We report the adiabatic elastic constants of single-crystal thorium dioxide over a temperature range of 77–350 K. Time-domain Brillouin scattering, an all-optical, non-contact picosecond ultrasonic technique, is used to generate and detect coherent acoustic phonons that propagate in the bulk perpendicular to the surface of the crystal. These coherent acoustic lattice vibrations have been monitored in two hydrothermally grown single-crystal thorium dioxide samples along the (100) and (311) crystallographic directions. The three independent elastic constants of the cubic crystal (C11, C12, and C44) are determined from the measured bulk acoustic velocities. The longitudinal wave along the (100) orientation provided a direct measurement of C11. Measurement of C44 and C12 was achieved by enhancing the intensity of quasi-shear mode in a (311) oriented crystal by adjusting the polarization angle relative to the crystal axes. We find the magnitude of softening of the three elastic constants to be ∼2.5% over the measured temperature range. Good agreement is found between the measured elastic constants with previously reported values at room temperature, and between the measured temperature-dependent bulk modulus with calculated values. We find that semi-empirical models capturing lattice anharmonicity adequately reproduce the observed trend. We also determine the acoustic Grüneisen anharmonicity parameter from the experimentally derived temperature-dependent bulk modulus and previously reported temperature-dependent values of volumetric thermal expansion coefficient and heat capacity. This work presents measurements of the temperature-dependent elasticity in single-crystal thorium dioxide at cryogenic temperature and provides a basis for testing ab initio theoretical models and evaluating the impact of anharmonicity on thermophysical properties.

1.
G.
Leibfried
and
W.
Ludwig
,
Theory of Anharmonic Effects in Crystals, in Solid State Physics
(
Elsevier
,
1961
), pp.
275
444
.
2.
T.
Barron
, “
Interatomic potentials in ideal anharmonic crystals
,”
Discuss. Faraday Soc.
40
,
69
75
(
1965
).
3.
R. A.
Cowley
, “
Anharmonic crystals
,”
Rep. Prog. Phys.
31
(
1
),
123
(
1968
).
4.
J.
Belle
and
R.
Berman
,
Thorium Dioxide: Properties and Nuclear Applications
(
USDOE Assistant Secretary for Nuclear Energy
,
1984
).
5.
S.
Bharadwaj
, “
Thoria-based nuclear fuels
,” in
Green Energy and Technology
,
1st ed.
, edited by
S. R. B.
Dasarathi Das
(
Springer
,
London
,
2013
).
6.
D. H.
Hurley
et al, “
Thermal energy transport in oxide nuclear fuel
,”
Chem. Rev.
122
(
3
),
3711
3762
(
2021
).
7.
R. K.
Behera
and
C. S.
Deo
, “
Atomistic models to investigate thorium dioxide (ThO2)
,”
J. Phys.: Condens. Matter
24
(
21
),
215405
(
2012
).
8.
M.
Cooper
,
M.
Rushton
, and
R.
Grimes
, “
A many-body potential approach to modelling the thermomechanical properties of actinide oxides
,”
J. Phys.: Condens. Matter
26
(
10
),
105401
(
2014
).
9.
V.
Kanchana
., “
First-principles study of elastic properties of CeO2, ThO2 and PoO2
,”
J. Phys.: Condens. Matter
18
(
42
),
9615
(
2006
).
10.
J.
Liu
et al, “
Lattice thermodynamic behavior in nuclear fuel ThO2 from first principles
,”
J. Nucl. Mater.
511
,
11
17
(
2018
).
11.
Y.
Lu
,
Y.
Yang
, and
P.
Zhang
, “
Thermodynamic properties and structural stability of thorium dioxide
,”
J. Phys.: Condens. Matter
24
(
22
),
225801
(
2012
).
12.
J.-J.
Ma
et al, “
Molecular dynamics study on thermal properties of ThO2 doped with U and Pu in high temperature range
,”
J. Alloys Compd.
627
,
476
482
(
2015
).
13.
H.
Nakamura
and
M.
Machida
, “
High-temperature properties of thorium dioxide: A first-principles molecular dynamics study
,”
J. Nucl. Mater.
478
,
56
60
(
2016
).
14.
J.
Park
,
E. B.
Farfán
, and
C.
Enriquez
, “
Thermal transport in thorium dioxide
,”
Nucl. Eng. Technol.
50
(
5
),
731
737
(
2018
).
15.
I.
Shein
,
K.
Shein
, and
A.
Ivanovskii
, “
Elastic and electronic properties and stability of SrThO3, SrZrO3 and ThO2 from first principles
,”
J. Nucl. Mater.
361
(
1
),
69
77
(
2007
).
