Thermodynamics and complex dielectric permittivity of mixed crystals of the Rb 1 − x ( NH 4 ) x H 2 PO 4 type

We propose a pseudospin model for proton glasses of the Rb1 x(NH4)xH2PO4(Rb1 x(ND4)xD2PO4) type, which takes into account the energy levels of hydrogens (deuterons) around the PO4 group, long-range interactions between the hydrogen bonds, and an internal random deformational eld. Within the framework of a cluster approximation and a mean eld approximation over the long-range interactions, we derive a system of equations for the state parameters for the regions which are in the ferroelectric and antiferroelectric states, as well as in the proton glass state. Within the Glauber dynamics approach, we obtain a system of equations for the frequency-dependent linear responses of polarization and the proton glass order parameter. We obtain a qualitative description of the temperature behavior of dielectric permittivities of the K1 x(ND4)xD2PO4 and Rb1 x(NH4)xH2AsO4 compounds at different frequencies. The origin of low-temperature peak in the imaginary part of dielectric permittivity in proton glasses is discussed.


Introduction
A notion of structural glasses (now called proton glasses) was introduced by E. Courtens in [1] during investigations of the temperature behavior of transverse and longitudinal permittivities (at ν = ω/2π=1 kHz) of the Rb 1−x (NH 4 ) x H 2 PO 4 system (T c =147.8 K for RbH 2 PO 4 , T N = 148 K for NH 4 H 2 PO 4 ) at x 0.35.At x 0.2 the studies of the real part of dielectric permittivity ε 11 (ω, T ) revealed that with temperature lowering, the system first undergoes a transition to a mixed state at a certain temperature T f (ω) (a deviation from the Curie-Weiss law is observed) and then, after reaching a plateau-like maximum (at T max (ω)), to a structural glass state at temperature T g (ω) (when ε 11 (ω, T ) starts to decrease).In [2] by analyzing an expression for dielectric susceptibility χ 33 (ω, T ) a rectangular approximation for distribution function of relaxation times g(τ, T ) was proposed.A maximal relaxation time τ m at x = 0.35 is well described by the Vogel-Fulcher law (T g (ω → 0) → T 0 ) Later, on the basis of dielectric [2][3][4][5][6][7][8], optical [9], and X-ray scattering [6,7,10] measurements a phase diagram of the Rb 1−x (NH 4 ) x H 2 PO 4 was constructed, in which the concentration range of about 0.20 < x < 0.75 corresponds to the proton glass state.Ferroelectric and antiferroelectric phase transitions were detected at lower and higher concentrations, respectively.
In [11] it was shown for Rb 0.53 (NH 4 ) 0.47 H 2 PO 4 that the spectrum of distribution function g(τ, T ) between 55 K and 35 K consists of two wide lines, and with temperature decreasing, a fast crossover of intensity from one band to the other takes place (a peak is observed at τ max ∼ 10 −5 s at T = 55 K and at τ max ∼ 10 2 s for T = 35 K).To explain this, the authors use a model of dynamically correlated domains (DCD), for which distribution functions of the domain number and of relaxation frequencies on the domain size are introduced.At freezing temperature some of the domains form an infinite percolation cluster, and the system goes to a non-ergodic state.
In [12], the studies of relaxation times of the 87 Rb NMR spectra for Rb 0.58 (ND 4 ) 0.42 D 2 PO 4 and Rb 0.5 (NH 4 ) 0.5 H 2 PO 4 revealed that at low temperatures T < 25 K for Rb 0.5 (NH 4 ) 0.5 H 2 PO 4 one should take into account the relaxation time related to phonon-activated tunneling τ Γ .Plateau of 87 Rb relaxation time in a pure RbH 2 PO 4 , which at these temperatures is in the ferroelectric ordering state, turned out to be 10 4 times higher than for the proton glass Rb 0.5 (NH 4 ) 0.5 H 2 PO 4 .This indicates strong differences in the relaxation mechanisms in pure systems with hydrogen bonds and in their mixtures.
