Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions
The problem of equation of state for ferroelectric-antiferroelectric mixed systems in the whole region of a concentration change (06n61) is discussed. The main peculiarity of the presented model turns out to be the possibility for the site dipole momentum to be oriented ferroelectrically in z-dire...
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irk-123456789-1213132017-06-15T03:04:33Z Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions Korynevskii, N.A. Solovyan, V.B. The problem of equation of state for ferroelectric-antiferroelectric mixed systems in the whole region of a concentration change (06n61) is discussed. The main peculiarity of the presented model turns out to be the possibility for the site dipole momentum to be oriented ferroelectrically in z-direction and antiferroelectrically in x-direction. Such a situation takes place in mixed compounds of KDP type. The different phases (ferro-, antiferro-, paraelectric, dipole glass and some combinations of them) have been found and analyzed. 2006 Article Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions / N.A. Korynevskii, V.B. Solovyan // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 187–191. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 75.10.Hk, 77.22.Ch, 77.84.F DOI:10.5488/CMP.9.1.187 http://dspace.nbuv.gov.ua/handle/123456789/121313 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The problem of equation of state for ferroelectric-antiferroelectric mixed systems in the whole region of a
concentration change (06n61) is discussed. The main peculiarity of the presented model turns out to be the
possibility for the site dipole momentum to be oriented ferroelectrically in z-direction and antiferroelectrically
in x-direction. Such a situation takes place in mixed compounds of KDP type. The different phases (ferro-,
antiferro-, paraelectric, dipole glass and some combinations of them) have been found and analyzed. |
| format |
Article |
| author |
Korynevskii, N.A. Solovyan, V.B. |
| spellingShingle |
Korynevskii, N.A. Solovyan, V.B. Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions Condensed Matter Physics |
| author_facet |
Korynevskii, N.A. Solovyan, V.B. |
| author_sort |
Korynevskii, N.A. |
| title |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions |
| title_short |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions |
| title_full |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions |
| title_fullStr |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions |
| title_full_unstemmed |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions |
| title_sort |
ferroelectric-antiferroelectric mixed systems. equation of state, thermodynamic functions |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121313 |
| citation_txt |
Ferroelectric-antiferroelectric mixed systems. Equation of state, thermodynamic functions / N.A. Korynevskii, V.B. Solovyan // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 187–191. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
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AT korynevskiina ferroelectricantiferroelectricmixedsystemsequationofstatethermodynamicfunctions AT solovyanvb ferroelectricantiferroelectricmixedsystemsequationofstatethermodynamicfunctions |
| first_indexed |
2025-07-08T19:38:12Z |
| last_indexed |
2025-07-08T19:38:12Z |
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1837108817421139968 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 187–191
Ferroelectric-antiferroelectric mixed systems. Equation
of state, thermodynamic functions
N.A.Korynevskii1,2, V.B.Solovyan1
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., Lviv 79011, Ukraine,
2 Institute of Physics, University of Szczecin,
15 Wielkopolska Str., 70451 Szczecin, Poland
Received August 19, 2005
The problem of equation of state for ferroelectric-antiferroelectric mixed systems in the whole region of a
concentration change (06n61) is discussed. The main peculiarity of the presented model turns out to be the
possibility for the site dipole momentum to be oriented ferroelectrically in z-direction and antiferroelectrically
in x-direction. Such a situation takes place in mixed compounds of KDP type. The different phases (ferro-,
antiferro-, paraelectric, dipole glass and some combinations of them) have been found and analyzed.
