On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds
The problem of microscopic description of low-temperature phases in ferro-antiferroelectric mixed compounds is discussed. Those phases (ferro-, antiferroelectric and dipole glass) are interpreted based on the specific distributions for single and pair correlation functions in a non-ergodic thermod...
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2009
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irk-123456789-1199852017-06-11T03:04:29Z On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds Korynevskii, N.A. Solovyan, V.B. The problem of microscopic description of low-temperature phases in ferro-antiferroelectric mixed compounds is discussed. Those phases (ferro-, antiferroelectric and dipole glass) are interpreted based on the specific distributions for single and pair correlation functions in a non-ergodic thermodynamic system. Phase diagram and dielectric susceptibility of a mixed system are obtained and analyzed. Обговорюється питання мiкроскопiчного опису низькотемпературних фаз у сегнето-антисегнетоелектричних змiшаних сполуках. Цi фази (сегнето-, антисегнетоелектрична i дипольного скла) iнтерпретуються на основi специфiчних розподiлiв для унарних i парних кореляцiйних функцiй в неергодичних термодинамiчних системах. Отримано i проаналiзовано фазову дiаграму та дiелектричну сприйнятливiсть змiшаної системи. 2009 Article On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds / N.A. Korynevskii, V.B. Solovyan // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 267-273. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 75.10.Hk, 77.22.Ch, 77.84.F DOI:10.5488/CMP.12.2.267 http://dspace.nbuv.gov.ua/handle/123456789/119985 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The problem of microscopic description of low-temperature phases in ferro-antiferroelectric mixed compounds
is discussed. Those phases (ferro-, antiferroelectric and dipole glass) are interpreted based on the specific
distributions for single and pair correlation functions in a non-ergodic thermodynamic system. Phase diagram
and dielectric susceptibility of a mixed system are obtained and analyzed. |
| format |
Article |
| author |
Korynevskii, N.A. Solovyan, V.B. |
| spellingShingle |
Korynevskii, N.A. Solovyan, V.B. On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds Condensed Matter Physics |
| author_facet |
Korynevskii, N.A. Solovyan, V.B. |
| author_sort |
Korynevskii, N.A. |
| title |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| title_short |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| title_full |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| title_fullStr |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| title_full_unstemmed |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| title_sort |
on the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/119985 |
| citation_txt |
On the problem of low-temperature ordering in ferro-antiferroelectric mixed compounds / N.A. Korynevskii, V.B. Solovyan // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 267-273. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT korynevskiina ontheproblemoflowtemperatureorderinginferroantiferroelectricmixedcompounds AT solovyanvb ontheproblemoflowtemperatureorderinginferroantiferroelectricmixedcompounds |
| first_indexed |
2025-07-08T17:01:35Z |
| last_indexed |
2025-07-08T17:01:35Z |
| _version_ |
1837098963816153088 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 267–273
On the problem of low-temperature ordering in
ferro-antiferroelectric mixed compounds
N.A.Korynevskii1,2, V.B.Solovyan1
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Institute of Physics, University of Szczecin, 15 Wielkopolska Str., 70451 Szczecin, Poland
Received March 6, 2009, in final form May 18, 2009
The problem of microscopic description of low-temperature phases in ferro-antiferroelectric mixed compounds
is discussed. Those phases (ferro-, antiferroelectric and dipole glass) are interpreted based on the specific
distributions for single and pair correlation functions in a non-ergodic thermodynamic system. Phase diagram
and dielectric susceptibility of a mixed system are obtained and analyzed.
