DIFFRACTION METHODS IN MATERIAL SCIENCE. PD Dr. Nikolay Zotov Lecture 4_2
|
|
- Agatha Webster
- 5 years ago
- Views:
Transcription
1 DIFFRACTION METHODS IN MATERIAL SCIENCE PD Dr. Nikolay Zotov Lecture 4_2
2 OUTLINE OF THE COURSE 0. Introduction 1. Classification of Materials 2. Defects in Solids 3. Basics of X-ray and neutron scattering 4. Crystal Symmetry 5. Diffraction studies of Polycrystalline Materials 6. Microstructural Analysis by Diffraction 7. Diffraction studies of Thin Films 8. Diffraction studies of Nanomaterials 9. Diffraction studies of Amorphous and Composite Materials 2
3 OUTLINE OF TODAY S LECTURE Lattices Symmetry elements and symmetry operations Crystal Classes Crystal Systems Space Groups 3
4 Symmetry and Lattices in Art, Architecture, Natural Life M.C. Escher ( ) Periodic Lattice Periodic repetition of points in space 4
5 Why is Symmetry Important Systematic description of objects, molecules, lattices Description/ Prediction of Atomic Structures Interpretation of Diffraction Data Description of Physical Properties of Crystalline Materials Optical Mechanical Electric and Magnetic 5
6 Symmetry Operations Symmetry operation is a transformation, which leaves an object the same. Point-Symmetry operations: Identity 1; (E) Rotational Axes n; (n-fold rotation axis) Mirror Plane m; (s, s v, s h ) Inversion -1; (i) Rotoinversion Axes n; (S n ) Point symmetry means that a given point remains fixed (untransformed) 6
7 Angles of rotation f n m = m360/n Crystallographic Rotational Axes 2-fold f = 180 o 3-fold f = 120, 240 o 4-fold f = 90, 180, 270 o 6-fold f = 60,120,180,240,300 o Crystallographic Rotations n = 1,2,3,4,6!!! 7
8 Crystallographic Rotational Axes z Symmetry operation X cos(f) -sin(f) 0 x Y = sin(f) cos(f) 0 y Z z Axis Z x y
9 Non-Crystallographic Rotational Axes 5-fold Quasi-crystals 7-fold 8-fold 10-fold Quasi-crystals 9
10 Mirror Planes m Symmetry operation X X m bc plane Y = Y Z Z m ac plane c b m ab plane a m bc 10
11 Centre of Symmetry Symmetry operation X X Y = Y Z Z Typical indicator in single crystals - Pairs of paralell planes 11
12 Rotoinversion axis Rotation + inversion 3-fold 4-fold 6-fold 12
13 Rotoinversion axis Rotation + inversion Symmetry operation X cos(f) -sin(f) 0 X Y = sin(f) cos(f) 0 Y Z Z Axis z c b a Matrix of inversion 4-fold rotoinversion axis parallel to c f = 90 o
14 Crystal System 32 Point Groups (Crystal Classes) Orientation rules!!! Triclinic 1, -1 Monoclinic 2; m, 2/m; b 2 Orthorhombic 222, mm2, 2/m 2/m 2/m ( a b c) Tetragonal 4, -4, 4/m, 422, 4mm, -42m, 4/m 2/m 2/m ( c b [110] ) Hexagonal 3, -3, 32, 3m, -32/m, 6, -6, 6/m, 622, 6mm, -6m2, 6/m 2/m 2/m ( c a [1-10] ) Cubic 23, 2/m 3, 432, -43m, 4/m -3 2/m ( a [111] [1-10] ) 14
15 Examples of Monoclinic Point Groups (Stereographic Projections) The comma defines the handedness 15
16 Examples of Orthorhombic Point Groups (Stereographic Projections) 2/m 2/m 2/m Full symbol 16
17 Examples of Tetragonal Point Groups (Stereographic Projections) 17
18 Examples of Hexagonal Point Groups (Stereographic Projections) Full symbol 6/m 2/m 2/m 18
19 Examples of Cubic Point Groups (Stereographic Projections) 19
20 2 m m 20
21 3 2 21
22 -6 m 2 22
23 Crystal Structure Periodic Lattice Translational Symmetry Basis (Atoms, Molecules in the Unit cell) Point Symmetry Crystal 23
24 Combination of Translations and Point Symmetry Operations 1. New Symmetry Operations 2. New types of lattices 3. Crystal Structures 24
