 1. Introduction
 2. New developments in macro strain/stress analysis by spherical harmonics
 3. Voigt ground state spherical harmonics representation: the strain pole distribution
 4. The selection rules
 5. WSODFs and the averaged strain and stress tensors
 6. The `short' representation of the strain pole distribution
 7. Implementation in Rietveld programs
 8. Conclusions
 Supporting Information
 References
 1. Introduction
 2. New developments in macro strain/stress analysis by spherical harmonics
 3. Voigt ground state spherical harmonics representation: the strain pole distribution
 4. The selection rules
 5. WSODFs and the averaged strain and stress tensors
 6. The `short' representation of the strain pole distribution
 7. Implementation in Rietveld programs
 8. Conclusions
 Supporting Information
 References
research papers
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Elastic macro strain and stress determination by powder diffraction: spherical harmonics analysis starting from the Voigt model
^{a}National Institute of Materials Physics, Atomistilor 105 bis, PO Box MG 7, Magurele, Ilfov 077125, Romania, ^{b}Department of Physics and Astronomy, University of Denver, 2112 East Wesley Avenue, Denver, CO 80208, USA, and ^{c}Los Alamos Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM, USA
^{*}Correspondence email: nicpopa@infim.ro
A new approach for the determination of the elastic macro strain and stress in textured polycrystals by diffraction is presented. It consists of expanding the strain tensor weighted by texture in a series of generalized spherical harmonics where the ground state is defined by the strain/stress state in an isotropic sample in the Voigt model. In contrast to similar expansions already reported by other authors, this new approach provides expressions valid for any sample and crystal symmetries and can easily be implemented in whole powder pattern fitting, including ). J. Appl. Cryst. 34, 187–195] reported a similar model, but with a spherical harmonics expansion around the hydrostatic strain/stress state of the isotropic polycrystal. The availability of several different models is beneficial in order to allow one to select the representation in which the ground state is the closest to the actual stress state in the sample.
An earlier article [Popa & Balzar (2001Keywords: strain; stress; textured polycrystals; powder diffraction; spherical harmonics.
1. Introduction
One of the oldest applications of powder diffraction is the investigation of the elastic macro strain and stress state in polycrystalline samples. A recent review of the main theoretical approaches in this field can be found in the book chapter by Popa (2008). According to the definitions and notations from that work, the elements of strain and stress tensors dependent on the crystallite orientation, called also the strain and stress orientation distribution functions (SODFs), are e_{i}(g), s_{i}(g) (i = 1,6) in the sample orthogonal reference system , and , (i = 1,6) in the crystal reference system , where is the triplet of Euler angles transforming into . The SODFs are connected by the singlecrystal Hooke equations, which in the system are the following:
Here C_{ij} and S_{ij} are the singlecrystal stiffness and compliance elastic constants and are the components of the vector (1,1,1,2,2,2). In the sample system () the Hooke equations and the singlecrystal elastic constants follow as
P_{ij} and Q_{ij} denote the matrices of transformation of strain/stress tensors from the crystal into the sample reference system and vice versa:
The quantity measured in a diffraction experiment on a polycrystal is the mean value of the strain along the
vectors fulfilling the ; here is the wavevector transfer. If and are the unit vectors of and in and , respectively, the measured strain can be written in two equivalent ways:The brackets denote averaging only over the crystallites satisfying the condition for Bragg reflection and the symbol () denotes the average of the two terms and . The function f(g) is the orientation distribution function (ODF) and is the reduced pole distribution: . Analogously, the function can be called the strain pole distribution. The vector components (E_{i}) and (F_{i}) in equation (5) are defined as follows: () = (a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, a_{2}a_{3}, a_{1}a_{3}, a_{1}a_{2}), () = ( b_{1}^{2}, b_{2}^{2}, b_{3}^{2}, b_{2}b_{3}, b_{1}b_{3}, b_{1}b_{2}), where (a_{1},a_{2},a_{3}) = (, , ) are the direction cosines of in and (b_{1},b_{2},b_{3}) = (, , ) are the direction cosines of in , the pairs and being the corresponding polar and azimuthal angles.
