A Diffraction Method of Study of Thermal Quasiorder in a Finite Two-Dimensional Harmonic Lattice

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A Diffraction Method of Study of Thermal Quasiorder in a Finite Two-Dimensional Harmonic Lattice

P. Aranda, B. Croset

To cite this version:

P. Aranda, B. Croset. A Diffraction Method of Study of Thermal Quasiorder in a Finite Two- Dimensional Harmonic Lattice. Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1213-1222.

�10.1051/jp1:1995192�. �jpa-00247130�

J. Phys. I France 5

(1995)

1213-1222 SEPTEMBERI995, PAGE1213

Classification Physics Abstracts

61.10Dp 68.35Ja

A Diffraction Method of Study of Thermal Quasiorder in

_a

Finite Two-Dimensional Harmonic Lattice

P. Aranda and B.

Croset(*)

Groupe de Physique des Solides, URA CNRS 17, Tour 23-13, 2 place Jussieu, 75251 Paris cedex 05, France

(Received

20 February 1995, revised 8 April 1995, accepted 17 May

1995)

Abstract. Due to the non-existence of long-range order, the diffraction peaks of 2D-solids

are considered _to have _a power-law shape

q)~~.

Taking into account the limite

size effects and calculating the powder _{average, we} show that this power-law behaviour appears Orly for high _qp

and then for very small intensities. It is therefore quite dificult and hazardous to characterise the quasiorder by _usmg this asymptotic behaviour. Although the shape of the central part

of the peak cannot be used to characterise the quasiorder, _we show that, for

a fairly good resolution, it is possible to deterrnine _~ using this central part. This deterJnination _cari be dore irrespectively with the other details of the system by comparing trie peak width to its value at low temperature, i-e-, at low value of _~. By _usmg two diffraction peaks, _{we propose} the simple

relation:

~(QBI )/Qài

"

~(QB~)/Qà~

_{as a} check of the two-diJnensional quasiorder.

Among

the extensive theoretical and

experimental

studies of order in two-dimensional

(2D)

sohds

[1-6]

^{and of} two-dimensional

melting [7-14] significant

contributions _were

brought by

diffraction

techniques. Nevertheless,

in contrast with three-dimensional

(3D) studies,

most of the 2D diffraction works _are based _on the

investigation

of trie first diffraction

peak only.

Whereas this restriction has important ^technical _reasons

(low intensity,

_presence of substrate

peaks

at

high Q),

it _seems quite

surprising especially

in comparison ^with 3D usual results:

1) On the _one

hand,

the thermal activation of

phonons

leads _{to a}

non-perfect long-range quasiorder.

It results for _an infinite

crystal

in the

replacement

of the

à(Q QB) singularity by

_a

power-law singularity

_(qp(~~~

= (Q

QBÎ~~~,

and

thereby

in _a

broadening

of trie 2D diffraction

peaks [1,2]

^{for finite}

crystals.

The linear

dependence

of _{~ on} the _square of the

peak position QB predicts

that the _structure factor of _a 2D solid should

look,

at least

qualitatively,

like _a 3D

liquid

_one. To _our

knowledge,

_no attempt to check this

prediction

has been made.

ii)

On the other

hand,

the

liquid (10) peak

width has often been

interpreted

_as the inverse of

a coherence

length,

L

[10,13];

such _an assumption

implies equal

width for ail the

peaks,

which is in strong contradiction both with the usual

shape

of the structure factor of 3D

liquids

and

with the

fitting

of the

intensity

of the

liquid (10) peak by

_a ^Ll~~ law.

(")

corresponding author

© Les Editions de Physique 1995

iii) Lastly,

Gunther et ai. [3],

Weling

et ai. [4] and Dutta et ai. [5] have calculated the influence of finite size eifects and their results show that the

power-law shape

cannot be

easily verified, being

valid

only

for

large qpL.

For these _reasons, it _seems

quite

useful to

study

at least _two diisraction

peaks during

2D

melting

and to check trie linear

dependence

of _{~ on}

Qà.

We will examine how _{one can} determine

~

independently

of trie precise nature of finite size eisects.

