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: 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/
