Hyperspherical harmonics for the representation of crystallographic texture

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Acta Materialia 56 (2008) 6141–6155 www.elsevier.com/locate/actamat

Hyperspherical harmonics for the representation of crystallographic texture J.K. Mason, C.A. Schuh * Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 30 May 2008; received in revised form 13 August 2008; accepted 14 August 2008 Available online 25 September 2008

Abstract The feasibility of representing crystallographic textures as quaternion distributions by a series expansion method is demonstrated using hyperspherical harmonics. This approach is refined by exploiting the sample and crystal symmetries to perform the expansion more efficiently. The properties of the quaternion group space encourage a novel presentation of orientation statistics, simpler to interpret than the usual methods of texture representation. The result is a viable alternative to the Euler angle approach to texture standard in the literature today. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Crystallographic texture; Orientation distribution functions; Quaternion distribution; Spectral decomposition

1. Introduction Significant advances have been made recently in the field of texture and texture analysis. For example, the combination of instrumented electron-backscatter diffraction with either focused ion beam [1] or metallographic serial sectioning [2] allows relatively rapid determination of the local crystallographic orientations throughout an entire threedimensional volume of a polycrystalline microstructure. Furthermore, these advanced capabilities facilitate the engineering of texture-dependent properties through emerging mathematical frameworks, including ‘‘microstructure sensitive design” [3,4]. Indeed, recent developments in materials design often focus upon the explicit manipulation of crystallographic texture to achieve properties dramatically surpassing those expected on the basis of material chemistry alone [5–8]. The measurement and subsequent analysis of orientation information is usually performed within the Euler angle parameterization of orientations. In a sense, this is the edifice upon which the entire field of texture measure*

Corresponding author. E-mail address: [email protected] (C.A. Schuh).

ment, representation and engineering is built; every aspect of this field benefits from the enormous effort invested by the scientific community in developing the mathematical aspects of orientation analysis using Euler angles. One of the principal strengths of this parameterization, particularly with respect to texture representation, is the ability to express an arbitrary orientation distribution defined over the space of the Euler angles as a linear combination of standard basis functions. This property is sometimes considered to be unique to Euler angles among the various parameterizations of three-dimensional rotations [9]. The profound utility of this expansion is demonstrated by, for example, Roe [10] and Bunge [11], and underlies a substantial portion of the completed and ongoing research into crystallographic texture [12–15]. Nevertheless, certain difficulties remain inherent in the Euler angle parameterization of three-dimensional rotations. Although three parameters suffice to characterize a rotation, the topological properties of SO(3) preclude the existence of a three-dimensional parameterization that includes every orientation and is simultaneously non-singular [16]. This is manifested in the Euler angles by the wellknown degeneracy wherein certain orientations, including the identity, are represented by an infinite number of points

1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.08.031

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in the group space [11,17,18]. A distortion of the metric tensor [19], the related degeneracy of the invariant volume element [11] and singularities in the equations of motion [16,17] follow as a result. This severe non-uniqueness causes some small changes in a physical orientation to correspond to abrupt jumps in the values of the Euler angles, posing an inconvenient complication to the calculation of disorientations and the tracking of orientations during texture evolution. Furthermore, the present authors along with other researchers [20–22] find that the Euler angles are not intuitive; a particular physical orientation does not intuitively relate to a triplet of Euler angles, partly due to the difficulty of visualizing three successive rotations, and partly to the asymmetric treatment of the coordinate axes [21–23]. The calculation of grain misorientations, by extension, suffers from the same complexities, and requires equations involving trigonometric functions, inverse trigonometric functions and singularities [24,25]. Despite the achievements of the field using the Euler angles (e.g., Refs. [11,26,27]), the shortcomings outlined above render them either difficult or impractical to use for certain situations in the analysis of crystallographic texture. The authors propose that a series expansion for orientation distributions similar to the existing one in terms of Euler angles, but using a different parameterization, would be valuable. Indeed, the crystallography community already employs a variety of other orientation parameterizations (e.g., rotation matrices, axis-angle pairs, Rodrigues vectors and quaternions) and exploits the distinct properties of each as necessary. To the extent that it is possible without detailed explanations or qualifications, a comparison of these parameterizations is presented in Table 1, with emphasis on a few specific properties that are desirable for texture representation and analysis; these include 1. A simple multiplication rule for combining successive rotations. 2. An intuitive physical interpretation. 3. The absence of singular orientations. 4. The ability to express distribution functions in an explicit mathematical form. At least with regard to the characteristics assembled in Table 1, the quaternion parameterization appears to offer many advantages. This parameterization is actually reasonably common in the field, having been used for texture

analysis [23,28–31], disorientation and mean orientation calculations [21,32–38], expression of certain specific textures [39–41] and in connection with the equations of texture evolution [19,22]. Quaternions appear in related fields of materials science as well, as in the development of the theory of coincident site lattices [33,42,43] and in the investigation of the symmetry groups for modulated crystals and quasicrystals [44]. The purpose of this paper is to advance the use of the quaternion parameterization as an alternative to the Euler angles in crystallographic texture analysis. First, the expression of an arbitrary orientation distribution within the quaternion parameterization is illustrated as a series expansion over the hyperspherical harmonics, analogous to the generalized spherical harmonics commonly used with the Euler angle parameterization. This expansion does not appear to be known within the materials science literature, with some authors [9] even lamenting the absence of an alternative to the series expansion expressed in terms of Euler angles. Nevertheless, at least since Fock [45], the hyperspherical harmonics and the related series expansion have frequently been used in other fields, e.g., Refs. [46– 53]; the expansion is presented here in order to explicitly point out its utility to the materials science community and to place the remainder of the paper in context. Second, symmetrized hyperspherical harmonics are constructed consistent with various crystal and sample symmetries to facilitate the use of this expansion for the representation of textures. The symmetrization of the hyperspherical harmonics in the context of crystallographic symmetry is novel to the authors’ knowledge, as are the symmetrized harmonics themselves. Finally, symmetrized harmonics are presented for samples of orthorhombic symmetry and crystals with the proper rotational symmetries of all the Laue groups, and methods of visualizing texture information more transparent than the conventional sections of Euler space used widely in the materials science literature are discussed. 2. Quaternions As suggested in Table 1, the quaternion parameterization offers many advantages of interest to the crystallography community. This section provides a brief overview of the properties of quaternions to support later developments; the quaternion parameterization is more completely described in, for example, Refs. [54,55].