25 New Symmetry Operations Translations+point symmetry operations Glide plane (g) Reflection from a plane plus translation (g) along a given direction parallel to the plane The vector g is not a translation of the lattice! Symmetry operation X X 0 Y = Y + 1/2 Z Z 0 ab b g b 25
26 Glide Planes Symbol Translational Component a, b, c a/2, b/2, c/2 n One from (a+b)/2, (b+c)/2, (a+c)/2 d One from (a+b)/4, (b+c)/4, (a+c)/4 26
27 New Symmetry Operations Translations+point symmetry operations Srew rotation n j Rotation n with angle (360j/n) plus translation along the axis j = 1,2,, n-1 2 1, 3 1, 3 2, 4 1, 4 2,4 3, 6 1, 6 2, 6 3 ; Translational Component: v = j/n t Symmetry operation 2-fold c 2 1 : 2 axis; Translation parallel to c; f = 180 o X X 0 Y = Y + 0 Z Z 1/2 c Crystallographic screw rotations n = 2,3,4,6!!! 27
28 3-fold Screw Axes t t t Translational part Symmetry operation X X 0 Y = Y + 0 Z Z 1/3t 28
29 6-fold Screw Axes 29
30 Comparison of normal and screw rotational axes right-hand screw axes left-hand screw axes 30
31 Geometrical Symbols of Symmetry Elements perpendicular In-plane Perpendicular Inplane In-plane 31
32 Crystal Systems Pyrite FeS 2 Rutile TiO 2 Beryl 32
33 Crystal Systems Topas Azurite 33
34 D-Spacing in Different Crystal Systems d hkl = 1/G(hkl); G(hkl) = ha* + kb* + lc* G hkl2 = (hkl) g* (hkl) T Cubic a* = bc/v = 1/a b* = ac/v = 1/a c* = ab/v = 1/a 1/a a* = ß* = g* = 90 o ; g* = 0 1/a /a 2 1/a h G 2 = (h k l) 0 1/a 2 0 k ; G = (h 2 + k 2 + l 2 ) 1/2 /a 0 0 1/a 2 l d hkl = a/(h 2 + k 2 + l 2 ) 1/2 34
35 D-Spacings in Different Crystal Classes 35
36 New Crystal Lattices To better describe the Crystal symmetry! 14 Bravis Lattices Trigonal (rhomboedrisch) hexagonal 36
37 What is a Group? Group (G) is a set of objects together with an operation (rule) how to combine two elements from the set to get a new element. The elements of a group fulfill the following properties: # closure the operation on the elements of the group always produces another element of the group (closed set); A.B = C, C is also element of G # the group contains an identity element E, which leaves every element of the group unchanged, when it is conbined with the identity. A.E = E.A = A # for every element A of the group there is an inverse elemen A -1, so that A.A -1 = E In point, plane and space groups the elements are the symmetry transformations! 37
38 Space Groups Combination of symmetry elements located at diffrent points of the unit cell There are only 230 different space group symmetries (space groups) in 3D (There are only 17 plane groups in 2D) 38
39 Space Groups Symbols L a b c L P primitive lattice F face-centered lattice I body-centered lattice A,B,C base-centered lattice Examples a b c - Point group symbol along the main symmetry directions discused before P 2/m P m m 2 I 4 F m -3 m 39
40 Plane Groups 40
41 Monoclinic Space Group PG Symmetry Operations Mirror plane in the projection plane 41
42 Orthorhombic Space Group Symmetry Operations International Tables of Crystallography 42
43 Tetragonal Space Group Symmetrie- Operationen n 43
44 Hexagonal Space Group Symmetrie operationen 44
45 Triclinic P1 P 1 Monoclinic P2, P2 1, C2, Pm, Pc, Cm, Cc P2/m, P2 1 /m, C2/m, P2/c, P2 1 /c, C2/c Orthorhombic P222, P222 1, P , P , C222 1, C222, F222, I222, I ,2 1, Pmm2, Pmc2 1, Pcc2, Pma2, Pca2 1, Pnc2, Pmn2 1, Pba2, Pna2 1, Pnn2, Cmm2, Cmc2 1, Ccc2, Amm2, Abm2, Ama2, Aba2, Fmm2, Fdd2, Imm2, Iba2, Ima2 Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmca, Cmmm, Cccm, Cmma, Ccca, Fmmm, Fddd, Immm, Ibam, Ibca, Imma Tetragonal P4, P4 1, P4 2, P4 3, I4, I4 1 P 4, I 4 P4/m, P4 2 /m, P4/n, P4 2 /n, I4/m, I4 1 /a P422, P42 1 2, P4 1 22, P , P4 2 