16.
V.
Sobolev
and
S.
Lemehov
, “
Modelling heat capacity, thermal expansion, and thermal conductivity of dioxide components of inert matrix fuel
,”
J. Nucl. Mater.
352
(
1–3
),
300
308
(
2006
).
17.
B.
Szpunar
and
J.
Szpunar
, “
Theoretical investigation of structural and thermo-mechanical properties of thoria up to 3300 K temperature
,”
Solid State Sci.
36
,
35
40
(
2014
).
18.
B.
Szpunar
,
J.
Szpunar
, and
K.-S.
Sim
, “
Theoretical investigation of structural and thermo-mechanical properties of thoria
,”
J. Phys. Chem. Solids
90
,
114
120
(
2016
).
19.
R.
Terki
et al, “
First principles calculations of structural, elastic and electronic properties of XO2 (X = Zr Hf and Th) in fluorite phase
,”
Comput. Mater. Sci.
33
(
1–3
),
44
52
(
2005
).
20.
B.-T.
Wang
et al, “
First-principles study of ground-state properties and high pressure behavior of ThO2
,”
J. Nucl. Mater.
399
(
2–3
),
181
188
(
2010
).
21.
M. A.
Mathis
et al, “
The generalized quasiharmonic approximation via space group irreducible derivatives
,”
Phys. Rev. B
106
, 014314 (
2022
).
22.
R. t.
Cottam
and
G.
Saunders
, “
The elastic constants of GaAs from 2 K to 320 K
,”
J. Phys. C: Solid State Phys.
6
(
13
),
2105
(
1973
).
23.
W.
Boyle
and
R.
Sladek
, “
Elastic constants and lattice anharmonicity of GaSb and GaP from ultrasonic-velocity measurements between 4.2 and 300 K
,”
Phys. Rev. B
11
(
8
),
2933
(
1975
).
24.
M. E.
Diederich
and
J.
Trivisonno
, “
Temperature dependence of the elastic constants of sodium
,”
J. Phys. Chem. Solids
27
(
4
),
637
642
(
1966
).
25.
E.
Gutman
and
J.
Trivisonno
, “
Temperature dependence of the elastic constants of rubidium
,”
J. Phys. Chem. Solids
28
(
5
),
805
809
(
1967
).
26.
J.
Smith
and
J.
Gjevre
, “
Elastic constants of yttrium single crystals in the temperature range 4.2–400 K
,”
J. Appl. Phys.
31
(
4
),
645
647
(
1960
).
27.
D.
Pederson
and
J.
Brewer
, “
Elastic constants of cadmium fluoride from 4.2 to 295 K
,”
Phys. Rev. B
16
(
10
),
4546
(
1977
).
28.
S.
Kim
and
H.
Ledbetter
, “
Low-temperature elastic coefficients of polycrystalline indium
,”
Mater. Sci. Eng. A
252
(
1
),
139
143
(
1998
).
29.
J. J.
Adams
et al, “
Elastic constants of monocrystal iron from 3 to 500 K
,”
J. Appl. Phys.
100
(
11
),
113530
(
2006
).
30.
F.
Chu
et al, “
Elastic properties of C40 transition metal disilicides
,”
Acta Mater.
44
(
8
),
3035
3048
(
1996
).
31.
A.
Haglund
et al, “
Polycrystalline elastic moduli of a high-entropy alloy at cryogenic temperatures
,”
Intermetallics
58
,
62
64
(
2015
).
32.
Y.
He
et al, “
Elastic constants and thermal expansion of single crystal γ-TiAl from 300 to 750 K
,”
Mater. Sci. Eng. A
239
,
157
163
(
1997
).
33.
D.
Isaak
et al, “
Elasticity of TiO2 rutile to 1800K
,”
Phys. Chem. Miner.
26
(
1
),
31
43
(
1998
).
34.
T.
Sonehara
et al, “
Temperature dependence of the Brillouin frequency shift in crystals
,”
J. Appl. Phys.
101
(
10
),
103507
(
2007
).
35.
D. H.
Kang
et al, “
Elastic properties of taurine single crystals studied by Brillouin spectroscopy
,”
Int. J. Mol. Sci.
22
(
13
),
7116
(
2021
).
36.
Y.
Varshni
, “
Temperature dependence of the elastic constants
,”
Phys. Rev. B
2
(
10
),
3952
(
1970
).
37.
S. C.
Lakkad
, “
Temperature dependence of the elastic constants
,”
J. Appl. Phys.
42
(
11
),
4277
4281
(
1971
).
38.
H.
Siethoff
and
K.
