tem workshop 2013: electron diffraction

$: TEM workshop 2013: Electron diffraction$
Electron Diffraction

At the end of this lecture you should be able to

(1) index SAED patterns in case the cell parameters are already known

(2) determine the possible space groups from SAED patterns

(3) determine possible point groups from CBED patterns

Combine (2) and (3) to find the space group.

Reflections: what do they represent? What is their origin? What information

can they give us?

Constructive vs. destructive interference reflection – no reflection

position distances between the planes (d-values) intensity occupation in the planes

both symmetry of the structure

Single phase?

Crystalline or amorphous?

Orientation crystals/film

?

Targeted phase?

Domain formation?

Crystal parameters?

Selected Area Electron Diffraction

SAED

One atom type A

a b

Basic cell, one plane

b>a

a=3 Å b=5 Å

One atom type A

a b


b>a

(100)

100

000

a=3 Å b=5 Å

One atom type A

a b


b>a

(100)

100

000

1/3Å

a=3 Å b=5 Å

One atom type A

a b


b>a

(100)

100

a*

a=3 Å b=5 Å

One atom type A

a b


b>a

(010)

a=3 Å b=5 Å

One atom type A

a b


b>a

(010)

010

a=3 Å b=5 Å

One atom type A

a b


b>a

(010)

010

a=3 Å b=5 Å

1/5Å

One atom type A

a b


b>a

(010)

010 b*

a=3 Å b=5 Å

One atom type A

010

100

a b

[001]

Basic cell EDP

b>a a*>b*

a*

b*

a=3 Å b=5 Å

One atom type A

010

100

a b

[001]

Basic cell EDP

b>a a*>b*

a*

b*

Central reflection

is always 000.

000

a=3 Å b=5 Å

One atom type A

010

100

a b

[001]

Basic cell EDP

a=3 Å b=5 Å

One atom type A

010

100

a b

[001]

Basic cell EDP

1/3Å 1/5Å

a=3 Å b=5 Å

010

100

[001]

Basic cell EDP

Indices all other reflections:

vector addition

110

020

200 210

120

1/3Å

1/5Å

Experimentally: the other way around:

010

100

[001]

*

*How?

Make a list of all reflections with hkl and their d-values.

• use Excel to make the list yourself

• use free software like Powdercell

• ...

d hkl

7.68 001

5.64 010

5.46 100

4.55 011

4.45 101

3.92 110

... ...

http://www.ccp14.ac.uk/ccp/web-mirrors/powdcell/a_v/v_1/powder/e_cell.html

010

100

Zone-index: [001]

obtained by vector multiplication.

Circle ACW around 000.

1 0 0 1 0 0 0 1 0 0 1 0

0*0-0*1 0*0-1*0 1*1-0*0

[ 0 0 1 ]

0 01 0

0 10 0

1 00 1

(010)

(100)

(110)

[001]

010

100 110

[001]

110 -

(110) -

Exercise: index the given patterns taken

from a CaF2 mineral

a

b

c

b

a

c

CaF2

cubic

a=5.46 Å

Fm3m -

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

We are going to index these patterns. They are obtained by tilting around the diagonal row.

(Online version: working page can be found at the end.)

We are going to index these patterns. They are obtained by tilting around the diagonal row.

(Online version: workpage can be found at the end.)

Start with easiest: highest symmetry or smallest interreflection distances = usually lower zone indices (“main zones”)

(Online version: workpage can be found at the end.)

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

Why go for smaller interreflection distances?

=higher d = less choices

First pattern:

Apparent symmetry: 4-fold

4,4,0

4,2,0

4,0,0

4,-2,0

4,-4,0

2,4,0

2,2,0

2,0,0

2,-2,0

2,-4,0

0,4,0

0,2,0

0,-2,0

0,-4,0

-2,4,0

-2,2,0

-2,0,0

-2,-2,0

-2,-4,0

-4,4,0

-4,2,0

-4,0,0

-4,-2,0

-4,-4,0

Zone axis : [0,0,1]

Go for highest symmetry

a

b

c

Along which direction does the 4-fold axis lie in a cubic system? <001>

<011> <111>

b

a

c

CaF2

a

b

c

Along which direction does the 4-old axis lie in a cubic system? <001>

<011> <111>

b

a

c

CaF2

probably this is <001>

(Cubic: [100], [010], [001] equivalent = <001>)

4,4,0

4,2,0

4,0,0

4,-2,0

4,-4,0

2,4,0

2,2,0

2,0,0

2,-2,0

2,-4,0

0,4,0

0,2,0

0,-2,0

0,-4,0

-2,4,0

-2,2,0

-2,0,0

-2,-2,0

-2,-4,0

-4,4,0

-4,2,0

-4,0,0

-4,-2,0

-4,-4,0

Zone axis : [0,0,1]

To do: measure the distances, compare to list d-hkl, index consistently.

