Statistical Theory For Disordered Quasi-One-Dimensional Ferroelectrics With Hydrogen Bonds

Thermodynamics and dynamics of the quasi-one-dimensional disordered ferroelectrics with hydrogen bonds are studied within the framework of disordered Ising model. We obtain thermodynamic potentials , polarization, static and dynamic q-dependent susceptibili-ties within the two-site cluster approximation for short-range interactions. Theoretical results are compared to dielectric meusurements in pure and partially deuterated ferroelectrics Cs(H 1?x D x) 2 PO 4 and PbH(D)PO 4. The model predicts the sort ordering in the former crystal: protons and deuterons tend to occupy the nearest neighbour H-bonds.


Introduction
Ferroelectrics of KDP family compels much attention due to their simple structure and interesting properties.Many investigations have been devoted to low-symmetric ferroelectrics, i.e.CsH 2 PO 4 (CDP), PbHPO 4 (LHP) and their deuterated isomorphs.
The structure of the cesium dihydrogen phosphate crystal is shown in gure 1.In the high-temperature paraelectric phase CDP crystal and its deuterated isomorph CsD 2 PO 4 (DCDP) are monoclinic and belong to space group P2 1 =m 1{3].The elementary cell has parameters a=7.906A, b=6.372A, c=4.In the paraelectric phase the protons of the shorter H-bonds occupy one of two equilibrium positions with equal probability.The cooling to T c = 156K brings CDP crystal in the ferroelectric phase 5,6] with the P2 1 symmetry.In this phase the H1 protons localize in the acentric ordered positions in H-bonds and the PO 4 tetrahedra slightly turn around the P | O2 bonds 7].Such ordering of the shorter H-bonds also takes place in the antiferroelectric phase of the CDP crystal, that replaces the ferroelectric one c R.R.Levitskii, R.O.Sokolovskii, S.I.Sorokov, 1997 ISSN 0452{9910.Condensed Matter Physics 1997 No 10 (67{92) 67 under hydrostatic pressure (P 330MPa).The protons H2 of the longer H-bonds are ordered at any temperature and hydrostatic pressure.An important role of protons in the phase transition is justi ed by the isotopic e ect: Curie temperature in the deuterated CDP raises to 268K 3,4].
It is possible to achieve the perfect (b; c)-plane chips of this crystal that shows a weakness of the interplane forces.Since during the phase transition into the ferroelectric phase the ordering of protons takes place only on the shorter H-bonds with the appearence of the polarization along chains, the CDP-type crystals are described as the quasi-one-dimensional in contrast to the three-dimensional tetragonal crystals of the KH 2 PO 4 group.
PbHPO 4 (LHP) and PbDPO 4 (DLHP) are also monoclinic crystals, which posess the ferroelectic phase transition of the second kind at temperatures 310K and 452K, respectively 8].At room temperature they belong to the space group Pc, whereas in the paraelectric phase their symmetry is P2=c.Both LHP and DLHP contain two formula units in the elementary cell.These crystals have only short hydrogen bonds O ?H O (R O O = 2:52 A), which connect PO 4 groups in the chains along the caxis.The hydrogen bonds direction in this crystals is almost parallel to the direction of the spontaneous polarization 8].The papers 9{12] show that protons (deuterons) in the LHP (DLHP) hydrogen bonds move in the double-well potential and their behaviour is similar to that of the protons in the shorter H-bonds of CDP.The large change of T c with deuteration shows that the phase transitions in these crystals are also closely related to the proton or deuteron ordering.Thus at this moment the notion has been established about the quasione-dimensional character of proton (deuteron) ordering in CDP and LHP crystals with strong intrachain and weak interchain interactions.It brings the idea to desribe the phase transition in these crystals on the basis of the quasi-one-dimensional Ising model.This idea was realized in the papers 5,9,13{23] (CDP) and 9,24,25] (LHP) where the phase transition, static and dynamic properties of the mentioned ferroelectrics was discussed in the framework of the microscopic theories 13,23,24,26,27]) based on the quasione-dimensional Ising model.It has been shown that this model provide a good description of polarization, dielectric permittivity, speci c heat and dynamic susceptibility of CDP and LHP crystals as well as of their deuterated analogs.Some authors 15,16,28,29] came to the conclusion that Ising model can not describe the isotopic e ect in CDP.They experienced the following problems.
