Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field
We calculated dynamic dielectric permittivity of the Rochelle salt within the Mitsui model, extended by piezoelectric interaction and transverse field. Calculations were based on the parameters derived earlier within the study of thermodynamic properties of Rochelle salt. The study of dynamic proper...
Saved in:
| Published in: | Condensed Matter Physics |
|---|---|
| Date: | 2010 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізики конденсованих систем НАН України
2010
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/32049 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field / R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13705: 1-16. — Бібліогр.: 38 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859637872249798656 |
|---|---|
| author | Levitskii, R.R. Andrusyk, A.Ya. Zachek, I.R. |
| author_facet | Levitskii, R.R. Andrusyk, A.Ya. Zachek, I.R. |
| citation_txt | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field / R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13705: 1-16. — Бібліогр.: 38 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | We calculated dynamic dielectric permittivity of the Rochelle salt within the Mitsui model, extended by piezoelectric interaction and transverse field. Calculations were based on the parameters derived earlier within the study of thermodynamic properties of Rochelle salt. The study of dynamic properties was performed within the Bloch equation method. We showed that taking transverse field into account allows for better description of the Rochelle salt relaxation dynamics. Furthermore, we showed that taking transverse field into account results in the appearance of a resonant component in dynamic permittivity like it is observed in experiment. However, in accordance with the calculations, resonant response reveals itself within infrared frequency range, whereas in experiment it is observed within submillimeter spectral region.
|
| first_indexed | 2025-12-07T13:18:07Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13705: 1–16
http://www.icmp.lviv.ua/journal
Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal
studied within the Mitsui model extended by
piezoelectric interaction and transverse field
R.R. Levitskii1, A.Ya. Andrusyk1, I.R. Zachek2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitsky Str., 79011 Lviv, Ukraine
2 Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Received February 10, 2010
We calculated dynamic dielectric permittivity of the Rochelle salt within the Mitsui model, extended by piezo-
electric interaction and transverse field. Calculations were based on the parameters derived earlier within the
study of thermodynamic properties of Rochelle salt. The study of dynamic properties was performed within
the Bloch equation method. We showed that taking transverse field into account allows for better description
of the Rochelle salt relaxation dynamics. Furthermore, we showed that taking transverse field into account
results in the appearance of a resonant component in dynamic permittivity like it is observed in experiment.
However, in accordance with the calculations, resonant response reveals itself within infrared frequency range,
whereas in experiment it is observed within submillimeter spectral region.
Key words: Rochelle salt, molecular field approximation, dynamic dielectric permittivity, relaxation dynamics,
resonant dynamics
PACS: 77.22.Ch, 77.80.Bh, 77.84.Fa
1. Introduction
Ferroelectric crystals of the order-disorder type with asymmetric double-well potential has
a number of unusual properties. A good example of such crystal is sodium potassium tartrate
tetrahydrate NaKC4H4O6·4H2O (Rochelle salt or Rs). The presence of two Curie points and the
existence of the ferroelectric phase within a narrow temperature range (between Tc1 = 255 K and
Tc2 = 297 K) is a prominent characteristic of this crystal. Both phase transitions are of the second
order [1].
Structural analysis occupies an important place in the study of the Rochelle salt. These studies
showed that crystalline structure of Rs proved to be complex. It is orthorhombic (space group
D3
2−P212121) in the paraelectric phases and monoclinic (space group C2
2 −P21) in the ferroelectric
phase [2]. Spontaneous polarization is directed along the a crystal axis; it is accompanied by a
spontaneous shear strain ε4. There are four formula units (112 atoms) in the unit cell of the
Rochelle salt. The unit cell is divided into four non-equivalent structural elements, each of which
produces a dipole moment both along ferroelectric x- and along y-axes. Consequently, a crystal
lattice is divided into four sublattices. Studies based on x-ray diffraction data [3] argued that these
were the order-disorder motions of O9 and O10 groups, coupled with the displacive vibrations of
O8 groups, which were responsible for the phase transitions in the Rochelle salt as well as for the
spontaneous polarization.
Inspite of the achievements in determining the of Rochelle salt structure, the arrangement
of hydrogen atoms in the crystal and their role in phase transition still remain unclear. Their
important role is testified by significant hydrogen/deuterium isotope effect. Such situation makes
ab-initio calculations impossible and requires the application of alternative approaches.
We study Rochelle salt properties within the semi-microscopic approach, which is based on
the assumption that phase transition in Rochelle salt is of order-disorder type. This assumption
c© R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek 13705-1
http://www.icmp.lviv.ua/journal
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
is supported by a system dielectric response of relaxation type, which was derived for microwave
region [4–6]. Besides that, neutron diffraction experimental data [7] support the presence of two
possible ionic configurations. The dielectric [8], spectroscopic (Raman spectrum) [9] and struc-
tural (neutron scattering) [10] studies, carried out in submillimeter range, indicate the presence of
resonant response, which is over-damped once one approaches the point of phase transition from
low-temperature paraelectric phase, and this is the evidence of phase transition of displacive type.
On the other hand, this resonant response may be caused by dynamic flipping of structural
elements between two equilibrium positions. One may expect that the introduction of additional
term like transverse field, responsible for dynamic flipping of structural elements, into a model
Hamiltonian will permit to make the description of this resonant response. Thus, a simple model
like the de Gennes one, can be applicable to the description of Rochelle salt properties.
Mitsui proposed a model of the order-disorder type adequate to Rochelle salt [11]. This model
is based on the assumption that structural elements move in asymmetric double well potential
and hence have two equilibrium positions. Occupation of one of the equilibrium states results in
emergence of a dipole moment. Each cell contains two dipoles. The dipoles form two co-penetrating
sublattices with local potentials, which are the mirror reflections of each other. Therefore, even
though dipoles in each sublattice are ordered (sublattice polarization is available), the total polar-
ization at certain temperatures may be equal to zero.
The study of the Mitsui model, formulated in terms of pseudospin operators [5, 12], reveals
several types of temperature behavior of the model. Depending on the values of model param-
eters, it can undergo, for instance, a single second order phase transition from ferroelectric into
paraelectric phase (observed in RbHSO4), two second order phase transitions (Rs, dRs), one low
temperature first order and one high temperature second order phase transition1 (NH4HSO4), etc.
The spontaneous polarization, dielectric permittivity and other thermodynamic characteristics of
the crystals mentioned above were calculated and studied within the Mitsui model in the molecular
field approximation [12–15].
