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AFCRL-TN-60-687 ^^ Waverly Plaee, New York 3, N. Yj


^ I |J |Y ^ Institute of Mathematical Sciences

;^ >->^ X Division of Electromagnetic Research


Bounds on Scattering Phase Shifts
For Compound Systems


Contract Nos. AF 19(604)4555

DA 30-069-ORD-2581
Nonr-285(49), NR 012-109

fS OCTOBER, 1 960

Institute of Mathematical Sciences
Division of Electromagnetic Research
Research Report No. CX-52


Leonard Rosenberg and Larry Spruch

The research reported in this paper has
heen Jointly sponsored by the Geophysics
Research Directorate of the Air Force
Cambridge Research Center^ Air Research
and Development Command, liijder Contract
No. AF 19(6o4)lj-555, and the Office of
Ordnance Research ^mder Contract No. DA-
5O-O69-ORD-258I, Project No. 236O, and
the Office of Naval Research under Con-
tract Nonr-285(^9), NR 012-109-

Leonard Rosenberg

Larry Spruch

Morris Kline, Director


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


Department of Defense contractors must be established for ASTIA services
or have their 'need-to-knov' certified by the cognizant military agency
of their project or contract. All other persons and organizations apply
to the:



An extension of recently developed methods detennlnes a rigorous
upper iDound on (-k cot -t]) , where y\ is the phase shift, for the general
one channel scattering process. The method unfortunately requires trun-
cation of the various potentials, but It should generally be possible, in
practice, to so truncate the potentials that the difference between the
phase shifts of the original problem and of the problem for which a bound
is obtained is Insignificant.

In the course of the development it is necessary to introduce, for
compound system scattering, an absolute definition of the phase shift,
not simply a definition modulo n. The definition chosen is to take the
projection of the full scattering wave f\mction on the ground state wave
function of the scattering system, and to treat the resultant one coordi-
nate wave function as if it were the scattering wave function for a par-
ticle on a center of force. Though Irrelevant with regard to the deter-
mination of a bound on cot t), it is Interesting that at least for some
simple cases this definition automatically increases the phase shift by
at least « whenever the Paul! prinicple introduces a spatial node into
the scattering wave function. The triplet scattering of electrons by H
atoms provides an example.

Table of Contents


1. Introduction 2

2. Definition of the Phase Shift 5
An Application: Triplet Electron Hydrogen Scattering 6

5. Definition of the Problem for vhlch the Boimd is to

be Obtained 8

h-. Bound on k cot(Ti-0) ^^'^ on t) ,11

A - The Associated Potential Strength Eigenvalue Problem 11

B - The Conditional Inequality ik

5. The Use of Energy Eigenf unctions 17

References 22

In a series of papers , it has been shovn that for the one channel
problem^ to which the present paper is restricted^ it is possible to re-
place certain variational principles for scattering theory by much more
powerful minimum principles. More specifically, a rigorous upper bound
on the scattering length. A, was first obtained for the relatively simple
case of the (zero energy) scattering of a particle by a static central
potential which is not sufficiently attractive to bind the particle . Us-
ing the Hylleraas-Undheim theorem , it proved to be possible to extend
the method to the scattering of a particle by a static central potential
(henceforth to be denoted simply as one body scattering) when bound states
do exist. (The interaction will the electromagnetic field is assumed to
have been turned off, so that capture can not take place.) The generali-
zation to the scattering of one compound system by another, for zero init-

ial relative kinetic energy of the two scattering systems, was trivial .

The method was then extended to treat positive energy scattering .

The presentation was there restricted to the one body problem, taking

into account bound states when they exist. The quantity bounded from

above in this case is (-k cot t^) , where r\ is the phase shift; this of

course reduces to A as k goes to zero. Unlike the situation at zero

energy, it was unfortunately necessary to restrict the potentials to

those which vanish identically beyond some given point, R. While the

rigorous bounds obtained do not then generally apply directly to the real

problem of interest (account must be taken of the truncated portion of

the potential) it should be emphasized that the effects of the artificial

restrictions on the potentials may be made to be quite small; indeed in

principle we may come as close as desired to the true problem, by choosing
the point R beyond which the potential must vanish to be further and fur-
ther out. In practice the necessary labor increases as the point is moved
out, but it increases sufficiently slowly so that it should ordinarily be
possible to choose the point R far enough out so as not to have introduced
any serious truncation error without having unduly increased the work re-

It is the purpose of the present paper to provide the further exten-
sion to the problem of the positive energy scattering of one compoiind system
by another. (For the one channel scattering with which we are presently
concerned, this is the final possible extension. ) As for positive energy
one-body scattering, the various potentials must be truncated.

