Phonon spectrum, thermal expansion and heat capacity of UO$_2$ from first-principles

Abstract

We report first-principles calculations of the phonon dispersion spectrum,
thermal expansion, and heat capacity of uranium dioxide. The so-called direct
method, based on the quasiharmonic approximation, is used to calculate the
phonon frequencies within a density functional framework for the electronic
structure. The phonon dispersions calculated at the theoretical equilibrium
volume agree well with experimental dispersions. The computed phonon density of
states (DOS) compare reasonably well with measurement data, as do also the
calculated frequencies of the Raman and infrared active modes including the
LO/TO splitting. To study the pressure dependence of the phonon frequencies we
calculate phonon dispersions for several lattice constants. Our computed phonon
spectra demonstrate the opening of a gap between the optical and acoustic modes
induced by pressure. Taking into account the phonon contribution to the total
free energy of UO$_2$ its thermal expansion coefficient and heat capacity have
been {\it ab initio} computed. Both quantities are in good agreement with
available experimental data for temperatures up to about 500 K.

Full text

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Phonon spectrum, thermal expansion and heat capacity of UO2from first-principles
Y. Yun,1, ∗D. Legut,2,3and P.M. Oppeneer1
1Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden
2Nanotechnology Centre, VSB-Technical University of Ostrava,
17. listopadu 15, CZ-708 33 Ostrava, Czech Republic
3Atomistic Modeling and Design of Materials, University of Leoben, Leoben, Austria
(Dated: October 6, 2011)
We report first-principles calculations of the phonon dispersion spectrum, thermal expansion,
and heat capacity of uranium dioxide. The so-called direct method, based on the quasiharmonic
approximation, is used to calculate the phonon frequencies within a density functional framework for
the electronic structure. The phonon dispersions calculated at the theoretical equilibrium volume
agree well with experimental dispersions. The computed phonon density of states (DOS) compare
reasonably well with measurement data, as do also the calculated frequencies of the Raman and
infrared active modes including the LO/TO splitting. To study the pressure dependence of the
phonon frequencies we calculate phonon dispersions for several lattice constants. Our computed
phonon spectra demonstrate the opening of a gap between the optical and acoustic modes induced
by pressure. Taking into account the phonon contribution to the total free energy of UO2its thermal
expansion coefficient and heat capacity have been ab initio computed. Both quantities are in good
agreement with available experimental data for temperatures up to about 500K.
PACS numbers: 63.20.dk, 65.40.Ba, 65.40.De
I. INTRODUCTION
Over the last few decades UO2has been one of most
widely studied actinide oxides due to its technological
importance as standard fuel material used in nuclear re-
actors. There exists currently considerable interest in un-
derstanding the behavior of nuclear fuel in reactors which
is a complex phenomenon, influenced by a large num-
ber of materials’ properties, such as thermomechanical
strength, chemical stability, microstructure, and defects.
Especially, knowledge of the fuel’s thermodynamic prop-
erties, such as specific heat, thermal expansion, and ther-
mal conductivity, is essential to evaluate the fuel’s per-
formance in nuclear reactors.1–5These thermodynamic
quantities are directly related to the lattice dynamics of
the fuel material.6–8
Dolling et al.9were the first to measure phonon disper-
sion curves of UO2, using the inelastic neutron scattering
technique in 1965; their seminal article has become the
standard reference for uranium dioxide’s phonon spec-
trum. Later the vibrational properties of UO2were in-
vestigated in detail by Schoenes,10using infrared and Ra-
man spectroscopic techniques. A good agreement with
phonon frequencies obtained from inelastic neutron scat-
tering was observed.10More recently, Livneh and Sterer11
studied the influence of pressure on the Raman scattering
in UO2and Livneh12demonstrated the resonant coupling
between longitudinal optical (LO) phonons and U4+crys-
tal field excitations in a Raman spectroscopic investiga-
tion. A theoretical investigation of the phonon spectra of
UO2was reported recently by Yin and Savrasov13who
employed a combination of a density-functional-theory
(DFT) based technique and a many-body approach. Ac-
cording to their results, the low thermal conductivity
of UO2 stems from the large anharmonicity of the LO
modes resulting in no contribution from these modes in
the heat transfer. Goel et al.14,15investigated the phonon
properties of UO2 using an empirical interatomic po-
tential based on the shell model and observed that the
calculated thermodynamic properties including the spe-
cific heat are in good agreement with available exper-
imental data. Devey16employed recently the general-
ized gradient approximation with additional Coulomb U
(GGA+U) to compute the main phonon mode frequen-
cies at the Brillouin zone center which were in reason-
able agreement with experimental data. Very recently,
Sanati et al.17used the GGA and GGA+U approaches
to investigate phonon density of states and elastic and
thermal constants, which were found to be in reason-
ably good agreement with experimental data. In spite of
the already performed studies, further investigations are
needed. Especially, the full dispersions of the phonons
in reciprocal space have not yet been considered. Also,
important quantities such as the thermal expansion coef-
ficient and heat capacity are directly related to the lattice
vibrations but these quantities have not yet been studied
ab initio from the calculated phonon spectrum.
