Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 352-356   | 1 | 2 |

Section Symmetry determination of quasicrystals

M. Tanakaf Symmetry determination of 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. Decagonal quasicrystals

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The first decagonal quasicrystal was found by Bendersky (1985[link]) in an alloy of Al–Mn using the electron-diffraction technique. This phase has periodic order parallel to the tenfold axis, like ordinary crystals, but has quasiperiodic long-range structural order perpendicular to the tenfold axis. The diffraction peaks were indexed by one vector parallel to the tenfold axis and four independent vectors pointing to the vertices of a decagon. Thus, the decagonal quasicrystal is described in terms of a regular crystal in five dimensions.

Two space groups, P105/m and P105/mmc, have been proposed for the alloy by Bendersky (1986[link]) and by Yamamoto & Ishihara (1988[link]), respectively. However, owing to the low quality of the specimens, CBED examination of the alloy could not determine whether the point group is 10/m or 10/mmm. Furthermore, identification of the space-group symmetry was not possible because observation of dynamical extinction caused by the screw axis and/or the glide plane was difficult. The Al–M (M = Mn, Fe, Ru, Pt, Pd, …) quasicrystals found at an early stage were thermodynamically metastable. Subsequently, thermodynam­ically stable decagonal phases were discovered in the ternary alloys Al65Cu15Co20 (Tsai et al., 1989[link]a), Al65Cu20Co15 (He et al., 1988[link]) and Al70Ni15Co15 (Tsai et al., 1989[link]b). However, space-group determination was still difficult due to their poor quasicrystallinity.

Tsai et al. (1989[link]c) succeeded in producing a metastable but good-quality decagonal quasicrystal of Al70Ni15Fe15. This alloy was found to be the first decagonal quasicrystal that could tolerate symmetry determination using CBED. The space group was determined to be [P\overline{10}m2] by Saito et al. (1992[link]).

Fig.[link](a) shows a CBED pattern of Al70Ni15Fe15 taken with an incidence parallel to the fivefold axis (c axis). The pattern clearly exhibits fivefold rotation symmetry and a type of mirror symmetry, the total symmetry being 5m. The slowly varying intensity distribution in the discs indicates that the pattern is formed by the interaction between ZOLZ reflections. Thus, the projection approximation should be applied to the analysis of the pattern. Patterns that were related to Fig.[link](a) by an inversion were observed when the illuminated specimen area was changed, indicating the existence of inversion domains. Table[link] shows possible pentagonal and decagonal point groups, which are constructed by analogy with the trigonal and hexagonal point groups (Saito et al., 1992[link]).

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Pentagonal and decagonal point groups constructed by analogy with trigonal and hexagonal point groups

This table is taken from Saito et al. (1992[link]) with the permission of the Japan Society of Applied Physics.

[Scheme scheme1] [Scheme scheme2]
[Scheme scheme3]
[Scheme scheme4] [Scheme scheme5]
[Scheme scheme6] [Scheme scheme7]
[Scheme scheme8] [Scheme scheme9]
[Scheme scheme10]
[Scheme scheme11] [Scheme scheme12]

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CBED patterns of metastable Al70Ni15Fe15 taken from a 3 nm diameter area. (a) Electron incidence along the decagonal axis: symmetry 5m. (b) Electron incidence along direction A indicated in (a): symmetry m perpendicular to the decagonal axis. (c) Electron incidence along direction B indicated in (a): symmetry 2mm. This alloy is found to be noncentrosymmetric.

