DOCSLIB.ORG

Crystal Structure and Scattering Experiments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Crystal Structure and Scattering Experiments Crystal structure and scattering experiments Chris J. Pickard How can we tell where the atoms are? ² We have see the rich variety of bonding and structures that collections of atoms can adopt ² But how can we ever know where the atoms actually are? ² We must experimentally probe the Amphibole structural information Possible Experimental Probes Hit something and see what it, or the probe, does Scattering ² Probe atomic structure by \bouncing" particles o® it, and measuring changes in momentum etc. ² May be X-ray, neutron or electron scattering and elastic or inelastic A scattering experiment: The ² X-ray and neutron facilities European Synchrotron Radiation are massive pieces of scienti¯c Facility infrastructure Crystal Lattices ² In order to interpret the scattering experiments we need a model of where the atoms might be ² There are simply too many atoms in a solid for each's coordinates to be determined ² For crystalline solids we introduce the concept of a Bravais lattice The Bravais Lattice A ² The Bravais lattice describes the underlying periodic structure: not the crystal structure itself B a2 ² An in¯nite array which looks identical a1 if viewed from any point ² R = i niai, ni are all integers, and ai arePnot in the same plane A 2D Bravais Lattice A = a1 + 2a2 and B = ¡a1 + a2 ² The primitive vectors ai generate or span the lattice Finite Crystals ² All point are equivalent in a Bravais lattice: it must be in¯nite, but real crystals are ¯nite ² Finite size e®ects, or surface e®ects, may not be important: the Bravais lattice is a useful approximation ² It can be useful to consider a ¯nite Gold crystallite portion: R = i niai; 0 · ni < Ni P Coordination Number ² The Bravais lattice points closest to a given point are the nearest neighbours ² Because the Bravais lattice is periodic, all points have the same number of Diamond: 4 Simple Cubic: 6 nearest neighbours or coordination number. It is a property of the lattice ² Can be extended to arrays of points that are not Bravais lattices (the diamond lattice is not a Bravais BCC: 8 FCC: 12 lattice) Primitive Unit Cell ² Volume which when translated by all vectors in Bravais lattice just ¯lls space ² Not uniquely de¯ned ² The set of points r = x1a1 + x2a2 + x3a3, 0 · xi < 1 is an obvious choice ² Does not display the full symmetry of Two choices the lattice The Conventional Unit Cell ² A unit cell just ¯lls space when translated through a subset of Bravais lattice vectors ² The conventional unit cell is chosen to be larger than the primitive cell, but with the full symmetry of the Bravais lattice ² The size of the conventional cell is given by the lattice constants { a FCC Bravais lattice single a in this case The Wigner-Seitz Primitive Cell ² The Wigner-Seitz cell is a primitive cell with the full symmetry of the Bravais lattice ² Constructed by selecting a lattice point and taking the volume closer to that point than any others Wigner-Seitz cell for a 2D lattice ² Algorithm: draw lines from the lattice point to all others, bisect each line with a plane and take the smallest polyhedron containing the point The Wigner-Seitz Primitive Cell BCC FCC Wigner-Seitz cells for BCC and FCC 3D lattices Crystal structure ² A crystal structure is a basis, or physical unit, translated by each vector of the Bravais lattice ² Also known as a \lattice with a basis" ² A monatomic Bravais lattice has a basis consisting of a single atom 2D Bravais lattice with ² A Bravais lattice is a lattice with two point basis a basis when a non-primitive cell is chosen The Diamond Structure ² Two interpenetrating FCC Bravais lattices, displaced by 1/4 length of body diagonal ² FCC cubic lattice with two-point a/4(x+y+z) basis: 0 and a/4(x+y+z) z x ² The diamond lattice is not a Bravais a y lattice Conventional cubic cell ² Zincblende structure with two species Hexagonal Close Packing ² The hexagonal close packed (HCP) structure is not a Bravais lattice ² Many pure elements (about 30) a3 crystallise in this way ² The HCP structure consists of two interpenetrating simple hexagonal a2 1/3a1+1/3a2+1/2a3 a1 Bravais lattices Hexagonal close packed crystal structure ² Other stackings are possible: ...ABCABC... is FCC The Reciprocal Lattice ² The reciprocal lattice arises from the relationship between the Bravais a2 £ a3 ik¢r b1 = 2¼ lattice and plane waves, e a1 ¢ (a2 £ a3) ² For certain k the plane waves will have a3 £ a1 b2 = 2¼ the periodicity of the lattice. These a1 ¢ (a2 £ a3) are the points of the reciprocal lattice ¢ a1 £ a2 K and eiK R = 1 b3 = 2¼ a1 ¢ (a2 £ a3) ² The reciprocal lattice is a Bravais lattice, and the reciprocal lattice of the reciprocal lattice is the original b ¢ a = 2¼± i j ij direct lattice Some Examples a a a a1 = ax^; a2 = ay^; a3 = az^ a1 = (y^+z^); a1 = (z^+x^); a1 = (x^+y^) 2 2 2 2¼ 2¼ 4¼ 1 4¼ 1 b1 = x^; b2 = y^ b1 = (y^+z^¡x^); b2 = (z^+x^¡y^) a a a 2 a 2 2¼ 4¼ 1 b3 = z^ b3 = (x^ + y^ ¡ z^) a a 2 The Simple Cubic Bravais Lattice The FCC Bravais Lattice The First Brillouin Zone ² The ¯rst Brillouin zone is the Wigner- Seitz primitive cell of the reciprocal lattice ² Higher Brillouin zones exist, and are important in the theory of electronic FCC levels in a periodic potential ² Take care: the ¯rst BZ of a FCC BCC lattice is the Wigner-Seitz primitive The ¯rst Brillouin zone cell of the BCC lattice (and vice versa) Lattice Planes ² Families of lattice planes can be classi¯ed in terms of the reciprocal lattice ² For any reciprocal lattice vector K there is a family of planes normal to K and separated by a distance d Two families of lattice planes in a ² The length of the shortest reciprocal 2¼ simple cubic Bravais lattice lattice vector parallel to K is d Miller Indices ² The reciprocal lattice provides a a3 a3 convienient way to label lattice planes a1 a1 ² The Miller indices are the a2 a2 (010) (110) coordinates of the shortest reciprocal a3 lattice vector normal to the plane a1 ² A plane with Miller indices h,k,l is a2 normal to K = hb1 + kb2 + lb3 (111) Lattice planes and Miller indices in a simple cubic Bravais lattice ² h,k,l are integers with no common factor Specifying Directions: some conventions ² Lattice planes: (h; k; l) ! (hkl) with a3 a3 ¡n ! n¹ e.g. (2; ¡1; 4) ! (21¹4) a1 a1 ² Directions in the direct lattice: use a2 a2 (010) (110) square brackets e.g. [111] a3 ² Planes equivalent by symmetry: a1 (100), (010), and (001) are equivalent a2 in a cubic crystal, and we write f100g (111) Lattice planes and Miller indices in a simple cubic Bravais lattice ² For directions: [100], [010], [001], [100]¹ , [010]¹ , [001¹] ! h100i Beyond Translational Symmetry in Crystals ² There can be other symmetries in addition to translational symmetry ² The subject of crystallography systematises the classi¯cation of these di®erent symmetries The symmetries of a cube are identical ² The number of possibilities can be to an octahedron, but not a shown to be ¯nite for crystals, and tetrahedron have been ennumerated Seven Crystal Systems and Fourteen Bravais Lattices Cubic a Hexagonal Tetragonal a a Cubic a Trigonal Orthorhombic a a Trigonal c Monoclinic c Triclinic a c a a b Tetragonal Monoclinic Hierarchy of Symmetries a a a Hexagonal c c ² e.g. the cubic crystal system includes a b a Triclinic the simple cubic, BCC and FCC b Orthorhombic The Seven Crystal Systems Bravais lattices Point and Space Groups of Bravais Lattices and Crystal Structures Bravais Lattice Crystal Structure (basis of spherical symmetry) (basis of arbitary symmetry) Number of 7 32 point groups (crystal systems) (crystallographic point groups) Number of 14 230 space groups (Bravais lattices) (space groups) ² Adding a basis considerably complicates the situation ² Nonsymmorphic groups account for many of the 230: these