Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quasicrystals as cluster aggregates


Quasicrystals are solids that exhibit symmetries long thought forbidden in nature. Since their discovery in a rapidly solidified Al–Mn alloy in 1984, the central issue in the field has been to understand why they form. Are they energetically stable compounds or stabilized by entropy? In recent years, major strides have been made in determining atomic structure, largely by direct imaging using advanced electron microscopy. One system is now known to be energetically stabilized, and quasicrystals are therefore firmly established as a new physical state of matter. They represent a unique packing of atomic clusters some tens of atoms in size, with substantial localized fluctuations, referred to as phasons. Understanding phasons may in future allow their unique macroscopic properties to be tailored for useful materials applications.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Morphologies of faceted single grains of icosahedral quasicrystals.
Figure 2
Figure 3: Generation of 1D quasiperiodic order from a 2D square lattice.
Figure 4: Atomic configurations commonly found in Cd6X intermetallic compounds.
Figure 5: Atomíc-resolutíon ADF-STEM images of decagonal Al72Ni20Co8 with cluster models.


  1. 1

    Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metalic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).

    CAS  Article  Google Scholar 

  2. 2

    Dubost, B., Lang, J.-M., Tanaka, M., Sainfort, P. & Audier, M. Large AlCuLi single quasicrystals with triacontahedral solidification morphology. Nature 324, 48–50 (1986).

    CAS  Article  Google Scholar 

  3. 3

    Tsai, A. P., Inoue, A. & Masumoto, T. A stable quasicrystal in Al-Cu-Fe system. Jpn J. Appl. Phys. 26, L1505–L1507 (1987).

    CAS  Article  Google Scholar 

  4. 4

    Ohashi, W. & Spaepen, F. Stable Ga-Mg-Zn quasi-periodic crystals with pentagonal dodecahedral solidification morphology. Nature 330, 555–556 (1987).

    CAS  Article  Google Scholar 

  5. 5

    Tsai, A. P. in Physical Properties of Quasicrystals (ed. Stadnik, Z. M.) 5–50 (Springer, 1999).

    Book  Google Scholar 

  6. 6

    Stephens, P. W. & Goldman, A. I. Sharp diffraction maxima from an icosahedral glass. Phys. Rev. Lett. 56, 1168–1171 (1986); ibid 57, 2331 (1986).

    CAS  Article  Google Scholar 

  7. 7

    Pauling, L. Apparent icosahedral symmetry is due to directed multiple twinning of cubic crystals. Nature 317, 512–514 (1986); ibid So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal. Phys. Rev. Lett. 58, 365–368 (1987).

    Article  Google Scholar 

  8. 8

    Levine, D. & Steinhardt, P. J. Quasicrystals: A new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984).

    CAS  Article  Google Scholar 

  9. 9

    Desiraju, G. R. In search of clarity. Nature 423, 485 (2003).

    CAS  Article  Google Scholar 

  10. 10

    Bak, P. Icosahedral crystals: Where are the atoms? Phys. Rev. Lett. 56, 861–864 (1986).

    CAS  Article  Google Scholar 

  11. 11

    Janssen, T. Crystallography of quasi-crystals. Acta Crystallogr. A 42, 261–271 (1986).

    Article  Google Scholar 

  12. 12

    Yamamoto, A. Crystallography of quasiperiodic crystals. Acta Crystallogr. A 52, 509–560 (1996).

    Article  Google Scholar 

  13. 13

    Elser, V. & Henley, C. L. Crystal and quasicrystal structures in Al-Mn-Si alloys. Phys. Rev. Lett. 55, 2883–2886 (1985).

    CAS  Article  Google Scholar 

  14. 14

    Audier, M. et al. Structural relationships in intermetallic compounds of the Al-Li-(Cu, Mg, Zn) system. Phil. Mag. B 60, 437–466 (1989).

    CAS  Article  Google Scholar 

  15. 15

    Hiraga, K., Sugiyama, K. & Ohsuna, T. Atomic cluster arrangements in cubic approximant phases of icosahedral quasicrystals. Phil. Mag. A 78, 1051–1064 (1998).

