Lattices

• Lattice


  An infinite array of points in space, in which each point has identical surroundings to all others
• Lattice Point


  Any point in a lattice which are infinitesimal points in space
• Unit Cell


  The simplest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal
• Crystal Structure


  The periodic arrangement of atoms in the crystal
• Two-dimensional packing of spheres (atoms)

  • Primitive packing (low space filling)
  • Close packing (high space filling)
• In 1850, Auguste Bravais showed that crystals could be divided into 14 unit cells, which meet the following criteria:
• Criteria for Bravais unit cells

  • The unit cell is the simplest repeating unit in the crystal
  • Opposite faces of a unit cell are parallel
  • The edge of the unit cell connects equivalent points
• 14 Bravais unit cells

  • Cubic (Simple, Body-centered, Face-centered)
  • Tetragonal (Simple, Body-centered)
  • Monoclinic (Simple, End-centered)
  • Orthorhombic (Simple, Body-centered, Face-centered, End-centered)
  • Rhombohedral
  • Hexagonal
  • Triclinic
• Simple cubic (sc)


  One lattice point is present at each of the eight corners of a cube
• Body-centered cubic (bcc)


  There is one host atom (lattice point) at each corner of the cube and one host atom in the center of the cube
• Face-centered cubic (fcc)


  There is one host atom at each corner and one host atom in each face
• Hexagonal close packed (hcp)


  Instead of stacking hexagonal closest packed planes directly above one another, they can be stacked such that atoms in successive planes nestle in the triangular "grooves" of the adjacent plane
• Simple Cubic


  • In a metal, the atoms are all identical, and most are spherical. Metals thus tend to adopt relatively simple structures
  • We can think of this lattice as layers of square packed spheres. The layers are stacked so that each sphere is directly above the one in the layer beneath
• Hexagonal Close Packed (hcp)


  • This is a close-packed array, but the symmetry is different
  • The hexagonal close packed structure can be made by piling layers in the ABABAB sequence
• Cubic Close Packing


  • Also known as Face-Centered cubic (fcc)
  • This cell is made by inserting another atom into each face of the simple cubic lattice – hence the "face-centered cubic" name
  • The layers are in the ABCABC sequence. No other type of packing can exceed the packing efficiency of cubic close packing. There are however, other types of packing with the same efficiency
• Octahedral Holes


  Looking at the right-pointing holes in the blue layer, we see that the gold layer does not nestle into them as it does the left-pointing ones. These cavities are surrounded by 6 atoms in octahedral geometry. This is a 6-coordinate octahedral interstitial site or octahedral hole

  
• Tetrahedral Holes


  Under each gold atom is a small space surrounded by 4 atoms in a tetrahedral arrangement. The tetrahedral arrangement is called a 4-coordinate tetrahedral interstitial site or a tetrahedral hole
• This image is the property of IBM Corporation.
• Comparison of Properties of Metals, Salts and Non-Metallic Elements

  • Metals: Close packed, C.N. = 8-12, Infinite structures, Good conductors, Malleable, ductile, elastic, Lustrous, hard, high tensile strength, Large range of densities and melting points, Tend to form cations, Reducing agents, More electropositive liberate H2 from water, Form alloys with other metals, Electropositive react with O2 to form (mostly ionic, basic) oxides

  Salts: Close packed, C.N. = 4-8, Infinite structures, Insulators as solids, good conductors molten, Brittle, Non-reflective, low tensile strength, Large range of densities; high melting points, Poor conductors, Brittle (can be hard), Non-reflective, low tensile strength, Large range of densities and melting points, Combination of ions, Simple salts are neither oxidizing or reducing, Never liberate H2 from water, Do not react with metals, Do not react with oxygen, Tend to form anions, Oxidizing agents, Never liberate H2 from water, Form salts with metals, React with O2 to form covalent (mostly acidic) oxides

  Non-Metallic Elements: Loose packed, low connectivity, Molecular, chain, layer or infinite structures
• Metallic Lattice Structures


  • Individual metal atoms sit on lattice sites, Metallic crystals tend to be very dense and have high melting points
• Ionic Lattice Structures


