Bound States and Low-Energy Neutron Scattering

Solutions are given for the $S$ states of the potential $V\left(r\right)=-\frac{V_0}{[1+e^{\alpha \mathrm{}(r-a)\mathrm{}}]}$. These are used to calculate the position of the giant $S$-wave neutron resonances at low (zero) energy and the bound states of the potential. If one chooses $V_0=39.8\mathrm{}$ Mev, $\alpha =2\mathrm{}\times {10}^{13}$ ${\mathrm{cm}}^{-1\mathrm{}}$ and $a=1.35\mathrm{}{A}^{\frac{1}{3}}\times {10}^{-13\mathrm{}}$ cm, the potential binds the right number of particles. $S$-wave maxima in the elastic scattering cross sections then occur at $A=13, 64, \mathrm{and} 183$. The experimental curves show a rise in the total scattering cross section in the region $A\sim 60 \mathrm{and} 180$ for small neutron energies.