Determination of hyperfine splittings and Land\'{e} $g_J$ factors of $5s{}^2{S}_{1\mathrm{/}2}$ and $5p{}^2{P}_{1\mathrm{/}2,3\mathrm{/}2}$ states of ${}^{111,113}$Cd${}^{+}$ for a microwave frequency standard

Regarding trapped-ion microwave-frequency standards, we report on the determination of hyperfine splittings and Land'{e} $g_J$ factors of ${}^{111,113}$Cd${}^{+}$. The hyperfine splittings of the $5p{}^2{P}_{3\mathrm{/}2}$ state of ${}^{111,113}$Cd${}^{+}$ ions were measured using laser-induced fluorescence spectroscopy. The Cd${}^{+}$ ions were confined in a linear Paul trap and sympathetically cooled by Ca${}^{+}$ ions. Furthermore, the hyperfine splittings and Land'{e} $g_J$ factors of the $5s{}^2{S}_{1\mathrm{/}2}$ and $5p{}^2{P}_{1\mathrm{/}2,3\mathrm{/}2}$ levels of ${}^{111,113}$Cd${}^{+}$ were calculated with greater accuracy using the multiconfiguration Dirac–Hartree–Fock scheme. The measured hyperfine splittings and the Dirac–Hartree–Fock calculation values were cross-checked, thereby further guaranteeing the reliability of our results. The results provided in this work can improve the signal-to-noise ratio of the clock transition and the accuracy of the second-order Zeeman shift correction, and subsequently the stability and accuracy of the microwave frequency standard based on trapped Cd${}^{+}$ ions.