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