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Chapter 5 Structure of Solids

Chapter 5 Structure of Solids. 6 Lectures. Solids. Crystalline. Noncrystalline. Long-range periodicity. No long-range periodicity. Gives sharp diffraction patterns. Does not give sharp diffraction patterns. Does not have a sharp meliing point. Has sharp melting point.

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Chapter 5 Structure of Solids

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  1. Chapter 5 Structure of Solids 6 Lectures

  2. Solids Crystalline Noncrystalline Long-range periodicity No long-range periodicity Gives sharp diffraction patterns Does not give sharp diffraction patterns Does not have a sharp meliing point Has sharp melting point Has a lower density Has higher density

  3. Factors promoting the formation of noncrystalline structures • Primary bonds do not extend in all three directions and the secondary bonds are not strong enough. • The difference in the free energy of the crystalline and non crystalline phases is small. • The rate of cooling from the liquid state is higher than a critical cooling rate. Metallic Glass: 106 K s-1

  4. Inorganic Solids Covalent Solids Metals and Alloys Ionic Solids Silica: crystalline and amorphous Polymers Classification Structure Crystallinity Mechanical Behaviour

  5. 7th. Group (Halogens): single covalent bonds Diatomic molecules Weak van der Waals bond between molecules F2, Cl2: Gas; Br2: Liquid; I2: orthorhombic xl

  6. 6th. Group: two covalent bonds: long zig-zag chains Weak van der Waals bonds between chains mostly noncrystalline

  7. 5th. Group: Three covalent bonds: Puckered sheets Weak van der Waals bond between sheets Mostly noncrystalline

  8. 4th. Group: Carbon

  9. Allotropes of C Graphite Diamond Buckminster Fullerene1985 Graphene2004 Carbon Nanotubes1991

  10. Graphite Sp2 hybridization  3 covalent bonds  Hexagonal sheets a = 2 d cos 30° = √3 d y x d = 1.42 Å a = 2.46 Å =120 b=a a

  11. Graphite a = 2.46 Å c = 6.70 Å Lattice: Simple Hexagonal Motif: 4 carbon atoms A c B y x A www.scifun.ed.ac.uk/

  12. Graphite Highly Anisotropic: Properties are very different in the a and c directions Uses: Solid lubricant Pencils (clay + graphite, hardness depends on fraction of clay) carbon fibre www.sciencemuseum.org.uk/

  13. Diamond Sp3 hybridization  4 covalent bonds  Tetrahedral bonding Location of atoms: 8 Corners 6 face centres 4 one on each of the 4 body diagonals

  14. Diamond Cubic Crystal: Lattice & motif? y 0,1 0,1 R M D C y M R L S N P Q Q T D 0,1 L S K C N K T A x B B A P x 0,1 0,1 Projection of the unit cell on the bottom face of the cube Diamond Cubic Crystal = FCC lattice + motif: 000; ¼¼¼

  15. Diamond Cubic Crystal Structure FCCLattice 2 atomMotif = + Crystal Structure = Lattice + Motif There are only three Bravais Lattices: SC, BCC, FCC. Diamond Cubic Lattice

  16. There is no diamond cubic lattice.

  17. Diamond Cubic Structure Coordination number 4 Face Inside Corners Effective number of atoms in the unit cell = Relaton between lattice parameter and atomic radius Packing efficiency

  18. Diamond Cubic Crystal Structures C Si Ge Gray Sn a (Å) 3.57 5.43 5.65 6.46

  19. Equiatomic binary AB compounds having diamond cubic like structure y 0,1 0,1 IV-IV compound: SiC III-V compound: AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, InSb II-VI compound: ZnO, ZnS, CdS, CdSe, CdTe I-VII compound: CuCl, AgI 0,1 S 0,1 0,1

  20. USES: Diamond Abrasive in polishing and grinding wire drawing dies Si, Ge, compounds: semiconducting devices SiC abrasives, heating elements of furnaces

  21. Inorganic Solids Covalent Solids Metals and Alloys Ionic Solids Silica: crystalline and amorphous Polymers Classification Structure Crystallinity Mechanical Behaviour

  22. Metals and Alloys 1. Metallic bond: Nondrectional (Fact) As many bonds as geometrically possible (to lower the energy)  Close packing 2. Atoms as hard sphere (Assumption) 3. Elements (identical atoms) 1, 2 & 3  Elemental metal crystals: close packing of equal hard spheres

  23. Close packing of equal hard spheres Arrangement of equal nonoverlapping spheres to fill space as densely as possible Sphere packing problem:What is the densest packing of spheres in 3D? Kepler’s conjecture, 1611 Kissing Number Problem What is the maximum number of spheres that can touch a given sphere? Coding Theory Internet data transmission

  24. Close packing of equal hard spheres 1-D packing A chain of spheres 2 P.E.= Kissing Number= =1 Close-packed direction of atoms

  25. Close packing of equal hard spheres 2-D packing A hexagonal layer of atoms Close-packed plane of atoms Close-packed directions? 3 Kissing Number=6 P.E.= 1940 L. Fejes Toth : Densest packing of circles in plane

  26. Close packing of equal hard spheres 3-D packing First layer A A A Second layer B A A C C C B B B A A Third layer A or C A A C C C B B B A A A A C C C B B B A A A A Close packed crystals: …ABABAB… Hexagonal close packed (HCP)…ABCABC… Cubic close packed (CCP)

