Branch : Computer Science and Engineering
Subject : Fundamental of Electronic Devices
Planes and Directions: Miller indices
In discussing crystals it is very helpful to be able to refer to planes and directions within the lattice. The notation system generally adopted uses a set of three integers to describe the position of a plane or the direction of a vector within the lattice.
Planes and Directions:
We first set up an xyz coordinate system with the origin at any lattice point (it does not matter which one because they are all equivalent!), and the axis are lined up with the edges of the cubic unit cell. The three integers describing a particular plane are found in the following way:
 Find the intercepts of the plane with the crystal axis and express these intercepts as integral multiples of the basis vectors (the plane can be moved in and out from the origin, retaining its orientation, until such an integral intercept is discovered on each axis).
 Take the reciprocals of the three integers found in step 1 and reduce these to the smallest set of integers h, k, and l, which have the same relationship to each other as the three reciprocals.
 Label the plane (hkl).
Miller indices:
 The three integers h, k, and I are called the Miller indices; these three numbers define a set of parallel planes in the lattice.
 One advantage of taking the reciprocals of the intercepts is avoidance of infinities in the notation.
 One intercept is infinity for a plane parallel to an axis; however, the reciprocal of such an intercept is taken as zero.
 If a plane contains one of the axis, it is parallel to that axis and has a zero reciprocal intercept.
 If a plane passes through the origin, it can be translated to a parallel position for calculation of the Miller indices.
 If an intercept occurs on the negative branch of an axis, the minus sign is placed above the Miller index for convenience,such as (hkl).

From a crystallographic point of view, many planes in a lattice are equivalent; that is, a plane with given Miller indices can be shifted about in the lattice
simply by choice of the position and orientation of the unit cell.
 The indices of such equivalent planes are enclosed in braces {} instead of parentheses.
 For example, in the cubic lattice of Fig. just given below all the cube faces are crystallographically equivalent in that the unit cell can be rotated in various directions and still appear the same.
 The six equivalent faces are collectively designated as {100}.
 Two useful relationships in terms of Miller indices describe the distance between planes and angles between directions.
 The distance d between two adjacent planes labeled (hkl) is given in terms of the lattice constant, a, as
The angle θ between two different Miller index directions is given by