Soft mode characteristics in the KTN ferroelectric transition

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Soft mode characteristics in the KTN ferroelectric transition

J.M. Courdille, J. Dumas, S. Ziolkiewicz, J. Joffrin

To cite this version:

J.M. Courdille, J. Dumas, S. Ziolkiewicz, J. Joffrin. Soft mode characteristics in the KTN ferroelectric transition. Journal de Physique, 1977, 38 (12), pp.1519-1525. �10.1051/jphys:0197700380120151900�.

�jpa-00208727�

SOFT MODE CHARACTERISTICS IN THE KTN FERROELECTRIC TRANSITION

J. M.

COURDILLE,

^J.

DUMAS,

S. ZIOLKIEWICZ

Laboratoire d’Ultrasons

(*),

Université

Pierre-et-Marie-Curie,

^Tour

13, 4, place Jussieu,

75230 Paris Cedex

05,

^France

and J. JOFFRIN

Institut

Laue-Langevin,

¹⁵⁶

X,

^{Centre de}

tri,

38042 Grenoble

Cedex,

^France

(Reçu

^{le 21}

juin 1977,

^{révisé le 19}

juillet 1977, accepté

^{le 19 août}

1977)

Résumé. ²⁰¹⁴ Nous

présentons

^{des mesures de}

propagation

ultrasonore au voisinage de la transition

ferroélectrique

^de

K(TaxNb1-x)O3

pour deux valeurs de concentration _x, donnant

respectivement

une transition du 1er et du 2e ordre. Les coefficients critiques ^de

champ

moyen qui ^rendent compte de l’atténuation et de la variation de la constante

élastique

C11 au

voisinage

de la transition du 2e ordre sont respectivement n ⁼ 1,57 ± 0,05 ^{et m =} 0,48 ± 0,05.

Deux mécanismes différents sont

analysés

^pour expliquer ^ce comportement. ^{A notre} fréquence ^de

mesure, seul le couplage

quadratique

^entre la déformation et les fluctuations du paramètre ^d’ordre permet ^{de rendre} compte

quantitativement

^{de nos} résultats. La valeur des coefficients m et n montre que la relation de

dispersion

^{du mode mou est à} trois dimensions.

Abstract. ²⁰¹⁴ We present ultrasonic measurements in the

vicinity

^{of the}

KTaxNb1-xO3

ferroelectric transition for two values of _x,

giving respectively

^a first and a second order phase transition. The variations of the attenuation and the C11 elastic ^{constant are} governed

by

the mean-field critical exponents ^{n = 1.57 ±} 0.05 and m ⁼ 0.48 ± ^0.05

respectively.

Two different mechanisms are

analysed

^to

explain

^this behavior, ^{but in our}

frequency

range, ^our data are very well

explained by considering only

^a

quadratic

coupling between the strain and the order parameter fluctuations. These values of the m and n indices indicate the three-dimensional

nature of the soft mode

dispersion.

Classification Physics ^Abstracts

77.80 ^- 62.65 ^- 64.70 K - 43.35

1. Introduction. ^- A

great

^{number of structural}

phase

transitions in

perovskite-type ^crystals

^{have been}

studied in the past five years.

The

pretransitional

^{behavior of}

^KTa03,

^{the dis-}

placive

transition

of SrTi03

^and

KMnF3,

^{as well as the}

ferroelectric transitions of

BaTi03, KNb03,

^{have all}

been examined.

The

properties

^{of these}

crystals

^{have been}

explored by

very different methods

leading

^to observations of dielectric

properties,

^{Raman or infrared}

frequencies, phonon dispersion

^{curves as a} function of

temperature,

and

dimensionality

of the soft mode in the

vicinity of To.

Among

^these

crystals,

^{the mixed}

crystal (KTa,,,Nbl-.,03)

^{is of}

special interest,

because the character of the transition _may be

arbitrarily changed

(*) Associated with the Centre National de la Recherche Scien-

tifique.

by altering

^the

proportion

^{of Ta} and Nb : KTN

samples

with less than

33 %

niobium have a second- order transition. Thus KTN is a

good

^{candidate to}

test the

properties

^{of a} second-order ferroelectric

phase

transition.

