We report on the precision measurement of the unperturbed ${}^3{D}_1$ microwave transition frequencies in ${}^{176}\mathrm{Lu}^{+}$ to a fractional uncertainty of $4\times {10}^{-14}$. We find the $F=8$ to $F=7$ hyperfine transition frequency to be $10491519945.22882\left(38\right)\mathrm{Hz}$ and the $F=7$ to $F=6$ transition frequency to be $11290004289.88161\left(36\right)\mathrm{Hz}$. At this precision we are able to observe the hyperfine-mediated effects in the ratio of the quadrupole shifts, from which we can directly infer the residual quadrupole moment after ${}^{3}D_1$ hyperfine averaging. We find a residual quadrupole moment of $-2.48\left(23\right)\times {10}^{-4}e{a}_0^2$, consistent with a previous assessment using a different and less direct method. With the unperturbed microwave frequencies accurately known, the residual quadrupole shift for a ${}^{176}\mathrm{Lu}^{+}$ (${}^{3}D_1$) optical frequency standard can henceforth be readily evaluated to $<{10}^{-20}$ uncertainty by routine ${}^3{D}_1$ microwave spectroscopy.

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• Atomic, optical & lattice clocks
• Fine & hyperfine structure
• Trapped ions
• Electric moment
• Spectroscopy