Modeling decagonal quasicrystals : random assembly of interpenetrating decagonal clusters

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Modeling decagonal quasicrystals : random assembly of interpenetrating decagonal clusters

S. Burkov

To cite this version:

S. Burkov. Modeling decagonal quasicrystals : random assembly of interpenetrating decagonal clusters.

Journal de Physique I, EDP Sciences, 1992, 2 (5), pp.695-706. �10.1051/jp1:1992160�. �jpa-00246577�

Classification Physics Abstracts

05.50 61.50E

Modeling decagonal quasicrystals

_:

random assembly ^of interpenetrating _decagonal clusters

S. E. Burkov

Comell

University, Laboratory

of Atomic and Solid State

Physics,

Ithaca, NY14853-2501, U-S-A-

Landau Institute for Theoretical

Physics,

Moscow, U-S-S-R-

(Received it February 1991, revised 26 November1991, accepted 22 January 1992)

Abstract. Conventional

tiling

models of

quasicrystals imply

the existence of _{two or more} elementary cells (tiles). A _new

approach

is

proposed

^{that allows} a

quasicrystal

to be

thought

of _as

a random assembly ^of identical

interpenetrating

atomic cIusters. This model is shown _to be

equivalent

to _a decagonal binary

tiling.

On

applying

a random

tiling hypothesis, originally postulated by

Elser, _to the present cluster model it is found that the free energy as a function of the alloy

composition

has _a cusp at a

point exactly corresponding

to the

decagonal quasicrystal.

This fact

helps

_to

explain

an old mystery,

namely,

why a system ^is phase locked in _a

quasicrystalline

state even

though

it is incommensurate.

Introduction.

The _recent

experimental

_progress made in the

study

of the

thermodynamically

stable

decagonal quasicrystals

Al-Cu-Co and Al-Ni-Co

(believed

to be

isostructural)

has revived _an interest in the

decagonal phase.

These materials have been

probed by high

resolution electron

microscopy (HREM) [4, 15, 16], scanning tunneling microscopy [3], powder [2]

and

single crystal X-ray

diffraction

[17-19]. However,

the aim of the

present

^{article is} not to determine the atomic structure of these

particular alloys,

Alcuco and

AlNico,

but rather _to

develop

_a

mathematical framework which

might

^be used in

determining

their structure as well _as that of other

possible decagonal quasicrystals.

The atomic model for Alcuco and AlNico has been

recently developed [20] making

_use of the above mentioned

experimental

data and the

general approach

described below.

Although

the

decagonal

cluster shown in

figure

I of the present paper is _a cluster

successfully describing Alcuco,

I will not discuss below

why

this is so,

leaving

the discussion _to

[20].

This is done,

first,

not to

qbscure geometry by crystallography and, second,

to

emphasize

that Alcuco

might

be

merely

one

particular example

of _a class of

decagonal quasicrystals. Besides,

the

general approach

itself _was

developed

before the

experimental

works

[15-19]

had been

published.

The paper consists of _two almost

independent

parts. ^The

first, geometric,

^section

provides

_a

fresh

insight

into

tiling

models. The conventional

interpretation

of

tiling

models that decorates the _two Penrose tiles

[5]

_seems unconvenient to

explain

the

high

resolution electron

micrograms [4].

On

contemplating

^these

experimental images

I discovered _{a new}

approach

to their

interpretation

and _to the atomic model itself. The _new model

proposed

in this paper appears to be

mathematically equivalent

to one

particular tiling

model, I.e. _a

decagonal binary tiling [6-8].

Nevertheless, I believe that the present ^{cluster model is} more

justified

from

a

physical point

of view and is _more attractive because it _uses

only

_one

decagonal

atomic cluster instead of _{two or more} rhombic tiles. In other

words, by starting

with _{a more}

general proposition

than is usual

[5]

the

binary tiling

_can be

derived,

in contrast with

previous

^works

in which it _was taken _{as an} axiom. Section I is devoted

only

to geometry ^and may be

applied

to either _« random

tiling

_»

[6-9, 12]

or « ideal _»,

phason free, lo, 14] quasicrystals.