    CAS  Article  Google Scholar 

  16. 16

    Janot, C. & de Boissieu, M. Quasicrystals as a hierarchy of clusters. Phys. Rev. Lett. 72, 1674–1677 (1994).

    CAS  Article  Google Scholar 

  17. 17

    Ishihara, K. N. & Yamamoto, A. Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns. Acta Crystallogr. A 44, 508–516 (1988).

    Article  Google Scholar 

  18. 18

    Bendersky, L. Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis. Phys. Rev. Lett. 55, 1461–1463 (1985).

    CAS  Article  Google Scholar 

  19. 19

    Hiraga, K. in Advances in Imaging and Electron Physics (ed. Hawks P. W.) 37–98 (Academic, London, 1998).

    Google Scholar 

  20. 20

    Abe, E., Takakura, H. & Tsai, A. P. Ho arrangement in the Zn6Mg3Ho icosahedral quasicrystal studied by atomic-resolution Z-contrast STEM. J. Electron Microsc. 50, 187–195 (2001).

    CAS  Google Scholar 

  21. 21

    Beeli, C. & Horiuchi, S. The structure and its reconstruction in the decagonal Al70Mn17Pd13 quasicrystal. Phil. Mag. B 70, 215–240 (1994).

    CAS  Article  Google Scholar 

  22. 22

    Tsuda, K. et al. Structure of Al-Ni-Co decagonal quasicrystals. Phil. Mag. A 74, 697–708 (1996).

    CAS  Article  Google Scholar 

  23. 23

    Penrose, R. The role of aesthetics in pure and applied mathematical reserach. Bull. Inst. Math. Applic. 10, 266–271 (1974).

    Google Scholar 

  24. 24

    Burkov, S. Structure model of the Al-Cu-Co decagonal quasicrystal. Phys. Rev. Lett. 67, 614–617 (1991); ibid Modeling decagonal quasicrystals: random assembly of interpenetrating decagonal clusters. J. Phys. 2, 695–706 (1992).

    CAS  Article  Google Scholar 

  25. 25

    Henley, C. L. in Quasicrystals: The State of the Art (eds DiVincenzo, D. & Steinhardt, P. J.) 429–524 (World Scientific, Singapore, 1991).

    Book  Google Scholar 

  26. 26

    Joseph, D., Ritsch, S. & Beeli, C. Distinguishing quasiperiodic from random order in high-resolution TEM images. Phys. Rev. B 55, 8175–8183 (1997).

    CAS  Article  Google Scholar 

  27. 27

    Ritsch, S. et al. Highly perfect decagonal Al-Co-Ni quasicrystal. Phil. Mag. Lett. 74, 99–106 (1996).

    CAS  Article  Google Scholar 

  28. 28

    Abe, H. et al. Atomic short-range order in an Al72Ni20Co8 decagonal quasicrystal by anomalous X-ray scattering. Jpn J. Appl. Phys. 39, L1111–L1114 (2000).

    CAS  Article  Google Scholar 

  29. 29

    Pennycook, S. J. & Boatner, L. A. Chemically sensitive structure imaging with a scanning transmission electron microscope. Nature 336, 565–567 (1988).

    CAS  Article  Google Scholar 

  30. 30

    Pennycook, S. J. & Jesson, D. E. High-resolution Z-contrast imaging of crystals. Ultramicroscopy 37, 14–38 (1991); ibid High-resolution incoherent imaging of crystals. Phys. Rev. Lett. 64, 938–941 (1990).

    Article  Google Scholar 

  31. 31

    Saitoh, K. et al. Structural study of an Al72Ni20Co8 decagonal quasicrystal using the high-angle annular dark-field method. Jpn J. Appl. Phys. 36, L1400–1402 (1997).

    CAS  Article  Google Scholar 

  32. 32

    Yan, Y., Pennycook, S. J. & Tsai, A. P. Direct imaging of local chemical disorder and columnar vacancies in ideal decagonal Al-Ni-Co quasicrystals. Phys. Rev. Lett. 81, 5145–5148 (1998).

    CAS  Article  Google Scholar 

  33. 33

    Steinhardt, P. J. et al. Experimental verification of the quasi-unit-cell model of quasicrystal structure. Nature 396, 55–57 (1998); correction Nature 399, 84 (1999).

