On Tuesday, 20 January 2015 10:18:35 PM Chen Zhao wrote:
> Dear all,
>
> I am sorry about this slightly off-topic question. I am now a graduate TA
> for crystallography course and one student asked me a question that I
> didn't ask myself before. I don't have enough knowledge to precisely answer
> this question, so I am seeking for help here.
>
> The question is, as I rephrased it, assuming we are able to measure the
> diffraction pattern of a single molecule with acceptable accuracy and
> precision (comparable to what we have now for the common crystals), is it
> better than we measure the diffraction spots from a crystal, given that the
> spots are just a sampling of the continuous pattern from a single molecule
> and there is loss of information in the space between the spots that are
> not sampled by the lattice?
While it is true that there is a loss of information because of the space
between the Bragg reflections, this is not as bad as you might think.
The Nyquist theorem tells us that we can reconstruct a Fourier term exactly
if we can sample at one half the period of that term.
So for any given resolution of Bragg spots, the continuous transform
to half that resolution can be reconstructed. Here "can be reconstructed"
implicitly includes "... if we know the phase". So it comes back to the
phase problem. If we could measure the phase, it would only matter to a
factor of 2 in resolution that we are not measuring the continuous transform.
By the way, as Jacob Keller alluded to earlier, XFEL diffraction from
nanocrystals introduces a situation half way between the two cases.
Because there are only a small number of unit cells in each direction,
the observed diffraction pattern indeed contains information in between
the Bragg peaks. One approach to interpreting this data is to treat the
measured diffraction pattern as a continuous transform of a single particle,
where that single particle just happens to be a nanocrystal containing
a small number of identical unit cells.
Ethan
> Of course this is more of a thought experiment,
> so we don't need to consider that all measurement is discrete in nature
> owing to the limitation of the pixel size. I kinda agree with him and I
> have a feeling that this is related to the sampling theorem. I do
> appreciate your valuable comments. If this is not true, why? If this is
> true, what is its effect on electron density?
>
> Thank you so much for your attention and your help in advance!
>
> Best,
> Chen
--
mail: Biomolecular Structure Center, K-428 Health Sciences Bldg
MS 357742, University of Washington, Seattle 98195-7742
