scholarly journals Structure determinations for random-tiling quasicrystals

2000 ◽  
Vol 215 (10) ◽  
Author(s):  
C.L. Henley ◽  
V. Elser ◽  
M. Mihalkovic
Keyword(s):  
Structural Information ◽  
Direct Method ◽  
Dimensional Space ◽  
Waller Factor ◽  
Data Set ◽  
3 Dimensional ◽  
Random Tiling ◽  
Debye Waller Factor ◽  
Higher Dimensional ◽  
Structure Determinations

How, in principle, could one solve the atomic structure of a quasicrystal, modeled as a random tiling decorated by atoms, and what techniques are available to do it? One path is to solve the phase problem first, obtaining the density in a higher dimensional space which yields the averaged scattering density in 3-dimensional space by the usual construction of an incommensurate cut. A novel direct method for this is summarized and applied to an i(AlPdMn) data set. This averaged density falls short of a true structure determination (which would reveal the typical unaveraged atomic patterns.) We discuss the problematic validity of inferring an ideal structure by simply factoring out a "perp-space" Debye-Waller factor, and we test this using simulations of rhombohedral tilings. A second, "unified" path is to relate the measured and modeled intensities directliy, by adjusting parameters in a simulation to optimize the fit. This approach is well suited for unifying structural information from diffraction and from minimizing total energies derived ultimately from ab-initio calculations. Finally, we discuss the special pitfalls of fitting random-tiling decagonal phases.

Physical properties

2018 ◽  
Author(s):  
Ted Janssen ◽  
Gervais Chapuis ◽  
Marc de Boissieu
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Waller Factor ◽  
Hydrodynamic Theory ◽  
Dimensional Representation ◽  
Debye Waller Factor ◽  
Higher Dimensional ◽  
Experimental Findings ◽  
Aperiodic Crystals

Physical properties of aperiodic crystals present some theoretical challenges due to the lack of three-dimensional periodicity. For the description of the structure there is a periodic representation in higher-dimensional space. For physical properties, however, this scheme cannot be used because the mapping between interatomic forces and the high-dimensional representation is not straightforward. In this chapter methods are described to deal with these problems. First, the hydrodynamic theory of aperiodic crystals and then the phonons and phasons theory are developed and illustrated with some examples. The properties of electrons in aperiodic crystals are also presented. Finally, the experimental findings of phonon and phason modes for modulated and quasicrystals are presented. The chapter also discusses diffuse scattering, the Debye–Waller factor, and electrical conductivity.

Dynamical Variation of Weierstrass-Mandelbrot Function in Higher Dimensional Space

2013 ◽  
Vol 470 ◽  
pp. 767-771
Author(s):  
L. Zhang ◽  
Shu Tang Liu
Keyword(s):  
Hurst Exponent ◽  
Dimensional Space ◽  
Dynamical Behavior ◽  
Fractal Dimensions ◽  
Statistical Characteristics ◽  
3 Dimensional ◽  
Higher Dimensional ◽  
Complex Phenomena ◽  
Real Complex

Many real complex phenomena are related with Weierstrass-Mandelbrot function (WMF). Most researches focus on the systems as parameters fixed, such as calculations of its different fractal dimensions or the statistical characteristics of its generalized form and so on. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these systems as parameters change is important. However, there is few results about this aim. In this paper, we propose simulated results for the effects of parameters changeably on the graph of WMF in higher dimensional space. In addition, the relationships between the Hurst exponent of WMF and its parameters dynamically in 2-and 3-dimensional spaces are also given.

Structural Studies on Phenylalanine Hydroxylase and Implications Toward Understanding and Treating Phenylketonuria

PEDIATRICS ◽  
2003 ◽  
Vol 112 (Supplement_4) ◽  
pp. 1557-1565
Author(s):  
Heidi Erlandsen ◽  
Marianne G. Patch ◽  
Alejandra Gamez ◽  
Mary Straub ◽  
Raymond C. Stevens
Keyword(s):  
Molecular Basis ◽  
Phenylalanine Hydroxylase ◽  
Structural Information ◽  
Missense Mutations ◽  
Structural Studies ◽  
Gene Encoding ◽  
X Ray ◽  
3 Dimensional ◽  
Substrate Analogs ◽  
Structure Determinations

