An introduction to X-ray Absorption Fine Structure Spectroscopy

Transcription

1 An introduction to X-ray Absorption Fine Structure Spectroscopy Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 1

2 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

3 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 3

4 Main X-ray based techniques Two undamental X-ray-matter interactions: photoelectric absorption scattering (elastic, inelastic) Two amilies o experimental techniques: spectroscopy electronic structure, local structure o matter absorption emission inelastic scattering elastic diusion microscopic geometric structure diraction (crystalline solids) scattering (amorphous solids, liquids) 4 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

5 The Absorption Coeicient m I I t I = I exp[-mt] linear absorption coeicient mt = ln [ I / I ] S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 5

6 The Absorption Coeicient m synchrotron source monochromator incident lux monitor transmitted lux monitor t polychromatic X-rays monochromatic X-rays I sample I 1. Measure I and I as a unction o E X. Calculate: m t = ln [I /I ] S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 6

7 m/r [barns/atom] The Absorption Coeicient m m depends strongly on X-ray energy E and atomic number Z, and on the density r and atomic mass A m r Z 4 A E 3 m has sudden umps (absorption edges) which occur at energies characteristic o the element. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 7

8 m [cm -1 ] Germanium total absorption coeicient 1 photoelectric absorption E(eV) elastic scattering inelastic scattering Photoelectric absorption dominates the absorption coeicient in this energy range 8 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

9 Photoelectric Absorption X-rays (light with wavelength.6 l 1 Å or energy 1 E kev) are absorbed by all matter through the photoelectric eect: An x-ray is absorbed by an atom when the energy o the x-ray is transerred to a core-level electron (K, L, or M shell) which is eected rom the atom. The atom is let in an excited state with an empty electronic level (a core hole). Any excess energy rom the x-ray is given to the eected photoelectron. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 9

10 De-excitation: Fluorescence and Auger Eect When x-rays are absorbed by the photoelectric eect, the excited core-hole will relax back to a ground state o the atom. A higher level core electron drops into the core hole, and a luorescent x-ray or Auger electron is emitted. X-ray Fluorescence: Auger Eect: An x-ray with energy = the dierence o the core-levels is emitted. An electron is promoted to the continuum rom another core-level. 3p p 1s K a : LK, K b : M K X-ray luorescence and Auger emission occur at discrete energies characteristic o the absorbing atom, and can be used to identiy the absorbing atom. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 1

11 XAS measurements synchrotron source monochromator I I F I sample XAS measures the energy dependence o the x-ray absorption coeicient μ(e) at and above the absorption edge o a selected element. μ(e) can be measured in several ways: Transmission: The absorption is measured directly by measuring what is transmitted through the sample: I = I e μ (E)t μ(e) t = ln (I/I ) Fluorescence: The re-illing the deep core hole is detected. Typically the luorescent x-ray is measured. μ(e) ~ I F / I S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 11

12 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 1

13 Absorption Absorption coeicient m What is XAFS? X-ray Absorption Fine Structure: oscillatory variation o the X-ray absorption as a unction o photon energy beyond an absorption edge As K-edge in InAsP E(eV) E Proximity o neighboring atoms strongly modulates the absorption coeicient S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 13

14 Absorption Absorption coeicient m EXAFS and XANES XAFS is oten broken into regimes: XANES X-ray Absorption Near-Edge Spectroscopy EXAFS Extended X-ray Absorption Fine-Structure which contain related, but slightly dierent inormation about an element s local coordination and chemical state. As K-edge in InAsP XANES EXAFS E (ev) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 14

15 Absorption Absorption coeicient m EXAFS and XANES XANES: transitions to unilled bound states, nearly bound states, continuum local site symmetry, charge state, orbital occupancy EXAFS: 5-1 ev ater edge due to transitions to continuum local structure (bond distance, number, type o neighbors.) XANES EXAFS E (ev) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 15

16 EXAFS qualitatively isolated atom condensed matter E kin e - e - E l R E The kinetic energy o the eected photoelectron E kin is: p k E kin = E E = = l = p/k m m S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 16

