Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type with fcc and HCP structures in harmonic, classical and anharmonic approximations

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0038 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 94-103 This paper is available online at ANALYTIC EXPRESSION OF THERMODYNAMIC QUANTITIES FORMOLECULAR CRYOCRYSTALS OF NITROGEN TYPEWITH FCC AND HCP STRUCTURES IN HARMONIC, CLASSICAL AND ANHARMONIC APPROXIMATIONS Nguyen Quang Hoc1, Mai Thi La1, Vo Minh Tien2 and Dao Kha Son2 1Faculty of Physics, Hanoi National University of Education 2Faculty of Physics, Tay Nguyen University Abstract. The analytic expressions of thermodynamic quantities such as the Helmholtz free energy, the internal energy, the entropy, the molar heat capacity at constant volume for molecular cryocrystals of N2 type with face-centered cubic (FCC) and hexagonal close-packed (HCP) structures in harmonic, classical and anharmonic approximations are obtained by combining the statistical moment method (SMM) and the self-consistent field method (SCFM). Keywords: Statistical moment method, self-consistent field method, cryocrystal. 1. Introduction Molecular crystals, comprising a vast and comparatively little investigated class of solids, are characterized by a diversity of properties. Up to now only solidified noble gases have systematically been investigated and this is due to the availability of relevant theoretical models and the ease of comparing theories with experimental results. Recently, experimental data have been obtained for simple non-monoatomic molecular crystals as well, and this in turn has stimulated the appearance of several theoretical papers on that subject. This paper deals with the analysis of thermodynamic properties of the group of non-monoatomic molecular crystals including solid N2 and CO that have similar physical properties. These crystals are formed by linear molecules and in their ordered phase, the molecular centers of mass are situated at the site of a face-centered cubic (FCC) pattern, the molecular axes being directed to the four spatial diagonals of a cube (space group Pa3). The characteristic feature of the intermolecular interaction in such crystals is that the non-central part of the potential results from quadrupole forces and from the part of valence and dispersion forces having analogous angular dependence as quadrupole forces, and further, that dipole interaction either does not exist (N2) or is negligible (CO) to influence the majority of thermodynamic properties. In addition, all crystals considered have a common feature, namely their intrinsic rotational temperatures B = ~2/(2I) (I is the momentum of inertia of the corresponding molecule) are small compared to the energy of non-central interaction. Received December 11, 2014. Accepted October 1, 2015. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 94 Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type... In the low-temperature range, it is reasonable to apply an assumption successfully used by the authors [1, 2] that translational motions of the molecular system are independent. As shown [3] there are two types of excitations in molecular crystals - phonons and librons and, furthermore, the thermodynamic functions can be written as a sum of two independent terms corresponding to each subsystem. In such a treatment, the translational–orientational interaction leads to a renormalization of the sound velocity and of the libron dispersion law only. The investigation of the librational behavior of molecules is usually carried out within the framework of the harmonic approximation. However, anharmonic effects for the thermodynamic properties are essential at temperatures substantially lower than the orientational disordering temperature. The effect of molecular rotations in N2 and CO crystals not restricted by the assumption of harmonicity of oscillations has been calculated numerically in the molecular field approximation by Kohin [4]. Full calculations on thermodynamic properties of molecular crystals of nitrogen type are given by the statistical moment method (SMM) in [5] and by self-consistent field method (SCFM) in [6]. This paper represents the analytic expressions of thermodynamic quantities for molecular cryocrystals of nitrogen type with FCC and hexagonal close-packed (HCP) structures such as the free energy, the energy, the entropy and the heat capacity at constant volume in harmonic, classical and anharmonic approximations. 2. Content 2.1. Analytic expression of thermodynamic quantites for crystals of N2 type from the combination of SMM and SCFM 2.1.1. Free energy By combining the SMM and the SCFM, the free energy of molecular crystals of N2 type with FCC and HCP structures is the sum of the vibrational free energy and the rotational free energy. In harmonic approximation (harmonic approximation of lattice vibration and pseudo-harmonic approximation of molecular rotational motion) for FCC crystal [6, 7], ψfcc,har = ψfcc,harvib + ψ fcc,har rot , ψfcc,harvib = V fcc 0 + ψ fcc 0vib = V fcc 0 + 3Nθ [ xfcc + ln(1− e−2xfcc) ] , ψharrot = kBN { 2T ln [ 4 sinh ξ 2T ] − U0η + U0η 2 2 } , kfcc ≡ 1 2 ∑ i ( ∂2ϕi0 ∂u2iα ) eq ≡ mωfcc2,α = x, y, z, x = ~ω fcc 2θ , θ = kBT, V fcc 0 = N 2 ∑ i ϕi0 (2.1) and for HCP crystal [7, 8], ψhcp,har = ψhcp,harvib + ψ har rot , ψhcp,harvib = V hcp 0 +ψ hcp 0vib = V hcp 0 +2Nθ [ xhcp + ln ( 1− e−2xhcp )] +Nθ [ xhcpz + ln ( 1− e−2xhcpz )] , khcpx ≡ 1 2 ∑ i ( ∂2ϕi0 ∂u2ix + ∂2ϕi0 ∂uix.