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# Space lattice and crystal structure,miller indices PEC UNIVERSITY CHD

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### Space lattice and crystal structure,miller indices PEC UNIVERSITY CHD

1. 1. PRESENTATION ON SPACE LATTICE AND CRYSTAL STRUCTURE Submitted By Manmeet Singh 13209015 ME Mechanical Submitted To Dr. Sanjeev Kumar Mechanical Deptt.
2. 2. MATERIALS SOLIDS 1. Crystalline a. Single crystalline b. Poly crystalline “High Bond Energy” and a More Closely Packed Structure 2. Non Crystalline(Amorphous) These have less densely packed lower bond energy “structures”
3. 3. • Non dense, random packing • Dense, ordered packing Dense, ordered packed structures tend to have lower energies & thus are more stable. ENERGY AND PACKING Energy r typical neighbor bond length typical neighbor bond energy Energy r typical neighbor bond length typical neighbor bond energy
4. 4. CRYSTALS GEOMETRY  Space lattice  Crystal structure  Crystal directions and planes
5. 5. SPACE LATTICE  Arrangement of atoms taken as point periodically repeating in infinite array in 3 dimensional space such that every point has identical surrounding.  Infinite  Identical surrounding  In a space lattice we can have more than one kind of cell, shape of cell, size of cell…..
6. 6. Meaning of identical surrounding
7. 7. TO DEFINE MOVEMENT  2d – two vectors which are non collinear  3d – three vectors which are non coplanar
8. 8. PRIMITIVE CELL SMALLEST CELL WITH LATTICE POINTS AT EIGHT CORNERS HAS EFFECTIVELY ONLY ONE ATOM IN THE VOLUME OF CELL PARAMETERS OF PRIMITIVE CELL  3D lattices can be generated with three basis vectors  6 lattice parameters  3 distances (a, b, c)  3 angles (, , )
9. 9. PARAMETERS OF PRIMITIVE CELL a,b,c adjacent sides of cell α,β,γ interfacial angles Points to note---- 1. Knowing actual values of (a,b,c) and (α,β,γ) ---form and size of cell. 2. Knowing (α,β,γ) but only rations of (a,b,c)---we can only know shape of cell not size.
10. 10. UNIT CELL NOT NECESSARY TO BE PRIMITIVE, CAN BE BIGGER THAN PRIMITIVE AS LONG AS IT SHOWS ALL POSSIBLE MAXIMUM SYMMETRIES It is characterized by:  Type of atom and their radii,  Cell dimensions,  Number of atoms per unit cell,  Coordination number (CN)–closest neighbors to an atom  Atomic packing factor, APF Atomic packing factor (APF) or packing fraction is the fraction of volume in a crystal structure that is occupied by atoms.
11. 11. CHOOSING UNIT CELL--- BRAVAIS SPACE LATTICES  Smallest size  Maximum possible symmetry  Symmetry 1. Translational symmetry (inherent in definition of space lattice is that identical surrounding) 2. Rotation symmetry
12. 12. So there are only 14 bravais space lattices which belong to 7 crystal classes. Depending on minimum size and maximum symmetry.
13. 13. CRYSTAL STRUCTURE  Crystal– considered as consisting of tiny blocks which are repeated in 3D pattern.  Tiny block--- UNIT CELL Unit cell---arrangement of small group of atoms . It is that volume of solid from which entire crystal can be constructed by repeated translation in 3D.  Lattice point --- each atom in cell is replaced by point.  Space lattice(infinite lattice array) --- arrangement of lattice in 3D.every point has identical surrounding.  Lattice spacing--- distance between atom points.
