An introduction to X-ray Absorption Fine Structure Spectroscopy Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France sakura@esrf.fr
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

口X-ray Absorption Spectroscopy口X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description □ EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
2
□ X-ray Absorption Spectroscopy口X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description □ EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

□ Two fundamental X-ray-matter interactions: ●  photoelectric absorption ●  scattering (elastic, inelastic) □ Two families of experimental techniques: ❏ spectroscopy → electronic structure, local structure of matter ● absorption ● emission ● inelastic scattering ■  elastic diffusion →  microscopic geometric structure ● diffraction (crystalline solids) ● scattering (amorphous solids, liquids)
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

I = Io exp[-μ†] linear absorption coefficient μ† =  In [ I 0 / I ]
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

synchrotron source monochromator incident flux monitor transmitted flux monitor Measure I o and I as a function of E x 1. μ † = In [Io/I] 2. Calculate:
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

depends strongly on X-ray energy E and atomic number μ Z, and on the density ρ and atomic mass A ll ~ A E 3 has sudden jumps (absorption edges) which occur at energies μ characteristic of the element. ρΖ 4 ,
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
7 u [cm-1] E(eV) total absorption coefficient photoelectric absorption elastic scattering inelastic scattering Photoelectric absorption dominates the absorption coefficient in this energy range
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

X-rays (light with wavelength 0.06 < 200 < E < 200 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.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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.
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 in several ways: Transmission: The absorption is measured directly by measuring what is transmitted through the sample: I = Lo e -v (E)t µ(E) t = − In (I/Io) Fluorescence: The re-filling the deep core hole is detected. Typically the fluorescent x-ray is measured. L / Lo ? S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan) 11 Outline □ X-ray Absorption Spectroscopy □ X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description □ EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

X-ray Absorption Fine Structure: oscillatory variation of the X-ray absorption as a function of photon energy beyond an absorption edge. Proximity of neighboring atoms strongly modulates the absorption coefficient S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan) 13
XAFS is often broken  into 2 regimes: XANES  X-ray Absorption Near-Edge Spectroscopy EXAFS  Extended X-ray Absorption Fine-Structure which contain related, but slightly different information about an element's local coordination and chemical state. S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
XANES: transitions to unfilled bound states, nearly bound states, continuum → local site symmetry, charge state, orbital occupancy EXAFS: 50 - 1000 eV after edge due to transitions to continuum → local structure (bond distance, number, type of neighbors….) S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
isolated atom condensed matter The kinetic energy of the ejected photoelectron Ekin is: Ekin = E-E o =
λ = 2 π/k
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

Due to a quantistic effect, the autointerference of photoelectron wave modifies the absorption coefficient value: 1. As E is scanned above Eo, Ekin is varied, and consequently k and λ. 2. The outgoing and backscattered parts of the wave interfere either constructively or destructively, depending on the ratio between λ and R. 3. It is the interference between outgoing and incoming waves that gives rise to the sinusoidal variation of μ(Ε)口 frequency ~ distance from neighbors ■ amplitude ~ number and type  of neighbors
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
17

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: We subtract off the smooth " bare atom" background  μο(E), and divide by  the "edge step" △μο(Eo), to give the oscillations  normalized to 1 absorption  event.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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 χ (k) is often shown weighted by k² or k³ to amplify the oscillations at high-k:
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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.
χ (k) is the sum of contributions χ i (k) from backscattered wavelets:
Each χ i (k) can be approximated by a damped sine wave of the type: χ (k) = A (k)sin | φ (k) | The larger the number the signal 2kR + δ (k) The stronger the scattering amplitude, the larger the signal Damping of the amplitude at large k, due to static Each shell contributes a sinusoidal signal which oscillates more rapidly the larger the distance 22 S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
The sum over paths in the EXAFS equation includes many shells of atoms (1st neighbor, 2nd  neighbor, 3rd  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  ⇒ f = 2 R1 MS  ⇒ f = 4 R 1 MSニュー   g3(r) f = R 1 + R 2 + R 3 MS ⇒ f = 2R1+ 2R3 EXAFS can give information on the n-body distribution functions g,(r).
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

