Woods-Saxon potential

The Woods-Saxon potential is a mean-field Potential for the nucleons in an atomic nucleus, motivated by the observation that nuclei are not perfect spheres with sharply defined surfaces. Rather, the charge density smoothly falls off as the surface is approached instead of cutting out abruptly. This potential is meant to be used in the nuclear shell model to replace the more basic quantum harmonic oscillator.

Mathematically, it reads

where

• $r$ is the distance from the nuclear center;
• $V_{0}$ is the potential well depth, typically $\sim50\text{ MeV}$;
• $a$ is the nuclear surface thickness, typically $\sim0.5\text{ fm}$;
• $R$ is the mean nuclear radius, $\sim1.25A^{1/3}$, and $A$ is the atomic mass number.

This form is derived from describing the charge distribution of the nucleus through a two-parameter Fermi-Dirac distribution

where $c$ is the radius at which $\rho(c)=\rho(0)/2$ and $a$ is the aforementioned nuclear surface thickness.