A 6-D structural model for the icosahedral (Al, Si)-Mn
quasicrystal
J.W. Cahn, D. Gratias, B. Mozer
To cite this version:
J.W. Cahn, D. Gratias, B. Mozer.
A 6-D structural model for the icosahedral (Al, Si)-Mn quasicrystal.
Journal de Physique, 1988, 49 (7), pp.1225-1233.
<10.1051/jphys:019880049070122500>. <jpa-00210805>
HAL Id: jpa-00210805
https://hal.archives-ouvertes.fr/jpa-00210805
Submitted on 1 Jan 1988
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Classification
Physics Abstracts
61.10
61.50E
-
(1988)
1225-1233
JUILLET
1988,
1225
.
-
61.55H
-
64.70E
A 6-D structural model for the icosahedral
J. W. Cahn
(1),
D. Gratias
(2)
and B. Mozer
(1) Institute for Materials Science and
U.S.A.
Engineering
(Al, Si)-Mn quasicrystal
(1)
National Bureau of
(2) C.E.C.M./C.N.R.S., 15 rue G. Urbain, 94407 Vitry,
(Reçu le 24 fgvrier 1988, accepté le 25 mars 1988)
Standards, Gaithersburg MD-20899
France
Un modèle périodique 6-dimensionnel est proposé pour décrire la phase quasipériodique
icosaédrique Al-Mn-Si. Ce modèle est construit à partir de la représentation à 6 dimensions de la phase
cristalline cubique approximante 03B1. Dans le formalisme de Janner-Janssen-Bak, il consiste en trois couronnes
sphériques concentriques, l’une de manganèse et les deux autres d’aluminium centrées autour des n0153uds du
réseau hypercubique à 6 dimensions, et deux couronnes additionnelles d’aluminium centrées au milieu des
diagonales principales de l’hypercube. Ce modèle vérifie les données des diagrammes de diffraction X de
poudre avec un facteur d’accord résiduel de 0,128.
Résumé.
2014
A 6-dimensional (6-D) periodic model is proposed for the Al-Mn-Si icosahedral quasiperiodic
Abstract.
crystal. The model results from an embedding of the periodic cubic 03B1 structure in 6-D. In the Janner-JanssenBak description, it consists of three concentric spherical shells of respectively Mn, Al and Al aligned in
perpendicular space around the lattice nodes and two additional shells of Al around the body centers. This
model is shown to match the X-ray powder diffraction data with a satisfactory residual R-factor of 0.128.
2014
1. Introduction.
One of the principal problems in the study of the
recently discovered quasiperiodic crystals (quasicrystals) [1, 2, 3] is the determination of their structure,
a prescription for the localization of the atoms. For
periodic crystals, the description of the structure of a
single unit cell suffices ; for aperiodic crystals, the
algorithm must include additional information. Because quasiperiodic structures can be described by a
known irrational cut of a periodic higher-dimensional structure, this additional information is contained
in a single higher-dimensional unit cell.
We propose here a 6-D structure of an (Al, Si)Mn icosahedral quasicrystal, suggested by the results
of a Patterson analysis performed in both 3 and
6 dimensions with neutrons and X-ray powder diffraction data [4, 5]. The 6-D auto-correlation or
Patterson function (PF) was found to be surprisingly
simple (Figs. la, b) : only two structured peaks are
found in the unit cell, one around the nodes and one
around the body centers, implying that all distance
vectors in 3-D belong to two sets : one set that is
close to being quasilattice translations, i.e. projections of 6-D translation vectors (nodes to nodes) and
the other being motif translations, that are all close
to being projection of body-centering translation
vectors (nodes to body-centers). Comparison of Xray with neutron PF’s shows that these peaks have
well defined chemical structure : the body centering
translations are mostly heteroatomic distances as are
the outer reaches of the nodal peaks (Fig.1b). Such
a simple result suggests that all atom positions in the
3-D structure are either on or near quasilattice nodes
(coming from projections of nearby 6-D lattice
nodes), or on points that are near projections of
nearby body
centers.
