Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation
Temperature dependences of the optical path difference, variable part of the refractive index and thickness of triglycine sulphate crystal for three crystallophysic directions are studied in the temperature range of 39–70 °C, including the ferroelectric phase transition at Tc=49 °C, using the Jamen...
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irk-123456789-1210092017-06-14T03:06:07Z Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation Andriyevsky, B.V. Myshchyshyn, O.Ya. Romanyuk, M.O. Temperature dependences of the optical path difference, variable part of the refractive index and thickness of triglycine sulphate crystal for three crystallophysic directions are studied in the temperature range of 39–70 °C, including the ferroelectric phase transition at Tc=49 °C, using the Jamen type optical interferometer. Temperature dependences of the spontaneous changes of the characteristics studied in the range of 39–49 °C are fitted by the power-like low Y ∼ τ²β with doubled averaged effective critical indices 2β=0.87–0.95. The 2β values being different from the unity is explained by the essential temperature dependence of the coefficients of electrooptic, inverse piezoelectric and electrostriction effects in the range close to the phase transition point. Досліджено температурні залежності оптичної різниці ходу, змінної частини показника заломлення і товщини кристала тригліцинсульфату для трьох кристалофізичних напрямів в області температур 39– 70 °C, що містить температуру Tc=49 °C сегнетоелектричного фазового переходу, використовуючи оптичний інтерферометр типу Жамена. Температурні залежності спонтанних змін досліджуваних характеристик в області 39–49 °C апроксимовані степеневими залежностями Y ∼ τ²β з подвоєними засередненими ефективними критичними індексами 2β = 0.87 − 0.95. Відмінність 2β від одиниці пояснюється суттєвою температурною залежністю поблизу точки фазового переходу коефіцієнтів електрооптичного, оберненого п’єзоелектричного ефектів та електрострикції. 1999 Article Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation / B.V. Andriyevsky, O.Ya. Myshchyshyn, M.O. Romanyuk // Condensed Matter Physics. — 1999. — Т. 2, № 4(20). — С. 693-702. — Бібліогр.: 7 назв. — англ. 1607-324X DOI:10.5488/CMP.2.4.693 PACS: 77.80.Bh, 77.84.Fa, 78.20.Ci http://dspace.nbuv.gov.ua/handle/123456789/121009 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Temperature dependences of the optical path difference, variable part of the refractive index and thickness of triglycine sulphate crystal for three crystallophysic directions are studied in the temperature range of 39–70 °C, including the ferroelectric phase transition at Tc=49 °C, using the Jamen type optical interferometer. Temperature dependences of the spontaneous changes of the characteristics studied in the range of 39–49 °C are fitted by the power-like low Y ∼ τ²β with doubled averaged effective critical indices 2β=0.87–0.95. The 2β values being different from the unity is explained by the essential temperature dependence of the coefficients of electrooptic, inverse piezoelectric and electrostriction effects in the range close to the phase transition point. |
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Andriyevsky, B.V. Myshchyshyn, O.Ya. Romanyuk, M.O. |
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Andriyevsky, B.V. Myshchyshyn, O.Ya. Romanyuk, M.O. Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation Condensed Matter Physics |
| author_facet |
Andriyevsky, B.V. Myshchyshyn, O.Ya. Romanyuk, M.O. |
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Andriyevsky, B.V. |
| title |
Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation |
| title_short |
Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation |
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Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation |
| title_fullStr |
Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation |
| title_full_unstemmed |
Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation |
| title_sort |
anisotropy of critical indices of ferroelectric phase transition in tgs crystals by the optical interference investigation |
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Інститут фізики конденсованих систем НАН України |
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1999 |
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Anisotropy of critical indices of ferroelectric phase transition in TGS crystals by the optical interference investigation / B.V. Andriyevsky, O.Ya. Myshchyshyn, M.O. Romanyuk // Condensed Matter Physics. — 1999. — Т. 2, № 4(20). — С. 693-702. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T19:01:01Z |
| last_indexed |
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1837106476540231680 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 4(20), pp. 693–702
Anisotropy of critical indices of
ferroelectric phase transition in TGS
crystals by the optical interference
investigation
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
The Ivan Franko National University of Lviv,
8 Kyryla and Mefodiya Str., 79005 Lviv, Ukraine
Received January 13, 1999, in final form September 14, 1999
Temperature dependences of the optical path difference, variable part of
the refractive index and thickness of triglycine sulphate crystal for three
crystallophysic directions are studied in the temperature range of 39–70 ◦C,
including the ferroelectric phase transition at Tc=49 ◦C, using the Jamen
type optical interferometer. Temperature dependences of the spontaneous
changes of the characteristics studied in the range of 39–49 ◦C are fitted by
the power-like low Y ∼ τ2β with doubled averaged effective critical indices
2β=0.87–0.95. The 2β values being different from the unity is explained by
the essential temperature dependence of the coefficients of electrooptic,
inverse piezoelectric and electrostriction effects in the range close to the
phase transition point.
