Study of the SA-SC* phase transition by a dielectric method

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Submitted on 1 Jan 1990

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method

C. Legrand, J.P. Parneix

To cite this version:

Study

of the

_SA-SC*

_phase

transition

_by

a

dielectric method

C.

_Legrand

and J. P. Parneix

_(*)

Centre

_{Hyperfréquences}

et Semiconducteurs

(**),

Bât. P4, Université de Lille-Flandres-Artois,

59655 Villeneuve

_d’Ascq

Cedex, France

(Reçu

le 3 août 1989, révisé le 5

_janvier

1990,

accepté

le 8

_janvier

₁₉₉₀₎

Résumé. 2014

L’utilisation d’un

_{dispositif expérimental précédemment publié}

nous a

_permis

de réaliser l’étude

_{diélectrique}

à

_large

bande de

_fréquences

d’un cristal

_{liquide ferroélectrique}

orienté dans la

_{géométrie planaire}

_{(c’est-à-dire}

axe de l’hélice

_{perpendiculaire}

au

_champ

électrique

de

_mesure).

En

_plus

des mécanismes de relaxation

_classiques

observés dans les cristaux

liquides,

les données

_{expérimentales}

montrent l’existence de deux modes de relaxation

_(mode

de Goldstone et mode « mou

_»)

_{caractéristiques}

de la transition de

_phase

_SA-SC*.

_{L’amplitude}

et la

fréquence

critique

des mécanismes de relaxation liés à ces deux modes ont été obtenus dans toute

la gamme de

température

étudiée. Dans le cas du mode « mou », ceci a été

possible

dans la

phase

SC*

grâce

à la forte

_polarisation

du matériau et en _superposant au

_{champ électrique}

de mesure un

champ électrique

continu suffisamment élevé pour d6rouler l’hélice. Les résultats

expérimentaux

sont discutés à l’aide d’un modèle

_théorique

basé sur un

_{développement}

de type Landau de la variation de la densité

_d’énergie

libre à la transition

_SA-SC*.

Les

_principaux

désaccords entre les données

_{expérimentales}

et les données

_théoriques

sont liés au fait que le modèle

_théorique

ne

prédit

pas l’évolution en

_température

_{du pas de l’hélice. Des évaluations}

_{quantitatives}

des coefficients de friction ont été déduits de ces mesures

_{diélectriques.}

Abstract. 2014 A

_previously

described

_{experimental procedure}

has been used to

_perform

the

large

frequency

range dielectric

study

of a ferroelectric

_{liquid crystal}

oriented in

_planar

_geometry

_(i.e.

helix axis

_{perpendicular}

to the measurement electric

_field).

In addition to the usual relaxation mechanisms observed in

_liquid

_crystals,

the

_experimental

data show the existence of two

relaxation modes

_(Goldstone

mode and soft

_mode)

characteristic of the

_SA-SC*

_phase

transition. The dielectric

_strength

and the critical

_frequency

of the relaxation mechanisms connected with these two modes were

obtained throughout

the measurement _temperature domain. In the case of

the soft mode, this was

_possible

in the

_SC*

phase

thanks to the

_{high polarization}

of the

_sample

and the

_{superimposition}

on the measurement electric field of a DC electric field

_{high enough}

to

unwind the helix. The

_experimental

data are discussed in terms of a theoretical model based on a

Landau-type

expansion

of the free energy

density

variation at the

_SA-SC*

_phase

transition. The main

_{disagreements}

between the theoretical and the

_experimental

data are connected with the fact that the theoretical model fails in

_predicting

the _temperature

_dependence

of the helix

pitch.

Quantitative evaluations of the friction coefficients were deduced from the data.

Classification

Physics

Abstracts 77.20 - 77.40 2013 77.80

(*)

Permanent address : Ecole Nationale

_Supérieure

de Chimie

_Physique,

351 cours de la Libération,

33405 Talence _Cedex, France.

(**)

U.A. C.N.R.S. 287.

1. Introduction.

The

_study

of the

_physical

_properties

of ferroelectric

_{liquid crystals}

has caused a

_great

amount

of interest since the

_discovery

of their

_{potential applications}

in the field of

_{high speed}

visualisation

_[1].

