A simple example of a polyhedral network or polynet, constructed from truncated octahedra and hexagonal prisms.
|
|
- Sheila Ford
- 6 years ago
- Views:
Transcription
1 Polyhedral Nets
2 A simple example of a polyhedral network or polynet, constructed from truncated octahedra and hexagonal prisms. The building process indicated produces a labyrinth. The labyrinth graph happens to be a diamond net, with truncated octahedron centres at the nodes and the prism axes along the edges. This particular example gives a representation of the zeolite framework FAU (faujasite). The edges of the structure form a 4-connected net; in faujasite Si and Al atoms are centered at the polyhedron vertices and the polyhedron edges correspond to the oxygen linkages. Note the large voids and channels in the structure, the characteristic feature of zeolites.
3 An icosahedron with octahedra on eight of its faces can be extended to a polynet with icosahedral nodes centered on a bcc lattice. The labyrinth graph is 8-connected. In Al 12 Mo aluminium atoms are centered at the vertices of this structure. Molybdenum atoms are at the centers of the icosahedra.
4 Alpha manganese Another intricate bcc structure is α-mn. The structural units are centered truncated tetrahedra as in a Laves phase. They occur both as nodes at the bcc positions and also as the links between the nodes. Recall that, in the regular space filling of tetrahedra and truncated tetrahedra (equivalently: in a Laves phase) the hexagonal faces are shared by pairs of polyhedra related by inversion. In α-mn this this does not occur. Instead, the contiguous polyhedra are related by reflection in hexagonal or in triangular faces. The pictures show two ways in which diamond type polynets can be built with Friaufs as both nodes and links. In α-mn these are combined to give an 8-connected net...
5 Basic polynet structure of α-mn. The centers of the white units are at the vertices and at the center of the cubic unit cell. The net extended further. The central atoms of all the polyhedra are 16-coordinated. The atoms at the centers of the nodal units (white) are have 12 neighbours at the polyhedron vertices and 4 in the centers of neighbouring polyhedra as in a Laves phase. The centers of link polyhedra (purple) are coordinated to the 12 vertices and to 4 vertices of nearby polyhedra. Most of the polyhedron vertices are 12-coordinated. Sadoc, J. F. & Mosseri, R. Geometrical Frustration. Cambridge Univ. Press (1999).
6 How and why manganese atoms arrange themselves in such an intricate pattern is a mystery. Another manganese phase, β-mn is just as intricate but quite different: A stereo pair of images illustrating the β-mn structure. It has been identified by Nyman, Carroll & Hyde as a packing of helical rods of tetrahedra. Nyman, H., Carroll, C. E. & Hyde, B. G. Rectilinear rods of face-sharing tetrahedra and the structure of β-mn. Z. Kristallogr. 196 (1991)
7 Polyhedral D-nets The structure of very many crystalline materials can be described in terms of polyhedral nets. Those with labyrinth graphs of diamond type are common. Left: unit cell of a pair of complementary diamond type nets (D-nets). In polynets with D-nets as labyrinth graphs the two complementary structures may or may not be equivalent. Moreover, the nodes of a D-net may be of two varieties in an alternating arrangement (as in the zinc blende ZnS). Right: A portion of a polynet of D-type. Inthis example the nodes and the links connecting them are octahedra. This structure occurs in the mineral pyrochlore.
8 The structure of the silicate β-crystobalite. A D-net of vertex-connected tetrahedra. Silicon atoms lie at tetrahedron centers, each coordinated to four oxygen atoms at the vertices. The tetrahedra in the uninodal tiling of space by tetrahedra and truncated tetrahedra form this pattern. The basic structure of pyrochlore can be described as a pair of complementary (interwoven) D-type polynets. One formed of octahedra as both nodes and links, the other formed of vertex-connected tetrahedra as in β- crystobalite. Nyman & Andersson have identified and described several materials with this basic structure. A pyrochlore unit. Four octahedra round a central octahedron. Nyman, H. & Andersson, S. the pyrochlore structure and its relatives. J. Solid State Chem. 26 (1978)
9 A unit cell of the D-net of octahedra in the pyrochlore structure. Nodal octahedra omitted for clarity (and because in pyrochlore they lack a central atom). In the generalised pyrochlore geometry identified in a variety of materials, a D-net of stella quadrangulae replaces the D-net of tetrahedra.
10 The structure of W 2 Fe 3 C Two interwoven D-nets. One of octahedra and one of vertex-connected stella quadrangulae. (The edge length of the tetrahedra is 4/5 the edge length of the octahedra.)
