AMORPHOUS STRUCTURAL MODELS USING REGULAR TESSELLATION OF CURVED SPACES

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AMORPHOUS STRUCTURAL MODELS USING

REGULAR TESSELLATION OF CURVED SPACES

J. Sadoc

To cite this version:

J. Sadoc.

AMORPHOUS STRUCTURAL MODELS USING REGULAR TESSELLATION

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JOURNAL DE Ph'ISIQUE CoZZoque C8, supplgment au n08, Tome 41, aoat 1980, page C8-326

AMORPHOUS STRUCTURAL MODELS U S I N G REGULAR TESSELLATION OF CURVED SPACES

J. Sadoc

Laborafoire de Physique d e s S o l i d e s asso&& au C.N.R.S., Universite' Paris-Sud

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BZtiment 510, 91405 Orsay, France

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INTRODUCTION

The existence of amorphous material is related to the presence of a well defined local order uncompatible with a crystalline periodfcit;-. So, to understand the structure of amorphous systems, a first step is the study of the local order wi- thout respect to the strain coming from the need oL filling the space. For example, we have shown the importance of the icosahedral structure in amorphous close packing structures''), however complete space filling with icosahedra is impossible. A fruitful approach of the geometrical study in amorphous structures is to look at a kind of non-euclidian 3-d space, which will be able to be tessellated with a studied local order. The mapping of the non-euclidian space on the cartesiav space is the next step.

This mapping introduces defects in the structure. In a previous paperC2), we have generalize2 some concepts of crystallography to curved spaces and described how they are usable for the amorphous structure description. In this paper, we would like to show some applications of this method to real structures.

TETRAHEDRAL PACKING

It is well known that tiling of the space with only tetrahedra is impossible : it is possibie to put 5 tetrahedra together around one edge, but it remains a small unfilled dihedral angle (fig I).

If the space is curved with a positive curvature, this small angle disappears. Tessellation of a curved space with tetrahedra is possible. The obtained figure is a polytope (which is equivalent in the 4-d space to a polyhedron in the euclidian 3-d space). The polytope is inscribed on a 3-sphere defined in 4-d euclidian space. Its radius is

1.6180 if the tetrahedron edge length is one. It has 600 cells and 120 vertices. Each vertex is surrounded by 12 connected vertices in icosahedral positions. If each vertex is replaced by an atom, one obtains a very regular close packing structure

(although in curved space). The R.D.F. of this structure is presented of fig 2-a. On this R.D.F., peaks are Dirac peaks, because the structure is strictly regular. To obtain an amorphous structure, it requires to map the 3-sphere on the 3-d euclidian space. Such a mapping will necessarily intro- duce non-homogeneous distortions in lengths. A 123

F i g 1

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F i v e t e t r a h e d r a w i t h a common e d g e l e a v e

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a s m a l l u n f i l l e d d i h e d r a l a n g l e . F i g 2

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a ) RDF o f t h e r e g u l a r 6 0 8 - c e l l p o l y t o p e .

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The e d g e l e n g t h i s 0 = 2 . 5 A , t h e f i r s t i n t e r - a t o m i c d i s t a n c e is 1.0160, i f it i s d e f i n e d i n the c u r v e d s p a c e .

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b ) ZDF o f the c e n t r a l a t o m o f the model ( f i g 3 ) a f t e r mapping o n t h e r e a l s p a c e . L e n g t h d i s t o r t i o n s b r o a d e n D i r a c p e a k s .

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atom aggregate is obtained, limited by a surface which is the image, after mapping, of cuts defined on the 3-sphere. (Pealing an orange i; a practical example of such a cut on a 2-sphere). This surface is stellated and looks like a sea-urchin (fig 3). The whole space filling is achieved by adding new replicas of the polytope map close to the first one. This induces two kinds of defects :

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elastic distortions which keep the local topology unchanged ;

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walls, image of cut surfaces, where local order is destroyed. (Perhaps these walls can be relaxed and give rise to parallel disclination lines).

The fig 2-b shows the R.D.F. of the central atoms of the sea-urchin (fig 3). We observe the ef- fect of distortions induced by mapping on the peaks. A general discussion of tetrahedral packing is presented in ref 3.

APPLICATION TO DEFERENT STWCTUIIES

Starting from the 120 vertex polytope, it is possible to describe or predict some structures. This is achieved in the same way used to obtain ar- chimedian polyhedra : putting vertices in the center of faces or edges, and only keeping the new vertices. With polytope, there is one more possibility: to put vertices at a center of a cell.