Table 1 Overview of the parameterizations of rotations frequently used within the crystallography community; generally, discrete orientations are parameterized using rotation matrices, and continuous orientation distributions are parameterized using the Euler angles. Characteristics

Rotation matrices

Euler angles

Axis-angle

Rodrigues vectors

Quaternions

Simple multiplication rule Intuitive interpretation Absence of singular orientations Series expansion

Yes No Yes No

No No No Yes

No Yes No No

Yes Yes No No

Yes Yes Yes Present work

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A quaternion q is often represented as a vector in a fourdimensional vector space over the field of real numbers, that is q ¼ q0 þ q1 i þ q 2 j þ q 3 k

ð1Þ

where the unit quaternions 1, i, j and k form the basis. Henceforth, the term quaternion will refer specifically to a normalized quaternion, i.e., q20 þ q21 þ q22 þ q23 ¼ 1, which describes a point on the surface of the unit sphere in fourdimensional space. A quaternion is related to an active rotation of three-dimensional space by the angle x about the unit vector n, pointing along the axis of rotation, by q ¼ ðq0 ; qÞ ¼ ½cosðx=2Þ; n sinðx=2Þ

ð2Þ

where q0 is conventionally referred to as the scalar part, and q as the vector part. The relationship between the axis-angle and quaternion parameterizations appearing in Eq. (2) is depicted explicitly in Fig. 1. Exactly as changing the sign of the components of n in the axis-angle parameterization inverts a rotation, changing the sign of the components of q of a quaternion q forms the quaternion corresponding to the inverse rotation or, more simply, the inverse quaternion q1. However, while increasing the angle of a rotation by 2p results in an orientation indistinguishable from the original one, the above correspondence indicates that this increase in rotation angle changes the sign of every component of q. More explicitly, a particular orientation of three-dimensional space corresponds to the two distinct quaternions +q and q, and the corresponding rotation operations result in the same orientation, differing in rotation angle by 2p. Therefore, although rotation angles in the domain 0 6 x 6 p are sufficient to describe every unique orientation of three-dimensional space, the above construction requires the domain 0 6 x 6 2p to describe the rotation operations corresponding to every distinct quaternion. The multiplication law for quaternions follows directly from the definition of multiplication for the basis quaternions 1, i, j and k, and is a linear function with respect to

Fig. 1. Relationship shared by the axis-angle parameterization of a rotation, the quaternion parameterization of a rotation and the parameterization of a quaternion by three angles. (a) A three-dimensional rotation by the angle x = 2a about the unit vector n, pointing along the axis of rotation. The direction of n is specified by the angles h and /. (b) The vector part q of the quaternion q corresponding to the rotation in (a). The vectors q and n point in the same direction, though the length of q is sin(x/2) rather than one.

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all the quaternion parameters [54]. While other parameterizations exhibit a bilinear composition rule, the quaternion parameterization is the representation of the smallest dimension for which this is the case [16]. The composition rule is often written as a vector equation containing the dot and cross-products in the form ½p0 ; p½q0 ; q ¼ ½p0 q0  p  q; p0 q þ q0 p þ p  q

ð3Þ

and provides the single rotation equivalent to the rotation q followed by the rotation p.1 As defined, quaternions operate in the same order as rotation matrices, from right to left. The unusual bilinearity of this composition allows the formulation of equivalent expressions using four-byfour orthogonal matrices of unit determinant [56], which will be made use of later in this paper. As described by Frank [23], a number of related parameterizations of rotations may be formed by scaling the unit vector n, pointing along the axis of rotation, by an elementary function of x, the rotation angle, thereby giving a three-dimensional parameterization. The advantages of these parameterizations include, for example, a definition in terms of readily recognizable physical quantities [25], and a symmetric treatment of the coordinate axes [23]; these properties enable the straightforward visualization of the rotation’s physical effect, as reflected in Table 1 for the axis-angle, Rodrigues vector and quaternion parameterizations. Note that the relevant three-vector of the quaternion parameterization, for purposes of interpretation at least, is construed as the vector part q of the quaternion q. Later in this paper, an intuitive representation of the quaternion group space is discussed in more detail, in the context of texture analysis. Contrasting markedly with other parameterizations currently in use, the quaternion parameterization includes no singular points [16]. As the quaternion parameterization handles rotations about all three axes equally in the sense of having a complete set of infinitesimal generators [25], quaternions avoid the singularity in the vicinity of the identity operation that is present for the Euler angles. Meanwhile, the singularities and discontinuities in the values of the axis-angle and Rodrigues parameters in the vicinity of rotations by p do not appear in the quaternion parameterization. This is actually intimately related to the correspondence of two quaternions to a single orientation; specifically, the quaternion group is isomorphic to the simply connected covering group SU(2), and is related to SO(3) by a 2-to-1 homomorphism [55]. The absence of singularities is not only convenient, but practically important for numerical calculations. 3. Hyperspherical harmonics The expansion of an arbitrary square-integrable function defined on the unit circle in two-dimensional space 1 ‘‘The rotation q” is occasionally written as an abbreviation for ‘‘the rotation corresponding to the quaternion q”.

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using a Fourier series is well known, as is the analogous expansion on the unit sphere in three-dimensional space using spherical harmonics. Similarly, a probability distribution of crystalline orientations, expressed as quaternions and hence residing on the unit sphere in four-dimensional space, may be expanded as a linear combination of the hyperspherical harmonics. These functions, while not as familiar to the general scientific community as spherical harmonics, frequently appear in a complex-valued form in the physics literature on the quantum theory of angular momentum [47,49,57,58]. Exactly as a point on a circle is conveniently described by an angle, and a point on a sphere by a pair of angles, a quaternion residing on the unit sphere in four-dimensional space is parameterized by three angles; namely, a hyperspherical angle a and the spherical angles h and /, constrained to the values 0 6 a 6 p, 0 6 h 6 p and 0 6 / < 2p. These relate to the coordinates of the normalized four-vector q by [46,50,53,59] q0 ¼ cos a q1 ¼ sin a sin h cos / q2 ¼ sin a sin h sin / q3 ¼ sin a cos h

ð4Þ

0

0

0

Although the construction is straightforward, these real forms do not, to the authors’ knowledge, appear in the literature. Explicitly, they are given by  1=2 L ðL  MÞ! ðN þ 1ÞðN  LÞ! LþM 2 L! MC ð2L þ 1Þ Z NL ¼ ð1Þ p ðL þ MÞ! ðN þ L þ 1Þ! M ðsin aÞL C Lþ1 N L ðcos aÞP L ðcos hÞ cosðM/Þ  1=2 L ðL  MÞ! ðN þ 1ÞðN  LÞ! LþM 2 L! ð2L þ 1Þ ¼ ð1Þ p ðL þ MÞ! ðN þ L þ 1Þ! L

M ðsin aÞ C Lþ1 N L ðcos aÞP L ðcos hÞ sinðM/Þ

ð6Þ

ð7Þ

¼ d dMM 0 dLL0 dNN 0 II 0

where the index I stands for either C or S, and d is the Kronecker delta. The expansion of an orientation distribution function f(a, h, /), or a distribution of quaternions on the unit sphere in four-dimensional space expressed in terms of a, h and /, is given as a linear combination of the real hyperspherical MC MS and fNL by harmonics above with real coefficients fNL " # 1 N L X XX  0C 0C MC MC MS MS fNL Z NL þ fNL Z NL þ fNL Z NL f ða; h; /Þ ¼ N ¼0 L¼0