The objective of this study is to contribute to a detailed
understanding of the lattice vibrations of UO2. Using
the first-principles approach, based on the DFT we have
calculated phonon dispersion curves and phonon density
of states of UO2. The calculated phonon properties are
compared with the available experimental data from in-
elastic neutron scattering and Raman spectroscopy along
with a detailed discussion. Furthermore, several thermo-
dynamic properties have been computed taking the influ-
ence of lattice vibrations into account. Here, we report
the lattice contribution to the heat capacity as function
of temperature as well as temperature and volume (in the
quasiharmonic approximation). The dependence of the
arXiv:1110.0984v1 [cond-mat.mtrl-sci] 5 Oct 2011

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total free energy on the lattice constant of UO2as a func-
tion of temperature has calculated, from which we derive
the thermal expansion coefficient. The thermal expan-
sion coefficient as well as lattice heat capacity compare
favorably to available experimental data up to 500 K,
which is the temperature range in which the influence of
anharmonicity can be neglected.
II.COMPUTATIONAL METHODOLOGY
The electronic structure of UO2has been discussed in
the past years.18–27DFT calculations within the general-
ized gradient approximation (GGA) underestimate the
influence of the strong on-site Coulomb repulsion be-
tween the 5f electrons. An improved 5f electronic struc-
ture description can be obtained with the GGA+U ap-
proach, in which a supplementary on-site Coulomb repul-
sion term is added; this approach correctly gives the elec-
tronic band gap of UO2.18,22,28–30While the GGA+U ap-
proach would appear preferable for description of UO2’s
electronic structure, we encountered specific problems
when using this method. Some of the phonon branches
became negative away from the zone center. This artifact
might be related to the occupation matrix of 5f states
that would require an additional stabilizing constraint
in the GGA+U method.24,25Using conversely the spin-
polarized GGA approach, we found that such difficul-
ties did not occur. The phonon dispersion spectrum pre-
sented below is hence computed with the GGA exchange-
correlation for antiferromagnetically ordered UO2and is
found to be in good agreement with experiment.9
Here, we have determined the phonon dispersion
curves and density of states (DOS) in the quasiharmonic
approximation using the direct method.31,32By displac-
ing one atom in a supercell (of 96 atoms) from its equilib-
rium position, non-vanishing Hellmann-Feynman forces
were generated. Due to the high symmetry of the face-
centered cubic (fcc) lattice of UO2, only one atom for
uranium (U) and for oxygen (O) was needed to be dis-
placed. The actual shift of the atoms in the supercell had
an amplitude of 0.03˚ A and was taken along the [001] di-
rection only, on account of the cubic symmetry of UO2.
In the calculation of the resulting forces we employed
the projector augmented wave (PAW) pseudopotential
approach within the Vienna Ab-initio Simulation Pack-
age (VASP).33,34The PHONON code31,32has been used
to extract the force constant matrix from the Hellmann-
Feynman forces and to subsequently calculate the phonon
dispersion curves and DOS.
