A phenomenological theory of possible sequences of ferrotoroidal phase transitions in boracites

A phenomenological theory of the sequence of two second-order phase transitions with close temperatures is considered; such transitions occur in the Ni-Br boracite. The thermodynamic potential is written as a function of polarization Pi, magnetization Mi, and toroidal moment Ti vectors and fields Ei and Hi; Ti is treated as an order parameter. It is assumed that only one coefficient of Ti2passes through zero as T decreases. The possibility of a sequence of two proper ferrotoroidal phase transitions along the T1 and T2 components is demonstrated. Spontaneous Ti, Pi, and Mi vector values and equations for susceptibility tensors (dielectric χij = dPi/dEj, magnetic kij = dMi/dHj, and magnetoelectric αij = dPi/dHj = dMj/dEi) were obtained for three phases. Some of these values have well-defined anomalies in the vicinity of transitions. All possible sequences of ferrotoroidal phase transitions in boracites are considered. Depending on two potential coefficient values, these sequences may consist of one, two, or three such transitions.