Dear Bart,
Thank you for further analysing this idea of "inherence", which is so
striking in this case.
I just wanted to point out that the most natural setting for Fourier
transform theory is the space L2 of square-integrable functions (although L1
is the more natural one for convolution-related properties). Both of these
are spaces of complex-valued functions from the start, so I am not aware of
a restriction to real-valued functions anywhere in Fourier theory, and no
initial framework thus needs to be broken to accommodate a complex-valued
electron density. Real-valued functions in L1 or L2 are rather oddities, a
little bit like centric reflections for us. Their Fourier transforms have
the extra property of having Hermitian symmetry (the generic mathematical
term for what we call Friedel symmetry). What happens when anomalous
scattering gets us back to a general complex-valued electron density
function is simply that we lose that extra property, not that the framework
of Fourier theory has to be extended.
This is a minor point, but worth bearing in mind.
With best wishes,
Gerard.
--
On Mon, Mar 22, 2010 at 05:00:23PM -0600, Bart Hazes wrote:
> Hi Jabob,
>
> Mathematics is abstract and does not cause anything (well maybe headaches).
> It describes behaviors of real-world phenomena and probably a lot of other
> things that have no tangible interpretation.
>
> What Gerard meant when he said that "Fourier transform is at the heart of
> diffraction" is not that it causes diffraction but that the properties of
> the Fourier transform form directly capture the properties of the physical
> phenomenon of diffraction. Unless our understanding of diffraction turns
> out to be wrong, like the early astronomers were wrong about the center of
> the universe, the Fourier transform will remain the natural mathematical
> model for this process.
>
> What does happen frequently is that a simpler mathematical model needs to
> be replaced by a more general model once more data becomes available. It is
> conceivable, at least to me, that some day we need a more generalized model
> for diffraction. For instance, our typical Fourier transforms assume that
> electron density can be treated as a real value, but heavy atoms also cause
> a phase shift of the diffracted wave and thus need to be modeled as an
> imaginary value. That doesn't mean that the initial model was wrong, just
> that it is only valid in a certain "domain", and outside that domain we
> need to elaborate the model (which in this case is still a Fourier
> transform). In many (all?) cases the old model ends up being a special case
> of the more general variant, just like real numbers are just a special
> subset of the imaginary numbers.
>
> Bart
>
>
>
> Jacob Keller wrote:
>> ----- Original Message ----- From: "Gerard Bricogne"
>> <g...@globalphasing.com>
>> To: <CCP4BB@JISCMAIL.AC.UK>
>> Sent: Friday, March 19, 2010 2:32 PM
>> Subject: Re: [ccp4bb] Why Do Phases Dominate?
>>
>>
>>> Dear Marius,
>>>
>>> Thank you for pointing this out - I was about to argue in the same
>>> direction, i.e. that the Fourier transform is at the heart of diffraction
>>> and is not just a convenient, but perhaps renegotiable, procedure for
>>> analysing diffraction data.
>>
>> I wonder how one can establish that a certain mathematical function is "at
>> the heart" of a phenomenon? Does mathematics cause phenomena, or consitute
>> the essence of a phenomenon? Many believe that it does, and I am not
>> saying that I do not feel this way about some relationships between
>> mathematics and phenomena--but there seems to be a gradation. On one side,
>> the trajectory of the stream of water from a garden hose is
>> all-too-obviously essentially a parabola, and on the other side, "laws"
>> like Moore's law seem completely descriptive and not at all causal or
>> essential. A gray-area case for me is whether the manifest world is
>> fundamentally Euclidean, or other such questions. (It certainly *feels*
>> Euclidean, but...) But what I am unsure about is what standard we use to
>> decide whether a given phenomenon is *inherently* tied to a given
>> mathematical function. A troubling thought is that there are many
>> historical examples of phenomena being *fundamentally* one way
>> mathematically--and unthinkably otherwise--and later we have revised our
>> "certainty." One thinks of the example Ian Tickle cited of negative
>> numbers being meaningless, or also of the Earth's being the center of the
>> universe and orbits being perfectly circular. A medieval philosopher
>> wanted once to emphasize the certainty of his conclusions, and he wrote
>> that they were as clear and certain as the Earth's being the center of the
>> universe! (Ergo: how certain can we be, then, about *his*
>> conclusions...?). Anyway, one could speculate that there be an alternative
>> model to diffraction which does not involve the Fourier synthesis, it
>> seems. But would that be just a model, and the Fourier-based one the
>> reality?
>>
>>
>>
>>
>>>
>>> Another instance of such natural "hardwiring" of the Fourier
>>> transform
>>> into a physical phenomenon is Free Induction Decay in NMR. There,
>>> however,
>>> one can measure the phases, as it is the Larmor precession of the
>>> population
>>> of spins that is measured along two orthogonal directions and gives the
>>> real
>>> and imaginary parts of the FID signal. Equal opportunity for real and
>>> imaginary part: doesn't that make a crystallographer dream ... ?
>>>
>>>
>>> With best wishes,
>>>
>>> Gerard.
>>>
>>> --
>>> On Fri, Mar 19, 2010 at 08:15:05PM +0100, Marius Schmidt wrote:
>>>> 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
>>>> else).
>>>> 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
>>>> Fourier amplifier!!!
>>>> 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).
>>>>
>>>> Marius
>>>>
>>>>
>>>> >> 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/
>>>> >>
>>>>
>>>> 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/
>>>
>>> --
>>>
>>> ===============================================================
>>> * *
>>> * Gerard Bricogne g...@globalphasing.com *
>>> * *
>>> * Global Phasing Ltd. *
>>> * Sheraton House, Castle Park Tel: +44-(0)1223-353033 *
>>> * Cambridge CB3 0AX, UK Fax: +44-(0)1223-366889 *
>>> * *
>>> ===============================================================
>>
>
> --
>
> ============================================================================
>
> Bart Hazes (Associate Professor)
> Dept. of Medical Microbiology & Immunology
> University of Alberta
> 1-15 Medical Sciences Building
> Edmonton, Alberta
> Canada, T6G 2H7
> phone: 1-780-492-0042
> fax: 1-780-492-7521
>
> ============================================================================
