1. crystallography lv

2. Useful concept for crystallography & diffraction
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 sets of planes in lattice d
d -
interplanar
spacing

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

26. 2 intersecting lattice planes form a zone
Zones
2 intersecting lattice planes form a zone
zone
axis
zone
axis
zone axis [uvw] is ui + vj + wk
i j k
h1 k1 l1
h2 k2 l2
plane (hkl) belongs to zone [uvw] if hu + 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. zone axis [uvw] is ui + vj + wk
Zones
Example: zone axis for
(111) & (100) - [011]
zone axis [uvw] is ui + vj + wk
i j k
h1 k1 l1
h2 k2 l2
i j k
(100)
(111)
[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. Real space lattice - basis vectors
Reciprocal lattice
Real space lattice - basis vectors a a

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

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

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

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

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

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

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

37. Still more reciprocal lattice points
(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