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Acta Materialia 61 (2013) 3068–3081
www.elsevier.com/locate/actamat
Representation of single-axis grain boundary functions
Srikanth Patala a,b,⇑, Christopher A. Schuh a
a
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
b
Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA
Received 3 October 2012; received in revised form 30 January 2013; accepted 30 January 2013
Available online 27 February 2013
Abstract
The ability to describe continuous functions on the space of grain boundary parameters is crucial for investigating the functional relations between the structure and the properties of interfaces, in analogy to the way that continuous distribution functions for orientations
(i.e. texture information) have been used extensively in the optimization of polycrystalline microstructures. Here we develop a rigorous
framework for the description of continuous functions for a subset of the ﬁve-parameter grain boundary space, called the “single-axis
grain boundary” space. This space consists of all the boundary plane orientations for misorientations conﬁned to a single axis, and is
relevant to the method of presenting boundary plane statistics in widespread current use. We establish the topological equivalence
between the single-axis grain boundary space and the 3-sphere, which in turn enables the use of hyperspherical harmonics as basis functions to construct continuous functions. These functions enable the representation of statistical distributions and the construction of
functional forms for the structure–property relationships of grain boundaries.
Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Grain boundaries; Crystal symmetry; Grain boundary planes; Distribution functions
1. Introduction
The importance of grain boundaries to the properties of
polycrystalline materials is widely appreciated and is
quickly becoming a cornerstone of the modern materials
design paradigm. There are numerous instances where
grain boundary distributions have been manipulated to
improve the functional and mechanical properties of polycrystalline materials [1–8]. While the primary focus of
many of these studies has been to tailor the grain boundary
misorientations, there has been a recent emphasis on
manipulating the grain boundary plane distributions to
obtain better properties [9–13]. These investigations have
beneﬁted greatly from experimental advances in the characterization of grain boundaries in full crystallographic
detail, inclusive of all ﬁve macroscopic parameters deﬁning
their geometry [14–19]. The focus for the future of grain
boundary engineering is shifting towards simultaneously
tailoring the ﬁve parameters.
One signiﬁcant obstacle to the investigation of the distributions of the ﬁve grain boundary parameters is a lack of
analytical tools to describe the distributions of quantities
involving both the misorientation and the boundary-inclination aspects of grain boundaries. This is because the
ﬁve-parameter space is vast and has a complicated topology due to various constraints. Owing to symmetries of
the boundary and the crystals abutting it, there are some
duplicate sets of distinct parameters that describe the same
physical boundary, and hence are symmetrically equivalent. Such symmetry constraints can be expressed as equivalence relations, and have been developed in detail
elsewhere [20–22]. It is useful to reiterate these here:
h
i ðM;~
nÞ ðS i Þ1 MðS j Þ; g ðS i Þ1 ~
n where i; j 2 f1; ... ;ng
⇑ Corresponding author at: Department of Materials Science and
Engineering, Northwestern University, Evanston, IL 60208, USA.
E-mail address: srikanth.patala@gmail.com (S. Patala).
1
1
nÞÞ
ðM;~
nÞ ðM ; g½M ð~
2
0
0
~
~
n; n 2 S
ðI;~
nÞ ðI; n Þ 8~
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.actamat.2013.01.067
ð1aÞ
ð1bÞ
ð1cÞ
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
In these expressions, M denotes the misorientation, ~
n
represents the boundary-plane normal vector, and S represents a point symmetry operation of the crystallographic
point group of order n. The operation g simply outputs
the 3 3 matrix equivalent to the rotation operation in
its argument, regardless of the parameterization used. I
represents the identity matrix corresponding to the zeromisorientation angle. These three equivalence relations
capture some important physical concepts about grain
boundary crystallography: (i) rotating one or both of the
crystals through one of their symmetry operations does
not change the boundary; (ii) the boundary is physically
the same when viewed from either of the two grains at
the boundary (the “grain exchange symmetry”); and (iii)
if there is no misorientation, then there is no unique boundary plane either (the “no-boundary singularity”).
To be able to analyze and exploit the full potential of the
vast amounts of grain boundary data that can now be
obtained from microstructural analysis, it is crucial to
develop tools that help resolve or remove some of the complexities of the grain boundary space. This is largely an
open problem at present [23]. In Ref. [24], we addressed a
simple version of the problem, for one-dimensional boundaries between two-dimensional (2-D) crystals. By appropriately transforming the 2-D grain boundary parameters, the
no-boundary singularity was resolved, and by including the
grain exchange symmetry, the space of grain boundary
parameters was shown to be equivalent to the 2-sphere with
appropriate equivalence relations (S2/E). The analysis of
the 2-D grain boundary space emphasized the necessity
of a new parameterization that naturally accounts for the
no-boundary singularity and simpliﬁes the equivalence
relation associated with the grain exchange symmetry.
In this paper, we present developments that resolve a
subset of the ﬁve-parameter grain boundary space, the single-axis grain boundary (SAGB) space. The SAGB space is
the collection of grain boundary parameters with the misorientation axis conﬁned to lie along any speciﬁc crystal
direction ~
b (with certain exceptions discussed in Section.
2). This subset of the complete grain boundary space is particularly relevant because, in the experimental literature,
grain boundary planes are often analyzed for misorientations along a speciﬁc symmetry axis of the crystal [25–
31]. This space also describes the collection of grain boundaries of perfect ﬁber-textured materials, and has direct relevance to, for example, thin ﬁlms and severely extruded
metals.
2. Mapping the single-axis grain boundary space onto the
hypersphere (S3)
As mentioned previously, the SAGB space is the collection of all boundary-plane orientations corresponding to
disorientations (i.e. the misorientations lying in the fundamental zone of interest) along a ﬁxed crystal direction ~
b.
The boundary inclination space is the unit-sphere in three
dimensions (2-sphere, S2) since any normal vector can be
3069
represented as a point on the unit-sphere. Therefore, from
a topological perspective the SAGB space is the product
space [0, xmax] S2, where xmax is the maximum disorientation angle along the axis ~
b in the fundamental zone. More
precisely, the SAGB space is equivalent to [[0, xmax] S2]/
E, where E is the equivalence class representing all possible
symmetries of the boundary-plane spaces (i.e. Eq. (1)). In
Ref. [22], we have enumerated these symmetries for disorientations belonging to all the crystallographic point groups.
The ﬁrst objective of this article is to ﬁnd a suitable
transformation of the boundary parameters that maps
the SAGB space ([[0, xmax] S2]/E) to the 3-sphere S3
4
(with
P4 2 coordinates (x1, x2, x3, x4) in R such that
i¼1 xi ¼ 1) with a convenient set of equivalence relations.
Before considering these mappings and symmetries in
detail, certain conventions are introduced. The symmetries
of the boundary-plane spaces are denoted by G. As
observed in Ref. [22], when x belongs to the interval
(0,xmax), all the boundary-plane spaces S2 exhibit symmetries of a single point group (denoted as G1) and we denote
the symmetries of the boundary-plane space when
x = xmax as G2 (where G1 # G2). In this paper, the quaternion (q) parameterization is used to represent misorientations (M) and the grain boundary parameters are hence
denoted as ðq;~
nÞ. The mapping of the ðq;~
nÞ parameters
to the 3-sphere is obtained through the following steps:
(a) The ﬁrst equivalence relation that needs to be
addressed is that of the no-boundary singularity Eq.
