REPRESENTATIONS OF POLYCRYSTALLINE MICROSTRUCTURE BY n-POINT CORRELATION TENSORS

An important class of representations of polycrystalline microstructure consists of the n-point 
correlation tensors. In this paper the representation theory of groups is applied to a consideration of 
symmetries in the n-point correlation tensors. Three sources of symmetry are included in the 
development: indicial symmetry in the coefficients of tensors, symmetry associated with the crystal 
lattice, and statistical symmetries in the microstructure induced by processing. The central problem 
discussed here is the residence space, or the space of minimum dimension occupied by correlation 
tensors possessing such symmetries. In addition to the general case of correlation tensors possessing 
such symmetries, a model microstructure is also considered which embodies an assumption of no 
spatial coherence of lattice orientation between neighboring grains or crystallites. It is shown that the 
model microstructure generally results in residence spaces of lower dimension.


INTRODUCTION
Microstructure refers to any of the myriad of features observed by the experimentalist when probing the internal arrangement of components of solid materials.In polycrystals microstructure refers to aspects of its constituent grains, including the distribution and spatial correlation of material points by chemical phase and lattice orientation, the size and shape distribution of the crystallites, their topological connectivity, and the density and distribution of lattice defects.Observations of microstructure are always limited by probing instruments to a restricted set of accessible features.Experiments yield transformed or projected views of the true microstructure since probing is always limited in resolution (spatial, wavelength, etc.).A consequence of the fact that most solids are opaque is a recourse to certain principles of stereology in order to infer information about the three-dimensional microstructure from two-dimensional plane sectioning.Even if the observer had an unobscured access to the microstructure, he is limited in understanding the relevance of microstructural observations to particular properties and behavior of interest.Deep physical insight into the associations between microstructural features and behavior is required to understand their importance.
It is in this complicated setting that we shall consider a particular and specialized class of representations of polycrystalline microstructure that fre- quently appears in property bounding theories.N-point correlation tensors naturally occur in estimates and bounding theories of linear properties (cf.Beran and Molyneux, 1966;Hashin and Shtrikman, 1962;Kr6ner, 1986;McCoy, 1981;Willis, 1981).More recently the same class of representations has been shown to occur in some nonlinear estimates and bounds (cf.Willis 1983, Talbot andWillis 1985, Ponte-Castefiada 1991, Adams and Field 1991).It would appear that representation by n-point tensors has significant breadth of application in materials science.The goal of the paper is to carefully consider three classes of symmetry which appear in such representations: the indicial symmetry of selected material tensors, symmetries which arise in the crystal lattice, and symmetries arising during thermomechanical processing of the polycrystal.
Representation of microstructure refers to any mathematical classification of features present in microstructure.Such representation is always limited in scope.Complete representation of microstructure in polycrystalline materials is neither possible nor is it desirable.Rather, we must seek to find a minimal representation of microstructure which correlates, to the desired resolution, with those properties and behaviors of interest.It is in this context that consideration of symmetry becomes an important factor, since symmetry analysis helps identify the minimal required representation.More precisely, the goal is to define the smallest space of representation of the microstructure (consistent with the indicial, lattice and processing symmetries present in the material) and the related minimum set of experimental measurements required to construct the representation.
In the sections which follow we first define the n-point correlation tensors in the context of materials science and the modern quantitative texture theory of microstructure.Subsequently, a section is furnished containing a few results of the representation theory of finite groups and compact Lie groups.We make no attempt to derive these results; our primary goal is to establish some definitions and vocabulary that will be required in the subsequent development.Readers not familiar with representation theory of groups will find an extensive published literature (cf.Gel'fand, Minlos and Shapiro 1963;Broecker and tom Dieck 1985).
The main section of the paper describes a general approach to symmetry considerations in the n-point tensors.Prescriptions for application of the general approach to specific cases is somewhat varied, and we shall not attempt to describe it here.Rather, we elect to give results for a selected set of problems ranging from the most elementary to examples which have substantial technologi- cal importance.
