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The Woods–Saxon potential is a mean field
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
for the
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
s (
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s and
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s) inside the
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
, which is used to describe approximately the forces applied on each
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
, in the
nuclear shell model In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model utilizes the Pauli exclusion principle to model the structure of atomic nuclei in terms of energy levels. The first shell model was proposed by Dmitri Ivanenk ...
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 The mass number (symbol ''A'', from the German word: ''Atomgewicht'', "atomic weight"), also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. It is appro ...
. 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 A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is cap ...
. 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 In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interactions. It confines quarks into protons, neutrons, and other hadron particles, an ...
. 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 In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
. 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 The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
*
Particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes the movement of a free particle in a small space surrounded by impenetrable barriers. The model is mainly used a ...
*
Yukawa potential Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns. People * Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1 * Hideki Yukawa (1907–1 ...
*
Nuclear force The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between hadrons, most commonly observed between protons and neutrons of atoms. Neutrons and protons, both ...
*
Nuclear structure Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. Models The cluster model The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups (e.g., alpha particl ...
* Shell model


References

* *


External links


http://nucracker.volya.net/ Woods–Saxon Solver
Nuclear physics {{nuclear-stub Quantum mechanical potentials Exactly solvable models