To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach

Kinetic equations for the dielectric properties of CsH₂PO₄ type ferroelectrics are derived within the nonequilibrium statistical operator method proposed by D.N.Zubarev. These equations are solved with two different approaches. In the ﬁrst approach the closure relation in the spirit of a two-site c...

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Datum:	1998
Hauptverfasser:	Levitskii, R.R., Zachek, I.R., Sokolovskii, R.O.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1998
Schriftenreihe:	Condensed Matter Physics
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Zitieren:	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach / R.R. Levitskii, I.R. Zachek, R.O. Sokolovskii // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 849-872. — Бібліогр.: 52 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1198922025-02-09T21:09:01Z To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach До теорії релаксаційних явищ у квазіодновимірних сегнетоелектриках з водневими зв’язками. Підхід нерівноважного статистичного оператора Levitskii, R.R. Zachek, I.R. Sokolovskii, R.O. Kinetic equations for the dielectric properties of CsH₂PO₄ type ferroelectrics are derived within the nonequilibrium statistical operator method proposed by D.N.Zubarev. These equations are solved with two different approaches. In the ﬁrst approach the closure relation in the spirit of a two-site cluster approximation is used. In the second approach the exact relation between the static correlation functions is extrapolated to a dynamic case and used as a closure relation. The both approaches give similar results. The obtained dynamic characteristics are close to those of the Glauber model and are shown to describe the dynamic dielectric permittivity of CsH₂PO₄ and CsD₂PO₄ crystals. В рамках методу нерівноважного статистичного оператора Д.М.Зубарєва виведені кінетичні рівняння для діелектричних характеристик сегнетоелектриків типу CsD₂PO₄. Ці рівняння розв’язані в наближенні двочастинкового кластера та шляхом розщеплення ланцюжка рівнянь для функцій розподілу. Обидва методи дають близькі результати. Показано, що розраховані динамічні характеристики є близькими до тих, що отримуються в рамках кінетичної моделі Глаубера та дозволяють описати динамічну діелектричну проникність кристалів CsH₂PO₄ і CsD₂PO₄. 1998 Article To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach / R.R. Levitskii, I.R. Zachek, R.O. Sokolovskii // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 849-872. — Бібліогр.: 52 назв. — англ. 1607-324X DOI:10.5488/CMP.1.4.849 PACS: 64.60.Cn https://nasplib.isofts.kiev.ua/handle/123456789/119892 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Kinetic equations for the dielectric properties of CsH₂PO₄ type ferroelectrics are derived within the nonequilibrium statistical operator method proposed by D.N.Zubarev. These equations are solved with two different approaches. In the ﬁrst approach the closure relation in the spirit of a two-site cluster approximation is used. In the second approach the exact relation between the static correlation functions is extrapolated to a dynamic case and used as a closure relation. The both approaches give similar results. The obtained dynamic characteristics are close to those of the Glauber model and are shown to describe the dynamic dielectric permittivity of CsH₂PO₄ and CsD₂PO₄ crystals.
format	Article
author	Levitskii, R.R. Zachek, I.R. Sokolovskii, R.O.
spellingShingle	Levitskii, R.R. Zachek, I.R. Sokolovskii, R.O. To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach Condensed Matter Physics
author_facet	Levitskii, R.R. Zachek, I.R. Sokolovskii, R.O.
author_sort	Levitskii, R.R.
title	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach
title_short	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach
title_full	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach
title_fullStr	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach
title_full_unstemmed	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach
title_sort	to the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. nonequilibrium statistical operator approach
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	https://nasplib.isofts.kiev.ua/handle/123456789/119892
citation_txt	To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach / R.R. Levitskii, I.R. Zachek, R.O. Sokolovskii // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 849-872. — Бібліогр.: 52 назв. — англ.
series	Condensed Matter Physics
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last_indexed	2025-11-30T20:50:00Z
