Dynamic properties of NH$_3$CH$_2$COOH$\cdot$H$_2$PO$_3$ ferroelectric

Using a modified pseudospin model of NH$_3$CH$_2$COOH$\cdot$H$_2$PO$_3$ ferroelectric taking into account the piezoelectric coupling with strains $\varepsilon_i$, $\varepsilon_4$, $\varepsilon_5$ and $\varepsilon_6$ within Glauber method in two-particle claster approximation, we have calculated components of dynamic dielectric permittivity tensor and relaxation times of the model. At the proper set of theory parameters, frequency and temperature dependences of the components of permittivity and temperature dependences of the relaxation times are studied. A satisfactory agreement of the theoretical results with experimental data for longitudinal permittivity is obtained.


Introduction
The problem of investigation of physical properties of ferroelectric materials has occupied one of the central places in condensed matter physics for a long time. The presence of different classes of these materials with different crystal structure and chemical composition requires elaboration of universal methods for investigation of phase transition mechanisms. It is also necessary to develop concrete microscopic theories for them, which could explain the observed experimental data for thermodynamic and dynamic characteristics and anomalies in the behaviour of these characteristics in the phase transition region.
Granting this, glycinium phosphite NH 3 CH 2 COOH·H 2 PO 3 (GPI) is of special interest due to the combination of structure elements typical of different classes of ferroelectric crystals. In [1][2][3] basing on the analysis of structural data [4] it was determined that the main role in the phase transition in GPI is played by two structurally nonequivalent types of O-H. . .O hydrogen bonds of different length, which connect phosphite groups HPO 3 in the chains along the crystallographic c-axis. As a result, in [1,3] there was proposed a model of GPI crystal with proton ordering, within which the main peculiarities of its dielectric permittivity were explained qualitatively. Later, this model was supplement by taking into account the piezoelectric coupling of proton and lattice subsystems [5], which made it possible to calculate thermal, piezoelectric and elastic characteristics of GPI. At the proper set of theory parameters, a good agreement of the obtained theoretical results with corresponding experimental data for the crystals of this type was obtained.
In order to better understand the mechanism of phase transition in these crystals and their physical properties, the effects of transverse electric fields [6] and uniaxial pressures [7] on the static physical properties of GPI were calculated within the model proposed in [5]. A good agreement of the obtained theoretical results with the available experimental data was obtained. This confirms the key role of proton ordering on the above mentioned bonds. It should be noted that several results obtained in these papers may be interpreted as predictions which will be a stimulus for further experimental investigations.
The aim of this paper is to study the relaxation phenomena in GPI and explain the available experimental data [8][9][10][11] for longitudinal dynamic characteristics within the proton ordering model proposed in [5].

Model of GPI crystal
The pseudospin model proposed in [5] considers the system of protons in GPI, localized on O-H. . .O bonds between phosphite groups HPO 3 , which form chains along the crystallographic c-axis of the crystal (figure 1). Dipole moments d q f = µ µ µ f σ q f 2 are ascribed to the protons on the bonds. Here, q is a primitive cell index, f = 1, . . . , 4; σ q f 2 are pseudospin variables that describe the changes connected with reorientation of the dipole moments. The Hamiltonian of a proton subsystem of GPI, which takes into account the short-range and longrange interactions and the applied electric fields E 1 , E 2 , E 3 along the positive directions of the Cartesian axes X, Y and Z (X ⊥ (b, c), Y b, Z c) can be written in such a way: where N is the total number of primitive cells. The first term in (2.1) is the "seed" energy, which relates to the heavy ion sublattice and does not explicitly depend on the configuration of the proton subsystem. It includes elastic, piezoelectric and dielectric parts expressed in terms of electric fields E i and strains ε i : The fourth term in (2.1) describes the interactions of pseudospins with an external electric field: (2.10) Here, µ µ µ 1 = (µ x 13 , µ y 13 , µ z 13 ), µ µ µ 3 = (−µ x 13 , µ y 13 , −µ z 13 ), µ µ µ 2 = (−µ x 24 , −µ y 24 , µ z 24 ), µ µ µ 4 = (µ x 24 , −µ y 24 , −µ z 24 ) are the effective dipole moments per one pseudospin.

13704-3
The two-particle cluster approximation for short-range interactions is used for the calculation of thermodynamic characteristics of GPI. In this approximation, thermodynamic potential is given by: (2.11) q f are two-particle and one-particle Hamiltonians: where such notations are used: The symbols ∆ f are the effective cluster fields created by the neighboring bonds from outside the cluster. Minimizing the thermodynamic potential (2.11) with respect to the cluster fields ∆ f and to the strains ε i , and expressing ∆ f through the equilibrium order parametersη 1 =η 3 =η 13 ,η 2 =η 4 =η 24 , we have obtained a system of equations for the equilibrium order parameters and strains for the case of zero mechanical stresses and fields: D = cosh(ỹ 13 +ỹ 24 ) + a 2 cosh(ỹ 13 −ỹ 24 ) + 2a coshỹ 13 + 2a coshỹ 24 + a 2 + 1,

Theoretical calculations of dynamic dielectric permittivity of mechanically clamped GPI crystal
To calculate the dynamic properties we use an approach based on the ideas of a stochastic Glauber model [12]. Using the methods developed in [13], we obtain the following system of Glauber equations for time dependent correlation functions of the pseudospins:

13704-4
where parameter α determines the time scale of dynamic processes, ε q f ′ (t) is the local field acting on the f ′ -th pseudospin in q-th cell. We use a two-particle cluster approximation in order to obtain a closed system of equations. In this approximation, local fields ε q f (t) are coefficients at σ q f /2 in two-particle and one-particle Hamiltonians (2.12), (2.13). Correspondingly, these fields are presented in a two-particle approximation: and in a one-particle approximation: As a result, from (3.1) we obtain a system of equations for mean values of pseudospins σ q f = η f in a two-particle approximation: and in a one-particle approximation: where the following notations are used: Let us restrict ourselves to the case of small deviations from equilibrium state to solve the equations (3.4) and (3.5). For this case we write η f and effective fields y f ,ȳ f in the form of a sum of equilibrium values and their deviations from equilibrium values (a mechanically clamped crystal): (3.6) Here, ∆ 13 = ∆ 1 = ∆ 3 , ∆ 24 = ∆ 2 = ∆ 4 are equilibrium effective cluster fields, and ∆ f t are their deviations from equilibrium values. Parameters ν ± i describe long-range interactions. We decompose the coefficients P f and L f in a series of y f t 2 limited by linear items: where the following notations are used: Substituting (3.6), (3.7) into (3.4), (3.5) and excluding parameter ∆ f t , we obtained the following differential equations for sums and differences of proton unary distribution functions: Solving the equations (3.8)-(3.10), we obtained time-dependent unary distribution function of protons. The components of dynamic susceptibility of GPI clamped crystal can be written as: The obtained susceptibilities consist of the "seed" part and two relaxational modes: where In (3.12), (3.13) γ = "+" for i = y and γ = "-" for i = x, z.
Components of dynamic dielectric permittivity of proton subsystem of GPI is as follows: (3.14)
At the frequencies ν ≫ ν s , the dielectric permittivity behaves as a purely lattice contribution. It corresponds to the frequency region ν > 10 10 Hz on the frequency dependences ε 22 (ν) in figure 3. ; [15]. Starting from frequency ν s ≈ 10 7 , a depression-minimum appears instead of a maximum of ε ′ 22 (ν, T), and this minimum decreases with an increase of frequency.
From figures 3-5 one can see that the proposed theoretical model satisfactorily describes the experimental data for the frequency and temperature dependences ε ′ 22 (ν, T) and ε ′′ 22 (ν, T) of GPI crystal in the paraelectric phase, with the exception of [10], and less satisfactorily in the ferroelectric phase. A disagreement of the theoretical curves with the experimental data in the low-frequency region in the ferroelectric phase is connected with an essential role of domain processes in this region [16], which are not taken into account in the proposed theory. Let us discuss the transverse dynamic characteristics. Transverse relaxation frequencies ν x,z s and transverse relaxation times τ x 2 and τ z 2 are calculated at the same α as longitudinal ν y s and τ y 2 . The frequencies ν x,z s are higher than ν y s and they also decrease at approaching the phase transition temperature (figure 6), and take on a nonzero value at T = T c . The transverse relaxation times τ x,z 2 in contrast to τ y 2 are finite at T = T c . This results in the frequency dependences of ε 11 (ν) (figure 7) and ε 33 (ν) (figure 8) at different ∆T that are qualitatively similar to the frequency dependences of ε 22 (ν), but the region of dispersion exists at higher frequencies and at weaker changes with temperature. However, in the temperature dependences of ε ′ 11 and ε ′ 33 , only the angle of the curve fracture in the point T c changes (figures 9, 10) instead of a depression near the phase transition temperature. The maximum value of ε ′ 11,33 (T, ν) at T = T c decreases with an increase of frequency. Values of ε ′′ 11,33 (T, ν) at T = T c increase with an increase of frequency up to 1.5 · 10 10 Hz. At higher frequencies, the maximum values of ε ′′ 11,33 (T, ν) decrease and shift to the region of higher temperatures. Experimental investigations of transverse dynamic characteristics of GPI are very important to verify the obtained theoretical results in this regard. It is necessary to note that experimental data in figures 9 and 10 are measured at frequency 1 kHz. They are close to static permittivities at such a small frequency.   The results of calculation of Cole-Cole curves (figure 11) witness for monodispersivity of dielectric permittivity in the crystals studied. The results of measurements of Cole-Cole curves for the longitudinal permittivity, presented in [9][10][11], disagree with each other. The calculated curves well agree with the results of [9] for longitudinal permittivity.

Conclusions
Using the modified GPI model, the components of dynamic dielectric permittivity tensor and relaxation times are calculated in a two-particle claster approximation. A satisfactory agreement of the theoretical results with experimental data for longitudinal permittivity is obtained, with the exception of low-frequency region in the ordered phase, inasmuch as the proposed theory does not take the domain processes into account, which can give a contribution into the above mentioned frequency region.
It is determined that the dynamic dielectric permittivity at low frequencies behaves as static; at the frequencies comparable with an inverse relaxation time, a relaxational dispersion is observed; at high frequencies, only a lattice contribution to permittivity reveals itself. The region of longitudinal dispersion in GPI shifts to the low frequencies at temperature approaching the phase transition point, which is connected with a considerable increase of relaxation time at approaching the temperature T c . The region of transverse dispersion lies at higher frequencies and weakly depends on temperature.
The obtained results for transverse characteristics bear the character of predictions and can be a stimulus for further experimental investigations.