Antiferroelectric surface layers in a liquid crystal as observed by synchrotron X-ray scattering

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Antiferroelectric surface layers in a liquid crystal as observed by synchrotron X-ray scattering

E.F. Gramsbergen, W.H. de Jeu, J. Als-Nielsen

To cite this version:

E.F. Gramsbergen, W.H. de Jeu, J. Als-Nielsen. Antiferroelectric surface layers in a liquid crys- tal as observed by synchrotron X-ray scattering. Journal de Physique, 1986, 47 (4), pp.711-718.

�10.1051/jphys:01986004704071100�. �jpa-00210252�

E. F.

Gramsbergen,

^{W. H. de Jeu}

(*)

Solid State Physics Laboratory, Melkweg 1, ^{9751 EP} Groningen, ^The Netherlands and J. Als-Nielsen

Risø National Laboratory, ^DK-4000 Roskilde, ^Denmark

(Rep ^le 28 octobre 1985, accepté ^{le 10 dgcembre} 1985)

Résumé. - Nous étudions la réflectivité aux rayons X de la surface d’un cristal liquide dont les molécules ont des groupes terminaux polaires (cyano-); l’instrument ^{est un} spectromètre ^{3 axes à} haute résolution combiné à

une source de rayonnement synchroton. ^A la surface de la phase smectique Al, ^{nous observons le} développement

de quelques doubles couches antiferroélectriques qui peuvent ^être distinguées ^{de la structure en} simples ^couches

du matériau massif. Nous développons ^{un modèle} qui sépare ^{la densité} électronique ^{en un facteur} de forme molé- culaire et des facteurs de structures de mono- et bicouches. Avec un nombre restreint de paramètres ajustables,

ce modèle rend compte ^{de la courbe de réflexion} observée, qui ^{est fort} complexe. ^Il permet de conclure _que (i)

la première ^couche moléculaire est disposée ^avec les queues plutôt que les têtes ^vers la surface, (ii) ^le paramètre

d’ordre smectique ^{de la} première ^{mono- et bicouche est saturé,} (iii) l’organisation antiferroélectrique disparait abruptement ^{et non} pas exponentiellement.

Abstract. ^- The X-ray reflectivity ^form the surface of ^a liquid crystal ^with terminally polar (cyano substituted)

molecules has been studied using ^a high-resolution triple-axis X-ray spectrometer ⁱⁿ combination with a syn- chrotron source. It is demonstrated that at the surface of the smectic A1 phase ^{a few} antiferroelectric double layers develop ^{that can be} distinguished ^{from the bulk} single layer structure. A model is developed ^that separates ^the electron density ^{in a} contribution from the molecular form factor, ^{and from the structure} factor of the mono-

and the bilayers, respectively. ^With only ^{a few} adjustable parameters ^{it accounts for} the rather complex ^observed

reflection ^curve. It shows that (i) ^{the first molecular} layer ^{has tails} up rather than heads _up, (ii) the smectic order parameter ^{of the first mono- and} bilayer ^is saturated, (iii) ^the antiferroelectric bilayering ^does decay ^rather abruptly

and not exponentially.

1. Introduction.

Since the first observation of a smectic A-smectic A

phase

transition

[1],

many

examples

^of

SA polymor- phism

^{have been}

reported.

^{These cases}

always

^involve

strongly asymmetric elongated

^{molecules with a}

large

^terminal

dipole

^moment

(-CN

^or

-N02

^end

substituents). Today

^{at least four «} exotic >>

SA phases

are known

[2].

^{The most common} type ^of

phase

^with

these

compounds

^{is the socalled}

SAd

phase.

Disregard- ing

^diffuse

scattering

^it

gives

ⁱⁿ

X-ray experiments

^a Bragg

peak

^at

Qo

⁼

(2 7C//’)

^{n, where n is the director,}

1 the molecular

length

^{and I l’ 2 L}

Apart

^{from a}

possible

^{reentrant nematic}

phase,

^with

decreasing

temperature, ^the

phases SAII Sx

^and

SA2

^{may be}

observed. The

SA2 phase

^has

long-range antiparallel

correlation of the molecular orientation in

neigh- bouring layers (Fig. la).

^{It is an} antiferroelectric

phase

that

gives

^{rise to two}

Bragg peaks

^at

Qo

⁼

(2 n/0

ⁿ

and at

approximately Qo/2.

