Quasicrystals transformation

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

Alan Mackay made the connection with crystallography [139], He designed a pattern of circles based on a quasi-lattice to model a possible atomic structure. An optical transformation then created a simulated diffraction pattern exhibiting local tenfold symmetry (see, in the Introduction). In this way, Mackay virtually predicted the existence of what was later to be known as quasicrystals, and issued a warning that such structures may be encountered but may stay unrecognized if unexpected ... 

A. Csanady, K. Papp, M. Dobosy, M. Bauer, Direct Observation of the Phase Transformation of quasicrystals to Al6Mn Crystals. Symmetry 1990, 1, 75-79. 

Quasicrystals are solid materials exhibiting diffraction patterns with apparently sharp spots containing symmetry axes such as fivefold or eightfold axes, which are incompatible with the three-dimensional periodicity associated with crystal lattices. Many such materials are aluminum alloys, which exhibit diffraction patterns with fivefold symmetry axes such materials are called icosahedral quasicrystals. " Such quasicrystals " may be defined to have delta functions in their Fourier transforms, but their local point symmetries are incompatible with the periodic order of traditional crystallography. Structures with fivefold symmetry exhibit quasiperiodicity in two dimensions and periodicity in the third. Quasicrystals are thus seen to exhibit a lower order than in true crystals but a higher order than truly amorphous materials. 

The structure of amorphous metals, quasicrystals, and crystalline inter-metallic compounds can be modelled by atom clusters with icosahedral arrangement [3.113-117]. The differences between the various phases result from a different arrangement of the individual atom clusters. Therefore, it is evident that there exists a close relation between the different states of matter, and that the different phases corresponding to minima of the free enthalpy can be quite easily transformed into each other. For example, rapid cooling from the melt results in an amorphous alloy for high quenching rates, and a quasicrystalline... 

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