17 Where do the oscillations come rom? Due to a quantistic eect, the autointererence o photoelectron wave modiies the absorption coeicient value: 1. As E is scanned above E, E kin is varied, and consequently k and l.. The outgoing and backscattered parts o the wave interere either constructively or destructively, depending on the ratio between l and R. 3. It is the intererence between outgoing and incoming waves that gives rise to the sinusoidal variation o m(e) p k E kin = E E = = m m l R e - requency ~ distance rom neighbors amplitude ~ number and type o neighbors S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 17

18 m (E) Absorption The EXAFS signal c We re interested in the energy dependent oscillations in μ (E), as these will tell us something about the neighboring atoms, so we deine the EXAFS as: c E = m E m m E E We subtract o the smooth bare atom background μ (E), and divide by the edge step μ (E ), to give the oscillations normalized to 1 absorption event m (E) m E (ev) E(eV) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 18

19 c (k) k c (k) EXAFS : c (k) XAFS is an intererence eect, and depends on the wave-nature o the photoelectron. It s convenient to think o XAFS in terms o photoelectron wavenumber, k, rather than x-ray energy m k = E E c (k) is oten shown weighted by k or k 3 to ampliy the oscillations at high-k: k (A -1 ) k (A -1 ) 19 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

20 c c(k) Amplitude o FT Qualitative picture o local coordination in R space The requencies contained in the EXAFS signal depend on the distance between the absorbing atom and the neighboring atoms (i.e. the length o the scattering path). A Fourier Transorm o the EXAFS signal provides a photoelectron scattering proile as a unction o the radial distance rom the absorber..1.5 R 1 (SS). -.5 R (SS), R 3 (SS), MS k (A -1 ) k(a -1 ) R(A) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

21 Amplitude o FT Quantitative structural determination Structural determinations depend on the easibility o resolving the data into individual waves corresponding to the dierent types o neighbors (SS) and bonding conigurations (MS) around the absorbing atom. absorber As atom R(A) In As As In P As As As In In In As As InAs x P 1-x As P As In 1 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

22 c : sum o damped waves c (k) is the sum o contributions c (k) rom backscattered wavelets: c k A k sin k c = k = c k Each c (k) can be approximated by a damped sine wave o the type: The larger the number o neighbors, the larger the signal N k e k kr k The stronger the scattering amplitude, the larger the signal Damping o the amplitude at large k, due to static and thermal disorder Each shell contributes a sinusoidal signal which oscillates more rapidly the larger the distance S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

23 Amplitudes shape o the envelope o each wave indicative o nature o backscatterer atom: InAs x P 1-x absorber As atom 1.5 As In. As P As.6 As.6.4. As.4. In k(a -1 ) k(a -1 ) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 3

24 Frequencies: Single and Multiple Scattering paths The sum over paths in the EXAFS equation includes many shells o atoms (1 st neighbor, nd neighbor, 3 rd neighbor,... ), but can also include multiple scattering paths, in which the photoelectron scatters rom more than one atom beore returning to the central atom. SS g (r) = R 1 MS g (r) = 4 R 1 MS g 3 (r) R 1 R 1 = R 1 + R + R 3 R 1 R 33 MS g 3 (r) R = R 1 + R 3 EXAFS can give inormation on the n-body distribution unctions g n (r). S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 4

25 XAFS vs Diraction Methods Diraction Methods (X-rays, Neutrons) Crystalline materials with long-range ordering -> 3D picture o atomic coordinates Materials with only short-range order (amorphous solid, liquid, or solution) -> 1D RDF containing interatomic distances due to all atomic pairs in the sample. XAFS 1D radial distribution unction (centered at the absorber) Element selectivity Higher sensitivity to local distortions (i.e. within the unit cell) Charge state sensitivity (XANES) Structural inormation on the environment o each type o atom: distance, number, kind, static and thermal disorder 3-body correlations Investigation o matter in the solid (crystalline or amorphous), liquid, solution or gaseous state with same degree o accuracy. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 5

