Temperature coefficients of crystalline-quartz elastic constants over the cryogenic range [4 K, 15 K]

Abstract

This paper brings out results of a measurement campaign aiming to determine the temperature coefficients of synthetic quartz elastic constants at liquid helium temperature. The method is based on the relationship between the resonance frequencies of a quartz acoustic cavity and the elastic constants of the material. The temperature coefficients of the elastic constants are extracted from experimental frequency-temperature data collected from a set of resonators of various cut angles, because of the anisotropy of quartz, measured on the very useful cryogenic range [4 K-15 K]. The knowledge of these temperature coefficients would allow to further design either quartz temperature sensors or conversely frequency-temperature compensated quartz cuts. With extremely low losses, lower than 10 −9 for the best ones, key applications of such devices are ultra-low loss mechanical systems used in many research areas including frequency control and fundamental measurements. The Eulerian formalism is used in this study to identify the temperature coefficients.

Cryogenic optomechanic cavity in low mechanical loss material

“…Indeed, since the elastic coefficients, density, and thickness of the acoustic cavity depend on temperature, the acoustic-resonance frequency is also temperature dependent, and it becomes mandatory to select the appropriate cut angles exhibiting a frequency-temperature turning point in the [3 K-15 K] range. The temperature coefficients of the stiffness coefficients have recently been measured 20 and frequency-temperature compensated cuts have then been identified: devices are currently under test to check theanglesensitivity.…”

Optomechanical Coupling in a Quartz Crystal Resonator for Cryogenic Clocks

“…a cut whose frequency-temperature curve has a turning point inside the range [3 -15 ]. To identify such a cut a campaign has been carried out to measure the temperature sensitivity of the elastic coefficients of quartz within the operating temperature range [7]. In a second step, those results have been used to calculate the temperature coefficients of a selected resonance frequency with respect to cut angles.…”

Temperature coefficients of quartz at cryogenic temperature

“…parasitic stresses and then frequency changes), and consequently a kind of guarantee to qualify the material regardless of the device structure. However, the modelling is complicated and the resulting frequency is as follows, according to the Stevens-Tiersten model [5], one of the most accurate theories: Nevertheless, several assumptions can be made by considering some orders of magnitude [6], and calculation proceeds according to the described process.…”