On phase transitions in NH4HSeO4 and ND4DSeO4

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On phase transitions in NH4HSeO4 and ND4DSeO4

V. Dvořák, M. Quilichini, N. Le Calvé, B. Pasquier, G. Heger, P. Schweiss

To cite this version:

V. Dvořák, M. Quilichini, N. Le Calvé, B. Pasquier, G. Heger, et al.. On phase transitions in NH4HSeO4 and ND4DSeO4. Journal de Physique I, EDP Sciences, 1991, 1 (10), pp.1481-1498.

�10.1051/jp1:1991221�. �jpa-00246430�

Classification

Physics

Abstracts

63.20 61.12

On phase bansitions in NH4HSe04 and ND4DSe04

V. Dvofhk (~,

*),

M.

Quilichini ('),

N. Le Calvb (~), ^B.

Pasquier

_~2), G.

Heger (')

and P. Schweiss

(', *)

(')

Laboratoire Ldon Brillouin

(CEA-CNRS), CEN-Saday,

91191 Gif-sur-Yvette Cedex, France

f)

Laboratoire de

Spectrochimie

IR et Raman, CNRS, Thiais, France

(Received16 April

1991,

accepted

in

final feral

2

July 1991)

Rksumk. Nous proposons une

phase

prototype

hypothbtique

(de _groupe d'espace Immm) fi

partir

de

laquelle

_{on peut} dbduire toutes les

phases

observbes dans NH4HSe04 et

ND4DSe04

par l'introduction de

paramdtres

d'ordre ayant une

symbtrie

dbfinie.

D'aprds

cette

hypothdse

le groupe

d'espace

de la

phase superionique

doit dtre

P2/n

ce

qui

est en dbsaccord _avec des rbsultats

expbrimentaux

rbcents. Pour

chaque

transition de

phase

_on befit

l'bnergie

libre de type ^{Landau I}

partir

de

laquelle

elle peut ^{dtre dbcrite. La} grande anomalie

diblectrique

_au

voisinage

de 252 K dans le

composb

NH4HSe04 est discutbe de faqon dbtaillbe. Dans le

composb

Nd4DSe04, _nous

avons btudib par diffusion

blastique

de neutrons la transition de

phase

du

premier

ordre

qui

transforme le cristal de la _structure

P2j2j21

dans la

phase

commensurable

d'accrochage (de

vecteur d'onde k~

(0,

^{0, I}

^).

^{Nous avons montrb que la}

^symbtrie

^{de cette}

^phase

^est

^PI12i,

^ce

c

qui

est en accord _{avec nos}

prbvisions thboriques.

Abstract. We propose a

hypothetical

prototype

phase

(space _group Immm) ^from which all observed

phases

in

NH4HSe04

and

ND4DSe04

_can be deduced

by introducing

order parameters of definite

symmetries. Following

this

hypothesis

the symmetry of the

superionic phase

should be

P2/n

in

disagreement

with recent

experimental

results. Free

energies

of Landau type are derived

by

_means of which

particular phase

transitions could be described. The

large

dielectric

anomaly

near 252K in

ND4HSe04

is discussed in _some detail. The first order

phase

transition in

NIJ4DSe04

from the _room temperature

phase

P2j2j2j into the commensurate lock-in phase

(with

the wave-vector k~

(0,

^{0, " has been}

investigated by

neutron elastic

scattering

and the

c

symmetry of the latter has been found to be

Pi12j

in agreement ^with our theoretical

prediction.

1. Introduction.

Ammonium

hydrogen

selenate

NH4HSe04 (AHSe)

_as well _as the deuterated

compound

ND~DSeO~ (ADSe)

have been

subjects

of many

experimental investigations

because of their

(*)

Pernlanent address.- Institute of

Physics

of the Czechoslovak

Academy

of Sciences,

Prague,

Czechoslovakia.

(**)

Permanent address _:

Kemforschungszentrum

Karlsruhe, J-N-F-T. 7500 Karlsruhe and

University

of

Marburg,

Institut [fir

Mineralogie,

3550

Marburg, Germany.

very

interesting properties (see [I]

for _a

review). They

_possess various

phases,

I-e-

superionic, ferroelectric,

incommensurate

(IC). Moreover,

_some

phases

_are

metastable,

_{some may} coexist. The kinetics of

phase

transformations is rather

complicated

and not yet

completely

understood. Phase

diagrams

of AHSe and ADSe _are discussed in detail in

[2, 3]. Although

a

rich

experimental

material is available now,

surprisingly

no

phenomenological theory,

which should

precede

_any

nficroscopic theory, describing

the

phase

_sequence in these selenates has been

systematically developed

_so far. The aim of tills _paper is to make first

steps

in this

direction.

Several

predictions

_come out from the

phenomenological theory,

for

example

what the space-group symmetry of _a

particular

should be. In order to prove or

disprove

_some of these

predictions

_we have

performed

elastic _neutron

scattering investigations

of ADSe. At the _same time _we have studied in _more detail the

phase diagram

from the

room-temperature phase

to an IC

phase

in this material.

