Polar Smectic subphases: phase diagrams, structures and $\mathbf{X}$-ray scattering

We analyze a discrete phenomenological model accounting for phase transitions and structures of polar Smectic-$C^*$ liquid crystalline phases. The model predicts a sequence of phases observed in experiment: antiferroelectric Sm$C^*_A$ - ferrielectric Sm$C^*_{FI1}$ - antiferroelectric Sm$C^*_{FI2}$ (three and four layer periodic, respectively) - incommensurate Sm$C^*_{\alpha}$ - Sm$A$. We find that in the three-layer Sm$C^*_{FI1}$ structure both the phase and the module of the order parameter (tilt angle) differ in smectic layers. This modulation of the tilt angle (and therefore of the layer spacing $d$) must lead to $X$-ray diffraction at the wave vectors $Q_s=2\pi s/d$ ($s=n\pm{1}/{3})$ even for the non-resonant scattering.