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NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

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

^E ET PR^

A^^ k^ yx r^ NEW YORK UNIVERSITY

^ I |J |Y ^ Institute of Mathematical Sciences

;^ >->^ X Division of Electromagnetic Research

RESEARCH REPORT NO. CX-52

Bounds on Scattering Phase Shifts

For Compound Systems

LEONARD ROSENBERG and LARRY SPRUCH

Contract Nos. AF 19(604)4555

DA 30-069-ORD-2581

Nonr-285(49), NR 012-109

fS OCTOBER, 1 960

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-52

BOUNDS ON SCATTERING PHASE SHIFTS

FOR COMPOUND SYSTEMS

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

OCTOBER i960

Requests for additional copies by Agencies of the Department of Defense,

their contractors, and other Government agencies should be directed to

the:

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

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:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

Abstract

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 Â« 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

Page

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

1 - INTRODUCTION

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-

2

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

k

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-

quired.

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,

8

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.

5.

2 - DEFINITION OF THE PHASE SHIFT

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

9

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

6.

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

function

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 â€ž (i) is the spin function for neutron i. The integral is

DO

z

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 â€” > oo

(m)

where q_^^ is the m'th zero in g(q. )

We note that from the relation

? â€” > const Xjj^(i)F. sin(kq_^ + â– (]) / q.

22 1 J- J-

it follows that g(q.^, ) â€” > const sin(kq_^, + t\) as qi. â€” > 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

DO It

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

o o

has at least one node. Here q and q are the magnitudes of the electron

position vectors, 9 â€ž 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

GO GO OO It

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

5 - DEFINITION OF THE PROBLEM FOR WHICH THE BOUND

IS TO BE OBTAINED

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,

9.

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

10.

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,

12

such as the scattering of one compound system by another , where each nu-

ll

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)Â¥^dT

- (2^/â– 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,

^â– ^ 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

11.

^\^/2ti = E-E

(a.)

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

â– (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

where

-/< ^^ / (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

12.

elsewhere. J satisfies the boundary conditions

f

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

15.

t

where t^^ is defined by the equations

1

k cot(Tij^ -Â©) = 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 )Â«

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

ik

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

t

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

XJ

while rigorous, may be too low by a multiple of Â« 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

16.

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

R

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

o

R R rt

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

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

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

^E ET PR^

A^^ k^ yx r^ NEW YORK UNIVERSITY

^ I |J |Y ^ Institute of Mathematical Sciences

;^ >->^ X Division of Electromagnetic Research

RESEARCH REPORT NO. CX-52

Bounds on Scattering Phase Shifts

For Compound Systems

LEONARD ROSENBERG and LARRY SPRUCH

Contract Nos. AF 19(604)4555

DA 30-069-ORD-2581

Nonr-285(49), NR 012-109

fS OCTOBER, 1 960

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-52

BOUNDS ON SCATTERING PHASE SHIFTS

FOR COMPOUND SYSTEMS

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

OCTOBER i960

Requests for additional copies by Agencies of the Department of Defense,

their contractors, and other Government agencies should be directed to

the:

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

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:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

Abstract

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 Â« 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

Page

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

1 - INTRODUCTION

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-

2

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

k

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-

quired.

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,

8

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.

5.

2 - DEFINITION OF THE PHASE SHIFT

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

9

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

6.

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

function

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 â€ž (i) is the spin function for neutron i. The integral is

DO

z

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 â€” > oo

(m)

where q_^^ is the m'th zero in g(q. )

We note that from the relation

? â€” > const Xjj^(i)F. sin(kq_^ + â– (]) / q.

22 1 J- J-

it follows that g(q.^, ) â€” > const sin(kq_^, + t\) as qi. â€” > 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

DO It

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

o o

has at least one node. Here q and q are the magnitudes of the electron

position vectors, 9 â€ž 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

GO GO OO It

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

5 - DEFINITION OF THE PROBLEM FOR WHICH THE BOUND

IS TO BE OBTAINED

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,

9.

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

10.

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,

12

such as the scattering of one compound system by another , where each nu-

ll

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)Â¥^dT

- (2^/â– 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,

^â– ^ 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

11.

^\^/2ti = E-E

(a.)

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

â– (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

where

-/< ^^ / (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

12.

elsewhere. J satisfies the boundary conditions

f

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

15.

t

where t^^ is defined by the equations

1

k cot(Tij^ -Â©) = 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 )Â«

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

ik

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

t

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

XJ

while rigorous, may be too low by a multiple of Â« 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

16.

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

R

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

o

R R rt

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

1 2

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