Dear Jacob, That's a good question, as you can see from the amount of debate you've inspired. There have been a lot of good answers already.
Depending on how your intuition works, you might prefer Gerard's explanation in terms of convolutions, or Bart's in terms of the size of errors in the electron density. The argument Bart uses is the same as one I used for a chapter in Methods in Enzymology, which was then revised for volume F of the International Tables. And it actually turns out that the mathematical analysis isn't that difficult. Basically, from Parseval's theorem, we know that the rms error of an electron density map is proportional to the rms error of the structure factors (including both amplitude and phase). It's not that difficult to work out the rms error that is introduced in a structure factor either by taking the phase randomly from a uniform distribution of phases or by taking the amplitude randomly from a Wilson distribution of amplitudes. The random choice of phases will introduce structure factor errors proportional to the square root of 2 (approximately 1.41) times the rms structure factor amplitude, whereas the random choice of amplitudes will introduce structure factor errors proportional to the square root of [(4-Pi)/2] (approximately 0.66) times the rms structure factor amplitude. In comparison, if you just set all the amplitudes to zero (in which case you could use anything for the phases), the rms error in the structure factors would simply be equal to the rms value of the structure factors. To summarise this, the random phase map looks *less* like the real structure than a completely flap map (Fourier transform of all zeros), but the random amplitude map looks *more* like the real structure than the flat map. There's a schematic illustration of this on our web page (http://www-structmed.cimr.cam.ac.uk/Course/Fourier/Fourier.html#Parseval), but I see I could add a bit about the simple integrals that give the results above. Regards, Randy Read On 18 Mar 2010, at 20:57, Bart Hazes wrote: > Hi Jacob here is another stab at it. Not as forme > > When you create an electron density map you can think of each amplitude F and > phase P as forming a vector Vmap that consists of two sub-vectors: one that > represents the true vector Vtrue and the other representing the vector, > Vdiff, that connects Vtrue to Vmap. Just try to picture this. > > You can now think of calculating three maps based on Vtrue, Vdiff, and Vmap. > Since FFTs are additive you can consider the map you would normally > calculate, Vmap, as being the sum of the two others; Vtrue being reality and > Vmap being noise. > > If you have a phase error of 60 degrees Vdiff will actually already be of the > same magnitude as Vtrue. If you have random phases you will, on average, be > 90 degrees off, and Vdiff will be 1.41 times as large as Vtrue (sqrt(2)). > Even relatively small phase errors give significant Vdiff/Vtrue ratios > (2*sin(half-the-phase-error) if I'm right) > > You mentioned "Amplitudes as numbers presumably carry at least as much > information as phases, or perhaps even more, as phases are limited to 360deg, > whereas amplitudes can be anything." > > But in reality amplitudes cannot be anything since they follow a Wilson's > distribution which has the bulk of the amplitudes cluster near the peak of > the distribution. The real mathematicians can probably tell you the expected > amplitude error in such a scenario but that would certainly go beyond an > intuitive explanation and my own math skills. If you look at experimental > errors in the amplitudes it is even much less. Rmerge tends to be in the > 3-10% range and that is on intensities, it will be considerable less on > amplitudes. > > Bart > > > Jacob Keller wrote: >> Dear Crystallographers, >> >> I have seen many demonstrations of the primacy of phase information for >> determining the outcome of fourier syntheses, but have not been able to >> understand intuitively why this is so. Amplitudes as numbers presumably >> carry at least as much information as phases, or perhaps even more, as >> phases are limited to 360deg, whereas amplitudes can be anything. Does >> anybody have a good way to understand this? >> >> One possible answer is "it is the nature of the Fourier Synthesis to >> emphasize phases." (Which is a pretty unsatisfying answer). But, could there >> be an alternative summation which emphasizes amplitudes? If so, that might >> be handy in our field, where we measure amplitudes... >> >> Regards, >> >> Jacob Keller >> >> ******************************************* >> Jacob Pearson Keller >> Northwestern University >> Medical Scientist Training Program >> Dallos Laboratory >> F. Searle 1-240 >> 2240 Campus Drive >> Evanston IL 60208 >> lab: 847.491.2438 >> cel: 773.608.9185 >> email: [email protected] >> ******************************************* >> > > -- > > ============================================================================ > > Bart Hazes (Associate Professor) > Dept. of Medical Microbiology & Immunology > University of Alberta > 1-15 Medical Sciences Building > Edmonton, Alberta > Canada, T6G 2H7 > phone: 1-780-492-0042 > fax: 1-780-492-7521 > > ============================================================================ ------ Randy J. Read Department of Haematology, University of Cambridge Cambridge Institute for Medical Research Tel: + 44 1223 336500 Wellcome Trust/MRC Building Fax: + 44 1223 336827 Hills Road E-mail: [email protected] Cambridge CB2 0XY, U.K. www-structmed.cimr.cam.ac.uk