The characteristic

_parameters

_(dielectric

_strength

and critical

_frequency)

of the relaxation mechanisms connected with the existence of the

_SA-Sé

_phase

transition are

strongly dependent

on

_{important physical}

_parameters

such as the

_{macroscopic polarization,}

the

_viscosity

of the

_sample

and the helix

_pitch.

The dielectric method is

_expected

to be of

particular

interest in

_studying

this

_phase

transition

_[2-7].

The

_experimental

data also

_represent

a

_good

tool to discuss the

_validity

of theoretical models

_[8-10].

In this

work,

we

_performed

a

large

frequency

range dielectric

study

of a ferroelectric

_{liquid crystal}

_[11]

whose chemical formulae and

_phase

_sequence are

_given

in

_figure

1. The

_sample

was oriented in

_planar

geometry

(i.e.

helix axis

_{perpendicular}

to the measurement electric

_field)

in order to measure

the

_{perpendicular}

dielectric constant

_s1.

The main interest of this

_compound

for a dielectric

study

is to have a

_high

_{macroscopic polarization}

in the

_St

_phase

_(P

50 nCb

/cm 2

T = Tc - 10 °C).

Fig.

1. - Chemical formulae and

phase

sequence of the

investigated compound.

2.

_{Experimental.}

The

_complex

dielectric

_permittivity

_{(ê * = ê’ - j ê" ) is}

obtained from the measurement of the

impedance

of an

_experimental

dielectric cell filled with the

_sample

to be characterized. We used a

_previously

described

_{experimental procedure}

[12].

2.1 IMPEDANCE MEASUREMENTS. - The

_impedance

measurements can be made in the

frequency

range

10-1

Hz-109

Hz

by using

a low

_frequency

system

(10-1

Hz-102

Hz)

and two

impedance analyzers

(5

Hz-109

_Hz).

The whole

system

is

_{automatically}

driven

_by

a

_computer.

This allows real time

_analysis

of the

_{complex permittivity}

on a

_printer

or on a

_plotter

and data

may be stored on a

_floppy

disk for further treatments. A _{DC electric field may be}

superimposed

on the AC measurement electric field.

2.2 EXPERIMENTAL CELL. - The

_experimental

cell is

_equivalent

to a

_capacity

filled with the

sample

in the

_{isotropic phase}

_{(capillarity filling).}

The main characteristics of this cell are the

following :

- in-situ

checking

of the

_sample

orientation

_{by using}

a

_polarizing

_{microscope ;}

- control of the cell thickness

by

an electronic

calibrating

translation

stage

(0.1

um

resolution).

This has been

_{possible by using glass}

electrodes coated with

_transparent

conductive I.T.O.

electrodes

_consisting

in P.V.A.

_coating

and

rubbing.

A

_{microstrip-line}

is achieved on each slide

_{by using}

a

_{technological}

_process to avoid effects on the measured value of orientation

defects and to connect

_easily

the cell with the

_{impedancemeter}

via a standard SMA

_connector.

The

_temperature

of the cell is maintained constant within 0.05 °C in the

_temperature

_range 20 °C-200 °C

_{by using}

an electronic

_temperature

_regulation

_stage,

_heating

resistors and a

thermal

_probe

fixed on the cell. A

_possible

_temperature

_gradient

is reduced with anticalorific filters. A

_{limiting high frequency}

_{appears connected with the conductive}

_coating

resistance. This

_{limiting frequency}

is

_{high enough}

_{(= 106}

_Hz)

to observe

clearly

all the relaxation mechanisms connected with the existence of

_{ferroelectricity.}

The cell thickness was about 75 um to obtain a

_planar

wounded

geometry

in the

_Sc*

phase.

The measurement electric field

was chosen as small as

_possible

to

_{slightly perturbe}

the helix structure in the

_St

_phase

(E

1.5

_{mV /J.Lm).}

3. Theoretical model.

In this

_section,

the main results of the theoretical model

_{[9, 10]}

used to discuss our

data,

are

presented.

This model is based on a

_Landau-type

_expansion

of the free energy

_density

variation at the

_SA-St

_phase

transition. The order

_parameters

are the tilt

_angle

and the

spontaneous

polarization.

The

_applied

electric field is assumed to be

uniform,

of small

amplitude

and

_parallel

to the smectic

_planes

of the

_sample.

