By Ted Janssen, Gervais Chapuis, Marc de Boissieu
Until eventually the Seventies all fabrics studied consisted of periodic arrays of unit cells, or have been amorphous. within the final a long time a brand new type of good nation topic, known as aperiodic crystals, has been came upon. 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, below a variety of pressures and temperatures. due to the loss of periodicity the standard recommendations for the learn of constitution and actual homes now not paintings, and new options must be constructed. This e-book offers with the characterisation of the constitution, the constitution choice and the examine of the actual houses, specifically dynamical and digital homes of aperiodic crystals. The therapy is predicated on an outline in an area with extra dimensions than 3, the so-called superspace. this permits us to generalise the normal crystallography and to seem otherwise on the dynamics. the 3 major sessions of aperiodic crystals, modulated levels, incommensurate composites and quasicrystals are handled from a unified standpoint, which stresses similarities of some of the structures. The publication 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 sphere.
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Extra resources for Aperiodic crystals: from modulated phases to quasicrystals
Another type is the well-known uniform shift of all particles: xn —> xn + c. The invariance of the energy under this transformation is related to the conservation of momentum of the crystal. For the transformation by renumbering and shifting, there is a similar conserved quantity. Displacements in a crystal may be described in terms of the normal modes of the crystal, characterized by the irreducible representations of the space group. This gives a way to describe the modulation functions. R|t} such that Kk = k (up to a reciprocal lattice vector).
Therefore, all diffraction peaks are at positions which are linear combinations of the six basis vectors. Generally, these six vectors a*vi (y = 1,2, i = 1,2,3) are not independent. One may choose a basis of rank n < 6 vectors b* such that each diffraction wave vector belongs to the Fourier module generated by these n vectors. One has Fig. 11. A two-dimensional model for an incommensurate composite with b/a = A/2. For a simple model with two chains the positions olthe atoms are (Fig. 11) with f(x) = f ( x + 6), and g(y) = g(y + a).
This is the property of almost periodicity. A closely related property is the following. If one renumbers the particles and shifts them according to xn —> xn+p —pa then the new positions are na + u(qna + > p ), where the phases >p = qpa (mod 1) cover the unit interval in a uniform way. The result is a shifted modulation wave, but the shifting only involves a renumbering and a common shift. These do not change the distances, and consequently the potential energy will not change. Here we have a new type of symmetry: the positions of the particles are not transformed into each other, but the energy remains invariant.