Scaling properties of vibrational spectra and eigenstates for tiling

Scaling properties of vibrational spectra and eigenstates for tiling models of icosahedral quasicrystals J. Los, Janssen, F. G¨ahler

To cite this version: J. Los, Janssen, F. G¨ahler. Scaling properties of vibrational spectra and eigenstates for tiling models of icosahedral quasicrystals. Journal de Physique I, EDP Sciences, 1993, 3 (1), pp.107134. .

HAL Id: jpa-00246704 https://hal.archives-ouvertes.fr/jpa-00246704 Submitted on 1 Jan 1993

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Phys.

J.

I

France

(1993)

3

107-134

1993,

JANUARY

107

PAGE

Classification

Physics

Abstracts

63.20D

Scaling properties models J.

(I),

Los

T.

Institute

(~

(I

Jaiissen

and

Gililer

F.

D4partement

(~)

CH-1211

de

Geiibve,

tiling

for

Nijmegen,

of

Nijmegen,

ED

6525

Abstract.

Th40rique,

accepted

199?,

study of

A

quasicrystals

Physicjue

in

final

form

dynamics

lattice

the

presented, both in approximation commensurate

the

Universit4

Genbve,

de

is

this

is

states

at

are

periodic the

more

very

states

compared

with

majority

The

analysis

localization those

is for

the

of the

end

upper

localized.

multifractal

a

the

Also

structures.

modelling icosahedral approximations. different three types of tilings, namely: It turns that the density of states out as multifractal analysis of the spectrum A

approximations

done

for

eigenvectors

spectrum of

the

tilings

3-dimensional

of

commensurate

randomized symmetrized and approximants. function of frequency is smoothed by random12ation. a that mainly at high frequencies the scaling behaviour shows for

Ansermet,

Ernest

October1992)

I

perfect,

that

quai

24

Switzerland

(Received11 August

by

eigenstates

and

(~)

University

Physics,

Theoretical

for

spectra

quasicrystals

Netherlands

The

In

vibrational

of icosahedral

of

are

have

relatively

a

scale

states

of the

examined normal

as

cluster

and

spectrum

and

small

it

appears

the

not a

eigenvectors

extended

for

only the fraction, I.e.

states,

is

as

Also, for paper

from

different that

participation

systematic approximants. Throughout the enhanced by randomization. chain. I-dimensional quasicrystal, the Fibonacci

of

is

the

shown of

most

results

are

Introduction.

1.

The

calculations

first

tals

were

published

3-dimensional icosahedral The

I-PT

of the in

lattice

reference

iii.

dynamics Here

we

of

3-dimeiisional

present

tilings (I-PT'S), quaJsicrystals, discovered by Shechtman lattice periodicity has no 3-dimensional icosahedral

Penrose

further

a

which et

al.

but is

are

in

models of icosahedral quasicrysstudy of the lattice dynamics of closely related to the structute of 1984 [2].

quasiperiodic.

Therefore

calculations

much difficult than in physical properties, such as phonons and electronic states, are more calculations numerical of the phonon ordinary, periodic In ill ,ve have presented structures. and rational clusters approximants in a simple dynamical model. In spectrum of syiumetrized but considered. This will the also the eigenvectors are the present only spectrum not paper constructed from perfect quasiperiodic randomized tilings, which are also be done for so-called atomic positions. Differences tilings or rational approximants by repeatedly flipping certain

of

JOURNAL

108

between

the

vibrational

distinguish

random tilings, quasicrystals [3, 4].

or

density

The between

of

w

of

and

w

(DOS)

states

is

These

singularities

where

w,

(k)

are

orthogonal

to

with

more

and

times

the

number

surface

at

inside

atoms

more

of

of

phonon branch

branch

the

in

atoms

D(w)dw

that

only

with

only points in the phonon

function

due to

is the s~~

such

defined

is

ordinary crystals

For

smooth

a

randomized tilings might give us a mean to perfect and experimentally. It is known that the randomized tilings, usually called, are very well acceptable as models for real

are

D(w)

states

+ dw.

N°1

structures

they

as

I

of

spectra

such

between

PIIYSIQUE

DE

the

with

w

as

k.

a

function

I-PT

The

the

unit

unit

cell,

cell a

a

branches of the

large

for

limit

thus, number

which vector

wave

is the and

the

to

in the

atoms

singularities (van

few

a

equal

is

few

of

since

a

the

n

number

unit

states

density

Hove singularities). i7~w,(k) 0 holds,

k and sequence

number

=

u

is

of of

singularities might

of

of

the

cell

a

unit

vector

approximants branches occur

is 3 in

the

singularities will not be strong however, since the Brillouin zone is small for with large unit cells, and thus only in the fine irregular behaviour of the structures structure an DOS can be expected. Globally one expects that the DOS is smeared randomized In the out. tilings the local order is partially lost. It is likely that this loss of order is also visible in the spectrum.