  • The atoms are held together by electrostatic forces (ionic bonds), Ionic crystals are hard and have relatively high melting points, Sodium chloride (NaCl) is an example of this type of crystal
• Ionic compounds generally have more complicated structures than metals
• Coordination Number (CN)


  The coordination number (CN) of a specified atom in a chemical species is the number of other atoms directly linked to that specified atom
• Cesium Chloride (CsCl)


  Cesium chloride crystallizes in a cubic lattice, The unit cell may be depicted as two interpenetrating simple cubic cation and anion sublattices
• Coordination Number of CsCl

  Each Cs+ is surrounded by 8 Cl- at the corners of each cube, COORDINATION NUMBER (CN) OF Cs+ = 8

  Each Cl- is also surrounded by 8 Cs+ at the corners of a cube, COORDINATION NUMBER (CN) OF Cl- = 8
• Sodium Chloride (NaCl)


  NaCl adopts a fcc lattice because the smaller Na+ ions can fit into the cavities whereas the interstitial sites are not large enough to accommodate the Cs+ ions. However, the Na+ also end up in the octahedral holes. There are 6 Cl surrounding each Na, and 6 Na around each Cl.
• Lithium Chloride (LiCl)


  Simple salts such as lithium chloride adopt a ccp structure. Lithium is the smallest of all the atoms with the exception of hydrogen, and the big chlorine atoms just pack together with the ccp structure, leaving the small lithium atoms to squeeze into the octahedral holes.
• Zinc Blende (ZnS)


  The Zn atom prefers to occupy these tetrahedral holes, where it is surrounded by only four S atoms. Sulfurs form ABC layers, Zincs fall in tetrahedral holes, CN of Zinc is 4
• Wurtzite (ZnS)


  An alternative form of zinc sulphide. The sulphur ions are arranged in a hexagonal close-packed type of lattice with zinc ions in half the tetrahedral holes.
• Calcium Fluorite (CaF2)


  When the cations are larger, such as those of calcium, the more common CaF2 structure is favoured, with the sites of the cations and anions interchanged. The fluorite structure is favoured when the cations are so big that they need eight anions to cover them.
• Covalent Lattice Structures


  • There are true covalent bonds between all of the atoms in the crystal, You can think of a covalent crystal as one big molecule, Many covalent crystals have extremely high melting points
• Molecular Lattice Structures


  • There are recognizable molecules within their structures, These molecules are held together by non-covalent interactions, like van der Waals forces or hydrogen bonding, Molecular crystals tend to be soft with relatively low melting points
• Counting Atoms in 3D Cells

  Vertex atom shared by 8 cells = 1/8 atom per cell

  Edge atom shared by 4 cells = ¼ atom per cell

  Face atom shared by 2 cells = ½ atom per cell

  Body unique to 1 cell = 1 atom per cell

  
• Why is Atomic Structure Important?
• Physical Properties and Crystal Types

  • Ionic: Hard; high melting points; non-conductors of electricity as solids but good conductors when molten

  Molecular: Soft; low melting points; non-conductors of electricity in both solid & liquid states

  Covalent: Very hard; very high melting points. Nonconductors of electricity

  Metallic: Ranges from very hard to soft; melting points range from high to low; conduct electricity well; have characteristic luster
• ALL compounds are solids under suitable conditions of temperature and pressure. Many exist only as solids.
• Why Study Solids?


  • Appearance: Precious and Semi-precious gemstones of many varieties

  Mechanical Properties: Metals/Alloys, Cement/Concrete, Ceramics, Lubricants, Abrasives

  Electrical Properties: Metallic Conductors, Semiconductors, Superconductors, Electrolytes, Piezoelectrics

  Magnetic Properties: e.g. CrO2, Fe3O4 for recording technology

  Optical Properties: Pigments, Phosphors, Lasers

  Catalysts: Zeolites used in petroleum refining and also in the conversion of methanol to octane

  Sensors: Oxygen sensor e.g. ZrO2/CaO solid solution
• Solids (especially crystals) have always been fascinating.
• Coordination Number


  In an ionic compound, the radius of the positive ion, r+, divided by the radius of the negative ion, r-. The number of anions that can be grouped around a cation will depend on the relative size of the cations and anions.
• Coordination Number (CN): the number of anions that can fit around a cation. This number increases as the radius ratio increases.