Interaction of Radiation with Matter. Particles: probes. Two process of interaction. Absorption and scattering. k d 2q. I k i

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1 Interaction of Radiation with Matter «Element of modern x-ray physics» J. Als-Nielsen et D. McMorrow «Processus d interaction entre photons et atomes» C. Cohen-Tannoudji, Particles: probes Two process of interaction Absorption and scattering dz dw I0 I k i k d 2q l

2 Characteristics of particles Three types of particles Are used in condensed matter physics Tender and hard X-ray photons: kev Low or high energy electrons: 150 ev-100 kev Hot, thermal or cold neutrons: mev Interference effects: Wave length of particle must be smaller than interatomic distances 2d sin θ = mλ λ 2d

3 Characteristics of particles X Photons Neutrons Electrons Description Electromagnetic field E = E 0 e i(k r ωt) Particle y ~ exp(i k.r) Particle y ~ exp(i k.r) Energy E E=hn=hc/l l(å)=12398/e(ev) l=1 Å, E=12.4 kev n= Hz (EHz) E 2 =p 2 c 2 +m n2 c 4 ; E=p 2 /2m n l(å)=0.286/e 0.5 (ev) l=1 Å, E=81.8 mev v n = 4000 m/s E=p 2 /2m e l(å)=12.265/e 0.5 (ev) l=1 Å, E=150 ev v e = 7274 km/s Momentum p p=hk=hn/c p=hk (=mv) p=hk (=mv k B T/E 300K << 1 ~ 1 ~ 10-5 Interaction Charge s th ~ Z 2 barn Moments magnétiques s d ~ 10-6 barn Noyaux (forte) s d ~ 5 barn Moments magnétiques s d ~ 3 barn Potentiel electrostatique s d ~ 10 8 barn Absorption 4700 barn (Z=28, 1,5 Å) Typique : 0,1-1 barn -

4 Absorption cross section After going through matter of width dz, beam intensity decreases by di dz di = I z μdz I = I 0 e μl I 0 I m attenuation coefficient (cm -1 ) Beer-Lambert law F 0 : flux incident particles (s -1 /cm 2 ), F = I/S Number of absorbed particles dn q per time unit l dn q = φ z N(dz)σ a s a : absorption cross section, expressed in barn = cm 2 The cross section depends on the element, its environnement (RX) and on the particle energy Ex: 2D lattice Unit cell 0.3 nm Surface per atom is s~10-15 cm 2 N dz = ρ a Sdz μ = σ a ρ a

5 Scattering process number of scattered particles dn d = φdω dσ dω Scattering differential cross section Wave function of the scattered particle Scattering cross section k i q k d 2q dw ψ d r = b(q) eik dr Differential cross section r b(q): scattering length Neutrons: b independent on q dσ ቇ = k d b 2 dσ ቇ = b 2 dω s k i dω s

6 Characteristics of particles X Photons Neutrons Electrons Description Electromagnetic field E = E 0 e i(k r ωt) Particle y ~ exp(i k.r) Particle y ~ exp(i k.r) Energy E E=hn=hc/l l(å)=12398/e(ev) l=1 Å, E=12.4 kev n= Hz (EHz) E 2 =p 2 c 2 +m n2 c 4 ; E=p 2 /2m n l(å)=0.286/e 0.5 (ev) l=1 Å, E=81.8 mev v n = 4000 m/s E=p 2 /2m e l(å)=12.265/e 0.5 (ev) l=1 Å, E=150 ev v e = 7274 km/s Momentum p p=hk=hn/c p=hk (=mv) p=hk (=mv k B T/E 300K << 1 ~ 1 ~ 10-5 Interaction Charge s th ~ Z 2 barn Magnetic moments s d ~ 10-6 barn Noyaux (forte) s d ~ 5 barn Magnetic moments s d ~ 3 barn Electrostatic potential s d ~ 10 8 barn Absorption 4700 barn (Z=28, 1,5 Å) Typique : 0,1-1 barn -

