Larry Spruch. # Low energy scattering by a compound system: positrons on hydrogen online

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Strictly speaking, one should study the integral, rudou^dTi , in the

limit of large b , with the parameters c and d always adjusted to give

the correct hydrogenic energy, e j the error involved in changing the

S

potential should then be shown to be arbitrarily small for b sufficiently

large .

-In-

sufficiently small, a may be chosen large enough so that the neglect

of the potential which may exist beyond r-i = a introduces a negligible

error in the phase shift. If k is zero, we may let a become infinite,

in which case we introduce no distortion of the positron-hyxiiK?gen

potentials. In this case, the validity of the second condition stated

above, namely ">? < 6 < n, is deduced by assuming that no three-body

bound state exists for the positron-hydrogen system (none has been

found either experimentally or theoretically) '- -' . We then apply a

theorem which is the generalization to many body systems of a theorem proved

by Levinson^ '-' for the scattering from a static potential. It states

that when the exclusion principle is not in effect the phase shift

for zero energy scattering is nn where n is the number of bound states

of the composite system'- -^ . From this we conclude that 17 = for k = 0.

A rigorous proof of the above theorem has not been given even for

potentials which fall off rapidly. A proof for the slowly decaying

Coulomb forces will be even more difficult. We nevertheless apply the

theorem with some confidence, for its generalization, which applies to

electron scattering from atoms, has proved to be correct in all cases

fPl!

where it has been checked"- -â– . That the theorem should be applicable to

the Coulomb case (assuming it applies to rapidly decaying potentials)

is perhaps made more reasonable by the observation that the " effective

interaction" of the positron or electron with the atomic system falls

off fairly rapidly. For e"*" on H, e.g., it goes as Vr , Actually, as

we will show in Section ^, the results we have obtained do not depend on

the validity of the above mentioned theorem.

With the choice & = n/2 and -A s lim "yj/k , -A = lim l^/k,

k->o â€¢â€¢ k^o ^

u Â« lijB *'(i/2)n^Â» *Â® ^^"^^ **^ following bound on the scattering length, A,

- 16 -

UCM u d^

(3.6) A < A -

J

where u has the asymptotic form

One of the strong features of the Kato method is that in general

it provides a different bound for each choice of 9. This is not true

at k = C, however, where the identical result is obtained for any &

other than 9=0, The choice & = will lead to a valid bound on A

only if "Yi approaches zero from below as k goes to zero, i.e, , only if

A is positive. In this case we have

(3.8) A -^ > A"^+ A"^ tlcTu dr

A -1 > A-^ . A-2 J {IjTu

If u is sufficiently close to the true function such that A and A

are of the same sign and such that |llÂ«iA|, where

i^jsC^dtf

then Eq. (3.8) may be rewritten as

(3.8) A< A - I + A~^ i^ + ..., |i|Â«|aI.

It then follows (since A > O) that Eq. (3.6) is superior to Eq. (3.8),

Therefore, for the special case of zero energy scattering, where no

composite bound state exists, the problem of the optimum choice of Â©

is particularly simple. One should use Eq. (3.6), which follows from

any choice of 9 satisfying < & < n ,

- 17 -

k. Numerical Calculation

A. No Polarization Approximation, k = 0.

To begin with, we consider the class of trial functions of the

form

(it.l) u(r^,r2,r^2^ ^ g(r2)f(r^) .

The required asymptotic form fixes g(r,) as

gv^2

We also have

(rj =R(rJ = (2/a 3)1/2 e-^2Ao.

(U.2) f(o) =

f(r, ) -> A- r- , as r^ -> oo.

A trial function in this product form corresponds to the approximation

(variously referred to as the no polarization approximation, the static

approximation, and the one body approximation) in which the hydrogen

atom is not polarized by the incoming positron. We substitute the above

expression for u into Eq. (3.6), and integrate over r^ and .p. Dropping

the subscript on r^ we find

fOO

(Li.3) A < A - f(r) !â€¢ This

[22I

lower bound can be improved quite easily. Bargmann*- J has shown

that a necessary condition for the existence of a bound state for

a potential U(r) is

rOO

(U.9)

,00

I r Iu(r)l dr> 1.

With U(r) = -(1-|J,)W (r), this leads to oa/A^ > 5/3.'