2. REPRESENTATION OF MICROSTRUCTURE BY n-POINT CORRELATION TENSORS Some fundamental aspects of microstructure in polycrystalline materials can be described in terms of an ideal reference crystal.This reference crystal is modeled as the repetition of a particular structural basis upon a reference, lattice of points in R3.The lattice of points is defined by the vectors v vtai where {ai) is the set of basic vectors of the lattice and v e ;7, the set of integers.For the moment we shall consider that no particular point symmetry is present in the lattice.Consider affine transformations of the ideal reference crystal.Such transforma- tions alter the lattice of points of the reference crystal according to the following formula: v----; where b is a translation vector, and is a second-order tensor.It is known that affine transformations take points into points, lines into lines, and planes into planes.(Such transformations are also known as homogeneous deformations.) The tensor of transformation, , can always be expressed as the product of two second-order tensors: ga, where a is a positive-definite, real, and symmetric, and g belongs to the three-dimensional group of real, orthogonal transformations, 0(3).In the following we shall be most interested in tensors of transformations in which a can be considered to differ negligibly from the second-order unit tensor; thus 3. g. (This is equivalent to ignoring elastic strain in the lattice.) The polycrystal is modeled as an aggreagate of crystallites filling a region of space, R.These crystallites have finite volume, and each material point belonging to any particular crystallite carries local structure associated with a specific atiine transformation of the ideal reference crystal.The polycrystal will also contain material points for which this association is not possible, but we shall assume that this subregion has negligible volume.
Partition 0(3) into subsets digi such that [._J dig/= O(3), tSg tq 6g 0 for =/=j. (2) At point x in R associates with 6g/ when there exists a neighborhood of x, N(x) R, which carries structure related to the ideal reference crystal by an orthogonal transformation g e dig/, and a translation b which shall not be of further interest.Define the characteristic function associated with dig to be 1 if x associates with tSg Z(C)(x)= 0 otherwise (3) The volume fraction of crystallites associated with lattice orientation 6gi in the polycrystal is just Our primary interest lies not in representation of individual polycrystals, but rather in ensembles of polycrystals sharing a common thermo-mechanical processing path.We shall hereafter assume statistical homogeneity in the ensemble.For such an ensemble of polycrystals, each of which has volume V, the expected volume fraction of crystallites associated with lattice orientation dig is given by (5)   where (Zn(x)) is independent of position x.
2.1 Definition of the n-Point Probability Density Functions of Lattice Orientation The expectation value (Z(x)) is closely related to the crystallite orientation distribution function f" 0(3) --through the relation where d/z is the invariant probability measure in 0(3).In the limit that our partitioning of 0(3) obtains infinitesimally small subsets dg of measure/(dg) d/t, containing the orthogonal transformation g, Eq. ( 6) can be expressed as (V(dg)) [V (g(x)) =f(g) d/. (7) The orientation distribution function is normalized to unity: (3) The preceding development can easily be extended to higher-order repre- sentations.For example, the expectation of go and g separated by r in the ensemble of polycrystals is given by (o(x),,(x + r,)> -f(g0, g, It1) dpo dp,, (9) where ) is known as the two-point orientation coherence function (cf.Adams, et al. 1987;Adams, Wang and Morris, 1988;Adams and Onat, 1991).This can be generalized to consider the (n + 1)-point probability density functions of lattice orientation since (lg0(X) gn(x "-rn)) --fn+l(gO,'', gn i'l, l'n) d0""" do- (10) These functions have the property fn+l(O,'--, gn tl,--', t,) d/t, =f,(go,..., g,-, r,,..., l'n-1)-( 11) -/0(3) Furthermore, they are normalized according to f l fn+l(g0,..., rl,.rn) d/t0---d/ 1, (12) gn O(3)1 n+ where the integration is over the product of n + 1 copies of 0(3).We are now in a position to define the n-point correlation tensors.A large class of physical and mechanical properties of crystals can be represented by tensors.Let T denote a tensor of order k representing a material property of the ideal reference crystal.Relative to an orthonormal basis {ei} the reference material tensor can be expressed in polyadic form as: (13) where repeated indices imply summation according to the Einstein convention.In crystal, with its lattice transformed by g relative to the ideal reference crystal, the same material property is represented by a tensor T (k) Pg'I(k), T <) Ti...ike, ''" ei, PT ) T...ge, --ge gid''" gikj T...]ei, ''' eik, (14) where we have set ge gei.Thus the coefficients of the property tensor in the transformed crystal are related to those of the ideal reference cstal by the relation ,...i gi,," gir,....
