34.2: Mathematics

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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.

Diagram of part of reciprocal lattice

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