Crystal Representation in the Reciprocal Space

TL;DR

The paper addresses the non-uniqueness of direct-space crystal representations and introduces a 4D reciprocal-space representation defined by $(oldsymbol{q}, ar{I}(oldsymbol{q}))$ to encode periodicity and symmetry via structure factors $F_{hkl}$.To achieve rotation invariance, it derives a power-spectrum descriptor $P_{nl}(d)= extstyle\sum_{m=-l}^{l}|A_{nlm}(d)|^2$, built from spherical-harmonic coefficients weighted by orthonormal radial bases $R_n(d)$, resulting in a rotation- and translation-invariant crystal signature.The approach enables robust crystal structure matching (via a distance $D(s_1,s_2)$) and reconstruction by optimizing a combined objective with the power spectrum and RDF, demonstrated on diamond and quartz prototypes, with practical implementation in the PyXtal package.This framework supports improved generative design and inverse problems by providing a stable, continuous descriptor that respects physical invariances and captures both angular and radial structural information.

Abstract

In crystallography, a structure is typically represented by the arrangement of atoms in the direct space. Furthermore, space group symmetry and Wyckoff site notations are applied to characterize crystal structures with only a few variables. While this representation is effective for data records and human learning, it lacks one-to-one correspondence between the crystal structure and its representation. This is problematic for many applications, such as crystal structure determination, comparison, and more recently, generative model learning. To address this issue, we propose to represent crystals in a four-dimensional (4D) reciprocal space featured by their Cartesian coordinates and scattering factors, which can naturally handle translation invariance and space group symmetry with the help of structure factors. In order to achieve rotational invariance, the 4D coordinates are then transformed into a power spectrum representation under the orthogonal spherical harmonic and radial basis. Hence, this representation captures both periodicity and symmetry of the crystal structure while also providing a continuous representation of the atomic positions and cell parameters in the direct space. Its effectiveness is demonstrated by applying it to several crystal structure matching and reconstruction tasks.

Figures (6)

• Figure 1: Atomic form factors $f(|\mathbf{q}|)$ as functions of scattering vector magnitude for C and Si atoms according to Ref. brown2006intensity.
• Figure 2: Schematic of diamond crystal's representation at (a) the real space and (b) the reciprocal space. In the reciprocal space, the size of each point represents the relative intensity $I(\mathbf{q})$ at that reciprocal lattice vector $\mathbf{q}$.
• Figure 3: First five Bessel functions (top) and Chebyshev polynomials (bottom) used as the radial basis.
• Figure 4: The computed power spectrum $P_{nl}$ for various representative crystals.
• Figure 5: The computed power spectrum $P_{nl}$ (left) and radial distribution function $G(d)$ (right) for various diamond structures listed in Table \ref{['tab:diamond-sym']}. The $P_{nl}$ is computed using the Bessel functions as the radial basis functions and spherical harmonics up to degree 10. The $G(d)$ is computed by summing the intensity over all directions at each binned distance $d$. The noise levels are indicated in the legend.