ELE 362: Structures of Materials - Department of .crystalline materials are the unit cells ... specify

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  • ELE 362: Structures of Materials

    James V. Masi, Ph.D.


    1028 atoms/m3 Crystallites approach size of unit cell, i.e. A (10-10 m)

    Regular arrangementBasic building blocks areunit cells NO LONG RANGE ORDER

    Portions perfect--polycrystalline--grain boundaries

    Crystalline solids have of the order of 1028 atoms/cubic meter and have a regular arrangement of atoms. Amorphous solids have about the same density of atoms, but lack the long range order of the crystalline solid (usually no greater than a few Angstroms). Their crystallites approach the order of the unit cell. The basic building blocks of crystalline materials are the unit cells which are repeated in space to form the crystal. However, most of the materials which we encounter (eg. a copper wire, a horseshoe magnet, a sugar cube, a galvanized steel sheet) are what we call polycrystalline. This word implies that portions of the solid are perfect crystals, but not all of these crystals have the same axis of symmetry. Each of the crystals (or grains) have slightly different orientations with boundaries bordering on each other. These borders are called grain boundaries.

  • Crystalline and amorphous solids

  • Polycrystals and grain boundaries


  • Polycrystal G.B. Movement

  • Graphite


    Buckminsterfullerene Crystal


    Covalent bonding within layers. Van der Waals bonding between layers. Hexagonal unit cell.

    Covalently bonded network. Diamond crystal structure.

    Covalently bonded C60 spheroidal molecules held in an FCC crystal structure by van der Waals bonding.

    Electrical And Thermal Properties

    Good electrical conductor. Thermal conductivity comparable to metals.

    Very good electrical insulator. Excellent thermal conductor, about 5 times more than silver or copper.

    Semiconductor. Compounds with alkali metals (e.g. K3C60) exhibit superconductivity.

    Mechanical Properties

    Lubricating agent. Machinable. Bulk graphite: Y 27 Gpa, = 2.25 g cm-3

    The hardest material. Y = 827 GPa = 3.5 g cm-3

    Mechanically soft. Y 18 GPa = 1.65 g cm-3


    Stable allotrope at atmospheric pressure.

    High pressure allotrope.

    Laboratory synthesized. Occurs in the soot of partial combustion.

    Uses, Potential Uses

    Metallurgical crucibles, welding electrodes heating elements, electrical contacts, refractory applications.

    Cutting tool applications. Diamond anvils. Diamond film coated drills, blades, bearings etc. Jewelry. Heat conductor for ICs. Possible thin film semiconductor devices as the charge carrier drift mobilities are large.

    Possible future semiconductor or superconductivity applications.

    Crystalline allotropes of carbon (r is the density and Y is the elastic modulus or Young's modulus)

  • Pictorial of carbon allotropes

    NOTE: there is also aQuasi-allotrope calledDiamond-like carbon

  • Crystalline Sic-Si

    Amorphous Si a-Si

    Hydrogenated a-Si a-Si:H


    Diamond cubic.

    Short range order only. On average, each Si covalently bonds with four Si atoms. Has microvoids and dangling bonds

    Short range order only. Structure contains typically 10% H. Hydrogen atoms passivate dangling bonds and relieve strain from bonds.

    Typical Preparation

    Czochralski technique

    Electron beam evaporation of Si

    Chemical vapor deposition (CVD) of silane gas by RF (radio frequency) plasma.

    Density g cm-3


    About 3-10% less dense.

    About 1-3% less dense

    Electronic Applications

    Discrete and integrated electronic devices.


    Large area electronic devices such as solar cells, thin film transistor arrays in flat panel displays and photoconductor drums used in photocopying.

    Crystalline and amorphous silicon

  • Crystal Systems

  • Bravais Lattices

    In 1850, Auguste Bravaisshowed that crystals could be divided into 14 unit cells, which meet the following criteria.

    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.

    The 14 Bravais unit cells are shown in the figure adjacent

  • THE UNIT CELLIn order to further our knowledge and ability to apply the principals of atomic and crystalline structure to the problems at hand, we must first define a concept on which hinges the rest of crystallography. This concept is the unit cell. The unit cell is defined as the smallest group of atoms which, by repeated translation in three dimensions, builds up the whole crystal. This means that the unit cell consists only of those atoms which exclusively belong to that basic building block. The concept is better shown in pictorial fashion for three of the cubic crystal types

  • Number of Atoms per Unit Cell

    In a mathematical way, for the cubic lattice, the number of atoms per unit cell is given by

    N = N(c)/8 + N(i) + N(f)/2

    where N is the number of atoms per unit cell, N(c) is the number of corner atoms (shared by eight), N(i) is the number of internal atoms (shared by none), and N(f) is the number of face atoms (shared by two). So, for the BCC and FCC, the number of atoms per unit cell are 2 and 4 respectively. Similar calculations can be performed for other lattices.

