Fundamentals of X-ray Absorption Fine Structure

UNDERGRADUATE SUMMER SCHOOL, ESRF, SEPTEMBER 16 Fundamentals of X-ray Absorption Fine Structure Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France Page Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES X-ray Absorption Page 3 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Main X-ray based techniques o Two fundamental X-ray-matter interactions: l photoelectric absorption l scattering (elastic, inelastic) o Two families of experimental techniques: q spectroscopy exchange of energy (electronic structure, local structure of matter) absorption (XAS, EXAFS, XANES,..) emission (XES, HERFD,..) inelastic scattering (IXS, RIXS, X-ray Raman, etc..) q elastic diffusion no exchange of energy (microscopic geometric structure) diffraction for crystalline solids (XRD, GIXRD,.) scattering for amorphous solids, liquids (XRS, WAXS, SAXS, ) Page 4 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES The Absorption Coefficient m I I t I = I exp[-mt] linear absorption coefficient mt = ln [ I / I ] Page 5 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES The Absorption Coefficient m synchrotron source monochromator incident flux monitor transmitted flux monitor t polychromatic X-rays monochromatic X-rays I sample I 1. Measure I and I as a function of E X. Calculate: m t = ln [I /I ] Page 6 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES The Absorption Coefficient m m/r [barns/atom] m depends strongly on X-ray energy E and atomic number Z, the density r and atomic mass A m» r Z 4 A E 3 m has sudden jumps (absorption edges) which occur at energies characteristic of the element. Page 7 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES 1 6 1 4 Germanium total absorption coefficient m [cm -1 ] 1 1 photoelectric absorption 1-1 -4 1 1 3 1 4 1 5 E(eV) elastic scattering inelastic scattering Photoelectric absorption dominates the absorption coefficient in this energy range Page 8 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Photoelectric Absorption X-rays (light with wavelength.6 l 1 Å or energy 1 E kev) are absorbed by all matter through the photoelectric effect An X-ray is absorbed by an atom when the energy of the X-ray is transferred to a core-level electron (K, L, or M shell) which is ejected from the atom. The atom is left in an excited state with an empty electronic level (a core hole). Any excess energy from the X-ray is given to the ejected photoelectron. Page 9 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES De-excitation: Fluorescence and Auger Effect When X-rays are absorbed by the photoelectric effect, the excited core-hole will relax back to a ground state of the atom. A higher level core electron drops into the core hole, and a fluorescent X-ray or Auger electron is emitted. X-ray Fluorescence: An X-ray with energy = the difference of the core-levels is emitted. Auger Effect: An electron is promoted to the continuum from another core-level. X-ray fluorescence and Auger emission occur at discrete energies characteristic of the absorbing atom, and can be used to identify the absorbing atom. Page 1 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES synchrotron source XAS measurements monochromator I F I I sample XAS measures the energy dependence of the X-ray absorption coefficient μ(e) at and above the absorption edge of a selected element. μ(e) can be measured two ways: Transmission: The absorption is measured directly by measuring what is transmitted through the sample: μ (E)t I = I e μ(e) t = ln (I/I ) Fluorescence: The re-filling the deep core hole is detected. Typically the fluorescent X-ray is measured: Page 1 Undergraduate summer school Sakura Pascarelli 15 September 16 μ(e) ~ I F / I

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XAFS X-ray Absorption Fine Structure Page 1 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XAFS What is XAFS? X-ray Absorption Fine Structure: oscillatory variation of the X-ray absorption as a function of photon energy beyond an absorption edge..4 Absorption coefficient m Absorption.. 1.8 1.6 1.4 1. 1..8 As K-edge in InAsP 1 14 18 13 E (ev) E(eV) Proximity of neighboring atoms strongly modulates the absorption coefficient Page 13 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XAFS EXAFS and XANES XAFS is also referred to as X-ray Absorption Spectroscopy (XAS) and is broken into regimes: XANES EXAFS X-ray Absorption Near-Edge Spectroscopy Extended X-ray Absorption Fine-Structure which contain related, but slightly different information about an element s local coordination and chemical state. XANES : transitions to unfilled bound states, nearly bound states, continuum - low energy photoelectrons EXAFS: ~ 5 1 ev from edge, transitions to continuum high energy photoelectrons Page 14 Absorption coefficient m Absorption.4.. 1.8 1.6 1.4 1. 1..8 Undergraduate summer school Sakura Pascarelli 15 September 16 1 14 18 13 E E(eV) Mean Free Path (A) 1 1 1 1 1 1 1 1 Energy of photoelectron (ev)

