  We present a simple theory for structural transitions in displacive
ferroelectrics of the perovskite type. In our theory, the competition between
the elastic energy cost for the displacement of the homopolar ion from the
centrosymmetric position, and the energy gain due to a ferroelectric ordering
of the dipoles formed by the ionic displacements, leads naturally to a
first-order transition from a paraelectric to a ferroelectric phase. This
transition takes place at a certain temperature $T_c(L)$ as the temperature is
decreased and, at a certain size $L_c (T)$ as the size of the system is
increased. The transition temperature $T_c (L)$ is suppressed as the sample
size is reduced, and vanishes for samples below a certain size. For
order-disorder ferroelectrics, our theory shows that the suppression of $T_c$
by a reduction in system size is not appreciable, a result that is borne out by
experiments.

Materials Science