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We have measured the quadratic Zeeman coefficient for the ${^{1}S_{0}\leftrightarrow {^{3}P_{0}}}$ optical clock transition in $^{27}$Al$^{+}$,$C_{2}=-71.944(24)$~MHz/T$^{2}$, and the unperturbed hyperfine splitting of the$^{25}$Mg$^{+}$ $^{2}S_{1/2}$ ground electronic state, $\Delta W / h =1~788~762~752.85(13)$~Hz, with improved uncertainties. Both constants arerelevant to the evaluation of the $^{27}$Al$^{+}$ quantum-logic clocksystematic uncertainty. The measurement of $C_{2}$ is in agreement with aprevious measurement and a new calculation at the $1~\sigma$ level. Themeasurement of $\Delta W$ is in good agreement with a recent measurement anddiffers from a previously published result by approximately $2\sigma$. With theimproved value for $\Delta W$, we deduce an improved value for thenuclear-to-electronic g-factor ratio $g_{I}/g_{J} = 9.299 ~308 ~313(60) \times10^{-5}$ and the nuclear g-factor for the $^{25}$Mg nucleus $g_{I} = 1.861 ~957~82(28) \times 10^{-4}$. Using the values of $C_{2}$ and $\Delta W$ presentedhere, we derive a quadratic Zeeman shift of the $^{27}$Al$^{+}$ quantum-logicclock of $\Delta \nu / \nu = -(9241.8 \pm 3.7) \times 10^{-19}$, for a biasmagnetic field of $B \approx 0.12$~mT.

Measurements of $^{27}$Al$^{+}$ and $^{25}$Mg$^{+}$ magnetic constants for improved ion clock accuracy