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- 1. CRYSTAL STRUCTURE Prof. H. K. Khaira Professor Deptt. of MSME M.A.N.I.T., Bhopal
- 2. Topics Covered • Crystal Structure • Miller Indices • Slip Systems
- 3. Types of Solids • Solids can be divided into two groups based on the arrangement of the atoms. These are –Crystalline –Amorphous
- 4. Types of Solids • In a Crystalline solid, atoms are arranged in an orderly manner. The atoms are having long range order. – Example : Iron, Copper and other metals, NaCl etc. • In an Amorphous solid, atoms are not present in an orderly manner. They are haphazardly arranged. – Example : Glass
- 5. ENERGY AND PACKING • Non dense, random packing Energy typical neighbor bond length typical neighbor bond energy • Dense, regular packing r Energy typical neighbor bond length typical neighbor bond energy r Dense, regular-packed structures tend to have lower energy. 5
- 6. Some Definitions Crystalline material is a material comprised of one or many crystals. In each crystal, atoms or ions show a long-range periodic arrangement. Single crystal is a crystalline material that is made of only one crystal (there are no grain boundaries). Grains are the crystals in a polycrystalline material. Polycrystalline material is a material comprised of many crystals (as opposed to a single-crystal material that has only one crystal). Grain boundaries are regions between grains of a polycrystalline material. 6
- 7. Atomic PACKING Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Adapted from Fig. 3.18(a), Callister 6e. Noncrystalline materials... Si Oxygen • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.18(b), Callister 6e. From Callister 6e resource CD. 7
- 8. Space Lattice A Space LATTICE is an infinite, periodic array of mathematical points, in which each point has identical surroundings to all others. LATTICE A CRYSTAL STRUCTURE is a periodic arrangement of atoms in the crystal that can be described by a LATTICE 8
- 9. Space Lattice Important Note: • Lattice points are a purely mathematical concept, whereas atoms are physical objects. • So, don't mix up atoms with lattice points. • Lattice Points do not necessarily lie at the center of atoms. In Figure (a) is the 3-D periodic arrangement of atoms, and Figure (b) is the corresponding space lattice. In this case, atoms lie at the same point as the space lattice. 9
- 10. Crystalline Solids: Unit Cells Unit Cell: It is the basic structural unit of a crystal structure. Its geometry and atomic positions define the crystal structure. Fig. 3.1 Atomic configuration in Face-Centered-Cubic Arrangement R R R A unit cell is the smallest part of the unit cell, which when repeated in all three directions, reproduces the lattice. R a 10
- 11. Unit Cells and Unit Cell Vectors Lattice parameters axial lengths: a, b, c interaxial angles: α, β, γ unit vectors: a b c c b a All unit cells may be described via these vectors and angles. 11
- 12. Possible Crystal Classes 12
- 13. Possible Crystal Classes 13
- 14. The 14 Bravais Lattices! 14
- 15. Unit Cells Types A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition. • Primitive (P) unit cells contain only a single lattice point. • Internal (I) unit cell contains an atom in the body center. • Face (F) unit cell contains atoms in the all faces of the planes composing the cell. • Centered (C) unit cell contains atoms centered on the sides of the unit cell. Primitive Body-Centered Face-Centered End-Centered KNOW THIS! 15
- 16. Orderly arrangement of atoms in Crystals
- 17. Orderly arrangement of atoms in Crystals Atom Translation Vectors a1, a2 ,a3 a3 a2 a1 17
- 18. CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT AMORPHOUS CRYSTALLINE There exists at least one crystalline state of lower energy (G) than the amorphous state (glass) The crystal exhibits a sharp melting point “Crystal has a higher density”!!
- 19. Factors affecting the formation of the amorphous state When the free energy difference between the crystal and the glass is small ⇒ Tendency to crystallize would be small Cooling rate → fast cooling promotes amorphization • “fast” depends on the material in consideration • Certain alloys have to be cooled at 106 K/s for amorphization • Silicates amorphizes during air cooling
- 20. Types of Lattices
- 21. Types of Lattices • There are 14 lattice types • Most common types: – Cubic: Li, Na, Al, K, Cr, Fe, Ir, Pt, Au etc. – Hexagonal Closed Pack (HCP): Mg, Co, Zn, Y, Zr, etc. – Diamond: C, Si, Ge, Sn (only four)
- 22. Unit Cell • Unit Cell is the smallest part of the lattice which when repeated in three directions produces the lattice. • Unit Cell is the smallest part of the lattice which represents the lattice.