    CAS  Article  Google Scholar 

  34. 34

    Gummelt, P. Construction of Penrose tilings by a single aperiodic protoset. Geometriae Dedicata 62, 1–17 (1996).

    Article  Google Scholar 

  35. 35

    Steinhardt, P. J. & Jeong, H.-C. A simpler approach to Penrose tiling with implications for quasicrystal formation. Nature 382, 433–435 (1996).

    Article  Google Scholar 

  36. 36

    Abe, E. et al. Quasi-unit cell model for an Al-Ni-Co ideal quasicrystal based on clusters with broken tenfold symmetry. Phys. Rev. Lett. 84, 4609–4612 (2000).

    CAS  Article  Google Scholar 

  37. 37

    Yan, Y. & Pennycook, S. J. Chemical ordering in Al72Ni20Co8 decagonal quasicrystals. Phys. Rev. Lett. 86, 1542–1545 (2001).

    CAS  Article  Google Scholar 

  38. 38

    Mihalkovic, M. et al. Total-energy-based prediction of a quasicrystal structure. Phys. Rev. B 65, 104205 (2002).

    Article  Google Scholar 

  39. 39

    Goedecke, T. et al. Isothermal sections of phase equilibria in the Al-AlCo-AlNi system. Z. Metallkd. 89, 687–698 (1998).

    CAS  Google Scholar 

  40. 40

    Hume-Rothery, W. Researches on the nature, properties, and conditions of formation of intermetallic compounds, with special reference to certain compounds of tin.-I.-V. J. Inst. Met. 36, 295–361 (1926).

    Google Scholar 

  41. 41

    Ritsch, S. et al. The existence regions of structural modifications in decagonal Al-Co-Ni. Phil. Mag. Lett. 78, 67–75 (1998).

    CAS  Article  Google Scholar 

  42. 42

    Hiraga, K. et al. Structural characteristics of Al-Co-Ni decagonal quasicrystals and crystalline approximants. Mater. Trans. 42, 2354–2367 (2001).

    CAS  Article  Google Scholar 

  43. 43

    Bak, P. Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys. Phys. Rev. Lett. 54, 1517–1519 (1985).

    CAS  Article  Google Scholar 

  44. 44

    Levine, D. et al. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett. 54, 1520–1523 (1985).

    CAS  Article  Google Scholar 

  45. 45

    Socolar, T., Lubensky, T. & Steinhardt, P. J. Phonons, phasons and dislocations in quasicrystals. Phys. Rev. B 34, 3345–3360 (1986).

    CAS  Article  Google Scholar 

  46. 46

    Urban, K. & Feuerbacher, M. Structurally complex alloy phases. J. Non-Cryst. Solids 334–335, 143–150 (2004).

    Article  Google Scholar 

  47. 47

    Lubensky, T. C. et al. Distortion and peak broadening in quasicrystal diffraction patterns. Phys. Rev. Lett. 57, 1440–1443 (1986).

    CAS  Article  Google Scholar 

  48. 48

    Jaric, M. V. & Nelson, D. R. Diffuse scattering from quasicrystals. Phys. Rev. B 37, 4458–4472 (1988).

    CAS  Article  Google Scholar 

  49. 49

    Ishii, Y. Phason softening and structural transitions in icosahedral quasicrystals. Phys. Rev. B 45, 5228–5239 (1992).

    CAS  Article  Google Scholar 

  50. 50

    de Boissieu, M. et al. Diffuse scattering and phason elasticity in the AlPdMn icosahedral phase. Phys. Rev. Lett. 75, 89–92 (1995).

    CAS  Article  Google Scholar 

  51. 51

    Coddens, G. & Steurer, W. Time-of–flight neutron-scattering study of phason hopping in decagonal Al-Co-Ni quasicrystals. Phys. Rev. B 60, 270–276 (1999).

    CAS  Article  Google Scholar 

  52. 52

    Francoual, S. et al. Dynamics of phason fluctuations in the i-AlPdMn quasicrystal. Phys. Rev. Lett. 91, 225501 (2003).

