Page 45 - Instant notes

Diffraction by solids 31

The subjects covered in this topic are indifferent to the nature of the radiation used,
and the arguments may be applied to all types of diffraction study.

Bragg equation

In crystallographic studies, the different lattice planes which are present in a crystal are
viewed as planes from which the incident radiation can be reflected. Diffraction of the
radiation arises from the phase difference between these reflections. For any two parallel
planes, several conditions exist for which constructive interference can occur. If the
radiation is incident at an angle, θ, to the planes, then the waves reflected from the lower
plane travel a distance equal to 2d sinθ further than those reflected from the upper plane
where d is the separation of the planes. If this difference is equal to a whole number of
wave-lengths, nλ, then constructive interference will occur (Fig. 1). In this case, the
Bragg condition for diffraction is met:
n λ=2d sinθ

In all other cases, a phase difference exists between the two beams and they interfere
destructively, to varying degrees. The result is that only those reflections which meet the
Bragg condition will be observed. In practice, n may be set equal to 1, as higher order
reflections merely correspond to first order reflections from other parallel planes which
are present in the crystal.
For most studies, the wavelength of the radiation is fixed, and the angle θ is varied,
allowing d to be calculated from the angle at which reflections are observed (Fig. 2).

Fig. 2. Diffraction due to reflections
from a pair of planes. The difference in
path length between reflected beams a
and b is equal to 2d sinθ. If this is
equal to a whole number of
wavelengths, nλ, then constructive
interference occurs.