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Polyhedral Clusters
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
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 1 + 12 + 20
(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, 1 + 44 vertices (atom positions).
Bergman, G., Waugh, J. L. T. & Pauling, L. Crystal structure of the intermetallic compound Mg 32(Al, Zn)49 and related phases. Nature 169 (1952) 1057-1058;
The crystal structure of the metallic phase Mg32(Al, Zn)49, Acta Cryst. 10 (1957) 254-259.
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 12 + 20 + 12 = 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) 936-945.
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 Mg 32(Al, Zn)49 and related phases. Nature 169 (1952)
1057-1058; The crystal structure of the metallic phase Mg32(Al, Zn)49, Acta Cryst. 10 (1957) 254-259.
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) 136-142.
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) 437-488.
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) 181-194.
Lord, E. A., Ranganathan, S. & Kulkarni, U. D. Tilings, coverings, clusters & quasicrystals. Curr. Sci. 78 (2000) 64-72.
Lord, E. A., Ranganathan, S. & Kulkarni, U. D. Quasicrystals: tiling versus clustering. Phil. Mag. A 81 (2001) 2645-2651.
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) 401-413.
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 ‘fiverings’ around the
edges of the inner
tetrahedron. Gives a
cluster of 17
tetrahedra, 14 vertices.
Then...
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.
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).
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, 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) 335-343.
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.
Frank and Kasper [1958]
considered coordination shells
in the intermetallic phases now
known as
Frank-Kasper phases
12 vertices
20 tetrahedra
0 disclinations
15 vertices
26 tetrahedra
3 disclinations
The coordination shells of
atoms in these alloys are
triangulated, containing either
12, 14, 15 or 16 atoms. They
are therefore polytetrahedral
structures.
14 vertices
24 tetrahedra
2 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) 184-190.
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) 1916-1918.
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 vertexsharing 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) 15-36.
Kreiner, G. & Schäpers, M. A new description of Samson’s Cd 3Cu4 and a model of icosahedral i-CdCu. J. Alloys and Compounds 259 (1997) 83-114.
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) 123-131.
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) 265-270.
Li, X. Z. & Frey, F. Structure of Al-Mn decagonal quasicrystal. II. A high-dimensional description. Acta Cryst. B51 (1995) 271-275.
Lord, E. A. & Ranganathan, S. The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001) 531-539.
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) 531-539.
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) 593-619.
Lord, E. A. & Ranganathan, S. The Gummelt decagon as a quasi unit cell. Acta Cryst. A57 (2001) 531-539.