As confirmed by our PF’s [4], the periodic cubic
a phase [6] with 138 atoms per unit cell (in 11 orbits)
of the same elements is known to have an atomic
arrangement quite similar to that of the icosahedral
phase [7-11]. Embedding the atoms of the a structure on the rational 3-D planes in 6-D shows that all
atoms can be associated with either a nearby cell
node or body center. Therefore, we will assume in
this paper that, in 6-D, crystal and quasicrystal have
essentially the same structure, that the difference
between them arises solely from the cut orientation,
rational for the periodic crystal and irrational for the
quasicrystal, and thus we propose a 6-D model that
is consistent with both the crystal structure and the
simple two-peak 6-D PF of the quasicrystal. With a
simple one parameter refinement procedure, the Xray intensities are fit with a residual R-factor of
0.128.
In the process of starting with diffraction data, and
Fourier transforming it to obtain PF’s, we were
naturally led to the higher dimensional structural
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070122500
1226
microscopy diffractions that would
of glides mirrors and/or screw
reveal the presaxes in 6-D, we
assumed the 6-D space group of the structure to be
the direct product of m35 with the simple hypercubic
6-D lattice. Details of the X-ray data analysis are
reported elsewhere [22].
ence
The average width of the peaks as measured both
by X-rays and neutrons leads to an average correlation length better than 40 nm. The peak widths
depend on both the perpendicular and the parallel
components of the 6-D K-vectors in good agreement
with the frozen-in phason model of disorder originally proposed by Bak [23] and elaborated by others
[24-26].
3.
Fig. 1.
X-ray (a) and neutron (b) Patterson functions
in
displayed the 5-fold plane spanned by [1, 0, 0, 0, 0, 0]
and [0, 1, 1, 1, 1, T] basis vectors. The dashed lines in the
neutron map represent negative contours and depict
atomic distances that are primarily heteroatomic.
-
of the Janner-Janssen-Bak [12-15] type,
rather than those descriptions that are based on
decoration of tilings [16-18].
descriptions
2.
Experimental.
X-ray and neutron powder diffraction data show that
the finely powdered rapidly solidified A173Mn21Si6
used is almost entirely icosahedral phase [19] containing only 2 % of the hexagonal crystalline {3 phase
and 1 % of f.c.c. Al [20]. On heating at 30 °C/hr, the
alloy eventually transforms entirely into the {3
hexagonal phase at 700 °C. A fine powder specimen
of the « cubic phase Al73Mn16Si11 was also prepared
and studied by both X-ray and neutron diffraction.
The positions of the X-ray reflections of the
icosahedral phase have been shown to fit a 6 integer
indexing [21] in 3-D, related to a 6-D icosahedral
primitive lattice of parameter A 0.6497 nm at
=
temperature, with a relative accuracy better
than 5 x 10- 3. X-ray integrated intensities have
been carefully formulated in absolute units (square
of a number of electrons per A3) using a well defined
standard of Ni3Fe alloy. The patterns were indexed
according to the scheme proposed by Cahn,
Shechtman and Gratias [21]. Since no systematic
extinctions were found in either X-ray and electron
room
Crystallography in 6-D.
It is necessary to choose between two structural
descriptions of atoms in the 6-D unit cell. In one, the
atoms are assumed to be localized at points in 6-D,
as they are in 3-D. The 3-D structure is then
obtained by collecting all the atoms in a carefully
defined neighbourhood of a 3-D plane and projecting
them onto this plane [27-29]. This is the construction
for obtaining the quasilattice from 6-D, and also for
generating aperiodic tilings from higher dimensional
lattice points.
In the other description, the atoms in 6-D are
characterized by 3-D surfaces [30] ; the 3-D structure
is obtained by cutting the 6-D structure with a 3-D
plane, the points of intersection of the physical 3-D
plane with the atom surfaces in 6-D identify the
actual locations of the atoms in the physical 3-D
space. This simple cut-without-projection is the
standard mathematical construction relating a quasiperiodic function to a higher dimensional periodic
function [31]. It is, for example, how the 3-D PF is
obtained from the 6-D PF. Although these two
descriptions can often be made equivalent, the latter
method lends itself more naturally both to analysing
the experiments and the subsequent modeling of the
structure : the experimental procedure is more naturally understood as a collection of data in reciprocal
space issuing from a 6 to 3-D projection (corresponding to a cut in direct space) rather than cuts in
reciprocal space that would seem to obliterate inaccessible data.
