Role of hyperfine interaction in Land\'{e} $g$-factors of ${}^3\text{\hspace{0.17em}⁣}{P}_0^o$ clock states

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 $3s3p{}^3\text{\hspace{0.17em}⁣}{P}_0^o$ clock state in ${}^{27}$Al${}^{+}$ and $5s5p{}^3\text{\hspace{0.17em}⁣}{P}_0^o$ 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.