  Hyperfine-structure constants of odd Ra$^{+}$ due to the interactions of
nuclear magnetic dipole, electric quadrupole, and magnetic octupole moments
with the electrons are investigated in the framework of relativistic
coupled-cluster method within single- and double-excitation approximation. The
calculated energies and magnetic dipole hyperfine-structure constants $A$
exhibit a good agreement with available experimental values. Combining with the
experimental electric quadrupole hyperfine-structure constant, we also
extracted the electric quadrupole moments $Q$ of $^{209,211,221,223}$Ra. Our
$Q$($^{221}$Ra) and $Q$($^{223}$Ra) are consistent with the referenced values
from 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 $0.48(4)$ by
about 30\%. Furthermore, we also performed a procedure for assessing the
contributions of magnetic octupole moment to the hyperfine splitting. The
sensitivity of hyperfine-structure interval measurements in $^{223}$Ra$^{+}$
that can reveal the effect caused by the nuclear octupole moment are found to
be on the order of kHz.
