Conventional crystallographic unit cells

The unit cells described above are conventional crystallographic unit cells. However, the method of unit cell construction described is not unique. Other shapes can be found that will fill the space and reproduce the lattice. Although these are not often used in crystallography, they are encountered in other areas of science. The commonest of these is the Wigner-Seitz cell. 

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

Quasicrystals or quasiperiodic crystals are metallic alloys which yield sharp diffraction patterns that display 5-, 8-, 10- or 12-fold symmetry rotational axes - forbidden by the rules of classical crystallography. The first quasicrystals discovered, and most of those that have been investigated, have icosahedral symmetry. Two main models of quasicrystals have been suggested. In the first, a quasicrystal can be regarded as made up of icosahedral clusters of metal atoms, all oriented in the same way, and separated by variable amounts of disordered material. Alternatively, quasicrystals can be considered to be three-dimensional analogues of Penrose tilings. In either case, the material does not possess a crystallographic unit cell in the conventional sense. 

The choice of unit cell shape and volume is arbitrary but there are preferred conventions. A unit cell containing one motif and its associated lattice is called primitive. Sometimes it is convenient, in order to realise orthogonal basis vectors, to choose a unit cell containing more than one motif, which is then the non-primitive or centred case. In both cases the motif itself can be built up of several identical component parts, known as asymmetric units, related by crystallographic symmetry internal to the unit cell. The asymmetric unit therefore represents the smallest volume in a crystal upon which the crystal s symmetry elements operate to generate the crystal. 

The cluster method has been implemented in the program MOPAC. In conventional solid-state methods, the translation vector is defined in terms of the crystallographic unit cell sizes a, b, and c and angles a, and y. For chemical calculations, it is more convenient to define the translation vector in terms of the atomic coordinates of the polymer, either Cartesian or internal. 

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

As noted above, when a is irrational, the incommensurate modulation occurs, and this is shown schematically in Figure 1.53. The exact description of incommensurately modulated structure is impossible using only conventional crystallographic symmetry in the unit cell of any size smaller than the crystal. The periodicity of the structure can only be restored by using two different periodic functions. The first function is the conventional crystallographic translation, and the second one is the sinusoidal modulation function with certain period, which is incommensurate with the corresponding translation, and amplitude. 

Regardless of the nature of the diffraction experiment, finding the unit cell in a conforming lattice is a matter of selecting the smallest parallelepiped in reciprocal space, which completely describes the array of the experimentally registered Bragg peaks. Obviously, the selection of both the lattice and the unit cell should be consistent with crystallographic conventions (see section 1.12, Chapter 1), which impose certain constraints on the relationships between unit cell symmetry and dimensions. 

For all RMC methods of modelling crystalline materials the configuration must consist of an integral number of unit cells along each axis, i.e. it must be a supercell. The unit cell must therefore be predetermined by conventional crystallographic techniques. 

Figure 1. Drawing of the cellulose I unit cell, according to the standard crystallographic convention in ref. 7, by Bill Garner, Martin Marietta Corporation.

New work on cellulose should draw upon some of the observations herein. Unit cells should conform to crystallographic convention. The temperature factor Is a valuable gauge of the validity of the data set. Perhaps some electron diffraction studies of the hkO plane will be able to give precise values of the a and b dimensions to aid In the stereochemical analyses. 

Coordinates of the crystallographic positions (in units of bottom) of all ions are given. The symbol Z stands for one conventional unit cell... 

An important aim of simulation methods is to be able to predict crystal structures in advance. This is perhaps a bit ambitious for the time being as it requires both interatomic potentials which are reliable over a wide range of distances and methods which can sample vast regions of conformation space. However, an aim which can, and is, being achieved is the solution of structures given a unit cell and composition, both of which can normally be readily obtained even when a structure proves difficult to solve completely by conventional crystallographic means. 

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