Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 352-354   | 1 | 2 |

Section Icosahedral quasicrystals

M. Tanakaf Icosahedral quasicrystals

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Penrose (1974[link]) demonstrated that a two-dimensional 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[link]) 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 three-dimensional space-filling method was later completed by Ogawa (1985[link]). These studies, however, remained a matter of design or geometrical amusement until Shechtman et al. (1984[link]) discovered an icosahedral symmetry presumably with long-range structural order in an alloy of Al6Mn (nominal composition) using electron diffraction. Since then, the term quasicrystalline order, a new class of structural order with no translational symmetry but long-range structural order, has been coined. Levine & Steinhardt (1984[link]) showed that the quasilattice produces sharp diffraction patterns and succeeded in reproducing almost exactly the diffraction pattern obtained by Shechtman et al. (1984[link]) using the Fourier transform of a quasiperiodic icosahedral lattice. When analysing X-ray and electron-diffraction 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[link]). A quasicrystal is produced by the intersection of the six-dimensional crystal with an embedded three-dimensional hyperplane (the cut-and-projection 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[link]) prepared such a less-strained quasicrystalline Al71Mn23Si6 alloy and determined its point group. They obtained fairly good zone-axis 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 [m\bar 3\bar 5]. Figs.[link](a)–(f) show three pairs of CBED patterns taken from an area about 100 nm thick and about 3 nm in diameter of an Al74Mn20Si6 quasicrystal at an accelerating voltage of 60 kV (Tanaka, Terauchi & Sekii, 1987[link]). This quasicrystal was found to have much better ordering than Al71Mn23Si6. 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.[link]b, 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.


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Three pairs of ZOLZ [(a), (c) and (e)] and HOLZ [(b), (d) and (f)] CBED patterns taken at 60 kV from an area of Al74Mn20Si6 about 3 nm in diameter and about 10 nm thick (Tanaka, Terauchi, Suzuki et al., 1987[link]). Symmetries are (a) 10mm, (b) 5m, (c) 6mm, (d) 3m, (e) 2mm and (f) 2mm.

The whole pattern of Fig.[link](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.[link](b), formed by HOLZ reflections, shows a fivefold rotation symmetry and a type of mirror plane, the resultant symmetry being 5m. Figs.[link](c) and (d) show symmetries 6mm and 3m, respectively. Figs.[link](e) and (f) show symmetry 2mm. There are two icosahedral point groups, 235 and [m\bar 3\bar 5] (see Table[link] in IT A, 2005[link]). The former is noncentrosymmetric with no mirror symmetry but the latter is centrosymmetric. Table[link] 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 incident-beam directions are omitted because they can be seen in Table[link]. The whole-pattern symmetries observed for better-quality images of Al74Mn20Si6 have confirmed the result of Bendersky & Kaufman (1986[link]), i.e. the point group [m\bar 3\bar 5]. Fig.[link](a) shows a zone-axis CBED pattern taken at an electron incidence along the threefold axis. Figs.[link](b) and (c) show ±DPs taken when tilting the incident beam to excite a low-order 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 2R directly proves that the quasicrystal is centrosymmetric, again confirming the point group as [m\bar 3\bar 5]. 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 [Pm\bar 3\bar 5].

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Diffraction groups and CBED symmetries for two icosahedral point groups

Point groupDiffraction groupBPWPDP±DP
235 5mR 5m 5 1 1
m2 1
(Projection) 5m1R 10mm 5m 2 = 1R 1
2mvm2 1
[m\bar 3 \bar 5] 10RmmR 10mm 5m 1 2R
m2 2Rm2
mv 2Rmv
(Projection) 10mm1R 10mm 10mm 2 21R
2mvm2 21Rmv

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CBED patterns of Al74Mn20Si6 taken with an electron incidence along the threefold axis. (a) Zone-axis pattern showing symmetry 3m. (b,c) ±DP showing translational symmetry or 2R, indicating that the quasicrystal is centrosymmetric.

Quasicrystals of Al–Mn alloys have been produced by the melt-quenching method and are thermodynamically metastable. Tsai et al. (1987[link]) discovered a stable icosahedral phase in Al65Cu20Fe15. This alloy has larger grains and is much better quality with less phason strain than Al74Mn20Si6. 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 face-centred (F) lattice. All the icosahedral quasicrystals known to date belong to the point group [m\bar 3\bar 5]; none with the noncentrosymmetric point group 235 have been discovered.


International Tables for Crystallography (2005). Vol. A. Space-Group 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 convergent-beam 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 3-dimensional 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 long-range 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 convergent-beam electron diffraction. Ultramicroscopy, 21, 245–250.
Tsai, A. P., Inoue, A. & Masumoto, T. (1987). A stable quasi-crystal in Al–Cu–Fe system. Jpn. J. Appl. Phys. Lett. 26, L1505–L1507.

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