Key words: ferroelectrics, solid mixtures, phase diagram, dipole glass
PACS: 75.10.Hk, 77.22.Ch, 77.84.F
The recently introduced and analysed model [1,2] for microscopic investigation of ferroelectric-
antiferroelectric mixed compounds with hydrogen bonds predicts numerous solutions for single
and pair correlation functions of interactive particles. The single correlation functions are directly
connected with an order parameter (ferro- and antiferroelectric), but pair correlation functions
describe a possible dipole glass phase and in its origin they are close to a well-known Edwards
and Anderson spin glass parameter [3]. In the limit of small (n 6 0, 45) and large (n > 0, 85)
concentrations of ferroelectric component, the investigated system possesses antiferroelectric and
ferroelectric properties up to the very low temperatures, respectively. However, for intermediate
concentration (0, 45 < n < 0, 85) there arises a specific ordered phase with only short range cor-
relations between particles. The number of different types of correlations (the number of different
solutions for pair correlation functions) increase with temperature falling down [4]. Each of these
pair correlation functions describe some specific arrangement of the nearest neighbours in disor-
dered ferro-antiferroelectric systems. Since such systems are usually prepared due to high-speed
cooling the distribution of such correlation functions is quite random. So, in order to calculate ther-
modynamic functions, the averaging procedure should be performed not for a partition function,
as in Gibbs system, but for its logarithm, directly connected with the observed quantities.
Our approach differs from the well-known methods of investigating the mixed KDP-type com-
pounds (see [5–8]). Those methods are based on the random fields taken into account, that are
induced by competition between ferro- and antiferroelectric types of interaction in hydrogen bonds
lattice. In this way, the dynamical properties of a mixed system were investigated too [9]. We
have proposed a model of randomly distributed interacting groups (Rb–PO4 and NH4–PO4) which
form a regular crystalline lattice. So, the microscopic proton-proton interactions are not considered
directly. Such an approach essentially simplifies the calculations and provides good physical results.
For rubidium ammonium dihydrogen phosphate – Rbn(NH4)1−nH2PO4 (RADP) and isomor-
phous rubidium ammonium dihydrogen arsenate – Rbn(NH4)1−nH2AsO4 (RADA) mixed com-
pounds, this scheme of calculations (using replica method [10]) has shown the existence of at least
two areas of different non-trivial solutions for pair correlation functions [4]. Namely, a high temper-
ature paraelectric phase with only one type of pair correlation function; the second one: dipole glass
phase G1 with three types of them; the third one: dipole glass phase G2 with five types of them. In
every phase, one type of solution is “trivial”, so it exists for any temperature, including only the
c© N.A.Korynevskii, V.B.Solovyan 267
N.A.Korynevskii, V.B.Solovyan
temperature of paraelectric phase. The aim of the present paper is a more precise investigation of
our model in order to find new dipole glass phases and to analyse the behaviour of thermodynamic
functions in a large scale of temperatures and concentrations.
We regard a regular crystalline lattice of KDP-type system, each site of which is occupied (quite
randomly) by rubidium or ammonium groups, that is by Rb or NH4. The dipole momenta of Rb–
AsO4 group are directed along z-axis of the crystal and the dipole momenta of NH4–AsO4 group
are directed along the x-axis. In the first case we can observe a ferroelectric ordering (below certain
temperature) and in the second case the antiferroelectric ordering is possible. Due to competition
between ferroelectric and antiferroelectric types of ordering, the crystal undergoes frustration: no
type of ordering is preferential and instead of long range order only short range correlations between
particles remain. At low temperature and at intermediate concentration, those correlations form a
specific state of a crystal, i. e., a dipole glass phase.
The Hamiltonian of ferroelectric-antiferroelectric mixed system is as follows [1,2]:
H = −
1
2
∑
i,j
∑
m,m′
{
V mm′
ij n̂in̂jS
z
imSz
jm′ + Umm′
ij (1 − n̂i)(1 − n̂j)S
x
imSx
jm′
}
. (1)
Here V mm′
ij is the intensity of interaction between m-th “dipole particle” Rb–AsO4 in i-th site
with m′-th “dipole particles” Rb–AsO4 in j-th site and Umm′
ij is the same for NH4–AsO4 “dipole
particles”. So, the first term in (1) is responsible for ferroelectric ordering and the second term is
responsible for antiferroelectric ordering in the mixed system. Sx, Sz are projections of the unit
classical vector ~S on the x and the z axis, respectively; the site occupation operators n̂i have the
following eigenvalues:
ni =
{
1, when i-th site is occupied by Rb,
0, when i-th site is occupied by NH4 .
