Local rules and growth in quasicrystals

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Local rules and growth in quasicrystals

Zeev Olami

To cite this version:

Zeev Olami. Local rules and growth in quasicrystals. Journal de Physique I, EDP Sciences, 1991, 1

(1), pp.43-62. �10.1051/jp1:1991103�. �jpa-00246303�

Classification Physics Abstracts

61.50 61.70

Local rules and growth in quasicrystals

Zeev Olami

(*)

Racah Institute of Physics, Hebrew

University,

Jerusalem

(Received

5 April 1990, revised 26 June 1990, accepted17 September 1990)

Abstract. The

problem

of local definition of

general projected quasicrystals (QC)

in two and three dimensions is studied. It is shown that local order in

projected QC

is _a result of the existence of

QC

fines of

points

within it

(which

_can be connected to the incommensurate constants of the QC and its

symmetry)

and of a

special

definition of the

strips.

The

shape

of the

strip

will determine whether strong, weak _{or no} local rules Will exist in such _a QC. ^For the _non

crystallographic

lattices with n-fold symmetry and for the icosahedral lattice _we construct lattices with strong local rules and _prove that strong ^local rules indeed exist. We suggest ^{that those results}

can serve to

explain

the

growth

_process and the relative

stability

of

quasicrystals.

I. Introducdon.

One of the fundamental

problems

raised

by

^the

discovery

of

quasicrystals (QC)

that does not exist for

periodic crystals

is the

problem

of how

they

_grow.

Quasicrystals

can be

represented

as cuts

through periodic

structures in _a space which has _a

higher

dimension

[2-6].

This

description explains

_many of the basic

properties

of icosahedral and

decagonal phases.

The diffraction

picture,

the symmetry ^{and the atomic} structure of those

phases

can be

explained using

this

representation.

Nevertheless the way in which

QC

can grow is not

clearly

understood. For _a

periodic

lattice this

problem

does not exist because

repetition

of the same

unit many times will

always give

_a

periodic,

well ordered lattice. For

quasicrystals

_no

periodicity

exists. There are at least two different tiles in the lattice.

Clearly, therefore,

_no

simple growth

_process will generate an ideal

QC.

An obvious

approach

to the

understanding

of the

growth

_process of

QC

is to

generalize

from the

growth

_process of

periodic crystals

and define the

growth

_process

using

the different local environments of the

QC.

A natural way of

defining growth

rules in _a

quasicrystal

is to

consider the set of all the

possible quasicrystal point

environments up ^to some radius R.

Using

this set, which _we call the

quasicrystal

_map, we can

try

to build _a model for the

growth

_process. In other

words,

with this

QC

_{map we can}

_try

to continue the

growth

_process

by completing

environments in the

boundary using

the map of the

QC

we want to build. A

basic demand from such _an

algorithm

is

locality

the

growth

at one

point

cannot

depend

_on

what is

going

_on

(or

not

going on)

in far away

points.

If _a

quasicrystal

_can be grown in such _a way, a more basic condition must obtain _: the

quasicrystal

must be defined

by

its local environments.

Any

structure that has the _same map is

equivalent

to the

original quasicrystal [7].

(*)

Present address _: Brookhaven National

Laboratory,

Dept, of

Physics, Upton,

New York 11973, U-S-A-

44 JOURNAL DE PHYSIQUE I M

For the Penrose lattice

[6,

_8] the

matching

rules define the lattice

(Fig. I)

_so that this lattice is defined

by

its local environments. One should note,

however,

that the Penrose

matching

rules do not define _a local

growth procedure

because freedom of choice exists for _some local environments of the Penrose lattice. Two distant free choices of this kind

might

be inconsistent with the

quasicrystalline long

_range order. As _a result of such inconsistent

choices,

disorder and breaks _occur in the

growing

structure. This

problem

can be solved

by demanding

that all the deterministic choices _are made first and that _{a non} deterministic choice will be made

only

when there is _no deterministic choice to be made. This

demand, however,

is

a

long

_range demand and not _a local rule. A different

algorithm

which _uses

only

deterministic rules _was

suggested by

Onoda but this

algorithm

also _{uses a} non-local

growth procedure

because

again

certain choices _are made

only

when other choices far from the

growth point

can no

longer

be done

[9].

The

previous algorithm

also _uses

extremely complex laws,

we would like to find out how

simpler algorithms

_can be defined based _{on a}

simpler

definition of the laws.

Another

approach

to this

problem

is to consider random

growth

with certain rules and then to consider the statistical

properties

of _a lattice which is grown this way. Such

growth might

lead to a random structure with

properties

which _can be correlated to actual structures but

z ³

1 3

~ _,

z

t

3

3

1

Fig.

