Set of supplementary notes.

3.5. DIFFRACTION CONDITIONS AND BRILLOUIN ZONES 45
Figure 3.8:
Illustration of Bragg scattering from a crystal
Define the scattered wavevector
and the momentum transfer
we then have for the waveform
k = k o
r
r
(3.13)
q = k o − k (3.14)
[
]
ψ ∝ e iko·r eiq·R
1 + cf(ˆr)
r
. (3.15)
Now sum over all the identical sites in the lattice, and the final formula is
[
ψ ∝ e iko·r 1 + c ∑ ]
eiq·R i
f i (ˆr) . (3.16)
r
i
Away from the forward scattering direction, the incoming beam does not contribute, and
we need only look at the summation term. We are adding together terms with different phases
q · R i , and these will lead to a cancellation unless the Bragg condition is satisfied
q · R = 2πm (3.17)
for all R in the lattice, and with m an integer (that depends on R). The special values of q ≡ G
that satisfy this requirement lie on a lattice, which is called the reciprocal lattice. 4
One can check that the following prescription for the reciprocal lattice will satisfy the Bragg
condition. The primitive vectors b i of the reciprocal lattice are given by
b 1 = 2π a 2 ∧ a 3
a 1 · a 2 ∧ a 3
and cyclic permutations . (3.18)
3.5 Diffraction conditions and Brillouin zones
For elastic scattering, there are two conditions relating incident and outgoing momenta. Conservation
of energy requires that the magnitudes of k o and k are equal, and the Bragg condition
requires their difference to be a reciprocal lattice vector k − k o = G. The combination of the
two can be rewritten as
k · G
2 = (G 2 )2 . (3.19)
4 We can be sure that they are on a lattice, because if we have found any two vectors that satisfy (3.17), then
their sum also satisfies the Bragg condition.