Each elastic strain/stress tensor has two components: the averaged macro strain/stress tensor and the intergranular tensor . Here stands for any of the tensors , , , . The elastic intergranular strains/stresses have various origins but can be grouped in only two terms, namely the elastically induced and the plastically induced intergranular strain/stress. The principal aim of stress analysis for many years was to determine only the average tensors. The determination was based on the assumption that the plastically induced strains/stresses are zero and the elastically induced strain/stress tensor elements dependent on the crystallite orientation are given as linear combinations of the averaged macro stress/strain tensor elements. This assumption leads to the following strain pole distribution: , where R_{j} are called diffraction stress factors. This formula is typical for classical models like Voigt (1928), Reuss (1929) and Kroner (1958). For isotropic (not textured) samples, this formula becomes linear in and and is the basic equation of the traditional `' method (Hauk, 1952; Christenson & Rowland, 1953). This method is appropriate to process most of the experimental data.
For textured samples the relation between the peak shifts and becomes nonlinear, but sometimes, especially in metals after plastic deformation, the departure from linearity cannot be explained only by texture. To explain such a large departure, the whole intergranular strain/stress must be accounted for and the exact relations (2) between strain and stress tensors considered. One possibility is to calculate the plastically induced part of the stress starting from the models of the of crystallites. The second possibility is to obtain SODFs directly, by inverting the strain pole distributions measured for several peaks in a large number of sample directions. In this case it is not necessary to assume any particular model for the elastic or plastic interactions of crystallites.
The representation of SODFs by generalized spherical harmonics has been proposed as a method for inverting the strain pole distribution. Three different approaches were reported: by Wang et al. (1999, 2001, 2003), by Behnken (2000) and by Popa & Balzar (2001). Wang and coworkers represented by spherical harmonics the stress tensor in the sample reference system s_{i}(g), and separately the ODF, and used the Clebsch–Gordon coefficients to express the product of the SODF and the ODF. Behnken (2000) proposed to expand in spherical harmonics both e_{i}(g) and s_{i}(g), independently, and calculate numerically the integrals in equation (5b). For determination of the coefficients of expansion, both Wang et al. (1999, 2001, 2003) and Behnken (2000) used a stabilized leastsquares method.^{1} Finally, these authors considered only the case of the cubic crystal symmetry and orthorhombic sample symmetry. The third approach, reported by Popa & Balzar (2001), is similar to those of Wang et al. (1999, 2001, 2003) and Behnken (2000), but with an important difference that makes the problem of determination of the strain tensor equivalent to the texture problem. In place of the SODF, e_{i}(g) and s_{i}(g), the analysis by spherical harmonics was performed on the product between , the SODF in the crystal reference system and the ODF, in other words on the SODF weighted by texture (WSODF). Equations (5) express the projection of the WSODF, with the condition , from the Euler space into the space . The WSODF is the quantity measured in an experiment, as the d spacings can be measured only along the grain orientations present in the sample. On the other hand, the ODF can be independently derived, from the integrated intensities of the diffraction peaks. Therefore, once the WSODF is known, the SODF can be readily calculated from it. The major advantage of expanding the WSODF in place of the SODF is that the integrals in equations (5) are solved analytically, resulting in an open (convergent infinite series) analytical expression for the strain pole distribution, similar to the texture pole distribution (see Von Dreele, 1997), and thus appropriate for easy implementation into Rietveld (1969) programs, allowing constraints during the refinements. Finally, such expressions were derived for all crystal and sample symmetries.
2. New developments in macro strain/stress analysis by spherical harmonics
In terms of the mathematical nomenclature, the open representation of a function, in other words, representation by an infinite convergent series, appears as a perturbation of the ground state represented by the coefficient of order zero. The further from the ground state the function is, the larger the number of terms necessary to reach convergence. If more open representations are known for a given function, it is rational to use the `shortest' for its calculation. The strain pole distribution supports multiple open spherical harmonics representations. As was proved by Popa (2008), the ground state of representation derived by Popa & Balzar (2001), starting from the WSODF formed with the strain tensor in the crystal reference system , is the hydrostatic strain/stress state of the isotropic polycrystal. The ground state will be of the Voigt (1928) or of the Reuss (1929) type if the object of the spherical harmonics analysis is the WSODF formed with the strain or the stress tensor in the sample reference system, respectively. The derivation for the first case is presented in this article, a separate article being planned for the second case. In contrast to already existent (Wang et al., 1999; Behnken, 2000) approaches, these new representations will be analytical, extendable to any sample and crystal symmetries, and implementable in Rietveld programs.