Since the unit cell for KDP-type crystal contains two formula units (there are also two sublattices
below a phase transition point [11]), the numbers m, m′ run from 1 to 2. So, we can use the
two-particle cluster approximation in which the interaction of particles belonging to the same
site is taken exactly, but for particles belonging to different sites the self-consistent field [4] is
sufficient. Another great problem connected with the calculation of thermodynamic functions is
configuration averaging over randomly distributed “dipole particles” (Rb–AsO4 and NH4–AsO4)
with z- and x-oriented dipole momenta all over the sites of the crystalline lattice. This problem can
be solved using the replica method [4,10]. Since we have two sets of variables {Sz
im} and {Sx
im},
the introduction of n replica variables σk
im for Sz
im and n replica variables ξk
im for Sx
im is necessary.
With the accuracy up to the second order of symmetric replica expansion (variables σk
im and
ξk
im reduce to σim, ξim) and using the two-particle cluster approximation, the following expansion
for −βH has been obtained [12]:
−βH =
∑
i
{A1σi1σi2 + B1(σi1 + σi2) + A2ξi1ξi2 + B2(ξi1 − ξi2)} − C, (2)
where A, B, C are some coefficients connected with initial parameters V mn′
ij , Umm′
ij of the Hamil-
tonian and “order” parameters P , q, gz, gx:
A1 = J1 + J2
[
V 2
f − V 2
af
V 2
+
(
1 +
(
Vf − Vaf
V
)2
)
gz −
U
V
gx
]
,
A2 = (tanhβV − J1)
U
V
+ J2
[
U2
f − U2
af
V 2
+
(
(
U
V
)2
+
(
Uf − Uaf
V
)2
)
gx −
U
V
gz
]
,
268
On the problem of low-temperature ordering . . .
B1 = J1
Vf
V
P + 4J2
[
(
Vf
V
)2
P 3 −
Vf
V
Uaf
V
Pq2
]
,
B2 = (tanhβV − J1)
Uaf
V
q + 4J2
[
(
Uaf
V
)2
q3 −
Vf
V
Uaf
V
P 2q
]
,
C =
{
J1
Vf
V
P 2 + (tanhβV − J1)
Uaf
V
q2 +
J2
2
[
−2 +
(
gz −
U
V
gx
)2
+
(
Vf + Vaf
V
)2
(1 + gz)
2
+
(
Uf + Uaf
V
)2
(1 + gx)
2
+ 24
(
Vf
V
P 2 −
Uaf
V
q2
)2
]}
;
Vf =
1
2
N
∑
j=1
(
V 11
ij + V 12
ij
)
, Vaf =
1
2
N
∑
j=1
(
V 11
ij − V 12
ij
)
,
Uf =
1
2
N
∑
j=1
(
U11
ij + U12
ij
)
, Uaf =
1
2
N
∑
j=1
(
U11
ij − U12
ij
)
,
V = V 12
ii , U = U12
ii . (3)
J1 =
∞
∑
t=1
(−1)t2 ln
(
1 − n + ne−2βV t
)
,
J2 =
∞
∑
t=1
(−1)t4t ln
(
1 − n + ne−2βV t
)
; (4)
P = 〈σi1〉, q = 〈ξi1〉, gz = 〈σi1σi2〉, gx = 〈ξi1ξi2〉; (5)
β = 1
kT
, k is the Boltzman constant, T is the absolute temperature. Here 〈. . . 〉 denotes thermal
averaging for a given distribution of z- and x-oriented dipole momenta of Rb–AsO4 and NH4–AsO4
groups, respectively, whereas a bar denotes a stochastic averaging over different distributions.
We shall use the random distribution of Rb and NH4 groups all over the crystal. P and q are
simple ferroelectric and antiferroelectric order parameters, respectively. gz and gx are the nearest
neighbours pair correlation functions (like Edwards and Anderson spin glass parameter).
