Transition sum rules in the shell model

TL;DR

This paper derives general formulas for sum rules in the nuclear shell model, enabling efficient evaluation of transition strength centroids without calculating all daughter states, and applies these to various nuclides.

Contribution

It introduces new general formulas for matrix elements of sum rules in the shell model, simplifying the calculation of transition centroids.

Findings

Sum rules follow smooth secular behavior with initial energy.

Electric dipole sum rule aligns with Thomas-Reiche-Kuhn.

Systematic behaviors observed for ground state E2 centroids in sd-shell.

Abstract

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the double-commutator have been to special cases, we derive general formulas for matrix elements of both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy, as well as compare the electric dipole (E1) sum rule against the famous Thomas-Reiche-Kuhn version. We also find surprising systematic behaviors for ground state electric quadrupole (E2) centroids in the $sd$-shell.