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

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

Scattering-Diffraction When do we talk about Scattering? 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 influenced by 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 Without scattering there would be no mechanism to create TEM images or DP and no sorce for spectroscopic data.

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 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.

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

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 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

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. 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.

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

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 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 http://en.wikipedia.org/wiki/File:ScatteringDiagram.svg

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) 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 2 4 6 Sin(θ)/λ (nm-1) Au Cu Al f(θ) (nm)

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

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.

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

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.

Auger electrons or x-rays EELS?

X-ray spectrum

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

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

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.

Cathodoluminescence Conduction band Valence 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. The oscillations are called plasmons. 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 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

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 Diffraction patterns: Picture of the distribution of scattered electrons Fig. 2.13 Williams and Carter