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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 kd I ki l 2q Characteristics of particles Three types of particles Are used in condensed matter physics • • Tender and hard X-ray photons: 3-100 keV • Low or high energy electrons: 150 eV-100 keV • Hot, thermal or cold neutrons: 120-25-10 meV • Interference effects: Wave length of particle must be smaller than interatomic distances 2𝑑 sin 𝜃 = 𝑚𝜆 𝜆 ≤ 2𝑑 Characteristics of particles Description X Photons Neutrons Electrons Electromagnetic field Particle Particle 𝑬 = 𝑬0 𝑒 𝑖(𝒌∙𝒓−𝜔𝑡) Energy Momentum kBT/E E=hn=hc/l l(Å)=12398/E(eV) l=1 Å, E=12.4 keV n=3.1018 Hz (EHz) E p p=hk=hn/c 300K 3.10-6 << 1 y ~ exp(i k.r) y ~ exp(i k.r) E2=p2c2+mn2c4; E=p2/2mn E=p2/2me l(Å)=0.286/E0.5(eV) l(Å)=12.265/E0.5(eV) l=1 Å, E=81.8 meV l=1 Å, E=150 eV vn = 4000 m/s ve = 7274 km/s p=hk (=mv) ~ 1 p=hk (=mv ~ 10-5 Interaction Charge sth ~ Z2 barn Moments magnétiques sd ~ 10-6 barn Noyaux (forte) sd ~ 5 barn Moments magnétiques sd ~ 3 barn Potentiel electrostatique sd ~ 108 barn Absorption 4700 barn (Z=28, 1,5 Å) Typique : 0,1-1 barn - Absorption cross section • After going through matter of width dz, beam intensity decreases by dI dz 𝑑𝐼 = −𝐼 𝑧 𝜇𝑑𝑧 ⟹ 𝐼 = 𝐼0 𝑒 −𝜇𝑙 I0 • m attenuation coefficient (cm-1) • Beer-Lambert law l • F0: flux incident particles (s-1/cm2), F = I/S Number of absorbed particles 𝑑𝑁𝑞 per time unit 𝑑𝑁𝑞 = 𝜙 𝑧 𝑁(𝑑𝑧)𝜎𝑎 • sa: absorption cross section, expressed in barn = 10-24 cm2 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 cm2 I 𝑁 𝑑𝑧 = 𝜌𝑎 𝑆𝑑𝑧 𝜇 = 𝜎𝑎 𝜌𝑎 Scattering cross section • Scattering process dW • number of scattered particles 𝒒 𝑑𝜎 𝑑𝑁𝑑 = 𝜙𝑑Ω 𝑑Ω ki Scattering differential cross section • Wave function of the scattered particle 𝑒 𝑖𝑘𝑑 𝑟 𝜓𝑑 𝒓 = −𝑏(𝒒) 𝑟 𝑏(𝒒): scattering length Neutrons: b independent on • Differential cross section 𝑑𝜎 𝑑Ω 𝑠 𝑘𝑑 = 𝑏 𝑘𝑖 2 𝑑𝜎 𝑑Ω = 𝑏 𝑠 2 q kd 2q Characteristics of particles Description X Photons Neutrons Electrons Electromagnetic field Particle Particle 𝑬 = 𝑬0 𝑒 𝑖(𝒌∙𝒓−𝜔𝑡) Energy Momentum kBT/E E=hn=hc/l l(Å)=12398/E(eV) l=1 Å, E=12.4 keV n=3.1018 Hz (EHz) E p p=hk=hn/c 300K 3.10-6 << 1 y ~ exp(i k.r) y ~ exp(i k.r) E2=p2c2+mn2c4; E=p2/2mn E=p2/2me l(Å)=0.286/E0.5(eV) l(Å)=12.265/E0.5(eV) l=1 Å, E=81.8 meV l=1 Å, E=150 eV vn = 4000 m/s ve = 7274 km/s p=hk (=mv) ~ 1 p=hk (=mv ~ 10-5 Interaction Charge sth ~ Z2 barn Magnetic moments sd ~ 10-6 barn Noyaux (forte) sd ~ 5 barn Magnetic moments sd ~ 3 barn Electrostatic potential sd ~ 108 barn Absorption 4700 barn (Z=28, 1,5 Å) Typique : 0,1-1 barn - Scattering length for particles Solve the Schrödinger equation of the particle in an interaction potential 𝑉 𝒓 ℏ2 𝑘 2 Stationary states of energy: 𝐸 = 2𝑀 « Mécanique quantique 2, chap.VIII » Cohen-Tannoudji, Diu, Laloë with 𝑉 𝑟 = ℏ2 𝑈(𝒓) 2𝑀 Born aprox. + 𝑘𝑟 ≫ 1 Scattering length = FT of potential ℏ2 ℏ2 𝑘 2 ∆+𝑉 𝒓 𝜑 𝒓 = 𝜑 𝒓 2𝑀 2𝑀 ∆ + 𝑘 2 − 𝑈(𝒓) 𝜑 𝒓 = 0 𝑖𝑘𝑟 𝑒 𝜑 𝒓 ~𝑒 𝑖𝒌𝑖 ∙𝒓 + 𝑏(𝒒) 𝑟 𝑏 𝒒 =− 𝟏 𝟒𝝅 𝑈(𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓 Scattering length X-rays: FT of the electron density 𝑏 𝒒 = −𝑟0 𝑓 𝒒 = −𝑟0 𝜌 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓 Rayons X Phase shift 𝜋 … r0 = 2,82 10-15 Å Neutrons: FT of Fermi pseudo-potential. It is a constant because (𝓥 𝒓 ~ δ 𝒓 ) 𝑀 𝑏 𝒒 =𝑏= 𝓥 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓 2 2𝜋ℏ Phase shift 0 ou 𝜋 … Electrons: TF of potential 𝑉(𝒓) 𝑀 −𝑖𝒒∙𝒓 𝑑 3 𝒓 𝑏 𝒒 =− 𝑉(𝒓)𝑒 2𝜋ℏ2 Electron 𝑏 𝒒 depends n l’énergie Phase shift 𝛿(q) Fadley, Physica Scripta, T17,39,1987 Optical theorem Mécanique quantique II, p. 940 C. Cohen-Tannoudji, B. Diu, Frank Laloë 𝜎𝑡𝑜𝑡 = 𝜎𝑎 +𝜎𝑑 = − 4𝜋 Im(𝑏 𝑘 0 ) 𝑒 𝑖𝑘𝑑 𝑟 𝜓𝑑 𝒓 = −𝑏(𝜃) 𝑟 Shadow: 𝜓𝑖 𝒓 = 𝐴𝑒 𝑖𝑘𝑖 𝑟 Interference between incident wave and scattered wave Absorption Origin of neutrons absorption • Neutrons weaklly absorbed • Absorbed through nuclear reactions 3He+n 3H-+p sa Ni Pb Gd 520 2100 74000 4.6 0.17 6Li 10B Detectors and shields Energy dependance: 𝑘0 𝜎𝑎 𝑘 = 𝜎𝑎 𝑘 𝑘0 = 34,947 nm−1 Origin of photons absorption Free electron energy 𝐸 2 = 𝑚2 𝑐 4 + 𝑝2 𝑐 2 𝑣 ≪ 𝑐 𝐸 = 𝑝2 /2𝑚 Photon energy (p,E) 𝐸𝑝ℎ = 𝑝𝑐 E E ? EO=511 keV EO=511 keV EO-EL Dp.Dr p Free electron: no absorption p Bound electron absorption X-ray absorption Hard X-rays UV VUV XUV Soft X-rays • Absorption At energies smaller than 1000 keV Gamma LEAD Z=82 Photoelectric effect X-ray absorption For E < 1000 keV photoelectric effect is dominant • Photoelectric effect • Photon is absorbed if hn > EI (EI binding energy of e-) • Excitation: Photoelectron is emitted ( E=hn - EI -F ) F: work function ~1 eV • De-excitation: fluorescence photon (hn = EI -EII ) Auger electron ( E= EI -EII -EIII) Fluorescence photon Photoelectron Continuum Fermi level Auger electron -EF M (2p3/2)4 L (2p1/2)2 (2s)2 hn Ka Core levels -EII Kb K (1s)2 -EI Excitation De-excitation Absorption of photons Emission of photons and electrons Order of magnitude X-rays: l = 1.542 Å sa Ni Pb Li B Gd 5,7 36 78300 4760 79800 Neutrons: 1.8 Å sa Ni Pb Gd 520 2100 74000 4.6 0.17 6Li 10B 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)… Scattering Scattering: atome-particle system changes of state Initial state, ei Final state, ef Elastic scattering: Does not change the nature or the internal state of the particles and the target Photon scattering • Rayleigh scattering: • Low energy elastic scattering • hn << EI , EI -EII ; Fi = Ff ; light scattering, blue sky • Raman/Brillouin scattering: • Low energy inelastic scattering • hn << EI ; Fi Ff ; scattering on optical/acoustical phonons • Thomson scattering: • High energy elastic scattering • hn >> EI ; Fi = Ff ; X-ray scattering • Compton scattering: • High energy inelastic scattering • hn >> EI ; Fi Ff ; X-ray scattering Photons scattering (pi ,Ei ) (pf ,Ef ) E 𝑝2 𝐸𝑒 = 𝑚 EO E EO EO-EL p Free electron (e- mass m) Compton scattering 𝑝2 𝐸𝑒 = 𝑀 p Bound electron (atom, crystal mass M»m) Thomson scattering Compton scattering Refraction 𝑅 𝑟 𝑆 A consequence of scattering 𝐷 Wave travelling through a plate of width Δ Phase shift: 𝑛𝑘Δ-𝑘Δ 𝑅0 Δ 𝑅2 = 𝑅0 2 + 𝑟 2 𝑅𝑑𝑅 = 𝑟𝑑𝑟 𝑖 𝑛−1 𝑘Δ 𝜓 𝑃 = 𝜓 𝑃 𝑒 0 𝑃 = 𝜓0 𝑃 (1 + 𝑖 𝑛 − 1 𝑘Δ) ∞ 𝑖𝑘𝑅 𝑒 𝜓 𝑃 = 𝜓0 𝑃 +𝜓0 (𝑆)𝑒 𝑖𝑘𝐷 −𝑏 (2𝜋𝑟𝑑𝑟Δ)𝜌𝑑 𝑅 0 ∞ 2𝜋𝑏Δ𝜌𝑑 𝑖𝑘𝐷 𝑖𝑘𝑅 = 𝜓0 𝑃 −𝜓0 𝑆 2𝜋𝑏Δ𝜌𝑑 𝑒 𝑒 𝑑𝑅 = 𝜓0 𝑃 (1 − 𝑖 ) 𝑘 𝑅0 2𝜋𝑏𝜌𝑑 𝑛 =1− 𝑘2 Absorption 𝑅Δ −1 𝜌𝑑 ~1𝑒Å−3 , 𝑏~ 𝑍 3. ∞ 10−5 k~4 Å −𝜇Å, 𝑒 𝑅0 𝑒 𝑖𝑘𝑅 𝑑𝑅 −5 𝛿 ~ 10 𝑅0 Refraction a n ki Refraction index kr kt 𝑛 = 𝑛𝑟 + 𝑖𝛽 a’ Phase shift and absorption 𝑒 𝑖𝑛𝑘𝑧 = 𝑒 𝑖𝑛𝑟𝑘𝑧 𝑒 −𝛽𝑘𝑧 For X-rays and neutrons 2𝜋𝑏𝜌𝑑 𝜇 𝑛 =1− + 𝑖𝛽 = 1 − 𝛿 + 𝑖 2 𝑘 2𝑘 Snell’s law 𝑛𝑟 cos 𝛼′ = cos 𝛼 Existence of a critical angle above which total reflection 𝛼𝑐 = 2𝛿 ac ki kr Stationnary wave Measure of the sign of b (holography) Experimental techniques EMISSION : X-ray EMISSION (par rayons X) : •Fluorescence (Chemical analysis) R • ayons X Electron Spectroscopy Fluorescence • (Analyse chimique) Electrons • Photoelectrons, Auger electrons (analysis) Photo-électrons, • électrons Auger (Spectrométrie, analyse) Photoelectron diffraction (structure (local structure) Diffraction • de photo-électrons locale) Photo-émission • (Structure de bande, surface Fermi) de 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 :Diffuse scattering (Disorder, liquids, soft matter) • Diffraction (Structures); R • ayons X Compton scattering (electronic structure) Diffraction • (Etude des structures) Diffusion • diffuse (Etude du désordre dans nano-particles, les cristaux, liquides,proteins) cristaux liquides) Small angle scattering (Polymer, liquid crystal, D • iffusion Compton (Structure électronique) Magnetic, inelastic, surface, coherent diffraction Diffusion • aux petits angles (Polymères, cristaux(synchrotrons) liquides, agrégats, grandes mailles) Diffusion • magnétique, inélastique, cohérente… (synchrotrons) Neutrons Neutrons • Diffraction, Diffuse scattering (Structures, Hydrogen, Diffraction, • Diffusion diffuse (Structures, Hydrogène,contrast) contraste différent) Inélastique • (Excitations élémentaires, phonons, dynamique) Inelastic scattering (phonons, dynamics, excitations) Magnétique • (Structures magnétique, magnons) Magnetic (magnetism, magnons) Electrons • Diffraction, • LEED, RHEED (Etude des surfaces) Electrons Low- or high-energy electron diffraction (surfaces, thin samples)