11 Surprisingly, the generalised pyrochlore structure just described is a D-net of γ- brass clusters! The pictures above show a gamma brass cluster composed of four interpenetrating icosahedra with minimal distortion, and one with slightly more distortion in which the hexagonal regions indicated in grey have an exact inversion symmetry. These polyhedra can pack together with the grey hexagons in contact, to form a D-type polynet. In what follows, the cluster of four interpenetrating icosahedra will be referred to simply as a γ-unit.
12 Another view of the γ-unit, emphasising its internal stella quadrangula. The tetrahedral cluster of five units that constitutes the node structure of the D-net.. The complementary D-net is the D-net of octahedra. The picture on the left shows how a pyrochlore unit and an icosahedron belonging to a γ-unit fit together.
13 Four pyrochlore units attached to a γ-unit. The generalised pyrochlore structure can be described as a space filling of octahedra and γ-units. This structure occurs in a large number of complex alloys of Ti 2 Ni type. In Ti 2 Ni the nickel atoms are at the stella quadrangulae vertices, the rest are titanium. There are 96 atoms per cubic unit cell.
14 Nyman and Anderson have analysed and described the amazingly intricate structure of Mg 3 Cr 2 Al 18. The following pictures illustrate its subtle geometry. A D-net in which the nodes are alternately pyrochlore units and truncated tetrahedra. Two interwoven (complementary) nets of pyrochlore units and truncated tetrahedra. Observe that they do not touch. Nyman, H. & Andersson, S. the pyrochlore structure and its relatives. J. Solid State Chem. 26 (1978)
15 Left: the two interwoven D-nets, distinguished by color for greater clarity. Right: the gaps between close pairs of pyrochlore units are nearly exact regular icosahedra! Left: one of the D-nets and the related set of icosahedra. Right: The configuration of icosahedra with the D- nets removed for clarity. A new way of understanding the structure emerges...
16 Recall the L-unit introduced by Kreiner & Franzen: a tetrahedron of vertex-sharing icosahedra. Kreiner, G. & Franzen, H. F. J. Alloys and Compounds 221 (1995) The structure of Mg 3 Cr 2 Al 18 is a D-net of Kreiner-Franzen L- units linked by their vertices like the tetrahedra in a Laves phase.
17 NaZn 13 The stella quadrangula or tetrahedral star is a cluster of five tetrahedra made by placing a tetrahedron on each face of a tetrahedron. Two icosahedra can be joined by a stella quadrangula. Below, left: a network of icosahedra, centered on a primitive cubic lattice, linked in this way. The complement of this polynet is an array of snub cubes:
18 In NaZn 13 the icosahedra are 13-atom Zn clusters; the Na atoms are at the centers of the snub cubes. A large number of intermetallics have this kind of geometry. Haüssermann, U., Svensson, C. and Lidin, S. Tetrahedral stars as flexible basis clusters in sp-bonded intermetallic frameworks and the compound BaLi 7 Al 6 with the NaZn 13 structure. J. Am. Chem. Soc. 120 (1998)
19 Polynets with dodecahedra The dihedral angle of a regular dodecahedron close to 120. Hence three dodecahedra sharing an edge can be slightly deformed to bring them into face contact. The vertex angle in a regular pentagon is 108 close to the tetrahedral coordination angle A tetrahedral cluster of four face-sharing almost regular dodecahedra is possible: The structure can be extended to produce a D-net a portion of which is shown on the right. Pearce, P. Structure in Nature is a Strategy for Design. MIT press (1978).
20 The complementary labyrinth of the D-net of dodecahedra is a D-net of hexakaidodecahedra. This 16-faced polyhedron has 12 pentagonal faces and 4 hexagonal faces. It is one of the typical shapes of the Voronoi cells surrounding the atoms of a Frank- Kasper phase. The dodecahedra and 16-hedra in this space filling arrangement represent the Voronoi regions surrounding the atoms in a Laves phase. The 4- connected network of polyhedron vertices and edges is the zeolite framework MTN.
21 Voronoi cells of Frank-Kasper phases From left to right: Dodecahedron. 12 pentagons. Maximal symmetry icosahedral. 14-hedron. 12 pentagons and two hexagons. Maximal symmetry 12m2 15-hedron. 12 pentagons and 3 hexagons. Maximal symmetry 16-hedron. 12 pentagons and 4 hexagons. Maximal symmetry tetrahedral.
22 A space filling of dodecahedra and 14-hedra. Rods built of 14-hedra (grey) in three mutually perpendicular directions are packed together. The voids are dodecahedra (yellow) centered on a bcc lattice. A unit cell is outlined in green. The edges form a 4-connected network. The hydrogen bonds in chlorine hydrate form such a network. (The vertices correspond to water molecules; chlorine atoms are at centers of the 14- hedra. This polyhedral space-filling also represents the pattern made by the Voronoi regions of beta-tungsten (β-w).
