OPTICAL PROPERTIES OF BLUE PHASE I : MEASUREMENTS OF THE ORDER PARAMETER

HAL Id: jpa-00224623

https://hal.archives-ouvertes.fr/jpa-00224623

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

OPTICAL PROPERTIES OF BLUE PHASE I : MEASUREMENTS OF THE ORDER PARAMETER

R. Barbet-Massin, P. Piera_ski

To cite this version:

R. Barbet-Massin, P. Piera_ski. OPTICAL PROPERTIES OF BLUE PHASE I : MEASUREMENTS OF THE ORDER PARAMETER. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-61-C3-79.

�10.1051/jphyscol:1985306�. �jpa-00224623�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome *6, mars 1985 page C3-61

O P T I C A L P R O P E R T I E S O F B L U E P H A S E I : M E A S U R E M E N T S O F T H E ORDER P A R A M E T E R

R. Barbet-Massin and P. Pierariski

Laboratoire de Physique des Solides, Univevsite Faris-Sud, Bat. 510, 91405 Ovsay, France

Résumé - Nous appliquons la théorie dynamique de la diffraction à la Phase Bleue I. Utilisant des spectres obtenus sur de gros monocristaux sans défaut nucléés sur des surfaces de verre, nous les comparons aux spectres théoriques et en déduisons la première mesure directe de la composante de Fourier dans la direction <110> du paramètre d'ordre de la Phase Bleue I. Enfin, nous montrons que ces résultats sont en accord avec la théorie développée par R.M. Hornreich et al.

Abstract - We apply the dynamical theory of diffraction to Blue Phase I.

Using spectra we have got viith large monocrystals without defect growth on glass surfaces, we compare them to the theoretical ones and obtain the first direct measurement of the Fourier component in the direction <110> of the order parameter of BPI. At least, we show that those results are in agreement with the theory proposed by R.M. Hornreich et al.

I - INTRODUCTION

Blue Phases I and II, occurring ina small temperature range between an isotropic and a cholesteric phase, have cubic structures with lattice constants of the order of wavelength of visible light /1-6/. On these scales (0.1-1 um-) the visible light is, like in collofdal crystals, a very convenient way of studying the crystal-line struc- ture. However, in contrast with ordered suspensions where one (bec) cubic unit-cell contains only two colloTdal particles, one cubic unit-cell of the Blue Phase I (or

II) contains about 107 rodshaped chiral molecules. Moreover, the molecular order in Blue Phases is not positional but orientational what means that these molecules can move freely (like in a liquid) through the cubic lattice and only their orientations are correlated with their positions in the lattice. These characteristics, very different from usual colloTdal crystals, allow one to consider Blue Phases as a continuum where the dielectric tensor e(r) is a smooth function of the position r.

Because of the periodicity of the lattice one can develoo e in a Fourier series /7,8/ :

e(r) = el + J eT e i 2" . r ( 1 )

T

where x is a reciprocal lattice vector of BPI (in this paper we are only interested in BPI).

The zeroth-compotjent is proportional to the unit-tensor 1 because of non birefrin- gence of BPI. e(r) - e1 may be choosen as the order parameter of the phase, because it is a traceless tensor which is zero in the isotropic phase /7,8/.

The aim of the present paper is to apply the general theory of dynamical light dif- fraction to Blue Phases. It will be shown in the next section that this can be done in a very convenient way if one takes into account (1°) the lack of birefringence Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985306

C3-62 JOURNAL DE PHYSIQUE

(E0 = E ? ) c h a r a c t e r i s t i c o f Blue Phases and (2") a well-known experimental f a c t t h a t B1 ue Phases r e f l e c t s e l e c t i v e l y almost u n i o u e l y a c i r c u l a r y p o l a r i z e d l i g h t o f one c h i r a l i t y . As a t h e o r e t i c a l r e s u l t we w i l l show how t o g e t i n f o r m a t i o n s about t h e components ET o f t h e o r d e r parameter from t h e s p e c t r a l shapes o f Bragg r e f l e c t i o n s . P r a c t i c a l l y , t h e a p p l i c a t i o n o f d y n a ~ i c a l th e o r y o f l i g h t s c a t t e r i n g i s "ab i n i t i o "

reserved t o t h e case o f d e f e c t - f r e e monocrystals. C l e a r l y , t h e development o f t h e order parameter i n a F o u r i e r s e r i e s (1) nakes sense o n l y when t h e c r y s t a l i s p e r f e c t ; any d i s l o c a t i o n would break t h e phase r e l a t i o n s h i p between d i f f e r e n t regions o f t h e c r y s t a l .

Very r e c e n t l y , ire have shown /9/ t h a t t h i s necessary c o n d i t i o n can be s a t i s f i e d i n Blue Phases. I n f a c t , we have developed a method o f n u c l e a t i o n (on glass surfaces) and o f growth o f d e f e c t l e s s monocrystals w i t h well-known o r i e n t a t i o n w i t h respect t o the glass surface. I n a presence o f a small temperature gradient, these monocrystals can be made f l a t ( l i m i t e d by two s t r i c t l y p a r a l l e l c r y s t a l faces) and l a r g e , so they have i d e a l shapes s u i t e d f o r s t u d i e s o f dynamical l i g h t d i f f r a c t i o n .

I n s e c t i o n 111, we r e p o r t experimental snectra obtained i n b a c k - r e f l e c t i o n (under normal incidence) from such p e r f e c t c r y s t a l s o f a well-known thickness L. These spectra a r e compared w i t h t h e t h e o r e t i c a l ones and (knowing t h e thickness L ) t h e amplitude o f t h e F o u r i e r component i s determined.

I n t h e l a s t section, we s h a l l procede t o a c o n f r o n t a t i o n o f these experimental r e s u l t s w i t h t h e Landau type t h e o r e t i c a l approach by Hornreich e t a1 /7/.

