(proper) rotations are represented (special) orthogonal ... · • there is one-to-one...
TRANSCRIPT
• There is one-to-one correspondence between object's orientations and
rotations.
• An orientation of an object can be represented an orthogonal matrix.
Composition of rotations corresponds to multiplication of representing
them orthogonal matrices.
(Proper) rotations are represented (special) orthogonal matrices.
• There is two-to-one correspondence between unit quaternions
and special orthogonal matrices.
• Rotation/orientation parameterizations:
• Axis and angle
• Euler angles
• Rodrigues parameters
• Miller indices as orientation descriptors
Description of orientations and misorientations
in the presence of symmetry
Institute of Metallurgy and Materials Science
A.Morawiec
Polish Academy of Sciences
• Crystallographic point groups
• Asymmetric domains
• Misorientation angle distributions
• Misorientation axis distributions
Outline
3
Orientation analysis is influenced by objects’ symmetries.
Crystals exhibit symmetries.
Symmetry operation – object’s configurations w/r to its environment
before and after the operation are identical.
An object is symmetric if there exists a non-trivial symmetry operation.
Symmetry operations of an object constitute a group.
Point group – there exists a point fixed under all symmetry
operations of the group.
Symmetry operations of a point group are rotations.
4
Point groups
- the family of cyclic groups
- the family of dihedral groups
- tetrahedral group
- octahedral group
- icosahedral group
Proper rotations
)3(SO
nC
nD
T
O
I 5
Point groups - example
4C
6
100
001
010
100
010
001
100
010
001
100
001
010
)2/,( zeO ),( zeO )2/3,( zeOI
ze
All rotations
nC + reflection w/r to h
nC + reflection w/r to v
nC + (/n) & (reflection w/r to h)
nD + reflection w/r to h
nD + reflection w/r to d
T + reflection w/r to d
T + inversion
O + inversion
I + inversion
)3(O
nhC
nvC
nS2
nhD
ndD
dT
hT
hO
hI 2SCi
hs CC 1
Point groups
7
Crystallographic point groups
Crystallographic point group = a point group of a lattice
Crystallographic point groups have finite orders.
In 3D there are 32 (geometric classes) of crystallographic
point groups.
.32643234
64326462
643264321
dhdvvvvhs
hhhhdhhhhi
TDDCCCCCSC
OTDDDDCCSCC
OTDDDDCCCCC
8
h
hhh
dhds
vvvv
hhhdhhi
CS
TCCC
TCCS
TDDC
CCCC
ODDDDCC
ODDDDCC
34
643
646
32
6432
643221
64322
Triclinic
Monoclinic
OrthorhombicTrigonal
Tetragonal
HexagonalCubic
Crystal systems
9
0 360
90k
Asymmetric domains
Asymmetric domain – the smallest closed subset of the orientation
space such that symmetrical images of the domain cover the space
0 360
)23(306090 lklk
0 0 0
1 1 -1
2 0 1
3 1 0
4 0 2
5 1 1
6 2 0
7 1 2
8 2 1
9 3 0
10 2 2
11 3 1
lk 23 k l
Asymmetric domains
Asymmetric domain – the smallest closed subset of the orientation
space such that symmetrical images of the domain cover the space
SOC sample
symmetry
crystal
symmetry O
O + Crystal symmetry - CO
Two crystallites: ,11OC 22OC
Misorientation
TTTTOCCCOOCOCOC 1211221122 )(
12 COC crystal
symmetry
crystal
symmetry O
Asymmetric domains
12
0 3600 360
One phase:
Multi-phase:
- natural reference misorientation
- additional equivalence
for indistinguishible objects
1 - 2
2 - 1
Asymmetric domains
13
Asymmetric domains Euler angles
1
2
20 1
20 2 0
1
2
20 1
2/0 2
0
100
001
010
)2/,( ze
)2/,,(),,()2/,( 2121 OOez
)2/,,(),,( 2121 k
4C (tetragonal)
14
1 2
4D (tetragonal)
tetragonal - orthorhombic
24 DD
12
SOC
12
1/3
cubic - orthorhombic 2DO
sample
symmetry
crystal
symmetry O
Asymmetric domains Euler angles
15
1
52 102 152
302
402 502 552
602
802
702 752
852
02
202 252 352
452
652
902 16
Orientation
components
),,( 21 ff
1
52 102 152
302
402 502 552
602
802
702 752
852
- cube
- Goss
- shear- S
- copper
- Dillamore
- brass
17
Example experimental ODF
Orientation density
function of FeSi after
primary recrystalization
1 projection18
Example experimental ODF
Orientation density function
of extruded Ti-a (Grade 2)
1 projection19
Cubic crystal symmetry
Simultaneous permatations
in (...) and [...].
