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

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Presentation on theme: "(Based on chapter 2, 3, 4 in Williams and Carter) Scattering and diffraction."— Presentation transcript:

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

2 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

3 Scattering-Diffraction When do we talk about A)Scattering? B)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? – T he mean free path, λ Does the scattering event cause the electrons to lose energy or not? – Distinguishing elestic and inelastic scattering

7 Some definitions Single scattering: 1 scattering event Plural scattering: 1-20 scattering events Multiple scattering: >20 scattering events Forward scattered: scattered through < 90 o Bacscattered: scattered through > 90 o 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 – 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 (<1 o )) – Elastic scattering to higher angles (>~10 o )

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). 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. σ atom =πr 2 r has different value for each scattering process and depends on E 0 Elastic scattering from an isolated atom: Radius of the scatteing field of the nucleus and the electron : r e =e/Vθ r n =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 section θ dθdθ Ω Incident beam Scattered electrons Unscattered electrons dΩdΩ 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: σ total =Nσ atom = N o σ atom ρ/A N= atoms/unit volume N o : 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: tσ total = N o σ atom ρt/A

14 Some numbers For keV – The elastic cross section is almost always the dominant component of the total scattering. – 100keV: σ elastic = ~ m 2 σ inelastic = ~ m 2 – Typical scattering radius: r ~ 0.01 nm See examples of σ in Figure 4.1

15 Mean free path λ λ = 1/σ total = A/N o ρσ atom The mean free path for a scattering process is the average distance travelled by the primary particle between scattering events. Material10kV20kV30kV40kV50kV100kV200kV1000kV C (6) Al (13) Fe (26) Ag (47) Pb (82) U (92) Mean free path (nm) as a function of acceleration voltage for elastic electron scattering more than 2 o.

16 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:

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 180 o

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 180 o 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 A diagram of a scattering process Differential scattering cross section i.e. the probability for scattering in a solid angle dΩ: dσ/dΩ = 2πb (db/dΩ) b= (Ze 2 /4πε o mv 2 )cotanθ/2 dσ/dΩ = -(mZe 2 λ 2 /8πε o h 2 ) 2 (1/sin 4 θ/2) Impact parameter: b Solid angle: Ω= 2π(1- cosθ)

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

22 The scattering process kIkI ψ= ψ 0 exp2πik I r Scattered amplitude: ψ sc = ψ 0 f(θ)(exp2πikr)/r The incomming wave: The scattering process can be described by: ψ= ψ 0 (exp2πik I r + if(θ)(exp2πikr)/r) NB! There is a phase shift of 90 o between the incident and the scattered beams. (see page 46, chapter 3 for more info) θ Constructive interference

23 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 a crystal may result in ZERO scattered intensity. The intensity: IF(θ)I 2 A cel =(exp2πikr)/r Σf i (θ)exp2πiK.r i K=? and r i = ?

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 Collective oscillations Non- localized Localized processes Non- localized SE

25 Valence K L M Electron shell 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 1s 2 2s 2 2p 2 2p 4 3s 2 3p 2 3p 4 3d 4 3d 6 Auger electron or x-ray Electron Ionization of inner shells

26 Auger electrons or x-rays EELS?

27 X-ray spectrum

28 K L M Photo electron x-ray Fluorescence

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

30 Secondary electrons Secondary electrons (SEs) are electrons within the specimen that are ejected by the beam 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 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 Valence band Conduction 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. Plasmon frequency: ω=((ne 2 /ε o m)) 1/2 Energy: E p =(h/2π)ω E p ~ eV, λ p,100kV ~150 nm n: free electron density, e: electron charge, ε o : dielectric constant, m: electron mass

33 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

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


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