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Eikonal theory of electron-atom collisions
COMPUTER PHYSICS COMMUNICATIONS 6(1974) 358—371. ~ NORTH-HOLLAND PUBLISHING COMPANY
EIKONAL THEORY OF ELECTRON-ATOM COLLISIONS ~ Charles J. JOACHAIN Physique Théorique et Mathématique, Faculté des Sciences, Université Libre de Bruxelles, Brussels, Belgium and Insritut de Physique Corpusculaire, Université de Louvain, Louvain-la-Neuve, Belgium
The application of eikonai methods to electron—atom scattering is reviewed. After recalling the basic features of the eikonal approximation, we discuss the Glauber theory, the eikonal—Born series method, the eikonal optical model theory and the eikonal distorted wave approach. We also give a survey of recent results obtained for electron—hydrogen and electron—helium collisions at intermediate energies.
1. Introduction Originally introduced in quantum collision theory by Moliere [1], the eikonal approximation has been considerably developed by Glauber [2], who proposed a very convenient many-body generalization of the method. Numerous applications to intermediate and high-energy hadronic collisions followed the work of Glauber [3]. More recently, atomic scattering processes have also been studied by means of the eikonal approximation. Leaving aside the equally interesting case of atom—atom scattering [4], we shall survey the application of the eikonal method to electron—atom collisions. After recalling in section 2 a few basic properties of the eikonal approximation in non-relativistic potential scattering, we discuss in section 3 various generalizations of the eikonal method to many-body collisions. Applications to electron—hydrogen and electron—helium scattering where extensive calculations have been made recently are described in section 4. We emphasize that the present paper only gives an outline of the subject. A more comprehensive account of multiple scattering expansions in several particle dynamics, including the application of the eikonal method to electron—atom scattering, may be found in the recent review of Joachain and Quigg [5]. —
—
2. Eikonal approximation in potential scattering Let us consider the non-relativistic scattering of a spinless particle of mass m by a potential V(r) of range a. We denote by k~and kf the initial and final wave vectors of the particle and introduce the reduced potential U(r) = 2m V(r)1h2 The energy of the particle is E = flk2/2m, where k is its wave number. The stationary scattering wave function ~ji~ which corresponds to an incident plane wave of momentum p 1 = hk~and exhibits the behaviour of an outgoing spherical wave, satisfies the Lippmann—Schwinger equation .
~~lr)=lIk1(r) *
+fG(~)(r,r’)U(r’) ~i~(r’) dr’,
Invited paper presented at the conference on “Numerical Methods in Electron—Atom Collisions”, 9—12 July 1973, Paris, France.
(1)
Cf. Joachain, Eikonal theory
359
where we choose the “normalization” in such a way that t~k.(r)= (2ir)312 exp(ik1’ r) and the Green’s function G~(r,r’)is given by 4 0~ or G~(r,r’)= ±e~k1T_P~ (2a,b) G~(r,r’)= —(2i )_31e 2 dK, eThe scattering amplitude is then given by —
f —2ir~(4kfIUI ~ with 4kf(r)
=
(3)
(2ir)3/2 exp(ikf’ r)’. We now assume that the short wavelength (semi-classical) condition
ka~’l,
(4)
is satisfied, together with the condition V
2’(
(5) 2 If these two conditions are satisfied -
=
2mV0/h
(6)
where we have made the change of variable Q= K—k and has “linearized” the denominator of the integrand by 2 term. Performing the integral in (6),1 one neglecting the Q ~ (r, r’) ~ (i/2k) euJ~(z_z’) ~2 (b_b’) 0 (z—z’), (7) where we have adopted a cylindrical coordinate system such that r = tion 0(x)1
if
x>0,
and
0(x)=0
if
b + zk~,r’ = b’ + z’k~and 0
is
the step func-
x<0.
(8)
The linearized propagator (6)—(7) clearly describes forward propagation between successive interactions with the potential. Upon insertion in the Lippmann—Schwinger equation (1) it leads to the eikonal wave function ~1’E (r)
=
(2ir)3/2 exp(iki~r
—
(i/2k)
f U(b, z’) dz’) or
~‘E (r)
=
(2ir)3/2 exp(iki~r (i/hvj)f
—=
_oo
v(b, z’) dz’),
(9a,b)
where v~= hk~/mis the incident velocity and the integral is evaluated along a straight line parallel to k.. Using the eikonal wave function (9) in the integral representation (3) of the scattering amplitude, we then obtain the eikonal scattering amplitude fE
=
Here A
_(1/47r)f exp(iA’r) U(r) exp(_(i/2k)
f U(b,z’) dz’) dr.
(10)
k~ kf is the momentum transfer of length ~ = 2k sin 40, 0 being the scattering angle between k —
1 and
kf. We have pointed out above that the phase of the eikonal wave function (9) is obtained by integrating along a straight line parallel to k~.In fact, since the actual phase of the corresponding semi-classical wave function is evaluated along a curved trajectory, it is reasonable to expect that an improvement on eq. (10) may be obtained by performing the z’ integral along a direction parallel to the bisector of the scattering angle (i.e. perpendicular to A). This suggestion, first made by Glauber [2] leads directly to the eikonal scattering amplitude 2b, (11) fE ~‘(k/2iri)fexp(iAb) {exp[ix(b)] 1 }d —
360
C.J. Joachain, Eikonal theory
where we have written r=b+z2,
21A
and the vector A is contained in the plane of “impact parameters” b. Furthermore, the eikonal phase shift tion x(b) is given by x(b)=—(1/2k)
(12) func-
f U(b,z)dz.
(13)
It is also convenient to introduce the quantity
r’(b) = 1
—
exp[ix(b)J,
(14)
in terms of which we may write ~~(ik/27r)fexp(iA.b)r(b)d2b. We note that for potentials which possess azimuthal symmetry eq. (11) reduces to the expression -fE
fE
(k/i)fJo(i~b){exp[i~(b)]—l}bdb,
(15)
(16)
which is known as the Fourier—Bessel representation of the scattering amplitude. For high-energy, small-angle scattering the eikonal phase x(b) appearing in eq. (16) may be related to the phase shift 6~of the partial wave method as X(k,b)’61(k),
l~_kb.
(17)
Let us state two important properties of the eikonal approximation. Firstly, it is equally valid for real and complex (optical) potentials. Secondly, within its range of validity, the eikonal amplitude satisfies the Optical theorem [2], in contrast with the first Born approximation. We now turn to the comparison of the eikonal method with the Born series. We first write the exact scattering amplitude as
fEf~~,
(18)
where the Born term of order n, 2 (cI~k~ IUG~~ U~.-~ UI’~k~)~ (19) fBnexpression —2ir in which the interaction appears n times and the propagator ~ (n—l) times. Hence the Born is an series (18) may be visualized as a multiple scattering series in which the projectile interacts repeatedly with the potential. We shall also introduce the notation fBI
= ~
(20)
to denote the jth order Born approximation to the scattering amplitude. Similarly, we may define an eikonal multiple scattering series by expanding the quantity r(b) appearing in eq. (15) in powers of the interaction. That is,
C.J. Joachain, Eikonal theory
fEE
361
(21)
1t~n’
with
2b.
—(ik/2ir) (i”/n!) fexp(iA. b)[~(b)}’~d In particular, for potentials having azimuthal symmetry, one has
(22)
=
fEn
=
f
—ik(i’~/n!) J 0(Ab) [~(b)]’~b db.
(23)
We remark that in the case of a real potential the objects fEn given by eq. (23) are alternatively real and purely imaginary. We shall also write (24)
fEJ~1~~•
Let us now compare in more detail the terms of the eikonal and Born series. We first note that [2] fBi fEl• (25) for all momentum transfers The relationships between the higher terms of the eikonal and Born series have only been explored recently [6—9].We shall concentrate here on real, central potentials. For superposition of Yukawa potentials of the form ~.
U(r)
=
U0
1’
j
p(a)
e~” da,
(26)
—
a0>0
Byron, Joachain and Mund [7, 8] have shown, on the basis of a careful analysis of the first few terms of the Born and eikonal series (up to n = 4), that the following relations hold: if one writes for large k j~~(k,z~)ABfl(/~)/k’~’1~O(k_n),
(27)
then defining A~(~)by (28)
fEfl(k,~)=A~(z~)/k”’, one has A~(~)=A~(~),
(29)
for all values of the momentum transfer. A general proof of eq. (29) has been given recently by Swift [9]. For the case n = 2 a straightforward calculation [8] shows that the relations (29) only hold for small-angle scattering when the interaction the exponential form of a gaussian potential 2 + d2)2. The case has of the potential U(r) = or U a Buckingham polarization potential U(r) = U0 X (r 0 exp(—or) is similar’to that of the Yukawa discussed above. We may also rephrase the relations (27—29) in the following way. For sufficiently large values ofk, one has ImjB2(k,
~) = Imj~2(k,~),
Re 7B3(k, i~)=j~3(k,~),
(30a, b)
Imj~~(k,~s)Im7~(k,~), n even
(30c)
RefBfl(k,~)’fp~(k,L~),
(30d)
nodd
362
C.J. Joachain, Eikonal theory
and we recall that for the real, central interactions which we are considering one has RefE2=0,
ImfE3=O,---
while
RefB20,
Imfg3ZO,
etc.
