Destructive interferences between diffuse scatterings due to disorder and displacive modulation (X-ray “white line" effect)

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Destructive interferences between diffuse scatterings due to disorder and displacive modulation (X-ray “white

line” effect)

S. Ravy, Jean Pouget, R. Comes

To cite this version:

S. Ravy, Jean Pouget, R. Comes. Destructive interferences between diffuse scatterings due to disorder

and displacive modulation (X-ray “white line” effect). Journal de Physique I, EDP Sciences, 1992, 2

(6), pp.1173-1190. �10.1051/jp1:1992203�. �jpa-00246596�

Classification

Physics

Abstracts

61.IOD 72.15N 61.70R

Destructive interferences between diffuse scatterings due to disorder and displacive modulation (X-ray

_«

white line

_»

effect)

S.

Ravy,

J. P.

Pouget

and R. Comes

(*)

Laboratoire de

Physique

des Solides

(CNRS-URA

2), Bit. 510, Universitd Paris-Sud, 91405

Orsay

Cedex, France

(Received

J5 October J99J,

accepted

J6 December

J99J)

Rdsumd. Dans des solutions solides de sels h transfert de charge ^{de la} famille

TTF~TCNQ

_un effet de diffraction

original

est observd. Los

figures

de diffraction de tels matdriaux font

appardtre

de

larges

diffusions diffuses _sur

lesquelles

_on observe de fines

lignes

diffuses

« blanches _{» en} Q ₊ 2 k~ alors

qu'en

Q _{2 k~ on} retrouve

l'aspect

habituel des

lignes

noires.

Nous _{proposons une}

interprdtation simple

de cette

asymdtrie

d'intensitd ₊ 2

k/

2 k~ en

couplant

en

quadrature

_une onde de

ddplacement

et une onde de

composition

de mEme vecteur d'onde. La cohdrence entre le ddsordre de substitution _et le

ddplacement

dtant h l'origine de l'effet, _nous effectuons ensuite _un calcul de l'intensitd diffusde _au

premier

ordre _en

d£placement

_pour un

modble

d'alliage

unidimensionnel dans

lequel

toutes les

impuretds

accrochent _une distorsion de rdseau h 2 k~, avec la mEme

phase.

Ces rdsultats _nous permettent ^de mettre en Evidence les trois

parambtres pertinents

du

problkme

_: la diffdrence entre les facteurs de _structure des entitds substitudes, la

phase

de l'onde accrochde _par rapport ^h

l'impuretd

et sa

polarisation.

Nous

proposons que les

lignes

blanches observdes

expdrimentalement

dans

(HMTTF)o,05

(HMTSF)o,95-TCNQ

rdsultent des distorsions de rdseau

coupldes

_aux oscillations de

charge

induites par des

impuretds

_sur les chaines donneuses.

Abstract. In tile solid solutions of

charge

transfer salts of the

TTF-TCNQ family,

a new

diffraction effect is observed. The

X-ray

pattems ^{of such materials show} a broad diffuse scattering

on which fine _« white _» diffuse lines at Q + 2k~ are observed, whereas at Q _{2 k~ tile} diffuse

scanering

has the usual aspect. ^We propose a

simple interpretation

of this

+2k/-2k~

asymmetry of

intensity by

_a

coupling

in

quadrature

between a

displacement

_wave and _a

composition

_wave of _{same wave vector.} As the effect

originates

from _a coherence between substitutional disorder and the

displacement

_{waves, we}

perform

_a calculation of the diffracted

intensity

to first order in

displacement,

for _a one-dimensional

alloy

in which all the

impurities pin

a lattice distortion of wave vector 2 k~, with the _same

phase.

We

emphasize

the three relevant parameters of the calculations _: the difference of structure factors of the substituted entities, the

phase

of the

displacement

_wave with respect to the

impurity

and its

polarization.

We then propose that the white lines

experimentally

observed in

(HMTTF)o.05(HMTSF)095-TCNQ, originate

from the 2k~ distortions induced

by

charge oscillations around the substituted HMTTF molecules

on tile donor stacks.

(*) Present Address _: LURE, Bit. 209 D, Universitd Paris-Sud, 91405

Orsay

Cedex, France.

1174 JOURNAL DE

PHYSIQUE

I N° 6

1. Introduction.

The

organic charge

transfer salts of the

TTF-TCNQ family

have been

extensively

studied in the past

[Il. They

_are

composed

of

segregated

one-dimensional

(lD)

chains of donors

(TTF,

HMTTF _or

HMTSF)

and acceptors

(TCNQ),

on which electrons _are delocalized. These ID

conductors _are known _to be unstable with respect to the formation of

charge density

wave at

twice the Fermi wave vector

(2ki).

Below

T~,

^{the Peierls transition} temperature, a

2

lq periodic

lattice distortion is stabilized and _a gap is

opened

at the Fermi

level,

the

gain

of electronic energy

being greater

^{than the loss of elastic} energy. ^In a

large temperature

_range above

T~,

a

regime

of lD

pretransitional

fluctuations is observed.

X-ray

diffuse

scattering

is _a

powerful technique

to

study

this lD

regime [2].

The lattice distortion

coupled

of the electron gas

gives

rise to diffuse

planes

located

symmetrically

_{at ±} 2

ki

^{of the}

layers

^of

Bragg

spots

perpendicular

to the lD direction.

