Inelastic scattering at high energy

Volume 69A, number 2

PHYSICS LETTERS

27 November 1978

INELASTIC SCATFERING AT HIGH ENERGY Adolph BAKER University of Lowell, Lowell, MA 01854, USA

and P.N. HOULE East Stroudsburg State College, East Stroudsburg, PA 18301, USA Received 28 June 1978

A derivation by Byron of the Glauber wavefunction for inelastic scattering without the explicit introduction of the nonmoving bound particle assumption is shown in fact to have this restriction implicitly.

The Glauber wavefunction ‘I’(R,r)

=

exp

r z 1 Li~i~ (i/flu) f V(R’~r)dZ’ju(r)~ —

where R is the incident particle coordinate, r the

The exact wavefunction in terms of a complete set of target energy eigenstates is 4’(R ,r) = ~m ~I1m(R) X U~fr), where

(1)

bound particle coordinate (both relative to the target nucleus), un(r) the initial state of the target system, and k~and v the incident wave vector and velocity magnitude, respectively, has been applied and tested in atomic scattering. For example, Gau and Macek [1,2] calculated inelastic electron—hydrogen atomic scattering in an “unrestricted” Glauber approxima. tion previously investigated by Byron [3Jand compared it to the conventional Glauber approximation calculations of Thomas and Gerjuoy [4]. Eq. (1) was originally obtained by Glauber [5] in time-dependent formalism, and then generalized to this time-independent solution. Two crucial approximations in the derivation were his neglect, first of &/E, the ratio of energy transfer to incident energy, and second, of vt/u, the ratio of bound to incident particle velocity. What is interesting and suprising about Byrons’s [3] time-independent derivation of eq. (1), on the other hand, is that it never explicitly introduces the second approximation, namely, neglect ofbound particle velocity. To see how this comes about, we reproduce his result, but in the position representation.

~

(R)

=

e

ik~•R ‘ ~mn

—

(1/4

1r)f

ikmlR_R’I IR — R ~ Fm (R’) dR’,

2 Fm(R) = (2M/h2)fum(r)V(R,r)41(R,r)dr, 2k~]1/2 km = k1[l + 2M(c~ Cm)/h in terms of initial target energy state 6n and incident mass M. In Glauber theory one looks for a solution of the form Pm(R) = øm(R)exp[ikiR] ,where Øm(R) øm(1” Z) is to be determined. Following Byron, let øm(1’, Z) = e1t~mP~am(b,Z), where qrnn = k~ km. (This differs slightly from Byron’s definition of but agrees to first order in ~e/E, qm~~M(m 1~2k~.) Then ~ (b Z) = (2M/4~fl2) —

—

—

—

m

‘

mn

>< r~X~,[~k IR —R’I J

X Z~ exp ~

m

—

ik

(R —R’)]/IR —R’J

m

Z’)am,(b, Z’)dR’, (3)

m

where km is a vector of magnitude km in the Z direc. tion, and ~mm’ = (m~VIm’).Eq. (3) may be integrated

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PHYSICS LETTERS

by parts over angles just as for elastic scattering [5] with the result that to first order in (1/k1) am (b, Z)

=

~mn — (i/flu) ~

mel

,

fexp [iq~~~Z’]

X Q9,~,(b,Z’)crm(b, Z’) dZ’.

(4)

To obtain eq. (4), which is Byron’s result, the close-coupling assumption has been explicitly incorporated by summing over only those intermediate states m’cI, in which the set I includes states for which a(em Cm~)/hV~ 1, where a is the range of the potential. This corresponds to Glauber’s first approximation (applied now also to intermediate states), since a~e/hv= (k~a)(&/2E),and k~a~ 1. To obtain the factor (i/flu) in eq. we note 1 (4), to first order.that The (kmY~ = (k~ ~ close-coupling (k1y restriction is that implication of the “unimportant” states not meeting Glauber’s first assumption are uncoupled from the solution and make relatively little contribution to the scattering amplitude. To solve eq. (4) Byron replaces the exponential by unity, relying on this same approximation. Since for the close-coupled terms the exponents are ~ 1, this is perfectly reasonable. However, he then uses closure to obtain the solution, which, as may be yenfled by substituting in eq. (4) after the latter has been differentiated with respect to Z, is r z 1 cxm(R) = (mjexp [(~-~i/flu) V(R’~r)dZ’jIn). —

