Abstract. Thermodynamic properties of rare-earth metals have been
studied using staistical moment method. The analytic expressions of
the Helmholtz free energy and thermodynamic quantities were obtained.
Present SMM results of nearest neighbor distance, linear thermal expansion
coefficient and specific heats at constant pressure for Th and Ce metals are
compared with the experimental results.

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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2011, Vol. 56, No. 7, pp. 58-64
THERMODYNAMIC PROPERTIES OF SOME RARE-EARTH METALS
Vu Van Hung(∗)
Hanoi National University of Education
Dang Thanh Hai
Vietnam Education Publishing House
(∗)E-mail: bangvu57@yahoo.com
Abstract. Thermodynamic properties of rare-earth metals have been
studied using staistical moment method. The analytic expressions of
the Helmholtz free energy and thermodynamic quantities were obtained.
Present SMM results of nearest neighbor distance, linear thermal expansion
coefficient and specific heats at constant pressure for Th and Ce metals are
compared with the experimental results.
Keywords: Thermodynamic, rare-earth metals, staistical moment method.
1. Introduction
Recently, there has been a great interest in the study of rare-earth metals since
they provide us a wide variety of academic problems as well as the technological ap-
plications. Various theoretical studies of simple, noble and transition metals [1-3]
have been made so far based on pseudopotential theory but relatively few attempts
have been made on rare-earth and actinide elements such as La, Yb, Ce and Th
[4-6] by the same method. Rosengren et al [7] have investigated the lattice dynam-
ics of thorium using pseudopotential due to Krasko and Gurskii [8]. N.Singh and
S.P.Singh [9] have calculated the phonon dispersion of La, Yb, Ce and Th. Recently,
Pandya et al [10], have investigated the phonon dispersion curves, phonon density
of states, Debye-Waller factor, mean square displacements and equation of state for
thorium. J.K.Baria and A.R.Jani [11] also have calculated the phonon dispersion
curves, phonon density of states, Debye temperature, Gru¨neisen parameters and
dynamic elastic constants for La, Yb, Ce and Th.
Most of the previous theoretical studies, however, are concerned with the ma-
terials properties of rare-earth metals 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 some rare-earth metals using the analytic statistical
moment method (SMM) [12-15]. The thermodynamic quantities are derived from
the Helmholtz free energy.
58
Thermodynamic properties of some rare-earth metals
2. Content
2.1. Theory
To derive the temperature dependence of the thermodynamic properties of
rare-earth metals, we use the statistical moment method. This method allows us
to take into account the anharmonicity effects of thermal lattice vibrations on the
thermodynamic quantities in the analytic formulations.
The essence of the SMM scheme can be summarized as follows: for simplicity,
we derive the thermodynamic quantities of crystalline materials with cubic symme-
try, taking into account the higher (fourth) order anharmonic contributions in the
thermal lattice vibrations going beyond the quasi-Hamonic (QH) approximation.
The extentions for the SMM formalism to non-cubic systems is straightforward.
The basic equations for obtaining thermodynamic quantities of the given crystals
are derived in a following manner: the equilibrium thermal lattice expansions are
calculated by the force balance criterion and then the thermodynamic quantities are
determinded for the equilibrium lattice spacings. The anharmonic contributions of
the thermodynamic quantities are given explicitly in terms of the power moments
of the thrmal atomic displacements.
Let us first define the lattice displacements. We denote ~uil the vector defining
the displacement of the ith atom in the lth unit cell, from its equilibrium position.
The potential energy of the whole crystal U(~uil) is expressed in terms of the positions
of all the atoms from the sites of the equilibrium lattice. We use the theory of small
atomic vibrations, and expand the potential energy U as a power series in the
cartesian components, uIiα of the displacement vector ~uil around this point.
For the evaluation of the anharmonic contributions to the free energy ψ, we
consider a quantum system, which is influenced by supplemental forces αi in the
space of the generalized coordinates qi. For simplicity, we only discuss monatomic
metallic systems and hereafter omit the indices l on the sublattices. Then, the
Hamiltonian of the crystalline system is given by
Ĥ = Ĥ0 −
∑
i
αiqˆi, (2.1)
where Ĥ0 denotes the crystalline Hamiltonian without the supplementary forces
αi and upper huts ∧ represent operrators. The supplementary forces αi are acted
in the direction of the generalized coordinates qi. The thermodynamic quantities
of the anharmonic crystal (harmonic Hamiltonian) will be treated in the Einstein
approximation.