By

intersection to the Ewald

sphere

these

planes yield

diffuse lines _on the

X-ray pattem.

^A

thorough study

of these lines

gives

direct information _on the in-chain correlation

length f

of the lattice fluctuations and _on the

susceptibility

of the

coupled electron-phonon

_system.

The effects of

disorder, especially singular

in chain-like

materials,

_were

investigated by studying

the solid solutions

(TTF)~(TSF)i_~TCNQ

and

(HMTTF)~(HMTSF)i-XTCNQ,

were selenium- and sulfur-based donor molecules _are

alloyed.

An

X-ray

pattem ^{obtained for} the

(HMTTF )oos(HMTSF)O~STCNQ alloy

is shown

figure

^{16 and}

compared

to that of pure

HMTSF-TCNQ (Fig. la).

In

figure16

_{a very}

striking

_asymmetry in the intensities of the

± 2 k~ ^{diffuse lines is observed. At} + 2 k~, the lines have _an

intensity

minimum visible at the expense of the

background (white arrows)

and at 2 k~ an

intensity

maxima

(black arrows).

It

was

suggested [3]

that the 2 k~ ^{white lines}

originate

from _a substraction of _a diffuse

intensity

term, due _to lattice

fluctuations,

from the so-called _« monotonic _» Laue

scattering

due _to the

HMTTF/HMTSF

disorder. Such

negative

and

positive

interferences _were assumed to result from _a

phase

coherence between the

position

of the

impurities

and the lattice distorsions. This paper presents calculations which confJrrrJ these ideas. In the first

section,

_we recall that in _a system ^{where there is} a 90°

out-of-phase coupling

between _a

displacement

_wave and _a

composition

wave of _{same wave} vector q, an asymmetry ^{of the}

intensity

of the

±q satellite reflections located _on each side of the

Bragg

reflections is

expected.

This

asymmetry

^is a

key

to

understanding

the white line effect. An exact calculation of first-order

a) b)

Fig.

I. a)

X-ray

_pattem from

HMTSF-TCNQ

at 25 K

showing

_{2 k~} diffuse lines (black arrows). b) X- ray pattem ^{from the} solid solution (HMTTF

)o_05(HMTSF)O~STCNQ

at 20 K

showing

the white diffuse lines at + 2 k~ (white arrows) and the black diffuse lines at 2 k~ (black arrows) for f

> 0. Both

X-rays

pattems ^{have been taken with about tile} same

crystal

orientation, with the _c axis horizontal and the b axis vertical. The

wavelengh

used _was A~~~ ₌ 1.542

A.

terms in

displacement,

which _are

responsible

for the

effect,

is

performed

in section 2 for _a

model

system

^{in which each}

impurity pins

_a

displacement

_{wave on} a coherence

length

f,

with the _same relative

phase.

The relevant parameters are then discussed and

experimental

evidence is

briefly

shown.

2. The

origin

of the white lines.

2,I THE _{ASYMMETRY OF SATELLITE} REFLECTIONS. ln order to

clearly emphasize

the

physical origin

of this

effect,

it is useful _to reconsider _a

simple

calculation first

performed by

Guinier

[4].

Let _us consider _a one-dimensional lattice of

period

_a,

composed

of

N atoms, the atomic

scattering

factor

f~

of which varies from site _to site _{as a} sine function

fn

~

f11

+ 'l Sin

(2

_{Wqna +}

~f)i, (1)

where q is the wave vector of the

modulation,

7~ its

amplitude

and ~ ^its

phase.

This

expression

can, for

example,

account for the

spatial dependence

of the _mean structure factor of _an atomic

occupation

_wave in _a

binary alloy.

More

generally

it _can represent ^the

generic

term of the Fourier transform of _any

composition

modulation.

Now,

let _us consider _a sinusoidal

displacement

_{wave as} follows

x~ = na + u sin

(2

_wqna + _q~

), (2)

where _u is the

amplitude

of the

displacement.

This modulation has the _{same wave vector} q as the

composition

modulation but a different

phase,

_q~. We define W _as the

phase

difference between the two modulations _:

wmq~-~. (3)

The diffracted

amplitude A(s)

is

given by

A(s)

₌

£ f~

_exp

(-

2 I

wsx~)

,

(4)

n

where the index _{n runs}

through

the N sites and _s represents ^the

scattering

_wave vector. We have then _to calculate the

expression

£ f(1

_{+ 1~} sin

(2

_wqna +

$r))

_exp

(-

2

iws(na

+ u sin

(2

_wqna + _q~

))). (5)

n

The _exact calculation is

performed

in

appendix A,

but for the sake of

simplicity

the values

of the

peak

intensities up ^to the second order in _{u are} indicated in tables I and

II,

with the usual

approximation

2 _{wsu «} I.

Figure

2 summarizes the

following

situations _:

In the _case of _a pure

composition

_wave, the

reciprocal

_space consists of nodes at

s ₌

Q

=

hla

(h integer)

and of

pairs

of satellites located _{at s}

=

Q

± q, the intensities of which

are

proportionnal

to 7~~/4. Characteristic of such _a modulation is the fact that these intensities

are

independent

of the

scattering angle,

when the s-variations of the

scattering

factor

f

_are

ignored.

