An Elementary Approach to the Crystallography of Twins in Icosahedral Quasicrystals

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An Elementary Approach to the Crystallography of Twins in Icosahedral Quasicrystals

O. Radulescu

To cite this version:

O. Radulescu. An Elementary Approach to the Crystallography of Twins in Icosahedral Quasicrystals.

Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.719-728. �10.1051/jp1:1995101�. �jpa-00247097�

J. Phys. I fronce 5

(1995)

719-728 _JUNE 1995, PAGE 719

Classification

Physics Abstracts

61.44+p 61.50Em 61.72Mm

An Elementary Approach to the Crystallography of Twins in Icosahedral Quasicrystals

O. lladulescu

Laboratoire de recherche _sur les matériaux, Université de Marue-la-Vallée, 2 _rue de la Butte Verte, 93166 Noisy le Grand Cedex, France

(Received

8 November1994, revised 30 January 1995, accepted 21 February1995)

Abstract. The properties of icosahedral modules and 6D hypercubic lattices

are used in

order to show that values of coincidence ratios for twins of icosahedral quasicrystals _are of the form 5x~ y~, with _{x,y E} lZ, and to assiglr _a geometricalmeaning to this form. Some physical

consequences are inferred.

1. Introduction

Quasicrystals

_are

long-range ordered,

but

aperiodic,

solids. Both

theory

and experiment ^showed

that,

_as for

periodic crystals,

extended defects _can also be defined and characterised for qua-

sicrystals.

Tue

challenge

to grow centimetre-sized _mono-

quasicrystals

raises tue need _to

study

tue structure and formation of

grain

boundaries of

quasicrystals.

Tue

thecry

of

special

_grain boundaries of

quasicrystals

bas

made,

recent progress

by

borrow-

ing

from normal

crystallography

and

generalising

concepts such _as tue coincidence site lattice

(CSL).

Tue

problem

of coincidences _in

quasicrystals

_was introduced

by Warrington. il,

_{2] and}

independently

discussed in _more

generality,

[3, 4]. ^Here we

give

some results for the _case of

icosahedral

quasicrystals.

Similar results _were obtained [5],

by using prime

factorisation in the

ring

of icosians.

A toy ^{model of} icosahedral

quasicrystal

is the

tiling

model. In finis

model,

the icosahedral

quasicrystal

is _a

tiling

of the friree-dimensional _space, made with two types ^of

files,

_an oblate and _a

prolate rhombohedron,

and

having global

icosahedral symmetry _[6]. Tue

edges

of the files form _a star of 6 vectors

e)

₌ 1,6

(Fig. l).

The

position

r" of _any vertex in tue

tiling

_con be

uniquely expressed

_{as an}

integer

combination of these _vectors: e"

=

£)~~ zie(',

_{zi E} 2L. One

con thus index ail vertices of tue

tiling

with 6

integers

_z,,

= 1,

6,

and consider them _as

being projections

onto tue 3D

physical space(~

of vertices of _a 6D

hypercubic

lattice. Tue vertices of tue

tiling

do not exhaust ail tue

possible

values of

integer

indices _z,

(a

rule is used to Select

only

a part ^{of tue} vertices of tue

hypercubic

lattice and

project

them _as vertices of tue

tiling,

_see

(~)Throughout

this _paper, vectors and matrices with superscript ^Il are in the physical _space.

Q Les Editions de Physique 1995

z

~

(iioooo)

c~

G

joolooij

Aj

x H

Fig. l. Trie module of _an icosahedral quasicrystal is generated by trie vectors

e)',

_i

= 1,6. There _are four types

(colours)

of module planes orthogonal to [110000j: type A

(black)

containing Ai, A2, A3.

A4, and trie origin O

(r(

= 0), type B containing Bi, 82, ^and having the origin Bi shifted by

r(

=

e)

with respect to O, type C containing Ci, C2

(r$

=

e)),

and type ^D containmg D

(r[

=

ej

₊

e().

Ref.

