therefore small scattering angles (see Section 1,C). In their comparative study of N,O and CO,, Farrar et al. (1973) found a larger forward inelastic peak in N,O presumably due to the mass asymmetry and dipole moment of N,O. For CS, , Blais et al. (1976a) found roughly twice as much energy transfer as for CO, . This is attributed to the greater polarizability and smaller energy level spacing. They also find that the fractional energy losses follow the relations Xe z K r 2 Ar > Ne. b. Vibrational excitation in Li collisions. With the same apparatus used in the Li+-N,/CO studies Eastes et al. (1977) carried out a comparative study of Li+-COz/N20 at CM collision energies between 3 and 8 eV. Figure 34 compares TOF spectra for the two systems for identical collision energies and scattering angles. In both cases the vibrationally elastic peak is nearly coincident with the location expected for pure elastic scattering. A detailed analysis of the results indicates that the rotational excitation in CO, corresponds to an increase from about jmp = 4 (T,,, 'v 27"K, Gallagher and Fenn, 1974) to only about j' = 8 (T,,, 1: 100°K). This result is surprising especially when compared with the results for Li+-N, at similar energies +

SCATTERlNG STUDIES OF EXClTATION MOLECULES I

I

,

I

000'

'

'

30 1

I

500

N

0

1

2

3

L

5

6

7

Flight timd difference (psec)

FIG.34. Comparison of TOF spectra for collisions of Li' with CO, and N,O both at

ECM= 4.72 eV and 9 = 17" (Eastes et al., 1977). At this energy the rainbow angle is located at about 10"CM. Only a small amount of rotational excitation, which is somewhat larger for N,O, is observed. Thus the transitions are attributed to nearly pure vibrational excitation.

and those just discussed for NzO and COz in collisions with the rare gases. The spectra show considerable structure, which coincides with that expected for excitation of the bending and symmetric stretch vibrations with about the same small amount of rotational excitation as found in the vibrationally elastic peak. NzO shows a somewhat similar behavior as COz with, however, 30% smaller vibrational transition probabitities and three to four times more rotational energy transfer. These results cannot be simply explained by the optical (infrared) transition matrix elements, since those would only favor the optically active OOO-+OlO and OOO+OOl transitions, of which the OOO+OOl does not appear to have been observed. The results have been explained by classical trajectories calculated for a simple potential model in which the Li+ ion interacts with the fractional charges on the atoms of the molecule. The fractional charges are calculated from the known quadrupole and, in the case of NzO, dipole moments of the molecule. Figure 35 illustrates the interaction and shows how the resulting Coulomb forces lead to the excitation of the bending vibrations. Whereas in COz the forces are symmetric with respect to the center of mass of the molecule, the dipole

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M . Faiibel and J . P . Toennies

FIG.35. Schematic diagram showing the electmstatic forces on CO, and N,O in collisions with Li+ ions. Classical trajectory calculationss b w that these charge distributionsexplain the observed vibrational ,excitation and the small .amount of rotational exatation. The dipole moment of N,O leads to a torque about the center of mass and explains the greater rotational excitation compared to CO, .

,moment in NzO leads to a toque about the center 0s mass. Thus the ,observed differencesin rotational excitation between N20and C 0 2 are also compatible with this model. c. 'Vibrational excitation in H + collisions. Krutein and Linder (1977) have very recently observed vibrational quantum excitation in H+-C02 collisions in the energy range ELAB( N E&J = 13-30 eV and scattering angles 0 (= 9) = 0-12". The apparatus was simihr to that described in 'Section V:B,3. At the lowest energy the spectra are quite similar to those observed in Li+-COz with the important digerence that the OOO~OO1transition, not seen in the Li+ collisions, is clearly resolved. The strength of this transition increases with energy and angle until ,at 50 eV and 7" the. 001 transition is the most probable event. At low energies the authors show that the results are nearly consistent with a Born approximation calculation and the optical matrix elements. At the other extreme of high energies and impact parameters they feel that chemical valence forces similar to those *found in H'-& are also important. This interpretation is confirmed by a recent ab initio calculation, which indicates that the proton in the 'H+-O-C-0 complex is bonded to an 0 atom (Green et al., 1976).Thus this .strong interaction could lead to the excitation of the asymmetric vibrations. The authors do not, however, discuss the possible effect of charge transfer, !which should be highSy probable since the ionization potentials are nearly

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identical [I(H) = 13.61 eV; I ( C 0 , ) = 13.79 eV]. Charge transfer has already been invoked by Gentry ef al. (1975)to explain the unexpectedly high energy transfer of AE = 1.8 eV in H+-CH, collisions at E,, = 20 eV and 9 = I 0 . In this case the charge transfer configuration H-CH: is more stable by about 0.5 to 1.0 eV. The results on CO, seem to indicate that three types of forces are doniinant depending on the collision partner and collision energy. I n the lowenergy rare gas collision in which only rotational excitation is observed. the short-range nonchemical exchange forces are important. In Li ' collisions a different type of interaction due to intermediate range dynamical Coulomb forces explains thc vibrational excitation. Finally in H ' collisions optical sclrction rules and valence forces possibly coupled with charge transfer lead to a different energy loss spectrum.

D.

SCATTERING IKOM THE

ALKALIHALIDES

I . Molrciilur Properties of the 4lkali Halides

Of all diatomic molecules the alkali halides have the largest dipole moments ranging from 5.8 (LiF) to 12.1 D (Csl).The quadrupole moments vary over a wide range from - 2 x (NaF) to - 18 x 10-" esu (CsF) (Stogryn and Stogryn, 1966). The rotational and vibrational energy level spacings are small. For the i = 1-2 transition AE is between 1.2 x 10 (Csl) and 0.66 meV (LiF). Because of the close rotational level spacing and high source temperatures (800-1800 K ) needed to vaporize the alkali halides. the most probable rotational state in an effusive molecular beam is between 20 (LiF) and 100 (Csl). However, by using a large excess of a rare gas the rotational temperature in the beam has been lowered in CsF to about 6 K with a most probable state j = 3. The alkali halides can be detected with nearly 100",, efficicncy and complete discrimination against the background gas by means$ofa Langmuir Taylor surface ionization detector. Moreover, because of.t\ieir large dipole moments individual (j,m )states can be selected and focused in electrostatic quadrupole fields (see Table VIII). Essentially became of these attractive features, a number of scattering experiments with state-selected alkali-halide beams have been performed. These include measurements of'the dependence of integral total cross sections on the direction of polarization of the molecule with rcspect to the collision direction (Bennewitz cf ul.. ,1964; Reuss, 1975). state to state inelastic scattering experiments (Twnnics, 1962, 1965), and more recently total differential cross section meaumements for non-state-selected beams (Reed and Wharton, 1977)and for polarized state-

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M . Faiibel and J . P . Toennies

selected beams (Tsou et al., 1977); both with very good angular resolution. State selection experiments with alkali halides are reviewed by Pauly and Toennies (1968) and in Toennies (1976), and we will not comment on them further. Despite these and a number of other detailed studies little is known about the range and well depth of the potential hypersurface for most of the alkali halides.

2. Energy Change E?cperiments A number of energy change experiments have been carried out over a wide range of scattering angles in a further study of the inelastic scattering of the alkali halides from the rare gases. These experiments can be conveniently characterized by the ratios of the collision energy to the average initial internal energy E,,,. The experiments of Armstrong et al. (1975) explore collisions with E,-M < Ein,and ECM 2: Einr, those of Loesch and Herschbach (1972) and King et al. (1973) and King (1974) EcM > Einl,and finally the experiments of Crim et al. (1973, 1974) ECM E,,,. The experiments of Armstrong et al. have a vertical plane geometry and a very good velocity resolution of about Au/u ‘v- 5%. The other two experiments were in-plane experiments with a relatively poor velocity resolution (230%). Quantum transitions could not be resolved in these experiments. This is not surprising in view of the large number of rotational states excited in the hot ( T 2: 900°K) incident non-state-selected beams, which are estimated to have a total internal energy somewhat less than 0.2 eV. Figure 36 shows a comparison of some of the data for scattering of CsI from A r taken from the first two experiments. Since the primary beam is heavier than the target, two maxima are observed. They are attributed to forward and backward scattering in the CM system. To permit an easy comparison of the data, the reported curves have been replotted in such a way that the forward and backward elastic maxima appear at the same location. Thus the reduced velocity scales at the bottom are different in each of the curves. Furthermore, to help in estimating the energy changes the expected shape and location of the forward and backward elastic maxima (dashed curves) are also shown. Comparison of the spectra reveals an interesting trend. At small collision energies and scattering angles (top curve) only small energy changes are observed. In the middle curve thecollision energy is comparable to Ein,and the energy loss spectrum differs significantly from that estimated for elastic scattering. Quite striking is the increased intensity at u/u,,,> 1, which is attributed to a significant amount of deexcitation. For lack of resolution it is not possible to determine the relative contribution of rotational versus vibrational deexcitation. Moreover, the region between the fast and slow

-=

Relative laboratory velocity v/vmax

FIG.36. Comparison of velocity distributions measured in energy change scattering experiments on Csl-Ar collisions. The flux density (intensity divided by scattered beam velocity) is plotted against the fractional scattered beam velocity. The dashed curve shows the estimated location and shape of the elastic maxima. The velocity abscissas have been scaled so that the forward and backward peaks, indicated by F and B, respectively, appear at the same place. The ordinates have been arbitrarily normalized to unity at the maximum. Curves (a) and (b) are measured in the perpendicular plane (Armstrong et al., 1975), while curve (c)is measured in the in-plane arrangement (King, 1974).

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M . Fauhel and J . P . Toennies

elastic peaks can be attributed to excitation in either forward or backward scattering. The gradual dropoff at relative velocities less than 0.88 is attributed to backward CM scattering. Whereas the above behavior is reminiscent of that observed in other systems, the result in the bottom curve is totally unexpected. Instead of finding a further washing out of the elastic structure a new maximum is observed. Since the velocity of this maximum coincides quite closely with the expected velocity of the center of mass, the Jacobian makes this peak appear especially large. In these collisions roughly 90% of the initial relative kinetic energy is converted into internal motion. From the relative height of the peak the integral cross section for this extremely inelastic process is estimated to be about 7.5 A’ (King et al., 1973). King et al. (1973) have presented convincing experimental data that indicate that the observed peak is actually due to strongly inelastic events and not to an artifact of the apparatus. Although a completely satisfactory explanation of these results is not available at the present time, the observation of very efficient energy transfer is consistent with experiments on the dissociation of the alkali halides to ions where, for example, for CsI-Xe a dissociation cross section of = 10 A’ has been measured at 5.4eV [D: (CsI) = 4.35eV] (Tully et al., 1971). The other extreme of Ei,, > ECMwas achieved in the experiments of Crim et al. (1973). These authors used the crossed molecular beam reaction of K on Br, to reactively produce KBr, which is highly vibrationally excited (E,,, ‘Y 2 eV) but rotationally relatively cold (Erd 0.3 eV). The KBr product beam was then crossed directly (without velocity selection) with a third beam. Typically the collision energy was only about 0.05eV. The scattered KBr was then velocity selected by a low-resolution mechanical velocity selector (Au/u = 0.25) in the plane of the incident KBr and target beam. The velocity distributions were compared with those measured in scattering of KBr from a conventional source. For scattering from the rare gases and simple molecules such as N, , CO, and C 0 2 the velocity distributions obtained in the first experiment were shifted to higher velocities, indicating a transfer of internal vibrational energy to relative translational energy in addition to transfer to the vibrational degrees of freedom of the target molecule (V-V transfer) (Crim and Fisk, 1976). The energy transfer cross sections were found to be about 20 A’. In addition to the rare gases a large number of polyatomic molecules were used as scattering partners. For H 2 0 , NH3, CH30H, C3HS,C z H 5 0 H , and (CH3)2O a more complex behavior was observed, which suggests that an energy-randomizing complex is formed with a large inelasticcross section ofthe order of 300 With the first three partners the velocity distributions suggest that the internal vibrational modes of the molecule d o not participate in the energy randomization

wz.

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process. In the case of C H 3 N 0 , ( p = 3.5 D), however, there is definite evidence that the internal vibrational modes contribute to the energy randomization process. These results indicate the importance of collision complex formation in energy transfer, an aspect that has been only little explored so far in scattering studies.

VI. Summary During the last decade the development of experimental and theoretical methods for the investigation of pure rotational and vibrational excitation of molecules in their ground electronic states has reached a level that allows the quantitative study of differential cross sections for state to state transitions. Despite the extreme requirements on the experimental resolution and the resulting low scattering intensities, energy level resolutions of the order of 10 meV in energy change experiments and as low as 0.1 meV in state selection experiments have been achieved with cross section sensitivities between lo-' and lo-, A2/sr corresponding to transition probabilities as small as The accuracies are generally about 102, but in some cases have reached 1%. This progress coupled with theoretical advances has made it possible to carry out a comprehensive comparison of measured cross sections sufficiently well resolved to reveal all quantum features with ab initio results for the system Li+-H2. The agreement to about 202, in all experimentally observed details demonstrates that the experimental and theoretical techniques are sufficiently reliable to be useful in probing the potential hypersurface. Of the 60 systems summarized in Tables VI-VIII about a dozen studies are sufficiently detailed that the results could be compared with theory to yield infoimation on the potentials. In the case of systems such as Li+-H2, Li+-N, /CO, Ne-HD, Ar-CO, and the rare gas-alkali halides rotational excitation appears to be the most important inelastic process at angles less than the rainbow. This result is in accord with the long-standing concept that the atoms and ions interact with rigid rotors by way of an anisotropic potential. The same behavior would have been anticipated for systems such as Li+-C02/N,O, but here the experimental results show a predominance of vibrational excitation and a negligible amount of rotational excitation. This has led to the realization that point charge coulomb interactions are important in ion collisions with larger molecules. In the potentially reactive system H+-H2 chemical forces also lead to an unexpected enhancement of vibrational excitation at the expense of rotational excitation. These and other results, such as the observation of superinelastic collisions in Ar-CsI at

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M. Fauhrl ant1 J . P. Tornriies

z 1 eV and in H+-CH4 point to the diversity of interactions that can be studied in a quantitative way in inelastic scattering experiments. In the future, work along three broad avenues appears desirable: (1) It will be necessary to further improve experimental techniques to make possible (a) the resolution of rotational quantum transitions in energy change experiments on nonhydrogenic systems such as N, and CO, ,and (b) the resolution of Am transitions, averaged over in most available results. Little is known about A m transitions since they are the last frontier of resolution. There is some reason to believe that they may be crucial in extracting information on anisotropic potentials from cross sections. (2) It will be necessary to study more complex systems in the hope of uncovering new interaction mechanisms and of developing rules by which to classify systems according to mechanisms. As soon as sufficiently powerful tunable lasers become available in the ultraviolet and infrared regions, we expect that laser spectroscopy techniques will be instrumental in achieving both goals. (3) Finally, the availability of high-resolution experiments points to the great need for faster methods for calculating cross sections. These should prove helpful in developing ways to invert inelastic cross sections to obtain quantitatively precise potential hypersurfaces. This in turn will stimulate the development of faster reliable approximations for calculating potential hypersurfaces. Inversion procedures will also help in designing more efficient experiments. Inversion methods for handling inelastic cross sections from reactive systems appear more difficult. Once available, however, the study of inelastic collisions could well prove to be a much more efficient method for probing many regions of the potential hypersurfaces of reactive systems than the direct study of reactive cross sections.

ACKNOWLEDGMENTS We thank P. C. Hanharan (Bochum), J. Schaefer (Munich), and K. Walachewski (Gottingen) for their help in preparing the tables and figures of Section 11. We are very grateful to our colleagues U. Buck. W. Eastes. W. D. Held. H. D. Meyer, and D. Micha, as well as to J. Schaefer and L. Thomas (Munich) for reading and commenting on the manuscript. Finally, we are indebted to Ch. Bertram, U. Gierz, 1. Siadat, and B. von Stutterheim for their careful and quick secretarial services.

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3 10

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LOW-ENERGY ELECTRON SCATTERING BY COMPLEX ATOMS: THEORY AND CALCULATIONS* R . K . NESBET IBM Research Laboratory San Jose, California I. Introduction . . .

. . . . . . . . . . . . , 315

11. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... ........ A. Structure of the Wave Function 9. Cross Sections . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . .

3 18 318 32 1 C. Polarization Potentials and Pseudostates . . . . . . . . . . . . . . . . . . . . . . . 323 324 D. Resonances.. . ........... 330 F. Excitation of Autoionizing States . . . . . . . . . . . . . . . 335 Ill. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 338 343 346 349 349 355 360 . . . . . . . . . . . . . . . 362 370 E. Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 74 References . . . . . . . . . . . . . . . . . . . . . 378

I. Introduction In a recent review article (Nesbet, 1975a). theoretical methods were discussed for the quantitative description of electron scattering by atoms with more than one electron, in the low-energy region where ionization is either energetically impossible or unimportant. The present article will survey results of calculations by these methods. Emphasis will be placed on the * Supported in

part by the Office of Naval Research, Contract No. N00014-72-C-0051. 315

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R. K . Nrshet

qualitative and quantitative theory of identifiable structural features (resonances and threshold effects)and of inelastic processes (excitation and deexcitation, particularly of metastable states). Electron scattering by helium provides an excellent comparison between theory and experiment, because many experimental data are available, and relatively accurate theoretical calculations are feasible. In its excited states, helium becomes an open-shell atom, and requires theoretical methods appropriate to complex atoms. For these reasons, results for e--He scattering in the energy range of the n = 2 singly excited states will be presented here in detail. Electron scattering by the light open-shell atoms carbon, nitrogen, and oxygen is important in astrophysics and atmospheric physics, but very difficult to study in the laboratory. Recent theoretical calculations predict structural features and quantitative values for cross sections and excitation rates. These results will be presented here and should help to stimulate new experimental work. Atomic hydrogen and the " one-electron " alkali atoms will not be considered here. Except for helium and carbon, nitrogen, and oxygen, theoretical studies of low-energy electron scattering by other atoms are much less developed and are primarily limited to elastic scattering. The present survey will be limited to the atoms mentioned above. The quantitative theory of electron scattering by a complex atom requires consistent treatment of the N-electron target atom and of the (N + 1)-electron scattering system. For a neutral target atom, low-energy scattering is dominated by the long-range polarization potential due to dynamical distortion of the atom by the incident electron. At short range, the external electron becomes part of a transient negative ion, whose specific states produce resonance structures in scattering cross sections. Much of the observed energy-dependent structure in electron-atom scattering arises from such resonances. Energy can be transferred between the incident electron and target atom, inducing transitions from the initial atomic state. As the energy of the incident electron is increased, successively higher excited states of the atom become energetically accessible. Characteristic scattering structures can occur at each excitation threshold. Observation and interpretation of such threshold structures is a subject of current experimental interest. Lowenergy inelastic electron scattering is not subject to the electric dipole selection rules that govern electromagnetic radiation. Excited atomic states that are metastable against radiative decay can be produced by electron impact. These states are important carriers of energy in plasmas. Theory and computational methods incorporating the essential physics of the scattering processes outlined above will be described in Sections I1 and

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317

111. Specific applications and comparisons with available experimental data are given in Section IV. A thorough treatment of the formal quantum theory of scattering has been given by Newton (1966). Early developments and applications of electron-atom scattering theory are surveyed by Mott and Massey (1965). Geltman (1969) and Brandsden (1970) cover basic scattering theory relevant to electron-atom collisions. A brief review of more recent developments has been given by Burke (1972). A recent treatise is by Joachain (1975). At low energies, only a relatively small number of partial waves (incident electron angular momentum states) contribute significantly to electron scattering, This number increases with incident electron energy and with the strength of the dominant long-range interaction potential. The methods considered here are all based on a partial wave expansion of the electron scattering wave function. Inelastic scattering processes require explicit representation of two or more states of the target atom. Expansion of the scattering wave function in a series of such states gives the close-coupling expansion, which is basic to all methods considered here. Systematic approximations must be introduced to extend these methods into the intermediate energy range, above the first ionization threshold of the target atom, since the number of open scattering channels becomes infinite. The formalism and practical computational techniques of close-coupling theory have been reviewed by several authors (Burke, 1965, 1968; Burke and Seaton, 1971; Smith, 1971; Seaton, 1973). Electron impact excitation of positive atomic ions (theory and applications), which will not be considered here, has been reviewed by Seaton (1975). Theoretical computations of excitation cross sections have recently been reviewed by Rudge (1973). A collection of papers reviewing various computational methods relevant to low-energy electron-atom scattering has recently been published (van Regemorter, 1973). This collection includes summaries of two currently important methods for applications to low-energy inelastic scattering by complex atoms: the R-matrix (Burke, 1973) and matrix variational (Nesbet, 1973a) methods. A more detailed review of the R-matrix method has recently been published by Burke and Robb (1975). The polarized orbital method, for elastic scattering, is reviewed by Callaway (1973), and in more detail by Drachman and Temkin (1972). A very promising new method, based on formal Green’s function theory, is described by Thomas et al. (1974a), following an earlier review of the formalism (Csanak et al., 1971). A critical review of experimental measurements of total electron-atom scattering cross sections has been published by Bederson and Kieffer (1971). Experimental work on low-energy differential elastic cross sections has been reviewed by Andrick (1973). A review of available total cross section and

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forward elastic cross section data for rare gas target atoms has been given by de Heer (1976). Both Andrick (1973) and de Heer (1976) emphasize e--He scattering. Theory and experimental results for electron-atom scattering resonances have been reviewed by Taylor (1970) and by Schulz (1973).

11. Theory A. STRUCTURE OF THE WAVEFUNCTION A stationary state Schriidinger wave function for electron scattering by an N-electron atom can be expressed in the form

where 0,is a normalized N-electron target state wave function, $, a oneelectron channel orbital antisymmetrized into 0,by the operator d ,and QB one of an assumed orthonormal set of (N 1)-electron Slater determinants constructed from quadratically integrable orbital (one-electron) functions. The orbital angular momentum of $, is I, and its asymptotic linear momentum in atomic units is k,, corresponding to kinetic energy &ki in Hartree units. This asymptotic energy is positive for open scattering channels and negative for closed channels at total energy E. By definition,

+

$ki = E - E ,

(2) where E , is the energy mean value of target state 0,. For closed channels k, is replaced by i K p with K, > 0. Channel orbital functions are of the form

4bmS

(3 ) wheref, satisfies the usual boundary conditions at r = 0, and $, is orthogonal (by construction) to all orbital functions used in constructing 0,and a,,.For open channels, the asymptotic form off, is * p =fp(r)Kml(e,

fp(r)

-

k;''2r-'

sin(k,r - fl,n

+ 6,)

(4)

for single-channel scattering by a neutral atom. For Coulomb or dipole scattering this functional form must be suitably modified. The normalization corresponds to unit flux density for a free electron. For multichannel scattering

-

k;'/2r-'[sin(k,r - flpn)0lOp + cos(k,r - #,n)al,] (5) such that scattering matrices and cross sections are determined by the coefficients aip, i = 0, 1. fp(r)

LOW-ENERGY ELECTRON SCAlTERING

3 19

Closed-channel radial functionsf, must satisfy the same boundary condition at r = 0 and the same orthogonality conditions as open-channel functions, but the closed-channel radial functions are quadratically integrable. They vanish as r -+ in a way determined by the detailed solution of the Schrodinger equation. It should be noted that the term in exp( - IC, I) arising from analytic continuation of Eq. ( 5 ) below a threshold at which k, vanishes is dominated asymptotically by terms in reciprocal powers of r due to longrange interchannel multipole potentials. The orthogonality conditions imposed on f p ( r )ensure that each term in the first summation in Eq. (1) is orthogonal to the Hilbert space of functions {a,,},used to construct the quadratically integrable function P

The coefficients c, in YHare determined variationally. The Slater determinants aPcan be defined in terms of virtual excitations of an N-electron reference determinant (Do, itself defined as an antisymmetrized product of N orthonormal occupied orbital functions 4i, tPj, . . . . Virtual excitations are defined by replacing some n specified occupied orbitals of Oo by n + 1 one-electron functions drawn from a set of unoccupied orbitals fp,, , 4 b , . . . that are mutually orthonormal but orthogonal to the occupied set. The orbitals are all quadratically integrable functions of the space and spin variables of a single electron. A denumerable set of orbitals {$i;4,,}generates a uniquely defined basis {a,,)for the (N + 1)-electron Hilbert space. A typical Slater determinant 0,can be denoted by

qf.:",

i <j < < N c a c b c c < (7) where the notation implies that (4,,, + b , 4,, ...).replace (4i, 4 j , ...) in reference determinant (Do, in the order specified. Appropriate normalization of OPis implied. An oscillatory function of nonvanishing asymptotic amplitude cannot be represented as a finite superposition of quadratically integrable orbitals. For this reason, the open-channel terms do,t,bp in Eq. (1)remain distinct from the Hilbert space component for any calculation using a finite orbital basis. In contrast, closed-channel terms can be included either explicitly or in the Hilbert space component YH. An implicit variational solution for the coefficients c,, in YHcan be incorporated into a modified Schrodinger equation for the channel functions do, t,bp by a partitioning technique, used in the resonance theory of Feshbach (1958, 1962). If projection operator Q is defined by

R . K. Nrsbet

320

and if operator P is the orthogonal complement of Q, then

and the modified Schrodinger equation is

where M denotes H - E if H is the ( N + 1)-electron Hamiltonian. The operator Mid is an ( N + 1)-electron linear integral operator with kernel

Equation (10) provides a common basis for the computational methods to be considered here. Effective one-electron equations are derived as matrix components of Eq. (10) with respect to target states 0,, integrating over angular momentum and spin factors of the channel orbitals t , ! ~ ~to obtain coupled equations for their radial factors f,(r). When suitable normalizing factors are included

(@,I

'YP) =

9,

(12)

The matrix operator acting on channel orbitals is mpq

I

1

= (0, Mkp

'

(13)

The terms in MLParising from (MQQ)- define a matrix optical potential that acts on the channel orbitals. This operator describes correlation and polarization effects. The wave function Y can be taken to be an eigenfunction of L2,S2, and parity n,for nonrelativistic scattering by light atoms. Then instead of simple Slater determinants, the Hilbert space wave function YHand the target atom functions 0, can be expanded in antisymmetrized LS-eigenfunctions. Since these in turn can be expressed as linear combinations of Slater determinants, the expansion indicated in Eq. (1) is completely general. To simplify practical calculations, LS-eigenfunctions are used, and the antisymmetrizing operator .d in Eq. (1) is extended in definition to include angular-momentum coupling. When several target states 0, included in Eq. (1) have the same LSn quantum numbers, it will be assumed that the N-electron Hamiltonian matrix among these states has been diagonalized. Then the energy values E , in Eq. (2) are eigenvalues of this matrix, and all nondiagonal elements vanish. Different functions 0, are assumed to be orthogonal.

32 1

LOW-ENERGY ELECTRON SCATTERING

B. CROSS SECTIONS At given total energy E , with N , open channels, there are N , linearly independent solutions of the Schrodinger equation for Y. Each solution Y q is characterized by a vector whose elements are the coefficients aip(i = 0, 1) of Eq. (5). A matrix a can be defined as the 2N, x N , rectangular matrix with elements aip, , consisting of N , linearly independent column vectors, one for each solution. Any nonsingular linear transformation of these vectors produces a physically equivalent set of solutions. Thus a given matrix a can be multiplied on the right by an N , x N , nonsingular matrix to produce equivalent solutions in a specified canonical form. The reactance matrix K is defined (Mott and Massey, 1965) by the canonical form

,

a. = I,

(14)

a1 = K

where, in a matrix notation with open-channel indices p,q suppressed and with matrices and vectors segmented according to the indices i = 0, 1 defined by Eq. (5), N , x N , square matrix a. is the upper half of a, a1 is the lower half, and I is the N , x N , unit matrix. An arbitrary solution matrix a can be reduced to this form by multiplying on the right by a; if a. is not singular. Then, in general K = a,a;' (15) Alternatively, if a, is not singular,

',

K - ' = CC~C~,' (16) For exact solutions, K is real and symmetric, with real eigenvalues that are tangents of " eigenphases." The corresponding eigenvectors define " eigenchannels " as linear combinations of physical open channels. In terms of eigenchannels, indexed by p, Nc

K,, = ,=11x p p x q ptan 6 , ,

= 1,

Nc,

(17 1

where 6 , is an eigenphase and x,, a component of a normalized eigenchannel column eigenvector. Matrix functions of K are most easily computed by substituting the corresponding function of 6, into Eq. (17). The scattering matrix (Mott and Massey, 1965)is expressed in terms of the reactance matrix by S = ( I + iK)(I - i K ) (18) substituting exp(2i6,) for tan 6, in Eq. (17). The transition matrix T can be defined by 1

T = I ( S - I) = K(Z - i K ) - ' 2i

(19)

322

R . K . Nesbet

substituting exp(i6,) sin 6, for tan 6, in Eq. (17). The partial cross section for scattering from channel q to channel p is

The total cross section for unpolarized scattering is obtained by summing over degenerate final states and averaging over initial states. Equivalently, the cross section can be summed over all states of the ( N + 1)-electron scattering system and divided by the degeneracy of the initial state. Since the T-matrix is diagonal in the total quantum numbers LSx and independent of total M L and M s , the sum is over LSx only, with each term weighted by the M L , M s degeneracy factor (2L + 1)(2S + 1). If target atom quantum numbers are denoted by (LSn)),,the total cross section for transition y -+ y’ is CJ

where 1, 1’ are orbital angular momentum quantum numbers of the incident and scattered electron, respectively. Equation (22) is a special case of formulas due to Blatt and Biedenharn (1952) and to Jacob and Wick (1959). These authors also give formulas for differential cross sections and for cross sections with selected angular momentum sublevels (polarized electrons, polarized target or recoil atom). The differential cross section for scattering from state yp (where p denotes M L,M, , m,) to state y‘p’, for electron deflection angles (0, c$), in units a: per steradian if k , is in atomic units a i I , is do,,,

y l , m

I

I

= (Y’P’ f(07

4 )I YP)l2

(23)

where [Blatt and Biedenharn, 1952, Eq. (3.14)]

(Y’P’I f IYP)=

c c “21’ + w +w2

LSn 11‘

il-1’

d1‘mpO ( ~ ) e i m ” ( T yl /ky) ~~,

I

x (LyJGy1’m’ W L ) ( L , M L , l O

IW

L )

x (S,~Msy;tm:,ISMs)(S,Ms,~~,IS~,)

Here, and

dL,o(0)ei*’4= (4x/2l’ + 1)’’’ &,,,(&#J) M L = ML, = ML,, + m’,

M s = Msy

+ m, = Ms,. + m:

(24)

LOW-ENERGY ELECTRON SCATTERING

323

The total differential cross section for the process y + y’ is da -

1

dR - 2(2L, + 1)(2S, + 1 ) 1 1 I (Y‘P’ I f’ I YP) l2 p,

Explicit formulas for the sums appearing in Eq. (25), expressed in compact form in terms of angular momentum recoupling coefficients, are given by Blatt and Biedenharn (1952). When contributions of a large number of partial waves must be combined, it may be more efficient to compute the amplitudes ( y ’ p ’ l f l y p ) of Eq. (24) directly than to use the summed crosssectional formulas of Blatt and Biedenharn. C. POLARIZATION POTENTIALS AND PSEUDOSTATES

1

The Hilbert space component @,c, of Y, Eq. (l), interacts with the explicit open-channel terms to describe electronic correlation effects. The polarization mutually induced between the external electron and the target atom is an effect of this kind. In the limit of large r this mutual polarization is equivalent to a “polarization potential” acting on the external electron. For electric dipole virtual excitations this potential is -a/2r2, where a is the electric dipole polarizability of the target atom. In general, virtual excitations of the target atom with multipole index 1 produce a multipole polarization potential of the asymptotic form -a1/2r2‘+2,where a, is a generalized polarizability. For a neutral atom in a spherically symmetrical state ( L = 0), the dipole polarization potential is the term of longest range in the effective scattering potential. This term dominates low-energy scattering behavior. The asymptotic form of the polarization potential can be derived most directly by writing Y in the close-coupling form, =

