Abstract. The equation of state, the absolute stability temperature of crystalline
state and the melting temperature for N2, CO, CO2 and N2O molecular cryocrystals
under pressure are determined using the statistical moment method and are
compared with the experimental data and other calculations.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 67-75
This paper is available online at
EQUATION OF STATE AND MELTING TEMPERATURE FOR N2, CO, CO2
AND N2OMOLECULAR CRYOCRYSTALS UNDER PRESSURE
Nguyen Quang Hoc1, Dinh Quang Vinh1, Bui Duc Tinh1 and Nguyen Duc Hien2
1Faculty of Physics, Hanoi National University of Education
2Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province
Abstract. The equation of state, the absolute stability temperature of crystalline
state and the melting temperature for N2, CO, CO2 and N2Omolecular cryocrystals
under pressure are determined using the statistical moment method and are
compared with the experimental data and other calculations.
Keywords:Molecular cryocrystal, statistical moment method, limiting temperature,
absolute stability.
1. Introduction
Molecular crystals are characterized by their strong intramolecular forces and much
weaker intermolecular forces. High-pressure spectroscopic studies provide useful data for
refining the various model potentials which are used to predict the physical properties of
such systems as well as the formation of various crystalline phases.
In the most cases, the melting temperature of crystals is described by the empirical
Simon equation ln (P + a) = c lnT + b, where a, b and c are constant and P and T,
respectively, are the melting pressure and the melting temperature [1]. However, this
equation cannot be used for crystals at extremely high pressure.
On the theoretical side, in order to determine the melting temperature we must use
the equilibrium condition of the liquid and solid phases. However, a clear expression of
the melting temperature has not yet been obtained in this way.
Notice that the limiting temperature of absolute stability for the crystalline state at a
determined pressure is not far from the melting temperature. Therefore, some researchers
had identified the melting curve with the curve of absolute stability for the crystalline
state. In order to better determine the limiting temperature of absolute stability for the
crystalline state, the correlation effects are calculated using the one-particle distribution
function method [4, 10]. Because the difference between these two temperatures is large
Received January 7, 2014. Accepted September 30, 2014.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
67
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien
at high pressure, this approximation is effective only at low pressure. Other researchers
have concluded that it is impossible to find the melting temperature using only the limit
of absolute stability for the solid phase because the obtained results on the basis of
the self-consistent phonon method and the one-particle distribution function method are
larger than the corresponding melting temperatures by a factor of 3 to 4 and 1.3 to 1.6,
respectively [5].
On the basis of the statistical moment method (SMM) in statistical mechanics, some
authors have determined the limiting temperature of absolute stability for the crystalline
state at various pressures and then they adjust this temperature in order to find the melting
temperature [9, 10]. The melting temperature is obtained by this way for low as well as
high pressures. The calculated results for the inert gas crystals agree rather well with the
experimental data [2].
In the present study, we apply the SMM to investigate the equation of state and
the melting temperature of solid N2, CO, CO2 and N2O. We will calculate the pressure
dependence of the lattice constant and the melting temperature of these crystals.
2. Content
2.1. Equation of state, limiting temperature of absolute stability and
melting temperature for molecular cryocrystals
The equation of state of a crystal with a face-centered cubic (fcc) structure can be
written in the following form [2]:
Pv = −a
6
(
∂u0
∂a
)
T
+ 3γTGθ,
v =
√
2
2
a3, u0 =
∑
i
ϕi0 (|⃗ai|) ,γTG = −
a
6k
∂k
∂a
X,X ≡ x coth x, θ = kBT, x = ~ω
2θ
,
k =
1
2
∑
i
(
∂2ϕi0
∂u2iβ
)
eq
≡ mω2, β = x, y, z. (2.1)
here P is the hydrostatic pressure, v is the volume of the fcc lattice, a is the nearest
neighbor distance of the fcc crystal, a⃗i is the vector determining the equilibrium position
of the ith particle, ϕi0 is the interaction potential between the ith particle and the 0th
particle, γTG is the Gruneisen constant and kB is the Boltzmann constant. From the limiting
condition of absolute stability for the crystalline state(
∂P
∂v
)
T
= 0, i.e.
