Abstract. The analytic expressions of thermodynamic quantities such as the Helmholtz
free energy, the internal energy, the entropy, the molar heat capacity at constant volume
for molecular cryocrystals of N2 type with face-centered cubic (FCC) and hexagonal
close-packed (HCP) structures in harmonic, classical and anharmonic approximations are
obtained by combining the statistical moment method (SMM) and the self-consistent field
method (SCFM)

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0038
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 94-103
This paper is available online at
ANALYTIC EXPRESSION OF THERMODYNAMIC QUANTITIES FORMOLECULAR
CRYOCRYSTALS OF NITROGEN TYPEWITH FCC AND HCP STRUCTURES
IN HARMONIC, CLASSICAL AND ANHARMONIC APPROXIMATIONS
Nguyen Quang Hoc1, Mai Thi La1, Vo Minh Tien2 and Dao Kha Son2
1Faculty of Physics, Hanoi National University of Education
2Faculty of Physics, Tay Nguyen University
Abstract. The analytic expressions of thermodynamic quantities such as the Helmholtz
free energy, the internal energy, the entropy, the molar heat capacity at constant volume
for molecular cryocrystals of N2 type with face-centered cubic (FCC) and hexagonal
close-packed (HCP) structures in harmonic, classical and anharmonic approximations are
obtained by combining the statistical moment method (SMM) and the self-consistent field
method (SCFM).
Keywords: Statistical moment method, self-consistent field method, cryocrystal.
1. Introduction
Molecular crystals, comprising a vast and comparatively little investigated class of solids,
are characterized by a diversity of properties. Up to now only solidified noble gases have
systematically been investigated and this is due to the availability of relevant theoretical models and
the ease of comparing theories with experimental results. Recently, experimental data have been
obtained for simple non-monoatomic molecular crystals as well, and this in turn has stimulated the
appearance of several theoretical papers on that subject.
This paper deals with the analysis of thermodynamic properties of the group of
non-monoatomic molecular crystals including solid N2 and CO that have similar physical
properties. These crystals are formed by linear molecules and in their ordered phase, the molecular
centers of mass are situated at the site of a face-centered cubic (FCC) pattern, the molecular axes
being directed to the four spatial diagonals of a cube (space group Pa3). The characteristic feature
of the intermolecular interaction in such crystals is that the non-central part of the potential results
from quadrupole forces and from the part of valence and dispersion forces having analogous
angular dependence as quadrupole forces, and further, that dipole interaction either does not exist
(N2) or is negligible (CO) to influence the majority of thermodynamic properties. In addition,
all crystals considered have a common feature, namely their intrinsic rotational temperatures
B = ~2/(2I) (I is the momentum of inertia of the corresponding molecule) are small compared
to the energy of non-central interaction.
Received December 11, 2014. Accepted October 1, 2015.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
94
Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
In the low-temperature range, it is reasonable to apply an assumption successfully used
by the authors [1, 2] that translational motions of the molecular system are independent. As
shown [3] there are two types of excitations in molecular crystals - phonons and librons and,
furthermore, the thermodynamic functions can be written as a sum of two independent terms
corresponding to each subsystem. In such a treatment, the translational–orientational interaction
leads to a renormalization of the sound velocity and of the libron dispersion law only.
The investigation of the librational behavior of molecules is usually carried out within the
framework of the harmonic approximation. However, anharmonic effects for the thermodynamic
properties are essential at temperatures substantially lower than the orientational disordering
temperature. The effect of molecular rotations in N2 and CO crystals not restricted by the
assumption of harmonicity of oscillations has been calculated numerically in the molecular field
approximation by Kohin [4]. Full calculations on thermodynamic properties of molecular crystals
of nitrogen type are given by the statistical moment method (SMM) in [5] and by self-consistent
field method (SCFM) in [6]. This paper represents the analytic expressions of thermodynamic
quantities for molecular cryocrystals of nitrogen type with FCC and hexagonal close-packed (HCP)
structures such as the free energy, the energy, the entropy and the heat capacity at constant volume
in harmonic, classical and anharmonic approximations.