The reader should note that the mean and correlation tensors can in principle be obtained directly from experimental measurements (viz, from microdiffraction methods; cf.Wright and Adams 1992; Kunze, WriSt and Adams 1992).
Consider an ensemble of N polycstals.Measure the tensor T at position x in each polycstal.e mean tensor is then formed according to N (T(x)) Lim T(x),/N N n=l where T(x) denotes the measurement in the polycstal denoted by n. e n-point correlation tensor is formed in analogous fashion by N )(r,..., m-l) Lim T(x) T(x + rl), ---T(x + rn_)./N. (19)n=l In statistically homogeneous ensembles the mean and correlation tensors are independent of position x.It should be evident that correlations of all types of mixed tensors, such as (T S T)(r; rE), are defined in the same manner.
It is important to note that it is not generally possible to constct mean and correlation tensors for a second material tensor (e.g. of differing rank) from those of a first material tensor.For this reason the n-point probability density nctions f are considered to be more fundamental representations of microstcture than the correlation tensors themselves.However, in specific problems it may be easiest to deal with the particular n-point correlation tensors directly.
2.3 Model Microstructures Exhibiting Only Crystallite-Scale Auto-Coherence It is possible that in some microstructures the geometrical features of the constituent crystallities may be statistically uncorrelated with lattice orientation.
In other words, grain size and shape, and other geometrical features of the microstructure may be statistically decoupled from lattice orientation.It is also possible that some microstructures may exhibit no measureable coherence between neighboring crystallites.An example of such microstructure exhibiting these two features was recently described by Wang, Morris and Adams (1990).
When such microstructure is present a major simplification occurs in the structure of the probability density functions of lattice orientation.

REPRESENTATIONS OF GROUPS
In this section we recall some aspects of the theory of group representations which are required in the subsequent development.We focus on finite and compact groups.
Let G be a group.We shall say that we have an n-dimensional matrix representation g--> M(g) of G if to each element g e G there associates an invertible linear n x n matrix M(g) such that the product of elements of the group corresponds to the product of their matrices.That is to say, M(gl)M(g2)= M(gg2).It is also required, for the identity element of the group g id, that there corresponds the identity transformation M(id)= I, which is the n x n diagonal matrix consisting of l's down the diagonal and O's elsewhere.It follows that M(g-1) (M(g))for all g e G.
If g--> M(g) is a matrix representation of G, then for any invertible (n x n) matrix A we can construct another representation /t/(g)= A-1M(g)A.Since M and/ can be obtained from one-another by simple conjugation, their properties are the same, and they are called equivalent (isomorphic).In representation theory, equivalent representations are considered to be the same representation.Now consider a linear action of the group G in a vector space V of dimension n.Thus, to every element g e G is associated a non-singular linear transformation Tg" V--> V with properties TglTg2 "-Tglg2 and To idv where idv is the identity element of V. Taking any basis B of V, we can express the operator Tg by its matrix [Tg] a= M(g) in this basis; note that thereby we define an n-dimensional matrix representation of G. Representations obtained by means of different bases are equivalent since for any pair of bases B1 and B2 there is defined a transition matrix Sma2 such that ME(g)= -1 S IIB2MT (g)S/l/2.
Two operations are defined for matrix representations: the direct sum, and the tensor product.Given two representations M and Me of dimensions n and n2, respectively, their direct sum, M M Me, is constructed according to The new representation M is of dimension n + n2.Notice that, strictly speaking M M = M2 M; however, these two representations are obviously isomor- phic, and therefore indistinguishable in representation theory.
The tensor product of two matrix representations obtains from the Kronecker cross-product of two matrices: m"')(g)M2(g) m"'2)(g)M2(g).., m",",)(g)M2(g) where m#)(g) denote the i] entry of M(g).As in the case of the direct sum, the tensor product is not commutative: M (R) M2 M2 (R) M. In the context of representation theory, however, they are isomorphic.
