Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 354-356   | 1 | 2 |

Section Decagonal quasicrystals

M. Tanakaf 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]).


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.
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.
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.
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., 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. (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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