HAL Id: jpa-00212397
https://hal.archives-ouvertes.fr/jpa-00212397
Submitted on 1 Jan 1990
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Comparison of HREM images and contrast simulations
for dodecagonal Ni-Cr quasicrystals
C. Beeli, F. Gähler, H.-U. Nissen, P. Stadelmann
To cite this version:
Comparison
of HREM
images
and
contrast
simulations
for
dodecagonal
Ni-Cr
quasicrystals
C. Beeli
(1),
F. Gähler(2),
H.-U. Nissen(1)
and P. Stadelmann(3)
(1)
Laboratorium fürFestkörperphysik, ETH-Hönggerberg,
CH-8093 Zürich, Switzerland(2)
Dépt
dePhysique Théorique,
Université de Genève, 24quai
Ernest Ansermet,CH-1211 Genève 4, Switzerland
(3)
Inst.Interdépartemental
deMicroscopie Electronique,
I2M,
EcolePolytechnique
FédéraleLausanne, CH-1015 Lausanne, Switzerland
(Requ
le 11 octobre 1989,accepte
sousforme définitive
le 7 décembre1989)
Résumé. 2014 Du fait de la similarité de contraste entre
micrographies électroniques
hauterésolution des
quasi-cristaux dodécagonaux
Ni-Cr et celles deplusieurs phases périodiques
voisines, il a été
suggéré
que lesquasi-cristaux dodécagonaux
peuvent être décrits comme unedécoration d’un pavage
quasi-périodique
dodécagonal
renfermant les mêmes unités structurales que cellesqui
interviennent dans cesphases
périodiques.
Dans le but de corroborer cettehypothèse,
des simulations de contraste enmicroscopie électronique
utilisant de tels modèles structuraux sontprésentées.
Les modèles considérés sont basés sur différents pavagesquasi-périodiques
desymétrie
12 ainsi que surplusieurs
pavagespériodiques
conduisant aux structuresdes
phases
périodiques
mentionnées ci-dessus. Le contraste simulé est en excellent accord avec lesimages expérimentales
demicroscopie électronique
aussi bien pour les structurespériodiques
quequasi-périodiques.
Il reflète essentiellement la structure du pavagesous-jacent
aussi pour les structuresquasi-périodiques.
On en conclut donc quel’interprétation
habituelle desimages
destructure haute résolution est valable non seulement pour les structures
périodiques,
mais aussipour les structures
dodécagonales
nonpériodiques
et que, parconséquent,
le schéma de décoration utilisé décrit correctement la structureatomique
duquasi-cristal dodécagonal.
Abstract. 2014 Sincehigh-resolution
electronmicrographs
ofdodecagonal
Ni-Crquasicrystals
aresimilar in contrast to those of several
closely
relatedperiodic phases,
it has beenargued
thatdodecagonal quasicrystals
can be described as a decoration of adodecagonal
quasiperiodic tiling
with the same structural units as occur in these
periodic phases.
In order to corroborate thishypothesis,
electronmicroscopic
contrast simulationsusing
such model structures arepresented.
The models considered are based on different
quasiperiodic, twelvefold-symmetric tilings
as wellas on several
periodic tilings leading
to the structures of theperiodic phases
mentioned above. The simulated contrast is in excellent agreement with theexperimental
electronmicroscopic
images,
both for theperiodic
and for thequasiperiodic
structures. Itessentially
reflects the structure of theunderlying
tiling, also for thequasiperiodic
structures. It is therefore concluded that the usualinterpretation
ofhigh-resolution
structureimages
is valid not only forperiodic
but also for thenonperiodic dodecagonal
structures, and that therefore the decoration scheme usedcorrectly
describes the atomic structure of thedodecagonal quasicrystal.
~lassification
Physics
Abstracts60.50E-61.55H-61.14
Introduction.
Since the
discovery
ofquasicrystals
[1]
in1984,
alarge
number ofcompositionally
andstructurally
differentquasicrystalline phases
have been found. Besidesquasicrystals
with icosahedral symmetry,dodecagonal
[2],
decagonal
[3]
andoctagonal
[4] quasicrystals
havealso been discovered.
