Nucleon Energy Levels in a Diffuse Potential

The level sequence and shell structure for the bound single-particle states of nucleons moving in the spherically symmetric potential $V\left(r\right)=-\frac{V_0}{[1+\exp \alpha \mathrm{}(r-a\left)\right]}$ 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 $\alpha =1.45\mathrm{}\times {10}^{+13\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ (implying a surface layer of approximately 3×${10}^{-13\mathrm{}}$ cm, which is constant for all $A$), a spin-orbit coupling 39.5 times the Thomas term and $a=1.3\mathrm{}{A}^{\frac{1}{3}}\times {10}^{-13\mathrm{}}$ 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 $V_0=42.8\mathrm{}$ 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 $A=56\mathrm{} \mathrm{and} 166$ respectively. The neutron and proton distributions in ${}_{79}{}^{}{}_{}{}^{}\mathrm{Au}_{}^{197}{}_{}{}^{}$ are examined. With the values of $\alpha$, $r_0$, and $\lambda$ given above, the thickness of the surface layer on the proton distribution is 1.92×${10}^{-13\mathrm{}}$ cm and the radius is 6.77×${10}^{-13\mathrm{}}$ cm.