Role of hyperfine interaction in Land\'{e} $g$-factors of $^3\!P^o_0$ clock states

TL;DR

This paper derives a general formula for hyperfine-induced Landé g-factors and applies it to specific clock states in aluminum and strontium, achieving results that match experimental data closely.

Contribution

It introduces a new theoretical expression for hyperfine-induced Landé g-factors and demonstrates its accuracy through calculations for specific atomic states.

Findings

Calculated g-factors agree with experimental data

Derived a general formula for hyperfine-induced g-factors

Predicted g-factors for other atomic systems

Abstract

In the weak-magnetic-field approximation, we derived a general expression of hyperfine-induced Land\'{e} $g$-factors. By using this formula and the multi-configuration Dirac-Hartree-Fock theory, the $g$-factors were calculated for the $3s3p~^3\!P^o_0$ clock state in $^{27}$Al$^+$ and $5s5p~^3\!P^o_0$ in $^{87}$Sr. The present results, $\delta g^{(1)}_{\rm hfs}(^3\!P^o_0) = -1.183(6) \times 10^{-3}$ for $^{27}$Al$^+$ and $\delta g^{(1)}_{\rm hfs}(^3\!P^o_0) = 7.78(30) \times 10^{-5}$ for $^{87}$Sr agree with experimental values very well. Our theory is also useful to predict hyperfine-induced Land\'e $g$-factors for other atomic systems.