The great thing with diffraction, from crystals and
from objects in microscopy is THAT this is
A NATURALLY OCCURRING FORM of Fourier transform once
one accepts that light is a wave (could be something
If Fourier transform would not have been invented with
another problem from engineering, then it would
have emerged NATURALLY from diffraction.
Diffraction is an analog (not a digital) Fourier transform.
A crystal is a low-noise, analog, natural
If you want to build the fastest Fourier transform
of the world, you could represent your function, which
you want to Fourier transform, as
density fluctuation and scatter from it, or, you
could amplify scattering into certain direction
by putting this, your, function in a unit cell of a
1-D, 2-D or even 3-D lattice.
The Patterson function is also a special Fourier-transform,
the convolution of one Fourier with itself.
Yes there are other functions that are also conceivable.
They also map real space (E-density) to reciprocal
space (structure factor). For example, manifold embedding
techniques might never ever even refer to a Fourier transform and
other highly flexible functions are used for this mapping. But
the physics behind it is scattering of waves (as long
as you believe that there are waves, of course).
>> 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
>> could be used, instead of the Fourier synthesis?
>> Simply: are phases *inherently* more important than amplitudes, or is this
>> merely a Fourier-thinking bias?
>> 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 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).
>>> 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]
Dr.habil. Marius Schmidt
University of Wisconsin-Milwaukee
Department of Physics Room 454
1900 E. Kenwood Blvd.
Milwaukee, WI 53211
email: [log in to unmask]