Electrostriction in quasicrystals and in the icosahedral liquid crystalline Blue Phases

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Electrostriction in quasicrystals and in the icosahedral liquid crystalline Blue Phases

H.-R. Trebin, W. Fink, H. Stark

To cite this version:

H.-R. Trebin, W. Fink, H. Stark. Electrostriction in quasicrystals and in the icosahedral liq- uid crystalline Blue Phases. Journal de Physique I, EDP Sciences, 1991, 1 (10), pp.1451-1468.

�10.1051/jp1:1991219�. �jpa-00246427�

J.

Phys.

1France1

(1991)

1451-1468 ocToBRE1991, PAGE 1451

Classification Physics Abslracls

77.60 61.50E 61.30

Electrostriction in quasicrystals and in the icosahedral liquid crystalline Blue Phases

H.-R.

Trebin,

W. Fink and H. Stark

Institut ffir Theoretische und

Angewandte Physik

der Universitht

Stuttgart, Pfaffenwaldring

57, D-7000

Stuttgart

80, Gerrnany

(Received

8 March 1991, accepted in

final form

J4 June 1991)

Rksomk. Sous l'effet de

l'61ectrostriction,

les cristaux sont d6forrn6s _{par un}

champ 61ectrique

en second ordre. Les coefficients du tenseur d'dlectrostriction rattachant le

champ dlectrique

_aux composantes du tenseur des ddformations _ont 6t6 mesur6s pour les

m6sophases cubiques

Phase Bleuel et II. Dans _un

systdme quasi p6riodique,

_par

exemple

dans les

quasi-cristaux icosa6driques

Almn _ou Alcufe, ou dans le moddle

icosa6drique

de la Phase Bleue III le

champ 61ectrique

_peut aussi _{causer une} d6formation des

phasons.

Ici nous d6terminons le tenseur

d'61ectrostriction

g6n6ralis6

_pour des

phases icosa6driques

et

d6cagonales

et nous le mettons en

rapport avec les tenseurs

g6n6ralis6s 61astiques

et

61asto-optiques.

Il _{y a} des valeurs du

champ 61ectrique

off les

systdmes quasi p6riodiques

peuvent devenir

p6riodiques

_ou bien _en direction du

champ

61ectrique

_ou bien perpendiculaire I lui.

Abstract. In the electrostriction process

crystals

_are deforrned by an electric field in

quadratic

order. The components of the

corresponding

electrostriction tensor, which relates the _square of the electric field components to the components ^{of the strain} tensor, ^{have been measured}

recently

for the cubic

liquid crystalline

Blue Phases. In

quasiperiodic

_systems, like the icosahedral metallic

alloys

Almn _or Alcufe, _or in the icosahedral model for the Blue Phase III the electric field _can also induce

phason

strain. Here _we determine the corresponding

generalized

electrostriction

tensor for icosahedral and

decagonal phases

and relate it to the

generalized

elastic And elasto-

optic

tensors. For certain values of the field,

quasiperiodic

_{systems can} become

periodic

either

along

the field direction _or

perpendicular

to it.

1. Inhoducfion.

The response of _an elastic medium to _an

applied

electric field E iS

expressed

in lowest orders

by

the

piezoelectric

tensor y and the electrostriction tensor R. These

provide

the

expansion

coefficients for the strain tensor e :

e;y ⁼ y;~~

E~

+

R,j~t E~ Et

I,

j, k, I

_e

(1, 2,

3

(1)

In

geometrical

linearization the strain tensor is related to the

displacement

_vector field

u(x) by

j 3U;

3Uj

~'J ~

2

a~

^~

ax;

1452 JOURNAL DE

PHYSIQUE

I bt 10

For

periodic crystals

the number of

independent piezoelectric

and electrostriction coefficients is determined

by

the

point

_{symmetry group} of the

crystal through simple

_arguments of the

theory

of group

representations.

A cubic

crystal,

for

example,

does not

display piezoelectricity

but electrostriction with three

independent

coefficients.

Incommensurate _or

quasicrystalline

_{systems carry} also

phase degrees

of

freedom,

denoted

phason elasticity,

in addition to the translational

degrees

of

freedom, expressed by

the direct space

displacement

field

u(x).

Their structure

commonly

is elucidated

by adding

to the

physical

space of dimension d _an

orthogonal

_space of _n

dimensions,

where _n counts the incommensurate modulation

frequencies. Formally

the

system

^{is described} as a cut of _an

(n

₊ d

)-dimensional hypercrystal by

_a d-dimensional

hyperplane,

^the

physical

_space. ^Phason

strain is due _{to an} n-dimensional

displacement

field

w(x)

^{of the}

points

of the

hypercrystal along

the

orthogonal

_space and is

quantified by

_an

(n

_x

d)-phason

strain matrix

E and

x

_are

sually packed together to an ((n + d)

x -matrix ~ of

generalized

For [1, 2] the

dimension of

_real space _is

d

⁼ _3, and of

space

is n

= 3.

