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:

$V(r) = -\frac$

where ''V''₀ (having dimension of energy) represents the potential well depth,
''a'' is a length representing the "surface thickness" of the nucleus, and $R = r_0 A^$ is the nuclear radius where and ''A'' is the mass number.

Typical values for the parameters are: , .

There are numerous optimized parameter sets available for different atomic nuclei.

For large atomic number ''A'' this potential is similar to a potential well. It has the following desired properties
* It is monotonically increasing with distance, i.e. attracting.
* For large ''A'', it is approximately flat in the center.
* Nucleons near the surface of the nucleus (i.e. having within a distance of order ''a'') experience a large force towards the center.
* It rapidly approaches zero as ''r'' goes to infinity (), reflecting the short-distance nature of the strong nuclear force.

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

$u(r)= \frac 1r y^\nu (1-y)^\mu _2F_1 (\mu+\nu, \mu+\nu+1; 2\nu+1; y)$

where $y = \dfrac$, $\mu = i\sqrt$, $\dfrac=-\nu^2$, $\nu <0$ and $\dfraca^2=\gamma^2$. Here $_2F_1(a,b;c;z) = \sum_^\infty \frac \frac$ 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.

It is also possible to give analytic formulas of the Fourier transformation of the Woods-Saxon potential which makes it possible to work in the momentum space as well.

See also

* Finite potential well
*Quantum harmonic oscillator
*Particle in a box
*Yukawa potential
*Nuclear force
*Nuclear structure
* Shell model

References

*
*

External links

http://nucracker.volya.net/ Woods–Saxon Solver

Nuclear physics

{{nuclear-stub
Quantum mechanical potentials
Exactly solvable models