According to Ising model, the dielectric permittivity in paraelectric phase is " = 1 + A exp(?2K= ) ?J ; (1.1) where = k B T, k B is a Boltzman constant, K and J are the nearest neighbour coupling and q = 0 Fourier-transform of long-range interaction, respectively.Authors of the papers 9,15] found values of the parameters A, K and J that gave the best t of the permittivity (1.1) to their experimental data.They got that the Curie constant A = 4 n 2 (n is a number of quasispins in the unit volume, being a dipole moment of the quasispin along the ferroelectric axis) is less in the deuterated crystal than in the ordinary CDP (see table 1.).Then Ising model suggests that saturated value of the spontaneous polarization P s = n lowers with deuteration.The experiment instead shows that P s increases.Weakening of the long-range interaction J also contradicts to the P s rise, because the main contribution to J is given by the dipole-dipole interaction, which is proportional to 2 .
These inconsistencies are weakened within the transverse eld Ising The authors of the papers 30,31] have also shown that if the tunnelling frequency is small (2 K) then the dynamic susceptibility of the transverse eld Ising model is of the relaxational type and model describes the dynamic dielectric permittivity of CDP and DCDP.But recently Sorokov and Levitskii 32] have displayed the incorrectness of the approximations used in 30,31] for the temperatures > .On the basis of an improved approximation they show that this model does not describe the high-frequency behaviour of "(!) in CDP and can not describe Debye-type relaxation at all.In order to remove the Curie constant inconsistency Kojyo and Onodera 28,29] suggested quasispin model which attaches dipole moment to the PO 4 groups.In this model H-bonds, since they are oriented almost perpendicularly to the ferroelectric axis, induce polarization indirectly, in uencing deformation of the PO 4 tetrahedra.Parameters of this model suggested by the authors are free of the mentioned above inconsistencies, but the advantages of the model remain doubtful, because the Curie constant found from the permittivity data is one and half times larger than that obtained from P s .The authors of the papers 22,23] insist on the applicability of the Ising model to the description of thermodynamic and dynamic properties of CDP and DCDP.They have found the model parameters which are free of the mentioned above inconsistencies and provide a satisfactory description for a large set of experimental data within the wide range of temperatures and frequencies.In the same papers a good description of the partially deuterated crystals Cs(H 1?x D x ) 2 PO 4 is achieved on the basis of an assumption that the parameters of Ising model are linear functions of the deuterium concentration x (mean crystal approximation).This latter result is quite strange because in the systems with strong short-range correlations a disorder brought with deuteration should produce more pronounced e ects.
In this paper we aim to describe both the thermodynamics and the dynamic properties of disordered Ising model on the basis of the same consistent approach.This approach alows us to study the e ect of disorder in quasi-one-dimensional ferroelectrics with hydrogen bonds.

The Hamiltonian. Formulation of the problem
Following the ideas of the papers 22,23,25] we try to describe the properties of the quasi-one-dimensional ferroelectrics on the basis of Ising model with strong nearest neighbour interaction in chains K ij and weak long-range interaction J ij H = ?X (ij) K ij S i S j ? 1 2 X ij J ij S i S j ?X i i S i : (2.1) Here S i = 1 is an operator of the z-component of quasispin, that corresponds to the two possible equilibrium positions of proton (deuteron) in H-bond; K ij is a coupling between the nearest neighbours in chains; J ij denotes an e ective long-range interaction between quasispins that contains indirect interaction of protons (deuterons) via lattice vibrations; i = Ed i , where E is an external electric eld, and d i is an e ective dipole moment that accounts also a contribution of the other elements of the lattice.Partial deuteration brings chaos into the system and requires to consider the disordered Ising model.We assume the interaction parameters to be locally dependent on the sort con guration Operators X i describe the sort con guration: X i = 1 if site i contains the quasispin of sort , otherwise X i = 0.In the partially deuterated ferroelectric the hydrogen bond may contain proton or deuteron, therefore the sort indices (greek letters) take two values: = 1(D) or 2(H).A site index is two-component: i = l a , in the nominator we write the index of elementary cell (l = 1 : : :N), denominator contains sublattice index (the elementary cells of CDP and LHP contain two quasispins (short H-bonds): a = A, B).