I. The Penrose lattice

(from

Ref. [6]).

exhibit too much disorder. In this

study

_we shall not pursue this direction. We shall consider

only growth

_processes of ideal

QC [10].

Quasicrystals

_are described _{as a}

projection

from lattice in _a

higher

dimensional _space. In _n dimensional _space R~ _we define _a

periodic

lattice L. A

general point

in this lattice _can be written _{as :}

a ₌

£m~

_a~

(I)

n

where m~ are

integers

and a~ ^{are n}

independent

vectors in R~.

The space R~ is

composed

of two

subspaces

RP-the

physical

_space and R°-the

orthogonal

space.

Any point

_a in R~ _can be written _{as :}

a =

(a~, a~) (2)

where

a~

^{is the}

physical component

of the vector a and a~ the

orthogonal

_component.

When _{we use} the

projection

method any

point

_a in L which is contained in _some

orthogonal strip

S

(acceptance domain)

is _a

QC point.

We _can write this condition for the vector a =

(a~, a~)

^{in the}

following

_way :

A

point _a~

is _a

QC point

if

a~ e S

(3)

where a~ is the

orthogonal

_component of _a and S is the

orthogonal strip region

the acceptance domain.

Equivalent

lattices _can be

generated by moving

the

strip

in the

orthogonal

direction

~phason shifts). Any

structure that is _grown

by using

the local rules of the _map of _a

quasicrystal

_or that has this map can be

represented

in the superspace

representation

L of the

original quasilattice

_as a set of lattice

points

within this n-dimensional lattice. The

points

of this structure will

usually

define _a surface in the dimension of the

QC

in R~.

The

problem

of local definition _was discussed in _a

systematic

_way

by

L. S. Levitov

[7]

for the standard

quasicrystals

which _are defined

by grids.

Levitov introduced two

important

definitions which _we will _use in _our paper

I)

_a

QC

has

strong

local rules if _{any structure} that has the _same map is the _same

QC

_or

equivalent

to the

original QC by orthogonal

movement. In other words this structure is

generated by

the _same

strip

2)

a

quasicrystal

satisfies weak local rules if any

point

structure which has the _same map can

be

represented

in L _{as a} bounded set of

points

in R°

(I.e.

for any

point

in this lattice

(~P ~P ^" C

(C-a

real

number)

Levitov also introduced another definition _: if for three

grids

in _a

grid _pattern

_one of the

grid

lines passes

through

the line of intersections of two other sets of

grid

lines then _{we say} that those three

grids satisfy

the SI condition.

We

present

some of Levitov's main results. Two dimensional

QC

which _are

generated by

four

grids

that

satisfy

the SI condition between them have weak local rules. Grid

QC

^which

satisfy

those conditions _are defined

only by algebraic

numbers of the second

degree (of

the

form

(m /k

₊

/ _/q).

Furthermore the SI condition exists for any Grid

QC

with

strong

^local rules. Levitov has made _some limited

generalizations

from those results for 3-dimensional

lattices. General

projected QC

and conditions for

strong

local rules _were not discussed at all.

JOURNAL DE PHYSIQUE I T I,M I, JANViER lwl 4

46 JOURNAL DE

PHYSIQUE

I M

In all this discussion and also in other works _on

matching

rules of

QC just

_one kind of

QC

was

discussed, namely

the standard

QC generated by grids.

Most discussions refer to a

specific QC

and discuss its local

properties.

In this paper we are interested in the

properties

of

general projected QC (that obviously

include those

specific grid QC).

We present

general

results _on

strong

and local rules _on

general projected QC

in two and three dimensions.

We

present general

_{arguments on} when weak local definition and strong ^{local definition} exist. We find that the existence of both types ^{of rules}

depend

in

general

2-d

QC

on the

existence of

QC

lines

through

the lattice

(QC

lines _are lines within the

QC

that have _an infinite number of

points

within

them).

If lines

exist,

and those fines

keep

certain

laws,

the existence of weak and strong ^{laws will}

depend

_on the form of the

strip.

If the

strip

boundaries

are not in the direction of the

QC

lines

(I.e.

the _set of

orthogonal projections

of the

points

whose

parallel projection

generate ^the

physical QC line)

_no local order will exist _at all. If

they

are

along

line

directions,

weak local order will exist. If _any

possible

movement of the

strip

will create a two-level system ^of

points,

then

strong

^{local order will exist.}

Using

those results _we show that lattices with inflation symmetry ^{and all the}

noncrystallographic

lattices with n-fold symmetry can have weak and

strong

local rules. We _construct

explicitly

the

strips

that will

provide

strong ^{local rules.}

The results _are

generalized easily

to the _case of the icosahedral lattice. All these statements can be

proved

and _we show that for _some

strips

that generate new

QC

strong definition exist.

