addenda and errata addenda and errata

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1 addenda and errata Journa of Appied Crystaography ISSN addenda and errata Eastic strain and stress determination by Rietved refinement: generaized treatment for textured poycrystas for a Laue casses. Second corrigendum N. C. Popa a and D. Bazar b * a Nationa Institute for Materias Physics, PO Box MG 7, Bucharest, Romania, and b Department of Physics and Astronomy, University of Denver, 11 East Wesey Avenue, Denver, CO, USA. Correspondence e-ma: bazar@du.edu This corrigendum (C) repaces the first corrigendum (C1; Popa & Bazar, 1), which was pubished to correct errors in the artice by Popa & Bazar (1). The basic equation considered in C1 is the integra of spherica harmonics over the crystaites contributing to the Bragg refection. In C1, this integra is represented by equation (1), which is the expression (14.16) originay cacuated by Bunge (19). In C1, we added the factor ð 1Þ n to that expression. This factor resuted from a coding error that, unfortunatey, we discovered ony recenty in the computer routine used to cacuate such integras. With the correct coding, the factor ð 1Þ n disappears, and the Bunge (19) equation (14.1) is correct. The necessary corrections in the paper by Popa & Bazar (1) are the foowing: (1) Tabe 1: mutipy the right side of each equation by. () Tabe 15: deete the factor in the right side of the equations for R and R 1. (3) Severa errors in Tabe 16. The correct tabe is shown as Tabe 1 of the present artice. (4) The first unnumbered equation after equation (4) becomes g jk ¼ P 6 ¼1 C j g k. References Bunge, H. I. (19). Texture Anaysis in Materias Science. London: Butterworth. Popa, N. C. & Bazar, D. (1). J. App. Cryst. 34, Popa, N. C. & Bazar, D. (1). J. App. Cryst. 45, Tabe 1 The corrected version of Tabe 16. k ¼ : w ij ¼ 1=3 for i; j ¼ 1; 3 k ¼ 1 : w ij1 ¼ 1=3 for i; j ¼ 1; ; w ij1 ¼ =15 for ð3; 3Þ; w ij1 ¼ 1=15 for ð1; 3Þ; ð; 3Þ; ð3; 1Þ; ð3; Þ k ¼ : w ij ¼ 1=3ð3=Þ 1= for ð5; 1Þ; ð5; Þ; w ij ¼ 1=15ð3=Þ 1= for ð5; 3Þ k ¼ 3 : w ij3 ¼ 1=3ð3=Þ 1= for ð4; 1Þ; ð4; Þ; w ij ¼ 1=15ð3=Þ 1= for ð4; 3Þ k ¼ 4 : w ij4 ¼ 1=3ð3=Þ 1= for ð1; 1Þ; ð1; Þ; w ij4 ¼ 1=15ð3=Þ 1= for ð1; 3Þ; w ij4 ¼ 1=3ð3=Þ 1= for ð; 1Þ; ð; Þ; w ij4 ¼ 1=15ð3=Þ 1= for ð; 3Þ k ¼ 5 : w ij5 ¼ 1=3ð3=Þ 1= for ð6; 1Þ; ð6; Þ; w ij5 ¼ 1=15ð3=Þ 1= for ð6; 3Þ k ¼ 6 : w ij6 ¼ 1=15ð3=Þ 1= for ð1; 5Þ; ð; 5Þ; w ij6 ¼ =15ð3=Þ 1= for ð3; 5Þ k ¼ 7 : w ij7 ¼ 1=1 for ð5; 5Þ k ¼ : w ij ¼ 1=1 for ð4; 5Þ k ¼ 9 : w ij9 ¼ 1=1 for ð1; 5Þ; w ij9 ¼ 1=1 for ð; 5Þ k ¼ 1 : w ij;1 ¼ 1=1 for ð6; 5Þ k ¼ 11 : w ij;11 ¼ 1=15ð3=Þ 1= for ð1; 4Þ; ð; 4Þ; w ij;11 ¼ =15ð3=Þ 1= for ð; 4Þ k ¼ 1 : w ij;1 ¼ 1=1 for ð5; 4Þ k ¼ 13 : w ij;13 ¼ 1=1 for ð4; 4Þ k ¼ 14 : w ij;14 ¼ 1=1 for ð1; 4Þ; w ij;14 ¼ 1=1 for ð; 4Þ k ¼ 15 : w ij;15 ¼ 1=1 for ð6; 4Þ k ¼ 16 : w ij;16 ¼ 1=3ð3=Þ 1= for ð1; 1Þ; ð; 1Þ; w ij;16 ¼ 1=3ð3=Þ 1= for ð1; Þ; ð; Þ; w ij;16 ¼ 1=15ð3=Þ 1= for ð3; 1Þ; w ij;16 ¼ 1=15ð3=Þ 1= for ð3; Þ k ¼ 17 : w ij;17 ¼ 1= for ð5; 1Þ; w ij;17 ¼ 1= for ð5; Þ k ¼ 1 : w ij;1 ¼ 1= for ð4; 1Þ; w ij;1 ¼ 1= for ð4; Þ k ¼ 19 : w ij;19 ¼ 1= for ð1; 1Þ; ð; Þ; w ij;19 ¼ 1= for ð1; Þ; ð; 1Þ k ¼ : w ij; ¼ 1= for ð6; 1Þ; w ij; ¼ 1= for ð6; Þ k ¼ 1 : w ij;1 ¼ 1=15ð3=Þ 1= for ð1; 6Þ; ð; 6Þ; w ij;1 ¼ =15ð3=Þ 1= for ð3; 6Þ k ¼ : w ij; ¼ 1=1 for ð5; 6Þ k ¼ 3 : w ij;3 ¼ 1=1 for ð4; 6Þ k ¼ 4 : w ij;4 ¼ 1=1 for ð1; 6Þ; w ij;4 ¼ 1=1 for ð; 6Þ k ¼ 5 : w ij;5 ¼ 1=1 for ð6; 6Þ J. App. Cryst. (14). 47, 113 doi:1.117/s # 14 Internationa Union of Crystaography 113

2 Journa of Appied Crystaography ISSN 1-9 Eastic strain and stress determination by Rietved refinement: generaized treatment for textured poycrystas for a Laue casses Received 16 October Accepted 3 January 1 N. C. Popa a,b and D. Bazar a,c * a Materias Science and Engineering Laboratory, Nationa Institute of Standards and Technoogy, Bouder, CO 35, USA, b Nationa Institute for Materias Physics, PO Box MG-7, Bucharest, Romania, and c Department of Physics, University of Coorado, Bouder, CO 39, USA. Correspondence e-ma: bazar@bouder.nist.gov # 1 Internationa Union of Crystaography Printed in Great Britain ± a rights reserved A nove approach to mode diffraction ine shifts caused by eastic residua or appied stresses in textured poycrystas is proposed. The mode yieds the compete strain and stress tensors as a function of crystaite orientation, as we as the average vaues of the macroscopic strain and stress tensors. It is particuary suitabe for impementation in Rietved re nement programs. The requirements on re nabe parameters for a crysta Laue casses are given. The effects of sampe symmetry are aso incuded and the conditions for strain invariance to both the sampe symmetries (texture and stress/strain) are discussed. 