Scalebar = R (in mm)

Step 1: Use the scalebar for the conversion factor to 1/d-values.

equal to 1/0.08 nm R.d=L

L

42 43 44

34.4 mmÅ 53.8 mmÅ 0.02 mmÅ

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

4,4,0

4,2,0

4,0,0

4,-2,0

4,-4,0

2,4,0

2,2,0

2,0,0

2,-2,0

2,-4,0

0,4,0

0,2,0

0,-2,0

0,-4,0

-2,4,0

-2,2,0

-2,0,0

-2,-2,0

-2,-4,0

-4,4,0

-4,2,0

-4,0,0

-4,-2,0

-4,-4,0

Zone axis : [0,0,1]

5.46 Å 3.15 Å 2.73 Å

1 2

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

Keep in mind: d-values valid 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 with your choice.

100

010

200 020

Step 5: calculate the zone-index

[100] [010] [001]

010

200 020

Step 5: calculate the zone-index

[100] [010] [001]

200 020

Next zone

Which one would be easiest next?

2 3 4

Next zone: with reflections closest to the central beam.

Reflections closer to the central beam: higher d-values

smaller amount of possible matches of hkl to this d

5,5,5

5,3,3

5,1,1

5,-1,-1

5,-3,-3

5,-5,-5

4,4,4

4,2,2

4,0,0

4,-2,-2

4,-4,-4

3,5,5

3,3,3

3,1,1

3,-1,-1

3,-3,-3

3,-5,-5

2,4,4

2,2,2

2,0,0

2,-2,-2

2,-4,-4

1,5,5

1,3,3

1,1,1

1,-1,-1

1,-3,-3

1,-5,-5

0,4,4

0,2,2

0,-2,-2

0,-4,-4

-1,5,5

-1,3,3

-1,1,1

-1,-1,-1

-1,-3,-3

-1,-5,-5

-2,4,4

-2,2,2

-2,0,0

-2,-2,-2

-2,-4,-4

-3,5,5

-3,3,3

-3,1,1

-3,-1,-1

-3,-3,-3

-3,-5,-5

-4,4,4

-4,2,2

-4,0,0

-4,-2,-2

-4,-4,-4

-5,5,5

-5,3,3

-5,1,1

-5,-1,-1

-5,-3,-3

-5,-5,-5

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

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

Make the indexation in a consistent manner.

1 2

Point 2 should be indexed as 200 020 200 all are correct

-

Consistency: This is a tilt series...

...so the common row needs to have the same indices in all patterns

200 200

200 200

Consistency:

5,5,5

5,3,3

5,1,1

5,-1,-1

5,-3,-3

5,-5,-5

4,4,4

4,2,2

4,0,0

4,-2,-2

4,-4,-4

3,5,5

3,3,3

3,1,1

3,-1,-1

3,-3,-3

3,-5,-5

2,4,4

2,2,2

2,0,0

2,-2,-2

2,-4,-4

1,5,5

1,3,3

1,1,1

1,-1,-1

1,-3,-3

1,-5,-5

0,4,4

0,2,2

0,-2,-2

0,-4,-4

-1,5,5

-1,3,3

-1,1,1

-1,-1,-1

-1,-3,-3

-1,-5,-5

-2,4,4

-2,2,2

-2,0,0

-2,-2,-2

-2,-4,-4

-3,5,5

-3,3,3

-3,1,1

-3,-1,-1

-3,-3,-3

-3,-5,-5

-4,4,4

-4,2,2

-4,0,0

-4,-2,-2

-4,-4,-4

-5,5,5

-5,3,3

-5,1,1

-5,-1,-1

-5,-3,-3

-5,-5,-5

200 1

Point 1 should be indexed as 111 111 111 all of the above are allowed

- - -

Consistency:

200 111

111 -

If 111 for point 1 -

311 -

Point 3 is 311 111

Consistency:


311 -

Point 3 is 311 111

200 1

Consistency:

200 111

111 -

1 200

3


point 3 = 311 -

Consistency:

200 111

111 -

1 200

3


point 3 = 311 -

d311 ≠ dpoint3 - but

Consistency:

200 111

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

h3+2 = h1 k3+0 = k1 l3+0 = l1

(also possible 111 and 111, make a choice and stick to it for the following patterns) - - -

= h1 k1 l1

Consistency:

200 111

111 -

Sum = 022.

Indeed consistent: if 𝑔 perpendicular to 𝑔 200 then type of reflection needs to be 0kl.