It can be seen that the point groups that satisfy the observed symmetry 5m in the projection approximation are 52, 5m and [\overline{10}m2]. Point group 52 is a possibility because the horizontal twofold rotation axis is equivalent to the vertical mirror plane in the projection approximation. Figs.[link](b) and (c) were taken with beam incidences A and B, respectively, as denoted in Fig.[link](a). Mirror symmetry perpendicular to the c axis is seen in Fig.[link](b) and (c). Since the mirror symmetry requires a twofold rotation axis or a mirror plane perpendicular to the c axis, point groups 52 and [\overline{10}m2] remain as possibilities. Fig.[link](c) exhibits symmetry 2mm. Mirror symmetry parallel to the c axis requires the existence of a mirror plane parallel to the axis (a twofold rotation axis is not possible because the fivefold rotation axis already exists.). Since the mirror plane does not exist in point group 52 but does exist in [\overline{10}m2], the point group of the alloy is determined to be [\overline{10}m2]. Examination of the ordinary diffraction patterns of the alloy revealed that the lattice type is primitive with a periodicity of 0.4 nm in the c direction and no dynamical extinction was observed. Thus, the space group of Al70Ni15Fe15 was determined to be [P\overline{10}m2] (Saito et al., 1992[link]) by full use of the potential of CBED. This is the first quasicrystal with a noncentrosymmetric space group. High-resolution electron-microscope images revealed that the quasicrystal is composed of specific pentagonal atom clusters 2 nm in diameter (Tanaka et al., 1993[link]). Dark-field microscopy revealed the existence of inversion domains with an antiphase shift of c/2, the polarity being perpendicular to the c direction (Tsuda et al., 1993[link]).

Quasicrystals of Al70Ni10+xFe20−x (0 ≤ x ≤ 10) were investigated by CBED and transmission electron microscopy (Tanaka et al., 1993[link]). The change in space group takes place at x = 7.5 upon a sudden decrease of the size of the inversion domains or a rapid mixing of the atom clusters with positive and negative polarities. As a result, the average structure becomes centrosymmetric. A CBED pattern of Al70Ni20Fe10 taken at an incidence along the c axis shows tenfold rotation symmetry (Fig.[link]a). CBED patterns taken at incidences A and B (shown in Fig.[link]a) exhibit two mirror symmetries parallel and perpendicular to the c axis (Figs.[link]b and c). Thus, the point group of this phase is determined to be 10/mmm. Fig.[link](d) shows a CBED pattern taken by slightly tilting the incident beam to the c* direction from incidence A. Dynamical extinction lines (arrowheads) are seen in the odd-order reflections along the c* axis. This indicates the existence of a 105 screw axis and a c-glide plane. No other reflection absences were observed, implying the lattice type to be primitive. Therefore, the space group of Al70Ni20Fe10 is determined to be centrosymmetric P105/mmc. It was found that the alloys with 0 ≤ x ≤ 7.5 belong to the noncentrosymmetric space group [P\overline{10}m2] and those with 7.5 < x ≤ 15 belong to the centrosymmetric space group P105/mmc, keeping the specific polar structure of the basic clusters unchanged.


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CBED patterns of metastable Al70Ni20Fe10 taken from a 3 nm diameter area. (a) Electron incidence along the decagonal axis: symmetry 10mm. (b) Electron incidence along direction A indicated in (a): symmetry 2mm. (c) Electron incidence along direction B indicated in (a): symmetry 2mm. (d) Reflections 00l (l = odd) show dynamical extinction lines. This alloy is determined to have the centrosymmetric space group P105/mmc.

Another phase was found in the same alloys with 15 < x ≤ 17. This phase showed the same CBED symmetries as the phase with 7.5 < x ≤ 15. The space group of the phase was also determined to be P105/mmc. However, high-angle annular dark-field (HAADF) observations of the phase with 15 < x ≤ 17 showed that each atom cluster has only one mirror plane of symmetry (Saitoh et al., 1997[link], 1999[link]). This implies that the structure of the specific cluster is changed from that of the phase with 7.5 < x ≤ 15. The clusters are still polar but take ten different orientations, producing centrosymmetric tenfold rotation symmetry on average, which was confirmed by HAADF observations (Saitoh, Tanaka & Tsai, 2001[link]).

These three phases have been found for the similar alloys Al–M1–M2, where M1 = Ni and Cu, and M2 = Fe, Co, Rh and Ir (Tanaka et al., 1996[link]). Subsequently, decagonal quasicrystals were found in Al–Pd–Mn, Zn–Mg–RE (RE = Dy, Er, Ho, Lu, Tm and Y) and other alloy systems (Steurer, 2004[link]). There are seven point groups in the decagonal system (Table[link]). However, only two point groups, [\overline{10}m2] and 10/mmm, and two space groups, [P\overline{10}m2] and P105/mmc, are known reliably in real materials to date, though a few other point and space groups have been reported.