    CAS  Article  Google Scholar 

  53. 53

    Edagawa, K., Suzuki, K. & Takeuchi, S. High resolution transmission electron microscopy observation of thermally fluctuating phasons in decagonal Al-Cu-Co. Phys. Rev. Lett. 85, 1674–1677 (2000).

    CAS  Article  Google Scholar 

  54. 54

    Abe, E., Pennycook, S. J. & Tsai, A. P. Direct observation of a local thermal vibration anomaly in a quasicrystal. Nature 421, 347–350 (2003).

    CAS  Article  Google Scholar 

  55. 55

    Takakura, H., Yamamoto, A. & Tsai, A. P. The structure of decagonal Al72Ni20Co8 quasicrystal. Acta Crystallogr. A 57, 576–585 (2001).

    CAS  Article  Google Scholar 

  56. 56

    Abe, H. et al. Anomalous Debye-Waller factor associated with an order-disorder transformation in an Al72Ni20Co8 decagonal quasicrystal. J. Phys. Soc. Jpn 72, 1828–1831 (2003).

    CAS  Article  Google Scholar 

  57. 57

    Cervellino, A., Haibach, T. & Steurer, W. Structure solution of the basic decagonal Al-Co-Ni phase by the atomic surfaces modeling method. Acta Crystallogr. B 58, 8–33 (2002).

    Article  Google Scholar 

  58. 58

    Keppens, V. et al. Localized vibrational modes in metallic solids. Nature 395, 876–878 (1998).

    CAS  Article  Google Scholar 

  59. 59

    Cohn, J. L. et al. Glasslike heat conduction in high-mobility crystalline semiconductors. Phys. Rev. Lett. 82, 779–782 (1999).

    CAS  Article  Google Scholar 

  60. 60

    Mizutani, U., Takeuchi, T. & Sato, H. Atomic structure determination, electronic structure calculations and interpretation of electron transport properties of various 1/1–1/1–1/1 approximants. J. Phys. Condens. Matter 14, R767–R788 (2002).

    CAS  Article  Google Scholar 

  61. 61

    Macia, E. May quasicrystals be good thermoelectric materials? Appl. Phys. Lett. 77, 3045–3047 (2000).

    CAS  Article  Google Scholar 

  62. 62

    Gomez, C. P. & Lindin, S. Comparative structural study of the disordered MCd6 quasicrystal approximants. Phys. Rev. B 68, 024203 (2003).

    Article  Google Scholar 

  63. 63

    Fisher, I. R. et al. Growth of large-grain R – Mg – Zn quasicrystals from the ternary melt (R = Y, Er, Ho, Dy and Tb). Phil. Mag. B 77, 1601–1615 (1998).

    CAS  Article  Google Scholar 

  64. 64

    Tsai, A. P., Guo, J. Q., Abe, E., Takakura, H. & Sato, T. J. A stable binary quasicrystal. Nature 408, 537–538 (2000).

    CAS  Article  Google Scholar 

  65. 65

    Weickenmeier, A. & Kohl, H. Computation of absorptive form factors for high-energy electron diffraction. Acta Crystallogr. A 47, 590–597 (1991).

    Article  Google Scholar 

Download references


We are grateful to A. P. Tsai, K. Saitoh, P. J. Steinhardt, H.-C. Jeong and H. Takakura for collaboration, on which the present article is based. We also thank T. J. Sato, M. Widom, C. L. Henley, M. Miharcovic, W. Steurer, M. de Boissieu, A. Yamamoto, N. Tanaka, K. Ishizuka and H. Inui for valuable comments and discussions. E.A. acknowledges support from the CREST-JST 'Fundamental properties of quasicrystals' project (1996-2001, Project leader: A. P. Tsai). Y.Y. and S.J.P. acknowledge support from the US department of Energy under contract numbers DE-AC36-99GO10337 and DE-AC05-00OR22725.

Author information



Corresponding authors

Correspondence to Eiji Abe or Stephen J. Pennycook.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Abe, E., Yan, Y. & Pennycook, S. Quasicrystals as cluster aggregates. Nature Mater 3, 759–767 (2004).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing