Studies in jj -coupling. I. Classification of nuclear and atomic states

Stimulated by the many successes of the recent jj-coupling shell-model of nuclear structure which is based upon the assumption of strong spin-orbit coupling, the present work aims at obtaining allowed terms and energies of systems of many equivalent particles in jj- coupling for various types of two-body interaction. In this first paper the methods of group theory are employed to enumerate and classify the states of the nuclear configuration (lj)n. Classification of states according to the irreducible representations of groups of linear transformations leads naturally to the introduction of certain quantum numbers which then characterize the states. Invariance under the group of unitary transformations in the space of the 2j + 1 individual-particle functions introduces the isotopic spin, while invariance under the subgroup of symplectic transformations introduces the concepts of seniority and reduced isotopic spin. The complete classification and enumeration of the states of the nuclear configurations jn, carried through for j = 3/2, 5/2 and 7/2, is presented in table 6; the states of the corresponding atomic configurations are included, these being states of maximum isotopic spin. As a result of the classification, certain degeneracies observed in the energy matrix of the electrostatic interaction in jj-coupling are shown to be intimately connected with the properties of the symplectic groups. Finally, the spins of nuclear ground states are briefly discussed in the light of the jj-coupling shell-model.