Solution of the crystallographic phase problem by iterated projections

TL;DR

This paper introduces an iterative projection algorithm that solves the crystallographic phase problem by enforcing atomicity in real-space and intensity constraints in Fourier-space, successfully reconstructing complex crystal structures.

Contribution

The paper presents a novel real-space projection method combined with a difference map for solving the phase problem without relying on Fourier-space atomicity assumptions.

Findings

Successfully solved structures with over 400 atoms

Effective for both x-ray and neutron data

Does not require positivity constraint for neutron data

Abstract

An algorithm for determining crystal structures from diffraction data is described which does not rely on the usual Fourier-space formulations of atomicity. The new algorithm implements atomicity constraints in real-space, as well as intensity constraints in Fourier-space, by projections which restore each constraint with the minimal modification of the scattering density. To recover the true density, the two projections are combined into a single operation, the difference map, which is iterated until the magnitude of the density modification becomes acceptably small. The resulting density, when acted upon by a single additional operation, is by construction a density which satisfies both intensity and atomicity constraints. Numerical experiments have yielded solutions for atomic resolution x-ray data sets with over 400 non-hydrogen atoms, as well as for neutron data, where positivity of the density cannot be invoked.