Paraelectric - Ferroelectric Phase Transitions in Particles of the Sphere Shape

A transition in a spheroidal particle from the paraelectric to the ferroelectric phase as well as dynamic susceptibility are studied without approximation in the paraphase. It is assumed that the surface charge is compensated and the boundary condition for the polarisation is P=0, i. e. with zero polarisation at the surface of the particle. There is an infinite number of resonance frequencies in the dynamic dielectric function within the quasistatic approximation. The paraphase properties of the dielectric response of the particle are discussed. The transition temperature decreases with decreasing diameter d of the particles inverse quadratically. There exists such a critical diameter that for the particles with the diameter below the critical one the ferroelectric phase is absent. Comparison of the experiment with theoretical results is carried out. Introduction of a dead layer thickness leads to a very good agreement of theory with the experiment for $PbTiO_{3}$ and to a good agreement with the experimentfor $BaTiO_{3}$.