Energy of the $^{229}$Th Nuclear Clock Isomer Determined by Absolute $γ$-ray Energy Difference

Abstract: The low-lying isomeric state of $^{229}$Th provides unique opportunities for high-resolution laser spectroscopy of the atomic nucleus. We determine the energy of this isomeric state by taking the absolute energy difference between the excitation energy required to populate the 29.2-keV state from the ground-state and the energy emitted in its decay to the isomeric excited state. A transition-edge sensor microcalorimeter was used to measure the absolute energy of the 29.2-keV $\gamma$-ray. Together with the cross-band transition energy (29.2 keV$\to$ground) and the branching ratio of the 29.2-keV state measured in a recent study, the isomer energy was determined to be 8.30$\pm$0.92 eV. Our result is in agreement with latest measurements based on different experimental techniques, which further confirms that the isomeric state of $^{229}$Th is in the laser-accessible vacuum ultraviolet range.