Even permatation
add even number of ‘-’.
Odd permatation
add odd number of ‘-’.
( -3 2 1 )[ 4 3 6 ]
( -3 -1 2 )[ 4 -6 3 ]
( 2 -3 -1 )[ 3 4 -6 ]
( 1 3 -2 )[ 6 -4 -3 ]
( -3 1 -2 )[ 4 6 -3 ]
( 3 -1 -2 )[ -4 -6 -3 ]
( -3 -2 -1 )[ 4 -3 -6 ]
( -1 -3 -2 )[ -6 4 -3 ]
( 2 1 -3 )[ 3 6 4 ]
( -1 2 -3 )[ -6 3 4 ]
( 1 -2 -3 )[ 6 -3 4 ]
( -2 -1 -3 )[ -3 -6 4 ]
( 1 2 3 )[ 6 3 -4 ]
( 2 -1 3 )[ 3 -6 -4 ]
( 2 3 1 )[ 3 -4 6 ]
( -1 -2 3 )[ -6 -3 -4 ]
( 3 -2 1 )[ -4 -3 6 ]
( -1 3 2 )[ -6 -4 3 ]
( 3 1 2 )[ -4 6 3 ]
( -2 1 3 )[ -3 6 -4 ]
( -2 -3 1 )[ -3 4 6 ]
( 1 -3 2 )[ 6 4 3 ]
( -2 3 -1 )[ -3 -4 -6 ]
( 3 2 -1 )[ -4 3 -6 ]
Asymmetric domains Miller indices
20
Besides equivalences of
cubic crystal symmetry:
This does not make sense
when orientations of individual crystallites are considered.
It is common to omit bars, e.g., ( 1 2 3 )[ 6 3 4 ]
24( 1 2 3 )[ 6 3 -4 ]
( 1 2 -3 )[ 6 3 4 ]
( 1 2 3 )[ -6 -3 4 ]
( 1 2 -3 )[-6 -3 -4 ]
Cubic - orthorhombic symmetry
Asymmetric domains Miller indices
all simultaneous permatations in (...) and [...] +
all sign changes not violating hu+kv+lw=0
21
Orientation
components
1
52 102 152
302
402 502 552
602
802
702 752
852
- cube
- Goss
- shear- S
- copper- Dillamore
- brass
22
Miller indices
’cube’ (0 0 1) <1 0 0>
’brass’ (0 1 1) <2 1 1>
’copper’ (1 1 2) <1 1 1>
’Goss’ (0 1 1) <1 0 0>
’S’ (2 1 3) <3 6 4>
’shear’ (0 0 1) <1 1 0>
’Dillamore’ (4 4 11) <11 11 8>
Cubic-orthorhombic
symmetry
2/tan ii nr
Rodrigues parameters and orientations
at the same distance to two distinguished orientations
1r
2r
C
1A
2A
B
planes
),(),( 21 CACA
),(),( 21 BABA
Asymmetric domains Rodrigues space
23
1r
2r
Orientations closer
to A than to B
Distinguished points
Asymmetric domains Rodrigues space
24
2r
1r
A
B
2r
1r
2r
1r
Construction of the asymmetric domain for CO
Voronoï cell around I
Asymmetric domains Rodrigues space
25
3D O
Asymmetric domains Rodrigues space
26
6D
Asymmetric domains, example texture
27
Extruded Ti-a
12OCCDomains in the case
1. No common symmetry operations
Voronoï cell around I = asymmetric domain
2. There are common symmetry operations
Voronoï cell around I = ‘large cell’ asymmetric domain
There are equivalent points within the large cell.
Asymmetric domains Rodrigues space
28
33, DD OO,
Asymmetric domains Rodrigues space
29
OO,
- additional equivalence
for indistinguishible objects
1 - 2
2 - 1
2,OO
Asymmetric domains Rodrigues space
30
66 , DD
Asymmetric domains Rodrigues space
31
Extruded Ti-a
Previous considerations:
proper rotations and proper symmetry operations
)3(SO
)3()3( SOO second component
Standard approach in textures:
enatiomorphic crystals – different phases
Proper rotations Improper rotations
Asymmetric domains
32
Misorientation angle distribution
Mis(dis)orientation –you must be disoriented
Disorientation angle - the smallest of angles of all rotations
leading from one orientation to the other
[2 1 0], 131.81deg
= [1 1 1], 60.00deg
Question: what is the frequency of occurence of these smallest
angles when the crystalites are randomly distributed?
