Surprises in diffuse scattering

Höffe, Moritz; Baake, Michael

doi:10.1524/zkri.2000.215.8.441

Cited by 39 publications

(47 citation statements)

References 11 publications

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“…As an example from [13], we mention the binary Rudin-Shapiro chain in one dimension, which is deterministic, versus the binary Bernoulli chain, which is stochastic. They define a homometric pair of point sets that even differ in entropy, being 0 versus log(2) in this example.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Homometric model sets and window covariograms

Baake¹,

Grimm²

2007

Zeitschrift Für Kristallographie

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Abstract. Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set Λ within a given cut and project scheme is a Delone set that is defined through a window W in internal space. The autocorrelation measure of Λ is a pure point measure whose coefficients can be calculated via the so-called covariogram of W . Two windows with the same covariogram thus result in homometric model sets. On the other hand, the inverse problem of determining Λ from its diffraction image ultimately amounts to reconstructing W from its covariogram. This is also known as Matheron's covariogram problem. It is well studied in convex geometry, where certain uniqueness results have been obtained in recent years. However, for non-convex windows, uniqueness fails in a relevant way, so that interesting applications to the homometry problem emerge. We discuss this in a simple setting and show a planar example of distinct homometric model sets.

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Section: Discussion and Outlookmentioning

confidence: 99%

Homometric model sets and window covariograms

Baake¹,

Grimm²

2007

Zeitschrift Für Kristallographie

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“…Lattices and regular model sets [55,8] are examples with γ = ( γ) pp , while the Thue-Morse and the Rudin-Shapiro sequence show singular continuous and absolutely continuous components, respectively; compare [35] and references given there. Absolutely continuous components appearing as a result of stochastic influence are the main theme below.…”

Section: Some Recollections From Fourier Analysis and Diffraction Theorymentioning

confidence: 99%

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Baake

Birkner

Moody

2009

Commun. Math. Phys.

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Abstract. Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.

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“…Baake and several co-workers [32][33][34][35][36] are currently performing a systematic study whose purpose is to charac-terize which distributions of matter diffract to produce a pure point component in their spectrum, and thus can qualify as possessing long-range order. In some cases it is even difficult to determine whether the Fourier transform of a structure exists, in the sense that it has a unique infinite volume limit.…”

Section: What Else Is Crystalline?mentioning

confidence: 99%

What is a crystal?

Lifshitz¹

2007

Zeitschrift Für Kristallographie

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Abstract. Almost 25 years have passed since Shechtman discovered quasicrystals, and 15 years since the Commission on Aperiodic Crystals of the International Union of Crystallography put forth a provisional definition of the term crystal to mean "any solid having an essentially discrete diffraction diagram." Have we learned enough about crystallinity in the last 25 years, or do we need more time to explore additional physical systems? There is much confusion and contradiction in the literature in using the term crystal. Are we ready now to propose a permanent definition for crystal to be used by all? I argue that time has come to put a sense of order in all the confusion.

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Surprises in diffuse scattering

Cited by 39 publications

References 11 publications

Homometric model sets and window covariograms

Homometric model sets and window covariograms

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

What is a crystal?

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