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Hi Dirk, My interpretation of your question is what is the impact of resolution given by the individual diffraction spots from the electron density sampling and the Nyquist theorem. My explanation would be that the Nyquist theorem gives an upper limit to the frequency information that can be obtained, in the case of crystallography, the highest resolution spot that is possible. Everything with lower resolution, or smaller index, is at a lower "frequency" than the nyquist limit. The nyquist limit would come from the sampling done in the fourier transform of the frequency domain, which in this case is the transform of reciprocal space to real space. The sampling that is done in real space is limited by the interaction of the X-rays with the electron density of the individual molecules in the lattice. That interaction is nearly continuous across a molecule, leading to a very "high/fast" sampling rate. The limit of this interaction would be due to the wavelength (~lambda/2) which would result in the diffraction limit in reciprocal space (limiting the largest index that is observable). This my understanding, but I too would like to have a more intuitive understanding of this fundamental limitation. Brett 2011/4/15 Dirk Kostrewa <[log in to unmask]> > 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 >> <[log in to unmask]> 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: [log in to unmask] >>> 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: [log in to unmask] > WWW: www.genzentrum.lmu.de > ******************************************************* >