2. Content
2.1. Analytic expression of thermodynamic quantites for crystals of N2
type from the combination of SMM and SCFM
2.1.1. Free energy
By combining the SMM and the SCFM, the free energy of molecular crystals of N2
type with FCC and HCP structures is the sum of the vibrational free energy and the rotational
free energy. In harmonic approximation (harmonic approximation of lattice vibration and
pseudo-harmonic approximation of molecular rotational motion) for FCC crystal [6, 7],
ψfcc,har = ψfcc,harvib + ψ
fcc,har
rot ,
ψfcc,harvib = V
fcc
0 + ψ
fcc
0vib = V
fcc
0 + 3Nθ
[
xfcc + ln(1− e−2xfcc)
]
,
ψharrot = kBN
{
2T ln
[
4 sinh
ξ
2T
]
− U0η + U0η
2
2
}
,
kfcc ≡ 1
2
∑
i
(
∂2ϕi0
∂u2iα
)
eq
≡ mωfcc2,α = x, y, z, x = ~ω
fcc
2θ
, θ = kBT, V
fcc
0 =
N
2
∑
i
ϕi0
(2.1)
and for HCP crystal [7, 8],
ψhcp,har = ψhcp,harvib + ψ
har
rot ,
ψhcp,harvib = V
hcp
0 +ψ
hcp
0vib = V
hcp
0 +2Nθ
[
xhcp + ln
(
1− e−2xhcp
)]
+Nθ
[
xhcpz + ln
(
1− e−2xhcpz
)]
,
khcpx ≡
1
2
∑
i
(
∂2ϕi0
∂u2ix
+
∂2ϕi0
∂uix.∂uiy
)
eq
≡ mωhcp2x , xhcp =
~ωhcpx
2θ
, V hcp0 =
N
2
∑
i
ϕi0,
95
Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
khcpz ≡
1
2
∑
i
(
∂2ϕi0
∂u2iz
)
eq
≡ mωhcp2z , xhcpz =
~ωhcpz
2θ
. (2.2)
In classical approximation for FCC crystal [6, 7],
ψfcc,cla = ψfcc,clavib + ψ
cla
rot,
ψfcc,clavib ≈ V fcc0 + ψfcc0vib + 3N
{
θ2
kfcc2
(
γfcc2 − γfcc1
)
+
4θ3
kfcc4
[
γfcc22 − 3
(
γfcc21 + 2γ
fcc
1 γ
fcc
2
)]}
,
ψfcc0vib = Nθ
[
xfcc + ln(1− e−2xfcc)
]
,
xfcc ≈ 0.283, ψ
cla
rot − ψcla0rot
kBN
= −U0η + U0η
2
2
+ 2T ln
(√
6BU0η
T
)
,
γfcc1 ≡
1
48
∑
i
(
∂4ϕi0
∂u4iα
)
eq
,γfcc2 ≡
1
8
∑
i
(
∂4ϕi0
∂u2iα∂u
2
iβ
)
eq
,α 6= β, α, β = x, y, z (2.3)
and for HCP crystal [7, 8],
ψhcp,cla = ψhcp,clavib + ψ
cla
rot, ψ
hcp,cla
vib = V
hcp
0 + ψ
hcp
0vib
+
Nθ2
4
[
6τhcp5 + τ
hcp
6
khcp2x
+
3τhcp1
khcp2z
+
2τhcp2
khcpx k
hcp
z
]
+
3Nθ3
4khcp2z
.
[
τhcp21
khcp2z
+
τhcp22
9khcp2x
]
− Nθ
4
36khcp3z
[
33τhcp31
khcp3z
+
2τhcp32
khcp3x
]
,
kz ≡ 1
2
∑
i
(
∂2ϕio
∂u2iz
)
eq
, τ1 ≡ 1
12
∑
i
(
∂4ϕi0
∂u4iz
)
eq
, τ2 ≡ 1
2
∑
i
(
∂4ϕi0
∂u2ix∂u
2
iz
)
eq
,
τ5 ≡ 1
12
∑
i
(
∂4ϕi0
∂u4ix
)
eq
, τ6 ≡ 1
2
∑
i
(
∂4ϕi0
∂u2ix∂u
2
iz
)
eq
. (2.4)
In anharmonic approximation (anharmonic approximation of lattice vibration and
self-consistent libron approximation of molecular rotational motion) for FCC crystal [6, 7],
ψfcc,anh = ψfcc,anhvib + ψ
anh
rot ,
ψfcc,anhvib = V
fcc
0 + ψ
fcc
0vib + 3N
{
θ2
kfcc2
[
γfcc2 X
fcc2 − 2γ
fcc
1
3
(
1 +
Xfcc
2
)]
+
2θ3
kfcc4
[
4
3
γfcc22 X
fcc
(
1 +
Xfcc
2
)
− 2
(
γfcc21 + 2γ
fcc
1 γ
fcc
2
)(
1 +
Xfcc
2
)(
1 +Xfcc
)]}
,
96
Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
ψanhrot
kBN
= 2T ln
[
4 sinh
ξ
2T
]
− ξ
2
coth2
ξ
2T
− B
2
− U0η
2
2
(2.5)
and for HCP crystal [7, 8],
ψhcp,anh = ψhcp,anhvib + ψ
anh
rot ,
ψhcp,anhvib = V
hcp
0 + ψ
hcp
0vib +
Nθ2
12
[
6τhcp5 + τ
hcp
6
khcp2x
(
Xhcp + 2
)
+
3τhcp1
khcp2z
(Xhcpz + 2)
+
τhcp2
khcpx k
hcp
z
(Xhcp +Xhcpz + 4)
]
+
Nθ3
24khcp2z
[
τhcp21
khcp2z
(
Xhcpz + 2
)
(Xhcpz + 5)
+
τhcp22
9khcp2x
(
Xhcp + 2)(Xhcp + 5
)]
− Nθ
4
108khcp3z
[
τhcp31
khcp3z
(
Xhcpz + 2
)