For the thermodynamic quantities we consider the to-
tal free energy of UO2, including the phonon contribu-
tion,
F(?,T) = U(?) + Fphon(?,T) + Fel(?,T),(1)
where F(?,T) is the Helmholtz free energy at a given
strain ?. The phonon free energy contribution Fphonis
expressed as
Fphon(?,T) =
?∞
+kBT ln(1 − e−¯ hω/kBT)?,
0
dω g(ω,?)?¯ hω/2
(2)
where g(ω,?) is the phonon DOS, computed as mentioned
above. We note that the free electronic energy, Fel(?,T),
is not considered in the present study, because the ther-
mal electronic contribution is known to be negligible in
the temperature range up to 1000K, which is the range of
interest in this work.35,36The static lattice energy U(?)
appearing in Eq. (1) can be expressed as
U(?) = U0+ V
?
ij
Cij?i?j,(3)
where U0is the static lattice energy at zero strain, Cij
are the elastic constants, and V is the equilibrium volume
at T = 0 K. The static lattice energies have also been
calculated using the VASP code.33,34
In our calculations we have used a 2×2×2 su-
percell containing 96 atoms with a 4×4×4 k-point
mesh in the Brillouin zone (BZ). The Perdew-Wang
parametrization37of the GGA functional was used. The
kinetic energy cut-off for the plane waves was set at 600
eV and the energy criterion used for convergence was
10−7eV. The force acting on each ion was converged until
less than 0.01 eV/˚ A. Once the phonon DOS has been cal-
culated, the thermal expansion of UO2can be evaluated
straightforwardly. First, the phonon DOS with static lat-
tice energy is calculated for several volumes around the
T = 0 K equilibrium volume. Subsequently, the total
free energies are calculated for these different volumes at
constant temperature using Eqs. (1)-(3). After the free
energy has been calculated its minimum gives the corre-
sponding equilibrium volume at the considered tempera-
ture. By repeating the process for different temperatures,
the thermal expansion coefficient α defined by
α(T) ≡1
a
da
dT=
1
3V
dV
dT,
(4)
is obtained; here a is the lattice constant.
A further thermodynamic quantity, the lattice contri-
bution to the specific heat can be derived fromδF(V,T)
a fixed temperature in the quasiharmonic approximation.
δT
at
III.RESULTS AND DISCUSSION
A.Ground-state properties of UO2
UO2crystallizes in the cubic fluorite structure (CaF2)
belonging to the Fm3m space group (no. 225) and there
are three atoms per primitive unit cell with one U atom
(Wyckoff position 4a) and two inequivalent O atoms
(Wyckoff positions 8c). Therefore, there are generally

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TABLE I: Calculated equilibrium lattice constant a (in˚ A)
and bulk modulus B (in GPa) of UO2. Theoretical values
obtained in this work are compared to values from molecu-
lar dynamics simulations7(MD), as well as to experimental
data39–42(Exp.).
a (˚ A)
5.406
5.4727
5.4739,40
B (GPa)
184
1827
This work
MD (300K)
Exp. (300K) 192,41198,3920740
nine phonon branches. Before turning to the descrip-
tion and analysis of the calculated lattice dynamics let us
briefly consider the ground-state properties of UO2. As
mentioned above the employed DFT framework is that
of the spin-polarized GGA. The calculated equilibrium
lattice constant a and bulk modulus B, which we have
obtained by a Birch-Murnaghan 3rdorder fit,38are pre-
sented in Table I, where these are compared to experi-
mental lattice properties.39–42Our calculated equilibrium
volume as well as the bulk modulus compare reasonably
well with the experimental data and with results from
molecular dynamics simulations.7,8Compared to the ex-
perimental lattice constant the lattice constant computed
here is 1.2% smaller. This can be attributed to a too
strong binding of the 5f orbitals which become too much
delocalized in the spin-polarized-GGA approach. In the
GGA+U approach the 5f orbitals are more localized and
their contribution to the bonding reduced,18,21,24which
leads to a theoretical lattice parameter which is larger
than the experimental one.27
B.Phonon spectrum of UO2
Figure 1 shows the measured and ab initio calculated
phonon dispersion curves of UO2 along high-symmetry
points of the fcc BZ. The black lines are the calculated
phonon dispersions obtained for the different lattice pa-
rameters, the blue open symbols are the experimentally
measured data from inelastic neutron scattering,9and
red closed symbols are data from Raman scattering.12
Note that M denotes the additional point (1, 1, 0) that
was included by Dolling et al.,9but that is outside of
the first BZ. The calculated phonon dispersions along
the high-symmetry lines X-W-Γ have been added to il-
lustrate the whole dispersion of phonon states in the BZ.