(1c), which is crucial to the mapping between the
product space [0, xmax] S2 and S3. According to
this singularity, the space S2 corresponding to the
zero misorientation angle needs to be collapsed to a
single point. This is achieved by the following mapping from the ðq;~
nÞ to the ðq;~
rÞ parameterization:
ðq;~
rÞ ¼ Cðq;~
nÞ ¼ ðq; ½CðqÞ ~
nÞ
ð2Þ
where C(q) is a scalar function deﬁned as:
CðqÞ ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 maxfððq GÞ0 Þ2 g
ð3Þ
G is the point group symmetry of the crystal. q G is
the left co-set of G in SO(3) and ðq GÞ0 is the set of
the ﬁrst quaternion components of the left co-set
q G. To state it simply, C(q) is a continuous function
on the quaternion space and takes the value zero
when the boundary misorientation is either the identity or symmetrically equivalent to the identity. In
the case of the grain boundary space of a crystal with
rotational point group symmetry
C1, G
ﬃ = C1 = {I},
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
(q G)0 = q0 and thus CðqÞ ¼ 1 q20 . In general,
for
allﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃcrystallographic
point
groups,
ﬃ
p
CðqÞ ¼ 1 q20 ¼ sin x2 if q belongs to the fundamental zone.This mapping results in a parameterization ðq;~
rÞ that is compatible with the no-boundary
singularity by collapsing all the boundary normal vectors corresponding to the zero misorientation angle to
3070
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
a single point. In the ðq;~
rÞ parameterization, the space
[0, xmax] S2 is transformed into a solid ball with the
origin corresponding to the zero-misorientation
case
and each sphere with radius sin x2 corresponding to
the boundary-inclination space of misorientation
angle x. This solid parametric ball in ðq;~
rÞ parameterization can be easily extended to a 3-sphere if none of
the boundary-plane symmetries are considered. However, as shown in Ref. [22], the boundary-plane spaces
may exhibit a wide array of point group symmetries.As mentioned already, the collection of boundary-plane spaces corresponding to a single ﬁxed axis
exhibits the symmetries of point group G1 when
x 2 (0, xmax) and exhibits the symmetry of point
group G2 when x = xmax. In the ðq;~
rÞ parameterization obtained using Eq. (2), the SAGB space is equivalent to a solid ball with the points in the interior of
the solid ball and at a constant radius identiﬁed
according to the symmetry operations of the point
group G1 and the points on the surface of the solid
ball identiﬁed according the point group G2.
(b) In general, the symmetry axes of the point group G1
change as x changes (discussed in detail in Ref.
[22]). For example, consider the symmetries
h of thei
boundary-plane spaces along the axis ~
b ¼ p1ﬃﬃ ; p1ﬃﬃ ; 0
2
2
for the crystallographic point group D4. The quaternions corresponding
x 1 to this
x axis are of the type
x p1ﬃﬃ
ﬃﬃ
p
ðcosÞ 2 ; 2 sin 2 ; 2 sin 2 ; 0Þ, where x 2 0; p2 .
In the domain (0, xmax), the boundary-plane spaces
exhibit symmetries corresponding to the G1 = C2v
point group with the twofold axis along ~
b and one
of the mirror-planes
perpendicular
to
the axis
x 1
x
x
1ﬃﬃ
ﬃﬃ
p
p
ðq2 ; q1 ; q0 Þ ¼
sin 2 ; sin 2 ; cos 2
(refer
2
2
to Ref. [22]), which implies the dependence of the
position of the mirror-plane on the misorientation
angle x.If the ðq;~
rÞ parameterization obtained in
Eq. (2) is used and the solid-parametric ball so
obtained is mapped to the 3-sphere, the equivalence
relations will depend on the coordinates xi (i.e. the
position of the point on the 3-sphere). Such relations
cannot be used to derive symmetrized hyperspherical
harmonics. Therefore, the ðq;~
rÞ parameterization is
further modiﬁed, to facilitate symmetrization of the
basis functions, as follows:
(
ðq; g½pðqÞ ~
rÞ
if q0 – 1
~Þ ¼ Pðq;~
ðq; m
rÞ ¼
ðð1; 0; 0; 0Þ; ~
0Þ if q ¼ 1
0
ð4Þ
where p(q) takes the form:
1
pðqÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ q0 ; q1 ; q2 ; q3 Þ
2ð1 þ q0 Þ
ð5Þ
It is necessary to ensure that the above mapping is
topologically consistent (satisﬁes uniqueness and
continuity conditions) with the ﬁve-parameter grain
boundary space. This parameterization is closely
related to the so-called symmetric parameterization
deﬁned in Ref. [32]. However, the symmetric parameterization is lacking due to a discontinuity in the
mapping, which is explained in Appendix A. Also
described in Appendix A are the mathematical
aspects (A2) and the equivalence relations (A3) of
~Þ paramethe ðq; ~
mÞ parameterization. Using the ðq; m
terization, the symmetry axes of the point groups G1
and G2 align and do not vary as the misorientation
angle x varies (as discussed in the examples that
follow).
(c) Even though the dependence of the symmetry axes on
~Þ
the misorientation angle x is removed using the ðq; m
parameterization, the symmetry axes need not be
positioned in a manner convenient for the symmetrization of the hyperspherical harmonics. For example,
consider again theh crystallographic
point group D4
i
with the axis ~
b ¼ p1ﬃﬃ ; p1ﬃﬃ ; 0 . As will be shown in an
2
2
example in the following sections, the twofold axis
of G1 = C2v is parallel to p1ﬃﬃ2 ; p1ﬃﬃ2 ; 0 and one of the
mirror planes is perpendicular to (0,0,1) in the
~Þ parameterization. For the symmetrization of
ðq; m
the hyperspherical harmonics, it is useful to have
the symmetry axes of the point groups parallel to
standard Cartesian axes ð^ex ; ^ey ; ^ez Þ. This is obtained
by using an additional rotation operation (denoted
by the matrix R), which aligns the symmetry elements
of the point groups G1 and G2 along the relevant
Cartesian axes. Therefore, we deﬁne parameters
ðq;~
lÞ such that:
~Þ
ðq;~
lÞ ¼ ðq; R m
ð6Þ
(d) Now, coordinates (x1, x2, x3, x4) of the 3-sphere are
deﬁned such that:
2 3
2 3
x2
l1
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
6 7
6 7
4 x3 5 ¼
4 l2 5; x1 ¼ 1 x22 x23 x24
sinðxmax =2Þ
x4
l3
ð7Þ
It is observed that x22 þ x23 þ x24 ¼
sin ðx2 Þ
2
lies in the
Þ
range [0, 1]. Therefore, the collection of parameters
(x2, x3, x4) belongs to a solid ball of outer radius unity
in R3 . The relation x21 þ x22 þ x23 þ x24 ¼ 1 implies that
the coordinates (x1, x2, x3, x4) indeed belong to a 3sphere of unit radius (although Eq. (7) only deﬁnes
points in one half of the 3-sphere corresponding to
x1 2 [0, 1]).The points on a sphere of radius
sin ðx2 Þ
correspond to the boundary-plane vectors
r ¼ sin xmax
ð 2 Þ
of boundary misorientation x (related by Eqs. (2),
(4), (6) and (7)). If this boundary-plane space
sin ð
xmax 2
2
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
(corresponding to the misorientation angle x and axis
~
b) has the symmetry G1, the following equivalence
relation should hold:
0 2 31 0
2 31
x2
x2
B 6 7C B
6 7C
ð8Þ
@x1 ; 4 x3 5A @x1 ; g½g1 4 x3 5A 8g1 2 G1
x4
x4
The sphere corresponding to x1 = 0 has a radius r = 1
and describes the surface of the solid ball. The points
on this surface correspond to boundary-plane vectors
at the misorientation angle xmax. According to Eq.
(8), by substituting x1 = 0, these points exhibit symmetries of the point group G1. However, the boundary-plane space corresponding to misorientation
angle xmax should exhibit the symmetry of point
group G2. Therefore, the condition that the points
(0, x2, x3, x4) need to satisfy is:
0 2 31 0
2 31
x2
x2
B 6 7C B
6 7C
ð9Þ
@0; 4 x3 5A @0; g½g2 4 x3 5A 8g2 2 G2
x4
x4
Since G1 # G2, Eq. (8) does not violate the condition
in Eq. (9) but it is not suﬃcient. An additional equivalence relation has to be imposed on the 3-sphere such
that Eq. (9) is satisﬁed. This additional equivalence
relation is as shown below (h is deﬁned in the next
paragraph):
0 2 31 0
2 31
x2
x2
B 6 7C B
6 7C
ð10Þ
@x1 ; 4 x3 5A @x1 ; g½h 4 x3 5A
x4
x4
By imposing the above equivalence relation, Eq. (9)
will be satisﬁed and no additional relations will be
introduced if the groups G1 and G2 are such that:
G1 is a normal subgroup of G2 (denoted as
j G2 =G1 j 6 2); and
the number of elements in the quotient group G2/
G1 is less than or equal to two (i.e. j G2 =G1 j 6 2).