26 EXAFS: typical applications Element selectivity Local structure sensitivity Local structure in non-crystalline matter Local environment o an atomic impurity in a matrix o dierent atomic species Study o systems whose local properties dier rom the average properties Detection o very small distortions o local structure S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 6

27 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 7

28 Kr gas Rh metal S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 8

29 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 9 Absorption coeicient ˆ ) ( i H I m photoelectron core hole continuum 1s electron continuum ˆ in principle, all electrons are involved multi body process single electron 1 1 ˆ N i N i r r A A ˆ ˆ dipole ˆ ˆ i r 1 1 N N i S sudden ˆ i r S I r A H

30 Consequences o dipole approximation: Selection rules l = ± 1 s = = ± 1 m = For 1-electron transitions: edge initial state inal state K, L 1 s (l=) p (l=1) L, L 3 p (l=1) s (l=), d (l=) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 3

31 Absorption coeicient photoelectron ˆ i = 1s Ei continuum s electron = p E continuum core hole = E i m( ) ˆ i r Approx: dipole + single electron + sudden : photon polarization r: electron position i > relatively easy ground state o atom; i.e. 1s e - waveunction > very complicated inal state strongly inluenced by environment S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 31

32 Isolated atom: atomic absorption coeicient e - photoelectron ree to travel away undisturbed = h outgoing spherical wave originating rom the absorbing atom m r ˆ i m ˆ i r d r * r r overlap integral o initial and inal state waveunctions: monotonically decreases as unction o E S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 3

33 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 33 Non-isolated atom sum o the outgoing and all the incoming waves, one per each neighboring atom. = ˆ i r m * ˆ r r r r r d i m * * * * ˆ ˆ ˆ Re ˆ r r r d r r r r r r r d r r r r d r i i i i m EXAFS e - h

34 c k Origin o EXAFS c : ractional change in m introduced by the neighbors Re d r m= m 1 c * * r ˆ r r r ˆ r r i i * d r r ˆ i r r (1) Intererence between outgoing waveunction and backscattered wavelets Dominant contribution to integral comes rom spatial region close to absorber atom nucleus, where the core orbital waveunction i. The region where i represents simultaneously the source and the detector or the photoelectron that probes the local structure around the absorber atom S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 34

35 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 35

36 The EXAFS equation To model the EXAFS, we use the EXAFS Equation c k = k k sin [ S N e kr k] kr where (k) and (k) are photoelectron scattering properties o the neighboring atom. (The sum is over shells o similar neighboring atoms). I we know these properties, we can determine: R distance to neighboring atom. N coordination number o neighboring atom. mean-square disorder o neighbor distance. The scattering amplitude (k) and phase-shit (k) depend on atomic number Z o the scattering atom, so we can also determine the species o the neighboring atom. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 36

37 The EXAFS equation: simple description With spherical wave or the propagating photoelectron: and a scattering atom at a distance r = R, we get: ~ Region III: amplitude o backscattering on B e ikr e ikr [i e ic ] (k) e is(k) e ic ] kr kr e ikr kr I II III A R B Region I: amplitude o outgoing wave Region II: amplitude o wave arriving on B Region II: amplitude o ingoing wave, backscattered rom B where the neighboring atom gives the amplitude (k) and phase-shit s (k) to the scattered photoelectron. Substituting into equation (1) and ater some math we get: c k = S k kr sin [kr ] k or 1 scattering atom. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) k = s k k c 37

38 Development o the EXAFS equation For N scattering atoms, and with a thermal and static Gaussian disorder o, giving the mean square disorder in R*, we have c k = S k kr sin [kr ] k N e k A real system will have neighboring atoms at dierent distances and o dierent types. We add all these contributions to get a version o the EXAFS equation: c k = k k sin [ S N e kr k] kr To obtain this ormula we used a spherical wave or the photoelectron: * EXAFS takes place on a time scale much shorter than that o atomic motion, so the measurement serves as an instantaneous snapshot o the atomic coniguration S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) e ikr kr 38

