Shape of the unit cell

Therefore, any point in a three-dimensional lattice can be described by a vector, q, defined in Eq. 1.1, where u, v, and w are integer numbers 

The first three parameters (a, b and c) represent the lengths of the unit cell edges, and the last three (a, p and y) represent the angles between them. Thus, a is the angle between b and c, p is the angle between a and c, and y is the angle between a and b. 

Unit cell dimensions are usually quoted in angstroms (A, where 1 A = 10 ° m = 10 cm), nanometers (nm, 1 nm = 10 m), or picometers (pm, 1 pm = 10 m) for the lengths of the unit cell edges, and in degrees ( ) for the angles between basis vectors. To differentiate between basis vectors (a, b, c), which appear in bold, the lengths of the unit cell edges a, b, c) always appear in italic. 

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In the powder diffraction technique, a monochromatic (single-frequency) beam of x-rays is directed at a powdered sample spread on a support, and the diffraction intensity is measured as the detector is moved to different angles (Fig. 1). The pattern obtained is characteristic of the material in the sample, and it can be identified by comparison with a database of patterns. In effect, powder x-ray diffraction takes a fingerprint of the sample. It can also be used to identify the size and shape of the unit cell by measuring the spacing of the lines in the diffraction pattern. The central equation for analyzing the results of a powder diffraction experiment is the Bragg equation... 

The size and shape of the unit cell is determined, usually from rotation photographs and... 

The amount of structural information obtainable by the morphological Study of skeletal crystals is naturally very limited, especially when they are distorted. In order to be able to deduce the shape of the unit cell it is necessary to have well-fotmed polyhedral crystals. The faces of such crystals are, as we have already seen, related in some simple way to the unit cells. We must now define more closely what is meant by the last phrase— related in some simple way to the unit cells —and to do this it is necessary to give some account of the accented nomenclature of crystal planes. 

Fig. 15. Determination of the probable shape of the unit cell from interfacial angles.

At an earlier point (p. 185) the unit cell of a crystalline structure was described as one of a large number of identical prisms, which, when oriented in the same way and stacked together in three dimensions, form a perfect crystal. The corners of an array of unit cells put together in this way are said to be the points of a space lattice the surroundings about each point of the lattice must be identical to the surroundings about every other point. Additional lattice points may sometimes be put at the face centers or at the body centers of the unit cells in crystalline sodium chloride (Fig. 12-2), for example, chloride ions are located both at the corners and the face centers of the unit cell, and an observer at a corner would have the same surroundings as one at a face center. The description of the structure of a crystalline solid is then a description of the size and shape of the unit cell and of the locations of the atoms within it. 

The structure of a crystal is solved in three steps (Cullity, 1956). Firstly, the size and shape of the unit cell (a crystal lattice consists of identical unit cells) is found from the angular distribution of the diffraction beams. Secondly, the number of molecules per unit cell is computed from the size and shape of the unit cell, the chemical composition of the sample and the sample s measured density. Lastly, the positions of the molecules within the unit cell are deduced from the relative intensities of the diffraction beams. Data analysis, which is complex, is described by Woolfson and Fan (1995) and Clegg (2001). 

Crystals of the material are grown, and isomorphous derivatives are prepared. (The derivatives differ from the parent structure by the addition of a small number of heavy atoms at fixed positions in each — or at least most — unit cells. The size and shape of the unit cells of the parent crystal and the derivatives must be the same, and the derivatization must not appreciably disturb the structure of the protein.) The relationship between the X-ray diffraction patterns of the native crystal and its derivatives provides information used to solve the phase problem. 

What can we leaxn about the internal arrangement of atoms, ions, or molecules from the appearance and physical properties of a crystal The macroscopic physical properties of crystals are related to the arrangements of atoms within them. A careful examination of these physical properties leads to much useful information on the symmetry of the atomic arrangement, on the shape of the unit cell, and on the overall arrangement of molecules within the unit cell. Selected examples of such studies are described in this Chapter. 

FIGURE 21.6 Shapes of the unit cells in the seven crystal systems. The... 

The lattice itself, including the shape of the unit cell, may be chosen in an infinite number of ways. As an example, a second alternative lattice with a different unit cell is shown in Figure 1.3. Both the origin of the lattice and the shape of the unit cell have been changed when compared to Figure 1.1, but the content of the unit cell has not - it encloses the same molecule. 

As seen in Table 1.10, there is one powder Laue class per crystal system, except for the trigonal and hexagonal crystal systems, which share the same powder Laue class, 6/mmm. In other words, not every Laue class can be established from a simple visual analysis of powder diffraction data. This occurs because certain diffraction peaks with potentially different intensities (the property which enables us to differentiate between I ue classes 4/m and 4/mmm 3, 3m, 6/m and 6/mmm m3 and m3m) completely overlap since they are observed at identical Bragg angles. Hence, only Laue classes that differ from one another in the shape of the unit cell (see Table 1.11, below), are ab initio discernible from powder diffraction data without complete structure determination. 

Since all the cells of the lattice shown in Fig. 2-1 are identical, we may choose any one, for example the heavily outlined one, as a unit cell. The size and shape of the unit cell can in turn be described by the three vectors a, b, and c drawn from one corner of the cell taken as origin (Fig. 2-2). These vectors define the cell and are called the crystallographic axes of the cell. They may also be described in terms of their lengths a, b, c) and the angles between them (a, P, y). These lengths and angles are the lattice constants or lattice parameters of the unit cell. 

The shape and size of the unit cell are deduced from the angular positions of the diffraction lines. An assumption is first made as to which of the seven crystal systems the unknown structure belongs to, and then, on the basis of this assumption, the correct Miller indices are assigned to each reflection. This step is called indexing the pattern and is possible only when the correct choice of crystal system has been made. Once this is done, the shape of the unit cell is known (from the crystal system), and its size is calculable from the positions and Miller indices of the diffraction lines. 

Fig. 2.6. Illustration of the internal rearrangements that attend the structural transformation in Zr02 (adapted from Finnis et al. (1998)). Although the overall shape of the unit cell can be described in terms of an affine deformation characterized by a constant deformation gradient F, the individual internal atoms do not transform accordingly.

FIGURE 15.3 Crystal systems. Shown is the shape of the unit cell for each system, the number N of different Bravais lattices in each system, geometrical characteristics of the unit cell, and optical anisotropy (birefringence). Some examples of crystals of various materials are given. 

Scattering from crystalline materials irradiated by X-rays or neutrons will be strong in some directions (constructive interference) and much less (almost perfect destmctive interference) in other directions. The directions for the constructive interference are determined from the size and shape of the unit cells. The intensities of the constructive interference peaks are given by the... 

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