Let V, W be vector spaces.Let ,..., be a basis of V and w,..., w, be a basis of W. The tensor product V (R) W is defined as the space whose basis is we can define the cross product v (R) w e V (R) W by v w ., aibjvi (R) wj. (26 If there is a linear action of a group G in V and in W, we can naturally define an action of G in V (R) W by g(vi (R) Note that this action is independent of the choice of basis in V and W, although a choice has been used in the definition.It can also be seen that the matrix representation of this action is equal to the tensor product of matrix repre- sentations of the actions of G in V and W: M M1 (R) M2.
The foregoing construction enables a definition of the power V (R)M of a vector space V, and an action of G in V(R)M, assuming one exists in V.A basis of V (R)u is {oil (R).--(R) vj,} where {vA is a basis of V. Define the Mth symmetric power of V as a subspace SUV c V b which consists of all possible linear combinations where aj,...,, is symmetric with respect to permutations of its indices.Obviously, S'V is an invariant subspace under any linear action of a group G in V (R)u arising from an action of G in V.It follows that if we have a linear action of G in V, we in SV.The dimension of Slav is (M+n-1 where n is the dimension of V. A representation M is said to be reducible if it is possible to find an equivalent representation M such that/l(g) for all g belonging to the group G has the form /17/(g)= [Ml(g) Y(g)] where M(g) and ME(g) are (n x nl) and (n2xn2) matrix representations, respectively.M(g) is called a subrepresentation of M(g), and M2(g) is called the factor representation of M by M1.If it is not possible to find an equivalent representation of this form, the representation is called irreducible.For the groups of interest in this paper (i.e., finite and compact groups) any repre- sentation is isomorphic to a direct sum of irredubile representations.This decomposition is unique up to the order of the summands.In other words, Eq. ( 28) can always be reduced to the form of Eq. ( 23) with Y(g)= 0, where Ml(g) and M2(g) are new representations of G.
We conclude this general discussion by mentioning invariants.Suppose we have a linear action g--> Tg of a group G in a vector space V. Let Z be another group (this may be G itself or any subgroup of G, or a group of different nature) that also acts linearly on V: o--* So:V---> V. Then we can form a particular subspace of V, V r'={veVlfor all o,, o.v=v}.It is called the subspace of Z-invariants in V. Assume that S commutes with T; that is, So T TgSo for all o e and g e G. Then the subspace V will be invariant under the action of G.It gives rise to a subrepresentation M of the matrix representation M of the action of Gin V.
3.1 Representations of 0(3) One of the central problems of representation theory is to classify the irreducible representations of a given group.For finite and compact groups this problem is generally solved.Here we shall state a few results of this classification for the orthogonal group in 3, 0(3).
It is known (el.Broecker and tom Dieck 1985) that for any integer m _> 0 there exists exactly two non-isomorphic irreducible representations of 0(3) having dimension k 2m + 1.Even dimensional irreducible representations do not exist.
Let us denote these two representations by M and M where k is now any positive odd integer.Also, let inv denote the inversion element of the orthogonal group: In the representation Mthe group element inv maps to I, the k-dimensional identity matrix.For this reason we can think of Mas being the k-dimensional representation of the quotient group O(3)/H SO(3) where H is the group of two elements {id, inv}.(O(3)/II is the manifold (collection) of all physically distinctive (right or left) cosets of II in O(3), and SO(3) is the three-dimensional special orthogonal group of transformations which have determinant equal to +1.)The representations Mare obtained from M, for all g e G, by the relation M-(g) M-(g) det(g).
An important example of a reducible matrix representation of 0(3) is the representation in tensors of rank 1 on 3.This representation associates to an element g 0(3) the matrix connecting the coefficients of any tensor T of rank 1 and the transformed tensor PT (cf.Eq. ( 15)).This representation has dimension 3t, and it is isomorphiC: to M--/-times-Mf(M) (R).
(31) This leaves 21 + 1 independent components.Now consider transformations of the form T pgO, g O(3), as described in Eq. ( 15).If T o is completely symmetric and traceless, then it is easily demonstrated that T will also be completely symmetric and traceless.The matrix of coefficients connecting the independent components of T with those of the transformed tensor T satisfies all of the required properties to qualify as a 21 + 1 dimensional irreducible representation of 0(3).It is isomorphic to M+I if 1 is even, and to M+ if I is odd.