[6]). By considering

ail mteger combinations of tue form

£)~~ _z,e)'

_{one gets a} set whicu

densely

fills 3D

puysical

_space, and is called Iimit-module [4], or Bravais module _[3]

(from

_now

on referred _{to as} tue

module).

Tue module represents tue

complete projection

of trie

hypercubic

lattice onto tue

puysical

_space.

Tiling

tue 3D space witu two types ^of ruombohedral tiles is _a

puzzle

with _an

infinity

of dilferent solutions, tuat _are

quasiperiodic, periodic,

_or random. Ail these

tilings belong

tu the _same

module,

their

specificity being

_given

by

the selection rule for

tiling

vertices.

Two

quasicrystalline grains

_in a

special

disorientation

(for

which coincidences

exist)

have coincidences bath in their

tilings

and in their modules. The set of coincidences _in the module form the coincidence module, that is the

projection

enta the 3D

physical

_space of _a 6D CSL of the

hypercubic

Iattice [3]. ^{The set of} coincidences _in the

tiling

form the coincidence

quasilattice

[3]. ^{The rotations that}

produce

coincidences _are called

quasirational

rotations [3]. Coincidence

ratios La,

L,

are defined _as the

reciprocal

volume densities of

coincidences,

for the

tiling

and for the

module, respectively.

Refer _{to [3]} for the connection between L and La.

The outline of this paper ^is the

following:

_in Section 2 the group ^structure of the quasir- ational rotations _is

presented

for icosahedral modules. The rotations

through

_{7r are} used _as generators

(a

similar idea _can be found _{in [7]} for cubic and

hypercubic Iattices).

In Section 3

we

develop

_{a new} method to compute coincidence ratios and show that for rotations

through

7r the coincidence ratios have _a

geometrical

_meaning. Section 4 presents twins of

quasicrystals

in correction with

experimental

results and Section 5 discusses the symmetry aspects ^{of the} set of comcidences.

N°6 TWINS IN ICOSAHEDRAL QUASICRYSTALS 721

2.

Group

Structure of

Quasirational

Rotations A basic concept, ^which was used in reference

[3] ^to

analyse coincidences,

is tuat of module direction. A module direction is tue

analogue

of _a lattice direction of _a

periodic crystal,

and represents trie set of ail

points

of tue

module,

collinear to a

given

vector. For _an icosahe- dral

module,

_a module direction _cari be

specified by

tue 6

integer

indices of _a vector in finis

direction(~).

As shown in reference _[3]

quasirational

rotations

produce

coincidences in ail module direc- tions, ^and

consequently,

_an

equivalent

definition of

quasirational

rotations is that

they

trans- form _an

arbitrary

module direction _{into a} module direction which is commensurate to it. We colt _two module directions commensurate if there _{are two}

equipollent

vectors, _one in tue first

direction,

tue orner in tue second direction. This

implies

that rotations

through

_7r about _ar-

bitrary

module directions _are

quasirational. Generally,

_a

quasirational

rotation is either _{a 7r} rotation about _a module

direction,

_{or a}

product

of two such _7r rotations. In order _{to prove}

this,

consider tue

quasirational

rotation

R".

_{Then tuere}

are two

equipollent

module _vectors

d), d),

orthogonal

to tue rotation _axis, sucu tuat

R" dl

=

d).

_{As tue}

_angle

_between

d)

₊

dl

_and

_d)

is ualf tue rotation

angle

^of

R",

tuen

R"

_{is tue}

product

of _7r rotations

R)

_about

d)

and

R)~

about

d)

₊

dl R"

=

R)~R) (sec Fig. 2).

~II

d2

II

0/2

Fig. 2. Tue rotation R" _{of axis} d" and angle 0 is the product of two _7r rotations,

R)

about

d)

and

R)~

about

d)

₊

d(.

_{To show this, notice that}

R)~R)d"

=

d",

and

R)~R)d)

=

d(.

Hence,

tue _7r rotations about

arbitrary

module directions form _a set of generators ^{G of tue} group of

quasirational

rotations, but these generators _are non free and

satisfy

tue

following

relations:

Gl) g~=1,foranygEG G2)

_g3

" gig2, if gi, g2> g3 E G and bave

mutually orthogonal

axes.

