Investigation of AlPO4 by Brillouin Spectroscopy from 300 to 1100 K

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Investigation of AlPO4 by Brillouin Spectroscopy from 300 to 1100 K

N. Magneron, Y. Luspin, G. Hauret, E. Philippot

To cite this version:

N. Magneron, Y. Luspin, G. Hauret, E. Philippot. Investigation of AlPO4 by Brillouin Spec- troscopy from 300 to 1100 K. Journal de Physique I, EDP Sciences, 1997, 7 (4), pp.569-580.

�10.1051/jp1:1997176�. �jpa-00247345�

J.

Phys.

I £Fance 7

(1997)

569-580 APRIL 1997, PAGE 569

Investigation of AIP04 by Brillouin Spectroscopy

from 300 to 1100 K

N.

Magneron _(~~~,*),

Y.

Luspin _(~~~),

G. Hauret (~~~) and E.

Philippot _(~)

(~ Centre de Recherche _sur la

Physique

des Hautes

TempAratures, CNRS,

45071 Orldans Cedex 2, France

(~) ^Universitd

d'orldans,

45067 Orldans Cedex 2, France

(~) Laboratoire de

Physicochimie

des lfatdriaux

Solides,

CNRS. Universitd de Montpellier II, 34095

Montpellier

Cedex 5, France

(Receiied

9 October 1996. receii~ed in final form 17 December 1996, accepted 24 December1996

)

PACS.64.70.Rh Commensurate-incommensurate transitions P-~CS.62.20.-x Mechanical properties of solids

PACS.78.35.+c Brillouin and

Ra,~leigh scattering;

other

light scattering

Abstract. The elastic constants of AIP04

(berlinite)

have been measured at _room temper-

ature. The

longitudinal

and _transverse acoustic _waves related to

Ci,

_{C33 and}

CL

_{have been}

investigated

_over the temperature range 300-l100 K which includes the _{a ~} inc. _~

fl

transitions

near 860 K.

Large

anomalies of

Ci

_and C33,

together

with

broadening

of lines, _are observed

near the trawitions, with _a

significant

temperature range of pretransitionnal features. The

C)4

constant exhibits _a discontinuous

drop

at the lock-in transition. The results

concerning

the lon-

gitudinal

modes _can be

qualitatively

interpreted in term of _a third order anharmonic

coupling

between the acou~tic modes and the soft modes. The range of the incommensurate

phase

is found to be T Tc _+~ 3 K, in agreement ^u~ith previous data. The

broadening

of lines related to

longitudinal

modes allo,ved the determination of the constant A of the relaxation time for the

amplitudon

_T~

=

A/(T~ T);

_we obtained A

= I-I _x 10~~~ K _s.

Rdsumd. Une dtude par diffusion Brillouin _{nous a}

permis

de _mesurer les constantes Alas-

tiques de -tP04

(berlinite)

h temp6rature ambiante. Les ondes acoustiques

longitudinales

et

transverses correspondant _aux constantes

Ct.

_{C33 et}

C$,

_{ont dtA suit.ies}

sur un intervalle de

tempArature

300-l100 K qui inclut les transitions _o ~ inc. ~

fl proches

de 860 K. PrAs de _ces transitions, des anomalies ont AtA obser,~Aes _pour les constantes

Ci,

_{C33. ainsi}

_qu'un Alargissement

des raies Brillouin. La constante

C)4

montre une discontinuitA h la transition de

"lock-in". Les rdsultats

correspondant

_aux modes

longitudinaux

_peuvent @tre interprAtAs quali- tativement _en termes de

couplage

du troisiAme ordre entre les modes acoustiques et les modes

mous. Cette dtude _{nous a}

dgalement

permis de vArifier _que la

phase

incommensurable s'dtend _sur

3 K. Une Atude

plus approfondie

des

largeurs

de raies Brillouin.

correspondant

_aux ondes lon-

gitudinales,

permet ^la dAtermination de la constante A du temps de relaxation de

l'amplitudon

T~ =

A/(T, T)

_on obtient A

= I-I _x 10~~~ K _s.