Mutations in the gene encoding for phenylalanine hydroxylase (PAH) result in phenylketonuria (PKU) or hyperphenylalaninemia (HPA). Several 3-dimensional structures of truncated forms of PAH have been determined in our laboratory and by others, using x-ray crystallographic techniques. These structures have allowed for a detailed mapping of the >250 missense mutations known to cause PKU or HPA found throughout the 3 domains of PAH. This structural information has helped formulate rules that might aid in predicting the likely effects of unclassified or newly discovered PAH mutations. Also, with the aid of recent crystal structure determinations of co-factor and substrate analogs bound at the PAH active site, the recently discovered tetrahydrobiopterin-responsive PKU/HPA genotypes can be mapped onto the PAH structure, providing a molecular basis for this tetrahydrobiopterin response.

scholarly journals Issues in clustering algorithm consistency in fixed dimensional spaces. Some solutions for k-means

2021 ◽  
Author(s):  
Mieczysław A. Kłopotek ◽  
Robert A. Kłopotek
Keyword(s):  
Clustering Algorithm ◽  
Dimensional Space ◽  
Clustering Algorithms ◽  
Practical Importance ◽  
Axiomatic System ◽  
One Dimension ◽  
Data Set ◽  
Data Transformations ◽  
Higher Dimensional ◽  
Consistency Property

AbstractKleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed, two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is, in fact, inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters. Two types of remedy are proposed: gravitational consistency property and dataset consistency property which both hold for k-means and hence are suitable when developing the mentioned testbeds.

Atomic Structure Studies of Zirconia Solid Solutions by EXAFS

1993 ◽  
Vol 307 ◽  
Author(s):  
I-Wei Chen ◽  
Ping Li ◽  
James Penner-Hahn
Keyword(s):  
Solid Solutions ◽  
Structural Information ◽  
Waller Factor ◽  
Cation Ordering ◽  
Atomic Structures ◽  
Local Environments ◽  
Debye Waller Factor ◽  
Characteristic Features ◽  
Special Case ◽  
Shed Light

ABSTRACTWe have investigated, using EXAFS, the local atomic structures of four zirconia polymorphs and their solid solutions with Ca2+, Ga3+, Fe3+, y3+, Gd3+, Ge4+, Ti4+, Ce4+ and Nb5+. Structural information up to 9Å, and in one special case up to 10.9Å, from the absorbing atom has been obtained. The characteristic features of local environments of both the host Zr and the dopant cations, and their variations leading to cation ordering and symmetry evolutions, are elucidated in terms of a dopant size effect and an oxygen vacancy effect. The dynamic aspects of the EXAFS Debye-Waller factor are used to shed light on the connection between phase transition, dopant stabilization, and cation-anion correlation.

Solving a 3-dimensional quasicrystal structure in 6-dimensional space using the direct method

1993 ◽  
Vol 206 (1) ◽  
Author(s):  
Fu Zheng-qing ◽  
Li Fang-hua ◽  
Fan Hai-fu
Keyword(s):  
Diffraction Data ◽  
Direct Method ◽  
Dimensional Space ◽  
Dimensional Structure ◽  
Test Results ◽  
Phase Problem ◽  
3 Dimensional ◽  
Structure Factors ◽  
A Direct Method ◽  
Quasicrystal Structure

AbstractA direct method was tested in solving the structure of a 3-dimensional quasicrystal in 6-dimensional space. Theoretical 3-dimensional diffraction data were used which contain Gaussian distribution errors with a mean error of about 20% for intensities. The diffraction data were firstly converted to a set of 6-dimensional structure factors. The window fuhction used for the conversion was measured from the Patterson origin peak in pseudo space. A direct method was then applied to solve the phase problem in 6-dimensional space. Test results showed that the procedure is very efficient.

EELFS AND EXFAS ELECTRON SPECTROSCOPIES: A COMBINED STRUCTURAL INVESTIGATION

1995 ◽  
Vol 02 (02) ◽  
pp. 255-268 ◽  
Author(s):  
L. LOZZI ◽  
M. PASSACANTANDO ◽  
P. PICOZZI ◽  
S. SANTUCCI ◽  
M. DE CRESCENZI
Keyword(s):  
Fine Structure ◽  
Nearest Neighbor ◽  
Structural Information ◽  
Mean Free Path ◽  
Waller Factor ◽  
Auger Transition ◽  
X Ray ◽  
Debye Waller Factor ◽  
Electron Energy Loss ◽  
X Ray Absorption

