See attached file for more details.
1. A plane wave in incident upon single atom regular lattice. Calculate the scattered amplitude from all the atoms.
3. Define the Brillouin zone and demonstrate that all wave vectors which start from the origin and satisfy the condition for maximum will terminate on the Brillouin zone boundary.
4. For a two dimensiona crystal with its cell vectors given, find the reciprocal lattice and draw it.
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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
The incoming plane wave is:
It is scattered by the atom at location .
Due to symmetry we assume that the scattered amplitude from each lattice point is the same and proportional to A.
When a plane wave hits the scattering pint, it is reflected with a different wave vector k, so from the atom to point rD, ...
The 7 pages file shows in great detail how to apply first principles to the problems and how to solve them.