The final stages of polarization and monodomain state stability of the Rochelle salt crystals
Based on the direct domain observations in the Rochelle salt crystals, the final stages of polarization and the dependence of polarized state stability on the electric field strength E and on the electric pulse duration is investigated. The conclusion is made that there exists a high electric fi...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | The final stages of polarization and monodomain state stability of the Rochelle salt crystals / M.O. Romanyuk, M.M. Romanyuk // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 751-760. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1206872017-06-13T03:03:44Z The final stages of polarization and monodomain state stability of the Rochelle salt crystals Romanyuk, M.O. Romanyuk, M.M. Based on the direct domain observations in the Rochelle salt crystals, the final stages of polarization and the dependence of polarized state stability on the electric field strength E and on the electric pulse duration is investigated. The conclusion is made that there exists a high electric field strength, at which small domains are displaced. At higher fields, electrooptical effects are obviously developed and the invisible network of defects is changed. The stability of the polarized state and the degree of polarization are determined by the electric field pulse, the duration τ and the electric strength of which E are connected by equation ln τ ∼ k/E + b . For incomplete polarization, the rate of polarization decay when the pulse is finished is proportional to the deviation of the domains from the initial state of the sample. На основі прямого спостереження доменів кристалів сегнетової солі досліджено прикінцеві стадії поляризації та залежність стійкості поляризованого стану від напруженості поля і тривалості імпульса. Зроблено висновок, що існує високе поле витіснення дрібних доменів, після чого явно проявляються електрооптичні ефекти та зазнає змін невидима сітка дефектів. Стійкість поляризованого стану та ступінь поляризації визначаються імпульсами поля, тривалість т і напруженість E якого пов’язані співвідношенням ln τ ∼ k/E + b. При неповній поляризації зразка втрата останньої при знятті поля пропорційна до відхилення доменів від вихідного стану зразка. 2002 Article The final stages of polarization and monodomain state stability of the Rochelle salt crystals / M.O. Romanyuk, M.M. Romanyuk // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 751-760. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 75.60.-d, 77.22.Ej, 61.72.Bb, 75.60.Ch DOI:10.5488/CMP.5.4.751 http://dspace.nbuv.gov.ua/handle/123456789/120687 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Based on the direct domain observations in the Rochelle salt crystals, the
final stages of polarization and the dependence of polarized state stability
on the electric field strength E and on the electric pulse duration is
investigated.
The conclusion is made that there exists a high electric field strength, at
which small domains are displaced. At higher fields, electrooptical effects
are obviously developed and the invisible network of defects is changed.
The stability of the polarized state and the degree of polarization are determined
by the electric field pulse, the duration τ and the electric strength
of which E are connected by equation ln τ ∼ k/E + b . For incomplete
polarization, the rate of polarization decay when the pulse is finished is
proportional to the deviation of the domains from the initial state of the
sample. |
| format |
Article |
| author |
Romanyuk, M.O. Romanyuk, M.M. |
| spellingShingle |
Romanyuk, M.O. Romanyuk, M.M. The final stages of polarization and monodomain state stability of the Rochelle salt crystals Condensed Matter Physics |
| author_facet |
Romanyuk, M.O. Romanyuk, M.M. |
| author_sort |
Romanyuk, M.O. |
| title |
The final stages of polarization and monodomain state stability of the Rochelle salt crystals |
| title_short |
The final stages of polarization and monodomain state stability of the Rochelle salt crystals |
| title_full |
The final stages of polarization and monodomain state stability of the Rochelle salt crystals |
| title_fullStr |
The final stages of polarization and monodomain state stability of the Rochelle salt crystals |
| title_full_unstemmed |
The final stages of polarization and monodomain state stability of the Rochelle salt crystals |
| title_sort |
final stages of polarization and monodomain state stability of the rochelle salt crystals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120687 |
| citation_txt |
The final stages of polarization and
monodomain state stability of the
Rochelle salt crystals / M.O. Romanyuk, M.M. Romanyuk // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 751-760. — Бібліогр.: 18 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T18:24:20Z |
| last_indexed |
2025-07-08T18:24:20Z |
| _version_ |
1837104173041057792 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 4(32), pp. 751–760
The final stages of polarization and
monodomain state stability of the
Rochelle salt crystals
M.O.Romanyuk 1 , M.M.Romanyuk 2
1 The Ivan Franko National University of Lviv,
8 Kyrylo i Mefodii Str., 79005 Lviv, Ukraine
2 National University “Lvivska Politekhnika”,
12 Bandera Str., 79013 Lviv, Ukraine
Received August 30, 2002, in final form October 11, 2002
Based on the direct domain observations in the Rochelle salt crystals, the
final stages of polarization and the dependence of polarized state stabil-
ity on the electric field strength E and on the electric pulse duration is
investigated.