Comparison with Eq. (2) reveals that the spherical angles h and / indicate the direction of the rotation axis n, and a is half the rotation angle x; this is also reflected in Fig. 1b. The various definitions of the complex hyperspherical harmonics Z M NL appearing in the literature are given as functions of these angles and generally agree up to a phase factor. One of the more common definitions is given in Refs. [46,50,53,59], though the convention of this paper differs from these. Nevertheless, for the expansion of a realvalued probability distribution of orientations, the use of complex functions and coefficients is unnecessary; the expansion is potentially simpler as a linear combination of real functions with real coefficients. The real hyperspherMS ical harmonics Z MC NL and Z NL for M – 0 are defined here in terms of the complex hyperspherical harmonics by the linear transformations   pffiffiffi M M L M Z MC NL ¼ i ð1Þ Z NL þ Z NL = 2 ð5Þ   pffiffiffi M L1 M Z MS ð1Þ Z M NL ¼ i NL  Z NL = 2

Z MS NL

with integer indices 0 6 N, 0 6 L 6 N and 1 6 M 6 L; for L 0 0C 0C M = 0, Z 0S NL vanishes, and Z NL is defined as Z NL ¼ i Z NL . These functions involve a Gegenbauer polynomial and an associated Legendre function C Lþ1 N L ðcos aÞ PM L ðcos hÞ; definitions of these consistent with the present usage appear in Appendix A. Notice particularly that the harmonics cleanly separate into a function of the rotation angle and a function of the spherical angles of n. The funcMS tions Z MC NL and Z NL are normalized and form an orthonormal basis with respect to the inner product, i.e. Z 2p Z p Z p 0 I 0 MI 2 ZM N 0 L0 Z NL ðsin aÞ da sin h dh d/

M¼1

ð8Þ Suitable coefficients for the function f(a, h, /) are determined by the inner product, which behaves exactly as a projection operator, with the appropriate harmonic. Explicitly, the MC of the even basis function Z MC coefficient fNL NL and the coeffiMS are provided by cient fNL of the odd basis function Z MS NL Z 2p Z p Z p 2 MC fNL ¼ Z MC NL f ða; h; /Þðsin aÞ da sin h dh d/ MS fNL

¼

Z

0 2p 0

Z

0 p 0

Z

0 p 2 Z MS NL f ða; h; /Þðsin aÞ da sin h dh d/ 0

ð9Þ 4. Symmetrization of the harmonics With a sufficient number of coefficients, the expansion in Eq. (8) allows an orientation distribution to be expressed as a continuous quaternion distribution on the unit hypersphere with arbitrary accuracy. Practically speaking, the labor required to calculate the coefficients requires truncation of the expansion at Nmax, a previously determined maximum value of N. Following Bunge [11], this constraint is to a certain extent relieved by defining a set of symmetrized hyperspherical harmonics conforming to the crystal and sample symmetry as linear combinations of the regular hyperspherical harmonics. These symmetrized functions constitute a set of orthonormal functions over which an arbitrary orientation distribution may be expanded significantly more efficiently. The symmetrized harmonics for crystals of each of the Laue symmetry groups and for samples of orthorhombic symmetry are determined using the procedure explained below. Despite the use of well-known group theoretical tech-

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niques, certain aspects of the present symmetrization procedure, as well as the symmetrized harmonics themselves, appear to be unique within the literature. Although the following is not intended as a general discussion of group theory, some results indispensable for the development of a series expansion of functions on the hypersphere will be briefly reviewed within the more familiar context of three-dimensional rotations. For a more complete presentation refer to, for example, Ref. [18]. 4.1. Representations of a group The handling of the group of proper rotations in three dimensions, or SO(3), requires an analytical realization of the group to give the elements an explicit form. Any realization to a set of matrices is more specifically called a representation of the group, and the matrices R(gi) which represent the rotations gi are called the representatives. Note that matrix multiplication proceeds from right to left, and the same convention is adopted for the result of rotation gj followed by gi gi gj ¼ gk ) Rðgi ÞRðgj Þ ¼ Rðgk Þ

ð10Þ

It is further noted that a similarity transformation of the representation, with matrices T1R(gi)T formed from R(gi) and an invertible linear transformation T, may be considered simply as a change of basis of the representation. The canonical realization of SO(3) is, of course, the group of real, three-dimensional orthogonal matrices with unit determinant, but the prevalence of this representation does not preclude the existence of other representations by matrices of different dimension. Indeed, an infinite number of representations exist, but the majority may be decomposed into a direct sum of representations of smaller dimension. The representations for which decomposition is not possible are said to be irreducible, and possess a number of quite useful properties. 4.2. Irreducible representations of SO(3) Irreducible representations for SU(2), the covering group of SO(3), exist for all integer dimensions. The matrix elements of these representatives are given by the expression [18,26,54] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rlm0 ;m ða; bÞ ¼ ðl þ m0 Þ!ðl  m0 Þ!ðl þ mÞ!ðl  mÞ! 

X k

lm0 k

0

k

alþmk ða Þ bm mþk ðb Þ 0 ðl þ m  kÞ!ðl  m  kÞ!ðm0  m þ kÞ!k!

ð11Þ though the cited references differ subtly in the use and meaning of this representative. This study follows the interpretation of Altmann [54]. The dimension of this representative is (2l + 1), where l is restricted to positive integer or half-integer values. The index m0 labels the rows of the matrix sequentially from l to l, and m labels the columns sequentially from l to l. The index k ranges over all values

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for which the factorials are finite. The proper rotation corresponding to the representative above is identified by the Cayley–Klein parameters a and b, related to the components of the corresponding quaternion q by [54] ð12Þ a ¼ q0  iq4 ; b ¼ q3  iq2 * Finally, the operation indicates the complex conjugate. Eq. (11) then gives an infinite number of equally valid representatives of all integer dimensions for a single rotation, corresponding to the quaternion q. As SO(3) is a subgroup of SU(2), the irreducible representatives of SO(3) must appear among the irreducible representatives of SU(2); they are found by restricting the Rl to integer values of l. Sets of the (2l + 1) complex spherical harmonics Y ml , with a single value of 0 6 l and every l 6 m 6 l, form a complete set of orthonormal basis functions for these (2l + 1)-dimensional representatives of SO(3). As such, the rotation expressed by the representative Rl transforms a linear combination of harmonics of a given value of l into a different linear combination of the same harmonics, and no others. This is more commonly expressed in the physics literature as the principle of conservation of angular momentum [18,26,54]. Although the definitions of the complex spherical harmonics vary subtly with the field in which they are used, this paper follows the definitions provided in Refs. [18,26,54]. That an arbitrary function on the unit sphere in threedimensional space may be expressed as an infinite linear combination of the complex spherical harmonics is well known. Specifically, an arbitrary complex square-integrable function f(h, /) may be expanded as f ðh; /Þ ¼