口 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 RDF containing interatomic distances due to all atomic pairs in the sample.
口XAFS
口 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. 25
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

Element selectivity Local structure sensitivity口 Local structure in non-crystalline matter □ Local environment of an atomic impurity in a matrix of different atomic species □ Study of systems whose local properties differ from the average properties ❏ Detection of very small distortions of local structure
□ X-ray Absorption Spectroscopy □ X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description口EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

h ωミ

S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

●   For 1-electron transitions:

S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

photoelectron μ(ω) ≈ ∑|(ψ i | ˆ ε· r |ψ f | Approx: dipole + single electron + sudden r: electron position ε : photon polarization |¥¡> relatively easy   →  ground state of atom; i.e. 1s e- wavefunction |Ψƒ› very complicated →  final state strongly influenced by environment
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
31
photoelectron free to travel away undisturbed
outgoing spherical wave originating from the absorbing atom
→ overlap integral of initial and final state wavefunctions: monotonically decreases as function of E
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
32

sum of the outgoing and all the incoming waves, one per each neighboring atom.
| d ¯ | ψ j ( ¯ ) ê · ¯ ψ j ( ¯ ) | ² + 2 Re [ d ¯ ¯ [ψ j ( ¯ j ) ê · ¯ ψ j ( ¯ j [ ψ j (
EXAFS +
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

μ=μ 0 | 1+ x | χ  : fractional change in μ introduced by the neighbors 2 Re
v i (r) ê · r v i * (r) |  | v i * (r) ê · r  δv i (r)
d ¯ (1) Interference between outgoing wavefunction and  backscattered wavelets Dominant contribution to integral comes from spatial region close to absorber atom nucleus, where the core orbital wavefunction ψ i ≠ 0. The region where ψ i ≠ 0 represents simultaneously the source and the detector for the photoelectron that probes the local structure around the absorber atom
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
34

□ X-ray Absorption Spectroscopy口X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description口EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
35

To model the EXAFS, we use the EXAFS Equation χ(k)=- S 2 Σ i N i
p-2k²σ j² sin [2kR」+ δ i (k)] l where i (K) and δ(K) are photoelectron scattering properties  of the neighboring atom. (The sum is over "shells" of similar neighboring atoms). If we know these properties, we can determine: R distance to neighboring atom. N coordination number of neighboring atom. σ² mean-square disorder of neighbor distance. The scattering amplitude f(K) and phase-shift δ(K) depend on atomic number Z  of the scattering atom, so we can also determine the species of the neighboring atom.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
36

With spherical wave for the propagating photoelectron: and a scattering atom at a distance r = R, we get: eikr kr Region III: amplitude of backscattering on B Region I: Region II: amplitude amplitude of Region II: amplitude of ingoing wave, backscattered from B of outgoing wave wave arriving on B where the neighboring atom gives the amplitude | f(K) | and phase-shift δ (K) the scattered photoelectron. Substituting into equation (1) and after some math we get: sin [2kR+ δ(k)] δ(k)=δ(k) for 1 scattering atom.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

For N scattering atoms, and with a thermal and static Gaussian disorder of σ², giving the mean square disorder in R*,  we have
A real system will have neighboring atoms at different distances and of different types. We add all these contributions to get a version of the EXAFS equation:
◄ eikr To obtain this formula we used a spherical wave for the photoelectron: kr * EXAFS takes place on a time scale much shorter than that of atomic motion, so the measurement serves as an instantaneous snapshot of the atomic configuration S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
38