Key words: ferroelectrics, phase transition, optical properties, critical
indices
PACS: 77.80.Bh, 77.84.Fa, 78.20.Ci
1. Introduction
It is known, that the critical behaviour of the spontaneous polarization Ps at the
2nd order phase transition (PT) in a crystal can be described by the critical index
β,
Ps ∼ (Tc − T )β, (1)
where Tc is the PT temperature [1]. Here, index β can be treated as asymptotic or
effective one [2]. The effective critical index βeff is determined to be the derivative
βeff(τ) = d[ln(Ps)]/d[ln τ ], (2)
taken at some temperature T < Tc. Here, τ = (Tc − T )/Tc. The effective critical
index βeff is becoming the asymptotic one, when the value τ is approaching zero
c© B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk 693
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
(τ < 10−3) [2].
On the other hand, experimental temperature dependences of crystal parameters
in the range of PT can be also presented with the help of coefficients of thermo-
dynamic potential expansion. According to the Landau theory, the elastic Gibbs
energy G1 in the case of uniaxial ferroelectric can be expressed in the polynomial
form
G1 =
1
2
AD2 +
1
4
BD4 +
1
6
CD6, (3)
where D is electric displacement, A, B and C are coefficients [3]. Here, the energy
is measured from the non-polar phase and the polynomial is arbitrarily terminated
at D6. Dielectric state equation E = ∂G1/∂D takes the following form
E = AD +BD3 + CD5, (4)
where E is electric field. In the first approximation, which is in many cases satisfying,
we assume:
A = A0(T − Tc). (5)
Coefficient A0 is usually determined from the measurements of dielectric constant
as a function of temperature in the paraelectric (PE) phase. In the ferroelectric (FE)
phase the coefficients B and C can be determined from the temperature dependence
of Ps (E=0,D=Ps). The equation of dielectric state (3) can be now written as follows
P 2
s = (B/2C)
{
[
1 + 4A0(Tc − T )C/B2
](1/2)
− 1
}
. (6)
To characterize the order of PT, the parameter V = B2/A0C can be used [3]. The
physical sense of this parameter is clearly visible in the description of the first-order
PT. The polarized state becomes stable at the temperature T0 = Tc+3/16(B2/A0C),
whereas the upper limit of a superheating is T1 = Tc + B2/4A0C. When the first-
order PT comes near the second-order PT, the absolute value of V decreases at the
tricritical point T0 = T1 = Tc.
Both sets of parameters, index 2β in formulae (1), and coefficients A, B, and C
in formulae (6), can be used for the approximation of temperature dependences of
spontaneous polarization of ferroelectrics near the PT point. In the first five columns
of table 1 the parameter V as well as the parameters A0, B, C of the equation of
state for some ferroelectric crystals are presented [3].
