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/
