# Liquid drop model

The liquid drop model is a model in nuclear physics which treats the nucleus as a drop of incompressible nuclear fluid, first proposed by George Gamow. The fluid is made of nucleons, and is held together by the strong nuclear force.

This is a crude model that does not explain all the properties of nuclei, but does explain the spherical shape of most nuclei.

It also helps to predict results in the field of nuclear fission by calculating the variation of binding energy necessary to change the shape of the drop, and then comparing it to the energy given by a neutron joining this nucleon. If it is sufficient, the drop "breaks" and this is called fission.

Mathematical analysis of the theory delivers an equation which attempts to predict the binding energy of a nucleus in terms of the numbers of protons and neutrons it contains. This equation has five terms on its right hand side. These correspond to the cohesive binding of all the nucleons by the strong nuclear force, the electrostatic mutual repulsion of the protons, a surface energy term, an asymmetry term (derivable from the protons and neutrons occupying independent quantum momentum states) and an exchange term.

If we consider the sum of the following five types of energies, then the picture of a nucleus as a drop of incompressible liquid roughly accounts for the observed variation of binding energy of the nucleus :

Volume Energy. When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has 12 other nucleons in contact with it. So, this nuclear energy is proportional to the volume.

Surface Energy. A nucleon at the surface of a nucleus interacts with fewer other nucleons that one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.

Coulomb Energy. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy.

Asymmetry Energy. An energy associated needed as a correction when the number of neutrons is greater than the number of protons.

Pairing Energy. An energy which is a correction term that arises from the tendency of proton pairs and neutron pairs to occur.

These terms are added, and coefficients are derived experimentally to obtain the following rule:

[itex]

E_B(MeV) = 15.5 A - 23 (N-Z)^2/A - 0.72 Z^2/A^{1/3} - 16.8 A^{2/3} \pm 34A^{3/4} [itex] where [itex]N[itex] is the number of neutrons, [itex]Z[itex] is the number of protons, and [itex]A=N+Z[itex] is the atomic mass. [itex]E_B[itex] is the binding energy in MeV. The final term is added if the number of protons and neutrons are both even, subtracted if the number of protons and neutrons are both odd, and omitted of only one is odd.

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