MOLAR HEAT CAPACITY UNDER CONSTANT VOLUME OF MOLECULAR CRYOCRYSTALS OF NITROGEN TYPE WITH HCP STRUCTURE CONTRIBUTION FROM LATTICE VIBRATIONS AND MOLECULAR ROTATIONAL MOTION

MOLAR HEAT CAPACITY UNDER CONSTANT VOLUME OF MOLECULAR CRYOCRYSTALS OF NITROGEN TYPE WITH HCP STRUCTURE: CONTRIBUTION FROM LATTICE VIBRATIONS AND MOLECULAR ROTATIONAL MOTION NGUYEN QUANG

MOLAR HEAT CAPACITY UNDER CONSTANT VOLUME OF MOLECULAR CRYOCRYSTALS OF NITROGEN TYPE WITH HCP STRUCTURE: CONTRIBUTION FROM LATTICE

VIBRATIONS AND MOLECULAR ROTATIONAL MOTION

NGUYEN QUANG HOC Hanoi National University of Education, 136 Xuan Thuy Street,

Cau Giay District, Hanoi NGUYEN NGOC ANH, NGUYEN THE HUNG, NGUYEN DUC HIEN

Tay Nguyen University, 456 Le Duan Street, Buon Me Thuot City

NGUYEN DUC QUYEN University of Technical Education, 1 Vo Van Ngan Street,

Thu Duc District, Ho Chi Minh City

Abstract The analytic expression of molar heat capacity under constant volume of molecular cryocrystals of nitrogen type with hcp structure is obtained by the statistical moment method and the self-consistent field method taking account of the anharmonicity in lattice vibrations and molec-ular rotational motion Numerical results for molecmolec-ular cryocrystals of N 2 type (β-N 2 ,β-CO) are compared with experiments.

I INTRODUCTION The study of heat capacity for molecular cryocrystals of nitrogen type is carried out experimentally and theoretically by many researchers For example, the heat capacity of solid nitrogen is measured by Giauque and Clayton [1], Bagatskii, Kucheryavy, Manzhelii and Popov [2] The heat capacity of solid carbon monoxide is determined by Clayton and Giauque [3], Gill and Morrison [4] Theoretically, the heat capacity of solid nitrogen and carbon monoxide is investigated by the Debye heat capacity theory, the Einstein heat ca-pacity theory, the self-consistent phonon method (SCPM), the self-consistent field method (SCFM), the pseudo-harmonic theory and the statistical moment method (SMM) [5, 6, 7] In [5, 6] the heat capacities at constant volume and at constant pressure of β−N2 and β−CO crystals are calculated by SMM only taking account of lattice vibration and the obtained results only agreed qualitatively with experiments The heat capacity at constant volume of crystals of N2 type in pseudo-harmonic approximation is considered by SCFM only taking account of molecular rotations [8] In this report we study the heat capacity

at constant volume of α−N2 and α−CO crystals in pseudo-harmonic approximation by combining SMM and SCFM taking account of both lattice vibrations and molecular rota-tions In section 2, we derive the heat capacity at constant volume for crystals with hcp structure taking into account lattice vibrations by SMM and for crystals of N2 type taking into account molecular rotations by SCFM Our calculated vibrational and rotational heat capacities for β−N2 and β−CO crystals are summarized and discussed in section 3

II THEORY 2.1 The heat capacity at constant volume of crystals with hcp structure by

SMM

The displacement of a particle from equilibrium position on direction x (or direction

y) is given approximately [6] by:

ux0≈

4

X

i=1

"

γθ (kx+ kxy)2

#i

ai, where:

a1= kxy

kx

(1 − X) − X, a2= 3γ

kx

a1



X + 2kx− kxy

kx+ kxy

 , a3 = 18γ

2

kx(kx+ kxy)a

2 1

 2X −3kx+ 2kxy

kx+ kxy

 ,

a4= − 108γ

3

kx(kx+ kxy)2a

3

1(X − 1) , X ≡ x coth x, θ = kBT, kx≡ 1

2 X

i

 ∂2ϕi0

∂u2ix



eq

≡ mωx2, x = ~ωx

2θ ,

kxy ≡ 1

2 X

i



∂2ϕi0

∂uix∂uiy



eq

, γ ≡ 1 4 X

i





 ∂3ϕi0

∂u3ix



eq

3ϕi0

∂uix∂u2iy

!

eq



Here kB is the Boltzmann constant, T is the absolute temperature, m is the mass of

particle at lattice node, ωx is the frequency of lattice vibration on direction x (or y),

kx, kxy and γ are the parameters of crystal 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 and uiαis the displacement of ith particle

from equilibrium position on direction α(α = x, y, z)

