X-ray scattering by 1D molecular Guinier-Preston zones in ordered smectic phases

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X-ray scattering by 1D molecular Guinier-Preston zones in ordered smectic phases

Patrick Davidson, Elisabeth Dubois-Violette, Anne-Marie Levelut, Brigitte Pansu

To cite this version:

Patrick Davidson, Elisabeth Dubois-Violette, Anne-Marie Levelut, Brigitte Pansu. X-ray scattering by

1D molecular Guinier-Preston zones in ordered smectic phases. Journal de Physique I, EDP Sciences,

1992, 2 (6), pp.899-913. �10.1051/jp1:1992167�. �jpa-00246610�

Classification

Physics

Abstracrs

61.30 61.50 61.708

X-ray scattering by lD molecular Guinier.Preston

_zones

in ordered smectic phases

Patrick

Davidson,

Elisabeth

Dubois-Violette,

Anne-MaRe Levelut and

Brigitte

Pansu

Laboratoire de

Physique

des Solides (*), B£timent 510, Universit6 Paris-Sud, 91405

Orsay

Cedex, France

(Received 29 January J992,

accepted

in

final form

28 February J992)

Rdsumk. Los

phases smectiques

G et B sont des

phases

ordonndes h trois dimensions. Bien

souvent _{ce ne} sont pas des cristaux 3D

parfaits beaucoup

de d6fauts y sont

pr6sents.

Dans _un

article

pr£cddent,

_{nous avons}

sugg£r6, d'aprbs l'analyse

de l'intensit6 de la diffusion diffuse des rayons X, I'existence de nombreuses lacunes. Dans _cet article, _nous

pr£sentons

_un modme

simple

de lacunes associ£es h des d£formations

61astiques

de taille finie, s'6tendant dans la direction de

glissement

facile des mol£cules

(parallle

h leur

grand

axe). Ce modble

simple

d£ctit bien [es

lignes

diffuses blanches et nodes observ6es _sur [es clichds de diffraction des rayons X par des

£chantillons de TBBA _en

phase smectique

G. C'est _un

premier

_{pas pour} une

description plus g£n6rale qui

tiendrait compte ^d'un

champ

de d6formation

61astique

h trois dimensions.

Abstract. Smectic G and smectic B

phases

_are 3D ordered

phases,

but very often

they

are not

perfect crystals

_{: many} kinds of defects do exist in these

crystals.

In _a

previous

_{paper, we}

suggested

that _a

large

number of molecular vacancies _were

generally

_present

leading

to a diffuse

X-ray

scattering. Here _we consider the

scattering

induced

by

a vacancy associated with _an elastic deformation of finite size, similar to _a Guinier-Preston _zone. We present a

simple

model where the elastic distortion

coupled

to the vacancy

only

extends in the direction of easy

glide

of

molecules,

parallel

to their

long

axis. This

simple

model

explains

the main features of the diffuse

intensity

observed in

X-ray scattering experiments performed

_on TBBA smectic G

samples.

This model is _a rust step ^towards a

complete description

which should take into account a 3D

elasticity inducing

_{a more}

sophisticated

deformation field.

1. Introduction.

Smectic

liquid crystalline phases

of rod-like molecules have _a lamellar

organization (I- dimensionally periodic phase)

in which the

aliphatic

end chains of the

mesogenic

molecules

are in _a molten state

ill.

Some of these

phases

_are ordered in the

layer plane

and may even

present

a

crystalline

3-dimensional

(3D)

order.

Among them,

the smectic G

(SmG)

and

smecticB

(SmB) phases

can be considered _as

orientationally

disordered

crystals (the

(*) Assoc16 _au CNRS.

molecule _rotates around its

long axis)

with _a

hexagonal

_symmetry

(crystal SmB)

or a

monoclinic

(pseudo hexagonal)

_symmetry

(SmG) [2].

These two

phases

_are

usually

considered to be built of

weakly

correlated 2-dimensional

(2D) crystalline layers. Actually,

when the

positional

correlations between

layers vanish,

the 2D

intra-layer positional

order has _a finite range while the _mean

(pseudo) hexagonal symmetry

^is

preserved (quasi-long

range bond orientational

order).

^These _are the so-called hexatic

phases (hexatic

SmB and

SmF)

which _can therefore be considered _{as true}

liquid-crystalline phases [3, 4].

Most of the

studies

performed

on SmG and

crystal

SmB have

emphasized

their lamellar

(smectic)

character,

for

example by

measurements of the shear elastic _constant related _to the

sliding

mode of the

layers

_{over one}

another,

which appears very weak

[5].

The

large density

of

stacking

faults is another consequence of the weak

interlayer

correlations. However, the

analysis

of the

X-ray

diffraction pattems of these

phases

also

gives

^evidence of I -dimensional

(ID)

fluctuations

(or defects)

of the molecular

positions along

the _c axis. These ID defects

involve about 5-10 molecules in _{a row}

parallel

to the

long

molecular axis

(c axis) [6].

