Pure point diffraction

Baake, Michael; Moody, Robert V.

doi:10.1016/j.jnoncrysol.2003.11.016

Cited by 4 publications

(3 citation statements)

References 21 publications

(39 reference statements)

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“…The justification for this method, however, became clear only recently. This is described (mathematically rigorous) in [6] (also see the review articles [8,7]). …”

Section: Cut-and-project Schemesmentioning

confidence: 99%

Deformed model sets and distorted Penrose tilings

Sing¹,

Welberry²

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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In this article, we show how the mathematical concept of a deformed model set can be used to gain insight into the diffraction pattern of quasicrystalline structures. We explain what a deformed model set is, what its characteristic features are and how it relates to certain disorder phenomena in solids. We then apply this concept to distorted Penrose tilings, i.e., Penrose tilings where we apply size-effect-like distortions. While the size effect in crystals only operates on the diffuse scattering, there is also an intensity transfer on the Bragg peaks in distorted Penrose tilings. The persistence of pure point diffraction in distorted Penrose tilings can be explained by interpreting such tilings as deformed model sets.

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“…The justification for this method, however, became clear only recently. This is described (mathematically rigorous) in [6] (also see the review articles [8,7]). …”

Section: Cut-and-project Schemesmentioning

confidence: 99%

Deformed model sets and distorted Penrose tilings

Sing¹,

Welberry²

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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show abstract

“…Replacing Sr by Ca or adding F to the compound thus results in different structures, which will be compared and discussed. In a recent study [1] using Monte Carlo (MC) simulation of a model quasicrystal it was shown that the kinds of local distortions that result from 'size-effect'-like relaxations are in fact very similar to some mathematical constructions called 'deformed model sets' [2][3][4]. Of particular interest is the fact that it has been proven that these deformed model sets are pure 'point diffractive', i.e.…”

mentioning

confidence: 99%

Deformed Penrose tilings

Welberry

Sing

2006

Acta Cryst Sect A

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Complex manganese oxides A x MnO 2 (x < 1) show a tendency to form complex tunnel structures. The major structural unit of the tunnel walls is a rutile-type chain of edge sharing MnO 6 octahedra. A variety of structures arises due to the ability of these chains to form double and triple chain blocks by sharing octahedral edges. We will present the structure determination of three such tunnel compounds with approximately the same A/Mn ratio, SrMn 3 O 6 [1], CaMn 3 O 6 and Sr 0.9 Mn 3 (O,F) 6 , where the shape of the channels and the ordering pattern of the A-cations vary. We have solved their structures with transmission electron microscopy (TEM) and then refined them from XRD and/or NPD data. The HREM images taken along the direction of the tunnel propagation clearly show that the three compounds have different shapes of tunnels. In the case of SrMn 3 O 6 the tunnel structure is of a ''figure-of-eight'' shape. The electron diffraction patterns of SrMn 3 O 6 show satellite reflections due to an incommensurate modulation caused by the ordered Sr-distribution over the available positions in the tunnels, only 67% of the tunnel sites are filled with cations. This ordering also affects the positions of the oxygen, making one Mn-O distance substantially longer, resulting in a pyramidal instead of an octahedral coordination for some of the Mn-cations. Local areas were found with a range of different modulations, corresponding to slightly different Sr-contents. Although it has nominally the same composition as SrMn 3 O 6 , the CaMn 3 O 6 structure contains smaller six-sided tunnels comparable to those of CaFe 2 O 4 . Each tunnel comprises a single string of the Ca-cations in which every third A-position is vacant. The empty sites in neighbouring Ca-strings are shifted relative to each other by one repeat period along the c-axis of the CaFe 2 O 4 subcell giving rise to a monoclinic distortion. The analysis of the Mn-O distances allows to speculate that there is Mn +3 /Mn +4 charge ordering with the Mn +4 cations located near the cation vacancies in the A-sublattice. Introducing fluorine into the anion sublattice drastically changes the connectivity scheme of the octahedral strings. The Sr 0.9 Mn 3 (O,F) 6 compound shows large todorokite-type 3x3 channels in which four Sr strings are placed. The electron diffraction patterns show satellite reflections indicating a composite structure with two different repeat periods for the Sr-and octahedral sublattices due to the ordering of Sr in the tunnels. Replacing Sr by Ca or adding F to the compound thus results in different structures, which will be compared and discussed.

show abstract

“…In a recent study [1] using Monte Carlo (MC) simulation of a model quasicrystal it was shown that the kinds of local distortions that result from 'size-effect'-like relaxations are in fact very similar to some mathematical constructions called 'deformed model sets' [2][3][4]. Of particular interest is the fact that it has been proven that these deformed model sets are pure 'point diffractive', i.e.…”

mentioning

confidence: 99%