# Slow and fast relaxation times of quantum lattice model with local multi-
well potentials: phenomenological dynamics for Sn2P2S6 ferroelectric crystals

R. Erdem  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, S. Özüm  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, N. Güçlü  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author:
<EMAIL_ADDRESS>
(Received June 29, 2022, in final form October 16, 2022)

###### Abstract

As a continuation of the previously published work [Velychko O. V., Stasyuk I.
V., Phase Transitions, 2019, 92, 420], a phenomenological framework for the
relaxation dynamics of quantum lattice model with multi-well potentials is
given in the case of deformed Sn2P2S6 ferroelectric lattice. The framework is
based on the combination of statistical equilibrium theory and irreversible
thermodynamics. In order to study these dynamics in a connected way we assume
that the dipole ordering or polarization ($\eta$) and volume deformation ($u$)
can be treated as fluxes and forces in the sense of Onsager theory. From the
linear relations between the forces and fluxes, the rate equations are derived
and characterized by two relaxation times ($\tau_{S},\tau_{F}$) which describe
the irreversible process near the equilibrium states. The behaviors of
$\tau_{S}$ and $\tau_{F}$ in the vicinity of ferroelectric phase transitions
are studied.

Key words: quantum lattice model, ferroelectric crystals, Sn2P2S6, relaxation
times, Onsager theory

###### Abstract

Ó ïðîäîâæåííÿ îïóáëêîâàíî¿ ðàíøå ðîáîòè [Velychko O. V., Stasyuk I. V., Phase
Transitions, 2019, 92, 420], äëÿ äåôîðìîâàíî¿ Sn2P2S6 ñåãíåòîåëåêòðèíî¿ ðàòêè
íàâåäåíî ôåíîìåíîëîãíó ñõåìó äëÿ ðîçãëÿäó ðåëàêñàöiéíî¿ äèíàìêè êâàíòîâî¿
ðàòêîâî¿ ìîäåë ç áàãàòîìíìóìíèìè ïîòåíöàëàìè. Ñõåìà áàçóòüñÿ íà ïîäíàíí
ñòàòèñòèíî¿ ðâíîâàæíî¿ òåîð¿ òà íåîáîðîòíî¿ òåðìîäèíàìêè. Ç ìåòîþ îòðèìàííÿ
óçãîäæåíîãî îïèñó ïðèïóñêàòüñÿ, ùî äèïîëüíå âïîðÿäêóâàííÿ è ïîëÿðèçàöþ
($\eta$) òà îá’ìíó äåôîðìàöþ ($u$) ìîæíà ðîçãëÿäàòè ÿê ïîòîêè òà ñèëè â
ðîçóìíí òåîð¿ Îíçàãåðà. Ç ëíéíèõ ñïââäíîøåíü ìæ ñèëàìè òà ïîòîêàìè îòðèìàíî
ðâíÿííÿ äèíàìêè, ÿê õàðàêòåðèçóþòüñÿ äâîìà àñàìè ðåëàêñàö¿
($\tau_{S},\tau_{F}$), ùî îïèñóþòü íåîáîðîòíé ïðîöåñ ìæ ðâíîâàæíèìè ñòàíàìè.
Âèâåíî ïîâåäíêó $\tau_{S}$ $\tau_{F}$ ïîáëèçó ñåãíåòîåëåêòðèíèõ ôàçîâèõ
ïåðåõîäâ.

Ключов слова: êâàíòîâà ðàòêîâà ìîäåëü, ñåãíåòîåëåêòðèí êðèñòàëè, Sn2P2S6, àñè
ðåëàêñàö¿, òåîðÿ Îíçàãåðà

##  1 Indroduction

Sn2P2S6 (SPS) ferroelectrics (FEs) is one of the prospective materials which
presents spontaneous polarization determined by anharmonic potentials for the
order parameter fluctuations. Intensive experimental and theoretical efforts
have been devoted to the study of SPS crystals [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16]. Among the theories, quantum lattice model (QLM) is of
special interest for the SPS investigations [17, 18, 19, 20, 21, 22, 23, 24,
25]. It is a lattice system with three-well local potentials related to Blume-
Emery-Griffiths model [26]. Based on the QLM, the thermodynamics of the
deformed SPS crystals has recently been studied in the presence of external
pressure by Velychko and Stasyuk [24]. For the monoclinic SPS-FEs, they
obtained a set of mean-field self-consistent equations which describe stable
states at a given mechanical stress and determined self-consistency parameters
using the experimental data. From the numerical solutions, the first- and
second-order phase transitions between the ferroelectric (F) and paraelectric
(P) phases were obtained as well as the tricritical point (TCP).

