Band 215, Heft 10

How, in principle, could one solve the atomic structure of a quasicrystal, modeled as a random tiling decorated by atoms, and what techniques are available to do it? One path is to solve the phase problem first, obtaining the density in a higher dimensional space which yields the averaged scattering density in 3-dimensional space by the usual construction of an incommensurate cut. A novel direct method for this is summarized and applied to an i(AlPdMn) data set. This averaged density falls short of a true structure determination (which would reveal the typical unaveraged atomic patterns.) We discuss the problematic validity of inferring an ideal structure by simply factoring out a "perp-space" Debye-Waller factor, and we test this using simulations of rhombohedral tilings. A second, "unified" path is to relate the measured and modeled intensities directliy, by adjusting parameters in a simulation to optimize the fit. This approach is well suited for unifying structural information from diffraction and from minimizing total energies derived ultimately from ab-initio calculations. Finally, we discuss the special pitfalls of fitting random-tiling decagonal phases.

State-of-the-art X-ray diffraction techniques allow quasicrystals to be characterised quantitatively. Based on two-dimensional scans of Bragg reflections with high spatial resolution twinned approximants or nanodomain structures can be reliably distinguished from quasiperiodic average structures. Linear phason strain as well as mosaicity can directly be detected in situ under non-ambient conditions. Accurate cell parameters can be obtained. The experimental limitations are discussed. Methods to determine the space group and the optimum n-dimensional unit cell are shown. The phases of the reduced structure amplitudes can be obtained by several techniques. Highly resolved electron density distributions are derived from a combination of Patterson deconvolution techniques with maximum entropy methods. Special emphasis is put on the achievable resolution of experimentally determined electron density distributions. Based on a subtle n-dimensional atomic surface modelling the structure can be described in an accurate way according to the standards in conventional crystallography. All relevant structural parameters are determined quantitatively by an n-dimensional structure refinement. Different structure models are distinguished by their likelihood. The reliability of a quantitative structure determination is examined.

Methods based on X-ray area detectors are presented for acquiring and analysing single-crystal X-ray diffraction data from quasicrystals and approximants. These methods are complementary to high-resolution diffractometry and fully exploit the possibilities of modern two-dimensional X-ray detector systems (imaging plates and charge coupled devices) such as the on-line read-out capability and the possibility to collect large volumes of reciprocal space in a quantitative and rapid way. In contrast to a classical point detector, area detectors allow the measurement of a very large number of Bragg reflections in a reasonable time scale, as is shown with a large unit-cell periodic approximant in the Al-Co-Ni alloy system. For large approximants exhibiting strong pseudosymmetry, direct methods for structure solution usually fail. Recently implemented Patterson deconvolution techniques are shown to succeed. The extension of techniques for area detectors – beyond the standard application of collecting integrated Bragg intensities – to the quantitative acquisition of continuous and diffuse scattering data is presented with particular emphasis on quasicrystals. One of the key techniques is the quantitative reconstruction of undistorted layers and volumes in reciprocal space. This is crucial for developing and comparing possible models for structural disorder in quasicrystals. Examples for decagonal Al-Co-Ni and icosahedral Al-Mn-Pd are given in respect to possibilities and limitations of X-ray area detectors.

We review the use of neutron scattering in the structure determination of quasicrystals. The contrast between X-ray and neutron cross-section gives a useful complementary information. When feasible, contrast variation techniques allow to disentangle the different atomic contribution. Diffuse scattering studies are also simplified, thanks to the possibility of an energy analysis. We discuss results obtained in the case of the i-AlPdMn phase.

Structural investigations, using a transmission electron microscope (TEM), have been highly beneficial for the analysis of icosahedral and decagonal quasicrystals. Many structural properties of quasicrystals can be recognized and quantified by electron diffraction patterns alone. Nevertheless, a much more complete understanding of the real structure of quasicrystals can be achieved by the analysis of high-resolution transmission electron microscopy (HRTEM) images. Compared to diffraction-based techniques, HRTEM offers several advantages, however, special points which must be considered in the interpretation of HRTEM images will be discussed. For the case of 2-dimensional quasicrystals, e.g. the decagonal phases, HRTEM images can directly reveal the (projected) atomic structure as well as the long-range translational order. This is possible because 2-dimensional quasicrystals are periodic along one direction and the atoms thus form periodic atom columns as in crystalline materials. Therefore, the imaging theory established for crystals can similarly be applied to electron micrographs of 2-dimensional quasicrystals taken with the electron beam parallel to the unique periodic axis. It is evident, that possible limitations due to projection effects along the electron beam direction have to be considered in the interpretation of the images. HRTEM images of decagonal quasicrystals have frequently been employed to differentiate between disorder and order, i.e., to infer the difference between a random and perfect quasiperiodic tiling, respectively.

The applications of the high-angle annular dark-field scanning transmission electron microscope (HAADF-STEM) and convergent-beam electron diffraction (CBED) methods to decagonal quasicrystals and its approximant are presented by giving some examples. The HAADF method can show the atomic arrangement mainly composed of transition metal (TM) atoms in Al-TM systems. The CBED method can determine the super point- and space-groups of quasicrystals, and furthermore can be used to refine the structural parameters.