Various mixed hydrogen bonded ferroelectrics of this type have been studied: K In this work we shall explore dielectric properties of Rb 1−x (ND 4 ) x D 2 PO 4 and Rb 1−x (NH 4 ) x H 2 AsO 4 ferroelectrics.In the first one, a rather high value of T g ∼ 60 K is observed; also in deuterated compounds one can neglect tunneling effects.In contrast to a symmetric behaviour over x phase diagram of Rb 1−x (ND 4 ) x D 2 PO 4 , the diagram of a Rb 1−x (NH 4 ) x H 2 AsO 4 mixture (T g ∼ 30 K) is strongly asymmetric.For Rb 1−x (NH 4 ) x H 2 AsO 4 we can find out how important it is to take tunneling into account in calculations of dielectric characteristics.
Investigations of the state of proton glasses in Rb 1−x (ND 4 ) x D 2 PO 4 started in [13] for x = 0.55 and [14] for x = 0.44 using the NMR method.It was shown [14] that a satisfactory description of temperature dependence of Edwards-Anderson parameter q EA (T ) is possible within the framework of a model which takes into account fluctuations of interactions between hydrogen bonds and chaotic deformational field (the random-field random-bond Sherrington-Kirkpatrick Ising model).
In [15] the Edwards-Anderson q EA (T ) parameter was studied using the NMR method for x = 0.22, 0.44, 0.78 at different orientations of external magnetic field with respect to the c axis of the crystal.
Temperature and frequency curves of transverse ε 11 (ω, T ), ε 11 (ω, T ) and longitudinal ε 33 (ω, T ), x (H 1−y D y ) 2 PO 4 (0.25 x 0.75, 0 y 1.0) were measured at 4.2 K T 300 K, 1 kHz ν 1 GHz in [16].It was shown that the experimental data for the permittivities are well described by the distribution function of relaxation times g(τ, T ) calculated from phenomenological relation between τ and Gaussian distribution w E, Ē, σ E of the activation energy In [17] a phase diagram was presented; it was also shown that the system dynamics on the proton glass state (x = 0.5, 0.7) is thermally activated, in contrast to a soft mode behavior in the regions with predominantly ferroelectric and antiferroelectric orderings.
In [18] the transverse static ε 11 (T ) and low-frequency dynamic ε 11 (ω, T ) permittivities of the Rb 1−x (ND 4 ) x D 2 PO 4 mixture at different compositions (x = 0.15 − 0.70) were measured.For the distribution function of relaxation times g(τ, T ) a linear approximation with respect to the z = ln ω a τ parameter was proposed.A maximal relaxation time τ m diverges with temperature according to the Vogel-Fulcher law.Interpretation of separation lines T c (x) between EPG (Ergodic Proton Glass) and EF (Ergodic Ferromagnetic) phases and T N (x) between EPG and EAF (Ergodic Antiferromagnetic) phases, as well as the freezing lines T 0 (x) ≈ 33 K±0.5 K between EPG and NPG (Nonergodic Proton Glass) were performed using a random-field random-bond Ising model.Here the freezing line T 0 = T 0 (x) is interpreted as an Almeida-Thoulesse instability line, below which the replica-symmetric (RS) solutions are incorrect.This work did not contain a theoretical description of ε 11 (T ), ε 11 (ω, T ) temperature curves.
In [21], the NMR studies of Rb 0.22 (ND 4 ) 0.78 D 2 PO 4 showed that proton glass state is formed due to gradual freezing of deuterons on hydrogen bonds.It also leads to deformation and rotation of ND 4 groups.
In [22], the dielectric properties of Rb 0.5 (ND 4 ) 0.5 D 2 PO 4 as well as microwave and millimeter frequency range were explored.The distribution function of relaxation times g(τ, T ) at 120 K< T <170 K calculated from the experimental data is well described by the theoretical curve calculated using a Gaussian averaging of phenomelogical relation for τ with Gaussian distribution for activation energy E and parameter of the potential well asymmetry.
In [25], by measuring transverse dielectric permittivity of Rb 1−x (NH 4 ) x H 2 AsO 4 , the occurrence of proton glass state was revealed at low temperatures in the regions of ferroelectric phase (small x) and antiferroelectric phase (large x).Later [27], the proton glass state was detected even at x = 0.01.
Let us now briefly review some theoretical works on the thermodynamic and dielectric properties of the Rb 1−x (NH 4 ) x H 2 PO 4 type proton glasses.
To the best of our knowledge, the first theory for the Rb 1−x (NH 4 ) x H 2 PO 4 mixture was proposed in [30].A pseudospin Hamiltonian was used to describe the energy levels of hydrogens around a PO 4 group; a cluster approximation was used to calculate the transition lines T c (x), T N (x).A qualitative description of the phase diagram was obtained.
