ElasticElastic scatteringscattering

LectureLecture 22 PartPart 11 ParticleParticle vs.vs. wavewave

Particle: – Scattering cross section & differential scattering cross section – Scatter to particular angles – Columbic interaction with the nucleus – Used as primary description when considering spectroscopies Wave: – Consider the wave to be diffracted by atoms or scattering centers – Strength of scatter depends on ‘atomic form factor’ – Used as primary description when considering diffraction & imaging BriefBrief reviewreview ofof particleparticle modelmodel …… ElasticElastic scatteringscattering singlesingle electronelectron // isolatedisolated atomatom InteractionInteraction betweenbetween electronelectron && atomatom isis CoulombicCoulombic – Incident electron & electron cloud – Incident electron & nucleus In a prior lecture, we discussed – Interaction cross section – Mean free path – Differential cross section ElasticElastic scatteringscattering singlesingle electronelectron // isolatedisolated atomatom

Interaction cross section expresses the probability of a given scattering event. Generally: σ = πr2 Elastic scattering radius has the form: e Ze r = ; r = electron Vθ nucleus Vθ Z = atomic weight e = charge of the electron V = potential of the electron Implications: θ = angle V ⇑ σ ⇓ Z ⇑ σ ⇑ θ ⇑ σ ⇓ ElasticElastic scatteringscattering throughthrough specimenspecimen thicknessthickness

Consider instead specimen of N atoms / unit thickness. Total cross section for scattering from specimen:

Noσ Tρ QT = Nσ T = A N = # atoms / unit volume r = density A = atomic weight Scattering probability for a sample of thickness t: N σ ρt Q t = o T ρt Ä ‘mass-thickness’ T A Can re-arrange to give another useful concept: 1 A Mean free path: mfp or λ mfp = = Q Noσ Tρ OnOn toto thethe wavewave perspectiveperspective …… WhatWhat isis aa wave?wave?

λ

v

A wave: periodic disturbance in both space and time Characteristics: – Wavelength (λ), velocity (v), frequency (f): v = f λ 1 – Wave number (k): k = * λ

– Angular frequency (ω): ω = f k * Sometimes written as: k = 2π λ WaysWays thatthat wewe’’llll depictdepict waveswaves

This is our plane wave …

Here we show it propagating, with Wave number k can be the lines being the location of e.g. expressed as a vector k, the peaks of the wave … which then captures both the direction of the wave k and it’s wavelength This is thus an equivalent representation of the wave … MathematicsMathematics ofof waveswaves

A wave: periodic disturbance in both space and time, u(x,y,z,t): 2 2 2 2 2 ⎡ d u d u d u ⎤ d u Wave Eqn. v + + = ⎢ 2 2 2 ⎥ 2 ⎣dx dy dz ⎦ dt (2nd order ∂2u differential v2∇2u = ∂t2 equation) In one dimension, u(z,t), “plane wave”: ∂2u 1 ∂2u = ∂z2 v2 ∂t2 Solutions: u(z,t) = sin( kz +ωt) via Euler’s Equations Preferred u(z,t) = exp ⎡−i ωt −kz ⎤ form ⎣ ( )⎦ MathematicsMathematics ofof vibrationsvibrations

AA vibration:vibration: timetime--dependentdependent oror spacespace--dependentdependent periodicperiodic disturbancedisturbance

SpaceSpace dependentdependent periodicity,periodicity, u(xu(x):): d2u a + bu = 0 dx2

AA solution:solution: u = Aexp⎡iαx⎤ + Bexp⎡ -iαx⎤ ⎣ ⎦ ⎣ ⎦ b α= a ElectronElectron inin freefree spacespace

ElectronElectron moving:moving: – in absence of a potential field (V=0) – independent of t h2 ∂2ψ + Eψ=0 2m ∂z2 Solution:Solution: ψ=Aexp ikx + Bexp -ikx ( ) ( ) 2mE with k = h2 ElectronElectron inin freefree spacespace

TimeTime independentindependent propagationpropagation onlyonly oneone arbitraryarbitrary directiondirection (+r):(+r): 2mE exp ikr with k = Plane wave: ψ=ψo ( ) 2 h

EasyEasy toto addadd timetime dependencedependence backback in:in: ⎡ E ⎤ Ψ=Ψ exp ikr exp⎢−i t⎥ o () h ⎣ ⎦ HaveHave aa wave,wave, withwith energyenergy givengiven by:by: h2k2 k Ä fixed E = 2m p Ä infinite (unknown) ElectronElectron inin freefree spacespace

SphericalSpherical wave:wave: – Isotropic wave propagating outwards from a point WaveWave EqnEqn:: ∂2 ()rΨ ∂2 (rΨ) = v2 ∂t2 ∂r2 Solution:Solution: Time independent exp(ikr −ωt) exp(ikr) Ψ=Ψ ψ=ψ o r o r CoherenceCoherence && IncoherenceIncoherence

