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Hyperfine-structure constants of odd Ra$^{+}$ due to the interactions ofnuclear magnetic dipole, electric quadrupole, and magnetic octupole momentswith the electrons are investigated in the framework of relativisticcoupled-cluster method within single- and double-excitation approximation. Thecalculated energies and magnetic dipole hyperfine-structure constants $A$exhibit a good agreement with available experimental values. Combining with theexperimental electric quadrupole hyperfine-structure constant, we alsoextracted the electric quadrupole moments $Q$ of $^{209,211,221,223}$Ra. Our$Q$($^{221}$Ra) and $Q$($^{223}$Ra) are consistent with the referenced valuesfrom a semi-empirical analysis (Z. Phys. D: At., Mol. Clusters 11, 105 (1988)),but $Q(^{211}$Ra)=$0.33(2)$ is smaller than the referenced value $0.48(4)$ byabout 30\%. Furthermore, we also performed a procedure for assessing thecontributions of magnetic octupole moment to the hyperfine splitting. Thesensitivity of hyperfine-structure interval measurements in $^{223}$Ra$^{+}$that can reveal the effect caused by the nuclear octupole moment are found tobe on the order of kHz.

Relativistic coupled-cluster-theory analysis of the hyperfine interaction of Ra$^{+}$ isotopes