Later, the cluster approach was used in [31,32].In [31] to describe the Rb 1−x (NH 4 ) x H 2 PO 4 mixtures, a pseudospin model was employed which took into account a configurational energy of proton bonds cluster around PO 4 group, as well as the long-range interaction.The free energy was presented as a sum of the energies of three phases with the probabilities p + (x) for ferroelectric phase, p − (x) for antiferroelectric phase, and p 0 (x) for neutral phase.By analyzing the coefficients of free energy expansion in the S 1 + S 3 ; S 1 − S 3 parameter, the regions of ferroelectric (0 < x < 0.2 at T = 0) and antiferroelectric (0.75 < x < 1 at T = 0) phases were found, which were close to experimental data.This model was also used [32] to describe the phase diagram in the proton glass region (0.2 < x < 0.75 at T = 0).A RS approximation was employed for averaging the system free energy; an explicit expression for T g (x) was obtained.Thus, in [31,32] not a single approach for description of all states of this type of compounds was developed.
Experimental data are most often interpreted using the random-field random-bond Ising model (also with transverse field) within the framework of RS approach [33].This model does not take into account a real structure of the crystals.A Gaussian distribution of random interactions with infinite correlation range ( , and the RS-approximation are used.It is shown that T g (protonglass transition temperature) exists at h 2 i c = 0 only and corresponds to the peak on a temperature curve of χ(T ).Internal chaotic field ( h 2 i c = 0) gives rise to the occurrence of the proton glass state at a rather high temperature above T g (q > 0, q −→ T →∞ 0) and smears out the peak of the χ(T ) temperature curve.
This model was successfully used in [34] to describe a Tl 2+ EPR spectrum in Rb 0.52 (NH 4 ) 0.48 H 2 PO 4 .The Edwards-Anderson parameter q is proportional to the second moment of the EPR distribution function and tends to zero at T 200 K.This confirms the occurrence of significant static random fields in Rb 1−x (NH 4 ) x H 2 PO 4 compounds.In [35], the peculiarities of low-temperature (slow motion) dynamics are taken into account within the Glauber approach.The observed shape of the line contains an integral with the distribution function W (p) of local polarization (τ 0 is the relaxation time related to the two-well potential).The theoretical curves satisfactorily describe the shape of the line (a single peak at high temperatures and two peaks at low temperatures) in a wide temperature range (T = [10 K,150 K]).
In [36] the random-field random-bond Ising model with phonon thermostat is used to calculate dielectric permittivity of Rb 1−x (ND 4 ) x D 2 PO 4 and Rb 1−x (ND 4 ) x D 2 AsO 4 .The theory yields Debye type relaxation, with the relaxation time considered as a phenomenological parameter taken according to the Arrhenius law.A quantitative comparison is performed for the temperature dependence of ε (ω) peak; the entire temperature curves of permittivity ε (ω, T ), ε (ω, T ) are not described.
In [37,38] a model Hamiltonian is used, the parts of which describe ferroelectric ordering along the Z axis (S z components of a classical spin) and antiferroelectric ordering (S x component).Instead of the generally accepted Edwards-Anderson parameter q EA , single-site correlators S z i1 S z i2 , S x i1 S x i2 for sublattices 1, 2 are used as an order parameter for the proton glass phase.The phase diagram for Rb n (NH 4 ) 1−n H 2 AsO 4 constructed within RS approximation qualitatively correctly describes some experimental phase boundaries.
Hence, a theoretical description of thermodynamic and dielectric properties of the hydrogen bonded compounds of Rb 1−x (NH 4 ) x H 2 PO 4 type, which can undergo a transition into the proton glass state, that takes into account structural peculiarities and different types of interactions, is still an important and unresolved problem of statistical physics.Especially important is a microscopic description of relaxational properties of this type of mixtures.It is interesting to describe temperature curves of real and imaginary parts of longitudinal and transverse dielectric permittivities at different frequencies, in particular, the low-temperature portion of the imaginary part of the permittivity, whose peak indicates a transition to a non-ergodic state.