The relationships (29)—(30) have important consequences. Let us first consider the weak coupling limit IV0Ia/hv1
=
IU0Ia/2k ~ 1
(31)
-
In this case the Born series converges rapidly and the relations (29) imply that the eikonal amplitude fE is a poorer approximation to the exact amplitude than the second Born quantity fB2~This is due to the fact that for ka ~ 1 we may write the exact amplitude for Yukawa-type potentials as f(k,~)=fBl(~)+~
+
k I fB2
I
~)+i~?~) 2 k3
+...,
(32)
\k fB3
while the eikonal amplitude has the form fE(k,~)=fBl(~)+i~+~+-... k k2
(33)
Hence we see that neitherfB 2. However, since A is proportional to U~while Cis norfE are the correct to order k proportional to U~,it is clear2 that when potential strength ~U 0Iis small the quantity fB2 should be more precise than fE As the coupling increases we expect from the above discussion that the eikonal method should improve steadily. This is confirmed by detailed numerical calculations [7, 81. Finally, let us briefly consider the strong coupling situation, for which IV0LaIhv> 1 and 1V01/E> 1. In this case the Born series is useless. Despite the fact that the condition (5) is violated, the eikonal approximation is still quite accurate at small angles for a variety of interaction potentials [8]. This is undoubtedly related to the unitary character of the eikonal approximation. Before leaving these basic aspects of the eikonal approximation, we mention that various other forms of the method have been proposed in recent years. A few important papers dealing with this subject and with additional properties of eikonal methods are listed in refs. [10—21].
3. Eikonal approximation for several particle collisions 3.1. Glauber theory The extension of the eikonal approximation to many-body scattering problems was first proposed by Glauber in connection with high-energy, small-angle hadron—nucleus collisions. Consider a point particle A incident on a composite target B (for example an atom) which contains N scatterers. We assume that the relative motion of A and B is fast with respect to the internal motion of the particles in the target. Furthermore, we assume that the incident particle interacts with the target scatterers via two-body spin-independent interactions. The Glauber scattering amplitude for a small-angle direct collision leading from an initial target 10 > to a final state Im) is given in the center-of-mass system by 2b, (34) F~0= (k1/211i)f exp(iA b) (mI{ exp[ix~t(b, b1, -.., bN)]—l }I0) d the corresponding differential cross section being [2]
da~
2. 0/d~ = (k~/k1)IF~0I
(35)
Cf. Joa chain, Eikonal theory
363
Here A is the center of mass wave vector transfer, while r = b+z2 is the initial relative coordinate and
=
b1
+ Z~2
are the coordinates of the target particles (relative to the target center of mass). As in the case of potential scattering we shall choose the z axis along the bisector of the scattering angle. The total Glauber phase shift function x~t(b,bl,.-.bN)=~
~(b—~),
(36)
is the sum of the phase shifts x1 contributed by each of the target scatterers. If we define the quantity 1’~0~(b, b1,
...,
bN) = 1
—
exp [ix~~(b,
b1,
.-.,
bN)]
(37)
then eq. (34) becomes
2b.
(38)
F~0= (ik/2iT)f exp(iA. b) (m I r~0~(b, b1, ..., bN)tO ) d Introducing the “single particle” quantities F 1(b—1~,i)= 1 —exp[ix1(b—b1)],
(39)
we may write eq. (37) in the form
or
r~0~(b,b1, -..,bN) =
—
j=1
1
—~
[1 —r1(b—~)],
r1 r1 + --- +
j~=l
(_~-‘
H
(40) (41)
~1.
j=1
Substitution of this expression into eq. (38) leads directly to an interpretation of the collision in terms of a multiple scattering expansion [2]. The term linear in I’~leads to the single scattering (impulse) contribution, while the next terms provide double, triple, --~scattering corrections. We note that the order of the multiple scattering can at most be N, reflecting the fact that the scattering is concentrated in the forward direction. It is important to note that the above generalization of the eikonal method, proposed by Glauber [2] makes no reference to interaction potentials. It is therefore of particular interest in high-energy hadronic collisions. If the basic two-body interactions are known, as in atomic physics, one can actually write the many-body eikonal wave function in terms of these interaction potentials and gain further insight into the scattering amplitude. For example, in the case of non-relativistic electron—atom scattering, we may write the full eikonal wave function (using atomic units) as exp (ikrr_Wki)
f V~(b,z’,X)dz’) ~0(X),
(4~)
where V~is the full interaction potential in the initial channel, X denotes collectively the target coordinates and is the initial bound-state wave function of the target. For a direct collision process (no exchange) such that V~= V~= V we then obtain the many-body eikonal scattering amplitude (corresponding to a transition I0)—~im>) as F~ =
—
(2~ 1 fexp (i A r) (m V(b,z, X) exp
(—
f V(b, z’, X) dz’)
(i/k~)
2b dz. 0
(43)
d
For elastic scattering processes, and if we choose the z axis along the bisector of the scattering angle, we may
364
Cf. Joachain, Eikonal theory
perform the z integral in eq. (43) to obtain directly the Glauber result F~k/27ri)fexp(iAb)(0I{exp[iX~t(b,bi, “-,bN)]—l}I0>d2b, with k =
1k
1!
=
~~44)
kf! and
x~t=—(lIk)fV(b,z,X)dz.
(45)
On the other hand, for inelastic processes, the simple Glauber result (34) can only be “derived” from eq. (43) neglecting the longitudinal momentum transfer. This neglect is not too serious for high-energy hadronic collisions (excluding violent inelastic events such as production reactions), but it leads to difficulties in electron— atom collisions, as we shall see below. by
3.2. Comparison of the Glauber and Born series. The eikonal—Born series method As in the case of potential scattering, we can expand the eikonal Glauber expression (34) in powers of the interaction potential and compare with the Born series. We shall consider here the case of elastic scattering. Starting from eq. (44), we write
F~=~~Gfl,
(46)
where 2b,
(47)
(k/27ri) We shall also write(i’~/n!)fexp(iA.b) (0I[x2~t]”l0>d =
FGJ
(48)
~‘Gn-
=
The Born term of order n in the interaction V is given by ~‘Bn
=
where ~ write
(2ir)2 (kf, 0 I VG~~ V... V1k =
FB,=~
(E
—
1, 0>, (49) H0 + iE)~is the unperturbed Green’s function of the system, with H0 = H— V. We shall also
(50)
~‘Bn~
Assuming that the Born series converges, the exact, direct (no exchange) elastic scattering amplitude is given by F~=~~Bfl.
(51)
For electron—atom scattering, the total Glauber phase shift function X~tmay be obtained explicitly. One finds 2)], x~t(lIk)Elog[l —2(b1/b)cos(~i—Ø)+(b~!b
(52)
Cf. Joachain, Eikonal theory
where we have written b
F~= (k/i)
365
(b, 0) and b1 (b~,Ø~).Hence
f J0(~b)(UI [exp(iX~t)—1] 10> b db,
(53)
f .J0(z~b)(OI[X~~]nIO>bdb.
(54)
and 5/n!) =
(k/i)(i’
With the choice of z axis which we have adopted, it is a simple matter to show that for all scattering angles FB1 =FG1.
(55)
In addition, we_also note from eq. (54) that, as in the case of potential scattering, one has Re FG2 = 0, Im FG3 = 0, while Re FB2 #0, ImFB3 * 0. Thus important terms (in an expansion in powers of k’) are missing from the Glauber amplitude. As we shall illustrate in section 4, this fact, together with an unphysical divergence of Fel in the forward direction, implies that the Glauber method is unreliable for the treatment of electron—atom collisions. However, Byron and Joachain [22] have shown recently that it is possible to combine the Born and the eikonal Glauber series in order to keep all the terms which contribute to a given order in k~ in the differential cross section. Their eikonal—Born series (EBS) method, which also includes exchange effects, will be discussed in more detail below. 3.3. The eikonal optical model We now turn to eikonal methods based on the optical model formalism. We begin by considering the elastic scattering of electrons by atoms at intermediate and high energies. Neglecting first exchange effects, we write the optical potential through second order in the full interaction V as [23, 24] =
+
~
2~ = E
(OIVln>(nIVIO)
with v~’~ =(OIVIO> and v~ n#0 E K_(Wn_Wo) In these expressions the electron—atom interaction is given by V_Z/r+E
- ,
c—k
(56)
o~.
(57,58)
+1
1/tr—r 11,
(59)
and the summation appearing in eq. (58) runs over all the intermediate stateselectron of the target initial state) 2 is the incident energy(except and K the its kinetic having eigen-energies wn. Furthermore, the quantity E = ~k energy operator. At intermediate and high energies one may replace in eq. (58) the energy differences wn w 0 by an average excitation energy i~.We then obtain in configuration space the non-local, complex, second-order expression 2~Ir’>= G~7~(k’,r,r’)A(r,r’), (60) —
(rIv~
where G (k’, r,byr’) is the free propagator corresponding to the wave number k’ A(r,r’) is1~ given
2 =
—
2i~)1/2and the object
(k
A(r,r’)=2(OIV(r,X)V(r’,X)I0)—(OIV(r,X)I0)(0IV(r’,X)I0). The Schrödinger equation satisfied by the scattering wave function
(61) is then seen to be
366
Cf. Joachain, Eikonal theory
[K + V(’)(r)
—
~k2]
ii4~+fG~~)(k’, r, r’)A(r, r’) ~,14~(r’)dr’ = U.
(62)
A study of this equation in the eikonal approximation has been made by Joachain and Mittleman [25,261 who showed that absorption effects due to the imaginary part of the potential (or, in other words, induced by unitarity from the open channels) play an important role in small angle elastic electron—atom scattering at intermediate energies. The scattering amplitude obtained by Joachain and Mittleman has the form Fei
r(k/2iri)fexp(iA.b) {exp[i~~t(b)]—l}
d~b,
(63)
f dz f dz’ exp[—i(k—k’)(z—z’)] A(b, z; b, z’).
