Line shape Lorentzian function

In most cases in which line profiles are completely resolved by the infrared spectrometer both collision and Doppler broadening contribute to the line shape. The function that describes the composite line profile is a convolution of a Gaussian and a Lorentzian function,... 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

A plot of v vs. T2(a>o co) is shown in Figure 5.1. Equation (5.14) corresponds to the classical Lorentzian line shape function and the absorption curve of Figure 5.1 is a Lorentzian line . The half-width at half-height is easily found to be ... 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

However, the Lorentzian form of the dipolar broadening function, which has the advantage of mathematical simplicity, is not suitable for an interpretation in terms of second moments it is replaced with a Gaussian dipolar function S(oa, AG), where the parameters AG correspond to the appropriate fractions of the square root of the intra-group rigid lattice second moments. With appropriate values for AG, calculated and experimental line shapes I(oo) are found to be in a good agreement for cross-linked polyethylene oxide) swollen in chloroform 1U). 

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

Since NMR spectra are not sequences of lines representing discrete Larrnor frequencies but sequences of Lorentzian frequency distributions f(to) (Fig. 1.9), eq. (2.10) must be replaced by eq. (2.11) M0 sin c is multiplied by the frequency function f(to), where a> represents the difference between the frequency ojx and the Larrnor frequency distribution con + Aw, w = co1 — (w0 + Aro). Further, Mosin0f(tu)e must be integrated over the Larrnor frequency distribution. Given a Lorentzian line shape as in Fig. 1.9, the limits of integration are oo ... 

Molecular Weight Dependence of Phase Structure. Similar line shape analysis was performed for samples with molecular weight over a very wide range that had been crystallized from the melt. In some samples, an additional crystalline line appears at 34.4 ppm which can be assigned to trans-trans methylene sequences in a monoclinic crystal form. Therefore the spectrum was analyzed in terms of four Lorentzian functions with different peak positions and line widths i.e. for two crystalline and two noncrystalline lines. Reasonable curve fitting was also obtained in these cases. The results are plotted by solid circles on the data of the broad-line H NMR in Fig. 3. The mass fractions of the crystalline, amorphous phases and the crystalline-amorphous interphase are in good accord with those of the broad, narrow, and intermediate components from the broad-line NMR analysis. 

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... 

Fig. 7.6. (a) Energy dependence of a Lorentzian line-shape function with width KT centered at the resonance energy (Ei + 6E). (b) Partial photodissociation cross sections a(E,0) as given by (7.23). All of them have the same width hT the values at the maximum scale like the partial decay rates Tp. 

The Fourier transformation of this autocorrelation function, eiEt/h x( ) x(t))dt, gives an energy spectrum with lines centered at E°, and line shapes of a Lorentzian form. The widths of these lines are Tn = h/Tn and the rate constants can then be obtained as kn = l/r . 

Here kv is the absorption coefficient at frequency v, Nc is the number of absorbing centres per cubic centimeter, v is the frequency of absorption, and S(v) is the line shape function. For our estimates we shall assume that the line shape is Lorentzian having half width 6. If one evaluates the absorption cross section when the absorption is maximum the above expression takes the form... 

The FID from the 50 50 mixture is shown in Fig. 11. The decay is clearly nonexponential, and thus the line shape is not a pure Lorentzian. The decomposition of the line shape into Lorentzian and Gaussian components (132) is shown as a function of concentration in Fig. 10. Although there is some uncertainty in such a decomposition, the major features are clear. 

Figure 10 Width (FWHM) of the isotropic Raman line of the sym-methyl stretch in CH3I as a function of concentration in CDCI3 ( ). Voight fits give Lorentzian (O) and Gaussian (A) contributions to the line shape. The Lorentzian component is consistent with a concentration independent fast-modulation process. The Gaussian component suggests an additional contribution from slow concentration fluctuations. (From Ref. 4.)...

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