Most of the conventional concepts used in 3-D
crystallography remain valid in the higher dimensional space. The space group is defined as the set of the
6-D isometries which superimpose the equivalent
atom surfaces ; the little group of the atomic site is
the normalizer of the corresponding atom surface,
i.e. the subset of the symmetry elements of the 6-D
space group which leaves the atom surfaces globally
invariant. The atomic sites of the j-th orbit are
defined by their associated 3-D surface Vj, which, in
the general case are functions of both the physical
1227
space (usually called parallel space and denoted by
its basis vectors Ell) and the complementary space
(more often called perpendicular space and denoted
by E_L). The Fourier coefficient of a reflection K of
the 6-D reciprocal lattice is obtained by :
The strong similarity between the PFs of the
crystal and the quasicrystal suggests that we
examine the a crystal in 6-D. In the rational
approximant geometry [32, 33], T (=1.618... ) is
replaced by 1 in the equations for the cut plane [21],
each of the three cubic basis vectors becomes a
specific 6-D vector : for example the (1, 0, 0) basic
vector of the a structure becomes (1,1,0, 20131,
1, 0 ) in 6-D, and atoms localized at (x, y, a ) in 3-D
where fl is the volume of the elementary 6-D unit
cell, k¡ and kl are respectively the parallel and
perpendicular components of the 6-D momentum
transfer vector K, fj are the atomic form factors,
which depend only on k¡, and uj is a 6-D space
variable running over the atom surfaces Vi. For the
special case where the atom surfaces are aligned
along the perpendicular space, the points uj that
belong to Vj can be decomposed into a parallel
component, say rj, and a perpendicular component
ul ; the relation (1) transforms into :
appear at
where the
integration is now over a volume in the
perpendicular space.
Because of the irrationality of the cut, each point
’ contained in the 6-D unit cell will be explored once
and only once in the real structure extended to
infinity. The stoichiometry and the density of the
real structure
therefore those of the 6-D model.
the multiplicity (defined as the
index of the normalizer of a representative atom of
the j-th orbit onto the point group of the 6-D space
group), one obtains the atom fraction cj of the
species in the j-th orbit by the relation :
are
Designating by 03BC
a
in 6-D. In this way, all 138 atoms (in 11 orbits) of the
known a structure can be placed as points on
rational planes in 6-D. Table I shows that 8 of the
11 orbits, including both Mn orbits, project along
El onto the vicinity of nearby nodes, while the
remaining 3 Al orbits project closely to nearby body
centers. The parallel and perpendicular components
of the distances of each orbit from these points is
also given in this table. Because there is a small
parallel component for each orbit, we conclude that
all atoms of the a structure in 6-D are indeed fit into
a space that is close to being entirely perpendicular
to nodes and body centers (the three orbits around
the body centers could have been fit around edge
centers, but that would not have been consistent
with the PF). Since both the PF and the a structure
revealed that the atom surfaces in 6-D are approximately confined to the 3-D perpendicular subspace
surrounding these points, we assume that the orbit
surfaces are 3-D volumes in El with icosahedral
symmetry. The simplest shapes consistent with these
facts, thick concentric spherical shells in El , are the
basis of the model. From each orbit of the a crystal,
we obtain :
whether the shell from this orbit is centered
node or body center ;
(b) the radius of the atoms from the center, which
we will take as the mid-point of its shell ; and
(c) the thickness of the shell, which will be taken
as proportional to the multiplicity of that orbit. The
total volume of all orbits is fixed by the density and
composition of the icosahedral phase, rather than
that of the a phase.
