bunge notation

1

Texture Components and Euler Angles: part 2

13th January 05

27-750

Spring 2005

A. D. (Tony) RollettCarnegie

Mellon

MRSEC

2

Lecture Objectives

• Show how to convert from a description of a crystal orientation based on Miller indices to matrices to Euler angles

• Give examples of standard named components and their associated Euler angles

• The overall aim is to be able to describe a texture component by a single point (in some set of coordinates such as Euler angles) instead of needing to draw the crystal embedded in a reference frame

• Part 2 provides mathematical detail

Obj/notation AxisTransformation Matrix EulerAngles Components

3

(Bunge)Euler Angle Definition


4

Euler Angles, Ship Analogy• Analogy: position and

the heading of a boat with respect to the globe. Co-latitude () and longitude () describe the position of the boat; third angle describes the heading () of the boat relative to the line of longitude that connects the boat to the North Pole.

Kocks vs. Bunge angles:to be explained later!


5

Meaning of Euler angles

• The first two angles, 1 and , tell you the position of the [001] crystal direction relative to the specimen axes.

• Think of rotating the crystal about the ND (1st angle, 1); then rotate the crystal out of the plane (about the [100] axis, );

• Finally, the 3rd angle (2) tells you how much to rotate the crystal about [001].


6

Euler Angles, Animated

[010]

[100]

[001]

Crystal

e1=Xsample=RD

e2=Ysample=TD

e3=Zsample=ND

Sample Axes

RD

TD

e”2

e”3

=e”1

2nd position

ycrystal=e2’’’

xcrystal=e1’’’

zcrystal=e3’’’=

3rd position (final)

e’1

e’2

e’3=

1st position

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Brass component, contd.

RD

ND

TD

(110)

[112]

(100) (111) (110)

(110)[1 12]Obj/notation AxisTransformation Matrix EulerAngles Components

• The associated (110) pole figure is very similar to the Goss texture pole figure except that it is rotated about the ND. In this example, the crystal has been rotated in only one sense (anticlockwise).

8

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].

• (1,,2) = (35°,45°,0°).


9 Table 4.F.2. fcc Rolling Texture Components: Euler Angles and Indices

Name Indices Bunge(1,,2)RD= 1

Kocks(,,)RD= 1

Bunge(1,,2)RD= 2

Kocks(,,)RD= 2

copper/1st var.

{112}111̄ 40, 65, 26 50, 65, 26 50, 65, 64 39, 66, 63

copper/2nd var.

{112}111̄ 90, 35, 45 0, 35, 45 0, 35, 45 90, 35, 45

S3* {123}634̄ 59, 37, 27 31, 37, 27 31, 37, 63 59, 37, 63S/ 1st var. (312)<0 2 1> 32, 58, 18 58, 58, 18 26, 37, 27 64, 37, 27S/ 2nd var. (312)<0 2 1> 48, 75, 34 42, 75,34 42, 75, 56 48, 75, 56S/ 3rd var. (312)<0 2 1> 64, 37, 63 26, 37, 63 58, 58, 72 32, 58, 72brass/1st var.

{110}1̄12 35, 45, 0 55, 45, 0 55, 45, 0 35, 45, 0

brass/2nd var.

{110}1̄12 55, 90, 45 35, 90, 45 35, 90, 45 55, 90, 45

brass/3rd var.

{110}1̄12 35, 45, 90 55, 45, 90 55, 45, 90 35, 45, 90

Taylor {4 4 11}11 11 8̄ 42, 71, 20 48, 71, 20 48, 71, 70 42, 71, 70Taylor/2nd var.

{4 4 11}11 11 8̄ 90, 27, 45 0, 27, 45 0, 27, 45 90, 27, 45

Goss/1st var.

{110}001 0, 45, 0 90, 45, 0 90, 45, 0 0, 45, 0

Goss/2nd var.

{110}001 90, 90, 45 0, 90, 45 0, 90, 45 90, 90, 45

Goss/3rd var.

{110}001 0, 45, 90 90, 45, 90 90, 45, 90 0, 45, 90


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Meaning of “Variants”

• The existence of variants of a given texture component is a consequence of (statistical) sample symmetry.

• If one permutes the Miller indices for a given component (for cubics, one can change the sign and order, but not the set of digits), then different values of the Euler angles are found for each permutation.

• If a pole figure is plotted of all the variants, one observes a number of physically distinct orientations, which are related to each other by symmetry operators (diads, typically) fixed in the sample frame of reference.

• Each physically distinct orientation is a “variant”. The number of variants listed depends on the choice of size of Euler space (typically 90x90x90°) and the alignment of the component with respect to the sample symmetry.

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Notation: vectors, matrices• Vector-Matrix: v is a vector, A is a matrix• Index notation: explicit indexes (Einstein convention):

vi is a vector, Ajk is a matrix (maybe tensor)• Scalar (dot) product: c = a•b = aibi; zero dot product

means vectors are perpendicular. For two unit vectors, the dot product is equal to the cosine of the angle between them.

• Vector (cross) product: c = ci = a x b = a b = ijk ajbk; generates a vector that is perpendicular to the first two.


12

Miller indices to vectors

45

Axes

Sample Direction 1 = RD = e1

Sample Direction 2 = TD = e2

^

^

Sample Direction 3 = ND = e3^2

1

Direction Cosine 1 = cos(1)

[u v 0]

Origin = (0,0,0)


• 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.