If we choose

-nr/a -qr/a

(U.IO) f(r)=A(l-e Â°) -r(l+Be Â°)

we find, with n = 1.5 and q = 2, that

(U.ll) 0.5762a^ o, the Born approximation yields

1... ' - v|

,-00 ^

(['..16) k cotV)^ â€ž -> - 1/ ( r W (r)dr = l/a,

I J-o wig (I

It therefore follows that

(li.lY) (k cofy"_^)j^^^ > -Sa^"-"- + 0.10560 a^""'"

and

(U.18) Ig = -(k coth_g);;i^ > 0.5273 a^

While this result is not as good as the one given previously, it was

obtained without constructing a trial function.

Summarizing the results of thj.s sub-section, we find that an

analysis of the static problem loads to an upper bound or the tnte

scattering length, A, but does not provide any information on a lower

bound for ~. Further, the least upper bound obtainable in the static

approximation, namely that given by Eq. (h.3)> is positive so that we

do not obtain an upper or a lower bound on o'(k=o). Actually, as we

will show in the next sub-section, T is negative so that the static

approximation is totally inadequate in this problem.

B. Polarization Considered, k = o.

A trial function of the form u = g(r-, jr^) may take into account,

to some extent, the polarization of the hydrogen atom. As we can see.

- 22 -

using arguments quite similar to those amployed in sub-section UA,

this type of trial function leads to a positive scattering length.

Thus, substituting g(r, ^r^) into Eq. (3.6) and integrating over p we

find

;00 ,00

where

,2

(U.20) Â£, = -^ ^ -K ^ r/ ^ . ^(A . J. . i) . e

* ar^2 r/ ^^2 2 Â°^2 ^o ^2 ^> ^1 ^

Here r^ is the larger of r, and r^. Since no bound state exists for the

operator k- , none exists for Ju. . This is a consequence of the fact that

the ground state energy satisfies a minimum principle and the space

on which cC, is defined is a sub-space of that forju. (The same result

also follows directly from the fact that l/r - l/r, is always negative l- ^-',)

Therefore, the exact scattering length, TT , for for sufficiently small non-zero

energies the cut-off point may be made large enough so that the

difference between rj and vj" (the phase shift in the presence of the

cut-off potentials) is negligibly small. To illustrate the procedure

we describe here a determination of a loirer bound on Y) for ka = 0.2.

(c

To begin with we ignore the question of obtaining a bound and

perform an ordinary variational calculation for the true problem. We

use a trial function which reduces, as k -> 0, to the zero energy

function, Eq. (i|.23), namely

(U

.27) ^/2V(^1'^2'^12^'^ " (2Ao-^)^'^^[Ae"'"^ ^Â° (l-e"'^ ^Â°) cos kr^

- e sm kr, A + Br-,e

-tr^2/% - ^^2/^0 ^ -^12/^^0 - 2^1/^0

+ Gr-e + Dr-|_e

] â–

- 28 -

From the asymptotic fom of this function we see that A = -tanlTA-

The variational principle is now

r

(U.28) k tan-h ^ k tan>i + ^/2\n "^ "6./^ ^'^

where cC is given by Eq. (2.2). The result of the calculation is

T] = 0.l6a5 ,

^ =J ^4^^/2>^^ = -0.0017/a^

and 11 5^ 0.156. As a point of comparison we note that a variational

ri2i

calculation (including polarization) performed by Massey and Moussa'- -"

for the same scattering energy gave 77 7^ - O.O98 â€¢

Since the trial function is based on the zero energy function which

we expect to be quite good, we have obtained what we believe to be an

accurate estimate of TJ , although not necessarily a lower bound.

Before considering the calculation involving the cut-off potentials

we recall that the cut-off point, a , is limited by the condition ka < n-9,

In order to be able to pick the largest value of a (for fixed k) we

wish to choose the smallest value of 6 consistent with the condition

TJ < &. (Note that we desire that this relation be satisfied, in order

that our bound be a useful one. As discussed previously, the bound will

be valid in either case.) The trial function, Eq. (l4.27)> is noimalized

with Â© " (1/2V1. This would lead to a â– 7.9a . In order to obtain a

variational estimate of f) corresponding to the use of a trial function

with different (and smaller) 9 - nonnalization, we may simply choose

(a. 29)

^ = ^/2y Â°osysin(|-9)

- 29 -

so that

(U.30) k cot ffi-e)A::k cot(7j-9) - cos 19 /sin ("n - 9)