Another

important question

that _was not addressed is the

following

_: is the entire map needed for _a proper definition of _a lattice that has local rules ? Can essential features of the map be extracted _so that _a smaller set of

defining

rules _can be found ? This is indeed the _case for the Penrose lattice. If such _a set of rules _can be

found,

can we use it to generate ^local

growth

?

Our result

gives

_{a very}

simple general

definition of rules that

provide

weak local definition.

The

QC

line rules of any lattice of the above kind _are

enough

to define it

weakly.

An addition of _map rules up ^to the _range of the two-level system

jumps

will be

enough

to

provide

_strong

local rules. Those kind of rules _can be used

effectively

to define the

growth

and to

provide energetic

considerations for the

stability

of

QC'S.

In section 2 _we discuss _one dimensional

QC

_{as an} introduction to the next sections and show that _no weak rules exist for them

(this

result is well

known).

In section 3 _we discuss

QC

lines within the two and three dimensional

QC.

We discuss conditions for the existence of

QC

lines within _a 2-d _or 3-d

QC,

show the

geometrical meaning

of these conditions and show that such demands in _a four dimensional lattice lead to the definition of

quadratic

incommensurate constants. We present a

representation

of n-fold lattices and show that

QC

lines will exist for _any 2-d

QC

with n-fold

symmetry.

^{We also show} that the existence of _an inflation transformation leads to the existence of

QC

lines. In _a

general QC

the existence of sets of

QC

lines will

play

a role

analogous

to the SI condition in _a

grid QC.

In section 4 _we associate the conditions for the existence of

QC

fines with two level systems

and their structures.

Using

this _{we can} formulate demands _on the

strip

form and the

incommensurate _constants if local rules exist. We show that the non-existence of fines leads _to the absence of weak local rules. We prove that the existence of four

independent QC

lines within the

QC

under _some conditions will define the

phason

strain of the

QC

as zero and lead to the existence of weak local rules. We show that weak local rules exist for the n-fold lattices and lattices which _are

projected

from 4-d lattices. We also show how definition of the

strip boundary

_can lead in these _cases to the absence of local rules.

In section 5 _we show how

strong

^{local rules arise from} a

specific

definition of the

strip

form and the local laws. We construct lattices with strong ^{local definition for all lattices with n-fold}

symmetry ^and prove that

they

indeed have it

using

_{a new} method.

Since _our

geometric approach highly simplifies

the

proofs

_{we can}

easily generalize

our

results _to 3-d

QC

in section 6. In 3 dimensions the results _are

analogous

to the results in 2 dimensions.

Our

approach provide

_a

general

and very

simple

_way of

defining

local order within _a

QC.

Definition of line directions and rules of line distances will be

enough

to

provide

weak local rules where

they

exist.

We _use these results in section 7 to discuss the

growth

of

QC

and _we

suggest that, though

_an ideal

QC

cannot grow

locally,

relaxation of the demand of

perfection

from the

growing

structure

might

enable local

growth

to _occur. Growth rules that

provide only

a weak

definition of the

QC

_can

give

_a

satisfactory growth procedure

in which mistakes do not

destroy

the

global

order. Global correlations _can be

achieved,

for

example, by keeping straight QC

lines

passing through

the

points

of the

QC.

2. One dimensional

quasilattices.

The

problem

of local definition is very

simple

for _one dimensional

QC

lines

[7].

It _can

easily

be

proved

for these lattices that _no weak local rules _can exist. Therefore _no

growth

rules _can be defined for such _a line. Nevertheless _we discuss this _case since _some of the results and the

ideas _{we use} here will be used later for the _case of the 2-dimensional

QC.

We will then find out

that, though

_no weak local rules exist for _{a one} dimensional

lattice,

local rules exist _{as a} result of the existence of _one dimensional

QC

lines within _a two dimensional

QC.

We discuss _a one-dimensional lattice in two-dimensional space

R~ (Fig. 2). Physical

_space is defined

by

_a two-dimensional vector rP

=

(I,

_p

),

the one-dimensional

orthogonal

_space is in the direction r°

=

(-

_p, I

).

We define _some radius R _as the radius of the local environment map and then define the _map

using

all the environments _up to this radius in the lattice.

E

e e

E

o o m

o e o o o

.~

O O O . O

Fig.

2. A _one dimensional

quasicrystal projected

from _a two dimensional _square lattice.

48 JOURNAL DE PHYSIQUE ^{I M I}

In _a small movement of the

strip

in the

orthogonal direction,

_an infinite number of

points

will appear and

disappear

in the

QC.