I 1

-

GENERAL THEOEY OF DYNAMICAL DIFFRACTION BY PERFECT B? MCNOCRYSTALS 11.1. S o l u t i o n s o f ~ l a x u e l l ' s equations i n a p e r i o d i c medium

A l l t h e beginning o f t h a t t h e o r e t i c a l work has been f i r s t made by Belyakov and r e l a t e d i n a very i n t e r e s t i n g paper /8/. I n t h e present paper, we have choosen Belyakov's way o f c o n s i d e r i n g BPs, and t r i e d t o develop t h e theory furthermore, using some i n - t e r e s t i n g experimental r e s u l t s t h a t g r e a t l y s i m p l i f y t h e theory.

Ke consider an e l e c t r i c f i e l d o f a plane cave :

i n c i d e n t on t h e c r y s t a l , i n such c o n d i t i o n s t h a t t h e r e i s o n l y one Braag r e f l e c t i o n t h a t occurs i n s i d e t h e c r y s t a l . We assume moreover t h a t t h i s r e T l e c t i o n i s a back- r e f l e c t i o n and i s symmetric (Fig. l a ) . This i s a very r e s t r i c t i v e c o n d i t i o n t h a t must be c a r e f u l l y c o n t r o l l e d f o r each experimental data. For instance, i f one looks a t a Bragg peak i n b a c k - r e f l e c t i o n w i t h normal incidence corresponding t o t h e f a m i l y o f planes (220), one c o u l d g e t i n t h a t peak a double successive r e f l e c t i o n on planes (200) a t 45" w i t h t h e planes (220) and t h e two-wave approximation i s thus n o t a t a l l v a l i d .

I f t h i s approximation i s v a l i d , we can s o l v e Plaxwell's equations /10/ c o n s i d e r i n g t h e e l e c t r i c f i e l d amp1 i t u d e i n s i d e the c r y s t a l as :

We then o b t a i n t h e system o f equations ( c o n s i d e r i n g o n l y t h e r e f l e c t i o n due t o t h e T-component o f t h e r e c i p r o c a l space o f t h e l a t t i c e ) /8/ :

where C-, = i: because i s r e a l and E = n2, w i t h n the i n d i x o f r e f r a c t i o n .

0 4 0

.+

I f

11

and

8;

are two perpendicular u n i t vectors so t h a t

(o:,

e2, kg) i s a d i r e c t t r i e d r a , and i)l 1 and -8; two perpendicular u n i t - v e c t o r s so t h a t

G:. 2:,

cl) i s a l s o a d i r e c t one ( F i g l b ) , u.e can p r o j e c t ( 2 ) along t h e d i r e c t i o n s

2:

and

3;

and ( 3 ) along those o f 3; and

3

and o b t a i n :

w i t h (Fol),, =

zY*

^i, and d40)ik = (iol)ti = " C -7

L:

Combining those equations, i f we s e t :

.. -

then we f i n d t h a t :

k: E;

= c 2 ( 1

- - l

( l

--l

( 2)

x2

_{x2 E.}

So t h e very important r e s u l t o f t h a t c a l c u l a t i o n i s t h a t

to

and

?,

a r e eigenvectors of So and S, f o r t h e

same

eigenvalue.

L e t us c a l l and

S'

t h e two unit-eigenvectors o f

i0

^{and and}

$'

the two c o r - responding ones of S,. The former a r e i n t h e plane perpendicular t o

KO

and t h e l a t t e r i n t h e plane perpendicular t o

K1.

I f we p r o j e c t t h e two equations ( 2 ) and ( 3 ) along those i n t r i n s i c d i r e c t i o n s o f p o l a r i z a t i o n , then we o b t a i n unconnected equations between E. and El.

Along the a d i r e c t i o n s ( z i f o r (2) and f o r ( 3 ) ) : ( 1

-

^--) k; ^{E; + Fa E: = 0}

X

=

F;: E;

+

( l

-

-)

f

E; = 0 X

q*

^C

zy

w i t h F a Z E

C3-64 JOURNAL DE _PHYSIQUE

Along t h e o 1 d i r e c t i o n s (G' f o r ( 2 ) and

q'

^{f o r} ( 3 ) ) :

- W 1 *

- -p!

w i t h Fo, =

no

^{O E . ~} 1

E

The d i s p e r s i o n equations are :

One p o l a r i z a t i o n g i v e s than two couples (cO, kl) (by permutation) and we c a l l them + ( % ( l ) , ? ( l ) ) and (t:(2), ?(2)), and t h e sane f o r o l .

So, i n s i d e t h e c r y s t a l we g e t by s u p e r p o s i t i o n o f t h e s o l u t i o n s : E(F,t) eiwt = ; C;(; exp [i2nRE(l).;] + B:

$7

exp [i?nq(l).;])

o +o

+ : C;(; exp [i2+%;(2).q + g3;, nl exp [i?nE;(2).3)

I I

+ C;' ($' exp [ i 2 n ~ : ' ( l ) . a + 87 7 . exp [ i 2 n q 1 ( 2 ) . 8 )

+ C ' ($I exp

[izntg1

(2).;] + 8;'

$7'

exp [i2nky1 ( 2 1 . 3 )

If we s e t : p = 1 f o r ( o , l ) ; p = 2 f o r ( 0 , ~ ) ; p = 3 f o r ( o l , l ) and p = 4 f o r (01,2) then :

4

E(;,t) eiot =

r

C (bp exp [i2nPp.;I + Bp

iiy

exp [ i ~ n R y . a )

p = l p O . ( l 6 1

11.2. Boundary c o n d i t i o n s

We know t h a t on t h e uppeg surface o f the c r y s t a l , t h e i n c i d e n t e l e c t r i c f i e l d o f t h e plane wave i n d i r e c t i o n X i s E , and on t h e l o v e r surface, t h e r e i s no j n c i d e n t wave w i t h d i r e c t i o n k l /l l/.

