The "Bragg planes" are a contrivance of our invention to make the
math simpler and allow us to converse in shorthand terms like "Bragg's
Law". The photon's wave function interacts with the wave functions
of every electron it overlaps with, which is many unit cells because
our photons have quite diffuse wave functions, reasonably approximated
by plane waves. One writes out the integral over all these interactions
to get the amount of net scattering in any direction, but the
mathematicians have done this and told us that you can get the same
result, for our case single wavelength illumination, by calculating
a Fourier Transform to get the amplitudes and phases, while the
reciprocal lattice construction along with Ewald's sphere will tell
us the directions of the diffracted beams.
Unfortunately text books usually start and end with Bragg Planes
so their descriptions are confusing to people who start thinking
about continuous electron density. The problem is that the real
math is rather involved and the discussion requires knowledge of
optics that is beyond, and probably uninteresting, to most of the
people who want to solve protein structures.
If your sample is not crystalline, the Fourier Transform and
Ewald's sphere still works, but you then have a continuous function
instead of spots and your life will be hard.
The best book I've seen on this topic, but by no means have
absorbed, is "The Optical Principles of the Diffraction of X-Rays",
by R. James.
Dale Tronrud
Michel Fodje wrote:
Would it be taking it too far to suggest that one could go all the way
and consider that each electron diffracts not as groups in a plane but
as individual electrons and a photon impinging on an electron with with
a specific phase will be diffracted in a specific direction. However the
lattice arrangement of the electrons will statistically accumulate
photons which impinged on electrons on a specific family of planes in
one direction at the detector. Such that the crystal is a phase sorter.
In which case diffraction is not based on constructive or destructive
interference but on conservation of some property of the photon, such as
angular momentum? IANAM either.
On Fri, 2007-08-24 at 15:36 -0700, James Stroud wrote:
Without resorting to a circular argument? You are asking too much.
However, this probability distribution is perfectly described by
considering a component wave model wherein coherence of the component
waves correlates with peaks in the probability distribution--i.e.
Bragg's Law.
IANAM (I am not a mathematician), but, if pressed, I would posit that
one could decompose the fun description just a little bit and consider
the lattice not as *groups* of reflecting planes, but as individual
planes. In such a case, each single reflecting plane would contribute a
probability distribution with an angular dependence. The total
probability distribution would then be the sum of the probability
distributions for every plane in the lattice.
Your next question might be, "what's the probability distribution for a
single plane". Well, I would imagine that it has a maximum where the
angle of incidence equals the angle of reflection and that the phase of
a component probability distributions is spatially (i.e. angularly)
directly related to the phase of the originating photon.
The sum distribution of the reflected photon takes into account the
angular phase dependence of its components and so one gets positive and
negative interference between component distributions.
James
Jacob Keller wrote:
Yes, but why should the directions of diffraction conditions be most probable
(one of your premises)?
==============Original message text===============
On Fri, 24 Aug 2007 4:54:53 pm CDT James Stroud wrote:
Here's a fun way to think of it:
A photon hits a crystal and will diffract off in a certain direction
with the same energy as the original photon. The direction is subject to
a probability distribution based on the lattice, with angles at the
diffraction conditions being most likely and the broadness of the peaks
in the distribution arising from imperfections in the lattice. The
photon propagates as this probability distribution and then is forced to
select from the distribution because we stuck a detector up. The
diffraction pattern we observe is the sum of many such photons
interacting with the crystal.