022 200 022

Calculate the zone-index

The zone-index is: [011] [011]

200 111

111 -

022

-

...this helps to index the two remaining patterns!!!

Consistency: This is a tilt series...

a

b

c

The crystallite is threedimensional.

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.

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.

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: -

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]: -

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.

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.

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.

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.

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.

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.

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

Zone axis : [0,0,0]

We can also see this in projection

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

Zone axis : [0,0,0]


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]


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

Zone axis : [0,0,0]

We can also see this in projection which is easier to draw manually...

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

Zone axis : [0,0,0]

[001]

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

Zone axis : [0,0,0]

[001]

[015] -

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

Zone axis : [0,0,0]

[001]

[015] -

[013] -

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

Zone axis : [0,0,0]

[001]

[015] -

[013] -

[012] -

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

Zone axis : [0,0,0]

[001]

[015] -

[013] -

[012] -

[035] -

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

Zone axis : [0,0,0]

[001]

[015] -

[013] -

[012] -

[035] -

[011] -

Right upper zone:

Point 2 d

1.22 Å 1.11 Å 1.05 Å


5,5,1

5,-5,-1

4,0,0

3,5,1

3,-5,-1

2,0,0

1,5,1

1,-5,-1

-1,5,1

-1,-5,-1

-2,0,0

-3,5,1

-3,-5,-1

-4,0,0

-5,5,1

-5,-5,-1

We already know the first point: 200.

200 2

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

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.

Calculate the zone-index

The zone-index is: [0 2 10] [0 1 5] [0 1 5]

200 151

-

[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

041 021

[025] -

052 [014] -

[023] - [034] -

When indexed correctly, the patterns in between have to give you one of these as zone-index.

011

Pattern bottom left:

5,3,1

5,-3,-1

4,0,0

3,3,1

3,-3,-1

2,0,0

1,3,1

1,-3,-1

-1,3,1

-1,-3,-1

-2,0,0

-3,3,1

-3,-3,-1

-4,0,0

-5,3,1

-5,-3,-1

Point 2 d


200 2

1.65 Å 1.58 Å 1.37 Å

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 042 131 153

Point 2 d

Point 2 hkl

200 2

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

Point 2 d

Point 2 hkl

200 2

042 131 153

The indexation is indeed consistent.

200 131 062

131 -

200 131 062

131 -

[013] -

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”

So now you have indexed these four patterns.

200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

...indexed patterns give you info on fase, orientation, cell parameters,...

200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

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.

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

200 111

200 020

[001] [015] -

[013] -

[011] -

Reflection conditions (SAED)

Space group?

P no reflection conditions

F h+k=2n, k+l=2n, h+l=2n

I h+k+l=2n

A/B/C k+l=2n/h+k=2n/h+k=2n

Point Group (CBED)

glide planes conditions on hk0/h0l/0kl

screw axes conditions on h00/0k0/00l

mirror planes, inversion centre, rotation axes

no extra conditions

CBED

SAED

+

Reflection conditions can be looked up in tables in International Tables for Crystallography Vol. A

Or using freeware such as Space Group Explorer

http://it.iucr.org/A/

http://www.calidris-em.com/spacegroupexplorer.php

Be careful: forbidden reflections can occur because of dynamical diffraction

Incident electron wave

When reflection conditions say this:

For example possible

020

100

Can see this:

010

100

020

100

F(100)≠0

F(1 10) ≠0

F(010)=0

Need to tilt to remove these paths...

Destroy double diffraction paths by tilting.

If becomes

If stays

then extinct, was due to DD

then not extinct.

You will need this table (from IT volume A)

Figure out reflection conditions for these sets.

hkl: h+k+l=2n h+k, k+l, h+l=2n h+k=2n

Step 1: determine the reflection conditions from the patterns.

200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

hkl: h+k+l=2n h+k, k+l, h+l=2n h+k=2n


200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

For these patterns both would be good....!?

This means we do not have sufficient information.

we missed [012], which will make the difference. -

By coincidence

4,4,2

4,0,0

4,-4,-2

2,4,2

2,0,0

2,-4,-2

0,4,2

0,-4,-2

-2,4,2

-2,0,0

-2,-4,-2

-4,4,2

-4,0,0

-4,-4,-2

200

042

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

Zone axis : [0,0,0]

[001]

[015] -

[013] -

[012] -

[035] -

[011] -


we missed [012], which will make the difference. -

By coincidence

4,4,2

4,0,0

4,-4,-2

2,4,2

2,0,0

2,-4,-2

0,4,2

0,-4,-2

-2,4,2

-2,0,0

-2,-4,-2

-4,4,2

-4,0,0

-4,-4,-2

200

042 hkl:

h+k+l=2n h+k, k+l, h+l=2n h+k=2n


we miss [012], which will make the difference. -

By coincidence

4,4,2

4,0,0

4,-4,-2

2,4,2

2,0,0

2,-4,-2

0,4,2

0,-4,-2

-2,4,2

-2,0,0

-2,-4,-2

-4,4,2

-4,0,0

-4,-4,-2

200

042 hkl:

h+k+l=2n h+k, k+l, h+l=2n h+k=2n

For only h+k=2n there is no reason why 021 would be absent.

