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Crystal Structures
Crystal Structures Academic Resource Center Crystallinity: Repeating or periodic array over large atomic distances. 3-D pattern in which each atom is bonded to its nearest neighbors Crystal structure: the manner in which atoms, ions, or molecules are spatially arranged. Unit cell: small repeating entity of the atomic structure. The basic building block of the crystal structure. It defines the entire crystal structure with the atom positions within. Lattice: 3D array of points coinciding with atom positions (center of spheres) Metallic Crystal Structures FCC (face centered cubic): Atoms are arranged at the corners and center of each cube face of the cell. FCC continued Close packed Plane: On each face of the cube Atoms are assumed to touch along face diagonals. 4 atoms in one unit cell. a 2R 2 BCC: Body Centered Cubic • Atoms are arranged at the corners of the cube with another atom at the cube center. BCC continued • Close Packed Plane cuts the unit cube in half diagonally • 2 atoms in one unit cell 4R a 3 Hexagonal Close Packed (HCP) • Cell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half- hexagon of 3 atoms. • There are two lattice parameters in HCP, a and c, representing the basal and height parameters Volume respectively. 6 atoms per unit cell Coordination number – the number of nearest neighbor atoms or ions surrounding an atom or ion. For FCC and HCP systems, the coordination number is 12. For BCC it’s 8. -
Sodium Chloride (Halite, Common Salt Or Table Salt, Rock Salt)
71376, 71386 Sodium chloride (Halite, Common Salt or Table Salt, Rock Salt) CAS number: 7647-14-5 Product Description: Molecular formula: NaCl Appearance: white powder (crystalline) Molecular weight: 58.44 g/mol Density of large crystals: 2.17 g/ml1 Melting Point: 804°C1 Density: 1.186 g/ml (5 M in water)2 2 Solubility: 1 M in H2O, 20°C, complete, clear, colorless 2 pH: 5.0-8.0 (1 M in H2O, 25°C) Store at room temperature Sodium chloride is geologically stable. If kept dry, it will remain a free-flowing solid for years. Traces of magnesium or calcium chloride in commercial sodium chloride adsorb moisture, making it cake. The trace moisture does not harm the material chemically in any way. 71378 BioUltra 71386 BioUltra for molecular biology, 5 M Solution The products are suitable for different applications like purification, precipitation, crystallisation and other applications which require tight control of elemental content. Trace elemental analyses have been performed for all qualities. The molecular biology quality is also tested for absence of nucleases. The Certificate of Analysis provides lot-specific results. Much of the sodium chloride is mined from salts deposited from evaporation of brine of ancient oceans, or recovered from sea water by solar evaporation. Due to the presence of trace hygroscopic minerals, food-grade salt has a small amount of silicate added to prevent caking; as a result, concentrated solutions of "table salt" are usually slightly cloudy in appearance. 71376 and 71386 do not contain any anti-caking agent. Applications: Sodium chloride is a commonly used chemical found in nature and in all body tissue, and is considered an essential nutrient. -
A Parallelepiped Based Approach
3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach Marta Wilczkowiak, Edmond Boyer, and Peter Sturm MOVI–GRAVIR–INRIA Rhˆone-Alpes, 38330 Montbonnot, France, [email protected], http://www.inrialpes.fr/movi/people/Surname Abstract. In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric con- straints frequently present in man-made environments and allow cam- era calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characterize rigidity con- straints: parallelepipeds. It has been shown previously that the intrinsic metric characteristics of a parallelepiped are dual to the intrinsic charac- teristics of a perspective camera. Here, we generalize this idea by taking into account additional redundancies between multiple images of multiple parallelepipeds. We propose a method for the estimation of camera and scene parameters that bears strongsimilarities with some self-calibration approaches. Takinginto account prior knowledgeon scene primitives or cameras, leads to simpler equations than for standard self-calibration, and is expected to improve results, as well as to allow structure and mo- tion recovery in situations that are otherwise under-constrained. These principles are illustrated by experimental calibration results and several reconstructions from uncalibrated images. 1 Introduction This paper is about using partial information on camera parameters and scene structure, to simplify and enhance structure from motion and (self-) calibration. We are especially interested in reconstructing man-made environments for which constraints on the scene structure are usually easy to provide. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Evaporationin Icparticles