By calculating

the

iuteusity

diisracted

by

_a

powder

in various cases, we shall show

firstly

that trie

asymptotic

behaviour of trie

peak profiles

_cau be used neither _{as a} check of trie thermal

quasiorder

_{nor as a measure} of _~,

secoudly

that the thermal

quasiorder

leads _{to a}

broadeniug

of the central part of the

peaks

which _can be used in trie determination of _~

independently

of trie

precise intensity profiles.

For _a 2D harmonic

crystal

with

periodic boundary conditions,

trie diifracted

intensity

_can be written _as

1(Qp)

₌

~jexp(-iop (Iti Rj))

_exp <

(Qp.u(Rz Rj))~ >l

~~

2

where

Qp

is trie

projection

of trie diffusion vector

Q

in trie

crystal plane,

~1(r) ^{is trie relative} thermal

displacement

of _atoms distant from _r.

To be able _to

perform

_an

analytical calculation,

_we must _assume that

I(Qp)

_can be wntten as a sum of

peak profiles

centred around each vector

QB

of the

reciprocal

lattice:

Ijop)

=

LIQ~ jqp)

=

L L expj-iqp.jJt~ Rj))

_exp

(-j

_<

(QB.ujRz Rj))~ >)

QB QB 1,J

with qp =

Qp QB.

This

assumption

is

questionable

for broad diffraction fines but it allows the continuum ap-

proximation

for the _two direct lattice _{sums so} that

IQ~

is written

(for

the sake of

simplicity,

we will _omit the index

QB

for

IQ~

_):

I(qp)

=

~°~) / / _{exp(iqp r)} f(r)c(r)dr,

L

where

f(r)

is the

generalised Debye-Waller

function,

c(r)

describes the cut-off in the direct space and accounts for the number of the

pair

of

crystalline

sites

separated by

_r. Within _a

Debye

model and

neglecting

the diiference between transverse and

longitudinal

sourd [6],

f(r)

is written:

kn

f(r)

_{= exp -~}

L ^d~~~~

_[1

Jo(~r)]

with _~

=

[kBTQ[(3~1+

_À)]

/[4àr~1(2~1+

À)], _~1 ^and are the Lame coefficients.

In order to calculate

I(qp),

both

Weling

et ai. and Dutta et ai. substitute for

f(r)

its

asymptotic

behaviour for

large

_r. For _an infinite

crystal

this

"asymptotic" approximation gives f jr)

=

jr laD)~~

with _aD the lattice

spacing. Together

with

c(r)

= 1, this

approximation

allows _us to write

I(qp)

=

qp~+~,

^{the well-known}

^{power-law shape.}

^For an infinite

crystal

the

asymptotic approximation

is

quite

valid for distances greater ^{than second}

neighbour.

For _a limite

crystal,

limite _size eifects have consequences on the _one hard _on the behaviour of

f(r)

and _on the other hard in

c(r)

which is _no

longer

constant.

Doing

the

"asymptotic"

approximation, Weling

et

ut.,

then Dutta et ai. have shown that the variation of

c(r)

dominates the behaviour of

I(qp)

and that

f(r)

_may be

replaced by

its value for _an infinite

crystal.

To

produce

_more realistic fine

shapes

than

Weling

et

ut.,

Dutta et ut. made the choice

c(r)

₌

exp(-r~ /L~),

_a

simple

but

surprising

choice _{as we} will _see. This choice allows them to

N°9 A DIFFRACTION METHOD OF STUDY OF 2D QUASIORDER 1215

show that the

approximation

of

f(r) by

its value for _an infinite

crystal

is reasonable. In other

words,

except _near the

edges,

the atoms

really

present ^{in the}

crystal

of dimension L vibrate like the atoms of _a cluster of dimension L immersed in _an infinite

crystal.

This conclusion _may

seem

surprising

when _one thinks of the existence of surface modes in _a 3D

crystal.

A correct calculation without

periodic

conditions of the vibration modes will exhibit

edge

modes. These

edge

modes _are Dot due to the limite size but

only

to the existence of _an

edge

and will _appear for _a semi-infinite

crystal.

At the very most,

they

have the

only

consequence ^to reduce the

apparent size of the

crystal.

Approximating f(r) by jr laD)~~, I(qp)

is written

Iiqp)

=

iLlaD)~-~rii n/2)i~/fi)-~ _ifiii

_~J/2;1;

_-q(L~/4)

where ifi is the confluent

hypergeometric

function.

