Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations*

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

LOW-ENERGY ELECTRON SCATTERING

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

3 18

R. K. Nesbet

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
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R . K. Nrsbet

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

Here, and

IW

L )

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

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

1

da -

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/,+ P

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

(26)

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 (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 r$ /r$+

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) where

6 ( E ) = do@)

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) where

a/a, = ( E

+ q ) y (1 +

E2)

(31)

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 (36)

c,= ( E - E , ) - ' ( a l H l u )

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,

where A, = - [a I ( H

- E,)G(H -

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

E,) I a]

I H Iw o )

(431

(44) (45)

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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

328

R. K . Nesbet

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

where M denotes H multichannel case,

c 4

- E,

M(Oq,B’,

c Mu,

u(E - Ea - A,)-

(0p)P

P

(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

in terms of a total width

ragiven by

c

M(Oq,,. u

4

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

2

(55)

2

p ’ 4

(56)

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

n

E - E,, = - rbl y$ cot[S;(E) 2 p=1 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 0.43

0.44

33s 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

+

This implies

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

- 2kB

k’ - (a’

+ p’)

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

r = 2k,p,

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 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 = AMM,

KMn

= K n M = AMnklI2,

= 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), KMM

K M M

= AMM - AMnAnM

+0(KZ)

Knn

(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

k=O

=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 kl

+ O(k2)]

(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

r the width of the compound state, and

(94) (95)

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

where q is real, and &

+ 4)/(&+ 4

= 2(E - El

- Eth)/r

(99) (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 =

’0

1

.m

‘ro

-*(4 64’ - 4’ 6$)ro

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

(106)

(107)

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

1

.m

’ ro

- E)+

d r +.6c$(ro)4’(ro)

6u(H - E)u dr - 6u(ro)u’(ro)

(108)

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 )

Ppq(r)

=

= P(rob'b.0)

1

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 =

Ca Cb Ca Qab Cb - 5 Ca Ca 4'Aro )u'(ro) 1

(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)

R. K. Nesbet

342

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

and Eq. (141) becomes

6ao = 0,

(142)

6a, = 6 K

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

or

+ mllK, = 0

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)

346

R. K. Nesbet

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.

R . K . Neshet

358 x10-19 1.5 c L

-6

N

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

E (eV)

;

8

9

I0

1'1

12

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

10.0 -

I

-

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.

370

R . K . Nesbet

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 -

60 -

G\

Single Configuration

2

NO

40-

CI

2o

-

I

-\\ A

Henry

A a .

0

0

MCC

/C'

2

..1 . . 1

+I

A

D

+,ZP

4

A-

*

+,I~PI

-

,

6 EkVI

1' ,

: f. ,

a

I

..,

c _ _

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

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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.

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