Hi Jacob here is another stab at it. Not as forme
When you create an electron density map you can think of each amplitude
F and phase P as forming a vector Vmap that consists of two sub-vectors:
one that represents the true vector Vtrue and the other representing the
vector, Vdiff, that connects Vtrue to Vmap. Just try to picture this.
You can now think of calculating three maps based on Vtrue, Vdiff, and
Vmap. Since FFTs are additive you can consider the map you would
normally calculate, Vmap, as being the sum of the two others; Vtrue
being reality and Vmap being noise.
If you have a phase error of 60 degrees Vdiff will actually already be
of the same magnitude as Vtrue. If you have random phases you will, on
average, be 90 degrees off, and Vdiff will be 1.41 times as large as
Vtrue (sqrt(2)). Even relatively small phase errors give significant
Vdiff/Vtrue ratios (2*sin(half-the-phase-error) if I'm right)
You mentioned "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."
But in reality amplitudes cannot be anything since they follow a
Wilson's distribution which has the bulk of the amplitudes cluster near
the peak of the distribution. The real mathematicians can probably tell
you the expected amplitude error in such a scenario but that would
certainly go beyond an intuitive explanation and my own math skills. If
you look at experimental errors in the amplitudes it is even much less.
Rmerge tends to be in the 3-10% range and that is on intensities, it
will be considerable less on amplitudes.
Bart
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
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