The designation of crystal planes and faces

Knowing that a crystal i s built u pb y the repetition o f a unit cell, we can explain the development of faces of various kinds Fig. 27.21(a) illustrates this in two dimensions. The faces and F2 are formed by the bottoms and sides of the unit cells. Other faces F3 and F are possible, formed by the corners of the unit cells. Since the unit cell is of atomic size, we do not see the little steps but see only another face of the crystal. Because the crystal is built in this special way an important relation exists between the axial intercepts of any face and those of any other face. We compare the intercepts on the axes of the face F3 with those of a possible face F. The geometry of these faces is shown in Fig. 27.21(b). The line PL is produced until it intersects the x-axis at a and the y-axis at b. In intercept form the equation of the line is 

But there are other points on F3 x = ma, y = nb, where m and n are integers. Equation (27.2) must be satisfied at these points. Hence 

Therefore, if to a given face, F, of a crystal we assign intercepts, a, b, c on the coordinate axes, then the intercepts Uj,, Cj of any possible f ace, such as F4, of the crystal are rational multiples of a, b, c. This is a fundamental law of crystallography, the law of rational intercepts. 

Instead of describing a given face of the crystal by multiples of standard intercepts, we use the reciprocals of these multiples. That is, in terms of the intercepts a, b, c of the ref erence face, the intercepts of any face are given by 

If we measure distances in terms of the reference intercepts, x in units of a, y in units of b. 

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