Interaction of Radiation with Matter « Element of modern x-ray physics »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  dd kiki kdkd I0I0 I l dz

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

Characteristics of particles X Photons Electromagnetic field E=h =hc/  Å  =12398/E(eV)  Å, E=12.4 keV = Hz (EHz) p=hk=h /c << 1 Charge  th ~ Z 2 barn Moments magnétiques  d ~ barn 4700 barn (Z=28, 1,5 Å) Neutrons Particle  ~ exp(i k.r) E 2 =p 2 c 2 +m n 2 c 4 ; E=p 2 /2m n  Å  =0.286/E 0.5 (eV)  Å, E=81.8 meV v n = 4000 m/s  p=hk (=mv) ~ 1 Noyaux (forte)  d ~ 5 barn Moments magnétiques  d ~ 3 barn Typique : 0,1-1 barn Description Energy E  Momentum p k B T/E 300K Interaction Absorption Electrons Particle  ~ exp(i k.r) E=p 2 /2m e  Å  =12.265/E 0.5 (eV)  Å, E=150 eV v e = 7274 km/s p=hk (=mv ~ Potentiel electrostatique  d ~ 10 8 barn -

Absorption cross section After going through matter of width dz, beam intensity decreases by dI  attenuation coefficient (cm -1 ) Beer-Lambert law I0I0 I l dz  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~ cm 2

Scattering cross section Scattering process number of scattered particles  dd kiki kdkd Scattering differential cross section Wave function of the scattered particle Differential cross section

Characteristics of particles X Photons Electromagnetic field E=h =hc/  Å  =12398/E(eV)  Å, E=12.4 keV = Hz (EHz) p=hk=h /c << 1 Charge  th ~ Z 2 barn Magnetic moments  d ~ barn 4700 barn (Z=28, 1,5 Å) Neutrons Particle  ~ exp(i k.r) E 2 =p 2 c 2 +m n 2 c 4 ; E=p 2 /2m n  Å  =0.286/E 0.5 (eV)  Å, E=81.8 meV v n = 4000 m/s  p=hk (=mv) ~ 1 Noyaux (forte)  d ~ 5 barn Magnetic moments  d ~ 3 barn Typique : 0,1-1 barn Description Energy E  Momentum p k B T/E 300K Interaction Absorption Electrons Particle  ~ exp(i k.r) E=p 2 /2m e  Å  =12.265/E 0.5 (eV)  Å, E=150 eV v e = 7274 km/s p=hk (=mv ~ Electrostatic potential  d ~ 10 8 barn -

Scattering length for particles Scattering length = FT of potential « Mécanique quantique 2, chap.VIII » Cohen-Tannoudji, Diu, Laloë

Scattering length Rayons X Electron Fadley, Physica Scripta, T17,39,1987

Optical theorem Mécanique quantique II, p. 940 C. Cohen-Tannoudji, B. Diu, Frank Laloë Shadow: Interference between incident wave and scattered wave

Absorption

Origin of neutrons absorption Neutrons weaklly absorbed Absorbed through nuclear reactions 3 He+n  3 H - +p 6 Li B 2100 Gd Ni 4.6 Pb 0.17 aa Detectors and shields Energy dependance:

Origin of photons absorption Free electron energy Photon energy E p E O =511 keV E p Free electron: no absorption Bound electron absorption ? E O -E L p.r  p.r   (p,E)

X-ray absorption Absorption Hard X-rays Gamma Tender X-rays Soft X-rays UVVUVXUV At energies smaller than 1000 keV Photoelectric effect LEAD Z=82

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

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

Electrons mean free paths From A. Zangwill, ‘Physics at Surfaces’, Cambridge Univ. Press. Distance between two inelastic collisions with Plasmons Valence electrons 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 Elastic scattering: Initial state,  i Final state,  f Does not change the nature or the internal state of the particles and the target

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

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

Refraction A consequence of scattering

Refraction n ktkt kiki krkr Refraction index For X-rays and neutrons  ’’ Snell’s law Existence of a critical angle above which total reflection Stationnary wave kiki krkr cc Phase shift and absorption Measure of the sign of b (holography)

Experimental techniques DIFFUSION : Rayons X Diffraction (Etude des structures) Diffusion diffuse (Etude du désordre dans les cristaux, liquides, cristaux liquides) DiffusionCompton (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) EMISSION (par rayons X) : Rayons X Fluorescence (Analyse chimique) Electrons Photo-électrons, électrons Auger (Spectrométrie, analyse) Diffraction de photo-électrons (structure locale) Photo-émission (Structure de bande, surface de Fermi ) WAVES/PARTICLES X-Rays Neutrons Electrons EMISSION : X-ray Fluorescence (Chemical analysis) Electron Spectroscopy Photoelectrons, Auger electrons (analysis) Photoelectron diffraction (local structure) Photoemission (band structure) REFRACTION : X-ray, neutrons Reflectrometry (surfaces, interfaces) Stationnary waves (surfaces) ABSORPTION : X-ray XAS, EXAFS, XANES (local order) Dichroism (Magnetism, surfaces) SCATTERING X-rays 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) Crystal Liquid, liquid crystal Polymer Surface