parameters to be determined, $A$-$D$ and the $\eta_{mn}$. However, since all of the terms included in our model are tensors of rank $k\!\le\!4$, there is a degeneracy in Eqs. (\[eq:DeltaW12\]) and a proper linear combination of any four of the measurements can be used to predict the fifth. While this reduces the number of experimentally determined quantities to $k_\mathrm{max}\!=\!4$, it will provide a test of the model presented above and way to detect and reject systematic effects in the experiment.

Within the 3 second order terms ($\eta_{mn}$), since the energy difference between the ${}^2\!F_{J^\prime\!=\!5/2}^o$ and ${}^2\!F_{J\!=\!7/2}^o$ states is known, if we assume that these are the only terms that contribute, they contain only 2 unknowns: $\mu \langle \gamma J || T^e_1 || \gamma J^\prime \rangle$ and $Q \langle \gamma J || T^e_2 || \gamma J^\prime \rangle$. Further, the coefficient $A$ can be determined from existing experimental data [@Fisk1997; @Werth1987; @Taylor1999], $$\label{eq:Acoeffs}
    A^{(173)}_{^2F_{7/2}^o} = \frac{A^{(173)}_{^2S_{1/2}}\,\, A^{(171)}_{^2F_{7/2}^o}}{A^{(171)}_{^2S_{1/2}}} = -250 \mbox{ MHz}.$$ Here, we have extracted $A_{{}^2F_{7/2}^o}^{(171)}$ from the measured energy splitting $\Delta W_3^{(171)}=3.620\mbox{ GHz}$ [@Taylor1999] via $$A_{{}^2F_{7/2}}^{(171)} = \frac{\Delta W_3^{(171)}}{4} + \frac{1}{144}\left( \frac{\mu^{(171)}}{\mu^{(173)}}\right)^2 \eta_{\mu\mu} \approx \frac{\Delta W_3^{(171)}}{4}$$ and therefore neglected the contribution (tens of Hz) of the second order correction to the 