Representation of Orientation
Representation of Orientation
Lecture Objectives
- Representation of Crystal Orientation
Stereography : Miller indices, Matrices
3 Rotations : Euler angles
Axis/Angle : Rodriques Vector, Quaternion
- Texture component by a single point
Representation of Orientation
Crystalline Nature of Materials
• Many solid materials are composed of
crystals joined together:
• In metals and other materials, the
individual grains may fit together
closely to form the solid:
• EBSD can determine
the orientation of individual grains and characterize grain boundaries.
Representation of Orientation
Crystalline Nature of Materials Grains
• A grain is a region of discrete crystal
orientation within a polycrystalline
material.
• The grain boundary is the
interface between individual
grains.
• Grain boundaries have a
significant influence on the
properties of the material,
dependant on the misorientation
across boundaries.
Representation of Orientation
Crystalline Nature of Materials Grain Boundaries
• A grain boundary is the interface
between two neighbouring grains.
• Classified by the misorientation - the difference in
orientation between two grains.
• Grain boundaries are regions of comparative disorder,
but, Special grain boundaries exist where a significant
degree of order occurs
• These are termed ‘CSL’ boundaries which have special
properties
Representation of Orientation
Crystalline Nature of Materials CSLs
• Coincident Site Lattice boundaries (CSL's).
•
A significant degree of order occurs at a CSL boundary, which leads
to special properties
Schematic representation of a Σ3 boundary. Where the
two grain lattices meet at the boundary, 1 in every 3
atoms is shared or coincident - shown in green.
Schematic representation of a Σ5 boundary. 1 in
every 5 atoms is shared or coincident.
Representation of Orientation
Crystalline Nature of Materials Properties
• Isotropic & Anisotropic Properties
•
If properties are equal in all directions, a material is termed 'Isotropic'.
•
If the properties tend to be greater or diminished in any direction, a
material is termed 'Anisotropic'.
•
Many/most materials are anisotropic
•
Anisotropy results from preferred
orientations or 'Texture'
In an isotropic polycrystalline
material, grain orientations are
random.
Representation of Orientation
Crystalline Nature of Materials Texture
• Texture may range from slight to highly
developed
Representation of Orientation
Orientation by Stereography
(Miller Index & Matrix)
Representation of Orientation
Basic Crystallography - Bravais Lattices
• There are 14 Bravais Lattices:
• From these, 7 crystal systems are derived
Representation of Orientation
Basic Crystallography - Crystal System
• Geometry of the unit cell
Æ Repeated structure throughout the crystal.
• There are seven Crystal systems:
1. Cubic
2. Hexagonal
3. Trigonal
4. Tetragonal
5. Orthorhombic
6. Monoclinic
7. Triclinic
The lengths of the sides of the unit cell are shown below as a, b and c. The
corresponding angles are shown as alpha, beta and gamma.
Representation of Orientation
Basic Crystallography –
Crystallographic Direction
• A crystallographic direction describes the
intersection of specific faces or lattice planes
• Miller Indices can be used to describe directions
For a cubic material the plane and the normal to the
plane have the same indices
Representation of Orientation
Basic Crystallography - Miller Indices
• For a cubic
example, Miller
indices can be
derived to describe
a plane
• Consider a cubic
unit cell with sides a,
b, c, with an origin
as shown
Representation of Orientation
Miller indices of a pole
Representation of Orientation
Stereographic Projection
• Uses the inclination of
the normal to the
crystallographic plane:
the points are the
intersection of each
crystal direction with a
(unit radius) sphere.
Representation of Orientation
Projection from Sphere to Plane
• Projection of spherical
information onto a flat
surface
– Equal area projection
(Schmid projection)
– Equiangular projection
(Wulff projection,
more common in
crystallography)
Representation of Orientation