The basic structure of
quasicrystals
cansatisfactorily
be describedby quasiperiodic tilings.
However,
it is still difficult to determine the atomic decoration of thegeometric
units of thesetilings.
Several methods haverecently
beenapplied
to solve thisproblem.
Forexample,
for icosahedralquasicrystals
the Patterson function has been calculated in 3dphysical
space[5]
aswell as in 6d space
[6, 7],
and from this information the atomic decoration can inprinciple
beobtained
[8].
Such a Pattersonanalysis
howeverrequires
agood quality X-ray
powder
diffraction
pattern,
for which asufficiently large single phase specimen
ofgood quality
isnecessary.
Since such
specimens
are notalways
available,
different methods have beenapplied
as well. One of these is theanalysis
ofhigh-resolution
electronmicroscope
(HREM)
images,
whichcan be obtained from a
single
quasicrystalline grain.
Given aconjectural
model structure,these
images
can be simulated andcompared
to theexperimental images ;
in this way themodel can either be
rejected
or used for furtherapplication, depending
on thequality
of thefit between the calculated and the
experimental images.
Several of these simulations have been
published
for icosahedralquasicrystals (see
e.g.[9,
10]);
however,
the simulatedimages
were very similar for different assumeddecorations,
sothat no discrimination between different decorations could be made in this way. In the case of
quasicrystals
which areperiodic
in one direction andquasiperiodic
in theplane
perpendicular
to this
direction,
i.e. for thedodecagonal,
decagonal
andoctagonal
quasicrystals,
this method appears much morepromising
however,
at least forimages
taken with the electron beamnormal to the
quasiperiodic
plane.
HREMimages
areessentially
determinedby
theprojected
potential
of the structure, and since theprojection
direction is in this case aperiodic
one, it iseasier to draw conclusions about such a structure than for structures
aperiodic
in all threedimensions,
especially
if theperiod length
isrelatively
short. The best candidate for theapplication
of thistechnique
is thedodecagonal quasicrystal
which occurs in thesystems
Ni-Cr[2,
11,
12]
and Ni-V-Si[13]
and has aperiod length
ofonly
0.46 nm.Apart
from the shortperiod length,
thedodecagonal quasicrystal
is apromising
candidate foryet
another reason. Ittypically
occurs associated with other well knownperiodic alloy phases,
such as the0’-phrase,
the
H-phase,
and theA15-type
structure. These threeclosely
relatedperiodic phases,
whosestructure can be described
by periodic tilings
decorated with square andtriangular prisms,
show HREMimages
very similar to thedodecagonal quasicrystal.
Therefore it may beconjectured
that the latter iscomposed
of aquasiperiodic
arrangement
of the same basicstructural units. This is in fact the
proposal
madeoriginally by
Ishimasa et al.[2] (compare
also[14]).
The purpose of this paper is to test the correctness of this
proposal by
detailed contrastcalculations. Electron
microscopic
contrast simulations for the threeperiodic
structuresmentioned above as well as for different
quasiperiodic
model structures arepresented
and arecompared
with each other as well as withexperimental images.
Thequasiperiodic
modelstructures are obtained as decorations of different
quasiperiodic tilings
with the basicstructural units taken from the
periodic phases.
It will be shown that the simulated contrast, which is in excellentagreement
with theexperimental images,
isbasically
determinedby
thelocal
arrangement
of thetiles,
and that this holds forperiodic
as well as fornonperiodic
interpretation
of the structureimages
is also valid for thedodecagonal
quasicrystal
structure.This
strongly
supports theproposal
madeby
Ishimasa et al. for the structure ofdodecagonal
quasicrystals.
Model structures.
The
dodecagonal
quasicrystal
structure(also
termedcrystalloid
[2])
has first been observed inNi-Cr small
particles
madeby
the gasevaporation technique
[2].
Theingot
used for theevaporation
had a bulkcomposition
ofCr~oNi3a.