1. Introduction Stress state (where the stress can be both appied and residua, that is, resident in the materia after the appication of externa force is removed) in uences many different materia properties, which is especiay important in engineering and technoogica appications. X-ray and neutron diffraction are the most accurate and widey used methods for stress determination in crystaine materias. There is a substantia amount of iterature on this subject. For more information, the reader shoud consut recent monographs by Noyan & Cohen (197) and Hauk (1997). A common approach to strain and stress determination empoys the so-caed sin method (Christenson & Rowand, 1953), where the strain is derived from directiona measurements of the interpanar spacing d as a function of the ange between the diffraction vector and an arbitrary direction in the specimen. Recenty, it was proposed that the strain/stress orientation distribution function (SODF), which is de ned as a strain/stress tensor component as a function of crystaite orientation, can be determined simary to the crystaite orientation distribution function (CODF), through the expansion in a series of generaized spherica harmonics (Wang et a., 1999, ; Behnken, ). It was assumed in such work that the crysta symmetry acts on the SODF in the same manner as on the CODF. In this context, it means that a six components of the strain tensor in the sampe coordinate system are invariant to the point-group symmetry operations. However, consider, for instance, a poycrystaine assemby of non-spherica crystaites under the appied stress. A crysta symmetry operation on a crystaite coud in genera produce different physica strain states because of interactions with other crystaites in the aggregate, athough two orientations are crystaographicay equivaent. Therefore, strain/stress tensor eements are not invariant to the crysta symmetry operations in genera. Contrary, we postuate that the observabe quantity measured by diffraction in a given sampe direction, that is, the interpanar spacing d, averaged around the diffraction vector, is invariant to the point-group symmetry operations. In this paper, we w use this invariance condition to derive the seection rues in SODF harmonic representation for a Laue casses. An aternative approach to a traditiona sin method of strain determination has been described by Ferrari & Lutterotti (1994) and Bazar et a. (199). It incudes the re nement of strain- and stress-reated parameters in a Rietved (1969) re nement program. An advantage of this approach is that a avaabe Bragg re ections are used simutaneousy to obtain the strain tensor. Even if the strain/ stress determination is not of interest, diffraction ine shifts caused by residua stresses w generay be crysta-direction dependent. These diffraction ine shifts shoud be corrected for, in order to carry out an accurate structure determination and re nement using the Rietved approach, poe- gure (texture) measurements, and simar tasks. However, for a successfu appication in the Rietved re nement, the chaenge ies in the accurate modeing of strain and stress dependence on the crystaographic direction and the abity to hande arbitrary crysta symmetry. In a recent paper, Popa () presented a method for modeing diffraction ine shifts for a Laue casses within the frame of the Voigt (19) and Reuss (199) approximations. The aim here is to propose an aternative method to mode diffraction ine shifts accuratey in a Rietved re nement program for a Laue symmetries without making Voigt or Reuss approximations. This is accompished by expanding the strain and stress tensor J. App. Cryst. (1). 34, 17±195 Popa and Bazar Eastic strain and stress 17

3 components in a series of spherica harmonics, simar to the texture modeing as described by Popa (199) and impemented in the Rietved re nement program GSAS by Von Dreee (1997). Thus, strain and stress are determined by the re nement of the respective coef cients in the east-squares re nement procedure, simutaneousy with other re nabe parameters. The genera intent in Rietved re nement is to minimize the number of re nabe parameters. Hence, we aso describe an aternative approach, where expansion of strain/stress tensor eements into direction cosines repaces the usua description in terms of harmonic components, which signi canty owers the number of re nabe parameters. Here we present ony the methodoogy, with the emphasis on introduction of this mode into a Rietved re nement program. An appication of the method and a comparison with a traditiona sin anaysis w be described esewhere (Bazar & Popa, 1).. The measured strain, the average strain and