Abstract. The elastic moduli as the isothermal elastic modulus BT , the adiabatic elastic
modulus BS, the Young modulus E, the bulk modulus G, the rigidity modulus K, the
elastic constants C11, C12, C44, the longitudinal and transverse wave velocities vl and vt
for crystals with face-centered cubic (FCC) structure are derived by the statistical moment
method. The numerical results are applied to N2 and CO molecular cryocrystals and are
compared with the experimental data.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0040
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 115-120
This paper is available online at
ELASTIC MODULI AND VELOCITY OF ELASTIC WAVE
FOR N2 AND COMOLECULAR CRYOCRYSTALS
WITH FACE-CENTERED CUBIC STRUCTURE
Nguyen Quang Hoc1, Nguyen Thi Tam1, Bui Duc Tinh1 and Nguyen Duc Hien2
1Faculty of Physics, Hanoi National University of Education
2Mac Dinh Chi High School, Gia Lai Province
Abstract. The elastic moduli as the isothermal elastic modulus BT , the adiabatic elastic
modulus BS , the Young modulus E, the bulk modulus G, the rigidity modulus K, the
elastic constants C11, C12, C44, the longitudinal and transverse wave velocities vl and vt
for crystals with face-centered cubic (FCC) structure are derived by the statistical moment
method. The numerical results are applied to N2 and CO molecular cryocrystals and are
compared with the experimental data.
Keywords: Molecular cryocrystal, elastic moduli, elastic wave, elastic constants.
1. Introduction
Due to influence of the external force, the solid is deformed (its form and volume are
changed). The mechanical properties, including the plasticity, the firming and the elastic moduli,
are of special interest [1]. For the Hookean deformation, the relation of normal stress and elastic
deformation is obtained. Murnaghan [2] also derived Hooke’s generalized law and the relations
of elastic moduli E, G and K by expanding the elastic energy in terms of the elastic strain. Using
the characteristic values of monocrystal, Voight [3] and Reuss [4] considered the elastic moduli
of isotropic polycrystal. Hill [5] used geometrical or arithmetical averaging of the results counted
using Voight and Reuss’s method for investigation of the elastic moduli of the polycrystalline
body. Using the Voight-Reuss-Hill method, the obtained results agree with the experimental data
for many cases.
In this paper, by using the statistical moment method (SMM) we can derive the analytic
expressions for the isothermal elastic modulus BT , the adiabatic elastic modulus BS , the Young
modulus E, the bulk modulus G, the rigidity modulus K, the elastic constants C11, C12, C44, and
the longitudinal and transverse wave velocities vl and vt for face-centered cubic (FCC) crystals.
The temperature dependence of the above mentioned quantities is obtained and their calculations
are compared with the experimental data in the case of N2 and CO molecular cryocrystals.
Received November 30, 2014. Accepted October 18, 2015.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
115
Nguyen Quang Hoc, Nguyen Thi Tam, Bui Duc Tinh and Nguyen Duc Hien
2. Content
2.1. Elastic property and velocity of elastic wave for FCC crystals
* The nearest neighbor distance
The displacement of a particle from equilibrium position is determined by [6]
y0 =
√
2γθ2
3k3
A,A = a1 +
6∑
i=2
(
γθ
k2
)i
ai,
k ≡ 1
2
∑
i
(
∂2ϕi0
∂u2iβ
)
eq
= mω2, x =
~ω
2θ
, γ ≡ 1
12
∑
i
(∂4ϕi0
∂u4iβ
)
eq
+ 6
(
∂4ϕi0
∂u2iβ∂u
2
iγ
)
eq
,
θ = kBT,X = xcothx, β, γ = x, y, z, β 6= γ,
a1 = 1 +
1
2
X, a2 =
13
3
+
47
6
X +
23
6
X2 +
1
2
X3,
a3 = −
(
25
3
+
121
6
X +
50
3
X2 +
16
3
X3 +
1
2
X4
)
,
a4 =
43
3
+
93
2
X +
169
3
X2 +
83
3
X3 +
22
3
X4 +
1
2
X5,
a5 = −
(
103
3
+
749
6
X +
363
2
X2 +
391
3
X3 +
148
3
X4 +
53
6
X5 +
1
2
X6
)
,
a6 = 65 +
561
2
X +
1489
3
X2 +
927
2
X3 +
733
3
X4 +
145
2
X5 +
31
3
X6 +
1
2
X7, (2.1)
where kB is the Boltzmann constant, m is the mass of particle at the lattice node, uiβ(β = x, y, z)
is the displacement of the ith particle from equilibrium position in the direction β, ϕi0 is the
interaction potential between the 0th particle and the ith particle, and k, γ are the parameters of
FCC crystal. The nearest neighbor distance is equal to a = a0 + y0, where a0 is the nearest
neighbor distance at 0 K. The lattice constant of FCC crystal is
√
2a.