In the _case of _{a pure}

displacement

_wave,

pairs

of satellites appear ^at s =

Q

_±mq

(m integer),

^with intensities

proportional

to

(J~(2 w(Q

_±

mq) u))~,

where

J~

is the Bessel function of m-th order.

Generally,

the

amplitudes

of distortion _are small and the

high

order

( )m

_~ l

)

satellite intensities _are very small.

Contrary

to the

previous

_case, the

intensity

of the satellite reflections increases with the

scattering angle.

1176 JOURNAL DE

PHYSIQUE

I N° 6

Table

Pure

composition

_wave Pure

displacement

_wave

s =

Q N~ f~

N ^~

f~(I (wou )~)

s =

Q

_+q

N~f~l~14 N~f~(«(Q +q)U~)

S

=Q-q N2f2~2/4 N2f2(«(Q-q)u~)

Table II

Coupled composition

and

displacement

_waves

S ₌

Q N~ f~ N~ f~(wou)~ (l~~cos~

W

2)

s

=Q+q N2f2~2/4+N2f2~«(Q+q)usin4~+N~ «(Q+q)u ((1 1~~(2

+ cos 2

W))/4)

s=Q-q N~ ~l~"- ~l«(Q-q)Usind~+N~f((«(Q-q)U)~

(l

_1~

(2

_{+ cos} 2

W))/4)

-q +q

(a)

-

q

~2q+2q

(b)

- I

-q

(C)

-

ig.

mposition

q. _of

large ^h c) Same

representation

for a

coupling

between

and

displacive

modulations

of

same wave vector with

1~(Q

+

q )

u

sin _q~

positive.

For h

side of a Bragg

reflection

to the

ther side, inducing an symmetry

+

q,

-

q on the

When both modulations _are

present,

a 7~u bilinear term appears in the

expression

of the first order satellite

intensity.

The

sign

of this term is

opposite

for the two

Q

± q satellites

and, provided

that sin W #

0,

the

intensity

becomes

asymmetric

with respect to the nodes of the

average lattice. The

N~f~7~w(Q _+q)u

sin W

intensity

is transferred from _one satellite

reflection to the other one

(Fig. 2).

As first

emphasized by

Guinier

[4]

this

phenomena

is characteristic

of

_a correlation between _a

composition

and _a

displacement

_wave.

Let _us discuss _more

precisely

the

importance

of the bilinear _term

7~w(Q

_{± q}

)

u sin W. If

su « _7~ and sin W # 0 the classical second order term is

neglected

and the intensities of the satellites _are

given by

1(Q±q) =N~f~7~~/4±N~f~l~w(Q±q)usin

4l.

(6)

These intensities

crucially depend

_on the

sign

of the bilinear _term which itself

depends

_on

7~,

w(Q ±q)

u and sin 4l. The influence of the two first terms will be discussed in the

following

but the role of the latter _term is

quite straightforward.

If the modulated _{waves are} in

phase

or 180°

out-of-phase,

the bilinear term vanishes. The

satellites intensities _are

given by

the second order term and _no

asymmetry

appears. If

4l

#0

_{or w,} the satellites have _not the _same

intensity

except ^{around the}

origin

of the

reciprocal

_space

Q=0

and the relative difference of

intensity

is maximum when 4l

= ± w/2. This latter situation is of

physical importance.

For

example

it

corresponds

_to the so-called size effect

[5]

in

alloys

where the _s variation of the diffuse

scattering

arises from lattice distortion

resulting

from the different size of the

constituting

atoms. We shall _see that the white line effect has _a similar

origin.

2.2 DISTORTION _{WAVE PINNED TO AN IMPURITY.}

Now,

let _us consider _a lattice of

N ₊ I atoms B with _a

foreign

atom A _at the _n

=

0

position.

The atomic

positions

_are

given by

x~ = na + u sin

(2

_{wqna +}

q~

), (7)

as shown in

figure

3.

This very

simple

model represents a

displacement

modulation

pinned

to the

impurity

A.

The

scattering

factors of such _a lattice _can be written _as

fn

"

fB

+

WA fB) 8n, (8)

where

fB

and

f~

are the

scattering

factor of the B and A atoms

respectively.

The

8~ function,

defined

by 8~

₌ ^if n = 0 and

8~

=

0

elsewhere,

_can be

developed

in _a Fourier

series,

which

gives

fn

₌

fB

₊

(f~ f~) £ exp(2 iwq'na) q'

=

'~

,

m ₌

N/2,

,

N/2

N +

~.

a(N

+

1)

