Dear colleagues of the CCP4BB,

many thanks for all your replies - I really got lost in the trees (or 
wood?) and you helped me out with all your kind responses!

I should really leave for the weekend ...

Have a nice weekend, too!

Best regards,

Dirk.

Am 15.04.11 13:20, schrieb Dirk Kostrewa:
> Dear colleagues,
>
> I just stumbled across a simple question and a seeming paradox for me 
> in crystallography, that puzzles me. Maybe, it is also interesting for 
> you.
>
> The simple question is: is the discrete sampling of the continuous 
> molecular Fourier transform imposed by the crystal lattice sufficient 
> to get the desired information at a given resolution?
>
> From my old lectures in Biophysics at the University, I know it has 
> been theoretically proven, but I don't recall the argument, anymore. I 
> looked into a couple of crystallography books and I couldn't find the 
> answer in any of those. Maybe, you can help me out.
>
> Let's do a simple gedankenexperiment/thought experiment in the 
> 1-dimensional crystal case with unit cell length a, and desired 
> information at resolution d.
>
> According to Braggs law, the resolution for a first order reflection 
> (n=1) is:
>
> 1/d = 2*sin(theta)/lambda
>
> with 2*sin(theta)/lambda being the length of the scattering vector 
> |S|, which gives:
>
> 1/d = |S|
>
> In the 1-dimensional crystal, we sample the continuous molecular 
> transform at discrete reciprocal lattice points according to the von 
> Laue condition, S*a = h, which gives |S| = h/a here. In other words, 
> the unit cell with length a is subdivided into h evenly spaced 
> crystallographic planes with distance d = a/h.
>
> Now, the discrete sampling by the crystallographic planes a/h is only 
> 1x the resolution d. According to the Nyquist-Shannon sampling theorem 
> in Fourier transformation, in order to get a desired information at a 
> given frequency, we would need a discrete sampling frequency of 
> *twice* that frequency (the Nyquist frequency).
>
> In crystallography, this Nyquist frequency is also used, for instance, 
> in the calculation of electron density maps on a discrete grid, where 
> the grid spacing for an electron density map at resolution d should be 
> <= d/2. For calculating that electron density map by Fourier 
> transformation, all coefficients from -h to +h would be used, which 
> gives twice the number of Fourier coefficients, but the underlying 
> sampling of the unit cell along a with maximum index |h| is still only 
> a/h!
>
> This leads to my seeming paradox: according to Braggs law and the von 
> Laue conditions, I get the information at resolution d already with a 
> 1x sampling a/h, but according to the Nyquist-Shannon sampling theory, 
> I would need a 2x sampling a/(2h).
>
> So what is the argument again, that the sampling of the continuous 
> molecular transform imposed by the crystal lattice is sufficient to 
> get the desired information at a given resolution?
>
> I would be very grateful for your help!
>
> Best regards,
>
> Dirk.
>

-- 

*******************************************************
Dirk Kostrewa
Gene Center Munich, A5.07
Department of Biochemistry
Ludwig-Maximilians-Universität München
Feodor-Lynen-Str. 25
D-81377 Munich
Germany
Phone: 	+49-89-2180-76845
Fax: 	+49-89-2180-76999
E-mail:	[log in to unmask]
WWW:	www.genzentrum.lmu.de
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