I f we make the assumption t h a t vie are i n normal incidence, o r t h a t t h e index of r e f r a c t i o n o f t h e o u t s i d e r e g i o n i s n e a r l y equal t o t h a t o f the i n s i d e region, then the e l e c t r i c f i e l d must be continuous a t t h e surfaces between t h e two regions. I n t h e experiments described i n n e x t section, we a r e always i n such cases. A c l a s s i c a l c a l - c u l a t i o n /10,11/ g i v e s then f o r boundary c o n d i t i o n s :

-

A t the upper s u r f a c e : 2 C exp (i2nZ:.?)

6;

= -+ so exp (i2ng.;)

n P

and so Z P - + - X +

xmp

^{+ S} ( f o r

c!.;

⁼

;.:

on the surface) ( 1 7 )

and + S O = C C ;p

P P O

(18)

-

A t t h e lower s u r f a c e : 1 _p C _? B exp (i2atY.;) _P

zy

⁼

⁸

with

if;

⁼

+ ? = 2 ^{+ 3 + ?}

from Bragg law.

I f t h e o r i g i n i s taken on t h e upper s u r f a c e , with a c r y s t a l of t h i c k n e s s L we g e t then :

I & B exp ( i ~ n x + ~ ~ c o s e ~ ) =

8

P P P

with y = exp ( i 2 a x + P c o s 8 0 ~ )

P (21

We now s e t : k: =

x

(1 + 5;) k y =

x

(1 + 5:)

3

and 6 = ^T

(?

+ 22)

2 ( t h a t measures t h e departure from Bragg law).

Then, from equations (17) and (19) we obtain : 5" = +P cos e

0 0

57 =;- EgP

The d i s p e r s i o n equations (14,15) then become : E; ($

-

E;) = q 1 F, F; f o r p = I , ?

6 1

5: (;T

- $ 1

= F,, F;, ^{f o r p = 3 , 4}

6 1

So 5 i y 2 = ? (6'

-

4 1 ~ , ( ' ) ~ ' ~ i f 161 ) 21Fa(

6 i 2 1/2

tAy2 = ? (41F0,

1 1

i f 161

6

2lF,l and

6 1 2 1 / 2

53y4 ₀ = ? ( 6 2

-

4 ] F a ,

1

^{) i f 161 ) 21FD1}

1

6 i

<iy4

^{= ? (41FUI}

^1' -

^{62)1'2 i f 161}

<

^ZIF,,

I

11.3. Meaning of t h e experimental d a t a of BP1

All t h e 3ragg r e f l e c t i o n s from Blue Fhases observed up t o d a t e seemed t o be c h i r a l : i f one uses right-handed c i r c u l a r l y polarized l i g h t under normal i n c i d e n c e , i t i s s t r o n g l y r e f l e c t e d , while t h e left-handed r e f l e c t i o n has not been y e t detected. So in following we w i l l suppose t h a t left-handed c i r c u l a r l y polarized l i g h t corresponds t o an i n t r i n s i c p o l a r i z a t i o n t h a t i s not r e f l e c t e d a t a l l . In t h e c a l c u l a t i o n t h i s i s obtained by making F, = 0 ( o r F,, = 0 ) . Then only one i n t r i n s i c p o l a r i z a t i c n i s r e f l e c t e d .

Furthermore, when e o = 0 (normal i n c i d e n c e ) , observing i n back r e f l e c t i o n ( e 2 = O), t h e r e f l e c t e d p o l a r i z a t i o n i s c i r c u l a r so i f F, = 0, we a l s o g e t :

ny' = (g! -+ i e i n .

Using those two c o n d i t i o n s and t h e boundary c o n d i t i o n s , we can then c a l c u l a t e t h e r e f l e c t e d i n t e n s i t y , i n t h e c a s e e o = 0.

C3-66 JOURNAL DE PHYSIQUE

11.4. Reflected i n t e n s i t y with normal incident l i g h t If the incident e l e c t r i c f i e l d has the polarization :

+ _e

+o

= ccsa

z0

⁺ e i B sina e

1 2

Me s e t :

A = ^det(no,no

_

Then a f a s t i d i o u s calculation using the above conditions gives, i f the incident in- t e n s i t y i s I O :

For

a

= 0, t h i s gives :

ch [ ~ T X L

I

^F~

I] -

1 lmax = I ~ A ~ ( ~ , B )

ch [2sxL

I

Fn,

I]

^{+ 1}

And f o r 161 = 2 ) F u I

I

t h i s gives :

If L i s small enough ( l e s s than hundred times the l a t t i c e constant) the spectra

the well-known Pendel l5sung beats, with minima related by : cos [ T X L ( ~ ~ - ~

1

F ~ , 1 2 ) 1 ~ ~ ~ : 1

or 6' = 4 1

I 2

~+ ~ ~with k integer.

