REFINEMENT OF SEVERAL STRUCTURES ON THE SAME DIFFRACTION PATTERN

Henrik Birkedal, Marc Hostettler, Wlodzimierz Paciorek, and Dieter Schwarzenbach

Institute of Crystallography, University of Lausanne, BSP, Dorigny, CH-1015 Lausanne, Switzerland, email: Henrik.Birkedal@ic.unil.ch

Keywords: Refinement, Disorder, OD-structures, Stacking Faults, Twinning, Non-merohedral Twinning, Diffraction

The diffraction pattern from a crystalline sample does not always correspond to one unique structure, but reveals rather a superposition of two or more different structures. Examples of this include stacking faulted systems, near twinning. For stacking faulted systems, two more or less ordered structures may occur. The domain sizes are relatively small and the diffraction pattern normally shows diffuse scattering. Near twinned crystals are systems in which the contributing domains are so large that no diffuse scattering is observed, but the individual diffraction patterns are not exactly related by symmetry operation.

The typical approach in the refinement of such structures is to index the diffraction pattern using a single cell and refine a superposition of the contributing structures using population parameters. Consider a case where there are two contributing structures, A and B. For a reflection containing contributions from both structures the absolute square of the structure factor is

|F|² = c²|F_A|² + (1-c)²|F_B|² + c(1-c)²(F_AF_B^* + F_A^*F_B)

where c is the population parameter. However, in the limit where the domain sizes are so large that one may talk about Bragg diffraction from distinct structures, the interference term, c(1-c)(F_AF_B^* + F_A^*F_B), vanishes. With this in mind, we have developed a refinement program able to refine a superposition of a number of structures, where every structure has its own lattice and space group. The only requirement is that the entire diffraction pattern can be indexed using a single cell. The absolute square of the structure factor is written as

|F( h)|² = S^N_s=1 v_s|F_s( R_s h)|² c_s(R_sh)

where N is the number of contributing structures, v_s is the volume fraction of structure s, R_s is a matrix transforming indices of the reference cell to the cell of structure s, F_s(R_sh) is the corresponding structure factor, and c_s(R_sh) is a function that determines if reflection h has a contribution from structure s. The volume fractions are constrained to S^N_s=1 v_s = 1. The program refines positional, displacement, and population parameters for all atoms and allows a rich choice of restraints and constraints.

The application of the new refinement scheme to several systems will be discussed. The advantage of the present approach is that it gives easy access to, even small, differences in the contributing structures. In traditional refinement strategies this information may not be accessible at all.