Back-scattering amplitude

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. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. The phase- and amplitude-corrected Fourier transform becomes ... 

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. 

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/r 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 11.17 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 single element, N is a fractional coordination number, i.e. the product of the real coordination number and the concentration of the element involved. Also, the EXAFS information is an average over the entire sample. As a consequence, meaningful data on supported catalysts are only obtained when the particles have a monodisperse size distribution. 

Fig. 13. Back-scattering amplitudes of Br, Mn, Cl, and N plotted as a function of k. Notice the distinct differences in the functions between the low-Z (N) and the higher Z elements (Cl, Mn, and Br). The high-Z elements have maxima at higher k values. [From V. J. DeRose, Ph.D. Dissertation, University of California, Berkeley, Lawrence Berkeley Laboratory Report, LBL 30077 (1990).]...

The periodicity of this modulation of the absorption will be dependent upon the inteiatomic distance between the absorbing and back-scat tei i ng atoms, R, and the phase shifts, 5. , encountered when the photoelect i on expeiiences the potentials at these centres. Its intensity will be governed by the number of back-scatterers, Nj and their back-scattering amplitudes, F (k). Finally the amplitude is dampened by disorder (thermal and static), a in the interatomic distance and any inelastic piocesses (related to the mean free path of the election, j)- The recognition of the stiuctural information intrinsic to this phenomenon and the derivation of a tractable formula for the estimation of interatomic distances was the result of the work of Sayers, Lytle and... 

Some discontinuities may be identified by a conventional two-dimensional ultrasonic technique, from which the well-known C-scan image is the most popular. The C-scan technique is relatively easy to implement and the results from several NDE studies have been very encouraging [1]. In the case of cylindrical specimens, a circular C-scan image is convenient to show discontinuity information. The circular C-scan image shows the peak amplitude of a back-scattered pulse received in the circular array. The axial scan direction is shown as a function of transducer position in the circular array. The circular C-scan image serves also as an initial step for choosing circular B-scan profiles. The latter provides a mapping between distance to the discontinuity and transducer position in the circular array. 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

With the f-matrix and the Green function we can represent any multiple-scattering path explicitly. For instance, scattering by atom 1 to atom 2, then back to atom 1 and on to final scattering by atom 3 yields a scattering amplitude of... 

This unconventional nonadiabatic protection from decoherence is attributed to the ability of the periodically alternating detuning from the PBG edge to augment the interference of the emitted and back-scattered photon amplitudes, thereby increasing the probability amplitude of the stable state. This may pave the way to new methods of controlling decay and decoherence in spectrally structured continua [Viola 1999 Facchi 2001 (c) Wu 2002 Zanardi 2003],... 

Subscript 1 refers to M-N parameters and subscript 2 refers to M-S or M-Cl parameters. The functions Fj(kj) and 0j(kj) used in this study are the theoretically calculated amplitude and phase functions for the j back-scatterer (4H. Four parameters are least-squares refined for each term the scale factor, B the Debye-Waller thermal parameter, a the interatomic distance, r and the energy threshold, AEq. 

---