But the photoelectron can also scatter inelastically*, and may not be able to get back the absorbing atom.   Also: The core-hole has a finite lifetime**, limiting how far the photoelectron can go. ikr -r/ λ(k) Using a damped wave-function: where λ(k) is the photo electron's mean free path (including core hole lifetime), the EXAFS equation becomes:
The mean free path λ depends on k. For the EXAFS k range, λ < 25 Å. The λ and R-² terms make EXAFS  a local atomic prob * Electrons that have suffered inelastic losses will not have the proper wave vector to contribute to the interference process. ** the photoelectron and core hole exist simultaneously
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

Another important Amplitude Reduction Term is due to the relaxation of all the other electrons in the absorbing atom to the hole in the core level: where Ψ i N-1 accounts for the relaxation of the other N-1 electrons relative to these electrons in the unexcited atom: Ψ o N-1 . Typically S o² is taken as a constant: 0.7 < So² < 1.0 which is found for a given central atom, and simply multiplies the XAFS χ. Note that So² is completely correlated with N. This, and other experimental and theoretical issues, make EXAFS amplitudes (and therefore N) less precise than EXAFS phases (and therefore R). Usually So² is found from a "standard" (data from a sample with well-known structure) and applied to a set of unknowns as a scale factor.
40
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

The scattering amplitude f (k) and phase-shift δ (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 δ(k) shows sharp changes for very heavy elements. These scattering functions 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 2014  (Japan)

These days, we can calculate f (k) and δ (k) easily using different software codes These programs take as input: 1. a list of atomic x,y,z coordinates for a physical structure 2. a selected central atom. The result is a set of files containing the f (k),  and δ (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. Outline □ X-ray Absorption Spectroscopy口X-ray Absorption Fine Structure口EXAFS: Simple Theoretical Description □ EXAFS: Derivation of EXAFS Equation □ XANES: Qualitative Interpretation
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

XANES: transitions to unfilled bound states, nearly bound states, continuum → local site symmetry, charge state, orbital occupancy EXAFS: 50 - 1000 eV after edge due to transitions to continuum → local structure (bond distance, number, type of neighbors….) S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
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 molecular orbitals band-structure multiple-scattering regular, distorted octahedral, tetrahedral, . . . p-d orbital hybridization, crystal-field theory, . . . the density of available electronic states 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... 45
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

Mn: [Ar] 3d5 4s² Excited States
Mn: [Ar] 3d5 4s²
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 sometimes used to determine Fe³+/Fe²+ ratios.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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.
49
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

On centrosymm only weak quadrupole channel Ty non-centrosymm p-d mixing dipole channel The XANES of Cr³+ and Cr°+ 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.
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

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 . . . .
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
More information: web links International XAFS Society: http://ixs.iit.edu/ Tutorials and other Training Material:  http://xafs.org/Tutorials Software Resources EXAFS: http://xafs.org/Software http://leonardo.phys.washington.edu/feff http://gnxas.unicam.it/
Fundamentals of  XAFS Introduction to XAFS: A Practical Guide to X-ray Absorption Fine Structure Spectroscopy, G. Bunker, Cambridge University Press, 2012 X-ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS, and XANES, in Chemical Analysis 92 D. C. Koningsberger and R. Prins, ed., John Wiley & Sons, 1988. Basic Principles and Applications of EXAFS, Chapter 10 in Handbook of Synchrotron Radiation, pp 995-1014. E. A. Stern and S. M. Heald, E. E. Koch, ed., North-Holland, 1983
FEFF
Theoretical approaches to x-ray absorption fine structure J. Rehr et al., Rev. Mod.  Phys. 72, 621 - 654 (2000)
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 52, 15122  (1995)
MXAN
Geometrical fitting of experimental XANES spectra by a full multiple-scattering procedure M.Benfatto and S. Della Longa J. Synchr. Rad. 8 , 1087 (2001)
I(†) e-No。 = e-u t I(0)
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

: µ is related to the atomic cross section:
in general you find tabulated the mass absorption coefficient μ/ρ:
for a generic sample P x Qy.....:

S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
55
I(†) e –No I(0) e-μ † 1. Total absorption above the edge must not be too high: I / Io ~ 0.14  ⇒ 0.007 U above edge t = 2 ⇒ 5 ideally || above edge † =  2-3 2i Contrast at edge must be as large as possible: [ ↓ above edge - ↓ below edge ] ↑ > 0.1 ideally μ above edge - μ below edge ] † =   1 If  absorber is very dilute, and matrix absorbs a lot, then this is not possible → fluorescence detection 56
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

energy density u carried by X-ray beam is: linear absorption coefficient μ measures the energy density reduction due to the interaction with the system of atoms:
U (W
ε o E 2 ε, ω²Α² 2 2 1 du μ(ω u dx S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan) 57
Lets consider the interaction between: monoatomic sample Monochromatic X-ray beam (ω = 2πν) EM field (classic) atom (quantistic) + (semi-classical description) || ~ || photoelectric absorption for_1 < E < 50 keV Qualitatively, interaction process is: continuum or 1s electron
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

●  μ(φ) depends on: - atomic density n - transition probability W۽of atom from | Ψ̣> to | Ψ̣> 2h (1) ●  to calculate W۽: time-dependent perturbation theory based on power series of EM field - atom interaction potential. ● The interaction is in general WEAK 2π | Ψ i | Ĥ i | Ψ f ⟩|² ρ ( E f (2) ● Can limit series to 1st order: Golden Rule Â , EM field - atom interaction hamiltonian operator Matrix element of H t between initial and final state PlE f Density of final states, compatible with energy conservation:ﾋｨ= Ei + h ω ● the interaction hamiltonian for photoelectric absorption (see Appendix 1) is (to 1st order):
●  the transition probability for photoelectric absorption of a
monochromatic, polarized and collimated photon beam is [(3) into (2)]:

60
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

●  further simplification:
o transition probability in dipole approximation:
o alternative and equivalent expression : Wif
o finally one gets [(5) into (1)]:
(6)
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
61
●  if |Ψ i >  and |Ψ i >  are known (if wavefunctions and energies can be calculated): 1) calculate W۽2) calculate μ(φ) ● in practice, one is interested in inverse process: 1) measure μ(φ) 2) extract  EXAFS 3) obtain information on local structure through |Ψ۽> ● but, to obtain structural info, one still needs to calculate |Ψ; >  and |Ψ۽> or at least be able to express their structural properties in parametric form -   |Ψ; > relatively easy - ground state of atom -   |Ψ۽> in general very complicated - in principle, all electrons are involved -> multi body process - final state strongly influenced by environment S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)
62

●  large part of μ 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 O  remaining part of μ 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 μ(ω) =    μel(ω)  + where ¹ ψ, | ˆ ˆ ˆ ˆ ˆ ˆ |Ψ Slater determinant of  "passive" electrons' wavefunctions W , r , E f Wavefunction, position vector, final energy of "active" electron
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

-  if photoelectron energy is sufficiently high (E > few 10 eV above edge) □ time to exit atom  <<  relaxation time of passive electrons □ its state not influenced by passive electrons relaxation
(7) where 2
- Allows to reduce interpretation of EXAFS to the calculation of the final state of ONLY the photoelectron
S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)

μ(ω) 〜 L|\vi|ê·r|v f \ dipole operator X-ray prop direction α = x, y , z q = +1, 0, -1 polarization states (qħ photon angular momentum) electron position vector r = xex + yey + zez photon polarization vectors linear polarization circular polarization with k // z dipole operator in terms of spherical harmonics N Y1,0, S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan) 65
μ(ω) ≈ matrix elements factor into spin, radial and angular parts By looking at the non-zero matrix elements we get the dipole selection rules =l' - l= +1, =m,-m,-g=0,±1, As=s'—s=0, — IN : — 0. Ams = m', where qň is the X-ray angular momentum S. Pascarelli - An Introduction to XAFS - Cheiron School 2014  (Japan)