Bragg case geometry

If the quantitative texture analysis is not of interest the sample is not rotated on a goniometer and only one or a small number of patterns are recorded. Because the number of points in the space (T, y) is not sufficient, one expects that the refined harmonic coefficients give only a rough description of the texture, even if the texture correction is very good. An extreme case is the Bragg-Brentano geometry. In this case in Equations (41-43) we must take (0 = 9, x =

[Pg.348]

Absorption of X-rays by the sample is also an important factor during pattern decomposition or refinement. In the case of low-absorption material such as clathrate hydrates, the sample thickness should be sufficient taking into account the wavelength of the X-ray source because the sample should be completely opaque to X-rays for the Bragg-Bientano geometry. 

This very useful method also has the advantage that the equations do not contain anything about the material or diffraction conditions other than the Bragg angle and geometry. The independence from material parameters arises because the refractive index for X-rays is very close to unity. The equations are, of course, similar to those for optical interference from thin films, since the physics is the same, but in the optical case we do need to know the refractive index. 

Eq. 2.72 can be solved analytically for all geometries usually employed in powder diffraction. For the most commonly used Bragg-Brentano focusing geometry the two limiting cases are as follows ... 

Figure 3.9. Transmission geometry in the case of flat (left) and cylindrical (right) samples. F - focus of the x-ray source, DS - divergence slit, RS - receiving slit, 0 - Bragg angle.

Figure 3.10. Synchronization of the goniometer arms the x-ray source is stationary while the sample and the detector rotations are synchronized to fulfill the 0-20 requirement (left) the sample is stationary while the source and the deteetor arms are synchronized to realize the 0-0 condition (middle) - this geometry is in common use at present only the detector arm revolves around the goniometer axis in the case of a cylindrical sample (right). F - focus of the x-ray tube indicating the position of the x-ray source arm, D - detector arm, 0 - Bragg angle. The common goniometer axis (which is perpendicular to the plane of the projection) around which the rotations are synchronized is shown as the open circle in each of the three drawings. The location of the optical axis is shown as the dash-dotted line.

This example illustrates the derivation of a crystal structure based on a suspected analogy with related compounds followed by geometry optimization to enhance and improve the deduced structural model. Such a complex approach in this case has been adopted because of poor crystallinity of the material, which results in a low resolution of its powder diffraction pattern (see Figure 6.33), where the full widths at half maximum range from 0.25 to 0.55°. Furthermore, the pattern is relatively complex, with as many as 255 Bragg reflections possible for 20 < 37.5° when Mo Ka radiation is employed. 

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. 

In a previous paper (De Chiara and Prontera, 1992) we derived the expression of the reflectivity of a mosaic crystal in Bragg geometry in the general case of linearly polarized X-rays and we discussed the optimization criteria of the hard X-ray reflectivity of mosaic crystals. 

For being able to make a specific evaluation, a plane-parallel geometry of the crystal is chosen, looking for the solutions of the propagation for the two consecrated cases, respectively the Laue and Bragg diffractions, schematically represented in Figure 5.25. 

---