Jan 1, 1993 - atoms in the unit cell, a large number of singularities might occur in the ..... the vertices of a true. I-PT, made of rhombohedral tiles, we must take care that .... cubic approximants the difference in the corresponding values of n.
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