Effective range parameters of separable potentials

Nuclear Physics

A236 (1974) 302-306;

Not to be reproduced

EFFECTIVE

RANGE

by photoprint

@

North-Holland

Publishing

Co., Amsterdam

or microfilm without written permission from the publisher

PARAMETERS

OF SEPARABLE

POTENTZALS

M. J. ENGLEFIELD Department

of Mathematics,

Monash

University,

Received

20 August

(Revised

4 October

Victoria,

3168, Australia

1974 1974)

Abstract: Formulas are obtained which express the effective range parameters separable potential as non-singular integrals of the form factors. These results allow for a Coulomb interaction.

of a rank-two are extended to

1. Introduction The description of interactions by separable potentials is popular ‘) owing to the simplifications which result in the three-body problem. For calculations on a three‘, “) that the separable potential model should fit body bound state, it is important well the low-energy two-body data, which is summarized by the values of the coefficients in the expansion kcot6 = -a-l++rk*+. . .. (1) It is well known that for a separable potential k cot 6 may be expressed in terms of quadratures involving the form factors. If these integrals can be evaluated analytically, expanding the result in powers of k gives the effective range expansion coefficients in terms of the potential parameters. However such formulas are usually very complicated 3), except in certain rank-one examples. A general formula for the scattering length a has been given 4), but the extension to the higher coefficients in (1) does not seem to be in the literature +. The object of this paper is to present formulas allowing the calculation of the effective range r without first evaluating k cot 6, and to extend to charged particles the results for both a and r. Since all integrals in this paper are from 0 to cc, taking a principal value if necessary, it will be sufficient to write j meaning P f,“. 2. Rank-two potentials Suppose a rank-two s-wave potential has an attraction and a repulsion factors g and h respectively. If z = g2, y = h’, and x = gh then ‘) kcot6

=

(l+H)(l-G)+M*

(2)

z(l+H)-y(l-G)-2xM’ t Since

this paper

was submitted,

the author

has seen a formula 302

with form

for r in ref. I’).

SEPARABLE

POTENTIALS

303

where :nG

=

z(p)dp + k2

s

z(p)dp/(p2

s

- k2),

(3)

while H and A4 are given by (3) with z replaced by y and x respectively. Since “) in the second integral in (3) one can replace z(p) by z(p) -z(O), and Sdp/(p’ -k2) = 0, then putting k = 0 in the denominator does not give a divergent integral. Defining T and Z by +nT =

+7cz = J‘dp:z(p)-z(0)~lp2

z(p)dp,

s

= /dpz’(p)lp,

(4)

one gets the expansion G = T+Zk2+. . .. Similarly H = S+ Yk2+. . ., A4 = Q+Xk’+. . ., where S and Y are given by (4) with y replacing z, and Q and X are given by (4) with x replacing z. If also g(p) then a straightforward a=

= %+92P2+.-

h13+h2p2+...,

of (2) in powers of k gives

expansion

h,E+goF D

h(p)=

.>

+r =

YE2+2XEF+ZF2

2h2E+2g2F Da2

’

-

a2D2

9

(5)

where D = (l+S)(l-T)+Q’,

Putting potential.

E =

h = 0 (g = 0) gives

F = h,Q-g,(l+S).

h,(l-T)+g,Q,

formulas

for

an

attractive

(repulsive)

(6) rank-one

The validity of these results has also been verified by solving the zero-energy equation in coordinate space. The momentum space form factors g and h are essentially the sine transforms of coordinate space form factors w and u .Thus kg(k)

=

s

dr w(r) sin kr,

and h, u can replace g, W. Define U(Y) and t(r) by (primes u ” = w> u’=u=t’=t=()

(7)

denote

derivatives)

t” = 0, at

y=m.

(8)

Then the determinant r

t(0) - t(r)

u(O)- u(r)

t(0)

1+s

Q

4’)

Q

T-l

gives the zero-energy wave function, from which a and r can be derived. In this treatment, the integrals S, T, Q, X, Y and Z appear expressed in terms of the coor-

304

M. J. ENGLEFIELD

dinate space form factors, for example T =

s

1rf2dr,

Z =

s

u(r){u(r)-22u(O)}dr.

Properties of the sine transform show from (7) and (8) that these integrals are the same as in (4). 3. Coulomb effects Phase shifts relative to a Couiomb potential ZZ’e’fr are still given by (2) if the form factors are modified. If F(r) is the s-wave regular Coulomb radial function, then (7) is replaced by “) kg,(k)

= /dr w(r)F(r).

