Abstract. The α - β (fcc-hcp) phase transition temperature under pressure can
be investigated theoretically by combining the self-consistent field method and
the statistical moment method and only the statistical moment method which was
developed by the authors. Theoretical results are applied to N2 and CO molecular
cryocrystals and our calculated results 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. 93-100
This paper is available online at
STRUCTURAL PHASE TRANSITION TEMPERATURE
OF N2 AND COMOLECULAR CRYOCRYSTALS UNDER PRESSURE
Nguyen Quang Hoc1, Bui Duc Tinh1, Dinh Quang Vinh1 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 - (fcc-hcp) phase transition temperature under pressure can
be investigated theoretically by combining the self-consistent field method and
the statistical moment method and only the statistical moment method which was
developed by the authors. Theoretical results are applied to N2 and CO molecular
cryocrystals and our calculated results are compared with the experimental data
and other calculations.
Keywords: Self-consistent field, statistical moment, molecular cryocrystal.
1. Introduction
Electronic transitions lie at the heart of many high-pressure phenomena including
structural stability and metallization [1]. Such changes in the underlying electronic
structure drive concomitant sequences of structural phase transitions and there has
recently been intense interest in such changes for the alkali and alkaline earth metals.
He and Xe transform from face-centered cubic (fcc) structures to hexagonal close-packed
(hcp) structures at high pressures, but corresponding transitions have not been reported
for Ne, Ar or Kr.
The properties of rare-gas solids at high pressures are of fundamental interest
because they provide an ideal system due to their closed-shell electronic configurations,
allowing fruitful comparison between experiment and theory.
The polymorphous transformation is the phase transition between the different
configurations of a crystal. Because this transformation temperature is smaller than the
melting temperature, in order to investigate it, Kotenok [2] applied the one-particle
distribution method (OPDM) of Bazarov [3] in the quasiharmonic approximation of the
Received August 19, 2014. Accepted October 21, 2014.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
93
Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien
self-consistent field method and found that the calculated phase transition temperatures
for solid nitrogen and carbon monoxide were quite near that of the experimental data.
This phase transition temperature is still determined by the self-consistent phonon
method (SCPM) [4], the self-consistent field method (SCFM) [5] taking into account
the molecular rotational motion and the statistical moment method (SMM) [6]. However,
in these works the change of the thermodynamic quantities of molecular cryocrystals at
the structural phase transition temperature was not calculated.
The α - β structural phase transition is a Type I phase transition because the volume
and the entropy change suddenly at this phase transition. Experimental data diverge
significantly for important characteristics of this phenomenon such as the lattice constant
at the phase transition, the phase transition pressure and the change of entropy. Theoretical
methods such as the lattice dynamics method, the thermodynamic method and the
one-particle distribution method, including the calculation of many-particle interaction,
apply the harmonic approximation and do not take into account the correlations between
particles.
The present paper presents a theoretical study of the α - β structural phase transition
temperature under pressure based on the SMM and the SMM combining with the SCFM.
Theoretical results are applied to N2 and CO molecular cryocrystals and our calculated
results are compared with experiments and other calculations.
2. Content
2.1. Theoretical formulation
2.1.1. Determination of the α - β phase transition temperature by combining the
SCFM and the SMM
In a quasiharmonic approximation of the SCFM, the free energy per particle at the
lattice node has the form [5]
ψ
N
= −3
2
θ ln(2πθ) +
θ
2
lnD +
u0
2
(2.1)
where u0 =
∑
i
φi0 (|⃗ai|) is the potential energy of a particle at the lattice node, ϕi0 is the
interaction potential between the ith particle and the 0th particle, D is the determinant
of the matrix
∂2u0∂u1α∂u1β
and is equal to the product of the diagonal elements of this
matrix belonging to the principal axis, θ = kBT with kB being the Boltzmann constant.