Several remarks _must be

pointed

out:

For q = 0,

1(qp)

has _a Gaussian

shape

because of the choice of

c(r).

Within the

logarithmic approximation,

the whole

dependence

_{on aD appears} in the in-

tensity prefactor.

Within the

logarithmic approximation,

the

I(qp) profile

is _a universal function of

qpL.

The relative

broadening

of this

profile compared

to its value at q = 0

(T

= 0

K)

is

independent

of

L,

_as

long

_as the

logarithmic approximation

is valid. This

paradoxical

result is

mainly

due to the

logarithmic divergence

ofu~

jr)

in _a small

crystal,

the farthest

pairs

_are ^{better correlated than} in _a

big

_one but

they

_are in

larger

proportion

relatively

to ail other pairs.

The

asymptotic

^{form of} the

degenerate hypergeometric

function allows _us to find the

qp~+~

^law established

by Imry

et ut. Nevertheless this behaviour is observed

only

at

large qpL

and with intensities _near 1% of the _maximum.

In his seminal paper

[15],

^{Warren chose} a Gaussian approximation ^for

c(r),

_a choice also made

by

Dutta et

ut.;

this _is Dot the

only possibility leading

to

simple analytical

calculations.

However,

_in

spite

of its

simplicity,

_a Gaussian

shape

is Dot _a

good approximation

either for

sin~Lz/sin~~,

_or for its average on diiferent

L,

since it does Dot accourt for the

qp~ asymptotic

behaviour of these "raturai"

profiles.

This

profile

_appears to be

satisfactory

in diffraction

techniques only

when the resolution is _worse than the limite size eifect

broadening. Generally,

in the opposite _case where raturai

profile

is

observed,

_a Lorentzian

shape

is

preferred

_{as an} approximation.

To discuss the consequences of the choice for

c(r),

_we will calculate the

intensity

observed in

studying

diffraction

by

_a

powder

of 2D

crystallites.

To obtain

J(Q),

the

intensity

diifracted

by

_a

powder,

two

integrations

_on

Q

orientations must be

performed:

«/2 2« «/2

JIQ)

"

1/Q / _dll / _dTIlQp)Qp

"

1/Q / _dllBlop)

where _~1 is the

angle

of

Q

with the normal to the

crystal plane

and _T the

directing angle

of

Qp.

The

integral

on ~1

gives

the

typical

saw-tooth

shape

of 2D

peaks

in

powder technique

_[15] and must

usually

be

numerically performed.

The

integral

_{on T cari} be

performed analytically by

doing

the

tangential approximation

which is usual in

crystallography

[16]: ^the

integral along

the circle of radius

Qp

^is

replaced by

the

integral along

its

tangent.

^This

approximation

is valid if

ôq

is small with respect to

Qp,

1-e-, if q is trot too close to 2. Then:

B(qp)

₌

(uD)~ Î

ce

exp(iqpr)r~~c(r)dr

While

I(qp)

is _a two-dimensional Fourier

transform, B(q~)

is _a one-dimensional _one. In

fact,

the

experiment

does not

give

direct _{access to}

J(Q)

but to

K(Q),

the convolution of

J(Q)

with the

experimental

resolution

R(Q).

K(Q)

=

J(Q)

~p

R(Q)

=

i/Q

£~~ ^(qp)d~lj

^~p

^R(Q)

o

By reversing

the order of

integration

_{on q} and

~1 and

by noting T(r)

the inverse Fourier transform of

R(Q), K(Q)

is written:

K(Q)

₌

1/Q ~~~(B(qp)19

R(qp))dtl= 1/Q ~~~

M(qp)dtl

with

ce ce

M(qp)

=

(aD)~ cos(iqpr)r~~c(r)T(r)dr

=

(aD)~ cos(iq~r)r~~W(r)dr

Î Î

The

profile

of the diifracted intensity

depends

Dot

only

_{on q} but also _on

W(r)

which certains at once the limite size eifects and the limite resolution _ores. In the

following

part, we will try to propose a method

allowing

the _measure of _{q, as}

independent

of

W(r)

details _as

possible.