3071
If the number of elements in the quotient group G2/
G1 is one, then G2 is equal to G1 and h is simply the
identity operation {e}. If the number of elements of
the quotient group G2/G1 is two, then it can be written
in the form {G1, h G1}, where G1 and h G1 are disjoint sets and h is one of the elements (can be any element) of the co-set not equal to G1. Using the
properties of quotient groups, the group G2 can be
written as G2 = G1 [ h G1. Combining the equivalence relations in Eqs. (8) and (10) deﬁnes all the symmetries on the 3-sphere. For constant x1, the points
(x2, x3, x4) are related through the point group symmetry G1 and for x1 = 0, they are related through
the point group symmetry G2. Eq. (10) also helps
deﬁne the 3-sphere completely; Eq. (7) only deﬁnes
one half of the 3-sphere where x1 2 [0, 1]. In Eq.
(10) the points in one half (x1 2 [0, 1]) of the 3-sphere
are related to the other half (x1 2 [1, 0]).
Table 1 shows the cases in which the combination of
point groups G1 and G2 do not satisfy the above-mentioned conditions (G1 E G2 and j G2 =G1 j 6 2). For
these SAGB spaces, the formalism developed here
may not be applied. However, the number of cases
for which this condition is violated is small compared
to the large number for which it works. The complete
list of SAGB spaces, G1, G2 and the operation h for
diﬀerent crystallographic point groups is provided in
the online Supplementary Information.
2.1. Examples
The four steps described above for the mapping from
any SAGB space (not considering the exceptions listed in
Table 1) to the 3-sphere are now illustrated using the following examples.
2.1.1. Crystallographic point group D4, axis ~
b ¼ p1ﬃﬃ2 ; p1ﬃﬃ2 ; 0 ,
xmax is equal to p2 ; G1 ¼ C 2v and G2 = D2d
The quaternions (q0, q1, q2, q3) that lie along the axis ~
b
x x pﬃﬃﬃ
are such that q0 ¼ cos 2 ; q1 ¼ q2 ¼ sin 2 = 2 and
Table 1
List of single-axis grain boundary (SAGB) spaces for which a mapping to the 3-sphere cannot be obtained using the formulation developed in this paper.
The SAGB spaces are listed by specifying the crystallographic point group, the point where the axis intersects the outermost surface of the corresponding
fundamental zone (speciﬁed as conditions satisﬁed by the quaternions in the “Geometry” column) and the boundary-plane symmetries G1(x 2 (0, xmax))
and G2(x = xmax). The proper rotational point group and the corresponding Laue group are grouped together.
Point group
Geometry
G1
G2
C3/C3i
Cn/Cnh (n = 4, 6)
S4 (proper/improper)
S6 (proper/improper)
Dn/Dnh (n = 2, 4, 6)
T/Th
q3 = q0 = 0
q3 = q0 = 0
q3 = q0 = 0
q3 = q0 = 0
p
q = (cos h, cos h, sin h, sin h), where h ¼ 2n
q1 þ q2 þ q3 ¼ q0 ; q3 – 2p1 ﬃﬃ3
q1 = 0; q2 – 0
q2 = 0; q1 – 0
q0 ¼ q3 ¼ p1ﬃﬃ2 ; q1 ¼ q2 ¼ 0
q ¼ 2p1 ﬃﬃ2 1k ; 1; 1; k
C1/Ci
Cs/C2h
Cs/C2
C2/Cs
C1/Ci
Cs/C2h
C3v/D3d
Cnv/Dnh
D2d
D3h
D2/D2h
C3v/D3d
C2v/D2h
Cs/C2h
Td/O
D2d/D4h
T/Th
O/Oh
3072
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
q3 = 0. xmax ¼ p2 is obtained by using the condition
pﬃﬃﬃ
2q0 ¼ q1 þ q2 (deﬁning the boundary intersecting the axis
~
b in the D4 fundamental zone), q1 = q2 and q3 = 0. The axes
of the point group G1 = C2v are such that one of the mirror-planes is perpendicular to the axis (q2, q1, q0) and
~Þ
the twofold rotation axis is along p1ﬃﬃ ; p1ﬃﬃ ; 0 . If the ðq; m
2
2
parameterization is used, the equivalence relation is modiﬁed such that the mirror-plane is perpendicular to the axis
(0, 0, 1) as described in the equation below.
ðq;~
nÞ ðq; g½ðp;~
aÞ ð~
nÞÞ; where
~
a ¼ ðq2 ; q1 ; q0 Þ and q ¼ ðq0 ; q1 ; q2 ; 0Þ
) ðq;~
rÞ ðq; g½ðp;~
aÞ ð~
rÞÞ
From Eq. (4),
~Þ ) ~
~
ðq;~
rÞ ¼ ðq; ½g½pðqÞ1 m
r ¼ ½g½pðqÞ1 m
~Þ ðq; g½ðp;~
aÞ ½g½pðqÞ1 ð~
mÞÞ
) ðq; ½gðpðqÞÞ1 m
1
~Þ ðq; ½g½pðqÞ g½ðp;~
mÞÞ
) ðq; m
aÞ ½g½pðqÞ ð~
~Þ ðq; g½ðp; ½0; 0; 1Þð~
) ðq; m
mÞÞ
ð11Þ
where
½g½pðqÞ g½ðp;~
aÞ ½g½pðqÞ1 ¼ g½ðp; ½0; 0; 1Þ
relation (x1, x2, x3, x4) (x1, x2, x3, x4) to the above
set of relations.1
2.1.2. Crystallographic point group C1, axis ~
b such that
bz P 0,xmax is equal to p, and G1 = C1 and G2 = Cs
xmax = p is obtained by using the condition q0 = 0
(deﬁning the boundary intersecting the axis ~
b in the C1 fundamental zone). The mirror-plane in the G2 = Cs point
group is perpendicular to the axis ~
b. In this case, the symmetry axes for the Cs point group are not modiﬁed in the
~Þ parameterization and the mirror-plane remains perðq; m
pendicular to ~
b. However, the disorientation dependence
of the symmetry axes is not a problem here since
G1 = C1. A rotation R is deﬁned such that axis ~
b is rotated
parallel to the z-axis.
p
p ~
ð14Þ
ðq;~
lÞ ¼ q; g½h; cos u ; sinðu Þ; 0 m
2
2
where (h, u) are the polar coordinates of the axis ~
b. The
parameters (x1, x2, x3, x4) and the equivalence relations
on the 3-sphere can now be described as:
2 3
2 3
l1
x2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
6 7
6 7
1
4 x3 5 ¼ sinðp=2Þ 4 l2 5; x1 ¼ 1 x22 x23 x24
ð15Þ
x4
l3
if q3 ¼ 0
~Þ parameterization is used, the
Therefore, if the ðq; m
symmetry axes of the point group C2v are transformed such
that one of the mirror-planes is perpendicular
to thez-axis
(0, 0, 1) and the twofold axis is parallel to p1ﬃﬃ2 ; p1ﬃﬃ2 ; 0 . The
rotation matrix R is used to rotate the axes p1ﬃﬃ2 ; p1ﬃﬃ2 ; 0 and
(0, 0, 1) such that they coincide with (0, 0, 1) and (1, 0, 0),
respectively. R is deﬁned by the product
of two
axis-angle
parameters as g p2 ; ð0; 1; 0Þ g p
;
ð0;
0;
1Þ
. Hence, the
4
~
parameters ðq; lÞ are:
hp
i hp
i
~
; ð0; 0; 1Þ m
ðq;~
lÞ ¼ q; g ; ð0; 1; 0Þ g
2
4
ð12Þ
The parameters (x1, x2, x3, x4) and the equivalence relations on the 3-sphere can now be described as:
2 3
2 3
x2
l1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
6 7
6 7
1
4 x3 5 ¼ sinðp=4Þ
4 l2 5; x1 ¼ 1 x22 x23 x24
x4
l3
2 03
2 3
x1
x1
6 7
6 7
ðx1 ;x2 ;x3 ;x4 Þ x1 ;x02 ;x03 ;x04 4 x02 5 ¼ g½g1 4 x2 5 8g1 2 G1 ¼ C 2v
x03
x3
2 03
2 3
x1
x1
6 7
6 7
ðx1 ;x2 ;x3 ;x4 Þ x1 ;x002 ;x003 ;x004 4 x02 5 ¼ g½h 4 x2 5 h ¼ Z p2 1
0
x3
x3
ð13Þ
where Z p2 is a rotation operation of angle p/2 along the ^z
axis and 1 is the inversion operation (i.e. the negative of
an identity matrix). This above example can be easily extended to the Laue group D4h by adding the equivalence
ðx1 ; x2 ; x3 ; x4 Þ ðx1 ; x2 ; x3 ; x4 Þ
due to the point group symmetry of G2 ¼ C s
For Laue group Ci, the relation (x1, x2, x3, x4) (x1,
x2, x3, x4) is considered in addition to the above
relation.