2. The commensurate

phases.

The basic

problem

of

Landau-type theory

is to find out whether _a prototype

phase

exists from

which all observed

phases

could be deduced

by

means of order parameters ^{of definite}

symmetries.

For this _purpose let _us first _enumerate the known

symmetries

of commensurate

phases.

2.I MONOCLINIC PHASE 82

[4].

We denote the vectors of the conventional

non-prinfitive

unit cell _{as a}

=

19.7

A,

_b

=

4.6

A,

c ₌ 7.6

A

_and

y =

102~5'.

For the search of _a prototype

phase

the

following

two facts _are very

important. First,

the monoclinic

phase

is in fact of

pseudo-orthorhombic

non-standard space-group symmetry ¹² with the vectors a'

= a + b

(=

19.3

A _{), b,}

c of the conventional

non,prinfitive

unit cell and

y'

=

89°54'

(cf. Fig. I). Second,

there is _a

quasiperiod

a' in the direction of a' _seen in

X-ray

3

diffraction

pattern [4]. Indeed,

it is _{easy to} show that

by

small

displacements

of atoms from their known

positions

in the 12

phase

_{we recover} the translation

period

a'm §. This is _not

3

surprising

because § is the shortest distance between chains of

Se04

tetrahedra

(linked by hydrogen bonds) parallel

to b in the _same

plane perpendicular

to c

(cf. Fig. I).

'/~

Fig.

1. Structure of the

phase

82 of AHSe

projected

on the

(a, b) plane

after [41. ^{A half-cell is shown.}

Wavy lines denote

hydrogen

bonds. Protons (not

shown)

in the

hydrogen

bonds _are disordered.

j.2

ORTHORHOMBIC _PHASE

P2j2j2j.

The vectors of the

primitive

unit cell _are

)~" ^~'~'

2.3 TRICLINIC PHASE Pl

[7]

which

essentially

results from _a small distortion of the 82 cell _or of the

pseudo-orthorhombic12

cell but has _a

tripled period

3c

[2].

2.4 SUPERIONIC MONOCLINIC PHASE

presumably

of

P2j/b symmetry [8]

^with _{a~ =} 7.8

A,

b~ =

7.7

A,

_c~

= c, y =

112.51 As far _as lattice

parameters

are concerned this

phase

is close to a

hexagonal

one.

3. The

prototype phase.

Obviously

_none of the

phases

described in section 2

provides

_a

prototype phase

for all others.

It is evident that the basic translational

periods along

the orthorhombic axes

(which

_we

identify

with _our coordinate _{axes x, y,}

z)

of

P2j2j2j phase

are

§, b,

_c. Since the

phase12

is

body centered,

the

prototype

^{unit cell} cannot be

primitive

but should be

body

centered too.

We show _now that the lattices of all observed

phases

_are

specific

linear combinations of the

following

unit-cell vector e; of the prototype orthorhombic

body

centered unit cell

We denote the unit cell volume _as V.

The unit-cell vectors p, of the

pseudo-orthorhombic

12

phase

_are

p~=~(a'-b+c) _=2e~+e~,

2

~3~j(~'+~~C) ~~2+~~3' ~'p~~~" (1)

The unit-cell vectors o, of the orthorhombic

P2j2j2j phase

_{are :}

oj ₌

~ a'

=

2

(e~

+

e~)

3 ,

o~=b=ej+e~,

o~=c=ej+e~,

Vo=4V. (2)

We

prefer

to describe the triclinic

phase

Pl _{as a} small distortion of the 12

phase,

I-e- _we shall

use a « non-standard _{space group} Il _» with _a

tripled period along

the _c axis. We choose the unit-cell vectors

t;

as follows _:

~'~~~ ~'~~~~~~~~~'

_~~'

~2"((~'~b+~C) "~l+~~2+~3,

t~=~(a'+b-3c) _{=-ej+2e~, V~=3Vp=9V.}

(3)

In order to get ^the apparent

hexagonal

lattice of the

superionic phase

_we have to choose the unit-cell vectors si as follows

(cf. Fig. 2)

_:

sj =

§+b=ej+e~+2e~

s~=-§+b=ej-e~

s~=c=ej+e~,

V~=4V. (4)

Then

[sj

₌

[s~[

=

7.9

A

_and

y =

108.6° which is close to the

experimental

values

[8].

b

S] 52

Fig. 2.

(T,

b ) of the ^rototype conventional unit-cell.

Now what is the space group of the

prototype phase

? It must be _a supergroup of the _space groups listed in section

2,

I.e. _one of the _{space groups} in the

crystal

class _mmm

(D~ J.