Although a few investigations are devoted to the statics of the QLM-based SPS
crystals up till now, there has been no study to observe the relaxation
dynamics of the system. In this work, we have made use of a simple
phenomenological way to define the slow and fast relaxation times near the
ferroelectric phase transitions in the SPS crystals. Such a study of
relaxation dynamics has already been presented to observe the time dependent
behaviours near the phase transitions and critical phenomena in various
systems [27, 28, 29, 30, 31, 32, 33, 34, 35].

In this study, giving a brief description of the QLM with local multi-well
potentials and its phase transitions for Sn2P2S6 crystals under the MFA and
being motivated by [31, 32, 33, 34, 35], we have focused on the simple
relaxation study of the system using phenomenological approach. Particularly,
we observe the dynamic effects appearing in the regions of first- and second-
order phase transitions as well as in the vicinity of the tricritical point.
The large positive values of relaxation time near the F-P phase transitions
once again confirm the divergence singularity at the continuous phase
transition observed in other cooperative phenomena.

##  2 The model description and static properties

The quantum lattice model with multi-well potentials is simply described by
the Hamiltonian [24]

$\widehat{H}=\sum_{i}\widehat{H}_{i}+\widehat{H}_{1}+\widehat{H}_{2},$ (2.1)

where the first, second and third terms are known as the single-site, the
interaction and the deformation energy parts, respectively. Since these terms
are explicitly given in [24] and [25], we skip the details here and proceed
with mentioning the properties of the system at equilibrium. These are
determined self-consistently using Gibbs free energy calculations. Letting,
$N,J,\eta,\nu,c_{0},u,\theta,\varepsilon,D$ and $\sigma_{S}$ be the number of
lattice points, the effective field acting on dipoles, dipole ordering
parameter, volume related with one formula unit, volume elastic constant,
deformation (or relative volume change), reduced temperature, the
renormalization of the energy gap due to the deformation, the constant of an
electron-deformational interaction and mechanical stress, respectively, the
mean-field Gibbs free energy per site is given by the following formula [24]

$f_{\text{MF}}=\frac{G_{\text{MF}}}{N}=\frac{1}{2}J\eta^{2}+\frac{1}{2}\nu
c_{0}u^{2}-\theta\ln\left[1+2\exp\left(-\frac{\varepsilon+Du}{\theta}\right)\cosh\left(\frac{J\eta}{2\theta}\right)\right]-\nu
u\sigma_{s}.$ (2.2)

Here, we define $J=\sum_{j}J_{ij},\eta=\langle s_{i}\rangle$ ($s_{i}$ variable
related to the local dipole moment, $\langle...\rangle$ denotes the thermal
expectation value), $\theta=kT$ ($k$ denotes the Boltzmann constant, $T$ means
the absolute temperature), $\sigma_{S}=-p$ ($p$ is the hydrostatic pressure).
The minimization of equation (2.2) with respect to the variables $\eta$ and
$u$ are treated as two independent variational parameters and using (2.2) in
the MFT give the following conditions

$\frac{\partial f_{\text{MF}}}{\partial\eta}=0,\quad\frac{\partial
f_{\text{MF}}}{\partial u}=0.$ (2.3)

For the system at equilibrium, the self-consistent equations are expressed in
the form

$\eta=\frac{\mathrm{e}^{-y}z}{1+2\mathrm{e}^{-y}x},$ (2.4)