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XAFS EXAFS and XANES 1. Approximations can be used to interpret EXAFS, that are not valid for XANES. Cannot be analized in the same way 3. Different information content. XANES: local site symmetry, charge state, orbital occupancy EXAFS: local structure (bond distance, number, type of neighbors.) Page 15 Absorption coefficient m Absorption.4.. 1.8 1.6 1.4 1. 1..8 Undergraduate summer school Sakura Pascarelli 15 September 16 1 14 18 13 E E(eV) Mean Free Path (A) 1 1 1 1 1 1 1 1 Energy of photoelectron (ev)

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS EXAFS qualitatively isolated atom condensed matter E kin e - e - E l R E The kinetic energy of the ejected photoelectron E kin is: p h k E kin E - E = = m m = l = p/k Page 16 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Where do the oscillations come from? Due to a quantum effect, the autointerference of photoelectron wave modifies the absorption coefficient value: 1. As E is scanned above E, E kin is varied, and consequently k and l. p h k E kin = E - E = = m m. The outgoing and backscattered parts of the wave interfere either constructively or destructively, depending on the ratio between l and R. 3. It is the interference between outgoing and incoming waves that gives rise to the sinusoidal variation of m(e) l R e - p p frequency ~ distance from neighbors amplitude ~ number and type of neighbors Page 17 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS l = R/3 l = R/3.5 l = R/4 photoelectron energy increases wavelength l decreases The probability of absorption oscillates due to constructive and destructive interference Page 18 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS 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 define the EXAFS as: c ( E) = m ( E) - m ( E) Dm ( E ) We subtract off the smooth bare atom background μ (E), and divide by the edge step Dμ (E ), to give the oscillations normalized to 1 absorption event..4 Absorption m (E).. 1.8 1.6 1.4 1. 1. Dm m (E).8 1 14 18 13 E (ev) E(eV) Page 19 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS EXAFS: c (k) XAFS is an interference effect, and depends on the wave-nature of the photoelectron. It s convenient to think of XAFS in terms of photoelectron wavenumber, k, rather than X-ray energy m k = ( ) E-E h c (k) is often shown weighted by k or k 3 to amplify the oscillations at high-k: Page Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Qualitative picture of local coordination in R space The frequencies contained in the EXAFS signal depend on the distance between the absorbing atom and the neighboring atoms (i.e. the length of the scattering path). A Fourier Transform of the EXAFS signal provides a photoelectron scattering profile as a function of the radial distance from the absorber. Amplitude of FT R 1 R R 3 R(Å) Page 1 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Quantitative structural determination Structural determinations depend on the feasibility of resolving the data into individual waves corresponding to the different types of neighbors (SS) and bonding configurations (MS) around the absorbing atom. absorber As atom Amplitude of FT 8 6 4 1 3 4 5 6 7 8 Page R(A) In -4 1 3 4 5 6 7 8 Undergraduate Undergraduate summer summer school school Sakura Sakura Pascarelli Pascarelli 15 17 September September 15 16 8 6 4 - As As In P As As As In In In As As As P InAs x P 1-x In As

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS c: sum of damped waves c (k) is the sum of contributions c j (k) from backscattered wavelets: ( k) = c ( ) Path expansion c j k implies photoel w/short mean free path true only for EXAFS Each c j (k) can be approximated by a damped sine wave of the type: j c j [ ] ( k) = A ( k) sin j ( k) j j The larger the number of neighbors, the larger the signal N j f j ( k) e -k s k R j + d j ( k) The stronger the scattering amplitude, the larger the signal Damping of 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 Page 3 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Amplitudes ( k) A j ~ N f ( k) shape of the envelope of each wave indicative of nature of backscatterer atom: j j e -k s absorber As atom 1.5 1.5 As In..1 As P InAs x P 1-x -.5-1 -.1-1.5 -.6 4 8 1 16 As -..6 4 8 1 16 As.4. As.4. In -. -. -.4 -.6 Page 4 4 8 1 16 Undergraduate summer school Sakura Pascarelli 15 September 16 -.4 -.6 4 8 1 16 k(a -1 ) k(a -1 )