(
∂P
∂a
)
T
= 0, (2.2)
68
Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals...
we find the corresponding expression of the limiting temperature Ts as follows:
Ts =
2Pv + a
2
6
(
∂2u0
∂a2
)
T
3kB
[
a
(
∂γTG
∂a
)
T
− γTG
] . (2.3)
If we take the values of the parameters a, k, ω at the same limiting temperature of
absolute stability Ts then (2.3) can be transformed into the form [2]
Ts =
4k2
kB
(
∂k
∂a
)2
T
{
1
6
(
∂2u0
∂a2
)
T
+
~ω
4k
[(
∂2k
∂a2
)
T
− 1
2k
(
∂k
∂a
)2
T
]
+
2Pv
a2
}
. (2.4)
In the case of P = 0, it gives
Ts =
4k2
kB
(
∂k
∂a
)2
T
{
1
6
(
∂2u0
∂a2
)
T
+
~ω
4k
[(
∂2k
∂a2
)
T
− 1
2k
(
∂k
∂a
)2
T
]}
. (2.5)
The nearest neighbor distance a is determined by a = a0 + ux0, where ao denotes
the distance a at temperature 0 K and is determined from the experimental data. The
displacement ux0 of a particle from the equilibrium position is calculated by
ux0 =
√
2γθ2
3k3
A, γ =
1
12
∑
i
(∂4ϕi0
∂u4iβ
)
eq
+ 6
(
∂4ϕi0
∂u2iβ∂u
2
iγ
)
eq
, β
̸= γ, β, γ = x, y, z, A = a1 +
6∑
i=2
(
γθ
k2
)i
ai. (2.6)
where ai(i = 1−6) is determined in [2]. The equation for calculating the nearest neighbor
distances at pressure P and at temperature 0 K has the form [2]
y2 = 1.1948 + 0.1717y4 − 0..0087Pσ
3
ε
y5 + 0.0021
Pσ3
ε
y7, y =
(a
σ
)3
. (2.7)
We notice that the nearest neighbor distance am corresponding to the melting
temperature Tm of the crystal is approximately equal to the nearest neighbor distance as
corresponding to limiting temperature Ts. In addition, from (2.1) we see that temperature
T is a function of nearest neighbor distance a when pressure P is constant, i.e. T = T(a).
Therefore, we can expand temperature Tm according to the distance difference am - as
and keep only the first approximate term [2]
Tm ≈ Ts + am − as
kBγsG
(
Pvs
as
+
1
18
∂u0
∂as
+ as
∂2u0
∂a2s
)
, (2.8)
69
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien
where am = a (Tm.P ) , as = a (Ts, P ) , vs =
√
2
2
a3s,
∂u0
∂as
=
(
∂u0
∂a
)
a=as
, ∂
2u0
∂a2s
=
(
∂2u0
∂a2
)
a=as
,
the Gruneisen parameter γsG is regarded as invariable in the interval from T to Tm because
it changes very little and γsG = −
(
a
6k
∂k
∂a
x coth x
)
T=Ts,a=as
.
The equation of state of a crystal with a hexagonal close-packed (hcp) structure can
be written in the following form [9]:
Pv = −a
4
∂u0
∂a
− c
2
∂u0
∂a
+ 12γTGθ,
v =
√
3
2
a2c, γTG = −
X
24kx
(
a
∂kx
∂a
+ 2c
∂kx
∂c
)
− Xz
48kz
(
a
∂kz
∂a
+ 2c
∂kz
∂c
)
,
X ≡ x cothx,Xz ≡ xzcthxz,
x =
~ωx
2θ
, xz =
~ωz
2θ
, kx =
1
2
∑
i
[(
∂2ϕi0
∂u2ix
)
eq
+
(
∂2ϕi0
∂uix∂uiy
)
eq
]
= mω2x,
kx =
1
2
∑
i
(
∂2ϕi0
∂u2iz
)
eq
= mω2z . (2.9)
here a and c are the lattice constants of the hcp crystal, v is the volume and γTG is the
Gruneisen constant.
From the limiting condition of absolute stability of the crystalline state(
∂P
∂v
)
T
= 0, i.e.