$\sigma=uc_{0}+\frac{D}{\nu}\left(\frac{2\mathrm{e}^{-y}x}{1+2\mathrm{e}^{-y}x}\right),$
(2.5)

where $x=\cosh\left(\frac{J\eta}{2\theta}\right)$,
$y=\frac{\varepsilon+Du}{\theta}$,
$z=\sinh\left(\frac{J\eta}{2\theta}\right)$. In order to have an insight into
the phase transitions undergoing in the SPS-FEs on the monoclinic lattice
structure, Velychko and Stasyuk [24] firstly solved the above equations using
the parameter values of $J=0.14$ eV, $\nu=0.23\times 10^{-21}$ cm3,
$c_{0}=5\times 10^{11}$ erg/cm3, $\nu c_{0}=71.8$ eV, $D=-1.1$ eV. They
obtained the polarization $\eta$ and deformation $u$ as functions of the
energy gap $\varepsilon$ and pressure $p$ at different temperature values
$\theta$. Recently, Erdem [25] improved their results by adding $\eta$ vs.
$\theta$ and $u$ vs. $\theta$ at a fixed pressure to represent the effect of
temperature. Here, we summarize the basic results from these references for
the convenience of our later discussions as follows: a finite jump of $\eta$
and $u$ at the first-order (discontinuous) phase transition from F phase to P
phase accompanied by compression of the lattice occured. These jumps reduce as
the TCP is approached. Above this point the phase transition becomes of the
second-order (continuous), and thus the jumps of the parameters $\eta$ and $u$
vanish.

##  3 Theoretical framework

In order to study the relaxation dynamics of the above system we now assume
that a small external stimulation is applied removing the system slightly from
equilibrium. The Gibbs free energy produced in the irreversible process
($\Delta f_{\text{MF}}$) is calculated to observe how rapidly the system
relaxes back to equilibrium. In this case, the Gibbs free energy near the
equilibrium states will be

$f_{\text{MF}}(T,\eta,u)=f^{(0)}_{\text{MF}}(T,\eta_{0},u_{0})+\Delta
f_{\text{MF}},$ (3.1)

where $f^{(0)}_{\text{MF}}(T,\eta_{0},u_{0})$ is the equilibrium free energy
[found from (2.2) by setting $\eta=\eta_{0},u=u_{0}$]. $\Delta f_{\text{MF}}$
is given by the terms in a Taylor series expansion of $f_{\text{MF}}$ with
respect to the spontaneous equilibrium point $\eta=\eta_{0},u=u_{0}$ as

$\Delta
f_{\text{MF}}=\frac{1}{2}\left[\phi_{\eta\eta}\left(\eta-\eta_{0}\right)^{2}+2\phi_{\eta
u}\left(\eta-\eta_{0}\right)\left(u-u_{0}\right)+\phi_{uu}\left(u-u_{0}\right)^{2}\right].$
(3.2)

From (3.2), the expressions given below for $\phi_{\eta\eta},\phi_{\eta
u},\phi_{uu}$ are explicit functions of the known equilibrium quantities
$\eta_{0},u_{0}$ and take the form

$\phi_{\eta\eta}=\left(\frac{\partial^{2}f_{\text{MF}}}{\partial\eta^{2}}\right)_{\text{eq}},\quad\phi_{\eta
u}=\left(\frac{\partial^{2}f_{\text{MF}}}{\partial\eta\partial
u}\right)_{\text{eq}}=\phi_{u\eta}=\left(\frac{\partial^{2}f_{\text{MF}}}{\partial
u\partial\eta}\right)_{\text{eq}},\quad\phi_{uu}=\left(\frac{\partial^{2}f_{\text{MF}}}{\partial
u^{2}}\right)_{\text{eq}}.\quad$ (3.3)

Here, the subscribe ‘eq’ means equilibrium. The generalized forces
($X_{\eta},X_{u}$) can be described using the derivatives with respect to
deviations ($\eta-\eta_{0},u-u_{0}$), respectively:

$X_{\eta}=-\frac{\partial(\Delta
f_{\text{MF}})}{\partial(\eta-\eta_{0})}=-\phi_{\eta\eta}\left(\eta-\eta_{0}\right)-\phi_{\eta
u}\left(u-u_{0}\right),$ (3.4)

$X_{u}=-\frac{\partial(\Delta
f_{\text{MF}})}{\partial(u-u_{0})}=-\phi_{u\eta}\left(\eta-\eta_{0}\right)-\phi_{uu}\left(u-u_{0}\right).$
(3.5)

