Eigenfunctions for a Spherical and a Deformed Saxon-Woods Potential

Abstract

The eigenfunctions for spherical and deformed Saxon-Woods potentials with a Thomas-type spin-orbit coupling are calculated for mass number $A=185$. The expansion coefficients into oscillator functions and the energies are tabulated for the spherical Saxon-Woods functions for $N=4 \mathrm{and} 5$ proton shells and for $N=5 \mathrm{and} 6$ neutron shells. For states with a binding energy >2 MeV the first six coefficients contain more than 99% of the Saxon-Woods function. The overlap integral for these states with the corresponding oscillator functions is always greater than 0.92. The basis for the spheroidal nuclei is confined to a single $N$ shell. The deformed potential reproduces the experimental level sequence in the rare-earth nuclei using the optical-model parameters. The well depth, however, is fitted using the binding energy of the last particle. The differences between the quadrupole matrix elements in the Saxon-Woods and oscillator potentials are as large as 60% for states with a binding energy greater than 2 MeV. The $\ensuremath{\gamma}$-ray transitions between states of small binding energy and more than one node are enhanced by several oscillator units. The quadrupole matrix elements and the mixing coefficients for the rare-earth nuclei are tabulated and the expressions for the transition probabilities are given in the ($\mathrm{jlK}$) representation.