In the extension of the formalism for positive energy scattering from
the one body problem to the case of compound system scattering certain new
features arise which were not present in the zero energy case and which re-
qiiire some study. The development of a bound in the method that we have
used always effectively involves the expansion of the difference function,
the difference between the trial function and the exact function, in tenns
of some complete set of functions. (Equivalently, the question is always
whether or not the difference function satisfies the boundary conditions
which are necessary if it is to be possible to use it as one of the trial
functions in the application of the Hylleraas-Undheim theorem.) For one
body scattering, it was shown that the complete set could be taken to be
the eigenfunctions either of an associated potential strength eigenvalue
problem or of an associated energy eigenvalue problem. The former was
found to be preferable, but both gave rigorous bounds. It will be demon-

strated in Sec. h that the potential strength eigenvalue approach admits
of a straightforward generalization to include the case of compoimd system
scattering. The energy eigenvalue approach; on the other hand^ will not
in general be applicable for compound system scattering. It is possible to
make the energy eigenvalue approach applicable, but unfortunately only by
introducing certain rather restrictive boundary conditions on the trial
f unction. (See Sec. 5)- The origin of the necessity of these restrictive
conditions, for many body scattering, lies in the fact that for any given
R, the wave functions of the virtually excited states, while decaying, have
not vanished identically.

The technique for getting bounds on phase shifts using the associated
potential strength eigenvalue problem was first given by Kato for the one

body problem and later extended to some restricted cases of scattering by

7 +

compound systems ; in this latter work numerical calculations for e H scat-
tering were included. We note that using the method of the present paper
it would now be quite feasible to perform the e H calculation for higher
values of the kinetic energy of the incident positron.

It might be noted that the same remarks are applicable for compound
system scattering with regard to the error introduced by truncation as
were applicable for one body scattering. As examples, consider the scat-
tering of electrons or positrons by Hydrogen. An estimate based on the

method of the present paper is that for R of the order of 15 Bohr radii,

only about' three or four eigenstates need by accounted for , right up to

the threshold energies for inelastic scattering, i.e., 7 ev for e H and
10 ev for e~H.



The fonnal development of a bound is "based on a consideration of the
associated potential strength eigenvalue problem noted above. One here
encounters the necessity of having a definition of the phase shift which
is unairibiguous . This is in contrast with the normal requirements where
one need only know some trigonometric function of r\, that is, one need
merely know rj modulo rt . Despite the fact that theorems have been sunnised
which involve the value of t) (not simply of r\ modulo it), and despite the
fact that phase shift values are often discussed for scattering by compound
systems;, to our knowledge no definition of T) has been given which is appli-
cable to general compound system scattering. While the above theorems and

values undoubtably have some meaning , and certainly so within the context

of some approximation, such as the static approximation, it is clear that
an \inambiguous definition of the phase shift is very much called for. One
would of course like to choose a definition which is the most natural pos-
sible generalization of the definition for one body scattering, but this
general question doesn't arise in our present concern in obtaining a bound
on cot T\.

We now propose a definition of the phase shift for the scattering of
one compound systems by another. (As always in the present paper, we are
concerned with systems and energies for which the open channels can be
decomposed and analyzed in terms of uncoupled channels.) To avoid irrela-
vant kinematical complications we give the definition for the particular
case of the zero orbital angular momentimi scattering of a neutron by a
nucleus of angular momentum I. We further assvmie, purely for convenience,
that the total angular momentum J, and its z projection, J , satisfy


J = J = I + ^. (in Sec. h the description of the method for obtaining a
bound on k cot t] will be given in terms of this system. ) We define the

8(0^) = j^F. Xi^(i)^dT. (2.1)

vhere q. is the distance between neutron i and the center of mass of the
nucleus which consists of all the particles except neutron i. F. is the
ground state wave function of this nucleus, ^ is the full scattering wave
function^ and X „ (i) is the spin function for neutron i. The integral is


over all coordinates except q. , and is understood to represent a siimmation

over all spin indices as well. The phase shift is defined by treating
g(q. ) as a one body wave function and applying one of the standard defini-
tions of the phase shift for static potential scattering, namely

T^ = lim ( mjt - kq. I
m — > oo


where q_^^ is the m'th zero in g(q. )
We note that from the relation

? — > const Xjj^(i)F. sin(kq_^ + ■(]) / q.