7 Scattering length for particles Solve the Schrödinger equation of the particle in an interaction potential V r Stationary states of energy: E = ħ2 k 2 2M «Mécanique quantique 2, chap.viii» Cohen-Tannoudji, Diu, Laloë ħ 2 2M + V r φ r = ħ2 k 2 2M φ r with V r = ħ2 2M U(r) + k 2 U(r) φ r = 0 Born aprox. + kr 1 φ r ~e ik i r + b(q) eikr r Scattering length = FT of potential b q = 1 4π U(r)e iq r d 3 r

8 Scattering length X-rays: FT of the electron density b q = r 0 f q = r 0 න ρ r e iq r d 3 r Rayons X Phase shift π r 0 = 2, Å Neutrons: FT of Fermi pseudo-potential. It is a constant because (V r ~ δ r ) b q = b = M 2πħ 2 V r e iq r d 3 r Phase shift 0 ou π Electrons: TF of potential V(r) b q = M 2πħ 2 න V(r)e iq r d 3 r b q depends n l énergie Phase shift δ(q) Electron Fadley, Physica Scripta, T17,39,1987

9 Optical theorem Mécanique quantique II, p. 940 C. Cohen-Tannoudji, B. Diu, Frank Laloë σ tot = σ a +σ d = 4π k Im(b 0 ) ψ d r = b(θ) eik dr r Shadow: ψ i r = Ae ik ir Interference between incident wave and scattered wave

10 Absorption

11 Origin of neutrons absorption Neutrons weaklly absorbed Absorbed through nuclear reactions 3 He+n 3 H - +p s a 6 Li B 2100 Gd Ni 4.6 Pb 0.17 Detectors and shields Energy dependance: σ a k = σ a k 0 k k 0 = 34,947 nm 1

12 Origin of photons absorption (p,e) E Free electron energy E 2 = m 2 c 4 + p 2 c 2 v c E = p 2 /2m Photon energy E ph = pc E? E O =511 kev E O =511 kev p E O -E L Dp.Dr p Free electron: no absorption Bound electron absorption

13 UV VUV XUV Soft X-rays Hard X-rays Absorption X-ray absorption Gamma At energies smaller than 1000 kev Photoelectric effect LEAD Z=82

14 X-ray absorption For E < 1000 kev photoelectric effect is dominant Photoelectric effect Photon is absorbed if hn > E I (E I binding energy of e - ) Excitation: Photoelectron is emitted ( E=hn - E I -F ) F: work function ~1 ev De-excitation: fluorescence photon (hn = E I -E II ) Auger electron ( E= E I -E II -E III ) Photoelectron Fluorescence photon Auger electron Continuum Fermi level M -E F hn (2p 3/2 ) 4 L (2p 1/2 ) 2 (2s) 2 K a K b -E II Core levels K (1s) 2 -E I Excitation Absorption of photons De-excitation Emission of photons and electrons

15 Order of magnitude X-rays: l = Å s a Li 5,7 B 36 Gd Ni 4760 Pb Neutrons: 1.8 Å s a 6 Li B 2100 Gd Ni 4.6 Pb 0.17

Electrons mean free paths Distance between two inelastic collisions with Plasmons Valence electrons From A. Zangwill, Physics at Surfaces, Cambridge Univ. Press.

16 Electrons mean free paths Distance between two inelastic collisions with Plasmons Valence electrons From A. Zangwill, Physics at Surfaces, Cambridge Univ. Press. After this distance (attenuation length), electrons loose their coherence. Low energy electron diffraction (LEED) is a surface technique Only surface photoelectrons and Auger electrons escape from the sample Importance in X-Ray Absorption (XAS)

17 Scattering

18 Scattering: atome-particle system changes of state Initial state, e i Final state, e f Elastic scattering: Does not change the nature or the internal state of the particles and the target

