Dirk Another way of looking at it See slide 7 in http://www.aps.anl.gov/Science/Future/Workshops/Frontier_Science_Using_Soft_Xrays/Presentations/WeierstalTalk.pdf
sampling interval 1/W (Bragg sampling) is Shannon sampling if complex Fraunhofer wavefield of object with width W is recorded. If only Fraunhofer intensity of object with width W is recorded, then the FT of the intensity is the autocorrelation with width 2W and the correct (Shannon) sampling interval is 1/2W. Additional issues are present for 2D and 3D but the above gives the basic idea. > the sampling of the continuous molecular transform imposed by the crystal > lattice is sufficient to get the desired information at a given resolution? Yes, if you have phased amplitudes Regards Colin > -----Original Message----- > From: CCP4 bulletin board [mailto:CCP4BB@JISCMAIL.AC.UK] On Behalf Of > Dirk Kostrewa > Sent: 15 April 2011 12:20 > To: CCP4BB@JISCMAIL.AC.UK > Subject: [ccp4bb] Lattice sampling and resolution - a seeming paradox? > > 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 > *******************************************************
