Positive and negative deflection-angle coulomb scattering

Volume 57B, number 3

PHYSICS LETTERS

7 July 1975

POSITIVE AND NEGATIVE DEFLECTION-ANGLE COULOMB SCATTERING * R.C. FULLER Brookhaven National Laboratory, Upton, New York 119 73, USA Received 1 April 1975 The decomposition of the Rutherford amplitude into positive and negative deflection-angle contributions shows how Fraunhofer diffraction for charged-particle scattering is moved away from forward angles. In the semi-classical approach to elastic scattering [ 1] the partial-wave summation is replaced by integration after substituting large4 approximations for both the partial-wave amplitudes and the Legendre polynomials. Integration requires interpolation of the amplitudes between integer angular momenta which for potential scattering is generally effected by a W.K.B. approximation. In the presence of absorption the W.K.B. approximation is, at best, awkward. This note discusses a method for separating the positive and negative deflection-angle contributions to elastic scattering which uses the partial-wave amplitudes at integer lvalues; e.g., one can work directly with optical-model amplitudes. Positive and negative deflection angle contributions are associated, respectively, with the two travelingwave components ~!(-) (cos 0) and ~ t +) (cos 0) of the Legendre polynomial where for large l, ~t±)(cos 0) "" (2hi sin 0) -1/2

(1)

,

" l-1 ~ 0 ~ Ir - 1-1 . !l)vo,ol.o. to.

For all l the traveling ~

then given by fN(0) = ~ (2/+ I) al(k)~.~-)(cos O) fF(O) = ~ ( 2 / +

1)al(k)~.i (+)(cos 0)

(3a) (3b)

and the total amplitude satisfies

f(O) =fF(O) + fN(0).

(4)

Interference between fF(0) and fN(0) gives rise to Fraunhofer diffraction at small angles. The elastic-scattering amplitude for charged particles is generally considered as

f(O) = fc(0) + f'(0),

(5)

where the partial-wave series for the "nuclear" amplitude, 1~(0), converges and its far-near decomposition is given by eqs. (3). The point-Coulomb amplitude fc(0) is given by fc(0) = fc0r) exp (-i~/In sin 2 ~ 0)/sin 2 ~0

(6)

"I x "-l-i~ ,

Legendre equation are given by

1 FP, (cos 0) ~,i 2QI (cos 0)~. (cos o) = 2t.'

(2)

In heavy-ion applications there is virtually no transmission through the interaction region and only strongly damped orbiting around it.It is then reasonable [2] to callpositive (negative) deflection-angle scattering "near" ("far"). If a partial-wave expansion is appropriate, the "near" and "far" amplitudes are

performed under the auspices of Energy Research and Development Administration.

Work

To decompose the Coulomb amplitude, for which a partial-wave decomposition is inappropriate, it is necessary to generalize the above approach. Notation in the following argument may ,be found in ref. [3]. Consider the complex z-plane where for Im (z) = 0 and -I ~ Re (z) ~ I, z = x = cos 0. Introduce the function Qt(z),defined on the complex zplane cut from -l to I where

Ql(x + ie) - Ql(x - ie) = -ilrPt(x ),

(7)

which satisfies the dispersion relation

217

Volume 57B, number 3

PHYSICS LETTERS

fC, N(O)/fc(O) = (1 -- e - 2 n q ) - I

1

1

fdt

fi(z) = ~

7 July 1975

~ -1 t /,t(t)

(8)

-1 for any z not on the real axis between - 1 and 1. The Legendre function of the second kind in eq. (2) is defined as the average o f Qi(z) across the cut;

QI(x) - - ~ [QI(x + ie) + QI (x 1

ie)]

(9)

Eqs. (7) and (9) may then be combined with eq. (2) to show i ~.}+)(x) = + -~Ql(X ± ie). (10)

.fC,F(O)ffc(O) = --e-2~n(1 i

~: S

I

I

~

I

I

[

-,~

................. ................. ............

,= £

~

(15a)

- - e - 2 ~ ) -1 +iC(n, 0), (15b)

0~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:-2L~~

-- iC(n, O)

-

~- ....

o o 8~

We now define the amplitude,fQ(z), in terms of the physical scattering amplitude, / ( x ) , by analogy with eq. (8); (11)

I'\'\\

-1 and identify the near and far amplitudes as the continuations of this amplitude to the cut from above and below;

o~- [~

fQ(z) = ~-

dt

f(t)

•

-2

,

,

-1

1

= ~/c(-

(1 + i~)k k!

X [~(k + I), @(k+ i +i.)_ in 2z]k 1

(14) (l_~)k

is continuous across the cut. Eqs. (12) then give the positive and negative deflection-angle Coulomb amplitudes

"

"•%•,,,.........,,,.,. ......