The

density

of these

points

in the lattice _can be made

arbitrarily

small

by defining

a small

enough orthogonal

movement. If _a 2-d lattice vector

connects the

strip boundaries,

these

points

will appear as two level systems a

point

will

disappear

and at a fixed distance from this

point

another

point

will appear. If _no 2-d lattice

vector connects the

points

then isolated

points

will appear and

disappear

with _as

large

_a

distance between them _{as we} wish. In both _cases isolated

changes

in the lattice _are

possible

without _a violation of the local rules. Therefore

obviously

_{no strong} local rules _can exist in such a lattice.

Furthermore _{we can use} this

technique

of

introducing

isolated

changes

without a map violation in such _a lattice to create _a

phason

strain in the lattice _or to induce disorder of unlimited

phason length.

This _can

easily

be done. Since the

changes

_are isolated _{we can}

perform

_an

orthogonal

movement at _one side of the isolated

change and,

without _a violation

of the map, obtain _a line which is built of two half

QC

lines which _are removed in the

orthogonal

direction. We _can continue this kind of process ^at any

point

where _an isolated

change

is

possible

and obtain

orthogonal

movement of any

magnitude.

The

phason

strain of these

changes

will be limited

by

the range of the laws of the map and the incommensurate

constant and

thereby

limits the maximal

change

in the strain. We _can of _course

generate

random lattices which _are not

QC

_or

periodic approximants

in this _way.

The conclusion from this

argument

^{is that} the absence of strong local order leads in this _case to the _non existence of weak local order. As _we will show later this is also true for lattices with

a

higher

dimension under certain conditions. It does not _mean also that in other _cases weak local rules do not exist. This _was not realized

by

Levitov for the two dimensional lattice.

The

possible phason

strain _can be defined

by

the

phason change

in the

orthogonal component

^a° and the

physical

distance aP _over which the

change

_occurs. The

phason

strain is defined

by

the ratio

AR = a%aP .

(4)

This _{number defines the hason} strain of this lattice. We _notice that

this

is an

average

umber and

that

it is accurate up to _{the scale}

of

the strip of _the original

a° is large we _can

consider this number as exact

(and we shall _do it later).

We

also

^note that if

the

acceptance

domain

is _defined

by

a lattice

vector

and _therefore level _systems exist in the lattice, any hange in the

length of the _acceptance domain will

3. Existence of

quasicrystal

lines in 2 and 3 dinlensional

quasicrystals.

A

QC

line is _a

quasiperiodic

line in the

QC

that contain _an infinite number of

points.

The existence of

quasicrystalline linqs

ⁱⁿ a

higher

dimensional

QC

is _a result of the existence of _a 2-d

(or more)

sublattice

planes

of the

original periodic

lattice in the

higher

dimensional

periodic hyperspace

that _can be

decomposed

into both

parallel

and

orthogonal

spaces. If _no such

plane

exists and

only planes

that cut

parallel

_space at one

point

exist in this

lattice,

then

no

quasicrystal

lines will exist in the lattice since

only

a finite number of

points

of this

plane

will appear in the

QC.

If such _a

plane

indeed exists the

orthogonal projections

of the

points

of this

plane

will

generate

a dense line of

points

in the

orthogonal

_space.

If this 2-d lattice

plane

is

spanned by

two lattice vectors k and _q we must be able to construct vectors which _are in the

parallel

direction and also vectors in the

orthogonal

direction from these two vectors. This will be obtained

only

when

~~

^~

"~~ (5)

k

=

pq

If

(4)

holds then the two dimensional lattice

generated by

k and q

generates

a one

dimensional

QC

in the

original quasilattice (another possibility

which _we shall not discuss here is such _a 3-dimensional space because then

only

_one kind of _a line _can exist in the

lattice).

In _our discussion _we will _use here the incommensurate

representation

of

orthogonal

and

parallel

lattice introduced

by

S. Alexander

[11]

since this

representation

clarifies the incommensurate character of the lattice. In this

representation

the lattice is

represented using

the

separate

incommensurate constants p.

