Nuclear Excitation by Near-Resonant Electron Transition in $^{229}$Th$^{39+}$ Ions

Theoretical considerations are made for the nuclear excitation from the ground state to the 8 eV $^{229m}$Th isomer via near-resonant electron transitions in Sb-like ($q=39+$) thorium ions. The energy of the first excited atomic state ($J=7/2$) in the $^{229}$Th$^{39+}$ ion is estimated to be 8.308$\pm$0.011 eV, which is very close to the new reference value for the $^{229m}$Th nuclear isomer energy, 8.338$\pm$0.024 eV [Kraemer et al., Nature 617, 706 (2023)]. It was found that within the uncertainty range of both atomic and nuclear excitation energies, the rate of nuclear excitation by electron transition can vary by more than 20 orders of magnitude. Our results indicate that the upper theoretical limit for the $^{229m}$Th isomer excitation rate reaches an enormous value of $1.05\times10^{16}$ s$^{-1}$ at resonance ($\Delta=0$ meV). Additionally, it was shown that using an electron beam ion trap (EBIT), the production of the $^{229}$Th isomer can reach rates ranging from tenths to approximately $6\times10^{19}$ s$^{-1}$. Thus, EBIT experimental studies could serve as an extremely sensitive tool for the independent verification of the excitation energy of the isomeric state in the $^{229}$Th nucleus.