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Polyhedral Clusters

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**Regular tetrahedra will not pack to fill Euclidean space**

Regular tetrahedra will not pack to fill Euclidean space. Slight deviations from strict regularity will allow tilings of 3D Euclidean space by tetrahedra. In materials with polytetrahedral structure, the atoms are in a closely packed arrangement, at the vertices of a packing of ‘almost regular’ tetrahedra. 5 regular tetrahedra sharing an edge. The gaps between faces are less than 1.5°. Slight distortion to close the gaps gives a five-ring of tetrahedra in face contact 20 regular tetrahedra sharing a vertex. A regular icosahedron consists of 20 almost regular tetrahedra

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**The Bergman cluster built from almost regular tetrahedra.**

(1) A tetrahedron placed on every face of an icosahedron (which itself can be thought of as a packing of 20 tetrahedra). Number of vertices (2) Thirty more tetrahedra, over the icosahedron edges, completing the rings of five tetrahedra around the icosahedron edges. No new vertices. (3) Finally, a five-ring can be placed in each of the concavities. 12 new vertices. Total: 130 tetrahedra, vertices (atom positions). Bergman, G., Waugh, J. L. T. & Pauling, L. Crystal structure of the intermetallic compound Mg32(Al, Zn)49 and related phases. Nature 169 (1952) ; The crystal structure of the metallic phase Mg32(Al, Zn)49, Acta Cryst. 10 (1957) Bergman, G., Waugh, J. L. T. & Pauling, L. Crystal structure of the intermetallic compound Mg32(Al, Zn)49 and related phases. Nature 169 (1952) ; The crystal structure of the metallic phase Mg32(Al, Zn)49, Acta Cryst. 10 (1957)

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**Inner icosahedral shell, 12 atoms; **

Observe how the configuration of the outermost 32 atoms in the Bergman cluster approximate to a rhombic triacontahedron. The cluster is, accordingly, sometimes called the Pauling triacontahedron. The Samson cluster. Twenty truncated tetrahedra (Friauf polyhedra) packed in an icosahedral arrangement. With a central ‘atom’ in each, and atoms at the vertices, we get a cluster of 104 atoms: Inner icosahedral shell, 12 atoms; centers of Friaufs forming a dodecahedron, 20 atoms; a larger icosahedral shell (centers of the white depressions in the picture), 12 atoms. This gives a Bergman cluster = 44 This is surrounded by the outer shell, a truncated icosahedron, 60 atoms. Samson, S. Complex cubic A6B compounds. II. The crystal structure of Mg6Pd. Acta Cryst. B28 (1972)

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With only slight deformation, a truncated icosahedron {5, 62} can be enclosed inside a truncated octahedron {4, 62} so that all 60 vertices lie in the faces of the truncated octahedron. The description of the structure of Mg32(Al, Zn)49 given by Bergman, Waugh & Pauling is essentially a bcc tiling of space by truncated octahedra, each enclosing a 104-atom Samson cluster. Observe that in this arrangement every atom of the 60-atom truncated icosahedral shell is shared by two of the samson clusters. Half the vertices of the truncated octahedra are also occupied. Bergman, G., Waugh, J. L. T. & Pauling, L. Crystal structure of the intermetallic compound Mg32(Al, Zn)49 and related phases. Nature 169 (1952) ; The crystal structure of the metallic phase Mg32(Al, Zn)49, Acta Cryst. 10 (1957)

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The structure of the R-phase is very similar to that discovered by Bergman et al. for Mg32(Al, Zn)49. A Samson cluster lies inside a triacontahedron, with two of the 60 vertices of the outer shell lying in each triacontahedral face. No distortion is needed. Observe how the fivefold vertices of the triacontahedron complete the five-rings of tetrahedra so that the 104-atom cluster Samson cluster is augmented to 116-atom icosahedron. Audier, M., Pannetier, J., Leblanc, M., Janot, C., Lang, J. M. & Dubost, B. An approach to the structure of quasicrystals: A single crystal X-Ray and neutron diffraction study of the R-Al5CuLi3 phase. Physica B, (1988) Audier, M., Janot, Ch., de Boissieu, M. & Dubost, B. Structural relationships in intermetallic compounds of the Al-Li-(Cu, Mg, Zn) system. Phil. Mag. B 60 (1989)