We _cari

already

notice that the

analytic shape

of

M(qp)

is very similar to that of

I(qp).

Ail the remarks _on the

universality

in

qpL

of the

I(q~) profile apply

to

M(qp),

within the

validity

limits of the

approximations.

Because _a

least-squares

fit of

K(Q)

at low temperatures ^{is the} only _way to obtain

W(r),

its

shape

for small _r

depends mainly

_on the far tait of

M(qp,

_{q =}

0)

and therefore is _{in some} way

arbitrary

from _an

experimental point-of-view.

Nevertheless for

large

values of _q the whole

peak profile depends

_on this

shape

at small _r, because of the

rapid

decrease of r~~. To circumvent this

difficulty,

_we have

analytically

calculated the _curves

M(qp,q)

for three choices of

W(r),

we have

numerically

determined the relative

broadening

à'

=

à(q) /à(0)

for each choice and

we have looked for _a universal transformation

g(à')

almost

obeying

the relation

g(à')

_{m q} for any

W(r).

a) W(r)

=

exp(-r IL),

the "Lorentzian case".

Such _a choice

leads,

for _q

= 0, to _a Lorentzian

profile:

M(qp,

_{q =}

0)

₌

L/(q(L~

+

1).

As _we have _seen

before,

this

profile

is the raturai _one, 1-e-, the

profile

obtained in the absence of thermal vibrations

(q

₌

0)

^for a

disperse powder

of 2D

crystallites

^of

general shape. However,

the choice of _an

exponential

for

W(r)

is also

suggested by

Nelson et ut. for the hexatic

phase I?i.

For _{q non zero, we} have:

~~QP'~Î~ (qÎÎÎ

₊

ÎÎÎ2-1/2

^~°~

^((~ 4)~~~~~~(QP~))

P

~

b) W(r)

₌

exp(-r~ /L~),

the "Gaussian" _case.

N°9 A DIFFRACTION METHOD OF STUDY OF 2D QUASIORDER 1217

Such _a choice

leads,

for _q

= 0, to a Gaussian

profile:

M(qp,

_{q =}

0)

=

exp(-q(L~ /4).

As _we have _seen

before,

this

profile

is the instrumental one, 1-e-, ^the

profile

obtained in the absence of thermal vibrations when resolution is broader than the raturai

shape. Although

every

experimentalist

tries to avoid such _a

situation,

_we have studied this _case after Dutta et ut.

For _{q non zero, we} have:

M(qp, q)

=

r(1/2 q/2)L~~~

iFi

Il /2 q/2; 1/2; -q(L~ /4).

c) W(r)

=

il

_{+ r}

IL) exp(-r/L),

the

"square-Lorentzian"

_case.

Such _a choice

leads,

for _q

= 0, to _a

square-Lorentzian profile:

M(qp,

_q

=

0)

=

2L/(q(L~

+

1)~.

This

shape

is very similar to _a Lorentzian except in the tait and _cari be

regarded

_{as an} intermedi- ary

shape

between Lorentzian and Gaussian. For

good

resolutions,

W(r)

is _an autocorrelation

function: its discontinuities for _r

= 0

depends

_on the nature,

abrupt

_or trot, of the coherence

disappearance.

This is

why,

to accourt for less

abrupt

situations than

crystallites

of finite

size with

Sharp edges,

_we have chosen _a function without discontinuities of the derivative for

r = 0 but

preserving

the

exponential

behaviour for

high

_r. This is the _case of the function

W(r)

=

il

₊

r/L) exp(-r/L).

