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Multiplicity


"Multiplicity" refers to the multitude of X-ray or neutron reflexions that contribute to a single peak observed in a powder pattern.

When a single crystal is exposed to X-rays or neutrons, diffraction may occur. The directions of the scattered radiation beams (reflexions) depend on the orientation and spacing of lattice planes in the crystal. Different sets of lattice planes scatter in different directions. Thus, reflexions from lattice planes with identical spacings (e.g., the lattice planes with Miller Indices 100 and 010 in a cubic crystal) can be measured independently of each other as their scattered beams occur in different directions.

In powder diffraction, the sample is polycrystalline, with many crystal grains arranged in all kinds of orientations. Reflections from many different grains become merged into cones of intensity, arranged around the straight-through radiation beam direction. The cone semi-angle is the scattering angle (two-theta). Discrete cones of intensity can be related to particular interplanar spacings ("d-spacings") in the crystal, but it is no longer possible to separate-out reflexions from planes which have the same d-spacings, but different orientations (e.g., 100 and 010 in the example above).

Thus, in a powder diffraction pattern, a diffraction peak may be caused by the diffraction from a multiplicity of lattice planes in different crystallographic orientations, but with the same d-spacings. The number of different sets of lattice planes (and hence the number of different sets of Miller Indices) determines the multiplicity of a powder reflexion.


Example: in a cubic crystal, the lattice planes (1 0 0), (-1 0 0), (0 1 0), (0 -1 0), (0 0 1), and (0 0 -1) all have the same d-spacing and would therefore appear on top of each other in a powder diffraction pattern. The multiplicity of a "1 0 0" reflexion is therefore 6.


The concept of multiplicity is important when attempting to relate a diffraction pattern to an underlying structure, and one has to be aware that different powder "reflexions" may have different multiplicities, which must be corrected for. CrystalDiffract displays the multiplicity (m) of all reflexions.


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