(3Xhcp2z + 17X
hcp
z + 13)
+
τhcp32
9khcp3x
(
Xhcp + 2)2(Xhcp + 5
)]
,
Xhcp ≡ xhcp cothxhcp,Xhcpz ≡ xhcpz coth xhcpz . (2.6)
In above mentioned expressions, kB is the Boltzmann constant, T is the absolute
temperature, m is the mass of particle at lattice node, ωfcc, ωhcpx , ω
hcp
z are the frequencies of lattice
vibration, kfcc, γfcc1 , γ
fcc
2 , k
hcp
x , k
hcp
z , τ
hcp
1 , τ
hcp
2 , τ
hcp
5 , τ
hcp
6 are the parameters of FCC and HCP
crystals depending on the structure of crystal lattice and the interaction potential between particles
at nodes, ϕi0 is the interaction potential between the ith particle and the 0th particle, uiα is the
displacement of the ith particle from equilibrium position in direction α and N is the number of
particles per mole or the Avogadro number, U0 is the barrier which prevents the molecular rotation
at T = 0 K, B = ~2/(2I) is the intrinsic rotational temperature or the rotational quantum or the
rotational constant, ξ is the energy of rotational excitation and η is the ordered parameter.
In the harmonic, classical and anharmonic approximations, the rotational free energy, the
rotational energy, the rotational entropy and the rotational heat capacity at constant volume of
HCP crystal are identical to that of FCC crystal. This is a common property of crystals of nitrogen
type [7].
In the harmonic approximation of lattice vibration, the quantities such as
V fcc0 , V
hcp
0 , k
fcc, khcpx , k
hcp
z are expressed in terms of the nearest neighbor distance a0 at 0
K. The pseudo-harmonic approximation of molecular rotational motion corresponds to the
condition T√
U0Bη
<< 1 [7].
In the classical approximation of lattice vibration, xcthx ≡ X = 1, xzcthxz ≡ Xz = 1
or x = xz ≈ 0.283. Then the formulae of ψ,E, S,CV obtained in anharmonic approximation
97
Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
give the classical results. From that we can determine the temperature Tlim ≈ 1.3494.10−11ω
corresponding to that where the quantum effects can be neglected [6]. The classical approximation
of molecular rotational motion corresponds to the condition T√
U0Bη
>> 1 [7].
In the anharmonic approximation of lattice vibration, the quantities such as
kfcc, γfcc1 , γ
fcc
2 , k
hcp
x , k
hcp
z , τ
hcp
1 , τ
hcp
2 , τ
hcp
5 , τ
hcp
6 are expressed in terms of the nearest neighbor
distance afcc = afcc0 + u
fcc
x0 , a
hcp = ahcp0 + u
hcp
x0 , where a
fcc
0 , a
hcp
0 respectively are the nearest
neighbor distance of FCC and HCP crystals at 0 K and the displacements ufccx0 , u
hcp
x0 of a particle
from the equilibrium position are calculated by
ufccx0 =
√
2γfccθ2
3kfcc
3 A, γ
fcc ≡ 1
12
∑
i
(∂4ϕi0
∂u4iβ
)
eq
+ 6
(
∂4ϕi0
∂u2iβ∂u
2
iγ
)
eq
, β 6= γ, β, γ = x, y, z,
A = a1 +
6∑
i=2
(
γfccθ
kfcc
2
)i
afcci , u
hcp
x0 =
6∑
i=1
γhcpθ(
khcpx
)2
i
ahcpi ,
γhcp ≡ 1
4
∑
i
(∂3ϕi0
∂u3ix
)
eq
+
(
∂3ϕi0
∂uix∂u2iy
)
eq
, (2.7)
where afcci , a
hcp
i (i = 1 − 6) are determined in [8]. The anharmonic approximation of molecular
rotational motion corresponds to the condition T√
U0Bη
≈ 1[7].
