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 > ******************************************************* >
