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I know the general idea behind x-ray crystallography is to take a high quality crystal and place it in the path of an x-ray beam. Areas of high electron density will diffract the beam and lead to spots on a detector screen where photons have constructively interfered with each other (and nothing appears where destructive interference has occurred). The crystal is then rotated and another diffraction pattern image is taken and the cycle repeats until there are enough views to reconstruct the atomic structure of whatever you crystallized. My question is what is the general process for doing this without computer software? How was this done in the 20s? The only equation I know for x-ray diffraction is Bragg's Law but is this the only equation used to interpret the data? Surely there must be others? How do you translate the spots on a detector to electron density plots using Bragg's law?

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    $\begingroup$ Though it is a fascinating question, it is also quite a broad. I'd recommend to get André Authier's book Early Days of X-ray Crystallography which gives a nice overview of the problems that had to be tackled as well as historical background; brief history on Wikipedia; also, there is a series of free access articles by IUCr; and, of course, Bragg's X-Ray Crystallography (1968). $\endgroup$ – andselisk Aug 3 at 2:53
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    $\begingroup$ To put it simply, the bare-bone math apparatus was pretty much ready at that time, but the calculations had to be done manually, by hand. That's why crystal structure determination of a single compound used to be a topic for a Ph.D. and might took several years. $\endgroup$ – andselisk Aug 3 at 2:55
  • $\begingroup$ In x-ray diffraction images it is the intensity of spots that contain the information. In the past Beevers and Lipson Strips were used to help with the Fourier transforms; not easy but a clever way of determining intensities on a photographic plate. Wikipedia gives a little more information. $\endgroup$ – porphyrin Aug 3 at 8:07
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I agree with @andselisk that this question is quite broad. I will focus on two specific questions asked

The only equation I know for x-ray diffraction is Bragg's Law but is this the only equation used to interpret the data? [...] How do you translate the spots on a detector to electron density plots using Braggs law?

Apart from Bragg's law (which tells you where the diffraction spots are for a known orientation of a crystal with know unit cell), it was also known that real space (electron density) and reciprocal space (diffraction pattern) are related by 3D Fourier transform. For structures containing one or two atoms, just knowing the unit cell parameters and the symmetry is enough to get the entire structure. For anything slightly more complicated, the Fourier transform of amplitudes or later, of intensities (Patterson methods, http://reference.iucr.org/dictionary/Patterson_methods), had to be used. The first crystal structure analyses were of crystals with centrosymmetry, where the phase problem is easier to solve.

My question is what is the general process for doing this without computer software?

Diffraction data was measured on film, with gray-scales to assess intensity of signals. To calculate a Fourier transform, pre-computed tables were used, such as the Beevers-Lipson strips. As Andselisk commented, Fourier transform was used late in the 20s, and initially for problems that were one- or two-dimensional. There is a nice account by P.P. Ewald available here: https://www.iucr.org/publ/50yearsofxraydiffraction/full-text/structure-analysis This was written at a time when the myoglobin structure had just been solved, after "50 years of X-ray diffraction".

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    $\begingroup$ I believe Fourier synthesis wasn't used extensively until late 1930s, even though it was proposed in 1925 and Zachariasen first applied 2D Fourier projections in 1929, and 3D Fourier transformation is definitely not about the 20s:) $\endgroup$ – andselisk Aug 3 at 3:25
  • $\begingroup$ I also remember an empirical plots useful to identify the crystallographic type or something. Forgot the name of the compiler. Basically one moved his/her stripe to find a correspondence. Do you remember that? $\endgroup$ – Alchimista Aug 3 at 10:30
  • $\begingroup$ @Alchimista We had to find out the crystalligraphic type and cell dimensions from diffractions patterns in our advanced inorganic lab course. With a bit of practice (I TA'd the course later), you could guess quite many of the films' types at first glance. Typical patterns. (Of course the films were chosen to be rather easy, and I seems to remember that some types were missing, because they're impossible to tell apart by any student?) $\endgroup$ – Karl Aug 3 at 22:30
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Not just in the '20s but up to the '90s at least, the d-spacings were estimated by hand measurements of diffractometer peaks or film lines and applying the Bragg formula. d-spacings were then ranked from the most intense down to the least intense. Starting with the 2 most intense d-spacing values - and allowing +/- 0.02 angstrom error margin to each - a process of searching the Hanawalt Search Manuals was done till a chemically plausible phase with these two values was found. If no plausible phase was found for these two most intense d-spacings, the second value was replaced with each of the third/fourth/fifth/etc most intense d-spacing value and another search made till a chemically plausible phase was found. Once located, the remaining less intense d-spacings listed for this phase were compared with the remaining calculated d-spacings and if found this was supporting evidence for this phase. Remaining d-spacings were relisted according to intensity and the process repeated. It always seemed daunting at first but, with a little initial guidance from a more experienced person (usually another postgrad student) and with a shortlist of all plausible phases plus their d-spacings beside you, it got easy enough in metallurgy and ceramics. It must have been much harder in biochemistry but I guess those working in that métier had their own shortcuts.

For more detailed work, e.g. for residual stress calcs, grain texture effects, part-crystalline/part-amorphous phases, exploration of short-range ordering in amorphous phases, etc one referred to texts by Cullity and Azaroff. Beyond that you had to look at papers by other researchers, talk to x-ray lab technicians and above all try to look at the problem in the way that was most natural to oneself or most amenable to the nature of the problem.

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