These mixed

crystals

^{are not}

completely unknown ; they

^{have been}

synthesized by

^various

people [1]

^who

have

given

^some indication of the critical concentra- tion

[2, 3, 4].

^{Also a}

study

^{of the}

dispersion

relation of the

phonon

modes has been obtained

by

^neutron

diffraction on a

crystal undergoing

^a first-order transi- tion

[5] :

^{it was} shown that the transverse

optical

^soft

mode is _very

anisotropic

^near

the 100 )

direction and remains

overdamped

^{in the}

vicinity

^of

To.

^This

is,

^at

first

sight,

^{a common} feature of

crystals having

^a

first-order transition.

Some new neutron diffraction

experiments [6]

^on

crystals

with second-order transitions have also shown that there is a linear

coupling

between the

optical

^soft

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380120151900

1520

mode and the acoustic

branch, resulting

^{in a} very clear interference

pattern

^of

diffraction ; consequently,

^{it is}

reasonable to

hope

that critical

properties

^{are exhibited}

clearly by

the acoustic behaviour of the

crystals.

In this _paper we

report

^new

experiments

^{on the}

critical acoustic

properties

^{of two}

crystals

of different

compositions.

^{The first}

crystal

^{had a} concentration of 28

% Nb

and had a second-order transition with

To

^{= - 22.5}

°C ;

the other had a

composition

^of

40 %

^Nb and the transition is of first order and occurs at 23 °C.

In section

2,

^{we first}

give

^a

description

^{of the}

growth

of the

samples

^{which were used.}

In section

3,

^we

present

^our

experimental

^{data. The}

theoretical

interpretation

follows in section

4,

^{with the}

presentation

^{of two}

possible

mechanisms

leading

^to

critical behaviour of acoustic

properties.

The conclusions are

given

in section 5.

2.

Crystal growth.

^- The method used is

top-

seeded

growth

from the melt. A seed oriented

along 100 )

^{axis is rotated}

(20 revolutions/minute)

^and

translated

(1 mm/day)

^{at the}

surface

^{of a melt of}

Ta20-5, Nb20_,

^and

K2C03.

^The crucible contains two

regions :

^{the lower one contains a solid}

phase

^{with a}

Ta/Nb

^{ratio of}

62/38,

^covered

by

the melt with a

Ta/Nb

^{ratio of}

43/68. Following

^a

thermodynamic analysis,

^{the solid}

part

which is the hoter one feeds the melt to

keep

^its

composition perfectly

^{constant. The}

homogeneity

^{is obtained}

by component

^diffusion between the two

regions

of the crucible.

Two

parameters

^{need to} be controlled _very

carefully during crystal growth :

^the

homogeneity

^{of the}

components

^{at a} certain level in the melt and the

temperature

^{in the}

growing

^zone. The transition tem-

perature

^{of KTN is}

strongly dependent

^{on the ratio}

Nb/Ta,

^{and since a}

slight

variation of the

composition

inside the

crystal

induces . in

particular

very

large

variations in electroacoustic and

electro-optic

pro-

perties,

^{it is}

clearly important

that the ratio

Nb/Ta

^be

uniform if

significant

^{results are to be obtained.}

With our

experimental conditions, homogeneous

and

transparent

^KTN

crystals

^are

obtained ;

^the

samples

^are

single crystals

^{as shown}

by X-ray

^diffrac-

tion and the

perfection

of the lattice ^was determined at the Institut

Laue-Langevin

^{with a}

high

^resolution

y-diffractometer (the

mosaic is about

1/100°).

The

crystals

^{obtained are colorless and have a}

high

electrical

resistivity (about

^{5 x}

^{1 O13} Q.cm).

^Their

dimensions are about 1

cm3

with all natural faces

clearly

^visible.

