The ferroelectric phase transition in (CH3)4NCdBr3 (TMCB) studied by means of group theory, Raman scattering and calorimetry

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The ferroelectric phase transition in (CH3)4NCdBr3 (TMCB) studied by means of group theory, Raman

scattering and calorimetry

G. Aguirre-Zamalloa, J. Igartua, M. Couzi, A. Lopez-Echarri

To cite this version:

G. Aguirre-Zamalloa, J. Igartua, M. Couzi, A. Lopez-Echarri. The ferroelectric phase transition in (CH3)4NCdBr3 (TMCB) studied by means of group theory, Raman scattering and calorimetry.

Journal de Physique I, EDP Sciences, 1994, 4 (8), pp.1237-1257. �10.1051/jp1:1994251�. �jpa-00246981�

Classification Phy.çi<..ç Ah.çn.ac.t.ç

64.70 65.40 78.30

The ferroelectric phase transition in (CH~)4NCdBr~ (TMCB)

studied by

_means

of _group theory, Raman scattering and

calorimetry

G.

Aguirre-Zamalloa Il),

J. M.

Igartua (2),

M. Couzi

(')

and A.

Lopez-Echarn (2) (')

Laboratoire de

Spectroscopie

Moléculaire et Cristalline (±j. Université Bordeaux 351 _cours de la Libération, 33405 Talence Cedex, France

(2)

Departamento

de Fisica de la Materia Condensada, Facultad de Ciencias, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao,

Spain

(Receii~ed 22 Jwie /993, iei>i.çed 28 Febiuaiy /994, a<.<.epted /5 April /994)

Résumé. La transition de

phase

ferroélectrique impropre se produisant ^dans les cristaux de (CH~&NCdBr~ (TMCB) _{à T~ =} 160 K est étudiée à l'aide de la théorie des groupes, à partir ^des données structurales établies antérieurement à la fois pour la phase

paraélectnque

(P6~/m, Z

= 2) et pour la phase ferroélectrique

(P6j(P6~),

Z 6) _en

particulier,

les variables de

pseudo-

spin décrivant les réorientations des groupes (CH~)4N~ (TMA) _{agissant comme} paramètres d'ordre sont déterminées. Les spectres Raman indiquent _que cette transition est sans aucun doute du type ordre-désordre, liée _aux réorientations des groupe~ TMA, _{ce qui} de plus est confirmé par

1importante

entropie de transition déterminée par calorimétrie (AS =2,17±0,20R). Les résultats expérimentaux sont comparés avec les

prévisions

de la théorie de Landau.

Abstract.-A group theoretical

investigation

of the improper ferroelectric

phase

transition occurring in crystals of (CH~)4NCdBr~ (TMCB) at T~i160 K is developed, _on the basis of

previous structural determinations of bath

paraelectric

(P6~/m, Z

= 2) and ferroelectric

(P6j(P6~),

Z= 6) phases; ⁱⁿ particular, ^the pseudo-spin variables attached _to the reorientations of the (CH~)4N~ (TMA) _groups,

acting

_as

order-parameters,

_are considered. The Raman spectra indicate that the transition i~ undoubtedly of order-disorder type, as a result of reonentations of the TMA groups. This _is further confirrned by ^the large transition entropy as measured by calorimetry (AS=2.17±0.20R). The

experimental

results _are

compared

with classical Landau theory

predictions.

I. Introduction.

The

tetramethylammonium

tribromocadmate

(CHI)4NCdBr3

ITMCB for

short) belongs

to the

well-known

family

of

(CH~)~NMX~ crystals

(M: Mn, Cd _; X: Cl,

Br)

of which the

(*) U-R-A- 124 C.N.R.S.

compound (CH~)~NMnCI~ (TMMC)

has

probably

^{been the} most

widely

studied because of its magnetic

properties

and the _occurrence of structural

phase

transitions (see _e-g-

[1-6]

and references cited

therem).

^These

crystals

_are

hexagonal

at room temperature, ^with space-group

P6~/m

and Z

= 2 formula units per

primitive

unit-cell

(phase I).

The _structure exhibits _a strong one-dimensional character it is built up from infinite linear chains made of

face-sharing MXô

octahedra

running

in

parallel

to the hexad direction. The space between chains _is

occupied by

the

tetramethylammonium (TMA)

_groups,

exhibiting

orientational disorder

[2-4]

of

dynamic

nature

[7-9]

this disorder

merely

_comes ^{from the}

incompatibility

between the C~~ site

symmetry ^and the

T~

symmetry ^{of the free TMA} groups, so that orientational disorder _restores

statistically

the site symmetry. ^Ail

compounds

in this

family undergo

_a number of structural

phase

transitions

leading

to different ordered low temperature

pha8es,

which _are ferroelastic in the _case of TMMC and

(CH~)~NCdCI~ (TMCC) [2-6, 10-12]

or ferroelectric _in the _case of

TMCB

[13j.

Actually,

two structural

phase

transitions have been

reported

_in TMCB. The

high

temperature ^transition

occurring

at

- 390 K

[14] probably

leads to the prototype

(disordered)

structure with space group

P6~/mmc

and Z

= 2

(phase I')

such _as that found in TMMC and TMCC

[4-6].

_{At T~}

- 160

K,

another

phase

transition of the first-order takes

place [14],

from the _room temperature

phase (P6~/m,

Z

= 2

[15-16])

to an ordered

improper

ferroelectnc

phase [13]

with space-group

P6j-P65

and Z

=

6

(phase II) [1Si.

