scholarly journals Modeling of quasicrystal lattices with a 4-fold symmetry axis

2018 ◽  
Vol 26 (2) ◽  
pp. 49-54
Author(s):  
O. V. Smolyakov
Keyword(s):  
Symmetry Axis ◽  
Flat Surface ◽  
Rotational Symmetry ◽  
Reciprocal Lattice ◽  
Lattice Points ◽  
Hypercubic Lattice ◽  
Indexing System ◽  
Icosahedral Quasicrystals ◽  
Quasiperiodic Structures ◽  
Projection Approach

The construction method of a quasilattice with a four-fold rotational symmetry axis is proposed. The described method is based on the recurrent generation of the initial group of lattice points, which are a set of vertices of a square. The aperiodic crystal reciprocal lattice modeling algorithm is analyzed. Used modeling technique is compared with conventional projection approach. The orthogonal basis of a four-dimensional hypercubic lattice is proposed. This lattice produces two-dimensional quasicrystal with a four-fold symmetry axis after it projection on a flat surface. It is shown that the indexation of diffraction pattern of similar quasiperiodic structures can be carry out using 3 integer indexes, which is analogous to the indexing system proposed by Cahn for application to icosahedral quasicrystals.

Crystal truncation rod X-ray scattering: exact dynamical calculation

2008 ◽  
Vol 41 (1) ◽  
pp. 18-26 ◽  
Author(s):  
Václav Holý ◽  
Paul F. Fewster
Keyword(s):  
Reciprocal Lattice ◽  
Dynamical Theory ◽  
Lattice Points ◽  
Space Distribution ◽  
Matrix Formalism ◽  
X Ray ◽  
Transmission Coefficients ◽  
X Ray Scattering ◽  
Crystal Truncation Rod ◽  
Scattering Geometry

A new method is presented for a calculation of the reciprocal-space distribution of X-ray diffracted intensity along a crystal truncation rod. In contrast to usual kinematical or dynamical approaches, the method is correct both in the reciprocal-lattice points and between them. In the method, the crystal is divided into a sequence of very thin slabs parallel to the surface; in contrast to the well known Darwin dynamical theory, the electron density in the slabs is constant along the surface normal. The diffracted intensity is calculated by a matrix formalism based on the Fresnel reflection and transmission coefficients. The method is applicable for any polarization of the primary beam and also in a non-coplanar scattering geometry.

scholarly journals Nonlinear Optical Properties of Spheroidal Metallic Inclusions in a Dielectric Medium

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Bernard de Dormale ◽  
Vo-Van Truong
Keyword(s):  
Optical Properties ◽  
Nonlinear Optical ◽  
Symmetry Axis ◽  
Rotational Symmetry ◽  
Strong Dependence ◽  
Wavelength Region ◽  
Nonlinear Optical Properties ◽  
Linear Absorption ◽  
Direction Parallel ◽  
Depolarization Factor

A model for linear and nonlinear optical properties of a composite material consisting of spheroidal metal inclusions embedded in a host medium has been formulated using an effective medium approach. Both aligned and randomly oriented spheroids have been considered, and the results obtained showed a considerable difference between the two situations. Numerical calculations for metallic Au inclusions in a glass matrix have shown that the linear absorption in the case of aligned spheroids with their symmetry axis parallel to the z-axis is largely dependent on the depolarization factor, exhibiting an absorption in the vicinity of 500 nm when the depolarization factor in the direction parallel to the rotational symmetry axis is small. This structure shifts progressively to higher wavelengths when this depolarization factor is increased. In the case of randomly oriented spheroids, contributions from the different particle depolarization factors are present and prominent structures in the linear absorption appear in the long wavelength region, beyond 700 nm. Nonlinear optical properties for both aligned and randomly oriented spheroids also show a strong dependence on the depolarization factor and significant enhancements of these properties can be observed, suggesting possible tailoring of composite properties for various applications.

scholarly journals Quasicrystalline 30° twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling

2018 ◽  
Vol 115 (27) ◽  
pp. 6928-6933 ◽  
Author(s):  
Wei Yao ◽  
Eryin Wang ◽  
Changhua Bao ◽  
Yiou Zhang ◽  
Kenan Zhang ◽  
...  
Keyword(s):  
Electronic Structure ◽  
Graphene Layer ◽  
Rotational Symmetry ◽  
Reciprocal Lattice ◽  
Double Resonance ◽  
Interlayer Coupling ◽  
Bilayer Graphene ◽  
Lattice Vector ◽  
Graphene Layers ◽  
Twisted Bilayer Graphene