The relaxation dynamics of the Rs, dRs and RbHSO4 was explored within the Mitsui model
too [12, 15–19], where the temperature dependence of relaxation times and dynamic permittivity
of these crystals were calculated. Concerning Rochelle salt, it was obtained that relaxation time,
exhibiting a critical slowing down at the Curie points, actually diverges at these points. However,
experiments [5] indicate that the values of relaxation time are large but remain finite. Moreover, the
theoretically calculated and experimental static dielectric permittivities are essentially different:
the former diverges at the phase transition points whereas the latter remains finite. Naturally, such
inconsistency stems either from the model incompletion or from the defect of the approximation
used. Later the Mitsui model was extended by means of the accounting tunnelling effects [12,
16]. Also the Mitsui model was studied within two-particle cluster approximation for the short-
range interactions [20]. However, these improvements failed to provide for correct temperature
dependence of relaxation time at the phase transition points.
The problem of incorrect temperature dependence of relaxation times and dynamic permit-
tivity for Rs crystal at the Curie points was successfully solved by taking into account piezoelectric
interaction of pseudospin subsystem with lattice deformation [21]. Finite relaxation time at the
phase transition points was obtained within the extended Mitsui model. This approach allows to
experimentally explain the detected difference of the static permittivities of clamped and free crys-
tals. Generally, good agreement of theory and experiment for dielectric permittivities of clamped
and free crystals was derived [21]. The unique theory parameter set, providing acceptable agree-
ment of theory and experiment for numerous physical characteristics of the Rochelle salt: elastic,
longitudinal dielectric (static and dynamic) and piezoelectric, was first derived in this work.
Recent works deal with new theoretical consideration of the effect produced by external factors
like field [22], hydrostatic [23] and uniaxial σ4 [24] pressures on physical properties of Rs. The effect
of partial deuteration on Rochelle salt was studied within the extended Mitsui model [25] as well as
piezoelectric resonance in Rs [26]. Taking into consideration a real structure of Rochelle salt crystal,
the four-sublattice Mitsui model instead of the two-sublattice Mitsui model was studied [27, 28].
1Here the additional assumption that the model parameters slightly depend on temperature is needed.
13705-2
Dynamics of the Rochelle salt
This model explains transverse dielectric properties of Rochelle salt. Also, experiments [29] were
carried out for the effect of humidity, annealing, stresses, electric field on dielectric permittivity of
Rochelle salt. Modern experimental techniques, applied therein, updated the experimental data,
formerly derived for Rs crystal.
Notwithstanding the significant advances in studying the Rs crystal, some problems in theoreti-
cally explaining the physical properties of Rochelle salt still remain unsolved. Thus, the theoretically
calculated value of spontaneous polarization appeared to be significantly lower than the experi-
mental one along the whole temperature range. Among other things, it has been found that the
theoretically calculated real part of dielectric permittivity in low-temperature paraelectric phase
is considerably higher than the experimental one. Also, as it was mentioned above, the dielec-
tric response is of resonant type in submillimeter range, and the Mitsui model with piezoelectric
interaction is not capable of explaining this fact.
The first problem was successfully solved by taking into account the possibility of dynamic
flipping of structural elements between two equilibrium positions [30]. For this purpose Mitsui
Hamiltonian with piezoelectric interaction was supplemented with the term of transverse field type.
The set of theory model parameters was obtained and on the base thereof some improvement in
the description of static dielectric, piezoelectric, and elastic properties was achieved. Besides that,
we managed to gradually improve the accordance between theory and experiment for temperature
dependence of spontaneous polarization.
In present work we tried to solve two other problems. We carried out our research within Mitsui
model with the same consideration of piezoelectric interaction and transverse field. The research was
based on the theory parameters derived under consideration of thermodynamics [30]. Besides, we
compared our results with the same ones, derived within Mitsui model without transverse field [21],
and found out the effect of the transverse field on the dynamic properties of the Rochelle salt.
2. Dynamic physical characteristics of the Mitsui model with piezoelectric
interaction and transverse field
We give consideration of the ferroelectric order-disorder type with an asymmetric double-well
potential. Hamiltonian of such a system is referred to as the Mitsui Hamiltonian. We assume that
this system has an essential piezoelectric interaction which should be taken into account. Besides,
the model requires that the transverse field should be taken into account (this term describes the
possibility of dynamic ordering units flipping between two equilibrium positions). We suppose the
polarization is directed along x-axes and arises due to the structural units ordering in one of the
two possible equilibrium positions. We consider the situation when ε4 component of strain tensor
effects the energy of these equilibrium positions. Precisely this case occurs in Rs. The resulting
Hamiltonian is of the following form:
H =
∑
q
[
1
2
vcE0
44 ε4
2 − ve014E1qε4 −
1
2
vχε0
11E
2
1q
]
−
∑
q,q′
[
Jqq′
2
(Sz
q1S
z
q′1 + Sz
q2S
z
q′2) +Kqq′Sz
q1S
z
q′2
]
−
∑
qf
[
ΩSx
qf + (∆f − 2ψ4ε4 + µE1q)S
z
qf
]
. (1)
Three terms in the first sum of equation (1) represent the elastic, piezoelectric, and electric
energies attributed to a host lattice, in whose potential the pseudospin moves (with the “seed”
elastic constant cE0
44 , the coefficient of piezoelectric stress e014, and dielectric susceptibility χε0
11); v is
the volume of a cell, containing a pair of pseudospins (ordering units or dipoles) of one lattice site
q and different sublattices f = 1, 2 (further we will call it a unit cell2). The second sum describes
a direct interaction of the ordering units: Jqq′ = Jq′q and Kqq′ = Kq′q are interaction potentials
between pseudospins belonging to the same and to different sublattices, respectively. The first term
in the third sum is the transverse field; the second term describes a) the energy, associated with
2Actual unit cell of the Rochelle salt crystal contains two pairs of pseudospins of two lattice sites and different
sublattices; therefore, we should set the value of the model unit cell volume to be half of the crystal unit cell volume.
13705-3
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
asymmetry of the potential, where ∆f is asymmetry parameter: ∆1 = −∆2 = ∆, b) interaction
energy of pseudospin with the field, arising due to the piezoelectric deformation and ψ4 is the
parameter of piezoelectric interaction, c) interaction energy of pseudospin with external electric
field E1q , where µ is the effective dipole moment of the model unit cell.
We carry out the study of the system that is described by Hamiltonian (1) within the Bloch
equation method. For this purpose we first of all should derive the Hamiltonian within the mean
field approximation (MFA).