Standard (001) Projection
Representation of Orientation
Stereographic, Equal Area Projections
Stereographic
(Wulff)
Projection*:
OP’=R tan(θ/2)
Equal Area
(Schmid)
Projection:
OP’=R sin(θ/2)
* Many texts, e.g. Cullity, show the plane touching the sphere at N: this changes the magnification
factor for the projection, but not its geometry.
Representation of Orientation
Pole Figure Example
• If the Diffraction goniometer is set for {100} reflections,
then all directions in the sample that are parallel to <100>
directions will exhibit diffraction.
Æ The Sample shows a crystal oriented to put all 3 <100>
directions approximately equally spaced from the ND.
Representation of Orientation
Miller Index of a Crystal Orientation
• 3 orthogonal directions as the reference frame.
Æ a set of unit vectors called e1 e2 and e3.
• In many cases we use the names Rolling Direction
(RD) // e1, Transverse Direction (TD) // e2, and
Normal Direction (ND) // e3.
• We then identify a crystal (or plane normal) parallel
to 3rd axis (ND) and a crystal direction parallel to
the 1st axis (RD), written as (hkl)[uvw].
Representation of Orientation
Basic Crystallography – Miller Index
• In this example
the orientation of
the crystal
shown can be
written as:
{100}<110>
Representation of Orientation
Basic Crystallography - Orientation Matrix
• The orientation
matrix describes
the absolute
orientation of the
crystal with respect
to the sample axes.
Representation of Orientation
Basic Crystallography - Euler Angles
• Euler Angles are the
three rotations about
the main crystal
axes
• Euler angles are one
possible means of
describing a crystal
orientation
Representation of Orientation
Basic Crystallography – Axis/Angle
Misorientation
• Misorientation is
the expression of
the orientation of
one crystal with
respect to another
crystal.
Representation of Orientation
Orientation by 3 Rotations
(Euler Angles :Mathematical Approach)
Representation of Orientation
Euler Angle Definition (Bunge)
Representation of Orientation
Euler Angles, Animated
[001]
e’3= e3=Zsample=N D
zcrystal=e3’’’
= e”3
φ1
φ2
Crystal
[010]
ycrystal=e2’’’
e”2
3rd position (final)
2nd position
1st position
e’2
e2=Ysample=TD
RD
Φ
TD
Sample Axes
xcrystal=e1’’’
e’1 =e”1
e1=Xsample=R D
Representation of Orientation
[100]
Euler Angles, Ship Analogy
Analogy: position and the heading of a boat with respect to the globe.
• Latitude (Θ) and longitude (ψ)
: Position of the boat on Earth
• third angle (φ)
: heading of boat relative to the
line of longitude that connects
the boat to the North Pole.
Representation of Orientation
Meaning of Euler angles
• First two angles, φ1 and Φ,
the position of the [001] crystal direction
relative to the specimen axes.
• Think of rotating the crystal about the ND (1st
angle, φ1); then rotate the crystal out of the
plane (about the [100] axis, Φ);
• 3rd angle (φ2) tells
Rotation of the crystal about [001].
Representation of Orientation
Euler Angle Definitions
Bunge and Canova are inverse to one another
Kocks and Roe differ by sign of third angle
Bunge rotates about x’, Kocks about y’ (2nd angle)
Representation of Orientation
Conversions
Conv ention
1st
2nd
3rd
Kock s
(symmetric)
Bung e
Matthies
Ψ
Θ
φ
φ1-π/2
α
Φ
β
π/2−φ 2
π−γ
x
Roe
Ψ
Θ
π−Φ
y
Representation of Orientation
2nd ang le
abou t axis:
y
y
Miller indices to vectors
• Need the direction
cosines for all 3 crystal
axes.
• A direction cosine is the
cosine of the angle
between a vector and a
given direction or axis.
• Sets of direction cosines
can be used to construct
a transformation matrix
from the vectors.
Axes
[u v 0]
Sample Direction 1 = RD = ^e1
α1
Sample Direction 2
= TD = ^e2
α2
Sample Direction 3 = ND = e^3
Direction Cosine 1 = cos(α1)
Representation of Orientation
Origin = (0,0,0)
45
Rotation of axes in the plane:
x, y = old axes; x’,y’ = new axes
y
y’
v
x’
 cos θ sin θ 
v
v′ = 
 − sin θ cos θ 
θ
N.B. Passive Rotation/ Transformation of Axes
Representation of Orientation
x
Definition of an Axis Transformation:
e = old axes; e’ = new axes
Sample to Crystal (primed) ^
e’3
aij = eˆi′ • eˆ j
 a11 a12
=  a21 a22