\I

-4

""'".%

\ \\

(c) i ?'~

f.y

il I'

o

=~

\ \

=3

1) [Dr(sinhfflT)-l(Z21)-l-in+s(~,z)],

k>0

218

~

(13)

where SO?, z)

,".7

-e

fC,Q(Z)=~fC(-1) f dt

(b)

L"%

°

fN(X)=~fQ(x+ie), fF(X)=--~fQ(x-ie). (12a, b) Iff (x) is given in terms of a partial-wave expansion, eqs. (3) for the near and far amplitudes follow upon interchanging the order of summation and integration in eq. (I I) and using eq. (10). To obtain the Coulomb decomposition ~e firstsubstitute/c(X) into eq. (1 I);

"~"X

30 l,

60 90 I

e~.(de¢

I

mo

I

t

)5o

Fig. 1 (a) The ratios of the positive-deflection-angle (dot-dash) and negative-deflection-angle (dot) Coulomb angular distributions to Rutherford for q = 1.496 (139 MeV a on SSNi). Co) Comparison of 139 MeV ~-SSNi optical potential fat-side angulat distributions calculated assuming no Coulomb contribution (dot-dash) and the Coulomb contribution given by eq. (15b) (dot). (c) Fat (dot), neat (dot-dash), and total (solid) optical-potential angular distributions for 139 MeV a on SSNi. In the mid-angular region fat and total angulat distributions coincide and ate indicated by the dotted curve.

Volume 57B, number 3

PHYSICS LETTERS

where

C(rh 0) = (sin 2 t 0 ) l +inS(r/, x)]21r.

(16)

This result differs qualitatively from the partialwave far-near decomposition given by eqs. (3). For such an amplitude 1

"~'

TN(0)/f(0) = 2 + iC(0), ffF(0)/ff(0)= ¼ - i ? ( 0 ) , (17a, b) while for the infinite-range Coulomb interaction the relative maguitude of the terms of eqs. (15) is the often-quite-small Gamow factor, e -2un; the relative probability of finding the particles at the origin for a repulsive and attractive Coulomb interaction. The preponderance of positive defiection-angle repulsiveCoulomb scattering at forward angles is responsible for moving the Fraunhofer diffraction structure in the elastic angular distribution of strongly absorbed particles away from the forward direction where it is most prominant for neutral-particle scattering. In most heavy-ion tandem experiments the Coulomb force is so strong the near-side contribution completely dominates and one sees only Fresnel diffraction [4]. This and other effects will be discussed in a forthcoming publication. One can have e -2~rn ,¢ I for a repulsive Coulomb interaction which leads to fC,N(0) ~/C(0) and IfC,F(0)l ~ IfC,N(0)t for all but the most back angles. This is illustrated in fig. la which shows the angular distributions for the amplitudes of eqs. (15) for a rela. tively weak Coulomb force, ~ = 1.496; e.g., 139 MeV a-scattering from 58Ni. Although repulsive Coulomb scattering is dominantly near-sided, the corrections in eq. (15) to total "near-sidedness" can be significant. Fig. 1b compares the far-side angular distributions for a potential used

7 July 1975

by Goldberg et al. [5] in their discussion of 139 MeV a elastic scattering on 58Ni calculated with eq. (15b) and the total "near-sidedness" assumption;fc, N(0) =fc(O),fC,F(O) --- 0, which is seen to be a good approximation at forward angles. Fig. lc shows do[dO as well as both doN/d0 and doF/d0 for the same potential. Eqs. (3) and (15)were used to calculate fiN(0), ffF(O),fC, N(O), andfc, N(0) respectively. The diffraction structure near the forward direction clearly arises from the far-side contribution passing through the near-side contribution which is dominant in the forward direction. For 0 ~ 45 ° the near-side contribution is dominant although it again interferes with the far-side at back angles. The structure in doF/d0 for 0 < 70 ° is characteristic of nuclearrainbow in the presence of absorption as Goldberg et al. [6] have argued. This will also be discussed in a forthcoming publication. I would like to thank K.W. McVoy for providing the original impetus for this work by suggesting fC, N(0 )

=f(O),fC,F(O ) = O. References

[1] K.W. Ford and J.A. Wheeler, Ann, Phys. (N.Y.) 7 (1959) 259. [2] R.C. Fuller and K.W. McVoy, Phys. Lett. (in press). [3] Handbook of mathematical functions, eds. M. Abramowitz and I. Stegun, National Bureau of Standards (1955). [4] W.E. Frahn, Phys. Rev. Lett. 26 (1971) 568. [5] D.A. Goldberg, S.M. Smith and G.F. Burdzik, phys. Rev.

C 10 (1974) 1362. [6] D.A. Goldberg and S.M. Smith, Phys. Rev. Lett. 33 (1974)

715.

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