For _{q non-zero, we} have:

~~~P'~~ q)ÎÎ

+

ÎÎÎ~ ^Î/2

^~°~

^((~ 4)~~~~~~(QP~))

+ ~~

iq(ÎIÎ i)/~ÎÎ

^{~ COS}

ⁱ¹² ~l)arctallioPL) i

For ail these cases, we cari see that:

The critical value of _q, for which the

broadening diverges,

is q = 1 instead of 2 for

I(q~).

For these values of _q, the

tangential approximation

_{may seem}

questionable

but the

validity

of the

approximation depends

_on the width of

I(qp)

and trot _on the width of

B(qp).

Nevertheless for small

L,

the

tangential approximation

should be invalid.

For every value of _{q trot}

equal

to zero, the

asymptotic

behaviour of

M(qp, q)

is

qj~~.

For q = 0, there is _a

discontinuity

in the

asymptotic

behaviour

(as

_an

example

the

asymptotic

behaviour of

M(qp,

q =

0)

^is

qp~

^{in the Lorentzian}

case).

This

discontinuity already

existed in the

profile

calculated

by

Dutta et ut.: their

profile

had _a Gaussian

behaviour for _q

= 0 and _a power law

shape

_{for q non-zero.} In practice, this result _means the

following:

the

validity

domain of the

asymptotic

^behaviour in

qpL

is

pushed

_away to

infinity

for the low values of q.

In the central part ^of the

peak,

the thermal

quasiorder

leads to a

broadening

of the

peak.

While

hardly

^visible _{on a}

log-log plot,

this

broadening

is

easily

noticeable _{on a} linear

plot

and is

strongly dependent

_{on q as cari} be _seen in

Figure

2. This

broadening

must be taken into accourt in order _{to measure} the coherence

length

of _a hexatic _or

liquid phase

from the

(10) peak

width and invalids the direct interpretation of this

peak

width

as the inverse of coherence

length.

The increase of the

broadening

with _q and then with

0.

z~

'( _O.Oi

'

j

O.OOI

~'~~~Î _{o i oo i ooo}

q~L

Fig. l. Comparison between M computed in the Lorentzian _case and the asymptotic behaviour for ~ = ù-1 in

a log-log plot.

M(qp,~

= ù-1); ): asymptotic behaviour

(qpL)~°'~;

): best fit of

M(qp,~

= ù-1) between qpL ₌ 10 and qpL ₌ 10ù by _a

(qpL)~"

law with

ce = 1.19

(such

_a fit provides _{a wrong} and unphysical value of

~).

QB

allows _us to expect

liquid peaks

of

increasing

width with

QB

in accordance with 3D results.

No dear characteristic of the

peak shape

in its central part _seems to be attributed to thermal

quasiorder.

A tentative way to characterise the 2D thermal

quasiorder

and to

measure q should be to _use the

qj~~ asymptotic

behaviour of

M(qp, q)

since this behaviour

does not

depend

_on

c(r).

But _our

profile

calculations show that it _seems

problematic

to characterise the thermal

quasiorder by

the

asymptotic

behaviour of the

profiles.

As

can be _{seen in}

Figure

1, the

validity

domain of the

asymptotic

behaviour starts _on and after

qpL

₌ 100.

Moreover,

in _a

log-log diagram,

_a

quasi-linear intermediary

behaviour

leading

to _a wrong evaluation of _{q appears.}

Lastly,

_{as can} ^be _seen in

Figure

2, the _sum of

a Lorentzian and _a fine of base fits well the

profiles

obtained

induding

thermal

disorder,

for low q.

(The

fine of base does not exceed 7% of the _maximum intensity for _q

=

0.2).

This

diiliculty

to

identify

without any

ambiguity

the nature of the disorder from the

shape

of _{a unique} diffraction

peak

_was also mentioned

by Heiney

et ut.

[loi

in their

study

of the

melting

of Xe

monolayer

adsorbed _on ZYX. These authors obtained _as

good

results

with Lorentzian

profiles

_as with Dutta's

profiles.

The very low value of the intensity in the

asymptotic

domain

explains

the strong

dispersion

of their _{q measure.}

This

diiliculty

_suggests

measuring

_q

by

_use of the central part ^{of the} diffraction

peaks

and

characterising

thermal

quasiorder by

_use of two diffraction

peaks

and

by

the check of the linear

dependence

of _{q on}

Q[.

This method needs the observation of _two

peaks

for which the relation:

1(QBI)/1(QB2)

"

QÎI/QÎ2

should be verified. It also needs to find _a relation

g(à')

between the relative

broadening

and _q.

To find _g, the first step is to look for the

asymptotic

behaviour of the à'

divergence

for _q

= 1.

It is easy ^to see, both in the "Lorentzian" _case and in the

"square-Lorentzian"

_case, that:

lim (1

q) In(à'~ +1))

= 2In 2

~-i

N°9 A DIFFRACTION METHOD OF STUDY OF 2D QUASIORDER 1219

0.8

1

0.6 ~~~'~

~l=0.2

(

0.4

0.2

°0 2 4 6 8 10 12 14

q~L

Fig. 2. Comparison between M computed in the Lorentzian _case and the best fit with the _sum of

a Lorentzian and _a fine of base, for three dilferent values _{~: ~}

= o, _{~ =} ù-1 and _~

= 0.2. ):

M(qp, ~)

for _~

= 0, _{~ =} ù-1 and _~

= 0.2 in the Lorentzian case; ): ^{best fit with the} sum of _a

Lorentzian and _a fine of base.

0.8

é ^~~ ,v"Î"

~

0.4

_.1°

o

_~ u

0.2 ^~ .°'°

0.2

0A

0.6 0.8

~

Fig. 3. Companson between

g(~)

and ~ for three dilferent _cases.

(-

_. -); Lorentzian _case;

(-V-)

square-Lorentzian _case;

(-o-):

Gaussian _case; ): _g = ~ fine.

For q = 0, we have

by

construction à'

= 1 and (1

q) In(à'~

_{+ 1)}

= In 2.

If the transformation _g is chosen to be

g(à')

=

