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Crystral structure

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  • 1. A SEMINAR ON CRYSTAL STRUCTURE PRESENTED BY K. GANAPATHI RAO (13031D6003) Presence of Mr. V V Sai sir
  • 2. CONTENT  INTRODUCTION.  TRANSLATION VECTOR.  BASIS & UNIT CELL.  BRAVAIS & SPACE LATTICES.  FUNDAMENTAL QUANTITIES.  MILLER INDICES.  INTER-PLANAR SPACING.
  • 3. Materials Solids Liquids Gasses
  • 4. Solids Crystalline Single Poly Amorphous
  • 5. TRANSLATION VECTOR
  • 6. LATTICE PARAMETERS & UNIT CELL
  • 7. Bravais Lattice System Possible Variations Axial Distances (edge lengths) Axial Angles Examples Cubic Primitive, Bodya=b=c centred, Face-centred α = β = γ = 90° NaCl, Zinc Blende, Cu Tetragonal Primitive, Bodycentred a=b≠c α = β = γ = 90° White tin, SnO2,TiO2, CaSO4 Orthorhombic Primitive, Bodycentred, Facea≠b≠c centred, Base-centred α = β = γ = 90° Rhombic sulphur,KNO3, BaSO4 Rhombohedral Primitive a=b=c α = β = γ ≠ 90° Calcite (CaCO3,Cinnaba r (HgS) Hexagonal Primitive a=b≠c α = β = 90°, γ = 120° Graphite, ZnO,CdS Monoclinic Primitive, Basecentred a≠b≠c α = γ = 90°, β ≠ 90° Monoclinic sulphur, Na2SO4.10H2O Triclinic Primitive a≠b≠c α ≠ β ≠ γ ≠ 90° K2Cr2O7, CuSO4.5H2O,H3BO3
  • 8. 1 Cubic P  Cube I  F  C I P a b c Lattice point 90 F
  • 9. P 2 Tetragonal Square Prism (general height) I  F C  I P a b c 90
  • 10. P 3 Orthorhombic Rectangular Prism (general height) I F C     One convention a b c I P a b c 90 F C
  • 11. P 4 Trigonal Parallelepiped (Equilateral, Equiangular) I  Rhombohedral a b c 90 Note the position of the origin and of ‘a’, ‘b’ & ‘c’ F C
  • 12. P 5 Hexagonal  120 Rhombic Prism a b c 90 , A single unit cell (marked in blue) along with a 3-unit cells forming a hexagonal prism 120 I F C
  • 13. P 6 Monoclinic  Parallogramic Prism One convention a b c a b c 90 Note the position of ‘a’, ‘b’ & ‘c’ I F C 
  • 14. P 7 Triclinic  Parallelepiped (general) a b c I F C
  • 15. FUNDAMENTAL QUANTITIES • NEAREST NEIGHBOUR DISTANCE (2R). • ATOMIC RADIUS (R). • COORDINATION NUMBER (N). • ATOMIC PACKING FACTOR.
  • 16. 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) (Courtesy P.M. Anderson)
  • 17. ATOMIC PACKING FACTOR (APF):SC APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 atoms unit cell a R=0.5a APF = 1 4 3 a3 close-packed directions contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.24, Callister & Rethwisch 8e. volume atom (0.5a) 3 volume unit cell
  • 18. 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 (Courtesy P.M. Anderson) Adapted from Fig. 3.2, Callister & Rethwisch 8e. 2 atoms/unit cell: 1 center + 8 corners x 1/8
  • 19. ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68 3a a 2a Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. atoms R Close-packed directions: length = 4R = 3 a a 4 2 unit cell 3 APF = ( 3a/4) 3 a3 volume atom volume unit cell
  • 20. 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 Adapted from Fig. 3.1, Callister & Rethwisch 8e. (Courtesy P.M. Anderson) 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
  • 21. ATOMIC PACKING FACTOR: FCC • APF for a face-centered cubic structure = 0.74 maximum achievable APF Close-packed directions: length = 4R = 2 a 2a a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell atoms 4 4 unit cell 3 APF = ( 2a/4) 3 a3 volume atom volume unit cell
  • 22. MILLER INDICES • PROCEDURE FOR WRITING DIRECTIONS IN MILLER INDICES • DETERMINE THE COORDINATES OF THE TWO POINTS IN THE DIRECTION. (SIMPLIFIED IF ONE OF THE POINTS IS THE ORIGIN). • SUBTRACT THE COORDINATES OF THE SECOND POINT FROM THOSE OF THE FIRST. • CLEAR FRACTIONS TO GIVE LOWEST INTEGER VALUES FOR ALL COORDINATES
  • 23. MILLER INDICES • INDICES ARE WRITTEN IN SQUARE BRACKETS WITHOUT COMMAS (EX: [HKL]) • NEGATIVE VALUES ARE WRITTEN WITH A BAR OVER THE INTEGER. [hkl] • EX: IF H<0 THEN THE DIRECTION IS •
  • 24. MILLER INDICES • CRYSTALLOGRAPHIC PLANES • IDENTIFY THE COORDINATE INTERCEPTS OF THE PLANE • THE COORDINATES AT WHICH THE PLANE INTERCEPTS THE X, Y AND Z AXES. • IF A PLANE IS PARALLEL TO AN AXIS, ITS INTERCEPT IS TAKEN AS . • IF A PLANE PASSES THROUGH THE ORIGIN, CHOOSE AN EQUIVALENT PLANE, OR MOVE THE ORIGIN • TAKE THE RECIPROCAL OF THE INTERCEPTS
  • 25. Miller Indices for planes (0,0,1) z y (0,3,0) x (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)
  • 26. MILLER INDICES • CLEAR FRACTIONS DUE TO THE RECIPROCAL, BUT DO NOT REDUCE TO LOWEST INTEGER VALUES. • PLANES ARE WRITTEN IN PARENTHESES, WITH BARS OVER THE NEGATIVE INDICES. [hkl] • EX: (HKL) OR IF H<0 THEN IT BECOMES
  • 27. z z y y x x Intercepts → 1 Plane → (100) Intercepts → 1 1 Plane → (110) z y Intercepts → 1 1 1 x Plane → (111) (Octahedral plane)
  • 28. INTER-PLANAR SPACING • FOR ORTHORHOMBIC, TETRAGONAL AND CUBIC UNIT CELLS (THE AXES ARE ALL MUTUALLY PERPENDICULAR), THE INTER-PLANAR SPACING IS GIVEN BY: h, k, l = Miller indices a, b, c = unit cell dimensions • For cube a = b = c than a d hkl h2 k 2 l 2
  • 29. REFRENCES • APPLIED PHYSICS BY P.K. PALANISAMY • http://en.wikipedia.org/wiki/crystal_structure • http://journals.iucr.org/c/ • http://www.scirp.org/journal/csta/ • http://www.asminternational.org/portal/site/www/subje ctguideitem/?vgnextoid=ad7cdc8cc359d210vgnvcm10 0000621e010arcrd

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