A reader not interested in

tiling

geometry may

skip

the

beginning

of section

I,

_{up to} definition 3.

~ o~.

O

O .

o . .

~ O

~ O

~ *a ^{. .} co ^a.

O~

O~~~*O

_O

° O

~

OO

Fig.

I. One of the

possible examples

of _a

decagonal

atomic cluster, _a cluster found in Alcuco and AWiCO. Circles _: Al, _squares T-metal _; white _{: z}

= 0, black _{: z}

= c/2 layer. Presence of two

layers

and,

respectively,

of two colors is _not essential for the present

theory.

Section 2 addresses the

problem

of

phase locking

in _a

quasicrystalline

state. The random

tiling approach

^is

adopted

^{in this} section.

Although

the results of section I _are used _to derive _a cusp in the free energy, this is done

only

to make calculations

tangible.

The main

result, regarding

the existence of a range of chemical

potentials corresponding

to a

quasicrystalline phase,

_can be rederived for

virtually

_any random

tiling models,

not

necessarily decagonal

and

not

necessarily

two-dimensional.

Moreover,

I believe the result is still valid _even

beyond

the framework of

tiling

models,

provided

that _some

generalized

version of entropy stabilization

hypothesis [12]

holds.

Real materials _are three-dimensional

layered

structures.

Stacking

of

decagonal layers along

the 10-fold _axes is

usually presumed periodic. Although

there _are certain doubts in

stacking periodicity [21]

this

topic

will _not be discussed below. In Alcuco there _{are two}

decagonal layers

_per vertical

period

_c

=

4.15

I _(they

are shown

by

black and white in

Fig. I)

however I will

completely disregard

this

subtlety

and consider _a

purely

two-dimensional model.

1.

Geometry

of

overlapping decagonal

clusters.

Below is the

description

of the model. Atoms _are allowed _to form

only

one

decagonal

cluster

(Fig.

I is _an

example,

the cluster found in Alcuco

[20]).

One does not care about the

particular positions

of atoms, all _one needs is

decagonal

_or at least

pentagonal

_symmetry for the atomic

configuration

inside the cluster. Clusters may intersect _or share _a side. The latter

case will be referred to as a

particular

_case of the former. The

positions

of atoms inside _a

region

of

intersection, although

similar _to those of _a sole

cluster,

_may not be

exactly

those.

However~ there is

only

_one atomic decoration for each

type

of intersection

region (Fig. 2).

OaO O~.

O'~ ~"O O~ _~"O

. . O . ~.~ .

©

~,a.°,

_

.©, a

~

~~ ~, ,~ ~f

^~

.~O

~

.~.

~

°

~~

O . . O~. .

a) ^{~ ~ ~}

.a. ~ .a.

~.~

^~ ~

~. .~

° o ^°

o' 'o

o ^°

° .U" .°. °

. .u u. .u u.

~ ~« «~ ~« «~

~ a~a a~a

~

~ ^o ~

: o

~ ^o ~

.o on a.

. ~.~ ^{o .} ~.~ ^o

b)

Fig.

2.-a) ar/5-intersection, b) 3ar/5-intersection.

DEFINITION I. A ^"

,

3

f -covering

is _a

covering ofa plane by

identical clusters such

5 5

that

I) Every point of

the

plane belongs

to at least _one cluster.

ii) If

two clusters intersect then it is either _a ^" _{or a} 3 ^" intersection

(Fig. 2).

5 5

The

(ar/5,

3

ar/5)-coverings

will also be referred to as allowed

coverings. Every

allowed

covering (Fig. 3)

is characterized

by

^the set r of the cluster centers. One _can examine this set and the Voronoi

decomposition

associated with it. Let the cluster radius be

equal

to I.

According

to definition I the distances between

intersecting

clusters take two values

only (Fig. 2)

_:

I

= 2 _cos

(gr/10)

_{or s}

=

2 _cos

(3 gr/10). (I)

Note that

(Is

= T =

(/

+

1)/2

= 1.618..

,

the

golden

mean.