The

crystals

^are very insensitive to

rapid cooling

^or

heating :

^{we never observed} any crack in the core of the

crystal ;

ultrasonic and dielectric

properties

^{are very}

reproducible

^{after numerous}

cycles

^of

heating

^and

cooling.

Different

samples

taken from the same bulk show the same transition

temperature

^{as measured}

by

^the

dielectric constant : this is a

good

^{test of}

crystal homogeneity.

By varying

^the

temperature

of the cold and hot

regions,

^{we are able to obtain}

single crystals

^with

différent

compositions

^{of Ta} and Nb. Thus

samples having

^{either a} first order or second order

phase

^tran-

sition can be

produced.

3.

Experiments.

^{- In its}

high temperature phase,

KTN is cubic and thus there is no

major problem

ⁱⁿ

propagating

pure ultrasonic

longitudinal

^{waves in a}

high symmetry

direction. We chose to work

along

^the

four-fold axis. The

frequency

of the acoustic wave was 540 MHz. The

specimen crystal,

of thickness 2.42

min.,

^was mounted between two quartz

crystals,

the first one

serving

^as generator and the second one as

detector. Two other

(100)

^faces

perpendicular

^{to the}

axis of

propagation

^were

polished

flat and afterwards metallized in order to measure the dielectric constant of the

sample simultaneously

with the acoustic

signal.

This

procedure

is essential if a

good

estimate of the transition

temperature

^{is to} be obtained.

The

sensitivity

^{of the}

absorption

measurement was 0.3

dB,

^{and we were able to measure}

velocity

variations of about

10- 4.

In

figure

^{1 we have}

plotted

the attenuation as a

function of

temperature

^{in the}

sample

^which

undergoes

a second-order transition. The measurements were

performed

^{in a} very

large temperature

range above

T,,

in order to be able to subtract a

good

^{estimate of the}

non-critical

background.

The critical

(1) part

^{of the} attenuation is

plotted

ⁱⁿ

figure

^{2 on a}

log-log

scale : the critical exponent ^{is n = 1.57} + ^0.05.

In

figure

^{3 we show the}

temperature dependence

^of

the

velocity.

The non-critical

part

^{is also}

shown ;

^{it is} obtained

by adjusting

^the

background

^{in such a} way

FIG. 1. ^- Attenuation of longitudinal ^waves prbpagating along

the [100] ^{axis near} the 2-order transition in KTN, ^{as a function} of T - To.

(1) ^{In what} follows, ^{we are} using ^{the term critical} by convenience in the sense of variation related with the phase transition ; ^we observe, in fact, the Landau mean-field region of the transition and not the critical one in the sense of Ginzburg, ^{as we shall see.}

FIG. 2. ^- Dtermination of the critical exponent n ^{= 1.57 from} the curve of figure ^1.

FIG. 3. ^- Variation of the velocity ^of longitudinal ^waves propa-

gating along [100] ^{axis near} the 2-order transition in KTN, ^as

a function of T - To.

FIG. 4. ^- Determination of the critical exponent m ^{= - 0.48} from the curve of figure ^3.

that the

log-log plot

^{of the}

velocity

^{as a} function of AT ⁼ T -

7.

^{is a}

straight

^line

(Fig. 4).

^{The cri-}

tical exponent

resulting

from this

procedure

^is

m ⁼ 0.48 ± 0.05. The

temperature

of transition is not considered here as an

adjustable

parameter ^{because it}

was estimated

by measuring simultaneously

^the

dielectric constant which follows a standard Curie-

Law er

⁼

c

^{with C = 104}

^{700 +}

^{100 K and}

T - To

^_

To

^{= -} 22.5 °C for T >

To (Fig. 5).

^{Below the tran-}

sition

temperature

^{the Curie constant is}

exactly C/2.

FiG. 5. ^- Measurement of the inverse of the dielectric constant

FI ^{1 near} the 2-order ferroelectric transition in KTN.

