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    The evaluation of grain orientation data
    Orientation Microscopy

    The prime objective of orientation microscopy is the acquisition of grain orientation data. They are usually represented by three Euler angles or by the rotation matrix which rotate the coordinate system of the specimen in the coordinate system fixed to the particular crystal lattice.

    A clear graphical representation of the microstructure is obtained by constructing Crystal Orientation Maps (COM), or in short Orientation Maps (OM) [1]. Hereby the orientation parameters in the measured grid points are assigned in an image to unique shades of three basic colors that are red, green and blue. Orientation parameters in crystal orientation maps may be two crystallographic directions (hkl)[uvw] for two sample reference directions ("Miller maps"), the Euler angles (φ1, Φ, φ2) ("Euler maps"), or the Rodrigues vector R ("Rodrigues maps"). Finally an orientation map of the sample is obtained which illustrates the morphology of microstructure and the spatial orientation distribution of the grains. If abutting points in a region have the same or similar orientations as their neighbors, they are identified as a grain, and the intersections of such regions are called grain respectively phase boundaries. Grains with an orientation closely in common are represented by similar colors. It is so possible to quantitatively represent the orientations and misorientations in materials on the sub-micron scale.

    The misorientations between neighboring raster points are calculated and, by assuming specified threshold values, grain boundaries (GB) as well as phase boundaries can be marked out in the COM. However, a grain boundary in a planar sample section forms a closed perimeter line which must not „leak“. A special „path finding“ algorithm along the grain boundary segments is applied to fill missing spots where indexing may have locally failed, and a (binary) grain boundary network of the microstructure is obtained. The lines may be further skeletonized to one pixel in width. Grain boundaries are commonly marked by the Rodrigues vector, R, the axis-angle parameters of the misorientation, Δg, or the Σ character according to the CSL model. Other local properties or parameters can be represented graphically by color maps of the microstructure in a similar way, such as dislocation density, the predominant glide systems or twin systems in the grains, the Schmid factor or the residual deformation energy in the individual grains [2].

    The advantages of GB COM over conventional light microscopical images of grain boundary networks are for stereological evaluation:
    + The GB COM is based quantitatively on measured crystallographic orientations rather than on tricky methods of surface (GB) etching and imaging.
    + A high contrast on a flat background is obtained over the whole GB COM due to a yes/no discrimination.
    + The data are available in a digital (binary) format on a regular raster grid.
    Therefore, standard methods of quantitative metallography can directly be applied to skeletonized GB COM. In many labs sophisticated stereological programs are already available. 2D microstructure parameters can so be determined such as the area fraction, planar size, average grain size, shape and arrangement of grains, and their statistical distributions. If the phases in the material have sufficiently different lattice constants or differ in their element composition such that the phases can be differentiated – with the aid of simultaneous EDS analysis – in every pixel, quantitative metallography can be extended to a phase discriminating stereology. Under usual stereological assumptions, 2D stereology is often extended to 3D stereology to calculate such parameters as volume fraction, average grain volume, 3D shape and arrangement, contiguity of phases, and their statistical distributions. Approaches are made to reveal the true 3D microstructure on a grain specific scale by combining BKD with in-situ serial sectioning in a FIB&SEM instrument.

    Since a grain in the GB COM is represented by the group of pixels with similar grain orientations within the GB loop, its area fraction is simply measured by counting the raster points which are enclosed by the grain boundary perimeter line. This point-counting method is superior over the line-intercept method because it is not affected by concave sections of the grain boundary line nor by „islands“ formed by a second grain underneath which may shine through the sample surface.   A rapid but less precise alternative for estimating size and shape of grains is the visual comparison of a GB COM with standard grain charts or reticules.

    Color code of Miller maps (cubic symmetry) For orientation maps using Miller indices as orientation parameters, a color triangle is overlaid on the stereographic standard triangle of the crystal lattice under consideration. In case of cubic crystal symmetry red is commonly assigned to directions near the (001) corner, green to directions near the (011), and blue to the directions near the (111) corner. Using this legend, colored Miller maps provide an illuminating display of the spatial and angular distribution of crystallographic directions related to reference directions in the sample, e.g. the sheet normal direction ND and the rolling direction RD in rolled sheets. Two Miller maps are required, one for the {hkl} in-surface planes (i.e., in case of cubic symmetry the <hkl> directions perpendicular on the specimen surface), and one for the <uvw> in a reference direction in the specimen surface, to fully represent the grain orientations in the surface. When both Miller maps are considered, the spatial and angular distribution of crystallographic orientations is clearly visualized.

    The unique colors, on the other hand, can be interpreted in terms of orientation parameters by comparison with a color legend. The full grain orientation may be reconstructed from a set of two colored Miller maps in the following way: From the symmetrically equivalent orientations only those have been used for imaging whose Miller indices (hkl) of the in-surface planes fall in the standard triangle 001-011-111. One of the permutations of the Miller indices, [uvw], of an orthogonal direction is given in the standard triangle 001-011-111. For a full description of crystal orientation the sign and sequence of the Miller indices [uvw] is determined considering the condition of orthogonality between (hkl) and [uvw]. If this condition is fulfilled for more than one of the permutations of <uvw>, these solutions are symmetrically equivalent. They cannot be discriminated physically from each other, because the choice of the position of the axes of the elementary lattice cell is arbitrary.

      It is worth noting that conventional light or scanning electron microscopy images may display abutting grains by the same color or gray shade. Therefore, Quantitative Metallography may fail when deriving grain size distributions or other statistical parameters from microscopy images. In crystal orientation maps, however, grains (and phases) are discriminated unambiguously by indicating their crystal orientation and lattice structure. Since crystal orientation maps are available in digital form by their way of construction, the derivation of statistical parameters (such as the distributions of grain size, length of grain boundaries, grain size as a function of grain orientation, and the fractions of Σ grain boundaries) is simply reduced to pixel counting.  

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    [1] Download  D. Gerth and R.A. Schwarzer: Graphical representation of grain and hillock orientations in annealed Al-1%Si films. Textures and Microstructures 21 (1993) 177-193
    [2] Download  R.A. Schwarzer: Review Paper: Automated crystal lattice orientation mapping using a Computer-controlled SEM. Micron 28 (1997) 249-265

     


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