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Crystal structures

Crystal structures. Unit-I. Hari Prasad Assistant Professor MVJCE-Bangalore. Learning objectives. After the chapter is completed, you will be able to answer: Difference between crystalline and noncrystalline structures Different crystal systems and crystal structures

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Crystal structures

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  1. Crystal structures Unit-I Hari Prasad Assistant Professor MVJCE-Bangalore Hari Prasad

  2. Learning objectives • After the chapter is completed, you will be able to answer: • Difference between crystalline and noncrystalline structures • Different crystal systems and crystal structures • Atomic packing factors of different cubic crystal systems • Difference between unit cell and primitive cell • Difference between single crystals and poly crystals Hari Prasad

  3. What is space lattice? • Space lattice is the distribution of points in 3D in such a way that every point has identical surroundings, i.e., it is an infinite array of points in three dimensions in which every point has surroundings identical to every other point in the array. Hari Prasad

  4. Common materials: with various ‘viewpoints’ Graphite Glass: amorphous Ceramics Crystal Metals Polymers

  5. Common materials: examples • Metals and alloys  Cu, Ni, Fe, NiAl (intermetallic compound), Brass (Cu-Zn alloys) • Ceramics (usually oxides, nitrides, carbides)  Alumina (Al2O3), Zirconia (Zr2O3) • Polymers (thermoplasts, thermosets) (Elastomers) Polythene, Polyvinyl chloride, Polypropylene Based on Electrical Conduction • Conductors  Cu, Al, NiAl • Semiconductors  Ge, Si, GaAs • Insulators  Alumina, Polythene* Based on Ductility • Ductile  Metals, Alloys • Brittle  Ceramics, Inorganic Glasses, Ge, Si * some special polymers could be conducting

  6. The broad scientific and technological segments of Materials Science are shown in the diagram below. • To gain a comprehensive understanding of materials science, all these aspects have to be studied. MATERIALS SCIENCE & ENGINEERING Science of Metallurgy ELECTRO- CHEMICAL TECHNOLOGICAL MECHANICAL PHYSICAL • Extractive • Casting • Metal Forming • Welding • Powder Metallurgy • Machining • Structure • Physical Properties • Deformation • Behaviour • Thermodynamics • Chemistry • Corrosion

  7. Definition 1 Crystal = Lattice + Motif Motif or Basis: typically an atom or a group of atoms associated with each lattice point Latticethe underlying periodicity of the crystal Basis Entity associated with each lattice points Latticehow to repeat Motif what to repeat Lattice Crystal Translationally periodic arrangement of points Translationally periodic arrangement of motifs

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  9. Space Lattice A lattice is also called a Space Lattice An array of points such that every point has identical surroundings • In Euclidean space  infinite array • We can have 1D, 2D or 3D arrays (lattices) or Translationally periodic arrangement of points in space is called a lattice

  10. Unit cell: A unit cell is the sub-division of the space lattice that still retains the overall characteristics of the space lattice. Primitive cell: the smallest possible unit cell of a lattice, having lattice points at each of its eight vertices only. A primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell. Hari Prasad

  11. Materials and Packing noncrystalline SiO2 (Glass) Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 (Quartz) Oxygen Si Non-crystallinematerials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline Hari Prasad

  12. Crystal Systems Unit cell:smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants Hari Prasad

  13. The Unite Cell is the smallest group of atom showing the characteristic lattice structure of a particular metal. It is the building block of a single crystal. A single crystal can have many unit cells. Hari Prasad

  14. Crystal systems Hari Prasad

  15. --diamond single crystals for abrasives --turbine blades Some engineering applications require single crystals: Hari Prasad

  16. What is coordination number? • The coordination number of a central atom in a crystal is the number of its nearest neighbours. What is lattice parameter? • The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. • Latticesin three dimensions generally have three lattice constants, referred to as a, b, and c. Hari Prasad

  17. Simple Cubic Structure (SC) • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) Hari Prasad

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  21. Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe (), Tantalum, Molybdenum • Coordination # = 8 2 atoms/unit cell: 1 center + 8 corners x 1/8 Hari Prasad

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  24. Atomic Packing Factor: BCC a 3 a 2 Close-packed directions: R 3 a length = 4R = a atoms volume 4 3 p ( 3 a/4 ) 2 unit cell atom 3 APF = volume 3 a unit cell • APF for a body-centered cubic structure = 0.68 a Hari Prasad

