Icosahedral quasicrystals (i-QCs) are long-range ordered solids that show non-crystallographic symmetries such as five-fold rotations. Their detailed atomic structures are still far from completely understood, because most stable i-QCs form as ternary alloys suffering from chemical disorder. Here, we present the first detailed structure solution of i-YbCd 5.7 , one of the very few stable binary i-QCs, by means of X-ray structure determination. Three building units with unique atomic decorations arrange quasiperiodically and fill the space. These also serve as building units in the periodic approximant crystals. The structure is not only chemically feasible, but also provides a seamless structural understanding of the i-YbCd 5.7 phase and its series of related i-QCs and approximant crystals, revealing hierarchic features that are of considerable physical interest.Icosahedral quasicrystals (i-QCs) are the only class to show quasiperiodicity in three dimensions 1,2 . Obviously, structural knowledge is essential for understanding the physical properties, stability and tailoring applications of these exotic materials 3 . However, in contrast to other types of QCs that show periodic order in at least one direction, i-QCs cannot make effective use of two-dimensional (2D) imaging techniques such as high-resolution electron microscopy or high-angle annular dark-field scanning transmission electron microscopy for their structural characterization 4 . The i-QCs' structure determination is best achieved in the context of hyperspace crystallography 5,6 , where the structure can be described as a periodic crystal in higher dimensions. For i-QCs, the periodic space is 6D and decomposes into two orthogonal 3D subspaces: the parallel (physical) space and the perpendicular (complementary) space. The 6D unit cell is decorated by 3D objects known as 'occupation domains' (OD), the 3D QCs being obtained as a section of this decorated 6D lattice. This approach allows modelling and refinement of the structure against experimental diffraction data in a way similar to that achieved for 3D periodic crystals 6 . Although much progress has been achieved recently, for instance in the i-AlPdMn phase 7 , the models proposed so far are still being debated 8 . Indeed, the amount of observed diffraction data is in general rather limited, which precludes a detailed refinement of the chemical order in ternary QCs. Therefore, the atomic order in i-QCs remains a challenging and outstanding question. The recent discovery of the first stable binary icosahedral YbCd 5.7 QCs 9,10 has been a breakthrough and led to discoveries of a whole series of related ternary i-QCs 11 . This i-QC offers a unique opportunity for the structural analysis of i-QCs. Indeed, the i-YbCd 5.7 phase can be obtained as high-quality single grains. Furthermore, it is binary and exhibits very good X-ray contrast between Cd (Z = 48) and Yb (Z = 70) atoms.Finally, there is a series of periodic 'approximant crystals' (ACs) to the QC, having almost the same chemical composition and for...

The aim of this review is to describe many approaches to modulated crystals and quasicrystals developed in two decades after the introduction of higher-dimensional crystallography in a unified way. Much attention is focused on higher-dimensional crystallography of quasicrystals, which is under development. After discussions on symmetries of modulated crystals and methods of their structure analysis, many subjects on the analysis of quasicrystals are discussed, which include methods of generating quasiperiodic tilings, their diffraction patterns, similarity transformations, indexing problems, point density of quasicrystals, phason distortion, relations between quasicrystals and their crystalline approximants, model constructions, Patterson, refinement and maximum-entropy methods for quasicrystals, and superstructures in quasicrystals. In particular, the theories for octagonal, decagonal, dodecagonal and icosahedral quasicrystals are given in detail.

A general structure-factor formula for n-dimensionally modulated structures (n = 1, 2, ...) has been derived to have both density (or substitutional) and displacive modulations. This structure-factor formula covers any modulated structure described by a (3 +n)-dimensional space group. Future papers show the applications of this formula to one-and three-dimensional modulations.

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A new large radius imaging plate diffraction camera for high-resolution and high-throughput synchrotron x-ray powder diffraction by means of multiple exposures has been developed for an insertion device beamline of SPring-8, Japan. The new imaging plate camera consists of a large radius cylindrical shape imaging plate cassette that is 400 mm in length and 954.9 mm in cylinder radius. The cassette is designed to be mounted on the 2 theta arm of the diffractometer of BL15XU in SPring-8. One imaging plate covers 24 degrees and several times of exposure changing the 2 theta-setting angle is necessary to obtain whole powder diffraction data up to a high angle region. One pixel of the imaging plate corresponds to 0.003 degrees in 2 theta when the readout pixel size is 50 microm squares. Separately collected data are translated to 2 theta-intensity format and are connected by comparing the peak and background intensity included in the overlapped area. The exposure time is less than 120 s for most samples and the readout time is about 3 min; thus, the total measurement time for one powder diffraction pattern is less than 20 min. The measurement time is the same order as the continuous 2 theta-scanning method of the third generation synchrotron powder diffractometer. The angular resolution of the new imaging plate camera was evaluated by comparing the full width at half maximum of the 111 reflection of NBS-Si. The observed angular resolution is not so high as a powder diffractometer with a Si or a Ge analyzer monochromator in the third generation synchrotron facility but higher than a powder diffractometer with a Ge analyzer monochromator at a bending magnet beamline of the second generation synchrotron. The Rietveld analysis of NBS-CeO2 was successfully carried out with the data taken by the new imaging plate camera.

The structure of a decagonal Al72Ni20Co8 quasicrystal with space group P10(5)/mmc has been determined on the basis of a single-crystal X-ray data set using the five-dimensional description. The best-fit model structure based on a cluster model having lower symmetry than the decagonal symmetry with 103 parameters gives wR = 0.045 and R = 0.063 for 449 reflections. The structure is well described by the hexagon, boat and star tiling with an edge length of 6.36 A and is very consistent with recent high-resolution electron-microscopy images. The refined structure is compared with previously discussed model structures including cluster-based models having 20 A tenfold symmetric clusters.

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The modulated structure of the NC-type pyrrhotite Fe~_xS with x = 0.09 has been determined. The analysis is based on a four-dimensional space group, wp~a21 and an anharmonic modulation model with 53 qql ' positional, two thermal, and eight occupational parameters including up to fourth-order harmonics, a = 11.952, b = 6.892, c = 5.744/~. The final R for 404 observed reflections is 0.097. The modulation wave for occupation probability of Fe shows a strong anharmonicity and has a low value in a narrow range corresponding to successive Fe layers perpendicular to the c axis of the fundamental NiAs-type structure. From this and the four-dimensional symmetry, it is concluded that Fe vacancies are distributed spirally along the c axis. The relationship between the usual superstructure model and this four-dimensional description is discussed.