34.2: Mathematics

Relation 1

The distance, r_hkl, on the pattern between the spot hkl and the spot 000 is related to the interplanar spacing between the hkl planes of atoms, d_hkl, by the following equation: (Derivation)

\[r_{h k l}=\frac{\lambda L}{d_{h k l}}\]

where L is the distance between the sample and the film/screen.

We can therefore say that the diffraction pattern is a projection of the reciprocal lattice with projection factor λL, because reciprocal lattice vectors have length 1/d_hkl.

Relation 2

Since the diffraction pattern is a projection of the reciprocal lattice, the angle between the lines joining spots h₁k₁l₁ and h₂k₂l₂ to spot 000 is the same as the angle between the reciprocal lattice vectors [h₁k₁l₁]* and [h₂k₂l₂]*. This is also equal to the angle between the (h₁k₁l₁) and (h₂k₂l₂) planes, or equivalently the angle between the normals to the (h₁k₁l₁) and (h₂k₂l₂) planes. This angle is θ in the diagram below.

Using these two relations between the diffraction pattern and the reciprocal lattice, we are now able to index the electron diffraction pattern from a specimen of a known crystal structure.

The two pages linked to here refer only to indexing the central region of the diffraction pattern - the rest will be dealt with later.

Indexing with the orientation of the electron beam known

Indexing with the orientation of the electron beam unknown