Applying these rules, we find that the 11 orbits in
the crystal can be merged into just 5 orbits in 6-D ; a
Mn and two (Al, Si) shells around the node, and two
(Al, Si) shells around the body center. All inner and
outer shell radii are completely determined by the
stoichiometry, the density of the a structure and the
above rules. There should have been a sixth orbit
from the Al(7) Wyckoff position of the a structure,
but we found that it made very little difference if this
was distributed among the Al(I) and Al(II) 6-D
orbits. Density and composition (see relations (2)
and (3) give two constraints. The remaining par-
(a)
on
and the
where
mass
Mj
density
p :
is the atomic
weight
of the
species
j.
4. A tentative model.
The
simple
6-D PF suggests
atom surfaces
a
model in which all
aligned in E 1. and centered
VJ
about either nodes or body-centers, and are
stretched out entirely in E 1. with icosahedral symmetry. This is, of course, an idealization that ignores
the small displacement in parallel space away from
such symmetric positions found in the EXAFS
[9, 10], the a crystal and confirmed in our PF’s.
are
.
1228
Table I.
atoms
The 6-D embedding of the cvstructure showing the closest approach to node and body center of
projected along E 1. ; bold face highlights the parallel and perpendicular distances from these points to
-
the attached a-orbits.
ameters for the initial model come from the crystal
orbit radii and the apportionment of atoms among
the orbits, as shown in table II and figure 2.
The two manganese orbits of the crystal merge
into a single nodal shell. Five aluminum orbits merge
to form a contiguous shell surrounding the manganese shell ; thus 96 of the 138 atoms in the a structure
belong to a single compound shell (manganese on
the inside and aluminum on the outside) around the
6-D nodes, comprising the entire second Mackay
[34] shell plus 6 aluminum atoms from one of the aorbits (Al(7)) that has been called glue. Around this
compound shell is another aluminum shell containing
12 atoms from the glue orbit Al(6) plus the remaining
6 of the previous Al(7) orbit.
The remaining 30 aluminum atoms are merged
into two orbits about the body centers. One of them,
with 24 atoms, arises from the inner Al shell of the
Mackey icosahedra. Although these atoms in 3-D
are approximately half-way between the central
vacancy and the Mn icosahedron, it is important to
emphasize that this orbit is not associated with the
midpoint of an edge. The last orbit of 6 Al atoms
that belongs to the body center originates from the
remaining glue atoms in the a structure.
5. Refinement parameters.
In the present model, the orbit surfaces are taken as
the volumes within 3-D spherical shells in El ,
centered on points of high symmetry, and having
unit occupancy. The orbit is specified with just two
parameters, the inner and outer radii, that are taken
from the a structure and not obtained from optimization. As such, the model does not allow for
disorder, which is an interesting features of these
quasicrystals. A simple way to handle disorder is to
introduce Gaussian terms in the structure factors,
like static Debye-Waller terms (SDW), which spread
the orbits as in ordinary crystallography, but, in 6-D,
take on additional meaning. While the spread in
parallel space displaces the atoms by small amounts,
the perpendicular components of the spread displaces atoms in 3-D by a large quasilattice translation
vector having a small perpendicular component. The
effect is a gradual decrease in site occupancy especially at the fringes of the orbit. This method of
introducing the SDW into the model specifies which
kind of disorder is expected in 3-D : when the SDW
causes orbits of two chemical species to spread into
each other, a chemical disorder results, including the
1229
Table II.
-
The initial 6-D model
showing
the
merging of the eleven ce-orbits
into
five
6-D orbits.
(1) Cooper and Robinson notation [5].
(2) Adjusted for density p 3.587 glcm3 and stoichiometry Al79Mn21.
(3) In 6-D lattice parameter units (A 0.6495 nm ).
=
=
Since the experimental intensities were converted
into absolute units, no additional parameter was
required for scaling. Both primary and secondary
anomalous absorption terms AF’ and AF" were
introduced in the calculated Fs.
The minimization was performed using the following functional:
The proposed 6-D structure seen in the 5-fold
Fig. 2.
plane ; the glue atoms, found around the edges of the main
nodal Mn-Al shells (gray region), can be viewed as bridges
between adjacent nodal shells.
-
possibility of Mn close neighbors ; when the shells
spread into empty space, it leads to an occupancy
disorder. The disorder would mostly affect those
vacancies and atoms generated by cuts at the fringes
of the 3-D surfaces.
