Ideal AlMnSi quasicrystal : a structural model with icosahedral clusters

J.

Phys.

France 50

Classification Physics Abstracts 61.50E 61.55H -

(1989)

-

Ideal AlMnSi clusters

135-146

(Reçu

quasicrystal :

1989,

135

a

structural model with icosahedral

Oguey

Physique Théorique,

le 7

JANVIER

64.70E

M. Duneau and C. Centre de

15

juillet 1988, accepté

Ecole sous

Polytechnique,

forme définitive

91128 Palaiseau, France

le 16

septembre 1988)

Nous proposons un modèle, pour la phase quasicristalline de l’AlMn(Si), caractérisé Résumé. par un ordre à longue distance purement quasipériodique et la symétrie icosaédrique. La densité et les concentrations coincident avec les valeurs expérimentales. Les distances interatomiques sont compatibles avec les données de la cristallographie et sont proches de celles de la phase 03B1 voisine. Le modèle comprend une forte proportion d’icosaèdres de Mackay. -

A model for Al(Si)Mn quasicrystals is proposed. It is characterized by pure order with icosahedral symmetry. The density and concentrations fit to The interatomic distances are compatible with crystallographic data and compare well to the related 03B1 phase. A large number of icosahedral clusters (Mackay icosahedra) are present in the model. Abstract.

-

quasiperiodic long range the experimental values.

1. Introduction.

First reported by Shechtman et al. [1] the quasicrystalline phase of AlMn alloys with icosahedral symmetry is now experimentally available at a very high degree of purity. In particular, the Al73Mn2lSi06 compound shows coherent diffraction effects revealing long range order of both orientational and positional types [2]. Very early, it was noticed that quasiperiodic tilings such as the Penrose ones would provide suitable templates to account for this long range order [3, 4]. However, on a local scale, no simple recipe accounts for the atomic positions within such templates. We shall describe our model by a section technique. Familiar to modulated crystals [5], this technique is now known to work for tilings [6, 7] and related quasiperiodic structures [4, 8]. It insures that the derived model is quasiperiodic (with pure point spectrum in the Fourier transform). The icosahedral symmetry requires a suitable embedding of the physical 3D-space E in a 6D-space where the group G Ih has a crystalline representation [4]. Before we get into the detailed description, it is convenient to make a « tour d’horizon » of the already existing tentative models. The first proposal, for the atomic positions, was provided by Cahn and Gratias [9]. Their idea consisted in building the quasicrystal on the basis of an icosahedral structural unit made of 12 Mn and 42 Al and known as Mackay’s Icosahedron (MI) ; the MI’s would then be located on a quasiperiodic set of sites provided by =

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005002013500

136 a standard section technique. This matrix of MI is present in our model as a framework providing o-- 2/3 of the total number of atoms. The remaining ones the linking atoms will be provided by our building suitable atomic surfaces. Yamamoto and Hiraga gave an alternative proposal for the distribution of the linking atoms [10]. Their solution is partially as we’ll see probabilistic but it contains very unrealistic local clusters. Guyot and Audier based a model on local atomic configurations similar to the crystalline euphase [11]. However, with an underlying tiling of unit edge 12.70 À, their model desagrees with diffraction data (1). An analysis of the a phase in terms of rhombohedra similar to the tiles of the icosahedral packing was carried out by Elser and Henley [12]. Indeed a number of local configurations are common - up to slight metrical distortions to the a phase and our model for the Ico phase. Recently, Cahn, Gratias and Mozer [20b] proposed an interesting model built by a section method, similar to ours, but involving spherical atomic surfaces. Their model contains only incomplete Mackay icosahedra and a few abnormally short bonds. As we show, polyhedral atomic surfaces allow to totally avoid unphysical short bonds without loosing in densities. Moreover, we designed the model so as to involve a large proportion of MI’s. The model is presented as follows. First, we briefly recall the section method (Sect. 2) together with the settings required by icosahedral symmetry (Sect. 3). The 3D icosahedral tiling will be used, when possible, as a reference frame to fix positions and tao describe local patterns. This is why we recall some of its properties in section 4. Various standard polyhedra are then analyzed (Sect. 5) as windows in the section method. This will justify our choice of atomic surfaces presented in section 6. Then section 7 contains a systematic investigation of the nearest neighbors bonds. The diffraction spectra of a finite sample are presented in -