 a31 a32
a13 
a23 

a33 
^
e3
^
e’2
^
e2
^
e1
Representation of Orientation
^
e’1
Geometry of {hkl}<uvw>
Sample to Crystal (primed)
^
Miller index
e’3
[001]
notation of
texture component
specifies direction
cosines of xtal
directions || to
^
sample axes.
e1 || [uvw]
t = hkl x uvw
^
e3 || (hkl)
Representation of Orientation
[010]
^
e’2
^
^
e2 || t
^
e’1
[100]
Form matrix from Miller Indices
nˆ =
( h, k , l )
h +k +l
2
ˆ
ˆ
×
n
b
tˆ =
nˆ × bˆ
2
aij
2
bˆ =
( u , v , w)
u +v +w
 b1

= Crystal  b2
b
 3
2
2
Sample
Representation of Orientation
t1
t2
t3
2
n1 

n2 

n3 
Bunge Euler angles to Matrix
Rotation 1 (φ1): rotate axes (anticlockwise)
about the (sample) 3 [ND] axis; Z1.
Rotation 2 (Φ): rotate axes (anticlockwise)
about the (rotated) 1 axis [100] axis; X.
Rotation 3 (φ2): rotate axes (anticlockwise)
about the (crystal) 3 [001] axis; Z2.
Representation of Orientation
Bunge Euler angles to Matrix
1
 cos φ1 sin φ1 0
Z1 =  − sin φ1 cos φ1 0 , X =  0



0
 0
0
1
 cos φ2
Z2 =  − sin φ2

 0
sin φ 2
cosφ 2
0
0
0

1
0 
cosΦ sin Φ  ,

− sinΦ cosΦ
0
A=Z2XZ1
Representation of Orientation
Matrix with Bunge Angles
[uvw]
A = Z2XZ1 =
(hkl)
sinϕ1 cos ϕ2

 cosϕ1 cosϕ 2
 − sin ϕ1 sin ϕ 2 cosΦ + cos ϕ1 sin ϕ 2 cosΦ sinϕ 2 sin Φ 




 − cos ϕ1 sin ϕ 2
− sin ϕ1 sin ϕ 2
cosϕ 2 sin Φ



 − sin ϕ1 cos ϕ 2 cosΦ + cos ϕ1 cos ϕ 2 cosΦ




sinϕ1 sinΦ

− cos ϕ1 sinΦ
cosΦ 
Representation of Orientation
Matrix, Miller Indices
• The general Rotation Matrix, a, can be represented
as in the following:
[100] direction
[010] direction
[001] direction
 a11
a
 21
 a31
a12
a22
a32
a13 

a23 
a33 
• Where the Rows are the direction cosines for [100],
[010], and [001] in the sample coordinate system
(pole figure).
Representation of Orientation
Matrix, Miller Indices
•
The columns represent components of three other unit vectors:
[uvw]≡RD
TD
ND≡(hkl)
 a11
a
 21
 a31
a12
a22
a32
a13 

a23 
a33 
• Where the Columns are the direction cosines (i.e. hkl or uvw) for
the RD, TD and Normal directions in the crystal coordinate system.
Representation of Orientation
Compare Matrices
[uvw]
aij
 b1

