Fundamental translation vector

The points on a lattice are defined by three fundamental translation vectors, a, b, and c, such that the atomic arrangement looks the same in every respect when viewed from any point r as it does when viewed at point r ... 

The pattern points associated with a particular lattice are referred to as the basis so that the description of a crystal pattern requires the specification of the space lattice by ai a2 a3 and the specification of the basis by giving the location of the pattern points in one unit cell by K, i= 1,2,. .., (Figure 16.1(b), (c)). The choice of the fundamental translations is a matter of convenience. For example, in a face-centred cubic fee) lattice we could choose orthogonal fundamental translation vectors along OX, OY, OZ, in which case the unit cell contains (Vg)8 + (l/2)6 = 4 lattice points (Figure 16.2(a)). Alternatively, we might choose a primitive unit cell with the fundamental translations... 

The primitive rhombohedral cell in Figure 16.6 can be specified by giving the length a ai and the angle a between any pair of the fundamental translation vectors ab a2, 83. Choose ei along the projection of ai in the xy plane 0 is the angle made by ai with e3. 

Aperiodic crystal is described by two entities, the lattice and the basis. The (translational) lattice is a perfect geometrical array of points. All lattice points are equivalent and have identical surroundings. This lattice is defined by three fundamental translation vectors a,b,c. Starting from an arbitrarily chosen origin of the lattice, any other lattice point can be reached by a translation vector r that satisfies... 

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