Sensitivity of the $^{229m}$Th clock transition to the fine-structure constant in a Skyrme-Hartree-Fock-BCS approach

Abstract

The sensitivity of the $^{229m}$Th isomer transition frequency to the possible temporal variation of the fine-structure constant $α$ is investigated theoretically. We evaluate both the Coulomb as well as the isomer energies in a selfconsistent Hartree-Fock plus Bardeen-Cooper-Schrieffer (BCS) approach with Skyrme energy density functional, taking a detailed account of the nuclear shape deformation and pairing correlations. Our results show that by a fine tuning of the pairing strengths and considering axial octupole deformation, the model can predict a very low isomer energy below the keV limit, which is not accessible in the presence of imposed reflection symmetry. Furthermore, in the model solution with octupole deformation, the difference between the Coulomb energy in the isomeric and ground states is one order of magnitude larger than for the case with imposed reflection symmetry. These results allow for a fully theoretical prediction of the sensitivity of the isomer transition frequency to $α$ in the framework of a microscopic nuclear model, yielding values of $K\sim 10^3-10^4$, and for a deeper understanding of its underlying physical conditions.