= Crystal  b2
b
 3
(hkl)
Sample
t1
t2
t3
n1 

n2 
n3 
[uvw]
(hkl)
sinϕ1 cos ϕ2
 cosϕ1 cosϕ 2

ϕ
sin
Φ
sin
2
 − sin ϕ sin ϕ cosΦ + cos ϕ sin ϕ cosΦ

1
2
1
2




 − cos ϕ1 sin ϕ 2
− sin ϕ1 sin ϕ 2
cosϕ 2 sin Φ


−
sin
ϕ
cos
ϕ
cosΦ
+
cos
ϕ
cos
ϕ
cosΦ


1
2
1
2




sinϕ1 sinΦ

− cos ϕ1 sinΦ
cosΦ 
Representation of Orientation
Miller indices from Euler angle matrix
h = n sin Φ sin ϕ 2
k = n sin Φ cos ϕ 2
l = n cos Φ
u = n′(cos ϕ1 cos ϕ 2 − sin ϕ1 sin ϕ 2 cos Φ )
v = n′(− cos ϕ1 sin ϕ 2 − sin ϕ1 cos ϕ 2 cos Φ )
w = n′ sin Φ sin ϕ1
n, n’ = factors to make integers
Representation of Orientation
Euler angles from Miller indices
cos Φ =
Inversion of
the previous
relations:
cos ϕ 2 =
sin ϕ1 =
l
h2 + k 2 + l 2
k
h2 + k 2
w
h2 + k 2 + l 2
u 2 + v 2 + w2
h2 + k 2
Caution: when one uses the inverse trig functions, the range of result is limited to
0°≤cos-1θ≤180°, or -90°≤sin-1θ≤90°. Thus it is not possible to access the full 0360° range of the anlges. It is more reliable to go from Miller indices to an
orientation matrix, and then calculate the Euler angles. Extra credit: show that the
following surmise is correct. If a plane, hkl, is chosen in the lower hemisphere, l<0,
show that the Euler angles are incorrect.
Representation of Orientation
Euler angles from Orientation Matrix
Notes:
The range of cos-1 is
0-π, which is sufficient
for Φ from this, sin(Φ)
can be obtained
The range of tan-1 is
0-2π, (must use the
ATAN2 function)
which is required for
calculating φ1 and φ2.
(a33 )
−1  (a13
ϕ 2 = tan 
Φ = cos
−1
sin Φ )
(a23

−1  (a31 sin Φ )
ϕ1 = tan 
(a32


sin Φ )