The conclusion is made that there exists a high electric field strength, at
which small domains are displaced. At higher fields, electrooptical effects
are obviously developed and the invisible network of defects is changed.
The stability of the polarized state and the degree of polarization are deter-
mined by the electric field pulse, the duration τ and the electric strength
of which E are connected by equation ln τ ∼ k/E + b . For incomplete
polarization, the rate of polarization decay when the pulse is finished is
proportional to the deviation of the domains from the initial state of the
sample.
Key words: domains, polarization, stability of the polarized state, defects
PACS: 75.60.-d, 77.22.Ej, 61.72.Bb, 75.60.Ch
1. Introduction
The attention to the twin structure of ferroelectrics has been renewed lately [1–
5]. Apart from its own significance, this is important for the investigation of the laws
of polarization of ferroics, and for measurements of fundamental characteristics of
materials (spontaneous polarization, electric permittivity, dielectric losses, coercive
field, piezoelectric, piezooptical and other parameters). Besides high-frequency pro-
cesses [6], a particular attention is now paid to the slow ones, in which appreciable
changes are possible in the defect network of the sample. In particular, for DKDP
crystals, a set of relaxation times of polarization (∼ 1 hour) is detected, and the
c© M.O.Romanyuk, M.M.Romanyuk 751
M.O.Romanyuk, M.M.Romanyuk
absence of an accurate value of coercive field [1] is stated. The investigation of the
domain motion has been started not only under the effect of an electric field, or a
mechanical stress associated with spontaneous polarization of the sample, but un-
der the effect of several stresses (ferrobielastics) [5]. Corresponding results can be
also important for classic investigations of polarization for the samples with some
internal mechanic stress.
Innovative solutions to the issues of domain formation and motion under the
effect of external fields are presented in [2,3] in spite of the numerous papers devoted
to these problems at the initial stages of twins investigation [7–14]. Specifically,
repolarization is interpreted as a phase transition of the second type.
As before, most of these papers have an indirect relation to the domains and to
the defects, because they do not deal with them directly.
2. Experimental techniques
In this paper the laws of growth, saturation, and spontaneous decay of the po-
larized state of the Rochelle salt (RS) crystals, were investigated by means of pho-
tographic and photo-electric registration of polarization process dynamics [10,11].
Using both of these techniques made it possible to enlarge the polarization time du-
ration interval from 104 to 10−3 s. The upper limit of polarization time duration is
difficult to reliably realize using a routine electronic registration of domain changes
due to the conductivity of the sample.
An example of photographic and photo-electric registration of the changes of the
domain structure of the Rochelle salt crystals is presented in figure 1.
Figure 1a illustrates the change of the domain areas during the decay of the
monodomain state under the effect of different shearing stresses Yz (marks 1, 2, 3).
Figure 1. An examples of photographic (a) and photoelectric (b) registration
of the time dependence of domains during the polarization of the Rochelle salt
crystals (dots on (b) – are time marks). Here K = Sl/(Sl +Sd), I – photocurrent,
which is proportional to K. Curves 1, 2, 3 – the kinetics of depolarization under
the action of the different shearing stresses Yz .
752
The final stages of polarization. . .
Figure 1b illustrates the changes of the normalized area of the light domains of
the sample under the effect of the Ex electric field, and the spontaneous decay of
the polarized state after the removal of the electric pulse. The general picture is
created by the shift of the corresponding sections of the oscillograph record along
the abscissa axis. The time distance between the time marks is 10−4 s. The similar
pictures are observed under the effect of the other fields, but with another time
duration between the time marks. In the presented figure, the duration of the whole
process is 10−2 s.
The field dependence of domain changes is described by the relationship between
the areas of light and dark components of the twins, or by changes of the photocur-
rent I, caused by the light flux, which passed through the system of the crossed
nicols and the investigated sample, which is put between the nicols in a diagonal
position in accordance with the following relation
I = I0
Sl
Sl + Sd
sin2 2ϕ sin2
πd (n2 − n3)
λ
. (1)
Here, Sl and Sd – areas of light and dark (installed for the extinction of light)
components of the twins, ϕ – an angle of spontaneous turn of the crystallophysic axis
of domain, d – thickness of sample, λ – wavelength of light, (n2−n3) – birefringence of
beams of light which propagate in the crystal along the polar axis X1, I0 – intensity
of the incident beam of light.
