Đang tải... (xem toàn văn)
The α - β (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.
JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol 59, No 7, pp 93-100 This paper is available online at http://stdb.hnue.edu.vn STRUCTURAL PHASE TRANSITION TEMPERATURE OF N2 AND CO MOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1 , Bui Duc Tinh1 , Dinh Quang Vinh1 and Nguyen Duc Hien2 Faculty of Physics, Hanoi National University of Education Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province Abstract The α - β (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 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 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 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 θ u0 = − θ ln(2πθ) + ln D + N 2 where u0 = ∑ (2.1) φi0 (|⃗ai |) is the potential energy of a particle at the lattice node, ϕi0 is the i interaction potential between the ith particle and the 0th particle, D is the determinant ∂ u0 and is equal to the product of the diagonal elements of this of the matrix ∂u1α ∂u1β matrix belonging to the principal axis, θ = kB T with kB being the Boltzmann constant The polymorphous transformation temperature or the α - β phase transition temperature θα−β Tα−β = is determined using the equation system kB 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 ∂ψβ =− ∂vα ∂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 ≈ atm we can suppose P = in (2.2) and then, θα−β = u0β − u0α ( ) , α ln D Dβ ∂ψβ ∂ψα = = ∂vα ∂vβ (2.4) At T = Tα−β , the structure is unstable and we have a structural phase transition Dα We can calculate u0β − u0α and from the SMM We apply equation (2.4) in order to Dβ 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 + P V, (2.6) where ψ is the free energy, V = N v is the volume, N is the number of particle and v is the volume per particle, we have ψα + Pα−β N vα = ψβ + Pα−β N vβ , (2.7) a3 where vα = √α is the volume of the α phase with the fcc structure, aα is the nearest √ neighbour distance in the fcc lattice, vβ = a cβ is the volume of the β phase with the β 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 K and is determined by the experimental data The displacement ux0α of a particle from the equilibrium position is calculated from ( √ ) ( ) 4 ∑ ∂ ϕi0 2γα θ ∂ ϕi0 , β ̸= γ, β, γ = x, y, z, A, γα = +6 ux0α = 3kα 12 i ∂uiβ ∂u2iβ ∂u2iγ eq A = a1 + )i ( ∑ γα θ i=2 kα2 ( 1∑ , kα = i eq ∂ ϕi0 ∂u2iγ ) = mωα2 , xα = eq ωα , 2θ (2.8) where (i = − 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 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 ( )i [( ( ) ) ] ∑ γβ θ ∂ ϕi0 ∂ ϕi0 1∑ ux0β = , kxβ = + = mωxβ , 2 k ∂u ∂u ∂u ix iy ix xβ eq eq i=1 i i 1∑ γβ = i [( ∂ ϕi0 ∂u3ix ( ) + eq ∂ ϕi0 ∂uix ∂u2iy ) ] , (2.9) eq where (i = − 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 uz 0β of a particle from the equilibrium position in the direction z is calculated from [ ( )i ]1/2 ( ) 1∑ θ ∑ ∂ ϕi0 uz0β ≈ = mωzβ , (2.10) bi , kzβ = i=1 kzβ i ∂u2iz eq where bi (i = − 6) is determined in [6] The free energy of the phase with the fcc structure in the harmonic approximation is determined by {u [ ( )]} 0α −2xα (2.11) ψα = 3N + θ xα + ln − e The free energy of the β phase with the hcp structure in the harmonic approximation is determined by { } ( )] [ ( )] u0β [ −2xβ −2xzβ ψβ = 3N + θ xβ + ln − e + θ xzβ + ln − e (2.12) 3 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 = 6kB u0β − u0α [ ]} 1−e−2xα xα − xβ − xzβ + ln 2/3 1/3 (1−e−2xβ ) (1−e−2xzβ ) (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 (√ √ 3) u0β − u0α + P aβ cβ − 2aα ]} { [ (2.14) Tα−β = 1−e−2xα 6kB xα − xβ − xzβ + ln 1/3 2/3 (1−e−2xβ ) (1−e−2xzβ ) 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 [( ) ] σ 12 ( σ )6 − ϕ(r) = 4ε , (2.15) r r where