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. 352356
Section 2.5.3.5. Symmetry determination of 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.
The first decagonal quasicrystal was found by Bendersky (1985) in an alloy of Al–Mn using the electrondiffraction technique. This phase has periodic order parallel to the tenfold axis, like ordinary crystals, but has quasiperiodic longrange 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, P10_{5}/m and P10_{5}/mmc, have been proposed for the alloy by Bendersky (1986) and by Yamamoto & Ishihara (1988), 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 spacegroup 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 Al_{65}Cu_{15}Co_{20} (Tsai et al., 1989a), Al_{65}Cu_{20}Co_{15} (He et al., 1988) and Al_{70}Ni_{15}Co_{15 }(Tsai et al., 1989b). However, spacegroup determination was still difficult due to their poor quasicrystallinity.
Tsai et al. (1989c) succeeded in producing a metastable but goodquality decagonal quasicrystal of Al_{70}Ni_{15}Fe_{15}. This alloy was found to be the first decagonal quasicrystal that could tolerate symmetry determination using CBED. The space group was determined to be by Saito et al. (1992).
Fig. 2.5.3.26(a) shows a CBED pattern of Al_{70}Ni_{15}Fe_{15} 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. 2.5.3.26(a) by an inversion were observed when the illuminated specimen area was changed, indicating the existence of inversion domains. Table 2.5.3.15 shows possible pentagonal and decagonal point groups, which are constructed by analogy with the trigonal and hexagonal point groups (Saito et al., 1992).

It can be seen that the point groups that satisfy the observed symmetry 5m in the projection approximation are 52, 5m and . Point group 52 is a possibility because the horizontal twofold rotation axis is equivalent to the vertical mirror plane in the projection approximation. Figs. 2.5.3.26(b) and (c) were taken with beam incidences A and B, respectively, as denoted in Fig. 2.5.3.26(a). Mirror symmetry perpendicular to the c axis is seen in Fig. 2.5.3.26(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 remain as possibilities. Fig. 2.5.3.26(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 , the point group of the alloy is determined to be . 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 Al_{70}Ni_{15}Fe_{15} was determined to be (Saito et al., 1992) by full use of the potential of CBED. This is the first quasicrystal with a noncentrosymmetric space group. Highresolution electronmicroscope images revealed that the quasicrystal is composed of specific pentagonal atom clusters 2 nm in diameter (Tanaka et al., 1993). Darkfield 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).
Quasicrystals of Al_{70}Ni_{10+x}Fe_{20−x} (0 ≤ x ≤ 10) were investigated by CBED and transmission electron microscopy (Tanaka et al., 1993). 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 Al_{70}Ni_{20}Fe_{10} taken at an incidence along the c axis shows tenfold rotation symmetry (Fig. 2.5.3.27a). CBED patterns taken at incidences A and B (shown in Fig. 2.5.3.27a) exhibit two mirror symmetries parallel and perpendicular to the c axis (Figs. 2.5.3.27b and c). Thus, the point group of this phase is determined to be 10/mmm. Fig. 2.5.3.27(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 oddorder reflections along the c* axis. This indicates the existence of a 10_{5} screw axis and a cglide plane. No other reflection absences were observed, implying the lattice type to be primitive. Therefore, the space group of Al_{70}Ni_{20}Fe_{10} is determined to be centrosymmetric P10_{5}/mmc. It was found that the alloys with 0 ≤ x ≤ 7.5 belong to the noncentrosymmetric space group and those with 7.5 < x ≤ 15 belong to the centrosymmetric space group P10_{5}/mmc, keeping the specific polar structure of the basic clusters unchanged.
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 P10_{5}/mmc. However, highangle annular darkfield (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, 1999). 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).
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). 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). There are seven point groups in the decagonal system (Table 2.5.3.15). However, only two point groups, and 10/mmm, and two space groups, and P10_{5}/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) and Steurer (2004), and to a review of more theoretical aspects by Yamamoto (1996).
References
International Tables for Crystallography (2005). Vol. A. SpaceGroup Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.Bendersky, L. A. (1985). Quasicrystal with onedimensional 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 convergentbeam 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 Al_{65}Cu_{20}Mn_{15}, Al_{65}Cu_{20}Fe_{15}, Al_{65}Cu_{20}Co_{15} or Al_{65}Cu_{20}Ni_{15}. 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 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.
Saito, M., Tanaka, M., Tsai, A. P., Inoue, A. & Masumoto, T. (1992). Space group determination of decagonal quasicrystals of an Al_{70}Ni_{15}Fe_{15} alloy using convergentbeam electrondiffraction. Jpn. J. Appl. Phys. 31, L109–L112.
Saitoh, K., Tanaka, M. & Tsai, A. P. (2001). Structural study of an Al_{73}Ni_{22}Fe_{5} decagonal quasicrystal by highangle annular darkfield 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 Al_{72}Ni_{20}Co_{8} decagonal quasicrystal using the highangle annular darkfield method. Jpn. J. Appl. Phys. 36, L1400–L1402.
Saitoh, K., Tsuda, K., Tanaka, M. & Tsai, A. P. (1999). Structural study of an Al_{70}Ni_{15}Fe_{15} decagonal quasicrystal using highangle annular darkfield scanning transmission electron microscopy. Jpn. J. Appl. Phys. 38, L671–L674.
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.
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 convergentbeam electron diffraction. Ultramicroscopy, 21, 245–250.
Tanaka, M., Tsuda, K. & Saitoh, K. (1996). Convergentbeam 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. NonCryst. 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 quasicrystal 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 Al_{70}Ni_{15}Fe_{15}. 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.