(9)

or,

~~

N l

l~ ~~ ~~~~~

_~

~~~ ~

~

f~

~A/(~~~

2)

^{~~" ~~}

"~'~~

~

~~~~

q'=

^'~

,

m=0,. ,N/2 (10)

a(N+1)

f~ given by (10)

is _{a sum} of functions defined in

(I)

with

~

~

~~~ ~~~)~ ^~~~

'

'~

(fB ~A/(~~~ ₂₎₎

^~"~

^~

^~~~

l178 JOURNAL DE

PHYSIQUE

I N° 6

2n/q

~~°

o

u

.

~

o . .

~

~

qmW2

)

° ° _$

@ ~

Fig.

3. Atomic

displacements

for _a transverse lattice distortion of _wave vector q,

pinned

to _an

impurity A with relative phases _q~

=

0

(top)

^and _{q~ =} ^gr/2 (bottom).

For all the _{non zero} values of

q',

this variation of the

scattering

factor will

give

_a set of

pairs

of satellites around each node. For _a

sufficiently high

value of

N,

this _set will appear as a

continuous

background

of

intensity

IB

₌

(fA fB)~, _(ii)

which is

just

the so~called

« monotonic _» Laue diffuse

scattering.

The bilinear terms

f~ _7~w(Q

_±

q')

_iu sin (q~

w/2)

are non zero

only

when

q'=

_q, I,e.

when the two modulation _waves do interfere. Their values _are

lS

"

4

fB ~fA fB

_"

(Q

± q u cos q~

(12)

Assuming

that

(fA-fB)/f~»(w(Q±q)u)

and

cosq~#0,

the intensities at the

Q

± q

Positions

_are

1(Q

+ q

)

"

(~fA fB)~

4

fB~fA fB)

_"

(Q

+ q

)

_{U CDS 9~}

(13)

1(Q

_~

"

(~fA fB)~

₊ 4

fB~fA fB)

_"

(Q

_~

)

_{U CDS ~}

(14)

When _u

=

0,

the

reciprocal

_space of this lattice consists of nodes at

Q

₌ hla and of _a Laue

« monotonic

»

background

due _to the

impurity.

^When a

displacive

modulation is

added,

_an

asymmetry

^{of the satellite}

intensity

is observed around each node. If

Is ~0,

the satellite

intensity

at the

Q

_{+ q}

(Q

_q

position

is

greater (weaker)

than that of the Laue

scattering background.

Thus

Q

_-q

«negative»

satellite reflections will appear on the diffraction pattem

(Fig. 4).

The

position

of this

negative

satellite reflections

(I,e,

white

line)

with respect

to the main

Bragg

spots

depends

on the

phase

_q~ ^of the lattice distortion and of the

scattering

factors of the _atoms

alloyed.

Let _us remark that _our calculation satisfies the Friedel rule

because

expressions (13)

and

(14)

_are

interchanged by

inversion symmetry

(Q--Q

around the

origin

of the

reciprocal

_space.

3. Calculation of the

X.ray

diffuse

scattering intensity.

3,I FORMULATION _{OF THE PROBLEM.} Let _{us now} consider _an

alloyed

chain

composed

^of

two

species

A and B

(atoms

or

molecules)

of concentration xA and x~, with _structure factor _:

FAj~~

=

£ fj

_exp

(-

2 I

wry

s

), (15)

J

~q ~q

Laue

scattering

^V=n

Q+q Q-q

-q +q

- ~

o+q ^""

o-q

Fig.

4. Schematic

representation

of the

intensity

scanered

by

_a chain in which an

impurity pins

_a

lattice distortion of _wave vector q. In the _case where

fA

_>

fB,

two extrem situations

corresponding

to the relative

phases

_q~

= w and

q~ =

0 _are shown on top and bottom

respectively.

where

f~

^{is the}

scattering factor,

assumed to be

real,

of the

j-th

atom of the A

(B)

unit of

position

_r~. For the sake of

simplicity,

_we shall _assume that the structure is

centrosymmetric

and that the molecules themselves _are

centrosymmetric

in order to _assure that the _structure factors _are real

quantities.

The model system ^{studied is}

schematically represented

in

figure

5. We _assume that the A atoms _are the

impurities

and that around each

impurity,

located _at

position ka,

_a

displacement

wave is

pinned.

It is characterized

by

its _wave vector q and its correlation

length

f. The

displacement

of the n-th _atom due _to the k-th

impurity

is written

u$

₌

£ _u~,

sin

(2 wq'(n

k _{a + q~}

) (16)

q.

where

q'belongs

to the first Brillouin _zone

(B.Z.)

and

u~> is _a function of

q'centred

around q.

Moreover,

_we shall _assume that 2 _w

)q) f

» I which _ensures that

£ _u~

=

uq,

_=o = 0.

(17)

n

It is also useful to define the

~]

and

~f

variables

by ~]

= +

l, ~f

=

0 if the site

n is

occupied by

the molecule

A,

and

~)

=

0, ~f

= + I if there is _a B molecule at

n-

Fig.

5. One-dimensional model used for the calculation of the diffuse

intensity.

All the

impurities

A

pin displacive

modulations of _{wave vector q} and

spatial

extension f. A and B

species

are

represented by

large

and small dots

respectively.

l180 JOURNAL DE

PHYSIQUE

I N° 6

Following

these definitions the

displacement

of _a molecule located at _na is

~

j~ ^~A

^~k

⁽¹⁸⁾

n k _n

k

and its _structure factor

F(

₌

F~

_exp

(-

2 I _ws. u~

)

=

(~) _FA

₊

~f _F~)

_exp

_(-

_{2 ins}

_u~)

,

(19)

where in

(18)

the summation is

performed

_over all the

impurity

sites k.

3.2 CALCULATION _{OF THE DIFFRACTED INTENSITY.} In the kinematic

approximation (see

for

example

Ref.

[4])

the total diffracted

intensity

^is

given by

J

(s)

= N

~)~

^~ ~ y

(s)

,

(20)

where V is the volume of the

crystal,

N is the total number of unit cells,

3(s)

is the Fourier

transform of the form factor

v(r)

for the

crystal (v(r)=

I inside the

crystal

and

v(r)

₌ 0

outside)

^{and Y}

(s)

is

Y

(s)

=

£ (F( _{F( (} ~) exp(2

I _ws

ma) (21)

m

~

(F( _F(( ~)

~

is the average of

F( F((

~

over all the sites of the

crystal,

that _we suppose to be

homogeneous.

This total

intensity

contains _two

contributions,

the

sharp Bragg

reflection term due to the

periodicity

of the average

lattice,

Y~~~~(s )

=

£

F ^~ exp

(2

ins

ma), (22)

m

and the diffuse

scattering

term due to deviations from

perfect

order

Y~,~~(s =

£ (F F( (F

^*

F( (

~

))

~

exp

(2

ins

ma), (23)

where F

=

(F()~

^{is the} average atomic

(molecular)

structure factor.

The calculation of the average in

(23)

is difficult because the correlation of structure factors of _two

species

not

only depends

_on their _nature and their

position

but also _on the

position

of all the

impurities

in the

crystal.

As first shown

by Cowley [6],

such _an average contains

high-

order correlations terms.

Up

to first order in

displacement,

we can write F

=

(F~)

2 in

(F~ u~)

,

(24)

and

I(F _-Fl)(F* -Fl$m))~

"

((F ~Fn)(F* -Fl+m))~

₊

+ ~ _"~

l~n(F

*

Fz+

m

)

_Un

~z+

m

(F Fn

_Un

+

ml

~

(25)

In order to calculate these terms, _we have to introduce

pair

and

triplet

correlation coefficients. Let _us define the two-sites

probabilities PA~(m) (resp. P~A(m))

to have B

(resp. A)

at distance _ma of A

(resp.B)

and the three-sites

probabilities PARR(m, k),

P

A~A

(m,

k

),

P

~A~

(m,

k and P