(64)
where
x~~(b) = —(1/k)
f V(’)(b, z) dz
+
(i/2kk’)
The first term on the right of eq. (64) is simply the phase shift function arising from the static potential ~ The second term accounts for polarization and absorption effects. It is interesting to note that the Glauber theory of section 3.1 yields the result =
(k/2iri) fexp (iA
b) [exp [ix~~~(b)]—1] d2b,
(65)
with exp[ix~~t(b)}‘~(0Iexp[i~~t(b,bi, -.-, b~)]IU>,
(66)
and
f
X~~t(b)~(l/k) V~t(b,z)dz—(l/k)
f V(1)(b,z)dz+(i/2k2)
fdz
fdz’A(b,z;b,z’)+-.-.
As we should expect from the analysis of section 3.1, the Glauber expression (65) suffers from serious deficiencies. It leads to a forward direction divergence of the scattering amplitude and does not account properly for the long range (polarization) effects. A careful analysis of eqs. (63)—(64), recently made by Byron and Joachain [27] shows that none of these difficulties are present in the eikonal optical model. 3.4. The eikonal distorted wave method The optical model method may also be used together with the two-potential formalism of Gell—Mann and Goldberger [28] to analyze inelastic or rearrangement atomic processes [29—31].The basic idea is to split the interactions V 1 and Vf appearing respectively in the initial and final channel as V~=U~1-W~,
VfUf+Wf,
(68a,b)
where U~and Uf are distorting potentials which are usually chosen to be optical potentials describing elastic scattering in the initial and final channel. To lowest order in the “primary” interactions W~or Wf the transition matrix element Tba for a npn-elastic collision process leading from the state la>~1k1, 0> to the state lb>~Ikf, n>is then given by the DWBA (Distorted wave Born approximation) expression T~WBA ~
(69)
where x~and ~ are the distorted waves corresponding to the distorting potentials U1 and Uf. If we replace the distorted waves by their approximate eikonal expressions, we obtain the eikonal DWBA matrix element of Chen et al. [29], namely
Cf. foachain, Eikonal theory
T~= (2if)3 fexp(i A r) exp{i[A~(b,z)
+ Af(b,
z)]
367
}V 00 (b, z) dr,
(70)
where A1(b~z)=—(1/k~)fU~(b,z’)dz’,
Af(b,z)=—(l/kf)f Uf(b,z’)dz’,
(72)
and Vn0(b,z)
=
(nIVIU>.
(73)
It is worth noting that this method leads to reasonably simple expressionswhich take into account explicitly the longitudinal momentum transfer, allow the evaluation of exchange effects and may be applied to complex target atoms. Before we conclude this section, we want to mention the very interesting approach developed recently by Bransden et al. [31—34] to analyze the scattering of charged particles by atoms. Starting from the set of closecoupling equations, Bransden et al. retain explicitly a certain number of important states in a truncated expansion of the wave function. Second-order potentials, similar to those discussed in section 3.3, are then introduced to account for the coupling with the remaining states. This method, which uses the eikonal approximation to simplify the resulting equations, has already been applied successfully to the scattering of electrons and protons by atomic hydrogen and helium.
4. Electron scattering by atomic hydrogen and helium at intermediate and high energies 4.1. Electron scattering by atomic hydrogen We begin by considering elastic collisions, and follow the treatment of Byron and Joachain [22] who have carried out a detailed comparison of the Born and Glauber eikonal series. We first recall that the Glauber elastic amplitude (44) diverges logarithmically as the magnitude ~ of the momentum transfer tends to zero [35,36]. Moreover, one has FBi = FG1 at all scattering angles [see eq. (55)]. Let us now compare the higher terms of the Born and Glauber series. By using an average excitation energy ~ it is possible to reduce the second Born term FB2 to an expression which can be evaluated in a straightforward way. In fact, for simple target atoms one may even include a few states exactly in the summation on intermediate states which appears in FB2 finds [37]. that Of particular interest the limit of Im large values of klike for k~ which, scatter2),one Re FB2 varies likeis k~, while FB2 behaves logatk.small We remark ing (0 < 2i~/k thatangles this behaviour of FB2 is different from that found in section 2 for the case of potential scattering. This is due to the role played by the long-range forces at small scatteringangles. In particular, we stress that Re FB2 now gives the dominant correction to the first Born differential cross section at small angles. At larger angles one obtains the “potential scattering” behaviour of section 2. That is, Re FB2 varies like k2 and Im FB2 like k~ for large values of k. Again the term Re FB2 is essential to correct in a consistent way the first Born term FBi. Since the term Re FB2 is completely missing from the Glauber series we expect the Glauber method to be unreliable for this problem. Let us now pursue our comparison of the Born and the Glauber series. The second-order term FG2, which is purely imaginary, diverges logarithmically at ~ = 0. The corresponding quantity i Im FB 2 is finite at ~ = U but would diverge in the forward direction if i~(the average excitation energy) were set equal to zero. In fact, although the quantities Im FG2 and Im FB2 differ substantially at very small momentum transfers, a detailed study [22] of these two quantities shows that otherwise they agree very well, even at 0 = 1800 and for rather low values of k. This is not surprising in view of the fact that the large-angle scattering is mainly governed by the static po-
368
Cf. foachain, Eikonal theory
tential V~’~ = (UI yb>, for which the relation (30a) holds. For n > 3, the terms EGO of the Glauber series (46) are finite, even at ~ = 0. It is therefore very likely that these terms should agree with the corresponding terms of the Born series (i.e. FG3 with Re FB3, FG4 with i Im FB4, etc.). Since the direct evaluation of the quantity Re FB3 (which yields an important contribution of order k2 to the differential cross section) is a formidable task, it is therefore reasonable to use FG3 in place of Re FB3. Thus Byron and Joachain [22] have suggested to use for the dfrect amplitude F~the eikonal-Born series expression
FFBl+ReFB2+i~G3+iImfB2+....
(74)
Before we compute differential cross we recall that the leading (Ochkur) term [38] of the firstoforder 2 sections, at small momentum transfers. Therefore, a consistent calculation the exchange amplitudecross is also of order elastic differential section dae k 2 (or E~,where E is the incident electron energy) re1/d~lthrough order k by GOch. For an unpolarized incident electron beam quires the inclusion of this Ochkur term, which we denote and target atom, and if no attempt is made to distinguish the final spin states, we then have, through order k2 dGel/d~Z~IEe1+GOchI2 +~IFei GOch!2,
(75)
where F~is given by eq. (74). A detailed comparison of the above eikonal—Born series (EBS) method with other approximations (first Born, Glauber) and with the experimental results of Teubner et a!. [39, 59] has been made by Byron and Joachain [22, 60]. They showed that the Born and Glauber angular distributions differ substantially at intermediate energies from the EBS expression (75). Moreover, the Born and Glauber methods predict the same angular distribution for electron and positron scattering, whereas the EBS method yields significant differences (at small angles) between the electron and positron curves. Unfortunately, the measurements of Teubner et al. [39] are not absolute and have been made at rather large angles, where the differences in the slopes of the various theoretical predictions are rather small. The more recent experimental data of Teubner, Lloyd and Wiegold [40], at an incident electron energy of 50 eV, and relative to molecular hydrogen, cover a wider angular range. They are in good agreement with the EBS results [41]. We now consider briefly some inelastic transitions induced in atomic hydrogen by electron impact. Calculations using the Glauber approximation (34) have been made by several authors [40—45]and reviewed some time ago by Gerjuoy [46]. It is clear from the analysis of section 3.2 that the Glauber amplitude (34) also suffers from severe deficiencies when it is applied to inelastic electron—atom collisions. For example, in the case of a transition is 2s, the Glauber amplitude diverges for ~ = 1k 1 k~-I= 0. Although this value of the momentum transfer is unphysical, it is clear that at intermediate and high energies the presence of this divergence influences the shape of the Glauber differential cross section near the forward direction. Moreover, we may also develop the Glauber amplitude (34) as a Glauber series in powers of the interaction [as in eq. (47)]. Comparison with the corresponding Born series shows again that important terms are missing from the Glauber amplitude. Finally, the Glauber expression (34) neglects the longitudinal momentum transfer, an approximation which leads to undesirable features such as: L~.m= ±1 selection rule for s p transitions and identical results for the excitation of atoms by electron and positron impact. Using the more general Glauber expression (44) (which includes the longitudinal momentum transfer), together with Monte-Carlo integration techniques, Byron [47]has found significant differences between the excitation of the 2s state of hydrogen by electron and positron impact. The excitation of the n = 2 states of atomic hydrogen has also been studied recently by used the eikonal DWBA method [29, 30]. Chen et al. [29]have used static distorting potentials, while Joachain and Vanderpoorten [30] have included absorption effects by using Glauber optical potentials. As one should expect on physical grounds, the eikonal 2p DWBA particularly reliable for transitions. example, the totalwith crossthe sections for excitastate isobtained by Joachain andallowed Vanderpoorten is inFor excellent agreement experimental data tionLong of the of et al. [48]. On the other hand, the eikonal DWBA method yields rather poor values for the polarization of the light emitted from the 2p states. An extension of the method, which includes all the important final state inter—~
—
—~
Cf. foachain, Eikonal theory
369
actions (as in the method of Bransden eta!. [31—34])would be very desirable. Finally, we note that the eikonal DWBA method and the approach of Bransden et al. predict differences between electron and positron scattering. 4.2. Electron—helium collisions Let us now consider the scattering of fast electrons by helium, where absolute measurements of differential cross sections have recently become available [49—51]. We begin by analyzing elastic electron—helium scattering and follow again the EBS treatment of Byron and Joachain [22]. The direct amplitude is still obtained from eq. (74), but the differential cross section is now given by
(76) dciei/d&2 = IF~ GOchl2. A detailed comparison of the EBS results with the small-angle data of Bromberg [49], Chamberlain et al. [50] and Crooks and Rudd [51] has been made in ref. [22] for incident electron energies ranging from IOU eV to 500 eV. This paper also compares the EBS values with those obtained from the first Born approximation and with the Glauber results of Franco [52]. The EBS results, which agree very well with the data of Bromberg at 500 eV, are consistently better than the Born or Glauber results. More recently, Byron and Joachain [53] have extended their analysis to include large angle scattering. As one should expect, the scattering is mainly governed in this angular region by the static potential y(l). It is gratifying to note that the EBS method also yields good results at large angles. However, this last feature depends delicately on cancellations between higher order terms of perturbation theory, which play a significant role for large angle scattering. These cancellations, although present in the elastic electron—helium case cannot be expected to occur in other situations. We now consider eikonal optical model calculations of elastic electron—helium scattering. By solving eq; (62) within the framework of the eikonal approximation, Joachain and Mittleman [25—26] showed explicitly that absorption effects play an important role in small angle elastic electron—helium scattering at intermediate energies. More recently, Byron and Joachain [27] have analyzed in detail the role of long range forces in the eikonal optical model. Their eikonal optical model calculations for elastic electron—helium scattering, which are free of any phenomenological parameters, are in excellent agreement at small angles with the EBS calculations and the experimental data. Let us now describe briefly a few inelastic electron—helium processes. For the process e + He (1 1S) e + He(21 S) the most reliable results are the four-channel calculations of Berrington et a!. [34]. The eikonal DWBA results of Joachain and Vanderpoorten [54] agree with the experimental results [50, 55] only at high incident electron energies. Moreover, the 21S_ 21P coupling, which strongly influences the angular distribution at small angles [34] is missing in the eikonal DWBA method. In the case of the process e + He (1 1S) ~ e + He (21 P) the strong 11S 21 P coupling completely dominates, so that reliable results are obtained from the eikonal DWBA method [54]. The eikonal calculations of Byron [47], using eq. (43) are also in good agreement with the experimental results [50, 55—57]. Finally, we come to a process of considerable interest, namely the excitation of triplet states of helium by electron impact. For example, let us consider the reaction —
—
e
+He(i15)-+e
+He(23S).