Therefore,

this reveals that _rows of molecules _can

easily glide

_{over one}

another,

I-e- that the

corresponding

shear elastic constant is

weak,

too.

Recently

we reexamined the nature of these lD defects and gave a

description,

based upon

some features of the diffraction pattem, ^such as ID Guinier Preston Zones

(GP Zones) [7].

The _zone is made of _a localized vacancy associated with _an elastic deformation

(a compressed part

^of a row of

molecules).

We have demonstrated that the existence of these defects is

general

for SmG and

crystal

SmB

phases

and _we gave a

simplified description

which fits

qualitatively

with the features of the diffraction pattem.

However,

_a

slightly

distinct behaviour of the

intensity profiles,

induced

by

structural

differences,

_can be found between different

compounds.

Indeed, the elastic deformation is

strictly

I-dimensional in the SmG

phase (this

^is not the _case in the SmB

phase).

In order _to take these differences into _account

we should examine _more

precisely

the

shape

of the stress field around _a

single

_vacancy and then establish _a relation between the

X-ray

diffraction

intensity profiles

and the elastic

properties

^of a

given compound.

In this paper, our

study

is focused _on the _case of

strictly

I -dimensional defects which appear

typical

of the SmG

phase and,

more

particularly,

_{on one} of the most

extensively

studied

example

of this

phase,

the

terephthal

bis

p-butyl

aniline

(TBBA).

In the first

part

^{of this} _paper, we recall the structural and elastic

properties

of the SmG

phase

of TBBA and then

give

a

quantitative description

of the

intensity profiles.

In the

following

_{part, we} discuss the contribution of

impurities

and

point

defects to the

X-ray

scattered

intensity

for cubic

crystals.

Then _we

give

_an extension of this model to the _case of smectic

phases

: a

simple

lD model is

presented

and

compared

to the

experimental

data. In the last

section,

we discuss the conditions of

validity

of the model.

2.

Experiments.

TBBA presents a rich

polymorphism

in which several 3D

periodic

structures have been

identified. When

heating

the

compound

from _room temperature, ^the

crystalline phase

is

followed

by

a SmG and then

by

the fluid

SmC,

SmA and nematic

phases.

On

cooling

from the SmG

phase,

two metastable

phases

_are observed before

reaching

the stable

crystalline phase [8].

The unit cell of the SmG

phase

is monoclinic C2/m

(a=10,lh, b=5,18h,

c ₌ 28.6

h, _fl

=

l19°,

Z=

2,

at 125

°C).

The _c lattice _constant is

nearly equal

to the molecular

length,

the

long

molecular axis is

parallel

to the _z direction and the molecules have

a

quasi isotropic

rotational motion around this axis

[9] (Fig, la)

_;

therefore,

the network is

nearly hexagonal

in _a

plane perpendicular

to the _c axis. All the measurable

Bragg

reflections

hexaqonal

^axh

lattice

c

b

a

a)

w

~ w w "

m

m

w m m

~ w

w

w "

~ ~ m

~ ~ m

w

~ m

~ w m

~ ~ m

« w

~ w

b)

Fig.

I. a) Scheme of the unit cell of TBBA in the SmG

phase,

b) Molecular

representation

of the

herring

bone lattice short range order _: the

ellipses

_are the molecule _cross sections

perpendicular

to their

long

axis the _{crosses are} their centers of

gravity

which _are located _on the nodes of the

hexagonal

lattice.

Three orientations of the different domains of rectangular centered symmetry are

represented.

are located in _a volume

=( ±4a*, ±2b*,

±3

c*). However,

their width is resolution

limited

(at

least in _a classical

X-ray

diffraction

experiment

with _a Guinier

camera).

The

X-ray

diffraction

pattem (Fig. 2)

shows _zones of diffuse scattered

intensity

indicative of _some fluctuations _:

~.

l-K i~ _,

%~/ _~ i

'- ~' ~

~) a~ ^"'

~+~~ 4,w

[fig S,

,-~ ^~

t" -~~(

~~.

'i T

I

4> _j '~

''~~~ ~l'

., '(I..Q '-j

'~ ~ _'( ~~*~$

jli~ ^~~ fl

))@'_

"@j- '~+ ft'$~?

j'

~ ~_

( ~ '.

~~

'$~

h

a)

b)

Fig.

2.- X-ray diffraction pattems of TBBA in its SmG

phase

in _a fixed

crystal experiment.

A

=

1.54

A,

_{a) The}

arrow shows the

approximative

direction of the c* axis. The white diffuse line is

perpendicular

to the c* axis and goes

through

the center of the pattem. ^{A series of black diffuse lines}

parallel

to the white _one can also be _seen. (The c* axis of the

crystal

was not exactly

perpendicular

to the

X-ray

beam _so that the pattem ^is not

perfectly symmetrical).

b)

X-ray

diffraction pattem ^{in the} (a*, b*)

plane.