X 2 L 2

W have e T =

-

and X = ¹ so with equation (24) vie pet : 'max

A A 2 k 2 A i a x IFU1l2

vihere A A = A

-

A,,,

(+ ^rnax =

-

^{+ -}

11.5. Discussion of theoretical r e s u l t s

Using equations (33) and ( 3 4 ) , we have computed nunerically the spectra of the r e l a t i v e i n t e n s i t y I(A)/IOA2 f o r several values of the crystal thickness L ( n L = IQ, 30, 50 and 70 urn) and f o r four d i f f e r e n t values of the paraneter F u , (Fu,=mx0.0145 ; m = 1 , 2, 3, 4). The central wavelength A,,, = 567 nn was chosen t o correspond t o the (110) reflection observed experimentally (section 111.2). These theoretical spectra, ( i n Fig. 2 ) show the Penselassung beats which, as indicated by equation (37) a r e denser e i t h e r f o r l a r g e r thickness L o r f o r l a r g e r amplitudes sT ( F o I i s related t o by equation ( 1 3 ) ) of the d i e l e c t r i c constant. The central peak, f o r large values of l- or F u l has a f i n i t e width and shows a f l a t t e n i n g of i t s top. The general shapes of these spectras, are i n agreement with previous theoretical works concerning Blue Phases (Belyakov /8/ /14/) cholesteric l i q u i d c r y s t a l s (Chandrasekhar /12/) or X ray

Fig. 1

-

( l a ) Conditions o f t h e t h e o r e t i c a l c a l c u l a t i o n

( I b ) T r i e d r a s r e f e r r e d t o i n c i d e n t and r e f l e c t e d plane waves i n s i d e t h e c r y s t a l

d i f f r a c t i o n by o r d i n a r y c r y s t a l s (Zachariasen /13/). I t i s important t o emphasize t h a t t h i s s i m i l a r i t y w i t h t h e case o f X-ray d i f f r a c t i o n r e s t s on t h e c o n d i t i o n t h a t o n l y one o f eigen p o l a r i z a t i o n s , a '

,

was supposed t o be r e f l e c t e d .

I n general, i n t h e case o f t h e Blue Phases, t h e r e f l e c t i v i t y c o e f f i c i e n t Fa of. the o t h e r e i ~ e n p o l a r i z a t i o n a i s c e r t a i n l y much s m a l l e r than Fa, (Fo/FoI

<

1 0 - l ) b u t i n t h e l i m i t o f a thickness L l a r g e enough, i t could g i v e r i s e t o a measurable Bragg r e f e c t i o n .

I f one supposes t h a t t h e i n c i d e n t beam i s n o t p o l a r i z e d , then t h e r e f l e c t e d i n t e n s i t y can be c a l c u l a t e d as a simple s u p e r p o s i t i o n o f two spectra I a l ( x ) and I a ( x ) c o r r e s - ponding t o t h e same parameters o f equations (33) and (34) except f o r Fa which should be taken much s m a l l e r than F,,

.

The t o t a l i n t e n s i t y I ( x ) = I a l ( x ) + I,(A) should then show a narrow peak, corresponding t o F,, s i t t i n g on t h e l a r g e peak corresponding t o Fol, as depicted i n F i g u r e 3. This r e s u l t i s i n agreement w i t h an analogeous p l o t shown i n F i g u r e 2 o f t h e paper by Belyakov /8/.

I11

-

EETERMIP4ATION OF THE ORDER PARAMETER COMPCNE?!TS 111.1. Nucleation and growth o f p e r f e c t BP1 monocrystals

I n a r e c e n t a r t i c l e /g/, we have p o i n t e d o u t t h a t l a r g e p e r f e c t f a c e t t e d n o n o c r y s t a l s o f BPI, c o e x i s t i n g w i t h t h e i s o t r o p i c phase, can be grown i n a m a t e r i a l being a m i x t u r e o f 58,8 % CB 15 i n ZLI 1840. I n presence o f a small temperature g r a d i e n t , t h e e q u i l i b r i u m shape o f monocrystals grown on glass surfaces i s w e l l adaoted f o r t h e present study o f Bragg r e f l e c t i o n . I n these c o n d i t i o n s , t h e f a c e t a a r a l l e l t o t h e glass surface (perpendicular t o t h e temperature g r a d i e n t ) i s much l a r g e r than a l l o t h e r f a c e t s so t h a t t h e monocrystal a c t s on l i g h t as a l a m e l l a o f a e r f e c t l y u n i f o r m thickness L. Moreover, t h e i r o r i e n t a t i o n w i t h respect t o t h e glass surface can be determined w i t h o u t ambiguity from t h e i r shapes. Ke have found t h a t BP1 m o n o c r y s t a ? ~ a r e o r i e n t e d p r e f e r e n t i a l l y w i t h (110) o r (211) f a c e t s p a r a l l e l t o t h e g l a s s surface S.

C3-68 JOURNAL DE PHYSIQUE

Fig. 2

-

T h e o r e t i c a l spectra o f t h e r e f l e c t e d i n t e n s i t y when F, = 0. Each f i g u r e i s made f o r a given value o f t h e thickness L o f t h e c r y s t a l :

A. L = 7300 nm ; B. L = 21900 nm, C. L = 36500 nm ; D. L = 51100 nm and f o u r d i f f e - r e n t values o f

IF,II

: a.

IF,II

⁼ 1.45~10-' ; b .

IF,# I

⁼ 2 . 9 ~ 1 0 - ~ ; c. IF,rl = 4.35~10-' d.

IF,# I

^{= 5.8x10-'.}

The wavelength o f t h e maximum i s h = 567 nm l i k e i n experiments.

We have drawn R = I vs.

ax

where t h e o r i g i n o f R i s craving o f 0.1 when passing from

=F

one spectrum t o t h e f o l l o w i n g one.

~ i3

-

~T h e o r e t i c a l s p e c t r a o f t h e r e f l e c t e d i n t e n s i t y w i t h .

I

F,

I

⁼

m

1

I

F,'

I

f o r two d i f f e r e n t v a l u e s o f t h e t h i c k n e s s L o f t h e c r y s t a l : A. L = 21900 nm ; B. L = 43800 nm and f o r t h e same f o u r d i f f e r e n t v a l u e s o f IFu,

I

as on F i g . 2.