The

rounding,

^near

To,

^of

1 /Er

^takes

place

^{in an}

interval of temperature

of only

^{20 :} this is smaller than the zone accessible to the acoustical

experiment,

^{due to}

the

large

attenuation of the acoustic wave. This esti- mate of

To

^is

considerably

^{more accurate than the}

value which could have been obtained from ultrasonic

experiments only. Using

the value of

To

determined

by

, this _means, the

experimental

results for the attenuation and the

change

in the elastic constant

AC11/Cll

^are

given

^{at our}

frequency by

^the

following

^{formula :}

Following

^{the same}

procedure,

^we

give

ⁱⁿ

figure

⁶

the results of the critical

velocity

^{in the}

crystal having

^a

first-order transition. The

temperature

of the transi-

tion,

^measured

by

^the dielectric constant, ^was 23 °C and the critical index for the sound

velocity

^was

n ⁼ +

0.50,

^not very different from the second-order

case. Such a small difference can be

explained

^{in the}

following

way : in the series of KTN

samples,

^the

change

^from second order to first order occurs at about

10 °C,

^{which is} very close to the transition tem-

perature

of the second

sample :

^the second order character is not

yet completely

^{removed for this}

concentration. The attenuation measurements on the

sample

^{were not} very

reproducible

^{and are} not,

accordingly, given

^here.

Moreover, n

^{= 0.5 seems to}

indicate,

^{as we shall} see, that the

dispersion

^{curve is}

isotropic

ⁱⁿ wave-vector space ; it seems that this is not

1522

Fic. 6. ^- Variation of the velocity ^{of a} longitudinal ^waves pro-

pagating along [100] ^{axis near} the Ist order transition in KTN,

as a function of T - To.

FIG. 7. ^- Determination of the critical exponent ^{m = 0.51 from} the curve of figure ^6.

generally

^{the case at} first order transitions

[5] :

^{This is}

another indication that this

sample

^{is too close to the}

critical concentration to show first order transition effects.

4.

Theory

^and

interprétation.

^{- The} modifications of the acoustical

properties

of a material near its

phase transition,

^have very often been

presented. (See

^for

instance

[7].)

In this _paper we focus attention on the ferroelectric transition of

KTN, by applying arguments

^used

previously by

several authors

[8, 9].

KTN,

^{in its}

high temperature phase,

^{is centro-}

symmetric ; consequently

^{there cannot exist a term}

coupling linearly

the strain and the

polarization,

^at

least,

^{in the}

long wavelength

^{limit. As a}

result, only

^two

possibilities

^{remain to}

explain

the critical behaviour of the acoustical

properties.

We now consider them

successively.

(A) The first corresponds

^{to the} anharmonic

coupl- ing

between the strain and the _square of the fluctua- tions : the

temperature dependence

of the real and

imaginary

parts of the elastic constant comes from the Fourier transform of the

time-dependent four-point

correlation function

[8, 9]

of the order parameter. ^This is a

special

feature of acoustical

experiments

ⁱⁿ

general.

If we

call q

^{the order} parameter

(the polarization),

ôF the

density

of the free _energy due to the

coupling,

ôF ⁼

G81]2,

^{where G is the}

coupling

^{constant with the}

elastic strain 8 ; we have

previously

^shown

[9] by using

a

decoupling procedure

that when the critical mode is

overdamped (Q q « ^T)

the variation AC of the elastic constant is

and the attenuation a is

When the critical mode is

underdamped (Qq ^r)

^we

have

In these formulas

Q q

^{is the}

^frequency

of the soft mode which is

temperature dependent ; v

^{is the}

velocity

^of

sound far from the transition. r is the

damping

^fre-

quency of the soft mode and is

expected

^{to be a function}

of (T - To) and q

^as

suggested

ⁱⁿ

[10, 11, 12].

^How-

ever, these indications of the functional form of T are

slightly

^too

rough

^{and we} will instead consider it to be

a constant,

keeping

in mind that this

expression

^{is an}

effective

damping frequency

^which

^is,

ⁱⁿ

fact,

^{a mean}

value over all wave vectors we will take into account.

