On the effects of non-local potentials

2.1

I

Nuclear Physics A90 (1967) 140--144; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON THE EFFECTS OF NON-LOCAL P O T E N T I A L S Y. N. S R I V A S T A V A and M. S. WEISS

Northeastern University, Boston, Massachusetts Received 4 July 1966 Abstract: The results of Austern on the Percy effect have been re-derived under more general assumptions for one dimension and are extended to three dimensions. We propose a mechanism to explain the over-estimation of calculated photonuclear cross sections.

Perey 1) found numerically that if scattering data are fitted by two sets of potentials, one non-local and the other local, then the corresponding non-local wave function is smaller in the nuclear interior than the local one. Austern has recently given an analytic derivation of this so-called Perey effect for a one-dimensional problem 2). Also, Saxon and Perey have, in the effective mass approximation tried to obtain a similar behaviour 3). In this note we would like to show that the Austern-SaxonPerey results can be derived under very general assumptions. In particular, we do not require the "local exponential approximation" invoked in ref. 2), only a shortrange non-locality and the reality of the equivalent local potential. We extend the argument to three dimensions and find that the local potential so generated is l (i.e. angular momentum) independent. An explanation of the frequent theoretical overestimation of the magnitude of photonuclear cross sections is suggested. (i) Consider one-dimensional, "non-local" and "local" Schr/Sdinger equations (we follow the notation in ref. 2)), hereafter called I, for ease of comparison) - ~,"(~) + i ( x ) = k 2 ~ ( ~ ) ,

(1)

- ~L'(x) + VL(~)~L(~) = k 2~L(~),

(2)

where

l(x) =

U

H(x-x')T(x')dx'.

(3)

co

Here U and H are real. Define

v(x) _= F(x)~'L(x),

(4)

where

Ixl-~ ~ ,

F ( x ) ~ 1,

for the same scattering to occur in both cases. * W o r k supported in part by a grant from the National Science Foundation. 140

(5)

NON-LOCAL POTENTIALS

141

F(x) UL(x ) = l(x) -- F"(x) ~UL(X) -- 2F'(x) 7JL(X)

(6)

Then one simply obtains

IJ~/L(X )

Here we depart from I. Rather than make the "local exponential approximation", we utilize the reality of the equivalent local potential UL(X). Let us write

[~L(X)leiO(x).

t//L(X ) =

(7)

If we insert (7) into (6) and demand that both UL(x ) and F(x) be real, we get 2F'(x) =

'

f_

dtU(x+½t)H(t)F(x+t)

[(IYL(X+ t)l sin [O(x + t)-- O(x)] ]~L(X)] O'(x)

(S)

Eq. (8) may be further simplified, if as in I, we assume that F is slowly varying with respect to H, or, equivalently, that the region of non-locality is small. Then, keeping terms to lowest order in F, (8) yields

F'(x) ~ ½F(x

U(x +½t)H(t)tdt,

(9)

Ct3

from which

F(x) = exp

dx'

dt U x'+½t)H t)t ,

10)

which satisfies the boundary condition (5). This F(x) is clearly < 1 in the region of an attractive potential. The first correction to (9) is of the form

½F'(x)~

t 2U(x + ½t)H(t)dt.

If we formally Taylor expand H(t) in terms of 3-functions, we see that this term is proportional to 5"(0 and higher derivatives in H(t). This provides an a posteriori check that the corrections are indeed small for a small region of non-locality. It may easily be checked using these F's that the equivalent local potentials UL(x ) of (5) have no "pathologies"; i.e., they do not blow up at the zeros of ~(x) (ref. 1)). Specifically, using eq. (6), we may derive the following form for UL(X) in the spirit of eq. (9): UL(X) m ( ' ° U(x +½OH(t) F(x + t) d t - F"(x) J- ~o F(x) F(x)

(11)

Since F(x) lies between 0 and 1, UL(x) is non-singular on the x-axis. If we further approximate (11) by neglecting the F" term and replacing F(x+ t) by F(x) inside the integral, we obtain UL(X) ~ o~

U(x+½t)H(t)dt.

(12)

142

Y. N . SRIVASTAVA A N D M. S. "vVEISS

Now, we consider scattering of spinless particles in a central potential in three dimensions. Again, we define for the lth partial wave, 7Jt(r) = e,(r)q,t(r),

(13)

where ~ and ~b are the solutions to the radial Schr/Sdinger equation for the non-local

and the local potentials, respectively. Following our steps outlined above, the reality of the equivalent local potential Vc, ~ and F¢ requires: F;(r) - - - ~ ½fo ° V(r, r ' ) ( r ' - r ) d r ' ,

Ft(r)

(14)

which obtains an F independent of l. Also,

o°l/(r, r')F(r')dr'- F'(r)