111.2. Measurements o f Fa, c o r r e s p o n d i n g t o (110) r e f l e c t i o n

Using a m i c r o s p e c t r o s c o p e ( see t h e p r e s e n t proceedings ) we have measured t h e s p e c t r a o f l i g h t r e f l e c t e d under normal i n c i d e n c e f r o m such f l a t m o n o c r y s t a l s o f t h e o r i e n t a t i o n ( 1 10)//S. U s i n g a nionochromator o f a bandwidth 6A = 2 nm, we have o b t a i n e d s p e c t r a , shown i n F i g . 4, f o r c r y s t a l s o f t h i c k n e s s r a n g i n g between 1 C pm and 30 pm. A l l s p e c t r a show b o t h t h e Pendellosung b e a t s and t h e f i n i t e w i d t h o f t h e c e n t r a l peak p r e d i c t e d t h e o r e t i c a l l y i n t h e p r e v i o u s s e c t i o n .

I n o r d e r t o d e t e r m i n e t h e a m p l i t u d e o f t h e o r d e r parameter, we have proceeded as f o l l o w s :

F i r s t , by p l o t t i n g (%l2 vs. k 2 ( i k i s t h e wavelength o f t h e kth minimum) we have 'may,

v e r i f i e d t h a t t h e t h e o r e t i c a l r e l a t i o n s h i p g i v e n b y e q u a t i o n (37) i s s a t i s f i e d ( f i g u r e 5 ) . From t h e s l o p e s o f t h e l i n e a r p l o t s s e have determined t h e v a l u e s o f t h e p r o d u c t xmaxL f o r each o f t h e s p e c t r a .

Then, knowing xmaXL, we have c a l c u l a t e d t h e v a l u e o f F,, on each s p e c t r a by a succes- s i o n o f a p p r o x i m a t i o n s : a v a l u e o f F,, g i v e s us 1 2 F u '

l

and we f i n d on t h e g i v e n

'max

spectrum t h e c o r r e s p o n d i n o w i d t h

. .

AAF. The i t e r a t i v e processus was w e l l c c n v e r g i n g and n A ~

we stopped t h e i t e r a t i o n when ⁼

I

F,,

I +

1 %. The average v a l u e f o r F,, was max

found :

JOURNAL DE PHYSIQUE

This value of F a , , d e t e m i n e d experimentally, was used p r e c i s e l y i n computing t h e

>

E z

Y l- z

-

t h e o r e t i c a l s p e c t r a shorrn previously i n Fig. 2. When comparing t h e s e s p e c t r a with experimental ones, one observes f i r s t of a l l t h a t PendelSssung S e a t s a r e l e s s deep in experiments than i n theory. This e f f e c t , due t o t h e f i n i t e moncchromator band- width, can be c a l c u l a t e d by a convolution of t h e t h e o r e t i c a l s p e c t r a (equations

(33) and ( 3 4 ) ) with a t r i a n g u l a r shaped pass-band of t h e monochromator. The r e s u l t i n g convoluted s p e c t r a , shown in f i g u r e 6 a r e i n a much b e t t e r agreement with t h e

experinental ones.

The second observation ccncerns t h e shape of t h e top of t h e c e n t r a l peak. I t does not seem t o have a tendency t o f l a t t e n f o r l a r g e thicknesses but on t b e o t h e r hand, i t does not show any marked narrow peak which could be a t t r i b u t e d t o t h e c o n t r i b u t i o n of t h e second eigen p o l a r i z a t i o n o. We concluded t h a t t h e value of F. irust be l e s s than Fa, by a f a c t o r l a r g e r than 10.

500 550 h 1 n m ) 5 0 0 550 A l n m )

-

500 550 A ( n m )

Fig. 4

-

Experimental s p e c t r a f o r s i x d i f f e r e n t values of t h e thickness L of t h e c r y s t a l : A. L = 9800 nm ; B. L = 11400 nm ; C. L = 12900 nm ; D. L = 17900 nm ; E. L = 21400 nm ; F. L = 28100 nm obtained f o r t h e (110) r e f l e c t i o n on a sample of 58,8 % CB15 i n ZLI 1840. The v e r t i c a l s c a l e of i n t e n s i t y i s t h e same f o r a l l t h e s p e c t r a .

D

4

111.3. Calculation of ElIG i n t h e O8 space group

In r e f . 8, Belyakov has given cn Table 1 t h e r e s t r i c t i o n s imposed by t h e symmetry on t h e components of E f o r d i f f e r e n t cubic space groups. Following Eelyakov, l e t us

E

l'

F

A

Fig. 5

-

Experimental v e r i f i c a t i o n of Eq. (37) f o r t h e f i v e f i r s t s p e c t r a ( A , E , C , w o o f f i g u r e 4 and determination of IFb,

l.

choose t h e coordinate frame in t h e following manner : t h e axes X , y , z a r e n a r a l l e l t o the edges of t h e cubic u n i t - c e l l , and t h e o r i g i n l i e s on a t h r e e f o l d a x i s (Fig. 7a)-

Fig. 6

-

Theoretical s p e c t r a of Figure 2 convoluted with t h e t r i a n g u l a r shaped pass-band of 2 nm of t h e nonochromator, f o r t h e two values of t h e thickness L of the c r y s t a l : A. L = 21900 nm ; B. L = 51100 nm.

Most of experimental and t k o r e t i c a l s t u d i e s i n d i c a t e t h a t BP1 has a cubic centered Bravais l a t t i c e and i t s space group assignment is 08. This comes from Laudau theory approach /7/ a s well a s from t h e experiments on c r y s t a l h a b i t /g/. In t h i s paper, we w i l l t h e r e f o r e i n t e r p r e t our r e s u l t s of t h e dynamical l i g h t s c a t t e r i n g a s due t o a

C3-72 JOURNAL DE PHYSIQUE

d i e l e c t r i c t e n s o r E of symmetry 08.

The general form of t h e Fourier component i l 1 0 compatible with t h e O8 space group, given by Belyakov, i s :

where I , 11, R a r e r e a l parameters t o be determined.