(* ^{Author for} correspondence

(e-mail: magneronsiadmin.cnrs-orleans.fr)

©

Les

#ditions

_de

Physique

1997

570 JOURNAL DE

PHYSIQUE

I N°4

1. Introduction

Aluminium

phosphate AIP04 (berlinite) belongs

to the

family

of

quartz-like

materials. It is known that _a

strong analogy

exists between its

physical properties

^{and those of} quartz

[lj.

Several

investigations

have been

performed

in order to _assess the

potential

of

AIP04

_{as an} alternative to

quartz

in ultrasonic device

applications

with _a

larger

electromechanical _cou-

pling [2-4j.

AIP04 undergoes

_{an a} ~

fl

transition _near 860 K

[5j.

Between the _a and

fl phases,

_an

intermediate incommensurate

phase

exists _{over a narrow}

temperature

range of 3 K

[6j.

^This sequence of

phases

is

quite

similar to that in

quartz [7j, corroborating

the

analogy

between the two

compounds.

At _room

temperature,

the elastic

properties

of

AIP04

have been determined

by

ultrasonic methods

[2-4j

and Brillouin

scattering [8j.

^{Unlike those of} quartz, these

properties

have not

generally

been

investigated

_{over a} temperature range which includes the transitions.

Only

the elastic constant

C33

has been studied

by

Brillouin

scattering [8j.

^However, measurements in the

fl phase

have been limited to 12 K above the

transitions,

due to the _appearance of

opalescence

in

samples.

This deterioration of the

sample

has also been observed in other

experiments [6, 9j

and it is _a

likely

_reason for the lack of

experiments by light scattering.

In the

present

_paper, we report new results of Brillouin

scattering investigations

in

AIP04.

We have been

supplied

with

samples having

an excellent

optical quality

at room tempera- ture which is

only- slightly

affected

by heating significantly

above the transitions. Thus, the

measurements have been

performed

from 300 K to 1100 K.

2.

Experimental

and

Crystal

Data

Brillouin

scattering experiments

_were

performed

with _a spectrometer ^{which is} a pressure

scanned, triple passed plane Fabry

Perot interferometer

(effective

finesse

70, resolving

_power

760000).

The

spectra

_are

frequency

checked

by

a Michelson interferometer in

parallel.

The

light

_source is the

~o

_" 514.5 _nm line of _a

single frequency

Ar-ion

laser,

the

frequency

of,vhich is controlled

by

_an iodine cell.

The

samples

_were cut from

monocrystals

of

AIP04

which _,vere grown

by

the

hydrothermal

method. This method has

required preliminary

studies of berlinite

solubility

in different media with different concentrations. These studies have shown _a

retrograde solubility

which is much

higher

in

sulphuric

acid than in the other _ones

[10j.

^{All the}

experiments

were conducted

through

the _reverse

temperature gradient

method

[llj using

375

cm~ platinum-lined

autoclaves,vith

battle apertures ^of

8%

and _a

filling

of 80%.

Systematic investigations

into

crystal growth

conditions

[11,12j

have

emphasized

that the sulfuric acid medium is the best _one to obtain

crystals

with the lowest OH

impurity

concentration.

Then,

both

crystals

used in this

study

have been obtained from x-seed in

H2S04

4.5 mol. solvent for _a

growth temperature

of Tc ₌ 240 ^°C.

At _room

temperature,

the

dextrogyre crystals

of

AIP04

show the

trigonal _symmetry (class

32)

with _space group

P3121 la phase).

The

parameters

of the

primitive hexagonal

cell _are

a = b

= 4.942

I

_and

c = 10.97

I [2,13j;

this cell contains three

AIP04

_groups. The struct,ure, close _to that of

quartz,

is derived from that of the latter

by replacing

t,vo Si atoms,vith _one Al _atom and _one P

atom,

^which results in _a

doubling

of the cell

along

the _c axis.

Near 860

K, AIP04 undergoes

an ^" o ~

fl

transition. Evidence for _an intermediate

phase

came from several

experiments [9,14-16j.

Neutron

scattering experiments

_[6j have

gi,>en

a

direct verification of the incommensurable structure of this

phase,

which lies _{over a narrow} range

Tj Tc

₌ 3 K, Ti ^and

Tc being

the

fl

_~ inc. and lock-in transition

temperatures

respectively.