_For cagonal uasicrystals [3] _which

are

in

one and

quasiperiodic in the

planes orthogonal to it, the numbers are _d

=

3, n ₌

additional

dimensions will

be

discussed in ection 3. Now the ew phenomenon is

that

an

electric

field can give rise both to _the conventional formation («

phonon »

strain) phason strain.

7~~~=y~~~E~+R~y~tE~Et,

ae

(1,2,..,n+d ); j,k,ie (1,2,..,d). (5)

Electrostriction has been

investigated

_very

intensively

in most exotic

crystals, namely

the cubic

liquid crystalline

Blue Phases I and II

(BP

I and

II) [4].

These _are

liquids

of

organic, elongated molecules,

where the

long

_axes of the molecules form _a

complex

orientational

pattern

^of cubic _{space group}

symmetries ^O~

and

02, respectively.

For _a review _see reference

[5].

The cubic lattice _constant is of the order of several hundred _nm. Electrostriction is

observed

by

_a

wavelength

shift of

Bragg-reflected electromagnetic

radiation in the visible

[6, 7, 8]

or

by

deformation of Kossel

diagrams,

also in the visible

[7].

There is a third Blue

Phase,

BP

III,

which appears

amorphous

and reflects

light

in _a broad band. Its _{response to an} electric field has been

investigated [9-13]

with the

general feature,

that the band _narrows and

increases its

intensity drastically

with much shorter relaxation times than _are found for the

wavelength

shifts in the cubic Blue Phases. For the yet unknown _structure of BP III several models have been

proposed. Theoretically

most

appealing

is _a

quasiperiodic

model of

icosahedral

point symrnetry [14-17].

^In such _a model also the

phase degrees

of freedom

respond

to the electric field such that at certain field values the

system

becomes

periodic along

the field direction

(in

structure similar to the

decagonal T-phase)

_or in the

plane

perpendicular

to the field direction.

In section 2 _we calculate the

independent

_components of the

generalized

electrostriction

tensor

by

_group

theory

for the _cases of icosahedral and

decagonal symmetry. Then,

in

section

3,

_we

investigate

in

reciprocal

_space, how the _structure of

quasicrystals

and icosahedral

liquid crystalline

Blue Phases

changes

under the influence of _an electric field and

bt 10 ELECTROSTRICTION IN

QUASICRYSTALS

1453

determine field values at which

pedodicity

is induced. In the final section _we relate the

generalized

electrostriction tensor to

generalized

elastic and

elasto-optic

tensors.

2.

Components

of the

genera1izell

elechoshicfion tensor.

We _now determine the

components

^{of the}

generalized

electrostriction tensor

by

methods of the

theory

of group

representations.

Let _us consider first the _case of icosahedral

quasicrystals.

We regroup the components 7~~j ^of the combined

phonon

and

phason

strain field into several sets 7~)~~ which form bases for irreducible

representations

_x of the icosahedral _group

Y~.

This group consists of the 120

operations (including spatial inversion)

which leave _an

icosahedron invariant. The

regrouping

is also

performed

for the

components E~

of the electric field to sets

E)~~

and its

dyadic product E~ Et

to sets

[E @ E])~~

In the bases of irreducible

representations

the

piezoelectric

and electrostriction coefficients _are

simply proportionality

factors between sets of the _same irreducible

representation (the proof

is

straightforward

and

uses Schur's lemma

[18]).

The

components

^{of the} vector fields uy,

E~,

^{and the}

gradient

? transform

according

to the

representation l'~~,

the

components

^of vectors in

orthogonal b,

space, like

wj,

^transform

according

to

r]~ [19, 20].

The

symmetrized product

_e;~ of

? and _u; divides into the irreducible

representations a,

(~3u 4§ ~3u)s

"

~lg

~

~5g

>

(6)

which _{are one-} and

five-dimensional, respectively (s-

and d-like

just

_as in _case of full rotational

syrnmetry).

The

product _x;y

of ^? and w; divides into

a,-

~3u 4§ ~ju

~

~4g

fl~

~5g (7)

The vector field

E; transforming according

to

r~~

^does not find _a

partner

among the

representations (6)

and

(7),

and therefore

piezoelectricity

is not

present.

^The

argument

^is

unchanged

for the icosahedral group Y without

inversion,

where for the irreducible

representations

the suffixes _u and g are

dropped.

The transformation

properties

of the

quadratic

field tern

E; _Ey

_are the _{same as} for the strain _tensor e,~. ^Thus we see, that there _are two electrostriction coefficients for the

phonon

strain

(from

the

point

of

symrnetry

^identical to the Lamb-constants for

isotropic media),

and

one for

phason

strain.

For

phonon strain,

the

proportionalities

_are :

For the irreducible

representation rj~.