In this simple model, 20 optimization parameters
are possible : each orbit has a maximum and a
minimum radius and could have two SDW terms.
Since there are relatively few peaks in the powder
diffraction spectra that are well defined and not
affected by either the f3 hexagonal phase or the f.c.c.
Al (among the 39 reflections that have been detected, only 17 have significant intensities and not
altered by the other phases), we introduced only one
global SDW, denoted Bll, into the refinement by :
where u F2 = 21 Flu F, and 0- F is the standard deviation of the reflection F. If we make no correction
for background intensity, 0-F is well approximated
by -11P
-A 1 L-P,with /.k
as
the
multiplicity and Lp
as
the
polarization factor [35]. For a direct comparison with what is conventional in crystallography,
we also calculated the residual R-factor defined by
Lorentz
[36] :
and a more meaningful
w R defined by :
weighted
The results of the refinement
residual factor
procedure
are
given
1230
in table III and displayed in figure 3. The optimized
3.14 A2 giving
value of the SDW parameter is Bll
wR
0.257. Calresidual factors of R
0.128 and
culating the intensities for all the possible reflections
=
=
=
.
with
kl
less
2 (where A is the 6-D lattice
A
than 2,/-,-
.
parameter) revealed no missing experimental peaks
compared to the calculated intensities in the experimentally accessible range of kll. A reconstruction of
the electron density map in 6-D along the 5-fold
plane has been performed by Fourier transforming
the experimental structure factors, each being given
Table III.
General characteristics of the final refined model compared
notations for indexing the peak are defined in reference [21] of the text).
-
(1)
(2)
(3)
Defined
as
k; = 2 sinA 0
In number of
AF =
with A
electrons/ Å 3 .
(Fobs - F cal ) .
y Fobs
=
1.79028 A
(CoKa radiation).°
to
experimental
data
(N and
M
1231
Fig.
the
3.
-
sign
The observed and refined calculated
of its
corresponding
X-ray
structure factors versus the
theoretical value
(Fig. 4).
We examined the sensitivity of X Z with respect to
changing orbit radii, and to the introduction of a
global perpendicular SDW factor. Although the
gradient of X 2 with respect to the perpendicular
SDW was slightly negative, none of these additional
variables had a strong enough effect on X 2 to
warrant
their introduction into the minimization.
Fig. 4.
Density map obtained by Fourier transforming
the experimental structure factors with the signs obtained
from the calculated ones.
-
6. Discussion.
This model was suggested by two observations on
Patterson maps of the icosahedral phase and the
prototypic crystalline a structure, and the assumption that these two structures are simply different
cuts of a single 6-D structure :
(1) finding only two 6-D peaks in the quasicrystal
PF implied that all interatomic distance vectors in 3D could be associated with two sets ; quasilattice
translations, that are projections of 6-D node to
node vectors, and motif translationsthat are projections of 6-D vectors from nodes to body centers ;
(2) finding that
( 2 nm) there was
parallel
for
a
momentum transfer.
short
distance
one-to-one
vectors
correspondence
between the interatomic distance vectors in the
known a structure, and those in the icosahedral
quasicrystal.
Embedding the a structure in 6-D revealed that it
too was consistent with only two peaks in the PF
and, knowing how we wanted to group the atoms in
6-D, permitted us to use the atom positions in the
a structure for constructing a model. The simplification of the a structure in 6-D is remarkable.
The satisfactory agreement between calculated
and observed diffraction intensities is an indication
that this model is a good starting point for possible
further refinement. Solid spheres aligned in perpendicular space are only approximations to forms
having icosahedral symmetry to be used as cut
surfaces to give structures in which identical atoms
decorate vertices of a 3-D generalized Penrose tiling.