Connecting

the _centers of the

intersecting

clusters is _a

graph (Fig. 3)

with bonds of the _two

lengths, ^I

and _s. We shall call this

graph

_an

^I-s graph.

^Note that the above

given

definition of both the gr/5 and 3 gr/5 intersections

implies

that if _two clusters intersect then their _axes of symmetry are

parallel,

This is

possible if,

and

only if,

the

angles

between the bonds _are of the form

grn/5,

I-e- the bonds

of

the

I-s graph

are oriented

along

_a

symmetric

5-star.

It is useful to

distinguish

between _« real _» and _« fake _»

(-bonds.

If _a

gr/5-intersection

does not

belong

to a 3

gr/5-intersection

then the

corresponding ^I-bond

is called real

(long

solid lines in

Fig. 3a).

The interval

perpendicular

to a real

I-bond

_at its center is _a side of _some Voronoi

domain. If _a gr/5-intersection

belongs

to a 3

gr/5-intersection

then the

corresponding ^(-bond

is called fake

(dashed

lines in

Fig. 3a).

The

perpendicular

to a fake

I-bond

_{is not}

a side of _a

Voronoi

domain,

it is _« beaten _»

by nearby

s-bonds. For this _reason fake

(-bonds play

no

significant

role and will be

ignored

in what follows. It is clear that there _{are no} fake

s-bonds,

I-e- _an interval

perpendicular

to any s-bond _at its _center is the side of _a Voronoi domain.

,

a) b)

Fig.

3. (w/5, 3

gr/5)-covering.

a) I-s

graph

(s-bonds and real (-bonds _are shown

by

solid lines, fake I- bonds-dashed fines) and Voronoi

partition

(thin lines), b) f-s

graph

(fake bonds _are not shown) and binary

tiling

(dashed lines).

Below _{are some} useful

properties

of

I-s graphs

which could be

easily

observed in

figure

3. If

contemplating figures

is _not

convincing

strict mathematical

proofs

can be found in the

preprint

version

[22]

of the

present

^work.

I)

The Voronoi

decomposition

^associated with _any

I,

₃ ^"

_covering

_is constructed

of

5 5

only

6 Voronoi domains shown in

figure

4.

ii)

The

only

allowed

multiple

cluster intersection other than _a

single point

is _a

triple

intersection.

iii)

An

I-s graph corresponding

to any

( gr/5,

3

gr/5) covering forms

the

tiling ofa plane by

three tiles _{: a}

triangle,

_a

trapezoid

and _a

regular

pentagon

(Fig. 3).

Triangle-trapezoid-pentagon tiling

^is not convenient for _our purposes.

However,

this

tiling

is known to be

equivalent [I I]

to

different,

_more convenient

tiling, namely

a

binary tiling.

One may recall _a definition of the latter

[6].

Consider _a

tiling

of _a

plane by

two Penrose tiles

(a

thin rhombus with WI10 and 4 «II 0

angles

and _a fat rhombus with 2 WI10 and 3 WI10

angles).

DEFINITION 2. A vertex

ofa tiling

is called _even

(odd) if

all

angles

made

by

tiles

meeting

at this vertex are

of

the

form

N

gr/10,

where N is _even

(odd).

A

tiling ofa plane by

two Penrose

tiles such that any ^vertex is either _{even or} odd is called _a

bina~y tiling.

The central result of section I

(and

the

only

_{one we} will need in the

following)

is _:

Proposition

I. A _set

of

all ^'~

,

3

I coverings

is

equivalent

to the set

of

all

binary

5 5

tilings.

The

mapping

between the two models is illustrated

by figure

3b

(if Fig.

3b is not

convincing

strict mathematical

proofs

_can ^be found in the

preprint

version

[22]

of the present

work)

odd vertices of a

binary tiling

_are centers of

decagonal

clusters _;

even vertices of _a

binary tiling

_{are comers} of

decagonal clusters,

_common for _two

overlapping

clusters

(Fig. 2)

_;

Fig.

4. All allowed Voronoi domains and local environments.

binary tiling's

^side

length equals

the cluster radius _;

gr/5 overlappings

_are associated with thin Penrose tiles

(Fig. 2),

true

I-bonds being long diagonals

of thin tiles _;

3 gr/5

overlappings

are associated with thick Penrose tiles, s-bonds

being

short

diagonals

of thick tiles.

