Larry Spruch.

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»S y/averly Place, New York 3, N. Y.

^EET PR^^ NOV IS 1959



^ I ^ 1 1 ^ Institute of Mathematical Sciences


"^ Division of Electromagnetic Research


Low Energy Scattering by a Compound System:
Positrons on Hydrogen


Contract No. AF 19(604)4555






Institute of Mathematical Sciences
Division of Electromagnetic Research

Research Report No. Cl-hO


Larry Spruch

Leonard Rosenberg

Larry Spruch
Leonard Rosenberg

Sidney Borowitz '
. Acting Project Director

February 1959

The research reported in this article has been sponsored
by the Geophysics Research Directorate of the Air Force
Cambridge Research Center, Air Research Development
Command, under Contract No. AF19(60U)U555.

New York 1959

Requests for additional copies by Agencies of the Department of Defense,
their contractors, and other Govenunent agencies should be directed to the:


Department of Defense contractors must be established for ASTIA services or

have their 'need-to-know' certified by the cognizant military agency of their

project or contract.

All other persons and organizations should apply to the:


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The formali^sm, developed by Kato, which gives upper and lower

bounds on the phase shift for the scattering of a particle from a

center of force is shown to have useful applications when the

scatterer is a compound system. In particular, the problem of

low energy positron scattering from, atomic hydrogen, with zero

total orbital angular momentum, is studied. It is shown that at

zero energy the ordinary variational calculation, which ignores

second order contributions, provides a bound on the scattering

length, from which a bound on the cross section is deduced. For

non-zero energies a bound on the phase shift may similarly be

obtained, but for a fictitious problem \-ilth cut-off potentials.

If the energy is sufficiently small (less than 3 e.v., say) the

error thus incurred is expected to be negligibly small. Numerical

calculations performed at k = and ka =0.2 lead to the result


that the effects of polarization are large enough to cause the
positjxsn to be on the whole attracted to the hydrogen atom.

It is shown, independently of the Kato formalism, that in
general a bound on the scattering length may be obtained from the
ordinary variational calculation provided no composite bound
state exists for the system.

A preliminary report was given at the Washington Meeting Ox the
American Physical Society in April, 1958 [Bull. Am. Phys. Soc,
Ser. II, 3, 171 (1958)]

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Table of Contents


1. Introduction 1

2. The Kato Method

A. The true problem 6

B. The associated eigenvalue problem 11

3. Application to Positron Scattering from Hydrogen 13
U, Numerical Calculation

A. No polarization approximation, k = 1?

E. Polarization considered, k = 21

C. Fblarization consider-ed, k y' 27
0. Discussion - . 33
Appendix A 36
Appendix B 39
References U3

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1. Introduction

It is somewhat disconcerting that more than thirty years after the
advent of quantum theory there is in common use no general method
that may be applied v/ith confidence to the calculation of the cross
section for scattering by a compound system. This is true even for
compound system.s consisting of only two particles. In the few cases
in which the calculation is on finn theoretical grounds, such as the
low energy scattering of neutrons from chemically bound protons'- ■* j
the reliability of the calculation depends on some quite special
properties of the forces and of the scattering system.

It is of course true that there are a few scattering problems
involving compound scatterers which do not possess any special properties
for which the calculation can be accepted with a high degree of
confidence. An example is the scattering of electrons by hydrogen
atoms '- -^ ' '•-^ ( fo r restricted ranges of scattering energies).
The fact remains, nervertheless, that these calculations do not
give any indication cf their reliabilityj they do not provide any
estimate of the error nor, contrarj' to the situation in the evaluation
of the energy of the ground state, do they provide an upper or a lower
bound on the cross section. Rather, our confidence is based on the
fact that variational calculations performed with a variety of trial
functions lead to results which do not differ appreciably among
themselves, nor with a result based on the accurately determined bound
state wave function of H"*-^, The use of a variational method is in
itself by no means sufficient to ensure any accuracy whatever. Since

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the effects of polarization can be profound, it may be quite difficult
to determine a suitable form for the trial function, and of course even
in a variational calculation a poor trial function generally gives
a poor result. Further, the inclusion of more parameters in the trial
function need not improve the result. In applications of the Kohn
and Hulthen forms of the variational principle certain consistency
criteria are frequently used to check the validity of the calculation' - ^.

It is Quite clear, however, that these criteria are by no means
completely satisfactory. The knowledge that the consistency ratio is
very different from unity is, to be sure, a useful if negative piece
of information; the difficvJ.ty is that a consistency ratio close to
unity may iiriply notl'dng more than an accurate calculation within the
limitations, however severe and inappropriate, of the assumed form of
the wave function. Thus, Bransden, Dalgamo, John, and Seaton"- -" offer
the example of a trial function for the electron-hydrogen scattering
problem in the form of a properly symmetrized product of one-particle
functions. The assumed form gives rise to certain (approximate)
one-body equations. If the one-particle functions are determined
exactly as the solutions of these equations, perfect consistency will
be achieved even if the resultant trial function bears no relationship
to the time function, h still simpler example of the failure of the
consistency criteria is found in the present e H problem. A trial
function involving no polarization of the hydrogen atom will lead to
perfect consistency if the positron wave function is the exact solution
of the static problem, and yet polarization plays a crucial role. The
weakness of the consistency checks is well recognized, but though all

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agree with s previous author"- -' that "a foolish consistency is the
hobgoblin of little minds", these checks have nevertheless been
frequently applied simply because no better standards have been available.