Key words: ferroelectrics and antiferroelectrics, mixed systems, equation of state
PACS: 75.10.Hk, 77.22.Ch, 77.84.F
The present paper is a continuation of our recent investigations of ferroelectric-antiferroelectric
mixed systems with hydrogen bonds [1–3]. There is an abundant experimental material concerning
phase diagrams and the behaviour of thermodynamic functions for this mixed crystals [4–8]. The
phase diagrams demonstrate the coexistence of pure and mixed ferro-, antiferro-, paraelectric,
dipole glass phases in a wide region of temperatures and concentrations. A theoretical description
of such systems is in its initial stage so far. The main approach is based on the statistic replica
method for calculating the thermodynamic functions for a mixed system with non-equilibrium
distribution of differently occupied sites, similar to the ones exploited in mixed ferromagnetic
system investigations [9].
It is well known that short range interparticle interactions play an important role in ferro-
electrics (especially in hydrogen bonded ones). So, in order to adequately describe the ferroelectric-
antiferroelectric mixed systems we should take into account both the short and long range inter-
actions. An effective method was proposed in [2], where the model Hamiltonian includes two
subsystems of particles with dipole momenta perpendicular to each other, with short and long
range interaction between them and which depend on the probability of occupying each lattice
site. In the second order of replica expansion for −βH (β = (kT )
−1
, k is the Boltzmann constant
and T is the absolute temperature) the free energy of the system investigated has been obtained.
The equations for ferroelectric P = 〈Sz〉, antiferroelectric ξ = 〈Sx〉 order parameters and dipole
glass gz = 〈Sz
miS
z
m′i〉, gx = 〈Sx
miS
x
m′i〉 parameters were found. Here 〈· · ·〉 denotes thermal averaging
for a given distribution of z- and x-oriented dipole momenta of particles whereas a bar denotes a
stochastic averaging over different distributions; m, m′ denotes the nearest neighbours of particles
in i-th size. The obtained phase diagram (the temperature of the corresponding order parameter
appearing versus concentration) shows a possibility of different phases occurring: ferro-, antiferro-,
paraelectric, dipole glass and some combinations of them.
The purpose of the present paper is to perform a continual study of ferroelectric-antiferroelectric
mixed system taking into account the interaction processes between both subsystems, i.e. ferro-
electric and antiferroelectric ones. In this approach and using the replica method [9,10] we will
obtain a phase diagram, the equation of state and expressions for base thermodynamic functions
c© N.A.Korynevskii, V.B.Solovyan 187
N.A.Korynevskii, V.B.Solovyan
of the system investigated. We will compare our results to the known experimental data concerning
mixed Rubidium Ammonium Dihydrogen Arsenate Rbn(NH4)1−nH2AsO4 (RADA).
Taking the two-particle cluster approximation (see [11]) for short range interaction among
particles and the self-consistent field approximation with the accuracy up to the second order of
replica expansion for long range one we have for −βH the expression which contains the terms
describing the correlation between ferroelectric and antiferroelectric subsystems
−βH =
N
∑
i=1
{
A
(i)
1 Sz
1iS
z
2i + B
(i)
1 (Sz
1i + Sz
2i) + A
(i)
2 Sx
1iS
x
2i + B
(i)
2 (Sx
1i − Sx
2i) − J1
Vf
V
P 2
− (tanhβV − J1)
Uaf
V
ξ2 −
J2
2
[
−2 +
(
gz −
U
V
gx
)2
+
+
(
Vf + Vaf
V
+
Vf − Vaf
V
gz
)2
+
(
Uf + Uaf
V
+
Uf − Uaf
V
gx
)2
+24
(
Vf
V
P 2 −
Uaf
V
ξ2
)2
]}
. (1)
Here
J1 =
∞
∑
t=1
(−1)t2 ln
(
1 − n + ne−2βV t
)
,
J2 =
∞
∑
t=1
(−1)t4t ln
(
1 − n + ne−2βV t
)
(2)
are the functions which describe a relative contribution of each phase into the general properties
of a mixed system (J1) and a correlation between those phases (J2), n is a concentration of
ferroelectrically ordered particles.
