Scattering curve three dimensional

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Figure 17. Angular distribution of autoionization electrons from Ne (2p 3s ) D, excited by He impact. A-A shows the distribution in the scattering plane, B-B in the plane perpendicular to the He -beam direction, and C-C in a plane perpendicular to the scattering plane, tilted by 45 with respect to the He -beam direction. Solid curves represent the result of a fit calculation. Parameters obtained from the fit are used to calculate the complete distribution, which is shown in the three-dimensional view. 

Diffraction methods depend on interference effects, and therefore obtaining structural information from the observed patterns also involves Fourier transformations. For crystal lattices a three-dimensional FT is used to convert between the recorded diffraction scattering pattern (in reciprocal space) and the crystallographic lattice (in real space) (Sections 3.4 and 10.2). Similarly, in gas-phase electron diffraction a one-dimensional FT converts between the (reciprocal space) diffraction data and the (real space) radial distribution curves, which are one-dimensional plots of increasing distances separating pairs of atoms in the stmcture. 

Fig. 4.10 Experimental and theoretical DCS of the HF(n = 2, j = 6) product of the F( P3/2) + HD( /o = 0) reaction in the backward scattering direction. The solid circles are experimental data the red curve, the result of full quantum dynamics calculations convoluted with the experimental resolution and shifted 0.03 kcal/mol lower in energy. The error btirs in the experimental data are the estimated measurement errors (la) for the HF(u = 2, j = 6) product peak intensity in the collision energy scan. The three petiks are assigned to the ptirtitd waveFeshbach resonances of / = 12, 13, and 14 in the F +HD—> HF + D reaction, as explained in the text. The three-dimensional DCS shown was measured at 1.285 kcal/mol, with F and B indicating the forward- and backward-scattering, respectively, directions for HF with respect to the F-atom beam direction...

Fig. 6.13. Small-angle x-ray scattering curves (a) and one-dimensional (1) and three-dimensional (2) correlation functions for polymers 5.2 (I) and 5.3 (II. Table 6.4) (b) [50].

Another difficulty arises because in XPS spectra it is often not straightforward to separate the intrinsic, elastic part of the spectrum (peak area) from the inelastic contributions (Section 3.2.2.1.2). This contributes uncertainties to the sample volume in the direction perpendicular to the surface, defined by the inelastic mean free path in, especially in the case of substrate emission. Moreover, f is strongly energy dependent in three-dimensional soHds and only approximately described by universal curves found in the literature [11] (see also Chapter 3.2.3.2). Litde quantitative information is available for inelastic scattering of electrons at interfaces and within molecular layers. 

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