23 A tiling of space with 12-, 14- and 15-hedra. Columns of 14-hedra run parallel to the hexagonal symmetry axis. Layers of 12- and 15-hedra alternate. Williams, R. The Geometrical Foundations of Natural Structure. Dover (1979).
3-D Crystal Lattice Images
3-D Crystal Lattice Images All of the following images are crossed-stereo pairs. To view them, cross your eyes and focus. Author's note this material has been expanded and updated, and can be found at
More informationLecture 4! ü Review on atom/ion size! ü Crystal structure (Chap 4 of Nesseʼs book)!
Lecture 4! ü Review on atom/ion size! ü Crystal structure (Chap 4 of Nesseʼs book)! 15 C 4+ 42 Si 4+ Size of atoms! Hefferan and O Brien, 2010; Earth Materials Force balance! Crystal structure (Chap. 4)!
More informationChapter 3. The structure of crystalline solids 3.1. Crystal structures
Chapter 3. The structure of crystalline solids 3.1. Crystal structures 3.1.1. Fundamental concepts 3.1.2. Unit cells 3.1.3. Metallic crystal structures 3.1.4. Ceramic crystal structures 3.1.5. Silicate
More informationRemember the purpose of this reading assignment is to prepare you for class. Reading for familiarity not mastery is expected.
Remember the purpose of this reading assignment is to prepare you for class. Reading for familiarity not mastery is expected. After completing this reading assignment and reviewing the intro video you
More informationCHAPTER 4. Crystal Structure
CHAPTER 4 Crystal Structure We can assume minerals to be made of orderly packing of atoms or rather ions or molecules. Many mineral properties like symmetry, density etc are dependent on how the atoms
More informationCrystallographic structure Physical vs Chemical bonding in solids
Crystallographic structure Physical vs Chemical bonding in solids Inert gas and molecular crystals: Van der Waals forces (physics) Water and organic chemistry H bonds (physics) Quartz crystal SiO 2 : covalent
More informationAtomic Arrangement. Primer Materials For Science Teaching Spring
Atomic Arrangement Primer Materials For Science Teaching Spring 2016 31.3.2015 Levels of atomic arrangements No order In gases, for example the atoms have no order, they are randomly distributed filling
More informationCondensed Matter A Week 2: Crystal structure (II)
QUEEN MARY, UNIVERSITY OF LONDON SCHOOL OF PHYSICS AND ASTRONOMY Condensed Matter A Week : Crystal structure (II) References for crystal structure: Dove chapters 3; Sidebottom chapter. Last week we learnt
More informationExperiment 7: Understanding Crystal Structures
Experiment 7: Understanding Crystal Structures To do well in this laboratory experiment you need to be familiar with the concepts of lattice, crystal structure, unit cell, coordination number, the different
More informationAtomic Arrangement. Primer in Materials Spring
Atomic Arrangement Primer in Materials Spring 2017 30.4.2017 1 Levels of atomic arrangements No order In gases, for example the atoms have no order, they are randomly distributed filling the volume to
More informationCrystal Models. Figure 1.1 Section of a three-dimensional lattice.
Crystal Models The Solid-State Structure of Metals and Ionic Compounds Objectives Understand the concept of the unit cell in crystalline solids. Construct models of unit cells for several metallic and
More information4. Interpenetrating simple cubic
2 1. The correct structure t of CsClCl crystal is 1. Simple cubic 2. Body centered cubic 3. Face centered cubic 4. Interpenetrating simple cubic If corner as well as the particle at the center are same
More informationBonding and Packing: building crystalline solids
Bonding and Packing: building crystalline solids The major forces of BONDING Gravitational forces: F = G m m 1 2 F = attractive forces between 2 bodies G = universal graviational constant (6.6767 * 10
More informationERIC A LORD, ALAN L MACKAY AND S RANGANATHAN
ERIC A LORD, ALAN L MACKAY AND S RANGANATHAN New Geometries for New Materials Recent advances in materials science have given rise to novel materials with unique properties, through the manipulation of
More informationBonding. Bringing the atoms together
Bonding Bringing the atoms together More than one atom Until now, we have been consumed with describing individual atoms of elements. However, isolating individual atoms in most elements is an arduous
More informationCrystal Structure and Chemistry
Crystal Structure and Chemistry Controls on Crystal Structure Metallic bonding closest packing Covalent bonding depends on orbital overlap and geometry Ionic bonding Pauling s Rules Coordination Principle
More informationSilicate Structures. Silicate Minerals: Pauling s s Rules and. Elemental Abundance in Crust. Elemental Abundance in Crust: Pauling s s Rules
Silicate Minerals: Pauling s s Rules and Silicate Structures February 6, 2007 Elemental Abundance in Crust Fe Ion O 2- Si 4+ Al 3+, 3+ Ca Na + K + Mg mol % 2.6 1.4 mol% x charge 4.8 3.8 2.6 1.4 3.8 Sum
More informationMolecular Dynamics Investigation of Crystallization in the Hard Sphere System