stress tensors The diffraction method directy measures the interpanar spacing d aong the direction of the diffraction vector, which must be parae to a reciproca-attice vector H for an (hk) diffracting pane. Measured strain is then de ned as an average change in the interpanar spacing from a reference vaue d : hdi=d 1 ˆhdi=d ˆ hhi=h ˆh" h y i; where the averaging is performed by the rotation for! around h = H/H, which is parae to y, the direction of the diffraction vector in the sampe. If " i are the strain tensor eements in the crystaite coordinate system (in the condensed Voigt notation), (1) can be written as (Popa, ) R h" h y i ˆ P6 E i d! f ' 1; ;' " i ' 1; i R ;' : iˆ1 d! f ' 1; ;' Averaging is weighted by the CODF f(' 1,, ' ). Here, (' 1,, ' ) are the Euer anges transforming the sampe orthogona coordinate system (y 1, y, y 3 ) into the crystaite orthogona coordinate system (x 1, x, x 3 ), as de ned by Bunge (19). The integration in () is evauated over ony vaues of the Euer anges (' 1,, ' ) that fu the condition h y. They depend on the vaues of the poar and azimutha anges of h in (x 1, x, x 3 ), denoted as (, ), of y in (y 1, y, y 3 ), denoted as (, ), and on the rotation ange!. The pairs of anges (, ) and (, ) give the direction cosines of h and y, respectivey, in their coordinate systems: A 1 ; A ; A 3 ˆ cos sin ; sin sin ; cos ; B 1 ; B ; B 3 ˆ cos sin ; sin sin ; cos : In (), we used the foowing abbreviations: E 1 ;...; E 6 ˆ A 1; A ; A 3; A A 3 ; A 1 A 3 ; A 1 A ; 1 ;...; 6 ˆ 1; 1; 1; ; ; : 1 3a 3b 4 The average strain and stress tensors in the sampe coordinate system, e i and s i (i = 1, 6), respectivey, are de ned as (Popa, ) and e i ˆ 1= R R R e i ' 1 ; ;' s i ˆ 1= R R R s i ' 1 ; ;' : d' 1 d d' sin f ' 1 ; ;' d' 1 d d' sin f ' 1 ; ;' If we denote by i the stress tensor eements in the crystaite coordinate system, Hooke's aw hods for the eastic part: i ˆ P6 iˆ1 C ij j " j ; where C ij are the monocrysta eastic stiffness modui. The foowing inear reations ink the strain and stress tensor components in the two coordinate systems: s i ˆ P6 jˆ1 e i ˆ P6 jˆ1 P ij j ; P ij " j ; a b where the eements of the matrix P are sums of the products of two Euer matrix eements. Both the Euer matrix and the matrix P have been given expicity by Popa (). 3. The strain expansion in generaized spherica harmonics We now foow an approach simar to that of Wang (1999, ) and Behnken (), but with an important distinction that makes the probem of determination of components of the strain tensor equivaent to the texture probem and signi canty simpi es the mathematica formaism. This is accompished by repacing the product of the SODF and CODF by the SODF weighted by texture (WSODF): " i ' 1 ; ;' ˆ" i ' 1 ; ;' f ' 1 ; ;' : 9 The product (9) expicity appears in () and impicity in (5) and (6) through (7) and (). This is an additiona advantage in directy determining WSODF instead of SODF, because the texture-weighted strain is actuay measured in the diffraction experiment and is required in order to cacuate other properties of interest, in particuar average strain and stress tensors. With (9), the integra in the numerator of () becomes simar to the expression for cacuating the poe distribution function P h (y): P h y ˆ 1= R f ' 1; ;' d!: 1 1 Popa and Bazar Eastic strain and stress J. App. Cryst. (1). 34, 17±195

4 Tabe 1 The reations between the coef cients + ib a ˆ a m m Hence, the probem of strain becomes equivaent to the probem of texture. The measurabe strain h" h (y)i is for WSODF what the poe distribution P h (y) is for CODF. This is usefu for impementation in Rietved re nement programs, because the texture and strain impementations become equivaent, ony the seection rues being different. Now we foow the procedure used by Bunge (19) for texture, by deveoping " i in generaized spherica harmonics: " i ' 1 ; ;' ˆP1 P P ˆ mˆ nˆ exp in' 1 : c exp im' P 11 Then, for the integra over! in the numerator of (), we have (cf. equation of Bunge, 19) R d!" i ' 1; ;' ˆP1 4= 1 Š P ˆ P mˆ nˆ P m exp in P n : As " i is a rea quantity, the coef cients c compex numbers fu ing the condition c m; n c exp im = a + ib 1 are ˆ 1 m n c : 13 With x = cos, the functions P and P m are de ned as foows: and. = ˆ a m ˆ b m P P m x ˆ x ˆ 1 m i n m m! 1= m! n! m! n! 1 x n m = 1 x n m = d n dx n 1 x m 1 x m Š 1= 1= m! 1 1 m m!! 1 x m= d m dx 1 m x : The functions P are rea for m + n even and imaginary for m + n odd. They have the foowing properties: P nm ˆ P ˆP m; n ; 14a P ˆ 1 m n P : 14b There is an obvious reation between the functions P and P m : P m n ˆ a n ˆ a ˆ b 1 n a m; n 1 n b m; n, ˆP m ˆi m = 1 Š 1= P m : By introducing this reation in (14), one obtains the foowing