Crystal structures in material science

Chapter describes the crystalline and non-crystalline solid's structure & properties

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

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’ Glass: amorphous Ceramics Crystal Graphite Polymers Metals

5. Metals and alloys  Cu, Ni, Fe, NiAl (intermetallic compound), Brass (Cu-Zn alloys) Ceramics (usually oxides, nitrides, carbides)  Alumina (Al 2 O 3 ), Zirconia (Zr 2 O 3 ) Polymers (thermoplasts, thermosets) (Elastomers)  Polythene, Polyvinyl chloride, Polypropylene Common materials: examples 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. MATERIALS SCIENCE & ENGINEERING PHYSICAL MECHANICAL ELECTRO- CHEMICAL TECHNOLOGICAL Extractive Casting Metal Forming Welding Powder Metallurgy Machining Structure Physical Properties Science of Metallurgy Deformation Behaviour Thermodynamics Chemistry Corrosion 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.

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

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9. 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) Space Lattice Translationally periodic arrangement of points in space is called a lattice or A lattice is also called a Space 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. • atoms pack in periodic, 3D arrays Crystalline materials... -metals -many ceramics -some polymers • atoms have no periodic packing Non-crystalline materials... -complex structures -rapid cooling crystalline SiO 2 (Quartz) " Amorphous " = Noncrystalline Materials and Packing Si Oxygen • typical of: • occurs for: noncrystalline SiO 2 (Glass) Hari Prasad

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

13. Hari Prasad 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.

14. Crystal systems Cubic Three equal axes, mutually perpendicular a=b=c ===90 ˚ Tetragonal Three perpendicular axes, only two equal a=b ≠ c ===90 ˚ Hexagonal Three equal coplanar axes at 120 ˚ and a fourth unequal axis perpendicular to their plane a=b ≠ c == 90 ˚ =120 ˚ Rhombohedral Three equal axes, not at right angles a=b=c == ≠ 90 ˚ Orthorhombic Three unequal axes, all perpendicular a ≠ b ≠ c ===90 ˚ Monoclinic Three unequal axes, one of which is perpendicular to the other two a ≠ b ≠ c ==90 ˚ ≠  Triclinic Three unequal axes, no two of which are perpendicular a ≠ b ≠ c  ≠  ≠  ≠ 90 ˚ Hari Prasad

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

16. What is coordination number? T he 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 . Lattices in three dimensions generally have three lattice constants , referred to as a, b, and c. Hari Prasad

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

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

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

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

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

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

29. A B + + FCC = 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 A B C A B C C

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

31. APF for HCP Hari Prasad c a A sites B sites A sites 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 h a Area of triangle = Area of hexagon = Volume of HCP= APF= 6   a =2r APF =0.74

32. SC-coordination number Hari Prasad 6

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

34. BCC-coordination number Hari Prasad 8

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

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

39. Theoretical Density, r where n = number of atoms/unit cell A = atomic weight V C = Volume of unit cell = a 3 for cubic N A = Avogadro’s number = 6.023 x 10 23 atoms/mol Density =  = V C N A n A  = Cell Unit of Volume Total Cell Unit in Atoms of Mass Hari Prasad

40. Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2  theoretical a = 4 R / 3 = 0.2887 nm r actual a R  = a 3 52.00 2 atoms unit cell mol g unit cell volume atoms mol 6.023 x 10 23 Theoretical Density, r = 7.18 g/cm 3 = 7.19 g/cm 3 Hari Prasad

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

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

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

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

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

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

47. 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) (2,0,0) (0,3,0) (0,0,1) Miller Indices for planes

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

49. Hari Prasad 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)

50. Hari Prasad 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 p arallel 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. IMPORTANT FEATURES OF MILLER INDICES

51. Hari Prasad Worked Example: Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes. The intercepts are 2, - 3 and 4 Step 1: The intercepts are 2, -3 and 4 along the 3 axes Step 2: The reciprocals are Step 3: The least common denominator is 12. Multiplying each reciprocal by lcd, we get 6 -4 and 3 Step 4: Hence the Miller indices for the plane is

52. Intercepts → 1   Plane → (100) Family → {100} → 3 Intercepts → 1 1  Plane → (110) Family → {110} → 6 Intercepts → 1 1 1 Plane → (111) Family → {111} → 8 (Octahedral plane)

53. Hari Prasad Miller Indices :   (100)

54. Hari Prasad Intercepts :    a , a , ∞ Fractional intercepts :   1 , 1 , ∞ Miller Indices :   (110)

55. Hari Prasad Intercepts :    a , a , a Fractional intercepts :   1 , 1 , 1 Miller Indices :   (111)

56. Hari Prasad Intercepts :   ½  a , a , ∞ Fractional intercepts :   ½ , 1 , ∞ Miller Indices :   (210)

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58. Hari Prasad (101) Z Y X

59. Hari Prasad (122)

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61. Hari Prasad Crystallographic Directions The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly , the crystallographic planes are fictitious planes linking nodes. The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a , b , and c .

62. Hari Prasad To find the Miller indices of a direction , Choose a perpendicular plane to that direction. Find the Miller indices of that perpendicular plane. The perpendicular plane and the direction have the same Miller indices value. Therefore, the Miller indices of the perpendicular plane is written within a square bracket to represent the Miller indices of the direction like [ ].

63. Summary of notations Symbol Alternate symbols Direction [ ] [uvw] → Particular direction < > <uvw> [[ ]] → Family of directions Plane ( ) (hkl) → Particular plane { } {hkl} (( )) → Family of planes Point . . .xyz. [[ ]] → Particular point : : :xyz: → Family of point *A family is also referred to as a symmetrical set

64. Hari Prasad For each of the three axes, there will exist both positive and negative coordinates. Thus negative indices are also possible, which are represented by a bar over the appropriate index. For example, the 1 The above image shows [100 ], [110], and [111] directions within a unit cell

65. Hari Prasad The vector, as drawn, passes through the origin of the coordinate system, and therefore no translation is necessary. Projections of this vector onto the x , y , and z axes are, respectively,1/2, b , and 0 c , which become 1/2, 1 , and 0 in terms of the unit cell parameters (i.e., when the a , b , and c are dropped). Reduction of these numbers to the lowest set of integers is accompanied by multiplication of each by the factor 2.This yields the integers 1, 2, and 0, which are then enclosed in brackets as [120].

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67. Hari Prasad Worked Example Find the angle between the directions [2 1 1] and [1 1 2] in a cubic crystal. The two directions are [2 1 1] and [1 1 2] We know that the angle between the two directions,

68. Hari Prasad In this case, u 1 = 2, v 1 = 1, w 1 = 1, Type equation here. u 2 = 1, v 2 = 1, w 2 = 2 (or) cos  = 0.833  = 35° 35  30  .  

69. Hari Prasad http:// core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/indexing_a_plane_embed.swf http:// core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/drawing_lattice_planes.swf http:// core.materials.ac.uk/repository/doitpoms/tlp/miller_indices/miller.swf Reference