Extend X-ray absorption Fine Structure

Extended x-ray absorption fine structure (EXAFS) refers to the oscillatory variation of the x-ray absorption as a function of photon energy beyond ∼40 eV of an absorption edge. Although the extended fine structure has been known for a long time (Kronig, 1931, 1932) [4, 5], its structural content was not fully recognized until the work of Stern, Lytle, and Sayers [6, 7] (1974, 1975). In addition, the

availability of synchrotron radiation has resulted in the establishment of EXAFS as a practical structure tool particularly through the work of Eisenberger [8] (1975) and Kincaid [9, 10, 11]. This technique is especially valuable for structure analysis of chemical or biological systems where conventional diffraction methods are not applicable.

EXAFS spectroscopy refers to the measurement of the x-ray absorption coef-ficient µ as a function of photon energy E above the threshold of an absorption edge. Fig. 2.2 shows schematically one edge of an absorber. In a fluorescence mode experiment, µ and µx (x is the sample thickness) is calculated by

µ_x = ln(If/I0), (2.12)

where If and I0 are intensities of the fluorescence and incident beams, respectively.

EXAFS spectrum generally refers to the region 40-1000 eV above the absorp-tion edge. From an EXAFS spectrum the type of neighbors, the coordinaabsorp-tion number, the distance between the central and the neighboring atoms, and the Debye-Waller factors (DW) can be deduced by data analysis. Near or below the edge, there generally appear absorption peaks due to excitation of core electrons to some bound states (1s to nd, (n+1)s, or (n+1)p orbital for K edge, and 2s for LI edge , 2p for LII, LIII edges to the same set of vacant orbital, etc)[12].

This pre-edge region contains valuable bonding information such as the energy of virtual orbital, the electronic configuration, and the site symmetry. The edge po-sition also contains information about the charge on the absorber. In between the

NEXAFS

I_f

I0

Edge

E0

EXAFS Pre-edge

E

Figure 2.2: Schematic representation of the fluorescence yield mode and the re-sulting x-ray absorption spectrum µ_x vs E.

pre-edge and the EXAFS regions is the near x-ray absorption fine structure (NEX-AFS) which we have discusses in the previous section arises from effects such as many-body interactions, multiple scattering, distortion of the excited state wave-function by the Coulomb field, band structures, etc.

EXAFS is a final state interference effect involving scattering of the outgoing photoelectron from the neighboring atoms. From a qualitative viewpoint, the probability that an x-ray photon will be absorbed by a core electron depends on both the initial and the final states of the electron. The initial state is the localized core level corresponding to the absorption edge. The final state is that of the ejected photoelectron which can be represented as an outgoing spherical wave originating from the x-ray absorbing atom. If the absorbing atom has a neighboring atom, the outgoing photoelectron wave (solid line in Fig. 2.3) will be backscattered by the neighboring atom, thereby producing an incoming electron wave (dashed line).

The final state is then the sum of the outgoing and all the incoming waves, one from each neighboring atom. The coefficient µ of EXAFS is the interference between the outgoing and the incoming waves that give rise to the sinusoidal-like variation.

The x-ray absorption coefficient µ is a function of photon energy E above the threshold of an absorption edge, and can be clearly revealed after the subtraction of the atomic absorption, i.e. the absorption of the central atom in the absence of

Figure 2.3: The Model of interference effect of EXAFS.

neighboring atoms. For reasonably high energy (≥ 60 eV) and modulate thermal or static disorders, the modulation of absorption rate in EXAFS, normalized to the “background” absorption (µ₀) is given by

χ(E) = µ(E)− µ0(E)

µ(E) . (2.13)

In order to relate χ(E) to structural parameters, it is necessary to convert the energy E into the photoelectron wavevector k via Eq. (2.14):

k = r2m

~² (E− E⁰). (2.14)

Here E is the incident photon energy and E0 is the threshold energy of that particular absorption edge. This transformation of χ(E) in E space gives rise to χ(k) in k space where Here Fj(k) is is the backscattering amplitude from each of the Nj neighboring atoms of the jth type with a Debye-Waller factor of σj(k) to account for thermal vibration (assuming harmonic vibration) and static disorder (assuming Gaussian pair distribution) and at a distance rj away. ϕ_ij(k) is the total phase shift, which contains contributions from both the absorb and the backscatter and affects the origins of the sine wave and its frequency. The term exp(−2r^j/λj(k))is due to in-elastic losses in the scattering process (due to neighboring atoms and the medium

in between) with λj(k) being the electron mean free path. Si(k) is an amplitude reduction factor due to many-body effects such as shake-up/shake-off processes at the central atom (denoted by i). The term 1/kr_j² presents that the outgoing or incoming wave is a spherical surface wave. It is clear that each EXAFS wave is determined by the backscattering amplitude (NjFj(k)) , modified by the reduc-tion factors Si(k), exp(−2σ²j(k)k²), and exp(−2r^j/λj(k)), and the 1/kr²_j distance dependence, and the sinusoidal oscillation which is a function of interatomic dis-tances (2krj) and the phase shift (ϕ_ij(k)). So the sinusoidal oscillation of EXAFS is caused by interference sin(2kr) term with a frequency 2r in k space.

EXAFS can be applied in both crystalline and amorphous materials and it can be also applied to detect local lattice distortions. The common detection schemes in the soft x-ray range are the total electron yield (TEY) and fluorescence yield (FLY) modes. The TEY mode is surface sensitive and meticulous surface preparation and cleaning are required in order to avoid contributions from surface contaminants. The FLY mode is bulk sensitive and is thus better suited when the bulk material properties are sought. The fluorescence yield mode can suffer from self-absorption effects that affect mostly the amplitude of the EXAFS spectrum and thus the calculated coordination numbers and the Debye-Waller factors.