properties for P m : n,, ˆ b n ˆ b ˆ a and c = 1 n b m; n 1 n a m; n P m ˆ 1 m P m ; 15a P m ˆ 1 m P m : 15b With equations (15) and (13), we can rearrange (1) and consequenty () in a more convenient form for our purpose, with ony positive indices m, n. Taking into account (4), () becomes h" h y ip h y ˆP1 = 1 ŠI h; y ; 16 where ˆ I h; y ˆA 1t 1 h; y A t h; y A 3t 3 h; y A A 3 t 4 h; y A 1 A 3 t 5 h; y A 1 A t 6 h; y ; 17 with t h; y ˆA y P P A m y ˆ m P P nˆ1 B m y ˆ m P P nˆ1 The coef cients coef cients c mˆ1 A m y cos m B m y sin mšp m ; 1 cos n sin n P n m ˆ ; ; cos n sin n P n m ˆ 1; : 19,, and are obtained from the by the inear transformations given in Tabe 1. They can be directy re ned in the Rietved program to yied the WSODF and the average strain tensor. The average stress tensor can aso be determined if monocrysta eastic stiffness modui C ij are known. The required number of re ned coef- cients to achieve the desired precision of the WSODF, strain and stress tensors w depend on the crysta and sampe symmetries, as we as on the magnitude and the gradient of the strain and the texture. 4. The seection rues for a Laue casses Equation (16) is a genera formua for strain (diffraction ine shift) determination, vaid for tricinic crysta symmetry. For a given vaue of, the tota number of the coef cients for every i is ( +1), where the number takes ony even vaues because of Friede's aw (see x4..1). If the crysta and sampe symmetries are higher than tricinic, the number of coef cients,, and in (19) and () is reduced, some coef cients being zero and some being correated. To nd the seection rues for a Laue casses, we appy the invariance condition to the measured strain, h" h (y)i, as outined in the x1. We denote two operators for the crysta and sampe point group by X and Y, respectivey. Because the terms in the sum (16) are independent, the invariance condition has to appy to every I in (16): J. App. Cryst. (1). 34, 17±195 Popa and Bazar Eastic strain and stress 19

5 Tabe The seection rues imposed by the sampe non-cubic symmetries. r denotes the r-fod axis in the direction =. =m; 3; 4=m; 6=m : =mmm; 3m; 4=mmm; 6=mmm : m ; m ; >< m ; m ; ; ; ; n ˆ rk ; n ˆ rk; n even ; ; n ˆ rk; n odd Tabe 3 The seection rues imposed by the cubic sampe symmetry for 4. The foowing constraints must be added to the seection rues for the orthorhombic /mmm Laue cass and the tetragona 4/mmm Laue cass, respectivey (Tabe ). m3 : m3m : i ; i ˆ; n ˆ ; m4 i4 ; i4 m4 ˆ 1=7 1= m i4 ; i4 m m i ; i m ˆ m4 i4 ; i4 m4 ˆ 1=7 1= m i4 ; i4 m Tabe 4 The seection rues for the monocinic /m Laue cass. i ˆ 1; ; 3; 6 : A A m ; B m ; m ˆ k i ˆ 4; 5 : A m ; B m ; m ˆ k 1 Tabe 5 The seection rues for the tetragona 4/m Laue cass. i ˆ : A 1 A m 1; B m 1; m ˆ k A ˆ A 1 A m ˆ 1 k A m 1; B m ˆ 1 k B m 1; m ˆ k A 3 A m 3; B m 3; m ˆ 4k i ˆ 4 : A m 4; B m 4; m ˆ k 1 i ˆ 5 : A m 5 ˆ 1 k 1 B m 4; B m 5 ˆ 1 k A m 4; m ˆ k 1 i ˆ 6 : A m 6; B m 6; m ˆ 4k Tabe 6 The seection rues for the trigona 3 Laue cass. i ˆ : A 1 A m 1; B m 1; m ˆ 3k ; 3k 1; 3k >< A ˆ A 1 A m ˆ A m 1; B m ˆ B m 1; m ˆ 3k A m ˆ A m 1; B m ˆ B m 1; m ˆ 3k ; 3k 1 A 3 A m 3; B m 3; m ˆ 3k i ˆ 4 : A m 4; B m 4; m ˆ 3k ; 3k 1 A m 5 ˆ B m 4; B m 5 ˆ A m 4; m ˆ 3k i ˆ 5 : A m 5 ˆ B m 4; B m 5 ˆ A m 4; m ˆ 3k 1 A m 6 ˆ B m 1; B m 6 ˆ A m 1; m ˆ 3k i ˆ 6 : A m 6 ˆ B m 1; B m 6 ˆ A m 1; m ˆ 3k 1 I X h; Y y ˆI h; y : 4.1. Seection rues imposed by the sampe symmetry 1 For textured sampes under stress, two sampe symmetries must be distinguished: texture and stress/strain sampe symmetry. Sometimes they are identica, but generay the strain sampe symmetry can be ower than the texture sampe symmetry. Furthermore, according to (16), the texture sampe symmetry operations must form a supergroup of the strain sampe symmetry point group because P h (y) must be invariant to both symmetry operations. An exampe is the dependence of the observed strain on the direction in a sampe with randomy orientated crystaites, derived in the Reuss (199) approximation by Popa (). Here the texture has a spherica symmetry but the dependence of h" h (y)i on y, in genera, shows tricinic symmetry and ony for hydrostatic stress does it become independent of y (isotropic). Moreover, consider the uniaxia stress acting on the sampe with a cubic sampe texture. The symmetry for strain is tetragona if the stress axis is aong the cube axis, trigona if the stress axis is aong the body diagona, but tricinic if this axis is oriented in an arbitrary direction. For strain sampe symmetry higher than tricinic, the invariance reation (1) for any h, with X = E (where E is the identity operator), requires the invariance of the functions A m y and B m y in (1) independenty for every m. In other words, the coef cients of different strain tensor components are independent. This impies that the seection rues that appy to the coef cients,, and are identica to those for the texture of the same sampe symmetry. These rues were previousy given by Popa (199) and are summarized in Tabes and 3. Therefore, athough strain and texture sampe symmetries may be different, the seection rues for the same symmetry are identica, which greaty simpi es impementation in the Rietved re nement programs. 