I

where Tfl and I have already been determined. Vie expect that this

variational estimate of TJ" should change very slowly as 9 is decreased

from(l/2)n. It is found, in fact, that to three significant figures

the result is unchanged for 9 as small as O.3U. (Smaller values of 9

lead to appreciably lower estLmates of tT. ) We have therefore chosen

9 = 0.3U00 which allows us to pick a = liia â€¢

To obtain a bound on"^ we must now perform a similar calculation

with oo replaced by Aj which contains the cut-off potentials, Eq. (3.5)Â»

The trial function, vu , must now satisfy the additional requirement

that Jo Ug = for r-, > a", while lu must of course be continuous

in slope ana value. Such a function may be written as

('4.31) u^^pfcos -y/sinCy-d)! = iUy2V (Eq. (U.27)) g(r^)

+ (2/a^3)l/2^_g(^_^)) ^""2 \in(kr3^+|)/cosTj

where

(a.32)

g(o) = 1

g'(a) = o

g(r^) =0 , r^ > a

If g(r, ) is close to unity throughout the region r, < a we might

expect that the results obtained will not differ very much from those of

This may be seen from the fact that In order to obtain Pa = 00, p must

v.anish beyond r^ = a. However, J (JCt. )^p'''^dV should exist.

- 30 -

the above variational calculation. With this in mind we chose

(U.33) g^^i) =

1 - e

- ^(1-r^/a)

1 - e

<

r, < a

r, > a

where ^ is arbitrary. If we let Y, take on a Isr^e, positive value

fe Â«i' 5a, say), g(r^) will have the desired shape.

We may write , ' . .

(U.3U) "a . Â» Â»

where p is the nth zero (not including the zero at the origin) in the

wave function. Now consider the point r' = (ii-'h)A:. From the

assumed restriction onil it is clear that r > a. Now the wave

function takes en its asymptotic fonn beyond the point _a, and must

therefore vanish at r'. But from Eq. (U.36), r' = p, , so that we

have shown that the fii^^t zero- occurs beyond a. In the present problem,

under similar conditions, ( i.e . , W. + W_^ =0 for r, > a, and

ka + T7 < n) it may be shown that the function

\

n

2

(li.37) g(r-j^) 5 T^ dr2 dp V.{r^) u(r^,r2,p)

J

is nodeless for r, < a. For s_ large enough it seem.s reasonable (we

limit of large b , with the parameters c and d always adjusted to give

the correct hydrogenic energy, e j the error involved in changing the

S

potential should then be shown to be arbitrarily small for b sufficiently

large .

-In-

sufficiently small, a may be chosen large enough so that the neglect

of the potential which may exist beyond r-i = a introduces a negligible

error in the phase shift. If k is zero, we may let a become infinite,

in which case we introduce no distortion of the positron-hyxiiK?gen

potentials. In this case, the validity of the second condition stated

above, namely ">? < 6 < n, is deduced by assuming that no three-body

bound state exists for the positron-hydrogen system (none has been

found either experimentally or theoretically) '- -' . We then apply a

theorem which is the generalization to many body systems of a theorem proved

by Levinson^ '-' for the scattering from a static potential. It states

that when the exclusion principle is not in effect the phase shift

for zero energy scattering is nn where n is the number of bound states

of the composite system'- -^ . From this we conclude that 17 = for k = 0.

A rigorous proof of the above theorem has not been given even for

potentials which fall off rapidly. A proof for the slowly decaying

Coulomb forces will be even more difficult. We nevertheless apply the

theorem with some confidence, for its generalization, which applies to

electron scattering from atoms, has proved to be correct in all cases

fPl!

where it has been checked"- -â– . That the theorem should be applicable to

the Coulomb case (assuming it applies to rapidly decaying potentials)

is perhaps made more reasonable by the observation that the " effective

interaction" of the positron or electron with the atomic system falls

off fairly rapidly. For e"*" on H, e.g., it goes as Vr , Actually, as

we will show in Section ^, the results we have obtained do not depend on

the validity of the above mentioned theorem.

