Real crossing INDEX

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

It is well known that for lossless media, all squared effective indexes are real, and for any transversally limited structure they form a discrete nongrowing sequence. The field distributions / (O and /i (0 of the m-th mode are mutually orthogonal, have the same phase at each point of the cross-section, and the set of functions corresponding to all modes is complete. The orthogonality relations can be taken in the form... 

The real and imaginary parts of the complex refractive index satisfy Kramers-Kronig relations sometimes this can be used to assess the reliability of measured optical constants. N(oj) satisfies the same crossing condition as X(w) N (u) = N( — u). However, it does not vanish in the limit of indefinitely large frequency lim JV(co) = 1. But this is a small hurdle, which can be surmounted readily enough by minor fiddling with JV(co) the quantity jV(co) — 1 has the desired asymptotic behavior. If we now assume that 7V( ) is analytic in the top half of the complex [Pg.28]

Figure 10. Dielectric model of the protein. Within this model, the protein medium (i.e., the medium with the refractive index of n = 1.2) is represented with a set of cylinders. The cross section of these cylinders is shown with white circles. The real location of the transmembrane part of a-helices in PSI are indicated by coiled structures. Chlorophylls are presented as Mg-chlorin rings, lacking the phytyl tail. Chlorophyll Mg atoms are shown in van der Waals representation. See color insert.

X quantifies all second-order NLO effects such as SHG, electro-optic effect (Pockel) and frequency mixing, x is representative of third-order NLO effects such as THG, optical Kerr effect and two-photon absorption (TEA). The real part of 7 describes the nonlinear refractive index and its imaginary part the two-photon cross section (<72). 

Rolla et ah, used microwave dielectric measurements to monitor the polymerization process of mono functional n-butyl acrylate as well as 50/50 w/w blends with a difunctional hexane-diol diacrylate that gave highly cross-linked networks. In these real time cure experiments the decreasing acrylate monomer concentration was studied via a linear correlation with the dielectric loss index at microwave frequencies. This correlation is a result of the largely different time scales for dipolar polarization in the monomer on one hand and in the polymerized reaction product on the other hand. 

Fig. 5. a - theoretically found spectral dependences of optical constants of porous model medium, due to the presence of "ideal" silver particles (1 = 8 nm) and "real" particles (1 2 nm) y and An are, respectively, absorption constant and refractive index, b -dependence of phase incursion (AnT/X), due to the presence of silver particles in the medium, on silver coverage, C, of the studied sample experimental measurements with the use of developed holography film plates PFG-02 (dots) calculations by measured attenuation spectra with the use of dispersion Kramers-Kronig relations (crosses). 

It is apparent from Figure 7 that the dependence of P on the thickness of plate courses t is non-linear. Reliability index P for nominal thickness of the real tank (cross) is approximately constant with the exception of the top shell course 9, which has a very high reliability p 10.4. The reliability indices P for the first two shell courses evaluated according to API 653 (blue dots) are P 6.3, see Figure 7. The minimum acceptable thickness evaluated for the first two bottom shell courses according to standard API 653 (blue dots) is higher than the nominal value of plate thickness of the assessed tank. Strictly speaking, the tank does not comply with the requirements of standard API 653, however, the difference is so small that the tank can be practically considered reliable (blue dot practically overlaps the cross). 

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