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No. 4(16), p. 849–872 To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine 1 Svientsitskii St., 290011 Lviv–11, Ukraine Received December 27, 1997 Kinetic equations for the dielectric properties of CsH2PO4-type ferroelec- trics are derived within the nonequilibrium statistical operator method pro- posed by D.N.Zubarev. These equations are solved with two different ap- proaches. In the first approach the closure relation in the spirit of a two-site cluster approximation is used. In the second approach the exact relation between the static correlation functions is extrapolated to a dynamic case and used as a closure relation. The both approaches give similar results. The obtained dynamic characteristics are close to those of the Glauber model and are shown to describe the dynamic dielectric permittivity of CsH2PO4 and CsD2PO4 crystals. Key words: nonequilibrium statistical operator, kinetic equation, order-disorder ferroelectrics, dynamic permittivity PACS: 64.60.Cn 1. Introduction The Ising model (IM) appears to be very effective for the formulation and solu- tion of problems in the theories for electrically or magnetically ordered compounds, binary alloys, etc. [1–8]. In the case of low-dimensional structures, the model al- lows an accurate calculation of the thermodynamic quantities (order parameters, static susceptibility, specific heat) and the spin correlation functions [7–9]. Accu- rate solutions of the two-dimensional IM [2,9] and the one-dimensional Kac model make a substantial contribution to the understanding of the nature of the second order phase transitions. Great interest is given to the dynamics of the IM. The study of Ising systems’ kinetics was initiated by Glauber’s paper [10], where a one-dimensional (1D) model c© R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii 849 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii was treated. The papers [11–13] extend Glauber’s approach to 2D and 3D Ising lattices. As in these cases the problem becomes rather intractable, the mean field approximation [11] and the cluster approximation [12,13] had to be used. Paper [13] differs from the preceding work [12] due to the account of a long-range interaction. Glauber’s idea also inspired studies of the model with several modifications [14,15]. Within Glauber’s approach one can derive [32] a chain of coupled equations for time-dependent distribution functions (DF). Their solution can be found only for 1D model. For pseudo 1D model Zumer [20] suggested the simplest decoupling of the chain, which leads to some interpolative equation for one-site DF that yields the generally known exact result for static susceptibility. But Zinenko [30] dis- covered that approximations of papers [21,22] lead to incorrect results in the low- temperature region. Paper [32] suggests a detailed analysis of the coupled equations for time-dependent DF obtained from the Glauber master equation. Several de- coupling schemes are suggested. One of them yields thermodynamic and dynamic quantities of the model similar to those obtained within a cluster approach devel- oped in papers [28,29]. These theoretical results were extensively used for the dis- cussion and analysis of experimental data on ferroelectric of CsH2PO4 type [16–32]. The crystal structure of CsH2PO4 can be described as “PO4 tetrahedrons con- nected by hydrogen bonds of two types”. The shorter H-bonds connect PO4 groups forming zig-zag chains along the b-axis. In the paraelectric phase the protons of these bonds are distributed symmetrically with respect to the centre of an H-bond, whereas in the ferroelectric phase there appears a spontaneous asymmetry. The protons of the longer H-bonds are ordered in both phases. Deuteration signifi- cantly changes the temperature of the phase transition, the dielectric and thermal properties of CsH2PO4 crystal [16–29]. It is the evidence of the important role of H-bonds in the ferroelectricity of these crystals. Ferroelectric crystals PbHPO4 contain only short H-bonds connecting PO4 groups along the c-axis [17,18,19,33,34,35]. The H-bonds in these crystals are al- most parallel to the direction of the spontaneous polarization. The phase transition in these crystals is also closely connected with the proton ordering. However, the effects of deuteration in PbHPO4 crystals are much less pronounced in comparison with CsH2PO4 crystals. It is nowadays commonly accepted that the ordering of protons (deuterons) in the ferroelectrics of CsH2PO4 type is of a quasi-one-dimensional nature implying strong intrachain and weak interchain interactions. This results in some specific properties of such compounds which should be explained on a microscopic level. Papers [16–27] are focused on the discussion of the selected experiment and the suggested model parameters which provide the best fit to those data. Such a one-sided approach sometimes leads to incorrect conclusions. Thus, the authors of papers [22–25], comparing predictions of the model with such parameters with other measured quantities, found a marked disagreement and concluded that the model cannot describe the compounds under investigation. However, papers [28,29, 32] show that the theory [28,32] which allows for short- and long-range interactions, describes thermodynamics and dynamics of these compounds with a satisfactory 850 Nonequilibrium statistical operator approach accuracy. Glauber’s approach, however, has several deficiencies. This is, first of all, lack of substantiation for the master equation, which contains phenomenologically chosen spin-flip probabilities and unknown kinetic parameters that should be provided by a separate ab-initio theory. Such drawbacks are not intrinsic to the kinetic equations method [33–43]. There exists a number of approaches in this field. In this paper we shall use a nonequilibrium statistical operator (NSO) method suggested by D.N.Zubarev. In the presence of a small parameter, the NSO method allows one to obtain kinetic equations for the model [41–43]. We intend to use the method in order to describe the dynamics of quasi-1D ferroelectrics with hydrogen bonds. In section 2 a kinetic equation is derived. It is solved in section 3 within a two-site cluster approximation. In section 4 a chain of coupled equations for time-dependent DFs is derived and a decoupling scheme is proposed, which supports the results of the cluster approximation. If the kinetic parameters are independent of frequency, the NSO method yields the results that coincide with those [28,32] obtained within the Glauber master equation approach. 