Dear James, what an interesting discussion!
Am 30.01.2009 um 19:42 schrieb James Holton: ...
I think the coherence length is related to how TWO different photons can interfere with each other, and this is a rare event indeed. It has nothing to do with x-ray diffraction as we know it. No matter how low your flux is, even one photon per second, you will eventually build up the same diffraction pattern you get at 10^13 photons/s. Colin is right that photons should be considered as waves and on the length scale of unit cells, it is a very good approximation to consider the electromagnetic wave front coming from the x-ray source to be a flat plane, as Bragg did in his famous construction.
This is also my current understanding, since no matter what the longitudinal coherence (spectral purity) or transversal coherence (size of the source and detector distance) of the X-ray beam is, there is no time coherence in the beam, neither for rotating anode generators, nor for undulator beamlines (see Lengeler, Naturwissenschaften, Vol. 88, p 249-260; it is in English). Apparently, even a single photon "sees" the whole crystal as a wave and deposits its energy as a particle with a probability according to Bragg's law.
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Now, if a perfect crystal is really really small (much smaller than the interaction length of scattering), then there is no opportunity for the re-scattering and extinction and all that "weird stuff" to happen. In this limiting case, the scattered intensity is simply proportional to the number of unit cells in the beam and also to |F| ^2. This is the basic intensity formula that Ewald showed how to integrate over all the depleting beams and re-scattering stuff to explain a large perfect crystal.
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I'm not sure where this rumor got started that the intensity reflected from a mosaic block or otherwise perfect lattice is proportional to the square of the number of unit cells. This is never the case. The reason is explained in Chapter 6 of M. M. Woolfson's excellent textbook, but the long and short of it is: yes the instantaneous intensity (photons/steradian/s) at the near- infinitesimal moment when a mosaic domain diffracts is proportional to the number of unit cells squared, but this is not useful because x-ray beams are never perfectly monochromatic nor perfectly parallel.
Hmm, I don't have Woolfson's book at hand, so I can't read this chapter. My current understanding is: if N unit cells scatter in phase, the scattered total amplitude is N times the scattered amplitude of the unit cell in that direction. Since the recorded intensity is proportional to the square of the amplitude, the scattered total intensity is proportional to N^2 (for simplicity, I assume perfect sources, crystals and detectors, so I don't discuss spot shapes). Now, if the coherence lengths is limited by the size of the mosaic blocks, each block scatters like a tiny crystal independent of the other mosaic blocks. Thus, if we have m < N mosaic blocks (of equal size), each block results in a scattered intensity ~(N/m)^2, and the m blocks add their intensities, yielding a total intensity ~N^2/m. Dependent of the perfection of the crystal, the total scattering intensity for the two extremes between m=N (which wouldn't make any sense) and m=1 is proportional to between N and N^2, respectively. Please, correct me, if I'm wrong.
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