The Orientation Distribution

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The Orientation Distribution

• A. D. Rollett
• Advanced Characterization Microstructural
  Analysis
• Lecture 2 part A

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High Temperature Superconductors an example
Theoreticalpole figuresfor c? a ?
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YBCO (123) on various substrates
Various epitaxialrelationshipsapparent fromthe
pole figures
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Scan with ?a 0.5, ?b 0.2
Azimuth, b
Tilta
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Dependence of film orientation on deposition
temperature
Ref Heidelbach, F., H.-R. Wenk, R. E.
Muenchausen, R. E. Foltyn, N. Nogar and A. D.
Rollett (1996), Textures of laser ablated thin
films of YBa2Cu3O7-d as a function of
deposition temperature. J. Mater. Res., 7,
549-557.
Impact superconduction occurs in the
c-planetherefore c? epitaxy is highly
advantageous tothe electrical properties of the
film.
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Lecture Objectives

• Introduce the concept of the Orientation
  Distribution
• Explain the connection between Euler angles and
  pole figure representation.
• Present an example of an OD (rolled fcc metal).
• Begin to explain the effect of symmetry.

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Concept of OD

• Orientation Distribution (OD) is a central
  concept in texture analysis and anisotropy.
• Probability distribution in whatever space is
  used to parameterize orientation, i.e. a function
  of three variables, e.g. 3 Euler angles
  f(f1,F,f2). f ? 0 (very important!).
• Probability of finding a given orientation
  (specified by all 3 parameters) is given by f.

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Orientation Distribution Function

• Literature mathematical function is always
  available to describe the (continuous)
  orientation density known as orientation
  distribution function.
• From probability theory, however, remember that,
  strictly speaking, distribution function is
  reserved for the cumulative frequency curve (only
  used for volume fractions in this context).

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Normalization of OD

• If random then OD has the same value everywhere,
  i.e. 1 (since a normalization is required)
• Normalize by integrating over the space of the 3
  parameters (as for pole figures).
• Sin(F) corrects for volume of the element.

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Discrete OD

• Real data is available in discrete form.
• Normalization also required for discrete OD
• Define a cell size (typically ?angle 5) in
  each angle.
• Sum the intensities over all the cells.

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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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(Bunge)Euler Angle Definition
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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!
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Area, Volume Element

• Spherical coordinates result in an area element
  whose magnitude depends on the declinationdA
  sinQdQdyVolume element dAdf sinQdQdydf.

Q
dA
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Complete orientations in the Pole Figure
f1
f2
N.B. loss ofinformationin adiffractionexperime
nt!
F
f1
F
f2
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Complete orientations in the Inverse Pole Figure
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Need for 3 Parameters

• Another way to think about orientation rotation
  through q about an arbitrary axis, n this is
  called the axis-angle description.
• Two numbers required to define the axis, which is
  a unit vector.
• One more number required to define the magnitude
  of the rotation.
• Reminder! Positive rotations are anticlockwise
  counterclockwise!

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Meaning of an OD

• Each point in the orientation distribution
  represents a specific orientation or texture
  component.
• Some (most!) properties depend on the complete
  orientation, therefore must have the OD to
  predict properties.
• Can use the OD information to determine
  presence/absence of components, volume fractions,
  predict properties of polycrystals.

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Grains, Orientations, and the OD

• Given a knowledge of orientations of discrete
  points in a body with volume V, OD given
  byGiven the orientations and volumes of the N
  (discrete) grains in a body, OD given by

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Volume Fractions from Intensity in the OD
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Intensity from Volume Fractions
Objective given information on volume fractions
(e.g. numbers of grains of a given orientation),
how do we calculate the intensity in the OD?
General relationships
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Intensity from Vf, contd.

• For 5x5x5 discretization, particularize to

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Representation of the OD

• Major problem!
• Typical representation Cartesian plot
  (orthogonal axes) of the intensity in Euler
  angle space.
• Unfortunate choice Euler angles are inherently
  spherical (globe analogy).
• Recall the Area/Volume element points near the
  origin are distorted (too large area).

