W0223

Difference Fourier Synthesis in Fiber Diffraction. Shyamsunder Baskaran, Rick Millane, Whistler Center for Carbohydrate Research and Computational Science and Engineering Program, Purdue University, West Lafayette, IN 47907-1160 USA

Polymer structures in polycrystalline fibers determined by x-ray fiber diffraction analysis sometimes contain ordered guest molecules (such as solvent or cations) that are not located in the initial structure determination. These may be searched for using difference Fourier synthesis. However, because fiber x-ray data are linear combinations of the squares of certain sets of structure factors, enough data is not available to directly calculate a Fourier difference map. This problem is usually circumvented by making assumptions about the distributions of the structure amplitudes within a set, but this produces noise in the maps and can lead to incorrect assignments.

Expressions are derived for difference map coefficients that minimize the mean-square-error in the estimate of the electron density, conditioned by a prior model. The prior model is a uniform distribution of missing atoms within the unit cell. The solution consists of weighted individual structure factors that are estimated from each observed datum. These weighted coefficients are compared with the coefficients used in current difference syntheses. Application of this new difference Fourier method leads to cleaner maps than do current methods.