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CHAPTER 3

CHAPTER 3. THE STRUCTURE OF CRYSTALLINE SOLIDS. 3.2 FUNDAMENTAL CONCEPTS. SOLIDS. AMORPHOUS. CRYSTALLINE. Atoms in a crystalline solid are arranged in a repetitive three dimensional pattern Long Range Order. Atoms in an amorphous solid are arranged randomly- No Order.

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CHAPTER 3

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  1. CHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS

  2. 3.2 FUNDAMENTAL CONCEPTS SOLIDS AMORPHOUS CRYSTALLINE Atoms in a crystalline solid are arranged in a repetitive three dimensional patternLong Range Order Atoms in an amorphous solid are arranged randomly- No Order All metals are crystalline solids Many ceramics are crystalline solids Some polymers are crystalline solids

  3. LATTICE Lattice -- points arranged in a pattern that repeats itself in three dimensions. The points in a crystal lattice coincides with atom centers 

  4. 3-D view of a lattice

  5. 3.3 UNIT CELL • Unit cell -- smallest grouping which can be arranged in three dimensions to create the lattice. Thus the Unit Cell is basic structural unit or building block of the crystal structure

  6. Unit cell & Lattice

  7. Unit Cell Lattice

  8. 3.4 METALLIC CRYSTALS • Tend to be densely packed. • Have several reasons for dense packing: -Typically, only one element is present, so all atomic radii are thesame. -Metallic bonding is not directional. -Nearest neighbor distances tend to be small in order to have lower bonding energy. • Have the simplest crystal structures. Let us look at three such structures... 4

  9. FACE CENTERED CUBIC STRUCTURE (FCC)

  10. FACE CENTERED CUBIC STRUCTURE (FCC) Al, Cu, Ni, Ag, Au, Pb, Pt

  11. BODY CENTERED CUBIC STRUCTURE (BCC)

  12. BODY CENTERED CUBIC STRUCTURE (BCC) Cr, Fe, W, Nb, Ba, V

  13. HEXAGONAL CLOSE-PACKED STRUCTURE HCP Mg, Zn, Cd, Zr, Ti, Be

  14. SIMPLE CUBIC STRUCTURE (SC) Only Pd has SC structure

  15. Number of atoms per unit cell BCC 1/8 corner atom x 8 corners + 1 body center atom =2 atoms/uc FCC 1/8 corner atom x 8 corners + ½ face atom x 6 faces =4 atoms/uc HCP 3 inside atoms + ½ basal atoms x 2 bases + 1/ 6 corner atoms x 12 corners =6 atoms/uc

  16. Relationship between atomic radius and edge lengths For FCC: a = 2R√2 For BCC: a = 4R /√3 For HCP a = 2R c/a = 1.633 (for ideal case) Note: c/a ratio could be less or more than the ideal value of 1.633

  17. Face Centered Cubic (FCC) r 2r a0 r a0

  18. Body Centered Cubic (BCC) a0

  19. Coordination Number • The number of touching or nearest neighbor atoms • SC is 6 • BCC is 8 • FCC is 12 • HCP is 12

  20. ATOMIC PACKING FACTOR • APF for a simple cubic structure = 0.52 6

  21. ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68 a = 4R /√3 8

  22. FACE CENTERED CUBIC STRUCTURE (FCC) • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. • Coordination # = 12

  23. ATOMIC PACKING FACTOR: FCC • APF for a face-centered cubic structure = 0.74 a = 2R√2

  24. 3.5 Density Computations • Density of a material can be determined theoretically from the knowledge of its crystal structure (from its Unit cell information) • Density= mass/Volume • Mass is the mass of the unit cell and volume is the unit cell volume. • mass = ( number of atoms/unit cell) “n” x mass/atom • mass/atom = atomic weight “A”/Avogadro’s Number “NA” • Volume = Volume of the unit cell “Vc”

  25. THEORETICAL DENSITY

  26. Example problem on Density Computation Problem:Compute the density of Copper Given: Atomic radius of Cu = 0.128 nm (1.28 x 10-8 cm) Atomic Weight of Cu = 63.5 g/mol Crystal structure of Cu is FCC Solution:  = n A / Vc NA n= 4 Vc= a3 = (2R√2)3 = 16 R3 √2 NA = 6.023 x 1023 atoms/mol • = 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 cm)3 x 6.023 x 1023 atoms/mol Ans = 8.98 g/cm3 Experimentally determined value of density of Cu = 8.94 g/cm3

  27. 3.6 Polymorphism and Allotropy • Polymorphism  The phenomenon in some metals, as well as nonmetals, having more than one crystal structures. • When found in elemental solids, the condition is often called allotropy. • Examples: • Graphite is the stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures. • Pure iron is BCC crystal structure at room temperature, which changes to FCC iron at 912oC.

