Woods–Saxon potential

Woods–Saxon potential for A = 50, relative to V₀ with a = 0.5 fm and ${\displaystyle R=4.6fm}$

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:

${\displaystyle V(r)=-{\frac {V_{0}}{1+\exp({r-R \over a})}}}$

where V₀ (having dimension of energy) represents the potential well depth, a is a length representing the "surface thickness" of the nucleus, and ${\displaystyle R=r_{0}A^{1/3}}$ 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

• 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 r ≈ R within a distance of order a) experience a large force towards the center.
• It rapidly approaches zero as r goes to infinity (r − R >> a), 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

${\displaystyle u(r)={\frac {1}{r}}y^{\nu }(1-y)^{\mu }{}_{2}F_{1}(\mu +\nu ,\mu +\nu +1;2\nu +1;y)}$

where ${\displaystyle y={\dfrac {1}{1+\exp \left({\frac {r-R}{a}}\right)}}}$, ${\displaystyle \mu =i{\sqrt {\gamma ^{2}-\nu ^{2}}}}$, ${\displaystyle {\dfrac {2mE}{\hbar ^{2}}}=-\nu ^{2}}$, ${\displaystyle \nu <0}$ and ${\displaystyle {\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma ^{2}}$.[4] Here ${\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}}$ is the hypergeometric function.

It is also possible to analytically solve the eignenvalue problem of 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.

See also

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

References

1. Dudek, J.; Szymanski, Z.; Werner, T. (1980). "Woods-Saxon potential parameters optimized to the high spin spectra in the lead region". Phys. Rev. C. 23: 940. doi:10.1103/PhysRevC.23.920.
2. Schwierz, N.; Wiedenhover, I.; Volya, A. "Parameterization of the Woods-Saxon Potential for Shell-Model Calculations". arXiv:0709.3525.
3. Gan, L.; Li, Z.-H.; Sun, H.-B.; Hu, S.-P.; Li, E.-T.; Zhong, J. (2021). "Systematic study of the Woods-Saxon potential parameters between heavy-ions". Chinese Physics. 45 (5): 054105 – via 10.1088/1674-1137/abe84f.
4. Flügge, Siegfried (1999). Practical Quantum Mechanics. Springer Berlin Heidelberg. pp. 162ff. ISBN 978-3-642-61995-3.
5. Erkol, H.; Demiralp, E. (2007). "The Woods–Saxon potential with point interactions". Physics Letters A. 365 (1–2): 55–63. doi:10.1016/j.physleta.2006.12.050.
6. Hlope, L.; Elster, Ch.; Johnson, R.C.; Upadhyay, N.J.; Nunes, F.M.; Arbanas, G.; Eremenko, V.; et, all (2013). "Separable representation of phenomenological optical potentials of Woods-Saxon type". Phys. Rev. C. 88: 064608. arXiv:1310.8334. doi:10.1103/PhysRevC.88.064608.

• Woods, R. D.; Saxon, D. S. (1954). "Diffuse Surface Optical Model for Nucleon-Nuclei Scattering". Physical Review. 95 (2): 577–578. Bibcode:1954PhRv...95..577W. doi:10.1103/PhysRev.95.577.
• Flügge, Siegfried (1999). Practical Quantum Mechanics. Springer Berlin Heidelberg. pp. 162ff. ISBN 978-3-642-61995-3.

External links

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