DOCSLIB.ORG

Symmetry and Groups, and Crystal Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetry and Groups, and Crystal Structures CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL STRUCTURES Sarah Lambart RECAP CHAP. 2 2 different types of close packing: hcp: tetrahedral interstice (ABABA) ccp: octahedral interstice (ABCABC) Definitions: The coordination number or CN is the number of closest neighbors of opposite charge around an ion. It can range from 2 to 12 in ionic structures. These structures are called coordination polyhedron. RECAP CHAP. 2 Rx/Rz C.N. Type Hexagonal or An ideal close-packing of sphere 1.0 12 Cubic for a given CN, can only be Closest Packing achieved for a specific ratio of 1.0 - 0.732 8 Cubic ionic radii between the anions and 0.732 - 0.414 6 Octahedral Tetrahedral (ex.: the cations. 0.414 - 0.225 4 4- SiO4 ) 0.225 - 0.155 3 Triangular <0.155 2 Linear RECAP CHAP. 2 Pauling’s rule: #1: the coodination polyhedron is defined by the ratio Rcation/Ranion #2: The Electrostatic Valency (e.v.) Principle: ev = Z/CN #3: Shared edges and faces of coordination polyhedra decreases the stability of the crystal. #4: In crystal with different cations, those of high valency and small CN tend not to share polyhedral elements #5: The principle of parsimony: The number of different sites in a crystal tends to be small. CONTENT CHAP. 3 (2-3 LECTURES) Definitions: unit cell and lattice 7 Crystal systems 14 Bravais lattices Element of symmetry CRYSTAL LATTICE IN TWO DIMENSIONS A crystal consists of atoms, molecules, or ions in a pattern that repeats in three dimensions. The geometry of the repeating pattern of a crystal can be described in terms of a crystal lattice, constructed by connecting equivalent points throughout the crystal. Step1: 2D crystal lattice CRYSTAL LATTICE IN TWO DIMENSIONS Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal. The environment about any lattice point is identical to the environment about any other lattice point. The choice of reference lattice point is arbitrary. One choice of a reference point CRYSTAL LATTICE IN TWO DIMENSIONS Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal. The basic parallelogram (parallelepiped in three dimensions) constructed by connecting lattice points defines a unit cell. A lattice constructed from the chosen point. CRYSTAL LATTICE IN TWO DIMENSIONS Lattice Points and Unit Cell Because the choice of reference lattice point is arbitrary, the location of the lattice relative to the contents of the unit cell is variable. Regardless of the reference point chosen, the unit cell contains the same number of atoms with the same geometrical arrangement. The same lattice and unit cell defined from a different reference point. CRYSTAL LATTICE IN TWO DIMENSIONS Unit Cell - The unit cell is the basic repeat unit from which the entire crystal can be built. - A primitive unit cell contains only one lattice point. The same lattice and unit cell defined from a different reference point. CRYSTAL LATTICE IN TWO DIMENSIONS Unit cell: The unit cell of a mineral is the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral. The unit cell is defined by three axes or cell edges, termed a, b, and c and three inter- axial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b. CRYSTAL LATTICE IN TWO DIMENSIONS Alternate lattice and choice of the unit cell CRYSTAL LATTICE IN TWO DIMENSIONS Rules to choose a unit cell Smallest repeat unit Highest possible symmetry (with the most 90° angles) CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS Unit cell: The unit cell of a mineral is the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral. The unit cell is defined by three axes or cell edges, termed a, b, and c and three inter- axial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b. CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered Each corner is F: Face-centered shared between 8 C: Side-centered cells CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered F: Face-centered C: Side-centered CRYSTAL LATTICE Unit cell in 3 D: 4 type of unit cells: P: primitive I: Body-centered Each face is shared F: Face-centered between 2 cells C: Side-centered 7 CRYSTAL SYSTEMSc c c b a2 a b 1 P or C a R Monoclinic Hexagonal Rhombohedral Triclinic a ο ο ο α ≠ β ≠ γ α = γ = 90 ≠ β α = β = 90 γ =120 α = β = γ ≠ 90ο a ≠ b ≠ c = ≠ a ≠ b ≠ c a1 a2 c a1 = a2 = a3 a 3 c c a2 a1 b a 2 a a1 Orthorhombic Tetragonal Isometric (or cubic) ο ο ο α = β = γ = 90 a ≠ b ≠ c α = β = γ = 90 a1 = a2 ≠ c α = β = γ = 90 a1 = a2 = a3 7 CRYSTAL SYSTEMSc c c b a2 a b 1 P or C a R Monoclinic Hexagonal Rhombohedral Triclinic a ο ο ο α ≠ β ≠ γ α = γ = 90 ≠ β α = β = 90 γ =120 α = β = γ ≠ 90ο a ≠ b ≠ c = ≠ a ≠ b ≠ c a1 a2 c a1 = a2 = a3 a 3 c c a2 a1 b a 2 a a1 Orthorhombic Tetragonal Isometric (or cubic) ο ο ο α = β = γ = 90 a ≠ b ≠ c α = β = γ = 90 a1 = a2 ≠ c α = β = γ = 90 a1 = a2 = a3 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES a 3 Isometric or cubic α = β = γ = 90 ο a1 = a2 = a3 a2 a1 P F I CN = 6 CN = 12 CN = 8 14 BRAVAIS LATTICES 14 BRAVAIS LATTICES Example: Sphalerite (ZnS) 14 BRAVAIS LATTICES Example: Sphalerite (ZnS) Basis: S2- + Zn2+ 14 BRAVAIS LATTICES Example: Halite (NaCl) 14 BRAVAIS LATTICES Example: Halite (NaCl) SYMMETRIES “… is the hardest thing for student to understand, appreciate or visualize.” SYMMETRIES 4-fold rotational symmetry. SYMMETRY OPERATIONS A Symmetry operation is an operation on an object that results in no change in the appearance of the object. There are 3 types of symmetry operations: rotation, reflection, and inversion. ROTATIONAL SYMMETRY 1 fold rotation axis = no rotational symmetry 1 1 2 fold rotation axis: identical after a rotation of 180° (360/180 = 2) 2 symbol: filled oval or A2 2 2 ROTATIONAL SYMMETRY 3 fold rotation axis = identical after a rotation of 120° (360/120 = 3) symbol: filled triangle or A 3 3 3 4 fold rotation axis: identical after a rotation of 90° (360/90 = 4) 4 4 symbol: filled square or A4 ROTATIONAL SYMMETRY 6 fold rotation axis = identical after a rotation of 60° (360/60 = 6) symbol: filled hexagon or A6 6 6 IMPROPER ROTATIONAL SYMMETRY 5 fold, 7 fold, 8 fold or higher: does not exist in crystals because cannot fill the space MIRROR SYMMETRY A mirror plan is something that gives you the reflection that exactly reflects the other side: same distance, same component. The plane of the mirror is an element of symmetry referred to as a mirror plane, and is symbolized with the letter m. MIRROR SYMMETRY MIRROR SYMMETRY MIRROR SYMMETRY CENTER OF SYMMETRY A center of symmetry is an inversion through a point, symbolized with the letter "i". ROTOINVERSIONS Combinations of rotation with a center of symmetry. __ 1 fold rotoinversion axis = A1 center of symmetry symbol: A1 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry m = mirror perpendicular to the axis symbol: A2 ROTOINVERSIONS 1) Rotation 360° 2) Inversion ROTOINVERSIONS Combinations of rotation with a center of symmetry. __ 1 fold rotoinversion axis = A1 center of symmetry symbol: A1 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry m = mirror perpendicular to the axis symbol: A2 ROTOINVERSIONS 1) Rotation 180° 2) Inversion ROTOINVERSIONS 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry symbol: A3 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry symbol: A4 ROTOINVERSIONS 1 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry symbol: A3 1 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry symbol: A4 ROTOINVERSIONS 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry 2 symbol: A3 4 fold rotoinversion axis = 1) 90° rotation, 2) center of symmetry 2 symbol: A4 ROTOINVERSIONS 6 fold rotoinversion axis = 1) 60° rotation, 2) center of symmetry = 3 fold rotation axis + 1 perpendicular mirror plan symbol: A6 CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS CRYSTAL LATTICE IN TWO DIMENSIONS COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes.
