Line profile Lorentz

Blickensderfer et al. (119) have recently extended the Samson theory to include various vessel geometries under conditions of pure Doppler, pure Lorentz, and Voigt (a combination of Doppler, Lorentz, and natural broadening) line profiles. The quenching cross section can be obtained also by a dynamic method in which the decay of the fluorescence intensity is measured after the exciting light is cut off. If the initial fluorescence intensity is / and the intensity after a lime f is If, we obtain... 

This can be substituted into Equation 1.17 in the same way as was the Lorentz function, to give a value for the absorption coefficient of the Doppler broadened line profile. 

Collisions between molecules are the greatest cause of line broadening at the pressures normally employed for MMW spectrometry. In the Lorentz theory (ref 2, p. 338) the lifetime of the rotational state involved in the transition is ended abruptly by collision with another molecule which stops the rotation. When the molecule starts to rotate again, its phase with respect to the other molecules is random. For an assembly of molecules this will give rise to an absorption line profile with a FWHM of Xjlm, where r is the mean time between collisions. This is the linear sum of two terms, one for the upper and one for the lower state, having the shape of the Lorentz function (Figure 1.4) when Av [Pg.12]

This imcertainty in frequency, which is inversely proportional to the lifetime, generates a line profile of Lorentz shape, centered at vq, with a width Using the relation AAn = (A /c) Ai/N the so-called natural line width AAn is obtained and the corresponding wavelength-dependant intensity distribution /n(A) of the area-normalized profile is given by ... 

Figure 2.3 Comparison of Gauss (blue line) and Lorentz (green line) curves of equal area and same FWHM, and a Voigt (red line) profile produced by convoluting the other two curves...

Figure 6.1 Comparison of 26 — 6 scan profiles obtained by a monochromatized (pure Cu kal) parallel beam configuration (hybrid x-ray mirror) and a conventional parallel beam configuration achieved by divergence slit (ds) module measured at 001/100 (a), 002/200 (b), 003/300 (c), 004/400 (d) of 500nm-thick Pb(Zro.B4Tio.46)03 thin film. Dotted lines represent the second derivative of the profiles, indicating the peak positions. Note that the profiles are simulated fitted profiles for obtained spectrum using pseudo-Voight function (mixed Lorentz and Gauss function).

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26],... 

Equations 1.16 and 1.17 are based on the assumption that all the molecules undergoing the transition m —> do so at the same frequency v. In reality they will have slightly different transition frequencies centred around the centre frequency Vq due predominately to collisional interactions between molecules. Doppler broadening also makes a small contribution giving a Gaussian shape to the line (Figure 1.4), but the overall result is a profile approximated by the Lorentz shape function 5 (v) ... 

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... 

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,... 

The observable profile of a spectral line is, in general, neither a pure Lorentz nor a pure Gauss distribution but a combination of both, known as a Voigt profile. If it is assumed that Doppler and collision broadening are independent processes, the Voigt profile is the result of the convolution of the Lorentz distribution with AAc and the Gauss distribution with AAd. Since the Voigt profile cannot be obtained analytically, numerical convolution procedures have to be applied. A parameter often used for profile characterization is the... 

The FWHM of the Voigt profile, the so-called Voigt line width AAy, cannot be obtained by simple addition of the Doppler and Lorentz widths, but can be approximated by an empirical formula ... 

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. 

Figure 2.14 Normalized, shot-noise determined, minimnm detectable absorbance for Gauss- and Lorentz-shaped absorption lines as function of AAinst rectangular instrument profile assumed...

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... 

---