Electromagnetic and weak transitions tell us a great deal about the structure of atomic nuclei. Yet modeling transitions can be difficult: it is often easier to compute the ground state, if only as an approximation, than excited states. One alternative is through transition sum rules, in particular the non-energy-weighted and energy-weighted sum rules, which can be computed as expectation values of operators. We investigate by computing sum rules for a variety of nuclei, comparing the numerically exact full configuration-interaction shell model, as a reference, to Hartree–Fock, projected Hartree–Fock, and the nucleon pair approximation. These approximations yield reasonable agreement, which we explain by prior work on the systematics of transition moments.
For gamma-ray bursts (GRBs) with durations greater than two seconds (so-called long GRBs), the intrinsic prompt gamma-ray emission appears, on average, to last longer for bursts at lower redshifts. We explore the nature of this duration–redshift anticorrelation, describing systems and conditions in which this cosmological evolution could arise. In particular, we explore its dependence on the metallicity of a massive star progenitor, because we can securely count on the average stellar metallicity to increase with decreasing redshift. Although stars with higher metallicity/lower redshift lose mass and angular momentum through line-driven winds, in some cases these stars are able to form more extended accretion disks when they collapse, potentially leading to longer-duration GRBs. We also examine how this duration–redshift trend may show up in interacting binary models composed of a massive star and compact object companion, recently suggested to be the progenitors of radio-bright GRBs. Under certain conditions, mass loss and equation-of-state effects from massive stars with higher metallicity and lower redshift can decrease the binary separation. This can then lead to spin-up of the massive star and allow for a longer-duration GRB upon the massive star’s collapse. Finally, the duration–redshift trend may also be supported by a relatively larger population of small-separation binaries born in situ at low redshift.
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