Observations on the Energies of Single-Particle Neutron States

The available information on location of neutron single-particle levels is analyzed and the following conclusions are reached: (1) There is strong evidence that the depth of the shell-model potential well varies with symmetry energy, getting shallower as the neutron excess increases. (2) There is a stronger than average interaction between nucleons with the same orbital angular momentum; this causes a level to move down in energy as it fills (self-binding effect), or as a proton state of the same $l$ fills, but the effect seems to be weaker on the ($l-\frac{1}{2}$) neutron state as the ($l+\frac{1}{2}$) neutron state fills. (3) Spin-orbit splittings are extra large when the members of the doublet are in different shells, one full and the other empty; this is attributed to the self-binding effect. (4) The rate of change of binding energy with mass number for a given level, $\frac{\mathrm{dE}}{\mathrm{dA}}$, is considerably smaller than calculations would indicate; this may be explained as a decrease in potential well depth with $A$, or as a velocity dependence giving an effective mass of nucleons in nuclei somewhat larger than the free nucleon mass. (5) The spacings between oscillator shells is somewhat smaller than in the harmonic oscillator potential, and in available calculations for a Saxon potential; this again may indicate an effective mass greater than the free nucleon mass. (6) The $l$ dependence of the energies of shell-model levels is much smaller than given by Nilsson when the levels are empty, but the Nilsson term gives reasonable agreement when the levels are full; this indicates that the self-binding effect increases with increasing $l$. All of these effects are discussed and quantitative estimates of their magnitudes are given.