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Woods–Saxon potential

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The Woods–Saxon potential is a <a href="Mean_field" class="mw-redirect" title="Mean field">mean field</a> <a href="Potential" title="Potential">potential</a> for the <a href="Nucleon" title="Nucleon">nucleons</a> (<a href="Proton" title="Proton">protons</a> and <a href="Neutron" title="Neutron">neutrons</a>) inside the <a href="Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a>, which is used to describe approximately the forces applied on each <a href="Nucleon" title="Nucleon">nucleon</a>, in the <a href="Nuclear_shell_model" title="Nuclear shell model">nuclear shell model</a> for the structure of the nucleus. The potential is named after Roger D. Woods and <a href="David_S._Saxon" title="David S. Saxon">David S. Saxon</a>.

The form of the potential, in terms of the distance r from the center of nucleus, is:

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<annotation encoding="application/x-tex">{\displaystyle V(r)=-{\frac {V_{0}}{1+\exp({r-R \over a})}}}</annotation>

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</math><img src="./f55f561232e9a01456c55db644122842b07628fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.856ex; height:6.843ex;" alt="{\displaystyle V(r)=-{\frac {V_{0}}{1+\exp({r-R \over a})}}}" loading="lazy">

where V0 (having dimension of energy) represents the potential well depth,

a is a length representing the "surface thickness" of the nucleus, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=r_{0}A^{1/3}}">

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<annotation encoding="application/x-tex">{\displaystyle R=r_{0}A^{1/3}}</annotation>

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</math><img src="./3411ac30b088fae7f8699599aad6e5d43889c5c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.407ex; height:3.176ex;" alt="{\displaystyle R=r_{0}A^{1/3}}" loading="lazy"> is the <a href="Nuclear_size" class="mw-redirect" title="Nuclear size">nuclear radius</a> where r0 = 1.25 <a href="Femtometre" title="Femtometre">fm</a> and A is the <a href="Mass_number" title="Mass number">mass number</a>.

Typical values for the parameters are: V0 ≈ 50 <a href="Electronvolt" title="Electronvolt">MeV</a>, a ≈ 0.5 fm.

There are numerous optimized parameter sets available for different atomic nuclei.<a href="#cite_note-1">[1]</a> <a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

For large atomic number A this potential is similar to a <a href="Potential_well" title="Potential well">potential well</a>. It has the following desired properties

<ul><li>It is monotonically increasing with distance, i.e. attracting.</li>

<li>For large A, it is approximately flat in the center.</li>

<li>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.</li>

<li>It rapidly approaches zero as r goes to infinity (r − R >> a), reflecting the short-distance nature of the <a href="Strong_nuclear_force" class="mw-redirect" title="Strong nuclear force">strong nuclear force</a>.</li></ul>

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

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<annotation encoding="application/x-tex">{\displaystyle u(r)={\frac {1}{r}}y^{\nu }(1-y)^{\mu }{}_{2}F_{1}(\mu +\nu ,\mu +\nu +1;2\nu +1;y)}</annotation>

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</math><img src="./81aec12466dda0cb0fc2835086073bbc80386386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.758ex; height:5.176ex;" alt="{\displaystyle u(r)={\frac {1}{r}}y^{\nu }(1-y)^{\mu }{}_{2}F_{1}(\mu +\nu ,\mu +\nu +1;2\nu +1;y)}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\dfrac {1}{1+\exp \left({\frac {r-R}{a}}\right)}}}">

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<annotation encoding="application/x-tex">{\displaystyle y={\dfrac {1}{1+\exp \left({\frac {r-R}{a}}\right)}}}</annotation>

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</math><img src="./ac0561b233f38e96830a78ce558eb9fdfa153783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:19.524ex; height:7.843ex;" alt="{\displaystyle y={\dfrac {1}{1+\exp \left({\frac {r-R}{a}}\right)}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =i{\sqrt {\gamma ^{2}-\nu ^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu =i{\sqrt {\gamma ^{2}-\nu ^{2}}}}</annotation>

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</math><img src="./d37a5be5ccb1e49abeb59a7e2397a5b8cc5f72e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.112ex; height:4.843ex;" alt="{\displaystyle \mu =i{\sqrt {\gamma ^{2}-\nu ^{2}}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {2mE}{\hbar ^{2}}}=-\nu ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {2mE}{\hbar ^{2}}}=-\nu ^{2}}</annotation>

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</math><img src="./7364a27fdb7ff8ba998c6bd57bde89040a4f1f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.033ex; height:5.509ex;" alt="{\displaystyle {\dfrac {2mE}{\hbar ^{2}}}=-\nu ^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu <0}">

