Line profile Gaussian

The corresponding halfwidth of such a Gaussian line profile, with its center at wavenumber t/Q, can be calculated according to... 

These examples illustrate that in the visible and UV regions, the Doppler width exceeds the natural linewidth by about two orders of magnitude. Note, however, that the intensity / approaches zero for large arguments (v — vq) much faster for a Gaussian line profile than for a Lorentzian profile (Fig. 3.7). It is therefore possible to obtain information about the Lorentzian profile from the extreme line wings, even if the Doppler width is much larger than the natural linewidth (see below). 

Effect of the geared monomer rotation Ry on the dimer vibrational spectrum of (SiF4)2. The stick spectrum indicates the calculated frequencies and transition strengths, the envelope is obtained by adopting Gaussian line profiles, fwhm 6 cm. The effect of the relaxation of R is included in the spectra, but it is hardly visible. 

The first two broadening mechanism produce Gaussian line profiles. 

Fig.3.5. Comparison between Lorentzian and Gaussian line profile of equal halfwidths...

Magnification 2 MX. Composition range is shown on the intensity scale at RHS (Reproduced with permission by Carpenter et al. 1999). (B) Line profile extracted from Figure 5.24a averaged over 30 pixels parallel to the boundary. The solid curve is a Gaussian distribution fitted to the data (Reproduced with... 

Figure 5.25. (A) Quantitative Cu map of an Al-4wt% Cu film at 230 kX, 128 x 128 pixels, probe size 2.7nm, probe current 1.9 nA, dwell time 120 msec per pixel, frame time 0.75 hr. Composition range is shown on the intensity scale (Reproduced with permission by Carpenter et al. 1999). (B) Line profile extracted from the edge-on boundary marked in Figure 5.25a, averaged over 20 pixels ( 55 nm) parallel to the boundary, showing an analytical resolution of 8nm FWTM. Error bars represent 95% confidence, and solid curve is a Gaussian distribution fitted to the data (Reproduced with permission by Carpenter...

Application. In practice, three or more peak orders must be observed. The shape of the lines should be Gaussian or Lorentzian. On these premises it is promising to carry out line profile analysis. Corresponding graphical separation methods are readily derived from the breadth relations that have just been discussed, after that the breadths B ( (x)) and B (Hdq,) (s)) of a polycrystalline ensemble have been related to structure and substituted in Eq. (8.29) or Eq. (8.30). For the size term the structure relation is... 

Two samples of the same phosphor crystal have quite different thicknesses, so that one of them has a peak optical density of 3 at a frequency of vo. while the other one has a peak optical density of 0.2 at vq. Assume a half width at half maximum of Av = IGHz and a peak wavelength of 600 nm, and draw the absorption spectra (optical density versus frequency) for both samples. Then show the absorbance and transmittance spectra that you expect to obtain for both samples and compare them with the corresponding absorption spectra. (To be more precise, you can suppose that both bands have a Lorentzian profile, and use expression (1.8), or a Gaussian line shape, and then use expression (1.9).)... 

A method known as Rietveld analysis has been developed for solving crystal structures from powder diffraction data. The Rietveld method involves an interpretation of not only the line position but also of the line intensities, and because there is so much overlap of the reflections in the powder patterns, the method developed by Rietveld involves analysing the overall line profiles. Rietveld formulated a method of assigning each peak a gaussian shape and then allowing the gaussians to overlap so that an overall... 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

The conventional Lorentzian, Gaussian, etc., profiles mentioned above are all symmetric, g(—v) = g(v). In contrast to this symmetry, the individual line profiles in collision-induced absorption have early been recognized as being quite asymmetric, roughly as [120, 215, 188]... 

Fig. 2. Emission line profiles of M1-5, M1-9, K3-66, and K3-67. Crosses and real lines are observational results, broken lines are Gaussian components.

Gaussian Laser Profile-Voigt Atom Profile. This case turns out to be a better approximation of our experimental situation, i.e., the laser FWHM is fairly broad compared to the absorption line width and the absorption profile of atoms in an atmospheric combustion flame is described by a Voigt profile. Here the laser is assumed to have a Gaussian spectral profile as well as a Gaussian atomic absorption profile. In this case, convolution of two Gaussian functions is still a Gaussian function. Evaluation of the ratio n2/nT, and the fluorescence radiance. Bp, allows determination of the half width of the fluorescence excitation profile, 6X... 

A numerical calculation of the line profiles due to the combined effect of the natural lifetime, the light-shift and the saturation has been performed taking into account all possible trajectories of atoms inside the metastable beam. Actually, the study of experimental linewidths shows there are some other stray effects responsible for the broadening of the lines. He have considered their contribution by making a convolution of the line profile with a gaussian curve. 

Therefore, the fractional column density (and the total column density N according to Eq. (20)) can only be obtained from the optical depth if Tex is known. If a gaussian line shape is assumed, integration of the observed line profile yields... 

For optical depth t 1 the observed interstellar molecular lines usually have a gaussian shape. This is to be expected, since collisional broadening, which causes the Lorentzian line profile, should be negligible and become important only at gas densities 1012 — 1014 cm- 3. If thermal motions of the molecules were the only source of line broadening the line half power width (i.e. the width between half power points) would be given by... 

The line shapes are described by Voigt functions, which reflect the Lorentzian line profiles due to natural line width and Gaussian profiles due to Doppler broadening. The instrumental broadening by the rocking curve of the crystal, de-focusing and the finite resolution of the detector is described well by a Voigt profile shape too [3[. 

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