EXAFS amplitude

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as... 

Whereas there is little that one can do to overcome the effects of static disorder, the effects of thermal vibration can be significantly decreased by performing experiments at low temperatures, and, in fact, many solid samples are typically run at liquid nitrogen temperatures just to minimize such effects. An example of the effect of thermal vibration can be ascertained in Fig. 8 A, where the EXAFS amplitude decreases precipitously due to the large vibrational amplitude of the Cu—O bond. In general, failure to consider the effects of thermal vibration and static disorder can result in large... 

The reason for multiplying with a k weighting factor is to compensate for the decrease of the EXAFS amplitudes at high k values due to the Debye-Waller factor, the backscattering amplitude, and the k 1 dependence of the EXAFS (see, e.g., Ref. (21)). 

Multiple scattering, where the photoelectron wave samples severd scatteiers before returning to the absorbCT, is only important in KAFS for cases where two scatterers and the absc rber are nearly coUinear. In such cases, the EXAFS amplitude of the outer scatterer will be significantly enhanced. See Teo, B.K., /. Am. Chem. Soc., 1981,103,3990-4001. 

One SEXAFS specific feature is the polarisation dependence of the amplitude. This derives from the high anisotropy of the surface and of ultrathin interfaces, that we may consider as quasi two dimensional systems. The relative orientation of the X-ray electric vector with respect to the surface (interface) normal does represent a preferential excitation for those atom pairs aligned along the electric vector e.g. with the electric vector perpendicular to the surface (interface) plane the EXAFS amplitude will be maximum for the atom pairs aligned normal, or almost normal to the surface (interface). The electric vector can be also aligned, within the surface plane, along different crystallographic directions. 

Figure 21. Illustration of the thickness effect in X-ray absorption spectroscopy. The actual absorbance vs. energy is shown at the bottom. Due to these thickness effects, the measured signal (right) is related to the actual absorbance via a sub-linear transfer curve (Saturation). Two specific points along the curves are picked out with dotted lines and arrows, showing how the pre-edge features are raised relative to the edge. Notice also that the EXAFS amplitude in the Measured curve is reduced compared to its actual value. The Actual curve is transmission data for a Ti foil and the Measured curve is the fluorescence data for the same sample (6 pm, 45° incidence and exit angle). The Saturation curve comes from a fit between the Actual and Measured curves.

Figure 22. Micro-XANES data of a ferromanganese crust (Hlawatsch et al. 2001) recorded in transmission mode in spots of the same composition but differing thickness. The pre-edge is enhanced and the EXAFS amplitude is reduced by hole effect.

Figure 23 shows a contour map of St as a function of / and jurt. For a thick sample, even a small fraction of beam skimming by can affect the EXAFS amplitude. Similarly, if there are harmonics in the beam, they go through the sample with less absorption than the fundamental, leading to the same effect as if there were a hole (Stern and Kim 1981). While the sample may not have holes in it, it is common to encounter particles smaller than the beam. Suppose one is dealing with a primary mineral such as magnetite ( 04),... 

Figure 23. Reduction of the EXAFS amplitude as a function of the sample s hole fraction and the absorption edge jump. Percentages correspond to the ratio of the amplitude of the measured EXAFS wiggle to the actual signal.

The term exp [-2 RjlAj k)] describes the attenuation of the EXAFS amplitude caused by inelastic scattering processes experienced by the excited electrons outside of the absorbing atom. The mean free path Aj k) of electrons in condensed matter is rather small, typically between 5 andl 0 A. The k dependence of X is often linear. Experimental Xj k) curves are known for only very few substances. 

All the interatomic distances were fixed at 2.2 A, and used the same (chemically reasonable) cf value of 0.0025 A. The larger number of electrons around the sulfur, compared with an oxygen, is reflected in the approximately two-fold larger Mo-S EXAFS amplitude. The Mo-S and Mo-O EXAFS are approximately 180° out of phase and simple to discriminate while the Mo-O and Mo-N are very similar and discriminating these based on EXAFS alone would be problematic. 

Because quantities of protein are typically of limited availability, metallopro-tein EXAFS data acquisition can be challenging. As we have discussed above, the application of A -weighting in data analysis is a means to approximately equalize the EXAFS amplitude throughout the A -range of the data. In order... 

The EXAFS amplitude falls off as 1 /R. This reflects the decrease in photoelectron amplitude per unit area as one moves further from the photoelectron source (i.e., from the absorbing atom). The main consequence of this damping is that the EXAFS information is limited to atoms in the near vicinity of the absorber. There are three additional damping terms in Equation (2). The 5 q term is introduced to allow for inelastic loss processes and is typically not refined in EXAFS analyses. The first exponential term is a damping factor that arises from the mean free path of the photoelectron (A(k)). This serves to limit further the distance range that can be sampled by EXAFS. The second exponential term is the so-called Debye-Waller factor. This damping reflects the fact that if there is more than one absorber-scatterer distance, each distance will contribute EXAFS oscillations of a... 

This gives A7 >0.13A for data to A max= 12A However, this estimate is generally too optimistic as illustrated by Figure 11, which shows pairs of simulated EXAFS spectra for one shell (dashed lines) and two shells (solid lines).The top trace shows the simulation for a pair of scatterers separated by 0.25 A (7 i = 1.75 A, R2 = 2.00 A). There is an obvious beat in the EXAFS amplitude at which distinguishes these data from the EXAFS for a single shell at the... 

In summary, the EXAFS results show that there is no difference between pure iron and the catalyst in the average local structure. Furthermore, there are no iron atoms present in the catalyst in significant quantities which exhibit a different geometric environment to the average, i.e., there is only one type of iron atom present. This is strong evidence against the paracrystallinity theory and is more consistent with a macroscopic distribution of the promoter phases, which affects fewer iron atoms than can be detected with the X-ray absorption technique. The porosity of the reduced catalyst may account for a certain lack of EXAFS amplitude. Any significant reduction in the coordination number of the iron in the catalyst would require a cluster-like microstructure for which no evidence has yet been found. 

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