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  Indexing a Powder Pattern     6 of 7

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 direct beam.

• For back reflections, i.e. where 2q > 90 you can measure S2 as the distance from the beam entry point.

Powder diffraction image

• 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 from:

Equation image  or   Equation image

• We know Bragg's Law: nl = 2dsinq  

and the equation for interplanar spacing, d, for cubic crystals is given by: 

Equation image   where a is the lattice parameter

this gives:

Equation image

• 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+ k2+ l2 values. Integer values are then assigned.

Podwer diffraction table image

• The integer values of h2+ k2+ l2 are then equated with their hkl values to index each arc, using the table shown below:

Powder indices table image

• 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.



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