It is possible to draw the wrong conclusions if you do not have

enough zones!

0kl: k=2n k,l=2n k+l=2n


200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

200

042

hhl: h=2n h,l=2n h+l=2n


200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

200

042

00l: no condition l=2n l=4n


200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -

200

042

Step 2: look up the matching extinction symbol in the International Tables of

Crystallography.

? ?

200 and 020 could be due to double diffraction...

200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -


Tilt around 200 until all other reflections gone except h00 axis:

200 131

200 151

200 111

200 020

[001] [015] -

[013] -

[011] -


Tilt around 200 until all other reflections gone except h00 axis:

200 does not disappear It is not double diffraction 00l: l=2n not 00l: l=4n

Step 2: look up the matching extinction symbol in

the International Tables of Crystallography.

From the reflection conditions you get the extinction symbol:

F - - -


F - - -

This still leaves 5 possible space groups

F23 Fm3 F432 F43m Fm3m

- - -


F - - -

This still leaves 5 possible space groups

F23 Fm3 F432 F43m Fm3m

Only difference: rotation axes and mirror planes

cannot be derived from reflection conditions

need CBED

- -

Convergent Beam Electron Diffraction

CBED

6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m

Table redrawn from B.F. Buxton, J.A. Eades, J.W. Steeds, G.M. Rackham: Phil. Trans. R. Soc. London, 281 (1976) 171

Diffraction groups vs. Point groups

For CBED you need sufficiently thick crystals:

10 nm 20 nm 30 nm

40 nm 50 nm

Table from: J.A. Eades, Convergent beam diffraction, in: Electron Diffraction Techniques, volume 1, ed. J. Cowley, Oxford University Press, 1992

Symmetry is 4mm.

Whole pattern projection symmetry

6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m



[111]

6mm

[111]

6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m



CBED

SAED

So, sometimes just whole pattern projection symmetry is enough if you combine it with the

reflection conditions from SAED.

Try it yourself on example SnO2

SAED CBED

Example: rutile-type SnO2

Projection whole pattern symmetry [001]

4 4mm 2mm


Projection WP: 4mm

Projection diffraction group: Table Eades

4 4RmmR

4mm1R

Projection diffraction group: 4mm1R

Possible diffraction groups:

4mRmR

4mm 4RmmR

4mm1R


6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m


What will be useful to narrow it down further?

look at the bright field symmetry look at the whole pattern symmetry

[...] [...]

Whole Pattern projection symmetry

Whole pattern symmetry

WP symmetry

4 4mm 2mm

[...] [...]

6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m

Projection whole pattern [101]

2 m 2mm

(smaller cond.ap.)

Possible projection diffraction group:

21R

m1R

2mm1R

Projection whole pattern:

2mm

Whole pattern [101]

2 m 2mm

(smaller cond.ap.)

Diffraction group:

2mm

2RmmR

2mm1R

Whole pattern m

6mm1R

3m1R

6mm

6mRmR

61R

31R

6

6RmmR

3m

3mR

6R

3

4mm1R

4RmmR

4mm

4mRmR

41R

4R

4

2mm1R

2RmmR

2mm

2mRmR

m1R

m

mR

21R

2R

2

1R

1

-1 1 2 m 2/m

22

2

mm

2

mm

m

4 -4 4/m

42

2

4m

m

-42

m

4/m

mm

3 -3 32

3m

-3m

6 -6 6/m

62

2

6m

m

-6m

2

6/m

mm

23

m3

43

2

-43

m

m3

m

Possible point groups

4/mmm

m3m

What would make a difference further?

For example: -cell parameters -look for a third zone etc. -SAED for reflection conditions

For example, if you need

cell parameters a=b= 4.72 Å, c=3.16 Å

to be able to index all patterns,

the point group is

4/mmm

m3m

CBED

SAED

Space Group P42/mnm

Then you would combine this again with reflection conditions (not derived in this exercise) to get the space group.

At the end of this lecture you should be able to

(1) index SAED patterns in case the cell parameters are already known

(2) determine the possible space groups from SAED patterns

(3) determine possible point groups from CBED patterns

Combine (2) and (3) to find the space group.

Working page for indexing

tem workshop 2013: electron diffraction

Education