Based on the Onsager reciprocity theorem (ORT) [36, 37], a linear relation
between the currents ($\dot{\eta},\dot{u}$) and forces ($X_{\eta},X_{u}$) is
introduced as follows in terms of a matrix of phenomenological rate
coefficients (or Onsager constants):

$\displaystyle\begin{bmatrix}\dot{\eta}\\\ \dot{u}\end{bmatrix}$
$\displaystyle=-$ $\displaystyle\begin{bmatrix}\gamma_{\eta}&\gamma\\\
\gamma&\gamma_{u}\end{bmatrix}$ $\displaystyle\begin{bmatrix}X_{\eta}\\\
X_{u}\end{bmatrix},$ (3.6)

where the corresponding off-diagonal elements on the two sides of the main
diagonal have the same signs (the matrix is symmetric since both $\eta$ and
$u$ are even variables under time inversion), and the dot denotes a derivative
with respect to time $t$. The above matrix equation yields, upon using
equations (3.4) and (3.5), the rate equations:

$\dot{\eta}=\frac{\mathrm{d}\eta}{\mathrm{d}t}=-\Phi_{\eta\eta}\left(\eta-\eta_{0}\right)-\Phi_{\eta
u}\left(u-u_{0}\right),$ (3.7)

$\dot{u}=\frac{\mathrm{d}u}{\mathrm{d}t}=-\Phi_{u\eta}\left(\eta-\eta_{0}\right)-\Phi_{uu}\left(u-u_{0}\right).$
(3.8)

The coefficients are defined by:

$\Phi_{\eta\eta}=\gamma_{\eta}\phi_{\eta\eta}+\gamma\phi_{\eta
u},\quad\Phi_{\eta u}=\gamma_{\eta}\phi_{\eta
u}+\gamma\phi_{uu},\quad\Phi_{u\eta}=\gamma\phi_{\eta\eta}+\gamma_{u}\phi_{\eta
u},\quad\Phi_{uu}=\gamma_{u}\phi_{uu}+\gamma\phi_{\eta u}.$ (3.9)

Assuming a solution of the form $\exp(-t/\tau)$ for equations (3.7) and (3.8),
one obtains the secular equation

$\displaystyle\begin{vmatrix}\tau^{-1}-\Phi_{\eta\eta}&-\Phi_{\eta u}\\\
-\Phi_{u\eta}&\tau^{-1}-\Phi_{uu}\end{vmatrix}=0,$ (3.10)

which yields two inverse relaxation times [38]:

$\frac{1}{\tau_{S}}=\frac{1}{2}(\Phi_{\eta\eta}+\Phi_{uu})-\frac{1}{2}\big{[}(\Phi_{\eta\eta}-\Phi_{uu})^{2}+4\Phi_{\eta
u}\Phi_{u\eta}\big{]}^{1/2},$ (3.11)
$\frac{1}{\tau_{F}}=\frac{1}{2}(\Phi_{\eta\eta}+\Phi_{uu})+\frac{1}{2}\big{[}(\Phi_{\eta\eta}-\Phi_{uu})^{2}+4\Phi_{\eta
u}\Phi_{u\eta}\big{]}^{1/2}.$ (3.12)

Here, $\tau_{S}$ corresponds to a slower relaxation process while $\tau_{F}$
corresponds to the faster one. They characterize the dipole ordering parameter
and deformation (or relative volume change) relaxations, respectively, in the
phenomenological approach using the ORT.

##  4 Numerical results

Figure 1: (Colour online) (a), (b) Slow relaxation time $\tau_{S}$ and (c),
(d) fast relaxation time $\tau_{F}$ vs. energy gap $\varepsilon$ at
$\theta=0.0230,0.0179,0.0151$ eV for $\gamma_{\eta}=\gamma_{u}=1$ eV-1s-1,
$\gamma=10^{-5}$ eV-1s-1.

