Nuclear potential wells

In the last chapter, we set out a semi-empirical theory for the binding energy of an atomic nucleus, and quantum-mechanical considerations came in only rather indirectly. Experimental atomic masses show deviations from the smooth curve given by the semi-empirical mass formula, deviations which we said were of quantum-mechanical origin. Since a nucleus in its ground state is a quantum system of finite size, it has angular momentum J, with quantum number j which is some integral multiple of ½. If j ≠ 0 the nucleus will have a magnetic dipole moment, and it may have an electric quadrupole moment as well.

The nuclear angular momentum and magnetic moment manifest themselves most immediately in atomic spectroscopy, where the interaction between the nuclear magnetic moment and the electron magnetic moments gives rise to the hyperfine structures of the electronic energy levels. In favourable cases both j and the magnetic moment may be deduced from this hyperfine splitting.

The observed values of nuclear angular momenta give strong support to the validity of a simple quantum-mechanical model of the nucleus: the nuclear shell model. In this model, each neutron moves independently in a common potential well that is the spherical average of the nuclear potential produced by all the other nucleons, and each proton moves independently in a common potential well that is the spherical average of the nuclear potential of all the other nucleons, together with the Coulomb potential of the other protons.