target="_blank" rel="nofollow" href="#ulink_c2207465-24cc-5afb-a0de-96fc28d70d7f">a The alternating axis is a combination of rotation (n‐fold) and reflection perpendicular to the rotation axis. It is little used in crystallography.
1.2.2 Quasicrystals
The discovery of a new state of matter, the quasicrystalline state, by Schechtman and colleagues, published in 1984 (and which led to the Nobel Prize in Chemistry in 2011) appeared at first sight to violate the rules concerning allowable rotational symmetries in crystal lattices. From their single‐crystal diffraction patterns, rotational symmetries such as n = 5 but also n = 10 and 12 were observed whereas, as shown in Fig. 1.4(c), a regular crystal lattice exhibiting fivefold rotational symmetry cannot exist. The answer to this conundrum is that quasicrystals do not have regularly repeating crystal structures based on a single unit cell motif. Instead, they have fully ordered but non‐periodic arrays constructed from more than one motif or building block.
Elegant examples of quasisymmetry are found in so‐called Penrose tiling, as shown in Fig. 1.5. In this example, space is filled completely by a combination of red and blue diamonds; such a tiling pattern has many local areas of fivefold symmetry but the structure as a whole is not periodic, does not exhibit fivefold symmetry and a regular repeat unit cannot be identified. Quasicrystals have since been discovered in a wide range of alloy systems and also in organic polymer and liquid crystal systems; they have been discovered in Nature in an Al–Cu–Fe alloy named icosahedrite that was believed to have been part of a meteorite and had existed on Earth for billions of years. It is probably just a matter of time before they are discovered also in inorganic oxide materials, natural or synthetic.
Figure 1.5 Two‐dimensional Penrose tiling constructed by packing together two different sets of parallelograms. C. Janot, Quasicrystals: A Primer, Oxford University Press (1997). Penrose was co‐recipient of the 2020 Nobel Prize in physics, in a totally different area to quasicrystals and Penrose tiling, for ‘discovery that black hole formation is a robust prediction of the general theory of relativity’.
Figure 1.6 Hypothetical twinned structure showing fivefold symmetry.
Adapted from J. M. Dubois, Useful Quasicrystals, World Scientific Publishing Company (2005).
In the early days of work on quasicrystals, an alternative explanation for possible fivefold symmetry was based on twinning, as shown schematically in Fig. 1.6. Five identical crystalline segments are shown, each of which has twofold rotational symmetry in projection. Pairs of crystal segments meet at a coherent interface or twin plane in which the structures on either side of the twin plane are mirror images of each other. The five crystal segments meet at a central point which exhibits fivefold symmetry as a macroscopic element of point symmetry but the individual crystal segments clearly do not exhibit any fivefold symmetry. Schechtman showed conclusively that twinning such as shown in Fig. 1.6 could not explain the quasicrystalline state.
The discovery of quasicrystals without long range periodicity and an identifiable unit cell, has forced the International Union of Crystallography to reconsider what is meant by a ‘crystal’. Since well‐prepared quasicrystals have highly ordered structures, many of which are thermodynamically stable and give sharp single crystal diffraction patterns, quasicrystals need to have a home in a wider definition of crystallinity. The requirement for crystallinity to be associated with long range periodicity has served the scientific community well for nearly a century; however, it is now necessary to include quasicrystals that are highly ordered but lack long range periodicity. The terms ‘classical crystal’ and ‘aperiodic crystal’ have been suggested to distinguish between crystals that do, and do not, exhibit long range periodicity.
The consequences that aperiodicity within ordered structures has for the electronic structure and optoselectronic properties of quasicrystals are still being evaluated. Periodicity has provided an important starting point in the development of theoretical concepts such as Brillouin zones and the band theory of solids. It has also underpinned theories of dislocations and explanations about the mechanical properties of crystalline materials, especially metals and alloys. However, it is probably fair to say that, at present, although quasicrystals are much more than a scientific curiosity, they do not, as yet, fit into the general panorama of mainstream solid state sciences, perhaps because of the challenges and difficulties associated with evaluating and presenting aperiodic crystallography in terms of higher dimensional space.
1.2.3 Mirror symmetry
A mirror plane, m, exists when two halves of, for instance, a molecule can be interconverted by carrying out the imaginary process of reflection across the mirror plane. The silicate tetrahedron possesses six mirror planes, one of which, running vertically and perpendicular to the plane of the paper, is shown in Fig. 1.7(a). The silicon and two oxygens, 1 and 2, lie on the mirror plane and are unaffected by reflection. The other two oxygens, 3 and 4, are interchanged on reflection. A second mirror plane lies in the plane of the paper; for this, Si and oxygens 3, 4 lie on the mirror but oxygen 2, in front of the mirror, is the image of oxygen 1, behind the mirror.
The photograph in Fig. 1.7(d) shows the British comedian Harry Worth creating a mirror image of half of his body by posing at the end of a high street shop window. Half of the picture is of his body and half is its mirror image. It therefore exhibits perfect mirror symmetry which is never achieved by human bodies in reality!
1.2.4 Centre of symmetry and inversion axes
A centre of symmetry,
The inversion axis,
