DOCSLIB.ORG

Space-Group Symmetry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Space-Group Symmetry SPACE GROUPS International Tables for Crystallography, Volume A: Space-group Symmetry Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain Bilbao Crystallographic Server http://www.cryst.ehu.es C´esar Capillas, UPV/EHU 1 SPACE GROUPS Crystal pattern: A model of the ideal crystal (crystal structure) in point space consisting of a strictly 3-dimensional periodic set of points Space group G: The set of all symmetry operations (isometries) of a crystal pattern Translation subgroup H G: The infinite set of all translations that are symmetry operations of the crystal pattern Point group of the The factor group of the space group G with respect to the translation space groups PG: subgroup T: PG ≅ G/H INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY VOLUME A: SPACE-GROUP SYMMETRY Extensive tabulations and illustrations of the 17 plane groups and of the 230 space groups •headline with the relevant group symbols; •diagrams of the symmetry elements and of the general position; •specification of the origin and the asymmetric unit; •list of symmetry operations; •generators; •general and special positions with multiplicities, site symmetries, coordinates and reflection conditions; •symmetries of special projections; GENERAL LAYOUT: LEFT-HAND PAGE General Layout: Right-hand page Primitive and centred lattice basis in 2D {a1, a2}: two translation vectors, linearly independent, form a lattice basis Primitive basis: If all lattice vectors are expressed as integer linear combinations of the basis vectors If some lattice vectors are expressed as Centred basis: linear combinations of the basis vectors with rational, non-integer coefficients Conventional centred basis (2D): c Number of lattice points per primitive and centred cells Crystal families, crystal systems, conventional coordinate system and Bravais lattices in 2D 3D-unit cell and lattice parameters lattice basis: {a, b, c} C B unit cell: the parallelepiped A defined by the basis vectors primitive P and centred unit cells: A,B,C,F, I, R number of lattice points per unit cell Crystal families, crystal systems, lattice systems and Bravais lattices in 3D HEADLINE BLOCK Short Hermann- Crystal class Mauguin symbol Schoenflies (point group) Crystal symbol system Number of Full Hermann- Patterson space group Mauguin symbol symmetry HERMANN-MAUGUIN SYMBOLISM FOR SPACE GROUPS Hermann-Mauguin symbols for space groups The Hermann–Mauguin symbol for a space group consists of a sequence of letters and numbers, here called the constituents of the HM symbol. (i) The first constituent is always a symbol for the conventional cell of the translation lattice of the space group (ii) The second part of the full HM symbol of a space group consists of one position for each of up to three representative symmetry directions. To each position belong the generating symmetry operations of their representative symmetry direction. The position is thus occupied either by a rotation, screw rotation or rotoinversion and/ or by a reflection or glide reflection. (iii) Simplest-operation rule: pure rotations > screw rotations; pure rotations > rotoinversions ‘>’ means ‘has priority’ reflection m > a; b; c > n 14 Bravais Lattices crystal family Symmetry directions A direction is called a symmetry direction of a crystal structure if it is parallel to an axis of rotation, screw rotation or rotoinversion or if it is parallel to the normal of a reflection or glide-reflection plane. A symmetry direction is thus the direction of the geometric element of a symmetry operation, when the normal of a symmetry plane is used for the description of its orientation. Hermann-Mauguin symbols for space groups Example: Hermann-Mauguin symbols for space groups secondary tertiary primary direction direction direction PRESENTATION OF SPACE-GROUP SYMMETRY OPERATIONS IN INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY, VOL. A