The polymorphous transformation temperature or the α - β phase transition temperature
Tα−β =
θα−β
kB
is determined using the equation system
94
Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure
θα−β ln
(
Dα
Dβ
)
+ 2P (vα − vβ) = u0β − u0α, P = −∂ψα
∂vα
= −∂ψβ
∂vβ
. (2.2)
The real solutions of the equation system (2.2) at determining P or T give us the
temperature or pressure of the polymorphous transformation. The volume discontinuity
∆v − vβ − vα per particle and the corresponding heat quantity of the transition Q are
determined from (2.2):
∆v = vβ − vα, Q = u0β − u0α + P (vβ − vα) . (2.3)
At pressure P ≈ 1 atm we can suppose P = 0 in (2.2) and then,
θα−β =
u0β − u0α
ln
(
D
D
) , ∂ψα
∂vα
=
∂ψβ
∂vβ
= 0. (2.4)
At T = Tα−β, the structure is unstable and we have a structural phase transition.
We can calculate u0β − u0α and Dα
Dβ
from the SMM. We apply equation (2.4) in order to
find the phase transition temperature Tα−β in solid nitrogen.
2.1.2. Determination of the α - β phase transition temperature from the SMM
The equilibrium condition of the α and β structural phases at the Tα−β phase
transition temperature has the form
Tα = Tβ = Tα−β, Pα = Pβ = Pα−β, µα = µβ, Gα = Gβ, (2.5)
where T is the temperature, P is the pressure, µ is the chemical potential and G is the
Gibbs thermodynamic potential. From (2.5) and because
G = ψ + PV, (2.6)
where ψ is the free energy, V = Nv is the volume, N is the number of particle and v is
the volume per particle, we have
ψα + Pα−βNvα = ψβ + Pα−βNvβ, (2.7)
where vα =
a3α√
2
is the volume of the α phase with the fcc structure, aα is the nearest
neighbour distance in the fcc lattice, vβ =
√
3
4
a2βcβ is the volume of the β phase with the
hcp structure, aβ and cβ are the lattice constants of the hcp lattice.
95
Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien
The nearest neighbour distance aα is determined by aα = a0α + ux0α, where a0α
denotes the distance aα at temperature 0 K and is determined by the experimental data.
The displacement ux0α of a particle from the equilibrium position is calculated from
ux0α =
√
2γαθ2
3k3α
A, γα =
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
(
γαθ
k2α
)i
ai, kα =
1
2
∑
i
(
∂2ϕi0
∂u2iγ
)
eq
= mω2α, xα =
~ωα
2θ
, (2.8)
where ai(i = 1− 6) is determined in [6].
The nearest neighbour distance or the lattice constant aβ is determined by aβ =
a0β + ux0β, where a0β denotes the distance aβ at temperature 0 K and is determined by
the experimental data. The displacement ux0β of a particle from the equilibrium position
in the direction x or y is calculated from
ux0β =
6∑
i=1
(
γβθ
k2xβ
)i
i
ai, kxβ =
1
2
∑
i
[(
∂2ϕi0
∂u2ix
)
eq
+
(
∂2ϕi0
∂uix∂uiy
)
eq
]
= mω2xβ,
γβ =
1
4
∑
i
[(
∂3ϕi0
∂u3ix
)
eq
+
(
∂3ϕi0
∂uix∂u2iy
)
eq
]
, (2.9)
where ai(i = 1− 6) is determined in [6].
The lattice constant c is determined by c = c0 + uz0, where c0 denotes the distance
c at temperature 0K and is determined by the experimental data. The displacement uz0β
of a particle from the equilibrium position in the direction z is calculated from
uz0β ≈
[
1
3
6∑
i=1
(
θ
kzβ
)i
bi
]1/2
, kzβ =
1
2
∑
i
(
∂2ϕi0
∂u2iz
)
eq
= mω2zβ, (2.10)
where bi(i = 1− 6) is determined in [6].