~(~~.=tre=RjE~=Rj[E@E]~~~ ₍₈₎

For

rs~

~ l I

~

^{T Eli + m E22 + T + E33}

2 3 _r T

~~~ ~~~~ ^~

~~~

~~~'~~°~°~~ _"

/

E12 ~

/

e~~

/

e~j

JOURNAL DE PHYSIQUE I T i, M 10, OCTOBRE <WI 57

1454 JOURNAL DE

PHYSIQUE

I M 10

2 2

1~2 lj~~j

@

^~

~l~/

~~

~~j

3

j _{El rEj}

+

E))

=

R~ / _{Ej E2}

=

R~[E

@

El

^(~~

(9)

/ _{E~ E~}

/ _{E~ Et}

The irreducible

representations

have been chosen

according

to reference

[19],

_r is the

golden

mean

(l

₊

/). _Solving equations (8)

and

(9)

for the components

e~~ of the strain tensor, 2

we obtain the electrostriction _tensor in the standard

Voigt

notation _:

ejj

(Rj

₊ 2

R2) (Rj R2) (Rj R~)

0 0 0

E)

E22

(Ri R2) (Ri

₊ 2

R~) (Ri R~)

o o o

E]

e~~

(RI R2) (Rj R~) (Rj

+ 2

R~)

0 0 0

E)

E23 ° ° °

R2

^° °

E2 E3

E31 ° ° ° °

R2

^°

E3 El

E12 ° ° ° ° ° R2

El E2

(10)

The

phason

electrostriction constant R~ connects the

following

^basis functions for the

irreducible

representation l's~.

jj _(X

Ii

X22)

~ _(Xii

2

+ X22 ~

X33)

6

~~~'~~~~°~~ ~

~

₃ TX21

X12)

^~

T

i i

/

₃ ^{TX 32 X23}

T

i i

/

^~~~~

^i~~~

~ _~~~~~

~~~~~ ^~

^~

^~~

El rEj

₊

2 _r

E))

= R~

/ _{Et E~}

=

R~[E @ E](~~ (l I)

/ _{E~ E~}

/ _{E~ Ej}

ti lo ELECTROSTRICTION IN

QUASICRYSTALS

1455

whereas the basis functions for the irreducible

representation r~~

are not influenced

by

the field _:

~

~~~~ ~ ~~~ ~

~~~~

l I

~q p

^{X21 + TX12}

~(~~

=

~

= 0.

(12)

~q

₃ ^{X32 + TX 23}

r

I

/

₃ ^{X13 + TX31}

r

Solving equations (I I)

and

(12)

for the

components

_x;j ^of the

phason

strain tensor _we obtain _:

~~j

"

~ ^{~ ~( ~} _('3)

33 3

T I

T

~~~l" _dl3° ~ _~~~~j

~

~~ ^~~ _('4)

X21

l~l1~2

_X12

The

generalized

electrostriction _tensor for

decagonal quasicrystals

is derived in the

Appendix.

3. Structural

changes

of

quasicrystals

and icosahedral Blue Phases under the influence of _an electric field.

3.I THE _{PHASON DEGREES oF FREEDOM.} The presence of

phason degrees

of freedom is

most

clearly

seen in

reciprocal

_space

[21].

We therefore describe the

reciprocal

lattice of

quasicrystals

in _some detail and evaluate its

changes

under field induced

phonon

and

phason

strain. The

long-range positional

order of

periodic crystals

_or

quasicrystals

reveals itself

by sharp

diffraction

patterns

ⁱⁿ

X-ray

_or electron

scattering.

The Fourier transform of the _mass

density

p(x)

=

jjp(k)exp(ik.x) (15)

only

contains wavevectors k out of _a countable set in

reciprocal

_space. For

liquid crystalline

Blue Phases the scalar _mass

density

_p

(x)

and its Fourier coefficients _p

~k)

have to be

replaced by

the

traceless, symmetric quadrupole alignment

tensors

Q(x)

_or

Q(k) respectively [22].

The

reciprocal

_space for

periodic

three-dimensional

crystals

is _a Bravais

lattice, spanned by

three basic vectors k~,

3

k=

£ njkj, n~ell. (16)

j=1

If such _a

crystal

is deformed

elastically

due to _a

displacement

field _u

(x),

then it is described

by

1456 JOURNAL DE

PHYSIQUE

I ti 10

a mass

density p (x),

which at the

point

_{x + u}

(x) equals

the undistorted _mass

density

at

point

X~

P(X

_{+ U}

(X))

" P

(X) (17)

For _a constant strain tensor _e the

displacement

field is

(up

to

rigid

rotations _or

translations) u(x)

