A review on the Parameter Space Concept and its use for crystal structure determination

TL;DR

PSC reframes crystal-structure determination as a geometric search in a high-dimensional Parameter Space, bypassing the Fourier phase problem by mapping observations to isosurfaces and locating their intersections. It introduces three complementary solution strategies (grid-based direct search, isosurface intersection, and inequality-based reduction) and leverages 1D projections to render high-dimensional problems tractable, enabling pm-scale resolutions from limited diffraction data. The method accommodates both equi-point and energy-dependent scattering models, supports resonant contrast to distinguish true versus pseudo-solutions, and provides practical workflows through grid and linearization approaches, including a Python implementation on GitHub. While PSC can reveal multiple homometric and non-homometric solutions and offers high interpretability, it remains computationally intensive and currently limited in symmetry handling and validation, necessitating further theoretical and software development for broader, robust application in 3D crystallography.

Abstract

We present a comprehensive review of the emerging crystal structure determination method Parameter Space Concept (PSC), which solves and refines either partial or complete crystal structures by mapping each experimental or theoretical observation as a geometric interpretation, bypassing the conventional Fourier inversion. The PSC utilizes only a few \xray (equivalently neutron) diffraction amplitudes or intensities and turns them into piecewise analytic hyper-surfaces, called isosurfaces, embedded in a higher-dimensional orthonormal Cartesian space (Parameter Space (PS)). It reformulates the crystal determination task into a geometric interpretation, searching for a common intersection point of different isosurfaces. The art of defining various kinds of isosurfaces, based on signs, amplitude or intensity values, normalized ratios, and ranking of reflections, offers multiple choices of adapting methods in PSC based on available experimental or theoretical observations. The elegance of PSC stems from one-dimensional projections of atomic coordinates, enabling the construction of full three-dimensional crystal structures by combining multiple projections. By these means, the user may explore homometric and non-homometric solutions within both centric and acentric structures with spatial resolution remarkably down to pm, even with a limited number of diffraction reflections. Having demonstrated the potential of PSC for various synthetic structures and exemplarily verified direct application to realistic crystals, the origin and development of PSC methods are coherently discussed in this review. Notably, this review emphasizes the theoretical foundations, computational strategies, and potential extensions of PSC, outlining the roadmap for future applications of PSC in the broader context of structure determination from experimental diffraction observations.

Figures (11)

• Figure 1: Development of PSC over the last decades. The final goal is to have PSC available as a complete, user-friendly simulation package.
• Figure 2: Example for typical isosurfaces in parameter space concept, generated for an arbitrary centrosymmetric structure consisting of two atoms. The atomic coordinates $(x_1, x_2)$ are varied in the limits $[0, 1/2]$, i. e. covering half of the unitcell of the crystal structure. The isosurfaces $\mathcal{G}(h, g(h))$ of magnitudes $g$ equal to (a) 1.04, (b) 0.75, and (c) 1.45 for reflections $h=1$, $h=2$, and $h=3$, respectively, are highlighted by the lines. The solid and dashed lines represent the positive and negative amplitudes. The color map represents the calculated amplitude for a combination of atomic coordinates within the assumed limits. Recreated from Ref. VNZ2024 with permission.
• Figure 3: An example of typical isosurfaces in the parameter space concept, generated for an arbitrary 1-dim. acentric structure consisting of three atoms. The atomic coordinates $(x_1, x_2, x_3)$ are varied in the limits $[0, 1]$, i. e. covering the full unitcell of the crystal structure. For simplicity, the coordinate of one atom, along the $x_3$ direction, is restricted at $x_3=0$ (top row) and $x_3=0.35$ (bottom row). The isosurfaces of magnitudes $|\mathcal{F}|$ 1.00 (solid line), 1.50 (dashed line), and 2.00 (dash-dotted line) are highlighted for each combination of $f_i$ and $h$ (assumed values are given at top). The color map represents the calculated intensity for a combination of atomic coordinates within the range of 0 and 1. The color bar indicates the values of computed intensity.
• Figure 4: Basic explanation of structure solving options available in PSC. Each option may invoke further constraints to define the problem.
• Figure 5: The linear approximation offers routines for single-segment (SS) and double-segment (DS) treatment within both EPA and non-EPA frameworks. Note that the SS method is generally applicable to all PS of $m~\geq~2$, while DS so far is exclusively implemented for PS of $m~=~2$code. The elegance of treating realistic structures with non-equal atomic scattering factors $f(E, \mathbf{Q})$ (non-EPA), elevates the previous restriction of approximating all atoms solely based on the geometric structure