In agreement with t h e experimental condition of t h e normal incidence ( e o = 0 ) we take (Fig. 7b) :

The above mentionned experimental f a c t t h a t only one ( U ' ) of t h e eigen p o l a r i z a t i o n s i s s t r o n g l y r e f l e c t e d , reduces t h e number of t h e independent parameters. The theore- t i c a l approximation F,, = 0 l e a d s t o t h e following r e l a t i o n :

Also t h e f a c t t h a t t h e r e f l e c t e d eigen p o l a r i z a t i o n U ' i s c i r c u l a r g i v e s a second r e l a t i o n :

With t h e one independent parameter l e f t vie g e t :

i / 2

-

i / 2

-

i/21/Q

i~~~ = IO

[ -

^{i / 2 l / n}

^-

^{i / 2}

^{I/n - -}

^{l/v? i}

Using t h e d e f i n i t i o n of F,,, (equation ( 1 3 ) ) we c a l c u l a t e : I1

I

F , , ,

1

^{= 2 = 0.0145 f I f 3}

l

Fig. 7

-

( 7 a ) coordinate frame used f o r c a l c u l a t i o n s of i l 1 0 and 6200

(7b.c) Vectors

g:

of t h e wave planes o f t h e i n c i d e n t and r e f l e c t e d waves i n s i d e t h e c r y s t a l f o r (11C) r e f l e c t i c n (b) and (200) r e f l e c t i o n ( c ) .

111.4. Calculation of i200 i n t h e O8 space Group

From Belyakov, t h e general form of

eZO0

compatible with t h e O8 space group i s :

when R and I a r e r e a l paraneters. (42)

Again i n agreement with t h e experimental condition

e 0

= 0, we t a k e (Fig. 7c) :

;:

^;

^; "

^=;

The condition t h a t Fu = 0 implies t h a t R = I (This condition corresponds t o t h e s a r e c h i r a l i t y of r e f l e c t i o n a s t h a t f o r (110) r e f l e c t i o n i n t h e above paragraph).

Then ny' i s automatically a c i r c u l a r p o l a r i z a t i o n . Ke have then from equation (13)

So i f we would g e t s p e c t r a s from a p e r f e c t monocrystal with t h e s u i t a b l e o r i e n t a t i o n (planes (200) p a r a l l e l t o t h e g l a s s s u r f a c e ) , we could obtain t h e value of R. Here t h e r e s u l t s about p o l a r i z a t i o n implied a d r a s t i c s i n p l i f i c a t i o n of t h e component i700, with j u s t one independent pararceter l e f t (and t h e same f o r i I l 0 ) . Unfortunately, up t o d a t e , t h e s p e c t r a s we g o t from p e r f e c t m o n o c r y s t a l l i t e s w i t h ' t h i s o r i e n t a t i o n d i f f e r from t h e ideal t h e o r e t i c a l shapes f o r two main reasons : ( I 0 ) t h e c r y s t a l S of such an o r i e n t a t i o n have been only nucleated i n volume so t h a t t h e i r o r i e n t a t i o n chan- ges very e a s i l y under Brownian r o t a t i o n a l motion ; (2') t h e i r forms, a s seen from

(100) d i r e c t i o n a r e not f l a t because t h e (100) f a c e t s a r e missing /g/. Consequently, t h e i n t e r p r e t a t i o n of t h e s e s p e c t r a i n view of determining t h e value of i200 i s not possible using t h e p r e s e n t theory.

IV

-

CURFRONTATION WITH LANDAU THEORY

IV.1. Components (110) and (200) of t h e order warameter

As we have shown in t h e previous s e c t i o n , experimental r e s u l t s concerning p o l a r i z a - t i o n of r e f l e c t e d l i g h t a t normal incidence, applied t o t h e C8 group, l e a d t o a d r a s t i c s i m p l i f i c a t i o n of t h e components E l and d Z O 0 (eq. (41) and (43))of t h e o r d e r parameter.

Other Fourier components of t h e s e r i e s ( 1 ) corresponding t o t h e same type of r e c i p r o - cal l a t t i c e v e c t o r , can be obtained from i l 1 0 and i200 by applyin'j t h e syvmetry operations of t h e O8 group (Fig. 8 ) . For i n s t a n c e , i f one makes a t r a n s l a t i o n of the o r i g i n from (O,O,O)~to (1/4,3/4,1/4) and a r o t a t i o n of axes of 90" around

f:

i n the new coordinates frame i020 i s the same a s was iZo0 i n t h e o r i g i n a l frame. ay t h i s mean, we can c a l c u l a t e c o n t r i b u t i o n s of a l l terms of type (1 10) ((1701, (01 l ) ,

. .

^{. )}

id e s t 12 terms, t o the o r d e r parameter :

JOURNAL DE PHYSIQUE

Fig. 8

-

A cubic u n i t - c e l l o f t h e O8 symmetry space group showing a l l t h e axes o f symmetry.

where Cl = cos qx,

. . .

S = s i n qx,

...

^{and q =} 21~/a, a being t h e l a t t i c e constant 1

and c 2 ⁼ SIC2

+

S2C3 + S3C,

The expression (44) i s e x a c t l y t h e same as t h a t found by Hornreich and a l . /7/, keeping o n l y t h e lowest energy harmonic (1 10) i n a Landau theory approach.

The c o n t r i b u t i o n o f t h e s i x terms o f type (200) i s :

where C { = cos 2qx

...

S i = s i n 2qx

...

and i s a l s o i n agreement w i t h t h a t found by Hornreich e t a l . /7/. So f a r , t h e o p t i - c a l theory, t a k i n g i n t o account experimental datas, f i t s v e r y w e l l t h e Landau theory i n t h e sense t h a t o n l y the s p h e r i c a l harmonics o f lowest energy (m = 2) a r e present i n each component iT o f t h e o r d e r parameter.