2.1.2. Energy
The energy of molecular crystals of N2 type with FCC and HCP structures is the sum of the
vibrational energy and the rotational energy. In harmonic approximation for FCC crystal [6, 7],
Efcc,har = ψfcc,har − T
(
∂ψfcc,har
∂T
)
V
= Efcc,harvib +E
har
rot ,
Efcc,harvib = V
fcc
0 + E
fcc
0vib = V
fcc
0 + 3NθX
fcc,
Eharrot = −NkBU0
[
1− 3B
ξ
coth
ξ
2T
]
+
NkBU0
2
[
1− 3B
ξ
coth
ξ
2T
]2
−NkB
(
T
∂ξ
∂T
− ξ
)
coth
ξ
2T
+
3NkBU0BT
2ξ2
[
1 +
3B
ξ
coth
ξ
2T
]{
ξ
2T 2
(
T
∂ξ
∂T
− ξ
)[
1− coth2 ξ
2T
]
− ∂ξ
∂T
coth
ξ
2T
}
(2.8)
and for HCP crystal [7, 8],
Ehcp,har = Ehcp,harvib + E
har
rot , E
hcp,har
vib = V
hcp
0 + E
hcp
0vib = V
hcp
0 +Nθ(2X
hcp +Xhcpz ). (2.9)
In classical approximation for FCC crystal [6, 7],
98
Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
Efcc,cla = Efcc,clavib + E
cla
rot, E
fcc,cla
vib = V
fcc
0 + 3Nθ
[
1 +
θ
kfcc2
(
γfcc1 − γfcc2
)]
,
Eclarot = ψ
cla
0rot +NkB
[
−U0η + 1
2
U0η
2 + 2T ln
(√
6BU0η
T
)]
− T ∂
∂T
{ψ0rot + kBN [−U0η
+
1
2
U0η
2 + 2T ln
(√
6BU0η
T
)]}
= Ecla0rot + kBNU0η
(η
2
− 1
)
+
kBNTU0
2
∂η
∂T
(1− 2η)
−2kBNT
(
T
2η
∂η
∂T
− 1
)
, η =
1
2
(
1−
√
1− 4T
U0
)
,
∂η
∂T
= − 2
U0
√
1− 4TU0
(2.10)
and for HCP crystal [7, 8],
Ehcp,cla = Ehcp,clavib + E
cla
rot,
Ehcp,clavib = V
hcp
0 + E
hcp,cla
0vib −
Nθ2
4
(
6τhcp5 + τ
hcp
6
)
khcp2x
+
3τhcp1
khcp2z
− τ
hcp
2
khcpx k
hcp
z
. (2.11)
In anharmonic approximation for FCC crystal [6, 7],
Efcc,anh = Efcc,anhvib + E
anh
rot ,
Efcc,anhvib = V
fcc
0 + 3NθX
fcc +
3Nθ2
kfcc2
[
γfcc2 X
fcc2 +
γfcc1
3
(
2 + Y fcc2
)
− 2γfcc2 XfccY fcc2
]
,
Eanhrot = NkB
{
2T
[
ln
(
4 sinh
ξ
2T
)]
− ξ
2
coth2
ξ
2T
− B
2
− U0η
2
2
}
−kBTN ∂
∂T
{
2T
[
ln
(
4 sinh
ξ
2T
)]
− ξ
2
coth2
ξ
2T
− B
2
− U0η
2
2
}
=
kBN
2
coth
ξ
2T
(
ξ coth
ξ
2T
+ T
∂ξ
∂T
)
−kBN
[
B
2
+ U0η
(
1
2
− T ∂η
∂T
)]
− kBN
4
(
T
∂ξ
∂T
− ξ
)(
ξ
T
1
sinh2 ξ2T
− 1
)
(2.12)
and for HCP crystal [7, 8],
Ehcp,anh = Ehcp,anhvib + E
anh
rot , E
hcp,anh
vib = V
hcp
0 +E
hcp
0vib
−Nθ
2
12
[
6τhcp5 + τ
hcp
6
khcp2x
(
2 + Y hcp2
)
+
3τhcp1
khcp2z
(
2 + Y hcp2z
)
+
τhcp2
khcpx k
hcp
z
(
1 + Y hcp2 + Y hcp2z
)]
,
(2.13)
99
Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
2.1.3. Entropy
The entropy of molecular crystals of N2 type with FCC and HCP structures is the sum of the
