¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

3. Mathematical Representation of Crystal
Orientation, Misorientation
last updated: ADR: 26 th Aug. 09
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3. Mathematical Representation of Crystal Orientation and Misorientation
A. Representation by transformations, second-rank orthogonal tensors
Bi. Miller Index representation
Bii. Orthogonal Matrix Representation
C. Euler-angle representation
D. Axis-angle representation
E. Rodrigues vectors
F. Quaternion representation
Gi. Numerical issues in working with orientations
Gii. Examples of Texture Components
H. The role of crystal and sample symmetries on representations: fundamental
zones
J. Symmetry Operators (Matrices)
K. Groups, Rotation Groups for Symmetry
L. Effect of Symmetry on Orientation, especially in Euler-angle space
Mi. Effect of Symmetry on Misorientations, especially in Rodrigues space
Mii. Effect of Symmetry in Quaternion space (misorientations)
N. Orientation Distributions, Pole Figures and Inverse Pole Figures
O. Projection Geometry
P. Misorientation Distributions
3 Introduction
This chapter focuses mainly on the various different types of mathematical
representation of orientation and misorientation and emphasizes the connection to
physical representation. At its core is the idea of describing an orientation, or texture
component, by a single point in whatever (three-dimensional) space one chooses to
represent orientations. Given a unique, mathematical description of an orientation, one
can then use it to convert tensor quantities that describe (anisotropic) properties, and
fields (stress, strain, current etc.) from a crystal frame of reference to a sample frame and
vice versa. This ability to work with orientations in both mathematical and numerical
form is essential for quantitative predictions of the anisotropy of polycrystalline
materials.
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It is useful to collect together the various types of representation of orientation or
misorientation. In this context there is no difference between orientation and
misorientation! Both these quantities are represented as rotations and all the various
representation methods are useful, depending upon the problem to be solved. Although
Euler angles are conventionally used to describe textures, and Rodrigues vectors are
typically used to describe grain boundary textures, one can use any representation of
rotations for either of these quantities. Historically, Euler angles were favored because of
the availability of series expansions (in terms of generalized spherical harmonics) in this
space that had long ago been devised for the representation of electron orbitals. Recently,
however, Schuh [Mason, J. K. and C. A. Schuh (2008). “Hyperspherical harmonics for
the representation of crystallographic texture”, Acta materialia 56 6141-6155] has
published an account of hyperspherical harmonics that can be used with the quaternion
representation. Of course, the physical entities represented are different: grains and the
boundaries between them (or different phases) are physically different things. It is also
very useful to keep in mind that an Orientation Distribution, for example, is really a
probability distribution because it tells you the probability of finding a grain within a
certain range of orientation. Be careful, however, of the distributions used in materials
science because they are not, strictly speaking probability densities because they are
normalized to have unit average density, regardless of the parameterization of rotations
(as opposed to the integral of the density over the space being set equal to unity).
A very useful visual aid for this chapter is a Tinkertoy set! This allows you to
build a ball and stick model for the reference axes and for the crystal axes and move one
with respect to the other.
The major difference from a mathematical perspective between orientation and
misorientation lies in the symmetries that must be applied to them. A misorientation
always has to take account of the crystal symmetry on either side of the boundary. An
orientation has to take account of the crystal symmetry at one “end” of the rotation, and
the sample symmetry at the other “end” of the rotation. This can be expressed in the
following pair of equations, where S is a symmetry operator, and g is a rotation denoting
either orientation or misorientation. In these expressions, we have introduced a new
notation, ← ⎯ = → , which signifies “is physically indistinguishable from” in order to make
the point that two quantities involving symmetry operators may produce the results that
cannot distinguished from one another in the physical world (though they are not
mathematically identical). The first g is an orientation, and the second is a
misorientation, often written as ∆g, where the “∆” denotes difference in orientation.
g ← ⎯ = → S sample gS crystal
crystal crystal
g ← ⎯ = → SB gSA
The crystal symmetry is fixed by the packing of the atoms that comprise the material and
thus determine the crystal lattice. We shall use only point group symmetry and, in fact,
only the symmetry operators that are proper rotations (with determinant equal to +1).
Note however that distortions of the lattice (e.g. elastic distortions) or the presence of
defects can lower the actual crystal symmetry. Also, the sample symmetry is a
statistically based symmetry and applies only in an average sense to a polycrystal.
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3.A Orthogonal Tensors of Rank 2.
Recall that a transformation of axes can be constructed as a matrix, a, of direction
cosines between two sets of (orthogonal) axes, e, and e’.
a
ij = e i′ • e ˆ j
The equivalent active rotation is obtained as the transpose of the axis transformation,
A = a T . The approach here is exactly that found in standard Mathematics texts, as for
example in chapter 10 of “Mathematical Methods in the Physical Sciences” by M.L.
Boas. Please note that later in the text, orientations will generally be written as “g” which
is exactly the same as “a”, i.e. an axis transformation. More details on the mathematics
used here can be found in Ch. 2. For making diagrams of transformations, we follow the
mathematical convention of directing x to the right (horizontal), y up (vertical) and z
coming out of the plane of the plot in order to have a right-handed reference frame.
Angles are positive when directed anti-clockwise.
The typical choice of axes is to assign an orthogonal set of axes in the specimen to
the unprimed set as a reference frame, and another set of orthogonal axes in the crystal
lattice to the primed set. If the sample has a rectangular shape with a large enough aspect
ratio, then a plane normal can be associated with the face with largest area, and a
direction with the longest dimension. In the case of rolled sheets, these two are the
rolling plane normal (ND, or RPN // axis 3) and the rolling direction (RD // axis 1),
respectively. In the case of geological specimens, the typical choice is “up” (sample z
axis), “North” (sample x axis) and “East” (sample –y axis: logically, one should use
“west” for positive y). For thin films, there is an obvious choice of the film plane normal
as the third axis but the first and second axes are arbitrary unless the substrate happens to
be crystalline in which case it is sensible to fix the reference axes with respect to
directions in the substrate. In the crystal, the standard choice of axes is to fix them on the
edges of the unit cell. However, in the case of crystals with non-orthogonal axes,
additional work must be done to relate the axes of the crystallographic unit cell to the
Cartesian set used to describe orientation. We will confine this discussion to cubic
systems for simplicity. Thus we have:
e 1 // RD
e 2 // TD
e 3 // ND
e 1’ // [100]
e 2’ // [010]
e 3’ // [001]
Note that successive rotations (in either the active or the passive sense) can be
combined in the matrix representation by performing matrix multiplications. If a 1 = a 2•a 3,
then the operation of combination is simply the first rotation, a 3, pre-multiplied by the
second one, a 2.
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3.B.1 Miller index representation of orientation.
Miller indices can be used to represent an orientation and this is in fact the
standard method in the materials literature to describe particular (“ideal”) orientations
(“texture components”). The representation takes two forms, one for a full description of
orientation (sometimes referred to as “bi-axial alignment” in the thin film literature), and
the second for an implied fiber texture. In the first form, a sample is assumed to have a
rectangular shape with large enough aspect ratio that a place normal can be associated
with the face with largest area, and a direction with the longest dimension. In the case of
rolled sheets, these two are the rolling plane normal (ND, or RPN) and the rolling
direction (RD), respectively.
To describe an orientation, the convention is to specify the crystallographic plane
normal that is parallel to the specimen normal (e.g. the ND) and a crystallographic
direction that is parallel to the long direction (e.g. the RD). Looking ahead, this is akin to
working in the space of an inverse pole figure, in that we specify which crystal direction
can be found when we look along a fixed sample direction.
(hkl) // ND, [uvw] // RD, or (hkl)[uvw]
In effect, one is writing down the direction cosines † of the specimen’s third and first axes
with respect to the crystallographic axes. Think about this carefully because, although
one has used a crystallographic description, the quantities described are associated with
the reference frame fixed in the specimen.
† See Ch. 2 for definitions of quantities such as direction cosines
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Fig. 3.B.1. Diagram showing the relationship between a pair of reference frames. This is
particularized such that the unprimed set is aligned with the sample, and the primed set of
axes is aligned with the crystal. Note that crystals with non-orthogonal axes will require
an additional relationship between the Cartesian frame fixed in the crystal and the actual
crystallographic axes.
Now we can convert this to an orthogonal matrix representation (axis transformation).
Write n for the plane, b for the direction (unit vectors) and form a third direction as the
cross-product of the two, taking care to use the right-hand rule to obtain a right-handed
set of vectors, ˆ t = ˆ n × ˆ b
n ˆ × ˆ
. Then we form the matrix, a, that describes an axis
b
transformation from sample axes to crystal axes as follows, where the columns are the
components of (unit) vectors b, t, and n, respectively.
Sample
⎛
b t n
⎞ ⎛ b t n ⎞
⎜ 1 1 1⎟
1 1 1
a = ⎜ b t n
2 2 2
⎟ ≡ Crystal⎜
b t n ⎟
2 2 2
⎜
⎝
b t n ⎟ ⎜ ⎟
3 3 3⎠
⎝ b t n
3 3 3⎠
(3B.1.1)
It should be evident that the Miller index form can be recovered from a matrix
description by copying the first and third columns and scaling them to integers. Each i
€
th
row of the matrix represents the coefficients of the corresponding crystal axis expressed
in terms of the sample coordinate system: for example, the first row gives the crystal
