Real part spectrum

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. 

In the pioneering work the same information was extracted from the extremum position assuming it is independent of y [143]. This is actually the case when isotropic scattering is studied by the CARS spectroscopy method [134]. The characteristic feature of the method is that it measures o(ico) 2 not the real part of Ko(icu), as conventional Raman scattering does. This is insignificant for symmetric Lorentzian contours, but not for the asymmetric spectra observed in rarefied gas. These CARS spectra are different from Raman ones both in shape and width until the spectrum collapses and its asymmetry disappears. In particular, it turns out that... 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

By the very definition of the GF, the real parts of the poles of its frequency Fourier component correspond to natural frequencies of the system (see, for example, Eqs. (A1.23) or (A1.55)). Consequently, the spectrum of natural frequencies of the perturbed system, cop, should fit the equation... 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Figure 33. The stability of yeast glycolysis A Monte Carlo approach. A Shown in the distribution of the largest positive real part within the spectrum of eigenvalues, depicted from above (contour plot). Darker colors correspond to an increased density of eigenvalues. Instances with > 0 are unstable. B The probability that a random instance of the Jacobian corresponds to an unstable metabolic state as a function of the feedback strength 0, . The loss of stability occurs either via in a saddle node (SN) or via a Hopf (HO) bifurcation.

At this point, our notion and implications of the term stability must be clarified. At the most basic level, and as utilized in Section VILA, dynamic stability implies that the system returns to its steady state after a small perturbation. More quantitatively, increased stability can be associated with a decreased amount of time required to return to the steady state as for example, quantified by the largest real part within the spectrum of eigenvalues. However, obviously, stability does not imply the absence of variability in metabolite concentrations. In the face of constantperturbations, the concentration and flux values will fluctuate around their... 

Figure 42. The distribution of the largest real part within the spectrum of eigenvalues for the model of the Calvin cycle described in Section VIII. F. Only a minority of sampled models correspond to a stable steady state. See also Fig. 37 for convergence in dependence of the number of samples.

Figure 44. The correlation coefficient of the saturation parameters with stability, here identified with the largest real part within the spectrum of eigenvalues. Models (Jacobians) of the Calvin cycle are iteratively sampled and the correlation coefficient between each saturation parameter and the largest real part of the eigenvalues are evaluated. Negative values imply a negative correlation, that is, small saturation parameters correspond to a higher probability of instability.

Figure 6.5a illustrates part of a real mass spectrum of a radioactive waste solution in the mass range 180-250 u measured by double-focusing sector field ICP-MS (ELEMENT, Thermo Fisher Scientific) at low mass resolution. In this mass spectrum, atomic ions of Re at mass 185 u and 187 u with isotope abundances of 37.4 % and 62.6 %, respectively (Re was used for "Tc precipitation),... 

If the real part v(a>) of the NMR spectrum is computed in the absorption (r) mode, the imaginary part is usually displayed in the dispersion (h) mode. The magnitude spectrum is therefore related to the t and u modes as indicated in eq. (1.37). 

Fig. 2.13 (b, c) illustrates a phase correction. For correcting the phase, either the real or the imaginary part of the spectrum can be used. Correction of the real part for the absorption mode yields the dispersion mode in the imaginary part and vice versa (Fig. 2.13). 

In a typical situation we are interested in the absorption mode of a dynamic spectrum,/abs (co), which equals the real part of the complex function f(co) given by equation (145). In most cases of unsaturated spectra the relaxation matrix which describes single-quantum transitions can be replaced by a constant — E/T2(effective) which is characteristic of the experimental conditions involved and reflects the inhomogeneity of the external magnetic field B0. The absorption mode spectrum is given by ... 

Since the polarizers discussed above involve light reflection combined with the real part of the refractive index tensor, they can be used effectively over a broad spectral range about a central wavelength. Calcite Glan-Thompson polarizers, for example, operate successfully over the entire visible spectrum. When fabricated of crystalline quartz, these polarizers can be used to polarize ultraviolet light as well as visible light. 

Fig. 44. (a) Schematic picture of the real part of the self-energy for a valence hole, together with graphical solutions , and of the Dyson equation (Eq. (15)). (b) is meant to represent a typical outer-valence hole spectrum while (c) and (d) describe the possible behaviour of inner-valence holes. (bHd) are connected with the solutions -( ) resp. Note that in principle the self-energy is different for different valence holes, contrary to what is suggested in (a)... 

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