Miller indices and crystal directions How to describe a particular crystallographic plane and direction ? not necessarily of equal length Primitive translation vectors a1, a2, a3 : not necessarily right angles Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions n1,n2,n3 here 1,3,1 Miller indices (h,k,l) specify set of equivalent planes Take the reciprocals 1/n1,1/n2,1/n3 here 1,1/3,1 Clear fractions with the smallest possible integer here 3,1,3

Simplified example for cubic system

negative intercept is denoted by a bar : the symbol denotes all planes equivalent to (h,k,l) example for a plane that cuts the a-axis at

Set of four Miller indices for hexagonal crystals first three Miller indices add up to zero

Crystal directions Lattice vector: T=n1a1+n2a2+n3a3 direction defined by [n1 n2 n3] If [n1 n2 n3] have a common factor, the latter is removed E.g., [111] instead of [222] Note: [hkl] is in general not normal to the (hkl) plane