(77)
This process is a pure rearrangement (knock-out) collision provided that very small spin-dependent interactions are neglected. Although this reaction has already received a large amount of attention, no satisfactory explanation was given for the forward peak observed [55] in the differential cross section for incident electron energies ranging from 100 eV to 225 eV. In particular, the first Born and the Ochkur approximations completely fail. The reasons for this failure have been given recently by Byron and Joachain [58], who have also performed a many-body eikonal calculation (using the Monte-Carlo integration method) and obtained encouraging preliminary results.
370
Cf. foachain, Eikonal theory
However, much work remains to be done before a complete understanding of simple rearrangement collisions at intermediate and high energies will be achieved. It is to be hoped that eikonal methods will help to solve these problems, which are among the most difficult (and fascinating) ones of several particle dynamics.
References [1] G. Mo1i~re,Z. Naturforsch. 2A (1947) 133.
[21 R.J. Glauber, in Lectures in theoreticsl physics, edited by W.E. Brittin and L.G. Dunham (Interscience Pub!., New York, 1959) Vol. 1, p. 315. [3] See for example R.J. Glauber, Theory of high-energy hadron—nucleus collisions, in “High Energy Physics and Nuclear Structure” (Plenum Press, New York, 1970); K. Gottfried and J.D. Jackson, Nuovo Cimento 34 (1964) 375; A.S. Goldhaber and C.J. Joachain, Phys. Rev. 171 (1968) 1566. [4] See for example J.C.Y. Chen and K.M. Watson, Phys. Rev. 174 (1968) 152; 188 (1969) 236; J.C.Y. Chen, C.J. Joachain and K.M. Watson, Phys. Rev. AS (1972) 1268. 15] C.J. Joachain and C. Quigg, National Accelerator Laboratory Report NAL-THY-99, 1973; Rev. Mod. Phys. (to be published). [61 R.J. Moore, Phys. Rev. D2 (1970) 313. [71 F.W. Byron and C.J. Joachain, Physica 66 (1973) 33~ [8] F.W. Byron, Ci. Joachain and E.H. Mund, Phys. Rev. D8 (1973) 2622. [9] A. Swift, Phys. Rev. D (to be published). [10] BJ. Malenka, Phys. Rev. 95 (1954) 522. [11] L.I. Schiff, Phys. Rev. 103 (1956) 443. [12] D.S. Saxon and L.I. Schiff, Nuovo Cimento 6 (1957) 614. [13] R. Blankenbecker and M.L. Goldberger, Phys. Rev. 126 (1962) 766. [14] T. Adachi and I. Kotani, Progr. Th. Phys. Suppi. Extra no. 316 (1965), T. Adachi. Progr. Th. Phys. 35 (1966) 463, 483. [15J L. Wilets and S.J. Wallace, Phys. Rev. 169 (1968) 84. [16] R.L. Sugar and R. Blankenbecker, Phys. Rev. 183 (1969) 1387. [17] M. Levy and J. Sucher, Phys. Rev. 186 (1969) 1656. [181 H.D.I. Abarbanel and C. Itzykson, Phys. Rev. Letters 23 (1969) 53. [19] S.J. Wallace, Phys. Rev. Letters 27 (1971) 622. [20] A. Baker, Phys. Rev. D6 (1972) 3462. [21] S.J. Wallace, Ann. Phys. (N.Y.) 78 (1973) 190. [22] F.W. Byron and C.J. Joachain, Phys. Rev. A8 (1973) 1267. [231 M.H. Mittleman and K.M. Watson, Phys. Rev. 113 (1959) 198; Ann. Phys. (N.Y.) 10(1960) 268. [24] M.L. Goldberger and K.M. Watson, Collision theory (J. Wiley and Sons, New York, 1965), Chapter 11. [25] C.J. Joachain and M.H. Mittleman, Physics Letters 36A (1971) 209. [26] C.J. Joachain and M.H. Mittleman, Phys. Rev. A4 (1971) 1492. [27] F.W. Byron and C.J. Joachain, Phys. Rev. A (to be published). [28] M. Gell-Mann and M.L. Goldberger, Phys. Rev. 91(1953) 398. [291 J.C.Y. Chen, C~J.Joachain and K.M. Watson, Phys. Rev. AS (1972) 2460. [30J C.J. Joachain and R. Vanderpoorten, J. Phys. B6 (1973) 622. [31] BH. Bransden and J.P. Coleman, J. Phys. B5 (1972) 537. [32] B.H. Bransden, J.P. Coleman and i. Sullivan, J. Phys. B5 (1972) 546. [33] J. Sullivan, i.P. Coleman and B.H. Bransden, i. Phys. B5 (1972) 2061. [34] K.A. Berrington, B.H. Bransden and J.P. Coleman, J. Phys. B6 (1973) 436. [351 F.W. Byron and C.J. Joachain, unpublished (1967). [36] V. Franco, Phys. Rev. Letters 14 (1968) 709. [37] See for example A.R. Holt, J.L. Hunt and B.L. Moiseiwitsch, i. Phys. B4 (1971) 1318; M.J. Woouings and M.R.C. McDowell, J. Phys. B5 (1972) 1320; F.W. Byron and C,i. Joachain, ref. [22]. [38] V.!. Ochkur, Zh. Eksp. Teor. Fig. 45 (1963) 734. (Eng. trans.: Soviet Physics JETP 18 (1964) 503). [39] P.J.O. Teubner, K.G. Williams and J.H. Carver, quoted in H. Tai, P.i. Teubner and R.H. Bassel, Phys. Rev. Letters 22 (1969) 1415; 23 (1969) 453 (E). [40] A.S. Ghosh and N.C. Sil, Indian J. Phys. 43 (1969) 490. [41] A.S. Ghosh, P. Sinha and N.C, Sil, J. Phys. B3 (1970) L58.
Cf. foachain, Eikonal theory [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60]
371
H. Tai, R.H. Bassel, E. Gerjuoy and V. Franco, Phys. Rev. Al (1970) 1819. K. Bhadra and A.S. Ghosh, Phys. Rev. Letters 26 (1971) 737. V.B. Sheorey, E. Gerjuoy and B.K. Thomas, i. Phys. B4 (1971) 657. E. Gerjuoy, B.K. Thomas and V.B. Sheorey, J. Phys. B5 (1972) 321. E. Gerjuoy, in The physics of electronic and atomic collisions, edited by T.R. Govers and F.J. de Heer (North-Holland Publ. Cy, Amsterdam, 1972) p. 243. F.W. Byron, Phys. Rev. A4 (1971) 1907. R.L. Long, D.M. Cox and S.S. Smith, i. Res. Nat. Bur. Stand. 72A (1968) 521. J.P. Bromberg, J. Chem. Phys. 50 (1969) 3906. G.E. Chamberlain, S.R. Mielczarek and C.E. Kuyatt, Phys. Rev. A2 (1970) 1905; see also L. Vriens, CE. Kuyatt and S.R. Mielczarek, Phys. Rev. 170 (1968) 163. G.B. Crooks and M.E. Rudd, Bull. Amer. Phys. Soc. 17 (1972) 131; G.B. Crooks, Thesis, University of Nebraska (unpublished). V. Franco, Phys. Rev. Al (1970) 1705. F.W. Byron and C.i. Joachain, Phys. Rev. A (to be published). C.i. Joachain and R. Vanderpoorten, J. Phys. B (to be published). L. Vriens, LA. Simpson and SR. Mielczarek, Phys. Rev. 165 (1968) 7. i.P. de Jongh and J. Van Eck, in Abstracts of papers of the ViIth ICPEAC, edited by L.M. Branscomb et al. (North-Holland Publ. Cy, Amsterdam, 1971) p. 701. F.G. Donaldson, M.A. Helder and i.W. McConkey, J. Phys. B5 (1972) 1192. F.W. Byron and Cl. Joachain, Physics Letters 38A (1972) 185. P.l.O. Teubner, C.R. Lloyd and E. Weigold, I. Phys. B6 (1973) L134. F.W. Byron and Ci. Joachain, I. Phys. B (to be published).