2. Nonequilibrium statistical operator. Kinetic equation In this section we shall consider relaxation phenomena in systems with the Hamiltonian H = H0 + V ; V̂ = Ĥ ′(s, t), (2.1) where H0 = H1(s) +H2(t). (2.2) Here Ĥ1(s) denotes the Hamiltonian of the quasispin subsystem, Ĥ2(t) is a Hamil- tonian of the dissipative subsystem, H ′(s, t) describes their interaction. Non-spin degrees of freedom constitute a heat bath, i.e. their heat capacity is so large that any perturbation caused by the quasispin subsystem cannot push the heat bath out of the equilibrium. The nonequilibrium statistical operator of the entire system satisfies the Liouville equation ∂ ∂t ρ̂(t) + iL̂ρ̂(t) = 0, (2.3) where iL is a Liouville operator defined by iL̂Â = − 1 ih̄ [Â, Ĥ]. (2.4) In order to calculate NSO for the system with a small parameter, the following form of (2.3) is convenient [43]: ∂ ∂t ln ρ̂(t) + iL̂ ln ρ̂(t) = 0. (2.5) Solution of the Liouville equation (2.5) (or (2.3)) requires boundary conditions. Following the Zubarev NSO method [40] we shall look for such solutions of (2.5) 851 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii which depend on time only via several observable variables. These variables of a reduced description set the nonequilibrium state of the system. The desired solutions can be selected, including an infinitely small source in the right–hand side of equation (2.5) [40]: ∂ ∂t ln ρ̂(t) + iL̂ ln ρ̂(t) = −ε[ln ρ̂(t)− ln ρ̂q(t)]. (2.6) Here ρ̂q(t) is a quasiequilibrium statistical operator that has to minimize the en- tropy of the system at fixed variables of a reduced description and the normal- ization Spρ̂q(t) = 1. The source in the right-hand side of (2.6) violates the time- reversal symmetry of the equation and selects retarded solutions corresponding to a reduced description. After the thermodynamic limit one has to set ε = 0 and to obtain the solution of the equation (2.5). The described above approach is based on Bogolubov’s idea that the nonequi- librium dynamics of the macroscopic physical system is characterized by a hierar- chy of relaxation times [33]: parts of the system reach the equilibrium long before the entire system does. Therefore, these parts can be described by a small number of quasiintegrals of motion which slowly change under the action of weaker interac- tions that equilibrate all the subsystems. Thus, for times that are longer than some relaxation times, the nonequilibrium dynamics of the system can be described by a small number of parameters. In such a way a reduced description becomes suffi- cient [33]. Let P̂m be a set of reduced description variables. The quasiequilibrium statistical operator takes the form [40]: ρ̂q(t) = exp{−Φ(t)− ∑ m Fm(t)P̂m}, (2.7) where Φ(t) = ln Sp exp{− ∑ m Fm(t)P̂m} (2.8) ensures the normalization of ρ̂q(t), Fm(t) being the corresponding Lagrange pa- rameters that have to be found from self-consistency conditions 〈P̂m〉 = 〈P̂m〉q, (2.9) where 〈· · ·〉 = Sp(ρ̂(t) · · ·); 〈· · ·〉q = Sp(ρ̂q(t) · · ·). (2.10) These conditions lead to δΦ(t) δFm(t) = −〈P̂m〉q = −〈P̂m〉. (2.11) It means that Fm(t) are conjugated to 〈P̂m〉. An expression for entropy clarifies the physical meaning of the parameters Fm(t): S(t) = −〈ln ρq(t)〉q = Φ(t) + ∑ m Fm(t)〈P̂m〉. (2.12) 852 Nonequilibrium statistical operator approach This leads to the thermodynamic relations δS(t) δ〈P̂m〉 = Fm(t). (2.13) Equation (2.6) can be written in terms of ∆ ln ρ̂(t) = ln ρ̂(t)− ln ρ̂q(t): { ∂ ∂t + iL̂+ ε } ∆ ln ρ̂(t) = − ( ∂ ∂t + iL̂ ) ln ρ̂q(t). (2.14) The formal solution of equation (2.14) is of the form: ∆ ln ρ̂(t) = − 0 ∫ −∞ eετeiL̂τ ( ∂ ∂t + iL̂ ) ln ρ̂q(t+ τ)dτ (2.15) or ln ρ̂(t) = ln ρ̂q(t)− 0 ∫ −∞ eετeiL̂τ ( ∂ ∂t + iL̂ ) ln ρ̂q(t+ τ)dτ. (2.16) If we introduce the entropy operator S(t) = Φ(t) + ∑ m Fm(t) P̂m = − ln ρq(t) (2.17) and use the identity Ŝ(t) = Φ(t) + ∑ m Fm(t) P̂m, (2.18) then integral equation (2.16) takes the form: ρ̂(t) = exp { − Ŝ(t) + 0 ∫ −∞ eετeiL̂τ 1 ∫ 0 ρ̂(λ)q (t+ τ) ( ∂ ∂τ + iL̂ ) Ŝ(t+ τ)ρ̂(1−λ) q (t+ τ)dλdτ } = exp { − Φ(t)− ∑ m Fm(t)P̂m (2.19) + 0 ∫ −∞ eετeiL̂τ 1 ∫ 0 ρ̂(λ)q (t+ τ) { ∂ ∂τ + ∑ m ∂ ∂τ Fm(t+ τ)P̂m + ∑ m Fm(t+ τ) ˙̂ Pm } ρ̂(1−λ) q (t+ τ)dλdτ } , where ˙̂ Pm = iL̂P̂m. This expression for NSO coincides with that obtained by Pokrovskii [41], if the following relation is taken into account: eiL̂τ P̂m = P̂m(τ) = eiĤτ/hP̂me −iĤτ/h. (2.20) 853 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii Using NSO (2.19) one can get general expressions for the average values of interest d dt 〈P̂m〉 = 〈 ˙̂ Pm〉. (2.21) The evolution of the free system is determined by quantities aml [Ĥ0, P̂m] = ∑ l aml P̂l . (2.22) Transfer equations can be written using these parameters: d dt 〈P̂m〉 = − 1 ih̄ ∑ l aml〈P̂l〉+ 1 ih̄ 〈[P̂m, V̂ ]〉. (2.23) If