A 6-D structural model for the icosahedral (Al, Si)-Mn quasicrystal

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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�

A 6-D structural model for the icosahedral (Al, Si)-Mn quasicrystal

J. W. Cahn (1), ^{D. Gratias} (2) ^{and B. Mozer} (1)

(1) Institute for Materials Science and Engineering National Bureau of Standards, Gaithersburg ^MD-20899

U.S.A.

(2) C.E.C.M./C.N.R.S., ^{15 rue G.} Urbain, ⁹⁴⁴⁰⁷ Vitry, ^France (Reçu ^{le 24} fgvrier 1988, accepté ^{le 25 mars} 1988)

Résumé.

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.

Abstract.

A 6-dimensional (6-D) periodic ^{model is} proposed for the Al-Mn-Si icosahedral quasiperiodic crystal. ^The model results from an embedding ^{of the} periodic cubic 03B1 structure in 6-D. In the Janner-Janssen- Bak 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.

Classification

Physics ^Abstracts

61.10

61.50E

61.55H

64.70E

1. Introduction.

One of the principal problems ^{in the} study ^{of the} recently discovered quasiperiodic crystals (quasicrys- tals) [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. Be-

cause quasiperiodic structures can be described by ^a

known irrational cut of a periodic higher-dimension-

al 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 ^dif-

fraction 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. projec-

tions 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 X-}

ray 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 struc-}

ture 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 X-}

ray 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

Fig. ^1.

X-ray (a) ^{and neutron} (b) Patterson functions

displayed in 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.

descriptions of the Janner-Janssen-Bak [12-15] type, rather than those descriptions ^{that are based on}

decoration of tilings [16-18].

2. Experimental.

X-ray ^{and neutron} powder diffraction data show that the finely powdered rapidly solidified A173Mn21Si6

used is almost entirely icosahedral phase [19] ^con- taining 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}

room 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

microscopy diffractions that would reveal the pres-

ence of glides mirrors and/or screw axes 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].

The average width of the peaks ^{as measured both} by X-rays ^{and neutrons leads to an} average ^corre- lation 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} origi- nally proposed by ^Bak [23] ^and elaborated by ^others [24-26].

3. 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 quasi- periodic ^{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 nat-} urally understood as a collection of data in reciprocal

space issuing ^{from a 6 to 3-D} projection (correspond- ing ^{to a cut in direct} space) rather than cuts in

reciprocal space that would seem to obliterate inac- cessible data.

Most of the conventional concepts used in 3-D

crystallography ^{remain valid in the} higher ^dimension-

al 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

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 :

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 :

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 are therefore those of the 6-D model.

Designating by _03BC ^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 :}

and the mass density p :

where Mj is the atomic weight ^{of the} species j.

4. A tentative model.

The simple ^{6-D PF} suggests ^a model in which all atom surfaces V J ^{are aligned in E 1.} and centered about either nodes or body-centers, ^{and are}

stretched out entirely ⁱⁿ E 1. ^with icosahedral _sym- metry. ^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.}

The strong similarity ^{between the} PFs of the

a 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}

appear ^at

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 approxi- mately ^{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 :

(a) whether the shell from this orbit is centered

on 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-

1228

Table I.

The 6-D embedding of ^the cvstructure showing the closest approach ^{to node and} body ^center of

atoms 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 mangan-

ese 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 a-

orbits (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 optimi-

zation. 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 ^dis- places ^{atoms in 3-D} by ^a large quasilattice translation vector having ^{a small} perpendicular component. ^The effect is a gradual ^{decrease in site} occupancy espe-

cially ^{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

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 ).

Fig. ^2.

^The proposed ^{6-D structure seen} in the 5-fold

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 de-

tected, only ^{17 have} significant intensities and not

altered by ^{the other} phases), ^we introduced only ^one global SDW, ^denoted Bll, ^{into the} refinement by :

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 follow-

ing functional:

where _{u F2 =} 21 Flu F, ^and 0- F is the standard devi- ation of the reflection F. If we make no correction for background intensity, 0-F is well approximated

by -11 -A 1 L-P,with /.k ^{as the} multiplicity ^and Lp ^{as the}

P

Lorentz polarization ^factor [35]. ^{For a direct com-} parison with what is conventional in crystallography,

we also calculated the residual R-factor defined by [36] :

and a more meaningful weighted ^{residual factor}

w R defined by :

The results of the refinement procedure ^are given

1230

in table III and displayed ⁱⁿ figure ^{3. The} optimized

value of the SDW parameter ^is Bll

^3.14 A2 giving

residual factors of R

0.128 and wR

0.257. Cal-

culating the intensities for all the possible reflections

.

2

with kl ^less than 2,/-,- ^{(where A} is the 6-D lattice A

parameter) ^{revealed no} missing experimental peaks compared ^{to the} calculated intensities in the exper-

imentally 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 ^to experimental ^data (N ^{and M}

notations for indexing ^the peak ^are defined ⁱⁿ reference [21] of ^the text).

(1) ^{Defined as} k; = 2 ^{sin 0} _A ^{with A}

^{1.79028 A} (CoKa radiation). ° (2) In number of electrons/ Å 3 .

(3) ^{AF =} (Fobs - F cal ) .

y ^Fobs

Fig. ^3.

The observed and refined calculated X-ray ^{structure factors versus the} parallel ^{momentum transfer.}

the sign ^{of its} corresponding 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 assump- tion 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 3- D could be associated with two sets ; quasilattice translations, ^{that are} projections ^{of 6-D node to}

node vectors, ^and motif translations that ^are projec-

tions of 6-D vectors from nodes to body centers ;

(2) finding ^{that for short distance vectors} ( ² nm) ^{there was a} one-to-one 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 simplifi-

cation 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 perpen- dicular 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 perpendicu-

lar 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} result, all distances are of the form In +-mr, ^with

special restrictions on n and m. For instance the Mn 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.}

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 formu-} lated on the mathematical theorem that all quasi- periodic ^{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 ^struc-

ture, the curving ^{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 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 inti-

mately ^{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}

are few decagons ^{in a 2-D Penrose} tiling ; instead,

there are many recognizable fragments ^{that inter-}

penetrate. ^{The initial model} is, ^of course, exactly quasiperiodic ; ^even the way the SDW factor intro- duces 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.

There have been a number of models that consider

packing 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 coordi- nates that describe a general Wyckoff position ^{in 3-}

D 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 par-

ameters) ^{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 _sym- metry. ^{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 ^dimen-

sions remains to be seen.

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.

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