International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 352354
Section 2.5.3.5.1. Icosahedral quasicrystals
M. Tanaka^{f}

Penrose (1974) demonstrated that a twodimensional plane can be tiled with thin and fat rhombi to give a pattern with local fivefold rotational symmetries but with no translational symmetry. Mackay (1982) extended the tiling to three dimensions using acute and obtuse rhombohedra, which also resulted in the acquisition of local fivefold rotational symmetries and in a lack of translational symmetry. The threedimensional spacefilling method was later completed by Ogawa (1985). These studies, however, remained a matter of design or geometrical amusement until Shechtman et al. (1984) discovered an icosahedral symmetry presumably with longrange structural order in an alloy of Al_{6}Mn (nominal composition) using electron diffraction. Since then, the term quasicrystalline order, a new class of structural order with no translational symmetry but longrange structural order, has been coined. Levine & Steinhardt (1984) showed that the quasilattice produces sharp diffraction patterns and succeeded in reproducing almost exactly the diffraction pattern obtained by Shechtman et al. (1984) using the Fourier transform of a quasiperiodic icosahedral lattice. When analysing Xray and electrondiffraction data for a quasicrystal, the diffraction peaks can be successfully indexed by six independent vectors pointing to the vertices of an icosahedron. It was then found that the icosahedral quasicrystal can be described in terms of a regular crystal in six dimensions (e.g. Jarić, 1988). A quasicrystal is produced by the intersection of the sixdimensional crystal with an embedded threedimensional hyperplane (the cutandprojection technique).
Addition of several per cent of silicon to Al–Mn alloys caused a great increase in the degree of order of the quasicrystal. Bendersky & Kaufman (1986) prepared such a lessstrained quasicrystalline Al_{71}Mn_{23}Si_{6} alloy and determined its point group. They obtained fairly good zoneaxis CBED patterns that showed symmetries of 10mm, 6mm and 2mm in the ZOLZ discs and 5m, 3m and 2mm in HOLZ rings. From these results, they identified the point group to be centrosymmetric . Figs. 2.5.3.24(a)–(f) show three pairs of CBED patterns taken from an area about 100 nm thick and about 3 nm in diameter of an Al_{74}Mn_{20}Si_{6} quasicrystal at an accelerating voltage of 60 kV (Tanaka, Terauchi & Sekii, 1987). This quasicrystal was found to have much better ordering than Al_{71}Mn_{23}Si_{6}. The fact that Kikuchi bands are clearly seen in the HOLZ patterns and the profiles of the bands are symmetric with respect to their centre indicates (Figs. 2.5.3.24b, d and f) that the quasicrystal has sufficiently good quality or highly ordered atomic arrangements to perform reliable symmetry determination. Each pair of CBED patterns consists of a ZOLZ pattern and a HOLZ pattern. The former is produced solely by the interaction of ZOLZ reflections, showing distinct symmetries in several discs.
The whole pattern of Fig. 2.5.3.24(a), formed by ZOLZ reflections, exhibits a tenfold rotation symmetry and two types of mirror symmetry, the resultant symmetry being expressed as 10mm. The whole pattern of Fig. 2.5.3.24(b), formed by HOLZ reflections, shows a fivefold rotation symmetry and a type of mirror plane, the resultant symmetry being 5m. Figs. 2.5.3.24(c) and (d) show symmetries 6mm and 3m, respectively. Figs. 2.5.3.24(e) and (f) show symmetry 2mm. There are two icosahedral point groups, 235 and (see Table 10.1.4.3 in IT A, 2005). The former is noncentrosymmetric with no mirror symmetry but the latter is centrosymmetric. Table 2.5.3.14 shows the diffraction groups expected from these point groups with the incident beam parallel to the fivefold or tenfold axis, and their symmetries appearing in the WP, BP, DP and ±DP. Projection diffraction groups and their symmetries, in which only the interaction between ZOLZ reflections is taken into account, are given in the second row of each pair. Diffraction groups obtained for the other incidentbeam directions are omitted because they can be seen in Table 2.5.3.3. The wholepattern symmetries observed for betterquality images of Al_{74}Mn_{20}Si_{6} have confirmed the result of Bendersky & Kaufman (1986), i.e. the point group . Fig. 2.5.3.25(a) shows a zoneaxis CBED pattern taken at an electron incidence along the threefold axis. Figs. 2.5.3.25(b) and (c) show ±DPs taken when tilting the incident beam to excite a loworder strong reflection. The pattern of disc +G agrees with that of disc −G when the former is superposed on the latter with a translation of −2G. This symmetry 2_{R} directly proves that the quasicrystal is centrosymmetric, again confirming the point group as . The lattice type was found to be primitive and no dynamical extinction was observed. Thus, the space group of the alloy was determined to be .

Quasicrystals of Al–Mn alloys have been produced by the meltquenching method and are thermodynamically metastable. Tsai et al. (1987) discovered a stable icosahedral phase in Al_{65}Cu_{20}Fe_{15}. This alloy has larger grains and is much better quality with less phason strain than Al_{74}Mn_{20}Si_{6}. The discovery of this alloy greatly accelerated the studies of icosahedral quasicrystals. It was found that the lattice type of this phase and of some other Al–Cu–TM (TM = transition metal) alloys is different from that of Al–Mn alloys. That is, Al–Cu–TM alloys display many additional spots in diffraction patterns of twofold rotation symmetry. The patterns were indexed either by all (six) even or all (six) odd, or by a facecentred (F) lattice. All the icosahedral quasicrystals known to date belong to the point group ; none with the noncentrosymmetric point group 235 have been discovered.
References
International Tables for Crystallography (2005). Vol. A. SpaceGroup Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Bendersky, L. A. & Kaufman, M. J. (1986). Determination of the point group of the icosahedral phase in an Al–Mn–Si alloy using convergentbeam electron diffraction. Philos. Mag. B53, L75–L80.
Jarić, M. Y. (1988). Editor. Introduction to Quasicrystals, Vol. 1. New York: Academic Press.
Levine, D. & Steinhardt, P. J. (1984). Quasicrystals – a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480.
Mackay, A. L. (1982). Crystallography and the Penrose pattern. Physica (Utrecht), 114A, 609–613.
Ogawa, T. (1985). On the structure of a quasicrystal 3dimensional Penrose transformation. J. Phys. Soc. Jpn, 54, 3205–3208.
Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271.
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with longrange orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953.
Tanaka, M., Terauchi, M. & Sekii, H. (1987). Observation of dynamic extinction due to a glide plane perpendicular to an incident beam by convergentbeam electron diffraction. Ultramicroscopy, 21, 245–250.
Tsai, A. P., Inoue, A. & Masumoto, T. (1987). A stable quasicrystal in Al–Cu–Fe system. Jpn. J. Appl. Phys. Lett. 26, L1505–L1507.