The lattice constant on direction x (or y) is determined by a = a0 + ux0,where a0

is the distance a at temperature 0K and is determined from experiments

The displacement of a particle from equilibrium position on direction z

approxi-mately is as follows [6]:

uz0≈



1

3

P6

i=1



θ

k z

i

bi

1/2

, where :

b1= τ2 +τ 3

k z u2x0, b2= τ1

k z + (1 − b1) C, b3 = −hτ1

k zC + (1 − b1) C2i, b4 = τ1

k zC2+ (1 − b1) C3,

b5= −

h

τ 1

k zC3+ (1 − b1) C4

i , b6= τ1

k zC4+ C5, C ≡ τ1

k z (Xz+ 1) + τ2

3k x (X + 2) , X ≡ x coth x,

Xz ≡ xzcoth xz, kz ≡ 1

2

P

i



∂2ϕ io

∂u 2 iz



eq = mωz2, xz= ~ω z

2θ ,

τ1≡ 1

12

X

i

 ∂4ϕi0

∂u4 iz



eq

, τ2 ≡ 1 2 X

i



∂4ϕi0

∂u2

ix∂u2 iz



eq

, τ3 ≡ 1 2 X

i



∂4ϕi0

∂uix∂uiy∂u2

iz



eq

, (2)

Here ωz is the frequency of lattice vibration on direction z and kz, τ1, τ2 and τ3 also are

the parameters of crystal

The lattice constant on direction z is determined by c = c0+ uz0,where c0 is the

distance c at temperature 0K and is determined from experiments

The free energy of crystals with hcp structure has the form [6]:

ψ ≈ U0+ ψ0+ N θ2kxy γ 2

k 3

x (k x +k xy )

h

X2− kxy

k x +k xyX −kx (k x +2k xy )

3(k x +k xy )2 −1

3

i + +N θ4k2

z

h

τ 1

k z (Xz+ 2) + τ2

3k x(X + Xz+ 4) + 6τ5 +τ 6

3k z (X + 2)i+ +N θ123

n

X z +5

k z

hτ 3

3k 4 (Xz+ 1) + (τ2 +τ 3 )γ 2

k 4

i +X+5k2 x

h τ 2

18k 2 (X + 2) +(2τ4 +6τ 5 +τ 6 )γ 2

k 3

io + +N θ124n15kτ58(Xz+ 1)2+3kτ25γ2

x k 2 X2(X + 2) + X+52  + (4τ4 +6τ 5 +3τ 6 )γ 4

k 8

+12kN θ56γ2

x k 2X2

nτ 2

3

h

3γ 2

k 2

x − τ2

9k z (X + 2) (X + 5)

i +τk2γ22

x X2

o

−N θ54k6τ93γ4

x k 3 X4(X + 2) ,

τ4 ≡ 1

2

X

i



∂4φi0

∂u3

ix∂uiy



eq

, τ5≡ 1 12 X

i

 ∂4φi0

∂u4 ix



eq

, τ6 ≡ 1 2 X

i

∂4φi0

∂u2

ix∂u2 iy

!

eq

, where:

U0= N

2 X

i

φi0, ψ0 = N θ2 x + ln 1 − e−2x
 + xz+ ln 1 − e−2xz
 (3)

Applying the Gibbs-Helmholtz relation and using (3), we find the expression for the energy

of crystal

E ≈ U0+ E0+N θ122 hn 4kxy γ 2

k 3

x (k x +k xy )1 +kx (k x +2k xy )

(k x +k xy )2 + 3kxy

k x +k xyY2+ 3X X − 2Y2
i

−

−6τ 5 +τ 6

k 2

x 2 + Y2
−3τ 1

k 2 2 + Y2
− τ 2

k x k z 2 + Y2+ Y2
o

−

−N θ123 n2τ3k35 3Xz+ XzYz2+ 3Yz2+ 5
+(τ 2 +τ 3 )γ 2

k 4

x k 2 XX −Xz+ Yz2
+ Y2 X − 2Y2
 + +18kτ22

x k 2 7X + 2XY2+ 7Y2+ 20
+ 2τ 4 +6τ 5 +τ 6

k 5

x X −X2+ 3XY2+ 10Y2
o

, where:

E0 = N θ (2X + Xz) , Y ≡ x

sinh x, Yz ≡

xz

sinh xz

The vibrational molar heat capacity at constant volume is determined by [6]:

CVvib ≈ N kBn2Y2+ Yz2+ kxy γ 2 θ

k 3

x (k x +k xy )

h

2

3 +2kx (k x +2k xy ) 3(k x +k xy )2 + 2kxy

k x +k xy − 5XY2− Y2 Y2− 3X2
i

−

−θ

6

 6τ5+ τ6

k2 x

2 + XY2
+ 3τ1

k2 2 + XzYz2
+ τ2

kxkz

4 + XY2+ XzYz2



(5)

2.2 The heat capacity at constant volume of crystals with hcp structure by

SCFM

Using SCFM, only taking account of molecular rotation, the rotational free energy of

crystals with fcc and hcp structures in pseudo-harmonic approximation (when U0η/B >>

1 or T /√U0Bη << 1)is determined by [7]:

ψrot

kBN = 2T ln

h

4 sinh ε

2T

i

− U0η + U0η

2

2 , ε =

p

where U0is 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

The rotational molar heat capacity at constant volume in pseudo-harmonic approximation

is determined by the following expression [7]:

CVrot= −T ∂2ψrot

∂T2



V

= N kb 2

(ε/T )2 sinh2(ε/2T )