Molecular Dynamics Investigation of Crystallization in the Hard Sphere System Brendan O Malley B. App. Sci. (Hons.) RMIT Department of Applied Physics Faculty of Applied Science Royal Melbourne Institute
More informationStructure of Crystalline Solids
Structure of Crystalline Solids Solids- Effect of IMF s on Phase Kinetic energy overcome by intermolecular forces C 60 molecule llotropes of Carbon Network-Covalent solid Molecular solid Does not flow
More informationE8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron Frank Dodd (Tony) Smith Jr
E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron Frank Dodd (Tony) Smith Jr. - 2013 The E8 Physics Model (vixra 1108.0027) is based on the Lie Algebra E8. 240 E8 vertices = 112
More informationChemical bonds. In some minerals, other (less important) bond types include:
Chemical bonds Chemical bond: force of attraction between two or more atoms/ions Types of bonds in crystals: Ionic bond: electrostatic attraction between two oppositely charged ions. This type of bond
More informationChapter 2. Atomic Packing
Chapter 2. Atomic Packing Contents 2-1. Packing of directional bonding atoms 2-2. Packing of indirectional bonding in same size atoms 2-3. Packing of indirectional bonding in different size atoms 2-4.
More informationAdvanced Ceramics for Strategic Applications Prof. H. S. Maiti Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Advanced Ceramics for Strategic Applications Prof. H. S. Maiti Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture -3 Crystal Structure Having made some introductory
More informationSorosilicates, Colors in Minerals (cont), and Deep Earth Minerals. ESS212 January 20, 2006
Sorosilicates, Colors in Minerals (cont), and Deep Earth Minerals ESS212 January 20, 2006 Double tetrahedron Sorosilicate is defined by the Si 2 O 7 group. Three groups of minerals, commonly, Epidote Zoisite
More informationSolids / Crystal Structure
The first crystal analysis proved that in the typical inorganic salt, NaCl, there is no molecular grouping. The inference that the structure consists of alternate ions of sodium and chlorine was an obvious
More informationCation Exchange Capacity, CEC
Cation Exchange Capacity, CEC The basic building blocks of clay minerals are: silicon atoms surrounded by four oxygen atoms (tetrahedra), and aluminium atoms surrounded by six hydroxide groups (dioctahedra),
More informationS.No. Crystalline Solids Amorphous solids 1 Regular internal arrangement of irregular internal arrangement of particles
Classification of solids: Crystalline and Amorphous solids: S.No. Crystalline Solids Amorphous solids 1 Regular internal arrangement of irregular internal arrangement of particles particles 2 Sharp melting
More informationPractice Problems Set II
P1. For the HCP crystal structure, (a) show that the ideal c/a ratio is 1.633; (b) show that the atomic packing factor for HCP is 0.74. (a) A sketch of one-third of an HCP unit cell is shown below. Consider
More informationEarth and Planetary Materials
Earth and Planetary Materials Spring 2013 Lecture 3 2013.01.14 14 1 Close Packed Anion Arrays Closest Packing Coordination number (C.N.) : number of anions bonded to a cation larger cation, higher C.N.
More informationRam Seshadri MRL 2031, x6129, These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings
Crystals, packings etc. Ram Seshadri MRL 2031, x6129, seshadri@mrl.ucsb.edu These notes complement chapter 6 of Anderson, Leaver, Leevers and Rawlings The unit cell and its propagation Materials usually
More informationPART I Chapter 8 Conclusions. With evolution all capsid structures are topologically related
68 PART I Chapter 8 Conclusions With evolution all capsid structures are topologically related We have seen how three concepts dominate the scene one begins with the snub dodecahedron forming the smallest
More informationregular tetrahedron. In the sodium chloride crystal each atom of one kind is
VOL. 51, 1964 CENTENNIAL: SECOND SCIENTIFIC SESSION 977 energies. The nuclear phenomena which we study in the laboratory are mostly manmade. Hence the natural place of nuclear processes would be the center
More informationChapter 12: Structures of Ceramics
Chapter 12: Structures of Ceramics Outline Introduction Crystal structures Ceramic structure AX-type crystal structures A m X p -type A m B n X p - type Silicate ceramics Carbon Chapter 12 - Ceramics Two
More informationCHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS
CHAPTER THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS Fundamental Concepts.1 What is the difference between atomic structure and crystal structure? Atomic structure relates to the number of protons
More information305 ATOMS, ELEMENTS, AND MINERALS
DATE DUE: Name: Instructor: Ms. Terry J. Boroughs Geology 305 ATOMS, ELEMENTS, AND MINERALS Instructions: Read each question carefully before selecting the BEST answer. Use GEOLOGIC VOCABULARY where APPLICABLE!