EIKONAL THEORY OF ELECTRON-ATOM COLLISIONS ~ Charles J. JOACHAIN Physique Théorique et Mathématique, Faculté des Sciences, Université Libre de Bruxelles, Brussels, Belgium and Insritut de Physique Corpusculaire, Université de Louvain, Louvain-la-Neuve, Belgium
The application of eikonai methods to electron—atom scattering is reviewed. After recalling the basic features of the eikonal approximation, we discuss the Glauber theory, the eikonal—Born series method, the eikonal optical model theory and the eikonal distorted wave approach. We also give a survey of recent results obtained for electron—hydrogen and electron—helium collisions at intermediate energies.
1. Introduction Originally introduced in quantum collision theory by Moliere [1], the eikonal approximation has been considerably developed by Glauber [2], who proposed a very convenient many-body generalization of the method. Numerous applications to intermediate and high-energy hadronic collisions followed the work of Glauber [3]. More recently, atomic scattering processes have also been studied by means of the eikonal approximation. Leaving aside the equally interesting case of atom—atom scattering [4], we shall survey the application of the eikonal method to electron—atom collisions. After recalling in section 2 a few basic properties of the eikonal approximation in non-relativistic potential scattering, we discuss in section 3 various generalizations of the eikonal method to many-body collisions. Applications to electron—hydrogen and electron—helium scattering where extensive calculations have been made recently are described in section 4. We emphasize that the present paper only gives an outline of the subject. A more comprehensive account of multiple scattering expansions in several particle dynamics, including the application of the eikonal method to electron—atom scattering, may be found in the recent review of Joachain and Quigg [5]. —
—
2. Eikonal approximation in potential scattering Let us consider the non-relativistic scattering of a spinless particle of mass m by a potential V(r) of range a. We denote by k~and kf the initial and final wave vectors of the particle and introduce the reduced potential U(r) = 2m V(r)1h2 The energy of the particle is E = flk2/2m, where k is its wave number. The stationary scattering wave function ~ji~ which corresponds to an incident plane wave of momentum p 1 = hk~and exhibits the behaviour of an outgoing spherical wave, satisfies the Lippmann—Schwinger equation .
~~lr)=lIk1(r) *
+fG(~)(r,r’)U(r’) ~i~(r’) dr’,
Invited paper presented at the conference on “Numerical Methods in Electron—Atom Collisions”, 9—12 July 1973, Paris, France.
(1)
Cf. Joachain, Eikonal theory
359
where we choose the “normalization” in such a way that t~k.(r)= (2ir)312 exp(ik1’ r) and the Green’s function G~(r,r’)is given by 4 0~ or G~(r,r’)= ±e~k1T_P~ (2a,b) G~(r,r’)= —(2i )_31e 2 dK, eThe scattering amplitude is then given by —
f —2ir~(4kfIUI ~ with 4kf(r)
=
(3)
(2ir)3/2 exp(ikf’ r)’. We now assume that the short wavelength (semi-classical) condition
ka~’l,
(4)
is satisfied, together with the condition V
2’(
(5) 2 If these two conditions are satisfied -
=
2mV0/h
(6)
where we have made the change of variable Q= K—k and has “linearized” the denominator of the integrand by 2 term. Performing the integral in (6),1 one neglecting the Q ~ (r, r’) ~ (i/2k) euJ~(z_z’) ~2 (b_b’) 0 (z—z’), (7) where we have adopted a cylindrical coordinate system such that r = tion 0(x)1
if
x>0,
and
0(x)=0
if
b + zk~,r’ = b’ + z’k~and 0
is
the step func-
x<0.
(8)
The linearized propagator (6)—(7) clearly describes forward propagation between successive interactions with the potential. Upon insertion in the Lippmann—Schwinger equation (1) it leads to the eikonal wave function ~1’E (r)
=
(2ir)3/2 exp(iki~r
—
(i/2k)
f U(b, z’) dz’) or
~‘E (r)
=
(2ir)3/2 exp(iki~r (i/hvj)f
—=
_oo
v(b, z’) dz’),
(9a,b)
where v~= hk~/mis the incident velocity and the integral is evaluated along a straight line parallel to k.. Using the eikonal wave function (9) in the integral representation (3) of the scattering amplitude, we then obtain the eikonal scattering amplitude fE
=
Here A
_(1/47r)f exp(iA’r) U(r) exp(_(i/2k)
f U(b,z’) dz’) dr.
(10)
k~ kf is the momentum transfer of length ~ = 2k sin 40, 0 being the scattering angle between k —
1 and
kf. We have pointed out above that the phase of the eikonal wave function (9) is obtained by integrating along a straight line parallel to k~.In fact, since the actual phase of the corresponding semi-classical wave function is evaluated along a curved trajectory, it is reasonable to expect that an improvement on eq. (10) may be obtained by performing the z’ integral along a direction parallel to the bisector of the scattering angle (i.e. perpendicular to A). This suggestion, first made by Glauber [2] leads directly to the eikonal scattering amplitude 2b, (11) fE ~‘(k/2iri)fexp(iAb) {exp[ix(b)] 1 }d —
360
C.J. Joachain, Eikonal theory
where we have written r=b+z2,
21A
and the vector A is contained in the plane of “impact parameters” b. Furthermore, the eikonal phase shift tion x(b) is given by x(b)=—(1/2k)
(12) func-
f U(b,z)dz.
(13)
It is also convenient to introduce the quantity
r’(b) = 1
—
exp[ix(b)J,
(14)
in terms of which we may write ~~(ik/27r)fexp(iA.b)r(b)d2b. We note that for potentials which possess azimuthal symmetry eq. (11) reduces to the expression -fE
fE
(k/i)fJo(i~b){exp[i~(b)]—l}bdb,
(15)
(16)
which is known as the Fourier—Bessel representation of the scattering amplitude. For high-energy, small-angle scattering the eikonal phase x(b) appearing in eq. (16) may be related to the phase shift 6~of the partial wave method as X(k,b)’61(k),
l~_kb.
(17)
Let us state two important properties of the eikonal approximation. Firstly, it is equally valid for real and complex (optical) potentials. Secondly, within its range of validity, the eikonal amplitude satisfies the Optical theorem [2], in contrast with the first Born approximation. We now turn to the comparison of the eikonal method with the Born series. We first write the exact scattering amplitude as
fEf~~,
(18)
where the Born term of order n, 2 (cI~k~ IUG~~ U~.-~ UI’~k~)~ (19) fBnexpression —2ir in which the interaction appears n times and the propagator ~ (n—l) times. Hence the Born is an series (18) may be visualized as a multiple scattering series in which the projectile interacts repeatedly with the potential. We shall also introduce the notation fBI
= ~
(20)
to denote the jth order Born approximation to the scattering amplitude. Similarly, we may define an eikonal multiple scattering series by expanding the quantity r(b) appearing in eq. (15) in powers of the interaction. That is,
C.J. Joachain, Eikonal theory
fEE
361
(21)
1t~n’
with
2b.
—(ik/2ir) (i”/n!) fexp(iA. b)[~(b)}’~d In particular, for potentials having azimuthal symmetry, one has
(22)
=
fEn
=
f
—ik(i’~/n!) J 0(Ab) [~(b)]’~b db.
(23)
We remark that in the case of a real potential the objects fEn given by eq. (23) are alternatively real and purely imaginary. We shall also write (24)
fEJ~1~~•
Let us now compare in more detail the terms of the eikonal and Born series. We first note that [2] fBi fEl• (25) for all momentum transfers The relationships between the higher terms of the eikonal and Born series have only been explored recently [6—9].We shall concentrate here on real, central potentials. For superposition of Yukawa potentials of the form ~.
U(r)
=
U0
1’
j
p(a)
e~” da,
(26)
—
a0>0
Byron, Joachain and Mund [7, 8] have shown, on the basis of a careful analysis of the first few terms of the Born and eikonal series (up to n = 4), that the following relations hold: if one writes for large k j~~(k,z~)ABfl(/~)/k’~’1~O(k_n),
(27)
then defining A~(~)by (28)
fEfl(k,~)=A~(z~)/k”’, one has A~(~)=A~(~),
(29)
for all values of the momentum transfer. A general proof of eq. (29) has been given recently by Swift [9]. For the case n = 2 a straightforward calculation [8] shows that the relations (29) only hold for small-angle scattering when the interaction the exponential form of a gaussian potential 2 + d2)2. The case has of the potential U(r) = or U a Buckingham polarization potential U(r) = U0 X (r 0 exp(—or) is similar’to that of the Yukawa discussed above. We may also rephrase the relations (27—29) in the following way. For sufficiently large values ofk, one has ImjB2(k,
~) = Imj~2(k,~),
Re 7B3(k, i~)=j~3(k,~),
(30a, b)
Imj~~(k,~s)Im7~(k,~), n even
(30c)
RefBfl(k,~)’fp~(k,L~),
(30d)
nodd
362
C.J. Joachain, Eikonal theory
and we recall that for the real, central interactions which we are considering one has RefE2=0,
ImfE3=O,---
while
RefB20,
Imfg3ZO,
etc.