4.. Seection rues imposed by the crysta symmetry With Y = E in (1), the seection rues are obtained by soving the system of equations for crysta symmetry operators 19 Popa and Bazar Eastic strain and stress J. App. Cryst. (1). 34, 17±195

6 Tabe 7 The seection rues for the hexagona 6/m Laue cass. i ˆ : A 1 A m 1; B m 1; m ˆ 6k 4; 6k ; 6k >< A ˆ A 1 A m ˆ A m 1; B m ˆ B m 1; m ˆ 6k A m ˆ A m 1; B m ˆ B m 1; m ˆ 6k 4; 6k A 3 A m 3; B m 3; m ˆ 6k i ˆ 4 : A m 4; B m 4; m ˆ 6k 5; 6k 1 A m 5 ˆ B m 4; B m 5 ˆ A m 4; m ˆ 6k 5 i ˆ 5 : A m 5 ˆ B m 4; B m 5 ˆ A m 4; m ˆ 6k 1 A m 6 ˆ B m 1; B m 6 ˆ A m 1; m ˆ 6k 4 i ˆ 6 : A m 6 ˆ B m 1; B m 6 ˆ A m 1; m ˆ 6k Tabe The seection rues for a twofod axis in the direction = /, =. i ˆ 1; ; 3; 4 : i ˆ 5; 6 : >< A A m ; m ˆ k B m ; m ˆ k 1 B m ; m ˆ k A m ; m ˆ k 1 Tabe 9 The seection rues for the orthorhombic /mmm Laue cass. i ˆ 1; ; 3 : A A m ; m ˆ k i ˆ 4 : B m 4; m ˆ k 1 i ˆ 5 : A m 5; m ˆ k 1 i ˆ 6 : B m 6; m ˆ k X in an arbitrary sampe direction y. A crysta symmetry operator X acts on both the functions t (h, y) and the coef cients A i A j in (17). As a resut, seection rues for the harmonic representation of the WSODF are more compex than seection rues for the harmonic representation of the CODF The inversion center. The inversion center is imposed by Friede's aw. It transforms (, ) into (, )and A i into A i. From the condition (1), one obtains the foowing equations: t (,, y)=t (,, y). By using (1) and (15b), these equations become: ( 1) t (,, y) = t (,, y). Therefore, we must take = as an even number The r-fod axis in the direction U =,r =,3,4,6. An r-fod axis in the direction = transforms into +/r Tabe 1 The seection rues for the tetragona 4/mmm Laue cass. i ˆ : A 1 A m 1; m ˆ k A ˆ A 1 A m ˆ 1 k A m 1; m ˆ k A 3 A m 3; m ˆ 4k i ˆ 4 : B m 4; m ˆ k 1 i ˆ 5 : A m 5 ˆ 1 k 1 B m 4; m ˆ k 1 i ˆ 6 : B m 6; m ˆ 4k Tabe 11 The seection rues for the trigona 3m Laue cass. There are two distinct situations: for m even, at the eft side of the vertica bar, and for m odd, at the right side of the bar. i ˆ : A 1j A m 1jB m 1; m ˆ 3k ; 3k 1; 3k >< A ˆ A 1j A m ˆ A m 1jB m ˆ B m 1; m ˆ 3k A m ˆ A m 1jB m ˆ B m 1; m ˆ 3k ; 3k 1 A 3j A m 3jB m 3; m ˆ 3k i ˆ 4 : A m 4jB m 4; m ˆ 3k ; 3k 1 B m 5 ˆ A m 4jA m 5 ˆ B m 4; m ˆ 3k i ˆ 5 : B m 5 ˆ A m 4jA m 5 ˆ B m 4; m ˆ 3k 1 B m 6 ˆ A m 1jA m 6 ˆ B m 1; m ˆ 3k i ˆ 6 : B m 6 ˆ A m 1jA m 6 ˆ B m 1; m ˆ 3k 1 Tabe 1 The seection rues for the hexagona 6/mmm Laue cass. i ˆ : A 1 A m 1; m ˆ 6k 4; 6k ; 6k >< A ˆ A 1 A m ˆ A m 1; m ˆ 6k A m ˆ A m 1; m ˆ 6k 4; 6k A 3 A m 3; m ˆ 6k i ˆ 4 : B m 4; m ˆ 6k 5; 6k 1 A m 5 ˆ B m 4; m ˆ 6k 5 i ˆ 5 : A m 5 ˆ B m 4; m ˆ 6k 1 B m 6 ˆ A m 1; m ˆ 6k 4 i ˆ 6 : B m 6 ˆ A m 1; m ˆ 6k and (A 1, A ) into [A 1 cos(/r) A sin(/r), A 1 sin(/r) + A cos(/r)]. After appying (1), one obtains a system of six J. App. Cryst. (1). 34, 17±195 Popa and Bazar Eastic strain and stress 191

7 Tabe 13 The seection rues for the cubic m3 Laue cass. The foowing constraints must be added to the seection rues for the orthorhombic /mmm Laue cass (Tabe 9). ˆ : A 3 ˆ A ˆ A 1 ˆ : A 1 ˆ =3 1= A 1 A 3 A ˆ =3 1= A A 3 A 3 ˆ =3 1= A 1 A B 1 4 A 1 5 B 6 ˆ 3= 1= A 1 A A 3 = B 1 4 A 1 5 = ˆ 4 : A 4 14 ˆ 3 =35 1= A 14 =35 1= A 34 A 14=7 1= A 4 4 ˆ 3 =35 1= A 4 =35 1= A 34 A 4=7 1= B 3 44 ˆ A 14 3A 4 =35 1= 7=4 A 34=35 1= 1= 6A 4 5A 34 =14 1= 3B A 1 54 =7 1= 1=4 A 4 34= 1= A 3 54 ˆ 3A 14 A 4 =35 1= 7=4 A 34=35 1= 1= 6A 14 5A 34 =14 1= 4B A 1 54 =7 1= 1=4 A 4 34= 1= B 64 ˆ =5 1= A 14 A 4 A 34 A 14 A 4 = 1= B 1 44 A 1 54 B 4 64 ˆ =35 1= A 14 A 4 A 14 A 4 3A 34= =7 1= =7 1= B 1 44 A The cubic groups. To obtain seection rues for the cubic groups m3 and m3m, we need to add a threefod axis on the body diagona of the orthorhombic /mmm and tetragona 4/mmm unit ces, respectivey. This axis transforms (A 1, A, A 3 ) into (A, A 3, A 1 ). However, the anges (, ) are no onger ineary transformed, which makes the use of the same procedure as in xx4.. and 4..3 practicay impossibe. We proceed here in a different way, by evauating the expression (1) in terms of direction cosines A 1, A, A 3 for m3 from orthorhombic /mmm, and for m3m from tetragona 4/mmm symmetry. The resuting expression