(Based on chapter 2, 3, 4 in Williams and Carter)

Scattering and diffraction

Learning outcome• Know what is :– 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

Scattering-Diffraction

• When do we talk about A) Scattering?B) Diffraction?

Incident beam

Scattered/diffracted beam

Direct beam

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 influencedby its thickness, density, crystallinity, angle of the incident beam.

• Why are electrons scattered in the specimen?

• How can the scattering process affect the energy and the coherency of the incident electrons?

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

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

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

Electron scattering • 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

• Coherent– Elastically scattering electrons are usually coherent

• Incoherent– Inelastic electrons are usually incoherent (low angles (<1o))– Elastic scattering to higher angles (>~10o)

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).

When divided by the actual area of the atom the it represents the probability that a scattering event will occure.

σatom=πr2

r has different value for each scattering process and depends on E0

Elastic scattering from an isolated atom:Radius of the scatteing field of the nucleus and the electron : re=e/Vθ rn=Ze/Vθ

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.

d σ/dΩ :Differential cross section

θ

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

Scattering form the specimenTotal 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 volumeNo: 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

Some numbers

• For 100-400 keV– The elastic cross section is almost always the

dominant component of the total scattering.– 100keV: • σelastic = ~10-22 m2

• σinelastic = ~10-22 - 10-26 m2

– Typical scattering radius: r ~ 0.01 nm

See examples of σ in Figure 4.1

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.

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) 0.15 0.6 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

Mean free path (nm) as a function of acceleration voltage for elastic electron scattering more than 2o.

Electron scattering

• 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

The probability of scattering is described in terms of either an “interaction cross-section” or a mean free path.

Mote Carlo simulations: http://www.matter.org.uk/TEM/electron_scattering.htm#

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

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

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

A diagram of a scattering process http://en.wikipedia.org/wiki/File:ScatteringDiagram.svg

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)

Impact parameter: b

Solid angle:Ω= 2π(1- cosθ)

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

Atomic scattering factor f(θ)

Incident beams

Scattered/diffracted beams

ANGLE VARIATIONBoth the differential cross section and the scattering factor are simply measures of how the electron-scattering intensity varies with θ.

1.2

1.0

0.8

0.6

0.4

0.2

2 4 6 Sin(θ)/λ (nm-1)

Au

Cu

Al

f(θ) (nm)

The scattering process

kI

ψ= ψ0exp2πikIr

Scattered amplitude:ψsc= ψ0f(θ)(exp2πikr)/r

The incomming wave:

The scattering process can be described by:

ψ= ψ0(exp2πikIr + if(θ)(exp2πikr)/r)

NB! There is a phase shift of 90o betweenthe incident and the scattered beams.(see page 46, chapter 3 for more info)

θConstructive interference

The structure factor F(θ)

F(θ) is a measure of the amplitude scattered by a unit cell of a crystal structure

Under specific conditions, electrons scattering in acrystal may result in ZERO scattered intensity.

The intensity: IF(θ)I2

N

jjhklg fFF

1

2exp( ))( jjj lwkvhui

Acel=(exp2πikr)/r Σfi(θ)exp2πiK.ri

K=? and ri= ?

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 exitationsCollective oscillationsNon- localized

Localized processes

Non- localized SE

Valence

K

L

M

Electronshell

Characteristic x-ray emitted or Auger electron ejected after relaxation of inner state. Low energy photons (cathodoluminescence)when relaxation of outer stat.

K

L

M

1s2

2s22p2

2p43s2

3p2

3p4

3d4

3d6

Auger electron or x-ray

Electron

Ionization of inner shells

Auger electrons or x-rays

EELS?

X-ray spectrum

K

L

M

Photo electron

x-ray x-ray

Fluorescence

Continuous and characteristic x-rays

http://www.emeraldinsight.com/journals.htm?articleid=1454931&show=html

Continous x-rays du todeceleration of incident electrons.

The cut-off energy forcontinous x-rays corresponds to the energy of the incident electrons.

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 electronsThe secondary emission coefficient:

δ=number of secondary electrons/numbers of primary electrons

Dependent on acceleration voltage.

Cathodoluminescence

Valence band

Conduction band

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.

Plasmon frequency: ω=((ne2/εom))1/2 Energy: Ep=(h/2π)ω Ep~ 10-30 eV, λp,100kV ~150 nm

n: free electron density, e: electron charge, εo: dielectric constant, m: electron mass

Phonon excitation

Equivalent to specimen heating

The effect in the diffraction patterns:-Reduction of intensities (Debye-Waller factor)-Diffuce bacground between the Bragg reflections

Energy losses ~ 0.1 eV

EELS

Sum of several losses

Thin specimens

Fraunhofer and Fresnel diffraction

• Far-field diffraction• Near-field diffraction

Diffraction from slits and holes

• Young`s slitt experiment• Phasor diagram

• Airy disk

Angles and diffraction patterns

• Figure 2.12– Beam convergence angle, α– Collection angle, β– Scattering semiangle, θ

Fig. 2.13 Williams and Carter

Diffraction patterns: Picture of the distribution of scattered electrons