~~~ ~

+ l then the relation

In(à'2

₊

1)

+ In 2

g(à')

= q is

obeyed

for _q

= 0 and _q

= in the last _{two cases} mentioned.

Figure

3

displays

g(à')

_{as a} function of _q for the latter and the "Gaussian" _case. The

linearity

is very

good

for the whole domain in the "Lorentzian" _case while _{it is not too} bad for other _cases. It then appears that _we achieved _our purpose of

finding

_a

quasi-universal g(à')

function and that this

measure is _very

good

for

M(qp,

_{q =}

0)

close to a Lorentzian.

The method that _we

proposed

for _{measunng q} suifers from _two defects:

firstly

it requires

a measure of the

peak

width at

suiliciently

low q in order to define the relative

broadening;

secondly

it does _Dot exhibit _any

particular

feature _{in a}

single profile allowing

the thermal

quasiorder

to be characterised. This second defect is _{common in}

crystallography Ii?i,

and is

generally

_overcome

by comparing

the width of two diiferent

peaks.

This

comparison

is very

simple

_{in our case} since it should lead _to:

g(/~~) l(QBI) QÎI

g(/~~)

SÎ(QB2)

QÎ2

To

identify

trie thermal

quasiorder,

_{we propose} to use

g(à'),

the _measure of _q, for two

peaks

and to

verify

this relation.

In conclusion several

points

must be discussed:

i)

^{Dur whole}

study

_uses the universal

dependence

of the

profile M(qp)

on

qpL, irrespective

of the value of L. It may seem worthwhile both _to

explain

this

universality

and to

study

the

approximations leading

to such _a result.

To understand the

dependence

of the

profile M(q~)

_on

L,

the best _way is to _come back to its Fourier transform representation:

M(q~)

=

/ ~ Cos(iq~r)(r/uD)~~W(r)dr

By using general properties

of Fourier

transforms, M(qp)

can be written _as the convolution of two functions: _one

being

^the

intensity profile

for _a

powder

of infinite

crystals presenting

quasiorder

observed with _an infinite resolution

Mr~(q~)

^{and the other}

including

ail the eifects of finite _size and finite resolution

T(q~).

M(qp)

=

Moe(qp)

19

T(qp)

with

~ ~

Mr~(qp)

₌

Î cas(iq~r) f(r)dr

₌

Î cos(iq~r)(r/uD)~~dr T(Lqp)

₌

/

ce

cas(iqpr)W(r)dr

o

Scaling

arguments

clearly

show that the _power law

dependence

with

r/uD

of

f(r)

results

in _a power law

dependence

_{on uDqp} of

Mr~(qp).

On the other

hard, W(r)

and

T(qp)

are

characterised

by

_a coherence

length

L

(or

_a

peak

width

àqp

=

1IL)

and _a

shape

_SO that

T(qp)

may be written _as

S(Lqp).

Because of the _power law

shape

of

Mr~(qp)

and of the

properties

of the convolution

product,

it is dear that the

shape

of

M(qp)

in _a

1IL

scale

depends only

on

S(Lqp)

whatever the ratio

LlaD.