Vf =
1
2
N
∑
j=1
(
V 11
ij + V 12
ij
)
, Uf =
1
2
N
∑
j=1
(
U11
ij + U12
ij
)
,
Vaf =
1
2
N
∑
j=1
(
V 11
ij − V 12
ij
)
, Uaf =
1
2
N
∑
j=1
(
U11
ij − U12
ij
)
(3)
and V and U are the energies of intrasite interactions between particles for both subsystems
A
(i)
1 = J1 + J2
[
V 2
f − V 2
af
V 2
+
(
1 +
(
Vf − Vaf
V
)2
)
gz −
U
V
gx
]
,
B
(i)
1 = J1
Vf
V
P + 4J2
[
(
Vf
V
)2
P 3 −
Vf
V
Uaf
V
Pξ2
]
,
A
(i)
2 = (tanhβV − J1)
U
V
+ J2
[
U2
f − U2
af
V 2
+
(
(
U
V
)2
+
(
Uf − Uaf
V
)2
)
gx −
U
V
gz
]
,
B
(i)
2 = (tanhβV − J1)
Uaf
V
ξ + 4J2
[
(
Uaf
V
)2
ξ3 −
Vf
V
Uaf
V
P 2ξ
]
. (4)
Performing the trace operation in the formula for free energy
F = −
1
β
lnTre−βH (5)
188
Ferroelectric-antiferroelectric mixed systems
over cluster states of spin operators Sz
fi, Sx
fi and minimizing the obtained expression with respect
to all order parameters P , ξ, gz, gx the set of equations of state can be written
P = Z−1eA1 sinh 2B1
(
eA2 + e−A2 cosh 2B2
)
,
ξ = Z−1e−A2 sinh 2B2
(
e−A1 + eA1 cosh 2B1
)
,
gz = −Z−1
(
e−A1 − eA1 cosh 2B1
) (
eA2 + e−A2 cosh 2B2
)
,
gx = Z−1
(
e−A1 + eA1 cosh 2B1
) (
eA2 − e−A2 cosh 2B2
)
,
Z =
(
e−A1 + eA1 cosh 2B1
) (
eA2 + e−A2 cosh 2B2
)
. (6)
To solve the set (6) in general form is a rather complicated problem. But the phase diagram of
the system investigated can be obtained as a line dividing different possible phases in coordinates:
concentration – temperature. Such diagram (approximated to a real RADA system) is presented
in figure 1. One can observe the existence of paraelectric (P ), ferroelectric (F ), antiferroelectric
(AF ), dipole glass (G) phases and some combinations thereof: ferroelectric-dipole glass (F − G),
antiferroelectric-dipole glass (AF −G). It should be noted that the occurrence of dipole glass phase
is fixed as a point of bifurcation of the number of roots of the system (6) for correlation functions of
the nearest neighbours gz and gx. So, at low temperatures a uniform kind of short range correlation
changes into the nonuniform one described by a set of functions gz, gx.
Figure 1. Phase diagram of RADA built for the model Hamiltonian parameters; Vf = 70 K,
Vaf = −80 K, Uf = −40 K, Uaf = 120 K, V = 65 K, U = −210 K.
The comparison of the obtained phase diagram with the ones presented in [2,3] and with the
experimental ones [7,12] shows a better coincidence of curves in figure 1 with the experiment for
RADA than the corresponding curves in [2,3]. However, there is some quantitative discrepancy in
the critical points of concentration where the ferroelectric phases appear. Namely, this theoretical
critical value is equal to 0.76–0.80 whereas the experimental one is only 0.82–0.90. At the same time
the theoretical and experimental points of the occurrence of concentration for antiferroelectric phase
are practically the same (0.50–0.52). It must be noted that the theoretical and experimental critical
concentrations completely coincide for deuterated material DRADA. The second fact which should
be noted is a somewhat larger (about 20–25%) calculated temperature of dipole glass compared to
the experimental one. However, in our opinion, this difference in the calculated and in the observed
values is not of great importance.
