Abstract. Using the statistical moment method (SMM), we calculated the Helmholtz free
energy of hcp, bcc and fcc phases at zero pressure of some rareearth metals at a wide
range of temperatures. The stable phase of Gd, Tb, Sc, Y and Yb metals can be determined
by examining the Helmholtz free energy at a given temperature, i.e. the phase that gives
the lowest free energy will be stable. These findings on the structural phase transition
temperature are in agreement with previous experiments.
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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/23541059.20160038
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 106111
This paper is available online at
STUDY STRUCTURE TRANSITION TEMPERATURE
OF SOME RAREEARTH METALS
Dang Thanh Hai and Vu Van Hung
Vietnam Education Publishing House
Abstract. Using the statistical moment method (SMM), we calculated the Helmholtz free
energy of hcp, bcc and fcc phases at zero pressure of some rareearth metals at a wide
range of temperatures. The stable phase of Gd, Tb, Sc, Y and Yb metals can be determined
by examining the Helmholtz free energy at a given temperature, i.e. the phase that gives
the lowest free energy will be stable. These findings on the structural phase transition
temperature are in agreement with previous experiments.
Keywords: structural transition temperature, rareearth metals, statistical moment method.
1. Introduction
The structural transformation temperature of materials is of great interest to those working
with solid state sciences and technologies. The structural and electronic properties [1] show that
the 17 rareearth metals exist in one of five crystal structures. At room temperature, nine exist in
a hexagonal closepacked (HCP) structure, four in a double caxis hcp (DHCP) structure, two in
a facecentered cube (FCC), and one in both a bodycentered cube (BCC) and rhombic (Smtype)
structures. This distribution changes according temperature and pressure as many of the elements
go through a number of structural phase transitions.
Many scientists have been interested in the study of thermodynamic properties and the
elasticity of rare earth metals. In this study [2], the phonon spectrum and isothermal elastic
constants of the rare earth metals La, Yb, Ce, and Th were calculated. Born M and J.R
Oppenheimer [3] used the method of densityfunctional theory (DFT) and the equation of state at
various pressures to calculate the lattice constant, molar volume, c/a ratio, the total energy and the
temperature of the structural phase transition of Ce, Th, Pu, Am and their alloys. Rao R. S, Godwal
B. K, and Sikka S. K investigated the electronic structure of thorium, and its FCCBCC phase
transition at high pressure [4]. Baria and Jani studied phonon dispersion curves, phonon density,
Debye temperature, the Gru¨neisen constant and elastic constants, and compared the atomic volume
depending on pressure, and the ionion interactions of La, Yb, Ce and Th [5]. In the research [6],
it was shown that the heat capacity depends on the temperature of Ce, and the atomic volume
Received August 15, 2016. Accepted September 14, 2016.
Contact Dang Thanh Hai, email address: dthai@nxbgd.vn
106
Study structure transition temperature of some rareearth metals
depends on pressure. Landa A and Soderlind P created a phase diagram for Ce, Th and Pu using
the density functional theory (DFT) [7]. However, few scientists have looked at the structure transit
temperature of rare earth metals.
The authors of this study have looked at the structural transformation of rareearth metals
using the moment method in quantum statistical mechanics, hereafter referred to as the statistical
moment method (SMM) [813]. This work focuses on the transition structure temperature of Gd,
Tb, Sc, Y and Yb metals using the statistical moment method.
2. Content
2.1. Theory
