Lorentzian-shaped peak

The SPR is then also called Mie resonance. For simple metals, the SPR absorption band has a Lorentzian shape peaked at oi p, the width of which is directly proportional to the collision constant E introduced in the Drude description of the metal dielectiic constant (Eq. 2). Of course, for noble metals the absorption due to interband transitions has to be taken into account in order to obtain the complete spectrum. 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

The observed cross sections for the 18s (0,0) collisional resonance with v E and v 1 E are shown in Fig. 14.12. The approximately Lorentzian shape for v 1 E and the double peaked shape for v E are quite evident. Given the existence of two experimental effects, field inhomogeneties and collision velocities not parallel to the field, both of which obscure the predicted zero in the v E cross section, the observation of a clear dip in the center of the observed v E cross section supports the theoretical description of intracollisional interference given earlier. It is also interesting to note that the observed v E cross section of Fig. 14.12(a) is clearly asymmetric, in agreement with the transition probability calculated with the permanent electric dipole moments taken into account, as shown by Fig. 14.6. 

The resonance phase shift Sr produces much time delay of Lorentzian shape if T is small. The peak value of the time delay at E = E, is 4h/ r = 4r if the background term is neglected. The Lorentzian time delay averaged over the probability density p(E) is hf p2(E)dE = 2r, of which r is spent for the formation of the QBS and the other r for its decay. 

Here, T is the observed line width (Av << F), 7d is the peak-to-valley intensity in the difference spectrum, and To is the peak height of the Raman line. Although this equation is for Lorentzian-shaped bands, the results are approximately the same for Gaussian-shaped bands (the constant 0.385 becomes 0.350). In the case of carbon disulfide-benzene mixtures, the smallest shift observed was -0.06 cm-1, and the associated error was 0.02 cm-1 (77). A convenient rotating system that can be used for (1) difference spectroscopy, (2) normal rotating sample techniques (solid and solution), and (3) automatic scanning of the depolarization ratios as a function of the wave number has been designed (45). 

The resulting cross-section is shown in Fig. 9.11(a). The distinguishing feature of Fig. 9.11(a) is that every peak or shoulder in the cross-section can be lined up uniquely with one of the bars in Fig. 9.11(a), i.e. every feature of the cross-section (9.4.6) can be assigned uniquely to one of the resonances Ek. Another typical feature is the characteristic Lorentzian shape of the isolated resonances in a. 

W21 beamline and diffractometer [33]. A large number, 366 of which 267 were non-equivalent, of in-plane reflections arising from the reconstruction were measured. All peaks were exactly centred at the expected positions to within 0.001° of azimuthal rotation, which showed that the surface reconstruction is perfectly commensurate with the underlying bulk lattice. Their width and Lorentzian shape indicated an exponential decay in correlations with the decay length of -500 A. Several reconstruction diffraction rods were also measured. The absence of symmetry of the rod intensity with respect to =0 showed that the reconstruction has the minimal hexagonal symmetry p3. 

Scattered electron intensities have been calculated for all 27 normal modes of the propane molecule. The electronic ground state of the molecule was described by the restricted Hartree-Fock wave function within Sadlej s basis set23. Cross sections obtained from DMR calculations were converted to a theoretical band spectrum by assuming Lorentzian shape for all bands with a half-width taken from the width of the observed elastic peak. The bands were centered at the positions of observed vibrational frequencies. 

Filters with resonance absorption peaks, which absorb neutrons only in a narrow range of energies are used to determine E. Two filters have been used in the measurements, either a gold filter with Nd = 7.35 x 1019cm-2, which defines a final energy E = 4908 meV, with an approximately Lorentzian shape of half width at half maximum (HWHM) AE] 140 meV, or a uranium filter with Nd = 1.46 x 102° cm-2, E = 6671 and an approximately... 

If the process is symmetrical (xf = Xb), then the relaxation time in each direction is equal to x/2. In writing equation (7.9.13) it is assumed that the absorption peaks have a Lorentzian shape. In addition, the line-broadening effects illustrated in fig. 7.18 are those for a symmetrical process. 

An important assumption in this theory is that there is no correlation between successive jumps and this is generally a good assumption at low H concentrations. However, at larger concentrations, this is not strictly true because correlation effects become significant, i.e. if an atom jumps from a filled site to an empty one, the site vacated is, at this instant, empty whereas the other sites are occupied with a probability c where c is the overall fractional occupation, so that the chance of jumping back is enhanced. This effect was first considered by Ross and Wilson [37] who showed by Monte Carlo simulation that, at finite concentrations, the quasi-elastic peak deviates from the Lorentzian shape. This was the first example of the need to resort to Monte Carlo simulation of the diffusion process to obtain G (r, t) in situations where the diffusion process becomes significantly complicated. This is likely to be important in efforts to understand the diffusive process in complex hydride stores. 

In the preceding sections, we have assumed that an absorption line has a Lorentzian shape. If this is not true, then the linewidth cannot be defined as the full width at half maximum intensity. Transitions from the ground state of a neutral molecule to an ionization continuum often have appreciable oscillator strength, in marked contrast to the situation for ground state to dissociative continuum transitions. The absorption cross-section near the peak of an auto-ionized line can be significantly affected by interference between two processes (1) direct ionization or dissociation, and (2) indirect ionization (autoionization) or indirect dissociation (predissociation). The line profile must be described by the Beutler-Fano formula (Fano, 1961) ... 

Spectra of Neat and Middle Phases. Typical spectra of the neat and middle phases are shown in Figures 12 and 13. These spectra were obtained at 100°C. from systems of 70% surfactant (neat phase) and 40% surfactant (middle phase) in D20. The small peak, or shoulder, visible on the low-field side of the spectra arises from the residual HDO in the solvent. Shown also in the figures, by means of dots, are calculated Lorentzian lines having the same heights and widths as the experimental lines. The experimental lines have the super-Lorentzian shapes discussed above. 

Figure 25. Rocking curve scans for the barite(001)-water interface at selected values of Lr u = QJ(2 / < ). Note that the peaks are narrow and Gaussian shaped at L 1 and L 2, while is the peak is broader and Lorentzian shaped near L 1.5. [Reproduced with permission from J. Phys. Chem. 2001, 105, 8112-8119. Copyright 2001 Am. Chem. Soc.]...

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