22, P , P4 3 22, P , I422, I P4mm, P4bm, P4 2 cm, P4 2 nm, P4cc, P4nc, P4 2 mc, P4 2 bc, I4mm, I4cm, I4 1 md, I4 1 cd P 4 2m, P 4 2c, P m, P c, P 4 m2, P 4 c2, P 4 b2, P 4 n2, I 4 m2, I 4 c2, I 4 2m, I 4 2d P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4 2 /mmc, P4 2 /mcm, P4 2 /nbc, P4 2 /nnm, P4 2 /mbc, P4 2 /mnm, P4 2 /nmc, P4 2 /ncm, I4/mmm, I4/mcm, I4 1 /amd, I4 1 /acd Trigonal P3, P3 1, P3 2, R3 P 3, R 3 P312, P321, P3 1 12, P3 1 21, P3 2 12, P3 2 21, R32 P3m P 3 1m, P 3 1c, P 3 m1, P 3 c1, R 3 m, R 3 c Hexagonal P6, P6 1, P6 5, P6 2, P6 4, P6 3 P 6 3D Space Groups P6/m, P6 3 /m P622, P6 1 22, P6 5 22, P6 2 22, P6 4 22, P P6mm, P6cc, P6 3 cm, P6 3 mc P 6 m2, P 6 c2, P 6 2m, P 6 2c P6/mmm, P6/mcc, P6 3 /mcm, P6 3 /mmc Cubic P23, F23, I23, P2 1 3, I2 1 3 Pm 3, Pn 3, Fm 3, Fd 3, Im 3, Pa 3, Ia 3 P432, P4 2 32, F432, F4 1 32, I432, P4 3 32, P4 1 32, I P 4 3m, F 4 3m, I 4 3m, P 4 3n, F 4 3c, I 4 3d Pm 3 m, Pn 3 n, Pm 3 n, Pn 3 m, Fm 3 m, Fm 3 c, Fd 3 m, Fd 3 c, Im 3 m, Ia 3 d 45
46 Classification according to Crystal Systems 46
47 47
48 48
49 49
50 Determination of Point Symmetry from Space Group Symbol Replace all symmetry elements with translational component(s) with point group elements I P Pccn = P 2 1 /c 2 1 /c 2/n 2/m 2/m 2/m m m m The orientation of the symmetry operations remains the same! 50
51 Additional Sources Point Groups International Tables of Crystallograophy Changes No Practical No Lecture Practical in 3P Practical in 3P2 51
GENERATORS AND RELATIONS FOR SPACE GROUPS
GENERATORS AND RELATIONS FOR SPACE GROUPS Eric Lord Bangalore 2010 Contents Introduction.... 1 Notation.... 2 PART I. NON-CUBIC GROUPS Explanation of the figures.... 3 Triclinic.... 5 Monoclinic.... 5
More informationChapter 1. Introduction
Chapter 1. Introduction 1. Generation of X-ray 1) X-ray: an electromagnetic wave with a wavelength of 0.1 100 Å. 2) Types of X-ray tubes a) A stationary-anode tube b) A rotating-anode tube 3) X-ray tube:
More information3.1. Space-group determination and diffraction symbols
International Tables for Crystallography (2006). Vol. A, Chapter 3.1, pp. 44 54. 3.1. Space-group determination and diffraction symbols BY A. LOOIJENGA-VOS AND M. J. BUERGER 3.1.1. Introduction In this
More informationCrystallographic Symmetry. Jeremy Karl Cockcroft
Crystallographic Symmetry Jeremy Karl Cockcroft Why bother? To describe crystal structures Simplifies the description, e.g. NaCl structure Requires coordinates for just 2 atoms + space group symmetry!
More informationInfluence of Chemical and Spatial Constraints on the Structures of Inorganic Compounds
381 Acta Cryst. (1997). B53, 381-393 Influence of Chemical and Spatial Constraints on the Structures of Inorganic Compounds I. D. BROWN Brockhouse Institute of Materials Research, McMaster University,
More informationAppendixes POINT GROUPS AND THEIR CHARACTER TABLES
Appendixes APPENDIX I. POINT GROUPS AND THEIR CHARACTER TABLES The following are the character tables of the point groups that appear frequently in this book. The species (or the irreducible representations)
More informationSymmetry Crystallography
Crystallography Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern In 3-D, translation defines operations which move the motif into infinitely repeating patterns
More informationThe structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures
Describing condensed phase structures Describing the structure of an isolated small molecule is easy to do Just specify the bond distances and angles How do we describe the structure of a condensed phase?
