Sampling element, phase shift

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 backscattering 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 ... 

For a pure Px tip state, the tunneling matrix element for the sample wavefunction at F is negligible, whereas for sample wavefunctions at K, a phase shift of 90° in the x direction is obtained. The tunneling current is ... 

Ellipsometry grants the independent determination of two results per resolution element without changing the experimental geometry this technique is described in Sec. 6.4.4.2. More often however, the information on the phase shift is abandoned and just the reflectance is determined. Usually it is measured by comparing the signal caused by the reflected intensity with the one obtained when the sample is replaced by a mirror. Even without correcting for its reflectance such results can often be used for further mathematical evaluation. The Fresnel reflection coefficient r in the form given by Eq. 

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. 

FigureS-l. (a) Schematic SAXS setup, (b) X-ray beamptUhs from the source to the detector, both elements locatedfar away from the sample. The segment AB + BC is the optical path difference from which the phase shift is determined.

The contribution of the sampling element is evident, in that the natural period of a dead-time process with integral control is 4t( without sanpling. Sampling adds 2 At to the period. The phase shift introduced by the sampling element can then be related to the period of the loop ... 

In all these time-resolved experiments the principles of pump-probe laser spectroscopy are the key element in the experimental design. A laser pulse is optically split into two components of unequal amplitude. The intense fraction, acting as a pump piilse, is directed towards the target or sample cell to trigger the molecular event under study. The much attenuated probe pulse monitors the absorption, raman scattering, polarization, coherence, or phase shift, which is linked explicitly to the dynamical observable under investigation. Extremely precise time... 

A more fundamental breakthrough emerged by using X-ray phase contrast. This technique is attractive because the interaaion aoss section of the X-ray phase shift is about a 1000 times larger than that of absorption for low-Z elements. Polymer samples consisting of low-Z elements generate clear phase contrast while the absorption contrast is insufficient. The use of X-ray phase information is therefore essential in the tomographic observation of polymers. 

Figure 9.1 Inverse partial melting problem in the three-dimensional space of elements 1, 2, 3 when the source is known. Projection of the source onto the sample subspace provides the mass-fraction of each phase of the molten source. If one phase is at the origin (sterile phase), every representative point can be shifted by a constant vector.

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