= ex, and hence _:

~~~~~~~~~~ jizP~i~Ji~exPli~~nJkJ.

(x-ex)j

=

jj

_p

(n j) )

_exp

13

I

jj

_{n~ k~} x ,

(18)

nj~Z j=1

where

ij 13

=

(I e) kj

^{and where} we have denoted _p k

=

jj

_n~

kj by

_p

( (nj) ). Thus,

the

j =1

reciprocal

_space of _a

uniformly

deformed system ^{is deformed in} an inverse way.

In the

density-wave picture (and similarly

in the

plane-wave expansion

of the

quadrupole tensor),

the deformation _can

simply

be viewed _{as a} space

dependent phase

shift

q~j(x)

₌

-k~ u(x)

=

kj

ex

(19)

3

for each basic mode in

equation (16),

and of

£ _{nj q~~(x)}

for each harmonic.

j i

The _wave vectors of the diffraction spectrum ^{and hence of}

reciprocal

_space for three-

dimensional icosahedral

quasicrystals

_{are sums} and differences of six of the twelve vertex vectors of _an icosahedron

(Fig. I,

the

remaining

six _wave vectors differ

only by sign).

These

I

/

,

/

'

/~

yi

X'

6

Fig.

I. Icosahedron in

fhysical

^{space E~. The six indicated vertex vectors are}

^integrally independent

and _serve as basis

(k(, k~,

,

k

()

of the

reciprocal

lattice.

bt 10 ELECTROSTRICTION IN

QUASICRYSTALS

1457

are incommensurate.

If,

for

example,

_we

project

all vertex vectors onto a threefold

symmetry

axis

passing through

_a

triangular

surface

element,

_we obtain two

lengths

_a and b of the

irrational ratio

r~ (Fig. 2).

The

partial

_waves

along

this direction therefore _superpose to _a

quasiperiodic

instead of _a

periodic density

function. A

projection

of the vertex vectors

along

the threefold symmetry ^axis

(Fig. 2) yields equilateral triangles

of the two

edge lengths

_c and

d,

which _are of ratio _r. In the

plane, therefore,

the

pattern

^{also is}

aperiodic.

As _now there _are six

independent phase

shift fields and hence three _more

degrees

^of

freedom,

_we

formally

add _{a space}

E~

_of three dimensions to

physical

_space, which is denoted

parallel

_space

E'.

_{The vertex} wavevectors k~

=.kj

are

supplemented by orthogonal components

k/

thus that their _sums

,

=

kj

+

k/, j

₌ 1,

...,

6, point

to the vertices of _a six-dimensional

hypercube.

The inner

product

of the lifted

position

vector

(=x' _{+x~, x":=x,}

_with

<y:

<y

(

= k

x'

+

k/

^x~

(20)

is identical to the

phase

factor

ky.x

of each basic mode _as

long

_{as we} remain in

E",

I-e- _as

long

_as x~ vanishes. Thus the

quasi-periodic

_mass

density

function p

(x) 16

=

£

_p

(n j) )

_exp I

£

_n

j

kj

x

(21)

nj~Z j=1

is _a three-dimensional _cut

through

the

corresponding periodic

_mass

density

function p

(~~(() 16

=

£

_p

(nj))

exp ^I

jj

_{n~ <y}

( (22)

nj~Z

j

Fig.

2. Vertex vectors of the icosahedron

along

a threefold symlnetry axis. The

projection

of the vectors onto the axis

yields

two

lengths

_a and b of irrational ratio

r~,

the

projection parallel

to the axis

yields

two types ^of

equilateral triangles

of incoIninensurate ratio of the sides _{c :} d

= r.

1458 JOURNAL DE

PHYSIQUE

I bt 10

Phonon and

phason

deformation _{now causes} six

phase

shift functions _q~

j(x)

_as in

equation (19)

and _can be

expressed through displacement

fields

u(x)

in

E"

_and

w(x)

in E~

q~~

(x)

=

kj

_u

(x) k/

_w

(x) (23)

Under

homogeneous phonon

and

phason deformations,

_we have _u

(x)

= ex, ^w

(x)

= xx, and

with

equation (4)

Wj(x)

_{= <~ ~x.}

(24)

Thus,

the

phase

factors of the six basic _mass

density

_waves

(21) change

from

kj

x to

k~.x+wj(x)

=

(kj-~~<~).x=.i~.x ₍₂₅₎

with

ij

=

(I

_E

) k) x' kh'i(~°"°"

₊

ijhas°" ⁽²⁶⁾

3.2 STRUCTURAL CHANGES OF ICOSAHEDRAL BLUE PHASES _IN AN ELECTRIC FIELD. In

the framework of standard

electrostriction,

where the electric field

only

_causes

spatial

~phonon-like) deformation,

_a system ^of icosahedral

point symmetry

^behaves

exactly

like _an

isotropic _system (Eq.(10)), govemed by

two electrostriction coefficients

Rj

and

R~.

According

to

equations (10)

and

(26)

_an electric field scales the

reciprocal

_space of the system

along

the field direction

uniformly by

the factor

ii

=

1-

(Ri

₊ 2

R~) E~ (27)

and

orthogonal

to it

by

the factor

f~

₌

(Rj R~) ^E~ (28)

respectively, expressed

in compact ^form

by

I(~°~°~

=

If i(k k)

₊

_f2(1 k _@ #)) _k)

,

(29)

where

k

=

El

E

Thus the irrational ratios of the _wave vector

projections

as shown in

figures

2 and 3 _are not

changed by

the field.

The

phason

shift of the wavevectors is

according

to

equation (23)

I)~~S°~ = X~

k) (30)

with _X

being provided

b~ e~uations ⁽¹³⁾

^and

^(14).

Since

q

<; ₌ 0 and

k) k;

=

k/ k)

for I #

j,

^the

three-component

vectors

k)

_can be

interpreted

_as vertex vectors of _an icosahedron dual _{» to} the

original

_one

(see Figs.

I and

4)

with the three

orthogonal

basis _vectors in direct and

orthogonal

_space

being

chosen _as in reference

[19],

the

three-component

vertex vectors of the icosahedron and its dual _are related

by

:

kt k(

=

kl k(

= =

kt k(

= °

(31)

The

phason

_parts of the electrostriction

(26)

_{now are} able to

change

the ratios of the wavevector

projections

onto _or

parallel

to certain _axes.

bt 10 ELECTROSTRICTION IN QUASICRYSTALS 1459

-/

C5

1

Fig.

_3.

-Projection of the vertex vectors _of

onto

the

,,

~

4

Fig. 4. - orthogonal kf

the « dual »

cosahedron in

E~.

Combining ^quations (10), (13),

(14)

and

along

he threefold xis

r~fj+QiR~E~

~ 4R~ ~

1460 JOURNAL DE

PHYSIQUE

I bt 10

for the

projections

of the icosahedral vertex vectors onto the field direction. For certain values of E the ratio _can be made

rational,

for

example

for

F~~

j

~2

³

Qi Fk

^~

2

R3 r~(2

r

1)

^'

where

