Partial conservation law in a schematic single j shell model

Abstract

We report the discovery of a partial conservation law obeyed by a schematic Hamiltonian of two protons and two neutrons in a [Formula: see text] shell. In our Hamiltonian, the interaction matrix element of two nucleons with combined angular momentum [Formula: see text] is linear in [Formula: see text] for even [Formula: see text] and constant for odd [Formula: see text]. It turns out that in some stationary states, the sum of the angular momenta [Formula: see text] and [Formula: see text] of the proton and neutron pairs is conserved. The energies of these states are given by a linear function of [Formula: see text]. The systematics of their occurrence is described and explained.

Schematic interactions with many degeneracies

“…In previously published work [5,6] with the 0123, 01234, and 012345 interactions for the f 7/2 , g 9/2 and h 11/2 shells respectively, we find critical angular momentum beyond which we get equally spaced spectra. The values are respectively 6, 8, and 10 with an obvious generalization to higher shells.…”

“…[6]. We also note work by K. Neergärd ref [7], which was important in getting analytical results in ref [5]. We call this the 0123 interaction.…”

“…That is to say The 12-10 splitting is the same as the 10-8 splitting is the same as the 8-6 splitting. This was discussed extensively in Ref [5].…”

“…This new state would not be orthogonal to a T=2 state and hence would be a mixture of T=0 and T=2. Some more details can be found in Rule 2 of the previous publication [5].…”

“…We then considered [5] a schematic interaction E(J)=0 for odd J (T=0) and E(J)= J for even J (T=1). For convenience, we here change from J to J/2 as was done in ref.…”