∂uiy ) eq ≡ mωhcp2x , xhcp = ~ωhcpx 2θ , V hcp0 = N 2 ∑ i ϕi0, 95 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son khcpz ≡ 1 2 ∑ i ( ∂2ϕi0 ∂u2iz ) eq ≡ mωhcp2z , xhcpz = ~ωhcpz 2θ . (2.2) In classical approximation for FCC crystal [6, 7], ψfcc,cla = ψfcc,clavib + ψ cla rot, ψfcc,clavib ≈ V fcc0 + ψfcc0vib + 3N { θ2 kfcc2 ( γfcc2 − γfcc1 ) + 4θ3 kfcc4 [ γfcc22 − 3 ( γfcc21 + 2γ fcc 1 γ fcc 2 )]} , ψfcc0vib = Nθ [ xfcc + ln(1− e−2xfcc) ] , xfcc ≈ 0.283, ψ cla rot − ψcla0rot kBN = −U0η + U0η 2 2 + 2T ln (√ 6BU0η T ) , γfcc1 ≡ 1 48 ∑ i ( ∂4ϕi0 ∂u4iα ) eq ,γfcc2 ≡ 1 8 ∑ i ( ∂4ϕi0 ∂u2iα∂u 2 iβ ) eq ,α 6= β, α, β = x, y, z (2.3) and for HCP crystal [7, 8], ψhcp,cla = ψhcp,clavib + ψ cla rot, ψ hcp,cla vib = V hcp 0 + ψ hcp 0vib + Nθ2 4 [ 6τhcp5 + τ hcp 6 khcp2x + 3τhcp1 khcp2z + 2τhcp2 khcpx k hcp z ] + 3Nθ3 4khcp2z . [ τhcp21 khcp2z + τhcp22 9khcp2x ] − Nθ 4 36khcp3z [ 33τhcp31 khcp3z + 2τhcp32 khcp3x ] , kz ≡ 1 2 ∑ i ( ∂2ϕio ∂u2iz ) eq , τ1 ≡ 1 12 ∑ i ( ∂4ϕi0 ∂u4iz ) eq , τ2 ≡ 1 2 ∑ i ( ∂4ϕi0 ∂u2ix∂u 2 iz ) eq , τ5 ≡ 1 12 ∑ i ( ∂4ϕi0 ∂u4ix ) eq , τ6 ≡ 1 2 ∑ i ( ∂4ϕi0 ∂u2ix∂u 2 iz ) eq . (2.4) In anharmonic approximation (anharmonic approximation of lattice vibration and self-consistent libron approximation of molecular rotational motion) for FCC crystal [6, 7], ψfcc,anh = ψfcc,anhvib + ψ anh rot , ψfcc,anhvib = V fcc 0 + ψ fcc 0vib + 3N { θ2 kfcc2 [ γfcc2 X fcc2 − 2γ fcc 1 3 ( 1 + Xfcc 2 )] + 2θ3 kfcc4 [ 4 3 γfcc22 X fcc ( 1 + Xfcc 2 ) − 2 ( γfcc21 + 2γ fcc 1 γ fcc 2 )( 1 + Xfcc 2 )( 1 +Xfcc )]} , 96 Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type... ψanhrot kBN = 2T ln [ 4 sinh ξ 2T ] − ξ 2 coth2 ξ 2T − B 2 − U0η 2 2 (2.5) and for HCP crystal [7, 8], ψhcp,anh = ψhcp,anhvib + ψ anh rot , ψhcp,anhvib = V hcp 0 + ψ hcp 0vib + Nθ2 12 [ 6τhcp5 + τ hcp 6 khcp2x ( Xhcp + 2 ) + 3τhcp1 khcp2z (Xhcpz + 2) + τhcp2 khcpx k hcp z (Xhcp +Xhcpz + 4) ] + Nθ3 24khcp2z [ τhcp21 khcp2z ( Xhcpz + 2 ) (Xhcpz + 5) + τhcp22 9khcp2x ( Xhcp + 2)(Xhcp + 5 )] − Nθ 4 108khcp3z [ τhcp31 khcp3z ( Xhcpz + 2 ) (3Xhcp2z + 17X hcp z + 13) + τhcp32 9khcp3x ( Xhcp + 2)2(Xhcp + 5 )] , Xhcp ≡ xhcp cothxhcp,Xhcpz ≡ xhcpz coth xhcpz . (2.6) In above mentioned expressions, kB is the Boltzmann constant, T is the absolute temperature, m is the mass of particle at lattice node, ωfcc, ωhcpx , ω hcp z are the frequencies of lattice vibration, kfcc, γfcc1 , γ fcc 2 , k hcp x , k hcp z , τ hcp 1 , τ hcp 2 , τ hcp 5 , τ hcp 6 are the parameters of FCC and HCP crystals 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, uiα is the displacement of the ith particle from equilibrium position in direction α and N is the number of particles per mole or the Avogadro number, U0 is 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, ξ is