On periodic and non-periodic space fillings ofEmobtained by projection

Krämer, Peter; Neri, R.

doi:10.1107/s0108767384001203

Cited by 407 publications

(126 citation statements)

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“…From the point of view of the diffraction properties an important feature of this pattern is that the vertices of the rhombus-shaped tiles lie on families of parallel lines with spacings in the ratio 1:~" in a non-periodic sequence that resembles the sequence that we have considered in previous sections. It has been shown (de Bruijn, 1981;Kramer & Neri, 1984) that a threedimensional structure whose projection onto two dimensions resembled the Penrose tiles would have a diffraction pattern with five-or tenfold symmetry and a pattern of spacings and intensities similar to that shown in Fig. 1.…”

Section: The Relation Of Structure To Penrose Tilesmentioning

confidence: 99%

Diffraction patterns from tilings with fivefold symmetry

Prince¹

1987

Acta Cryst Sect A

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A procedure involving projection from sixdimensional to three-dimensional space to describe objects that give sharp diffraction with fivefold symmetry can be reduced to the easier problem of projection from two dimensions to one dimension. This result is used to derive an explicit formula for the quasilattice contribution to the diffracted intensity for an arbitrary size and shape of the selection region. The predictions of this formula are compared with the electron diffraction patterns obtained from rapidly solidified aluminium-manganese alloys, and it is concluded that the edges of the rhombic faces of the three-dimensional objects from which models for these alloy structures may be constructed is larger than that used in previous analyses by a factor of z 3, where r is the golden mean. It is shown that the quasilattice density is proportional to the volume of the selection region in the complementary threedimensional space into which a lattice point in sixdimensional space must project in order for the point to be included in the direct space; this results in important constraints on the possible structures of these alloys.

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Section: The Relation Of Structure To Penrose Tilesmentioning

confidence: 99%

Diffraction patterns from tilings with fivefold symmetry

Prince¹

1987

Acta Cryst Sect A

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“…But these three lattices are just Z Z 6 , D 6 , and D R 6 , so we are back to root lattices and their reciprocals (cf. [6] and [7] for tiling models based on Z Z 6 and D 6 , respectively). Let us now focus on 2-D quasilattices with rotational symmetries of order 5, 8, 10, and 12, which occur in nature in form of sections through so-called T-phases [8], perpendicular to the symmetry axis.…”

Section: For Details) One Obtains the Listmentioning

confidence: 99%

Root lattices and quasicrystals

Baake

Joseph

Krämer

et al. 1990

J. Phys. A: Math. Gen.

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“…Similarly to the modulated structures, quasicrystals can be described either in higherdimensional space (HDS) [1][2][3] or, alternatively, in only physical space [4]. In the multidimensional analysis, modulated structures become periodic and the atoms get some elongated inner structure called atomic surface (AS), which can be also interpreted as a distribution of all possible structural configurations.…”

Section: Introductionmentioning

confidence: 99%