In [39], on the basis of the cluster expansion method, a single approach is suggested to a theory of proton ordering in ferroelectrics and antiferroelectrics of the orthophosphate type where the configuration short-range, dipole long-range interactions are taken into account.In [40,41] a theory of static properties of model proton glasses with an arbitrary range of competing interactions was proposed.Thermodynamic properties of the model with essential short-range and weak long-range interactions were explored within the two-particle cluster approximation.In the present paper, the approaches developed in [39][40][41] are used to describe thermodynamic and dielectric properties of hydrogen bonded

Thermodynamic properties of the compounds of Rb
In order to describe possible configurations in a mixed Rb 1−x (NH 4 ) x H 2 PO 4 crystal, let us first consider a structure of pure RDP-RbH 2 PO 4 and ADP-NH 4 H 2 PO 4 crystals (in figure 1 a unit cell of the isomorphic to RDP for the KDP-KH 2 PO 4 crystal is shown).
A primitive cell of RDP-RbH 2 PO 4 contains one tetrahedron of the "A" type and one tetrahedron of the "B" type, two Rb atoms and four protons on four hydrogen bonds.In the ferroelectric phase, the total dipole moment of the cell, created by displacements of different ions and by deformation of a PO 4 group, is directed along the c axis.A triggering mechanism for ionic displacements is proton ordering (their positions are described by spin variables S f = ±1, f = 1 − 4) on double potential wells of the hydrogen bonds.The bond dipole moments lie almost within the ab plane; the total dipole moment of the protons for RDP is zero It is well known that in order to describe the thermodynamic characteristics and dielectric properties (in a certain frequency range) of these crystals within the pseudospin-phonon model, the ionic variables can be excluded in the static approximation.The system description is then performed within the framework of a pseudospin model with renormalized moments of hydrogen bonds d f,α (α = + for RDP, α = − for ADP) Here, we introduced an effective dipole moment of a tetrahedron P α ; . . .stand for a thermodynamic average; summation f = A(B) is carried out over the bonds, on which the protons order close to the given tetrahedron A(B).For RDP, the tetraderon polarization can have two opposite values along the c axis, when two protons are ordered close to the upper edge of the tetrahedron (η f = η) and close to the lower one (η f = −η) For ADP-NH 4 H 2 PO 4 , the primitive cell is twice as large as for RDP, and in addition to "A", "B" tetrahedra, it contains "A ", "B " tetrahedra.Since their polarizations are opposite to those of "A", "B", the total cell polarization is zero: For an ADP-NH 4 H 2 PO 4 crystal the change of sign of η at transition to the "A ", "B " tetrahedra can be taken into account as (here n is the RDP primitive cell vector; k z * is the vector at the Brillouin zone boundary directed along Z) η Hence, in the cases of both ADP and RDP we use a primitive cell with "A" and "B" tetrahedra.
Hamiltonian of a mixed Rb 1−x (NH 4 ) x H 2 PO 4 system can be written as Here S nf = ±1 are spin operators describing the position of a proton on the f = 1, 2, 3, 4 hydrogen bond in the n cell at a R tetrahedron; E is an external uniform electric field; G n is an internal random deformational field; J nf,n f is the long-range interaction between protons; H A (n), H B (n) are the configurational energies of the "A", "B" tetrahedra.In this work we take into account two configurational states of a tetrahedron (α = +, −) In the state +, the energy states of a tetrahedron are analogous to those in a pure RDP crystal with the ground level In the state -(ADP) we use the same relations for V α , U α , Φ α but with different values of ε α , w α , w 1α .
