I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions.
According to Claude,
When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand showed mathematically that the diffracted amplitude on the n-th layer line is proportional to a Bessel function of the first kind, order n:
Jn(2π r R)
where r is the radius of the helix and R is the distance out from the center (the meridian) in the diffraction pattern.
The citation for this paragraph is a paper from 1952 [2] that amazingly is behind a paywall.
Update: Evan pointed out something fishy in the expression above: “Isn’t there a dimensional problem with J(2π r R)? Maybe 2πr/R instead?”
This was Claude’s response:
- r is the radius of the helix, a real-space length (say, in Å).
- R is the radial coordinate in reciprocal space — it has units of inverse length (Å⁻¹), because that’s how X-ray diffraction/Fourier space works (spatial frequency, not distance).
So Evan and Claude were both right. There is an implicit division happening, but crystallographers do write r R.
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[1] Watson and Crick didn’t “discover DNA” as is commonly said. DNA was discovered in 1878. Watson and Crick discovered the structure of DNA in 1953.
[2] Cochran, W., Crick, F. H. C., & Vand, V. (1952). “The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix.” Acta Crystallographica, 5(5), 581–586.
Isn’t there a dimensional problem with J(2π r R)? Maybe 2πr/R instead?
Good catch! It would help to be able to see the original paper.