Frequency-dependent spectra

We now proceed to the spectrum, or frequency-dependent response [44, 42]. The power, or rate of energy absorption, is given by... 

Galica G E, Johnson B R, Kinsey J L and Hale M O 1991 Incident frequency dependence and polarization properties of the CH I Raman spectrum J. Phys. Chem. 95 7994-8004... 

Kastner et al. [25] also reported Raman spectra of cathode core material containing nested tubules. The spectral features were all identified with tubules, including weak D-band scattering for which the laser excitation frequency dependence was studied. The authors attribute some of the D-band scattering to curvature in the tube walls. As discussed above, Bacsa et al. [26] reported recently the results of Raman studies on oxidatively purified tubes. Their spectrum is similar to that of Hiura et al. [23], in that it shows very weak D-band scattering. Values for the frequencies of all the first- and second-order Raman features reported for these nested tubule studies are also collected in Table 1. 

Fig. 12b). Since practically the same spectral shape is obtained at Q-band (35 GHz) (Fig. 12c), the commonly used criterion stating that the shape of an interaction spectrum is frequency-dependent fails to apply in this case. Actually, outer lines arising from the exchange interaction are visible on the spectrum calculated at Q-band (Fig. 12c), but these lines would be hardly detectable in an experimental spectrum, because of their weak intensity and to the small signal-to-noise ratio inherent in Q-band experiments. In these circumstances, spectra recorded at higher frequency would be needed to allow detection and study of the spin-spin interactions. 

Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

Figure A2.1 Microwave frequency dependent HF-EPR spectra of aqueous Cr2+ (0.1-0.2 m), sulfate counterion. Experimental conditions temperature 10 K microwave frequency as indicated. In the spectrum taken at 329 GHz a sharp signal from aqueous Cr3+ impurity at g = 2 is indicated and the resonances due to Cr2+ are labeled (Figure A2.2).

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows power law frequency dependence as follows129 130... 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

The relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

Chapter 4, presents details of the absorption and reflectivity spectra of pure crystals. The first part of this chapter coimects the optical magnimdes that can be measured by spectrophotometers with the dielectric constant. We then consider how the valence electrons of the solid units (atoms or ions) respond to the electromagnetic field of the optical radiation. This establishes a frequency dependence of the dielectric constant, so that the absorption and reflectivity spectrum (the transparency) of a solid can be predicted. The last part of this chapter focuses on the main features of the spectra associated with metals, insulators, and semiconductors. The absorption edge and excitonic structure of band gap (semiconductors or insulator) materials are also treated. 

The combined effects of the dipolar and exchange interactions produce a complex frequency-dependent EPR spectrum, which can however be analysed by performing numerical simulations of spectra recorded at different microwave frequencies. When centre A is a polynuclear centre, the value of its total spin Sa = S, is determined by the strong exchange coupling between the local spins S, of the various metal sites. In this case, the interactions between A and B consist of the summation of the spin-spin interactions between Sb and all the local spins S, (Scheme II). The quantitative analysis of these interactions can therefore yield the relative arrangement of centres A and B as well as information about the coupling within centre A. 

A chemical reaction occurs above 1.5 GPa The sample turns black, new peaks develop in the Raman spectrum, and the absorption edge moves below 11,000cm. The recovered material has an optical band gap of 1.39eV, smaller than the band gap of polyacetylene. From the analysis of the Raman spectrum, it is seen that the C=C stretching mode completely disappears in the reaction product, while the C=N stretching band is present but at a different frequency than in cyanocetylene. In addition, the Raman bands of polyacetylene are observed with their characteristic frequency dependence on the wavelength... 

The sampling theorem tells us that we must sample the signal at a rate equal to or greater than twice the highest-frequency component in the Fourier transform of the signal. For an actual spectrum of absorption lines, this maximum frequency depends on the selected scanning rate in a direct way. In addition, the bandpass of the electronics must be established in such a way that the maximum signal frequency will be minimally attenuated. 

In the previous section several equations were described that can be used to calculate MCD spectra. If the spectra are to be calculated using the transition-based approach described in Sections II.A.1-II.A.4, a number of quantities must be evaluated. These include the perturbed and unperturbed excitation energies, the perturbed and unperturbed transition moments between the ground and excited states, and/or the magnetic moment of the ground state. If an MCD spectrum is to be calculated with the imaginary Verdet approach described in Section II.A.6, then the first-order correction to the frequency-dependent polarizability due to a magnetic field is required. 

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