3. Voigt ground state spherical harmonics representation: the strain pole distribution
Following an approach similar to that of Popa & Balzar (2001), the WSODF defined by the elements of the macro strain tensor in the sample reference system is expanded in generalized spherical harmonics:
Here, are Legendre functions defined according to Bunge (1982) and c_{il}^{mn} are complex coefficients fulfilling the condition resulting from the real character of the series:
The definitions and properties of the Legendre functions P_{l}^{mn} and P_{l}^{m} are reviewed by Popa & Balzar (2001). Integrating equation (6) in the Euler space and using the orthogonality of spherical harmonics, one obtains ; therefore, the free term of expansion is the strain given by the Voigt model of an isotropic polycrystal.
The next step in calculating the strain pole distribution is to substitute equation (6) into equation (5b) and to integrate according to formula (1) from Popa & Balzar (2012). For the average of the two terms (), this formula becomes
In the resulting expression, the sums are permuted to have the following sequence of indices: (l,i,n,m). This sequence is imposed by the derivation of the selection rules, involving in this case the direction cosines of in the reference system .^{2} Finally, rearranging to have only positive indices, accounting for the condition (7) and using some properties of the Legendre functions P_{l}^{m}, one obtains
The coefficients , , , are obtained from c_{il}^{mn} = a_{il}^{mn} + ib_{il}^{mn} by the relations given in Table 1. They can be refined in any leastsquares program, including programs, fitting the strain pole distribution measured for a number of peaks in a number of sample directions by the parameterized equations (9)–(13). The is performed by taking into account the selection rules for coefficients imposed by sample and/or crystal symmetries higher than triclinic and truncating the series (9) at a value l = l_{0}. Finally, gathering the coefficients , , , into a vector in the sequence given by equation (14) below, equations (11)–(13) can be written in the compact form (15):
The functions are paired with the coefficients according to Table 2.
4. The selection rules
The number of coefficients for triclinic crystal and sample symmetries is , l = even, and decreases for higher symmetries. The selection rule for a given is derived from the invariance of to this operation. This quantity has a structure identical to those for spherical harmonics expansion of the hydrostatic ground state but with the dependences on the vectors and permuted, as can be seen comparing equations (10)–(13) with (17)–(20) from Popa & Balzar (2001).^{3} Consequently, to obtain the selection rules for the actual expansion, the selection rules derived in that paper for crystal and sample symmetries should be interchanged, taking care to set correctly the indices m,n. For sample Laue groups 2/m and mmm the selection rules are given in Tables 3 and 4, higher sample symmetry being improbable. The selection rules for crystal noncubic Laue groups are given in Table 5 and those for the cubic groups in Table 6.



5. WSODFs and the averaged strain and stress tensors
According to Appendix A and using the refined parameters, the strain tensor weighted by texture in terms of generalized spherical harmonics in real space is the following:
The stress tensor weighted by texture is obtained by substituting equation (16) into equation (2a):
Here C_{ij}(g) are the singlecrystal elastic stiffness constants in the sample reference system given by equation (3a).
Integrating equations (16) and (17) in the Euler space, one finds the averaged strain and stress tensors. Because the spherical harmonics are orthogonal, for l = 0, and for the rest, we have
Let us now introduce the Fourier coefficients of C_{ij}(g) in the basis :
Because the elements of the matrices P_{ij} and Q_{ij} in equation (3a) are sums of products of two Euler matrix elements, one expects that for l = 0,2,4 and for some values of i, j, but for , and then
The matrix of 3852 elements is very sparse, but it is not possible to predict which elements are different from zero. The matrix can be calculated by numerical integration requiring the singlecrystal stiffness constants as input data. A program fulfilling this task is provided as supplementary material.^{4} Note that the elements of the matrix represent the Voigt averaged stiffness constants for isotropic polycrystal [see equations (12) of Popa (2000)]. Keeping only the term l = 0, equation (20) becomes the basic Hooke equation in the Voigt model: .
6. The `short' representation of the strain pole distribution
If both the WSODFs and the averaged strain/stress tensors are not required, an alternative representation of strain pole distribution can be used to fit the measured distribution. This alternative representation may have a smaller number of parameters than the original one, namely equations (9)–(13), and can be derived by replacing with the direction cosines (b_{1},b_{2},b_{3}) in equation (11). One obtains
In equation (21) are homogenous polynomials of degree l in the direction cosines, invariant to the sample symmetry operations. They are listed by Popa & Balzar (2001). The refinable parameters are , from equation (22); this equation should be invariant to the crystal symmetry operations. The selection rules for these parameters are those from Tables 5 and 6.
