X-ray Crystallography

Introduction

Most people interested in science have heard of x-ray crystallography. Hearing of is one thing, but like me until very recently, one may only have a very limited idea of what a x-ray crystallography experiment involves. Here is what I might have told you before having a chance to inform myself: we take a crystalline sample (I suppose it must be possible to make some proteins into crystals then), shine a beam of x-rays through the sample at different angles, measure a set of diffraction patterns produced, and then use the diffraction patterns to deduce the 3D structure of the sample. At a surface level this is a correct enough overview, but the correctness obscures three horrors:

  1. I didn’t know what a diffraction pattern was.

  2. As for the magic final step, I was at a total loss.

  3. I wasn’t certain what “crystalline” meant in these circumstances. Furthermore why does our sample need to be a crystalline?

In this post I address these three horrors, mathematically of course.

Diffraction Patterns

Recovering the Structure

Why Crystals?

Final Comments

I hope that these short sections have given you some idea of the maths x-ray crystallography involves.

Interestingly, crystal samples aren’t strictly necessary for diffraction based imaging techniques, ptychography being a key example. Background reading on ptychographic numerical methods is what made me realise I knew so little about x-ray crystallography!