for t (h, y) contains poynomias of degree in A 1, A, A 3. The invariant I is a poynomia of degree + in these variabes, the coef cients being inear combinations of A m and B m. Furthermore, by using the invariance condition for I for the threefod axis, one nds a homogenous system of equations for A m and B m. The matrix rank of this system is aways smaer than the number of coef cients A m and B m, and we nd inear Tabe 14 The seection rues for the cubic m3m Laue cass. The foowing constraints must be added to the seection rues for the tetragona 4/mmm Laue cass (Tabe 1). Tabe 15 Functions R k (' 1,, ' ). x = cos in the isted functions. ˆ : A 3 ˆ A 1 ˆ : A 1 ˆ =3 1= A 1 A 3 B 6 ˆ 3= 1= A 1 A 3= B 1 4 ˆ 4 : A 4 14 ˆ 3 =35 1= A 14 =35 1= A 34 A 14=7 1= B 3 44 ˆ A 14=35 1= 7=4 A 34=35 1= 3A 14=14 1= 7 1= B =4 A 4 34= 1= B 64 ˆ =5 1= A 14 A 34 A 14 1= B 1 44 P inear equations of the foowing form: t (, +/r, y) = 6 kˆ1f ik t k (,, y), where f ik is the matrix with its eements determined by r. Furthermore, with (1), one obtains a system of homogenous equations for the functions A m and B m. This system has a non-trivia soution ony for certain vaues of m. This non-trivia soution is vaid for any y; therefore, these are the seection rues for the corresponding coef cients, and,. For the Laue casses /m, 4/m, 3 and 6/m, the seection rues are given in Tabes 4 to 7. For brevity, we write seection rues for the functions A m, B m rather than for the corresponding coef cients,,, Twofod axis in the direction U = p/, b =. This axis transforms into, into, and (A, A 3 ) into ( A, A 3 ). Then we proceed simary as in x4.. and obtain the seection rues given in Tabe. Combining this tabe with Tabes 4 to 7, one obtains the seection rues for the Laue casses /mmm, 4/mmm, 3m and 6/mmm. These are given in Tabes 9 to 1. R ' 1 ; x;' ˆQ x R 1 ' 1 ; x;' ˆQ x R ' 1 ; x;' ˆsin ' 1 Q 1 x R 3 ' 1 ; x;' ˆ cos ' 1 Q 1 x R 4 ' 1 ; x;' ˆcos ' 1 Q x R 5 ' 1 ; x;' ˆsin ' 1 Q x R 6 ' 1 ; x;' ˆsin ' Q 1 x R 7 ' 1 ; x;' ˆ cos ' ' 1 Q 11 x cos ' ' 1 Q 11 R ' 1 ; x;' ˆ sin ' ' 1 Q 11 x sin ' ' 1 Q 11 R 9 ' 1 ; x;' ˆ sin ' ' 1 Q 1 x sin ' ' 1 Q 1 R 1 ' 1 ; x;' ˆ cos ' ' 1 Q 1 x cos ' ' 1 Q 1 R 11 ' 1 ; x;' ˆcos ' Q 1 x R 1 ' 1 ; x;' ˆ sin ' ' 1 Q 11 x sin ' ' 1 Q 11 R 13 ' 1 ; x;' ˆ cos ' ' 1 Q 11 x cos ' ' 1 Q 11 R 14 ' 1 ; x;' ˆ cos ' ' 1 Q 1 x cos ' ' 1 Q 1 R 15 ' 1 ; x;' ˆ sin ' ' 1 Q 1 x sin ' ' 1 Q 1 R 16 ' 1 ; x;' ˆcos ' Q x R 17 ' 1 ; x;' ˆ sin ' ' 1 Q 1 x sin ' ' 1 Q 1 R 1 ' 1 ; x;' ˆ cos ' ' 1 Q 1 x cos ' ' 1 Q 1 R 19 ' 1 ; x;' ˆ cos ' ' 1 Q x cos ' ' 1 Q R ' 1 ; x;' ˆ sin ' ' 1 Q x sin ' ' 1 Q R 1 ' 1 ; x;' ˆ sin ' Q x R ' 1 ; x;' ˆ cos ' ' 1 Q 1 x cos ' ' 1 Q 1 R 3 ' 1 ; x;' ˆ sin ' ' 1 Q 1 x sin ' ' 1 Q 1 R 4 ' 1 ; x;' ˆ sin ' ' 1 Q x sin ' ' 1 Q R 5 ' 1 ; x;' ˆ cos ' ' 1 Q x cos ' ' 1 Q 19 Popa and Bazar Eastic strain and stress J. App. Cryst. (1). 34, 17±195

8 Tabe 16 The non-zero eements of the matrix w. k ˆ : w ij ˆ =3 for i; j ˆ 1; 3: k ˆ 1 : w ij1 ˆ 1=15 for i; j ˆ 1; ; w ij1 ˆ 4=15 for 3; 3 ; w ij1 ˆ =15 for 1; 3 ; ; 3 ; 3; 1 ; 3; : k ˆ : w ij ˆ 1=3 3= 1= for 5; 1 ; 5; ; w ij ˆ 1=15 3= 1= for 5; 3 : k ˆ 3 : w ij3 ˆ 1=3 3= 1= for 4; 1 ; 4; ; w ij3 ˆ 1=15 3= 1= for 4; 3 : k ˆ 4 : w ij4 ˆ 1=3 3= 1= for 1; 1 ; 1; ; w ij4 ˆ 1=15 3= 1= for 1; 3 ; w ij4 ˆ 1=3 3= 1= for ; 1 ; ; ; w ij4 ˆ 1=15 3= 1= for ; 3 : k ˆ 5; 6 : w ijk ˆ 1=3 3= 1= for 6; 1 ; 6; ; w ijk ˆ 1=15 3= 1= for 6; 3 : k ˆ 7 : w ij7 ˆ 1=6 for 5; 5 : k ˆ : w ij ˆ 1=6 for 4; 5 : k ˆ 9 : w ij9 ˆ 1=1 for 1; 5 ; w ij9 ˆ 1=1 for ; 5 : k ˆ 1 : w ij;1 ˆ 1=1 for 6; 5 : k ˆ 11 : w ij;11 ˆ 1=15 3= 1= for 1; 4 ; ; 4 ; w ij;11 ˆ =15 3= 1= for 3; 4 : k ˆ 1 : w ij;1 ˆ 1=6 for 5; 4 : k ˆ 13 : w ij;13 ˆ 1=6 for 4; 4 : k ˆ 14 : w ij;14 ˆ 1=1 for 1; 4 ; w ij;14 ˆ 1=1 for ; 4 : k ˆ 15 : w ij;15 ˆ 1=1 for 6; 4 : k ˆ 16 : w ij;16 ˆ 1=3 3= 1= for 1; 1 ; ; 1 ; w ij;16 ˆ 1=3 3= 1= for 1; ; ; ; w ij;16 ˆ 1=15 3= 1= for 3; 1 ; w ij;16 ˆ 1=15 3= 1= for 3; : k ˆ 17 : w ij;17 ˆ 1= for 5; 1 ; w ij;17 ˆ 1= for 5; : k ˆ 1 : w ij;1 ˆ 1= for 4; 1 ; w ij;1 ˆ 1= for 4; : k ˆ 19 : w ij;19 ˆ 1= for 1; 1 ; ; ; w ij;19 ˆ 1= for 1; ; ; 1 : k ˆ : w ij; ˆ 1= for 6; 1 ; w ij; ˆ 1= for 6; : k ˆ 1 : w ij;1 ˆ 1=15 3= 1= for 1; 6 ; ; 6 ; w ij;1 ˆ =15 3= 1= for 3; 6 : k ˆ : w ij; ˆ 1=1 for 5; 6 : k ˆ 3 : w ij;3 ˆ 1=1 for 4; 6 : k ˆ 4 : w ij;4 ˆ 1=1 for 1; 6 ; w ij;4 ˆ 1=1 for ; 6 : k ˆ 5 : w ij;5 ˆ 1=1 for 6; 6 : Tabe 17 The invariant poynomias J k,+ for the orthorhombic Laue casses. Additiona terms that shoud be added for the monocinic Laue casses are encosed in square brackets. ˆ : A 1; A ; A 3; A 1 A ˆ : A 4 1; A 4 ; A 4 3; A A 3; A 1A 3; A 1A ; A 3 1A ; A 1 A 3 ; A 1 A A 3 ˆ 4 : A 6 1; A 6 ; A 6 3; A 4 1A ; A 1A 4 ; A 4 1A 3; A 1A 4 3; A 4 A 3; A