-

-

-

-

section 8. 2. The section method.

through higher dimensional periodic structures provide a canonical way to build quasiperiodic structures. The natural setting for quasicrystals is as follows. The physical space E, of dimension 3, is embedded in the 6-dimensional vector space R6 as the range of the projector p : R6 --+ E defined below. A typical atomic surface A is a bounded subset (2) a polyhedron of the orthogonal complement El of E in R6 (despite the word « surface », A is 3-dimensional). When set anywhere in R6, say at g, the intersection En (A + e) of such a surface with E is either empty (when pl (e) falls outside - A) or a single point {x=p(g)} (when pl.(g) falls inside - A). The subset () such that p_L (e ) is in A} of R6 is known as the strip. It is a 6-D cylinder (or prism) with a 3-D basis in El. called the « window » or « profile » of the strip. Such intersection points will be atomic sites in the model, the species Al or Mn being specified by the atomic surface itself. This means that there will be two types of distinct surfaces, called Mn and Al as well, prescribing a Mn atom when E cuts Mn + e, and an Al or Si atom (with probability 73/79 for Al, 6/79 for Si) Affine cuts

-

-

-

when E cuts Al + e. The atomic surfaces are disposed periodically throughout R6. To describe such a periodic set we need to specify 1° the translation lattice, 2° the set A of atomic surfaces associated to a fundamental domain. The lattice will be the simple hypercubic Z6 generated by the canonical basis si, ... , 86 of R6. The atomic surfaces are described in section 6.

(1) This edge length is T24.85 Â (4.85 Â being the radius of a Mn icosahedron). Even the discrepancy 4.85-4.60 Â is away from experimental errors. (2) For sets A and B of vectors, we write A ± B for the set {a ± b1 a in A, b in B} . In particular, - A is the symmetric of A.

137

The model in the real space is obtained the atomic surfaces :

as

the intersection X of E with the

periodic set of all

The relative shift a (a vector of R6) of the periodic set with respect to the section space E indexes equivalent models, degenerate from an energetic point of view. 3. Icosahedral

symmetry.

The icosahedral symmetry related to quasiperiodic patterns has been systematically analysed (see [3]). The group Ih may be represented as signed permutations of El’ E6 (we stick to the conventions of [13]) leaving two 3D-subspaces E and El. invariant. E is the range of the ...

following projector

whereas El. is its kernel. The complementary projector pl = 1 - P projects R6 onto El. with kemel E. A sufficient condition for the final structure to satisfy icosahedral symmetry is that the atomic surfaces be symmetric, as subsets of R6, modulo Z6. 4. The standard icosahedral

tiling.

The standard unit cell of Z6, namely the 6D unit cube, projects onto a regular triacontahedron TR in E1. If this polyhedron is taken as atomic surface (« vertex » surface in this case), then it leads, via the section algorithm, to the set of all the vertices of a Penrose-type tiling of E. The 3-dimensional Penrose tiling (3DPT) of the Euclidean space by means of an oblate and a prolate rhombohedron is icosahedrally symmetric. How to build regular packing of parallelohedra matching face to face by section techniques was explained in [6]. Here, we are only concerned with the vertices as a discrete set of sites. Comparison of the Fourier transform of the 3DPT with the diffraction spectra for AIMnSi fixes the edge length to e =lei1 4.60 Â [14, 12]. Since the edges of both rhombohedra are 6.5 Â in the projected basis vectors ei p(Ei), we get a unit length equal to 1 Ci[ = e. All our metrical data will be given in units of 6D lattice ; thus e unless another (conventional) unit is explicitly written. Worth remembering is the inflation property of Penrose tilings. The following modular =

=

1 et(

matrix

=

Ipl. ( Ci )[

N/2 e

=

138

leaves

Z6 invariant and

commutes with the 6D

representation

of

Ih since, actually,

B/5)/2.