For

deciding

which _one it is

important

to realise that in the 12

phase

Z

=

3

[4]

and hence the

primitive

unit cell of the prototype

phase

should contain

just

_one molecule. To meet these conditions _we have to

displace slightly

all atoms to _more

symmetrical positions

and the

prototype

space group ^must be such that its

symmetry

elements do not

produce

_any new

atomic

positions. Using

^the

experimental

data of

[4]

it is _easy to find that these

symmetrical positions

of Se and N atoms

(which

_are the centers of

Se04

tetrahedra and

NH~ cations) expressed

in

(§,

b,

c)

are :

Se(0,0,0);

N

~, ~,0)

Inspecting

Intemational Tables for

X-ray Crystallography

_we determine

unambiguously

the

prototype

space group as Immm.

The unit-cell vectors of the

phase

discussed in sections 2 and 3 _are summarhed in table1.

4. The

superioldc phase.

Since _a

SeO~

tetrahedron does _{not possess} the centre of inversion I it is obvious that both Immm and

P2j16

_may describe _an

averaged symmetry.

^It has been found [9] that the

high conductivity

in the

highest temperature superionic phase

is due to diffusional motion of the

protons ^{and ammonium} groups. The destruction of

hydrogen

bonds

finking

the

SeO~

tetrahedra makes their

rapid isotropic

reorientational motion

possible. Consequently,

the

superionic phase effectively

exhibits _a centre of inversion.

What is the

symmetry

^{of the order}

parameter describing

the

hypothetical phase

transition from the prototype to the

superionic phase

? From

(4)

_we conclude that it should transform

according

to _an irreducible

representation (irrepre)

of Immm with the star of wavevectors

l~~'~~~~~ ~j'~'~j'~~~~~~~~~~~~* ~'~ l'~j~

where e;* are the unit-cell vectors of the

reciprocal prototype

lattice. The

irrepres

at tills

point

of the

prototype

» BZ _are

presented

in table II. We _{can now} determine in _an usual _way what

.t

£ §

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JOURNAL DE PHYSIQUE I T I, M 10, OCTOBRE 1991

Table II. Four

irrepres «$,

p at the

point _(k~j

_;

_k~~) of

the BZ

of

the space group Immm. E denotes the unit matrix, A

=

° '

l 0

).

e 2~ 2~ 2= ¹ m~ m~ m~

«$

E A A E ±E ±A ±A ±E

«)

E A -A -E ±E ±A ±A ±E

will be _{space groups} when different _{waves we} denote their real

amplitudes Pi,

P~

Of

«$,

~

symmetries

with the wavevectors

k~j,

_k~~ are frozen in the prototype lattice. We remind that both _waves must condense

simultaneously

in order to get ^the observed

translational symmetry

[8].

When

Pi

=

P~

the

irrepres «$, _«jp

lead to Cmmm and

«Q,

«j

to Cmrna which is not the _case of the

superionic phase.

When

Pi

#

P~

_we get monoclinic _{space groups :}

«$, _«j give P2/m

and

«j, «) give P2/n. Unfortunately

_{we can}

never obtain the space group

P2j16 because,

for

example,

there is _no translation lost

along

the

c-ads.

However,

it should be

pointed

out that _an

X-ray

diffraction

study

of the

superionic phase

Was made _on the

powder

of AHSe and determination of _a

particular

_{space group}

(P2j/b)

within the

crystal

class

2/m

has not been considered _as definitive.

Actually,

the

experimental

diffraction data _are

compatible equally

well with the extinction rules for both

P2j16

and

P2/n

_{space groups.} At

present

we consider the symmetry ^P2

In

to be the most

likely

and in

figure

³ we

present

^{the modes}

transforming according

to

wk.

^For

simplicity

the

displacements

of

SeO~

tetrahedra

only

_are shown but the _same

picture

could be drawn for

NH~

_groups. Two modes of

«j symmetry

^{which induce the} same

symmetry

as

«Q represent

rotations of

Se04

tetrahedra around d _or b axis.

~

5

~

~

~ _~~~

~~*~

~2

_{~~ '} ^'~

j

~ta O

Td~

_o

-~ o ~-+- o H%-~

o

K~~

_o

~f~

_o

~-4- o H+-~ o ~-+-

~

ill ill

,"~

~

ksj (I')

_'°

Fig.

3. The

displacements

of

Se04

tetrahedra in two modes of «Q symmetry

leading

to the structure

P2/n.

The sublattice of Se04 tetrahedra is

projected

onto the

plane perpendicular

to the c-axis. 0 and

1/2

are z-coordinates in term of the

period

_c. The full and dashed _arrows drawn

conventionally

in the

± T direction represent in fact the

displacement along

± c-axis. The arrows of

equal

_type

(full

_or dashed)

are of the _same length.

Finally

_we

point

out

that,

if indeed the

superionic phase

arises from _a

hypothetical prototype phase Immrn,

there should exist 2 ferroelastic and 4 translational domains.

For

completeness

_we

present

^{the free} energy

F~ describing

the

hypothetical phase

transition from the prototype

phase

Immrn to the

superionic phase.

The form of

F~

is the same for all

irrepres.

+(Yl(~'~~+~)~()+' +8i(~~+~i)Uii+8(~~~~~)Uxy> (5)

where u~~ denote strain

components.