The three branches in the low frequency region are the
transverse acoustic (TA) and longitudinal acoustic (LA)
modes that belong to vibrations of the U atom with its
relatively heavy mass. The other branches are optical
modes with higher frequencies that are mainly associated
with lattice vibrations of O atoms and can be labeled as
TO1, LO1 and TO2 and LO2, respectively (see Fig. 1), as
there are two inequivalent O atoms in the unit cell. The
three panels in Figure 1 illustrate the volume dependence
of the phonon frequencies, which have been calculated at
0
5
10
15
20
0
5
10
15
20
Frequency (THz)
0
5
10
15
20
M
Γ
W
Γ
LX
a=5.470 Å
a=5.406 Å
a=5.330 Å
TA
LA
LO2
TO2
LO1
TO1
FIG. 1: (Color online) Calculated (full lines) and measured
phonon dispersion curves (open symbols, from Ref. 9) along
high-symmetry directions in the fcc Brillouin zone. The mea-
sured Raman and infrared modes (Ref. 12) at the Γ point are
depicted by solid (red) circles. The notation of the special
points is M: (1,1,0), Γ: (0,0,0), X: (1,0,0), W: (1,
and L: (1
1
2, 0),
2,1
2,1
2).
three different lattice constants, the experimental one,
a = 5.47˚ A, the optimized theoretical one, a = 5.406˚ A,
and a = 5.33˚ A, a selected smaller lattice constant.
In the surrounding of the Γ point the agreement be-
tween the calculated and measured phonon frequencies
is very good.A first discrepancy between measured
and computed dispersions is observed along the M−Γ
high-symmetry direction. Our calculations predict three
acoustic branches whereas in experiment there are two
branches. It could be that a splitting of the low-lying
TA branch near the M point could not be sufficiently
resolved in the experiment.
the inelastic neutron scattering technique, which might
be affected by the size of samples and as well as a rela-
tively low neutron flux (see, e.g., Refs. 43 and 44). Also,
as mentioned by Dolling et al. the frequency measure-
ments of the phonons with the neutron technique might
be impeded in the zone boundary regions. The LO and
TO branches agree reasonably well with experiment, in
particular for the optimized theoretical lattice parame-
ter. There are some discrepancies in the positions of the
branches at the zone boundary X and at the M point.
One of the TO1 branches turning up from the Γ point
This might be related to

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to the M point has not been detected in the experiment.
Along Γ−L the agreement is quite good. The top-most
(LO2) branch deviates most between calculation and the
experiment. This high-lying branch has the largest ex-
perimental uncertainty. Nonetheless, the ab initio cal-
culated branch has more dispersion than present in the
measurement and, except for the zone center, it falls out-
side of the experimental error bar.
The phonon frequencies calculated at a = 5.406˚ A and
a = 5.47˚ A are very similar to each other and agree rea-
sonably well with the measurement data. A notable dif-
ference between the two sets of dispersion curves appears
however in the frequency gap at the zone boundaries of
M and X. When the lattice constant decreases with pres-
sure from a = 5.47˚ A to a = 5.33˚ A a pressure-induced
phonon softening occurs.The frequency gap between
LA and TO1 modes at the zone boundary X point and
at the M point is increased as the pressure increases. At
a = 5.47˚ A the LA and TO1 modes almost approach
each other, whereas at a = 5.406˚ A, a gap is predicted to
exist between the LA and TO1 modes. A small or van-
ishing gap between these modes is in accordance with the
measurement. We also note that with increased lattice
constant the negative slope of the LO1 branch is remark-
ably increased along the Γ−X symmetry line. This find-
ing suggests that the propagation of the LO1 phonon is
significantly restrained as the lattice constant increases.
