Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Baake, Michael; Birkner, Matthias; Moody, Robert V.

doi:10.1007/s00220-009-0942-x

Cited by 30 publications

(97 citation statements)

References 53 publications

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“…Note that, in order to apply the strong law of large numbers, we first have to split the last sum in the first line into a finite number of sums over points with non-overlapping surroundings of radius |z| + R. The law of large numbers is then applied to each sum separately, each of which a.s. converges to the same limit, compare [2,13,22] for related results.…”

Section: Influence Of Disordermentioning

confidence: 82%

Absence of Singular Continuous Diffraction for Discrete Multi-Component Particle Models

Baake

Zint

2007

J Stat Phys

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Particle models with finitely many types of particles are considered, both on Z d and on discrete point sets of finite local complexity. Such sets include many standard examples of aperiodic order such as model sets or certain substitution systems. The particle gas is defined by an interaction potential and a corresponding Gibbs measure. Under some reasonable conditions on the underlying point set and the potential, we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density. In particular, no singular continuous part is present.

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Section: Influence Of Disordermentioning

confidence: 82%

Absence of Singular Continuous Diffraction for Discrete Multi-Component Particle Models

Baake

Zint

2007

J Stat Phys

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“…This provides an alternative way to define the autocorrelation of the system. It is possible to include cases such as crystallographic systems or model sets into this scheme, 88,89 and it was recently also shown 6 how to use this approach in a systematic way for systems with various kinds of disorder. Since the theory of point processes is a highly developed branch 36,37 of modern probability theory, the use of these methods looks rather promising.…”

Section: Discussionmentioning

confidence: 99%

“…The vertex point set can be described as a three-component model set 81,113 based on a cut and project scheme (9) with physical and internal space R 3 . The corresponding lattice L is the embedding of M F in R 6 , which is similar to the root lattice D 6 . The vertices of the four types of (topological) octahedra (thus disregarding their centres) separate into three different types, which stem from different cosets of the embedding lattice.…”

Section: Icosahedral Model Setsmentioning

confidence: 99%

“…Due to the linear arrangement, the ensemble is well under control by elementary methods from probability theory. In particular, one can either invoke the ergodicity of the Bernoulli (coin tossing) chain 17 or the renewal theorem 6 to show that the random Dirac comb obtained this way almost surely leads to the diffraction measure with the Radon-Nikodym density function…”

Section: Random Tilingsmentioning

confidence: 99%

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Mathematical diffraction of aperiodic structures

Baake

Grimm

2012

Chem. Soc. Rev.

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Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details. IntroductionDiffraction techniques have dominated the structure analysis of solids for the last century, ever since von Laue and Bragg employed X-ray diffraction to determine the atomic structure of crystalline materials. Despite the availability of direct imaging techniques such as electron and atomic force microscopy, diffraction by X-rays, electrons and neutrons continues to be the method of choice to detect order in the atomic arrangements of a substance; see Cowley's book 35 and references therein for background.In its full generality, the diffraction of a beam of X-rays, electrons or neutrons from a macroscopic piece of solid is a complicated physical process. It is the presence of inelastic and multiple scattering, prevalent particularly in electron diffraction, which makes it essentially impossible to arrive at a complete mathematical description of the process. Here, we restrict to kinematic diffraction in the far-field or Fraunhofer limit. In this case, powerful tools of harmonic analysis are available to attack the direct problem of calculating the (kinematic) diffraction pattern of a given structure.In contrast, the inverse problem of determining a structure from its diffraction intensities is extremely involved. A diffraction pattern rarely determines a structure uniquely, as there can be homometric structures sharing the same autocorrelation (and hence the same diffraction). 9,54,57,103 We are far away from a complete understanding of the homometry classes of structures, in particular if the diffraction spectrum contains continuous components. At present, a picture is emerging, based on the analysis of explicit examples, which highlight how large the homometry classes may be.Originally, much of the effort concentrated on the pure point part of diffraction, also called the Bragg diffraction, for † Part of a themed issue on Quasicrystals in honour of the 2011 Nobel Prize in Chemistry winner, Professor Dan Shechtman.

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“…But the randomness is not in the individual point sets themselves (though that is not disallowed, e.g. [14] or [1]) but rather in the manner in which we choose them from X and the way in which the measure µ of the dynamical system on X can be viewed as a probability measure. The primary building blocks of the topology on X are the cylinder sets A of point sets Λ that have a certain colour pattern in a certain finite region of space, and µ(A) is the probability that a point set Λ, randomly chosen from X, will lie in A.…”

Section: Introductionmentioning

confidence: 99%

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

Deng

Moody

2008

Journal of Geometry and Physics

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Abstract. The setting is an ergodic dynamical system (X, µ) whose points are themselves uniformly discrete point sets Λ in some space R d and whose group action is that of translation of these point sets by the vectors of R d . Steven Dworkin's argument relates the diffraction of the typical point sets comprising X to the dynamical spectrum of X. In this paper we look more deeply at this relationship, particularly in the context of point processes.We show that there is an R d -equivariant, isometric embedding, depending on the scattering strengths (weights) that are assigned to the points of Λ ∈ X, that takes theWe examine the image of this embedding and give a number of examples that show how it fails to be surjective. We show that full information on the measure µ is available from the weights and set of all the correlations (that is, the 2-point, 3-point, . . . , correlations) of the typical point set Λ ∈ X.We develop a formalism in the setting of random point measures that includes multi-colour point sets, and arbitrary real-valued weightings for the scattering from the different colour types of points, in the context of Palm measures and weighted versions of them. As an application we give a simple proof of a square-mean version of the Bombieri-Taylor conjecture, and from that we obtain an inequality that gives a quantitative relationship between the autocorrelation, the diffraction, and the ǫ-dual characters of typical element of X. The paper ends with a discussion of the Palm measure in the context of defining pattern frequencies.

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Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Cited by 30 publications

References 53 publications

Absence of Singular Continuous Diffraction for Discrete Multi-Component Particle Models

Absence of Singular Continuous Diffraction for Discrete Multi-Component Particle Models

Mathematical diffraction of aperiodic structures

Dworkin’s argument revisited: Point processes, dynamics, diffraction, and correlations

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