1 Ld@,rC/,+

~~~q(p,rC/q(p)l

(26)

P

where 04(p) is a pseudostate that represents the first-order perturbation of open-channel state 0, in an external dipole field (Damburg and Karule, 1967; Damburg and Geltman, 1968; Burke et al., 1969b; Geltman and Burke, 1970; Burke and Mitchell, 1974; Vo Ky Lan et al., 1976).The energy mean value E,(,, is usually above the ionization threshold, corresponding to a closed-channel state for low-energy scattering. If E b special provision must be made to avoid effects due to nonphysical open channels and spurious resonances. In particular, physical target atom states that interact strongly through dipole matrix elements should be included explicitly in the close-couplingexpansion. Since the residual polarized pseudostate is orthogonalized to all such target states, its energy will tend to a higher value. Coupled integrodifferential equations for the channel orbitals $, and )I¶(,

324

R. K. Nesbet

are obtained by taking matrix elements of the (N + lhelectron Schrodinger These ). equaequation with respect to the N-electron functions 0,and 04(, tions are solved explicitly in the close-coupling method (Burke and Seaton, 1971; Smith, 1971; Seaton, 1973). Potential functions in these equations come from the two-electron Coulomb interaction in matrix elements of the IH I@J. When expanded in the spherical polar coordinates of two form (0, electrons, the Coulomb potential l/r12 depends on the two radial coordinates rl, r2 through a factor r$ /r$+

(27) that multiplies spherical harmonics of degree 1.Here r < is the lesser of rlr r2 and r, is the greater. For a neutral target atom, at large r terms with 1= 0 drop out, due to complete screening of the nuclear Coulomb field by the electrons. For a target atom state with L > 0, (@,I H 10,)contains static multipole potential terms proportional to l/r"+' for even 1 such that 0 < L < 2L. The dominant term is the electric quadrupole potential, asymp totically proportional to r-3. Nondiagonal matrix elements of spherical harmonic index 1 connect target atom states such that L,C,and 1satisfy the triangular condition

C = ( L - L I , IL-ll + 2 , . . . , L + 1

(28) In the close-coupling equations, such matrix elements produce an offdiagonal potential asymptotically proportional to l/ra+ l , connecting external channel orbitals 9, and ,,+(, . By transformation of these equations, this offdiagonal matrix element contributes quadratically to an effective potential in channel p, proportional to l/r2"'. When 1= 1, this effective potential is the electric dipole polarization potential.

D. RESONANCES An electron scattering resonance appears as a rapid variation of cross sections in a small energy interval of magnitude 2r, where r is the width of the resonance. Reviews of theory and experimental data have been given by Burke (1968),Taylor (1970),and Schulz (1973).A resonance is characterized by a rapid increase through A rad of the sum of eigenphases for a particular scattering state (quantum numbers UA). The physical origin of a resonance is a nearly bound state that interacts with a scattering continuum, implying a finite lifetime in a time-dependent formalism. The simplest example is a particle in a potential well enclosed by a barrier of finite height. States with energy levels above the external asymp totic energy but below the barrier are known as shape resonances. The resonance energy E, is approximated by an energy level of a particle

LOW-ENERGY ELECTRON SCATTERING

325

confined within the barrier, and the width r (inversely proportional to the lifetime) is determined by the matrix element of the Hamiltonian operator connecting this unperturbed state with the adjacent continuum. In electron-atom scattering the state that describes an electron bound to an excited state of the target atom lies in the scattering continuum. The resulting state is a resonance. If the resonance lies below the corresponding excitation threshold it is narrow, because the adjacent continuum arises from lower states of the target atom. Such narrow resonances immediately below an excitation threshold are referred to as Feshbach or closed-channel resonances (Burke, 1968), or as core-excited resonances of type l(CE1) (Taylor, 1970). If the external electron has l > 0, an effective rotational barrier can lead to resonance energies above the corresponding excitation threshold. Such resonances, of type CE2 (Taylor, 1970), are analogous to shape resonances. For an isolated resonance in a single open channel, the energy variation of the phase shift 6 of Eq. (4)is (Feshbach, 1958, 1962) 6 ( E ) = do@)

where tan 6,@) =

+ 6,(E)

-1 / = ~ -3r/(E

(29)

- E,)

(30) The background phase shift do is a slowly varying function of E near E,. If 6, is constant, Eq. (29) gives the resonant lineshape formula (Fano, 1961; Fano and Cooper, 1965) a/a, = ( E

+ q ) y (1 +

E2)

(31)

where q = -cot 6,

(32) Equation (31) gives a characteristically unsymmetrical resonance shape that goes to zero when E = - q. In electron scattering, this partial cross section is superimposed on other partial wave contributions that in general would not show resonance structure at the same energy E,. Equation (29) can be derived by application of the partitioning theory of Feshbach (1958, 1962)as in Eqs. @)-(lo). A simple example is provided by a shape resonance due to a one-dimensional potential function V ( r )consisting of a potential well inside a finite barrier. The Hamiltonian is

If 4ais defined by an eigenvector of a finite matrix representation of H,with eigenvalue E a , the scattering wave function can be expressed as $ =u +4aca

(34)

R. K . Nesbet

326

where u is orthogonal to 4, and satisfies the modified Schrodinger equation

(H- E)u - +,(aIH - E l u ) = -(H - E , ) ~ , c ,

(35) This is an example of Eq. (lo), if the Q-space consists of the single function 4,. The coefficient c, is explicitly c,= ( E - E , ) - ' ( a l H l u )

(36)

The background continuum function wo orthogonal to integrodifferential equation

4, satisfies the

(H-E)wo- 4,(aIH-Elwo)=0

(37)

and an irregular solution w1 of the same equation can be defined. The asymptotic normalization is wo

-

k-1'2 sin(kr + b,),

w1

-

/c-'/~

cos(kr + 6,)

(38)

In terms of the projection operator P , which orthogonalizes functions to 4,, Eq. (37) is P ( H - E)wo = 0

(39)

Then the solution of Eq. (35) is u = W O - PG(H - E , ) ~ , c ,

where the integral operator G, the formal right inverse of P(H - E)P, has the Green's function kernel r < r' r > r'

2wo(r)wl(r'), g(r, r') = 2wl(r)wo(r'),

An equivalent expression for the kernel of PGP is the principal value integral (Fano, 1961)

From Eq. (40), (++)=

(aIHlwo)+A,c,

(431

where A, = - [a I ( H

- E,)G(H -

E,) I a]

(44)

Then from Eqs. (36) and (43), c, = ( E - E , - A,)- ' ( a

I H Iw o )

(45)

327

LOW-ENERGY ELECTRON SCATTERING

By definition, tan 6, is the asymptotic coefficient of w1 in $(r). From Eqs. (40) and (41), since 4, vanishes asymptotically, this is

tan 6,, = - 2(w0I H I a)C, = -2(wolHIa)(E

(46) - E, - Aa)-l(alHlwo)

(47)

This verifies Eq. (30),with

qr, = 2(w01 H Ia)’

(48)

+

Era = E , Aa (49) The width r, and shift A, are functions of E. An eigenvalue E, of the bound-bound Hamiltonian matrix HQQ corresponds to a resonance only if r, and A, vary slowly near E, and if A, is small. Since the number of eigenvalues below E, increases indefinitely as the finite basis for HQQ is increased, most eigenvalues cannot be identified with resonances. In practice, to be so identified, an eigenvalue and eigenvector of HQg must be insensitive to changes in the basis. These properties are examined in the stabilization method (Taylor, 1970; Hazi and Taylor, 1970; Fels and Hazi, 1971, 1972). In electron-atom scattering, resonances are usually associated with well-defined electronic configurations and quantum states of the negative ion. The derivation given above can be generalized for multichannel resonances. Background eigenchannel states, orthogonal to the entire Hilbert space {@,}, are defined as solutions of Eq. (lo), omitting the matrix optical potential due to a specified eigenvector {c,}, energy E,, of the bound-bound matrix H,, (or MQQ).With definitions as in Eqs. (l),(5), and (17), for background eigenchannels /3 with eigenphases 6, and eigenvector components x p p ,the asymptotic radial channel orbitals are W(0p)B

-

(wop cos 6 , + W l p sin 6,)xp,

(-wop sin 6 , + W l p cos 6,)xpfl (50) The Green’s function linear operator becomes a matrix G ! , with a kernel, antisymmetrized into functions 0 , and 0,, whose radial factor, multiplying appropriate orbital angular and spin functions, is W(lP)/I

z,

When interaction with state a is taken into account, the asymptotic radial open-channel functions become

R. K . Nesbet

328

defining a perturbing reactance matrix E? as a generalization of tan 6, in Eq. (29). The argument leading to Eq. (47) follows as before, giving

R,,

= -2

c

M(Oq,B’,

c Mu,

u(E - Ea - A,)-

(0p)P

P

4

where M denotes H multichannel case,

- E,

(53)

as in Eq. (10). From Eq. (44),extended to the

Since I?,,, is a matrix, the appropriate generalization of Eq. (30) is obtained by defining a unit column vector with components Ypa = 2l-i

c

M(Oq,,. u

(55)

4

in terms of a total width

ragiven by -1r a = 2 c ( c M ( o q ) p . a )

2

2

(56) p ’ 4

a generalization of Eq. (48). Then Eq. (53) becomes

with E, defined in terms of r, and A, as in Eqs. (30) and (49). In the basis of background eigenchannels, Eq. (15) for the perturbed K matrix becomes

K

= (sin 6

+ cos 6 R)(cos 6 - sin 6 R)-

(58)

cos 6, yy+ cos 6 + &

(59)

in a matrix notation. Here sin 6 and cos 6 are real symmetric matrices defined as in Eq. (17) by the background eigenphases {a;} and eigenchannel vectors. From Eqs. (18) and (59) the S-matrix is S = (cos 6 + i sin 6)

LOW-ENERGY ELECTRON SCATTERING

329

If the background phase matrix S and the width and shift functions vary slowly near E, and if A, is small, this formula describes a multichannel resonance. The general properties of multichannel resonances follow from Eq. (60), which Macek (1970) has derived in an eigenchannel representation. McVoy (1967)has reviewed the mathematical theory of resonances, as developed in nuclear physics, with many illustrative examples from model calculations. Equation (60) shows that the S-matrix has a pole at E = - i in the vicinity of a resonance. If r and A are constant the pole is at

+ A, - +ir,

E = E,

(61)

displaced below the physical E-axis in the complex plane. The determinant of S is exp(2i 6), where C 6 is the sum of eigenphases. It can easily be shown that

defining a resonance phase a,(&) that increases from 0 to K rad as E varies from - 00 to + 00. If the background matrix 6 remains constant, it follows from Eqs. (60) and (62) that the sum of perturbed eigenphases, given by

increases by K rad as E traverses the resonance. Here (6;) are the background eigenphases. By writing S in the form

s = sA+ e2iS&3 S B

(64)

where, from Eq. (60),

sA= eia(l - yyt)e'd,

sB= eiayytei'

(65)

it can be seen that each matrix element S,, describes a circle of phase 26&) in the complex plane as E increases. The background S-matrix is S A + S , , or exp(2iS). It follows from Eq. (60) that a resonance affecting n coupled ehannels is described by the n eigenphase solutions S,(E) of the equation (Macek, 1970) i

1

n

E - E,, = - rbl y$ cot[S;(E) 2 p=1

- S,(E)]

Typical behavior for a narrow resonance is shown in Fig. 1 (Oberoi and Nesbet, 1973b).

R . K . Nesbet

330

0.0

1

0.42

FY;l 33s 0.43

0.44

0.45

I

46

k (a.u.)

FIG. 1. e-He. *Po eigenphases and their sum.

E. THRESHOLD EFFECTS Scattering cross sections have characteristic functional forms near thresholds. The formal mathematical theory of scattering is helpful in interpreting the resulting structural features. Geltman (1969) summarizes the theory of resonance and threshold structures, including effects of the long-range potentials typical of electron-atom scattering. The discussion here follows McVoy (1967). For single-channel scattering, the S-matrix is a function of complex k, S(k) = eZid(') =f(- k)/f(k)

(67) where f(k) is the Jost function, whose zeroes determine poles of S(k). The upper half-plane of complex k is mapped onto the entire complex energy plane, with a branch cut along the positive E-axis. The lower half-plane of k maps onto a second Riemann surface of complex E, the " nonphysical sheet." Zeroes off(k) in the upper half-plane can occur only on the positive imaginary axis, at points k,=irc, K>O (68) and correspond to bound states. Zeroes off(k) in the lower half-plane occur either as pairs of roots k,=u-iP, -k;= -a-i/3, u,p>O (69) corresponding to a resonance pole of S, or as single points on the negative imaginary axis k,= -iK, which correspond to virtual states.

K>O

(70)

LOW-ENERGY ELECTRON SCATI’ERING

33 1

At a resonance, S contains the factor

+

1 i tan 6, - ( - k - k,)(-k + k,*) 1 - i tan 6, (k - k,)(k + k,*) This implies tan 6, =

- 2kB

k’ - (a’

+ p’)

so that, comparing with Eq. (30), the resonance parameters are

E , = *k: = *(a’ + p’) (73) If the scattering potential is varied so that a resonance approaches the threshold ( k = 0), p must vanish at least as rapidly as k , , and r vanishes at least quadratically with k , . For a resonance due to the centrifugal barrier in partial wave 1 > 0, r varies as k:”’ near threshold (McVoy. 1967). If 1 = 0, there is no centrifugal barrier, and a zero o f f ( k )approaches k = 0 as a single point on the negative imaginary axis as the scattering potential is varied. From Eq. (70), this is a virtual state, which becomes a bound state. Eq. (68), as the potential is further varied. Thus a virtual state is a phenomenon associated with s-wave scattering just above a threshold. For a virtual state, the phase shift 6, is given as in Eq. (71) by

r = 2k,p,

1 + i tan 6, - - k - k, k - k, 1 - i tan 6,

(74)

tan 6, = k/K

(75)

or, from Eq. (70),

If K is small, the phase shift rises rapidly from threshold, but the total increase is limited to 1112 rad. For s-wave elastic scattering, the partial cross section, given by Eq. (21), is cr = 4n/(k2

+ K’)

(76)

which is finite at threshold. Similarly, for a bound state at energy - ~ ’ / 2just below threshold, tan S, = - k/K

(77)

from Eq. (68). The phase shift descends linearly in k from a value at threshold that can be taken to be a rad, representing the contribution of the bound state as a resonance of zero width. The pa’rtial cross section is again given by Eq. (76), finite at threshold. Thus the scattering effect of a virtual state at “energy” -k-’/2 is the same as a true bound state at the same

332

R . K . Nrshet

energy, but the virtual state is displaced onto the nonphysical sheet of the complex E-plane. When extended to multichannel scattering, these analytic properties of S ( k ) apply to the eigenphase associated with the new open channel at an excitation threshold. Elastic scattering in the new channel is affected by a resonance below threshold as if it were a bound state, giving a finite threshold cross section through Eq. (76). The effects of long-range potentials in single-channel scattering were studied by O'Malley et al. (1961) and by Levy and Keller (1963). This analysis has been extended to multichannel scattering (Bardsley and Nesbet, 1973). At a specified excitation threshold, let the old channels be labeled by 01, p, ... and new channels, opening at the threshold, by p , q, .... Then k P = k 4 = ... , and the dependence of elements of the K-matrix on k, is to be determined. For any short-range interaction, K,, , which describes elastic scattering in the new channels, varies as k)+'q+' near threshold. If the interaction potential VPq(r)contains a long-range term of the form CY-', and if s < 1, + I, + 3, the threshold behavior is changed to kSp-2 (Bardsley and Nesbet, 1973). Elements K,, , which describe inelastic scattering, vary as k!' l i 2 , and this behavior is not modified by long-range interactions. From this rule for K p g ,the inelastic cross section for the excitation process c( -+ p varies as k ; ) p + at threshold. An abrupt onset occurs only for 1, = 0, when the excitation cross section is proportional to k , or ( E - E,h)"', where E t h is the threshold energy. The initial slope is infinite. This behavior is mirrored in the partial wave elastic cross section oaaor inelastic cross , can have infinite slope at threshold (Wigner, 1948; Baz, section o P b which 1957). The resulting structure is a Wigner cusp, or an apparent abrupt step. Such cusps or steps should occur only if I , = 0 for the electron in the new channel (the inelastically scattered electron). If (Lsn),, are the target atom quantum numbers for the excited state defining this channel, cusp structure is possible only for partial wave states with total quantum numbers L,, S, k 3, 7cp . For example, for electron scattering by alkali metal atoms in the ' S ground state, the first excitation threshold is 'Po. Cusps can appear at this threshold only for incident p-waves, for elastic scattering in states 'Po or 3P0.Because of this selectivity, prominent cusp structure can be expected only when the observed cross section is dominated by a few partial wave states. The theory of threshold cusp effects has been summarized by Geltman (1969) and by Brandsden (1970). The basic mathematical theory is described by Newton (1966, Chapter 17). Although the S-matrix can be defined for both open and closed channels and can be continued analytically through an excitation threshold, only the open-channel submatrix is unitary. The K matrix, Eqs. (14) and (17), is determined by this submatrix of S . Since its

333

LOW-ENERGY ELECTRON SCATTERING

dimension increases at an excitation threshold, the definition of the K matrix changes at the threshold. New elements of K occur, and the old elements cannot be followed through threshold by analytic continuation. This discontinuity in the definition of the K-matrix produces the cusp effects mentioned above. For orbital s-waves in n channels opening at a given threshold, with M channels open below this threshold and all specified channels coupled, the K-matrix changes from dimension M x M to N x N at the threshold, if N = M + n. Above threshold,

Below threshold (Dalitz, 1961), the correct form is R M M

=KMM +jKMn(I - j p -

1 ~ n M

(79) where the right-hand member is defined by analytic continuation from above to below threshold. Equation (79) follows from the general formula

(I

+ iT)(I - iK) = I

(80) which defines K in terms of the open-channel submatrix of the T-matrix. From its definition in terms of the S-matrix, Eq. (19), this submatrix of Tcan be continued analytically through a threshold. From Eq. (80), I - i K M M= (1 + iTMM)-I= { [ ( I - i K " ) - 1 l M M } - 1 (81) This formula defines T" above threshold in terms of K", and then defines K M Min terms of the submatrix T M MIn. general, for a partitioned matrix A". ( A - ~ ) M M = [AMM - ~ M n ( ~ n n ) - 1 ~ n M ] - l (82) unless Ann is singular. When applied to Eq. (81), this results in Eq. (79) (Nesbet, 1975b). For s-waves in the n channels opening at threshold, the submatrices of K N Ncan be expressed in the form KMM

= AMM,

KMn

= K n M = AMnklI2,

Knn

= Annk

(83) The matrices A can be continued analytically as real functions through the threshold. Below threshold, k becomes iK, where K 2 0. From Eq. (79), K M M

= AMM - AMnAnM

+0(KZ)

(84) Analytic continuation is probably not reliable beyond the linear region, since long-range potentials imply irregular dependence of the matrices A on variable k , beyond the leading terms indicated in Eqs. (83). K

334

R . K . Neshet

It is generally convenient to represent scattering data in terms of eigenphases 6, and eigenchannel vectors x,,, as in Eqs. (17). For Eqs. (83) and (84) to be valid, certain conditions must be satisfied by analytic forms chosen to fit the eigenvectors x , (Nesbet, 1975b). Above threshold XMM

= aMM,

XMn

= aMnkli2 1

XnM

= anMkl/2

(85)

where the matrix of eigenvectors is not symmetric. The matrices a may contain terms linear in k. The first of Eqs. (83) will be free of terms linear in k, and hence real below threshold, only if

;

XPP%P ad,] cos’ 6, 8k

1

=o

k=O

(86)

Together with the orthonormality conditions x, . xp’ = 6,,,

(87)

Equations (86) determine the terms in x M Mand 6 7 linear in k, given aMn,anM, and the constant leading terms of x M Mand 6.: This analysis leads to well-known formulas when M = 1, N = 2. Let eigenphase 6, correspond to the open channel below threshold and have the value 6 at threshold. The second eigenphase Sb is proportional to k just above threshold. The eigenvectors are defined by a mixing coefficient x such that, for the open channel below threshold denoted by index 1 and the new channel by index 2, x 2 , = Xk“’, x l b = -Xk”2 (88) Sufficiently near threshold that x can be assumed constant, Eqs. (84) and (86) lead to below threshold 6 - q2sin’ 6 + o ( K ~ ) , 6,= (89) 16 + kX’ sin 6 cos 6 + O ( k 2 ) , above threshold The discontinuity of slope apparent here becomes a corresponding discontinuity of slope in the sum of eigenphases when there are more than two channels. The coefficient of k in 6 , is not constrained by Eq. (86). The elastic cross section below threshold is

4n

o I 1= kfsin2 6[1 - ~ K X ’ cos 6 sin 6

+ O(K’)]

Above threshold, =11-

- %sin2 k; 6[1 - 2kz2 sin’ 6

+ O(k’)]

LOW-ENERGY ELECTRON SCATTERING

335

and the excitation cross section is c I 2=

472 . sinZ 6[kx2

+ O(k2)]

kl

(92)

Equations (90) and (91) show the discontinuity in slope that characterizes a Wigner cusp or rounded step. It should be noted that this discontinuity in c I 1is of the same magnitude as o12, but the total cross section above threshold, 4n crI1 c 1 2= ,sin2 6[1 + kx2 cos 26 O ( k 2 ) ] (93) kl

+

+

also differs in slope from c I 1just below threshold. Thus cusp effects can appear in measurements of total cross sections.

F. EXCITATION OF AUTOIONIZING STATES Autoionizing states of the target atom lie above the ionization threshold and decay with a certain lifetime by emitting an electron, acting as resonances in the scattering continuum of the positive ion. They have the structure of an electron attached to an excited state of the ion. Electron impact excitation of these states is of current experimental interest. Threshold excitation of autoionizing states of rare gas atoms has recently been reviewed by Read (1975). In the threshold region, if the autoionization takes place sufficiently rapidly, the ejected and inelastically scattered electrons can interact. This is known as a postcollision internctiort (Hicks et al., 1974). The scattered electron, of energy El after the inelastic collision, loses energy AE to the ejected electron. Since AE increases as El approaches zero, this leads to an apparent upward shift of the autoionization threshold if very low energy electrons are detected. When AE exceeds El, the scattered electron is recaptured, resulting in indirect excitation of excited target states (Read, 1975). A satisfactory quantitative theory of this process has not yet been worked out. This involves correlation between two electrons in continuum states and is beyond the scope of present methods in the intermediate-energy range. Under the approximation that the effect of ionization on other scattering is small, the theory presented here can be applied qualitatively to the excitation of autoionizing states, although the theory is not strictly applicable above the ionization threshold. A very long-lived autoionizing state can be expected to produce threshold excitation effects similar to those due to bound states. The modification of the theory of Wigner cusp effects due to a finite lifetime (width r > 0) of the target state excited at a given threshold has been considered in scattering

R . K. Nesbet

336

theory for high-energy physics (Baz, 1961; Nauenberg and Pais, 1962; Fonda and Ghirardi, 1964). The essential result is that the analytic structure of the S-matrix near a compound-state threshold (autoionizing state in the present context) is modified by moving the threshold branch point off the real k-axis to the position of the compound state resonance pole. Thus the scattered electron momentum is to be defined by the complex value

k = [2(E - Eth)+ ir]’l2 = r1/2(& + i)l/z

where Eth is the energy and

(94) (95)

r the width of the compound state, and

as in Eq. (30). The effect is to spread out the onset of excitation and threshold cusp structure over the width r (Baz, 1961). Nauenberg and Pais (1962) use the picturesque term “wooly cusp” to describe this modified threshold structure. Formulas for cross sections near threshold can be derived by substituting Eq. (95)for k into the appropriate expression for the T-matrix, then evaluating the cross section from Eq. (20). In the case of two interacting channels, one of which opens at the specified threshold, T-matrix elements for elastic scattering and excitation, respectively, are All

- i k ( A 1 1 4 2 - -4% - i(All + kA,,)

T1l = 1 - k(A11A2, - &)

(97)

where the matrix elements of A are defined as in Eq. (83). These expressions are valid in the vicinity of E = 0 for both positive and negative E, with complex k defined by Eq. (95). The postcollision interaction is not taken into account in this approximation, which should be valid in the limit r 0. At energies well above the threshold region, the excitation of autoionizing states leads to resonance structures described by the theory of Fano (1961). For electron-impact excitation, when the energy loss E - El is close to the energy of an autoionizing state, the T-matrix for the background excitation process producing electrons of energy El is modified by a factor of the form (E

+ 4)/(&+ 4

(99)

where q is real, and &

= 2(E - El

- Eth)/r

(100)

with Eth and r the energy and width of the autoionizing state as in Eq. (96).

LOW-ENERGY ELECTRON SCATTERING

337

The partial cross section for the background process is modified by a resonance lineshape factor of the form given in Eq. (3l), with parameter q defined by Eq. (99). This theory applies also to resonances observed in photoionization (Fano and Cooper, 1968).

111. Methods Experience with the computational methods considered here indicates that a quantitative theory of electron scattering by complex atoms below the ionization threshold must correctly include the effect on the incident electron of target atom polarizability and of short-range electronic correlation. A multichannel formalism is required to describe excitation processes. As discussed in Section II,C, in multichannel variational methods, target atom polarization is taken into account by the introduction of polarization pseudostates or their equivalent in the closed-channel or Hilbert space component of the scattering wave function. The first method to derive the target atom polarization response beyond the simple asymptotic polarization potential was the polarized orbital method (Temkin, 1957). This method, which is limited to single-channel scattering, will not be described here in detail. Formalism and applications have recently been reviewed by Drachman and Temkin (1972) and by Callaway (1973). In the polarized orbital method, the external closed-channel orbital $ 4 ( p ) , Eq. (26), is replaced by the functional form x $ ~where , x is a modulating factor that can depend on the coordinates of two electrons. In its simplest form (Temkin and Lamkin, 1961)the factor x contains the multipole interaction factor l/r"+' and a cutoff factor E ( X , r), which vanishes when the external coordinate r is less than the internal coordinate x. Through analysis equivalent to the general derivation of an optical potential, as in Eqs. (10) and (13), this functional ansatz leads in channel p to a polarization potential of the correct asymptotic form (Temkin, 1957, 1959; Sloan, 1964). Calculations with various forms of this method have been extensively applied to electron-atom scattering. Multichannel variational methods differ among themselves in the choice of either explicit closed-channel components of the scattering wave function or of physically equivalent terms in the Hilbert space component, Eq. (6). These alternative partitionings of the wave function result in different definitions of the effective optical potential in the resulting set of coupled integrodifferential equations for the channel orbital functions, Eqs. (13). A second major distinction between alternative methods is the choice of technique for solving these coupled equations. In variations of the closecoupling method, these equations are solved explicitly. In the rytrix uaria-

338

R. K. Nesbet

tional method, a matrix representation of these equations is used to compute a variational estimate of the K-matrix. The close-coupling method has been adequately described in several review articles and monographs (Burke and Seaton, 1971; Smith, 1971; Seaton, 1973). Details of the method do not need to be presented here. Recent close-couplingcalculations have included polarization pseudostates. Inclusion of short-range correlation effects in close-coupling calculations has been much less systematic. Detailed results will be discussed below. Another general approach has been developed from the quantum field theory formalism of Schwinger (Csanak et al., 1971; Thomas et al., 1973). This method develops a hierarchy of coupled equations for n-particle Green’s functions (Schwinger, 195 l), which have immediate physical significance as transition amplitudes or as functions describing the response of a many-electron system to external perturbations. Truncation of this hierarchy leads to self-consistent approximations at successive levels of complexity. A generalized optical potential for inelastic electron-atom scattering has been derived from this formalism by Csanak et al. (1973). The lowest-order approximation has been applied to calculations of excitation cross sections for helium in the intermediate energy range (Thomas et al., 1974a; Chutjian and Thomas, 1975). These results will be discussed below. Csanak and Taylor (1972, 1973) have analyzed this formalism in terms of widely used models and approximate methods, and have shown that it provides a rationale for making internally consistent approximations in such methods.

A. R-MATRIXMETHOD This method combines matrix techniques for expansion of the wave function within a boundary radius ro with numerical solution of coupled differential equations outside ro (Burke et al., 1971; Burke and Robb, 1972; Fano and Lee, 1973 ; Burke, 1973). Formalism and applications have recently been reviewed by Burke and Robb (1975). This method has the advantage of allowing matrix manipulation of algebraic equations containing matrix elements of nonlocal operators within ro while exploiting the simple asymptotic form of close-coupling equations (without exchange) outside ro . The R-matrix method was developed in nuclear physics (Wigner and Eisenbud, 1947; Lane and Thomas, 1958). As usually presented, the theory makes use of Green’s theorem to relate value and slope of the channel orbitals at ro ,expanding these functions for r < ro as linear combinations of basis functions satisfying fixed boundary conditions at ro . The true logarithmic derivative (or its multichannel generalization, the reciprocal of the Rmatrix) is computed from Green’s theorem, despite the use of basis functions with a fixed but arbitrary value of this quantity. Because of the inherent

LOW-ENERGY ELECTRON SCATTERING

339

discontinuity of the boundary derivative this expansion tends to converge slowly, but this can be corrected by an approximate method due to Buttle (1967). The theory of the method can be understood most clearly in terms of a variational formulation. This can be stated as an application of the wellknown variational principle of Kohn (1948) to a variational trial function whose derivative at ro is discontinuous. The R-matrix, which connects multichannel vectors defined by the value and derivative of the external channel orbital wave functions at ro ,is determined by the variational principle. The R-matrix represents the results of solution of Schriidinger’s equation within ro in terms of the boundary conditions at ro required to determine the external solutions. This variational derivation has been presented in somewhat different form by several authors (Jackson, 1951; Bloch, 1957; Oberoi and Nesbet, 1973a, 1974; Schlessinger and Payne, 1974).The clear advantage of this approach is that it makes possible the use within ro of basis functions that are not constrained by an arbitrary boundary condition. Model calculations (Oberoi and Nesbet, 1973a) show that convergence is greatly improved by dropping this constraint, and that very simple basis functions can be used without the Buttle correction. A calculation by Kracht and Chang (1975) shows rapid convergence of the R-matrix for a model of elastic scattering by helium. The method will be derived here for the single-channel problem defined by Eq. (33). The general multichannel case is treated by Oberoi and Nesbet (1973a, 1974). As in Eq. (34) the wave function $ is separated into two terms, but they are defined in distinct regions: r < ro (101) *(‘I= r 3 ro (102) It is assumed that 4(ro) = 4 . 0 ) (103) but 4’(ro)and u’(ro)may differ. The asymptotic form of u(r) is

::: :1

u

-

w0

+ w1 K

(104)

where wo and w 1 are the two exact solutions for r 2 ro with asymptotic forms specified by Eq. (38). For a single channel, K is tan 6. The possible discontinuity of $’ introduces a term at ro into the variational functional

E=

Lm$(H

-

Jwdr

4(H - E ) 4 dr + $(&$’ - uu‘),,

+1

.m

ro

u(H - E)u dr

(105)

340

R . K . Nesbet

Evaluation of the variation of E requires boundary terms, obtained by partial integration, lr0[$(H- E ) 6 4 - 6 4 ( H - E ) 4 ] dr =

-*(4 64’ - 4’ 6$)ro

(106)

’0

1

.m

[u(H - E ) 6u - du(H - E)u] dr = 3 6 K + 4(u 6u’ - u’ 6u),,

(107)

‘ro

Here Eq. (38) has been used. From Eq. (104), 6u is asymptotically equal to w1 6 K . These equations can be combined to give

6(E - 3 K ) = 2

.*o

1

‘0

+2

6+(H

- E)+

d r +.6c$(ro)4’(ro)

.m

1

6u(H - E)u dr - 6u(ro)u’(ro)

(108)

’ ro

Since Eq. (103) holds for all variations, it follows immediately that the Kohn functional

[K]= K - 2 s

(109) is stationary if u is an exact solution for r > ro , if 4 is an exact solution for r < r o , and if

4“ro) = U’b-0)

(1 10)

The formal equation for 4 is ( H - E ) 4 = 3 6 ( r - ro)[u’(ro)- 4’(ro)], r d ro (111) but this can be satisfied in the closed interval r < ro only if the right-hand member vanishes, which implies Eq. (110). If 4 is approximated by a finite expansion in linearly independent basis functions {qa},

then variation of the Kohn functional with respect to the coefficients c, gives the matrix equations QobCb = !Pla(ro)~‘(rO) (113)

1 b

where Qd is a matrix element of the operator (Bloch, 1957)

LOW-ENERGY ELECTRON SCAlTERING

34 1

This replaces the kinetic energy integrand -&a$ by ;&&, to make the matrix of Q symmetric. The approximate solution 4 is where

From continuity at

yo,

Eq. (115) gives

(117) the desired relation between value and slope of the external function at the boundary ro . The R-matrix is defined as r; 'p(ro). For a multichannel Hamiltonian H , , the appropriate generalization of Eq. (116) is 4 - 0 )

= P(rob'b.0)

1

Ppq(r)

=

5 Ca Cb ~ 3 r ) ( Q '-) % ~ 8 ( r o )

(118)

where {q."(r)}is the set of basis functions for channel p . The boundary conditions for external channel orbitals at ro are

for the external orbital in channel p corresponding to the solution of index q of the multichannel Schrodinger equation. The K-matrix is obtained by integrating the coupled equations for r > ro , using these boundary conditions at ro , It can easily be verified that the variational functional Z vanishes when 4pq(r)is computed by this method. In the single-channel case, when u(r)is an exact solution for r 2 ro , Eq. (105) reduces to 1

=

Ca Cb Ca Qab Cb - 5 Ca Ca 4'Aro )u'(ro)

(120)

from Eqs. (112) and (1 14). This reduces to zero when Eq. (113) is satisfied. It follows from Eq. (109) or its multichannel generalization that the computed K-matrix is stationary with respect to variations of 9 within the linear space of basis functions. In the standard R-matrix method (Lane and Thomas, 1958; Burke and Robb, 1975), the basis functions qa are required to satisfy boundary conditions yo v'(r0)

=M

ro)

(121)

342

R. K. Nesbet

Fano and Lee (1973) and Lee (1974) adjust the parameter B so that consistent solutions of Eq. (111) can be obtained. In the standard method, Eq. (113) is solved in the form

Cb Mabcb

= ha(rO)[u'(rO)-

(B/rO)u(rO)l

(123)

where {Mab} is the matrix of H - E in the range r ,< r o . This matrix is symmetric because of the boundary condition at ro . With the definition

the formula corresponding to Eq. (117) is u(r0) = p(ro"'(r0) - ( W 0 ) 4 r o ) I (125) If B = 0 this is equivalent to Eq. (113), except that a constraint has been imposed on the basis orbitals. As before, the R-matrix is r;'p(ro) or its matrix equivalent. In the method described by Burke and Robb (1975), n basis functions qo, with the specified boundary condition at ro are obtained as eigenfunctions of a model Hamiltonian H o . The matrix Hd is diagonalized to give Mabas (Ea- E) dab, whose eigenvectors define orthonormal functions qa over the interval 0 < r < ro as linear combinations of the orthonormal basis functions. An additional function 4 0 ( r , E) is obtained by integrating the Schrodinger equation for H o outward to ro at the given value of E. Then the R-matrix for H o is

from Eq. (125). When these results are combined with Eq. (124), the Buttlecorrected (Buttle, 1967) R-matrix is

When the Buttle correction is included, the approximate function +(r) is a linear combination of the n basis functions {qo,}, and the additional function t$o(r, E). Zvijac et al. (1975) point out that a further improvement can be obtained by determining the final linear combination of these functions variationally. In the variational formalism described here, this result can be obtained more directly by adding cPo(r, E) to the basis set before solving Eq. (113).