The 6

Bragg

spots characterize the

pseudo hexagonal

order. One _{can see} white diffuse lines

going through

these reflections but the

intensity

of the lines is _not

symmetrical

around the

Bragg

spots.

Around

Bragg peaks,

^the usual thermal

scattering

has been

quantitatively analysed by

coherent inelastic neutron

scattering experiments [10]

_; in

fact, only

the

longitudinal phonons

of _wave vector in the

(a*, b*) plane

_were detected in the SmG

phase.

Diffuse

spots

are located in forbidden

positions

of reflection in the

equatorial (hk0) reciprocal plane

_; these

spots

are due _{to a} local order which breaks the C centered

symmetry. ^This means that the molecular sections in the

(a, b) plane

are

locally

ordered with

a 2D pgg symmetry,

forming

_a

herring-bone

_array

(Fig, lb).

In other words the molecule

reorients itself around its

long

axis with _a six fold symmetry. ^The reorientational motion of

neighbouring

molecules is

only

correlated _{on a} short

scale,

about 20-30

h.

_{The life time of the}

herring-bone

domains is about

10~"

s

[10].

Another kind of diffuse

scattering

is localized in the

(00 I) reciprocal planes

and arises from the ID defects which _{we want to} discuss here.

Let _us describe _more

precisely

the

intensity profile

I

(s~) along

^the c* axis _{at a} short distance si from it

(Fig. 3),

_s is the diffusion vector of modulus _s

=

2 sin o/A where 2 o is the

scattering angle,

_s~ is the _s

component along

the c* axis and si is its

perpendicular component.

^The

intensity

starts from _a rather low value _: this

corresponds

to the white central line which is

perpendicular

to the c* axis _on the pattem. ^{Then the}

intensity

increases and

presents strong

^maxima

(black

diffuse

lines)

around the 00

I reciprocal spots

^{the widths}

(full

width at half maximum _:

FWHM)

of the minimum and of the maxima are

comparable.

^When

we leave the c* axis

(on increasing si),

the

intensity profile

remains

unchanged

_up to

si = a ^*

1~

o C*

s

z

Fig.

3. Curve of the

intensity profile

J

(sz)

in _a direction almost

parallel

to the c* axis at _a short distance s~ = 3 _x 10~~

A~~

_{from it. The}

arrow

points

to the central white line.

In order _to compare

quantitatively

_our

experimental

^results to the

prediction

^{of the}

model,

we first need _to estimate the molecular form factor. This _can be done

through

_an

analysis

of the diffuse spots ^due to the

herring-bone

local array which is _not correlated among

adjacent layers. Figure

4 shows _an

intensity

_scan of such _a diffuse

spot along

_a direction

parallel

to c*. The molecular form factor decreases from its maximum value

fo

at the _center of

reciprocal

space to 0.9

fo

for _s~

= c ^* and 0.5

fo

_{for s~}

= 2 _c*. Therefore the effect of the form factor will be

negligible

close to s~ = 0.

However,

this factor will

slightly change

the

intensity

ratio of the first black diffuse line _to that of the

background by

a factor of 0.9 and will _more

strongly

affect this ratio around 2 c*

id

o c°

s

z

Fig.

4.

Squared

molecular form factor

f~(sz)

derived from the

intensity profile

of the diffuse spots located in the (a*, b*) plane which _are due _to the

herring

bone lattice fluctuations.

The characteristic features of the diffuse

intensity

described above _are reminiscent of the

predictions

and results about

X-ray

diffraction

by impurities

or

point

defects in _a cubic

crystal

which _we shall _now recall.

3.

Comparison

with the _case of _a metal cubic

phase.

Isolated

point

defects

(vacancies, interstitials,

_or

foreign atoms)

_can be considered _{as a} gas where each

impurity

scatters

X-rays independently

of the others. An atom B

(the defect)

which

simply replaces

an atom A in the

crystalline phase

^of A will

provide

_a scattered

intensity proportional

to

~f~ f~)~

where

f~

and

f~

_are the atomic

scattering

factors of atoms A and B

ill ],

this is the so-called Laue

background

which is

proportional

to the defect

concentration and _to the contrast between _{an atom} of the matrix and the defect.

Actually,

the defect

generally

has _a volume different from that of the host matrix atoms, and the distorsion of the lattice around the defect must also be considered. If the _stress strain relation is

purely

elastic,

then the strain decreases _as

I/r~.

