Coulomb Effect on the Proton-Proton Low Energy Scattering Parameters1

M. W. KERMODE~, Y. NOGAMI, A N D W. VAN D I J K ~ Depnrttnetlt of'Physics. McMnster Utliversity, Hntniltotl, O~ltrrrio U S 4M1

Received September 30, 1974

The effect of the Coulomb force on low energy proton-proton scattering is examined when the nuclear interaction is represented by a rank one separable potential of Tabakin's type. The scattering length which is obtained by switching off the Coulomb force is found to be -20.0 F. This value is appreciably different from those for local realistic potentials. This anomaly is similar to the one which was recently found by Sauer.

On examine I'effet de la force de Coulomb sur la diffusion proton-proton a basse energie, lorsque ['interaction nucleaire est representee par un potentiel separable de rang un de type Tabakin. On trouve que la longueur de diffusion obtenue lorsqu'on enleve la force de Coulomb est Cgale a -20.0 F. Cette valeur differe appriciablement de celles qu'on obtient avec des potentiels realistes locaux. Cette anomalie est semblable a celle qui a recemrnent etC trouvee par Sauer. [Traduit par le journal] Can. J . Phys., 53,207(1975)

There is a long history concerning the test of charge independence and charge symmetry of nuclear forces by means of the comparison of the low energy scattering parameters for pp, pn, and nn in the ' S o state (Henley 1969; and references quoted therein). For this comparison one has to subtract the effect of the Coulomb force for the pp parameters. This is done by first fitting the pp data using V, + 'J,, and then cal- culating the scattering length a, and effective range r, using V, alone. Here Vc (= e2/r) is the Coulomb potential and V, is the rest of the pp potential. The suffix s refers to the strong inter- action4. Using several local realistic potentials for V,, Heller et al. (1964) found

indicating that a, was quite insensitive to the choice of V,.

Recently Sauer (1974) pointed out, however, by using a family of potentials which are related to Reid's potential (Reid 1968) by a phase shift conserving unitary transformation, that the

'Work supported by the National Research Council of Canada.

'Permanent address: Department of Applied Mathe- matics and Theoretical Physics, University of Liverpool, Liverpool L69 3BX, England.

3Permanent address: Dordt College, Sioux Center, Iowa 51250.

4Since V, is the Coulomb potential between point protons, Vs contains effects of finite size of proton, magnetic interaction, etc. When theseeffects are neglected, charge symmetry says that V, is the neutron-neutron potential.

value of a, strongly depends on the form of the nuclear potential at short distances. For Sauer's potentials which result in a large deviation from [I], the wave function has one or more additional nodes at short distances than in the case of more conventional potentials, and hence the prob- ability of finding two protons close together is enhanced. This reminds us of a nonlocal separ- able potential of rank one proposed by Tabakin (1968), for which the wave function has an addi- tional node at a short distance. Motivated by this similarity we have examined a, and r, for Tabakin's separable potential of rank one.

The potential V, for two protons in the ' S o state in the momentum space is assumed to be

[2] V,(k, k') = - (h2/2n2m)g(k)g(k')

where m is the proton mass (h2/m = 41.50 MeV.F2) and

We deal with two phase shifts, 6, and 6,,; 6, is due to V, alone while 6,, arises from V, + Vc. The phase shift 6, is determined by

[4 1 tan 6,(k) = kg2(k) /~(k)

where

In determining the parameters in V,, we impose Tabakin's condition

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208 CAN. J. PHYS. VOL. 53, 1975

[6 1 D(kc) = 0 where

so that D(k) and hence tan 6,(k) changes sign at Ic = kc. Note that g(kc) = 0. The scattering length a, and effective range r, are defined by k cot 6,(k) = -a,-' + (1/2)r,k2 + ...

The Schrodinger equation for Vs + V, can be solved in momentum space using the method given by Harrington (1965). However, it is quite straightforward to integrate the Schrodinger equation for a local plus nonlocal potentials numerically in coordinate space, using a simple extension of the Numerov method as described briefly below.

The Fourier transform of V,(lc, k') is given by

[7] Vs(r, r ') = (27~)- j dk dk' Vs(k, k')

x exp [i(k. r + k'. r')]

= -(h2/m)w(r)w(r')/(4.rr')

where

and

The Schrodinger equation for u(r) = r\lr(r) is

where v,(r, r') = w(r)w(rl) and v, = (m/h2)VC. If h is the step length and r,, = nh, then un = u(rn) satisfies the recursion formula for n = 1, 2, ..., N:

s = - sow w(rl)u(r') dr'

With uo = 0 and ul = 1, [ l 1 ] gives us N equa- tions for N unknowns (u,, u,, ..., uN+ l). s is known when u's are known. We can eliminate s by combining two adjacent equations, to obtain

Hence, given u,, u,, and u,, we can solve for any u, and hence find the phase shift. However, we do not know u2, so we integrate twice, starting with uo(') = 0, u,(') = 1, u2(') = 2, and then u , ( ~ ) = 0, u , ( ~ ) = 0 9 u 2 ( 2 ) = 1. At the same time we calculate s(') and d 2 ) . The actual wave function and s have the form u = u(') + A d 2 ) and s = s(') + A d 2 ) respectively. Substituting these for u and s in [l 1 ] with 11 = I, we have a consistency condition for A. In principle, we can repeat the process with u2(" = 2 + A, but this was found to be unnecessary, sufficient accuracy being obtained already.