[100] direction in terms of sample coordinates. From the notation used above, it is a little
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more obvious that each column provides the equivalent in terms of expressing a sample
direction with coefficients based on the crystal frame of reference.
It is very important to remember that this definition of an orientation should be
interpreted as a transformation of axes (passive rotation) that converts tensor quantities
from the sample reference frame to the crystal frame. This means, for example, that
quantities known in the crystal frame, such as anisotropic elastic moduli, must be
converted to the sample frame with the inverse transformation. For the matrix
description provided above, the inverse is simply the transpose of the orientation matrix.
Be aware that no simple method exists of combining rotations when expressed in Miller
index form. Moreover, Miller index notation does not provide a continuous description
of rotations and is therefore not suitable for numerical descriptions of orientation.
Similarly axis-angle notation can provide a continuous description if the axis is described
as a unit vector with three real numbers, but not if the axis is described in terms of integer
Miller indices (with a finite number of digits).
3.B.2 Number of variables needed for an orientation.
We only need 3 variables to describe an orientation. This is hard to see in a 3x3
matrix with nine entries, though counting up the various constraints on the coefficients of
the matrix does provide the answer (orthogonality gives one set of 3 in terms of the other
6, i.e. reduces the number from 9 to 6; then each set of three direction cosines is
normalized, which reduces the number from 6 to 3). In fact, a matrix that describes a
rotation is termed an orthogonal matrix. It is much easier, however, to see the
requirement in terms of rotation angles and, specifically, Euler angles. Later on we will
show how axis-angle pairs, Rodrigues vectors, and quaternions are related to these
descriptions. An orthogonal matrix, a, is one for which one can write the following set of
six equations:
3.C Euler angles
2
2 ∑ aij =1, aij i
j
∑ =1 (3B.2.1)
Euler angles are the € most commonly used basis for representation of textures. They were
adopted early on in the development of texture analysis and it is convenient to develop a
familiarity with the locations of texture components of special interest for interpreting
experimental and simulation results. The reason for adopting them was that there was
already a large body of work on the physics of electrons and atoms that uses generalized
spherical harmonics. These (orthonormal) functions are most conveniently expressed in
terms of Euler angles. One very confusing point, however, is that different communities
have adopted different conventions for the definition of Euler angles (Bunge, Roe,
Canova, Kocks, etc.).
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3.C.1 Bunge Euler angles
We will use the dominant Bunge convention here, which turns out to be the same that
you can find in the Wikipedia pages on Euler angles. The Euler angles are illustrated by
starting with the crystal and sample frames in coincidence. The transformations by
rotation then take the crystal frame away from the sample frame. Some cautionary
remarks must be made here.
Caution 1: there are several different definitions, or conventions for Euler angles in
current use: see below for more detail.
Caution 2: most of us first encounter angles as a measure of separation of lines about a
point of intersection, i.e. as a difference in inclination: it is critically important to
remember (in the context of texture) that angles associated with rotations/transformations
have a sense or sign associated with them and the (mathematical) convention is that
anticlockwise ≡ positive). As before, note that these are passive rotations for the
transformation of axes, not active/rigid body/vector rotations (as commonly used in solid
mechanics).
Start with the Bunge convention for Euler angles. The three transformations (rotations)
are as follows.
Rotation 1 (φ 1): rotate axes (anticlock) about the (sample) 3 [ND] axis; Z1.
Rotation 2 (Φ): rotate axes (anticlock) about the (rotated) 1 axis [100] axis; X.
Rotation 3 (φ 2): rotate axes (anticlock) about the (crystal) 3 [001] axis; Z2.
⎛ cosφ1 sinφ1 0⎞
⎛ 1 0 0 ⎞
⎜
⎟ ⎜
⎟
Z1 = ⎜ −sinφ1 cosφ1 0⎟
, X = ⎜ 0 cosΦ sinΦ ⎟
⎜
⎟ ⎜
⎟
⎝ 0 0 1⎠
⎝ 0 −sin Φ cosΦ⎠
, Z2 =
⎛ cosφ2 sinφ2 0⎞
⎜
⎟
⎜ −sinφ2 cosφ2 0⎟
⎜
⎟
⎝ 0 0 1⎠
These three transformations (rotations) are composed by matrix multiplication to give a
single transformation matrix. The resulting transformation (rotation) matrix in the
following form, derived from a = Z2XZ1:
⎛ cosϕ1 cosϕ 2 − sin ϕ1sin ϕ2 cosΦ
a(φ1,Φ,φ2) =
sinϕ1 cosϕ 2 + cosϕ1sinϕ 2 cosΦ sin ϕ2 sinΦ ⎞
⎜
−cosϕ1 sinϕ 2 −sin ϕ1 cosϕ 2 cosΦ
⎝ sinϕ1 sinΦ
−sinϕ1 sinϕ 2 + cosϕ1 cosϕ2 cosΦ
−cosϕ1 sinΦ
cosϕ 2 sinΦ ⎟
cosΦ ⎠
The reader should verify that this matrix is indeed the set of direction cosines (rows) of
each crystal (new) axis in terms of the sample (old) axes (columns). Caution: the Z-axis
is special for both sample and crystal axes. All variants of the Euler angles share this
(arbitrary) feature. It does not mean that other choices of transformations (rotation) are
not possible. The following figure is reproduced from Bunge’s book on Texture
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Analysis. He labels the crystal axes as K A and the crystal axes as K B; the specimen axes
are X, Y & Z, and the crystal axes are X’, Y’ & Z’.
3.C.2 Correspondence between Matrices
Given the orientation matrix derived from Euler angles and the matrix derived from
direction cosines, one can immediately see how to convert between the various
descriptions because each pair of corresponding entries must be identical to each other.
That is to say, b 1 in the first matrix = cosφ 1 cosφ 2− sinφ 1 sinφ 2 cosΦ, etc.
b1 Sample
t1 n1 ⎛ ⎞
⎜ ⎟
aij = Crystal
⎜
b2 t2 n2 ⎟
⎜ ⎟
⎝ b3 t3 n3 ⎠
⎛ cosϕ1 cosϕ 2 − sinϕ1 sinϕ 2 cosΦ sinϕ1 cosϕ 2 + cosϕ1 sinϕ 2 cosΦ
⎜
≡
⎜
−cosϕ1 sinϕ 2 − sinϕ1 cosϕ 2 cosΦ −sinϕ1 sinϕ 2 + cosϕ1 cosϕ 2 cosΦ
⎜
€
⎝ sinϕ1 sinΦ −cosϕ1 sinΦ
sinϕ 2 sinΦ⎞
⎟
cosϕ 2 sinΦ
⎟
cosΦ ⎠
This permit Euler angles to be converted to Miller indices by extracting the first and third
€ columns of the orientation matrix, and re-scaling each (unit) vector to have suitable
integer values.
More conversions are given below.
3.C.3 Other definitions € of Euler angles
h = n sin Φ sinϕ 2
k = n sin Φ cosϕ 2
l = n cosΦ
( )
u = n ′ cosϕ1 cosϕ 2 − sinϕ1 sinϕ 2 cosΦ
v = n ′ ( − cosϕ1 sinϕ 2 − sinϕ1 cosϕ 2 cosΦ)
w = n ′ sin Φ sinϕ1 The other conventions are those of Roe [Roe, R. J. (1965). “Description of
Crystallite Orientation in Polycrystalline Materials. 3. General Solution to Pole Figure
Inversion”, Journal of Applied Physics 36 2024], who developed an analysis in parallel
with Bunge), Matthies and Kocks. The main difference between the Bunge convention
and the others is that the second rotation (Θ above) is about the (rotated) X-axis, whereas
in all the other conventions it is about the (rotated) Y-axis. From the (Russian) physics
literature, Borisenko and Tarapov (Vector and Tensor Analysis, Dover) call them the
precession, nutation and pure rotation angles, respectively, and follow the Bunge
convention. Other physics literature follows the Roe convention; it would be interesting
to know how the west and east came to take different approaches to Euler angles!
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Figure 3.C.1 Diagrams showing the successive positions of the crystal axes after
transformations by each Euler angle. The first diagram (a) shows the two frames in
coincidence, followed by the first rotation, φ1, to give (b), followed by the second
rotation, Φ, to give (c) and finally the third rotation, φ2, to give the final position (d). The
reference frame is labeled as “KA” in the figure, and the crystal frame as “KB”. Each
diagram shows successive rotations, more properly thought of as transformations of axes.
After Bunge [1982, Texture Analysis in Materials Science, Butterworths.]
The range of the Euler angles is 0 ≤ φ 1 ≤ 360°, 0 ≤ Φ ≤ 180°, and 0 ≤ φ 2 ≤ 360°. This
allows any arbitrary proper rotation to be described in terms of Euler angles. As we shall
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see below, the application of sample and crystal symmetry permits a reduction in the
range of the angles that are required.
3.C.2 Kocks and Roe Euler angles
In Kocks, or “symmetric” angles [Kocks, U. F. (1988), “A symmetric set of Euler angles
and oblique orientation space”, Eighth International Conference on Textures of Materials
(ICOTOM-8), Santa Fe, New Mexico, USA, TMS, Warrendale, Pennsylvania, pp 31-36],
the transformation matrix is as follows. The term symmetric refers to the fact that in this
convention, the first rotation is positive (anticlockwise) with respect to the local frame,
no matter whether one starts with the sample or the crystal frame. This point will become
more apparent when we consider graphical representation of orientations.
⎛ −sin Ψsinφ − cosΨcosφ cosθ cos Ψsinφ − sinΨcosφ cosθ cosφ sinθ ⎞
a(Ψ,θ,φ) = ⎜
sinΨ cosφ − cosΨsinφ cosθ
⎝ cos Ψsinθ
−cos Ψcosφ − sinΨsinφ cosθ
sin Ψsinθ
sinφ sinθ ⎟
cosθ ⎠
(3.C.2.1).
In Roe angles, the transformation matrix is as follows.
a(ψ,θ,φ) =
⎛ −sinψ sinφ + cosψ cosφ cosθ cosψ sinφ + sinψ cosφ cosθ −cosφ sinθ⎞
⎜
−sinψ cosφ − cosψ sinφ cosθ
⎝ cosψ sinθ
cosψ cosφ − sinψ sinφ cosθ
sinψ sinθ
sinφ sinθ ⎟
cosθ ⎠
(3.C.2.2).
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Fig. 3.C.2. Kocks Euler angles illustrated by analogy to the position and the heading of a
boat with respect to the globe. Latitude (Θ) and longitude (ψ) describe the position of the
boat; third angle describes the heading (φ) of the boat relative to the line of longitude that
connects the boat to the North Pole.
The following table describes the different conventions for Euler angles and the
way in which they area related.
Table 3.1. Definitions of Euler Angles and Conversions
Convention 1st 2nd 3rd 2nd angle
about axis:
Kocks
(symmetric)
Ψ Θ φ y
Bunge
Matthies
φ1-π/2 α
Φ
β
π/2−φ2 π−γ
x
y
Roe Ψ Θ π−Φ y
Fig. 3.C.3. Diagrams illustrating the differences between Kocks, Roe, Bunge and Canova
Euler angle definitions. The first row, (a), shows pole figures (crystal z axis in relation to
sample axes) and the second row, (b), shows inverse pole figures (sample Z in relation to
crystal axes). Bunge and Canova are inverse to one another
Kocks and Roe differ by sign of third angle
Bunge rotates about x’, Roe/Kocks about y’ (2nd angle rotation).
An important aspect of Euler angle representation is that when the second angle is
exactly zero, then the other two angles can be combined together, i.e. there is a
singularity near the origin of the space. This leads to the idea that there may be
circumstances under which it is helpful to use the sum and difference of the first and third
angles. These were introduced by Bunge [(1988). Calculation and representation of the
complete ODF. ICOTOM-8, Santa Fe, TMS, Warrendale, PA.] and Helming et al.
[Helming, K., S. Matthies, et al. (1988). ODF representation by means of sigma sections.
8/27/09 11