~~~~

is the k-th rational

Fibonacci-approximation

of the

golden

_{mean r} with

F~

F~~

i ⁼

F~

₊

F~_

j,

Fo

=

0,

and

Fj

= I.

If R~ ~

0,

then for real electric fields

only

the values

~~

~ a r, I-e- the _upper

approxirnants F~

2, ~, (

,

of the series 1,

2, (, _~,

^~

,

~~,

_are allowed

(for

_R~

~ 0

only

the lower

ones).

5 8

For R~ _~ ⁰

(R~

_~

0)

_we obtain

a :b

=

2

~~

~ + l _{a r} ^~

(respectively

_{s r}

,

F~ (33)

I-e- 5

(3)

for the first allowed

approximant

and the strongest ^{suitable field.}

Hence,

in this _case

periodicity

is induced

along

the threefold _axes while

quasiperiodicity pertains

in the normal

plane. Systems

of this kind _are known _as

T-phases

in

multicomponent

metallic

alloys (e.g.

Alcuco, AlcuNi).

Other fields

yield

rational ratios for the

edge

sizes of the

projected triangles,

r9f~-2QiR~E~ _4R~

~~~ ~

9f~+r2/R~E~

^~

3Qi~~~~ ^~~'

for

example,

for

~

Ft

~ j

~~ ~

r

(2

_r

~l

~~~~

Here for

R~~0 only

the

approximants ~~~'

ST, I.e, 1,

~, ~,..

are

allowed,

for

Ft

5

R~ _~ ⁰

only

the upper ones. For

R~

_~ 0

(R~

_~

0)

_we obtain

c : d

=

~~

~ s r

(respectively

_a

r) (35)

Ft

Thus in the

plane

_a

hexagonal periodic phase

_appears whereas

quasiperiodicity pertains along

the threefold axis.

Systems

of tl~is latter type are called Fibonacci

crystals.

A field

applied along

a

pentagonal

symmetry ^axis

(see Fig. 3)

does _not

change

the fivefold

syrnmetry,

^thus not

lifting quasiperiodicity

in the normal

plane.

The two

lengths along

the axis

change according

to

3fj+QiR~E~ _{4R~ j}

~~~

~~~ ~~'3fj-/R~E~ ^~~~ ~~~~Qi~

^~~~~

bt 10 ELECTROSTRICTION IN

QUASICRYSTALS

1461

and thus _can be made rational

again

with _:

jf~~j

~2 / Fk

^~

~2R~ _(2r-1)

to become

F~~j

~'~~~

F~

^~'

4. Relation between the

genera1izell

elechostdction tensor and the

generalized

elastic and

elasto-opfic

tensors.