Below _{are some} useful

properties

of

binary tilings. Any binary tiling

_can be lifted to _a five- dimensional space in the standard _manner

[5-10, 14]. Any

vertex of the

tiling

can be

presented

in the form _:

s

R

=

£

_{X~ A~}

(2)

j =1

where

A~ denotes five _vectors directed

along

the tile sides (A~ ^{form a}

regular 5-star)

and where X~ ^{denotes five}

integers. So,

_any vertex of _a

tiling

is

mapped unambiguously

to a

point

of

Z~.

_The

tiling

itself _can be viewed _{as a}

projection

of _a two-dimensional lattice surface in _a five-

dimensional space ^onto the

tiling (physical) plane M).

^One may

project

vertices of this lattice

surface onto a space

Ml

which is

perpendicular

_to

_M).

Since the _sum of five vectors

A~ is zero, an

integer

vector B

=

(I,

I,

I,

I, I

belongs

to

Ml.

This _means that

projections

^of vertices onto

Ml

cannot fill

Ml everywhere densely, projected points

_are situated _on

layers perpendicular

to B _:

s

~~

~

/ _j~j ^~~

^~~~

Proposition

2.

(Widom, Deng, Henley [9])

The

projection of

_any odd vertex

of

_a

binary tiling

onto a vector B

=

(I, I, I, I,

I _e

Ml equals

zero

(under

_proper choice

of

the

origin).

The

projection of

_any even vertex onto B

equals

_±

Ill-

« One may show that in _a thin tile the _sum of the

X~'s

^{is the} same for both its vertices with _a gr/10

angle.

In

fact,

the coordinates -of the second _vertex may be derived from the coordinates of the first _one

by adding Aj A~,

_A~

A~

or,

generally,

_{A~ A~.} After

lifting

to a 5-D space this

corresponds

to

changing

X

by

_a vector

consisting

of three

0's,

_{one +} I and _one I.

So,

the _sum of the

X~'s

^is

unchanged.

An

analogous

statement holds for _a fat tile both vertices with _a 3 WI10

angle

have the _{same sum} of the

X~'s.

^If an odd _vertex of _a

binary tiling

is taken _as the

origin

then the _sum of the

X~'s

at that

point

is _zero. Since this is _a

binary tiling, only

the 3 gr/10

angle

^of a fat tile and the WI10

angle

of _a thin tile _can meet at the

origin.

Therefore,

the _{zero sum} of the

X~'s spreads

to

adjacent

gr/10 and 3 gr/10

vertices,

which _are also odd. Then

considering

new tiles

adjacent

to the latter vertices _{one sees} that the _{zero sum} of the

X~'s ^spreads

out over all odd vertices

(Fig. 3b).

Even vertices _are obtained

by adding

_or

subtracting

one A vector from _{a nearest} odd

neighbor

whose _sum of the

X~'s

^is zero, which in 5-D

corresponds

to the addition of _a vector with four 0's and _{one +} I or four 0's and _one

I. _»

The present ^{model allows} a

generalization

useful for

applications [20].

Consider clusters of

decagonal shape

but decorated

by

atoms so that the decorated cluster possesses

only

5-fold symmetry, as is the case for Alcuco

(Fig. I).

Such _a symmetry ^reduction may be viewed _{as a}

coloring

of cluster _corners, in which black and white _comers altemate _{as one} goes around the cluster. Since cluster _{comers are}

mapped

to even vortices of _a

binary tiling,

this

procedure

creates a colored

binary tiling.

^The

problem

of existence arises

immediately

_: is it

possible

to

color _a

given binary tiling

_so that when _one goes around any odd _vertex the comers' colors alternate ? The _answer is affirmative. All _even vertices for which the _sum of the five

integer

coordinates

equals

+ I _must be colored, _say,

white,

whereas, the others, whose coordinate

sum

equals

I, are colored black. Since _two

opposite

_even vertices of _a tile

always

have different _sums

(one

is + I, the other is

I,

_see the

proof

of

proposition 2) they

will be colored in different colors. Two _even vertices _are either _nearest

neighboring

_comers of _a cluster

(gr/5

intersection)

_or third

neighbors (3 gr/5 intersection),

which

implies

that the altemation rule is fulfilled

(Figs,

1,

3).

