The duMond diagram

Many types of beam conditioners have been designed, since the original double-ciystal experiments of Ehrenberg, Ewald and Mark in the early 1920s. These 

We now add another reflection. The sense of a reflection in relation to the previous reflection is important. The notation used for multiple reflections is as follows  

The next reflection is given the symbol n if the planar spacing is identical to the previous reflection (same material, same plane), and wi if it is different. This reflection is given the positive sign if it deflects the beam in the same sense as does the first crystal, the negative sign if the opposite sense. 

Thus the commonly used channel-cut crystal with two reflections has the notation (+ n, - n). The formal duMond diagram notation is as follows  

The ( ) plot for the first crystal is drawn starting from the origin. This defines the direction of the incident beam and hence the zero for angular rotations of the crystals. 

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If the second reflection is -m then the plots diverge and a rotation is necessary for the second crystal to diffract (Figure 2.10). The reflecting range of the system comprising source, aperture and first crystal is defined as discussed above, and diffraction from the second crystal occurs when its curve on the duMond diagram overlaps with some part of this range. 

The intensity of the combined reflection is proportional to the area of overlap of the rocking ciuves in the duMond diagram if the source intensity is uniform across the relevant wavelength region. 

The dispersive (+ n, - m ) mode has already been seen clearly with the duMond diagrams, Figure 2.10. Here, the curves are no longer identical and the crystals must be displaced from the parallel position in order to get simultaneous diffraction. As the crystals are displaced, so the band of intersection moves up and down the curve. When the curves become very different, the K 1 and K 2 intensities are traced out separately. Then the peaks are resolved in the rocking curve, and if no better beam conditioner is available it is important in such cases to remove the K 2 component with a slit placed after the beam conditioner. A slit placed in front of the detector, with the detector driven at twice the angular speed of the specimen, also works very well. This is in effect a low resolution triple-axis measurement. 

In the general multiple-crystal beam conditioner case, there is no universal formula for broadening. Rather, the duMond diagram is constructed for the beam conditioner and the shape of the passed band in and is determined. The specimen crystal is then represented on the duMond diagram and scanned... 

Figure 2.6 Si 220 reflection, (a) duMond diagram showing the wavelength-angle coupling imposed by the Bragg law, (b) the corresponding real-space geometry...

Figure 2.11 The (+n, +n) setting, (a) Real-space geometry, (b) duMond diagram when the second crystal is parallel to the first and does not diffract, (c) duMond diagram when the second crystal is rotated anticlockwise by 2 b and the second crystal diffracts. Si 220 with CuK...

The intensity at any angrrlar setting restrlting from the two reflections corresponds approximately to the area trader the overlap of the two perfect crystal duMond diagrams, Figrtre 2.12. More precisely, the intensity of each... 

As stated earlier, the postmonochromator optics should prepare the incident beam in a way that avoids angle and wavelength averaging effects that would smear (or reduce the fringe visibility of) the XSW. Figure 12a shows the X vs. DuMond diagram for the APS undulator source at E7 = 12.50 keV, the Si(lll) monochromator, and the pair of Si(004) postmonochromator reflections. The slanted stripes represent the conditions where Bragg diffraction is allowed on the basis of dynamical diffraction theory. The... 

Figure 12. 2 vs. DuMond diagrams for the optics of the postmonochromator (from Rodrigues 2000). 

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