Selection Rules and Practical Crystallography
Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation:
where is the lattice spacing of the cubic crystal, and, and are the Miller indices of the Bragg plane. Combining this relation with Bragg's law:
One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will be given as is.
| Bravais lattice | Example compounds | Allowed reflections | Forbidden reflections |
|---|---|---|---|
| Simple cubic | Po | Any h, k, l | None |
| Body-centered cubic | Fe, W, Ta, Cr | h + k + l = even | h + k + l = odd |
| Face-centered cubic | Cu, Al, Ni, NaCl, LiH, PbS | h, k, l all odd or all even | h, k, l mixed odd and even |
| Diamond F.C.C. | Si, Ge | all odd, or all even with h+k+l = 4n | h, k, l mixed odd and even, or all even with h+k+l ≠ 4n |
| Triangular lattice | Ti, Zr, Cd, Be | l even, h + 2k ≠ 3n | h + 2k = 3n for odd l |
These selection rules can be used for any crystal with the given crystal structure. KCl exhibits a fcc cubic structure. However, the K+ and the Cl− ion have the same number of electrons and are quite close in size, so that the diffraction pattern becomes essentially the same as for a simple cubic structure with half the lattice parameter. Selection rules for other structures can be referenced elsewhere, or derived.
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