ICOTOM-8, Santa Fe, TMS, Warrendale, PA.] to address this issue. The first and third
angles can be replaced by the following, where ν is equivalent to a parameter β used
previously by Williams.
ν = (Ψ-φ)/2
µ = (Ψ+φ)/2 (3.C.2.3)
These are related to the oblique section parameters of Bunge (φ + ,φ - ) and Helming (σ,δ) by
φ + = σ = ν+π/2, and φ - = δ+π/2 = µ.
Note that when two rotations are expressed in the form of Euler angles, there is no simple
method for combining them. It is usually sensible to convert them to matrix form before
performing manipulations on them. The action of symmetry operators on orientations
can, however, be expressed in reasonably simple terms.
3.C.3 Conversions
To convert from Miller index description to Euler angles, one may use these formulae.
ϕ 1 = sin −1
Φ = cos −1 l
h 2 + k 2 + l 2
⎛
⎜
⎝
⎞
⎟
⎠
ϕ2 = cos −1 k
h 2 + k 2
⎛
⎜
⎝
⎞
⎟
⎠
w
u 2 + v 2 + w 2
h 2 + k 2 + l 2
h 2 + k 2
⎛
⎜
⎝
⎞
⎟
⎠
(3.C.3.1)
A limitation of these formulae from a numerical perspective is the range of inverse
trigonometric functions. The only such function that can cover the range 0-2π is the
inverse tangent € function with two arguments, generally written as ATAN2 (lower case
for C programs). Considerable care with this inverse trigonometric function is required
because in some software packages such as Excel, the two arguments are ordered as
{x,y}, whereas in C and Fortran, they are ordered in the opposite fashion as {y,x}. Also,
if both arguments approach zero, the result is undefined and the function will return an
error. This means that the programmer has to detect this condition and take special
measures, which can easily happen for orientations close to the reference position (cube
component). Therefore it is better to write the orientation (rotation) matrix in terms of
the Miller indices, and then convert from matrix to Euler angles, as given here:
€
( )
Φ = cos −1 a33 ϕ 2 = tan −1⎛ ( a13 ⎜
⎝
sin Φ)
( a23
⎞
sin Φ
⎟
) ⎠
ϕ1 = tan −1⎛ ( a31 sin Φ)
⎜
⎝ ( −a32
⎞
sin Φ
⎟
) ⎠
(3.C.3.2)
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If the second angle is near to zero or to 180°, then these formulae become inaccurate (or
unusable when it is exactly equal to zero). In this circumstance, the following formulae
may be used.
if a 33 ≈1, Φ = 0, ϕ 1 =
tan −1 a ⎛
⎜ 12
⎝
2
about what € values to assign to the first and third angles.
a 11
⎞
⎟
⎠
, and ϕ2 = ϕ1 (3.C.3.3)
There is one last special case, which is for orientations (symmetrically related to the exact
cube) where a11 and a12 are both close to zero. In this an arbitrary decision must be made
3.D Axis-angle pair representation
Another very useful description of rotation is in terms of a single rotation axis and an
associated rotation angle where the location of the axis is arbitrary (as opposed to axes
that are fixed on one of the reference axes). This is not used for orientation (texture)
because it reveals little of interest about the nature of the orientation. It is much used for
grain boundaries, however, because the magnitude of the rotation is the primary means of
distinguishing low angle boundaries from high angle boundaries. Note that this method
still only requires three independent parameters because the axis represents two
parameters (three direction cosines constrained to be a unit vector) and the angle is a third
parameter. This is commonly written as ( r ˆ ,θ ) or as (n,ω). The following figure
illustrates the effect of a rotation about an arbitrary axis, OQ (equivalent to r
ˆ and n)
through an angle α (equivalent to θ and ω).
Figure 3.D.2 Showing rotation about an arbitrary axis OQ through α. After Spencer
[Spencer, A. J. M. (1980). Continuum Mechanics. New York, Longman.]
8/27/09 13

The resulting rotation can be converted to a matrix (passive rotation) by the following
expression, where δ is the Kronecker delta and ε is the permutation tensor.
a ij = δ ij cosθ + r i r j 1− cosθ
( ) + ε ijk r k sinθ
k =1,3
Note that the active rotation is obtained by changing the sign of the third term on the
RHS in order to obtain the transpose. The axis-angle pair can be recovered from a matrix
by finding the eigenvector associated with the only real eigenvalue of the matrix, and
examination of the trace of the matrix. The quantity norm is equal to 2sinθ. Note that
from a numerical perspective, the cosine formula for the angle is more accurate at large
angles, whereas the sine formula is more accurate for small rotation angles. Also, when
working with the inverse cosine function in computer programs, it is good practice to
check the magnitude of the argument and force it to lie between -1 and +1.
3.E Rodrigues vectors
norm = [ a(2,3) − a(3,2) ] 2 + [ a(1,3) − a(3,1) ] 2 + [ a(1,2) − a(2,1) ] 2
( r1 ,r2 ,r3 ) = ( a(2,3) − a(3,2),a(1,3) − a(3,1),a(1,2) − a(2,1) )/ norm
θ = cos −1 tr[ a]
−1
2
Another choice in more general use than Euler angles is that of Rodrigues vectors,
as popularized by Frank [Frank, F. (1988). “Orientation mapping.” Metallurgical
Transactions 19A: 403-408.], hence the term Rodrigues-Frank space for the set of
vectors. It has turned out to be quite useful for representation of misorientations for the
reason that many of the boundary types that correspond to a high fraction of coincident
lattice sites (i.e. low sigma values in the CSL model) occur on the edges of the Rodrigues
space and their coefficients are reciprocals of integer values. Some authors [Neumann, P.
(1991). “Representation of orientations of symmetrical objects by Rodrigues vectors”,
Textures and Microstructures 14-18 53-58] have pointed out that it has advantages for
texture representation also. Any set of points related by a common axis lie on a straight
line in Rodrigues space, for example, which has advantages for numerical manipulation
of orientation data. As a consequence of its use in misorientation representation, the
rotation matrix is denoted by ∆a, where the ∆ denotes a difference in orientation, i.e.
misorientation.
This representation can be thought of as an axis-angle representation in which the
direction of the R-F vector is parallel to the rotation axis and the magnitude of the vector
is the tangent † of the semi-angle (θ/2). The convenient features of R-F space include the
† Note that this scaling of the vector corresponds to the stereographic projection used in pole
figure representation.
∑
8/27/09 14

esult that all vectors corresponding to the same rotation axis (i.e. a zone axis) fall on a
straight line that passes through the origin. The formulae for conversion are as follows.
The Rodrigues vector, ρ, is defined as follows, where r ˆ and θ are the rotation axis (unit
vector) and angle as described above:
3.E.2. Conversions
€
ρ = tan(θ/2) r
ˆ
(3.E.1)
To convert from [Bunge] Euler angles to a Rodrigues vector, one may apply these
formulae:
θ = 2tan −1
cos( Φ/2)cos
2 ( ( [ ϕ1 + ϕ2] /2)
−1)
ρ1 = tan( θ 2)sin(
[ ϕ1 −ϕ 2]
/2)
/cos( [ ϕ1 + ϕ2] /2)
. (3.E.2.1)
ρ2 = tan( θ 2)cos(
[ ϕ1 −ϕ 2]
/2)
/cos( [ ϕ1 + ϕ2] /2)
ρ3 = tan [ ϕ1 + ϕ2] /2
( )
A simpler set of conversion formulae is due to Morawiec:
[ ]
⎛ ρ ⎞ 1
⎜ ⎟
⎜
ρ2 ⎟
⎜ ⎟
⎝ ρ3 ⎠
=
⎡ (a23 − a32 )/ 1+ tr(a) ⎤
⎢
⎥
⎢ (a31− a13 )/ [ 1+ tr(a) ] ⎥ , (3.E.2.2)
⎣ ⎢ (a12 − a21 )/ [ 1+ tr(a) ] ⎦ ⎥
where tr(a) denotes the trace of the matrix a, as usual.
Note that all these € quantities could be re-expressed in terms of the sum and difference of
the first and third Euler angles. This has an interesting connection to the singularity that
exists at the origin of Euler space but which can be eliminated by use of the sum and
difference angles: see the discussion in Kocks, Wenk and Tomé and also above. One
advantage of Rodrigues space is that this singularity is eliminated, as is evident from the
way in which (the functions of) the Euler angles are combined. Another convenient
feature is that misorientations that share a common rotation axis lie on a straight line in
Rodrigues space; thus all rotations that share a axis lie along the first axis of the
space.
3.E.3 Combination of Rodrigues vectors
Two Rodrigues vectors combine to form a third as follows where ρ2 follows after ρ1.
(ρ1, ρ2) = {ρ1 + ρ2 - ρ1 x ρ2}/{1 - ρ1•ρ2} (3.E.3.1)
8/27/09 15