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In an R-phase the triacontahedral clusters are centered at bcc positions. Neighbouring clusters share faces along [100] directions and overlap along [111] directions. As can be seen in the picture, the overlap region is an oblate rhombohedron. Also shown is the way the threefold vertices of a triacontahedral cluster coincide with vertices of the smaller Pauling triacontahedron inside neighbouring clusters. The R-phase is structurally closely similar to the quasicrystalline T-phase. Audier, M. & Guyot, P. The structure of the icosahedral phase atomic decoration of the basic cells. In: Quasicrystalline Materials: Proc. ILL/ Codest Workshop, Grenoble 1988 (Ch. Janot & J. M. Dubois, eds) World Scientific (1988) Lord, E. A., Ranganathan, S. & Kulkarni, U. D. Tilings, coverings, clusters & quasicrystals. Curr. Sci. 78 (2000) Lord, E. A., Ranganathan, S. & Kulkarni, U. D. Quasicrystals: tiling versus clustering. Phil. Mag. A 81 (2001)

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**A block of Friaufs arranged as in a Laves phase.**

A ring of five Friauf polyhedra is a fundamental building block in a many complex crystalline structures. Below we illustrate a clustering of these units identified by Samson in the highly complex phase of Mg2Al3 A block of Friaufs arranged as in a Laves phase. More Friaufs attached, producing six ‘five-rings’. Samson, S. The crystal structure of the phase of Mg2Al3. Acta Cryst. 19 (1965)

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The -brass cluster The Bergman cluster is a polytetrahedral structure systematically built around a single vertex. The -brass cluster starts from a single tetrahedron. A tetrahedron on each face of a central tetrahedron. The stella quadrangula: 5 tetrahedra, 8 vertices. 12 more tetrahedra completing the ‘five-rings’ around the edges of the inner tetrahedron. Gives a cluster of 17 tetrahedra, 14 vertices. Then...

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The five-rings around the edges of the four outer tetrahedra of the stella quadrangula are completed, giving a cluster of 41 tetrahedra, 26 vertices. This models the fundamental atomic cluster of the -alloys 16 more tetrahedra can be inserted without adding any more vertices. The -brass cluster can thus be described as 4 interpenetrating icosahedra sharing a common tetrahedron.

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**The augmented -brass cluster**

Add 40 more tetrahedra. This introduces 12 more vertices. (indicated by orange triangles) These correspond to positions of atoms in neighbouring 26-atom clusters. We get a cluster of 97 tetrahedra with 38 vertices. In the -alloys, the clusters are centered on a bcc lattice. Remove the green tetrahedra. The augmented cluster is then a cluster of four icosahedra in face contact with a central tetrahedron and with each other. This configuration is the Pearce cluster. Pearce, P. Structure in Nature is a Strategy for Design. MIT press (1978). E A Lord & S Ranganathan. The -brass structure and the Boerdijk–Coxeter helix. J. Non-Crystalline Solids (2004) (to be published).

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In space filling by tetrahedra deviations from strict regularity are necessary because five regular tetrahedra around an edge must be slightly stretched to fit. The distortions build up unless the stretching is counteracted by having some edges shared by six tetrahedra, squashed to fit. These edges are disclination lines. They constitute a disclination network. The polytope {3, 3, 5} Perfectly regular tetrahedra can be packed together in a spherical space S3. On a hypersphere embedded in Euclidean space E4 the vertices are those of the regular polytope {3, 3, 5}. It has 120 vertices, 720 edges, 1200 equilateral triangular faces and 600 regular tetrahedral cells. There are 5 tetrahedra around every edge and twenty around every vertex, forming a regular icosahedron. Sadoc and Mosseri [1999] have developed an approach to understanding polytetrahedral structures. Their methods are based on introducing disclinations into the polytope {3, 3, 5} to ‘flatten’ it to fit into 3D Euclidean space. Sadoc, J. F. & Mosseri, R. Geometrical Frustration. Cambridge Univ. Press (1999). Coxeter, H. S. M. Regular Polytopes. Macmillan (1963); Dover (1973). Lord, E. A & Ranganathan, S. Sphere packing, helices and the polytope {3,3,5}. EPJ D 15 (2001)