The relationships (29)—(30) have important consequences. Let us first consider the weak coupling limit IV0Ia/hv1
=
IU0Ia/2k ~ 1
(31)
-
In this case the Born series converges rapidly and the relations (29) imply that the eikonal amplitude fE is a poorer approximation to the exact amplitude than the second Born quantity fB2~This is due to the fact that for ka ~ 1 we may write the exact amplitude for Yukawa-type potentials as f(k,~)=fBl(~)+~
+
k I fB2
I
~)+i~?~) 2 k3
+...,
(32)
\k fB3
while the eikonal amplitude has the form fE(k,~)=fBl(~)+i~+~+-... k k2
(33)
Hence we see that neitherfB 2. However, since A is proportional to U~while Cis norfE are the correct to order k proportional to U~,it is clear2 that when potential strength ~U 0Iis small the quantity fB2 should be more precise than fE As the coupling increases we expect from the above discussion that the eikonal method should improve steadily. This is confirmed by detailed numerical calculations [7, 81. Finally, let us briefly consider the strong coupling situation, for which IV0LaIhv> 1 and 1V01/E> 1. In this case the Born series is useless. Despite the fact that the condition (5) is violated, the eikonal approximation is still quite accurate at small angles for a variety of interaction potentials [8]. This is undoubtedly related to the unitary character of the eikonal approximation. Before leaving these basic aspects of the eikonal approximation, we mention that various other forms of the method have been proposed in recent years. A few important papers dealing with this subject and with additional properties of eikonal methods are listed in refs. [10—21].
3. Eikonal approximation for several particle collisions 3.1. Glauber theory The extension of the eikonal approximation to many-body scattering problems was first proposed by Glauber in connection with high-energy, small-angle hadron—nucleus collisions. Consider a point particle A incident on a composite target B (for example an atom) which contains N scatterers. We assume that the relative motion of A and B is fast with respect to the internal motion of the particles in the target. Furthermore, we assume that the incident particle interacts with the target scatterers via two-body spin-independent interactions. The Glauber scattering amplitude for a small-angle direct collision leading from an initial target 10 > to a final state Im) is given in the center-of-mass system by 2b, (34) F~0= (k1/211i)f exp(iA b) (mI{ exp[ix~t(b, b1, -.., bN)]—l }I0) d the corresponding differential cross section being [2]
da~
2. 0/d~ = (k~/k1)IF~0I
(35)
Cf. Joa chain, Eikonal theory
363
Here A is the center of mass wave vector transfer, while r = b+z2 is the initial relative coordinate and
=
b1
+ Z~2
are the coordinates of the target particles (relative to the target center of mass). As in the case of potential scattering we shall choose the z axis along the bisector of the scattering angle. The total Glauber phase shift function x~t(b,bl,.-.bN)=~
~(b—~),
(36)
is the sum of the phase shifts x1 contributed by each of the target scatterers. If we define the quantity 1’~0~(b, b1,
...,
bN) = 1
—
exp [ix~~(b,
b1,
.-.,
bN)]
(37)
then eq. (34) becomes
2b.
(38)
F~0= (ik/2iT)f exp(iA. b) (m I r~0~(b, b1, ..., bN)tO ) d Introducing the “single particle” quantities F 1(b—1~,i)= 1 —exp[ix1(b—b1)],
(39)
we may write eq. (37) in the form
or
r~0~(b,b1, -..,bN) =
—
j=1
1
—~
[1 —r1(b—~)],
r1 r1 + --- +
j~=l
(_~-‘
H
(40) (41)
~1.
j=1
Substitution of this expression into eq. (38) leads directly to an interpretation of the collision in terms of a multiple scattering expansion [2]. The term linear in I’~leads to the single scattering (impulse) contribution, while the next terms provide double, triple, --~scattering corrections. We note that the order of the multiple scattering can at most be N, reflecting the fact that the scattering is concentrated in the forward direction. It is important to note that the above generalization of the eikonal method, proposed by Glauber [2] makes no reference to interaction potentials. It is therefore of particular interest in high-energy hadronic collisions. If the basic two-body interactions are known, as in atomic physics, one can actually write the many-body eikonal wave function in terms of these interaction potentials and gain further insight into the scattering amplitude. For example, in the case of non-relativistic electron—atom scattering, we may write the full eikonal wave function (using atomic units) as exp (ikrr_Wki)
f V~(b,z’,X)dz’) ~0(X),
(4~)
where V~is the full interaction potential in the initial channel, X denotes collectively the target coordinates and is the initial bound-state wave function of the target. For a direct collision process (no exchange) such that V~= V~= V we then obtain the many-body eikonal scattering amplitude (corresponding to a transition I0)—~im>) as F~ =
—
(2~ 1 fexp (i A r) (m V(b,z, X) exp
(—
f V(b, z’, X) dz’)
(i/k~)
2b dz. 0
(43)
d
For elastic scattering processes, and if we choose the z axis along the bisector of the scattering angle, we may
364
Cf. Joachain, Eikonal theory
perform the z integral in eq. (43) to obtain directly the Glauber result F~k/27ri)fexp(iAb)(0I{exp[iX~t(b,bi, “-,bN)]—l}I0>d2b, with k =
1k
1!
=
~~44)
kf! and
x~t=—(lIk)fV(b,z,X)dz.
(45)
On the other hand, for inelastic processes, the simple Glauber result (34) can only be “derived” from eq. (43) neglecting the longitudinal momentum transfer. This neglect is not too serious for high-energy hadronic collisions (excluding violent inelastic events such as production reactions), but it leads to difficulties in electron— atom collisions, as we shall see below. by
3.2. Comparison of the Glauber and Born series. The eikonal—Born series method As in the case of potential scattering, we can expand the eikonal Glauber expression (34) in powers of the interaction potential and compare with the Born series. We shall consider here the case of elastic scattering. Starting from eq. (44), we write
F~=~~Gfl,
(46)
where 2b,
(47)
(k/27ri) We shall also write(i’~/n!)fexp(iA.b) (0I[x2~t]”l0>d =
FGJ
(48)
~‘Gn-
=
The Born term of order n in the interaction V is given by ~‘Bn
=
where ~ write
(2ir)2 (kf, 0 I VG~~ V... V1k =
FB,=~
(E
—
1, 0>, (49) H0 + iE)~is the unperturbed Green’s function of the system, with H0 = H— V. We shall also
(50)
~‘Bn~
Assuming that the Born series converges, the exact, direct (no exchange) elastic scattering amplitude is given by F~=~~Bfl.
(51)
For electron—atom scattering, the total Glauber phase shift function X~tmay be obtained explicitly. One finds 2)], x~t(lIk)Elog[l —2(b1/b)cos(~i—Ø)+(b~!b
(52)
Cf. Joachain, Eikonal theory
where we have written b
F~= (k/i)
365
(b, 0) and b1 (b~,Ø~).Hence
f J0(~b)(UI [exp(iX~t)—1] 10> b db,
(53)
f .J0(z~b)(OI[X~~]nIO>bdb.
(54)
and 5/n!) =
(k/i)(i’
With the choice of z axis which we have adopted, it is a simple matter to show that for all scattering angles FB1 =FG1.
(55)
In addition, we_also note from eq. (54) that, as in the case of potential scattering, one has Re FG2 = 0, Im FG3 = 0, while Re FB2 #0, ImFB3 * 0. Thus important terms (in an expansion in powers of k’) are missing from the Glauber amplitude. As we shall illustrate in section 4, this fact, together with an unphysical divergence of Fel in the forward direction, implies that the Glauber method is unreliable for the treatment of electron—atom collisions. However, Byron and Joachain [22] have shown recently that it is possible to combine the Born and the eikonal Glauber series in order to keep all the terms which contribute to a given order in k~ in the differential cross section. Their eikonal—Born series (EBS) method, which also includes exchange effects, will be discussed in more detail below. 3.3. The eikonal optical model We now turn to eikonal methods based on the optical model formalism. We begin by considering the elastic scattering of electrons by atoms at intermediate and high energies. Neglecting first exchange effects, we write the optical potential through second order in the full interaction V as [23, 24] =
+
~
2~ = E
(OIVln>(nIVIO)
with v~’~ =(OIVIO> and v~ n#0 E K_(Wn_Wo) In these expressions the electron—atom interaction is given by V_Z/r+E
- ,
c—k
(56)
o~.
(57,58)
+1
1/tr—r 11,
(59)
and the summation appearing in eq. (58) runs over all the intermediate stateselectron of the target initial state) 2 is the incident energy(except and K the its kinetic having eigen-energies wn. Furthermore, the quantity E = ~k energy operator. At intermediate and high energies one may replace in eq. (58) the energy differences wn w 0 by an average excitation energy i~.We then obtain in configuration space the non-local, complex, second-order expression 2~Ir’>= G~7~(k’,r,r’)A(r,r’), (60) —
(rIv~
where G (k’, r,byr’) is the free propagator corresponding to the wave number k’ A(r,r’) is1~ given
2 =
—
2i~)1/2and the object
(k
A(r,r’)=2(OIV(r,X)V(r’,X)I0)—(OIV(r,X)I0)(0IV(r’,X)I0). The Schrödinger equation satisfied by the scattering wave function
(61) is then seen to be
366
Cf. Joachain, Eikonal theory
[K + V(’)(r)
—
~k2]
ii4~+fG~~)(k’, r, r’)A(r, r’) ~,14~(r’)dr’ = U.
(62)
A study of this equation in the eikonal approximation has been made by Joachain and Mittleman [25,261 who showed that absorption effects due to the imaginary part of the potential (or, in other words, induced by unitarity from the open channels) play an important role in small angle elastic electron—atom scattering at intermediate energies. The scattering amplitude obtained by Joachain and Mittleman has the form Fei
r(k/2iri)fexp(iA.b) {exp[i~~t(b)]—l}
d~b,
(63)
f dz f dz’ exp[—i(k—k’)(z—z’)] A(b, z; b, z’).
(64)
where
x~~(b) = —(1/k)
f V(’)(b, z) dz
+
(i/2kk’)
The first term on the right of eq. (64) is simply the phase shift function arising from the static potential ~ The second term accounts for polarization and absorption effects. It is interesting to note that the Glauber theory of section 3.1 yields the result =
(k/2iri) fexp (iA
b) [exp [ix~~~(b)]—1] d2b,
(65)
with exp[ix~~t(b)}‘~(0Iexp[i~~t(b,bi, -.-, b~)]IU>,
(66)
and
f
X~~t(b)~(l/k) V~t(b,z)dz—(l/k)
f V(1)(b,z)dz+(i/2k2)
fdz
fdz’A(b,z;b,z’)+-.-.