  28. POLYMORPHISM AND ALLOTROPY BCC (From room temperature to 912 oC) Fe FCC (at Temperature above 912 oC) 912 oC Fe (BCC)Fe (FCC)

  29. 3.7 Crystal Systems • Since there are many different possible crystal structures, it is sometimes convenient to divide them into groups according to unit cell configurations and/or atomic arrangements. • One such scheme is based on the unit cell geometry, i.e. the shape of the appropriate unit cell parallelepiped without regard to the atomic positions in the cell. • Within this framework, an x, y, and z coordinate system is established with its origin at one of the unit cell corners; each x, y, and z-axes coincides with one of the three parallelepiped edges that extend from this corner, as illustrated in Figure.

  30. The Lattice Parameters Lattice parameters a, b, c, , , are called thelattice Parameters.

  31. Seven different possible combinations of edge lengths and angles give seven crystal systems. • Shown in Table 3.2 • Cubic system has the greatest degree of symmetry. • Triclinic system has the least symmetry.

  32. The Lattice Parameters Lattice parameters are a, b, c, , , are called thelattice Parameters.

  33. 3.7 CRYSTAL SYSTEMS

  34. 3.8 Point Coordinates in an Orthogonal Coordinate System Simple Cubic

  35. 3.9 Crystallographic Directions in Cubic System • Determination of the directional indices in cubic system: Four Step Procedure (Text Book Method) • Draw a vector representing the direction within the unit cell such that it passes through the origin of the xyzcoordinate axes. • Determine the projections of the vector on xyz axes. • Multiply or divide by common factor to obtain the three smallest integer values. • Enclose the three integers in square brackets [ ]. e.g. [uvw] u, v, and w are the integers

  36. Crystallographic Directions in Cubic System [111] [120] [110]

  37. Crystallographic Directions in Cubic System

  38. Head and Tail Procedure for determining Miller Indices for Crystallographic Directions • Find the coordinate points of head and tail points. • Subtract the coordinate points of the tail from the coordinate points of the head. • Remove fractions. • Enclose in [ ]

  39. Indecies of Crystallographic Directions in Cubic System Direction A Head point – tail point (1, 1, 1/3) – (0,0,2/3) 1, 1, -1/3 Multiply by 3 to get smallest integers 3, 3, -1 A = [33Ī] Direction B Head point – tail point (0, 1, 1/2) – (2/3,1,1) -2/3, 0, -1/2 Multiply by 6 to get smallest integers _ _B = [403] C = [???] D = [???]

  40. Indices of Crystallographic Directions in Cubic System Direction C Head Point – Tail Point (1, 0, 0) – (1, ½, 1) 0, -1/2, -1 Multiply by 2 to get the smallest integers  C = [0I2] Direction D Head Point – Tail Point (1, 0, 1/2) – (1/2, 1, 0) 1/2, -1, 1/2 Multiply by 2 to get the smallest integers  D = [I2I] B= [???] A = [???]

  41. Crystallographic Directions in Cubic System [210]

  42. Crystallographic Directions in Cubic System

  43. Crystallographic Directions in Cubic System

  44. Indices of a Family or Form

  45. 3.10 MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES • Miller Indices for crystallographic planes are the reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell. • 4-Step Procedure: • Find the intercepts that the plane makes with the three axes x,y,z. If the plane passes through origin change the origin or draw a parallel plane elsewhere (e.g. in adjacent unit cell) • Take the reciprocal of the intercepts • Remove fractions • Enclose in ( )

  46. Miller Indecies of Planes in Crystallogarphic Planes in Cubic System

  47. Drawing Plane of known Miller Indices in a cubic unit cell Draw ( ) plane

  48. Miller Indecies of Planes in Crystallogarphic Planes in Cubic System Origin for A Origin for B Origin for A A = (IĪ0) B = (I22) A = (2IĪ) B = (02Ī)

  49. CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT CELLSMiller-Bravais indices -- same as Miller indices for cubic crystals except that there are 3 basal plane axes and 1 vertical axis.Basal plane -- close packed plane similar to the (1 1 1) FCC plane.contains 3 axes 120o apart.

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