More informationCrystallography Reading: Warren, Chapters 2.1, 2.2, 2.6, 8 Surface symmetry: Can be a clue to underlying structure. Examples:
Crystallography Reading: Warren, Chapters 2.1, 2.2, 2.6, 8 Surface symmetry: Can be a clue to underlying structure. Examples: Snow (SnowCrystals.com) Bismuth (Bao, Kavanagh, APL 98 66103 (2005) Hexagonal,
More informationMultipole Superconductivity and Unusual Gap Closing
Novel Quantum States in Condensed Matter 2017 Nov. 17, 2017 Multipole Superconductivity and Unusual Gap Closing Application to Sr 2 IrO 4 and UPt 3 Department of Physics, Kyoto University Shuntaro Sumita
More informationTables of crystallographic properties of double antisymmetry space groups
Tables of crystallographic properties of double antisymmetry space groups Mantao Huang a, Brian K. VanLeeuwen a, Daniel B. Litvin b and Venkatraman Gopalan a * a Department of Materials Science and Engineering,
More informationSymmetry in 2D. 4/24/2013 L. Viciu AC II Symmetry in 2D
Symmetry in 2D 1 Outlook Symmetry: definitions, unit cell choice Symmetry operations in 2D Symmetry combinations Plane Point groups Plane (space) groups Finding the plane group: examples 2 Symmetry Symmetry
More informationLecture course on crystallography, 2015 Lecture 5: Symmetry in crystallography
Dr Semën Gorfman Department of Physics, University of SIegen Lecture course on crystallography, 2015 Lecture 5: Symmetry in crystallography What is symmetry? Symmetry is a property of an object to stay
More informationCrystallography basics
Crystallography basics 1 ? 2 Family of planes (hkl) - Family of plane: parallel planes and equally spaced. The indices correspond to the plane closer to the origin which intersects the cell at a/h, b/k
More informationn-dimensional, infinite, periodic array of points, each of which has identical surroundings.
crystallography ll Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points A2 ("bcc") structure lattice points Lattice n-dimensional,
More informationBasics of crystallography
Basics of crystallography 1 Family of planes (hkl) - Family of plane: parallel planes and equally spaced. The indices correspond to the plane closer to the origin which intersects the cell at a/h, b/k
More informationApplications of X-ray and Neutron Scattering in Biological Sciences: Symmetry in direct and reciprocal space 2012
Department of Drug Design and Pharmacology Applications of X-ray and Neutron Scattering in Biological Sciences: Symmetry in direct and reciprocal space 2012 Michael Gajhede Biostructural Research Copenhagen
More informationSymmetry. 2-D Symmetry. 2-D Symmetry. Symmetry. EESC 2100: Mineralogy 1. Symmetry Elements 1. Rotation. Symmetry Elements 1. Rotation.
Symmetry a. Two-fold rotation = 30 o /2 rotation a. Two-fold rotation = 30 o /2 rotation Operation Motif = the symbol for a two-fold rotation EESC 2100: Mineralogy 1 a. Two-fold rotation = 30 o /2 rotation
More informationIntroduction to crystallography The unitcell The resiprocal space and unitcell Braggs law Structure factor F hkl and atomic scattering factor f zθ
Introduction to crystallography The unitcell The resiprocal space and unitcell Braggs law Structure factor F hkl and atomic scattering factor f zθ Introduction to crystallography We divide materials into
More informationBasic Crystallography Part 1. Theory and Practice of X-ray Crystal Structure Determination
Basic Crystallography Part 1 Theory and Practice of X-ray Crystal Structure Determination We have a crystal How do we get there? we want a structure! The Unit Cell Concept Ralph Krätzner Unit Cell Description
More informationresearch papers Homogeneous sphere packings with triclinic symmetry 1. Introduction 2. Sphere packings corresponding to lattice complex P11 1a
Acta Crystallographica Section A Foundations of Crystallography ISSN 008-7673 Received 3 May 2002 Accepted 7 July 2002 Homogeneous sphere packings with triclinic symmetry W. Fischer and E. Koch* Institut
More informationPX-CBMSO Course (2) of Symmetry
PX-CBMSO Course (2) The mathematical description of Symmetry y PX-CBMSO-June 2011 Cele Abad-Zapatero University of Illinois at Chicago Center for Pharmaceutical Biotechnology. Lecture no. 2 This material
More informationNove fizickohemijske metode. Ivana Radosavljevic Evans Durham University, UK
Nove fizickohemijske metode Ivana Radosavljevic Evans Durham University, UK Nove fizickohemijske metode: Metode zasnovane na sinhrotronskom zracenju Plan predavanja: Difrakcione metode strukturne karakterizacije
More informationChemical Crystallography
Chemical Crystallography Prof Andrew Goodwin Michaelmas 2014 Recap: Lecture 1 Why does diffraction give a Fourier transform? k i = k s = 2π/λ k i k s k i k s r l 1 = (λ/2π) k i r l 2 = (λ/2π) k s r Total
More informationUNIT I SOLID STATE PHYSICS
