Aperiodic Crystals: From Modulated Phases to Quasicrystals by Ted Janssen, Gervais Chapuis, Marc de Boissieu

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 reliable country topic, referred to as aperiodic crystals, has been came across. it's a lengthy variety ordered constitution, yet with out lattice periodicity. it really is present in a variety of fabrics: natural and anorganic compounds, minerals (including a considerable section of the earths crust), and steel alloys, less than a variety of pressures and temperatures. due to the loss of periodicity the standard concepts for the examine of constitution and actual homes not paintings, and new suggestions must be constructed. This e-book bargains with the characterisation of the constitution, the constitution choice and the examine of the actual homes, in particular dynamical and digital houses of aperiodic crystals. The remedy relies on an outline in an area with extra dimensions than 3, the so-called superspace. this enables us to generalise the normal crystallography and to appear another way on the dynamics. the 3 major sessions of aperiodic crystals, modulated levels, incommensurate composites and quasicrystals are taken care of from a unified viewpoint, which stresses similarities of a few of the structures. The booklet assumes as a prerequisite an information of the elemental innovations of crystallography and the speculation of condensed subject, 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 (International Union of Crystallography Monographs on Crystallography)

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C. 3 Summary It has become clear that the ground state of solid matter is not necessarily lattice periodic. This new paradigm has the consequence that a crystal should be defined in another way than just by requiring lattice periodicity. Another consequence is that a new approach is needed for those crystals that are not lattice periodic. The usual techniques for studying physical properties and for describing the structure are heavily based on the assumption of lattice periodicity. 30 INTRODUCTION Perfect quasiperiodic crystals have, just as periodic crystals, sharp Bragg peaks, on the positions of a Fourier module.

13) with T = (i/5 + l)/2 and i = 1 , . . , Q. With respect to this basis, a five-fold and a three-fold rotation are given by the matrices QUASIPERIODIC CRYSTALS 25 These two matrices generate, together with the total inversion (the opposite of the identity), a point group with 120 elements, the symmetry group of an icosahedron (or a dodecahedron). This group contains 60 rotations: 6 five- fold axes (giving 24 rotations), 10 three-fold axes (for 20 rotations), 15 two-fold axes (for 15 rotations), and the identity.

Such a function has a Lourier transform consisting of delta peaks at k* = h/a, that is on the reciprocal lattice. ,d. The function g(x) = cos(27rx/a)+cos(27rx/6) has a simple Lourier decomposition with wave vectors ±l/a and ±1/6. If a/6 is a rational number it is a periodic function with as period the smallest common multiple of a and 26. However, when a/6 is irrational the function is aperiodic (it has no periodicity), but it is by no means a chaotic function. It is an example of a quasiperiodic function.

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