S. The

pseudo-orthorhombic phase

12.

From

(I)

it follows that the translational

symmetry

of the

phase12

_can be achieved

by

^modes with the wavevectors

kjj~

=

.(- e(

+

et

+

et

=

~ _"

,

0,

0 _; k

ij~ It _can be shown

3 3 a

that _none of

irrepres

of Immm at this

point

leads to II12. On the other

hand,

if _we start from the space

group1222,

_a frozen mode of

«jj~,

_~

symmetry (cf.

Tab.

II)

induces Il12

provided

the

phase 4

of the mode

complex amplitude Q=Re~'

attains values

n"

_where

3

n is _an

integer.

We _{are now} in the situation when _{we can} derive the

superionic phase (which

contains _a centre of

inversion)

from the

prototype phase

Immrn but _we cannot achieve either the

phase

Il12 _or the

phase P212121 (see

Sect.

7).

This fact

strongly

_suggests that between

these

phases

and the prototype

phase

there should exist another

phase (1222)

in which the centre of

symmetry

has been lost. In section 6 _we discuss in detail the

physical

mechanism

leading

to the loss of the centre of inversion.

It is easy ^to find out that the

irrepre

_«ij~

~ leads _to 2

pyroelectric

domains of Il12

phase

_:

n=0, 2,4 correspond

to 3

translationil

_{domains of}

one

pyroelectric

domain and

n =

1, 3,

5 of the other. It _can be shown that for

4

=

(2

_{n +} I

)

^" _{we get} the symmetry ^{Ii 21}

6

and for

general

values of

4

the triclinic

phase

II. For the form of the free _energy

describing

these transitions _see

(7)

in section 8.

6. A _new orthorhombic

phase

^-1222 ?

Since _we cannot achieve the 12

phase directly

from the prototype ^Immm

phase

_we have to

assume that there should exist yet ^another

phase

between

them,

I-e- of1222 symmetry. ^When

lowering

^the

temperature protons

^which in the

superionic phase perform

diffusional

motion,

establish

hydrogen

bonds between

Se04

tetrahedra

preventing

thus their

isotropic

rotation.

Obviously,

the creation of bonds will

strongly

influence the

frequencies

of internal vibrations of

Se04

tetrahedra.

Indeed,

_an

abrupt change

of these

frequencies

has been

recently

found in

Raman and infrared spectra and the existence of _{a new}

phase

both in AHSe and ADSe has been

anticipated [10].

Its space-group

symmetry, however,

has _not been determined

experimentally

_so far.

At _a

given hydrogen-bond length

there are

4configurations

of

neighbouring Se04

tetrahedra related

by

symmetry

operations

of the

point

_{group mrnm}

(cf. Fig. 4).

From

figure

4

it _seems that b thermal

agitation configuration @

can be most

easily

transformed into

configuration).

It would cost _more energy ^to get

configuration @

or

@

_{because the}

tetrahedra should turn

by right angle

_or the

hydrogen

bond should be

drastically

shortened

ix;2y

_1;m~

mx;my

O Z @ I

~ '

,, f ' ,,

, , ',

f '

~ ~ ~ ~

b

_:

j

...,_ _{_;}

_~~j

..._

,, f ' ',

f '

f '

~ ~

Fig.

4. Four

configurations

of

neighbouring

_Se04 tetrahedra

projected

onto a

plane perpendicular

to

the c-axis-

Symmetry

elements which

produce

_a

particular configuration

from the

configuration @

are

given.

Dashed lines

join

lower vertices of tetrahedra. Dotted fines represent ^the

hydrogen

bonds.

during

the transformation _process. Therefore

by

formation of

hydrogen

bonds _a disordered

phase

is created with two

possible configurations ~

_and

Q

of

Se04

tetrahedra

(or @

_and

@

_which

_represent

_another orientational

domain)

of

equal probabilities

and

consequently

the centre of inversion is lost. As the order

parameter

p

(with

k

=

0) describing

the transition Immrn -1222 _{we can} choose the difference between the

probabilities

of

occupation

of the states

~

or

@

_{and of the states}

3

or

@. _Obviously,

p transforms

according

to the

irrepre A~

of the

point

_{group mrnm.}

Now the

phase

transition 1222

- Il12 _can be treated _as the order-disorder transition with

the order

parameter

^of _«jj~,p symmetry ^which describes the difference between the

populations

of

configurations ~

_and

~. _Formally

_{the order}

_parameter

can be

interpreted

as rotation around the c-axis of

Se04

tetrahedra from middle

positions

between

configurations

~

_and

~.

_The

_pattern

_{of the modes of}

«jj~ _~ symmetry

(in

the domain _n

=

0)

is drawn in

figure

5. It is

compatible

with the structure oil11 2

phase

determined

experimentally _[4]

_as it should

(compare Figs.

I and

5).

~ j

~-

-i

^'

-+ # -+ +----4

j j

b '» _-~ ~---»

~-+

# _--

, (,

In ~

~)

[

i 35' '

~- -4 _-+ # _-+ +---4

iii iii iii

Fig.