^{In the}

SA phase

^{in addition}

the

density

modulation shows

phase

fluctuations,

leading

^{to a}

regular

^{lattice of «} kinks » in the

plane

± n

(Fig. lb). Consequently,

^the

Bragg

spot ^at

Qo/2

is

split

^{into two} spots ^{situated off axis}

[3].

^The

SAI phase

^{shows a}

Bragg

spot

only

^at

Qo.

^{This can be}

interpreted

^{as a «} classical >>

SA phase

^{in which the}

up-down

distribution of the molecular orientation is random.

Alternatively

^{it can be} considered as a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704071100

712

Fig. ^1. - (a) ^Smectic layering with the orientation of the

aliphatic tail-polar head molecules alternating ^from layer

to layer. ^{This smectic} phase ^{is denoted} SA2. (b) ^{In the} Sx phase ^{there is a} regular ^domain pattern ^of up- and down- domains in each layer. (For ^{the sake of} clarity ^the pictures

are drawn for perfect orientational and positional order).

variation on the

SX

structure, in which the lattice of

« kinks » is not

regular.

This _paper is concerned with an X-ray

study

^{of the}

surface of a

liquid crystal

around the nematic-smectic Al

phase

transition. It is shown that ^- contrary

to the bulk ^- at the surface of ^the

SAl phase

^a

Bragg

peak

develops

^at

Qo/2,

^{which means that at} the surface

a series of antiferroelectric double

layers

^{exist. The}

results are discussed

using

^a model for the electron

density profile

^{that can be}

separated

^into contributions of the molecular from-factor, ^{and structure factors}

of mono- and

double-layers, respectively.

^The

plan

of the _paper is as follows : after ^a short

description

of the

experiment

^{in the next section,} in section 3 the results are

given.

^{In section 4 the model is} described,

including

^the type ^{of fits to the data that can be} obtained. Section 5 contains a

concluding

discussion.

2.

Experimental.

The

compound

^{studied is}

which has the

following

sequence of

phase

^transi-

tions

[4] :

A

sample

^is

prepared

^{in the form of a}

droplet covering

f"Ott.I 2 cm2 of a

glass plate

^{treated such that n is perpen-}

dicular to the

glass plate

^{as well as to the upper}

liquid crystal-air

^interface.

The

experiments

^{have been carried out at HASY-}

LAB

(Hamburg),

^{with a modified}

high-resolution triple-axis

spectrometer ^{that has been described} elsewhere

[5, 6].

^The geometry ^{shown in}

figure

^{2 is}

such that the

liquid

^surface remains horizontal under all circumstances. The horizontal

synchrotron

^beam

Fig. ^{2. -} Scattering diagram ^and resolution volume for

specular reflection. The angular divergence, e91, ^{of the}

incident radiation is determined by ^{the source} height (- ² mm) ^{and the} source-to-sample ^distance (20 m) ^whereas e62 ^is determined by the detector slit height (0.20 mm) ^and

the sample-to-detector ^distance (620 mm).

is deflected downwards

by

^an

angle 0, (spread

^{in the}

vertical direction

A01 ) by Bragg

reflection from a tilted Ge

monocrystal

^to

produce

the incident wave vector

kl.

^The

height

^{of the}

sample

^is controlled such that the beam

always

^{strikes a fixed}

position

^{of the}

sample.

The detector is

placed

^{behind a narrow slit that} accepts ^{radiation at}

angles 02 in

the vertical direction and

t/J2 in

^the horizontal

(out

^of

plane)

direction, ^with

a

spread A02

^and

A4/2, respectively.

^{As a} consequence of translational invariance ⁱⁿ the horizontal

plane,

the

scattering

cross-section for

X-rays

^{that are either}

specularly

^{reflected from the flat surface or}

Bragg

scattered from the smectic

layers parallel

^{to the} surface, ^is

proportional

^to

6(Q.,) 6(Qy) ^[7].

^{The reso-}

lution of the spectrometer ^{tuned to this radiation is} indicatetd

by

^the

parallelogram

ⁱⁿ

figure

^{2, with}

02 = 81

^and

t/J2

^{= 0.}

Scattering

^from the bulk material

can be monitored

by moving

^{the detector in the x or z}

direction such that

QZ

^{= 2}

k,

^{sin 0 does not intersect}

with the resolution

parallelogram.

3. Results.

The

X-ray reflectivity

of the surface has been measured

as a function of

Q.

