Lorentzian broadening function

Figure 1. 205T1-NMR spectrum (solid line) at 5 K. The intensity is plotted in a linear scale. The thin solid line depict the histogram at particular local fields of the Readfield pattern. The dotted line represents the simulation spectrum convoluted with Lorentzian broadening function. The filled circles show the frequency dependence of 205 f,1 1 at the T1 site. The inset shows the image of the field distribution in the vortex square lattice center of vortex core (A), saddle point (B) and center of vortex lattice (C).

The results of the lineshape analysis for the observed Si spectra at -120, 25, 60, 90 and 120°C are shown in Fig. 17.10, where the Si signal is deconvoluted with Lorentzian broadening functions. The chemical shifts, linewidths and peak intensities at temperatures from -120 to 120°C, determined from the lineshape analysis, are given in Table 17.3. As shown in Table 17.3, the halfwidths of these peaks decrease as the temperature is increased from -120 to 120°C. This means that the decrease in linewidths for the Si resonances comes mainly from the increase of mobility as the temperature is increased. All the Si peaks shift to high frequency as the temperature is increased from -120 to 120°C. According to the calculations, this means that the populations of the GG and GT conformations increase. 

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. 

These powder line shapes have been convoluted with Lorentzian broadening functions of different widths of A = [cri i — (T22] >o, where cuq is the NMR frequency. 

We will first consider the error bounds for a spectral density broadened by a Lorentzian slit function, Eq. (15), describing the response to an exponentially damped perturbation. In this case the broadened spectrum,... 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

Another approach to estimating spectral densities, which has the advantage of guaranteeing that the approximate functions are positive, can be based on the error bounds constructed in Section III-A for the spectral density broadened by a Lorentzian slit function. If we had a sufficient number of moments to make the error bounds very precise, then we could reduce the broadening as much as we like, so that the broadened distribution of spectral density becomes as close as we like to the true distribution. In order to estimate these higher moments, we should need to take advantage of some special feature of the distribution. For example, in the case of the harmonic vibrations of a crystalline solid, the distribution of frequencies lies between limits — co,nax and +comax, and is zero outside... 

However, the Lorentzian form of the dipolar broadening function, which has the advantage of mathematical simplicity, is not suitable for an interpretation in terms of second moments it is replaced with a Gaussian dipolar function S(oa, AG), where the parameters AG correspond to the appropriate fractions of the square root of the intra-group rigid lattice second moments. With appropriate values for AG, calculated and experimental line shapes I(oo) are found to be in a good agreement for cross-linked polyethylene oxide) swollen in chloroform 1U). 

Figure 31 Calculated imaginary part of the dielectric function of Si and Ge quantum films for different orientations and thicknesses. The curves are convolved with Lorentzian broadening of 0.1 eV.

Calculation of X-ray profiles was performed in steps of 0.04° throughout the 6-50° 20 angular region by applying the procedure used in the structure determination of H-BOR-D (6). In this procedure the instrumental broadening was simulated by convoluting the sample profiles with two Lorentzian line functions, with a 2 1 intensity ratio and a full width at half maximum of 0.1° 20, representing the contribution of Ka. and Ka lines, respectively. 

The width (fwhm) of the electronic origin peak at 23,836 cm is 6.5 cm" at T = 1.2 K. Its lineshape can be well approximated by a Lorentzian Hneshape function, while a fit to a Gaussian lineshape is unsuccessful [74]. This indicates that the peak is homogeneously broadened [103,104]. Thus, the width of 6.5 cm" can be related to the lifetime of this Sj state to r(Si) = 8 x 10 s. (Compare Eq. (2).) Eurther, the high phosphorescence intensity that is observed from the Tj state when the Sj state is excited, indicates that the lifetime of this Sj state is largely determined by the intersystem crossing process from Sj to Tj. This would correspond to the rate of kjs = 1/r(Si) = 10 s. A value of the same order of magnitude has also been determined for Pd-phthalocyanine [144]. 

So the first thing we get by allowing H to have complex diagonal elements is that state populations (diagonal elements of the density matrix) can decay exponentially with the expected decay rate, (1 /tj = r /fi), and this exponential decay corresponds to a Lorentzian broadened level (as opposed to a (5-function) centered at e3 with the expected full width at half maximum... 

Lorentzian line broadening function. The apodization function most commonly used to emphasize the signal-rich initial portion of a digitized FID. 

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... 

Fig. 21. Full curves 4f-contribution to the XPS and BIS spectra as a function of ej. A Lorentzian broadening of 1 eV FWHM is used. Dashed curves Total energy difference e, between the ground state and the final eigenstates as a function of E( (see fig. 20) (Imer and Wuilloud 1987).

Fig. 13. The core level XPS spectrum as a function of A. The shape of V(zY was chosen to simulate a compound with a transition element and A refers to the maximum of %V z). The average value of V(sf is A — OMA. For details about K(s) see fig. 6 in Gunnarsson and Schonhammer (1983b). The spectra are normalized to the height of the f peak and a 1.8eV (FWHM) Lorentzian broadening has been introduced.

FIGURE 8.8 Radial distribution function g r) for the octathio[8]circulene crystal in supercells of dimension 3 x 3 x 3 A. Note that a Lorentzian broadening (F) of 0.06 is used for smoothening the crystalline 5-functions at the peak positions (dotted histogram). 

A product presentation for harmonic oscillator lineshapes with Lorentzian broadening is employed for the calculation of the lattice contribution to the dielectric function of uniaxial crystals, such as wurtzite-structure GaN [37] ... 

Another approach to resolution enhancement is the use of Fourier self-deconvolution, which is useful when the spectral features are broader than the instrument resolution [97]. This technique is based on the fact that the breadths of the peaks are the result of the convolution of the intrinsic line shape with a broadening function. Deconvolution allows the removal of the broadening. The broadening function is Lorentzian, and deconvolution with a Lorentzian function will lead to a substantial, physically significant narrowing [98]. 

---