Lorentz profile

The practically observed combined Doppler plus Lorentz profile is called the Voigt profile and can be computed numerically if required, to resolve the measured linewidth into its contributory components (ref 6, p. 167). 

Where, y - width of Lorentz profile, w - width of Gauss pa-ofile and Wo - central position of absorption band. To take into account the individual rate of thermal decomposition of absorbing Lorentz profiles of aggregates, we need to include in the Voight function the equation describing the thermal decay curve, depending on the energy activation distribution. The equation for thermal decomposition as follows from (4) shown in the above curve is... 

S.N. Dobryakov, Y.S. Lebedev Analysis of spectral lines whose profile is described by a composition of Gaussian and Lorentz profiles. Sov. Phys. Dokl. 13,... 

Figure 2.3 shows Gauss and Lorentz profiles of equal area and FWHM as well as the resulting Voigt distribution. While the Lorentz portion dominates at the line wings, the Gauss portion determines the shape in the line core. 

Next, the synthetic Cu profiles were convoluted by a Lorentz profile with 1.1pm FWMH representing the collisional broadening in a 2600 K flame at normal pressure, and for an assumed collisional cross-section of 2 10 m (see Figure 2.2). 

Figure 2.7 Convolution of the synthetic Doppler profile from Figure 2.6 with a Lorentz profile corresponding to 2600 K, normal pressure, and a collisional cross-section of 2 10 m A Doppler profile, B convoluting Lorentz profile with 1.1pm FWHM, C resulting Voigt profile...

Table 2.1 Comparison of measured with calculated FWHM values for Lorentz profiles...

The process of FSD is illustrated stepwise in Figure 6.6, where, in accordance with a common practice, a Lorentz profile is assumed for the bandshape in the condensed phase. An observed bandshape M v) is generally broader than its true shape E v) for various reasons. If a function G(v) is used to express the broadening of a bandshape, M(v) is expressed as a convolution of G(v) and E(v) (see Sections 4.4.1.1 and D.2) as... 

Let us make an assumption for a band located at vo that its true bandwidth is infinitely narrow. Such a hypothetical band can be expressed by a delta function <5(v - vq) (see Section D.3.2), which corresponds to a line spectrum with a line at Vq. By using the Lorentz profile... 

The formula transformation from the middle to the right-hand side of this equation is explained in Section D.4.3. The Lorentz profile has a FWHM of 2a, and its area is equal to % but it is normalized to unity in Equation (6.2). Thus, M(v) in Equation (6.2) expresses a Lorentz profile with its peak wavenumber at Vq shown in Figure 6.6a. 

The right-hand side of this equation is a mere cosine function as shown in Figure 6.6d, and its Fourier transform gives a delta function as shown in Figure 6.6e. This means that in an ideal case the assumed true bandshape is recovered by the above series of computations. What is needed in a practical case is an operation to narrow the bandwidth of the Lorentz profile to an appropriate degree. In more general terms, FSD is an operation to narrow (deconvolve) the bandwidth of a band by computations involving the (inverse) Fourier transform of the bandshape function itself (FSD). 

The exponential decay and the Lorentz profile are referred to in the descriptions of the method of Fourier self-deconvolution in Chapter 6. The function fix) expressing an exponential decay is defined as... 

Figure D.2 (a) An exponential decay function and (b) the Lorentz profile corresponding to...

The right-hand side of this equation represents a curve centering about v = 0, which is shown in Figure D.2b. This curve is called the Lorentz profile which is known to fit approximately the shapes of infrared absorption bands. The integrated area under this curve is 3t, and its full width at half maximum (FWHM) is 2a. 

In the descriptions of Fourier self-deconvolution and Figure 6.6, the convolution theorem in Equation (D6b) and the shift theorem for convolution with the delta function in Equation (D25) are repeatedly used. In Equation (6.2), the Lorentz profile centering at V = 0 is shifted to the same profile centering at v = vq by the shift property of the delta function. 

The relation between the exponential decay and the Lorentz profile and that between the cosine function and the delta function as the Fourier transform pairs are described, respectively, in Sections D.3.4 and D.3.5. The Fourier transform of the triangular function shown in Figure 6.6f is a sine function squared as described in Section D.3.6. 

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