Crystal Representation in the Reciprocal Space

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.

💡 Research Summary

The paper addresses a fundamental limitation of traditional crystal representations, namely the lack of a one‑to‑one mapping between a crystal structure and its description in direct space (fractional coordinates, CIF files, etc.). Because the same lattice can be described with different origins, cell choices, or space‑group settings, downstream tasks such as similarity search, structure reconstruction, or generative modelling suffer from ambiguity and poor generalisation. To overcome this, the authors propose a representation built in four‑dimensional reciprocal space. Each point is defined by the Cartesian components of a reciprocal‑lattice vector q = (qₓ, qᵧ, q_z) together with the normalised scattering intensity Ī(q) = |F(q)|² / |F(0)|², where F(q) is the structure factor that already incorporates atomic form factors and phase information. This 4‑D point cloud naturally encodes translation invariance: any internal translation of atoms or a uniform scaling of the cell either leaves the set of q vectors unchanged or adds/removes zero‑intensity points, which can be discarded without affecting the representation. Moreover, because atomic form factors decay rapidly with |q|, the intensity becomes negligible beyond a modest |q| cutoff, guaranteeing convergence and allowing the use of a finite set of low‑index Miller planes for practical calculations. The representation is also smooth: small perturbations of atomic positions or lattice parameters lead to proportionally small changes in both the coordinates of q and the intensities, which is essential for gradient‑based learning. Although the 4‑D description resolves translation issues, it is still orientation‑dependent; rotating the crystal yields a different set of q vectors. To achieve full rotational invariance, the authors transform the 4‑D point cloud into a rotation‑invariant power‑spectrum. They first express each point in spherical coordinates (θ, φ, d = |q|). The angular dependence is expanded using orthogonal spherical harmonics Yₗᵐ(θ, φ), while the radial dependence is captured by an orthonormal radial basis Rₙ(d), specifically a spherical Bessel basis (with Chebyshev polynomials explored as alternatives). The combined expansion yields coefficients aₗₙₘ; the power spectrum Pₗₙ = Σₘ |aₗₙₘ|² discards the phase associated with rotation, leaving a set of scalar features that are invariant under any 3‑D rotation. By selecting sufficient angular degree ℓ and radial order n, the power spectrum can approximate the original intensity distribution arbitrarily well, preserving detailed structural information while being compact and continuous. The authors validate the approach on three fronts. First, they compute similarity metrics between crystals using the power‑spectrum and demonstrate superior discrimination compared with traditional radial distribution functions (RDF) or powder X‑ray diffraction (PXRD) patterns, which often conflate distinct hkl families. Second, they show that the intensity data alone (without phase) can be used to reconstruct the original atomic coordinates and lattice parameters via Patterson methods and direct methods, confirming that the 4‑D reciprocal representation is information‑complete. Third, they embed the power‑spectrum into a variational auto‑encoder (VAE) and train it on a dataset of diverse crystals. The resulting latent space is smooth, clusters similar structures together, and maps all rotated or translated copies of a crystal to the same latent point, addressing the over‑fitting and mode‑collapse issues observed in generative models that rely on direct‑space inputs. The paper also discusses computational trade‑offs: higher ℓ and n increase expressive power but also raise the cost of spherical harmonic transforms; however, the authors argue that the benefits for downstream tasks outweigh this overhead. In conclusion, the work introduces a mathematically rigorous, translation‑ and rotation‑invariant, continuous crystal descriptor based on 4‑D reciprocal space and spherical‑harmonic power spectra. This descriptor bridges the gap between crystallographic physics and modern machine‑learning pipelines, enabling robust structure matching, accurate reconstruction, and reliable generative design of materials.