We report a detailed study of the size dependence of the ferroelectric transition in an ensemble of ${\mathrm{PbTiO}}_3$ nanoparticles produced by coprecipitation. The phase transition was monitored by dielectric measurements, variable temperature x-ray diffraction, and differential scanning calorimetry (DSC). Size effects were found to become important only below ≊100 nm (coherently diffracting x-ray domain size). The tetragonal distortion of the unit cell (which is related to the spontaneous polarization) decreases exponentially with size and vanishes at 7 nm. The ${\mathit{T}}_{\mathit{c}}$ decreases gradually but the transition becomes increasingly diffuse with a decrease in the size from 80 to 30 nm, below which there is no peak in the dielectric constant or the heat flow (DSC), though ferroelectric ordering probably persists down to ≊7 nm.

• Received 24 May 1995

DOI:https://doi.org/10.1103/PhysRevB.52.13177