In the simulations thiscomposition
has been assumed to be theapproximate
composition
of both thequasiperiodic
structure and the associatedperiodic
structures, and it was assumed that no chemicalordering
is present. Theperiodic phases occurring
together
withdodecagonal
quasicrystals
can be describedby
decorations of the
tilings
shown infigure
1. The decoration of thesetilings
with atoms isFig. 1. - Periodic tilings and the
corresponding crystal
structures related tododecagonal quasicrystals
shown in ac-projection. (a) ~ phase, (b) H-phase
and(c)
A15-structure.illustrated in a
c-projection
infigure
1(top).
Theo--phase
and theH-phase
structures can bedescribed
by
periodic
arrangements
of square andtriangular prisms,
whereas theA15-structure consists of a
periodic
arrangement
of the squareprisms only.
Note that there are twovariants for the decoration of the
triangular prisms, depending
on the orientation of thetriangle.
In theH-phase
only
one variant occurs, whereas inthe ~ phase
both are present.For the
image
simulation of thedodecagonal quasicrystal,
structures based on two differentdodecagonal tilings
shown infigure
2 were used. Thetiling
offigure
2a has been obtained[14,
Fig.
2. - Presentation of twoquasiperiodic, dodecagonal tilings
with twelvefold Fourier spectrum :(a)
truly
quasiperiodic tiling, (b) quasiperiodic tiling involving
some randomness in the construction(see
text), (c)
decoration of the basic structural units in thequasicrystal
(for
anexplanation
ofsymbols
seeFig. 1 ).
characteristic feature of this
tiling
is the occurrence of many rhombic tiles with a 30°angle.
These tiles are
interpreted
as defects of thequasicrystal
structure and do not occur asfrequently
in the observedimages
asthey
do in the modeltiling.
Note that the same rhombicunits also occur as defects in
the a-phase
of Fe-Cr[16].
The squares andtriangles
of thetiling
areagain replaced by
square andtriangular prisms
which are decorated in the same way as forthe
periodic
structures discussed above. The decoration of theremaining
rhombicprisms
isthen
completely
enforcedby
that of thesurrounding
ones. The decoration of these basic unitsis shown in
figure
2c. The structure obtained in this way has beenanalysed
in detail in aprevious
paper[14].
It has thenon-symmorphic
space group12~/mmc
which contains a screwaxis and a set of
glide
mirrorplanes.
Figure
2b shows adodecagonal tiling consisting
of squares andtriangles only,
whichcorresponds
moreclosely
to the real structure than thetiling containing
many rhombic tiles. It has beengenerated
by
an iterated inflation process due toStampfli
[17].
It can be shown[18]
that it is
essentially
quasiperiodic
andtwelvefold-symmetric, although
thearrangement
involves a certain amount of randomness. Moreprecisely,
in thistiling
the orientations of thehexagons
inside thedodecagons
are selected at random so as to result in a structure with atwelvefold
symmetric
diffractionpattern.
This random choice of the orientation of thehexagons
has to be made at allstages
of the inflation process. The tiles areagain replaced by
prisms
which are decorated inexactly
the same way as for the structures discussed above.Contrast simulation
For the contrast simulations of the
high-resolution
structureimages
thefollowing
electronoptical
parameters for a JEOL JEM200CX electronmicroscope
have been used : accelerationvoltage
200kV,
spread
of focus 15 nm,spherical
aberration constantCs
= 1.2 mm, halfangle
of convergence
1.2 mrad,
radius ofobjective
aperture 6.7 nm-1.All
image
simulations have beenperformed
with the EMS softwarepackage
[19].
Themultislice
technique
has been used to simulate the contrast of structureimages
of thecr-phase,
theH-phase
and theA15-type
structure as well as thedodecagonal quasicrystal
structure.a-phase
(a
= 0.88 nm, c = 0.46nm),
with 128 x 256 beams for theH-phase (a =
1.717 nm,b = 0.46
nm, c = 0.46
nm)
and with 64 x 64 beams for the A15-structure(a
= 0.46nm).