one calculates the last term in (2.23) using a linear in interaction V̂ approxima- tion for NSO (for the details of that calculation the reader is referred to [41]), the following kinetic equations for 〈P̂m〉 are obtained (to the second order approxima- tion with respect to V̂ ): d dt 〈P̂m〉 = L̂(0) m + L̂(1) m + L̂(2) m , (2.24) where L̂(0) m = − 1 ih̄ ∑ l aml〈P̂l〉q, (2.25) L̂(1) m = − 1 ih̄ 〈[V̂ , P̂m]〉q, L̂ ′(2) m + L̂ ′′(2) m = − 1 h̄2 0 ∫ −∞ dτeετ 〈[V̂ (τ), [V̂ , P̂m] + ih̄ ∑ l P̂l δL(1) m δ〈P̂l〉q ]〉q. (2.26) The kinetic equation (2.24) can be rewritten in the form: d dt 〈P̂m〉 = −i〈[P̂m, Ĥ1(s) + Ĥ ′(s, f)]〉q +R1(P̂m) + R2(P̂m), (2.27) R1(P̂m) = 0 ∫ −∞ dt1e εt1〈[Ĥ ′(t1), [P̂m, ˆ̄H ′]]〉q, R2(P̂m) = 0 ∫ −∞ dt1e εt1〈[ ˆ̄H ′(t1), Pk ∂〈[P̂m, ˆ̄H ′]〉q ∂〈Pk〉 ]〉q. (2.28) Now we must be more explicit and write the quasispin-heat bath interaction term H ′(s, t) of the Hamiltonian. Let us consider only one-particle interactions Ĥ ′(s, f) = ∑ j ∑ α uα j (f)V̂ α j , (2.29) 854 Nonequilibrium statistical operator approach where Uα j (f) are operators related to the heat bath, and 〈Uα j (f)〉 = 0, V 0 j = Sz j , V ±1 j = S± j = Sx j ± iSy j . Let us introduce (the equilibrium) statistical operator of the heat bath ρf = e−βĤ2(f) Sp e−βĤ2(f) (2.30) and operator ˆ̄H ′ (s, f) defined by ˆ̄H(s, f) = Ĥ ′(s, f)− 〈Ĥ ′(s, f)〉f , (2.31) 〈Ĥ ′(s, f)〉f = Spf Ĥ ′ρf , (2.32) whose evolution is of the form: ˆ̄H ′(t) = eit[Ĥ1(s)+Ĥ2(f)] ˆ̄H ′e−it[Ĥ1(s)+Ĥ2(f)]. (2.33) Using (2.29) one finds that ˆ̄H ′(t) = ∑ jα ūα j (t)V̂ α j (t), (2.34) V̂ α j (t) = eitĤ1(s) V̂ α j e−itĤ1(s), ūα j (t) = eitĤ2(f) ūα j (t)e −itĤ2(f); ūα j (f) = 〈uα j (f)〉f . (2.35) With the help of the time Fourier transformation V̂ α j (t) = ∑ µ V̂ α j (Ω− µα) exp(−itΩα µ) (2.36) we obtain R1(P̂m) = ∑ ij ∑ µν ∑′ αα′ 0 ∫ −∞ dt1e εt1−iΩα µt1 × {〈ūα j (t1)ū α′ i 〉q〈V̂ α j (Ω α µ)[P̂m, V̂ α′ i (Ωα′ ν )]〉q (2.37) − 〈ūα′ i ūα j (t1)〉q〈[P̂m, V̂ α′ i (Ωα′ ν )]V̂ α j (Ω α µ)〉q}. Hereafter we assume that fluctuations of heat bath variables at different sites do not correlate. Using also a translational invariance of the model we obtain: 〈ūα j (t)ū α′ j 〉q = δij〈ū α j (t)ū α′ j 〉q, (2.38) 〈ūα j (t)ū α′ j 〉q = 〈ūα j ū α′ j (−t)〉q = γαα′(t), Re γαα′(t) = mαα′(t) = mα′α(−t), (2.39) Im γαα′(t) = nαα′(t) = −nα′α(−t). If there exists a diagonal basis for Ĥ2(f), where matrix elements of operator Uα(f) are real, the following relations hold [44]: mαα′(t) = mαα′(−t); nαα′(t) = −nαα′(−t). (2.40) 855 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii In this case γαα′(t) = mαα′(t)− inαα′(t) = mα′α(t)− inα′α(t), (2.41) γα′α(−t) = mα′α(−t) − inα′α(t) = mα′α(t) + inα′α(t) and (2.37) can be written in the form R1(P̂m) = − ∑ j ∑ µν ∑ αα′ {[Kα′α(Ω α µ) + iMα′α(Ω α µ)]Q − jνµα′α(P̂m) (2.42) + [−Lα′α(Ω α µ) + iNα′α(Ω α µ)]Q + jνµα′α(P̂m)}. where the following notations are used: Q∓ jνµα′α(P̂m) = 〈[[P̂m, V α′ j (Ωα′ ν )], V α j (Ω α µ)] ∓〉q, (2.43) Kα′α(Ω α µ) = ∞ ∫ µ dte−εtmα′α(t) coshΩ α µt = ∞ ∫ µ dte−εtRe γα′α(t)e −iΩα µ t, Lα′α(Ω α µ) = ∞ ∫ 0 dte−εtnα′α(t) sinhΩ α µt = ∞ ∫ 0 dte−εtImγα′α(t)e −iΩα µt, Mα′α(Ω α µ) = ∞ ∫ 0 dte−εtmα′α(t) sinhΩ α µt, (2.44) Nα′α(Ω α µ) = ∞ ∫ 0 dte−εtnα′α(t) coshΩ α µt. Using a spectral representation for time-dependent correlation functions in (2.43) γα′α(t) = ∞ ∫ −∞ J(Ωµ)e iΩα µtdΩα µ, γαα′(−t) = ∞ ∫ −∞ J(Ωµ)e βΩα µteiΩ α µtdΩα µ, (2.45) J(−Ωα µ) = J(Ωα µ)e βΩα µ , one gets: i Imγα′α(Ω α µ) = − tanh (1 2 βΩα µ ) Reγα′α(Ω α µ), (2.46) Lα′α(Ω α µ) = − tanh (1 2 βΩα µ ) Kα′α(Ω α µ). If correlations in the dissipative subsystem have a Markovian character andMα′α(t) takes the form [7] mα′α(t) = 1 2 〈[ūα′ , uα]〉q e−t/τfα , (2.47) where τfα are correlation times, then Mα′α(Ω α µ) = Ωα µτfαKα′α(Ω α µ) (2.48) 856 Nonequilibrium statistical operator approach coefficients N and K can be related using (2.43), (2.46) and (2.47). We obtain: nα′α(t) = − 1 τfαΩα µ tanh (βΩα µ 2 ) mα′α(t), (2.49) Nα′α(Ω α µ) = tanh ( β 2τfα ) Kα′α(Ω α µ). Finally, R1(P̂m) = − ∑ j ∑ µν ∑ αα′ {[1 + iΩα µτfα]Q − jνµα′α(P̂m) (2.50) + [tanh βΩα µ 2 − i tanh β 2τfα ]Q+ jνµα′α(P̂m)}Kα′α(Ω α µ). The terms with Ωα µ 6= Ωα′ ν give contributions of order Kα′α(Ω)/Ω to the solution of the kinetic equation. Such contributions are of no importance in many cases [7]. Condition Ωα µ = Ωα′ ν , in its turn, allows one to keep only the terms with α = −α′. The last condition can be proven in a very general form [7,42,43]. The relaxation term (2.50) takes the simplest form if the set {P̂m} contains only diagonal oper- ators. In this case imaginary terms in (2.50) and R2(P̂m) vanish, and the kinetic equation (2.27) takes the form [7,42,43]: d dt 〈P̂m〉 = − ∑ j ∑ µα { Q− jµα(P̂m) + tanh βΩµα 2 Q+ jµα(P̂m) } Kα µ . (2.51) Here the following notations are used: Kα µ = K−αα(Ω α µ); Q∓ jµα(P̂m) = Q∓ j,µµ−αα(P̂m). (2.52) It is equation (2.51) that will be used in the following section in order to describe the relaxation dynamics of deuterated quasi-1D ferroelectrics with hydrogen bonds. 3. Dynamics and thermodynamics of deuterated quasi-1D fer- roelectrics with hydrogen bonds in the two-site cluster approximation We shall consider a system of protons located at O-H. . .O bonds constituting zig-zag chains. Elementary cells of such compounds contain two PO4 tetrahedrons with two short H-bonds related to one of them (tetrahedron of type A). H-bonds related to the other (B-type) tetrahedron belong to two nearest structural elements surrounding it [28]. The Hamiltonian of the proton subsystem of the considered ferroelectrics has the form [28]: Ĥ = − 1 2 w ∑ ij σz Ri1 σz Rj2 [δRiRj + δRi+r2,Rj ] (3.1) − 1 8 ∑ if ∑ jf ′ Jff ′(Ri −Rj)σ z Rif σz Rjf ′ − 1 2 µ̃E ∑ if σz Rif ; σz Rif = 2Sz Rif . 