Ferrielectric smectic with layer-by-layer change of the two-component order parameter

One of the most remarkable properties of smectics is the wide variety of possible equilibrium structures. In this paper, based on Landau theory of the phase transitions, the transitions between ferroelectric and antiferroelectric phases and structure formed by smectic layers with different azimuthal and polar orientation of molecules were calculated. This unique structure has been predicted (P. V. Dolganov et al., Pis'ma v ZhETF 76, 579 (2002)) using the minimization of the free energy with respect to the phase and modulus of the two-component order parameter, but never detected before. Recently non-resonant Bragg reflection, consistent with the predictions of the model, was found (P. Fernandes et al., Eur. Phys. J. E 20, 81 (2006)) in ferrielectric smectic $C_{FI1}^*$ ($\mathrm{Sm}C_{FI1}^*$) phase. In the 3-layer ferrielectric structure with a macroscopic helical pitch the modulus of the order parameter is larger in anticlinic-like layers and smaller in layers with mixed ordering. The values of the interlayer interactions were determined for smectic liquid crystalline materials forming different polar structures.