By Ted Janssen, Gervais Chapuis, Marc de Boissieu
Till the Seventies all fabrics studied consisted of periodic arrays of unit cells, or have been amorphous. within the final many years a brand new category of sturdy country subject, referred to as aperiodic crystals, has been discovered. it's a lengthy diversity ordered constitution, yet with out lattice periodicity. it truly is present in a variety of fabrics: natural and anorganic compounds, minerals (including a considerable section of the earths crust), and metal alloys, less than numerous pressures and temperatures. end result of the loss of periodicity the standard options for the learn of constitution and actual homes now not paintings, and new ideas need to be constructed. This ebook offers with the characterisation of the constitution, the constitution decision and the examine of the actual houses, in particular dynamical and digital houses of aperiodic crystals. The therapy relies on an outline in an area with extra dimensions than 3, the so-called superspace. this permits us to generalise the traditional crystallography and to seem in a different way on the dynamics. the 3 major periods of aperiodic crystals, modulated stages, incommensurate composites and quasicrystals are taken care of from a unified perspective, which stresses similarities of many of the platforms. The publication assumes as a prerequisite an information of the elemental options of crystallography and the idea of condensed topic, and covers the literature on the leading edge of the sector.
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Extra info for Aperiodic crystals: from modulated phases to quasicrystals
For example, in CuAuII the roles of the Cu and Au positions are exchanged with a period of approximately 11 lattice constants in one direction. The term species may also refer to the orientation of a molecule, as in NaNC>2. Then the difference between displacive and occupational modulation is no longer clear-cut. The density function of the occupation modulated crystal is When the functions p(r) have Fourier components with wave vectors q, the Fourier transform of p(r) has a Fourier module with main reflections and satellites.
If the functions Uj have several Fourier components expressible in a number of basic modulation vectors as q = J^m hmqm, the expression for the Fourier transform p becomes a sum of Bragg peaks at positions belonging to the Fourier module spanned by the basis vectors a* of the reciprocal lattice of the basic structure and the vectors q in the Fourier transform of the periodic modulation. Therefore, Because we want to deal also with one- or two-dimensional aperiodic crystals, we consider a more general case.
Relatively dense' means that there are for each e values RI and R-2 such that every sphere of radius R-2 contains at least one a satisfying the condition, and in every sphere with radius RI around any translation a satisfying the condition, there is no other translation but a. A function satisfying this condition is called almost periodic. The theory of almost periodic functions was developed by H. Bohr (cf. ). Every quasiperiodic function is almost periodic. rax) if a is irrational. Then the Fourier module consists of (m + no) and a basis of the module is 1 and a.
Aperiodic crystals: from modulated phases to quasicrystals by Ted Janssen, Gervais Chapuis, Marc de Boissieu