investigation of thermodynamic properties of cerium dioxide by statistical moment method

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Investigation of thermodynamic properties of cerium

dioxide by statistical moment method

Vu Van Hung

a

, Jaichan Lee

a

, K. Masuda-Jindo

b,

*

a_{Department of Materials Science and Engineering, Center for Advanced Plasma Surface Technology,}

Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Suwon, 440-746, South Korea
b

Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 226-8503, Japan
Received 21 April 2005; received in revised form 6 September 2005; accepted 16 September 2005

Abstract

The thermodynamic properties of the cerium dioxide (CeO2) are studied using the statistical moment method, including the anharmonicity

effects of thermal lattice vibrations. The free energy, linear thermal expansion coefficient, bulk modulus, specific heats at the constant volume and
those at the constant pressure, CVand CP, are derived in closed analytic forms in terms of the power moments of the atomic displacements. The

temperature dependence of the thermodynamic quantities of cerium dioxide is calculated using three different interatomic potentials.
The influence of dipole polarization effects on the thermodynamic properties and thermodynamic stability of cerium dioxide have been studied
in detail.

q2005 Elsevier Ltd. All rights reserved.

Keywords: A. Inorganic compounds; D. Anharmonicity; D. Thermodynamic properties

1. Introduction

Cerium dioxide (CeO2) has been the subject of the recent

studies in catalysis, fuel cell electrolyte materials, gas sensors,
optical coatings, high-Tc superconductor structures, high

storage capacity devices, and other related applications[1–6].
For instance, owing to the remarkable redox properties
ceria-based mixed oxides are active components of three-way
automotive catalysts (TWC) [7,8], in which CeO2 play an

important role for enhancing the removal of carbon monoxide
(CO), hydrocarbon (HC) and nitrogen oxide (NOx) pollutants.

Ceria is able to store oxygen under lean conditions and to release
it when the O2concentration in the gas phase becomes virtually

nil. A large number of experimental and theoretical studies have
been carried out on catalytic[10,11], lattice vibrational[12,13],
and structural [14–16] properties of cerium dioxides.
Theo-retical study on the structure, stability and morphology of
stoichiometric ceria crystallites has been done using the
simulation method[9]. Recently, Tsunekawa et al.[14]have

shown that the anomalous lattice expansion in the monodisperse
ceria (CeO2Kx) nanoparticles are caused by the valence

reduction of Ce ions, i.e. decrease of the electrostatic forces.
The reduction of the valance induces an increase in the lattice
constant due to the decrease in electrostatic forces, but the
increase in the valence does not always lead to lattice shrinkage.
This behavior is in contrast to a decrease of the lattice constant
often observed in the metal nanoparticles with decreasing

particle size. In this respect, it is of great importance to take into
account the electrostatic Coulomb interactions between the
constituent ions in the calculations of stabilities and
thermo-dynamic quantities of CeO2.

Most of the previous theoretical studies, however, are
concerned with the materials properties of cerium dioxides at
absolute zero temperature, and temperature dependence of the
thermodynamic quantities has not been studied in detail. The
purpose of the present article is to investigate the temperature
dependence of the thermodynamic properties of cerium dioxide
using the analytic statistical moment method (SMM)[17–19].
The thermodynamic quantities are derived from the Helmholtz
free energy, and the explicit expressions of the thermal lattice
expansion coefficient, unit cell volume, specific heats at
constant volume and those at the constant pressure CV and

CP, and elastic modulus are presented taking into account

the anharmonicity effects of the thermal lattice vibrations.
In the present study, the influence of dipole polarization effects

www.elsevier.com/locate/jpcs

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jpcs.2005.09.100

* Corresponding author. Permanent address: Dept. of Physics, Hanoi
National Petagogic University, Hanoi, Vietnam.

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on the thermodynamic properties have also been studied, using
three different interatomic potentials. Each set of potentials is
based on a fully ionic description of the fluorite (CaF2) lattice.

We will compare the results of the present calculations with
those of the previous theoretical calculations as well as with the
available experimental results.