One central problem of representation theory is the decomposition of the tensor product.If Ml(g) and ME(g) are irreducible representations of a group G, the representation Ml(g)(R) ME(g) is usually not irreducible.It follows that it can be decomposed into a direct sum of irreducible representations.Any particular irreducible repesentation may occur several times in this direct sum.The number of times it occurs is called its multiplicity.The multiplicities of occurrence of irreducible representations in tensor products are called Clebsch-Gordan coefficients, and the corresponding decomposition is called the Clebsch-Gordan decomposition.
The Clebsch-Gordan decomposition for 0(3) is well known.Given two representations M: and M, the decomposition is where the combination of +, + or -, on the left-hand side of the equation results in + in all terms on the fight-hand side, otherwise appears in all terms on the fight-hand side.The multiplicities are either 0 or 1 when considering 0(3).
3. 2 Representations In the Space of Square-lntegrable Functions, L2(X) There is a class of infinite-dimensional representations that will be of interest.Of interest is the action of 0(3) in spaces of square-integrable functions, LE(x), defined over the domain X O(3)/F where F is a closed subgroup of 0(3).The action of 0(3) in LE(x) is defined by Pgf(X) =f(g-ix), g O(3), x e X.
(33) The relationship between f(x) and Pgf(X) is an infinite dimensional matrix M in any given basis of LE(x).It gives rise to an infinite dimensional matrix representation of 0(3).A well known result of representation theory insures that by appropriate change of the basis it is possible to reduce this matrix to a block-diagonal matrix consisting of blocks of finite size.Stated another way, the action of 0(3) on L2(X) reduces to an infinite sum of finite, odd-dimensional, irreducible representations.Thus, mk MEk+l, (34) k=O where m: is the multiplicity of occurrence of the representation + M2k+ in M(R).
3. 3 Tensorial Representations off: X-.The decomposition of L2(X) into irreducible representations enables a Fourier analysis on the space X O(3)/F.It is known (see Adams, Guidi, Boehler and Onat 1992, and Guidi, Adams and Onat 1992) that any function f: X--can be expressed as where the vector q0 'k'+ transforms according to the Mk+l component of M under the action of 0(3).This decomposition may not be unique since the multiplicities m: will generally exceed 1, but a decomposition always exists.q0 i'k' {q0i'_',..., q0**'+/-} can be interpreted as vector functions on X with values in the space of completely symmetric and traceless tensors of rank k on 3.They are easily constructed from the tensor transformation expressed in Eq. ( 15).
It is also readily shown that the coefficients C'""-(f) are also associated with symmetric and traceless tensors ci'k'+(f) of order k.Then relation (35) can be rewritten as (x) + E c""'- (D. (35') where the product C. q9 is understood to be the contraction of tensors.
With regard to the choice of decomposition, we impose orthogonality and normalization conditions on the invariant integration of products of functions: Any set of tensor valued functions satisfying these conditions constitutes a coordinate-free, tensorial, orthogonal representation of L2(X).(If multiplicities >1 are involved, it is still not unique, however.)

REPRESENTATIONS OF CORRELATION TENSORS
We now formulate the central problem of the paper.Let T be a tensorial observable of rank k.Form the correlation functions (TM)(rl,..., rM_l), which are tensors of rank Mk.Let Z denote the indicial symmetry characteristics of T (YcSk, the symmetric group of degree k).By F we denote the symmetry subgroup of the crystal lattice (F c O(3)), and by P we indicate the symmetry subgroup associated with thermomechanical processing (P 0(3)).The physical meaning of P is that processing often induces statistical symmetries in the microstructure reflecting symmetries in the mechanical, thermal and chemical forces employed in processing.The question of focal interest here is to determine the number of independent parameters that fix the value of (T)(rl,..., r_) 3 for specified r,..., r_ e In other words, we seek to find the appropriate "residence space" for such correlation tensors.
