K. Klementiev – Introduction to XAFS theory 1 of 11 Introduction to X-ray Absorption Spectroscopy: Introduction to XAFS Theory K. Klementiev, Alba synchrotron - CELLS

K. Klementiev – Introduction to XAFS theory 2 of 11 Introduction In the present lecture: Channels of interaction between x-rays and matter Discussion on Fermi’s Golden Rule General steps in derivation of EXAFS formula, early formulation Modern formulation Qualitative picture of XANES

K. Klementiev – Introduction to XAFS theory 3 of 11 Interaction cross sections Two principal channels: absorption and scattering. The cross sections are Z- and energy- dependent. Photoelectric process is the most probable in the synchrotron energy range (the range of the ALBA-CLÆSS beamline is marked by green). Electron-positron pair production and photonuclear absorption occur at E >1MeV (not shown). The shown cross-sections are for a single atom. The collective effects, like Bragg peaks, can be more intense. data from physics.nist.gov/PhysRefData/Xcom/Text/XCOM.html

K. Klementiev – Introduction to XAFS theory 4 of 11 Fermi’s Golden Rule Fermi’s Golden Rule in one-electron approximation:  i  is an initial deep core state (e.g. | 1s  = 2Z 3/2 e -Zr /√ 4  ): strongly localized.  f  is an unoccupied state in the presence of a core hole [ a collective response of the other electrons which is effectively described as a single particle of a positive charge called 'hole' ], H int is the electron transition operator: H int = p·A(r ) ; The photon is taken to be a classical wave: A(r) = eA 0 e ik·r : For deep-core excitations e ik·r ≈ 1 (dipole approximation) because r is small due to the strong localization of the initial state The next term +ik·r (quadrupole approximation) is ~ Z/(2·137) times weaker, and for heavy elements like Pb, Au, Pt is not negligible (but anyway is normally neglected) equivalent representations: momentum form p·e and position form (ħ  /m)r·e. For example, consider a photon propagating along z with e||x and its K-absorption: Then r·e = x = r sin  cos   Y 1 ±1 (  ) and  ( E )  |   f  Y 1 ±1 Y 0 0  d  . Hence for K absorption the final states  f  can only be of Y 1 ±1 (i.e. p) symmetry (in general,  l =±1 ).

K. Klementiev – Introduction to XAFS theory 5 of 11 Fermi’s Golden Rule. Summary Photo-electric absorption is the main process in the x-ray range of photon energies (apart from coherent effects). XAS is element specific because the photon energy is tuned to a specific absorption edge. All elements can be selected, there are no spectroscopically silent ones. Due to the selection rules the empty states can be selected (via selecting the absorption edge). A common error: angular momenta are always about the origin! A p orbital on a neighbor is not a p orbital with respect to the central atom! XAS is sensitive to the filling of the final state bands (because we sum over the final states) and thus sensitive to valence. The scalar product r·e means sensitivity to anisotropy in respect to photon polarization. Oriented samples provide more information on low symmetry sites.

K. Klementiev – Introduction to XAFS theory 6 of 11 Simple derivation of EXAFS General steps in wave function approach (early EXAFS history): The final states are perturbed by neighboring atoms:  f  =  f 0 +  f , where  f 0  is a pure atomic state and is found as spherical Hankel function with a phase shift found from its asymptotic behavior at infinity.  is now factored:  0 (1+  ), where and    f |( r·e )| i  The scattering part  f  is found by expansion in spherical harmonics about the origin with retaining the proper symmetry (e.g p for K-absorption). Different derivations find  f  differently. Assumptions: The central atom is accounted for by a phase shift, i.e. at the neighboring atoms its potential is neglected. The criterion is k >> Z/a 0. ( E >> 50 or even 100 eV ) Spherically symmetric central potential. Will not work in asymmetric environment. The response of the environment to the absorption (core-hole potential) is weak.

K. Klementiev – Introduction to XAFS theory 7 of 11 Early EXAFS expression For each coordination shell j: r j, N j,  2 j are the structural parameters ( distance, coord. number and distance variance ),  ( k ) is the phase shift due to (only!) the central atom, f(k) is the global scattering factor. Note: There was no shell-attributed phase shift and the amplitude was global. Nevertheless, first Fourier analysis was successfully applied to invert the EXAFS equation: Stern, Sayers, Lytle, Phys.Rev.Lett (beginning of modern EXAFS history). Finally, in the photoelectron momentum space, k = [2(E–E F )] ½, the  function was parameterized as [Stern, Sayers, Lytle]:

K. Klementiev – Introduction to XAFS theory 8 of 11 Modern EXAFS expression, FEFF Rewrite golden rule squared matrix element in terms of real-space Green’s function and scattering operators [Ankudinov et al. PRB 58, 7565]: Expand GF in terms of multiple scattering from distinct atoms Initial atomic potentials generated by integration of Dirac equation; modified atomic potentials generated by overlapping (optional self-consistent field) Complex exchange correlation potential computed (gives mean free path) Scattering from atomic potentials described through k -dependent partial wave phase shifts for different angular momenta l Radial wave function obtained by integration to calculate  0 Unimportant scattering paths are filtered out f eff for each path calculated and finally for the complex photoelectron momentum p :

K. Klementiev – Introduction to XAFS theory 9 of 11 Modern EXAFS expression, FEFF There are still limitations in the modern (FEFF) theory: muffin-tin approximation is coarse. In the near-edge regime, the intra-atomic excitations (shake-up, shake-off, resonances) do not have quantitative description. Many-body effects are not quantitatively understood. Self-energy is based on a simplistic electron-gas model (inaccurate mean free path) See Rehr&Albers, Rev.Mod.Phys for a big review For each path  : R , N ,  2  are the structural parameters (path half-length, coord. number and distance variance), f eff is the effective scattering amplitude (is complex, thus also includes the phase), S 0 2 accounts for many-electron excitations. Possible scattering paths: single scattering multiple (3-leg) scattering

K. Klementiev – Introduction to XAFS theory 10 of 11 2) “shake-up” and “shake-off” Qualitative Picture of XANES (difficulties) E x-ray E photoelectron K 1s L 2 2p 1/2 L 3 2p 3/2 L 1 2s M1–M5M1–M5 continuum EFEF a) pre-edge peak b) white line 3) “shake-down” EFEF 1) resonances shifted downwards due to core-hole

K. Klementiev – Introduction to XAFS theory 11 of 11 Conclusions The fact that already the old derivation worked acceptably well tells us that when first applied to a reference material and tuned, EXAFS can give reliable results. Remember about references: this is an important logical step even when using modern theory (yes, there are still some manually tweaked inputs) The modern EXAFS calculations (e.g. with FEFF) are quite reliable. The amplitude factors still have some uncertainty due to inaccurate self-energy, simplified many- body effects and weak energy dependence of the neglected quadrupolar contribution. The phases are more reliable. There are more and more XANES calculations appearing showing success also in the near-edge region. To my point of view, quantitative agreement is still exception rather than a rule. I haven’t seen a single example where a good agreement was got after a fully automatic calculation, without good manipulation in inputs in these ‘ab-initio’ codes. The qualitative picture of XANES is well understood. XANES is mostly used for fingerprint analysis (symmetric-asymmetric, oxidized-reduced) and for analysis of mixtures.