  In this paper, we investigate the mass spectra of bottom-charm baryons
systematically, where the relativistic quark model and the infinitesimally
shifted Gaussian basis function method are employed. Our calculation shows that
the $\rho$-mode appears lower in energy than the other excited modes. According
to this feature, the allowed quantum states are selected and a systematic study
of the mass spectra for $\Xi_{bc}^{'}$ ($\Xi_{bc}$) and $\Omega_{bc}^{'}$
($\Omega_{bc}$) families is performed. The root mean square radii and quark
radial probability density distributions of these baryons are analyzed as well.
Next, the Regge trajectories in the $(J,M^{2})$ plane are successfully
constructed based on the mass spectra. At last, we present the structures of
the mass spectra, and analyze the difficulty and opportunity in searching for
the ground states of bottom-charm baryons in experiment.