5. The pattern of the modes

inducing

the Ii12 structure in the _same scheme _as in

figure

3. The

full _arrows, all of equal

length,

_are

displacements

of Se04 tetrahedra in the b-direction- The dashed

arrows, short and

long

_ones being in the

ratio1/2,

in the _± d direction

respectively,

represent ^{in fact} rotation of

Se04

_group around the _± c-axis from their fictive middle

positions

in the

phase1222 (cf.

Sect.

6).

7. The orthorhombic

phase P212121.

The translational

change (2)

of the

body-centered

orthorhombic lattice is due to frozen modes

with wavevectors

kjj4

=

(- et

₊

et

₊

et )

=

"

,

0,

0 k

jj4

Again

_as with the

4 d

Table III. Two

irrepres

_«~,

~, p ^at the

point (k~

= p

(- et

₊

e?

₊

et

_;

k~

(0

^{~ p ~} ₂

^of

^{the BZ}

^of

^{the space group 1222.}

e 2~ 2~ 2~

«~

~

E E A A

«~ _p ^{E -E -A A}

phase

12 the orthorhombic

phase P2j2j21

cannot be achieved

directly

from the

prototype phase

Imrnm but

only

from the

phase

1222

by

_means of the

irrepre

_«ij4

p

(cf.

Tab.

III).

Let _us denote the

complex amplitude

of the mode which transforms

accoriing

_{to this}

_irrepre

as

q =

re~'

_For

₄

= n

"

and _n odd _we get ^the symmetry

P212j21

with 4 translational 4

domains. The domain _n

=

I is

depicted

in

figure

6. If _n is _even, the

resulting

_symmetry is

P2j22.

For

general

values of

4

we

get

^the

symmetry P2111. Among possible phases

^related to the

irrepre

_«jj4,~ the

phase P2j2j21 only

has been found _so far.

ill ill

l~

-

i-~ e-j ^I-~

- b _-~

~+-l

__

j j ~,((

.

0

0)

-

~-~

~-

l i-~

iii iii

Fig.

6. The modes

leading

to the symmetry

P212j2i

The full _arrows denote rotations of

Se04

_groups around the b-axis

(or displacements along

the

b-axis)

but the dashed _arrows represent ^rotations (or

displacements)

around the c-axis which is

perpendicular

to the

figure.

The _arrows of

equal

_type

(full

_or

dashed)

_are of the _same

length.

The free energy

Fo describing

these transition reads _:

Fo

_{~ "}

qq*

₊

fl (qq

^* )~ +

fl

i

(qq*

)~ + Y

(q~

+

q*

_{~) +}

+

Yi(q~

₊

q*~)

₊ +18

(q~- q*~) Px +181

_P

(q~- q*~)

_{uyz +}

81qq*

_u

«

(6)

where P~ denotes the

polarization component.

^{In order} to

get

^the

phase P2jll

_we have to include the

eight-order anisotropy

_energy. This form of

Fo

is in fact invariant of the prototype

phase

Immm in the

phase1222

_we

simply put

_p

= ± I.

Before

discussing

the modulated

phases

of AHSe and ADSe let _us

briefly

summarize _our main

points.

We _{assume a}

hypothetical prototype phase

Immm from which the monoclinic

phase P2/n develops.

It should be

pointed

out that _our

prediction

of the

superionic phase

symmetry differs from that

proposed

ⁱⁿ

[8] (P21/b). When,

at lower temperature

hydrogen

bonds between

Se04

tetrahedra _are fornled _a first order

phase

transition _to the

phase1222

characterized

by partial ordering

of

Se04

_groups takes

place. By

further

ordering

the symmetry can be reduced _to

orthorhombic,

monoclinic and tridinic

phases including

to those observed

experimentally,

I-e- 12 and

P2j2j2j.

The relevant wavevectors of the order parameters ^{of these}

phases

_are summarized in table1.

8. Modulated

phases

of AHSe and ADSe.

Various

phases

modulated

along

the c-axis have been observed in ammonium selenates

[2].

Modulation of structures is described

by

_a wavevector k= = k

(et

₊

et et)

=

k

(0, ^0,

~ "

c

and _can be IC _or locked in _a rational value of k

(~ ). ^Obviously,

translational

periods

of 2

the

prototype phase

Immm in the

plane perpendicular

_{to c are not} affected

by

this

modulation,

I-e-

they

_{are e~ + e~}

=

(a, 0,

0 and ej + e~ =

(0, b,

0

).

The

period along

the _c- axis is lost unless k

='~

_where

m,n are

integers.

Then the _new

period

is

).c

and

n

nc for _{n even} and

odd, respectively.

It _can be shown

that,

when _n is

odd,

the type ^{of the lattice} remains

body

centered I while for _{n even} it

changes

into _a

primitive _type

P. In

particular,

for k

= the unit cell vectors have been

already presented

in

(3).