The

_complex

dielectric constant is found to be the sum of the different contributions of the normal modes of the

_system.

Two

normal modes are

_expected

in the

_{SA phase :}

the « soft mode » and the « hard mode »

connected,

respectively,

with « in

_{phase »}

and « out of

_{phase »}

fluctuations of the order

parameters

[8].

These two modes

_degenerate

in four modes in the

_St

_{phase :}

the « soft mode » of the

_{SA phase}

_degenerate

in the Goldstone mode

_(«

in

_{phase »}

orientation fluctuations of the order

_parameters)

and in the « soft mode »

_(«

in

_{phase » amplitude}

fluctuations of the order

_{parameters) ;}

the « hard mode »

degenerate

in two « out of

_{phase »}

modes. In the case of a

_{time-dependent}

electric

_field,

a

_{purely damped regime}

is assumed for

the normal modes and each mode is translated into the

_expression

of the dielectric constant

_by

a

_Debye

_type

relaxation mechanism whose characteristic

_parameters

_(dielectric

_strength

and relaxation

_frequency)

are

_numerically

calculated. The dielectric

_strength

of the relaxation mechanisms connected with the « out of

_{phase »}

modes are found to be much smaller than the « in

_{phase »}

ones

_[10].

_Then,

later in this section we will be concerned

_only

with the Goldstone and the soft modes.

The dielectric

_permittivity

as a function of the

_frequency

F reduces to :

where eoo

is the dielectric

_permittivity

considered at

_frequencies

much

_higher

than the relaxation

_frequencies

of the ferroelectric relaxation

mechanisins

and much lower than the relaxation

_frequencies

of the classical

_dipolar

relaxation mechanisms

_(i.e.

rotation around the

Fig.

2. - Dielectric

strength

of the

sôft

mode and of the Goldstone mode and static

_permittivity

versus

reduced _temperature

_{A / Q 2.}

Fig.

3. -

Relaxation

_frequencies

of the soft mode and of the Goldstone mode versus reduced

temperature

A/Q2.

where

_7c

is the transition

_temperature,

_{q= 2 ir/p}

is the critical wave vector, Xoo =

(4 -ff )-’

1

(E 00 - 1), K

is the renormalized twist elastic constant, C is the

_{piezoelectric}

constant, and a occurs in the

_temperature

_dependent

coefficient

a = a (T - T*) of

the free

energy

density

expansion.

In this

simplified

model,

the

pitch p

of the helix in the

St

phase

is found to be

_temperature

_independent

which

_disagrees

with

_experimental

data

[14].

The influence of the flexoelectric

_coupling

_parameter

_{(8 = ILq/C,u}

is the flexoelectric

coupling

term)

in the

_S/Î

_phase

is also

_{reported (full lines).}

The

_frequencies

_FTi

are the

frequencies

related to the friction coefficients ri of the different modes

(i

= 1 for the ’soft

mode in the

_{SA phase}

and i = ₊ 1 and - 1 for the Goldstone and the soft mode

respectively

in

the

_{St phase).}

-3.1 GOLDSTONE MODE. - In the absence of flexoelectric

_coupling

_{({3 = 0),}

the dielectric

strength

and the relaxation

_frequency

of

the

Goldstone mode are found to be

_temperature

independent, :

At the

transition,

the

_egality

_{F al F T +}

_{1 =}

_{F si F T -1}

1 is found.

In the case of flexoelectric

_coupling

_{(13 #= 0),}

the dielectric

_strength

and the relaxation

frequency

of the Goldstone mode are

_{température dependent}

near

_Tc

(Figs.

2 and

3)

and

FalFT

:

FslFT .

3.2 SOFT MODE. - Close to

_7c

and without flexoelectric

_coupling

in the

_St

_phase,

the characteristic

_parameters

of the soft mode are :

SA phase :

Sc*

phase,

no

flexoelectric

coupling f3 =

0

A

_{discontinuity}

of the dielectric

_strength

of the soft

_{mode ËS}

is observed at the transition. From

_equations

₍₄₎

and

_(5),

the linear evolutions of the inverse of the dielectric

_strength

Ês-1

i _versus

_temperature

are obtained near

_T,

in both the

_SA

and

_Sc*

_phases

_(Fig.