is the golden mean (1 + when Thus, operated upon the periodic set of atomic surfaces, M reduces homothetically A factor - T - 3 whereas the section tiling undergoes a transformation which scales the by a linear a tiles by (linear) factor r 3 and changes the ordering only through a natural isomorphism a - a ’ [3, 13]. Therefore, from measurements in the physical space, the lengthscales in the 6D space can be fixed only up to such an inflation-deflation symmetry. where

T

5. Standard

polyhedra.

polyhedra having icosahedral symmetry. Starting with the 3DPT, we changes related to replacing TR by other polyhedra as atomic surfaces (3) (see Tab. I, below, for the definitions of typical lengths). 1) In the 3DPT the minimal distance between vertices is dr = 0.5628 ; the next neighboring 1.0514. distances are the edges e = 1 and the small diagonals of the facets df TR is bounded by 30 standard rhombuses orthogonal to the a2 axis (see Fig. 1). TR can be obtained as the Dirichlet domain associated to the 30 points of the orbit of an We

investigate

various

comment on the successive

=

Various polyhedra entering the construction of the atomic surfaces : the standard Fig. 1. triacontahedron TR is in light grey ; the large triacontahedron LTR truncated by the large dodecahedron gives A (white) ; sA, homothetic to A, is in dark grey. -

(3) e-L ei ,

The

...,

et .

polyhedra

are

assumed to have

common

symmetry axis a2, a3, a5, with a5 along

139

a2 vector of length D f + df. Consequently, the a2 diameter of TR is D f + df. This property implies that the distance Df - df = 0.6498 never appears in the 3DPT albeit it being larger than the actual shortest distance d,. (Indeed, if two points, say x and x + t, occurred in Xa at a distance Il t Il D f - df, it would mean that both xl and xl + t.L belong to TR, which is generically impossible since, as easily seen, Il t’ Il Df + df diam (TR) ; see [13, 16].) As a consequence, the acceptance domain TR can be enlarged, to some extent, so as to preserve the same minimal distance in the (now denser) structure. The first new points occurring with a small increase of TR appear on the large a2 diagonal of Rhombic Dodecahedra. More precisely, they appear on the large diagonal of an intemal facet at distance df from one vertex and at distance D f - df from the other vertex. One could ask whether it is possible to decorate each facet of the 3DPT in the above manner without introducing a new minimal distance. Since the density of facets in the 3DPT is 3 times larger than the density of sites, one could expect from this decoration a final structure 4 times more compact. The answer is no, because such a systematic decoration yields the new 2 (i.e. 1.086 Â), parallel to a a5 axis, in the decoration of obtuse rhombohedra. distance =

=

=

J5 -

2) LTR denotes the triacontahedron T times larger than TR (see Fig. 1). Since LTR is a Dirichlet domain associated to a2 axis, with an a2 diameter 2 . D f + df, the a2 distance which is forbidden is 2. df - Df = 0.4017 (i.e. 1.85 However, the a5 diameter of LTR is than thus the corresponding structure involves which 2. (T + 1 ) = is + + 3, 2; larger the new minimal distance 2 along a5 which is unacceptable. Actually the structures associated to LTR can be viewed as 3DPT with all their facets decorated in the above manner, plus some other points.

J5

J5

Â).

J5 -

3) LD denotes the regular dodecahedron whose diameter along a5 is J5 + 2. This is the Dirichlet domain associated to a 12-star of points at the distance J5 + 2 from the origin. Consequently, the structures corresponding to such an acceptance domain will not show any distance smaller than 1 on the a5 axis. However, since the a2 diameter of LD is larger than that of LTR, forbidden distances 2. df - Df = 0.4017 (i.e. 1.85 Â) will appear on the a2 axis. 4) A LTR n LD is the large triacontahedron LTR truncated by the dodecahedron LD (see Fig. 1). It follows from the above discussion that the structures corresponding to an acceptance domain A will show neither the distance 2 . df - Df = 0.4017 (i.e. 1.85 Â) on the a2 axis, nor the distance B/5-2 (i.e. 1.086 Â) on the a5 axis. A is almost the largest connected acceptance domain ensuring the minimal distance dr = 0.5628. (An optimal domain is given in [15] ; it looks much like A but for the pentagonal facets which are ruffed decagons instead of pentagons : the relative improvement in the volume, and thus in the density, is of order 10-4. ) Even if we explicitly use A, mainly for graphical and numerical simplicity, it should be understood that, everywhere in our construction, A can and should be replaced by the non-convex optimal polyhedron. sA is the of A by a factor r 3 J5 - 2 (see Fig. 1). Since sA is related to A homothetic 5) =

«

»

-

-

by the standard inflation parameter T3, the structures associated to an acceptance domain sA will show characteristic distances r3 larger than those occurring in the A-structures. Consequently, the minimal distances are J5 + 2 (i.e. 19.4859 Â) on a5 axis, T. Df = 2.7528 (i.e. 12.6627 Â) along a2 axis, and DR = 2.3839 (i.e. 10.9662 Â) on the a3 axis. Again, sA is close to the largest connected domain for which the minimal distance is DR. These distances are close to the distances between Mackay Icosahedra in the crystalline a phase of AIMnSi [9].

6) sD is the regular dodecahedron whose diameter along a5 provided by A is a fairly loose packing : packing fraction 0.6. =

is 1. Notice that the structure Part of the holes may be filled

140

by adding a new atomic surface sD to the body centers of Z6. These new sites are not closer to A-sites than 7"-1 = 0.618 and are therefore compatible with hard cores of diameter dr 0.563. The consequences of such a new atomic surface on the body centers can be understood from the general discussion given in [16]. =

Table I.

-

Some metrical data,

given

is the small

w.r.t

the

diagonal

is the

edge

e

diagonal

is the

large

standard

tiling (e

=

4.60

Â).

of the facets.

large diagonal

is the small a3

of the

a3

of the facets.

of the oblate rhombohedra.

diagonal

of the

prolate

rhombohedra.

6. The atomic surfaces.

The model is built upon two main atomic surfaces (per cell in Z6). The first one the is centered on the nodes of polyhedron A truncated in the vicinity of its a3 vertices Z6. It is divided into three concentric domains described below. The central polyhedron defines the sites of MI ; the intermediate shell provides Mn sites whereas the outer shell defines the Al atoms. The second atomic surface is a small dodecahedron sD centered at the body centers of the lattice Z6. Its species is Mn in agreement with Patterson diagrams [2] and stoichiometry. -

-

6.1 ON

THE VERTICES OF

Z6 (see Fig. 2).

The MI domain. - The atomic surface sA defines the unoccupied centers of the Mackay Icosahedra. To each point x given by the cut of a translate sA + e (§ in Z6) of sA will correspond a complete MI centered at x. On one hand, as we’ll see, the outer 12-Al and 30Mn shells are filled by means of the neighboring atomic surfaces. On the other hand, the inner Al-shell, which consists of a 12-star of Al at a radius (4) a e/2 2.3 Â from x, is to be considered as a cluster which decorates, in a systematic way, all the sites of the type sA. An equivalent way of producing this inner shell is to build a « star » of 12 atomic surfaces sA ± a. e i, i = 1, 6, around each vertex of Z6. The outer shells of the Ml’s are now obtained in the following way. =

=

...,

(4) a

is

an

open parameter, close to 0.5. Its exact value is to be matched to the measured radius.

141

Fig. Z6.

2.

-

A section

orthogonal

to an a2 axis of the atomic surface A attached to the vertices of

The Mn domain. - The 12 translates sA ± e/ of sA are assigned Mn atoms while the 30 translates sA + D f are Al atoms. In the physical space, this provides a 12-star of Mn of radius 4.6 Â from the center of the MI and a 30-star of Al of radius df = 4.84 Â. Note that the 12 translates sA:t et are all inside the standard triacontahedron TR corresponding to the 3DPT ; but the 30 translates sA + D f only slightly overlap TR ; it tums out that a simple and convenient choice, compatible with stoichiometry, consists in defining the Mn atomic surface as the triacontahedron TR from which we remove the MI domain sA and the 30 intersections TR n (sA + Df). The volume of this atomic surface is 12.3107 - 0.6234 - 30*0.060719 9.8657. =

