D. Experiencias y lecciones
D.1 Experiencias y lecciones en la formulación de proyectos
The initial development and application of EXAFS have been impeded by a great deal of confusion about its correct theoretical description. The question arose as to whether a long‐range order theory formulated in terms of Bloch waves [215] or a short‐range order theory in terms of scattering from nearby atoms [216, 220] was more appropriate. With the advent of synchrotron X‐ ray sources in the 1970s, a quantitative comparison between theory and experiment became possible for the first time. Since then it is generally established that the single‐electron short‐ range‐order description is adequate in most cases to explain the theory of EXAFS, although both approaches can be reconciled when appropriate broadening is introduced [213].
Within this picture, EXAFS can be phenomenologically described by the so‐called EXAFS equation. Its derivation can be abundantly found in the literature [68, 204, 206]. Here we prefer to focus on the physical meaning of its various components and how the equation is used to extract structural information from the XAS spectrum. The EXAFS spectrum is defined phenomenologically as the normalized, oscillatory part of the absorption coefficient above a given absorption edge, i.e. Equation 2‐8
where is the smoothly varying atomic‐like background absorption of an "embedded atom" (in the absence of neighboring scatterers but incorporated in the lattice of neighboring potentials). In order to relate to structural parameters, we need to make the transformation from (absolute) E‐space into (relative) k‐space. The kinetic energy of the photoelectron is given by the difference in energy E of the X‐ray photon and the energy E0 necessary to expel the electron from its core level (the IP). By using [Equation 2‐2] and [Equation 2‐3] the electron wave vector is expressed as
2
Equation 2‐9
which can be used to transform into (from now on, the wave vector subscript referring to the photoelectron is omitted). The latter is given by the EXAFS equation, expressed as a scattering path expansion generalized to include MS pathways [213]:
| |
sin 2 2
Equation 2‐10
with the scattering path index, the number of equivalent scattering paths , the halfpath distance and the squared Debye‐Waller (DW) factor . In the case of SS, represents the direct interatomic distance between absorber and scatterer. In addition, | | is the complex backscattering amplitude for path , is the central atom phase shift of the final state, is the energy‐dependent mean free path and is the overall amplitude reduction factor, which was added later to account for many‐body (shake‐up, shake‐off) effects [(§2.1.5)].
In principle, Equation (2‐10) contains all of the key elements that a correct theory must entail and it provides a convenient parameterization that allows fitting the local atomic structure around the absorbing atom to the experimental EXAFS data [(see §2.2.4.1)]. The relation between the oscillatory fine structure and the interatomic distances is clearly reflected by the sin 2 term. The finite lifetime of the excited state consisting of the photoelectron together with the core hole where it came from is captured by the exponential damping term exp( 2 . The strength of the reflected waves depends on the type and number of the neighboring atoms via the backscattered amplitude | |. The phase factors and reflect the quantum‐mechanical nature of the scattering process (the factor of 2 is due to the fact that the photoelectron encoun‐ ters the potential of the central atom twice). These phases account for the difference between the measured and geometric interatomic distances, which are typically a few tenths of an and must be corrected for by using an experimental or theoretical reference [68]. Finally, the damping factor is partially due to thermal effects causing the atoms to jiggle around their equilibrium
atomic positions with a root‐mean‐square deviation of . Thus the contributions of these atoms to the interference will not all be exactly in phase. Effects of static disorder are similar and they give rise to an additional contribution to . The Debye‐Waller damping term is most pronounced for high energies making it essential for EXAFS, but its energy dependence is often negligible in XANES. In the case of a simple diatomic vibration and scattering along the interatomic axis, the squared DW factor can be expressed as [221]
8 coth 2
Equation 2‐11
where is the reduced mass of the vibration, T is the temperature, and is the frequency of the vibration. As increases with T, the fine structure tends to "melt" at high temperatures, con‐ fining the spectrum to lower regions of energy.
2.2
Experimental Methods: Synchrotron radiation experiments
2.2.1 Steady‐state XAS setup
Both methods of detection (transmission and fluorescence) are implemented in the time re‐ solved data acquisition used in our experiments. The detectors used for each mode are described in [(§ 2.2.1.1) and (§2.2.1.2)].
Figure 2‐6: Schematic setup for fluorescence (right) and transmission (left) measurement modes in a X‐ray absorp‐ tion experiment. The incident beam I0 is partially absorbed by the sample of the thickness d. The sample is set at an
angle 45° with respect to the incident beam to increase the fluorescence yield IF , while this geometry has no
influence on the transmission IT but the increase in path length z through the sample.
The difference between the fluorescence and transmission detection mode is visible at first glance in the positioning of the detectors (Figure 2‐6). While the transmission detector is located in the direct X‐ray beam, the total fluorescence is usually collected by a detector at the side of the sample, usually at about 90° to the incident beam.