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Frequencies: Single and Multiple Scattering paths The sum over paths in the EXAFS equation includes many shells of atoms (1 st neighbor, nd neighbor, 3 rd neighbor,... ), but can also include multiple scattering paths, in which the photoelectron scatters from more than one atom before returning to the central atom. SS g (r) f = R 1 MS g (r) f = 4 R 1 R1 RR 3 3 MS g 3 (r) f = R 1 + R + R 3 R 1 R 1 R MS g 3 (r) f = R 1 + R 3 EXAFS can give information on the n-body distribution functions g n (r). Page 5 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Scattering Amplitude and Phase-Shift: f (k) and d (k) The scattering amplitude f (k) and phase-shift d (k) depend on atomic number. The scattering amplitude f (k) peaks at different k values and extends to higher-k for heavier elements. For very heavy elements, there is structure in f (k). The phase shift d(k) shows sharp changes for very heavy elements. These scattering functions can be accurately calculated (i.e. 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. Page 6 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS Calculating f (k) and d (k) These days, we can calculate f (k) and d (k) easily using different software codes. These programs take as input: 1. a list of atomic x,y,z coordinates for a physical structure. a selected central atom The result is a set of files containing the f (k), and d (k) for a particular scattering shell or scattering path for that cluster of atoms. Many analysis programs use these files 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 refine distances and coordination numbers. Page 7 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS XAFS vs Diffraction Methods q Diffraction Methods (X-rays, Neutrons) Crystalline materials with long-range ordering -> 3D picture of atomic coordinates Materials with only short-range order (amorphous solid, liquid, or solution) -> 1D radial distribution function containing interatomic distances due to all atomic pairs in the sample. q XAFS 1D radial distribution function (centered at the absorber) Element selectivity Higher sensitivity to local distortions (i.e. within the unit cell) Charge state sensitivity (XANES) Structural information on the environment of each type of atom: distance, number, kind, static and thermal disorder 3-body correlations Investigation of matter in the solid (crystalline or amorphous), liquid, solution or gaseous state with same degree of accuracy Page 8 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: EXAFS EXAFS: typical applications Element selectivity Local structure sensitivity q Local structure in non-crystalline matter q Local environment of an atomic impurity in a matrix of different atomic species q Study of systems whose local properties differ from the average properties q Detection of very small distortions of local structure Page 9 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES XANES Page 3 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES XANES Analysis: Oxidation State and Coordination Chemistry The XANES of Cr 3+ and Cr 6+ shows a dramatic dependence on oxidation state and coordination chemistry. For ions with partially filled d shells, the p-d hybridization changes dramatically as regular octahedra distort, and is very large for tetrahedral coordination. This gives a dramatic pre-edge peak absorption to a localized electronic state. Page 31 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES Edge Shifts and Pre-edge Peaks in Fe oxides The shift of the edge position can be used to determine the valence state The heights and positions of pre-edge peaks can also be reliably used todetermine Fe 3+ /Fe + ratios (and similar ratios for many cations). Page 3 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES XANES Analysis: Oxidation State The Normalized XANES from several Fe compounds: XANES can be used simply as a fingerprint of phases and oxidation state. XANES Analysis can be as simple as making linear combinations of known spectra to get compositional fraction of these components. Page 33 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES XANES Interpretation The EXAFS Equation breaks down at low-k, and the mean-free-path goes up. This complicates XANES interpretation: We do not have a simple equation for XANES. XANES can be described qualitatively (and nearly quantitatively ) in terms of coordination chemistry regular, distorted octahedral, tetrahedral,... molecular orbitals p-d orbital hybridization, crystal-field theory,... band-structure the density of available electronic states multiple-scattering multiple bounces of the photoelectron These chemical and physical interpretations are all related, of course: What electronic states can the photoelectron fill? XANES calculations are becoming reasonably accurate and simple. These can help explain what bonding orbitals and/or structural characteristics give rise to certain spectral features. Quantitative XANES analysis using first-principles calculations are still rare, but becoming possible... Page 34 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: XANES XANES: Conclusions XANES is a much larger signal than EXAFS XANES can be done at lower concentrations, and less-than-perfect sample conditions. XANES is easier to crudely interpret than EXAFS For many systems, the XANES analysis based on linear combinations of known spectra from model compounds is sufficient. XANES is harder to fully interpret than EXAFS The exact physical and chemical interpretation of all spectral features is still difficult to do accurately, precisely, and reliably. This situation is improving, so stay tuned to the progress in XANES calculations.... Page 35 Undergraduate summer school Sakura Pascarelli 15 September 16

UNDERGRADUATE SUMMER SCHOOL, ESRF, SEPTEMBER 15 Examples of Applications Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France Page 36 Undergraduate summer school Sakura Pascarelli 15 September 16

LATTICE DISTORTIONS AROUND IMPURITIES IN DILUTE ALLOYS systematic study of substitutional impurities in fcc and bcc metals important shifts in first shell bond length detected Ni Ni % Mn in Ni Ni Mn Ni Mn K edge Page 37 Undergraduate summer school Sakura Pascarelli 15 September 16

