Crystal imperfect

Diffusion of ions or molecules in solids is preliminary to reaction. It takes place through the normal crystal lattices of reactants and products as well as in channels and fissures of imperfect crystals. It is slow in comparison with that in fluids even at the elevated temperatures at which such reactions have to be conducted. In cement manufacture, for instance, reaction times are 2 to 3 h at 1,200 to 1,500°C (2,192 to 2,732°F) even with 200-mesh particles. 

In the concepts developed above, we have used the kinematic approximation, which is valid for weak diffraction intensities arising from imperfect crystals. For perfect crystals (available thanks to the semiconductor industry), the diffraction intensities are large, and this approximation becomes inadequate. Thus, the dynamical theory must be used. In perfect crystals the incident X rays undergo multiple reflections from atomic planes and the dynamical theory accounts for the interference between these reflections. The attenuation in the crystal is no longer given by absorption (e.g., p) but is determined by the way in which the multiple reflections interfere. When the diffraction conditions are satisfied, the diffracted intensity ft-om perfect crystals is essentially the same as the incident intensity. The diffraction peak widths depend on 26 m and Fjjj and are extremely small (less than... 

Kroeger, F.A. (1974) The Chemistry of Imperfect Crystals, 2 volumes (North Holland, Amsterdam). 

So far we have discussed the surface of a perfect crystal. But for an imperfect crystal there is another possibility to provide a step source. This is due to the screw dislocation. Assume that one cuts a crystal half-way from one side into the center, and slides the freshly created two faces against each other in... 

A crystal may be defined as an orderly three-dimensional array of atoms, and all metals are aggregates of more or less imperfect crystals. In considering the structure of metals, therefore, it is convenient to start with the arrangement of atoms in a perfect metal crystal and then to proceed to the imperfections which are always present in the crystal structure. 

A. Guinier, Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies, Dover, New York, 1963. 

The two extremes of ordering in solids are perfect crystals with complete regularity and amorphous solids that have little symmetry. Most solid materials are crystalline but contain defects. Crystalline defects can profoundly alter the properties of a solid material, often in ways that have usefial applications. Doped semiconductors, described in Section 10-, are solids into which impurity defects are introduced deliberately in order to modify electrical conductivity. Gemstones are crystals containing impurities that give them their color. Sapphires and rubies are imperfect crystals of colorless AI2 O3, red. 

See e.g. Kroger FA (1973) The chemistry of imperfect crystals, North-Holland, Amsterdam Brixner LH (1987) Mat Chem Phys 16 253... 

On the basis of advances of the chemistry of imperfect crystals [209] the role of point defects existing in the volume and on the surface of oxide adsorbents and their effect on electrophysical characteristic during adsorption-caused change is being elucidated [32]. 

To state clearly the problem at hand it is necessary to introduce initially a detailed notation for the composition of a crystal. For much of the later manipulations it is possible to use a very much simpler, abbreviated version of the notation. From the point of view of thermodynamics, the composition of an imperfect crystal is specified when the number of atoms of each of the different chemical species present is given. Let atoms which appear in a perfect crystal be denoted by a subscript 0, and let N0 denote the 2V0 atoms of a different species (2V01, N02,. . ., N0a), all of which species appear in the perfect crystal, i.e. 

Then the composition of an imperfect crystal is given thermodynamically by the set of numbers N = N0 + N0. For each species there is a chemical potential thus ft0s is the chemical potential for an atom of species Os, p0 denotes the set of a such quantities, and similarly for pa. 

We turn now to the microscopic description of an imperfect crystal. The various defects in any imperfect crystal can be imagined to be formed from a corresponding perfect crystal by one or more of the following processes (a) remove an atom of species Os from the crystal leaving a vacant lattice site, (b) remove an atom of species Os from the crystal and replace it by an atom of a different species (either Oi or at), (c) add to the crystal an atom of any species to a site on a sublattice unoccupied in the perfect crystal. We refer to the latter as atoms in interstitial positions. Let B be a set of numbers such that Br is the number of sites on sublattice number r in the perfect crystal, and let be the number of sublattices in the crystal (including interstitial sublattices not occupied in the perfect crystal). The total number of sites of all kinds in the perfect crystal is then... 

Thus atoms of species Os may be found in an imperfect crystal in their normal lattice positions, occupying sites on the wrong sublattice (that is a sublattice occupied by an atom of a different species in the perfect crystal), or in interstitial positions. Let the numbers of such atoms be Ng N, Ng, respectively, so that... 

If Mtr is the ratio of the number of sites in sublattice number q to the number of sites in sublattice number r, both lattices being one of the 6 sublattices occupied by atoms in a perfect crystal, then for the imperfect crystal we have... 

Kroger, F. A., Chemistry of Imperfect Crystals, North-Holland Publishing Company, Amsterdam, 1964. 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

As a result of the relatively poor spatial resolution in X-ray topographs, there can be confusion as to whether uniform contrast results from a highly perfect or a highly imperfect crystal. There are, however, very important tests which can be applied, none more so than for the section topograph where the contrast from a perfect crystal is far from uniform. 

In an imperfect crystal or amorphous material the wavenumber k is not a good quantum number, and if Ak/k becomes comparable to unity then the concept of a Fermi surface has little meaning. Nevertheless, at zero temperature a sharp Fermi energy must still exist. 

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