Scattering amplitude of diffracted beam by a crystal

[Andreasdreas]

Homework Statement

The scattering amplidtude, F, of a, by a crystal, diffracted beam is defined to be:

(1) $F=\sum{_\vec{G}}\int n_{\vec{G}}e^{i(\vec{G}-\delta\vec{k})\cdot \vec{r}}\mathrm{d}V$

The integral is over all the volumeelements, dV, of the crystal.
$n_{\vec{G}}$ is the local electron concentration of the crystal in dV, $\vec{G}$ is a reciprocal lattice vector and the sum is over the set of all the reciprocal lattice vectors. $\vec{r}$ is the position vector of dV
further $-\delta\vec{k}=\vec{k}-\vec{k'}$ where $\vec{k}$ is the wave vector of the incomming beam and $\vec{k'}$ is the wave vector of the outgoing scattered beam.

The beam could be
fotons, electrons neutrons etc. It is descibed as a plane wave.

It is easy to see that $F=n_{\vec{G}}V$ when $\delta \vec{k}=\vec{G}$

But how can it be shown that F is negligibly small when $\delta \vec{k}$ differs sginificantly from any $\vec{G}$?

Homework Equations

I thought some expression for the summation could be used.

Maybe

(2) $\sum{_m=0}^{M-1}x^m=\frac{1-x^{M}}{1-x}$

The Attempt at a Solution

I can't find a away to make (1) fit into (2). And if i just look at (1) i really am lost.

Homework Equations

The Attempt at a Solution

 

[member 553083]

I thought of using the fact that the sum only goes over all the reciprocal lattice vectors and not \delta \vec{k}. This means that for \delta \vec{k} to be significantly different from any \vec{G}, then F should be small for the term in the summation that involves \delta \vec{k}.But how can one prove this mathematically?