Performing the trace operation in the expression for free energy
F = −
1
β
ln Tr
{
e−βH
}
(6)
over cluster states of spin-like variables σ and ξ, and configuration averaging using an effective
version of replica method [10], the final formula for F has been obtained:
F = −
1
β
ln
{
4
(
e−A1 + eA1 cosh 2B1
) (
eA2 + e−A2 cosh 2B2
)}
+
1
β
C. (7)
Minimization procedure of (7) under all the introduced “order” parameters P, q, gz , gx gives a set
of equations for their determination:
P = Z−1eA1sinh2B1
(
eA2 + e−A2cosh2B2
)
,
q = Z−1e−A2sinh2B2
(
e−A1 + eA1cosh2B1
)
,
gz = Z−1
(
e−A1 − eA1cosh2B1
) (
eA2 + e−A2cosh2B2
)
,
gx = Z−1
(
e−A1 + eA1cosh2B1
) (
eA2 − e−A2cosh2B2
)
,
Z =
(
e−A1 + eA1cosh2B1
) (
eA2 + e−A2cosh2B2
)
. (8)
The solutions of (8) may be found by numerical calculations only, because all parameters A1, A2, B1, B2
are strongly non-linear functions of temperature and concentration. The basic solutions of (8) have
269
N.A.Korynevskii, V.B.Solovyan
been obtained in [2], where one area for the existence of simulations of three different roots for gz, gx
was found, and in [4], where an additional area for five different roots for gz, gx was detected. So,
it was shown that dipole glass phase is characterized by a set of different roots for pair correlation
functions of the nearest neighbours.
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
180
200
n
G
4
G
3
G
2
G
1 F
P
AF
T,K
Figure 1. Phase diagram of RADA type compound built for the model Hamiltonian parameters:
V = 65K, U = −210K, Vf = 70K, Uf = −40K, Vaf = −80K, Uaf = 120K.
Using a highly precise method for numerical solution of a set of non-linear equations (8) we
have found five regions with different numbers of roots for gz, gx, starting with region P (one
root), and the next are: region G1 (three roots), region G2 (five roots), region G3 (seven roots)
and region G4 (nine roots) (see figure 1). So, for different values of temperature and concentration
(at sufficiently low temperature and for intermediate concentration) there is a set of local states in
ferroelectric-antiferroelectric mixed system. The lower is the temperature, the more complicated is
the dipole glass phase (the bigger number of different pair correlation functions is realized).
Comparing figure 1 to the classical experimental results for RADA [11,13,14] one may find
that regions G2, G3, G4 completely coincide with the dipole glass phase observed experimentally.
However, the region G1 lies in sufficiently higher temperature range. What’s the matter? In our
opinion the region G1 (three roots for gz, gx are realized) present the phase of precursor dipole
glass state. Three different types of interparticle correlations show only a slight deviation of pure
paraelectric phase (P in figure 1) from homogenous state. But in regions G2−G4 such deviations are
substantial, the distribution of interparticle correlations in them is sufficiently non-ergodic. They
really present the dipole glass phase. The possibility of precursor of dipole glass phase existence
was claimed in [15,16], where the polar nano-clusters which appear at rather high temperatures
and remain to exist down to low temperatures were detected using Raman scattering method.
The recent investigations [17,18] also confirm the existence of precursor dipole glass phase at
temperatures about 100-120K in mixed RADA materials.
It is important to note the coexistence of ferroelectric-dipole glass (F −G) and antiferroelectric-
dipole glass (AF − G) phases (see figure 1), which take place practically in the whole region of
concentration for ferroelectric phase, but only in rather small region of concentration for antifer-
roelectric phase. This fact is in good agreement with experiment [11,14,17].
In order to perform a more detailed investigation of different ordered and disordered phases of
RADA in a wide region of temperatures and concentrations the analysis of basic thermodynamic
functions is essential. In the dipole glass phase, the free energy (7) possesses a set of local minima
according to different solutions of equations (8) [19]. Those minima are separated by potential bar-
riers with non-limited height in the thermodynamic limit N → ∞. Due to these potential barriers,
the investigated mixed system remains near some minima for a sufficiently long time. So, we have a
remarkable example of non-ergodic system, because the time averaging values do not coincide with
270
On the problem of low-temperature ordering . . .
the ensemble averaging values for them. In order to correctly calculate thermodynamic functions,
e. g., the free energy, all potential minima should be taken into account. We shall use “nonuniform”
mean values, because the states with lower energy contribute mostly to the observed quantities.