1 X l X l¼0 m¼l

flm Y ml ¼

1 X

l l Y ll . . . Y l l jfl . . . fl



ð13Þ

l¼0

where the flm are the complex coefficients of the expansion. Although an unusual use of the notation, here hY ll . . . Y l l j indicates the row vector of complex spherical harmonic functions Y ml with a specific value of l arranged from highest to lowest m, and jfll . . . fll i indicates the column vector of the corresponding complex coefficients arranged from highest to lowest m. The inner product of these vectors, exl l pressed by hY ll . . . Y l l jfl . . . fl i, is equivalent to the summation over m in Eq. (13). The result of an active rotation of a vector, written as a linear combination of orthonormal basis vectors, may be written either as a linear combination of the transformed basis vectors with the original coefficients, or as a linear combination of the original basis vectors with transformed coefficients. Analogously, the result of an active rotation of f(h, /) is written either as a linear combination of the transl formed basis functions hY ll . . . Y l l jR with the original coefl l ficients jfl . . . fl i, or as a linear combination of the original basis functions hY ll . . . Y l l j with the transformed coefficients Rl jfll . . . fll i. That is 1 X l

l l l ð14Þ Y l . . . Y l Rf ðh; /Þ ¼ l jR jfl . . . fl l¼0

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When necessary, this is interpreted as a transformation of the coefficients. It is noted here that a set of coefficients that remains unchanged by the application of Rl describes a function that is invariant under that rotation operation; this observation will be important in the following sections. While this expansion is valid for both real and complexvalued functions, functions that are known to be real-valued, e.g., the probability distribution of crystalline axes, may instead be expanded in terms of real spherical harmonics with real coefficients. The real spherical harmonics are constructed from the complex spherical harmonics p byffiffiffi m ½ð1Þ Y ml þ Y m the linear transformations Y mc lpffiffi¼ l = 2 ffi m m m and Y ms l ¼ i½ð1Þ Y l  Y l = 2 for integer indices 0 0 6 l and 1 6 m 6 l, and by Y 0c l ¼ Y l for m = 0. This linear transformation may be written as a unitary matrix Ul; this matrix transforms a row vector of complex spherical harmonics with a specific value of l into a row vector of the corresponding real spherical harmonics, i.e. l lc ls 0c hY ll . . . Y l l jU ¼ hY l ; Y l . . . Y l j

ð15Þ

The invariance of the function with respect to expansion over the basis of the complex or real spherical harmonics requires 1 X l

l l Y l . . . Y l f ðh; /Þ ¼ l jfl . . . fl l¼0 1 X l

l ly l l ¼ Y l . . . Y l l jU U jfl . . . fl l¼0 1 X lc ls

lc ls 0c ¼ Y l ; Y l . . . Y 0c l jfl ; fl . . . fl

ð16Þ

l¼0

where Ul  is the adjoint of Ul, or the complex conjugate transpose. This indicates that the corresponding transformation of the coefficients from the complex expansion to the real expansion is effected by



ð17Þ U ly jfll . . . fll ¼ jfllc ; flls . . . fl0c The irreducible representative in the basis of the real spherical harmonics is therefore Ul RlUl, as Rf ðh; /Þ ¼

1 X

l l l Y ll . . . Y l l jR jfl . . . fl



l¼0

¼

1 X

l ly l l ly l l Y ll . . . Y l l jU U R U U jfl . . . fl



l¼0

¼

1 X

ly l l lc ls 0c ls 0c Y lc l ; Y l . . . Y l jU R U jfl ; fl . . . fl



l¼0

ð18Þ While the choice of phase of the real spherical harmonics and the ordering of the rows and columns of Ul should not change the physical result, the convention followed above results in the three-dimensional matrix U1 R1U1 corresponding exactly to the usual three-dimensional rotation matrix of real space.

The material included in this section has followed, for the most part, from well-known results and methods of group theory. For example, the physics literature [18,26,54] provides derivations and explanations of the irreducible representatives of SU(2), appearing in Eq. (11), and their action on the spherical harmonics, as indicated by Eq. (15). The similarity transformation to the real spherical harmonics, given in Eq. (16), follows directly from the definition of the real spherical harmonics. In what follows, this notation is used, and these techniques are extended to symmetrize the hyperspherical harmonics. 4.3. Symmetries and four-dimensional rotations The convention of this paper is that a crystallographic orientation is described by an active rotation q of a reference crystal from an orientation aligned with the sample to an orientation aligned with the observed crystal. Because an initial rotation of the reference crystal by a crystal symmetry operation r must result in a symmetrically indistinguishable initial orientation of the crystal, the composition of the rotation q with the prior symmetry operation r results in the symmetrically equivalent rotation u given by the quaternion product [q0, q][r0, r] = [u0, u]. This is readily written as a multiplication of the quaternion components by an orthogonal four-dimensional matrix of unit determinant [56] 2 32 3 2 3 q0 u0 r0 r1 r2 r3 6r 76 q 7 6 u 7 r r r 0 3 2 76 1 7 6 1 6 17 ð19Þ 6 76 7 ¼ 6 7 4 r2 r3 r0 r1 54 q2 5 4 u2 5 r3

r2

r1

r0

q3

u3

Once the crystal is oriented, a subsequent rotation of the sample by a sample symmetry operation p must result in a statistically indistinguishable final orientation of the crystal, as every orientation of this type occurs with equal frequency in the sample. That is, the composition of the rotation q with the subsequent symmetry operation p results in the symmetrically equivalent rotation v, given by [p0, p][q0, q] = [v0, v]. This may be written as a multiplication by a similar orthogonal four-dimensional matrix of unit determinant, though with the signs of the off-diagonal components of the lower three-by-three section reversed [56] 2 32 3 2 3 q0 v0 p0 p1 p2 p3 6p 7 6 7 6 7 6 1 p0 p3 p2 76 q1 7 6 v1 7 ð20Þ 6 76 7 ¼ 6 7 4 p2 p3 p0 p1 54 q2 5 4 v2 5 p3

p2

p1

p0

q3

v3

As orthogonal four-dimensional matrices with unit determinants, each of these is a rigid rotation of four-dimensional space, though not an unrestricted rotation; each matrix is determined by only three parameters, e.g., the angles a, h and /, and a general four-dimensional rotation is determined by six independent parameters [55]. Neverthe-