  25. Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 Hari Prasad

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  27. Atomic Packing Factor: FCC Close-packed directions: 2 a length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a atoms volume 4 3 p ( 2 a/4 ) 4 unit cell atom 3 APF = volume 3 a unit cell • APF for a face-centered cubic structure = 0.74 maximum achievable APF Hari Prasad

  28. B B B B C C C A A A B B B B B B B A sites C C C C C C B B sites sites B B B B C sites A B C FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection • FCC Unit Cell

  29. Putting atoms in the B position in the II layer and in C positions in the III layer we get a stacking sequence  ABC ABC ABC….  The CCP (FCC) crystal = + + C A B FCC A A B B C C

  30. Hexagonal Close-Packed Structure (HCP) A sites Top layer c Middle layer B sites A sites Bottom layer a • ABAB... Stacking Sequence • 3D Projection • 2D Projection 6 atoms/unit cell • Coordination # = 12 ex: Cd, Mg, Ti, Zn • APF = 0.74 • c/a = 1.633 Hari Prasad

  31. A sites c B sites A sites a APF for HCP C=1.633a Number of atoms in HCP unit cell= (12*1/6)+(2*1/2)+3=6atoms Vol.of HCP unit cell= area of the hexagonal face X height of the hexagonal Area of the hexagonal face=area of each triangle X6 a=2r Area of triangle = Area of hexagon= Volume of HCP= APF= 6 a h a APF =0.74 Hari Prasad

  32. SC-coordination number 6 Hari Prasad

  33. • Coordination # = 6 (# nearest neighbors) Hari Prasad

  34. BCC-coordination number 8 Hari Prasad

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  36. FCC-coordination number 4+4+4=12 Hari Prasad

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  38. HCP-coordination number 3+6+3=12 Hari Prasad

  39. nA VCNA Mass of Atoms in Unit Cell Total Volume of Unit Cell  = Theoretical Density, r Density =  = where n = number of atoms/unit cell A =atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol Hari Prasad

  40. R a 2 52.00 atoms g  = unit cell mol atoms 3 a 6.023x1023 mol volume unit cell Theoretical Density, r • Ex: Cr (BCC) A =52.00 g/mol R = 0.125 nm n = 2 a = 4R/ 3 = 0.2887 nm theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3 Hari Prasad

  41. iron system liquid 1538ºC -Fe BCC 1394ºC -Fe FCC 912ºC BCC -Fe Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism)  titanium , -Ti carbon diamond, graphite Hari Prasad

  42. Miller indices Miller indices: defined as the reciprocals of the intercepts made by the plane on the three axes. Hari Prasad

  43. Procedure for finding Miller indices Determine the interceptsof the plane along the axes X,Y and Z in terms of the lattice constants a, b and c. Step 1 Hari Prasad

  44. Determine the reciprocals of these numbers. Step 2 Hari Prasad

  45. Step 3 Find the least common denominator (lcd) and multiply each by this lcd Hari Prasad

  46. Step 4 The result is written in parenthesis. This is called the `Miller Indices’ of the plane in the form (h k l). Hari Prasad

  47. Miller Indices for planes (0,0,1) (0,3,0) (2,0,0) • Find intercepts along axes → 2 3 1 • Take reciprocal → 1/2 1/3 1 • Convert to smallest integers in the same ratio → 3 2 6 • Enclose in parenthesis → (326)

  48. Z Plane ABC has intercepts of 2 units along X-axis, 3 units along Y-axis and 2 units along Z-axis. C B Y A X Hari Prasad

  49. DETERMINATION OF ‘MILLER INDICES’ Step 1: The intercepts are 2, 3 and 2 on the three axes. Step 2: The reciprocals are 1/2, 1/3 and 1/2. Step 3: The least common denominator is ‘6’. Multiplying each reciprocal by lcd, we get, 3,2 and 3. Step 4:Hence Miller indices for the plane ABC is (3 2 3) Hari Prasad

  50. IMPORTANT FEATURES OF MILLER INDICES • For the cubic crystal especially, the important features of Miller indices are, • A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity (). • Therefore the Miller index for that axis is zero; i.e. for an intercept at infinity, the corresponding index is zero. • A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts. • All equally spaced parallel planes have same ‘Miller indices’ i.e. The Miller indices do not only define a particular plane but also a set of parallel planes. • Thus the planes whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all represented by the same set of Miller indices. Hari Prasad

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