sin Φ )
 a12 


tan
a

11 
if a 33 ≈ 1, Φ = 0, ϕ1 =
, and ϕ 2 = −ϕ1
2
Representation of Orientation
−1
Complete orientations in the Pole Figure
(φ1,Φ,φ2) ~
(30°,70°,40°).
φ1
φ2
1 Rotation
2 Rotation
3 Rotation
φ1
Φ
Φ
φ2
Note the loss of information in a diffraction experiment if each set of
poles from a single component cannot be related to one another.
Representation of Orientation
Complete Orientations in Inverse Pole Figure
Note the loss of information in
a diffraction experiment if each
set of poles from a single
component cannot be related to
one another.
The same as Pole figure
Experiment
Representation of Orientation
Other Euler angle definitions
• Very Confusing Aspect of Texture Analysis is that
there are multiple definitions of the Euler angles.
• Definitions according to Bunge, Roe and Kocks are in
common use.
• Roe definition is Exactly Classical definition by Euler
• Components have different values of Euler angles
depending on which definition is used.
• The Bunge definition is the most common.
• The differences between the definitions are based on
differences in the sense of rotation, and the choice of
rotation axis for the second angle.
Representation of Orientation
Orientation by 3 Rotations
(Euler Angles :Texture Components)
Representation of Orientation
Cube Component = {001}<100>
{100}
{110}
{111}
Representation of Orientation
Cube Texture (100)[001]:cube-on-face
• Observed in recrystallization of fcc metals
• The 001 orientations are parallel to the three
ND, RD, and TD directions.
ND
TD
RD
{100}
{001}
{010}
Representation of Orientation
Sharp Texture (Recrystallization)
• Look at the (001) pole figures for this type of
texture: maxima correspond to {100} poles in the
standard stereographic projection.
RD
ND
TD
RD
TD
{100}
{001}
{010}
Representation of Orientation
Euler angles of Cube component
• The Euler angles for this component are simple,
but not so simple!
• The crystal axes align exactly with the specimen
axes, therefore all three angles are exactly zero:
(φ1, Φ, φ2) = (0°, 0°, 0°).
• Due to the effects of crystal symmetry:
aligning [100]//TD, [010]//-RD, [001]//ND.
Æ evidently still the cube orientation
Æ Euler angles are (φ1,Φ,φ2) = (90°,0°,0°)!
Representation of Orientation
{011}<001>: the Goss Component
• Goss Texture : Recrystallization texture for FCC
materials such as Brass, …
• (011) plane is oriented towards the ND
and the [001] inside the (011) plane is along the RD.
ND
(110)
TD
[100]
RD
Representation of Orientation
{011}<001>: cube-on-edge
• In the 011 pole figure, one of the poles is oriented
parallel to the ND (center of the pole figure) but the
other ones will be at 60° or 90° angles but tilted 45°
from the RD!
ND
RD
(110)
TD
[100]
TD
{110}
Representation of Orientation
RD
Euler angles of Goss component
• The Euler angles for this component are
simple, and yet other variants exist, just as for
the cube component.
• Only one rotation of 45° is needed to rotate
the crystal from the reference position (i.e. the
cube component); this happens to be
accomplished with the 2nd Euler angle.
• (φ1,Φ,φ2) = (0°,45°,0°).
Other variants will be shown when symmetry
is discussed.
Representation of Orientation
Brass component
• Brass Texture : a rolling texture component
for materials such as Brass, Silver, and
Stainless steel.
ND
(110)[1 12]
TD
(110)
RD
[112]
Representation of Orientation
Brass component
• 30o Rotation of the Goss texture about the ND
(100)
(110)
Representation of Orientation
(111)
Brass component: Euler angles
• The brass component is convenient
because we can think about performing
two successive rotations:
• 1st about the ND, 2nd about the new
position of the [100] axis.
• 1st rotation is 35° about the ND; 2nd
rotation is 45° about the [100].
• (φ1,Φ,φ2) = (35°,45°,0°).
Representation of Orientation
Table 4.F.2. fcc Rolling Texture Components: Euler Angles and Indices
{112}〈111̄〉
Bunge
(ϕ1,Φ,ϕ2)
RD= 1
40, 65, 26
Kocks
(ψ,Θ,φ)
RD= 1
50, 65, 26
Bunge
(ϕ1,Φ,ϕ2)
RD= 2
50, 65, 64
Kocks
(ψ,Θ,φ)
RD= 2
39, 66, 63
{112}〈111̄〉
90, 35, 45
0, 35, 45
0, 35, 45
90, 35, 45
{123}〈634̄〉
(312)<0 2 1>
(312)<0 2 1>
(312)<0 2 1>
{110}〈1̄12〉
59, 37, 27
32, 58, 18
48, 75, 34
64, 37, 63
35, 45, 0
31, 37, 27
58, 58, 18
42, 75,34
26, 37, 63
55, 45, 0
31, 37, 63
26, 37, 27
42, 75, 56
58, 58, 72
55, 45, 0
59, 37, 63
64, 37, 27
48, 75, 56
32, 58, 72
35, 45, 0
{110}〈1̄12〉
55, 90, 45
35, 90, 45
35, 90, 45
55, 90, 45
{110}〈1̄12〉
35, 45, 90
55, 45, 90
55, 45, 90
35, 45, 90
{4 4 11}〈11 11 8̄〉
{4 4 11}〈11 11 8̄〉
42, 71, 20
90, 27, 45
48, 71, 20
0, 27, 45
48, 71, 70
0, 27, 45
42, 71, 70
90, 27, 45
{110}〈001〉
0, 45, 0
90, 45, 0
90, 45, 0
0, 45, 0
{110}〈001〉
90, 90, 45
0, 90, 45
0, 90, 45
90, 90, 45
{110}〈001〉
0, 45, 90
90, 45, 90
90, 45, 90
0, 45, 90
Name
Indices
copper/
1st var.
copper/
2nd var.
S3*
S/ 1st var.
S/ 2nd var.
S/ 3rd var.
brass/
1st var.
brass/
2nd var.
brass/
3rd var.
Taylor
Taylor/
2nd var.
Goss/
1st var.
Goss/
2nd var.
Goss/
3rd var.
Representation of Orientation
Summary
• Conversion between different forms of
description of texture components
described.
• Physical picture of the meaning of Euler
angles as rotations of a crystal given.
• Miller indices are descriptive, but
matrices are useful for computation, and
Euler angles are useful for mapping out
textures (to be discussed).
Representation of Orientation
Orientation by Axis/Angle
(Rodrigues Vectors, Quaternions)
Representation of Orientation
Objectives
• Introduction of Rodrigues* vector as a
representation of rotations, orientations and
misorientations (grain boundary types).
• Introduction of quaternion and its relationship
to other representations
*French mathematician active in the early part of the 19th C.
Representation of Orientation
Rodrigues vector
• Rodrigues vectors were popularized by Frank
[Frank, F. (1988). “Orientation mapping.”
Metallurgical Transactions 19A: 403-408.], hence
the term Rodrigues-Frank space for the set of
vectors.
• Most useful for representation of misorientations,
i.e. grain boundary character; also useful for
orientations (texture components).
• Fibers based on a fixed axis are always straight
lines in RF space (unlike Euler space).
Representation of Orientation
Rodrigues vector
• Axis-Angle representation :
OQ
Rotation axis r =
Rotation Angle about Axis : α
OQ
ρ = r tan (α / 2)
The rotation angle is α,
and the magnitude of the
vector is scaled by the
tangent of the semi-angle.
BEWARE: Rodrigues vectors do NOT obey the
parallelogram rule (because rotations are NOT commutative!
Representation of Orientation
Orientation, Misorientation
• Orientation
:g
• Misorientation : ∆g
• Given two orientations (grains) gA and gB
• Misorientation between A and B Orientation
Æ ∆g
= gBgA-1
∆g
A
gA
gB
ND
TD
RD
Representation of Orientation
B
Conversions: Matrix→ RF vector
• Conversion from rotation (misorientation) matrix : ∆g=gBgA-1
θ [∆g (2,3) − ∆g (3,2)] / norm

