Lagrangean versus classical formulation of frequency temperature problems in quartz resonators

Equations for calculating the Lagrangean temperature derivatives of the elastic and piezoelectric constants of quartz using their classical temperature derivatives are derived and presented. In the classical formulation, the resonator geometry and hence the reference frame changes with temperature, while in the Lagrangean formulation the reference frame is fixed at a certain temperature, say 25°C. The immediate consequence of changing the reference frame in the classical formulation would be that the temperature coefficients of the material constants are referred to a reference frame which is itself a function of temperature. Another consequence is the difficulty in maintaining the conservation of mass at all temperatures. Hence the theoretical foundation of the classical method is unsound. For certain crystal symmetries there are similarities between the two formulations; however, in general there are significant differences between them, and going forward the Lagrangean formulation should be employed. The Lagrangean-classical relationships presented here will allow us to calculate the Lagrangean temperature derivatives of material constants such as the elastic and piezoelectric constants from the existing and published classical temperature coefficients of the said constants. Results are shown for the temperature derivatives of the elastic and piezoelectric constants of alpha quartz. Simple one-dimensional vibration problems are used to illustrate the similarities and differences between the two formulations