Equilibrium

electrostriction is the result of _a

competition

between

elasto-optic

and pure elastic terms. Under the influence of

independent

strain- and electric fields the free energy of

a standard

periodic crystal

is increased

by [23]

F

=

c;jkt e;j

^ski

£ ^p;jkt _{E; Ej}

ski + K;y

E, Ej

+

d,jkt ^E; Ej Ek Et (37)

Here

c;j~t

^{denote the} coefficients of the elastic tensor, p;j~t ^{of the}

elasto-optic

tensor, x,y of the linear dielectric tensor, and

d;j~t

^{of the nonlinear dielectric tensor.}

For icosahedral

quasiperiodic crystals

the elastic and

elasto-optic

tensors have to be

supplemented by

the

phason degrees

of freedom

(the

linear dielectric tensor is

diagonal)

_:

F

=

c~jpt

_{7~~~ 7~pi} +

p,y~~

E; Ej

7~~~ +

xE~

₊

d;j~t E, Ej E~ Et (38)

8 _gr 6

The material tensors c, p and d _are

represented

in _a

straightforward

_{way as}

(15 x15)-, (6 x15)-,

and

(6

_x

6)-matrices,

if _we arrange the

components

_7~~j ^and

E;E~,

^{into 15-}

component

and

6-component

column vectors

respectively

_:

~(')

(5,phonon)

~1 "

~ j~~

(39)

~1

~(5,phason) [E @ El

=

I~°

4#

~°

~~) (40)

[E @ El

and write the

free-energy

_{as sum} of bilinear forms _:

F=~a~.ca~- ([E@E].pq+«E~+~[E@E].d[E@E]). ⁽⁴¹⁾

2 8ar 6

Then, by

arguments ^similar to those in section 2 the matrices _{are :}

o('

x5)

~jl

x 4)

~(l

_x5)

o(5

x 1) ~

~ i⁽⁵x5)

~(5

x4) ~ i^{(5 x5)}

~ ~(4 x 1)

~(4

x5) ~ ~⁽⁴x4)

~(4

x5) ^' ~~~~

4

o(5

x 1) ~

~ i j5 x5)

~(5

x4) ~

~ i^j5 x 5)

1462 JOURNAL DE PHYSIQUE I M 10

where

0~~~~~

_{denotes the}

(n

_{x m}

)-matrix

of _zeros, and

11~~~~

the

(n

_{x n}

)-unit

matrix _;

ii

^and A~ are the

phonon

Lamb constants, A~ ^and A~ are

phason-elasticity

_constants, and A~ a constant

coupling phonon

and

phason elasticity.

The latter allows to excite

phason

strain

by

_pure

phonon-like

_stress.

The matrices for the

elasto-optic

and the nonlinear dielectric tensor _{are :}

~

01~~

'