_{On this basis the diffracted}

intensity

can be

decomposed

into four _terms which have been estimated

by

B. Bone

[12]

_:

the above mentioned Laue

background

a corrective _{term to} the Laue

background taking

into _account the volume difference between the defect and the matrix atom

a

phonon-like

term, issued from the diffraction

by

the distorted _area,

corresponds

to some

intensity

scattered around the

Bragg

reflections

(but

with _an asymmetry

resulting

from the interferences between the diffraction

by

the distorted _area and the Laue

term)

_;

simultaneously,

the

Bragg peak intensity

decreases

by

_an additive

Debye-Waller

term

taking

the

displacements

into account.

If the

impurities

are no

longer randomly arranged

in the

crystal,

then the

intensity dependence

_{versus s} reflects the correlation function of the

impurity

distribution.

Starting

from _a random _{one, a}

slight tendency

to

impurity clustering

will enhance the scattered

intensity

_near the _center of

reciprocal

_space and around each

Bragg peak.

For _a

simple cluster,

the width of the

corresponding

diffuse _area is

inversely proportional

to the cluster size. On the

contrary,

ⁱⁿ

quenched

aluminium

alloys doped

with zinc _or

silver,

for the first stages ^of

ageing,

the

intensity

increases from _{zero at} the _center of

reciprocal

_space to reach its

maximum _{at a} small value of _s

(=

0.03

h~~).

Therefore _{one sees a} diffuse

ring

around

s = 0 and around all

Bragg peaks.

^This

ring

results from the

scattering by

_a

complex

_zone

made of _a very small cluster of

impurities

surrounded

by

_a

depletion

_zone

II 3].

The number of

foreign

atoms in the cluster is

equal

to the number of _atoms which _were inside the

depletion sphere

before the

beginning

of the

clustering (Fig. 5).

The radius of the

sphere corresponds

to the diffusion

length

of the

impurities

at the stage ^{where the}

sample

is studied. This model is

only

valid if the _{zones are}

randomly

distributed. At _a further

stage

^{of the}

ageing

_process, this is _no

longer

the _{case :} the

intensity profile

^has still the _same

shape

but it reflects a modulation of the

impurity

distribution induced

by spinodal decomposition

instead

[14].

One goes from _a

three-level electronic

density profile, (the

_core and the extemal _area of _a GP _zone immersed in the

matrix),

^towards a two-level

profile

^with clusters rich in

impurities

ⁱⁿ a

quite

_pure

matrix.

~G~A6

~

_o

0

4

~

ze°

o jo ~~ ~~ ~~ ^{~A) ~} ~ ~ ~

a) b)

Fig.

5. Guinier-Preston _zones in the rust stages of

ageing

of the

Ag-Al alloy

(from Ref. ii3]). a) 3- level

Ag-Ag pair

correlation

probability

_{as a} function of the distance in direct space, b)

X-ray

diffuse

scattering intensity

_versus

scattering angle.

Note that this

intensity

decreases at small

angles.

In the _case of smectics B and

G,

the

intensity

distribution

along

c* starts from _{zero at} the

center of the

reciprocal

_space and increases

quickly,

_as in the _case of GP _zones. In _our

experiments,

more

precisely,

the

intensity begins

to increase from _s m±c*/10. At

larger

values of s~, one detects _an

increasing background

which extends up to s = ± c*. The

intensity

modulation is

independent

of si ; then it

corresponds

to a I dimensional disturbance of the

perfect

lattice. The black lines show _a

periodicity along

_s~ close to c* This suggests a defect with _a structure reminiscent of that of the

perfect crystal.

The width of the diffuse lines proves that the defect is of small extension. The

similarity

of behavior with GP _{zones, as we} have

JOURNAL DE PHYSIQUE I -T 2, N'6, JUNE >992 35

already explained [7], suggests

an

interpretation

in _terms of _a defect of _a molecular size

(vacancy)

associated with _a

compressed part

^{of N} ₌ ^{5-10 molecules in} a row

parallel

to _c and of

length

L

= Nc. Since the diffuse

intensity

vanishes _at the _center of

reciprocal

_space, then this

compressed

part must have _a

density higher

than the _mean

density

in order _to

keep

the number N of molecules _{constant over} the

length

L. In _{our case,} the linear GP _zone is _a

vacancy-interstitial pair

ⁱⁿ which the vacancy is localized _{on one} lattice site while the associated interstitial may be _{seen as} delocalized _over 5-10 lattice sites. This is reminiscent of the classical dumb-bell interstitial found in metals

[15].

In _our

previous

_paper, the mechanism of vacancy creation _was

already

^discussed but the

length

of the associated

compressed

_{zone was} not

justified.

In _an elastic ID

model,

the molecules _are

displaced along

the _z direction and in _a static

model,

the deformation

length

would be infinite. Therefore the characteristic

length

of the defect must be

explained

in another way. The stress field range could be limited

by

interactions between

neighbouring

vacancies then

assuming

_a non-random distribution of vacancies. If it _{were so,} the _mean distance between _two

neighbouring

vacancies

along

_c would be of the order of the defect total

length (5-10

molecular

sites).

