About nonlinear effects in the quartz homeotypes single crystals

The influence of electric field on the properties of piezoelectric single domain crystal was precisely specified by means of non-linear electro-elastic differential equations [1]. An equivalent treatment, i.e. using the thermodynamically defined material constants for the thickness shear mode resonators, is presented in this paper. Based on the Tierstenpsilas relations [2], we attempt to specify the 4
^th
order effective elastic stiffness tildec
₆₆₆₆
for the Y-cut quartz resonators, and gallium ortho-phosphate (GaPO
₄
) resonators. It was discovered that the nonlinear effective stiffness tildec
₆₆₆₆
consists of five basic stiffnesses of the 4
^th
order. The tildec
₆₆₆₆
was measured for five different cuts of piezoelectric resonators and the constants A
₁
, A
₂
, A
₃
, A
₄
, A
₅
were specified for each of the cuts. A system of five equations was obtained. By solving this particular system of equations values of basic elastic stiffnesses of the 4
^th
order c
₅₅₅₅
, c
₅₅₅₆
, c
₅₅₆₆
, c
₅₆₆₆
, and c
₆₆₆₆
were obtained. In addition, the GaPO
₄
crystals were used to study the nonlinear amplitude-frequency-, and inter-modulation effects, and to determine the tildec
₆₆₆₆
. The measured data were used for modeling the nonlinear A(F) effect by nonlinear electric equivalent circuit of the BAW GaPO
₄
resonator vibrating in the thickness-shear mode of vibrations. Additional important nonlinear effect such as influence of the electric field to the SAW velocity was used for design of a new SAW transducer. The basic equations describing this effect are presented in the paper.