14. 14. DEFINING CRYSTAL STRUCTURE A crystal structure is a unique arrangement of atoms or molecules in a crystalline solid. A crystal structure describes a highly ordered structure, occurring due to the intrinsic nature of molecules to form symmetric patterns.  Inter atomic spacing  Number of atoms and their kind  Orientation in space  MOTIF –kind of atoms associated with each lattice point  in polymeric and protein crystals there can be more than 10,000 atoms in a motive
15. 15. ARRANGEMENT OF LATTICE POINTS IN SPACE LATTICE 1. Primitive simple cubic 8 points or atoms at each corner 2. 8 corner atom and one body centered
16. 16. 3. 8 atoms at each corner and 6 atoms at center of each face. 4. 8 atoms at each corner and 2 atoms at centers of opposite face.
17. 17. 4. Hexagonal with 12 corner atoms and 2 center atoms
18. 18. 7 CRYSTAL CLASSES  FIRST CRYSTAL CLASS  CUBIC CRYSTAL CLASS Symmetry --- THREE -4 fold 1. Simple cubic 2. Body centered cubic 3. Face centered cubic a = b= c α = β = γ = 90° Only one parameter need to be defined No end center as they do not show maximum symmetry
19. 19. • Rare due to low packing density (only Po – Polonium -- has this structure) • Coordination No. = 6 (# nearest neighbors) for each atom as seen SIMPLE CUBIC STRUCTURE (SC)
20. 20. • APF for a simple cubic structure = 0.52 APF = a3 4 3 p (0.5a) 31 atoms unit cell atom volume unit cell volume ATOMIC PACKING FACTOR (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres close-packed directions a R=0.5a contains (8 x 1/8) = 1 atom/unit cell Here: a = Rat*2 Where Rat’ atomic radius
21. 21. • Coordination # = 8 • Atoms touch each other along cube diagonals within a unit cell. --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)
22. 22. ATOMIC PACKING FACTOR: BCC a APF = 4 3 p ( 3a/4)32 atoms unit cell atom volume a3 unit cell volume length = 4R = Close-packed directions: 3 a • APF for a body-centered cubic structure = 0.68 a R a2 a3
23. 23. • 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 ½) + (8 corners x 1/8)
24. 24. • APF for a face-centered cubic structure = 0.74 ATOMIC PACKING FACTOR: FCC The maximum achievable APF! APF = 4 3 p ( 2a/4)34 atoms unit cell atom volume a3 unit cell volume Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x1/2 + 8 x1/8 = 4 atoms/unit cella 2 a (a = 22*R)
25. 25. SECOND CRYSTAL CLASS TETRAGONAL CRYSTAL CLASS  Symmetry– One 4 fold 4…Simple tetragonal 5…Body centered tetragonal a = b not equal to c α = β = γ = 90°  Two parameters to define a and c.  One might suppose stretching face-centered cubic would result in face-centered tetragonal, but face- centered tetragonal is equivalent to body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more fundamental, so that is the standard terminology
26. 26. THIRD CRYSTAL CLASS ORTHORHOMBIC CRYSTAL CLASS  Symmetry–Three 2 fold 6…Simple 7…Body centered 8…Face centered 9…End centered a≠ b ≠ c α = β = γ = 90°  Three parameters to define a and c.
27. 27. 4. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)  Symmetry–One 6 fold 10…Simple Hexagonal a = b ≠ c α = β = 90° γ = 120°  Two parameters to define a and c.
28. 28. • Coordination # = 12 • ABAB... Stacking Sequence • APF = 0.74 • 3D Projection • 2D Projection 4. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)4 6 atoms/unit cell ex: Cd, Mg, Ti, Zn c a A sites B sites A sites Bottom layer Middle layer Top layer
29. 29. FIFTH CRYSTAL CLASS Rhombohedral crystal class 11…Simple rhombohedral Symmetry– One 3 fold (120°) a = b = c α = β = γ ≠ 90°  Two parameters to define a and angle α. SIXTH CRYSTAL CLASS Monoclinic crystal class Symmetry– One 2 fold 12… Simple monoclinic 13…End centered Monoclinic (A and B not C) a ≠ b ≠ c α = β = 90° ≠ γ  Four parameters to define a ,b ,c and angle γ.