We already know that the matrix representation of the action of 0(3) on the space V(k) of tensors of rank k is isomorphic to M (R)k.The tensor T that is of interest to us is invariant with respect to the action of Y, and it resides in the space ft-(k, Z) =def V(k) r" {v e V(k) If or all o e Z, Soy v}. (37) Since the actions of 0(3) and Y. in V(k) commute, V(k) r" inherits a linear O(3)-action from V(k), and the matrix representation of this action is (Mf(R)k)r'.
The structure of this representation can be easily determined for all cases of interest.Its decomposition into a direct sum of irreducible representations has the where the sign of M, on the fight hand side, is positive when k is even, otherwise it is negative.The coefficients C(k, .)are positive integers, or zero.
We consider material tensors T which are symmetric with respect to a subgroup F characteristic of the crystal lattice.It follows that T lies in the space of F-invariants of (k, Y.); or in other words, in the space -(k, )r.-(k, y)l-is not, in general, an O(3)-invariant subspace since the actions of F and 0(3) in -(k, Y) need not commute.We shall therefore restrict our attention to the O(3)-invariant subspace W(k, ,, F) of -(k, Y.) that is generated by the subspace if(k, )r.In other words, W(k, ,, F) is defined to be the smallest invariant subspace of 0(3) in -(k, Y) which contains -(k, y)r.
There is an easy rule which allows us to construct W(k, ,, F) from (k, ).
MEk+l is "empty' with respect to F if the matrices Mk+I(y), Let us say that +/- , e F, do not have a non-zero common invariant vector.A criterion for Mk+ to be empty is that m: =0, where m: is the multiplicity of occurrence of Mk+l in LE(x), and X O(3)/F.The procedure for constructing W(k, Y, F) is as follows: (1) Decompose -(k, Y.) into the direct sum of irreducible representations of 0(3) using the Clebsch-Gordan rule, then (2) Eliminate any terms which are empty with respect to F. The result is exactly W(k, Y., F).The matrix representation of the action of 0(3) in the subspace, denoted by d/IW(k, Y, F) decomposes according to As we extend these ideas to the correlation tensor (TM)(rl,... ,rM_l) it is natural to regard it as an element of the space W(k, Y., F) (R).This is the residence space for correlation tensors (TM) when M > 2.
When M 1 we must take into account the processing symmetry, and the residence space of (T) is W(k, Y., F) e.When M 2, if the processing symmetry group P contains the element inv, then the space where ('IT)(r) lies will be S2W(k, ,, ['), and For the model microstructure described in Section 2.3 the number of independent parameters can be further reduced.As before the tensor (T) lies in the space -(k, Y)(R), and its independent parameters can be found as described above.However, if M--> 2, some of these parameters occur in for N < M.More precisely, use the notation (T) to represent the mean value of the Mth tensor product of T: (TM)o (T(x) (---M times---@ T(x)).
Thus, the only additional information required by the two-point correlation tensor, beyond the simple mean value of the tensor itself, is the mean value of the tensor squared (i.e.(T2)).For the three-point correlation tensor the only additional information required, beyond what is required for the two-point correlation tensor, is the mean value of the tensor cubed (i.e.(T)).This trend obviously repeats itself for higher-order correlation tensors.The only new information required by the Mth correlation tensor, beyond the (M-1)th and lower-order correlation tensors, is (Ta*).
Mean values of the form (TU) occurring in the model microstructure are defined by many fewer independent parameters than (Ta*).In particular, there is no dependence upon r,.. ,r_.It is also clear that (TU) is symmetric with respect to permutations of its M factors.Moreover, since T lies in (&, X) r, T lies in S((&, x)r), so (T) must lie in the minimal O(3)-subrepresentation of SW(i, X, F) that contains S((&, x)r). is minimal subrepresentation we denote by U(&, X, F, M).As before, in order to find U, we can decompose SW(&, X, F) into a direct sum of irreducible subrepresentations, and then eliminate those which are empty with respect to F. In general, this will still leave us with a larger subrepresentation than U; there may be some non-empty components of SuW that do not occur in U. Indeed, for instance if (&, X)'-is 1-dimensional, every irreducible O(3)-representation (even non-empty) can occur in the decomposition of U no more than once.This illustrates that finding the space U for nontfivial F may require a case-by-case consideration, and cannot be reduced to a general rule.