σ is the distance in which ϕ(r) = 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 ε = 95.05 K and σ = 3.698.10−10 m for α − N2 , = 110.07 K and σ = 3.59.10−10 kB kB ε ε m for α − CO, = 95.05 K and σ = 3.698.10−10 m for β − N2 , = 100.1 K kB kB 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: ] ( )6 ( )6 [ σα 4εα σα kα = 265.298 − 64.01 aα aα aα ] ( )6 [ ( )6 16εα σα σα γα = 4410.797 − 346.172 aα aα aα ] ( )6 [ ( )6 σα 4εα σα − 40.547 803.555 γ1α = aα aα aα ] ( )6 [ ( )6 4εα σα σα γ2α = − 305.625 3607.242 aα aα aα 97 Nguyen Quang Hoc, Bui Duc Tinh, Dinh Quang Vinh and Nguyen Duc Hien ( u0α = 4εα σα aα ] )6 [ ( )6 σα 12.132 − 14.454 aα (2.16) and the crystal parameters for β-N2 and β-CO as follows: ] )6 [ ( )6 σβ kxβ 614.6022 − 162.8535 , aβ ] ( )6 [ ( )6 σβ 4εβ σβ 286.3722 − 64.7487 , kzβ = aβ aβ aβ ] ( )6 [ ( )6 4εβ σβ σβ γβ = − 161.952 − 24.984 , aβ aβ aβ ] ( )6 [ ( )6 4εβ σβ σβ τ1β = 6288.912 − 473.6748 , aβ aβ aβ ] ( )6 [ ( )6 σβ 4εβ σβ τ2β = 11488.3776 − 752.5176 , aβ aβ aβ ] ( )6 [ ( )6 σβ 4εβ σβ τ3β = 8133.888 − 737.352 , aβ aβ aβ ] ( )6 [ ( )6 σβ 4εβ σβ 43409.3184 − 4550.04 , τ4β = aβ aβ aβ ] ( )6 [ ( )6 4εβ σβ σβ τ5β = 11315.6064 − 1006.0428 , aβ aβ aβ ] ( )6 [ ( )6 σβ 4εβ σβ τ6β = 40782.6048 − 4189.6536 , aβ aβ aβ 4εβ = aβ ( σβ aβ ( u0β = 4εβ σβ aβ )6 [ ] ( )6 σβ 11.648 − 14.1601 aβ (2.17) Using the definitions of u0α , u0β , kα , kxβ , kzβ , Dα , Dβ and 73 coordination spheres, we have the following results [6] ] ( )6 )6 [ σβ 11.648 − 14.1601 u0β − u0α = 4εβ aβ ] ( )6 [ ( )6 σα σα − 4εα 12.232 − 14.154 , aα aα ( 98 σβ aβ Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure Dα k3 = α Dβ kxβ kzβ ]}3 265.298 − 64.01 ={ ( ) [ ]}2 { ( ) [ ]} ( )6 ( )6 6 4εβ σβ σβ 4εβ σβ σβ 385.071 aβ − 99.2106 286.5978 aβ − 64.1154 aβ aβ a2 a2 { 4εα a2α ( σα aα )6 [ ( σα aα β )6 β (2.18) where we not use (2.9) and (2.17) but we apply kxβ 1∑ = ( ∂ ϕi0 ∂u2ix ) = mωxβ eq 4εβ = aβ ( σβ aβ )6 [ ( σβ 385.071 aβ ] )6 − 99.2106 (2.19) ε ε = 95.1 K and σ = 3, 708.10−10 m for α − N2 [7] and = 96.57 kB kB K and σ = 3.73.10−10 m 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 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] Using Table Phase transition temperature Tα−β of solid nitrogen Method EXPT [8] EXPT [9] CAL by combining SMM and SCFM according to formula (2.4) of this paper CAL [8] from SCPM using the free rotation molecule model CAL [8] from SCPM using the disordered direction distribution model CAL.[9] using molecular field theory (MFT) CAL.[10] using classical ordered theory (COT) CAL [2] using OPDM Tα−β , K 35.6 38.3 39.1 47 37.4 54.4 74.9 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 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, MGU Bulletin, 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 Librational motion in solid α − N2 J Low Temp Phys 7, No 5/6, pp 449-458 [6] Nguyen Quang Hoc, Do Dinh Thanh and Nguyen Tang, 1996 The limiting of absolute stability, the melting and the α − β phase transition temperatures for solid nitrogen Communications in Physics 6, No 4, pp 1-8 [7] B I Verkina, A Ph Prikhotko (ed.), 1983 Cryocrystals, Kiev, pp.1-526 (in Russian) [8] A F Schuch and R L Mills, 1970 Crystal structure of the three modifications of nitrogen 14 and nitrogen 15 at high pressure J Chem Phys 52, No.12, pp 6000-6008 [9] T A Scott, Phys, Rev C 27, No (1976) 40 [10] B Kohin, 1960 Molecular rotation in crystals of N2 and CO J Chem Phys 33, No 3, pp 882-889 100 ... 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. .. (2.12) 3 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),... [ ( )6 σα σα − 4εα 12.232 − 14.154 , aα aα ( 98 σβ aβ Structural phase transition temperature of N2 and CO molecular cryocrystals under pressure Dα k3 = α Dβ kxβ kzβ ]}3 265.298 − 64.01 ={ (