2.1.3. Crystallographic
point group O and
~
b ¼ p1ﬃﬃ3 ; p1ﬃﬃ3 ; p1ﬃﬃ3 ; xmax is equal to p/3, and G1 = C3v and
G2 = D3h
The axes of the point group G1 = C3v are such that one of
the
mirror-planes is perpendicular to the axis
pﬃﬃﬃ
q0 þq1 q1 q0
pﬃﬃ ; pﬃﬃ ; 2q
1 and the threefold rotation axis is along
2
2
p1ﬃﬃ ; p1ﬃﬃ ; p1ﬃﬃ ,
where q0 ¼ cos x2
and q1 ¼ q2 ¼ q3
3
3
3
x pﬃﬃﬃ
~Þ parameterization, the sym¼ sin 2 = 3. Using the ðq; m
metry axes are transformed
such that the mirror-plane is per1ﬃﬃ p
1ﬃﬃ
p
; 2 ; 0 and the threefold axis is along
pendicular to
2
p1ﬃﬃ ; p1ﬃﬃ ; p1ﬃﬃ . The symmetry axes for the point group
3
3
3
~Þ parameterization are also such that
G2 = D3h in the ðq; m
1ﬃﬃ
;
0
the vertical mirror-plane is perpendicular to p1ﬃﬃ2 ; p
2
and threefold axis is along p1ﬃﬃ3 ; p1ﬃﬃ3 ; p1ﬃﬃ3 . The rotation matrix
1ﬃﬃ
; 0 parallel to the x-axis
R is used to rotate the axis p1ﬃﬃ2 ; p
2
and the axis p1ﬃﬃ3 ; p1ﬃﬃ3 ; p1ﬃﬃ3 parallel to z-axis. R is deﬁned by
1
As shown in Ref. [22], the boundary-plane symmetries of a Laue group
can be obtained from the boundary-plane symmetries of the proper
rotational point group by adding an inversion center ð1Þ operation.
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
the product h of two
i parameters as
axis-angle
p
1ﬃﬃ
1ﬃﬃ p
1ﬃﬃ
p
p
g 4 ; ð0; 0; 1Þ g acos 3 ; 2 ; 2 ; 0 . Hence, the parameters ðq;~
lÞ are:
h
i
p
1
1 1
~
ðq;~
lÞ ¼ q; g ;ð0;0;1Þ g½acos pﬃﬃﬃ ; pﬃﬃﬃ ; pﬃﬃﬃ ;0 m
4
3
2 2
ð16Þ
2
ð17Þ
2.2. Equivalence relations
To summarize the above developments, the SAGB space
for any ﬁxed axis ~
b (except for the list in Table 1) can be
mapped to the 3-sphere (S3) with a collection of equivalence relations. The equivalence relations are in fact matrices that belong to the group O(4) (the group of 4 4
orthogonal matrices). Any equivalence relation described
in this section can be expressed as a 4 4 orthogonal
matrix as shown in the following equation:
3 2
32 3
1 0 0 0
x1
x1
6 x 7 6 0 g g g 76 x 7
B 6 7C
6 27 6
6 7
11
12
13 76 2 7
@x1 ; 4 x2 5A ðx1 ;g½g 4 x2 5Þ () 6 7 6
76 7 ð18aÞ
4 x3 5 4 0 g21 g22 g23 54 x3 5
x3
x3
x4
0 g31 g32 g33
x4
32 3
2 3 2
0 2 31 0
2 31
x1
1 0 0 0
x1
x1
x1
6 x 7 6 0 g g g 76 x 7
B 6 7C B
6 27 6
6 7C
11
12
13 76 2 7
@x1 ; 4 x2 5A @x1 ;g½g 4 x2 5A () 6 7 6
76 7
4 x3 5 4 0 g21 g22 g23 54 x3 5
x3
x3
x4
0 g31 g32 g33
x4
ð18bÞ
2
x1
31
2
x1
2
3
The relationship (x1, x2, x3, x4) (x1, x2, x3, x4) is
obtained when g = I3 in Eq. (18b) and (x1, x2, x3, x4) (x1, x2, x3, x4) is obtained by substituting g with
I3 in Eq. (18a). Using this notation, the equivalence relations on the 3-sphere can be summarized as:
2
3
x1
6x 7 1
6 27
6 7
4 x3 5
½031
x4
3
x1
7
½013 6
6 x2 7
6 7
gðg1 Þ 4 x3 5
2
x4
8g1 2 G1
3
6 x 7 1
6 27
6 7
4 x3 5
½031
ð19aÞ
2 3
x1
7
½013 6
x
6 27
6 7
gðhÞ 4 x3 5
ð19bÞ
x4
such that G2 = G1 [ h G1, G1 E G2 and j G2 =G1 j 6 2.
The complete set of equivalence relations forms a subgroup G4SAGB of O(4) and is obtained by using the closure
property of groups. The generators for the group G4SAGB
can be enumerated using the generators of
the group G1
and the single symmetry operation h. If g11 ; g21 ; . . . ; gn11
are the generators of the group G1, then 4 4 orthogonal
matrices F i1 and F2 are deﬁned as:
1
¼
½031
1
F2 ¼
½031
F i1
where Z p3 is a rotation operation of angle p/3 along the ^z
axis and 1 is the inversion operation. This above example
can be easily extended to the Laue group Oh by adding
the equivalence relation (x1, x2, x3, x4) (x1, x2, x3, x4) to the above set of relations.
0
x1
x4
The parameters (x1, x2, x3, x4) and the equivalence relations on the 3-sphere can now be described as:
3
2 3
x2
l1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
4 x3 5 ¼ 1 4 l2 5; x1 ¼ 1 x2 x2 x2
2
3
4
sinðp=6Þ
x4
l3
2 03
2 3
x2
x2
ðx1 ; x2 ; x3 ; x4 Þ x1 ; x02 ; x03 ; x04 4 x03 5 ¼ g½g1 4 x3 5 8g1 2 G1 ¼ C 3v
x204 3
2x4 3
x002
x2
ðx1 ; x2 ; x3 ; x4 Þ x1 ; x002 ; x003 ; x004 4 x003 5 ¼ g½h 4 x3 5 h ¼ Z p3 1
x004
x4
2
3073
½013
i 2 f1; 2; . . . ; n1 g
g gi1
½013
where G2 ¼ G1 [ h G1
gðhÞ
ð20Þ
The experimentally measured SAGB parameters can
now be mapped to the parameters (x1, x2, x3, x4) of the 3sphere (S3) using the transformations provided in this section. The group of equivalence relations on this 3-sphere is
the group G4SAGB with generators deﬁned in Eq. (20). The
existence of a mapping from the SAGB space to the quotient space of the 3-sphere (S3/E) suggests that functions
of these parameters, such as statistical distributions, energy
or mobility, can be described using continuous distribution
functions.