The values of the area of the light beam (Sl + Sd) and the intensity I0 were kept
constant for the whole set of experiments.
The values Sl, ϕ and (n2 − n3) depend on the electric field strength. Maximal
changes of intensity are related with Sl(E). The change of the axis of the domain
optical indicatrix ϕ is defined by the linear electrooptic effect (EOE) [15]
∆ϕ =
r41 · n
2
2
· n2
3
· E1
n2
2 − n2
3
, (2)
and the change of the main refractive indices (n2 − n3) – by relationship
∆(n2 − n3) ' −
n3
2
(R2211 − R3311) · E
2
1
, (3)
where r41 – an actual coefficient of the linear electrooptic effects; R – corresponding
coefficients of the quadratic electrooptic effect, E1 – the electric strength of the
external electric field, applied along the polar axis of the sample X1. The field
dependence of the sample’s thickness is neglected.
3. Field and time dependences of domains
The field dependences of polarization, described by the change of the twins areas,
or by the photocurrent I, are presented in figure 2 and figure 3. Here, the origination,
i.e., a visible side growth, and general mass domain changes at coercive field take
753
M.O.Romanyuk, M.M.Romanyuk
Figure 2. Polarization of the RS crystals in a weak fields for different pulse
duration: 1 – 2 s, 2 – 1 s, 3 – 0.4 s, 4 – 0.15 s, 5 – 0.07 s, 6 – 0.03 s. Temperature
15 ◦C.
Figure 3. An example of polarization of
RS crystals in the strong electric fields.
Areas 1, 2, 3 – correspond to the differ-
ent field changes of the domains. Ec –
coercive field, Es – saturation field.
place (figure 2), and then we observe
linear sections in the vicinity of satu-
ration field, and fixing of monodomain
state fields (figure 3). At relatively low
fields (< 200 V/cm) it is possible to ne-
glect the field changes of optical indica-
trix, but in the higher fields just these
changes are registered. In this case, it is
necessary to take into consideration the
clamping degree of the disappearing do-
mains, which is a simultaneous action of
electric and mechanic fields. Vanishing-
ly small slope of the middle area of the
plot in figure 3 can be caused by some
factors. Among them there are different
signs of electrooptic and piezooptic co-
efficients r41 and π2323 [16], and fixing of
the small domains on the defects, when
the area of domains is not changed.
At higher fields (E > 500 V/cm),
obviously the full monodomainization
takes place, and now the main factor be-
comes the deformation of the optical in-
dicatrix of one twin component.
754
The final stages of polarization. . .
Table 1. Relationship of amplitude E, polarizing pulse duration τi and the time
constant β(E) for the Rochelle salt crystals at 15 ◦C.
E, 50 80 130 163 195 235 315 500 500 1000
V/cm
τi, 7 · 103 360 2 1 0.4 0.07 0.03 5 · 10−4 15 · 10−4 25 · 10−5
s
β(E), 63 · 10−5 0.01 1.72 3.45 8.5 50 106 7 · 10−3 2500 14 · 103
s−1
The slope being a bit higher than the predicted slope of the last area of the plot
of polarization can be caused by the quadratic electrooptic effect. This explanation
is confirmed by the fact, that for different samples this area of the plot has a little bit
different slope. It is connected with the one of the three components of the relative
increase ∆I/I, which follows from (1) and depends on the difference in the optical
beam path in the sample:
− cot
πd(n2 − n3)
λ
{
πd
λ
·
n3
2
(R2211 − R3311) · E
2
1
}
.
Thus, at least in the strong fields, the polarization of RS crystals is connected not
only with the change of twin areas relationship but with the change of the angle of
monoclinity of the separate component of the twins over the value, typical of the
present temperature.
Using a single voltage pulses with a large period of repetition we exclude the link
of the observed effects with heating of the sample. Moreover, the heating may lead
to the opposite changes of the light beam intensity.
The presence of two linear areas (figure 3) indicates that there exist two critical
fields: the first – coercive Ec, at which there appears a massive domain motion, and
the second – saturation field Es, a field, in which the small domain disappears. The
second field, first of all, produces an effect on the precision of the measurement of
spontaneous polarization and the related values.