With the choice & = n/2 and -A s lim "yj/k , -A = lim l^/k,

k->o â€¢â€¢ k^o ^

u Â« lijB *'(i/2)n^Â» *Â® ^^"^^ **^ following bound on the scattering length, A,

- 16 -

UCM u d^

(3.6) A < A -

J

where u has the asymptotic form

One of the strong features of the Kato method is that in general

it provides a different bound for each choice of 9. This is not true

at k = C, however, where the identical result is obtained for any &

other than 9=0, The choice & = will lead to a valid bound on A

only if "Yi approaches zero from below as k goes to zero, i.e, , only if

A is positive. In this case we have

(3.8) A -^ > A"^+ A"^ tlcTu dr

A -1 > A-^ . A-2 J {IjTu

If u is sufficiently close to the true function such that A and A

are of the same sign and such that |llÂ«iA|, where

i^jsC^dtf

then Eq. (3.8) may be rewritten as

(3.8) A< A - I + A~^ i^ + ..., |i|Â«|aI.

It then follows (since A > O) that Eq. (3.6) is superior to Eq. (3.8),

Therefore, for the special case of zero energy scattering, where no

composite bound state exists, the problem of the optimum choice of Â©

is particularly simple. One should use Eq. (3.6), which follows from

any choice of 9 satisfying < & < n ,

- 17 -

k. Numerical Calculation

A. No Polarization Approximation, k = 0.

To begin with, we consider the class of trial functions of the

form

(it.l) u(r^,r2,r^2^ ^ g(r2)f(r^) .

The required asymptotic form fixes g(r,) as

gv^2

We also have

(rj =R(rJ = (2/a 3)1/2 e-^2Ao.

(U.2) f(o) =

f(r, ) -> A- r- , as r^ -> oo.

A trial function in this product form corresponds to the approximation

(variously referred to as the no polarization approximation, the static

approximation, and the one body approximation) in which the hydrogen

atom is not polarized by the incoming positron. We substitute the above

expression for u into Eq. (3.6), and integrate over r^ and .p. Dropping

the subscript on r^ we find

fOO

(Li.3) A < A - f(r) !â€¢ This

[22I

lower bound can be improved quite easily. Bargmann*- J has shown

that a necessary condition for the existence of a bound state for

a potential U(r) is

rOO

(U.9)

,00

I r Iu(r)l dr> 1.

With U(r) = -(1-|J,)W (r), this leads to oa/A^ > 5/3.'

If we choose

-nr/a -qr/a

(U.IO) f(r)=A(l-e Â°) -r(l+Be Â°)

we find, with n = 1.5 and q = 2, that

(U.ll) 0.5762a^ o, the Born approximation yields

1... ' - v|

,-00 ^

(['..16) k cotV)^ â€ž -> - 1/ ( r W (r)dr = l/a,

I J-o wig (I

It therefore follows that

(li.lY) (k cofy"_^)j^^^ > -Sa^"-"- + 0.10560 a^""'"

and

(U.18) Ig = -(k coth_g);;i^ > 0.5273 a^

While this result is not as good as the one given previously, it was

obtained without constructing a trial function.

Summarizing the results of thj.s sub-section, we find that an

analysis of the static problem loads to an upper bound or the tnte

scattering length, A, but does not provide any information on a lower

bound for ~. Further, the least upper bound obtainable in the static

approximation, namely that given by Eq. (h.3)> is positive so that we

do not obtain an upper or a lower bound on o'(k=o). Actually, as we

will show in the next sub-section, T is negative so that the static

approximation is totally inadequate in this problem.

B. Polarization Considered, k = o.

A trial function of the form u = g(r-, jr^) may take into account,

to some extent, the polarization of the hydrogen atom. As we can see.

- 22 -

using arguments quite similar to those amployed in sub-section UA,

this type of trial function leads to a positive scattering length.

Thus, substituting g(r, ^r^) into Eq. (3.6) and integrating over p we

find

;00 ,00

where

,2

(U.20) Â£, = -^ ^ -K ^ r/ ^ . ^(A . J. . i) . e

* ar^2 r/ ^^2 2 Â°^2 ^o ^2 ^> ^1 ^

Here r^ is the larger of r, and r^. Since no bound state exists for the

operator k- , none exists for Ju. . This is a consequence of the fact that

the ground state energy satisfies a minimum principle and the space

on which cC, is defined is a sub-space of that forju. (The same result

also follows directly from the fact that l/r - l/r, is always negative l- ^-',)

Therefore, the exact scattering length, TT , for for sufficiently small non-zero

energies the cut-off point may be made large enough so that the

difference between rj and vj" (the phase shift in the presence of the

cut-off potentials) is negligibly small. To illustrate the procedure

we describe here a determination of a loirer bound on Y) for ka = 0.2.