~~~

(m,

k

),

where P

~~A

(m,

k

)

is for

example

the

probability

to have A at site _na, B _at site

(m

+ n

)

a and A _at site

(k

+ n a. With _our notations

~AB(~ll

"

(~( _~~+m)~

Some of the

properties

of these

pair

and

triplet

coefficients used here _are recalled in

appendix

B

[7, 8]

for _a

centrosymmetric

structure.

By using

these

definitions, (24)

becomes

~

"

(~~)~ ^~A

+

(~~)~ ^~B

^~ l "S

~£ ^{(~~ ~/}

^U~)

^~A

⁺

^{(~~ ~l U~) B)}

, n

or, after _a k

- k _{+ n}

substitution,

with the

equality u(

^+~

=

u(

F

= x~

F~

+ x~

F~

2 ins.

£ (P~~ (k ) F~

+

P~~ (k F~ ) u(

,

(27)

k

which

yields

F

= x~

F~

+ x~

F~

2 ins.

(F~ F~) £P~~ (k u( ₍₂₈₎

,

using (17)

and the

relationship P~~(k)

+

P~~(k)

= x~.

Up

to first order in

displacement,

^the intensities of the

Bragg _spots

_are thus

given by

F ^~

=

(x~ F~

+ x~

F~

)~

(29)

with _our

hypothesis

that the molecules _are

centrosymmetric.

The first term of

(25)

substituted into

(23) gives

the diffuse

scattering

due to short range order

(S.R.O.)

which is

expressed

_as

(see

for

example

Warren

[5])

Ys

~ o

(s)

= x~ x~

(FA F~

)~

£

_a

(m )

_exp

(2

ws ma

) (30)

m

PBA(m

in which the coefficients _a

(m

m I _are the so-called

Warren-Cowley

S.R.O.

XA XB

parameters. ^{If the} system ^is

completely disordered,

_a

(0)

=

and a

(m

=

0 for _m #

0, (30) gives

the so-called Laue

scattering.

~S.R,O.

(S)

~ XA XB

(~A ~B

_)~

(31)

By using (18)

and

(19),

the second _term of

(25)

is

rewritten,

~ _{l "S}

l~£ ^{~~ ~(+}

m

~l ~)

~A (~

^*

~A ~£ ^~~+

m

~~ ~l

_U~

+ m

~A (~ ~A

~ ⁿ ~ ⁿ

+

I ~~ ~~+

m

~/

_U~

_~A (F

*

FB £ ~~+

m

~~ ~/

_U~

+ m

~A (~ ~B

~ ⁿ ~ ⁿ

~

j~ ^{~B ~A ~A~k}

^j/

^(j/*

^jz

⁾ £~B ^{~A ~A~L}

j/

(j/

j/

n n+m k _n B A _{n+m n n+m} B A

~ n j n

~

~i ^{~~ ~~+}

m

~/

U~

~B (~

^*

~B ) ~£ ^~~+

_m

^{~~ ~l}

^U~_{+ m}

^{~B (F FB} )1(32)

k ⁿ , ⁿ

II 82 JOURNAL DE

PHYSIQUE

I N° 6

By

the _use of the

relationships (26)

and _a k

- k _{+ n}

substitution, (32)

becomes

2 ins

£ (P

AAA

(m,

k

FA (F

^*

FA ) u(

P

AAA

(m,

k

FA (F FA u$

~

+

£ (P

A~~

(m,

^k

) FA (F

^*

F~ ) u(

P

~AA

(m,

k

FA (F F~ ) u$

k

+

£ _(PB~~ (m,

^k

) F~ (F

^*

F~ ) u(

P

~~~

(m,

k

) F( (F F~ ) u$)

k

+

£ _(P~~A (m,

k

F~ (F

^*

F~ u(

P

~~~

(m,

k

) F~ (F F~ ) u$ (33)

~

By considering

the four

symmetry relationships

of the type

P~~~(m,k)=

P~~~ (m,

m k

), (33)

becomes

2 ins.