It is shown that the time dependence of the array of the increasing twin compo-
nent Sl is described by the equation
Sl = S [1 − exp{−β(E)t}] , (4)
where E – electric field strength; t – polarization time, and β(E) – depends sharply
on the field strength, time and temperature. Information about its field dependence
is given in table 1, where only those pairs E and τi are presented, to which there
corresponds practically the same polarization state (equal value of the index of
exponent). Theoretical papers predict exponential dependence of polarization rate
on E−1. As it is seen from figure 4, such a dependence can be presented by three
linear areas, described by the following relations
ln τi = kE−1
i + bi .
755
M.O.Romanyuk, M.M.Romanyuk
Figure 4. The relation between the duration and the amplitude of the pulses,
capable of providing the same polarization level of the sample.
The numerical values of the constants are: k = 371 V/cm, b = 1.43 (E > 500 V/cm;
k = 1046.9 V/cm, b = −6.78 (500 > E > 190 V/cm), and k = 2238.2 V/cm,
b = −11.0 (for E < 190 V/cm). These data are consistent with the number of areas,
predicted in [13], and with the field dependence of the number of new polarity domain
nuclei [2,3]. There is a correlation of coefficients k for weak and middle fields as well.
Logarithmic time dependences are also obtained during the investigations of ageing
of the triglycine sulphate group crystals [14].
4. Stability of the polarized state
The direct observations of the domains showed that the stability of the polarized
state depends on the initial geometry of domain structure and the regime of the
sample’s polarization [11]. It is shown, that the polarized state easily decays when
small domains of new polarity, areas of joint b- and c-domains and mechanical defects
are present. Apart from visible changes of the domain structure, which are registered
by microphotographs, or by photocurrent (figure 5), during the polarization of the
sample, some invisible changes take place, probably – of the defect network, and
the internal fields associated with them. Just this can explain the dependence of
stability of monodomain state upon the polarization pulse duration and the value
of the electric field.
Actually, it was shown that at the same polarization level after the end of the
polarizing pulse, the polarization decays more quickly if it had been created by a
more short voltage pulse.
At a given temperature, the initial rate of decay of the induced, not-fully polar-
ized state, depends on the level of deformation of the initial domain structure, and
is defined by the following relations (figure 6):
756
The final stages of polarization. . .
Figure 5. Examples of spontaneous decay of the monodomain RS crystals state
after the end of the polarizing pulse for different polarizing pulse durations: 1 –
0.04 s, 2 – 0.34 s, 3 – 2.4 s, 4 – 10 s, 5 – 40 s, 6 – 80 s. Measured at E = 300 V/cm,
T = 16 ◦C.
Figure 6. The dependence of the spon-
taneous polarization decay Id on Ip. The
decay is registered in 0.06 s after the
end of the polarizing pulse. The dura-
tion times of polarization processes are
placed near the corresponding plots
Id = k(Ip − I0),
Id ∼ 1 − exp{β(t) · t},
α = b − A ln τi .
Here I0 corresponds to the initial state
of the domain structure, Ip – to polar-
ized state, A and b – some constants.
Thus, here the forces act, which are sim-
ilar to quasi-elastic, and initial and final
stages differ from linearity.
Repolarization under the effect of in-
ternal fields has a different time depen-
dence than the polarization, caused by
an external field. Time constant of this
process is a function of the logarithm
of polarizing pulse duration τi at a con-
stant field, or a function of the logarithm
of electric field – when τ is constant (fig-
ure 7). That is, already at polarization
pulse duration 10−2 . . . 102 s, the de-
pendence of stability of polarized state
from polarization pulse duration is dis-
played.
757
M.O.Romanyuk, M.M.Romanyuk
Figure 7. The dependence of the time of the half-decay of polarization τp from (1
– 1250 V/cm, 2 – 700 V/cm, 3 – 300 V/cm, 4 – 156 V/cm) the pulse duration
τi, and (1 – 60 s, 2 – 10 s, 3 – 2.4 s, 4 – 0.63 s, 5 – 0.08 s, 6 – 0.04 s) electric field
strength amplitude E. Measured at 16 ◦C.
Considering the time constant β(t) as a function of time, we can come to a
conclusion about the time changes of the defect structure of the crystal and the
related internal fields.
A strong field at small τi, changes mainly the domains, while a network of defects,
being more stable and inertial, remains practically unchanged. That is why we have
a fast decay of the polarized state (domains adjustment to the old defect fields).
If the weak field during the short pulses does not change noticeably, neither the
domains, nor the defects network, then in the initial moment the decay is fast as
well (but the general changes are small, and are difficult to be registered).
With the growing of the polarization time, both the domains and the defects are
changed. That is why the stability of the new polarized state increases, and after
the monodomainization of the sample only the defects suffer some changes, and the
stability of their new configuration grows.
