James Holton wrote:
marc.schi...@epfl.ch wrote:
The elastically scattered photons (which make up the Bragg peaks) also
do not not retain the momentum of the incident photon.
Although technically true to say that photons traveling in different
directions have different momenta, all elastically scattered photons
have the same wavelength (momentum) as the incident photon. Otherwise,
I would definitely avoid to amalgamate wavelength and momentum, as is
more-or-less suggested in the final part of this statement. Momentum is
a vector quantity, although it is true that the NORM of the momentum
vector of a particle is related to its energy (by the De Broglie
wavelength relation). In X-ray diffraction, the momentum of the
elastically scattered photon does change, while its energy does not. In
X-ray physics, the change in momentum is actually called the "momentum
transfer" : \vec{Q} = \vec{k'} - \vec{k}. The word says it all.
they would not interfere constructively to form Bragg peaks and they
would be called Compton-scattered photons. The small change in energy
required to preserve wavelength upon a change in direction during
elastic scattering is contributed by the entire crystal as a "recoil"
phonon. Arthur Compton wrote a paper about this:
http://www.pnas.org/cgi/reprint/9/11/359.pdf
Very interesting paper, but I see no mention of a "recoil phonon" and I
would be surprised if that is what Compton really meant. No mention
about lattice dynamics (phonons) can be found in this paper. The crystal
is implicitly assumed to be a rigid body. In fact, what the paper nicely
demonstrates is that the conservation of wavelength (i.e. photon energy)
between incident and diffracted rays is achieved in the limiting case
when the total mass of the crystal is very large with respect to the
mass of one photon - a condition which, I presume, is always satisfied
in X-ray crystallography, even when going towards microcrystals.
This is really the same situation as a tennis ball that bounces
(elastically) off the surface of the earth. In principle, we must assume
that some of its energy is transferred to the earth during the
collision. But because the mass of the earth is so vastly superior to
the mass of the tennis ball, the transfer of energy is vanishingly
small. It certainly can not be measured. The change of momentum of the
tennis ball, however, is not negligible and can be measured.
Back to X-ray diffraction, the reciprocal lattice is just a
representation of momentum transfer vectors \vec{Q} = 2\pi \vec{h}. You
may never have thought of it like this, but when we index an X-ray
pattern, we are really measuring the change in momentum of the photons
which were scattered into the various Bragg peaks. But we can not
measure their change in energy, as it is practically zero.
The situation becomes somewhat different if we take into account lattice
dynamics (phonons) as it is now possible to measure the energy transfer
of a scattered X-ray photon upon phonon creation in the crystal. But
these are very difficult measurements (much easier with neutrons) and
are certainly of no relevance for macromolecular X-ray crystallography.
It is anyway called inelastic scattering.
which probably contributed to his Nobel four years later. This is a
classic example of the confusion that can arise from the particle-wave
duality.
It seems to me that the confusion here is between energy and momentum.
--
Marc SCHILTZ http://lcr.epfl.ch
