The easiest way to do diffraction measurements is to take a crystal, shoot an X-ray beam through it and measure the direction of outgoing waves. However, it is highly unlikely that you will fulfil the Laue condition for any set of crystal planes. There does exist a more practical experiment - **powder diffraction**.
By crushing the crystal into a powder, the small crystallites are now orientated in random directions. This highly improves the chances of fulfilling the Laue condition for a fixed direction incoming beam. The experiment is illustrated in the figure above. The result is that the diffracted beam exits the sample via concentric circles at discrete **deflection angles** $2 \theta$.
To deduce the values of $\theta$ for a specific crystal lets put the Laue condition into a more practical form.
Where we use $|\mathbf{k'}| = |\mathbf{k}|$ in the second line and insert the Laue condition in the third line. During the exercises for this week, you will derive an important relation $G_{hkl} = \frac{2}{d_{hkl}}$ where $d_{hkl}$ is the spacing between $(hkl)$ miller planes. With this, one can finally derive **Bragg's Law**:
$$ \lambda = 2 d_{hkl} \sin(\theta) $$
where $\phi = \theta - \pi/2$.
## Summary
* We described how to construct a reciprocal lattice from a real-space lattice.
* Points in reciprocal space that differ by a reciprocal lattice vector are equivalent --> Band structure can be fully described by considering 1st Brillouin zone.
