Schematic of the x-ray beam, sample, and detector. We use a single detector and move it sequentially to the positions shown. Diffraction from one particular grain is shown together with notation ( and ) for specifying the direction of the diffracted wave vector, , with . Note that a circular grain in the illuminated plane generates an elliptical spot on the detector due to the projection along . The coordinate system origin is at the intersection of the rotation axis and the beam plane and the beam is incident in the direction.
Schematic diagram of the simulation geometry. The sample is represented by a hexagonal grid with each element assigned its own crystallographic phase and orientation, . For elements that generate scattering at this (dark triangles), the grid element vertices are projected along a unit vector, , parallel to the scattered wave vector, , onto the detector (black dots in magnified view at right). Any detector pixel inside or intersected by the projected triangle is considered to be illuminated by this Bragg peak (filled squares).
Demonstration of imaging of a single triangular area element ( side length) and the effect of incident beam models (see text). Each image contains scattering generated over a interval as observed at three sample-to-detector distances (6, 8, 10 mm in red, green, and blue, respectively); the detector covers a area. The nominal energy is 50 keV. The projection of the incident beam is shown near the bottom (centered on pixel row 1000). Each row of images shows intensity at the same three successive positions. (a)–(c) show model (i), (d)–(f) model (ii), and (g)–(i) model (iii).
A typical detector image obtained from an aluminum polycrystal. The gray scale is in CCD counts per pixel. The uniform background is slightly under 600 counts. Diffraction spots are horizontally extended (10–50 pixels) darker regions. The direct beam projects onto the image at , below the long horizontal streak that is stray scattering out of the direct beam. Note that the most intense points (those that saturate the gray scale) are isolated “hot” pixels rather than diffraction spots. The maximum intensity in the image is 3997. The following figures show the effects of image processing steps on the boxed region.
Expanded view of the subregion indicated in Fig. 4 in three stages of image analysis. (a) Raw image (gray scale 0–1600; this scale saturates some pixels since the maximum in the figure is 2700), (b) background (600 counts) subtracted image (gray scale 0–1000), and (c) median filtered and background subtracted image (gray scale 0–1000).
Binary images of data from Fig. 5. Each region of interest has been thresholded at a fraction of the maximum intensity in the region. Thresholds are (a) and (b) 0.25 of the peak height.
An illustration of the counting scheme used in determining overlap between simulated scattering and the experiment. The square grid shows pixels in a small region of the detector. The black dots represent the projection of vertices of a triangular sample area element. Experimentally illuminated pixels not hit by the simulation are black; pixels struck in the simulation but not the experiment are yellow; overlapping intensities are red. In this example, (yellow plus red), (red), so . The contribution to the Monte Carlo cost function [Eq. (3)] from this region is , which is the number of “mistakes” (black and yellow pixels). Ideally, a finer sample space grid will remove simulated intensity from the yellow pixels and neighboring area elements will cover the remaining black pixels.
Skeletal overview of fitting procedures. If exceeds the number of members of , then element is tagged as generating no scattering and the next element, , is processed.
Horizontal beam profile and horizontal and vertical diffraction spot profiles from a silicon wafer. (a) Circles show the horizontal incident beam profile after summing across the thin vertical width. The solid line is a guide to the eye. The full width at half-maximum is . The downward, square, and upward pointing symbols are horizontal profiles of a (311) Bragg spot measured at , and , respectively. All profiles have been shifted to the same center and the incident beam has been normalized to the Bragg spots which have been scaled by . Since the single-crystal sample is wider than the beam, the horizontal extent of these spots is essentially that of the beam. (b) The vertical profile across a (311) Bragg spot [symbols as in (a)]. The light line indicates the detector response to the direct incident beam; the heavy line is a Gaussian guide to the eye showing that the vertical width of the peak is resolved. (c) Vertical profile of a higher-order, (553) or (731) Bragg peak showing the increased width due to larger deflection in the vertical direction.
Sample-space maps of the illuminated “slot” through the silicon wafer obtained under data reduction and convergence criteria of Table I. The maps are aligned so that the beam is incident from the left and the wafer is in the plane at . The hexagons, with sides, indicate the simulated region of sample space. The sign indicates the simulated rotation axis. The dashed lines show the region of sample space illuminated by the experimental incident beam as the sample is rotated . The solid diamond outlines the region that remains in the beam throughout. White space inside the hexagons has been determined to not satisfy the fitting criteria for any crystallographic orientation and have been assigned .
Diffracted beam images at nominal orientation with fitted intensity corresponding to Fig. 10(a). In (a), three images, at , have been superimposed. Experimental data are shown as black points. Colors (or gray scale) show simulated intensity: red, green and blue when overlapping experimental data, aqua when not. The line at the bottom indicates the projected position of the incident beam at ; the associated dot indicates the detector origin at . Bragg spots are seen to radiate from the incident beam position at different scattering angles, , and orientations, . (b) and (c) show expanded views of individual Bragg spots; tic marks correspond to 20 pixel intervals; spot positions have been displaced for ease of comparison. The reduced height in (c) relative to (b) is due to the smaller vertical displacement causing a more anisotropic projection in (c). The at-most subtle broadening of experimental spots with is consistent with an energy bandwidth .
Diffracted beam images at nominal orientation with fitted intensity corresponding to Fig. 10(b). All conventions are as in Fig. 11. Here, due to use of more stringent convergence criteria, the simulated intensity lies almost entirely within the experimental Bragg spots.
Parameters used and results of three fits to silicon single-crystal data. Corresponding maps are shown in Fig. 10.
Experimental parameters obtained in the fit shown in Fig. 12.
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