For a full answer to all your questions, I refer you to the classic
textbook of M. M. Woolfson "an introduction to x-ray crystallography" by
Cambridge University Press. This book has been quite helpful to me of
late. Unlike some similar texts I find it easy to read. There are even
examples! With real numbers! Woolfson begins by describing scattering
as it was originally derived by J. J. Thomson from classical mechanics
of a charged mass (the electron) vibrating in an electric field (the
photon), and takes you, step-by-step all the way to Bragg scattering
from a rotating 3-D crystal.
You have received many comments so far, so I will try not to repeat
those in answering your questions here:
1) yes, x-ray sources are "non-coherent", but the photons ARE coherent
on a short length scale. Specifically, this is about 0.7 micron if the
x-rays are 1 A in wavelength and have a spectral dispersion of 1/7000,
such as with a Si(111) monochromator. That is, after traveling 0.7
micron, two photons that were once in phase will no longer be, because
they have different wavelengths. This is approximately the "coherence
length" in the direction of propagation. The other two directions are
... more complicated. It is actually quite difficult to make an x-ray
source with a very short spatial coherence. I've heard "talk" about
shaping x-ray wavefronts in next-generation accelerators, such as the
ERL planned at CHESS. Thus would have the advantage of changing the
phase relationship between atoms as you described and (potentially)
getting phase information directly. However, I don't think anybody has
actually done this yet.
2) In general, photons do not interfere with each other. Have you ever
heard of photon-photon scattering? Neither have I. However, there
actually is a body of work on multi-photon correlations in scattering.
Apparently, two photons can have correlated wave functions, and this
leads to correlations in the arrival time of photons at the detector:
something called the Hanbury-Brown and Twiss effect. The math behind it
it a bit beyond me. However, theoretically, such correlated scattering
events could be used to get phase information if you have a fast
detector and a REALLY fast source. My colleague Ken Frankel
<[log in to unmask]> can tell you more about it if you are interested.
2b) WRT the "all electrons scattering together" question. Yes, they do
all scatter "together" because they are all confined in the same atom.
The total scattering is explained by integrating Thomson's scattering
formula (including the phase shift) over all of the electron positions.
Yes, a single photon can interact with more than one electron, just as a
single photon can pass through two slits in the famous experiment by
Thomas Young. For any given photon, the positions of each electron will
(in the classical view) be at some well-defined position, but we
experimentally integrate over a LOT of photons.
If no two photons ever see a consistent arrangement of electron
positions (such as when you shoot x-rays at a free electron beam) then
the scattering is "incoherent" and the scattered intensity distribution
you see simply follows Thomson's classical scattering formula for one
electron, with the intensity multiplied by the number of electrons in
the x-ray beam.
However, if the "electron density" is not random but instead has
some kind of consistent feature from photon to photon, then the
interference of the scattered waves will also be consistent as it builds
up on your detector. This "binding effect" is only significant on the
length scale of the electron confinement (the size of an atom in this
case), so it falls off with increasing scattering angle. Remember, the
phase of the scattered wave depends on the total path length traversed
by the incident and scattered photons. The phase shift between
scattering at any two points in the sample will always become
vanishingly small at a small scattering angle because the difference in
path length from one "side" of the atom to the other "side" becomes
smaller and smaller at low scattering angle. For larger scattering
angles there is a larger phase shift, and the interference becomes more
destructive. The quantitative angular dependence of scattering from any
atom is called the "atomic form factor", and it is tabulated in
${CLIBD}/atomsf.lib. The "form factor" is why high-angle spots are
weaker than low-angle spots, even if all the B-factors are zero. Atomic
form factors are derived from scattering measurements on gasses (which
don't have B-factors), and theoretical electron-distribution
calculations have been found to be consistent with these observations.
The upper limit to constructive interference from a single atom is
if all of the electrons in the atom are scattering in phase. This will
always occur at vanishingly small scattering angles (forward
scattering). Since there is no phase shift the amplitudes add, and you
square the amplitude to get intensity. Experimentally, the forward
scattering intensity (that is: low-angle scattering extrapolated to zero
angle) from any particle is equal to Z^2 multiplied by Thomson's formula
(where Z is the number of electrons in the particle). This is true for
everything from He gas to protein molecules in solution (as observed by
SAXS).