A 4 3; A 1A A 3; A 5 1A ; A 1 A 5 ; A 3 1A 3 ; A 3 1A A 3; A 1 A 3 A 3; A 1 A A 4 3 Tabe 1 The invariant poynomias J k,+ for the tetragona 4/mmm Laue cass. Additiona terms that shoud be added for the tetragona 4/m Laue cass are encosed in square brackets. ˆ : A 1 A ; A 3 ˆ : A 4 1 A 4 ; A 4 3; A 1 A A 3 ; A 1A ; A 1 A A1 A ˆ 4 : A 6 1 A 6 ; A 6 3; A 4 1 A 4 A 3 ; A 1 A A 4 3 ; A 1A A 3; A 1 A A 1 A ; A 1 A A1 A A 3; A 4 1 A 4 A1 A constraints between these coef cients. Except for =, more than two coef cients de ne the constraints. Unfortunatey, the probem cannot be soved in genera, but ony for every vaue of independenty. For cubic m3m symmetry, additiona tetragona constraints are appended to those for the orthorhombic crysta system with respect to the seection rues for the m3 Laue cass. The constraints obtained in this way for =,, 4 are given in Tabes 13 and Determination of average strain and stress tensors For the cacuation of both average eastic strain and stress tensors, e i and s i, ony the coef cients,, and with = and = are needed. This is easy to see by combining (11) and (b) into (5) for strain, and (11), (7) and (a) into (6) for stress. The integras of the terms with = 1 and > are zero because the eements of the matrix P are sums of J. App. Cryst. (1). 34, 17±195 Popa and Bazar Eastic strain and stress 193

9 products of two Euer matrix eements and the generaized harmonics are orthogona. So, keeping in (11) ony the terms with = and =, and rearranging to have ony positive indices m, n, in pace of " i we have the foowing: Tabe The invariant poynomias J k,+ for the hexagona 6/mmm Laue cass. Additiona terms that shoud be added for the hexagona 6/m Laue cass are encosed in square brackets. " i ' 1 ; ;' ˆP5 kˆ g ik R k ' 1 ; ;' : The functions R k (' 1,, ' ) are inear combinations of cos(m' n' 1 )Q (cos ) or sin(m' n' 1 ) Q (cos ) terms, where Q = P for m + n even and Q = ip for m + n odd. They are tabuated in Tabe 15. The vector g i is de ned as foows (the index t stands for transposed): 1 g t i ; i ; 1 i ; 1 i ; i ; i ; 1 i ; 11 i ; 11 i ; 1 i ; 1 i ; i i 1 ;i 11 ; 11 i ;i 1 ; 1 i ; i ; 1 i ; 1 i ; i ; i ; A: i ;i 1 ; 1 i ;i ; i By introducing () in (b) and, consequenty, (b) in (5), one obtains e i ˆ P6 P 5 jˆ1 kˆ w ijk g jk : 3 Simary, for stress one obtains after combining (), (7), (a) and (6), and rearranging the terms, the foowing: s i ˆ P6 P 5 jˆ1 kˆ w ijk g jk; where g jk = P 6 ˆ1C j g k. In (3) and (4), w is w ijk ˆ 1= R R R R k ' 1 ; ;' : d' 1 d d' sin P ij ' 1 ; ;' 4 In principe, these integras coud be cacuated anayticay, but because there are 936 such integras, we obtained components of w by numerica integration, with better than 1 6 precision, by using Gauss quadrature over and Simpson quadrature over ' 1 and '. Ony 73 eements are different from zero and many are interreated by an integer factor. The exact vaues of unique eements were cacuated anayticay and a non-zero eements are given in Tabe 16. Tabe 19 The invariant poynomias J k,+ for the trigona 3m Laue cass. Additiona terms that shoud be added for the trigona 3 Laue cass are encosed in square brackets. ˆ : A 1 A ; A 3 ; ˆ : A 1 A A 4 3 ; A 1 A A 3 ; 3A 1 A A A 3 ; A 1 3A A1 A 3 3; ˆ 4 : A 1 A A 6 A 3 ; A 1 A 3 ; A 1 A A 4 3 ; A 1 A 3A 1 A A A 3 ; 3A 1 A A A 3 3; A A 4 1A 15A 1A 4 A 6 ; A 1 A A 1 3A A1 A 3 ; A 1 3A A1 A 3 3; 3A 1 A A 1 3A A1 A ˆ : A 1 A ; A 3 ; ˆ : A 1 A A 4 3 ; A 1 A A 3 3; ˆ 4 : A 1 A A 6 A 3 ; A 1 A 3 ; A 1 A A 4 3 ; A A 4 1A 15A 1A 4 A 6 ; 3A 1 A A 1 3A A1 A Tabe 1 The invariant poynomias J k, + for the cubic Laue casses. ˆ : 1 ˆ : A 4 1 A 4 A 4 3; A A 3 A 1A 3 A 1A ˆ 4 : A 6 1 A 6 A 6 3; A 1A A 3; m3m : A 4 1A A 4 A 3 A 4 3A 1 A 4 1A 3 A 4 A 1 A 4 3A m3 : A 4 1A A 4 A 3 A 4 3A 1; A 4 1A 3 A 4 A 1 A 4 3A 6. Aternative approach ± corrections for diffraction ine shifts When nding the WSODF and the average strain and stress tensors during the Rietved re nement is not of interest, one can choose a different approach that corrects ony for the ine shifts caused by stress. In this case, an aternative representation for I, with fewer re nabe parameters, is possibe. To accompish this, the orientation anges in the crysta system (, ) are repaced in (1) by the direction cosines (A 1, A, A 3 ). After introducing in (17) and rearranging, one obtains I h; y ˆPk 194 Popa and Bazar Eastic strain and stress J. App. Cryst. (1). 34, 17±195 kˆ1 M k ; J k; A 1 ; A ; A 3 ; 5 where J k,+ are homogeneous poynomias of degree +in the variabes A 1, A, A 3, invariant to the Laue cass symmetry operations. The functions M k (, ) are inear combinations of A m (, ) andb m (, ): M k ; ˆ kp P nˆ1 n k cos n n k sin n P n : 6 The coef cients n k, n k can be re ned in the Rietved program in the same way as the coef cients,, and from the aternative approach. For sampe symmetry higher than tricinic, the coef cients n k, n k foow seection rues identica to those for, from Tabes and 3. The maximum number k of functions M k in the series expansion (5) must be equa to or smaer than the tota number of functions A m, B m in (17) and (1), but for crysta symmetry higher