Performing identical transformation
Sz
qf =
〈
Sz
qf
〉
+ ∆Sz
qf (2)
and neglecting the quadratic fluctuations, we derive the initial Hamiltonian (1) within MFA:
HMFA =
∑
q
[
1
2
vcE0
44 ε4
2 − ve014E1qε4 −
1
2
vχε0
11E
2
1q
]
+
∑
qq′
[
1
2
Jqq′
(〈
Sz
q1
〉 〈
Sz
q′1
〉
+
〈
Sz
q2
〉 〈
Sz
q′2
〉)
+Kqq′
〈
Sz
q1
〉 〈
Sz
q′2
〉
]
−
∑
qf
Hqf Sqf , (3)
where Hqf are the mean local fields producing the effect on the pseudospins Sqf :
Hx
qf = Ω, Hy
qf = 0, Hz
qf = εqf ,
εq1 =
∑
q′
[
Jqq′
〈
Sz
q′1
〉
+Kqq′
〈
Sz
q′2
〉]
+ ∆ − 2ψ4ε4 + µE1q , (4a)
εq2 =
∑
q′
[
Jqq′
〈
Sz
q′2
〉
+Kqq′
〈
Sz
q′1
〉]
− ∆ − 2ψ4ε4 + µE1q . (4b)
Within MFA we can calculate the mean equilibrium values of the pseudospin operators [30]:
〈Sqf 〉 =
1
2
Hqf
Hqf
tanh
Hqf
2kBT
, (5)
where Hqf ≡ |Hqf | = λqf , and kB is Boltzmann constant.
We study the dynamic properties of the system with Hamiltonian (1) within the Bloch equations
method
~
d〈Sqf 〉t
dt
= 〈Sqf 〉t × Hqf (t) −
~
T1
[
〈Sqf 〉t‖ − 〈Sqf 〉t
]
−
~
T2
〈Sqf 〉t⊥ . (6)
The right hand part of this equation consists of three terms.
The first term is Heisenberg part of the equation of motion, calculated within random phase
approximation (RPA), where “×” denotes the vector product and Hqf (t) are the instantaneous
values of the local fields3:
Hx
qf (t) = Ω, Hy
qf = 0, Hz
qf (t) = εqf (t), (7)
εq1(t) =
∑
q′
[
Jqq′
〈
Sz
q′1
〉
t
+Kqq′
〈
Sz
q′2
〉
t
]
+ ∆ − 2ψ4ε4 + µE1q(t), (8a)
εq2(t) =
∑
q′
[
Jqq′
〈
Sz
q′2
〉
t
+Kqq′
〈
Sz
q′1
〉
t
]
− ∆ − 2ψ4ε4 + µE1q(t). (8b)
3Original Heisenberg part of the equation of motion is −i〈[Sqf , H]〉. Within RPA we calculate the commutator
of the pseudospin operators with MFA-Hamiltonian (3). Doing necessary calculations one can derive that
−i〈[Sqf ,H]〉t = 〈Sqf 〉t × Hqf (t).
13705-4
Dynamics of the Rochelle salt
We restricted our consideration to sufficiently high frequencies (above the piezoelectric resonance),
where the crystal can be regarded to be “clamped” [21]. Mathematically “clamping” means that
deformation ε4 is time independent.
The second and the third terms describe the relaxation of pseudospin towards quasiequilibrium
state which is defined by instantaneous values of molecular fields. In particular, the second term
describes relaxation of the pseudospin component 〈Sqf 〉t‖ (longitudinal to the instantaneous value
of the local field) towards its quasiequilibrium value with a characteristic time T1. The third term
describes the decay process of the transverse component of pseudospin 〈Sqf 〉t⊥ with a characteristic
time T2.
Quasiequilibrium mean values 〈Sqf 〉t are defined as (see equation (5)):
〈Sqf 〉t =
1
2
Hqf (t)
Hqf (t)
tanh
(
1
2
βHqf (t)
)
. (9)
As we are interested in the linear response of the system to the small external electric field
δE1q(t), (E1q(t) = E1 + δE1q(t)) ,
it is sufficient to present 〈Sqf 〉t as a sum of constant term 〈Sqf 〉0 (mean equilibrium value, calcu-
lated in MFA) and time dependent small deviation δ 〈Sqf 〉t:
〈Sqf 〉t = 〈Sqf 〉0 + δ 〈Sqf 〉t .
Similarly:
Hqf (t) = H
(0)
f + δHqf (t), 〈Sqf 〉t = 〈Sqf 〉0 + δ〈Sqf 〉t .
Now, we can linearize the motion equation (6) by retaining the terms, which are linear in deviations
δ 〈Sqf 〉t, δHqf (t), δ〈Sqf 〉t:
~
dδ〈Sqf 〉t
dt
= δ〈Sqf 〉t × H
(0)
f + 〈Sqf 〉0 × δHqf (t)
−
~
T1
[
δ〈Sqf 〉t‖ − δ〈Sqf 〉t‖
]
−
~
T2
[
δ〈Sqf 〉t⊥ − δ〈Sqf 〉t⊥
]
, (10)
where
δHx
qf (t) = 0, δHy
qf (t) = 0,
δHz
q1(t) =
∑
q′
Jqq′δ〈Sz
q′1〉t +
∑
q′
Kqq′δ〈Sz
q′2〉t + µδE1q(t),
δHz
q2(t) =
∑
q′
Kqq′δ〈Sz
q′1〉t +
∑
q′
Jqq′δ〈Sz
q′2〉t + µδE1q(t).