Firstly, the behavior of slow ($\tau_{S}$) and fast ($\tau_{F}$) relaxation
times [in seconds ($s$) unit] as a function of the energy gap $\varepsilon$
(in electronvolts (eV) unit) is shown in figure 1 for several values of
$\theta$ when there is no external pressure $p=0$. For Onsager rate
coefficients, we choose $\gamma_{\eta}=\gamma_{u}=1$ eV-1s-1, $\gamma=10^{-5}$
eV-1s-1. The temperature values on the figures are for the second-order phase
transition, the first-order phase transition and TCP. The vertical dotted
lines refer to the phase transition values of $\varepsilon$, i.e.,
$\varepsilon_{C}$, $\varepsilon_{D}$ and $\varepsilon_{\text{TCP}}$. In this
case, $\tau_{S}$ increases rapidly while rising (lowering) $\varepsilon$ value
on left (right) side of $\varepsilon_{C}/\varepsilon_{\text{TCP}}$ and
diverges to infinity, illustrated by the blue/red curves in figure 1(a). On
the contrary, it presents a large and abrupt drop across the discontinuous
phase transition point [see figure 1(b)]. Former singularity in $\tau_{S}$ is
an expected property known as the signature of continuous phase transitions
and hence well agrees with earlier investigations [31, 32]. However, the
latter one is a novel property which is not found before. As for the fast
relaxation time $\tau_{F}$, it also increases with $\varepsilon$ but remains
scarcerly varied just around the continuous phase transition (tricritical
point), as also shown via blue (red) curves in figure 1(c). Hence, both
critical and tricritical behaviours of $\tau_{F}$ are different from
$\tau_{S}$, just a cusp singularity. Beyond $\varepsilon_{C}$, a maximum of
$\tau_{F}$ is seen but it disappears at $\varepsilon_{\text{TCP}}$. It should
be stressed that cusps also occurred for $\tau_{F}$ during the discontinuous
phase transition, indicated in figure 1(d). These findings are also in
agreement with previous relaxation studies [32, 35].

Figure 2: (Colour online) (a), (b) Slow relaxation time $\tau_{S}$ and (c),
(d) fast relaxation time $\tau_{F}$ vs. energy gap $\varepsilon$ in the
absence and presence of the off-diagonal rate coefficient when
$p=0,\theta=0.0230$ eV.

Figure 3: (Colour online) (a), (b) Slow relaxation time $\tau_{S}$ and (c),
(d) fast relaxation time $\tau_{F}$ vs. pressure $p$ at
$\theta=0.0225,\;0.0175,\;0.0125\;eV$ for $\gamma_{\eta}=\gamma_{u}=1$
eV-1s-1, $\gamma=10^{-5}$ eV-1s-1.

In order to show the effects of rate constants on the relaxation process, we
illustrate in figure 2 the energy gap $\varepsilon$ variation of $\tau_{S}$
and $\tau_{F}$ in the absence and in the presence of the off-diagonal Onsager
constant for $p=0$ and $\theta=0.0230$ eV. As seen in figure 2(a), on both
sides of $\varepsilon_{C}$, illustrated by a vertical dotted line, the
divergence of $\tau_{S}$ gets pushed away from the critical energy gap value
as we decrease the values of $\gamma_{\eta}$. This means that the rising of
$\gamma_{\eta}$ leads to a speed up of the whole relaxation process. The
opposite is valid in the case of off-diagonal Onsager coefficient $\gamma$ in
figure2(b). $\tau_{F}$ vs. $\varepsilon$ is shown for different values of
$\gamma_{\eta}$ and $\gamma$ for $p=0$ and $\theta=0.0230$ eV in figures 2(c)
and 2(d), respectively. It is evident that $\tau_{F}$ curves are independent
of $\gamma_{\eta}$ and $\gamma$ values, shown in figures 2(c) and 2(d).