LOW-ENERGY ELECTRON SCATTERING

343

An alternative method proposed by Lane and Robson (1969) gives a variational estimate of the S-matrix, but the result is not stationary. B. MATRIX VARIATIONALMETHOD

This method is an extension to electron scattering problems of computational techniques that have been very useful in applications to bound states of complex atoms. The background of this method has been reviewed by Harris and Michels (1971), by Truhlar (1974),and by Nesbet (1973a, 1975a). The partitioning of the scattering wave function is used explicitly as indicated in Eqs. (10) and (13),so that a quite general variational expansion of the Hilbert-space component of Y is folded into an effective matrix optical potential acting in the space of channel orbitals. These equations are then solved by a matrix technique developed from the variational theory of Kohn (1948).Computational details of a particular implementation of this method are given by Lyons et al. (1973). The method was first used for quantitative calculations of e-H scattering by Schwartz (1961). A systematic procedure is used to build up the Hilbert-space wave function. The equations solved can be described as the variational equivalent of a hierarchy of n-electron continuum Bethe-Goldstone equations (Mittleman, 1966; Nesbet, 1967). Details of this procedure have recently been reviewed (Nesbet, 1975a) and will not be repeated here. Typical calculations are carried out in the first order of the hierarchy of Bethe-Goldstone equations, equivalent to solving the two-electron continuum equations for electron pairs consisting of the external electron and each target atom valence electron. In general, this is accomplished by including in the Hilbert space basis all (N + ltelectron configurations containing Slater determinants of types W and in the notation defined by Eq. (7). The orbital basis in each case is augmented until computed results appear to be stabilized. The physical effects included at this level of approximation include the static-exchange interaction augmented by all dynamic effects of target atom multipole polarizability, assumed to be additive among subshells of occupied orbitals (Nesbet, 1973a, 1975a). Electronic pair correlation of negative-ion states is also included at this level of approximation, excluding pair correlation in the target atom except for specific near-degeneracy effects (Thomas et al., 1974b). For open-shell atoms, the balance of correlation effects between target atom and negative ion states requires special attention. This subject will be discussed in more detail below. In the matrix variational method, radial channel orbitals for open channels are approximated by

e,

fp

= W o p ~ o p+ W l p E l p

(128)

R . K . Nesbet

344 where

wpp wlp

-

k; 1'2r-1 sin(k,r - $ 1 , ~ )

k;1'2r-1 cos(k,r

-$1,~)

(129)

so that the asymptotic form off$-) agrees with Eq. (5). The functions w i p , i = 0, 1, are constructed to be regular at the origin and orthogonal to all Hilbert space basis orbital radial factors for 1 = 1,. The functions wipprovide

a basis set for variational solution of the reduced equations

C4 mpq$,

= 0,

all p

(130)

in the linear space of channel orbitals, as defined by Eqs. (10) and (13). Since closed-channel functions can be included with the Hilbert space component in Eq. (lo),which defines the matrix optical potential in Eq. (130), it will be assumed that all indices p,q here refer to open channels. From Eq. (130), the coefficients a must satisfy the matrix equations

C rnpj"aj,= 0, k

all i,p

(131)

For N c open channels, these equations should have Nc linearly independent solutions. From Eqs. (lo), (ll), and (13), the matrix in Eq. (131) is rnpg u = Mtq -

Cc

v

M i ,p ( M - l ) p v M vj q,

(132)

The matrices here are the bound-bound matrix M p v

= (@,IH - El@")

(133)

(@,I H - E Id @ p $ i p )

(134)

the bound-free matrix, M p , ip

=

and the free-free matrix (non-Hermitian) M n = (d@,$i,( H - E I dOs$jq)

(135)

where @ i p is defined by Eq. (3) for radial factor wip(r).The matrix r n c g is computed explicitly from these formulas (Lyons et al., 1973; Nesbet, 1973a). A triangular factorization algorithm is used to evaluate matrix elements involving (M-l),,v for matrices of large dimension (Nesbet, 1971). Equation (132) involves an implicit solution for the coefficients c, in Eq. (1) that reduces the variational functional (YsI H - E 1 yl,) to the form

345

LOW-ENERGY ELECTRON SCATTERING

In the matrix notation of Section II,B this is

-

+ mo1al) + a ~ ( m l o a+o m l l a l )

G = atma = a$(mooao

(137)

and for an exact scattering solution, Eq. (131) is

ma=

(:: :::)(I:)=o

(138)

Here the symbol (7) denotes an Hermitian adjoint, or transpose of a real matrix. In general, for approximate wave functions, Eq. (138) has no nontrivial solutions, and a variational method must be used. For infinitesimal variations of the matrix of coefficients a,

6= = Gatma + (rna)+ba+ at(m - mt)sa

(139) The non-Hermitian part of rn arises from the free-free matrix Eq. (135),and, with channel orbitals satisfying Eq. (129), reduces to a surface integral that gives mol - mIo = (140) When this is substituted into Eq. (139),

68 = Gatma + (rna)t6a + $(a$6al - af6ao)

(141) This equation leads to various multichannel forms of the variational principle of Kohn (1948). If the reactance matrix K is defined as in Eq. (14) and the matrix elements of K are treated as variational parameters, then 6ao = 0,

6a, = 6 K

(142)

and Eq. (141) becomes

6E = 6Kt(mlo + rnllK) + (mlo + mllK)'6K

+ 36K

(143)

It follows immediately that the Kohn functional [ K ] = K , - 28,

(144)

is stationary for variations about a trial matrix K , chosen so that mlo

+ mllK, = 0

or

K , = -m;,'mlo Substitution of this into Eq. (137), making use of Eq. (140) to express mOl in terms of mIo, gives the Kohn formula

[KI = -2(moo

- mlom;llm10)

(147)

R. K. Nesbet

346

Anomalous singularities are inherent in this formula, occurring at singular points of the matrix m , , as a function of E or of parameters in the variational wave function. In analogous formulas for K - anomalous singularities occur at singular points of moo. The anomalies can be avoided by alternative use of [K] near singularities of mooand of [ K - '3 near singularities of m , , (Nesbet, 1968, 1969). Truhlar (1974) surveys several of the available alternative variational methods. Nesbet and Oberoi (1972) concluded that the most satisfactory method that could be based on the Kohn formalism is the optimized anomuly-free (OAF) method. In this method, the unsymmetric matrix m is transformed to upper triangular form, which can be done by orthogonal transformation unless complex eigenvalues are encountered (Nesbet and Oberoi, 1972). This transformation defines column vector matrices u and fl such that

mio = B+ma= 0

(148)

The Kohn functional in the transformed linear space is [K'] = -(mbI)-'rnbo

(149)

and the reactance matrix is

[KI = (a1 + f l l [ K ' l ) b O + 8o[K'I)(150) In an approximate calculation this matrix is not necessarily symmetrical and must be symmetrized. This method has been incorporated into computer programs used for calculations to be discussed here (Nesbet, 1973a). C. COMPARISON OF METHODS In the close-coupling expansion, the atomic states 0,in Eq. (1) are used in the sense of an approximation to a complete set, and each channel orbital I(lp is solved for exactly. Hilbert-space functions aflare ordinarily included only as required for consistency with orthogonality constraints. In general, the set {(D,,} includes the (N 1)-electron configurations that can be constructed from orbitals belonging to occupied subshells in the functions @, . The niunber of coupled integrodifferential equations increases with the number of target states 0,. In practice this greatly restricts the basic expansion. Many calculations have been carried out in the static-exchangeapproximation, which includes only a single state of the target atom. Inclusion of polarization pseudostates, described in Section II,C, is a relatively recent development. Results to be discussed here show that it apparently succeeds quite well in treating the electric dipole polarization potential for ground states of complex atoms, without making accurate calculations impractical.

+

LOW-ENERGY ELECTRON SCAlTERING

347

Within the close-coupling formalism, the resulting coupled equations for channel orbitals can be solved either directly or by the R-matrix method. The matrix variational method can also be used as in the algebraic closecoupling method (Seiler et d.,1971). These methods obtain identical results, in principle. Calculations with the R-matrix method have remained within this close-coupling framework (Burke and Robb, 1975). For open-shell atoms, the number of coupled equations increases rapidly with the number of target states, since all alternative vector-coupling schemes contribute to the total wave function. In work on complex atoms, this has severely limited the close-coupling expansion, not yet going beyond dipole polarization pseudostates for the target atom ground state, although excited target states are of course included for inelastic scattering. Specific short-range correlation terms have usually been neglected, except for those required for consistency with occupied orbitals. Because of these limitations it is difficult in practice to judge the convergence of the close-coupling expansion by internal criteria. In the matrix variational method, effects of electronic correlation and polarization are represented by the Hilbert space component of the wave function, the last term in Eq. (1).This allows more flexibility than the closecoupling expansion, since matrix methods developed in bound-state theory can be used for very large matrices. At the level of calculations carried out so far (equivalent to solving two-electron continuum Bethe-Goldstone equations), this approximation describes dynamical effects of multipole polarizabilities for all target atom states included in the open-channel expansion (Nesbet, 1973a, 1975a). The disadvantage of this great flexibility is that several terms in the variational wave function are required to approximate a single term d O , $ , in the close-coupling expansion. The orbital basis set is required to represent three distinct aspects of the wave function: occupied target orbitals and the corresponding polarization functions; the inner portion of each external open-channel orbital; and each external closed-channel orbital in its entirety. The latter requirement indicates that the orbital basis should include functions with asymptotic character r-'-' sin(kr + 6). This has been done by Seiler et al. (1971) for calculations on e--H scattering, but a cruder approximation in terms of exponential functions has been used in most of the work discussed here. In the R-matrix method as currently used (Burke and Robb, 1975), Eq. (124)is solved by diagonalizing the matrix Mab. This provides the values of q i ( r o )and E, required in Eq. (127) for all values of E. Only Ro(E),for the Buttle correction, requires separate computation for each energy. This has the very great advantage of producing the R-matrix as an explicit function of E from a single calculation. The work of obtaining a solution of the scatter-

348

R. K. Nesbet

ing equations at given E reduces to solving the asymptotic close-coupling equations, for which rapid and accurate methods are available (Norcross, 1973). In the implementation of the matrix variational method by Lyons et al. (1973), the large bound-bound matrix H,, is independent of E, but subsequent steps in the calculation are relatively time consuming, and the grid of energy values used in reported calculations is rather coarse compared with R-matrix results. An inherent difficulty with the R-matrix method is that the boundary radius ro must be large enough to justify dropping exchange terms and nonlocal interactions outside it. This makes it difficult to extend calculations on valence states to higher atomic excited states, since the radius of a Rydberg orbital nl is proportional to n2. The matrix variational method, by making use of a basis expansion for all values of r, avoids this particular difficulty with target atom Rydberg states. Another practical advantage of this method is that the unperturbed continuum functions are included explicitly in the variational wave function. As k + 0 in any open channel, these functionsgive the partial wave Born approximation, and ensure that calculations are numerically accurate just above thresholds. The limit of small k is a source of difficulty for direct integration techniques (Norcross, 1973). The balance of electronic correlation energy between target atom and negative ion bound states or resonances is important for complex atoms at low energies, since resonance structures dominate the cross sections. The polarizedfrozen-core model (LeDourneuf et al., 1976; LeDourneuf, 1976)has been remarkably successful in maintaining this balance, as shown by Rmatrix computations of negative ion binding energies. Results obtained with the matrix variational method (Thomas et al., 1974b; Nesbet and Thomas, 1976)show that this balance is sensitive to the pattern of choice of electronic configurations within the general formal scheme of a Bethe-Goldstone hierarchy. Thomas and Nesbet (1975c,d) treat the residual correlation energy difference at a given level of calculation as an adjustable parameter. Recent work indicates that a specific pattern of target atom open-shell configurations is implied for consistency with negative ion configurations included in the variational wave function. This development is being explored in calculations on negative ion states (Nesbet et al., 1976). When exact target atom wave functions are used in the close-coupling expansion, theoretical lower bounds to phase shifts or upper bounds to elements of K - I can be established (Hahn and Spruch, 1967). Since exact target wave functions are available only for hydrogen, these theories cannot be applied rigorously to complex atoms. Hahn (1971) has formulated a " quasi-minimum " variational principle that relaxes the rigorous bound in

LOW-ENERGY ELECTRON SCA'ITERING

349

order to allow variational approximation to the wave functions. In practice, it is found that computed K-matrix elements or phase shifts show stationary behavior as functions of variational parameters when the overall wave function is relatively accurate.

IV. Applications A. HELIUM, RESONANCES Excellent agreement has been achieved between theory and experiment for e-He scattering in the elastic scattering region, below the 2jS threshold at 19.818 eV. A narrow resonance of ' S symmetry occurs in this region, near 19.36 eV. Experimental data for low-energy e-He scattering have been reviewed by Andrick (1973) and by de Heer (1976). In the energy interval 2-19 eV, differential cross sections were measured with a relative accuracy better than 5 % and used for a phase shift analysis (Andrick and Bitsch, 1975). These results for the s- and p-wave phase shifts are compared with theoretical values in Fig. 2 (de Heer, 1976). The calculations of Sinfailam and Nesbet

2.5

-e U

0

10

2.0

0

5

10

15

E (eV) FIG.2. e-He. Comparison of elastic s- and p-wave phase shifts (de Heer. 1976. Fig. 8, with permission).

(1972) (-) used the matrix variational method in the electron pair BetheGoldstone (BG) approximation. This includes dynamical multipole polarization effects and is in excellent agreement .with experiment. The calculations by Burke and Robb (1972) (- - -), also shown in Fig. 2, used the R-matrix method, but in the static exchange approximation of the close-

R . K. Nesbet

350

coupling expansion. This neglects the polarization potential and clearly underestimates the p-wave phase shift. The BG p-wave phase shift is very close to polarized orbital calculations of Duxler et al. (1971).The theoretical results extend down to zero energy, below the range of the Andrick and Bitsch data (O),and have been taken by de Heer (1976) to provide the best available estimate of the e-He cross section in the low-energy region. The analysis of Andrick and Bitsch (1975) showed that up to 19 eV the d-wave phase shift is in close agreement with the partial-wave Born approximation. For a dipole polarization potential this gives the simple formula 6,

.m k 2 = (21 + 3)(21+ 1)(2l - 1)'

l>O

in the limit k2 -+ 0, where c1 is the electric dipole polarizability of the target atom. If Eq. (151) is valid for 1 = 2 it must hold for higher 1-values. The scattering amplitude is then given by a simple formula (Thompson, 1966) obtained by summing partial wave amplitudes for 1 2 2, leaving only do and d l to be determined. In these circumstances,Andrick and Bitsch (1975) were able to deduce the whole set of phase shifts from the observed angular dependence of the differential cross section. Since the phase shifts determine the absolute value of the total cross section, no external calibration is needed. As shown in Fig. 2, these results are confirmed by theory. At energies below 5 eV, comparison of the BG results (Sinfailam and Nesbet, 1972) with the momentum transfer cross section oM,obtained from measurements of the drift velocity of electrons moving through helium gas in an applied electric field (Crompton et al., 1967, 1970), indicates that there is a residual discrepancy of several percent. Bederson and Kieffer (1971) use computed phase shifts to evaluate a/aMand then use this ratio to convert the drift velocity data to an estimated total cross section. Below 5 eV this gives a total cross section several percent greater than the theoretical curve. This small discrepancy may be due to the failure of variational wave functions to include target atom electronic correlation. The free atom is represented only in the Hartree-Fock approximation in present calculations. The BG calculations (Sinfailam and Nesbet, 1972) locate the 'S resonance at 19.42 eV, with background phase shifts do = 104.9",6, = 18.1", d2 = 3.2" (Nesbet, 1975a), in substantial agreement with values deduced from resonance scattering (Andrick, 1973): d, = 105", d, = 18", 6 , = 3.2". From observations of threshold structures in e-He elastic scattering at the 23S and 2% thresholds, 19.818 eV and 20.614 eV, respectively,Cvejanovic et al. (1974) calibrate the 'S resonance at 19.367 k 0.008 eV. The width measured both by Gibson and Dolder (1969) and by Golden and Zecca (1971) is 0.008 eV. There is an unresolved discrepancy between this value and the results of variational calculations. Temkin et al. (1972) find

LOW-ENERGY ELECTRON SCATTERING

351

r = 0.0144 eV at E , = 19.363 eV, while Sinfailam and Nesbet (1972) find = 0.015 eV at E , = 19.42 eV. These variational calculations neglect electronic correlation in the helium ground state. Since correlation must reduce the coefficient of the dominant 1s' configuration, it will tend to reduce the integrals indicated in Eq. (48) that determine the width of the ' S resonance. The dominant configurations ls2s2 and ls2p2 of the ' S resonance state S ground state contininteract directly with the configuration ls'ks of the ' uum, but not with perturbing configurations (nl)'ks unless n = 2. New calculations are needed to explore this expected effect of target atom correlation on the resonance width. In the matrix variational method, resonances appear explicitly associated with eigenvalues of the bound-bound matrix, Eq. (133). A search for resonances, no matter how narrow, amounts only to counting eigenvalues and examining the behavior of eigenphases in the immediate vicinity of such eigenvalues. An automatic resonance search procedure using this formalism is described by Nesbet and Lyons (1971). In the calculations of Sinfailam and Nesbet (1972), this search procedure indicated that for partial wave states up to 'F0 the ' S resonance near 19.36 eV is the only e-He scattering resonance below the 23S threshold. If this ' S resonance can be described as dominated by configurations 1~2.9'and ls2p2, other resonances of structure (ls2s2p)'P0, (ls2p2)'D, and the upper ' S resonance from ls2s' and ls2p2 might be expected at higher energy. Calculations by Oberoi and Nesbet (1973b), continuing the BG calculations of Sinfailam and Nesbet (1972) above the 23S threshold, summarize this region of energy by the results shown in Fig. 3. Sums of eigenphases for the principal doublet scattering states are plotted against k, the momentum relative to the 23S threshold. The main structural features in these computed curves correspond to observed features in the various elastic and inelastic cross sections coupling the five target states in this energy region (llS, z3S,2% 23P0at 20.964 eV, 2lPo at 21.218 eV). These structural

FIG.3. e-He. Sums of eigenphases for ' S . 'Po, and 'D partial waves.

352

R. K . Nesbet

features are in substantial agreement with earlier five-state close-coupling calculations (Burke et al., 1969a), and with more recent R-matrix calculations (Berrington et al., 1975a; Sinfailam, 1976) using the five-state expansion, augmented in some cases by 'Po and 'D pseudostates. The Feshbach resonances just below the n = 3 threshold (3% at 22.72 eV) found in the BG calculations could not be described by these close-coupling wave functions. As mentioned above, inclusion of the n = 3 parent states of these resonances increases the atomic radius by a factor 9/4 from the n = 2 levels, and makes accurate R-matrix calculations more difficult because of the resulting increase in the boundary radius y o . The theory of multichannel resonances outlined in Section II,D shows that such a resonance corresponds to a rise through n: rad of the sum of eigenphases for some symmetry component of the scattering wave function. From Fig. 3, the computed eigenphases show a broad 'Po resonance between the 2% and 2% thresholds, a broad 2Dresonance near the 23P0 threshold, and several narrow (Feshbach) resonances just below the 3% threshold. The smooth descent of the 'S eigenphase sum from the 2% threshold is characteristic of threshold behavior in elastic scattering when a true bound state lies just below threshold, as indicated in Eq. (77) in the theoretical discussion. In the present case, two 'S channels are open, but the narrow resonance near 19.36 eV has the same analytic effect on the channel opening at the 2% threshold (19.818 eV) as a similarly displaced bound state would have on a single elastic-scattering channel. Ehrhardt et al. (1968) give a detailed argument based on Eq. (77) and on the assumption of weak coupling between the two 2S eigenchannels near threshold to account for s-wave structure observed near the 23S threshold in their experimental measurements of the differential excitation cross section. Other threshold structures apparent in Fig. 3 will be discussed below. Computed results for the ~ ' S - F ~excitation ~S cross section are shown in Fig. 4 (Berrington et al., 1975a). The curves are labeled BG (Oberoi and Nesbet, 1973b), RM (Berrington et al., 1975a), and exp for experimental results of Brongersma et al. (1972). Similar experimental data have been obtained by Hall et al. (1972). The error bar shown indicates the large uncertainty in the absolute normalization of the experimental cross section. The three prominent peaks, in order of increasing energy, correspond to the broad 'Po and 'D resonances and to the cluster of narrow resonances below the n = 3 threshold. If the experimental data were normalized to either of the theoretical results, there would be excellent agreement, except for the upper peak, missing from the R-matrix results because of inadequacy of the fivestate close-coupling model, as discussed above. Similar results for the 1'S-.2'S excitation are shown in Fig. 5 (Berrington et al., 1975a). The curves are labeled as before: BG (Oberoi and Nesbet,

0.07

6

0.06

5

0.05

0.02 1

0.01 0

0 Electron Energy (eV)

FIG.4. e-He. 1%

+ 2%

excitation cross section (Berrington er a/., 1975a, Fig. 5).

0.04

0

21

22

23

Electron Energy (eV) FIG.5. e-He. 1's +2IS excitation cross section (Berrington er al., 1975a. Fig. 6).

354

R. K . Neshet

1973b), RM (Berrington et al., 1975a), and exp (Brongersma et al., 1972). Contributions to the RM total curve from individual partial wave states are shown as separate curves. The calculations of Oberoi and Nesbet (1974) were not carried out for points near enough to the 2 % threshold to give the very narrow peak shown in the RM cross section in Fig. 5. When the matrix variational calculation is carried out on a finer energy grid, it gives the 'S excitation peak shown in Fig. 6 (Nesbet, 1975b). This peak is qualitatively E(eV1 20.614

20.615

20.619

20.626

20.636

20.648

20.663

20.681

20.701

0.04

0.03

u (nail 0.02

0.01

0

-22 0.01

0.02

0.03

0.04

k(a:

0.05

0.06

0.07

I

8

1

FIG.6. e-He. 1 IS + 2's excitation cross section, near threshold.

similar to that found by Berrington et al. (1975a), but it is both higher and narrower, peaking within one millivolt above threshold. Details of the threshold scattering structure will be discussed below. At higher energies, the RM results shown in Fig. 5 rise above the BG results, in part because the BG calculations include only scattering states *S, 'Po, and 'D, while the RM results also include 'F0 and 'G, which contribute increasingly to the total excitation cross section as the energy increases. The experimental data follow this increasing trend of the RM results. At energies above 22 eV, structure appears in the BG curve due to Feshbach resonances below the 33S threshold. This structure is missing from the RM calculations, which

LOW-ENERGY ELECTRON SCATTERING

355

cannot describe these resonances for reasons given above. In more recent experimental data (Brunt et al., 1977) these resonances appear much more prominently than they do in the data of Brongersma et al. Recent data on resonance series with n 2 3 have been summarized by Heddle (1976). The BG calculations should be augmented by additional partial wave states, and the RM results must be extended to include the Feshbach resonances with n = 3 parent states. Both theory and experiment need refining near the 23P threshold, where the detailed curves are noticeably different. Experimental data on e-He resonances obtained prior to 1973 are summarized by Schulz (1973). The n = 3 Feshbach resonances are not yet completely characterized. The BG calculations of Oberoi and Nesbet (1973b) locate two 2S resonances at 22.44 and 22.53 eV, respectively, with widths 0.15 and 0.03 eV. A 'Po resonance is computed at 22.45 eV with width 0.022 eV. These resonances may account for structures in total metastable production observed by Pichanick and Simpson (1968) at 22.44 and 22.55 eV, with widths estimated to be approximately 0.1 eV. If the BG calculations are correct, the n = 3 resonances form a pattern quite different from that of the n = 2 resonances. More precise experimental data are needed, especially differential cross sections in order to determine the angular quantum numbers of these resonances.

B. HELIUM, EXCITATION AND THRESHOLD STRUCTURES Threshold effects are evident in the sums of eigenphases shown in Fig. 3. The very sharp rise of the *S eigenphase sum at the 2% threshold is characteristic of a virtual state. As indicated in Eq. (75), the new eigenphase rises from threshold as k / K , but the rise is limited to n/2 rad. Observed structure in e-He scattering was attributed to this virtual state near the 2% threshold by Ehrhardt er al. (1968). The virtual state was first identified by Burke et al. (1969a) and confirmed by subsequent calculations (Oberoi and Nesbet, 1973b; Berrington et al., 1975a). Calculations carried out very close to this threshold give the narrow 2% excitation peak shown in Fig. 6 (Nesbet, 1975b). The theory of cusp effects outlined in Section II,E shows that Wigner cusp or rounded step structures can occur in the 'S scattering state at lS3Sexcitathresholds. Figure 3 shows a tion thresholds and in the 2Postate at lS3P0 prominent Wigner cusp in the 2Poeigenphase sum at the 23P0 threshold. A similar structure occurs at the 2'P0 threshold, but is not visible on the scale of the figure. These cusps appear clearly in the 2's excitation cross section computed by Oberoi and Nesbet (1973b), and in a recent high-resolution study of total metastable production (Brunt et al., 1976).As shown in Fig. 5, the BG and RM calculations differ in detail at the 23P0and 2lPo thresholds.

356

R. K. Nesbet

In the original publication (Oberoi and Nesbet, 1973b) the BG calculation shows true cusps at the variationally computed threshold energies, slightly displaced from their experimental values. Detailed calculations have been carried out for energies close to the 23S and 2lS thresholds, where cusp structure can occur in the ' S scattering state (Nesbet, 1975b). At the 23S threshold, a rounded step structure is computed in the total elastic cross section, in quantitative agreement with parameters used to fit the observed structure in the differentialcross section (Cvejanovic et al., 1974). Theoretical differential cross sections were computed by Sinfailam (1976) from the R-matrix results (Berrington et al., 1975a). The 90" The experelastic cross section is shown in Fig. 7 (Sinfailam, 1976) (-). x10'9 200

199

--

'l?

N

-._sE

..

198

i e

YI

-

u .Z

197

E

r

2

n 196

195

+

"

'

1

19.7

'

"

"

'

1

'

1

19.8

. 19.9

Electron Energy ( e V l

FIG. 7. e-He. Differential elastic cross section at 90" near 2% threshold (Sinfailam, 1976, Fig. 1. Copyright of the Institute of Physics).

imental points are from Cvejanovic et al. (1974)normalized to the calculated threshold cross section, and the dashed curve is a parameterized R-matrix calculation by Herzenberg and Ton-That (1975), who computed only the ' S partial cross section. At 90", the 2Poscattering state does not contribute. The agreement between theory and experiment shown in Fig. 7 is excellent. The 90"differential 23S excitation cross section is shown in Fig. 8 (Sinfai-

357

LOW-ENERGY ELECTRON SCAITERING x10-19 7

1

0

20.0

20.5

21.0

21.5

22.0

Electron Energy (eV)

FIG.8. e-He. Differential 23S excitation cross section at 90" (Sinfailam, 1976, Fig. 4. Copyright of The Institute of Physics).

lam, 1976). The R-matrix computed curves are shown with and without the 2D component (curves A and B, respectively) and are compared with the BG results (Oberoi and Nesbet, 1973b) and with the absolute differential crosssectional data of Pichou er al. (1975, 1976).The dip below the 2's threshold is shown here to be due to interference between the 'S and 'D components of the wave function. Near the 2jPo threshold the cross section is dominated by the 'D shape resonance. All three curves are in excellent agreement below the 2'P0 threshold, but the BG curve follows the trend of the experimental data more closely above the threshold. Algebraic close-coupling calculations by Wichmann and Heiss (1974) of the 23S differential excitation cross sections are in good agreement with experiment. R-matrix results for the 90" differential 2lS excitation cross section are shown in Fig. 9 (Sinfailam, 1976), and compared with absolute differential cross-section data of Joyez et al. (1975, 1976). Curves A and B show the R-matrix results with and without the 2Dcomponent, respectively. The pure 'S narrow peak just above threshold corresponds to the structure shown in Fig. 6, from BG calculations, due to the 'S virtual state. As in Fig. 5, the R-matrix cross section appears to follow the rising trend of the experimental data, but the theoretical cross section above the 2lP0 threshold is significantly larger.

358

R . K . Neshet x10-19

1.5 c L

N

-6 C

.-

li '.O I

e

u -m .-c

$ 0.5

r

LC

n

0 20.5

21 .o

21.5

22.0

Electron Energy (eV)

FIG.9. e-He. Differential 2's excitation cross section at 90" (Sinfailam, 1976, Fig. 5. Copyright of The Institute of Physics).

The multichannel threshold theory outlined in Section II,E was used (Nesbet, 1975b)to analyze e-He cross sections near the 2% and 2lS thresholds. Cusp and step structures are found in all 'S partial cross sections. The most striking result is the extremely rapid rise of the 2% excitation cross section at threshold, due to the 'S virtual state. The 2 % excitation cross section rises much more rapidly from threshold than does the 23S excitation cross section.This agrees with high-resolution experiments on helium threshold excitation (Cvejanovic and Read, 1974), which indicate that the 2's threshold excitation peak becomes progressively larger than the corresponding 2% peak as experimental energy resolution is improved. Figure 10 presents the computed 2's and 2% excitation cross sections on a common energy scale (Nesbet, 1975b).R ( A E ) is defined as the ratio of the integrals of these cross sections up to energy AE above threshold. This ratio is plotted for comparison with trapped-electron experiments. Cvejanovic and Read (1974) find that the 2's to z3S threshold peak ratio, which corresponds to R(AE) for an experimental acceptance width AE, increases with experimental resolution. The peak ratio 2.6 corresponds to the best attainable resolution, with acceptance width 16 meV (width at half-height of the extraction efficiency curve). From Fig. 10, the computed ratio R for this

359

LOW-ENERGY ELECTRON SCATTERING

0

0.02

0.04 0.06 AE (eV above threshold)

-

4.0

-

R

0.08

FIG. 10. e-He. Excitation cross sections and ratio R ( A E ) of their energy integrals.

value of AE is 2.5, in quantitative agreement with the experiment. This ratio drops to values near unity for AE in the range 50-100 meV, in agreement with earlier experiments of apparently lower resolution. By implication, the computed function R(AE) can be used to calibrate the resolution of future trapped-electron experiments. From Eq. (92), the rate of increase from threshold of an s-wave excitation cross section is determined by the parameter b = 21' sin2 6

(152) BG calculations (Nesbet, 1975b) indicate that the helium 2% excitation cross section remains nearly linear in k, including both 2S and *Pocontributions, up to k = 0.10~; '. Hence for the reverse process (electron impact deexcitation), the rate is (153) nearly constant in this energy range. The parameter b at the helium 2% threshold was determined from rounded-step structure in the observed ground-state elastic cross section (Cvejanovic et al., 1974) to be k2 021

=2~b/3

b = (13 k 3) x

m

(154)

The computed value (Nesbet, 1975b) is b = 0 . 1 9 6 8 ~=~10.4 x

m (155) within the experimental error limits. Nesbet et al. (1974) showed that the rate coefficient for deactivation by thermal electrons can be estimated by k2 c~~~ in units nacai for temperature T = ki/3kB

(156)

360

R. K . Nesbet

where c1 is the fine-structure constant and kB the Boltzmann constant a.u./K). Then the theoretical value of b, Eq. (155), gives an (3.1667 x estimate of the 2% metastable deactivation rate for thermal electron impact, from Eq. (153), T G 1053 K (157) This rate constant is an important parameter in plasma dynamics, and has not been measured directly. The analysis given here shows that it can be determined indirectly by accurate measurements of threshold cusp or step structures in elastic scattering. Cvejanovic et al. (1974) found rounded steps in the 90" elastic cross section at both 2% and 2% thresholds. The observed structures were similar in form, but the height of the resulting step appeared to be greater for 2%. With reference to the parametric formulas, Eqs. (90)-(92) here, this was attributed to similar values of the background phase shift, with a larger mixing coefficient for 2% due to the more rapid rise from threshold of the excitation cross section. These qualitative results are confirmed by theoretical calculations (Nesbet, 1975b). The 2% threshold structures have been studied by Huetz (1975), in differential elastic and 23S excitation cross sections. Total cross sections obtained by extrapolating the angular data agree qualitatively with theory, but there is an unresolved discrepancy at angles below 90" between the observed data and the parametric theory. Detailed theoretical calculations of differential cross sections near this threshold are needed. Cross sections for other elastic and inelastic processes connecting the helium n = 2 states have been computed (Oberoi and Nesbet, 1973b; Berrington et al., 1975a; Nesbet, 1975b). Details may be found in the original publications.

KH,(2, 1) = 0.1312nolca~= 2.525 x

cm3/sec,

C. HELIUM, INTERMEDIATE ENERGIES Above the ionization threshold, the methods considered here are no longer strictly valid, because the number of open channels becomes infinite. However, in the low intermediate energy range, an intuitive approach can be based on neglecting the specific effects of target atom ionization. In the close-coupling formalism, pseudostates can ,be added to simulate the flux loss to open ionization channels. In calculations of e-H 2s and 2p excitation, Burke and Webb (1970) introduced functions 3s and 3p for this purpose, added to the three-state basis Is, 2s, 2p. For energies up to 54.4 eV their results were significantly closer to experiment than either the threestate close-coupling results or the Born approximation (which is not expected to be valid in this case for energies below 100 eV). In a more

LOW-ENERGY ELECTRON SCATTERING

36 1

