Inter-Bragg crystallographic phase retrieval from shape transforms, stacking faults and substitutional disorder.

One of the brilliant ideas of John Spence when he saw the first diffraction patterns from the Linac Coherent Light Source was that one could solve the crystallographic phase problem by utilising the intensities between Bragg peaks. Because these intensities are due to the Fourier transform of the shape of the crystal, the approach came to be known as "shape-transform phasing." Shape-transform phasing was developed over the next ten years and formed the basis for many other interesting ideas and pursuits. Here we describe the current best implementation of the original idea using a lattice occupancy formalism and show that certain types of crystal defects can also be modelled via this approach, allowing the molecular structure to be recovered from the additional information offered by the inter-Bragg intensities from these crystal defects.

A Metropolis Monte Carlo algorithm for merging single-particle diffraction intensities.

Single-particle imaging with X-ray free-electron lasers depends crucially on algorithms that merge large numbers of weak diffraction patterns despite missing measurements of parameters such as particle orientations. The expand-maximize-compress (EMC) algorithm is highly effective at merging single-particle diffraction patterns with missing orientation values, but most implementations exhaustively sample the space of missing parameters and may become computationally prohibitive as the number of degrees of freedom extends beyond orientation angles. This paper describes how the EMC algorithm can be modified to employ Metropolis Monte Carlo sampling rather than grid sampling, which may be favorable for reconstruction problems with more than three missing parameters. Using simulated data, this variant is compared with the standard EMC algorithm.

A new solution to the curved Ewald sphere problem for 3D image reconstruction in electron microscopy.

We develop an algorithm capable of imaging a three-dimensional object given a collection of two-dimensional images of that object that are significantly influenced by the curvature of the Ewald sphere. These two-dimensional images cannot be approximated as projections of the object. Such an algorithm is useful in cryo-electron microscopy where larger samples, higher resolution, or lower energy electron beams are desired, all of which contribute to the significance of Ewald curvature.

Shape transform phasing of edgy nanocrystals.

Diffraction patterns from small protein crystals illuminated by highly coherent X-rays often contain measurable interference signals between Bragg peaks. This coherent `shape transform' signal introduces enough additional information to allow the molecular densities to be determined from the diffracted intensities directly, without prior information or resolution restrictions. However, the various correlations amongst molecular occupancies/vacancies at the crystal surface result in a subtle yet critical problem in shape transform phasing whereby the sublattices of symmetry-related molecules exhibit a form of partial coherence amongst lattice sites when an average is taken over many crystal patterns. Here an iterative phase retrieval algorithm is developed which is capable of treating this problem; it is demonstrated on simulated data.