AMORPHOUS Si STRUCTURE MODELLING

For example, a new polytope is obtained by replacing vertices by tetrahedron centers. This is a regular polytope (dual of the last one), with 600 vertices and 120 dodecahedral cells. Each vertex

is surrounded by 4 connected vertices in tetrahedral positions. This polyrope can be a model for a-Si structure, if atoms are replaced on all vertices. All Si atoms are surrounded by 4 atoms. Bonds between Si atoms along edges b f the polytope describe 5 membered rings. This structure is exactly the old structure presented by Grigorovici (4) with the dodecahedral "amorphon".

It is also possible to have an other polytope for an a-Si structure modelling with only 6 membered rings. This polytope is obtained by the same rule, which allows to get diamond structure starting-from f.c.c. structure : to put new point in the center of some tetrahedral sites. 17e start: from the 600-cell polytope and we put new vertices in the center of one of the five tetrahedra surrounding each edge. We have now 120 vertices (the old

Fig 3

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120 atom model obtained by mapping of the polytope on the real space. The 3-sphere has been cut in four lobes. Three of these four lobes are visible on the figure.

Fig 4

In Laves phase, TM atoms are on vertices of truncated

tetrahedra. RE atoms are in centers. These truncated tetrahedra are joined by a common hexagonal face in staggered configuration (a)

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In the dodecahedra1 structure, they are joined in

b

eel ipsed configuration (b)

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c8-328 JOURNAL DE PHYSIQUE

ones) and 60015 = 120 new vertices. Placing atoms on these 240 vertices and mapping on the 3-d space gives an a-Si model with only 6-mexbered rings. AMORPHOUS LAVES PRASE S

The Lavesphase is a structure often observed for alloys between small and large atoms. For example : rare earth (R E) and transition metal (TM) with the composition RE-TM2. Experimentally, it is possible to prepare these alloys in the amorphous state for various compositions. So, it is important to know if there are structures in curved spaces able to model amorphous alloys with a local order similar to the Lave phase local order.

In the Lave phase, RE atoms are disposed like Si atoms in a diamond structure. Holes in this structure are filled by TM atoms groupad by 4 as tetrahedra. RE are coordinated with 4 RE and 12TM atoms ; Ttf atoms are coordinated with 6 TI1 and

6 atoms.

DODECAHEDRAL STRUCTURE (D. STRUCTURE)

Using the 120-cell polytope, which is a packing of dodecahedra, it is possible to insert a centered icosahedron inside each dodecahedron. If RE atoms are replaced on the dodecahedron vertices, and TM atoms on the centered icosahedron vertices, we have built a structure with an attractive local order :

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RE atoms are exactly disposed like in Lave phases

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there is two kinds of TlI atoms :

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TF! atoms coordinated with 5 RE atoms

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and 7 M atoms in distorted icosahedral positions.

There are 600 RE atoms and 1 560 (13 x 120) TI1

atoms ; the composition is RE0.277TM0.722. If RE-RE first distance is d

RE-RE = 1 , other distances are

d ~ = 0.740 or 0.775, dT14-RE ~ - ~ = 0.951. For exam- 0

ple, if TI1 is nickel with dT eF,-Il = 2. 5 A,

"

3.4 A

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d ~ = 3.2 A. These distances - ~ ~ are similar to distances observed in RE-TM alloys. CENTERED TETRAHEDRAL STRUCTUFX (C.T. STRiiCTURE)

Using the second polytope describing a-Si structure, it is always possible to insert M atoms in holes and to replace RE atoms in place of Si atoms. A new structure is obtainea with interatomic first disiances dRE-* = I

,

= 0.608

,

dm-RE = 0.93 in a curved space. There are 240 RE atoms and 480 TM atoms. The composition is RE-TM2 Like in Lave phase. Nevertheless, the geometrical configuration is different from the crystalline

one, or from the D-structure. In these two cases, TM atoms are on vertices of a truncated tetrahedron with regular hexagonal faces (fig 4). The two structures are obtained by a packing of truncated tetrahedra joined by co-n hexagonal faces in eclipsed (D. structure) or staggered (Lave phase) configuration. For the C.T. structure, truncated tetrahedra are twisted and hexagonal faces become not plane. TI4 atoms are coordinated with only 4 T1i atoms and 6 RE atoms. Their coordination number is lower than in Lavesphase. In this structure, the first distance is greater than in Laves phase, but it remains smaller than in Haucke phase (CaCu5). This is due to the high symmetry of the 6-membered ring of RE atoms surrounding a TM atom. InLavesphase, there is place for 4 TPI atoms on one side of this ring and for 2 Tfi atoms on the other side. In the CT structure, it is necessary to have the same number on the two sides. From this geometrical study, we suggest that the proba- bility to have a local order similar to the Laves phase one in amorphous alloy is greater near the composition REoez7

ni0.73' OTHER EXAMPLES

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G_ctahed_ron-_and-L-etr_ak~dron~~acking