For _a four dimensional incommensurate space the basis vectors _are :

for the

physical

_space

et

=

fP~(10

_p

0) e(

=

~P~,(0

⁰

^p')

^~~~

and for the

orthogonal

_space

We write _our condition in _a detailed _way

using

this base _:

i lilt]i]i]il ⁽⁸⁾

for the

parallel

_components

for the

orthogonal

_components

~~

~~~ ^~~~~ ~)~~ ₍₁₀₎

P2 R P4 "

fl (~2

_R

~4)

We notice that for _any lattice in which _an inflation

symmetry

^exists

(like

the

pentagonal

and the Icosahedral

lattices)

_our conditions

(9-10)

will be obtained since for such _a lattice _a

symmetry operation

T _on the lattice exists for which

Tp

= q, det

(T~

=

I

pP

=

~yqP (ii)

p°

=

I

laq°

Therefore _{we can span a} lattice

plane

which will generate a one dimensional

quasicrystal

for

any ^vector p. The _same argument ^is

right

also in

higher

dimensional _spaces.

If

special

conditions

exist,

_any incommensurate number _can be defined. If

pi = P2 = q3 " q4 or p3 " P4 = qi = q2 any incommensurate number _can be defined. An

example

of such a lattice is the square incommensurate lattice. In any other _{case we} obtain two combined

equations

for the incommensurate _constants

by dividing (9) by (10).

50 JOURNAL DE PHYSIQUE I M

~Pi

+

aP3) (qi

+

aq3)

~P2 +

a'P4) (q2

₊

a'q4)

(12)

~Pi aP3) (qi aq3)

~P2

a'P4) (q2 a' q4)

After _some calculation _we get

RR'~P3

_~4

~3P4)

"

(~l

_{P2 PI}

~2) (l~)

If the coefficients of this

equation (12)

are zero, the

equations

for _p become linear

equations

and the lattice is not incommensurate. If

they

_are not zero we get a

quadratic equation

with

integer

coefficients for the incommensurate constants. Therefore _{p can} be

written _as

p =

(m+ /)/1. ₍₁₄₎

m, n, I

integers

For _a lattice with _one incommensurate constant

(p

=

p')

the result is

p =

(m

₊

/)li. ₍₁₅₎

m, n, I

integers

This condition

puts

a severe limitation _on the

QC

and its incommensurate constants. An

example

is the

non-crystallographic

2-d lattices with _{an n} fold

symmetry C~.

^The incommen-

surate numbers of these lattices _are defined

by

the irrational numbers sin ~"

),

n

cos

~ _"

These numbers _are

always algebraic

numbers of _a

degree

which is

dependent

_on

n

the Euler number

j (n) ~loiven [I I]).

The ratios between these incommensurate numbers _are

quadratic only

for _n

=

5, 8, 10,

12. These lattices _can be

represented

in 4 dimensional space and therefore will

always

have

QC

lines.

We _now

give

a

general proof

that for _a lattice with symmetry

C~ QC

^{lattice lines} appear.

We first

give

the

general

construction of these lattices. Let L~ be _an n-dimensional cubic

lattice in n-dimensional _space R~ whose basis vectors _are the _n orthonormal vectors

a~. The _n fold

operation

is

represented by

the n-dimensional

operation

that permutes ^the vectors a~

cyclically.

We define _a

j (n)

dimensional _space

by

the

following

base

e~i =

(sin

~"U ))

n

e~2 =

COS

^{~ "~J}

~~~~

n

where I

= I..,n and

j

are all the

integer

numbers smaller than

n/2

whose _common

denominator with _n is I. It is easy to see that _a

pair

of vectors of number

j

defines _a 2- dimensional

subspace Rj

that is invariant under the n-fold

operation

^{and transform}

according

to _one of the two dimensional

representations

of the _group under the rotation. We choose

Ri

to be the

physical

_space and define

strips

in all the other

subspaces _Ry.

It is _easy to _see that

an inflation transformation exists for this lattice. An

example

is the _n dimensional

transformation

010... 0 01

0 0 0... 0 0

T= 01010...

(17)

I..

_{0 0}

whose fines _are the

permutations

of the first line

by

the n-fold rotation. Therefore it is _easy to see that there _are lattice lines that _are

generated by ^~ ~~

dimensional lattice _{as we} showed before in the four dimensional _case. An

important

class of lines are the fines in the directions of the

projections

of the unit vectors on

physical

_space. These lines _are

projected

from _a

subspace

which is

generated by

the

projections

of the unit vectors _on the lattices

l§.

The

QC

lines will be defined

by

one dimensional

strips

within the

orthogonal

acceptance domain which _{are cuts} of the

planes

with

Ry.

^One can

easily

_see

that,

for any

domain,

_an infinite number of different _one dimensional domains will exist. Therefore _an infinite number of different kinds of lines will exist within any 2-d

QC.

We _now discuss the conditions for _a 3-dimensional

QC generated

from _a 6-dimensional

periodic

lattice.

Again

_we will consider

only

the _cases of

periodic

lattice

planes

that _are

composed

of _an

orthogonal

and

parallel

parts.

