The existence of closed shells in nuclei is indicated by the particular stability and abundance of nuclear systems with certain numbers of neutrons and of protons. An interesting correspondence exists between these numbers and the degeneracy of energy levels in the model of free particles in a simple rectangular potential well. The empirical relations suggest the addition of a central depression to the well for light nuclei ($N$ or $Z\leqq 20$) and a central elevation (Elsasser's wine bottle potential) for heavy nuclei. The qualitative physical explanation of these modifications is that for light nuclei, the particle density, and thus the nuclear interaction energy, is greatest at the center of the nucleus; for heavy nuclei, the Coulomb repulsion between protons produces a particle density varying from a minimum at the center to a maximum near the boundary, and therefore a similarly varying interaction energy. In the free particle model, levels with small angular momentum and nodes within the nucleus are displaced upward by the central elevation and crossing of levels in the desired direction occurs, though not exactly as required to explain the complete empirical list of closed shell numbers. It is apparent that a single particle model of the nucleus is an insufficient approximation; however qualitative arguments based on configuration interaction enable one to formulate, with little ambiguity, a shell model in agreement with the empirical facts.

Experimentally known spins, magnetic moments and quadrupole moments of nuclei with a small number of particles outside of closed shells, or missing from closed shells, tend to corroborate the model. The agreement is particularly good for odd nuclei with one kind of nucleon forming a closed shell±one particle (and still better if the even group of nucleons constitutes a closed shell). Under these conditions, the orbit of the odd particle, as given by the shell model, usually determines the state of the nucleus, which is checked for consistency with the known spins and moments. In some cases, the reasoning is reversed as a means of fixing the crossover points between energy levels in the theory.

Islands of isomerism appear in a table of odd nuclei arranged according to the odd member of the pair $N$, $Z$. Since isomerism occurs when large spin differences exist between the ground state and the excited state, the correlation of isomerism with the crossing or close spacing of energy levels in the shell model is readily seen. Possible paired configurations representing ground state and isomeric state are assigned to various isomeric transitions. Spins and moments (where known), and parity relationships given by Wiedenbeck's plot of half-life vs. energy, help to decide the preferred configurations and to clarify the details of the shell model.

Beta-decay transitions are classified empirically into allowed and forbidden categories on the basis of calculated $\mathrm{ft}$ values. The configurations of the parent and daughter nuclei, as given by the shell model, determine the parity and range of possible spin values for each. Using the Gamow-Teller selection rules, the consistency of the empirical allowed or forbidden classification with the possibilities of the shell model are investigated. In many cases the ambiguity of spin in the parent or daughter nucleus is lessened. The empirical transition classification breaks down in several cases, chiefly those in which transitions that are theoretically first forbidden ($\Delta I=\pm 2$, change in parity) have $\mathrm{ft}$ values that place them in the empirical second forbidden category, indicating a very small nuclear matrix element. An extensive table of allowed (unfavored) transitions shows no trend of $\mathrm{ft}$ with atomic mass over the range $19\leqq A\leqq 140$.

• Received 27 December 1948

DOI:https://doi.org/10.1103/PhysRev.75.1877