Abstract
The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation behaviour known under the term hyperuniformity. Here, we consider one-dimensional systems with pure point, singular continuous and absolutely continuous diffraction spectra, which include perfectly ordered cut and project and inflation point sets as well as systems with stochastic disorder.
Highlights
Di raction and scalingThroughout this article, we use the notation and results from [11] on di raction theory, as well as some more advanced results on the Fourier transform of measures from [12, 33]
Starting from the idea to use the degree of ‘(hyper)uniformity’ in density fluctuations in many-particle systems [39] to characterise their order, the scaling behaviour of the di raction near the origin has emerged as a measure that captures the variance of the long-distance correlations
Our approach to the scaling behaviour of the di raction measure near the origin requires a number of di erent methods
Summary
Throughout this article, we use the notation and results from [11] on di raction theory, as well as some more advanced results on the Fourier transform of measures from [12, 33]. Note that the term ‘measure’ in this context refers to general (complex) Radon measures in the mathematical sense. The Fourier transform of γ is the di raction measure γ , which is a positive measure This is the measure-theoretic formulation of the structure factor from physics and crystallography, which is better suited for rigorous results. This approach defines the di erent spectral components by means of the Lebesgue decomposition γ = γpp + γsc + γac of γ into its pure point, singular continuous and absolutely continuous parts; see [11, section 8.5.2] for details. It is natural to replace Z(k) by Z(k)/I(0), where I(0) = γ {0} is the intensity of the central di raction (or Bragg) peak Note that this just amounts to a di erent normalisation.
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