The scattering parameters a,, and r,, are ob- tained from 6,, by the standard formula

For the parameters in the potential, we take kc = 1.7 F- ' which corresponds to a center of mass (CM) energy of 120 MeV, and then as- suming a value for a, we determine b, d, and ci so that the experimental values of a,, and r,, are reproduced, with the constraint of [6]. For the experimental data we take (Slobodrian 1968)

[I51 a,, = -7.786 F; r,, = 2.840 F

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KERMODE E T AL.: COULOMB EFFECT 209

TABLE 1. Parameters in the potential, and anomalous results in the three body system. results. The experimental value of 6-c (60 MeV) Beam (1969) using Tabakin's potential obtained

is 20.69" (MacGregor et a/. 1969) 355 MeV for the binding energy of the triton.

a (F-') 4.05 3.52 Beam attributed this anomaly to there being a b (F-l) 0.95425 0.77016 zero width resonance at k = kc. We have veri- d (F-l) 1.47434 0.99814 fied the existence of this resonance by inte- a2 (Fb2) 412.9056 310.3240 grating the Schrodinger equation (for V, alone) as (F) -20.54 - 19.95

2.955 for various values of cc2 ranging from zero to the

rs (F) 2.948 (60 MeV) 18.07" 20.68" value given in Table 1. The width of the reso-

nance decreased to zero at the tabulated value. In the presence of Vc, although tan S,(k) varies

For a we consider two values; a = 4.05 F- ' which is the same as Tabakin's, and a = 3.52 F- ' which fits the phase shift S,, at 60 MeV (CM) very well.

The results are summarized in Table 1. Note that the values of a, obtained are well outside of the range of [I]. This means that the effect of the Coulomb force is considerably larger for Tabakin's potential than for those potentials considered for [l]. As shown by Tabakin (1968) the wave function at zero energy for this separ- able potential has a node at r. z 0.5 F, and hence \lr2 is quite appreciable for r < 0.5 F, and this is why the Coulomb effect is enhanced. It was shown before (Vo Dai and Nogami 1970) that the rank two separable potential of Naqvi (1964) leads to a, = - 18.15 F, and this was inter- preted as due to the wave function which is relatively large at short distances. The effect is greater for Tabakin's potential due to the addi- tional node of the wave function. Some of the wave functions obtained by Sauer (1974) also have one or more additional nodes. In fact close scrutiny reveals that his wave function can be enormously large at short distances, and this is why the Coulomb effect is drastically enhanced in his case.

Finally it must be pointed out that potentials of the type considered in this note have given

smoothly around lc = kc, tari6,,(k) has a dis- continuity which corresponds to a resonance of a very small but finite width. Actually we should have adjusted the parameters in the potential so that tan SSc(k) rather than tan S,(k) varies smoothly around k = kc, but the effect of this readjustment on the results will be negligible.

Acknowledgments

We would like to thank Don Sprung for helpful discussions. M. W. Kermode and W. van Dijk are grateful to the physics depart- ment of McMaster University for the kind hospitality extended to them during their visit in the summer of 1974.

BEAM, J . E. 1969. Phys. Lett. B,30,67. HARRINGTON, D. R. 1965. Phys. Rev. 139, B691. HELLER, L., SIGNELL, P., and YODER, N. R. 1964. Phys.

Rev. Lett. 13,577. HENLEY, E. M. 1969. I n Isospin in nuclear physics (North

Holland Publishing Co., Amsterdam), p. 15. MACGREGOR, M. H. , ARNDT, R. A., and WRIGHT, R. M.

1969. Phys. Rev. 182, 1714. NAQVI, J. H. 1964. Nucl. Phys. 59,289. REID, R. V. 1968. Ann. Phys. 50.41 1. SAUER, P. U. 1974. Phys. Rev. Lett.32,628. SLOBODRIAN, R. J . 1968. Phys. Rev. Lett. 21,438. TABAKIN, F. 1%8. Phys. Rev. 174, 1208. V o DAI, T.and NOGAMI, Y. 1970. Nucl. Phys. A, 141,369.

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