The

fl

~ inc. t,ransition is induced bv the condensation of soft modes with _,vave

N°4 BRILLOUIN SPECTROSCOPY IN AIP04 FROM 300 TO l100 K 571

vector

ki

= +0.03a* and

equivalent by symmetry,

which

corresponds

to the

vicinity

of the Brillouin _zone centre. The

temperature

_range of the incommensurate

phase

is

greater

^than twice that in

quartz [17j.

^The structure of the

fl phase

^is

hexagonal,

class

622,

with space

group

P6222.

At _room

temperature,

we have measured the refractive indices _at

~o

and obtained the value

na = 1.5276 and _nc

= 1.5369. The _mass

density,

deduced from cell

parameters

at _room

temperature,

_{is p =} 2.618 g

cm~~

ivhen _we

computed

the elastic constants from Brillouin shifts at any

temperature,

we used the _room temperature ^values of both _mass

density

and refractive indices. No data _on indices

versus

temperature

_are available. An

investigation

^of the lattice parameters ^{has been made}

from 200 K to 1100 K

[2j,

^{but the data} are not

sufficiently

detailed in the

vicinity

of the transitions. It is also to be noticed that the variations in temperature of the

density

and of the indices _may

partially

_compensate in the calculation.

Brillouin

spectra

^{have been}

performed only

_on

heating

the

samples

in order to avoid eventual

hysteresis

at the transitions and

optical degradation

after _a first

heating

_run. Some

samples

break in the

vicinity

of the lock-in transition

Tc

but without any consequences on

spectrum collecting

since

large regions keep

their

optical quality;

_on further

heating,

this

quality

is

practically unchanged

_up to the

highest in,>estigated

temperature. After

cooling

down to _room temperature, a

slight opacity

can be

observed;

the initial spectra are recovered but with _a decreased cross-section of the Brillouin line

by

_a factor 2 and _an increase of the elastic

Rayleigh peak (factor r~10).

Thus the same

sample

_{was not} used in _more than _one

heating

run.

3. Brillouin

Experiments

we used

right angle scattering geometry

for the

experiments;

in this _case the

velocity

l~ of _an acoustic _wave is related to the

corresponding

Brillouin

frequency

shift _v

by:

l~

=

~ov/(n)

₊

n))~/~

where _ni and _{ns are} the refractive indices for the incident and scattered

light

beams respec-

tively.

For the measurements of elastic constants at _room temperature, we used the

scattering geometries

which _are

reported

in Table I.

The

piezoelectricity

of the

trigonal phase

introduces _no distinction between elastic constants at constant electric

displacement C(,

and elastic constants at electric field

C(

for

C33.

On the

contrary,

^the distinction holds for all other constants. In the

fl phase,

which is also

piezoelectric,

this distinction

only

holds for

C44.

The

piezoelectric

moduli _en and _e14 have been measured

by

several authors at _room tem-

perature [2-4j

with

widespread

results.

probably

due to _a different

origin

of the

samples [3j.

Therefore,

_we will

give

below the results for the elastic

constants, C(

_or

C(,

which _are the most

directly

obtained from the Brillouin

shifts, except

^when not

possible.

4. Results

4.I. ELASTIC CONSTANTS _AT Room TEAIPERATURE. We have obtained the

following

values in

(GPa):

Cl

= 71.7, C,13 _" 91.I,

C)~

= 44.6,

C$

=

29.2, Ci =13.3, C)~

=

19.4, C)~

₌ ^-12.7 with _{an accuracy} of

I$l

for the first four constants,

3$l

for

Cl

_and

10%

for

Cl

and

Ct.

The calculation of

C)~

^and

C)~ requires

the values of

Cl

we have estimated the

piezoelectric

572 JOURNAL DE

PHYSIQUE

I N?4

Table I. Characteristics

of

the

investigated scattering geometries.

The usual convention

for

_wave vector and

polarization

direction

of

the incident and scattered

light

beams has been

used. _q is the acoustic _wave vector.

