Plane-wave approximation

The Mo K-edge EXAFS spectra for the catalysts and reference compounds (MoSj and NajMoOJ were measured on the BL-lOB instruments of the Photon Factory at the National Laboratory for High Energy Physics by using a synchrotron radiation. The EXAFS spectra were obtained at room temperature without exposing the sample to air by using an in situ EXAFS cell with Kapton windows [12]. Data analysis was earned out assuming a plane wave approximation. 

Full derivations of the theory were presented by Lee and Pendry and Ashley and Doniach in 1975. They showed that a complete quantitative description of th EXAFS process was possible and that accurate bond lengths and coordination numbers could be extracted from the analysis of EXAFS data. Lee and Pendry also showed that at high photoelectron energies, the curvature of the electron wave can be neglected and thus the theory can be greatly simplified into what has beeome known as the plane-wave approximation. This approximation results in an expression equivalent to that derived by Stem semi-empirically ... 

In general, therefore, there is a longitudinal propagating component of the electric field in the vacuum. However, in the plane-wave approximation used here, there occurs the relation... 

In the plane-wave approximation, all electromagnetic effects are derived from the structured time-like potential difference... 

It has been shown that the electromagnetic field in Whittaker s view originates in the vacuum, and in the plane wave approximation, in the equation... 

In general, the vacuum current density has a definite structure in the vacuum that is much richer than in the first plane-wave approximation a structure that has to be computed because analytical solutions to Eq. (183) are not available. 

In this section, we illustrate the self-consistent calculation of these charge current densities in the plane-wave approximation, using plane waves in the X, Y, and Z directions. In general, the solution of the field equation (459) must be found numerically, and it is emphasized that the plane-wave approximation is a first approximation only. In the internal space, there is the real vector ... 

Here /jn(f) is the intensity of the incident radiation and 0 is the phase of the interferometer in the dark. The functions N(< >) and M(< >) relate the intensities of the transmitted and intracavity fields to that of the incident light. The function 7ref (0 corresponds to the intensity of radiation from an additional source, which is very likely to be present in a real device to control the operating point. This description is valid in a plane-wave approximation, provided that we neglect transverse effects and the intracavity buildup time in comparison with the characteristic relaxation time of nonlinear response in the system. It has been shown that the Debye approximation holds for many OB systems with different mechanisms of nonlinearity. 

In the plane-wave approximation, the second-order correlation function (20) can be written as... 

The multiple scattering pathways where the angle between the outgoing wave and the reflected wave is 90° (for example the n = 3 contribution for the M—O—O—M pathway) has zero amplitude in the plane wave approximation as can be seen from formula (12) in Sect. 3.1. Therefore the pathways for atoms in collinear configurations provide the main contribution to multiple scattering. 

The data analysis was performed with use of the classical EXAFS formula (1) within the plane-wave approximation. The amplitude and phase shifts of Teo et al. were used except when otherwise specified in the text. When tabulated and experimental parameters are used on the same data, slight differences appear in the final results (Tables 1 and 3 for CUC2O4 for example). 

This is known as the plane wave approximation, i.e. the electron wave is approximated by a plane wave. The latter is only applicable when the radius of the emitting atom is small in comparison to the curvature of the outgoing wave which is the case when the photoelectron energy is high. A more accurate description, often known as exact curved wave theory has been given by Lee and Pendiy (1975). For a review of the theories and procedures see Gurman (1990). 

For the sake of comparison with the ESCA spectrum of Nj, and to provide a more critical test of the EOM method, we calculate the peak intensities as well as the peak positions. The details of the intensity calculations are left to the Appendix. The dipole approximation is invoked, as the calculation is based on Fermi s golden rule. A plane wave approximation is employed for the outgoing electron. The cross section for the ejected electron is averaged over all possible molecular orientations and polarizations of the incident photon as described by Ellison. ... 

One of the important attributes of synchrotron X-ray beams is that the radiation is horizontally polarized in the plane of the electron orbit (Koningsberger and Prins 1988 Hazemann et al. 1992). This property can be used to enhance the sensitivity of EXAFS to the various metal species tied up to layered minerals in performing angular measurements on textured samples. In P-EXAFS, the amplitude of the EXAFS contribution of an atomic shell S coordinated to an element E can be written in the plane wave approximation and at the E K-edge... 

Another frequently employed simplification is the ID assumption according to which all variables depend only on one coordinate, namely, the one transversal to the plane of the nematic layer, say, the coordinate. This means that the incident fight should be an infinite plane wave (hence this approximation is often called the infinite plane wave approximation), and, by virtue of incompressibility and the boundary conditions, v = Vx z, t), Vy z, t), 0) which... 

The bifurcation scenario discussed above was actually observed in the experiment. Although a good qualitative agreement between theory and experiment was found [40], there are quantitative discrepancies. In the experiment, the measured onset of the nutation-precession motion turns out to be about 20% lower than predicted by theory. Moreover, the slope of the precession frequency versus intensity predicted by theory turned out to be different from that observed in the experiment. One of the two possible reasons could be the use of finite beam size in the experiment (that is typically of the order of the thickness of the layer), whereas in theory the plane wave approximation was assumed. Actually, the ratio 5 between diameter of the beam and the width of the layer is another bifurcation parameter (in the plane wave approximation, 6 oo) and was shown to play crucial role on the orientational dynamics [13]. There and in [44] the importance of the so called walk-off effect was pointed out which consists of spatial separation of Pointing vectors of the ordinary and extraordi-... 

The plane-wave approximation allows the emission of the secondary electron to the final state, taking into account its single scattering by a neighboring atom, to be presented in the form of a simple scheme (Fig. 8a). 

APW augmented plane wave (approximation) CERN European Center for Nuclear Research,... 

Calculations based on density functional theory and the plane waves approximation allow one to determine the energy distribution of electrons. Figure 9.1 shows the density of states for silver and platinum. One can see that the electron density is non-zero at the Fermi energy for both elements. An electric field applied to the material will accelerate electrons to higher energies than when there is no field. As a result both elements are conductors. Silver has a greater interval of states concentrated above the Fermi level than platinum. This seems to be a cause of difference in resistivity between silver and platinum. The resistivity of silver (1.6 x 10 Q m) is by the order of magnitude less than that of platinum (10.7 x 10 Q m). 

Again the delta-function approximation and the plane-wave approximation can be used. For example, using the delta-function approximation, we obtain... 

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