DiffractedDiffracted wavewave isis eithereither coherentcoherent oror incoherentincoherent DependsDepends onon thethe phasephase relationship:relationship:

Ψ(z,t) =Ψ exp ⎡−i ωt − kz ⎤ o ⎣ ( )⎦ velocity phase ∂z ω v = = phase ∂t k Coherence:Coherence: phasephase relationshiprelationship preservedpreserved afterafter scatteringscattering Incoherence:Incoherence: phasephase relationshiprelationship destroyeddestroyed afterafter scatteringscattering CoherenceCoherence && IncoherenceIncoherence

These waves have the same phase (are coherent) at these angles CoherenceCoherence && IncoherenceIncoherence ElasticElastic && InelasticInelastic NoteNote cancan have:have: – Coherent elastic scattering – Incoherent inelastic scattering – Coherent inelastic scattering – Incoherent inelastic scattering ForFor now,now, wantwant toto focusfocus onon coherentcoherent elasticelastic scattering,scattering, asas itit isis responsibleresponsible forfor diffractiondiffraction CoherenceCoherence && IncoherenceIncoherence

These waves have the same phase (are coherent) at these angles CoherentCoherent elasticelastic scatteringscattering

Detector r’s - position vectors -ki k’s - wave vectors K r kd r - r’

kd θ r’

ki r r r r Incident plane wave:Ψ = exp⎡ 2πi(k ⋅r′ −ωt)⎤ = exp⎡ 2πi(k ⋅r)′ ⎤ incident ⎣ i ⎦ ⎣ i ⎦ Scattered wave: r r r r r r exp⎡ 2πik ⋅ r − r′ ⎤ exp⎡ 2πik ⋅ r − r′ ⎤ ⎣ d ( )⎦ ⎣ d ( )⎦ Ψscattered = f ()ki,kd r r = f ()θ r r r − r′ r − r′

AtomicAtomic formform factorfactor ff ((θθ)) ““scatteringscattering factorfactor forfor electronselectrons””

What is f (θ)? – Measures the strength of the scattering event Elastic scatter has a strong angular dependence Remember: “scattering cross section” used to describe strength of scatter earlier See now that we are also concerned with the angular dependence Called “differential cross section” AtomicAtomic formform factorfactor ff ((θθ)) ““scatteringscattering factorfactor forfor electronselectrons”” f (θ) is measure of amplitude of electron wave scattered from the atom – Is thus the tendency of the scattered wave to interfere in a constructive manner I = Ψ × Ψ* with the incident wave (more on this later) f 2(θ) is proportional to the scattered intensity†

2 dσθ( ) f ()θ = dΩ † * “Differential cross section” I=Ψ × Ψ DifferentialDifferential crosscross sectionssections

The area offered by the scatter (dσ) for scattering the incident electron into a particular increment of solid angle, dΩ Differential cross sections can be found by solving Schrödinger Eqn inside the atom (!) There are three primary models used to do this to find differential cross sections: – “Screened Columbic” – Thomas-Fermi / Rutherford – Mott Lots of heavier physics in this, which I’ll ignore – See both Reimer and Fultz & Howe texts – Ask me if you are interested, and we can discuss one on one DifferentialDifferential crosscross sectionsection

Has general form shown to the right For lower Z & lower V use the “Screened Relativistic Rutherford Cross Section” dσθ() λ 4 Z2 = R 2 dΩ ⎛ 2 ⎞ 2 θ ⎛ θ⎞ 64π4 a ⎜ sin2 + ⎟ ()o ⎜ 2 ⎝⎜ 2⎠⎟ ⎟ ⎝ ⎠

For heavier elements & Screened Relativistic Rutherford For heavier elements & Cross Section vs. Angle larger voltages the “Mott Cross Section” is more accurate AtomicAtomic formform factorfactor ff ((θθ)) ““scatteringscattering factorfactor forfor electronselectrons””

Now, with dσ/dΩ in hand can find f(θ): 2 dσθ( ) f θ = () dΩ

2 ⎛ ⎞ 1+ EE ⎜ λ ⎟ f θ = o Z − f () 2 ⎜ ⎟ ( x ) 8π a θ ⎜ sin ⎟ ⎝ 2 ⎠ Implications: θ ⇑ f ⇓

λ⇑ f ⇓ Z ⇑ f ⇑ CoherentCoherent elasticelastic scatteringscattering

Detector r’s - position vectors -ki k’s - wave vectors K r kd r - r’

kd θ r’

ki r r r r Incident plane wave:Ψ = exp⎡ 2πi(k ⋅r′ −ωt)⎤ = exp⎡ 2πi(k ⋅r)′ ⎤ incident ⎣ i ⎦ ⎣ i ⎦ Scattered wave: r r r r exp⎡ 2πik r − r′ ⎤ exp⎡ 2πik r − r′ ⎤ ⎣ d ⎦ ⎣ d ⎦ Ψscattered = fk()i,kd r r = f ()θ r r r − r′ r − r′