Lorentzian broadening, resolution

For the purpose of comparison with the measured absorption coefficient, the theoretical spectra are convoluted with a Lorentzian broadening function F(E). This function is the sum of two terms. The first takes account of the core hole width and the second term is the width of the excited band energy, which is a function dependent on the mean free path of the excited electrons, and takes account of the photoelectron inelastic scattering which is energy dependent and varies for each material as shown in Fig. 1. Note that in this theory any broadening effect due to the experimental resolution and many-body effects, such as the influence of the core hole on the band states, are not included. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

A high-resolution spectrum of the clock transition is shown in Fig. 2. The clock-laser power was reduced to 30 nW to avoid saturation broadening. The fit with a lorentzian curve results in a linewidth of 170 Hz (FWHM), corresponding to a fractional resolution bv/v of 1.3 10-13. A spectral window of 200 Hz width contains 50% of all excitations. According to our present experimental control of the ion temperature, electromagnetic fields and vacuum conditions, no significant Doppler, Zeeman, Stark or collisional broadening of the absorption spectrum of the ion is expected beyond the level of 1 Hz. The linewidth is determined by the frequency instability of the laser and the lineshape is not exactly lorentzian... 

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[. 

Figure 4.7. Observed and curve-fitted Si spectra of three erystalline forms of the silica polymorph tridymite. A. Room-temperature ordered monoclinic form, showing resolution of nine of the twelve Si sites, fitted to 12 pseudo-Voight lines of equal area with a Gaussian Lorentzian ratio of 0.3. B. Orthorhombic form at 142°C, fitted to six lines broadened by an ineommensurate plane-wave modulation. C. Orthorhombic form at 202°C, fitted to a single line simulated with a non-linear incommensurate modulation. Adapted from Kitchin et al. (1996), by permission of the...

Doppler broadening has a Gaussian lineshape, and its convolution with the Lorentzian natural lineshape yields a Voigt profile. In typical experiments, this effect can be neglected since the Doppler width is usually much smaller than the resolution of the apparatus. Collisional line broadening is also Lorentzian, and the Lorentzian component of measured lines must be carefully extrapolated to zero pressure. 

Relatively weak vibronic modes are evident in the tails which broaden the bands. Even when these are eorrected, however, the best fit is unequivocally Lorentzian. The inadequaey of the spectrophotometer to give eorrect contours at the tops of the peaks and between the two major peaks becomes evident in the resolutions. 

The type of the line-broadening function (LBF) may be either Lorentzian or Gaussian. The appropriate type depends on the nature of the line broadening mechanism. If in doubt it is recommended to start with a Lorentzian shape. After you have selected a file and wavenumber range as usual in the dialog box (Fig. 10.50), specify on the Adjust Parameter page a deconvolution factor and a suppression factor for the noise, or alternatively, a factor for the bandwidth and the resolution enhancement. Start the function using the Deconvolute button. 

The photoemission data of Er and Sm have been reported by Brod6n (1972), and the ODS curve for Er is shown in fig. 3.47. The theoretical density of states curve in the comparison was obtained by broadening the calculated curve of Keeton and Loucks (1968) with a Lorentzian resolution function with a width of... 

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