More informationExperiment 2a Models of the Solid State*
Experiment 2a Models of the Solid State* *This lab is adapted from solid-state labs offered at Purdue and Rice Universities. Introduction The structures of metals and simple ionic solids are prototypes
More informationCold Fusion and Palladium Atom Clusters
Cold Fusion and Palladium Atom Clusters In Palladium cold fusion processes, Ken Shoulders and Steve Shoulders consider micron (1,000 nanometer) size clusters of electrons and how they interact with the
More informationChapter 12: Structures & Properties of Ceramics
Chapter 12: Structures & Properties of Ceramics ISSUES TO ADDRESS... Bonding and structure of ceramic materials as compared with metals Chapter 12-1 Atomic Bonding in Ceramics Bonding: -- Can be ionic
More informationPhysics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces)
Physics of Materials: Classification of Solids On The basis of Geometry and Bonding (Intermolecular forces) Dr. Anurag Srivastava Atal Bihari Vajpayee Indian Institute of Information Technology and Manegement,
More informationLecture 05 Structure of Ceramics 2 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000.
MME 467 Ceramics for Advanced Applications Lecture 05 Structure of Ceramics 2 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000. Prof. A. K. M. Bazlur Rashid Department of MME, BUET, Dhaka
More informationSOLID STATE CHEMISTRY
SOLID STATE CHEMISTRY Crystal Structure Solids are divided into 2 categories: I. Crystalline possesses rigid and long-range order; its atoms, molecules or ions occupy specific positions, e.g. ice II. Amorphous
More informationReport Form for Experiment 6: Solid State Structures
Report Form for Experiment 6: Solid State Structures Note: Many of these questions will not make sense if you are not reading the accompanying lab handout. Station 1. Simple Cubic Lattice 1. How many unit
More informationContinuous symmetry and shape measures, or how to measure the distance between polyhedra representing molecules
Continuous symmetry and shape measures, or how to measure the distance between polyhedra representing molecules Pere Alemany Institut de Química Teòrica i Computacional de la Universitat de Barcelona (IQTCUB)
More informationWholemovement of the circle Bradford Hansen-Smith
Wholemovement of the circle Bradford Hansen-Smith 4606 N. Elston #3, Chicago IL 60630, USA brad@synasoft.com Wholeness is the most practical context in which to process information. The circle is Whole.
More informationStructures of Solids. Unit Cells - Not(?) Chapter 4 Ionic and Other Inorganic Solids. CHEM 462 Wednesday, September 22 T.
Chapter 4 Ionic and Other Inorganic Solids CHEM 462 Wednesday, September 22 T. Hughbanks Structures of Solids Many dense solids are described in terms of packing of atoms or ions. Although these geometric
More information1 8 =1 8 8 =1 6 =3. Unit cell Atoms at corner Atoms at faces Atoms at centre. Total no. of atoms per unit cell. bcc. fcc
Q. No. Amorphous substances show () Short and long range order (2) Short range order (3) Long range order (4) Have no sharp M.P. Option and 3 are correct Option 2 2 and 3 are correct Option 3 3 and 4 are
More information300 ATOMS, ELEMENTS, AND MINERALS
DATE DUE: Name: Instructor: Ms. Terry J. Boroughs Geology 300 ATOMS, ELEMENTS, AND MINERALS Instructions: Read each question carefully before selecting the BEST answer. Use GEOLOGIC VOCABULARY where APPLICABLE!
More informationClassification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava
Classification of Solids, Fermi Level and Conductivity in Metals Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus
More information305 ATOMS, ELEMENTS, AND MINERALS
DATE DUE: Name: Instructor: Ms. Terry J. Boroughs Geology 305 ATOMS, ELEMENTS, AND MINERALS Instructions: Read each question carefully before selecting the BEST answer. Use GEOLOGIC VOCABULARY where APPLICABLE!
