An Extensible Julia Toolkit for Symmetry-Aware Dual Space Phasing in Arbitrary Dimensions

TL;DR

The toolkit is specifically designed for aperiodic crystals and quasicrystals, supporting general space-group symmetries in arbitrary dimensions, and includes the symmetry-breaking anti-aliasing sampling scheme, optimized for computational efficiency when working with strongly anisotropic diffraction data.

Abstract

We present an open-source Julia-based software toolkit for solving the phase problem using dual-space iterative algorithms. The toolkit is specifically designed for aperiodic crystals and quasicrystals, supporting general space group symmetries in arbitrary dimensions. A key feature is the symmetry-breaking anti-aliasing sampling scheme, optimized for computational efficiency when working with strongly anisotropic diffraction data, common for quasicrystals. This scheme avoids sampling redundancy caused by symmetry constraints, imposed during phasing iterations. The toolkit includes a reference implementation of the charge flipping algorithm and also allows users to implement custom phasing algorithms with fine-grained control over the iterative process.

Figures (7)

• Figure 1: Effect of applying a windowing function to normalized diffraction amplitudes. The charge density profiles were obtained using the reference implementation of the algorithm on simulated data for a one-dimensional crystal with a single atom per unit cell and 32 Bragg reflections (vertical scale arbitrary). Left: Reconstruction using raw normalized amplitudes. The algorithm converges to the optimal solution under its objective function --- maximizing the minimal density --- as evidenced by multiple local minima all reaching the same level. However, this solution does not correspond to the correct one (a sinc-shaped peak). Right: Result obtained with a Bartlett window applied to the amplitudes. In this case, the algorithm converges to the correct solution.
• Figure 2: Illustration of the importance of breaking the symmetry of the charge sampling grid. Both panels show a unit cell of a two-dimensional P4m lattice. The boldly outlined triangle denotes the fundamental domain. Empty circles indicate sampling points located outside the fundamental domain; filled circles show the corresponding positions folded back inside. Left panel: with a symmetric sampling grid, 7 out of 8 points are redundant. Right panel: breaking the grid symmetry yields an eightfold increase in sampling density within the fundamental domain.
• Figure 3: Illustration of the alias-free grid selection method. The figure shows reciprocal space. The lattice of small dots represents $L^*$, and the large black circles indicate its sublattice $\Lambda^*$. The small solid black dots denote the set $M$ of wavevectors for which amplitudes are known, while the thick black ellipse corresponds to the Löwner ellipsoid of $M$. The sublattice $\Lambda^*$ is constructed such that translated copies of $M$ (shown as hollow dots), shifted by vectors of $\Lambda^*$, do not overlap --- thus ensuring alias-free sampling of the charge density.
• Figure 4: Ratio of the number of known amplitudes to the number of sampling grid points. Histograms were obtained from 100 independent runs of the algorithm described in Section \ref{['sec:antialias']}, using the datasets of takakura2007atomic and buganski2020atomic.
• Figure 5: Evolution of the minimum, maximum, and all deciles of the charge density values on the sampling grid during the execution of the reference implementation of the algorithm, using the data of takakura2007atomic. The run was initialized with $\alpha = 0.8$ and a decrement factor of 0.995.