Higher Dimensions of Clusters of Intermetallic Compounds

2019 ◽  
Vol 4 (1) ◽  
pp. 8-25 ◽  
Author(s):  
Gennadiy V Zhizhin
Keyword(s):  
Diffraction Pattern ◽  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters

The author has previously proven that diffraction pattern of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this paper, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters is obtained, from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three - dimensional, have a higher dimension. The Euler-Poincaré equation was used, the internal geometry of clusters was investigated.

Higher Dimensions of Clusters of Intermetallic Compounds

2021 ◽  
pp. 31-57
Keyword(s):  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Translational Symmetry ◽  
Higher Dimension ◽  
Internal Geometry ◽  
Metallic Nanoclusters ◽  
Diffraction Patterns

The author has previously proved that diffraction patterns of intermetallic compounds (quasicrystals) have translational symmetry in the space of higher dimension. In this chapter, it is proved that the metallic nanoclusters also have a higher dimension. The internal geometry of clusters was investigated. General expressions for calculating the dimension of clusters are obtained from which it follows that the dimension of metallic nanoclusters increases linearly with increasing number of cluster shells. The dimensions of many experimentally known metallic nanoclusters are determined. It is shown that these clusters, which are usually considered to be three-dimensional, have a higher dimension. The Euler-Poincaré equation was used, and the internal geometry of clusters was investigated.

Joint Normal Partitions and Hierarchical Filling of N-Dimensional Spaces

2021 ◽  
pp. 261-272
Keyword(s):  
Three Dimensional ◽  
Translational Symmetry ◽  
Higher Dimension

The structures arising in spaces of various dimensions with simultaneous normal partitioning of spaces and their hierarchical fillings are considered. The conditions for the appearance of translational symmetry in these structures are investigated. It is shown that simultaneous hierarchical filling and normal tiling in three-dimensional spaces do not lead to the formation of translational symmetry. Such consistent transformations lead to many elements of translational symmetry in spaces of higher dimension. The higher the dimension of space, the more complex the emerging structure and the more symmetry the elements.

Prediction of internal geometry and tensile behavior of 3D woven solid structures by mathematical coding

2021 ◽  
pp. 152808372110013
Author(s):  
Vivek R Jayan ◽  
Lekhani Tripathi ◽  
Promoda Kumar Behera ◽  
Michal Petru ◽  
BK Behera
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Areal Density ◽  
Woven Fabrics ◽  
Tensile Behavior ◽  
Geometrical Parameters ◽  
Fiber Volume ◽  
Acceptable Error ◽  
Internal Geometry ◽  
Unit Cells

The internal geometry of composite material is one of the most important factors that influence its performance and service life. A new approach is proposed for the prediction of internal geometry and tensile behavior of the 3 D (three dimensional) woven fabrics by creating the unit cell using mathematical coding. In many technical applications, textile materials are subjected to rates of loading or straining that may be much greater in magnitude than the regular household applications of these materials. The main aim of this study is to provide a generalized method for all the structures. By mathematical coding, unit cells of 3 D woven orthogonal, warp interlock and angle interlock structures have been created. The study then focuses on developing code to analyze the geometrical parameters of the fabric like fabric thickness, areal density, and fiber volume fraction. Then, the tensile behavior of the coded 3 D structures is studied in Ansys platform and the results are compared with experimental values for authentication of geometrical parameters as well as for tensile behavior. The results show that the mathematical coding approach is a more efficient modeling technique with an acceptable error percentage.

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

scholarly journals Gold’s Structural Versatility within Complex Intermetallics: From Hume-Rothery to Zintl and even Quasicrystals

2013 ◽  
Vol 1517 ◽  
Author(s):  
Gordon J. Miller ◽  
Srinivasa Thimmaiah ◽  
Volodymyr Smetana ◽  
Andriy Palasyuk ◽  
Qisheng Lin
Keyword(s):  
Complex Networks ◽  
Intermetallic Compounds ◽  
Chemical Bonding ◽  
Three Dimensional ◽  
The Other ◽  
Icosahedral Quasicrystal ◽  
Bulk Solids ◽  
Atomic Properties ◽  
Polar Intermetallics ◽  
Structural Versatility

ABSTRACTRecent exploratory syntheses of polar intermetallic compounds containing gold have established gold’s tremendous ability to stabilize new phases with diverse and fascinating structural motifs. In particular, Au-rich polar intermetallics contain Au atoms condensed into tetrahedra and diamond-like three-dimensional frameworks. In Au-poor intermetallics, on the other hand, Au atoms tend to segregate, which maximizes the number of Au-heteroatom contacts. Lastly, among polar intermetallics with intermediate Au content, complex networks of icosahedra have emerged, including discovery of the first sodium-containing, Bergman-type, icosahedral quasicrystal. Gold’s behavior in this metal-rich chemistry arises from its various atomic properties, which influence the chemical bonding features of gold with its environment in intermetallic compounds. Thus, the structural versatility of gold and the accessibility of various Au fragments within intermetallics are opening new insights toward elucidating relationships among metal-rich clusters and bulk solids.

Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

2008 ◽  
Vol 372 (11) ◽  
pp. 1851-1855 ◽  
Author(s):  
Nazar Savytskyy ◽  
Oleg Tymoshchuk ◽  
Oleh Hul ◽  
Szymon Bauch ◽  
Leszek Sirko
Keyword(s):  
Three Dimensional ◽  
Translational Symmetry ◽  
Nodal Domains

Combined method forab initiostructure solution from powder diffraction data

1999 ◽  
Vol 32 (5) ◽  
pp. 864-870 ◽  
Author(s):  
H. Putz ◽  
J. C. Schön ◽  
M. Jansen
Keyword(s):  
Global Optimization ◽  
Potential Energy ◽  
Diffraction Pattern ◽  
Intermetallic Compounds ◽  
Powder Diffraction ◽  
Diffraction Data ◽  
Powder Diffraction Data ◽  
Combined Method ◽  
The Difference ◽  
Space Method

A new direct-space method forabinitiosolution of crystal structures from powder diffraction diagrams is presented. The approach consists of a combined global optimization (`Pareto optimization') of the difference between the calculated and the measured diffraction pattern and of the potential energy of the system. This concept has been tested successfully on a large variety of ionic and intermetallic compounds.

Spatial Models of Sugars and Their Compounds

2021 ◽  
Vol 11 (2) ◽  
pp. 9-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Convex Bodies ◽  
Three Dimensional ◽  
Spatial Models ◽  
Higher Dimensions ◽  
Three Dimensional Images

The images of saccharide and polysaccharide molecules in spaces of various dimensions are considered. A method has been developed for obtaining simplified three-dimensional images of sugar molecules and their chains based on their images in spaces of higher dimensions. It was found that three-dimensional images of furanose and pyranose molecules fundamentally differ from each other to form convex and, accordingly, non-convex bodies. This leads to fundamental differences in the structure of polysaccharides from these molecules.

The phase problem and electron-density maps

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Crystal Structure ◽  
Fourier Series ◽  
Diffraction Pattern ◽  
Electron Density ◽  
Bragg Reflection ◽  
Three Dimensional ◽  
Density Maps ◽  
Structure Factors ◽  
Unit Cells ◽  
Phase Angles

In order to obtain an image of the material that has scattered X rays and given a diffraction pattern, which is the aim of these studies, one must perform a three-dimensional Fourier summation. The theorem of Jean Baptiste Joseph Fourier, a French mathematician and physicist, states that a continuous, periodic function can be represented by the summation of cosine and sine terms (Fourier, 1822). Such a set of terms, described as a Fourier series, can be used in diffraction analysis because the electron density in a crystal is a periodic distribution of scattering matter formed by the regular packing of approximately identical unit cells. The Fourier series that is used provides an equation that describes the electron density in the crystal under study. Each atom contains electrons; the higher its atomic number the greater the number of electrons in its nucleus, and therefore the higher its peak in an electrondensity map.We showed in Chapter 5 how a structure factor amplitude, |F (hkl)|, the measurable quantity in the X-ray diffraction pattern, can be determined if the arrangement of atoms in the crystal structure is known (Sommerfeld, 1921). Now we will show how we can calculate the electron density in a crystal structure if data on the structure factors, including their relative phase angles, are available. The Fourier series is described as a “synthesis” when it involves structure amplitudes and relative phases and builds up a picture of the electron density in the crystal. By contrast, a “Fourier analysis” leads to the components that make up this series. The term “relative” is used here because the phase of a Bragg reflection is described relative to that of an imaginary wave diffracted in the same direction at a chosen origin of the unit cell.

Rare Earth-rich Cadmium Compounds RE4TCd (T = Ni, Pd, Ir, Pt)

2008 ◽  
Vol 63 (9) ◽  
pp. 1127-1130 ◽  
Author(s):  
Falko M. Schappacher ◽  
Ute Ch. Rodewald ◽  
Rainer Pöttgen
Keyword(s):  
Rare Earth ◽  
Transition Metal ◽  
Single Crystal ◽  
Intermetallic Compounds ◽  
Three Dimensional ◽  
Type Structure ◽  
Space Group ◽  
X Ray ◽  
Structure Space ◽  
Cadmium Compounds

New intermetallic compounds RE4TCd (RE = Y, La-Nd, Sm, Gd-Tm, Lu; T = Ni, Pd, Ir, Pt) were synthesized by melting of the elements in sealed tantalum tubes in a highfrequency furnace. They crystallize with the Gd4RhIn-type structure, space group F 4̄3m, Z = 16. The four gadolinium compounds were characterized by single crystal X-ray diffractometer data: a = 1361.7(1) pm, wR2 = 0.062, 456 F2 values, 19 variables for Gd4NiCd; a = 1382.1(2) pm, wR2 = 0.077, 451 F2 values, 19 variables for Gd4PdCd; a = 1363.6(2) pm, wR2 = 0.045, 494 F2 values, 19 variables for Gd4IrCd; a = 1379.0(1) pm, wR2 = 0.045, 448 F2 values, 19 variables for Gd4PtCd. The rare earth atoms build up transition metal-centered trigonal prisms which are condensed via common corners and edges, leading to three-dimensional adamantane-related networks. The cadmium atoms form Cd4 tetrahedra which fill voids left in the prisms’ network.

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