Darwin width

These experimental results and their interpretations clearly showed that the study of the diffracted intensity makes it possible to accurately determine the nature and the positions of the atoms inside the crystal cell. Nevertheless, the link between stmctural arrangement and the value of the total diffracted intensity was not proven. This aspect will be studied in detail by Darwin, who showed in two famous articles [DAR 14a, DAR 14b] that, on the one hand, the intensity is not concentrated in one point (defined by the Laue relations), but that there is a certain intensity distribution around this maximum (referred to as the Darwin width) and, on the other hand, that real crystals show a certain mosaicity that can account for the values of the diffracted intensity measured experimentally. These considerations were based on a description similar to that used for visible light in optics and constituted a preamble to the dynamic theory of X-ray diffractiort, the core ideas of which were later established by Ewald [EWA 16a, EWA 16b]. 

Most of the current optics using Synchrotron Radiation diffracts in the vertical plane and thus is sensitive to vertical bouncing of the beam. The horizontal optical plane of the dispersive scheme combines this extra advantage which helps to keep superior energy resolution since the orbit seems to show a better stability in the horizontal direction. Owing to the horizontal polarization of S.R., one must consider the cos (20) attenuation factor which reduces the Darwin width of the crystal. This results in a lower reflectivity and an improved energy resolution as well. 

Owing to the good spatial resolution of our position sensitive detector (better than 50 pm), the dominant term is usually the Darwin width. Nevertheless, the Darwin width can be reduced by going to a higher order and/or asymmetric reflections and hence, the effect of the spatial resolution of the detector can become as important. 

Another important parameter that may affect the resolution is the higher harmonic contribution from the Bragg reflector. A fused quartz mirror behind the monochromator has been currently used to reject this high harmonic contribution. Therefore, the energy resolution of the spectrometer is just limited by the Darwin width of the rocking curve and the spatial resolution of the position-sensitive detector [8]. 

The rocking width or Darwin width mono of a perfect crystal is derivable from dynamical diffraction theory as... 

This is the Darwin width of the reflectivity curve or rocking curve. ... 

On the other hand, one can make XSW applications under certain conditions where the stringent requirements of the conventional theory are relaxed. One of such condition is when a Bragg reflection occurs near the back-reflection geometry, typically when 87° <6b< 90°. Under this condition, Equation (11) breaks down, and the intrinsic Darwin width is magnified to milliradians. Another condition is when the crystal is very thin under this condition the kinematical theory can replace the dynamical theory in calculating Bragg reflection. Both of these special-cases, which will be discussed briefly below, are important alternatives when the conventional XSW technique cannot be applied. 

Figure 7. Comparison of two measured muscovite rocking curves for the (006) Bragg reflection acquired by using (a, open circles) an undulator synchrotron beamline with a Si(l 11) monochromator at Er = 7.4 keV, and (b, solid circles) Cu Kax X-rays from a Cu fixed-anode source (Er = 8.04 keV ) followed by a Si(lll) four-bounce high-resolution monochromator and high-resolution diffractometer. The difference between the two-curves is primarily due to the size of the incident beam slit, which is 0.2 mm x 0.05 mm for (a) and 2 mm x 1 mm for (b). Only a source with the brightness of the undulator could be slitted down to such a small size as for (a) and still has sufficient flux on the sample for XSW measurements. Also shown is the best fit to the rocking curve of (a) (solid line). At 7.4 keV, the Darwin width of Si(l 11) is 37 / -ad and that of the muscovite (006) is 20 /rrad.

The systems of cracks and microcracks in cement-based materials may be analyzed at different scales. An extensive discussion of scales of cracks is presented, for example, by Darwin et al. (1995) and by Ringot and Bascoul (2001). The methods of observation of cracks in specimens and elements are adapted to the selected scale and related to the purpose. At different scales, different systems of cracks and microcracks exist and may be observed and measured. For material design, testing and applications in building and civil engineering structures, but also in non-structural elements, the attention is concentrated at microcracks of width from approximately 10 pm to cracks a few millimetres wide. All features are strongly influenced by the sensitivity of observation and measurements. The lower bound of the crack width cannot be specified without ambiguity, because very thin cracks and thin crack tips disappear in pores and cavities of the cement paste structure. 

Because of the large optical anisotropy of liquid crystals, the relative width A2/2 of this Darwin band is much larger than in X-ray crystallography where the modulation of the refractive index (for X-rays) is of the order of 10 . Experimentally, the measure of the Darwin band in cholesteric liquid crystals requires production of perfectly aligned samples and the use of a spectrometer. This can hardly be done during a lecture. 

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