Approximate Analytical Wave Functions for the Nuclear Independent-Particle Model

Abstract

The wave functions, normalization constants, and certain important diagonal matrix elements have been calculated for the diffuse boundary potentials previously described. By taking advantage of the variety of well parameter combinations which give rise to the dimensionless potentials and by utilizing a simple perturbation method, one may extend the usefulness of these wave functions to a large variety of potentials, and it should be possible to generate approximate independent-particle-model wave functions for any proton or neutron state in any nuclear species with $A>10$.Relative spin-orbit energy coefficients are calculated for a Thomas-type energy. The requirement that large energy discontinuities exist at the magic numbers $N=20, 28, 50, 82, \mathrm{and} 126$ restricts the spin-orbit term to be about 45 times the Thomas expression.The radii of neutron and proton orbitals are calculated, the latter being obtained after consideration is given to an attractive proton potential which is needed to account for the general trends of proton binding energies. Radii of neutron and proton distributions are also calculated. It is found that the equivalent uniform radii are ${R}_{n}=1.30{A}^{\frac{1}{3}}$ and ${R}_{p}=1.24{A}^{\frac{1}{3}}$ (in units of ${10}^{\ensuremath{-}13}$ cm). These are to be compared with an equivalent nuclear force radius ${R}_{w}=1.38{A}^{\frac{1}{3}}$. The attractive nuclear anomaly leads one to expect a thin (0-0.4) neutron membrane around nuclei.