The JapaneseAssociationJapanese Association for Crystal Growth (JACG){JACG) Small Metallic ParticlesProduced by Evaporation in lnert Gas at Low Pressure Size distributions, crystal morphologyand crystal structures KazuoKimoto physiosLaboratory, Department of General Educatien, Nago)ra University Various experimental results ef the studies on fine 1. Introduction particles produced by evaperation and subsequent conden$ation in inert gas at low pressure are reviewed. Small particles of metals and semi-metals can A brief historical survey is given and experimental be produced by evaporation and subsequent arrangements for the production of the particles are condensation in the free space of an inert gas at described. The structure of the stnoke, the qualitative low pressure, a very simple technique, recently particle size clistributions, small particle statistios and "gas often referred to as evaporation technique"i) the crystallographic aspccts ofthe particles are consid- ered in some detail. Emphasis is laid on the crystal (GET). When the pressure of an inert is in the inorphology and the related crystal structures ef the gas range from about one to several tens of Torr, the particles efsome 24 elements. size of the particles produced by GET is in thc range from several to several thousand nm, de- pending on the materials evaporated, the naturc of the inert gas and various other evaporation conditions. One of the most characteristic fea- tures of the particles thus produced is that the particles have, generally speaking, very well- defined crystal habits when the particle size is in the range from about ten to several hundred nm. The crystal morphology and the relevant crystal structures of these particles greatly interested sDme invcstigators in Japan, 4nd encouraged them to study these propenies by means of elec- tron microscopy and electron difliraction. -
Crystal Structure of a Material Is Way in Which Atoms, Ions, Molecules Are Spatially Arranged in 3-D Space
Crystalline Structures – The Basics •Crystal structure of a material is way in which atoms, ions, molecules are spatially arranged in 3-D space. •Crystal structure = lattice (unit cell geometry) + basis (atom, ion, or molecule positions placed on lattice points within the unit cell). •A lattice is used in context when describing crystalline structures, means a 3-D array of points in space. Every lattice point must have identical surroundings. •Unit cell: smallest repetitive volume •Each crystal structure is built by stacking which contains the complete lattice unit cells and placing objects (motifs, pattern of a crystal. A unit cell is chosen basis) on the lattice points: to represent the highest level of geometric symmetry of the crystal structure. It’s the basic structural unit or building block of crystal structure. 7 crystal systems in 3-D 14 crystal lattices in 3-D a, b, and c are the lattice constants 1 a, b, g are the interaxial angles Metallic Crystal Structures (the simplest) •Recall, that a) coulombic attraction between delocalized valence electrons and positively charged cores is isotropic (non-directional), b) typically, only one element is present, so all atomic radii are the same, c) nearest neighbor distances tend to be small, and d) electron cloud shields cores from each other. •For these reasons, metallic bonding leads to close packed, dense crystal structures that maximize space filling and coordination number (number of nearest neighbors). •Most elemental metals crystallize in the FCC (face-centered cubic), BCC (body-centered cubic, or HCP (hexagonal close packed) structures: Room temperature crystal structure Crystal structure just before it melts 2 Recall: Simple Cubic (SC) Structure • Rare due to low packing density (only a-Po has this structure) • Close-packed directions are cube edges. -
Tetrahedral Coordination with Lone Pairs