Note that

they

_are

compatible

3

with the wavevectors

kjj~

=

(- et

+

e?

+

et)

and k~ =

(et

+

e? et).

On the other

3 3

hand for

kjj~

^{and k~} =

(et

+

e? et

the unit-cell vectors

f~

should be chosen _{as :} 4

fj

=

a'

=

3

(e~

+

e~)

,

f~=b=ej+e~,

f~=2c=2(ej+e~);

V

I=12

V.

The

group-theoretical analysis

of the modulated

phases

has

already

been made

[2].

Nevertheless _we

repeat

^{it here} now in _a

simpler

form since in _some

respects

we get ^different results. It has been shown

[2]

that the

pseudo-orthorhombic phase12 provides

_{a parent}

phase

for all modulated

phases.

Two

physically irrepres

of this group ^at the

point

_{k~ are}

given

in table IV. From this table _{wan can}

easily

find space-group

symmetries

for lock-in _commensur- ate

phases. Obviously,

the repre

d~

_j leads to Pl12

(for

_n

even)

_or Il12

(n odd) independently

Table IV. Two

physically irrepres

_d~,

of

12 with the reducible star

k~

=

k

(0, ^0,

^{~ "}

^k~

c

e 2, _c

~

jl 0j jl _0j

~'~ 0 0

~-14nk

o

~

14nkj

~

jl 0j j-1 0j

~~~ 0 0 -1

of the value of k. The repre

d~,~

^{leads to Pl}

(n even)

_or Il

(n odd)

_except when k

='~

=

~" _~

(a, fl

denote

integers).

In this _case the

period

is 2

flc

and the lost

n 4

fl

translation

flc

is

represented by

the matrix-E

(cf.

Tab.

IV).

This

partial

translation

produces

the _screw axis and thus the symmetry becomes

Pl121.

In

particular

this is the _case of

k

= and at this

point

_our result differs from that of

[2].

We shall _now discuss the lock-in 4

phases

which

actually

_occur in AHSe and ADSe.

8.I AHSe.-

According

to

[2]

^there is _an IC

phase

below

1j

₌ ^{262K which locks in} at

k

= at T~ m 252 K. Below T~ ^{down to 98 K}

ill]

the

phase

with

triple period along

the c-axis 3

is of Il symmetry

[7].

^From this it has been concluded

[2]

that the order parameter

~~(k)= p~e~~~ describing

these transitions should transform

according

to the _repre

d~,~ (cf.

Tab.

IV~.

^The

phase

_e~

being arbitrary

in the IC

phase,

in the lock-in

phase

is

determined

by anisotropy

_energy for k

=

The lowest order term is of the sixth

order,

I-e- 3

(~(+ ~?~)

_as in

A~BX4-type crystals.

The lowest order

coupling

terms of the order

parameter

^with

polarization P~

and strain tensor u~~ components are :

1(~/ ~?~) _~y

_Or

Uxz>

(~/

+

~?~) ~x Uyz>1(~~~ ~?~) _~z

_Uxy

(~)

and

non-symmetry-breaking

terms

~~ ~?.u~~(I

_{= x, y,}

z).

All these terms _are invariants

even in the 1222

phase.

In the12

phase,

which _we treat _{as a} small deformation of the 1222

phase,

several _new terms occur, for

example

(Q~

₊

Q*~)s (~/

₊

~?~) _~y

where

Q

is the order parameter

describing

the 1222 -12 transition

(cf.

Sect.

5)

and the index _s denotes its spontaneous ^{value. We} shall

neglect

such terms _as

higher

order effects. So far the spontaneous

P~ only

has been detected

[12]

and _as it follows from

(7)

P~

~ p

(

sin 3 _e~

(8)

Accordingly

_a Curie-Weiss

type anomaly

of the

permittivity

_e~~ has been observed _near T~

II 3].

Such

anomaly

could be due to _a contribution _to

e~~ ^{of the}

phason

mode

II 4].

A

large temperature anomaly

