Overview: graduate material

1
Overview graduate material

• 27-750, Advanced Characterization and
  Microstructural Analysis,
• Spring 2003, A. D. Rollett, H. Garmestani

2
Objective

• The objective of this lecture is to lay out the
  topics to be covered at the graduate level in
  Advanced Characterization and Microstructural
  Analysis, 27-750.
• This complements the general overview lecture.

3
Orientations, Rotations, Math

• One important aspect is the use of many of the
  different methods of representing orientation,
  all of which relate back to the mathematics of
  rotations and symmetry.
• Another key aspect is working with distributions,
  both in discrete form, and as continuous
  functional forms.
• Certain aspects of atomic structure of boundaries
  are important, e.g. for understanding Coincident
  Site Lattice theory.

4
Lecture List

• Vectors, Matrices and Rotations
• Tensors, Dyads, Products
• Groups, Group Theory
• Misorientation and Symmetry
• Rodrigues Vectors, Quaternions

5
Vectors, Matrices, Rotations

• Topics
• Definitions of vectors, scalar product, vector
  product
• Coordinate systems, basis vectors, handedness of
  coordinates
• Direction cosines, transformation of axes,
  orthogonal matrices, rotation matrices
• Vector realization of rotation
• Derivation of angle, axis from rotation matrix

6
Tensors, Dyads

• Topics
• Definition of dyad
• Dyadic product
• Inner product
• Unit dyads
• Tensors, symmetric and skew-symmetric
• Transformations of tensors
• Eigenanalysis
• Characteristic equation, principal values
• Diagonalization
• Invariants
• Polar Decomposition

7
Group Theory

• Topics
• Definition of Group
• Axioms
• Notation
• Multiplication Tables
• Group Properties
• Representation of Groups
• Rotation symmetry elements, groups
• Relationship to orientation analysis

8
Misorientation Symmetry

• Topics
• Contrast Active vs Passive Rotations
• Matrix representation
• Effect of symmetry
• Order of applying symmetry operators
• Notation
• How many equivalent representations?
• Switching symmetry for grain boundaries
• Finding the disorientation angle
• Franks formula for grain boundaries, dislocation
  structure, net Burgers vector
• O-lattice

9
Rodrigues vectors Quaternions

• Relationships to CSL structure
• Orientation vs. misorientation
• Relationship to axis-angle description
• Conversions
• Combining rotations in RF-form
• Shape of Rodrigues space, effect of symmetry,
  fundamental zones
• Quaternions, history
• Conversions, combination of quaternions
• Symmetry and quaternions

10
Rotations

• Miller Indices (hkl)uvw
• Euler angles variants of Euler angles (Bunge,
  Kocks, Roe, sumdifference ) f1, F, f2.
• Axis-angle n, f
• Rodrigues vector R1, R2, R3
• Quaternion q1, q2, q3, q4
• Rotation Matrix lij
• Passive (axis transformation) versus Active
  (vector rotation) rotations.

11
Matrix, Miller Indices

• The general Rotation Matrix, a, can be
  represented as in the following
• Where the Rows are the direction cosines for
  100, 010, and 001 in the sample coordinate
  system (pole figure).

100 direction
010 direction
001 direction
12
Matrix, Miller Indices

• The columns represent components of three other
  unit vectors
• Where the Columns are the direction cosines (i.e.
  hkl or uvw) for the RD, TD and Normal directions
  in the crystal coordinate system.

TD
ND?(hkl)
uvw?RD
13
Compare Matrices
uvw
(hkl)
uvw
(hkl)
14
Rodrigues vector, contd.

• We write the axis-angle representation as ( ,q)
• From this, the Rodrigues vector is r
  tan(q/2)

15
Conversions matrix?RF vector

• Conversion from rotation matrix, ?ggBgA-1

16
Conversion from Bunge Euler Angles

• tan(q/2) v(1/cos(F/2) cos(f1 f2)/22 1
  
• r1 tan(F/2) sin(f1 - f2)/2/cos(f1
  f2)/2
• r2 tan(F/2) cos(f1 - f2)/2/cos(f1
  f2)/2
• r3 tan(f1 f2)/2

P. Neumann (1991). Representation of
orientations of symmetrical objects by Rodrigues
vectors. Textures and Microstructures 14-18
53-58.
17
Quaternion definition

• q q(q1,q2,q3,q4) q(u sinq/2, v sinq/2, w
  sinq/2, cosq/2)
• Alternative notation puts cosine term in 1st
  positionq (cosq/2, u sinq/2, v sinq/2, w
  sinq/2).

18
Conversions matrix?quaternion
19
Symmetry

• Meaning of symmetry as an operation that leaves
  an object unchanged.
• Representation of symmetry as
• Physical operation, e.g. rotation
• Orthogonal matrix
• Quaternion
• Rodrigues vector
• Effect of symmetry on orientations, orientation
  space
• Fundamental Zones

20
Euler spacePoints related by triad
symmetryelement on lt111gt(triclinicsample
symmetry)
21
Rodrigues spacesymmetry

• Fundamental zones in Rodrigues space (a) cubic
  crystal symmetry with no sample symmetry (b)
  orthorhombic sample symmetry (c) cubic-cubic
  symmetry for disorientations. after Neumann,
  1990

22
Quaternion symmetry operators
23
Switching Symmetry
gB
?ggBgA-1? gAgB-1
Switching symmetryA to B is indistinguishable
from B to A because there is no difference in
grainboundary structure
gA
24
Summary

• Much of the material to be covered at this level
  can be found in mathematics texts but is not
  related back to the application area of crystals
  and grain boundaries.