—

f

—=

But closure requires a complete set of basis vectors. The uncoupled terms must therefore be reinstated, but now without the exponential, despite the fact

27 November 1978

example, that the matrix elements ~mm’ initially fall off rapidly enough with increasing m’ that truncation of the series can be properly accomplished so as to include only terms that satisfy az~c/hv~ 1; however, eventually the decline in magnitude of matrix elements may proceed to level off, in which case the exponential, with its large exponent and rapid fluctuations, takes over, causing the series to converge to a result not markedly different from that of the truncated series. Replacing the exponential by unity may disrupt this process, forcing the infinite sum of “unimportant” terms to converge to a wrong answer, if indeed it converges at all. Thus an answer obtained by such a procedure is of uncertain reliability. In any case, when the resulting wavefunction i/i(R, r) in termsamplitude, of the am exponentials is finally introduced into the scattering again appear, and again are replaced by unity; once more closure is inyoked by Byron in the same way by reinstating the “unimportant” terms stripped of their exponentials, and the result of this summation over m is that the wavefunction as it appears in the scattering amplitude finally becomes equivalent to that of eq. (1). It is now possible to recognize Glauber’s additional requirement of negligible target particle velocity. If instead of following the somewhat questionable procedure described above, before carrying out the double summation over m and m’ we let k~ term by term in such a way that aLXe/hv 1 continues to be satisfied even while ~&eincreases also, then all the exponents do in fact go to zero, and eq. (1) is precisely obtained by means of the subsequent use of closure. But it does not follow that the same result would be obtained if the double summation were carried out first, and the limit k 1 taken afterwards. The limiting procedure described above may be seen to correspond to Glauber’s stationary target approximation. If the incident particle passes through the scattering center at infinite speed (and the infinity is taken in the appropriate manner), it is not surprising that the target particle influences the scattering as if it were frozen in position. Thus such a special limiting process is equivalent to both of Glauber’s assumptions, not just the first one. Since the restriction ut/v 1 is generally more stringent (by roughly the square root) than L~/E~ 1, this should not be overlooked. Thus in Gau and Macek’s calculations [2] for electrons incident on hydrogen at 50 eV for example, Ae/E-~1/5, -+ °°

~

—~ °°

that the exponent is no longer small. In fact, some of the dropped terms in the expansion of the exponentials become larger in magnitude than the term (unity) which has been retained. One could, of course, argue that since it is “unimportant” terms m’~Ifor which this happens, and the assumption is that these make negligible contribution to the answer, it makes no difference that they are incorrectly included. But the fact that these terms (of which there is an infinite number) make a negligible contribution when correctly evaluated provides no assurance that they remain unimportant when incorrectly summed, i.e., with exponentials replaced by unity. It could be, for 74

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Volume 69A, number 2

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which may reasonably be neglected compared to unity, whereas ut/u representing a fairly significant error. ~,

References

27 November 19Th

[2] i.N. Gau and J. Macek, Phys. Rev. A12 (1975) 1760. [3] F.W. Byron, Phys. Rev. A4 (1971) 1907. [4] B.K. Thomas and E. Gerjuoy, Math. Phys. 12 (1971) 1567. [5] R.J. Glauber, in: Lectures in theoretical physics, eds. W.E. Britten et al. (Interscience, New York, 1959) Vol. 1, p. 315.

[1] J.N. Gau and J. Macek, Phys. Rev. AlO (1974) 522.

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