Splitting sensitivity of the ground and 7.6 eV isomeric states of ${}^{229}\mathrm{Th}$

The lowest-known excited state in nuclei is the 7.6 eV isomer of ${}^{229}\mathrm{Th}$. This energy is within the range of laser-based investigations that could allow accurate measurements of possible temporal variation of this energy splitting. This in turn could probe temporal variation of the fine-structure constant or other parameters in the nuclear Hamiltonian. We investigate the sensitivity of this transition energy to these quantities. We find that the two states are predicted to have identical deformations and thus the same Coulomb energies within the accuracy of the model (viz., within roughly 30 keV). We therefore find no enhanced sensitivity to variation of the fine-structure constant. In the case of the strong interaction the energy splitting is found to have a complicated dependence on several parameters of the interaction, which makes an accurate prediction of sensitivity to temporal changes of fundamental constants problematical. Neither the strong- nor Coulomb-interaction contributions to the energy splitting of this doublet can be constrained within an accuracy better than a few tens of keV, so that only upper limits can be set on the possible sensitivity to temporal variations of the fundamental constants.

Figure 1

Calculated quasiparticle energies for the three lowest quasiparticle states (relative to the $[633,5/2^{+}]$ ground state, which therefore has zero energy for all deformations) in ${}^{229}\mathrm{Th}$ as functions of ${\epsilon }_2$. All three states remain almost degenerate at all values of ${\epsilon }_2$ near the common minimum at ${\epsilon }_2=0.170.$

Figure 2

Calculated quasiparticle energies of the ${3/2}^{+}$-${5/2}^{+}$ doublet and of a $7/2^{-}$ state and a $13/2^{+}$ state in ${}^{229}\mathrm{Th}$ as functions of ${\epsilon }_2$. The doublet states remain almost degenerate for a wide range of values of ${\epsilon }_2$. Note the change in energy scale from Fig. 1.