(15)

which is also l-independent. Hence, this reality requirement on the equivalent potential not only yields the Perey effect in three dimensions but leads naturally to a potential which does not depend upon the angular momentum l. These results lead us to suggest that the Perey effect may explain the existing theoretical over-estimation of the magnitude of the photonucleon giant resonanc: cross section. This over-estimate is a factor of 1.5 to 2 (Mikeska 4) on 160 and Balashov et aI. s) on 2°8pb) whereas the calculated relative cross sections are in much better agreement with experiment 6). To understand this effect, let us consider an idealized example of such a calculation. For a given A, one would construct a finite well whose bound states and resonances were chosen from assumed single particle states in the A _ 1 nuclei 7). The motivation is to construct a potential which will have the properties associated with the HartreeFock potential for that A. In addition to fitting the single-particle energies, filling the lowest A orbitals of this well must represent the ground state of the nucleus, and this imposes the additional constraint that the ground state "size" be correct. The giant-resonance calculation would then proceed by configuration mixing the particlehole states of that well to generate the nuclear eigenstates. Now generally one performs this calculation using a local well. If it were repeated using a non-local well, which would be a more realistic representation of the HartreeFock well because the Hartree-Fock well must be non-local due to the Pauli principle, we would expect the scattering states of the non-local well to be of the Perey type. The bound states would, at least to a first approximation, not be altered, because of the size requirement on the ground state. The configuration mixing would also not be altered to a first approximation because F is slowly varying and independent of both l and energy. There would be a difference in the magnitude of the calculated cross

143

NON-LOCAL POTENTIALS

section because the dipole matrix element would be reduced by some average F due to the reduction in the value of the scattering state wave function in the interior of the nucleus where it would have its overlap with the bound state. If we employ Perey's F, which is about 0.8, the cross section will be multiplied by ~ 2. There are at least three criticisms one can make of this proposal. The first is that we have generated a paradox. We argue that replacing the local potential with a non-local potential reduces the magnitude of the cross section, whereas one might think that changing from a local to a non-local must increase the sum rule as one will have an extra term not commuting with the dipole operator. This is fallacious because the sum rule depends upon the dipole operator's commutator with the total nuclear Hamiltonian. The Hartree-Fock potential is really added and subtracted from this Hamiltonian

tt = E i

E Vii -- Y

i
i

E i

E V,j- Z

i
i

and in going l?om a local to a non-local Ui one must correct for the residual interaction. The second objection is that our extrapolation of Perey's F-function to the HartreeFock and our assumption about the independence of F on 1 and energy may not be correct. This is certainly a valid criticism. We have no knowledge of the Hartree-Fock F and it really is dependent upon l and energy, the independence in our deriw~tion was shown only to lowest order in the range of non-locality. Still, Perey's results do appear quite general, and Muthukrishnan and Baranger s) claim their Hartree-Fock potential is quite non-local. Moreover, we are only drawing qualitative conclusions, which we would not expect to be altered by some slight l and energy dependence of F. The third objection is that the Perey effect on the bound states might wash out our depletion argument. If one applies the F-operator to bound states of the local well in order to get the bound states of the non-local well, the new bound states will not (necessarily) still be normalized because 0 < F < 1. If the bound states of the local well were entirely inside the nucleus, then after multiplying by F and then renormalizing them, we would come back to essentially the same wave function in the local and non-local well. If one considered a bound state so bizarre that only a negligible amount of the wave function was inside the well, then the Perey effect would not alter the wave function, but there would be no reduction in the calculated dipole cross section. We are assuming the former case is qualitatively valid for real nuclei. For a nucleus with some bound state wave function exterior to the nucleus, applying the F-operator and a subsequent renormalization will tend to reduce the amount of wave function inside the well relative to the amount outside. However, if one demands that the size, which is essentially the expectation value of r 2, be essentially the same for the local and non-local wells, the bound states cannot be too different. We would like to thank P. Nath for bringing this problem to our attention and for an illuminating discussion. Thanks are also due to N. Austern, G. E. Brown and A. Cromer for helpful criticisms of the manuscript.

144

Y. N .

SRIVASTAVA A N D M. S. WEISS

References 1) F. G. Perey, in Direct interactions and nuclear reaction mechanisms, ed. by E. Clementel and C. Villi (Gordon and Breach, New York, 1963) p. 125 2) N. Austern, Phys. Rev. 137 0965) B752 3) F. G. Perey and D. S. Saxon, Phys. Lett. 10 (1964) 107 4) H. J. Mikeska, Z. Phys. 177 (1964) 441 5) V. V. Balashov, V. G. Shevchenko and N. P. Yudin, JETP (Sov. Phys.) 14 (1962) 1371 6) R. L. Bramblett, J. T. Caldwell, R. R. Harvey and S. C. Fultz, Phys. Rev. 133 (1964) B869; R. R. Harvey, J. T. Caldwell, R. L. Bramblett and S. C. Fultz, Phys. Rev. 136 0964) B126 7) G. E. Brown, Unified theory of nuclear models (North-Holland Publ. Co., Amsterdam, 1965) 8) R. Muthukrishnan and M. Baranger, Phys. Lett. 18 0965) 160