LabAdviser/314/Microscopy 314-307/SEM/Nova/Transmission Kikuchi diffraction: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 12: / Line 12: @@
 = Indexing of Kikuchi patterns =
-The automated determination of the position of Kikuchi lines in a diffraction pattern (and consequently of lattice planes in the crystal) has enabled the practical use of Kikuchi diffraction for the study of micro- and nanomaterials. An algorithm known as the Hough transform was introduced by Krieger Lassen to recognize the edges of the Kikuchi bands [1,2]. The Hough transform is a mathematical tool which can be used to isolate features of a particular shape, such as lines, circles and ellipses, within an image. The main advantage of the Hough transform is that it is relatively unaffected by image noise. The essential idea of the the Hough transform consists of applying to each pixel of the Kikuchi pattern the equation: ''ρ''i = xcos�i + ysin�i (3.1) where (x, y) are the coordinates of a pixel in the original image and (�i, �i) are the parameters of a straight line passing through (x, y). From Eq. 3.1 it is evident that the set of possible lines passing through the chosen pixel is represented by a sinusoid in the Hough space.
+The automated determination of the position of Kikuchi lines in a diffraction pattern (and consequently of lattice planes in the crystal) has enabled the practical use of Kikuchi diffraction for the study of micro- and nanomaterials. An algorithm known as the Hough transform was introduced by Krieger Lassen to recognize the edges of the Kikuchi bands [1,2]. The Hough transform is a mathematical tool which can be used to isolate features of a particular shape, such as lines, circles and ellipses, within an image. The main advantage of the Hough transform is that it is relatively unaffected by image noise. The essential idea of the the Hough transform consists of applying to each pixel of the Kikuchi pattern the equation:
-An example is shown in Fig. 3.4. Consider four pixels along a Kikuchi line. For each pixel in the line, all possible values of � are calculated for � ranging from 0° to 180° using Eq. 3.1, which produces four sinusoidal curves. These curves intersect at a point with (�, �) coordinates (green rectangle), corresponding to the angle of the line (�) and its position relative to the origin (�). The grey-scale intensity I(xi; yi) of the image pixels is accumulated in each quantized point H(�; �) of the Hough space, so that I(xi; yi) serves as a measure of evidence for a line passing through the pixel (x, y). In this way, a line in the pattern space transforms to a point having a certain intensity
+''ρ''i = xcos''θ''i + ysin''θ''i (1)
-in the Hough space, which is easily detected by the indexing software over the at background. For a correct indexing, the crystal system, chemical composition, unit cell dimensions and atomic positions of the material must be supplied to the analysis software.
+where (x, y) are the coordinates of a pixel in the original image and (''ρ''i, ''θ''i) are the parameters of a straight line passing through (x, y). From Eq. 1 it is evident that the set of possible lines passing through the chosen pixel is represented by a sinusoid in the Hough space.
+An example is shown in Fig. 3. Consider four pixels along a Kikuchi line. For each pixel in the line, all possible values of ''ρ'' are calculated for ''θ'' ranging from 0° to 180° using Eq. 1, which produces four sinusoidal curves. These curves intersect at a point with (''ρ'', ''θ'') coordinates (green rectangle), corresponding to the angle of the line (''θ'') and its position relative to the origin (''ρ''). The grey-scale intensity I(xi; yi) of the image pixels is accumulated in each quantized point H(''ρ''; ''θ'') of the Hough space, so that I(xi; yi) serves as a measure of evidence for a line passing through the pixel (x, y). In this way, a line in the pattern space transforms to a point having a certain intensity in the Hough space, which is easily detected by the indexing software over the at background. For a correct indexing, the crystal system, chemical composition, unit cell dimensions and atomic positions of the material must be supplied to the analysis software.
+In a Kikuchi pattern, a set of orientations is obtained from a triplet of bands by measuring the interplanar angles between the bands. These values are compared against theoretical values of all angles formed by various planes for a given crystal system. When the (h, k, l ) values of a pair of lines are identified, they give information about the pair of planes, but that does not permit to distinguish between the two planes. At least three sets of lines are required to completely identify the individual planes and hence to find the orientation of the sample. For a given number of bands, n, used for pattern indexing, the number of band triplets is determined by the following formula:
+N.triplets = n!/(n - 3)! * 3! (2)
-In a Kikuchi pattern, a set of orientations is obtained from a triplet of bands by measuring the interplanar angles between the bands. These values are compared against theoretical values of all angles formed by various planes for a given crystal system. When the (h, k, l ) values of a pair of lines are identi�ed, they give information about the pair of planes, but that does not permit to distinguish between the two planes. At least three sets of lines are required to completely identify the individual planes and hence to �nd the orientation of the sample. For a given number of bands, n, used for pattern indexing, the number of band triplets is determined by the following formula:
-N:triplets =
-n!
-(n 􀀀 3)! � 3!
-(3.2)
 Typically from 3 to 9 detected bands are used for automatic indexing in commercial softwares.