The processing symmet reduces the space U(&, X, F, M) to the space of invariants U(&, X, F, M) e, which is exactly the space of independent parameters of (T). is answers the fundamental question addressed by this paper.
4.1 Examples from Representations of Material Tensors of Order Two 4.1.1.Consider symmetric material tensors T of order two when no lattice symmetry is present.Thus, k 2, X $2 (the group of permutations of two objects), and F is the trivial group I consisting of the identity element.The space W(2, $2, I) is just the space of symmetric tensors of rank two.The representation of 0(3) in this space is isomorphic to M M and its dimension is 6. (The reader may find it convenient to think of the stress or strain tensors which are symmetric, and can be expressed in terms of a five-dimensional deviatoric tensor and a one-dimensional trace.)4.1.1.a.When the.processing symmetry group is trivial, then the residence space of (TM) is the 6" dimensional space W(2, $2, I) (R)M whose independent para- meters are the entries (TM)iljlizi2...idu where < jr" The remaining entries of the correlation tensor are obtained by symmetry.4.1.1.b.When the processing symmetry group is the orthotropic group, P DEh then for M > 2 the answers are precisely the same as when P is equal to the trivial group I.However, when M 1 the residence space of (T) is W(2, $2, I) .T his space is three-dimensional, and the corresponding DEh-invariant tensors are all diagonal tensors.The independent parameters are (T)n, (T)22, and (T)33 with all other entries being zero.When M 2 the residence space of (T2) is $2W(2, S2, I), recalling that S2V c. V (R) V, S:'V {E aiv, (R) vi Iv,, vi e. V, aii= ai,}.
(44) It is evident that $2W(2, S2, I) is 21-dimensional.In this case the independent parameters are (TE)ijkl, with i<_j and k-<l, and [ij] <-[kl] which means i-< k or i=k but ]<-I (lexicographic order).The remaining entries retrieve from consideration of index symmetry.The reader will note that the additional symmetry (T2) ijkl (T2) kli appears due to processing. 4.1.1.c.When the processing symmetry is "wire-like", the symmetry group is cos q sin q 0 cos sin q 0 P=D,h= --sinq0 cos 0 sinq --cosq 0 (45) It follows that when M > 2 the answer obtained is identical to that given in 4.1.1.a.When M 2, the answer obtained is identical to that given in 4.1.1.b.
When M= 1 the residence space of (T) is W(2, $2, I)(R)h, which is two- dimensional.The elements of this space are tensors of diagonal form with (T)I (T)22; the independent parameters are (T)ll and (T)33.
4.1.2.Consider symmetric material tensors T of order 2 with no crystal symmetry, and assuming no orientation correlation between distinct grains according to the model described in 2.3.Then the space U(k, Z,, F, M) defined above is just SM-(2, Z) and it has dimension [M + 5 (M + 1)(M + 2)(M / 3)(M + 4)(M + 5) (46) M ] 120 4.1.2.a.If there is no processing symmetry (i.e.P is trivial) then the residence space of (TM) is U(2, Z,F,M), and the independent parameters are o ih T )iljl""iktjM W t i <j, 1 < n < M, and [injn] < [in+ljn+l]..1.2.b.If P is the orthotropic symmetry group D2h the residence space of (T) is (S-(2, y.))oThe independent parameters are now (T) with ijt"'ij indices satisfying the same conditions as enumerated in 4.1.2.a, plus an additional condition: there must be an even number of l's, an even number of 2's, and an even number of 3's among the indices, otherwise the corresponding component is zero.This condition is equivalent to DEh-invariance.The remaining components are retrieved by symmetries.