^with

Qx

⁼

Qy

^{= 0, for five tempe-}

ratures around the

NSA1 phase

transitions. The

signal

from the bulk, ^obtained

by making

^{the same scans with}

a

slight

offset in

Q"

^was subtracted to ’obtain the surface

reflectivity R(6z)

^{In order to} facilitate com-

parison

^{with the} theoretical

reflectivity

^{from the model}

to be described below, ^e result has been devided

by

^the

reflectivity calculated

^{from Fresnel’s law,}

RF(Q.)-

^This

quantity gi s

^the

reflectivity

^{for radia-}

tion incident on a

plan r

dielectric

discontinuity,

The

reflectivity

^divided

by

^Fresnel’s law,

RZ(Qz)I RF(Qz)

^is

depicted

ⁱⁿ

figure

^{3. In this}

figure Qz

is the value of the wavevector transfer inside the

liquid crystal,

which is lower than the external value accord-

ing

^to

Q’in = Qi out - ^{Q 2.}

^The

^peak

^at

6,/6o = 1

with

Qo = 0.215 A-1 corresponds

^to the smectic

layers

^{as a}

pretransitional

effect in the nematic

phase

when the

NSAL phase

transition is

approached.

^The

corresponding periodicity

^is

approximately equal

^to

the

length

^{of a} stretched molecule. The

layers

^extend

to a

depth çL(T)

^{below the surface,}

with ÇL diverging

at the transition temperature. ^{The scan at the lowest}

Fig. ^{3. -} The measured reflectivity R(Q), ^{relative to the} calculated Fresnel reflectivity RF(Q), ^{versus wavevector}

transfer Q ⁼ 2 k sin 0 relative to Qo = ² x/L ^{= 0.215} A-1,

Notice the logarithmic ^{scale on the ordinate. Data for}

each temperature ^{have been} displaced by ^{one decade. The}

full line is the best least _squares fit of a model described in the text.

at

Q.IQO

^{= 0.55. Since all}

peaks

^are

asymmetric

^each

of them _may be

parameterized by

^its

position, height,

width and skewness. This would lead

altogether

^to

12 parameters ^{to fit the data at each} temperature.

We shall in the

following

section describe ^a

physical

model with much fewer

adjustable

parameters.

4. Model.

We shall first

give

^a

qualitative interpretation

^{of the}

data.

At the interface between

liquid

^and vapour there will be a

pronounced tendency

^{for the molecules to}

be oriented so the

aliphatic

^{tail is}

pointing

^towards

the _vapour,

consequently

there will be an oriented top

layer

^{of molecules.}

The nematic

phase

^is

susceptible

^{to smectic A}

layering

^{and since one}

layer

^is

already

^{formed at the} top, ^{it is followed}

by

^another

layer

below, ^a

layer

below that, ^{and so on. The}

penetration depth

^{of such}

layers, ÇL’

^{increases as the} temperature

approaches

the nematic-smectic A transition temperature ^because the

susceptibility

^for

layering

^{becomes more and more}

pronounced.

^This

layering

appears in the reflection

curve

R(Q)IRF(Q)

^{as a}

peak

^around

Q/Qo

^{= 1 where}

Qo

^{= 2}

7c/(one layer thickness).

^The

peak

gets

higher

and

sharper

^as

ÇL

^{increases in} accordance with the data around

Q/Qo

^{= 1 in}

figure

^3.

However, ^the top

layer

is oriented. If there is a

tendency

^{for molecules to} stick their heads

together

rather than head-tail, ^{then the second}

layer

^{will be}

oriented

opposite

^to the first

layer.

^{The two}

layers together

^{form a}

bilayer

^and

bilayering

^will penetrate into the bulk with another

penetration depth çp,

where the index P indicates

polarization,

^whereas

index L in

ÇL

^indicates

layering.

^{Since there is no} particular response ^to

polarization

ⁱⁿ the nematic

phase

^{one must} expect

çp

^to be smaller

than ÇL

^and

also to be

roughly independent

^of temperature. ^Since the

period

^of

bilayering

^{is twice as} large that for the smectic

layering,

^it appears ^{as a}

peak

in the reflected

wave around

Q/Qo

⁼ 0.5. Since this wave can interfere either

constructively

^or

destructively

^{with the}

ordinary

Fresnel reflected _wave, one can with the _proper

phases

account for a

peak

^around

Q/Qo

0.5 and an anti-

peak »

^for

Q/Qo >

^{0.5 as observed in the reflection}

curve.

