During these two decades since the first report of an icosahedral quasicrystal, a large quantity of researches on quasicrystals has been accumulated in order to understand various aspects of the new class of material, i.e., their atomic structure, the origin of their stability, their specific physical properties and their possible applications. In this review, we reflect on the development of the researches on various aspects of quasicrystals, focusing mainly on pioneering contributions in Japan.

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Recently a software package for the structure analysis of quasicrystals has been released, giving a better environment for determining guasicrystal structures. Therefore we can analyze their structures if we know data collection and indexing methods and a theory of structure analysis. At this occasion, their structure analysis and its recent development have shortly been reviewed.

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Stable icosahedral quaicrystals (i-QC) have been identified in a large number alloys. Stability of the i-QCs has been understood in terms of Hume-Rothery rules i.e., atomic size factor and valency electron concentration (e/a) . The stable i-QCs are electron compounds which have strict e/a, and hence most stable i-QCs were discovered along the criterion of e/a. According to the structures of icosahedral clusters derived from the corresponding approximants, the stable i-QCs are classified into three classes ; Al-Mn-Si, Zn-Mg-Al and Cd-Yb. All the i-QCs commonly have their corresponding e/a, but the dependence of stability on the atomic size factor is different among the three classes. Qualitatively, the most relevant criterion for the formation of the stable i-QC is that the alloys should have a definite e/a. On top of this, a higher stability of the i-QC is expected when the atomic size factor is favored.

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Recent progress on a new group of icosahedral quasicrystals is reviewed. This group is characterized by Tsai-type atomic cluster as a local structural unit, which is different from Mackay and Bergman-types. This group can be regarded as largest including two Cd-based binary, seven Zn-based and ten Cd-based ternary stable phases as well as Cu- and Ag-based phases. They are all P-type, and some of them exhibit higher degree of structural perfection than the “perfect” Al-Pd-Mn quasicrystal. In the Tsai-type quasicrystals, there are substitution rules in the constituent elements. Hume-Rothery rules, namely the equality in e/a and in the ratio of atomic radii, play important role in the substitution rules.

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Atomic short-range order (SRO) relating to a phason is key to resolve phase stabilities in Al-Ni-Co decagonal quasicrystals. Complicated SRO was determined by anomalous X-ray scattering. At the same time, anisotropic distributions of diffuse scattering appeared around Bragg reflections. Moreover, the anomalous Debye-Waller factor was seen in an order-disorder phase transition at high temperature.

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Originating in the quasiperiodic order, quasicrystals have a special type of elastic degrees of freedom, termed as phason degrees of freedom. Quasicrystals are accompanied by the phason elastic field in addition to the phonon (conventional) elastic field. The elastic theory of quasicrystals is constructed in terms of the two types of elastic fileds. Then, the elasticity of quasicrystals comprises the three parts : pure phonon elasticity, pure phason elasticity and phonon-phason coupling. In this article, we review the elastic theory of quasicrystals and present our recent study on experimental evaluation of phonon-phason coupling constants in icosahedral quasicrystals.

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Symmetry of icosahedral quasicrystals of Cd-Mg-Yb and Cd-Yb has been investigated by the convergent-beam electron diffraction (CBED) method.
The CBED pattern symmetries of the Cd-based quasicrystals are not consistent with any icosahedral point groups, indicating that the structures of the alloys have no icosahedral symmetry. Selected-area electron diffraction patterns of the alloys show regular icosahedral symmetry, indicating that the alloys has a regular icosahedral lattice with a long-range quasiperiodic order. We consider that the structure of the Cd-based quasicrystals is formed by a quasiperiodic arrangement of a non-icosahedral atom-cluster with the same orientation.

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The structure of a W- (AlCoNi) approximant phase allows us to discuss a local atomic arrangement for a variety of Al-Co-Ni decagonal phases, by serving an ideal frame of the pentagonal columnar structure of 2 nm in diameter. The present approach also demonstrates that the combinatory analysis of the single crystal X-ray diffraction coupled with electron microscopy such as HAADF-STEM and HRTEM is one of powerful tools for the structural analysis of complicated structures.

