## Theoretical Models

A goal of nuclear physics is to account for the properties of nuclei in terms of mathematical models of their structure and internal motion. Three important nuclear models are the Liquid Drop Model, the Shell Model (developed by Maria Goeppert-Mayer and Hans Jensen), which emphasizes the orbits of individual nucleons in the nucleus, and the Collective Model (developed by Aage Bohr and Ben Mottleson), which complements the shell model by including motions of the whole nucleus such as rotations and vibrations.

The Liquid Drop Model treats the nucleus as a liquid. Nuclear properties, such as the binding energy, are described in terms of volume energy, surface energy, compressibility, etc.–parameters that are usually associated with a liquid. This model has been successful in describing how a nucleus can deform and undergo fission.

The Nuclear Shell Model is similar to the atomic model where electrons arrange themselves into shells around the nucleus. The least-tightly-bound electrons (in the incomplete shells) are known as valence electrons because they can participate in exchange or rearrangement, that is, chemical reactions. The shell structure is due to the quantum nature of electrons and the fact that electrons are fermions–particles of half-integer spin. Particles with integer spin are bosons. A group of bosons all tend to occupy the same state (usually the state with the lowest energy), whereas fermions with the same quantum numbers do just the opposite: they avoid each other. Consequently the fermions in a bound system will gradually fill up the available states: the lowest one first, then the next higher unoccupied state, and so on up to the valence shell. In atoms, for example, the electrons obey the Pauli Exclusion Principle, which is responsible for the observed number of electrons in each possible state (at most 2) characterized by quantum numbers n, l, and m. It is the Pauli Principle (based on the fermionic nature of electrons) that gives the periodic structure to both atomic and nuclear properties.

Since protons and neutrons are also fermions, the energy states the nucleons occupy are filled from the lowest to the highest as nucleons are added to the nucleus. In the shell model the nucleons fill each energy state with nucleons in orbitals with definite angular momentum. There are separate energy levels for protons and neutrons. The ground state of a nucleus has each of its protons and neutrons in the lowest possible energy level. Excited states of the nucleus are then described as promotions of nucleons to higher energy levels. This model has been very successful in explaining the basic nuclear properties. As is the case with atoms, many nuclear properties (angular momentum, magnetic moment, shape, etc.) are dominated by the last filled or unfilled valence level.

The Collective Model emphasizes the coherent behavior of all of the nucleons. Among the kinds of collective motion that can occur in nuclei are rotations or vibrations that involve the entire nucleus. In this respect, the nuclear properties can be analyzed using the same description that is used to analyze the properties of a charged drop of liquid suspended in space. The Collective Model can thus be viewed as an extension of the Liquid Drop Model; like the Liquid Drop Model, the Collective Model provides a good starting point for understanding fission.

In addition to fission, the Collective Model has been very successful in describing a variety of nuclear properties, especially energy levels in nuclei with an even number of protons and neutrons. These even nuclei can often be treated as having no valence particles so that the Shell Model does not apply. These energy levels show the characteristics of rotating or vibrating systems expected from the laws of quantum mechanics. Commonly measured properties of these nuclei, including broad systematics of excited state energies, angular momentum, magnetic moments, and nuclear shapes, can be understood using the Collective Model.

The Shell Model and the Collective Model represent the two extremes of the behavior of nucleons in the nucleus. More realistic models, known as unified models, attempt to include both shell and collective behaviors.

last updated: August 9, 2000 webmaster