Several models of the nucleus have evolved that fit certain aspects of nuclear behavior, but no single model has successfully described all aspects. One model is based on the fact that certain properties of a nucleus are similar to those of a drop of incompressible liquid. The liquid-drop model has been particularly successful in explaining details of the fission process and in evolving a formula for the mass of a particular nucleus as a function of its atomic number and mass number, the so-called semiempirical mass formula.
Another model is the Fermi gas model, which treats the nucleons as if they were particles of a gas restricted by the Pauli exclusion principle, which allows only two particles of opposite spin to occupy a particular energy level described by the quantum theory. These particle pairs will fill the lowest energy levels first, then successively higher ones, so that the "gas" is one of minimum energy. There are actually two independent Fermi gases, one of protons and one of neutrons. The tendency of nucleons to occupy the lowest possible energy level explains why there is a tendency for the numbers of protons and neutrons to be nearly equal in lighter nuclei. In heavier nuclei the effect of electrostatic repulsion among the larger number of charges from the protons raises the energy of the protons, with the result that there are more neutrons than protons (for uranium-235, for example, there are 143 neutrons and only 92 protons). The pairing of nucleons in energy levels also helps to explain the tendency of nuclei to have even numbers of both protons and neutrons.
Neither the liquid-drop model nor the Fermi gas model, however, can explain the exceptional stability of nuclei having certain values for either the number of protons or the number of neutrons, or both. These so-called magic numbers are 2, 8, 20, 28, 50, 82, and 126. Because of the similarity between this phenomenon and the stability of the noble gases, which have certain numbers of electrons that are bound in closed "shells," a shell model was suggested for the nucleus. There are major differences, however, between the electrons in an atom and the nucleons in a nucleus. First, the nucleus provides a force center for the electrons of an atom, while the nucleus itself has no single force center. Second, there are two different types of nucleons. Third, the assumption of independent particle motion made in the case of electrons is not as easily made for nucleons. The liquid-drop model is in fact based on the assumption of strong forces between the nucleons that considerably constrain their motion. However, these difficulties were solved and a good explanation of the magic numbers achieved on the basis of the shell model, which included the assumption of strong coupling between the spin angular momentum of a nucleon and its orbital angular momentum. Various attempts have been made, with partial success, to construct a model incorporating the best features of both the liquid-drop model and the shell model.