Load more

Recommended publications
  • In Order for the Figure to Map Onto Itself, the Line of Reflection Must Go Through the Center Point
    3-5 Symmetry State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. 1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has reflectional symmetry. In order for the figure to map onto itself, the line of reflection must go through the center point. Two lines of reflection go through the sides of the figure. Two lines of reflection go through the vertices of the figure. Thus, there are four possible lines that go through the center and are lines of reflections. Therefore, the figure has four lines of symmetry. eSolutionsANSWER: Manual - Powered by Cognero Page 1 yes; 4 2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. ANSWER: no 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure has reflectional symmetry. The figure has a vertical line of symmetry. It does not have a horizontal line of symmetry. The figure does not have a line of symmetry through the vertices. Thus, the figure has only one line of symmetry. ANSWER: yes; 1 State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.
    [Show full text]
  • Crystal Structure of a Material Is Way in Which Atoms, Ions, Molecules Are Spatially Arranged in 3-D Space
    Crystalline Structures – The Basics •Crystal structure of a material is way in which atoms, ions, molecules are spatially arranged in 3-D space. •Crystal structure = lattice (unit cell geometry) + basis (atom, ion, or molecule positions placed on lattice points within the unit cell). •A lattice is used in context when describing crystalline structures, means a 3-D array of points in space. Every lattice point must have identical surroundings. •Unit cell: smallest repetitive volume •Each crystal structure is built by stacking which contains the complete lattice unit cells and placing objects (motifs, pattern of a crystal. A unit cell is chosen basis) on the lattice points: to represent the highest level of geometric symmetry of the crystal structure. It’s the basic structural unit or building block of crystal structure. 7 crystal systems in 3-D 14 crystal lattices in 3-D a, b, and c are the lattice constants 1 a, b, g are the interaxial angles Metallic Crystal Structures (the simplest) •Recall, that a) coulombic attraction between delocalized valence electrons and positively charged cores is isotropic (non-directional), b) typically, only one element is present, so all atomic radii are the same, c) nearest neighbor distances tend to be small, and d) electron cloud shields cores from each other. •For these reasons, metallic bonding leads to close packed, dense crystal structures that maximize space filling and coordination number (number of nearest neighbors). •Most elemental metals crystallize in the FCC (face-centered cubic), BCC (body-centered cubic, or HCP (hexagonal close packed) structures: Room temperature crystal structure Crystal structure just before it melts 2 Recall: Simple Cubic (SC) Structure • Rare due to low packing density (only a-Po has this structure) • Close-packed directions are cube edges.
    [Show full text]
  • Types of Lattices
    Types of Lattices Types of Lattices Lattices are either: 1. Primitive (or Simple): one lattice point per unit cell. 2. Non-primitive, (or Multiple) e.g. double, triple, etc.: more than one lattice point per unit cell. Double r2 cell r1 r2 Triple r1 cell r2 r1 Primitive cell N + e 4 Ne = number of lattice points on cell edges (shared by 4 cells) •When repeated by successive translations e =edge reproduce periodic pattern. •Multiple cells are usually selected to make obvious the higher symmetry (usually rotational symmetry) that is possessed by the 1 lattice, which may not be immediately evident from primitive cell. Lattice Points- Review 2 Arrangement of Lattice Points 3 Arrangement of Lattice Points (continued) •These are known as the basis vectors, which we will come back to. •These are not translation vectors (R) since they have non- integer values. The complexity of the system depends upon the symmetry requirements (is it lost or maintained?) by applying the symmetry operations (rotation, reflection, inversion and translation). 4 The Five 2-D Bravais Lattices •From the previous definitions of the four 2-D and seven 3-D crystal systems, we know that there are four and seven primitive unit cells (with 1 lattice point/unit cell), respectively. •We can then ask: can we add additional lattice points to the primitive lattices (or nets), in such a way that we still have a lattice (net) belonging to the same crystal system (with symmetry requirements)? •First illustrate this for 2-D nets, where we know that the surroundings of each lattice point must be identical.
    [Show full text]
  • The Metrical Matrix in Teaching Mineralogy G. V
    THE METRICAL MATRIX IN TEACHING MINERALOGY G. V. GIBBS Department of Geological Sciences & Department of Materials Science and Engineering Virginia Polytechnic Institute and State University Blacksburg, VA 24061 [email protected] INTRODUCTION The calculation of the d-spacings, the angles between planes and zones, the bond lengths and angles and other important geometric relationships for a mineral can be a tedious task both for the student and the instructor, particularly when completed with the large as- sortment of trigonometric identities and algebraic formulae that are available (d. Crystal Geometry (1959), Donnay and Donnay, International Tables for Crystallography, Vol. II, Section 3, The Kynoch Press, 101-158). However, such calculations are straightforward and relatively easy to do when completed with the metrical matrix and the interactive software MATOP. Several applications of the matrix are presented below, each of which is worked out in detail and which is designed to teach you its use in the study of crystal geometry. SOME PRELIMINARY COMMENTS We begin our discussion of the matrix with a brief examination of the properties of the geometric three dimensional space, S, in which we live and in which minerals and rocks occur. For our purposes, it will be convenient to view S as the set of all vectors that radiate from a common origin to each point in space. In constructing a model for S, we chose three noncoplanar, coordinate axes denoted X, Y and Z, each radiating from the origin, O. Next, we place three nonzero vectors denoted a, band c along X, Y and Z, respectively, likewise radiating from O.
    [Show full text]
  • Primitive Cell Wigner-Seitz Cell (WS) Primitive Cell
    Lecture 4 Jan 16 2013 Primitive cell Primitive cell Wigner-Seitz cell (WS) First Brillouin zone The Wigner-Seitz primitive cell of the reciprocal lattice is known as the first Brillouin zone. (Wigner-Seitz is real space concept while Brillouin zone is a reciprocal space idea). Powder cell Polymorphic Forms of Carbon Graphite – a soft, black, flaky solid, with a layered structure – parallel hexagonal arrays of carbon atoms – weak van der Waal’s forces between layers – planes slide easily over one another Miller indices Simple cubic Miller Indices Rules for determining Miller Indices: 1. Determine the intercepts of the face along the crystallographic axes, in terms of un it ce ll dimens ions. 2. Take the reciprocals 3. Clear fractions 4. Reduce to lowest terms Simple cubic d100=? Where does a protein crystallographer see the Miller indices? CtlCommon crystal faces are parallel to lattice planes • Eac h diffrac tion spo t can be regarded as a X-ray beam reflected from a lattice plane , and therefore has a unique Miller index. Miller indices A Miller index is a series of coprime integers that are inversely ppproportional to the intercepts of the cry stal face or crystallographic planes with the edges of the unit cell. It describes the orientation of a plane in the 3-D lattice with respect to the axes. The general form of the Miller index is (h, k, l) where h, k, and l are integers related to the unit cell along the a, b, c crystal axes. Irreducible brillouin zone II II II II Reciprocal lattice ghakblc The Bravais lattice after Fourier transform real space reciprocal lattice normaltthlls to the planes (vect ors ) poitints spacing between planes 1/distance between points ((y,p)actually, 2p/distance) l (distance, wavelength) 2p/l=k (momentum, wave number) BillBravais cell Wigner-SiSeitz ce ll Brillouin zone .
    [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]
  • Chapter 4, Bravais Lattice Primitive Vectors
    Chapter 4, Bravais Lattice A Bravais lattice is the collection of all (and only those) points in space reachable from the origin with position vectors: n , n , n integer (+, -, or 0) r r r r 1 2 3 R = n1a1 + n2 a2 + n3a3 a1, a2, and a3 not all in same plane The three primitive vectors, a1, a2, and a3, uniquely define a Bravais lattice. However, for one Bravais lattice, there are many choices for the primitive vectors. A Bravais lattice is infinite. It is identical (in every aspect) when viewed from any of its lattice points. This is not a Bravais lattice. Honeycomb: P and Q are equivalent. R is not. A Bravais lattice can be defined as either the collection of lattice points, or the primitive translation vectors which construct the lattice. POINT Q OBJECT: Remember that a Bravais lattice has only points. Points, being dimensionless and isotropic, have full spatial symmetry (invariant under any point symmetry operation). Primitive Vectors There are many choices for the primitive vectors of a Bravais lattice. One sure way to find a set of primitive vectors (as described in Problem 4 .8) is the following: (1) a1 is the vector to a nearest neighbor lattice point. (2) a2 is the vector to a lattice points closest to, but not on, the a1 axis. (3) a3 is the vector to a lattice point nearest, but not on, the a18a2 plane. How does one prove that this is a set of primitive vectors? Hint: there should be no lattice points inside, or on the faces (lll)fhlhd(lllid)fd(parallolegrams) of, the polyhedron (parallelepiped) formed by these three vectors.
    [Show full text]
  • Chapter 1 – Symmetry of Molecules – P. 1
    Chapter 1 – Symmetry of Molecules – p. 1 - 1. Symmetry of Molecules 1.1 Symmetry Elements · Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation every point of the molecule is coincident with an equivalent point. · Symmetry element: Geometrical entity (line, plane or point) which respect to which one or more symmetry operations can be carried out. In molecules there are only four types of symmetry elements or operations: · Mirror planes: reflection with respect to plane; notation: s · Center of inversion: inversion of all atom positions with respect to inversion center, notation i · Proper axis: Rotation by 2p/n with respect to the axis, notation Cn · Improper axis: Rotation by 2p/n with respect to the axis, followed by reflection with respect to plane, perpendicular to axis, notation Sn Formally, this classification can be further simplified by expressing the inversion i as an improper rotation S2 and the reflection s as an improper rotation S1. Thus, the only symmetry elements in molecules are Cn and Sn. Important: Successive execution of two symmetry operation corresponds to another symmetry operation of the molecule. In order to make this statement a general rule, we require one more symmetry operation, the identity E. (1.1: Symmetry elements in CH4, successive execution of symmetry operations) 1.2. Systematic classification by symmetry groups According to their inherent symmetry elements, molecules can be classified systematically in so called symmetry groups. We use the so-called Schönfliess notation to name the groups, Chapter 1 – Symmetry of Molecules – p. 2 - which is the usual notation for molecules.
    [Show full text]
  • Pose Estimation for Objects with Rotational Symmetry
    Pose Estimation for Objects with Rotational Symmetry Enric Corona, Kaustav Kundu, Sanja Fidler Abstract— Pose estimation is a widely explored problem, enabling many robotic tasks such as grasping and manipulation. In this paper, we tackle the problem of pose estimation for objects that exhibit rotational symmetry, which are common in man-made and industrial environments. In particular, our aim is to infer poses for objects not seen at training time, but for which their 3D CAD models are available at test time. Previous X 2 X 1 X 2 X 1 work has tackled this problem by learning to compare captured Y ∼ 2 Y ∼ 2 Y ∼ 2 Y ∼ 2 views of real objects with the rendered views of their 3D CAD Z ∼ 6 Z ∼ 1 Z ∼ Z ∼ 1 ∼ ∼ ∼ ∞ ∼ models, by embedding them in a joint latent space using neural Fig. 1. Many industrial objects such as various tools exhibit rotational networks. We show that sidestepping the issue of symmetry symmetries. In our work, we address pose estimation for such objects. in this scenario during training leads to poor performance at test time. We propose a model that reasons about rotational symmetry during training by having access to only a small set of that this is not a trivial task, as the rendered views may look symmetry-labeled objects, whereby exploiting a large collection very different from objects in real images, both because of of unlabeled CAD models. We demonstrate that our approach different background, lighting, and possible occlusion that significantly outperforms a naively trained neural network on arise in real scenes.
    [Show full text]
  • 12-5 Worksheet Part C
    Name ________________________________________ Date __________________ Class__________________ LESSON Reteach 12-5 Symmetry A figure has symmetry if there is a transformation of the figure such that the image and preimage are identical. There are two kinds of symmetry. The figure has a line of symmetry that divides the figure into two congruent halves. Line Symmetry one line of symmetry two lines of symmetry no line symmetry When a figure is rotated between 0° and 360°, the resulting figure coincides with the original. • The smallest angle through which the figure is rotated to coincide with itself is called the angle of rotational symmetry. • The number of times that you can get an identical figure when repeating the degree of rotation is called the order of the rotational Rotational symmetry. Symmetry angle: 180° 120° no rotational order: 2 3 symmetry Tell whether each figure has line symmetry. If so, draw all lines of symmetry. 1. 2. _________________________________________ ________________________________________ Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. 3. 4. _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 12-38 Holt Geometry Name ________________________________________ Date __________________ Class__________________ LESSON Reteach 12-5 Symmetry continued Three-dimensional figures can also have symmetry. Symmetry in Three Description Example Dimensions A plane can divide a figure into two Plane Symmetry congruent halves. There is a line about which a figure Symmetry About can be rotated so that the image and an Axis preimage are identical.
    [Show full text]
  • PHZ6426 Crystal Structure
    SYMMETRY Finite geometric objects (“molecules) are symmetric with respect to a) rotations b) reflections (symmetry planes, symmetry points) c) combinatons of translations and rotations (screw axes) 2π If an object is symmetric with respect to rotation by angle φ = , it is said to have has an “n-fold rotational axis” n n=1 is a trivial case: every object is symmetric about a full revolution. Non-trivial axes have n>1. 1 Symmetries of an equilateral triangle 2 2 2 180-rotations (flips) 3’ 1’ about 22’ 3 1 3 1 2’ about 33’ about 11’ 1 3 1 2 2 3 3 2 2π 1 120 degree= rotations 3 3 2 1 3 3 3 4 2 2 1 1 An equilateral triangle is not symmetric with respect to inversion about its center. But a square is. 5 Gliding (screw) axis Rotate by 180 degrees about the axis Slide by a half-period 6 2-fold rotations can be viewed as reflections in a plane which contains a 2-fold rotation axis 7 Crystal Structure Crystal structure can be obtained by attaching atoms or groups of atoms --basis-- to lattice sites. Crystal Structure = Crystal Lattice + Basis Crystal Structure 88 Partially from Prof. C. W. Myles (Texas Tech) course presentation Building blocks of lattices are geometric figures with certain symmetries: rotational, reflectional, etc. Lattice must obey all these symmetries but, in addition, it also must obey TRANSLATIONAL symmetry Which geometric figures can be used to built a lattice? 9 10 Crystallographic Restriction Theorem: Crystal lattices have only 2-,3, 4-, and 6-fold rotational symmetries 11 B’ ma A’ n-fold axis φ-π/2 φ=2π/n a A B 1.
    [Show full text]
  • Lattice Vibrations - Phonons
    Lattice vibrations - phonons So far, we have assumed that the ions are fixed at their equilibrium positions, and we focussed on understanding the motion of the electrons in the static periodic potential created by the ions. But, of course, the ions are quantum objects that cannot be at rest in well-defined positions { this would violate Heisenberg's uncertainty principle. So they must be moving! In the following, we will assume that we are at temperatures very well below the melting temperature of the crystal, so that the ions perform small oscillations about their equilibrium positions. Of course, with increasing T the amplitude of these vibrations increases and eventually becomes large enough that the ions start moving past one another and the lattice order is progressively lost (the crystal starts to melt); our approximations will not be valid once we reach such high temperatures. Dealing with small oscillations is very nice (as you may remember from studying Lagrangian/Hamiltonian dynamics) because in the harmonic approximation, i.e. where we expand potentials up to second- order in these displacements, such classical problems can be solved exactly by mapping them to a sum of independent harmonic oscillators. Going to the quantum solution is then trivial, because it will simply be the sum of independent quantum harmonic oscillators. In this chapter we dis- cuss this solution and its implications, and for the time being we will completely forget about the electrons. We will return back to them in the next section. I assume you have already studied phonons in your undergraduate solid state physics course.
    [Show full text]