Liquid drop model
From Wikipedia, the free encyclopedia
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.
Contents |
[edit] The Bethe-Weizsäcker formula
- Main article: Semi-empirical mass formula
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 (also called Pauli 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 to obtain the following rule (the Weizsäcker formula):
where N is the number of neutrons, Z is the number of protons, and A = N + Z is the atomic mass. EB is the binding energy in MeV. The final term's sign depends on whether the number of protons and neutrons are both even, or both odd. It is defined to be zero for the case where one value is even and one is odd.
Coefficient values depend on the domain over which the curve fit is being performed. Some experimentally determined values are:
| Wapstra | Rohlf | |
|---|---|---|
| av | 14.1 | 15.75 |
| as | 13 | 17.8 |
| ac | 0.595 | 0.711 |
| aa | 19 | 23.7 |
| δ (even-even) | -33.5 | +11.18 |
| δ (odd-odd) | +33.5 | -11.18 |
| δ (even-odd) | 0 | 0 |
- Wapstra: Atomic Masses of Nuclides, A. H. Wapstra, Springer, 1958
- Rohlf: Modern Physics from a to Z0, James William Rohlf, Wiley, 1994
[edit] References
- RADIOCHEMISTRY and NUCLEAR CHEMISTRY, Gregory Choppin, Jan-Olov Liljenzin, and Jan Rydberg, 3rd Edition, 2002, the chapter on nuclear stability (PDF)
[edit] External links
- Liquid drop model in the hyperphysics online reference at Georgia State University.
- Liquid drop model with parameter fit from First Observations of Excited States in the Neutron Deficient Nuclei 160,161W and 159Ta, Alex Keenan, PhD thesis, University of Liverpool, 1999 (HTML version).
