Crystalline solids Bravais lattices

Bravais lattice — used to describe atomic structure of crystalline -> solid materials [i,ii], is an infinite array of points generated by a set of discrete translation operations, providing the same arrangement and orientation when viewed from any lattice point. A three-dimensional Bravais lattice consists of all points with position vectors R ... 

A solid surface is intrinsically an imperfection of a crystalline solid by destroying the three-dimensional (3D) periodicity of the structure. That is, the unit cell of a crystal is usually chosen such that two vectors are parallel to the surface and the third vector is normal or oblique to the surface. Since there is no periodicity in the direction normal or oblique on the surface, a surface has a 2D periodicity that is parallel to the surface. By considering the symmetry properties of 2D lattices, one obtains the possible five 2D Bravais lattices shown in Figure 2. The combination of these five Bravais lattices with the 10 possible point groups leads to the possible 17 2D space groups. The symmetry of the surface is described by one of these 17 2D symmetry groups. 

Taking into account such nonprimitive unit cells, the structure of any crystalline solid can be represented in terms of one of 14 possible basic types called Bravais lattices (Figure 6.14). 

A central tenet of materials science is that the behavior of materials (represented by their properties) is determined by their structure on the atomic and microscopic scales (Shackelford, 1996). Perhaps the most fundamental aspect of the structure-property relationship is to appreciate the basic skeletal arrangement of atoms in crystalline solids. Table 2.21 illustrates the fundamental possibilities, known as the 14 Bravais lattices. All crystalline structures of real materials can be produced by decorating the unit cell patterns of Table 2.21 with one or more atoms and repetitively stacking the unit cell structure through three-dimensional space. 

In case of harmonic solids with Debye-like low-frequency dynamics and vibrational isotropy, that is, polycrystalline sample, or single-crystalline one with a cubic Bravais lattice, the Debye velocity of sound vd can be precisely determined from the expression [107]... 

The existence of crystals has provided a tremendous boost to the study of solids, since a crystalline solid can be analyzed by considering what happens in a single unit of the crystal (referred to as the unit cell), which is then repeated periodically in all three dimensions to form the idealized perfect and infinite solid. The unit cell contains typically one or a few atoms, which are called the basis. The points in space that are equivalent by translations form the so called Bravais lattice. The Bravais lattice and the basis associated with each unit ceU detennine the crystal. This regularity has made it possible to develop powerful analytical tools and to use clever experimental techniques to study the properties of solids. 

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