EXAFS Debye-Waller term

As was the case for XANES, here electronic structure calculations including solvent effects are used to draw conclusions on the most probable coordination isomer and from that structure the interatomic bond distances can be directly compared to the computed EXAFS spectra [150-152]. As EXAFS spectroscopy probes the structure of a statistically averaged system, the most appropriate way of comparing theoretical EXAFS data to experimental ones is to use molecular dynamics trajectories to sample the configuration space, select snapshots and finally compute a statistically average spectrum [89,153] with a direct estimate of the mean square relative disorder (MSRD, also called or the EXAFS Debye-Waller term). However, as discussed in Section 11.2.3, the relevance of MD simulations hinges on how accurate intermolecular interactions are, given that these are usually obtained at DFT level (in Car-Parinello MD simulations) or with force-fields (in classical MD simulations). 

Cd, and others. For example, nickel EXAFS of carbon monoxide dehydrogenase (CODH) from Clostridium thermoaceticum strain DSM 521 were collected and the Fourier filtered data was best fit to sulfur at 0.216 nm. A reasonable fit to oxygen/nitrogen could only be obtained by setting the zIEq to 40 and the Debye-Waller term to 5xl0 nm. A reasonable Ni-S fit was also obtained by assuming two Ni-S bond lengths at 0.222 and 0.211 nm, respectively. Addition of a Ni-M (M=Fe, Ni, Zn) term at 0.325 nm, also improved the... 

Figure 7 (a) Cd K-edge X-ray absorption spectrum of rat liver Cd5Zn2MT-l. (b) Cd K-edge EXAFS (solid line) and a theoretical simulation (dashed line) with 4 Cd-S distances at 2.53 A and a Debye-Waller term, of... 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

The Debve-Waller factor is an amplitude term in any scattering experiment that takes account of the movements of the scatterers about their average positions. This results in attenuation of the scattering which increases with scattering vector. For EXAFS analysis the appropriate Debye-Waller factor takes account of variations in the absorber-scatterer distance, and thus depends on how much the motion of this pair of atoms is correlated. 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

In EXAFS experiments as well as in other EXAFS-like methods, variations in the sample temperature are well described by the Debye-Waller factor and lead to the exponential attenuation of line structure when the sample temperature increases. The temperature dependence of the SEFS spectrum is also described by the Debye-Waller factor— more exactly, by two Debye-Waller factors corresponding to the interference terms of the final and intermediate states [Eq. (38)). Since these interference terms are determined by different wave numbers, p and q, the change of the sample temperature results in a change of the relative intensity of the oscillating terms, which reveals itself in the unusual dependence behavior of SEFS. 

Another important factor is the Debye-Waller factor e °. This accounts for thermal and static disorder effects concerning the movement/position of atoms around their equilibrium/averaged position. A point to stress is that the nature of this term is different to the counterpart term in XRD." Since vibrations increase with temperature, EXAFS spectra are usually acquired at low temperature (below 100 K) in order to maximise information. Spectra at different temperatures may, on the other hand, allow decouple thermal and static contributions to DW. The DW term smears the sharp interference pattern of the sinusoidal term and cuts off EXAFS at sufficiently... 

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... 

---