Aperiodic materials

In addition to the crystalline periodic state of matter, a class of materials exists that lacks 3-D translational symmetry and is called aperiodic. Aperiodic materials cannot be described by any of the 230 space groups mentioned above. Nevertheless, they show another type of long-range order and are therefore included in the term crystal . This notion of long-range order is the major feature that distinguishes crystals from amorphous materials. Three types of aperiodic order may 

Material Structure type Pearson symbol Space group a (A) MgAl204 MgAl204, spinel cF56 Fd3m (No. 227) 8.174(1)  

In a modulated structure, periodic deviations of the atomic parameters from a reference or basic structure are present. The basic structure can be understood as a periodic structure as described above. Periodic deviations of one or several of the following atomic parameters are superimposed on this basic structure  

Let the period of the basic structure be a and the modulation wavelength be the ratio a/X may be (1) a rational or (2) an irrational number (Fig. 1.3-7). In case (1), the structure is commensurately modulated we observe a qa superstructure, where q= /X. This superstructure is periodic. In case (2), the structure is incommensurately modulated. Of course, the experimental distinction between the two cases is limited by the finite experimental resolution, q may be a function of external variables such as temperature, pressure, or chemical composition, i. e. = f T, p, X), and may adopt a rational value to result in a commensurate lock-in stmcture. On the other hand, an incommensurate charge-density wave may exist this can be moved through a basic crystal without changing the internal energy U of the crystal. 

When a 1-D basic structure and its modulation function are combined in a 2-D hyperspace R = 0 

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Steurer, W. (1995) Experimental aspects of the structure analysis of aperiodic materials. In Beyond Quasicrystals, eds. Axel, F. and Gratia, D. (Les Editions de Physique-Springer, Berlin, Germany), p. 203. 

The structural study of aperiodic material is a field that is still evolving. Their detailed study at the atomic level will possibly deliver the origin of the mechanism leading to aperiodicity rather than periodicity. We can also expect to use the structural properties of incoinmensurabilities as a probe to improve our knowledge of the chemical and physical aspects of atomic interactions. In addition, we can also expect to discover and take advantage of some specific properties associated with the aperiodicities of crystalline material. A series of studies on the NMR and NQR properties of incommensurate crystals already revealed some specific characteristics of these systems. 

Quasicrystals represent the third type of aperiodic materials. Quasiperiodicity may occur in one, two, or three dimensions of physical space and is associated with special irrational numbers such as the golden mean r = (1 -h /5)/2, and = 2 -F V3. The most remarkable feature of quasicrystals is the appearance of noncrystal-lographic point group symmetries in their diffraction patterns, such as 8/mmm, lO/mmm, l2/mmm, and 2lm35. The golden mean is related to fivefold symmetry via the relation r = 2 cos( r/5) r can be considered as the most irrational number, since it is the irrational number that has the worst approximation by a truncated continued fraction,... 

In between the ideal crystalline and the purely amorphous states, most real crystals contain degrees of disorder. Two types of statistical disorder have to be distinguished chemical disorder and displacive disorder (Fig. 1.3-14). Statistical disorder contributes to the entropy S of the solid and is manifested by diffuse scattering in diffraction experiments. It may occur in both periodic and aperiodic materials. 

Designing a specific material architecture. 3D hierarchical carbon [79,80], 3D aperiodic [79,81,82] or highly-ordered hierarchical carbons are representative samples with multimodal pore structure to optimize the performance of the capacitors. The micropore, mesopore and macropore structure of such three-dimensional hierarchical carbons are generally perfectly interconnected. 

While periodically-poled materials are typically characterized by very narrow spectral acceptance bandwidths (< 1 nm), aperiodically-poled structures (characterized by a linear gradient in grating period) have the advantage of providing sufficiently broad spectral acceptance bandwidths to utilize more of the spectrum associated with picosecond and femtosecond" pulses. They can also simultaneously provide some pulse compression of sufficiently pre-chirped incident pulses ... 

Impurity and Aperiodicity Effects in Polymers.—The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),103 Dean s method in its SCF form,51 the Hartree-Fock Green s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations... 

As we established in Chapter 1, crystal lattices, used to portray periodic three-dimensional crystal structures of materials, are constructed by translating an identical elementary parallelepiped - the unit cell of a lattice -in three dimensions. Even when a crystal structure is aperiodic, it may still be represented by a three-dimensional unit cell in a lattice that occupies a superspace with more than three dimensions. In the latter case, conventional translations are perturbed by one or more modulation functions with different periodicity. 

One of the interesting implications of this section is that the walks, and hence moments, may be generated without any recourse to the translational synunetry of the solid, or the point group of the molecule. In the context of extended arrays therefore, the moments method may be used in the study of aperiodic systems such as are found " in amorphous materials and in surface phenomena. In this article we shall exclude such areas fom discussion, and will concentrate on structural problems in molecules and crystalline solids. 

Ideas on the three-dimensional structure of proteins were no less preliminary. It was agreed that they were big molecules - macromolecules - probably composed of linear chains of amino acids of up to 100 or more. But just how their primary sequences were determined, how they were synthesized, and how they arrived at their native three-dimensional conformations was a complete mystery (Hunter, 2000, ch. 11). It is a testimony to how little was known about protein function that a majority of biochemists considered that it was proteins, rather then nucleic acids, that played the primary role in heredity and formed the basic aperiodic molecule that made up the material structure of the gene (McCarty, 2003j. The question of whether proteins represented a potentially infinite set of Lego -like assemblages, largely unconstrained by physical law and determined by natural selection, or whether they represented a finite set of natural forms determined mainly by natural law and therefore were antecedent to life and evolution (like a set of atoms or crystals), was simply impossible to answer. 

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