When the radius is 1.14, the sphere has the same
volume as the cut triacontahedron that gives the
vertices of the 3-D Penrose tiling in which the edges
of the rhombuses are along 5-fold axes and have
length 0.46 nm. The hollow spherical shells with
outer radii less than this, e.g. the Mn shell, can be
interpreted in the 3-D structure as such tilings with
an ordered decoration of atoms and vacancies. Most
of the vacant sites arise from the central hollow,
which itself is an approximate tiling in which the
shortest distances are 1.09 nm along the 3-fold axes,
and 1.24 nm along the 2-fold axes. These are the
distances between vacant centers of Mackay
icosahedra in the a structure. With respect to the
basic tiling, these sites correspond to high local
symmetry vertices which are centers of a tesselation
of 20 prolate rhombi ([37] Henley, Duneau and
Katz, private communications). There are also
aluminum atoms substituting for manganese atoms
owing to the difference in the outer radius and 1.14.
If this were an abrupt edge to the shell, it would give
rise to an ordered set involving all quasilattice
1232
distances, but
we suspect it to be the result of
disorder and accounted for by a possible perpendicular SDW factor. Similarly, the larger radii of the
nodal Al orbits can be interpreted as tilings with
shorter edge vectors.
Because the 6-D embedding of the « structure
does locate atoms as points and not as 3-D surfaces,
there is considerable latitude in the construction of
the atomic surface in a 6-D structure of a that is also
a model for the quasicrystal. For the simplest model,
we chose spherical shells aligned in perpendicular
space, centered on either nodes or body centers.
Because we forced all atom surfaces to lie along
perpendicular space in 6-D, atoms in 3-D are on
exact projections of nodes or body centers. As a
with
result, all distances are of the
the
Mn
For
instance
restrictions
on
n
and
m.
special
atoms in the a crystal are 0.46 nm from the center,
and the inner Al atoms at 0.28 nm instead of 0.48
and 0.24 nm respectively as reported by Cooper and
Robinson [6] for the a structure.
form In +-mr,
Our treatment of the data led us naturally to
choosing the Janner-Janssen-Bak description of ideal
quasiperiodic structures, i.e. those that have discrete
diffraction patterns with a finite basis, and formulated on the mathematical theorem that all quasiperiodic functions can be described as planar cuts of
higher dimensional periodic functions [30]. This
description is thus completely general for all ideal
quasiperiodic structures. The descriptions based on
a finite set of decorated tilings are less general and
represent a first approximation where only local
environments (those within each tile) have been
taken into account. In a general quasiperiodic struccurving of the atom surfaces permits
infinitely varied configurations. Because we aligned
our surfaces, we did not, in this initial model, avail
ourselves of this generality : our structure can be
considered as a relatively complex superposition of
different tilings resulting from projections of a 6-D,
mostly vacant, body-centered cubic lattice ordered
ture, the
to
give
a
primitive
translation group.
The atom surfaces in this model are disjoint, but
there are obvious regions in 6-D of close approach
and it is there that the « glue » atom orbits appear.
Glue atoms have recently been shown to be intimately involved in third and to some extent shared
shells in the structural description of the a crystal
[38]. The place of closest approach in the icosahedral
phase is seen, in the 5-fold plane, to involve all three
glue orbits. Here, glue atoms could link adjacent
nodal shells, just as these atoms do in the a
structure. Such linkages are apparent in the PF. One
of the glue orbits seems to be the outer part of the
large 6-D shell containing all of the second shell of
the Mackay icosahedra. The gap between the two
nodal
glue orbits may just be a way of representing
decreasing site occupancy. The body centering glue
orbits are partially inserted into the region of closest
approach between two nodal orbits.
The present model includes all atoms and has the
correct density. Examination of the predicted 3-D
structure (without the SDW disorder) reveals rela-
tively few complete Mackay icosahedra just as there
few decagons in a 2-D Penrose tiling ; instead,
there are many recognizable fragments that interpenetrate. The initial model is, of course, exactly
quasiperiodic ; even the way the SDW factor introduces disorder in the optimization, leaves discrete
peaks, with a Laue monotonic background and no
other diffuse scattering. The refined value of the
SDW in parallel space is rather large, approximately
four times what is usually observed in good quality
crystals, which is the confirmation of the appreciable
disorder expected in quasicrystals.
are
There have been a number of models that consider
of identical units because such models have
factorable structure factors. In an earlier paper [11],
we proposed a model with a strictly quasiperiodic
array of perfect two-shell Mackay icosahedra, with
the remaining Al (glue) atoms omitted. The density
was therefore low. The fit with the spectra was poor,
but still an improvement over that calculated from
models that place a single scatterer on a 0.46 nm 3-D
Penrose tiling. There are other factorable models
that pack Mackay icosahedra in 3-D, most notably a
random instead of a quasiperiodic packing [39, 40]
with the principle concern being the prediction of
line broadening.
