Theoretical hyperfine splittings of $^{7,9}$Be$^{2+}$ ions for future studies of nuclear properties

The hyperfine structures of the $2,^3!S_1$ and $2,^3!P_J$ states of $^7$Be$^{2+}$ and $^9$Be$^{2+}$ are investigated within the framework of the nonrelativistic quantum electrodynamics, including relativistic and radiative corrections up to order $malpha^6$. The uncertainties of the calculated hyperfine splittings are on the order of tens of ppm, and for $^9$Be$^{2+}$ our results improve the previous theoretical and experimental values by at least two orders of magnitude. The improved sensitivity of the hyperfine splittings of $^{7,9}$Be$^{2+}$ to the nuclear Zemach radius and electric quadrupole moment opens the way to future measurements to extract the atomic physics values of these two nuclear properties to an accuracy of 5\% or better.