> Perhaps this was really my question: > > Do phases *necessarily* dominate a reconstruction of an entity from phases > and amplitudes, or are we stuck in a Fourier-based world-view? (Lijun > pointed out that the Patterson function is an example of a reconstruction > which ignores phases, although obviously it has its problems for > reconstructing the electron density when one has too many atoms.) But > perhaps there are other phase-ignoring functions besides the Patterson > that > could be used, instead of the Fourier synthesis? > > Simply: are phases *inherently* more important than amplitudes, or is this > merely a Fourier-thinking bias? > > Also, > > Are diffraction phenomena inherently or essentially Fourier-related, just > as, e.g., projectile trajectories are inherently and essentially > parabola-related? Is the Fourier synthesis really the mathematical essence > of the phenomenon, or is it just a nice tool?
In far-field diffraction from a periodic object, yes, diffraction is inherently Fourier-related. The scattered amplitudes correspond mathematically to the Fourier coefficients of the periodic electron density function. You can find this in a solid state physics textbook, like Kittel, for example. > > Jacob > > ******************************************* > 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: j-kell...@northwestern.edu > ******************************************* > > ----- Original Message ----- > From: "Marius Schmidt" <marius.schm...@ph.tum.de> > To: <CCP4BB@JISCMAIL.AC.UK> > Sent: Friday, March 19, 2010 11:10 AM > Subject: Re: [ccp4bb] Why Do Phases Dominate? > > >> You want to have an intuitive picture without >> any mathematics and theorems, here it is: >> >> each black spot you measure on the detector is >> the square of an amplitude of a wavelet. The amplitude >> says simply how much the wavelet goes up and down >> in space. >> Now, you can imagine that when you have many >> wavelets that go up and down, in the average, they >> all cancel and you have a flat surface on a >> body of water in 2D, or, in 3-D, a constant >> density. However, if the wavelet have a certain >> relationship to each other, hence, the mountains >> and valleys of the waves are related, you are able >> to build even higher mountains and even deeper valleys. >> This, however, requires that the wavelets have >> a relationship. They must start from a certain >> point with a certain PHASE so that they are able >> to overlap at another certain point in space to form, >> say, a mountain. Mountains are atomic positions, >> valleys represent free space. >> So, if you know the phase, the condition that >> certain waves overlap in a certain way is sufficient >> to build mountains (and valleys). So, in theory, it >> would not even be necessary to collect the amplitudes >> IF YOU WOULD KNOW the phases. However, to determine the >> phases you need to measure amplitudes to derive the phases >> from them in the well known ways. Having the phase >> you could set the amplitudes all to 1.0 and you >> would still obtain a density of the molecule, that >> is extremely close to the true E-density. >> >> Although I cannot prove it, I have the feeling >> that phases fulfill the Nyquist-Shannon theorem, since they >> carry a sign (+/- 180 deg). Without additional assumptions >> you must do a MULTIPLE isomorphous replacement or >> a MAD experiment to determine a unique phase (to resolve >> the phase ambiguity, and the word multiple is stressed here). >> You need at least 2 heavy atom derivatives. >> This is equivalent to a sampling >> of space with double the frequency as required by >> Nyquist-Shannon's theorem. >> >> Modern approaches use exclusively amplitudes to determine >> phase. You either have to go to very high resolution >> or OVERSAMPLE. Oversampling is not possible with >> crystals, but oversampled data exist at very low >> resolution (in the nm-microm-range). But >> these data clearly show, that also amplitudes carry >> phase information once the Nyquist-Shannon theorem >> is fulfilled (hence when the amplitudes are oversampled). >> >> Best >> Marius >> >> >> >> >> >> >> >> Dr.habil. Marius Schmidt >> Asst. Professor >> University of Wisconsin-Milwaukee >> Department of Physics Room 454 >> 1900 E. Kenwood Blvd. >> Milwaukee, WI 53211 >> >> phone: +1-414-229-4338 >> email: m-schm...@uwm.edu >> http://users.physik.tu-muenchen.de/marius/ >
