This paper brings out 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 within the very useful cryogenic range [4 K–15 K] from a set of resonators of various cut angles, because of the anisotropy of quartz.

1.
S.
Galliou
,
J.
Imbaud
,
M.
Goryachev
,
R.
Bourquin
, and
Ph.
Abbé
, “
Losses in high quality quartz crystal resonators at cryogenic temperatures
,”
Appl. Phys. Lett.
98
,
091911
(
2011
).
2.
S.
Galliou
,
M.
Goryachev
,
R.
Bourquin
,
Ph.
Abbé
,
J. P.
Aubry
, and
M. E.
Tobar
, “
Extremely low loss phonon-trapping cryogenic acoustic cavities for future physical experiments
,”
Sci. Rep.
3
,
2132
(
2013
).
3.
A.
Lo
,
P.
Haslinger
,
E.
Mizrachi
,
L.
Anderegg
,
H.
Müller
,
M.
Hohensee
,
M.
Goryachev
, and
M. E.
Tobar
, “
Acoustic Tests of Lorentz Symmetry Using Quartz Oscillators
,”
Phys. Rev. X
6
,
011018
(
2016
).
4.
S.
Kotler
,
R. W.
Simmonds
,
D.
Leibfried
, and
D. J.
Wineland
, “
Hybrid quantum systems with trapped charged particles
,” e-print arXiv:1608.02677v1 [quant-ph].
5.
M.
Aspelmeyer
,
T. J.
Kippenberg
, and
F.
Marquardt
, “
Cavity optomechanics
,”
Rev. Mod. Phys.
86
,
1391
(
2014
).
6.
M.
Goryachev
,
E. N.
Ivanov
,
F.
Van Kann
,
S.
Galliou
, and
M. E.
Tobar
, “
Observation of the fundamental Nyquist noise limit in an ultra-high Q-factor cryogenic bulk acoustic wave cavity
,”
Appl. Phys. Lett.
105
,
153505
(
2014
).
7.
A. G.
Smagin
, “
Frequency-temperature characteristics of quartz crystal units of different cuts operating over a wide temperature range including helium temperatures
,” in
IEEE International Frequency Control Symposium
(
1995
).
8.
M.
Goryachev
,
Ph.
Abbé
,
B.
Dulmet
,
R.
Bourquin
, and
S.
Galliou
, “
Measurements of elastic properties of langatate at liquid helium temperatures for design of ultra low loss mechanical systems
,”
Appl. Phys. Lett.
104
,
261904
(
2014
).
9.
T. H. K.
Barron
,
J. F.
Collins
,
T. W.
Smith
, and
G. K.
White
, “
Thermal expansion, Grüneisen functions and static lattice properties of quartz
,”
J. Phys. C
15
,
3311
3326
(
1982
).
10.
B.
Dulmet
and
R.
Bourquin
, “
Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. II. Applications and numerical values for quartz
,”
J. Acoust. Soc. Am.
110
,
1800
(
2001
).
11.
D. S.
Stevens
and
H. F.
Tiersten
, “
An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation
,”
J. Acoust. Soc. Am.
79
,
1811
(
1986
).
12.
B. A.
Auld
,
Acoustic Fields and Waves in Solids
(
Wiley
,
1973
), Vol. I.
13.
D.
Royer
and
E.
Dieulesaint
,
Elastic Waves in Solids I, Free and Guided Propagation
(
Springer
,
2000
).
14.
R.
Bechmann
, “
Elastic and piezoelectric constants of alpha-quartz
,”
Phys. Rev.
110
,
1060
(
1958
).
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