- 23. Metals • Most of the metals are crystalline in solid form. They normally have the following crystal structures. – Body Centered Cubic (BCC) – Face Centered Cubic (FCC) – Hexagonal Close Packed (HCP) or Close Packed Hexagonal (CPH)
- 24. Unit Cell of BCC Lattice •Fe (Up to 9100 C and from 14010C to Melting Point), W, Cr, V,
- 25. A less close-packed structure is Body-Centered-Cubic (BCC). Besides FCC and HCP, BCC structures are widely adopted by metals. • Unit cell showing the full cubic symmetry of the BCC arrangement. • BCC: a = b = c = a and angles α = β =γ= 90°. • 2 atoms in the cubic cell: (0, 0, 0) and (1/2, 1/2, 1/2). 25
- 26. Unit Cell of FCC lattice • Al, Cu, Ni, Fe (9100 C-14010 C)
- 27. ABCABC.... repeat along <111> direction gives Cubic Close-Packing (CCP) • Face-Centered-Cubic (FCC) is the most efficient packing of hard-spheres of any lattice. • Unit cell showing the full symmetry of the FCC arrangement : a = b =c, angles all 90° • 4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0) Self-Assessment: Write FCC crystal as BCT unit cell. 29
- 28. FCC Stacking A Highlighting the stacking B C Highlighting the faces 30
- 29. FCC Unit Cell Highlighting the ABC planes and the cube. Highlighting the hexagonal planes in each ABC layer. 31
- 30. FCC CLOSE PACKING A = + + B Note: Atoms are coloured differently but are the same C FCC
- 31. Unit Cell of Hexagonal closed packed (HCP) • Zn, Mg
- 32. ABABAB.... repeat along <111> direction gives Hexagonal Close-Packing (HCP) • Unit cell showing the full symmetry of the HCP arrangement is hexagonal • Hexagonal: a = b, c = 1.633a and angles α = β = 90°, γ = 120° • 2 atoms in the smallest cell: (0, 0, 0) and (2/3, 1/3, 1/2). 34
- 33. HCP Stacking A Highlighting the stacking B A Layer A Layer B Highlighting the cell Figure 3.3 Self-Assessment: How many atoms/cell? Layer A 35
- 34. Comparing the FCC and HCP Planes Stacking Looking down (0001) plane FCC HCP Looking down (111) plane! 36
- 35. Important Properties of unit cell • Number of atoms required to represent a unit cell • Effective number of atoms per unit cell • Coordination number • Atomic packing factor
- 36. Number of atoms required to represent a unit cell • It is the number of atoms required to show a unit cell
- 37. Number of atoms required to represent a unit cell of BCC = 9
- 38. Number of atoms required to represent a unit cell of FCC = 14
- 39. Number of atoms required to represent a unit cell of HCP = 17
- 40. Effective Number of Atoms • It is the total number of atoms belonging to a unit cell
- 41. Counting Number of Atoms Per Unit Cell Counting Atoms in 3D Cells Atoms in different positions are shared by differing numbers of unit cells. • Corner atom shared by 8 cells => 1/8 atom per cell. • Edge atom shared by 4 cells => 1/4 atom per cell. • Face atom shared by 2 cells => 1/2 atom per cell. • Body unique to 1 cell => 1 atom per cell. Simple Cubic 8 atoms but shared by 8 unit cells. So, 8 atoms/8 cells = 1 atom/unit cell How many atoms/cell for Body-Centered Cubic? And, Face-Centered Cubic? 43
- 42. Effective Number of atoms per unit cell for BCC It will be 8*1/8 + 1*1 = 2
- 43. Effective Number of atoms per unit cell for FCC It will be 8*1/8 + 6*1/2 = 4 William D. Callister, Jr., Materials Science and Engineering, An Introduction, John Wiley & Sons, Inc. (2003)
- 44. Effective Number of atoms per unit cell for Hexagonal closed packed (HCP) •It will be 6*1/6 + 2*1/2 + 3*1 = 6 William D. Callister, Jr., Materials Science and Engineering, An Introduction, John Wiley & Sons, Inc. (2003)
- 45. Coordination Number • It is the number of nearest equidistant neighbours of an atom in the lattice
- 46. Coordination Number of a Given Atom Consider a simple cubic structure. Six yellow atoms are are touching the blue atom. Hence, the blue atom is having six nearest neighbors. Therefore, for a Simple cubic lattice: coordination number, CN = 6 48
- 47. Coordination Number
- 48. Atomic Packing Factor • Atomic Packing Factor is the fraction of volume occupied by atoms in a unit cell.