Theslice thickness
corresponded
to thelength
of one c-axisparameter
of 0.46 nm. For thecontrast simulations of the
dodecagonal quasicrystal
structure, the same slice thickness wasused. Since no unit cell can be defined for the
nonperiodic
quasicrystal
structure, the use of anartificial
supercell
was necessary for multislice calculations. In order to reduce the effect ofperiodic boundary
conditions on the simulated contrast, a ratherlarge
supercell
dimension wasrequired.
Thesupercells
were selected in such a way thatperiodic
boundary
conditionswere
possible
without any rearrangement of atoms. This resulted in thefollowing
four different sizes of thesupercells :
1.72 nm x 1.72 nm, 3.43 nm x 3.43 nm, 4.69 nm x 4.69 nmand 6.41 nm x 6.41 nm. For the multislice calculations 256 x 256 beams for the smallest
supercell
and 512 x 512 beams for thelarger
ones were used. Detailedimage
simulations arepresented
for one of thelarger
supercells
(a =
4.69nm )
shown infigure
2. For all calculationspresented,
theabsorption
was assumed to be zero.However,
calculations with realisticabsorption
were also made and did not show anysignificant
differences in theimage
contrast.The effect of a smaller slice thickness
(0.23
nm)
on theimage
contrast wasinvestigated
for theA15-structure,
theo-phase
as well as for the smallestsupercell
of thedodecagonal
structure.A
comparison
with the calculationsusing
0.46 nm slice thickness did not reveal anydifferences in the
image
contrast. It is therefore concluded that the slice thickness chosen issufficiently
small.Besides the multislice
technique
the Bloch wave method can also beapplied
to theimage
simulation ofquasicrystals
[10, 20].
However,
the Bloch wave method is limited totruly
quasiperiodic
structures.Moreover,
if one cannot make use of ahigh
point
group symmetry, avery
large
number ofindependent
beams has to be taken into account in order to obtain reliable results. If such ahigh point
groupsymmetry
is notpresent,
theapplication
of the Bloch wave method results in aprohibitively high
demand of computercapacity.
Since oursecond model structure is neither
truly
quasiperiodic
norexactly
twelvefoldsymmetric,
theBloch wave method cannot be
applied.
For this reason, all ourimage
simulations have beenmade
using
the multislicetechnique.
All these contrast simulations are then
compared
tohigh-resolution
structureimages
obtained from Ni-Cr small
particles.
Theseimages
have been takenby
a JEOL JEM200CXhigh-resolution
electronmicroscope.
Since the Ni-Cr smallparticles change
their orientation after two to three exposures, nothrough
focus series could be obtained.Results.
For the Ni-Cr
cr-phase
in the second thicknessfringe
visible in 200 kV electronmicrographs,
itwas found that the calculated contrast agrees well with the observed one. The second thickness
fringe
corresponds
to 24 nm to 26 nm, and thicknesses in this range were selectedfor contrast simulations.
Equivalent
resultsusing
Bloch wave calculations have been obtainedby
Ishimasa et al.[16a, 21]
for the Fe-Crcr-phase.
The contrast ofthe 7-phase
structure in thec-projection
consists of fourbright
spots per unit cell situated at thepositions
of the atomswith coordinates z = ±
1/4 ;
seefigure
3a. Infigures
3b-c the contrast calculations for theH-phase
and the A15-structure arepresented.
All calculations infigure
3correspond
to athickness of 25.3 nm. The
images
on the leftcorrespond
to a defocus value of 59 nm, and theimages
on theright
to the Scherzer defocus value of 67 nm.Figure
4 presents twoexperimental high-resolution images
to Ni-Cr smallparticles
containing the a-phase,
theH-phase
and the A15-structure. The contrast simulations show that for thicknesses betweenFig.
3. - Contrast simulations for thea-phase
(a),
theH-phase (b)
and the A15-structure for twodefocus values : 59 nm
(left)
and 67 nm(right).
The thicknesscorresponds
to 25.3 nm in all calculations.defocus
conditions)
the simulated contrast is in excellentagreement
with the observedcontrast.
