Diffraction from a translationally periodic object

1.4 Diffraction from a translationally periodic object

Very quickly it was realised that diffraction was particularly valuable for the study of crystalline materials. It is worth us considering how the derived expressions will be affected when applied to a crystalline system.

So, how does the translationally periodic crystal effect the diffraction we observe from the material?

A similar diagram setup to that used in Section 1.1 and Figure 1.1, is shown below in Fig. 1.4 for a crystal with orthogonal axes (α = β = γ = 90◦). The only differences from the general object is that the volume units (i.e the unit cells) are now all identical and that the vector r is restricted to integral multiples of the repeat vectors a, b and c.

Figure 1.4: Scattering from an Object composed of Unit Cells

Starting with Equation (1.13), the total scattering from the object is given as:

The reciprocal lattice

Bragg Diffraction Pattern

The net result of this is that we have sharp diffraction peaks at the positions of the reciprocal lattice- the so called Bragg reflections. This creates diffraction patterns such as those shown in Fig. 1.5 - either single crystal or powder, depending on the type of sample and the technique used.

Figure 1.5: Comparison of a Powder diffraction pattern and a single crystal pattern

Thus far, our consideration of materials has been limited to crystalline materials, i.e. those that have translational symmetry. However, such a consideration is obviously limited as it excludes a vast proportion of potential materials that require characterisation. In this section we will consider what we might expect to see if a non-periodic structure is exposed to incident radiation or if there are local variations from the average structure

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