The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. The potential is named after Roger D. Woods and David S. Saxon.
The form of the potential, in terms of the distance r from the center of nucleus, is:
V(r)=-
| V0 | |
| 1+\exp({r-R\overa |
)}
where V0 (having dimension of energy) represents the potential well depth,a is a length representing the "surface thickness" of the nucleus, and
R=r0A1/3
Typical values for the parameters are:, .
For large atomic number A this potential is similar to a potential well. It has the following desired properties
The Schrödinger equation of this potential can be solved analytically, by transforming it into a hypergeometric differential equation. The radial part of the wavefunction solution is given by
u(r)=
| 1r | |
| y |
\nu(1-y)\mu{}2F1(\mu+\nu,\mu+\nu+1;2\nu+1;y)
where
y=\dfrac{1}{1+\exp\left(
| r-R | |
| a |
\right)}
\mu=i\sqrt{\gamma2-\nu2}
\dfrac{2mE}{\hbar2}=-\nu2
\nu<0
| 2}a | |
| \dfrac{2mV | |
| 0}{\hbar |
2=\gamma2
{}2F1(a,b;c;z)=
| infty | |
| \sum | |
| n=0 |
| (a)n(b)n | |
| (c)n |
| zn | |
| n! |