For free energy we have
F =
∑p
l=1 Fie
−βFl
∑p
l=1 e−βFl
, (9)
where Fl is the expression (7) for l-th solution of set of equations (8), l run from 1 to p, p being
a number of different roots of (8) in separate regions (P, F, AF, Gk , see figure 1). For example, at
concentration n ≈ 0, 6 and temperature T ≈ 50 K p = 7, because there are 7 roots in the region G3.
Static dielectric susceptibility χ =
(
∂P
∂E
)
E=0
we shall calculate using the thermodynamic iden-
tity:
dF = −SdT − PdE, (10)
where E is an external electric field.
To obtain free energy depending on E, the term
β
2
∑
i
{E(σ11 + σ21) + E(ξ1j + ξ2i)} , (11)
describing the energy of the investigated system in the electric field, should be introduced into
expression (2). As a result, for the “partial” free energy Fl, instead of the expression (7), a new
one takes place:
Fl = −
1
β
ln
{
4
[
e−A
(l)
1 + eA
(l)
1 cosh
(
2B
(l)
1 + βE
)]
[
eA
(l)
2 coshβE + e−A
(l)
2 cosh2B
(l)
2
]}
+
1
β
C(l). (12)
Here A
(l)
1 , A
(l)
2 , B
(l)
1 , B
(l)
2 , C(l) are formulae (3) with P (l), q(l), g
(l)
x , g
(l)
z parameters forming one root.
The general expression for χ now takes the form:
χ =
∑
l
[
∂P (l)
∂E
+ β
(
P (l) − P
)
P (l)
]
e−βFl
∑
l e
−βFl
∣
∣
∣
∣
∣
∣
E=0
, (13)
where
P =
∑
l P
(l)e−βFl
∑
l e
−βFl
. (14)
For the regions of concentration and temperature, where long range ordering in mixed ferro-
antiferroelectric system does not exist, the expression (13) may be substituted by:
χ = −
∑
l
∂2Fl
∂E2 e−βFl
∑
l e
−βFl
. (15)
The temperature dependencies of inverse dielectric susceptibility (in arbitrary units) calculated
based on (14) for some concentration 0, 55 6 n 6 0, 85 are presented in figure 2. According to
figure 1 only paraelectric and pure dipole glass phases exist in this region. Moreover, the dipole
glass phase is divided into several zones, from G1 – precursor of dipole glass state to G4 – the state
with strongly non-ergodic distributions of interparticle correlations.
The behaviour of 1/χ in general is close to the law predicted by Curie, especially at low
and high temperatures. However, for temperatures corresponding to transitions between different
Gl regions, there arise some perturbations in the linear behaviour of 1/χ. Most peculiarities of
χ take place when the precursor of dipole glass phase (G1) appears. At low temperatures, the
appearance of a new dipole glass phase (G4 in figure 1 for example) insufficiently contribute into
the dielectric susceptibility because the total non-ergodic character of the investigated system
internal arrangement changes insignificantly in this case.
271
N.A.Korynevskii, V.B.Solovyan
Figure 2. Inverse dielectric susceptibility of RADA type compound at different concentrations
of Rb atoms. The model Hamiltonian parameters are the same as in figure 1.
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272
On the problem of low-temperature ordering . . .
До питання про низькотемпературне впорядкування у
сегнето-антисегнетоелектричних змiшаних сполуках
М.А.Кориневський1,2, В.Б.Солов’ян1
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
2 Iнститут фiзики Щецiнського унiверситету, Польща, 70451 Щецiн, вул. Вєлькопольська, 15
Отримано 6 березня 2009 р., в остаточному виглядi – 18 травня 2009 р.
Обговорюється питання мiкроскопiчного опису низькотемпературних фаз у сегнето-антисегнето-
електричних змiшаних сполуках. Цi фази (сегнето-, антисегнетоелектрична i дипольного скла) iн-
терпретуються на основi специфiчних розподiлiв для унарних i парних кореляцiйних функцiй в неер-
годичних термодинамiчних системах. Отримано i проаналiзовано фазову дiаграму та дiелектричну
сприйнятливiсть змiшаної системи.
Ключовi слова: сегнетоелектрики, твердi розчини, фазова дiаграма, дипольне скло
PACS: 75.10.Hk, 77.22.Ch, 77.84.F
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