L, T, FL,

PT _are related to the nature

of

acoustic _wave:

longitudinal,

transverse,

pseudo-longitudinal, pseudo-transverse respectively.

wave

Scattering geometries Expression

of pl~~

q direction nature

I

j-x

+

y>[z,

z]

ix

₊

y)

[100] ^L

Ci

2

(v z) lx, xl (v

+

z)

_1°°11 L C33

3

(v z)lx. Iv

+

z)llz

₊

v)

_1°°11 T

CL

4

1-v

+

z)i(v

₊

z), xl(v

+

z)

_1°1°1 T

CA

=

ICI Ci)/2

5

(~Y

+ Z)[Xi Xj(Y + Z) [010j ^PL

(Gil

₊

^C~

+

[(Gil C)4)~

_~

~(C)4)~j~~~ /~

6

(~Y

+ Z)[X,Xj(Y + Z) [010j ^PT

(Gil

~

C~ ((Gil C)4)~

~

~(~)4)~j~~~ /~

l

/4 (C)1

+ C33 +

2(C)4 C)4)

~

(Y)(~, ~j(~)

_(°~~j ~~

+[(C)1

C33

2C~)~

+4(C)3

₊

C)4

_C)4)~j~~~

1/4 (C)1

+ C33 ₊

2(C$ C)4)

~

(Y)(~, ~j(~)

_(~~~j ~~

~((C)1

C33

2C)4)~

+4(C)3

₊

_C)4

_C)4)~j~~~

correction

C( C)i

=

e(i fell

with the data of

Bailey

et al.

[3j,

^{since those authors}

give

the values which _are the closest to _ours among the different sets of results available about elastic constants. The

sign

of

C)~

has been attributed

follo,ving

_a

previous

result for which the IRE standard _on

piezoelectric crystals

has been taken into account

[2j.

Our results agree

satisfactorily

with

previous

ones obtained

by

Brillouin

scattering

[8j ^and ultrasonic methods

[2-4j.

4. 2. INVESTIGATIONS _wiTH TEMPERATURE. The Brillouin lines related to the elastic _waves

of _cases

1,

2 and 3

(Tab. I)

have been

investigated

versus

temperature,

from _room

temperature

up to r~l loo K. The results for

C(, _C33

and the full widths at half maximum

(FWHM) ri

and

r3

of the related

lines,

after

deconvolution,

_are

reported

in

Figures 1,

2 and 3.

Large

anomalies _are to be found for the elastic constants

(Fig. I).

Considerable

dips

_occur

on both sides of the

temperature

range of the _a ~ inc. _~

fl

transitions. Pretransitionnal effects _are present at temperatures as low _as 300 K and still exist at I loo K. The result for

C33

_agrees with _a

previous study [8j.

^{As shown} on an

expanded

scale

(Fig. 3),

the bottoms of

dips correspond

to _a continuous

change

which shows _a rounded

upward step

on

heating.

It is to be

remarked,

that _a

change

in the evolution of the increase of both elastic constants takes

place

in the

vicinity

^of 860 K which will be

interpreted

_as the

temperature T

of the inc. _~

fl

transition.

The linewidths

ri

and

r3

also show _a continuous evolution in the

vicinity

of the transition

(Fig. 3). ri

and

r3

take _a maximum value at _a temperature

TM,

the _same for both _curves.

TM

is 1.5 K below the value 860 K which has been noted above.

N°4 BRILLOUIN SPECTROSCOPY IN AIP04 FROIf 300 TO l100 K 573

110

c~~

~

loo ^'~

i

_.'

>

Z .°"

~

_j"

~ 90 _.

(

I

~

(

i

80

o i

,

*

70 '

j

$

#

60

°;

50

290 490 690 890 1090

TEMPERATURE (K)

Fig.

I. Temperature dependence of elastic constants

Ci,

C33.

The

general

behaviours for

C(

and

C33

are consistent with those found in

quartz [18j.

^On

an

expanded scale,

the temperature

dependence

of

C(

_{and of}

_{C33 [18,19j}

_is

qualitatively

similar to the results _we have obtained in

AIP04.

The linewidths

ri

and

r~

in quartz

[19, 20j

also show _a maximum below

T; however,

discontinuities of these linewidths _are observed _at

Tc by

further

cooling, contrary-

to the _case of

AIP04.

There is another difference between

quartz and

AIP04 [18-20j.