LATTICE DISTORTIONS AROUND IMPURITIES IN DILUTE ALLOYS XAFS k c Fourier Transform of k c Mn shifts 1 Ni nearest neighbors outwards by:.3 ±.4 Å Fit of 1 st shell (1 % of distance) Page 38 Undergraduate summer school Sakura Pascarelli 15 September 16

LATTICE DISTORTIONS AROUND IMPURITIES IN DILUTE ALLOYS Comparison to band structure calculations Band structure calculations XAS N. Papanikolau et al., Phys. Rev. B 55, 4157 (1997) Displacements d has two contributions: 1. Valence difference between impurity and host - change of charge density in impurity cell - parabolic dependence d(z). Magnetoelastic contribution Cr, Mn, Fe in Cu majority and minority bands are split - large magnetic moment - low DOS at Fermi level - low binding energy - increased interatomic distance Page 9 Undergraduate summer school Sakura Pascarelli 15 September 16

CHEMISTRY OF XENON AT EXTREME CONDITIONS Understand abundance of Xe in atmospheres of giant planets Deficiency in Earth and Mars atmospheres BM3 Xe [Kr] 4d1 5s 5p6 Rare gases reluctant to form bonds Xe oxides not stable at ambient P, T Xe-O interactions at HP None of the predicted phases observed XRD alone unable to determine oxide structure Normalized Absorption k c(k) (A - ) 1 8 GPa after laser heating before laser heating 345 3465 348 - Energy (ev) 5 1 15 Wavenumber (A -1 ) Xe O 5 P4/ncc space group Dewaele Nature Chemistry 16 c(r) (A -3 ) Xe-Xe Xe-O 1 4 6 Distance (A) Page 4 Undergraduate summer school Sakura Pascarelli 15 September 16

MORE INFORMATION Web links International XAFS Society: http://ixs.iit.edu/ Tutorials and other Training Material: http://xafs.org/tutorials http://gbxafs.iit.edu/training/tutorials.html Software Resources EXAFS: http://xafs.org/software http://leonardo.phys.washington.edu/feff http://gnxas.unicam.it/ Page 41 S. Pascarelli HERCULES 16

MORE INFORMATION Fundamentals of XAFS Books and Review Articles 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 of EXAFS, SEXAFS, and XANES, in Chemical Analysis, D. C. Koningsberger and R. Prins, ed., John Wiley & Sons, 1988 Recent Review XAS XES: X-Ray Absorption and X-Ray Emission Spectroscopy: Theory and Applications Copyright 16 John Wiley & Sons, Ltd Editor(s): Jeroen A. Van Bokhoven, Carlo Lamberti Published Online: JAN 16 9:48PM EST DOI: 1.1/978111884443 FEFF Theoretical approaches to x-ray absorption fine structure J. Rehr et al., Rev. Mod. Phys. 7, 61-654 () GNXAS X-ray absorption spectroscopy and n-body distribution functions in condensed matter (I): theory of the GNXAS data-analysis method A. Filipponi, A. Di Cicco and C. R. Natoli, Phys. Rev. B 5, 151 (1995) MXAN Geometrical fitting of experimental XANES spectra by a full multiple-scattering procedure M. Benfatto and S. Della Longa J. Synchr. Rad. 8, 187 (1) Page 4 S. Pascarelli HERCULES 16