(~) Actually, the indices of vectors _in the _same module direction form _a two-dimensional sub-lattice of trie 6D hypercubic lattice

[3,6,8]).

We could choose

a single representative vector, with _a convention that _we propose m [9].

JOURNAL DE FHYSIQIJE1- T.5,N°6,lUNE 1995 28

G3)

_{gi g2}

= g3g4, if _gi, g2, g3, g4 ^E G and bave

coplanar

rotation _axes

d)

,

d), d(, d(

tuat

satisfy £(d), dl)

=

£(dj,dj).

Tue rotation axis of _a

quasirational

rotation _can be any module

direction,

and tue rotation

angle

_is twice tue

angle

between two

arbitrary

module directions _in tue

plane orthogonal

to tue rotation axis.

3. Values of trie Coincidence Ratios L

The method used in this section to compute coincidence ratios consists in

reducing

the 3D coincidence

problem

to _a 2D coincidence

problem

_in module

planes orthogonal

to the rotation axis. A module

plane

is

analogous

to the Iattice

plane

of _a normal

crystal,

and consists of aII points of the

module, coplanar

with _a

given

one.

The method works for

quasicrystals,

but aise for normal

crystals,

as

following.

In the Bravais Iattice of _a normal 3D

crystal,

there is _an

infinity

of Iattice

planes orthogonal

to a given Iattice

direction,

but

only

_a finite number of types of

planes

that cannot be obtained from _one another

by

_a translation with _a Iattice vector

parallel

to the _common normal. For instance, for _a 3D

cubic

Iattice,

^there _are three types ^of

planes (A,B,C) orthogonal

to a

[iii]

direction

(Fig. 3).

A rational rotation about the

[iii]

direction

produces

the _same

planar

coincidence ratio in

planes

of the _same type. ^This coincidence ratio

depends

_on the relative position of the rotation axis with respect to the Iattice

plane,

and _as shown in reference [3], cari be eituer infinite

(no coincidences)

_or

equal

to _a finite value

Lo.

In tue _case of tue rotation

turougu

_7r about tue

iii ii

_axis of tue cubic

Iattice,

tue

planar

coincidence ratio _is infinite for

planes

of type B,C and

equal

to 1 for

planes

of tue type ^{A. Tue}

global

3D coïncidence ratio _is L

= 3.

Similarly,

_in tue module of _an icosauedral

quasicrystal,

tuere is _an

infinity

of module

planes

which stack

orthogonally

to _an

arbitrary

module direction

d",

_but

_only

a finite number of types that cannot be obtained from _one another

by

a translation with _a module vector

parallel

to

d".

We say that each type ^of

plane

has _a certain

"colour",

see

Figure

1. A

quasirational

rotation about d"

produces

coincidences with the

planar

coincidence ratio Lo for _no crieurs of trie total number of _n

crieurs,

and _no comcidences _at aII for the rest of the crieurs. The 3D coincidence ratio L is:

jiiii

B

C

C

'

A Î B

/ ~~

~

B C

Fig. 3. There

are three types of lattice planes

(A,B,C)

stacking orthogonally _to

[III]

in _a 3D cubic lattice.

N°6 TWINS IN ICOSAHEDRAL QUASICRYSTALS 723

z =

~°~

(i)

no

n is tue total number of crieurs

(number

of types of

planes)

and

depends only

_on tue rotation axis. In tue

appendix

_we show that for _a P icosahedral

module(3):

n =

fl~ (2)

where fl is tue _area of tue

primitive

unit ceII of tue 2D lattice which

corresponds

in 6D to tue rotation _axis.

no and

Lo depend

both _on tue rotation axis and _on tue rotation

angle.

For _a rotation

through

_7r, Lo "1 and

(see appendix):

Il,

^{for odd rotation axes}

~°

4,

^for _even rotation _axes ~~~

6 6

A module direction _{is even} if it contains

only

vectors

£ _zie)'

_with

even

~j

_zi, and odd if it

1=1 1=1

6

also contains vectors with odd

~j

zi.

1=1

This shows that for rotations

through

_7r, trie coincidence ratios _are:

Q~,

^{for odd rotation} axes

~ Q2

(~)

-, for _even rotation _axes 4

As shown in the

appendix,

^fl~ is of the form 5x~

g~,

with _x, y

integers.

This and Tl

(see appendix)

shows that Z is of trie form 5àJ~

y~,

^for ail rotations

through

7r. This result is

compatible

with reference [5], ^that

predicts

coincidence ratios of the form p~ _{+ pq}

q~.

Tue two forms 5x~ g~ and p~ + pq q~,

though

non

equivalent,

_generate tue _{same sets} of

integers [loi.

Without

possessing

a

proof,

_{we expect} that the values of coincidence ratios for _any

quasir-

ational rotation _are

products

of coincidence ratios for _7r

rotations,

and therefore _{are repre-} sentable

(~)

as 5x~

y~.

Similar results _can be found for _a 3D cubic lattice. Tue number of types ^of

planes orthogonal

to a lattice direction [h,

k,

_{ii is n =} ^h~ + k~ ₊ l~

(tue

_square of the

period along _{[h, k, ii} ).

For

7r rotations

Lo

_" 1. Also _no

= for odd rotation _axes

(h

+ k +1

odd)

and _no

= 2 for _even

rotation _axes

(h

₊ k +1

even) (see

also Ref.

[7]).

4. Twin Disorientations and Interfaces

An interface is

geometrically

characterised

by

two main features: tue

disorientation,

^which

is tue relative rotation of the

grains,

and tue inclination of tue interface

plane

with respect

to tue

grains (we

do Dot take into accourt here the relative translation of tue

grains).

For twin interfaces

iii],

tue disorientation _can be

represented

_{as a 7r} rotation about tue twin axis

(here

_we refer to trie

simpler

_case of

centro-symmetric grains),

and tue interface

plane

(~) ^There _are ^three types of rank 6 modules witu icosahedral symmetry: P, F, and1 [12]. Only P and F icosahedral Bravais modules bave been found experimentally. Trie results of this paper are proven

in trie P _case, but they _can be extended to trie F and I _cases also.

(~) The product of two integers of the form 5x~ _{y~ is}

an integer of the _sonne form [21].

(twin plane)

is

oTthogonal

to the twin axis. It must be stressed that all rotations of tue form

Si RSp~,

where

Si 52

are elements of the

point

symmetry group of the

grains, produce

the

same disorientation of the

grains.

We say that _a disorientation has _a twin

representation

^if

among the rotations which

produce

the

disorientation,

there _are also rotations

through

_7r.

Nol all

special

disorientations have _a twin

representation,

but those with small values of the coincidence ratio, do.

In tue _case of the icosauedral

quasicrystals,

_{one can} show

by

_computer searcu [9] tuat tue disorientation witu tue smallest coincidence ratio wuicu bas _no twin

representation

is L209

=

iii _x L19. In certain _cases tuere _{are two}

non-equivalent

twin _axes for tue _same disorientation

(two

twin

representations),

wuicu

correspond

to twin

planes uaving

dilferent densities. Tuis

occurs wuen _a twin axis is

orthogonal

to a two-fold symmetry axis of tue icosauedron. Tuen

tuere is _a second twin axis for tue _same

disorientation, orthogonal

to tue first _one, and _to trie two-fold symmetry ^{axis of trie} icosahedron. This second twin axis is

non-equivalent

to the first _one, except for trie _case wuen it is of tue type

(001111).

In

Figure

1 DG is _an

arbitrary

twin axis,

orthogonal

to tue two-fold symmetry ^axis

[110000],

and OH is tue second twin axis,

orthogonal

to DG and to

[l10000].

OG is

equivalent

to OH if and

only

ii it is the mirror image of OH in tue

plane Ozy,

_or tuis is true

only

for tue two directions

making

_an

angle

of +45°

witu

Oy.

Tuese directions _are botu from tue

family (ooTlll)

and _are

equivalent

twin _axes for tue L4 disorientation.