3.F Quaternions
A close cousin to the Rodrigues vector is the quaternion. An excellent description of
quaternions, their history and properties can be found in Altmann’s book. They have also
been much used in robotics for describing rotations. It is defined as a four component
vector in relation to the axis-angle representation as follows, where [uvw] are the
components of the unit vector representing the rotation axis, and θ is the rotation angle.
q = q(q1,q2,q3,q4) = q( u sinθ/2, v sinθ/2, w sinθ/2, cosθ/2) (3.F.1)
Note that many authors put the fourth component in the first position, i.e.
q = ( cosθ/2, u. sinθ/2, v sinθ/2, w sinθ/2). This set of components was obtained by
Rodrigues prior to Hamilton’s invention of quaternions and their algebra. Some authors
refer to the Euler-Rodrigues parameters for rotations in the notation (λ,Λ) where λ is
equivalent to q4 and Λ is equivalent to the vector (q1,q2,q3). The particular form of the
quaternion that we are interested in has a unit norm (√{q1 2 +q2 2 +q3 2 + q4 2 }=1) but
quaternions in general may have arbitrary “length”. Yet another notation writes the unit
quaternion q is an ordered set of four real numbers of the following form,
3
2
q = (q0 ;q) = (q0 ;q1 ;q2 ;q3 ), satisfying ∑ qi =1. (3.F.2)
The quaternion q = { q0
, −q
} is associated with the inverse rotation,
€
3.F.1 Conversions
i= 0
−1
g .
If one has the rotation in (orthogonal) matrix form (g or a), it can be converted to a
quaternion via the following formulae.
ε ijk Δg jk
qi = ±
4 1+ tr Δg
⎛ q ⎞ ⎡
1 sinθ
⎜ ⎟ 2
[a(2,3) − a(3,2)]/ norm
⎤ ⎡
⎢
⎥ ±[Δg(2,3) − Δg(3,2)]/2 1+ tr( Δg)
⎤
⎢
⎥
⎜ q2⎟ ⎢ sinθ
⎜ ⎟ =
2
[a(1,3) − a(3,1)]/ norm ⎥
⎢
⎥
⎢ ±[Δg(3,1) − Δg(1,3)]/2 1+ tr( Δg)
⎥
⎜ q €
=
3⎟
⎢ sinθ
⎜
⎝ q
⎟
2
[a(1,2) − a(2,1)]/ norm ⎥ ⎢
⎥ (3.F.3)
±[Δg(1,2) − Δg(2,1)]/2 1+ tr Δg
⎢
⎥ ⎢
( ) ⎥
4⎠
⎣ ⎢ cosθ
2 ⎦ ⎥
⎣
⎢ ± 1+ tr( Δg)
/2 ⎦
⎥
( ) (3.F.3)
norm = [ a(2,3) − a(3,2) ]
€
2 + [ a(1,3) − a(3,1) ] 2 + [ a(1,2) − a(2,1) ] 2 (3.F.4)
cos θ 2 = 1 2
1+ tr( a)
(3.F.5)
8/27/09 16

sinθ 2 = 1 2
3− tr a ( ) (3.F.6)
For representing rotations, only quaternions of unit length are considered. Among many
€
other attractive properties, they offer the most efficient way known for performing
computations on combining rotations. This is because of the small number of floating
point operations required to compute the product of two rotations. The algebraic form is
given as, where qB follows qA:
qC = qA • qB
qC1 = qA1 qB4 + qA4 qB1 - qA2 qB3 + qA3 qB2
qC2 = qA2 qB4 + qA4 qB2 - qA3 qB1 + qA1 qB3
qC3 = qA3 qB4 + qA4 qB3 - qA1 qB2 + qA2 qB1
qC4 = qA4 qB4 - qA1 qB1 - qA2 qB2 - qA3 qB3 (3.F.6A)
The alternative description of quaternion composition,
where q0 is the cosine term (q4 in this text).
€
q A ⋅ q B , is given by the following,
q
€
A ⋅ q B A B A B A B B A A B
= {q0 q0 − q ⋅ q , q0 q + q0 q + q ⋅ q }. (3.F.6B)
Also, to get a misorientation in the local frame (defined below),
q C = q A • q -1
B
qC1 = qA1 qB4 - qA4 qB1 + qA2 qB3 - qA3 qB2
qC2 = qA2 qB4 - qA4 qB2 + qA3 qB1 - qA1 qB3
qC3 = qA3 qB4 - qA4 qB3 + qA1 qB2 - qA2 qB1
qC4 = qA4 qB4 + qA1 qB1 + qA2 qB2 + qA3 qB3 (3.F.6C)
It is also useful to be aware of the formulation in terms of Euler-Rodrigues parameters,
which is essentially the same as 3.F.6B above.
3.F.2 Positive vs. Negative Rotations
λ C = λ Bλ A - Λ B.Λ A
Λ C = λ AΛ B + λ BΛ A - Λ A x Λ B (3.F.7)
One curious feature of quaternions that is not obvious from the definition is that they
allow positive and negative rotations to be distinguished. This is more commonly
described in terms of requiring a rotation of 4π to retrieve the same quaternion as you
started out with but for visualization, it is more helpful to think in terms of a difference in
the sign of rotation. Let’s start with considering a rotation of θ about an arbitrary axis, r.
8/27/09 17

From the point of view of the result one obtains the same thing if one rotates backwards
by the complementary angle, θ-2π (also about r). Expressed in terms of quaternions,
however, the representation is different! Setting r = [u,v,w] again,
q(r,θ) = q( u sinθ/2 , v sinθ/2 , w sinθ/2 , cosθ/2 )
q(r,θ−2π) = q( u sin(θ−2π)/2 , v sin(θ−2π)/2 , w sin(θ−2π)/2 , cos(θ−2π)/2 )
q(r,θ−2π) = q(-u.sinθ/2,-v.sinθ/2,-w.sinθ/2,-cosθ/2)= -q(r,θ) (3.F.7)
The result, then, is that the quaternion representing the negative rotation is the negative of
the original (positive) rotation. This has some significance for treating dynamic problems
and rotation: angular momentum, for example, depends on the sense of rotation. For
static rotations, however, the positive and negative quaternions are equivalent or, more to
the point, physically indistinguishable, q ← ⎯ = → -q. This equivalence will recur in
discussions of symmetry operators.
3.F.2 Rotation of Vectors with Quaternions
-
[Figure from Altmann: Rotations, Quaternions and Double Groups]
The active rotation of a vector from X to x is given by:
8/27/09 18

xi = (q4 2 -q1 2 -q2 2 -q3 2 )Xi + 2qiΣjqjXj - 2q4ΣjXjΣkεijkqk (3.F.8)
3.F.3 Conversion of Quaternions to Matrix
The conversion of a quaternion to a rotation matrix is given by:
aij = (q4 2 -q1 2 -q2 2 -q3 2 )δij + 2qiqj + 2q4Σk=1,3εijkqk (3.F.9)
Note the similarity to the above formula.
Considering two quaternions, q and
given by,
€
k
q on the unit sphere, the Euclidean distance is
k
k −1
2
2 1
q − q = I − q q = 4sin
ω k .
4
The square of the distance between two orientations is related to scalar product of the
k
k
quaternion, q q + q ⋅ q ,
0
0
2 2 1
dk = 4 sin
4 ωk = 2(1− cos 1
2 ωk ) = 2[1− (q0q k k
0 + q ⋅ q )].
3.G.1 Computational Effort to Combine Rotations
(3.F.10)
(3.F.11)
This table makes € it immediately obvious that the number of operations required to form
the product of two rotations represented by quaternions is 16 multiplies and 12 additions,
with no divisions or transcendental functions. This should be compared with matrix
multiplication which requires 3 multiplications and 2 additions for each of nine
components, for a total of 27 multiplies and 18 additions. For the Rodrigues vector, the
product of two rotations requires 3 additions, 6 multiplies & 3 additions (cross product),
3 multiplies & 3 additions, and one division, for a total of 10 multiplies and 9 additions.
Thus we see that calculating the product of two rotations (or composing two rotations as
one, as Morawiec puts it) requires the least work with Rodrigues vectors. All these
estimates assume that no loops are used, which require additional additions for the loop
counters. This is not the whole story concerning computational efficiency, however, as
we see next.
3.G.2 Computational Effort to Determine Minimum Rotation Angle
Looking ahead, it interesting to note that finding the minimum misorientation angle or
disorientation angle is simpler in quaternions than with either matrices or Rodrigues
vectors. Since the fourth component is directly related to the magnitude of the angle (and
8/27/09 19

independent of the axis), one merely needs to sort the components into the order required
by the fundamental zone and then choose the particular version that maximizes the fourth
component. The details are given in Sutton & Balluffi on pages 19-20 [Sutton, A. P. and
R. W. Balluffi (1995), Interfaces in Crystalline Materials, Oxford, UK, Clarendon Press].
If Rodrigues vectors are use, then the magnitude of each vector that represents a
physically equivalent rotation must be calculated, which requires 3 multiplies and 2
additions. If matrices are used, the trace is sufficient to determine the rotation angle, i.e.
the largest trace corresponds to the smallest angle. Computing the trace only requires
two additions but the prior matrix multiplications are far less efficient, as discussed
above.
3.Gii. Examples of Texture Components
3.Gii.1 The “Cube” Texture Component
The Cube Texture (001)[100], also known as cube-on-face, often found as a product of
(primary recrystallization in fcc metals). It is the simplest orientation to describe in that
the crystal axes, are parallel to the three sample axes, i.e. the ND, RD, and TD
directions.
Fig. Gii.1. Diagram of the “cube” component, showing the crystal axes aligned with the
sample axes. In fact the labeling on the diagram indicates that the Euler angles are not all
zero because [100] is parallel to ND, [001] is parallel to RD and [010] is parallel to –TD.
8/27/09 20

Fig. Gii.2. Schematic {100} pole figure for the cube component, showing strong intensity
aligned with all three sample axes. In reality one could not measure such a pole figure
unless reflection and transmission methods were combined.
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:
(φ 1, Φ, φ 2) = (0°, 0°, 0°).
⎛ 1 0 0⎞
⎜ ⎟
Matrix:
⎜
0 1 0
⎟
⎜ ⎟
⎝ 0 0 1⎠
Rodrigues vector: [0,0,0]
Unit quaternion: [0,0,0,1]
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 (φ1, Φ, φ2) = (90°,0°,0°).
3.Gii.2 The “Goss” Texture Component
The Goss Texture {110}, also known as cube-on-edge, often found as a product of
secondary recrystallization in both bcc metals. It is famous as the soft magnetic
orientation that is deliberately grown in transformer steels to make an essentially single
crystal sheet product. The typical orientation spread (a.k.a. mosaic spread) is a few
degrees and the precise amount of orientation spread is critically important to the
performance of the material in terms of electrical losses. The orientation puts a {110}
plane parallel to the rolling plane.
8/27/09 21