After the action of the supplementary forces αi the system passes into a new
equilibrium state. If the 0th atom in the lattice is affected by a supplementary force
αβ , then the total force acting on it must be zero, and one gets the force balance
59
Vu Van Hung and Dang Thanh Hai
relation as
1
2
∑
i,α
(
∂2ϕi0
∂uiα∂uiβ
)
eq
〈uiα〉+ 1
4
∑
i,α,γ
(
∂3ϕi0
∂uiα∂uiβ∂uiγ
)
eq
〈uiαuiβ〉
+
1
12
∑
i,α,γ,η
(
∂4ϕi0
∂uiα∂uiβ∂uiγ∂uiη
)
〈uiαuiβuiη〉 − αβ = 0 (2.2)
The thermal averages of the atomic displacements 〈uiαuiβ〉 and 〈uiαuiβuiη〉 (called
as second and third - order moments) at given site ~Ri can be expressed in terms of
the first moment 〈uiα〉 with the aid of the recurence formula [12, 14].
Then equation (2.2) is transformed into the new differential equation:
γθ2
d2y
dα2
+ 3γθy
dy
dα
+ γy3 + ky +
γθ
k
(xcothx− 1)y − α = 0, (2.3)
where
k =
1
2
∑
i
(∂2ϕi0
∂u2iβ
)
eq
≡ mω2,
γ =
1
12
∑
i
[(
∂4ϕi0
∂u4iβ
)
eq
+ 6
(
∂4ϕi0
∂u2iβ∂u
2
iγ
)
eq
]
, (2.4)
θ = kBT ; x =
~ω
2θ
; y = 〈uiα〉,
with kB is the Boltzmann constant and ω is the atomic frequence.
Then, the solutions of the non-linear differential equation (2.3) can be expanded
in the power series of the supplemental force α as
y = ∆r + A1α + A2α
2. (2.5)
Here, ∆r is the atomic displacement in the limit of zero of supplemental force α.
After a bit of algebra, it can be shown that the atomic displacement ∆r in cubic
systems is given by [12]
∆r =
√
2γθ2
3k3
A (2.6)
Once the thermal expansion ∆r in the lattice is found, one can get the Helmholtz
free energy of the system in the following form
ψ = u0 + ψ0 + ψ1 (2.7)
60
Thermodynamic properties of some rare-earth metals
where ψ0 denotes the free energy in the harmonic approximation and ψ1 the anhar-
monic contribution to the free energy. We calculate the anharmonic contribution to
the free energy ψ1 by applying the general integral formula [12]
ψ = u0 + ψ0 +
∫ λ
0
〈V̂ 〉λdλ (2.8)
where λV̂ represents the Hamiltonian corresponding to the anharmonicity contribu-
tion. Then the free energy of ther system is given by
ψ = u0 + 3Nθ
[
x+ ln(1− e−2x)
]
+ 3N
{
θ2
k2
[
γ2x
2coth2x− 2
3
γ1
(
1 +
xcothx
2
)]
+
2θ3
k4
[4
3
γ22xcothx
(
1 +
xcothx
2
)
− 2γ1(γ1 + 2γ2)
(
1 +
xcothx
2
)
(1 + xcothx)
]}
(2.9)
where the second term denotes the harmonic contribution to the free energy.
With the aid of the ”real space” free energy formula ψ = E − TS, one can
find the thermodynamic quantities of given systems. The thermodynamic quantities
such as specific heats and elastic modul at temperature T are directly derived from
the free energy ψ of the system. For instance, the isothermal compressibility χT is
given by
χT =
3
( a
a0
)3
2P +
1
3N
√
2
a
(∂2ψ
∂r2
)
T
, (2.10)
where
∂2ψ
∂r2
= 3N
{
1
6
∂2u0
∂r2
+ θ
[xcothx
2k
∂2k
∂r2
− 1
4k2
(∂k
∂r
)2(
xcothx+
x2
sinh2x
)]}
, (2.11)
here P is the pressure and a is the nearest neighbor distance at temperature T
a = a0 +∆r, (2.12)
where a0 is the nearest neighbor distance at zero temperature.