Ahlborn
, “
The dependence of the Debye temperature on the elastic constants
,”
Phys. Status Solidi B
190
(
1
),
179
191
(
1995
).
39.
P.
Macedo
,
W.
Capps
, and
J.
Wachtman
, Jr.
, “
Elastic constants of single crystal ThO2 at 25 °C
,”
J. Am. Ceram. Soc.
47
(
12
),
651
(
1964
).
40.
K.
Clausen
et al, “
Inelastic neutron scattering investigation of the lattice dynamics of ThO2 and CeO2
,”
J. Chem. Soc., Faraday Trans.
83
(
7
),
1109
1112
(
1987
).
41.
J. B.
Watchman
, Jr.
et al, “
Exponential temperature dependence of Young's modulus for several oxides
,”
Phys. Rev.
122
(
6
),
1754
(
1961
).
42.
R.
Wolfe
and
S.
Kaufman
,
Mechanical Properties of Oxide Fuels (LSBR/LWB Development Program)
(
Bettis Atomic Power Lab
,
Pittsburgh, PA
,
1967
).
43.
K. K.
Phani
and
D.
Sanyal
, “
Elastic properties of porous polycrystalline thoria—A relook
,”
J. Eur. Ceram. Soc.
29
(
3
),
385
390
(
2009
).
44.
S.
Spinner
,
L.
Stone
, and
F.
Knudsen
, “
Temperature dependence of the elastic constants of thoria specimens of varying porosity
,”
J. Res. Natl. Bur. Std.
67
(
2
),
93
100
(
1963
).
45.
V. E.
Gusev
and
P.
Ruello
, “
Advances in applications of time-domain Brillouin scattering for nanoscale imaging
,”
Appl. Phys. Rev.
5
(
3
),
031101
(
2018
).
46.
K.
Yu
et al, “
Brillouin oscillations from single Au nanoplate opto-acoustic transducers
,”
ACS Nano
11
(
8
),
8064
8071
(
2017
).
47.
H.
Ledbetter
, “
Sound velocities, elastic constants: Temperature dependence
,”
Mater. Sci. Eng. A
442
(
1–2
),
31
34
(
2006
).
48.
M.
Mann
et al, “
Hydrothermal growth and thermal property characterization of ThO2 single crystals
,”
Cryst. Growth Des.
10
(
5
),
2146
2151
(
2010
).
49.
D.
Hurley
et al, “
Coherent control of gigahertz surface acoustic and bulk phonons using ultrafast optical pulses
,”
Appl. Phys. Lett.
93
(
11
),
113101
(
2008
).
50.
Y.
Wang
et al, “
Nondestructive characterization of polycrystalline 3D microstructure with time-domain Brillouin scattering
,”
Scr. Mater.
166
,
34
38
(
2019
).
51.
Y.
Wang
and
M.
Khafizov
, “
Shear wave generation by mode conversion in picosecond ultrasonics: Impact of grain orientation and material properties
,”
J. Am. Ceram. Soc.
104
(
6
),
2788
2798
(
2021
).
52.
D. G.
Cahill
, “
Analysis of heat flow in layered structures for time-domain thermoreflectance
,”
Rev. Sci. Instrum.
75
(
12
),
5119
5122
(
2004
).
53.
H.
Fujiwara
,
Spectroscopic Ellipsometry Principles and Applications
(
John Wiley & Sons
,
2007
).
54.
M.
Khafizov
et al, “
Subsurface imaging of grain microstructure using picosecond ultrasonics
,”
Acta Mater.
112
,
209
215
(
2016
).
55.
Y.
Wang
et al, “
Imaging grain microstructure in a model ceramic energy material with optically generated coherent acoustic phonons
,”
Nat. Commun.
11
(
1
),
1
8
(
2020
).
56.
V. E.
Gusev
et al, “Theory of time-domain Brillouin scattering for probe light and acoustic beams propagating at an arbitrary relative angle: Application to acousto-optic interaction near material interfaces,” arXiv:2107.05294 (2021).
57.
A.
Mock
et al, “
Band-to-band transitions and critical points in the near-infrared to vacuum ultraviolet dielectric functions of single crystal urania and thoria
,”
Appl. Phys. Lett.
114
(
21
),
211901
(
2019
).
58.
D. B.
Leviton
,
B. J.
Frey
, and
T. J.
Madison
, “
Temperature-dependent refractive index of CaF2 and Infrasil 301
,” in
Proc. SPIE 6692, Cryogenic Optical Systems and Instruments XII
( SPIE, 2007), p. 669204.
59.
R. A.
Heaton
and
C. C.
Lin
, “
Electronic energy-band structure of the calcium fluoride crystal
,”
Phys. Rev. B
22
(
8
),
3629
(
1980
).
