Advanced Solid State Physics

Now we expand the electron densityn(r)in a fourier series (this is possible because the periodic-
ity ofn(r)is the same as for the crystal).

n(r) =

∑

G

nGexp (iG·r) =

∑

G

nG(cos (G·r) +isin (G·r)) (36)

To expand in terms of complex exponentials the factornGhas to be a complex number, because a real
function like the electron density does not have an imaginary part. If the fundamental wavelength of
the periodic structure isathen there would be a vectorG=^2 aπin reciprocal space. So we just sum
over all lattice vectors in reciprocal space which correspond to a component of the electron density.

We combine the scattering amplitudeFand the electron densityn(r)and get

F=

∑

G

∫

dV nGexp (i(G−∆k)·r)

Thus, the phase factor depends onGand on∆k. If the conditionG−∆k= 0is fullfilled for every
positionr, the phase factor is 1. For all other values of∆kit’s a complex value which is periodic inr,
so the integral vanishes and the waves interfere destructively. We obtain thediffraction condition

G= ∆k=k′−k

Figure 7: Geometric interpretation of the diffraction condition

Most of the time in diffraction scattering is elastic. In this case the incoming and the outgoing wave
have the same energy and because of that also the same wavelength. That means|k|=|k′|because
of^2 λπ=k. So for elastic scattering the difference betweenkandk′must be a reciprocal lattice vector
G(as shown in fig. 7). We apply the law of cosines and get a new expression for the diffraction
condition.

k^2 +G^2 − 2 kGcos (θ) =k^2 → 2 k·G=G^2 −→:4 k·

G

2

=

(

G

2

) 2

The geometric interpretation can be seen in Fig. 8. The origin is marked as 0. The point C is in
the direction of one of the wave vectors needed to describe the periodicity of the electron density. We
take the vectorGChalf way there and draw a plane perpendicular on it. This plane forms part of a
volume.