(c

To begin with we ignore the question of obtaining a bound and

perform an ordinary variational calculation for the true problem. We

use a trial function which reduces, as k -> 0, to the zero energy

function, Eq. (i|.23), namely

(U

.27) ^/2V(^1'^2'^12^'^ " (2Ao-^)^'^^[Ae"'"^ ^Â° (l-e"'^ ^Â°) cos kr^

- e sm kr, A + Br-,e

-tr^2/% - ^^2/^0 ^ -^12/^^0 - 2^1/^0

+ Gr-e + Dr-|_e

] â–

- 28 -

From the asymptotic fom of this function we see that A = -tanlTA-

The variational principle is now

r

(U.28) k tan-h ^ k tan>i + ^/2\n "^ "6./^ ^'^

where cC is given by Eq. (2.2). The result of the calculation is

T] = 0.l6a5 ,

^ =J ^4^^/2>^^ = -0.0017/a^

and 11 5^ 0.156. As a point of comparison we note that a variational

ri2i

calculation (including polarization) performed by Massey and Moussa'- -"

for the same scattering energy gave 77 7^ - O.O98 â€¢

Since the trial function is based on the zero energy function which

we expect to be quite good, we have obtained what we believe to be an

accurate estimate of TJ , although not necessarily a lower bound.

Before considering the calculation involving the cut-off potentials

we recall that the cut-off point, a , is limited by the condition ka < n-9,

In order to be able to pick the largest value of a (for fixed k) we

wish to choose the smallest value of 6 consistent with the condition

TJ < &. (Note that we desire that this relation be satisfied, in order

that our bound be a useful one. As discussed previously, the bound will

be valid in either case.) The trial function, Eq. (l4.27)> is noimalized

with Â© " (1/2V1. This would lead to a â– 7.9a . In order to obtain a

variational estimate of f) corresponding to the use of a trial function

with different (and smaller) 9 - nonnalization, we may simply choose

(a. 29)

^ = ^/2y Â°osysin(|-9)

- 29 -

so that

(U.30) k cot ffi-e)A::k cot(7j-9) - cos 19 /sin ("n - 9)

I

where Tfl and I have already been determined. Vie expect that this

variational estimate of TJ" should change very slowly as 9 is decreased

from(l/2)n. It is found, in fact, that to three significant figures

the result is unchanged for 9 as small as O.3U. (Smaller values of 9

lead to appreciably lower estLmates of tT. ) We have therefore chosen

9 = 0.3U00 which allows us to pick a = liia â€¢

To obtain a bound on"^ we must now perform a similar calculation

with oo replaced by Aj which contains the cut-off potentials, Eq. (3.5)Â»

The trial function, vu , must now satisfy the additional requirement

that Jo Ug = for r-, > a", while lu must of course be continuous

in slope ana value. Such a function may be written as

('4.31) u^^pfcos -y/sinCy-d)! = iUy2V (Eq. (U.27)) g(r^)

+ (2/a^3)l/2^_g(^_^)) ^""2 \in(kr3^+|)/cosTj

where

(a.32)

g(o) = 1

g'(a) = o

g(r^) =0 , r^ > a

If g(r, ) is close to unity throughout the region r, < a we might

expect that the results obtained will not differ very much from those of

This may be seen from the fact that In order to obtain Pa = 00, p must

v.anish beyond r^ = a. However, J (JCt. )^p'''^dV should exist.

- 30 -

the above variational calculation. With this in mind we chose

(U.33) g^^i) =

1 - e

- ^(1-r^/a)

1 - e

<

r, < a

r, > a

where ^ is arbitrary. If we let Y, take on a Isr^e, positive value

fe Â«i' 5a, say), g(r^) will have the desired shape.

We may write , ' . .

(U.3U) "a . Â» Â»

where p is the nth zero (not including the zero at the origin) in the

wave function. Now consider the point r' = (ii-'h)A:. From the

assumed restriction onil it is clear that r > a. Now the wave

function takes en its asymptotic fonn beyond the point _a, and must

therefore vanish at r'. But from Eq. (U.36), r' = p, , so that we

have shown that the fii^^t zero- occurs beyond a. In the present problem,

under similar conditions, ( i.e . , W. + W_^ =0 for r, > a, and

ka + T7 < n) it may be shown that the function

\

n

2

(li.37) g(r-j^) 5 T^ dr2 dp V.{r^) u(r^,r2,p)

J

is nodeless for r, < a. For s_ large enough it seem.s reasonable (we

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