TETRAHEDRAL COORDINATION WITH LONE PAIRS In the examples we have discussed so far, the shape of the molecule is defined by the coordination geometry; thus the carbon in methane is tetrahedrally coordinated, and there is a hydrogen at each corner of the tetrahedron, so the molecular shape is also tetrahedral. It is common practice to represent bonding patterns by "generic" formulas such as AX4, AX2E2, etc., in which "X" stands for bonding pairs and "E" denotes lone pairs. (This convention is known as the "AXE method") The bonding geometry will not be tetrahedral when the valence shell of the central atom contains nonbonding electrons, however. The reason is that the nonbonding electrons are also in orbitals that occupy space and repel the other orbitals. This means that in figuring the coordination number around the central atom, we must count both the bonded atoms and the nonbonding pairs. The water molecule: AX2E2 In the water molecule, the central atom is O, and the Lewis electron dot formula predicts that there will be two pairs of nonbonding electrons. The oxygen atom will therefore be tetrahedrally coordinated, meaning that it sits at the center of the tetrahedron as shown below. Two of the coordination positions are occupied by the shared electron-pairs that constitute the O–H bonds, and the other two by the non-bonding pairs. Thus although the oxygen atom is tetrahedrally coordinated, the bonding geometry (shape) of the H2O molecule is described as bent. There is an important difference between bonding and non-bonding electron orbitals. Because a nonbonding orbital has no atomic nucleus at its far end to draw the electron cloud toward it, the charge in such an orbital will be concentrated closer to the central atom. -
Types of Lattices
Types of Lattices Types of Lattices Lattices are either: 1. Primitive (or Simple): one lattice point per unit cell. 2. Non-primitive, (or Multiple) e.g. double, triple, etc.: more than one lattice point per unit cell. Double r2 cell r1 r2 Triple r1 cell r2 r1 Primitive cell N + e 4 Ne = number of lattice points on cell edges (shared by 4 cells) •When repeated by successive translations e =edge reproduce periodic pattern. •Multiple cells are usually selected to make obvious the higher symmetry (usually rotational symmetry) that is possessed by the 1 lattice, which may not be immediately evident from primitive cell. Lattice Points- Review 2 Arrangement of Lattice Points 3 Arrangement of Lattice Points (continued) •These are known as the basis vectors, which we will come back to. •These are not translation vectors (R) since they have non- integer values. The complexity of the system depends upon the symmetry requirements (is it lost or maintained?) by applying the symmetry operations (rotation, reflection, inversion and translation). 4 The Five 2-D Bravais Lattices •From the previous definitions of the four 2-D and seven 3-D crystal systems, we know that there are four and seven primitive unit cells (with 1 lattice point/unit cell), respectively. •We can then ask: can we add additional lattice points to the primitive lattices (or nets), in such a way that we still have a lattice (net) belonging to the same crystal system (with symmetry requirements)? •First illustrate this for 2-D nets, where we know that the surroundings of each lattice point must be identical. -
Self-Diffusion of Sodium, Chloride and Iodide Ions in Methanol-Water Mixture
Self-Diffusion of Sodium, Chloride and Iodide Ions in Methanol-Water Mixture E. Hawlicka* Institute of Applied Radiation Chemistry, Technical University, Wroblewskiego 15, 93-590, Lodz, Poland Z. Naturforsch. 41 a, 939-943 (1986); received April 25, 1986 The self-diffusion coefficients of Na+, C l- and I- in methanol-water solutions at 35 ± 0.01 °C have been measured in their dependence on the salt molarity in the range 1 • 10-4— 1 • 10-2 mol dm -3. The ionic self-diffusion coefficients in infinitely diluted solutions have been computed. The influence of the solvent composition on the solvation of the ions is discussed. A preferential hydration of Na+, Cl - and I “ ions in water-methanol mixtures has been found. In spite of the great interest in the porperties of aqueous solution with a Nal(Tl) scintillation crystal water-organic solvent electrolyte solutions, data on of the well-type (2 x 2"). the ionic mobilities in such systems are scarce. For the self-diffusion measurements the open-end Usually the ionic mobility is calculated from the capillary method was used. The details of the equivalent conductance