In the case of a mixed Rb 1−x (NH 4 ) x H 2 PO 4 crystal, ionic positions are occupied by Rb with probability c + = 1 − x and by NH 4 with probability c − = x.Hence, the distribution function of a strongly random energy parameter ε α (and similarly for w α , w 1α ) can be qualitatively written as A state of a dipole moment on the bond d f,αα f is determined by the states α, α f of two tetrahedra connected by this bond.In the mean field approximation over the bonds, the average on configuration moment of a tetrahedron P B c reads In the present work we consider only two realizations of the sets of the averaged over configurations values of ηf = η; −η B 1,− = ηB 2,− = ηB 3,− = −η B 4,− = η, which correspond to ferroelectric and antiferroelectric ordering.This permits to use the primitive cell of RDP with 2 tetrahedra and 4 hydrogen bonds.Then the mean free energy per primitive cell F can be written as follows: where k * = 0 * for ferroelectric ordering = F (1)  c .We use the following notations for averages over different random fields of the single-particle F (0) f and cluster F [0000] 1234 generating functions Here, we introduce notations for the average values of cluster ϕ f and long-range ϕ L,f fields, and Averaging is performed over random cluster fields with dispersion q and over random deformational fields with dispersion The expressions for the single-particle function F (0) f and its derivatives F (n) f are as follows: = th (βξ f ) f , F The cluster function F [0000] 1234 and its derivatives F [n1n2n3n4] 1234 read

0.5L
[0000] 1234 Here the partition function 0.5L ({ξ} ||R α ) is calculated with the cluster Hamiltonian We shall use the same model dependence of the average eigenvalues of the long-range interaction matrix as for the dipole moment of a hydrogen bond: For these values of k * the long-range interaction matrix Jf,f = J f,f k * c and the unitary transformation matrix U = {u µf } read From the condition of the free energy extremum we find an expression for the average ηf = S nf c , reduced Edwards-Anderson parameter Q EA,f and an equation for unknown quantities φL,f , φf , q In the absence of external field and for the ferroelectric ordering we obtain the following expressions for the free energy, for the average η = ηf , reduced Edwards-Anderson parameter Q EA = Q EA,f and for equations for φL , φ, q 2) = F [2000] , φL = ν1 ( 0 * )η.
In the case of an antiferroelectric ordering in the absence of external field the free energy, for the average η = −η 1 = η2 , reduced Edwards-Anderson parameter Q EA = Q EA,f and equations for φL , φ, q read As numerical calculations for the free energy show, the antiferroelectric state is realized in the region close to the x = 1−c → 1 limit, the ferroelectric state is realized in the region 1−x = c → 1, and a proton glass state ( φ = φL = 0, q > 0) takes place at intermediate compositions.
The static susceptibility of the system reads (v e is the cell volume) Here we used the fact that after the unitary transformation with Û , the matrix of η µµ = δ µµ η µ correlators is diagonal for ferroelectric and antiferroelectric orderings, and We shall be interested in the longitudinal χ 33 and transverse χ aa susceptibilities of the system 3. Relaxational dynamics of the Rb 1−x (NH 4 ) x H 2 PO 4 type mixtures Dynamics of the mixed system Rb 1−x (NH 4 ) x H 2 PO 4 will be described on the basis of Glauber equations for the n-th order correlation functions: Here we introduce operator fields ϕ + j , ϕ − j exerted on the bond j by the two tetrahedra connected by this bond The seed relaxation time τ 0,j can be expressed via correlation functions of an ionic subsystem.Within the cluster approximation given below, the bonds i, j belong to the same tetrahedron (as an example we use the "B" tetrahedron with indices f, f = 1 ÷ 4).The averaging over configurations will also be performed for each tetrahedron with its hydrogen bonds independently.Therefore, in our equations there is no primitive cell index.

Discussion
In this section we analyze temperature dependencies of the Edwards-Anderson parameter Q EA (T ) and transverse ε 11 (ω, T ) permittivities for different compositions, as well as phase diagrams of Rb 1−x (ND 4 ) x D 2 PO 4 and Rb 1−x (NH 4 ) x H 2 AsO 4 compounds.It should be noted that the choice of the model parameters proposed below provides a good agreement with experimental data for specific heat, sublattice polarization, transverse and longitudinal permittivities for pure ND 4 D 2 PO 4 and RbH 2 AsO 4 , NH 4 H 2 AsO 4 [42] crystals in wide temperature and frequency ranges.
For the Rb 1−x (ND 4 ) x D 2 PO 4 mixture (T N (x = 1) = 242 K) the set of the model parameters is in table 1 (we use everywhere w 1α = ∞).Dashes in table mean, that the given parameter is averaged over two states only (excluding the neutral state of the tetrahedron 0 (Glass)).We take into account the difference between d x α (G) (effective dipole moment of bond in proton glass phase) and d x α (F ) (effective dipole moment of bond in phases with microscopic polarization).It should be noted that crystal structure of Rb(H 1−y D y ) 2 PO 4 is isomorphic to structure of KH 2 PO 4 only at y < 0.90.Thus, our results for crystal RbD 2 PO 4 are purely theoretical and can differ from experimental data.