systematic examination of this approach, Burke and Mitchell (1973) studied the e-H 2s excitation cross section, using pseudostate s-functions from a discrete complete set (constant exponent). They found a spurious resonance below each pseudostate threshold, but the cross section appeared to converge to a reasonable limit if the resonance structure were smoothed out. Two-state close-couplingcalculations of e-He 2'P0 excitation cross sections were carried out in the intermediate energy range by Truhlar et al. (1973). In the matrix variational method, open channels can be suppressed simply by omitting the corresponding channel functions from the scattering wave function. Resonances below the omitted thresholds can still occur, since they are described by the Hilbert space component, the second summation in Eq. (1). In this formalism, calculations with a selected subset of strongly interacting open channels constitute a strong-coupling approximation. The helium 23S differential excitation cross section was computed in this way by Thomas and Nesbet (1974a). The variational wave function represented the 1'S ground state of helium and all four n = 2 states as open channels, with virtual excitation structure equivalent to the Bethe-Goldstone approximation of Oberoi and Nesbet (1973b). The differential excitation cross section computed at 29.6 eV was in reasonably quantitative agreement with experiment (Trajmar, 1973),and showed a pronounced dip at 125", evident in the experimental data. This structure had not been obtained in any previous calculation. This strong-coupling approximation included the dynamical polarization effect of the 23S-23P0 interaction, which apparently cannot be neglected in the low intermediate energy range. This multichannel effect is not included in methods that include explicitly only the initial and final target state for a particular transition. Such methods have been successful in this energy range for the dipole-allowed 2'P0 excitation. In particular, the distorted wave calculations of Madison and Shelton (1973) have given good results at 40 eV, and the first-order Green's function method is still in good agreement with experiment for both 2'P" and 2's differential excitation cross sections at 29.6 eV (Thomas r r d.,1974a). Similar results have been obtained for n = 3 excitations (Chutjian and Thomas. 1975). These methods become more accurate as energy increases. The calculation of the 2% excitation cross section i n the matrix variational strong-coupling approximation was successful only because pseudostate resonances were not present at 29.6 eV. An attempt to apply the same method to 2lP" excitation failed for this reasbn (Thomas and Nesbet, 1974b). In the case of 2' PO, many overlapping pseudoresonances occurred, precluding a meaningful calculation without some systematic procedure for removing the resonances or averaging over them. This problem of removing effects of spurious resonances remains the prin-

362

R . K . Nesbet

cipal barrier to extending multichannel methods into the low intermediate energy range. The problem of accounting for the flux loss into ionization channels is also unresolved, although at low energies it may suffice to ignore this effect or to add pseudostate open channels as done by Burke and Webb (1970) and by Burke and Mitchell (1973).

D. OXYGEN Electron scattering by open-shell atoms C, N, and 0 is difficult to study experimentally. The only available results of reasonably high precision are observations of narrow resonance structures in the n = 3 excitation range of e-0 scattering. The few experiments at low energies have been reviewed by Bederson and Kieffer (1971). The general trend of total ground state cross sections at low impact energies has been measured for e-N scattering (Neynaber et al., 1963; Miller et al., 1970)and for e-0 scattering (Neynaber et al., 1961; Sunshine et al., 1967). Dehmel et al. (1974, 1976) have measured the ratio of forward to backward e-0 scattering from 3 to 20 eV and the total differential cross section at 5 and 15 eV. There are no comparable data for e-C scattering. Because of the open shell structure (2s22p") of the ground state configurations of carbon, nitrogen, and oxygen there are several distinct target states in the low-energy range. Electric dipole polarizabilities are small compared to the alkali metals, but polarization potentials still dominate the low-energy scattering. The 3P ground states of carbon and oxygen have static quadrupole moments, which couple partial waves 1 and 1 2. The first application of the polarized orbital method (Temkin, 1957) was to e-0 elastic scattering, originally including only the 1 = 0 partial wave and the 2p+d part of the dipole polarizability. Henry (1967)completed this work by including2p+s and 2s-+p polarization effects. Contributions from partial waves with 1 > 2 were estimated from the Born formula, Eq. (151). Figure 11 shows the total elastic cross section computed by Henry, in comparison with other theoretical calculations and with experimental total cross-sectional data (Sunshine et al., 1967). The experimental points, with error estimated to be 20%, do not define a smooth curve. The very low values of Henry's computed cross section near threshold are compatible with the value deduced from shock tube measurements, 2 x an2 (2.37~;)at 0.5 eV (Lin and Kivel, 1959). Close-coupling calculations, including all states of the ground configuration, were first reported for e-0 scattering by Smith et al. (1967). An error in this work was corrected subsequently by Henry et al. (1969). These results were verified by an algebraic close-coupling calculation, in the single configuration (SC) approximation (Thomas et af., 1974b). This SC

363

12 -

Single Configuration

cross section is shown in Fig. 11. This approximation neglects target atom polarizability. The resulting cross section is much larger than the polarized orbital result and above the experimental points up to 6 eV. The curve labeled CI in Fig. 11 is a matrix variational calculation (Thomas et al., 1974b) that includes only the polarization and correlation effects due to near-degeneracy of the 2p and 2s orbitals. All states of configurations 2s22p4,2s2p5, and 2p6 were included in the target state variational basis. The configuration 2s2p5 provides dipole polarization pseudowhich contribute to the polarizability ofall three states 3P, ID, 's states 1.3P0, of the 2s22p4 ground configuration. The configuration 2p6 interacts with 2s22p4 to bring the 'S state down in energy relative to the other states. In Fig. 11, ('S) denotes the computed threshold (relative to 'P) in the SC approximation, while 'S denotes the relative threshold in the CI approximation. The CI total cross section is reduced noticeably from the SC curve, but only a fraction of the distance to the polarized orbital curve. Recent calculations have clarified this situation. Figure 12 shows the previous results, together with the total cross section obtained in the electron-pair Bethe-Goldstone (BG) approximation by matrix variational calculations (Thomas and Nesbet, 1975a,b).Effects of 2s+np virtual excitations, beyond n = 2, were found to be small, and the calculations were simplified by omitting this class of virtual excitations. The BG approximation includes all significant effects of electron pair correlation of the external electron with valence electrons of the target atom. Basis orbitals and partial waves with 1 < 3 were included in the calculations. The computational procedure (Lyons et al., 1973) includes in the variational wave function all vector-coupling schemes possible for any configuration generated by the

R. K . Nesbet

364 16

1

I

I

,

I

1

I

I

I

I

1

-

Sunshine et al.

14

10

Polarized orbital

'D

OO

; 1'

'S

3

:

5

6

;

8

9

I0

1'1

12

E (eV)

FIG. 12. e - 0 . Total cross section.

Bethe-Goldstone structural algorithm or by coupling unoccupied orbitals to configurations used for target atom states. Hence electron pair correlation of the external electron with the 'D and 'S excited states is automatically included. As can be seen from Fig. 12, the remarkable effect of this systematic inclusion of electron pair correlation and polarization is to bring the BG cross section into close agreement with the polarized orbital calculation of Henry (1967).

365

LOW-ENERGY ELECTRON SCAmERING 11.0

I

I

I

I

I

I

-

0

10.0 0

0

-

9.0 0

0

8.0

0

S

-

7.0

C ._

4.0 0 Sunshine et al.

3.0

Neynaber et al.

2.0

l.O 0

A Lin and Kivel

1 0

2.0

4.0

6.0

8.0

10.0

13.60

Energy in eV

FIG. 14. e-0. Total elastic cross section (Tambe and Henry, 1976a, Fig. 2).

Figure 12 shows the experimental data of Sunshine et al. (1967), the shock-tube result of Lin and Kivel(1959), and the least-square value used by Neynaber et al. (1961) to represent their data. The BG results are consistent with all of these data. An additional experimental test is shown in Fig. 13 (Thomas and Nesbet, 1975a), which compares computed and observed values of the ratio of forward to backward scattering (Dehmel et al., 1974). The BG results lie within the experimental error bars. The forward/backward ratio differs from unity at zero energy because of the electric-quadrupole potential of the 3Ptarget ground state. Several recent close-coupling calculations of e-0 scattering have augmented the single configuration wave function with correlation terms and polarization pseudostates (Saraph, 1973; Rountree et al., 1974; Tambe and Henry, 1976a,b). The computed total elastic cross sections are shown in Fig. 14. Curves SC and BG are labeled as in Fig. 12. Curve R (Rountree et al., 1974)is labeled “close coupling” in Fig. 13; S refers to Saraph (1973).PS refers to the polarization pseudostate calculations of Tambe and Henry (1976a,b), who include pseudostates constructed from S and d polarized orbitals but omit specific short-range correlation terms. They find the effect of polarized orbital p to be small and omit it, as did Thomas and Nesbet (1975a). In general, the various close-couplingresults lie between the CI and BG curves shown in Fig. 12.

366

R. K . Nesbet

An important approximation in the PS calculation of Tambe and Henry (1976a,b), which makes their theoretical model structurally different from the BG calculation, is that each polarization pseudostate is retained as a unit in the close-coupling expansion, defining a single closed-channel state. Thus only three pseudostate channels are included in the calculations, one each for 3S0, 3P0,and 3D0,even though these states each have several different components expressed as states of 0’ coupled to polarized orbitals. In the BG expansion, each component of the polarization pseudostate is coupled to all available orbital functions, and the coefficient of each resulting function in the Hilbert space basis is determined independently. Thus the matrix variational method, as used in the BG approximation, involves complete uncoupling of closed-channel states. This is beyond the present capability of the close-coupling method or of its R-matrix equivalent. Recent R-matrix calculations (LeDourneuf et al., 1975; LeDourneuf, 1976), in the framework of the “polarized frozen core” approximation (LeDourneuf et al., 1976) are structurally equivalent to the work of Tambe and Henry (1976a,b), but include 2s+2p effects and g, p, and d polarized orbitals. The e-0 elastic cross sections computed below 6 eV are quite similar. Both results lie above the BG cross section down to the elastic threshold. The BG results show some effect of an imbalance between electronic correlation energy computed for the target atom and for negative ion or electron-scattering states. The consequences of such an imbalance were examined for the ’Po component of the e-0 scattering wave function by Thomas et a!. (1974b), by parameterizing the residual correlation energy difference between the 0 - (*Po)state of configuration 2s22p5and the O(3P) target state. Variation of this parameter by k0.5 eV had an insignificant effect in the elastic scattering region. An exploratory calculation by LeDourneuf (1976) indicated that uncoupling the pseudostate functions of given symmetry into separate closed-channel terms tended to reduce the computed cross section. Comparison of the calculations of Rountree et al. (1974) (R in Fig. 14) with those of Tambe and Henry (1976a,b) (PS in Fig. 14) indicates a similar effect, since the Rountree et al. calculations treated individual polarization components as separate closed-channel terms. This work differs from Tambe and Henry’s at higher energies because polarization terms are included only for 1 = 0 partial waves. The experimental differential cross section at 5 eV (Dehmel et al., 1976) is shown in Fig. 15 (Tambe and Henry, 1976b). The curves PS and BG are labeled as before; DW indicates a polarized-orbital distorted wave calculation by Blaha and Davis (1975). Since the DW calculation cannot describe short-range correlation effects, but includes the full atomic polarizability as an ad hoc polarization potential, it provides a model calculation with physi-

367

LOW-ENERGY ELECTRON SCAITERING 2.0

1.5

1 .o

0.5

0 0

30

60

90

120

150

180

0 . Deg.

FIG. 15. e-0. Differential cross section at 5 ev (Tambe and Henry, 1976b, Fig. 1).

cal content similar to the pseudostate approximation (PS here). The BG differential cross section falls below the experimental point at 30°, although, as shown in Fig. 13, the integrated forward/backward ratio falls within the experimental error bars (Dehmel et al., 1974). The apparent structure in the experimental data between 30 and 90" is not found in any of the computed curves. More precise experimental data are needed at low impact energies in order to make a conclusive choice among the theoretical results. In view of these considerations, the BG calculations probably provide the most reliable current estimate of the true e-0 cross section at low energies. The computed K-matrix (Thomas and Nesbet, 1975a) was used by LeDourneuf and Nesbet (1976) to compute collision strengths for fine-structure transitions 3PJ-3PJ# of atomic oxygen induced by thermal electron impact. This was done by transforming the K-matrix, computed in LS-coupling, to jj-coupling (Saraph, 1972), and then shifting the excitation cross sections to energy thresholds defined by the fine-structure energy levels. The collision strength for p - q is defined as

a,,, = k:o,a,,

= ki(o,aqp

(158)

where w is the degeneracy factor of the initial state of the transition. Figure 16 shows the computed collision strength (BG) for the 3P2-3P1 transition, as a function of electron temperature. The BG result is compared with a full jj-coupling calculation (THD) that used a simplified closecoupling wave function (Tambe and Henry, 1974) and with the PS calculation (THA) transformed from LS to jj-coupling, but without shifting the fine-structure threshold (Tambe and Henry, 1976a). There is substantial

368

R. K. Nesbet

T

(OK)

FIG.16. e - 0 . 'P2-'P, collision strength.

agreement between the BG curve for Q(2, 1) and the two results of Tambe and Henry in their respective ranges of validity. Formulas fitted to this BG curve have been used to recompute the thermal rate constant for electron cooling by excitation of fine-structurelevels of atomic oxygen (Hoegy, 1976). This process is important in the dynamics of the ionosphere. The new cooling rate is significantly smaller than that currently used in upper-atmosphere models. Cross sections for excitation of the 'D and 'S valence states and for scattering from these states have been computed by Thomas and Nesbet (1975a,b), BG calculations up to 10 eV; and by LeDourneuf et al. (1975) and LeDourneuf (1976), R-matrix polarized frozen core calculations up to 45 eV. The BG calculations included differential cross sections for excitation of 'D and 'S from the ground state. The two sets of theoretical results are in general agreement, but there are no experimental data for comparison. Details are given in the original publications. The one case of apparent disagreement is the 3P+'S excitation cross section. The cross section given by LeDourneuf (1976) rises to a maximum of approximately 0.03aai at 8.7 eV,while the BG cross section continues to increase up to 11 eV. Calculations at more k-values are probably needed to define the BG curve more precisely in this energy range. Figure 17 shows the BG excitation cross compared with earlier calculations by Vo Ky Lan et al. sections (-), (1972) (- - -), who augmented the single-configuration close-coupling expansion with a multichannel polarized orbital wave function.The agreement is excellent for all three cross sections. - Resonances formed by electron attachment to n = 3 levels of oxygen have been observed by colliding 0-with helium (Edwards et al., 1971; Edwards and Cunningham, 1973). Spence and Chupka (1974) and Spence (1975) have

369

LOW-ENERGY ELECTRON SCATTERING

2

3

4

5

6

7 8 EkV)

9

1

0

1

1

FIG. 17. e-0. Cornpanson of total excitation cross sections.

observed 0 resollances directly by electron scaritding lrom a beam or partially dissociated molecular oxygen. Except for a close-coupling study by Ormonde et al. (1973) limited to the 'Po partial wave state, there have been no detailed calculations of these resonances by collision theory. The resonance energies have been computed by Matese et d. (1973; Matese, 1974). The method uses bound-state variational wave functions containing configurations built from a fixed ion core function with two external electrons described by orbitals orthogonalized to the core orbitals. Observed resonances correspond to 0 ' core states 4S0, 'Do, 'Po and external orbital dominant configurations 3s2, 3s3p, 3p2. The general agreement between theory and experiment is very good (Spence, 1975), but the theory does not provide information on resonance widths. Matese (1974) has carried out similar calculations for the analogous C- resonances. Electron impact excitation cross sections for the ( 2 ~ ~ 3states ~ ) of~ ~ ~ s ~ oxygen have been measured by Stone and Zipf (1971, 1974). A two-state close-coupling calculation by Rountree and Henry (1972) for 3S0excitation disagreed in the energy range up to 20 eV in both shape and magnitude with the original experimental data. The computed cross section showed resonance structure near threshold. Smith (1976) carried out improved calculations. A five-state expansion was used for the dominant partial wave states, and energy differences within the calculation were taken relative to the computed excited state energy. This moved the resonance structures below threshold, in agreement with bound-state calculations of Matese (1974). When adjusted for cascade effects, the resulting cross section agrees with experiment within rather generous expected error.

R . K . Nesbet

370

E. CARBON Like oxygen, carbon has a 3P ground state and valence excited states 'D and 'S, but it also has a low-lying state ( 2 ~ 2 ~ ~ )The ~ s low-energy '. states of C- are more complex than 0-, which has the single bound state (2s22p5)'P0 at - 1.462 eV (Hotop and Lineberger, 1975),relative to O(3P). The 2s22p3ground configuration of C- has three states, 4S0, 'DO, and 'Po. The lowest of these is bound, at - 1.268 eV, and the 'Do state is weakly bound, at -0.035 eV (Ilin, 1973). The remaining state, 'Po, would be expected to appear as a resonance in the scattering continuum of the neutral ground state. This C- resonance should produce prominent resonance structure in the scattering cross section, while the unique 0-bound state is too far below threshold to have a strong influence. Henry (1968) used the polarized orbital method to compute the cross section shown in Fig. 18. As usually formulated, the polarized orbital method cannot describe multistate resonances, and the curve shows no resonance structure. Single-configuration (SC) close-coupling calculations were carried out by Smith et al. (1967)and by Henry et al. (1969).The latter work is verified by an algebraic close-coupling calculation in the SC approximation (Thomas et al., 1974b; Nesbet and Thomas, 1976). The SC cross section, shown in Fig. 18,shows resonance structure corresponding to a 'Do state of C- near 0.5 eV and a 2Poresonance near 2 eV. The computed 'S threshold in the SC approximation is indicated as ( ' S ) in the figure. The SC

Ci I

"0

..

,

,

I...

2

4

6

0

E (eV)

FIG. 18. e-C. Total cross section.

10

12

LOW-ENERGY ELECTRON SCAlTERING

371

and polarized orbital cross sections are strikingly different. Both are qualitatively incorrect, since the 2Doresonance should be a bound state just below threshold, and the 'Po resonance should appear as a scattering structure. The correct behavior of the low-energy polarized orbital cross section in e-0 scattering is seen to be a fortuitous consequence of the absence of 0 states in the scattering region. The first qualitatively correct theoretical result obtained for the lowenergy e-C cross section (Thomas et al., 1974b;Nesbet and Thomas, 1976)is shown as the curve labeled CI in Fig. 18. The CI calculation is similar to that described above for e-0 scattering. The variational wave function includes 2s-2p near-degeneracy effects, including polarization, but omits other contributions to polarization and to short-range correlation. Matrix variational calculations in the Bethe-Goldstone (BG) approximation (Thomas and Nesbet, 1975d,e)give the total elastic cross section shown in Fig. 19. The polarized orbital cross section (Henry, 1968) is also shown. All single virtual excitations of the 2p orbital subshell are included in the BG wave function, but 2s virtual excitations other than 2s+2p were omitted after tests showed them to be unimportant. It has been found in bound-state calculations (Moser and Nesbet, 1971) that three-electron correlation energy differences can significantly influence the computed electron affinities of complex atoms. Since the inclusion of such terms in the scattering problem is impractical in any existing formalism, there will be a residual error in any feasible ab initio calculation of

E(eW

FIG.19. e-C. Total elastic cross section.

372

R. K. Nesbet

negative ion energies relative to neutral states. A parameter A can be intraduced (Nesbet, 1973b) as an adjustable correction for this residual net correlation energy difference. Calculations in the CI approximation showed that this parameter, used to bias the energy mean value of terms of the negative ion ground-state configuration, could be varied over a rather wide range with internally consistent results (Thomas et al., 1974b). Energies of resonances were found to vary linearly with A, and the widths approached zero smoothly as the resonances approached threshold. These properties are in accord with the analytic theory of resonances near thresholds (Section I1,E) and indicate that A is suitable for adjustment of computed cross sections by interpolation of resonance energies to experimentalvalues. This can also be used in the inverse sense. Experimental and computed cross sections can be matched to determine a best value of A, which then fixes the position of a resonance or bound negative ion state that might be experimentally inaccessible (Nesbet, 1973b). The value of A indicated in Fig. 19 was determined by adjusting the 'Do state energy to its experimental value (Ilin, 1973).The same value was used for both 2Doand 2Pocomponents of the scattering wave function. With this value of A, the expected 2P0shape resonance appears in the scattering cross section, as shown in Fig. 19. The resonance peak is 51.6aaa at E = 0.461 eV, width r = 0.233 eV. In the absence of experimentaldata, these results can be taken to predict a broad 2Poresonance peak in e-C scattering between 0.4 and 0.6 eV (Thomas et al., 1974b). with Excitation cross sections computed in the BG approximation (-), A = 0.530 eV, are shown in Fig. 20 (Thomas and Nesbet, 1975d), and compared with close-coupling results in the SC approximation (Henry et al., 1969) (---). There is reasonable agreement for 013 and 023. The leading peak in a12(SC)is twice the BG peak, probably due to the 'Po resonance, which is displaced upward to about 2 eV in the SC approximation.Differential cross sections up to energy 7 eV for all processes connecting the first three states are included in the BG publication (Thomas and Nesbet, 1975d). Thermal rate constants for deexcitation of the 'D, 'S, and ' S o states by electron impact are estimated from the variation of the excitation cross sections near their thresholds. Recent R-matrix calculations in the polarized frozen core (PFC) approximation (LeDourneuf et al., 1975, 1976)essentially confirm these results. The PFC calculations (LeDourneuf, 1976) show a 2P0resonance peak in the elastic cross section of height 44na: at E = 0.68 eV. These calculations include total cross sections for all processes connecting the four lowest states. They extend to energies beyond the ionization threshold and show resonance structuresdue to all negative ion states of configuration 2s2p4 : 4P,'D, 2s, 2P.The PFC and BG excitation cross sections are in good agreement. In

LOW-ENERGY ELECTRON SCA'ITERING

-

3P

'D

'S

373

5S0

FIG. 20. e-C. Excitation cross sections u l # P + uZ3('D 'S).

'D), o , , ( ~ P - S ) , u , ~ ( ~ P 'So), - + and

comparison with the BG results shown in Fig. 20, the PFC cross section o1 is nearly identical, oI3is somewhat higher at its peak and less irregular in shape, oI4is similar, showing a rapid rise to a 4Presonance peak of height 0.95nai, and 023is roughly 20% greater at the initial peak or shoulder (LeDourneuf, 1976). The general agreement between these results supports their validity, since they are derived by different methods, each incorporating the essential elements of the scattering process, but organized differently and independently in almost every computational detail. It should be pointed out here, for the record, that the final calculation of the BG curve c I 2shown in Fig. 20 was corrected for inadequate convergence of the orbital basis set after comparison with preliminary results from the PFC calculations (LeDourneuf et al., 1975). An important aspect of the PFC calculations is that the energies of nega-

374

R. K. Nesbet

tive ion bound states and resonances have been computed with the same theoretical model (LeDourneuf and Vo Ky Lan, 1974, 1977; LeDourneuf et al., 1976;LeDourneuf, 1976).The model is essentially that used by Tambe and Henry (1976a,b), described above. Closed channels are defined by coupled polarized pseudostates, one for each symmetry type, and other open or closed channels are limited to states of the atomic ground-state configuration (plus 2s2p3 for carbon). Polarized orbitals 5, p, d are computed by preliminary variational calculations of the ground-state static polarizability and short-range correlation is included at various levels of approximation to the extent that it can be described by (N + 1)-electron configurations constructed from these orbitals (Vo Ky Lan et al., 1976; LeDourneuf et al., 1976). This model is quite successful in computing negative ion energies. In the simplest approximation (no specific short-range correlation terms) the electron affinity of O(’P) is computed to be 1.480 eV and that of C(’P) to be 1.406 eV (LeDourneuf and Vo Ky Lan, 1974, 1977; LeDourneuf, 1976).The experimental values are 1.462 and 1.268 eV, respectively (Hotop and Lineberger, 1975).The C-(zDo)state is computed to be bound by 0.008 eV. The same level of approximation is used for the scattering calculations cited here. The predicted resonance energies can be expected to be of accuracy comparable to that of the negative-ion binding energies. In comparison with these results, it is apparent from the rather large value of the parameter A used in the BG calculations that for open-shell atoms the latter formalism has a less satisfactory balance between core atom and negative-ion correlation energies, despite the use of a much more flexible variational wave function. Some progress has been made in restructuring the algorithm used to select configurations in the BG method. Preliminary results indicate that more satisfactory negative-ion binding energies can be obtained without restricting the generality of the BG variational wave function (Nesbet et al., 1976).

F. NITROGEN The ground configuration of nitrogen is 2s22p3,with states 4S0 (0.00 eV), ’Do (2.38 eV), and ’Po (3.58 eV). The ground configuration of N- is 2s22p4, with states ’P,‘D, and ‘ S , none of which have been observed. Electron affinities of open-shell atoms including nitrogen were computed by Moser and Nesbet (1971), using a variational formalism that included threeelectron correlation effects. The computed energies for C- and 0-,relative to neutral ground states, were - 1.29 and - 1.43 eV, respectively, in good agreement with observed values and with extrapolated values of EdlCn (1960). The computed energy of N-(’P) was 0.12 eV, in the scattering

375

LOW-ENERGY ELECTRON SCATTERING

80 -

G\

60 -

Single Configuration

-

NO

2

40-

CI

2o

-

I

-\\ A

Henry

A a .

0

0

MCC

/C'

2

A

..1 . . 1

+I

D

+,ZP

4

A-

-

*

+,I~PI

,

6 EkVI

..,

c _ _

1' ,

: f. ,

a

I

10

, 12

FIG.21. e-N. Total cross section.

continuum. This disagreed with Edlen's extrapolation (-0.05 eV). The 3P state of N- is indicated to be either very weakly bound, or a low-lying electron scattering resonance. Figure 21 (Thomas et al., 1974b) shows the e-N cross section computed by the polarized orbital method (Henry, 1968). As for oxygen and carbon atoms, there is no resonance structure. Single configuration (SC) closecoupling calculations (Smith et al., 1967; Henry et d., 1969) and the algebraic close-coupling SC curve shown in Fig. 21 (Thomas et al.,1974b)show a strong ' P resonance near 1.0 eV. As in the case of carbon, matrixvariational CI calculations (Thomas et al., 1974b) bring this resonance into agreement with qualitative expectations for the negative ion, in this case placing the 3P state of nitrogen very close to threshold. In Fig. 21, the 2P threshold energy computed in the CI approximation is indicated by 2P,and the SC computed threshold by ('P). In the CI calculations, the ' P state could be placed either above or below threshold by varying the residual correlation energy parameter A in a small range about zero (Thomas et al., 1974b).Figure 21 also shows results of a multiconfigurational close-coupling (MCC) calculation by Orrnonde ef al. (1973) and experimental data of Neynaber et al. (1963). Similar conclusions were reached in R-matrix calculations that included polarization pseudostates in the close-coupling expansion (Burke et al., 1974). The 'P resonance moved very close to threshold when polarization and correlation terms were added to the single configuration wave function. A six-state calculation gave the resonance position as 0.06 f 0.05 eV. This

376

R. K . Nesbet

FIG.22. e-N. Total elastic cross section. cross section is shown in Fig. 22 (- - -). The figure also shows experimental data by Miller et al. (1970) (e),never published in detail and with no error estimates, which might indicate the existence of a low-lying resonance. Matrix variational calculations in the electron-pair Bethe-Goldstone (BG) approximation were carried out by Thomas and Nesbet (1975c,e). Several values of the parameter A were used. The computed elastic scattering cross sections are shown in Fig: 22. The value A = 0.575 eV gave the closest fit to the data of Miller et al. (1970) and placed the resonance peak at 0.105 eV, in good agreement with the bound-state calculations of Moser and Nesbet (1971). A somewhat larger energy value. 0.19 eV, for the 3Pstate of N-, is estimated by Sasaki and Yoshimine (1974)by extrapolating the residual error in elaborate variational calculations. R-matrix calculations in the PFC formalism (LeDourneuf et al., 1976; LeDourneuf, 1976) place the 3P state of N- at 0.057 eV in the approximatior, used for scattering calculations, and at -0.004 eV if two-electron correlation terms are included for both core states and polarization pseudostates. Total and differential cross sections up to 9 eV for processes connecting the 4S0, 'Do,and 'Po states were computed in the BG approximation with parameter A = 0.575 eV (Thomas and Nesbet, 1975~).The excitation cross sections .are shown in Fig. 23 (-), and compared with MCC results

LOW-ENERGY ELECTRON SCA'ITERING

S ' 'DO), u,,(~S' FIG.23. e-N. Excitation cross sections V , ~ ( ~ +

-+

377

'Po), and oZ3('Do-+ 'PO).

(Ormonde et al., 1973) (---). Agreement is reasonable for o12and 0 1 3 , but the MCC 023 cross section apparently lacks the h11 effect of the 'Po scattering state, which dominates the rise from threshold and produces a shape resonance near 10 eV. This resonance corresponds to the N- state of configuration 2s2p5 and is also found near 10 eV in R-matrix calculations (Berrington et al., 1975b; LeDourneuf et al., 1975). The R-matrix results include total elastic and inelastic cross sections in a wider energy range than the BG calculations.The most recent results use the PFC formalism (LeDourneuf et al., 1975; LeDourneuf, 1976). Except for the elastic scattering range, where the dominant 3P resonance structure depends on the precise location of the resonance, the results are in general agreement. With reference to Fig. 23, the PFC excitation cross section o12is lower at the peak (approximately 0.8nai), g I 3is similar to the MCC curve, and oZ3 rises above threshold to a higher value, approximately0.63nai. There are no experimental data for comparison. As in the case of carbon, thermal rate constants for electron-impactdeexcitation of the 'Do and 2Postates of nitrogen were estimated by Thomas and Nesbet (197%) from threshold formulas for the excitation cross sections.

318

R . K . Neshet

The computed cross sections show no effect of the excited states 'D and ' S of the N- ground configuration. If the 'D state lies below the N(2Do) threshold at 2.38 eV, it is metastable for autodetachment (Hotop and Lineberger, 1975).It cannot interact directly with the e-N(4So) continuum, which consists of states 3?30, 3.5P, 3.sD0,etc. Similarly, the ' S state of N cannot interact directly with this continuum. It also cannot interact with the '*3PDF,etc. However, e-N(2Do)continuum, which consists of states lQ3Do, thee-N(2Po)continuum contains states of symmetry types 'Sand 'D. Hence the 'D state of N- would be metastable if it lay below the 2Do threshold (2.38 eV) and the ' S state would be metastable if below the 2Po threshold (3.58 eV). The locations of there missing states of N- were estimated (Thomas and Nesbet, 1975c) by extrapolating known levels of the isoelectronic series. The values found, adjusted relative to the 3P state, taken to be at 0.10 eV, were 1.44 eV for 'D and 2.88 eV for 'S. This indicates that these states should be metastable for autodetachment, with no effect on electron scattering cross sections. ACKNOWLEDGMENTS The author wishes to thank J. D. Lyons, R. S. Oberoi, A. L. Sinfailam, and ,. D. Thomas for their collaboration in developing and implementing the matrix variational method and in obtaining many of the results presented here. He is indebted to K. A. Berrington, J. N. H. Brunt, P. G. Burke, R. J. W. Henry, W. R. Hoegy, M. LeDourneuf, Vo Ky Lan, and W. D. Robb for communicating results in advance of publication and for cooperating in verification of preliminary results; to the authors cited in the text for permission to reproduce illustrations from their publications; and to the Office of Naval Research for support of this project under Contract No. N0014-72-C-0015.

REFERENCES Andrick, D. (1973). Adu. At. Mol. Phys. 9, 207. Andrick, D., and Bitsch, A. (1975). J. Phys. B 8, 393. Bardsley, J. N., and Nesbet, R. K. (1973). Phys. Rev. A 8, 203. Baz, A. 1. (1957).Zh. Eksp. Teor. Fiz. 33,923 [Sou. Phys.-JETP 6, 709 (1958)l. Baz, A. I. (1961). Zh. Eksp. Teor. Fiz. 40, 1511 [Sou.Phys.-JETP 13, 1058 (1961)l. Bederson, B., and Kieffer, L. J. (1971). Reu. M o d . Phys. 43, 601. Berrington, K. A., Burke, P. G.,and Sinfailam, A. L. (1975a).J. Phys. B 8, 1459. Berrington, K. A., Burke, P. G., and Robb, W. D. (1975b).J. Phys. B 8,2500. Blaha, M.,and Davis, J. (1975). Phys. Rev. A 12, 2319. Blatt, J. M., and Biedenharn, L. C. (1952). Reu. Mod. Phys. 24, 258. Bloch. C. (1957). Nucl. Phys. 4, 503. Brandsden, 9. H. (1970). "Atomic Collision Theory." Benjamin, New York. Brongersma H. H., Knoop, F. W. E., and Backx, C. (1972). Chem. Phys. Lett. 13, 16. Brunt, J. N. H.. King, G. C., and Read, F. H. (1977).J . Phys. B 10,433.

LOW-ENERGY ELECTRON SCATTERlNG

3 79

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LOW-ENERGY ELECTRON SCATTERING

38 1

Moser, C. M., and Nesbet, R. K. (1971). Phys. Rev. A 4, 1336. Mott, N. F., and Massey, H. S. W. (1965). “The Theory of Atomic Collisions.” Oxford Univ. Press, London and New York. Nauenberg, M., and Pais, A. (1962). Phys. Rev. 126, 360. Nesbet, R. K. (1967). Phys. Rev. 156, 99. Nesbet, R. K. (1968). Phys. Rev. 175, 134. Nesbet, R. K. (1969). Phys. Rev. 179, 60. Nesbet, R. K. (1971). J. Comput. Phys. 8,483. Nesbet. R. K. (1973a). Comput. Phys. Commun. 6, 275. Nesbet, R. K. (1973b). Bull. Am. Phys. SOC. 18, 1501. Nesbet, R. K. (1975a). Ado. Quantum Chem. 9, 215. Nesbet, R. K. (1975b). Phys. Rev. A 12, 444. Nesbet, R. K., and Lyons, J. D. (1971). Phys. Rev. A 4, 1812. Nesbet, R. K., and Oberoi, R. S. (1972). Phys. Rev. A 6, 1855. Nesbet, R. K., and Thomas, L. D. (1976). I n “Electron and Photon Interactions with Atoms” (H. Kleinpoppen and M. R. C. McDowell, eds.), p. 27. Plenum, New York. Nesbet, R. K., Oberoi, R. S., and Bardsley, J. N. (1974). Chem. Phys. Lett. 25, 587. Nesbet, R. K., Moser, C. M., and Cornille, M. (1976). Unpublished. Newton, R. G. (1966). “Scattering Theory of Waves and Particles.” McGraw-Hill, New York. Neynaber, R. H., Marino, L. L., Rothe, E. W., and Trujillo, S. M. (1961). Phys. Rev. 123, 148. Neynaber, R. H.. Marino, L. L., Rothe, E. W., and Trujillo, S. M. (1963). Phys. Rev. 129,2069. Norcross, D. W. (1973). Comput. Phys. Commun. 6, 257. Oberoi, R. S., and Nesbet, R. K. (1973a). Phys. Rev. A 8, 215. Oberoi, R. S., and Nesbet, R. K. (1973b). Phys. Rev. A 8, 2969. Oberoi, R. S., and Nesbet, R. K. (1974). Phys. Rev. A 9, 2804. O’Malley, T. F., Spruch, L., and Rosenberg, L. (1961). J. M a t h . Phys. 2, 491. Ormonde, S., Smith, K., Torres, B. W., and Davies, A. R. (1973). Phys. Rev. A 8, 262. Pichanick, F. M. J., and Simpson, J. A. (1968). Phys. Rev. 168, 64. Pichou, F., Huetz, A., Joyez, G., Landau, M., and Mazeau, J. (1975). J. Phys. B 8, L236. Pichou, F., Huetz. A,, Joyez, G., Landau, M., and Mazeau, J. (1976). J. Phys. B 9, 933. Read. F. H. (1975). I n “The Physics of Electronic and Atomic Collisions” (J. S. Risley and R. Geballe, eds.), p. 176. Univ. of Washington Press, Seattle. Rountree, S. P.. and Henry, R. J. W. (1972). Phys. Rev. A 6, 2106. Rountree, S. P., Smith, E. R., and Henry, R. J. W. (1974). J. Phys. B 7, L167. Rudge, M. R. H. (1973). Ado. At. Mol. Phys. 9, 47. Saraph, H. E. (1972). Comput. Phys. Commun. 3,256. Saraph, H. E. (1973). J . Phys. B 6, L243. Sasaki, F., and Yoshimine, M. (1974). Phys. Rev. A 9, 26. Schlessinger, L., and Payne, G. L. (1974). Phys. Rev. A 10, 1559. Schulz, G. J. (1973). Rev. Mod.Phys. 45, 378. Schwartz, C. (1961). Phys. Rev. 124, 1468. Schwinger. J. (1951). Proc. Natl. Acad. Sci. U.S.A. 37,452. Seaton, M. J. (1973). Comput. Phys. Commun. 6, 247. Seaton, M. J. (1975). Adv. At. Mol. Phys. 11, 83. Seiler. G. J., Oberoi, R. S., and Callaway, J. (1971). Phys. Rev. A 3, 2006. Sinfailam, A. L. (1976). J. Phys. B 9, L101. Sinfailam, A. L., and Nesbet, R. K. (1972). Phys. Rev. A 6, 2118. Sloan, 1. H. (1964). Proc. R. Soc. London, Ser. A 281, 151. Smith, E. R. (1976). Phys. Rev. A 13, 65. Smith, K. (1971). “ The Calculation of Atomic Collision Processes.” Wiley (Interscience), New York. Smith, K., Henry, R.J. W., and Burke, P. G. (1967). Phys. Rev. 157, 51.