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Cartesian Euler Space
f1
F
f2
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Sections
f2 5
f2 15
f2 0
f2 10
F
f1
f2
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Example of OD in Bunge Euler Space
f1
Section at 15
OD is represented by aseries of sections, i.e.
one(square) box per section. Each section shows
thevariation of the OD intensityfor a fixed
value of the thirdangle. Contour plots
interpolatebetween discrete points.
F
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Example of OD in Bunge Euler Space, contd.
This OD shows the textureof a cold rolled copper
sheet.Most of the intensity isconcentrated
along a fiber. Think of connect the
dots!Technically, this is the beta fiber.
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Numerical lt-gt Graphical
f1
F
f2 45
Example of asingle section
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OD lt-gt Pole Figure
f2 45
f1
F
C Copper
B Brass
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Euler Angle Conventions
Different conventions for Euler angles
developedhistorically based on conventions
adopted by western, eastern math, physics
communities. Bunge (Germany), Roe (US) developed
orientationdistribution simultaneously (late
60s).
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Euler Angle Conventions
Bunge and Canova are inverse to one anotherKocks
and Roe differ by sign of third angleBunge
rotates about x, Kocks about y (2nd angle)
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Texture Components

• Many components have names to aid the memory.
• Specific components in Miller index notation have
  corresponding points in Euler space, i.e. fixed
  values of the three angles.
• Lists of components the Rosetta Stone of texture!

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Miller Index Map
Bunge, p23 et seq.
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45 section
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Miller Index Map, contd.
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(Partial) Fibers in fcc Rolling Textures
C Copper
f1
f2
B Brass
F
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Variants and Symmetry

• An understanding of the role of symmetry is
  essential in texture.
• Two separate and distinct forms of symmetry are
  relevant
• CRYSTAL symmetry
• SAMPLE symmetry
• Typical usage lists crystal-sample, e.g.
  cubic-orthorhombic.

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Stereographic projectionsof symmetry elements
and general poles in thecubic point groupswith
Hermann-Mauguinand Schoenflies
designations. Note the presence of four triad
symmetry elements in all these groups on
lt111gt. Cubic metals mostlyfall under m3m.
Groups topicfor graduate section
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Sample Symmetry
Torsion, shearMonoclinic, 2.
Rolling, plane straincompression, mmm.
Otherwise,triclinic.
Axisymmetric C?
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Symmetry Issues

• Crystal symmetry operates in a frame attached to
  the crystal axes.
• Based on the definition of Euler angles, crystal
  symmetry elements produce relations between the
  second third angles.
• Sample symmetry operates in a frame attached to
  the sample axes.
• Sample symmetry produces relations between the
  first second angles.

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Sample Symmetry Elemente.g. diad on
ND(associated with f2)
Crystal Symmetry Elemente.g. rotation on
001(associated with f2)
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Choice of Section Size

• Quad, Diad symmetry elements are easy to
  incorporate, but Triads are highly inconvenient.
• Four-fold rotation elements (and mirrors in the
  orthorhombic group) are used to limit the third,
  f2, (first, f1) angle range to 0-90.
• Second angle, F, has range 0-90 (diffraction
  adds a center of symmetry).

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Section SizesCrystal - Sample

• Cubic-Orthorhombic 0?f1 ?90, 0?F ?90, 0?f2
  ?90
• Cubic-Monoclinic 0?f1 ?180, 0?F ?90, 0?f2
  ?90 Cubic-Triclinic 0?f1 ?360, 0?F ?90, 0?f2
  ?90
• But, these limits do not delineate a fundamental
  zone.

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Fundamental Zone

• The fundamental zone is that region of
  orientation space that contains one and only one
  physically distinguishable orientation.
• Example standard stereographic triangle
• Conventional choice of angle limits means that
  for textures, three copies of each physically
  distinguishable component are present, as a
  consequence of the triad crystal symmetry
  element.

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Points related by triad symmetryelement on
lt111gt(triclinicsample symmetry)
f2
F
f1
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Section Conventions