Reference distribution33
0 45 62.8
Mackenzie distribution
The case of (O,O)
Mackenzie, 1958, Handscomb, 1958
Misorientation angle distribution
34
const
The frequency of occurence of misorientation angles
is related to the area of the surface =const in the
asymmetric domain (based on the smallest angles)
invariant volume (density of random orientations)
+ shape of the large cell
misorientation angle distribution
Misorientation angle distribution
35
0
0.002
0.004
0.006
0.008
0.01
0.012
0 20 40 60 80 100 120 140 160 180
in degrees
6C
1C2C
3C
4C
Cyclic symmetries
Misorientation angle distribution
36
6D
0
0.005
0.01
0.015
0.02
0.025
0 20 40 60 80 100 120
in degrees
4D
2D3D
6D
Dihedral symmetries
Misorientation angle distribution
37
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 10 20 30 40 50 60 70 80 90
in degrees
O
Y
T
Misorientation angle distribution
38
Misorientation angle distribution
39
0.00
0.01
0.02
0.03
0.04
0 10 20 30 40 50 60 70 80 90 100
Num
ber Fra
ctio
n
Misorientation Angle [degrees]
Misorientation Angle
CorrelatedUncorrelatedRandom
66 , DD
Extruded Ti-a
0 10 20 30 40 50
Multi-phase cases
Misorientation angle distribution
40
Axis corresponding to the smallest of angles
of all rotations leading from one orientation
to the other.
The cubic case – Mackenzie, 1964
1r
2r
Question: what is the frequency of occurence
of these these axes when the crystalites are
randomly distributed?
Reference distribution
Misorientation angle distribution
41
Misorientation axis distribution
In the Rodrigues space, the distance of the planes
bounding the asymmetric domain is related to the
frequency of occurence of misorientation axes.
constn
invariant volume (density of random orientations)
+ shape of the large cell
misorientation axis distribution 42
Misorientation
axis distributions
Od
3Ca
3Db
Tc
Ye
43
CSL misorientations
44
Coincident lattices
Coincident Site Lattice
G.Friedel, (1926).
Kronberg, M. L. and F. H. Wilson (1949),
Trans. Met. Soc. AIME, 185, 501 (1947).
36.8699° <100>
S - reciprocal of the fraction
of overlapping lattice nodes.
CSL misorientations
45
S Angle Axis
1 0.0000
3 60.0000 <111>
5 36.8699 <100>
7 38.2132 <111>
9 38.9424 <110>
11 50.4788 <110>
13a 22.6199 <100>
13b 27.7958 <111>
15 48.1897 <210>
17a 28.0725 <100>
17b 61.9275 <221>
19a 26.5254 <110>
19b 46.8264 <111>
21a 21.7868 <111>
21b 44.4153 <211>
23 40.4591 <311>
25a 16.2602 <100>
25b 51.6839 <331>
27a 31.5863 <110>
27b 35.4309 <210>
S Angle Miller indices of axis
7 21.7868 <0 0 1>
10 78.4630 <2 1 0>
11 62.9643 <2 1 0>
13 27.7958 <0 0 1>
14 44.4153 <2 1 0>
17 86.6277 <1 0 0>
18 70.5288 <1 0 0>
19 13.1736 <0 0 1>
22 50.4788 <1 0 0>
25 23.0739 <2 1 0>
26 87.7958 <8 0 1>
27 38.9424 <1 0 0>
29 66.6372 <16 0 3>
31a 17.8966 <0 0 1>
31b 56.7436 <5 1 0>
34 53.9681 <4 0 1>
35a 34.0477 <2 1 0>
35b 57.1217 <2 1 0>
37a 9.4300 <0 0 1>
37b 72.7047 <7 2 0>
hcpcub
CSL misorientations and ORs
46
Cubic symmetry
Summary
Orientation analysis is influenced by symmetries of objects
Asymmetric domains:
Euler angles – nice for low symmetries
Rodrigues parameterization – Voronoï tessellation
Crystal symmetry and statistical sample symmetry
Reference misorientation angle/axis distributions