The symmetrization procedure of the hyperspherical
harmonics has been well developed and used for the expansion of both orientation [33] and misorientation distribution functions [34]. The same formulation can be directly
extended to symmetrize the hyperspherical harmonics to
represent distributions on the SAGB space using appropriate symmetries. In the case of orientation distribution functions, the ﬁrst step in the symmetrization process is
concerned with obtaining the rotation operation in SO(4)
that represents the simultaneous action of both the crystal
and sample symmetries. In the case of SAGB space, the
equivalence relation can be directly expressed as operations
of O(4) as shown in Eq. (20). The steps necessary for the
symmetrization of hyperspherical harmonics such that
functions of the SAGB space inherit the symmetries G1
and G2 are described in detail in Appendix B. To illustrate
the process of constructing continuous distribution functions for SAGB spaces, a case study describing all the
essential steps is discussed in the following section.
3. Case study
In this section we provide an example for representing
statistics of SAGB parameters using the hyperspherical
harmonic formulation. The representation of the statistics
has an added layer of complexity as compared to, for
example, a function of grain boundary energies. The
3074
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
statistics of SAGB parameters are more intuitively represented on the product space S1 S2. However, the noboundary singularity makes it diﬃcult to use the basis
functions of S1 S2 to construct continuous distribution
functions. Functions on the 3-sphere (S3) provide a convenient alternative to deal with the singularity, since we have
shown above that the space S1 S2 with the no-boundary
singularity is equivalent to the 3-sphere. The hyperspherical
harmonic expansions are especially well-suited for constructing functions that vary continuously to a singlebounded value at the zero-misorientation case (energies,
mobility, etc.). In the case of constructing functions for statistics, the added complexity comes from the deﬁnition of
the uniform distribution case and the convenience in having a constant uniform distribution function.
3.1. The uniform distribution case
The uniform distribution refers to an ideal polycrystal
whose individual grain orientations (sampled from the
space of rotations) and the normal vectors of the interfaces
(sampled from the points on a 2-sphere) are uniformly distributed and there is no correlation between grain orientations and interface normals. Let us consider a triclinic
crystal (with point group C1) for simplicity. In the case of
grain boundaries, there is a small-angle cutoﬀ below which
the grains cannot be physically distinguished. For this theoretical case, let the cutoﬀ xc be equal to zero. Since the
choice of the interface normal is completely independent
of the misorientation, the interface normals for any given
misorientation are uniformly distributed. For any given
axis of interest, if the orientations are uniformly distributed, the misorienation angle x is also uniformly distributed. Therefore, the probability distribution (p(x)) of
boundaries as a function of misorienation angle x (along
a ﬁxed axis) is a ﬁnite-constant for all
x 2 (0,
p] and zero
for x = 0. Suppose the function U x2 ; h; u (constructed
as a summation of hyperspherical harmonics) represents
the uniform distribution, then the following condition is
satisﬁed:
Z p Z 2p x2
x
; h; u sin
pðxÞ ¼
U
ðsin hÞ dh du ð21Þ
2
2
0
0
Fig. 1. A simulated set of discrete boundary-plane orientations for misorientations along the [1 0 0] axis for crystals with Oh point group. The boundaryplane orientations are represented as points on a sphere and the top (left) and bottom (right) halves of the sphere are projected using the area-preserving
projection scheme. The orientation of the Cartesian coordinate axes is shown on the top, suggesting a high population of boundary orientations along the
twist axis and the [0 0 1] crystal direction. The red dotted lines represent mirror-plane symmetries exhibited by the boundary-plane distributions. The blue
ellipses, squares and octagons represent the locations of the two, four and eightfold axes. The circled dot (in green) at the origin represents the presence of
an inversion-center symmetry. The boundary-plane symmetries are D4h point group when the misorientation anlge x < p/4 and D8h when x = p/4. (For
interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
3075
Fig. 2. Projections of the single-axis grain boundary function constructed for the data set shown in Fig. 1. It suﬃces to project only the top half of the
sphere due to the presence of the inversion-center symmetry. The red dotted lines represent the mirror-plane symmetries present in the boundary-plane
space. The blue ellipse, square and octagon represent the locations of the two, four and eightfold axes. The circled dot (in green) at the origin represents the
presence of an inversion-center symmetry. The location of the symmetry elements depends on the misorientation angle. The boundary-plane symmetries
are D4h point group when the misorientation angle x < p/4 and D8h when x = p/4. (For interpretation of the references to color in this ﬁgure legend, the
reader is referred to the web version of this article.)
Since p(x) is a constant, U x2 ; h; u is proportional to
1
2 . Therefore, the uniform distribution function is repsin ðx2 Þ
resented by a function that is not a constant on the
3-sphere. Instead of using U x2 ; h; u , we can deﬁne the
2
function U 0 x2 ; h; u ¼ U x2 ; h; u ðsin x2 Þ as the constant
function describing the probability distribution function
on the space S1 S2. This can be generalized to any distribution function f(a, h, u) such that:
Z p=2 Z p Z 2p
2
fða; h; uÞðsin aÞ ðsin hÞ da dh du
0
¼
Z
0
0
p=2
0
Z
p
0
Z
2p
f 0 ða; h; uÞðsin hÞ da dh du ¼ 1
ð22Þ
0
where f0 (a, h, u) = f(a, h, u)(sin a)2. The uniform distribution for triclinic crystals is simply:
1
a–0
2
ð23Þ
f 0 ¼ 2p
0 a¼0
This formulation can be used for any SAGB distribution
(with exceptions listed in Table 1) and the constant describing the uniform distribution will be used to deﬁne the multiples of random distribution (MRD) scale.
3.2. Point group Oh with misorientation axis (1 0 0)
The ﬁxed axis of misorientation is the fourfold symmetry axis in the Oh point group. A simulated set of data,
shown in Fig. 1, is used for the purpose of this example.
The boundary-plane normals for any given angle of misorientation are chosen to be concentrated around the z-axis
and the x-axis (twist axis). The population of points close
to the z-axis, on average, is about six times higher than
the population of points around the x-axis. The number
of interfaces for any given misorienation angle x is proportional to sin(x). This following set of data is chosen specifically to compare the distribution function with the
experimental distribution shown in Fig. 14 in Ref. [26].
The mappings and the symmetries required to construct
the SAGB distribution function for crystals with Oh point
group symmetry and with misorientations lying along the
fourfold axis [1 0 0] are obtained by using the formulation
in Section. 2. For this high-symmetry SAGB space, xmax is
equal to p/4. The boundary-plane symmetries are such that
G1 = D4h and G2 = D8h as mentioned in Ref. [22] and
detailed in the Supplementary Information. The axes of
the point group D4h are such that one of the mirror-planes
is perpendicular to ~
a1 ¼ ð0; q0 ; q1 Þ and the fourfold axis is
3076
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
Fig. 3. A simulated set of discrete boundary-plane orientations for misorientations along the [1 1 0] axis for crystals with Oh point group. The orientation
of the Cartesian coordinate axes is shown on the top, suggesting a high population of boundary orientations along the twist axis and the [0 0 1] crystal
direction. The distribution of points is identical to the distribution shown in Fig. 1 except for the twist axis, which is now located along the [1 1 0] axis. The
red dotted lines represent the mirror-plane symmetries present in the boundary-plane space. The blue ellipses represent the locations of the twofold axes.
The circled dot (in green) at the origin represents the presence of an inversion-center symmetry. The symmetry exhibited by the boundary-plane
orientations is D2h. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 4. Projections of the single-axis grain boundary function constructed for the data set shown in Fig. 3. It suﬃces to project only the top half of the
sphere due to the presence of the inversion-center symmetry. The red dotted lines represent the mirror-plane symmetries present in the boundary-plane
space. The blue ellipses represent the locations of the twofold axes. The circled dot (in green) at the origin represents the presence of an inversion-center
symmetry. The location of the symmetry elements depends on the misorientation angle. The symmetry exhibited by the boundary-plane orientations is
D2h. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
along ~
a2 ¼ ð1; 0; 0Þ, where q0 ¼ cos x2 and q1 ¼ sin x2 .