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less, the product of these matrices produces a general orthogonal four-dimensional matrix of unit determinant [60,61]. That is, the quaternion w as calculated from [p0, p][q0, q][r0, r] = [w0, w] is related to the quaternion q by a single four-dimensional rotation, which simultaneously performs the crystal symmetry operation r and the sample symmetry operation p. A necessary though not sufficient condition for this to occur is that the group space be Euclidian, which is certainly the case for quaternions interpreted as points on the unit sphere in a Euclidian fourdimensional space. Notably, this does not occur for the Euler angle parameterization. A symmetrized function may now be defined as one that is invariant under the application of every distinct fourdimensional rotation corresponding to the simultaneous action of a crystal and a sample symmetry operation. The individual enforcement of every symmetry operation in the complete crystal and sample symmetry groups is avoided by making use of the group generators, as a function that is invariant under the application of these generators will remain invariant under the application of any operation in the full rotation group. To symmetrize the hyperspherical harmonics in Eq. (6), then, requires a method of constructing the real irreducible representatives of the four-dimensional rotation group SO(4) in order to find sets of coefficients in the expansion of Eq. (8) that remain invariant under the generators of the crystal and sample symmetry groups. These sets of coefficients will then describe the basis of symmetrized functions used to write a texture, for given crystal and sample symmetries, as a linear combination of symmetrized basis functions. 4.4. Irreducible representations of SO(4) The procedure for finding the irreducible representations of SO(4) requires a well-known theorem from group theory. For a certain set of conditions, when the elements gi of one group of transformations commute with the elements hj of a second group of transformations, the set of products gihj formed by combinations of these elements form a group as well, known as the direct product group. Given the irreducible representatives G(gi) and H(hj) of the elements gi and hj, respectively, the direct product G(gi)H(hj) of these matrices forms an irreducible representative of the element gihj of the direct product group [18,61]. For some quaternion q, consider prior multiplication by the quaternion r and subsequent multiplication by the quaternion p as distinct transformations. Then the associativity of quaternion multiplication requires that the quaternion w, resulting from q by [p0, p][q0, q][r0, r] = [w0, w], should not depend on the order of application of r and p; that is, the associativity of quaternion multiplication requires that these transformations commute. As outlined above, the combination of right multiplication by r and left multiplication by p corresponds to a single four-dimensional rotation of q, and Eq. (11) provides

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the irreducible representations Rl of SU(2) corresponding to r and p individually. The above theorem then indicates that the irreducible representatives of the four-dimensional rotation group may be uniquely determined, up to a similarity transformation, by the direct product of two of the Rl, one corresponding to the quaternion r and the other to the quaternion p. This is equivalent to the statement that the direct product group SU(2)SU(2) is locally isomorphic to the group SO(4), as is widely recognized in the literature on the four-dimensional rotation group, e.g., [46,47,50,60]. More precisely, the direct product group is related to SO(4) by a 2-to-1 homomorphism, as evidenced by the ambiguity in the common sign of the Rl [58,61]. While the direct product of irreducible representatives of SU(2) does provide the irreducible representatives of SO(4), the bases of these representatives do not initially correspond to the real hyperspherical harmonics as defined in Eq. (6). For that reason, they may not initially be used to perform a rotation of a function in the form of Eq. (8) through a transformation of the coefficients of the linear combination; a suitable similarity transformation of the representatives is first necessary to bring them into the basis of the real hyperspherical harmonics. This similarity transformation is determined by beginning with the transformation properties of the irreducible representatives of SU(2). The rotation r transforms a function f(r) into the function R(r)f(r), where the operator R(r) is a function of the quaternion components of r. The parenthesized superscript (r) is simply a label denoting that f(r) resides in the threedimensional space on which r operates. Since f(r) is defined on the unit sphere, it may be expanded as an infinite linear mðrÞ combination of the complex spherical harmonics Y l with mðrÞ complex coefficients fl , as in Eq. (13). The rotated function R(r)f(r) is written in terms of the coefficients of the expansion and the irreducible representatives Rl (r) of the rotation r, similar to Eq. (18) 1 D E X lðrÞ lðrÞ lðrÞ lðrÞ Y l . . . Y l jRly ðrÞjfl . . . fl ð21Þ RðrÞf ðrÞ ¼ l¼0

Similarly, the rotation p transforms a function f(p) into the function R(p)f(p), where f(p) is defined on the unit sphere; as before, the parenthesized superscript (p) is a label indicating that the rotation p operates on the three-dimensional space in which f(p) resides. When the function f(p) is expanded as a linear combination of the complex spherical   mðpÞ mðpÞ harmonics Y l with complex coefficients fl , the rotated ðpÞ function RðpÞf is given in terms of the coefficients of this  expansion and the irreducible representatives Rl ðpÞ of the rotation p by RðpÞf ðpÞ ¼

1 D E X lðpÞ lðpÞ  lðpÞ lðpÞ Y l . . . Y l jRl ðpÞjfl . . . fl

ð22Þ

l¼0

Notice that only the Rl with integer values of l appear in these expansions.

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The commutation of r and p, in the sense discussed above, indicates that R(r) and RðpÞ operate on distinct functions residing in distinct spaces, as emphasized by the use of the superscripts (r) and (p) and the differentiation of the indices by the use of an overbar. The meaning of the symbol f ðrÞ f ðpÞ , then, must not be interpreted as the product of f ðrÞ and f ðpÞ in the usual sense, but as a single function, one part of which is operated on by RðrÞ and the other part of which is operated on by RðpÞ. Since f ðrÞ f ðpÞ defines values at points in spaces with separate transformative properties, this function must be expanded over a similar set of basis functions;  mðrÞ mðpÞ mðrÞ namely, the basis functions Y l Y l , where Y l resides in  mðpÞ the space operated on by the rotation r, and Y l resides in the space operated on by the rotation p. The corresponding mðrÞ mðpÞ  coefficients are denoted by fl fl . The expansion of the transformed function RðrÞRðpÞf ðrÞ f ðpÞ is then given in terms of the expansion of f ðrÞ f ðpÞ and the irreducible representatives  formed by the direct product of Rly ðrÞ and Rl ðpÞ, vis-a`-vis the above theorem, as 1 X 1 D X lðrÞ lðpÞ lðrÞ lðpÞ Y l Y l ...Y l Y l j RðrÞRðpÞf ðrÞ f ðpÞ ¼ l¼0 l¼0 E l lðrÞ lðpÞ lðrÞ lðpÞ ly R ðrÞ  R ðpÞjfl fl ...fl fl

 mðpÞ

 mðrÞ mðpÞ

where the basis functions Y l Y l and coefficients fl fl are ordered in decreasing values of m, and in decreasing val for a particular value of m. Although perfectly reaues of m  sonable representatives Rly ðrÞ  Rl ðpÞ may be formed with independent values of l and l, the complex hyperspherical harmonics form the basis of only those irreducible representatives given by a similarity transformation of the  Rly ðrÞ  Rl ðpÞ with l ¼ l [46,47,50,62]. This constraint is therefore applied to the function in Eq. (23), and the double summation becomes a single summation. While distinct, the functions f (r) and f (p) cannot inhabit completely isolated spaces; as noted before, independent rotations of these functions are related to a single four-dimensional rotation that operates on f (r) and f (p) simultaneously. The change of basis necessary to emphasize the nature of this transformation as a single operation performed on a single function is accomplished through the Clebsch–Gordan coefficients [46,50], which appear in an explicit form in, for example,  , Ref. [63]. They are denoted here in index notation as C l;m;lm;L;M   The with integer indices jl  lj 6 L 6 l þ l and M ¼ m þ m. Clebsch–Gordan coefficients relate the basis of Eq. (23), mðrÞ mðpÞ  where each basis function Y l Y l is constructed as a pair of separately rotating spherical harmonics, to the coupled basis, where each basis function Z M NL is a single hyperspherical harmonic in four dimensions2