1 −T ε

∂ε

∂T



The total molar heat capacity at constant volume in pseudo-harmonic approximation is determined by the following expression:

CV = CVvib+ CVrot= −T ∂2ψ

∂T2



V

= −T ∂2ψvib

∂T2



V

− T ∂2ψrot

∂T2



V

III NUMERICAL RESULTS AND DISCUSSION

In order to apply the theoretical results in Section 2 to molecular cryocrystals of nitrogen type, we use the Lennard-Jones (LJ) potential

ϕ(r) = 4ε1



σ r

12

−σ r

6

where ε1/kB = 95.1 K, σ = 3.708.10−10m for β−N2; ε1/kB= 100.1 K, σ = 3.769.10−10m for β−CO [6] In the approximation of two first coordination spheres, the crystal param-eters are given by [6]:

kx = 4ε1

a 2 σ

a

6h

384.711 σa
6

− 95.9247i, kxy = 4ε1

a 2 σ a

6h 229.8912 σa
6

− 66.9288i,

kz = 4ε1

a 2 σ

a

6h

286.3722 σa
6− 64.7487i, γ = −4ε1

a 3 σ a

6h 161.952 σa
6− 24.984i,

τ1= 4ε1

a 4 σ

a

6h

6288.912 σa
6

− 473.6748i, τ2= 4ε1

a 4 σ a

6h 11488.3776 σa
6

− 752.5176i,

τ3= 4ε1

a 4

σ

a

6h

8133.888 σa
6− 737.352i, τ4= 4ε1

a 4

σ a

6h 43409.3184 σa
6− 4550.04i,

τ5= 4ε1

a 4 σ

a

6h

11315.6064 σa
6

− 1006.0428i, τ6 = 4ε1

a 4 σ a

6h 40782.6048 σa
6

− 4189.6536i,

(10) where a is the nearest neighbour distance (or the lattice constant on direction x or y)

at temperature T The LJ potential has a minimum value corresponding to the posi-tion r0 = σ√62 ≈ 1.2225σ However, since there is interaction of many particles, the nearest neighbour distance a0 in the lattice is smaller than r0 It is equal to a0 =

r0pA6 12/A6 ≈ 1.0865σ, where A6 and A12 are the structural sums and they have the values A6 = 14.1601, A12 = 11.648 for a hcp crystal [6] From the above mentioned results, we obtain the values of crystal parameters at 0 K From that, we calculate the nearest neighbour distances of the lattice, the vibrational molar heat capacities at constant volume in different temperatures by SMM as in [5, 6] The values of B, U0 and the values

of η at various temperatures are given in Tables 1, 2 and 3 Our calculated results for the lattice constant a and the molar heat capacities CVvib, CVrot, CV for β−N2and β−CO crystals are shown in Figures 1-3 In comparison with experiments, the heat capacity

calculated by both SMM and SCFM is better than the heat capacity calculated by only SMM or only SCFM Both the lattice vibration and the molecular rotation have impor-tant contributions to thermodynamic properties of molecular cryocrystals of nitrogen type

Table 1 Values of B and U 0 for β−N 2 and β−CO crystals

Crystal β−N 2 β−CO

B (K) 2.8751 2.7787

U 0 (K) 325.6 688.2

Table 2 Values of η at various temperatures for β−N 2 crystal

η 0.8633 0.8617 0.8544 0.8404 0.8244 0.8038

Table 3 Values of η at various temperatures for β−CO crystal

η 0.9100 0.9099 0.9060 0.8942

Figure 1 Nearest neighbour distances a at various temperatures

for β−N 2 and β−CO crystals

Figure 2 Heat capacities at constant volume C Vvib, C Vrot, C V

in different temperatures for β−N 2 crystal

Figure 3 Heat capacities at constant volume C vib

V , C rot

V , C V

in different temperatures for β−CO crystal

This paper is carried out with the financial support of the HNUE project under the code SPHN-12-109

REFERENCES [1] W F Giauque and J O Clayton, J Amer Chem Soc 55, 12, 4875 (1933)

[2] M.I Bagatskii, V A Kucheryavy, V G Manzhelii and V A, Popov, Phys Status Solidi 26, 453 (1968)

[3] J O Clayton and W F Giauque, J Amer Chem Soc 54, 7, 2610 (1932)

[4] E K Gill and J A Morrison, J Chem Phys 45, 5, 1585 (1966)

[5] N Q Hoc and N Tang, Communications in Physics 4, 4, 157 (1994)

[6] N Q Hoc, PhD Thesis, Hanoi National University of Education (1994)

[7] B I Verkina, A Ph Prikhotko, Kriokristallu, Kiev (1983) (in Russian)

[8] A P Brodianskii and I N Krupskii, UPJ 19, 5, 862 (1974) (in Russian)

[9] I P Krupskii, A I Prokhvatilov and A I Erenburg, PNT 1, 3, 359 (1975) (in Russian) [10] I P Krupskii, A I Prokhvatilov, A I Erenburg and A P Isakina, PNT 1, 9, 1148 (1975) (in Russian)

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