The 3-d conditions have the _same structure as the 2-d conditions and _we

again get

^the same

result: the lattice must have

quadratic

incommensurate constants with certain relations between them. As before _any inflation transformation will lead to the existence of

QC

lines in any direction in the lattice.

We notice that the

geometrical origin

of _our conditions and the _so called SI condition of Levitov _are the _same.

4. Existence of

quasicrystalline

lines and local order.

In this section _we discuss the

relationship

between local

rules,

the existence of

quasicrystalline

lines within the

QC

and the

geometric

form of the

acceptance

^domain.

Let _us

begin

with _some definitions. We define _a two-level

system acceptance

^domain as an

acceptance

domain for which any movement in the

orthogonal

direction that will take _a

point

out of the

strip

will

bring

into the

strip

_a

point

with _a limited distance from the

original

_one.

One

simple example

of such domains _are

Voronoy

constructions but there is _an unlimited number of

possibilities.

An

acceptance

^{domain is} a line

acceptance

^{domain if its}

boundary planes

_are defined

by projections

of

point planes

that generate a

QC

line.

Using

these definitions _{we can} formulate the conditions for the existence of local rules.

A

QC

that has _no lines _or does not have _a two level system ^line

acceptance

^{domain does} not have weak local rules.

To prove this statement _we suppose that _some set of local rules _up to _a radius R is defined in the lattice and that this lattice is not _a two-level system ^line lattice.

If the acceptance domain is _a line acceptance domain _any

orthogonal projected hyperlattice point

_near the

boundary

of the

acceptance

^{domain will}

belong

to a dense line of such

points

which will be

parallel

to the

boundary

line

(Fig. 3). Any phason

shift of this domain will introduce such _a dense line of

points

into the

strip

since the line is

parallel

to the

strip

_{so a new}

QC

line will appear in the

QC.

However if the

boundary

of the

acceptance

^{domain is} not

defined

by

_a

projection

of such _a

plane

_any

phason

shift will define isolated

changes

in the

QC

with _as

large

a distance _{as we} wish. This is because there is _a finite number of _vectors

along

the

52 JOURNAL DE PHYSIQUE I M

Qc line points

strip /~

,....

:J...

~l

The strip fc>ra Qcline

Fig.

3. The

orthogonal projections

of the lattice

points

of _a

QC

line _near the

QC boundary.

Note that if the

boundary

is not

parallel

to the

points

_a

phason

shift will introduce

single points

within this line into the lattice. Note also that the intersection of these fines _on the

strip

define _one dimensional

strips.

Since these fines _can be

phason

shifted there is _a continuum of

lengths

of the

strips,

between _some maximal and _some minimal

lengths.

boundary

of the acceptance ^domain.

Only points

connected

by

these vectors will appear or

disappear

_{as a} result of _a small movement. When the acceptance ^domain is transformed

by

_a

small

orthogonal

_movement,

only

these

changed

environments with distances

largen

than R will appear. If _any two environments appear with _a smaller distance than R between

them,

one of them will

disappear

if _we make the

orthogonal

movement

smaller,

since the _vector that connects them is not on the

boundary

of the

acceptance

domain. The _same is true also if the

acceptance

^{domain is} not _a two-level system domain.

In such _a lattice _{we can} therefore

always

define _a small

enough orthogonal

movement

e(R)

of the

acceptance

^{domain such that}

only

isolated

changes

with _a distance

larger

than R between them will appear in the lattice

(this

kind of argument was used

by

Levitov for the

same

purpose).

Such _a lattice _cannot have strong ^{local rules} because _{we can}

independently change

each environment

separately

without

causing

_any violation of the rules of the

QC.

Such _a

crystal

has _no weak local rules either. To show this _we make the

following

construction. We find _a small

enough orthogonal

translation vector

e(R)

such that the distances between

changed

environments will be

larger

than R. Since the radius of these environments is limited _{we can}

always

find their maximal radius X. We then

perform

_a cut

through

the

QC

between these environments and in _a certain direction this cut will

give

two half

planes. (It

is

always possible

to do such _a cut

though

the line that defines it will not be _a

straight line).

We _now transform _one half

plane by

the

orthogonal

movement and leave the other half

unchanged.

The _new lattice is consistent with the map of the

original

and _{we can}

continue this process ^at a distance R ₊ 2 X from the

original

cut. Since this

procedure

_can be carried _on

indefinitely

the

orthogonal

shift in the coordinates of the lattice is unbounded while the local map is the _same. This

is, however,

inconsistent with weak local order. Therefore _no local order exists in such _a lattice.

This kind of argument

appfies

both to lattices without lines and to lattices which do not have

a proper acceptance ^{domain. For the second} kind it is

possible

to do _a modification if _a subset of

points

with

strong

local order exists in this lattice and creates _a lattice with weak local rules.