The model can be considered in the two limiting cases.In the case of annealed system the equilibrium sort con guration is realized and the thermodynamic averaging implies going over all spin and sort con gurations: h i eq = Z ?1 Sp fS;Xg exp(?H= )( ); Z = Sp fS;Xg exp(?H= ): (2.3) In the case of quenched system the sort con guration is xed, and the thermodynamic averaging h i H contains a trace over spin con gurations h i H = Sp fSg x (fSg)( )] (2.4) x (fSg) = Z ?1 x exp(?H= ) (2.5) Z x = Sp fSg exp(?H= ) (2.6) In order to calculate the observable quantities in quenched system one has to perform also an averaging over the sort con gurations h i x = Sp fXg (fXg)( ); (2.7) where distribution x (fSg) is determined by the conditions of the system freezing.For example, if the system was prepared at very high temperatures and then quickly cooled then there is no correlations in sort distribution hX i X j i x = hX i i x hX j i x = c c ; (2.8) where c is a fraction of the sort quasispins in the mixture.
In the case of the equilibrium disorder the properties of the system are a ected by the \nonexchange" interaction between quasispins ?X (ij) (2.9) that should be included in the Hamiltonian (V ij denotes the nearest neighbour interaction, I ij is the long-range interaction).In the case of deuterated ferroelectrics such terms in the Hamiltonian may describe the elastic energy of the lattice deformation that appears after the replacement of proton to deuteron.
The annealed model within the grand canonical ensemble is described by the density matrix (fS; Xg) = Z ?1 exp(?H= ); H = H ? X N ; (2.10) Chemical potentials have to be found from the conditions hX i i H = c ; (2.14) where h( )i H = Sp fS;Xg (fS; Xg)( )]: (2.15) The sort con guration relaxation time in the considered compounds is unknown at this time, therefore we do not know what type of disorder is observed in the dielectric 15,16] and calorimetric 20,21] experiments.This situation requires both types of disorder to be considered theoretically and compared with experimental data.
Ising models with the annealed or quenched disorder compel attention of theoreticians for many years, because they are simple and nontrivial microscopic models for the processes of chemical, ferroelectric or magnetic ordering in alloys of magnets, magnets with nonmagnetic impurities, solid solutions of ferroelectrics, etc.For the one-dimensional model with the nearest-neighbour interactions some exact results are obtained: for both annealed and quenched systems the thermodynamic potentials and correlation functions have been calculated 33] (for quenched system only the zero eld results have been found), for the ideal (one-sort) system without eld Glauber's kinetic equation has been solved (dynamic pair correlation function has been calculated 34]).Percolation phenomena and universality problems inspire new Monte-Carlo experiments with the diluted (noninteracting impurities) quenched system.When the couplings K i ;j can take values of di erent signs, one faces the problems of frustration and spin glass, which we do not consider in this paper, because herein we assume K i ;j > 0.
In the annealed model the coupling between sort and spin ordering is of great interest 35,36].The model provides also a test for the approximate approaches of statistical physics.For example, the mean eld approximation (MFA) does not distinguish between the properties of the annealed and quenched ferroelectrics, does not retrieve percolation phenomena, does not describe the in uence of sort ordering on the quasispin alignment 35,37].The main reason for this result is that MFA neglect uctuations of the concentration and spin moment.Therefore, the most straightforward way to improve the theory that is based on an e ective eld can be reached by the account of local spin and sort moments uctuations within the cluster approach.Di erent cluster approximations suggested in papers 36,38,39] give qualitatively correct results, which can be systematically improved in the higher order approximations.The papers 38] (AS) and 36] (QS) have presented phase diagrams of the model in some cluster approximations, which are quite close to Monte-Carlo results.An E ective Field Approximation of Kaneyoshi et.al. provides good results for average value of spin, static susceptibility and critical temperature of the diluted quenched system 40].In this paper we shall use the previously developed cluster approach 41{43], that gives us the analytical expressions for the thermodynamic potentials, q-dependent static and dynamic correlation functions of the model.We shall brie y mention the main points of the method and generalize it on the two-sublattice model appropriate for the CDP-type ferroelectrics.

