EXAFS amplitude term

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. 

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. 

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... 

The second term in (6-9) expresses that nearest and next nearest neighbors dominate scattering contributions to the EXAFS signal, while contributions from distant shells are weak. The dependence of the amplitude on 1/r2 reflects that the outgoing electron is a spherical wave, the intensity of which decreases with the distance squared. The term exp(-2r/X) represents the exponential attenuation of the electron when it travels through the solid, as in the electron spectroscopies of Chapter 3. The factor 2 is there because the electron has to make a round trip between the emitting and the scattering atom in order to cause interference. 

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. 

N,Sq is a term that modifies the amplitude of the EXAFS signal. The subscript i denotes that this value can be different for each path of the photoelectron. For a single scattering path, N, represents the number of coordinating atoms within a particular shell (at the same radial distance from the absorber). For multiple scattering, N, represents the number of identical paths. The passive electron reduction factor, Sg, usually has a value between 0.7 and 1.0. It accounts for the slight relaxation of the remaining electrons in the presence of the core hole vacated by the photoelectron. 

A straightforward Fourier transform of the EXAFS signal does not yield the true radial distribution function. First, the phase shift causes each coordination shell to peak at the incorrect distance. Second, due to the element specific back-scattering amplitude, the intensity may not be correct. Third, coordination numbers of distant shells will be too low mainly because of the term 1/r2 in the amplitude (10.10) and also because of the small inelastic mean free path of the photoelectron. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. Figure 10.17 illustrates the effect of phase and amplitude correction on the EXAFS of a Rh foil [38]. Note that unless the sample is that of a... 

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. 

Because the R term, and the mean-free path of backscattered photoelectrons is small (usually < 25 A), typically the total number of shells rarely exceeds 7. The backscattering amplitude, Fj(k), and phase shift, dj(k), for the absorber-neighbor pair may either be extracted from the EXAFS of reference materials or calculated theoretically using widely available codes such as the FEFF developed by John Rehr s group at the University of Washington. ... 

The advantage of EXAFS is that it can often be employed where other structure probes cannot. The standard EXAFS equation has a term for the loss of amplitude with increasing scattering vector. It has the form... 

Because of the similarity rn the amplitude functions of platinum and iridium, we do not separate the backscattering contributions of platinum and iridium atoms in the analysis of EXAFS data on platinum-iridium clusters or alloys. In our quantitative treatment of EXAFS arising from nearest neighbor atoms of platinum and iridium, the EXAFS function of Eq. 4.3 consists of only one term, as will be seen in the following discussion. 

This fact explains the results in Fig. 10 where X3(E) has as large an amplitude up to high energies as the EXAFS term x lE). Also, the attenuation of the higher terms X4.(E) and Xs(E) with increasing kinetic energy is much less than that observed for tetrahedral clusters. 

In Eq. (1), k is the photoelectron wave vector relative to Eq (k = 0) N is the the number of neighboring atoms of the same kind at a distance r., of is the mean-square relative displacement (MSRD) of the absorber-scatterer atom pair from their equilibrium inter-atomic distance or in molecular spectroscopy terminology, the mean-square amplitude of vibration other terms have their usual meaning Using standard Fourier transform and curve fitting procedures, we can derive the coordination number, bond length and local dynamics (MSRD) from EXAFS. 

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