In _{a recent}

X-ray

diffraction

study

of TMCB

single-crystal,

the _room temperature

(paraelectric)

and low temperature

(ferroelectric)

structures have been determined

[17]

in

particular,

it _was shown that the ferroelectric

phase

is related _to the

paraelectnc

one

by

a

trebling

of the lattice parameter

along

the _c

(hexad)

direction. It _was also established

[17]

that the order-disorder process due _to the TMA groups

can be described within the frame of Frenkel type ^{mortels of}

complex

nature,

according

to

which the TMA'S reorientate between

equivalent potential

wells.

Then,

the

freezing

of

appropriate pseudo-spin

coordinates leads to the ordered state as observed in the ferroelectnc

phase (see

Sect.

3).

A

phenomenological approach

within the Landau

theory

^framework was

developed [15],

which accounted, at least _on

qualitative grounds,

for the behaviour of the spontaneous

polarisation

and of the static

susceptibilities

across the first-order transition.

In this paper, we first present

(Sect.2)

a group theoretical

analysis

of the

P6~/m

(Z

= 2

- P6

j-P6~ (Z

=

6 transition of

TMCB, including

discussions of the

order-parameter

symmetry properties, of the Landau

free-energy

_expansion and of the lattice vibrations. In section

3,

much attention _is

paid

to

pseudo-spin

variables acting as

order-parameters, developed

within the frame of Frenkel models.

Then,

_in sections 4 and 5, Raman

scattering

and adiabatic

calonmetry

measurements are

reported,

and the apparent

inadequacy

of the

former

phenomenological

mortel

[15]

with the _new data is discussed in section 6.

2.

Group-theory.

2.1 THE SYMMETRY PROPERTIES OF THE ORDER-PARAMETERS AND LANDAU FREE-ENERGY

EXPANSION. The

hexagonal

unit-cell of the ferroelectric

phase,

with lattice parameters such

as

(a,

_a, 3

c)

determmes _a lattice

instability

_occurrmg at a point

d(0, 0, «)

inside the

hexagonal

Brilloum _zone of the

paraelectnc phase (a,

_a,

c) [18]

_; the

triphcation along

c is

obtained for the

particular

value

a = 1/3. In the

P63/m (C(~)

space-group, the wave-vector group for ail points

d(0, 0,

a

) lymg

on the r-A fine _is

6(C5),

of which the four little representations are denoted as

Aj,

A~

(one-dimensional)

^and

(A~, A~), (A5, Aô) (complex

conjugate physically degenerate representations) [18]. Now,

the fuit representations

(Ai P6~/m ), (A~ P6~/m

are of dimension two and

(A2,

_A~

P6~/m

),

((d5, Aô) P6~/m

of dimension

four,

since there are two arms in the _star of the wave-vector at

point

A

[18]. Following

classical

group-theoretical procedures [19],

it tutus out that the

P6j(P65)

space-group is induced

by

the

((A~, Ai)1P6~/m) representation

at

point

A 0,

0,

_; the

3

corresponding

matrices _are

given

in table Al of the

appendix.

It follows that the pnmary

order-parameter (O.P.)

for the

P61/m

_- P6

(P6~ phase

transition has four components qj, q~, q~, q~, such that qj =

q/

and q~ =

qÎ

(see tab. Al la)), so that we

put:

~' ~~

~'

_{~~ ~~ ~'~}

.

Il)

q4 = q

'

'~,

_q3

=

Q'e

^'~

The

P61

space-group

corresponds

to such solutions _as qj #0,

q~#0, q~=qi=0

(q

# 0,

q'

=

0 and

P65

to

equivalent

solutions where q~ # 0, _q~ # 0, _qj

= qi =

0

(q

=

0, q'

# 0 ).

On the other

hand,

since the

phase

transition of TMCB

produces

_a

change

of the

crystalline

Mass from

Côh

to

Cô (ferroïc transition),

_{it can} be

easily

established that the zone-centre

(point r(0,

0,

0)) A~

representation ^{induces the}

totally symmetric

A

representation

in

Cô (for

^the

zone-centre

representations,

we shall refer _to Wilson's notation

[20]

which is _more

commonly

used

by spectroscopists).

Since the _z component of a vector transforms

according

to

A~

in the Cô~

point-group,

^this representation is associated with _a

secondary

O.P.

(referred

to _as

p)

responsible

for the spontaneous

polarisation F=

observed in the ferroelectnc

phase (improper ferroelectric) [13].

For convenience, we shall _{use in} the

following

O.P.'s _{wntten in} real form, 1-e-

7J # q ^COS q7

, 7J~ #

q'COS

_w

l~

7~~ = q sin q~

, 7~1 =

q'sin

_w

Under these

conditions,

the Landau

free-energy developed

_up to fourth-order _{terms can} be

wntten as

[15]

AW

=

a(T- To)(7~)+ 7~j+ 7~Î+ 7~Î)+ 2b(7~/+ 7~Î-

_{3 7~j}

7~j-

3 _7~~

7~()+

+ 2

b'(IJÎ +'ÎÎ 31J)1J4 31JÎ'Î3)

₊

C[(IJÎ +'ÎÎ)~

₊

l'ÎÎ +'ÎÎ)~l

+gl'ÎÎ~'ÎÎ~'ÎÎ+'ÎÎ)F~+h(1J)+IJÎ+'ÎÎ+'ÎÎ)(F(+F))+. ₍₃₎

where

Xo(

^{Î and}

Xo( 1)

are the free dielectnc

susceptibilities parallel

and

perpendicular

_to the

c axis,

respectively,

and

F~, F~ (Ei~ symmetry)

and

F~ (A~ symmetry)

are the three

components ^{of the}

polanzation

vector.

