4 X-ray absorption spectroscopy 102461100Cross section (cm2g)Photon energy (keV)K edgeL3, L2, L1 edgesGeComptonThomsonK s ® pL s ® pL p1/2 ® s, dL p3/2 ® s, dZ dependence Þatomic selectivityxPhoton fluxhnFig. 1 – Photoelectric absorption coefficient of Germanium (continuous red line) as a function of photon energy; clearly visible are the L and K absorption edges. Also shown for comparison are the cross sections for scattering, both elastic (Rayleigh, dashed green line) and inelastic (Compton, dotted brown line).Fig 2 – X-ray absorption coefficient at the K edge of amorphous germanium. EXAFS is clearly visible as a sinusoidal oscillation above the edge energy.hnsampleX-ray energy range KeV
5 EXAFS analysisDetectorsStorage ringDouble - crystalmonochromatorSampleI0Iphoton fluxes; x sample thickness, C depends on the detectors efficiency.88009200960010000-2.0-1.00.01.0Ln ( I / I0 )E (eV)
7 Which method for which application? The most important criterion:The best signal to noise ratio for theelement of intereste-Ie-Always transmission, if possibleMost accurate method, best overall S/Ncounting statistics of about 10-4 frombeamlines with more than 108 photons/s)Fluorescence for very diluted samplesA specific signal reduces the large background (but maximum tolerable detector count-rate can result in very long measuring times).Total electron yield (TEY)for surface sensitivity and surface XAFS (adsorbates on surfaces)TEY for thick samples that cannot be made uniform.XEOL X-ray excited optical luminescenceVIS/UV detection from luminescent samples
8 Measurement sensitivity in transmission mode Sample = Species (A + B)A = solute, B = SolventxBy changing the solute absorption coefficientForandStatistically:so thatAssuming that the solute X-ray cross section is almost equal to that of the solvent:R= Dilution Ratio
9 Evaluation of the absorption coefficient n = atomic densityA0 = vector potential amplitudeTransition probability GifINITIAL STATEINTERACTIONFINAL STATECore hole +excitation or ionizationGround stateGOLDEN RULE (weak interaction, time dependent perturbation theory (1st order))is the density of the final states compatiblewith the energy conservation principleatomic initial and final states
10 Interaction Hamiltonian j = electrons; = polarization unit vector; = radiation wavevectorAn equivalent expression often used is:Electric dipole approximationThis approximation is valid if:i.e. for the K edges for energies up to K eV and for the L edgesSelection rules:
11 The direct and inverse problem Direct problem:Inverse problemIn practice one is interested to the inverse problem.Given: m, experimentally determined, one wants to know from XAFS, thefinal stateBut also in this case one needs to know the initial and final states or, at least , to express in parametric form the interesting structural properties.is known because it is the fundamental state of the atom.The calculation ofis complicated because the absorption process because:Many bodies interactionsThe final state is influenced by the environment around the absorption atom
12 One electron approximation Elastic transitions:1 core electron excitedN-1 passive electrons (relaxed)Anelastic transitions:1 core electron excitedOther electrons excited; shake up and shake offHigh photoelectron energy ® sudden approximation1 active electronN-1 electrons= mTOT(w) if So2 = 1= 0.7 ¸ 0.9Evaluation of the final state!