Crystallographic symmetry operations characteristics: fixed points of isometries (W,w)Xf=Xf geometric elements Types of isometries preserve handedness identity: the whole space fixed translation t: no fixed point x˜ = x + t ◦ rotation: one line fixed φ = k × 360 /N rotation axis screw rotation: no fixed point screw axis screw vector do not Types of isometries preserve handedness centre of roto-inversion fixed roto-inversion: roto-inversion axis inversion: centre of inversion fixed reflection: plane fixed reflection/mirror plane glide reflection: no fixed point glide plane glide vector Matrix formalism linear/matrix translation part column part matrix-column Seitz symbol pair Space group Cmm2 (No. 35) How are the symmetry operations Diagram of symmetry elements represented in ITA ? 0 b a Diagram of general position points General Position General position ~ (i) coordinate triplets of an image point X of the original point X= x under (W,w) of G y z ~ -presentation of infinite image points X under the action of (W,w) of G (ii) short-hand notation of the matrix-column pairs (W,w) of the symmetry operations of G -presentation of infinite symmetry operations of G (W,w) = (I,tn)(W,w0), 0≤wi0<1 Space Groups: infinite order General position Coset decomposition G:TG (I,0) (W2,w2) ... (Wm,wm) ... (Wi,wi) (I,t1) (W2,w2+t1) ... (Wm,wm+t1) ... (Wi,wi+t1) (I,t2) (W2,w2+t2) ... (Wm,wm+t2) ... (Wi,wi+t2) ... ... ... ... ... ... (I,tj) (W2,w2+tj) ... (Wm,wm+tj) ... (Wi,wi+tj) ... ... ... ... ... ... Factor group G/TG isomorphic to the point group PG of G Point group PG = {I, W2, W 3,…,Wi} Example: P12/m1 Coset decomposition G:TG Point group PG = {1, 2, 1, m} General position ¯ TG TG 2 TG 1 TG m (I,0) (2,0) ( 1¯ ,0) (m,0) (I,t1) (2,t1) (1¯ , t1) (m, t1) (I,t2) (2,t2) (1¯ , t2) (m,t2) ... ... ... ... (I,tj) (2,tj) (1¯ , tj) (m, tj) ... ... ... ... -1 n1 n1/2 at 1 n2/2 -1 n2 n3/2 inversion centres (1,t): -1 n3 EXAMPLE Coset decomposition P121/c1:T Point group ? General position (I,0) (2,0 ½ ½) (1,0) (m,0 ½ ½) (I,t1) (2,0 ½ ½+t1) (1 ,t1) (m,0 ½ ½ +t1) (I,t2) (2,0 ½ ½ +t2) (1 ,t2) (m,0 ½ ½ +t2) ... ... ... ... ... ... (I,tj) (2,0 ½ ½ +tj) (1 ,tj) (m,0 ½ ½ +tj) ... ... ... ... ... ... inversion centers (1 ,p q r): 1 at p/2,q/2,r/2 (2,0 ½+v ½) 21screw axes (2,u ½+v ½ +w) (2,u ½ ½ +w) Symmetry Operations Block TYPE of the symmetry operation SCREW/GLIDE component ORIENTATION of the geometric element LOCATION of the geometric element GEOMETRIC INTERPRETATION OF THE MATRIX- COLUMN PRESENTATION OF THE SYMMETRY OPERATIONS Example: Cmm2 General position Diagram of symmetry elements 0 b TG TG 2 TG my TG mx (I,0) (2,0) (my,0) (mx,0) (I,t1) (2,t1) (my,t1) (mx, t 1) a (I,t2) (2,t2) (my,t2) (mx,t2) Diagram of ... ... ... ... general position points (I,tj) (2,tj) (my,tj) (mx, t j) Space group P21/c (No. 14) EXAMPLE Matrix-column presentation Geometric interpretation Bilbao Crystallographic Server Problem: Matrix-column presentation Geometrical interpretation GENPOS space group 14 Example GENPOS: Space group P21/c (14) Space-group symmetry operations short-hand notation matrix-column presentation Geometric interpretation Seitz symbols General positions ITA data SEITZ SYMBOLS FOR SYMMETRY OPERATIONS short-hand description of the matrix-column presentations of Seitz symbols { R | t } the symmetry operations of the space groups - specify the type and the order of the symmetry rotation (or linear) operation; part R - orientation of the symmetry element by the direction of the axis for rotations and rotoinversions, or the direction of the normal to reflection planes. 