Tue smallest value of L for icosauedral

quasicrystals

is

4,

wuicu

corresponds

to twin _axes of tue type

(roll

ii

(fl~

₌

16).

Tuere _{are no}

experimental

reports ^{of this} type ^of twin, a fact tuat may be

explained by

tue

relatively

low

density

of tue twin

plane.

For P icosauedral

modules,

tue

density

of tue twin

plane

scales like

1fil

_{[13] and} is greater for tue next twin, ^{L5. Tuere} are

two twin _axes in tuis _case, < looooo > with fl~

= 5 and

(loolll)

witu fl~

= 20.

Experiments

suowed tue existence of L5 tlvins in Alfecu

quasicrystals [14,15]

^{with twin axis} < looooo _>.

In

increasing

order of the values of

L,

the

following

twins _are L9

(not

_yen

reported),

with two

non-equivalent

twin _axes

(ooolll),

fl~

=

9,

and _< olllol >, fl~

= 36, and two

nonequivalent

disorientations iii, with the twin axis _< lo2001>, fl~

= 44, and L11*

(reported

in Ref. [16]

in an Almn

quasicrystalline sample),

with the twin axis

(210001),

^fl~

= 44.

Tuere _are two remarks to be made. First, wuen there _are two non-equivalent twin _axes for tue

same

disorientation,

tue odd _one would

probably correspond

to _a lower energy interface because tue twin

plane

is two times denser for odd axis and _same L

(see Eq. (5)). Second,

the ratio of

the densities of the twin

planes

_{are :} for L

=

4,

5 9 II

respectively only

/À vÎ v§ /À

^{' ' '}

for

P-type

icosahedral modules. For

F-type

icosahedral modules

(which

is the _case of

Alfecu,

AlPdmn

quasicrystals),

the above Tatios must be corrected to _:

~

and L4

2/À vÎ v§ /ù'

is even more

strongly

unfavoured relative to L5. The conclusion which follows is that L4, if it

exists,

must be searched in the

P-type

icosahedral

quasicrystals (Almn, AlLicu, GamgZn, etc.).

5. Point

Symmetry

of trie Coincidence Module and

Quasilattice

When it exists, symmetTy ^has

always

consequences theTeioTe it is

important

to

speciiy

the

point

symmetry of the coincidence module and

quasilattice.

For

planar quasicrystals

the

quasirational

rotations commute witu tue elements of tue point symmetry group of tue

quasicrystal (ail

two-dimensional rotations

commute),

uence tue coin- cidence module and

quasilattice

bave tue _same symmetry as the

quasicrystal.

In tue 3D case, tue rotations do not commute

generically,

except in the

iollowing

_cases:

N°6 TWINS IN ICOSAHEDRAL QUASICRYSTALS 725

a)

rotations

uaving

a common axis.

b)

180° rotations with

orthogonal

_axes.

For _an icosahedral

quasicrystal,

ail elements of the icosahedral _group, whicu _commute with

one of the

quasirational

rotations which

produce

the

disorientation,

_are

symmetries

of tue coincidence module and

quasilattice. According

to

a)

and

b)

tue coincidence module and

quasilattice

will bave tue

following

_symmetry elements:

A) Two-fold, three-fold,

_or rive-fold symmetry axis if tue rotation axis of

SIRSp~

is respec-

tively

_a

two-fold, three-fold,

_or rive-fold symmetry axis of tue

icosahedron,

for _some Si 52 E Y

(tue

symmetry _group ^of tue

icosahedron).

B)

Two-fold symmetry axis if the disorientation has _a twin representation with the twin axis

orthogonal

to _a two-fold symmetry axis of the icosahedron.

The order of _a symmetry axis can be doubled in _case

A)

for three-fold and five-fold axis, when coincidences _are

produced only

in "black"

planes orthogonal

to the rotation axis

(module planes containing points

wuicu _are botu in tue module and _on tue rotation

axis).

This _comes from tue fact

that, orthogonally

to a rive-fold _{or a} three-fold axis, tue planes of tue "black"

type ^bave

ten-fold,

_or six-fold symmetry

respectively.