It is possible to put vertices on the midpoint of each edge of the 600-cell polytope. Each vertex of this polytope being surrounded by 12 vertices, the midpoints of edges around one vertex build an icosahedron. Cells of the 600-cell polytope are tetrahedra : the midpoint of edges of these tetrahedra build octahedron. If wekeep,as new vertices, only the midpoint of edges of the 600-cell poly-

tope, we obtain a new polytope which is a packing of 120 icosahedra with 600 octahedra. If A atoms are replaced on icosahedron vertices and B atoms in the center of icosahedra, a close structure with 120 B atoms and 720 A atoms is obtained. B atoms are coordinated with 12 A atoms in icosahedral positions. A atoms are coordinated with 10 B and 2 A atoms drawing a polyhedron with 5 square and 10 triangular faces (fig 5).

First distances between atoms are equals. This structure can possibly describe alloys with the composition A.857 having the same atomic size and a tendancy to heterocoordination. A good example is the a-Gd-Y alloy.

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If the interatomic distance is 1, space curvature is 0.309 (112 x 1.618). This suggests a compa- rison with other structures obtaine6 by packing of tetrahedra with octahedra. If there are only tetrahedra in the packing, the structure is described in a more curved space (600-cell polytope with curvature 0.618 = 111.618).

If there is 1 octahedron for 2 tetrahedra in the packing (f .c.c. structure), curvature is zero (euclidian space). The (T4.0) structure is an intermediate case between f.c.c. structure and the amorphous tetrahedral close packing.

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unre&ular-ter~ahedrsL-~asking

We have centered cells of the 600-cell polytope to get the 120 cell polytope, or we have centered edges to get the T4-0 polytope. It is also possible to center faces. These new vertices draw an other polytope which is a packing of tetrahedra

(centers of 4 faces of a tetrahedron form a smaller tetrahedron) with icosidodecahedron (fig 6) (which is the polyhedron drawn by the 30 centers of faces having a common vertex).

It is pbssible to build an atomic aggregate with atoms in icosidodecahedral positions. start from 13 atoms building an icosahedron ; we add new atoms, layer by layer, on triangular faces. The first layer is a dodecahedron, the secnild a big icosahedron, the third (obtained by adding atoms on the triangular faces formed by 2 vertices of the dodecahedron and a vertex of the big icosehe- dron) is the f osidodecahedron. This cluster contains 13 + 12 + 20 + 30 = 75 atoms;

All sites are tetrahedra, but irregular. By packing together these clusters in a way to have atoms of the last layer disposed on the vertices of the described polytope, a new tetrahedral structure is obtained. It contains 6 600 atoms in a space with a low curvature (1/4x1..618 = 0.1545). This structure, including length distortions befo- re the mapping on the euclidian space, is useful if very big models are required.

CONCLUSION

We have described some structures which account for amorphous metallic alloys.'~he inte- rest of this method is to be able to predict possible structures, to suggest some particular compositions more stable in amorphous form, and to have

rules for classification of amorphous structures. If a perfect structure is defined, it is possible to define a defect as divergence from the refe- rence order.

We have shown that these defects, induced by mapping of a curved space on the euclidian spkce, are of two kinds :

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topological defects due to cuts,

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length distortions.

But we have absolutely excluded, from this discussion, chemical disorder and concentration fluctua- tions. These two effects may relax strains induced by mapping and stabilize amorphous structures.

Fig 6

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The icosidodecahedron. There are

3 0 vertices, 1 2 pentagonal faces and 2 0 triangular faces.

BIBLIOGRAPHY

(1) J.F. SADOC, J. DIWIIER, A. GUINIER, J.N.C.S., 12, 46 (1973).

(2) M. KLEMAN, J.F. SADOC, Journ. de Phys.,

&,

L-59 (1979).

(3) submitted to J.N.C.S.