research papers 154 doi:10.1107/S1600576713029208 J. Appl. Cryst. (2014). 47, 154–159 Journal of Applied Crystallography ISSN 1600-5767 Received 14 July 2013 Accepted 23 October 2013 # 2014 International Union of Crystallography Elastic macro strain and stress determination by powder diffraction: spherical harmonics analysis starting from the Voigt model Nicolae C. Popa, a * Davor Balzar b and Sven C. Vogel c 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 e-mail: 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 Rietveld refinement. An earlier article [Popa & Balzar (2001). 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. 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 theore- tical 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 ðgÞ, s i ðgÞði ¼ 1; 6Þ in the sample orthogonal reference system ðy 1 ; y 2 ; y 3 Þ, and " i ðgÞ, ' i ðgÞði ¼ 1; 6Þ in the crystal reference system ðx 1 ; x 2 ; x 3 Þ, where g ¼ð’ 1 ;  0 ;’ 2 Þ is the triplet of Euler angles transforming ðy i Þ into ðx i Þ. The SODFs are connected by the single-crystal Hooke equations, which in the system ðx i Þ are the following: ' i ðgÞ¼ P 6 j¼1 C ij & j " j ðgÞ; ð1aÞ " i ðgÞ¼ P 6 j¼1 S ij & j ' j ðgÞ: ð1bÞ Here C ij and S ij are the single-crystal stiffness and compliance elastic constants and ð& i Þ are the components of the vector (1; 1; 1; 2; 2; 2). In the sample system (y i ) the Hooke equations and the single-crystal elastic constants follow as s i ðgÞ¼ P 6 j¼1 C ij ðgÞ& j e j ðgÞ; ð2aÞ e i ðgÞ¼ P 6 j¼1 S ij ðgÞ& j s j ðgÞ; ð2bÞ C ij ðgÞ¼ & 1 j P 6 m¼1 P 6 k¼1 P im ðgÞC mk & k Q kj ðgÞ; ð3aÞ S ij ðgÞ¼ & 1 j P 6 m¼1 P 6 k¼1 P im ðgÞS mk & k Q kj ðgÞ: ð3bÞ 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: ðe i ; s i Þ¼ P 6 j¼1 P ij ð" j ;' j Þ; ð4aÞ ð" i ;' i Þ¼ P 6 j¼1 Q ij ðe j ; s j Þ: ð4bÞ The quantity measured in a diffraction experiment on a polycrystal is the mean value of the strain along the reciprocal lattice vectors H fulfilling the Laue equations Q ¼H; here Q is the wavevector transfer. If h and y are the unit vectors of H and Q in ðx i Þ and ðy i Þ, respectively, the measured strain can be written in two equivalent ways: " h ðyÞ   ¼ P 1 h ðyÞ P 6 i¼1 E i & i ð1=2%Þ R hjjy d!" i ðg  Þ f ðg  Þ ; ð5aÞ