Using

tue above discussion _{one con} show that tue coincidence module and

quasilattice

L4 bave tetrahedral symmetry Th> L5 bas

decagonal

symmetry Dioh> ^{L9 bas tue} symmetry

D6h,

iii and iii ^* bave

D5d

symmetry. It

might

bave _some

significance,

but this is Dot dear yet

(see

also Ref.

il?]),

that tue coincidence

quasilattice

and module

bave,

for tue twin

25,

tue _same point symmetry as tue

decagonal phase.

This

phase, periodic along

tue ten-fold symmetry axis and

quasiperiodic

in tue

plane orthogonal

to this axis, competes with tue icosahedral

quasicrystalline phase

in _some

alloy

systems.

We also _note that when _a cubic

phase

transforms into tue icosahedral

phase, by keeping

tue

common point symmetry _group

Th,

two

possible products

occur in _a 24 disorientation relation.

6. Conclusion

Plain

geometric

arguments ^allow coincidence ratios of tue form 5x~ y~ to be

predicted

for twins of tue icosauedral

quasicrystals.

Geometric criteria for low _energy interfaces [18],

namely

tue low coincidence ratio criterion and tue

uigu

twin

plane density criterion,

favour the 25 twin and tue

[looooo]

twm axis, with respect to L4 and L9. Tue 25 twin bas been

experimentally

observed in F icosauedral

quasicrystals.

24 _twm boundaries _are

strongly

disfavoured

by

tue

density

criterion in F icosauedral

quasicrystals,

but

tuey

_may bave _a chance to appear in P icosauedral

quasicrystals.

Nothing

is yet known of tue

origin

^of trie observed twins in

quasicrystals.

It bas been

suggested

that these _are

growth

twins and that

twinning

^{mechanisms which work for}

crystals,

should also

apply

to

quasicrystals il?].

The symmetry considerations in Section 5 suggest ^the

possibility

of _a

special

kind of trans- formation twin. Transformation twins _occur from

phase

transitions when several

equivalent

results _are

possible. Usually,

a

group-subgroup

symmetry ^relation exists between trie initial and the final

phase,

and

twinning

is

produced

_{one-way, 1-e-,} when trie transition goes from _more

symmetry to less symmetry.

Quasicrystals taught

_us that transformations with _{no group-}

subgroup

symmetry relation

(but preserving

_{a common} symmetry

subgroup

of the two

phases)

are also

possible.

In this _case

twinning

_{can occur} both _ways: each

phase

bas symmetry ^el-

ements that trie other

phase

has not. 24 and L5 _twins could thus

possibly

_occur from the

transitions

Oh

_{- Yh>} Dioh _- Yh,

respectively.

Appendix

A

For _an icosahedral

quasicrystal

the vectors in _a module direction d"

=

£ _zie('

_are trie

projec-

tions onto trie

physical

_space of vectors of _a 2D lattice

L[

= L n

E~

_{L is trie 6D}

hypercubic lattice,

of canonical basis

(e,) ((eÎ

are trie

projections

of

(e,)

onto the

physical space),

and

E~

_is

a 2D

plane spanned by

d

=

£

_z;e, and Sd. S is the

following integer

matrix:

1 1 1 1 1

1 o 1 -1 -1 1

S =

~ ~

IA-1)

-1 -1 1 o

-1 -1 1 o

Let

(ai >a2)

be _a

primitive

basis of

L[, L[

=

(ziai

₊ z2a2,zi>z2 E

2L).

Let

L[

be trie

projection

of L onto E~. If r~

= giai + g2a2 is the

projection

of _{a vector r e} L _onto

E~,

then

(r r~,

_ai

"

(r r~, _a2)

" o, where

(*,*

is for dot

product

in 6D. We have _a system

of _two

equations

for _{yi> y2.} Hence

L[

₌

(yiai

₊

g2a2)>

with:

vi =

j _{lzi(a~, a~) z~(ai, a~)1,}

g~ ⁼

j _{lz~(ai, ai) zi(ai,}

a~)1

(A.2)

fl ₌

[(ai,

_ai

_(a2

_a2

(ai

_a2)~j^~~~ is the _area of the

primitive

cell of

L[.

