Lecture for 6 Feb 01

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Lecture for 6 Feb 01

• Symmetry, Rotations

2
Objectives

• To illustrate the effect of symmetry on the Euler
  space required for unique representation of
  orientations.
• To point out the special circumstance of cubic
  crystal symmetry and the presence of 3 equivalent
  points in the 90x90x90 box.
• To describe fibers as they appear in rolled fcc
  bcc metals.

3
Effect of Symmetry

• Illustration

4
Rotations definitions

• Rotational symmetry elements exist whenever you
  can rotate a physical object and result is
  indistinguishable from what you started out with.
• Rotations can be expressed in a simple
  mathematical form as unimodular matrices.
• Rotations are transformations of the first kind
  determinant 1.

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Determinant of a matrix

• Multiply each set of three coefficients taken
  along a diagonal top left to bottom right are
  positive, bottom left to top right negative.
• a a11a22a33a12a23a31a13a21a32-
  a13a22a31-a12a21a33-a11a32a23ei1i2inai11ai22ai
  NN

-

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Nomenclature for rotation elements

• Distinguish about which axis the rotation is
  performed.
• Thus a 2-fold axis about the z-axis is known as a
  z-diad, or C2z, or L0012
• Triad about 111 as a 111-triad, or,
  120-lt111gt, or, L1113 etc.

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Other symmetry operators

• Symmetry operators of the second kind these
  operators include the inversion center and
  mirrors determinant -1.
• The inversion simply reverses any vector so that
  (x,y,z)-gt(-x,-y,-z).
• Mirrors operate through a mirror axis. Thus an
  x-mirror is a mirror in the plane x0 and has the
  effect (x,y,z)-gt(-x,y,z).

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Examples of symmetry operators
Inversion Center(2nd kind)

• Diad on z(1st kind)
• Mirror on x(2nd kind)

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How to use a symmetry operator?

• Convert Miller indices to a matrix
• Perform matrix multiplication with the symmetry
  operator and the orientation matrix.
• Convert the matrix back to Miller indices.
• The two sets of indices represent (for xtal
  symmetry) indistinguishable objects.

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Example

• Goss 110lt001gt
• Pre-multiply by z-diad
• which is
  -1-10lt001gt

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Order of Matrices

• Order depends on whether crystal or sample
  symmetry elements are applied.
• For an operator in the crystal system, Oxtal, the
  operator pre-multiplies the orientation matrix
  (first transform into xtal coordinates, then
  apply crystal symmetry once in crystal
  coordinates).
• For sample operator, Osample, post-multiply.

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Symmetry Relationships

• Note that the result of applying any available
  operator is physically indistinguishable from the
  starting configuration (not mathematically equal
  to!).

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Symmetry and Properties

• For later when you use a material property (of
  a single crystal, for example) to connect two
  physical quantities,then applying symmetry means
  that the result is unchanged. In this case there
  is an equality. This equality allows us to
  decrease the number of independent coefficients
  required to describe an anisotropic property
  (Nye).

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Anisotropy

• Given an orientation distribution, f(g), one can
  write the following for any tensor property or
  quantity, t, where the range of integration is
  over the fundamental zone of physically
  distinguishable orientations, SO(3)/G

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Matrixrepresentation of the rotation point groups
Kocks Ch. 1 Table II
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How many equivalent points?

• Each symmetry operator relates a pair of points
  in orientation (Euler) space.
• Therefore each operator divides the available
  space by a factor of the order of the rotation
  axis. In fact, order of group is significant.
• This suggests that the orientation space is
  smaller than the general space by a factor equal
  to the number of general poles.

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Cubic symmetry

• O(432) has 24 operators (i.e. order24) O(222)
  has 4 operators (i.e. order4) why not divide
  the volume of Euler space (8p2, or,
  360x180x360) by 24x496 to get p2/12 (or,
  90x30x90)?
• Answer we leave out a triad axis, so divide by
  8x432 to get p2/4 (90x90x90).

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Crystallite Orientation Distribution
Sections at constantvalues of the third
angleKocks Ch. 2 fig. 36
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Sample Orientation Distribution
Sections at constantvalues of the first angle
Kocks Ch. 2 fig. 37
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Sample Symmetry Relationships in Euler Space
f10
360
180
270
90
diad
F
mirror
mirror
2-fold screw axis changes f1by p
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Sample symmetry, detail
Tables for Texture Analysis of Cubic Crystals,
Springer Verlag, 1978
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Crystal Symmetry Relationships in Euler Space
3-fold axis
f20
360
180
270
90
F
4-fold axis
mirroracts on f1 also
Note points related by triad (3-fold) have
different f1 values.
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Crystal symmetry detail
24

• Diagram showing the relationship between
  coordinates in square (Cartesian) sections, polar
  sections with Bunge angles, and polar sections
  with Kocks angles.