As mentioned above, the almost optimal connected acceptance domain The Al domain. for which the minimal distance is the small a3 diagonal dr = 2.59 Â is the intersection A LTR n SD. So we could think of defining the Al domain as the polyhedron A minus the MIdomain sA, and minus the Mn-domain ; we observe that the domain for the outer Al shell of the Ml’s, given by the 30 translates sA + D f, is entirely contained in A ; this ensures that the sites corresponding to the atomic surface sA are actually surrounded by complete Mackay icosahedra (5). However, this domain is too large for the centers of Ml’s have dr nearest neighbors in the A-structure ; more precisely, since the a3 radius of A is the large diagonal DR, the 20 translates sA + DR do overlap A (see Fig. 2) ; each such intersection A n (sA + DR) has a volume = 0.2182 ; so, taken within the Al surface, these intersections would give rise to Al atoms at distance dr from the centers of Ml’s (--+ dist. 0.346 from inner but in it can such exist be checked that the don’t Ml’s. true Otherwise, Al), neighbors -

=

=

(5) The whole domain AB (sA + atoms, with an occupation probability 1/2.

{O, et, Df} )

was

proposed by

Yamamoto et al. for the

linking

142

translates sA + DR do not intersect the translates sA + D f ; thus the best choice for the Al domain is obtained by removing those intersections from A. The corresponding volume is 47.3938 - 12.3107 - 20*0.2182 + 30*0.0607 = 32.540195. As mentioned in section 5, the sites coming from an sD 6.2 ON THE BODY CENTERS OF Z6. attached to the body centers of Z6 occur at distances r - 1 = 0.618 from their nearest neighbors on a5 axis ; they can be seen to be compatible with the Mackay Icosahedra associated to sA (i.e. not falling too close to MI). The atomic surface sD will specify Mn atoms : the first reason for this choice is the fit with the stoichiometry : (AI/Si)79.,Mnw.g ; a second reason follows from an analysis of the X-ray and neutron Patterson function [2] which shows that pairs of atoms for the distance T -1 are mainly hetero-atomic ; actually, the atoms associated to sD have = 0.25 % Mn neighbors and o-- 0.75 % Al neighbors provided by the main atomic surface described above. -

10.5594 and The volume of the Mn atomic surface is 9.8656 + 0.69379 6.3 DENSITIES. the volume of the AISi atomic surface is 32.5402 + 12*0.623495 40.02214, leading to a total volume of 50.5815. Densities in the direct space are related to volumes in the complementary space through =

-

=

The factor 1/8 is the

density

of

Z6 with1 si1

=

B/2.

With specific masses mMn 54.938, masi (73 mMn + 6 mMn)/79 27.066 e-3 = 0.01706 mollcm3), we get the following number and mass densities : =

=

In particular, the calculated stoichiometry is Mn : 20.9, AISi : 79.1 close to measured densities [17, 20b].

=

(at %).

g/mol (and

These values

are

7. Local patterns.

Apart from the inner Al shell of the MI’s, all the atomic sites are in the projection of the 6D BCC lattice Z6 U (Z6 + 0/2) where 6 (1,1, 1,1,1,1) is the main diagonal of the 6-cube. The centers of MI are projected from Z6 as well and are thus included in the following discussion, considered as the sites of the structural unit made of 12 Al. What happens individually for those Al atoms will be discussed at the end of this section. A pair of sites (x, x + t) occurs in the structure if and only if 1° both x and x + t are projected from the BCC lattice, 2° their conjugated points xl and xl + tl in the complementary space E.l fall into their respective windows. If we let Dl be the domain associated to the species of the atom at x and D2 the domain of the one at x + t (it can be Al, Mn, or MI), then the translation vector t is projected from the BCC lattice and its complementary image t’ in E.l belongs to the intersection Dl n (D2 - t.l). Such an intersection is the existence domain of the pairs (x, x + t ) with the respective species ; in particular its volume in E.l yields the density of such pairs in the direct space [6, 13]. The first inter-sites translations are listed in table II in order of increasing lengths. Those vectors tum out to be on symmetry axis of the Ih group as mentioned in the table. The table =

143

Table II.

