Woods–Saxon potential
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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 (nucleon number). Until the 1960s, nucleons were ...
s ( protons and
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
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 based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
, 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 (nucleon number). Until the 1960s, nucleons were ...
, in the
nuclear shell model In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The first shell m ...
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'' tomic 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 approxima ...
. Typical values for the parameters are: , . 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 The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
. 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.


See also

* Finite potential well * Quantum harmonic oscillator *
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 a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
*
Yukawa potential In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: :V_\text(r)= -g^2\frac, where is a ...
*
Nuclear force The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...
* Nuclear structure *
Shell model The SHELL model is a conceptual model of human factors that clarifies the scope of aviation human factors and assists in understanding the human factor relationships between aviation system resources/environment (the flying subsystem) and the huma ...


References

* * *


External links


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