The free energy of the phase with the fcc structure in the harmonic approximation
is determined by
ψα = 3N
{u0α
6
+ θ
[
xα + ln
(
1− e−2x)]} . (2.11)
The free energy of the β phase with the hcp structure in the harmonic approximation
is determined by
ψβ = 3N
{
u0β
6
+
2
3
θ
[
xβ + ln
(
1− e−2x)]+ 1
3
θ
[
xzβ + ln
(
1− e−2xz)]} . (2.12)
96
Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure
At pressure P = 0, from Gα = Gβ we have ψα = ψβ. From that and using (2.11)
and (2.12), we find the α− β phase transition temperature
Tα−β =
u0β − u0α
6kB
{
xα − 23xβ − 13xzβ + ln
[
1−e−2x
(1−e−2x)2=3(1−e−2xz)1=3
]} . (2.13)
At pressure P , the α - β phase transition temperature is determined by combining
(2.7), (2.11) and (2.12) and is equal to
Tα−β =
u0β − u0α + P
(√
3
2
a2βcβ −
√
2a3α
)
6kB
{
xα − 23xβ − 13xzβ + ln
[
1−e−2x
(1−e−2x)2=3(1−e−2xz)1=3
]} . (2.14)
2.2. Numerical results and discussion
For solid N2 and CO, 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]
, (2.15)
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 by the following experimental data:
ε
kB
= 95.05 K and σ = 3.698.10−10 m for α − N2, ε
kB
= 110.07 K and σ = 3.59.10−10
m for α − CO, ε
kB
= 95.05 K and σ = 3.698.10−10 m for β − N2, ε
kB
= 100.1 K
and σ = 3.769.10−10 m for β-CO [7]. Therefore, using two coordination spheres and
the results in [6], we obtain the values of the crystal parameters for alpha-N2 and α-CO
as follows:
kα =
4εα
a2α
(
σα
aα
)6 [
265.298
(
σα
aα
)6
− 64.01
]
γα =
16εα
a4α
(
σα
aα
)6 [
4410.797
(
σα
aα
)6
− 346.172
]
γ1α =
4εα
a4α
(
σα
aα
)6 [
803.555
(
σα
aα
)6
− 40.547
]
γ2α =
4εα
a4α
(
σα
aα
)6 [
3607.242
(
σα
aα
)6
− 305.625
]
97
Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien
u0α = 4εα
(
σα
aα
)6 [
12.132
(
σα
aα
)6
− 14.454
]
(2.16)
and the crystal parameters for β-N2 and β-CO as follows:
kxβ =
4εβ
a2β
(
σβ
aβ
)6 [
614.6022
(
σβ
aβ
)6
− 162.8535
]
,
kzβ =
4εβ
a2β
(
σβ
aβ
)6 [
286.3722
(
σβ
aβ
)6
− 64.7487
]
,
γβ = −4εβ
a3β
(
σβ
aβ
)6 [
161.952
(
σβ
aβ
)6
− 24.984
]
,
τ1β =
4εβ
a4β
(
σβ
aβ
)6 [
6288.912
(
σβ
aβ
)6
− 473.6748
]
,
τ2β =
4εβ
a4β
(
σβ
aβ
)6 [
11488.3776
(
σβ
aβ
)6
− 752.5176
]
,
τ3β =
4εβ
a4β
(
σβ
aβ
)6 [
8133.888
(
σβ
aβ
)6
− 737.352
]
,
τ4β =
4εβ
a4β
(
σβ
aβ
)6 [
43409.3184
(
σβ
aβ
)6
− 4550.04
]
,
τ5β =
4εβ
a4β
(
σβ
aβ
)6 [
11315.6064
(
σβ
aβ
)6
− 1006.0428
]
,
τ6β =
4εβ
a4β
(
σβ
aβ
)6 [
40782.6048
(
σβ
aβ
)6
− 4189.6536
]
,
u0β = 4εβ
(
σβ
aβ
)6 [
11.648
(
σβ
aβ
)6
− 14.1601
]
. (2.17)
Using the definitions of u0α, u0β, kα, kxβ, kzβ, Dα, Dβ and 73 coordination spheres,
we have the following results [6].
u0β − u0α = 4εβ
(
σβ
aβ
)6 [
11.648
(
σβ
aβ
)6
− 14.1601
]
− 4εα
(
σα
aα
)6 [
12.232
(
σα
aα
)6
− 14.154
]
,
98
Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure
Dα
Dβ
=
k3α
k2xβkzβ
=
{
4ε
a2
(
σ
a
)6 [
265.298
(
σ
a
)6
− 64.01
]}3
{
4ε
a2
(
σ
a
)6 [
385.071
(
σ
a
)6
− 99.2106
]}2{
4ε
a2
(
σ
a
)6 [
286.5978
(
σ
a
)6
− 64.1154
]}
(2.18)
where we do not use (2.9) and (2.17) but we apply
kxβ =
1
2
∑(∂2ϕi0
∂u2ix
)
eq
= mω2xβ =
4εβ
a2β
(
σβ
aβ
)6 [
385.071
(
σβ
aβ
)6
− 99.2106
]
. (2.19)
Using
ε
kB
= 95.1 K and σ = 3, 708.10−10 m for α − N2 [7] and ε
kB
= 96.57
K and σ = 3.73.10−10m for β-N2 [6] and calculating approximately aα ≈ aα(T = 0),
aβ ≈ aβ(T = 0), we find the phase transition temperature Tα−β of solid nitrogen. Our
calculated result is compared with experiments and other calculations in Table 1. Our
result based on combining SMM and SCFM is in very good agreement with experimental
results [9] and calculation [8]. Our result is better than other calculations in [2, 8-10].
