Geometric interpretation of the effect of the quadrupole force in the collisions of deformed nuclei

Abstract

The effect of a quadrupole force on a set of degenerate states of a rotational band with arbitrary spin projection along the symmetry axis K is studied. Analytic expressions for the eigenvalues and eigenvectors are obtained in terms of a set of orthogonal polynomials. This is applied to the collision of a spherical nucleus with a deformed one in which the coupling to a given set of rotational states is allowed, ignoring excitation energies. The elastic S-matrix, transition amplitudes, and the fusion cross sections are obtained as a weighted average of the magnitudes corresponding to a set of definite orientations of the axis of the deformed nucleus with respect to the relative coordinate. That weighted average corresponds to approximate the extreme sudden result, consisting of an integral over all the orientations, by a generalized Gaussian quadrature