Then the

intensity

would present a strong ^{maximum for}

s ₌ ± c */10 which has _not been detected

experimentally. Moreover,

within this

assumption

of

a model with

only

two electronic

density

^levels

(I.e.

two kinds of unit cell

content),

about 10 fb of the lattice sites would be empty. ^{This would}

imply

an

equivalent

relative difference

(10fb)

between the molecular volume deduced from

dilatometry experiments

and that deduced from the

X-ray

diffraction

experiments.

In fact these _two

quantities

_agree within I fb

[16].

This

justifies

the model of isolated defects.

We have underlined above that the defect characteristic

length

cannot be

explained _by

_a

purely

static elastic model. The _stress field induced

by

the vacancy has _a limited extension the

origin

of which may be found in the vacancy creation process. In _our

previous

_paper we have

suggested

that the

packing

of molecules is disturbed at the

boundary

of two

herring

bone array domains with different orientations.

Therefore,

_a molecule _can be

expelled

in the next

layer. Consequently

the vacancy is located _{at one} end of the defect and _{not at} the center, as was assumed in _our

previous

_paper,

Considering

the life time of _a

herring

bone domain and the time for rotational diffusion of _a

single

molecule, we must admit that the vacancy has _a very short life-time m10~ _s. Then the defect

length

_{appears as} the diffusion

length

of _a

compression

_wave

along

_c within this time. The

corresponding length

_can be estimated from

coherent neutron

scattering experiments.

^These

experiments provide

_an estimate of the

diffusion coefficient for _a

longitudinal

_wave

propagating along

a*

[17]

:

D~m2.8x

10~ ^~

m~

_s~ ^' _at II7 °C. The

resulting length (Dt )"~

is 53

h

_{which is in fair agreement with the}

defect size deduced from the diffuse line width.

Besides,

let _us also remark that the self- diffusion coefficient

D~

^{which has been measured}

by

other

techniques (NMR,

incoherent inelastic

scattering experiments) [18],

^is seven orders of

magnitude

lower than

D~,

which confirms _our

hypothesis

of _a localized vacancy.

Taking

all the

previous points

into account, the diffracted

intensity

_can be estimated within the

following

frame _{: we} shall consider _a

purely

I dimensional defect and _assume that there

are no correlations among them. As indicated

by

the process of creation of the

vacancies,

the

compressed

_zone will

only

extend _{on one} side of the vacancy.

4. The I dimensional model.

The existence of these ID defects is due to the easy

glide

of molecules

parallel

to their

long

axes c. The calculation of the scattered

intensity

is

performed by considering

deformations in the _z direction and

depending only

on z.

This ID defect extends _{on a} finite part,

conceming

N

molecules,

in _{a row.} In the

perfect

crystal

this

corresponds

to a

length

L

= Nc. Let _us introduce the parameter _a ^{which describes} the size of the vacancy, ^{I.e. the}

displacement

of the first molecule in the distorted _zone

(Fig. 6).

Since the defect is of finite size this

implies

that the

displacement

_u~ of the

n'~

_{molecule satisfies the}

boundary

conditions _:

u~~~ ^=o,

_~-______-____l__---

d<c

c

~

~

c/2+a

~ c/2 o

~_---~--.

6a 6b

Fig. 6. Schematic

representation

of _{one row} of molecules

along

the _c axis. a) In the

perfect crystal

with _a

period

_c. b) In the ID Guinier-Preston _zone. The vacancy sits at the

origin

0 and the

drawing

was

made for _a

= 0.5 _c. In this lD elastic model, the

compressed

part ^of

length

L is asymmetric ^and presents a

pseudo period

d

< c.

A

typical

ID deformation

satisfying

the

elasticity equation

is _:

u~ = a +

(n I)

_K

,

(I)

where _K

= al

(N

I

this leads _to the _new

position x(

of the

n~

_{molecule referred to the}

origin

O shown in

figure

6.

x(=c/2+a+(n-I)d,

with d=c+K.

This ID elastic model is rather

singular

since it reveals _{a new}

period

^d inside the defect. A

3D elastic deformation would not lead _to such _{a new} well defined

period.

^The

amplitude

Aj v(s)

^scattered

by

this _zone is _:

A i v

(s)

=

f (s) e~"'~~~

~ " sin

(N

_ars~

d)/sin (ars~ d) (2)

where

f(s)

is the molecular form factor. The total

amplitude A~(s)

scattered

by

^the

crystal

with the defect _can be

expressed

_{as :}

A~(s)=A~o(s)-Ajp(s)+Ajv(s)

where

A~~

is the

amplitude

of _a

perfect

3D

crystal

of

periods (a,

b,

c) generating

the

Bragg spots.