Table 1. Phase transition temperature T− of solid nitrogen
Method Tα−β , K
EXPT. [8] 35.6
EXPT. [9] 38.3
CAL. by combining SMM and SCFM according to formula (2.4) of this paper 39.1
CAL. [8] from SCPM using the free rotation molecule model 47
CAL. [8] from SCPM using the disordered direction distribution model 37.4
CAL.[9] using molecular field theory (MFT) 54.4
CAL.[10] using classical ordered theory (COT) 74.9
CAL. [2] using OPDM 43.8
At pressure P = 0, our calculated phase transition temperatures from the formula
(2.13) are equal to Tα−β = 17.3 K for solid N2 and Tα−β = 48 K for solid CO. These
results are far from the experimental data Tα−β = 35.6 K for solid N2 and Tα−β = 61.6
K for solid CO. Our phase transition temperatures from the formula (2.14) at pressure
P = 50 bar are equal to Tα−β = 29.2 K for solid N2 and Tα−β = 57.6 K for solid CO
and at pressure P = 100 bar are equal to Tα−β = 41.2 K for solid N2 and Tα−β = 79.7
K for solid CO. When the pressure increases, the phase transition temperature increases.
That is in agreement with experimental results [7].
99
Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien
3. Conclusion
In this paper, we derive the analytic expressions (2.4), (2.16) - (2.19) of the
α - β structural phase transition temperature based on combining SCFM and SMM.
These results are applied to cryocrystal of N2 and our numerical calculation is in very
good agreement with experiments [9] and calculation [8]. Our result is better than
other calculations in [2, 8-10]. We still give the analytic expressions (2.13), (2.14)
of the α - β structural phase transition temperature based on SMM. The numerical
calculations from these results for N2 and CO cryocrystals at pressures 0, 50 and 100 bar
also are compared with experiments. When the pressure increases, the phase transition
temperature increases. That is in agreement with experiments [7]. Our calculated results
from combining SCFM and SMM are better than our calculated results only from SMM.
REFERENCES
[1] R. J. Hemley and N. W. Asheroft, 1998. The revealing role of pressure in the
condensed-matter sciences. Phys. Today 51, No. 8, pp. 26-52.
[2] N. V. Kotenok, 1972. Theory of polymorphic transformation, MGUBulletin, Physics
and Astronomy Series No. 1, pp. 40-43 (in Russian).
[3] I. P. Bazarov, 1972. Statistical theory of the crystalline state. MGU (in Russian).
[4] J. C. Raich, N. S. Gillis and T. R. Koehler, 1974. Self-consistent phonon calculation
of the β phase and the α − β transition in solid nitrogen. J. Chem. Phys. 61, No. 4,
pp. 1411-1414.
[5] J. C. Raich and R. D. Etters, 1972. Librational motion in solid α−N2. J. Low Temp.
Phys. 7, No. 5/6, pp. 449-458.
[6] Nguyen Quang Hoc, Do Dinh Thanh and Nguyen Tang, 1996. The limiting of
absolute stability, the melting and the α− β phase transition temperatures for solid
nitrogen. Communications in Physics 6, No. 4, pp. 1-8.
[7] B. I. Verkina, A. Ph. Prikhotko (ed.), 1983. Cryocrystals, Kiev, pp.1-526
(in Russian).
[8] A. F. Schuch and R. L. Mills, 1970. Crystal structure of the three modifications
of nitrogen 14 and nitrogen 15 at high pressure. J. Chem. Phys. 52, No.12, pp.
6000-6008.
[9] T. A Scott, Phys, Rev. C 27, No. 3 (1976) 40.
[10] B. Kohin, 1960. Molecular rotation in crystals of N2 and CO. J. Chem. Phys. 33,
No. 3, pp. 882-889.
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