the energy of rotational excitation and η is the ordered parameter. In the harmonic, classical and anharmonic approximations, the rotational free energy, the rotational energy, the rotational entropy and the rotational heat capacity at constant volume of HCP crystal are identical to that of FCC crystal. This is a common property of crystals of nitrogen type [7]. In the harmonic approximation of lattice vibration, the quantities such as V fcc0 , V hcp 0 , k fcc, khcpx , k hcp z are expressed in terms of the nearest neighbor distance a0 at 0 K. The pseudo-harmonic approximation of molecular rotational motion corresponds to the condition T√ U0Bη << 1 [7]. In the classical approximation of lattice vibration, xcthx ≡ X = 1, xzcthxz ≡ Xz = 1 or x = xz ≈ 0.283. Then the formulae of ψ,E, S,CV obtained in anharmonic approximation 97 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son give the classical results. From that we can determine the temperature Tlim ≈ 1.3494.10−11ω corresponding to that where the quantum effects can be neglected [6]. The classical approximation of molecular rotational motion corresponds to the condition T√ U0Bη >> 1 [7]. In the anharmonic approximation of lattice vibration, the quantities such as kfcc, γfcc1 , γ fcc 2 , k hcp x , k hcp z , τ hcp 1 , τ hcp 2 , τ hcp 5 , τ hcp 6 are expressed in terms of the nearest neighbor distance afcc = afcc0 + u fcc x0 , a hcp = ahcp0 + u hcp x0 , where a fcc 0 , a hcp 0 respectively are the nearest neighbor distance of FCC and HCP crystals at 0 K and the displacements ufccx0 , u hcp x0 of a particle from the equilibrium position are calculated by ufccx0 = √ 2γfccθ2 3kfcc 3 A, γ fcc ≡ 1 12 ∑ i (∂4ϕi0 ∂u4iβ ) eq + 6 ( ∂4ϕi0 ∂u2iβ∂u 2 iγ ) eq  , β 6= γ, β, γ = x, y, z, A = a1 + 6∑ i=2 ( γfccθ kfcc 2 )i afcci , u hcp x0 = 6∑ i=1  γhcpθ( khcpx )2  i ahcpi , γhcp ≡ 1 4 ∑ i (∂3ϕi0 ∂u3ix ) eq + ( ∂3ϕi0 ∂uix∂u2iy ) eq , (2.7) where afcci , a hcp i (i = 1 − 6) are determined in [8]. The anharmonic approximation of molecular rotational motion corresponds to the condition T√ U0Bη ≈ 1[7]. 2.1.2. Energy The energy of molecular crystals of N2 type with FCC and HCP structures is the sum of the vibrational energy and the rotational energy. In harmonic approximation for FCC crystal [6, 7], Efcc,har = ψfcc,har − T ( ∂ψfcc,har ∂T ) V = Efcc,harvib +E har rot , Efcc,harvib = V fcc 0 + E fcc 0vib = V fcc 0 + 3NθX fcc, Eharrot = −NkBU0 [ 1− 3B ξ coth ξ 2T ] + NkBU0 2 [ 1− 3B ξ coth ξ 2T ]2 −NkB ( T ∂ξ ∂T − ξ ) coth ξ 2T + 3NkBU0BT 2ξ2 [ 1 + 3B ξ coth ξ 2T ]{ ξ 2T 2 ( T ∂ξ ∂T − ξ )[ 1− coth2 ξ 2T ] − ∂ξ ∂T coth ξ 2T } (2.8) and for HCP crystal [7, 8], Ehcp,har = Ehcp,harvib + E har rot , E hcp,har vib = V hcp 0 + E hcp 0vib = V hcp 0 +Nθ(2X hcp +Xhcpz ). (2.9) In classical