Woods–Saxon potential
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:
where V0 (having dimension of energy) represents the potential well depth, a is a length representing the "surface thickness" of the nucleus, and
is the nuclear radius where r0 = 1.25 fm and A is the mass number.
Typical values for the parameters are: V0 ≈ 50 MeV, a ≈ 0.5 fm.
There are numerous optimized parameter sets available for different atomic nuclei.
[1] [2][3] 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
ν
( μ + ν , μ + ν + 1 ; 2 ν + 1 ; y )
ν
ν
ν < 0
is the hypergeometric function.
It is also possible to analytically solve the eigenvalue problem of the Schrödinger equation with the WS potential plus a finite number of the Dirac delta functions.
[5] It is also possible to give analytic formulas of the Fourier transformation[6] of the Woods-Saxon potential which makes it possible to work in the momentum space as well.
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