Systematics of strength function sum rules

Abstract

Sum rules provide useful insights into transition strength functions and are often expressed as expectation values of an operator. In this letter I demonstrate that non-energy-weighted transition sum rules have strong secular dependences on the energy of the initial state. Such non-trivial systematics have consequences: the simplification suggested by the generalized Brink-Axel hypothesis, for example, does not hold for most cases, though it weakly holds in at least some cases for electric dipole transitions. Furthermore, I show the systematics can be understood through spectral distribution theory, calculated via traces of operators and of products of operators. Seen through this lens, violation of the generalized Brink-Axel hypothesis is unsurprising: one \textit{expects} sum rules to evolve with excitation energy. Furthermore, to lowest order the slope of the secular evolution can be traced to a component of the Hamiltonian being positive (repulsive) or negative (attractive).

5 pages, 4 figures; minor revisions; references updated; title revised; matches accepted version

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• Nuclear level densities and $\gamma$-ray strength functions in ${}^{120,124}$Sn isotopes: impact of Porter-Thomas fluctuations
• Transition sum rules in the shell model
• A new software implementation of the Oslo method with rigorous statistical uncertainty propagation
• Magnetic dipole excitation and its sum rule in nuclei with two valence nucleons
• A modified Brink-Axel hypothesis for astrophysical Gamow-Teller transitions
• New experimental constraint on the ${}^{185}$W($n,\gamma$)${}^{186}$W cross section
• QRPA calculations for M1 transitions with the noniterative finite amplitude method and the application to neutron radiative capture cross sections
• Exact sum rules with approximate ground states
• Gamow-Teller strength and lepton captures rates on 66-71Ni in stellar matter
• Validity of Brink Axel Hypothesis for calculations of allowed stellar weak rates of heavy nuclei
• Impact of the Brink Axel Hypothesis on Unique First Forbidden \b{eta} transitions for r process nuclei
• Lepton capture rates due to isotopes of vanadium in astrophysical environment
• Possible existence of pygmy dipole resonance built on excited states in a neutron-rich ${}^{80}$Ge nucleus
• Neutrino cooling rates due to nickel isotopes for presupernova evolution of massive stars
• Angela and the electric dipole response -- giant and pygmy, hot and cold, isoscalar and isovector
• Comprehensive test of the Brink-Axel hypothesis in the energy region of the pygmy dipole resonance
• Systematic analysis of double Gamow-Teller sum rules