As we know, the structural phase transition between the aphase and the bphase when a
chemical potential is µ balances
µa = µb ⇔ ψa + PV a = ψb + PV b. (2.1)
In formula (2.1), ψa and ψb are the free energy of the metal structure in the aphase and bphase.
Free energy ψa, ψb are determined using statistical moment methods.
At zero pressure, the formula (2.1) is rewritten as follows:
ψa = ψb. (2.2)
In expression (2.2), free energy ψa, ψb corresponding to structure aphase and bphase is a function
of temperature T . Thus, we can determine the structural phase transition temperature by plotting
the dependence of the free energy of the aphase and bphase on temperature with the same
coordinate axes. The intersecting point of two line graphs allow us to determine the transition
temperature Tc of the two phases.
On the other hand, the free energy is calculated using the formula:
ψ = U − TS, (2.3)
where S is the entropy.
At the transition temperature T = Tc structures, the free energy ψ of the aphase and
bphase are equal, and using eq.(2.3), we have
Ua − U b = T (Sa − Sb)⇔ ∆U = T∆S. (2.4)
Thus, another way that we can determine the structural phase transition temperature
between the aphase and bphase is plotting∆U, T∆S the dependence of temperature on the same
coordinate axes. The twoline intersect allows us to identify structural phase transition temperature.
Free energy ψ, interaction U and entropy S of the crystal are determined using the statistical
moment method.
According to thermodynamic theory, the phase with the lowest free energy will be favorable
at each temperature. With the aid of the free energy formula ψ, at a wide range of temperatures T
107
Dang Thanh Hai and Vu Van Hung
for the aphase and bphase for some rareearth metals, we evaluate the components of ψ which
consist of the internal energy U and entropy S:
∆ψ = ψa − ψb = (Ua − U b)− T (Sa − Sb). (2.5)
At the temperature where ∆ψ > 0, the aphase structure of a rareearth metal is stable. At
the temperature where ∆ψ < 0, the bphase structure of a rareearth metals is also stable.
The temperature when ∆ψ = 0 or in other words when ∆U = T∆S, is called the transition
temperature and both phases coexist. Therefore, if we plot the difference of these components, for
example ψb = ψb for the rareearth metal, as the functions of temperature, their interceptive point
will denote the transition temperature.
We determine the free energy, entropy and potential interactions of metal using statistical
moment methods. From the results of referents [10, 11], we deduce the free energy of the metal
atoms N using the following expression:
ψ = U0 + ψ0 + 3N
{
θ2
k2
[
γ2x
2coth2x− 2γ1
3
(
1 +
xcothx
2
)]
+
+
2θ2
k4
[
4
3
γ22xcothx
(
1 +
xcothx
2
)
− 2(γ21 + 2γ1γ2)
(
1 +
xcothx
2
)
+ (1 + xcothx)
]}
, (2.6)
where
ψ0 = 3Nθ
[
x+ ln
(
1− e−2x)] (2.7)
U0 =
N
2
∑
i
ϕi0(ri) (2.8)
k =
1
2
∑
i
(
∂2ϕi0
∂u2iβ
)
. (2.9)
γ1 =
1
48
∑
i
(
∂4ϕi0
∂u4iβ
)
, γ2 =
6
48
∑
i
(
∂4ϕi0
∂u2iα∂u
2
iβ
)
, (2.10)
and the entropy S of the metal of the form:
S = S0 +
3NkBθ
k2
[
γ1
3
(
4 + xcothx+
x2
sinh2x
)
− 2γ2x
3cothx
sinh2x
]
, (2.11)
where
S0 = 3NkB
[
xcothx− ln(2sinhx)]. (2.12)
108
Study structure transition temperature of some rareearth metals
2.2. Results and discussions
To calculate the transition structrure temperature, we use the LennardJones (LI) type of
pair potentials:
φ(r) =
D0
n−m
{
m
(
r0
r
)n
− n
(
r0
r
)m}
. (2.13)
Table 1. Parameters n,m,D and r0 determined using the experimental data [14]
Metal n m D/kB ,K r0(A)
Gd 8.76 3.02 8059.11 3.5800
Tb 8.64 2.91 7867.74 3.5200
Sc 7.72 4.48 7659.47 3.2500
Y 9.61 3.55 8546.35 3.5846
Y b 8.78 3.98 3112.36 3.8800
Using the Maple program with the values of parameters D and r0 determined by
experimental data [14], we calculate the Helmholtz free energy ψ of the HCP and BCC phases
for Sc, Y, Gd and Tb metals (or FCC and BCC for Yb metal).
1300 1400 1500 1600 1700 1800
9.6
9.4
9.2
9.0
8.8
8.6
F
r
e
e
e
n
e
r
g
y
o
f
G
d
(
1
0