Let _us take two

examples

to illustrate this fact: _on the

one

hand,

_suppose the

crystal

size L~r to be quite

large,

the intensity

profile

will be the _same in _a qp

/àQ

scale _as

long

_as the resolution

àQ

is

large compared

with

1/L~r

and _a resolution improvement ^will not make easier the

study

of the 2D

quasiorder;

_on the other hand, _suppose the resolution to be quite

good,

the

intensity profile

will be the _same in _a

L~rqp

^scale as

long

as the

crystals

size is small

compared

with

1/àQ

and _an increase of the

crystals

size will not make the

study

of thermal

quasiorder

easier. These assumptions ^have limitations and

they

are correct in the limit of

validity

of the

approximations

that _we have made in _our

analytical

calculation. These approximations _are

mainly

three: the

asymptotic approximation

^for the thermal motion of atoms, the

tangential approximation

_m the calculation of the

powder

_average and the

separability

of the diffraction

peaks.

Ail these

approximations

_{are wrong} for _very broad

peaks,

1-e-, ^for low L _or very bad resolution. In practice, the asymptotic

approximation

is

good

for _any

crystal

size _smce

f(r)

is _near its

asymptotic logarithmic

branch for

rlaD equal

to 2. This

large validity

domain of the asymptotic approximation is the _same in 3D calculations for which

f(r)

reaches its asymptotic constant value for quite ^{low values of}

rlaD leading

to a validity of

N°9 A DIFFRACTION METHOD OF STUDY OF 2D QUASIORDER 1221

the well-known

Debye-Waller

^formula for _any

crystal

_size. The

tangential approximation

_may be renounced at the price of _more detailed calculations. But to _renounce the

separability

of the

peaks

is _{to renounce} the concept ^of

quasiorder

itself.

ii) Though

the

g(q)

function that _we

proposed

_seems to be

quite satisfactory

_{as a measure}

of _q for _any low temperature

profile (Lorentzian,

_square

Lorentzian, Gaussian)

when examined in the whole domain of variation [0,

Ii,

the

discrepancy

to

linearity

_rnay be

problematic

when

verifying

the

q(QBI )/n(QB~

"

Qà~ /Qà~

^law in the "Gaussian" _case: for

Qà~ /Qà~

" 3

(which

corresponds

to the first two

peaks

of _a

hexagonal structure), g(q(QBI ))/g(n(QB~ ))

take values between 2.5 and

5.8,

its average value

being

3.9. Three _{ways may} be used to overcome this default.

Firstly

_a numerical determination of _a

special

measure of _{q in} the "Gaussian" _{case may} be clone

using

the theoretical

profile

_we have determined

ifi Il /2 q/2;1/2; -q(L~/4),

this method _seems to be realistic

only

in the _case of _very bad resolution.

Indeed,

in the _case of _mean

resolution,

the _use of _a Gaussian

profile

at low _q will

neglect

the

high

_qp tait of the

intensity profile

due to finite size eifects and this

high

_qp tait will have consequences on the

broadening

for

large

values of _~/.

Secondly,

when _ever

possible,

trie ratio

g(~/(QBI ))/g(~1(QB~ ))

should be studied _{on a} wide _range of

~/ values and its

good,

_even not

perfect,

fit with its theoretical

value, Qà~/Qà~>

may be considered _{as not} fortuitous and therefore _{as a}

good

test of the thermal

quasiorder

when

compared

^with other disorders

(paracrystalline order,

_pure finite size

eifects, etc.). Thirdly

because the q measure that _we propose is quite

good

bath in the "Lorentzian"

case and in the

"square-Lorentzian"

_{case, 1-e-,} when the finite size eifects dominate the

profile,

the determination of _q and the check of the thermal

quasiorder

will be

highly

facilitated

by

a resolution

improvement.

This result is

quite physical

and

stimulating

^for

experimentalists.

Nevertheless,

the

fairly good

agreement ^of

gin)

with _q in the _pure "Gaussian" _case

dearly

shows that the _measure of _q does _net need _an excellent resolution and _{a very} fine

analysis

of the

profile.

Acknowledgments

The authors _are

grateful

to C.

Aslangul,

C. Marti and C. Simon for

helpful

discussions.

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