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NEW YORK UNJVERSI TY

INSTITUTE OF MATHEMATICAL SQENCE?

AFCRC-TN-59.259 LIBRARY

ASTIA DOCUMENT NO. 212556

Â»S y/averly Place, New York 3, N. Y.

^EET PR^^ NOV IS 1959

A.'^w -r-^ Tr% NEW YORK UNIVERSITY

Wl

^ I ^ 1 1 ^ Institute of Mathematical Sciences

'^^cccxx+

"^ Division of Electromagnetic Research

RESEARCH REPORT No. CX-40

Low Energy Scattering by a Compound System:

Positrons on Hydrogen

LARRY SPRUCH and LEONARD ROSENBERG

Contract No. AF 19(604)4555

FEBRUARY, 1959

N8W YOWS UNIVBHaTY

INSTITUTE OF MATHEMATICAL SCIEMCES

LiBKARY

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. Cl-hO

IDW ENERGY SCATTERING BY A COMPOUND SYSTEM:

rosiTRDNS ON HYDROGEN

Larry Spruch

and

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:

ARhED SERVICFS TECmiCAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTO'M HALL STATION

ARLINGTON 12, VIR.GINIA

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:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C,

- 1 -

Abstract

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

o

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)]

- ii -

Table of Contents

Pare

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

- 1 -

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'- â– * 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 '- -^ ' 'â€¢-^ ( 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

- 2 -

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

- 3 -

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* ^^ ^^^Â®

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

^'â€¢'^ ^"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) .jâ€žA>^i= ^ J- ^i'^*T^ -/ J:

r^ 1 1 rg ^ -i

2 ar,â€ž ^12 ar,o r,r,o bj^hr

r^^- 12 ^^ Â°^12 'â– L'12 "^n-^12

(r 2+r ^-r M 2

^2^12 ^^2^^12

- 9 -

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(râ€ž) is the exact ground state wave function for the bound

particle, normalized such that

^2 â€”' 00

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

The normalization parameter, Â©, 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^, ^â– ]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

2

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

and where

(2.7) ^Q^^2.f^2'^12^ "â– '%^^1'^2*^12^"^^^1Â»^2'^12^*

- 10 -

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 uâ€ž 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[-Â©) 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

- 11 -

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) (Â£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

n,"

ind

(2.10) k"^

y

0â€ž a 0â€ž a P dr = 6 , in,n = 0, Â±1, Â±2, ...

n,w m,w nm

where 5 is the Kronecker 5-symbol, such that

ran

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

n,fcy ^1 -L

(2.11) 0^ ^ -> , as Tg -> 00

^n,e = Â° ' for r^ = .

We denote the smallest positive eigenvalue of the set p. â€ž ^y ^â– o ^""^ ^^'^

smallest (in magnitude) negative eigenvalue by -Pq. The Kato inequalities

are

(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.

- 12 -

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, â€ž, at least for r, and/or r^ -> oo.

Then E^. will be independent of \

-^

NEW YORK UNJVERSI TY

INSTITUTE OF MATHEMATICAL SQENCE?

AFCRC-TN-59.259 LIBRARY

ASTIA DOCUMENT NO. 212556

Â»S y/averly Place, New York 3, N. Y.

^EET PR^^ NOV IS 1959

A.'^w -r-^ Tr% NEW YORK UNIVERSITY

Wl

^ I ^ 1 1 ^ Institute of Mathematical Sciences

'^^cccxx+

"^ Division of Electromagnetic Research

RESEARCH REPORT No. CX-40

Low Energy Scattering by a Compound System:

Positrons on Hydrogen

LARRY SPRUCH and LEONARD ROSENBERG

Contract No. AF 19(604)4555

FEBRUARY, 1959

N8W YOWS UNIVBHaTY

INSTITUTE OF MATHEMATICAL SCIEMCES

LiBKARY

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. Cl-hO

IDW ENERGY SCATTERING BY A COMPOUND SYSTEM:

rosiTRDNS ON HYDROGEN

Larry Spruch

and

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:

ARhED SERVICFS TECmiCAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTO'M HALL STATION

ARLINGTON 12, VIR.GINIA

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:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C,

- 1 -

Abstract

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

o

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)]

- ii -

Table of Contents

Pare

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

- 1 -

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'- â– * 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 '- -^ ' 'â€¢-^ ( 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

- 2 -

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

- 3 -

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* ^^ ^^^Â®

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

^'â€¢'^ ^"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) .jâ€žA>^i= ^ J- ^i'^*T^ -/ J:

r^ 1 1 rg ^ -i

2 ar,â€ž ^12 ar,o r,r,o bj^hr

r^^- 12 ^^ Â°^12 'â– L'12 "^n-^12

(r 2+r ^-r M 2

^2^12 ^^2^^12

- 9 -

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(râ€ž) is the exact ground state wave function for the bound

particle, normalized such that

^2 â€”' 00

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

The normalization parameter, Â©, 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^, ^â– ]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

2

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

and where

(2.7) ^Q^^2.f^2'^12^ "â– '%^^1'^2*^12^"^^^1Â»^2'^12^*

- 10 -

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 uâ€ž 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[-Â©) 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

- 11 -

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) (Â£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

n,"

ind

(2.10) k"^

y

0â€ž a 0â€ž a P dr = 6 , in,n = 0, Â±1, Â±2, ...

n,w m,w nm

where 5 is the Kronecker 5-symbol, such that

ran

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

n,fcy ^1 -L

(2.11) 0^ ^ -> , as Tg -> 00

^n,e = Â° ' for r^ = .

We denote the smallest positive eigenvalue of the set p. â€ž ^y ^â– o ^""^ ^^'^

smallest (in magnitude) negative eigenvalue by -Pq. The Kato inequalities

are

(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.

- 12 -

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, â€ž, at least for r, and/or r^ -> oo.

Then E^. will be independent of \

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