Thermodynamics and correlation functions of the model
As well as by the density matrices (2.5) and (2.10) the complete statistical description of a system is provided by the generating functions QS : F(f g) = hln Z x i x ; (3.1) AS : F(f ; g) = ln Z: Nonuniform elds i , i allow us to calculate the correlation functions of the model with the use of the di erentiation of the generating functions @ i1 1 @ @ in n @ @ j1 1 @ @ jl l F(f ; g); were the superscript c denotes the cumulant averaging, and S i = S i X i .
Let us introduce the short notations for these correlation functions QS : m (n) i1 1 in n = hhS i1 1 S in n i c H i x ; (3.4) AS : M (n) i1 1 in n = hS i1 1 S in n i c H ; (3.5) sort index in the case of AS takes three values: In the nal expressions one should account the translational invariance of the system in uniform eld (UF) and assume the elds and chemical potentials to be independent of elementary cell index: l a UF = a ; l a UF = a : Having found the generating functions, one obtains the free energy (QS) F = ?F(f g) and the grand thermodynamic potential (AS) G = ?F(f ; g).
For long-range interactions the mean led approximation yield satisfactory results.Therefore we adopt MFA for interactions V i ;j , I i ;j and the problem reduces to the calculation of the thermodynamic potentials of the model with short-range interactions in some self-consistent eld 41] (the reference system).The static and dynamic correlation functions of the model can be analytically expressed via the corresponding values for the reference system.For the system with short-range interactions the approach of cluster expansions is natural 43].Having performed the cluster expansion 44] of the free energy (two-site cluster) and restricting that expansion to the rst therm 41,43] one obtains the result of the rst order cluster approximation.Thus obtained generating functions contain only one-and two-site contributions QS : F(f g) = ?z 0P i hF i i x + P (ij) hF ij i x ; (3.7) AS : F(f ; g) = ?z 0P i F i + P (ij) F ij ; (3.8) where z 0 = z ? 1, z is the rst coordination number, P (ij) denotes the sum over pairs of the nearest neighbour sites.The one-site contributions F i = ln Z i ; Z i = Sp Si e ?Hi= ; H i = ?i S i ; and the two-site ones F ij = ln Z ij ; Z ij = Sp SiSj e ?Hij= ; H ij = ?j i S i ?i j S i ?K ij S i S j ; (3.11) Statistical theory for disordered quasi-1D ferroelectrics 75 j i = i + X r6 =j r ' i ; F ij = ln Z ij ; Z ij = Sp SiXiSjXj e ?Hij= ; H ij = ?j i S i ?i j S j ?K ij S i S j ?j i ?i j ?V ij ; (3.12) contain e ective elds j ' i ; j i , that describe the in uence of the site j on its nearest neighbour i.We assume these elds to be locally sort dependent j ' i = X j ' i X i ; j i = X j i X i : (3.13) In the uniform eld limit they lose their dependence on the elementary cell index The quantities ' and are variational parameters that must provide extremum of the generating functions QS : @F(f g) @ r ' i = 0; AS : @F(f ; g) @ r ' i = 0; @F(f ; g) @ r i = 0: ( These conditions lead to the following expressions for the rst moments QS : m (1) D hS i i Hij E x ; (3.17) i = Z ?1 i e ?Hi= ; ij = Z ?1 ij e ?Hij= ; AS : M (1)  i = hS i i Hi = hS i i Hij ; (3.18) i = Z ?1 i e ?Hi= ; ij = Z ?1 ij e ?Hij= The equations (3.17) and (3.18) for the elds ', give the selfconsistency conditions, that require equality of the rst moments found with the onesite i and cluster ij density matrices.In the case of the simple lattice and uniform eld we obtain the generally known results of the Bethe approximation and the rst-order cluster variation method 37,39].Nonuniform elds allows us to obtain the correlation functions by the di erentiation of the generating functions, and it is the advantage of our approach.For example, after di erentiation of the equation (3.17) with respect to j we obtain the Ornstein-Zernike type equation for the pair correlation function.This equation can be written in the matrix form QS : Ûii m(2) ij = ij + X r Ĝir m(2) rj ; (3.19) where Ûii = ?z 0( F(2) i ) ?1 ?X r h F( 11) ir ( F(20) ri ) ?1 F(11) ri ?F(20) ir i ?1 ; G ir = h F(20) ri ( F(11) ir ) ?1 F(20) ir ?F (11) ri i ?1 ; (3.20) r is the nearest neighbour of the site i (and therefore belong to the other sublattice), matrices F are the one-site and intracluster correlation functions Hi i x : In the uniform eld the pair correlation function become invariant with respect to the translation on a lattice vector m (2) i a j b = m (2)  a b (R i ; R j ) UF = m (2)  a b (R i ?R j ); (3.22) and the equation (3.19) can be solved in the Fourier representation m (2)  a b (R i ?R j ) = V e (2 ) 3 Z d 3 q e ?iq(Ri?Rj) m (2)   a b (q); ( where V e is a volume of the elementary cell, R i is a cordinate of the ith elementary cell. The solution has the form m(2) AA (q) m( 2) AB (q) m( 2) BA (q) m(2) BB (q) != ÛAA Ŷq Ŷ q ÛBB !?1 (3.24)where ÛAA = ?