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Spence, D. (1975). Phys. Rev. A 12. 721. Spence, D.. and Chupka. A. (1974). Phys. Rev. A 10. 71. Stone, E. J., and Zipf. E. C. (1971). Phys. Rev. A 4, 610. Stone, E. J., and Zipf, E. C. (1974). J. Chem. Phys. 60,4237. Sunshine. G., Aubrey, B. B., and Bederson, B. (1967). Phys. Rev. 154, 1. Tambe, B. R., and Henry. R. J. W. (1974). Phys. Rev. A 10, 2087. Tambe, B. R., and Henry, R. J. W. (1976a). Phys. Rev. A 13, 224. Tambe. B. R.,and Henry, R. J. W. (1976b). Phys. Rev. A14. 512. Taylor, H. S. (1970). Adv. Chem. Phys. 18, 91. Temkin, A. (1957). Phys. Rev. 107, 1004. Temkin, A. (1959). Phys. Reu. 116, 358. Temkin, A., and Lamkin, J. C. (1961). Phys. Rev. 121, 788. Temkin, A., Bhatia, A. K.. and Bardsley, J. N. (1972). Phys. Rev. A 5, 1663. Thomas. L. D., and Nesbet, R. K. (1974a). Int. Con/: At. Phys., 4rh. 1974. p. 468. Thomas. L. D., and Nesbet, R. K. (1974b). Unpublished. Thomas, L. D., and Nesbet, R. K. (1975a). Phys. Rev. A 11, 170. Thomas, L. D., and Nesbet, R. K. (1975b). Phys. Rev. A 12, 1729. Thomas, L. D., and Nesbet. R. K. (1975~).Phys. Rev. A 12, 2369. Thomas, L. D., and Nesbet, R. K. (1975d). Phys. Rev. A 12, 2378. Thomas, L. D., and Nesbet, R. K. (1975e). Int. Conl: Phys. Electron. A t . Collisions, 9th, 1975, p. 637. Thomas, L. D., Yarlagadda, B. S., Csanak, Gy., and Taylor, H. S. (1973). Comput. Phys. Commuri. 6, 3 16. Thomas. L. D., Csanak, Gy., Taylor, H. S., and Yarlagadda, B. S. (1974a). J . Phys. B 7, 1719. Thomas, L. D., Oberoi, R. S., and Nesbet, R. K. (1974b). Phys. Rev. A 10, 1605. Thompson, D. G. (1966). Proc. R . Sac. London, Ser. A 294, 160. Trajmar, S. (1973). Phys. Rev. A 8, 191. Truhlar, D. G. (1974). Ado. Chem. Phys. 25, 211. Truhlar, D. G., Trajmar, S.. Williams, W.. Ormonde, S., and Torres, B. (1973). Phys. Rev. A 8, 2475. van Regemorter, H. (1973). Comput. Phys. Commun. 6, 245. Vo Ky Lan, Feautrier, N., LeDourneuf, M.,and van Regemorter, H. (1972). J . Phys. B 5, 1506. Vo Ky Lan, LeDourneuf, M., and Burke, P. G. (1976). J . Phys. B 9, 1065. Wichmann, E., and Heiss, P. (1974). J . Phys. B 7 , 1042. Wigner, E. P. (1948). Phys. Rev. 73, 1002. Wigner, E. P., and Eisenbud, L. (1947). Phys. Rev. 72, 29. Zvijac, D. V., Heller, E. J., and Light, J. C. (1975). J. Phys. B 8, 1016.

MICRO WAVE TRANSITIONS OF INTERSTELLAR ATOMS AND MOLECULES W. B. S O M E R V I U E Department of’ Physics and Astronomy University College London, England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

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

385

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

389

B. Deuterium . .

D. Other Atoms

.................................... 391 A. Hydrogen Atom B. Other Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 V. Recombination Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 . . . . . . . . . . . . 391

...........

. . . . . . . . . . . . 422

. . . . . . . . . . . . . . . 425 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Several hundred rf spectrum lines have been detected in radiation from interstellar matter, most of them in emission but some also in absorption. Wavelengths range from just over 1 mm to several tens of centimeters. The observations have been discussed elsewhere (Somerville, 1977a), where refer383

384

W B. Sornerville

ences are given to origmal papers and to more detailed review articles on individual topics. Hyperfine structure is observed in atomic hydrogen-the celebrated 21 cm line-and possibly also in deuterium. Fine-structure transitions have been considered but not yet detected. Recombination lines are seen for hydrogen, helium, carbon, and with less certainty for sulfur and probably some other heavier species. The number of identified interstellar molecules has been increasing steadily with time; including those few detected in the visible and ultraviolet, 43 have been identified by early 1977. Most of the molecules are detected by their rotational transitions in the microwave radio region. It is found that abundances of the elements by number of nuclei are fairly constant in the Sun and stars, in interstellar clouds, and apart from a deficiency in lighter elements, on Earth. The possibility that some atom or molecule is present in interstellar regions, and perhaps detectable, depends on the cosmic abundances although only crudely because a whole complex of processes of formation and destruction needs to be considered. Some values are given in Table I (Allen, 1973). Here we do not consider the observations, and abundances only indirectly. We are concerned with the atoms and molecules themselves, their structures, the types of transition that can occur, and the theory of their radiative transition probabilities. For atoms we can give a complete discussion, considering not only those which are observed but also all those which might conceivably be observable. The discussion of diatomic molecules includes all those which have been observed and a small selection of others. For polyatomic molecules it is necessary to be selective and brief, mainly for reasons of space. Necessary information such as dipole moments is not yet available for all the molecules detected. The standard work on microwave spectroscopy has long been the book by Townes and Schawlow (1955); a more recent book on applications to molecules is by Gordy and Cook (1970). Herzberg (1945,1950, 1966,1971) has given an exhaustive treatment of molecular structure and spectra. Recent discussions of the experiments are given by Winnewisser et al. (1972) and, in an introductory way, by Carrington (1974). Kovacs (1969) has given a thorough theoretical discussion of rotational structure and line strengths for diatomic molecules in intermediate coupling, with extensive tables; for doublet molecules see also Bennett (1970). Molecular constants and transitions have been tabulated in the NBS microwave tables (Wacker et al., 1964-1968), in the Landolt-Bornstein tables (Hellwege and Hellwege, 1967, 1974), and in the compilation edited by Rosen (1970). An important series of critical survey papers is appearing in Journal of Physical and Chemical Reference Data; there are papers on individual astrophysical molecules and also a survey of diatomic molecules (Lovas and Tiemann, 1974).

385

INTERSTELLAR MICROWAVE TRANSITIONS TABLE I COSMIC ABUNDANCES OF THE ELEMENTS"

Element

Symbol

H

Atomic number

Ground state

E (eV)b

I'

Log abundance

13.60 24.59 13.62 11.26

0 0 0

12.00 10.93 8.82 8.52

0 0 0

Hydrogen Helium Oxygen Carbon

He 0 C

1 2 8 6

Nitrogen Neon Iron Silicon

N Ne Fe Si

7 10 26 14

14.53 21.56 7.87 8.15

Magnesium Sulfur Argon Aluminium

Mg S Ar

7.65 10.36 15.76 5.99

0 0 0

Al

12 16 18 13

Calcium Nickel Sodium Potassium

Ca Ni Na K

20 28 11 19

6.11 7.64 5.14 4.34

0 0

2s1/2

'SO

3P2 Po

1

$

5

7.96 7.92 7.60 7.52 7.42 7.20 6.8 6.39 6.30 6.30 6.25 4.95

' Abundances are given by number of nuclei, relative to hydrogen = 10l2, for all elements of abundance greater than lo6 and also for potassium (Allen, 1973). * E, ionization potential (ground state). ' I, nuclear spin (principal isotope).

11. Spectroscopic Formulas Three quantities are in common use to describe the intrinsic strength of a spectrum line, the transition probability A, the oscillator strength f, and the line strength S (Condon and Shortley, 1935). The line strength appears naturally in the theory; it is the square of the interaction matrix element. For the spontaneous emission of radiation from upper level 2, statistical weight g 2 , to lower level 1, statistical weight g, A,, = (64z4/3hc3g2)v3S

The line strength is symmetrical in the upper and lower levels: s(1,2) = S(2,l)

(2)

The frequency v of the transition is generally expressed in hertz or megahertz

386

W B. Somerville

(formerly Mc/s). The relation to other frequency, energy, and equivalent units is

1 MHz = lo6 Hz =

GHz

= 3.33564 x 10-5 cm-1 = 4.13570 x

lo-’ eV

= 3.03966 x lo-’’ rydbergs (Ryd) = 6.62618 x lo-’’ ergs = 6.62618 x

10-28joules (J)

(3 1 with uncertainties in most cases of a few ppm (Cohen and Taylor, 1973). A wavelength of 1 mm corresponds to 299.792 GHz. With the physical constants given by Cohen and Taylor, = 4.79927 x

O K

A,, = 7.51976 x 10-38v3vg, sec-’

(4)

for electric dipole transitions and A,, = 1.00342 x 10-42v3S/g2 sec-I

(5)

for magnetic dipole transitions, with S in atomic units [units aie’ and1 pi(g,/2)2 respectively] and with v in hertz. The use of modern values for the physical constants changes some of the transition probabilities by a few percent, compared with values published previously. Unless there is some unforseen large change, the numbers given here should survive future refinements of the constants. The oscillator strengthf,, for absorption from level 1 to level 2 and the corresponding emission oscillator strength f, (which is a negative number) are related to A,, and S by

,

A, 1 = ( 8 n 2 e W m c 3 ) h/gz)fiz = 7.42165 x 10-22(g1/g2)vzfi2 sec-’ g, flz =

(6)

- g, f,,= (8x2mv/3he2)S

=i

1.01322 x l o - ’ % , (electric dipole) (7) 1.35201 x lO-”vS, (magnetic dipole) (8) with v in hertz and Sin atomic units; m is the electron mass. The convention The numerical values in (5) and (8) for magnetic dipole transitions are for the common case of a hyperfine transition arising from an electron spin interaction. For an orbital angular momentum interaction the numbers should be divided by ( ~ , / 2 = ) ~1.00232.

INTERSTELLAR MICROWAVE TRANSITIONS

387

has been adopted here of writing the subscripts o n f i n the order (initial, final);f has the particular advantage of being dimensionless. Another useful parameter of the transition is the natural linewidth. In the absence of external broadening influences such as thermal or turbulent motions (Doppler broadening) or pressure effects, the shape of a spectrum line is determined by fundamental quantum processes, and the whole width of the line at half-maximum is (91 where r is the sum of all spontaneous transition probabilities from the upper and lower levels of the line. Av is usually designated the half-width. AV

= r/2a

HZ

111. Atomic Hyperfine Structure Hyperfine structure is produced by the interaction between electronic angular momenta and the nuclear spin. In LS coupling, which is followed closely for light atoms, the total electron spin S and orbital angular momentum L couple with the nuclear spin I as

J=L+S, F=J+I (10) There are (25 + 1)or (21 + 1)values of the quantum number F, whichever is the smaller. Hyperfine structure separations are measured with high precision, using microwave techniques. The transition probabilities must be determined theoretically. Consider an atomic S state (L = 0). For the angular momentum matrix element we may use equation Z92 b of Condon and Shortley (1935). The transition probability is given by (5). where the line strength

+ S)(F + I - S ) ( I + S + F + 1) x ( I + S - F + l)/F

S(F, F - 1)= ( F - I

(11)

Also, 92 = 2F2

+1

is the statistical weight of the upper state; F , can be either F or F -

(12) 1.

A. HYDROGEN ATOM The ground state ls2S,,, of the hydrogen atom has L = 0, J = S = i, I = 4, and F = 0, 1. The triplet level F = 1 has higher energy than F = 0; the separation has the frequency, wavelength, and transition probability given in

388

W B. Soinerville

sec-' (Wild,

Table 11. Values previously quoted for A are 2.85 x sec-' (Spitzer, 1968). 1952) and 2.868 x TABLE I1 ATOMICHYPERFINE STRUCTURE TRANSITIONS

Atom

I

State

[H

t

2D

1

3Het

5

lS2S1/z lS2SI,, 1s2s1/,

14N

1

2P34s3/z

Transition F 0- 1 I-J 2 2

2 2

V

I,

A

(MH4

(cm1

(sec- 1y

1420.41 327.384 8665.65 26.1273

21.11 91.57 3.460 1147

2.876, -15 4.695, - 1 7 1.959, - 12 4.295, -20

1912

1.289, -20

15.6764

' The negative index is a decimal exponent.

In this case, the lower level is the ground one and this line itself is the only emission from the upper level. The natural half-width (9) is

AV = 4.577 x HZ (13) very much smaller than the Doppler width in any source, and it may always be neglected. The 21 cm line was the first spectrum line in radio astronomy and for some years the only one. It has been studied extensively to give information about the properties and distribution of the interstellar gas in our own and other galaxies. It is not to be expected that interstellar hyperfine transitions will be detected from any excited electronic states of atomic hydrogen. In H I regions, where temperatures are low, virtually all the hydrogen atoms present are in n = 1. At higher temperatures where a reasonable fraction of the neutral atoms present is in n = 2, most atoms are ionized and so the excited state populations are still low. B. DEUTERIUM The deuteron has nuclear spin I = 1, which in the ground state combines with J = S = 4 to give F = i,3. The frequency was measured by Wineland and Ramsey (1972). The transition probability in Table 11, from (1 l),agrees of Field (1958) used by Weinreb (1962) but with the value 4.65 x disagrees with some other values (Shklovskii, 1960; Syunyaev, 1967). With

INTERSTELLAR MICROWAVE TRANSITIONS

389

our constants, Weinreb's formula relating optical depths of deuterium and hydrogen becomes t D

= 0.3202~ NDINH

(14)

There has been a tentative interstellar identification of this line.

C. HELIUM As indicated in Table I helium is the second most abundant element in the universe, about 8 % of the total by number of atoms. However, the 4He isotope has nuclear spin I = 0 and so no hyperfine structure. The isotope 3He has 1 = f, but the ground electronic state is 1s' 'So, with zero electronic angular momentum. There is a hyperfine structure in the ground state of the 3He+ one-electron positive ion. In hot interstellar H I1 regions, the helium is predominantly ionized and essentially all the ions are in the ground state: the first excitation potential is 40.8 eV. 3He+ is a hydrogenic system with 2 = 2. The ground state is IS'S,,, and with 1 = f t h e structure is similar to that of the hydrogen atom, except that the hyperfine levels are inverted: the singlet state F = 0 is the higher. The transition has the values given in Table I1 (Schuessler et al., 1969; Goldwire and Goss, 1967). Observation so far has failed to detect this transition.

D. OTHERATOMS Various other atoms have ground-state hyperfine structure at microwave frequencies, but their cosmic abundances (Table I ) combined with low transition probabilities militate against their detection. Of the four species next "C, and in abundance after hydrogen and helium, the major isotopes l60, "Ne all have I = 0; also the ground states of carbon ('Po) and neon ('So) have J = 0. But the major nitrogen isotope 14N has nuclear spin I = 1 and its ground state is 2p3 4S3,2.,and so it does have a hyperfine structure. Shklovskii (1960) has discussed the prospects for interstellar 14N and concludes that the lines could be detectable. I do not know of any attempt to find them. They are at wavelengths near the upper end of the radio window in Earth's atmosphere, an unfashionable region, and the A values are small. Values for the two magnetic dipole transitions are from laboratory measurements by Crampton et al. (1970).The A values calculated using (1 1 ) are slightly different from those given by Shklovskii. The hyperfine splittings are very small because in strict LS coupling there is no splitting at all in this case. For this atom there is a small departure from LS coupling, producing

the splittings, a configuration interaction that mixes excited states into the ground-state wave function. After 'D and 'He, the most abundant minor isotope with nuclear spin is 13C ( I = i).The ratio of the '3C/'2C abundance is 1/89 in the solar system, but in galactic sources studied by molecular microwave lines it can vary and in some sources is more than twice this value (Bertojo ef ul., 1974).The ratio is sensitive to the history of the material in nuclear processes within stars and could in theory be as high as about 1/4. The I3C atom has a hyperfine transition at 372.596 MHz (Wolber rt ul., 1969) but this is in the finestructure level 'P, , which is 43.5cm- above the ground state and unlikely to be populated in normal H I regions. The ion 13C+is a more interesting possibility for it has hyperfine structure in the ground state 2p2Pl,,, 3 , 2 . C t is found quite widely in the Galaxy, through its radio recombination lines; it is capable of being ionized in H I regions. The 2 P 3 ; 2 , although it has an excitation energy of 64 cm-', could be reasonably populated in hotter regions where carbon is ionized. However, the transitions are at rather low energy. They have not been measured in the laboratory and only a theoretical estimate is available (Schaefer and Klemm, 1970). Considering the atoms in Table I in order of abundance, the next isotopic atoms with ground-state hyperfine structure are 27Al and 2'Na, with transitions at 1506.10 and 1771.63 MHz, respectively (Harvey eta!., 1972; Chan et al., 1970).The abundances being down by factors of almost 10- in comparison with neutral hydrogen and the transition probabilities being of the same order, the prospect of observing either of these does not seem good. Also, the ionization potentials are low, so that both these atoms will be predominantly ionized in interstellar H I regions by the general flux of starlight, in both cases to give ions with ground state 'So.

'

IV. Atomic Fine Structure There are very few atomic fine-structure transitions at microwave frequencies; most of the transitions are in the infrared. There is a reasonable hope of detecting microwave fine structure in the ti = 2 levels of atomic hydrogen, and a much poorer hope for higher states in hydrogen. The n = 2 population should be quite high in an H I1 region because of trapping of the Ly a line (Field and Partridge, 1961). The next best possibilities are transitions in metastable states of H e + ,N, and 0'. The oxygen and nitrogen lines suffer in comparison with hydrogen and helium because of the lower cosmic abundance and because they involve magnetic dipole rather than electric dipole transitions, but they have the advantage of being in low-lying metastable states. The helium lines involve excited states of an ion whose production requires high temperatures.

39 1

INTERSTELLAR MICROWAVE TRANSITIONS

For electric dipole transitions, the selection rules require that the electron configuration must change. It is only for one-electron atoms that transitions of this sort are likely to be in the microwave region; for each 11 the splitting is purely relativistic and states with different I are close together. A.

HYDROGEN ATOM

In the n = 2 levels of hydrogen, the separation of 2pIl, and 2p3,, is the fine structure proper; the 2p,,,-2s1,, splitting is the Lamb effect, arising from quantum electrodynamical corrections to the magnetic moment of the electron. The theory and experiment of the fine structure and the Lamb effect have been reviewed by Taylor rt ul. (1969). The electric dipole transition probability is given by (4) with line strength S(nLJ; n, L - 1, J - 1) = 9n2(n2 - L2)(4J2- 1)/16J

S(nLJ; 12, L - 1, J ) = 9nz(n2- L2)(2J + 1)/16J(J + 1)

(15)

as calculated by Wild (1952) from formulas of Condon and Shortley (1935). In this case, g, = 25

+1

(16)

Details of the hydrogen lines are in Table 111. Frequencies for n = 3 and 4 are theoretical values quoted by Fabjan et al. (1971); their own experimental values, which are less accurate, are in substantial agreement. These lines have hyperfine structure. The values of v for n = 2 are center of mass values, TABLE 111 FINE-STRUCTURE TRANSITIONS I N HYDROGEN Transition

v

(MHz)

I (cm)

A (sec-l)

A\# (MHz)

1058

28.3 3.02

8.01,- 10 6.59.- 7

99.6 99.6

95.2

3 1.2 31.2 40.4 40.4 13.6 13.6 17.3 17.3 6.6 6.6

991I

315 2935 3245 1078

9.24 27.8

1.27, - 10 1.03, -7 8.67.-8 3.81, -9

133 1238 1369 455 456 228

225 24.2 21.9 65.9 65.7 132

3.19.-11 2.57,-8 2.78.-8 1.22,-9 7.20,- 10 9.60,- I I

10.2

392

W B. Somerville

obtained on the assumption that upper levels are populated according to their statistical weights. Hyperfine structures for n = 3 and 4 are very much smaller and may be neglected. For the hyperfine components, the line strengths are given by the standard formulas for multiplets, for example, equations 2’2 of Condon and Shortley (1935) with the substitution (LSJ -+ J I F ) . For each of the lines in n = 2, the selection rules permit three components, and in terms of the line strengths (15) we have s(P312 = 2; s112 = l ) = $s(P3/2 s1/2) 7

9

s(P3/2 s(P3i2

9

= 1;

s112

9

7

= 1;

sl/2

9

I

9

= O ) = ;s(P3/2 =

7

(17)

s1/2) ’

1) = $s(P3/2 s112) 9

1

and S(P,/2 F = 1; s112 F = 1 ) = s(P,/2 s112) 7

7

9

s(Pl/, F = 1 ; s1/2 F = 0) 9

7

! i

(18)

F = 0; s112 F = 1) = H P l / 2 S l i 2 ) The relative line strengths are thus 5 : 2 : 1 and 2 : 1 : 1 for the two cases. Such ratios are frequently referred to as “relative intensities,” but the intensities actually observed are never more than approximately related in this way. The transition probabilities are given by (4)with (12) and using the frequency v of the individual hyperfine component. We obtain the numbers in Table IV. All 2p states have the same transition probability A = 6.258 x 10’ sec- for decay to the ground state, and so all the hyperfine components have the same half-width, 99.60 MHz, which is substantial. The total intensity of a spectrum line, summed over all directions, is the energy emitted in unit volume and is related to the A coefficient by = s(P112

1

7

9

(19) where N 2 is the number density of atoms in the upper level. For transitions with the same upper level, (19) immediately gives the relative intensities but for other transitions it is necessary to know the relative populations. The simplest assumption is that the levels are populated in proportion to the statistical weight g2, true if the populations are determined by collisions, the Boltmann factor e-AE/kTbeing negligibly different from 1. The results appear as the final column in Table IV. For s112-p312they are not very different from the relative line strengths 5 : 2 : 1, but for pliZ-sli2 they are a lot different from 2 : 1 : 1, because of the differences in v. It is not really obvious that under interstellar conditions the populations will be proportional to the statistical weights. I21

= hv21A21N2

393

INTERSTELLAR MICROWAVE TRANSITIONS

TABLE IV

HYPERFINE COMPONENTS OF THE 11

=2

FINE-STRUCTURE LINESI N HYDROGEN

Transition

v (MHz)

g2

2s1,, , F = 1 ; 2p3,, , F = 2 F=O F=1 F=l F=l

9876 10,030 9852

5

2p1,,. F = 1 ; 2s1,,, F = 1 F=O F=l F= 1 F=O

1088 1147 910

3 3

3 3 1

A (sec-l)

Relative intensity

6.52, - 7 4.55. - 7 2.16, -1

5.05 2.15

5.80, - 10 3.40, -10 5.10, -10

4.08 2.52 1 .oo

1 .oo

States in hydrogen with n = 3,4, . . . are not metastable. Also, they d o not share the pumping mechanism, which can maintain a large population in 2p. However, the possibility of detecting some of these fine-structure lines in a recombination spectrum has been considered (Wild, 1952; Myers and Barrett, 1972; Dickinson and Penfield, 1972) and so values are included in Table 111. To conclude this section, we may note an odd property of line strengths. In (17) and (18), the factors on the right-hand sides add up to 2 in each case: thus, the line strength for a line is not equal to the sum of the line strengths for its components. The line strength for the transition between two states is clearly and unambiguously defined, but if each “state” is subdivided, for example by including hyperfine structure, then the definition changes. This is because the statistical weights have changed. For 2p3,,, the statistical weight when hyperfine structure is ignored is 25 1 = 4;when hyperfine structure is included, the total statistical weight is

+

2

g=

C (2F + 1 ) = (21 + 1)(25 + 1 ) = 8

(20)

F=l

It is the factor (21 + 1) that produces the 2 in the sum of equations (17). The transition probability is related to the line strength with a factor l/g and so the extra factor is cancelled when the total A is calculated. The line strength is not directly observable and so this property has no physical significance. However, care is needed if one is to be consistent in the calculations.

B. OTHER ATOMS

For the deuterium atom, the fine structure in n = 2 is very similar to that of hydrogen, with a different hyperfine structure because of the different

394

W B. Somerville

nuclear spin. The frequencies of the two lines are 1059 and 9913 MHz, respectively, and the transition probabilities and half-widths are similarly close to the hydrogen values. The hyperfine splittings are much smaller. In ionized helium, He', the fine structure is larger than in hydrogen because of the excess nuclear charge. The dominant isotope, 4He', has zero nuclear spin and there is no hyperfine structure. The fine-structure splitting has been reviewed by Taylor et al. (1969). For the transition probabilities we may use (4). We obtain v(2p1,, - 2 ~ ' , ~ )14.04 = GHz,

A = 1.87 x

sec-'

(21)

v(2s1,, - 2p3,,) = 161.6 GHz,

A = 2.85 x

sec-'

(22)

The only situation where one could expect many atoms in the n = 2 states of He' is in a very hot H I1 region, where helium is doubly ionized and there is recombination to form He+. Lines of the recombination spectrum in He' have not been detected; this puts an upper limit of about 10% on the population of He2+in any H I1 region (Churchwell et al., 1974). For the seven-electron atom N or O', the ground configuration 2p3 has terms 4 s 3 1 2 , ZD,12, 3 , 2 , 'P312, 112 in increasing order of energy. Both doublet terms are inverted. Transitions within this one configuration are forbidden as electric dipole, but can occur as magnetic dipole or electric quadrupole (Garstang, 1962, 1968); thus the 'D and 2P levels are metastable. The finestructure splittings of the doublet terms are in the microwave region. The calculation of transition probabilities for these forbidden transitions in complex atoms is a major task. For both N and O', the 2P transition probabilities are very small. For 0' the 2D splitting, at I = 0.5 mm, lies outside the observable range. The only prospect among these lines is of 2D in N, which has (Radford and Evenson, 1968; Garstang, 1968)

v ( ~ DJ, = i-;)= 260

1 GHz,

I = 1.2 mm, A = 1.3 x lo-' sec-' (231 2D5,2is 19,223 cm-' above the ground state 4 s 3 , 2 . All these lines have very small half-widths because the states are all highly metastable. This encourages the hope that in astronomical locations the states, once populated, would remain so for a long time. The forbidden optical lines S-D and D-P of 0 ' and S-D of N are prominently seen in emission from gaseous nebulae.

V. Recombination Lines Radio recombination lines are emitted after an electron has recombined with a positive ion. They are transitions in which the electron jumps between states with neighboring high values of the principal quantum number n

395

INTERSTELLAR MICROWAVE TRANSITIONS

(similar lines with low n are observed in the optical spectrum). They are seen in emission from hot interstellar regions (H I1 regions) where the atoms are predominantly ionized. The lines are seen generally in neutral atoms (after recombination) but in principle also in positive ions. They appear after an electron rejoins any positive ion, from H + to the heaviest positive molecular ion. In the formula for the microwave frequencies, the only difference between species is in the reduced mass factor. This is because for the states concerned the outer electron is very far out and the effect of structure in the core is negligible. For n = 137 in hydrogen, the Bohr radius is 1 pm. Interstellar densities are very low (typically lo4 atoms ~ m - ~ - t h a t is, atomic nuclei-in an H I1 region) and so such large atoms can exist there. A further consequence is that the distribution among 1 states can be assumed statistical and therefore may be ignored, a considerable simplification in the theory (Brocklehurst, 1971). Lines have been detected for hydrogen, helium, and carbon over a wide range of frequencies. These have been extensively studied and the astrophysical consequences discussed. There are also a few observations of heavier atoms, including sulfur. They are harder to study because the atoms are less abundant and the frequencies for different atoms become very close together. He+ was discussed in the previous section. The transition n + 1 -+ n is designated nu; n + 2 -,n, n p ; n + 3 -,n, ny; and so on; this is preceded by the chemical symbol for the atom concerned, as for example H109u, He137P. For the transition between states n1 and n , , with n2 > n,, in an atom with effective nuclear charge 2, the hydrogenic formula is v

= Z2Rc(1 - m/M,)( l/n: -

l/n;)

(24)

where 2 is the net charge on nucleus plus inner electrons; for the cases of interest Z = 1, except for He+ for which Z = 2. M, is the mass of this inner core of the atom in question and R is the Rydberg energy for infinite nuclear mass. With the 1973 values of the constants, v = 3.289842 x 109Z2(l- m/M,)(l/n: - l / n i )

MHz.

(251

Lilley and Palmer (1968) have given an extensive tabulation of frequencies for transitions in hydrogen (Hnu to Hnc) and helium (He nu, He np), for IZ = 40 to 889. Their tables are not affected by changing to the 1973 constants. A few numbers from their tabulation are given in Table V, not selected for any particular astronomical reason, with values for carbon and for infinite nuclear mass derived using an extrapolation formula that they give. It is seen how for increasing M , the lines for different atoms become closer together.

396

W B. Somerville TABLE V RECOMBINATIONLINEFREQUENCIES (MHz)

n

50

150

H na He nu C na

51,072 51,092 51,097 51,099

1929.2 1929.9 1930.1 1930.2

418.36 418.53 418.57 418.59

99,225 99,266 99,279

3820.4 3821.9 3822.5

831.75 832.09 832.20

144,679 187,631

5674.6 7492.9

m nu

H nfl

Hens 00

nfl

H ny

H nfi

250

1240.2 1643.9

Transitions between states of different n are allowed as electric dipole transitions. We are concerned only with the total transition, averaged over orbital angular momentum states for the upper n with the assumption that populations are proportional to 2(21 + 1). Transition probabilities are calculated using standard formulas for hydrogenic atoms, discussed in detail by Menzel and Pekeris (1935). Large tables of oscillator strengths are given by Goldwire (1968) and Menzel (1969).In this case there is particular point in using the oscillator strengths f because they are the same for all emitters, regardless of Z or M,. Transition probabilities are related to the oscillator strengths by (6) with gn = 2n2

A selection of values is given in Table VI. Thefvalues are large; as Menzel (1969) explains, the ,f sum rule C f 1 2 = 2

1

(27)

is satisfied,because the sum includes the large negative values for emission (2 below 1) as well as the positive values for absorption (2 above 1). Because of the small energies of transition, the A values are quite small. Small transition probabilities imply narrow energy levels and small natural linewidths. In the case of n = 50 in hydrogen, the total probability for all emissions turns out to be 288.8 sec-'. The largest contribution is 40.1 sec- for the Lyman series transition to the ground state Is, an interesting result because the transition is allowed only from the 1 = 1 sublevel 50p, which contains only 2(21+ 1)/2nZ= 0.0012 of the atoms. The total

INTERSTELLAR MICROWAVE TRANSITIONS

397

TABLE VI WAVELENGTHS AND TRANSITION PROBABILITIES FOR RECOMBINATION LINESIN HYDROGEN

Line

H ~OGL H 50j H 5@)

H 506

1 (cm)

.I'

0.5870 0.3021 0.2072 0.1598

9.824 1.395 0.4416 0.1955

18.28 9.427 6.106 4.378

A (sec- ')

H 150a H l50p H is@; H 1506

15.54 7.847 5.283 4.001

28.90 4.029 1.252 0.5447

7.878, - 2 4.250, -2 2.877. - 2 2.153. - 2

H 250a H 250p H 250p H 2506

71.66 36.04 24.17 18.24

47.98 6.662 2.063 0.8939

6.183. - 3 3.366, - 3 2.299, - 3 1.737, - 3

transition probability from n = 5 1 being similar, we have the natural half-width

-

Av(H 50a) 90 Hz (28) This is negligible in comparison with the effect of Stark broadening and even much smaller than the fine structure. For transitions involving higher values of n, the natural widths are still smaller than this.

VI. Structure in Diatomic Molecules We consider only transitions within the ground electronic state and, with few exceptions, within the ground vibrational state also. Interstellar molecules generally are found in clouds of low kinetic temperature, where in thermal equilibrium any excited state populations will be negligibly small. Most of the astronomical observations involve rotational lines. In laboratory microwave spectroscopy, spontaneous emission is too slow to be a significant process. Under interstellar conditions it is often the most important one. It is thus convenient to work with the Einstein A-coefficient, given by (4)with an appropriate line strength S. Within ground electronic and vibrational states the only spontaneous emissions are the microwave lines themselves, perhaps with some infrared rotational lines. The natural linewidths (9) are therefore negligibly small. In a diatomic molecule there are three basic components of angular

398

W B. Somerville

momentum: the total electron spin S, quantum number S; the component Ak of electron orbital angular momentum along the nuclear axis, where A = 0, 1,2, . . .; and the nuclear rotation 0.Angular momentum is expressed in units of the reduced Planck constant h. These three can combine in different ways. We need consider only Hund’s coupling schemes (a) and (b) (Herzberg, 1950, p. 219).2 In case (a) coupling

C=S*k,

R = IA+Cl, J = 0 + Rk,

C=-S,-S+l,

..., S (29)

J = R, R + 1, ...

and in case (b)

N=O+Ak,

N = A , A + l , ...

J=N+S, J = I N - S I ,..., N + S (30) For most light molecules, to a good approximation case (b) applies if A is zero and case (a) or occasionally case (b) if A is nonzero. Coupling intermediate between the two cases frequently is found. The term symbol is C, n, A, @, . . ., for A = 0, 1,2,3, .. ., with the multiplicity 2 s + 1 as a left superscript. For nonzero A in case (a) or intermediate coupling, the value of A + C is often added as a right subscript, for example 2111,2, 2113,2. For diatomic molecules in general, the vast majority of chemically stable molecules have ‘C ground states. Electronic states have two symmetry properties that affect selection rules:

1. Positioe-negative symmetry, for A = 0 only. Each C state is either C+ or C- . Allowed transitions connect states of the same symmetry. 2. Even-odd symmetry, for homonuclear molecules only. Each state for a homonuclear molecule is either g or u, as 211sor 211u. Allowed transitions connect states of opposite symmetry; thus for a homonuclear molecule transitions within a single electronic state are forbidden. The ground state of H2 is lC;; the ground state of CO is ‘ C + ; the ground state of OH is 211. A. ROTATION For ‘Z states both S and A are zero and Hund’s cases (a) and (b) are identical. The rotation is described by angular momentum J, of magnitude [J(J + 1)]1’2, with quantum number J = 0, 1,2, . . . . The rotational energies are E(u, J ) = B,J(J + 1) - D , J 2 ( J + 1)2 (31)

+

The notation in general use has been changed since Herzberg’s book was published. The main changes are that 0 (or sometimes R) replaces N,and N replaces K.

INTERSTELLAR MICROWAVE TRANSITIONS

399

where D, is much smaller than B , . The selection rule for electric dipole rotational transitions in 'C states is AJ= +1 (32) We consider next the 'C and 3C terms. In case (b) coupling, for each N there are (2s+ 1) fine-structure levels, given by different values of J according to equation (30);3 the J splitting is smaller than the N splitting. The energy levels for ' C are

+ 3)= B , N ( N + 1) - D , N 2 ( N + 1)2 + *jN E(v, N , J = N - 3)= B , N ( N + 1) - D , N 2 x ( N ,+ 1)2 - i y ( N + l), N #0

E(u, N , J = N

(33)

(34) where y is the coefficient of the interaction between electron spin and nuclear rotation. The Hamiltonian contains a term HSN= $4

N

(35) In the case of 3C this interaction is present and there is also a much stronger interaction between the two unpaired electron spins, coefficient I , which introduces departures from pure case (b) coupling. I includes also a contribution from the distortion of the X state by neighboring ll states (Miller and Townes, 1953). The three energy levels are given by formulas similar to (33) and (34). For electric dipole rotational transitions in 2C or 'C states, in case (b) coupling, *