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<annotation encoding="application/x-tex">{\displaystyle \nu <0}</annotation>

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</math><img src="./1671306e07fbec887a9879be574366723e8d2a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu <0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma ^{2}}</annotation>

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</math><img src="./95566b05aef3528781172d3daf617669d01a12af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.165ex; height:5.676ex;" alt="{\displaystyle {\dfrac {2mV_{0}}{\hbar ^{2}}}a^{2}=\gamma ^{2}}" loading="lazy">.<a href="#cite_note-4">[4]</a> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}}</annotation>

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</math><img src="./40b9e5a2fe04ccbc4c3511f1bce3351bae765dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.941ex; height:6.843ex;" alt="{\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}}" loading="lazy"> is the <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>.

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.<a href="#cite_note-5">[5]</a>

It is also possible to give analytic formulas of the Fourier transformation<a href="#cite_note-6">[6]</a> of the Woods-Saxon potential which makes it possible to work in the <a href="Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> as well.

<ul><li><a href="Finite_potential_well" title="Finite potential well">Finite potential well</a></li>

<li><a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></li>

<li><a href="Particle_in_a_box" title="Particle in a box">Particle in a box</a></li>

<li><a href="Yukawa_potential" title="Yukawa potential">Yukawa potential</a></li>

<li><a href="Nuclear_force" title="Nuclear force">Nuclear force</a></li>

<li><a href="Nuclear_structure" title="Nuclear structure">Nuclear structure</a></li>

<li><a href="Nuclear_shell_model" title="Nuclear shell model">Shell model</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFDudekSzymanskiWerner1980" class="citation journal cs1">Dudek, J.; Szymanski, Z.; Werner, T. (1980). <a rel="nofollow" class="external text" href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.23.920">"Woods-Saxon potential parameters optimized to the high spin spectra in the lead region"</a>. Phys. Rev. C. 23: 940. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevC.23.920">10.1103/PhysRevC.23.920</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSchwierzWiedenhoverVolya" class="citation web cs1">Schwierz, N.; Wiedenhover, I.; Volya, A. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/0709.3525">"Parameterization of the Woods-Saxon Potential for Shell-Model Calculations"</a>. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0709.3525">0709.3525</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFGanLiSunHu2021" class="citation journal cs1">Gan, L.; Li, Z.-H.; Sun, H.-B.; Hu, S.-P.; Li, E.-T.; Zhong, J. (2021). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1674-1137/abe84f/pdf">"Systematic study of the Woods-Saxon potential parameters between heavy-ions"</a>. Chinese Physics. 45 (5): 054105 – via 10.1088/1674-1137/abe84f.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFFlügge1999" class="citation book cs1">Flügge, Siegfried (1999). Practical Quantum Mechanics. Springer Berlin Heidelberg. pp. 162ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-61995-3</bdi>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFErkolDemiralp2007" class="citation journal cs1">Erkol, H.; Demiralp, E. (2007). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/S0375960106020007">"The Woods–Saxon potential with point interactions"</a>. Physics Letters A. 365 (1–2): 55–63. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physleta.2006.12.050">10.1016/j.physleta.2006.12.050</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFHlopeElsterJohnsonUpadhyay2013" class="citation journal cs1">Hlope, L.; Elster, Ch.; Johnson, R.C.; Upadhyay, N.J.; <a href="Filomena_Nunes" title="Filomena Nunes">Nunes, F. M.</a>; Arbanas, G.; Eremenko, V.; et, all (2013). <a rel="nofollow" class="external text" href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.88.064608">"Separable representation of phenomenological optical potentials of Woods-Saxon type"</a>. Phys. Rev. C. 88: 064608. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1310.8334">1310.8334</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevC.88.064608">10.1103/PhysRevC.88.064608</a>.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFWoodsSaxon1954" class="citation journal cs1">Woods, R. D.; Saxon, D. S. (1954). "Diffuse Surface Optical Model for Nucleon-Nuclei Scattering". <a href="Physical_Review" title="Physical Review">Physical Review</a>. 95 (2): 577–578. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954PhRv...95..577W">1954PhRv...95..577W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.95.577">10.1103/PhysRev.95.577</a>.</cite></li>

<li><cite id="CITEREFFlügge1999" class="citation book cs1">Flügge, Siegfried (1999). Practical Quantum Mechanics. Springer Berlin Heidelberg. pp. 162ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-61995-3</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://nucracker.volya.net/">http://nucracker.volya.net/ Woods–Saxon Solver</a></li></ul>

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