Two-Body Forces and Nuclear Saturation. I. Central Forces

Abstract

The problem of nuclear saturation for the rapidly varying and nonmonotonic potentials of pseudoscalar meson theory has been investigated. In these potentials, variational methods using independent-particle trial functions are grossly inadequate. Although the problem can be approached using more general variational functions with interparticle correlation, the evaluation of the resulting expressions is very difficult since indirect correlations involving many more than two particles become important at high densities. An alternative procedure has been developed which allows a rather straightforward evaluation of the many-body energy even when the potentials are of great complexity. This method depends on a treatment of the coherent particle motion which is exact in the limit of very many scatterers, and treats the incoherent motion as a perturbation. In this case the many-body potential energy can be simply expressed in terms of the low-energy scattering amplitudes. This method has been applied to the two-body potentials given by pseudoscalar meson theory when the effects of nucleon pair formation are assumed to be small. In this approximation the many-body forces of the theory are negligible. These potentials given an excellent fit to the low-energy scattering parameters of the two-nucleon system and also an approximately correct description of scattering up to 90 Mev. They are characterized by repulsive cores of radii 0.3-0.4\ensuremath{\Elzxh}/\ensuremath{\mu}c and quite weak interactions in odd states. The many-body energy has been evaluated neglecting the tensor contributions which average to zero in first approximation. The result shows saturation at an energy per particle (neglecting Coulomb energy) of 12 Mev at a nuclear radius of $1.15\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{A}^{\frac{1}{3}}$ cm. The method has also been applied to potentials of the L\'evy type in which the odd-state potentials are rather strong and attractive. To give saturation with these near normal density, a 3-body force of the type given by the pair terms in the pseudoscalar coupling with a coupling constant $\frac{{g}^{2}}{4\ensuremath{\pi}}\ensuremath{\sim}3$ is required. Finally, the method can easily be extended to the determination of the elastic interaction of a slow neutron with a nucleus. The resulting "Weisskopf" potential has a depth of 35 Mev.