13 X-ray Absorption Fine Structure (XAFS) 102461100Cross section (cm2g)Photon energy (keV)K edgeL3, L2, L1 edgesGeComptonThomsonmx0.00.40.81.21111.512a-GeXAFS=XANES + EXAFSXANESEXAFSXANES = X-ray absorptionNear Edge StructureEXAFS = Extended X-ray Absorption Fine StructureX-ray photon energy (keV)
14 XANES: pre edge structure Fermi Golden ruleEm½f >½i >EmArctangent curve½f >}Inflectionpoint½i >EEm½i >E
15 m Energy Experimental K- edge of metallic Iridium and its simulation Ir metal (Experimental signal)Experimental K- edge of metallic Iridium and its simulationobtained by the superposition of an arctang function and a Lorentian2Energy11600.511.5mTheoretical edgeLorentianArctangent
16 Single particle binding energy (eV) XANESWLm xAr K1s®np, n>3Single particle binding energy (eV)1s2s2pspd50100EXAFSXANESEDGEX-ray edgesVisibleUVArctangentcurve0.58 eV2468Energy eVm x1s®2pNe KThe solid curve represents the K absorption spectrum (uncorrected)of gaseous Ar. The resonance lines and the main absorption edge (arctangentcurve) at the series limit are shown by dashed curves.4p5p866868870eVEnergy eV
17 K-shell photoabsorption of N2 molecule C.T. Chen and F. Sette, Phys. Rev. A 40 (1989)K-shell photoabsorption of gas-phase N2Absorption Intensity (arb.units)N 1s® 1pg*N 1s ®Rydberg seriesDoubleexcitationsShaperesonancex10400405410415420Photon energy (eV)(a)400401402N 1s®1pg*(a) 8 vibrational levels observed in the absorpttion spectrum: N 1s®1pg*414415Double excitations(c)(c) Double excitations in the N2 spectrum Rydberg associated to the N 1s®1pg* transition.Absorption Intensity (arb. Units)406407408409410Photon Energy (eV)(b)N 1s®Rydberg series13245678910111213(b) N 1s®Rydberg series
18 m XANES Chemical information: oxidation state CuO Cu2O KCuO2 Cu E (eV) 897089900.51Y-Ba-Cu-OOxidation Numbers (formal valences)I Cu2OII CuOIII KCuO2Higher transitions energy are expected for higher valence states.X-ray absorption near-edge structure in the vicinity of the Cu K edge for Cu2O, CuO and KCuO2 and for Y-Ba-Cu-O (dotted line). The energy scale is referenced to the Cu K edge energy, E0=8980 eV. The near-edge spectra are the same for each of the threee samples studied and are independent of temperature from 4.2 to 688 K.(J.B. Boyce et al. Phys. Rev. B 1987)
19 SbSI XANES: projected density of state X-rays Sb L1 I L1 Absorbance (a.u.)E - E0 (eV)Total DOS (a.u.)-55151020Sb L3I L3E2s ® pX-rays2p ® p, dFine structure al the L1 edges of antimony and Iodine in SbSI and total DOS of the conduction band calculated by Nakao and Balkanski (blu line).Fine structures at the L3 edges od Antimony and Iodine in SbSI and total DOS of the conduction band calculated by Nakao and Balkanski (blu line). The vertical bars show the positions of the first two maxima of the first derivative of the experimental spectra.DOS(G. Dalba et al., J.of Condensed Matter, 1983)
20 phenomenological interpretation EXAFS:phenomenological interpretatione-ABABAutointerference phenomenon ofthe outgoing photoelectron with its parts that arebackscattered by the neighboring atoms
21 phenomenological interpretation EXAFS:phenomenological interpretationAtomsKrmx14.215.014.6E (KeV)e-AOutgoing wave:MoleculesPositiveinterferenceNegativeinterference50-501234Br2XANESPhoton energy (eV)1340013800mx1420014600krABABSchematic explanation of the origin of EXAFS oscillations. A x-ray photon is absorbed by atom A (left). A photoelectron is emitted by atom A as an outgoing spherical wave (centre). The photoelectron can be back-scattered by atom B, generating a new incoming spherical wave (right), which interferes with the original outgoing wave.rf smoothly varying in the EXAFS region
22 Scattering phases of the photoelectron BBABABAxBAR
23 Standard EXAFS formula Approximations1) Inelastic scattering effect:Electron mean free path2) Thermal disorder:Standard EXAFS formula
24 EXAFS formula For several coordination shells: 1 st 2 nd 3 rd 1 st2 nd3 rdCoordination shellsRFrom ab-initio calculations or from reference compoundsCoordinationnumberDebye WallerfactorInteratomic distance
25 Multiple scattering EXAFS: single scattering processes R XANES: multiple scattering processes
26 RAB Single scattering Double scattering Double scattering Triple scattering
27 Multiple scattering 8 7 6 5 Absorption 4 3 2 1 50 100 150 Energy E-E SiK-edge in c-SiExperiment8765Absorption4321m50100150Energy E-E(eV)FComparison between the experimental Si K-edge XAFS for c-Si and calculations carried out with the FEFF code.The total multiple scattering in the first 8 shells around the absorption atom have been considered. m0 is the atomic absorption coefficient.
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