Now we transform the equation (10) into a form involving single variable δ 〈Sqf 〉t, then Fourier
transform to k-space, introduce new variables
δ 〈Sk1〉t =
δξk(t) + δσk(t)
2
, (11a)
δ 〈Sk2〉t =
δξk(t) − δσk(t)
2
, (11b)
and introduce parameters
R̃+
k =
J̃k + K̃k
2
, R̃−
k =
J̃k − K̃k
2
,
J̃k = Jk/kB , K̃k = Kk/kB . (12)
13705-5
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
Upon applying these transformations, the Bloch equation (10) is reduced to the system of linear
differential equations of the following matrix form:
~
kB
dxk(t)
dt
= Akxk(t) − µ̃δE1k(t)b, (13)
where µ̃ = µ/kB ,
Ak =
a11 a12 −Ω̃ 0 a15 a16
a21 a22 0 −Ω̃ a16 a15
a31 a32 − 1
T ′
2
0 a35 a36
a41 a42 0 − 1
T ′
2
a36 a35
a51 a52 −a35 −a36 a55 a56
a61 a62 −a36 −a35 a56 a55
, xk(t) =
δξz
k(t)
δσz
k(t)
δξy
k(t)
δσy
k(t)
δξx
k (t)
δσx
k (t)
, b =
b1
b2
ξx
σx
b5
b6
. (14)
Components of matrix Ak:
a11 = −1/T ′
2 + U1 + R̃+
k (Q1 +G1), a12 = U2 + R̃−
k (Q2 +G2), a15 = V1, a16 = V2,
a21 = U2 + R̃+
k (Q2 +G2), a22 = −1/T ′
2 + U1 + R̃−
k (Q1 +G1),
a31 = Ω̃ − R̃+
k ξ
x, a32 = −R̃−
k σ
x, a35 = −
[
R̃+
0 ξ
z − 2ψ̃4ε4
]
, a36 = −
[
R̃−
0 σ
z + ∆̃
]
,
a41 = −R̃+
k σ
x, a42 = Ω̃ − R̃−
k ξ
x,
a51 = V1 + R̃+
k H1, a52 = V2 + R̃−
k H2, a55 = −1/T ′
2 +W1, a56 = W2,
a61 = V2 + R̃+
k H2, a62 = V1 + R̃−
k H1; (15)
components of vector b:
b1 = −(Q1 +G1), b2 = −(Q2 +G2), b5 = −H1 , b6 = −H2 , (16)
where the following notations are used:
Q1,2 =
1
2T ′
2
(
ξz + σz
ε̃1
±
ξz − σz
ε̃2
)
, U1,2 =
1
2
(
1
T ′
2
−
1
T ′
1
)
(
ε̃21
λ̃2
1
±
ε̃22
λ̃2
2
)
,
V1,2 =
1
2
(
1
T ′
2
−
1
T ′
1
)
(
Ω̃ε̃1
λ̃2
1
±
Ω̃ε̃2
λ̃2
2
)
, W1,2 =
1
2
(
1
T ′
2
−
1
T ′
1
)
(
Ω̃2
λ̃2
1
±
Ω̃2
λ̃2
2
)
,
G1,2 = K1
ε̃21
λ̃2
1
±K2
ε̃22
λ̃2
2
, H1,2 = K1
Ω̃ε̃1
λ̃2
1
±K2
Ω̃ε̃2
λ̃2
2
,
K1,2 =
1
T ′
1
1
4T
(
cosh
λ̃1,2
2T
)2 −
1
2T ′
2
ξz ± σz
ε̃1,2
,
Ω̃ =
Ω
kB
, ∆̃ =
∆
kB
, ψ̃4 =
ψ4
kB
, ε̃1,2 =
ε1,2
kB
, λ̃1,2 =
λ1,2
kB
. (17)
Relaxation times T ′
1 and T ′
2 have the following form:
T ′
1 =
kB
~
T1 , T ′
2 =
kB
~
T2 . (18)
After applying the time Fourier transformation we transform equation (13) to the form
(
Ak − i ~
kB
ωI
)
xk(ω) = µ̃δE1k(ω)b, (19)
where I is identity matrix. It is convenient to present the solution of equation (19) in the form
xk(ω) = µ̃δE1k(ω)
[
(
Ak − i ~
kB
ωI
)−1
b
]
, (20)
13705-6
Dynamics of the Rochelle salt
where we denote the inverse matrix of
(
Ak − i ~
kB
ωI
)
by
(
Ak − i ~
kB
ωI
)−1
.
Expression for fluctuation of polarization
δP1k(ω) = χε0
11(k)E1k +
µ̃
ṽ
δξz
k(ω)
and solution (20) allows us to come up with the final expression for permittivity:
ε11(k, ω) = ε11(k,∞) + 4π
µ̃2
ṽ
F1(k, iω), (21)
F1(k, iω) =
[
(
Ak − i ~
kB
ωI
)−1
b
]
1
, (22)
where ε11(k,∞) = 1 + 4πχε0
11(k) is a high frequency contribution into the dielectric permittivity
and ṽ = v/kB. If k = 0 we have ε11(0,∞) ≡ ε11(∞) and ε11(∞) = 1+4πχε0
11, where χε0
11 is a “seed”
dielectric susceptibility. Index “1” on the right side of equation (22) means the first component of
the vector derived by multiplying matrix
(
Ak − i ~
kB
ωI
)−1
and vector b (index “1” arises from the
fact that δξz
k(ω) is the first component of vector xk(ω)).
The analysis shows that susceptibility ε11(0, 0), obtained according to equation (21) equals
the static susceptibility obtained earlier for the Mitsui model with piezoelectric interaction and
transverse field at k = 0, ω = 0, and at any relaxation times T1 and T2. If we set the transverse field
equal to zero and identify T1 with parameter α, which is a characteristic time of the pseudospins
chaning their states within Glauber approach, we derive the dynamic permittivity which strictly
corresponds to the one derived earlier [21].
It can be easily seen that the function F1(k, iω) is a rational function of iω, where numerator
is a polynomial function of the degree not higher than 5 and denominator is polynomial function
of the degree 6. Therefore, we can decompose the function F1(k, iω) into partial fractions:
F1(k, iω) =
n
∑
i=1
kiτi
1 + iωτi
+
m
∑
j=1
Mj(iω) +Nj
(iω)2 + pj(iω) + qj
. (23)
Here coefficients ki, τi, Mj , Nj , pj , qj are real numbers, n is the number of real (equal to −1/τi)
eigenvalues and 2m is the number of complex eigenvalues of matrix Ak. The number of real and
the number of complex eigenvalues of matrix Ak is defined by theory parameters and temperature.
However, it is clear that matrix Ak has 6 eigenvalues in total. The first sum in equation (23) is a
contribution of Debye (relaxation) modes into the dynamic dielectric permittivity, and the second
sum is a contribution of resonant modes.
In paraelectric phase the dynamic dielectric permittivity is as follows:
ε
(0)
11 (k, ω) = ε11(k,∞) + 4π
µ̃2
ṽ
F
(0)
1 (k, iω), (24)
F
(0)
1 (k, iω) =
[
(
A
(0)
k − i ~
kB
ωI
)−1
b(0)
]
1
. (25)
Here matrix A
(0)
k and vector b(0) are as follows4:
A
(0)
k =
a11 −Ω̃ a16
a31 − 1
T ′
2
−ε̃
a61 ε̃ a55
, b(0) =
b1
b3
b6
, (26)
4In paraelectric phase we have ξz = 0 and σx = 0. Accordingly, matrix Ak in paraelectric phase has two invariant
subspaces of dimension 3 and fluctuations δξz
k
(ω) are defined by matrix A
(0)
k
which produces the effect only on one
of the subspaces.
13705-7
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
where
a11 = −1/T ′
2 + U + R̃+
k (Q+G), a31 = Ω̃ − R̃+
k ξ
x,
a16 = V, a61 = V + R̃+
k H, a55 = −1/T ′
2 +W ; (27)
vector b(0) components:
b1 = −(Q+G), b3 = ξx, b6 = −H, (28)
and the following notations are used:
Q =
1
T ′
2
σz
ε̃
, U =
(
1
T ′
2
−
1
T ′
1
)
ε̃2
λ̃2
, V =
(
1
T ′
2
−
1
T ′
1
)
Ω̃ε̃
λ̃2
,
W =
(
1
T ′
2
−
1
T ′
1
)
Ω̃2
λ̃2
, G = 2K
ε̃2
λ̃2
, H = 2K
Ω̃ε̃
λ̃2
,
K =
1
T ′
1
1
4T
(
cosh λ̃
2T
)2 −
1
2T ′
2
σz
ε̃
, λ̃ =
√
Ω̃2 + ε̃2 , ε̃ = R̃−
0 σ
z + ∆̃ . (29)
Hence, depending on the eigenvalues of matrix A
(0)
k , the dynamics in paraelectric phase is described
either by three relaxation modes or one relaxation and one resonant mode.
