In the lecture on X-ray diffraction we stated that the sum over the scattered waves of every atom will only be nonzero if the scattered waves interfere constructively. However I am wondering about two things:
There are many atoms, so why is it a problem if one of the scattered waves interfered destructively with another wave? Won’t there still other scattered waves from other atoms in the crystal? The crystal may or may not be isotropic so the waves can scatter of the atoms in a multitude of ways
If the answer to the first question is something along the lines of "we model the crystal as a tiled sum of the situation we described in the unit cell so if we describe the interaction in a pair of atoms, we’ve done it for all the atoms, then I pose the following question. How come the sum is only non-zero if they interfered constructively? The interference of these waves can be something between totally destructive and totally constructive so it seems to be a very crude approximation to just ignore all these possibilities of interference that yield a non-zero wave. How can we justify this approximation?
Regarding your first question, the phase difference between the incoming wave is dependent on the position of the atoms in the lattice (\Delta \phi = (\mathbf{k}-\mathbf{k'})\cdot \mathbf{R}). If \mathbf{k}-\mathbf{k'} \neq \mathbf{G}, then the argument of the exponent has a phase factor dependent on the real-space lattice points. Because we sum over each of these lattice points, each argument has a different phase. Summing over all these phases results in an average amplitude of 0, resulting in no intensity peaks in the spectrum.
To answer your second question. A crystal can be described by a lattice and a proper basis. In other words, if we have defined a proper unit cell and a corresponding basis, we can build up the crystal by repeating the unit cell over and over again. Indeed, for a single lattice the interference can be somewhere between fully constructive and destructive. However, the amplitude of the outgoing wave does not depend on a single unit cell, but of all unit cells in the lattice.