I f measurements o f t h e r a t i o Io/R were possible, then we should be a b l e t o compare i t t o t h e r a t i o obtained by t h e Landau theory o f m i n i m i z a t i o n o f t h e f r e e energy.

IV.2. V i s u a l i z a t i o n o f t h e o r d e r parameter

Although t h e a n a l y t i c expressions (44) and ( 4 5 ) o f t h e order parameter

~ ~ ( $ 1

have been d e r i v e d above, i t i s s t i l l n o t easy t o imagine what i s a 3D o r i e n t a t i o n a l order

corresponding t o them. Therefore we have developed a graphic representaJion o f the t e n s o r i a l order parameter. A t each p o i n t o f t h e u n i t c e l l , t h e tensor i ( r ) i s diago- n a l i z e d and reprgsented by p l o t t i n g a b r i c k , which o r i e n t a t i o n s are eigen d i r e c t i o n s o f t h e tensor i ( r ) and dimensions are p r o p o r t i o n a l t o t h e corresponding eigen values.

Using o n l y the f i r s t m a t r i x (eq. ( 4 4 ) ) , corresponding t o t h e c o n t r i b u t i o n o f terms o f (110) type, we have g o t t h e r e s u l t s o f f i g u r e 9. This v i s u a l i z a t i o n i s a very convenient way t o f i n d t h e p l a c e o f d i s c l i n a t i o n l i n e s i n t h e l a t t i c e and t o d e t e r - mine o f what k i n d they are. I f one looks i n (111) d i r e c t i o n (Fig. 9b) and t u r n s around such a t h r e e f o l d a x i s , one can see t h e r e i s a S =

-

112 d i s c l i n a t i o n near t o it. The t h r e e - f o l d a x i s i s n e c e s s a r i l y a p l a c e where t h e d i e l e c t r i c t e n s o r i s u n i - a x i a l , so t h e d i s c 1 i n a t i o n l i n e (where t h e tensor must a l s o be u n i a x i a l ) i s s i t u a t e d p r e c i s e l y on t h e t h r e e - f o l d axis.

Fig. 9a

-

A cubic u n i t - c e l l where o n l y planes o f t y p e (100) a r e drawn. Binary axes are i n d i c a t e d by (b), screw axes o f t h e same s i g n as t h a t o f c h o l e s t e r i c t w i s t a r e i n d i c a t e d by (+), screw axes o f t h e o p p o s i t e s i g n by ( - ) .

I n t h e O8 group o f symmetry, t h e r e are two s o r t s o f screw axes o f opposite s i g n p a r a l l e l t o t h e ( l o o ) , (010) and (001) d i r e c t i o n s (Fig. 8). One o f those signs i s

JOURNAL DE PHYSIQUE

Fig. 9b

-

A view l o o k i n g along

L l l l ]

d i r e c t i o n . Around t h e [llll t h r e e - f o l d a x i s , one can see a S =

-

1/2 d i s c l i n a t i o n . On t h e screw a x i s o f opposite s i g n from t h a t o f t h e c h o l e s t e r i c t w i s t , t h e b r i c k s are t u r n i n g round t w i c e f a s t e r than j u s t o f f the a x i s , and i n t h e opposite sense.

a l s o t h e s i g n o f t h e n a t u r a l t w i s t o f t h e c h o l e s t e r i c m a t e r i a l . Around t h a t type o f axis, one cannot see any d i s c l i n a t i c n , b u t around t h e o t h e r type o f screw-axis, t h e r e i s a d i s c l i n a t i o n o f s t r e n g t h

-

1/2 (Fig. 9c). I f one l o o k s a t t h e m a t r i x , one can see t h a t along such a screw-axis, t h e t e n s o r i s

not

u n i a x i a l , thus t h e d i s c l i n a t i o n l i n e i s n o t on i t . Moreover, on f i g . 9b, one can see t h a t t b e tensor t u r n s t h e r g t w i c e f a s t e r than j u s t o f f t h e a x i s , and i n t h e opposite sense. By computing E ( r ) i n t h e neighbourhood o f t h i s screw a x i s , we have found t h a t t h e o r d e r parameter i s u n i a x i a l on a s p i r a l l i n e t h a t t u r n s t h r e e times around t h e screvi a x i s over a l a t t i c e period, and i n t h e sense o f t h e c h o l e s t e r i c t w i s t (which i s the opposite s i g n o f t h a t o f t h e screw a x i s i t s e l f ) . Over t h a t period, t e n s o r on t h e screvi a x i s t u r n s o f

-

^IT, w h i l e o f f the a x i s i t t u r n s o f + IT. The r o t a t i o n o f 6a o f t h e s p i r a l l i n e around t h e screw a x i s i s t h e minimum r o t a t i o n imposed by t h e symmetry correspor- ding t o t h e screw a x i s , and o v e r a period, the d i s c l i n a t i o n t u r n s o f 16a/3 around the s p i r a l l i n e t o a l l o w the opposite r o t a t i o n s c f t h e t e n s o r described j u s t above

(Fig. 10). This r e s u l t i s q u i t e an i n t e r e s t i n g one, you cannot f i n d by o n l y l o o k i n g a t t h e m a t r i x (44), so v i s u a l i z a t i o n o f such tensors i s very h e l p f u l 1 f o r comprehen- s i o n o f t h e o r i e n t a t i o n a l order i n EPI i n such a model.

