Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 93-100 This paper is available online at STRUCTURAL PHASE TRANSITION TEMPERATURE OF N2 AND COMOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1, Bui Duc Tinh1, Dinh Quang Vinh1 and Nguyen Duc Hien2 1Faculty of Physics, Hanoi National University of Education 2Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province Abstract. The  -  (fcc-hcp) phase transition temperature under pressure can be investigated theoretically by combining the self-consistent field method and the statistical moment method and only the statistical moment method which was developed by the authors. Theoretical results are applied to N2 and CO molecular cryocrystals and our calculated results are compared with the experimental data and other calculations. Keywords: Self-consistent field, statistical moment, molecular cryocrystal. 1. Introduction Electronic transitions lie at the heart of many high-pressure phenomena including structural stability and metallization [1]. Such changes in the underlying electronic structure drive concomitant sequences of structural phase transitions and there has recently been intense interest in such changes for the alkali and alkaline earth metals. He and Xe transform from face-centered cubic (fcc) structures to hexagonal close-packed (hcp) structures at high pressures, but corresponding transitions have not been reported for Ne, Ar or Kr. The properties of rare-gas solids at high pressures are of fundamental interest because they provide an ideal system due to their closed-shell electronic configurations, allowing fruitful comparison between experiment and theory. The polymorphous transformation is the phase transition between the different configurations of a crystal. Because this transformation temperature is smaller than the melting temperature, in order to investigate it, Kotenok [2] applied the one-particle distribution method (OPDM) of Bazarov [3] in the quasiharmonic approximation of the Received August 19, 2014. Accepted October 21, 2014. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 93 Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien self-consistent field method and found that the calculated phase transition temperatures for solid nitrogen and carbon monoxide were quite near that of the experimental data. This phase transition temperature is still determined by the self-consistent phonon method (SCPM) [4], the self-consistent field method (SCFM) [5] taking into account the molecular rotational motion and the statistical moment method (SMM) [6]. However, in these works the change of the thermodynamic quantities of molecular cryocrystals at the structural phase transition temperature was not calculated. The α - β structural phase transition is a Type I phase transition because the volume and the entropy change suddenly at this phase transition. Experimental data diverge significantly for important characteristics of this phenomenon such as the lattice constant at the phase transition, the phase transition pressure and the change of entropy. Theoretical methods such as the lattice dynamics method, the thermodynamic method and the one-particle distribution method, including the calculation of many-particle interaction, apply the harmonic approximation and do not take into account the correlations between particles. The present paper presents a theoretical study of the α - β structural phase transition temperature under pressure based on the SMM and the SMM combining with the SCFM. Theoretical results are applied to N2 and CO molecular cryocrystals and our calculated results are compared with experiments and other calculations. 