We call 3-dimensional a critical mode which is

equally

soft in all directions around some defined

point

in the Brillouin zone. In that case :

and

qmax is a cut-off wave vector

giving

^the

region

^{outside of}

which the

parabolic dispersion

^{curve is no}

longer

correct. In

(5)

^{we have}

neglected

^the

higher-order

harmonic terms.

Similarly,

^we call 2-dimensional a critical mode which is soft

along

^one line in the Brillouin zone : for

instance,

^its

dispersion

surface is :

By analogy,

^{it is} easy ^to obtain the definition of one-dimensional critical modes.

We

present

^here

only

^the calculation in the 3-dimen- sional case.

The attenuation and the rate of

change

ⁱⁿ

velocity

of the acoustic wave can be written as :

and

Introducing

the dimensionless

parameter

^{u defined}

b

^{yR’ =}

Q0

^{tg u, the}

^integrals

ⁱⁿ

⁽⁷⁾

^and

⁽⁸⁾

^become

,lb

^{g g}

⁷⁾

respectively :

and

Finally :

A

good approximation

^{of a and}

àClC

^{can be}

obtained

by taking

qmax ⁼

Qollb-

^{as an upper value}

for _q. This

gives

Umax ⁼

n/4

^and

The soft mode

dispersion

law and the cut-off rule used are both

special

^{cases, and we} checked if other laws

give

^{the same}

temperature dependence

^{in acoustic}

measurements.

We found that the influence of this choice is

negli- gible

^{on the critical}

exponents

^{but a numerical constant} of order

of unity

may be introduced in the calculation of

f1C/C

^{and a}

depending

^{on the}

dispersion

^{law taken}

into account.

However,

the model taken is not unreasonable and it is

likely

^{that it}

corresponds exactly

^to the actual case.

An

equivalent

calculation could be

performed

^for

other dimensions of the critical mode

dispersion

^curve.

Under the

assumption

^that

Q’ (T - To)

^{we can}

draw the

following

table for the critical

exponents :

TABLE 1

Theoretical values

of

the critical exponents

of

^the

ultrasonic

velocity

^and attenuation in the

vicinity of

^a

second-order

phase transition,

^{as a}

function of

^the

soft

mode characteristics.

In

principle, then,

^a

comparison

^{of the}

experimental

values of the critical

exponents m

and n with the values

quoted

^{in table I}

helps

^to decide without

ambiguity

what are the characteristics of the soft mode.

In that

respect

^our

experimental

^results

(n N 1.5,

m N

0.5)

indicate that the soft mode is

overdamped

and that its

dimensionality,

^{in the sense defined}

previously,

is 3. This is in

agreement

^{with the}

preli- minary

^{results obtained}

by

^neutron diffraction.

From the theoretical and

experimental

results for

dC/C

^{one can extract an} estimated value of G. We use

the characteristics of the soft mode

dispersion

^curve

determined from

KTa03 [13, 14] :

We find a

coupling

^{constant :}

Finally

^the

comparison

^{between a and}

ACIC gives

^a

value off

equal

^{to 8 x}

¹⁰¹³ rad/s (L--

⁵³

meV).

^This

result is of a greater ^{order of}

magnitude

^than

expected

form neutron

scattering experiments ^but,

^{as indicated}

above,

^{r is a mean value obtained}

by heavily weight- ing

the small values of the wave vector q, and we do not have sufficient

experimental

^{data to make a}

good comparison.

In these calculations we used a Landau formalism which

is justified only

^{if the}

approach

^to the transition is not too close.

Following Ginzburg [15],

^we evaluate the true

critical

region

^AT

^within

which the calculation

presented

^{above is no}

longer

^correct,

by

^{a direct}

comparison

between the mean value of the electrical

polarization

and the fluctuations of the same

quantity ;

AT is evaluated in the low

temperature phase

^{and is}

assumed to be the same in the

high temperature phase.