Using the expression of the free energy ψ from (2.9), after a bit of algebra, the
specific heat at constant volume Cv is derived as
Cv = 3NkB
{
x2
sinh2x
+
2θ
k2
[(
2γ2 +
γ1
3
)x3cothx
sinh2x
+
+
γ1
3
(
1 +
x2
sinh2x
)
− γ2
( x4
sinh4x
+
2x4coth2x
sinh2x
)]}
(2.13)
61
Vu Van Hung and Dang Thanh Hai
Then the specific heat at constant pressure Cp is given from the thermodynamic
relation as:
Cp = Cv +
9TV α2T
χT
, (2.14)
where the linear thermal expansion coefficient αT is given by
αT = −
√
2kBχT
3a2
1
3N
∂2ψ
∂θ∂a
(2.15)
For simplicity, we take the effective pair interaction energy in rare-earth metals
as the power law, similar to the Lennard - Jones
ϕ(r) =
D
(n−m)
[
m
(r0
r
)n
− n
(r0
r
)m]
, (2.16)
where D, r0 are determined to fit to the experimental data (e.g., cohesive energy
and elastic modulus). Using the effective pair potentials of Equation (2.16), it is
straightforward to get the interaction energy u0 and the parameter k, γ in the crystal
as
u0 =
1
2
∑
i
ϕi0(ri) =
D
(n−m)
[
mAn
(r0
r
)n
− nAm
(r0
r
)m]
, (2.17)
k =
1
2
∑
i
(∂2ϕi0
∂u2iβ
)
=
Dnm
2a2(n−m)
[
(n+ 2)A
a2ix
n+4 − An+2
](r0
a
)n
−
− Dnm
2a2(n−m)
[
(m+ 2)A
a2ix
m+4 − Am+2
](r0
a
)m
= m0ω
2
0, (2.18)
γ =
1
12
∑
i
[(
∂4ϕi0
∂u4ix
)
eq
+ 6
(
∂4ϕi0
∂u2ix∂u
2
iy
)
eq
]
≡ 4(γ1 + γ2)
=
Dnm
12a4(n−m)
{[
(n+ 2)(n+ 4)(n+ 6)
(
A
a4ix
n+8 + 6A
a2ixa
2
iy
n+8
)
−
− 18(n+ 2)(n+ 4)Aa2ixn+6 + 9(n+ 2)An+4
](r0
a
)n
−
[
(m+ 2)(m+ 4)(m+ 6)
(
A
a4ix
m+8 + 6A
a2ixa
2
iy
m+8
)
−
− 18(m+ 2)(m+ 4)Aa2ixm+6 + 9(m+ 2)Am+4
](r0
a
)m}
(2.19)
where m0 is the mass of particle, ω0 is the frequency of lattice vibration, and
An, Am, ... are the structural sums for the given crystal.
62
Thermodynamic properties of some rare-earth metals
2.2. Results and discussions
Using the moment method in statistical dynamics, we calculated the thermo-
dynamic properties of rare-earth metals Ce and Th. The potential parameters are
listed in Table 1.
Table 1. Parameter D and r0 determined by the experimental data [18]
Metal n m D/kB(K) r0(A˘)
Th 4.0 3.5 4458.6 3.5898
Ce 17.0 12.0 1966 3.6496
Table 2. Temperature dependence of thermodynamic quantities of Th metal
T (K) 300 400 500 600 700 800
a(A˘) 3.1310 3.1349 3.1388 3.1428 3.1468 3.1508
αT (10
−6K−1) 12.58 12.61 12.66 12.72 12,79 12.86
Exp[17] 11.1 11.9 12.5 13.1 13.7 14.2
Cp(cal/mol.K) 6.08 6.16 6.22 6.29 6.35 6.42
Exp[17] 6.53 7.00 7.45 7.90 8.36 8.81
Table 3. Temperature dependence of thermodynamic quantities of Ce metal
T (K) 300 400 500 600 700 800
a(A˘) 3.4496 3.5525 3.5555 3.5585 3.5615 3.5647
αT (10
−6K−1) 8.28 8.37 8.50 8.66 8.84 9.05
Exp[17] - 6.0 6.1 6.3 6.8 7.6
Cp(cal/mol.K) 5.94 6.15 6.30 6.43 6.55 6.66
Exp[17] 6.80 7.30 7.70 8.10 8.50 8.90
We present in Tables 2 and 3 the linear thermal expansion coefficient αT ,
nearest neighbor distance a and specific heats at constant pressure Cp of Th and
Ce metals calculated by the present SMM, together with those of the experimental
results [17]. The calculated thermal expansion coefficients of Th and Ce metals are
in good agreement with the experimental results. The thermal expansion coefficient
of Th and Ce metals are also calculated as a function of the temperature T .
The calculated specific heat at constant pressure Cp of Th and Ce metals are
presented in Tables 2 and 3. As shown in these Tables, the specific heat Cp depends
strongly on the temperature. The caculated lattice specific heat Cp at constant
pressure are in good agreement with the experimental results.
63
Vu Van Hung and Dang Thanh Hai
3. Conclusion
The SMM calculations of thermodynamic quantities of Th and Ce metals were
performed. Present SMM results of the linear thermal expansion coefficient αT ,
specific heats at constant pressure Cp are in good agreement with the experimental
data.
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