A maximum entropy method for determining column-length distributions from size-broadened X-ray diffraction profiles

Abstract

We present a novel application of the maximum entropy (MaxEnt) method for solving the double-inverse problems of removing instrument broadening from X-ray diffraction profiles and calculating the column-length distribution of the crystallites. The MaxEnt approach is shown to have compelling advantages over the conventional methods it replaces: it is stable and robust, incorporates noise anda prioriinformation into the solution, preserves positivity of the solution, and can be applied successively. We also show how uncertainties in the derived profiles and column distributions can be determined and used in subsequent calculations, including integral breadth, Fourier coefficients, column-length distributions and apparent particle sizes. Calculations are performed on simulated X-ray diffraction profiles for a range of particle sizes, with a detailed study of the sensitivity of the results to background-level estimates and the use of an incorrect instrument response function.