60.
D.
Taylor
, “
Thermal expansion data
,”
Trans. J. Br. Ceram. Soc.
83
(
2
),
32
37
(
1984
).
61.
R.
Singh
,
S.
Mitra
, and
C.
Rao
, “
Temperature variations of the elastic constants of CaF2 and SrF2 crystals
,”
Phys. Rev. B
44
(
2
),
838
(
1991
).
62.
O. G.
Brandt
and
C. T.
Walker
, “
Temperature dependence of elastic constants and thermal expansion for UO2
,”
Phys. Rev. Lett.
18
(
1
),
11
(
1967
).
63.
J.
Garber
and
A.
Granato
, “
Theory of the temperature dependence of second-order elastic constants in cubic materials
,”
Phys. Rev. B
11
(
10
),
3990
(
1975
).
64.
O. L.
Anderson
, “
A simplified method for calculating the Debye temperature from elastic constants
,”
J. Phys. Chem. Solids
24
(
7
),
909
917
(
1963
).
65.
D.
Chung
and
W.
Buessem
, “
The elastic anisotropy of crystals
,”
J. Appl. Phys.
38
(
5
),
2010
2012
(
1967
).
66.
P.
Zhang
,
B.-T.
Wang
, and
X.-G.
Zhao
, “
Ground-state properties and high-pressure behavior of plutonium dioxide: Density functional theory calculations
,”
Phys. Rev. B
82
(
14
),
144110
(
2010
).
67.
M.
Ali
and
P.
Nagels
, “
Evaluation of the Debye temperature of thorium dioxide
,”
Phys. Status Solidi B
21
(
1
),
113
116
(
1967
).
68.
D. W.
Osborne
and
E. F.
Westrum
, Jr.
, “
The heat capacity of thorium dioxide from 10 to 305 K: The heat capacity anomalies in uranium dioxide and neptunium dioxide
,”
J. Chem. Phys.
21
(
10
),
1884
1887
(
1953
).
69.
B.
Willis
, “
Neutron diffraction studies of the actinide oxides I. Uranium dioxide and thorium dioxide at room temperature
,”
Proc. R. Soc. London A
274
(
1356
),
122
133
(
1963
).
70.
L. A.
Girifalco
,
Statistical Physics of Materials
(
Wiley-Interscience
,
1973
).
71.
R.
Schwarz
and
J.
Vuorinen
, “
Resonant ultrasound spectroscopy: Applications, current status and limitations
,”
J. Alloys Compd.
310
(
1–2
),
243
250
(
2000
).
72.
S.
Ganesan
and
R.
Srinivasan
, “
Lattice dynamics of calcium fluoride: Part I: Lyddane, sachs, teller formula, diffuse x-ray scattering, and specific heat
,”
Can. J. Phys.
40
(
1
),
74
90
(
1962
).
73.
R.
Srinivasan
, “
Elastic constants of calcium fluoride
,”
Proc. Phys. Soc.
72
(
4
),
566
(
1958
).
74.
M.
Elcombe
and
A.
Pryor
, “
The lattice dynamics of calcium fluoride
,”
J. Phys. C: Solid State Phys.
3
(
3
),
492
(
1970
).
75.
A.
Togo
and
I.
Tanaka
, “
First principles phonon calculations in materials science
,”
Scr. Mater.
108
,
1
5
(
2015
).
76.
A.
Ward
et al, “
Ab initio theory of the lattice thermal conductivity in diamond
,”
Phys. Rev. B
80
(
12
),
125203
(
2009
).
77.
K.
Brugger
and
T.
Fritz
, “
Grüneisen gamma from elastic data
,”
Phys. Rev.
157
(
3
),
524
(
1967
).
78.
C. A.
Dennett
et al, “
The influence of lattice defects, recombination, and clustering on thermal transport in single crystal thorium dioxide
,”
APL Mater.
8
(
11
),
111103
(
2020
).
79.
M.
Jin
et al, “
Assessment of empirical interatomic potential to predict thermal conductivity in ThO2 and UO2
,”
J. Phys.: Condens. Matter
33
(
27
),
275402
(
2021
).
80.
S.
Zhou
et al, “Improving empirical interatomic potentials for predicting thermophysical properties by using an irreducible derivatives approach: The case of thorium dioxide,” arXiv:2204.13685 (2022).
81.
S.
Zhou
et al, “
Capturing the ground state of uranium dioxide from first principles: Crystal distortion, magnetic structure, and phonons
,”
Phys. Rev. B
106
(
12
),
125134
(
2022
).
You do not currently have access to this content.