Handout: Bragg Planes and Miller Indices
Biochemistry 102a
Fall 2002
Some nomenclature and mathematical notation to keep things straight when we are discussing threedimensional crystals and their diffraction patterns:
Unit cell axes. The directions of the three edges of the unit cell are denoted by the unit
$ , and c$ . Remember that these are not necessarily at right angles to one another.
vectors a$ , b
Unit cell dimensions. The lengths of the three edges of the unit cell are denoted a, b, and c.
Crystallographic coordinate system. Any position in the unit cell can be identified as (u, v, w),
where the coordinates refer to the fraction of the distance along each of the three unit-cell edges.
Thus, 0 # {u, v, w} < 1. I try to always use parentheses for coordinates and square brackets for Miller
indices. A more precise mathematical definition is that the point (u, v, w) is defined by the vector
 u  $  v  $  w $
 a +  b +  c
 a
 b
 c
Bragg planes. A set of planes through the crystal are Bragg planes if they are parallel, evenly
spaced, and contain every unit cell vertex. Each set of Bragg planes corresponds to a single spot in
the diffraction pattern. This spot is observed when the incoming and outgoing x-ray beams make
equal angles 2 = sin-1(8 / 2d) with the Bragg planes, where d is the perpendicular spacing of the
Bragg planes. (The equation is just Bragg’s Law with n = 1; n is taken to be 1 by convention when
matching spots to planes.)
Miller indices. Every set of Bragg planes/spot in the diffraction pattern is uniquely identified by the
Miller indices [h k l] (note: square brackets). To determine the Miller indices of a set of planes, look
at the plane closest to the origin of a unit cell but which does not pass through the origin. This plane
will pass through all three of these points in the unit cell: (1/h, 0, 0), (0, 1/k, 0), and (0, 0, 1/l), where
[h k l] are the Miller indices of this set of planes. Special case: planes that are parallel to one or two
unit cell edges have Miller indices of zero for the direction(s) of those edges. For example, all sets of
$ plane have Miller
Bragg planes parallel to the a$ , b
indices [0 0 l].
In the two-dimensional example shown here, plane 1
passes through the origin of the unit cell shown. Plane 2
passes through (1/4, 0) and (0, 1/2); therefore, the planes
shown are the [4 0] planes. Note that Miller indices that
are interger multiples of one another refer to sets of
Bragg planes that are parallel but have different
spacings. For example, [1 1 1] and [2 2 2] have the same
orientation, but the spacing of the [1 1 1] planes is twice
that of the [2 2 2] planes.