Relationship between exchange and relativistic energies defining linear magnetoelectric effect in Cr$_2$O$_3$ antiferromagnet

The linear magnetoelectric effect in $\mathrm{Cr_2O_3}$ 3 within the framework of a phenomenological approach has been considered. The longitudinal (an external field is parallel to the three-fold axis) and the transverse part of the effect in $\mathrm{Cr_2O_3}$ are shown to be dictated by the exchange and relativistic interactions, respectively. This explains the change in the magnetoelectric effect tensor $\alpha_{ij}$ under the spin-flop transition and determines the toroidal moment $\textbf{T}$ vector components which are other than zero in the $\mathrm{Cr_2O_3}$ spin-Hop phase.