With two parameters per orbit, the 6-D spherical
shell description is comparable to the three coordinates that describe a general Wyckoff position in 3D crystals. Instead of 11 orbits with more than 20
coordinate parameters needed to give the 3-D atom
positions in a, we found that, because of the
merging of the orbits in 6-D, five shells (ten parameters) and a single global SDW fitting parameter
sufficed to give a good fit of the diffraction spectra of
the icosahedral phase. Considering that we are
modeling an aperiodic phase, these represent few
parameters. We did not use the radii as fitting
parameters, but took them directly to conform with
the known a structure. They could have been used
to improve the fit with data. It is noteworthy that the
6-D description of the a crystal also has an economy
of parameters. The 3-D a-orbits calculated from the
6-D structure are a good fit, with the Mackay
icosahedra having m3 rather than icosahedral symmetry. A similar 6-D CsCI type description was.
successful for the R-phase in the Al-Cu-Li system.
Whether or not other types of Frank-Kasper phases,
especially those with higher coordination numbers,
can also be economically depicted in higher dimensions remains to be seen.
packing
1233
Acknowledgments.
We
gratefully acknowledge our colleagues Dr Y.
Calvayrac, S. Lefebvre, M. Bessiere and A. Quivy
to have allowed us to use their X-ray data prior to
publication. We thank Drs P. Bak, M. Duneau, C.
L. Henley, M. Jaric, A. Katz, S. C. Moss and E.
Prince for their many stimulating suggestions and
discussions on the subject. Finally, we would like
especially to thank Dr J. Bigot, Mr A. Dezellus and
Mrs S. Peynot who prepared the samples, in a
remarkably reliable and reproducible way.
References
[1] SHECHTMAN, D. and BLECH, I., Metall. Trans. 16A
(1985) 1005-1012.
[2] SHECHTMAN, D., BLECH, I., GRATIAS, D. and
CAHN, J. W., Phys. Rev. Lett. 53 (1984) 19511953.
complete bibliography, see MACKAY, A. L.,
Int. J. Rapid Solidification 2 (1987) S1-S41.
[4] CAHN, J. W., GRATIAS, D. and MOZER, B., Phys.
[3]
For
M., LEFEBVRE, S., QUIVY, A. and GRATIAS,
D., submitted to Philos. Mag. Lett.
J. W., SHECHTMAN, D. and GRATIAS, D., J.
CAHN,
[21]
Mater. Res. 1 (1986) 13-26.
[22] GRATIAS, D., CAHN, J. W., BESSIERE, M.,
(to appear).
[5] GRATIAS, D., CAHN, J. W. and MOZER, B., Phys.
Rev. B (to appear).
[6] COOPER, M. and ROBINSON, K., Acta Crystallogr. 20
(1966) 614-617.
[7] GUYOT, P. and AUDIER, M., Philos. Mag. B 52
(1985) L15-L19.
[8] ELSER, V. and HENLEY, C. L., Phys. Rev. Lett. 55
(1985) 2883-2886.
[9] MA, Y., STERN, A. and BOULDIN, C. E., Phys. Rev.
Lett. 59 (1988) 1611-1614;
Rev. B
also STERN, E. A., MA, Y., BAUER, K. and
BOULDIN, C. E., J. Phys. Colloq. France 47
(1986) C3-371-377.
SADOC, A., FLANK, A. M., LAGARDE, P., SAINFORT,
P. and BELLISSENT, R., J. Phys. France 47
(1986) 873-877.
CAHN, J. W. and GRATIAS, D., J. Phys. Colloq.
France 47 (1986) C3-415-424.