(
∂P
∂a
)
T
= 0or
(
∂P
∂c
)
T
= 0 (2.10)
we find the corresponding expression of the limiting temperature Ts as follows:
Ts =
Pv + a
2
4
.∂
2uo
∂a2
− c
2
∂uo
∂c
12kB
[
a
(
∂γTG
∂a
)
− γTG
] . (2.11)
If we take the values of the parameters a, c, kx, kz, ∂kx∂a ,
∂kx
∂a
, ... at the same limiting
temperature of absolute stability Ts then (2.11) can be transformed into the form [9]
Ts =
Pv + a2.∂
2uo
∂a2
− 2c∂uo
∂c
+ a2
(
~ωx
kx
∂2kx
∂a2
+ ~ωz
2kz
∂2kz
∂a2
)
− 2c
(
~ωx
kx
∂kx
∂c
+ ~ωz
2kz
∂kz
∂c
)
kBa
[
2
k2x
∂kx
∂a
(
a∂kx
∂a
+ 2c∂kx
∂c
)
+ 1
k2z
∂kz
∂a
(
a∂kz
∂a
+ 2c∂kz
∂c
)] .
(2.12)
In the case of P = 0, it gives
Ts =
a2.∂
2uo
∂a2
− 2c∂uo
∂c
+ a2
(
~ωx
kx
∂2kx
∂a2
+ ~ωz
2kz
∂2kz
∂a2
)
− 2c
(
~ωx
kx
∂kx
∂c
+ ~ωz
2kz
∂kz
∂c
)
kBa
[
2
k2x
∂kx
∂a
(
a∂kx
∂a
+ 2c∂kx
∂c
)
+ 1
k2z
∂kz
∂a
(
a∂kz
∂a
+ 2c∂kz
∂c
)] . (2.13)
70
Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals...
The nearest neighbor distance or the lattice constant a is determined by a =
a0 + ux0, where a0 denotes the distance a at temperature 0 K and is determined from
the experimental data. The displacement ux0 of a particle from the equilibrium position
in direction x or y is calculated from
ux0 ≈
6∑
i=1
[
γθ
kx
2
]i
ai,γ ≡ 1
4
∑
i
[(
∂3ϕi0
∂u3ix
)
eq
+ 6
(
∂3ϕi0
∂uix∂u2iy
)
eq
]
, (2.14)
where ai(i = 1 − 6) is determined in [9]. Lattice constant c is determined by c =
c0 + uz0, where co denotes the distance c at temperature 0 K and is determined from
the experimental data. Displacement uz0 of a particle from the equilibrium position in
direction z is calculated from
uz0 ≈
[
1
3
6∑
i=1
(
θ
kz
)i
bi
]1/2
,
τ1 ≡ 1
12
∑
i
(
∂4φi0
∂u4iz
)
eq
, τ2 ≡ 1
2
∑
i
(
∂4φi0
∂u2ix∂u
2
iz
)
eq
, τ3 ≡ 1
2
∑
i
(
∂4φi0
∂uix∂uiy∂u2iz
)
eq
.
(2.15)
here bi(i = 1 − 6) is determined in [9]. The equation for calculating nearest neighbor
distances at pressure P and at temperature 0 K has the form [9]
y2 = 0.9231 + 0.3188y4 − 0.0015Pσ
3
ε
y5 − 0.0316y6 + 0.0007Pσ
3
ε
y7 − 0.0001Pσ
3
ε
y9,
y =
(a
σ
)3
. (2.16)
The melting temperature Tm of the hcp crystal is approximately equal to [9]
Tm ≈ Ts+am − as
6kBγsG
[
2Pvs
as
+
1
8
(
∂u0
∂as
+ as
∂2u0
∂a2s
)]
+
cm − cs
6kBγsG
[
2Pvs
cs
+
1
4
(
∂u0
∂cs
+ as
∂2u0
∂c2s
)]
.
(2.17)
where cm = c (Tm.P ) , cs = c (Ts, P ) , vs =
√
3
2
a2scs and other quantities are as in (2.8).
