1. Scattering and diffraction
(Based on chapter 2, 3, 4 in
Williams and Carter)

2. Learning outcome Know what is : Possible scattering processes
Elastic scattering, coherent scattering, incident beam, direct beam, cross section, differential cross section, mean free path, Airy disc, major semiangles, Fraunhofer and Fresnel diffraction
Possible scattering processes
Typical scattering angles, effect of Z and U etc

3. Scattering-Diffraction
When do we talk about
Scattering?
Diffraction?
Incident beam
Scattered/diffracted beam
Direct beam

4. Scattering and diffraction
Particles are scattered/deflected
Waves are diffracted
A single scattering event is dependent on U and Z
Scattering from a specimen is influenced
by its thickness, density, crystallinity, angle of
the incident beam.

5. Why are electrons scattered in the specimen?
How can the scattering process affect the energy and the coherency of the incident electrons?

6. Electron scattering
What is the probability that an electron will be scattered when it passes near an atom?
The idea of a cross section, σ
If the electron is scattered, what is the angle through which it is deviated?
Used to control which electrons form the image
What is the average distance an electron travels between scattering events?
The mean free path, λ
Does the scattering event cause the electrons to lose energy or not?
Distinguishing elestic and inelastic scattering
Without scattering there would be no mechanism to create TEM images or DP and no sorce for spectroscopic data.

7. Some definitions Single scattering: 1 scattering event
Plural scattering: 1-20 scattering events
Multiple scattering: >20 scattering events
Forward scattered: scattered through < 90o
Bacscattered: scattered through > 90o
As the specimen gets thicker more electrons are back scattered

8. X-rays versus electrons
X-rays are scattered by the electrons in a material
Electrons are scattered by both the electron and the nuclei in a material
The electrons are directly scattered and not by an field to field exchange as in the case for X-rays
The scattering process is not important for diffraction

9. Electron scattering Elastic Inelastic Coherent Incoherent
The kinetic energy is unchanged
Change in direction relative to incident electron beam
Inelastic
The kinetic energy is changed (loss of energy)
Energy form the incident electron is transferred to the electrons and atoms in the specimen
Coherent
Elastically scattering electrons are usually coherent
Incoherent
Inelastic electrons are usually incoherent (low angles (<1o))
Elastic scattering to higher angles (>~10o)
Each scattering event might be elastic or inelastic. The scattered electron is most likely to be forward scattered but there is a small chance that it will be backscattered.
When the solid specimen is thicker than about twice the mean free path, plural scattering is likely. This can be modelled using the Monte Carlo technique. The important features are the fraction of electron scattering forward and backwards and the volume of the specimen in which most of the interactions (scattering events) take place.

10. Interaction cross section
The chance of a particular electron undergoing any kind of interaction with
an atom is determined by an interaction cross section (an area).
σatom=πr2
r has different value for each scattering process and depends on E0
When divided by the actual area of the atom the it represents
the probability that a scattering event will occure.
Elastic scattering from an isolated atom:
Radius of the scatteing field of the nucleus and the electron :
re=e/Vθ rn=Ze/Vθ

11. Differential cross section
d σ/dΩ
The differential cross section dσ/dΩ describes the
angular distribution of scattering from an atom,
and is a measure of the probability for scattering in a
solid angle dΩ.

12. d σ/dΩ :Differential cross sectionθ dθ Ω
Incident beam
Scattered electrons
Unscattered
electrons dΩ
Ω= 2π (1-cosθ)
dΩ= 2π sinθ dθ
dσ/dΩ = (1/2π sinθ) dσ/dθ
Calculate σ by integration.
σ decreases as θ increases

13. Scattering form the specimen
Total scattering cross section/The number of scattering events
per unit distance that the electrons travels through the specimen:
σtotal=Nσatom= Noσatom ρ/A
N= atoms/unit volume
No: Avogadros number, ρ: density of pecimen,
A: atomic weight of the scattering atoms
If the specimen has a thickness t the probability of scattering
through the specimen is:
tσtotal=Noσatom ρt/A

14. Some numbers
For keV
The elastic cross section is almost always the dominant component of the total scattering.
100keV:
σelastic = ~10-22 m2
σinelastic = ~ m2
Typical scattering radius: r ~ 0.01 nm
See examples of σ in Figure 4.1

15. Mean free path λ λ = 1/σtotal = A/Noρσatom
The mean free path for a scattering process is the average distance travelled by the primary particle between scattering events.
λ = 1/σtotal = A/Noρσatom
Material
10kV
20kV
30kV
40kV
50kV
100kV
200kV
1000kV
C (6)
5.5 22 49 89
140
550
2200
55000
Al (13)
1.8
7.4 17 29 46
180
740
18000
Fe (26)
0.15
0.6
2.9
5.2
8.2 30
130
3000
Ag (47)
1.3
2.3
3.6 15 60
1500
Pb (82)
0.08
0.34
0.76
1.4
2.1 8 34
800
U (92)
0.05
0.19
0.42
0.75
1.2 5 19
500
Higher density areas will scatter more, The target becomes smaller when the bullet becomes faster. How thick can the sample be?
For all forms of scattering the the total cross section decreases as Eo incrases.
Mean free path (nm) as a function of acceleration voltage for elastic electron
scattering more than 2o.

16. Electron scattering Elastic Inelastic The kinetic energy is unchanged
The probability of scattering is described in terms of either
an “interaction cross-section” or a mean free path.
Mote Carlo simulations:
Elastic
The kinetic energy is unchanged
Change in direction relative to incident electron beam
Inelastic
The kinetic energy is changed (loss of energy)
Energy form the incident electron is transferred to the electrons and atoms in the specimen
Each scattering event might be elastic or inelastic. The scattered electron is most likely to be forward scattered but there is a small chance that it will be backscattered.
When the solid specimen is thicker than about twice the mean free path, plural scattering is likely. This can be modelled using the Monte Carlo technique. The important features are the fraction of electron scattering forward and backwards and the volume of the specimen in which most of the interactions (scattering events) take place.

17. Elastic scattering Major source of contrast in TEM images
Scattering from an isolated atom
From the electron cloud: few degrees of angular deviation
From the positive nucleus: up to 180o

18. Scattering
Eleastic scattering is the major source of contrast in TEM images
Scattering from an isolated atom
From the electron cloud: few degrees of angular deviation
From the positive nucleus: up to 180o
Fig. 3.1 Williams and Carter

19. Elastic scattering process
Rutherford scattering (Coulomb scattering)
Coulomb interaction between incident electron and the electric charge of the electron clouds and the nuclei.
Elastic scattering
Differential scattering cross section
i.e. the probability for scattering in a solid angle dΩ:
dσ/dΩ = 2πb (db/dΩ)
b= (Ze2/4πεomv2)cotanθ/2
dσ/dΩ = -(mZe2λ2/8πεoh2)2(1/sin4θ/2) 
Solid angle:
Ω= 2π(1- cosθ)
Impact parameter: b
A diagram of a scattering process

20. Atomic scattering factor f(θ)
| f(θ)|2=dσ/dΩ
f(θ) is a measure of amplitude of an electron wave scattered from an isolated atom
| f(θ)|2 is proportional to the scattered intesity

21. Atomic scattering factor f(θ)
ANGLE VARIATION
Both the differential cross section and the scattering factor are simply measures of how the electron-scattering intensity varies with θ.
Incident beams
Scattered/diffracted beams
1.2
1.0
0.8
0.6
0.4
0.2
Sin(θ)/λ (nm-1) Au Cu Al
f(θ) (nm)

22. The scattering process
The incomming wave:
ψ= ψ0exp2πikIr
The scattering process can be described by:
ψ= ψ0(exp2πikIr + if(θ)(exp2πikr)/r) kI
Scattered amplitude:
ψsc= ψ0f(θ)(exp2πikr)/r
NB! There is a phase shift of 90o between
the incident and the scattered beams.
(see page 46, chapter 3 for more info) θ
Constructive interference

23. The structure factor F(θ)
Acel=(exp2πikr)/r Σfi(θ)exp2πiK.ri
K=? and ri= ?
F(θ) is a measure of the amplitude scattered by a
unit cell of a crystal structure
The amplitude (and hence its square, the intensity)
of scattering is influenced by the type of atom (f(y)),
the position of the atom in the cell (x,y,z), and the
specific atomic planes (hkl) that make up the crystal
structure.
The intensity: IF(θ)I2
Under specific conditions, electrons scattering in a
crystal may result in ZERO scattered intensity.

24. Inelastic scattering processes
Ionization of inner shells
Auger electrons
X-rays
Light
Continuous X-rays/Bremsstrahlung
Exitation of conducton or valence electrons
Plasmon exitation
Phonon exitations
Localized processes
Non- localized SE
Collective oscillations
Non- localized

25. Ionization of inner shells
Electron
Auger electron or
x-ray
Valence K L M
Electron
shell K L M
1s2
2s2
2p2
2p4
3s2
3p2
3p4
3d4
3d6
Characteristic x-ray emitted or Auger
electron ejected after relaxation of inner
state.
Low energy photons (cathodoluminescence)
when relaxation of outer stat.

26. Auger electrons or x-rays
EELS?

27. X-ray spectrum

28. Fluorescence
x-ray
x-ray M L K
Photo electron

29. Continuous and characteristic x-rays
The cut-off energy for
continous x-rays corresponds
to the energy of the incident
electrons.
Cut-off energy for continous x-rays correspond to the energy of the incident electrons.
Continous x-rays du to
deceleration of incident
electrons.

30. Secondary electrons Secondary electrons (SEs) are electrons within
the specimen that are ejected by the beam electrons.
Electrons from the conduction or valence band.
E ~ 0 – 50 eV
Auger electrons
The secondary emission coefficient:
δ=number of secondary electrons/numbers of primary electrons
Dependent on acceleration voltage.

31. Cathodoluminescence
Conduction band
Valence band

32. Plasmon excitations The oscillations are called plasmons.
The incoming electrons can interact with electrons in the ”electron gas”
and cause the electron gas to oscillate.
The oscillations are called plasmons.
Plasmon frequency: ω=((ne2/εom))1/2
Energy: Ep=(h/2π)ω Ep~ eV, λp,100kV ~150 nm
n: free electron density, e: electron charge, εo: dielectric constant, m: electron mass

33. Phonon excitation Equivalent to specimen heating
Energy losses ~ 0.1 eV
These losses has little practical importance in TEM at the moment. Exp(-M),
The effect in the diffraction patterns:
-Reduction of intensities (Debye-Waller factor)
-Diffuce bacground between the Bragg reflections

34. EELS
Sum of several losses
Thin specimens

35. Fraunhofer and Fresnel diffraction
Far-field diffraction
Near-field diffraction

36. Diffraction from slits and holes
Young`s slitt experiment
Phasor diagram
Airy disk

37. Angles and diffraction patterns
Figure 2.12
Beam convergence angle, α
Collection angle, β
Scattering semiangle, θ
Diffraction patterns:
Picture of the distribution
of scattered electrons
Fig Williams and Carter