A N = +1, A J = O , )1 (36) However, for 'X in intermediate coupling transitions with AN = + 3 are also allowed (Powell and Lide, 1964). It should be noted that the extensive tables of line strengths given by Kovacs (1969) do not apply to intermediate coupling from this cause, but only to that arising from a spin-orbit interaction (39), a larger effect for A # 0. Figure 1 shows the energy levels of 211 in both case (a) and case (b) coupling. For light molecules, case (a) applies for low rotational levels, with movement toward case (b) with increasing rotation. In case (a) coupling there are two series,

Note that Herzberg's (1950) Fig. 101 contains a minor error for 'Z, carried into his 1971 book (Fig. 23, cf. Fig. 18) and also-such is Herzberg's influence-copied by Gordy and Cook (1970, Fig. 4.8) and by Levine (1970, Fig. 3.21). For the same 3C case Johnson (1949, Fig. 70) went more seriously wrong.

W B. Somerville J

712

512 2nl,2

3/2 112

?I2

912

2n,,*

312

Case (al(raqu!orl

Case (aNinverled)

1. Correlation of the energy levels for a 'Il molecule in case (a) and case (b) coupling (Bennett, 1970).

According as 2111/2J= 3 or zl13/2J= 3 is the lower, the structure is described as regular or inverted. The spin-rotation interaction (35) is present for 211 molecules; interactions with the coefficient 1 are absent. More important than (35) is the electron spin-orbit interaction

Hso = AS hk

(39)

The coefficientA is much larger than y and actually larger than B, even for a light molecule such as OH. For electric dipole transitions in 'll in intermediate coupling the selection rules are A J = U , +1,

AN=O, +I, k2

(40)

The second of these is impbed by the first. In pure case (b) coupling, only the transitions with AN = 0, f1 are allowed; in pure case (a) coupling An = 0 only; in each case with the same selection rule (40) on J . In most 211 molecules of reasonable possible interstellar abundance, the rotational transitions are in the infrared rather than the microwave region.

401

INTERSTELLAR MICROWAVE TRANSITIONS

However, all these rotational energy levels show a fine-structure, A-doubling (discussed in the next section) and in most molecules there is hyperhe structure as well. So transitions within a given rotational level are possible. For rather heavier molecules such as NO, rotational transitions do come into the microwave. As an indication of the relative sizes of the coefficientsinvolved, values for some particular molecules of the various types are given in Table VII. TABLE VII

MOLECULAR ROTATIONALCONSTANTS IN MHZ, FOR PRINCIPAL ISOTOPICSPECIPS IN u = 0 OF THE GROUND ELECTRONIC STATE' Bn

CO('Z)"

CN('Z) SO('ZY OH(%) a

57635.969(3) 56693.583(2003 21 523.561(5) S56037(6)'

*n

0.18357(7) 0.192d 0.03421(9) 106.66(11)0

Yn

An

A"

-

-

-

217.571 (300p - 168.79(15) 3627(150r

-

-

158258.7(12) ~

-

-4 163 865 (280)'

The number in brackets is the stated uncertainty in units of the last figure given.

* Lovas and Krupenie (1974).

Turner and Gammon (1975). Poletto and Rigutti (1965). Tiemann (1974b). f Moore and Richards (1971). Destombes et al. (1975). Scar1 and Dalby (197 1).

@

B. LAMBDADOUBLING

In a I: state, the electronic wave function is symmetrical about the nuclear axis. The axis of molecular rotation is perpendicular to the nuclear axis, and the moment of inertia and the energy of rotation are the same about any such axis. In the case of nonzero A, the electronic wave function is not symmetrical about the nuclear axis and there are two independent principal rotation axes, at right angles to each other. Rotations about these axes have slightly different energies and so all the rotational energy levels, such as those shown in Fig. 1, are split in two. It is always a doubling, for any nonzero value of A. A-doubled levels are labeled using the positive-negative symmetry property (Herzberg, 1950, pp. 237ff.). Lower levels belong to 'lI+ and upper ones to 211- or vice versa, depending on the positions of neighboring I: electronic states. In OH they cross over, as N increases. 211+ behaves like I:+

402

W B. Somerville

and 211- like Z-,and in case (b) the rotational levels have symmetry as follows : N

'n+ W

Symmetry

even odd even odd

+

-

+

As shown in Fig. 1, the symmetry of the levels carries over in the transition to case (a). The symmetry has selection rule

+*-.

(41)

In a rotational transition there are therefore only two rather than four A-doubling components. Astrophysically, the transition within a single rotational state, between the A-doubled levels, is important. By (40) and (41) this transition is allowed. C. HYPERFINE STRUCTURE

If a nucleus has a magnetic dipole moment I( (I > 5) or electric quadrupole moment Q (I 2 1) these can interact with rotation or electronic angular momenta to produce hyperfine splittings. They include a very weak magnetic interaction of I with J; there are other small effects such as the interaction between the spins of two nuclei in the molecule. If there are nonzero electronic angular momenta, the magnetic dipole interactions are the stronger. However, in 'Zterms these are essentially zero and the quadrupole interaction dominates, with a smaller contribution from the interaction between I and J. Because most ground state molecules are 'X, the quadrupole interaction has received most attention from laboratory workers. For such a molecule, the hyperfine energies are given by

-

Ehf= -eqQf(I, J , F ) + C I ( I J )

(42)

where the algebraic functionf(1, J , F ) is called Casimir's function and ( ) is the matrix element. Experiments give the combined factor eqQ, and also the coefficient C,. These energies (42) are t o be added to the rotational energies (31). Tables of quadrupole energies and relative line strengths are given in textbooks. For example, in the 'C ground state of the interstellar isotopic molecule 12C33S(the 33Snucleus has I = 1) eqQ = 12.835 & 0.026 MHz, (Mockler and Bird, 1955).

C, = 0.019 & 0.015 MHz

(43)

403

INTERSTELLAR MICROWAVE TRANSITIONS

If the molecule has nonzero electronic angular momenta, the nuclear spin I combines with the other angular momenta to produce a new total F, but this can be done in various ways. Hund’s coupling schemes are subdivided (Townes and Schawlow, 1955, p. 197).We list here the “ P ” cases, where I is not coupled to the nuclear axis, which occur most frequently [cf. Eqs. (29) and (30)]: ag: R = IA+CI (44) J=O+Rk, F=I+J bpN:

N = 0 +Ak

bps:

+ I, F2 = I + S,

b,, :

J=N+S,

(45)

F=Fl+S

F, = N

F=N+F2

(46)

F=J+I

(47)

Coupling is frequently intermediate between two of these. The magnetic dipole hyperfine interaction energy can be written as Ehf= uA(I

*

-

k)

+ b(I

*

S)

+ ~ ( 1kS *

*

k)

+ y(S

*

N)

(48)

to first order. The S N interaction is included because it is of similar magnitude to the other terms, and it is evaluated in the same process of analysis of experiments. In such work the coefficient y of equation (35) is frequently written as d . Formulas for the matrix elements ( ) in the various coupling cases have been given by Frosch and Foley (1952). For example, in the ground state 211of OH, Radford (1962) found experimentally a = 86.0 f 0.6 MHz,

b = - 119.0

c = 133.2 & 1.0 MHz,

d = 56.5 f 0.4 MHz

0.4 MHz (49)

VII. Transitions in Diatomic Molecules The diatomic molecules that have been detected in interstellar space and some others that have been or might be considered, selected here on a somewhat arbitrary basis, are listed in Table VIII. Most of the molecules in Table VIII are well studied in the laboratory. The allocation of a 311ground state to Sic is by Lutz and Ryan (1974)from an accurate ab initio calculation. In a slightly earlier paper, Lovas (1974) supposed the ground state to be ‘C and proposed Sic as a candidate molecule for the unidentified interstellar line U89.2;4 in view of the result of The notation is U followed by the frequency in G H t

404

W B. Somerville TABLE VIII GROUND ELECTRONICSTATES OF SELECTED DIATOMIC MOLECULES' Homonuclear

Heteronuclear

'X

If, N, C,

CH'

22

Hl Ni 0, S, Si,

A10

'X

,n 'n

NH CH Sic

CO CN SH+ NO

CS

CO' SO NS

SiO S B SiN SiO+

OH SH

Interstellar molecules detected at microwave frequencies are in bold type, those detected at other wavelengths in italics.

Lutz and Ryan this had to be disregarded: and in any case it has more recently been established by laboratory measurement that U89.2 is produced by the ion HCO', always the favorite candidate. It is interesting to note that for many years it was believed that the ground state of C2 was 311, until an experiment by Ballik and Ramsay (1963) established that a 'C state has lower energy. C2 and Sic are electronically similar, both atoms having p2 configurations. The first interstellar molecules known were CH, CH', and CN, detected in the late 1930s in the visible and near ultraviolet in electronic transitions seen in absorption in the spectra of hot stars. Crutcher and Watson (1976) have recently similarly detected OH at 3078 8,.' Rocket studies (Spitzer and Jenkins, 1975) have detected ultraviolet absorption lines, in electronic bands, of H2 about 1108 A (Carruthers, 1970), CO about 1447 8, (Smith and Stecher, 1971), and O H about 1222 A (Snow, 1976). H2 has also been detected in the near infrared at around 2.12 ,urn (Gautier et al., 1976).This is the vibrational transition u = 1-0 within the ground electronic state, seen in emission: several rotational lines of which the S(1) line J = 3-1 is strongest. It is an electric quadrupole transition (Field et al., 1966). In this section, we consider general structural formulas and principles involved in calculating radiative transition probabilities. A few molecules of each type are discussed, except for 311 states, where Sic is the only molecule in question and its properties are not well established. Those discussed are intended more to serve as examples rather than as an attempt to give tabulations of data. In many cases, however, the numbers here should be more accurate than those published previously. The well-established convention is followed that, whereas for atoms the states in a transition are written lower-upper, for molecules they are written upper-lower. 1A

= o . ~nm.

405

INTERSTELLAR MICROWAVE TRANSITIONS

A. ROTATIONAL TRANSITIONS The electric dipole transition probability is given by (4). In molecular rotational spectroscopy it is usual to work not with the line strength S but with a related quantity SJ,also called the line strength and given by

s= p2sJ

(50) where p is the electric dipole moment of the molecule and depends on the electronic and vibrational state. The use of SJconveniently separates out the angular momentum factors, which can be evaluated algebraically, from the factor p2, which is determined experimentally. Consider the case of a ‘C molecule. The rotational state being given by quantum numbers (5, M ) , the strength of the transition is given by the matrix element of the dipole moment vector p between ( J , M ) and (J’, M ’ ) where J’ = J & 1. When the square of this is summed over the values M’ for the final states the dependence on M cancels and the result IP(J+J’)I2 =

c )(JMlPlJ’M’)12

(51)

M’

is referred to, a little ambiguously, as the square ofthe dipole moment matrix element. It also involves a sum over three perpendicular directions in space. This quantity has the disadvantage of not being symmetrical in the initial and final states, unlike the line strength. They are related by P2SJ =

c c I(JM M

I

121 J’M’)

l2

Mi

+ l)lp(J + J + 1)l’ = (25 + 3)lp(J + 1 + J ) I 2 = ( J + 1)p2 = (25

(52) (see Townes and Schawlow, 1955, Section 1.4; or Gordy and Cook, 1970, Section 2.6). From (4), A(J

+ 1+J)

= 1.16395 x 10-38v3p2SJ/g2sec-’;

g2 = 25

+3

(53)

with v in hertz and p in debyes (1 D = esu = 0.393427 a.u.). In case (a) coupling, the general formula is (e.g., Kovacs, 1969, p. 121)

sJ(!&J + 1,

J) = [(J f

+ 1)

- n2]1/(J

(54) which reduces to (52) for a ‘C term. Values for case (b) are given in Table IX. If the rotational line is split into several components, the separate line strengths are related to the total one by formulas similar to (17) and (18). Relative values are tabulated in most textbooks. These are generally referred to as relative intensities, but the actual intensity in emission is given by (19) to be v4!5 and so this description is valid only when the component v are

W B. Somerville

406

TABLE IX

LINESTRENGTHS FOR ELECTRIC DIPOLE ROTATIONAL TRANSITIONS IN CASE(b) COUPLING

(J

+ $. J + 1) - ( J + +,J )

(J

+ $, J + 1) - (J - 4, J )

(J

+ +,J ) - (J - +,J )

(J

+ $, J + 1) - ( J + +,J )

(25

+ 1)2(2J + 5 ) + 1)(2J + 3) (25 - 1)(2J + 3)2

+ 1)(2J + 3) 4(J

+

(21 4(J

1)

(25 + 1)(2J + 3) 4(J 1)

+

4(J

25 + 1 4J(J + 1)

+ 1)(2J + 1)

(25 - 1)(2J + 3) 1)(2J + 1)

445

+

4

0

(J

+ 1)(2J + 1)(2J + 3)

32 (J + 2, J

+ 1) - ( J + 1, J )

(J

+ 1, J + 1) - (J,J)

(J

+ 1, J ) - (J, J )

(J,J

+ 1) - (J - 1, J)

( J , J - 1 ) - (J - 1, J )

(J

+ 2)(2J + 1) 2J + 3

+

J(J 2) J+l

1 J + l

+ +

J ( 2 J 3) 25 1

1 4 2 5 - 1)(2J + 1)

very close together. It also requires the source to be optically thin in all the lines. Optical thickness makes a line relatively less strong than it would otherwise be; in the extreme case when all the lines are optically thick they all have the same intensity.

B. INDIVIDUAL C MOLECULES 1. ‘X Molecules

For the first four ‘X molecules that have been detected by interstellar microwave radiation, the spectrum lines that have been seen, or looked for

INTERSTELLAR MICROWAVE TRANSITIONS

407

and reported, are listed in Table X. For CO and CS, details are given for the different isotopic species. For SiO and SiS, only one species has been detected. In general, astronomical references in this and later tables are to the first detection of the transition whether in absorption or in emission. A comprehensive survey of laboratory results on the microwave spectra of CO, CS, and SiO is given by Lovas and Krupenie (1974). For a molecule with nuclear spin, there is a hyperfine structure. This can add structure to rotational transitions. There is also the possibility of hyperfine transitions within the rotational state. For I = 4 in a 'C molecule, the I J interaction is very weak and neither effect is significant. For I 2 1 the quadrupole interaction (42) also is present and the splitting can be detectable, although the pure hyperfine transitions will still be at very long wavelengths. CO. Frequencies in Table X are from the compilation by Lovas and Krupenie (1974). For all isotopes in J = 1-0 and for l 2 C l 6 0in J = 2-1 they are experimental values; for other 2-1 transitions they are deduced from the molecular constants. The dipole moment has recently been determined by Muenter (1975)to be

-

p = 0.10980(3) D

(55)

and this value has been used for all the calculations here of A values. However, this is the value for l2Cl60,and those for other isotopic molecules will be slightly different. This value is po ,averaged over the vibrational wave function for u = 0, and differs from the value p, at the equilibrium nuclear separation by about loo/,,a larger difference than normal and attributed to the small value of p. It may be expected that the isotopic changes will similarly be larger than normal, but probably still very much smaller than the present observational errors in the astronomical abundance ratios (Bertojo et al., 1974; Wannier et al., 1976). The isotope 7O has nuclear spin I = $ and so has a quadrupole moment, the only case for this molecule. Rosenblum and Nethercot (1957) measured the J = 1-0 triplet of 12C"0 to have components

'

I

;,

F = -z4, 12 2-2, 5

v

=

112,360.016k 0.015 MHz

v = 112,358.980k 0.015 MHz

(56)

v = 112,358.720& 0.100 MHz

The relative line strengths and the equilibrium relative intensities are 3 : 4 : 2, respectively. The transition probabilities all equal the average value, given in Table X.The astronomical measurements of Encrenaz et al. (1973) show some structure, but the signal-to-noise ratio did not permit them to

W B. Somrrdlr

408

TABLE X

ROTATIONAL THANSITIUNS IN INTEKSTELLAH 'z DIATOMIC MOLECULES" Astronomical reference

(MHz)

I (mm)

2-1 I-0t 2-It 1-0 2 1 1-0 2- I 1-0 2- 1 1-0 2-1

115,271.2044 230,537.9744 112.359.2684 224.7 14.35 1 109,782.182J 2 19.560.369 110,201.370~ 220,398.714 J 104,711.416J 209,419. I98 105,87 I. 110 21 1.738.505

2.60 1.30 2.67 1.33 2.73 1.37 2.72 1.36 2.86 1.43 2.83 1.42

7.164, 6.877, 6.635. 6.369, 6.189, 5.941, 6.260, 6.009, 5.370, 5.155, 5.551, 5.328,

-8 -7

d

1-0 2-1 3 -2 4-3 14f 2-lt 3-2t 1-0 2-1 3 -2 1-0 2- I 3-2

48.99 l.WJ 97,98 1.0074 146,969.039J 195,954.162 48,586.520 J 97.172.090 145,755.760 48,206.948 J 96,412.953 J 144.617.1 17 46,247.472 J 92,494.084J 138,738.97J

6.12 3.06 2.04 1.53 6.17 3.09 2.06 6.22 3.1 1 2.07 6.48 3.24 2.16

1.763, - 6 1.693, - 5 6.121, - 5 1.504, -4 1.720, - 6 1.651, - 5 5.970, - 5 1.680, -6 1.613, - 5 5.832, - 5 1.483, -6 1.424, - 5 5.149, - 5

j i

0

1-0

I

43,423.76 86,846.86d 130,268.494 173,687.98 43,122.03J 86,243.27J 129,363.12J 172,480.82 42,820.48J 85,640.30 128,458.66 42.519.34

6.90 3.45 2.30 1.73 6.95 3.48 2.32 1.74 7.00 3.50 2.33 7.05

3.049, -6 2.927, - 5 1.058, -4 2.602, -4 3.024, - 6 2.903, - 5 1.050, -4 2.580, - 4 2.998, -6 2.878, - 5 1.041, -4 2.973, -6

P

2- 1 3-2 4-3 1-0 2-1 3 -2 4-3

r

J

0

0

2

3

1-0

1-0 2- I

3-2 1-0

1'

A (sec-I)

-8 -7 -8 -7 -8 -7 -8 -7 -8

h

.l' e C C

.f g

-1

It

.i .I j .I j j

h

I k 0

m n

P P P

P

409

INTERSTELLAR MICROWAVE TRANSITIONS TABLE X (cotititlirctl)

ZSsiJZs

1;

J

0

1-0 2- 1 3 -2 4-3 5-4 6-5 7-6

v

(MHz)

18.154.88I 36,309.631 54,464.081 72,618.10 90,771.55~' 108,924.27 127,076.12

I (mm)

A (sec-I)

16.51 8.26 5.50 4.13 3.30 2.75 2.36

6.95, 6.67, 2.41, 5.93, 1.18. 2.08 3.34,

Astronomical reference

-8 -7 -6 -6 -5 -5 -5

4 q

4, Astronomically detected; 1-, mean of hyperfine components. Turner of a / . (1973). * Wilson Pt a/. (1970).

a

j

Penzias ri a/. (1971a). Schwartz and Wilson (1972). ' Encrenaz PI al. (1973). Phillips C I a/. (1973). Wannier et af. (1976). I, Penzias et a/. (I971b). Zuckerman t'i a/. (1972).

Wilson C t a / . (1971). Dickinson (1972). "' Snyder and Buhl (1974). " Davis c'r a/. (1974). " Thaddeus PI a/. (1974). Buhl Ct a/. (1974). Morris Pt a/. (1975).

study relative intensities of the components. The J = 2-1 line has nine hyperfine components, similarly close together. The 14Cisotope is unstable, with a half-life of 5730 years. It is not surprising that Schwartz and Wilson (1972) failed to detect 14C160.Their measurement puts an upper limit on the rate of nuclear processes of formation. CS. The frequencies of transitions have been obtained in laboratory studies by Mockler and Bird (1955) and Kewley et al. (1963); see also Lovas and Krupenie (1974). There are four isotopic forms; three are simple but the fourth has a quadrupole hyperfine structure. All four have been studied in interstellar emission. The 33S nucleus has spin I = 3.The hyperfine constants of 12C33Sare given in Eq. (43). There are three components of the 1-0 transition and eight components of 2-1. Turner et al. (1973) quote three numbers for the 2-1 frequencies; these are quite fictitious, being simply twice the 1-0 values. This may invalidate their upper limit for the one source in which they give a value for J = 2-1 in 12C33S. The dipole moment for J = 1-0 in v = 0 of 12C32Swas measured by Winnewisser and Cook (1968); after correction (Winnewisser et al., 1972) their value is p = 1.966(5) D (57) This value has been used for all the calculations here.

410

W B. Somervillr

SiO. Transition frequencies for J = 1-0 of u = 0 to 3 in 2*Si160and of u = 0 in 29Si160and 30Si160were measured by Torring (1968). Frequencies

for higher J transitions were calculated from the molecular constants given by Torring, by Dickinson and Gottlieb (1971), and by Wilson et al. (1971). Later calculations have been reported by Lovas (1974) and Lovas and Krupenie (1974); Buhl et al. (1974) allude to unpublished laboratory experiments. The astronomical observations are only of the principal isotopic species. Only the species 2 e S 1 7 0will have a quadrupole hyperfine structure, but this has not been studied astronomically or in the laboratory. Torring failed to resolve the I * J magnetic dipole hyperfine structure of 29Si160. The dipole moment in 28Si'60was measured by Raymonda et al. (1970) to be 10, U , = 3.0982 D 1,

p = 3.1178 D

2,

p = 3.1372 D

(3.

p = 3.1574

D

Notice the variation with u. The interesting feature of the astronomical spectra is the presence of several vibrationally excited states. With the 1973 physical constants, J = 1 of u = 1, 2, 3 correspond to thermal excitation temperatures of 1771, 3523, 5258"K, respectively. Evidently nonthermal processes are at work, with maser action in the excited u states (Buhl et al., 1975). SiS. The frequencies for the J = 1-0,2-1, and 3-2 transitions in SiS have been measured by Tiemann et al. (1972). Other frequencies here are calculated from the constants they give (Tiemann, 1976). Morris et al. (1975) quote frequencies calculated similarly from earlier measurements (Hoeft, 1965).The dipole moment was measured by Hoeft et al. (1969) to be /i

=

1.73(6) D

(59)

2. ' C Molecules C N . The only laboratory analysis of CN is from optical measurements on electronic transitions (Poletto and Rigutti, 1965). More accurate values of the rotational constants come from astronomical measurement of the microwave lines (Penzias et al., 1974; Turner and Gammon, 1975). From the one rotational transition observed ( N = 1-0 at 2.64 mm) the constants Bo and Do cannot be obtained separately; Turner and Gammon find 80-200 = 56693.199(200) MHz

(60)

41 1

INTERSTELLAR MICROWAVE TRANSITIONS

For the purpose of Table VII, the optical value for Do has been used. Turner and Gammon also have derived the hyperfine constants, h = -33.78(10),

c = 60.20(10),

= 217.571(300)

MHz (61)

CN is represented very well by case b,, coupling [Eq. (47)) In Table XI. frequencies are from the analysis by Turner and Gammon (1975) of their observations. The A values are calculated with the dipole moment p = 1.45(8) D

(62) of Thomson and Dalby (1968) and confirm those given by Turner and Gammon. T A B L E XI HYPERFINETRANSITIONS (J. F ) (upper)-(lower)

11

(MHz)

IN

N

s,

=

1-0 OF "C"N"

A

(set-1)

113,520.34 113.509.06 J 113,499.72 J 113,491.15J 113,488.39d

2/27 16/27 16/27 2 20127

1.33, 5.30, 1.06, 1.19. 6.62,

-6 -6 -5 -5

113,191.33J 113,170.87d 113,144.34J 113,123.83

20/27 16/27 16/27 2/27

6.57. 5.25. 1.05, 1.31,

-6 -6 -5 -6

Asironomical rcference

-6

' J, Astronomically detected. Jefferts or ul. (1970b). Penzias C I ul. (1974); Turner and Gammon (1975)

CN is one of the optical interstellar molecules and is observed in absorption from N = 1 as well as N = 0; the relative strengths correspond to an excitation temperature of 2.8"K, suggesting that the level population is maintained by the isotropic microwave background radiation (see the review by Thaddeus, 1972). Penzias et al. (1972) looked for the microwave line in a dark cloud where CN had been detected optically. They obtained a null result-detecting neither absorption nor emission-confirming that thc excitation is in equilibrium with the background radiation. (The sources where CN is detected in microwave emission are higher-density regions, where collisional excitation is more important; and the sources are associated with HI1 regions that supply energy to the clouds. These effects produce higher excited-state populations.)

412

W B. Someruille

C O + . Chemical reactions involving molecules and positive ions dominate the formation and destruction of molecules in interstellar clouds [see Somerville (1977a) and the detailed discussions by Dalgarno and Black (1976) and Watson (1976)l. The microwave spectrum of CO' has recently been determined in the laboratory (Dixon and Woods, 1975). It is only the second positive ion to have been studied; the first was H i , discussed in a later section. CO' occurs as an intermediate product in the formation of CO (Oppenheimer and Dalgarno, 1975), but is quite rapidly destroyed in reaction with atomic or molecular hydrogen,

CO'+H-+CO+H+ CO'

+ H,

-+

HCO'

+H

and so the equilibrium abundance of CO' may not be high. Interstellar CO is very abundant and HCO+ is also a well-known interstellar species, the former " X-ogen " (Section VIILA). The dipole moment of CO' is estimated to be quite large; Certain and Woods (1973) calculated that (65) With this the two components of N = 1-0 have the A values in Table XII. CO' has no hyperfine structure. p = 2.5(5) D

TABLE XI1

N

. I

= 1-0

v

TRANSITIONS IN 1zC160+

(MHz)

A (sec-')

Relative intensity

3.99, - 5 3.95, - 5

2.03 1.00

~~

i-4 4-i

118,101.99(05) 117,692.55(10)

3. 3CMolecules SO. Tiemann (1974a,b) has given a thorough analysis of the SO spectrum, including isotopic forms. The lowest-energy levels are shown in Fig. 2. Departures from case (b) coupling are large for small N and become smaller as N increases; according to Tiemann they may be neglected beyond about N = 8. The extent of the departures is seen in Fig. 2-in pure (b) coupling the J structure is smaller than the N structure and levels for different N do not overlap-and in Table XI11 where line strengths in case (b) are compared with those in intermediate coupling given by Tiemann.

INTERSTELLAR MICROWAVE TRANSITIONS

i 0

413

SO -

I

k,l,-l-it 1,Ol

FIG.2. Energy levels and transitions in SO. Microwave transitions are drawn as solid lines.

The dipole moment in 32S'60is po = 1.55(2)

D

(66) (Powell and Lide, 1964). Only the principal isotope has been detected astronomically. In intermediate coupling, transitions with AN = 3 are also allowed as electric dipole transitions. However, none has yet been detected in the laboratory or astronomically. It is common to describe the energy levels using not the notation ( N , J ) but instead ( K , J ) , written as J K .This is by analogy with notation used for the symmetric top in polyatomic molecules (Section VII1,B); however, K has different meanings in the two cases and the usage is potentially confusing. Indeed, this is why the change from K to N was made. C. HOMONUCLEAR MOLECULES 1. Rotational Transitions As discussed by Field et al. (1966) for H2, electric dipole transitions within a 'C electronic state for a homonuclear molecule are highly forbidden. First, they violate the even-odd electronic symmetry selection rule

g-u

(67)

N B. Soinewille

414

TABLE XllI MICHOWAVE THANSITIONS IN 32s'60"

v

(MHz)

-~-

SJ(b)

'J

A (sec-')

3o.oo1.5n 62.931.80

0.333 1.667

0.956 1.936

2.41, - 7 2.70. -6

13,043.704 86,093.954 99.299.874

0.500 1.500 2.800

1.435 1.500 2.933

2.97. - 8 5.35, -6 1.15. -5

36,201.82 109,252.104 129,138.85 138,178.60J

0.333 1.800 2.667 3.857

0.746 1.510 2.667 3.938

1.98, - 7 1.10, -5 2.29, - 5 3.23, -5

66,034.94 158.971.80 172.181.46 178.605.17

0.250 2.857 3.750 4.889

0.488 2.689 3.750 4.943

5.61, 4.32. 5.95. 7.16,

Astronomical reference

e

c,d b c

h

-7 -5 -5 -5

Electric dipole transitions are included for which SJ > 0.1. S,, is evaluated in intermediate coupling (Tiemann, 1974): S,(b) is the corresponding case (b) value, from Table IX. J, Astronomically detected. Gottlieb and Ball (1973). Kaih rt a / . (1974). Clark and Johnson (1974). " Clark and Johnson (1975).

Second, through (32) they violate the nuclear spin symmetry rule s-s

(68) (in H,,even J has a nuclear symmetry and odd J has s); third, the electric dipole moment is zero. Magnetic dipole transitions also do not occur-the magnetic dipole moment is zero and (68) is violated, although (67) is replaced by a-a,

g-g, u-u (691 Electric quadrupole transitions can occur: the quadrupole moment is nonzero, (69) applies, and (32) is replaced by A J = +2 which allows (68) to be satisfied. For H , , the transitions are in the infrared. The first line, J = 2-0, is at A = 28 pm and has A = 2.95 x lo-''

sec-'

(71)

INTERSTELLAR MICROWAVE TRANSITIONS

415

(Black and Dalgarno, 1976). A increases as J 5 and so higher levels have larger A ; but they have vastly smaller interstellar populations (Spitzer and Cochran, 1973). The corresponding J = 2-0 transitions in N, and C2 are at 11.9 and 10.9 cm- ', respectively, outside the microwave range which can be studied from the ground. The A values may be expected to be about lo-' smaller than (71), through the v 5 factor in A for a quadrupole transition. In studies related to the 3°K isotropic microwave background radiation (Thaddeus, 1972), no strong cosmic emission lines have been observed in this region, not surprising in view of these small A values. Interstellar N, is expected t o be abundant (Dalgarno and Black, 1976); it forms slowly but is hard to destroy. In isotopic molecules such as HD, electric dipole transitions can occur. The dipole moment is small but nonzero: for HD, p = 5.85 x 10-4

D (72) (Trefler and Gush, 1968); the nuclear symmetry rule (68) does not arise and by itself (67) is easily violated. The H D transitions are in the infrared and the molecules exist only in cold interstellar clouds, but they are potentially detectable (Bussoletti and Stasinska, 1975). Bunker (1974) has shown that for isotopic homonuclear diatomic ions the dipole moment is quite large-for HD' p = 0.87

D

(73) -and rotational transitions may occur strongly. For HD' the rotational transitions are far in the infrared, but this result could have important consequences for heavier species such as 14N15N+,which have transitions in the millimeter range and for which Bunker estimates p about a factor 10 smaller than (73). The oxygen molecule 0, has a 3C ground state, and so the magnetic dipole moment P m = - (L

+ 2S)PB

(74) is nonzero. Magnetic dipole transitions are permitted, by the selection rules AN = 0, f 2 , AJ = 0, + 1 (75) Only transitions with AN = 0 are in the accessible microwave region; those where N changes are in the infrared. Rudnitskii (1974) has examined the prospects for detecting interstellar 0,. He has evaluated A-coefficients in intermediate coupling, using the theory of Tinkham and Strandberg (1955). A few of his values are given in Table XIV. Rudnitskii concludes that the effect of 0, in Earth's atmosphere is so great that it would be necessary to study interstellar 0, in these lines

W B. Sornerville

416

from a space vehicle. Viala and Walmsley (1976) have discussed detailed processes involved in the interstellar formation of 0 2 . TABLE XIV MICROWAVE TRANSITIONS IN l602' N

J

v (MHz)

A (sec-')

1

2-1 0-1 4-3 2-3

56,264.766 118,750.343 58,446.580 62,486.255

5.84, - 10 4.46, - 9 7.66, - 10 9.12, - 10

3

a

Rudnitskii (1974).

2. Hyperjine Transitions Transitions between hyperfine levels of a rotational state have very small energies in 'C molecules. For example, in H2 the I J interaction leads in v = 0, J = 1 to two transitions of frequency 546 and 54.8 kHz; Field et al. (1966) coined the name " ultrafine structure " for this very small separation. With nonzero electronic angular momenta, magnetic hyperfine structure gives transitions of much greater energy-in the ground state of OH, for example, the hyperfine separation is 53 MHz. This still corresponds to a wavelength outside the range available to radio astronomy. The only molecule for which the pure hyperfine transitions have been considered seriously in an astronomical context is H i . In the ground state 'El the coupling scheme b,, is satisfied rather closely. Each nucleus has spin 4, and from symmetry the total I = 0 in states of even N,I = 1 in states of odd N. There is no structure for N = 0; a splitting into two for every other even N, J = N L- f from the S N interaction; and a more complicated splitting for odd N. The hyperfine splitting is given by (48). Expressions for the angular momentum matrix elements were given by Dalgarno et al. (1960) and Somerville (1968). The coupling coefficients b, c, and d have been calculated (Somerville, 1968, 1970; Kalaghan and Dalgarno, 1972; Cohen and McEachran, 1976) but are known more precisely from the laboratory measurements (Jefferts, 1968, 1969). Interstellar H l is rapidly destroyed by the process (76) Hi+Hz+H:+H (Bowers et al., 1969) and the equilibrium abundance is expected to be small. More could be present under nonequilibrium conditions. Attempts to detect

417

INTERSTELLAR MICROWAVE TRANSITIONS

interstellar H : by its microwave transitions have been unsuccessful (Shuter and Sloan, 1969; Jefferts et al., 1970a; Encrenaz and Falgarone, 1971). These have all considered transitions within N = 1 of the ground vibrational state u = 0. However, it has been suggested by Stecher and Williams (1969 and private communication) that Hi might best be formed by photoionization of H2,populating the vibrational states in relation to the Franck-Condon factors (Dunn, 1966; Villarejo 1968). This would give the largest population in u = 2 and reasonable populations in u = 1 to 4 with few ions in u = 0. A similar distribution follows from ionization of H, by fast cosmic-ray particles, now considered an important mechanism (Dalgarno and Black, 1976). The cross section for the process (76) decreases slightly with increasing u for the H : (Chupka et al., 1968); this acts in the same direction. There could therefore be a somewhat better hope of detecting vibrationally excited inter: . stellar H Figure 3 shows the energy level structure for N = 1. Transition frequencies and A-values are given in Table XV, from the experiments of Jefferts (1969) and the theory of Somerville (1968). TABLE XV FREQUENCIES (MHz) AND TRANSITION PROBABILITIES (SEC-I)FOR N = 1 IN H i "

V

1412.24 1.77,-15 1374.20 1.63, -15 1338.97 1.50,- 15 1306.38 1.39,- 15 1276.271 1.30,- 15 1248.509 1.21,-15 1222.971 1.14,- 15 1199.551 1.07.- 15 1178.159

A

1.01. - 15

V

A V

A V

A V

A V

A V

A V

A V

A

1404.35 3.65,- 15 1366.91 3.37,-15 1332.24 3.12,-15 1300.17 2.90. - 15 1270.550 2.71,- 15 1243.25I 2.54. - I5 1218.154 2.39,- 15 1195.156 2.26. - 15 1174.169 2.15,-15

1392.64 1.87,- 15 1355.70 1.73,- 15 1321.54 1.61,- 15 1289.99 1.50,- 15 1260.900 1.41.- 15 1234.128 1.32,- 15 1209.558 1.25.- 15 1187.090 1.18.-15 1166.642 1.13. - 15

13 15.79 4.02,- 16 1283.71 3.71, -16 1254.19 3.44,-16 1227.08 3.20.-16 1202.244 2.99. -16 1179.576 2.80. - 16 1158.982 2.64. -16 1140.387 2.50. - 16 1123.734 2.37. - 16

1296.18 2.73, - 15 1265.21 2.54, -15 1236.76 2.36, - 15 1210.69 2.21,- 15 1186.873 2.08. - 15 1 165.195 1.96, - 15 1145.569 1.86, - 15 1127.926 1.77. -15 I 112.217 1.70,-15

' For v = 4 to 8,frequenciesin the first two columns are from Jefferts (1969)and others are derived from his measurements by subtraction.Those for other u are from polynomial extrapolation. A-values are from the theory of Somerville (1968),using experimental frequencies and including state mixing.

W B. Somerville

418

1/2

112

3/ 2

FIG.3. Energy levels in N = 1 of Hi (Somerville, 1968).

D.

TRANSITIONS IN

’ll MOLECULES

1. Theory of the A-Doubling Transition

The calculation of the A-doubling transition probability in OH has an involved history, which teaches the need for care. There was an error in the original paper by Dousmanis et al. (1955) and the book by Townes and Schawlow [1955, Eqs. (2.16) and (7.20)]. Even after that was corrected it required some iteration to achieve agreed solutions (Goss and Weaver, 1966; Turner, 1966,1967; Lide, 1967; Carrington and Miller, 1967; Poynter and Beaudet, 1968; Burdyuzha and Varshalovich,6 1973). The correct formulas (81) and (82) in the case (b) limit are given here for the first time. OH has a magnetic hyperfine structure. Each A-doubled level is further split into two and there are four lines within the A doublet instead of one (Fig. 4). Ignoring this splitting for the present, the line strength in intermediate coupling is for A = I 25

+1

%(+, - ) = J ( J + 1)4Xz [( +x - 2 + np,+ (fX+ 2 - np,]’(77) Formulas for the coefficients called a,,,, b,,, are misprinted there; the numerical results are not affected.

419

INTERSTELLAR MICROWAVE TRANSITIONS H. F.S

A - DOUBLING

F =2 1

F:2 1

EQUILIBRIUM RELATIVE INTENSITY

t

L )(

1612

1721

v (MHz)

1665 1667

FIG.4. Structure and transitions in the lowest rotational level of OH (schematic).

with

X

= [(2J

+ 1)’ + I ( I - 4)]”’,

1 = A/B,

(78)

A is the coefficient of the electron spin-orbit interaction (39), which produces intermediate coupling. I is not to be confused with the coefficient of the spin-spin interaction, not present in ’ll molecules. Some authors use Y instead of I . The transition probability is given by (53) with statistical weight g, = 25

+1

(79)

replacing (25 + 3), and using an appropriate mean of the four hyperfine component frequencies. In (77), with R, = f and R, = $, upper signs are to be taken in case (b) when A is positive (as in CH) and when J = N + 4 and in case (a) for for l l 3 / 2 when A is negative (as in OH); lower‘ signs in case (b) when J = N - 4and in case (a) for nli2 when A is negative and for l l 3 , z when A is positive. The case (a) limit is obtained as ) I (+ 00 and is

SJ(+,- ) = ( 2 J +

l)W/J(J+ 1)

(80)

420

W B. Sornerville

while small rl gives the case (b) limit

+ 1)(2J + 1)

S,(J = N

- f) = 4JA2/(J

Sj(J

+ 4) = 4(J + 1)A2/J(2J+ 1 )

=N

(81)

(82) The formula that Dousmanis et al. (1955) state for case (b), with corrected outside factor [Gordy and Cook 1970, Eq. (4.7711,is the same as (SO) with A replacing R.It agrees with the correct formulas (81) and (82) in the limit of large J, in which case however it would be better to take Sj = 2A2/J. (831 It may be noted that for J = f the coupling is pure case (a), and (77) reduces to (80).The mixing is of states with the same J but different R,and there is only one state with J = f. In coupling schemes ap and bp, the line strengths for the hyperfine structure components are given by

+ 1) - ( F - I + J + 1)(F+ I - J + 1)(1 + J + F + 2)(1 + J - F ) 4(F + 1)(2J + 1)J(J + 1)

Sj(IJF, I J F

SJ(IJF, I J F )

-

sJ(+,

-) (84)

+

+ + + 111’ +

(2F + l)[F(F 1 ) - I(I 1) J ( J 4F(F + 1)(2J + 1)J(J 1)

sJ(+9

-1

(s5)

For I = f,as in OH and CH, these give for J = $ and J = 9 the well-known relative values 1 : 5 : 9 : 1 and 1 : 2 : 1, respectively (for J = f, F = 0-0 is forbidden). However, except for J = i, there are off-diagonal matrix elements in this representation, corresponding in the case (b) limit to departures from b,, coupling. Diagonalization modifies the line strengths (Somerville, 1977b). Hyperfine transition probabilities are given by (53) with