tan
ρ
 1 
2

 
 ρ 2  =  tan θ 2 [∆g (1,3) − ∆g (3,1)] / norm 

ρ   θ
 3   tan [∆g (1,2) − ∆g (2,1)] / norm 

norm =
2

[∆g (2,3) − ∆g (3,2)]2 + [∆g (1,3) − ∆g (3,1)]2 + [∆g (1,2) − ∆g (2,1)]2
cos θ
2
=
1 (cos θ + 1) = 1 1 + tr (∆g )
2
2
Representation of Orientation
Conversion from Bunge Euler Angles:
•
•
•
•
tan(θ/2) = √{(1/[cos(Φ/2) cos{(φ1 + φ2)/2}]2 – 1}
ρ1 = tan(Φ/2) sin{(φ1 - φ2)/2}/[cos{(φ1 + φ2)/2}]
ρ2 = tan(Φ/2) cos{(φ1 - φ2)/2}/[cos{(φ1 + φ2)/2}]
ρ3 = tan{(φ1 + φ2)/2}
“Representation of orientations of symmetrical objects by Rodrigues vectors.”
P. Neumann (1991). Textures and Microstructures 14-18: 53-58.
Representation of Orientation
Combining Rotations as RF vectors
• Two Rodrigues vectors combine to form a
third, ρC, as follows, where ρB follows after ρA.
Note : NOT parallelogram law for vectors!
ρC = (ρA, ρB) =
{ρA + ρB - ρA x ρB}/{1 - ρA•ρB}
vector product
scalar product
Representation of Orientation
Combining Rotations as RF vectors:
component form
(ρ
C
1
,ρ ,ρ
C
2
C
3
)
[
[
[
]
]
]
)
 ρ1A + ρ1B − ρ 2A ρ 3B − ρ 3A ρ 2B , 