1 and 1 identity and inversion m reflections 2, 3, 4 and 6 rotations 3, 4 and 6 rotoinversions translation part t translation parts of the coordinate triplets of the General position blocks EXAMPLE Seitz symbols for symmetry operations of hexagonal and trigonal crystal systems Glazer et al. Acta Cryst A 70, 300 (2014) EXAMPLE Matrix-column presentation Geometric interpretation Seitz symbols (1) {1|0} (2) {2010|01/21/2 } (3) {1|0} (4) {m010|01/21/2} EXERCISES Problem 2.16 (b) 1. Characterize geometrically the matrix-column pairs listed under General position of the space group P4mm in ITA. 2. Consider the diagram of the symmetry elements of P4mm. Try to determine the matrix-column pairs of the symmetry operations whose symmetry elements are indicated on the unit-cell diagram. 3. Compare your results with the results of the program SYMMETRY OPERATIONS SPACE-GROUPS DIAGRAMS Diagrams of symmetry elements 0 b c 0 three different settings a a permutations c of a,b,c 0 b Diagram of general position points Diagram of symmetry elements Space group Cmm2 (No. 35) 0 b conventional setting a General Position How many general position points per unit cell are there? Diagram of general position points EXAMPLE Space group Cmm2 (No. 35) Geometric interpretation glide plane, t=1/2a glide plane, t=1/2b 0 b at y=1/4, ⊥b at x=1/4, ⊥a Matrix-column General Position presentation of symmetry a operations x+1/2,-y+1/2,z -x+1/2,y+1/2,z Example: P4mm Diagram of symmetry Diagram of general elements position points ⎫ ⎭ ⎬ Symmetry elements Geometric Fixed points Symmetry element } elements } + Symmetry operations Element set that share the same geometric element Examples line All rotations and screw rotations with the same axis, the same Rotation axis st th } 1 , ..., (n-1) powers + angle and sense of rotation and all coaxial equivalents the same screw vector (zero for rotation) up to a lattice translation vector. plane Glide plane All glide reflections with the same } defining operation+ reflection plane, with glide of d.o. all coplanar equivalents (taken to be zero for reflections) by a lattice translation vector. Symmetry operations and symmetry elements Geometric elements and Element sets P. M. de Wolff et al. Acta Cryst (1992) A48 727 Diagram of symmetry elements Example: P4mm Element set of (00z) line A l l r o t a t i o n s a n d s c r e w Symmetry operations 1st, 2 nd, 3 rd powers + rotations with the same axis, that share (0,0,z) as all coaxial equivalents geometric element } the same angle and sense of rotation and the same screw vector (zero for rotation) up to Element set of (0,0,z) line a lattice translation vector.
Load more

Recommended publications
  • Glide and Screw
    Space Groups •The 32 crystallographic point groups, whose operation have at least one point unchanged, are sufficient for the description of finite, macroscopic objects. •However since ideal crystals extend indefinitely in all directions, we must also include translations (the Bravais lattices) in our description of symmetry. Space groups: formed when combining a point symmetry group with a set of lattice translation vectors (the Bravais lattices), i.e. self-consistent set of symmetry operations acting on a Bravais lattice. (Space group lattice types and translations have no meaning in point group symmetry.) Space group numbers for all the crystal structures we have discussed this semester, and then some, are listed in DeGraef and Rohrer books and pdf. document on structures and AFLOW website, e.g. ZnS (zincblende) belongs to SG # 216: F43m) Class21/1 Screw Axes •The combination of point group symmetries and translations also leads to two additional operators known as glide and screw. •The screw operation is a combination of a rotation and a translation parallel to the rotation axis. •As for simple rotations, only diad, triad, tetrad and hexad axes, that are consistent with Bravais lattice translation vectors can be used for a screw operator. •In addition, the translation on each rotation must be a rational fraction of the entire translation. •There is no combination of rotations or translations that can transform the pattern produced by 31 to the pattern of 32 , and 41 to the pattern of 43, etc. •Thus, the screw operation results in handedness Class21/2 or chirality (can’t superimpose image on another, e.g., mirror image) to the pattern.
