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Understanding Lattice Planes in Crystallography

Learn about lattice planes, Miller indices, zone axes, reciprocal lattice, and more concepts in crystallography for better understanding. Detailed exercises provided.

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Understanding Lattice Planes in Crystallography

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  1. crystallography lv

  2. d - interplanar spacing d Lattice planes Useful concept for crystallography & diffraction Think of sets of planes in lattice - each plane in set parallel to all others in set. All planes in set equidistant from one another Infinite number of set of planes in lattice

  3. Keep track of sets of planes by giving them names - Miller indices (hkl) Lattice planes

  4. Miller indices(hkl) Choose cell, cell origin, cell axes: origin b a

  5. Miller indices(hkl) Choose cell, cell origin, cell axes Draw set of planes of interest: origin b a

  6. Miller indices(hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin: origin b a

  7. Miller indices(hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes: 1,1,∞ origin b 1 a 1

  8. Miller indices(hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes 1,1,∞ Invert these to get (hkl) (110) origin b 1 a 1

  9. Lattice planes Exercises

  10. Lattice planes Exercises

  11. Lattice planes Exercises

  12. Lattice planes Exercises

  13. Lattice planes Exercises

  14. Lattice planes Exercises

  15. Lattice planes Exercises

  16. Lattice planes Exercises

  17. Lattice planes Exercises

  18. Lattice planes Exercises

  19. Lattice planes Exercises

  20. Lattice planes Exercises

  21. Lattice planes Exercises

  22. Lattice planes Exercises

  23. Lattice planes Exercises

  24. Lattice planes Two things characterize a set of lattice planes: interplanar spacing (d) orientation (defined by normal)

  25. a3 (110) (1120) a2 a1 Strange indices For hexagonal lattices - sometimes see 4-index notation for planes (hkil) where i = - h - k

  26. zone axis zone axis [uvw] is ui+ vj + wk zone axis i j k h1 k1 l1 h2 k2 l2 Zones 2 intersecting lattice planes form a zone plane (hkl) belongs to zone [uvw] ifhu + kv + lw = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

  27. (100) (111) [011] i j k h1 k1 l1 h2 k2 l2 i j k 1 1 1 1 0 0 Zones zone axis [uvw] is ui+ vj + wk Example: zone axis for (111) & (100) - [011] (011) in same zone? hu + kv + lw = 0 0·0 + 1·1 - 1·1 = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

  28. Reciprocal lattice Real space lattice

  29. Reciprocal lattice Real space lattice - basis vectors a a

  30. Reciprocal lattice Real space lattice - choose set of planes (100) planes n100

  31. Reciprocal lattice Real space lattice - interplanar spacing d (100) planes d100 1/d100 n100

  32. Reciprocal lattice Real space lattice ––> the (100) reciprocal lattice pt (100) planes d100 n100 (100)

  33. Reciprocal lattice The (010) recip lattice pt n010 (100) planes d010 (010) (100)

  34. Reciprocal lattice The (020) reciprocal lattice point n020 (020) planes d020 (010) (020) (100)

  35. Reciprocal lattice More reciprocal lattice points (010) (020) (100)

  36. Reciprocal lattice The (110) reciprocal lattice point (100) planes n110 d110 (010) (020) (110) (100)

  37. Reciprocal lattice Still more reciprocal lattice points (100) planes (010) (020) (100) the reciprocal lattice (230)

  38. Reciprocal lattice Reciprocal lattice notation

  39. Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

  40. Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

  41. Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

  42. Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

  43. Reciprocal lattice

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