This _can be done

by separating

the lattice

points

into _two kinds of

points

_{: one} set of

points

with strong ^{local order and the} second the rest of the

points

in the

QC.

If the first set has strong ^{local rules} one can orient the second set of

points

in relation to it and

obviously

the

possible orthogonal

shift will be bounded.

We _now claim that the existence of

independent QC

lines in _a lattice will define the

slope

of the lattice and prevent ^the

development

of

phason

strain.

Relatively

weak demands from the

local rules will be needed to obtain this result.

We first discuss the

pentagonal

lattice.

By

_our former discussion the

pentagonal

lattice has

QC

lines. We demand that this

pentagonal

lattice will

keep

_a few local rules _: four

QC

lines will _pass

through

each

point

and there will be _a minimal and maximal distance between

neighboring points

_on those lines.

If those rules _are

kept throughout

the

pentagonal lattice,

weak local rules will exist and the

phason

strain will be _zero.

The

pentagonal

lattice _can be

represented

_{as we} showed before for the

general

lattice

C~ by

_a four dimensional

periodic

lattice with _a 2-dimensional

orthogonal

and

parallel

_space.

The

following

directions _are chosen to _{serve as} the

pentagonal

directions in

parallel

and

orthogonal

_space

(Fig. 4)

_:

~

2 vi 2 vi

~ ~°~

5 ^'~~~ 5

~° l~°~ ~i~ ^~~~~ ~i~ II

_~~~~

The

pentagonal

_group in four dimensions transforms

parallel

and

orthogonal

_space

according

to different

representations

of the

pentagonal

_group. This is the _source of the

difference between the

orthogonal

and

parallel

vectors.

G°

p gP

G~ i g°

Goo

gP

o

~4

o

G2o

a) b)

Fig.

4. The two dimensional

pentagonal

star e'in

parallel

and

orthogonal

_space. Notice the difference in the directions between the

orthogonal

and

parallel projections.

54 JOURNAL DE

PHYSIQUE

I M I

In each of the five

QC

directions

QC

lines _exist. If the

slope

of the

quasi crystalline

is not

along

the

perfect QC

direction _an

orthogonal

strain

develops,

characterized _over

larger

distances _as

AR

=

c%cP (19)

where _c ° is the

orthogonal

strain and

cP is the

parallel

distance.

Even if such _a strain exists the

points

of _a certain line will still be _on the _same two- dimensional

plane.

Therefore _{we can} still

represent

^the

points

of the

QC

line _on two lines in

parallel

and

orthogonal

_space.

We _now suppose that such _a strain exists for _some line in the direction _eo. We _can construct in this _case two fines in

orthogonal

and

parallel

_space between the _two end

points

A and B of the

QC.

Since

by

our

assumption QC

lines pass

through

each

point

we can construct a

triangle

of

QC

lines

by adding

two lines in the directions _{e~, e~}

(Fig. 5a).

This will _{create a}

triangle

ABC in

orthogonal

_space and _a

triangle

ABC in

parallel

_space

(Figs. 5a, 5b). @Ve

^define

lengths

and

points

in

orthogonal

_space

by

bold

letters).

Since

large

scales _are involved _we

ignore

small mistakes _on the scale of the

QC

distances that _can

happen

in such _a construction.

The strain in the line AB is AB

/AB.

We construct scaled

triangles

in

orthogonal

and

parallel

space in the

following

_{manner : we use a} line in the direction e4 ^{to construct} the side CB' between C and the intersection of this line with AB. We then continue and build the scaled

triangle

AB'C' and the

triangle

AB'C' in

orthogonal

_space

(Fig. 5).

The result of the

c

B ~2

G~ c.

a go a. A

a)

A

e~

e~ e~

c e~ a

b)

Fig.

^5. The construction of scaled pentagonal triangles in parallel

(a)

and orthogonal _space (b).

difference in the directions _e in

parallel

and

orthogonal

_space is that the

triangles

ABC and ABC are scaled

differently by

this construction. This construction scales AB

by

the factor

(1/2

_cos

(36))~

and AB

by (1/2

_cos

(72))~

_so the _new

phason

strain is scaled

by ^r~

AR

^'

= r

~

AR (19)

AB _can

always

be selected great

enough

_so this

procedure

can be

repeated

^N times and _we get

Apjj

=

r~/~Ap. ₍₂₀₎

We showed in section 2 that the

phason

strain is bounded

by

the lattice rules but _our

procedure

_can create as

large

_a strain _{as we} wish. Therefore the strain should be _zero.

