In this contribution, a slight modification of the structure-factor equation for incommensurately modulated structures is proposed in order to enable structure description with non-spherical atomic form factors. A hypothetical (2 + 1)-dimensional structure in Flatland is used as an example to demonstrate the need for the modification when switching to a more advanced description of the electron density. The study is primarily concerned with molecular systems.
We present a machine learning (ML) method to determine unit cell parameters from powder X-Ray diffraction (XRD) data using a novel invariant lattice representation. In ML, the data representation used can have a substantial impact on the prediction quality. Previous approaches have directly predicted lattice parameters (a,b,c,α,β,γ) from XRD inputs. However, these parameters depend strongly on the unit cell reduction or convention used. In this work, we construct an invariant representation of the reciprocal lattice that is independent of primitive cell convention, based on the bispectrum–a descriptor built from spherical harmonic projections of lattice points. The calculation of the lattice bispectrum is differentiable, and we demonstrate how to invert it using gradient-based optimization initialized from a nearest-neighbor lookup. We show that with a fixed ML model architecture, using the lattice bispectrum as the ML target rather than the unit cell parameters leads to more accurate lattice parameter predictions. For example, using the MP-20 dataset, the bispectrum reduces length mean absolute percentage error (MAPE) from 11.08
A comprehensive study of the possible types of non-relativistic spin splitting of electronic bands in coplanar and non-coplanar magnetic structures is presented on the basis of spin-group theory. First, we tabulate all non-equivalent spin point groups (SpPGs) which can be expressed as a direct product of a nontrivial part and a spin-only group limited to be the intrinsic (trivial) one, or augmented by the time-reversal (TR) operation. This tabulation, which includes the listing of symmetry operations for 1249 nonequivalent SpPGs, is now available as an online database SPGENPOS in the Bilbao Crystallographic Server (BCS). This extends previous enumerations, in which the possible presence of TR in the magnetic point group was not taken into account, thus overlooking the full SpPG symmetry associated with the numerous magnetic structures which have a magnetic space group of type IV. For each of the listed coplanar and non-coplanar SpPGs, the spin-splitting that is symmetry allowed is analyzed in detail using the program STENSOR also in the BCS. Except for the SpPGs that include the operation 1', i.e., the combined operation of TR and space inversion, all other coplanar and non-coplanar SpPGs allow spin splitting at some order in a power expansion of the electron wave vector components. We find that, depending on the SpPG, spin-splitting terms can appear with the lowest-order monomials ranging from l=0 to 9, with the exception of l=8. This contrasts with the collinear case, where the lowest order is not higher than l=6, and where TR forbids any spin splitting. For the newly identified spin textures with powers l=5, 7, and 9, which are possible in some noncentrosymmetric SpPGs, the functional form of the spin splitting in terms of the components of the crystal momentum is given. One example of a real material, LaMnAu5, showing l=5 spin splitting is identified.
As an extension of previous works, we analyse regularity in disordered Al/Si distributions in phyllosilicates 2:1. The method is based on an analysis of maximal independent sets in a labelled quotient graph of the net. The results, similar to those obtained through the principle of homogeneous dispersion, are in quantitative agreement with published 29 Si MAS NMR intensity data for muscovite, phlogopite and vermiculite. The method also allows an easy construction of structural models considering disorder and non-stoichiometry.
The problem of how to split the 3D space into two identical triply periodic but disjointed objects, using a contiguous surface of minimal area, was solved by Hermann Schwarz. His minimal surface, later called P after the (two) underlying primitive cubic lattices used for the frameworks of the two labyrinths, has a formidable analytical definition based on elliptic functions and Weierstrass–Enneper integrals, although a simple trigonometric approximation also exists. Here, we derive the Schwarz minimal P surface from close tiling of the 3D space with regular polyhedra. This approach clearly exposes the Im 3 m symmetry of this surface and provides a starting point for its iterative generation through the minimization of the area of its crystallographic asymmetric unit.