We shall now cast this

qualitative picture

^{into a}

quantitative

model. To this end we recall the basic

714

relation between the

reflectivity

^{and the electron}

density :

We shall first consider the electron

density

^{in a}

single

molecule shown in

figure

^4a.

Fig. ^{4. -} (a) Position of atoms in the molecule shown in the plane ^{of the} molecular-axis and the two benzene rings

closest to the polar ^{head. The} third benzene ring ^is perpen- dicular to the drawing plane ^and appears ^{as a} cross-hatched

strip. The numbers coincide with the indices k in table I.

(b) The electron density around each atom is assumed to be a Gaussian distribution of width a. The resulting density

distribution along the molecular axis relative to the bulk

density ^{is shown.}

From

analogy

with similar chemical bonds it seems

plausible

^{that the two benzene}

rings

^{closest to the}

polar

^{head are}

approximately co-planar,

^{but the}

plane

of the third benzene

ring

^{is turned over about}

n/2.

^{With bond}

length

^{and bond}

angles

^{taken from}

space-filling

^molecular models, this leads to a

quanti-

tative

projection

of the molecule ^as shown in

figure

^4a.

However, the final results

depend only

^very

slightly

on the

assumptions concerning

the relative orienta- tions of the benzene

rings.

The electron

density projected

^on the molecular axis is

approximated by

^a

superposition

^of Gaussians,

Zk( 2 c Q) -1 exp[ - ( - (k)2/2 U2],

^{centred at the}

different atomic

positions (k

in the molecule and with

a

prefactor Zk equal

^{to the} atomic number

(6

^for C,

8 for

CH2,

^{7 for N}

etc.).

^{The width a of the Gaussian is}

taken to be 1

A.

The values

of (k

^and

Zk

^{are listed in}

table I. The

resulting density/(0

^{with -}

L/2

C

Table I. ^- Atomic numbers

Zk

and coordinates

Ck of

the atoms and groups in the molecules

investigated.

The indices k coincide with the numbers in

figure

^4a.

L/2 normalized to an average value of

unity,

^{is shown}

in

figure

^4b.

It should be noted that this

density

^also represents the

density

^{of a}

fully polarized layer.

^We shall decom-

pose/(0

^{into its} average value of

unity plus

^{a sym-}

metric part

f.(C) plus

^an

antisymmetric partJ:a(C) :

where L1 is

unity

^{for -}

L/2

(

L/2

^{and zero}

elsewhere. Such ^a

decomposition

^{is useful because the}

symmetric

part ^of

h«()

represents ^the

density

^{of a}

randomly

^oriented layer. ^It follows then that the anti-

symmetric

part ^of

f.(C)

^must represent ^{the difference} between a

completely polarized

^{and a}

randomly

oriented

layer.

Since the master formula for the

reflectivity

^is

expressed

^{in terms of the Fourier trans-}

form of the

density p(z)

^{it is}

intuitively

clear that we

The

integration

^{limits are between}

quotation

^marks

because the Gaussian

smearing implies

that the limits extend

beyond - ! Land!

^L,

respectively.

In

evaluating C(Q)

^and

S(Q)

the Gaussian model of the molecule is

particular

convenient.

Explicit

formulae ^are

given

^below and the functions

C(Q)

^and

S(Q)

^{are shown in}

figure

5. It is found that the

shape

of

S(Q)

^is

quite

sensitive for the

position

^{of the}

origo

of the

coordinates C

We shall now return to evaluation of the Fourier transform

O(Q),

^cf.

equation (1c). Assuming complete

Fig. ^{5. -} The deviation from unity of the normalized molecular electron distribution is decomposed ^{into its} symmetric ^and antisymmetric parts. ^The corresponding

cosine- and sine-transforms versus wavevector Q/Qo ^are

shown.

It is now assumed that the

shape

^{of the}

layer density is

invariant and

equal tof,(C)

⁺

( - 1)"f.(C)

^{but with}

decaying amplitudes

^{for mono- and}

bi-layer

structure :

We then find

The first term

originates

^{from the} average

density p (step-function),

^smeared

by

^{a Gaussian} surface fuzzi-

ness due to thermal vibrations and a

phase

^factor

exp[iQ,o]

that allows for a

displacement Co

^{of the} step ^{function relative to the}

origo

^{of the molecular}

coordinates.

We shall ^now discuss a convenient mathematical form of the

amplitude

^functions

As(n ; ÇL)

^and

Aa(n ; çp).