More informationCHEM Principles of Chemistry II Chapter 10 - Liquids and Solids
CHEM 1212 - Principles of Chemistry II Chapter 10 - Liquids and Solids 10.1 Intermolecular Forces recall intramolecular (within the molecule) bonding whereby atoms can form stable units called molecules
More informationAtommetrics: Another View of Atomic Structure Based on Electron Orbital Geometry Part 2
Forma, 22, 177 189, 2007 Atommetrics: Another View of Atomic Structure Based on Electron Orbital Geometry Part 2 Edward SUZUKI OERDT Edward Suzuki Associates, Inc., 1-15-23 Seta, Setagaya-ku, Tokyo 158-0095,
More informationH O H C H H N H H. Valence Shell Electron Pair Repulsion: Predicting Shape & Polarity
Valence Shell Electron Pair Repulsion: Predicting Shape & Polarity BJECTIVES Students will develop the ability to: 1. Predict the arrangement that valence e pairs assume around an atom (e pair geometry)
More information305 ATOMS, ELEMENTS, AND MINERALS
DATE DUE: Name: Instructor: Ms. Terry J. Boroughs Geology 305 ATOMS, ELEMENTS, AND MINERALS Instructions: Read each question carefully before selecting the BEST answer. Use GEOLOGIC VOCABULARY where APPLICABLE!
More informationIonic Coordination and Silicate Structures
Ionic Coordination and Silicate Structures Pauling s Rules A coordination polyhedron of anions forms around a cation Ionic distance determined by radii Coordination number determined by radius ratio. May
More informationI. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized usin
Accepted for publication in IEEE Trans. Inform. Thy., Mar. 2, 200 On Quantization with the Weaire-Phelan Partition * Navin Kashyap and David L. Neuho Department of Electrical Engineering and Computer Science
More informationMetal Structure. Chromium, Iron, Molybdenum, Tungsten Face-centered cubic (FCC)
Metal Structure Atoms held together by metallic bonding Crystalline structures in the solid state, almost without exception BCC, FCC, or HCP unit cells Bodycentered cubic (BCC) Chromium, Iron, Molybdenum,
More informationVERY SHORT ANSWER TYPE QUESTIONS (1 Mark)
UNIT I 10 Chemistry-XII THE SOLID STATE VERY SHORT ANSWER TYPE QUESTIONS (1 Mark) Q. 1. What do you mean by paramagnetic substance? Ans. Weakly attracted by magnetic eld and these substances are made of
More informationReview for Chapter 4: Structures and Properties of Substances
Review for Chapter 4: Structures and Properties of Substances You are responsible for the following material: 1. Terms: You should be able to write definitions for the following terms. A complete definition
More informationTetrahedra and Physics Frank Dodd (Tony) Smith, Jr. discussion with Klee Irwin
Tetrahedra and Physics Frank Dodd (Tony) Smith, Jr. discussion with Klee Irwin 1 - Start with a regular Tetrahedron in flat 3-dim space Tetrahedron Josephson Junction Quantum Computer Qubit 2 - Add 4 +
More informationBMT 2014 Symmetry Groups of Regular Polyhedra 22 March 2014
Time Limit: 60 mins. Maximum Score: 125 points. Instructions: 1. When a problem asks you to compute or list something, no proof is necessary. However, for all other problems, unless otherwise indicated,
More information305 ATOMS, ELEMENTS, AND MINERALS
DATE DUE: Name: Instructor: Ms. Terry J. Boroughs Geology 305 ATOMS, ELEMENTS, AND MINERALS Instructions: Read each question carefully before selecting the BEST answer. Use GEOLOGIC VOCABULARY where APPLICABLE!
More informationChemWiki BioWiki GeoWiki StatWiki PhysWiki MathWiki SolarWiki
Ashley Robison My Preferences Site Tools Popular pages MindTouch User Guide FAQ Sign Out If you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it
More informationCeramic Bonding. CaF 2 : large SiC: small
Recall ceramic bonding: - Mixed ionic and covalent. - % ionic character ( f ) increases with difference in electronegativity Large vs small ionic bond character: Ceramic Bonding CaF : large SiC: small
More informationMetallic & Ionic Solids. Crystal Lattices. Properties of Solids. Network Solids. Types of Solids. Chapter 13 Solids. Chapter 13
1 Metallic & Ionic Solids Chapter 13 The Chemistry of Solids Jeffrey Mack California State University, Sacramento Crystal Lattices Properties of Solids Regular 3-D arrangements of equivalent LATTICE POINTS
More informationThe Geometry of Root Systems. Brian C. Hall
The Geometry of Root Systems A E Z S Brian C. Hall T G R S T G R S 1 1. I Root systems arise in the theory of Lie groups and Lie algebras, but can also be studied as mathematical objects in their own right.
More information1.=:====.1 37/661 (2), Fort P.O., Trivandrum , Kerala, India
~ _ Transworld Research Network 1.=:====.1 37/661 (2), Fort P.O., Trivandrum-695 023, Kerala, India lecnt1n...llell-cl1iiiilllne SIIIIIs, 1[20011: 13-841S11:...18IIHI28-8 QuantilVing the interstitial structure
More informationThe Solid State. Phase diagrams Crystals and symmetry Unit cells and packing Types of solid
The Solid State Phase diagrams Crystals and symmetry Unit cells and packing Types of solid Learning objectives Apply phase diagrams to prediction of phase behaviour Describe distinguishing features of
More informationCHEM1902/ N-2 November 2014
CHEM1902/4 2014-N-2 November 2014 The cubic form of boron nitride (borazon) is the second-hardest material after diamond and it crystallizes with the structure shown below. The large spheres represent
More informationLecture 04 Structure of Ceramics 1 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000.