Page 43 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES How to calculate m energy density u carried by X-ray beam is: linear absorption coefficient m measures the energy density reduction due to the interaction with the system of atoms: u eo E eo w A = = 1 du m( w ) = - u dx m ( w) = - e w A du dx m m e w A ( w) = - [ hw n ] h e w A d dx d dx ( w) = - [ n ] ph ph m ( w) = h e w A n f W if n = atomic density W if = transition probability Page 44 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES X-ray Absorption I. Lets consider the interaction between: Monochromatic X-ray beam (w = pn) + monoatomic sample EM field (classic) + atom (quantistic) (semi-classical description) II. m ~ m photoelectric absorption for 1 < E < 5 kev III. Qualitatively, interaction process is: continuum or free energy level 3 3 1 1s electron 1 core hole Y i >, E i Y f >, E f = E i + h w Page 45 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Transition probability: Golden Rule l m(w) depends on: atomic density n transition probability W if of atom from Y i > to Y f > m ( w ) = e h w A n f W if (1) l l time-dependent perturbation theory (power series of EM field - atom interaction potential) The interaction is in general WEAK: can limit series to 1st order: Golden Rule W if p = Y Hˆ Y r () h f I i ( E ) f f Ĥ I Y Hˆ Y Matrix element of H I between initial and final state I i EM field - atom interaction hamiltonian operator r( E f ) Density of final states, compatible with energy conservation: E f = E i + hw Page 46 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES l the interaction hamiltonian for photoelectric absorption (see Appendix 1) is (to 1 st order): e H = ih m r A r r I j j r ( ) j ˆ (3) where r A( r j ) = A ê e r r ik r j l the transition probability for photoelectric absorption of a monochromatic, polarized and collimated photon beam is [(3) into ()]: W if = p he m A Y f j e r r ik j ˆ e r j Y i r ( E ) f (4) Page 47 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES l further simplification: l transition probability in dipole approximation: l alternative and equivalent expression (see Appendix ): l finally one gets [(5) into (1)]: ( ) 1! 1 @ - + = K r r r r r r j j r k i r k r k i e j ( ) f i j j f if E A m e W r e p ˆ Y Y = r h ( ) f i j j f if E r A e W r e w p ˆ Y Y = r h (5) (6) 1 << r j k if r r ( ) = f n e e w p w m ( ) f i j j f E r r e ˆ Y Y r Dipole approximation Page 48 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES ( w ) = l if Y i > and Y f > are known (if wavefunctions and energies can be calculated): 1) calculate W if ) calculate m(w) l in practice, one is interested in inverse process: 1) measure m(w) ) extract EXAFS p e e w n 3) obtain information on local structure through Y f > l but, to obtain structural info, one still needs to calculate Y i > and Y f > or at least be able to express their structural properties in parametric form m Y ˆ e r Y r( E ) f f j r j i f Y i > relatively easy ground state of atom Y f > in general very complicated in principle, all electrons are involved (multi body process, final state strongly influenced by environment) Page 49 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Single electron approximation l large part of m due to elastic transitions: only 1 electron out of N modifies 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 of core hole l remaining part of m due to inelastic transitions: primary excitation of core electron provokes successive excitations of other (external) electrons (shake up, shake off processes) excess energy distributed among all excited electrons where m m ( w ) = m ( w ) m ( w ) el + inel ( ) ( ) N -1 N -1 w Y y ˆ e Y y r r e el f f i i f r Y N -1 i y, Slater determinant of passive electrons wavefunctions, r e f Wavefunction, position vector, final energy of active electron Page 5 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Sudden approximation and overlap factor If photoelectron energy is sufficiently high (E > few 1 ev above edge) time to exit atom << relaxation time of passive electrons its state not influenced by passive electrons relaxation m el ( w) S y e r y r( e ) ˆ r f i f (7) where = Y Y N -1 N -1 S (S ~.7 1.) f i Allows to reduce interpretation of EXAFS to the calculation of the final state of ONLY the photoelectron Page 51 Undergraduate summer school Sakura Pascarelli 15 September 16

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES m( w)» f r y ˆ i e y f dipole operator a = x, y, z X-ray prop direction q = +1,, -1 polarization states (qh photon angular momentum) electron position vector photon polarization vectors linear polarization circular polarization with k // z dipole operator in terms of spherical harmonics Page 5 Undergraduate summer school Sakura Pascarelli 15 September 16 5

FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES m( w)» f r y ˆ e y f i matrix elements factor into spin, radial and angular parts By looking at the non-zero matrix elements we get the dipole selection rules where qh is the X-ray angular momentum Page 53 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 1 = Page 54 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 1 Page 55 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 1 Page 56 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 1 Page 57 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX Page 58 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX Page 59 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX Page 6 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 3 The absorption coefficient m t - di = I ( x) at/cm N dx s a t cm /at I I - t di I ( x) = N s t a t dx x ln I(t) - ln I( ) = - Ns a I(t) I( ) = e = e -Ns a -m t Page 61 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 3 I(t) I( ) = e -Ns -m t a = e m is related to the atomic cross section: m N Na Øcm ø Ø gr ø -1 ( w) = s ( w) = s ( w) r = [ cm ] a in general you find tabulated the mass absorption coefficient m/r: m r for a generic sample P x Q y..: t = a s a A N a A Œ º Øcm Œ º at at ø œ ß œ ß Ø at ø Œº mole œß Ø gr ø Œ º mole œ ß Ø at ø Œº mole œß Ø gr ø Œ º mole œ ß = Œ º cm Øcm Œ º gr 3 ø œ ß œ ß m Ł r ł tot = x m Ł r ł P AP M + y m Ł r ł Q AQ M + K Page 6 Undergraduate summer school Sakura Pascarelli 15 September 16

APPENDIX 3 Recipe for calculating t for transmission XAS I(t) I( ) = e e -Ns -m t a = 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 If absorber is very dilute, and matrix absorbs a lot, then this is not possible fluorescence detection Page 63 Undergraduate summer school Sakura Pascarelli 15 September 16