We notice that the essential

point

of _our

argument

is that since R° and RP transform

differently by

the

pentagonal

_{group no}

orthogonal

strain is

possible.

Our

proof

_can be carried out in the _{same way} for _a lattice with n-fold

symmetry C~

^{if the} proper fine directions _are chosen. For odd _{n we} choose the

following

lines in

physical

space

Ri

_: the lines in the directions of the

projections

of the orthonormal cubic base vectors ao, ai, a~ _i and ai

(where

I is the

integer _part

of _n

/4) (see

the notation Sect.

3).

It is very easy

to _see that _our construction will be valid also here for

Ri

and

R~.

For the 8-fold and 12-fold lattices this construction will not be efficient. A

good

choice will be to choose the set of lines in these directions and

also

in the directions between the

projections

of the unit vectors

°1

=

)

₊ ^~

_{)~ (21)}

For both lattices _{we can}

perform

_our construction

using

this set of

eight

_or twelve lines. For these lattices it is _more convenient to define the

projection differently

and

project

the unit vectors in these directions

(I.e.

to redefine the

angles

in

equation

16 _as half their values _so that the unit _vectors will be

projected

in n-directions and not in

n/2 directions).

In this _case all these lines will have the _same local rules and the

symmetry

^is

higher.

Other lattices with _even

n have _a

subgroup

that _we have

already

discussed _{so we} choose lines associated with this

subgroup

to

perform

_our construction. So _we show that all lattices with

symmetry C~

^{weak local rules exist.}

We _can also extend this discussion to other 4-dimensional lattices under the _same

conditions. To do this _we first discuss

QC

lines within _a 2-dimensional lattice.

We _assume that four

QC

lines exist in the lattice and pass

through

_any

point.

The first two

are defined

by

the four

independent

vectors n~

(I

= 1,

4).

n~

=

ant

~o

p~o

~

(22)

nf= a'n(

o

p

^{, o}

n4 " n3

Since

only

four

independent

vectors exist in such _a lattice the rest of the lines _can be defined

using

these _vectors. We define the four vectors

A, A', B,

B'

A=£a~n~ B=~b~n~

, ,

A'

=

£a)

_n~ B'

=

~ ^b)

n~.

(23)

, ,

56 JOURNAL DE PHYSIQUE ^{I M I}

These four vectors define two other lines and relations similar to

(19)

between A and A'

(B

and

B')

obtain. We _assume that these relations exist here. These four

QC

define four

directions in

parallel

and

orthogonal

_space.

We _can

always

build scaled

triangles

for any lattice which has four

QC

line directions _{as we}

already

did for the

pentagonal

_case. Two scaled

triangles

_can be built

(we

show them in

Fig. 6).

The

possibility

^of

defining

scale transformations of the

type

we used before will

depend

_on the ratios between the

lengths

of the sides of the

triangles A'B'C,

A'B"A and the

corresponding orthogonal triangles.

a

a

8' fi.

B"

n(

n( n(

C

A'

o

~

A' ~2

A A

a)

b)

Fig. 6. The construction of scaled triangles for _a general set of QC lines is described in orthogonal and

parallel

_space.

If AA'

/AC

=

AA'/AC

these transformed

triangles

will transform with the _same factor and therefore _no scale transformation will be

possible.

For the situation described in

figure

6 this condition will lead to the

following equivalence

_:

bi+b~aa~+a4a' bi+b~pa~+a4p'

ai+a~ab~+b4a'~ ai+a~pb~+b4p"

^~~~~

However,

this condition does not

usually apply.

If AA'

/AC

~

AA'/AC

_we construct the scaled

triangle

AA'B". The combined scale factor for the

orthogonal

strain will be AA' AC

/ (AC AA').

This number is

greater

than _one.

If AA'

/AC

_~

AA'/AC

the scaled

triangle

A'B'C is constructed and

again

_we

get

the scale factor

(A'C

AC

/ (AC

A'C which is greater than _one.

Therefore if

equation (24)

does not

apply

_{we can}

always

build _a scaled

orthogonal

strain with a factor greater ^than one and therefore the

only

consistent strain is

again

_zero.

We conclude that any

QC

^that is

generated

from _a 4-d lattice and contains _a set of four

QC

lines for which

equation (24)

does not

apply

has weak local rules

providing

^{that there} are some line rules _on the distances between

points

_on the fines. A subset of the map rules is

enough

to

provide

a weak definition of the

QC.

5.

Strong

local rules in two dbnensional

crystals.

We _{are now} interested in the

following problems

_: what _are the conditions for the existence of strong ^{local rules ? What is the} minimal _set of

defining

local rules that is sufficient to

provide

a

weak and

strong

^{definition of the}

QC

?