For

clarity

^{we shall for a moment} omit the indices

on A and 03BE. ^The

simplest

^{form of}

A(n ; Q

^{would be}

However the data, ⁱⁿ

particular

^{for the}

bi-layer

^cor-

relation function,

require

^{a more}

sophisticated decay

than

just

^a

simple exponential.

^{One reason is that the}

amplitude

^{is at its} saturation value of

unity

^{at the} top

layer

and it is therefore natural that the correlation function starts out with a horizontal tangent. ^One

possibility

^{is then the} function,

or more

general

The

properties

^{of this} function and its Fourier trans- form are discussed in

Appendix

^1.

716

With the

ingredients

of the model described above fits to the

experimental

^{data can be obtained of}

excellent

quality,

^{as shown as full lines in}

figure

^3.

The values of the parameters ^{used are}

given

^{in table II.}

Table II. ^- Least-squares

fit

^values

of

^the parameters in the model described in the text. The

layer

spacing ⁱⁿ

the bulk

SAt phase

^is

Lo

^{= 29.2 A. The value}

of 03BEL

labelled with an asterix is

in fact

the resolution width.

The index J

of

^the decay

function Yj(x)

^is given ^both

for layering (JL)

^and

polarization (Jp).

5. Discussion.

The first observations of smectic

layering

free surface in

liquid crystals

ⁱⁿ the nematic

phase [6-8]

^involved

only

^one

layer spacing

^{and in}

comparison

^to

figure

³

only

^{data in the}

region

^0.9

Q/Qo

^{1.1 were}

important.

^{These data were}

analysed by

^a

purely phenomenological

^{model and the}

question

^{of the}

degree

^of orientational, ^{or nematic} order, ^{at the}

surface was not addressed. In the present ^{work the}

knowledge

^{of the structure of the molecules is incor-}

porated

into the model of

analysis.

^For

comparison

with earlier work let us for a moment

neglect

^the

antiferroelectric

layering

^and

thereby

^{the third term}

in

equation (7).

^{The molecular}

modelling

^is

given by

the cosine transform

C(Q)

ⁱⁿ

equation (7)

^{and the}

amplitude

^function

As(n ; ÇL)

^{can then be}

interpreted directly.

^{We find that}

A,

^{=1 for n = 0, the first}

layer,

which means that the nematic order parameter ^is

fully

^{saturated at the surface. The} first and second term in

equation (7) gives

^{rise to} interference between the

ordinary

^{Fresnel wave}

(first term)

^{due to the dis-}

continuity

ⁱⁿ refractive index and the ^wave scattered from the

layered

^structure

(second term).

^{We have}

in the present

analysis

^{as well as in}

previous

^work

[8]

introduced a

phenomenological phase

^factor

exp[iQCO]

between these two waves. In

previous

^work

it turned out that the

phase

factor gave rise ^{to a}

strongly asymmetric peak

^around

Q

⁼

Qo

^{with cons-}

tructive interference for

Q Qo

and destructive inter- ference for

Q

>

Qo.

The data for the present material does not

display

^this

phenomenon

^around

Q

⁼

Qo.

However, ^the spectrum ^around

Q/Qo

^{= 0.5 does}

indeed exhibit strong interference ^effects

arising

^from

the antiferroelectric surface

layers.

^{The molecular}

model which has been introduced in the present work enters

solely by

^{its Fourier} transforms

given

ⁱⁿ

figure

^{5. With this model we find the}

physically

very

simple

^and

appealing

^{result that the} top

layer

^has

complete

orientational order

(all

molecules have tails

up)

and in the next

layer

^{all heads are} up. This antiferroelectric

degree

^{of order}

decays

^rather

rapidly

as one goes further _away from the surface. This

decay

is not

exponential

^{but is}

qualitatively

^{rather like the}

curve labelled J ⁼ 5 in

figure

^{6. The}

penetration depth 03BEp

^is

only weakly

temperature

dependent

^as

shown in table II.

Fig. ^{6. - The} penetration into the bulk of mono-layers

and bi-layers ^{is not a} simple exponential but has the charac- teristic shape ^{of a} logistic ^curve which is modelled by ^the

function Yj(x) ^{shown versus} x/J(J ⁺ 1). ^The definition of

YJ(x) ^is given ⁱⁿ equation (A .1 ) ^{in the text}

As far ^as smectic

layering, independent

^of orientation of the molecules within a

layer,

^is concerned, ^the

penetration

^{is much}

deeper

^{and exhibits a} strong temperature

dependence.