In the case of scattering by a static central potential, where the
cross section, cr, consists of a sum of partial cross sections, where r^
and r- are the position vectors cf the two particles measured from the
center of force. Defining e = (2mAi^)|E2 I, k^ = (2m^ )T^, and
W = -(2m/fi )V for all three " potentials" W,, W^ and W^g* ^^ ^^^®

(•? n\ P - ^ ^ JL. 3 ^ 2 _B_ ^ ,J^ . _1_ ^ e ,, 2^ 3

^'•'^ ^"i;r? ;7 ^ '2 ^^^ r2^^^^'-^^~p

.W^(r^).W2(r2)-W^2W2^ - k^ - e^ ,

where p is the cosine of the angle between r-, and r2. It is sometimes
more convenient to write the total kinetic energy operator, T, as

(2.3) .j„A>^i= ^ J- ^i'^*T^ -/ J:

r^ 1 1 rg ^ -i

2 ar,„ ^12 ar,o r,r,o bj^hr

r^^- 12 ^^ °^12 '■L'12 "^n-^12

(r 2+r ^-r M 2

^2^12 ^^2^^12

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The exect solution, iL, must satisfj"- the boundary conditions

^ _> Kr^) cos(kr^+e) + cotC'vr-e) 3in(kr^+&) , as Tt^ ->oo

(2.U) Uq -> , as r^

Uq = at r^ =

where R(r„) is the exact ground state wave function for the bound
particle, normalized such that

^2 —' 00

(2.5) 2 J r^^dr^ R^(r2) = 1.

The normalization parameter, ©, satisfies < & < n but is otherwise
arbitrary. Tlie true phase shift, "fT , is determined by (2.1), (2.2)
and (2, It). (We have here, and in the following, dropped the subscript
zero on'vi ,)

We now introduce a trial function, UgCrn* r^, ^■]2^ > which satisfies
the same boundary conditions as iL but with the true phase shift, 7) ,
replaced by a trial phase shift, >) . Note that this still involves
the exact wave function, R{rJ). The identity

(2.6) k cot(>i-9) = k cot(v|-e)- u^Xugdr + w^i^w^dT


then carries over from the one body case where now df = dr, r^ drp dp

and where

(2.7) ^Q^^2.f^2'^12^ "■ '%^^1'^2*^12^"^^^1»^2'^12^*

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Since w^j is a first order term, Eq. (2.6) constitutes a varistional
principle for cot (y)-9) upon dropping the last term.

The requirement that the asymptotic form of u„ be proportional
to the exact wave function R(r_) is evidently extremely restrictive.
It is, however, necessary in order that the decomposition

(2.8) ^^\ d*^ = Xu^oCugdr-J Wg^w^dt

may be used. If the requirement is dropped the integrals on the right
hand side of the above equation will not separately exist, each of
them diverging for large r,. (The sum of the two integrals is of course
still finite.) The required knowledge of the ground state wave
function formally eliminates the possibility of applying the Kato
method to practicalDy all scattering systems consisting of more than
two particles. This difficulty, present in the ordinary variational
problem as well, can fortunately be partially circumvented. Assuming
a knowledge of the ground state wave function we introduce into the
appropriate integral expressions a trial function which is the sum
of an "inside wave function" ^ '-J arid a term which provides the correct
asymptotic fonaj the operation of X on the asjTnptotic form is carried
out formally. At this stage, we can replace the ground state wave
function by some approximation to it without introducing any
infinities. The error involved will be reproduced in our estimate of
(or bound on) k cot (y[-©) to first order, i.e., this error is not
reduced by the fact that a variational formulation is being utilized.
Further, the question as to whether the bound had been preserved would
have to be examined. However, for the cases for which the ground state

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wave function is knoi'm with considerable precision, (He for example),
the replacement should be irrelevant.

B. The Associated Eigenvalue Problem.
Consider the equation

(2.9) (£0^ + i^g p 0Q =

where p is a non-negative function to be chosen such that the scattering
problem determined by Eq. (2.9) and by appropriate boundary conditions
is characterized for each value of ii,_ by one real phase shift, 6(iJ.g).

Then there exists an infinite set of discrete eigenvalues u a,

corresponding eigenfunctions r^ normalized by



(2.10) k"^


0„ a 0„ a P dr = 6 , in,n = 0, ±1, ±2, ...

n,w m,w nm

where 5 is the Kronecker 5-symbol, such that

^ -> const. R(rr,) sin(kr, + & + nn), as r, -> oo
n,fcy ^1 -L

(2.11) 0^ ^ -> , as Tg -> 00
^n,e = ° ' for r^ = .

We denote the smallest positive eigenvalue of the set p. „ ^y ^■o ^""^ ^^'^
smallest (in magnitude) negative eigenvalue by -Pq. The Kato inequalities

(2.12) ^^'^ j (JCuq)^ p"-^dr< I WqXwq d-t < Pq"^ J OCu^)^ p'-'-dr .

This formal development follows that for the one body problem,
mutatis mutandis > (Some questions of completeness are discussed in
Appendix B). However, the actual restrictions on p, referred to above.

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involve considerations peculiar to the many body problem. We observe first
a restriction which does carry over from the one body case, namely, that
p must vanish faster than l/r-, as r, becomes infinite in order that
normalization according to Eq. (2,10) be possible and the phase shift,
6(|j.), be defined. The choice p = p(rp), for example, is eliminated'"''.
To eliminate pick-up in the associated eigenvalue problem, which is
defined for an infinitely broad range of potential strengths, there are
two alternatives.

a) Choose p to be independent of r, „, at least for r, and/or r^ -> oo.
Then E^. will be independent of \

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