857 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii Here the first term describes short-range configurational interactions of protons in chains near the A-type tetrahedron (the first Kronecker symbol) and the B-type one (the second Kronecker); the second term is an effective long-range interaction between the protons including interactions mediated by lattice vibrations, the last term includes an interaction of protons with a longitudinal external field, σz Rif = ±1 is an operator of the z-component of the quasispin, which we juxtapose to the proton located in elementary cell Ri at the fth H-bond (f = 1, 2); r2 is a vector of a relative position of H-bonds in the cell; µ denotes an effective dipole moment of the elementary cell along the polar axis. We shall study thermodynamics and dynamics of the systems with the Hamiltonian (3.1) in the two-site cluster approximation [12, 13,28]. The approximation is based on the one-site and two-site Hamiltonians [28]: Ĥ (2) Rj = − w 2 σz Rj1 σz Rj2 − x 2β (σz Rj1 + σz Rj2 ), (3.2) Ĥ (1) Rjf = − x̄ 2β σz Rjf . Here x = β ( ∆+ ν1 2 η(1) + µ̃E ) , ∆ = ∆Rjf , ν1 = ∑ jf ′ Jff ′(Ri −Rj); η(1) = 〈σz Rjf 〉, (3.3) x̄ = β ( 2∆ + ν1 2 η(1) + µ̃ ) , ∆Rif denotes an effective field generated by those neighbouring H-bonds which are not included in the cluster, ν1 2 η(1) is a molecular field generated by remote H-bonds via a long-range interaction. The fields ∆Rif are variational parameters that have to be found via self-consistency equations. Following the Hamiltonian H(2) the evolution of quasispin operators is given by σα Rj1 (t) = σα Rj1 exp ( −itα[wσRj2 + x β ] ) , (3.4) σα Rj2 (t) = σα Rj2 exp ( −itα[wσRj1 + x β ] ) or in a frequency representation σα Rjf (t) = 1 ∑ µ=−1 σα Rjf (Ω(2) µ )e−itαΩ (2) µ , (3.5) where the eigenfrequencies are Ω (2) 1 = w + x β , Ω (2) −1 = −w + x β (3.6) and σα Rjf (Ω(2) µ ) = σα Rjf R̃Rjf (Ω (2) µ ), (3.7) R̃Rj1(Ω (2) µ ) = 1 2 (1 + µσz Rj2 ), R̃Rj2(Ω (2) µ ) = 1 2 (1 + µσz Rj1 ). (3.8) 858 Nonequilibrium statistical operator approach On the other hand, the Hamiltonian H (1) suggests σα Rjf (t) = σα Rjf R̃Rjf (Ω (1))e−itαΩ(1) . (3.9) In TCA, operators P̂m coming into kinetic equation (2.51) are PRjf = σz Rjf , PRj12 = σz Rj1 σz Rj2 . (3.10) Now one can calculate Q± Rjµα (Pm) and, taking into account the boundary condi- tions, 〈σz Rjf 〉q = 〈σz Rjf 〉 = η(1); 〈σz Rj1 σz Rj2 〉q = 〈σz Rj1 σz Rj2 〉 = η(2) (3.11) obtain a system of equations for one-site and two-site DFs within the “H (2) ap- proach”: d dt η(1) = T11η (1) + T12η (2) + T13, (3.12) d dt η(2) = T21η (1) + T22η (2) + T23, and an equation for one-site DF within the “H(1) approach” d dt η(1) = −2K(1)η(1) + 2K(1) tanh βΩ(1) 2 . (3.13) Here the following notations are used: T11 = K (2) 1 z (2) 1 −K (2) −1z (2) −1 −K (2) 1 −K (2) −1 , (3.14) T12 = K (2) −1 −K (2) 1 , T13 = K (2) 1 z (2) 1 +K (2) −1z (2) −1 , T21 = 2(K (2) 1 z (2) 1 +K (2) −1z (2) −1 −K (2) 1 +K (2) −1), (3.15) T22 = −2(K (2) 1 +K (2) −1), T23 = 2(K (2) 1 z (2) 1 −K (2) −1z (2) −1), K(1) = ∞ ∫ 0 dt coshΩ(1)t e−εtRe{〈ū−(t)ū+(t)〉q + 〈ū+(t)ū−(t)〉q}, K(2) µ = ∞ ∫ 0 dt coshΩ(2) µ t e−εt Re{〈ū−(t)ū+(t)〉q + 〈ū+(t)ū−(t)〉q}, (3.16) where z (2) 1 = tanh βΩ (2) 1 2 , z (2) −1 = tanh βΩ (2) −1 2 , (3.17) Thus, the TCA algorithm leads to a system of nonlinear differential equations (3.12), (3.13) for variables η(1), η(2), ∆. This system seems to be quite intractable. Therefore, we consider only the case of small deviations from equilibrium values η̃(1), η̃(2), ∆̃, Ẽ η(1) = η̃(1) + η (1) t , η(2) = η̃(2) + η (2) t , (3.18) ∆ = ∆̃ + ∆t, E = Ẽ +Et, (3.19) 859 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii and neglect nonlinear in η (1) t , η (2) t ,∆t, Et terms in the kinetic equations. This leads to separate equations for static quantities η̃(1) = sinh x̃ a+ cosh x̃ , η̃(2) = cosh x̃− a cosh x̃+ a , x̃ = 1 2 ln 1 + η̃(1) 1− η̃(1) + βν1 4 η̃(1), (3.20) and dynamic deviations − d dt η (1) t = a1η (1) t + a2η (2) t + a3 µ̃Et 2kBT , − d dt η (2) t = b1η (1) t + b2η (2) t + b3 µ̃Et 2kBT , (3.21) where a1 = a11 − Y (1) Y (1) − 2Y (0) (a11 − a31) + βν1 4 Y (0)Y (1) Y (1) − 2Y (0) , a2 = a12 − Y (1) Y (1) − 2Y (0) a12, a3 = − Y (0)Y (1) Y (1) − 2Y (0) , (3.22) b1 = a21 − Y (2) Y (1) − 2Y (0) (a11 − a31) + βν1 4 Y (0)Y (2) Y (1) − 2Y (0) , b2 = a22 − Y (2) Y (1) − 2Y (0) a12, b3 = − Y (0)Y (2) Y (1) − 2Y (0) , a11 = K̃ (2) 1 (1− z̃ (2) 1 ) + K̃ (2) −1(1− z̃ (2) −1); a12 = K̃ (2) 1 − K̃ (2) −1 , a21 = 2[K̃ (2) 1 (1− z̃ (2) 1 )− K̃ (2) −1 (1− z̃ (2) −1)]; a22 = 2(K̃ (2) 1 + K̃ (2) −1 ), Y (1) = K̃ (2) 1 z11(η̃ (1) + 1)− K̃ (2) −1z−11(η̃ (1) − 1), (3.23) Y (2) = 2[K̃ (2) 1 z11(η̃ (1) + 1) + K̃ (2) −1z−11(η̃ (1) − 1)], Y (0) = 2K̃(1) cosh−2 ( ˜̄x 2 ) ; ˜̄x = 1 2 ln 1 + η̃(1) 1− η̃(1) . z̃ (2) 1 = tanh (βw 2 + x̃ 2 ) = (1− a2 + 2a sinh x̃)y−1, (3.24) z̃ (2) −1 = tanh ( − βw 2 + x̃ 2 ) = (−1 + a2 + 2a sinh x̃)y−1, z̃ (2) 11 = 1− tanh2 (βw 2 + x̃ 2 ) = 4a{2a+ cosh x̃− sinh x̃+ a(cosh x̃+ sinh x̃)}y−2, (3.25) z̃ (2) −11 = 1− tanh2 ( − βw 2 + x̃ 2 ) = 4a{2a+ cosh x̃+ sinh x̃+ a(cosh x̃− sinh