Note that all the

decagonal

clusters _are colored the _same way.

Below, the

mapping

between

(gr/5,

3

gr/5)-coverings

and

binary tilings (proposition I)

and the

lifting

of

binary tilings

onto a 5,D space

((Eqs. (2), (3))

and

proposition 2)

is

exploited

to establish _{a one-to-one}

correspondence

between cluster

coverings

and two-dimensional lattice surfaces in a five-dimensional space.

DEFINITION 3. A two-dimensional lattice

surface

in _a

five-dimensional

_space is called _a

surface

with _an average

slope ifthere

exists _a two-dimensional

subspace _M(,

called _a

grid plane,

such that

lim _max

([h(R)) yR

= 0

(4)

R

- oJ R

where

h(R)e Ml

is the transversal deviation

of

the lattice

surface from

the

grid plane R(

at the

point

R e

R(.

It is useful _to choose the

following

normalized

orthogonal

basis of

R~

~~

~ _~~'

~ ~' ~' ~' ~ ~~ ^~

~~

~~

~i _~~'

~' ~' ~' _~~ ~

~~

Pi =

) _(2,

T, T~ _~, T~ _~,

T)

_e

Ml iB (5)

lo

~il

~ ~ll ~l + ~12 _{~2 +} ~ll PI + ~12 P2

~i2 ~

~21~l

+ ~22_~2 + ~21PI + ~22 P2

(6)

where the term

fib

is omitted because _{we are} interested in

binary tilings only

which,

according

to

proposition

2 do _not deviate from B. The a-matrix has to be chosen such that det

a =

I,

the best choice is

a~~ ⁼ 3~~. The 2 _x 2 matrix _e

defining

the

grid plane

is

usually

called the

phason

^{strain matrix. Zero} _e

corresponds

to

R(

=

R),

the

tilling

itself in this _case is called

a

decagonal binary tiling

because its diffraction pattem ^is

decagonally symmetric [6-9].

Proposition

3.

Ifthe

basis

Aj, _A~ ofthe grid plane R( ofa binary tiling

^is

given by equation (6)

then the cluster

density

_p

of

the

corresponding (gr/5,

3

gr/5) covering

is

~ det e,~

~ ~~~

5 sin

(gr/5)

^~ _{5 T~ sin}

(ar/5)

The cluster

densi~y of

a

decagonal tiling (R(

₌

^R),

e~~ ⁼

0)

is po =

T/5 sin

(gr/5).

« Let _p be the number of clusters per unit _area,

F,

the

density

of fat Penrose tiles

(72°-angle)

and

T,

the

density

of thin Penrose tiles

(36°-angle). Obviously A~

F ₊

AT

T

= I

,

A

~ = sin

72°,

_A~

= sin 36°

Since

A~/AT

₌ ^sin 72°/sin 36°

= T

(TF

₊

T) AT

=

(8)

Fat rhombi _are

projections

of 12,

23, 34,

45 and 51 two-dimensional facets of _a five- dimensional cube. Thin rhombi _are

projections

of

13, 24,

35, 41 and 52 facets. To calculate the

density

of the

ij rhombi,

one considers _a

large

but finite

portion

of the

grid plane

in the

shape

of _a

parallelepiped

with sides

along

the basis

Aj, _A~.

