>The Fourier transform of electron density is a complex scattering amplitude >that by the axiom of quantum mechanics is not a measurable quantity. What is >measurable is the module squared of it. In crystallography, it is called either F^2 (formally equal F*Fbar) or somewhat informally diffraction intensity, after one takes into account scaling factors. F*Fbar is the Fourier transform of an electron density autocorrelation function regardless if electron density is periodic or not. For periodic electron density the structure factors are described by sum of delta Dirac functions placed on the reciprocal lattice. These delta functions are multiplied by values of structure factors for corresponding Miller indices.
Okay, I may have been confused--I thought that the Fourier transform was essentially acting like an autocorrelation function (since generally Fourier transforms are similar to autocorrelation functions--not clear on the details right now), and I had thought I had heard stories of days of yore handwritten Fourier series calculations to make electron density maps. You're telling me they had to also back-calculate an autocorrelation function? Times were tough! Maybe someone from that generation can chime in about how they dealt with this? >This is interesting case of pseudocrystal, however because there is no crystal >lattice, it is not relevant to (1) or (2). In any case, pentagonal >quasilattices are probably not relevant to macromolecular crystallography. I tried a few simulations to show what I mean but ran out of time--sorry about that. I think I'll probably just drop this. NB Linus Pauling said more forcefully the same prediction about aperiodic crystals in general not existing, pentagonal or otherwise, but was proven dead wrong by now-Nobel laureate Dan Shechtman. Maybe someone will come across an aperiodic protein crystal, or already has and missed it, and stupefy us all. Someone mentioned to me once seeing personally a ten-fold symmetrical diffraction pattern from a protein crystal, but she dismissed it with exactly the argument that Pauling made, I think that it was a twinned cubic space group. >This is easy to test by analyzing diffraction patterns of individual crystals. In practice, the dominant contribution to angular broadening of diffraction peaks is angular disorder of microdomains, particularly in cryo-cooled crystals. However, exceptions do happen, but these rare situations need to be handled on case by case basis. I was thinking of this paper for example (see last line of abstract). Perhaps other crystals are different from lysozyme, though, as you mention. All the best, Jacob Keller Acta Crystallogr D Biol Crystallogr. 1998 Sep 1;54(Pt 5):848-53. A description of imperfections in protein crystals. Nave C. Author information Abstract An analysis is given of the contribution of various crystal imperfections to the rocking widths of reflections and the divergence of the diffracted beams. The crystal imperfections are the angular spread of the mosaic blocks in the crystal, the size of the mosaic blocks and the variation in cell dimensions between blocks. The analysis has implications for improving crystal perfection, defining data-collection requirements and for data-processing procedures. Measurements on crystals of tetragonal lysozyme at room temperature and 100 K were made in order to illustrate how parameters describing the crystal imperfections can be obtained. At 100 K, the dominant imperfection appeared to be a variation in unit-cell dimensions in the crystal. PMID: 9757100 [PubMed - indexed for MEDLINE]
