## POST #3: Theory Behind the Initial Solution

So far I have discussed most of the practical knowledge that a crystallographer needs, but I have not mentioned much of the theory. Crystallography requires the careful analysis of thousands of reflections by a computer, but at one point, it was all done by hand. The math behind the initial solutions is very tedious, but in short, it relies on Fourier transforms. Fourier transforms are used to break complicated oscillations into only sines and cosines. A Fourier transform lets us look at the repeating patterns of reflections (similar to a harmonic function) and determine what the unit cell looks like, as well as the arrangement of its contents.

## Modeling Crystal Structures

The idea behind x-ray crystallography is that focused, collimated x-rays diffract off of clouds of electron density in a calculated manner defined by the Bragg equation. By measuring the intensity of these reflected x-rays, we can reverse engineer a 3D lattice from sets of thousands of diffracted spots, called reflections. My last post detailed some of the more physical skills related to crystallography, but I completely neglected to mention one of the greatest programs ever created for crystallography: SHELX. SHELX was developed in the 1960s and 1970s. It was coded in fortran wth the goal that it could be completely self-contained. This occurred at a time when the Internet was in its infancy, so everything that crystallographers used had to be independent. Keep in mind, this was originally stored on perforated paper tapes! SHELXLE is the GUI that crystallographers use today for refining structures. Fortunately, it lets you see the 3D visualization after every iteration. (The original version required crystallographers to print out a paper with numbers in the exact location of each atom from a specified orientation, so this is much more user friendly.) The goal of refinement is to match the measured electron density with positions of molecules in the lattice, while realistically describing how atoms interact. This is very different from simply reading off a graph because the data is coming in three dimensions of space and it extends ad infinitum. It is important to remember that the refinements are attempting to match all of space with the average unit cell. Sometimes free variables must be added and refined to proportions that describe how likely a certain arrangement is.

## Crystallography Skills

Learning x-ray crystallography has been fascinating so far. Contrasting many other projects, my work does not require a wealth of research before beginning to learn and apply certain skills. I have spent the last four weeks honing my crystallography skills, and reading about theory in my spare time. This post will mostly cover the day to day challenges I’ve faced.