The internal field is revealed more distinctly in the crystals with the specially
introduced defects (doped and irradiated crystals). Here it is worthy to distinguish
such special features as:
1. Just after irradiation, and even more – after the repolarization cycle – do-
main walls in the RS crystals are not distinct. They become distinct after
10–20 minutes, that is an evidence of slowdown of relaxation processes.
2. The coercive field depends on the irradiation dose, and on the time of ageing
of the sample. After keeping a sample of RS with defects in the monodomain
758
The final stages of polarization. . .
state it can increase in dozens of times, unlike a pure sample, where the same
procedure gives the increase of the coercive field in 2–3 times only.
3. If the sample is irradiated in a paraelectric phase, or during the process of
repolarization of the irradiated crystals, the trapping of the rhombic areas in
monoclinic phase is possible, which creates an unstable phase. This phase is
capable of being polarized by the field of any sign, and looks like the mechanism
of repolarization of KDP crystals [17].
4. On the defect samples of RS unipolar and double hysteresis loops are observed,
the separation fields are displayed more distinctly, at which the domain motion
starts to be noticeable.
5. Conclusions
Thus, polarization of the RS crystals, except the known processes of nucleation
and growth of the domains of new polarity, is also connected with the slow displace-
ment of small components of twins, which is finished at some characteristic field.
When the electric field and polarization time increase, the deformation of domains
takes place, and the change of the sample’s defects network corresponding to the
relations ln τ ∼ k/E + b. This dependence is the evidence of the diffusive nature of
the related processes [18].
References
1. Gladkyj V.V., Kirikov V.A. // Fiz. Tverd. Tela, 2001, vol. 43, No. 1, p. 111–114 (in
Russian).
2. Kukushkin S.A., Osipov A.V. // Fiz. Tverd. Tela, 2001, vol. 43, No. 1, p. 80–87 (in
Russian).
3. Kukushkin S.A., Osipov A.V. // Fiz. Tverd. Tela, 2001, vol. 43, No. 2, p. 312–315 (in
Russian).
4. Kazuoki Kuramoto // J. Phys. Soc. Japan, 1987, vol. 56, No. 5, p. 1859–1867.
5. Dudnik O.F., Kushnarjov O.I., Duda V.M. // Ukr. Fiz. Zhurn., 2001, vol. 46, No. 4,
p. 321–323 (in Ukrainian).
6. Guilbert L., Salvestrini J.P., Fontana M.D., Czapla Z. // Phys. Rev. B, 1998, vol. 58,
No. 5, p. 2523–2528.
7. Turik A.B., Bondarenko E.I. // Fiz. Tverd. Tela, 1974, vol. 16, No. 4, p. 1240–1242
(in Russian).
8. Kudzin F.U., Pantchenko T.V., Judin S.P. // Fiz. Tverd. Tela, 1974, vol. 16, No. 8,
p. 2437–2440 (in Russian).
9. Fedotov V.N., Sydorkin A.S. // Fiz. Tverd. Tela, 1976, vol. 18, No. 7, p. 1967–1970
(in Russian).
10. Smolensky G.A. Ferroelectrics and Antiferroelectrics. Leningrad, Nauka, 1971 (in Rus-
sian).
11. Zheludev I.S., Romanyuk N.A. // Krystallografija, 1959, vol. 4, No. 5, p. 710–717 (in
Russian).
759
M.O.Romanyuk, M.M.Romanyuk
12. Romanyuk N.A., Zheludev I.S. // Krystallografija, 1960, vol. 5, No. 6, p. 904–911 (in
Russian).
13. Hayashi M. // J. Phys. Soc. Japan, 1972, vol. 33, No. 3, p. 616–628.
14. Stankowska J., Czosnowska E. // Acta Phys. Pol., 1973, vol. A 43, No. 4, p. 641–644.
15. Sonin A.S., Vasylevskaja A.S. Electrooptical Crystals. Moscow, Atomizdat, 1971 (in
Russian).
16. Narasimkhamurti T.S. Photoelastical and Electrooptical Properties of Crystals.
Moscow, Mir, 1984 (in Russian).
17. Balagurov A.M., Datt I.D., Savenko V.M., Shuvalov L.A. // Fiz. Tverd. Tela, 1980,
vol. 22, No. 9, p. 2735–2742 (in Russian).
18. Chebotin V.N. Chemical Diffusion in Solids. Moscow, Nauka, 1989 (in Russian).
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