In

quartz,

^the elastic anomalies which _are like those _we have observed in

AIP04

in the

vicinity

of the

transition,

still when

heating

_as

specified

in Section

2,

are obtained when

cooling

from the

fl phase

and also when

heating,

but

only

if the

sample

t,emperature ^is

kept

above

Tc [19j;

^when quartz ^{is heated from the} a

phase,

_a direct transition into the

fl phase

_occurs,

implying

that both the continuous increase of the elastic constants and the

broadenings

with their maxima cannot be observed.

Taking

into account the first order character of the lock-in

transition,

this second fact _can be

explained by

the narrowness

(r~l.3 K)

of the incommensurate

phase

in

quartz,

^with respect to the

hysteresis amplitude

at the lock-in transition.

The

temperature dependence

of

C)~

^is

reported

in

Figure

4. This _constant decreases from

room temperature _up to

(

₌ ^{857 K where} a

sharp

fall _occurs. Above 890

K,

_a linear increase is obtained _up to 1100 K. It _can be remarked that if _a discontinuous decrease of the _mass

density

existed at the _same

temperature

and _was taken into account

(according

to the data of [2j ^which

unfortunately

_are

incomplete

in the

,>icinity

of the

transition),

the fall of

C)~

^would be

enhanced;

thus the recorded fall should not be considered _{as an} artefact. No

broadening

of the

corresponding

line _can be observed

throughout

the

temperature

_range. ^A similar fall of

C)~

^{has also been observed in} quartz

[21j.

5. Discussion

5.I. ELASTIC CONSTANTS

C[, _C33

_AND RELATED BROADENINGS. The

changes

in those

quantities

with

temperature

are not unusual:

they

have been observed in many other _struc- tural transitions of ;ariable types such _as

improper

ferroelastic _or incommensurate _ones.

574 JOURNAL DE

PHYSIQUE

I N°4

5

4.5 ^~~

~3.5

~

1 ₃

)2.5

~

2

~l.5

I:

o.5

,/

_°,,

o

''

600 700 800 900 1000 11 00

~)

TE>IPERATURE(K)

3 r3

2.5

Q 2

$

~ _{l .5}

I I

3

0.5

j'l j'

0 ^°'

600 700 800 900 1000 l100

b)

TEMPERATURE (Kj

Fig.

2. Temperature

dependence

of linewidth ri

(a)

and r3 16) related to elastic constant~

Ci

and C33 respecti,~ely.

In

particular,

the results for

AIP04

_{are very.} much like those of

C33(T)

and related broad-

ening occurring

_near the normal-incommensurate transition in

K2Se04 [22-25j,

which has been among the most

thoroughly

studied

crystals

with _an incommensurate

phase.

These anomalies of elastic constants and linewidths _are induced

by

the

coupling geo~

which is linear in strain _e and

quadratic

to the order parameter

Q;

this

coupling

is

generally

the

lo,vest

order

_one when the bilinear

coupling

is forbidden

by symmetry.

In

AIP04,

the _star of the,vave vector related _to the soft mode in the

,@

phase

has six _arms

+ki, +k?

and

+k3 [6j.

^The

coupling geo~

is

effectively

the lowest order _one allowed by-

sy.mmetry [26].

^This

coupling

has been

widely analysed [22,

24, 25, 27,

28j. Therefore,

_,ve will

only

recall here the salient features and

apply

them to our results in

AIP04.

In the

fl phase.

~T°4 BRILLOUIN SPECTROSCOPY IN AIP04 FROM 300 TO l100 K 575

go 5

~~

c$

^4.5

~_

.ri .° 4

f

⁸⁰

~ ~

75

1'

~

i

~

ii

I _{70 2.5}

I I

)

65 ~

i

1.5 '

j

60

l

~~ "" ' ~'

~,

0.5

50 0

846 851 856 861 866

a)

TEMPERATURE (K>

85 3

C33

r~ x

~ ~

~ ~

j

~ ~°

~

~

§

i

₇₅ ~

t $

§

^l.5

~0 "

p £

q

.,~~ ^~

j

~~

~, ^..

0.5

60 0

846 851 856 861 866

b TEAIPERATURE (K)

Fig.

