Convergent beam electron diffraction FYS5310/FYS9320 Lecture 11 - - PowerPoint PPT Presentation

Convergent beam electron diffraction

FYS5310/FYS9320 Lecture 11 30.04.2015

Uses of CBED

• Symmetry determination
• Measurement of thickness
• Determination of structure factors

2

3

The structure factor

B A

Origin

k  k  ' k  ' k 

A

r 

B

r 

r k i

Ae r k

 

 



 

2

) , (



Incoming wave:

r k i r k i A r k r k r k i A r r k i r k i A A

e e f e f e e f r

A A A A A

              



           

   

' 2 2 ) ' ' ( 2 ) ( ' 2 2

) (

    

Wave scattered from atom j in : Equivalent expression for atom B

Total scattered wave detected far away:

r k i r k i B r k i A r k i r k i B r k i r k i A B A

e e f e f e e f e e f r r r

B A B A

             

  

              

        

' 2 2 2 ' 2 2 ' 2 2

} { ) ( ) ( ) ( '

      



     j r k i j k

j

e f F

    2

Can be calculated for all scattering vectors (angles) Δk, but usually we only concern ourselves with Δk=g (destructive interference in all

• ther cases)

The structure factor

In (parallell beam) electron diffraction, we very often concern ourselves only with:

– Position of peaks – Systematic absences

These are the «structure parts» of the structure factor. We can also look at the peak intensities, commonly done in XRD, to determine site occupancies

6



     j r k i j k

j

e f F

    2

What about f?

• f is the atomic form factor, also called the atomic

scattering factor.

• Usually expressed as a function of scattering angle s

f(s)

• 𝑡 =

sin 𝜄 𝜇

7

The atomic form factor for X-rays

• The electron cloud has a shape and

size

• Different sub-volumes of the

electron cloud scatter at different strengths

• The atomic scattering strength is

the sum of all of these sub- volumes. The Fourier-transform of the electron density

… and electrons

The Mott formula

0.00 5.00 10.00 15.00 20.00 25.00 30.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 f(s) [1/e] s

X-ray scattering factors

Co Fe Mn Cr V

• When the atoms participate in a solid, bonds

are formed between them.

• A reduction in total energy is achieved by re-

arranging the electrons

• This causes a change in the form factors
• And thereby the structure factors
• Example Fe  FeO or Fe2O3

0.00 5.00 10.00 15.00 20.00 25.00 30.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 f(s) [1/e] s

X-ray scattering factors

Co Fe Mn Cr V Fe2+ Fe3+

• The effects of bonding are most prominent at small scattering

angles

• In which compounds do you think the effect is largest?
• Both X-ray and electron diffraction can be used to study this
• But which one do you think is most sensitive to charge

redistribution?

Magnesium diboride

• Hexagonal structure
• Superconductivity

discovered in 2001

CBED on MgB2

Experiment Calculated from procrystal Best fit to experiment

Procrystal: no atomic interaction (bonding)

Conclusion

• Convergent beam electron diffraction can be

used to measure stucture factors – and more specifically: changes in scattering amplitude due to bond formation

• It is particularly well suited to low index

structurefactors (reflections)

• These are the reflections that are most

sensitive to bonding