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

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- 32 -

have not proved it) that a stronger ststement can be made, namely,

that u itself is nodeless for r^ < a. If we assume this statement

to be true, it may be used to test our trial function. With D = o

the function has nodes for r, < a but these nodes do in fact disappear

when the full trial function is used. Further, this im.provem.ent in

the trial function is accompanied by only a small improvement in the

bound on the phase shift. Both of these considerations suggest that

this bound may well be in the neighborhood of the true value.

We have previously noted that the shape independent approximation

can at best be expected to be applicable for ka < 0,1. If,

nevertheless, we apply Eq. ()i.26) to the present case, ka^ = 0.2,

we find tan ^ f^ 0.21. This differs considerably from the

variational result, tan ^-^0,16. The difference is ven - likely due

to the inapplicability of Eq. (li.26), but we may not exclude the

possibility that our calculated value of r is too large due to

inaccuracies in our zero energj'- wave function. It is of course

also possible, though less likely, that our variationally calculated

values of tan ^A for k = o and 0.2 are not close enough to the

correct values.

- 33 -

6. Discussion

We have shown, by providing a particular example, that it is

not only possible to obtain a bound on the cross section for scattering

hj a compound system but, under certain circumstances, that no additional

work beyond that of performing a variational calculation is required.

This is true in the case of zero energy scattering when no composite

bound state exists. For slightly larger energies the bound obtained

is no longer rigorous, since the interaction between the scattered

particle and the scattering system is neglected outside a certain

region, VJe believe, however, that in the example presented above,

where the cut-off point is at llta , the neglected potential contributes

practically nothing to the phase shift and that our bound is extremely

likely to be valid.

The fact that the variational principle for cot Tj" may, in special

cases, be a minimum principle as well has some interesting consequences.

Generalizing the results obtained previously, we may state that the

scattering length for positron scattering from a static atom is greater

than the true scattering length. A, provided that the positron and the

atom do not form a bound state. Also, a variational estimate of A

will necessarily be positive in this case unless the trial function

contains some dependence on the separation between tlie positron and one

or more of the electrons. (This is analogous to a result obtained by

Ore'- -^ in his investigation of the stability of simple positron compounds.)

Another result of this formalism which would be otherwise rather difficult

to prove is that under the conditions for which p/. /A = oo the Kohn

form of the variational principle for the phase shift (derivable from the

Kato form by setting 9 = n/2) has at least one important advantage over

- 3U-

the Hulthen formulation. That is, an increase in the number of

parameters in the trial function guarantees an improved result.

This remark applies, for example, to the problem of zero energy triplet

scattering of electrons from hydrogen.

The simple result that at zero energy the ordinary variational

calculation provides a bound if the system has no composite bound

state suggests that the Kato mechanism is not necessary in this case,

since the above conditions make no reference to the associated eigenvalue

problem. This is easily verified by observing that in the absence of

a bound state the operatorcC is negative definite in the space of

quadricatically integrable functions. Now for < & < n.

(5.1) w^ = lim w^A

k*o

has the form of a zero energy bound state wove function. That is

(5.2) iL -> const. R (rj) Â» for r-j^ -> oo.

Thus w~ may be considered to be the limit of a sequence of functions.

Wo(e), which have the asymptotic form

e

-cr,

(5.3) ^Q^^) "> const. ^ (r2) e , for r^ -> oo .

Since

(5.U)

r

Wa(e)Xwa(e)dr <

"â– 9""'^ â„¢e

we conclude that this is true for e = o as well. This, along with

Eq. (2.6), gives us the desired result. Note that

Wq -> const. R(r2)r^ , for large r^,

- 35-

so that the theorem is not true in general for this choice of 9 - normalization.

We had also come to this conclusion from the point of view of the Kato

formalism. (See Section 3).

While this proof merely reproduces a result previously obtained

it is still quite interesting. First, our conclusions are freed from '

any dependence on the validity of the generalization of Levinson's

theorem (see Section 3). More significantly, the type of reasoning

involved points directly to the extension which includes situations

in which bound states do exist. Here the results we have obtained do

net follow, in any simple way at least, from the Kato formalism and

may provide a considerable improvem.ent . Furthermore, we have been led

to the development of a method, valid at higher energies, which reproduces

all results of Kato for which p_= oo and which should otherwise provide

a useful supplement to the Kato method. This work will be reported on

in the future.

