Supplemental Lecture on the OD

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Supplemental Lecture on the OD

• Oblique Euler angles
• Invariant Measure

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Need for Oblique Euler angles?

• In addition to the distortion of the Euler space
  from plotting in a Cartesian frame, there is a
  singularity near the origin.
• When the second angle is zero, the first and
  third angles are not independent because their
  rotation axes are coincident.
• This leads to the idea of using sum and
  difference angles.

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References

• Bunge (1988). Calculation and representation of
  the complete ODF. ICOTOM-8, Santa Fe, TMS,
  Warrendale, PA.
• Helming et al. Helming, K., S. Matthies, et al.
  (1988). ODF representation by means of sigma
  sections. ICOTOM-8, Santa Fe, TMS, Warrendale, PA.

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Definitions

• Kocks n (Y-f)/2 µ (Yf)/2
• Bunge (f,f-)
• Helming (s,d)
• Convert f s np/2, f- dp/2 µ

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Advantage of Oblique angles?

• Textures close to the origin are not distorted,
  or smeared out over lines in the Euler space.
• Useful for textures with the cube component,
  001lt100gt.
• Also useful to combine two standard Euler angles
  with one oblique angle.
• Example rolled recrystallized Cu, Kocks, Ch.
  2, fig. 30.

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Example of Rolled and Partially Recrystallized
Copper

• Work of Carl Necker (Los Alamos) on
  recrystallization texture and kinetics in copper,
  PhD thesis, Drexel Univ. 1997.
• Sample exhibits weak remanent rolling texture
  with strong cube texture.
• Cube texture is present in all sections of Euler
  space, especially along the edge with F0.
• popLA analysis

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that cube component is confined to corners at 0
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Use of Oblique Angles

• Use oblique angles whenever you encounter
  textures with strong components near F0.
• Also useful when plotting in polar coordinates
  (as opposed to Cartesian).

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Topic no. 2 invariant measure

• In Euler space, as previously stated, the
  invariant measure is dg sinFdFdf1df2
• With a different choice of variables, a different
  invariant measure obtains.

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Axis-angle representation
Axis is written either as aunit vector (3
components,not independent) or as twoangles,
with an angle. g g(n,w) g( ,q) Figure
illustrates the effect of a rotation about an
arbitrary axis, OQ (equivalent to and n)
through an angle a (equivalent to q and w).
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Invariant Measure

• Invariant measure preserves volume element
  constancy.
• In Cartesian coordinates, the volume element is
  dV dr1dr2dr3, so I(r1r2r3)1.
• If we transform to spherical coordinates, dV
  r2sinqdqdrdy, so I(qry)r2sinq
• Contrast with invariant measure for rotations
  in Euler space, dg sinFdFdf1df2 , so I(F,f1,f2)
  sinF

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Invariant Measure, contd.

• If we use a rotation angle, w, and two angles to
  specify the axis, (q, y), we obtain dg 1/p2
  sin2(w/2) sinqdqdwdy
• How to convert from one set of parameters to
  another?

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Use of Jacobian

• Construct the Jacobian for the transformation
  between the two sets of parameters.
• The factor that relates a volume element in one
  system to that in another is then Ia,b,c
  J Ia,b,c

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Axis-Angle to Axis-radius

• Set q q, y y, r tan(w/2).
• dq dq, dy dy, dr dtan(w/2), dw
  2dr/(1r2).
• J 1/2 (cos2(w/2))
• I(q,y,r) I (q,y, w) / J
• I(q,y,r) 1/p2 sin2(w/2) sinq 2cos2(w/2)

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Axis-Angle to Axis-radius, contd.

• I(q,y,r) 1/p2 sin2(w/2) sinq 2cos2(w/2)
  2/p2 sinq tan2(w/2) cos2(w/2)2 2/p2 sinq r2
  /(1r2)2 Since 1/cos2x 1tan2x.

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Axis-radius to Cartesian-radius

• To pass from axis-radius to axis-cartesian, apply
  the relationship J r2 sinq .
• I(r1r2r3) I(q,y,r) / J 2/p sinq r2
  /(1r2)2 / r2 sinq 2/p 1 /(1r2)2

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Cartesian-radius to Cartesian-solid angle

• Solid angle W A/r2 ? d W dA/r2.
• dA r2 sinqdqdy
• Thus dg 2/p 1/(1r2)2 r2 sinqdqdydw 2/p
  1/(1r2)2 r2 dW dr

dA
dA