MME 467 Ceramics for Advanced Applications Lecture 04 Structure of Ceramics 1 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000. Prof. A. K. M. Bazlur Rashid Department of MME, BUET, Dhaka
More informationPhys 412 Solid State Physics. Lecturer: Réka Albert
Phys 412 Solid State Physics Lecturer: Réka Albert What is a solid? A material that keeps its shape Can be deformed by stress Returns to original shape if it is not strained too much Solid structure
More informationGeol /19/06 Labs 5 & 6 Crystal Chemistry Ionic Coordination and Mineral Structures
Geol 2311 9/19/0 Labs 5 & Crystal Chemistry Ionic Coordination and Mineral Structures Handout Oral Mineral Tray Report Samples Ionic Coordination Exercise Investigating Mineral Structures using XtalDraw
More informationEPSC501 Crystal Chemistry WEEK 5
EPSC501 Crystal Chemistry WEEK 5 Oxidation states of transition elements (many more in aqueous solutions than in the common rock-forming minerals) Notice that almost every transition metal has a +2 oxidation
More informationOne special case of interest is when we have a periodic partition of R k, i.e., a partition composed of translates of a fundamental unit consisting of
On the Potential Optimality of the Weaire-Phelan Partition avin Kashyap David. euho Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 4809-222 USA fnkashyap,neuhoffg@eecs.umich.edu
More information8.2 Hybrid Atomic Orbitals
420 Chapter 8 Advanced Theories of Covalent Bonding Answer: (a) is a π bond with a node along the axis connecting the nuclei while (b) and (c) are σ bonds that overlap along the axis. 8.2 Hybrid Atomic
More informationDiamond. There are four types of solid: -Hard Structure - Tetrahedral atomic arrangement. What hybrid state do you think the carbon has?
Bonding in Solids Bonding in Solids There are four types of solid: 1. Molecular (formed from molecules) - usually soft with low melting points and poor conductivity. 2. Covalent network - very hard with
More informationHigh Temperature Materials. By Docent. N. Menad. Luleå University of Technology ( Sweden )
Course KGP003 Ch. 12 High Temperature Materials By Docent. N. Menad Dept. of Chemical Engineering and Geosciences Div. Of process metallurgy Luleå University of Technology ( Sweden ) Ceramic materials
More informationThere are four types of solid:
Bonding in Solids There are four types of solid: 1. Molecular (formed from molecules) - usually soft with low melting points and poor conductivity. 2. Covalent network - very hard with very high melting
More information5.03 In-Class Exam 2
5.03 In-Class Exam 2 Christopher C. Cummins March 12, 2010 Instructions Clearly write your name at the top of this front page, but otherwise do not write on this front page as it will be used for scoring.
More informationFrustration and ice. Similarities with the crystal structure of ice I h : the notion of spin ice.
Frustration and ice The cubic (Fd-3m) structure of pyrochlore (CaNa)Nb 2 O 6 F [A 2 B 2 O 7 or A 2 B 2 O 6 Oʹ] The A site often has lone-pair cations (Pb 2+ or Bi 3+ ). Polar materials in this structure
More informationCopyright SOIL STRUCTURE and CLAY MINERALS
SOIL STRUCTURE and CLAY MINERALS Soil Structure Structure of a soil may be defined as the mode of arrangement of soil grains relative to each other and the forces acting between them to hold them in their
More informationLecture 6 - Bonding in Crystals
Lecture 6 onding in Crystals inding in Crystals (Kittel Ch. 3) inding of atoms to form crystals A crystal is a repeated array of atoms Why do they form? What are characteristic bonding mechanisms? How
More informationA Theory of Water and Ionic Solution, with Particular Reference to Hydrogen
A Theory of Water and Ionic Solution, with Particular Reference to Hydrogen and Hydroxyl Ions, J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1 (1933) 515-548. Ice-I h : a = 7.82 Å ; c = 7.36 Å P6 3 cm
More informationCrystal Structure. Dr Bindu Krishnan
Solid State Physics-1 Crystal Structure Dr Bindu Krishnan CRYSTAL LATTICE What is crystal (space) lattice? In crystallography, only the geometrical properties of the crystal are of interest, therefore
More informationEarth Materials I Crystal Structures
Earth Materials I Crystal Structures Isotopes same atomic number, different numbers of neutrons, different atomic mass. Ta ble 1-1. Su mmar y of quantu m num bers Name Symbol Values Principal n 1, 2,
More information12-neighbour packings of unit balls in E 3
