You are to measure the lattice parameters of the two crystal structures: (1) the fcc structure of aluminum and (2) the hexagonal structure of graphite. This means determining the length of the cube edge of the fcc structure and the length of the prism edge of the unit cell of graphite. Your basic data will be the diameter of the rings or the spacing of the spots as a function of the accelerating voltage of the electrons. At the same time you will be confirming the wave-like nature of particles in accordance with the DeBroglie theory.
Fig. 3 shows the scattering geometry. The angle between the
scattered beam and the incident beam is 
;
is the angle between
the incident beam and the crystal planes. Thus
where r is the radius of a given ring. The central spot is caused by the undeviated beam. The value of D is fixed and equal to 17.81 cm.
In the case of aluminum, measure the diameter of the rings
while holding the accelerating voltage constant. Since there is no
automatic regulation of the accelerating voltage you will have to
make minor adjustments of the HV knob during the course of a
measurement to keep it
at a selected value. Measure the ring diameters of as many rings as
you feel necessary for at least three values of the accelerating
voltage between 7500 and 10,000 volts. For each voltage calculate
the DeBroglie wavelength of the electrons.
From Eqn. 3 it is clear that the smallest ring corresponds to the
smallest set of 
; the next ring belongs to the the next largest
set, and so on. The smallest 
set which are either all odd or
all even is the set 
; therefore, these must be the values to
use with the smallest r value to calculate a. Other possible sets
of 
are 
, 
, 
, 
, etc. You should be
able to figure out which of these sets belongs to each of the diffraction
rings that you measure. For each one you should get the same value for
a, the edge-dimension of the unit cell
From all your data find the best value for a and
estimate your error. Compare your value to the accepted
value of 
determined from X-ray scattering.
Then, as a nice
confirmation, calculate a from the known density of aluminum, its
atomic weight, and Avagadro's number. For this you need to know that
each unit cell of the fcc structure contains 4 atoms.
=-1.5in =-9
In similar fashion for graphite, measure the
spacing of the spots as indicated on the figure at
right. In this case only one set of Miller indices
is involved, the set 
. Measure the distance r between as many spots
as seems reasonable for each accelerating voltage. Again determine
the spacing between reflecting planes and from that
calculate the edge dimension of the unit cell.
Be sure to take into account the fact that the edge of the unit
cell is not perpendicular to reflecting planes.
As shown below, d is the distance between planes of atoms, and
a is the length of the edge of the unit cell.
Compare your results with the value of 
given from X-ray measurements.