(F(2) A ) ?1 ?ÊAB ?ÊBA ; ÛBB = ?(F(2) B ) ?1 ?ÊAB ?ÊBA Ŷq = ?ĜAB e iqRf ?ĜBA e ?iqRfThis \partial" correlation functions contain trace over small number of spin and sort con gurations and can be easily calculated.We can go on with the di erentiation procedure and obtain the three-site and higher correlation functions 41,43].The same can be done for the annealed system.We come to the similar form M(2) AA (q) M( 2) AB (q) M( 2) BA (q) M( 2) BB (q) != ÛAA Ŷq Ŷ q ÛBB !?1 (3.28)where M(2) ab (q) are 3 3 matrices whose elements are the Fourier-transforms of the pair correlation functions Z d 3 q e ?iq(Ri?Rj) M( 2) ab (q) : ( Again our approach has expressed the model correlation functions via the \partial" ones F(20

Dynamic properties
The cluster approach of the previous sections can be applied to the description of the system's dynamics.Within the master equation approach the density matrix (fSg; t) is governed by the equation that describes some stationary Markov process d dt (fSg; t) = X fSg 0 Ŵ (fSg; t) (4.1) Kinetics of the system is determined by the transition matrix Ŵ.Within the Glauber's 45] approach the spin system interacts with a heat bath which induces spin ips (S i !?S i ).The state of the heat bath is assumed to be constant, therefore a probability of the spin ip per unit time W i (fSg) depends only on the spin con guration and is independent of time.Thus the master equation takes the form d dt x (fSg; t) = X i W i (: : : ?S i : : :) x (: : : ?S i : : :; t) ?X i W i (: : :S i : : :) x (: : :S i : : :; t) (4.2) A detail balance condition for the equilibrium density matrix of the system leads to the following relations for the transition matrix W i (: : :S i : : :) W i (: : : ?S i : : :) = x (: : : ?S i : : :) x (: : :S i : : :) = exp(?"i S i = ) exp(" i S i = ) = 1 ?S i tanh " i = 1 + S i tanh " i = ; where " i is a local eld acting on the ith spin " i = i + X j K ij S j : (4.4) Suzuki and Kubo 34] suggested the following form of the transition probability, which is compatible with (4.3) W i (: : :S i : : : where i is a constant which characterize an interaction of the ith spin with the heat bath and de ne the time scale of the relaxation processes.Expressions (4.2) and (4.5) lead to the following equation for an average value of spin D i;t hS i i H;t = htanh " i = i H;t ; (4.6) where h( )i H;t = Sp fSg (fSg; t)( ); D i;t = 1 + i d dt : (4.7)This equation can also be derived more rigirously from the Liouville equation with spin-phonon Hamiltonian.Nonequilibrium statistical operator method with an assumption that the phonon-phonon correlation functions are independent of frequency near !1= i 47] gives the same results as the Glauber's approach yields and provides microscopic expressions for the phenomenological constant i .The same can be done within the perturbation theory 46] on the basis of the quantum equation of motion for the spin moment.
In the case of the equilibrium disorder the sort con guration changes as well as the spin con guration during the observation time, thus in this case one should consider the sort kinetics also.This kinetics can also be formulated in terms of spin operators: i = X i1 ?X i2 ( i = 1), but the equation of Glauber type is invalid in this case, because the spin ip ( i !?i ), which is the basic process of equation (4.2), means the transformation of H ion to D ion (and vice versa) at ith hydrogen bond.It is obvious that the sort con guration changes due to another processes.For the annealed system the Kawasaki equation 48,49] is relevant, which considers elementary processes f i j g !f j i g (an exchange between the nearest neighbour sites).
