Short pitch structures in smectics with interactions over more than two layers

Phenomenological model of chiral polar smectics is introduced with interactions up to the fourth neighboring layers. The minimization of the free energy gives three stable structures: the ferroelectric Sm C${}^{\ast }$ phase, the antiferroelectric Sm C${}_A^{\ast }$ phase and the hellicoidally modulated structure of the Sm C${}_H^{\ast }$ phase. The phase Sm C${}_{\alpha }^{\ast }$ can be recognized as the Sm C${}_H^{\ast }$ phase below Sm A phase and as the Sm C${}_{FI2}^{\ast }$ phase or as the Sm C${}_{FI1}^{\ast }$ phase when appearing between the Sm C${}^{\ast }$ phase and the Sm C${}_A^{\ast }$ phase. Stability of these phases is analyzed and the phase diagram in the space of model parameters is presented.