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Nature can be tough with humans. It tells us that there are limits. For example, if we want to measure the energy of a particle (or a wave) very accurately we cannot do this in an 'ultrashort' time. Conversely, if we use a very short radiation pulse of length Δ*t* to determine its energy there will be a 'natural' error bar Δ*E* given by the well-known uncertainty relation Δ*E*Δ*t* ≥ 2.35^{2}*h*/4π where *h* is Planck’s constant. (The factor 2.35^{2} takes care of the fact that we consider the full width at half maximum of Gaussian distribution functions.) This is precisely what the paper is about: it describes how a crystal would respond to a hypothetical pulse of infinitely short duration. Because the crystal is perfect, the diffraction process is starting from results of dynamical theory obtained for the angular or energy response of a perfect crystal in the steady-state plane-wave approximation. The Darwin–Prins curve is the input for the subsequent calculations of kinematical type using a 'continuous superposition' of Green’s functions. The result is, as can be expected, a temporal smearing of the incident delta function. Numerical results were calculated for the 111 and 444 reflections from silicon in the symmetrical Bragg case. Does this semidynamical approach give physically reasonable results?

Delta-function-induced transient
reflected intensity at 8 keV from one and two Si(111) Bragg crystals of
thickness 10 μm.

There is another way of looking at the problem. For a given X-ray wavelength the beam has to penetrate a certain depth into the crystal to be fully reflected and to achieve the nominal monochromaticity of the reflection used. Although this penetration or extinction depth The theory was then applied to the output of a free-electron laser (FEL), a presently proposed fourth-generation source where a strongly bunched and highly intense electron beam is sent through a long undulator placed after a linear electron accelerator. Under certain challenging conditions self-amplified spontaneous emission takes place starting from noise. In this way the X-ray laser spouts out about 200 fs long bunch trains of many very short (≥fs), Fourier-transform-limited and extremely intense spikes with strongly fluctuating intensities. The radiation is laterally coherent. The second part of the paper describes what will happen to this hedgehog-like pattern when it is sent through a monochromator. As can be expected, the individual spikes are more or less washed out, according to the degree of monochromaticity achieved. Because the Fourier transform of the time spectrum is a spiky energy distribution, the intensity after transmission through a crystal monochromator shows strong fluctuations, 12% (r.m.s.) for silicon 111 and as much as 56% (r.m.s.) for silicon 444. The time and spectral distributions will vary because each bunch train is different.

All these results are of high interest for time-resolved X-ray diffraction and the future use of FELs. They show that a simple picture of short pulse diffraction can give very realistic results, even on a quantitative level. They also show that we will certainly want to have seeded FELs that would give smooth time and energy distributions. Of course, they confirm that experiments requiring very high energy resolution cannot be carried out on such sources. Finally, an amusing question arises: the diffraction process in the crystal gives priority to monochromaticity, why? It could have chosen to keep the pulse length constant and to smear out the bandpass … in full agreement with the uncertainty relation.

**References**

Sheppard, J.M.H., Lee, R.W. & Wark, J.S. (2001). *Proc. SPIE*, **4500**, 101–112.

Zachariasen, W.H. (1945). *The Theory of X-Ray Diffraction in Crystals*. New York: Wiley (reprinted by Dover in 1994).

European Synchrotron Radiation Facility, Grenoble, France

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