Gaussian line shape 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,... 

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. 

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. 

The line shapes of the resonance absorption curves were rather well described by a Gaussian shape, as is shown in Fig. 14. Two parameters are necessary in fitting a recorder derivative with a Gaussian shape function the maximum value of the derivative signal (dx"/3 f)m. and the width between points of maximum slope of the absorption, which are... 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

The Gaussian assumption yields the first equality. Assumption (25) is used in the second expression on the right-hand side. Thus the line shape function /(co) can also be written as... 

Here, we shall not go any further into the analysis of this sort. Instead, we show a few examples of line shape functions calculated by Eq. (72) for the Gaussian models, which are characterized by the operator T given by Eq. (52). The Gaussian models are not very adequate for the electric field in a plasma, as is seen by the fact that the Holtzmark distribution is far from Gaussian. Still, it may serve to give one an understanding of the... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Figure 1.1a shows the Gaussian function. The Lorentzian shape is similar to the Gaussian, but falls off more slowly. The Doppler shift of radiation from an emitting molecule is proportional to its velocity component in the direction of observation. The one-dimensional distribution of speeds in a gas is a Gaussian function. (See any physical-chemistry text.) Hence when Doppler broadening is dominant, we get a Gaussian-shaped line. 

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... 

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). 

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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