Electrostrictive Effect in Barium Titanate Ceramics

When a direct current bias is applied to a multicrystalline barium titanate ceramic, an alternating voltage can excite resonances in the ceramic. Four modes of motion have been excited---a longitudinal mode at right angles to the applied field, a radial mode of a circular plate at right angles to the applied field, a thickness longitudinal mode, and a thickness shear mode. The first three are excited when the a. c. field is applied in the same direction as the d. c. polarization, but the fourth is excited when the a. c. field is at right angles to the d. c. polarization. The amount of motion is larger than in magnetostrictive materials, and it appears that barium titanate may be an important electromechanical transducing element. All of these modes can be accounted for on the basis of a second-order electrostrictive effect. Two electrostrictive constants are involved and these have been evaluated as {Q₁₂, =-2. 1510^-12 (cm^{4}{ (statcoulombs) ^2}) }{Q₁₁, =+6. 910^-12 (cm^{4}{ (statcoulombs) ^2}). }Using these constants, the measured electromechanical coupling factors for the four modes are evaluated, and these compare well with the calculated values. A theoretical explanation of this effect is given which depends on the fact that when a given domain becomes ferro-electric it loses its cubic structure and becomes tetragonal. In this process it expands one percent along the tetragonal axis and contracts one-half percent along the other two axes. In the ceramic piece all directions for the tetragonal axis are equally probable, but an applied field can cause the domains in the direction of the field to grow at the expense of domains perpendicular to the field. This growth is accompanied by an increase in the thickness of the crystal and a decrease in radial dimensions. The measured ratio of 3 to 1, compared to the 2 to 1 ratio observed by x-rays for a single crystal, is accounted for by the nature of the ceramic material which does not join up for all grains. This does not prevent the ceramic from increasing in thickness but does cut down the radial contraction. Experimental measurements of the electrostrictive effect are given, and it is shown that the displacement is proportional to the square or products of the electric displacements in the ceramic.