2. Individual Molecules: OH and CH

For OH, the constants in Table VII lead to

I = -7.48847(60) (87) The dipole moment has been measured by Meerts and Dymanus (1973) to be p = 1.6676(9) D (88) For the total A-doubling transition, the frequency is obtained by weighting

42 1

INTERSTELLAR MICROWAVE TRANSITIONS

individual upper or lower levels according to their statistical weights, a procedure called the interval rule. This gives for the R = $, J = $ ground state V A = 1666.625 MHz (89) The transition probability is given by (53) with (79), A ( + , - ) = 8.631 x lo-'' sec-' (90) This is slightly different from values given previously, mainly because of the improved constants (87) and (88); it is also not clear that all previous workers used (89). Results of an extensive recalculation are given elsewhere (Somerville, 1977b) and in Table XVI values are presented for hyperfine components of the lowest J = 4 and J = $ levels only. Frequencies are from Lovas and Tiemann (1974). TABLE XVI

MICROWAVE TRANSITIONS IN OH AND CH

C2

OH

J

5 3

t i

v (MHz)

1, (cm)

A (sec-l)

Equilibrium relative intensity

1-2 1-1 2-2 2- I

1612.231 1665.402 1667.359 1720.530

18.59 18.00 17.98 17.42

1.302, - 1 1 7.176, - 1 1 7.778, - 1 1 9.496, - 12

0.87 4.98 9.00 1.13

0- 1 1-1 1-0

4660.242 4750.656 4165.562

6.43 6.31 6.29

1.092, -9 7.712, - 10 3.892, -10

0.93 2.00 1.01

1-0

3263.794 3335.481 3349.193

9.19 8.99 8.95

2.836, -10 2.018, -10 1.022, -10

0.92 2.00 1.02

Excitation energy (cm-')

F

0

t -

126

-+

For CH, the only laboratory studies are in the optical (Baird and Bredohl, 1971) and infrared (Evenson et al., 1971). Accurate values for the microwave frequencies have come from the astronomical observations; so far only the ground state 211,,2J = has been studied. Frequencies in Table XVI are from Rydbeck et al. (1974, 1976); negligibly different values have been derived by Zuckerman and Turner (1975). The dipole moment is p = 1.45 & 0.06 D

(91) (Phelps and Dalby, 1966; Scar1 and Dalby, 1974). For J =4, , Iis not required.

W 8.Somerville

422 3. Rotatiorz in NS

The molecule NS is heavier than O H or CH and the rotational transitions are in the microwave region. Molecular constants and frequencies are tabulated by Lovas and Tiemann (1974). As in CH, A is positive and 2111,2 J = 4 is the lowest state. The structure of each rotational transition is rather complicated, being split by A-doubling and by hyperfine structure from the spin I = 1 of l"N. Thus the lowest transition, J = $-+, has ten components near 69 GHz. Astronomically, several components of the R = 4, J = :-$ transition near 115 GHz have been detected (Gottlieb et al., 1975; Kuiper et al., 1975).

VIII. Rotation in Polyatomic Molecules A diatomic molecule is necessarily linear, but a molecule with three atoms can spread into two dimensions and one with more than three generally has a full three-dimensional structure. The pattern of energy levels depends on the symmetry of the molecular shape, which is classified in terms of the three principal moments of inertia: I A= 0, ]A

le = I ,

= IB = I C

1 A < !B = I C

< 1C

I A

=

]A

< 1B < IC

linear molecule spherical top molecule symmetric top molecule, prolate symmetric top molecule, oblate asymmetric top molecule

Names like symmetric rotor are also used. The rotational constants in the energy level formulas are given to a good approximation by A = h/8x2IA, B = h/8K21,, c = h/8x21C (92) For a diatomic molecule B = C and this is effectively the constant appearing in (31). This A is not related to the coefficient in (39). Spherical top molecules are very rare. The classic example is methane, CH4. The high symmetry is usually said to mean that the electric dipole moment is zero, but it has been shown that centrifugal distortion makes it nonzero although very small. For methane, Ozier (1971) has measured p = (5.38 0.10) x D (931 Fox (1974) suggested CH4 as a candidate for the unidentified interstellar line U 72.4. However, as indicated in Table XX,this line is now ascribed with some confidence to the transition 514-515 in H2C0. Methane should more

INTERSTELLAR MICROWAVE TRANSITIONS

423

easily be detectable by vibrational transitions in the infrared. However, in their recent detection of circumstellar acetylene, CzHz,at 2.4 pm (another molecule with zero dipole moment) Ridgway et al. (1976) have found only an upper limit for methane at 2.3 pm. The laboratory microwave spectrum of CH, has recently been studied by Holt et al. (1975). For CH,D, ,which has different symmetry, Hirota and Imachi (1975) have found a dipole moment p = 0.014 D.

A. LINEAR MOLECULES A linear polyatomic molecule has structure essentially similar to that of a diatomic molecule. The only major difference is that there are several modes of vibration. For interstellar molecules, in the lowest vibrational state, that does not matter. The interstellar linear molecules that have been detected are listed in Table XVII. Others that could be present include HSiN and NCO. The HCO' and HNC lines were first detected by chance and their conclusive identification came later with accurate laboratory measurement of the frequency (Woods et al., 1975; Saykally et al., 1976a; Blackman et al., 1976). HCO' was known, before this identification, as " X-ogen." TABLE XVII LINEAR INTERSTELLAR MOLECULES

HCN HCO' HC,N

ocs

HNC N2H+ C2H HC,N CJN

Hydrogen cyanide '' X-ogen "

Cyanoacetylene Carbonyl sulfide Hydrogen isocyanide Ethynyl Cyanodiacetylene Cyanoethynyl

Ground state

Wavelength first detected (mm)

'I: 'I: 'I: 'I: 'I:

3.38 3.36 32.9 2.74 3.3 1 3.22 3.43 28.1 3.03

lI: 21:

'I:

21:

Astronomical reference Snyder and Buhl (1971) Buhl and Snyder (1970) Turner (1971) Jefferts et al. (1971) Snyder and Buhl (1972) Turner (1974) Tucker et al. (1974) Avery et al. (1976) Guelin and Thaddeus (1977)

Energy levels for 'X and 'E are given by (31), (33), and (34), and the only complication is from hyperfine structure. For the normal isotopes in these molecules this comes from 14N (I = l), gving a quadrupole hyperfine structure, and from 'H (I = +), which gives structure only in 'C. Spectroscopic data for HCN and OCS are given in Table XVIII. Frequencies and dipole moments are from the compilation by Maki (1974).

W B. Somerville

424

TABLE XVIII IN~RS'IELLAR TRANSITIONS IN HCN AND

IH"CL4N

'H"C14N

.I

F

1-0

0-1 2-1 1-1

I

1 (mm)

A (sec-')

Astronomical reference

3.38

2.41,-5

c.b d

1.69

2.31,-4

3.47

2.22,-5

2-1 1-0 2-1 3-2

1.74 3.48 1.74 4.14 2.07 1.38

2.14,-4 2.20,-5 2.11,-4 1.32,-5 1 26. -4 4.57,-4

6 5 7-6 8-7 9-8 12-1 1

72,976.8 85,139.1 97,301.2 109,463.1 145,946.8

4.11 3.52 3.08 2.74 2.05

1.07,-6 1.71,-6 2.58,-6 3.70,-6 8.88. -6

2-1' 1-0

0-1 2- 1 1-1

2-1

16012~32s

(MHz)

88,633.9360 88,631.8473 88,630.4157 177,261.1 86,342.16 86,340.05 86,338.75 172,680.0 86,055.05 172,108.1 72,414.69 144,828.0 217,238.4

1H12C15N 1-0 *D1'C14N

v

ocs

Not yet detected. Hfs not resolved. ' Snyder and Buhl(l971). Zuckerman et al. (1972). ' Wilson et al. (1972). Kaifu et al. (1974).

t

cb

B

f e g> h

Bb ib

j k k

1

k

Jefferts et a/. (1973). Wilson et al. (1973). Phillips et al. (1974). j Akabane et al. (1974). I, Solomon et al. (1973). ' Jefferts et a!. (1971). @

C2H had previously been studied in ESR experiments in the solid state, and accurate values for the rotational and hyperfine constants have come from the astronomical measurements (Tucker et al., 1974). For N = 1-0 there are six hyperfine transitions, four of which are observed at frequencies around 87.3 to 87.4 GHz. Earlier than this detection, Barsuhn (1972) had calculated the 1-0 transition frequency and suggested C2H as a possible candidate for X-ogen, U89.2. This was ruled out by these observations. Turner (1974) observed the line U93.174 as a close triplet with intensity ratios which suggested a I4N atom as the source of the structure. From theoretical calculations, Green et al. (1974)tentatively identified the line as a transition in N2H+,showing that the further hyperfine structure from the inner I4N atom is too small to have been detected by Turner. The identification was nicely confirmed by Thaddeus and Turner (1973, who

INTERSTELLAR MICROWAVE TRANSITIONS

425

detected this finer structure in an astronomical source emitting particularly narrow lines. Final confirmation has come from laboratory measurement (Saykally et al., 1976b). B. SYMMETRIC TOP MOLECULES

Energy levels are described by two quantum numbers, J and the projection K of J onto the axis of symmetry. K takes the (25 + 1) values -J, -J + 1, .. . , J. The energies are E(J,) = BJ(J

+ 1) + ( A - B)KZ

(94)

for a prolate top (A > B) and E(J,) = BJ(J

+ 1) - ( B - C ) K Z

(95) for an oblate top, (B > C), neglecting higher-order terms [cf. Eq. (31)]. Except for K = 0 there is a double degeneracy. The quantum numbers are l . a fixed value of J, K = 0 has the lowest energy commonly written J I RFor in the prolate and K = +J in the oblate case. The selection rules are AJ = 0, & 1, AK = 0 (96) in the normal case that the dipole moment is directed along the axis of symmetry. The best known interstellar symmetric top is ammonia, N H 3 . For it the rotational transitions are in the infrared and it is detected by its inversion spectrum, discussed in Section IX. Methyl cyanide, CH,CN, was detected at 2.7 mm by Solomon et al. (1971), in several of the K components of J = 6-5; by (96) K does not change. Ulich and Conklin (1974) have detected two of the same rotational transitions, but vibrationally excited, in comet Kohoutek. The only other symmetric top detected so far is methylacetylene, CH3CzH, at 3.5 mm (Snyder and Buhl, 1972). C. ASYMMETRIC TOP MOLECULES

The structure of the asymmetric top is more complicated t h m in the symmetric case. The degeneracy for nonzero K is removed. The relations between energy levels for the asymmetric and symmetric top are shown schematically in Fig. 5. The subscripts on J are K - and Kl, the values of I K 1 in the prolate and oblate symmetric limits, respectively.A unique labeling actually needs only the difference z = K - - K l and there.is an alternative notation, in which J , is given. It is not possible to write the energies in closed form, corresponding to (94) and (95). Even within the ground vibra-

426 JK

33

-

W B. Somerville JK-I K,

JK

\'

\\'-

00 -___-ooo PROLATE SYMMETRIC

339-

'.

-___ ASYMMETRIC

--

00

OBLATE SYMMETRIC

FIG.5. Correlation of the.energy levels for symmetric and asymmetric top molecules.

tional state the energy structures are complex. The addition of vibration, in several modes, vastly increases the complexity; the vibrational splittings are, however, much larger than the rotational ones. The selection rule on J is the usual one (96). Selection rules on K - and K1 depend on the axis in the molecule of the electric dipole moment (Townes and Schawlow, 1955, Section 4.2; Gordy and Cook, 1970, Section 7.4). According to the symmetry, there may be two sets of energy levels, transitions between which are forbidden: as for diatomic molecules, these are called the ortho and para modifications, para having the lower nuclear spin statistical weight. Transitions in which J is constant and only one of the K changes are called K-doubling transitions. So far, 19 asymmetric top molecules have been detected in interstellar microwave radiation of rotational transitions (Somerville, 1977a). As examples, we consider here two of the most important molecules, H 2 0 and H2C0.

INTERSTELLAR MICROWAVE TRANSITIONS

427

FIG.6. The lowest rotational levels in the ground vibrational state of H,O

Figure 6 shows the energy level structure of H20. Table XIX contains a list of the H 2 0 lines of longest wavelength (De Lucia et al., 1974). The one observed line, 616-523, detected first by Cheung et al. (1969a) and exhibiting strong maser action, is uniquely special only because it has the longest wavelength. The 313-220 line, although at 1.64 mm, is not observable from the ground because of strong absorption by water vapor in the Earth's atmosphere (Snyder and Buhl, 1969). We may find rich results when we can study infrared spectrum lines of interstellar H20. TABLE XIX THELOWEST-FREQUENCY TRANSITIONS IN H,O

1. (mm)

Excitation energy (lower state) (cm-')

13.48 1.64 0.93 0.92

446.5 136.2 1282.9 315.8

JK.,K,

(upper-lower)

6i6-523 31,-22~ 1029-936 5 i 5-42 2

11

(MHz)

22.235 183.310 321.226 325,153

Because of differences in the moments of inertia, the energy levels of HDO are different and many lines are at observable wavelengths. Turner et al. (1975) have recently detected the low-lying 1,,-1,, transition of HDO at 80.6 GHz. Table XX contains a list of transitions observed in H 2 C 0 and its isotopes. Apart from HDCO, frequencies are taken from the compilation by Johnson et al. (1972). Later laboratory values for some lines are given by Tucker et al. (1972) and Nerf (1972).

W B. Somerville

428

TABLE XX MICROWAVE TRANSITIONS IN H,CO

(upper-lower)

Excitation energy (lower state) (cm- I )

V

I

(MHz)

(mm)

4,829.660 14,488.650 28,974.800 48,284.521" 72,409.099 101,332.991"

62.07 20.69 10.35 6.21 4.14 2.96

10.5 15.2 22.3 3 1.7 43.4 57.5

140,839.529 150,498.359 21 1,211.469" 225,697.787"

2.13 1.99 1.42 1.33

10.5 10.7 15.2 15.7

72,837,974 145,602.971 218,222.186'

4.12 2.06 1.37

0.0 2.4 7.3

2,483.408" 4,954.760" 8,884.870"

120.72 60.51 33.74

69.2 83.7 100.7

4,593.089 4,388.797 5,346.64"

65.27 68.31 56.07

JK.,K,

Not yet detected. Snyder et a/. (1969). Evans et a/. (1970). Welch (1970). Wilson et a/. (1973); Lovas (1974). Kutner et a/. (1971).

Astronomical reference b C

d e

f g

/I

9

Thaddeus et al. (1971). Akabane et al. (1974). Batchelor et a/. (1973). j Zuckerman et al. ( 1 969). Ir Gardner er al. (1971). ' Watson et al. (1975).

IX. Inversion in NH3 The three hydrogen atoms in ammonia lie at the corners of an equilateral triangle. The equilibrium position for the nitrogen atom is equidistant from the three hydrogen atoms and out of the plane that they define. There are two such equilibrium positions, on opposite sides of the plane (Fig. 7). In one vibrational mode, associated with quantum number u 2 , the nitrogen atom moves perpendicular to this plane and can pass between the two equilibrium positions.

INTERSTELLAR MICROWAVE TRANSITIONS

429

FIG.7. The ammonia molecule (schematic),showing the symmetrical equilibrium positions for the nitrogen atom.

For higher values of the vibrational quantum number u 2 , the nitrogen atom can pass relatively freely from one side of the plane t o the other, but for u2 = 0 and 1 it is faced with a potential barrier that it can penetrate only by quantum-mechanical tunneling. Vibration involving barrier penetration is described as a hindered motion. The potential in which the nitrogen atom moves has two minima, corresponding to the two equilibrium positions. In the two-minimum potential, all the energy levels are split in two (not if the rotational quantum number K = 0, for reasons of symmetry). The splitting is small for the ground vibrational state and is called the inversion splitting. The vibrational inversion splitting is combined with the usual rotational structure of an oblate symmetric top molecule. Each rotational level J K suffers the inversion splitting (unless K = 0), the split levels being labeled + and -. For u2 = 0 the inversion splitting is smaller than the rotational structure (Fig. 8). K = 3n (n = 0, 1, 2, . . .) are the ortho states and K = 3n k 1 the para. By internal molecular perturbations, transitions in

0

I

1

I

2

I

K-

3

I

4

1

5

6

FIG.8. Rotational energy levels for NH, in its ground vibrational state (Morris rt a/.. 1973, with permission of The University of Chicago Press). Ortho levels are marked by asterisks. Inversion splittings have been expanded by a factor of 10.

which K changes by 3 can occur as electric dipole transitions (Oka et ul., 1971). Table XXI contains a list of the lowest inversion transitions, several of which have been detected astronomically. Laboratory values are from Poynter and Kakar (1975). Levels with J = K are metastable. Levels with J > I K 1 are not and can decay quite rapidly to ( J - 1, K ) . To maintain sufficient interstellar populations in these levels, it is necessary to have rapid collisional excitation, implying high densities of molecular hydrogen in the cloud in question. Some of the ammonia inversion lines have tentatively been identified in emission from the atmosphere of Jupiter (Klein et al., 1975). In this case, the line strength

SJ(J,K ) = (25 + l)KZ/J(J+ 1).

(97)

TABLE XXI INVERSION TRANSITIONS I N NH,

(J,

K)

Excitation energy (lower state) (cm-') 16.12 44.58 85.39 138.55 204.04" 28 I .nx 55.88 104.21 164.86" 115.48"

Not yet detected.

* Cheung er a / . (1968).

81

(GHz) 23.6945 23.7226 23.8701 24.1394 24.5330 25.0560 23.0988 22.8342 22.6883 22.2345

(cm) 1.265 1.264 1.256 1.242 1.222 1.196 1.298 1.313 1.321 1.348

A (sec- ')

1.687. 2.257, 2.587, 2.853. 3.120, 3.419, 5.209, 1.006, 1.333. 2.323,

-7 -7 -7 -7 -7 -7 -8 -7 -7 -8

Astronomical reference h

h c

c c

d d c. d

' Cheung ri ul. (1969b). Zuckerman rf al. (1971).

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AUTHOR INDEX Numbers in italics refer to the pages on which the complete references are listed. A

Baird, K. M., 421.430 Baird, R. C., 22, 54 Aartsma, T. J., 90, 110 Baklanov, E. V., 86, 110 Abella, 1. D., 90, 96, 110, I l l Baht-Kurti, G. G., 239, 252. 309 Abragam, A,, 90. 110 Ball, J. A., 414, 417, 422, 432, 433 Abrdmowitz, M., 272, 308 Ballard, J. R., 384. 435 Adelman, S. A., 1 1 . 20, 32, 37. 51 Ballik, E. A,, 404, 431 Adler, F. T., 248, 310 Balling, L. C., 129, 224 Akabane, K..414,424,428,430,433 Baranger, M., 64, 105, 110 Albritton, D. L.. 41. 52 Baravian, G., 43.52 Alexander, M. H.. 242. 244, 250, 256. 277, Bardsley, J. N., 332, 350, 359, 378,381, 382 308,309, 313 Barg, G. D., 236,238,246,254, 255, 287,293. Alexseev, V. A., 62, 67, 86, 110 309 Allen, C. W., 384, 385,430 Barker, J. R., 203,224 Allen, L., 133, 136. 224 Barrett, A. H.. 393, 433 Allen, L. C., 12. 51 Barsuhn, J., 424,431 Allison, D. C. S., 18, 19, 51 Baskin, C. P., 242, 314 Alpher. R. A,, 22.51 Basting, D., 156,224 Altick. P. L., 1 I. 52 Batchelor, R. A,, 428,431 Amme, R. C., 233,309 Bates, D. R., 39,40, 43, 44. 45, 52 Amos, A. T., I3,52 Bauer, E., 204, 224 Anderson. P. W.. 64, 110 Baum, G., 42,55 Anderson, R. W., 156, 200,201, 202,224 Bauschlicher, C. W., Jr., 242, 309 Anderson, T. G., 423,425.434 Baz, A. 1.. 332, 336, 378 Andreeva, T. L., 62. 67. 86, 110 Beaudet, R. A,, 418,434 Andrick, D., 114,212,213,214,224. 317, 318, Beck. D., 275,299,309 349, 350,378 Bederson, B., 8, 18, 20, 27, 29, 31, 32, 33, 34, Angel, J. R. P., 41, j 2 35,36, 37,38,39,40,42,47,52,53,54,114, Apt, J., 106, 110 116, 153, 154, 165, 191, 192, 224, 225, 226, Armstrong, W. D., 275, 304, 305,309 227, 317, 350, 362, 365, 376, 378, 380, 382 Arthurs, A. M., 248, 309 Beenakker, J. J. M., 233, 311 Ashkin, A,, 152. 224 Bender. C. F., 242,309,314 Aubrey, B. B., 362, 365. 376,380,382 Benes, E., 15I , 225 Audibert, M. M., 233.309 Bennett, R. J. M., 384, 400, 431 Auerbach, D.. 304,314 Bennewitz, H. G., 303,309 Avery, L. W.. 423, 430 Ben-Reuven, A,, 66, 110 Aziz, R. A,, 233. 311 Bente, H. B., 275, 304, 310 Berard, E., 242, 244, 256,308 Eerg, H. C., 389,431 Bergmann, K., 151,224 B Berman, P. R.,59,60,62,63,66,67,91,93, Backx, C., 352,354,355,378 94, 96, 99, 102, 110, 112 Baede, L., 90. 110 Bernstein. R. B., 204, 227, 234, 250, 255, 262, Bagaev, C. N., 86, I10 275,286,309,312,314

431

43 8

AUTHOR INDEX

Berrington, K. A.. 352, 353, 354. 355, 356, 360. 366. 368, 372, 373, 375, 377,378, 379, 380 Berry, R. S., 306,314 Bershdder, D.,22, 23, 53 Bertojo, M., 390, 407, 431 Beterov, 1. M., 66.68. 69, 74, 101, 110 Bhaskdr, N. D., 43, 52, 116. 153, 154, 165, 191, 224 Bhatia, A. K., 350, 382 Bickes, R. W., 236,309 Biedenharn, L. C.. 322, 323, 378 Bielicz, E., 62, 110 Billing, G. D., 244. 309 Billingsley, F. P., 13. 14, 54 Binkley, J. S., 241, 313 Biraben, F., 101. 110 Bird, G. R., 402.409, 433 Bird, R. B., 3, 46,53 Birks, J . W., 242, 309 Bischel, W. K.. 100, 110, 234, 309 Bitsch, A., 349, 350,378 Bjerre, A., 204,224 Bjorkholm, J. E., 74, 110 Black, J. H., 412, 415, 417, 431 Blackman, G. L., 423.431 Blaha, M..366,378 Blair, G. N., 409,431. 435 Blais, N. C., 275, 299, 300,309 Blatt, J. M., 322, 323, 378 Bloch, C., 339, 340.378 Blum, K., 161. 164.226 Blum, P., 151, 225 Blythe, A. R., 275,309 Bolger, B., 90, 110 Borenstein, M., 77, 103, 110 Borkenhagen, U., 279,309 Bottcher, C., 167, 168, 193, 208,224 Bottner, R., 277, 281, 297, 298, 299, 309 Bowers, M. T., 416, 431 Brandsden, B. H., 317, 332,378 Brandt, S., 272,309 Brechignac, C., 101, 102, 111 Bredohl, H.. 421,430 Breene, R. G., 59, I l l Brewer, R. G., 68,69,91,93,96,102,110, 111, 112 Brink, D. M., 119, 120, 123, 159,224 Brochard, J., 93, 102, 111 Brocklehurst, M., 395,431

Brongersma, H. H., 352,354, 355,378 Brooks, J. W., 428,431 Broten, N. W., 423,431 Brown, P. J., 242, 309 Brown, R. D., 423, 428,431 Bruch, L. W., 20,52, 239,309 Brunner, W.,218,224 Brunt, J. N. H., 355,378 Buck, U., 114, 193. 196, 224, 234, 236, 275, 295,296,309 Buckingham, A. D., 5,7,52 Buhl, D., 409, 410, 423, 424, 425, 427, 428, 431,434,436 Bunker, P. R., 415,431 Burdyuzha, V. V., 4 18,431 Burgess, A.. 162,224 Burhop, E. H. S., 113, 175,236,227,312 Burke, P. G., 18,19,51,52, 317, 323, 324, 325, 338, 341, 342, 347, 349, 352, 353, 354, 355, 356, 360, 361, 362, 366, 368, 370, 372, 373, 374, 315, 377, 378,379,380, 382 Burkhard, D. G., 384,435 Burnham, D. C., 91,96,112 Burns, G., 242,244,314 Burshtein, A. I., 91, 96. 99, 111 Bussoletti, E., 415, 431 Buttle, P. J. A., 339. 342, 379

C Cagnac, B., 101,110, 156,224 Cahuzac, P., 86,93, 102, 111 Callaway, J., 317, 337, 379 Carlsten, J. L., 101, 105, I l l , 219, 225 Carney, G. D., 242,309 Carr, H. Y.,96.97, 111 Carrington, A., 384, 418,431 Carruthers, G. R., 404, 431 Carter, G. M., 116, 155, 168, 194, 195, 225, 22 7 Cartiaux, L., 188, 226 Casimir, H.9. G., 3, 52 Cattani, M., 62, 111 Caudle, G., 214,228 Caves, T. C., 17,52 Celotta, R. J., 40,53 Certain, P. R.,239, 242, 309, 312, 412, 431 Chamberlain, G. E., 25,27,52,55 Chan, Y.W., 390,431

AUTHOR INDEX Chang, E. S., 6. 12, 17,52, 248, 309 Chang, J. J., 339, 380 Chappell, W. R., 62, 111, 112 Chebotaev, V. P., 66. 68, 69, 74, 86, 101, 110 Chen, S. Y..59, 111 Cheung, A. C., 427,428,430,431.432 Chikada, Y.,424,428,431 Choi. B. H., 291,292,293,309 Chou, M. S., 275, 304,306,309,310 Chow, P., 275,310 Chu, F. Y.,194,227 Chui, M. F., 390,407,431 Chung, K. T., 8, 19, 29, 35, 52,53 Chupka, A., 368,382 Chupka. W. A., 417,431 Churchwell, E., 394,431 Chutjian, A., 338, 361, 379 Clark, F. O., 414, 431 Clarke, E. M., 213,214, 228 Clarke, J. F., 232, 309 Claverie, P., 241, 309 Cochran, W. D., 415.435 Coggiola, M. J., 236, 312 Cohen, E. R., 386,431 Cohen, H. D., 14.52 Cohen, M., 1 I , 52,416,431 Cohen, V. W., 390,431 Cohen-Tannoudji, C., 129, 131, 147,225 Cole, R. H., 21,54 Collins, F. S., 294, 309 Collins, R. E., 191,225,227 Comer, J., 335, 350, 356, 359, 360,379, 380 Condon, E. U., 8,52,385,387,391,392,431 Conklin, E. K., 425,435 Conley, R. J.. 275, 304, 305,309 Conte, S. D., 71, 111 Cook, R. L., 384,399,405,409,420,426,432. 436 Cooke. W . E.. 45.52 Cooper, B. F. C., 428,432 Cooper, J., 62, 111, 112 Cooper, J. W.,325, 337,352, 355,379 Coplan, M. A., 277, 313 Cord, M. S., 384,435 Cornille, M., 348, 374, 381 Cosby, P. C., 277,309,312 Crampton, S. B., 389. 431 Cravens, T. C., 193,224 Creaser, R. P., 275, 304, 305,309 Crim, F. F., 275,304, 306,309,310

439

Crompton, R. W., 350,379 Crosby, D. A., 25, 29,52 Cross, J. B., 215, 299, 300, 309 Cross, R. J., 236,310 Crutcher, R. M., 404,428,431,435 Csanak. Gy., 317, 338, 361, 366,379,382 Csizmadia, 1. G., 242. 310 Cummings, F. E., 15.52 Cunningham, D. C., 368,379 Curtis, C. F., 3. 46, 53, 248, 310, 311 Cvejanovic. S., 335, 350, 356, 358. 359, 360, 379, 380 Cyvin, S. J., 234,310 Czuchaj, E., 62, 110

D Dagdigian. P. J., 47, 52, 242, 250, 309 Dalby, F. W.. 401, 41 I , 421,434, 435 Dalgarno, A., 11, 12,23,24,52, 193,224,227, 228, 248, 309, 412. 415,416,417,431,433 Dalitz, R.H., 333, 379 Damburg, R. J., 323,379 Damgaard, A., 39,40,43,44,45,52 Das, T. P.. 6, 12, 17, 52, 53 David, R., 271,277,286. 310 Davidz, A., 12, 13,55 Davies, A. R., 369, 375, 377,381 Davis, J., 366, 378 Davis, J. H., 409, 431, 435 Davison, W. D., 47,52 Decomps, B., 147,225 de Heer, F. J., 318,349, 350,379 Dehmel, R. C., 362, 365,366, 367,379 Dehmelt, H. G., 389, 434 Dehmer, J. L., 15,52 DePristo, A. E., 242, 250,309 Delle Donne, M., 20, 54 De Lucia, F. C.,427,431 Demtroder, W., 151,224 Destombes, J. L., 401, 431 Dicke, R. H., 81, I l l Dickel, J. R.,428, 435 Dickinson, A. S., 256,310 Dickinson, D. F., 417, 433 Dickson, D. F., 393,409,410,431 Dillon, T., 62, 111, 112 Dimpfl, W. L., 277, 310 Ding, A. M. G., 233, 279,310

440

AUTHOR INDEX

Dixon, T. A., 412,423,425,432.434.436 Doering, J. P., 231, 277, 288. 310, 312 Doktorov, A. B., 91.96,99, I l l Dolder. K. T., 350, 380 Donohue, T., 275, 310 Dordn. M. B., IS, 52 Dousmanis, G . C.. 418.420,432 Dove. J. E., 233. 310 Drachman, R. J.. 317,337,379 Drechsler, M., 25, 52, 53 Dressler. K.. 404, 413. 416, 432 Drummond, D. L., 105, I l l Ducas, T. W., 116. 155, 168. 194, 195. 225 Ducloy. M., 91. 111. 147,225 Ducuing. J., 233. 309 Duff. J. W.. 252, 256, 310 Duguette, G . , 236. 309 ' Dumont. M., 147,225 Dunkrr, A. M.. 239,310 Dunn. G. H., 417,432 Diiren, R., 116. 152, 153. 168, 178. 179, 194, 196, 197, 198. 199. 200. 225

Dutton. N. C., 17, 53 Duven, R.. 260, 271,310 Duxler. W. M., 350. 379 Dymanus. A., 46,54420,433

E Eastes, W., 251, 271, 277, 300,310 Eberley, J. H..133. 136, 224 Edelstein. S. A.. 45, 46, 52 Edlen. 6.. 374,379 Edmonds, A. R., 119, 122,225 Edwards, A. K.. 368,37Y Efremov, I., 40,54 Ehrhardt, H., 352, 355,379 Einwohner. T. H.. 13 I , 156,225.228 Eisenbud, L., 338.382 Eisner, P. N., 362. 376,380 Elford, M. T.. 350,379 Ellder, J., 421.434 Elleman, D. D.. 416. 431 Eminyan, M., 164, 186. 190,225 Encrenaz, P. J., 407. 409, 417,432 English. T. C.. 41,46,52 Ennen, G., 234,310 Epstein, P. S.,4. 52 Epstein. S. T., 13, 55

Erlewein. W., 255, 310 Ertas, I., 233, 311 Evans, L., 390,432 Evans, N. J., 428,432 Evenson, K . M., 394,421.432, 434 Ezekiel, S., 219. 228

F Fabjan, C. W.. 391,432 Fabre, C.. 44, 52 Fain, V. M., 136. 225 Falcone, R. W., 216, 225, 226 Falgarone. E., 417, 432 Falkoff, D. L., 123,225 Fano. U., 118, 120, 122, 123, 124, 125, 145, 158. 162. 225, 248, 309, 325, 326, 336, 337, 338, 342, 379 Farrar, J. M., 275,299, 300,310 Faubel, M.. 231,271,273.277, 278, 281.283, 285,286,287,310,314 Feautrier. N., 368, 382 Feld, M. S., 68, 74, 89, 91. 111 Feldman. 8.J., 89. 111 Fels, M. F., 327, 379 Fenn, J. B., 233,300, 310 Feshbach, H., 319, 325,379 Field, G . B., 388, 390, 404.413. 416, 432 Figger, H., 390,436 Fineman, M. A., 362, 365, 366, 367.379 Fink, U., 404,432 Fisher, E. R., 204, 224,225 Fisk, C., 5 I , 54 Fisk, G. A., 275,304, 306,310 Fitz, D., 250, 310 Fiutak, J., 62, 110 Fluendy, M.A. D., 262,310 Flynn, G., 233, 314 Foley, H.M., 64,111,403. 432 Fonda, L., 336,379

Fontana, P. R., 25, 55 Forster, H., 275, 299, 309 Fortson, E. N., 389,434 Fourikis, N., 427,435 Fox, K., 422,432 Fraga, S.,51,54 Freeman, R. R., 46.52 Fremerey, H., 242, 251,310 Freund, S. M.. 233,310

AUTHOR INDEX Fried, B. D., 71, 111 Frosch, R. A,. 403,432

C

441

Gordon, J. P., 96, 111 Gordon, M. D.. 242,244,311 Gordon, R. G., 233, 239, 243. 246, 247. 251. 310.311,313 Gordon, R. J.. 236,312 Gordy, W., 384. 399.405.409, 420. 426. 432, 433 GOrZd, M. P., 147, 225 Goss, W. M., 389,418,432 Gottlieb. C. A,, 410, 414.422. 431, 432 Gottlieb, E. W., 422, 432 Could, H., 41. 52 Graff, J., 47, 52 Grasso. M. N., 8,53 Green, S.. 247, 255. 302, 311,424, 432 Green, W. R.. 216,225,226 Greene, E. F., 261, 264. 275, 304. 305. 309, 311 Greene. F. T., 28,29,41,53 Grosser, A. E.. 275,309 Grotrian, W., 42, 53 Grove, R. E., 219,228 Grynberg, G., 101,110 GuClin, M., 423, 432 Gulkis, S., 430, 433 Gunn, H. I., 423,431 Gush, H. P., 415,435

Galatry, L., 81, 111 Gallagher, A,, 105, 111. 216, 225 Gallagher. R . J., 233, 300, 310 Gallagher. T. G., 45.46, 52 Gallaher, D. F., 323,379 Gammon, R. H.. 401, 410,411,435 Gardner, F. F., 428,432 Garrett, W. R., 3. 52 Garrison. B. J., 247. 311 Garrison, J. C., 131, 156. 225, 228 Garstang, R. H., 394, 432 Gautier, T. N., 404. 432 Gavrila, M., 212, 214, 215. 225, 226 Geballe, R.. 368,379 Geltman,S., 212,225,317.323. 330,332,379. 380 Genack, A. 2..91, 102, 111 Gentry. W. R..236, 242, 244. 252. 271, 277, 287, 288. 289, 293, 294. 295, 303, 310. 314 Gerlach, G.. 156. 225 Gerlach, W., 23, 52 Gerritsen. H. J., 155, 225 Gerry, M. C. L., 423. 433 Gersten, J. I., 218, 219, 222. 225 H Gestermann, F., 236,309 Geurts, P. J. M.. 242, 243, 310 Haas, R. H., 384,435 Ghirardi, G. C., 336, 379 Haberstroh, R. A,, 390,436 Gibbs, H. M., 90,120 Hackam, R.,23,53 Gibson, J. R., 350,380 Hagstrom, S. A,, 20, 37, 54 Giese, C. F., 236,242, 244,252,271,275,277, Hahn, E., 96, 111 287, 288, 289, 293, 294, 295, 303, 310, 314 Hahn, E. L., 68, 69, 91, I l l Gilbody, H.B.,113, 175,227 Hahn, L., 218,219,220, 222,223,225 Gilmore, F. R., 204,224 Hahn, Y.,348,380 Gilmore, W.;409, 433 Hall, D. N. B., 423,434 Gioumousis, G., 248, 311 Hall, R. B., 275,304, 305,309 Glover, R. M., 19,52 Hall, R. I., 352,380 Godart, J., 43, 52 Hall, W. D., 25, 26, 27, 28, 37, 53 Goddard, T. P., 156, 200,201. 202, 224 Hammer, D., 151,225 Godfrey, P. D., 423,428, 431 Hansch, T. W., 65.67.68, 69, 74.93,94. 101, Goldberger. M. L., 250, 311 102, I l l , 112, 156,228 Golden, D. E., 350,380 Happer, W., 42,43,53,54, 119, 129, 225 Goldstein, M., 191, 225, 227 Hariharan, P. C., 243,31 I Goldstone, J., 17,52 Haroche, S., 44,52 Goldwire, H. C., 389, 396, 432 Harris, F. E., 343, 380

442

AUTHOR INDEX

Harris. S . E..156. 216,225,226,228 Hartrnann. S. R..90. 96. 110. 111 Harvey. J. S. M.. 390. 432 Harvey. K. C., 44. 53 Haus. H. A,. 86. 111 Hawkins. R. T.. 44.53 Hayes. E. F., 242. 256, 3OY. 311 Hazi. A . U.. 327.37Y, 380 Heddle, D. W. 0.. 355.380 Hedges. R . E. M.. 105. 111 Heer, C. V.. 91, 96.99, 111 Hefter. U., 151, 224 Heiss. P..357. 382 Helbing. R. K . B., 262. 311 Held, W. D.. 260. 271. 310 Heller. E. J., 342. 382 Hellwege, A . M.. 384, 432 Hcllwege. K. H., 384. 432 Helrninger, P.,427.431 Henry, R. J. W.. 362, 364, 365, 366, 367, 369, 370. 371. 372. 374, 375. 3x0, 381. 382

Hercher. H.. 219.227 Hering. P., 151. 224 Herrnann. H. W.. 179. 185, 186. 189,226 Herrnann. V.. 277.293.294,295,3/3 Herschbach. D. R., 2. 47. 53, 275, 279. 304,

Hofmann, H., 117, 149, 150. 151, 203. 205, 206,208, 209,210,226

Hoheisel, C., 241. 242. 245. 246, 312 Hohervorst, W.. 43. 53 Holmes. J. K., 214, 228 Holt. C. W.. 423, 433 Holt, H. K.. 89, 111 Hopf, F. A.. 91.93. 111 Hoppe, H . O., 116, 152. 153, 168, 178, 179, 196, 197, 198, 199, 200,225

Hotop. H., 370. 374, 378.380 Hsu. D. S. Y., 203,226 Huchtmeier, W., 394,431 Huetz, A,. 357, 360, 380, 381 Hughes, R. S., 156,226 Hughes, V. W.. 27.55 Huisken. F.. 275. 295.296.30Y Humphrey, L., 45.52 Hundhausen. E., 235,311 Hunding, A,, 244,309 Hunter. L. W.. 248, 311 Hurst, R. P.. 8. 12. 18, 19, 29, 35, 52, 53. 55 Husinsky, W., 151,225 Hutcheson. L. D.. 156, 226 Hyman, H . A., I I , 32,53

306.311.312

Hertel. I . V . , 116. 117. 118. 119. 122. 123. 131. 138. 140, 142. 143, 149. 150. 151, 155. 158, 163. 165. 167, 170. 172. 178. 179, 185. 186, 188. 189. 203. 205. 206. 208, 209, 2 10. 2 18, 219. 220. 222, 223. 225, 226, 227 Herzberg, G.. 384. 398. 399.401.432 Herzenberg. A,. 356. 380 Hess. S.. 62, 111 Heukels. W. F., 296, 314 Heydtrnann. H., 203, 226 Hibbert, A,, 18, 19, 52, 338. 379 Hicks, R. J., 335, 380 Hill. R. M.. 45. 46. 52 Hill. W . T.. 44. 53 Hinchen, J . J., 233, 311 Hirabayashi. H.. 424.428,43/ Hirota. E.. 423. 432 Hirschfelder, J . O., 3. 13, 46, 53, S5. 24/. 311 H.jalmarson, A,, 421, 434 Hobbs. R. H.. 233. 311 Hoeberling. R. F.. 25. 28. 53 Hoeli, J., 410. 432, 435 Hoegy, W. R.. 368, 380

I Ice. G . E.. 3. 53 Iguchi. T., 414.424.433 Ilin. R. N., 370, 372, 380 Imachi. M.. 423.432 Inokuti. M.. 15, 52 Irvine. W. M., 421. 434 Ishihara, T.. 17. 53

J Jackson, J. D.. 125,226 Jackson. J. L.. 339. 380 Jacob, M., 250,311, 322, 380 Jaduszliwer, B., 116, 153. 154. 165. 191, 224 Javan, A.. 68,74, 86, 111 JefTerts, K. B., 407, 409, 410, 41 I , 416, 417. 423, 424. 425, 428, 432. 433, 434, 435, 436 JefTries, J. T.. 59, 111 Jenkins, E. B., 404, 435 Jensen. H., 247.311

AUTHOR INDEX

Joachain, C. S., 188, 226 Johns, J. W., 233,310 Johnson, D. R.,409,410,414,427,431,433 Johnson, N. B., 25,27,28,53 Johnson, R. C., 399.433 Johnston, H. S.. 242.309 Jonkman, R. M.. 233,311 Jortner, J., 66, 110. 233, 312 Jory, R. L., 350.379 Journel. G., 401.431 Joyez. G., 352. 357,380,381

K Kaandorp, J. P. S., 214, 215, 226 Kagann. R. H., 41,52 Kaifu, N.. 414,424, 428,431,433 Kakar, R. K., 422,430.433.434 Kalaghan, P. M., 416,433 Kaneko, S., 12, 53 Kaplan, M.. 116, 168, 194. 195, 225 Kapral, R.. 242, 244, 314 Karl, G.. 47. 53, 203, 226 Karl, R. E., 242,310 Karo, A,, 242,311 Karplus, M.. 12. 17, 52, 53 Karule, E.. 323,379 Kasdan, A,, 192.226 Kaufman, S. E., 12.53 Kaul, R. D., 43.53 Keijser, R. A. J., 233, 311 Keil, R., 101, 111 Keilson, J., 77, 94, 111 Keller. J. B., 332. 380 Kelly, H. P., 16, 17, 53 Kelly, P. J., 100, 110 Kendall, G . M., 236. 238, 242, 246, 254, 255, 287, 293,309,310 Kennedy. J.. 186. 188. 226 Kerner. E. H., 252,311 Kessler. J., 114, 226 Kewley, R., 409.433 Khadjavi, A.. 42,43. 53 Khanin, YA. I., 136, 225 Kieffer. L. J.. 317, 350, 362, 378 Kim, Y. S., 246,247,311 King. D. L., 275, 304, 305, 306,311 King, G. C., 355.378 King, J., 416, 431 Kingston, A. E., 1 I , 23.52

443

Kinsey. J. L., 203,227 Kirchhoff, W. H.. 427. 431,433 Kivel, B.. 362, 365. 380 Klar, H., 248,311 Klein. L.. 66, I10 Klein, M . J.. 430. 433 Kleinmann. S. G., 423, 434 Kleinpoppen. H., 161. 164, 186. 190, 22-5.226. 227 Klernm. R. A,. 390.434 Klemperer. W.. 279, 233, 311, 312 Klemperer, W. A,, 410, 434 Kleppner, D., 46, 52, 53 Knaap. H. F. P.. 233,311 Knoop, F. W. E., 352, 354. 355.378 Kohn, W., 339, 343, 345,380 Kolchenko, A. P., 66, 86.90, 111 Kollberg, E., 421,434 Kolos. W.. 21. 46. 53 Kopfermann, H.. 42.53 Koski. W.S., 231, 311 Kouri, D. J., 251, 256. 311, 312 Kovacs, I., 384, 399. 405.433 Kowalski, F. V., 44. 53 Kracht, D., 339,380 Kramer, H. L., 2,47.53 Kramer, K. H.,303,309 Krause. L., 167,200,202,226 Krauss, M., 242, 243, 311,312 Kreek, H., 296,311 Krenos, J . R., 288. 289, 311 Kreutztrager, A,, 43.53 Krieger, J. B.. 12, 53 Kroll, N. M., 216,226 Kroto, H. W., 423,430 Kriiger, H., 212,214,226,228 Krupenie, P. H., 401,407,409,410,433 Krus, P., 203, 226 Krutein, J., 277, 302, 311 Kuiper, E. N. R., 422,433 Kuiper, T. B. H., 422, 433 Kukol, R. F., 384,435 Kuntz, P. J., 242, 311 Kuppermann, A.. 236,248,311,312 Kurnit, N. A., 86, 90, 96, 110, 111 Kutner, M. L., 409, 410, 423, 424, 428, 433, 435.436 Kutzelnigg, W., 241, 242, 243, 245, 246, 311, 312,314 Kwei, G. H., 275,299, 300,309

444

AUTHOR INDEX

L LaBahn, R. W., 350. 37Y LaBudde, R. A,. 255.312 Lada, C. J.. 422,432 Lam, S. H., 233,3/3 Lamb, W. E., 59. 62, 77, 89, 91, 103. 110.112 Lamkin. J . C.. 337, 382 Landau, M., 357,380,381 Lane, A. M., 338, 341,343.380 Lang. N . C., 234.312 Langenberg, D. N., 391, 394,435 Langendam. P. S. K.. 2 14. 2 15, 226 Langhans. L., 212.213,214,224.352,355.379 Langhoff. P. W., 12.53 Larson, H. P., 404. 432 Latshaw. W. S., 43.53 Lau. A. M. F.. 131, 216,226,227 Lau, M. H., 261,264,311 Lawley, K. P., 262. 310 Le Dourneuf, M.. 323, 348, 366, 367, 368, 372, 373, 374, 375, 376. 377, 379, 380. 382 Lee. C. M., 338,342,379,380 Lee, Y. T., 275, 300, 306,310, 314 Lehnen. A. P.,20,52 Leite, J. R. R., 91, 111 Lenz, W., 247. 312 LeRoy, J., 239, 312 LeRoy, R. J., 296.311 Lester, W. A., Jr.. 241, 242, 244, 245, 246, 247, 248, 250. 251. 252, 253, 255, 256, 257, 287, 311. 312. 313, 314 Levine, I. N., 399, 433 Levine, J., 38, 39, 40, 53,54 Levine, R. D., 204, 227, 228, 233. 234, 286, 309,312,313 Levy, B. R., 332,380 Levy, J. M., 93.96.99, 102, I10 Lew, H.. 390,432 Li, K. C.. 47,53 Liao, P. F., 74, 110 Lide, D. R., 399,413,418,433,434 Lidow, D. B., 216,226 Liepack. H., 25,53 Light, J., 232, 238,313 Light, J. C., 342,382 Lijnse, P. L., 203, 208, 227 Lilley, A. E., 395, 417, 433 Lin, B., 233, 312 Lin, M. C.. 203, 226

Lin, S. C., 362, 365,380 Linder, F., 277. 293, 294, 295, 302, 311, 313, 352, 355.37Y

Lindholm, E.. 64. 111 Lineberger, W. C.. 158, 227, 370, 374, 378, 380

Linke, R. A.. 407.409.435 Loesch, H. J.. 275,299,300. 304. 306,311,312 Lojko, M. S., 384,435 Lo Surdo, A,. 4,53 Lovas, F. J.. 384,401, 403,407, 409.410.421, 422,427,428,431. 432,433

Loy. M. M. T., 91, Ill Lukasik, J., 233, 3UY Lurio, A., 42, 43. 52, 53, 54 Lutz, B. L.. 403. 433 Lyons. J. D.. 343.344,348.351,363.380.381

M McAdam, K . B., 46,52,53. 164. 186, 190.225 McColm, D.. 42,53 McDdniel, E. W., 23. 53 McDowell, M. R. C.. 186, 188,226 McEachran, R., 416, 431 Macek, J., 118, 119, 122. 123, 124, 138, 140. 145, 158, 162, 167, 170, 172, 185, 188. 225, 227, 329,380 McGinn, G., 216,227 McGuire, P., 238. 251. 288, 289. 292, 293. 312 McKellar, A. R. W., 233. 310 MacLeod. J. M., 423, 431 McVoy, K., 329, 330. 33 I , 380 Madison, D. H..188, 227, 361, 380 Mahan. B. H., 277,310 Maki. A. G . . 423,433 Malthan, H., 279,309 Marcus. R. A., 251,252,310,312 Mariella, R. P., Jr., 279, 312 Marino, L. L.. 362. 365, 375. 381 Marliere. C., 401, 431 Marlow, W. C., 22,23,53 Marrus, R.. 42. 53 Massey, H. S. W.. 113, 175, 227, 236, 312. 317,321, 381 Matese, J. J., 369, 380 Mather, J., 409,435

445

AUTHOR INDEX