22 1 J- J-

it follows that g(q.^, ) — > const sin(kq_^, + t\) as qi. — > oo which guarantees
that r\ modulo it as defined by Eq, (2.1) is correct.

An Application: Triplet Electron Hydrogen Scattering

It has been stressed that for the purposes of obtaining a bound, the
question of the justification of our definition of the phase shift does not

arise. However, since the question of a useful definition of the phase
shift for compound system scattering is a very interesting one in its
own right, it may be worth noting that the definition given enables us
to see, in a few simple cases at least, how contributions to the zero
energy phase shift arise by virtue of the Pauli principle. (Such con-
tributions have been discussed previously on the basis of an approximate
model, the so-called static, or no-polarization approximation.) As an
example we consider the zero energy scattering of electrons by atomic
hydrogen in the triplet (spatially anti -symmetric) case. We wish to show
that the antisymmetry of the wave function implies that the phase shift is
at least n. According to our definition, as applied to this problem, we
need only show that the function


g((l^) = / q/dq^ J sin 9^2^9^/(q2)'^(qi^ ^2' ®12^ ^^'^^

o o

has at least one node. Here q and q are the magnitudes of the electron
position vectors, 9 „ is the angle between those vectors and F(q2) is the
hydrogenic ground state function. (The subscript on F is redundant in
this particular discussion and will not be retained here.) The spins have
been accounted for. We form the integral


J F(q^)g(q^)q^%^ = J q/dq^ J ^2^^l2 J ^^^ ®12*^®12
o o o o

X F{q.^)Fiq^)^{ci^,q_^,Q^^)^

This integral clearly vanishes since T is antisymmetric in the electronic
coordinates while F(q, )F(qp) is symmetric. The fact that the ground state
fimction F(q ) is nodeless leads to the desired result, namely t] g re

After the present work was completed, we learned from Dr. A Terakin
that he has considered the identical definition of the phase shift for com-
pound system scattering, and further that he has obtained the identical
result for the zero energy triplet e H phase shift.

An extension of the triplet e~H phase shift result to positive
energy scattering is discussed in Sec. k.B.


For purposes of clarity we begin with the case in which the mass of
one of the scattering systems can be effectively taken to be infinite. A
prototype problem would then be the scattering of an electron by a neutral
atom of atomic number Z. To describe the assumed potentials, we first de-
fine three regions in configuration space. The first region contains that
portion of configuration space for which all of the Z + 1 electrons are
within a sphere of radius R centered about the nucleus of the atom. Region
(2) is further subdivided into Z + 1 parts; region (2i) contains that por-
tion of space for which all but the i'th electron are within the sphere of
radius R, while the i'th electron is not, where i runs from 1 through Z + 1.
Region (5) consists of the rest of space, that is, the part for which two
or more electrons are outside of the sphere.

In region (l), the potentials are the true (in this case, Coiilomb)
potentials. In region (2i), all but the i'th electron interact as before,


but the i'th electron is assumed not to interact with the others. In
region (3), the potentials are taken to be infinitely repulsive.

We return now to the particular problem discussed in Sec. 2., namely,
the zero orbital angiaar momentum scattering of a neutron by a nucleus
consisting of Z protons and N neutrons. We again divide space into 3 re-
gions, but because the center of mass is not now fixed it is necessary to
introduce the auxiliary parameters, S. . These are defined, for any dis-
tribution of particles, as the radius of the smallest sphere whose center
is at the center of mass of the N + Z particles excluding neutron i and
which contains all these N + Z particles. (i of course now runs from Z + 1
through Z + W + 1.) The three regions are chosen to be

(1) q^ < R

for each neutron
S^ < [(N + Z - 1)/(N + Z + 1)]r

(2i) q^ >R

i = Z+1, Z+2, ... or Z+N+1
S. < [(N + Z - 1)/(N + Z + lyJR

(3) the rest of configuration space.

The limit on S. has been chosen such that the N + 1 regions which make up
region (2) are non- overlapping.

The potentials are assimied to satisfy the following requirements. In
region (l) the particles interact via two body central potentials which allow
for space and spin exchange. As discussed in ref. h we exclude tensor
forces since we are restricting ourselves to one channel processes, and we
are considering non-zero scattering energies. In region (2i) the potentials
are of the same form as in (l) except that the interaction between neutron i


and the rest of the system vanishes. The potentials are taken to be infi-
nitely repulsive in region (3).