Žekš [16] obtained the expression for relaxation time in paraelectric phase within the Mitsui
model with transverse field, though without piezoelectric interaction. The study was carried out
within Bloch equation method at 1/T2 = 0. The linear dependence of relaxation time on longitudi-
nal one-particle relaxation time T1 was derived. However, linear dependence is valid only for large
T1 and is not valid in general case. Besides, the expression for relaxation time of [16] contains an
error. In the Appendix we present the correct results for the relaxation time.
In the present work all calculations were carried out in the center of the Brillouin zone k = 0
and in the absence of external electric field E1 = 0.
3. Discussion
We need theory model parameters for particular calculations. We will use a model parameter
set derived earlier under consideration of the Rochelle salt thermodynamic properties [30]. These
parameters and parameters derived within the analogous model without transverse field [21] are
presented in table 1.
In order to calculate dynamic characteristics it is necessary to determine longitudinal and
transverse relaxation time T1, T2. The results of research show that the alteration of time T2
within interval (0.1T1, ∞) has hardly any effect upon the calculated dynamic characteristics; this
is illustrated in figure 1.
Weak dependence of the dielectric permittivity allows us to set T2 = ∞ in our calculations.
Thus, we completely neglect the decay of transverse component of pseudospin caused by external
factors (e.g. interaction with thermostat). However, in this case the decay of transverse component
along the whole temperature range occurs anyway (all poles of function F1(0, iω) have a negative
real part). But now it is caused exceptionally by the internal dynamics of the system and by the
relaxation of longitudinal component of pseudospin towards its quasiequilibrium value.
The value of parameter T1
T1 = 1.767× 10−13 s
was derived based on the agreement between theory and experimental data [5] for ε′11 in the upper
phase transition point at ν = 2.5 GHz: ε′11 = 164.53. For comparison, the value of parameter α,
which defines the time scale in Glauber model and which was obtained based on the same condition
is equal to 1.7× 10−13 s [21].
First, we shall present the structure of theoretically derived dynamic response of the Rs. The
structure of the dynamic dielectric permittivity is defined by poles of function F1(0, iω). This
13705-8
Dynamics of the Rochelle salt
Table 1. Optimal values of theory parameters for Rochelle salt. Two parameters sets for the
models with and without transverse field are presented.
Ω̃ = 0.0 K Ω̃ = 113.467 K
J̃0 (K) 797.36 813.216
K̃0 (K) 1468.83 1447.17
∆̃ (K) 737.33 719.937
ψ̃4 (K) –760.0 –720.0
cE0
44 (1010 N/m2) 1.28 1.224
e014 (10−2 C/m2) 3.336 31.64
χε0
11 0.318 0.0
µ(T ) = a+ k(T − 297)
a (10−30 Cm) 8.41 8.157
k (10−30 Cm/K) –0.022 –0.0185
240 260 280 300
60
75
90
105
120
135
150
292 294 296 298 300 302
84
86
88
90
'
T (K)
ε 11
'
T (K)
ε 11
Figure 1. Dependence of a real part of dielectric permittivity on temperature at frequency of
ν = 5.1 GHz. Solid line represents calculations at T1 = 1.767 × 10−13 s, T2 = ∞. Dashed line
represents calculations at T1 = 1.767×10−13 s, T2 = 0.1T1. Dots represent experimental data [5].
function has two real negative poles throughout the whole temperature interval, which determine
the two relaxation modes. Besides, this function has two pairs of complex conjugate poles with a
negative real part. Each pair of complex conjugate poles determines a resonant mode. However,
in the paraelectric phase, one relaxation and one resonant mode give zero contribution into the
dielectric permittivity. Hence, in ferroelectric phase dielectric permittivity has the form:
ε11(ω) = ε11(∞) +
k′1τ1
1 + iωτ1
+
k′2τ2
1 + iωτ2
+
M ′
1(iω) +N ′
1
(iω)2 + p1(iω) + q1
+
M ′
2(iω) +N ′
2
(iω)2 + p2(iω) + q2
,
and in paraelectric phases, the dielectric permittivity is as follows:
ε
(0)
11 (ω) = ε11(∞) +
k′1τ1
1 + iωτ1
+
M ′
1(iω) +N ′
1
(iω)2 + p1(iω) + q1
.
13705-9
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
Primed and non-primed values differ by the factor of 4πµ̃2/ṽ (see equations (21) and (23)). Such a
characteristic of dielectric response also remains unchanged in the case with finite T2. Within the
model without a transverse field, the dynamics is determined by only two relaxation modes. Both
modes have non-zero contribution within ferroelectric phase and only one of them has non-zero
contribution within paraelectric phase [21].
Temperature dependencies of two relaxation times τ1, τ2, obtained here, along with the same
ones derived within the model without transverse field [21] and data for relaxation time obtained on
the basis of analysis of experiment for ε′11(ν, T ) are presented in figure 2. As this figure illustrates,
240 260 280 300 320
2
4
6
8
10
240 260 280 300 320
6.8
7.0
7.2
7.4
7.6
7.8
8.0
τ 1-1
(
10
10
s
-1
)
T (K)
τ 2-1
(
10
12
s
-1
)
T (K)
Figure 2. Dependency of inverse relaxation times on temperature. Solid line corresponds to the
model with transverse field, dashed line corresponds to the model without transverse field [21].
Points correspond to experimental data: • ([4]), � ([5]), ◦ ([6]), � ([31]).
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
108 109 1010 10110
25
50
75
100
125
150
ν (Hz)
(f)(e)
(d)(c)
(b)(a)
ε' 11
ν (Hz)
ε' 11
ν (Hz)
ε' 11
ν (Hz)
ε' 11ε' 11
ν (Hz)
ν (Hz)
ε' 11
ν (Hz)
ε"
11
ν (Hz)
ε"
11
ν (Hz)
(e) (f)
(c) (d)
(b)(a)
ε"
11
ν (Hz)
ε"
11
ν (Hz)
ε"
11
ν (Hz)
ε"
11
Figure 3. The frequency dependence of the real and imaginary part of dielectric permittivity,
calculated within the model with transverse field (solid line) and without transverse field (dashed
line [21]) at different temperatures T (K): (a) 235, (b) 245, (c) 265, (d) 285, (e) 305, (f) 315.