3.

a)

Elastic constant

Ci

and linewidth ri _near the transitions.

b)

Elastic constant C33 and linewidth r3 _near the transitions.

this

coupling

_can be written _as follows:

~ §(q,k,k')e(q)Q(k)Q(k')d(q+k+ k')

q,k,k,

where

e(q)

is _one of the elastic strains

eij(q)

with

I, j

= I to 3.

Q(k)

and

Q(k').

which

are normal coordinates of modes _on the soft branch, _are to be taken into account

only

in the

vicinity

of

+ki.

Above

T,

this term induces anomalies for both elastic constants

(C~~,

^{I and}

j

= I to

3)

and linewidths

[29j

^which are

given by

the often

designated

fluctuation

integrals.

The

quantitative

calculation of these anomalies

requires

the

knowledge

uf the soft branch in the

vicinity.

of the critical

points +ki; unfortunately

those data _are not available

yet

as far _as

AIP04

is concerned.

576 JOURNAL DE

PHYSIQUE

I N°4

50 C~4

~45

£

(

40

35

',

(

,'

~ 30

~_,,l'

^{' '}

25

290 490 690 890 logo

a)

TEAIPERATURE @Q

35 c(~

34

133

~~~

^"

831

)

~30

~

29 '°°°°.

_~--S--

28

830 840 850 860 870 880

bj

TEMPERATURE jK)

Fig.

4.

a) Temperature dependence

of elastic constant

C)4. b) Expanded

temperature ^scale near

the transitions. The extrapolation of the linear variation of the fl phme allotvs the determination of

AC)4 (dashed lines).

For T _<

T,

in the incommensurate

phase.

the situation is _more

complex,

since the thermal

mean values of

Q(+k~)

become different from _zero. Therefore there _are two different

coupling

terms. A first _one, ,vhich _can be ~v.ritten _as follo~v.s:

fl _{glq, +ki}

₊

_{k, ~k~}

k

qlelq)Ql+k~

₊

k)Ql~k,

k

q)

q,k

is similar _to the

unique

term above

T;

it

gi,>es

a first contribution to elastic anomalies which

are similar to those ,vhich _occur above

2j.

The second

coupling

terra _can be written _as follows:

2

fl _{glq, +ki, ~ki} q)elq)Ql+ki)Ql~ki q).

qk

N°4 BRILLOUIN SPECTROSCOPY IN AIP04 FROM 300 TO l100 K 577

Through

the usual transformation into

amplitudon

and

phason;

it _can fall into two parts:

a

part

^related to the

amplitudon and,

if allowed

by symmetry [25],

a part ^{related to} the

phason.

These _two

parts

induce anomalies for elastic constants and

broadenings

of

lines, ACLK

and

rLK

which _are described

by

^the

usually

called Landau-Khalatnikov

(LK)

terms. The terms

related to the

amplitudon

_can be written in the form

[19, 25]

j~

~

_~ol~ki)~~

~~

Q](+ki)

1 ₊

uJ2r)

~~~

_~

~ol~ki)~~

_~a

fl](+ki)

1 ₊

uJ2r)

with _ra

=

ra/Q](+ki), _ra

and

Ha being

the

damping

and the

frequency

of the

amplitudon;

~ is the

angular frequency

of the

appropriate

Brillouin line. For T _<

Tc.

the

amplitudon

is to be

replaced by

the usual soft mode.

In the

simplified

framework of _mean field

approximation,

_one has

~Q(+ki)~~

_r~

jr T);

Q](+ki

r~

jr T)

and therefore _ra

=

Al jr T),

if

ra

is taken _as temperature

independent.

ACLK

describes _an

up,vard

step at

T,

when T

increases,

which _can be _{more or} less

rounded, depending

_on

ra.

The simultaneous

broadening rLK

is

expected

to show _a maximum at _a

temperature

T~I,

below

T,

when

ulna(TM)

= I.

The existence of such _a

maximum;

at a temperat,ure ^{in the}

vicinity

of which the elastic constant

sharply increases, unambiguously

indicates _a

significant

contribution of the LK _terms.

Below

T,

the various contributions to elastic anomalies ha,>e _to be added. An examina- tion of the

experimental

results shows that

they

_are in

qualitati,>e agreement

^{with the above}

predictions. Therefore,

_a few consequences can be drawn.