 Here, N ¼ l þ l, and C l;l is the unitary matrix formed by arranging the Clebsch–Gordan coefficients in accordance with the ordering of the row vectors in Eq. (24). The complex spherical harmonics are ordered in increasing values of L, and in decreasing values of M for a particular value of L. Following the procedure used in Eq. (16), it is found that E

 lðrÞ lðpÞ lðrÞ lðpÞ L L . . . fNL ð25Þ ¼ jfN00 . . . fNL C l;ly jfl fl . . . fl fl

where the coefficients are ordered similarly. Eqs. (24) and (25) and a procedure analogous to that of Eq. (18) indicate that the reformulation of Eq. (23) as a four-dimensional rotation of a function f, written as a linear combination of the complex hyper spherical harmonics, is given by Rðr; pÞf ¼

1 D X   l;ly ly Z 0N 0 . . . Z LNL . . . Z L R ðrÞ  Rl ðpÞC l;l NL jC N ¼0;2...



L L . . . fNL  fN00 . . . fNL

ð26Þ

where the direct product Rly ðrÞ  Rl ðpÞ is performed before   right multiplication by C l;l and left multiplication by C l;ly , and Rðr; pÞ is the operator that performs a single fourdimensional rotation equivalent to the action of RðrÞRðpÞ. Notice that N in Eq. (26) ranges over only the non-negative even integers, as is required by the relations N ¼ l þ l and l ¼ l for integer l and l. This may be viewed as a natural occurrence of performing transformations with elements of SO(3), rather than with elements of SU(2), in Eqs. (21) and (22). Alternatively, one may consider that the hyperspherical harmonics with odd values of N do not remain invariant following a rotation by 2p. Since this is a basic requirement of a crystal orientation distribution, the coefficients of these hyperspherical harmonics must vanish identically, and thus the expansion in Eq. (26) does not include the odd values of N. The unitary matrix relating the complex hyperspherical harmonics in Eq. (26) to the real hyperspherical harmonics is denoted by UN, and is defined by the relationships given in Eq. (5). This is applied to the row vector hZ 0N 0 . . . Z LNL . . . Z L NL j of the complex hyperspherical harmonics as N 0C LC LS 0C hZ 0N 0 . . . Z LNL . . . Z L NL jU ¼ hZ N 0 . . . Z NL ; Z NL . . . Z NL j

The relation between the basis of paired spherical harmonics and the hyperspherical harmonics is not necessarily a numerical equality; we only require that the transformative properties of these bases are related by the Clebsch–Gordan coefficients, which is known.

ð27Þ

and, as demonstrated by the procedure used in Eq. (16), the corresponding transformation of the coefficients of the expansion is given by



L L LC LS 0C ¼ jfN0C0 . . . fNL . . . fNL ; fNL . . . fNL U N y jfNL

2

ð24Þ



ð23Þ mðrÞ

D  lðrÞ lðpÞ lðrÞ lðpÞ Y l Y l . . .Y l Y l jC l;l ¼ Z 0N 0 . . . Z LNL . . . Z L NL j

ð28Þ

The basis functions and coefficients of Eqs. (27) and (28) are ordered in increasing values of L, and in decreasing values of M for a particular value of L, with C before S for a particular value of M. Applying this unitary transforma-

J.K. Mason, C.A. Schuh / Acta Materialia 56 (2008) 6141–6155

tion to Eq. (26) in accordance with the formulation of Eq. (18) results in the desired transformation 1 D X N y l;ly ly LC LS 0C Z 0C R ðrÞ Rðr; pÞf ¼ N 0 . . . Z NL ; Z NL . . . Z NL jU C N ¼0;2... 



LC LS 0C ; fNL . . . fNL Rl ðpÞC l;l U N jfN0C0 . . . fNL

E

ð29Þ

The (N + 1)2-dimensional irreducible representative of SO(4) appearing in Eq. (29) is understood to perform a transformation of the coefficients of the expansion, and is given in index notation by XXXX U NL0;M 0 ;L0 ;M 0 ;I 0 RNL0 ;M 0 ;I 0 ;L;M;I ðr; pÞ ¼  L;M  0 m;m L0 ;M 0 m0 ;m



 l l;l C l;m;lm 0 ;L0 ;M 0 Rl U NL;M;L;M;I  0 ;m  ðpÞC m0 ;m;L;M  m0 ;m ðrÞRm

ð30Þ

where the semicolons appearing among the indices of a given unitary transformation separate the indices characterizing the different bases related by the said unitary transformation. Whereas the irreducible representative of SO(3) changed a linear combination of the (2l + 1) spherical harmonics for a particular value of l into a different linear combination of the same harmonics, the irreducible representative of SO(4) changes a linear combination of the (N + 1)2 hyperspherical harmonics for a particular value of N into a different linear combination of the same harmonics. While a change in the phase of the real hyperspherical harmonics or the ordering of the rows and columns of the various transformations performed above should not change the physical result, following the present conven   tion causes the matrix U N y C l;ly Rly ðrÞ  Rl ðpÞC l;l U N for N ¼ 1 to correspond to the matrix form of the quaternion p in Eq. (20) when r is the identity, and to the matrix form of the quaternion r in Eq. (19) when p is the identity. 4.5. Eigenfunctions of the rotation operator Given a function f in the form of Eq. (8), the irreducible representatives of SO(4) determine the coefficients in the expansion of the function Rf, written in the same form, resulting from a four-dimensional rotation. We specifically want to find sets of coefficients defining linear combinations of real hyperspherical harmonics that remain invariant under the four-dimensional rotations corresponding to the simultaneous application of a crystal symmetry and sample symmetry operation. These are the components of the simultaneous eigenvectors of eigenvalue unity for the (N + 1)2-dimensional irreducible representatives of SO(4) corresponding to every unique combination of a generator from the crystal point symmetry group and the sample symmetry group. Explicitly, this process is performed via the following steps: 1. Enumerate the generators of the crystal point group symmetry and the sample point group symmetry.

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2. Determine the unique four-dimensional rotations resulting from the application of every combination of a crystal symmetry generator and a sample symmetry generator to an arbitrary rotation q. This may be performed by examining the product of the matrices corresponding to the symmetry operations r and p in Eqs. (19) and (20), respectively. 3. Find the representatives of the (N + 1)2-dimensional irreducible representations of SO(4) for every rotation found in Step 2. The necessary form of the representatives appears in Eq. (30). 4. Calculate the simultaneous eigenvectors with eigenvalues of unity for every representative found in Step 3. 5. Construct an orthonormal system from the resulting linearly independent eigenvectors. The components of these orthonormal vectors provide the coefficients necessary to find the symmetrized hyperspherical harmonics from Eq. (8). The symbol Z€_ kN denotes one of these symmetrized hyperspherical harmonics, constructed as a linear combination of the real hyperspherical harmonics with coefficients given by the kth linearly independent simultaneous eigenvector for a particular value of N. That is " # N L

_Z€ k ¼ X f€_ 0Ck Z 0C þ X f€_ MCk Z MC þ f€_ MSk Z MS ð31Þ N NL NL NL NL NL NL L¼0