- 49. Atomic Packing Factor (APF) APF = vol. of atomic spheres in unit cell total unit cell vol. Face-Centered-Cubic Arrangement Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell Depends on: • Crystal structure. • How “close” packed the atoms are. • In simple close-packed structures with hard sphere atoms, independent of atomic radius 52
- 50. Basic Geometry for FCC Geometry along close-packed direction give relation between a and R. 2 2R 2R 2 2R 2=R a 4 a a Vunit _ cell = a = (2 2 R ) = 16 2 R 3 3 4 Vatoms = 4 πR 3 3 3 Geometry: a = 2 2R 4 atoms/unit cell Coordination number = 12 53
- 51. Atomic Packing Factor for FCC Face-Centered-Cubic Arrangement APF = vol. of atomic spheres in unit cell total unit cell vol. How many spheres (i.e. atoms)? 4/cell What is volume/atom? 4π R3/3 What is cube volume/cell? a3 How is “R” related to “a”? 2=R a 4 atoms volume 4 3 π ( 2a/4) 4 unit cell atom 3 APF = volume a3 unit cell Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell = 0.74 Independent of R! 54
- 52. Atomic Packing Factor for BCC Again, geometry along close-packed direction give relation between a and R. 2a Geometry: 4R ≡ 3 a 2 atoms/unit cell a 4R ≡ 3 a V APF = atoms = Vcell a 3 3 4 2 π 3 4 a3 = 3π =0.68 8 55
- 53. Important Properties of unit cell SC BCC FCC HCP Relation between atomic radius (r) and lattice parameter (a) a = 2r Effective Number of atoms per unit cell 1 2 4 6 Coordination Number (No. of nearest equidistant neighbours) 6 8 12 12 0.52 0.68 0.74 0.74 Packing Factor 3a = 4r 2a = 4r a = 2r
- 54. Miller Indices • Miller Indices are used to represent the directions and the planes in a crystal • Miller Indices is a group of smallest integers which represent a direction or a plane z z c [111] [100] [110] x b [021] a [1 1 1] y y [01 2 ] x
- 55. Miller Indices of a Direction • Select any point on the direction line other than the origin • Find out the coordinates of the point in terms of the unit vectors along different axes • Specify negative coordinate with a bar on top • Divide them by the unit vector along the respective axis • Convert the result into the smallest integers by suitable multiplication or division and express as <uvw>
- 56. Miller Indices of directions z z c [111] [100] [110] x b [021] a [1 1 1] y y [01 2 ] x
- 57. Miller Indices of planes - Find the intercepts of the plane with the three axes: (pa, qb, rc) - Take the reciprocals of the numbers (p, q, r) - Reduce to three smallest integers (h, k, l) by suitable multiplication or division. - Miller indices of the plane: (hkl) - Negative indices are indicated by a bar on top - Same indices for parallel planes - A family of crystallographically equivalent planes (not necessarily parallel) is denoted by {hkl}
- 58. Index System for Crystal Planes (Miller Indices)
- 59. Example: determine Miller indices of a plane z • Intercepts: (a/2, 2b/3, ∞) • p= ½, q= 2/3, r= ∞ • Reciprocals: h = 2, k = 3/2, l = 0 • Miller indices: (430) c a x b y
- 60. Crystal Planes
- 61. Important planes in cubic crystals • {100} family: (100) (010) (001) • {110} family: (110) (101) (011) (1 1 0) (10 1 ) (01 1 ) • {111} family: (111) (1 1 1) ( 1 11) (11 1 ) {100} planes {110} planes
- 62. Hexagonal indices • A special case for hexagonal crystals • Miller-Bravais indices: (hkil) h= Reciprocal of the intercept with a1-axis k= Reciprocal of the intercept with a2-axis i = -(h + k) l= Reciprocal of the intercept with c-axis c c (1 1 00) (11 2 0) a2 a1 (0001) a2 a1
- 63. Packing Densities in Crystals: Lines and Planes Concepts FCC Linear and Planar Packing Density which are independent of atomic radius! Also, Theoretical Density 66
- 64. Linear Density in FCC LD = Number of atoms centered on a direction vector Length of the direction vector Example: Calculate the linear density of an FCC crystal along [1 1 0]. ASK a. How many spheres along blue line? b. What is length of blue line? ANSWER a. 2 atoms along [1 1 0] in the cube. b. Length = 4R 2atoms 1 LD110 = = 4R 2R XZ = 1i + 1j + 0k = [110] Self-assessment: Show that LD100 = √2/4R. 67
