with sufficiently high multiplicity $(2J \ge k)$ and the magnitude of the energy shift tends to decrease with increasing rank $k$. The measurement of HFS signatures of high rank $(k \ge 3)$ multipoles, therefore, requires a well-controlled atom in a high angular momentum state for precision and state of the art atomic structure theory for accuracy.

Here, we focus on the potential for measuring the rarely observed nuclear octupole ($\Omega, k\!=\!3$) and hexadecapole ($\Pi, k\!=\!4$) moments. These moments have been deduced for only a handful of nuclei and, in most cases, are in tension with nuclear theory (see Table \[table:NuclearMoments\]). For example, in $^{133}$Cs, the extracted [@GerDerTan03] nuclear octupole moment is 40 times larger that the nuclear theory value. This paper is motivated by the even more substantial disagreement for $^{173}\mathrm{Yb}$. Recently, @Singh reported a measurement of the nuclear octupole moment from their measurements of HFS in the ${^3P_2}$ state of neutral $^{173}\mathrm{Yb}$. However, this value $\Omega = -34.4 \,{\mathrm{b{\mu_N}}}$ is $10^4$ times larger than the nuclear theory prediction, $\Omega =0.003\,{\mathrm{b{\mu_N}}}$ [@Williams1962]. This striking four orders of magnitude disagreement calls for an independent measurement and analysis. Here, we investigate the prospects for extracting $\Omega$ and higher rank nuclear multipole moments 