We should like to underline a sharply asymmetric behaviour of a phase diagram in figure 1,
caused not only by the difference in the phase transition temperatures for pure crystals (for RDA
Tc = 110 K and for ADA TN = 216 K) but also, in our observations, due to the mixed system evi-
dently tending to antiferroelectric type of ordering instead of the ferroelectric one at corresponding
(small or large n) regions of concentration. As a result, the percolation phenomena for both these
phases take place at nonsymmetric points.
189
N.A.Korynevskii, V.B.Solovyan
Figure 2. Ferroelectric order parameters: ver-
sus temperature for some fixed concentrations;
curves: 1−n = 0.80, 2−n = 0.90, 3−n = 0.99.
The model Hamiltonian parameters are the
same as in figure 1.
Figure 3. Antiferroelectric order parameters
versus temperature for some fixed concentra-
tions; curves: 1 − n = 0.01, 2 − n = 0.10,
3 − n = 0.30. The model Hamiltonian param-
eters are the same as in figure 1.
The temperature dependencies of ferroelectric (P ) and antiferroelectric (ξ) order parameters
at certain fixed concentrations are presented in figure 2 and figure 3. It is natural that the tem-
perature of corresponding phase transitions (turning P or ξ into zero) decreases when the relative
concentration of ferroelectrically or antiferroelectrically ordering particles becomes lower. But one
can also observe the changes in critical behaviour of order parameters under concentration. For
both types of ordering those changes show a constant tendency: order parameters behave more
smoothly when the corresponding concentration increases. So, we constantly observe the second
order phase transitions for pure crystals and the second order phase transition close to the first
order one in a frustration region of concentration. It is interesting to note that short range inter-
particle interactions are similar to the effect of a partial omission of interacting sites of a crystal. In
a deluted crystal the role of short range forces is more important compared to the long range ones.
Figure 4. Ferroelectric order parameters ver-
sus concentration for some fixed temperatures;
curves: 1 − T = 98 K, 2 − T = 100 K,
3 − T = 102 K, 4 − T = 104 K. The mod-
el Hamiltonian parameters are the same as in
figure 1.
Figure 5. Antiferroelectric order parameters
versus concentration for some fixed tempera-
tures; curves: 1 − T = 185 K, 2 − T = 190 K,
3 − T = 195 K, 4 − T = 200 K. The mod-
el Hamiltonian parameters are the same as in
figure 1.
The concentration behaviours of both order parameters are presented in figure 4 and figure 5.
Those curves are built at fixed temperatures for which the corresponding ordered phases exist.
190
Ferroelectric-antiferroelectric mixed systems
The temperature and concentration behaviours of other thermodynamic functions of the system
investigated are strictly dependent on the behaviour of P and ξ.
References
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2. Korynevskii N.A., Baran O.R., Ferroelectrics, 2004, 300, 151.
3. Korynevskii N.A., Solovyan V.B., Ferroelectrics, 2005, 317, 19.
4. Trybula Z., Stankowski J., Blinc R., Ferroelectric Lett., 1986, 6, 57.
5. Kim S., Kwun S., Phys. Rev. B, 1990, 42, 638.
6. Trybula Z., Schmidt V., Drumheller J., Phys. Rev. B, 1991, 43, 1287.
7. Trybula Z., Stankowski J., Condens. Matter Phys., 1998, 1, 311.
8. Lanceros-Mendes S., Schmidt V., Shapiro S., Ferroelectrics, 2004, 300, 117.
9. Edwards S.F., Anderson P.W., J. Phys. E, 1975, 5, 965.
10. Bidaux R., Carton J.P., Sarma G., J. Phys. A: Math. Gen., 1976, 9, L87.
11. Yukhnovskii I.R., Korynevskii N.A., Phys. Stat. Sol. (b), 1989, 153, 583.
12. Trybula Z., Kaszynski J., Maluszynska H., Ferroelectrics, 2005, 316, 125.
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PACS: 75.10.Hk, 77.22.Ch, 77.84.F
191
192
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