_4).

At the

transition,

there is a reverse in the

_sign

of the

_slopes,

the

_slope

ratio

_{(slope St}

_phase/slope

SA

phase)

is - 4. The

observed at the transition where it is minimum. The ratio of the

_slopes

in the

SA phase

and in the

_St

_phase

is - 2.

In the case of flexoelectric

_coupling

(0 =F 0),

the

_{discontinuity}

_{of ÉS at}

the transition

disappears

(Figs. 2

and

_4).

The

slope

ratios for

_Ês-l(T)

and

_Fs(T)

are different from the

previous

case

_/3

= 0.

Fig.

4. - Inverse dielectric

strength

of the soft mode versus reduced _temperature

_{A / Q2.}

3.3 STATIC PERMITTIVITY. 2013

Equations

(6)

and

₍₇₎

_give

the

_expression

of the static

permittivity

(permittivity

at zero

_frequency)

in both the

_SA

and

_SC*

_{phases :}

SA phase :

Sc*

phase :

The static

_permittivity

is maximum at the transition

_(£s

= 4

-ux 2

c 21K q 2) .

There is no

contribution of the flexoelectric

_coupling

term to the static

_permittivity

_(see

_Fig.

_2).

4.

Experimental

results.

The

_{frequency dependence}

of the real

_(e’)

and of the

_imaginary

_{( e")}

_parts

of the

_complex

permittivity

are

_reported

for different

_temperatures

in the

_{SA phase}

(Fig. 5)

and in the

St

phase

(Fig. 6).

The

_frequency

_{range is limited} to 106 Hz connected with the

_{limiting high}

Fig.

5. - Real

part E’ and

_imaginary

_part e " of the

complex permittivity

versus

_frequency

in the

SA phase

(curve

1 : T -

_T,

= 0.9 °C, 2 : 1.4 °C, 3 : 2.2°C, 4 :

4.4 °C).

Fig.

6. - Real

part e’ and

imaginary

part e" of the

complex permittivity

versus

_frequency

in the

Sc*

phase

(curve 1 :

T -

_{Tc = -}

0.2 °C, 2 : - 3.5

_°C,

3 : - 13.5

_°C).

SA phase

In the

_{SA phase,}

the data show the existence of one relaxation mechanism whose relaxation

frequency

is in the HF range

(= 104

Hz ).

St

phase

In the

_Sc*

_phase,

the HF relaxation mechanism still

_{exists ;}

a second relaxation process

appears whose relaxation

frequency

is much lower than the

previous

one

(=

102

Hz).

As the

temperature decreases,

the dielectric

_strength

of this BF relaxation mechanism increases

strongly

to mask the other one

_completely.

For this reason, the characteristic

parameters

of the HF _{relaxation process} were not obtained in the

_Sc*

_phase.

The

temperature

dependence

of the dielectric

_strength

of the HF _{relaxation process in the}

_{SA phase}

and of the BF relaxation process

(in

the

_{St phase)}

are

_given

in

_figure

7. This

_parameter

was deduced from the

difference between the static and the infinite

_{permittivities}

_{(curve E’ (F )}

or Cole-Cole

diagram

c"(c’)).

An accurate determination of the dielectric

_strength

of the BF relaxation process is difficult : near

_T,,

the contributions of the two relaxation mechanisms need to be

separated and,

at lower

temperatures,

the Cole-Cole

_diagrams

were

_extrapolated

because the

dielectric

_spectra

are

_incomplete

(for

F 10

Hz).

In

_{figure 7,}

we also

_reported

twice the maximum value of the dielectric losses

(the

value of 8" at the relaxation

_frequency).

This parameter

represent

the dielectric

_{strength only}

in the case of a

_Debye-type

relaxation process

both the

_SA

and

_St

_phases.

The relaxation

_frequency

was deduced from the

_frequency

at

which the dielectric losses 6" are maximum

(curve 6"(F)).

The

_temperature

_dependence

of this

_parameter

is

_given

in

_figure

8.

The HF and the BF relaxation mechanisms are identified

_respectively

to the soft mode and

to the Goldstone mode described in the

_previous

section.

Fig.

