Question

Show that in a cubic crystal [HKL] direction is always perpendicular to a plane whose Miller indices are (HKL). Show the general proof and not example of specific plane and direction.

Answer 1

Consider a crystal lattice with unit cell edges a, b and c. A crystal direction [uvw] is parallel to the direction joining the origin of the crystal lattice with the point with coordinates (ua, vb, wc)

Planes

A plane with Miller indices (hkl) passes through the three points (a/h,0,0), (0, b/k,0) and (0,0, c/l) on the edges of the unit cell. The set of parallel lattice planes passes through all similar points in the lattice. The plane d-spacing is the perpendicular distance from the origin to the closest plane and also the perpendicular distance between successive planes. In materials with cubic symmetry the crystal direction [uvw] and the normal to the plane (uvw) .

fig. The plane with Miller indices (hkl) makes intercepts a/h, b/k and c/l on the edges of the unit cell. The plane shown here is (221) because the intercept in the a direction is ½a, in the b direction ½b and in c direction c.

This shows how the (100), (110) and (111) planes are defined

Then, given the three Miller indices h, k, ?, (hk?) denotes planes orthogonal to the reciprocal lattice vector:

For cubic crystals with lattice constant a, the spacing d between adjacent (hk?) lattice planes is (from above)

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