Activity dips in FC-cut resonators from interaction with A modes at twice the frequency

The FC-cut historically has been fraught with activity dips. However, interest in its use has been revived in a recent application where acceleration sensitivity has been found to be particularly low for this cut vs. other cuts. An analysis has been done to explore the source of the activity dips. Results for the A, B, and C modes of a 10 MHz, 3 ^rd overtone, plano-convex design indicate that the 3^rd overtone A mode family has slightly complex wavenumber and is in the vicinity of 20 MHz, which is 2× the desired 10 MHz operating frequency. Experimental measurements show that the 3^rd overtone A mode is indeed poorly energy trapped, but has a number of low Q anharmonic modes within a few kHz of 20 MHz. The A mode frequencies have a strong negative temperature coefficient and cross the exact 2× relationship with the 10 MHz 3^rd overtone C mode at several temperatures ranging from 25 to 100°C. It was possible to watch activity dips occur on the spectrum analyzer in the 10 MHz 3^rd overtone C mode signal during cooling or warming as the different 3^rd overtone A modes moved with temperature through the 2^rd relationship. Calculations of the anharmonic frequencies with changes in diopter follow experimental data obtained from units with different diopters. The experimental and theoretical data lead to the conclusion that it is possible to choose diopter designs to ensure ovenized crystal oscillators free of activity dips at oven temperatures. However, the results show that it is not practical to expect the FC-cut to operate without activity dips over large temperature ranges such as the warmup of an ovenized oscillator from room temperature. Crystal designers will have to take into consideration that a 2× relation between the desired mode and an interfering mode causes activity dips when designing other crystallographic orientations and calculate all three mode families to look for a 2× relationship