(9)

The function P can be expanded ‘) in powers of kZ after a factor kc(k) where C’(k) = x/kR {exp (?rjkR)- 11,

is removed,

and R = h2/2~Z2’e2 with p the reduced mass. Hence there is an expansion g,(k) = C(k)(g~+g~k’+. - .), and in the previous method of expanding (3), z(0) is replaced by gGC2(p). Defining T, and Z, by tnr,

=

s

:nz,

&p)dp,

=

dp:g,2(p)-g~C2(p))/p2, u1

(10)

one gets G(k) = (2s; k2/*)s C’(p)dp/(p” - k2) + T, + Z, k* + 0(k4), in which the integral can be evaluated “) in terms of the $+function: (2~~)~k2

s

= Re {$(- ~~2k~)~ +ln (2kR) = f(k).

C2(~)d~~(~’ -k’)

The termsf(k) thus arising are just those used to modify the effective range expansion for charged particles: C2k

cot 6 f (f/R)

= --a,

’ + $r,

k’ + O(k4).

It follows that c, and I’, are still given by (5) and (6) but with different expansion

coefficients now defined by g,(k) = C(k)(% +

92

kZ

-t-

* * .>,

h,(k)

=

C(k)(h,+h#

Jr

. . .),

and the integrals X, Y, 2 redefined by formulas like (10). However expressions for the modified form factors have only been obtained in two cases. If w(r) = le-@, g(k) = l(k2+p2)-1, (11)

SEPARABLE

POTENTIALS

30s

then *j

Kw- p pnyJk’“‘] .

f

g (kj = k2+B2ex Putting

(11) in (9) and differentiating

with respect

to the parameter

/I leads to the

result “) h,(k) correspondjng

= (Rk ‘- ~~)g~(k)~~(k’

+ p2),

to h(k) = lk*(k’

t p”)- 2

or

W(Fj

=

R(t+r)e?

4. Conclusion This paper has obtained formula (5) for the effective range of a separable potential, and shown that the same formulas for r and a are valid for charged particles if the form factors are modified according to (7) and (9). These results should be useful in determining form factors, for compared to a least squares fit to low energy phase shifts, they allow a faster and more accurate evaluation of Y. Also a fit could now be made to the experimental p-p effective range parameters. The formulas have been used to compute n-p and p-p singlet effective range parameters from most of the rank-two potentials which have been published. Values agreeing with those previously reported were obtained, except for Strobe13 singlet n-p potential lo), which gave u = 270 fm and I’ = 2.1 fm. However Strobel’s potential is better than these figures suggest, since a very small value of D makes a and r very sensitive to small changes in the potential parameters. Gaussian form factors ’ “) are a good example of the usefulness of the formulas derived here, since the integrals T. Z, etc. are given by very simple algebraic expressions, whereas k cot 6 cannot be evaluated in terms of elementary functions. The same methods may be used to expand off-shell quantities in powers of energy. For example, the coe~cient of k2 -p 2 in the half-off-shell extension function *‘j f’(y, h-) is - (sz F+fr, E)jaB (this parameter is often denoted by +A2 in the literature).

References 1) J. S. Levinger, Springer tracts in modern physics 71 (1974) 88 2) G. L. Schrenk,

V. K. Gupta

and A. N. Mitra,

Phys.

Rev. Cl (1970)

895;

B. F. Gibson and G. J. Stephenson, Phys. Rev. C8 (1973) 1222; R. van Wageningen, J. H. Stuivenberg, J. Bruinsma and G. Erens, Proc. Int. Conf. on few particle problems in the nuclear interaction. Los Angeles, 1972, ed. I. Slaus, S. A. Moszkowski, R. P. Haddock and W. T. H. van Oers (North-Holland, Amsterdam, 1972) p. 453; J. J. Benayoun, C. Gignoux, J. Gillespie and A. Laverne, Nuovo Cim. Lett. 8 (1973) 414; P. LJ. Sauer and J. A. Tjon, Nucl. Phys. A216 (1973) 541 3) N. M. Petrov, Sov. J. Nucl. Phys. 16 (1973) 733; J. H. Naqvi, Nucl. Phys. 58 (1964) 289; B. Bagchi and B. Muiiigan, Phys. Rev. Cl0 (1974) 126

306 4) 5) 6) 7) 8) 9) 10) 11)

M. J. ENGLEFIELD

F. Tabakin, Phys. Rev. 174 (1968) 1210 F. Tabakin, Ann. of Phys. 30 (1964) 51 J. Potter and K. Chadan, Compt. Rend. 248Q959) 1612 P. Swan, Nucl. Phys. A90 (1967) 412 D. R. Harrington, Phys. Rev. 139 (1965) B694 T. Vo-Dai and Y. Nogami, Nucl. Phys. A141 (1970) 369 G. L. Strobel, Nucl. Phys. All6 (1968) 465 H. P. Noyes, Phys. Rev. Lett. 15 (1965) 538; K. L. Kowalski, Phys. Rev. Lett. 15 (1965) 798 12) J. Bruinsma, W. Ebenhah, J. H. Stuivenberg and R. van Wageningen, Nucf. Phys. A228 (1974) 52