The number of the

ij,rhombi

inside this

region

is

proportional

to the _area of the

projection

of this

region

onto the

ij-plane. So,

the

density

of the

ij-rhombi

is

f

_~y

~'i~lj

~~~

~~

~21~2j

where _a is some coefficient

independent

of the I,

j

^indices.

Thus,

the densities of the fat and thin rhombi _{are :}

F

= «

(f12

₊

f23

₊

f34

₊

f45

₊

fsi

~

~ "

(f13

₊

f24

₊

f35

₊

f41

₊

f52) (1°)

Equations (9)

and

(10)

define bilinear

antisymmetric

forms for F and T _: F

= a

(Aj, FA~),

T

= a

(Aj, TA~). (I1)

The easiest way to find _a is _to

employ equation (8)

: a = iT

(At, FA~)

+

(Aj, TA~)1-

Substituting Aj,

A~ ^from

equations (5), (6)

and the operators ^{F and T from}

equations (9), (10)

into

equation (11)

_one obtains

F

=

~

+ det

~~)

A~ (12)

T +2 _T +2

T=

~+

^~

~dete,~) ^AT.

Next _one expresses the

density

of clusters _p in _terms of the densities of fat and thin tiles.

According

to

proposition

I _p

equals

the

density

of odd vertices of _a

binary tiling.

Both _even and odd vertices _are shared

by

fat and thin tiles. One 3 gr/5

angle

of _a fat tile contains

(3 gr/5)/2

_gr

= 3/10 ths of _an odd vertex, the whole fat tile contains

3/5

ths of _an odd _vertex.

Similarly,

_a thin tile contains 1/5 th of _an odd _vertex.

Therefore,

the cluster

density

is _:

p =

~

F ₊ T.

(13)

Finally, making

use of

equation (12)

_{one can} obtain _an

expression (7)

for the cluster

density.

_»

2. Random cluster

assembly

and

phase locking

in

quasicrystalline

state.

Below the statistical

physics

of

(gr/5,

3

gr/5)-coverings

is considered.

Following

Elser and

Henley,

_we

postulate

that all

coverings

have the _same energy,

which,

without any

restrictions,

can be taken to be _zero. This

assumption,

at least _{at a} first

glance,

does not look

justified

from

a

physical point

of view.

However,

there is evidence

[12, 6-9, 4, 15, 16]

that it

might

be _a reasonable

approximation.

The

dispute

between adherents of _« random

tiling

_» and _« match-

ing

rules _»

approaches

is far from

completion

and I _am not

going

to argue below for the random version. A final

theory

should account for both entropy ^and energy. The aim of the

present

^{section is} to demonstrate that _even when the energy is

neglected

the entropy ^alone

provides phase locking

in

quasicrystalline

state.

Consider _a statistical ensemble of all the

coverings entering

the

partition

function with

equal probability.

^{The free} _energy becomes F

=

TS,

where the entropy ^{S is} temperature

independent (both

F and S _are per unit

area). Proposition

I establishes _{a one-to-one}

correspondence

between

(gr/5,

3 gr/5

)-coverings

and

binary tilings,

the latter

being easily

lifted up ^to the five-dimensional space

((Eq. (2)), proposition 2). So,

_{one can} consider _an ensemble of the

corresponding

two-dimensional lattice surfaces in

f~.

_{To be}

precise,

_one has to

introduce the

entropy

of the ensemble of surfaces with fixed average

slope (definition 3).

The

grid plane R(

can be

specified by

four

independent coordinates, namely

the elements of the matrix of the

phason

strain

e~~ defined

by equation (6).

We denote this entropy ^henceforth as

s(Eij).

A _{RANDOM TILING HYPOTHESIS}

(Elser [121).

The entropy ^S_(e~~

) for

small

e~~ is _a

negative definite quadratic function of

the

phason

strain

e~~.

The

only

^{function of this} type ^{consistent with the} symmetry ^of

R)

is

[9]

_:

S(e~~)

=

So Ci e(

+

C~

det

~~j

(14)

2

where

So

^{is the} entropy ^of a random

decagonal binary tiling, Ci

^and

C~

_are called

phason

elastic constants. The random

tiling hypothesis

has not been proven

exactly,

but has been confirmed

by

_many numerical calculations

[6-9].

The elastic constant C

i has been found _to be

0.6,

the second constant

C~

has not yet ^been determined. The

quadratic

form

(14)

is

negative