The reason why you multiply the intensity by just Z (and not Z^2)
when the electrons are not confined to atoms is because out-of-phase
amplitudes add "in quadrature": Ftotal=sqrt(sum(F^2)) and I~Ftotal^2.
In-phase amplitudes just add: Ftotal=sum(F), and I~Ftotal^2.
In multi-atom particles (such as proteins) you have two (or more)
atoms that are constrained to be "near" one another. By "near" I mean
separated by a distance corresponding to a scattering angle that will
clear the beamstop. In this case, the scattering from one atom can
interfere with that from the other (just as the scattering from
electrons within the same atom interfere with each other). Taking the
vector sum of the form factors of the two atoms together explains the
total scattering. I have personally confirmed this with the scattering
from N2 gas from the cryo-stream at my beamline! N2 has 14 electrons
worth of forward scattering, not 7. If the distance between the two
atoms is not infinitely precise, then the Gaussian distribution of
atom-atom vectors in real space becomes a Gaussian in reciprocal space
that you multiply by the "perfect" two-atom form factor. There is some
debate over who first called this a "B factor", but the name has
certainly stuck.
3) As Dale pointed out, the energy goes into other reflections. Don't
forget F(0,0,0). That is a REAL reflection, and it is always on the
Ewald sphere. If you manage to orient a crystal so that no visible
Bragg peaks intersect the Ewald sphere, then all the elastically
scattered photons will go into F(0,0,0). Incidentally, for visible
light the interference of F(0,0,0) with the main beam gives rise to the
"index of refraction" effect. There is an index of refraction for
x-rays, but it is much smaller. These and other effects of conservation
of energy are accounted for in the "dynamical theory" of diffraction,
and this is what is used for the so-called "three beam" phasing
technique. The interaction of scattering and absorption gives rise to
anomalous dispersion, and this is also explained by the dynamical theory.
Bragg's Law and other familiar equations come to us from the
"kinematic approximation" to the dynamical theory. This approximation
involves ignoring small "violations" to conservation of energy and
conservation of momentum. For example, an elastically scattered photon
has changed direction (momentum) without changing energy (wavelength).
This approximation does ignore conservation of momentum, but the small
amount of momentum transferred from the crystal to the photon is
distributed evenly over all the atoms in the crystal, so the "recoil"
motion of the crystal is very small (and safe to ignore in
crystallography). Other things ignored by the kinematic approximation
are secondary scattering and depletion of the primary (and diffracted)
beam intensity as it looses photons to scattering. For protein
crystals, the kinematic approximation is very good since there are many
spots and they are all very weak when compared to the incident beam. I
bought a different book (not Woolfson) on the dynamical theory so that I
could understand all this more quantitatively. This book has served
well putting me to sleep every night...
-James Holton
MAD Scientist
Michel Fodje wrote:
> Dear Crystallographers,
> Here are a few paradoxes about diffraction I would like to get some
> answers about:
>
> 1. In every description of Braggs' law I've seen, the in-coming waves
> have to be in phase. Why is that? Given that the sources used for
> diffraction studies are mostly non-coherent.
>
> 2. Trying to derive the diffraction condition for a pair of non-coherent
> waves with a phase difference of 'y' where 0 < y < 2pi, I obtain the
> following diffraction condition
> y * (lambda/2pi) = 2d sin (theta)
> i.e. the phase difference y = 4pi * sin(theta) * d / lambda
> This seems to imply that diffraction will occur if the incident waves
> are not in phase but the phase difference still satisfies the above
> condition. One may be able to envision a case where for a given distance
> d, the diffracting condition will be met for various angles depending on
> the phase shift of the waves diffracting. Does this make sense? Has
> anyone looked at the significance of this relationship before? Any
> pointers will be welcome.
>
> 3. What happens to the photon energy when waves destructively interfere
> as mentioned in the text books. Doesn't 'destructive interference'
> appear to violate the first and second laws of thermodynamics? Besides,
> since the sources are non-coherent, how come the photon 'waves' don't
> annihilate each other before reaching the sample? If they were coherent,
> would we just end up with a single wave any how? With what will it
> interfere to cause diffraction?
>
> I'm sure some of these may have some really obvious answers I may be
> missing.
>
> Thanks,
>
> Michel
>
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