~Þ parameterization, the symmetry axes are
Using the ðq; m
a2 ¼ ð1 0 0Þ. The
transformed such that ~
a1 ¼ ð0 1 0Þ and ~
~Þ
symmetry axes for the point group G2 = D8h in the ðq; m
are also such that the mirror-plane is perpendicular to
(0 1 0) and the eightfold axis is along (1 0 0). The rotation
matrix R is used to rotate ~
a1 and ~
a2 parallelto the z- and
the x-axes respectively and is given by g p2 ; ð0; 1; 0Þ .
The parameters ðq;~
lÞ and (x1, x2, x3, x4) and the equivalence relations on the 3-sphere are described as:
~
ðq;~
lÞ ¼ q; g p2 ; ð0; 1; 0Þ m
2 3
2 3
l1
x2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
6 7
6 7
1
4 l2 5; x1 ¼ 1 x22 x23 x24
4 x3 5 ¼ sinðp=8Þ
x4
l3
2 03
2 3
x2
x2
6 7
6 7
ðx1 ; x2 ; x3 ; x4 Þ x1 ; x02 ; x03 ; x04 4 x03 5 ¼ g½g1 4 x3 5 8g1 2 G1 ¼ D4h
x04
x4
ðx1 ; x2 ; x3 ; x4 Þ ðx1 ; x2 ; x3 ; x4 Þ
2 00 3
2 3
x
x2
2
6 7
00 00 00 6 00 7
ðx1 ; x2 ; x3 ; x4 Þ x1 ; x2 ; x3 ; x4 4 x3 5 ¼ g½h 4 x3 5 h ¼ Z p4 1
x004
x4
ð24Þ
The generators for the symmetrization procedure are:
1
½013
g g11 ¼ gðM ^x Þ;
F 11 ¼
1
½031 gðg1 Þ
where M ^x is the mirror-plane perpendicular to the x-axis:
1
½013
2
F1 ¼
ð25Þ
½031 I 33
hp
i
1 ½013
; ð0; 0; 1Þ I
F2 ¼
gðhÞ ¼ g
4
½031 gðhÞ
The symmetry operation corresponding to p/2 rotation along the z-axis (for D4h point group) is not considered because F2 F2 represents this operation. Using the
above generators, the symmetrized hyperspherical harmonics are constructed for the simulated data shown in
Fig. 1 by ﬁtting to a continuous function obtained using
the summation of these symmetrized hyperspherical harmonics. The SAGB distribution function is plotted in
Fig. 2. A complex pattern in the distribution function
arises from a simple set of simulated data owing to the
arrangement and the dependence of the symmetry elements of the grain boundary plane spaces. For example,
the position of symmetry elements directly explains the
bimodal-like distributions that arise in the misorientation
angle range x 2 [25°, 35°]. The SAGB distribution function also closely resembles the experimental distributions
obtained in Ref. [26]. The signiﬁcant improvement in this
case, though, is that the relative magnitudes of populations can also be inferred across various misorienation
angles.
In order to elucidate the inﬂuence of boundary-plane
symmetries in the patterns observed, a simulated data set
similar to Fig. 1 is used to compute the probability density
3077
function with the misorientation axis along the [1 1 0] crystal direction. The simulated distributions binned in discrete
categories are shown in Fig. 3 with a high density of
boundary-plane orientations concentrated along the
[0 0 1] axis and along the [1 1 0] twist axis. The mappings
necessary to obtain the SAGB function along the [1 1 0]
misorientation axis for the octahedral point group are
detailed in Section. 3.1.9 of the Supplementary Information. The distribution function is plotted in Fig. 4 and a
bimodal-like distribution similar to the plot in Fig. 2 is
observed. However, the peaks are located perpendicular
to the [1 1 0] axis in Fig. 4 (as opposed to the location of
the peaks perpendicular to the [1 0 0] axis in Fig. 2). The
number of peaks perpendicular to the twist axis are also
diﬀerent: four and two in the distributions in Figs. 2 and
4, respectively. These diﬀerences arise solely due to the different point group symmetries exhibited by the boundaryplane distributions: D4h and D2h for [1 0 0] and [1 1 0] misorientation axes, respectively. Since the symmetrized
hyperspherical harmonics (described in Section 2.2) are
used, the distribution functions inherently represent these
symmetries.
4. Conclusions
A rigorous framework for constructing continuous functions of the single-axis grain boundary parameters has been
developed. If the misorientation axis is ﬁxed to be along a
crystal direction (with exceptions listed in Table 1), the
space of grain boundary parameters is shown to be topologically equivalent to the 3-sphere (S3) with appropriate
equivalence relations. Since the hyperspherical harmonics
form an orthonormal basis on the 3-sphere, any general
square-integrable function of the SAGB parameters can
be expressed as a series sum of these harmonics.
Owing to the symmetries of the boundary space, the
boundary-plane orientations exhibit a diverse array of
point-group symmetries. Therefore, a general method is
outlined to obtain symmetrized hyperspherical harmonics
to represent SAGB parameters. The equivalence relations
on the boundary parameters can be expressed as 4 4
orthogonal matrices and the complete set of symmetry
operations can be obtained by the set of generators
described in Eq. (20). In Section. 3, an example for constructing a continuous statistical distribution function of
boundary parameters along the ﬁxed fourfold symmetry
axis (1 0 0) (for point group Oh) is discussed. Rich patterns
in the boundary-plane orientations are possible because of
the dependence of the symmetry elements (n-fold axes, mirror-planes, etc.) on the corresponding misorientation.
The framework provided here is the ﬁrst of its kind with
respect to the fact that the continuous functions involve
simultaneously the misorientation angle and the boundary
orientations. These functions should be useful in the microstructural analysis of ﬁber-textured materials and, as
shown in Fig. 2, in representing statistical boundary-plane
distributions along a ﬁxed axis.
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S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
~Þ
Appendix A. Mathematical aspects of the ðq; m
parameterization
The mapping presented in Eq. (4) is inspired by the symmetric parameterization provided by Morawiec [32]. As
detailed there, a rotation matrix N is deﬁned, which corresponds to the same rotation axis as M, but with a rotation
angle half of that for M, such that N N ¼ M. This deﬁnition is well-deﬁned for rotation angles less than p. If M corresponds to a rotation of p along an axis ~
b, there exist two
equivalent descriptions ðp; ~
bÞ and ðp; ~
bÞ in axis-angle
parameters. Hence a convention is assumed for selection of
the axis if M corresponds to a rotation of p. The normal vector is deﬁned as ~
r ¼ N 1 ð~
nÞ. The advantage of this representation of grain boundaries is that the grain exchange
symmetry is considerably simpliﬁed, and is expressed as:
rÞ
ðN ;~
rÞ ðN 1 ; ~
ðA1Þ
Even though this parameterization is convenient for representing boundary parameters, the symmetric parameterization does not account for the “no-boundary” singularity,
since diﬀerent boundary inclinations linked to M = I give
distinct representations. Additionally, no matter what convention is used to select the rotation axis for misorientations
of angle p, there is a discontinuity in the mapping from ðM;~
nÞ
to ðN ;~
rÞ (unless an additional equivalence relation or a new
metric is introduced) as illustrated in the following section.
A.1. Discontinuity in the symmetric parameterization
The discontinuity in the symmetric parameterization
arises due to the convention necessary to deﬁne this representation. For example, consider two grain boundary parameters ðM 1 ; ~
nÞ ¼ ððx; ~
bÞ; ~
nÞ and ðM 2 ; ~
nÞ ¼ ððx; ~
bÞ; ~
nÞ. As
x ! p; ðM 1 ; ~
nÞ ! ðM; ~
nÞ ¼ ððp; ~
bÞ; ~
nÞ and ðM 2 ; ~
nÞ !