The interlayer coupling can be used to engineer the electronic structure of van der Waals heterostructures (superlattices) to obtain properties that are not possible in a single material. So far research in heterostructures has been focused on commensurate superlattices with a long-ranged Moiré period. Incommensurate heterostructures with rotational symmetry but not translational symmetry (in analogy to quasicrystals) are not only rare in nature, but also the interlayer interaction has often been assumed to be negligible due to the lack of phase coherence. Here we report the successful growth of quasicrystalline 30° twisted bilayer graphene (30°-tBLG), which is stabilized by the Pt(111) substrate, and reveal its electronic structure. The 30°-tBLG is confirmed by low energy electron diffraction and the intervalley double-resonance Raman mode at 1383 cm−1. Moreover, the emergence of mirrored Dirac cones inside the Brillouin zone of each graphene layer and a gap opening at the zone boundary suggest that these two graphene layers are coupled via a generalized Umklapp scattering mechanism—that is, scattering of a Dirac cone in one graphene layer by the reciprocal lattice vector of the other graphene layer. Our work highlights the important role of interlayer coupling in incommensurate quasicrystalline superlattices, thereby extending band structure engineering to incommensurate superstructures.

Crystallography and Diffraction

2021 ◽  
pp. 220-276
Author(s):  
Kannan M. Krishnan
Keyword(s):  
Rotational Symmetry ◽  
Reciprocal Lattice ◽  
Real Space ◽  
Crystalline Materials ◽  
Space Groups ◽  
Ewald Sphere ◽  
Point Groups ◽  
Diffraction Patterns ◽  
Point Line ◽  
Periodic Arrangement

Crystalline materials have a periodic arrangement of atoms, exhibit long range order, and are described in terms of 14 Bravais lattices, 7 crystal systems, 32 point groups, and 230 space groups, as tabulated in the International Tables for Crystallography. We introduce the nomenclature to describe various features of crystalline materials, and the practically useful concepts of interplanar spacing and zonal equations for interpreting electron diffraction patterns. A crystal is also described as the sum of a lattice and a basis. Practical materials harbor point, line, and planar defects, and their identification and enumeration are important in characterization, for defects significantly affect materials properties. The reciprocal lattice, with a fixed and well-defined relationship to the real lattice from which it is derived, is the key to understanding diffraction. Diffraction is described by Bragg law in real space, and the equivalent Ewald sphere construction and the Laue condition in reciprocal space. Crystallography and diffraction are closely related, as diffraction provides the best methodology to reveal the structure of crystals. The observations of quasi-crystalline materials with five-fold rotational symmetry, inconsistent with lattice translations, has resulted in redefining a crystalline material as “any solid having an essentially discrete diffraction pattern”

Protein crystals can be incommensurately modulated

2008 ◽  
Vol 41 (3) ◽  
pp. 600-605 ◽  
Author(s):  
Jeffrey J. Lovelace ◽  
Cameron R. Murphy ◽  
Lee Daniels ◽  
Kartik Narayan ◽  
Clarence E. Schutt ◽  
...  
Keyword(s):  
Diffraction Pattern ◽  
Reciprocal Lattice ◽  
Lattice Points ◽  
Chemical Crosslinking ◽  
X Ray Diffraction ◽  
X Ray ◽  
Periodic State ◽  
Orientation Matrix ◽  
Aperiodic Crystals ◽  
Future Work

For a normal periodic crystal, the X-ray diffraction pattern can be described by an orientation matrix and a set of three integers that indicate the reciprocal lattice points. Those integers determine the spacing along the reciprocal lattice directions. In aperiodic crystals, the diffraction pattern is modulated and the standard periodic main reflections are surrounded by satellite reflections. The successful indexing and refinement of the main unit cell andqvector usingTWINSOLVE, developed by Svensson [(2003). Lund University, Sweden], are reported here for an incommensurately modulated, aperiodic crystal of a profilin:actin complex. The indexing showed that the modulation is along thebdirection in the crystal, which corresponds to an `actin ribbon' formed by the crystal lattice. Interestingly, the transition to the aperiodic state was shown to be reversible and the diffraction pattern returned to the periodic state during data collection. It is likely that the protein underwent a conformational change that affected the neighbouring profilin:actin molecules in such a way as to produce the observed modulation in the diffraction pattern. Future work will aim to trap the incommensurately modulated crystal state, for example using cryocooling or chemical crosslinking, thus allowing complete X-ray data to be collected.