- 65. Planar Density in FCC PD = Number of atoms centered on a given plane Area of the plane Example: Calculate the PD on (1 1 0) plane of an FCC crystal. • Count atoms within the plane: 2 atoms • Find Area of Plane: 8√2 R2 a = 2 2R 4R Hence, 2 1 PD = = 8 2R 2 4 2R 2 68
- 66. Planar Packing Density in FCC PPD = Area of atoms centered on a given plane Area of the plane Example: Calculate the PPD on (1 1 0) plane of an FCC crystal. • Find area filled by atoms in plane: 2π R2 • Find Area of Plane: 8√2 R2 a = 2 2R 4R Hence, 2π 2 R π PPD = = =0.555 8 2R 2 4 2 Always independent of R! Self-assessment: Show that PPD100 = π /4 = 0.785. 69
- 67. Density of solids • Mass per atom: mA = atomic weight from periodic table (g/mol) 6.02 ×1023 (atoms/mol) 4 3 • Volume per atom: VA = πRA 3 • Number of atoms per unit cell (N) 2 for bcc, 4 for fcc, 6 for hcp Volume occupied by atoms in a unit cell = NVA • Volume of unit cell (VC) - Depends on crystal structure • Mass density: NmA ρ= VC Packing Factor = A NV V C
- 68. Theoretical Density # atoms/unit cell ρ= nA VcNA Volume/unit cell (cm3/unit cell) Atomic weight (g/mol) Avogadro's number (6.023 x 10 23 atoms/mol) Example: Copper • crystal structure = FCC: 4 atoms/unit cell • atomic weight = 63.55 g/mol (1 amu = 1 g/mol) • atomic radius R = 0.128 nm (1 nm = 10 cm)7 - Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3 Result: theoretical ρCu = 8.89 g/cm 3 Compare to actual: ρCu = 8.94 g/cm3 71
- 69. Characteristics of Selected Elements at 20 C Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008 Density (g/cm3) 2.71 -----3.5 1.85 2.34 -----8.65 1.55 2.25 1.87 -----7.19 8.9 8.94 -----5.90 5.32 19.32 ----------- Crystal Structure FCC -----BCC HCP Rhomb -----HCP FCC Hex BCC -----BCC HCP FCC -----Ortho. Dia. cubic FCC ----------- Atomic radius (nm) 0.143 -----0.217 0.114 -----Adapted from -----Table, "Charac0.149 teristics of 0.197 Selected 0.071 Elements", inside front 0.265 cover, Callister 6e. -----0.125 0.125 0.128 -----0.122 0.122 0.144 ----------72
- 70. DENSITIES OF MATERIAL CLASSES ρmetals > ρceramics > ρpolymers Metals have... • close-packing Metals/ Alloys 30 20 10 (metallic bonds) Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Ceramics have... • less dense packing (covalent bonds) • often lighter elements Polymers have... • poor packing ρ (g/cm3) • large atomic mass (often amorphous) 5 4 3 2 1 • lighter elements (C,H,O) Composites have... • intermediate values Titanium Aluminum Magnesium Graphite/ Ceramics/ Polymers Semicond Composites/ fibers Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass-soda Concrete Silicon Graphite Glass fibers PTFE Silicone PVC PET PC HDPE, PS PP, LDPE 0.5 0.4 0.3 GFRE* Carbon fibers CFRE* Aramid fibers AFRE* Wood Data from Table B1, Callister 6e. 73
- 71. Slip Systems • Slip system is a combination of a slip plane and a slip direction along which the slip occurs at minimum stress • Slip plane is the weakest plane in the crystal and the slip direction is the weakest direction in that plane. • The most densely packed plane is the weakest plane and the most densely direction is the weakest direction in the crystal.
- 72. Most Closely Packed Plane and directions
- 73. FCC CLOSE PACKING A = + + B Note: Atoms are coloured differently but are the same C FCC
- 74. An FCC unit cell and its slip system {111} - type planes (4 planes) have all atoms closely packed. Slip occurs along <110> - type directions (A-B,A-C,D-E) within the {111} planes. FCC has 12 slip systems - 4 {111} planes and each has 3 <110> directions
- 75. Shown displaced for clarity HCP + A + B = A A plane B plane A plane Note: Atoms are coloured differently but are the same HCP
- 76. * A crystal will be anisotropic (exhibit directional properties) if the slip systems are less than 5. It will be isotropic (have same properties in all the directions) if the slip systems are more than 5.

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