^A _jp ^{is the}

amplitude

diffracted

by

_one

perfect

_row of

period

c and size L. The diffuse

scattering intensity

then reads _:

I(s)

=

[Aiv -Aipl~ (3)

and may be written in the

following form,

in order to make its mathematical

analysis

easier _:

~

(sin

(N

_{ars~ c}

)

sin

(N

_{ars~ d} ²

~_~~

~

sin

(ars~ c)

sin

(ars~ d)

^~

~

sin

(N

_ars~

d)

sin

(N

_ars~

c)

+ 2

f (

I _cos ars~ a

) ) (4)

sin

(ars~ d)

sin

(

_ars~

c)

The diffuse

intensity

around _s~

= 0

(in

the limit of

large N~

is then

given by

:

l~in ^(N

arsz

c)

~

~~j~ (N

_arsz

c) ~)

~~~~~ _{_}

jjf2 ^cos (N«Sz C)

~~~

N «sz c

~

It vanishes

symmetrically

around sz = 0. A

typical

curve is shown in

figure

7. The central part

corresponds

to the white line of the

experimental

diffraction pattem

(Figs. 2, 3).

Its width does not

depend

_{on a} and scales as

(N )~

^' The behavior of the diffuse

intensity

in the

vicinity

of the other

Bragg peaks depends

_on the vacancy size _a. Indeed for _sz

=

ic* (I

=

integer),

and any si, one

gets

:

I

(ic*)

= 2

f~N

~( (l

^~~~

~"~"(~~ _)~

₊

2(1

_cos

(ariac*))) ⁽⁶⁾

w ac

Then for _a vacancy ^with a low

strength

_a there is _no

strong

contribution to the diffuse

intensity

close to a

Bragg peak.

On the

contrary,

^for

large

_a, ^this contribution _may become

significant

around

Bragg peaks

with I _# 0. This

explains

the presence of dark diffuse lines

perpendicular

to c* and

going through

the

Bragg peaks (I

_#

₀₎

observed _on the

experimental diagrams (Figs. 2, 3).

Their widths scale _as

(N )~

^' like that of the central white line.

We have considered several defect sizes I,e. N

=

2, 3, 4, 5,

6, 10 and different values of _a :

a/c

=

0,1, 0.5,

0.8. For _a

given

a the curves, for different values of

N,

all show _a similar

aspect

^between sz =

0 and s~ =

c*. The

intensity

vanishes around s~ = 0 and

presents

a first

maximum

I~m2.sa~f~/c~ corresponding

to

arNs~c=2.

For intermediate values of

s~(

^I/N « arsz c « I

),

the value

J~

of the diffuse

intensity

is

nearly

constant to _a ^~

f~/c~,

this is the Laue

background.

Therefore the ratio between the diffuse scattered

intensity

(Ij)

around the first

Bragg spot

^{and the Laue}

background intensity (I~)

is

1(s)

N = 4

12

7

2

0.5 1 1.5 2 2.5 3 ^~ ~~~~~ ~'~~

a)

Ils)

N ₌ 4

2

1

o

°.25°.50.75 1 1.251.51.75 ~~Pha = 0.8c

b)

Fig.

7.-a) Diffuse

intensity computed

for _a defect with N

=

4 molecules and _a vacancy size

a =

0.8 _c, b) ^The same

intensity

with _a

change

of the scale.

~' ~~~~~( (l

^~~~

~""C*)

)2

I~

arac* ⁺

2(1-

c~~ "ac ^~

))

f2

^{tY 2}

~ c

(7)

where

fi

and

f~

_are the molecular form factors for the

respective

wavevectors. This ratio

depends

both _on the

length

Nc of the defect and _on the vacancy

strength

_a. The average

length

L

= 4 _c is derived from the width of the diffuse lines of the

experimental profile,

_a is

then estimated from the

comparison

between the

experimental

ratio

Ij/I~=25 (after

substracting

the instrumental

background)

and the theoretical _ones

Ij/I~=125

for a/c

=

0, I and I

j/I~

₌ 30 for a/c

= I. This

comparison suggests

^{that the} vacancy

strength

_a is of the order of _c.

Figure

8 shows the theoretical _average

intensity profile

calculated with _a

distribution of defects N

=

3, 4, 5,

6. The main effect of this average is _to smooth the

background

oscillations.

This _exact calculation valid for any

displacements,

_even

large,

was made easier

by

the fact

that _we

only

considered ID

displacements.

Another

approach involving

continuous defor- mation modes

u~

^{would be} more convenient in the

general

frame of 3D

elasticity.

In the _case of small

displacements

_u~, the scattered

amplitude

is

easily

calculated with _use of the

following

expansion

for _s u~ ^« I _:

A

(s)

₌

if e~'"'~~~~~~~= ~j ^f e~~""(1

+ 2 I _ars

u~) (8)

n n

1is)

io

8

s Alpha ₌ 0.8c

0.5 1 1.5 2 2.5 3

a)

I (s)

0.250.50.75 1 1.251.51.75

2~

^{~~~~~ ~'~~}

b)

Fig.

