  Hyperfine structure (HFS) of atomic energy levels arises due to interactions
of atomic electrons with a hierarchy of nuclear multipole moments, including
magnetic dipole, electric quadrupole and higher rank moments. Recently, a
determination of the magnetic octupole moment of the $^{173}\mathrm{Yb}$
nucleus was reported from HFS measurements in neutral ${}^{173}\mathrm{Yb}$
[PRA 87, 012512 (2013)], and is four orders of magnitude larger than the
nuclear theory prediction. Considering this substantial discrepancy between the
spectroscopically extracted value and nuclear theory, here we propose to use an
alternative system to resolve this tension, a singly charged ion of the same
$^{173}\mathrm{Yb}$ isotope. Utilizing the substantial suite of tools developed
around $\mathrm{Yb}^+$ for quantum information applications, we propose to
extract nuclear octupole and hexadecapole moments from measuring hyperfine
splittings in the extremely long lived first excited state
($4f^{13}(^2\!F^{o})6s^2$, $J=7/2$) of $^{173}\mathrm{Yb}^+$. We present
results of atomic structure calculations in support of the proposed
measurements.
