Design and fabrication of a mode rectangular X-cut quartz resonator with zero temperature coefficient

Rectangular X-cut quartz crystal resonators with cut angles thetas > 5.0deg and aspect ratios R
_zy
(= width 2z
₀
/length 2y
₀
) from 0.3 to 0.5 are investigated. The resonators oscillate mode is a length-extensional mode. A semiempirical frequency equation was derived from the stress expressed in terms of the trigonometric and the hyperbolic transcendental functions with constants estimated by the regression curve fit to the stress simulated by the finite-element method (FEM). Contours on which a point satisfies a zero first order temperature coefficient condition are shown in a cut angle thetas and R
_zy
diagram. We proved that a fabricated resonator with R
_zy
= 0.400 and thetas = 16.0deg, whose design parameter is located in the area of the contour, had a zero temperature coefficient