 ρ 2A + ρ 2B − ρ 3A ρ1B − ρ1A ρ 3B , 
 A

B
A
B
A
B
 ρ 3 + ρ 3 − ρ1 ρ 2 − ρ 2 ρ1 


=
A B
A B
A B
1 − ρ1 ρ1 + ρ 2 ρ 2 + ρ 3 ρ 3
(
Representation of Orientation
Quaternions
• A close cousin to the Rodrigues vector
• A four component vector in relation to the
axis-angle representation as follows
Æ [uvw] : Unit vector of rotation axis
Æθ
: Rotation angle.
• q = q(q1,q2,q3,q4)
= q(u sinθ/2, v sinθ/2, w sinθ/2, cosθ/2)
Representation of Orientation
Why Use Quaternions?
• Quaternions offer a efficient way on combining
rotations, because of the small number of
floating point operations required to compute
the product of two rotations.
• The quaternion has a unit norm
(√{q12+q22+q32+ q42}=1) Æ Always Finite Value
Æ No Computational Overflow & Underflow
Representation of Orientation
Computation: combining rotations
• Quaternions
Æ16 multiplies and 12 additions
Æ with no divisions or transcendental functions.
• Matrix
Æ 27 multiplies and 18 additions.
• Rodrigues vector
Æ10 multiplies and 9 additions.
• The product of two rotations
Æ the least work with Rodrigues vectors
Representation of Orientation
Conversions: matrix→quaternion
θ [∆g (2,3) − ∆g (3,2)] / norm