As we should expect from the analysis of section 3.1, the Glauber expression (65) suffers from serious deficiencies. It leads to a forward direction divergence of the scattering amplitude and does not account properly for the long range (polarization) effects. A careful analysis of eqs. (63)—(64), recently made by Byron and Joachain [27] shows that none of these difficulties are present in the eikonal optical model. 3.4. The eikonal distorted wave method The optical model method may also be used together with the two-potential formalism of Gell—Mann and Goldberger [28] to analyze inelastic or rearrangement atomic processes [29—31].The basic idea is to split the interactions V 1 and Vf appearing respectively in the initial and final channel as V~=U~1-W~,
VfUf+Wf,
(68a,b)
where U~and Uf are distorting potentials which are usually chosen to be optical potentials describing elastic scattering in the initial and final channel. To lowest order in the “primary” interactions W~or Wf the transition matrix element Tba for a npn-elastic collision process leading from the state la>~1k1, 0> to the state lb>~Ikf, n>is then given by the DWBA (Distorted wave Born approximation) expression T~WBA ~
(69)
where x~and ~ are the distorted waves corresponding to the distorting potentials U1 and Uf. If we replace the distorted waves by their approximate eikonal expressions, we obtain the eikonal DWBA matrix element of Chen et al. [29], namely
Cf. foachain, Eikonal theory
T~= (2if)3 fexp(i A r) exp{i[A~(b,z)
+ Af(b,
z)]
367
}V 00 (b, z) dr,
(70)
where A1(b~z)=—(1/k~)fU~(b,z’)dz’,
Af(b,z)=—(l/kf)f Uf(b,z’)dz’,
(72)
and Vn0(b,z)
=
(nIVIU>.
(73)
It is worth noting that this method leads to reasonably simple expressionswhich take into account explicitly the longitudinal momentum transfer, allow the evaluation of exchange effects and may be applied to complex target atoms. Before we conclude this section, we want to mention the very interesting approach developed recently by Bransden et al. [31—34] to analyze the scattering of charged particles by atoms. Starting from the set of closecoupling equations, Bransden et al. retain explicitly a certain number of important states in a truncated expansion of the wave function. Second-order potentials, similar to those discussed in section 3.3, are then introduced to account for the coupling with the remaining states. This method, which uses the eikonal approximation to simplify the resulting equations, has already been applied successfully to the scattering of electrons and protons by atomic hydrogen and helium.
4. Electron scattering by atomic hydrogen and helium at intermediate and high energies 4.1. Electron scattering by atomic hydrogen We begin by considering elastic collisions, and follow the treatment of Byron and Joachain [22] who have carried out a detailed comparison of the Born and Glauber eikonal series. We first recall that the Glauber elastic amplitude (44) diverges logarithmically as the magnitude ~ of the momentum transfer tends to zero [35,36]. Moreover, one has FBi = FG1 at all scattering angles [see eq. (55)]. Let us now compare the higher terms of the Born and Glauber series. By using an average excitation energy ~ it is possible to reduce the second Born term FB2 to an expression which can be evaluated in a straightforward way. In fact, for simple target atoms one may even include a few states exactly in the summation on intermediate states which appears in FB2 finds [37]. that Of particular interest the limit of Im large values of klike for k~ which, scatter2),one Re FB2 varies likeis k~, while FB2 behaves logatk.small We remark ing (0 < 2i~/k thatangles this behaviour of FB2 is different from that found in section 2 for the case of potential scattering. This is due to the role played by the long-range forces at small scatteringangles. In particular, we stress that Re FB2 now gives the dominant correction to the first Born differential cross section at small angles. At larger angles one obtains the “potential scattering” behaviour of section 2. That is, Re FB2 varies like k2 and Im FB2 like k~ for large values of k. Again the term Re FB2 is essential to correct in a consistent way the first Born term FBi. Since the term Re FB2 is completely missing from the Glauber series we expect the Glauber method to be unreliable for this problem. Let us now pursue our comparison of the Born and the Glauber series. The second-order term FG2, which is purely imaginary, diverges logarithmically at ~ = 0. The corresponding quantity i Im FB 2 is finite at ~ = U but would diverge in the forward direction if i~(the average excitation energy) were set equal to zero. In fact, although the quantities Im FG2 and Im FB2 differ substantially at very small momentum transfers, a detailed study [22] of these two quantities shows that otherwise they agree very well, even at 0 = 1800 and for rather low values of k. This is not surprising in view of the fact that the large-angle scattering is mainly governed by the static po-
368
Cf. foachain, Eikonal theory
tential V~’~ = (UI yb>, for which the relation (30a) holds. For n > 3, the terms EGO of the Glauber series (46) are finite, even at ~ = 0. It is therefore very likely that these terms should agree with the corresponding terms of the Born series (i.e. FG3 with Re FB3, FG4 with i Im FB4, etc.). Since the direct evaluation of the quantity Re FB3 (which yields an important contribution of order k2 to the differential cross section) is a formidable task, it is therefore reasonable to use FG3 in place of Re FB3. Thus Byron and Joachain [22] have suggested to use for the dfrect amplitude F~the eikonal-Born series expression
FFBl+ReFB2+i~G3+iImfB2+....
(74)
Before we compute differential cross we recall that the leading (Ochkur) term [38] of the firstoforder 2 sections, at small momentum transfers. Therefore, a consistent calculation the exchange amplitudecross is also of order elastic differential section dae k 2 (or E~,where E is the incident electron energy) re1/d~lthrough order k by GOch. For an unpolarized incident electron beam quires the inclusion of this Ochkur term, which we denote and target atom, and if no attempt is made to distinguish the final spin states, we then have, through order k2 dGel/d~Z~IEe1+GOchI2 +~IFei GOch!2,
(75)
where F~is given by eq. (74). A detailed comparison of the above eikonal—Born series (EBS) method with other approximations (first Born, Glauber) and with the experimental results of Teubner et a!. [39, 59] has been made by Byron and Joachain [22, 60]. They showed that the Born and Glauber angular distributions differ substantially at intermediate energies from the EBS expression (75). Moreover, the Born and Glauber methods predict the same angular distribution for electron and positron scattering, whereas the EBS method yields significant differences (at small angles) between the electron and positron curves. Unfortunately, the measurements of Teubner et al. [39] are not absolute and have been made at rather large angles, where the differences in the slopes of the various theoretical predictions are rather small. The more recent experimental data of Teubner, Lloyd and Wiegold [40], at an incident electron energy of 50 eV, and relative to molecular hydrogen, cover a wider angular range. They are in good agreement with the EBS results [41]. We now consider briefly some inelastic transitions induced in atomic hydrogen by electron impact. Calculations using the Glauber approximation (34) have been made by several authors [40—45]and reviewed some time ago by Gerjuoy [46]. It is clear from the analysis of section 3.2 that the Glauber amplitude (34) also suffers from severe deficiencies when it is applied to inelastic electron—atom collisions. For example, in the case of a transition is 2s, the Glauber amplitude diverges for ~ = 1k 1 k~-I= 0. Although this value of the momentum transfer is unphysical, it is clear that at intermediate and high energies the presence of this divergence influences the shape of the Glauber differential cross section near the forward direction. Moreover, we may also develop the Glauber amplitude (34) as a Glauber series in powers of the interaction [as in eq. (47)]. Comparison with the corresponding Born series shows again that important terms are missing from the Glauber amplitude. Finally, the Glauber expression (34) neglects the longitudinal momentum transfer, an approximation which leads to undesirable features such as: L~.m= ±1 selection rule for s p transitions and identical results for the excitation of atoms by electron and positron impact. Using the more general Glauber expression (44) (which includes the longitudinal momentum transfer), together with Monte-Carlo integration techniques, Byron [47]has found significant differences between the excitation of the 2s state of hydrogen by electron and positron impact. The excitation of the n = 2 states of atomic hydrogen has also been studied recently by used the eikonal DWBA method [29, 30]. Chen et al. [29]have used static distorting potentials, while Joachain and Vanderpoorten [30] have included absorption effects by using Glauber optical potentials. As one should expect on physical grounds, the eikonal 2p DWBA particularly reliable for transitions. example, the totalwith crossthe sections for excitastate isobtained by Joachain andallowed Vanderpoorten is inFor excellent agreement experimental data tionLong of the of et al. [48]. On the other hand, the eikonal DWBA method yields rather poor values for the polarization of the light emitted from the 2p states. An extension of the method, which includes all the important final state inter—~
—
—~
Cf. foachain, Eikonal theory
369
actions (as in the method of Bransden eta!. [31—34])would be very desirable. Finally, we note that the eikonal DWBA method and the approach of Bransden et al. predict differences between electron and positron scattering. 4.2. Electron—helium collisions Let us now consider the scattering of fast electrons by helium, where absolute measurements of differential cross sections have recently become available [49—51]. We begin by analyzing elastic electron—helium scattering and follow again the EBS treatment of Byron and Joachain [22]. The direct amplitude is still obtained from eq. (74), but the differential cross section is now given by
(76) dciei/d&2 = IF~ GOchl2. A detailed comparison of the EBS results with the small-angle data of Bromberg [49], Chamberlain et al. [50] and Crooks and Rudd [51] has been made in ref. [22] for incident electron energies ranging from IOU eV to 500 eV. This paper also compares the EBS values with those obtained from the first Born approximation and with the Glauber results of Franco [52]. The EBS results, which agree very well with the data of Bromberg at 500 eV, are consistently better than the Born or Glauber results. More recently, Byron and Joachain [53] have extended their analysis to include large angle scattering. As one should expect, the scattering is mainly governed in this angular region by the static potential y(l). It is gratifying to note that the EBS method also yields good results at large angles. However, this last feature depends delicately on cancellations between higher order terms of perturbation theory, which play a significant role for large angle scattering. These cancellations, although present in the elastic electron—helium case cannot be expected to occur in other situations. We now consider eikonal optical model calculations of elastic electron—helium scattering. By solving eq; (62) within the framework of the eikonal approximation, Joachain and Mittleman [25—26] showed explicitly that absorption effects play an important role in small angle elastic electron—helium scattering at intermediate energies. More recently, Byron and Joachain [27] have analyzed in detail the role of long range forces in the eikonal optical model. Their eikonal optical model calculations for elastic electron—helium scattering, which are free of any phenomenological parameters, are in excellent agreement at small angles with the EBS calculations and the experimental data. Let us now describe briefly a few inelastic electron—helium processes. For the process e + He (1 1S) e + He(21 S) the most reliable results are the four-channel calculations of Berrington et a!. [34]. The eikonal DWBA results of Joachain and Vanderpoorten [54] agree with the experimental results [50, 55] only at high incident electron energies. Moreover, the 21S_ 21P coupling, which strongly influences the angular distribution at small angles [34] is missing in the eikonal DWBA method. In the case of the process e + He (1 1S) ~ e + He (21 P) the strong 11S 21 P coupling completely dominates, so that reliable results are obtained from the eikonal DWBA method [54]. The eikonal calculations of Byron [47], using eq. (43) are also in good agreement with the experimental results [50, 55—57]. Finally, we come to a process of considerable interest, namely the excitation of triplet states of helium by electron impact. For example, let us consider the reaction —
—
e
+He(i15)-+e
+He(23S).