We shall consider the dynamics of model near the frequencies of the ferroelectric dispersion.It seems obvious that the sort con guration can not change during the time 10 ?6 s.This implies that the sort distribution is frozen with respect to quick changes of the external eld (but can feel the slow changes of temperature and the quasistatic changes of the eld).
Therefore in the frequency region > 1MHz the spin kinetics of both annealed and quenched models is governed by the equations (4.2), (4.6).Their dynamic correlation functions, however, di er due to the di erences in the sort distribution.
When one expands tanh " i = in a power serie, he obtains moments hS i1 S in i H;t in the right-hand side of (4.6), where sites i 1 i n are the nearest neighbours of the site i.Expressions (4.2) { (4.6) yield an equation for these moments that in turn involves the second coordination sphere, therefore equation (4.6) is accurately solvable only for the one-dimensional models in zero eld (ideal chain 45,34,50], chain with impurity on one site 51]).For models with eld or chaos or in the lattice of higher dimensions the closures are used, which express the higher moments via lower.For example, the random phase approximation hS i1 S in i H;t hS i1 i H;t hS in i H;t provides a closure to equation (4.6) and lead to the results of MFA in statics.We can not restrict our consideration to MFA, because, as we mention before, it yield some qualitatively incorrect results for the disordered system with short-range interactions.The better results can be achieved in the two-site cluster approximation.This approximation with the long-range interactions accounted in MFA provides good agreement between the theory and the experimental data on the crystals CsD 2 PO 4 , CsH 2 PO 4 23,22] and PbDPO 4 , PbHPO 4 25] for spontaneous polarization, speci c heat, static and dynamic permittivity.
In our previous papers 41, 42,52] the two-site cluster approximation was developed for disordered Ising model and improved in order to obtain the q-dependent dynamic pair correlation function of the model.The exact results for one-dimensional lattice (dynamic pair correlator of ideal chain without eld 34], static correlation functions (3.24)) are partial cases of the obtained formula.In this paper we apply the same method for the twosublattice model, which are appropriate for the CDP-type ferroelectrics.
We construct the self-consistency equation,similar to the equation(3.17).
Within a one-site approximation we replace in " i the contributions of all spins by the e ective elds whereas in a two-site approximation the contribution of the one spin is considered explicitly In the case of the ideal system no more assumptions are needed in order to obtain the closed set of kinetic equations: D it hS i i H;t = htanh" 1] i = i H;t = tanh " 1] i = ; (4.10) D it hS i i H;t = htanh" 2] i = i H;t = P it hS i i H;t + L it ; where For the disordered system one should point out the dependence of the e ective elds r ' it on the sort con guration.We shall consider the elds r ' it to be di erent in the one-site and two-site approximations until the sort averaging is carried out.For the sort averaging procedure both the \one-site" and \two-site" elds will be assumed to be locally sort-dependent r ' it = X r ' i ;t X i : (4.12) We also consider the relaxation times i to be di erent for deuteron and proton i = X i 0 X i : The technique, similar to those used in the static case, yields us the dynamic pair correlation function of the model, as a response of the spin moment to weak perturbation of external eld with the frequency ! and wavevector q: m (2)  a b (q; !) = m (1)  a (q; !) b (q; !) : We obtain m AA (q; !) m(2) AB (q; !) m BA (q; !) m(2) BB (q; !) != ÛAA (q; !) Ŷq;! Ŷ q;! ÛBB (q; !) !?1 ; (4.15) where the matrices Û, Ŷ have the form (3.20) with the !-dependent cluster correlators in place of the matrices F .This correlators depend on the lowest moments of sort distribution hX i i x = c and hX i X j i x = w i ;j .The former is the fraction of the sort quasispins, the latter is a probability to nd the quasispins of sorts and at sites i, j respectively.c is de ned by the chemical content of the sample; in the quenched system w i ;j is a constant de ned by the conditions of the sample freezing, in the case of the annealed system w i ;j can be found from the static theory w i ;j = hX i X j i Hij

Comparison with the experiment
The theory developed above provides the analytic expressions for such observable quantities as the speci c heat, polarization (here h i denotes the averaging that corresponds to the type of disorder), static susceptibility (2)   a ; b (q); (5.3) (2)   a ; b (q); (5.))m (2)  a ; b (q; !); ( where r a is a position of the quasispin that belong to sublattice a in the elementary cell, d is a dipole moment of the sort quasispins along the ferroelectric axis. The dielectric permittivity of a ferroelectric crystal is " = " 1 +4 , " 1 is greater than 1 due to the contribution of the electron and other subsystems of the crystal.It does not depend on temperature or frequency in the region of the ferroelectric dispersion and can be easy estimated from the experimental data " 1 = "(1).
