Twist of rhombohedra, spin helix, and “giant” polarization in a CaMn$_{7}$O$_{12}$ multiferroic crystal

A qualitative assessment of axial vector $\mathbf{\Gamma}_{z}$ in invariant $P_{z}$($\mathbf{M}\operatorname{rot}\mathbf{M})\mathbf{\Gamma}_{z}$ proposed earlier for the description of the thermodynamic potential of a CaMn$_{7}$O$_{12}$ multiferroic material with electric polarization $P_z$ along the $z$ axis of the helix of magnetic moments M has been performed based on the fan-shaped structure of the atomic cell of the crystal. The dependence of $\mathbf{\Gamma}_{z}$ on the permanent dipole moment of MnO$_{6}$ octahedra (rhombohedra), which are inevitably distorted in the paraelectric phase due to the structural transformation, on the mechanical stiffness of the crystal lattice, and on the projected area of the cross section of rhombohedra with bevels undergoing a shift has been determined from dimensional considerations. It has been demonstrated how these factors contribute to the twist of rhombohedra.