Unit of pattern

Two pieces of information about the fundamental atomic pattern may be deduced from the actual shape of a crystal, provided this crystal shows a sufficient variety of faces and is large enough to permit measurements of the angles between the faces. One is a knowledge of the shape and relative dimensions of the unit of pattern. The other is a partial knowledge of the symmetries of the atomic arrangement. 

The unit of pattern ( unit cell ). A crystal consists of a large number of repetitions of a basic pattern of atoms. Just as in many textile... 

The only patterns of exactly repeated environments capable of indefinite extension are those in which successions of pattern-units lie on straight lines. Consider the shape of the unit of pattern, first of all in the simpler case of a plane pattern, such as that shown in Fig. 4. Mark any point such as A, and then mark other points whose surroundings are exactly the same (in orientation as well as geometrical character) as... 

Fig. 5 also shows an example of a molecular structure—that of hexa mothylbenzene. C6(CH3)6. The molecules, which can be represented as disks, are all stacked parallel to each other, and if the centre of each molecule is joined to those of its nearest neighbours, the structure is divided into a number of identical units of pattern, each of which is a non-reetangular box with all three sets of edges unequal in length. 

Diffraction of light by line gratings. Above grating of evenly spaced lines, with diffraction pattern. Below grating in which the unit of pattern is a pair of lines (repeat distance same as in the first), with diffraction pattern... 

The reason why it is not usually possible to employ this direct method for the solution of crystal structures has already been indicated at the beginning of Chapter VII it is that we do not usually know, and cannot determine experimentally, the phases of the various diffracted beams with respect to a chosen point in the unit of pattern. However, for certain crystals we can from the start be reasonably certain of the phase relations of the diffracted beams, or can deduce them from crystallographic evidence, and in these circumstances we can proceed at once to combine the information, either mathematically or by experimental methods in which light waves are used in place of X-rays. Otherwise, it is necessary to find approximate positions by trial, the approximation being taken as far as is necessary to be certain of the phases of a considerable number of reflections as soon as the phases are known, the direct method can be used. 

In the formation of crystals the constituent particles try to pack as closely as possible so as to attain a state of maximum possible density and stability. Since in different crystals, the units of pattern have different shapes and sizes, the actual mode of closest packing is different for different crystals. 

By far the greater proportion of incident X-radiation is transmitted by a crystalline sample. However, a small fraction is scattered (effectively reflected) in all directions by every motif in the material, without change in wavelength. The motif is the repeating unit of pattern in a crystal it is the TAG molecule in the case of a fat crystal. Motifs can be considered to be located at or near the intersections of an imaginary 3-dimensional grid called the crystal lattice and the intersections are called lattice points (Hammond, 1997). 

There are seven types of unit cell (see Tal)le 1 I) and therefore seven simple or primitive lattices with one unit of pattern at each cell corner. 

If the contents of a unit cell have symmetry, containing a number of units of pattern (atoms, molecules), the number of distinct types of space lattice becomes fourteen (Fig. 82) (Bravais, 1818). And when other symmetry operations are recognised (e.g. rotation of the lattice) there are found to be 230 distinct varieties of crystal symmetr ... 

The application of these symmetry elements is identical to that depicted in Figure 3.3. However, as we are ultimately aiming to explain patterns in two and three dimensions rather than solid shapes, it is convenient to illustrate the symmetry elements with respect to a unit of pattern called a motif, placed at a general position with respect to the rotation axis, (Figure 3.6). In this example the motif is an asymmetric three atom planar molecule . (In crystals, the motif is a group of atoms, see Chapter 5). 

In two dimensions there are five possible units of pattern, unit cells, which can build a repetitive pattern by translation in directions parallel to the edges (Fig. 27.13). The unit cell with the 120° angle is interesting because it permits a threefold or sixfold axis of symmetry at a point in the pattern, which is a permissible type of symmetry in two-dimensional patterns. 

It is also true that Wells [3, Chapter 1], perfectly well introduced a systematic and rigorous coding of the topology of tessellations and networks he worked with, which is now called the Wells point symbol notation, and that this was a simple coding scheme over the eireuitry and valences, about the vertices, in the unit of pattern of the tessellations and networks. The Wells point symbol notation was, however, nonetheless an important development for the rigorous mathematieal basis it put the tessellations and networks on, formally, as quasi-solutions (n, p) for the Schlafli relation shown as Eq. (2). 

Equation (4a) is the generic statement of the new elasticity law [59, Chapter 2], It states that elastic modulus is equal to an integral over the force density of a material undergoing elastic chemical bond deformations inside the unit of pattern of the material, in response to an applied stress. 

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