Coupling

terms with the _{stram tensor} components ^have been

omitted,

_since the transition is non-ferroelastic. _The Lifschitz _invanant and other

gradient

terms allowed

by symmetry

at

point

A

(0, ^0,

₃

^[15]

^{have also been}

^omitted,

^smce

^according

to Raman scattenng, calonmetric

(see

Sect.

5)

and dielectric

[13, 14]

expenments, ^there is no

hint of _existence of _an intermediate

(possibly incommensurate) phase.

Because of relations

(2)

A~P _can be _{wntten as :}

AW

=a(T-To)(q~+q'~)+2(bcos3

q~ -b's1n3

q~)q~+2(bcos3w -b'sin3w)q'~+

+c(q~+q'~)+dq~q'~+ ~Xi~(11)(Fl)+ ~Xô~(1)(F]+Fj)

+

g(q~ q'~)

F ₊

h(q2

₊

q'2)(F[

₊

F[)

₊

₍₄₎

Minimizing

A~P with respect to _q~ and _w

yields

tg ³ _q~ = tg ³ w =

~'

= constant

(5a)

and,

_as a result

q~ = w ± k 1

(k integer (5b)

For the

P61(P6~)

solution, _{i-e- q} # 0,

q'

=

0

(q

= 0,

q'#

0 the minimization equations

now

give

F,=F,.=0 (6)

p~

~

gx~~ 1) q2(p6j )1(7)

p

=

gx~( q'2(P6~)

which establish that

P6j

and

P6~

_are ferroelectnc _twins.

Then, the transition temperature T~ is given

by

T~=To+~~~Î" ₍₈₎

and the

equilibrium

value of the order parameter

by

:

q(T)(q'(T))=0

for

T~T~

q(T)(q'(T))

=

~/~

l

+

(1- ^{(~( )~}

^{for T~ T~ ~~~}

~

o)

where c'

= c g~

Xo(

^Î) il

0)

2

and q~ is the

jump

value of

q(T)(q'(T))

_{at T~}

(first-order

transition)

c&=

,

(bcos3q~

-b'sin3

q~).

Ill)

<.

For the sake of

completeness,

we should add that there _{exist two} other solutions with

monoclinic symmetnes ^induced

by

the

((A~, Ai) ÎP61/m)

representation,

namely P2j/m

(Z

₌ 6

corresponding

to _{7~ =} 7~~ #

0,

_7~i

= 7~4 # 0 and

P2j (Z

=

6) corresponding

to 7~ # 7~~ # 7~~ # 7~~ with _7~, # 0 (1 = 1, 2, 3,

4).

In those _cases

(which

have not been

observed

experimentally)

an elastic contribution should be added in

(3)

_or

(4),

and in

particular

the E~~ ^{strain tensor} components je _e~,

eô)

would act as

secondary

order parameters. ^Thus the

P6~/m (Z

=

2)

-

P2j/m (Z

=

6)

transition would be improper ferroelastic and

P6~/m (Z

=

2)

-

P21 (Z

=

6 would be

improper

ferroelectric-ferroelastic.

2.2 LATTICE viBRATioNs. -The _so called lattice vibrations

correspond

to ail vibrational modes of the

CdBrô

octahedron chains and to the externat

(translatory

^and

rotatory)

vibrations of the TMA groups considered _as

rigid

entities. In the

paraelectnc (P6~/m) phase,

^classical

factor-group analysis

at zone~centre (k ₌

0)

_gives :

ri~~~~~~=3A~e2B~e2Ei~e3E~~e3A~e4B~e4Ei~e3E~~. (12)

This enumeration includes the three acoustic modes

(A~eEj~)

of _zero

frequency

at

k =0.

Among

the even

parity (g)

modes, the

A~(a,,

+a,,,

a==), Eig(",=, ",=)

and

E~~(a,,

_{«,,, a,,} ones can be Raman active. The

corresponding symmetry-adapted

coordi- nates have

already

been described in detail

(see

_e-g-

[5, 21-23]).

At this stage, ^it is

important

to realize that the selection rules

given

_in

(12)

refer _to the

averaged

structure, 1-e- with the TMA units

having

the

required Cih

site symmetry. ^As stressed in section 1, this site symmetry now results from orientational disorder of the TMA

il 7],

_so that _a breakdown in the k

=

0 selection rule _can be

expected

to occur. In other words,

in addition to the allowed k

=

0 modes, the Raman spectra may exhibit _« disorder-induced _»

scattering,

e. may reflect _a

weighted frequency

distribution of

phonon

modes

throughout

the whole Brillouin-zone.

In the ferroelectnc

phase,

with

P61(P6~)

space-group, the TMA'S _are ordered in

general positions il 7].

The

rigorous

enumeration of lattice-modes at k

=

0 _now is

r~~~~~~~ ⁼ 18 A e 18 B e 18 E e 18

E~

11

3)

where acoustic modes

IA eEj)

have been included. Out of these lattiee modes, the

A

(a,,

_{+ a,,, a ==), E}

(a,-,

_«,= and E~

la,,

_{a,,,, a,,} ones can be Raman active. It should be

pointed

out that A and

Ei optical

modes _are

polar Ii.e. they

_are also infrared

active),

so that

they

can be

expected

to be

split

into _transverse and

longitudinal (TO-LO) pairs.

The

compatibility

relations between the symmetry

properties

of lattice modes _in the para

(P6~/m)

and ferroelectric

(P61) Phases

are given in table A2 of the

appendix.