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High pressure and low temperature X-ray diffraction study of Cd-Yb 1/1 crystalline approximant revealed that the innermost part of the atomic clusters of Cd4 tetrahedra exhibit various types of orientational ordering sensitively depending on pressure and temperature. Five ordered phases appeared in a P-T span up to 5.2 GPa and down to 10 K. The propagation direction of ordering alternated from [1 10] to ‹ 111› at about 1.0 GPa and again to [110] at 3.5-4.3 GPa. The primarily ordered phases that appeared by cooling to 210-250 K between 1-5.2 GPa further transformed to finely ordered ones at 120-155 K. Coexistence of two different types of interaction between Cd4 tetrahedra cause these complicated phase behavior.

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Al L-shell excitation spectra of Al-based quasicrystals of Al₇₅Cu₁₅V₁₀, Al₇₃Co₂₇ and Al₇₀Pd₂₀Mn₁₀ and related crystals were obtained from specified single crystalline areas by using a high energy-resolution electron energy-loss spectroscopy microscope. Intensity distributions of the spectra of not only the quasicrystalline materials but also related crystalline materials showed a presence of pseudogap at around the Fermi level. Onset energies of the spectra of the quasicrystals showed shifts to the higher energy side by 0.3 eV for Al₇₅Cu₁₅V₁₀ and Al₇₃Co₂₇ and 0.8 eV for Al₇₀Pd₂₀Mn 10 compared to that of aluminum. Those results show that a decrease of valence charge at Al sites is characteristic for Al-based quasicrystalline materials. It may be attributed to an increase of covalency in bonding nature of the quasicrystalline materials.

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First-principles calculation of electronic structures of approximant crystals is a reliable tool for studying cohesion mechanism and electronic properties of real quasicrystals. This article aims at explaining the first-principles calculation as methodology of quasicrystal reaseach. After brief introduction of approximant and methods of the first-principles calculations, we present several approaches, including the tight-binding method and the plane-wave-basis method, for Cd-M (M = Ca, Mg) binary systems.

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Al-based icosahedral quasicrystals and their corresponding approximants are characterized by large electrical resistivity, large thermoelectric power, and very small thermal conductivity. The factors leading to these characteristics were investigated by use of their crystalline approximants, which allow us to employ the structure analysis, band calculation, and phonon calculation well developed for crystalline materials. It is found that the quasiperiodic order does not play an exclusive role in providing the characteristic thermoelectric properties, while the icosahedral-symmetry and the characteristic quasi-Bllirouin-zone significantly contribute to these properties by deepening pseudogap at the Fermi level, reducing the mumber of acoustic phonons, and enhancing Umklapp scattering of phonons.

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This paper demonstrates phase transition phenomena of a Cd-based approximant Cd₆Ca for an icosahedral quasicrystal. A salient anomaly has been observed at 100 K in the magnetic susceptibility as well as in the electrical resistivity, which is attributed to temperature variation of the density of states at EF, possibly caused by lattice distortion. Diffraction experiment together with a group theoretical analysis based on the Landau theory suggests the space group symmetry of the low temperature phase is C2/c, and this result allows to propose possible superstructure with an antiparallel ordering of a Cd₄ tetrahedron located inside the icosahedral cluster, along a [110] direction of the high temperature phase.

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Owing to its quasiperiodicity, magnetic quasicrystals may possibly show unique spin dynamics differing from ordinary periodic crystalline magnets. In this article we will review our recent neutron inelastic scattering study on the f-Zn-Mg-RE (RE : rare-earth) magnetic quasicrystals. A particular focus is placed on a strongly localized inelastic mode in the RE=Tb system.

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Scanning tunneling microcopy (STM) studies of the fivefold surface of icosahedral Al-Cu-Fe (i-Al-Cu-Fe) quasicrystals are presented. A thorough analysis of high resolution images combined with a refined structure model leads to a rule for stable surface formation of Al-based icosahedral quasicrystals. Bi islands formed on this surface exhibit prominent stability at a specific thickness, showing evidence of quantum size effects.

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The thermoelectric properties of icosahedral AlPdRe quasicrystals obtained by replacing Re with transition metals have been studied in according with the guiding principle of Weakly Bonded Rigid Heavy Clusters (WBRHCs) . The dimensionless figure of merit (ZT) was increased 2.6 times from 0.08 to 0.21 by Fe substitution for Re in the icosahedral AIPdRe quasicrystals. It is then possible to improve the thermoelectric performance of i-AlPdRe by searching over a wider range of composition, including the addition of a fourth element or fourth and fifth elements.

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