Fig. Gii.3. Diagram of the “Goss” component, showing the how the crystal can be
thought of as being rotated 45° about one edge of the cube.
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) to (011)[100]; this happens to be
accomplished with the 2nd Euler angle.
(φ 1, Φ, φ 2) = (0°,45°,0°).
Other variants will be shown when symmetry is discussed.
⎛ 1 0 0 ⎞
⎜
⎟
Matrix:
⎜
0 1/ 2 1/ 2
⎟
⎜
⎟
⎝ 0 −1/ 2 1/ 2⎠
Rodrigues vector: [ tan(22.5°), 0 , 0 ]
Unit quaternion: [ sin(22.5°) , 0, 0, cos(22.5°)]
Note that, since there is only € one non-zero Euler angle, the rotation axis is obvious by
inspection, i.e. the x-axis. For more general cases, the rotation axis has to be calculated.
8/27/09 22

Fig. Gii.4. Schematic {110} pole figure for the Goss component, showing strong
intensity aligned with the ND and TD sample axes as a consequence of the 45° rotation
about the RD. AS above, one could not measure such a pole figure unless reflection and
transmission methods were combined.
3.Gii.2 The “Brass” Texture Component
The {110} component is known as the Brass Texture and occurs as a rolling
texture component for materials such as Brass, Silver, and Stainless steel. Its origin is
still mildly controversial because it can be obtained in models of plastic deformation but
the physical basis for the boundary conditions that produce it are not obvious.
Fig. Gii.5. Diagram of the “Brass” component.
8/27/09 23

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).
Fig. Gii.6. Schematic {001}, {111} and {110} pole figures, reading from left to right, for
the Brass component. The RD is vertical in these diagrams and the TD horizontal. Only
one variant of the Brass component is shown or clarity; the other variant (related by
sample symmetry) can be obtained by a rotation of 70° about the ND (for example).
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].
(φ 1, Φ, φ 2) = ( 35°, 45° , 0° ).
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 cubic
materials, 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 and the alignment of the component
with respect to the sample symmetry. A typical size of Euler space is 90x90x90°; this is
appropriate to a combination of cubic crystal symmetry and orthorhombic sample
symmetry. The space contains three copies, however, of the fundamental zone, which
leads to considerable potential for confusion, as will be discussed in detail below.
3.H Fundamental Zones: Summary
What is a “fundamental zone” or “irreducible space”? It is a set of physically distinct
orientations (or misorientations) and it is intimately linked to symmetry, both in the
crystal and in the sample. This approach illustrates that it is linked to representation of
physical objects, whether grains or grain boundaries (or interfaces….). In terms of the
mathematical representation, it is the range of the parameters the delineates the zone. In
graphical terms, it is the smallest region of the parameter space that encloses all of the
8/27/09 24

physically distinguishable objects. As an example of a familiar fundamental zone,
consider the stereographic projection of the holosymmetric point group, m3m, in standard
position. We speak of the unit triangle for plane normals because if we restrict the range
of Miller indices to h ≥ k ≥ l ≥ 0, then we have delineated all of the different kinds of
crystallographic plane.
3.H Fundamental Zones: Orientations
A fundamental zone is the range of the space used to describe orientations (or any
quantity based on rotational relationships) constructed such that it contains one and only
one copy for each physically distinct orientation. Defining fundamental zones is
impractical without first discussing symmetry, which follows in the next section of this
chapter. It should be intuitively clear that symmetry results in multiple fundamental
zones within the overall orientation space. Moreover, since each copy of version of the
zone contains exactly the same number of points, the volume associated with each one is
identically the same.
The fundamental zone in for orientations is generally defined in Euler angle space
because that is the space in which the practitioners of texture analysis are familiar with.
Nevertheless, one can always define a fundamental zone in any space associated with the
parameterization, be it Rodrigues vectors, unit quaternions etc. The awkward feature of
Euler angles is that the actual fundamental zone for cubic materials has a rather odd
shape, which does not lend itself to making attractive graphics. Therefore the convention
is to use Cartesian axes and show several copies of the fundamental zone in any given
plot. This is discussed in more detail in the next chapter on graphical representations of
texture.
3.H Fundamental Zones: Misorientations
Just as the representation of orientation has an irreducible set of orientations, each of
which represents a physically distinct relationship to the reference configuration, so too
there is a fundamental zone for misorientations. In the parlance adopted above, the
fundamental zone contains the entire set of physically distinct disorientations. The shape
of the zone depends on the representation chosen. Since an (mis-)orientation matrix has
nine coefficients, there is no simple geometrical shape available. For now, we will duck
the issue of how to develop the shape in the various representation spaces. Suffice it to
say that the shape of the fundamental zone for misorientation is usually defined (and
plotted) in R-F space because the boundaries are planes for all the Laue groups. The
fundamental zone is delimited by the following inequalities, where ρ = [ρ1,ρ2,ρ3].
ρ1 > ρ2 > ρ3 > 0
0 ≤ ρ1 ≤ (√2-1)
ρ2 ≤ ρ1
ρ3 ≤ ρ2
ρ1 + ρ2 + ρ3 ≤ 1
8/27/09 25

3.J Symmetry Operators (Matrices)
The standard way to approach symmetry operators is to represent them as matrices. First,
however, let’s recall some elementary ideas about symmetry operators from
crystallography. Rotational symmetry elements (which we will confine our attention to
here) exist whenever you can rotate a physical object and result is indistinguishable from
what you started out with. The following diagram is copied from Lecture Notes for 27-
201, Structures of Materials by Prof. M. De Graef (1996), fig. 9.2.
These rotations can be expressed in a simple mathematical form as unimodular matrices.
Except for the 3-fold axes in the trigonal and hexagonal groups, the effect of any of the 2fold
or 4-fold axes (or the 3-fold axes in the cubic point groups) is to permute the axes in
a simple fashion. 2-fold axes are called diads, 3-fold axes are triads and 4-fold axes are
quads. We will summarize the available operators in the table reproduced from the
Kocks/Wenk/Tomé book, Ch. 1, Table II.
Note how the essential features of the cubic point groups are based on the four 3-fold
axes on with the three 2-fold axes. Adding the 4-fold axes generates the more
highly symmetric cubic groups. Note also that the most symmetric group shown is 432,
rather than m3m, because the mirror elements have been omitted. The reasons for this
area discussed below. The symmetry elements have names and there are different names
for different contexts, as usual. In the context of crystallography, the operators are called
n-fold axes (or just n on diagrams) following the International notation. An alternative
notation is that of Schoenflies in which the operators are denoted by Cn, where n denotes
the order. This same notation occurs in discussions of group theory and quantum
n
mechanics. Yet another notation uses Luvw where uvw denotes the axis along which the
symmetry element operates, and n denotes the order of the axis.
For the purposes of discussing texture and misorientation, it is also important to
distinguish about which axis the rotation is performed. Thus a 2-fold axis about the z-
2
axis is known as a z-diad, or C2z, or L001 , a triad about [111] as a 111-triad etc.
Lastly, we need to mention the existence of symmetry operators of the second kind.
These operators include the inversion center and mirrors. The inversion simply reverses
any vector so that (x,y,z)->(-x,-y,-z). Mirrors operate through a mirror axis. Thus an xmirror
is a mirror in the plane x=0 and has the effect (x,y,z)->(-x,y,z). In matrix form,
the x-mirror is given by:
8/27/09 26

⎛ −1 0 0⎞
⎜ ⎟
⎜ 0 1 0⎟
⎜ ⎟
⎝ 0 0 1⎠
Mirrors and inversion centers have the effect of converting right-handed axes to lefthanded
and vice versa. Therefore we try to avoid using transformations of the second
kind whenever possible.
8/27/09 27

Fig. 3.J.2: table (from Kocks et al.) of symmetry operators expressed as orthogonal
matrices. Note that the a33 entry for the 1 st column, third matrix down (with “C2” against
it) should have a minus sign (to make it -1).
8/27/09 28

3.K Effect of Symmetry on Orientation, especially in Euler-angle space
The effect of symmetry on orientations can now be treated by reference to specific point
groups. For cubic materials, we use the O(432) point group for the crystal symmetry.
Similarly for the orthotropic sample symmetry typically observed in plane strain
compression we can use the mmm point group which contains three orthogonal mirrors.
This combination of mirrors is special in the sense that an inversion center is also present
which means that we can substitute three orthogonal diads for the mirrors and work with
proper rotations only.
The effect of the symmetry elements on the Euler angles varies in complexity with the
element. A diad on the (crystal) z-axis relates g(φ 1,Φ,φ 2) to g(φ 1,Φ,φ 2+π), for example.
A triad on [111] has a much more complex effect, however. Rather than explore this in
detail, the reader is directed towards more detailed discussion in Bunge, who in turn
refers to Pospiech [Pospiech, J., A. Gnatek, and K. Fichtner, Symmetry in the space of
Euler angles. Kristall und Technik, 1974. 4: p. 729.]. The 3-fold axis introduces a screwaxis
(by analogy) into the Euler space. The following table provides a partial listing of
the symmetry elements and their effect on Euler angles.
Table 3.2. Effect of Symmetry Elements on Euler Angles
Symmetry Element Bunge Symm
-etric
(Kocks)
2-fold axis on Sample X π-φ1 π-Φ φ2±π −Ψ π-Θ φ±π
2-fold axis on Sample Y
2-fold axis on Sample Z
-φ1 φ1-π π-Φ
Φ
φ2±π φ2 π-Ψ
Ψ±π
π-Θ
Θ
φ±π
φ
2-fold axis on Crystal x φ1±π π-Φ π-φ2 Ψ±π π-Θ -φ
2-fold axis on Crystal y
2-fold axis on Crystal z
3-fold axis on Crystal z
4-fold axis on Crystal z
6-fold axis on Crystal z
φ1±π φ1 φ1 φ1 φ1 π-Φ
Φ
Φ
Φ
Φ
−φ2 φ2±π φ2±2π/3 φ2±π/2 φ2±π/3 Ψ±π
Ψ
Ψ
Ψ
Ψ
π-Θ
Θ
Θ
Θ
Θ
π-φ
φ±π
φ±2π/3
φ±π/2
φ±π/3
The range of Euler angles used for representing textures depends, obviously on the
combination of symmetry subgroups appropriate for the systems of interest. Although
the cubic-orthorhombic combination (i.e. O(222) for sample symmetry, O(432) for
crystal symmetry) would permit a much smaller fundamental zone to be used, the actual
choice is typically 0≤φ 1≤90°, 0≤Φ≤90°, and 0≤φ 2≤90° with exactly the same ranges used
for any choice of Euler angle convention. The results in the subspace being three times
larger than the fundamental zone; the 3-fold axis that is not exploited produces peculiar
effects, however, in terms of which points are related to which others, as noted above.
More specifically, the number of physically equivalent copies of an orientation for
cubic/orthorhombic symmetry in any space is 4 x 24 = 96. Restricting the ranges of the
Euler angles as above means a reduction in the space of 4 x 2 x 4 = 32: the missing factor
of 3 is exactly the consequence of neglecting the triad symmetry element.
8/27/09 29

