Lorentzian Distributions

Our discussions so far have been limited to assuming a normal, Gaussian distribution to describe the spread of observed data. Before extending this analysis to multivariate measurements, it is worth pointing out that other continuous distributions are important in spectroscopy. One distribution that is similar, but unrelated, to the Gaussian function is the Lorentzian distribution. Sometimes called the Cauchy function, the Lorentzian distribution is appropriate when describing resonance behaviour, and it is commonly encountered in emission and absorption spectroscopies. This distribution for a single variable, x, is defined by 

Like the normal distribution, the Lorentzian distribution is a continuous function, symmetric about its mean, p., with a spread characterized by the 

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This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /inormal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

An isotropic extinction parameter, of type I and Lorentzian distribution (in the formalism of Becker and Coppens [16]), was also refined. The motions of the non-H atoms were described by anisotropic parameters, while those of the H atoms by isotropic B s. All these displacement parameters were included among the refinable quantities of the model, for a total of 1161 variables in a single least-squares matrix. 

We have seen in Chapter 2 that in EPR spectroscopy one usually varies the magnetic held instead of the frequency, because the use of a mechanically rigid micro-wave resonator dictates the frequency to be constant. For this reason, the Lorentzian distribution in Equation 4.5 is frequently rewritten as a distribution in resonance fields as... 

For the Lorentzian distribution, from equation 54-lb the first derivative is... 

For the Normal distribution, the exponential term has become a constant, and we see that the maximum magnitude of the derivative is inversely proportional to <72 (for the constant area expression) or inversely as a (for the constant height expression). This confirms our observation from figure 54-1. For the Lorentzian distribution, we see that the derivative decreases with the second power of the bandwidth. 

And substituting X - /x = 0 into equation 54-16 gives us the corresponding value for the Lorentzian distribution ... 

The error in equation 54-11 then propagated through to the rest of the equations for the Lorentzian distribution. The correct formulas are as follows ... 

Figure 5.1. Representations of double-exponential and bimodal Lorentzian distribution analyses of DPH fluorescent decay lifetimes in liver microsomal membranes. Results (see Table 5.2) are normalized to the major component. The double-exponential analysis, represented by the vertical lines, recovers lifetimes near the centers of the Lorentzian distributions. The width of the distributions represents contributions from a variety of lifetimes. (From Ref. 17.)...

The absorption curves given by coal macerals approached the horizontal (magnetic field strength) axis more slowly than a Gaussian distribution curve. Shape analysis (16) showed that over much of the curve, the form closely approximated a Lorentzian distribution curve, but both positive and negative deviations were found in the wings of the curves (that is, in various examples, the curves approached the axis either somewhat more or somewhat less rapidly... 

Here N designates the normalization factor. Clearly this equation in integrated form is the product of Gaussian and Lorentzian distribution functions 0mg and 0m define the line-widths of the two components, respectively. Here, the former represents Eq. (17) to a sufficient approximation for 0m 2 G and the latter was introduced to express the coupled rotational and/or the translational motion of proton pairs in the polymer, discussed by Pechhold53. ... 

Narrow Component. As discussed in Chapter II, the absorption spectrum for polyethylene cannot be described by a single Lorentzian even in the molten state. However, the deviation from one Lorentzian is not enhanced for well-fractionated samples in the melt and, furthermore, becomes negligible as the temperature decreases42. Accordingly, the differential form of a Lorentzian distribution can be used for the elementary spectrum of the narrow component ... 

Before these partial quantities are discussed further, an important comment has to be made unlike the partial transition rates, the partial level widths have no direct physical meaning, because even for a selected decay branch it is always the total level width which determines the natural energy broadening. The partial level width is only a measure of the partial transition rate. Both aspects can be inferred from the Lorentzian distribution attached to a selected decay branch, e.g., Auger decay, which is given by... 

The Lorentzian distribution arises from the decaying hole-state, but for an analysis of the observed shape of a photoline, the energy distribution of the incoming light and the spectrometer function must be known and taken into account. In general, the latter functions cannot be presented in a closed form however, quite often they are approximated by a Gaussian distribution ... 

Song et al.172 theoretically calculated the impedance spectra based upon Eq. (34) with such distribution functions of PSD as normal, lognormal, Lorcntzian, log Lorentzian distributions. They concluded that the impedance spectra simulated based upon the transmission line model (TLM) with different PSD functions share a common point that the wider PSD leads to the more frequency dispersion in the impedance spectra. 

Lorentzian line shapes are frequently used to take into account the interference between different vibrations, although other more complex line sh s are used. Unfortunately, the broad wings characteristic of a Lorentzian distribution often overestimate the amount of overlap, and therefore the amount of interference, between widely separated peaks. Particularly useful, although computationally more intensive, is the line shape profile described by Bain et al. [4,12]. 

Numerous authors have more recently expanded the expression for the cross section given by equation (32) in various ways. Most of this work is for determining the structure of adsorbed films on solid substrates in particular graphite. The details of these expansions can be found in Stephens etal.J Dutta etal and Dimon etal. The work of Dimon et al. is particularly noteworthy in that it finds that a Lorentzian distribution gives a more satisfactory fit to the lineshape. Our recent experimental evidence suggests that the... 

TABLE 1. Observed intensity of each component in F Ka fluorescent X-ray, ion induced X-ray [14] and KVV Auger [15] spectra, which were deconvoluted by the least-square fitting using the Lorentzian distribution functions for X-ray spectra and Gaussian distribution functions for Auger spectra, respectively. 

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