(77)
This process is a pure rearrangement (knock-out) collision provided that very small spin-dependent interactions are neglected. Although this reaction has already received a large amount of attention, no satisfactory explanation was given for the forward peak observed [55] in the differential cross section for incident electron energies ranging from 100 eV to 225 eV. In particular, the first Born and the Ochkur approximations completely fail. The reasons for this failure have been given recently by Byron and Joachain [58], who have also performed a many-body eikonal calculation (using the Monte-Carlo integration method) and obtained encouraging preliminary results.
370
Cf. foachain, Eikonal theory
However, much work remains to be done before a complete understanding of simple rearrangement collisions at intermediate and high energies will be achieved. It is to be hoped that eikonal methods will help to solve these problems, which are among the most difficult (and fascinating) ones of several particle dynamics.
References [1] G. Mo1i~re,Z. Naturforsch. 2A (1947) 133.
[21 R.J. Glauber, in Lectures in theoreticsl physics, edited by W.E. Brittin and L.G. Dunham (Interscience Pub!., New York, 1959) Vol. 1, p. 315. [3] See for example R.J. Glauber, Theory of high-energy hadron—nucleus collisions, in “High Energy Physics and Nuclear Structure” (Plenum Press, New York, 1970); K. Gottfried and J.D. Jackson, Nuovo Cimento 34 (1964) 375; A.S. Goldhaber and C.J. Joachain, Phys. Rev. 171 (1968) 1566. [4] See for example J.C.Y. Chen and K.M. Watson, Phys. Rev. 174 (1968) 152; 188 (1969) 236; J.C.Y. Chen, C.J. Joachain and K.M. Watson, Phys. Rev. AS (1972) 1268. 15] C.J. Joachain and C. Quigg, National Accelerator Laboratory Report NAL-THY-99, 1973; Rev. Mod. Phys. (to be published). [61 R.J. Moore, Phys. Rev. D2 (1970) 313. [71 F.W. Byron and C.J. Joachain, Physica 66 (1973) 33~ [8] F.W. Byron, Ci. Joachain and E.H. Mund, Phys. Rev. D8 (1973) 2622. [9] A. Swift, Phys. Rev. D (to be published). [10] BJ. Malenka, Phys. Rev. 95 (1954) 522. [11] L.I. Schiff, Phys. Rev. 103 (1956) 443. [12] D.S. Saxon and L.I. Schiff, Nuovo Cimento 6 (1957) 614. [13] R. Blankenbecker and M.L. Goldberger, Phys. Rev. 126 (1962) 766. [14] T. Adachi and I. Kotani, Progr. Th. Phys. Suppi. Extra no. 316 (1965), T. Adachi. Progr. Th. Phys. 35 (1966) 463, 483. [15J L. Wilets and S.J. Wallace, Phys. Rev. 169 (1968) 84. [16] R.L. Sugar and R. Blankenbecker, Phys. Rev. 183 (1969) 1387. [17] M. Levy and J. Sucher, Phys. Rev. 186 (1969) 1656. [181 H.D.I. Abarbanel and C. Itzykson, Phys. Rev. Letters 23 (1969) 53. [19] S.J. Wallace, Phys. Rev. Letters 27 (1971) 622. [20] A. Baker, Phys. Rev. D6 (1972) 3462. [21] S.J. Wallace, Ann. Phys. (N.Y.) 78 (1973) 190. [22] F.W. Byron and C.J. Joachain, Phys. Rev. A8 (1973) 1267. [231 M.H. Mittleman and K.M. Watson, Phys. Rev. 113 (1959) 198; Ann. Phys. (N.Y.) 10(1960) 268. [24] M.L. Goldberger and K.M. Watson, Collision theory (J. Wiley and Sons, New York, 1965), Chapter 11. [25] C.J. Joachain and M.H. Mittleman, Physics Letters 36A (1971) 209. [26] C.J. Joachain and M.H. Mittleman, Phys. Rev. A4 (1971) 1492. [27] F.W. Byron and C.J. Joachain, Phys. Rev. A (to be published). [28] M. Gell-Mann and M.L. Goldberger, Phys. Rev. 91(1953) 398. [291 J.C.Y. Chen, C~J.Joachain and K.M. Watson, Phys. Rev. AS (1972) 2460. [30J C.J. Joachain and R. Vanderpoorten, J. Phys. B6 (1973) 622. [31] BH. Bransden and J.P. Coleman, J. Phys. B5 (1972) 537. [32] B.H. Bransden, J.P. Coleman and i. Sullivan, J. Phys. B5 (1972) 546. [33] J. Sullivan, i.P. Coleman and B.H. Bransden, i. Phys. B5 (1972) 2061. [34] K.A. Berrington, B.H. Bransden and J.P. Coleman, J. Phys. B6 (1973) 436. [351 F.W. Byron and C.J. Joachain, unpublished (1967). [36] V. Franco, Phys. Rev. Letters 14 (1968) 709. [37] See for example A.R. Holt, J.L. Hunt and B.L. Moiseiwitsch, i. Phys. B4 (1971) 1318; M.J. Woouings and M.R.C. McDowell, J. Phys. B5 (1972) 1320; F.W. Byron and C,i. Joachain, ref. [22]. [38] V.!. Ochkur, Zh. Eksp. Teor. Fig. 45 (1963) 734. (Eng. trans.: Soviet Physics JETP 18 (1964) 503). [39] P.J.O. Teubner, K.G. Williams and J.H. Carver, quoted in H. Tai, P.i. Teubner and R.H. Bassel, Phys. Rev. Letters 22 (1969) 1415; 23 (1969) 453 (E). [40] A.S. Ghosh and N.C. Sil, Indian J. Phys. 43 (1969) 490. [41] A.S. Ghosh, P. Sinha and N.C, Sil, J. Phys. B3 (1970) L58.
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H. Tai, R.H. Bassel, E. Gerjuoy and V. Franco, Phys. Rev. Al (1970) 1819. K. Bhadra and A.S. Ghosh, Phys. Rev. Letters 26 (1971) 737. V.B. Sheorey, E. Gerjuoy and B.K. Thomas, i. Phys. B4 (1971) 657. E. Gerjuoy, B.K. Thomas and V.B. Sheorey, J. Phys. B5 (1972) 321. E. Gerjuoy, in The physics of electronic and atomic collisions, edited by T.R. Govers and F.J. de Heer (North-Holland Publ. Cy, Amsterdam, 1972) p. 243. F.W. Byron, Phys. Rev. A4 (1971) 1907. R.L. Long, D.M. Cox and S.S. Smith, i. Res. Nat. Bur. Stand. 72A (1968) 521. J.P. Bromberg, J. Chem. Phys. 50 (1969) 3906. G.E. Chamberlain, S.R. Mielczarek and C.E. Kuyatt, Phys. Rev. A2 (1970) 1905; see also L. Vriens, CE. Kuyatt and S.R. Mielczarek, Phys. Rev. 170 (1968) 163. G.B. Crooks and M.E. Rudd, Bull. Amer. Phys. Soc. 17 (1972) 131; G.B. Crooks, Thesis, University of Nebraska (unpublished). V. Franco, Phys. Rev. Al (1970) 1705. F.W. Byron and C.i. Joachain, Phys. Rev. A (to be published). C.i. Joachain and R. Vanderpoorten, J. Phys. B (to be published). L. Vriens, LA. Simpson and SR. Mielczarek, Phys. Rev. 165 (1968) 7. i.P. de Jongh and J. Van Eck, in Abstracts of papers of the ViIth ICPEAC, edited by L.M. Branscomb et al. (North-Holland Publ. Cy, Amsterdam, 1971) p. 701. F.G. Donaldson, M.A. Helder and i.W. McConkey, J. Phys. B5 (1972) 1192. F.W. Byron and Cl. Joachain, Physics Letters 38A (1972) 185. P.l.O. Teubner, C.R. Lloyd and E. Weigold, I. Phys. B6 (1973) L134. F.W. Byron and Ci. Joachain, I. Phys. B (to be published).