~~

~~

~~~

~~

~

~ ~~

l~

~~~

'

~~~~

l~ ^~(i

^x5)

~

"

o(511)

_{~~ ~(5x} 5)

(~)

The tensor of

generalized

elastic

compliances

_s, which is inverse to c so that _cs

= 11~~~ '~~ is

given by

~

~(l

_x5)

~(l

x4)

~(l

_x5)

~(5x

Ii

~

i(5x5) ~(5x4)

~

i(5x5)

~~ " ~ _"

o(4

x1)

[(4

x5)

s~1(4

^x4)

1(4

x51 ^' ~~~~

~(5 x 1)

~ ~(5x 5) ~(5x 4)

~ ~(5 x5)

3 5

with

~~

~j'

~~

i~i~~ I)~

^~~

i~i~~ _I)~

^~~

^4~

^~~

i~i~~ _I)

^~~~~

From the

equilibrium

condition ^~~

= 0 _we

immediately

derive the

generalized

electrostriction aa~

tensor in the

representation (39), (40)

_:

sj pi

o(i

x 5)

R:=

sp~

=

~~

~

~~

_{~~ ~}

fl

^{~~~~ ~~}

₍₄₇₎

8flT 8flT

o(

xl

o(x)

o(5

x ')

~p~_s~ + p

s~) ¹⁽⁵

^x5)

The electrostriction coefficients thus _are

given by

~~

8

ar

~~'~~~ ~ ~_~~~~

~~

8

ar ~~

_{~~ ~ ~~ ~~~}

5.

Summary.

In this

article,

_we have extended the notion of electrostriction _to the

phason degrees

of freedom in

quasicrystals

and have determined the components ^{of the}

generalized

electrostric- tion tensor for icosahedral _and

decagonal phases.

For certain values of the electric

field,

the

induced

phason

strain locks the

mutually

incommensurate basis _vectors of the

reciprocal

lattice into commensurate ratios and thus makes the

system periodic

either

along

_or

M 10 ELECTROSTRICTION IN QUASICRYSTALS 1463

perpendicular

to the field direction. If the icosahedral model

applies

to the Blue Phase

III,

this incommensurate to commensurate transition should be observable.

If the electric field is directed

along

_a threefold axis and the

phason

electrostriction coefficient is

positive,

then _a

periodic hexagonal phase

with lowest rational

approximants

c:d=1, ~,

^~

_{(Eq. (35))}

appears, when the dilation

R~E~ equals 0.31,

0.056 and

2 5

9.16

x10~~ respectively. Assuming

that

R~

^{is of the order of the}

phonon

electrostriction

~2

coefficients of BPI and

II,

I-e-

R~=10~~~~,

the

corresponding

field values _are

V E

= 5.6 _x

10~,

2.4 _x

10~,

9.6 _x

10~

^~

m

Provided _a thickness of the

liquid crystal

cell of10 _~m,

voltages

of U

=

56,

24 and 9.6 V have to be

applied.

As _we restricted ourselves to linear terms of the deformation tensor, the

theory

will

only roughly apply

to the

highest voltage

of U

=

56

V, although

in this range a

phase

transition from BP III to BP E of

yet

unknown structure _occurs

[10, 12, 9].

^If

R~

^is

negative,

the lowest rational

approximants

_{are c :} d

=

2, ~,

^~~ _with

_voltages

_of

_{44, 15,}

3 8

and 5.9V. The

voltage

_sequence allows to measure the

sign

of

R~.

Many experimental

tests have been

performed

to check whether BP III carries _an icosahedral _structure

[9-13].

No clear evidence has been found for the

model,

but also _no

sharp

rebuttal. The induction of _a

periodic phase by phason

electrostriction is _{a new} route which should be

pursued.

The fact that the broad reflection band of BP III

sharpens

in _an

electric field

alight

be _an indication that the

quasicontinuous reciprocal

lattice of the incommensurate system condenses to _a discrete lattice of the commensurate

phase. eve

cannot answer

presently, why

the increase in

intensity

is _so much

larger

in systems with

negative

dielectric

anisotropy

than in those with

positive anisotropy. Maybe

the electrostric- tion coefficients _are

significantly different).

Also indications have been

expressed,

that in

accordance with _our model BP E is _a

hexagonal phase [10].

As _our

theory

is _a very

general

_one, not restricted _to Blue

Phases, investigations

of the

specific

static and

dynamic

behaviour of BP III in

comparison

to the other Blue Phases

requires

_a closer

analysis

of the

mesoscopic

Landau-de Gennes

theory.

Such studies _are in progress.

Appendix.

Electrostriction in

decagonal quasicrystals.

The

reciprocal

lattice of

decagonal quasicrystals

is

generated by

five

rationally independent

wavevectors : first

kj, pointing along

the tenfold symmetry

axis,

which is the direction of discrete translational

periodicity,

and then

k~,...,k

~

perpendicular

to it. These four _are selected from the five vertices of _a

regular

pentagon

(the

fifth is

rationally dependent,

as the

sum of the vertex vectors is _zero. The five

together

with their inverse form _a

regular decagon).

Hence the dimension of direct _space is d=

3,

the dimension of

orthogonal

_space is

n = 2. Here _we

investigate

the tensors for the maximal

point

_group

Djo~

^{which is the dihedral}

point

_group

Djo

with inversion. The results _are

unchanged

for the

suigroup _Djo,

I-e- if the

suffixes _u and _{g are}

dropped.

The

components

of the

displacement

vector

u(x)

_e

El

_form

a reducible

representation,

which divides into the irreducible

representations r)]~

and

r)().

The components ^{of the} two- dimensional

phonon displacement

vector transform

according

to the irreducible _represen- tation

r)(~.

1464 JOURNAL DE

PHYSIQUE

I M 10

The components of the

symmetric phonon

strain tensor,

s,j,

^{and of the}

quadratic

field tensor,

E, E~,

^{divide into} irreducible

representations [3, 24]

(~j~~

_fl~

~~~~) @ (~j~~

_fl~

_~~~~))~

" 2

~)(~

fl~

~)(~ £ ~)(~.

The

(2

_x

3)-phason

strain _tensor falls into the irreducible

representations

rjj)

_{q~ ~mjj) q~}

~mj2)~ ~ ~mj2)q~ _~mj2)q~ ~mj2)

u g g g

With the

help

of Schur's lemma _we obtain _seven

independent

electrostriction

coefficients, connecting

the

following

irreducible

representations

_:

For

r)~~.