UNIT I SOLID STATE PHYSICS CHAPTER 1 CRYSTAL STRUCTURE 1.1 INTRODUCTION When two atoms are brought together, two kinds of forces: attraction and repulsion come into play. The force of attraction increases
More informationOverview - Macromolecular Crystallography
Overview - Macromolecular Crystallography 1. Overexpression and crystallization 2. Crystal characterization and data collection 3. The diffraction experiment 4. Phase problem 1. MIR (Multiple Isomorphous
More informationFundamentals of X-ray diffraction
Fundamentals of X-ray diffraction Elena Willinger Lecture series: Modern Methods in Heterogeneous Catalysis Research Outline History of X-ray Sources of X-ray radiation Physics of X-ray scattering Fundamentals
More informationSymmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES
Symmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES http://www.cryst.ehu.es AMPLIMODES Symmetry Modes Analysis Modes in the statics of low-symmetry distorted phases: Distorted
More informationTranslational symmetry, point and space groups in solids
Translational symmetry, point and space groups in solids Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy ASCS26 Spokane Michele Catti a = b = 4.594 Å; Å;
More informationCrystal Symmetry Algorithms in a. Research
Crystal Symmetry Algorithms in a High-Throughput Framework for Materials Research by Richard Taylor Department of Mechanical Engineering and Materials Science Duke University Date: Approved: Stefano Curtarolo,
More informationSPACE GROUPS. International Tables for Crystallography, Volume A: Space-group Symmetry. Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain
SPACE GROUPS International Tables for Crystallography, Volume A: Space-group Symmetry Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain SPACE GROUPS Crystal pattern: Space group G: A model of the
More informationNeutron Powder Diffraction Theory and Instrumentation
NTC, Taiwen Aug. 31, 212 Neutron Powder Diffraction Theory and Instrumentation Qingzhen Huang (qing.huang@nist.gov) NIST Center for Neutron Research (www.ncnr.nist.gov) Definitions E: energy; k: wave vector;
More informationLecture course on crystallography, 2015 Lecture 9: Space groups and International Tables for Crystallography
Dr Semën Gorfman Department of Physics, University of SIegen Lecture course on crystallography, 2015 Lecture 9: Space groups and International Tables for Crystallography UNIT CELL and ATOMIC POSITIONS
More informationTHE FIVE TYPES OF PLANAR 2-D LATTICES. (d) (e)
THE FIVE TYPES OF PLANAR 2-D LATTICES (a) (d) (b) (d) and (e) are the same (e) (c) (f) (a) OBLIQUE LATTICE - NO RESTRICTIONS ON ANGLES BETWEEN THE UNIT CELL EDGES (b) RECTANGULAR LATTICE - ANGLE BETWEEN
More informationCrystal Chem Crystallography
Crystal Chem Crystallography Chemistry behind minerals and how they are assembled Bonding properties and ideas governing how atoms go together Mineral assembly precipitation/ crystallization and defects
More informationX-ray analysis. 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods
X-ray analysis 1. Basic crystallography 2. Basic diffraction physics 3. Experimental methods Introduction Noble prizes associated with X-ray diffraction 1901 W. C. Roentgen (Physics) for the discovery
More informationSOLID STATE 18. Reciprocal Space
SOLID STATE 8 Reciprocal Space Wave vectors and the concept of K-space can simplify the explanation of several properties of the solid state. They will be introduced to provide more information on diffraction
More informationTILES, TILES, TILES, TILES, TILES, TILES
3.012 Fund of Mat Sci: Structure Lecture 15 TILES, TILES, TILES, TILES, TILES, TILES Photo courtesy of Chris Applegate. Homework for Fri Nov 4 Study: Allen and Thomas from 3.1.1 to 3.1.4 and 3.2.1, 3.2.4
More informationChapter 2 Introduction to Phenomenological Crystal Structure
Chapter 2 Introduction to Phenomenological Crystal Structure 2.1 Crystal Structure An ideal crystal represents a periodic pattern generated by infinite, regular repetition of identical microphysical structural
More informationTim Hughbanks CHEMISTRY 634. Two Covers. Required Books, etc.
CHEMISTRY 634 This course is for 3 credits. Lecture: 2 75 min/week; TTh 11:10-12:25, Room 2122 Grades will be based on the homework (roughly 25%), term paper (15%), midterm and final exams Web site: http://www.chem.tamu.edu/rgroup/
More informationINTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY
INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY SPACE-GROUP SYMMETRY (short overview) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain SPACE GROUPS Crystal pattern: infinite, idealized crystal
More informationMineralogy Problem Set Crystal Systems, Crystal Classes
Mineralogy Problem Set Crystal Systems, Crystal Classes (1) For each of the three accompanying plane patterns: (a) Use a ruler to draw solid lines to show where there are mirror planes on the pattern.