39 The photoelectron mean ree path But the photoelectron can also scatter inelastically*, and may not be able to get back the absorbing atom. Also: The core-hole has a inite lietime**, limiting how ar the photoelectron can go. Using a damped wave-unction: ikr e kr e r / l( k ) where l(k) is the photo electron s mean ree path (including core hole lietime), the EXAFS equation becomes: c k = The mean ree path l depends on k. For the EXAFS k range, l < 5 Å. S N kr k k R / lk e e sin [kr k ] The l and R - terms make EXAFS a local atomic probe. * Electrons that have suered inelastic losses will not have the proper wave vector to contribute to the intererence process. ** the photoelectron and core hole exist simultaneously S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 39

40 S : Amplitude Reduction Term Another important Amplitude Reduction Term is due to the relaxation o all the other electrons in the absorbing atom to the hole in the core level: S N 1 i N 1 where N-1 accounts or the relaxation o the other N-1 electrons relative to these electrons in the unexcited atom: N-1. Typically S is taken as a constant:.7 < S < 1. which is ound or a given central atom, and simply multiplies the XAFS c. Note that S is completely correlated with N. This, and other experimental and theoretical issues, make EXAFS amplitudes (and thereore N) less precise than EXAFS phases (and thereore R). Usually S is ound rom a standard (data rom a sample with well-known structure) and applied to a set o unknowns as a scale actor. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 4

41 Scattering Amplitude and Phase-Shit: (k) and (k) The scattering amplitude (k) and phase-shit (k) depend on atomic number. The scattering amplitude (k) peaks at dierent k values and extends to higher-k or heavier elements. For very heavy elements, there is structure in (k). The phase shit (k) shows sharp changes or very heavy elements. These scattering unctions can be accurately calculated (say with the programs FEFF, GNXAS, etc.), and used in the EXAFS modeling. Z can usually be determined to within 5 or so. Fe and O can be distinguished, but Fe and Mn cannot be. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 41

42 Calculating (k) and (k) These days, we can calculate (k) and (k) easily using dierent sotware codes These programs take as input: 1. a list o atomic x,y,z coordinates or a physical structure. a selected central atom. The result is a set o iles containing the (k), and (k) or a particular scattering shell or scattering path or that cluster o atoms. Many analysis programs use these iles directly to model EXAFS data. A structure that is close to the expected structure can be used to generate a model, and used in the analysis programs to reine distances and coordination numbers. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 4

43 Outline X-ray Absorption Spectroscopy X-ray Absorption Fine Structure EXAFS: Simple Theoretical Description EXAFS: Derivation o EXAFS Equation XANES: Qualitative Interpretation S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 43

44 Absorption Absorption coeicient m EXAFS and XANES XANES: transitions to unilled bound states, nearly bound states, continuum local site symmetry, charge state, orbital occupancy EXAFS: 5-1 ev ater edge due to transitions to continuum local structure (bond distance, number, type o neighbors.) XANES EXAFS E (ev) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 44

45 XANES Interpretation The EXAFS Equation breaks down at low-k, and the mean-ree-path goes up. This complicates XANES interpretation: We do not have a simple equation or XANES. XANES can be described qualitatively (and nearly quantitatively ) in terms o coordination chemistry regular, distorted octahedral, tetrahedral,... molecular orbitals p-d orbital hybridization, crystal-ield theory,... band-structure the density o available electronic states multiple-scattering multiple bounces o the photoelectron These chemical and physical interpretations are all related, o course: What electronic states can the photoelectron ill? XANES calculations are becoming reasonably accurate and simple. These can help explain what bonding orbitals and/or structural characteristics give rise to certain spectral eatures. Quantitative XANES analysis using irst-principles calculations are still rare, but becoming possible... S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 45

46 K-Edge XANES Mn: [Ar] 3d 5 4s Total electron energy 3d 1s Excited States p 4p 3d 3d 1s 1s Continuum Main edge Ground State 3d 1s Pre-edge Mn 3+ S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 46

