Texture Components and Euler Angles: part 1 11th January 05

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Texture Components and Euler Angles part 111th
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 1 will provide a mainly qualitative,
  pictorial approach part 2 will provide more
  mathematical detail

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Miller indices of a pole
Miller indices are a convenient way to represent
a direction or a plane normal in a crystal, based
on integer multiples of the repeat distance
parallel to each axis of the unit cell of the
crystal lattice. This is simple to understand
for cubic systems with equiaxed Cartesian
coordinate systems but is more complicated for
systems with lower crystal symmetry. Directions
are simply defined by the set of multiples of
lattice repeats in each direction. Plane normals
are defined in terms of reciprocal intercepts on
each axis of the unit cell.
When a plane is written with parentheses, (hkl),
this indicates a particular plane normal by
contrast when it is written with curly braces,
hkl, this denotes a the family of planes
related by the crystal symmetry. Similarly a
direction written as uvw with square brackets
indicates a particular direction whereas writing
within angle brackets , ltuvwgt indicates the
family of directions related by the crystal
symmetry.
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The 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.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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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)

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Standard (001) Projection
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Stereographic, Equal Area Projections
Stereographic(Wulff) Projection OPRtan(q/2)
Equal Area(Schmid) ProjectionOPRsin(q/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.
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Pole Figure Example

• If the goniometer is set for 100 reflections,
  then all directions in the sample that are
  parallel to lt100gt directions will exhibit
  diffraction. The example shows a crystal oriented
  to put all 3 lt100gt directions approximately
  equally spaced from the ND.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Cube Component 001lt100gt
100
111
110
Think of the q-2q setting as acting as a filter
on the standard stereographic projection,
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Miller Index Definition of a Crystal Orientation

• We use a set of three orthogonal directions as
  the reference frame. Mathematicians set up 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.

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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.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Sharp Texture (Recrystallization)

• Look at the (001) pole figures for this type of
  texture maxima correspond to 100 poles in the
  standard stereographic projection.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Euler angles of Cube component

• The Euler angles for this component are simple,
  and yet not so simple!
• The crystal axes align exactly with the specimen
  axes, therefore all three angles are exactly
  zero (f1, ?, f2) (0, 0, 0).
• As an introduction to the effects of crystal
  symmetry consider aligning 100//TD,
  010//-RD, 001//ND. This is evidently still
  the cube orientation, but the Euler angles are
  (f1,?,f2) (90,0,0)!

Obj/notation AxisTransformation Matrix
EulerAngles Components
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011lt001gt the Goss Component

• This type of texture is known as Goss Texture and
  occurs as a Recrystallization texture for FCC
  materials such as Brass,
• In this case the (011) plane is oriented towards
  the ND and the 001 inside the (011) plane is
  along the RD.

ND
(110)
TD
100
RD
Obj/notation AxisTransformation Matrix
EulerAngles Components
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011lt001gt 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! (Homework
  draw the (111) pole figure)

ND
(110)
TD
100
RD
110
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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.
• (f1,?,f2) (0,45,0).
• Other variants will be shown when symmetry is
  discussed.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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Brass component

• This type of texture is known as Brass Texture
  and occurs as a rolling texture component for
  materials such as Brass, Silver, and Stainless
  steel.

Obj/notation AxisTransformation Matrix
EulerAngles Components
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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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110lt112gt Brass component

• Think of rotating the Goss component around the
  ND. In this example, the xtal has been rotated
  in both senses (two variants).

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

• Analogy position and the heading of a boat with
  respect to the globe. 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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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
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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 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,
llt0, show that the Euler angles are incorrect.
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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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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.