Temperature-induced frequency shifts in quartz resonators

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Abstract
The thermal exchanges in a quartz crystal resonator are investigated. This leads to a temperature gradient distribution which is in the main plane of the crystal, and therefore does not exhibit any functional dependence (at least at the first order) along the plate thickness. Internal heat source due to the energy dissipation by the internal friction is considered. The corresponding in-plane thermal stresses are calculated in the isotropic approximation, and anisotropy is introduced by means of the stress-strain relations. Applying a perturbation method, the resulting frequency shifts are obtained for both static and dynamic thermal behaviors.