Texture%20Components%20and%20Euler%20Angles:%20part%202%2013th%20January%2005

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Texture Components and Euler Angles part 213th
January 05

• 27-750
• Spring 2005
• A. D. (Tony) Rollett

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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
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(Bunge)Euler Angle Definition
Obj/notation AxisTransformation Matrix
EulerAngles Components
4
Euler Angles, Ship Analogy

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

Kocks vs. Bunge anglesto be explained later!
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Meaning of Euler angles

• The first two angles, f1 and F, tell you the
  position of the 001 crystal direction relative
  to the specimen axes.
• Think of rotating the crystal about the ND (1st
  angle, f1) then rotate the crystal out of the
  plane (about the 100 axis, F)
• Finally, the 3rd angle (f2) tells you how much to
  rotate the crystal about 001.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Euler Angles, Animated
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Brass component, contd.

• 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).

(100)
(111)
(110)
Obj/notation AxisTransformation Matrix
EulerAngles Components
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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.
• (f1,?,f2) (35,45,0).

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Obj/notation AxisTransformation Matrix
EulerAngles Components
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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 ab 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
  eijk ajbk generates a vector that is
  perpendicular to the first two.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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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.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Rotation of axes in the planex, y old axes
x,y new axes
y
y
v
x
q
x
N.B. Passive Rotation/ Transformation of Axes
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Definition of an Axis Transformatione old
axes e new axes
Sample to Crystal (primed)


e3
e3

e2

e2


e1
e1
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Geometry of hklltuvwgt
Sample to Crystal (primed)


Miller indexnotation oftexture
componentspecifies directioncosines of
xtaldirections tosample axes.
e3
e3 (hkl)
001
010

e2


e2 t

e1 uvw

e1
t hkl x uvw
100
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Form matrix from Miller Indices
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Bunge Euler angles to Matrix
Rotation 1 (f1) rotate axes (anticlockwise)
about the (sample) 3 ND axis Z1. Rotation 2
(F) rotate axes (anticlockwise) about the
(rotated) 1 axis 100 axis X. Rotation 3 (f2)
rotate axes (anticlockwise) about the (crystal)
3 001 axis Z2.
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Bunge Euler angles to Matrix, contd.
AZ2XZ1
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Matrix with Bunge Angles
A Z2XZ1
uvw
(hkl)
Obj/notation AxisTransformation Matrix
EulerAngles Components
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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
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Matrix, Miller Indices

• The columns represent components of three other
  unit vectors

TD
ND?(hkl)
uvw?RD

• Where the Columns are the direction cosines
  (i.e. hkl or uvw) for the RD, TD and Normal
  directions in the crystal coordinate system.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Compare Matrices
uvw
(hkl)
uvw
(hkl)
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Miller indices from Euler angle matrix
Compare the indices matrix with the Euler angle
matrix.
n, n factors to make integers
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Euler angles from Miller indices
Inversion ofthe previousrelations
Caution when one uses the inverse trig
functions, the range of result is limited to
0cos-1q180, or -90sin-1q90. 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,
llt0, show that the Euler angles are incorrect.
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Euler angles from Orientation Matrix
NotesThe range of inverse cosine (ACOS) is 0-p,
which is sufficient for ?from this, sin(?) can
be obtainedThe range of inverse tangent is
0-2p, (must use the ATAN2 function) which is
required for calculating ?1 and ?2.
Corrected -a32 in formula for ?1 18th Feb. 05
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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).

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Supplementary Slides

• The following slides provide supplementary
  information.

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Complete orientations in the Pole Figure
f1
(f1,?,f2) (30,70,40).
Note the loss ofinformationin a
diffractionexperiment if each set of poles from
a single component cannot be related to one
another.
f2
F
f1
F
f2
Obj/notation AxisTransformation Matrix
EulerAngles Components
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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.
Obj/notation AxisTransformation Matrix
EulerAngles Components
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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.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Matrix with Kocks Angles
a(Y,Q,f)
(hkl)
uvw
Note obtain transpose by exchanging f and Y.
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Matrix with Roe angles
(hkl)
uvw
a(y,q,f)
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Euler Angle Definitions
Kocks
Bunge and Canova are inverse to one anotherKocks
and Roe differ by sign of third angleBunge
rotates about x, Kocks about y (2nd angle)
Obj/notation AxisTransformation Matrix
EulerAngles Components
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Conversions
Obj/notation AxisTransformation Matrix
EulerAngles Components