3.L Effect of Symmetry in Rodrigues space, with special reference to Misorientations,
or, how to get from one crystal to another (not necessarily adjacent):
3.L.1 Outline
We explain the difference between forming a misorientation from passive
transformations ( = ab•aa -1 ) which is (automatically) in a crystal frame, and forming
it as a rotation from crystal A to crystal B (∆g = gb•ga -1 ! g ˜
). Note that inverting the
misorientation (i.e. go from B to A) does not change the magnitude or the axis of
rotation, except that the axis is inverted (equivalent to reversing the sign of rotation: the
standard formulae effectively invert the axis, however). The latter (rotation) is in the
global reference frame, so you have to transform its axes to get the local frame,
! g
˜
= aa -1 • ∆g • aa = ga -1 •gb.
3.L.2 Active Rotations versus Axis Transformations (Passive Rotations)
Once again we must deal with the two complementary ways of considering texture that
are available to us. Recall that the standard texts (Bunge, Kocks/Tomé/Wenk) assume
that orientations are defined as transformations of axes. They provide the standard
definition wherein each coefficient of the transformation matrix, a, is a direction cosine
between the appropriate pair of new (crystal) and old (reference) basis (unit) vectors, e',
e, respectively.
aij = ei' • ej (3.M.2.i)
The standard texts typically use the notation g, for this same matrix but we reserve this
notation for the matrix that describes the active rotation operator customary in the
writings of Adams and his collaborators. As described above, the matrix that describes
the same active rotation (assuming orthogonal basis vectors) is simply the transpose of
the equivalent passive rotation. Thus if we describe the orientation of a grain in terms of
the transformation required to express a tensor quantity in grain (crystal) axes that is
given in terms of the reference (sample) axes with the matrix a, then the rotation that
would physically rotate the same grain from the reference configuration (the "cube"
component) to its actual position in space (neglecting translations) is g, where g = a T .
All this review may seem somewhat overly detailed but it is essential in order to avoid
confusion when comparing the derivations of different authors.
3.L.3 Misorientation (Axis Transformation)
Consider the misorientation to be the transformation that converts a tensor
quantity expressed in one set of crystal coordinates to those of another crystal. One can
think of two successive transformations where the first converts from the first crystal to
the reference frame, a1 T , and the second converts from the reference frame to the second
crystal, a2. Maintaining the definition for transformation matrices given above, the
misorientation, ∆a, is given by the following.
8/27/09 30

∆a = a2.a1 T (3.M.3.i)
Note that this form of the misorientation could equally well be defined by scalar products
between basis vectors for each crystal, ∆aij = 1 ei• 2 ej.
3.L.4 Misorientation (Active Rotation)
Now we must observe that one could equally well define misorientation as a rotation
between two crystal configurations. Following the same thought process, define the
misorientation as follows.
∆g = g2.g1 T (3.M.4.i)
This rotation is defined with respect to the reference frame, even though its physical
meaning is a rotation from one crystal configuration to another. If one performs an
eigenvalue/eigenvector extraction on ∆g, the results will be also in the reference frame.
As we shall see shortly, we often wish to determine the (common) rotation axis between
two crystals, which is equivalent to determining the appropriate (real) eigenvector of ∆g.
Therefore it is useful to transform the misorientation (rotation) from the reference frame
to the frame of one of the crystals.
3.L.5 "Local" Misorientation (Active Rotation)
Consider transforming the misorientation from the reference frame to the first crystal
frame. For generality, write this transformation as Q1. Then the transformed (second
rank tensor) "local misorientation", ! g ˜ , is obtained as follows.
= Q1 g g T T Q1
(3.M.5.i)
2 1
Now we apply some trickery! Note that Q is identical to a. Then recall that g = a T . This
then allows us to rewrite Eq. 11.4 as follows.
Eliminating the pair at the end,
! g ˜
! g ˜
= g T g2 g T g1
1 1
! g
˜
(3.M.5.ii)
= g1 T g2. (3.M.5.iii)
This then yields a definition of misorientation that is expressed with respect to the first
crystal frame, which is useful for finding the rotation axis in crystallographic terms. Note
that this development is (typically) ignored in developments based on transformation of
axes. Note also that it is perfectly possible to re-express the axis transformation form of
8/27/09 31

misorientation in global coordinates by performing the inverse transformation (from
crystal to reference frame) so that ∆a global = a1 T a2.
3.L.6 Effect of Crystal Symmetry on Misorientation
Now that we have established our definitions of misorientation, we can look at the
effect of symmetry operators. As before, we consider axis transformations first. Writing
crystal symmetry operators as O, we can write the general misorientation as follows,
where O1 is the symmetry operator applied to the first crystal, and O2 is the (crystal)
symmetry operator applied to the second crystal.
∆a' = O2 a2 a1 T O1 T (3.M.6.i)
The purpose here is to develop a method of finding all the physically equivalent
relationships between two lattices. [Caution - boundaries require at least two more
parameters in order to account for the inclination of the boundary, making five in total.]
Since the crystal symmetry operators can be regarded as re-labeling the crystal axes,
applying a symmetry operator to either crystal cannot change the physical nature of the
lattice relationship, or, by implication, of the boundary between the two lattices.
3.L.7 Switching Symmetry: Inversion of Rotation
Note that since trace(a) = trace(aT) the rotation angle is invariant with respect to
transposition of the rotation matrix. The rotation axis, however, reverses sign when you
transpose the rotation matrix. Therefore we can write that a( ,θ) = a( ,-θ) T = a(- ,θ) T
and that a( ,-θ) = a(- ,θ), or, a(- ,-θ) = a( ,θ). This result has an important physical
consequence. A given boundary formed as ∆a = a2.a1 T cannot be physically
distinguishable from the boundary formed by the transformation in the reverse sense, i.e.
∆a' ← = → ⎯ a1.a2 T . However, ∆a = (∆a') -1 = (∆a') T r ˆ r ˆ
r ˆ
r ˆ r ˆ
r ˆ r ˆ
. Therefore the misorientations
defined by a( r ˆ ,θ) and a( r
ˆ ,-θ) are physically equivalent. In more physical terms, there is
a switching symmetry, i.e. it does not make any difference to the boundary to switch the
order in which the grains are considered for constructing the misorientation.
This equivalence is useful when determining the fundamental zone in whichever space is
used to describe misorientation. In the Rodrigues space, for example, the application of
the (24 x 24) symmetry operators on both orientations, as in
∆a' = O2 a2.a1 T O1 T , (3.M.7.i)
results in a fundamental zone that contains two unit triangles in terms of rotation
directions. The switching symmetry, i.e. the equivalence of the positive and negative
rotation axes means, however, that the two triangles can be further reduced to a single
triangle. That is to say, two Rodrigues vectors that are equal in magnitude but opposite in
direction represent physically indistinguishable grain boundaries. This is important in the
determination of disorientation, 3.M.9.
8/27/09 32

3.L.9 Disorientation
The Disorientation is the representation of the misorientation that yields the
smallest available (misorientation) rotation angle, θ* in Eq. 3.M.9.i below. It is found by
searching the list of physically equivalent misorientations as generated by applying each
crystal symmetry operator in turn. Thus there are as many physically equivalent
misorientations as there are members of the group of rotations. Note that the sample
symmetry operators play a special role here: if they are included in the operators applied
then one is, in effect, finding the smallest possible rotation between any of the equivalent
orientations that include the statistical symmetry of the sample. This goes beyond simply
the relabeling of crystal axes implicit in the crystal symmetry operators. Note that if the
full set of 24 symmetry operators is used in the following equation to find θ*, 24 out of
the 1152 combinations (1152 = 24 x 24 x 2, i.e. 24 symmetry operators for each grain
together with the equivalence of inverting the order of labeling the two grains, section
3.M.7 above) will yield the same identical (minimum) rotation angle. The rotation axis,
however, will vary.
θ* = min[ cos -1 {0.5 (trace(O2 a2.a1 T O1 T )-1)}] (3.M.9.i)
From the point of view of physical grain boundaries, choosing to describe a boundary by
the disorientation is obviously sensible for identifying low angle boundaries. It makes
little sense to choose a misorientation with a rotation angle of, say, 95°, when the
boundary is a low angle boundary with a 5° misorientation. For high angle boundaries,
however, choosing to minimize the misorientation may not reveal the significant aspects
of the boundary. This is especially true if one is interested in the possibility that a
boundary corresponds to a symmetric tilt relationship, for example.
3.L.10 Switching Symmetries and Inclination
If boundary inclination is added to the description of an interface then an additional
switching symmetry is introduced. The discussion so far has been confined to rotations
that require 3 parameters. An inclination can be specified by a plane normal, which
therefore adds two more parameters to the description of a boundary. The same
switching symmetry exists for the labeling of the two grains, i.e. passing from grain A to
grain B is an equivalent description to passing from grain B to A. It must also be the
case, however, that describing the plane normal as pointing into grain B is an equivalent
description to having it point to grain A. Thus a second switching symmetry has been
introduced. To anticipate the graphical description of 5 parameter representations, the
plane normal associated with a particular disorientation can be either the positive or
negative of the vector.
8/27/09 33

4)
3)
2)
1)
B
B
A
A
ns ^
A !g BA
ns ^
B
B
ns
^
gB ns ^
!g A BA ns
+gB ns ^
-gA ns ^
!g AB
-gB ns ^
gA ns ^
!g AB
Figure 3.L.1. Showing the four physically equivalent variants of a grain boundary
description involving misorientation and inclination. The order in which the
misorientation is constructed can be switched and the sense of the boundary
normal can be switched (provided that the crystals are centro-symmetric).
^
-gB ns ^
gA ns ^
-gA ns ^
8/27/09 34