In Fig. 1 we furthermore depicted by the red solid
circles at the Γ point the frequencies obtained by Ra-
man and infrared measurements.10,12Using group the-
ory analysis these active Raman and infrared modes can
be decomposed into irreducible representations of the
(O5
h) point group, as 1T2g+ 2T1u. The U atom con-
tributes only to the infrared active mode (T1u), whereas
the O atom contributes to both, the infrared and Raman
mode, T1uand T2g, respectively. Both these modes are
triply degenerate. The frequencies of these modes are
summarized in Table II, together with results from MD
simulations15and experimental results.9,10,12To account
for the LO/TO splitting the Lyddane-Sachs-Teller45re-
lation was used on the basis of effective charges. These
were chosen to be 3 for the uranium atom and −1.5 for
oxygen. These values are consistent with values used in
the literature.14The macroscopic electric field in UO2
splits the infrared-active optical modes into TO and LO
components. Frequencies of TO modes are calculated in
a straightforward manner within the direct method but
the LO modes can only be obtained via introduction of
a non-analytical term46into the dynamical matrix. In
general, this term depends on the Born effective charge
tensor and the electronic part of the dielectric function
(high-frequency dielectric constant). Details regarding
this procedure can be found elsewhere, see e.g. Ref. 48.
The agreement of the computed Raman optical mode
as well as the infrared TO mode with available experi-
mental data is quite good, see Table II. The calculated
frequency of the infrared LO mode is somewhat smaller
than the experimental results, indicating that the effec-
TABLE II: The frequencies of the Raman (T2g) and infra-red
(T1u) vibrational modes of UO2 at the Γ-point. Theoretical
results (this work) are given for three lattice constants a and
compared to frequencies obtained from molecular dynamics
simulation15(MD) as well as experimental (exp.) frequencies
(given in THz).
Mode
a (˚ A)
T1u(TO)
this work
5.406
8.49
MD15
5.47
7.62
exp.
5.475.33
9.27
5.47
7.78 8.3447, 8.410
8.529
16.6847, 16.79
17.3410, 17.412
12.9347, 13.429
T1u(LO)16.5 15.8215.2617.28
T2g
14.22 13.412.8014.04
tive charge and dielectric constant taken from literature
do not fully account for the correct
dition there is a spread of ca. 0.7 THz in the measured
values of the LO infrared mode frequencies.10,47This is,
however, not sufficient to account for the underestima-
tion of the experimental values by about 2 THz that is
observed for a = 5.470˚ A (upper panel of Fig. 1). Using
the calculated theoretical volume (middle panel of Fig.
1) this deviation is reduced by 25%. A similar error of ca.
1.5 THz for the ab initio calculated infrared frequencies
of another insulating compound, CsNiF3, was reported
in a recent study.49
Figure 2 shows the calculated and measured phonon
DOS. The blue, red, and green lines with square, tri-
angle, and diamond symbols indicate the phonon DOS
computed for a=5.330˚ A, 5.406˚ A, and 5.470˚ A, respec-
tively. The experimental data9(a ≈ 5.47˚ A, T=296K)
are plotted with the black line. The U contribution to
the calculated phonon DOSs gives rise to a higher in-
tensity and narrower peak widths in the lower frequency
ε0
ε∞=
ω2
ω2
LO
TO. In ad-
05101520
Frequency (THz)
0
0.1
0.2
0.3
0.4
0.5
Density of states (THz-1)
a = 5.330 Å
a = 5.406 Å
a = 5.470 Å
experiment
FIG. 2: (Color online) The theoretical phonon density of
states (DOS) of UO2 computed for three lattice constants
a = 5.330, 5.406, and 5.406˚ A, compared to the measured9
DOS (for a ≈ 5.47˚ A, at T=296 K).