- 36 -

Appendix A

In the numerical evaluatio

integrals of the form

(A.l)

n of I ujb u

dti (Section UB) various

00 /OO /'I'T,

l(a,p,Yj i,m,n) 2 ] dr^ 1 drj

7

A^^2 ^r2-pr^-rr^2 Z m n

dr^2^ ^2 ^1 ^12

l^rgl

occur. These integrals may be generated from the basic integral

(A.2) I{a,?,r; 0,0,0) Â« (a^^)(0!yJ(a4Y) = |

using symmetry relations such as

(A. 3) I(a,p,Yj;2,m,n) = I(p,a,Y} m,J?,n)

and recursion relations such as

(A.h) l(a,p,Y; JP+l,m,n) = - -^ I(Â«i,P,Yj/,m,n)

and

r

(A. 5) l(a,p,Yj/-l,m,n) = I(a',|i,Yj i',m,n) da'

00

/

We list below expressions for those integrals, involving non-vanishing

values of y, which entered into the calculation.

- 37 -

(A.6)

l(a,n,Y; 1,0,0)= I [^-^]

l(a,%Y; 1,1,0) = -^ ~ + a + p + Y

1

I(a,-,Y; 0,1,1) = liÂ» -1. + a + p, + Y J

l(a,3,Y; 1,0,1) =li^|-|- + a + B+Y

7 l"^

i(a,.,Y;i,i,i)=^[2.^.:^.^]

i(a,?,Y; 1,-1,1)= - - j-^r^ lo

(^2:^ - p^ T^V?

In the calculation involving the non-vanisbj-n^ scattering energy

(Section hC) additional integrals had to be evaluated. The results

nay be stated in terms of the general form

(A. 7) l(a,p,Y; i,m,n; p,q) =

i

00 ^00 (^i^^2

dr.

dr.

^

dr^2 e-^^2-Pri-Yri2 rf r^^^g" cosPkr^sinV^ .

VrgI

Again we quote only those of the required integrals for which y j^ 0,

(A. 8) l(a,?,Y; 1,0,0; 0,1) = 2k\ ^^%^ | i^â€” ^ ^

(a^-r^)^ L(P+Y)^+k^ (a+p)W

, 1_ r 2(a-^p) "I

' a^-Y^ L((a+p)W)^ J

- 38 -

I(a,p,Y; 1,-1,1; 0,1) = iaWr'l^ tan^ ^ -J^^

+ a

have not proved it) that a stronger ststement can be made, namely,

that u itself is nodeless for r^ < a. If we assume this statement

to be true, it may be used to test our trial function. With D = o

the function has nodes for r, < a but these nodes do in fact disappear

when the full trial function is used. Further, this im.provem.ent in

the trial function is accompanied by only a small improvement in the

bound on the phase shift. Both of these considerations suggest that

this bound may well be in the neighborhood of the true value.

We have previously noted that the shape independent approximation

can at best be expected to be applicable for ka < 0,1. If,

nevertheless, we apply Eq. ()i.26) to the present case, ka^ = 0.2,

we find tan ^ f^ 0.21. This differs considerably from the

variational result, tan ^-^0,16. The difference is ven - likely due

to the inapplicability of Eq. (li.26), but we may not exclude the

possibility that our calculated value of r is too large due to

inaccuracies in our zero energj'- wave function. It is of course

also possible, though less likely, that our variationally calculated

values of tan ^A for k = o and 0.2 are not close enough to the

correct values.

- 33 -

6. Discussion

We have shown, by providing a particular example, that it is

not only possible to obtain a bound on the cross section for scattering

hj a compound system but, under certain circumstances, that no additional

work beyond that of performing a variational calculation is required.

This is true in the case of zero energy scattering when no composite

bound state exists. For slightly larger energies the bound obtained

is no longer rigorous, since the interaction between the scattered

particle and the scattering system is neglected outside a certain

region, VJe believe, however, that in the example presented above,

where the cut-off point is at llta , the neglected potential contributes

practically nothing to the phase shift and that our bound is extremely

likely to be valid.

The fact that the variational principle for cot Tj" may, in special

cases, be a minimum principle as well has some interesting consequences.

Generalizing the results obtained previously, we may state that the

scattering length for positron scattering from a static atom is greater

than the true scattering length. A, provided that the positron and the

atom do not form a bound state. Also, a variational estimate of A

will necessarily be positive in this case unless the trial function

contains some dependence on the separation between tlie positron and one

or more of the electrons. (This is analogous to a result obtained by

Ore'- -^ in his investigation of the stability of simple positron compounds.)

