Long range interactions and the sense of molecular long axis in ferroelectric smectics

The successive phase transitions in the ferro- and antiferro-electric smectics have been described hy the Ising model with competing interactions. The additional pseudo-spins are introduced newly to represent the sense of the molecular long axis. By taking the sum over the whole states of these pseudo-spins in the partition function, not only the nearest neighbour parameter of the Ising model are renormalized but additional long range interactions and many-body forces are shown to be induced. The effective second nearest neighbour interaction is shown to be antiferroelectric kind and the third nearest neighbour one is ferroelectric in some cases. Ln the ferroelectric phase and main antiferroelectric one the sense of the molecules is arranged such that the ordering is absent even in the ordered region in agreement with the symmetry of these phases, while in the other intermediate phases this ordering is proved to exist because of the bias due to the stacking of layer-orders.