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**Interaction X-rays - Matter**

Pair production > 1M eV Photoelectric absorption Transmission MATTER X-rays Scattering Compton Thomson Decay processes Fluorescence Auger electrons Primary competing processes and some radiative and non-radiative decay processes

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**X-ray attenuation: atomic cross section:**

hn sample X-ray attenuation: atomic cross section: Thomson Observed data Electron positron pairs Compton Photoelectric absorption Photonuclear absorption Cross section (barns/atom) 1 103 106 10 eV 1 KeV 1 GeV 1 MeV Cu Z=29 Energy Li Z=3 Ge Z=32 Gd Z=64 Energy (KeV) 100 102 104 s (Barns/atom)

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**Photoelectric absorption coefficient**

m m L3, L2, L1 edges m - mK m - mK - mL1 (cm2/g) m - mK - mL1- mL2 m - mK - mL1- mL2 - mL3 K edge Z=57 Lanthanum

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**X-ray absorption spectroscopy**

10 2 4 6 1 100 Cross section (cm2g) Photon energy (keV) K edge L3, L2, L1 edges Ge Compton Thomson K s ® p L s ® p L p1/2 ® s, d L p3/2 ® s, d Z dependence Þ atomic selectivity x Photon flux hn Fig. 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. hn sample X-ray energy range KeV

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EXAFS analysis Detectors Storage ring Double - crystal monochromator Sample I0 I photon fluxes; x sample thickness, C depends on the detectors efficiency. 8800 9200 9600 10000 -2.0 -1.0 0.0 1.0 Ln ( I / I0 ) E (eV)

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**XAFS measurements h e- Fluorescence X-rays TEY SAMPLE**

Incident X-rays Transmitted X-rays Visible light XEOL h e- X-ray energy XAFS spectrum

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**Which method for which application?**

The most important criterion: The best signal to noise ratio for the element of interest e- Ie- Always transmission, if possible Most accurate method, best overall S/N counting statistics of about 10-4 from beamlines with more than 108 photons/s) Fluorescence for very diluted samples A 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 luminescence VIS/UV detection from luminescent samples

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**Measurement sensitivity in transmission mode**

Sample = Species (A + B) A = solute, B = Solvent x By changing the solute absorption coefficient For and Statistically: so that Assuming that the solute X-ray cross section is almost equal to that of the solvent: R= Dilution Ratio

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**Evaluation of the absorption coefficient**

n = atomic density A0 = vector potential amplitude Transition probability Gif INITIAL STATE INTERACTION FINAL STATE Core hole + excitation or ionization Ground state GOLDEN RULE (weak interaction, time dependent perturbation theory (1st order)) is the density of the final states compatible with the energy conservation principle atomic initial and final states

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**Interaction Hamiltonian**

j = electrons; = polarization unit vector; = radiation wavevector An equivalent expression often used is: Electric dipole approximation This approximation is valid if: i.e. for the K edges for energies up to K eV and for the L edges Selection rules:

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**The direct and inverse problem**

Direct problem: Inverse problem In practice one is interested to the inverse problem. Given: m, experimentally determined, one wants to know from XAFS, the final state But 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 of is complicated because the absorption process because: Many bodies interactions The final state is influenced by the environment around the absorption atom

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**One electron approximation**

Elastic transitions: 1 core electron excited N-1 passive electrons (relaxed) Anelastic transitions: 1 core electron excited Other electrons excited; shake up and shake off High photoelectron energy ® sudden approximation 1 active electron N-1 electrons = mTOT(w) if So2 = 1 = 0.7 ¸ 0.9 Evaluation of the final state!