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region. The more broadened DOS with lower intensity
that occurs in the higher frequency region is mainly de-
rived from the oxygen atoms. A notable difference be-
tween the phonon DOS at the three lattice parameters
is the size and position of the phonon gap occurring for
frequencies of about 6 − 7 THz. For the experimental
lattice parameter a = 5.47˚ A the computed gap practi-
cally closes. The experimental phonon DOS spectrum at
this lattice parameter shows a minimum at about 6 THz,
in reasonable agreement, considering some experimental
broadening. We note that the trend of decreasing gap
with larger lattice constant continues, leading to a clos-
ing of the gap computed for larger lattice constants (not
shown). Overall, the computed phonons DOS of both
the theoretical equilibrium (a = 5.406˚ A) and the experi-
mental lattice parameter are in good accordance with the
measured spectrum. The phonon DOS of the theoretical
lattice parameter agrees best with the experimental data
at higher frequencies (7 to 13 THz), where the peaks co-
incide with the measured ones. As mentioned earlier, the
LO2 mode lies both lower in the computed spectra and
is more dispersive than in the measurements. We note
that the recent GGA+U calculations17provide a sharper
DOS peak at 17 THz, due to a flatter LO2 dispersion near
the zone boundaries. At the zone center the LO2 branch
lies however much deeper than in the experiment, at 10
THz (vs. 17 THz in experiment).
C. Thermal expansion of UO2
The calculated phonon DOS enables us to evaluate
some thermodynamic quantities which depend on the lat-
tice vibrations. We start with the thermal expansion.
The phonon contribution to the total free energy of UO2
increases with increasing temperature and hence becomes
progressively responsible for changes of the lattice pa-
rameters. To compute the thermal expansion of UO2
we have first computed the total free energy, including
the phonon contribution, for various lattice parameters,
from which we computed the temperature-dependent lat-
tice constant. Figure 3 (bottom) shows the calculated
variation of the lattice constant with temperature. The
red curve gives the spline interpolation of the calculated
lattice constants, shown by the symbols. The thermal ex-
pansion coefficient α(T) = a−1da/dT was subsequently
evaluated by differentiating the spline fit.
panel of Fig. 3 shows the calculated thermal expansion
coefficient, which is in good agreement with experimental
data50up to 500 K. The deviation between the calculated
and measured data slightly increases above 500 K and be-
comes significant at around 1000 K. This might be due
to an increased electronic contribution to the thermal ex-
pansion. At very low temperatures, in the region between
0 and 50 K, a deviation is also observed between the cal-
culated and measured data. The origin of this deviation
is not unambiguously clear. We note however that UO2
undergoes a magnetic phase transition at 31 K (see Ref.
The upper
0.0
5.0×10-6
1.0×10-5
1.5×10-5
2.0×10-5
α (1/K)
Experimental data
Calculated results
0200 400600 800
Temperature (K)
5.42
5.44
5.46
Lattice parameter (Å)
Calculated equilibrium lattice constant
Splines fit
UO2
FIG. 3: (Color online) Top: calculated and experimental ther-
mal expansion coefficient α(T) of UO2.
data are those of Taylor.50Bottom: Computed temperature-
dependent lattice parameter of UO2(open squares) and spline
fit function.
The experimental
51), which may add an additional influence on the lattice
parameter.
D. Thermodynamic properties of UO2
Next, we employ the Helmholtz free energy to compute
the lattice heat capacity at constant volume (CV) and at
constant pressure (CP). Fig. 4 shows theoretical results
for CV, computed with the harmonic approximation, and
CP, computed with the quasiharmonic approximation,
as well as experimental results52,54for CP up to 1000 K.