Next, $\tau_{S}$ and $\tau_{F}$ vs. pressure $p$ [in gigapascal (GPa) unit] is
displayed in figure 3 for three different temperatures using
$\gamma_{\eta}=\gamma_{u}=1$ eV-1s-1, $\gamma=10^{-5}$ eV-1s-1. Figure3 also
displays the pressure dependence of $\tau_{S}$ and $\tau_{F}$ where we adjust
the parameters so as to obtain an agreement with figures 6(a), 6(b) in [25].
Our calculated dependence of ($\tau_{S},\tau_{F}$) on $p$ demonstrates a good
correspondence to previous $\varepsilon$ dependence of ($\tau_{S},\tau_{F}$)
in figure 1. Our plots of the critical and tricritical cases are presented in
figures 3(a), 3(c) (blue and red curves) and the first-order case in figures
3(b), 3(d) (green curves). In other words, the overall critical and
tricritical behaviours of ($\tau_{S},\tau_{F}$) do not change so much at the
given $\varepsilon$ and $\theta$ values. However, the amount of the finite
jump in $\tau_{S}$ at the first-order phase transition point seen in figure
3(b) is very much smaller than that of figure 1(b). It is important to mention
that, because of the similar behavior of figure 2, the pressure $p$ variation
of $\tau_{S}$ and $\tau_{F}$ in the absence and in the presence of $\gamma$
for $p=0$ and $\theta=0.0230$ eV has not been drawn here.

Figure 4: (Colour online) (a), (b) Slow relaxation time $\tau_{S}$ and (c),
(d) fast relaxation time $\tau_{F}$ vs. temperature $\theta$ in the absence
and in the presence of the off-diagonal Onsager coefficient for
$\varepsilon=-0.011$ eV, $p=0.5$ GPa.

Finally, the dependence of $\tau_{S}$ and $\tau_{F}$ on the temperature
$\theta$ can be followed in figure 4. These quantities are plotted for both
$\gamma=0$ and $\gamma\neq 0$ characterizing the speed of the relaxation
process when $\varepsilon=-0.011$ eV, $p=0.5$ GPa. It is clearly visible that
the increases of $\gamma_{\eta}(\gamma$) result in much faster (slower)
increase of $\tau_{S}$ around the critical temperature during the critical
slowing down process [figures 4(a) and 4(b)]. These are in direct agreement
with previous results gained from dynamic studies for similar physical systems
[31, 32, 33, 34, 35]. It is quite evident from figures 4(c) and 4(d) that the
investigated plots of $\tau_{F}$ display a cusp-singularity at the critical
temperature regardless of the value of rate parameters. By comparison of the
colored curves in figures 4(c) and 4(d), it can be stated that $\tau_{F}$ is
not sensitive to any change in $\gamma_{\eta}$ when $\gamma=0$ and in $\gamma$
for $\gamma\neq 0$.

##  5 Conclusions

In this work, we have studied the near equilibrium relaxation dynamics of the
QLM for the ferroelectric SPS crystals by means of ORT. More particularly,
time derivatives of polarization $\eta$ and volume deformation $u$ are treated
as fluxes or currents conjugate to their appropriate forces. The forces are
found using the mean-field free energy production. A set of rate (or kinetic)
equations are derived from the linear relation between the forces and
currents. In terms of the phenomenological constants, the solutions of rate
equations are expressed as a set of two relaxations times denoted by
$\tau_{S}$ and $\tau_{F}$. Using the parameter values related to the SPS
crystals in [24, 25], we have plotted these two quantities as a function of
the energy gap, external pressure and temperature. Although the present
calculations on $\tau_{S}$ and $\tau_{F}$ display exactly the same critical
and tricritical behaviors as in the earlier investigations, we observe unusual
results during the discontinous phase transition. In other words, the slow
relaxation time $\tau_{S}$ shows a large jump in the $\tau_{S}$ vs.
$\varepsilon$ and $\tau_{S}$ vs. $p$ plots while fast relaxation time
$\tau_{F}$ has a cusp-singularity and small jump in the $\tau_{F}$ vs.
$\varepsilon$ and $\tau_{F}$ vs. $p$ plots, respectively, at the first-order
phase transition. In particular, comparing figures 1(b) and 1(d) with
corresponding plots in [32] it can be stated that this qualitative difference
distinguishes our results from the earlier ORT calculations.