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Rotation of axes in the plane:x, y = old axes; x’,y’ = new axes

x

y

x’

y’ v

v cos sin sin cos

v

N.B. Passive Rotation/ Transformation of Axes


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Definition of an Axis Transformation:e = old axes; e’ = new axes

aij ˆ e i ˆ e j

e1^ e’1

^

e2^

e’2^

e3^e’3

^Sample to Crystal (primed)

a11 a12 a13

a21 a22 a23

a31 a32 a33


15

Geometry of {hkl}<uvw>

e1 || [uvw]^

e’1^

e2 || t^

e’2^

e3 || (hkl)^e’3^

^

[001]

[010]

[100]

Miller indexnotation oftexture componentspecifies directioncosines of xtaldirections || tosample axes.

Sample to Crystal (primed)

t = hkl x uvwObj/notation AxisTransformation Matrix EulerAngles Components

16

Form matrix from Miller Indices

ˆ b (u,v, w)

u2 v2 w2ˆ n

(h, k, l)

h2 k2 l2

ˆ t ˆ n ˆ b

ˆ n ˆ b aij Crystal

Sample

b1 t1 n1

b2 t2 n2

b3 t3 n3


17

Bunge Euler angles to Matrix

Rotation 1 (): rotate axes (anticlockwise) about the (sample) 3 [ND] axis; Z1.

Rotation 2 (): rotate axes (anticlockwise) about the (rotated) 1 axis [100] axis; X.

Rotation 3 (): rotate axes (anticlockwise) about the (crystal) 3 [001] axis; Z2.


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Bunge Euler angles to Matrix, contd.

Z1 cos1 sin1 0

sin1 cos1 0

0 0 1

, X

1 0 0

0 cos sin 0 sin cos

,

Z2 cos2 sin2 0

sin2 cos2 0

0 0 1

A=Z2XZ1


19

Matrix with Bunge Angles

cos1 cos2

sin1sin2 cossin1 cos2

cos1sin2 cossin2 sin

cos1sin2

sin1cos2 cos sin1sin2

cos1cos2 coscos2 sin

sin1 sin cos1sin cos

A = Z2XZ1 = (hkl)[uvw]


20

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

a11 a12 a13

a21 a22 a23

a31 a32 a33


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Matrix, Miller Indices

• The columns represent components of three other unit vectors:

[uvw]RD TD ND(hkl)

a11 a12 a13

a21 a22 a23

a31 a32 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.

22

Compare Matrices

aij Crystal

Sample

b1 t1 n1

b2 t2 n2

b3 t3 n3

cos1 cos2

sin1sin2 cossin1 cos2

cos1sin2 cossin2 sin

cos1sin2

sin1cos2 cos sin1sin2

cos1cos2 coscos2 sin

sin1 sin cos1sin cos

[uvw] [uvw] (hkl)(hkl)


23

Miller indices from Euler angle matrix

Compare the indices matrix with the Euler angle matrix.

h nsin sin2

k nsin cos2

l ncosu n cos1cos2 sin1sin2 cos

v n cos1sin2 sin1 cos2 cos w n sin sin1

n, n’ = factors to make integersObj/notation AxisTransformation Matrix EulerAngles Components

24

Euler angles from Miller indices

Inversion ofthe previousrelations:

cos l

h2 k2 l2

cos2 k

h2 k2

sin1 w

u2 v2 w2

h2 k2 l2

h2 k2

Caution: when one uses the inverse trig functions, the range of result is limited to 0°≤cos-1≤180°, or -90°≤sin-1≤90°. Thus it is not possible to access the full 0-360° range of the angles. 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.

25

Euler angles from Orientation Matrix

cos 1 a33 2 tan 1 a13 sin

a23 sin

1 tan 1 a31 sin a32 sin

Notes:The range of inverse cosine (ACOS) is 0-π, which is sufficient for ;from this, sin() can be obtained;The range of inverse tangent is 0-2π, (must use the ATAN2 function) which is required for calculating 1 and 2.

if a33 1, 0, 1 tan 1 a12

a11

2, and 2 1

Corrected -a32 in formula for 1 18th Feb. 05

26

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).

27

Supplementary Slides

• The following slides provide supplementary information.

28

Complete orientations in the Pole Figure

1

1

2

2

Note the loss ofinformationin a diffractionexperiment if each set of poles from a single component cannot be related to one another.

(1,,2) ~ (30°,70°,40°).


29

Complete orientations in the Inverse Pole Figure

Think of yourself as an observer standing on the crystal axes, and measuring where the sample axes lie in relation to the crystal axes.


30

Other Euler angle definitions• A 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.• 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.


31

Matrix with Kocks Angles

sin sin cos cos cos

cos sin sincos cos

cos sin

sincos cos sin cos

coscos sin sin cos

sin sin

cos sin sin sin cos

a(,,) = (hkl)[uvw]

Note: obtain transpose by exchanging and .

32

Matrix with Roe angles

sin sincos cos cos

cos sinsin cos cos

cos sin

sin cos cos sin cos

cos cos sin sin cos

sin sin

cos sin sin sin cos

a(,,) = (hkl)[uvw]

33

Euler Angle Definitions

Bunge and Canova are inverse to one anotherKocks and Roe differ by sign of third angleBunge rotates about x’, Kocks about y’ (2nd angle)


Kocks

34

Conversions


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