DE WOLFF, P. M., Acta Crystallogr. A 30 (1974) 777 ;
see also JANSSEN, T., Acta Crystallogr. A 42 (1986)
261 ; J. Phys. Colloq. France 47 (1986) C3-85-94
and references therein.
BAK, P., Phys. Rev. Lett. 54 (1985) 1517-1519.
BAK, P., Phys. Rev. Lett. 56 (1985) 861-864.
BAK, P., J. Phys. Colloq. France 47 (1986) C3-135141.
HIRAGA, K., HIRABAYASHI, M., INOUE, A. and
MASUMOTO, T., J. Phys. Soc. Jpn 54 (1985)
4077.
SHEN, Y., POON, S. J., DMOWSKI, W., EGAMI, T.
and SHIFLET, G. J., Phys. Rev. Lett. 58 (1987)
1440.
ELSWIJK, H. B., DE HOSSON, J. Th. M., VAN
SMAALEN, S., and DE BOER, J. L., submitted to
Phys. Rev. Lett.
ROBERTSON, J. L., MISENHEIMER, M. E., MOSS, S.
C. and BENDERSKY, L., Acta Metall. 34 (1986)
2177.
New
Concepts in Physical Chemistry, Acqua
(September 1987) to be published.
[23] BAK, P., Phys. Rev. B 32 (1985) 5764-5772.
[24] LUBENSKY, T. C., RAMASWAMY, S. and TONER, J.,
Phys. Rev. B 32 (1985) 7444-7552.
[25] HORN, P. M., MALZFELDT, W., DIVINCENZO, D. P.,
TONER, J. and GAMBINO, R., Phys. Rev. Lett.
Fredda
57 (1986) 1444-1447.
[26] LEVINE, D., LUBENSKY, T. C., OSTLUND, S., RAMAS-
[27]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20] CALVAYRAC, Y., DEVAUD-RZEPSKI, J., BESSIERE,
WANY, S., STEINHARDT, P. J. and TONER, J.,
Phys. Rev. Lett. 54 (1986) 1520-1523.
DUNEAU, M. and KATZ, A., Phys. Rev. Lett. 54 (25)
(1985) 2688-2691 ;
KATZ, A. and DUNEAU, M., J. Phys. France 47
see
[10]
CAL-
Y., LEFEBVRE, S., QUIVY, A. and
MOZER, B., Proc. Nato Adv. Res. Workshop,
VAYRAC,
a
(1986) 181.
[28] KALUGIN, P. A., KITAYEV,
A. Y. and LEVITOV, L.
S.,J. Phys. Lett. France 46 (1985) L601-L607.
[29] ELSER, V., Acta Crystallogr. A 42 (1986) 36.
[30] BAK, P., Scr. Metall. 20 (1986) 1199-1204 ;
also in Scaling Phenomena in Disordered Systems,
Eds. R. Pynn and A. Skjetorpeds (Plenum,
N.Y.) (1986) 197-205.
BOHR, H., Acta Math. 45 (1924) 29 ; ibid. 46 (1925)
101 ; ibid. 47 (1926) 237.
HENLEY, C. L.,J. Non-Cryst. Solids 75 (1985) 91-96.
GRATIAS, D. and CAHN, J. W., Scr. Metall. 20 (1986)
see
[31]
[32]
[33]
1193-1198.
[34] MACKAY, A. L., Acta Crystallogr. 15 (1962) 916-918.
[35] PRINCE, E. and NICHOLSON, W. L., Structure and
Statistics in Crystallography, Ed. A. J. C. Wilson
(Adenine Press) 1985, 183-195.
[36] International Tables for Crystallography, II (Kynock
Press) 1955.
[37] LE LANN, A., Philos. Mag. B 56 (1987) 371-384.
[38] FOWLER, H., MOZER, B. and SIMS, J., Phys. Rev.
B 37 (March 1988) in press.
[39] STEPHENS, P. W. and GOLDMAN, A. I., Phys. Rev.
Lett. 56 (1986) 1168-1171.
[40] ELSER, V., Proc. Nato Adv. Res. Workshop, New
Concepts in Physical Chemistry, Acqua Fredda
(September 1987) to be published.