8. a) Diffuse

intensity computed

for _a mixture of defects with different sizes. N

= 3, 4, 5, 6.

a = 0.8 _c. b) The _same

intensity

with _a

change

of the scale.

where r~ is the

position

of the

n~

_molecule in the

perfect crystal.

The diffuse

amplitude

is driven

by

the last _term of

equation (8).

The continuous

displacement u(r)

_{= a}

(I

z/L c/c

corresponding

to that of

equation (I)

_can be

expanded

in Fourier modes _{as :}

u

(r )

=

lu~

^{e~ ~~"~'~}

^{d~q (9)}

with _:

( ~q

)2 )2 ^~

x ^~ Uq _y "

,

~ sin

("qz

L

~

+

(sin (

_"qz L )~ ~~~~

~~~~~

2

=

~)

~2 ^{~°~ ~~~ ~} "~~ ~

The diffuse

scattering intensity I'(s)

refers to the

intensity

calculated in the frame of this

expansion

and reads _:

1'(s)

=

~j4 ^ar~ f~s/(u~)/ (II)

where q = s-

ic*

_{In the}

vicinity

of _a

Bragg peak, only

_one term of this _sum

really

contributes. Around the

Bragg peak ^I

=

0,

this leads _to the

intensity I((s)

:

sin

(

_{"qz L}

)

^~

~

(sin (

_{Wqz L}

)

)~

l'

~j~~~

₌

) COS ("~z

~ ~~

«qz L

The diffuse

intensity Ii (s)

^for _{sz =}

c*,

I,e, sz =

c* ₊ qz, can be

expressed

as :

(sin ⁽

^{"qz L ~}

+

(

Sin

(

_"qz L

)

_~~ ~~~~

~ ~~~

~~

_~~~~~~~ ^C

*j

^{~z ~} _c°s

(

_«~z ^L

7r~z L

andfors=c*.

1((c*)

₌

gr~(a

^~

f~/c~)(L

_~/c~)

Around the _center of

reciprocal

_space

(I

=

0),

the

intensity I((s)

is the _{same as} in

(Eq. (5))

valid for any value of _a.

But,

around the 001

reciprocal node,

the

expansion

used in

equation (8)

which is

only

valid for small values of _s u~ leads _to

expression (6)

in the limit of small _a

(

_a « c

).

5. Discussion.

The

general

features such _as the black and white lines of the

experimental

_{curves are}

fairly

well described

by

the model.

However,

in between

(in

the intermediate

regions:

0.2 c* <sz < 0.8

c*),

_we notice _some difference. The Laue

background

is

experimentally

affected

by

an extra

scattering intensity

in this range. This is not

surprising since,

besides the defects which _we

consider,

_many other defects _are also present ⁱⁿ our system. ^{The first kind of}

defects,

related _to the

phase compressibility,

^which _may be

important

^is that of

longitudinal phonons

as in classical

crystals. They

should

give

_a

fairly

constant contribution in the

intermediate

region.

In order _to have the _same

intensity

induced

by

the vacancy Laue

background

one should need 10~^~ vacancies per site

[19].

The _same kind of effect in _a

liquid

would be

govemed by

its

compressibility.

It would

give

_a contribution IO times

larger

than that due _to the

phonons,

therefore

equivalent

to 10 ^~ vacancies per site

[20]

and

slowly increasing

with _s. Since the white central line is

clearly

_seen, then the concentration of vacancies in the SmG

phase

of TBBA is

larger

than 10~ ~, but less than 10~^~ _as

suggested by

the

dilatometry experiments.

However the vacancy Laue

background

is

superimposed

in the intermediate

region

to another

scattering intensity

^of an order of

magnitude

characteristic of _a

liquid,

In _some way, this suggests ^{that the SmG}

phase

of TBBA behaves _{as a}

crystal

at

long

range and in _{a more} disordered way at short range.

Indeed,

_at

large scale,

the system behaves like _a

crystal

in which the conformation of the molecules is

averaged. Conversely,

at short scale the differences of conformations and

positions

of

neighbouring

molecules constitute

another kind of defects. These

weakly

correlated motions also

give

substantial contribution _to the diffuse

scattering intensity

with _a

large

maximum around c*. Both kinds of defects _are

certainly

present since _we observe _a

significant Debye-Waller damping

^of the

Bragg

reflections.

Let _{us now} look _at the behaviour for

large

_{sz ~} 2 c*. We have

previously

mentioned that

our model introduces _{a new}

period (d~

in the

system.

This should lead to a

peculiar intensity profile

_: besides the

peaks

at sz =

ic*,

_additional

_peaks

_{should also appear at s~}

=

id*

=

ic

* I + ^"

).

^{Even if} we considered _{a more} realistic

non-periodic (liquid like) dependence

Nc

of the distance between two successive molecules in the

compressed

zone, we should still _see

some

broadening

of the black diffuse lines.

However,

this is _not observed

experimentally.

Actually,

in TBBA and in _a few other

compounds

_we

clearly

_see a set of about 10 black rather thin diffuse lines

equidistant

with _a

period

_c *. This

suggests