Thus, it is

established that the zone-centre modes _in the

P6j(P6~)

space-group are issued from those _at

point

^r in

P61/m,

_on the _one

hand,

and from those at

points

A

(0, ^0,

₃ ^{and d'}

(0, ^0,

₃

(or

equivalently A'(0,

^{0, ~ ), on the other hand, since ail these points are situated at zone-}

3

centre in the

tripled P61(P65)

uni-cell. Then the addition of the _two contributions (Tabs.

A2(b)

and

A2(c), respectively) merely

_gives the enumeration shown in

(13).

It would not be very

meaningful

to give here _a

complete description

of these modes _{in terms} of

symmetry-adapted

coordinates. Let _{us note} however that the _{transverse acoustic} modes

(TA(A))

at points

A

(0, ^0,

^and

A'(0,

^0,

^belong

^{to the}

^{(A~, Ai}

⁾

representation,

_{i e.}

_correspond

_to the

3 3

symmetry ^{of the} pnmary

order-parameter.

As _a consequence, the

«

freezing

_» of these modes

in the ferroelectnc

phase (obtained

for solutions such _as q #

0, q'=

0

(q

₌

0, q'# 0)

_as established in Sect.

2.1)

is

responsible

^for a static distortion,

resulting

in a helical

shape

^of the

CdBrô

octahedron chains

il 7].

3.

Pseudo~spin description

of the order~disorder processes in TMCB.

A _convenient way ^to descnbe onentational disorder in molecular

crystals

is

provided by

the

Frenkel mortel,

according

to which the molecules

(TMA

_{groups in our}

case) perform jumps

3

2

2

3

a)

2'

1'

5'

3'

4'

' 3'

6'

5'

1'

2'

b)

Fig.

rientations of two

fold

axi~ of _{the TMA} ntained in

theJii= (z _1/4, z = 3/4j

(in ddition, m coincides with _one

mimerplane of TMA'Sj ; b) the 6 : tu the

rientations

of _une two fold _{axis of}

the

_TMA group neneral oeitions. uitand dashed _are

related tu _{une another} by _{the n>_ mirror} plane, but, for

the

_sake of _clarity, they _{have been}

away _{tram each} other. _{Fui( arrows}

orrespond tu orientations « up » and ashed arrows tu onentation~

« ^down »

between _a finite number of

equiprobable

orientations. Such _a

description permits

the introduction of

pseudo-spin

variables that _can be

easily

handled with the

help

of classical group theoretical

techniques

and which _can be considered _as O.P"s in order-disorder

phase

transitions (see e. g.

[24, 25]).

Of _course, this

description

is _an

oversimplification

in those cases

where the

experimental

results rather suggest the existence of continuous distributions of orientations

(Pauling mortel).

In the _case of

TMCB,

structural determinations

il 7]

have shown that orientational disorder of the TMA

corresponds

to a

complex (intermediate)

_process situated in between the Frenkel and

Pauling

mortels. This situation _can be

approached

either

by

means of _{a «}

perturbed

_» three- well

(3 W)

^mortel or

by

_a six-well

(6 W)

mortel

[17].

In the pure 3 W Frenkel mortel, one

mirror

plane

of the TMA tetrahedron coincides with the m= ^site mirror

plane,

so that the

Ci

_axis of the site is achieved

statistically by superimposition

of three

equiprobable

orientations

(Fig. la).

The _«

perturbed

_» 3 W mortel consists in additional

large amplitude

librations of the TMA _out of the m=

plane,

_as has been shown

experimentally [17].

In the pure 6 W Frenkel mortel, the TMA groups are in instantaneous

general

positions, so that the _site symmetry is

achieved

statistically by supenmposition

of _six

equiprobable

orientations

(Fig. lb).

In the

ordered ferroelectric

phase,

the frozen in orientation found for the TMA _is somehow

intermediate between those determined from the pure 3 W and 6 W mortels

[17].

In this section, we intend to describe the

P6~/m

-

P6j (P65

transition of TMCB in terms of

pseudo-spin

^{variables denved from the} _pure ^{3 W} and 6 W Frenkel mortels,

respectively.

Then,

a number of remarks will be made conceming the

ordenng

_processes that _can take

place

_in this system.

3.1 THE PSEUDO-SPIN COORDINATES.

3.I.l The 3 W mode/. As shown _in

figure

la, the 3 W mortel _{consists in} TMA groups

reorienting

between three positions

labelled1, 2, 3, respectively.

In the disordered

P61/m phase,

the

C3h

site symmetry requires

equal occupation probabilities

_n, for each

position,

_i e.

~l

~~2~~3~) ^~j~>

~

Îj ⁽¹⁴⁾

,

The symmetry properties of the

corresponding pseudo-spin

coordinates

~

are

easily

established in the C~h

Point-group

_{as :}

R(site)

=

A'DE

E'

With the

help

of the

projection

_operator

technique,

one finds three

pseudo-spin

coordinates à

(A')

and

ô~,

ô~

(E')

determined from the transformation matnx

[25]

M

~

₌

~

_M~,

ôn,

,

À,é _,é

~

/~ Î~,~ _~~~~~

~ -l

À ,fi

where for convenience orientation has been taken _as the

angular

_{ongin in} the _{m- mirror}

plane.

Then, the mean

occupation probabilities,

_n,

=

1/3 ₊

ôn,

(1= 1, 2,

3),

obtained under the action of the

pseudo-spin

coordinates

~

are

given by

_:

~ 2

ni "

j+11+~

~2

,>'3

, 6

n~ = + ~~ ô ~~ ô~

~- _{ôi lsb)}

~

,"3 ,/6 ,,/2

ni = + ~~

ôj

^~ ô~ ₊

~-

_ô~

3

,l'3

,» 6

,l'2

Obviously

from the condition

~n~

⁼¹ one should have

ôj

₌ 0

(identity).