A first consequence of the above considerations is the

precise

localization of

T

on the _curve

Cl (T)

and

C33 IT).

The steep ^rises of

C(

and

C33

between 857 K and 860 K

ob,>iously

have to be identified ,vith the increase of the rounded

step

^described above.

Therefore,

the

temperature

at which the

change

of

slope

_occurs would coincide with

T.

Both _curves

give T

= 860 + 0.25 K.

This result agrees ,vith the fact that

T

must be above the

temperature

at ,vhich the maximum of the

broadening

_can be observed.

A second consequence is the determination of the constant A in _ra which _can be obtained

by:

4

=

jr

_TM

)/uJ.

If _{we assume} that the maximum of the total

broadening (including

each

contribution)

is

only

slightly displaced

,vith

respect

to

TM (relative

to

rLK), taking

into account the monotonic variations of other contributions, _we

obtain,

from both _curves

ri

and

r3 T -T,I

₌ 1.5+0.25 K.

For both

longitudinal

modes _we have

practically

the _same value

uJ/20r

₌ ^{20.9 GHz which}

gives

A = I.I _x

10~~~

_K

s ,vith _an accuracy of 16% in ~v.hich the

major

contribution _comes from

T.

5.2. ELASTIC CONSTANT

CL.

The strain _e4

being

not

totally symmetric,

the lowest order

coupling

terms ~v.ith modes of the soft branch _are

biquadratic:

fl

_h.4

_lq,

_-q,

_+k, ~k)e4 iq)e41-q) Ql+k)Ql~k).

q,k

The effect of this term upon the elastic constant

C)~

has also been

widely analysed [22, 24, 25].

Below

T,

it

gives

a static contribution to the

anomaly

of

C)~,

,vith respect to the bare constant

(C)~)o,

,vhich is

proportional

to the order parameter ^Inodulus

squared:

~~~ ~~ _(~~4)0

~

~Q(~~i)~~.

578 JOURNAL DE

PHYSIQUE

I N°4

If the linear variation of

C)~

in the

fl phase

is

extrapolated

below

Ti,

we can obtain

~C)~

and, consequently,

information about

~Q(+ki)~~

in the incommensurate and _a

phases.

The

sharp drop

for

C)~

_occurs at

(

₌ 857 K which is r~3 K below

Ti. Then,

between both these

temperatures,

we found

experimentally

that the order parameter

keeps

_a low value with respect to its values

just

^belo,v the

discontinuity.

This result

strongly

differs from that obtained in

K2Se04

for which, below

T,

_a continuous decrease of

C)~

is obtained _up to

Ti [22].

^The

non zero value of

AC)~

above

T (860 K)

_may be attributed to the fourth order anharmonic contribution involved in the

biquadratic coupling

term for k

# ki.

It

might

well be

thought

that the

drop

of

C)~,

which is the

sign

of _a

transition, corresponds

to the lock-in transition at

Tc,

since this

drop

_occurs

unambiguouslj~

belo,v

T

and that

only

the lock-in transition exists below

T.

This _can be

justified by

the

following

remarks.

In

AIP04,

_our result for

C)~ strongly

resembles that obtained about the

birefringence _[9],

for,vhich the

anomaly

is also

expected

to be

proportional

to the order

parameter squared.

The

curve for the

birefringence

shows _a

drop

to a low value, _{more or} less marked

depending

_on the

sample;

above the temperature of this

drop,

_a

temperature

_range of also 3 K.

corresponding

to the irreversible beha,~iour of

birefringence,

has been identified _as the incommensurate

phase.

It is worth

noting

that this

temperature

range of 3 K is also in

agreement

with the results of heat

capacity

measurements [6].

Therefore,

it _can be considered that the

drop

of

C)~

takes

place

at the lock-in

transition,

whose

temperature Tc

is thus available from Brillouin measurements.

5.3. REAIARKS. As _a first

remark,

_,ve will

emphasize that,

within

experimental

_accuracy,

no discontinuities appear ^at

Tc

for

C(, C33

and their related

broadenings,

in

spite

^{of the}

discontinuity-

of the order parameter modulus. The

following arguments

can be

proposed.

In the

simplified

_mean field

theory,

_~v~e ha,>e belo~v~

T [2j]: Q]

=

2b~Q(+ki)

_~~+

4c~Q (+ki

_~~+...,

where b and _{c are} the coefficients of the

expansion

^{of the} free energy related to fourth and sixth

po,ver of

~Q(+ki)~. Keeping only

the b term, the LK contribution to

C(

_or

C33

becomes:

~~~~

2b 1 +

uJ2r)

^{~~~~ ~~}

2b~Q(+k,)

~2

At T

=

Tc,

one has

Ti Tc

_m

2(T TM)

^and

~~rj

_m

1/4,

that reduces the effect of the

discontinuity

of

~Q(+ki)~

_upon

ACLK.

Furthermore. if the _c term is taken into account, we

have:

1 1

r~

~~~~

~ 2b +

4c~Q(+ki)~2

1+

uJ2r)

^{~~~~ ~~}

2b~Q(+ki)~2

+

4c~Q(+k~)~4

The t,vo terms in the denominator have _a

tendency

to compensate ^for a

change

of

~Q(+k~)~.

Similar

arguments

can be

developed

for the

broadenings.

An additional

possibility

is that the different

contributions, namely

the fluctuation

integral

and the LK _terms for the

amplitudon

and for the

phason,

have compensate ^{variations at}

Tc.

A second remark deals ~v~ith the order of the transitions. From the Brillouin

data,

it _can be deduced that the

fl

_~

inc.(Tj)

and lock-in

(Tc)

transitions _are of second and first order

respectively.

These results agree with other

experiments [6, 9].

^Similar conclusions have been drawn with

quartz [7], corroborating

the

analogy

of both

compounds.

A last remark is about the

temperature dependence

of the order parameter at Tc ^{and its}

particularly

low value in the incommensurate

phase.

obtained from the _constant

C)~.

We have _seen that this behaviour is consistent ,vith the

birefringence

measurements [9]. ^In quartz,

similar results _are obtained for the

birefringence _[19]

and for

C)~ [21];

moreo,;er, in _a recent Raman

study _[30],

the

intensity

of _a

line, expected

to ,>ary ^as the fourth _power of the order

N°4 BRILLOUIN SPECTROSCOPY IN AIP04 FROM 300 TO l100 K 579

parameter, shows _a

temperature dependence

which is

strikingly

similar to that of

AC)~

in

AIP04.

Thus, this

particular

behaviour of the order

parameter

_appears to be _common to this

family

of incommensurate materials.

6. Conclusion

In the present ^Brillouin

scattering study

of

AIP04.

_we report for the first time _an

in;estiga-

tion of the elastic

properties

over a

large

temperature range

(300-1100 K)

which includes the incommensurate

phase.

We have focused _on two

longitudinal

and _one transverse elastic _waves related to

Ct. _C33,

and

C)~ respectively.

Large

anomalies of both elastic constants and line

broadenings

related to the

longitudinal

waves are observed in the

vicinity

of the transitions with

pretransitional

features which _are

present

_up to both ends of the studied temperature range. The results _can be

qualitatively interpreted

_as the consequence of _a third order anharmonic

coupling

between strains and the soft modes. The trans;erse wa,>e, related to the

C)~

elastic

constant, yields

data about the

order parameter.

The results corroborate the temperature ^{extension of the} incommensurate

phase

_over r~3

K,

in

agreement

^,vith the results of other types ^of

investigations. They

also confirm the first and second order character of the lock-in and normal incommensurate transitions

respectively.

The

broadening

of lines allows _a determination of the relaxation time related to the

amplitudon.

Many

of these results _are similar to those obtained in

quartz,

^when available. The interest of

AIP04

is that the

temperature

_range of the incommensurate

phase

is _more than twice that of

quartz.

Therefore

experimental investigations

of this

phase

_are easier in

AIP04.

A

quantitative analysis

of the Brillouin data about the

longitudinal

modes

requires

data _on the soft

phonons

in each

phase.

&~ore

generally;

we think that _a widened

study

of the lattices

dynamics by

other

spectroscopies (IR,

Raman and neutrons

scattering

is suitable to get a

better

knowledge

of the transitions.

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