Interplanar Distance in a Cubic Crystal

Interplanar Distance in a Cubic Crystal

Consider a cube and a plane ABC as shown in Fig.0.24. Let the Miller indices of the plane be h, k and l. Now draw perpendicular OD from the origin of the cube to the plane ABC.

Let

a = the lattice constant of the cube,

dhkl = Perpendicular distance between the origin and the plane ABC i.e., the distance between two adjacent parallel planes having Miller indices (hkl),

α, β and γ = Angles which the perpendicular makes with x, y and z axes respectively, and

OA, OB and OC = Intercepts of the plane along x, y and z axes respectively.

We know that the Miller indices of a plane are the smallest integers of the reciprocals of its intercepts. Therefore, the intercepts may also be expressed as reciprocals of Miller indices.

From the geometry of the right angled triangles OAD, OBD and OCD, we get