ðM; ~
nÞ ¼ ððp; ~
bÞ; ~
nÞ. Since ðp; ~
bÞ ðp; ~
bÞ, which is the
trivial symmetry of the rotation space, the distance
dððM 1 ; ~
nÞ; ðM 2 ; ~
nÞÞ approaches zero as x ! p. d is the metric on the product space of SO(3) S2. If the mapping
f : ðM; ~
nÞ ! ðN ; ~
rÞ is continuous, then dðf ðM 1 ; ~
nÞ; f ðM 2 ;
~
nÞÞ > must approach zero as x ! p. But
p p
b ;
; ~
b ~
n
lim ðN 1 ; !
r1 Þ ¼ g ; ~
x!p
2
p
2p ðA2Þ
lim ðN 2 ; !
b ;
;~
b ~
n
r2 Þ ¼ g ; ~
x!p
2
2
And
lim dðfðM 1 ; ~
nÞ; fðM 2 ; ~
nÞÞ
x!p
¼ dððN 1 ; ~
r1 Þ; ðN 2 ; ~
r2 ÞÞ – 0
ðA3Þ
Since d(N1, N2) – 0 and d(r1, r2) – 0. Hence the mapping proposed for the symmetric representation is not continuous and is not topologically consistent (not a
homeomorphism) if the same metric d is used for the
parameterization ðN ;~
rÞ. One solution is to deﬁne a metric
d0 on the space ðN ;~
rÞ such that d0 (N1, N2) ! 0 as x ! p.
It is also possible to remedy these problems by using the
quaternion parameterization of the rotation space. In order
to obtain a rigorous and topologically consistent mapping
from the grain boundary space to the symmetric representation, the two copies of the rotation space, i.e. the complete 3-sphere with antipodal identiﬁcations, will be
considered. As mentioned in Section. 2, we also show that
the “no-boundary singularity” is essential for a topologically consistent deﬁnition of the symmetric representation.
~Þ parameterization
A.2. Topological consistency of the ðq; m
Using the quaternion parameterization, i.e. B ¼ ðq;~
nÞ,
the grain boundary space is equivalent to S3 S2/E, where
the set of equivalence relations E include:
ðq;~
nÞ ðq;~
nÞ
i 1
ðA4aÞ
i 1
j
ðq;~
nÞ ððS Þ qðS Þ; gððS Þ Þ ~
nÞ ðA4bÞ
ðq;~
nÞ ðq1 ; gðq1 Þ ð~
nÞÞ
ðð1; 0; 0; 0Þ;~
nÞ ðð1; 0; 0; 0Þ; n~0 Þ
ðA4cÞ
ðA4dÞ
ðA4Þ
The equivalence relation in Eq. (A4a) is this antipodal
symmetry which is also referred to as the trivial symmetry
of the rotation space. The relations in Eqs. (A4b–d) are
obtained by expressing Eq. (1) in the quaternion
parameterization.
Since the symmetric parameterization is related to the
axis-angle parameters of the misorientation, it will be useful to deﬁne the function that relates the axis-angle parameters to the quaternion space as A:[0, 2p] S2 ! S3:
x
x
x
x
Aðx; ~
bÞ ¼ cos
; bx sin
; by sin
; bz sin
2
2
2
2
ðA5Þ
The domain of the function A is [0, 2p] S2, since we let
x 2 [0, 2p] and the axis ~
b 2 S 2 . It is important to note that
A is not an injective map since Að0; ~
bÞ ¼ ð1; 0; 0; 0Þ and
Að2p; ~
bÞ ¼ ð1; 0; 0; 0Þ for any ~
b 2 S 2 . But if x – {0,2p},
the mapping A is injective. Now, we deﬁne the mapping
~Þ, where ðq; ~
P : ðq;~
rÞ ! ðq; m
mÞ is the new desired parameterization and ~
r is deﬁned as CðqÞ ð~
nÞ (Eq. (2)).
(
~Þ ¼ Pðq;~
ðq; m
rÞ ¼
ðq; g½pðqÞ ½~
rÞ
ðð1; 0; 0; 0Þ; ~
0Þ
if q0 – 1
if q0 ¼ 1
ðA6Þ
b
p(q), for q0 – ±1, is deﬁned such that gðpðqÞÞ ¼ g x2 ; ~
where ðx; ~
bÞ ¼ A1 ðqÞ. There is a one-to-one correspondence
between quaternions q and axis-angle parameters ðx; ~
bÞ
where x 2 (0,2p) and ~
b 2 S 2 . Hence, for all quaternions that
do not correspond to identity rotations (x – {0,2p} or
equivalently q0 – ±1), ðx; ~
bÞ ¼ A1 ðqÞ is well-deﬁned. In
terms of quaternion parameters, p(q) takes the form:
1
pðqÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð 1 þ q0
2ð1 þ q0 Þ
q1
q2
q3 Þ
ðA7Þ
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
Even though the function A1 is not well deﬁned for
x = 0, i.e. q0 = 1, the function p(q) is well-deﬁned. When
x = 0, the rotation axis ~
b is undetermined but this is not
an issue in the deﬁnition of matrix g[p(q)]. If misorientation q corresponds to a rotation angle x = 0, the rotation
p(q) corresponds to a rotation angle x2 ¼ 0, and hence it
is not necessary to determine the axis of rotation ~
b.
But when x = 2p, i.e. q0 = 1, the function p(q) is undeﬁned. In the deﬁnition of the function P, p(q) appears
only if q0 – 1. The abrupt change in the deﬁnition of
P in the neighborhood of q0 = 1 does not result in
any discontinuity of the mapping. To prove that the
mapping is well-deﬁned we prove the following
condition:
~Þ ¼ Pðð1; 0; 0; 0Þ;~
rÞ
lim Pðq; m
x!2p
¼ ðð1; 0; 0; 0Þ; ~
0Þ for all ~
b
2 S 2 where A1 ðqÞ ¼ ðx; ~
bÞ
ðA8Þ
Since P ðq;~
rÞ ¼ ðq; g½pðqÞ ½CðqÞð~
nÞÞ, we obtain
lim Pðq;~
rÞ ¼ lim ðq; g½pðqÞ ½CðqÞð~
nÞÞ
x!2p
hx i
b ½CðqÞð~
nÞ
¼ lim q; g ; ~
x!2p
2
¼ ðð1; 0; 0; 0Þ; g½p; ~
b ½ lim CðqÞð~
nÞÞ
x!2p
ðA9Þ
x!2p
x
x 1
~ ¼ gðp; ~
~
~0 ¼ g p ; ~
b ½g ; ~
b m
bÞ m
m
2
2
~Þ ðq; gðp; ~
~Þ
) ðq; m
bÞ m
3079
ðA12Þ
~Þ ðq; gðp; ~
~Þ is the
Hence, the relation ðq; m
bÞ m
trivial symmetry of the rotation space expressed in the symmetric parameterization.
The grain exchange symmetry can be derived in a similar
fashion. If q = (q0, q1, q2, q3), then there are two possibilities for q1 = ±(q0, q1, q2, q3), which are related to
each other by the trivial symmetry. In this formulation,
we
use
q1 = (q0, q1, q2, q3).
Hence,
if
1
~
A ðqÞ ¼ ðx; bÞ, then A1 ðq1 Þ ¼ ðx; ~
bÞ. The grain
exchange symmetry is now obtained as follows:
hx
i
b ~
rÞ
Pðq; ~
rÞ Pðq1 ; gðq1 Þ ~
rÞ ) ðq; g ; ~
2
h
i
x
ðq1 ; g p ; ~
b g½x; ~
b ~
rÞ
2
And
h
i
h
i1
~ ¼ g x2 ; ~
~
m
b ~
r )~
r ¼ g x2 ; ~
b
m
h
i
~0 ¼ g p x2 ; ~
b g½x; ~
b ~
r
m
h
i
h
i1
b g½x; ~
b g x2 ; ~
b
~
m
¼ g p x2 ; ~
ðA13Þ
ðA14Þ
¼ ~
m
The important requirement here is limx!2pC(q) = 0,
which is trivially satisﬁed since q0 ! 1 and
max{((q G)0)2} ! 1. Even though the matrix g½p; ~
b
depends on the choice of ~
b, the scaling function approaches
zero as x ! 0 and hence the mapping is continuous in the
neighborhood of q0 = 1.
