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?
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: [log in to unmask]
*******************************************
----- Original Message -----
From: "Marius Schmidt" <[log in to unmask]>
To: <[log in to unmask]>
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: [log in to unmask]
> http://users.physik.tu-muenchen.de/marius/
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