Why is it dark at night? This is Olber's Paradox. The modern explanation for it is that the Universe is expanding; but in 1922 C. V. L. Charlier suggested another, ingenious solution: the Universe might be infinite and hierarchical, with galaxies of galaxies of galaxies. There would be no privileged centre and all viewpoints would be quasi-equivalent, yet the density of this Universe would tend to zero with increasing scale. A similar idea might apply to the microcosmos of atomic arrangements: clusters of clusters are an alternative to strict crystalline arrangements, and could form a new type of condensed matter. On page 376 of this issue1, Hubert et al. report a structure with this theme — clustered icosahedral particles of boron suboxide.
The mathematical simplicity of infinite crystal lattices has beguiled crystallographers, with its absolute identity of lattice points. The classification system of crystals is based on the 230 symmetries allowed by that assumption — that an infinite number of asymmetric units are to be arranged so that their surroundings are absolutely identical. The assumption of crystallinity is necessary for the recovery of phase information in single-crystal X-ray diffraction, which is the principal tool of crystallography, so sharp diffraction patterns can be interpreted in terms of infinite crystals. But hierarchy could offer an alternative to lattice repetition, in providing an assembly of atoms with an infinite number of almost identical, or quasi-equivalent, sites.
This principle of quasi-equivalence was formulated by Caspar and Klug2 when they considered how polio virus (icosahedral particles of 60 subunits) could form crystals of cubic symmetry3 — as indicated by their diffraction pattern. The solution found by nature for polio is surprisingly like Linus Pauling's structure of Mg32(Al,Zn)49: the units are locally icosahedral clusters arranged on a translation lattice.
Perhaps this should not be surprising: identical units do not have to have identical surroundings, and natural materials are physical and not mathematical objects. Biological structures such as collagen are predominantly hierarchical in the same way as rope: fibres are twisted into strands, which are then twisted into bigger strands, and so on for several levels.
Hierarchy has now also appeared as a building principle in a class of inorganic materials, the quasicrystals. These are solids with fivefold symmetry (as betrayed by their diffraction patterns) — a symmetry impossible for a conventional crystal.
F. C. Frank noted that the densest packing of 12 spheres around a central sphere is icosahedral, and J. D. Bernal, following Charlier, asked whether this rule could be continued hierarchically. The problem is to find further rules to fill in the gaps so as to keep the overall density finite. Model-building experiments produced an icosahedral shell packing4 which, for the first few layers, was close to a packing of 13 icosahedral clusters, each of 13 spheres — and indeed β-rhombohedral boron was found in 1970 to contain such B12(B12)12units.
Another way to make icosahedra is found in multiply-twinned particles of silver and gold, where 20 face-centred-cubic regions are joined, with slight distortion, into icosahedra. These have been observed5 by electron microscopy from about 1964, and many other icosahedral clusters have since been found. Their size is limited by the increasing strain as successive layers are added, the spacing of spheres in the layers being some 5% greater than in close packing, so a transition from icosahedron to cuboctahedron will probably occur at a certain size.
The first theory of hierarchical packing came from the tiling of pentagons in two dimensions. Consideration of how the gaps were to be filled in led to Penrose tiling6 in two and three dimensions — a mathematical pattern that has the geometrical properties required of a quasicrystal. In three dimensions, Penrose tiles are obtuse and acute rhombohedra with angles of 116.6° and 63.4°.
Significantly, the 20 tetrahedra making up the multiple twin in gold and silver must have inter-edge angles at the centre of the cluster of 63.43° (instead of 60° for regular tetrahedra). And now Hubert et al.1 have found that boron suboxide crystals (B6O) are rhombohedral, with an inter-edge angle of 63.1°, and that these crystals occur as icosahedral twins. Twenty rhombohedral unit cells fit together snugly at a common vertex without dislocations at the interfaces, and the structure can continue outwards indefinitely, particles of some 1012 atoms being observed (Fig. 1).
Without dislocated grain boundaries, glide planes in this boron suboxide are locked and so the particles are very hard, promising technical applications. And these aggregates are far larger (up to 20 m) than those observed hitherto for multiply-twinned particles. B6O and various other boron compounds show marked tendencies towards icosahedral packing and it was thought, before quasicrystals of Al6Mn were reported, that boron would be the best candidate for forming a three-dimensional Penrose tiling, as the unit cell of B6O is the Penrose acute rhombohedron.
These boron suboxide particles are not quasicrystals, but they are an important step away from the 230 space groups towards a more general type of structure.
True quasicrystals can probably also be described as icosahedral clusters, themselves clustered icosahedrally in hierarchical levels, the gaps being filled by the overlapping of these clusters7. Quasicrystals are a further step away from conventional crystals, because they have many centres of local icosahedral symmetry, whereas a boron suboxide particle has only one.
As more varied structures appear — especially to the electron microscope, which does not depend on the assumption that many copies of the same unit can form only crystals — we can escape from the preconceptions engendered by the immense success of X-ray single-crystal structure analysis. We must expect many still more varied structures to lie outside the austere dominion of classical crystallography.
Hubert, H.et al. Nature 391, 376–378 (1998).
Caspar, D. L. D. & Klug, A. Cold Spring Harb. Symp. Quant. Biol. 27, 1–24 (1962).
Finch, J. T. & Klug, A. Nature 183, 1709–1714 (1959).
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Ogawa, S. & Ino, S. J. Vac. Sci. Tech. 6, 527-534 (1969); Adv. Epitaxy Endotaxy 183-226 (1971).
Mackay, A. L. Phys. Bull. Nov., 495-497 (1976).
Steinhardt, P. J. & Jeong, H.-C. Nature 382, 431–433 (1996).
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Journal of Physics: Condensed Matter (2008)
Journal of Physics: Condensed Matter (2008)
Journal of the Physical Society of Japan (2006)