  We report sympathetic cooling of $^{113}$Cd$^+$ by laser-cooled $^{40}$Ca$^+$
in a linear Paul trap for microwave clocks. Long-term low-temperature
confinement of $^{113}$Cd$^+$ ions was achieved. The temperature of these ions
was measured at $90(10)$ mK, and the corresponding uncertainty arising from the
second-order Doppler shifts was estimated to a level of $2\times10^{-17}$. Up
to $4.2\times10^5$ Cd$^+$ ions were confined in the trap, and the confinement
time constant was measured to be 84 hours. After three hours of confinement,
there were still $10^5$ Cd$^+$ ions present, indicating that this
Ca$^+$--Cd$^+$ dual ion system is surprisingly stable. The ac Stark shift was
induced by the Ca$^+$ lasers and fluorescence, which was carefully estimated to
an accuracy of $5.4(0.5)\times10^{-17}$ using a high-accuracy \textit{ab
initio} approach. The Dick-effect-limited Allan deviation was also deduced
because deadtimes were shorter. These results indicate that a microwave clock
based on this sympathetic cooling scheme holds promise in providing ultra-high
frequency accuracy and stability.