sin
 q1  
2

 
θ [∆g (1,3) − ∆g (3,1)] / norm 
sin

q
 2
2
=


q 
θ
g
g
norm
∆
−
∆
sin
[
(
1
,
2
)
(
2
,
1
)]
/
3


 
2
q  

θ
 4 
cos

2

norm =
[∆g (2,3) − ∆g (3,2)]2 + [∆g (1,3) − ∆g (3,1)]2 + [∆g (1,2) − ∆g (2,1)]2
cos θ = 1 1 + tr (g )
2
2
sin θ
2
=1
Representation of Orientation
2
3 − tr ( g )
Conversions: quaternion →matrix
• The conversion of a quaternion to a rotation matrix is
given by:
• eijk is the permutation tensor, δij the Kronecker delta
g ij = (q - q - q - q )δ ij
2
4
2
1
2
2
2
3
+ 2q i q j + 2q 4 ∑ eijk q k
k =1,3
Representation of Orientation
Bunge angles → Quaternion
• [q1, q2, q3, q4] =
[sinΦ/2 cos{(φ1 - φ2)/2 },
sinΦ/2 sin{(φ1 - φ2)/2},
cosΦ/2 sin{φ1 + φ2)/2},
cosΦ/2 cos{(φ1 + φ2)/2} ]
Note the occurrence of sums and differences of the 1st and 3rd Euler angles!
Representation of Orientation
Combining quaternions
• The algebraic form for combination of
quaternions is as follows, where qB follows
qA:
qC = qA qB
qC1 = qA1 qB4 + qA4 qB1 - qA2 qB3 + qA3 qB2
qC2 = qA2 qB4 + qA4 qB2 - qA3 qB1 + qA1 qB3
qC3 = qA3 qB4 + qA4 qB3 - qA1 qB2 + qA2 qB1
qC4 = qA4 qB4 - qA1 qB1 - qA2 qB2 - qA3 qB3
Representation of Orientation
Positive vs Negative Rotations
considering a rotation of θ about an arbitrary axis, r. If one rotates
backwards by the complementary angle, θ- 2π (also about r), in terms
of quaternions, however, the representation is different!
Representation of Orientation
Positive vs Negative Rotations
Rotation θ about an axis, r
q(r,θ)
= q(u sinθ/2, v sinθ/2, w sinθ/2, cosθ/2)
Rotation θ−2π about an axis, r
q(r,θ−2π)
= q(u sin(θ−2π)/2, v sin(θ−2π)/2,w sin(θ−2π)/2, cos(θ−2π)/2)
= q(-u sinθ/2, -v sinθ/2,-w sinθ/2, -cosθ/2)
= -q(r,θ)
• The quaternion representing the negative rotation is the
negative of the original (positive) rotation. The positive
and negative quaternions are equivalent or physically
indistinguishable, q ≡ -q.
Representation of Orientation
Negative of a Quaternion
• The negative (inverse) of a quaternion is
given by negating the fourth component,
q-1 = ±(q1,q2,q3,-q4); this relationship
describes the switching symmetry at grain
boundaries.
I = qA qB
I = (0,0,0,1)
qA =
-1
qB
Representation of Orientation
Summary
• Rodrigues vectors :
Æ Rotations with a 3-component vector.
• Quaternions form a complete algebra.
ÆIn unit length quaternions, they are very
useful for describing rotations.
ÆCalculation of misorientations in cubic
systems is particularly efficient.
Representation of Orientation
Matrix with Roe angles
a(ψ,θ,φ) =
(hkl)
[uvw]
 − sinψ sin φ
 + cosψ cosφ cosθ


 − sinψ cosφ

 − cosψ sin φ cosθ



cosψ sinθ
cosψ sin φ

− cosφ sin θ

+ sinψ cos φ cos θ


cosψ cos φ
sin φ sin θ 

− sinψ sin φ cos θ



sinψ sin θ
cosθ 
Representation of Orientation
Roe angles → quaternion
• [q1, q2, q3, q4] =
[-sinΘ/2 sin{(Ψ - Φ)/2 },
sinΘ/2 cos{(Ψ - Φ)/2},
cosΘ/2 sin{Ψ + Φ)/2},
cosΘ/2 cos{(Ψ + Φ)/2} ]
Representation of Orientation
Conversion from Roe Euler Angles:
•
•
•
•
•
tan(θ/2) = √{(1/[cos{Θ/2} cos{(Ψ + Φ)/2}]2 – 1}
ρ1 = -tan{Θ/2} sin{(Ψ - Φ)/2}/[cos{(Ψ + Φ)/2}]
ρ2 = tan{Θ/2} cos{(Ψ - Φ)/2}/[cos{(Ψ + Φ)/2}]
ρ3 = tan{(Ψ + Φ)/2}
See, for example, Altmann’s book on Quaternions,
where Ψ = α, Θ = β, Φ = γ. These formulae can be
converted to those on the previous page for Bunge
angles by substituting:
Ψ = φ1 − π/2, Φ = φ1 + π/2.
Representation of Orientation
Matrix with Kocks Angles
[uvw]
a(Ψ,Θ,φ) =
 − sin Ψsin φ
 − cos Ψ cosφ cosΘ


 sinΨ cos φ

 − cosΨ sin φ cosΘ


cos Ψ sinΘ

cosΨ sin φ
− sinΨ cos φ cosΘ
− cosΨ cos φ
− sinΨ sin φ cosΘ
sin Ψsin Θ
(hkl)

cosφ sin Θ


sin φ sin Θ 



cosΘ 

Note: obtain transpose by exchanging φ and Ψ.
Representation of Orientation