These

DOS. several I-dimensional quasiperiodic including also the For I-dimensional structures, quasicrystals, it has been sho,vn that excitation character different from that of spectra have a periodic crystals. In the latter case the spectrum consists of a number and is abof bands solutely All states continuous. extended Bloch-waves. It is known that for completely are random I-dimensional the structutes spectrum is a pure point spectrum with correspondingly localized exponential I-ewith decay of the wave function outside a bounded states, states an 2-diiuensional randoiu 3-dimensional region. This also holds for but not for structures, ranwhere dom of the be extended. Fibonacci normally For the chain, structures part states may which is a I-dimensional quasicrystal, it was shown IS, 6] that the vibrational spectrum is neither absolut.ely continuous so-called singularly point but continuous, with spectrum nor a correspondingly critical Examples of such states showing intermittent states. states an are behaviour, I-e- they are essentially localized on several disconnected bounded regions. have reconsidered the lattice As a comparison dynamics of the Fibonacci chain. The we of the and. the character have studied been using the numerical spectrum vectors wave same 3-diiuensional approach as for the tilings. The Fibonacci chain be considered Ican as a dimensional quasiperiodic sequence of long and short intervals, which remains invariant under rule: replace each long interval by a long and a short interval, replace the following inflation interval by a long interval. each short In our study of the dynamics of this system we simple placed atoms with mass 171 the vertices and connected neighbouring atoms by springs on =

with To

spring

constant

determine

the

=

I for

character

a

of

long

interval

spectra of

and

Ii

=

I-diiuensional

3 for

a

short

systems

interval.

Kohmoto

iii

introduced

a

thermodynaniical formalism. In this paper show how this on a we method be generalized for arbitrary in diiuension, and results presented for sj

7max

7min

[cc

12

13.82

1.40

Ii

12

18

II

Sym.

II

1.13 4

this

From than

for

table the

is

number

we

the

sense

in

closer

to

distribution

PF

932-approximants

randomized

two

that

iiieans

certain

a

configurations

randomized

A7 for the

which

one,

in

narrower,

that

see

perfect

tilings the

these

is

distribution

indeed

smaller

coordination

of

might explain why for the to the right (Fig. 9). 2/1-approximant in figure 8 with that of

the [cc is

case,

narrower

which

and

shifted

However, comparing the result for the symmetrized perfect 932-approximant we observe a considerable shift while the difference in the standard deviation of the could be found in the coordination is only very small. FIere the explanation

the

symmetrization which induces structural change, especially for We

have

appeared

also to

studied

have

a

the

rather

frequency plots versus participation fraction verse frequency for clusters with PF

JOURNAL

DE

PHYS>QUE

I

T

3. N' I.

extra

an

short

neiglibour

distance

and

leads

to

consirable

an

approxiniant. Ilowever, since most of the states of frequency. high PF, and only a few states have a considerably lower PF, the PF

are

this

as

a

not

low

function

very

infor1~lative.

(IPF),

w,hich is

fi.ee

clamped

JANUARY

or

>993

equal

to

boiiidary

Therefore

I/PF.

we

chose

Studying the

conditions

and

to

IPF

for all

the

in-

function

of

present as

a

kinds

of

rational 5

JOURNAL

126

approximants

found

we

for, respectively, the

a

5/3-approximant

metrized modes

the

at

very

I

N°1

shown In figure 10a, 10b and 10c the results are (7895 atoms) with free boundary conditions, the sym~ randomized tetragonal 932-approximant. In each case only have a relative high IPF, I,e. are of the spectrum more

results.

siiuilar

very

symmetrized

PHYSIQUE

DE

cluster

and

a

end

upper

localized. Just

spectrum one system,

the

for

as

enlargement

atic The

method

introduced

was

systems in

for

two

tending used by

and

dimensions

three

or

study the scaling behaviour of the eigenstates under systemmultifractal analysis. to the quasiperiodic limit, by a

can

of the

Kohmoto well.

as

pi

where

L

now

of the

N is the

=

which

is

number

partition

The

system.

of

o;'s

1-di1~lensional

each

pi

a

but

systems,

scaling

index

be

can

cvi is

used

by:

defined

())°'

=

in the

aton~s

the

over

for

For

which

system,

is

a

represented by

be

can

(26) good

for

measure

the'size'

'entropy'function

the

S(cv),

by:

defined

Q(cv)

L~(°J

(27)

=

between and cv + do. The S(cv)number of sites with scaling index o using the same ther1~lodynamical formalism as the spectral S(£)-function, only with Ai replaced by pi and £ by o. Consider again the mono-atoniic chain mentioned above. Taking a standing wave with ~" ~, number k that rational it be normalized with 'size'of the system equal wave so can a where

Q(cv)dcv is the

function

calculated

is

=

N

to

(Ii integer),

I 0 the 'partition corresponds to cv-values smaller than I. For fl < 0 the small pi dominate, which corresponds constraint normalization for the pi's, the support of the S(o)to cv bigger than I. Due to the the value Contributions with be compensated by function contain I. must > I must a a have

states

much

a

the

=

contributions