Another result of this formalism which would be otherwise rather difficult

to prove is that under the conditions for which p/. /A = oo the Kohn

form of the variational principle for the phase shift (derivable from the

Kato form by setting 9 = n/2) has at least one important advantage over

- 3U-

the Hulthen formulation. That is, an increase in the number of

parameters in the trial function guarantees an improved result.

This remark applies, for example, to the problem of zero energy triplet

scattering of electrons from hydrogen.

The simple result that at zero energy the ordinary variational

calculation provides a bound if the system has no composite bound

state suggests that the Kato mechanism is not necessary in this case,

since the above conditions make no reference to the associated eigenvalue

problem. This is easily verified by observing that in the absence of

a bound state the operatorcC is negative definite in the space of

quadricatically integrable functions. Now for < & < n.

(5.1) w^ = lim w^A

k*o

has the form of a zero energy bound state wove function. That is

(5.2) iL -> const. R (rj) Â» for r-j^ -> oo.

Thus w~ may be considered to be the limit of a sequence of functions.

Wo(e), which have the asymptotic form

e

-cr,

(5.3) ^Q^^) "> const. ^ (r2) e , for r^ -> oo .

Since

(5.U)

r

Wa(e)Xwa(e)dr <

"â– 9""'^ â„¢e

we conclude that this is true for e = o as well. This, along with

Eq. (2.6), gives us the desired result. Note that

Wq -> const. R(r2)r^ , for large r^,

- 35-

so that the theorem is not true in general for this choice of 9 - normalization.

We had also come to this conclusion from the point of view of the Kato

formalism. (See Section 3).

While this proof merely reproduces a result previously obtained

it is still quite interesting. First, our conclusions are freed from '

any dependence on the validity of the generalization of Levinson's

theorem (see Section 3). More significantly, the type of reasoning

involved points directly to the extension which includes situations

in which bound states do exist. Here the results we have obtained do

net follow, in any simple way at least, from the Kato formalism and

may provide a considerable improvem.ent . Furthermore, we have been led

to the development of a method, valid at higher energies, which reproduces

all results of Kato for which p_= oo and which should otherwise provide

a useful supplement to the Kato method. This work will be reported on

in the future.

- 36 -

Appendix A

In the numerical evaluatio

integrals of the form

(A.l)

n of I ujb u

dti (Section UB) various

00 /OO /'I'T,

l(a,p,Yj i,m,n) 2 ] dr^ 1 drj

7

A^^2 ^r2-pr^-rr^2 Z m n

dr^2^ ^2 ^1 ^12

l^rgl

occur. These integrals may be generated from the basic integral

(A.2) I{a,?,r; 0,0,0) Â« (a^^)(0!yJ(a4Y) = |

using symmetry relations such as

(A. 3) I(a,p,Yj;2,m,n) = I(p,a,Y} m,J?,n)

and recursion relations such as

(A.h) l(a,p,Y; JP+l,m,n) = - -^ I(Â«i,P,Yj/,m,n)

and

r

(A. 5) l(a,p,Yj/-l,m,n) = I(a',|i,Yj i',m,n) da'

00

/

We list below expressions for those integrals, involving non-vanishing

values of y, which entered into the calculation.

- 37 -

(A.6)

l(a,n,Y; 1,0,0)= I [^-^]

l(a,%Y; 1,1,0) = -^ ~ + a + p + Y

1

I(a,-,Y; 0,1,1) = liÂ» -1. + a + p, + Y J

l(a,3,Y; 1,0,1) =li^|-|- + a + B+Y

7 l"^

i(a,.,Y;i,i,i)=^[2.^.:^.^]

i(a,?,Y; 1,-1,1)= - - j-^r^ lo

(^2:^ - p^ T^V?

In the calculation involving the non-vanisbj-n^ scattering energy

(Section hC) additional integrals had to be evaluated. The results

nay be stated in terms of the general form

(A. 7) l(a,p,Y; i,m,n; p,q) =

i

00 ^00 (^i^^2

dr.

dr.

^

dr^2 e-^^2-Pri-Yri2 rf r^^^g" cosPkr^sinV^ .

VrgI

Again we quote only those of the required integrals for which y j^ 0,

(A. 8) l(a,?,Y; 1,0,0; 0,1) = 2k\ ^^%^ | i^â€” ^ ^

(a^-r^)^ L(P+Y)^+k^ (a+p)W

, 1_ r 2(a-^p) "I

' a^-Y^ L((a+p)W)^ J

- 38 -

I(a,p,Y; 1,-1,1; 0,1) = iaWr'l^ tan^ ^ -J^^

+ a

Online Library → Larry Spruch → Low energy scattering by a compound system: positrons on hydrogen → online text (page 3 of 3)