Notes on Orientations, Pole Figures, Orientation Distribution

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Notes on Orientations, Pole Figures, Orientation
Distribution

• A. D. Rollett
• Spring 2001

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Relation of PFs to OD

• A pole figure is a projection of the information
  in the orientation distribution.
• Equivalently, can integrate along a line in the
  OD to obtain the intensity in a PF.

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Fundamental Equation of Texture

• We can calculate the intensity in the pole
  figure, P(w), by summing over all the symmetry
  operators and integrating over the range of Euler
  angles (or as a sum if the OD is in discrete
  form).
• wposition in PF h (hkl) S symmetry

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• First rotate the crystal with thth(,q) so as to
  place the pole of interest, h, in the center of
  the hemisphere, i.e. at the North pole. This
  matrix can be obtained by first finding a
  rotation axis, , as the vector product of 001
  and h.

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(hkl)
001

t
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The rotation angle, q, is given by the inverse
cosine of the scalar product Cos(q)h3 From
this axis-angle pair, the rotation matrix, th,
can be constructed as follows.
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• Once the pole has been rotated onto the North
  pole, the crystal can then be rotated freely
  about the pole without affecting the position of
  the pole. This rotation matrix is similar to the
  intermediate rotation step described above for
  constructing the orientation matrix from the
  Euler angles.

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z
(hkl)
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• This is the key step in this development because
  this provides the free parameter over which we
  can integrate in order to find the intensity in
  the pole figure, based on a line of intensity in
  orientation space. A rotation, tf, about the
  North pole is simply tf tf(001,z), where z is
  the (free) rotation angle. Finally, we need to
  rotate the pole onto the position in the pole
  figure that is of interest. This is accomplished
  by successive rotations about 100/RD by c and
  then 001/ND by f.

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Pole Figure angles
TD
f
azimuth
c
declination
(hkl)
RD
ND
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The product of this set of 4 rotations must then
be equal to the orientation matrix, i.e.
g(f1,F,f2) (tftctzth)T tT(h,z,w) where the
matrix on the RHS contains only one free
parameter, i.e. the rotation by z about the pole
h.
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Sample Symmetry

• Add up intensities for locations in a pole figure
  related by the sample symmetry.
• For example, if orthorhombic sample symmetry
  (mmm) then, f,-f,p-f, and pf must give same
  intensity.

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Active versus Passive Rotations

• Passive rotation typical (unstated assumption in
  materials literature) axis transformation.
• Active rotation typical in solid mechanics (also
  unstated assumption) fixed coordinate frame,
  move vectors (tensors) through the space.

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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
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Rotation of vectors in the planev old vector
v new vector
y
v
v
q
x
N.B. Active rotation (in the plane)
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• Can write the following for the intensity, P, at
  w(c,f) in the pole figure.
• This is sometimes termed the fundamental equation
  of texture analysis because, for the standard
  situation in which only pole figure data are
  available, this is the equation that one seeks to
  invert in order to obtain the orientation
  distribution.

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Summary

• Aim of derivation show how a free rotation (as
  an integration parameter) is incorporated into
  the orientation description.
• Sum over variants related by crystal symmetry and
  by sample symmetry required.
• Similar analysis for inverse pole figures.