In figure 2 we show the temperature curves of the Q EA (T ) parameter for different compositions.At x = 0.22, the theoretical curve 3 well describes the experimental data of [15].At the same time, at x = 0.44, our results accord with the data of [14], but are lower than the values of [15].We think that the reason for this is an incorrect determination of composition in [15].16,2,43,44].In the cases of x = 0.24; 0.5, the experimental points for ε 11 (T ) from [18] are shown.A principal difference between static and dynamic results is observed in the proton glass region.While ε 11 (T ) has a plateau at temperature lowering, ε 11 (ω = 0, T ) sharply falls to zero, the peak position shifting to higher temperatures with ω increasing.However, at temperatures above the peak, the dynamic and static permittivities practically coincide.This fact enables us to talk about qualitative agreement or disagreement between the theoretical curves for ε 11 (T ) and experimental data for ε 11 (ω = 0, T ).For compositions x = 0.0 (curve 1); x = 0.1 (2); x = 0.2 (3) the theoretical results only are shown.For x = 0.24 (4), x = 0.50 (5), x = 0.62 (6) the theoretical data correlate well with experiment.At x = 0.50 the experimental values of ε 11 (ν = 10 kHz, T ) [16] are higher than the values of ε 11 (T ) [18] even at high T .This could indicate an incorrect determination of the composition in one of these experiments.At x=0.7 (7) the calculated values are lower than experimental data of [18] for 1 kHz, while at x = 1 (9) the calculated values are slightly higher than the experimental ones and than those calculated in [42] (9').In the latter paper, a relation is used, which is a partial case of ( 22), ( 23) and ( 49), (50) at x = 0, 1, but a different set of the model parameters is used.Our corrected values of the model parameters permit us to describe thermodynamic and dielectric characteristics in the total range of compositions x = [0, 1].The worst agreement with experiment is observed at concentrations close to x ∼ 0.2 and x ∼ 0.65, which are the transition regions between the pure phases (ferroelectric ordering, glass, antiferroelectric ordering), which we do not take into account in this work.[18]; 1 kHz - [18]; 10 kHz -2, [16]; 1 MHz - [18]; 10 MHz -3, [16]; 1 GHz -4, [16].
We also calculated ε 11 (ω, T ), ε 11 (ω, T ) in the regions of ferroelectric and antiferroelectric ordering.At low frequencies the real parts of dynamic permittivity ε 11 (ω, T ) in the high-temperature range (near and above T c , T N ) practically coincide with the static permittivities ε 11 (T ).At low temperatures, however, ε 11 (ω, T ) have peaks (inflection points of ε 11 (ω, T )), which correspond to the freezing line and indicate a possible occurrence of the proton glass phase at these values of x.On lowering x, positions of the low-temperature peak of ε 11 (ω, T ) practically do not change, but their maximal values fastly decrease.We failed to find this peak numerically at x < 0.15.A similar peak is detected in the antiferroelectric phase region at 0.65 < x < 0.70.At the phase diagram, the freezing lines in the ferroelectric and antiferroelectric phases are continuations of the freezing lines of the proton glass region (figure 5, dashed lines).
The phase diagram of Rb 1−x (ND 4 ) x D 2 PO 4 contains four regions.At high temperatures the system is in the ergodic proton glass state (EPG-ergodic proton phase).In this state the Edwards- - [17], - [18], • - [16], - [45], - [20].Anderson parameter is different from zero and decreases with temperature increasing.At x < 0.22 and x > 0.65 and at lowering temperature, the system goes to inhomogeneous ferroelectric (IFinhomogeneous ferroelectrics) or inhomogeneous antiferroelectric (IAF-inhomogeneous antiferroelectrics) states at T c (x) and T N (x), respectively (tetrahedron polarization appears at T c (x), T N (x)).In the central region 0.22 < x < 0.65 below the freezing lines T g,1 (x) ∼ const (this temperature corresponds to the position of the low-temperature peak of ε 11 (ω, T )) the system goes to the nonergodic proton glass state (NPG-nonergodic proton phase).This line continues in the regions x < 0.22, where the proton glass state can coexist with ferroelectric ordering.Let us note that the experimentally observed regions of phase coexistence in some Rb 1−x (NH 4 ) x H 2 PO 4 type mixtures (Rb 1−x (NH 4 ) x H 2 AsO 4 , K 1−x (NH 4 ) x H 2 PO 4 ) can be much wider, approaching the x = 0, x = 1 limits.