The
present
contrast simulations for thea-phase,
theH-phase
and the A15-structureindicate that in electron
microscopic
structureimages
nearoptimum
defocus conditions thebright
spots
correspond
to the vertices of thecorresponding
tiling composed
of squares andtriangles.
Contrast simulations for the
nonperiodic
structure based on thetiling
offigure
2b areshown in
figure
5,
in which calculations for two different defocus values arepresented.
On theright
hand side offigure
5 the atompositions
aresuperimposed
onto these contrastcalculations. These calculations
correspond
to aspecimen
thickness of 25.3 nm. Infigure
6calculations for the same two defocus values but for a thickness of 27.6 nm are
presented.
Thefigures
show that the white dots visible in the contrast calculationscorrespond
to the vertices of the basictiling.
Contrast simulations for a structure based on the
tiling
offigure
2a arepresented
infigure
7.They
correspond
to thicknesses of 25.3 nm and 27.6 nm and defocus values of 45 nmand 65 nm. The atom
positions
areagain superimposed
onto the calculated contrast. TheFig. 4. -
Experimental high-resolution
images ofcrystalline
Ni-Crparticles.
Regions with the a-phraseFig. 5. -
Fig.
6. -Image simulations of the Ni-Cr
quasicrystal
for a thickness of 27.6 nm and for two differentdefocus values : 45 nm
(top)
and 55 nm(bottom).
On theright
the atompositions
aresuperimposed
Fig.
7. -Image simulations of the Ni-Cr
quasicrystal
for two thicknesses : 25.3 nm(left)
and 27.6 nm(right)
and for two different defocus values : 45 nm(top)
and 65 nm(bottom).
The model structure isFig.
8. -(a) Experimental high-resolution image
of thedodecagonal
Ni-Crquasicrystal.
(b) Tiling
corresponding
to theimage
shown in(a).
Letters A and B indicate how tosuperimpose
thetiling
onto theimage.
The arrow in(a)
marks a rhombic unit.Figure
8apresents
ahigh-resolution
image
of a Ni-Crquasicrystal showing
thetypical
contrast in the second thickness
fringe.
Infigure
8b thetiling corresponding
tofigure
8a is shown. Thepoints
A and B marked in bothfigures
indicate how tosuperimpose
thetiling
onto
figure
8a. The simulated contrast of the structural units agrees well with the contrast in the observedimage, figure
8a. This is true for bothdodecagonal
model structures. The bestcorrespondence
is obtained for a defocus value of 60 nm and a thickness ofapproximately
26 nm ± 2 nm. Note that the calculated contrast of the rhombic
prisms
in excellent agreementwith the observations too. One such rhombic unit is marked
by
an arrow infigure
8.The
image
contrastessentially
consists ofbright
spots situated at the vertexpositions
of theunderlying tiling.
This is true for all theperiodic
as well as thequasiperiodic
structuresconsidered here. Therefore the
image
can be understood asbeing
composed
ofpieces
corresponding
to the tiles of theunderlying tiling.
Pieces whichcorrespond
to tiles of the sameshape
have identical atom decorations. These units in theimage
contrast thus represent the basic units of the model structures.It is not
surprising
that thebright
dots are located at the vertices of theunderlying tiling.
VanDyck et
al. have shown for variousalloy
systems with a column structure that the vertices of theunderlying tiling
appear asbright
dots[22].
For the
periodic
structures as well as for thesupercells representing
parts of thenonperiodic
structures, the
dependence
of theamplitude
and the relativephase
of the transmitted(000)-beam on the
specimen
thickness has been determined. This relation is identical for all thecases considered here. In
particular
no difference was found between theperiodic
and theFig.
9. - Simulated diffractionpatterns for supercells of size a = 1.72 nm
(a),
a = 4.69 nm(b)
anda = 6.41 nm
(c)
compared
to theexperimental
selected area electron diffraction pattern(d).
