Lattice geometrical points

A geometric interpretation follows from mapping the n x n fee-matrices B onto points P(B) in an n2-dimensional euclidean space. There D(B, E) is the Lt-distance, the city block -or taxi driver -distance, between P(B) and P(E). Thus an FIEM(A) corresponds to a lattice of points in an n2-dimensional euclidean space. The reaction matrices correspond to vectors between the fee-points [9,19,33]. 

An ideal crystal is an ordered state of matter in which the positions of atoms (or nuclei or ions or molecules) are repeated periodically in space. The periodicity of the atomic positions can also be described by means of a unit cell. This is a geometric figure constructed so that when a large number of them are placed with the same periodicity as the atoms, they fiU the space with no overlap and without any space between. The positions and types of atoms in the primitive unit cell are called the basis. The set of a translation, which generate the entire periodic crystal by repeating the basis, is a lattice of points in space called the Bravais lattice. There are 14 different types of the Bravais lattice. They are presented in Figure 4.1. 

In order to understand the idea of the geometric systematization, the general character of the symmetry elements will be explained by a simple example. Fig. 41 represents a crystal which possesses the property of self-coincidence on rotation through 180° about the axis A A. This is termed the symmetry property of a two-fold (digonal) axis of rotation it is determined by the fact that in this crystal lattice all points are translated into one another on rotation through 180° about the axis AA, as shown diagrammatically in Fig. 42. 1 changes into 2, 1 into 2, etc. 

These sites of the motifs, which are represented by simple geometrical points, have a special significance and they are known as Lattice. ... 

The two-dimensional infinite array of geometrical points symmetrically arranged in a plane where the different motifs may be placed to create the patterns is known as plane lattice. Figures 2.1 and 2.2 are the examples of plane lattices, where the neighborhood of every point is identical. Figures 2.1 and 2.2 are two types of plane lattices, where motifs are to be placed to generate the desired patterns [1]. 

Note The plane lattice of a two-dimensional pattern is that array of geometric points which specifies the scheme of repetition that is present in the pattern. There is no difference between the neighbors of one lattice site from any of its neighborhood. Actually the situation becomes so much identical that if the attention from one lattice point is removed, it then becomes impossible to identify and locate the same lattice point. 

A lattice is an array of geometrical points either in two or in three dimensions to make plane or space lattices. 

At this point, you may find that the subject of symmetry in a crysted structure to be confusing. However, by studying the terminology carefully in Table 2-2, one can begin to sort out the various lattice structures and the symbols used to delineate them. All of the crystal systems can be described by use of either Schoenflies or Hermaim-Mauguin S5mbols, coupled with the use of the proper geometrical symbols. 

One important difference between the solid and liquid phases is that the isotropy of the liquid is replaced by the anisotropy of the solid. The geometrical correlations that exist in a solid lattice may be expected to play an important part when reaction centers are localized and can react with their neighboring environment. In this section we point out some of these effects in the radiolysis of solid n-hexane the trans/cis hexene yields, the isomeric dimer pattern, and the fragmentation processes. 

Since the directions of reflected rays are obtained by joining the centre of the sphere to points on its surface, the crystal itself may be regarded as rotating in the centre of the sphere of reflection, while the reciprocal lattice of this same crystal rotates about a different point—-the point where the beam emerges from the sphere. If this seems odd, it must be remembered that the reciprocal lattice is a geometrical fiction and should not be expected to behave other than oddly the fact is, the reciprocal lattice is concerned with directions its magnitude and the location of its origin are immaterial. 

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