and the transference num experimental procedure have been described in [6], ber. Ionic transference numbers have been reported The labelling of the sodium ions with 22Na or for water and 17 organic solvents [1], but only for a 24Na and the iodide ions with i25I or 1311, respective few water-organic solvents mixtures. ly, did not make any difference in the results. Similar information can be obtained from the ionic self-diffusion coefficients, which have been reported for a few water-organic solvent systems Results [2-5], The aim of the present work was to determine the All self-diffusion experiments have been carried self-diffusion coefficients of sodium, chloride and out at 25.0 ± 0.05 °C. -
Abstract Spectroscopic Characterization of Fluorite
ABSTRACT SPECTROSCOPIC CHARACTERIZATION OF FLUORITE: RELATIONSHIPS BETWEEN TRACE ELEMENT ZONING, DEFECTS AND COLOR By Carrie Wright This thesis consists of two separate papers on color in fluorite. In the first paper, synthetic fluorites doped with various REEs (10-300 ppm) were analyzed using direct current plasma spectrometry, optical absorption spectroscopy, fluorescence spectrophotometry, and electron paramagnetic resonance spectroscopy before and after receiving 10-25 Mrad of 60Co gamma irradiation. The combined results of these techniques indicate that the irradiation-induced color of the Y-, Gd-, La- and Ce-doped samples are the result of a REE-associated fluorine vacancy that traps two electrons. Divalent samarium may be the cause of the irradiation-induced green color of the Sm- doped sample. In the second paper, fluorite crystals from Bingham, NM, Long Lake, NY, and Westmoreland, NH were similarly investigated to determine the relationship between sectorally zoned trace elements, defects, and color. The results indicate causes of color similar to those in the synthetic samples with the addition of simple F-centers. SPECTROSCOPIC CHARACTERIZATION OF FLUORITE: RELATIONSHIPS BETWEEN TRACE ELEMENT ZONING, DEFECTS AND COLOR A Thesis Submitted to the Faculty of Miami University In partial fulfillment of The requirements for the degree of Master of Science Department of Geology By Carrie Wright Miami University Oxford, OH 2002 Advisor_____________________ Dr. John Rakovan Reader______________________ Dr. Hailiang Dong TABLE OF CONTENTS Chapter 1: Introduction to the cause of color in fluorite 1 Manuscript 1-Chapter 2 29 “Spectroscopic investigation of lanthanide doped CaF2 crystals: implications for the cause of color” Manuscript 2-Chapter 3 95 “Spectroscopic characterization of fluorite from Bingham, NM, Long Lake, NY and Westmoreland, NH: relationships between trace element zoning, defects and color ii TABLE OF FIGURES Chapter 1 Figures 21 Figure 1a. -
Chap 1 Crystal Structure
Introduction Example of crystal structure Miller index Chap 1 Crystal structure Dept of Phys M.C. Chang Introduction Example of crystal structure Miller index Bulk structure of matter Crystalline solid Non-crystalline solid Quasi-crystal (1982 ) (amorphous, glass) Ordered, but not periodic 2011 Shechtman www.examplesof.net/2017/08/example-of-solids-crystalline-solids-amorphous-solids.html#.XRHQxuszaJA nmi3.eu/news-and-media/the-contribution-of-neutrons-to-the-study-of-quasicrystals.html Introduction Example of crystal structure Miller index • A simple lattice (in 3D) = a set of points with positions at r = n1a1+n2a2+n3a3 (n1, n2, n3 cover all integers ), a1, a 2, and a3 are called primitive vectors (原始向量), and a1, a 2, and a3 are called lattice constants ( 晶格常數). Alternative definition: • A simple lattice = a set of points in which every point has exactly the same environment. Note: A simple lattice is often called a Bravais lattice . From now on we’ll use the latter term ( not used by Kittel). Introduction Example of crystal structure Miller index Decomposition: Crystal structure = Bravais lattice + basis (of atoms) (How to repeat) (What to repeat) scienceline.ucsb.edu/getkey.php?key=4630 lattice basis 基元 (one atom, or a group of atoms) • Every crystal has a corresponding Bravais lattice and a basis. Note: Lattice is a math term, while crystal is a physics term. 晶格 晶體 Introduction Example of crystal structure Miller index Bravais lattice non-Bravais lattice = Bravais + p-point basis (p>1) Triangular (or hexagonal) lattice Honeycomb lattice Primitive vectors a2 basis a1 Honeycomb lattice = triangular lattice + 2-point basis (i.e. -
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.