2. Theory

Let us consider a quantum system given by the following
Hamiltonian:

^

H Z ^H0K
X

i

aiV^i; (1)

where ^H0 denotes the lattice Hamiltonian in the harmonic
approximation, and the second term is added due to the
anharmonicity of thermal lattice vibrations. ai denotes a

parameter characterizing the anharmonicity of lattice vibrations
and ^Vi the related operator. The Helmholtz free energy of the
system given by Hamiltonian (1) is formally written as
j Z j0K

X
i

ð
a_i

0

h ^V_iiadai; (2a)

with
h ^ViiaZ

Kvj
vai

; (2b)

where h ^Viia expresses the expectation value at the thermal
equilibrium with the (anharmonic) Hamiltonian ^H. The
expectation values in the above Eq. (2a) and (2b) are evaluated
with the use of the density matrix formalism

h ^ViaZ Tr½ ^r ^V
; (3)

where the density matrix ^_{r is defined, with qZk}BT, as

^

r Z exp jK ^H
q



: (4)

In the above Eq. (2a), j denotes the free energy of the system at
temperature T, and j0is the free energy of the harmonic lattice

corresponding to the Hamiltonian ^H0.

We will present the SMM formulation for the oxide material
with fluorite (CaF2) structure like CeO2, as schematically

shown inFig. 1. The concentrations of cerium and oxygen ions
are simply denoted by CCeZ(N1/N), and COZ(N2/N),

respectively. The free energy of cerium dioxide are then
written by taking into account the configurational entropies Sc,

via the Boltzmann relation as

J Z CCeJCeCCOJOKTSc; (5)

where JCeand JOdenote the free energy of Ce and O ions,

respectively. In the fluorite (CaF2) structure the Ce
4C

ions
occupy an fcc sublattice and O2Kions occupy the tetrahedral
interstitial sites forming a simple cubic sublattice of length
a0/2. However, the present thermodynamic calculations will be

done, assuming the completely ordered phase of CeO2, and the

thermodynamic properties and the phase stability of
non-stoichiometric CeO2 will be discussed in a forthcoming

publication.

Firstly, we expand the potential energy of the system in
terms of the atomic (ionic) displacements Uiof the atom i

U Z UCeCUOZN1
2

X
i

4CeioðjriCuijÞ

C
N2

2
X

i

4O_ioðjriCuijÞ Z
N1

2
X

i

4Ce_ioðjrijÞ


C
1
2

X
a_;b

v24Ceio
vuiavuib

 

eq
uiauib

C
1
6

X
a;b;g

v3₄Ce
io
vuiavuibvuig

 

eq

uiauibuig

C
1
24

X
a;b;g;h

v4₄Ce
io
vuiavuibvuigvuih

 

eq

uiauibuiguihC/

)

C
N2

2
X

i

4O_ioðjrijÞ C
1
2

X
a;b

v24O_io
vuiavuib

 

eq
uiauib
(

C
1
6

X
a;b;g

v34Oio
vuiavuibvuig

 

eq

uiauibuig

C
1
24

X
a_;b;g;h

v44Oio
vuiavuibvuigvuih

 

eq

uiauibuiguihC/
)

;

(6)

where riis the equilibrium position of the ith atom, uiadenotes

a-Cartesian component of the atomic displacement of ith atom,
and 4Ceio (or 4Oio) the effective interaction energy between the
zeroth and ith atoms, respectively. It should be reminded,
however, that the expansion of the above Eq. (6) is done for
deriving the criterion of the thermodynamical equilibrium of
the system, and in Eq. (6), the subscript eq means the quantities
calculated at the equilibrium state. Using Eq. (6), the thermal
average of the crystalline potential energy of the system is
given in terms of the power moments huni of the atomic
displacements and the harmonic vibrational parameter k,

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and three anharmonic parameters b, g1, and g2as

hUi Z UCe0 CU
O
0 C3N1

kCe
2 hu

2
Cei Cg

Ce
1 hu

4
Cei Cg

Ce
2 hu
2
Cei
2
 
C3N2
kO
2 hu
2

Oi CbohuOihu
2
Oi Cg

O
1hu

4
Oi Cg

O
2hu
2
Oi
2
 

C/
(7)
where

kCeZ1
2

X
i

v24Ceio
vu2

ia

 

eq

; kOZ1

2
X

i

v24Oio
vuO

ia

 

eq

(8a)

b0Z
1
2

X
i

v34Oio
vu_iavu_ibvu_ig

 

eq

; (8b)

gCe₁ _Z 1
48

X
i

v44Ce_io

vu4

ia

 

eq

; gO₁_Z 1
48

X
i

v44O_io
vu4

ia

 

eq

gCe2 Z
6
48

X
i

v4₄Ce
io
vu2

iavu2ib
!

