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.
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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: kostr...@genzentrum.lmu.de
WWW: www.genzentrum.lmu.de
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