# 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.

These methods attempt to label atoms in 3D space using only information in reciprocal space. “Reciprocal space” consists of the x-ray reflections measured by a detector that sweeps around the sample. In order to effectively capture a full sphere of information, the combination of sample and detector must have three degrees of freedom. The crystal sample can spin around the axis that it points through (think of pointing your finger at something and turning your wrist), and sweep out a circle around the normal to the stand (think how a toothpick would look if you poked it through the minute hand of a clock pointing toward the center of the clock). The detector can also spin around the sample, so even though the x-ray source cannot move, we can collect enough data to create a 3D representation of the lattice. By measuring the intensity of each reflection, and using the three angles listed above, Fourier transforms let us reverse engineer the lattice that would leave those patterns of spots. The size is directly proportional to the electron density, and with some knowledge of what the compound is made up, we can determine an initial solution of the structure.

A possible problem with this is if the crystallographer has no idea what atoms are present. I’ve learned that I must think of what molecules could be in the crystal structure based upon the synthesis of the crystal. Chemistry finally meets crystallography at this point. The behavior of certain atoms together must be accounted for, as well as if there are solvent molecules present in the crystal structure. Usually, knowledge of charge and polyatomic ions is enough to identify unknown blobs of electron density. However, there are also times when brainstorming with the chemist who made the sample–and some trial and error–are the only things a crystallographer can use to solve the structure.

Deciding which x-ray source to shine at a crystal relies on physics. Our XRD has two anodes (or sources of x-rays): the molybdenum anode and the copper anode. The former has a wavelength of 0.71 angstroms and the latter has a wavelength of 1.54 angstroms. The copper source is also about ten times as bright as the molybdenum one. Longer wavelengths diffract more through the lattice of crystals, so copper runs result in a larger spread of spots. This means these runs take much longer because more of the sphere around the crystal must be imaged. Molybdenum offers speed and precision, but unfortunately, some organic compounds don’t result in bright enough reflections to really learn about the structure under molybdenum. As long as the crystal does not contain heavy elements (below the third row of the periodic table), I can switch to the copper source and still find the necessary information for the initial solution. Lots of trade offs and decision making come from understanding the science behind crystallography, and so much of it is incredibly fascinating!