contain screw axes and glide planes ² SchÄonflies (e.g. D6h) and International (e.g. 6=mmm) notations exist Structure Determination by X-ray Diffraction ² Atoms are separated by distances of the order of Angstroms (10¡8cm): the wavelength must be comparable hc ² We need X-rays: ¹h! = 10¡8cm12:3 £ 103eV ² We now consider how the scattering of X-rays from rigid periodic arrays of ions reveals their structure λ The Bragg Formulation ² W.L. Bragg considered crystals to be made up of parallel planes of atoms θ θ ² The di®raction peak occurs if 1) θ θ d d sin θ d sin θ X-rays reflect specularly from the atoms in a plane and 2) the X-rays from successive planes constructively interfere ² The di®raction condition is n¸ = 2dsinθ, and n is the order of the Bragg reflection reflection The von Laue Formulation ² The crystal is considered to be made up of identical microscopic objects at O R, which reradiate in all directions 1/2 K 1/2 K k K ² The Laue condition: constructive k' interference occurs if K = k ¡ k0 is a reciprocal lattice vector The Laue Condition ² No assumption that crystal contains lattice planes is made or that the ² The von Laue picture is equivalent to reflection is specular the Bragg picture The Ewald Sphere ² Given an incident wave vector k, draw a sphere centered on the origin O ² If a reciprocal lattice vector K lies on k k' the surface, a di®raction peak will be O 0 K observed, and k is the Bragg reflected ray ² In general, this is not the case (hence The Ewald Construction \peak") Experimental Methods ² Laue { The direction of k is varied by { Use a non-monochromatic beam rotating the crystal (from ¸0 to ¸1) { All K between the two Ewald ² Powder or Debye-Scherrer spheres will be seen { The randomly oriented crystallites ² Rotating Crystal e®ectively rotate the crystal, and { A monochromatic beam is used vary the axis of rotation Structure and Atomic Form Factors 1 iK¢r fj(K) = ¡ dre ½(r) n e Z iK¢dj SK = e The Atomic Form Factor Xj=1 The Structure Factor ² Modulates the intensities of the di®raction peaks ² There can be forbidden reflections for lattices with a basis ² Can be used analytically n iK¢dj SK = fj(K)e ² Some species are more visible than Xj=1 others.
Load more

Recommended publications
  • 361: Crystallography and Diffraction
    361: Crystallography and Diffraction Michael Bedzyk Department of Materials Science and Engineering Northwestern University February 2, 2021 Contents 1 Catalog Description3 2 Course Outcomes3 3 361: Crystallography and Diffraction3 4 Symmetry of Crystals4 4.1 Types of Symmetry.......................... 4 4.2 Projections of Symmetry Elements and Point Groups...... 9 4.3 Translational Symmetry ....................... 17 5 Crystal Lattices 18 5.1 Indexing within a crystal lattice................... 18 5.2 Lattices................................. 22 6 Stereographic Projections 31 6.1 Projection plane............................ 33 6.2 Point of projection........................... 33 6.3 Representing Atomic Planes with Vectors............. 36 6.4 Concept of Reciprocal Lattice .................... 38 6.5 Reciprocal Lattice Vector....................... 41 7 Representative Crystal Structures 48 7.1 Crystal Structure Examples ..................... 48 7.2 The hexagonal close packed structure, unlike the examples in Figures 7.1 and 7.2, has two atoms per lattice point. These two atoms are considered to be the motif, or repeating object within the HCP lattice. ............................ 49 1 7.3 Voids in FCC.............................. 54 7.4 Atom Sizes and Coordination.................... 54 8 Introduction to Diffraction 57 8.1 X-ray .................................. 57 8.2 Interference .............................. 57 8.3 X-ray Diffraction History ...................... 59 8.4 How does X-ray diffraction work? ................. 59 8.5 Absent
    [Show full text]
  • Lecture Notes
    Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals .
    [Show full text]
  • Introduction to the Physical Properties of Graphene
    Introduction to the Physical Properties of Graphene Jean-No¨el FUCHS Mark Oliver GOERBIG Lecture Notes 2008 ii Contents 1 Introduction to Carbon Materials 1 1.1 TheCarbonAtomanditsHybridisations . 3 1.1.1 sp1 hybridisation ..................... 4 1.1.2 sp2 hybridisation – graphitic allotopes . 6 1.1.3 sp3 hybridisation – diamonds . 9 1.2 Crystal StructureofGrapheneand Graphite . 10 1.2.1 Graphene’s honeycomb lattice . 10 1.2.2 Graphene stacking – the different forms of graphite . 13 1.3 FabricationofGraphene . 16 1.3.1 Exfoliatedgraphene. 16 1.3.2 Epitaxialgraphene . 18 2 Electronic Band Structure of Graphene 21 2.1 Tight-Binding Model for Electrons on the Honeycomb Lattice 22 2.1.1 Bloch’stheorem. 23 2.1.2 Lattice with several atoms per unit cell . 24 2.1.3 Solution for graphene with nearest-neighbour and next- nearest-neighour hopping . 27 2.2 ContinuumLimit ......................... 33 2.3 Experimental Characterisation of the Electronic Band Structure 41 3 The Dirac Equation for Relativistic Fermions 45 3.1 RelativisticWaveEquations . 46 3.1.1 Relativistic Schr¨odinger/Klein-Gordon equation . ... 47 3.1.2 Diracequation ...................... 49 3.2 The2DDiracEquation. 53 3.2.1 Eigenstates of the 2D Dirac Hamiltonian . 54 3.2.2 Symmetries and Lorentz transformations . 55 iii iv 3.3 Physical Consequences of the Dirac Equation . 61 3.3.1 Minimal length for the localisation of a relativistic par- ticle ............................ 61 3.3.2 Velocity operator and “Zitterbewegung” . 61 3.3.3 Klein tunneling and the absence of backscattering . 61 Chapter 1 Introduction to Carbon Materials The experimental and theoretical study of graphene, two-dimensional (2D) graphite, is an extremely rapidly growing field of today’s condensed matter research.
    [Show full text]
  • Additive Effects of Alkali Metals on Cu-Modified Ch3nh3pbi3−Δclδ
    RSC Advances PAPER View Article Online View Journal | View Issue Additive effects of alkali metals on Cu-modified CH3NH3PbI3ÀdCld photovoltaic devices† Cite this: RSC Adv.,2019,9,24231 Naoki Ueoka, Takeo Oku * and Atsushi Suzuki We investigated the addition of alkali metal elements (namely Na+,K+,Rb+, and Cs+) to Cu-modified CH3NH3PbI3ÀdCld photovoltaic devices and their effects on the photovoltaic properties and electronic structure. The open-circuit voltage was increased by CuBr2 addition to the CH3NH3PbI3ÀdCld precursor solution. The series resistance was decreased by simultaneous addition of CuBr2 and RbI, which increased the external quantum efficiencies in the range of 300–500 nm, and the short-circuit current density. The energy gap of the perovskite crystal increased through CuBr2 addition, which we also confirmed by first-principles calculations. Charge carrier generation was observed in the range of 300– Received 25th April 2019 500 nm as an increase of the external quantum efficiency, owing to the partial density of states Accepted 24th July 2019 contributed by alkali metal elements. Calculations suggested that the Gibbs energies were decreased by DOI: 10.1039/c9ra03068a incorporation of alkali metal elements into the perovskite crystals. The conversion efficiency was Creative Commons Attribution 3.0 Unported Licence. rsc.li/rsc-advances maintained for 7 weeks for devices with added CuBr2 and RbI. PbI2 +CH3NH3I + 2CH3NH3Cl / CH3NH3PbI3 Introduction + 2CH3NH3Cl (g)[ (140 C)(2) Studies of methylammonium lead halide perovskite solar x y / cells started in 2009, when a conversion efficiency of 3.9% PbI2 + CH3NH3I+ CH3NH3Cl (CH3NH3)x+yPbI2+xCly 1 / [ was reported. Some devices have since yielded conversion CH3NH3PbI3 +CH3NH3Cl (g) (140 C) (3) efficiencies of more than 20% as studies have expanded This article is licensed under a 2–5 Owing to the formation of intermediates and solvent evap- globally, with expectations for these perovskite solar cells 6–9 oration, perovskite grains are gradually formed by annealing.
    [Show full text]
  • Pharmaceutico Analytical Study of Udayabhaskara Rasa
    INTERNATIONAL AYURVEDIC MEDICAL JOURNAL Research Article ISSN: 2320 5091 Impact Factor: 4.018 PHARMACEUTICO ANALYTICAL STUDY OF UDAYABHASKARA RASA Gopi Krishna Maddikera1, Ranjith. B. M2, Lavanya. S. A3, Parikshitha Navada4 1Professor & HOD, Dept of Rasashastra & Bhaishajya Kalpana, S.J.G Ayurvedic Medical College & P.G Centre, Koppal, Karnataka, India 2Ayurveda Vaidya; 3Ayurveda Vaidya; 43rd year P.G Scholar, T.G.A.M.C, Bellary, Karnataka, India Email: [email protected] Published online: January, 2019 © International Ayurvedic Medical Journal, India 2019 ABSTRACT Udayabhaskara rasa is an eccentric formulation which is favourable in the management of Amavata. Regardless of its betterment in the management of Amavata, no research work has been carried out till date. The main aim of this study was preparation of Udayabhaskara Rasa as disclosed in the classics & Physico-chemical analysis of Udayabhaskara Rasa. Udayabhaskara Rasa was processed using Kajjali, Vyosha, dwikshara, pancha lavana, jayapala and beejapoora swarasa. The above ingredients were mixed to get a homogenous mixture of Udayab- haskara Rasa which was given 1 Bhavana with beejapoora swarasa and later it is dried and stored in air-tight container. The Physico chemical analysis of Udayabhaskara Rasa before (UB-BB) and after bhavana (UB-AB) was done. Keywords: Udayabhaskara Rasa, XRD, FTIR, SEM-EDAX. INTRODUCTION Rasashastra is a branch of Ayurveda which deals with Udayabhaskara Rasa. In the present study the formu- metallo-mineral preparations aimed at achieving Lo- lation is taken from the text brihat nighantu ratna- havada & Dehavada. These preparations became ac- kara. The analytical study reveals the chemical com- ceptable due to its assimilatory property in the minute position of the formulations as well as their concentra- doses.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties
    Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 2 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7) Last week: • Crystals, • Crystal Lattice, • Reciprocal Lattice Today: • Types of bonds in crystals Diffraction from crystals • Importance of the reciprocal lattice concept Lecture 2 Andrei Sirenko, NJIT 2 1 (3) The Hexagonal Closed-packed (HCP) structure Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl • The HCP structure is made up of stacking spheres in a ABABAB… configuration • The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms • Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are not all perpendicular) • The number of nearest-neighbours is 12 • The ideal ratio of c/a for Rotated this packing is (8/3)1/2 = 1.633 three times . Lecture 2 Andrei Sirenko, NJITConventional HCP unit 3 cell Crystal Lattice http://www.matter.org.uk/diffraction/geometry/reciprocal_lattice_exercises.htm Lecture 2 Andrei Sirenko, NJIT 4 2 Reciprocal Lattice Lecture 2 Andrei Sirenko, NJIT 5 Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice 3 a1 = ax, a2 = ay, a3 = az V = a1·(a2a3) = a b1 = (2/a)x, b2 = (2/a)y, b3 = (2/a)z reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc lattice 1 1 a a x y z a2 ax y z 1 2 2 1 1 a ax y z V a a a a3 3 2 1 2 3 2 2 2 2 b y z b x z b x y 1 a 2 a 3 a Lecture 2 Andrei Sirenko, NJIT 6 3 2 got b y z 1 a 2 b x z 2 a 2 b x y 3 a but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc.
    [Show full text]
  • An Atomistic Study of Phase Transition in Cubic Diamond Si Single Crystal T Subjected to Static Compression ⁎ Dipak Prasad, Nilanjan Mitra
    Computational Materials Science 156 (2019) 232–240 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci An atomistic study of phase transition in cubic diamond Si single crystal T subjected to static compression ⁎ Dipak Prasad, Nilanjan Mitra Indian Institute of Technology Kharagpur, Kharagpur 721302, India ARTICLE INFO ABSTRACT Keywords: It is been widely experimentally reported that Si under static compression (typically in a Diamond Anvil Setup- Molecular dynamics DAC) undergoes different phase transitions. Even though numerous interatomic potentials are used fornu- Phase transition merical studies of Si under different loading conditions, the efficacy of different available interatomic potentials Hydrostatic and Uniaxial compressive loading in determining the phase transition behavior in a simulation environment similar to that of DAC has not been Cubic diamond single crystal Silicon probed in literature which this manuscript addresses. Hydrostatic compression of Silicon using seven different interatomic potentials demonstrates that Tersoff(T0) performed better as compared to other potentials with regards to demonstration of phase transition. Using this Tersoff(T0) interatomic potential, molecular dynamics simulation of cubic diamond single crystal silicon has been carried out along different directions under uniaxial stress condition to determine anisotropy of the samples, if any. -tin phase could be observed for the [001] direction loading whereas Imma along with -tin phase could be observed for [011] and [111] direction loading. Amorphization is also observed for [011] direction. The results obtained in the study are based on rigorous X-ray diffraction analysis. No strain rate effects could be observed for the uniaxial loading conditions. 1. Introduction potential(SW) [19,20] for their simulations.
    [Show full text]
  • Part 1: Supplementary Material Lectures 1-5
    Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2014 Part 1: Supplementary Material Lectures 1-5 Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option Contents 1 Frieze patterns and frieze groups 3 2 Symbols for frieze groups 4 2.1 A few new concepts from frieze groups . .5 2.2 Frieze groups in the ITC . .7 3 Wallpaper groups 8 3.1 A few new concepts for Wallpaper Groups . .8 3.2 Lattices and the “translation set” . .9 3.3 Bravais lattices in 2D . .9 3.3.1 Oblique system . 10 3.3.2 Rectangular system . 10 3.3.3 Square system . 10 3.3.4 Hexagonal system . 11 3.4 Primitive, asymmetric and conventional Unit cells in 2D . 11 3.5 The 17 wallpaper groups . 12 3.6 Analyzing wallpaper and other 2D art using wallpaper groups . 12 1 4 Fourier transform of lattice functions 13 5 Point groups in 3D 14 5.1 The new generalized (proper & improper) rotations in 3D . 14 5.2 The 3D point groups with a 2D projection . 15 5.3 The other 3D point groups: the 5 cubic groups . 15 6 The 14 Bravais lattices in 3D 19 7 Notation for 3D point groups 21 7.0.1 Notation for ”projective” 3D point groups . 21 8 Glide planes in 3D 22 9 “Real” crystal structures 23 9.1 Cohesive forces in crystals — atomic radii . 23 9.2 Close-packed structures . 24 9.3 Packing spheres of different radii . 25 9.4 Framework structures . 26 9.5 Layered structures .
    [Show full text]
  • Valleytronics: Opportunities, Challenges, and Paths Forward
    REVIEW Valleytronics www.small-journal.com Valleytronics: Opportunities, Challenges, and Paths Forward Steven A. Vitale,* Daniel Nezich, Joseph O. Varghese, Philip Kim, Nuh Gedik, Pablo Jarillo-Herrero, Di Xiao, and Mordechai Rothschild The workshop gathered the leading A lack of inversion symmetry coupled with the presence of time-reversal researchers in the field to present their symmetry endows 2D transition metal dichalcogenides with individually latest work and to participate in honest and addressable valleys in momentum space at the K and K′ points in the first open discussion about the opportunities Brillouin zone. This valley addressability opens up the possibility of using the and challenges of developing applications of valleytronic technology. Three interactive momentum state of electrons, holes, or excitons as a completely new para- working sessions were held, which tackled digm in information processing. The opportunities and challenges associated difficult topics ranging from potential with manipulation of the valley degree of freedom for practical quantum and applications in information processing and classical information processing applications were analyzed during the 2017 optoelectronic devices to identifying the Workshop on Valleytronic Materials, Architectures, and Devices; this Review most important unresolved physics ques- presents the major findings of the workshop. tions. The primary product of the work- shop is this article that aims to inform the reader on potential benefits of valleytronic 1. Background devices, on the state-of-the-art in valleytronics research, and on the challenges to be overcome. We are hopeful this document The Valleytronics Materials, Architectures, and Devices Work- will also serve to focus future government-sponsored research shop, sponsored by the MIT Lincoln Laboratory Technology programs in fruitful directions.
    [Show full text]
  • Electronic Structure Theory for Periodic Systems: the Concepts
    Electronic Structure Theory for Periodic Systems: The Concepts Christian Ratsch Institute for Pure and Applied Mathematics and Department of Mathematics, UCLA Motivation • There are 1020 atoms in 1 mm3 of a solid. • It is impossible to track all of them. • But for most solids, atoms are arranged periodically Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Outline Part 1 Part 2 • Periodicity in real space • The supercell approach • Primitive cell and Wigner Seitz cell • Application: Adatom on a surface (to calculate for example a diffusion barrier) • 14 Bravais lattices • Surfaces and Surface Energies • Periodicity in reciprocal space • Surface energies for systems with multiple • Brillouin zone, and irreducible Brillouin zone species, and variations of stoichiometry • Bandstructures • Ab-Initio thermodynamics • Bloch theorem • K-points Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell a2 a1 a2 a1 primitive cell Non-primitive cell Wigner-Seitz cell = + 푹 푁1풂1 푁2풂2 Christian Ratsch, UCLA IPAM, DFT Hands-On Workshop, July 22, 2014 Primitive Cell and Wigner Seitz Cell Face-center cubic (fcc) in 3D: Primitive cell Wigner-Seitz cell = + + 1 1 2 2 3 3 Christian푹 Ratsch, UCLA푁 풂 푁 풂 푁 풂 IPAM, DFT Hands-On Workshop, July 22, 2014 How Do Atoms arrange in Solid Atoms with no valence electrons: Atoms with s valence electrons: Atoms with s and p valence electrons: (noble elements) (alkalis) (group IV elements) • all electronic shells are filled • s orbitals are spherically • Linear combination between s and p • weak interactions symmetric orbitals form directional orbitals, • arrangement is closest packing • no particular preference for which lead to directional, covalent of hard shells, i.e., fcc or hcp orientation bonds.
    [Show full text]
  • Electronic and Structural Properties of Silicene and Graphene Layered Structures
    Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2012 Electronic and Structural Properties of Silicene and Graphene Layered Structures Patrick B. Benasutti Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Benasutti, Patrick B., "Electronic and Structural Properties of Silicene and Graphene Layered Structures" (2012). Browse all Theses and Dissertations. 1339. https://corescholar.libraries.wright.edu/etd_all/1339 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Electronic and Structural Properties of Silicene and Graphene Layered Structures A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by Patrick B. Benasutti B.S.C.E. and B.S., Case Western Reserve and Wheeling Jesuit, 2009 2012 Wright State University Wright State University SCHOOL OF GRADUATE STUDIES August 31, 2012 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVI- SION BY Patrick B. Benasutti ENTITLED Electronic and Structural Properties of Silicene and Graphene Layered Structures BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Physics. Dr. Lok C. Lew Yan Voon Thesis Director Dr. Douglas Petkie Department Chair Committee on Final Examination Dr. Lok C. Lew Yan Voon Dr. Brent Foy Dr. Gregory Kozlowski Dr. Andrew Hsu, Ph.D., MD Dean, Graduate School ABSTRACT Benasutti, Patrick.
    [Show full text]
  • Crystal Structure-Crystalline and Non-Crystalline Materials
    ASSIGNMENT PRESENTED BY :- SOLOMON JOHN TIRKEY 2020UGCS002 MOHIT RAJ 2020UGCS062 RADHA KUMARI 2020UGCS092 PALLAVI PUSHPAM 2020UGCS122 SUBMITTED TO:- DR. RANJITH PRASAD CRYSTAL STRUCTURE CRYSTALLINE AND NON-CRYSTALLINE MATERIALS . Introduction We see a lot of materials around us on the earth. If we study them , we found few of them having regularity in their structure . These types of materials are crystalline materials or solids. A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances; that is, long-range order exists, such that upon solidification, the atoms will position themselves in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest-neighbor atoms. X-ray diffraction photograph [or Laue photograph for a single crystal of magnesium. Crystalline solids have well-defined edges and faces, diffract x-rays, and tend to have sharp melting points. What is meant by Crystallography and why to study the structure of crystalline solids? Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The properties of some materials are directly related to their crystal structures. For example, pure and undeformed magnesium and beryllium, having one crystal structure, are much more brittle (i.e., fracture at lower degrees of deformation) than pure and undeformed metals such as gold and silver that have yet another crystal structure. Furthermore, significant property differences exist between crystalline and non-crystalline materials having the same composition. For example, non-crystalline ceramics and polymers normally are optically transparent; the same materials in crystalline (or semi-crystalline) forms tend to be opaque or, at best, translucent.
    [Show full text]