## References

  * [1] Eijt S. W. H., Currat R., Lorenzo J. E., Saint-Grégoire P., Katano S., Janssen T., Hennion B., Vysochanskii Y. M., J. Phys.: Condens. Matter, 1998, 10, 4811, doi:10.1088/0953-8984/10/22/008.
  * [2] Eijt S. W. H., Currat R., Lorenzo J. E., Saint-Grégoire P., Hennion B., Vysochanskii Y. M., Eur. Phys. J. B, 1998, 5, 169, doi:10.1007/s100510050431
  * [3] Gasgnier M., Szwarc H., Petit A., Mater. Res. Bull., 2003, 38, 1681, doi:10.1016/j.materresbull.2003.07.004.
  * [4] Haertle D., Caimi G., Haldi A., Montemezzani G., Günter P., Grabar A. A., Stoika I. M., Vysochanskii Y. M., Opt. Commun., 2003, 215, 333, doi:10.1016/S0030-4018(02)02251-4.
  * [5] Rushchanskii K. Z., Vysochanskii Y. M., Strauch D., Phys. Rev. Lett., 2007, 99, 207601,
doi:10.1103/PhysRevLett.99.207601.

  * [6] Yevych R. M., Vysochanskii Y. M., Condens. Matter Phys., 2008, 11, 417, doi:10.5488/CMP.11.3.417.
  * [7] Yevych R. M., Vysochanskii Y. M., Ferroelectrics, 2011, 412, 38, doi:10.1080/00150193.2011.542693.
  * [8] Glukhov K., Fedyo K., Banys J., Vysochanskii Y., Int. J. Mol. Sci., 2012, 13, 14356, doi:10.3390/ijms131114356.
  * [9] Ondrejkovic P., Guennou M., Kempa M., Vysochanskii Y., Garbarino G., Hlinka J., J. Phys.: Condens. Matter, 2013, 25, 115901, doi:10.1088/0953-8984/25/11/115901.
  * [10] Bilanych R. M., Yevych R. M., Kohutych A. A., Perechinskii S. I., Stoika I. M., Vysochanskii Y. M., Cent. Eur. J. Phys., 2014, 12, 611, doi:10.2478/s11534-014-0514-3.
  * [11] Rushchanskii K. Z., Bilanych R. M., Molnar A. A., Yevych R. M., Kohutych A. A., Perechinskii S. I., Samulionis V., Banys J., Vysochanskii Y. M., Phys. Status Solidi B, 2016, 253, 384, doi:10.1002/pssb.201552138.
  * [12] Rushchanskii K. Z., Molnar A., Bilanych R., Yevych R., Kohutych A., Vysochanskii Y. M., Samulionis V., Banys J., Phys. Rev. B, 2016, 93, 014101, doi:10.1103/PhysRevB.93.014101.
  * [13] Yevych R., Haborets V., Medulych M., Molnar A., Kohutych A., Dziaugys A., Banys J., Low Temp. Phys., 2016, 42, 1155, doi:10.1063/1.4973005.
  * [14] Oleaga A., Shvalya V., Salazar A., Stoika I., Vysochanskii Y. M., J. Alloys Compd., 2017, 694, 808, doi:10.1016/j.jallcom.2016.10.071.
  * [15] Yevych R., Medulych M., Condens. Matter Phys., 2018, 21, 23001, doi:10.5488/CMP.21.23001.
  * [16] Liubachko V., Oleaga A., Salazar A., Yevych R., Kohutych A., Vysochanskii Y., Phys. Rev. B, 2020, 101, 224110, doi:10.1103/PhysRevB.101.224110.
  * [17] Aoki T., J. Phys. Soc. Jpn., 1992, 61, 750, doi:10.1143/JPSJ.61.750.
  * [18] Yonemitsu K., Phys. Rev. B, 2008, 78, 205102, doi:10.1103/PhysRevB.78.205102.
  * [19] Mashiyama H., Mkam Tchouobiap S. E., Miyoshi T., Ferroelectrics, 2009, 378, 163,
doi:10.1080/00150190902848552.

  * [20] Rigol M., Singh R. R. P., Comput. Phys. Commun., 2009, 180, 540, doi:10.1016/j.cpc.2008.11.002.
  * [21] Zamaraite I., Yevych R., Dziaugys A., Molnar A., Banys J., Svirskas S., Vysochanskii Y., Phys. Rev. Appl., 2018, 10, 034017, doi:10.1103/PhysRevApplied.10.034017.
  * [22] Yevych R., Vysochanskii Y., Condens. Matter Phys., 2018, 21, 33001, doi:10.5488/CMP.21.33001.
  * [23] Pang S. Y., Muniandy S. V., Kamali M. Z. M., Int. J. Theor. Phys., 2019, 58, 4139,
doi:10.1007/s10773-019-04279-1.