Points represent experimental data: � ([5]), ◦ ([32] and [33]), + ([34]), H ([35]), • ([36]), � ([4]),
× ([6]), � ([31]), 4 ([37]), 5 ([38]).
13705-10
Dynamics of the Rochelle salt
the model with transverse field ensures a better agreement of the theory with the experiment in
comparison with the model without transverse field.
Figure 3 presents frequency dependencies of dielectric permittivity in dispersion region (109 −
1011 Hz) calculated within the models with transverse field and without it at different tempera-
tures along with experimental data. As figures show, both models ensure good agreement between
theory and experiment. However, the model with transverse field provides for better agreement of
theory with experiment at T = 235 K, T = 245 K (low temperature paraelectric phase). At other
temperatures the agreement between the results, obtained for both models and experimental data
are about the same.
The analysis shows that the contribution of the first relaxation mode consists of more than
99% of the total permittivity along the whole temperature range. Thus, the dynamics of Rs within
microwave region is of Debye relaxation type. The same conclusion was derived within the model
without transverse field [21].
220 240 260 280 300 320
0
50
100
150
200
220 240 260 280 300 320
0
25
50
75
100
125
150
T (K)
ε 11' ''
ε 11
T (K)
Figure 4. Temperature dependencies of the real and imaginary part of dielectric permittivity
at different frequencies. Points represent the results of measurements [5] at frequencies (GHz):
◦ 2.5, • 3, 4 3.9, N 5.1, O 7.05, H 8.25, � 9.45, � 11.96, � 12.95. Lines represent the results
of theoretical calculations carried out within the model with transverse field (solid line) and
without transverse field (dashed line [21]) at frequencies corresponding to experiment.
Figures 4, 5 illustrate temperature dependencies of dielectric permittivity ε11, calculated at
different frequencies, together with experimental data. Solid and dashed lines present results of
calculations derived within (this work) and without (results of work [21]) transverse field respec-
tively. In figure 4 we do not present an imagined part of dielectric permittivity, calculated within
the model without transverse field. However, it should be noted that in this case both models
provide for about the same agreement of theory with experiment. As figure 4 shows, the model
with transverse field provides for better agreement between theory and experiment for a real part
of dielectric permittivity. It should also be pointed out that transverse field being taken into ac-
count to some extent rectifies the problem of poor agreement between theory and experiment in
a low temperature paraelectric phase. Besides, if we assume that a real effective dipole moment
has a weaker dependence on temperature (lower in a lower phase transition point and higher in
an upper phase transition point), it becomes clear (see equation (24)) that we will reach a better
agreement between theory and experiment for a real part of this characteristic in low temperature
paraelectric phase. Earlier [30] it was mentioned that such an assumption may allow us to reach a
better agreement between theory and experiment for polarization and static dielectric permittivity
of a free crystal in a low temperature paraelectric phase.
Now, we calculate the temperature dependence of dielectric permittivity within a wide fre-
quency range and try to find out whether our model with transverse field is capable of describing
the resonant dynamics. Figure 6 shows both the experimental data and theoretical calculations for
13705-11
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
290 295 300 305 310
0
50
100
150
200
290 295 300 305 310
0
25
50
75
100
125
'
T (K)
ε 11 ''
T (K)
ε 11
Figure 5. The temperature dependence of the real and imaginary parts of the dynamic dielectric
permittivity at different frequencies. Points represent experiment [6] at frequencies (GHz): • 1,
4 2, � 2.5, � 3, � 4.5, N 7, H 8.25. Lines represent the results of theoretical calculations carried
out within the model with transverse field (solid line) and without transverse field (dashed
line [21]) at frequencies corresponding to experiment.
0 2 4 6 8
0
2
4
6
8
10
12
(a)
ε 11
a
d
c
b
e
ν (1011 Hz)
0 20 40 60 80 100
0
20
40
60
80
100
(b)
'' '' ε 11
ed
c
b
a
ν (1011 Hz)
Figure 6. The frequency dependence of the imaginary part of the dynamic dielectric permittivity
at different temperatures (K): a – 80, b – 114, c – 150, d – 187, e – 218. Experimental data (a)
of [8] and results of theoretical calculations (b) are presented.
frequency dependence of imaginary part of the dielectric permittivity. As one can see, the model
with transverse field has a resonant dielectric response at frequency ν = 8 × 1012 Hz (T = 80 K),
while the experiment shows a resonant peak at frequency ν = 6.6 × 1011 Hz. Besides, the theo-
retically calculated changes of the dielectric spectra with the temperature increase, differ from the
corresponding experimental data.
4. Concluding remarks
In this work we studied the dynamic properties of the Rochelle salt. We used the Mitsui model
with piezoelectric interaction extended by transverse field. Such an extension allows us to take into
account the dynamic flipping of the structural ordering units between two equilibrium positions.
The study was performed within the Bloch equation method.
Parameters, required for calculations, were taken from the study of the thermodynamic char-
13705-12
Dynamics of the Rochelle salt
acteristics (static dielectric, elastic and piezoelectric) [30], where good agreement between theory
and experiment was achieved. Two additional parameters (single-particle relaxation times T1 and
T2) which characterize dynamic processes were defined in the present work. In particular, it was
obtained that throughout the whole temperature range the change of T2 has a little effect on
dielectric permittivity in dispersion range and it can be selected as needed for calculation conve-
nience. By appropriate selection of the parameter T1 we managed to achieve a better agreement
between theory and experiment for dynamic dielectric permittivity in comparison with the model
that does not take transverse field into account. Therefore, within the Mitsui model with piezo-
electric interaction and transverse field, the unique parameter set was derived that well describes
both thermodynamic and dynamic properties of the Rochelle salt.
In addition, in this work we revealed the presence of the resonant response within certain fre-
quency range along with relaxation dynamics within microwave range. However, while experiment
shows resonant response on frequencies close to 6 · 1011 Hz, the theory shows resonant response on
frequencies 8 · 1012 – 10 · 1012 Hz. The character of changes of dielectric spectrum at temperature
increase is essentially different from the experimental data. Currently we have no explanation to
such discrepancy. One can notice that our approach has a free parameter T2. But the variation of
this parameter did not permit us to reach an acceptable agreement between theory and experiment
for resonant dynamics. Hence, one can conclude that modification of the Mitsui model by taking
the transverse field into account is not sufficient in order to properly explain the resonant dielectric
response. The additional degrees of freedom are probably essential in resonant response.
Acknowledgements
Authors are grateful to Prof. Ihor Stasyuk for his kind interest in the work and particularly for
discussing the nature of the transverse field in the Hamiltonian.