In contrast,

ô~

^and

ô~

are symmetry

breaking

coordinates

(E'),

so that order parameters

expressed

ⁱⁿ terms of occupation

probabilities

are obtained from

(15)

~2~)(2~i~~2~~3)"

^~C°S"

E'-

'~

(16)

~3 ~

) _{(~3 ~2)}

~ ~ _{SITÎ "}

,2

ô~ ^and

ôi

are partner coordinates

(physically degenerate)

and, in the low temperature

phase lé

#

0),

the

quantity

_« determines the direction taken

by

the O.P, _in the ni= mirror

plane

with respect to the

origin

chosen

(orientation1).

Whatever the value of _a, the _«

freezing

_» of (ô~,

ôi) always

induces the

point

_group

C,(m=),

but it should be noticed that different

situations arise for the _«

ground

state _»

expected

^when T

- 0

[25 ], depending

_on the value of _a

i-e- on the relative

amplitudes

of ô~ ^and

ô~.

^{We shall} come back _to this point in section 3.2.

In what follows, we shall descnbe the

ordenng

processes in a unit cell

P6j(P65)

containing

six TMA groups situated _{on sites} denoted

by

the index _À.

là

=1 to

6); namely,

= I.u

corresponds

to TMA at ₌

=

~

~j

On the other hand, _{smce we} limit the discussion

to a low temperature ^{unit-cell with} a lattice _con~tant

tnpled along

_c, it i~ convenient to denote the

pseudo-spin dephasings corresponding

to points ^A

(0,

^{0, and}

A'(0, ^{0, by}

^an

3 3

index _p

(p

= to

3)

which

specifies

for each _site k the directions that _can be

adopted by

the

degenerate

coordinates ô~ and

ôi

Thus, the site coordinates will be labelled _m

ô~

(1,

_{p so} that

il

= to 6)

~~~~'~~

~~

~~

_~~

~~~~

~~

Il 7j

~~~~'~~~

' 6

j~j

~~ ~~~

~ ~~

The primitive

P61/m

unit-cell _contains two disordered TMA'S, related to each other

by

inveoion symmetry. Thus, twice as much

pseudo-spin

coordinates _are

expected

at zone

centre, te-

R(rJ=A~eE~geBueEju

('8)

A~

^and

B~

are issued from

ôj(A')

and then, are not of

physical

interest. Now, the

Ej~

^and

Ej~ (symmetry breaking coordinates)

are

given by

6

&~~~(l)

⁼

~ ô~(k,1)

j ⁽

^'

9)

&~~

(2)

=

~ ôi(k,

1)

~~

~

i

ii

~

(

i- i _)À^{+ '}

ô~ik,

~~

~

'

120)

&~~~12) ⁼

z i-

1)~ ^{+ '}

ôi(k,

i =1

None of them transform

according

to the

A~ representation, corresponding

to the

secondary

order parameter p associated with the

P61/m

_-

P6j (P6~)

^transition

(see

^Sect.

2.1).

However,

&~~

Il

and &~~

(2

are involved in _cases of ferroelastic transitions

leading

to the

P21/m

or

P2j

sol~tions

_(see

àect. _3.2).

Each _one of the

pseudo

_spin coordinates _at point ^r generates a branch of

pseudo-spins.

Making

_use of table

A2(a)

(see

appendix),

it

merely

comes from

Il 8)

at

point

A

R(A)

=

Ai

e A~ e

lA~, Ai)

e

1~5, A~). j21)

Again, Ai

and A4 ^{issued from}

A~

^and B~

respectively,

are not of

physical

_interest. At points

A(0,

^{0, and}

A'(0, ^0,

^{there are now four}

pseudo-spin

variables of the _same

3 3

symmetry as the

primary

O.P. _7~, Ii ₌ to

4).

These variables _can be

expressed

in terms of the site

pseudo-spin coordinates,

and also

they

can be

directly

^{associated with} the O.P.

components _7~~ since all quantities are

developed

on the _same basis. Thus one obtains (see relations

(2))

_:

1,

7~j-&j(A)= ~ ô~(k,h)=qcosq~

i=1 6

~1? -

M?IA)

=

~ ô~(k,

.i) ₌

cl'cos

_w

'j (221

~1_1-

&~(A)

=

~ ô~(k,

s)

=

q'sin

_w

i-1 6

7~i -

&1(A)

=

~ ôi(k, h)

= q sin q~

i=

with h

=

(2 1)

mod. 3 and _.i

=

4 k mod. 3

3.12 Tfie 6 W Jiiode/. The 6 W model consi;ts in TMA groups

reorienting

between _six

position~,

labelled from _to 6'

(Fig.

lb), with

equal

_occupation

probabilities

RI ₌ 6 ii

= to

6)

in the disordered P61/m

phase.

Following

the _same

procedure

as for the 3 W model, _one obtains for the site

pseudo-spin

coordinate~

R(~itel=A'OEA"OEE'CEE" (23)

denoted _as à

IA'), à((A"), ôj, à((E'), à], à((E").

These coordinates

ôj

can be

expressed

ⁱⁿ

terms of the occupation

probabilities n,' by

_means of the transformation matrix _:

j ,j _,~ ,j ,) ,j

à1à fi _à fi

~~~~

l -1 -1 -1 -1

M'

=

,/à ,é 2,à

₂

,é

₂

À

₂

_,/3

~ ~ l

1

_{2 2 2}

À _,é

₂

_,é 2,é

₂

,é

₂

,à

~ ~ l

1

_{2 2 2}

Again à( (identity)

is not of

physical

interest _so that, at the zone-centre, the symmetry-

breaking

variables give

R(r)=BgeEigeE2geAuf9EiueE2u. ⁽²⁵⁾

One of these _now transforms

according

to the

A~ representation (secondary

order parameter

p)

