# 34.2: Mathematics

- Page ID
- 34935

## 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*, by the following equation: (Derivation)

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

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 **λ**L*, 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 h*_{1}*k*_{1}*l*_{1}* and h*_{2}*k*_{2}*l*_{2}* to spot 000 is the same as the angle between the reciprocal lattice vectors *[*h*_{1}*k*_{1}*l*_{1}]** and *[*h*_{2}*k*_{2}*l*_{2}]***. This is also equal to the angle between the (*h*_{1}*k*_{1}*l*_{1}) and (*h*_{2}*k*_{2}*l*_{2}) planes, or equivalently the angle between the normals to the (*h*_{1}*k*_{1}*l*_{1}) and (*h*_{2}*k*_{2}*l*_{2}) 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.