47 Mn: [Ar] 3d 5 4s Chemical Shit Total electron energy Excited States 4p 3d 1s 4p 3d 1s Mn 3+ Mn 4+ 3d 1s Ground State 3d 1s Mn O 3 MnO Mn 3+ Mn 4+ S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 47

48 Edge Shits and Pre-edge Peaks in Fe oxides XANES or Fe oxides and metal The shit o the edge position can be used to determine the valence state. The heights and positions o pre-edge peaks can also be sometimes used to determine Fe 3+ /Fe + ratios. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 48

49 XANES Analysis: Oxidation State The Normalized XANES rom several Fe compounds: XANES can be used simply as a ingerprint o phases and oxidation state. XANES analysis can be as simple as making linear combinations o known spectra to get compositional raction o these components. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 49

50 XANES Analysis: Coordination Chemistry O h centrosymm only weak quadrupole channel T d non-centrosymm p-d mixing dipole channel The XANES o Cr 3+ and Cr 6+ shows a dramatic dependence on oxidation state and coordination chemistry. For ions with partially illed d shells, the p-d hybridization changes dramatically as regular octahedra distort, and is very large or tetrahedral coordination. This gives a dramatic pre-edge peak absorption to a localized electronic state. S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 5

51 XANES: Conclusions XANES is a much larger signal than EXAFS XANES can be done at lower concentrations, and less-than-perect sample conditions. XANES is easier to crudely interpret than EXAFS For many systems, the XANES analysis based on linear combinations o known spectra rom model compounds is suicient. XANES is harder to ully interpret than EXAFS The exact physical and chemical interpretation o all spectral eatures is still diicult to do accurately, precisely, and reliably. This situation is improving, so stay tuned to the progress in XANES calculations S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan)

52 More inormation: web links International XAFS Society: Tutorials and other Training Material: Sotware Resources EXAFS: S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 5

53 More inormation: Books and Review Articles Fundamentals o XAFS Introduction to XAFS: A Practical Guide to X-ray Absorption Fine Structure Spectroscopy, G. Bunker, Cambridge University Press, 1 X-ray Absorption: Principles, Applications, Techniques o EXAFS, SEXAFS, and XANES, in Chemical Analysis 9 D. C. Koningsberger and R. Prins, ed., John Wiley & Sons, Basic Principles and Applications o EXAFS, Chapter 1 in Handbook o Synchrotron Radiation, pp E. A. Stern and S. M. Heald, E. E. Koch, ed., North-Holland, 1983 FEFF Theoretical approaches to x-ray absorption ine structure J. Rehr et al., Rev. Mod. Phys. 7, () GNXAS X-ray absorption spectroscopy and n-body distribution unctions in condensed matter (I): theory o the GNXAS data-analysis method A. Filipponi, A. Di Cicco and C. R. Natoli, Phys. Rev. B 5, 151 (1995) MXAN Geometrical itting o experimental XANES spectra by a ull multiple-scattering procedure M.Benatto and S. Della Longa J. Synchr. Rad. 8, 187 (1) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 53

54 How to measure the absorption coeicient m di = I x N dx t a at/cm cm /at t di I x = N t a t dx ln I(t) ln I( ) = N a I(t) I( ) = e = e N a m t S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 54

55 I(t) I( ) = e N a = e m t m is related to the atomic cross section: m at mole gr mole N Na cm gr 1 = = r = cm a t a A cm in general you ind tabulated the mass absorption coeicient m/r: at 3 m r = a N a A cm at at mole gr mole = cm gr or a generic sample P x Q y..: m r tot = x m r P A P M y m r Q A Q M S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 55

56 Recipe or calculating t or transmission XAS I(t) I( ) = e N a = e m t 1. Total absorption above the edge must not be too high: m above edge t = 5 I / I ~.14.7 ideally m above edge t = -3. Contrast at edge must be as large as possible: [ m above edge - m below edge ] t >.1 ideally [ m above edge - m below edge ] t = 1 I absorber is very dilute, and matrix absorbs a lot, then this is not possible luorescence detection S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 56