In Rb 1−x (NH 4 ) x H 2 AsO 4 mixtures (T c (x = 0) = 110 K, T N (x = 1) = 216 K), an essential role in the low-temperature dynamics is played by the processes of proton tunneling.To describe them we need to go beyond the limits of Glauber kinetic equations.However, as we shall see, the proposed approach can be used for a qualitative description of dynamic phenomena in this compound.
In figure 7 we compare the calculated static transverse permittivity ε 11 (T ) = ε 11 (ω = 0, T ) with experimental points for the real part of the transverse permittivity ε 11 (ω = 0, T ) at low frequencies ω.The phase diagram is strongly asymmetric, and the proton glass state occurs in the region x = [0.2;0.45].As seen, a quantitative or qualitative agreement with experiment takes place in the regions far from the phase boundaries (curves 1,2 for the ferroelectric state and curves 5, 6 for the antiferroelectric state).The value of x = 0.15 corresponds to the intermediate phases between the ferroelectric state and the proton glass state.Below the permittivity peak, the theoretical curve deviates from the experimental points.For x = 0.35 (approximately, a center of the proton glass phase) we compare the theoretical results with experimental data for ν=1 Hz, 30 kHz of [25].The theory yields a much faster decrease of ε 11 (ω, T ) with temperature and a very high and sharp curve of ε 11 (ω, T ).The shape of this line for Rb 1−x (NH 4 ) x H 2 AsO 4 sharper than for Rb 1−x (ND 4 ) x D 2 PO 4 is due to an essential contribution of proton tunneling to the low-temperature dynamics.This effect is not taken into account by the proposed theory.However, we think that the present approach can be used to describe a position of the imaginary part of the permittivity (freezing line).
The phase diagram of Rb 1−x (NH 4 ) x H 2 AsO 4 (figure 8), like in the case of Rb 1−x (ND 4 ) x D 2 PO 4 , contains four regions EPG, NPG, IF, IAF.The freezing line T g,1 (x) (dashed line) corresponds to the maximum of ε 11 (ν = 1 Hz, T ).According to experimental data [29,26] this line is observed in the ferroelectric phase region down to x = 0.01.With decreasing x, T g,1 (x) → 0. In calculations the decreasing line is observed down to x ∼ 0.15.Overall, the calculated phase diagram correctly describes the experimental data, though some disagreement is observed.Thus, the used values of the model parameters yield a somewhat wider proton glass region x ∼ [0.18; 0.46] than the one experimentally observed x ∼ [0.22; 0.42].This disagreement can be due to different methods of identifying the proton glass state.Experimentally, X-ray scattering, NMR, and Raman scattering data are used to identify the phases.

Figure 1 .
Figure 1.A unit cell (four formula units) for the KDP-KH2PO4 crystal.

Figure 3
Figure3contains a comparison of calculations of the transverse static permittivity ε 11 (T ) (ε 11 (T ) = ε 11 (ω = 0, T )) with experimental data for the real part of the permittivity ε 11 (ω = 0, T ) at low frequencies ω[18,16,2,43,44].In the cases of x = 0.24; 0.5, the experimental points for ε 11 (T ) from[18] are shown.A principal difference between static and dynamic results is observed in the proton glass region.While ε 11 (T ) has a plateau at temperature lowering, ε 11 (ω = 0, T ) sharply falls to zero, the peak position shifting to higher temperatures with ω increasing.However, at temperatures above the peak, the dynamic and static permittivities practically coincide.This fact enables us to talk about qualitative agreement or disagreement between the theoretical curves for ε 11 (T ) and experimental data for ε 11 (ω = 0, T ).For compositions x = 0.0 (curve 1); x = 0.1 (2); x = 0.2 (3) the theoretical results only are shown.For x = 0.24 (4), x = 0.50 (5), x = 0.62(6) the theoretical data correlate well with experiment.At x = 0.50 the experimental values of ε 11 (ν = 10 kHz, T )[16] are higher than the values of ε 11 (T )[18] even at high T .This could indicate an incorrect determination of the composition in one of these experiments.At x=0.7(7) the calculated values are lower than experimental data of[18] for 1 kHz, while at x = 1 (9) the calculated values are slightly higher than the experimental ones and than those calculated in[42] (9').In the latter paper, a relation is used, which is a partial case of (22), (23) and (49), (50) at x = 0, 1, but a different set of the model parameters is used.Our corrected values of the model