30. 30. SEVENTH CRYSTAL CLASS Triclinic crystal class 14… Simple triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90°  Six parameters to define a ,b ,c and anglesα,β ,γ.  HIGHLY UNSYMMETRIC CRYSTAL
31. 31. THEORETICAL DENSITY, R 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 Density =  = VC NA n A  = CellUnitofVolumeTotal CellUnitinAtomsofMass
32. 32. CRYSTALLOGRAPHIC DIRECTIONS, AND PLANES  Now that we know how atoms arrange themselves to form crystals, we need a way to identify directions and planes of atoms Why?  Deformation under loading (slip) occurs on certain crystalline planes and in certain crystallographic directions. Before we can predict how materials fail, we need to know what modes of failure are more likely to occur.  Other properties of materials (electrical conductivity, thermal conductivity, elastic modulus) can vary in a crystal with orientation
33. 33. MILLER INDICES USED TO SPECIFY DIRECTIONS AND PLANES. Vectors and atomic planes in a crystal lattice can be described by a three- value Miller index notation (hkl). The h, k, and l directional indices are separated by 90°, and are thus orthogonal.  Choice of origin arbitrary  Axes and their sense--once chosen not changed and choice is arbitrary.  Unit distance need not to be unity, it is distance between two lattice points in a particular direction in a particular crystal.
34. 34. CRYSTAL DIRECTIONS A crystallographic direction is defined as a line between two points, or a vector.
35. 35. CRYSTAL DIRECTIONS  Smallest integer reduction.  No separator used  Separator can be used when directions came with more than one digit number, example [1,14,7]  Bar is used for negative directions.  Characteristics of crystal in different directions is different that is why it is needed to give them different name.
36. 36. Negative Directions
37. 37. HCP CRYSTALLOGRAPHIC DIRECTIONS 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] [1120]ex: ½, ½, -1, 0 => dashed red lines indicate projections onto a1 and a2 axes a1 a2 a3 -a3 2 a2 2 a1 -a3 a1 a2 z
38. 38. FAMILY OF DIRECTIONS Directions which look physically identical not parallel, if it is parallel it is same direction o<011> Permuting it will give 12 different arrangements [110] [1-10] [-110] [011] [101] [0-11] [01-1] [-101] [10-1] [-1-10] [0-1-1] [-10-1]
39. 39. ex: linear density of Al in [110] direction a = 0.405 nm LINEAR DENSITY – CONSIDERS EQUIVALENCE AND IS IMPORTANT IN SLIP  Linear Density of Atoms  LD = a [110] Unit length of direction vector Number of atoms # atoms length 1 3.5 nm a2 2 LD - ==
40. 40. DEFINING CRYSTALLOGRAPHIC PLANES  Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.  Algorithm (in cubic lattices this is direct) 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl)  families {hkl}
41. 41. CRYSTAL PLANES
42. 42. CRYSTALLOGRAPHIC PLANES (HCP)  In hexagonal unit cells the same idea is used example a1 a2 a3 c 4. Miller-Bravais Indices (1011) 1. Intercepts 1  -1 1 2. Reciprocals 1 1/ 1 0 -1 -1 1 1 3. Reduction 1 0 -1 1 a2 a3 a1 z
43. 43. FAMILY OF PLANE  All planes which are physical identical is that arrangement of atoms is same  They are also not parallel if parallel this is same plane.
44. 44. PLANAR DENSITY OF (100) IRON Solution: At T < 912C iron has the BCC structure. (100) Radius of iron R = 0.1241 nm R 3 34 a = 2D repeat unit =Planar Density = a2 1 atoms 2D repeat unit = nm2 atoms 12.1 m2 atoms = 1.2 x 1019 1 2 R 3 34area 2D repeat unit
45. 45. COMPARISON OF FCC, HCP, AND BCC CRYSTAL STRUCTURES  Both FCC and HCP structures are close packed APF = 0.74.  The closed packed planes are the {111} family for FCC and the (0001) plane for HCP. Stacking sequence is ABCABCABC in FCC and ABABAB in HCP.  BCC is not close packed, APF = 0.68. Most densely packed planes are the {110} family.