57 How to calculate m energy density u carried by X-ray beam is: linear absorption coeicient m measures the energy density reduction due to the interaction with the system o atoms: u o E o A = = 1 m = u du dx m = A du dx m m d = n A d = n A dx dx ph ph m = n A S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) W i 57

58 X-ray Absorption I. Lets consider the interaction between: Monochromatic X-ray beam ( = pn + monoatomic sample EM ield (classic) + atom (quantistic) (semi-classical description) II. m ~ m photoelectric absorption or 1 < E < 5 kev III. Qualitatively, interaction process is: continuum or ree energy level s electron 1 core hole i >, E i >, E = E i + S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 58

59 Transition probability: Golden Rule m depends on: atomic density n transition probability W i o atom rom i > to > = n A to calculate W i : time-dependent perturbation theory based on power series o EM ield - atom interaction potential. The interaction is in general WEAK p W = Hˆ r E () Can limit series to 1st order: Golden Rule Ĥ I m W i i i (1) EM ield - atom interaction hamiltonian operator I ˆ Matrix element o H I between initial and inal state i H I r E Density o inal states, compatible with energy conservation: E = E i + S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 59

60 the interaction hamiltonian or photoelectric absorption (see Appendix 1) is (to 1 st order): H e = i A I m r (3) the transition probability or photoelectric absorption o a monochromatic, polarized and collimated photon beam is [(3) into ()]: W i p e ik r = A i e ˆ m r E (4) S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 6

61 S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 61 Dipole approximation urther simpliication: transition probability in dipole approximation: alternative and equivalent expression : inally one gets [(5) into (1)]: 1! 1 = r k i r k r k i e i i E A m e W r p ˆ = i i E r A e W r p ˆ = (5) (6) 1 r k i = n e p m i E r r ˆ

62 p e = n m ˆ r re i i i > and > are known (i waveunctions and energies can be calculated): 1) calculate W i ) calculate m in practice, one is interested in inverse process: 1) measure m ) extract EXAFS 3) obtain inormation on local structure through > but, to obtain structural ino, one still needs to calculate i > and > or at least be able to express their structural properties in parametric orm i > relatively easy - ground state o atom > in general very complicated - in principle, all electrons are involved -> multi body process - inal state strongly inluenced by environment S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 6

63 Single electron approximation large part o m due to elastic transitions: only 1 electron out o N modiies its state: leaves its deep core level all other N-1 passive electrons relax their orbitals to adapt to the new potential created by presence o core hole remaining part o m due to inelastic transitions: primary excitation o core electron provokes successive excitations o other (external) electrons (shake up, shake o processes) excess energy distributed among all excited electrons N 1 i where m m = m m el inel N 1 N 1 ˆ r el i i r Slater determinant o passive electrons waveunctions, r, Waveunction, position vector, inal energy o active electron S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 63

64 Sudden approximation and overlap actor i photoelectron energy is suiciently high (E > ew 1 ev above edge) time to exit atom << relaxation time o passive electrons its state not inluenced by passive electrons relaxation m el r S r i ˆ (7) where S = N 1 i N 1 (S ~.7 -.9) Allows to reduce interpretation o EXAFS to the calculation o the inal state o ONLY the photoelectron S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 64

65 The polarization dependent dipole operator m( ) ˆ i r dipole operator a = x, y, z X-ray prop direction q = +1,, -1 polarization states (q photon angular momentum) electron position vector photon polarization vectors linear polarization circular polarization with k // z dipole operator in terms o spherical harmonics S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 65

66 The dipole selection rules m( ) ˆ i r matrix elements actor into spin, radial and angular parts By looking at the non-zero matrix elements we get the dipole selection rules where q is the X-ray angular momentum S. Pascarelli An Introduction to XAFS - Cheiron School 14 (Japan) 66