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Hyperfine structure in $^{229g}$Th$^{3+}$ as a probe of the $^{229g}$Th $\rightarrow$ $^{229m}$Th nuclear excitation energy

We identify a potential means to extract the $^{229g}$Th $\rightarrow$ $^{229m}$Th nuclear excitation energy from precision microwave spectroscopy of the $5F_{5/2,7/2}$ hyperfine manifolds in the ion $^{229g}$Th$^{3+}$. The hyperfine interaction mixes this ground fine structure doublet with states of the nuclear isomer, introducing small but observable shifts to the hyperfine sub-levels. We demonstrate how accurate atomic structure calculations may be combined with measurement of the hyperfine intervals to quantify the effects of this mixing. Further knowledge of the magnetic dipole decay rate of the isomer, as recently reported, allows an indirect determination of the nuclear excitation energy.