_; it is of the form _:

6 6

~~

~~~~ Î~i

_~~~~~ ~~~~ ~~~~~

'i,~j

^~ ~~~ ~~~~

At point ^A

(0,

^{0, it comes}

R(A)

=

Ai

_OE A~OE

2(A~, A~)

e

2(d5, dô). (27)

There _{are now} two

pseudo-spin

variables attached with each

component

_7~, of the pnmary O.P.

(A~, A~), namely

:

6 6

'Il ~

81(A)= ~j ôj(k, ^{h)( 8i(A)=} ~j ^{ôi(k, h)}

I=i I=1

~ ~

~~ -

&j(A)

~

j~ ôj(k, ^{s) &j(A)}

=

j~ ôj(k, ^{s) (28)}

1=j ij

6 6

'Î3 ~

8i(A)

=

~j ^{ÙÎ(k, S) Mi (A)}

=

~j Ù~(k, S)

i i =1

6 6

'Î4 ~

Mi (A)

₌

~j ÙÎ(k, ^h) 81(À)

=

~j Ùi(k, ^h)

i i

with h

=

(2

k

1)

mod. 3 and _s

= 4 k mod. 3 and

ô](k,

^p )

=

,Î,~j

_°~

ô((k,

^p

3.2 THE ORDERING PRocEssEs.- As stressed

already

_in

[25],

_a

pseudo-spin description

subtends _a

microscopic

model of

interacting particles (TMA), probably

of

complex

nature in

the present case but

which,

in

pnnciple,

makes it

possible

to compute ^{the free} _energy ^{of the} system.

Indeed,

this free energy is _a function of all

independent pseudo-spin

variables and, _in

such _cases of multidimensional

pseudo-spins (3

W _or 6 W

models)

different scenarios _can

happen

as for the _{more or} less

complex

_serres of

phase

transitions that _can take

place, depending

on the order

according

to which the different

pseudo-spin

variables get ^{frozen with}

decreasing

temperature, on their relative

amplitudes

(in case of partner

coordinates)

^and on

their

respective

wave-vectors

[25].

We do _not intend to discuss here all

possible

scenarios issued from the 3 W _or 6 W models, but _we

just

consider the

particular

_case of the

pseudo-spin

variables with

(A~, Ai

symmetry (see Tab. A2c _in the

appendix)

of interest _in TMCB. In the framework of the 3 W model, when

&j(A)(7~j)

and

&~(A)(7~~)

are the first variables to get ^frozen at

T~~

(1.e. q#0, q'

₌

0,

see relations

(22)),

a low temperature

phase

with

P6j (Z

₌ 6) space-group is induced,

and this _{is true} whatever the value of _q~. In other

words,

the

degenerate eigenvectors

&j

(A

and &~

IA

_are undefined to within _a rotation in the (7~

j, 7~~)

plane.

However, _as

long

_as

a Frenkel model

involving

discrete

(and fixed)

orientations is considered [17], and that _a

perfect

ordered state is

expected

to take

place

at T

=

0, it _can be

easily

shown from relations

(15b),

Il

7)

and

(22)

that conditions _{on q~} must be enforced. Hence, two scenarios _are

possible

(1) if _q~

=

2 mw/3

(m

integer ^mod.

3),

the

P6j ground

state is

ordered,

i-e- for all sites k

(k

= to

6)

_one orientation Ii

= 1, 2, 3

depending

on k and m) is

occupied

with

probability

n)

=1, whereas the other two are forbidden. In this _{case no} further transition below

T~~ is necessary ;

(ii)

if _q~ #

2mw/3,

the symmetry ^{of the} system is still

P61,

but _{now more} than _one

n)

_are ^different from _zero in the

ground

state. So, another

phase

transition occurring ^at

T~~~T~~

is necessary,

governed by

the

pseudo-spin

variables that did _not

« freeze _» at T~~. Hence,

&~(A)

and

éJi(A) (~ee

relations

(22)) together

with

&~~~ Il ) and &~~

(2)

(see relations

(19)),

all of which

belonging

to the zone-centre

E2 representâiion

_in

spacel~roup _P6j

(see ^Tabs. A2b, 2c in the

appendix),

will _{act as} O.P.'s and will induce at

T~~ ^a

phase

with

P21 (Z

=

6

)

_{space group.} It _{can now,} be shown that the

P21 ground-state

can be ordered

only

if _one had at first

(1.e.

at T~~) ^{q~ =} (2 f ₊ 1) w/6

(1 integer

mod.

6). Otherwise,

there _{is no means} find

any combination of the E~ variables able to order the system in the

P2j

space-group.

Moreover,

there _{is no} further

possibility

left _since all available

independent pseudo-spin

coordinates _are frozen.

Apparently,

the

expenmental

data

[13-15, 17]

(see also Sect.

5)

indicate that the observed

transition in TMCB

corresponds

to scenano

Ii)

The latter _can be

easily

extended _to the 6 W

mortel. Indeed, the transition from

P6~/m

to

P6j (ordered) always implies

_q~

= 2 mw/3 but

now, in addition, the two

pairs

of coordinates

((&((A), &((A))

and

(&((A), &((A)) (see

relations

(28))

must freeze with

equal amplitude, respectively,

and _one should also have

&'(Au

_# 0 (see relations

(26))

this is

possible through

_a

coupling

of the form

7~

~ p which is allowed

by

symmetry

[15].

Let _{us come} back to the Landau free energy expansion AW

(3), (4)

(see Sect.

2.1).

Obviously,

this

phenomenological potential

cannot

pretend

to account for the 3 W _or 6 W models in their

entirety

indeed, it is limited _{to a}

description

of the observed

P61/m

-

P6j

transition,

_in which _we have considered that

phase P61

is disconnected from any other _ones that can «

potentially

_» exist in the

general

context of the 3 W _or 6 W models. Hence, _as

long

