Automated structure factor refinement from convergent-beam patterns

Abstract

An algorithm is described which automatically adjusts values of the low-order structure factors, crystal thickness and absorption coefficients for the best fit to experimental convergent-beam electron diffraction (CBED) patterns recorded in the systematics orientation. A fitting index is defined, by analogy with R factors used in neutron diffraction. A comparison of several least-squares optimization routines is made, and the “Simplex” method found to be most useful. An analysis of errors, background and noise is presented. Source-code listings of most of the program have been published elsewhere. This treats three-dimensional dynamical diffraction by the Bloch-wave method from non-centrosymmetric crystals with absorption. The use of perturbation methods for the final 33-beam refinement is found to reduce the computation time from 6 h to 30 min. An example of the use of the program is given. For MgO, we find U(200) = 0.05847 ± 0.00051 and U(400) = 0.02484 ± 0.00028. This algorithm may be used to measure and study any of the parameters (such as atomic coordinates, bonding effects or Debye-Waller factors) on which crystal structure factors depend. Similar methods allow lattice parameters to be refined to an accuracy of ∽ 0.05%.