'l~~~ " Eli ~

Rj [E

@ ^E_]~~~ +

R2[l~ @

_l~l~~~

~' ~~

_~

_{~~ ~~~}

_~

_~~~

'l~~~ " (E22 + E33) ~

R3[l~ @ ~l~~~

+

~4[~

@

~li~

=

R~ El

₊

_R~ (El

₊

El)

2

For

r)~~

For

r)~~

a~~6Ph°n°n~ ₌

~~jj ^~~j

=

R~IE © El

^~6~

~

~ ^(l~i~ ~)~

E~ E~

a~~~>P~~~°~~ =

j~~

~

~~)~

=

R~[E

@ E]~~~

)

~~ '~

For

rj~~

n~l~~ =

~~'

=

0 Xl_'

For

r)~~

~~~~

j'2 ^3jj

~.

i

^{22 13}

M 10 ELECTROSTRICTION IN QUASICRYSTALS 1465

Solving

these

equations

for s;j ^and

X;j

^{we obtain :}

ej_j

Rj R~ R~

0 0 0

El

E22 R~

(R~

₊

R~) (R~ R~)

0 0 0

Ei

e~~ R~

(R~ R~) (R~

₊

R~)

0 0 0

El

^'

s~~ 0 0 0

R~

^{0 0}

E2 E3

s~j 0 0 0 0

R~

0

E3 Ej

sj~ 0 0 0 0 0

R~ Ej E~

~'~

_{~'~ ~~~}

=

~

~

~~~ ~~~ ~~ ~~ .

~~~ ~~~ ~~3 0

E~ E~ (El _E~)~

The vectors of the

reciprocal

basis

change

under _a combined

phonon

and

phason

deformation

according

to

ki-ij

=

(1-e)ki

kj-ij= (I-e)kj+X~k/, _j=2,..,5,

where

k(:=kj

^and

k/

is the two-dimensional

orthogonal complement.

A field

applied along

the tenfold direction scales the

reciprocal

basis _{as :}

ii

=

(I Ri E~)kj

i~

=

I

R~ E~)

_k~

and does not eliminate the

quasiperiodicity perpendicular

to the tenfold axis.

If the field is

applied

in the

quasiperiodic plane

and

orthogonal

to one of the

edges

of the

decagon,

then there _are two characteristic

lengths

of the vectors

projected

onto the field

direction

(Fig. 5)

with ratio

a:b=T.

~~~~~~~~ _~~~

mT(I+jR~E~T(2T-1)),

E~(R~

₊

R~

+ T

~

R~)

which becomes

rational, namely

for

F~~

j T

~~

²

Fk

~R7 T~(2T-1)

1466 JOURNAL DE

PHYSIQUE

I M 10

The

projection

to the

orthogonal

direction

yields

three

length scales,

of ratios

(Fig. 5)

c:d:e

=

2(1

₂

^{E~(R~ R~} TR~))

T I

E~

(R~ ^R~

⁺

^R7

T

l

E~(R~ _R~

₊

T

3

~))

which cannot be made rational

simultaneously.

If the irreducible

components

^{of the}

generalized

deformation tensor are

aligned

to a twelve component ^column vector

~1~"~1)~~

ill

_~1)~~ ill _~1~~~ © ~1~~'~~°~°~~ e ~1~~~~~~°~~ e ~1~~~ e _~1~~~

,

the elastic free _energy in

quadratic approximation

is _a bilinear form

~ i

el

"j~l"C~I,

and the elastic tensor c contains nine

independent

coefficients

I i I

~

o(2

x2) j~

I~ i

~ ^~~~ 0~~~~)

o(2

x2)

0~~~ ~) A

~ i (2^x2) _~~~~

~ _"

~ 0~~~~)

o(2

x2)

isl~~~~)

_A

1(2x2)

0~~~ ~)

o(4

x2)

~

0~~~~)

o(4

x2) I ~l~~ ~~) A

~

i(2

x 2)

0~~^~~)

o(2

x 2) _~~ ~

0~~~~)

~(2

x2)

~ ^~

~8

l~~~~~ 0~~X2)

~~~~~~ 0~~~~) i ⁽² x2)

The inverse tensor, of elastic

compliances

_s, is of the _same form with

coefficients,

A~

-A~ ii

~~

~AjA~-A)~ ~~~AjA~-A)~ ~~~AjA~-A)~

~

l

~

17

~

~16

~

~4~ is17 il~

^~

_is17 il~

~

As

~

i

~

i

~

is17 il~

^~

_~8

^{~ ~}

₁₉

The structure of the

elasto-optic

tensor is the _{same as} of the electrostriction tensor, with _seven

independent

coefficients _{pi,..., p~.} An

argument

similar to

equation (47) shows,

that the electrostriction coefficients result from _a

competition

between the

elasto-optic

and elastic coefficients _:

M 10 ELECTROSTRICTION IN

QUASICRYSTALS

1467

Z~

u

~

«

yl

Fig.

5.

Reciprocal

basis vectors

k(, k[,

,

k for the

decagonal phase

and decagon

generated

by part

of them.

Projection

of the vertices yields distances of incommensurate ratios a:b=T and

c:d:e

= 2:r:

r

~2

_"

£ _(St

P3 +

S2P4)

,

~~

8

ar ~~~~

~

~~~~~

'

~

8

ar

~~~~~ ~ ~~

~~~

'

References

[Ii

KRAMER P. and NERI R., Acta

Cryst.

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580.

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