* Isothermal elastic modulus
The isothermal elastic modulus is determined by [6]
BT =
2P +
√
2
a .
1
3N .
(
∂2ψ
∂a2
)
T
3
(
a
a0
)3 ,
(
∂2ψ
∂a2
)
T
= 3N
{
1
6
∂2u0
∂a2
+ θ
[
X
2k
∂2k
∂a2
− 1
4k2
(
∂k
∂a
)2 (
X + Y 2
)]}
, Y ≡ x
sinhx
. (2.2)
116
Elastic moduli and velocity of elastic wave for N2 and CO molecular cryocrystals...
* Adiabatic elastic modulus
The adiabatic elastic modulus is determined by [6]
BS =
Cp
CV
BT ,
CV = 3NkB
{
Y 2 +
2θ
k2
[(
2γ2 +
γ1
3
)
XY 2 +
2γ1
3
− γ2
(
Y 4 + 2X2Y 2
)]}
,
Cp = CV +
9TV α2
χT
, α = −
√
2kBχT
3a2
.
1
3N
.
∂2ψ
∂θ∂a
,
1
3N
∂2ψ
∂θ∂a
=
1
2k
∂k
∂a
Y 2 +
2θ
k2
[
γ1
3k
∂k
∂a
(
2 +XY 2
) − 1
6
∂γ1
∂a
(
4 +X + Y 2
)− (2γ2
k
∂k
∂a
− ∂γ2
∂a
)
XY 2
]
,
(2.3)
where CV is the heat capacity at constant volume, Cp is the heat capacity at constant pressure and
α is the thermal expansion coefficient.
* Young modulus
The Young modulus is determined by [6, 7]
E =
1
π (ao + yo)A1
, A1 =
1
k
[
1 +
2γ2θ2
k4
(
1 +
X
2
)
(1 +X)
]
. (2.4)
* Bulk modulus
The bulk modulus is determined by [6, 7]
K ≈ E
3 (1− 2υ) , (2.5)
where υ is the Poisson’s ratio
* Rigidity modulus
The rigidity modulus is determined by [6-8]
G =
E
2 (1 + υ)
. (2.6)
* Longitudinal and transverse wave velocities
If all elastic deformations of body are small, the motion is small motion and we call it
the elastic vibration or the elastic wave according to the elastic theory. The longitudinal wave’s
velocity vl and the transverse wave’s velocity vt for cubic crystals are determined by [6-8]
vl =
√
2C44 + C12
ρ
, vt =
√
C44
ρ
, (2.7)
where ρ is the mass density.
