We shall now consider the powder patterns from a sample
crystal. The sample is known to have a cubic
structure, but we don't know which one.
We remove the film strip from the Debye camera after exposure, then develop and fix it.
From the strip of film we make measurements of the position of each diffraction line. From
the results it is possible to associate the sample with a particular type of cubic
structure and also to determine a value for its lattice parameter.
When the film is laid flat, S1
can be measured. This is the distance
along the film, from a diffraction line, to the centre of the hole for the transmitted
For back reflections, i.e. where 2q >
90° you can measure S2 as the
distance from the beam entry point.
The distance S1 corresponds to
a diffraction angle of 2q. The angle between the
diffracted and the transmitted beams is always 2q.
We know that the distance between the holes in the film, W, corresponds to a diffraction angle of q = p. So we can find q
We know Bragg's Law: nl
the equation for interplanar spacing, d, for
cubic crystals is given by:
- where a is the lattice parameter
From the measurements of each arc we can now generate
a table of S1,
q and sin2q.
If all the diffraction lines are considered, then
the experimental values of sin2q
should form a pattern related to the values of h,
k and l
for the structure.
We now multiply the values of sin2q
by some constant value to give nearly integer values for
all the h2+
values. Integer values are then assigned.
The integer values of h2+
are then equated with their hkl
values to index each arc, using the table shown below:
For some structures e.g. bcc, fcc, not all planes
reflect, so some of the arcs may be missing.
It is then possible to identify certain structures,
in this case fcc (- the planes have hkl
values: all even, or all odd in the table above).
For each line we can also calculate a value for
a, the lattice
parameter. For greater accuracy the value is averaged
over all the lines.