Bravais lattice vectors defined

We can construct vectors which connect all equivalent points in reciprocal space, which we call G, by analogy to the Bravais lattice vectors defined in Eq. (3.1) ... 

If we consider a primitive Bravais lattice with cell edges defined by vectors a, 82, and aj, the corresponding reciprocal lattice is defined by reciprocal vectors bj, b2, and bj so that... 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

Crystals are periodic repetitions of a unit cell in space in each of the three directions defined by the lattice vectors. A unit cell can be described as a parallelepiped (the description used by the conventional Bravais system of lattices) containing some number of atoms at given positions. The three independent edges of the parallelepiped are the lattice vectors, whereas the positions of the atoms in the unit cell form the basis. Defining crystals in this way is not unique, as any subset of a crystal which generates it by translations can be defined as a unit cell, for example, a Wigner-Seitz cell, which is not even necessarily a parallelepiped. 

The lattice types are labeled by P (simple or primitive), F (face-centered), I (body-centered) and A B,C) (base-centered). Cartesian coordinates of basic translation vectors written in units of Bravais lattice parameters are given in the third column of Table 2.1. It is seen that the lattice parameters (column 4 in Table 2.1) are defined only by syngony, i. e. are the same for all types of Bravais lattices with the point symmetry F and all the crystal classes F of a given syngony. 

The lattice vectors connect all equivalent points in space this set of points is referred to as the Bravais lattice . The PUC is defined as the volume enclosed by the three primitive lattice vectors ... 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

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