7. - Dielectric

strength

of the Goldstone mode

_(*)

and of the soft mode

_(e)

versus _temperature.

The parameter 2

_e"max

is also

reported

(dotted lines).

Fig.

8. - Relaxation

frequencies

of the Goldstone mode

(*)

and of the soft mode

₍₀₎

versus

temperature.

DC bias

The data are

_incomplete

in the

_Sé

_phase

for the soft

_mode,

this is connected with the

comparatively

high

dielectric

_strength

of the Goldstone mode. To obtain a

_complete

characterization of the soft mode in the

_Sé

_phase,

we have

_superimposed

a DC electric field to

helix

_{[14, 15].}

_Examples

of dielectric

_spectra

obtained in the

_Sê

_phase

are

_given

in

_figure

9. When the structure is unwound

_by

a DC

_bias,

the orientation fluctuations are

_hindered,

the Goldstone mode

_disappears

and

_only

one relaxation process

_{corresponding}

to the soft mode is observed. The measurements were also carried out in the

_SA

_phase.

In this

_phase,

the dielectric

_spectra

were found to _{be very similar} to those obtained in the

_St

_phase.

The results of the characterization of the soft mode under bias are

_presented

in

_figures

10 and 11. The dielectric

_spectra

are also distributed.

5. Discussion.

5.1 GOLDSTONE MODE. - The relaxation

_frequency

of this mode is

_expected

to be

temperature

independent

and

_slightly

modified

_by

the introduction of a flexoelectric

_coupling

term

_(insert

_Fig.

_3).

The

_experimental

_{data agree with these results}

_only

far below the

Fig.

9. - Real

part E’ and

_imaginary

_part e" of the

_{complex permittivity}

versus

_frequency

in the

Sc*

phase

in the case of

_{superimposition}

of a DC electric field V = 5 V

(curve

1 : T -

Tc = -

0.3 °C, 2 : - 0.7

°C, 3 : - 1.1 _°C, 4 : - 2.1

_°C).

Fig.

10. -

Fig.

11. - Relaxation

frequency

of the soft mode versus _temperature

_(Bias

= 5 V _{full lines,} bias = 0 V

(-)

transition

_temperature

_Tc

_{(Fig. 8).}

We assume that this is connected with the

_temperature

dependence

of the helix

_pitch

_[7]

which is not

_{predicted by}

the theoretical model. The increase of the

_{çritical frequency}

as we

_approach

the

_temperature

transition

_Tc

can be

_{explained by}

the increase of the term

Kq2

_{(Eq. (3)).}

The theoretical model

_predicts

the

temperature-independent

dielectric

_strength

of the Goldstone mode in the case of no flexoelectric

_coupling

term. The introduction of a flexoelectric

_coupling

term induces a

_strong

decrease in the dielectric

_strength

near

_7c

(Fig. 2).

The

_experimental

data show a decrease in the dielectric

strength

of the Goldstone mode near

_7c

(Fig. 7).

This decrease is

_{comparatively}

much

_greater

than the increase in the critical

_frequency.

This result is

_{explained by adding}

the influence of a

possible

flexoelectric

_coupling

term and the

_temperature

_dependence

of the helix

_pitch.

Far below

_Tc,

the characteristic

_parameters

become

_temperature

_independent

as

_{predicted by}

the

theoretical model. From the dielectric

_strength

of the Goldstone mode far below

Te,

the relative twist to the

_piezo-energy

_parameter

_Q2

can be estimated

_{(Eq. (3)) :}

êG ~

750 and £00 = 5 result in

Q2 ~

2.4 x

10- 3.

This low value is

_probably

connected with the

relatively high

value of the

_spontaneous

_polarization

_(high

value of the

_{piezoelectric}

coefficient

_C).

The friction

_frequency

_{F T +1}

₌ 80 kHz

_{corresponding}

to orientation fluctuations of the order

_parameters

is deduced from the relaxation

_frequency

of the Goldstone mode far below

_Tc

_{(Eq. (3), fG ~}

100

_Hz).

5.2 SOFT MODE. - The theoretical model

_predicts

a linear form with the

_temperature

of the

soft mode critical

_{frequency FS}

s and inverse dielectric

strength ê,-

i in the

vicinity

of the

SA-St

phase

transition

_(Figs.