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**X-ray Absorption Fine Structure (XAFS)**

10 2 4 6 1 100 Cross section (cm2g) Photon energy (keV) K edge L3, L2, L1 edges Ge Compton Thomson m x 0.0 0.4 0.8 1.2 11 11.5 12 a-Ge XAFS = XANES + EXAFS XANES EXAFS XANES = X-ray absorption Near Edge Structure EXAFS = Extended X-ray Absorption Fine Structure X-ray photon energy (keV)

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**XANES: pre edge structure**

Fermi Golden rule E m ½f > ½i > E m Arctangent curve ½f > } Inflection point ½i > E E m ½i > E

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**m Energy Experimental K- edge of metallic Iridium and its simulation**

Ir metal (Experimental signal) Experimental K- edge of metallic Iridium and its simulation obtained by the superposition of an arctang function and a Lorentian 2 Energy 1160 0.5 1 1.5 m Theoretical edge Lorentian Arctangent

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**Single particle binding energy (eV)**

XANES WL m x Ar K 1s®np, n>3 Single particle binding energy (eV) 1s 2s 2p s p d 50 100 EXAFS XANES EDGE X-ray edges Visible UV Arctangent curve 0.58 eV 2 4 6 8 Energy eV m x 1s®2p Ne K The solid curve represents the K absorption spectrum (uncorrected) of gaseous Ar. The resonance lines and the main absorption edge (arctangent curve) at the series limit are shown by dashed curves. 4p 5p 866 868 870 eV Energy eV

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**K-shell photoabsorption of N2 molecule**

C.T. Chen and F. Sette, Phys. Rev. A 40 (1989) K-shell photoabsorption of gas-phase N2 Absorption Intensity (arb.units) N 1s® 1pg* N 1s ® Rydberg series Double excitations Shape resonance x10 400 405 410 415 420 Photon energy (eV) (a) 400 401 402 N 1s®1pg* (a) 8 vibrational levels observed in the absorpttion spectrum: N 1s®1pg* 414 415 Double excitations (c) (c) Double excitations in the N2 spectrum Rydberg associated to the N 1s®1pg* transition. Absorption Intensity (arb. Units) 406 407 408 409 410 Photon Energy (eV) (b) N 1s®Rydberg series 1 3 2 4 5 6 7 8 9 10 11 12 13 (b) N 1s®Rydberg series

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**m XANES Chemical information: oxidation state CuO Cu2O KCuO2 Cu E (eV)**

8970 8990 0.5 1 Y-Ba-Cu-O Oxidation Numbers (formal valences) I Cu2O II CuO III KCuO2 Higher 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)

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**SbSI XANES: projected density of state X-rays Sb L1 I L1**

Absorbance (a.u.) E - E0 (eV) Total DOS (a.u.) -5 5 15 10 20 Sb L3 I L3 E 2s ® p X-rays 2p ® p, d Fine 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)

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**phenomenological interpretation**

EXAFS: phenomenological interpretation e- A B A B Autointerference phenomenon of the outgoing photoelectron with its parts that are backscattered by the neighboring atoms

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**phenomenological interpretation**

EXAFS: phenomenological interpretation Atoms Kr mx 14.2 15.0 14.6 E (KeV) e- A Outgoing wave: Molecules Positive interference Negative interference 50 -50 1 2 3 4 Br2 XANES Photon energy (eV) 13400 13800 mx 14200 14600 kr A B A B Schematic 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

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**Scattering phases of the photoelectron**

B B A B A B A x B A R

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**Standard EXAFS formula**

Approximations 1) Inelastic scattering effect: Electron mean free path 2) Thermal disorder: Standard EXAFS formula

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**EXAFS formula For several coordination shells: 1 st 2 nd 3 rd**

1 st 2 nd 3 rd Coordination shells R From ab-initio calculations or from reference compounds Coordination number Debye Waller factor Interatomic distance

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**Multiple scattering EXAFS: single scattering processes R**

XANES: multiple scattering processes

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**RAB Single scattering Double scattering Double scattering**

Triple scattering

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**Multiple scattering 8 7 6 5 Absorption 4 3 2 1 50 100 150 Energy E-E**

Si K -edge in c-Si Experiment 8 7 6 5 Absorption 4 3 2 1 m 50 100 150 Energy E-E (eV) F Comparison 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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