approximation for FCC crystal [6, 7], 98 Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type... Efcc,cla = Efcc,clavib + E cla rot, E fcc,cla vib = V fcc 0 + 3Nθ [ 1 + θ kfcc2 ( γfcc1 − γfcc2 )] , Eclarot = ψ cla 0rot +NkB [ −U0η + 1 2 U0η 2 + 2T ln (√ 6BU0η T )] − T ∂ ∂T {ψ0rot + kBN [−U0η + 1 2 U0η 2 + 2T ln (√ 6BU0η T )]} = Ecla0rot + kBNU0η (η 2 − 1 ) + kBNTU0 2 ∂η ∂T (1− 2η) −2kBNT ( T 2η ∂η ∂T − 1 ) , η = 1 2 ( 1− √ 1− 4T U0 ) , ∂η ∂T = − 2 U0 √ 1− 4TU0 (2.10) and for HCP crystal [7, 8], Ehcp,cla = Ehcp,clavib + E cla rot, Ehcp,clavib = V hcp 0 + E hcp,cla 0vib − Nθ2 4  ( 6τhcp5 + τ hcp 6 ) khcp2x + 3τhcp1 khcp2z − τ hcp 2 khcpx k hcp z  . (2.11) In anharmonic approximation for FCC crystal [6, 7], Efcc,anh = Efcc,anhvib + E anh rot , Efcc,anhvib = V fcc 0 + 3NθX fcc + 3Nθ2 kfcc2 [ γfcc2 X fcc2 + γfcc1 3 ( 2 + Y fcc2 ) − 2γfcc2 XfccY fcc2 ] , Eanhrot = NkB { 2T [ ln ( 4 sinh ξ 2T )] − ξ 2 coth2 ξ 2T − B 2 − U0η 2 2 } −kBTN ∂ ∂T { 2T [ ln ( 4 sinh ξ 2T )] − ξ 2 coth2 ξ 2T − B 2 − U0η 2 2 } = kBN 2 coth ξ 2T ( ξ coth ξ 2T + T ∂ξ ∂T ) −kBN [ B 2 + U0η ( 1 2 − T ∂η ∂T )] − kBN 4 ( T ∂ξ ∂T − ξ )( ξ T 1 sinh2 ξ2T − 1 ) (2.12) and for HCP crystal [7, 8], Ehcp,anh = Ehcp,anhvib + E anh rot , E hcp,anh vib = V hcp 0 +E hcp 0vib −Nθ 2 12 [ 6τhcp5 + τ hcp 6 khcp2x ( 2 + Y hcp2 ) + 3τhcp1 khcp2z ( 2 + Y hcp2z ) + τhcp2 khcpx k hcp z ( 1 + Y hcp2 + Y hcp2z )] , (2.13) 99 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son 2.1.3. Entropy The entropy of molecular crystals of N2 type with FCC and HCP structures is the sum of the vibrational entropy and the rotational entropy. In harmonic approximation for FCC crystal [6, 7], Sfcc,har = Efcc,har − ψfcc,har T = Sfcc,harvib +S har rot , S fcc,har vib = 3NkB [ Xfcc − ln ( 2 sinhxfcc )] , Sharrot = − NkBU0 T [ 1− 3B ξ coth ξ 2T ] + NkBU0 2 [ 1− 3B ξ coth ξ 2T ]2 −NkB T ( T ∂ξ ∂T − ξ ) coth ξ 2T + + 3NkBU0BT 2ξ2 [ 1 + 3B ξ coth ξ 2T ] × { ξ 2T 2 ( T ∂ξ ∂T − ξ )[ 1− coth2 ξ 2T ] − ∂ξ ∂T coth ξ 2T } − kBN T { 2T ln [ 4 sinh ξ 2T ] − U0η + U0η 2 2 } (2.14) and for HCP crystal [7, 8], Shcp,har = Shcp,harvib + S har rot , Shcp,harvib = NkB { 2 [ Xhcp − ln ( 2 sinhxhcp )] + [ Xhcpz − ln ( 2 sinhxhcpz )]} . (2.15) In classical approximation for FCC crystal [6, 7], Sfcc,cla = Sfcc,clavib + S cla rot, Sfcc,clavib = 3NkB [ 1− ln ( 2 sinhxfcc )] + 6NkBθ kfcc2 ( γfcc1 − γfcc2 ) , xfcc ≈ 0.283, Sclarot = S cla 0rot + kBNU0η T (η 2 − 1 ) + kBNU0 2 ∂η ∂T (1− 2η) − 2kBN ( T 2η ∂η ∂T − 1 ) −kBN T [ −U0η + U0η 2 2 + 2T ln (√ 6BU0η T )] (2.16) and for HCP crystal [7, 8], Shcp,cla = Shcp,clavib + S cla rot, Shcp,clavib = S hcp 0vib + NkBθ 2  ( 6τhcp5 + τ hcp 6 ) khcp2x − 3τ hcp 1 khcp2z − 2τ hcp 2 khcpx k hcp z  . (2.17) In anharmonic approximation for FCC crystal [6, 7], Sfcc,anh = Sfcc,anhvib + S anh rot , 100 Analytic expression of thermodynamic quantities for molecular cryocrystals of nitrogen type... Sfcc,anhvib = 3NkB [ Xfcc − ln ( 2 sinhxfcc )] + 3NkBθ kfcc2 [ γfcc1 3 ( 4 +Xfcc + Y fcc2 ) − 2γfcc2 XfccY fcc2 ] , Sanhrot = kBN 2T coth ξ 2T ( ξ coth ξ 2T + T ∂ξ ∂T ) −kBN T [ B 2 + U0η ( 1 2 − T ∂η ∂T )] − kBN 4T ( T ∂ξ ∂T − ξ )( ξ T 1 sinh2 ξ2T − 1 ) −kBN T { 2T ln [ 4 sinh ξ 2T ] − ξ 2 coth2 ξ 2T − B 2 − U0η 2 2 } (2.18) and for HCP crystal [7, 8], Shcp,anh = Shcp,anhvib + S anh rot , S hcp,anh vib = S hcp 0vib − NkBθ 12 [ 6τhcp5 + τ hcp 6 khcp2x ( 4 +Xhcp + Y hcp2 ) + 3τhcp1 khcp2z ( 4 +Xhcpz + Y hcp2 z ) + τhcp2 khcpx k hcp z ( 8 +Xhcp + Y hcp2 +Xhcpz + Y hcp2 z )] . (2.19) 2.1.4. Heat capacity at constant volume The heat capacity at constant volume of molecular crystals of N2 type with FCC and HCP structures is the sum of the vibrational heat capacity at constant volume and the rotational heat capacity at constant volume. In harmonic approximation for FCC crystal [6, 7], Cfcc,harV = −T ( ∂2ψfcc,har ∂T 2 ) V = Cfcc,harV vib + C har V rot, Cfcc,harV vib = 3NkBY fcc2, CharV rot = NkB 2 ( ξ T )2 sinh2 ξ2T ( 1− T ξ ∂ξ ∂T ) (2.20) and for HCP crystal [7, 8], Chcp,harV = C hcp,har V vib + C har V rot, C hcp,har V vib = NkB ( 2Y hcp2 + Y hcp2z ) . (2.21) In classical approximation for FCC crystal [6, 7], Cfcc,claV = C fcc,cla V vib + C cla V rot, C fcccla V vib = 3NkB [ 1 + 2θ kfcc2 ( γfcc1 − γfcc2 )] , CclaV rot = C cla V 0rot + kBNU0 (η − 1) ∂η ∂T + kBNU0 2 [ (1− 2η) ( ∂η ∂T + T ∂2η ∂T 2 ) − 2T ( ∂η ∂T )2] −2kBN { T 2 η ( ∂η ∂T − 1 ) + T 2η2 [ η ( ∂η ∂T − T ∂ 2η ∂T 2 ) − T ( ∂η ∂T )2]} (2.22) 101 Nguyen Quang Hoc, Mai Thi La, Vo Minh Tien and Dao Kha Son and for HCP crystal [7, 8], Chcp,claV = C hcp,cla V vib + C cla V rot, C hcp,cla V vib = NkB { 3− θ 2 [ 6τhcp5 + τ hcp 6 khcp2x + 3τhcp1 khcp2z + 2τhcp2 khcpx k hcp z ]} . (2.23) In anharmonic approximation for FCC crystal [6, 7], Cfcc,anhV = C fcc,anh V vib + C anh V rot, Cfcc,anhV vib = 3NkB { Y fcc2 + 2θ kfcc2 [( 2γfcc2 + γfcc1 3 ) XfccY fcc2 + 2γfcc1 3 − γfcc2 ( Y fcc4 + 2Xfcc2Y fcc2 )]} CanhV rot = NkB 2 ∂ξ ∂T [ coth2 ξ 2T + coth ξ 2T + 1 2T sinh ξ 2T ( T ∂ξ ∂T − ξ )] +kBNU0T [ η ∂2η ∂T 2 + ( ∂η ∂T )2] −kBN 4T 2 [ ξ coth ξ2T sinh2 ξ2T ( 2− 1 T ) + ( T ∂ξ ∂T − ξ )2(1 2 sinh ξ 2T − 1 sinh2 ξ2T )] + NkB 4 ∂2ξ ∂T 2 ( 3T coth ξ 2T − ξ sinh2 ξ2T ) , ∂ξ ∂T = ∂ξ ∂η . ∂η ∂T , ∂ξ ∂η = √ 6BU0 [ 1 2 √ η η + 2 3 + √ η 3 ] (2.24) and for HCP crystal [7, 8], Chcp,anhV = C hcp,anh V vib + C anh V rot, Chcp,anhV vib = NkB { 2Y hcp2 + Y hcp2z − θ 6 [ 6τhcp5 + τ hcp 6 khcp2x ( 2 +XhcpY hcp2 ) + 3τhcp1 khcp2z ( 2 +Xhcpz Y hcp2 z ) + τhcp2 khcpx k hcp z ( 4 +XhcpY hcp2 +Xhcpz Y hcp2 z )]} . (2.25) 3. 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