£ (C (m,

k

)

uI C

(m,

k

)

*

u$ (34)

k

with

C

(m,

k

=

(F

^*

FA (~

AAA

(m,

k

FA

₊ ~

ABA

(hl,

~

~B

+

(F

*

FB)(PBAA (m> k) FA

₊

PBBA (m, k) FB) (35)

If _we

neglect

the _u linear

dependence

of F

*,

which

gives

second order _terms in

displacement

in

(34), C(m, k)

becomes C

(m,

k

)

=

(FA F~ (xA ^(P

_~AA

(m,

k

F~

₊ P

~~~

(m,

k

FB)

xB

(P

~~~

(m,

k

) F~

₊ P

~~~

(m,

k

F~ )

,

(36)

which, by combining

with the

(83), (84)

and

(87) relationships

of

appendixB,

_may be

expressed

as

C

(m,

k

)

=

FB (F~ FB (PB~ (k )

_x~

x~)

+

(F~ F~

)~ P _~~~

(m,

m k

)

x~

(F~ F~

)~ P _~~

(m

k

(37)

Considering

that

uI

u$

=

£ _u~, (sin (2 wq'(k

_{m a} + _q~ sin

(2 wq'(m

k

)

_a + q~

))

q'

=

2 _{cos q~}

£ _u~,

sin

(2 wq'(k

_m

) a)

,

(38)

q'

the

expression (34)

then becomes

4 in _{cos q~s.}

£

C

(m,

k

£ _u~,

sin

(2 wq'(k m) (39)

~ ~.

Thus, by substituting

the second _term of

(25)

in

(23),

_one obtains

Yw

~

(s)

=

2 ins.

£ (F~ (F

*

Fi~

~

)

_u~

Fz~

~

(F F~)

_u~

~

~)

_exp

₍₂

I

wsma)

n m

which may be

expressed

as

Yw

~

(s)

=

4

ix~

_x~

F~(F~ F~)

_w cos q~ x

x s.

£u~, £ a(k)

sin

(2 wq'(k m) a)exp (2 iwsma)

q' _m,k

+ 4 I

(F~ F~)~

_{w cos q~ s}

£

_u~,

£ P~~~ (m,

m k

)

sin

(2 wq'(k

m

) a) exp(2

I

wsma)

q' m,k

4

ix~ (F~ FB

)~ w cos q~ s.

~j u~, £ P~~ (m k)

sin

(2 wq'(k m) a)

exp

(2

I

wsma).

q' _m,k

(40) Using

the

parity properties

of the

pair

and

triplet coefficients,

the third term of

(40)

vanishes and _one

gets

Yw

L

(s)

_{= 2 x~ x~}

FB (F~ FB )

w cos q~ s

£ _u~, £

_a

(k )

_exp

(2

I

wq'

ka

)

x

q, k

x

£ (exp(2

I

w(s

+

q')

_{ma exp}

(2

I

w(s q')

_ma

))

m

+

4(F~ F~)~

_{w cos q~ s.}

£ _u~, £ P~~~(m, k)

^sin

(2 wq'ka)

sin

(2 wsma). (41)

q' m,k

We _can check that the _s

dependence

of

(41)

is

antisymmetric

^about each

reciprocal

lattice node. The first _term

(which

is _{non zero}

only

when _s

=

q'± Q,

where

Q

^is a

reciprocal

lattice

vector)

is

analogous

to that obtained in section 2 and

originates

from the correlation between the

displacement

of _a molecule and the

position

of _one

impurity

A. This is the reason

why

it

depends only

_on

pair coefficients.

The second term, which contains

triplet coefficients,

is due

to the correlation between the

displacement

of _a molecule and the

position

of _two

impurities

A.

In order to evaluate the

respective

contributions of these _two terms to the diffuse

scattering,

let _us calculate Yw_~_

(s)

in _{a case} of _a

complete

disorder

(I.e.

for _an uncorrelated

system).

The

pair

and

triplet

coefficients _are

respectively equal

to P ~~

(0)

= 0 and P

~~

(m )

= x~ x~ for _m # 0

and _to

P~~~(m, 0)

= x~ x~,

P~~A (0, k)

=

PA~A (m,

_m

=

0 and

PA~A (m, k)

₌

xl

_x~ for

m # k # 0.

This

gives Yw

~

(s)

= 2 xA x~

FB(FA F~)

_w cos q~ x

x s.

£ _u~, £ (exp(2

in

(s

₊

q')

_ma

) exp(2

in

(s q') ma) )

q. _m

+ 2

x(

_xB

_(FA FB)~

" cos q~ s

£ _u~, £ (exp(2

in

(s

₊

q') ma) exp(2

in

(s q') ma) )

q' _m

(42)

where _{we can see} that if xA is small the second term is

negligible.

^In a

general situation,

_as

emphasized by Cowley [6],

the

magnitude

of _a term which contains

triplet

coefficients is difficult _to estimate. However in the _case of the

size-effect,

which also involves such

high

order terms,

Welbeny

shows

by optical

transform methods

[9]

that these terms have _a small influence _on the diffraction pattem. On

physical grounds,

in the framework of _our

model,

it

l184 JOURNAL DE

PHYSIQUE

I N° 6

seems

unlikely

that the second term of

(41)

can

significantly

contribute to the diffuse

scattering

if the average distances between

impurities A, a/xA,

is

larger

than the correlation

lengh f

of the

displacement

_wave induced

by

_an

impurity.

When the second term of

(41)

is

neglected,

the diffuse

scattering

is thus

given by l'Diff(S)

" YS.R.O.

(S)

_{+ YW.L.(S}

,

with

Ys.R.o. (s)

_{= xA}

xB(FA FB

_)~

i

_a

(m )

_exp

(2

ins

ma)

m

and

Yw.L. (S)

= 2 XA XB

FB (FA FB)

_{" cos q~ s.}

£ _u~, £

_a

(k

_exp

(2

I

wq'ka)

_x

q' k

x

£ (exp (2

in

(s

₊

q')

_ma

exp(2

in

(s q') ma) ) (43)

m

Yw.L.(s)

has _{a non-zero} value

only