A `short' representation of the strain pole distribution has been derived also in the frame of spherical harmonics analysis of the hydrostatic ground state. This comprises equations (25) and (26) of Popa & Balzar (2001). If used in the early stages of data processing, the `short' representations associated with different mathematical models will help select the most appropriate model for a given sample.
7. Implementation in Rietveld programs
To obtain a smoothed solution, it is worthwhile to implement equations (9)–(13) into a Rietveld program allowing constraints. In these programs, the Lagrange function is minimized. A simple choice for Λ is the WSODF index J, defined as follows:
By using equation (16) and the orthogonality conditions of the functions , this equation becomes
8. Conclusions
We introduce a new method for the determination of the elastic strain and stress in textured polycrystals. The approach is based on the expansion of the textureweighted strain orientation distribution function (WSODF) in generalized spherical harmonics. The difference between this and our earlier approach (Popa & Balzar, 2001) is in the ground (unperturbed) state, which is the hydrostatic strain/stress state of an isotropic sample in the earlier model and the state of an isotropic sample described by the Voigt (1928) model in this work. The choice of the model being employed will depend on the actual strain/stress state in the sample and can be established only by trials. The main advantage of this method over other existing approaches for the determination of strain and stress is that the model is applicable for an arbitrary crystal symmetry and texture. Moreover, it can be easily incorporated into whole diffraction pattern fitting programs, including those implementing Rietveld (1969) refinement.
APPENDIX A
Generalized spherical harmonics of real type
In the formalism introduced by Bunge (1982), which was used in this article and in numerous others, especially in the field of texture analysis, a real function defined in the Euler space is expanded in an infinite series, as in equation (6). In this series, the basis functions as well as the coefficients are complex quantities. For calculating the averaged strain and stress tensors as well as for further theoretical developments,^{5} it is useful to have spherical harmonics representations in terms of real functions of both basis and real coefficients. The real coefficients are just , , , defined before. For finding the corresponding general spherical harmonics in real space, we start from equation (6). Firstly, we arrange the sums in this equation in the sequence (l,n,m), then rearrange the terms to have only the positive indices; further we use condition (7) and some properties of the Legendre functions P_{l}^{mn} which are replaced by the real functions Q_{l}^{mn} = P_{l}^{mn} for m + n = even and Q_{l}^{mn} = iP_{l}^{mn} for m + n = odd. One obtains an expression containing only real functions and the coefficients . Finally, replacing these coefficients with , , , according to Table 1, equation (6) becomes
The functions in this equation are the generalized spherical harmonics in the real space that we are searching for. They are given in Table 7 in general form, valid for any harmonic index l. A table with these functions for l = 0,2 was already reported by Popa & Balzar (2001, 2012). These were derived starting from the sequence (l,m,n) for the sums in the expansion of the WSODF. As a consequence, the sequence of , , , in the vector is different from equation (14). It is the following:
Like the complex spherical harmonics, the real spherical harmonics are orthogonal:
Finally, the integral over the crystallites in the Bragg reflection condition is the following:
Supporting information
Fortran program. DOI: 10.1107/S1600576713029208/rw5054sup1.txt
Stiffness matrix. DOI: 10.1107/S1600576713029208/rw5054sup2.txt
Footnotes
^{1}A constrained leastsquares method was used to prevent the effect of the illposed character of the inversion problem; in place of χ^{2}, a Lagrange function constructed as a weighted sum of this quantity with the elastic free energy, or with the SODF index, is minimized.
^{2}In the hydrostatic ground state representation the sequence of sums is (l,i,m,n), because the direction cosines of in are involved in the derivation of the selection rules.
^{3}In the article by Popa & Balzar (2001) the direction cosines of in are denoted by capital letters A_{1},A_{2},A_{3}.
^{4}Supporting information for this article is available from the IUCr electronic archives (Reference: RW5054 ).
^{5}In the next article, a spherical harmonics analysis based on the Reuss type ground state of isotropic polycrystals will be provided.
Acknowledgements
This work was funded by the Romanian National Authority for Scientific Research through the contract PCE 102/2011.
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