M¼1

€_ MSk where the f€_ MCk NL and the f NL are the components of the kth linearly independent simultaneous eigenvector. With appropriate coefficients, the symmetrized hyperspherical harmonics form a complete orthonormal system of functions similar to the regular hyperspherical harmonics, i.e. Z 2p Z p Z p 0 2 ð32Þ Z€_ kN 0 Z€_ kN ðsin aÞ da sin h dh d/ ¼ dNN 0 dkk0 0

0

0

Any function f(a, h, /), expressing an orientation distribution that satisfies the requirements of crystal and sample symmetries, may be uniquely expressed as a linear combination of the symmetrized hyperspherical harmonics in the form KðN 1 XÞ _ X fNk Z€ kN ð33Þ f ða; h; /Þ ¼ N ¼0;2... k¼1

where K(N) indicates the number of linearly independent symmetrized hyperspherical harmonics for a given value of N. The coefficients fNk in this expansion are given by the inner product of the basis function Z€_ kN with f Z 2p Z p Z p k ð34Þ Z€_ kN f ða; h; /Þðsin aÞ2 da sin h dh d/ fN ¼ 0

0

0

As a demonstration of this procedure, the symmetrized harmonics were calculated for samples with orthorhombic symmetry and crystals with the proper rotational symmetries of all the Laue groups, such as would be required to describe textures in rolled sheet materials. At least 30 symmetrized hyperspherical harmonics are provided for each of these symmetries in the Online Supplement.

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5. Visualization There remains the matter of displaying orientation statistics in an intelligible two-dimensional format, suitable for the printed page. Since quaternions and hyperspherical harmonics reside on the surface of a unit sphere in fourdimensional space, this requires a projection from the four-dimensional space into two-dimensional space. The projections from four to three dimensions and from three to two dimensions are considered separately. As noted by certain researchers [22,65], a choice of projection from four to three dimensions is equivalent to a choice among the various three-dimensional neo-Eulerian mappings, each with their particular benefits. Of those appearing in the literature, three methods of projection are mentioned. One of these, the stereographic projection, preserves angles; as a consequence, this projection is frequently used to depict the angles between crystallographic directions. A second projection that has been the subject of considerable interest recently is the geodisc projection, defined as the projection that maps great circles on the four-dimensional unit sphere onto straight lines in threedimensional space. More simply, this projection maps a quaternion onto the corresponding Rodrigues vector, with the beneficial property that the resulting orientation and disorientation spaces of symmetric objects are bounded by planar surfaces. Explicit formulas for the stereographic and geodisc projections and some properties of the resulting parameterizations appear in, for example, Refs. [64– 66]. Meanwhile, the third projection, and the one that is used exclusively here, is the volume-preserving projection. The outstanding feature of this projection is that the resulting invariant measure is unity, meaning that the density of points within a particular region of the projected group space is directly proportional to the volume of crystalline material with that orientation. The projection relates a point with angular coordinates a, h and / on the surface of the four-dimensional unit sphere to a point with coordinates x, y and z in the interior of a three-dimensional solid ball by x ¼ r sin h cos / y ¼ r sin h sin /

scalar part q0, or angles 0 6 a 6 p/2, are considered in Eq. (35), without discarding any orientation information. The most convenient method of presenting a distribution in a three-dimensional volume on the printed page is by two-dimensional sections, though the ease of interpretation depends significantly on the method of sectioning. Because, as illustrated in Fig. 1, the radial distance of a point from the origin increases monotonically with rotation angle, the value of the orientation distribution on a spherical shell of radius r indicates the distribution of rotation axes for rotations by the angle x = 2a. The sectioning of the group space into concentric spherical shells therefore allows the orientation distribution to be viewed as a series of distributions of rotation axes for particular rotation angles, holding one of the physically relevant quantities constant and simplifying the interpretation. The equal-area projection is used for the projection of these spherical shells into two dimensions, for the same reason that the volumepreserving projection was used above. This projection relates a point with coordinates r, h and / on the surface of a three-dimensional spherical shell to a point with coordinates X and Y on a two-dimensional disk by X ¼ R cos / Y ¼ R sin /

ð36Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where R ¼ r 2ð1  j cos hjÞ. As above, these may be written in terms of the coordinates x, y and z or the coordinates

ð35Þ

z ¼ r cos h where r = (3/2)1/3(a  sin a cos a)1/3. This is identical to the homochoric projection of Refs. [22,23,66], though written in terms of the angular coordinates of a quaternion. Alternatively, these may be written in terms of the coordinates qi using Eq. (4). On the subject of the domain of a, note that a single orientation of three-dimensional space corresponds to the two distinct quaternions +q and q at antipodal points on the surface of the four-dimensional unit sphere. As one hemisphere contains a single point for every distinct three-dimensional orientation, all the information contained in a given orientation distribution is found in a single hemisphere. Therefore, only quaternions with a positive

Fig. 2. A random texture, corresponding to a uniform distribution of points on the surface of the unit four-dimensional sphere, presented using the volume-preserving (4D to 3D) and equal-area (3D to 2D) projections. The equal-area projection enforces the uniformity of the distribution for a particular rotation angle, and the volume-preserving projection enforces the uniformity of the distribution among the various rotation angles.

J.K. Mason, C.A. Schuh / Acta Materialia 56 (2008) 6141–6155

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Fig. 3. Examples of the symmetrized hyperspherical harmonics, as calculated from the coefficient tables in the Online Supplement. Specifically, these are sets of the three lowest-order, non-trivial symmetrized hyperspherical harmonics for orthorhombic sample symmetry and for the crystal point symmetries (a) 222, (b) 422 and (c) 432. In each of the projections, the z- and x-axes of the projections point out of the page and to the right, respectively.

qi through the use of Eqs. (35) and (4). Using this projection, the hemispheres defined by p/2 6 h 6 0 and 0 6 h 6 p/2 of the three-dimensional spherical shell generally contain distinct information for orthorhombic samples and crystal point symmetry of 6 or lower, and should be presented as separate projections. For orthorhombic samples and crystal point symmetry of 222 or higher, though,

these hemispheres contain identical information, and a projection of the positive hemisphere alone is sufficient. It is observed parenthetically that the space containing unique information (the fundamental zone) is generally smaller than even one hemisphere, but here it is intended only to clarify the use of Eq. (36) rather than to discuss the symmetric reduction of the group space.

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Fig. 4. Conventional methods of viewing a simulated cube texture. (a) (1 0 0) and (b) (1 1 1) pole figures are presented in stereographic projection, with the z- and x-axes pointing out of the page and up the page, respectively. (c) The blue and red regions in the Euler angle space indicate regions of positive and negative probability density, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 6. Conventional methods of viewing a simulated copper texture. (a) (1 0 0) and (b) (1 1 1) pole figures are presented in stereographic projection, with the z- and x-axes pointing out of the page and up the page, respectively. (c) The blue and red regions in the Euler angle space indicate regions of positive and negative probability density, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

The benefit of using this method of presentation is demonstrated in Fig. 2. This figure depicts a random texture, i.e., a uniform distribution of points on the surface of the four-dimensional sphere, for rotation angles from 0° to 180° in 15° increments. Inspection reveals that the apparent density of rotation axes is uniform not only within the projection of a spherical shell corresponding to a particular rotation angle, but among the projections of all the rotation angles as well; these properties result from the equalarea projection of Eq. (36), and from the volume-preserving projection of Eq. (35), respectively.