It should be noted that in the annealed model formulae (5.2) and (5.5) yield di erent results for the static susceptibility in the ferroelectric phase: (q) 6 = (q; 0).The reason for this is that in the dynamic theory we suppose sort con guration to be frozen with respect to the changes of the eld.Doing  2).The line 1 corresponds to the quenched system with the complete sort chaos.The lines for the annealed system and the quenched system with ordering almost coincide and are denoted by number 2. so, we exclude from (q; !) the corresponding contribution.This is correct at the frequencies of ferroelectric dispersion, but wrong at ! !0.
The one-dimensional model without a long-range interaction does not posess a phase transition.With the interchain interaction being accounted this model predicts the ferroelectric phase transition of the second kind.In the case of annealed system the sort ordering or segregation transition also appears.Description of the ordered or segregated phase does not introduce signi cant di culties into the theory, but we restrict our consideration to the uniform mixture.
Having accounted the symmetry of the CDP and LHP crystals, one obtains that the nearest neighbour interactions K i ;j , V i ;j do not depend on position in chain, q = 0 Fourier-transforms of the long-range interactions do not depend on the sublattice indices ( P j J i ;j = J , P j I i ;j = I ) therefore without the transverse electric eld sublattices A and B are equivalent and constitute one simple lattice.In this case the correlation functions do not depend on the sublattice indices, nal expressions become much simpler, and we present them below.The free energy of the system takes the form h cosh( 0 + 0 ) + a cosh( 0 ?0 ) i ; Z = 2 1 ?m (1) =c ?2 ; a = exp(?2K= ); 0 = 0 @ = + X J m (1) = + z 0 ln c + m (1)  c ? m (1)   1 A =z; ( AS : F=N = z 0 ln Z 2 c 2 ?z ; R = + q 2 + x(1 ? 2 ); = c 1 ?c 2 : Spin moments m (1) = hS i i should be found by minimization of the free energy.The cluster dynamic correlation functions read   (5.9) Parameters of the theory are: the short-range interaction in chains K DD , K DH = K HD , K HH ; the q = 0 Fourier-transform of the long-range interaction J DD , J DH = J HD , J HH ; the e ective dipole moments of deuteron and proton in hydrogen bond d D , d H ; the relaxation times of noninteracting proton and deuteron 0 D , 0 H .In the case of quenched disorder the probability w DH to nd proton and deuteron at the nearest neighbour H-bonds is another parameter, for the annealed system the parameter is a quantity Ṽ = V DD + V HH ?2V DH that de nes energetical advantage ( Ṽ < 0) or dis- advantage ( Ṽ > 0) of the sort con gurations, when the nearest neighbour H-bonds contain proton and deuteron.
The gures 3, 4 and 6, 7 show that the theory provides good description for the crystals without chaos: CDP, LHP and their completely deuterated analogs (model parameters are given in table 2).For the better description of the permittivity and relaxation time in the ferroelectric phase one should take somewhat di erent values for the quasispin dipole moment d and the bare relaxation time 0 as it was in the papers 22,23,25].
For CDP there exist also dielectric measurements of partially deuterated On the other hand, comparison of q-dependent static and dynamic correlation functions with experimental data (e.g.neutron scattering experiments) might also be very interesting.
At given parameters the theory predicts almost monodispersive relaxation in paraphase for all x.Such behaviour actually takes place in the experiment for Cs(H 1?x D x ) 2 PO 4 .Figure 8 shows, that in the case of the quenched disorder with complete sort chaos (w = c c ) the static permittivity di ers from that of Cs(H 1?x D x ) 2 PO 4 .The annealed model with Ṽ = 0 does not describe experiment also.Therefore we have rejected these (quite arti cial) restrictions and obtained good t to experiment within both annealed ( gure 9) and quenched models ( gure 8).Both models predict sort ordering in the system: great values of w 12 signify that hydrogens and deuterons tend to situate at the nearest neighbour H-bonds.
