Rodrigues–Frank (RF) space or Rodrigues space Population of R vectors, each of which represents an orientation or misorientation, can be represented in a 3-D space known as Rodrigues–Frank (RF) space or Rodrigues space Since the R vector is derived directly from the angle/axis pair, it can be described by a number of different crystallographically related solutions, with the precise number of solutions depending on the crystal system and symmetry The R vector is a function of θ, and so it follows that the smallest value of the vector is calculated from the smallest θ Such R vectors will lie closest to the origin of RF space, and therefore the crystallographically related solution of an orientation or misorientation having the smallest angle is the most convenient one to select for representation in Rodrigues space.
Fundamental Zone, FZ The entirety of Rodrigues space encompasses each one of the crystallographically related solutions of a rotation Since it is simpler for each orientation from a sample population to be represented only once, it is convenient to formulate the R vector from only one crystallographically related solution. That solution with the lowest rotation angle is chosen because it feeds through to give the smallest R vector. By use of the lowest-angle description, all orientations—represented by the endpoints of their R vectors—can be made to reside in a polyhedron known as the fundamental zone of Rodrigues space, which contains the origin of the space Any orientations that lie outside the fundamental zone can be re-expressed by equivalent points lying within it by re-choosing the crystallographically related solution to be the lowest angle one.
Fundamental Zone, FZ An orientation may lie on the surface of the fundamental zone and will be matched by an equivalent point on the opposite face The shape of the fundamental zone is governed by the crystal symmetry of the material, and the geometry of these polygons has been derived for each crystal system For cubic symmetry, the shape of the zone is a cube with truncated corners such that the bounding surfaces are six octahedra and eight equilateral triangles For the representation of orientations, the XYZ axes are aligned with the specimen axes. Alternatively XYZ can be aligned with the crystal axes, which is convenient for the representation of misorientations.
Fundamental Zone, FZ The fundamental zone of Rodrigues space for cubic symmetry, showing also the decomposition of the space into 48 subvolumes
Fundamental Zone, FZ The distance of the point where the reference axes intercept the zone surface is represented, in terms of crystal coordinates, by the angle/axis pair 45°/ 〈100〉 that gives R = (0.4142,0,0). The R vectors of the centers of the equilateral triangles that form the truncated corners of the zone are given by the angle/axis pair 60°/〈111〉 and a corresponding R vector (0.333,0.333,0.333). Another point that defines the shape of the fundamental zone for cubic crystals is an apex of the equilateral triangle, given by 62.8°/〈11( 2 –1)〉, which represents the greatest possible misorientation between two cubes, and yields an R vector of (0.4124,0.4124,0.1716).
Table of CSL values in axis/angle, Euler angles, Rodrigues vectors and quaternions Note: in order to compare a measured misorientation with one of these values, it is necessary to compute the values to high precision (because most are fractions based on integers).
Properties of Rodrigues Space Every straight line in Rodrigues space represents rotations about a certain, fixed axis
<111> r1 + r2 + r3 £ 1 <110> <111> <100>, r1 RF-space r1 + r2 + r3 £ 1 <110> <111> <100>, r1 The largest possible misorientation angle corresponds to the point marked by o. <110> <100>, r1
Sections through RF-space For graphical representation, the R-F space is typically sectioned parallel to the 100-110 plane. Each triangular section has R3=constant. Most of the special CSL relationships lie on the 100, 110, 111 lines. base of pyramid
Quaternions A unit vector is defined as a real vector in 4 Is misorientation angle and n is axis (unit vector)
Symmetrical tilt grain boundaries When the boundary plane represents the plane of the mirror symmetry of the crystal lattices of two grains, it is described by the same Miller indices from the point of view of both adjoining grains. This boundary is called symmetrical tilt boundaries.
Symmetrical tilt grain boundaries.....GB Plane
Asymmetrical tilt grain boundaries When the boundary plane does not represents the plane of the mirror symmetry of the crystal lattices of two grains, the boundary is called asymmetrical tilt boundaries. Symmetry plane Misorientation axis
Grain Boundary Plane The central tenet of the interface–plane scheme as applied to grain boundaries in polycrystals is to focus primarily on the crystallography of the plane, with the misorientation being a secondary consideration.
Five Parameter Analysis (a) Schematic shape of a single grain from a dense polycrystal. (b) Representation of a three grain junction within a polycrystal. The view is exploded so that the internal interfaces can be seen. The external surfaces are shaded and the internal surfaces are triangulated. The jth triangular facet on the ith grain is shaded and an enlarged view of this facet is shown in (c).
Five Parameter Analysis The parameters required to specify the crystallographic orientation of a single boundary plane (thin lines). The sample reference frame (thick lines) is xyz and the grain boundary trace vector on the specimen surface is l'ij, the ‘trace angle’ and the ‘inclination angle’ are α and β respectively, and the section depth is t.
Five Parameter Analysis Parameters required to measure the orientation of a surface. XYZ are the specimen axes, α and β are the angles relating the surface trace to the X axis, and T is the experimental section depth. For an exposed rather than an unexposed facet, the right-hand part of the diagram is missing. (b) Stereographic projection showing the relationship among XYZ, α, and β from (a), the crystal axes of one grain, and the boundary plane normal N.
Three-Dimensional EBSD
Five Parameter Analysis A superposition of the grain boundary traces from adjacent layers in the microstructure of MgO. The vertical separation is about 5 μm. (b) Illustration of the formation of triangles between traces on adjacent layers, with known lattice misorientation and orientation.
Five-Parameter Analysis using Stereological Techniques Illustration of the boundary trace reconstruction routine. (a) First reconstruction attempt by joining adjacent triple junctions. (b) Segmentation of the reconstructed trace. (c) Small map wherein reconstructed boundaries are superimposed on true boundaries.
Five-Parameter Analysis using Stereological Techniques Schematic illustration of the principle of the five-parameter stereology method. T is the boundary trace direction, and N is the boundary plane normal. If a sufficiently large number of observations are made, the true boundary plane will accumulate more than the false planes and form a peak in the distribution at N.
Stereographic projection of grain boundary plane normals from an annealed copper specimen. (a) All grain boundaries included. (b) All grain boundaries other than 3. Randle, V., Rohrer, G.S., Miller, H.M, Coleman, M., and Owen, G.T., 2008b, Five-parameter grain boundary distribution of commercially grain boundary engineered nickel and copper, Acta Mater., 56:2363.
Two-Surface Sectioning (a) Schematic illustration of a “sharp edge” specimen showing the approximate location (by red dotted line) where EBSD measurements were carried out. (b) Two IPF maps (with the sample edge direction as common reference direction) for the two mutual perpendicular surfaces near the sharp edge (boundary color code: 3, red; 9, blue; other HAGBs, black). (c–e) Stereographic projections for three representative GBs displaying the two abutting crystals, the boundary traces (black lines) and potential boundary normals (dashed blue lines); filled red circles and open blue circles in (c–e) represent all the potential symmetrically equivalent planes from the 1st and 2nd crystals, respectively, whereas the open black circle indicates the actual boundary plane. Scripta Materialia 81 (2014) 16–19