Magnetic hyperfine structure of the ground-state doublet in highly charged ions ${}^{229}\mathrm{Th}^{89+,87+}$ and the Bohr-Weisskopf effect

Background: The search for new opportunities to investigate the low-energy level in the ${}^{229}\mathrm{Th}$ nucleus, which is nowadays intensively studied experimentally, has motivated us to theoretical studies of the magnetic hyperfine (MHF) structure of the $5/2^{+}$ (0.0 eV) ground state and the low-lying $3/2^{+}$ (7.8 eV) isomeric state in highly charged ${}^{229}\mathrm{Th}^{89+}$ and ${}^{229}\mathrm{Th}^{87+}$ ions.

Purpose: The aim is to calculate, with the maximal precision presently achievable, the energy of levels of the hyperfine structure of the ${}^{229}\mathrm{Th}$ ground-state doublet in highly charged ions and the probability of radiative transitions between these levels.

Methods: The distribution of the nuclear magnetization (the Bohr-Weisskopf effect) is accounted for in the framework of the collective nuclear model with Nilsson model wave functions for the unpaired neutron. Numerical calculations using precise atomic density functional theory methods, with full account of the electron self-consistent field, have been performed for the electron structure inside and outside the nuclear region.

Results: The deviations of the MHF structure for the ground and isomeric states from their values in a model of a pointlike nuclear magnetic dipole are calculated. The influence of the mixing of the states with the same quantum number $F$ on the energy of sublevels is studied. Taking into account the mixing of states, the probabilities of the transitions between the components of the MHF structure are calculated.

Conclusions: Our findings are relevant for experiments with highly ionized ${}^{229}\mathrm{Th}$ ions in a storage ring at an accelerator facility.

Figure 1

The large radial component $g\left(x\right)$ of the $2s_{1/2}$ state (in a.u.) calculated in various models (see text for details). ${}^{229}\mathrm{Th}^{89+}$ is the hygrogenlike ion model for the $2s_{1/2}$ state (i.e., the model without screening). ${}^{229}\mathrm{Ra}^{87+}$ stands for the full screening of the nuclear charge +2 by two electrons in the $1s_{1/2}$ state. ${}^{229}\mathrm{Th}^{87+}$, solid line: The potential from two electrons in the $1s_{1/2}$ state is added to the nuclear potential. ${}^{229}\mathrm{Th}^{87+}$, dashed line: The ab initio DFT calculation [51, 52].

Figure 2

The magnetic hyperfine structure of the low-energy doublets in ${}^{229}\mathrm{Th}^{89+}$ and ${}^{229}\mathrm{Th}^{87+}$ and the $M1$ transitions between sublevels. The sublevel energy is counted from the energy of the level in the bare ${}^{229}\mathrm{Th}$ nucleus.