10 than tricinic, it is frequenty much smaer. For exampe, for the Laue cass 4/m and = 4, the tota number of A m, B m is 14 but k 4 =. This fact is important in Rietved re nement, as the tota number of re nabe parameters is kept to a minimum. On the other hand, if this approach is taken, there is no path to obtain the WSODF and the average strain and stress tensors from the coef cients n k, n k. Therefore, the choice of representation for I depends on the probem that we have to sove. If one is interested ony in correcting for ine shifts in the Rietved re nement caused by residua strain in the specimen, the approach empoying (5) and (6) is taken for any vaue of. Conversey, if one is interested in the WSODFs " (' 1,, ' ) and (' 1,, ' ) and the average strain and stress tensors e i and s i, then the approach empoying (17) to () is taken. Because determination of the average strain and stress tensors requires the terms ony for =,, a third `mixed' representation for h" h (y)i is aso possibe, which aows for correction of peak shifts and determination of e i and s i, but without reconstruction of the WSODF; because the sum terms (16) are independent, I can be deveoped according to (17) to () for and according to (5) and (6) for 4. The invariant poynomias J k,+ (A 1, A, A 3 ) are given in Tabes 17 to 1 for a Laue groups for 4. For higher vaues of, one can cacuate the poynomias in the same way as for 4. Aternativey, taking into account that the genera form of J k,+ in (5) is aready known, we can adopt another simper procedure: the invariance conditions are set on the genera expression I (A 1, A, A 3, y)= P iˆ P i jˆ M ij; y A i 1A j A i j 3 and the system of inear homogenous equations is soved for M ij; y. Finay, by rearranging a the terms with a common factor, which we denote by M k (y), one obtains (5). 7. Concusions A method for the determination of the texture-weighted strain orientation distribution function (WSODF) and average strain and stress tensors is presented. It is based on the deveopment of the texture-weighted strain tensor in a series of generaized spherica harmonics. The difference between this approach and recent descriptions of the strain/stress orientation distribution function (SODF) (Wang et a., 1999, ; Behnken, ) is fourfod: (i) this approach directy determines WSODF instead of SODF, which is in accord with the fact that a diffraction experiment yieds the texture-weighted strain measure, which is aso used to cacuate the average strain and stress tensors; (ii) it assumes that ony observabe strain is invariant to the symmetry operations of a crysta point group; (iii) the approach is extended for arbitrary crystaine symmetry; (iv) it is adopted speci cay for impementation in Rietved re nement programs. Impementation in a Rietved re nement program can be made in two ways: one set of re nabe parameters aows cacuation of strain and stress vaues, whe the aternative approach corrects ony for diffraction peak shifts, with generay fewer re nabe parameters, but without the possibity of estimating strains and stresses. References Bazar, D., Von Dreee, R. B., Bennett, K. & Ledbetter, H. (199). J. App. Phys. 4, 4±433. Bazar, D. & Popa, N. C. (1). In preparation. Behnken, H. (). Phys. Status Soidi A, 177, 41±41. Bunge, H. J. (19). Texture Anaysis in Materia Science. London: Butterworth. Christenson, A. L. & Rowand, E. S. (1953). Trans. ASM, 45, 63±676. Ferrari, M. & Lutterotti, L. (1994). J. App. Phys. 76, 746±755. Hauk, V. (1997). Structura and Residua Stress Anaysis by Nondestructive Methods. Amsterdam: Esevier. Noyan, I. C. & Cohen, J. B. (197). Residua Stress. New York: Springer-Verag. Popa, N. C. (199). J. App. Cryst. 5, 611±616. Popa, N. C. (). J. App. Cryst. 33, 13±17. Reuss, A. (199). Z. Angew. Math. Mech. 9, 49±5. Rietved, H. (1969). J. App. Cryst., 65±71. Voigt, W. (19). Lehrbuch der Kristaphysik. Leipzig: Teubner. Von Dreee, R. B. (1997). J. App. Cryst. 3, 517±55. Wang, Y. D., Lin Peng, R. & McGreevy, R. (1999). Proceedings of the Twefth Internationa Conference on Textures of Materias ICOTOM-1, Canada, August 9±13, pp. 553±559. Ottawa: NRC Research Press. Wang, Y. D., Lin Peng, R., Zeng, X. H. & McGreevy, R. (). Mater. Sci. Forum, 347±349, 66±73. J. App. Cryst. (1). 34, 17±195 Popa and Bazar Eastic strain and stress 195