It will now be obvious how to define the problem for other systems,

such as the scattering of one compound system by another , where each nu-
cleus may carry a net charge .

As shown in ref. 2 the generalization of the Kato identity for the

problem under consideration takes the form

k cot(Ti - 0) = k cot(Ti^- G) + (2|a/-fi^) / Y^*(H-E)¥^dT

- (2^/■fi^) / fi*(H-E)ndT, (5.1)

where now the wave function, m, which is a solution of (H-E)T = 0, satis-
fies the boundary conditions

* = (-l)'-(N+l)"^Xxi(i) JF.sin(k(i.+Ti)/[(i.sin(Ti-0)]

. 2 aF.(-)f.(-)(n)l,
^■^ a a 1 1 ^_T^ I

in region (2i), Z+l^iSZ+N+1,
^ = in region (5) .

Here satisfies S < n but is otherwise arbitrary. E is the total
energy, the s\jm of the relative kinetic energy and of E , where E is the
ground state energy of the target nucleus. The incident relative wave
niimber, k, therefore satisfies


^\^/2ti = E-E

The F. represent normalized excited state wave functions of the system

■ (cc)

which does not include the i th particle, while the f . ^ are free particle

decaying functions. If, for example, a denotes a nuclear state with zero

total angular momentum, then


-/< ^^ / (2^i) + E = E,

a a

with E representing the energy of the a'th nuclear excited state. If a re-
presents a nuclear state with angular momentum L, then F. f. will in-
volve a sum of products, each containing a pair of angular momentum eigen-
functions. T satisfies similar boundary conditions with \\ and a replaced
by T) and a , respectively. It is our purpose to obtain a lower bound on

the error integral, / Q (H-E)ndT, where n = Y - "i, thereby providing an

upper bound on k cot(T|-0).

h - BOUND ON k cot (t^-G) and on v^

A - The Associated Potential Strength Eigenvalue Problem
We consider the equation
(H-E)5 = |ip|
where p is positive^ but otherwise arbitrary, in region (l) and vanishes


elsewhere. J satisfies the boundary conditions


^ = (-1) (N+1)" 2Xjj^(i) \



where t^^ is defined by the equations


k cot(Tij^ -©) = right hand side of the one body equivalent of Eq. (^+.2)

' t t

T rt - kR < Ti^ < (T + Dit - kR.

This lower bound on t\, while rigorous ^ will be too low by roiighly (T - T )«
if T is less than T. (T can never be greater than T, for we can never
find more negative potential strength eigenvalues than actually exist.) The
lower bound then obtained will then be useless with regard to a comparison

with the experimental data, though it may still be useful as the starting

point for further theoretical calculation

The above developments can be taken over directly, with complete rigor,
for the many body case. There is nevertheless one respect in which the use
of the conditional inequality differs for the many body problem when iden-
tical particles are present from its use for the one body scattering prob-
lem. Thus, account must be taken of the additional nodes which arise from


the symmetry requirements on the wave function, or the lower bound, j] ,


while rigorous, may be too low by a multiple of « even when the nimiber of
eigenstates with negative eigenvalues \x has been correctly accounted for.
The effect of the Pauli principle can perhaps be better understood through
a comparison of one body scattering and the triplet (spatially antisymmetric)
scattering of electrons by hydrogen atoms. For both systems, we have that
5(-cr) ) = -kR. [For (j. = - co , the one body scattering function vanishes for
i q S R, as does the eqiiivalent one body scattering function, g(q.).3
However, while b{\i) is a continuous function of |i in the neighborhood of
|i = - 00 for one body scattering, it is not continuous for e H triplet
scattering. It is to be recalled that the e H problem xmder consideration


is the true problem modified by the introduction of a cutoff and a repul-
sive barrier. The proof given in Sec. 2 that rj S n at zero energy can
then be immediately extended for k / to show that kR + t) > rt. To see
this, we note that in the definition of g(q,), Eq. (2.2), the range of
integration, to oo, can be replaced by to R since F(qp) exists only
in that range. It then follows, using the same symmetry argument as was
used at zero energy, that the integral


/ F(q^)g((l^)q^ dq^ =


R R rt

r q^^dq^ / qg^dq^ / sin9^2


Online LibraryTony RandallBounds on scattering phase shifts for compound systems → online text (page 1 of 2)