crystal HIT presentation science slides.ppt

crystal

1. crystallography lv

2. Useful concept for crystallography & diffraction
Lattice planes
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:
a
b
origin

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

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

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,∞
a
b
origin
1
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)
a
b
origin
1
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. Strange indices
For hexagonal lattices - sometimes see 4-index
notation for planes (hkil)
where i = - h - k
a
1
a
2
a
3
(110)(1120)

26. Zones
2 intersecting lattice planes form a zone
zone
axis
zone
axis zone axis [uvw] is
ui + vj + wk
i j k
h
1
k
1
l
1


h
2
k
2
l
2
plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0
if (h
1
k
1
l
1
) and (h
2
k
2
l
2
) in same zone, then
(h
1
+h
2
k
1
+k
2
l
1
+l
2
) also in same zone.

27. Zones
zone axis [uvw] is
ui + vj + wk
(011) in same zone? hu + kv + lw = 0
0·0 + 1·1 - 1·1 = 0
if (h
1
k
1
l
1
) and (h
2
k
2
l
2
) in same zone, then
(h
1
+h
2
k
1
+k
2
l
1
+l
2
) also in same zone.
Example: zone axis for
(111) & (100) - [011]
i j k
h
1
k
1
l
1


h
2
k
2
l
2
i j k
1 1 1

1 0 0
(100)
(111) [011]

28. Reciprocal lattice
Real space lattice

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

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

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

32. (100)
planes
n
100
d
100
(100)
Reciprocal lattice
Real space lattice ––> the (100) reciprocal lattice pt

33. (010)
planes
n
010
d
010
(100)
(010)
Reciprocal lattice
The (010) recip lattice pt

34. The (020) reciprocal lattice point
(020)
planes
n
020
d
020
(100)
(010)(020)
Reciprocal lattice

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

36. The (110) reciprocal lattice point
(110)
planes
(100)
(010)(020)
n
110
d
110
(110)
Reciprocal lattice

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

38. Reciprocal lattice notation
Reciprocal lattice

39. Reciprocal lattice for hexagonal real space lattice
Reciprocal 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