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**Frank and Kasper [1958] considered coordination shells in the intermetallic phases now known as**

Frank-Kasper phases The coordination shells of atoms in these alloys are triangulated, containing either 12, 14, 15 or 16 atoms. They are therefore polytetrahedral structures. 12 vertices 20 tetrahedra 0 disclinations 14 vertices 24 tetrahedra 2 disclinations 15 vertices 26 tetrahedra 3 disclinations 16 vertices 28 tetrahedra 4 disclinations Frank, F. C. & Kasper, J. S. Complex alloy structures regarded as sphere packings. I. definitions and basic principles. Acta Cryst. 11 (1958)

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Some remarkable structures that occur in nature can be understood as packings of nearly regular tetrahedra and octahedra. If a regular octahedron is placed on every face of a regular icosahedron, the gap between faces of neighbouring octahedra is only 2.87. Slight deformation brings them into contact. The concavities in this structure can be filled by ‘five’rings’ of tetrahedra. The process can continue, adding at each stage a layer of octahedra and tetrahedra (purple and grey, respectively, in the figures below). The close packed arrangements of spheres centered on the vertices of these polyhedral packings are Mackay clusters Left: the 54-atom Mackay icosahedron. Shells are: Inner icosahedron, 12 vertices; icosidodecahedron, 30 vertices; outer icosahedron, 12 vertices. Right: the 146-sphere cluster. Mackay, A. L. A dense non-crystalline packing of equal spheres. Acta Cryst. 15 (1962)

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The i3 unit – identified by Kreiner and Franzen as a structural unit in a large number of complex alloys with trigonal and hexagonal symmetry. Observe how two octahedra neatly fill the void between the three vertex-sharing icosahedra. Below: another Kreiner et al. structural unit. Three interpenetrating icosahedra. Left: the Kreiner & Franzen L unit. A tetrahedral cluster of four vertex-sharing icosahedra Kreiner, G. & Franzen, H. F. A new cluster concept and its application to quasi-crystals of the i-AlMnSi family and closely related crystalline structures. J. Alloys and Compounds 221 (1995) Kreiner, G. & Schäpers, M. A new description of Samson’s Cd3Cu4 and a model of icosahedral i-CdCu. J. Alloys and Compounds 259 (1997)

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A cluster of five octahedra, with tetrahedral symmetry, gives a model of the pyrochlore unit. (In the mineral pyrochlore and related minerals the four outer octahedra would have central atoms.) The Kreiner and Franzen L-unit can be seen as a ‘pyrochlore unit’ with four icosahedra packed around it. Nyman, H. & Andersson, S. the pyrochlore structure and its relatives. J. Solid State Chem. 26 (1978)

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**Right: one of the towers in the decagonal phase of Al-Mn.**

The large clusters with icosahedral symmetry, of Bergman or Mackay type, are important structural units in icosahedral quasicrystals and their approximants. In general, decagonal quasicrystals can be thought of as towers or rods of 12- or 13-atom icosahedra, double icosahedra, ‘five-rings’ (‘decahedra’) and pentagonal antiprisms. These rods are packed in an aperiodic arrangement. A structural unit consisting of three towers, superimposed on a Gummelt decagon. A portion of the structure of a typical decagonal quasicrystal, viewed along the periodic axis. Right: one of the towers in the decagonal phase of Al-Mn. Li, X. Z. Structure of Al-Mn decagonal quasicrystal. I. A unit-cell approach. Acta Cryst. B51 (1995) Li, X. Z. & Frey, F. Structure of Al-Mn decagonal quasicrystal. II. A high-dimensional description. Acta Cryst. B51 (1995) Lord, E. A. & Ranganathan, S. The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001)

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The structure of decagonal Al-Mn showing how the towers are bonded to each other by octahedral linkages. Lord, E. A. & Ranganathan, S. The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001)

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**The 19-atom double icosahedron; a structural subunit in many decagonal phases.**

The structure of the decagonal phase of Al-Co. Towers of double icosahedra linked by octahedra. Cockayne, E. & Widom, M. Structure and phason energetics of Al-Co decagonal phases. Phil. Mag. A 77 (1998) Lord, E. A. & Ranganathan, S. The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001)

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