#&# References: &Tipler, Chap. 5, pages 166-173. &Kittel, Intro. to Solid State Physics, 4th Ed. Chaps. 1-2.
When a beam of photons (EM waves) or electrons (matter waves) falls on a crystal, the waves are scattered from the individual atoms of the crystal. Depending on the wavelength and direction of the incident beam and the spacing of the atoms, there will be directions with respect to the incident beam along which strong reflections take place. These directions are those for which constructive interference between the scattered waves occurs. A detailed analysis of this scattering process is too complicated to present here. There is a simple way to express the condition for constructive interference due to W. L. Bragg. His proposal was to consider planes of atoms within the crystal and to treat these planes as a "stack of mirrors" from which the incident waves are reflected. If the angle of incidence (equal to the angle of reflection), the spacing between planes, and the wavelength are all just right, the reflections from successive planes will constructively interfere and a strongly reflected (diffracted) beam will result. The Bragg condition is
where 
is the angle
of incidence, n is the order of the diffraction,
and d is the distance between crystal planes.
In crystallography, the various planes of atoms are denoted
by Miller indices 
. The Miller indices for a plane of atoms
are the reciprocals of the intersections of the plane with the
three axes of the unit cell associated with the crystal structure.
In order to denote a whole set of planes with only one set of
Miller indices, they are chosen as the set with the lowest whole
number ratio of the reciprocals of the intercepts. (See Kittel,
Chap. 1, for a good exposition of crystal structure.) The
perpendicular distance between planes with indices 
in a
structure with a cubic lattice is
where a is the length of the side of the cubic unit cell. The figure below shows a unit cube with two sets of Bragg planes for the face centered cubic (fcc) structure of aluminum. The Miller indices for the two sets of planes are also given.
=-2.25in =-8 Combining the two equations above gives an expression which relates the wave length of the electrons to the scattering angle.
where the order n is taken to be unity.
=-2.25in
=-4
Not all possible combinations of h, k
and l yield crystal planes from which scattering
will occur. Because of the crystal structure,
some sets 
give planes which will
give totally destructive interference when all the atomic sites are
taken together. The crystal structure factor is used
to determine which sets
of Miller indices correspond to planes which give strong
reflections. In the case of the fcc structure of aluminum, the
structure factor vanishes unless h, k, and l are either
all odd or all even. So sets (planes) such as (211) and (100) yield
no corresponding diffraction of the incident wave.
The ring pattern seen when the electron beam is directed at the
aluminum occurs because the target consists of many, tiny
microcrystals
with their axes oriented randomly with respect to the incoming
beam. Thus, for any given wavelength 
and
interplane spacing d,
there is somewhere in the target a microcrystal with its axes
oriented
just right so as to give an interference maximum at the angle
required by the Bragg equation. Each such microcrystal diffracts the
incident beam to one spot on a given circle. The locus of such
points resulting from many different microcrystals with different
azimuthal orientations with respect to the incident beam is the
observed circle. Each circle corresponds to a different set of
Miller indices. The values of 
and a are
the same for all rings.
The spot pattern (hexagonal array) seen when the electron beam
is directed at the graphite occurs because the target has regions
which consist of a single crystal. The crystal structure of graphite
is shown in Fig. 2. In each layer each atom has three nearest
neighbors, each at a distance of about
. The unit cell for
graphite is shown with a dotted outline
in both a perspective and a top view. A unit cell is
defined as the smallest structure which, when stacked together,
will produce the crystal structure. The lattice
for the structure is called hexagonal, but the unit cell
is prism shaped and each one contains four carbon atoms.
The unit cell
includes two layers of atoms because all the atoms in one layer are
not directly over those in the layer below. The positions are
reproduced in alternate layers.
The planes of atoms to be considered as
reflecting planes in this experiment are the set (100), that is the
planes parallel to the sides of the unit cell. Because of the
crystal symmetry there are three sets of rotationally equivalent
planes, each at an angle of 
with respect to the other
two sets.
Thus the incoming beam is diffracted into an hexagonal array of
spots, and the lines connecting the spots intersect at angles of
(
if you take the acute angles).