Discrete Transformations in the Thomson Problem

Total Page:16

File Type:pdf, Size:1020Kb

Discrete Transformations in the Thomson Problem Discrete Transformations in the Thomson Problem Tim LaFave Jr. Abstract A significantly lower upper limit to minimum energy solutions of the electrostatic Thomson Problem is reported. A point charge is introduced to the origin of each N-charge solution. This raises the total energy by N as an upper limit to each (N + 1)-charge solution. Minimization of energy to U(N + 1) is well fit with 0.5518(3/2)√N +1/2 for up to N=500. The energy distribution due to this displacement exhibits correspondences− with shell-filling behavior in atomic systems. This work may aid development of more efficient and innovative numerical search algorithms to obtain N-charge configurations having global energy minima and yield new insights to atomic structure. Keywords: “T. LaFave Jr. J. Electrostatics 72(1) 39-43 (2014)”. DOI: 10.1016/j.elstat.2013.11.007 1. Introduction within a continuous charged shell of unit radius hav- ing total charge, N. With this interpretation, the The Thomson Problem has drawn considerable half-integer terms may correspond to self-energies interest since the mid-1900s (1) having found use of N uniformly charged disks that are removed to in modeling fullerenes, (2,3) drug encapsulants, (4) yield the final minimized energy of discrete charges. spherical viruses, (5) and crystalline order on curved If the N 2/2 term is associated with the random surfaces. (6) In the past few decades several compu- distribution of point charges, the half-integer terms tational algorithms have yielded precise numerical may be related to correlation energies of surface solutions for many-N electron systems. (7–18) The Coulomb equilibrium states. (7) These ascriptions Thomson Problem has emerged as a benchmark of energy terms to physical entities have guided for global optimization algorithms (16,17) though its the development of fairly useful minimization al- general solution remains unknown. (19) gorithms. Numerical solutions of the Thomson Problem are Here, the discrete derivative of N 2/2 is shown those for which the total Coulomb repulsion energy, to correspond to the introduction of a single point charge, q0, at the origin of a given N-charge solu- N tion, and the discrete derivative of the remaining 1 U(N)= , (1) half-integer term(s) accounts for energy needed to ri rj displace q to the unit sphere. This yields each Xi<j | − | 0 subsequent (N + 1)-charge solution of the Thom- is a minimum for each N-charge system with ri and son Problem. In this manner, the upper limit of rj constrained to the surface of a unit sphere. each minimized (N + 1) energy solution is given by The initial condition of some minimization al- U(N)+ N. This represents a well-defined charge arXiv:1403.2592v1 [physics.class-ph] 22 Feb 2014 gorithms is the random distribution of N point configuration with a significantly lower energy than charges on the unit sphere. This sets a relatively random charge distributions and may be useful as (8) distant upper energy limit, Ur(N)= N(N 1)/2, initial conditions in relatively more efficient energy from which numerous iterations progress− toward a minimization algorithms. global minimum for each N-charge system. The Notably, the distribution of energy solutions of distribution of numerical solutions has been fit the Thomson problem is “systematic” (8) and not with empirical functions including U(N)= N 2/2+ random. Here, having ignored the linear term in aN 3/2 (7) and U(N) = N 2/2+ aN 3/2 + bN 1/2. (8) the discrete derivative, the remaining “systematic” The quadratic term is ascribed to energy stored distribution of energy associated with the displace- Preprint submitted to Elsevier June 15, 2021 ment of q0 to the surface of the unit sphere demon- 2 strably exhibits features uniquely correspondent N /2 with numerous shell-filling features throughout the N(N−1)/2 + periodic table of natural atomic systems. (20–22) U(N+1 ) + Hence, introduction of q0 to the Thomson Prob- U(N ) lem represents an intriguingly fresh perspective of + U(N+1) this extensively useful mathematical model and the Potential Energy U(N−1 ) classical underpinnings of atomic structure. U(N) Numerical solutions of the Thomson Problem for U(N−1) up to 500 charges used in this communication have N−1 N N+1 been obtained from a continually updated database Number of point charges maintained by Syracuse University. (23) Figure 1: The extreme upper energy limit of the Thomson 2. Discrete Derivatives Problem is given by N 2/2 for a continuous charge shell fol- lowed by N(N 1)/2, the energy associated with a random − + Two upper energy bounds of some interest are distribution of N point charges. Significantly lower, U(N ), those associated with a continuous charge shell (8) the energy of a given N-charge solution of the Thomson Problem with one charge at its origin is readily obtained consisting of infinite-many N charges such that by U(N) + N, where U(N) are solutions of the Thomson Problem. N 2 U∞(N)= (2) 2 imposing additional arguments. If these intermedi- ate values of f(N) should have physical usefulness, and the random distribution of N discrete point for instance if an effective fractional charge on the charges across the surface of the unit sphere unit sphere surface is admitted as a discrete charge approaches or leaves the surface, f(N) should have local minima at integer values of N. Though these N N 2 N Ur(N)= (N 1) = . (3) fit functions, Eqs. 4 and 5 have no local minima at 2 − 2 − 2 integer values of N, they are coarsely instructive. Plots of Eqs. 2 and 3 are shown in Fig. 1 together It is potentially more fruitful to design a fit with a few numerical solutions (open circles) of the function in accordance with the physical nature of Thomson Problem for illustration. electrostatic charge configurations. In particular, The minimized global potential energy solutions knowledge of f(N) at integer values of N, given for up to N=65 point charges were previously fit the discrete nature of point charges is paramount. using, (7) Consider the discrete derivative of f(N) at integer values of N, 2 N 3/2 f1(N)= + aN (4) ∆f (N) f(N + ∆N) f(N) 2 i = − (6) ∆N ∆N in which a = 0.5510, and for up to N 100 of Eq. 4, which yields, after binomial expansion of using (8) − ∼ the half-integer term, 2 N 3/2 1/2 f2(N)= + aN + bN . (5) 1 2 ∆f1(N) = N + 2 These smooth, continuous fit functions are gener- a 1 3 − 1 1 − 3 + 3N 2 + N 2 N 2 + (7) ally unrepresentative of the absolute minimum en- 2 4 − 8 ··· ergy configurations of discrete charges due to a vari- ety of issues. Among them, discrete charges cannot for ∆N = 1. For comparison, the derivative of be infinitesimally subdivided so all points, f(N), for Eq. 5, with binomial expansion of the half-integer non-integer N have no physical significance without terms, 2 [N+ ] [N+ ] 1 1 1 7 1 2 − 2 ∆f2(N) = N + + 3aN + (a + b)N 2 2 4 q 0 N 1 − 3 N + + (a b)N 2 + (8) U (N+1) 8 − ··· ∆ for ∆N = 1. The first set of parentheses in Eqs. 7 U’(N+1) ∆ [N+1] and 8 is the discrete derivative of N 2/2, the energy of a continuous charge shell, Eq. 2. Energy U(N+1) q ∆ 3. Discrete Transformation Energies 0 To understand the physical charge distribu- tion represented by the discrete energy differences, [N] Eqs. 7 and 8, consider an N-charge solution, such N N+1 as that of [N] shown in Fig. 2 for which the total Charge Number minimized electrostatic energy is U(N) as shown in Fig. 1. The discrete energy changes needed to Figure 2: A well-defined intermediate charge configuration, + obtain the solution U(N + 1) in Fig. 1 may be ob- [N ], with q0 at the origin of [N] linearly increases the + tained by introducing an (N + 1)th point charge, energy by N. Consequently, U(N ) is readily accessible. + Displacement of q0 to the unit sphere surface and global q0, to the origin denoted [N ] in Fig. 2. Its contri- minimization of energy yields the new [N + 1] solution with bution to the total energy, given equal interaction ∆U +(N + 1) √N. with all N charges on the unit sphere, is N. The ∝ total energy, U(N +), (Fig. 1) is U(N)+ N as illus- trated in Fig. 2. Thus, the first term of each dis- as shown in Fig. 2. Consequently, ∆U +(N + 1) crete derivative, ∆f (N), is due to the appearance i accounts for the half-integer terms in Eqs. 7 and of q at the origin. The remaining terms are due 0 8 and the +1/2 term associated with the discrete to the displacement of q to the unit sphere surface 0 derivative of N 2/2. and subsequent (or in situ) global minimization of To check the validity of the discrete transforma- energy, U(N + 1) in Fig. 1, to yield the (N + 1) tions from each N-charge solution of the Thomson configuration (Fig. 2). Problem to its adjacent (N + 1)-charge solution, In the strictest spirit of the Thomson Problem, ∆U +(N + 1) in Eq. 9 is fit with the remaining introduction of q to the origin of the unit sphere 0 terms of the discrete derivatives of Eqs. 7 and 8.
Recommended publications
  • Correspondences Between the Classical Electrostatic Thomson
    Correspondences Between the Classical Electrostatic Thomson Problem and Atomic Electronic Structure Tim LaFave Jr.1,∗ Abstract Correspondences between the Thomson Problem and atomic electron shell-filling patterns are observed as systematic non-uniformities in the distribution of potential energy necessary to change configurations of N ≤ 100 electrons into discrete geometries of neighboring N − 1 systems. These non-uniformities yield electron energy pairs, intra-subshell pattern similarities with empirical ionization energy, and a salient pattern that coincides with size-normalized empirical ionization energies. Spatial symmetry limitations on discrete charges constrained to a spherical volume are conjectured as underlying physical mechanisms responsible for shell-filling patterns in atomic electronic structure and the Periodic Law. 1. Introduction present paper builds on this previous work by iden- tifying numerous correspondences between the elec- Quantum mechanical treatments of electrons in trostatic Thomson Problem of distributing equal spherical quantum dots, or “artificial atoms”, 1 rou- point charges on a unit sphere and atomic electronic tinely exhibit correspondences to atomic-like shell- structure. filling patterns by the appearance of abrupt jumps Despite the diminished stature of J.J. Thomson’s or dips in calculated energy or capacitance distribu- classical “plum-pudding” model 19 among more ac- tions as electrons are added to or removed from the curate atomic models, the Thomson Problem has 2–10 system. Additionally, shell-filling
    [Show full text]
  • Assembly of One-Patch Colloids Into Clusters Via Emulsion Droplet Evaporation
    materials Article Assembly of One-Patch Colloids into Clusters via Emulsion Droplet Evaporation Hai Pham Van 1,2, Andrea Fortini 3 and Matthias Schmidt 1,* 1 Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, Universitätsstraße 30, D-95440 Bayreuth, Germany 2 Department of Physics, Hanoi National University of Education, 136 Xuanthuy, Hanoi 100000, Vietnam; [email protected] 3 Department of Physics, University of Surrey, Guildford GU2 7XH, UK; [email protected] * Correspondence: [email protected]; Tel.: +49-921-55-3313 Academic Editors: Andrei V. Petukhov and Gert Jan Vroege Received: 23 February 2017; Accepted: 27 March 2017; Published: 29 March 2017 Abstract: We study the cluster structures of one-patch colloidal particles generated by droplet evaporation using Monte Carlo simulations. The addition of anisotropic patch–patch interaction between the colloids produces different cluster configurations. We find a well-defined category of sphere packing structures that minimize the second moment of mass distribution when the attractive surface coverage of the colloids c is larger than 0.3. For c < 0.3, the uniqueness of the packing structures is lost, and several different isomers are found. A further decrease of c below 0.2 leads to formation of many isomeric structures with less dense packings. Our results could provide an explanation of the occurrence of uncommon cluster configurations in the literature observed experimentally through evaporation-driven assembly. Keywords: one-patch colloid; Janus colloid; cluster; emulsion-assisted assembly; Pickering effect 1. Introduction Packings of colloidal particles in regular structures are of great interest in colloid science. One particular class of such packings is formed by colloidal clusters which can be regarded as colloidal analogues to small molecules, i.e., “colloidal molecules” [1,2].
    [Show full text]
  • The Scottish Bubble and the Irish Bubble a Maths Club Project the Elementary Mathematics in an Advanced Problem
    The Scottish Bubble and the Irish Bubble A Maths Club Project The Elementary Mathematics in an Advanced Problem Overview The contents of this CD are intended for you, the leader of the maths club. The material is of 3 different kinds: history: the problems in their historical setting: how they arose, how people first tackled them. Many of you will be specialists in a science other than maths. I hope you find something to interest you personally among the references I have cited. I also hope you take this project as an opportunity to work with colleagues from across the curriculum. calculation: what students of school age can work out for themselves; something not for club-time but as a private challenge for the keen. This work is pitched at the 16-18 age range. However, elementary algebra, Pythagoras’ Theorem, trigonometric identities and their use in solving triangles is almost all the maths the students will need. experiment: what the students can do and make. This is the project core and can involve students from right across the secondary age range. With a weekly session of an hour and a half, the work will take a full term. As you see, these headings appear as possible bookmarks down the left side of the screen. Introduction When scientists turn to mathematicians for answers, they expect them to have ready-made solutions, something they can pull down off the shelf. In one area, packing, this has not been the case. The reason lies in the problems’ untidiness. To identify a packing as the best, one must exclude all others possible.
    [Show full text]
  • The Maximally Entangled Symmetric State in Terms of the Geometric Measure
    Home Search Collections Journals About Contact us My IOPscience The maximally entangled symmetric state in terms of the geometric measure This content has been downloaded from IOPscience. Please scroll down to see the full text. 2010 New J. Phys. 12 073025 (http://iopscience.iop.org/1367-2630/12/7/073025) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.63.180.147 This content was downloaded on 13/08/2014 at 08:38 Please note that terms and conditions apply. New Journal of Physics The open–access journal for physics The maximally entangled symmetric state in terms of the geometric measure Martin Aulbach1,2,3,6, Damian Markham1,4 and Mio Murao1,5 1 Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan 2 The School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK 3 Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, UK 4 CNRS, LTCI, Telecom ParisTech, 37/39 rue Dareau, 75014 Paris, France 5 Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo 113-0033, Japan E-mail: [email protected], [email protected] and [email protected] New Journal of Physics 12 (2010) 073025 (34pp) Received 22 April 2010 Published 22 July 2010 Online at http://www.njp.org/ doi:10.1088/1367-2630/12/7/073025 Abstract. The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement.
    [Show full text]
  • [Math.MG] 7 Oct 2008 Cat(F)Udrgatsh 1503/4-1
    EXPERIMENTAL STUDY OF ENERGY-MINIMIZING POINT CONFIGURATIONS ON SPHERES BRANDON BALLINGER, GRIGORIY BLEKHERMAN, HENRY COHN, NOAH GIANSIRACUSA, ELIZABETH KELLY, AND ACHILL SCHURMANN¨ Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima. [T]he problem of finding the configurations of stable equilibrium for a number of equal particles acting on each other according to some law of force...is of great interest in connexion with the rela- tion between the properties of an element and its atomic weight. Unfortunately the equations which determine the stability of such a collection of particles increase so rapidly in complexity with the number of particles that a general mathematical investigation is scarcely possible. J. J. Thomson, 1897 Contents 1. Introduction 2 1.1. Experimental results 4 1.2. New universal optima 8 2. Methodology 9 2.1. Techniques 9 2.2. Example 14 3. Experimental phenomena 15 arXiv:math/0611451v3 [math.MG] 7 Oct 2008 3.1. Analysis of Gram matrices 15 3.2. Other small examples 18 3.3. 2n + 1 points in Rn 20 3.4. 2n + 2 points in Rn 21 3.5. 48 points in R4 22 3.6. Hopf structure 23 3.7.
    [Show full text]
  • OPTIMA Mathematical Optimization Society Newsletter 100 MOS Chair’S Column
    OPTIMA Mathematical Optimization Society Newsletter 100 MOS Chair’s Column May 15, 2016. In June 1980, the chair of the MPS Publica- As of March 31, 2016, the society has $794,610 in total assets, tions Committee, Michael Held, wrote “Thus the decision was with $25,711 restricted for the Fulkerson Prize and $19,257 re- made to establish a new Newsletter – OPTIMA.” You can find stricted for the Lagrange Prize. This is a substantial increase over the text of his article in Optima 1, available on the Web page our balance of $550,261 at the end of 2012. The MOS is a learned www.mathopt.org/?nav=optima_details together with full copies of society, with no professional staff. Our main expense is an adminis- all 100 issues of Optima. It is great fun to click through the old copies trative fee paid to SIAM, to maintain our membership list, provide – give it a try. And please join me in thanking the current and past email service, and handle our financial matters. Over the past three Editors of Optima, Donald Hearn, issues 1–55 (!), Karen Aardal, is- years, the SIAM fee amounted to a total of $125,728. This was more sues 56–65, Jens Clausen, issues 66–72 , Andrea Lodi, issues 73–84, than offset by our $155,966 royalty income over the three-year pe- Katya Scheinberg, issues 85–93, and Volker Kaibel, issues 94–100. riod, derived mainly from the publication of MPA, MPB, and MPC. Their work has, of course, been strongly supported by a long line of Thanks again to everyone for your support over the past three Co-Editors, starting with Achim Bachem and continuing through the years.
    [Show full text]
  • Showing the Surprising Difficulty of Proving That a Circle Has The
    Showing The Surprising Difficulty of Proving That a Circle has the Smallest Perimeter for a Given Area, and Other Interesting Related Problems. Oliver Phillips 14th April 2019 Abstract Throughout this paper, I shall show why a circle has the minimum perimeter for a given area, using the isoperimetric inequality, and see if this is possible for a sphere. We shall also briefly discuss an unsolved problem which relates quite well to this topic and if solved could show, in a nice way, that a sphere has the smallest ratio between surface area and volume. 1 Introduction If I give you the following question: What shape or object would have to be put around a point to give the minimal perimeter for a unit area? If you’ve thought about this for a short time, some of you would have possibly thought about a circle, this is usually because a teacher in high school has told us of this, but how do we know this is a fact? We may think this is a relatively simple problem to prove, but a rigorous proof was only discovered in 1841 [9], and this was only for R2, a proof for higher dimensions was found almost a century later in 1919 [7]. Throughout this paper, we shall explore this problem and its surprising difficulty for the R2 plane and then its further applications within the R3 plane. 2 What is the Minimum Perimeter for a Unit Area in a 2D Plane? 2.1 Conditions and Notation To be able to understand why a circle is a very special shape, we shall look into regular n-gons and their ratios between area and perimeter.
    [Show full text]
  • Two-Dimensional Packing of Soft Particles and the Soft Generalized Thomson Problem
    Two-Dimensional Packing of Soft Particles and the Soft Generalized Thomson Problem William L. Miller and Angelo Cacciuto∗ Department of Chemistry, Columbia University 3000 Broadway, New York, New York 10027 We perform numerical simulations of purely repulsive soft colloidal particles interacting via a generalized elastic potential and constrained to a two-dimensional plane and to the surface of a spherical shell. For the planar case, we compute the phase diagram in terms of the system's rescaled density and temperature. We find that a large number of ordered phases becomes accessible at low temperatures as the density of the system increases, and we study systematically how structural variety depends on the functional shape of the pair potential. For the spherical case, we revisit the generalized Thomson problem for small numbers of particles N ≤ 12 and identify, enumerate and compare the minimal energy polyhedra established by the location of the particles to those of the corresponding electrostatic system. Introduction teractions. Surprisingly, the simple relaxation of the constraint of mutual impenetrability between isotropic components gives access to several non-close-packed crys- Understanding how nanocomponents spontaneously talline structures at high densities. Given the complex- organize into complex macroscopic structures is one of ity of these mesoparticles, their interactions are usually the great challenges in the field of soft matter today. In- extracted via an explicit coarse-grained procedure to ob- deed, the ability to predict and control the phase behav- tain ad-hoc effective pair potentials. What emerges is ior of a solution, given a set of components, may open the a great variety of nontrivial interactions, some of which way to the development of materials with novel optical, allow for even complete overlap among the components, mechanical, and electronic properties.
    [Show full text]
  • Long-Ranged Oppositely Charged Interactions for Designing New Types of Colloidal Clusters
    PHYSICAL REVIEW X 5, 021012 (2015) Long-Ranged Oppositely Charged Interactions for Designing New Types of Colloidal Clusters † ‡ Ahmet Faik Demirörs,* Johan C. P. Stiefelhagen, Teun Vissers, Frank Smallenburg, Marjolein Dijkstra, Arnout Imhof, and Alfons van Blaaderen§ Department of Physics, Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands (Received 23 October 2014; published 29 April 2015) Getting control over the valency of colloids is not trivial and has been a long-desired goal for the colloidal domain. Typically, tuning the preferred number of neighbors for colloidal particles requires directional bonding, as in the case of patchy particles, which is difficult to realize experimentally. Here, we demonstrate a general method for creating the colloidal analogs of molecules and other new regular colloidal clusters without using patchiness or complex bonding schemes (e.g., DNA coating) by using a combination of long-ranged attractive and repulsive interactions between oppositely charged particles that also enable regular clusters of particles not all in close contact. We show that, due to the interplay between their attractions and repulsions, oppositely charged particles dispersed in an intermediate dielectric constant (4 < ε < 10) provide a viable approach for the formation of binary colloidal clusters. Tuning the size ratio and interactions of the particles enables control of the type and shape of the resulting regular colloidal clusters. Finally, we present an example of clusters made up of negatively charged large and positively charged small satellite particles, for which the electrostatic properties and interactions can be changed with an electric field. It appears that for sufficiently strong fields the satellite particles can move over the surface of the host particles and polarize the clusters.
    [Show full text]
  • Locally and Globally Optimal Configurations of N Particles
    Locally and Globally Optimal Configurations of N Particles on the Sphere with Applications in the Narrow Escape and Narrow Capture Problems Wesley J. M. Ridgway,a) Alexei F. Cheviakov b) Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, S7N 5E6 Canada December 20, 2018 Abstract Determination of optimal arrangements of N particles on a sphere is a well-known problem in physics. A famous example of such is the Thomson problem of finding equilibrium config- urations of electrical charges on a sphere. More recently however, similar problems involving other potentials and non-spherical domains have arisen in biophysical systems. Many optimal configurations have previously been computed, especially for the Thomson problem, however few results exist for potentials that correspond to more applied problems. Here we numeri- cally compute optimal configurations corresponding to the narrow escape and narrow capture problems in biophysics. We provide comprehensive tables of global energy minima for N ≤ 120 and local energy minima for N ≤ 65, and we exclude all saddle points. Local minima up to N = 120 are available online. 1 Introduction n The problem of optimally distributing N points on the boundary of a bounded domain, D ⊆ R (n ≥ 2), has many applications in physical systems (see e.g. Ref. [1] and numerous references therein). Here, optimal refers to an arrangement of points, particles, etc. on the boundary, @D, N such that the configuration of particles fxigi=1 minimizes the pairwise `potential energy' N X H(x1; :::; xN ) = h(jxi − xjj) (1.1) i<j with the constraint xi 2 @D where h is the interaction energy between two particles.
    [Show full text]
  • Fullerenes: Applications and Generalizations Michel Deza Ecole Normale Superieure, Paris Mathieu Dutour Sikiric´ Rudjer Boskovi˘ C´ Institute, Zagreb, and ISM, Tokyo
    Fullerenes: applications and generalizations Michel Deza Ecole Normale Superieure, Paris Mathieu Dutour Sikiric´ Rudjer Boskovi˘ c´ Institute, Zagreb, and ISM, Tokyo – p. 1 I. General setting – p. 2 Definition A fullerene Fn is a simple polyhedron (putative carbon molecule) whose n vertices (carbon atoms) are arranged in n 12 pentagons and ( 2 − 10) hexagons. 3 The 2 n edges correspond to carbon-carbon bonds. Fn exist for all even n ≥ 20 except n = 22. 1, 2, 3,..., 1812 isomers Fn for n = 20, 28, 30,. , 60. preferable fullerenes, Cn, satisfy isolated pentagon rule. C60(Ih), C80(Ih) are only icosahedral (i.e., with symmetry Ih or I) fullerenes with n ≤ 80 vertices – p. 3 buckminsterfullerene C60(Ih) F36(D6h) truncated icosahedron, elongated hexagonal barrel soccer ball F24(D6d) – p. 4 45 4590 34 1267 2378 15 12 23 3489 1560 15 34 45 1560 3489 2378 1267 4590 23 12 Dodecahedron Graphite lattice 1 F20(Ih) → 2 H10 F → Z3 the smallest fullerene the “largest”∞ (infinite) Bonjour fullerene – p. 5 Small fullerenes 24, D6d 26, D3h 28, D2 28, Td 30, D5h 30, C2v 30, D2v – p. 6 A C540 – p. 7 What nature wants? Fullerenes Cn or their duals Cn∗ appear in architecture and nanoworld: Biology: virus capsids and clathrine coated vesicles Organic (i.e., carbon) Chemistry also: (energy) minimizers in Thomson problem (for n unit charged particles on sphere) and Skyrme problem (for given baryonic number of nucleons); maximizers, in Tammes problem, of minimum distance between n points on sphere Simple polyhedra with given number of faces, which are the “best” approximation of sphere? Conjecture: FULLERENES – p.
    [Show full text]
  • Application of Measured Directivity Patterns to Acoustic Array Processing
    Application of Measured Directivity Patterns to Acoustic Array Processing Mark R. P. Thomas Microsoft Research, Redmond, USA 1 My Background • 2011-present: Postdoctoral Researcher, Researcher (2013), Audio and Acoustics Research Group, Microsoft Research, Redmond, USA. • Microphone arrays (linear, planar, cylindrical, spherical). • Echo cancellation, noise suppression. • Head-related transfer functions. • Loudspeaker arrays. • http://research.microsoft.com 2 My Background • 2001-2002: Pre-University/Vacation Trainee, BBC Research & Development, Kingswood Warren, Tadworth, Surrey. • DAB data protocols, audio signal processing for HDTVs, TV spectrum planning, hardware for live TV streaming. • 2002-2010: MEng/PhD in Electrical and Electronic Engineering, Imperial College London. • MEng Thesis, “A Novel Loudspeaker Equalizer.” • PhD Thesis, “Glottal-Synchronous Speech Processing.” • 2010-2011: Research Associate, Imperial College London • EU FP7 project Self Configuring ENVironment-aware Intelligent aCoustic sensing (SCENIC) • Spherical microphone arrays, geometric inference, channel identification & equalization. 3 Directivity Patterns: Background • Directivity pattern is the response to a plane wave emerging from a known direction relative to the device under test. • Function of azimuth 휙 • Function of elevation / colatitude 휃 • Function of frequency 휔 • This is the ‘farfield’ response 푧 • Practically measured with a loudspeaker at a fixed distance of 1-2m. 푦 휃 푥 • Independent of reverberation 휙 4 Directivity Patterns: Background • All
    [Show full text]