V

-

CONCLUSIONS

I n t h i s paper, we have t r i e d t o develop t h e dynamical t h e o r y o f l i g h t d i f f r a c t i o n i n i t s a p p l i c a t i o n t o BPs i n our o r i g i n a l way, t a k i n g i n t o account t h e experimental re-sul t s on p o l a r i z a t i o n o f Bragg r e f l e c t e d l i g h t on BPs. From experimental spectras

F i g . 9c

-

The screw axes ( + ) e x h i b i t no d i s c l i n a t i o n around them, w h i l e t h e o t h e r screw axes

( - 1

show s =

-

1 / 2 d i s c 1 i n a t i o n s around 'them.

o b t a i n e d on l a r g e p e r f e c t m o n o c r y s t a l s , and f r o m t h e i r comparison w i t h t h e t h e o r e t i - c a l ones, we have deduced t h e f i r s t measurement o f t h e F o u r i e r comconent iIl0, and shown t h a t one o f t h e two e i g e n p o l a r i z a t i o n ~ was c e r t a i n l y v e r y weakly r e f l e c t e d w h i l e t h e o t h e r one was s t r o n g l y . F u r t h e r measurements of t h e p o l a r i z a t i o n o f t h e r e f l e c t e d l i g h t a r e necessary and b e i n g developed

,

t o e v a l u a t e t h e e x a c t c o n t r i - b u t i o n o f t h e ^U p o l a r i z a t i o n t o t h e Bragg r e f l e c t i o n s .

Measurements f o r o t h e r F o u r i e r components such as (ZOO), (211) and (220) need i m p o r t a n t m o d i f i c a t i o n o f t h e experience, because under normal i n c i d e n c e , m u l t i p l e r e f l e c t i o n s o c c u r t h a t f o r b i d easy i n t e r p r e t a t i o n o f e x p e r i m e n t a l s p e c t r a , w h i l e w i t h o b l i q u e i n c i d e n c e , i t i s p o s s i b l e t o a v o i d such m u l t i p l e r e f l e c t i o n . Such experiments w i l l be developed i n t h e n e a r f u t u r e .

Then a more p r e c i s e c o n f r o n t a t i o n w i t h t h e o r y t h a n t h a t done i n s e c t i o n I V . l w i l l be p o s s i b l e and perhaps w i l l a l l o w t o d i s t i n g u i s h w i t h i n t h e v a r i o u s models /15, 16/ what i s t h e most r e a l i s t i c .

kle w i s h t o acknowledge R.M. H o r n r e i c h f o r s t i m i l a t i n g d i s c u s s i o n s and P.P. Crooker f o r a p r i v a t e communication about m u l t i p l e r e f l e c t i o n s . L a s t , b u t n o t l e a s t , we want t o u n d e r l i n e t h a t t h i s work c o u l d n o t have been done w i t h o u t t h e c o n t r i b u t i o n o f P. C l a d i s who gave us t h e m a t e r i a l and s t i m u l a t e d o u r i n t e r e s t about t h a t study.

JOURNAL DE PHYSIQUE

Fig. 10

-

S p i r a l d i s c l i n a t i o n l i n e round a screw a x i s o f o p p o s i t e s i g n o f t h a t o f t h e c h o l e s t e r i c t w i s t . Along t h e screw a x i s t h e t e n s o r t u r n s o f

-

2a o v e r a p e r i o d w h i l e

o f f t h e a x i s , i t t u r n s of + II o v e r t h e same p e r i o d . The s p i r a l makes t h r e e t u r n s around t h e screw a x i s o v e r a p e r i o d because o f t h e 41 symmetry.

REFERENCES

/ l / For a review, s e e H. Stegemeyer and K. Bergmann, i n L i q u i d C r y s t a l s o f One and Two dimensional O r d e r , e d i t e d by W. H e l f r i c h and G. Heppke ( S p r i n g e r - V e r l a g , B e r l i n , 1980) p. 161.

/2/ Johnson D.L., F l a c k J.H. and Crooker P.P., Phys. Rev. L e t t .

45

(1980) 641 /3/ Meiboom S. and Sammon M., Phys. Rev. L e t t .

44,

(1980) 882

/4/ Marcus M., J . Phys. ( P a r i s ) 42 (1981) 61

/5/ Crooker P.P., Mol. C r y s t . L i F C r y s t . 9 8 (1983) 31

/ 6 / Marcus M. and Goodby J.H., Mol. C r y s t . 7 i q . C r y s t . L e t t . 71 (1982) 297 /7/ Grebe1 H . , Hornreich R.M. and S h t r i k n a n S., Phys. Rev. A , 7 8 (1983) 1114 /8/ Belyakov V.A., Dmitrenko V . E . and Osadchii S.M., Sov. ~ h y s 7 J E T p 56 (1982) 322 /9/ Barbet-Rassin R., C l a d i s P.E. and P i e r a n s k i P.

,

Ph s. Rev. F., 3 0 7 1 9 8 4 ) 1161 /ID/ Batterman B.W. and Cole H . , Rev. Of liod. Phys. 36 (1964) 681

-

/11/ Belyakov V.A., Sov. Phys. Usp. 10 (1975) 267 ^-

/12/ Chandrasekhar S. and S h a s h i d h a r a P r a s a d J . , Mol. C r y s t . and Liq. C r y s t .

14

(1971) 115

/13/ Z a c h a r i a s e n , i n Theory of X-Ray d i f f r a c t i o n i n C r y s t a l s , p. 123, Dover P u b l i c s t i o n s IPIC., F!ew York (1367)

/14/ I n t h e e x p r e s s i o n g i v e n by Belyakov i n ( 8 ) , p. 326, we have n o t i c e d t h a t P s h o u l d

A

-

1

be equal t o s i n 2 ( x k o ~ ) + s i n 2 ( x z u ~ )

/

and n o t t o

-

¹

s i n 2 ( x ~ , ~ ) 1 l b l * ; + s i n 2 ( x ~ , ~ ) /

.

/Is/

H o r n r e i c h R.M. e t a l , Phys. Rev. A ,

30

(1984) 3264

/16/ Meiboon~ S., Samrnon M. and Brinkrnann W.F., Phys. Rev. A - 27, (1983) 438