Dear Jacob,
This is a simple consequence of the convolution theorem, i.e. the
theorem that the Fourier transforms turns a convolution into a product and
conversely.
As each structure factor (which is a Fourier coefficient) is the
product of a real-valued, non-negative amplitude and of a phase factor (a
complex number of unit modulus), the Fourier transform will turn them into a
map that will be the convolution of the transform of the amplitudes and of
the transform of the phase factors.
Now, as all the amplitudes are non-negative real numbers, they all have
a phase of zero, so their transform will reach a maximum - and show a huge
dominant peak - at the origin. It will be, so to speak, a "dirty delta
function", with some side-lobes unlike a true delta function, but still very
peaky around the origin. Since the delta function is the unit of convolution
(i.e. the convolution of any function with delta is that function itself),
convoluting the transform of the phase factors with the dirty delta function
coming from the amplitudes will not make much difference to it - therefore
whatever features show up in the electron density map would have been
showing up in the transform of the phase factors in the first place. This is
just another way of saying that "Phases Dominate". This is a crude argument,
assuming for instance a primitive lattice.
I don't know whether you would call this argument intuitive, but that
is the simplest one I know of.
There might be other transforms that would emphasise amplitudes, but it
would remain to be seen whether they would produce anything useful if fed
with structure factor amplitudes ... .
With best wishes,
Gerard.
--
On Thu, Mar 18, 2010 at 12:51:03PM -0500, Jacob Keller wrote:
> Dear Crystallographers,
>
> I have seen many demonstrations of the primacy of phase information for
> determining the outcome of fourier syntheses, but have not been able to
> understand intuitively why this is so. Amplitudes as numbers presumably
> carry at least as much information as phases, or perhaps even more, as
> phases are limited to 360deg, whereas amplitudes can be anything. Does
> anybody have a good way to understand this?
>
> One possible answer is "it is the nature of the Fourier Synthesis to
> emphasize phases." (Which is a pretty unsatisfying answer). But, could
> there be an alternative summation which emphasizes amplitudes? If so, that
> might be handy in our field, where we measure amplitudes...
>
> Regards,
>
> Jacob Keller
>
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> Jacob Pearson Keller
> Northwestern University
> Medical Scientist Training Program
> Dallos Laboratory
> F. Searle 1-240
> 2240 Campus Drive
> Evanston IL 60208
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