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Cubic systems

Cubic systems. Paul Sundaram University of Puerto Rico at Mayaguez. Review. Seven crystal systems Fourteen Bravais lattices Cubic and Hexagonal systems: 90% of all metals have a cubic or hexagonal structure. Cubic system characteristics. Unit cell a=b=c, a = b = g =90˚

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Cubic systems

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  1. Cubic systems Paul Sundaram University of Puerto Rico at Mayaguez

  2. Review • Seven crystal systems • Fourteen Bravais lattices • Cubic and Hexagonal systems: 90% of all metals have a cubic or hexagonal structure

  3. Cubic system characteristics • Unit cell a=b=c, a= b = g =90˚ • Face diagonal and body diagonal • Number of atoms per unit cell • Coordination number:number of nearest neighbor atoms • Close-packed structures • Atomic Packing Factor (APF) APF=(vol.of atoms in unit cell)/(vol. of unit cell) • Atom positions, crystallographic directions and crystallographic planes (Miller indices) • Planar atomic density & linear atomic density

  4. Some concepts • Number of atoms per unit cell • Corner atom = 1/8 per unit cell • Body centered atom = 1 • Face centered atom = 1/2 Body diagonal= Face diagonal=

  5. Simple cubic(P)

  6. Simple cubic

  7. Simple cubic

  8. Body centered cubic(I)

  9. Real picture

  10. Body centered cubic

  11. Body centered cubic

  12. Face centered cubic(F)

  13. Real picture

  14. Face centered cubic

  15. Face centered cubic *Highest packing possible in real structures

  16. Questions

  17. Atomic Positions Z (1/2,1/2,1) (0,1,1) (0,0,1) (1/2,1/2,1/2) (1/2,0,1/2) Y (0,0,0) X

  18. Crystallographic directions Concept of a vector & components R R cos(90-f) f R cos(f)

  19. Components X:a cos 90=0 Y:a cos 90=0 Z:a cos 0=a Miller index:[001] Examples Components X:a cos 90=0 Y:a cos 0=a Z:a cos 90=0 Miller index:[010] Components X:a cos 0=a Y:a cos 90=0 Z:a cos 90=0 Miller index:[100] Components X:a cos 90=0 Y:a cos 0=a Z:a cos 90=0 Miller index:[010] Family <100> <010> <001>

  20. Components X: 0 Y: a Z: a Miller index:[011] Examples Components X: a Y: 0 Z: 1 Miller index:[101] Components X: a Y: a Z: 0 Miller index:[110]

  21. Components X: 0 Y: -a Z: -a Miller index:[0 1 1] Examples Components X: -a Y: 0 Z: -a Miller index:[1 0 1] Components X: -a Y: -a Z: 0 Miller index:[1 1 0] Family <110> <011> <101>

  22. Examples Components X: -a Y: -a Z: -a Miller index:[111] Components X: a Y: a Z: a Miller index:[111] Family <111>

  23. Crystallographic planes 1.Intersections with X,Y,Z axes  1 2. Take the inverse 1/ 1/ 1/1 Miller index(0 0 1) Z How to determine indices of plane 1.Intersections with X,Y,Z axes 1  2. Take the inverse 1/1 1/ 1/ Miller index(1 0 0) Y X 1.Intersections with X,Y,Z axes  1  2. Take the inverse 1/ 1/1 1/ Miller index(0 1 0) Family {100}

  24. Example Z How to determine indices of plane 1.Intersections with X,Y,Z axes 1 1  2. Take the inverse 1/1 1/1 1/  Miller index(1 1 0) Y X Family {110}

  25. Example Z How to determine indices of plane 1.Intersections with X,Y,Z axes 1 1 1 2. Take the inverse 1/1 1/1 1/1 Miller index(1 1 1) Y X Family {111}

  26. Examples Components X: 1/2 Y: 1/2 Z: 1 [1/2 1/2 1] [112] Components X: -1 Y: 1 Z: 1/2 [-1 1 1/2] [2 2 1] Components X: -1 Y: -1/2 Z: 1/2 [-1 -1/2 1/2] [2 1 1]

  27. Examples Intersections -1,-1,1/2 Inverse -1 -1 2 (1 1 2) Intersections 1/2,1,1/2 Inverse 2 1 2 (212) Intersections 1/6,-1/2,1/3 Inverse 6 -2 3 (6 2 3) Intersections -1/2,1/2,1 Inverse -2 2 1 (2 2 1)

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