-

Volumes

of the

existence domains

(for

1

direction).

the volume of the existence domain model « blind » to species.) The coordination number cab, defined as the ratio of the total number of pairs (a, b ) at distance[ tover the number of a, is given, in our case, by the volume of the corresponding existence domain for the pair (x, x + t ) multiplied by the multiplicity (12 for a5, 20 for a3, 30 for a2) and divided the volume of the a species. For completeness, table III contains explicitly the distances involving the inner Al of Ml’s. Since the neighbors at[ t1 1 of the inner Al are the Al and Mn involved in the outer shell of the MI and nothing else, the statistics is easily done by counting the number of times a certain bond lengths occurs in a single complete MI (such lengths are recognizable in that they depend on the parameter a set equal to 1/2 in the 2nd column) and multiplying it by the frequency of MI (= vol (sA)).

provides, for all different pairs of species, Dl n (D2 - tl ). (X stands for any atom, in the

Table III.

-

Coordinations.

If we take as first neighbors all the atoms separated by less than 0.68 (= 3.13 À), we get a coordination of 11.13 atoms around Mn at average distance 0.593. The Al atoms are surrounded by 10.82 neighbors at mean distance 0.610. These data compare well with those of the a phase [18] and to measurements on the ico phase [19].

(6)

For a

=

0.55, (d+ a2 - 4 aiT

-,15)m

=

0.894 whereas

(1

+

a2-4a /

0)112

=

0.900.

144

8. Calculated diffractions.

We give below the calculated diffractions, for both X rays and neutrons. The intensities were evaluated on a sample made of 5 504 Al and 1 363 Mn, involving 110 complete Mackay icosahedra. It was obtained by a straightforward use of the cut method generating the model presented in this paper. The X diffraction amplitudes are given in electronslA3 whereas the neutrons amplitudes are normalized so that the central peak gives 100. The reflections are indexed, following Cahn and Gratias, by two integers N and M which are quadratic forms in 6D with integer entries, commuting with the icosahedral group. The multiplicities of the orbits are given by 1£. These results should be compared to the experimental diffractions for instance in [20]. Table IV.

-

X and neutron

diffraction amplitudes.

Debye-Waller factor DW exp (- Bk2j4) is introduced in the calculated amplitudes comparison with observed diffractions. A best fit through a X2test [20b] A static

=

to allow

145

3. Section view of the atomic surfaces Mn, the thin ones to Al(Si).

Fig.

Fig.

-

4.

-

Observed

(black)

gave us an optimal B the calculated ones.

=

versus

3.04

along the (8, 66) plane.

calculated

(striped) X-ray

The bold segments

correspond to

intensities.

Â2. Figure 4 provides the observed diffraction intensities

versus

9. Conclusion. The model we have proposed in this paper for an ideal AIMnSi quasicrystal has quasiperiodic and icosahedral long range order. On an atomic scale, the local patterns are reminiscent of the known close crystalline a phase involving, e.g., disjoint Mackay icosahedra. Moreover, the nearest neighbors distances are exactly of the same order of magnitude as in the a phase and the predicted densities fit to the measured values within experimental error bars.

146

Notice that 66.6 % of the atoms are members of Ml’s. This is a slightly smaller proportion than in the a phase (78.3 %). Another difference between a and our model is the occurrence of Mn in the linking atoms whereas all the a Mn belong to Ml’s. This state of affairs is consistent with the discrepancy in the concentrations : 18.9 at % of Mn for a versus 21 % for Ico, and it forces our model to involve a few percents of nearest neighbor Mn (cf. Table II). Such nearest neighbor pairs, absent in the a phase, may give a structural origin to the particular magnetic properties observed in the Ico phase [21].

Acknowledgments. It is

a

pleasure

to thank D. Gratias for

frequent stimulating

discussions.

References

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[15] [16] [17] [18]

[21]

diffraction tells you where the atoms are, subm. to J. Phys. France. J. W., GRATIAS, D. and MOZER, B., J. Phys. France 49 (1988) 1225. CAHN, (b) BELLISENT, R., HIPPERT, F., MONOD, P., VIGNERON, F., Non uniform magnetism and properties of AlMnSi quasicrystal, to appear in Phys. Rev. B ; GARROCHE, P., private communication.

neutron

spin glass