_{z; =}

(r,

_{az are}

integers

^{and take} ail

possible

values [3], when _r describes L. As the transformation

(zi> z2)

_-

(pi y2)

bas _a determinant

equal

to

j,

_then

trie index of

L[

in

L[

is fl~.

Let

L)

be _a module

plane orthogonal

to d^I and

containing

_a

point

that is both in the module and

along

^d"

(we

call this type ^of

plane

"black" ). ^Tue

representative planes

of dilferent colours

are

L)'

=

L)

₊

r)',

₌ o, n -1 (1 = o is for

"black",

and

r)

= o, _see

Fig. l).

Also

r)' rj'

are Dot

parallel

to d" _for

# j.

_{LÎ' are}

projections

onto

puysical

_space of the 4D lattice

uyperplanes

Li ^of

L,

Li ₌

Lo

_{+ ri,}

=

o,

_n -1. For eacu _1, tue

projection

of Li onto

E~

_is

a

point rf, rf

belong

to

L[

and

satisiy rf r) )

L~ for1

# j,

uence eacu

rf

_{represents a coset} of

L[ /L[.

Tuis allows _a coset of

L[/L[

to be associated witu eacu "colour". The number of colours is

n =

((L[/L[)

=

fl~,

where is for cardinal.

As noticed

by

Elser [19], a

primitive

base of

L[

_con be taken of tue form:

ai "

dF

SdF,

for odd directions

~~ ~

( _~dF,

_for

even directions

~~'~~

A module _vector

~j _z;eÎ'

_{is even,}

or odd if

£)~~

_z, is

respectively

_{even or} odd. A module direction _{is even} if it

only

contains _even vectors, ^and odd if it also contains odd vectors.

dF

is any vector wuicu minimises tue form

f(d)

=

5(d,d)~ (d, Sd)~

restricted to

L[ (using

S~

= 5 _one shows tuat

f(d)

is tue square of tue _area of tue cell formed

by

d and

Sd).

In reierence [20] we called tue vectors

dF>

"Fibonacci" vectors, because in directions < lloooo >

tuey

_are of tue iorm

(Fn+i,Fn+i,Fn,0,o,Fn),

wuere Fn is tue Fibonacci series

(defined

_as

F»+i

= F~ + F~-i, Fo

= Fi

=1).

N°6 TWINS IN ICOSAHEDRAL QUASICRYSTALS 727

(A3) implies

tuat

~

5(àF> àF)~ (àF, SàF)~

(Or Odd direction

~ "

(5(dF,dF)~ ^(dF, SdF)~j ^/4

^{for even directions}

^l~'~~

The

iollowing

theorem is _a direct consequence of reference [21]: ^{Tl "If} n is _even and repre- sentable _as 5x~

y~,

witu _{x, y E} 2L tuen

n/4

is also

representable".

This shows that fl~ is

representable

_as 5x~

y~,

witu _{x, y E} 2L.

Let

R"

_be

a 180° rotation. Tuen

R"r)

=

-r),

where

r)

_are tue

projections

of

r('

^{onto tue}

plane orthogonal

to trie rotation _axis. R

produces

coincidences of trie colour

if,

and

only if,

3r)

sucu that

R"r)

_E

(r) +L)),

_1-e-,

r)

_E

L),

wuicu is trie _same

tuing

witu

rf

_E

L[.

There

2 2

are

only

black coincidences and _no is

equal

to when there is _no

rf ii # o) equal

to _one of the vectors

), _~~,

or

~~

( _~~,

i-e-, if _none of the above vectors

belong

to

L[.

2

Let _us show that ^~~ E

L[ if,

and

only if,

^d" is _even.