^{that the defect could have} a more

complicated

structure : in addition to the small

strongly

distorted

region (almost liquid)

the defect could also present ^another

region

which would be

only slightly

distorted but would

essentially

exhibit _a

purely displacive (d

= c disorder. This latter

region

is reminiscent of the

strings

of

displaced

molecules which _were first considered

[6]

to

explain

the

origin

of the black diffuse lines.

An evidence of the 3-dimensional character of the strained _{area can} be found in the

coupling

with

longitudinal phonons

of wavevectors located in the

(a*, b*) plane. Examining

this

plane,

we notice that the diffuse scattered

intensity

of thermal

origin

is concentrated around the 200 and 110

Bragg

reflections

(of

wavevector s~ =

2 a* _or a* ₊

b*)

characteristic of the

long

_range

pseudo hexagonal

lattice in the

layer plane. Scattering

is _more extended for

phonon

_wave vectors

perpendicular

to s~ than for

phonons parallel

to it. This _means that

transverse

phonons propagate

more

slowly

than

longitudinal

_ones. The

dissymmetry along

si of the scattered

intensity

is _more

surprising

: it is

negligible

for _s

< s~ and strong ^for

s~s~. This _can be understood

through

a

coupling

between

density

and

displacement

fluctuations _: if _a sinusoidal

density

wave of

wavelength

^{A is} associated with _a

displacement

wave of the _same wave vector

~fl,

the _two

corresponding

satellites at s~ I/A and

s~+I/A

will be of

unequal

intensities

[11].

Since the

intensity

of the satellite at s~ I/A is

lower,

then the

density

wave associated to each vacancy

displays

_a dilatation

around it.

6. Conclusion.

The aim of this paper ^{is to} propose a

simple

model

accounting

for the main characteristic features of the diffuse

intensity

observed in the

X-ray

diffraction

pattems

^{of TBBA in its SmG}

phase.

The

defect,

of finite extension L in the _c

direction,

is _{seen as a} vacancy of size

a

associated with a

compressed

zone of N molecules

(in

^the presence of the vacancy, the N molecules occupy a

length

smaller than in the

perfect crystal).

The presence ^of a white line at s~ = 0 reveals

positional

fluctuations _on the

length

L with conservation of the number of molecules _on this

length.

The existence of _a Laue

background

is

typical

of the vacancy. ^The black diffuse lines observed for sz =

ic*

_{are also}

typical

of the

scattering by

_an

object containing

N molecules. The

coupling

of the vacancy with

longitudinal phonons

of _wave

vectors located in the

(a *,

b

*) plane,

indicates that the molecules _are

repelled by

the vacancy in _a

plane perpendicular

to the lD defect.

So

far,

_we have

particularly

focussed _our

study

on the lD

periodic component

^{of disorder}

along

the molecular axis. Let _us remark that this aspect ^is

especially

_apparent in the systems ⁱⁿ

which the

layer

character is less marked. A

typical example

is that of the first derivatives

(short aliphatic chains)

of several

homologous

series which present a direct transition SmG or

SmB-Nematic without any SmA

phase

in between

[21].

Although

our

simple

model describes

experiments

in TBBA

samples

rather

well,

several

points

should be considered in _a next

approach.

Indeed the stress field around the vacancy is not

necessarily

restricted to _{a row}

parallel

to c,

especially

if _one remarks that the molecules may

undergo

_an ABAB

stacking

_as is observed in the

hexagonal

compact

(hcp)

structure found in many SmB

phases.

This raises the

question

of the elastic stress around

point

defects in _an

anisotropic

medium. Furthermore the

shape

and the size of the vacancies should also

depend

_on the elastic

properties

of the medium.

Moreover,

_we have _seen above that the defects have _a

dynamical origin

linked _to the short vacancy lifetime. All these considerations suggest to extend the

analysis

within the frame of _a 3D continuous model

[22] taking

the

viscoelastic

properties

of the medium into account.

References

ill DE GENNES P. G., The Physics ^of

Liquid Crystals, Chap.

7, Oxford Sciences

publications

(Clarendon Press Oxford, 1974).

[2] PERSHAN P., Structure of

Liquid Crystals.

World Scientific Lecture Notes in

Physics,

23 (Word Scientific

Singapore,

New

Jersey, Hong-Kong,

1988).

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Phys.

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Phys.

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