For further crystallographic aspects of quasicrystals, the reader is referred to the comprehensive reviews of Tsai (2003[link]) and Steurer (2004)[link], and to a review of more theoretical aspects by Yamamoto (1996[link]).


International Tables for Crystallography (2005). Vol. A. Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Bendersky, L. A. (1985). Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis. Phys. Rev. Lett. 55, 1461–1467.
Bendersky, L. A. (1986). Decagonal phase. J. Phys. Colloq. 47, C3, 457–464.
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.
He, L. X., Wu, Y. K. & Kuo, K. H. (1988). Decagonal quasicrystals with different periodicities along the tenfold axis in rapidly solidified Al65Cu20Mn15, Al65Cu20Fe15, Al65Cu20Co15 or Al65Cu20Ni15. J. Mater. Sci. Lett. 7, 1284–1286.
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.
Saito, M., Tanaka, M., Tsai, A. P., Inoue, A. & Masumoto, T. (1992). Space group determination of decagonal quasi-crystals of an Al70Ni15Fe15 alloy using convergent-beam electron-diffraction. Jpn. J. Appl. Phys. 31, L109–L112.
Saitoh, K., Tanaka, M. & Tsai, A. P. (2001). Structural study of an Al73Ni22Fe5 decagonal quasicrystal by high-angle annular dark-field scanning transmission electron microscopy. J. Electron Microsc. 50, 197–203.
Saitoh, K., Tsuda, K., Tanaka, M., Kaneko, K. & Tsai, A. P. (1997). Structural study of an Al72Ni20Co8 decagonal quasicrystal using the high-angle annular dark-field method. Jpn. J. Appl. Phys. 36, L1400–L1402.
Saitoh, K., Tsuda, K., Tanaka, M. & Tsai, A. P. (1999). Structural study of an Al70Ni15Fe15 decagonal quasicrystal using high-angle annular dark-field scanning transmission electron microscopy. Jpn. J. Appl. Phys. 38, L671–L674.
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.
Steurer, W. (2004). Twenty years of structure research on quasicrystals. Part 1. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Z. Kristallogr. 219, 391–446.
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.
Tanaka, M., Tsuda, K. & Saitoh, K. (1996). Convergent-beam electron diffraction and electron microscope studies of decagonal quasicrystals. Sci. Rep. RITU, A42, 199–205.
Tanaka, M., Tsuda, K., Terauchi, M., Fujiwara, A., Tsai, A. P., Inoue, A. & Masumoto, T. (1993). Electron diffraction and electron microscope study on decagonal quasicrytals of Al–Ni–Fe alloys. J. Non-Cryst. Solids, 153&154, 98–102.
Tsai, A. P. (2003). `Back to the future' – An account of the discovery stable quasicrystals. Acc. Chem. Res. 36, 31–38.
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.
Tsai, A. P., Inoue, A. & Masumoto, T. (1989a). A stable decagonal quasicrystal in the Al–Cu–Co system. Mater. Trans. Jpn. Inst. Met. 30, 300–304.
Tsai, A. P., Inoue, A. & Masumoto, T. (1989b). Stable decagonal Al–Co–Ni and Al–Co–Cu quasicrystals. Mater. Trans. Jpn. Inst. Met. 30, 463–473.
Tsai, A. P., Inoue, A. & Masumoto, T. (1989c). New decagonal Al–Ni–Fe and Al–Ni–Co alloys prepared by liquid quenching. Mater. Trans. Jpn. Inst. Met. 30, 150–154.
Tsuda, K., Saito, M., Terauchi, M., Tanaka, M., Tsai, A. P., Inoue, A. & Masumoto, K. (1993). Electron microscope study of decagonal quasicrystals of Al70Ni15Fe15. Jpn. J. Appl. Phys. 32, 129–134.
Yamamoto, A. (1996). Crystallography of quasiperiodic crystals. Acta Cryst. A52, 509–560.
Yamamoto, A. & Ishihara, K. N. (1988). Penrose patterns and related structures. II. Decagonal quasicrystals. Acta Cryst. A44, 707–714.

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