Exercise teaching students how to index electron diffraction patterns

1. Exercise:
indexing patterns of CaF2

2. h k l d I F
1 1 1 3.15349 83.73 61.89
2 0 0 2.731 0.11 3.07
2 2 0 1.93111 100 96.55
3 1 1 1.64685 31.44 46.49
2 2 2 1.57674 0.2 6.81
4 0 0 1.3655 12.69 74.25
3 3 1 1.25307 11.35 38.65
4 2 0 1.22134 0.54 8.67
4 2 2 1.11493 23.75 61.87
5 1 1 1.05116 6.88 34.2
3 3 3 1.05116 2.29 34.2
You need this table made for CaF2

3. We are going to index these patterns:
Start with easiest:
highest symmetry or smallest interreflection distances
= usually lower zone indices (“main zones”)

4. First pattern:
Apparent symmetry: 4-fold

5. a
b
c
Along which direction does the 4-old axis lie in a cubic
system? <001>
<011>
<111>
b
a
c
CaF2

6. probably this is <001>
(Cubic: [100], [010], [001] equivalent = <001>)

7. To do: measure the distances, compare to list d-hkl, index
consistently.
length scalebar = R (in mm)
here it depends on your print/screen
size, so let us suppose it is 45.5 mm
Step 1: Use the scalebar for the conversion
factor to 1/d-values.
equal to 1/0.08 nm
R.d=L
then L would be
36.4 mmÅ
53.8 mmÅ
0.02 mmÅ

8. Now calculate the L for your own case and use that for
the following slides.
Step 2: measure the distance of two reflections, not on
the same line, calculate the corresponding d-value
Point 1
d
5.46 Å
3.15 Å
2.73 Å
Point 2
d
5.46 Å
3.15 Å
2.73 Å
1
2

9. To do: measure the distances, compare to list d-hkl, index.
Step 3: look up in the table to which
reflection this corresponds
100
110
200
Point 1
d
Point 2
d
Point 1
hkl
Point 2
hkl
1
2
5.46 Å
3.15 Å
2.73 Å
5.46 Å
3.15 Å
2.73 Å
100
110
200

10. Keep in mind: d-values for all equivalent {hkl}!
Step 4: make the indexation consistent
100
010
1
2
If point 1 is 200 then point 2 is 020 or 002.
Choose and stick to your choice.

11. 100
010
200020

12. Step 5: calculate the zone-index
[100]
[010]
[001]
010
200020

13. Next zone
Which one would be easiest next?
2
3
4

14. Next zone: with reflections closest to the central beam.
Because reflections far from the central beam:
lower d-values
larger amount of possible matches of hkl to this d
difficult to conclude which one is correct index!

15. Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
Point 1
d
2.57 Å
2.75 Å
3.15 Å
Point 2
d
1 2
2.57 Å
2.73 Å
3.15 Å

16. Look up in the table to which reflection
this corresponds
110
200
111
110
200
111
Point 1
d = 3.15 Å
Point 2
d = 2.73 Å
hkl hkl
1 2

17. Make the indexation in a consistent manner.
1 2
Point 2 should be indexed as
200
020
200
all are correct
- (see next slide why)

18. Consistency:
This is a tilt series...
...so the common row needs to have the
same indices in all patterns
200 200
200 200

19. Consistency:
2001
Point 1 should be indexed as
111
111
111
all of the above are allowed
-
--
(see next slide why)

20. Consistency:
200111
111
-
1
200
3
1 and 3 have the same d-value
+
relation between 1 and 3 = vector 200
you need two indices such that
h1 k1 l1 = h3+2 k3 l3
(also possible 111 and 111, make a choice and stick to it for the following patterns)
- - - - -

21. Consistency:
200
111
111
-
Also: sum = 022.
022
Side-remark: if 𝑔 perpendicular to 𝑔200 then type
of reflection indeed needs to be of type 0kl.

