Calculation of the Piezo-electric Constants of α- and β-Quartz

Abstract

α-QUARTZ possesses only two piezo-electric constants, ε₁₁ and ε₁₄, given by the equations where p_x, p_y, p_z are the electric moments per unit volume developed on strain along the axes of the crystal, and u^xx, u^xy, etc., the components of strain, β-Quartz, which possesses a higher symmetry, has only one constant, ε₁₄, the other, ε₁₁ being zero. A method of deriving the above equations from the structure of the crystal, that is, the co-ordinates of the atoms in the unit cell, is not known. While the earlier work of Curie¹ (1882), Riecke² (1892) and Kelvin³ (1893) is purely speculative, Gibbs⁴ (1926) calculated the piezo-electric modulus δ₁₁ on the basis of his structure and obtained a value which is nearly five times too high. Moreover, he did not calculate the other constant, δ₁₄, and did not show how the above equations can be obtained from his co-ordinates. We have been able to derive the above equations from the structure of Gibbs and have in this way obtained values of ε₁₁ and ε₁₄ which agree completely with the observed values. The piezo-electric constant ε₁₄ of β-quartz and the variation of ε₁₁ and ε₁₄ with temperature in the case of α-quartz has also been calculated.