On 03/12/2014 04:15 PM, Keller, Jacob wrote:
>> For any sample, crystalline or not, a generally valid description of diffraction intensity is it being a Fourier transform of electron density autocorrelation function.
>
> I thought for non-crystalline samples diffraction intensity is simply the Fourier transform of the electron density, not its autocorrelation function. Is that wrong?
>

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.


>
> Anyway, regarding spot streaking, perhaps there is a different, simpler formulation for how they arise, based on the two phenomena:
>
> (1) Crystal lattice convoluted with periodic contents, e.g., protein structure in exactly the same orientation
> (2) Crystal lattice convoluted with aperiodic contents, e.g. n different conformations of a protein loop, randomly sprinkled in the lattice.
>
> Option (1) makes normal spots. If there is a lot of scattering material doing (2), then streaks arise due to many "super-cells" occurring, each with an integral number of unit cells, and following a Poisson distribution with regard to frequency according to the number of distinct conformations. Anyway, I thought of this because it might be related to scattering from aperiodic crystals, in which there is no concept of unit cell as far as I know (just frequent distances), which makes them really interesting for thinking about diffraction.
>

This formulation cannot describe aperiodic contents. The convolution of a 
crystal lattice with any function will result in electron density, which has a 
perfect crystal symmetry of the same periodicity as the starting crystal lattice.

> See the images here of an aperiodic lattice and its Fourier transform, if interested:
>
> http://postimg.org/gallery/1fowdm00/

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.

>
>> Mosaicity is a very different phenomenon. It describes a range of angular alignments of microcrystals with the same unit cell within the sample. It broadens diffraction peaks by the same angle irrespective of the data resolution, but it cannot change the length of diffraction vector for each Bragg reflection. For this reason, the elongation of the spot on the detector resulting from mosaicity will be always perpendicular to the diffraction vector. This is distinct from the statistical disorder, where spot elongation will be aligned with the crystal lattice and not the detector plane.
>
> I have been convinced by some elegant, carefully-thought-out papers that this "microcrystal" conception of the data-processing constant "mosaicity" is basically wrong, and that the primary factor responsible for observed mosaicity is discrepancies in unit cell constants, and not the "microcrystal" picture. I think maybe you are referring here to theoretical mosaicity and not the fitting parameter, so I am not contradicting you. I have seen recently an EM study of protein microcrystals which seems to show actual tilted mosaic domains just as you describe, and can find the reference if desired.

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.

Zbyszek

>> Presence of multiple, similar unit cells in the sample is completely different and unrelated condition to statistical disorder.
>
> Agreed!
>
> Jacob
>


-- 
Zbyszek Otwinowski
UT Southwestern Medical Center	
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