It means that the following ordering of H and D ions in chains takes place: DHDDDHDDHDHD at c H < c D (protons never meet together), HDHHDHHHHDHDH at c H > c D (deuterons never situate at neighbouring hydrogen bonds) and DHDHDH at c H = c D = 1 2 .

Conclusions
On the basis of disordered Ising model we develop the statistical theory for ferroelectric properties of pseudo-one-dimensional H-bonded ferroelectrics Cs(H 1?x D x ) 2 PO 4 and PbH 1?x D x PO 4 .We consider short-range intrachain interactions within two-site cluster approximation, long-range interactions are taken into account within mean eld approximation.Dynamics of the system is considered on the basis of master equation approach.Thermodynamic potentials, order parameters, static and dynamic correlation functions of the model are calculated.We develop the theories for quenched and annealed systems and compare their predictions to experimental data.The model provides good description of dielectric measurements in the crystals of CsH 2 PO 4 , PbHPO 4 and their deuterated isomorphs.We have not determined a type of disorder in Cs(H 1?x D x ) 2 PO 4 , because both quenched and annealed theories match with available experimental data.The experimental determination of T c (x), dielectric measurements in the ferroelectric phase and the measurements, related with q-dependent correlation functions, are desirable in this respect.The theory predicts a sort ordering in Cs(H 1?x D x ) 2 PO 4 crystals.This prediction as well as theoretical T c on x dependence are the most interesting features for experimental veri cation.
883 A, = 107:73 and contains two formula units.There are hydrogen bonds O ?H O of two types in this crystals 4{6].The shorter bonds O3 ?H1 O3 (R O O = 2:48 A) connect PO 4 groups in zigzag chains along the b-axis.The longer bonds O2 ?H2 O2 (R O O = 2:56 A) are almost parallel to the c-axis and connect b-chains of PO 4 groups in the (b; c)-plane.Between (b; c)-planes of PO 4 groups the Cs + ions are situated.

Figure 1 .
Figure 1.The structure of the CsH 2 PO 4 crystal.

Figure 2 .
Figure 2. The structure of the PbHPO 4 crystal in projection on the (a; c) plane.Two formula units in the primitive cell are related by the symmetry with respect to c-glide plane perpendicular to b-axis.Parameters of the elementary cell are a = 4:688 A, b = 6:649 A, c = 5:781 A, = 97:11 .In PbDPO 4 a = 4:6855 A, b = 6:6911 A, c = 5:7867 A, = 97:10 .
a base vector of the elementary cell along the chain.We have expressed the

Figure 4 .
Figure 4. Inverse dielectric permittivity and relaxation time of the crystals LHP and DLHP.Lines present the theory, symbols are experimental data 17].There is no good t for deuterated crystal near T c due to smearing of the phase transition in DLHP.The reason for such smearing out may be defects of crystal.Small spontaneous polarization in DLHP (see gure 3) is the evidence of these defects.

Figure 5 .
Figure 5.The dependence of the ferroelectric transition temperature T c on the degree of deuteration for CDP crystal (parameters are given in table2).The line 1 corresponds to the quenched system with the complete sort chaos.The lines for the annealed system and the quenched system with ordering almost coincide and are denoted by number 2.

Figure 7 .
Figure 7.The dynamic dielectric permittivity of the crystal CsH 2 PO 4 .Lines present theoretical results (parameters in table.2), symbols show experimental data 16].

Table 1 .
Parameters of Ising model that describe the experimental data on CDP/DCDP.CDP parameter is in the nominator, denominator contains DCDP parameter.

Table 2 .
The model parameters that describe experimental data on Cs(H 1?x D x ) 2 PO 4 : the nearest neighbour coupling K ; the long-range interaction J ; the e ec- Figure 9. Spontaneous polarization P, inverse dielectric permittivity 1=" 0 and inverse relaxation time 1= of the ferroelectric Cs(H 1?x D x ) 2 PO 4 at di erent degrees of deuteration.Squares correspond to experimental data 15], thin lines in the lower picture connect experimental points from the paper 16].Thick lines, denoted by numbers 1{5 are theoretical results for equilibrium model with parameters given in table 2 for concentrations x = 0, 0.2, 0.54, 0.88, 1.