as scenario

Ii)

is

valid,

the condition

q~ ⁼ 2 mw/3 las determined from symmetry ^consider-

ations) implies

that b'

=

0 (see relations

(5a)).

4.

Experimental

details.

TMCB

single crystals

have been

prepared

from saturated aqueous

solutions,

as described

previously [3,

4, 17,

23].

Colourless

single crystals

in form of pnsms

elongated along

the _c axis, with dimensions of about 2 _x 2 _x 5

mm~,

have been studied

by

_means of Raman

scattering. Crystal powder

was used for calonmetric measurements.

The Raman spectra ^{have been recorded} on a DILOR Z-24

triple-monochromator instrument, equipped

for detection with _a cooled Hamamatsu

photomultiplier coupled

with _a

photon

counting system. ^{The 514.5} nm emission line of _a SPECTRA-PHYSICS argon ion laser mortel 171 has been used for excitation with incident power (at

sample)

not

exceeding

100

mW,

in

order _to avoid

crystal damage.

The

spectral

slit width _was about 2 cm-' A DILOR C4N

nitrogen continuous flow cryostat was used _to

keep

the

sample

at different temperatures

between 80 and 300 K.

Spectra

have also been recorded down _to 20 K with _a CRYODINE

model 20 helium

refrigerator, equipped

with the

sample

^holder modification described in

[26].

In both _cases, the temperature

regulation

_was better than ±0.5K. Different scattenng

geometnes

(right angle

and

backscattenng)

have been

adopted

in order _to observe all Raman

tensor elements and to make it

possible

to discnminate the TO, LO and

quasi (oblique) phonon

components ^of

polar

modes _in the ferroelectric

phase.

The

laboratory

_axes,

X,

Y and Z have been chosen such _as Z is collinear with the _c hexad _axis, X with the _a

(or b) crystallographic

axis and Y is

perpendicular

to the

ac(bc) plane.

Specific

heat measurements have been

performed

on an automatic adiabatic calonmeter

described

previously [27, 28].

80.456 g of TMCB

powder

have been put ^into a copper

calonmetnc vessel filled with helium gas at

-10~

_Pa

in order _to

provide

a

good

thermal

conductivity

and _{to ensure a} fast achievement of thermal

equilibrium. Up

to 135 points ^have been measured

by

_using the conventional

pulse technique

between 50 K and 300 K

and,

_in

addition, continuous

heating thermograms

have been used around the transition temperature

(T~-160K)

for _{a more} precise determination of the

specific

heat behaviour _in this temperature range. Previous calibration of the

expenmental

system ensures a final

C~

accuracy of 0.1 9b.

5. ResuJts and discussion.

5, RAM AN SCATTERING.

5.1 Lattice vibi~ations _m trie

paiaelectiic phase.

Let _us

first,

recall the

low-frequency

Raman spectra ^{of TMMC and TMCC in the}

hexagonal

disordered

phase (P61/m)

^have

already

been

interpreted

into details

[3,

5,

23].

This facihtates the

assignment

of the

corresponding

spectra ^{of TMCB.}

Therefore the

A~(a==)

spectrum

(Fig. 2a)

is dominated

by

_a strong ^line at

ls6cm-~,

assigned

to the

totally symmetric «breathing»

vibration

(stretching-like)

of the

CdBrô

octadedron chains _; the

corresponding

mode in TMMC and TMCC lies around 250 cm-~

[23], essentially

due _to the _mass influence of chlorine _atoms with respect to the bromine _ones. At low

frequency

and _room temperature, a featureless _«

Rayleigh wing

_»

extending

from 0 _to

1 100 _{cm- is} observed

(Fig. 2a)

_; a similar

signal

_was also present ^{in TMMC and TMCC and}

it has been

assigned

to «disorder-induced

scattering» [5], supenmposed

_on allowed k

=

0A~ modes, which,

in the chlonne denvatives, _are situated around 80

cm-', (chain rotation)

and 50 _cm-

(TMA libration) [23].

In

TMCB,

at lower temperature, ^{this broad}

signal splits

into _two parts ^{centred around 40 and}

80cm-~, respectively;

this is

particularly

evidenced

just

above T~ - 160 K

(Fig. 2a).

From _a

simple

calculation

taking

^into account the

moment of inertia of octahedron chains in chlorine and bromine derivatives, the chain rotatory

mode in TMCB is

expected

to occur around 50

cm-~,

i-e- _{in a} close

vicinity

of the TMA

libration.

So,

both modes may contribute _to the broad

signal

at

-40cm~',

as for the

remaining

broad band _at

180cm~~,

_it

is

probably

due _{to a} disorder-induced

frequency

distribution of

optical phonon modes,

of which _a precise

assignment

_can

hardly

be made.

The

E~~(a,,.) spectrum (Fig. 2b)

exhibits three broad

peaks

at

56,

- 90

(shoulder)

and

105cm-~, assigned

to the

expected

TMA

translatory

mode

(56 cm-1)

and chain intemal

bending

modes

(90

and 105

cm-~)

_; the

corresponding frequencies

_in TMMC and TMCC _are 180, 1120 and -170 cm-~

respectively [23].

Note the presence of _a

polarization leakage

coming from the very strong

A~

^hne at 156 cm-~

(Figs.

2a,

2b).

ii14i iiiiii Illilf

~

~ ~

~ ~

j .©