X-Ray Investigation of Strain Relaxation in Short-Period SimGen Superlattices using Reciprocal Space Mapping

1993 ◽  
Vol 298 ◽  
Author(s):  
P. Hamberger ◽  
E. Koppensteiner ◽  
G. Bauer ◽  
H. Kibbel ◽  
H. Presting ◽  
...  
Keyword(s):  
Strain Relaxation ◽  
Reciprocal Lattice ◽  
Reciprocal Space ◽  
Lattice Points ◽  
Reciprocal Space Mapping ◽  
Double Crystal ◽  
X Ray ◽  
Short Period ◽  
Integrated Intensity ◽  
Strained Layer Superlattices

AbstractThe optoelectronic properties of SimGen strained layer superlattices (SLS's) depend strongly on the structural perfection. We used double crystal and triple axis x-ray diffractometry to characterize the structural properties of short period Si9Ge6 SLS's grown on about lμm thick step-graded SiGe alloy buffers. As grown SLS's and samples annealed subsequently at 550°C, 650°C and 780°C for 60 mmn were investigated. Precise strain data were extracted from two-dimensional reciprocal space maps around (004) and (224) reciprocal lattice points. These data were used as refined input parameters for the dynamical simulation of the integrated intensity along the qll[004] direction. Annealing causes interdiffusion as indicated by the decreasing superlattice (SL)-satellite peak intensities and by the change of the Si/Ge thickness ratio. However, the full width at half maximum of the SL satellite peaks does not change significantly with annealing up to 650°C. The in-plane SL lattice constant in both samples is increased only slighty by annealing (< 9×10−3 Å). Consequently the interface intermixing due to interdiffusion is the main cause for the shift of the luminescence energy to higher values in these annealed samples.

Elektronenbeugung an Vizinalflächen und an Kugelflächen von Einkristallen / Electron Diffraction of Vicinals and of Spherically Shaped Crystal Surfaces

1972 ◽  
Vol 27 (3) ◽  
pp. 420-425 ◽  
Author(s):  
H Melle ◽  
E Menzel
Keyword(s):  
Electron Diffraction ◽  
Fourier Transformation ◽  
Reciprocal Lattice ◽  
Lattice Points ◽  
Bulk Crystal ◽  
Crystal Surfaces ◽  
Spherical Crystals

Abstract The reciprocal lattice of a periodic terrace-ledge surface is calculated by Fourier transformation. This formalism is applied to spherical crystals. In that case the reciprocal lattice consists of bundles of finite streaks; their vertices are situated on the reciprocal lattice points of the bulk crystal. That confirms a model formerly used to discuss LEED experiments with spherical crystals. The calculation is generalized to describe terraces composed of several monolayers.

Spin and Orbital Magnetic Moment of Pd3Co Evaluated by X-Ray Magnetic Diffraction Experiment

2010 ◽  
Vol 459 ◽  
pp. 3-6
Author(s):  
Masahisa Ito ◽  
Yoshiaki Oba ◽  
Ayako Sato ◽  
Kosuke Suzuki ◽  
Tatsuki Tadenuma ◽  
...  
Keyword(s):  
Magnetic Moment ◽  
Reciprocal Lattice ◽  
Form Factors ◽  
Dipole Approximation ◽  
Magnetization Measurement ◽  
Lattice Points ◽  
Diffraction Experiment ◽  
X Ray ◽  
Orbital Magnetic Moment ◽  
Magnetic Form Factors

We have measured spin and orbital magnetic form factors of Pd3Co for various reciprocal lattice points by the X-ray magnetic diffraction. Observed spin and orbital magnetic form factors are fitted by theoretical curves of the isolated atomic model under the dipole approximation. By the fitting analysis we have evaluated separately the spin and orbital component of the magnetic moment of Pd3Co alloy as 2.19 µB/f.u. and 0.83 µB/f.u., respectively. The total magnetic moment 3.02 µB/f.u. is comparable to the value of 2.93 µB/f.u. obtained by the magnetization measurement.

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