Crystal structures & Packing Fraction

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Crystal structures & Packing Fraction

  1. 1. CRYSTAL“A crystal is a solid in which atoms are arranged insome regular repetition pattern in all directions.”“Aggregation of molecules with a definite internalstructure and the external form of a solid enclosedby symmetrically arranged plane faces.” STRUCTURES “Structure of anything is defined as the framework of its body.”
  2. 2. Crystal = Lattice+BaseMotif or basis: Typically an atom or a group of atoms associated witheach lattice point. Lattice  The underlying periodicity of the crystal Basis  Entity associated with each lattice points Lattice Crystal Translationally periodic Translationally periodic arrangement of points. arrangement of motifs.
  3. 3. Crystal = Lattice (Where to repeat) + Motif (What to repeat) Crystal a = Lattice a + Motif Note: all parts of the motif do not sit on the lattice a point 2
  4. 4. Let us construct the crystal considered before starting with aninfinite array of points spaced a/2 apartPut arrow marks pointing up and down alternately on the points:What we get is a crystal of lattice parameter „a‟ and not „a/2‟!And the motifis: +
  5. 5. A strict 1D crystal = 1D lattice + 1D motif The only kind of 1D motif is a line segment. An unit cell is a representative unit of the structure (finite part of a infinite structure) .  Which when repeated gives the whole structure.Lattice + Motif =Crystal
  6. 6.  2D crystal = 2D lattice + 2D motif Lattice +  Motifb  a
  7. 7. Crystal                     =                                   
  8. 8.  3D crystal = 3D lattice + 3D motifs CRYSTAL OR SPACE LATTICE  It is defined as an array of points in 3 dimensions in which every point has surroundings identical to every other point in array. According to BRAVAIS there are 14 possible types of space lattice in 7 basic crystal system
  9. 9. a = b= c = = = 90º • Simple Cubic (P) - SC • Body Centred Cubic (I) – BCC • Face Centred Cubic (F) - FCC Elements with Cubic structure → SC: F, O BCC: Cr, Fe, Nb, K, W FCC: Al, Ar, Pb, Ni, Ge
  10. 10. SIMPLE CUBIC STRUCTURE• Cubic unit cell is 3D repeat unit• Rare (only Po has this structure) • Coordination No. = 6 (# nearest neighbors)
  11. 11. a R=0.5a close-packed directions contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, Callister 6e. 0.52Lattice constant • APF for a simple cubic structure =
  12. 12. BODY CENTERED CUBIC STRUCTURE • Coordination No. = 8 (# nearest neighbors)
  13. 13. Unit cell c ontains: 1 + 8 x 1/8 = 2 atoms/unit cell R Adapted from a Fig. 3.2, Callister 6e.• APF for a body-centered cubic structure = 3/8 = 0.68
  14. 14. FACE CENTERED CUBIC STRUCTURE Atoms are arranged at the corners and center of each cube face of the cell. ◦ Atoms are assumed to touch along face diagonals
  15. 15. • Coordination No. = 12 (# nearest neighbors)
  16. 16. Unit cell c ontains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a• APF for a body-centered cubic structure = /(3 2) = 0.74
  17. 17. • FCC Unit Cell• ABCABC... Stacking Sequence• 2D Projection A B B C A A sites B B B C C B sites B B C sites
  18. 18. Ideally, c/a = 1.633 for close packingHowever, in most metals, c/a ratio deviates from this valu
  19. 19. • ABAB... Stacking Sequence • 3D Projection • 2D Projection A sites B sites A sites • Coordination NO.= 12 • APF = 0.74, for ideal c/a ratio of 1.633
  20. 20. Close packed crystals A plane B plane C plane A plane …ABCABCABC… packing …ABABAB… packing[Face Centered Cubic (FCC)] [Hexagonal Close Packing (HCP)]
  21. 21. Examples of elements with Cubic Crystal Structure Fe Cu Po n = 2 BCC n = 4 FCC/CCPn = 1 SC n = 8 DC C (diamond)
  22. 22. a=b c = = = 90º Simple Tetragonal Body Centred Tetragonal -BCT  Elements with Tetragonal structure → In, Sn
  23. 23. Example of an element with Body Centred Tetragonal Crystal Structure B C T
  24. 24. a b c = = = 90º  Simple Orthorhombic  Body Centred Orthorhombic  Face Centred Orthorhombic  End Centred OrthorhombicElements with Orthorhombic structure → Br, Cl, Ga
  25. 25. Element with Orthorhombic Crystal Structure
  26. 26. a=b c = = 90º =120º Simple Hexagonal Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn
  27. 27. Element with Hexagonal Crystal Structure
  28. 28. a=b=c = = 90º Rhombohedral (simple)Elements with Trigonal structure → As, B, Bi, Hg
  29. 29. Element with Simple Trigonal Crystal Structure
  30. 30. a b c = = 90º  Simple Monoclinic  End Centred (base centered) MonoclinicElements with Monoclinic structure → P, Pu, Po
  31. 31. a b c • Simple Triclinic
  32. 32. 14 Bravais Lattices divided into 7 Crystal Systems A Symmetry based concept „Translation‟ based concept Crystal System Shape of UC Bravais Lattices P I F C1 Cubic Cube   2 Tetragonal Square Prism (general height)  3 Orthorhombic Rectangular Prism (general height)    4 Hexagonal 120 Rhombic Prism 5 Trigonal Parallopiped (Equilateral, Equiangular) 6 Monoclinic Parallogramic Prism  7 Triclinic Parallopiped (general)  P Primitive I Body Centred F Face Centred C A/B/C- Centred
  33. 33. Face Centred Cubic (FCC) Lattice Two Carbon atom Motif (0,0,0) & (¼, ¼, ¼) + = Diamond Cubic Crystal

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