Miller Indices

A system of notation is required to identify particular direction(s) or plane(s) to characterize the arrangement of atoms in a unit cell

Rules for Miller Indices (Planes)

1. Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.

2. Take the reciprocals

3. Clear ractions

4. Reduce to lowest terms

5. For example, if the x-, y-, and z- intercepts are 2, 1, and 3, the Miller indices are calculated as:

6. Take reciprocals: 1/2, 1/1, 1/3

7. Clear fractions (multiply by 6): 3, 6, 2

8. Reduce to lowest terms (already there)

Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero. A generic Miller index is denoted by (hkl). A family of planes is represented by {hkl} If a plane has negative intercept, the negative number is denoted by a bar above the number. Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that the crystal may not have!

Miller Indices - Direction

•   A vector of convenient length is placed parallel to therequired direction

•    The length of the vector projection on each of three axes aremeasured in terms of unit cell dimensions

•    These  three  numbers  are  made  to  smallest  integer  values,  known  as  indices,  by multiplying or dividing by a commonfactor

•  The three indices are enclosed in square brackets, [uvw].

•  A family of directions is represented by <uvw>

Some General Principles

1.  If a Miller index is zero, the plane is parallel to that axis.

2. The smaller a Miller index, the more nearly parallel the plane is to the axis.

3. The larger a Miller index, the more nearly perpendicular a plane is to that axis.

4.  Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane

5.  Miller indices are almost always small.

Why Miller Indices?

1. Using reciprocals spares us the complication of infinite intercepts.

2. Formulas  involving  Miller  indices  are  very  similar  to  related  formulas  from  analytical geometry.

3. Specifying dimensions in unit cell terms means that the same label can be applied to any face with a similar stacking pattern, regardless of the crystal class of the crystal. Face 111 always steps the same way regardless of crystal system.