definite if and

only

if

Ci

+

C~

~ 0 and

Ci C~

_~ 0. These conditions have _{not even} been

checked

numerically.

But it will be shown below that for

physical applications

it is of _no

importance

whether _or not the form is

negative

definite and also that the value of

C~

is irrelevant. In

fact,

the free energy and the

entropy

are functions of the

density

of clusters p.

According

to

proposition

3

p = po + det eq, ^K =

5 _T^~ sin

(ar/5)

K

and, therefore,

if the

density

_p is fixed then the four variables

e~~ ^are not

independent.

Denoting Ap

= p po,

equation (14)

becomes _:

S~

(e~j )

=

So

C

i

e(

₊

C~

K

Ap

,

det

eq

=

K

Ap (15)

The distribution

(15)

is Gaussian.

Diagonalizing

the form

by

Ml " ~ll + ~22 ₁₁₃ " ~ll ~22

U2 ~ ~J2 E21 114 ~ E12 + E21

gives

S~(u~)=So-[~Cj(u)+uj+uj+u()+C~KApj, u)+uj-u)-u(=4KAp.(16)

4

Keeping Ap

fixed _{one averages}

equation (16)

_over the three

independent

_u~

weighted by

the

probability exp(- S(u~)

A

),

where A is the

sample

_area, to obtain.

Proposition

4. The

dependence ofthe jfee

_energy and the entropy on the cluster

densi~y

_p

has _a cusp at p = po, where po is the cluster

densi~y of

_a

decagonal quasicrystal

:

S(p)=so-(cijApj+c~Ap)K Ap=p-p~. (17)

It is worth

noting

that

only

C

j enters the distribution

(16)

in eamest, at fixed Ap the second constant

C~

is

completely

irrelevant. The

only

condition mandated

by

the

requirement

of

stability

is

Ci

~0. The

inequality Ci

~0 has _not been proven

analytically,

but has been checked

numerically

_:

Ci

= 0.6

[7-9].

As is

usually

the _case, the cusp in F

(p )

indicates _{a «}

phase locking

_» at p = po.

Considering

a

grand

canonical ensemble of clusters with _a chemical

potential

_{p :}

G

(p )

₌ ^min

(F (p )

_{p p}

(18)

and

substituting (17)

into

(18) gives

_:

Proposition

5. A

decagonal

random

tiling quasicrystal

is

thermodynamically

stable in _a

nonzero interval

of

the chemical

potential

p e

[(C~- Ci)KT, (C~+ Ci)KT]. (19)

Small non-harmonic corrections to

equation (14)

will

slightly change

the interval of

stability,

but _as

long

_as

they

_are small

they

will not wash _out the cusp.

Proposition

5 is the main result of this paper, it

gives

_a

plausible explanation

of _an old

mystery, namely, why

_a system ^is

phase

locked in _a

quasicrystalline

state _even

though

it is incommensurate.

Proposition

5 could be reformulated in _a

language

_more

appropriate

for

alloys

_; for the sake of

simplicity

we restrict ourselves _to the _case of _a

binary alloy.

^We assume that there is _a

unique,

fixed decoration of the cluster

by

atoms of _two

types

; A and B

(e,g.,

Al and TM of

Fig, I). Vacancies,

interstitial and substitutions _are

prohibited (which is, unfortunately,

far from the _case in

reality).

The atomic

composition

x ₌

N~/(N~

+

N~ (20)

is

obviously

_a rational function of the cluster

density.

To draw _a

phase diagram

for the

binary alloy

in

question

_one needs to know the free energy as a function of the

composition

_x.

Substituting

_p as a rational function of _x into the free energy

F(p)

=

TS(p),

where

S(p)

is

given by equation (17),

_{one sees} that

F(x)

also has _a cusp at x =xo, where xo is _a

composition corresponding

to a

decagonal quasicrystal.

Note that because of

proposition

3 the

decagonal quasicrystal _(ej~

= 0 _can

only

exist _{at p}

= po

and, consequently,

at x = xo.

Any change

in

composition

would result in _{a non-zero}

phason

strain

e~~ and _a

deviation from

decagonal symmetry.

We have found F

(x)

for _a

quasicrystalline phase.

As

usual,

information about _one

phase

is not

enough

to draw _a

phase diagram.

One needs to know

F(x)

for

competing crystalline phases.

^{This task} is

beyond

the reach of the present

theory.

For

example,

_some

simple crystalline

structure

might

have _an energy much lower than the energy of _a