19 Photon scattering Rayleigh scattering: Low energy elastic scattering hn << E I, E I -E II ; F i = F f ; light scattering, blue sky Raman/Brillouin scattering: Low energy inelastic scattering hn << E I ; F i F f ; scattering on optical/acoustical phonons Thomson scattering: High energy elastic scattering hn >> E I ; F i = F f ; X-ray scattering Compton scattering: High energy inelastic scattering hn >> E I ; F i F f ; X-ray scattering

20 Photons scattering (p i,e i ) E E e = p2 m (p f,e f ) E E O E O E O -E L E e = p2 M Free electron (e- mass m) Compton scattering p Bound electron (atom, crystal mass M»m) Thomson scattering Compton scattering p

21 Refraction A consequence of scattering r S D R 0 Δ R R 2 = R r 2 RdR = rdr Wave travelling through a plate of width Δ Phase shift: nkδ-kδ P ψ P = ψ 0 P ei n 1 kδ = ψ 0 P (1 + i n 1 kδ) b eikr ψ P = ψ 0 P +ψ 0 (S)e ikd න 0 R (2πrdrΔ)ρ d = ψ 0 P ψ 0 S 2πbΔρ d e ikd න e ikr dr = ψ 0 P (1 i 2πbΔρ d ) R 0 k n = 1 2πbρ d k 2 Absorption e μrδ ρ d ~1eA 3, b~ Z A, k~4 A 1 න R 0 e ikr dr δ ~ 10 5 R 0

22 a n Refraction Refraction index k i k r n = n r + iβ k t a Phase shift and absorption e inkz = e inrkz e βkz For X-rays and neutrons n = 1 2πbρ d k 2 + iβ = 1 δ + i μ 2k Snell s law n r cos α = cos α a c k i k r Existence of a critical angle above which total reflection α c = 2δ Stationnary wave Measure of the sign of b (holography)

23 Experimental techniques EMISSION : X-ray EMISSION (par rayons X) : Fluorescence Rayons X (Chemical analysis) Fluorescence (Analyse chimique) Photoelectrons, Electrons Auger electrons (analysis) Photo-électrons, électrons Auger (Spectrométrie, analyse) Photoelectron Diffraction de photo-électrons diffraction (structure (local structure) locale) Photo-émission (Structure de bande, surface Fermi) de Electron Spectroscopy Photoemission (band structure) WAVES/PARTICLES X-Rays Neutrons Electrons Crystal Liquid, liquid crystal Polymer Surface REFRACTION : X-ray, neutrons Reflectrometry (surfaces, interfaces) Stationnary waves (surfaces) ABSORPTION : X-ray XAS, EXAFS, XANES (local order) Dichroism (Magnetism, surfaces) SCATTERING X-rays DIFFUSION : Rayons X Diffraction (Etude des structures) Diffusion diffuse (Etude du désordre dans les cristaux, liquides, cristaux liquides) Diffusion Compton (Structure électronique) Diffusion aux petits angles (Polymères, cristaux liquides, agrégats, grandes mailles) Diffusion magnétique, inélastique, cohérente (synchrotrons) Neutrons Diffraction, Diffusion diffuse (Structures, Hydrogène, contraste différent) Inélastique (Excitations élémentaires, phonons, dynamique) Magnétique (Structures magnétique, magnons) Electrons Diffraction, LEED, RHEED (Etude des surfaces) Diffraction (Structures); Diffuse scattering (Disorder, liquids, soft matter) Compton scattering (electronic structure) Small angle scattering (Polymer, liquid crystal, nano-particles, proteins) Magnetic, inelastic, surface, coherent diffraction (synchrotrons) Neutrons Diffraction, Diffuse scattering (Structures, Hydrogen, contrast) Inelastic scattering (phonons, dynamics, excitations) Magnetic (magnetism, magnons) Electrons Low- or high-energy electron diffraction (surfaces, thin samples)

An Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney

$An Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney$ An Introduction to Diffraction and Scattering Brendan J. Kennedy School of Chemistry The University of Sydney 1) Strong forces 2) Weak forces Types of Forces 3) Electromagnetic forces 4) Gravity Types