International Union of Crystallography (IUCr)

Molecular replacement can be used for obtaining approximate phasing of an unknown structure from a known related molecule and for phase improvement as well as extension in the presence of noncrystallographic symmetry. Emphasis is placed on the latter procedure. It is shown that the real-space method of iterative electron density averaging and Fourier back transformation corresponds to iterative phase substitution in the right-hand side of expressions to give a set of improved phases. Analysis of these expressions (the 'molecular replacement equations') provides insight into the limits of possible phase extension, and the implications for the use of calculated structure factors when there are no observed amplitudes. It is shown that the percentage of observed data and inaccuracy of the observed amplitudes available for phase extension are compensated by the extent of noncrystallographic redundancy and the fraction of crystal cell volume that may be flattened because it is outside the control of noncrystallographic symmetry.

A new method to refine crystal structural parameters using convergent-beam electron diffraction (CBED), which is applicable to nanometre-size crystal structure analysis, is proposed. This method is based on the fitting between theoretical calculations and experimental intensities of energy-filtered two-dimensional CBED patterns containing higher-order Laue-zone (HOLZ) reflections. The use of HOLZ reflections is essential for the method because small displacements of atoms can be sensitively detected using HOLZ reflections with large reciprocal vectors. For this purpose, a new Ω-filter transmission microscope (JEM-2010FEF), which can take energy-filtered CBED patterns up to a high angle with a small distortion, and a new analysis program to refine structural parameters, which is based on many-beam Bloch-wave calculations and nonlinear least-squares fitting, have been developed. As a test example, a positional parameter and isotropic and anisotropic Debye–Waller factors of CdS have been refined. Two-dimensional CBED patterns calculated with the refined parameters show very good agreement with the experimental ones, and the refined values of the parameters also agree well with the result of a single-crystal X-ray diffraction experiment. Important problems of the analysis procedure are discussed item by item.