In the section that follows, the analysis and presentation methods developed in this paper are employed to represent continuous orientation distribution functions. As a preamble to this discussion, it is worthwhile briefly to examine a few of the symmetrized hyperspherical harmonics, as the calculation and use of these functions constitute the central aim of this paper. Fig. 3 illustrates the first three non-trivial symmetrized hyperspherical harmonics with orthorhombic sample symmetry and each of the crystal point symmetries 222 (Fig. 3a), 422 (Fig. 3b) and 432 (Fig. 3c). The specific nature of the symmetry present in these harmonics is not

Fig. 5. The current methods of viewing a simulated cube texture. The z- and x-axes of the projections point out of the page and to the right, respectively. (a) Projection of the discrete quaternions corresponding to the simulated texture, including all the rotations in the crystallographic point group 432. (b) Continuous orientation distribution corresponding to the discrete distribution of part (a), calculated using the first 37 symmetrized hyperspherical harmonics of orthorhombic sample symmetry and cubic crystal symmetry. Blue and red regions indicate regions of positive and negative probability density, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

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Fig. 7. The current methods of viewing a simulated copper texture. The z- and x-axes of the projections point out of the page and to the right, respectively. (a) Projection of the discrete quaternions corresponding to the simulated texture, including all the orientations described as part of this texture in Ref. [69]. (b) Continuous orientation distribution corresponding to the discrete distribution of part (a), calculated using the first 37 symmetrized hyperspherical harmonics of orthorhombic sample symmetry and cubic crystal symmetry. Blue and red regions indicate regions of positive and negative probability density, respectively. Notice particularly the close agreement between the discrete and continuous distributions.

necessarily obvious, owing to the fact that rotations do not combine linearly; for example, a symmetry rotation about the z-axis translates the point corresponding to the identity along the z-axis, but rotates certain of the points corresponding to 180° rotations about the z-axis. With the differences in sample and crystal symmetries, this leads to the marked chirality of some of the images. The exception to this difficulty is the identification of regions corresponding to transformations of the identity by symmetry operations, as allows the ready identification of symmetry 222 in the first harmonic of Fig. 3a, symmetry 422 in the third harmonic of Fig. 3b, and symmetry 432 in the first harmonic of Fig. 3c. 6. Texture representation As a demonstration of this technique, two simulated textures for cubic polycrystals in samples of orthorhombic symmetry are now considered: first, the ‘‘cube” texture, which is presented in Fig. 4 through the conventional means, namely, by (1 0 0) and (1 1 1) pole figures (Fig. 4a and b) and an orientation distribution function in the Euler angle parameterization (Fig. 4c), the blue and red regions of which indicate positive and negative probability density, respectively.3 The physical meaning of the high probability density regions in the Euler angle space is generally not intuitively clear, but the data in Fig. 4c can be identified as a cube texture by reference to a legend of common tex-

ture elements, as in Ref. [66]. Fig. 5a displays the discrete orientations of the simulated cube texture as projected quaternions, with the corresponding continuous orientation distribution function appearing in Fig. 5b. The continuous orientation distribution is calculated from Eq. (33) using the first 37 symmetrized hyperspherical harmonics with orthorhombic sample and cubic crystal symmetry, as presented in the Online Supplement. Although a depiction of actual orientation statistics would probably use finer sections and only cover the asymmetric unit, the full representation is used here to emphasize the symmetry; inspection of the figure reveals the presence of the expected 90° rotations about h1 0 0i axes, 120° rotations about h1 1 1i axes and 180° rotations about h1 0 0i and h1 1 0i axes. A more complex texture, the ‘‘copper” texture, is also considered, conventional and quaternion-based representations of which appear in Figs. 6 and 7, respectively. Whereas the symmetry of the cube texture permits an expansion in terms of a relatively small number of harmonics, including the first harmonic of Fig. 3c, the copper texture is considerably more asymmetric and therefore more thoroughly reflects the ability of the series representation method to represent arbitrary textures; notice the close agreement between the discrete orientation distribution of Fig. 7a and the continuous orientation distribution of Fig. 7b, calculated as before with the first 37 symmetrized hyperspherical harmonics with orthorhombic sample and cubic crystal symmetry. 7. Conclusion

3

The appearance of regions with negative probability densities is common when analyzing textures by a series expansion; numerous methods exist explicitly to enforce the positivity condition. For simplicity, the present paper does not consider this issue further, although in a future work the authors will discuss the positivity condition with respect to the use of hyperspherical harmonics.

The groundwork has been provided for, and the feasibility demonstrated of, representing crystallographic textures in the quaternion parameterization as an alternative to the Euler angle parameterization common in the literature

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today. Whereas quaternions have been used by the texture community for the analysis and manipulation of individual orientations, the present contribution is a series representation for textures expressed as continuous probability distribution functions of quaternions. This expansion is performed initially over the real hyperspherical harmonics and then over the symmetrized hyperspherical harmonics. The symmetrized hyperspherical harmonics account for the sample and crystal symmetries of the texture, and allow the expansion to be performed significantly more efficiently. These functions were calculated for orthorhombic sample symmetry and for the proper rotation groups corresponding to each of the Laue groups; the results are collected for use by the community in the Online Supplement. The approach developed in this paper is potentially more suitable as a series representation of texture than the conventional method in terms of Euler angles, and also offers advantages in terms of texture representation. The authors hope that the quaternion representation of texture will contribute not only to improved representation of textures in extant materials, but also to facilitate the integration of texture control into the modern materials design paradigm. Acknowledgement This work was supported by the US National Science Foundation under Contract No. DMR-0346848. Appendix A. Definition of functions Since the definitions of certain standard functions vary subtly with the field, definitions consistent with the remainder of this paper are provided. The associated Legendre functions P ml , used for the spherical harmonics, are defined using Rodrigues’ formula as P ml ðxÞ ¼ ð1Þ

m

lþm 1 m=2 d l ð1  x2 Þ ðx2  1Þ l dxlþm 2 l!

ð37Þ

Notice particularly the appearance of the Cordon–Shortly phase factor (1)m in the definition of this function; this is occasionally inserted into the definition of the complex spherical harmonics directly instead. Meanwhile, the Gegenbauer polynomials C vn , which appear in the definition of the hyperspherical harmonics, are defined as n

C vn ðzÞ ¼

ð2Þ Cðv þ nÞCð2v þ nÞ dn ð1  z2 Þ1=2v n ½ð1 n! dz CðvÞCð2v þ 2nÞ  z2 Þ

nþv1=2



ð38Þ

Further properties and relationships of these functions are given in, for example, Refs. [67,68]. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.actamat.2008.08.031.

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