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

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, 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 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: ux0 ≈ i=1 γθ (kx + kxy )2 where: 2kx − kxy kxy 3γ a1 = (1 − X) − X, a2 = a1 X + kx kx kx + kxy a4 = − i , , a3 = 3kx + 2kxy 18γ a21 2X − kx (kx + kxy ) kx + kxy 108γ ∂ ϕi0 a (X − 1) , X ≡ x coth x, θ = k T, k ≡ x B ∂u2ix kx (kx + kxy )2 i  ∂ ϕi0 ∂ ϕi0  ∂ ϕi0 kxy ≡ + ,γ ≡ ∂uix ∂uiy eq ∂u3ix eq ∂uix ∂u2iy i i ≡ mωx2 , x = eq ωx , 2θ  , (1) 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 uiα is 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]: uz0 ≈ i=1 θ kz 1/2 i bi , where : τ1 τ1 τ2 +τ3 2 kz ux0 , b2 = kz + (1 − b1 ) C, b3 = − kz C + (1 − b1 ) C b5 = − kτ1z C + (1 − b1 ) C , b6 = kτ1z C + C , C ≡ kτ1z (Xz + 1) ωz Xz ≡ xz coth xz , kz ≡ 21 i ∂∂uϕ2io = mωz2 , xz = 2θ , iz eq b1 = τ1 ≡ 12 i ∂ ϕi0 ∂u4iz , τ2 ≡ eq i ∂ ϕi0 ∂u2ix ∂u2iz , τ3 ≡ eq i , b4 = + τ2 3kx τ1 kz C + (1 − b1 ) C , (X + 2) , X ≡ x coth x, ∂ ϕi0 ∂uix ∂uiy ∂u2iz , (2) 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]: N θ kxy γ kxy kx (kx +2kxy ) X − kx +k X − 3(k − + kx3 (kx +kxy ) xy x +kxy ) τ2 +τ6 (X + Xz + 4) + 6τ3k (X + 2) + + N4kθz kτ1z (Xz + 2) + 3k x z 3 τ τ22 (2τ4 +6τ5 +τ6 )γ 2 )γ X + X+5 X + N12θ Xkz z+5 3k14 (Xz + 1) + (τ2 +τ 2 (X + 2) + k k 18k kx3 z x x z 2γ2 4 τ τ + N12θ 15k1 (Xz + 1)2 + 3k25 k2 X (X + 2) + X+5 + (4τ4 +6τk58+3τ6 )γ X + z x z x τ32 γ 2 τ22 3γ N θ τ23 γ 4 N θ5 γ τ2 − 9kz (X + 2) (X + 5) + k2 X − 54k9 k3 X (X + 2) , + 12k6 k2 X kx2 x z x x z ψ ≈ U0 + ψ0 + τ4 ≡ ∂ φi0 ∂u3ix ∂uiy i , τ5 ≡ eq 12 i ∂ φi0 ∂u4ix , τ6 ≡ eq i ∂ φi0 ∂u2ix ∂u2iy + , eq where: U0 = N + xz + ln − e−2xz φi0 , ψ0 = N θ x + ln − e−2x (3) i Applying the Gibbs-Helmholtz relation and using (3), we find the expression for the energy of crystal N θ2 12 4kxy γ kx3 (kx +kxy ) kx (kx +2kxy ) 3kxy + kx +k Y + 3X X − 2Y − xy (kx +kxy )2 − 6τ5k+τ + Y − 3τ + Yz2 − kxτ2kz + Y + Yz2 − kz2 x 2τ 3 )γ − N12θ 3k15 3Xz + Xz Yz2 + 3Yz2 + + (τ2k+τ X X −Xz + Yz2 + Y X − 2Y k2 z x z τ2 +τ6 X −X + 3XY + 10Y , + 18k22 k2 7X + 2XY + 7Y + 20 + 2τ4 +6τ kx5 x z E ≈ U0 + E0 + + + where: x xz , Yz ≡ sinh x sinh xz The vibrational molar heat capacity at constant volume is determined by [6]: E0 = N θ (2X + Xz ) , Y ≡ CVvib ≈ N kB 2Y + Yz2 + − θ kxy γ θ kx3 (kx +kxy ) + 2kx (kx +2kxy ) 3(kx +kxy )2 + 2kxy kx +kxy (4) − XY − Y Y − 3X 6τ5 + τ6 3τ1 τ2 + XY + 2 + Xz Yz2 + + XY + Xz Yz2 kx kz kx kz (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 U0 η/B >> or T / U0 Bη ... 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... the molecular rotation at T = 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. ..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