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How many equivalent points?

• For cubic-orthorhombic crystalsample symmetry,
  we use a range 90x90x90 for the three angles,
  giving a volume of 902 (or p2/4 in radians).
• In the (reduced) space there are 3 equivalent
  points for orientations unrelated to symmetry
  axes.

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Group theory approach

• Crystal symmetrya combination of 4- and 2-fold
  crystal axes (2x48 elements) reduce the range of
  F from p to p/2, and f2 from 2p to p/2.
• Sample symmetrythe 2-fold sample axes (4
  elements in the group) reduce the range of f1
  from 2p to p/2.
• Volume of 0 ? f1, F, f2 ? p/2 is p2/4.

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Example of 3-fold symmetry
The S component,123lt634gt has angles 59, 37,
63also 27,58,18,53,74,34 and occurs in
three related locations in Euler space. 10
scattershown about component. Regions I, II and
III are related by the triad symmetry
element, i.e. 120 about lt111gt.
Engler Randle, fig. 5.7
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Effect of 3-fold axis
section in f1 cuts through more than one subspace
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S component in f2 sections
Regions I, II and III are related by the triad
symmetry element, i.e. 120 about lt111gt.
Engler Randle, fig. 5.7
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Special Points
Copper 2 Brass 3 S 3 Goss 3 Cube 8 Dillamore
2
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Sample Symmetry Relationships in Euler Space
special points
f10
360
180
270
90
diad
F
mirror
mirror
Cube lies on the corners
Copper, Brass, Goss lie on an edge
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Special Points Explanations

• Points coincident with symmetry axes may also
  have equivalent points, often on the edge. Cube
  should be a single point, but each corner is
  equivalent and visible.
• Goss, Brass a single point becomes 3 because it
  is on the f20 plane.
• Copper 2 points because one point remains in the
  interior but another occurs on a face also the
  Dillamore orientation.

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Continuous Intensity Polar Plots
Brass
Copper
S
Goss
COD sections (fixed third angle) for copper cold
rolled to 58 reduction in thickness. Note that
the maximum intensity in each section is well
aligned with the beta fiber (denoted by a ""
symbol in each section).
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Partial Fibers

• Any line of uniform intensity threading
  orientation space can be identified as a fiber.
• Most deformation texture development occurs along
  fibers.
• Caution required in defining fibers location of
  maximum density often not on ideal fiber as
  defined in textbooks.

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The Beta fiber in rolled fcc metals

• The beta fiber in rolled fcc metals is defined as
  the line of high intensity that occurs between
  the Copper component 112lt11-1gt and the Brass
  component 110lt1-12gt.
• Three branches of the beta fiber exist because of
  the triad symmetry element.
• Typical to examine only the 45ltFlt90 branch.

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Fiber Plotsvarious rolling reductions (a)
intensity versus position along the fiber(b)
angular position of intensity maximumversus
position alongthe fiber
Kocks, Ch. 2
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Volume fraction vs. density
Volume fractionassociated with region
aroundthe fiber in a givensection. Vf
increases faster than density withincreasing
F. Location of max. density not at
nominallocation.
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The Alpha fiber in rolled fcc metals

• The alpha fiber in rolled fcc metals is defined
  as the line of high intensity that occurs between
  the Goss component 110lt001gt and the Brass
  component 110lt1-12gt.
• Three branches of the alpha fiber exist because
  of the triad symmetry element.
• Typical to examine only one branch.

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Fibers in rolled bcc metals
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bcc Fibers in Euler space
h
a
g
e
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bcc fibers the f2 45 section
e,lt110gtTD
g, lt111gtND
a, lt110gtRD
Goss
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f1 sections i.e. SOD f2 45 section single
CODsectionshows botha and g
Engler Randle, fig. 5.10
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(No Transcript)
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Fiber Plots bcc metals

• The gamma fiber only requires 30 along the
  length of the fiber because of the effect of the
  3-fold axis, with the 3 mirrors perpendicular to
  the 120-lt111gt symmetry axis.
• The alpha fiber requires the full 90 to
  represent it.
• Various rolling reductions shown.
• Caution intensity maximum not always at the
  ideal angular position.

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100 Pole figure for certain components of
rolled bcc metals
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Tables for Texture Analysis of Cubic Crystals,
Springer Verlag, 1978
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Tables for Texture Analysis of Cubic Crystals,
Springer Verlag, 1978
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Homework

• Copy the hkl2eul.f (Fortran) code onto a
  convenient machine, along with the symmetry
  files, cub.sym and ort.sym.
• Compile the code.
• Make up new symmetry files for triclinic symmetry
  (call it id.sym) and monoclinic symmetry (call it
  mono.sym)
• Explore the effect of symmetry by converting
  various Miller indices to angles.