4.1.2.c.When P is the "wire" symmetry group, Dooh, the residence space of (T) is just (Sff(2, y.))o.When M 2 we have the following decomposition- $2-(2, Y-i 2M 2M M. The group D(R)h has a unique invariant (to within M2/+.It follows that the residence space is a scalar multiplicative factor) in every + 5-dimensional.The independent parameters of (T2) (assuming that the wire axis is the 3-axis) are the coefficients of the decomposition 2 0 (T )i mn + + "-O[3('Vijmn "-Vmnq)"-  where v33 1 (all other vq =0) and Dqmn is a fourth-order symmetric and traceless tensor invariant under rotations about the 3-axis.The entries of D are D Dzz 3, D3333 8, D33 D33 D33 D33 -4, D Dill2 1, and all other entries are zero. The coefficients of the decomposition may be expressed in terms of (TZ) (=Z) in the following manner: aq (4Z# + 2Z33-2Z#33 3Z/I/I)/5, t$" 2 (Zjj33 Zj3j3 + 4Z/I/I 2Z//l)/10, tr3 Zljlj -l-Z]]33 Z#I-Z/3/3, t4 2Z#11 2Z#a3 + 3Z/3/3-3Z/l/l, t Z212-e2-(In these ex- pressions repeated j indices imply summation from I to 3.) All of these results hold for any crystal symmetry group F such that M is not empty, except the case M-> 3 under the no-correlation assumptions associated with the model microstructure; in this case a further reduction occurs in the number of independent parameters.The dihedral groups, for instance, retain these properties.When M; is empty with respect to F then -(2, N) is the one-dimensional space of scalar tensors, therefore all correlation tensors will be scalars.This occurs, for example, when F Oh, the symmetry group of the cube.
4.2.1.c.Now consider wire-like processing symmetry, P-Dh.Again we find that when M > 2 the results are identical to those described in 4.2.1.a.When M 2 the results described in 4.2.1.bapply.When M 1 the residence space of (T) (T) is (-(4, y.))o(R) which is 5-dimensional.The description of independ- ent parameters coincides with those described in 4.1.2.c for the second-order tensor with M 2. The reason for this coincidence is clear; fourth-order tensors with index symmetry characteristic of elastic properties lie in $2(2, 2) where -(2, Z2) is the space of symmetric tensors of order 2.
4.2.2.Consider the case where no symmetry of the crystal lattice is present, as before, but in addition lattice orientation between grains or crystallites is not present according to the model microstructure defined in 2.3.For this case U(4, Y., M, F) S-(4, Z) which has dimension (M//20).
4. 2. 2. b.Now consider the case of orthotropic processing symmetry, P D2h.The residence space of (TM) O is U(4, Y-, M, F) The independent components of (TU) are identical to those described in 4.2.2.a with an additional index condition: there must be an even number of 1's, 2's and 3's among ijsmsns for 1 -< s -< M. The dimension of the residence space (i.e. the number of independent components) is 75 if M 2 and 499 if M 3.
Since there are 10 such coefficients, they will satisfy one linear relation, again originating from the empty term M which has been eliminated.

SUMMARY
In the development described above, we have considered the n-point correlation tensors as a particular class of quantitative representation of polycrystalline microstructure.Although in general we must assume that the spatial coherence of lattice orientation extends beyond the scale of the crystallites themselves (i.e.into 1st and 2nd nearest neighbors, etc.), there is mounting experimental evidence that for some polycrystals it is reasonable to assume that a coherence persists only over distances associated with finite crystallite size.For such a "model" microstructure there obtains substantial simplification in the point statistics (cf.Eqs. (20) and (21) in the text) and in the residence space of the corresponding correlation tensors.
The central problem of this paper has been to show the action of various aspects of symmetry on the residence spaces of the n-point correlation tensors.
Beginning with a consideration of the action of 0(3) on the space of tensors of rank k, V(k), we have illustrated the reductions which occur in representation due to the presence of indicial symmetry in the coefficients of the tensor, the symmetries of the crystal lattice, and statistical symmetries occurring due to processing.The group-theoretical framework for reducing the residence space, from its general form to the smallest possible (least dimension) form, has been described in general, and in several specific examples.
Knowing the residence space for any particular correlation tensor can substan- tially reduce the computational labor required in forming properties estimates.In fact, the dimension of some spaces of interest would be completely out of the realm of current computational possibility when the reductions due to symmetry are not considered.There is another interest for such efforts in symmetrization, however, which seems equally important.This can be best described as a clarification of the physical nature of the models which incorporate the correla- tion tensors.Very often finding the smallest possible residence space can substantially clarify physical theory.

2. 2
Definition of the n-Point Correlation Tensors