Lorentz line shape

The two extensions of the Fourier series and Fourier transform are the Lorentz line shape and the autocorrelation function. 

In optics, the general solution of the differential equation of oscillators is given in terms of the Fourier series, 

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L22 (a) Sketch the first derivative of a Lorentz line shape, (b) Some ESR spectrometers record the second derivative (rather than the first derivative) of resonance lines. Sketch the second derivative of a Lorentzian line, indicating the position of Pq. 

For more information about Lorentzian distributions and Lorentz line shapes, see FabeUnskii (1968, pp. 218-222) and Dicke and Wittke (1960, pp. 275-278). 

Numerical integration of the 1st derivative spectrum to obtain the spin concentration of a spectrum with a Lorentz line-shape is more difficult and often less accurate than with a Gauss shape. The error in the integrated area of a Lorentz line is, for example, of the order 5% even by recording over a range of ca 50 times the peak-peak line-width of the 1st derivative spectrum [2]. Knowledge of the experimental line-shape can help to estimate and eventually reduce the error [16]. 

Fig. 9.2 ESR spectra of a GdCls water solution, (a) 2nd derivative with hi/h2 = 3.7 corresponding to ABl/ABq = 3.9. (b) 1st derivative experimental and simulated spectra with a Lorentz line-shape. The spectra were obtained from Dr. H. Gustafsson...

The experimental structure below 1 eV in Figures 1.10 and 1.11 corresponds to the free-electron behavior typically observed in metals. The lowest interband absorption starts to occur above 1.5 eV. In the non-noble (Cu) and noble metals (Ag, Au), Cooper et al. [18] indicated that the sharp rise in 2 at the lowest interband absorption edge is due to the fact that the transitions are from a very flat lower band to the Fermi surface and are not of the critical-point type. More strictly, these transitions occur between occupied states in band 5 and unoccupied states in band 6 as these cross the Fermi surface (5 —> 6 (Ep)) at the L point in the Brillouin zone. Here, the bands are numbered starting from the lowest band at a given k. Relatively poor agreement between the Lorentz line shape and the experiment observed at -1.5 eV in Figiues 1.10 and 1.11 may reflect this fact... 

Finally, we should mention the effect of far-wing absorption, an important but poorly understood aspect of line shape. Collision-induced opacity is observed in spectral regions well-separated by as much as 10 to 100 cm from the line center. This very weak absorption is not described by the wings of a simple Lorentz line shape discussed above it shows an exponentially decreasing dependence on the separation from the line center (see, e.g., Bimbaum, 1979). The anomalous far-wing absorption is due to inadequacies in the hard-sphere collision model used... 

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

Mathematically, this line shape is described by the Lorentz distribution... 

As seen, the spectral line of high-frequency local vibrations is of the Lorentz-like shape ... 

The interpretation of band progressions by the time dependent procedure is therefore identical with the Franck-Condon analysis and, in the low temperature limit, to the method of molecular distributions as well. The line shape function obtained on the basis of Eq. (52) (for E = hv) differs under this condition from that of Eq. (12) only in the line shape function of each vibrational member in the progression which in Eq. (52) is the delta function and in Eq. (12) has a Lorentz type distribution. 

Figure 30 shows the component analysis of the resonances of the methine and methyl carbons in the equilibrium DD/MAS 13C NMR spectrum. Here a Lorentz-ian function is assumed for each component. The rationality for this assumption was confirmed by examining the elementary line shapes for each component using the differences in the Tic and T2c values in a similar way to that described in preceding sections. The narrow Lorentzian components centered at 26.2 and 20.6 ppm, and 27.4 and 19.9 ppm are assignable to the methine and methyl carbons in the crystalline and amorphous phases, respectively, as discussed previously (see Table 13). In addition to these components, broad Lorentzian components are recognized centered at 26.6 and 21.1 ppm for the methine and methyl carbons. It was... 

The perturbational effects have been variously described as interruption broadening, resonance broadening, and statistical broadening. The perturbational line shape derived by Michelson was not correct even for pure interruption broadening because he neglected to average over all times between collisions. To do so results in the simplified Lorentz model. ... 

Fig. 7.46 ESR spectra of PA films observed at 296 K (a) and 9 K (b). The circles show the theoretical line shape (Lorentz type). The figure is adapted from [50] with permission from the American Institute of Physics...

ESR lines in solution can almost always be approximated by a Lorentz function. In the solid state the line-shape can in general be reproduced by a Gauss curve. In some instances a so-called Voigt profile can give a better approximation to the experimental line-shape. A Voigt line is a convolution of a Lorentz and a Gauss line. The shape is determined by the ratio ABi/ABg of the respective line-widths. The shapes of the 1st derivative lines of these types are given in Fig. 9.1. 

The line-shape of an experimental spectrum can in principle be determined by the procedure illustrated in Fig, 9.2. The 2nd derivative of the resonance line is then recorded. For a Gauss line the ratio hi/h2 between the minimum and maximum amplitudes of the 2nd derivative (Fig. 9.2(a)) equals 2.24 [18], while for a Lorentz shape it approaches the value 4. The hi/h2 ratio for a Voigt profile varies... 

Fig. 9.14 Homogeneous (a) and inhomogeneous (b) ESR lines. The spin-spin relaxation time T2 = 1/(y-ABl) is obtained from the line-width ABl measured at half height of the absorption amplitude of the Lorentz line (a). The inhomogeneous line (b) is an envelope of overlapping homogeneous line. The shape of the envelope approaches a Gauss curve...

The method to obtain relaxation times from CW microwave saturation measurements for an inhomogenously broadened line is based on assumptions given in the literature [77, 78], The ESR line shape is then expressed as a convolution of a Gauss and a Lorentz function [81] ... 

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