Hyperfine Structure of Trapped $^{137}$Ba$^{+}$ Ions as a Probe of the Nuclear Magnetic Octupole Moment

We present calculations of the 1$^{st}$ order hyperfine splittings in the $^{137}$Ba$^{+}$ ion in terms of the 3 hyperfine constants a, b, and c, and also two 2$^{nd}$-order energy effects, for the metastable states $D_{3/2}$ and $D_{5/2}$. It is shown that up to 2$^{nd}$-order, only one of these 2$^{nd}$ order effects (the dipole-quadrupole term) contributes to the measured value of the magnetic octupole constant c; the dipole-squared term does not contribute. Thus c can be determined purely from the measured energy splittings, with a small correction due to the dipole-quadrupole term. Using the octupole constants c$_{3/2}$ and c$_{5/2}$ of the two sets of levels, a particular linear combination of c$_{3/2}$ and c$_{5/2}$ will be completely independent of the 2$^{nd}$ order dipole-quadrupole correction as well, and expressed purely in terms of the measured energy level splittings. Together with atomic theory calculations of the electronic wavefunctions, this would provide a precise value of the nuclear magnetic octupole moment, which could be used as a test of nuclear models.