2. Content 2.1. Theoretical formulation 2.1.1. Determination of the α - β phase transition temperature by combining the SCFM and the SMM In a quasiharmonic approximation of the SCFM, the free energy per particle at the lattice node has the form [5] ψ N = −3 2 θ ln(2πθ) + θ 2 lnD + u0 2 (2.1) where u0 = ∑ i φi0 (|⃗ai|) is the potential energy of a particle at the lattice node, ϕi0 is the interaction potential between the ith particle and the 0th particle, D is the determinant of the matrix ∂2u0∂u1α∂u1β and is equal to the product of the diagonal elements of this matrix belonging to the principal axis, θ = kBT with kB being the Boltzmann constant. The polymorphous transformation temperature or the α - β phase transition temperature Tα−β = θα−β kB is determined using the equation system 94 Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure θα−β ln ( Dα Dβ ) + 2P (vα − vβ) = u0β − u0α, P = −∂ψα ∂vα = −∂ψβ ∂vβ . (2.2) The real solutions of the equation system (2.2) at determining P or T give us the temperature or pressure of the polymorphous transformation. The volume discontinuity ∆v − vβ − vα per particle and the corresponding heat quantity of the transition Q are determined from (2.2): ∆v = vβ − vα, Q = u0β − u0α + P (vβ − vα) . (2.3) At pressure P ≈ 1 atm we can suppose P = 0 in (2.2) and then, θα−β = u0β − u0α ln ( D D ) , ∂ψα ∂vα = ∂ψβ ∂vβ = 0. (2.4) At T = Tα−β, the structure is unstable and we have a structural phase transition. We can calculate u0β − u0α and Dα Dβ from the SMM. We apply equation (2.4) in order to find the phase transition temperature Tα−β in solid nitrogen. 2.1.2. Determination of the α - β phase transition temperature from the SMM The equilibrium condition of the α and β structural phases at the Tα−β phase transition temperature has the form Tα = Tβ = Tα−β, Pα = Pβ = Pα−β, µα = µβ, Gα = Gβ, (2.5) where T is the temperature, P is the pressure, µ is the chemical potential and G is the Gibbs thermodynamic potential. From (2.5) and because G = ψ + PV, (2.6) where ψ is the free energy, V = Nv is the volume, N is the number of particle and v is the volume per particle, we have ψα + Pα−βNvα = ψβ + Pα−βNvβ, (2.7) where vα = a3α√ 2 is the volume of the α phase with the fcc structure, aα is the nearest neighbour distance in the fcc lattice, vβ = √ 3 4 a2βcβ is the volume of the β phase with the hcp structure, aβ and cβ are the lattice constants of the hcp lattice. 95 Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien The nearest neighbour distance aα is determined by aα = a0α + ux0α, where a0α denotes the distance aα at temperature 0 K and is determined by the experimental data. The displacement ux0α of a particle from the equilibrium position is calculated from ux0α = √ 2γαθ2 3k3α A, γα = 1 12 ∑ i (∂4ϕi0 ∂u4iβ ) eq + 6 ( ∂4ϕi0 ∂u2iβ∂u 2 iγ ) eq  , β ̸= γ, β, γ = x, y, z, A = a1 + 6∑ i=2 ( γαθ k2α )i ai, kα = 1 2 ∑ i ( ∂2ϕi0 ∂u2iγ ) eq = mω2α, xα = ~ωα 2θ , (2.8) where ai(i = 1− 6) is determined in [6]. The nearest neighbour distance or the lattice constant aβ is determined by aβ = a0β + ux0β, where a0β denotes the distance aβ at temperature 0 K and is determined by the experimental data. The displacement ux0β of a particle from the equilibrium position in the direction x or y is