Polycrystalline powders, Bragg

For neutron diffraction experiments on polycrystalline powders it is usually more convenient to treat the differential cross-section as a function of d-spacing, d = 2lllQ), defined according to Bragg s law by... 

The appearance of eight diffi action cones when polycrystalline copper powder is irradiated by the monochromatic Cu Kaj radiation is shown schematically in Figure 2.32. All Bragg peaks, possible in the range 0 <29< 180°, are also listed with the corresponding Miller indices and relative intensities in Table 2.6. 

In powder diffraction, the scattered intensity is customarily represented as a function of a single independent variable - Bragg angle - 20, as modeled in Figure 2.33 for a polycrystalline copper. This type of the plot is standard and it is called the powder diffraction pattern or the histogram. In some instances, the diffracted intensity may be plotted versus the interplanar distance, d, the g-value Q = /d = d ) or sin0/A,. 

Figure 2.32. The schematic of the powder diffraction cones produced by a polycrystalline copper sample using Cu Ktti radiation. The differences in the relative intensities of various Bragg peaks (diffraction cones) are not discriminated, and they may be found in Table 2.6. Each cone is marked with the corresponding triplet of Miller indices.

There are many ways to build a model of the crystal structure of a polycrystalline material without first using the intensities of individual Bragg reflections, which are hidden in powder diffraction due to partial or complete overlapping. Most of the direct space approaches are, in effect, trial-and-error methods and they include some or all of the following components ... 

Preferred orientation is a problem encountered when using powder diffraction methods. It occurs when most of the crystallites in a polycrystalline sample have a nonrandom distribution of orientations, resulting in only a certain number of the Bragg reflections moving into the reflecting position and thus contributing disproportionately to the scattered intensity. Ideally, the crystallites should be randomly... 

The most precise structure determinations are performed using a single crystal sample where hundred to thousands of separate Bragg spots can be used to determine atomic positions to a precision of better than 0.001 A. In materials science, however, quite frequently the material is in a polycrystalline form and powder diffraction methods have to be employed. 

Figure 8 shows the relatively simple powder diffraction pattern of polycrystalline silicon, measured by time-of-flight neutron diffraction. Each allowed Bragg peak for the crystal structure is observed as a sharp peak in the diffraction pattern. A fundamental difference between single crystal and powder diffraction is that the single crystal diffraction pattern has the full directional information described by Equation [28], whereas for the powder diffraction pattern this information has been collapsed down into a onedimensional function. This difference has two important consequences firstly, the loss of directional information makes it very much more difficult to... 

Figure 1 Quantitative phase analysis of a polycrystalline mixture containing three phases performed by the Rietveld method. Selected portions of the observed (crosses) and calculated (solid line) X-ray powder diffraction patterns are shown. The vertical marks indicate the position of the single Bragg peaks.

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