    [Show full text]
  • Group Theory Applied to Crystallography
    International Union of Crystallography Commission on Mathematical and Theoretical Crystallography Summer School on Mathematical and Theoretical Crystallography 27 April - 2 May 2008, Gargnano, Italy Group theory applied to crystallography Bernd Souvignier Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen The Netherlands 29 April 2008 2 CONTENTS Contents 1 Introduction 3 2 Elements of space groups 5 2.1 Linearmappings .................................. 5 2.2 Affinemappings................................... 8 2.3 AffinegroupandEuclideangroup . .... 9 2.4 Matrixnotation .................................. 12 3 Analysis of space groups 14 3.1 Lattices ....................................... 14 3.2 Pointgroups..................................... 17 3.3 Transformationtoalatticebasis . ....... 19 3.4 Systemsofnonprimitivetranslations . ......... 22 4 Construction of space groups 25 4.1 Shiftoforigin................................... 25 4.2 Determining systems of nonprimitivetranslations . ............. 27 4.3 Normalizeraction................................ .. 31 5 Space group classification 35 5.1 Spacegrouptypes................................. 35 5.2 Arithmeticclasses............................... ... 36 5.3 Bravaisflocks.................................... 37 5.4 Geometricclasses................................ .. 39 5.5 Latticesystems .................................. 41 5.6 Crystalsystems .................................. 41 5.7 Crystalfamilies ................................. .. 42 6 Site-symmetry
    [Show full text]
  • The Cubic Groups
    The Cubic Groups Baccalaureate Thesis in Electrical Engineering Author: Supervisor: Sana Zunic Dr. Wolfgang Herfort 0627758 Vienna University of Technology May 13, 2010 Contents 1 Concepts from Algebra 4 1.1 Groups . 4 1.2 Subgroups . 4 1.3 Actions . 5 2 Concepts from Crystallography 6 2.1 Space Groups and their Classification . 6 2.2 Motions in R3 ............................. 8 2.3 Cubic Lattices . 9 2.4 Space Groups with a Cubic Lattice . 10 3 The Octahedral Symmetry Groups 11 3.1 The Elements of O and Oh ..................... 11 3.2 A Presentation of Oh ......................... 14 3.3 The Subgroups of Oh ......................... 14 2 Abstract After introducing basics from (mathematical) crystallography we turn to the description of the octahedral symmetry groups { the symmetry group(s) of a cube. Preface The intention of this account is to provide a description of the octahedral sym- metry groups { symmetry group(s) of the cube. We first give the basic idea (without proofs) of mathematical crystallography, namely that the 219 space groups correspond to the 7 crystal systems. After this we come to describing cubic lattices { such ones that are built from \cubic cells". Finally, among the cubic lattices, we discuss briefly the ones on which O and Oh act. After this we provide lists of the elements and the subgroups of Oh. A presentation of Oh in terms of generators and relations { using the Dynkin diagram B3 is also given. It is our hope that this account is accessible to both { the mathematician and the engineer. The picture on the title page reflects Ha¨uy'sidea of crystal structure [4].
    [Show full text]
  • COXETER GROUPS (Unfinished and Comments Are Welcome)
    COXETER GROUPS (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] October 10, 2018 1 2 Contents Preface 4 1 Regular Polytopes 7 1.1 ConvexSets............................ 7 1.2 Examples of Regular Polytopes . 12 1.3 Classification of Regular Polytopes . 16 2 Finite Reflection Groups 21 2.1 NormalizedRootSystems . 21 2.2 The Dihedral Normalized Root System . 24 2.3 TheBasisofSimpleRoots. 25 2.4 The Classification of Elliptic Coxeter Diagrams . 27 2.5 TheCoxeterElement. 35 2.6 A Dihedral Subgroup of W ................... 39 2.7 IntegralRootSystems . 42 2.8 The Poincar´eDodecahedral Space . 46 3 Invariant Theory for Reflection Groups 53 3.1 Polynomial Invariant Theory . 53 3.2 TheChevalleyTheorem . 56 3.3 Exponential Invariant Theory . 60 4 Coxeter Groups 65 4.1 Generators and Relations . 65 4.2 TheTitsTheorem ........................ 69 4.3 The Dual Geometric Representation . 74 4.4 The Classification of Some Coxeter Diagrams . 77 4.5 AffineReflectionGroups. 86 4.6 Crystallography. .. .. .. .. .. .. .. 92 5 Hyperbolic Reflection Groups 97 5.1 HyperbolicSpace......................... 97 5.2 Hyperbolic Coxeter Groups . 100 5.3 Examples of Hyperbolic Coxeter Diagrams . 108 5.4 Hyperbolic reflection groups . 114 5.5 Lorentzian Lattices . 116 3 6 The Leech Lattice 125 6.1 ModularForms ..........................125 6.2 ATheoremofVenkov . 129 6.3 The Classification of Niemeier Lattices . 132 6.4 The Existence of the Leech Lattice . 133 6.5 ATheoremofConway . 135 6.6 TheCoveringRadiusofΛ . 137 6.7 Uniqueness of the Leech Lattice . 140 4 Preface Finite reflection groups are a central subject in mathematics with a long and rich history. The group of symmetries of a regular m-gon in the plane, that is the convex hull in the complex plane of the mth roots of unity, is the dihedral group of order 2m, which is the simplest example of a reflection Dm group.
    [Show full text]
  • 1 NANO 704-Crystallography & Structure of Nanomaterials 3
    1 NANO 704-Crystallography & Structure of Nanomaterials 3. Space Groups Space lattices Lattice points are all equivalent by translational symmetry. We start with a primitive lattice, having points at rabcuvw uvw, where uvw,, ¢ . So lattice points exist at (0,0,0) and all equivalent positions. If we have a lattice point at xyz,, , then we also have a lattice point at xuyvzw,, . Suppose two lattice points exist at xyz111,, and xyz222,, . If xyz,, is a lattice point, then xyz,, x212121 x , y y , z z is also a lattice point. But this does not imply that for all xyz,, representing lattice points, the values of xyz,, are integers. In particular, it is often useful to represent some of them by half integers. A primitive cell has lattice points at 0,0,0 . Centered cells have additional lattice points. 11 An A-centered cell also has points at 0,22 , . (Center of the A face.) 11 A B-centered cell also has points at 22,0, . (Center of the B face.) 11 A C-centered cell also has points at 22,,0. (Center of the C face.) 11 11 11 An F (face)-centered cell also has points at 0,22 , , 22,0, , 22,,0. (Centers of all three faces.) 111 An I (body)-centered cell also has points at 222,, . (Center point of the unit cell.) Observations I. Suppose a cell is both A- and B-centered. The lattice points exist at 11 11 P1 : 0,0,0 , P2 : 0,22 , , and P3 : 22,0, and equivalent positions. P1 and P2 form a lattice row.
    [Show full text]
  • Crystallographic Symmetry Operations
    CRYSTALLOGRAPHIC SYMMETRY OPERATIONS Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain miércoles, 9 de octubre de 13 Bilbao Crystallographic Server http://www.cryst.ehu.es C´esar Capillas, UPV/EHU 1 SYMMETRY OPERATIONS AND THEIR MATRIX-COLUMN PRESENTATION miércoles, 9 de octubre de 13 Mappings and symmetry operations Definition: A mapping of a set A into a set B is a relation such that for each element a ∈ A there is a unique element b ∈ B which is assigned to a. The element b is called the image of a. ! ! The relation of the point X to the points X 1 and X 2 is not a mapping because the image point is not uniquely defined (there are two image points). The five regions of the set A (the triangle) are mapped onto the five separated regions of the set B. No point of A is mapped onto more than one image point. Region 2 is mapped on a line, the points of the line are the images of more than one point of A. Such a mapping is called a projection. miércoles, 9 de octubre de 13 Mappings and symmetry operations Definition: A mapping of a set A into a set B is a relation such that for each element a ∈ A there is a unique element b ∈ B which is assigned to a. The element b is called the image of a. An isometry leaves all distances and angles invariant. An ‘isometry of the first kind’, preserving the counter–clockwise sequence of the edges ‘short–middle–long’ of the triangle is displayed in the upper mapping.
    [Show full text]
  • Symmetry in Reciprocal Space
    Symmetry in Reciprocal Space The diffraction pattern is always centrosymmetric (at least in good approximation). Friedel’s law: Ihkl = I-h-k-l. Fourfold symmetry in the diffraction pattern corresponds to a fourfold axis in the space group (4, 4, 41, 42 or 43), threefold to a threefold, etc. If you take away the translational part of the space group symmetry and add an inversion center, you end up with the Laue group. The Laue group describes the symmetry of the diffraction pattern. The Laue symmetry can be lower than the metric symmetry of the unit cell, but never higher. That means: A monoclinic crystal with β = 90° is still monoclinic. The diffraction pattern from such a crystal will have monoclinic symmetry, even though the metric symmetry of the unit cell looks orthorhombic. There are 11 Laue groups: -1, 2/m, mmm, 4/m, 4/mmm, -3, -3/m, 6/m, 6/mmm, m3, m3m Laue Symmetry Crystal System Laue Group Point Group Triclinic -1 1, -1 Monoclinic 2/m 2, m, 2/m Orthorhombic mmm 222, mm2, mmm 4/m 4, -4, 4/m Tetragonal 4/mmm 422, 4mm, -42m, 4/mmm -3 3, -3 Trigonal/ Rhombohedral -3/m 32, 3m, -3m 6/m 6, -6, 6/m Hexagonal 6/mmm 622, 6mm, -6m2, 6/mmm m3 23, m3 Cubic m3m 432, -43m, m3m Space Group Determination The first step in the determination of a crystal structure is the determination of the unit cell from the diffraction pattern. Second step: Space group determination. From the symmetry of the diffraction pattern, we can determine the Laue group, which narrows down the choice quite considerably.
    [Show full text]
  • Group Theory 1 – Basic Principles
    Topological and Symmetry-Broken Phases in Physics and Chemistry – International Theoretical Basics and Phenomena Ranging from Crystals and Molecules Summer School to Majorana Fermions, Neutrinos and Cosmic Phase Transitions 2017 Group Theory 1 – Basic Principles Group Theory 2 & 3 – Group Theory in Crystallography TUTORIAL: Apply Crystallographic Group Theory to a Phase Transition Group Theory 4 – Applications in Crystallography and Solid State Chemistry Prof. Dr. Holger Kohlmann Inorganic Chemistry – Functional Materials University Leipzig, Germany [email protected] ©Holger Kohlmann, Leipzig University 1 Topological and Symmetry-Broken Phases in Physics and Chemistry – International Theoretical Basics and Phenomena Ranging from Crystals and Molecules Summer School to Majorana Fermions, Neutrinos and Cosmic Phase Transitions 2017 Group Theory 1 – Basic principles 1.1 Basic notions, group axioms and examples of groups 1.2 Classification of the group elements and subgroups Group Theory 2 & 3 – Group theory in crystallography 2 From point groups to space groups – a brief introduction to crystallography 3.1 Crystallographic group-subgroup relationships 3.2 Examples of phase transitions in chemistry TUTORIAL: Apply crystallographic group theory to a phase transition Group Theory 4 – Applications in crystallography and solid state chemistry 4.1 The relation between crystal structures and family trees 4.2 Complex cases of phase transitions and topotactic reactions ©Holger Kohlmann, Leipzig University 2 Topological and Symmetry-Broken Phases in Physics and Chemistry – International Theoretical Basics and Phenomena Ranging from Crystals and Molecules Summer School to Majorana Fermions, Neutrinos and Cosmic Phase Transitions 2017 Ressources • T. Hahn, H. Wondratschek, Symmetry of Crystals, Heron Press, Sofia, Bulgaria, 1994 • International Tables for Crystallography, Vol.
    [Show full text]
  • Crystallography: Symmetry Groups and Group Representations B
    EPJ Web of Conferences 22, 00006 (2012) DOI: 10.1051/epjconf/20122200006 C Owned by the authors, published by EDP Sciences, 2012 Crystallography: Symmetry groups and group representations B. Grenier1 and R. Ballou2 1SPSMS, UMR-E 9001, CEA-INAC / UJF-Grenoble, MDN, 38054 Grenoble, France 2Institut Néel, CNRS / UJF, 25 rue des Martyrs, BP. 166, 38042 Grenoble Cedex 9, France Abstract. This lecture is aimed at giving a sufficient background on crystallography, as a reminder to ease the reading of the forthcoming chapters. It more precisely recalls the crystallographic restrictions on the space isometries, enumerates the point groups and the crystal lattices consistent with these, examines the structure of the space group, which gathers all the spatial invariances of a crystal, and describes a few dual notions. It next attempts to familiarize us with the representation analysis of physical states and excitations of crystals. 1. INTRODUCTION Crystallography covers a wide spectrum of investigations: i- it aspires to get an insight into crystallization phenomena and develops methods of crystal growths, which generally pertains to the physics of non linear irreversible processes; ii- it geometrically describes the natural shapes and the internal structures of the crystals, which is carried out most conveniently by borrowing mathematical tools from group theory; iii- it investigates the crystallized matter at the atomic scale by means of diffraction techniques using X-rays, electrons or neutrons, which are interpreted in the dual context of the reciprocal space and transposition therein of the crystal symmetries; iv- it analyzes the imperfections of the crystals, often directly visualized in scanning electron, tunneling or force microscopies, which in some instances find a meaning by handling unfamiliar concepts from homotopy theory; v- it aims at providing means for discerning the influences of the crystal structure on the physical properties of the materials, which requires to make use of mathematical methods from representation theory.
    [Show full text]
  • Structure and Symmetry of Cus, (Pyrite Structure)
    American Mineralogist, Volume 64, pages 1265-1271, 1979 Structure and symmetryof CuS, (pyrite structure) Husnnt E. KING, JR.' eNo Cnenres T. PREWITT Departmentof Earth and SpaceSciences State Universityof New York StonyBrook, New York 11794 Abstract X-ray diffraction data collectedon a single-crystalspecimen of CuS2show that despiteits optical anisotropyCuS, apparentlyhas the cubic pyrite structure,with a: 5.7891(6)A.pre- cessionand Weissenbergphotographs fail to reveal any reflections which violate the require- ments for space group Pa3. Such reflections, however, Wereobserved in four-circle di-ffrac- tometer measurements,but they are shown to result from multiple di-ffraction,effects. Reflement of the structurein spacegroup Pa3 using 209 intensity data givesa weightedre- sidual of 0.014and x(S) : 0.39878(5).A comparisonof the refined structurewith other pyrite structuressuggests that copper in CuSz has a formal valenceof 2+ and three antibonding electrons.Also, the CuSeoctahedron is only slightly distorted,which is in contrastwith the square-planarcoordination usually found for Cu2*. Introduction spect to the X-ray diffraction studies on FeS'. Fin- Disulfides of the transition elementsMn through klea et al. (1976) found no deviations from cubic Zn crystalllz,ein the pyrite structure. The Mn, Fe, symmetry,but Bayliss (1977)concluded that at least Co, and Ni membersof this group occur as minerals, sone pyrite crystalsare triclinic. Becauseoptical ani- and their structures have been refined. CuS, and sotropy has always been observedfor CuS, we de- ZnS.' are not found in nature, but they have been cided to investigate its crystal structure to provide synthesizedat high temperaturesand pressures.This further information on this intriguing problem.
    [Show full text]
  • 2.3 Band Structure and Lattice Symmetries: Example of Diamond
    2.2.9 Product of representaitons Besides the sums of representations, one can also define their products. Consider two groups G and H and their direct product G × H. If we have two representations D1 and D2 of G and H respectively, we may define their product as D(g · h) = D1(g) ⊗ D2(h) ; (1) where ⊗ is the tensor product of matrices: (A ⊗ B)ij;kl = AikBjl. The dimension of such a representation equals the product of the dimensions of D1 and D2, and its character is given by the product of the two characters. Note that if both D1 and D2 are irreducible, then their product is also irreducible (as a representation of G × H). In other words, if a group is a direct product of two groups, then its table of irreducible representations can be obtained as the product of the tables of irreducible representations of its factors. One can also encounter a situation, where the two groups G and H are equal, and their product is also viewed as a representation of the same group: D(g) = D1(g) ⊗ D2(g) : (2) In this case, the product of two irreducible representations is not generally irreducible. For example, the product of two spin-1/2 representations of the rotation group is decomposed into a singlet and a triplet (which can be symbolically written as 2⊗2 = 1⊕3, if we mean by 1, 2, and 3 irreducible representations of SU(2) with the corresponding dimensions). 2.3 Band structure and lattice symmetries: example of diamond We now apply the general formalism developed in the last lecture to the example of the crystal structure of diamond.
    [Show full text]
  • Part 1: Supplementary Material Lectures 1-4
    Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2013 Part 1: Supplementary Material Lectures 1-4 Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option Contents 1 Frieze patterns and frieze groups 2 2 Symbols for frieze groups 3 2.1 A few new concepts from frieze groups . .4 2.2 Frieze groups in the ITC . .6 3 Wallpaper groups 8 3.1 A few new concepts for Wallpaper Groups . .8 3.2 Lattices and the “translation set” . .8 3.3 Bravais lattices in 2D . .9 3.3.1 Oblique system . .9 3.3.2 Rectangular system . .9 3.3.3 Square system . 10 3.3.4 Hexagonal system . 10 3.4 Primitive, asymmetric and conventional Unit cells in 2D . 10 3.5 The 17 wallpaper groups . 10 3.6 Analyzing wallpaper and other 2D art using wallpaper groups . 10 1 4 Point groups in 3D 12 4.1 The new generalized (proper & improper) rotations in 3D . 13 4.2 The 3D point groups with a 2D projection . 13 4.3 The other 3D point groups: the 5 cubic groups . 15 5 The 14 Bravais lattices in 3D 17 6 Notation for 3D point groups 19 6.0.1 Notation for ”projective” 3D point groups . 19 7 Glide planes in 3D 21 8 “Real” crystal structures 21 8.1 Cohesive forces in crystals — atomic radii . 21 8.2 Close-packed structures . 22 8.3 Packing spheres of different radii . 23 8.4 Framework structures . 24 8.5 Layered structures . 25 8.6 Molecular structures .
    [Show full text]