should also be

expected

in the elastic constant c~~.

The

hypothesis

that the transition to the lock-in

phase

k

= is driven

by

_an order

3

parameter

^of

d~

_~

symmetry

^has two weak

points. First,

since the

anisotropy

_energy is of the sixth

order,

I.e.

I

_small

one, the interval of the IC

phase

should be much

larger

than it

actually

is and _on the other hand the interval of the lock-in

phase

much smaller.

Secondly, according

to the

theory [15] e~~(T)

^{should exhibit} a

change

of

slope

at

7j.

No such effect has been detected _so far

[13].

We believe that _a small

jump

of

e~~

reported by

Gesi

[16]

is not related to

lj

since it _occurs 40 K above T~.

We could eliminate these difficulties

by assuming

that the repre d~,j is relevant for _our

problem.

In this _case the lock-in _energy is

large,

I-e-

(~(

+

~t~)

_and

_~j (k

=

is

coupled

3

to

P~ only,

I

(~( _~ t~). _P~.

Since the repre d~,j itself does not

give

the

phase

Il

[2]

_we shall

consider it _to be the last

phase

in the sequence 1222 -12 -Il

produced by

the order

parameter

Q (cf.

Sect.

5).

Such _a

scenario, however,

has another

difficulty

_{; we} have to

assume that the

temperature To

of the transition 12 _- Il and the

temperature T[

of the lock-in

transition into the

phase

k= _{are not too} apart so that the

coupling

between 3

Q

and

~i

^could

produce

_a first order transition at which

Q (corresponding

to the

phase Il)

and

~j

^condense

simultaneously.

For

discussing

the 12

- Il transition _we need the free energy F of the

phase

1222. It reads

F

=

aQQ*

₊

fl(QQ*)2

₊

yo(QQ*)3

₊

y'(Q6+ Q*6)

₊

+(x-iPi-j&(Q3-Q~3)Py (9)

where _x denotes the dielectric

susceptibility.

Since dielectric

properties along

the b axis

only

_are

anomalous,

_we omit

coupling

terms

(Q~

₊

Q~~). _{P~ u~~} I(Q~ Q~~) P~

_;

u~~. The

coupling

of u~~ with

Q

is the _{same as} the

coupling

of

P~

^with

Q

and therefore c~~ should exlfibit _an

anomaly

of the _same type as x After

eliminating P~

from

(9) by

_means

of the

equation

_:

P~

=

I _x8 (Q~ Q~~)

=

x3R~

_{sin 3}

_{4 (10)}

which detenrines the spontaneous

polarization

_we

_get

F

=

aR~+ flR~+

_yo

R~+ yR~cos

₆

_{4 (11)}

where _y

=

y'+ ^~~ 2~

It is _easy to show that the

phase

II12 characterized

by

the condition sin 6

4

= 0

(cf.

Sect.

5)

^{is stable}

provided

_y

~ 0. In order to

get

^the

phase

II with _an

arbitrary 4

_we have to add to

(I I)

the next

anisotropy

term

[17]

which is _y

j

R~~

_cos 12

4. Minimizing

6

then

(I I)

with

respect

to

4

_we find two solutions _: sin 6

4

=

0 and

sin~

₃

4

₌

(y

₊ 2 _y

j

R~)/(4

_yi

R~)

which

correspond

to the

phases

Il12 and

Il, respectively.

The

phase

Il12 is stable _as

long

as D

m y + 2 _y

i

R~

< 0. When D _~ 0

(

_y

j ^~

0)

II12 looses its

stability

and II becomes stable.

Obviously,

when

lowering

temperature ^the

amplitude R(T)

increases and

consequently

D may

change sign

at a temperature say To. ^{Therefore in the}

vicinity

of To we take

D(T~

in the form D

=

d(

_To

T), (d

~ 0

).

We _now obtain the Curie- Weiss behaviour of the

susceptibility

_xef

[13] (T~ To)

