The concept of the reference lattice has been thoroughly discussed in physical and in higher-dimensional spaces. Such an approach has been applied to di raction pattern analysis of di erent periodic, modulated and other aperiodic structures. It has been shown that two periodicities can be found in di raction patterns of the above structures, namely the periodicity of the envelope curves describing the di raction peak intensities, and the periodicity of peak positions ascribed to each envelope curve. The formula for the envelope curve has been given and it depends on the statistical moments of a specially de® ned real-space variable describing spatial¯uctuations in atomic positions. Full equivalence of the presented physical space approach to the higher-dimension analysis of perfect quasicrystals has been shown.

A comparative single-crystal X-ray diffraction structure analysis of the family of Al-Cu-Me (Me = Co, Rh and Ir) decagonal quasicrystals is presented. In contrast to decagonal Al-Cu-Co, the other two decagonal phases do not show any structured disorder diffuse scattering indicating a higher degree of order. Furthermore, the atomic sites of Rh and Ir can be clearly identified, while Cu and Co cannot be distinguished because of their too similar atomic scattering factors. The structure models, derived from charge-flipping/low-density elimination results, were refined within the tiling-decoration method but also discussed in the five-dimensional embedding approach. The basic structural building units of the closely related structures are decagonal clusters with 33 Å diameter, which are consistent with the available electron-microscopic images. The refined structure models agree very well with the experimental data.

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In elemental neodymium. a series of phase ansitions between multi-q modulated magnetic structures tnkes place below the magnetic ordering temperature TN. Based on extensive neulron diffraction studies of the temperature dependence of the length md orientation of the modulation vectors associated with these structures we suggest thal there exists a simple phenomenological relationship between the symmetry of the double hexagonal crystal lattice and the symmehy of the corresponding magnetic lattice. The model has a resemblance w the situation found for monolayer h l m on solid surfaces (orientational epitaxy) when considering the spin system as the adsorbate and the atomic structure as the subslrate. In one dimension, the modulaled magnetic structures are sequences of a commensurate to incommensurate Vansition Followed by incommensunle to incommensurate vansitions followed by an incommensurate lo commensmte transition. However, i n two dimensions. all the modulated magnetic sWcNres are equally well described 3s higher-order Commensmle to "mensurale msitions. where the magnetic unit cell is commensumte with the crystallographic unit cell, but rotated by some angle around the hexagonal axis with respect to the crystallographic unit cell. The rotation angle as well as the magnetic unit cell depend on temperature. The data suggest lhat the rotation angle is zem whenever there is B change from one type of multi-q structure to another. i.e.. in this case, the magnetic and the crystallogmphic unit cells are commensurate in both one and two dimensions. Previous and recent results for the light rare earth metals neodymium and praseodymium and alloys thereof, which lend supporl w this mterpremion. are reviewed.

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