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- Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type with fcc and hcp structures in harmonic, classical and anharmonic approximations
Xem mẫu
- 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 http://stdb.hnue.edu.vn
ANALYTIC EXPRESSION OF THERMODYNAMIC QUANTITIES FOR MOLECULAR
CRYOCRYSTALS OF NITROGEN TYPE WITH FCC AND HCP STRUCTURES
IN HARMONIC, CLASSICAL AND ANHARMONIC APPROXIMATIONS
Nguyen Quang Hoc1 , Mai Thi La1 , Vo Minh Tien2 and Dao Kha Son2
1 Faculty of Physics, Hanoi National University of Education
2 Facultyof 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
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- 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],
f cc,har f cc,har
ψ f cc,har = ψvib + ψrot ,
h f cc
i
f cc,har
ψvib = V0f cc + ψ0vib
f cc
= V0f cc + 3N θ xf cc + ln(1 − e−2x ) ,
har ξ U0 η 2
ψrot = kB N 2T ln 4 sinh − U0 η + ,
2T 2
1 X ∂ 2 ϕi0 ~ω f cc NX
k ≡f cc
2 ≡ mω f cc2
,α = x, y, z, x = , θ = kB T, V0f cc = ϕi0
2 ∂uiα eq 2θ 2
i i
(2.1)
and for HCP crystal [7, 8],
hcp,har
ψ hcp,har = ψvib + ψrothar
,
h i h i
hcp,har hcp −2xhcp
ψvib = V0hcp +ψ0vib
hcp
= V0hcp+2N θ xhcp + ln 1 − e−2x +N θ xhcp z + ln 1 − e z
,
hcp 1 X ∂ 2 ϕi0 ∂ 2 ϕi0 hcp2 hcp ~ωxhcp hcp N X
kx ≡ + ≡ mω x , x = , V0 = ϕi0 ,
2
i
∂u2ix ∂uix .∂uiy eq 2θ 2
i
95
- Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
1X ∂ 2 ϕi0 ~ωzhcp
kzhcp ≡ ≡ mωzhcp2 , xhcp
z = . (2.2)
2
i
∂u2iz eq 2θ
In classical approximation for FCC crystal [6, 7],
f cc,cla
ψ f cc,cla = ψvib cla
+ ψrot ,
f cc,cla θ 2 f cc
ψvib ≈ V0f cc + f cc
ψ0vib + 3N f cc2
γ2 − γ f cc
1
k
4θ 3 h f cc2 i
f cc2 f cc f cc
+ f cc4 γ2 − 3 γ1 + 2γ1 γ2 ,
k
h f cc
i
f cc
ψ0vib = N θ xf cc + ln(1 − e−2x ) ,
cla − ψ cla
√
f cc ψrot 0rot U0 η 2 6BU0 η
x ≈ 0.283, = −U0 η + + 2T ln ,
kB N 2 T
!
1 X ∂ 4 ϕi0 1X ∂ 4 ϕi0
γ1f cc ≡ 4
f cc
,γ2 ≡ 2 ∂u2 ,α 6= β, α, β = x, y, z (2.3)
48 ∂u iα eq 8 ∂uiα iβ
i i eq
and for HCP crystal [7, 8],
hcp,cla hcp,cla
ψ hcp,cla = ψvib cla
+ ψrot , ψvib = V0hcp + ψ0vib
hcp
" # " #
N θ 2 6τ5hcp + τ6hcp 3τ1hcp 2τ2hcp 3N θ 3 τ1hcp2 τ2hcp2
+ + hcp2 + hcp hcp + hcp2 . hcp2 + hcp2
4 kxhcp2 kz kx kz 4kz kz 9kx
" #
N θ4 33τ1hcp3 2τ2hcp3
− + hcp3 ,
36kzhcp3 kzhcp3 kx
4
1 X ∂ 2 ϕio 1 X ∂ 4 ϕi0 1X ∂ ϕi0
kz ≡ , τ1 ≡ , τ2 ≡ ,
2
i
∂u2iz eq 12
i
∂u4iz eq 2
i
∂u2ix ∂u2iz eq
4
1 X ∂ 4 ϕi0 1X ∂ ϕi0
τ5 ≡ 4 , τ 6 ≡ 2 ∂u2 . (2.4)
12 ∂u ix eq 2 ∂u ix iz eq
i i
In anharmonic approximation (anharmonic approximation of lattice vibration and
self-consistent libron approximation of molecular rotational motion) for FCC crystal [6, 7],
f cc,anh
ψ f cc,anh = ψvib anh
+ ψrot ,
( " #
f cc,anh f cc f cc θ2 f cc f cc2 2γ1f cc X f cc
ψvib = V0 + ψ0vib + 3N γ X − 1+
kf cc2 2 3 2
2θ 3 4 f cc2 f cc X f cc f cc2 f cc f cc X f cc f cc
+ f cc4 γ X 1+ − 2 γ1 + 2γ1 γ2 1+ 1+X ,
k 3 2 2 2
96
- Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
anh
ψrot ξ ξ ξ B U0 η 2
= 2T ln 4 sinh − coth2 − − (2.5)
kB N 2T 2 2T 2 2
and for HCP crystal [7, 8],
hcp,anh
ψ hcp,anh = ψvib anh
+ ψrot ,
"
hcp,anh N θ 2 6τ5hcp + τ6hcp hcp 3τ hcp
ψvib = V0hcp + hcp
ψ0vib + hcp2
X + 2 1
+ hcp2 (Xzhcp + 2)
12 kx kz
# "
τ2hcp N θ3 τ1hcp2 hcp
+ (X hcp + Xzhcp + 4) + Xz + 2 (Xzhcp + 5)
kxhcp kzhcp
24kzhcp2 kzhcp2
# "
τ2hcp2 N θ4 τ1hcp3 hcp
+ X hcp + 2)(X hcp + 5 − Xz + 2 (3Xzhcp2 + 17Xzhcp + 13)
9kxhcp2 108kzhcp3 kzhcp3
#
τ2hcp3 hcp
+ hcp3 X + 2)2 (X hcp + 5 ,
9kx
X hcp ≡ xhcp coth xhcp , Xzhcp ≡ xhcp hcp
z coth xz . (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, ω f cc , ωxhcp , ωzhcp are the frequencies of lattice
vibration, kf cc , γ1f cc , γ2f cc , kxhcp , kzhcp , τ1hcp , τ2hcp , τ5hcp , τ6hcp 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
V0 , V0hcp , kf cc , kxhcp , kzhcp are expressed in terms of the nearest neighbor distance a0 at 0
f cc
K. The pseudo-harmonic approximation of molecular rotational motion corresponds to the
condition √UT Bη
- 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 ≈ 1.3494.10−11 ω
corresponding to that where the quantum effects can be neglected [6]. The classical approximation
of molecular rotational motion corresponds to the condition √UT Bη >> 1 [7].
0
In the anharmonic approximation of lattice vibration, the quantities such as
kf cc , γ1f cc , γ2f cc , kxhcp , kzhcp , τ1hcp , τ2hcp , τ5hcp , τ6hcp are expressed in terms of the nearest neighbor
distance af cc = af0 cc + ufx0cc , ahcp = ahcp 0 + uhcp f cc hcp
x0 , where a0 , a0 respectively are the nearest
neighbor distance of FCC and HCP crystals at 0 K and the displacements ufx0cc , uhcp x0 of a particle
from the equilibrium position are calculated by
s ! !
2γ f cc θ 2 1 X ∂ 4ϕ
i0 ∂ 4ϕ
i0
ufx0cc = 3 A, γ
f cc
≡
4 +6 2 ∂u2
, β 6= γ, β, γ = x, y, z,
3kf cc 12 ∂u iβ ∂uiβ iγ
i eq eq
i
6 f cc i
X 6
X
γ θ γ hcp θ hcp
A = a1 + afi cc , uhcp
x0 = 2 ai ,
k f cc 2 hcp
i=2 i=1 kx
!
1 X ∂ 3 ϕi0 ∂ 3 ϕi0 ,
γ hcp ≡ + (2.7)
4
i
∂u3ix eq ∂uix ∂u2iy
eq
where afi cc , ahcp
i (i = 1 − 6) are determined in [8]. The anharmonic approximation of molecular
rotational motion corresponds to the condition √UT Bη ≈ 1[7].
0
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],
f cc,har f cc,har ∂ψ f cc,har f cc,har har
E =ψ −T = Evib + Erot ,
∂T V
f cc,har
Evib = =V0f cc + f cc
+ 3N θX f cc ,
E0vib V0f cc
har 3B ξ N kB U0 3B ξ 2
Erot = −N kB U0 1 − coth + 1− coth
ξ 2T 2 ξ 2T
∂ξ ξ
−N kB T − ξ coth
∂T 2T
3N kB U0 BT 3B ξ ξ ∂ξ 2 ξ ∂ξ ξ
+ 1+ coth T − ξ 1 − coth − coth
2ξ 2 ξ 2T 2T 2 ∂T 2T ∂T 2T
(2.8)
and for HCP crystal [7, 8],
hcp,har hcp,har
E hcp,har = Evib har
+ Erot , Evib = V0hcp + E0vib
hcp
= V0hcp + N θ(2X hcp + Xzhcp ). (2.9)
In classical approximation for FCC crystal [6, 7],
98
- Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
f cc,cla f cc,cla θ
E f cc,cla
= Evib + cla
Erot , Evib = V0f cc + 3N θ 1 + f cc f cc
γ1 − γ2 ,
kf cc2
√
cla cla 1 2 6BU0 η ∂
Erot= + N kB −U0 η + U0 η + 2T ln
ψ0rot −T {ψ0rot + kB N [−U0 η
2 T ∂T
√ η k N T U ∂η
1 2 6BU0 η cla B 0
+ U0 η + 2T ln = E0rot + kB N U0 η −1 + (1 − 2η)
2 T 2 2 ∂T
r !
T ∂η 1 4T ∂η 2
−2kB N T − 1 ,η = 1− 1− , =− q (2.10)
2η ∂T 2 U0 ∂T U 1 − 4T 0 U0
and for HCP crystal [7, 8],
hcp,cla
E hcp,cla = Evib + Erotcla
,
hcp hcp
Nθ 2 6τ 5 + τ 6 3τ1hcp
τ hcp
hcp,cla
Evib = V0hcp + E0vib
hcp,cla
− hcp2
+ hcp2 − hcp2 hcp . (2.11)
4 kx kz kx kz
In anharmonic approximation for FCC crystal [6, 7],
f cc,anh
E f cc,anh = Evib anh
+ Erot ,
" #
f cc,anh f cc f cc 3N θ 2 f cc f cc2 γ1f cc f cc2
f cc f cc f cc2
Evib = V0 + 3N θX + f cc2 γ2 X + 2+Y − 2γ2 X Y ,
k 3
anh ξ ξ 2 ξ B U0 η 2
Erot = N kB 2T ln 4 sinh − coth − −
2T 2 2T 2 2
∂ ξ ξ 2 ξ B U0 η 2
−kB T N 2T ln 4 sinh − coth − −
∂T 2T 2 2T 2 2
kB N ξ ξ ∂ξ
= coth ξ coth +T
2 2T 2T ∂T
!
B 1 ∂η kB N ∂ξ ξ 1
−kB N + U0 η −T − T −ξ −1 (2.12)
2 2 ∂T 4 ∂T T sinh2 ξ
2T
and for HCP crystal [7, 8],
hcp,anh hcp,anh
E hcp,anh = Evib anh
+ Erot , Evib = V0hcp + E0vib
hcp
" #
N θ2 6τ5hcp + τ6hcp 3τ1hcp τ hcp
− 2 + Y hcp2 + hcp2 2 + Yzhcp2 + hcp2 hcp 1 + Y hcp2 + Yzhcp2 ,
12 kxhcp2 kz kx kz
(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],
E f cc,har − ψ f cc,har h i
f cc,har f cc,har
S f cc,har = = Svib har
+Srot , Svib = 3N kB X f cc − ln 2 sinh xf cc ,
T
har N kB U0 3B ξ N kB U0 3B ξ 2
Srot = − 1− coth + 1− coth
T ξ 2T 2 ξ 2T
N kB ∂ξ ξ 3N kB U0 BT 3B ξ
− T − ξ coth ++ 1+ coth
T ∂T 2T 2ξ 2 ξ 2T
ξ ∂ξ ξ ∂ξ ξ
× 2
T − ξ 1 − coth2 − coth
2T ∂T 2T ∂T 2T
kB N ξ U0 η 2
− 2T ln 4 sinh − U0 η + (2.14)
T 2T 2
and for HCP crystal [7, 8],
hcp,har
S hcp,har = Svib har
+ Srot ,
n h i h io
hcp,har
Svib = N kB 2 X hcp − ln 2 sinh xhcp + Xzhcp − ln 2 sinh xhcp
z . (2.15)
In classical approximation for FCC crystal [6, 7],
f cc,cla
S f cc,cla = Svib cla
+ Srot ,
h i 6N k θ
f cc,cla B
Svib = 3N kB 1 − ln 2 sinh xf cc + f cc2 γ1f cc − γ2f cc , xf cc ≈ 0.283,
k
cla cla kB N U0 η η kB N U0 ∂η T ∂η
Srot = S0rot + −1 + (1 − 2η) − 2kB N −1
T 2 2 ∂T 2η ∂T
√
kB N U0 η 2 6BU0 η
− −U0 η + + 2T ln (2.16)
T 2 T
and for HCP crystal [7, 8],
hcp,cla
S hcp,cla = Svib cla
+ Srot ,
N k θ 6τ5hcp + τ6hcp 3τ hcp
2τ hcp
hcp,cla hcp B
Svib = S0vib + hcp2
− hcp2 1
− hcp2 hcp . (2.17)
2 kx kz kx kz
In anharmonic approximation for FCC crystal [6, 7],
f cc,anh
S f cc,anh = Svib anh
+ Srot ,
100
- Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type...
" #
h i 3N k θ γ f cc
f cc,anh B f cc
Svib = 3N kB X f cc − ln 2 sinh xf cc + f cc2 1
4 + X f cc + Y f cc2 − 2γ2 X f cc Y f cc2 ,
k 3
kB N
anh ξ ξ ∂ξ
= Srot coth ξ coth +T
2T 2T 2T ∂T
!
kB N B 1 ∂η kB N ∂ξ ξ 1
− + U0 η −T − T −ξ −1
T 2 2 ∂T 4T ∂T T sinh2 ξ
2T
kB N ξ ξ 2 ξ B U0 η 2
− 2T ln 4 sinh − coth − − (2.18)
T 2T 2 2T 2 2
and for HCP crystal [7, 8],
"
hcp,anh hcp,anh hcp N kB θ 6τ5hcp + τ6hcp
S hcp,anh = Svib anh
+ Srot , Svib = S0vib − 4 + X hcp
+ Y hcp2
12 kxhcp2
#
3τ1hcp τ hcp
+ hcp2 4 + Xzhcp + Yzhcp2 + hcp2 hcp 8 + X hcp + Y hcp2 + Xzhcp + Yzhcp2 . (2.19)
kz kx kz
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],
∂ 2 ψ f cc,har
CVf cc,har = −T = CVf cc,har
vib + CVhar
rot ,
∂T 2 V
2
ξ
N kB T T ∂ξ
CVf cc,har
vib = 3N kB Y f cc2
, CVhar
rot = 1− (2.20)
2 sinh2 ξ ξ ∂T
2T
and for HCP crystal [7, 8],
CVhcp,har = CVhcp,har
vib + C har
V rot , C hcp,har
V vib = N kB 2Y hcp2
+ Y z
hcp2
. (2.21)
In classical approximation for FCC crystal [6, 7],
2θ f cc
CVf cc,cla = CVf cc,cla
vib + CVclarot , CVf cccla
vib = 3N kB f cc
1 + f cc2 γ1 − γ2 ,
k
" #
∂η kB N U0 ∂η ∂2η ∂η 2
CVclarot = CVcla0rot
+ kB N U0 (η − 1) + (1 − 2η) +T − 2T
∂T 2 ∂T ∂T 2 ∂T
( " #)
T ∂η T ∂η ∂2η ∂η 2
−2kB N η −1 + 2 η −T −T (2.22)
2 ∂T 2η ∂T ∂T 2 ∂T
101
- Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son
and for HCP crystal [7, 8],
( " #)
θ 6τ5hcp + τ6hcp 3τ1hcp 2τ2hcp
CVhcp,cla = CVhcp,cla
vib + CVclarot , CVhcp,cla
vib = N kB 3− + + .
2 kxhcp2 kzhcp2 kxhcp kzhcp
(2.23)
In anharmonic approximation for FCC crystal [6, 7],
CVf cc,anh = CVf cc,anh
vib + CVanh rot ,
γ1f cc
CVf cc,anh
vib = 3N kB Y f cc2 2θ
+ kf cc2 f cc
2γ2 + 3 X f cc Y f cc2
2γ1f cc f cc f cc4 f cc2 f cc2
+ 3 − γ2 Y + 2X Y
N kB ∂ξ 2 ξ ξ 1 ξ ∂ξ
CVanh
=
rot coth + coth + sinh T −ξ
2 ∂T 2T 2T 2T 2T ∂T
" #
∂2η ∂η 2
+kB N U0 T η 2 +
∂T ∂T
" !#
ξ 2
kB N coth 2T 1 ∂ξ 1 ξ 1
− ξ 2− + T −ξ sinh −
4T 2 sinh2 2Tξ T ∂T 2 2T sinh2 2Tξ
! √
N kB ∂ 2 ξ ξ ξ ∂ξ ∂ξ ∂η ∂ξ p 1 η+2 η
+ 3T coth − , = . , = 6BU0 √ +
4 ∂T 2 2T 2 ξ
sinh 2T ∂T ∂η ∂T ∂η 2 η 3 3
(2.24)
and for HCP crystal [7, 8],
CVhcp,anh = CVhcp,anh
vib + CVanh
rot ,
( "
θ 6τ5hcp + τ6hcp
CVhcp,anh
vib = N k B 2Y hcp2
+ Y z
hcp2
− 2 + X hcp hcp2
Y
6 kxhcp2
#)
3τ1hcp τ hcp
+ hcp2 2 + Xzhcp Yzhcp2 + hcp2 hcp 4 + X hcp Y hcp2 + Xzhcp Yzhcp2 . (2.25)
kz kx kz
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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