  * [24] Velychko O. V., Stasyuk I. V., Phase Transitions, 2019, 92, 420, doi:10.1080/01411594.2019.1582051.
  * [25] Erdem R., Physica A, 2020, 556, 124837, doi:10.1016/j.physa.2020.124837.
  * [26] Blume M., Emery V. J., Griffiths R. B., Phys. Rev. A, 1971, 4, 1071, doi:10.1103/PhysRevA.4.1071.
  * [27] Tanaka T., Meijer P. H. E., Barry J. H., J. Chem. Phys., 1962, 37, 1397, doi:10.1063/1.1733295.
  * [28] Barry J. H., J. Chem. Phys., 1966, 45, 4172, doi:10.1063/1.1727475.
  * [29] Barry J. H., Harrington D. A., Phys. Rev. B, 1971, 4, 3068, doi:10.1103/PhysRevB.4.3068.
  * [30] Tanaka M., Takahashi K., J. Phys. Soc. Jpn., 1977, 43, 1832, doi:https://doi.org/10.1143/JPSJ.43.1832.
  * [31] Erdem R., Keskin M., Phys. Status Solidi B, 2001, 225, 145,
doi:10.1002/(SICI)1521-3951(200105)225:1<145::AID-PSSB145>3.0.CO;2-X.

  * [32] Erdem R., Keskin M., Phys. Rev. E, 2001, 64, 026102, doi:10.1103/PhysRevE.64.026102.
  * [33] Keskin M., Erdem R., Phys. Lett. A, 2002, 297, 427, doi:10.1016/s0375-9601(02)00264-5.
  * [34] Erdem R., Ekiz C., Keskin M., Phys. Status Solidi B, 2003, 240, 220, doi:10.1002/pssb.200301876.
  * [35] Gülpınar G., Demirhan D., Büyükkılıç F., Phys. Rev. E, 2007, 75, 021104, doi:10.1103/PhysRevE.75.021104.
  * [36] Onsager L., Phys. Rev., 1931, 37, 405, doi:10.1103/PhysRev.37.405.
  * [37] Onsager L., Phys. Rev., 1931, 38, 2265, doi:10.1103/PhysRev.38.2265.
  * [38] Pawlak A., Erdem R., Gülpınar G., J. Magn. Magn. Mater., 2019, 472, 86, doi:10.1016/j.jmmm.2018.09.115.

Ukrainian 0

àñè ïîâëüíî¿ òà øâèäêî¿ ðåëàêñàö¿ êâàíòîâî¿ ðàòêîâî¿ ìîäåë ç ëîêàëüíèìè
áàãàòîìíìóìíèìè ïîòåíöàëàìè: ôåíîìåíîëîãíà äèíàìêà â ñåãíåòîåëåêòðèíèõ
êðèñòàëàõ Sn2P2S6 [Ð. Åðäåì, Ñ. Îçþì, Í. Ãþäæëþ]Ð.
Åðäåì${}^{\textsf{\footnotesize{\mbox{?}}}}$, Ñ.
Îçþì${}^{\textsf{\footnotesize{\mbox{?}}}}$, Í.
Ãþäæëþð${}^{\textsf{\footnotesize{\mbox{?}}}}$

${}^{\textsf{\footnotesize 1}}$ Ôçèíèé ôàêóëüòåò, Óíâåðñèòåò Àêäåíç, 07058,
Àíòàëÿ, Òóðåèíà ${}^{\textsf{\footnotesize 2}}$ Ïðîôåñéíà øêîëà Àëàñà Àâí
Äæåëê, Óíâåðñèòåò Ãòò, 19600, îðóì, Òóðåèíà ${}^{\textsf{\footnotesize 3}}$
Ôàêóëüòåò ôçèíî¿ îñâòè, Óíâåðñèòåò Íåäæìåòòèíà Åðáàêàíà, 42090, Êîíüÿ, Òóðåèíà