A.3. Equivalence relations
The equivalence relations expressed using the symmetric
~Þ depend upon the type of symmetry under
parameters ðq; m
consideration. For example, the trivial symmetry of the
rotation space in the quaternion parameterization implies
that P ðq;~
rÞ P ðq;~
rÞ. This equivalence relation expressed
~Þ is:
in terms of the parameters ðq; m
x
Pðq; ~
rÞ Pðq;~
rÞ ) ðq; g ; ~
b ~
rÞ
2
x
q; g p ; ~
b ~
r
ðA10Þ
2
where ðx; ~
bÞ ¼ A1 ðqÞ. According to the deﬁnition of
~Þ, we observe that
ðq; m
~
m¼g
x
; ~
b
2
~
r )~
r ¼ ½g
b ~
r
and ~
m0 ¼ g p x2 ; ~
x
; ~
b
2
1
~
m
ðA11Þ
~0 , we obtain
By substituting ~
r in the equation of m
Therefore, the grain exchange symmetry is given by:
~Þ ððq0 ; q1 ; q2 ; q3 Þ; ~
ððq0 ; q1 ; q2 ; q3 Þ; m
mÞ
ðA15Þ
Similarly, the equivalence relations due to the rotational
symmetry operations can be expressed as:
r
Pðq;~
rÞ P ðS i Þ1 qðS j Þ; g½ðS i Þ1 ~
) ðq; g½pðqÞ ~
rÞ
h
i h
i 1
1
1
ðS i Þ qðS j Þ; g pððS i Þ qðS j ÞÞ g ðS i Þ
~
r
ðA16Þ
And
~ ¼ g½pðqÞ ~
~
m
r )~
r ¼ g½pðqÞ1 m
h
i h
i
1
i 1
j
0
r
m
~ ¼ g pððS Þ qðS ÞÞ g ðS i Þ ~
i
h
i h
1
i 1
j
~
¼ g pððS Þ qðS ÞÞ g ðS i Þ g½pðqÞ1 m
i
h
i h
i 1
j
i 1
j
) ðq;~
mÞ ðS Þ qðS Þ;g pððS Þ qðS ÞÞ g ðS i Þ1 g½pðqÞ1 m
~
ðA17Þ
Appendix B. Hyperspherical harmonics for the single-axis
grain boundary space
Any square-integrable function deﬁned on the 3-sphere
can be expanded as a linear combination of the hyperspherical harmonics [33], which form the standard basis functions on the 3-sphere (analogous to spherical harmonics
on the 2-sphere and the Fourier series on a unit-circle).
3080
S. Patala, C.A. Schuh / Acta Materialia 61 (2013) 3068–3081
The hyperspherical harmonics are provided here as a function of the angular coordinates (a, h, u), where x1 = cos(a),
x2 = sin(a) sin(h) cos(u), x3 = sin(a) sin(h) sin(u) and x4 = sin(a) cos(h). The complex Eq. (B1a) and the real hyperspherical harmonics Eq. (B1b and c) are as follows:
L
pﬃﬃ K expðiMhÞ
Z NL;M ða;h;uÞ ¼ ðiÞ
ðB1aÞ
2
h
i pﬃﬃﬃ
M
LþM
N
N
L
K cosðMuÞ
ðB1bÞ
Z NC
L;M ¼ i ð1Þ Z L;M þ Z L;M = 2 ¼ ð1Þ
h
i pﬃﬃﬃ
M
LþM
L1
ð1Þ Z NL;M Z NL;M = 2 ¼ ð1Þ
K sinðMuÞ
Z NS
L;M ¼ i
1
h
i
2
L
M
½sinaL C Lþ1
K ¼ 2 pL! ð2L þ 1Þ ðLMÞ!ðNþ1ÞðNLÞ!
NL ðcosðaÞÞ P L ðcosðhÞÞ ðB1cÞ
ðLþMÞ!ðNþLþ1Þ!
ðB1Þ
with integer indices 0 6 N, 0 6 L 6 N and L 6 M 6 L.
The Gegenbauer polynomial C Lþ1
NL and the associated
Legendre function P M
have
been
previously deﬁned in
L
Refs. [33,34]. Since probability distribution functions are
real-valued, it suﬃces to use the real harmonics Eq.
(B1b,c). A continuous probability distribution of SAGB
parameters may be expanded as a linear combination of
the hyperspherical harmonics Z NL;M ða; h; uÞ:
fða; h; uÞ ¼
"
1
N
X
X
NC NC
fL;0
Z L;0
N¼0;1;2... L¼0
#
L X
NC NC
NS NS
þ
fL;M
Z L;M þ fL;M
Z L;M
M¼1
ðB2Þ
NC
NS
The coeﬃcients fL;M
and fL;M
are determined from the
inner product of the function f(a, h, u) and the appropriate
harmonics. It is also necessary that the function f inherit
the symmetries of the boundary space.
B.1. Symmetrization procedure
The method of obtaining symmetrized hyperspherical
harmonics to describe orientation distribution functions
(ODFs) has been described in complete detail by Mason
and Schuh [33]. The symmetrization process consists of
ﬁrst obtaining rotation operations in SO(4) that are equivalent to applying the crystal and sample symmetries simultaneously. As developed in this paper, the equivalence
relations for SAGB spaces directly consist of 4 4 orthogonal matrices as shown in Eq. (20). The following steps
detail the symmetrization process of the hyperspherical
harmonic of order N:
(i) Given the axis and crystallographic point group
under consideration, the generators of the group
G4SAGB are ﬁrst enumerated.The subsequent steps mentioned here are already described in Ref. [33]. The
(N + 1)2-dimensional irreducible representatives of
the generators of the group G4SAGB are determined. If
the generator belongs to SO(4) the method outlined
in Ref. [33] is followed to obtain its irreducible representative. If the generator does not belong to SO(4),
it can be written out as a product of hinv and a matrix
g that belongs to the group SO(4). The irreducible
representative of hinv is described in Ref. [34]. The
irreducible representative of the generator that does
not belong to the group SO(4) can now be obtained
as a product of the irreducible representatives of hinv
and g.
(ii) The simultaneous eigenvectors of eigenvalues of unity
for all the irreducible representatives, of the generators obtained in the previous step, are calculated.
(iii) An orthonormal system is constructed using the linearly independent eigenvectors obtained in step (iii).
As mentioned in Ref. [33], the components of these
orthonormal vectors provide the coeﬃcients necessary to ﬁnd the symmetrized hyperspherical harmonic
of order N.
(iv) The symmetrized hyperspherical harmonic of order)N
corresponding to the kth eigenvector is denoted as Z Nk
[33] and is expressed as a linear combination of the
hyperspherical harmonic of order N as:
)
Z Nk ¼
N )
L )
X
X
)
;C;k N ;C
;S;k N ;S
N ;C
f NL;0;C;k Z L;0
þ
f NL;M
Z L;M þ f NL;M
Z L;M
L¼0
)
ðB3Þ
M¼1
)
;C;k
;S;k
and f NL;M
are the components of the kth linewhere f NL;M
arly independent simultaneous eigenvector. Any continuous distribution function on the SAGB space can be
expressed as a linear combination of symmetrized hypersphercial harmonics as:
f ða; h; uÞ ¼
1 KðN
X
XÞ ) )
f Nk Z Nk ða; h; uÞ
ðB4Þ
N¼0;1;... k¼1
where K(N) is the number of linearly independent) eigenvectors for a particular value of N. The coeﬃcients f Nk are obtained by )the inner product of the symmetrized basis
functions Z Nk and f. These coeﬃcients are enumerated in
the Supplementary Information for diﬀerent SAGB spaces
with the ﬁxed axis lying along a high-symmetry crystal
direction (e.g. a twofold roational symmetry axis in crystals
with point group D222(222)).
Appendix C. Supplementary data
Supplementary data associated with this article can be
found, in the online version, at http://dx.doi.org/10.1016/
j.actamat.2013.01.067.
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