eq
;

gO₂ _Z 6
48

X
i

v4₄O
io
vu2

iavu2ib
!

eq

(8c)

with asbZx,y, or z. U0Ce and U0O represent the sum of
effective pair interaction energies for Ce ion and Oxgen ion,
respectively,

U0CeZ
N1

2
X

i

4CeioðjrijÞ; U
O
0 Z
N2
2
X
i

4OioðjrijÞ: (9)

The Helmholtz free energy JCe for Ce ions can be derived

from the functional form of the potential energy of the above
Eq. (7) through the straightforward analytic integrations I1and

I2, with respect to the two anharmonicity ‘variables’ g1and g2.

Firstly, for Ce ions I1and I2are written in an integral form as

I₁_Z
ð

gCe
2
0
hu2
Cei
2_j
gCe
1Z0dg

Ce

2 ; I2Z
ð
gCe
1
0
hu4
Ceidg
Ce

1 : (10)

Using moment expansion formulas[17–19], one can find the
low-order moments hu2i and hu4_{i (see Appendix), and the final}

expression of the partial Helmholtz free energy JCeof Ce ion

sites in cerium dioxide is given by
J_Ce_{Z U}₀CeC3N1qẵx C ln1Ke

K2x_ị

C3N1
q2
k2

Ce
gCe2 x

2

coth2xK2g
Ce
1
3 a1



C
2q2
k4
Ce
4
3ðg
Ce
2 Þ
2

a1x coth xK2ððg
Ce

1 Þ

2

C2gCe1 gCe2 Þa1ð2a1K1Þ


;

(11)

where

a1Z 1 C xcoth
x

2; x Z

Zu_Ce

2q h Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kCe=mCe
p

2q : (12)

Similar derivation can be also done for finding the partial

Helmholtz free energy JO of oxygen ions by performing

analytic integrations I1, I2and I3as

I1Z
ð
gO

2

0
hu2

Oi2jbOZ0;gO1Z0dg

O

2; I2Z
ð
gO

1

0
hu4

OijbOZ0dg
O
1; and

I3Z
ð
bO

0
huOihu

2

OidbO: ð13Þ

The final expression of the partial Helmholtz free energy JOof

the oxygen ions is given by
JOZ UO0 C3N2qẵx C ln1Ke

K2x
ị

C3N₂ q

2
k2
O

gO₂x2coth2xK2g
O
1
3 a1



C
2q2
k4
O
4
3ðg
O

2Þ2a1xcothxK2ððgO1Þ2C2gO1gO2Þa1ð2a1K1Þ


Cq
bO
6KgO
kO
KK1


C
q2bO

K

2gO
3K3a1

 1=2

K

bOa1

9K2
"

C
bOkOa1

9K3 C

bO
6KkO

ðx coth xK1Þ


(14)
where

x ZZuO
2q h Z

ffiffiffiffiffiffiffiffiffiffiffiffiffi
kO=mO
p

2q ; (15a)

gOZ 4ðgO1CgO2Þ; K Z
kOKb2O

3gO : (15b)

In the above Eqs. (11) and (14), the harmonic contributions to
the Helmholtz free energies JCeand JOare derived using the

‘Einstein’ approximation. Therefore, for more quantitative
thermodynamic calculations at the low temperatures, one can
use the lattice dynamical model[20]by evaluating the matrix
elements of the dynamical matrix, Dab(q), in other words the

Fourier transform of the force constants between the
neighbor-ing atomic sites. The diagonal matrix elements (on-site energies
in the language of TB electronic theories[21]) are given by the
‘harmonic force constants’, and related to the maximum
frequencies wu2max=2. Therefore, the present formulation
gives the exact treatments in the moments expansion of all
orders coming from the harmonic phonon Hamiltonian. The
anharmonic contributions to the free energies are treated within
the fourth order moment approximation.