46. 46. ATOMIC DENSITIES WHY DO WE CARE? Properties, in general, depend on linear and planar density. Examples:  Speed of sound along directions  Slip (deformation in metals) depends on linear & planar density. Slip occurs on planes that have the greatest density of atoms in direction with highest density (we would say along closest packed directions on the closest packed planes)
47. 47. Symbol Alternate symbols Direction [ ] [uvw] → Particular direction < > <uvw> [[ ]] → Family of directions Plane ( ) (hkl) → Particular plane { } {hkl} (( )) → Family of planes Point . . .abc. [[ ]] → Particular point : : :abc: → Family of point
48. 48. POLYMORPHISM: ALSO IN METALS  Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium  (HCP), (BCC)-Ti BCC FCC BCC 1538ºC 1394ºC 912ºC -Fe -Fe -Fe liquid iron system: •Carbon (diamond, graphite,) •Silica (quartz, tridymite, cristobalite, etc.) •Iron (ferrite, austenite)
49. 49.  Anisotropic – Direction dependent properties Example carbon fibers, whiskers  Isotropic – Direction independent properties Each grain may be anisotropic ,but a specimen composed by grain aggregate behaves isotropically. The degree of anisotropy increases with decreasing structural symmetry. So TRICLINICS are highly anisotropic
50. 50. LOOKING AT THE CERAMIC UNIT CELLS
51. 51. CERAMIC CRYSTAL STRUCTURE  Broader range of chemical composition than metals with more complicated structures  Contains at least 2 and often 3 or more atoms.  Usually compounds between metallic ions (e.g. Fe,Ni, Al) called cations and non-metallic ions (e.g.O, N, Cl) called anions. How do Cations and Anions arrange themselves in space??? Structure is determined by two characteristics:  1. Electrical charge Crystal (unit cell) must remain electrically neutral .Sum of cation and anion charges in cell is 0  2. Relative size of the ions
52. 52. CERAMIC CRYSTAL STRUCTURE AX-TYPE CRYSTAL STRUCTURES Some of the common ceramic materials are those in which there are equal numbers of cations and anions. Example– NaCl, MgO,Feo,CsCl,Zinc blende Am Xp -TYPE CRYSTAL STRUCTURES Not equal number of charges Example – Fluorite CaF2, UO2, ThO2 Am Bn Xp -TYPE CRYSTAL STRUCTURES Barium titanate BaTiO3
53. 53. CESIUM CHLORIDE (CSCL) UNIT CELL SHOWING (A) ION POSITIONS AND THE TWO IONS PER LATTICE POINT AND
54. 54. SODIUM CHLORIDE (NACL) STRUCTURE SHOWING (A) ION POSITIONS IN A UNIT CELL, (B) FULL-SIZE IONS, AND (C) MANY ADJACENT UNIT CELLS.
55. 55. FLUORITE (CAF2) UNIT CELL SHOWING (A) ION POSITIONS AND (B) FULL-SIZE IONS.
56. 56. ARRANGEMENT OF POLYMERIC CHAINS IN THE UNIT CELL OF POLYETHYLENE. THE DARK SPHERES ARE CARBON ATOMS, AND THE LIGHT SPHERES ARE HYDROGEN ATOMS. THE UNIT- CELL DIMENSIONS ARE 0.255 NM × 0.494 NM × 0.741 NM.
57. 57. DENSITIES OF MATERIAL CLASSES metals > ceramics >polymers Why? (g/cm)3 Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers 1 2 20 30 *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix).10 3 4 5 0.3 0.4 0.5 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum Graphite Silicon Glass -soda Concrete Si nitride Diamond Al oxide Zirconia HDPE, PS PP, LDPE PC PTFE PET PVC Silicone Wood AFRE* CFRE* GFRE* Glass fibers Carbon fibers Aramid fibers Metals have... • close-packing (metallic bonding) • often large atomic masses Ceramics have... • less dense packing • often lighter elements Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values In general
58. 58. THANKYOU ANY QUESTION