if _{s ±}

q'

=

Q,

which becomes

finally

Yw.L. (Q

+

q')

= 2 xA x~

F~ (FA F~

_{w cos q~s}

_u~, £

_a

(k ) exp(2

I

wq' ka)

k

Yw.L. (Q

+

q')

= + 2 xA xB

FB (FA FB)

_{" cos q~s.}

_u~, £

_a

(k) exp(2 wq' ka) (44)

k

4. Discussion.

4,I RELEVANT _{PARAMETERS OF} THE WHITE LINE EFFECT.

First,

let _us recall that the

previous

results _are obtained for _an

alloyed

chain I.e. for _a one-dimensional model. The

generalization

of the calculations _{to a} three dimensional lattice is

straightforward

if _we suppose that the disorder and the distortion _are uncorrelated from chain _to chain. Each chain scatters

independently

and the diffuse

scattering intensity simply

adds. The

expressions (42)

and

(43)

^remain

unchanged.

Let _{us now} discuss the

significance

of these formulae. The diffuse

scattering intensity

calculated consists of two terms. The first _one Ys_~_o_

(s)

describes _a monotonic

scattering

with smooth variations in

reciprocal

_space

depending

_on the

degree

of order of the system. ^This

scattering

will appear as a modulated _«

background

_{» on}

X-ray

pattems. ^Due to the

pinning

of

displacement

waves to the

impurities,

new effects _occur via the second _term Yw_~_

(s).

If _we

assume that the _wave vector distribution of the

u~,

^{has a small width}

(2 wqf

» I

condition),

this effect will be observed

only

in the

vicinity

of the

Q

_q and

Q

+ q

reciprocal positions

in the form of diffuse streaks. If the

quantity F~(FA F~)

cos q~ s. u~> is

positive (negative)

the white lines will appear for _{wave vectors}

larger (smaller)

than those of the

Bragg

reflections.

As _an

exception,

_{if q}

=

a*/2,

the black and white diffuse lines

overlap

and Yw_~_

(s)

no

longer

contributes. Diffuse lines of weaker

intensity,

^due to the second order terms in

displacement

not considered

here,

will be observed.

Let _us remark that Yw_~_

(s)

contains the xA x~

(FA F~ ) £

_a

(k exp(2

I

wq' ka) factor,

k

which also _enters the

expression

of the short range order diffuse

scattering.

This

expected

result shows that the white line effect _occurs

only

for

scattering

vectors where the Laue

«

background

_» is _{non zero.}

The most

original

feature of this

effect,

due to the

linearity

with respect to the

amplitude

of the

displacement,

is the relevance of the

signs

of the

F~ (FA F~ ),

cos q~ and _s u~> factors _on the total scattered

intensity.

Let _us examine their effect

successively.

Bragg spot

a

fu

-u

u transverse _u longitudinal

Fig.

6. Locations of white and black diffuse lines for _a transverse modulation (left) and _a

longitudinal

modulations

(right).

The grey

background

_{represents a constant} Laue

scattering.

The

quantity F~(F~ F~)

cos q~ is

negative

in the upper

figures

and

positive

in the lower _ones.

The effect of the _s

u~.

^{term can be understood}

by studying

two

simple

_cases sketched in

figure 6,

when _a constant Laue

scattering

is assumed. If the distortion _wave has _a

longitudinal polarization (I.e,

if

u~,

^is

parallel

to

q),

each diffuse line is continuous and does _not

present

any inversion of

intensity.

On the other hand if the distortion is _transverse

(u~, perpendicular

to

q)

the _s, u~, ^term

changes sign

when the

scattering

vector s is

perpendicular

to u~.. ^{At these}

inversions,

the diffuse line

intensity

vanishes. This feature

provides

_{a way} to measure the

polarization

of the

displacement

_wave

by looking

at the locus of the inversions of

intensity.

To

study

the influence of the other terms _we shall consider the _case of _a

longitudinal polarization.

The

quantity F~(FA F~) only depends

on the _nature of the

scattering species.

^If

atoms were

considered,

this factor will

change sign according

to the

weight

of the

impurity

^A

with

respect

to the _atoms B.

Everything being equal,

_a lattice in which the A atoms are the

impurities gives

a diffuse

scattering

_pattem with white lines

positions

inverted with respect to the

pattem

^{obtained from the} same lattice where the B atoms are the

impurities.

^If the

scattering species

_are

complex molecules,

_we have to compute

carefully

the

F~(FA F~) quantity,

which _can

change sign

when _s varies. We shall _come back to its effect in the next