In addition to the contrast
simulations,
thecorresponding dynamically
calculated diffraction patterns have also been obtained. The diffractionpatterns
ofsupercells
of three different sizes(a =
1.72 nm,
4.69 nm and6.41 nm),
all taken from theexactly
quasiperiodic
modelstructure, are shown in
figures
9a-c. Acomparison
with theexperimental
image, figure
9d,
shows animprovement
in thequality
of the fit withincreasing
size of thesupercell.
Thecalculated diffraction pattern for the smallest
supercell
still shows a square arrangement of thereflections due the
underlying tetragonal supercell,
whereas for thelarger
supercells
thedeviations from a twelvefold
symmetric
arrangement of the maxima are smaller.Especially
thepositions
of the weaker reflections show smaller deviations for thelarger supercells.
Also theintensity
distribution shows animprovement
withincreasing
supercell
size. Our resultsDiscussion.
In the present
study
the electronmicroscopic image
contrast for two different modelstructures of the
dodecagonal
Ni-Crquasicrystal
has been calculated. One of them was basedon a
truly
quasiperiodic
dodecagonal
tiling,
while the construction of the other one involved somerandomness,
as described above. The decoration of the different tiles was the same asfor
the ~ phase,
theH-phase
and the A15-structure. Sincenothing
is known about thechemical
ordering
in thequasicrystal,
it was assumed in the calculationspresented
here that on each atomicposition
there are 0.7 Cr atoms and 0.3 Ni atoms. Calculations with apartially
ordered distribution of the two atomic
species
were also made. In theordering
scheme usedfor this calculation it was assumed that the
positions
with z-coordinates z = ± 1/4 areoccupied
by
Cronly.
This distribution isanalogous
to that of theNi-Fe-Cr ~ phase.
The results showedno detectable differences in the contrast from the calculations
assuming
total disorder.It is of interest to compare the thickness and defocus ranges at which the best fit is obtained for the electron
microscopic
contrast of thecrystalline
structures and of thequasicrystal
structure. The best fit of the calculated contrast to that observed in the 200 kV electron
microscope
was found for a thickness of 25 nm ± 2 nm and a defocus value of 60 nm ± 10 nmfor the
crystalline
structures, whereas for thequasicrystal
structure thecorresponding
valuesare 26 nm ± 2 nm
(thickness)
and 55 nm ± 10 nm(defocus).
It can therefore be concludedthat the best fit for
crystalline
andquasiperiodic
structures is obtained atapproximately
thesame thickness and defocus ranges.
A
comparison
of the contrast simulations for thequasicrystal
structures with those for thecrystalline
structures showed that the contrast of the structural units in thequasicrystals
is thesame as in the
periodic
structures. This means that the contrast of the structural units does notdepend
on theirspecific
localarrangement.
Our results therefore indicate that the structural units show the same contrast features even ifthey
arearranged according
to a randomtiling
[23].
This result is of greatimportance
since there are indications[12]
that the structure of thedodecagonal
quasicrystal
is best describedby
a randomtiling.
From the present contrast simulations we conclude that the usual
interpretation
ofhigh-resolution structure
images
nearoptimum
defocus conditions is valid also for two-dimensionalquasiperiodic
structures. Therefore the decoration schemepresented
herecorrectly
describes the structure of thedodecagonal quasicrystal.
Inspite
of thegeneral
lack ofexperimental
through
focus series we have thus been able to present evidence that the decoration of the structural unitsproposed
in this paper is correct.It has been
argued
[24]
that the electron diffractionpattern
ofdodecagonal quasicrystals
could beapproximated
by
that of a rather smallperiodic approximant
with ahexagonal
cell ofedge length approximately
1 nm. Thisargument
is basedexclusively
on the kinematicapproximation
of diffraction. It is wellknown, however,
that in the simulation of electrondiffraction
patterns
of metallic structures, inparticular
forspecimens
with therelatively large
thicknesses considered
here,
dynamic scattering
effects have to be included in order toproduce
realistic results. Suchdynamic
calculations of the electron diffraction patterns fordodecagonal
quasicrystals (using
the realistic decoration schemepresented
here)
can be foundin reference
[14].