Detailed extended oscillating features above the Cu M2,3VV Auger transition, recently named EXFAS (Extended Fine Auger Structure), and above the Cu M2,3 core edge, named EELFS (Electron Energy-Loss Fine Structure), on the polycrystalline Cu surface have been compared to assess the short-range nature of the EXFAS features. To obtain the structural information in terms of Debye-Waller factor, interatomic distance, anharmonic effects, backscattering amplitude, and phase-shift functions, the data analysis has been performed following the EXAFS (Extended X-ray Absorption Fine Structure) procedure. The intensity of the extended structures decreases strongly when the temperature increases. In both cases no difference, as a function of temperature, in the nearest-neighbor distance was observed but a sizeable increase of the Debye-Waller factor was observed. The Debye-Waller factor has been fitted, as a function of temperature, to obtain the Debye temperature. The main result shows that the EELFS spectroscopy mainly investigates the bulk properties because of the high mean free path of the analyzed electrons. On the contrary, the Debye-Waller factor obtained from the analysis on the EXFAS structures, which are due to the first 2–4 atomic layers, is greater than that obtained from the EELFS analysis because of the greater movement of the surface atoms with respect to the bulk atoms. The close analogy between the EELFS and EXFAS structural results confirms that the extended features above the Auger transition are dominated by a genuine autoionization effect rather than by a diffraction process and/or a density-of-state effects which modulate the background of the secondary emitted electrons. Our interpretation is confirmed by the complete lack of the extended Auger features in the electron yield spectrum, N(E), when a monochromatic X-ray source is used.

scholarly journals On the puzzling case of sodium saccharinate 1.875-hydrate: structure description in (3+1)-dimensional superspace

2020 ◽  
Vol 76 (1) ◽  
pp. 18-27
Author(s):  
Toms Rekis ◽  
Andreas Schönleber ◽  
Sander van Smaalen
Keyword(s):  
Present Model ◽  
Dimensional Space ◽  
Unit Cell ◽  
Fourth Dimension ◽  
X Ray Diffraction ◽  
Data Set ◽  
X Ray ◽  
Higher Dimensional ◽  
Structure Description ◽  
Additional Base

The structure of sodium saccharinate 1.875-hydrate is presented in three- and (3+1)-dimensional space. The present model is more accurate than previously published superstructures, due to an excellent data set collected up to a high resolution of 0.89 Å−1. The present study confirms the unusual complexity of the structure comprising a very large primitive unit cell with Z′ = 16. A much smaller degree of correlated disorder of parts of the unit cell is found than is present in the previously published models. As a result of pseudo-symmetry, the structure can be described in a higher-dimensional space. The X-ray diffraction data clearly indicate a (3+1)-dimensional periodic structure with stronger main reflections and weaker superstructure reflections. Furthermore, the structure is established as being commensurate. The structure description in superspace results in a four times smaller unit cell with an additional base centring of the lattice, resulting in an eightfold substructure (Z′ = 2) of the 3D superstructure. Therefore, such a superspace approach is desirable to work out this high-Z′ structure. The displacement and occupational modulation of the saccharinate anions have been studied, as well as their conformational variation along the fourth dimension.

The effect of temperature on high-resolution TEM image contrast

1995 ◽  
Vol 53 ◽  
pp. 640-641
Author(s):  
T. Geipel ◽  
W. Mader ◽  
P. Pirouz
Keyword(s):  
Form Factor ◽  
Inelastic Scattering ◽  
Accurate Method ◽  
Waller Factor ◽  
Effect Of Temperature ◽  
Specimen Thickness ◽  
Influence Of Temperature ◽  
Debye Waller Factor ◽  
Influence Of Temperature On ◽  
Atomic Form Factor

Temperature affects both elastic and inelastic scattering of electrons in a crystal. The Debye-Waller factor, B, describes the influence of temperature on the elastic scattering of electrons, whereas the imaginary part of the (complex) atomic form factor, fc = fr + ifi, describes the influence of temperature on the inelastic scattering of electrons (i.e. absorption). In HRTEM simulations, two possible ways to include absorption are: (i) an approximate method in which absorption is described by a phenomenological constant, μ, i.e. fi; - μfr, with the real part of the atomic form factor, fr, obtained from Hartree-Fock calculations, (ii) a more accurate method in which the absorptive components, fi of the atomic form factor are explicitly calculated. In this contribution, the inclusion of both the Debye-Waller factor and absorption on HRTEM images of a (Oll)-oriented GaAs crystal are presented (using the EMS software.Fig. 1 shows the the amplitudes and phases of the dominant 111 beams as a function of the specimen thickness, t, for the cases when μ = 0 (i.e. no absorption, solid line) and μ = 0.1 (with absorption, dashed line).

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