Based on the values of coefficients A, B and C presented in table 1, we have
calculated the temperature dependences of spontaneous polarization Ps(T ) by the
formula (6) in the range of 312–322 K. Then, using the dependence Ps(T ) obtained,
we have calculated the corresponding effective critical index βeff averaged in this
temperature range using the formula (2).
Experimental temperature changes of the optical path difference determined by
birefringence, D = l∆n, are frequentely identified in practice with the changes of
the birefringence ∆n. But the thickness l and birefringence ∆n can have different
temperature dependences. The temperature dependences of D(T ) and l(T ) are usu-
ally investigated in different experiments. This restricts the accuracy of determining
694
Anisotropy of critical indices
Table 1. Coefficients A, B and C of the equation of dielectric state (6), coeffi-
cient V [3], and the corresponding averaged effective critical index βeff for some
ferroelectrics calculated in the temperature range of (Tc − T )=10 K
Here: TGS – (NH2CH2COOH)3·H2SO4 ; TGSe – (NH2CH2COOH)3·H2SeO4 ;
DTGS47% – 47% deuterated TGS crystal ; DGN – (NH2CH2COOH)2·HNO3 ;
DMAAS – (CH3)2NH2Al(SO4)2·6H2O ; MAPBB – (CH3NH3)5Bi2 Br11 ; MAPCB
– (CH3NH3)5Bi2Cl11 ; TAAP – |Te(OH)6|·2|NH4H2PO4| ·|(NH4)2HPO4| ; CDP
– CsH2PO4.
Crystal A0 [Vm/K] B [Vm5/C3] C [Vm9/C5] V = B2/A0C βeff
TGS 3.7 · 107 7.5 · 1011 5 · 1015 3.03 0.38
TGSe 2.63 · 107 3.97 · 1010 3 · 1014 0.20 0.30
DTGS47% 3.4 · 108 3.1 · 1011 1 · 1015 2.76 0.30
DGN 1.26 · 107 6.1 · 1013 9.5 · 1015 3 · 103 0.50
DMAAS 3.69 · 107 3.5 · 1011 4.2 · 1015 0.79 0.33
MAPBB 5.57 · 107 1.64 · 1012 2.6 · 1016 1.88 0.36
MAPCB 11.56 · 107 7.2 · 1012 2.4 · 1017 1.88 0.37
TAAP 2.97 · 107 6.9 · 1011 1.5 · 1015 10.38 0.44
CDP 1.56 · 106 1.32 · 109 1.5 · 1013 0.07 0.28
the corresponding dependence of ∆n(T ) and l(T ), and makes it complicated to com-
pare these different parameters of crystal in the region of PT. We have suggested
the techniques of simultaneous determination of temperature dependences of the
variable part of the refractive index (n− 1) and the thickness l of a sample.
Temperature dependences of the refractive indices and the linear thermal expan-
sion of TGS in the range of PT have already been studied [4–6], but the correspond-
ing critical indices have not been determined. The goals of the present investigation
were the precise measurements of the temperature dependences of optical path dif-
ference (OPD) determined by the refractive index for TGS crystal in the range of
2nd order PT at 322K, calculating the temperature dependences of refractive in-
dices and linear thermal expansion for the principal crystallophysic directions, as
well as the study of these dependences using the corresponding effective critical
indices 2βeff .
2. Experimental
Temperature dependences of OPD, D = l(n − 1), determeined by the variable
part of refractive index (VPRI), (n − 1) = η, for two interfering beams, one of
which has passed through a sample studied, and the other one through the air, were
measured using the Jamen type home built interferometer (figure 1). In this case,
the OPD D and its temperature dependence D(T ) can be written as follows
D = l(n− 1) = lη, D(T ) = l(T )η(T ), (7)
695
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
where l is thickness of a sample. The laser light of the wavelength λ=632.8 nm was
used in the experiment.