MatSUbdI'd, c., 17. 53 Mattick, A. T.. 86. 111 Matyugin, Y.A., 66, 74, 101, 110 Mazeau. J.. 352, 357, 380, 381 Meerts. W. L., 420, 433 Meisel, G., 44. 53 Mendez-Moreno, R. M., 188.226 Menzel, D. H., 396. 433 Meyer. Th. W., 233. 312 Meyer. T. W., 86, I l l Meyer, W., 4, 14, 15, 16,20, 32, 37, 51,53,54, 55,239. 241,312. 313 Mezger, P. G., 394,431 Micha, D. A.. 251,312 Michels. H. H., 343, 380 Mies, F. H.. 193, 227, 243. 256, 311, 312 Miller, D. R., 362, 365, 366, 367. 379 Miller, J. A,, 17, 53 Miller, S. L., 399,433 Miller, T. A,. 418, 431 Miller, T. M.. 18. 20.29. 30. 31. 32, 33, 34, 35, 36, 37, 38, 39, 42, 47, 53, 54, 165. 191, 192. 224,226. 362,376.380 Miller, W. H., 252. 312 Milne, T. A.. 28, 29, 47. 53 Mincer, T. R., 279, 313 Mitchell. J. F. B., 323, 361, 362, 379 Mittleman, M. H.. 218, 219, 222, 225, 227, 343,380 Miyaji, T., 424, 428,430 Miyazawa, K., 424,428,430 Mizushima, M., 384. 435 Mockler. R. C., 402. 409.433 Molof, R. W., 18.20, 29, 30,31, 33, 34,35, 36, 37, 38, 39, 42. 47, 53, 54 Monchick, L., 255,311 Montgomery, J. A.. 424, 432 Moore, E. A,, 401,433 Moore, J. H., 277. 288, 312 Moores, D. L., 186, 188, 192,227 Moran, M . M., 421, 432 Moran, T . F., 271,277,312,313 Morimoto. M., 414, 424,428, 430, 435 Morris, M., 409,424, 427, 429,430, 433, 435, 436 Moser, C. M., 348, 371, 374, 376. 381 Mott. N. F., 317. 321, 381 Miiller. E. W., 25, 52 Miiller, W.. 155, 226 Muenter. J. S., 407.410. 433, 434

Mukamel, S., 66, 110 Muntz, E. P., 279,313 Musher. J. I., 12, 13,54 Myers, P. C.. 393,433

N Nagane. K.. 414,424,428,431,433 Nauenberg, M.. 336, 381 Nelissen, L.. 46. 54 Nerf, R. B., 427,433 Nesbet, R. K., 315, 317. 329, 332, 333. 334, 339, 343, 344, 346, 347. 348. 349, 350, 35 I . 352. 354, 355. 356. 357, 358. 359, 360, 361. 363, 365, 366, 367, 368, 370, 371, 372, 374, 315,376,377,378,380,381, 382 Nethercot, A. H., 407,434 Newell, A. C., 22. 54 Newton, R. G., 317, 332,381 Neynaber, R. H., 362, 365,375,381 Nieh, S. T. K., 156,228 Nienhuis, G., 155, 225 Nikitin, E. E., 203,204, 224. 227. 239, 312 Norbeck, J. M., 242,312 Norcross, D. W., 3, 18, 19. 37. 54, 186, 192. 227,348.381

0

Oberoi, R. S., 329. 339, 343, 346. 347, 348, 351, 352, 354, 355. 356. 357. 359, 360. 361: 362,363,366.370,371.372.375,381,382 Ochs, G., 294, 312 Oka. T., 233,310,312,423,430, 433 Olsen, E. T., 430, 433 Olson, R. E., 3,53 O'Malley, T. F., 3,54, 332.381 O'Neil, S. V., 242. 309, 314 Oppen, G.. 43,53 Oppenheim, 1.. 233,312 Oppenheimer, M., 412,433 Orcutt, R. H.. 21, 54 Ormonde, S., 233, 312, 352. 355. 361. 369. 375, 377,379,381, 382 Ottinger. Ch., 234. 310 Ouw, D., 156,224 Ozier, I., 422, 423, 433

446

AUTHOR INDEX

P Pack, R. T.. 21.23, 51,54. 241.247, 300, 312 Palmer, P.. 395. 409, 424. 427, 428, 429, 430. 433, 434. 435, 436 Park. J . T., 20, 29. 30. 3 I 47,54 Parker, G. A,. 247. 300,312 Parker, W. H.. 391, 394,435 Parrano, C., 156, 200, 201,202,224 Parson, J. M., 275. 300, 310 Partridge, R. B.. 390,432 PaSCdk. J., 195, 227 Patel. C. K. N., 90, 96, I l l , 112 Patterson. G. D., 101, 112 Patterson, T. N. L., 416. 431 Paul, H., 218. 224, 227 Paul, W., 42.53. 303,309 Pauly, H., 115, 116. 152, 153, 168, 178. 179, 193, 196, 197. 198. 199. 200,224,225, 227, 231, 234, 235. 236. 259, 262. 275, 295, 296. 304,309,311.312,313 Payne, G . L., 339,381 Pekeris, C. L.. 396, 433 Penfield, H., 393, 422, 431, 432 Pengelly, R., 1 I, 52 Penselin, S., 390, 436 Penzias, A. A., 407. 409. 410, 411, 417, 423, 424,425.428,432,433,434,435,436 Percival, I. C., 157, 162, 173, 224, 227, 249, 313 Pestov, G., 62, 112 Peterson, J. D., 384, 435 Petrasso, R., 41, 54 Petty, F.. 271, 277,312,313 Pfanzagl, J., 264,313 Phelps. D. H., 421,434 Phillips, T. G., 409. 424, 434 Phillips, W. D., 203. 227 Pichanick, F. M. J., 355, 381 Pichou, F., 357,380, 381 Picque, J. L.,152, 156, 227 Piosczyk, B., 43.55 Pipkin, F. M.,391,432 Player, M. A., 39.40, 41, 54 Podolsky, B., 23,54 Poe, R. T., 350.379 Polanyi, J. C., 203, 226, 233, 234, 242, 279, 286,310, 312,313 Polder, D., 3, 52

Poletto, G.. 401. 410, 434 Poll, J. D., 47. 53 Pollack, E.. 33, 34, 35. 38,54 Pople, J. A., 241, 313 Porter, R. N.. 242, 30Y Powell, F. X.,399, 413, 434 Poynter, R. L., 418,430.434 Prangsma. G. J., 233,311 Pratto, M. R.. 384, 435 Preston, R. K.,242, 288, 289, 309, 311 Pritchard, D. E., 106,110. 116, 155, 168, 194. 195,203,225,227,309 Procaccia, I., 234, 313 Pu, R. T., 6, 12. 17, 52 Pukhov, A. A.. 66.86.90, I l l Purcell, E. M., 96, 97, 111

R Rabitr. H., 233. 245. 252. 313. 314 Racah, G . , 125,225 Radford. H. E., 394, 403, 421, 432, 434 Rae, A. I. M., 247,313 Raff, L. M., 242, 286.313 Raith, W., 42, 55 Ramsauer, G., 42.53 Ramsay, D. A., 404, 431 Ramsey, A. T., 41,54 Ramsey, N. F., 26, 46, 53, 54, 388, 389, 431, 436 Rangarajan, R., 242,313 Rank, D. M., 427,430,431 Rankin, C. C., 343,344, 348, 363.380 Rautian, S. G., 62, 65, 66, 77, 81, 86, 90. 103, 111,112 Raymer, M. G., 105, 111, 219,225 Raymonda, J. W., 410,434 Read, A. W., 233,313 Read, F. H., 335, 350. 355,356,358, 359, 360, 378, 379,380,381 Rebentrost, R., 248,313 Reed, K. A,, 303, 313 Refaey, K., 417,431 Reid, R. H. G., 161, 167, 168. 193, 196, 227, 228 Reiland, W., 179, 185, 186, 189,226 Reinhardt, J., 352,380

447

AUTHOR INDEX

Reinhardt, W. P., 19, 54 Reinsch. E. A,. 14, 15, 16, 32, 51. 54. 239,313 Renwanz, E.. 410,435 Reuss, J., 46. 54. 239, 303, 313 Rhodes, C. K., 86, 100. 110. 111, 233. 234. 309,312 Ribes, J. C., 428.432 Richards, D., 256.310 Richards, W. G., 401. 433 Richter, G., 218, 224 Ridgway. S. T., 423, 434 Rigutti. M.. 401. 410. 434 Risley, J. S., 368. 379 Roach, A. C., 242.310 Robaux. O., 86. 111 Robb, M. A,. 242.310 Robb, W. D., 18, 19. 32. 51, 52, 54, 317, 338, 341, 342, 347. 349. 371. 378. 379 Roberts, R. E., 20,54 Robertson, A. G., 350.379 Robinson. E. J.. 8, 27, 38. 39.40, 52,54 Robinson, H.G., 389.431 Robson, D., 343.380 Rohart, F., 401.431 Roizen, S., 156,227 Roothaan. C. C. J., 14,52 Rose. M. E., 119,227 Rosen. B., 384,434 Rosenberg. L., 3.54, 332,381 Rosenblum, B., 407,434 Ross, J., 232, 238. 261, 264, 311, 313 Ross, K. J.. 158, 226 Ross, U., 271, 277, 281, 297, 298, 299, 300, 309,310 Rost. K. A.. 117, 149, 150, 151. 203,205,206, 208,209. 210.226 Rothe, E. W., 362, 365. 375.381 Rountree, S. P.. 365. 366. 369, 380. 381 Rousseau, D. L., 101. 112 Rubin, K., 40, 54, 191, 227 Rudge. M. R. H., 317,381 Rudnitskii, G. M.. 415, 416, 434 Rudolph, K., 271, 277, 286. 288, 290, 292, 293.310.312,313 Rulis, A. M.. 236, 309 Rumble, J. R., 20, 32, 37,54 Russell, M. E., 417, 431 Ryan, J. A,, 403,433 Rydbeck, 0. E. H., 421,434

S Salomaa. R.. 68, 74. 11-7 Salop. A,. 33, 34. 35. 54 Samson, J.. 245,313 Sandars. P. G. H.. 39. 40. 41, 52, 54 Sanders, T. M.. 418.420.432 Sandle. W. J., 43. 54 Saraph. H. E.. 365, 367,381 Sasdki. F., 376. 381 Sastry, K. V. L. N.. 409.433 Satchler. G . R., 119, 120. 123, 159,224 Sathyamurthy, N., 242,313 Saykally. R. J.. 423. 425,434. 436 Saxena, K. M. S., 51.54 Saxon, R. P., 15,52 Scarl, E. A,, 401.421.434 Schabert. A,, 101, 111 Schaefer. H. F., 111, 242. 30Y, 314. 390. 434 Schaefer, J., 241, 245. 246, 251. 252, 253. 255. 256,257,285.287.312,313 Schiifer. F. P.. 156, 175.224.227 Schatz. G. C.. 236. 248.311 Schawlow, A. L.,44,53,93,94, 102,111,384, 403,405,418,426.435 Schetfers, H., 23. 24. 25. 54 Schieder. R., 148, 152, 227 Schinke, R.. 256.313 Schlessinger. L..339. 381 Schleusener, J., 275, 295, 296,309 Schmidt, H., 277, 293. 294,295,313 Schmidt, J., 91, 96, 102, 112 Schmieder, R. W., 42, 54 Schor, H., 302.311 Schuessler, H. A,, 389, 434 Schulman, J. M., 13.54 Schulz, G. J., 114.227. 318. 324. 355.381 Schulz, M.. 212. 226 Schwartz. C.. 20,54, 343. 381 Schwartz, H. L., 18. 20. 29. 30, 31. 32, 33. 34. 35. 36, 37, 38, 39, 40. 42. 46. 47. 53. 54 Schwartz, L..264. 313 Schwartz, P. R., 409,434 Schwinger. J., 338.381 Scoles, G., 236, 309 Scott, P. B., 279,313 Scully. M., 91, 96, 112 Scully. M. O., 91. 93. 111 Seaton, M . J., 317. 324, 338. 379. 381

448

AUTHOR INDEX

Seaton, M . S., 157, 173, 227 Secrest, D.. 242. 244, 251. 252, 260, 271, 310, 311.313 Seeger, R.. 241,313 Seiler, G. J., 347, 381 Serry, J. A., 203, 227 Shafer. R., 239,243,313 Shahin, I. S., 93, 94, 102, I!! Shalagin, A. M., 65, 66, 77, 86, 90, 103, 111, 112 Sharma, R. D., 91. I l l Sharp, J. M.. 335, 3x0 Shea, R. F., 91.93, I l l Sheffield. R. L., 91, 111 Shelton, W. N.. 188,227, 361,380 Sheorey. V. B.. 192. 277 Shimizu, F. 0..430,433 Shimizu, T., 430, 433 Shklovskii, 1. S., 388, 389, 434 Shoemaker, R. L.,91.93, 96, 111, 112 Shortley, G. H., 8, 52, 385. 387, 391, 392.431 Shouda. F.. 149,219,227 Shuler, K., 232,238.313 Shuler, K. E., 233. 312 Shuter, W. L. H., 417,434 Siegbahn. P., 302, 311 Silsbee. H. B., 390,431 Silver. D. M., 242. 313 Silverman, M., 391, 432 Simpson. J. A,, 355,381 Sims, J. S.. 20, 32, 37,54 Sinfailam, A. L., 349, 350, 351, 352, 353, 354, 355, 356, 357. 358, 360, 370, 372, 375, 378, 380,381 Siskind, B., 277,313 Slevin, J., 164. 186, 190, 225 Sloan. D. S., 417,434 g o a n , 1. H., 337,381 Sloan, J. J., 234,312 Sloanaker, R. M., 428. 432 Slusher, R. E.. 90.96, 111,112 Smirnov. G. 1.. 65, 112 Smith, A. M., 404.434 Smith, A. V., 156,228 Smith, E. R.. 365, 366,369,381 Smith, E. W.. 62. 111, 112 Smith, G. K.. 204,225 Smith. I. W. M., 203, 226, 233, 239, 240. 313 Smith, K., 317, 324, 338, 362, 369. 370, 375, 377.381.382

Smith, K. M., 236,309 Smith, P. W., 67, 101 I12 Snow, R. L.. 247. 300,312 Snow, T. P., 404.434 Snyder, L. E., 409, 410, 423, 424. 425, 427, 428,431,434,436 Sobelman, I. I., 62. 67, 81. 86. 110. I12 Solomon, P. M., 409, 423,424, 425,433,434 Somerville, W. B., 383, 404, 412, 413, 416. 417, 418, 420, 421, 426, 431, 432. 434, 435 Spence, D., 368,369, 382 Spitzer, L., 388,404, 415. 435 Spruch. L.,3.54, 332, 348.3#0,38/ Srivastava, R. P.. 105. I12 Staemmler, V., 241. 242,245,246,312, 313 Stamatovic, A. S., 155. 226 StamatoviC, A., 179. 185. 186. 189, 226 Standage, M. C., 43, 54, 161, 164, 190, 225, 226,227 Stark, J., 4, 23, 24, 25, 54 Stasinka, G., 415, 43/ Stebbings, R. T., 43,54 Stecher, T. P., 404,417,434,435 Stegun, I. A., 272, 308 Steinfeld, J. I.. 233, 234, 311,313 Stenholm, S., 68, 74, 89, 91. 111, 112 Stephen, M. J., 91, 96, 112 Stern, O., 23, 52 Sternheimer, R., 16, 18, 19, 37,54 Stevens, R. M., 242,313 Stevens, W. J., 13, 14.54 Stewart, R. F., 32, 54 Stockdale, J., 40,54 Stogryn, A. P., 303, 313 Stogryn, D. E., 303,313 Stoll, W., 116, 118, 131, 142, 143, 155, 178, 179, 185, 186,226 Stone, E. J., 369,382 Storer, K. E., 77,94, 111 Strandberg, M. W. P., 415.435 Stroke, H. H., 152, 156,227, 228 Stroud, C. R.. 149.219.227 Stwalley, W. C., 2. 11, 32, 47.53, 54 Sultan, G., 43, 52 Sume, A., 421,434 Sunshine, G., 362, 365, 382 Svanberg, S., 43,44,53 Syunyaev, R. A., 388,435 Szabo, A., 1 I , 20, 32, 37, 51 Szanto, P. G., 423,425,434,436 ~

AUTHOR INDEX Sz0ke.A.. 101, 105, 111, 219,225

T Tabara, H., 424,428.430 Taguchi, R. T., 203, 226 Takagi, K., 414, 424, 433 Takeo, M., 59,111 Tambe, B. R., 365, 366, 367, 374,382 Tang, K. T., 242,244, 312,313 Tarr, S. M.. 245,313 Taylor, B. N., 386, 391, 394, 431,435 Taylor, D. J., 156, 228 Taylor, H. S., 317, 318, 324. 325, 327, 338, 361, 366,379,380. 382 Teachout, R. R., 21, 23, 51.54 Teitelbaum, H., 233, 310 Teloy, E., 294, 312 Temkin, A., 317, 337, 350, 362, 37Y, 382 Testard, 0.. 156, 227 Thaddeus, P., 255, 302, 311, 409, 410, 411, 415,423, 424.427, 428.431. 432,435, 436 Thomas, L., 299,314 Thomas, L. D., 317, 338, 343. 348, 361, 362, 363. 365, 366, 367. 368, 370. 371, 372. 375, 376,377,378,379,381,382 Thomas, R. G., 338, 341,380 Thompson, D. G., 350,382 Thomson, R., 41 I , 435 Thorhallsson, J.. 51, 54 Thornton, D. D., 427, 430,431 Tiemann, E.,384,401,410,412,414,421.422, 432,433,435 Tinkham, M., 415,435 Tischer, H., 179, 225 Toennies, J. P., 115, 193, 227, 231, 233, 234, 236. 238, 244, 246, 247, 251, 252. 254. 255, 256. 259. 262, 271, 277, 279. 281. 286. 287, 288,290, 292,293, 297, 298, 299, 300, 303, 304, 309, 310, 312, 313, 314 Tomasevich, G. R., 427,435 Tomlinson, W. J., 96, I l l Ton-That, D.. 356,380 Torres. B., 361, 382 Torres, B. W., 369, 375, 377,381 Torrey. H. C . , 91, 112 Torring, T., 410, 432, 435 Toschek, P., 101, 111 Toschek, P. E., 65.68. 69, 74. 101

449

Townes, C. H., 384. 390, 399, 403. 405, 407, 418.420,426,427. 430. 431, 432,435 Trajmar, S.. 361,382 Treffers. R. R.. 404, 432 Trefler. M.. 415,435 Tripathy, D. N., 338, 37Y Truhlar. D. G., 233, 244, 252, 256. 309. 310, 343. 346. 361,382 Trujillo, S. M.. 362, 365, 375. 381 Tsapline, B., 242, 243, 314 Tsou, L. Y., 304. 314 Tuan, D., F.-t.. 12, 13. 55 Tuck, D. G., 23. 51.55 Tucker. K. D., 423,424.427.435 Tully, F. P., 306, 314 J. c.,288,289,311 TUIIY, Turner, B. E., 401, 409, 410, 411, 418, 421. 423, 424. 427. 429, 430, 433, 435. 436 Turner, G. S., 277, 312

U Udseth, H., 277. 303, 310 Udseth, M., 271. 277. 293. 294, 295. 314 Uhlenbeck, G. E., 123, 225 Ulich, B. L., 425, 435

V

V. Oppen, G., 43.55 van den Bergh. H. E., 271,277,281,314 van den Meijdenberg, C. J. N., 236,3OY Vandeplanque. J., 195. 227 van der Avoird, A., 242. 243,310 van de Ree, J., 296,314 Van der Wiel, M. J., 214, 215. 226 van Kranendonk, J., 239, 312 van Montfort, J. Th.. 296, 314 van Raan, A. F. J., 42,55 van Regemorter, H., 317, 348. 366, 368. 372. 373. 376,380,382 Van Till, H., 409. 431 Varshalovich, D. A,. 418, 431 Verter. M. R., 233. 314 Vetter, R., 86, 111 Viala, Y. P., 416. 435 Vialle, I. L.. 152, 227

450

AUTHOR INDEX

Vilaseca, R., 233, 309 Villarejo, D., 417, 435 Vo Ky Lan, 323, 348,366,367. 368. 372, 373, 314, 315, 376, 311,379,380,382 von Seggern, M., 251,255,310,314

W Wacker, P. F., 384, 435 Wagner, A., 242,311 Wahl, A. C., 242,311 Walmsley, C. M., 416,435 Walther, H., 129. 148, 152,227,228 Wang, C. H., 96, 111 Wanner, J., 234,312 Wannier, P. G., 407,409,424,432, 434,435 Ward. J. F., 156,228 Warner, J., 156, 200. 201, 202, 224 Warnock, T. T., 262,3/4 Warrington, D. M.. 43. 54 Watson, J. K. G., 430,433 Watson, K. M., 216, 226,250,311 Watson, W. D., 404, 412, 428, 431,435 Way, K . A., 203,227 Weaver, H., 418,432 Webb. T.G., 360, 362,379 Wefel, W., 43, 53 Weinberger, D. A,, 423,434 Weingartshofer. A,, 213, 214,228 Weinhold, F., 19, 20, 52, 55 Weinreb, S., 388,435 Weis, G. H., 233, 312 Weitz, E., 233. 314 Welch, W. J.. 427,428.430, 431, 435 Wells, C. A,, 256, 311 Werner, H. J., 4. 14, 15, 20, 32, 37, 51, 55 Weston, R. E., 203,224 Wharton, L.. 47.52, 303. 304,313,314 White, D. R., 22, 51 White, J. C., 216, 225,226 Wichmann, E., 357,382 Wick, G. C., 250,311,322. 380 Wiersma. D. A., 90, 110 Wigner, E. P., 332, 338,382 Wild, J. P., 388, 391, 393, 435 Williams, D. A,, 417, 435 Williams, P. F., 101, 112

Williams, W., 361, 382 Willmann, K., 213,228 Wilson, A. D., 204,228 Wilson, C. W., Jr., 242,244, 314 Wilson, R. W., 407, 409, 410, 411, 423, 424, 425,428, 432,433, 434,435, 436 Wilson, W. J., 409.434 Wineland, D. J., 388, 436 Winnewisser, 9.P., 384,409,436 Winnewisser, G., 384,409, 436 Winnewisser, M., 384,409,433,436 Winter, T. G., 286, 313 Winters, K. H., 188, 226 Wofsy, S.,193, 228 Wojslaw, R. S., 423, 434 Wolber, G., 390, 436 Wolfgang, R., 288, 289, 311 Wolniewicz, L., 21, 46, 53 Wong, J., 131, 156, 225, 228 Woodall, K. B., 286, 313 Woods, R. C., 412, 423, 425, 431, 432, 434, 436 Wormer, P. E. S., 242, 243, 310 Woste, L., 152, 227 Wu, F. Y., 219,228

Y

Yao, Y. T., 42,55 Yaris, R., 317, 338, 379 Yarkony, D. R., 242, 314 Yarlagadda, 9.S., 317, 338, 361, 366,382 Yates, A. C., 242,314, 343,344, 348,363,380 Yellin, J.. 42, 53 Yoshimine, M., 12, 18.55, 316, 381 Young, J . F., 216,225,226

2

Zaidi, H. R., 62, 105, 112 Zarur, G., 233,313 Zecca, A., 350,380 Zipf, E. C., 369,382 Zorn, J. C., 25, 26, 21, 28, 29, 37, 52, 53, 55 Zuckerman, B., 409, 421, 422, 424, 427, 428, 429,430,433,434,435,436 Zvijac, D. V., 342, 382

B

A

Adiabatic collisions. 236 Algebraic close-coupling method, 347 Alkali halides molecular properties of, 303 scattering from. 303-307 Alkali metal atoms. polarizability values for. 37 Alkaline earth polarizability values, comparison of. 32 Ammonia, inversion in. 428-430 Atom(s) complex, see Complex atoms core polarizdbility of, 3 electric dipole polarizabihty of, 1 4 excitation of. see Atom excitation ground-state detection of by optical pumping, 148-151 Atom-atom scattering experiments, 193 Atom excitation. by laser optical pumping, 129-157 s w rrlso Laser optical pumping Atomic beam resonance experiments, 40 Atomic beam requirements, in crossed-beam collision experiments. 175-1 76 Atomic beam technique. 23-24 Atomic charge distribution, hyperpolarizability of, 5 Atomic fine structure. in interstellar microwave transitions. 390-394 Atomic hyperfine structure, in interstellar microwave transitions, 387--390 Atomic polarizability. in collision phenomena, 3 see u/so Polarizdbility Atomic scattering see NISO Scattering and coherent superposition of ground and laser-excited states in resonant atom excitation. 2 16-253 elastic atomic-exci led. 193-200 free-lree transitions and, 21 2-2 16 in presence of strong laser fields, 21 1-223 Atomic-molecular potentials, anisotropy and, 238-239 Autoionizing states. excitation and, 335-337

45 1

Beam experiments, limitations to. 58 Beam techniques atomic beam resonance experiments and. 40-4 1 N Y U electric deflection experiments and. 29-40 Bethe-Goldstone approximation. 349-350. 363, 37 1-372 Bethe-Goldstone structural algorithm. 364-366 Born approximation. in scattering multipole moments, 186. 190 Brueckner-Goldstone many-body theory. I7 Bulb-type collision studies, potential of, 59-60 Bulk experiments. in polarizability experimental measurements. 21-23 Bulk relaxation experiments. 232-233

C

Carbon. in interstellar microwave transitions. 389 Carbon scattering. in low-energy electron scattering applications, 370-374 Cdsimir-Polder effect, 2 Cesium, polarizability of. 3 Circularly polarized excitation, in inelastic electron scattering processes, 189-191 Circularly polarized light in atom-atom scattering, 200 in laser optical pumping, 139-140 scattering multipole moments and, 171I72 Clausius-Mosotti relation, 2 Close-coupling expansion, in inelastic scattering. 3 I7 Close-coupling method. in low-energy electron scattering. 337. 347 CM (center of mass) backward scattering. 280 CM cross sections, 264 CM scattering angle, 262, 265, 268, 275-280. 306

452

SUBJECT INDEX

CM system, 259 -262. 285-286. 298. 304 CM vector, 272 273 Collimation, in crossed-beam collision experiments. I75 Collision(s). 60.90 adiabatic, 61. 236 binary collision npproximation in, 61 collisional time rate of change oC density matrix in. 62 63 degenerate levels in. I05 I06 diabatic, 236 elastic. 103 105 free induction decay and. 91 -93 inelastic. 105 106 line shape and. 60-61 nondiabatic. 236 nondegenerate levels in. 103 - 105 off-diagonal density matrix elements and, I06 phase interrupting. 97 in strong laser fields, 21 I 223 in three-level systems, 65 -90 transient systems and. 90-99 two-level systems in, 89-90 two-pulse nutation in. 93-96 velocity-changing. 98-99 Collision broadening, in beam experiments. 58 Collision equation. analysis of, 63-65 Collision experiments crossed-beam, ser Crossed-beam collision experiments with laser-excited atoms i n crossed beams. 113-223 Collision phenomena. atomic polarizability in, 3 see ubo Polarizability Collision process. laser field and, 2 I 1 2 I2 Collision studies bulb type, scc Bulb-type collision studies delayed saturated absorption in. 93 96 experiment survey and. 100-106 photon echoes in, 96-99 steady-state experiments in, 100-I02 theoretical outlook for, 100-106 transients in, 102 Complex atoms. low-energy electron scattering by. 315-378 see crbo Low-energy electron scattering Core polarizability, defined. 3 .si’t’ d v o Polarizability Core polarization. defined. I5 ~

Crossed-beam collision experiments. I I3 117, 174-210 and inelastic electron-scattering processes from sodium in 32P, state, 179- 191 with laser-excited atoms. I13 223 radiation trapping and. 175-176 scattering geometry in. 176 179 Crossed beams. use ol; 58 Cusp erects. theory of, 330 335. 355

D DECENT model, in inelastic scattering, 252-253.287.294-295 Delayed saturated absorption. 93-96 Density matrix. collisional time rate of change of. 62-63 Deuterium fine structure of, 393-394 i n interstellar microwave transitons. 3 X X 389 Dtabatic collisions. 236 Diatomic molecules homonuclear molecules and. 413-417 hyperfine structure of, 402 403 individual sigma molecules and, 406-41 3 in interstellar space, 403 422 rotation in, 398-401 rotational transitions in. 4 0 5 4 0 6 structure of, 397 403 transitions in, 403-422 Distorted wave polarized orbital method. 186- 189 Doppler pumping, multidirectional, I55 Doppler-shifted fluorescence detector. 149151 DWPO (distorted wave-polarized orbital) method. 186-189

E E-H gradient balance method. in polarizability measurements, 2 9 4 0 Eigenchannels, 321 Eigenphases, 321 Elastic atom-atom scattering, 198-200 Elastic atom-excited scattering, at thermal energies, 193-200 Elastic scattering, of sodium 3’P,,, from

45 3

SUBJECT INDEX

mercury, I96 198 Electric deflection experiments, 24-29 Electric dipole moment. defined. 5 Electron collisions. inelastic and superelastic, 163- I65 .si'c* d s o Collision(s) Electronic-to-vibrational energy transfer. polarization effects in, 208 Electronic states. symmetry properties of. 398 Electron scattering, 3 I5--318 S P C NISOScattering inelastic. si'e Inelastic scattering with spin analysis, 165-167 Electron spin, 157 161 Electron spin analysis, 165- 167 Energy change experiments alkali halides and, 304-307 center ol' mass transformation lo laboratory system in, 261-262 design considerations in, 259-274 determination of experimental errors in, 272-274 evaluation of energy loss spectra in, 270272 nonidealized energy change apparatus in. 262-265 scattered beam intensities in. 265 270 velocity changes in. 261 Energy change method, in inelastic scattering measurements. 257 Energy-dependent collision times, vs. characteristic rotational and vibrational motions. 237 Energy loss spectra. evaluation of, 270-272 Energy transfer electronic to vibrational, 203-2 10 methods Tor studying. 231-234 Energy transfer spectra. 203-208 Even-scattering multipole moments, 161 Excitation molecules. scattering studies of. 229-308 Excited-state number density. examples of. 142-144 Excited states, in polarizability measurements. 42 46 ~

F

FID (Tree induction decay), 91 -93

Fine-structure-changing transitions in scattering inultipole moments. I67 I69 in heavy-particle.collisions.200.203 Fluorescence beam broadening. 154 Free-free transitions, in atomic scattering. 212 Free induction decay. 9 I 93

G Green's function theory. 31 7 Ground state preparation and detection of atoms in. 148- 15 I superposition of with laser-excited stales. 216 222 Ground-state scattering processcs. for sodium. 179 1x2

H

Hartree-Fock approximation. in perturbntion theory. 13 14 Hartree-Fock coupled method. I7 Hartree-Fock potential and wuvefunctions. 17 Hartrec perturbation equations. I6 Heavier lineiir molecules. scattering from. 1 9 6 303 Heavy particle collisions. fine-structurechanging transitions in, 200-203 . s i v (//.so Collision(s) Heavy-particle scattering. "elostic" theory of, 223 Helium tine structure of. 394 in interstellar microwave transitions, 3x9 Helium scattering, 316. 349-355 intermediate energies and. 360 362 threshold structures and. 355 360 Hilbert space wavefunction, 320 Hindered motion. 429 Honionuclear molecules hypertine transitions in, 416 417 rolational transitions in, 413-416 Hydrogen-ion collisions. vibrational excitation in, 302 Hydrogen ion-hydrogen molecule system, inelastic scattering and, 287- 295

454

SUBJECT INDEX

Hydrogen molecules, scattering from, 280 296 sw rrlso Scattering Hyperpolarizability. defined. 5 Hypersurfaces, potential. .sw Potential hypersurfaces

recombination lines in. 394397 Inversion splitting. 429

J

Jost function. 330 I

K Inelastic electron collisions. 179 Inelastic electron-scattering processes .sci’ d v o Inelastic scattering circularly polarized excitation and. 189191

and experiments with 3p -4 3d excitation, 188- 189

Inelastic scattering ilppMdtUS and energy loss spectra in. 179 I 82 approximate methods in, 252-254 close-coupling expansion in. 3 17 computational results in. 255 ~257 energy change method in, 257 experimental methods in. 257-274 of H + + H, system. 287-295 of HD-Ne, 295- 296 for linear triatomic molecules, 299 303 of lithium-hydrogen system, 280-287 of lithium-N,/CO system. 297-299 measurement of cross section in, 257 quantum mechanical “close coupling” method in. 248-252 “rainbow“ in, 235 recent experimental results in. 274-306 simple models of. 234 238 survey of experiments in, 274-280 theory of. 248-257 Infinite polarizability. 20-21 Interatomic potential. studies of. 193 Interstellar atoms and molecules, scr Interstellar microwtive transitions Interstellar microwave transitions. 383 -430 atomic fine structure and. 390 394 atomic hyperfine structure in, 387 -390 deuterium spin and. 388-389 diatomic molecular structure and. 397-403 i n diatomic molecules. 403-422 helium and, 389 hyperfine structure in, 402403 lambda doubling in, 401-402

Keilson-Storer kernel. 103

L Lambda doubling. 401402,418-420 Langevin capture cross section, 2 “Large shift” method. in electric-deflection experiments. 27 Laser-excited atoms basic theory of, 1 17- I28 collision experiments with. I13~-223 scattering experiment measurements and. 157-174 multipole moments and, 119-128 scattering by, 117-118. 157-174 scattering multipole moments and, 162174 Laser-excited beams, crossed-beam collision experiments and. 174-210 Laser-excited state, superposition of with ground state. 216-222 Laser fields atomic scatlering in presence of, 21 1-223 collision process and. 2 I 1 - 21 2 Laser optical pumping atom beam deflection by, 152-154 atom excitation by, 129-157 excited-state number density and, 142- 144 experimental aspects of. 142 -157 preparation and detection of atoms in ground state by, 148 151 rate equations and, 129- 133, 137-139 stationary condition in, 139-141 transition probabilities and. 134 Laser spectroscopy scc~N/SO Three-level collision systems collision studies with. 57- I10 differential scattering cross-sectional information from, 59

455

SUBJECT INDEX

transient systems in collision studies using. 90-99

Lester SCF potential. 287 Linearly polarized light in laser optical pumping. 140 multipole moments and. 161 scattering and. 172~174,182-186 Linear polyatomic molecules. 423 425 Linear triatomic moleculcs. inelastic scattering liom. 299-303 Line shape, in three-level collision systems. 67-76, 80 88 Lithium collisions. vibrational excitation in. 300 Lithium-hydrogen systems, inelastic scattering from, 280-287 Low-energy electron scattering applications of. 349-378 carbon and, 370-374 comparison of methods in. 346-349 cross sections in, 321-323 excitation or autoionizing states in. 335 337 helium applications in, 349 362 matrix variational method in. 343-346 methods in. 337-349 nitrogen and. 374-378 oxygen applications in. 362-374 polarization potentials and pseudostates in. 323-324 scattering resonances in, 3 2 4 3 2 9 for sodium 32P3,2,191-193 theory of. 31 8-337 threshold effects in. 330-335

M

Matrix elements, polarizability 01; 6-10 Matrix variational method, in low-energy electron scattering, 337-338. 343-346 Mean energy transl'er, defined, 272 Mercury, elastic scattering of sodium 3*P,,, l'rom. 196- I98 Microwave transitions. interstellar, see Interstellar microwave transitions Molecule( s) diatomic, .we Diatomic molecules electric dipole polarkability of. 1-4 rotational and vibrational excitation of. 229-308

Molecular polnrizability. for simple diatomic molecules, 46-47 , s w ((/.yo Pohrizdbility Monte Carlo traiectory. calculation ol: 287 Multichannel threshold theor ,330-335.358 Multipole moments examples of, 124- 128 I'rame transformation and. 21-124 lenguage of. 119-128 measurement ol: 144-148 optically excited. 144- 14X real. 121 124 spin polarization and. 124- I25 time developmcnt of. 141-142

N

Nd: YAG laser. optical parametric oscillator pumped with, 156 Neon. elastic scattering of sodium 3'P, rrom. 194-196 New York University, electric-deflection experiments at, 29 40 Nitrogen scattering. in low-energy electron scattering applications. 374-378 Nondiabatic collisions. 236 Nuclear spin uncoupling. 157-161

0 Optically excited multipole moments. measurements or. 144- 148 Optical parametric oscillator, N d : YAG laser and. 156 Optical pumping, sw Laser optical pumping Oscillator strengths. in polarizability calculations, 10-1 I Oxygen, electron impact excitation cross sections for. 369 Oxygen scattering, low-energy. 362-374

P

Percival-Seaton hypothesis. I57 161 Photon echoes. in collision studies, 96 99 2 l l molecules. transitions in, 41 8-422 PNO-CEPA (pseudo-natural-orbital coupled-electron-pair approximation). I4

456

SlJHJECT INDEX

PNO-CI (pseudo-notural-orbital conligurat i o n interaction). 14 Polarizability experimental. 21-47 future possibilities for. 47 51 infinite. 20- 21 intrinsic. I 7 matrix elements in, 6 10 nioinents in. 5 6 as second-rank tensor. 5 Polarizebility calculations. 10 21 eHective quantum numbers in, I I liwmula for. I 2 oscillator strengths and. 10 I I perturbation theory ;md. I 2 1 8 R-matrix calculiitions and. 18-19 statistical ciilculations and. 20- 21 valence electron polurizability and. 19 variational calculiitions and. 19-20 Polariiability meiisurenients. 2 I 4 7 utoniic beam resonance experiments and. 40 41

beam techniques in. 23 40 bulk experiments und. 2 I 23 E-H gradient balance method in. 29-40 electron deflection experiments in. 2 4 29 ~ excited states nnd. 42-46 indirect methods in. 23 N Y U experiments in. 29 -40 Rydberg states and. 43 -46 shock tube method in. 22-23 simple molecules and. 46 47 Polarization core. .w Core polarization of exciting light i n elastic atom-excited atom scattering. 198 200 Polarization erects, in clectronic to vibrational energy transfer. 208- 21 0 Polarization potentials, in low-energy electron scattering. 323-324 Polari7ed r'ro7en-core model. 348 Polarized light. in laser optical pumping. 139- 140 Polarized niolecular beams, scattering measurenients with. 239 Polerized orbital method. 337 Polyatomic molecules asymmetric top. 425-428 linear. 423-425 rotation in, 422-428 symmetric top. 425 ~

Postcollision internction, 33.5 Poten t ia I hyper sti r liices. 2 38 247 a b initio quentum chemical calculations for, 239 243 approximate methods in. 246~-247 potential models ol: 243-246 Potentials. bulk relaxation times and. 232 Pressure broadening. in beam experiments, 58

Pscud6-natural-orbital configuration interaction. 14 Pseudo-natural-orbitel coupled-electron-pair approximation. 14 Pseudostates. in low-energy electron scattering. 323-324

Q Quadratic Stark etkct. 4 Quantum chemical calculations. f o r potential hyper-siirl'aces. 239 -243 Quantum numbers. i n polarizability calculations. I I

R Radiation trapping. i n crossed-beurn collision experiments. I75 I76 Radio recombination lines. in interstellar microwave transitions. 394 397 Rainbow. i n inelastic scattering. 235 Rate equation. evaluation of in laser opticnl pumping. 137-139 Rayleigh scattering cross section, 2 Recoil techniques, in low-energy electron scattering for sodium. 191 -193 Recombination lines, in interstellar microwave transitions, 394-397 R-matrix method, in low-energy electron scattering. 338-343 Rotating wave approximation. i n nearresonant excitation. 218 219 Rotation i n diatomic molecules. 398 401 in polyatomic molecules. 422 428 Rotntional excitation in perturbation theory. 230 i n thermal energy scattering of atoms, 299 -300 ~

457

SUBJECT INDEX

Rotational transitions. in diatomic mole. cules. 405406 Rydberg states. measurements involving. 43 46

S Scattering for alkali halides, 303-307 electron, si'e Electron scattering Tor heavier linear molecules, 296-303 helium. s e e Helium scattering from hydrogen molecules. 280-296 inelastic. sei' Inelastic scattering low-energy. see Low-energy electron scattering multipole moments and, 119-128 polarized molecular beams in, 239 in rotational and vibrational excitation molecules, 229- 308 Scattering experiments atom-atom. 193 measurements in. 157-174 Scattering geometry, in crossed-beam collision experiments, 176-1 79 Scattering intensity. linear polarization of. 182- I86 Scattering multipole moments Born approximation in. 169- 171 circularly polarized light and, 171-172 determination ofq 171-174 tine-structure-changing transitions in. 167 169 linearly polarized light and, 161. 172-174 scattering amplitudes in, 162 174 for 3p .+ ns transitions, 186- 188 Scattering resonance, i n low-energy electron scattering, 325-329 Schrodinger equation. 320 in three-level collision states. 67 Schrodinger wavefunction. tbr electron scattering. 318 Short-lived states, lower-lying. 4 2 4 3 Sigma molecules. 4 0 6 4 1 3 "Small shift" method. in electric deflection experiments, 25 Sodium ground-state scattering processes for, 180I81 inelastic electron-scattering processes for,

179 191 Rydberg levels of. 45 Sodium 3'P, , elastic scattering of from neon. 194-196 Sodium atom scattering, analysis of. 204 Specific collision model. i n three-level collision system. 76 80 Spectroscopic formulas. 385-387 Spin dnalysis, in electron scattering. 165- 167 Stark effect. 4 Symmetric. rotor. 422 Symmetric top polyatomic moleculcs. 425 Symmetry properties of electronic states, 398

,

T Tensor polarizability. 39 Thermal energies. elastic atom-excited atom scattering at. 193-200 Three-level collision systems, 65-90 large detuning in, 72-73 line shape in. 67-74, 76-80 line shape analysis in. 80- 88 near-resonant tuning in. 73 74 specific collision model in, 76 -80 Time-dependent coupling system, 219-220 Time-of-flight analysis, in hydrogen molecule scattering. 281-287 Time-of-flight apparatus, in Li' - H, inelastic scattering studies. 28 1 - 3 2 Time-of-flight distribution, for lithiumnitrogen scattering, 297 Time-of-flight spectra. 271.274.290, 301 TOF analysis. scc Time-of-flight analysis Transient systems, in collision studies using laser spectroscopy. 90-99 Two-level collision systems. 89-90 Two-pulse nutation. in collision studies. 93-96

V

Van der Waals constant. 2 Variational calculations. in polarizability calculations. 19-20 Verdet constant, 2 Vibrational energy transfer. 203-208

458

SUBJECT INDEX

Vibrational excitation in H + collisions. 302 303 in Lit collisions, 300-302 Monte Carlo calculations in. 207 Vibrational inversion splitting. 429 Vibronic transitions. transition probabilities and, 339

W Wave [unction, structure or. 3 I 8 -320 Wignrr cusp, 332

X X-ogen, 423

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 . Bratmien 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 . Andrrs, and J . B. Fenn AUTHORINDEX-SUBJECT INDEX

The Quanta1 Calculation of Photoionization Cross Sections, A . L. Steivart Radiofrequency Spectroscopy of Stored Ions. I : Storage, H . G. Dehnielt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, F. Cliunoch 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 . Dalgarnoand W .D . Davison Thermal Diffusion in Gases, E. A . Mason, R . J . M u m . and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W . R. S. Gurtoti The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A . R . Samson The Theory of Electron-Atom Collisions, R . Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer Mass Spectrometry of Free Radicals, S . N . Foner AUTHORINDEX-SUBJECTINDEX 459

Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E. H . S . Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D . R . Bates und R . H . G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A . Buckingham und E . Gal Positrons and Positronium in Gases, P. A . Fruser Classical Theory of Atomic Scattering, A . Burgess and I. C . Percival Born Expansions, A . R. Holt und B..L. MoiseinVfsck Resonances in Electron Scattering by Atoms and Molecules, P. G . Burke

460

CONTENTS OF PREVIOUS VOLUMES

Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W . 0. Heddle and R . G . W . Keesing Some New Experimental Methods in Collision Physics, R. F. Stehbings Atomic Collision Processes in Gaseous Nebulae, M . J . Seaton Collisions in the Ionosphere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F. Boyd AUTHORINDEX-SUBJECT INDEX

Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A . S. KauJinan The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takajjanugi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A . Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenherg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Batesantf A . E. Kingston AUTHORINDEX-SUBJECTINDEX

Volume 5

Physics of the Hydrogen Maser. C . Autloin. J. P. Schermann, and P. Grivet Molecular Wave Functions : Calculation and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, H r m l Weinstein, Ruben Pauncz, 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, Nathan Wiser and A . J . Greenjrld AUTHORINDEX-SUBJECTINDEX

Flowing Afterglow Measurements of Ion-Neutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A . L. Schnieltekopf Experiments with Merging Beams, Roy H. Neynaher Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The ClassicalOscillator 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’s‘”pP4,C. D. H . Chisholm. A . Dalgarno, and F. R. Innes Relativistic 2 Dependent Corrections to Atomic Energy Levels, Holly Tlzomis Doyle AUTHORINDEX-SUBJECTINDEX Volume 6 Dissociative Recombination, J. N . Bards1e.v anti M . A. Bionrli

Volume. 7

Volume 8

Interstellar Molecules: Their Formation and Destruction, D. MrNally

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 Moleculr Beams, R. B. Cairns, Hulstead 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 Low Energy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C . Zorn and Thomas C.:English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McEIroy AUTHORINDEX-SUBJECTINDEX

Volume 10

A B C D F G J

Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong. Jr. and Serge Feneuille The First Born Approximation, K. L. Bell and A. E. Kingston 7 Photoelectron Spectroscopy, W . C . Price 8 9 Dye Lasers in Atomic Spectroscopy, W . Lunge, J. Luther, and A . Steudel 1 2 Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, s B. C. Fawcett

9

46 1

A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr. SUBJECTINDEX

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 Processes, P. G. Burke and w.D. Robb Role o f 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. Griem Chemiluminescence in Gases, M . F. Golde and B. A. Thrush AUTHORINDEX-SUBJECTINDEX

Volume 12

Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Buuche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M . Broyer. C. Gouedard, J. C. Lehmann, andJ. Vigub Highly Ionized Ions, Ivan A . Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid AUTHORINDEX-SUBJECTINDEX

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