More informationScattering and Diffraction
Scattering and Diffraction Andreas Kreyssig, Alan Goldman, Rob McQueeney Ames Laboratory Iowa State University All rights reserved, 2018. Atomic scale structure - crystals Crystalline materials... atoms
More information1/2, 1/2,1/2, is the center of a cube. Induces of lattice directions and crystal planes (a) Directions in a crystal Directions in a crystal are
Crystallography Many materials in nature occur as crystals. Examples include the metallic elements gold, copper and silver, ionic compounds such as salt (e.s. NaCl); ceramics, rutile TiO2; and nonmetallic
More informationSPACE GROUPS AND SYMMETRY
SPACE GROUPS AND SYMMETRY Michael Landsberg Electron Crystallography Workshop C-CINA, Basel, 1-7 Aug 2010 m.landsberg@uq.edu.au Averaging Why single molecule EM techniques are far superior in resolution
More informationCrystallographic Point Groups and Space Groups
Crystallographic Point Groups and Space Groups Physics 251 Spring 2011 Matt Wittmann University of California Santa Cruz June 8, 2011 Mathematical description of a crystal Definition A Bravais lattice
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 informationAxial Ratios, Parameters, Miller Indices
Page 1 of 7 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Axial Ratios, Parameters, Miller Indices This document last updated on 07-Sep-2016 We've now seen how crystallographic axes can
More information- A general combined symmetry operation, can be symbolized by β t. (SEITZ operator)
SPACE GROUP THEORY (cont) It is possible to represent combined rotational and translational symmetry operations in a single matrix, for example the C6z operation and translation by a in D 6h is represented
More informationHelpful resources for all X ray lectures Crystallization http://www.hamptonresearch.com under tech support: crystal growth 101 literature Spacegroup tables http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Crystallography
More informationChapter 4. Crystallography. 4.1 The crystalline state
Crystallography Atoms form bonds which attract them to one another. When you put many atoms together and they form bonds amongst themselves, are there any rules as to how they order themselves? Can we
More informationPhysical Chemistry I. Crystal Structure
Physical Chemistry I Crystal Structure Crystal Structure Introduction Crystal Lattice Bravis Lattices Crytal Planes, Miller indices Distances between planes Diffraction patters Bragg s law X-ray radiation
More informationSymmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES
Symmetry mode analysis in the Bilbao Crystallographic Server: The program AMPLIMODES Acknowledgements: Bilbao: M. Aroyo, D. Orobengoa, J.M. Igartua Grenoble: J. Rodriguez-Carvajal Provo, Utah: H.T. Stokes,
More informationPhys 460 Describing and Classifying Crystal Lattices
Phys 460 Describing and Classifying Crystal Lattices What is a material? ^ crystalline Regular lattice of atoms Each atom has a positively charged nucleus surrounded by negative electrons Electrons are
More informationThere are 230 spacespace-groups!
Xefg Symmetry for characteristic directions (dependent on crystal system) Non-symmorfe space groups (157 groups) Symmetry operations with translation: Screw-axis nm, 21,63, etc. Glideplane a,b,c,n,d Symmorfe
More informationWednesday, April 12. Today:
Wednesday, April 2 Last Time: - The solid state - atomic arrangement in solids - why do solids form: energetics - Lattices, translations, rotation, & other symmetry operations Today: Continue with lattices,
More informationAnalytical Methods for Materials
Analytical Methods for Materials Lesson 11 Crystallography and Crystal Structures, Part 3 Suggested Reading Chapter 6 in Waseda Chapter 1 in F.D. Bloss, Crystallography and Crystal Chemistry: An Introduction,
More informationMineralogy ( ) Chapter 5: Crystallography
Hashemite University Faculty of Natural Resources and Environment Department of earth and environmental sciences Mineralogy (1201220) Chapter 5: Crystallography Dr. Faten Al-Slaty First Semester 2015/2016
More informationLecture 2 Symmetry in the solid state -
Lecture 2 Symmetry in the solid state - Part II: Crystallographic coordinates and Space Groups. 1 Coordinate systems in crystallography and the mathematical form of the symmetry operators 1.1 Introduction
More informationFundamentals. Crystal patterns and crystal structures. Lattices, their symmetry and related basic concepts
Fundamentals. Crystal patterns and crystal structures. Lattices, their symmetry and related basic concepts Didactic material for the MaThCryst schools, France massimo.nespolo@univ-lorraine.fr Ideal vs.
More informationWe need to be able to describe planes and directions.
We need to be able to describe planes and directions. Miller Indices & XRD 1 2 Determining crystal structure and identifying materials (B) Plastic deformation Plastic deformation and mechanical properties
More informationPOINT SYMMETRY AND TYPES OF CRYSTAL LATTICE
POINT SYMMETRY AND TYPES OF CRYSTAL LATTICE Abdul Rashid Mirza Associate Professor of Physics. Govt. College of Science, wahdatroad, Lahore. 1 WHAT ARE CRYSTALS? The word crystal means icy or frozen water.
More informationResolution of Ambiguities and the Discovery of
ISST Journal of Applied hysics, Vol. 6 No. 1, (January - June), p.p. 1-10 ISSN No. 0976-90X Intellectuals Society for Socio-Techno Welfare Resolution of Ambiguities and the Discovery of Two New Space Lattices
More information5 Symmetries and point group in a nut shell
30 Phys520.nb 5 Symmetries and point group in a nut shell 5.1. Basic ideas: 5.1.1. Symmetry operations Symmetry: A system remains invariant under certain operation. These operations are called symmetry
More informationIntroduction to. Crystallography
M. MORALES Introuction to Crystallography magali.morales@ensicaen.fr Classification of the matter in 3 states : Crystallise soli liqui or amorphous gaz soli Crystallise soli : unique arrangement of atoms
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 informationThe Reciprocal Lattice
59-553 The Reciprocal Lattice 61 Because of the reciprocal nature of d spacings and θ from Bragg s Law, the pattern of the diffraction we observe can be related to the crystal lattice by a mathematical
More informationCRYSTALLOGRAPHIC SYMMETRY OPERATIONS. Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain
CRYSTALLOGRAPHIC SYMMETRY OPERATIONS Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION Mappings and symmetry operations Definition: A mapping
More informationTHE BILBAO CRYSTALLOGRAPHIC SERVER EXERCISES
INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY Volume A: Space-group Symmetry Volume A1: Symmetry Relations between Space Groups THE BILBAO CRYSTALLOGRAPHIC SERVER EXERCISES Mois I. Aroyo Departamento Física
More informationCrystal Structure SOLID STATE PHYSICS. Lecture 5. A.H. Harker. thelecture thenextlecture. Physics and Astronomy UCL
Crystal Structure thelecture thenextlecture SOLID STATE PHYSICS Lecture 5 A.H. Harker Physics and Astronomy UCL Structure & Diffraction Crystal Diffraction (continued) 2.4 Experimental Methods Notes: examples
More informationRoger Johnson Structure and Dynamics: The 230 space groups Lecture 3
Roger Johnson Structure and Dnamics: The 23 space groups Lecture 3 3.1. Summar In the first two lectures we considered the structure and dnamics of single molecules. In this lecture we turn our attention
More informationINTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY AND WORKSHOP ON STRUCTURAL PHASE TRANSITIONS. 30 August - 4 September 2017
INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY AND WORKSHOP ON STRUCTURAL PHASE TRANSITIONS 30 August - 4 September 2017 ROURKELA INTERNATIONAL CRYSTALLOGRAPHY SCHOOL BILBAO CRYSTALLOGRAPHIC SERVER
More informationMATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY. (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain
MATRIX CALCULUS APPLIED TO CRYSTALLOGRAPHY (short revision) Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain INTRODUCTION TO MATRIX CALCULUS Some of the slides are taken from the presentation Introduction
More informationChem 728 Introduction to Solid Surfaces
Chem 728 Introduction to Solid Surfaces Solids: hard; fracture; not compressible; molecules close to each other Liquids: molecules mobile, but quite close to each other Gases: molecules very mobile; compressible
More informationLecture Note on Crystal structures Masatsugu Sei Suzuki and Itsuko S. Suzuki Department of Physics, SUNY at Binghamton (Date: February 03, 2012)
Lecture Note on Crystal structures Masatsugu Sei Suzuki and Itsuko S. Suzuki Department of Physics, SUNY at Binghamton (Date: February 03, 2012) This is a part of lecture note on solid state physics (Phys.472/572)
More informationDirections Within The Unit Cell NCSU
Smmetr In Crstalline Materials SiO 2 1) Draw the unit cell, and label the internal rotational/ mirror/inversion smmetries. C 2 2) Show all positions of the molecule generated b smmetr (out to 4 unit cells).
More informationThe structure of planar defects in tilted perovskites
The structure of planar defects in tilted perovskites RichardBeanland Department of Physics, University of Warwick, Coventry, CV4 7AL, UK. E-mail: r.beanland@warwick.ac.uk Synopsis Continuity of octahedral
More informationLattices and Symmetry Scattering and Diffraction (Physics)
Lattices and Symmetry Scattering and Diffraction (Physics) James A. Kaduk INEOS Technologies Analytical Science Research Services Naperville IL 60566 James.Kaduk@innovene.com 1 Harry Potter and the Sorcerer
More informationInorganic materials chemistry and functional materials
Chemical bonding Inorganic materials chemistry and functional materials Helmer Fjellvåg and Anja Olafsen Sjåstad Lectures at CUTN spring 2016 CRYSTALLOGRAPHY - SYMMETRY Symmetry NATURE IS BEAUTIFUL The
More informationWALLPAPER GROUPS. Julija Zavadlav
WALLPAPER GROUPS Julija Zavadlav Abstract In this paper we present the wallpaper groups or plane crystallographic groups. The name wallpaper groups refers to the symmetry group of periodic pattern in two
More informationIntroduction to Materials Science Graduate students (Applied Physics)
Introduction to Materials Science Graduate students (Applied Physics) Prof. Michael Roth Chapter 1 Crystallography Overview Performance in Engineering Components Properties Mechanical, Electrical, Thermal
More informationEarth Materials Lab 2 - Lattices and the Unit Cell
Earth Materials Lab 2 - Lattices and the Unit Cell Unit Cell Minerals are crystallographic solids and therefore are made of atoms arranged into lattices. The average size hand specimen is made of more
More informationSymmetry considerations in structural phase transitions
EPJ Web of Conferences 22, 00008 (2012) DOI: 10.1051/epjconf/20122200008 C Owned by the authors, published by EDP Sciences, 2012 Symmetry considerations in structural phase transitions J.M. Perez-Mato,
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 informationCondensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras
Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture - 03 Symmetry in Perfect Solids Worked Examples Stated without prove to be in the lecture.
More informationUnderstanding Single-Crystal X-Ray Crystallography Exercises and Solutions
Understanding Single-Crystal X-Ray Crystallography Exercises and Solutions Dennis W. Bennett Department of Chemistry and Biochemistry University of Wisconsin-Milwaukee Chapter Crystal Lattices. The copper
More informationNMR Shifts. I Introduction and tensor/crystal symmetry.
NMR Shifts. I Introduction and tensor/crystal symmetry. These notes were developed for my group as introduction to NMR shifts and notation. 1) Basic shift definitions and notation: For nonmagnetic materials,
More information2. Diffraction as a means to determine crystal structure
2. Diffraction as a means to determine crystal structure Recall de Broglie matter waves: He atoms: [E (ev)] 1/2 = 0.14 / (Å) E 1Å = 0.0196 ev Neutrons: [E (ev)] 1/2 = 0.28 / (Å) E 1Å = 0.0784 ev Electrons:
More informationIntroduction to point and space group symmetry
Workshop on Electron Crystallography, Nelson Mandela Metropolitan University, South Africa, October 14-16, 2013 Introduction to point and space group syetry Hol Kirse Huboldt-Universität zu Berlin, Institut
More informationCHEM-E5225 :Electron Microscopy. Diffraction 1
CHEM-E5225 :Electron Microscopy Diffraction 1 2018-10-15 Yanling Ge Text book: Transmission electron microscopy by David B Williams & C. Barry Carter. 2009, Springer Outline Diffraction in TEM Thinking
More informationFinite Symmetry Elements and Crystallographic Point Groups
Chapter 2 Finite Symmetry Elements and Crystallographic Point Groups In addition to simple translations, which are important for understanding the concept of the lattice, other types of symmetry may be,
More information2. Diffraction as a means to determine crystal structure
Page 1 of 22 2. Diffraction as a means to determine crystal structure Recall de Broglie matter waves: 2 p h E = where p = 2m λ h 1 E = ( ) 2m λ hc E = hυ = ( photons) λ ( matter wave) He atoms: [E (ev)]
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 informationISOTROPY. Tutorial. July 2013
ISOTROPY Tutorial July 2013 Harold T. Stokes, Dorian M. Hatch, and Branton J. Campbell Department of Physics and Astronomy Brigham Young University with a contribution by Christopher J. Howard University
More informationX-ray, Neutron and e-beam scattering
X-ray, Neutron and e-beam scattering Introduction Why scattering? Diffraction basics Neutrons and x-rays Techniques Direct and reciprocal space Single crystals Powders CaFe 2 As 2 an example What is the
More informationBasic Crystallography Part 1. Theory and Practice of X-ray Crystal Structure Determination
Basic Crystallography Part 1 Theory and Practice of X-ray Crystal Structure Determination Course Overview Basic Crystallography Part 1 n Introduction: Crystals and Crystallography n Crystal Lattices and
More informationM\1any arguments have been concerned with what these symbols mean, and how they
SOME DESIRABLE MODIFICATIONS OF THE INTERNATIONAL SYMMETRY SYMBOLS* BY MARTIN J. BUERGER MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated August 21, 1967 With the publication of Hilton's Mathematical
More informationPreparation for the workshop. Part 1. Introduction to the Java applet
ISODISPLACE Workshop Harold T. Stokes International School in the Use and Applications of the Bilbao Crystallographic Server June 2009, Lekeitio, Spain isodisplace is a tool for exploring the structural
More informationGEOL. 40 ELEMENTARY MINERALOGY
CRYSTAL DESCRIPTION AND CALCULATION A. INTRODUCTION This exercise develops the framework necessary for describing a crystal. In essence we shall discuss how we fix the position of any crystallographic
More informationGroup theoretical analysis of structural instability, vacancy ordering and magnetic transitions in the system troilite (FeS) pyrrhotite (Fe1 xs)
Group theoretical analysis of structural instability, vacancy ordering and magnetic transitions in the system troilite (FeS) pyrrhotite (Fe1 xs) Authors Charles Robert Sebastian Haines a *, Christopher
More information