1
9
J
)
Temperature (K)
BCC
HCP
Gd
T
c
= 1599K
T
exp
= 1508K
1400 1500 1600 1700 1800 1900
9.6
9.4
9.2
9.0
8.8
8.6
F
r
e
e
e
n
e
r
g
y
o
f
T
b
(
1
0

1
9
J
)
Temperature (K)
BCC
HCP
Tb
T
c
= 1587K
T
exp
= 1562K
1400 1500 1600 1700 1800 1900 2000
8.6
8.4
8.2
8.0
7.8
F
r
e
e
e
n
e
r
g
y
o
f
S
c
(
1
0

1
9
J
)
Temperature (K)
BCC
HCP
Sc
T
c
= 1692K
T
exp
= 1610K
1400 1500 1600 1700 1800 1900 2000 2100
1.00
0.98
0.96
0.94
0.92
0.90
0.88
F
r
e
e
e
n
e
r
g
y
o
f
Y
(
1
0

1
9
J
)
Temperature (K)
BCC
HCP
Y
T
c
= 1834K
T
exp
= 1751K
Figure 1. Temperature dependence of the Helmholtz free energy for Gd, Tb, Sc and Y metals
109
Dang Thanh Hai and Vu Van Hung
Table 2. Transition structure temperature of Gd, Tb, Sc, Y and Yb metals
Metal Tc(K)[SMM] Texp(K)[15] Metal Tc(K)[SMM] Texp(K)[15]
Gd 1599 1508 Y 1834 1751
Tb 1587 1562 Y b 1031 1068
Sc 1692 1610
Figure 1 shows the temperature dependence of Helmholtz energies of Gd, Tb, Sc and Y
metals for BCC and HCP phases with the use of dashed and solid curves respectively. In Figure 1,
it is clearly shown that the Helmholtz energy of the HCP phase is lower than the Helmholtz energy
of the BCC phase in the temperature region lower than Tc and the BCC phase becomes more
stable than the HCP phase in the temperature region higher than Tc. The transition temperature
Tc of the HCP→ BCC transformation of Gd, Tb, Sc and Y metals in the presence of SMM are
Tc(Gd) = 1599K , Tc(Tb) = 1587K , Tc(Sc) = 1692K and Tc(Y) = 1834K . The present value
Tc of Gd, Tb, Sc and Y metals is still higher than the experimental value Texp(Gd) = 1508K ,
Texp(Tb) = 1562K , Texp(Sc) = 1610K and Texp(Y) = 1751K [15].
800 900 1000 1100 1200 1300 1400
5.7
5.6
5.5
5.4
5.3
5.2
5.1
5.0
4.9
4.8
4.7
F
r
e
e
e
n
e
r
g
y
o
f
Y
b
(
1
0

1
9
J
)
Temperature (K)
BCC
FCC
Yb
T
c
= 1031K
T
exp
= 1068K
Figure 2. Temperature dependence of the Helmholtz free energy for Yb metal
Figure 2 shows the temperature dependence of Helmholtz energies of Ybmetal for BCC and
FCC phases with the use of dashed and solid curves respectively. In Figure 2, it is clearly shown
that the Helmholtz energy of the FCC phase is lower than the Helmholtz energy of the BCC phase
in the temperature region lower than Tc(Yb) ≈ 1031K, and BCC phase becomes more stable than
FCC phase with temperature region higher than Tc(Yb) ≈ 1031K. The transition temperature Tc
of the FCC→ BCC transformation of Yb metal in the presence of SMM is 1031K. The present
value of Tc(Yb) = 1031K is still lower than the experimental value of Texp(Yb) = 1068K [15].
110
Study structure transition temperature of some rareearth metals
3. Conclusion
We use the statistical moment method in order to calculate the Helmholtz free energy and
components of Helmholtz free energy of the hcp and bcc phases at zero pressure for Gd, Tb, Sc
and Y metals, and the FCC and BCC phases for Tb metal at a wide range of temperatures. The
structural phase transition temperature of Gd, Tb, Sc, Y, and Yb metals at zero pressure calculated
using the SMM are generally in good agreement with the experimental results. The calculated
structural phase transition temperature of Gd, Tb, Sc, Y and Yb metals at high pressures will be
carried out in our future works.
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[4] Rao R. S, Godwal B. K and Sikka S. K., 1992. Phys. Rev., B46, p. 5780(R).
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[7] Landa A and Soderlind P., 2004. Condensed Matter Physics, Vol. 7, No. 2 (38), pp.247264.
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