3.L.11 Misorientation (Active Rotation) and (Crystal) Symmetry
Let us now expand the active rotation description of misorientation to include symmetry
operators, confining our attention to crystal symmetries. The form of the equation
becomes the following, where γ is a rotation drawn from the set of rotation operators in
the group of crystal symmetries (following the notation of Adams et al.).
∆g' = g γ (g γ ) 2 2 1 1 T = g γ γ T T g1
(3.M.10.i)
2 2 1
Note the different order of applying the symmetry operators as compared with Eq. 11.6
above. Note that the two symmetry operators can be combined to form a single operator
(assuming that we are dealing with a single-phase material) because they are both
elements of the same symmetry group. Thus the multiplicity of physically equivalent
operators is only 24 x 2 = 48 for misorientations described in the global frame (for cubic
crystals), in contrast to the 1152 variants available for misorientations described in the
local frame (section 3.L.9 above). If we now change to the local coordinate frame (i.e.
that of the first crystal) we obtain a result that looks similar to the axis transformation
version (except that the transposed rotation is in the first position).
! g
˜ "
3.L.12 Symmetry Operators in Rodrigues Space
= γ1 T g1 T g2 γ2 (3.M.10.ii)
The action of symmetry operators in Rodrigues space is derived from consideration of the
combination of Rodrigues vectors. The O(432) group can represented by Rodrigues
vectors as follows. Note that for 2-fold axes with 180° rotations, tan(θ/2) = tan(π) = ∞.
Table 3.3. Symmetry Operators Expressed as Rodrigues Vectors
Symmetry Operator Rodrigues Vector
2-fold on
2
L100 ∞(1,0,0)
4-fold on
2-fold on
4
L100 2
L110 (1,0,0)
∞(1,1,0)
3-fold on
3
L111 (1,1,1)
These operators immediately illustrate one of the challenges in working with the
Rodrigues space because it extends to infinity, at least in principle. In practice, however,
for any material possessing more than triclinic symmetry, we can confine our attention to
a box –1 ≤ x ≤ 1, −1 ≤ y ≤ 1, −1 ≤ z ≤ 1. The aspect that is more bizarre is the way in
which symmetry related points are connected by their symmetry operator. Sutton &
Balluffi discuss the example of the 2-fold axis which connects points close to the x=-1
plane to points close to the x = +1 plane, but with a 90° rotation. Frank likens this to a
Moebius strip-like connection.
8/27/09 35

Let’s take the example of the 2-fold symmetry on the x-axis which is best understood as
the limiting case of θ→∞ for [1,0,0]tanθ/2. If we consider a point lying on the symmetry
plane at x=-1, i.e. ρ A = [-1,ρ 2,ρ 3], then the action of the symmetry operator is to take it to
the following new point, ρ C:
ρC = −1,ρ 2 ,ρ [ 3]
+ tanθ / 2[ 1,0, 0]
− tan θ / 2[0, ρ3 ,−ρ 2 ]
1+ tanθ / 2
As we take this to the limit of θ→∞ then ρ C tends towards the point [+1,ρ 3,-ρ 2] which is
the original point reflected through x=0 but also turned through 90°! By considering a
point that is just inside the “back plane” (x=-1), i.e. ρ A = [-1+ε,ρ 2,ρ 3], then evaluation of
the same combination shows that the related point lies just outside the front plane, x=+1,
and vice versa. Note that the planes associated with the symmetry elements are not
mirror planes, as we are accustomed to seeing them in stereographic projections of point
groups, for example.
8/27/09 36

3.L.13 Range of Parameters for Rodrigues Space
For the combination of O(222) for sample symmetry and O(432) for crystal symmetry,
the limits on the Rodrigues parameters are given by the planes that delimit the
fundamental zone. These include six octagonal facets orthogonal to the
directions, at a distance of tan(π/8) (=√2-1) from the origin, and eight triangular facets
orthogonal to the directions at a distance of tan(π/6) (=√3 -1 ) from the origin. The
vertices have coordinates (√2-1, √2-1, 3-2√2) (and their permutations), which lie at a
distance (23-16√2) from the origin. This is equivalent to a rotation angle of 62.7994…°,
which represents the greatest possible rotation angle, either for a grain rotated from the
reference configuration, or between two grains. The shape of the fundamental zone for
disorientations is a subset of this polygon and is the truncated pyramid defined by
confining the rotation axis to a single unit triangle (see the limits specified in 3.H)
The shape of the fundamental zone(s) will be described in more detail in later sections
that deal with graphical representations of orientation and misorientation.
3.L.14 Explanation of Dividing Planes at Half the Rotation Angle
The discussion of (rotation) symmetry operators suggested that dividing planes exist at
half the angle of the symmetry operator. This occurs because of the rectilinear properties
of R-F space. That is, straight lines map into straight lines, which in turn means that
planes map to planes. First let’s see that there is such a thing as a plane on which all
points are equidistant from two other points. Let’s start with a point and its inverse, each
sitting at a distance tan(θ/4) from the origin. Clearly all the points on the bisecting plane
of the two vectors are equidistant from both points.
If we now apply the rotation ρ A to all points then we obtain a new map. The first point
maps onto a new point, ρ A’ (=ρ A • ρ A) which lies at a distance tan(θ/2) from the origin.
The property of planes mapping onto planes means that the bisecting plane maps onto
another bisecting plane (between 0 and ρ A’) that lies at a distance tan(θ/4) from the origin
(at closest approach).
Two Rodrigues vectors combine to form a third as follows where ρ2 follows after ρ1.
ρ A
-ρ A
8/27/09 37

(ρ1, ρ2) = {ρ1 + ρ2 - ρ1 x ρ2}/{1 - ρ1•ρ2} (3.L.14.i)
If one of the vectors defines a line (through the origin) by multiplying a scalar, e, by a
unit vector, m, then the product takes the form,
(ρ1, em) = ρ1 + e/{1 - eρ1•m}[ρ1(ρ1•m) + m - ρ1 x m}] (3.L.14.ii)
The term in the square brackets is a vector that does not depend on the scalar, e. The
term in the curly brackets is a scalar that does depend on e. Thus the rotation of the
straight line has produced another straight line. By extension, planes contain straight
lines and so planes also rotate into planes.
ρ A ’
3.L.14b Constructing the bisecting orientation between two orientations
To construct the bisecting orientation, ρbisect, we follow the development given by S&B
on p11. First form the misorientation between A and B, i.e. the rotation that carries one
from B to A.
0
∆g BA = (ρA,- ρB)
The rotation angle associated with this misorientation is defined as θ dis, and the rotation
axis is defined as Δ ˆ
g BA , given by:
tan(θdis/2) = |∆gBA| Δg ˆ BA = ∆gBA/|∆gBA| Then transform the misorientation by half of its rotation angle (in order to obtain equal
rotation angles between each orientation and the new, bisecting orientation) to obtain a
new rotation, ∆ρAB, where the “∆” merely signifies that we will use it to construct the
new orientation.
∆ρAB = tan(θ dis/4) Δ ˆ
g BA (3.L.14b.i)
8/27/09 38

Now, at last, we can apply this bisecting rotation to both of the original orientations to
find the rotation of interest, ρbisect:
ρbisect = (ρA, -∆ρAB) = (ρB, +∆ρAB)
R ′ ′
Note that S&B proceed as far as calculating ∆ρAB [ = ρA in their notation] but do not
provide the final step shown above to find the bisecting orientation.
3.L.15 Other Representations for Misorientation
Obviously other representations exist for misorientations beyond (orthogonal) matrices
and axis-angle pairs. Just as for orientations so Euler angles can be used to describe
misorientations, for which see [J. Zhao and B. L. Adams, Acta Cryst., A44, (1988)
p.326]. We do not discuss the Euler angle approach because there is no physical insight
to be gained with using Euler angles. The only practical use of Euler angles is to make a
simple parameterization of orientation or misorientation space with equal volume cells by
discretizing the cosine of the second angle. Volume element size is discussed elsewhere.
3.M Symmetry Operations with Quaternions
Table 3.4. Symmetry Operators Expressed as Quaternions
Symmetry Operator Quaternion
Representation
2-fold on
2
L100 ±(1,0,0,0),
±(0,1,0,0)
±(0,0,1,0)
4-fold on
4
L100 ±1/√2(±1,0,0,1),
±1/√2 (0, ±1,0,1)
±1/√2 (0,0, ±1,1)
2-fold on
2
L110 ±1/√2 (±1,1,0,0),
±1/√2 (0,1, ±1,0)
±1/√2 (±1,0,1,0)
3-fold on
3
L111 ±1/2 (±1,1,1,1),
±1/2 (1, −1, 1,1),
±1/2 (1,1,−1,1),
±1/2 (-1,−1, 1,1)
±1/2 (-1,1,−1,1),
±1/2 (1,−1,−1,1)
±1/2 (-1,-1,−1,1))
8/27/09 39

It is also useful to list the values of the important special misorientations based on the
CSL theory, and this done below in the next chapter (4). The negative (inverse) of a
quaternion is given by negating the fourth component, q -1 = ±(q1,q2,q3,-q4); this
relationship describes the switching symmetry at grain boundaries.
Table 3.5. Symmetrically Equivalent Quaternions for Cubic Symmetry
after Sutton & Balluffi, p19
Rotational symmetry operations are similarly simple: e.g. in the cubic crystal system, the
six diads about directions are represented as ±(1,0,0,0), ±(0,1,0,0) ±(0,0,1,0), the
three four-fold axes about are represented as ±1/√2(±1,0,0,1), ±1/√2 (0, ±1,0,1)
±1/√2 (0,0, ±1,1), the six diads about by ±1/√2 (±1,1,0,0), ±1/√2 (0,1, ±1,0) ±1/√2
(±1,0,1,0), and the four triads about by by ±1/2 (±1,1,1,1), ±1/2 (1, −1, 1,1), ±1/2
(1,1,−1,1), ±1/2 (-1,−1, 1,1) ±1/2 (-1,1,−1,1), ±1/2 (1,−1,−1,1) ±1/2 (-1,-1,−1,1). Sutton
& Balluffi quote the 24 equivalent representations in the 432 point group on page 19, see
table below. Note that each symmetrically equivalent quaternion has been formed as
8/27/09 40