section.

The influence of the

phase

term cos q~ has

already

been discussed in section 2.I. If the

phase

of the

displacement

_wave has the

special

values _± ar/2, the white line effect

disappears

and second order _terms become

dominant,

_as in _a pure

compound.

If the

phase

differs from these _two

values,

the white line's

position depends

_on the

sign

^of cos q~. Note that _as

expected

from _a kinematic _treatment of the

diffraction,

_we do not obtain information _on the absolute

phase

of the modulation _wave, but _on the relative

phase

between the distortion and the

composition

_waves.

Indeed,

_as

emphasized before,

this feature is also present ^{in the size-}

effect and

permits

_one to obtain local information of the deformation field around _an

impurity.

4.2 EXPERIMENTAL _EVIDENCE OF THE WHITE LINE EFFECT. Now _{we can come} back to the

X-ray pattems presented figure

I in the introduction.

HMTSF-TCNQ,

whose

constituting

1186 JOURNAL DE PHYSIQUE I N° 6

molecules _are

displayed figure

7,

crystallizes

in the

C~/~

^monoclinic space group with

a =

21.99

I,

_b

= 12.573

I,

_C

= 3.89

I

_and

_p

=

90.29°

[10],

_c

being

the chain axis. This

compound

exhibits _a

large regime

of lD fluctuations at 2 k~ = 0.37 c* visible in the form of diffuse lines

(Fig, la),

which condense into very weak satellites reflections at the reduced

wave vector

(0, 0.37, 0)

in the IO K~25 K temperature range

II.

The direction of

polarization

e of the lattice distortion in the

regime

of ID fluctuations has

already

been obtained

by Yamaji

_et al.

[ll]

from _a fit of the

intensity

of the _± 2 k~ diffuse

lines,

measured _on

X-ray

photographs

similar _to those of

figure

I. This direction _was found _to be _e

=

0,19(8)

a +

0.06(15)

b _{+ c.}

A solid solution

(HMTTF

_)~

(HMTSF )1_~ TCNQ

_can be obtained

by alloying

the HMTSF molecule with its sulfur

analogue

HMTTF also shown

figure

7.

Figure

16

presents

the

X-ray pattem

obtained at 25 K from _an

alloy

with _a nominal concentration of _x

=

0.25

(in

fact

x =

0.05 from

microprobe analysis).

It exhibits _two essential features, First, a modulated diffuse

background

is

visible,

with maxima located _on three broad diffuse

stripes (Fig, lb) running along

the c* direction and

secondary

weak maxima between the

f

= constant

layers

of

Bragg spots.

^{At the}

(f

₊

_0.37)

_c*

_scattering

_vectors, white lines appear at the expense of this

background,

without any inversion of

intensity along

the b* direction. The normal black

diffuse lines are observed at the

(f _0.37)

c*

scattering

vectors. These lines _are thin with half-width at half-maximum

(H.W.H.M.)

of 0.007

I-~.

In this _case, the short range order diffuse

scattering

is

proportional

to

x(I -x)

(F~MTT~-F~M~S~)~.

^The

quantity (F~MTT~-F~M~S~)

^has

complex

variations in

reciprocal

_space

which _are

mainly

due to the

Se/S

atomic substitution _on the donor molecule and

which,

to a

first

approximation, corresponds

to the Fourier transform of _a

rectangle

formed

by

« atoms »

of

scattering

factors

(fs fs~). Computer

simulations _are then needed _to evaluate the _s- variations of the Laue

scattering. They

show

that,

for the orientation of _our

crystal,

the

« monotonic _» Laue

scattering

forms

large stripes along

the c*

direction,

_as observed _on the

X-ray pattem

^of

figure

16. The

slight

increase of the diffuse

scattering midway

between the

f

= constant

Bragg layers

is the

signature

^of a weak short range ^{order in the} c

direction, probably

due to

repulsion

between

impurities.

In the framework of _our model, _{we propose} that around each HMTTF

impurity,

_a

periodic

lattice distortion is

pinned,

with the _q

= 2 k~ = 0.37 c* _{wave vector} and _a correlation

length

of

f

=

60A _{(obtained by assuming}

a Gaussian

experimental

resolution and _a Lorentzian

shape

^for the

u~,

correlation

function).

On the

X-ray pattem

of

figure 16,

which shows _a

part

of the

reciprocal

_space with small

h,

the white lines do not

present

_any inversion of

intensity.

Therefore the

polarization

of the

Se Se

Se Se

HMTSF

I$I

~m~

~ l$I

°~"

HMTTF TCNQ

Fig.

7. Molecular _structure of HMTSF, HMTTF, TCNQ.