Writing ( P >’

^{as -}

A(T - To)/2 B,

^{this true critical}

region

^is

given by

1524

Here

kB

is the Boltzmann

factor,

^{and t5 a constant}

which

depends

^{on the nature of the} interaction. It has been evaluated

by Ginzburg

^{for the} ferroelectric case, to be ô - 2 x

109 a2 MKS,

where a is the lattice constant. A is

given by

^our dielectric measurement

(2 A = 1/eo

^{C £}

¹⁰⁶ MKS)

^{and B comes} from Todd’s measurement of the variation

1/e

^{as a function}

of

p2 [2] (B --

^{2 x 108}

MKS).

With these

values, à TI To --

^{1.6 x}

^10-6

^K.

Our measurements are,

obviously,

outside this _range and the different

approximations

used in the

preceding

calculation

are justified.

In the same way, the temperature variation of the dielectric constant and the ratio

C/C’

^{= 2} of the Curie coefficients are consistent with the Landau

theory.

It is worth

noting

that in ferroelectric

crystals,

ⁱⁿ

which the interaction is of the

long

range Coulomb- type, ^{the width}

ATITO

^is

relatively

^small

(1.6

^{x 10- 6}

in

KTN, 4,3

^x

10-6

in

BaTi03 (values

^from

[16]),

3.9 x

10- 5)

ⁱⁿ

KNb03 (values

^from

[1])).

^{On the}

contrary,

ⁱⁿ

SrTi03,

where the distortion is

opposite

in two

neighboring

^{cells and}

consequently

^{of short}

range, the critical domain

T/To

^{is about}

2 x 10-1[17].

(B)

^A second mechanism _may be

envisaged

^to

explain

the critical behaviour of the elastic constants.

In the

previous

^case

(A)

^the

coupling

^{of an acoustic}

wave with the _square of the fluctuations of the order parameter ^is

possible

^{either at the center or at the}

boundary

of the Brillouin zone. The _process that we now discuss is

possible only

^{at the center of the zone.}

It is considered here

mainly

^{because neutron}

scattering experiments

^{show a}

large

interaction between acoustic and

optic

modes in KTN

[6]. Following

^{a remark}

[13]

put

^{forward some time} ago ^to

explain

^{the curvature of}

the transverse acoustic mode in

KTa03,

^{it is}

possible

to calculate the effect of this

coupling

^{which modifies}

the acoustic

dispersion

^{curve in the}

following

way :

In this

expression

vo is the sound

velocity

^{far from}

the

transition,

^and

F.(’) .,,,

^a

^coupling

coefficient between the

optical

^{soft mode} and the acoustic

phonon

^branch.

In addition it is _easy to take into account the lifetime

T -1 of

the soft mode : in

úJ2(q)

^{we substi-}

tute

Qg by Qg

^{+ 2 ifco.}

Finally

^{we have}

When T -

To, QO(T)

goes ^{to zero}

and,

ⁱⁿ

principle,

we

might

very well observe critical behaviour for w

or a

resulting

from this mechanism.

If Q’ (T - To),

then the critical

exponents

^{are :} for

àvlv : m

⁼

1,

and for a : n ⁼ 2.

Since these

predictions

^{do not}

correspond

^{with the}

experimental

values of the critical

exponents,

^it appears

already unlikely

that this mechanism

applies

in our case. It is instructive to calculate the order of

magnitude

of this effect for which we need to know the

following quantities :

e The

F.(’)

^ac

coefficient,

^{which can} be obtained from neutron

scattering experiments

ⁱⁿ

KTa03 [13]

^for-

mula (6) ; we adopt a

value of about 2.8 ^x

107 M2. S - 2.

e The line width of the critical

optical phonon ^F

which can be evaluated to be

roughly (2 meV) - ’.

e The

frequency

of the critical

optical phonon S2o,

which has

already

been estimated to be

Qo

^{= 9 x}

¹⁰¹¹ (T - TO) 1/2 rad/s-’ .

e The ultrasonic

frequency (co

^{= 31.4 x}

108 rad/s)

and the sound

velocity (vo

^{= 8 x}

¹⁰³ m/s).

