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The microwave clock frequency of the $|5s~^2S_{1/2}, F=0,m_F=0 \rangle\leftrightarrow |5s~^2S_{1/2}, F=1,m_F=0 \rangle$ transition in the$^{113}$Cd$^+$ ion has been reported as 15199862855.0192(10) Hz [Opt. Lett.{\bf 40}, 4249 (2015)]. Fractional systematic due to the black-body radiation(BBR) shift ($\beta$) arising from the Stark effect in the above clocktransition was used as $-1.1 \times 10^{-16}$ from our unpublished preliminaryestimation. We present here a precise value as $\beta=-1.815(77) \times10^{-16}$ by carrying out rigorous calculations of third-order polarizabilitiesof the hyperfine levels associated with the clock transition. This isdetermined by evaluating matrix elements of the magnetic dipole hyperfineinteraction Hamiltonian, electric dipole operator and energies between manylow-lying states of $^{113}$Cd$^+$. We employ all-order relativistic many-bodytheories in the frameworks of Fock-space coupled-cluster and relativisticmulti-configuration Dirac-Fock methods.

Estimation of BBR shift due to Stark effect for the Microwave $^{113}$Cd$^{+}$ Ion Clock