(proper) rotations are represented (special) orthogonal ... · • there is one-to-one...

• 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