Smooth Velocity-Dependent Potential and Nuclear Matter

A velocity-dependent potential of the form V(r,p)=-V₁(r)+p²e^-s²p²V₂(r)+V₂(r)p²e^-s²p², where V₁(r) and V₂(r) are Gaussians, is used to fit the singlet (0-320-MeV) phase shifts for elastic nucleonnucleon scattering. This exponential velocity dependence replaces the hard core with a short-range repulsion which is much softer than the case of quadratic ($s=0$) velocity dependence used by Green, Levinger, et al. The two-body scattering problem is solved in momentum space by numerical summation of the Born series; the ¹S₀ scattering length is calculated separately by a rapid matrix-inversion method. The applicability of ordinary many-nucleon perturbation theory for this interaction is tested by calculation of the first-order (P₁) and second-order (P₂) potential energy per particle of nuclear matter. A rapid singlet-even convergence rate of P₂P₁=4.3% at kF=1.5 F follows as a result of the reduced off-energy-shell matrix elements of this two-body interaction. One also finds qualitative agreement with the singlet-state potential energy per particle obtained by Sprung et al. in their complete nuclear-matter calculation. The harmonic-oscillator matrix elements required for Hartree-Fock calculations of spherical nuclei are evaluated quite simply for this potential. Also, since this potential readily separates into relative x, y, and z coordinates, it is well suited for Hartree-Fock calculations of deformed nuclei.