First the lattice contribution to the specific heat CV was
computed as a function of temperature for the equilib-
rium lattice constant, a = 5.406˚ A. Subsequently, the
specific heat at constant pressure was derived according
to
CP= CV+ 9α2Ba3T, (5)
where α(T), a, and B are the calculated linear thermal
expansion coefficient, the equilibrium lattice constant,
and the bulk modulus (see, e.g., Ref. 55).
lustrates that there is a very good agreement between
the computed heat capacity CP and the experimental
data.52–54Note that the sharp anomaly in the experi-
mental data at T ≈ 31 K is due to the aforementioned
magnetic phase transition,51which effect is not included
in the calculations. Clearly, the specific heat at constant
pressure is in much better agreement with the experi-
mental data at higher temperatures than CV, which is
mainly due to the thermal expansion of UO2. Conversely,
evaluating CPfor the theoretical equilibrium lattice con-
stant or for the experimental lattice constant a = 5.47
˚ A only gives very minor differences. The computed CP
curves fall somewhat below the experimental CP data
Fig. 4 il-

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0 100200 300400 500600700 800 9001000
Temperature (K)
0
10
20
30
40
50
60
70
80
90
Heat capacity (J.mol-1.K-1)
CP , exp. Gronvold
CP , exp. Huntzicker
CV (a = 5.406 Å)
CP
CP using exp. a
FIG. 4: (Color online) The lattice contribution to the heat ca-
pacity of UO2 at constant volume, CV, computed within the
harmonic approximation (full curve), and a constant pres-
sure, CP, computed with the quasiharmonic approximation
(dashed lines; for a = 5.406˚ A and the experimental a = 5.47
˚ A). The experimental data for lower and higher temperatures,
full circles and full squares, are taken from Huntzicker and
Westrum (Ref. 52) and Grønvold et al. (Ref. 53), respectively.
for temperatures in the range of 400 to 1000 K. The ura-
nium atoms in UO2 are in the paramagnetic state at
higher temperatures.10,42Therefore, the remaining dif-
ference between experimental and computed data above
T > 400 K could be due to the magnetic entropy con-
tribution to the specific heat or, alternatively, it could
be due to the anharmonic effects.
tropy contribution to the specific heat was investigated
for T = 200 to 300 K in Refs. 51 and 53. At higher
temperatures it saturates to approximately S = R ln3,
corroborating a magnetic J = 1 state on the uranium
atoms. It provides to a relatively small magnetic entropy
contribution that would lead to a small increase of the
computed CP data (by about 3 Jmol−1K−1).
The magnetic en-
IV. CONCLUSIONS
We have performed first-principles calculations to in-
vestigate the lattice vibrations and their contribution
to thermal properties of UO2.
culated phonon dispersions are in good agreement with
experimental dispersions9measured using inelastic neu-
tron scattering. Computing the phonon dispersions for
various lattice constants, we observed a softening of the
phonon frequencies with decreasing lattice constant. Fur-
thermore, the band gap between TA and LO modes at
high-symmetry zone-boundary points are found to de-
pend significantly on the volume. This gap almost closes
at a = 5.47˚ A, consistent with a pseudogap detected in
the inelastic neutron experiment. Also, Raman and in-
frared active modes have been determined as a function of
volume. The agreement with experimental data and with
results obtained with molecular dynamics simulations is
overall very good, with an exception of the infrared LO
mode that appears underestimated in our first-principles
calculations. Including the phonon contribution to the
free energy, the heat capacity and the thermal expansion
coefficient of UO2 have been computed. Both thermal
quantities are found to agree well with experimental data
for temperatures up to 500 K. The good correspondence
of the computed and measured thermal data exemplifies
the feasibility of performing first-principles modeling of
the thermal properties of the important nuclear fuel ma-
terial UO2.
We find that the cal-
Acknowledgments
This work has been supported by Svensk K¨ arnbr¨ ansle-
hantering AB (SKB), the Swedish Research Coun-
cil (VR), the Swedish National Infrastructure for
Computing(SNIC),and
CZ.1.07/2.3.00/20.0074 of the Ministry of Education of
the Czech Republic. D. L. thanks P. Pavone, University
of Leoben, Leoben, Austria, and U. D. Wdowik, Pedagog-
ical University, Cracow, Poland for fruitful discussions.
by researchproject No.
∗Present address: Laboratory of Reactor Physics ad Sys-
tems Behaviour, Paul Scherrer Institut, CH-5232 Villigen
PSI, Switzerland
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