misaligned crystals is a reflection with an approximately Gaussian profile and an
integrated intensity proportional to the square of the structure factor: I / |F|
2
(Fig. 4.15).
4.6.1 Perfect Crystal Diffraction
The reflections observed from the “perfect” single crystals used as monochromators
and analyzers are quite different—they exhibit nearly flat-topped profiles of almost
100% reflectivity with widths proportional to the structure factor: I / |F| (Fig. 4.16).
To explain this behavior, dynamical diffraction theory assumes one single crystal
and adds two new considerations: (a) the loss in amplitude of the incident wave as it
propagates through the crystal (due to reflection into the exit beam) and (b) the
possibility that the reflected beam is rescattered into the direction of the incident
beam (multiple scattering). Detailed descriptions and derivations of this theory can
be found in the references at the end of this chapter; here as usual we concentrate on
the results from those derivations.
As illustrated in Fig. 4.16, in dynamical theory with perfect crystals, it is possible
to achieve 100% reflectivity over a range of angles, as opposed to the Gaussian
profile of limited reflectivity that results from ordinary imperfect crystals. The reason
for such high reflectivity is simple—the beam keeps penetrating until it finally
scatters and leaves the crystal. The rationale for a width proportional to the structure
factor is also straightforward—stronger scattering means a smaller number of atomic
planes are required to scatter the beam (just as the width of a diffraction peak from a
grating vary inversely with the number of lines illuminated.) When absorption is
taken into account, the reflectivity drops slightly, and the rocking curve
(intensity vs. angle) becomes somewhat more asymmetric, but reflectivities of
>90% over a range of angles are not uncommon.
Fig. 4.15 Left: definition of terms in Bragg diffraction. Middle: an ideally imperfect crystal as a
mosaic assembly of crystal blocks. Right: the resulting broad diffraction profile peak
86
4 X-ray Optics and Synchrotron Beamlines
integrated intensity proportional to the square of the structure factor: I / |F|
2
(Fig. 4.15).
4.6.1 Perfect Crystal Diffraction
The reflections observed from the “perfect” single crystals used as monochromators
and analyzers are quite different—they exhibit nearly flat-topped profiles of almost
100% reflectivity with widths proportional to the structure factor: I / |F| (Fig. 4.16).
To explain this behavior, dynamical diffraction theory assumes one single crystal
and adds two new considerations: (a) the loss in amplitude of the incident wave as it
propagates through the crystal (due to reflection into the exit beam) and (b) the
possibility that the reflected beam is rescattered into the direction of the incident
beam (multiple scattering). Detailed descriptions and derivations of this theory can
be found in the references at the end of this chapter; here as usual we concentrate on
the results from those derivations.
As illustrated in Fig. 4.16, in dynamical theory with perfect crystals, it is possible
to achieve 100% reflectivity over a range of angles, as opposed to the Gaussian
profile of limited reflectivity that results from ordinary imperfect crystals. The reason
for such high reflectivity is simple—the beam keeps penetrating until it finally
scatters and leaves the crystal. The rationale for a width proportional to the structure
factor is also straightforward—stronger scattering means a smaller number of atomic
planes are required to scatter the beam (just as the width of a diffraction peak from a
grating vary inversely with the number of lines illuminated.) When absorption is
taken into account, the reflectivity drops slightly, and the rocking curve
(intensity vs. angle) becomes somewhat more asymmetric, but reflectivities of
>90% over a range of angles are not uncommon.
Fig. 4.15 Left: definition of terms in Bragg diffraction. Middle: an ideally imperfect crystal as a
mosaic assembly of crystal blocks. Right: the resulting broad diffraction profile peak
86
4 X-ray Optics and Synchrotron Beamlines