Miller Indices

The translation vectors of the convenient unit cell, e.g. the cube edges of the fcc structure (which is not primitive) are used for the coordinate axes. The plane shown intercepts at 3, 2, 2 on the axes. As, by definition, these three points are not co-linear, therefore they can adequately define the plane. As we shall see later, the reciprocal of these numbers are more useful. First, reciprocate these numbers thus getting 1/3, 1/2, 1/2. Miller indices are the smallest integers having the same ratio as these. The Miller indices of this plane is (hkl) = (2,3,3) By this definition, if the plane cuts an axis at infinity, the corresponding index will be zero. By convention, if the intercept has a negative value, the corresponding index is also negative. A minus sign is normally placed above that index in the bracket. Miller indices are also used to denote a set of planes which are parallel. For instance, the plane (200) is parallel to (100). The former cuts the x-axis at a /2. Also by symmetry, many sets of planes, e.g. all the faces of a cube, may be represented by a single set of Miller indices (100). In this case the curly bracket is used, hence {100}. In other words the {100} automatically includes the planes (100), (010) and (001). The direction of any general vector can also be expressed in terms of a set of indices h, k and l. These indices are the smallest integers that are proportional to the components of the vectors along the axes of the crystal (same as the translation vectors of the unit cell). Unlike the Miller indices above, these integers used for defining the direction of the general vector are not derived from the reciprocal. To distinguish from the Miller indices for planes, i.e. (hkl) or {hkl}, the direction indices are enclosed in square brackets, hence [hkl]. Also the indices [hkl] are used to denote many sets of equivalent directions similar to the plane designations above. For a cubic system, it can be shown that the vector [hkl] is normal to the planes defined by the Miller indices {hkl}. This fact happens to be extremely useful in the analysis of x-ray diffraction.