calculated from ux0β = 6∑ i=1 ( γβθ k2xβ )i i ai, kxβ = 1 2 ∑ i [( ∂2ϕi0 ∂u2ix ) eq + ( ∂2ϕi0 ∂uix∂uiy ) eq ] = mω2xβ, γβ = 1 4 ∑ i [( ∂3ϕi0 ∂u3ix ) eq + ( ∂3ϕi0 ∂uix∂u2iy ) eq ] , (2.9) where ai(i = 1− 6) is determined in [6]. The lattice constant c is determined by c = c0 + uz0, where c0 denotes the distance c at temperature 0K and is determined by the experimental data. The displacement uz0β of a particle from the equilibrium position in the direction z is calculated from uz0β ≈ [ 1 3 6∑ i=1 ( θ kzβ )i bi ]1/2 , kzβ = 1 2 ∑ i ( ∂2ϕi0 ∂u2iz ) eq = mω2zβ, (2.10) where bi(i = 1− 6) is determined in [6]. The free energy of the phase with the fcc structure in the harmonic approximation is determined by ψα = 3N {u0α 6 + θ [ xα + ln ( 1− e−2x)]} . (2.11) The free energy of the β phase with the hcp structure in the harmonic approximation is determined by ψβ = 3N { u0β 6 + 2 3 θ [ xβ + ln ( 1− e−2x)]+ 1 3 θ [ xzβ + ln ( 1− e−2xz)]} . (2.12) 96 Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure At pressure P = 0, from Gα = Gβ we have ψα = ψβ. From that and using (2.11) and (2.12), we find the α− β phase transition temperature Tα−β = u0β − u0α 6kB { xα − 23xβ − 13xzβ + ln [ 1−e−2x (1−e−2x)2=3(1−e−2xz)1=3 ]} . (2.13) At pressure P , the α - β phase transition temperature is determined by combining (2.7), (2.11) and (2.12) and is equal to Tα−β = u0β − u0α + P (√ 3 2 a2βcβ − √ 2a3α ) 6kB { xα − 23xβ − 13xzβ + ln [ 1−e−2x (1−e−2x)2=3(1−e−2xz)1=3 ]} . (2.14) 2.2. Numerical results and discussion For solid N2 and CO, the interaction potential between two atoms is usually used in the form of the Lennard-Jones pair potential ϕ(r) = 4ε [(σ r )12 − (σ r )6] , (2.15) 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 and σ are determined by the following experimental data: ε kB = 95.05 K and σ = 3.698.10−10 m for α − N2, ε kB = 110.07 K and σ = 3.59.10−10 m for α − CO, ε kB = 95.05 K and σ = 3.698.10−10 m for β − N2, ε kB = 100.1 K and σ = 3.769.10−10 m for β-CO [7]. Therefore, using two coordination spheres and the results in [6], we obtain the values of the crystal parameters for alpha-N2 and α-CO as follows: kα = 4εα a2α ( σα aα )6 [ 265.298 ( σα aα )6 − 64.01 ] γα = 16εα a4α ( σα aα )6 [ 4410.797 ( σα aα )6 − 346.172 ] γ1α = 4εα a4α ( σα aα )6 [ 803.555 ( σα aα )6 − 40.547 ] γ2α = 4εα a4α ( σα aα )6 [ 3607.242 ( σα aα )6 − 305.625 ] 97 Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien u0α = 4εα ( σα aα )6 [ 12.132 ( σα aα )6 − 14.454 ] (2.16) and the crystal parameters for β-N2 and β-CO as follows: kxβ = 4εβ a2β ( σβ aβ )6 [ 614.6022 ( σβ aβ )6 − 162.8535 ] , kzβ = 4εβ a2β ( σβ aβ )6 [ 286.3722 ( σβ aβ )6 − 64.7487 ] , γβ = −4εβ a3β ( σβ aβ )6 [ 161.952 ( σβ aβ )6 − 24.984 ] , τ1β = 4εβ a4β ( σβ aβ )6 [ 6288.912 ( σβ aβ )6 − 473.6748 ] , τ2β = 4εβ a4β ( σβ aβ )6 [ 11488.3776 ( σβ aβ )6 − 752.5176 ] , τ3β = 4εβ a4β ( σβ aβ )6 [ 8133.888 ( σβ aβ )6 − 737.352 ] , τ4β = 4εβ a4β ( σβ aβ )6 [ 43409.3184 ( σβ aβ )6 − 4550.04 ] , τ5β = 4εβ a4β ( σβ aβ )6 [ 11315.6064 ( σβ aβ )6 − 1006.0428 ] , τ6β = 4εβ a4β ( σβ aβ )6 [ 40782.6048 ( σβ aβ )6 − 4189.6536 ] , u0β = 4εβ ( σβ aβ )6 [ 11.648 ( σβ aβ )6 − 14.1601 ] . (2.17) Using the definitions of u0α, u0β, kα, kxβ, kzβ, Dα, Dβ and 73 coordination spheres, we have the following results [6]. u0β − u0α = 4εβ ( σβ aβ )6 [ 11.648 ( σβ aβ )6 − 14.1601 ] − 4εα ( σα aα )6 [ 12.232 ( σα aα )6 − 14.154 ] , 98 Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure Dα Dβ = k3α k2xβkzβ = { 4ε a2 ( σ a )6 [ 265.298 ( σ a )6 − 64.01 ]}3 { 4ε a2 ( σ a )6 [ 385.071 ( σ a )6 − 99.2106 ]}2{ 4ε a2 ( σ a )6 [ 286.5978 ( σ a )6 − 64.1154 ]} (2.18) where we do not use (2.9) and (2.17) but we apply kxβ = 1 2 ∑(∂2ϕi0 ∂u2ix ) eq = mω2xβ = 4εβ a2β ( σβ aβ )6 [ 385.071 ( σβ aβ )6 − 99.2106 ] . (2.19) Using ε kB = 95.1 K and σ = 3, 708.10−10 m for α − N2 [7] and ε kB = 96.57 K and σ = 3.73.10−10m for β-N2 [6] and calculating approximately aα ≈ aα(T = 0), aβ ≈ aβ(T = 0), we find the phase transition temperature Tα−β of solid nitrogen. Our calculated result is compared with experiments and other calculations in Table 1. Our result based on combining SMM and SCFM is in very good agreement with experimental results [9] and calculation [8]. Our result is better than other calculations in [2, 8-10]. Table 1. Phase transition temperature T− of solid nitrogen Method Tα−β , K EXPT. [8] 35.6 EXPT. [9] 38.3 CAL. by combining SMM and SCFM according to formula (2.4) of this paper 39.1 CAL. [8] from SCPM using the free rotation molecule model 47 CAL. [8] from SCPM using the disordered direction distribution model 37.4 CAL.[9] using molecular field theory (MFT) 54.4 CAL.[10] using classical ordered theory (COT) 74.9 CAL. [2] using OPDM 43.8 At pressure P = 0, our calculated phase transition temperatures from the formula (2.13) are equal to Tα−β = 17.3 K for solid N2 and Tα−β = 48 K for solid CO. These results are far from the experimental data Tα−β = 35.6 K for solid N2 and Tα−β = 61.6 K for solid CO. Our phase transition temperatures from the formula (2.14) at pressure P = 50 bar are equal to Tα−β = 29.2 K for solid N2 and Tα−β = 57.6 K for solid CO and at pressure P = 100 bar are equal to Tα−β = 41.2 K for solid N2 and Tα−β = 79.7 K for solid CO. When the pressure increases, the phase transition temperature increases. That is in agreement with experimental results [7]. 99 Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien 3. Conclusion In this paper, we derive the analytic expressions (2.4), (2.16) - (2.19) of the α - β structural phase transition temperature based on combining SCFM and SMM. These results are applied to cryocrystal of N2 and our numerical calculation is in very good agreement with experiments [9] and calculation [8]. Our result is better than other calculations in [2, 8-10]. We still give the analytic expressions (2.13), (2.14) of the α - β structural phase transition temperature based on SMM. The numerical calculations from these results for N2 and CO cryocrystals at pressures 0, 50 and 100 bar also are compared with experiments. When the pressure increases, the phase transition temperature increases. That is in agreement with experiments [7]. Our calculated results from combining SCFM and SMM are better than our calculated results only from SMM. REFERENCES [1] R. J. Hemley and N. W. Asheroft, 1998. The revealing role of pressure in the condensed-matter sciences. Phys. Today 51, No. 8, pp. 26-52. [2] N. V. Kotenok, 1972. Theory of polymorphic transformation, MGUBulletin, Physics and Astronomy Series No. 1, pp. 40-43 (in Russian). [3] I. P. Bazarov, 1972. Statistical theory of the crystalline state. MGU (in Russian). [4] J. C. Raich, N. S. Gillis and T. R. Koehler, 1974. Self-consistent phonon calculation of the β phase and the α − β transition in solid nitrogen. J. Chem. Phys. 61, No. 4, pp. 1411-1414. [5] J. C. Raich and R. D. Etters, 1972. 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