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the partial free energies JCe and JO, given by Eqs. (11) and

(14), respectively. Then, the specific heat at constant volume of
the cerium dioxide is given by

CVZ CCeC
Ce
V CCOC

O

V: (16)

We assume that the average nearest-neighbor distance of
cerium dioxide at temperature T can be written as

r1ðTÞ Z r1ð0Þ C CCey
Ce

0 ðTÞ C COy
O

0ðTÞ; (17)

in which yCe0 ðTÞ and yO0ðTÞare the atomic displacements of Ce
and O ions from the equilibrium position in the fluorite lattice

[22–24], and r1(0) is the distance r1at zero temperature. The

average nearest-neighbor distance at TZ0 K can be
deter-mined from the minimum condition of the potential energy of
the system composed of N1Ce ions and N2Oxygen ions:

vU
vr1
ZvU
Ce
0
vr1

C
vU0O

vr1
ZN1

2
v
vr1

X
i

4CeioðjrijÞ
!
C
N2
2
v
vr1
X
i

4O_ioðjrijÞ
!

Z 0:

(18)

From the definition of the linear thermal expansion coefficient,
it is easy to derive the formula

aTZ CCea
Ce
T CCOa

O

T; (19)

where
aCeT Z

kB
r1ð0Þ

vyCe₀

vq ; a

O
T Z

kB
r1ð0Þ

vyO₀

vq (20)

The bulk modulus of cerium dioxide is derived from the free
energy of Eq. (5) as

BTZKV0
vP
vV

 

T

ZKV0

v2J
vV2

 

T

Z CCeB
Ce
T CCOB

O
T (21)

where P denotes the pressure, V0 the lattice volume at zero

temperature, and the bulk moduli BCeT and BOT are given by
BCeT ZK

kB
3aCe

T

v2JCe
vVvq

 

; BOT ZK

kB
3aO

T

v2JO
vVvq

 

: (22)

Due to the anharmonicity of thermal lattice vibrations, the heat
capacity at a constant pressure, CP, is different from the heat

capacity at a constant volume, CV. The relation between CPand

CVis

CPZ CVKT
vV
vT

 2

P
vP
vV

 

T

Z CVC9a2TBTVT: (23)

3. Results and discussions

To calculate the thermodynamic quantities of CeO2 with

fluorite structure, we will use three different potentials [9],
which include the electrostatic Coulomb interactions and two
body terms to describe the short-range interactions. The two
body terms arise from the electronic repulsion and attractive
van der Waals forces, and they are described by a Buckingham
potential form. The internal energy of CeO2 compound can

then be written in the following form
U ZN1

2
X

i

4Ce_ioðriÞ C
N2

2
X

i

4O_ioðriÞ; (24)

where
4CeioðriÞ Z

qCeqi
ri

CACeKiexp K
ri
BCeKi

 

K
CCeKi

r6_i ; (25a)

and
4OioðriÞ Z

qOqi
ri

CAOKiexp K
ri
BOKi

 

K
COKi

r6_i : (25b)

Here qi(iZCe or O) denotes the ionic charge of ion i, rithe

distance between them and Aij, Bij and Cij are the potential

parameters between the ions i and j. The ionic Coulomb terms
in the above Eqs. (25a) and (25b) can be summed explicitly
using the Ewald method. In Eqs. (25a) and (25b), the

exponential term corresponds to electron cloud overlap and
the Cij/r6term originates from attractive Van der Waal’s force.

Potential parameters Aij, Bij and Cij are taken from Ref. [9],

cation–cation interactions are assumed to be purely ionic
Coulomb and the cation–anion interaction is considered to be
the form of Eqs. (25a) and (25b). The parameters of the three
potentials are presented inTable 1.