12-neighbour packings of unit balls in E 3 Károly Böröczky Department of Geometry Eötvös Loránd University Pázmány Péter sétány 1/c H-1117 Budapest Hungary László Szabó Institute of Informatics and Economics
More informationLone pairs in the solid state: Frustration
Lone pairs in the solid state: Frustration Bi 2 Ti 2 O 6 O, the pyrochlore analogue of perovskite PbTiO 3, is cubic down to 2 K. [Hector, Wiggin, J. Solid State Chem. 177 (2004) 139] Question: Is the absence
More informationActivity 5&6: Metals and Hexagonal Close-Packing
Chemistry 150 Name(s): Activity 5&6: Metals and Hexagonal Close-Packing Metals are chemicals characterized by high thermal and electrical conductivity, malleability and ductility. Atoms are the smallest
More information1 What Is a Mineral? Critical Thinking 2. Apply Concepts Glass is made up of silicon and oxygen atoms in a 1:2 ratio. The SiO 2
CHAPTER 5 1 What Is a Mineral? SECTION Minerals of Earth s Crust KEY IDEAS As you read this section, keep these questions in mind: What is a mineral? What are the two main groups of minerals? What are
More informationIonic Bonding. Chem
Whereas the term covalent implies sharing of electrons between atoms, the term ionic indicates that electrons are taken from one atom by another. The nature of ionic bonding is very different than that
More informationAn eightfold path to E 8
An eightfold path to E 8 Robert A. Wilson First draft 17th November 2008; this version 29th April 2012 Introduction Finite-dimensional real reflection groups were classified by Coxeter [2]. In two dimensions,
More informationLecture 6. Physical Properties. Solid Phase. Particle Composition
Lecture 6 Physical Properties Solid Phase Particle Composition 1 Questions What are tetrahedrons and octahedrons? How do silica tetrahedra bonds affect mineral weathering? Difference between primary and
More information9 Crystal Structures
9 Crystal Structures Supporting interactive 3D images of crystal structures and more advanced material may be found at:http://www-teach.ch.cam.ac.uk/links/3dindex.html www.xtremepapers.com A Introduction
More informationIntroduction to Engineering Materials ENGR2000 Chapter 12: Structures and Properties of Ceramics. Dr. Coates
Introduction to Engineering Materials ENGR2000 Chapter 12: Structures and Properties of Ceramics Dr. Coates 12.1 Introduction Ceramics Compounds between metallic & non-metallic elements Predominantly ionic
More informationThe exam time is 1hr45 minutes. Try to finish this practice exam in the same time.
Practice exam for final exam, Chem 1210, Dr. Wu Note: The exam time is 1hr45 minutes. Try to finish this practice exam in the same time. 1. Which of the following gases will exhibit the least ideal behavior?
More informationDifferentiaton: Redistribution of mass (elements) and energy by chemical & physical processes. Goal: Quantitative understanding of those processes.
Differentiaton: Redistribution of mass (elements) and energy by chemical & physical processes. Goal: Quantitative understanding of those processes. 1 2 What controls the periodicity of behavior of the
More informationPART 1 Introduction to Theory of Solids
Elsevier UK Job code: MIOC Ch01-I044647 9-3-2007 3:03p.m. Page:1 Trim:165 240MM TS: Integra, India PART 1 Introduction to Theory of Solids Elsevier UK Job code: MIOC Ch01-I044647 9-3-2007 3:03p.m. Page:2
More informationInorganic Exam 1 Chm October 2010
Inorganic Exam 1 Chm 451 28 October 2010 Name: Instructions. Always show your work where required for full credit. 1. In the molecule CO 2, the first step in the construction of the MO diagram was to consider
More informationEtudes on convex polyhedra. 5. Topological entropies of all 2907 convex 4- to 9-vertex polyhedra
24 References 1. Grünbaum, B. (1967). Convex Polytopes. New York: Springer. 2. Voytekhovsky, Y. L. (2016). Acta Cryst, A72, 582-585. 3. Voytekhovsky, Y. L. (2017). Acta Cryst, A73, 271-273. Etudes on convex
More informationMetallic and Ionic Structures and Bonding
Metallic and Ionic Structures and Bonding Ionic compounds are formed between elements having an electronegativity difference of about 2.0 or greater. Simple ionic compounds are characterized by high melting
More informationSolids. properties & structure
Solids properties & structure Determining Crystal Structure crystalline solids have a very regular geometric arrangement of their particles the arrangement of the particles and distances between them is
More information