Transition sum rules in the shell model

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.

3 figures (updated); paper updated to reflect accepted version

References in corpus (10)

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Cited by in corpus (8)

• Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics
• A modified Brink-Axel hypothesis for astrophysical Gamow-Teller transitions
• Magnetic dipole excitation and its sum rule in nuclei with two valence nucleons
• Exact sum rules with approximate ground states
• Validity of Brink Axel Hypothesis for calculations of allowed stellar weak rates of heavy nuclei
• Electric dipole response of sd-shell nuclei within the Configuration-Interaction Shell Model approach
• Impact of the Brink Axel Hypothesis on Unique First Forbidden \b{eta} transitions for r process nuclei
• Systematic analysis of double Gamow-Teller sum rules