Mosaic crystals

Fig. 4-9. This diagram shows the intensity variation with angle for a rock salt crystal in the region near the Bragg angle, 0q, for an incident monochromatic beam. The area under the mosaic crystal curve could be thirty times greater than the ideal. (After Renninger, Z. Krist. 89, 344.)...

A considerable amount of research has been conducted on the decomposition and deflagration of ammonium perchlorate with and without additives. The normal thermal decomposition of pure ammonium perchlorate involves, simultaneously, an endothermic dissociative sublimation of the mosaic crystals to gaseous perchloric acid and ammonia and an exothermic solid-phase decomposition of the intermosaic material. Although not much is presently known about the nature of the solid-phase reactions, investigations at subatmospheric and atmospheric pressures have provided some information on possible mechanisms. When ammonium perchlorate is heated, there are three competing reactions which can be defined (1) the low-temperature reaction, (2) the high-temperature reaction, and (3) sublimation (B9). 

The integrated intensity of reflection of an X-ray line from an extended face of a mosaic crystal is4)... 

The reciprocal lattice of a mosaic crystal is a three-dimensional periodic system of points, each of which characterized by a vector Hhu = ha -l- kb -l-Ic, where a, b, c are axial vectors and h,k,l, are point indices. 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

We can consider the total energy E(hkl) in a diffracted beam with reference to Darwins treatment for an ideally mosaic crystal rotating with constant angular velocity throu the reflecting position. 

Mosaic blocks (mosaic spread) Tiny blocks within a crystal structure that are slightly misoriented with respect to each other. As a result of such mosaic spread, Bragg reflections have a finite width. Extinction is weaker in a mosaic crystal than in a perfect crystal, and therefore the intensities can be predicted by the rules of kinematical diffraction. 

In order to test the dispersion theory per se the intensities of numerous reflections were measured by W. H. Bragg, Ehrenberg, Ewald, and others, for as ideal diamonds as could be got and for zinc blende. It turned out that the relationships actually are better reproduced if the intensity is taken as proportional to the structure factor S than if it is taken as proportional to the square of the structure factor. A proposal by W. L. Bragg that in mosaic crystals the reflected intensity p should be taken as proportional to... 

Diffraction by an ideal mosaic crystal is best described by a kinematical theory of diffraction, whereas diffraction by an ideal crystal is dynamical and can be described by a much more complex theory of dynamical diffraction. The latter is used in electron diffraction, where kinematical theory does not apply. X-ray diffraction by an ideal mosaic crystal is kinematical, and therefore, this relatively simple theory is used in this book. The word "mosaie" describes a crystal that consists of many small, ideally ordered blocks, which are slightly misaligned with respect to one another. "Ideal mosaic" means that all blocks have the same size and degree of misalignment with respect to other mosaic blocks. Most of this chapter is dedicated to conventional crystallographic symmetry, where three-dimensional periodicity is implicitly assumed. 

The kinematical approach is simple, and adequately and accurately describes the diffraction of x-rays from mosaic crystals. This is especially true for polycrystalline materials where the size of crystallites is relatively small. Hence, the kinematical theory of diffraction is used in this chapter and throughout this book. 

Extinction effects, which are dynamical in nature, may be noticeable in diffraction from nearly perfect and/or large mosaic crystals. Two types of extinction are generally recognized primary, which occurs within the same crystallite, and secondary, which originates from multiple crystallites. Primary extinction is caused by back-reflection of the scattered wave into the crystal and it decreases the measured scattered intensity Figure 2.51, left). Furthermore, the re-reflected wave is usually out of phase with the incident wave and thus, the intensity of the latter is lowered due to destructive interference. Therefore, primary extinction lowers the observed intensity of very strong reflections from perfect crystals. Especially in powder diffraction, primary extinction effects are often smaller than experimental errors however, when necessary they may be included in Eq. 2.65 as ... 

Figure 2.51. The illustration of primary (left) and secondary (right) extinction effects, which reduce intensity of strong reflections from perfect crystals and ideally mosaic crystals, respectively. The solid lines indicate actual reflections paths. The dashed lines indicate the expected paths, which are partially suppressed by dynamical effects. The shaded rectangles on the right indicate two different blocks of mosaic with identical orientations.

Secondary extinction Figure 2.51, right) occurs in a mosaic crystal when the beam, reflected from a crystallite, is re-reflected by a different block of the mosaic, which happens to be in the diffracting position with respect to the scattered beam. This dynamical effect is observed in relatively large, nearly perfect mosaic crystals it reduces measured intensities of strong Bragg reflections, similar to the primary extinction. It is not detected in diffraction from polycrystalline materials and therefore, is always neglected. 

The notion of the mosaic crystal dates from the early years of x-ray diffraction and depends on much indirect evidence, both theoretical and experimental. In the 1960s the electron microscope provided direct evidence. It showed that real crystals, whether single crystals or individual grains in a polycrystalline aggregate. 

Table 5.1. The intrinsic resolution of several Bragg reflections from symmetrically cut perfect and mosaic crystals.

In chapter 2 the relationship was given between the electron density Q(x,y,z), which describes the contents of the unit cell of the crystal, and the set of structure factors F(h). The amplitude component, F(h), is related to the intensity in the diffracted beam by a formula derived by Darwin (1914). The total energy in a diffracted beam, from a particular reflecting plane (hk ) for an ideally mosaic crystal rotating with constant angular velocity [Pg.245]

HARD X-RAY IMAGING VIA FOCUSING OPTICS WITH MOSAIC CRYSTALS... 

Key words Mosaic crystals - X-Ray imaging - Concentrators - X-ray Astronomy... 

Reflection of hard X-rays with mosaic crystals... 

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. 

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