Using (A2),

^~~ E

L[ if,

and

2 2

only if,

there exist

integers

_zi

"

~~~'~~~

and _z2

"

~~~'~~~

i-e

(see (A3))

if d" _is odd and

2 2

(dF>dF)

₊ 0

(mod 2), (dF>SdF)

₊ 0

(mod 2),

_or ^{if d"} is _even and

(dF>dF)

₊ 0

(mod 2), [(dF> dF)

₊

(dF

+

SdF))

₊ 0

(mod 4).

As z~ _{e z}

(mod 2),

then

(d, d)

e

£z, (mod 2).

If d"

is odd then

(dF, dF)

+ 1

(mod 2),

therefore

) _{) L[.}

_{If d" is}

even then

(dF>dF)

+ 0

(mod 2),

and because

(use Al) (d, d)

+

(d, Sd)

=

(£ z,)~ 4(z2z4

+ z2z5 + z3z5 + z3z6 +

z4z6),

then

[(dF, dF)

₊

(dF, SdF))

₊ 0

(mod 4).

In trie _sonne way we show that ^~~ E

L[,

^{~~ ~ ~~} E

L[

2 2

if,

and

only if,

d" _{is even.}

Hence for 180° rotations

"°

ÎÎÎ ÎÎÎn~ÎÎÎÎÎÎ~ÎS ^~~'~~

Acknowledgments

The author _is indebted to Dr. D.H.

Warrington

for the

suggestions

he made after

reading

^trie manuscript, ^and to Dr. M.

Kléman,

for useful and

encouraging

discussions. He is also

grateful

to tue Laboratoire de

Minéralogie-Cristallograpuie (Université

de Paris

6),

wuere _a part of tuis paper bas been written.

References

iii

Warrington D.H., "IIb92" Thessaloniki, Material Science Forum, Vol. 126-128

(1992)

_pp. 57-60.

[2] Warrington D.H. and Lück R., ^"Use of trie Wieringa Roof to Examine Coincidence Site Quasi-

lattices _m Icosahedral Quasicrystals", Aperiodic '94, 1994, Les Diablerets.

[3] Radulescu O. and Warrington D.H., "Arithmetic Properties of Module Directions _in Quasicrystals, Coincidence Modules and Coincidence Quasilattices", Acta Cryst. A, in _press.

[4] Pleasants P-A-B., Baake M. and Roth J., "Planar Coincidences for N-fold Symmetry", Tübingen preprint TPT-QC-94-05-1

(1994).

[5] ^Baake M., Pleasants P-A-B-, ^"The Coincidence Problem for Crystals and Quasicrystals", Aperi-

odic '94, 1994, Les Diablerets.

[6] Katz A. and Duneau M., J. Phys. France 47

(1986)

181.

[7] ^Liick R., Phys. Bl. 35

(1979)

72.

[8] Radulescu O., J. Phys. I France 3

(1993)

2099.

[9] Warrington D.H., Radulescu O, and Lück R., _in preparation.

[loi

Baake M., private communication.

iii]

Cahn R-W-, Ad~. Phys. 12

(1954)

202.

[12] ^Rokhsar D.S., Mermin N-D- and Wright D.C., Phys. Rm. B 35

(1987)

5487.

[13] ^{Holfmann S. and Trebin} H.-R., Phys. Star. Sol.

(b)174 (1992)

309.

[14] Dai M.X. and Urban K., ^Phu. Mag. Lent. 67

(1993)

67.

[15] Singh A. and Ranganathan S., J. Non- Cryst. Sohds 153&154 (1993) 86.

[16] Warrington D.H., Quasicrystalline Materials

(World

Scientific, 1988) _pp. 243-254.

[17] ^Mandai R-K-, Lele S. and Ranganathan S., Phil. Mag. Lent. 67 (1993) 301.

[18] ^{Sutton A.P, and Ballulli} R.W., Acta. Metall. 35

(1987)

2177.

[19] ^Elser V., Phys. Rm. B 32

(1985)

4892.

[20] ^Radulescu O., "Quasipériodicité, périodicité, et défauts dans les quasicristaux et leurs phases approximantes", Thèse, Orsay

(1994).

[21] ^Mordell L.J., Diophantine Equations

(Academic

Press, 1969) _pp. 164-173.