, c c

~ IX ^~

Î j

b _k

j .©

~ ù

#

j

"

w

~ ~

Z .t

Î

c c

fl i 1 Ill ii Jt 1i lfl

lR~%@i ltR

iq~çj

jçR

j ix~~<q

ii

_'i

a) b) c)

Fig. ^2.-Roman spectra ^{of TMCB} surgie crystal

through

the ferroelectric phase transition

(T~1

160 K) _: a) X(ZZ) Y (A~ - A) spectrum above T~, the ordinate scale has been

multiplied

by a

factors b)X(YX) Y

(E~-E~)

spectrum above T~, ^{the ordinate scale has been} multiplied by _a

factor 3 c) X(ZX Y (E~~

Î

_{E ) spectrum}

; above T~, ^{the ordinate scale has been} multiplied by _a factor 3.

The

Ei~(a,=, «,.=)

spectra

(Fig. 2c)

are almost featureless, _were it _not for the presence of weak

pol~rization leakages coming

from the modes _at 156 cm-Î

(A~)

^{and 105 cm-'}

(E2g).

^A

similar situation has been encountered in TMMC and TMCC

[23].

However, in TMCB, one

notices the presence of _a weak

plateau-like signal

at low

frequency, ending

in _a broad band _at 150 cm~ ~.

Certainly,

disorder induced

scattering

contributes to these spectra ^and

possibly,

the band _at

- 50 _cm could be due _to the

expected

TMA libration with

Ej~

symmetry

[4, 23].

5.1.2 Lattice vibrations _in the

ferroelectiic phase.

-Below

T~~160K, abrupt changes

occur in the Raman spectra ^{of TMCB}

(Figs.

2a,

b, c)

which _{are consistent} with the _occurrence

of _a first order

phase

transition.

Generally speaking,

the broad features charactenstic of the

paraelectric phase give

_use to narrow litres below T~ and numerous additional weak litres also take

place.

The latter

originate

from Raman inactive zone-centre modes of the

paraelectnc phase (see appendix,

Tab.

A2b),

and/or from

phonon

^modes at

points

A and A'

(Tab. A2c).

Anyway,

the number of observed lines _is far less than

expected (see

relation

(4)j.

It should also be noticed that _none of observed lattice vibration exhibits

significant softening,

thus

suggesting

that the

phase

transition is

essentially

of order-disorder type,

governed by pseudo- spin

coordinates (see ^Sect.

3) acting

as order parameters.

A detailed

interpretation

of the Raman spectra ^of ferroelectric TMCB is not

possible,

^due to the

complex

structure of this

phase. However,

_we would like _to stress the

following points

1) In the _case of the «,, spectra

(Fig. 2b)

where three E~~

(para)

-

E~(ferro)

modes can be

easily

followed in both

phases,

no

splitting

of these lines is observed, this is in agreement with the conservation of the

hexagonal

_system

through

the

phase

transition

il 7].

This _{is in} contrast

with the ferroelastic

(hexagonal

-

monoclinic)

transitions of TMMC and TMCC where a

splitting

of these modes has been observed

[5, 23],

_as

expected.

ii j After _numerous

experiments including

different

nght angle

^and

backscattering

_geomet-

ries,

_we conclude

that,

within the limits of

experimental

_{accuracy, no} TO-LO

sphtting

can be

shown for the

polar

modes with

Al:

and

Ej(.;, y)

symmetry. ^This means that the Raman

spectra ^reveal

only polar

modes with very weak oscillator

strength,

_i e. modes which _are

expected

to exhibit weak infrared

intensity.

In other words, _as far _as Raman scattenng is

concerned, the

departure

^from centrosymmetry ^is net so well marked. This _can

probably

be related _to the fact that TMCB is _a weak

(improper)

ferroelectric

il 3].

iii)

Because of Brillouin _zone

foldings

at

points

A 0, 0, and

A'(0,

^{0, (or}

3 3

equivalently

A' 0, 0, ^~ the _transverse acoustic

(TA)

and

longitudinal

acoustic

(LA)

modes 3

at those

particular points

in the

paraelectnc phase give

use to zone-centre

optical

modes _in the ferroelectnc

phase.

Therefore,

according

to table A2c

(see appendix), TA(A)

with

(A~, A~)

symmetry ⁱⁿ

P6~/m

generates two modes with A symmetry ^and a

degenerate

mode with E~

symmetry in

P6j(P65) LA(d),

with

Ai

symmetry in

P61/m yields

_a

degenerate

mode with

Ej

symmetry. ^The Al

«-=)

_spectrum at 20 K

(Fig.

2a) is charactenzed

by

a doublet observed _at very low

frequency

20-23 _cm- '. _The

dispersion

_curves measured with TMMC

[29]

show that

the _{transverse acoustic}

(TA)

mode at

points A(0, ^0,

₃ ^{lies around 26}

^cm-',

^a

^simple

calculation

taking

into account the mass difference between TMMC and TMCB shows that the

corresponding

mode _in TMCB is

expected

at 120 cm-' Then _we assign the 20-23 cm-~

doublet _to

TA(A).

Let _us recall that this mode is

responsible

for the helical

shape

of the

CdBrô

octahedron chains

[17] through

bilinear

coupling

with

pseudo-~pin

coordinates of the _same symmetry (see Sects. 2.2 and 3). As for the other modes issued from

TA(A)

and

LA(A),

with E~ and

Ej

symmetry,

respectively, they

cannot be

clearly

identified _on the

corresponding

spectra

(Figs.

2b, 2c),

probably

because

they

exhibit _very low intensities.