7D-folding integral in a density-dependent microscopic optical model potential for nucleon-nucleus scattering

Abstract

Microscopic optical model potentials, based on density-dependent effective interactions, involve multidimensional integrals to account for the full Fermi motion of the target struck nucleon throughout the nucleus. If a spherical matter distribution is assumed, then each matrix element of the optical potential requires the evaluation of seven-dimensional integrals. In this work we provide a full account of these integrals, retaining the genuine off-shell structure of the nucleon-nucleon effective interaction. The evaluation is based on the asymptotic separation of the optical model potential for nucleon-nucleus scattering in momentum space, where the potential is split into a free $t$-matrix contribution and another which depends exclusively on the gradient of the density-dependent $g$ matrix. The calculated potentials, based on the Paris nucleon-nucleon ($NN$) potential, are applied to proton elastic scattering from ${}^{16}$O and ${}^{90}$Zr at beam energies between 30 and 65 MeV. The results were compared with two approximations to the unabridged expression, revealing moderate differences among their scattering observables. When comparing with results based on the Argonne ${v}_{18}$ $NN$potential, these differences appear smaller than those attainable by the choice of the internucleon potential.