Relativistic coupled-cluster calculation of hyperfine-structure constants of $^{229}$Th$^{3+}$ and evaluation of the electromagnetic nuclear moments of $^{229}$Th

Abstract

229 Th is a promising candidate for developing a nuclear optical clock and searching the new physics beyond the standard model. Accurate knowledge of the nuclear properties of 229 Th is very important. In this work, we calculate hyperfine-structure constants for the first four states of 229 Th 3+ using the relativistic coupled-cluster method based on the Gauss basis set. The no-pair Dirac-Coulomb-Breit Hamiltonian with the lowest-order quantum electrodynamics (QED) correction is the starting point, together with all linear and non-linear terms of single and double excitations are included in coupled-cluster calculation. With the measured value of the hyperfine-structure constants [Phys. Rev. Lett. 106. 223001(2011)], we get the magnetic dipole moment, µ = 0.359(9), and the electric quadrupole moment, Q = 2.95(7), of the 229 Th nucleus. Our magnetic dipole moment is perfectly consistent with the recommended values, µ = 0.360( 7), from the all-order calculation by Safronova et. al.[Phys.Rev.A 88, 060501 (2013)], but our electric quadrupole moment is smaller than their recommended value, Q = 3.11(6), about 5%. Our results show that the nonlinear terms of single and double excitations, which were not included in the all-order calculation by Safronova et. al., are very crucial to produce a precise Q value of 229 Th. Additionally, we also present magnetic octupole hyperfine-structure constants and some important non-diagonal hyperfine transition matrix elements, which are required for further extracting the magnetic octupole moment Ω of 229 Th nucleus.