With these

values,

^formulas

(17)

^and

(18) give :

These formulas

give

^{for T -}

To

^{= 10}

K,

They

^{are much} smaller than the values observed at the same

temperature

^{which are :}

From this

comparison

^it appears very

unlikely

^that

the second mechanism ^can

explain

^the effects observed in ultrasonic

experiments :

^not

only

^{does the}

expected

effect seem a few orders of

magnitude

smaller than

experimental values,

but also the

temperature depen-

dence

given by

the critical exponents ^m and n does not agree ^at all with the

experimental

^results.

Our _range

of frequency

^{involves wave vectors of the}

order of 4 x

105 m-l ; they

^{are too} small relative to the size of the Brillouin zone

(qB l--

^{1.5 x}

^101° m-1)

to enable a clear observation of the direct

coupling

process between

optical

and acoustic

phonon

^branches.

This

coupling

^could

begin

^{to be}

important

^{for wave}

vectors at least ten times

larger

^{and this can}

hardly

^be

obtained

by

^acoustic

techniques ;

^{on the} contrary, ⁱⁿ Brillouin

experiments,

^this

coupling

mechanism could very well be

observed,

^{and due to} the différence in the

predicted

values of the critical

exponents,

it could be

distinguished

from other

types

^of

coupling ;

^{in addi-}

tion,

^neutron

scattering experiments

may also _prove to be _very useful in

studying

the critical behaviour of KTN.

5. Conclusion. ^- In this _paper we have been able

to show which of the two mechanisms

explains

^the

observed variation of the ultrasonic

properties.

^The

second _one, which was discussed because it has been used to

interpret

^the drastic effects observed in neutron

scattering experiments [6],

^{has been}

proved

^{to be}

irrelevant in the _range of

frequency

^{where ultrasonic}

experiments

^{can be}

performed.

The mechanism

responsible

for the critical beha-

viour of ultrasonic

velocity

and attenuation is thus the

quadratic coupling

between the ultrasonic strain and the _square of the order parameter fluctuations.

We have been able from our

analysis

^to

give

^the

range of

validity

^{of our} critical acoustic

phonon

branch model.

The results which we have

presented

^here are, to ^our

knowledge,

the first of their

type

^{to be made on KTN}

samples.

^{The reason for}

this,

is that it is far from

simple

^to

produce crystals

^of

sufficiently good quality

to enable reliable ultrasonic measurements to be made.

These results are

part

^{of a}

general study

^{of the}

influence of KTN

composition

^on ferroelectric transi- tion behaviour in

general,

and ultrasonic

properties

ⁱⁿ

particular.

References [1] TRIEBWASSER, S., Phys. ^{Rev. 114} (1959) ^63.

[2] TODD, ^{L. T.} Jr, ^M.I.T. Crystals Physics Labs Tech. Report ¹⁴ (1970).

[3] PERRY, C. H., ^Molecular Spectroscopy of ^{Dense Phases} (Else-

vier, Holland) 1976, 267.

[4] KIND, R., MÜLLER, ^K. A., ^Commun. Phys. ¹ (1976) ^223.

[5] YELON, ^W. B., COCHRAN, W. and SHIRANE, G., Ferroelectrics 2 (1971) ^261.

[6] Experiments ^{N° 04 02 030} (1976) ^{and N° 06 01 038} (1977)

at I.L.L., Grenoble (to ^be published).

[7] COURDILLE, J.-M., DEROCHE, ^{R. and} DUMAS, J., J. Physique

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[8] PYTTE, E., Structural Phase Transitions and Soft ^Modes (Ed. by ^{E. J.} Samuelson, ^E. Anderson and J. Feder, Universitets, Forlaget, Oslo) 1971, ^151.

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[13] AxE, ^J. D., HARADA, ^{J. and} SHIRANE, G., Phys. ^{Rev. B 1} (1970) ^1227.

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