~~~

y

+~~

i

R~

~T~

_~

o)

~~~~

From

(10)

_{we get} the

temperature dependence

of

P~

p

~-sin3tb~ (To-T)~'~>

i-e-,

_a usual behaviour for _a proper ferroelectric.

Indeed,

_{near To} this

simple

law is satisfied

[12].

^In the model of _a lock-in transition the temperature

dependence

of

P~

is

given by (8).

The temperature

dependence

of

pj(T)

has _not been determined

reliably

so far but it _seems

that very near T~ ^it behaves like

(T~- T)

which then would

give

another law for

P~(T~ (T~ T)~'~

Let _us discuss the

cooperation

of the lock-in transition

(k

⁼ ₃ ^{and the}

^phase

^transition

12 - II in _some detail. For this purpose we need at least two lowest-order

anisotropy

terms associated both with the order parameter

~j

= p _j

e~~')

and

Q(=

R

e~'

_{and their} interaction energy. This part

Fi

of the free energy reads

(cf. (ll))

+~Yi(Q~~+Q*~~)-(f(~/-~?~) ^(Q~+Q*~). (13)

Using

the

equilibriurn

conditions for the

phases

_ei and

4

_we

get

the

equation

sin~

3

#

= ~

d(

To

T)

₊

~~

_~

_~~

_{~ ~'}

(14)

4 yi R _{c +} 2 ci Hi cos 3 _e

j

Since the observed II

phase

has

triple period along

the c-axis we have to assume that To^{is lower than the} lock-in temperature

T[.

Now at 1~ ^{the second} term in the square bracket

might

attain such finite value that

sin~3# _jumps

_at

a

positive

value

already

at

T[,

I-e- order parameters

corresponding

to the II and lock-in

phases

condense

simultaneously

due to _a first order transition to the Il

phase

with

triple c-period.

The

qualitative description

of the

phase

transition

explained

above is over

simplified

since

we have considered the lowest-order interaction term

only.

A

rigorous phenomenological

treatment would not have _a

great

sense because of the

large

number of unknown constants involved. In

general,

whenever two order parameters

(each being sufficiently

«

soft»)

_are

coupled,

^model

parameters

can be chosen in such _a way that _a first order transition takes

place

at which both order parameters condense

simultaneously [18]. Actually

the

phase

transition in AHSe _at 252K is first order

[2, 13]. Clearly

detailed measurements of

e~~

P~~,

c~~ ^{as a} function of temperature ^could

distinguish

between the two models discussed in this section.

8.2 ADSe. The situation in _a deuterated

compound

is _more

complex [2, 3].

The

orthorhombic

phase P212j2j,

which is stable _{at room}

temperature,

^transforms at

high

temperature ^into an IC

phase

with _an average monoclinic 82 symmetry.

Then,

when

decreasing

the

temperature,

the IC

phase subsequently changes

into the lock-in

phases

with k

=

(, _~, (

_and

~.

^{As we have}

^{already pointed}

^{out, the}

^symmetry

^{of the}

^phase

k

= can never be reduced to the triclinic _one and it should be PI12 and

Pl12j

if the order

4

parameter

^has

d~,i

or

d~,~

symmetry,

respectively.

To determine the symmetry ^{of the 2} c-

superstructure (k

= we have undertaken _a neutron diffraction

experiment.

Also _we decided to

study

in _more detail the

phase diagram

from the _room

temperature

to the IC

phase.

9. Elastic neuwon

scaUedng invesdgadons

of ADSe.

9,I EXPERIMENTAL. The

sample

has been _grown

by

slow

evaporation

from _a saturated

deuterated solution and it contains 90 fb of deuterium. The

experiments

have been

performed

at the

Orphke

reactor

(Laboratoire

Won

Brillouin, CEN-Saclay, France).

Diffraction results

were obtained with _a 4-circle

spectrometer

^located on the hot _source

(A

=

0.83

A ).

Precise

superstructure

or satellite

positions

_were determined with the

triple-axis spectrometer

^4F2 on

the cold _source

(A

=

4.08

A _).

_{In both}

cases the

sample

_was mounted in _a fumace and data

were taken _{as a} function of

temperature

with _a 0.I K

stability.

9.2 TRIPLE-AXIS EXPERIMENT. We have started with _a

virgin sample (with

_a mosaic

spread

of

14')

and warmed _{it up}

slowly.

Between 295 K and 324 K the

temperature

was increased

by

steps ^of _m ^{5 K and the}

crystal

_was

kept

at each

temperature

^for at least 24 hours. In this

temperature

_range ^the

crystal

is stable and remains in the

P21212j phase.

Between 324 K and 329 K the

crystal undergoes

_a

strongly

first order

phase

transition and reaches _{a new}

phase

which is characterized

by

2

c-superstructure

reflections

(see

Ref.

[2]

and the

Fig. 7)

and stabilizes

slowly. Up

to _now this

phase

transition to the commensurate lock-in

phase

K K 3315K

o---~ J295K

o-.--...-..- 3245K

o. 315 K

o--~--- 311 K

o,... 304 K

-o -o,fi -o -o i -o -o i

q-10,0,11

Fig.

7. Elastic Q scans

along

the c~

direction,

around the

position

of the

(0,

2,

0.5)

superstructure reflection, at different temperature.

k

= has not been evidenced _on

heating

_run. At 329 K the

crystal

is

damaged

and

powder

4

peaks

_are observed. The

intensity

of _« main

Bragg

reflections settle down

rapidly

_(« main _»

means

Bragg

reflections of the

scattering plane ~b~,

c~ which _{are common} both to

P21212j

and 82 structures. The structural

relations

between the

P2j2j2j

and 82 groups are shown in

Fig.

3 of Ref.

[2]).

On the contrary ^it takes about 36 hours to the

intensity

of the 2c-

superstructure

peaks

to reach _a stable value. The

Bragg

reflections

belonging

to the

P2j2j2j

structure

only (like

the reflections

(012)

and

(021))

have also _an

intensity

which decreases with time and which stabilizes after 36 hours. The _same kinetic behaviour is observed when the

crystal temperature

is set

successively

at 331 K and 333 K. An anomalous increase of the lattice

parameters

b and _c has been observed between 329 K and 333 K

(cf. Fig. 8).

All these

experimental

facts

strongly _suggest

that in this

temperature

range

(329

K-333

K)

the

P2j2j2j

and the lock-in k

=

phase

coexist.

At 335 K the

crystal

_no

longer

exhibits _a time

dependent

behaviour. The mosaic

spread

is stable. The

(021)

and

(012) Bragg peaks disappear

and the 2

c-superstructure

reflections reach _a maximum

intensity

_{as soon} as the

temperature

of the

sample

is stabilized. From 335 K the

sample temperature

was increased

by

_steps of 3K. The

crystal

_was

kept

at each

temperature ^{about 36 hours.}

Taking

into account the

experimental precision

with which the superstructure

position

is defined _{we are} able to say that the 2

c-superstructure

is stable _up to T~ =

337 K. At

higher

temperatures the

position

of the superstructure

slowly

deviates from