vibrational entropy and the rotational entropy. In harmonic approximation for FCC crystal [6, 7],
Sfcc,har =
Efcc,har − ψfcc,har
T
= Sfcc,harvib +S
har
rot , S
fcc,har
vib = 3NkB
[
Xfcc − ln
(
2 sinhxfcc
)]
,
Sharrot = −
NkBU0
T
[
1− 3B
ξ
coth
ξ
2T
]
+
NkBU0
2
[
1− 3B
ξ
coth
ξ
2T
]2
−NkB
T
(
T
∂ξ
∂T
− ξ
)
coth
ξ
2T
+ +
3NkBU0BT
2ξ2
[
1 +
3B
ξ
coth
ξ
2T
]
×
{
ξ
2T 2
(
T
∂ξ
∂T
− ξ
)[
1− coth2 ξ
2T
]
− ∂ξ
∂T
coth
ξ
2T
}
− kBN
T
{
2T ln
[
4 sinh
ξ
2T
]
− U0η + U0η
2
2
}
(2.14)
and for HCP crystal [7, 8],
Shcp,har = Shcp,harvib + S
har
rot ,
Shcp,harvib = NkB
{
2
[
Xhcp − ln
(
2 sinhxhcp
)]
+
[
Xhcpz − ln
(
2 sinhxhcpz
)]}
. (2.15)
In classical approximation for FCC crystal [6, 7],
Sfcc,cla = Sfcc,clavib + S
cla
rot,
Sfcc,clavib = 3NkB
[
1− ln
(
2 sinhxfcc
)]
+
6NkBθ
kfcc2
(
γfcc1 − γfcc2
)
, xfcc ≈ 0.283,
Sclarot = S
cla
0rot +
kBNU0η
T
(η
2
− 1
)
+
kBNU0
2
∂η
∂T
(1− 2η) − 2kBN
(
T
2η
∂η
∂T
− 1
)
−kBN
T
[
−U0η + U0η
2
2
+ 2T ln
(√
6BU0η
T
)]
(2.16)
and for HCP crystal [7, 8],
Shcp,cla = Shcp,clavib + S
cla
rot,
Shcp,clavib = S
hcp
0vib +
NkBθ
2
(
6τhcp5 + τ
hcp
6
)
khcp2x
− 3τ
hcp
1
khcp2z
− 2τ
hcp
2
khcpx k
hcp
z
. (2.17)
In anharmonic approximation for FCC crystal [6, 7],
Sfcc,anh = Sfcc,anhvib + S
anh
rot ,
100
Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
Sfcc,anhvib = 3NkB
[
Xfcc − ln
(
2 sinhxfcc
)]
+
3NkBθ
kfcc2
[
γfcc1
3
(
4 +Xfcc + Y fcc2
)
− 2γfcc2 XfccY fcc2
]
,
Sanhrot =
kBN
2T
coth
ξ
2T
(
ξ coth
ξ
2T
+ T
∂ξ
∂T
)
−kBN
T
[
B
2
+ U0η
(
1
2
− T ∂η
∂T
)]
− kBN
4T
(
T
∂ξ
∂T
− ξ
)(
ξ
T
1
sinh2 ξ2T
− 1
)
−kBN
T
{
2T ln
[
4 sinh
ξ
2T
]
− ξ
2
coth2
ξ
2T
− B
2
− U0η
2
2
}
(2.18)
and for HCP crystal [7, 8],
Shcp,anh = Shcp,anhvib + S
anh
rot , S
hcp,anh
vib = S
hcp
0vib −
NkBθ
12
[
6τhcp5 + τ
hcp
6
khcp2x
(
4 +Xhcp + Y hcp2
)
+
3τhcp1
khcp2z
(
4 +Xhcpz + Y
hcp2
z
)
+
τhcp2
khcpx k
hcp
z
(
8 +Xhcp + Y hcp2 +Xhcpz + Y
hcp2
z
)]
. (2.19)
2.1.4. Heat capacity at constant volume
The heat capacity at constant volume of molecular crystals of N2 type with FCC and HCP
structures is the sum of the vibrational heat capacity at constant volume and the rotational heat
capacity at constant volume. In harmonic approximation for FCC crystal [6, 7],
Cfcc,harV = −T
(
∂2ψfcc,har
∂T 2
)
V
= Cfcc,harV vib + C
har
