Aperiodic Order

Baake, Michael; Grimm, Uwe

doi:10.1017/cbo9781139025256

Cited by 294 publications

(202 citation statements)

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“…This periodic approximant has one Tsai cluster per site in a bcc lattice, as described above for the Yb-Cd system. It was known (their refs [9,10,19]) that there is a critical temperature above which the central tetrahedron appears as disordered. Below that temperature, the crystal transforms to a low-temperature phase where the tetrahedra are ordered).…”

Section: Why Might Tdis Exist In the Melt?mentioning

confidence: 99%

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On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

et al. 2017

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Abstract:Where are the atoms in complex crystals such as quasicrystals or periodic crystals with one hundred or more atoms per unit cell? How did they get there? The first of these questions has been gradually answered for many materials over the quarter-century since quasicrystals were discovered; in this paper we address the second. We briefly review a history of proposed models for describing atomic positions in crystal structures. We then present a revised description and possible growth model for one particular system of alloys, those containing Tsai-type clusters, that includes an important class of quasicrystals.

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Section: Why Might Tdis Exist In the Melt?mentioning

confidence: 99%

“…The first of these two questions has given rise to a new field of mathematics, "long range aperiodic order" (see [9,10]). In this paper, we focus on self-assembly.…”

Section: Introductionmentioning

confidence: 99%

On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

et al. 2017

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“…Segments L and S follow a recurrence rule L → LS, S → L after each iteration step. Such sequence is fully aperiodic but perfectly deterministic and ordered [4,10,17]. Endpoints of the segments define nodes of the "lattice" which becomes a quasilattice of 1D quasicrystal.…”

Section: Auc For Quasicrystalsmentioning

confidence: 99%

“…A Thue-Morse sequence, defined by binary recurrence rule → , → with arbitrary and segments, can be used to construct a very special chain-structure which is aperiodic, but not chaotic, with singular-continuous component in the diffraction pattern [4,28,29]. The structure is perfectly ordered but completely aperiodic even in high dimensions.…”

Section: Auc For Thue-morse Sequencementioning

confidence: 99%

“…For example rotational symmetry axes, like 5-, 8-or 12-fold, are possible to appear in their diffraction patterns. Aperiodicity, on the other hand, does not exclude the long range order of the atomic structure, which is perfectly preserved in aperiodic crystals [2][3][4]. However, the symmetry of the diffraction pattern of quasicrystals is not the same as the symmetry of a structure itself.…”

Section: Introductionmentioning

confidence: 96%

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Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review

2016

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Abstract:In this paper, we show the fundamentals of statistical method of structure analysis. Basic concept of a method is the average unit cell, which is a probability distribution of atomic positions with respect to some reference lattices. The distribution carries complete structural information required for structure determination via diffraction experiment regardless of the inner symmetry of diffracting medium. The shape of envelope function that connects all diffraction maxima can be derived as the Fourier transform of a distribution function. Moreover, distributions are sensitive to any disorder introduced to ideal structure-phonons and phasons. The latter are particularly important in case of quasicrystals. The statistical method deals very well with phason flips and may be used to redefine phasonic Debye-Waller correction factor. The statistical approach can be also successfully applied to the peak's profile interpretation. It will be shown that the average unit cell can be equally well applied to a description of Bragg peaks as well as other components of diffraction pattern, namely continuous and singular continuous components. Calculations performed within statistical method are equivalent to the ones from multidimensional analysis. The atomic surface, also called occupation domain, which is the basic concept behind multidimensional models, acquires physical interpretation if compared to average unit cell. The statistical method applied to diffraction analysis is now a complete theory, which deals equally well with periodic and non-periodic crystals, including quasicrystals. The method easily meets also any structural disorder.

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Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

Maciá

2017

Annalen der Physik

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A spectral classi…cation of general one-dimensional binary aperiodic crystals (BACs) based on both their di¤raction patterns and energy spectrum measures is introduced along with a systematic comparison of the zeroth-order energy spectrum main features for BACs belonging to di¤erent spectral classes, including Fibonacci-class, precious means, metallic means, mixed means and period doubling based representatives. These systems are described by means of mixed-type Hamiltonians which include both diagonal and o¤-diagonal terms aperiodically distributed. An algebraic approach highlighting chemical correlation e¤ects present in the underlying lattice is introduced.Close analytical expressions are obtained by exploiting some algebraic properties of suitable blocking schemes preserving the atomic order of the original lattice. The existence of a resonance energy which de…nes the basic anatomy of the zeroth-order energy spectra structure for the standard Fibonacci, the precious means and the Fibonacci-class quasicrystals is disclosed. This eigenstate is also found in the energy spectra of BACs belonging to other spectral classes, but for speci…c particular choices of the corresponding model parameters only. The transmission coe¢ cient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the relative strength of the chemical bonds.

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Aperiodic Order

Cited by 294 publications

References 0 publications

On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

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