x̃)}y−2, xt 2 = β∆t 2 + βν1 4 η̃(1) + βµ̃Et 2 , x̃ 2 = β∆̃ 2 + βν1 4 η̃(1), (3.26) a = e−βw, y = 1 + a2 + 2a cosh x̃, 860 Nonequilibrium statistical operator approach K̃ (2) ± = ∞ ∫ 0 dt e−εt cos(±w + x̃ β )t Re{〈ū−(t)ū+〉q + 〈ū+(t)ū−〉q}, K̃(1) = ∞ ∫ 0 dt e−εt cos ˜̄x β t Re{〈ū−(t)ū+〉q + 〈ū+(t)ū−〉q}. (3.27) Equations (3.20) coincide with the results of the conventional static theory. The general solution of (3.21) is of the form: η (1) t = C1 e−t/τ1 + C2 e−t/τ2 + C(iω) µ̃Et 2kBT (3.28) with the relaxation times (τ I1,2) −1 = 1 2 { (a1 + b2)± √ (a1 + b2)2 − 4(a1b2 − a2b1) } , (3.29) and C(iω) = (a2b3 − a3b2)− a3iω (a1b2 − a2b1) + (a1 + b2)iω + (iω)2 . (3.30) The quantity C(iω) determines the dynamic susceptibility of the model χI(ν) = µ̃ va lim εt→0 dη (1) t αEt = 2 ∑ l=1 χ (−)I l 1 + i2πντ Il− , (3.31) where χI 1,2 = ± f̃ 2T ( 1 τ I1 − 1 τ I2 )−1 [τ I1,2(a3b2 − a2b3)− a3], f̃ = µ̃2 kBva . (3.32) Consequently, the dynamic permittivity is of the form: ε(ν, T ) = ε∞ + 4πχI(ν), (3.33) where ε∞ is a high-frequency contribution to permittivity. Let us note that, if K(2) µ = K(1) = K = 1 2α , the obtained in this section kinetic equations (3.12) coin- cide with that used in the stochastic Glauber model in [28]. Therefore, Glauberian equations describe a physical system where Fourier-transforms of the heat bath correlation functions do not depend on frequency [7,42,43]. In this case α is a constant of time dimensionality which characterizes an interaction of the spin sub- system with the heat bath. Now let us note that the cluster approach yields exact results (obtained in [5,7–10]) for the static properties (free energy, specific heat, static permittivity ε(0, T ) (see (3.33)), order parameter (3.20)), equilibrium distribution functions [32,37] of quasi-one-dimensional systems. For the results of the cluster approach for the systems of higher dimensions the reader is referred to [13]. In the case of a zero long-range interaction, formula (3.33) yields the result of [12]. To describe the property of some compound on the basis of the obtained above results one has to consider a mechanism of relaxation of that compound and cal- culate the kinetic parameters K(2) µ and K(1). We shall not discuss this problem 861 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii Table 1. The model parameters that describe experimental data on CsH2PO4 and CsD2PO4: the nearest neighbour coupling K; the long-range interaction J ; the effective dipole moment of a proton (deuteron) on a hydrogen bond µ; the parameter of the kinetic equation — relaxation time of the noninteracting proton (deuteron) α. CsH2PO4 CsD2PO4 K/kB, K 390 610 J/kB, K 1.05 2.85 µ, e.s.u 1.87×10−18 2.17×10−18 α, s 6.2×10−15 3×10−15 here. Figures 1 and 2 show that dynamic permittivity of CsH2PO4 and CsD2PO4 crystals in the paraelectric phase can be described within the Glauberian approach under certain choice of model parameters (see table 1). To describe the dynamic permittivity in the ferroelectric phase one should suppose that bare relaxation time α depends on temperature. In the theory of the CsH2PO4-type ferroelectrics it is important to describe dramatic isotope effects in these crystals. Partial substitution of hydrogens by deuterons can be described within the disordered Ising model. In such a way a problem of the calculation of dynamic properties of the disordered Ising model arises. Papers [49–52] treat the Glauberian dynamics of the model within the two- site cluster approximation. NSO dynamics of the disordered Ising model is an open problem. In the next section we shall consider alternative methods for the solution of the kinetic equation in order to verify the cluster approach developed here. 4. Relaxation dynamics of the quasi-one-dimensional Ising model For the sake of convenience we write the Hamiltonian of quasi-1D ferroelectrics of the order-disorder type in the following form [20,32] Ĥc(s) = − ∑ ij J‖S z i+1,jS z i,j − 1 2 ∑ ij ∑ mn J⊥mnS z i+m,j+nS z ij − ∑ ij µEijS z ij . (4.1) The first term describes the short-range intrachain interactions between the hy- drogen bonds, the second one accounts for their effective long-range interactions, the last term is an interaction of hydrogen bonds with an external longitudi- nal electric field; Sz ij is a quasispin operator corresponding to the hydrogen bond (Sz ij = 1 2 σz ij ; σz ij = ±1). The spatial position of each quasispin is described by two indices (ij). The first one denotes the sequential number of the quasispin in a chain, the second is a chain number. We shall treat the long-range interactions in the mean field approximation. In 862 Nonequilibrium statistical operator approach 1 10 100 1000 10000 100000 250 260 270 280 290 300 310 1 10 100 1000 10000 100000 250 260 270 280 290 300 310 ε' T K, ε" T K, Figure 1. The dynamic dielectric permittivity ε = ε ′−iε′′ of the crystal CsD2PO4 at the frequencies ν = 5.1, 72.4, 251, 423, 730, 1044, 2000, 1150 MHz: the lines are theoretical results that correspond to the model parameters of table 1, the symbols present the experimental data [48,25,47]. this case the effective Hamiltonian takes the form: Ĥc(s) = − ∑ ij J‖S z i+1,jS z ij − ∑ ij xij β Sz ij = − ∑ j Ĥjc(s). (4.2) Here the notations xij = β ∑ mn J⊥mn〈S z i+m,j+n〉+ βµEij (4.3) are used. The time evolution of the quasispin operator is defined by (2.33): Sα ij(t) = ∑ µ Sα ij(Ωµ)e i tΩµ , µ = 0,±1, (4.4) where Ω0 = 1 β xij, Ω1 = J‖ + 1 β xij , Ω−1 = −J‖ + 1 β xij , (4.5) 863 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii 10 100 1000 10000 100000 1000000 150 155 160 165 170 10 100 1000 10000 100000 1000000 150 155 160 165 170 ε' T K, ε" T K, ν, .4 . MHz 1 72 2512 1000 ν, .4 . MHz 1 12 72 2512 1000 Figure 2. The dynamic dielectric permittivity of the crystal CsH2PO4. The lines present theoretical results (parameters in table 1), the symbols show experimental data [25]. Sα ij(Ωµ) = Sα ijRij(Ωµ), (4.6) Rij(Ω0) = 1 4 [1− 4Sz i−1,jS z i+1,j ] = 1 2 (1− σz i−1,jσ z i+1,j), (4.7) Rij(Ω±1) = 1 4 (1± σz i−1,j)(1± σz i+1,j). The choice of operators Pk, which come into the kinetic equation, is determined by the system’s properties and the form of its Hamiltonian. In our case the following choice of Pk is convenient: Plj = ∏ i∈l σz ij . (4.8) The kinetic equation (2.51) takes the form: d dt 〈 ∏ i∈l σz ij〉 = − ∑ µ Kµ    〈 ∏ i∈l σz ij ∑ i∈l Rij(Ωµ)〉q − Zµ〈 ∏ i∈l σz ij ∑ i∈l Rij(Ωµ)〉q    . (4.9) 864 Nonequilibrium statistical operator approach where Zµ = tanh βΩµ 2 Kµ = K+1 µ +K−1 µ = ∞ ∫ 0 dt e−εt cos(Ωµt)Re[〈ū −(t)ū+〉q + 〈ū+(t)ū−〉q]. (4.10) Now, on the basis of (4.9) one can obtain a system of equations for time-dependent spin correlation functions d dt η = −a1ηq + a2η1q + a3η2q + a4η3q + a0, (4.11) d dt η1 = b1ηq − b2η1q + b3η2q + b4η3q + b0, where the following notations are introduced: a1 = K0 + 1 2 (K1 +K−1)− ā1, ā1 = K1Z1 −K−1Z−1, a2 = −(K1 −K−1), a3 = −K0Z0 + 1 2 (K1Z1 +K−1Z−1), a0 = K0Z0 + 1 2 (K1Z1 +K−1Z−1), a4 = K0 − 1 2 (K1 +K−1), (4.12) b1 = −(K1 +K−1)− b̄1, b̄1 = 2(K1Z1 +K−1Z−1), b2 = (K1 +K−1), b3 = −K1Z1 −K−1Z−1, b0 = K1Z1 −K−1Z−1, b4 = −(K1 −K−1); η = 〈σz ij〉, η1 = 〈σz ijσ z i+1,j〉 = 〈σz ijσ z i−1,j〉, η2 = 〈σz ijσ z i−2,j〉 = 〈σz ijσ z i+2,j〉, (4.13) ηm = 〈σz ijσ z i+m,jσ z i+(m+1),j〉 = ηm,m+1, η1,2 = η3, (4.14) ηq = 〈σz ij〉q, ηiq = 〈...〉q, i = 1, 2, 3. The system of equations (4.11) is not closed. Unfortunately, we do not know other relations between the nonequilibrium quasispin distribution functions of the model. The equilibrium spin distribution functions for the quasi-one-dimensional Ising model (4.1) were studied in papers [2,5,7–9]. The exact results for several correla- tions functions are known [7,8]: η0 = 〈σz ij〉0 = sinh x̃/2 √ sinh2 x̃/2 + a , x̃ = β 2 ν⊥η0 + βµE, a = e−βJ‖/2, (4.15) η0m = 〈σz ijσ z i±m,j〉0 = η20 + (1− η20)f m, f = η0 − tanh x̃/2 η0 + tanh x̃/2 , (4.16) η0mm+1 = 〈σz ijσ z i±m,jσ z i±(m+1),j〉0 = η0η01 + fm(1 + f)(1− η20) tanh x̃/2. (4.17) One can easily verify that all the correlation functions in (4.16) and (4.17) can be expressed in terms of the equilibrium averages η0 and η1 (see [7,8]): η0m = η20 + (η01 − η20) m(1− η20) 1−m, (4.18) η0mm+1 = η0 [ η01 + (1− η01) (η01 − η20 1− η20 )m] . 865 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii In view of relations (4.18) we can suggest closure relations for the kinetic equations (4.11): η2 = η2 + (η1 − η2)2 1 1− η2 , η3 = η { η1 + (1− η1) η1 − η2 1− η2 } . (4.19) One can see that (4.19) is consistent with the exact relations (4.18), though there is no guarantee that such a dependence among η, η1, η2 and η3 is preserved even in weakly nonequilibrium states. Using (4.19) and the condition 〈 ∏ i∈l Sz ij〉 = 〈 ∏ i∈l Sz ij〉q we derive a closed system of kinetic equations d dt η = (a4 − a1)η + a2η1 + a3 1− η2 (η2 + η21 − 2η2η1)− a4η 1− η2 (1− η1) 2 + a0, (4.20) d dt η1 = (b1 + b4)η − b2η1 + b3 1− η2 (η2 + η21 − 2η2η1)− b4η 1− η2 (1− η1) 2 + b0. The system of equations (4.20) is nonlinear and quite complex. Therefore, we re- strict further consideration to small deviations from the equilibrium and represent the distribution functions as sums of their equilibrium values and (small) time- dependent deviations: η = η0 + ηt, η1 = η01 + η1t, xij = x̃+ xijt. (4.21) In this case the following relations hold: Z0 = Z00 + [βν⊥ 4 ηt + βµEt 2 ] Z01, Z1 = Z10 + [βν⊥ 4 ηt + βµEt 2 ] Z11, (4.22) Z−1 = Z−10 + [βν⊥ 4 ηt + βµEt 2 ] Z−11, where Z00 = tanh x̃ 2 , Z10 = tanh (βJ⊥ 2 + x̃ 2 ) = 1− a4 + 2a2 sinh x̃ 1 + a4 + 2a2 cosh x̃ , (4.23) Z−10 = tanh ( − βJ⊥ 2 + x̃ 2 ) = −(1− a4) + 2a2 sinh x̃ 1 + a4 + 2a2 cosh x̃ , Z11 = 1− tanh2 (βJ⊥ 2 + x̃ 2 ) = 4a2[2a2 + (1 + a4) cosh x̃− (1− a4) sinh x̃] (1 + a4 + 2a2 cosh x̃)2 , Z−11 = 1− tanh2 ( − βJ⊥ 2 + x̃ 2 ) = 4a2[2a2 + (1 + a4) cosh x̃+ (1− a4) sinh x̃] (1 + a4 + 2a2 cosh x̃)2 , Z01 = 1− tanh2 x̃ 2 . (4.24) 866 Nonequilibrium statistical operator approach Substitution of (4.21) and (4.22) into (4.20) and linearization of the obtained system of equations with respect to ηi, ηit and Et allow one to obtain two systems of equations. The first one contains equilibrium correlation functions, and its solution yield relations (4.15) – (4.17). The second one contains a dynamic part of the correlation functions and has the form: d dt ηt + A1ηt + A2η1t = A3 µEt 2kBT , (4.25) d dt η1t +B1ηt +B2η1t = B3 µEt 2kBT , where A1 = a10 − a30 2η0(1− η01) 2 (1− η20)2 − a4 [ 1− (1 + η20)(1− η201) 2 (1− η20) 2 ] − βν⊥ 4 [ ā11η0 + a31 η20 + η201 − 2η20η01 1− η20 + a01 ] , (4.26) A2 = − [ a2 + a30 2(η01 − η0) 1− η20 + a4 2η0(1− η01) 1− η20 ] , A3 = ā11η0 + a31 η20 + η201 − 2η20η01 1− η20 + a01, B1 = −b10 − b30 2η0(1− η01) 2 (1− η20)2 − b4 [ 1− (1 + η20)(1− η201) 2 (1− η20) 2 ] − βν⊥ 4 [ b̄11η0 + b31 η20 + η201 − 2η20η01 