  • Determining the Unit Cell of a Crystal

    Atoms on the corners, edges, and faces of a unit cell (cubic) are shared by more than one unit cell, as shown in the figure below. An atom on a face is shared by two unit cells, so only half of the atom belongs to each of these cells. An atom on an edge is shared by four unit cells, and an atom on a corner is shared by eight unit cells. Thus, only one-quarter of an atom on an edge and one-eighth of an atom on a corner can be assigned to each of the unit cells that share these atoms.

  • Simple Cubic: unit cell

  • Body Centered Cubic: unit cell

  • Cubic Close Packed (FCC): unit cell

  • NaCl (salt shaker variety)

  • Hexagonal Close Packed: unit cell

  • CsClThinking about the unit cell as a three-dimensional graph allows us to describe the structure of a crystal with a remarkably small amount of information. We can specify the structure of cesium chloride, for example, with only four pieces of information.

    CsCl crystallizes in a cubic unit cell. The length of the unit cell edge is 0.4123 nm. There is a Cl- ion at the coordinates 0,0,0. There is a Cs+ ion at the coordinates 1/2,1/2,1/2.

    Because the cell edge must connect equivalent lattice points, the presence of a Cl- ion at one corner of the unit cell (0,0,0) implies the presence of a Cl- ion at every corner of the cell. The coordinates 1/2,1/2,1/2 describe a lattice point at the center of the cell. Because there is no other point in the unit cell that is one cell-edge length away from these coordinates, this is the only Cs+ ion in the cell.CsCl is therefore a simple cubic unit cell of Cl- ions with a Cs+ in the center of the body of the cell.

  • Diamond

  • Diamond cubic structure

  • Coordination numbercoordination number: the number of atoms closest to any given atom in a crystal.

    The arrangement of atoms in a crystal structure not only depends on the charge on the ion and type of bonding between atoms, but also on the size of the atoms or ions.In any given molecule or crystal structure each atom or ion will be surrounded by other atoms or ions. The number of ions or atoms that immediately surround an atom or ion of interest is called the coordination number, - C.N.

    As we shall see, the coordination number depends on the relative size of the atoms or ions. So, we must first discuss their sizes.

    At values of Rx/Rz < 0.155 the only way the smaller ion can be coordinated by the larger ions is to have 2 of the larger ions on either side. This 2-fold coordination is termed linear coordination.

    For triangular coordination, the coordination number is three, that is 3 anions surround the smallercation. The "no rattle" limit is reached for triangular coordination when Rx/Rz becomes less than 0.155.

  • Coordination numberFor Rx/Rz < 0.414 the structure goes into 4-fold coordination. Planes through the centers of the larger atoms in this case will form a tetrahedron, so 4-fold coordination is also called tetrahedral coordination.Six-fold coordination is also called octahedral coordination because the shape defined by drawing planes through the center of the larger ions is an octahedron. Octahedral coordination is stable when Rx/Rz , 0.732, but decreasing the radius of cation, Rx, will eventually reach a limit where again the smaller ion will rattle in its site.

    The no rattle limit can be determined by looking at the horizontal plane running through the ions labeled C and D. In this case we can write: (2Rz + 2Rx)2 = (2Rz)2 + (2Rz)2 = 2(4Rz2)2Rz +2Rx = 22RzRz + Rx = 2RzRx = (2 - 1)RzRx/Rz = 0.414

  • Coordination Numbers

    The table here summarizes thecation to anion radius ratios, Rx/Rz, for various coordination numbers and gives the name of the coordination polyhedron for each coordination number.

  • Perovskite Structure CaTiO3

  • Ferroelectric Barium Titanate:What happens when your marbles are loose

  • CeramicsCrystal structures

    oxygen larger generally in FCC latticecations go in lattice sites based on

    sizestoichiometrycharge balancebond hybridization

    no good slip planes brittle failure

    Silicatesbuilt up o