A sequence of two ferrotoroidal phase transitions in nickel-bromine boracite Ni$_3$B$_7$O$_{13}$Br

The phenomenological theory of a sequence of two second-order phase transitions in Ni-Br boracite is presented. Two different components of the toroidal moment vector $T_i$ are the order parameters of these transitions. Expressions are derived for the temperature dependences of the spontaneous values of $T_i$, polarization $P_i$, and magnetization $M_i$ and the dielectric $\chi_{ij}=dP_i/dE_j$, magnetic $k_{ij}=dM_i/H_j$, and magnetoelectric $\alpha_{ij}=dP_i/dH_j=dM_j/dE_i$ susceptibilities. Some of these susceptibilities display sharp temperature peaks in the vicinity of phase transitions.