22. Consistency:
200111
111
-
1
200
3
If 111 were point 1-
311
-
then point 3 would be
311
111

23. Consistency:
200111
111
-
1
200
3
If 111 were point 1-
point 3 = 311-

24. Consistency:
200111
111
-
1
200
3
If 111 were point 1-
point 3 = 311-
d311 ≠ dpoint3-
but

25. Consistency:
200111
111
-
1
200
3
If 111 for point 1-
point 3 = 311-
d311 ≠ dpoint3-
but

26. Calculate the zone-index
The zone-index is: [011]
[011]
200
111
111
-
022
-

27. ...this helps to index the two remaining patterns!!!
Consistency:
This is a tilt series...

28. a
b
c
The crystallite is threedimensional.

29. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0] So, the reciprocal lattice is threedimensional.

30. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
ED patterns are sections of reciprocal space.
[001]
This section is the [001] zone.

31. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[011]
-
This section is the [011] zone:
-

32. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[001]
[011]
-
We tilt from [001] to [011]:
-

33. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[001]
[011]
-
So the tilt series gives pattern of consecutive sections
between these two end zones
Closest is 020.

34. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[001]
[015]
-
[011]
-
x
051
Closest is 151.

35. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[011]
[001]
[013]
-x
031
Closest is 131.

36. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[011]
[001]
x
021
[012]
-
042
Closest is 042.

37. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[011]
[001]
[035]
-
x
053
Closest is 153.

38. 0,4,0
0,4,2
1,5,1
0,4,4
1,5,3
0,2,0
0,2,2
1,3,1
0,2,4
2,4,0
1,3,3
2,4,2
3,5,1
0,0,2
2,4,4
1,1,1
3,5,3
0,0,4
2,2,0
1,1,3
2,2,2
3,3,1
2,2,4
4,4,0
3,3,3
2,0,0
4,4,2
5,5,1
2,0,2
4,4,4
3,1,1
5,5,3
2,0,4
4,2,0
3,1,3
4,2,2
5,3,1
4,2,4
5,3,3
4,0,0
4,0,2
5,1,1
4,0,4
5,1,3
Zone axis : [0,0,0]
[001]
[011]
-
Closest is 111. Perpendicular is 022.

39. 0,4,0
0,4,2
0,4,4
0,2,0
1,5,1
0,2,2
1,5,3
0,2,4
1,3,1
0,0,2
1,3,3
0,0,4
2,4,0
1,1,1
2,4,2
1,1,3
2,4,4
2,2,0
3,5,1
2,2,2
3,5,3
2,2,4
2,0,0
3,3,1
2,0,2
3,3,3
2,0,4
4,4,0
3,1,1
4,4,2
3,1,3
4,4,4
4,2,0
5,5,1
4,2,2
5,5,3
4,2,4
4,0,0
5,3,1
4,0,2
5,3,3
4,0,4
5,1,1
5,1,3
We can also see this in projection
go through slides 39-42 in continuous manner

40. 0,4,4
0,4,2
0,4,0
1,5,3
1,5,1
0,2,4
0,2,2
0,2,0
1,3,3
1,3,1
0,0,4
0,0,2
2,4,4
2,4,2
2,4,0
1,1,3
1,1,1
3,5,3
3,5,1
2,2,4
2,2,2
2,2,0
3,3,3
3,3,1
2,0,4
2,0,2
2,0,0
4,4,4
4,4,2
4,4,0
3,1,3
3,1,1
5,5,3
5,5,1
4,2,4
4,2,2
4,2,0
5,3,3
5,3,1
4,0,4
4,0,2
4,0,0
5,1,3
5,1,1
We can also see this in projection

41. 0,4,4
0,4,2
0,4,0
0,2,4
0,2,2
0,2,0
0,0,4
0,0,2
1,5,3
1,5,1
1,3,3
1,3,1
1,1,3
1,1,1
2,4,4
2,4,2
2,4,0
2,2,4
2,2,2
2,2,0
2,0,4
2,0,2
2,0,0
3,5,3
3,5,1
3,3,3
3,3,1
3,1,3
3,1,1
4,4,4
4,4,2
4,4,0
4,2,4
4,2,2
4,2,0
4,0,4
4,0,2
4,0,0
5,5,3
5,5,1
5,3,3
5,3,1
5,1,3
5,1,1
Zone axis : [0,0,0]
We can also see this in projection

42. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
We can also see this in projection
which is easier to draw manually...

43. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
[001]
[015]
-

44. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
[001]
[015]
-
[013]
-

45. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
[001]
[015]
-
[013]
-
[012]
-

46. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
[001]
[015]
-
[013]
-
[012]
-
[035]
-

47. 0,0,4
0,0,2
0,2,4
0,2,2
0,4,4
0,2,0
0,4,2
0,4,0
1,1,3
1,1,1
1,3,3
1,3,1
1,5,3
1,5,1
2,0,4
2,0,2
2,2,4
2,0,0
2,2,2
2,4,4
2,2,0
2,4,2
2,4,0
3,1,3
3,1,1
3,3,3
3,3,1
3,5,3
3,5,1
4,0,4
4,0,2
4,2,4
4,0,0
4,2,2
4,4,4
4,2,0
4,4,2
4,4,0
5,1,3
5,1,1
5,3,3
5,3,1
5,5,3
5,5,1
xis : [0,0,0]
[001]
[015]
-
[013]
-
[012]
-
[035]
-
[011]
-

48. Right upper zone:
Point 2
d
1.22 Å
1.11 Å
1.05 Å
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
We already know the
first point: 200.
200
2

49. Look up in the table to which reflection this corresponds:
We know already it is either 151 or 131 or 042 or 153
1.05 Å 151
131
042
Point 2
d
Point 2
hkl
200
2

50. If this were not a tilt series...
Point 2 could have been at first sight
both 115 and 333...
In this case:
Can compare the experimental angles between reflections
to the theoretical angles
-either formulas from any standard crystallography work
-or simply simulate the different zones calculated for the
different options (JEMS, CrystalKit, Carine,...) to check this
Or in this particular case of 333: you would need to see 111 and 222 at 1/3 and 2/3 of the distance.

51. Calculate the zone-index
The zone-index is:
[0 2 10]
[0 1 5]
[0 1 5]
200
151
-

52. [001]
[015]-
[013]
-
[012]-
[035]-
[011]-
010
031 051
053
What if you didn’t know the material?
You would just need to check
more possibilities:
043
032
041021
[025]-
052 [014]-
[023]-
[034]-
When indexed correctly, the patterns in between
have to give you one of these as zone-index.
011

53. Pattern bottom left:
Point 2
d
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
200
2
1.65 Å
1.58 Å
1.37 Å

54. Look up in the table to which reflection this corresponds.
We know already it is either: 151 or 131 or 042 or 153
1.65 Å 113
131
311
Point 2
d
Point 2
hkl
200
2

55. The indexation is indeed consistent.
200
2
3
4
Same reasoning: if 2 is 131, then 3 is 131, and 4 is 062.
All consistent with each other
(cf. explanation previous pattern).
-

56. Calculate the zone-index = [013]
= in agreement with sections scheme
-
200
131
062

57. Make your analysis easier by not taking
ED patterns from separate crystals, but
taking different ED patterns from the
same crystallite, if possible.
=“Tilt series”

58. So now you have indexed these four patterns.
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-

59. ...indexed patterns give you info on fase,
orientation, cell parameters,...
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-

60. What if you do not have any prior
knowledge when you have to
index?
Analyse the patterns  try to propose basis vectors
(For example reflections closest to the central beam)
Same system as previous slides:
can you index all reflections?
If not, adapt your choice of
basis vectors and try again.

61. If we do not know the space group, the next step would
be to determine it!
(maybe you started from 0 or you had only cell parameters from XRD or...)
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-