V rot,
Cfcc,harV vib = 3NkBY
fcc2, CharV rot =
NkB
2
(
ξ
T
)2
sinh2 ξ2T
(
1− T
ξ
∂ξ
∂T
)
(2.20)
and for HCP crystal [7, 8],
Chcp,harV = C
hcp,har
V vib + C
har
V rot, C
hcp,har
V vib = NkB
(
2Y hcp2 + Y hcp2z
)
. (2.21)
In classical approximation for FCC crystal [6, 7],
Cfcc,claV = C
fcc,cla
V vib + C
cla
V rot, C
fcccla
V vib = 3NkB
[
1 +
2θ
kfcc2
(
γfcc1 − γfcc2
)]
,
CclaV rot = C
cla
V 0rot + kBNU0 (η − 1)
∂η
∂T
+
kBNU0
2
[
(1− 2η)
(
∂η
∂T
+ T
∂2η
∂T 2
)
− 2T
(
∂η
∂T
)2]
−2kBN
{
T
2
η
(
∂η
∂T
− 1
)
+
T
2η2
[
η
(
∂η
∂T
− T ∂
2η
∂T 2
)
− T
(
∂η
∂T
)2]}
(2.22)
101
Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
and for HCP crystal [7, 8],
Chcp,claV = C
hcp,cla
V vib + C
cla
V rot, C
hcp,cla
V vib = NkB
{
3− θ
2
[
6τhcp5 + τ
hcp
6
khcp2x
+
3τhcp1
khcp2z
+
2τhcp2
khcpx k
hcp
z
]}
.
(2.23)
In anharmonic approximation for FCC crystal [6, 7],
Cfcc,anhV = C
fcc,anh
V vib + C
anh
V rot,
Cfcc,anhV vib = 3NkB
{
Y fcc2 + 2θ
kfcc2
[(
2γfcc2 +
γfcc1
3
)
XfccY fcc2
+
2γfcc1
3 − γfcc2
(
Y fcc4 + 2Xfcc2Y fcc2
)]}
CanhV rot =
NkB
2
∂ξ
∂T
[
coth2
ξ
2T
+ coth
ξ
2T
+
1
2T
sinh
ξ
2T
(
T
∂ξ
∂T
− ξ
)]
+kBNU0T
[
η
∂2η
∂T 2
+
(
∂η
∂T
)2]
−kBN
4T 2
[
ξ
coth ξ2T
sinh2 ξ2T
(
2− 1
T
)
+
(
T
∂ξ
∂T
− ξ
)2(1
2
sinh
ξ
2T
− 1
sinh2 ξ2T
)]
+
NkB
4
∂2ξ
∂T 2
(
3T coth
ξ
2T
− ξ
sinh2 ξ2T
)
,
∂ξ
∂T
=
∂ξ
∂η
.
∂η
∂T
,
∂ξ
∂η
=
√
6BU0
[
1
2
√
η
η + 2
3
+
√
η
3
]
(2.24)
and for HCP crystal [7, 8],
Chcp,anhV = C
hcp,anh
V vib + C
anh
V rot,
Chcp,anhV vib = NkB
{
2Y hcp2 + Y hcp2z −
θ
6
[
6τhcp5 + τ
hcp
6
khcp2x
(
2 +XhcpY hcp2
)
+
3τhcp1
khcp2z
(
2 +Xhcpz Y
hcp2
z
)
+
τhcp2
khcpx k
hcp
z
(
4 +XhcpY hcp2 +Xhcpz Y
hcp2
z
)]}
. (2.25)
3. Conclusion
In this paper, we derive analytic expressions of thermodynamic quantities such as the
free energy, the energy, the entropy and the heat capacity at constant volume of molecular
cryocrystals of nitrogen type with FCC and HCP structures in harmonic, classical and anharmonic
approximations based on combining the SMM and SCFM.
In the next paper, we shall use theoretical results of this paper to calculate numerically the
thermodynamic properties for molecular cryocrystals of nitrogen type.
102
Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
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