117
Nguyen Quang Hoc, Nguyen Thi Tam, Bui Duc Tinh and Nguyen Duc Hien
2.2. Numerical results and discussion
It is known that the interaction potential between two atoms in the α phase of N2 and CO
molecular cryocrystals is usually used in the form of the Lennard-Jones pair potential
φ(r) = 4ε
[(σ
r
)12 − (σ
r
)6]
, (2.8)
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, σ are determined from experiments. ε/kB = 95.145 K, σ = 3.708.10−10 m for
α-N2 and ε/kB = 100.145 K, σ = 3.769.10−10 m for α- CO [8]. Therefore, using the coordinate
sphere method and the results in [8-6], we obtain the values of parameters for α-N2 and α-CO 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] , (2.9)
where a is the nearest neighbor distance at temperature T. Our calculated results for the elastic
moduli BT , BS , E, G, K, the elastic constants C11, C12, C44, the wave velocities vl, vt for α-N2
and α-CO at different temperatures and pressure P = 0 are shown in figures from Figure 1 to
Figure 8. When temperature increases, the nearest neighbor distance increases and the quantities
BT , BS , E, G, K, C11, C12, C44, vl and vt decrease.
Fig. 1. BT (T ), BS(T ) for α-N2 at P = 0
from SMM and EXPT[9]
Fig. 2. E(T ),K(T ), G(T ) for α-N2 at P = 0
from SMM and EXPT[9]
In order to calculate the elastic moduli BT , BS of α-N2, we use the following experimental
data: ε/kB = 95.05 K, σ = 3.698.10−10 m, and a0 = 4.0316.10−10 m [8]. The discrepancy in the
nearest neighbor distance of α-N2 that exists between our calculations and the experimental data
is from 0.4 to 0.6%. The discrepancy in the elastic moduli BT ,BS of α-N2 that exists between our
calculations and the experimental data is smaller than 10%. The discrepancy in the wave velocities
vl, vt of α-N2 that exists between our calculations and the experimental data is in the interval of
from 15 to 23%. Our calculations are in qualitative agreement with experiments for the elastic
moduli E, K, G and the elastic constants C11, C12, C44.
118
Elastic moduli and velocity of elastic wave for N2 and CO molecular cryocrystals...
Fig. 3. C11(T ), C12(T ), C44(T ) for α-N2
at P = 0 from SMM and EXPT[9]
Fig. 4. vl(T ), vt(T ) for α-N2 at P = 0
from SMM and EXPT[9]
Fig. 5. BT (T ), BS(T ) for α-CO at P = 0
from SMM and EXPT[10]
Fig. 6. E(T ),K(T ), G(T ) for α-CO at P = 0
from SMM and EXPT[10]
Fig. 7. C11(T ), C12(T ), C44(T ) for α-CO
at P = 0 from SMM and EXPT[10]
Fig. 8. vl(T ), vt(T ) for α-CO at P = 0
from (SMM and EXPT[10]
In order to calculate the elastic moduli BT ,BS of α-CO, we use the following experimental
data: ε/kB = 110 K, σ = 3.59.10−10 m and a0 = 3.9138.10−10 m [8]. The discrepancy in the
nearest neighbor distance of α-N2 that exists between our calculations and the experimental data is
119
Nguyen Quang Hoc, Nguyen Thi Tam, Bui Duc Tinh and Nguyen Duc Hien
smaller than 0.4%. The discrepancy in the elastic moduli BT , BS of α-N2 that exists between our
calculations and the experimental data is smaller than 11%. The discrepancy in the wave velocities
vl, vt of α-N2 that exists between our calculations and experimental data is from 17.5 to 21.5%.
Our calculations are in qualitative agreement with the experimental data for the elastic moduli E,
K, G and the elastic constants C11, C12, C44.
3. Conclusion
We derive the analytic expressions for the isothermal elastic modulus BT , the adiabatic
elastic modulus BS , the Young modulus E, the bulk modulus G, the rigidity modulus K, the elastic
constants C11, C12, C44, and the longitudinal and transverse wave velocities vl and vt for FCC
crystals based on the SMM. The theoretical results are applied to α-N2 and α-CO molecular
cryocrystals and are compared with the experimental data. In general, our calculations are in
qualitative agreement with the experimental data.
Acknowledgements. This work was carried out thanks to the financial support provided by the
National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under
Grant No. 103.01-2013.20.
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