1− η20 + b01 ] , (4.27) B2 = b2 − b30 2(η01 − η0) 1− η20 − b4 2η0(1− η01) 1− η20 , B3 = b̄11η0 + b31 η20 + η201 − 2η20η01 1− η20 + b01. Equations (4.25) yield the following expression for dynamic susceptibility of the system: χII(ω) = µ2 2vakBT (B2A3 − A2B3) + iωA3 (A1B2 − A2B1) + iω(A1 + B2) + (iω)2 (4.28) = χII 1 1 + iωτ II1 + χII 2 1 + iωτ II2 , where χII 1 = µ2 2vakB 1 T ( 1/τ II1 − 1/τ II2 ) [ τ II1 (A3B2 −A2B3)− A3 ] , (4.29) χII 2 = µ2 2vakB 1 T ( 1/τ II1 − 1/τ II2 ) [ τ II2 (A3B2 −A2B3)− A3 ] , 867 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii τ II1,2 = 2 { (A1 + B2)∓ √ (A1 + B2)2 − 4(A1B2 − A2B1) } . (4.30) In the paraelectric phase our results have a particularly simple form: χ(ω) = 2βµ2 va 1 4a− βν⊥ 1 1 + iωτ , (4.31) τ = (1 + a)2(1 + a2) [K0(1 + a2) + 2Ka](4a− βν⊥) . Like in the preceding section, in the case K0 = K1 = K−1, expressions (4.26) and (4.27) become simplified and we obtain the results of the Glauber approach derived in [32]. 5. Conclusions In this paper we study relaxational phenomena in quasi-one-dimensional fer- roelectrics of the order-disorder type. We use Zubarev’s NSO method in order to describe the kinetics of these compounds. Two closure relations are used to deal with the obtained kinetic equations. The first one uses the two-site cluster approxi- mation in order to calculate the dynamic and thermodynamic characteristics of the system. The second one uses suggested in [32] decoupling for a chain of equations for quasispin distribution functions. The both approaches give numerically close results. For the description of experimental data within the NSO method one must calculate kinetic parameters K(2) µ and K(1). In the course of such a calculation one must consider mechanisms of relaxation in given crystals. Such problems will be considered in our future works. Here we only note that if the kinetic parameters are independent of frequency, our results coincide with those obtained in [28,32] within the Glauber model and provide a good description of the experimental data on quasi-one-dimensional ferroelectric crystals CsH2PO4 and CsD2PO4. Appendix The obtained within the NSO method system of equations (4.11) for distribu- tion functions is not closed. Therefore, the problem of the correct decoupling of this system arises. The ones used in this paper are not unique. For example, in paper [20] in order to decouple the equation −α d dt 〈σz ij〉 = −〈σz ij〉+ P (1) 0 (〈σz i−1,j + σz i+1,j〉) + (M (1) 1 〈σz i−1,jσ z i+1,j〉+ L (1) 1 1 2 xij , (A.1) P (1) 0 = 1 2 1− a2 1 + a2 , M (1) 1 = 1 2 (1− a2 1 + a2 )2 , L (1) 1 = 1 2 1 + 6a2 + a4 (1 + a2)2 868 Nonequilibrium statistical operator approach from that for the pair distribution function the following approximation is used: 〈σz i−1,jσ z i+1,j〉 = Γ+ Γ′ xtij 2 , (A.2) where Γ = 2 tanh βJ‖/2 { 1− 2a2 − (1− tanh2 βJ‖/2)(βν⊥η0/4) 2 [a2 + (βν⊥η0/4)2]3/2 } − 1, Γ′ = 4 tanh βJ‖/2 β2βν⊥η0/4a 2 [a2 + (βν⊥η0/4)2]3/2 . (A.3) In this way one finds χIII(0, T ) = f 2t M(1) 1− 2P (1) 0 − βν⊥/4m(1) , (A.4) T>Tc= f 2t a+ a2 − a3 a2 − βν⊥/4(a+ a2 − a3) , τ III = α { 1− 2P (1) 0 − βν⊥ 4 M(1) }−1 , (A.5) T>Tc= α 1 + a2 2[a2 − βν⊥/4(a+ a2 − a3)] , where M(1) = L(1) + ΓM (1) + 1 4 βν⊥η0Γ ′M (1). (A.6) Numerical calculations show that these results are close to (4.29), (4.30) at high temperatures, but in this approach the static susceptibility and relaxation time become negative at T → Tc. In [21] the equation for the dynamic part of the distribution function is written in the form: α d dt 〈σz ij〉t = −〈σz ij〉t + P (2) 0 (〈σz i−1,j〉t + 〈σz i+1,j〉t) (A.7) + 1− 2P (2) 0 F (β 4 ∑ mn J⊥mn〈σ z i+m,j+n〉t + βµ 2 Etij ) , where P (2) 0 = 1 4 (Z10 − Z−10), F = (sinh2 βν⊥/4η0 + a2)3/2 a2 cosh βν⊥/4η0 . (A.8) The solution of this equation coincides with the results (3.32), (3.29) of the two-site cluster approximation, except that at T < Tc τ IV = α { 1− 2P (2) 0 − βν⊥ 4 (1− 2P (2) 0 ) 1 F } . (A.9) 869 R.R.Levitskii, I.R.Zachek, R.O.Sokolovskii References 1. Isyumov Yu.A., Kassan-Ogly F.A., Skriabin Yu.N. Field Methods in the Theory of Ferromagnetism. Nauka, 1974 (in Russian). 2. Isyumov Yu.A., Skriabin Yu.N. Statistical Mechanics of Magnetically Ordered Sys- tems. Nauka, 1987. 3. 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Підхід нерівноважного статистичного оператора Р.Р.Левицький, І.Р.Зачек, Р.О.Соколовський Інститут фізики конденсованих систем НАН України, 290011, Львів–11, вул. Свєнціцького, 1 Отримано 27 грудня 1997 р. В рамках методу нерівноважного статистичного оператора Д.М.Зу- барєва виведені кінетичні рівняння для діелектричних характеристик сегнетоелектриків типу CsH2PO4. Ці рівняння розв’язані в набли- женні двочастинкового кластера та шляхом розщеплення ланцюжка рівнянь для функцій розподілу. Обидва методи дають близькі резуль- тати. Показано, що розраховані динамічні характеристики є близьки- ми до тих, що отримуються в рамках кінетичної моделі Глаубера та дозволяють описати динамічну діелектричну проникність кристалів CsH2PO4 і CsD2PO4. Ключові слова: нерівноважний статистичний оператор, кінетичне рівняння, сегнетоелектрики типу лад-безлад, динамічна діелектрична проникність PACS: 64.60.Cn 872

To the theory of relaxation phenomena in quasi-one-dimensional ferroelectrics with hydrogen bonds. Nonequilibrium statistical operator approach

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