Molar Heat Capacity Under Constant Volume Of

Proc. Natl. Conf. Theor. Phys. 37 (2012), pp. 180-186

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 molecular rotational motion. Numerical results for molecular cryocrystals of N2 type (β-N2,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 capacity 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 rotations. 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. 181

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:

4 " #i X γθ ux0 ≈ 2 ai, i=1 (kx + kxy) where: 2 kxy 3γ 2kx − kxy 18γ 2 3kx + 2kxy a1 = (1 − X) − X, a2 = a1 X + , a3 = a1 2X − , kx kx kx + kxy kx (kx + kxy) kx + kxy 3 2 108γ 1 X ∂ ϕi0 ωx a = − a3 (X − 1) ,X ≡ x coth x, θ = k T, k ≡ ≡ mω2, x = ~ , 4 2 1 B x 2 ∂u2 x 2θ kx (kx + kxy) i ix eq   2 3 3 ! 1 X ∂ ϕi0 1 X ∂ ϕi0 ∂ ϕi0 kxy ≡ , γ ≡  3 + 2  , (1) 2 ∂uix∂uiy 4 ∂u ∂uix∂u i eq i ix eq iy 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 approximately is as follows [6]:

i 1/2 u ≈ 1 P6 θ b , where : z0 3 i=1 kz i h i b = τ2+τ3 u2 , b = τ1 + (1 − b ) C, b = − τ1 C + (1 − b ) C2 , b = τ1 C2 + (1 − b ) C3, 1 kz x0 2 kz 1 3 kz 1 4 kz 1 h i b = − τ1 C3 + (1 − b ) C4 , b = τ1 C4 + C5,C ≡ τ1 (X + 1) + τ2 (X + 2) ,X ≡ x coth x, 5 kz 1 6 kz kz z 3kx 2 1 P ∂ ϕio 2 ~ωz Xz ≡ xz coth xz, kz ≡ 2 i 2 = mωz , xz = 2θ , ∂uiz eq

4 4 4 1 X ∂ ϕi0 1 X ∂ ϕi0 1 X ∂ ϕi0 τ ≡ , τ ≡ , τ ≡ , (2) 1 12 ∂u4 2 2 ∂u2 ∂u2 3 2 ∂u ∂u ∂u2 i iz eq i ix iz eq i ix iy iz eq

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. 182

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

2 2 Nθ kxyγ h 2 kxy kx(kx+2kxy) 1 i ψ ≈ U0 + ψ0 + 3 X − X − 2 − + kx(kx+kxy) kx+kxy 3(kx+kxy) 3 2 h i + Nθ τ1 (X + 2) + τ2 (X + X + 4) + 6τ5+τ6 (X + 2) + 4kz kz z 3kx z 3kz 3 n h 3 2 i h 2 2 io Nθ Xz+5 τ1 (τ2+τ3)γ 2 X+5 τ2 (2τ4+6τ5+τ6)γ 2 + 4 (Xz + 1) + 4 X + 2 2 (X + 2) + 3 X + 12 kz 3kz kx kx 18kz kx 4 n 5 2 2 4 o Nθ τ1 2 τ2 γ 2 X+5 (4τ4+6τ5+3τ6)γ 4 + 8 (Xz + 1) + 5 2 X (X + 2) + + 8 X + 12 15kz 3kxkz 2 kx 5 2 n 2 h 2 i 2 2 o 6 3 4 Nθ γ 2 τ2 3γ τ2 τ3 γ 2 Nθ τ2 γ 4 + 6 2 X 2 − (X + 2) (X + 5) + 2 X − 9 3 X (X + 2) , 12kxkz 3 kx 9kz kx 54kxkz 4 4 4 ! 1 X ∂ φi0 1 X ∂ φi0 1 X ∂ φi0 τ4 ≡ 3 , τ5 ≡ 4 , τ6 ≡ 2 2 , 2 ∂u ∂uiy 12 ∂u 2 ∂u ∂u i ix eq i ix eq i ix iy eq where: N X U = φ , ψ = Nθ 2 x + ln 1 − e−2x + x + ln 1 − e−2xz . (3) 0 2 i0 0 z i Applying the Gibbs-Helmholtz relation and using (3), we ﬁnd the expression for the energy of crystal 2 Nθ2 hn 4kxyγ kx(kx+2kxy) 3kxy 2 2i E ≈ U0 + E0 + 3 1 + 2 + Y + 3X X − 2Y − 12 k (kx+kxy) (k +k ) kx+kxy x x xy o 6τ5+τ6 2 3τ1 2 τ2 2 2 − 2 2 + Y − 2 2 + Y − 2 + Y + Y − kx kz z kxkz z 3 n 3 2 Nθ 2τ1 2 2 (τ2+τ3)γ 2 2 2 − 5 3Xz + XzY + 3Y + 5 + 4 2 X X −Xz + Y + Y X − 2Y + 12 3kz z z kxkz z 2 o τ2 2 2 2τ4+6τ5+τ6 2 2 2 + 2 2 7X + 2XY + 7Y + 20 + 5 X −X + 3XY + 10Y , 18kxkz kx where: x xz E0 = Nθ (2X + Xz) ,Y ≡ ,Yz ≡ . (4) sinh x sinh xz The vibrational molar heat capacity at constant volume is determined by [6]:

2 vib n 2 2 kxyγ θ h 2 2kx(kx+2kxy) 2kxy 2 2 2 2i C ≈ NkB 2Y + Yz + 3 + 2 + − 5 XY − Y Y − 3X − V kx(kx+kxy) 3 3(kx+kxy) kx+kxy θ 6τ5 + τ6 2 3τ1 2 τ2 2 2 − 2 2 + XY + 2 2 + XzYz + 4 + XY + XzYz (5) 6 kx kz kxkz

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]: 2 ψrot h ε i U0η p = 2T ln 4 sinh − U0η + , ε = 6U0Bη, (6) kBN 2T 2 2 where U0is the barrier, which prevents the molecular rotation at T = 0 K, B = ~ /(2I)is the intrinsic rotational temperature or the rotational quantum or the rotational constant. 183

The rotational molar heat capacity at constant volume in pseudo-harmonic approximation is determined by the following expression [7]: 2 2 rot ∂ ψrot Nkb (ε/T ) T ∂ε CV = −T 2 = 2 1 − . (7) ∂T V 2 sinh (ε/2T ) ε ∂T The total molar heat capacity at constant volume in pseudo-harmonic approximation is determined by the following expression: 2 2 2 vib rot ∂ ψ ∂ ψvib ∂ ψrot CV = CV + CV = −T 2 = −T 2 − T 2 . (8) ∂T V ∂T V ∂T 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 σ 12 σ 6 ϕ(r) = 4ε − , (9) 1 r r −10 −10 where ε1/kB = 95.1 K, σ = 3.708.10 m for β−N2; ε1/kB = 100.1 K, σ = 3.769.10 m for β−CO [6]. In the approximation of two ﬁrst coordination spheres, the crystal parameters are given by [6]: h i h i 4ε1 σ 6 σ 6 4ε1 σ 6 σ 6 kx = a2 a 384.711 a − 95.9247 , kxy = a2 a 229.8912 a − 66.9288 , h i h i 4ε1 σ 6 σ 6 4ε1 σ 6 σ 6 kz = a2 a 286.3722 a − 64.7487 , γ = − a3 a 161.952 a − 24.984 , h i h i 4ε1 σ 6 σ 6 4ε1 σ 6 σ 6 τ1 = a4 a 6288.912 a − 473.6748 , τ2 = a4 a 11488.3776 a − 752.5176 , h i h i 4ε1 σ 6 σ 6 4ε1 σ 6 σ 6 τ3 = a4 a 8133.888 a − 737.352 , τ4 = a4 a 43409.3184 a − 4550.04 , h i h i 4ε1 σ 6 σ 6 4ε1 σ 6 σ 6 τ5 = a4 a 11315.6064 a − 1006.0428 , τ6 = a4 a 40782.6048 a − 4189.6536 , (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- 6 tion r0 = σ 2 ≈ 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 = p6 r0 A12/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 vib rot the lattice constant a and the molar heat capacities CV ,CV ,CV for β−N2and β−CO crystals are shown in Figures 1-3. In comparison with experiments, the heat capacity 184 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 U0 for β−N2and β−CO crystals

Crystal β−N2 β−CO B(K) 2.8751 2.7787 U0(K) 325.6 688.2

Table 2. Values of η at various temperatures for β−N2 crystal T (K) 36 38 40 42 44 45.354 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

T (K) 62 64 66 68 η 0.9100 0.9099 0.9060 0.8942

Figure 1. Nearest neighbour distances a at various temperatures for β−N2 and β−CO crystals 185

vib rot Figure 2. Heat capacities at constant volume CV ,CV ,CV in diﬀerent temperatures for β−N2 crystal

vib rot Figure 3. Heat capacities at constant volume CV ,CV ,CV 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. 186

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