Appendix. Relaxation time in paraelectric phase calculated within the Mitsui
model with transverse field
Here we study the dependence of the relaxation time on the theory model parameter T1 in para-
electric phase (ξz = 0, σx = 0). We consider the model with transverse field but neglect transverse
relaxation of the pseudospin (1/T2 = 0). Analogous model, but without piezoelectric interaction
was considered earlier [16]. It should be noted that in paraelectric phase, piezoelectric interaction
just effects the value of σz but does not change the analytic formula for the relaxation time.
Relaxation time is defined as
τ = −~/(kBΛ), (30)
where Λ is a real eigenvalue of the matrix A
(0)
k . In our case this matrix has the following form:
A
(0)
k =
−
ε̃2κk
T ′
1λ̃
2
−Ω̃ −
Ω̃ε̃
T ′
1λ̃
2
Ω̃βk 0 −ε̃
−
Ω̃ε̃κk
T ′
1λ̃
2
ε̃ −
Ω̃2
T ′
1λ̃
2
, (31)
where the following notations are used:
κk = 1 −
R̃+
k
2T
(
cosh λ̃
2T
)2 , βk = 1 −
R̃+
k
λ
tanh
λ̃
2T
, λ̃ =
√
Ω̃2 + ε̃2, ε̃ = R̃−
0 σ
z + ∆̃.
All eigenvalues of the matrix A
(0)
k are roots of the following characteristic polynomial:
W (x) = x3 +
1
T ′
1
a1x
2 + a2x+
1
T ′
1
a3 , (32)
13705-13
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
where constants a1, a2, a3 are independent of T ′
1 and have the following form:
a1 =
Ω̃2 + ε̃2κk
λ̃2
, a2 = ε̃2 + Ω̃2βk , a3 = Ω̃2βk + ε̃2κk .
Let us assume that a real root of polynomial W (x) at large T ′
1 is proportional to 1/T ′
1:
Λ = z/T ′
1. (33)
Then we can present W (x) in the following form:
W (x) = (x− z/T ′
1) (x2 + b1(T
′
1)x + b2) (34)
where the constant z is independent of T ′
1 and b1(T
′
1) is an unknown function of T ′
1. Expanding
last presentation of W (x) one can derive that
W (x) = x3 +
(
b1(T
′
1) −
z
T ′
1
)
x2 +
(
b2 −
b1(T
′
1)
T ′
1
)
x−
1
T ′
1
zb2 . (35)
Comparing the last terms of equation (32) and equation (35) one can conclude that
− zb2 = a3 (36)
and hence b2 is independent of T ′
1.
Comparing the coefficients near x2 of polynomial W (x) presented in forms (32) and (35) one
can derive the following result for b1(T
′
1):
b1(T
′
1) =
a1 + z
T ′
1
. (37)
Now comparing the corresponding coefficients of W (x) near x and taking into account equa-
tion (37) one can derive that
b2 = a2 +
a1 + z
T ′2
1
. (38)
Here the last term is a higher-order term that can be neglected. Hence, we can put
b2 = a2 . (39)
According to equations (36), (39) one can derive the constant z:
z = −
a3
a2
. (40)
According to equations (30), (33), and equation (40) the relaxation time at large T1 is presented
as:
τ = T1
1 + ϕ
1 + ϕ
(
1 −
R̃
+
k
2T
(
cosh λ̃
2T
)2
) , (41)
where
ϕ =
ε̃2
Ω̃2
(
1 −
R̃
+
k
λ̃
tanh λ̃
2T
) . (42)
Figure 7 presents the results of calculations of the dependence τ on T1 at T = 240 K and
k = 0. Both the results of direct calculations (relaxation time derived through numerical finding
of the real eigenvalue of the matrix A
(0)
k ) and the results derived based on equation (41) are
presented. As one can see, the result of direct calculations rapidly approaches the asymptote.
When T1 = 1.767× 10−13 s, approximation (41) almost perfectly agrees with the relaxation time
calculated directly. This result is valid for the whole paraelectric temperature range.
13705-14
Dynamics of the Rochelle salt
0 1 2 3 4
0
1
2
3
τ
(1
0-1
3 s
)
T
1
(10-15 s)
Figure 7. Dependence of the relaxation time on the model parameter T1 calculated at T =
240 K. Solid line presents the results of ‘direct’ calculations. Dashed line presents the results of
calculations based on (41).
References
1. Jona F., Shirane G., Ferroelectric Crystals. Pergamon Press Inc., Oxford, 1962.
2. Solans X., Gonzalez-Silgo C., Ruiz-Pérez C., J. Solid State Chem., 1997, 131, 350.
3. Kulda J., Hlinka J., Kamba S., Petzelt J., ILL: Annual Report, 2000 [www.ill.fr/AR-00/p-48.htm].
4. Müser H.E., Pottharst J., Phys. Status Solidi, 1967, 109.
5. Sandy F., Jones R.V., Phys. Rev., 1968, 168, 481.
6. Kolodziej H. – In: Dielectric and Related Molecular Processes, Vol. 2. The Chemical Society Burlington
House, London, 1975, p. 249–287.
7. Iwata Y., Mitani S., Shibuya O., Ferroelectrics, 1990, 107, 287.
8. Volkov A.A., Goncharov Yu.G., Kozlov G.V., Kryukova Ye.B., Petzelt J., Pis’ma Zh. Éksp. Teor. Fiz.,
1985, 41, 16 [JETP Lett., 1985, 41, 17].
9. Kamba S., Schaack G., Petzelt J., Phys. Rev. B, 1995, 51, 14998.
10. Hlinka J., Kulda J., Kamba S., Petzelt J., Phys. Rev. B, 2001, 63, 052102.
11. Mitsui T., Phys. Rev., 1958, 111, 1529.
12. Žekš B., Shukla G.G., Blinc R., Phys. Rev. B, 1971, 3, 2306.
13. Vaks V.G., Introduction into Microscopic Theory of Ferroelectrics. Nauka, Moscow, 1973 (in Russian).
14. Kalenik J., Acta Phys. Pol., 1975, A48, 387.
15. Mori K., Ferroelectrics, 1981, 31, 173.
16. Žekš B., Shukla G.G., Blinc R., J. Phys. Colloq., Suppl., 1972, 33, C2.
17. Levitskii R.R., Zachek I.R., Varanitskii V.I., Ukr. Fiz. Zh., 1980, 25, 1766 (in Russian).
18. Levitskii R.R., Zachek I.R., Varanitskii V.I., Fiz. Tverd. Tela (Leningrad), 1980, 22, 2750 [Sov. Phys.
Solid State, 1980, 22, 1603].