^In

previous

^work

[6]

^{it was}

found that the

penetration depth C;L

coincides with the correlation

range C;

for smectic fluctuations in bulk.

This is very

likely

^{to be true also in the} present case, but we have not yet

completed

^a

study

^{of the bulk}

properties.

^{In the bulk one can} also observe fluctuation of antiferroelectric double

layers.

^{It will be}

interesting

to compare the bulk correlation _range with

jp reported

here.

The r.m.s. values of the

smearing

parameters Us for the surface and a for the molecular electron

density

^are

roughly equal.

^{The values} obtained, ⁴ A

and 4.5

A, respectively,

^seem

quite

reasonable.

The value found for

jo

indicates, ^{somewhat sur-}

prisingly,

that the actual surface is elevated about 5 A above the first smectic

layer.

^Possible

explanations

are :

absorption

^{of water on the} surface, ^or the presence of

decomposition products

^{of the} molecules, ^{that are}

lying

^{on the} surface, ^{or a} different molecular confor- mation in the top

layer.

SAl phase closely SA2 phase,

which is of interest in context with some recent

theoretical models

[8].

^{Then the}

SAl phase

^{should be}

distinguished

^{from a «} classical >>

SA phase.

^A

possible

model is an

SAZ

^{structure with}

locally

^broken

up-down

symmetry ^{and in addition}

phase

fluctuations

similarly

as in the

SA phase,

^but

randomly

distributed.

Finally

^{it should be} noted that the present ^obser- vation of surface double

layers

^{is also}

highly

^relevant

for the

interpretation

^of

X-ray

^{results on free}

standing

smectic films

[9].

^{It indicates} that below a critical number of smectic

layers

^{the two} surfaces of the film

can induce a

phase

transition.

Acknowledgment.

The authors wish to thank Prof. G.

Heppke (TU, Berlin)

^for

providing

them with the substance. The excellent research conditions

provided by Hasylab

are

gratefully acknowledged.

This work form part of the research _program of the

« Stichting

^voor

Fundamenteel Onderzoek der Materie »

(Foundation

for Fundamental Research on Matter,

FOM)

^and

was made

possible by

^financial support ^{from the}

« Nederlandse

Organisatie

^{voor Zuiver}

Wetenschap- pelijk

Onderzoek »

(Netherlands Organization

^{for the}

Advancement of Pure Research,

ZWO).

Appendix.

Consider the function

Yj(x)

^defined

by

the finite series :

The derivatives of

Yj(x)

^{are :}

Therefore, ^{for J} > 0,

Yj(x)

starts out with a horizontal tangent ^{at x = 0 and}

decays

^{with convex curvature}

until at x ⁼ J the curvature becomes concave. With

increasing

^J

higher

^and

higher

derivatives of

Yj(x)

vanishes at x ⁼ 0 so in terms of the variable

x/(J + 1)

the

shape

gets ^{more and}

square-like.

Indeed, ^the

limiting shape

^{for J = oo is a} square ^as

immediately

inferred from the

identity

^e+x

= E Xj/j !

¹

where +

applies

^{for A =}

As and - applies

^{for A =}

Aa.

It is convenient to

interchange

^{the summation over j}

and n

with

in terms of the

complex

^variable

where

again

⁺

applies

^{for A}

= As

^{and - for A =}

Aa.

The summation in

equation (A. 5)

^{is carried out for}

increasing

^values

of j starting with j

^{= 0 :}

By differentiating (A. 8)

^with respect ^{to a on both} sides we obtain

or

By differentiating equation (A. 9)

^with respect ^{to a one} obtains

s2(Q),

^{and so on. The}

explicit expressions

^for

Thus with the

particular

^{choice of}

A(n) given by equations (A. 3)

^and (A .1 ) ^{the structure} factor, ^which

in

principle

^{is a sum over an}

infinite

^{number of}

layers,

is evaluated as a

finite

^{sum of}

(J

⁺

1)

^terms.

718

References

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(b) HARDOUIN, F., LEVELUT, A. M. and SIGAUD, G.,

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[2] ^{For a recent review} see, for example HARDOUIN, F., LEVELUT, ^A. M., ACHARD, ^{M. F. and} SIGAUD, G.,

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(b) COLLETT, J., PERSHAN, ^P. S., SIROTA, ^{E. B.} and SORENSEN, ^L. B., Phys. Rev. Lett. 52 (1984) ^356.