7 The Brillouin Condition

School of Chemistry

From the Laue condition, a different condition for diffraction can be derived.  It is again equivalent to both Laue and Bragg. 

k' - k = G and so k' = G + k

Taking the dot product of both sides,

(k' · k') = (G + k) · (G + k)

Hence k' · k' = G · G + 2G · k + k · k

But the magnitudes of k and k' are equal so k' · k' and k · k cancel.

Hence 2G · k = -G · G

If there is a reciprocal lattice point at the position G there is also one at -G so the minus sign is unnecessary.

Hence k · ½G = ½½G ½²       (7)

5 The Reciprocal Lattice

This result is called the Brillouin condition and shows that diffraction occurs about planes which are perpendicular bisectors of reciprocal lattice points.

Figure 16: planes about which diffraction occurs

It follows that there are enclosed volumes about each reciprocal lattice point which represent the various diffraction peaks.  The smallest such volume is the 'first Brillouin zone', the second smallest is the 'second Brillouin zone' and so on. 

Figure 17: The first Brillouin zone for a simple cubic lattice lattice in 2 dimensions; in 3 dimensions the first Brillouin zone is a cube.