Rodrigues–Frank (RF) space or Rodrigues space

Slides:



Advertisements
Similar presentations
INTRODUCTION TO CERAMIC MINERALS
Advertisements

Surface normals and principal component analysis (PCA)
Texture Components and Euler Angles: part 1 June 2007
L2: Texture Components and Euler Angles: part 1 14th January 08
ENTC 1110 Multiview Drawings.
Lecture 2: Crystal Symmetry
PH0101 UNIT 4 LECTURE 2 MILLER INDICES
When dealing with crsytalline materials, it is often necessary to specify a particular point within a unit cell, a particular direction or a particular.
Crystal Structure Continued! NOTE!! Again, much discussion & many figures in what follows was constructed from lectures posted on the web by Prof. Beşire.
Wigner-Seitz Cell The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to.
Lec. (4,5) Miller Indices Z X Y (100).
© Oxford Instruments Analytical Limited 2001 MODULE 2 - Introduction to Basic Crystallography Bravais Lattices Crystal system Miller Indices Crystallographic.
Volume Fractions of Texture Components
17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown on the five nets, 17-plane groups are derived.
Misorientations and Grain Boundaries
VECTORS AND THE GEOMETRY OF SPACE 12. VECTORS AND THE GEOMETRY OF SPACE A line in the xy-plane is determined when a point on the line and the direction.
Geometry Vocabulary 2-dimensional (2D) - a shape that has no thickness; a flat shape 3-dimensional (3D) - an object that has thickness (height, width and.
Introduction to Crystallography
CS-321 Dr. Mark L. Hornick 1 3-D Object Modeling.
Polygons Two-dimensional shapes that have three or more sides made from straight lines. Examples: triangles squares rectangles.
Equation A statement that two mathematical expressions are equal.
Unit 9 Understanding 3D Figures
Chapter 23 Mirrors and Lenses.
Part 6: Graphics Output Primitives (4) 1.  Another useful construct,besides points, straight line segments, and curves for describing components of a.
Chem Lattices By definition, crystals are periodic in three dimensions and the X-ray diffraction experiment must be understood in the context of.
Introduction to Crystallography and Mineral Crystal Systems PD Dr. Andrea Koschinsky Geosciences and Astrophysics.
Crystal Structure A “unit cell” is a subdivision of the lattice that has all the geometric characteristics of the total crystal. The simplest choice of.
PROJECTIONS OF SOLIDS & SECTIONS OF SOLIDS
Introduction. Like a house consisting of rooms separated by walls, a crystalline material consists of grains separated by grain boundaries. Within a grain,
Computer vision: models, learning and inference M Ahad Multiple Cameras
Key things to know to describe a crystal
Vocabulary for the Common Core Sixth Grade.  base: The side of a polygon that is perpendicular to the altitude or height. Base of this triangle Height.
GEOMETRY!!!. Points  A point is an end of a line segment.  It is an exact location in space.   It is represented by a small dot. Point A A.
Fundamentals of crystal Structure
Auxiliary Views Chapter 7.
Three Dimensional Viewing
Rigid Bodies: Equivalent Systems of Forces
Spatcial Description & Transformation
Properties of engineering materials
Unit 11: 3-Dimensional Geometry
SOLID STATE By: Dr.DEPINDER KAUR.
Crystallographic Points, Directions, and Planes.
SOLID STATE By: Dr.Bhawna.
Groups: Fill in this Table for Cubic Structures
Indira Gandhi Centre for Atomic Research, India
Introduction A chef takes a knife and slices a carrot in half. What shape results? Depending on the direction of the cut, the resulting shape may resemble.
Concepts of Crystal Geometry
Engineering Geometry Engineering geometry is the basic geometric elements and forms used in engineering design. Engineering and technical graphics are.
Section 13.2 The Ellipse.
BDA30303 Solid Mechanics II.
Unit 11: 3-Dimensional Geometry
SECTIONS OF SOLIDS Chapter 15
Three Dimensional Viewing
Introduction Think about the dissection arguments used to develop the area of a circle formula. These same arguments can be used to develop the volume.
Warm Up Classify each polygon. 1. a polygon with three congruent sides
Crystal Structure Continued!
Crystallographic Points, Directions, and Planes.
Depth Affects Where We Look
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
SOLIDS Group A Group B Cylinder Cone Prisms Pyramids
Three-Dimensional Object Representation
Orthographic Projection
MILLER PLANES Atoms form periodically arranged planes Any set of planes is characterized by: (1) their orientation in the crystal (hkl) – Miller indices.
Orthographic Projection
G13 Reflection and symmetry
MODULE 2 - Introduction to Basic Crystallography
CHAPTER 10 Geometry.
Volume 4, Issue 2, Pages (February 1996)
Presentation transcript:

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