0 2 4 6
0
2
4
6
8
1 0 9
1
9
2 3
9
8
9
3
7
999
4
10
�
Figure 1. Scheme of the experiment: 1 – He-Ne laser; 2 – polarizer; 3 – glass plates
of Jamen type interferometer; 4 – sample; 5 – diaphragms; 6 – photodiodes; 7 –
thermocouple; 8 – thermostat (furnace); 9 – electrical supply block; 10 – recording
block.
Proceeding from the relation (6), the temperature changes of relative OPD
∆D/D along three crystallophysic directions can be written in the form of a system
of linear equations
∆Dij
Dij
=
∆li
li
+
∆ηj
ηj
, (i, j = 1, 2, 3; i 6= j), (8)
where index i denotes the direction of light propagation, index j denotes the direc-
tion of light polarization. Based on the six temperature dependences ∆D ij/Dij mea-
sured we have determined the relative temperature changes of the thickness ∆li/li
and VPRI ∆ηj/ηj [7]. The results of the corresponding computer calculations have
shown that the relative errors of the temperature changes determination of thickness
δli/li and VPRI δηj/ηj caused by solving the system (8) did not exceed 5% of the
respective maximum magnitudes ∆li/li and ∆ηj/ηj for the case of TGS crystal. The
initial li and ηj values were measured independently at the initial temperature T0.
In our case (l ≈ 5 mm and n ≈ 1.5) the error of determining the interference order
was δm(T ) 6 1/4, which corresponds to the errors of δD/D ∼ δl/l ∼ δη/η ∼ 10−5.
3. Results and discussion
Temperature dependences of the relative changes of OPD ∆Dij/Dij for TGS
crystal are shown in figure 2. Refractive indices nj(T0) of TGS crystal were taken
from the paper [6]. The forms of temperature dependences of the thickness ∆li/li
696
Anisotropy of critical indices
20 30 40 50 60 70
-60
-40
-20
0
20
40
32
31
23
21
13
12
10
4 (
∆
D
i j
/
D
i j
)
T (oC)
Figure 2. Experimental temperature dependences of the relative changes of the
optical path difference ∆Dij/Dij of TGS crystal (indices ij indicate the corre-
sponding curves)
and VPRI δηj/ηj calculated from the system of equations (8) are characterized
by the similar anomalies at PT temperature. Based on the known relation for the
temperature changes of the order parameter p = Ps for 2nd order PT in the range
of T < Tc,
∆Ys ∼ P 2
s ∼ τ 2β =
(
Tc − T
Tc − Tmin
)2β
, (9)
we have calculated the double effective critical indices 2β averaged in the range of
39–49 ◦C, replacing P 2
s value by the spontaneous increases of ∆Ys(T )/∆Ys(Tmin)
(Y=D, l and η). Here Tc=49 ◦C is the temperature of PT, Tmin is the lower edge of
the temperature range studied (Tmin=39 ◦C in our case), ∆Ys(T ) and ∆Ys(Tmin) are
spontaneous increments corresponding to the Tc and Tmin temperatures. The double
critical indices 2β of TGS in the range of 39–49 ◦C are shown in table 2.
Table 2. Effective critical indices 2β, corresponding to the temperature depen-
dences of spontaneous increments of ∆Ds/D, ∆ls/l and ∆ηs/η for different crys-
tallophysic direction of TGS crystal
2β
(D)
12 2β
(D)
13 2β
(D)
21 2β
(D)
23 2β
(D)
31 2β
(D)
32
0.90 0.90 0.89 0.89 0.95 0.92
2β
(l)
1 2β
(l)
2 2β
(l)
3 2βη
1 2βη
2 2βη
3
0.91 0.90 0.92 0.88 0.93 0.88
The results obtained testify to an inexact fulfilment of the functional dependences
for quadratic electrooptic effect
∆ns ∼ P 2
s (10)
697
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
and electrostriction
∆ls ∼ P 2
s . (11)
If these effects were displayed in the form indicated, the double critical index 2β
would be equal to unity, 2β=1. Let us try to explain the experimental facts obtained.