Firstly, we calculate the lattice parameters and bulk moduli
of CeO2at 293 K (room temperature) using the three different

potentials. In Table 2, we compare the calculation results of
lattice parameters and bulk moduli of CeO2obtained by using

the SMM analytic formulae with the simulation results
(absolute zero temperature) of Ref. [9]. Here, it should be
noted that the three potentials are fitted to reproduce the
experimental lattice parameter 5.411 A˚ of CeO2compound at

the absolute zero temperature, without including quantum
mechanical zero point vibrations. It is well known that the
inclusion of thermal lattice vibrations (zero point vibrations)
leads to the lattice expansion of crystals at lower temperatures.
All three sets of potentials give reasonable lattice constants of
CeO2, near the experimental 5.411 A˚ even when we take into

accounts the contributions of thermal lattice vibrations. In

Table 2, we also compare the bulk moduli BTcalculated by the

present SMM with those by the simulation method [9]. The
bulk modulus of Ref. [9](with dipole polarization effects) is
larger than the experimental values while the present SMM
results of bulk modulus are smaller than the experimental ones.
The reason for the lower bulk modulus is due to the larger
lattice spacings of CeO2compound in the present calculations,

Table 1

The parameters of CeO2used in potentials 1, 2 and Butler potential
Interaction A/eV B/A˚ C/eVA˚6

O2_–O2K _9547.92 _0.2192 _32.00 _{Potential 1}
Ce4C_–O2K _1809.68 _0.3547 _20.40

O2_–O2K _9547.92 _0.2192 _32.00 _{Potential 2}
Ce4C–O2K 2531.5 0.335 20.40

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compared to those of Ref. [9], because in the present
calculations we take into account the contributions of thermal
lattice vibrations. The dipolar and quadrupolar parts of the
interatomic potentials of CeO2 are derived from ab initio

density functional calculations and available in the literature.
When both dipole and quadrupole effects (C8/r8) are added the

SMM calculations of the bulk modulus of cerium dioxide give
a larger value and a much better agreement with experiments.

In Fig. 2, we compare the temperature dependence of the
lattice parameters, a0, and unit cell volumes of cerium

dioxide calculated by using the potentials 1 and 2, with

the experimental results for temperature range

300 K&T&1300 K. The calculated lattice parameters and
unit cell volumes by the present theory are slightly larger than
the experimental values, but overall features are in good

agreement with experimental results[25,26]. One can see in

Fig. 2(a) and (b) that the calculated lattice parameter and unit
cell volume by potentials 1 and 2 are very similar. The small
difference between the two calculations simply comes from the
difference in cerium-oxygen interaction potentials, since the
ionic Coulomb contribution and the oxygen-oxygen potential
are the same for potentials 1 and 2. InFig. 3(a) and (b), we
show the lattice parameters of CeO2with and without including

the dipole polarization effects, respectively for wider
tempera-ture range of 100 K&T&2500 K. The calculated lattice
parameters by potentials 1 and 2 are almost identical, while
the Butler potential gives somewhat larger values, shifted
upwards about 1% at lower temperatures. Therefore,
tempera-ture dependence of the lattice parameters by three potentials
are similar for lower temperature region, but at higher

Table 2

Calculated lattice constants and bulk moduli of CeO2

a0(A˚ ) BT(GPa)

Method Pot 1 Pot 2 Butler Pot 1 Pot 2 Butler Ref.

Simulation 5.411 5.411 5.411 267.9 289.4 263.6 9

SMM (with dipolar) 5.4106 5.4156 5.4531 170.070 192.519 154.159

SMM (without dipolar) 5.4057 5.4108 5.4472 184.163 207.065 166.312

Expt. 5.411 236 20

Expt. 230G10 21

Fig. 2. Comparison of calculated lattice parameters a (in A˚ ) and unit cell
volumes (in A˚3) of CeO2with the experimental results for temperature range
293 K&T&1300 K.

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temperatures near the melting temperature, Butler potential
gives nonlinear increase of the lattice constants.

In contrast, the calculated bulk moduli depend sensitively
on the parameters of the potentials. In Fig. 4, we show the
calculated bulk moduli BT of the cerium dioxide, with and

without dipole polarization effects, as a function of the
temperature T. We have found that the bulk modulus BT

depends sensitively on the potential parameters and it is
decreasing function of the temperature T. The decrease of BT

with increasing temperature arises from the thermal lattice
expansion and the effects of the vibrational entropies. The
potential 2 gives the largest values of the bulk modulus, while
the Butler potential gives the lowest. The low bulk modulus
predicted by the Butler potential is due to the choice of the
oxygen–oxygen interaction potentials. The calculated lattice
constants by the SMM with dipole polarization effects are
greater than those without including dipole polarization effects.
The present SMM calculations of the bulk modulus, including
the dipole polarization effects, are smaller compared to the
experimental values. The lattice constants are increasing by
including dipole interactions (C/r6) for all three potentials, and
accordingly the bulk modulus becomes smaller. In the present
calculations of the bulk moduli of CeO2, we have seen that

there is a clear correlation between the lattice spacing
and calculated bulk modulus, and one can predict the relative

magnitudes of the bulk modulus. However, due to the nature of
the anharmonicity of the potential, the bulk modulus can be
calculated accurately by including the anharmonicity theory of
lattice vibrations (SMM).