quasicrystal,

calculated above. In this _case the

quasicrystal

would not exist at all. But if for _some

temperature region

the free

energies

_are like those shown in

figure 5a,

then

according

to the standard rules for the

thermodynamics

of

alloys [13]

the

decagonal quasicrystal

is _{a «} line

compound

_» in this

temperature region (Fig. 5b).

This _means that if _one mixes

together

A and B atoms with

composition

_x from _« d _{+ a »}

region

then the

sample

will separate ^into two

phases,

_one will be _a

decagonal quasicrystal

with the

composition

_xo and the other _a

crystal

_a

1 ~

a.

p a+j

_xp d+~ ~ X

T

j

j I+j i+oc

_~c

L,

X~ ^x

Fig.

5. a) Free energy as a function of the

alloy composition,

b) _« Line

compound

»

phase

diagram

(d-quasicrystalline phase,

_a and

p-competing crystalline phases).

with

composition

_x~

(Fig. 5b).

In _an

experiment nobody

_can mix

ingredients

with the

composition exactly equal

to xo

(which is, incidently,

irrational

!),

but the presence of the cusp in the free energy

(proposition 4),

which results in the

« line

compound

_» type ^{of the}

phase diagram,

offers _a

possible explanation

of

self-fine,tuning

to _an exact irrational

composition

xo, which is necessary for _a

decagonal quasicrystal

to exist in nature. It should be

emphasized that, although

the main result

regarding

_{a cusp} at x ₌ xo has been derived for the

particular

model of

overlapping decagonal clusters,

it is easy ^to believe that it should hold for

virtually

any random

tiling

model.

Indeed,

the

only things

_we need for the derivation _{are :}

I)

^the free energy is

quadratic

in

phason

strain tensor e~j

(random tiling hypothesis

_{or, may}

be,

_{even more}

general situation)

_;

it)

the atomic

composition

_x is rational in det

eq (obvious geometry)

iii)

the correct

thermodynamic

function for _an

alloy

^{is F}

(x),

not F

(e~j)

^{it is obtained}

^by averaging

over

components

^{of the} tensor

eq

^{while one variable x is}

kept

fixed. This process

would

always produce

_a term in

F(x) proportional

to

ix xo[.

I _{want to}

emphasize

that the conclusion of _{a «} line

compound

_»

phase diagram

has been

drawn _on the

assumption

that there _are neither substitutions _nor vacancies. The

theory

allowing

for such defects is _to be

developed. However,

it is not very difficult _to

imagine,

even

in that

theory,

that the cusp in the free energy ^would survive. If _so, the

phase diagram

would be similar to

figure 5b,

but the

decagonal quasicrystal

would be stable in _a

tiny

interval of

compositions

around xo. Recent

experimental

results

[23]

indicate that the

region

of

stability

of the best

quasicrystals

is

actually extremely small,

of the order of 0,I _at, f6 _{or even} less.

Recently

_new HREM

images

have become available

[15, 16]. They

show clusters of about 20

h

_{in diameter}

overlapping

in accordance with the present

theory. Moreover,

the

quality

of

Hiraga's images

allows _one to discem odd and _even vertices of the

binary tiling.

Reference

[16]

is devoted to rational

approximants

of the

decagonal tiling,

which _are also made of the

same clusters,

however, arranged periodically.

One _can also find

overlapping

clusters in the

atomic

images

obtained

by

the Patterson

analysis

^and

subsequent

refinement

[17, 18].

Employing

these

experimental

data and the above mathematical framework leads to the

developing

of _a rather successful atomic model of Alcuco/AlcoNi

[20].

Acknowledgements.

am thankful _to V. Elser and C.

Henley

for useful discussions and to

J.Poon,

Y.

He,

H. Chen and G. Shiflet for their HREM

images

which stimulated this work. I _am indebted to M. Oxborrow for his

help

with the

preparation

of the

manuscript.

This work _was

supported

in

part by

_{a grant} from the Sloan foundation.

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