Presentation on theme: "Scattering and diffraction"— Presentation transcript:
1 Scattering and diffraction (Based on chapter 2, 3, 4 inWilliams 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 diffractionPossible scattering processesTypical scattering angles, effect of Z and U etc
3 Scattering-Diffraction When do we talk aboutScattering?Diffraction?Incident beamScattered/diffracted beamDirect beam
4 Scattering and diffraction Particles are scattered/deflectedWaves are diffractedA single scattering event is dependent on U and ZScattering from a specimen is influencedby its thickness, density, crystallinity, angle ofthe 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 scatteringWhat is the probability that an electron will be scattered when it passes near an atom?The idea of a cross section, σIf the electron is scattered, what is the angle through which it is deviated?Used to control which electrons form the imageWhat is the average distance an electron travels between scattering events?The mean free path, λDoes the scattering event cause the electrons to lose energy or not?Distinguishing elestic and inelastic scatteringWithout 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 eventsMultiple scattering: >20 scattering eventsForward scattered: scattered through < 90oBacscattered: scattered through > 90oAs the specimen gets thicker more electrons are back scattered
8 X-rays versus electrons X-rays are scattered by the electrons in a materialElectrons are scattered by both the electron and the nuclei in a materialThe electrons are directly scattered and not by an field to field exchange as in the case for X-raysThe scattering process is not important for diffraction
9 Electron scattering Elastic Inelastic Coherent Incoherent The kinetic energy is unchangedChange in direction relative to incident electron beamInelasticThe kinetic energy is changed (loss of energy)Energy form the incident electron is transferred to the electrons and atoms in the specimenCoherentElastically scattering electrons are usually coherentIncoherentInelastic 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 withan atom is determined by an interaction cross section (an area).σatom=πr2r has different value for each scattering process and depends on E0When divided by the actual area of the atom the it representsthe 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/dΩThe differential cross section dσ/dΩ describes theangular distribution of scattering from an atom,and is a measure of the probability for scattering in asolid angle dΩ.
12 d σ/dΩ :Differential cross section θdθΩIncident beamScattered electronsUnscatteredelectronsdΩΩ= 2π (1-cosθ)dΩ= 2π sinθ dθdσ/dΩ = (1/2π sinθ) dσ/dθCalculate σ by integration.σ decreases as θ increases
13 Scattering form the specimen Total scattering cross section/The number of scattering eventsper unit distance that the electrons travels through the specimen:σtotal=Nσatom= Noσatom ρ/AN= atoms/unit volumeNo: Avogadros number, ρ: density of pecimen,A: atomic weight of the scattering atomsIf the specimen has a thickness t the probability of scatteringthrough the specimen is:tσtotal=Noσatom ρt/A
14 Some numbersFor keVThe elastic cross section is almost always the dominant component of the total scattering.100keV:σelastic = ~10-22 m2σinelastic = ~ m2Typical scattering radius: r ~ 0.01 nmSee 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/NoρσatomMaterial10kV20kV30kV40kV50kV100kV200kV1000kVC (6)5.5224989140550220055000Al (13)1.87.417294618074018000Fe (26)0.150.62.95.28.2301303000Ag (47)22.214.171.1245601500Pb (82)0.080.340.761.42.1834800U (92)0.050.190.420.751.2519500Higher 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 electronscattering more than 2o.
16 Electron scattering Elastic Inelastic The kinetic energy is unchanged The probability of scattering is described in terms of eitheran “interaction cross-section” or a mean free path.Mote Carlo simulations:ElasticThe kinetic energy is unchangedChange in direction relative to incident electron beamInelasticThe kinetic energy is changed (loss of energy)Energy form the incident electron is transferred to the electrons and atoms in the specimenEach 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 atomFrom the electron cloud: few degrees of angular deviationFrom the positive nucleus: up to 180o
18 ScatteringEleastic scattering is the major source of contrast in TEM imagesScattering from an isolated atomFrom the electron cloud: few degrees of angular deviationFrom the positive nucleus: up to 180oFig. 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 scatteringDifferential scattering cross sectioni.e. the probability for scattering in a solid angle dΩ:dσ/dΩ = 2πb (db/dΩ)b= (Ze2/4πεomv2)cotanθ/2dσ/dΩ = -(mZe2λ2/8πεoh2)2(1/sin4θ/2) Solid angle:Ω= 2π(1- cosθ)Impact parameter: bA diagram of a scattering process
20 Atomic scattering factor f(θ) | f(θ)|2=dσ/dΩf(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 VARIATIONBoth the differential cross section and the scattering factor are simply measures of how the electron-scattering intensity varies with θ.Incident beamsScattered/diffracted beams1.21.00.80.60.40.2Sin(θ)/λ (nm-1)AuCuAlf(θ) (nm)
22 The scattering process The incomming wave:ψ= ψ0exp2πikIrThe scattering process can be described by:ψ= ψ0(exp2πikIr + if(θ)(exp2πikr)/r)kIScattered amplitude:ψsc= ψ0f(θ)(exp2πikr)/rNB! There is a phase shift of 90o betweenthe incident and the scattered beams.(see page 46, chapter 3 for more info)θConstructive interference
23 The structure factor F(θ) Acel=(exp2πikr)/r Σfi(θ)exp2πiK.riK=? and ri= ?F(θ) is a measure of the amplitude scattered by aunit cell of a crystal structureThe 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 thespecific atomic planes (hkl) that make up the crystalstructure.The intensity: IF(θ)I2Under specific conditions, electrons scattering in acrystal may result in ZERO scattered intensity.
24 Inelastic scattering processes Ionization of inner shellsAuger electronsX-raysLightContinuous X-rays/BremsstrahlungExitation of conducton or valence electronsPlasmon exitationPhonon exitationsLocalized processesNon- localizedSECollective oscillationsNon- localized
25 Ionization of inner shells ElectronAuger electron orx-rayValenceKLMElectronshellKLM1s22s22p22p43s23p23p43d43d6Characteristic x-ray emitted or Augerelectron ejected after relaxation of innerstate.Low energy photons (cathodoluminescence)when relaxation of outer stat.
29 Continuous and characteristic x-rays The cut-off energy forcontinous x-rays correspondsto the energy of the incidentelectrons.Cut-off energy for continous x-rays correspond to the energy of the incident electrons.Continous x-rays du todeceleration of incidentelectrons.
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 eVAuger electronsThe secondary emission coefficient:δ=number of secondary electrons/numbers of primary electronsDependent on acceleration voltage.
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))1/2Energy: Ep=(h/2π)ω Ep~ eV, λp,100kV ~150 nmn: free electron density, e: electron charge, εo: dielectric constant, m: electron mass
33 Phonon excitation Equivalent to specimen heating Energy losses ~ 0.1 eVThese 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
35 Fraunhofer and Fresnel diffraction Far-field diffractionNear-field diffraction
36 Diffraction from slits and holes Young`s slitt experimentPhasor diagramAiry disk
37 Angles and diffraction patterns Figure 2.12Beam convergence angle, αCollection angle, βScattering semiangle, θDiffraction patterns:Picture of the distributionof scattered electronsFig Williams and Carter