2.2. Numerical results and discussion
For α-CO2 and α-N2O with a fcc structure and for β-N2 and β-CO with a hcp
structure, the interaction potential between two atoms is usually used in the form of the
Lennard-Jones pair potential
ϕ(r) = 4ε
[(σ
r
)12
−
(σ
r
)6]
, (2.18)
where σ is the distance in which ϕ(r) = 0 and ε is the depth of the potential well. The
values of the parameters ε and σ are determined from the experimental data. ε
kB
= 218.82
71
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien
K, σ = 3.829.10−10 m for α-CO2, εkB = 235.48 K, σ = 3.802.10
−10 m for α-N2O, εkB =
95.05 K, σ = 3.698.10−10 m for β-N2 and εkB = 100.1 K, σ = 3.769.10
−10 m for β-CO [3].
Therefore, using two coordinate spheres and the results in [2, 9], we obtain the values of
the crystal parameters for α-CO2 and α- N2O as follows:
k =
4ε
a2
(σ
a
)6 [
265, 298
(σ
a
)6
− 64, 01
]
, γ =
16ε
a4
(σ
a
)6 [
4410, 797
(σ
a
)6
− 346, 172
]
,
(2.19)
where a is the nearest neighbor distance of the fcc crystal at temperature T and the crystal
parameters for β-N2 and β-CO are as follows:
kx =
4ε
a2
(
σ
a
)6 [
614.6022
(
σ
a
)6 − 162.8535] , kz = 4εa2 (σa)6 [286.3722(σa)6 − 64.7487] ,
γ = − 4ε
a3
(
σ
a
)6 [
161.952
(
σ
a
)6 − 24.984] , τ1 = 4εa4 (σa)6 [6288.912(σa)6 − 473.6748] ,
τ2 =
4ε
a4
(σ
a
)6 [
11488.3776
(σ
a
)6
− 752.5176
]
, τ3 =
4ε
a4
(σ
a
)6 [
8133.888
(σ
a
)6
− 737.352
]
.
(2.20)
Our calculated results for the limiting temperature of absolute stability and the
melting temperature of α-CO2, α-N2O, β-N2 and β-CO at different pressures (low
pressures) are expressed in Figures 1-4.
Figure 1. The limiting temperature of absolute stability and the melting temperature
at different pressures for α-CO2
The discrepancy in the melting temperature of α-CO2 that exists between our
calculated results and the experimental data [7] is 5.3% at P = 0 and increases to 18%
at P = 1000 bar.
72
Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals...
Figure 2. The limiting temperature of absolute stability and the melting temperature
at different pressures for α-N2O
The discrepancy in the melting temperature of α-N2O that exists between our
calculated results and the experimental data [7] is 0.04% at P = 0 and increases to 7%
at P = 1000 bar.
Figure 3. The limiting temperature of absolute stability and the melting temperature
at different pressures for β-N2
73
Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien
The discrepancy in the melting temperature of β-N2 that exists between our
calculated results and the experimental data [8] is 2.46% and between our calculated
results and the experimental data [3] it is 2.69% at P = 0 and increases to 5% compared
with the experimental data [8] at P = 100 bar. Our calculation is better than that in [10].
Figure 4. The limiting temperature of absolute stability and the melting temperature
at different pressures for β-CO
The discrepancy in the melting temperature of β-CO that exists between our
calculated results and the experimental data [8] is 5.75% at P = 0 and increases to 7.68%
at P = 100 bar.
3. Conclusion
From the SMM and the limiting condition of absolute stability for the crystalline
state, we find the equation of state, the limiting temperature of absolute stability for the
crystalline state and the melting temperature for crystals with fcc and hcp structures at zero
pressure and under pressure, the equation for calculating the nearest neighbor distances
at pressure P and at temperature 0 K for fcc and hcp crystals. These results are analytic
and general.
Theoretical results are applied to determine the melting temperature for molecular
cryocrystals of nitrogen type (N2, CO, N2O, CO2) with fcc and hcp structures in the
interval of pressure from 0 to 100 bar for β-N2, β-CO and from 0 to 1000 bar for
α-CO2, α-N2O. In general, our numerical calculations are in good agreement with the
experimental data [3, 6-8] and other calculation [10], especially for α-N2O, β-N2 and
β-CO molecular cryocrystals. Our obtained results can be enlarged to cases in higher
pressures.
74
Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals...
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