19. Levitskii R.R., Antonyak Yu.T., Zachek I.R., Ukr. Fiz. Zh., 1981, 26, 1835 (in Russian).
20. Levitskii R.R., Sokolovskii R.O., Condens. Matter Phys., 1999, 2, 393.
21. Levitskii R.R., Zachek I.R., Verkholyak T.M., Moina A.P., Phys. Rev. B, 2003, 67, 174112.
22. Moina A.P., Slivka A.G., Kedyulich V.M., Phys. Stat. Sol. B, 2007, 244, 2641.
23. Levitskii R.R., Moina A.P., Andrusyk A.Ya., Slivka A.G., Kedyulich V.M., J. Phys. Stud., 2008, 11,
2603.
24. Levitskii R.R., Zachek I.R., Moina A.P., Verkholyak T.M., J. Phys. Stud., 2003, 7, 106.
25. Levitskii R.R., Zachek I.R., Moina A.P., Andrusyk A.Ya., Condens. Matter Phys., 2004, 7, 111.
26. Moina A.P., Levitskii R.R., Zachek I.R., Phys. Rev. B, 2005, 71, 134108.
27. Stasyuk I.V., Velychko O.V., Ferroelectrics, 2005, 316, 51.
28. Levitskii R.R., Zachek I.R., Vdovych A.S., Stasyuk I.V., Condens. Matter Phys., 2009, 12, 295.
29. Slivka A.G., Kedyulich V.M., Levitskii R.R., Moina A.P., Romanyuk M.O., Guivan A.M., Condens.
Matter Phys., 2005, 8, 623.
13705-15
R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek
30. Levitskii R.R., Zachek I.R., Andrusyk A.Ya. (unpublished)
31. Volkov A.A., Kozlov G.V., Lebedev S.P., Zh. Eksp. Teor. Fiz., 1980, 79, 1430 [Sov. Phys. JETP, 1980,
52, 722].
32. Poplavko Yu.M., Meriakri V.V., Pereverzeva L.P., Alesheckin V.V., Molchanov V.I., Fiz. Tverd. Tela
(Leningrad), 1974, 15, 2515 [Sov. Phys. Solid State, 1974, 15, 1672].
33. Poplavko Yu.M., Solomonova L.P., Bull. Acad. Sci. USSR, Phys. Ser. (Engl. Transl.), 1967, 1771.
34. Pereverzeva L.P. – In: Mechanisms of Relaxation Phenomena in Solids. Kaunas, 1974 (in Russian).
35. Deyda H., Z. Naturforschg, 1967, 22a, 1139.
36. Akao H., Sasaki T., J. Chem. Phys., 1955, 23, 2210.
37. Petrov V.M., Sov. Phys. Crystallogr., 1962, 7, 403.
38. Jäckle W., Z. Angew. Phys., 1960, 12, 148.
Дослiдження динамiчних властивостей сегнетової солi
NaKC4H4O6·4H2O в рамках моделi Мiцуї, що враховує
п’єзоелектричну взаємодiю та поперечне поле
Р.Р. Левицький1, А.Я. Андрусик1, I.Р. Зачек2
1 Iнститут Фiзики Конденсованих Систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1, Україна
2 Нацiональний Унiверситет “Львiвська Полiтехнiка”, 79013 Львiв, вул. Бандери, 12, Україна
В рамках моделi Мiцуї, що враховує п’єзоелектричну взаємодiю та поперечне поле розраховано
динамiчну дiелектричну проникнiсть сегнетової солi. Розрахунки проведено на основi параметрiв
теорiї, одержаних ранiше при дослiдженнi термодинамiчних властивостей сегнетової солi. Дослi-
дження проведено в рамках методу рiвнянь Блоха. Показано, що розширення моделi Мiцуї врахува-
нням поперечного поля та п’єзоелектричної взаємодiї дозволяє краще описати релаксацiйну дина-
мiку сегнетової солi. Крiм того показано, що врахування поперечного поля у моделi Мiцуї приводить
до появи резонансної складової у динамiчнiй проникностi так, як це спостерiгається експеримен-
тально. Проте згiдно проведених розрахункiв резонансний вiдгук проявляє себе у iнфрачервоному
частотному дiапазонi, тодi як на експериментi вiн спостерiгається у субмiлiметровому дiапазонi.
Ключовi слова: сегнетова сiль, наближення молекулярного поля, динамiчна дiелектрична
проникнiсть, релаксацiйна динамiка, резонансна динамiка
13705-16
Introduction
Dynamic physical characteristics of the Mitsui model with piezoelectric interaction and transverse field
Discussion
Concluding remarks
|
| id | nasplib_isofts_kiev_ua-123456789-32049 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:18:07Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Levitskii, R.R. Andrusyk, A.Ya. Zachek, I.R. 2012-04-06T18:08:12Z 2012-04-06T18:08:12Z 2010 Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field / R.R. Levitskii, A.Ya. Andrusyk, I.R. Zachek // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13705: 1-16. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 77.22.Ch, 77.80.Bh, 77.84.Fa https://nasplib.isofts.kiev.ua/handle/123456789/32049 We calculated dynamic dielectric permittivity of the Rochelle salt within the Mitsui model, extended by piezoelectric interaction and transverse field. Calculations were based on the parameters derived earlier within the study of thermodynamic properties of Rochelle salt. The study of dynamic properties was performed within the Bloch equation method. We showed that taking transverse field into account allows for better description of the Rochelle salt relaxation dynamics. Furthermore, we showed that taking transverse field into account results in the appearance of a resonant component in dynamic permittivity like it is observed in experiment. However, in accordance with the calculations, resonant response reveals itself within infrared frequency range, whereas in experiment it is observed within submillimeter spectral region. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field Article published earlier |
| spellingShingle | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field Levitskii, R.R. Andrusyk, A.Ya. Zachek, I.R. |
| title | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field |
| title_full | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field |
| title_fullStr | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field |
| title_full_unstemmed | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field |
| title_short | Dynamics of the Rochelle salt NaKC4H4O6·4H2O crystal studied within the Mitsui model extended by piezoelectric interaction and transverse field |
| title_sort | dynamics of the rochelle salt nakc4h4o6·4h2o crystal studied within the mitsui model extended by piezoelectric interaction and transverse field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32049 |
| work_keys_str_mv | AT levitskiirr dynamicsoftherochellesaltnakc4h4o64h2ocrystalstudiedwithinthemitsuimodelextendedbypiezoelectricinteractionandtransversefield AT andrusykaya dynamicsoftherochellesaltnakc4h4o64h2ocrystalstudiedwithinthemitsuimodelextendedbypiezoelectricinteractionandtransversefield AT zachekir dynamicsoftherochellesaltnakc4h4o64h2ocrystalstudiedwithinthemitsuimodelextendedbypiezoelectricinteractionandtransversefield |