Nucleon Energy Levels in a Diffuse Potential

Abstract

The level sequence and shell structure for the bound single-particle states of nucleons moving in the spherically symmetric potential $V(r)=\ensuremath{-}\frac{{V}_{0}}{[1+\mathrm{exp}\ensuremath{\alpha}(r\ensuremath{-}a)]}$ have been examined. For protons, a Coulomb potential was added corresponding to a uniform charge distribution out to the "nuclear radius," $a$, and the potential depth was increased to give sufficient binding energy for the last proton level in the nucleus under consideration. For $\ensuremath{\alpha}=1.45\ifmmode\times\else\texttimes\fi{}{10}^{+13}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ (implying a surface layer of approximately 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm, which is constant for all $A$), a spin-orbit coupling 39.5 times the Thomas term and $a=1.3{A}^{\frac{1}{3}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm, good shell structure is obtained for both neutrons and protons. The level sequences obtained are in close agreement with experiment except in the region of strong distortion from sphericity. For ${V}_{0}=42.8$ Mev the neutron binding energies are in reasonable agreement with experiment. With these parameters the $3s$ and $4s$ giant resonances in the low-energy neutron scattering cross section occur at $A=56 \mathrm{and} 166$ respectively. The neutron and proton distributions in $_{79}\mathrm{Au}^{197}$ are examined. With the values of $\ensuremath{\alpha}$, ${r}_{0}$, and $\ensuremath{\lambda}$ given above, the thickness of the surface layer on the proton distribution is 1.92\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm and the radius is 6.77\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm.