Types of Lattices
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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. -
C:\Documents and Settings\Alan Smithee\My Documents\MOTM
I`mt`qx1/00Lhmdq`knesgdLnmsg9Rbnkdbhsd This month’s mineral, scolecite, is an uncommon zeolite from India. Our write-up explains its origin as a secondary mineral in volcanic host rocks, the difficulty of collecting this fragile mineral, the unusual properties of the zeolite-group minerals, and why mineralogists recently revised the system of zeolite classification and nomenclature. OVERVIEW PHYSICAL PROPERTIES Chemistry: Ca(Al2Si3O10)A3H2O Hydrous Calcium Aluminum Silicate (Hydrous Calcium Aluminosilicate), usually containing some potassium and sodium. Class: Silicates Subclass: Tectosilicates Group: Zeolites Crystal System: Monoclinic Crystal Habits: Usually as radiating sprays or clusters of thin, acicular crystals or Hairlike fibers; crystals are often flattened with tetragonal cross sections, lengthwise striations, and slanted terminations; also massive and fibrous. Twinning common. Color: Usually colorless, white, gray; rarely brown, pink, or yellow. Luster: Vitreous to silky Transparency: Transparent to translucent Streak: White Cleavage: Perfect in one direction Fracture: Uneven, brittle Hardness: 5.0-5.5 Specific Gravity: 2.16-2.40 (average 2.25) Figure 1. Scolecite. Luminescence: Often fluoresces yellow or brown in ultraviolet light. Refractive Index: 1.507-1.521 Distinctive Features and Tests: Best field-identification marks are acicular crystal habit; vitreous-to-silky luster; very low density; and association with other zeolite-group minerals, especially the closely- related minerals natrolite [Na2(Al2Si3O10)A2H2O] and mesolite [Na2Ca2(Al6Si9O30)A8H2O]. Laboratory tests are often needed to distinguish scolecite from other zeolite minerals. Dana Classification Number: 77.1.5.5 NAME The name “scolecite,” pronounced SKO-leh-site, is derived from the German Skolezit, which comes from the Greek sklx, meaning “worm,” an allusion to the tendency of its acicular crystals to curl when heated and dehydrated. -
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. -
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 . -
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. -
Space Symmetry, Space Groups
Space symmetry, Space groups - Space groups are the product of possible combinations of symmetry operations including translations. - There exist 230 different space groups in 3-dimensional space - Comparing to point groups, space groups have 2 more symmetry operations (table 1). These operations include translations. Therefore they describe not only the lattice but also the crystal structure. Table 1: Additional symmetry operations in space symmetry, their description, symmetry elements and Hermann-Mauguin symbols. symmetry H-M description symmetry element operation symbol screw 1. rotation by 360°/N screw axis NM (Schraubung) 2. translation along the axis (Schraubenachse) 1. reflection across the plane glide glide plane 2. translation parallel to the a,b,c,n,d (Gleitspiegelung) (Gleitspiegelebene) glide plane Glide directions for the different gilde planes: a (b,c): translation along ½ a or ½ b or ½ c, respectively) (with ,, cba = vectors of the unit cell) n: translation e.g. along ½ (a + b) d: translation e.g. along ¼ (a + b), d from diamond, because this glide plane occurs in the diamond structure Screw axis example: 21 (N = 2, M = 1) 1) rotation by 180° (= 360°/2) 2) translation parallel to the axis by ½ unit (M/N) Only 21, (31, 32), (41, 43), 42, (61, 65) (62, 64), 63 screw axes exist in parenthesis: right, left hand screw axis Space group symbol: The space group symbol begins with a capital letter (P: primitive; A, B, C: base centred I, body centred R rhombohedral, F face centred), which represents the Bravais lattice type, followed by the short form of symmetry elements, as known from the point group symbols. -
Apophyllite-(Kf)
December 2013 Mineral of the Month APOPHYLLITE-(KF) Apophyllite-(KF) is a complex mineral with the unusual tendency to “leaf apart” when heated. It is a favorite among collectors because of its extraordinary transparency, bright luster, well- developed crystal habits, and occurrence in composite specimens with various zeolite minerals. OVERVIEW PHYSICAL PROPERTIES Chemistry: KCa4Si8O20(F,OH)·8H20 Basic Hydrous Potassium Calcium Fluorosilicate (Basic Potassium Calcium Silicate Fluoride Hydrate), often containing some sodium and trace amounts of iron and nickel. Class: Silicates Subclass: Phyllosilicates (Sheet Silicates) Group: Apophyllite Crystal System: Tetragonal Crystal Habits: Usually well-formed, cube-like or tabular crystals with rectangular, longitudinally striated prisms, square cross sections, and steep, diamond-shaped, pyramidal termination faces; pseudo-cubic prisms usually have flat terminations with beveled, distinctly triangular corners; also granular, lamellar, and compact. Color: Usually colorless or white; sometimes pale shades of green; occasionally pale shades of yellow, red, blue, or violet. Luster: Vitreous to pearly on crystal faces, pearly on cleavage surfaces with occasional iridescence. Transparency: Transparent to translucent Streak: White Cleavage: Perfect in one direction Fracture: Uneven, brittle. Hardness: 4.5-5.0 Specific Gravity: 2.3-2.4 Luminescence: Often fluoresces pale yellow-green. Refractive Index: 1.535-1.537 Distinctive Features and Tests: Pseudo-cubic crystals with pearly luster on cleavage surfaces; longitudinal striations; and occurrence as a secondary mineral in association with various zeolite minerals. Laboratory analysis is necessary to differentiate apophyllite-(KF) from closely-related apophyllite-(KOH). Can be confused with such zeolite minerals as stilbite-Ca [hydrous calcium sodium potassium aluminum silicate, Ca0.5,K,Na)9(Al9Si27O72)·28H2O], which forms tabular, wheat-sheaf-like, monoclinic crystals. -
Infrare D Transmission Spectra of Carbonate Minerals
Infrare d Transmission Spectra of Carbonate Mineral s THE NATURAL HISTORY MUSEUM Infrare d Transmission Spectra of Carbonate Mineral s G. C. Jones Department of Mineralogy The Natural History Museum London, UK and B. Jackson Department of Geology Royal Museum of Scotland Edinburgh, UK A collaborative project of The Natural History Museum and National Museums of Scotland E3 SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Firs t editio n 1 993 © 1993 Springer Science+Business Media Dordrecht Originally published by Chapman & Hall in 1993 Softcover reprint of the hardcover 1st edition 1993 Typese t at the Natura l Histor y Museu m ISBN 978-94-010-4940-5 ISBN 978-94-011-2120-0 (eBook) DOI 10.1007/978-94-011-2120-0 Apar t fro m any fair dealin g for the purpose s of researc h or privat e study , or criticis m or review , as permitte d unde r the UK Copyrigh t Design s and Patent s Act , 1988, thi s publicatio n may not be reproduced , stored , or transmitted , in any for m or by any means , withou t the prio r permissio n in writin g of the publishers , or in the case of reprographi c reproductio n onl y in accordanc e wit h the term s of the licence s issue d by the Copyrigh t Licensin g Agenc y in the UK, or in accordanc e wit h the term s of licence s issue d by the appropriat e Reproductio n Right s Organizatio n outsid e the UK. Enquirie s concernin g reproductio n outsid e the term s state d here shoul d be sent to the publisher s at the Londo n addres s printe d on thi s page. -
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. -
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. -
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. -
Crystal Systems and Example Minerals
Basics of Mineralogy Geology 200 Geology for Environmental Scientists Terms to Know: •Atom • Bonding • Molecule – ionic •Proton – covalent •Neutron – metallic • Electron • Isotope •Ion Fig. 3.3 Periodic Table of the Elements Fig 3.4A Ionic Bonding Fig 3.4B Covalent Bonding Figure 3.5 -- The effects of temperature and pressure on the physical state of matter, in this case water. The 6 Crystal Systems • All have 3 axes, except for 4 axes in Hexagonal system • Isometric -- all axes equal length, all angles 90ο • Hexagonal -- 3 of 4 axes equal length, three angles@ 90ο, three @ 120ο • Tetragonal -- two axes equal length, all angles 90ο (not common in rock forming minerals) • Orthorhombic -- all axes unequal length, all angles 90ο • Monoclinic -- all axes unequal length, only two angles are 90ο • Triclinic -- all axes unequal length, no angles @ 90ο Pyrite -- an example of the isometric crystal system: cubes Galena -- an example of the isometric crystal system: cubes Fluorite -- an example of the isometric crystal system, octahedrons, and an example of variation in color Garnet -- an example of the isometric crystal system: dodecahedrons Garnet in schist, a metamorphic rock Large masses of garnet -- a source for commercial abrasives Quartz -- an example of the hexagonal crystal system. Amethyst variety of quartz -- an example of color variation in a mineral. The purple color is caused by small amounts of iron. Agate -- appears to be a noncrystalline variety of quartz but it has microscopic fibrous crystals deposited in layers by ground water. Calcite crystals. Calcite is in the hexagonal crystal system. Tourmaline crystals grown together like this are called “twins”.