# 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.

SHELXLE allows crystallographers to manipulate how the electron density is labeled and interpreted at a glance. The most basic unit cell is the smallest repeating unit of the crystal lattice, but sometimes disordered molecules mean that the unit cell is not identical everywhere. A simple example of this is when a  can take two arrangements without changing the overall cell’s arrangment. Take for instance, CN-. This compound is called cyanide, and when CN- connects copper atoms in CuCN, the copper can bond to either the carbon or nitrogen in chains. There is no clear preference between which side bonds, so SHELXLE allows crystallographers to note this pattern, and make it easy to interpret for other chemists. Symmetry is absolutely imperative in understanding how unit cells link together in space, especially when it comes to connectivity and special positions. During the refinement process, it may become clear that an atom is on mirror plane. A mirror plane is exactly what it sounds like; everything on one side can be reflected exactly on the other. If we labeled said atom as being completely in the unit cell, then when we repeat the cell next to it, we will have twice as many of those atoms. Instead, we fix that atom to half occupancy. Then the unit cell can be repeated through space and accurately describe the chemical formula.

There are many other types of disorder and applications of symmetry that I am still trying to wrap my head around, but it is going to take a lot more practice of refining crystal structures to get there. This portion of research can be frustrating as well, but since it involves the most critical thinking, I find it incredibly satisfying when everything is accomplished.