Great job. But don't you have any better things to do (tm) ? s.
On 1 Apr 2008, at 18:59, James Holton wrote:
Dear CCP4BB,
I think it prudent at this point for me to announce what could
be a very old, but serious error in the fundamental mathematics of
crystallography. To be brief, I have uncovered evidence that the
"hand" of the micro-world is actually the opposite of what we have
believed since Bijvoet's classic paper in 1951.
Those of you who know me know that I have been trying to lay
down the whole of x-ray diffraction into a single program. This is
harder than it sounds. We all know what anomalous scattering is,
but a detailed description of the math behind translating this
"dynamical theory" effect all the way to the intensity of a
particular detector pixel is hard to find all in one place. Most
references in the literature about how anomalous scattering is
connected to absolute configuration point to the classic Nature
paper: Bijvoet et. al. (1951). Unfortunately, since this is a
Nature paper, it is too short to describe the math in detail. For
the calculations, the reader is referred to another paper by
Bijvoet in the Proc. Roy. Acad. Amsterdam v52, 313 (1949).
Essentially, the only new information in Bijvoet et. al. (1951) is
the assertion that Emil Fischer "got it right" in his initial
(arbitrary) assignment of the "R" and "S" reference compounds for
the absolute configuration of molecules.
I decided to follow this paper trail. The PRAA document was hard
to come by and, to my disappointment, again referenced the "real"
calculation to another work. Eventually, however, all roads lead
back to R. W. James (1946). This is the definitive textbook on
scattering theory (originally edited by Sir Lawrence Bragg
himself). It is extremely useful, and I highly recommend that
anyone who wants to really understand scattering should read it.
However, even this wonderful text does not go through the full
quantum-mechanical derivation of scattering, but rather rests on J.
J. Thompson's original classical treatment. There is nothing wrong
with this because the the exact value of the phase lag of the
scattering event does not effect anything as long as the phase lag
from all the atoms is the same. The only time it does become
important is anomalous scattering. Even so, changing the sign of
the phase lag will have no effect on any of the anomalous
scattering equations as long as all the anomalous contributions
have the same sign. The only time the sign of the phase lag is
important is in the assignment of absolute configuration.
Unfortunately, a full quantum mechanical treatment of the
scattering process DOES produce a phase lag with the opposite sign
of the classical treatment. This is not the only example of this
sort of thing cropping up. One you can find in any quantum text
book is the treatment of "tilting" a quantum-mechanical spin (such
as an electron). It was shown by Heisenberg that a "tilt" of 360
degrees actually only turns an electron upside-down. You have to
"tilt" it by 720 degrees to restore the initial state, or get it
"right-side-up" again. This is very counterintuitive, but true,
and unfortunately a similar treatment of scattering results in a
phase lag of +270 degrees to "restore" the electron after the
scattering event, not +90 degrees as was derived classically. To
be brief, there is a sign error.
Perhaps the reason why noone caught this until now is not just
that the quantum calculations are a pain, but that it was very
tempting to accept that the large body of literature following
Fischer's convention would not have to be "corrected" by inverting
the hand of every chiral center described up to that time.
Unfortunately, we now have an even larger body of literature
(including the PDB) that must now be "corrected".
It is an under-appreciated fact in chemistry that anomalous
scattering is arguably the only direct evidence we have about the
"hand" of the micro-world. There are other lines of evidence, such
as the morphology of macroscopic crystals and some recent STEM-type
microscope observations of DNA. However, as someone with a lot of
experience in motor control I don't mind telling you how easy it is
to make a sign error in the direction of an axis. This is
especially easy when the range of motion of the axis is too small
to see by eye. You end up just swapping wires and flipping bits in
the axis definitions until you "get it right". The "right"
configuration (we have all assumed) is the one asserted in Bijvoet
et. al. (1951). Apparently, the STEM observations fell prey to
such a "mistake". But can you blame them? Inverting the "hand of
the world" is going to be very hard for a lot of people to accept.
Indeed, if anyone can find an error in my math, please tell me! I
would really like to be wrong about this.
-James Holton
MAD Scientist
