The Description of a Crystal Structure

Atoms are not hard spheres, but are soft, so that by increased force they may be pushed more closely together (be compressed). This compression occurs, for example, when a copper crystal becomes somewhat smaller in volume under increased pressure. The sizes that are assigned to atoms correspond to the distances between the center of one atom and the center of a neighboring atom of the same kind in a crystal under ordinary circumstances. The distance from a copper atom to each of its twelve nearest neighbors in a copper crystal at room temperature and atmospheric pressure is 255 pm this is called the diameter of the copper atom in metallic copper. The radius of the copper atom is half this value. 

Chemists often make use of the observed shapes of crystals to help in the identification of substances. The description of the shapes of crystals is part of the subject of the science of crystallography. The method of studying the structure of crystals by the diffraction of x-rays, which was discovered by the German physicist Max von Laue (1879-1960) in 1912 and developed by the British physicists W. H. Bragg (1862-1942) and W. L. Bragg (1890-1971), has become especially valuable in recent decades. Much of the information about molecular structure that is given in this book has been obtained by the x-ray diffraction technique. 

The basis of the description of the structure of a crystal is the imit of structure. For cubic crystals the unit of structure is a small cube, which, when repeated parallel to itself in such a way as to fill space, reproduces the entire crystal. 

Arrangement of atoms in a plane. The unit of structure is a square. Small atoms have the coordinates 0, 0 and large atoms the coordinates i. 

The simple cubic arrangement of atoms. The unit of structure is a cube, with one atom per unit, its coordinates being 0, 0, 0. 

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The description of a crystal structure always refers to the dimensions of the unit cell, and the atomic positions within this space are described using the concept of fractional coordinates. A point within a unit cell is defined by fractions along the basis vectors a, b and c, and is thus defined as xa, yb, zc). For example, the center of a unit cell is and the point (1,1,1) is equivalent to the origin. 

The analysis of the resonant solution scattering data demands a different representation of the Debye equation (16). If the macromolecular structure had a spherical appearance, then the formalism of isomorphous replacement in single crystal diffraction outlined in the preceding section would apply. This is not surprising as the rotation of a spherical structure could not be noticed anyway. In more complicated, asymmetric macromolecular structures it is the spherical average of the structure which must be subjected to the phase analysis described above. As this statement is less trivial, we shall extend the description of a macromolecular structure beyond its spherical average by introducing an expansion of p(r) as a series of spherical harmonics Yj ((o)... 

Inorganic structures are said to be structures which do not contain any C-H bonds or C-C bonds in any residue and contain at least one nonmetal. In addition ICSD contains metal carbides and inoiganic frameworks including organic residues (e.g., zeolites including ethanol). The database tries to be complete for all papers from the beginning of X-ray diffraction in 1915 up to the present. The release 1997/II contains 45 914 entries. Each entry contains the primary data for a complete description of a crystal structure and some additional information. This is typical for this type of database but may differ in detail from database to database. 

In this chapter, general aspects and structural properties of crystalline solid phases are described, and a short introduction is given to modulated and quasicrystal structures (quasi-periodic crystals). Elements of structure systematics with the description of a number of structure types are presented in the subsequent Chapter 7. Finally, both in this chapter and in Chapter 6, dedicated to preparation techniques, characteristic features of typical metastable phases are considered with attention to amorphous and glassy alloys. 

According to Parthe et al. (1993), a standardization procedure is necessary in the presentation of the relevant data characteristic of a crystal structure (see also Parthe and Gelato 1984). A convenient description of the structure types is then possible using the Wyckoff sequence (the letters of the occupied Wyckoff sites). This allows a finer classification of structure types and offers suggestions not only for recognizing isotypic structures but also possible structural relationships like substitution, formation of vacancy or filled-in structure variants. 

The lattice approach has also been used for the systematic description of inorganic crystal structures (Wells 1975, pp. 119-55 Hyde and Andersson 1989, pp. 6-49), but the method is not just geometric and descriptive. It has a sound physical basis and can therefore be used for structure modelling. 

Since the submission of this article, further work by Halpern and co-workers (21) has led to the description of a chiraphos rhodium enamide crystal structure. 

The description of a given crystal by a lattice and a basis is not unique. One might e.g. choose to double the size of the unit cell by including more atoms into the basis. This would also lead to a different reciprocal lattice. This seams to lead to a contradiction, since the diffraction pattern should only depend on the crystal and not how we choose our description. As we will see in example A.2, the choice of a different basis leads to a change in the structure factor so that the combination of reciprocal lattice and structure factor always leads to the same diffraction pattern. 

In general, because the value of a crystal property depends on the direction of measurement, the crystal is described as anisotropic with respect to that property. There are exceptions for example, crystals having cubic symmetry are optically isotropic although they are anisotropic with respect to elasticity. For these reasons, a description of the physical behaviour of a material has to be based on a knowledge of crystal structure. Full descriptions of crystal systems are available in many texts and here we shall note only those aspects of particular... 

The internal structure or polymorphic state represents the molecular arrangement within a crystal and is manifested in the form of a definite heat of fusion (AHf) value. External structure or crystal habit is the outer description of a crystal and is described by its length, width, thickness, and surface appearance (roughness, smoothness, and porosity). Crystal growth may be impeded by adjacent crystals growing simultaneously or contacting container walls. As a result, the development... 

The crystal lattice is an abstract construction whose points of intersection describe the underlying symmetry of a crystal. To flesh out the description of a particular solid state structure, we must identify some structural elements that are pinned to the lattice points. These structural elements can be atoms, ions, or even groups of atoms as we see in this and the next chapter. We begin with some illustrative simple cases. Some of the chemical elements crystallize in particularly simple solid structures, in which a single atom is situated at each point of the lattice. 

Further symmetry occurs within the unit cell. This internal symmetry introduces relationships among the unit cell contents and can be due to rotation, reflection, and combinations of these with fractional translations. The internal symmetry of the unit cell allows the complete description of the entire contents on the basis of a unique structural unit, the asymmetric unit. Applying the internal unit cell symmetry to the asymmetric unit, followed by imposition of the lattice structure, leads to a complete description of the crystal. 

Quasiperiodic tilings are most widely used for the description of quasi-crystals. With appropriate atomic decorations of the vertices, they serve as structure models which explain physical properties of quasi crystals [39]. FVom a theoretical point of view, they are idealisations of real substances on which the usual models of statistical physics like the Ising model may be studied [40-42]. Quasiperiodic tilings arose before the discovery of quasi-crystals, however, more as an object of aesthetic interest in geometry [43,44]. 

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