3 and

_4).

The

_experimental

data show linear

_temperature

évolutions

of these

_parameters

in the whole measurement

_temperature

domain

_except

close to

Te

(Figs.

11 and

_12).

The

_slopes

of the linear

_parts

of these curves are 0.066/°C in the

SA phase

and 0.25/°C in the

_Sc*

_phase

for the dielectric

_strength

and 20 kHz/°C in the

SA phase

and 30 kHz/°C in the

_St

_phase

for the critical

_frequency.

The

_experimental

data agree with the theoretical results

except

close to

_Tc

if we consider that the measurement

ratio

_{F T -II F Tl}

in the case of the relaxation

_frequencies.

The values obtained from the

experimental

data are - 3.8 for the dielectric

_strength

and - 1.5 for the relaxation

frequencies.

To discuss these results

_further,

it would to _{be necessary} to consider that the data

on the soft mode were obtained under bias in the unwound structure

_[16]

whereas this is not

the case in the theoretical model. The theoretical model also

_predicts

that in the

SA phase

the

_frequency

_{F Tl}

is

_directly

connected with the

_{product Ês FS}

s

(Ês

F s

= 2

TT F Tl

X (0).

A similar result is obtained in the

_S*

_phase

in the case

₈

= 0

(ÊS FS

=

TT F T -

1 X (0).

The

experimental

temperature

evolution of this

_product

is

_given

in

_figure

13. In the

_{SA phase}

for

Fig.

12. - Inverse of the dielectric

_strength

of the soft mode versus _temperature

_(Bias

= 5 V full lines,

bias = 0 V

(-).

Fig.

13. -

T>

_Zc

+

1,

this

_parameter

is found to be

_{temperature-independent}

_{(Ês Fg}

= 400 x

103)

and

the friction

_frequency

_FT1

= 200 kHz far below

7c

is deduced

fromi

the data. In the

Sé

phase

with the

_hypothesis

of no flexoelectric

_coupling,

we obtain the same value

FT -1

= 200 kHz for T

Tc -

1

with

Fs

= 200 x

103.

As

_expected,

this

_frequency

corre-sponding

to

_amplitude

fluctuations of the order

_parameters

is found to be

_higher

than the

frequency

F T + 1.

6. Conclusion.

We

_performed

the

_complete

characterization of the Goldstone and of the soft modes of a

ferroelectric

_{liquid crystal.}

This was

_possible

for the soft mode in the

_St

_phase

thanks to

superimposition

to the measurement electric field of a DC electric field

_{high enough}

to

unwind the helix. The

_experimental

_{data agree with the theoretical model if} we consider that the model fails in

_predicting

the

_temperature

variation of the helix

_pitch.

On the basis of this

model,

quantitative

evaluations of the different friction coefficients were obtained from the

experimental

data. The dielectric method seems to be useful to

_study

ferroelectric

_liquid

crystals

and further

_experiments

on other

_compounds

are in progress to

_complete

these results.

References

[1]

CLARK N. A., LAGERWALL S. T.,

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₍₁₉₈₀₎

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[2]

PARMAR D. S., MARTINOT-LAGARDE P., Ann.

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3

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275-282.

[3]

LEVSTIK _A., ZEKS B., LEVSTIK _I., BLINC R., FILIPIC _C., J.

_Phys.

_Colloq.

France 40

(1979)

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In order to obtain a better _agreement with the

_experimental

data and also the

polarization

and the heat

_capacity,

_{it would be necessary} to use an extended model. The

_following

terms are then added to the _{free energy}

_{density expansion :}

c/6

(03B803B8*)3+ jd/2

(03B803B403B8*/03B4z-

03B8*03B403B8/03B4z)(03B803B8*)-j03A9/4(P03B8* - P*

03B8)2 + ~/4

(PP*)2.

CARLSSON T., ZEKS B., LEVSTIK A., FILIPIC C., LEVSTIK I., BLINC _R.,

_Phys.

Rev. A 36

₍₁₉₈₇₎

1484-1488.

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209-221.

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PAVEL J., GLOGAROVA M., Ferroelectrics 84

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GOUDA F., ANDERSSON G., CARLSSON, T., LAGERWALL S. T., SKARP K., STEBLER B., FILIPIC C.,