Homometric Point Sets and Inverse Problems

The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron’s covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.

💡 Research Summary

The paper tackles the inverse problem of diffraction theory: how to reconstruct the atomic positions of a solid from its diffraction pattern. From a mathematical standpoint the authors model an infinite atomic configuration as a point set Λ⊂ℝⁿ and associate to it the Dirac comb δ_Λ. The autocorrelation measure γ_Λ = δ_Λ ∗ ~δ_Λ encodes all pairwise distances and orientations; its Fourier transform is exactly the diffraction intensity measured experimentally (including both Bragg peaks and diffuse components). Two point sets that share the same autocorrelation are termed homometric. The central question becomes: given γ, can one uniquely recover Λ?

The authors focus on mathematical quasicrystals generated by a cut‑and‑project scheme. In this construction a higher‑dimensional lattice L⊂ℝ^{d+m} is projected onto physical space ℝ^d, while points are selected by a “window” W⊂ℝ^m in internal space. The autocorrelation of the resulting model set Λ is directly linked to the covariogram C_W(x)=vol(W∩(W+x)) of the window. Consequently, the homometry problem for model sets is equivalent to Matheron’s covariogram problem: does the covariogram uniquely determine the window? Known results guarantee uniqueness for convex, sufficiently regular windows (e.g., Gardner’s theorem). However, the paper provides explicit planar examples of non‑convex windows W₁ and W₂ that have identical covariograms, leading to distinct model sets Λ₁ and Λ₂ with exactly the same diffraction pattern. This demonstrates that, even in the idealised case of perfect Bragg scattering, the diffraction image may be insufficient to distinguish different atomic arrangements.

The second major theme concerns the presence of diffuse scattering. The authors construct a one‑dimensional “random phase modulation” model: starting from a deterministic point set Λ₀, each point is displaced by an independent random variable drawn from a prescribed distribution. By carefully choosing the distribution, the autocorrelation (and thus the diffraction) remains unchanged while the entropy of the ensemble varies dramatically. They exhibit two systems—one with almost no disorder (low entropy) and another with strong random displacements (high entropy)—that share an identical mixed spectrum (the same Bragg peaks plus the same continuous background). This shows that homometric structures can have fundamentally different statistical properties.

The paper then generalises this construction to higher dimensions. By combining multi‑window cut‑and‑project sets with independent random perturbations, one can generate families of structures that all produce the same diffraction image but possess arbitrarily different entropies and internal disorder. Numerical simulations confirm that the mixed spectrum is invariant under these variations.

In the discussion, the authors stress the practical implications for materials science. Diffraction is often regarded as the definitive probe of structure, yet the existence of homometric sets implies an intrinsic non‑uniqueness: the same diffraction data can correspond to distinct atomic configurations, different defect distributions, or even different thermodynamic states. Therefore, reliable structure determination must be complemented by additional experimental information (e.g., real‑space imaging, spectroscopic signatures, thermodynamic measurements) and by imposing physically motivated constraints (convexity of the window, statistical independence of displacements, etc.). The work clarifies precisely which extra assumptions restore uniqueness and highlights the need for multi‑modal approaches.

In summary, the paper (1) formalises the homometry problem via autocorrelation and covariogram theory, (2) provides concrete planar counter‑examples showing non‑uniqueness for non‑convex windows, (3) demonstrates that identical diffraction patterns can arise from structures with vastly different entropies when diffuse scattering is present, and (4) discusses how these mathematical insights translate into practical guidelines for interpreting diffraction data in the presence of unavoidable uncertainties.