Dear Dirk, You are getting confused about where the sampling occurs, and this is perhaps because we usually learn about the Shannon criterion from a certain way around (sampling in real/time space -> periodicity of the signal transform in frequency/reciprocal space). To see the Shannon criterion in crystallography, you have to look at it the other way around (sampling of the molecular transform in reciprocal space -> periodicity of the electron density in space). "Twice the signal bandwidth" becomes the physical width of the unique portion of your 1D electron density, which is equal to the unit cell repeat by definition. Hence, you are sampling the fourier transform at double the Shannon frequency.
Sampling of the electron density makes the sampled molecular transform periodic in reciprocal space, with interval 1/q, where q is your real-space sampling interval. If d is the minimum Bragg spacing, then your molecular transform lies between +/- 1/d in reciprocal space, i.e. has a full-width of 2/d. Thus, in order for the "ghost" copies of the molecular transform to not overlap, you must have q such that 1/q >= 2/d. i.e. q <= d/2. Hope that helps, Joe > Dear Ian, > > oh, yes, thank you - you are absolutely right! I really confused the > sampling of the molecular transform with the sampling of the electron > density in the unit cell! Sometimes I don't see the wood for the trees! > > Let me then shift my question from the sampling of the molecular > transform to the sampling of the electron density within the unit cell. > For the 1-dimensional case, this is discretely sampled at a/h for > resolution d, which is still 1x sampling and not 2x sampling, as > required according to Nyquist-Shannon. Where is my error in reasoning, > here? > > Best regards, > > Dirk. > > Am 15.04.11 14:25, schrieb Ian Tickle: >> Hi Dirk >> >> I think you're confusing the sampling of the molecular transform with >> the sampling of the electron density. You say "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". In fact the sampling of the molecular transform has nothing to do >> with h, it's sampled at points separated by a* = 1/a in the 1-D case. >> >> Cheers >> >> -- Ian >> >> On Fri, Apr 15, 2011 at 12:20 PM, Dirk Kostrewa >> <kostr...@genzentrum.lmu.de> wrote: >>> 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: kostr...@genzentrum.lmu.de >>> WWW: www.genzentrum.lmu.de >>> ******************************************************* >>> > > -- > > ******************************************************* > 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 > ******************************************************* >