Analytical relation of the observed temperature dependence of OPD ∆Ds/D
induced by spontaneous polarization can be presented in the most common form
∆Ds/D(τ) = a(τ)P 2
s (τ) = a(τ)τ, (12)
where a(τ) is the temperature dependent coefficient. It follows from the character
of experimental dependences of spontaneous increases of ∆Ds/D, ∆ls/l, and ∆ηs/η,
that the corresponding coefficients a(τ) are maximal in the region of PT (figure 3).
0.0 0.5 1.0
0.8
1.2
1.6
d(
∆ D
s3
2/
D
32
)/
dτ
τ=(T
c
-T)/T
c
Figure 3. Temperature dependence of the derivative d(∆Ds32/D32)/dτ in the
ferroelectric phase of TGS crystal
To obtain the additional proofs of the validity of this viewpoint, we have per-
formed experimental study of the artificially induced electrooptic effect in TGS crys-
tal in the temperature range of 30–65 ◦C. This investigation was carried out using
the same optical scheme. The external electric field of the magnitude E ≈ 3.5 kV/cm
was applied to the sample of TGS crystal at different temperatures along the [010]-
direction of spontaneous polarization Ps, and the corresponding induced increments
of the OPD ∆De/D were measured. The maximum-like temperature dependence of
∆De/D value (figure 4) correlates well with the temperature dependence of a(τ) in
the ferroelectric phase. This maximum-like character of the coefficient mentioned is
connected with the inequality 2β < 1.
Taking into account that the temperature dependences of spontaneous incre-
ments of ∆Ys/Y parameters can be presented in two forms, Y (τ) = τ 2β , and
Y (τ) = a(τ)τ , one can obtain the relation for the temperature dependence of a(τ)
coefficient
a(τ) = τ 2β−1. (13)
In the cases of 2β 6= 1 and τ 6= 1, the decreasing temperature dependence of the
coefficient a(τ) takes place in the ferroelectric phase at the removal from the PT
point τ = 0 (figure 3).
698
Anisotropy of critical indices
35 40 45 50 55 60 65
0.00
0.01
0.02
0.03
∆ D
e3
2/
D
32
T (oC)
Figure 4. Temperature dependence of the relative optical path difference
∆De32/D32 of TGS crystal induced by the constant electric field of 3.5 kV/cm
along the [010]-direction
Taking into account all the results obtained, we can summarize that the temper-
ature dependences of the coefficients of quadratic electrooptic and electrostriction
effects for TGS crystals take place (see relations (10) and (11)). The analysis of
table 1 testifies to some segregation of the [010]-direction of spontaneous polariza-
tion. Really, among the temperature changes of spontaneous increments ∆li and
∆ηj (i, j = 1, 2, 3), the dependence ∆l2(τ) is characterised by the least index 2β
while the dependence ∆n2(τ) is characterised by the greatest index (see table 2).
On the other hand, a proximity of the values 2β
(l)
2 ≈ 2βη
2 is observed (see table 1),
whereas obvious inequalities of similar characteristics for the other two crystallo-
physic directions 2β
(l)
1,3 > 2βη
1,3 take place (see table 2 and figure 4).
The latter features can be interpreted as a different rate of the ordering of two
subsystems. One subsystem relates to the electrons forming the refractive index n
while the other is connected with geometric parameters of the crystal unit cell for the
directions [100] and [001]. The equality 2β
(l)
2 ≈ 2βη
2 for the direction of spontaneous
polarization [010] can be interpreted as good correlation of the subsystems in TGS
crystal mentioned above. From such a viewpoint, the observable inequalities of the
indices 2β
(l)
1,3 > 2βη
1,3 testify to various rates of the temperature changes of the
corresponding subsystems of the crystal in the temperature range (∆T ∼ 10 ◦C)
below PT point. It is seen in figure 5, in case of z-direction of the crystal studied.
A crossing of the curves on figure 6 corresponds to two different indices β
(l)
3 and
βη
3 . Such a crossing will take place in all cases if the experimental temperature
dependence of the parameters studied (V = ∆Ds/D, ∆ls/l, ∆ηs/η) is described by
different indices β.
We suppose that such a peculiarity in the temperature dependence of different
parameters can be characteristic of the ordering of the other ferroelectric crystals.
699
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
n
3
l
3
δy
s(
T
)/
δy
s(
T
m
in
)
(T
c
-T)/(T
c
-T
min
)
Figure 5. Dependences of the normalized spontaneous changes of thickness (l3)
and changes of the variable part of refractive index (η3) of TGS for [001]-direction
on the normalised temperature (Tc − T )/(Tc − Tmin) in the range of 39–49 ◦C
0.0 0.5 1.0
0.5
1.0
1.5
2.0
dη
3
/dτ
dl
3
/dτ
dV
/d
τ
τ=(T
c
-T)/(T
c
-T
min
)
Figure 6. Temperature derivatives of the dependences from figure 5
700
Anisotropy of critical indices
4. Conclusion
1. The original laser interferometer techniques of Jamen type for measuring the
temperature change of the optical path difference of a transparent sample is
offered. The techniques makes it possible to define the temperature depen-
dences of thickness l(T ) and the variable part of refractive index [n(T ) − 1]
of the crystal based on the measurements of temperature dependences of the
optical path difference D(T ) = l(T )[n(T ) − 1] for different directions of light
propagation and polarization.
2. Deviation from the unity of the double effective critical index 2β for the tem-
perature dependence of the optical path difference induced by a spontaneous
polarization in TGS sample is explained by a significant temperature depen-
dence of the maximum-like character of the coefficient of electrooptic, inverse
piezooptic, and electrostriction effects.
3. An anisotropy of the critical indices 2β
(l)
i and 2β
(η)
i , and nonequality 2β
(l)
i 6=
2β
(η)
i testify to different rates of temperature changes of different crystal sub-
systems taking place at the ferroelectric ordering in the range of ∆T ∼ 10 ◦C
below Tc.
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Publ., 1991 (in Russian).
701
B.V.Andriyevsky, O.Ya.Myshchyshyn, M.O.Romanyuk
Анiзотропiя критичних iндексiв
сегнетоелектричного фазового переходу в
кристалах ТГС визначена з
оптико-інтерференційних досліджень
Б.В.Андрiєвський, О.Я.Мищишин, М.О.Романюк
Львівський національний університет ім. І.Франка,
79005 Львів, вул. Кирила і Мефодія, 8
Отримано 13 січня 1999 р., в остаточному вигляді –
14 вересня 1999 р.
Досліджено температурні залежності оптичної різниці ходу, змінної
частини показника заломлення і товщини кристала тригліцинсуль-
фату для трьох кристалофізичних напрямів в області температур 39–
70 ◦C, що містить температуру Tc=49 ◦C сегнетоелектричного фазо-
вого переходу, використовуючи оптичний інтерферометр типу Жа-
мена. Температурні залежності спонтанних змін досліджуваних ха-
рактеристик в області 39–49 ◦C апроксимовані степеневими залеж-
ностями Y ∼ τ2β з подвоєними засередненими ефективними кри-
тичними індексами 2β = 0.87 − 0.95. Відмінність 2β від одиниці по-
яснюється суттєвою температурною залежністю поблизу точки фа-
зового переходу коефіцієнтів електрооптичного, оберненого п’єзо-
електричного ефектів та електрострикції.
Ключові слова: сегнетоелектрики, фазові переходи, оптичні
властивості, критичні індекси
PACS: 77.80.Bh, 77.84.Fa, 78.20.Ci
702
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