Law of Rational Indices

If the intercepts of any other face of the crystal on the three axes are measured, the following generalizations observed which is called law of rational indices:

The intercepts made by any face of the crystal on the crystallographic axes are either:

(i)    Same as those of the unit plane, or
(ii)    Simple whole number multiples of those of the unit plane, or
(iii)    One or two intercepts mat be infinity, if the face is parallel to one or the two axes, i.e., the face does no cut one or the two axes.

For example, in the intercepts made by face LMN are 2a, 2b and 3c which are simple whole number multiples of those of the unit plane. Again consider the case of cube. Suppose the origin 0 of the crystallographic axes lies at the centre of symmetry and be axes OX, OY, OZ are parallel to the edges as shown in Suppose the axis OX cuts the face ABCD at the point L, Similarly, supple the axes OY and OZ cut the faces at M and N as shown in . If LMN is taken as the unit plane, the intercepts made by the unit plane on the three axes are OL, OM and ON. The face ABCD cuts the X-axis at L i.e., makes the same intercept as that made by the unit plane but it doses not cut Y=axis and Z-axis at all because these two axes are parallel to this face. In other words, the intercepts made by the face ABCD on the Y-AXIS AND Z-axis are infinity.

 
         Intercepts made by the
     facets of a cube on the axes.

In general if a face PQR makes intercepts p,q and r on the three axes and the intercepts makes by the unit plane are a,b,c then we have p = na, q=n’b and r=n”c where n,n’ where n” are simple whole numbers of some out of these may be infinity.

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