The linear thermal expansion coefficients of CeO2are also

calculated using the three potentials. InTable 3, we compare the

calculated linear thermal expansion coefficients of CeO2, with

and without dipole polarization effects, at room temperature
(293 K) and for higher temperature range (293–1300 K). Each
potential gives reasonable values of thermal expansion
coefficients compared with the experimental results. InFig. 5,
we also show the temperature dependence of linear thermal
expansion coefficients of CeO2, with and without the dipole

polarization effects. The calculated results by the present theory
are in good agreement with experimental results[25,27,28]and
the agreement is better for the SMM calculations with potentials
1 and 2, rather than the calculations by Butler potential. Again,
the different results obtained by potentials 1 and 2 from those by
the Butler potential arise from the parameterization of oxygen–
oxygen potentials. The thermal lattice expansion coefficients
calculated by including the dipole polarization effects are larger
(w10% at lower temperatures and w20% at higher
tempera-tures near the melting point) than those values without the
dipole polarization effects.

The calculated specific heats at constant volume Cv and

those at constant pressure Cpof CeO2compound are presented

in Fig. 6. As shown in Fig. 6, the specific heat CV depends

weakly on the temperature for the whole temperature region,
while the specific heat Cpdepends strongly on the temperature,

and becomes nonlinearly larger with increasing the
tempera-ture. It is surprising that the specific heats at the constant
volume Cv are almost independent of the nature of the

potentials used for CeO2for whole temperature range up to the

melting temperature. The three potentials (1, 2 and Butler type)
give the almost identical specific heats at constant volume CV,

while the Butler potential gives much larger CPthan those by

potentials 1 and 2 at higher temperatures. The stronger
nonlinear increase of CP values obtained by Butler potential

near the melting temperature is related to the stronger
anharmonicity of thermal lattice vibrations described by its

Table 3

Calculated thermal expansion coefficients of CeO2
aT(10

K6
KK1)

Method T (K) 293 293–1300

SMM(with dipolar)

CPotential 1 11.391 11.391–13.510

CPotential 2 11.163 11.163–13.399
CButler 12.991 12.991–16.522
SMM(without dipolar)

CPotential 1 10.322 10.322–11.826
CPotential 2 10.187 10.187–11.820
CButler 11.731 11.731–14.329

Expt. 11a 12.4b

11.2c
a _Ref._[26]_.

b _Ref._[22]_.
c _Ref._[23]_.

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potential. The anomalous nonlinear increase in the
thermo-dynamic quantities near the melting temperature are observed
in the strongly anharmonic solids like solid Xe and solid Ar

[17]. However, in the lack of the reliable experimental results
of specific heats of stoichiometric CeO2 compound near

the melting temperature, it is difficult to draw definite
conclusions on the validity of using the potentials 1, 2 or
Butler potential for the calculations of CP.

Summarizing the above-mentioned thermodynamic
calcu-lations done by the SMM, we would like to emphasize that it is
particularly useful for the systematic and fundamental

under-standing of the thermodynamic properties of cerium oxide. We
have presented the analytic expressions of thermodynamic
quantities of cerium dioxide compounds with fluorite structure
including the anharmonicity effects of thermal lattice
vibrations, which are straightforward and useful for the realistic
numerical calculations. The present calculations of
thermo-dynamic quantities of CeO2 (inorganic compound with the

fluorite structure) are much more efficient and straightforward
than those by using the simulation method, e.g. Monte Carlo or
quantum Monte Carlo simulations. The simulation methods
without including the thermal (anharmonic) vibration effects
are insufficient and often useless even for the calculations of the
low temperature (room temperature) thermodynamic
quan-tities. In addition to the thermodynamic calculations, the
present SMM can also be used for the parameterization of the
new quantum mechanical potentials derived from the fitting to
the ab initio density functional calculations, i.e. improved
version of the (empirically derived) interatomic potentials used
in the present study, for high temperature thermodynamic
quantities of the fluorite compounds.