q’=q•S, where S is the symmetry operator, with active rotations assumed.
3.M Finding the Disorientation Angle with Quaternions
Symmetry arguments allow for a very simple procedure for finding the disorientation
based on quaternions. The objective is to find the quaternion that places the axis in a
specified unit triangle (e.g. 0 q1>0, i.e. all four components positive and arranged in increasing order, then
the only three variants that need be considered are as follows because we are seeking the
minimum value of the rotation angle (i.e. the maximum value of the fourth component,
q4). (q1,q2,q3,q4)
(q1-q2, q1+q2, q3-q4, q3+q4)/√2
(q1-q2+q3-q4, q1+q2-q3-q4, -q1+q2+q3-q4, q1+q2+q3+q4)/2
Therefore with operations involving only changes of sign, a sort into ascending order,
four additions and a comparison, the disorientation can be identified. The contrast is with
the use of matrices where each symmetrically equivalent variant must be calculated
through a matrix multiplication and then the trace of the matrix calculated; each step
requires an appreciable number of floating point operations, as discussed above.
3.N.1 The Orientation Distribution
The Orientation Distribution (OD) is a central concept in texture analysis and anisotropy.
In qualitative terms, it is the relative frequency of occurrence of any given orientation (or
misorientation). In mathematical terms it is similar to a probability density in whatever
space is used to parameterize orientation, i.e. a function of three variables, f(φ 1,Φ,φ 2). It
is not the same as a probability density because for the latter, the integral of the function
over the range(s) of its parameters is defined to be unity. In materials science, however,
it is more convenient if a material is perfectly random then the OD has the same value
everywhere, i.e. 1 (since a normalization is required for a probability distribution),
regardless of the choice of parameters and their ranges. Therefore we must specify the
following.
1
8π 2
∫∫∫
( )
f ϕ 1 ,Φ,ϕ 2
sin Φdϕ 1 dΦdϕ 2 =1 3.N.1
The factor of 8π 2 is associated with the choice of radians and the range of the three
8/27/09 41

angles. For a completely orientation, the ranges of φ 1 and φ 2 are 0 through 2π, and the
range of Φ is π. Therefore when the expression in Eq. 3.N.1 above is integrated, the
volume is 8π 2 . Note that the choice of parameters dictates the form of the volume
element used to perform the integration. If we use degrees instead of radians, then the
normalization factor is 2*360 2 instead of 8π 2 . The volume element, or Haar measure,
changes depending on the parameters used.
3.N.2 Pole Figures projected from Orientation Distributions
Most of us encounter pole figures as the representation of experimental results obtained
on a special X-ray goniometer that permits the specimen to be rotated in two orthogonal
axes. A pole figure is a map of the intensity of diffraction from a particular set of
crystallographic planes as a function of declination/co-latitude (from the center of the
pole figure) and azimuth/longitude (sometimes labeled χ and φ, or Θ and φ). Having
developed the basis for an orientation distribution (OD), however, it is instructive to
consider how a pole figure can be constructed given a knowledge of the OD, i.e. a
mathematical definition. First we define a particular plane for which we would like to
construct a pole figure by the conventional Miller indices, (hkl). We then form a unit
vector, h, by normalizing the indices. Note that h represents a position on the unit sphere
so it is essentially spherical position information.
ˆ
h =
(h /a,k /b,l /c)
h 2 /a 2 + k 2 /b 2 + l 2 /c 2
We then need to relate the pole position to the corresponding position, w, in the pole
figure, which is € referred to specimen coordinates through the orientation of each grain
that contributes to (i.e. diffracts in) the pole figure. In this expression, R is a crystal
symmetry operator (unimodular matrix) and g-1 is the inverse (transpose) of the
orientation matrix described above.
€
w = g −1 ( ψ,Θ,φ )R (k) h
It is important to remember that this is a projection from a 3-parameter space into a 2parameter
space. This can be illustrated by reference to the standard mathematical
construction of a many-to-one mapping. Note that, although there are 24 proper rotation
operators in the cubic point group, the number of poles observed (on the upper
hemisphere) depends on the relationship of h to the symmetry elements ({001} coincide
with many of the rotation axes, e.g., and so only 3 distinct {001} poles are visible in the
upper hemisphere, unless one lies on the edge/equator); this effect is of course known as
multiplicity in crystallography. The relation is only simple for special choices of the
pole: the (001) pole, for example, eliminates the third Euler angle. In general to find the
range of values of Euler angles that correspond to a particular point in the pole figure (i.e.
fixing χ and φ) results in a relationship between the three angles. In fact a line in Euler
space is defined which is equivalent to a fiber.
8/27/09 42

To project the point w onto the page, a specific projection must be performed. Let us
take the stereographic projection as an example. It is simplest to first convert the (unit)
vector w into spherical angles (co-latitude and longitude), Θ and φ. These are given by:
tanϕ = w y /w x
cosΘ = w z
Then these spherical angles can be used to calculate the coordinates of the projected
point, p, (within a unit circle corresponding to the unit sphere of the pole figure). Recall
that the radius of the projected € point is equal to the tangent of the semi-angle:
px = tan Θ ( 2)sinϕ
py = tan Θ ( 2)cosϕ
More elegant and compact formulae are left up to the reader to write out.
Thus to construct a discrete € pole figure one first chooses the crystallographic pole (e.g.
100, 111 or 110), calculates all the crystallographically equivalent poles, transforms each
pole to the sample frame of reference, projects it using the selected projection (e.g.
stereographic) and finally plots the set of points. This procedure is then repeated for each
crystal in the set of crystals of interest. The set of crystallographically equivalent poles
can be generated with the aid of the symmetry group but it is usually more efficient to
make a list such as the six 100 equivalents: [1,0,0], [-1,0,0], [0,1,0], [0,-1,0], [0,0,1] and
[0,0,-1]. Transformation to the sample frame is easily accomplished by applying the
transpose of the standard orientation matrix:
8/27/09 43
€
h ′ = g T h (3.?.?)

If we write the OD as f(g) in the conventional manner, then 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 (also a sum if the OD is in discrete form). Note that the
coordinates in the pole figure are determined by the relation above. More specifically,
the three Euler angles in the expression below are not independent of one another and can
be written in terms of a single parameter. The functional relationships are to be
determined from the previous expression, i.e. that which relates the point, w, in pole
figure space to a line in orientation space.
( ( ) )
∑
P(w) = f a −1 ∫∫∫ ψ ,Θ,φ sinΘdψdΘdφ S (k) h
8/27/09 44
k

Pole Figure
Orientation
Distribution
(#,$,")
1
(#,$,")
2
(!,") (#,$,")
3
(#,$,")
4
(#,$,")
5
To make this approach more concrete and specific, consider a way in which this
relationship can be re-cast in terms of a single integration parameter. Start with the
crystal in the reference configuration and think in terms of active rotations (vector
transformations), i.e. we work in a fixed frame of reference corresponding to that used for
sample space (in which the pole figure is plotted). We first rotate the crystal with th=th(
t ˆ
h ,θ) 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, t ˆ
h , as the vector
product of [001] and h.
ˆ
t h =
[001] ! h
[001] ! h = h2 2 ["h2,h 1,0]
2 ) + h1 The rotation angle, θ, is given by the inverse cosine of the scalar product:
cos! = h 3
From this axis-angle pair, the rotation matrix can be constructed as follows.
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"
2
h3 + (1 ! h3 )h2 $
th =
$
!(1! h3 )h1 h2 $
h1 1 ! h
2
#
3
2
!(1! h3 )h1h2 !h1 1! h %
3
2
2 '
h3 + (1! h3 )h1 h2 1 ! h3 '
!h2 1 ! h
2
3 h
'
3 &
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.
# cos! "sin! 0&
t ! = %
sin!
$ 0
cos!
0
0 (
1'
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],ζ), where ζ 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] by χ and then [001] by φ.
# cos! "sin! 0&
t ! = %
sin!
$ 0
cos!
0
0 (
1'
,t ) =
# 1 0 0 &
%
0
$ 0
cos )
sin )
"sin ) (
cos) '
The product of this set of rotations must then be equal to the orientation matrix, i.e.
g(φ1,Φ,φ2) = tφ•tχ•tζ•th = t(h,ζ,w)
where the matrix on the RHS contains only one free parameter, i.e. the rotation by ζ
about the pole h. Alternatively we can write the following for the axis transformation.
a(φ1,Φ,φ2) = (tφ•tχ•tζ•th) T = t T (h,ζ,w)
This then allows us to write the following for the intensity, P, at w(χ,φ) in the pole figure.
8/27/09 46

( ) = f h,ζ,w
P h w
ζ = 2π
∫
ζ =0
( )dζ
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.
3.N.3 Inverse Pole Figures projected from Orientation Distributions
The relation for calculating an inverse pole figure, P inv , is similar, as might be expected.
In fact, the range of "inverse poles" or directions in sample space is generally restricted to
only the three (orthogonal) specimen axes, i.e. RD, TD or ND, equivalent to
[100]specimen, [010]specimen, or [001]specimen. S is a sample symmetry operator, i is a
sample direction (unit vector) as discussed above, and w' is a crystal direction defined by
two parameters (angles), α (latitude measured from [001]) and β (longitude measured
from [100]). Again, for certain choices of inverse pole figure, the relationship is simple:
for the ND inverse pole figure, α=Θ and β=φ. Geometrically, one can think of
constructing pole figures as collapsing the appropriate sections through the OD onto a
single plane, i.e. taking an average.
w ′ ( α,β ) = a( ψ ,Θ,φ)S
(k) i
( ( ) )
∑
P inv ( w ′ ) = ∫∫∫ f a ψ,Θ,φ sin ΘdψdΘdφ S (k) i
Again, in the second expression, the three Euler angles can each be expressed in terms of
a single parameter for which the relationships are obtained from the first expression
relating a point in the inverse pole figure (crystal axes) to a line in orientation space.
3P. Misorientation Distribution Function
3P.xx
The volume element, or Haar measure, in Rodrigues space is given by the following
formula [ρ = tan(θ/2)]:
€
ρ
dg =
1+ ρ 2
⎛ ⎞
⎜ ⎟
⎝ ⎠
2
dρd ˆ
n
We can also write this in terms of an azimuth, φ, and declination angle, χ:
8/27/09 47
k

The significance of the volume element is similar to that in Euler space. In Euler space,
the volume element increases in magnitude as the second angle increases such that the
density of a distribution € of random (uniformly distributed set of) orientations first
increases with distance from the origin and then decreases again, resulting in a sort of
doughnut-shaped density variation.
2
ρ
dg =
1+ ρ 2
⎛ ⎞
⎜ ⎟ dρsinχdχdφ
⎝ ⎠
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