How can crystal structures be determined using X-ray diffraction?

Question

You have the intensity peaks and the diffraction angles. Let's say you suspect the material is cubic, how do I find if it's simple cubic or BCC or FCC?

I've googled and all my textbooks just state that xrd is really nifty for determining crystal structure but no further information is presented.

Answer 1

X-ray diffraction ends up being a beautiful application of Fourier theory.

You are measuring the $k$-space (reciprocal lattice space) points with the x-ray diffraction. The Fourier transform of these points (provided you measured enough of them) gives you exactly the Bravais lattice.

The Bravais lattice is the crystal stucture! You then have to look at the Bravais lattice and construct a Wigner-Seitz unit cell, then that construction will determine your crystal type.

Alternatively, you could analyze the reciprocal lattice space "unit cell," find what it looks like, then Fourier transform that cell to get the Bravais lattice cell type. The Fourier transforms here is on an infinite periodic array, if that wasn't clear.

The point arrangement actually determines the FCC or BCC type...I have included a diagram of the two different types (FCC and BCC) shown in the reciprocal lattice space below, taken from Introduction to the Physics of Electrons in Solids by Henri Alloul. You can see that they are not the same...

Answer 2

For BCC and FCC some of the lines are missing, so if you manage to index all the lines in your diffraction pattern you can tell immediately what type of cubic lattice you have.

A quick Google found http://www.matter.org.uk/diffraction/x-ray/indexing_powder_pattern.htm and this is a pretty good introduction to the subject though anyone who has tried to do this in anger will knw that things are rarely as clear cut as the article suggests!