4. Conclusions

We have presented an analytic formulation for obtaining the
thermodynamic quantities of the inorganic compounds (cerium
dioxide) with fluorite structure using the statistical moment
method. The present formalism takes into account the
higher-order anharmonic vibrational terms in the Helmholtz free
energy and it enables us to derive the various thermodynamic

quantities in closed analytic forms. The thermodynamic
quantities, i.e. the unit cell volume, linear thermal expansion
coefficient, elastic modulus and the specific heats Cvand Cp, of

the cerium dioxide are calculated and compared with the
available experimental results. The interatomic potentials 1 and

Fig. 5. Temperature dependence of linear thermal expansion coefficients of
CeO2with and without dipole polarization effects: (a) potential 1, (b) potential
2 and (c) Butler potential.

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<span class='text_page_counter'>(8)</span><div class='page_container' data-page=8>

2 used in the present study give almost identical results in
thermodynamic quantities of CeO2 (except for the bulk

modulus), while the Butler potential gives the considerably
different results. This is originating from the different
parameter values used for the oxygen–oxygen interactions in
the potentials 1, 2 and Butler potential. In the present study, the
dipole polarization effects are taken into account, and the
inclusion of the dipolar terms is essential in order to understand
the characteristic thermodynamic properties of CeO2.

Acknowledgements

This work was supported by the Korea Science and
Engineering Foundation (KOSEF) through the Center for
Advanced Plasma Surface Technology (CAPST) and National
Research Laboratory (NRL) program, and the Core
Technol-ogy Development Program for Fuel Cells, the Ministry of
Science and Technology, Korea.

Appendix: Moments formulae in SMM

The present statistical moment method (SMM) has the
virtue of treating exactly the correlations, deviations from the
simple mean-field approximation, in the thermal averages huni
of the atomic displacements. For instance, (intra atomic)
correlations appearing in the lower order moments are given by
the following equations:

huiauigiaZhuiaiahuigiaCq
vhuiaia

vag
C

Zdag
2mucoth

Zu
2q

 

K
qdag
mu2;

(A1)
and

huiauiguihiaZ huiaiahuigiahuihiaCqPaghhuiaia
vhuigia

vah
Cq2 vhuiaia

vagvah
C

Zhuihiadag

2mu coth

Zu
2q

 

Kqhuihiadag

mu2 ; ðA2Þ

where Paghis 1 (aZgZh) or 0 (otherwise) depending on a, g

and h (Cartesian component) and u is the atomic vibration
frequency defined by Eqs. (12) and (15a), (uCe or uO), in the

text. Here it is important to note that the root mean square
atomic displacements hu2ii are evaluated exactly by using Eq.

(A1) and appropriate (ab initio) energetic for the interatomic
potentials in the solids.

However, the second moments, i.e. root mean square atomic
displacement hu2ii on atomic site i is different from the root
mean square relative atomic displacements (second cumulants)
which include the interatomic displacement–displacement
correlations as given by the following equation

s2jTị Z hẵujKu0ị R

2

i Z hu2
ji C hu

2

0iK2huju0i; (A3)
where, u0and ujare the atomic displacements of 0-th and j-th

sites from their equilibrium positions, respectively. ðR is an unit

vector at the 0-th site pointing towards the j-th site, and the
brackets denote the thermal average. Here, simple decoupling
scheme huju0izhujihu0i gives reasonable approximation. The

root mean square relative displacements are important
ingredients in the theory of EXAFS (X-ray-absorption fine
structure)[29,30], the treatments of melting transition of solids
by Lindemann’s criterion, and other related thermodynamic

calculations of solids. Using the exact moment formulae of Eqs.
(A1) and (A2), we find the second and fourth order moments,
which include both harmonic and anharmonic contributions. For
the evaluation of free energies, we only need moments (not
cumulants) up to fourth order in the present SMM formalism.

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