  27. Geometrical properties of ABAB stacking B b = a A A A A =120 C C C a A B B B A A A A c C C C B B B B A A A A A C C C B B B A A A A A and B do not have identical neighbours Either A or B as lattice points, not both Unit cell: a rhombus based prism with a=bc; ==90, =120 The unit cell contains only one lattice point (simple) but two atoms (motif) ABAB stacking = HCP crystal = Hexagonal P lattice + 2 atom motif 000 2/3 1/3 1/2

  28. c/a ratio of an ideal HCP crystal B A A A A C C C A B B B A A A A c C C C B B B B A A A A A C C C B B B A A A A A single B atom sitting on a base of three A atoms forms a regular tetrahedron with edge length a = 2R The same B atom also forms an inverted tetrahedron with three A atoms sitting above it c = 2  height of a tetrahedron of edge length a

  29. 3 a Geometrical properties of ABCABC stacking C B A A A A A C C C C B B B A A A A B C C C A B B B A A A A C C C B B B A A A A All atoms are equivalent and their centres form a lattice Motif: single atom 000 ABCABC stacking = CCP crystal = FCC lattice + single atom motif 000

  30. 3 a Geometrical properties of ABCABC stacking C B A C B A All atoms are equivalent and their centres form a lattice Motif: single atom 000 ABCABC stacking = CCP crystal = FCC lattice + single atom motif 000

  31. 3 a Geometrical properties of ABCABC stacking C B A A A A A A C C C C B B B A A A A A B C C C A B B B A A A A A C C C B B B A A A A A A A All atoms are equivalent and their centres form a lattice Motif: single atom 000 ABCABC stacking = CCP crystal = FCC lattice + single atom motif 000

  32. Body diagonal C Close packed planes in the FCC unit cell of cubic close packed crystal B A B A Close packed planes: {1 1 1}

  33. Stacking sequence? ABA: HCP

  34. Find the mistake in the following picture: http://www.tiem.utk.edu/~gross/bioed/webmodules/spherefig1.gif

  35. Table 5.1 Coordination Number and Packing Efficiency CrystalStructure Coordinationnumber Packingefficiency Diamond cubic (DC) 4 0.34 Simple cubic (SC) 6 0.52 Body-centred cubic 8 0.68 Face-centred cubic 12 0.74

  36. Voids in Close-Packed Crystals TETRAHEDRAL VOID OCTAHEDRAL VOID B A A A B C A A B B A A A No. of atoms defining 4 6 the void No. of voids per atom 2 1 Edge length of void 2 R 2 R Size of the void 0.225 R 0.414 R A Experiment 2 HW

  37. Location of Voids in FCC Unit cell

  38. Solid Solution A single crystalline phase consisting of two or more elements is called a solid solution. Substitutional Solid solution of Cu and Zn (FCC) Interstitial solid solution of C in Fe (BCC)

  39. Hume-Rothery Rules for Extensive Solid Solution (Unlimited solubility) Interstitial solid solution Substitutional solid solution • Structure factor Crystal structure of the two elements should be the same • Size factor: Size of the two elements should not differ by more than 15% 3. Electronegativity factor: Electronegativity difference between the elements should be small 4. Valency factor: Valency of the two elements should be the same

  40. TABLE 5.2 SystemCrystal Radius of Valency Electro- structureatoms, Ǻ negativity Ag-CuAg FCC 1.4411.9 AuFCC 1.4411.9 Cu-NiCu FCC 1.2811.9 Ni FCC 1.2521.8 Ge-SiGe DC 1.2241.8 SiDC 1.1841.8 All three systems exhibit complete solid solubility.

  41. BRASS Cu + Zn FCC HCP Unfavourable structure factor Limited Solubility: Max solubility of Cu in Zn: 1 wt% Cu Max Solubility of Zn in Cu: 35 wt% Zn

  42. Ordered and RandomSubstitutional solid solution Random Solid Solution Ordered Solid Solution

  43. Ordered and random substitutional solid solution β-Brass: (50 at% Zn, 50 at% Cu) Above 470˚C Disordered solid solution of β-Brass: Corner and centre both have 50% probability of being occupied by Cu or Zn 470˚C Below 470˚C Ordered solid solution of β-Brass: Corners are always occupied by Cu, centres always by Zn

  44. Intermediate Structures FCC Crystal structure of Cu: Crystal structure of Zn: HCP Crystal structure of random β-brass: BCC Such phases that have a crystal structure different from either of the two components are called INTERMEDIATE STRUCURES If an intermediate structure occurs only at a fixed composition it is called an INTERMETALLIC COMPOUND, e.g. Fe3C in steels.

  45. IONIC SOLIDS Cation radius: R+ Anion radius: R- Usually 1. Cation and anion attract each other. 2. Cation and anion spheres touch each other 3. Ionic bonds are non-directional 1, 2, 3 => Close packing of unequal spheres

  46. IONIC SOLIDS Local packing geometry • 1. Anions and cations considered as hard spheres always touch each other. • Anions generally will not touch, but may be close enough to be in contact with each other in a limiting situation. • As many anions as possible surround a central cation for the maximum reduction in electrostatic energy.

  47. Effect of radius ratio unstable Critically stable stable Anions not touching the central cation, Anions touching each other Anions touching the central cation Anions touching Anions touching central cation Anions not touching each other

  48. However, when tetrahedral coordination with ligancy 4 becomes stable Recall tetrahedral void in close-packed structure. Thus

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