It has been found thatgood
agreement with theexperimental images
could be obtainedonly
ifdynamic
effects areactually
included,
while apurely
kinematicalcalculation
gives only
very poor agreement withexperiments.
Similar reservations have also beenexpressed by
Kuo[25].
Moreover, as shown in this paper, HREM
images
ofdodecagonal quasicrystals
are a veryimportant
source of information which should not beneglected.
Infact,
these HREMimages
are
incompatible
with anyperiodic approximant
having
a unit cell notlarger
than a fewnanometers
only.
Therefore,
a structure such as the model structureproposed
by
Ho and Lican be excluded for the observed Ni-Cr and Ni-V-Si
dodecagonal
quasicrystals.
Acknowledgments.
We are very
obliged
to Dr. T. Ishimasa forstimulating
discussions,
as well as formaking
available the diffraction pattern and the
high-resolution
electronmicrographs
of Ni-Cr.References
[1]
SHECHTMAN D., BLECH I., GRATIAS D. and CAHN J. W.,Phys.
Rev. Lett. 53(1984)
1951-1953.[2]
ISHIMASA T., NISSEN H.-U. and FUKANO Y.,Phys.
Rev. Lett. 55(1985)
511-513.[3]
BENDERSKY L.,Phys.
Rev. Lett. 55(1985)
1461-1463.[4]
WANG N., CHEN H. and Kuo K. H.,Phys.
Rev. Lett. 59(1987)
1010-1013.[5]
CAHN J. W., GRATIAS D. and MOZER B.,Phys.
Rev. B 38(1988)
1638-1642.[6]
GRATIAS D., CAHN J. W. and MOZER B.,Phys.
Rev. B 38(1988)
1643-1646.[7]
GÄHLER F., PattersonAnalysis
of Face-Centered Icosahedral Al-Cu-FeQuasicrystals,
inpreparation.
[8]
CAHN J. W., GRATIAS D. and MOZER B., J.Phys.
France 49(1988)
1225-1233.[9]
ISHIMASA T., NISSEN H.-U.,Phys.
Scri. T 13(1986)
291-296.[10]
CORNIER M., ZHANG K., PORTIER R. and GRATIAS D., J.Phys. Colloq.
France 47(1986)
C3-447-456.[11]
ISHIMASA Y., FUKANO Y. and NISSEN H.-U., in :Quasicrystalline
Materials, Eds. Ch. Janot, J. M.Dubois
(World
Scientific)
1988,pp. 168-177.
[12]
ISHIMASA T., NISSEN H.-U. and FUKANO Y., Philos.Mag.
A 58(1988)
835-863.[13]
CHEN H., LI D. X. and Kuo K. H.,Phys.
Rev. Lett. 60(1988)
1645-1648.[14]
GÄHLER F., in :Quasicrystalline
Materials, Eds. Ch. Janot, J. M. Dubois(World Scientific),
272-284.
[15]
NIIZEKI K. and MITANI H., J.Phys. A :
Math. Gen. 20(1987)
L405-L410.[16]
ISHIMASA T., J. Sci. Hiroshima Univ. Ser. A 45(1981)
29-57 ;ISHIMASA T., KITANO Y. and KOMURA Y.,
Phys.
Stat. Sol. 66(1981)
703-715.[17]
STAMPFLI P., Helv.Phys.
Acta 59(1986)
1260-1263.[18]
GÄHLER F.,Quasicrystal
Structures from theCrystallographic Viewpoint,
Diss. ETH No. 8414(1988).
[19]
STADELMANN P.,Ultramicroscopy
21(1987)
131-145.[20]
GRATIAS D., CORNIER M. and PORTIER R., ActaCryst.
A 44(1988)
789-798.[21
ISHIMASA T., KITANO Y. and KOMURA Y., J. Solid State Chem. 36(1981)
74-80.[22]
VAN DYCK D., VAN TENDELOO G. and AMELINCKX S.,Ultramicroscopy
10(1982)
263-280 ;VAN TENDELOO G., VAN DYCK D. and AMELINCKX S.,