Lorentzian line function

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 Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

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. 

Fig. 8.4 Fe Mossbauer spectra of [Fe(2-pic)3]Cl2 C2H50H as a function of temperature source Co/Cu at room temperature). The outer two lines represent the quadrupole doublet of the T2 (HS) state, the inner two lines that of the Aj (LS) state. The solid lines are obtained by a least-squares computer fit of Lorentzian lines to the experimental spectra (from [11])...

A plot of v vs. T2(a>o co) is shown in Figure 5.1. Equation (5.14) corresponds to the classical Lorentzian line shape function and the absorption curve of Figure 5.1 is a Lorentzian line . The half-width at half-height is easily found to be ... 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

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).

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

Here N designates the normalization factor. Clearly this equation in integrated form is the product of Gaussian and Lorentzian distribution functions 0mg and 0m define the line-widths of the two components, respectively. Here, the former represents Eq. (17) to a sufficient approximation for 0m 2 G and the latter was introduced to express the coupled rotational and/or the translational motion of proton pairs in the polymer, discussed by Pechhold53. ... 

Since NMR spectra are not sequences of lines representing discrete Larrnor frequencies but sequences of Lorentzian frequency distributions f(to) (Fig. 1.9), eq. (2.10) must be replaced by eq. (2.11) M0 sin c is multiplied by the frequency function f(to), where a> represents the difference between the frequency ojx and the Larrnor frequency distribution con + Aw, w = co1 — (w0 + Aro). Further, Mosin0f(tu)e must be integrated over the Larrnor frequency distribution. Given a Lorentzian line shape as in Fig. 1.9, the limits of integration are oo ... 

Meiler and Pfeifer (493) measured 13C and H NMR spectra of carbon monoxide, carbon dioxide, and benzene adsorbed on ZSM-5 and silicalite. The 13C signal from benzene was a superimposition of two lines corresponding to relatively mobile molecules (narrow Lorentzian line) and strongly adsorbed molecules (broad asymmetric line similar to that in polycrystalline benzene). Quantitative interpretation of the spectrum was possible via the measurement of the transverse proton relaxation times, T2, as a function of temperature and coverage. Recent work involving 13C NMR studies of sorbed species is summarized in Table XX. 

Fig. 7.6. (a) Energy dependence of a Lorentzian line-shape function with width KT centered at the resonance energy (Ei + 6E). (b) Partial photodissociation cross sections a(E,0) as given by (7.23). All of them have the same width hT the values at the maximum scale like the partial decay rates Tp. 

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).

H-bonded systems may require additional diffuse or polarization functions. For example, the 6-311++G(d,p) basis set had been found to be suitable for H-bonded systems [78-81], It may be necessary to include Basis Set Superposition Errors (BSSE) [82] and Zero-Point-Energy (ZPE) corrections in evaluating the relative stabilities. Such corrections are often of the same magnitude as the energy differences among the dominant conformers. Moreover, the relative conformer energies may also differ noticeably with the basis sets used. All these factors will affect the Boltzmann factors predicted for different conformers and therefore the appearance of the population weighted VA and VCD spectra. Thus, an appropriate selection of DFT functionals and basis sets is very important for VCD simulations. A scale factor of 0.97-0.98 is usually applied to the calculated harmonic frequencies to account for the fact that the observed frequencies arise from an anharmonic force field instead of a harmonic one. A Lorentzian line shape is typically used in simulations of VA and VCD spectra. The full-width at half maximum (FWHM) used in the spectral simulation is usually based on the experimental VA line widths. 

FT theory shows that the effect of abrupt truncation of the FID is to convolve the normal Lorentzian line shape with a function of the form... 

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[. 

From the relative NMR areas obtained from the instrnmental integration, calculate the value of Kp and its uncertainty. If possible obtain the spectrum as an ASCII intensity file in order to obtain improved integration values by doing a nonlinear least-squares fit of the data. A suitable function based on Lorentzian line shapes is... 

These functions describe an absorption with a Lorentzian line shape for which n and k vary, as shown in Fig. 3.19. This is identical to the dispersion illustrated earlier in Fig. 2.4. Quite clearly this model will also apply to molecular vibrations that produce an oscillating dipole. Hence the dispersion associated with vibrations that give rise to infra-red absorption will be of the same form, see Fig. 2.4. 

Fig. 28. Excitation spectra of Pt(2-thpy)2 dissolved in n-octane (a) at zero magnetic field and T = 4.2 K and (b) for different magnetic fields at T = 1.5 K. Concentration = 10 mol/1. For detection, the energy of 16,444 cm with a band width of = 5 cm was used, in order to monitor simultaneously the 713 cm and 718 cm vibrational satellites that correspond to the emissions of the triplet substates I and II, respectively. Under magnetic fields (b), the detection energy is red-shifted according to the size of the field-induced red shift of these satellites. The total excitation spectrum is composed of different spectra. The spectral resolution of the equipment is = 5 cm and = 160 cm for energies below and above the vertical line near 22,300 cm respectively. Note, the halfwidth given in (b) refers to the fwhm of a Lorentzian line shape function which was fit to the red flank of the corresponding peak. (Compare also to the Refs. [74] and [95])...

In indirect methods, the resonance parameters are determined from the energy dependence of the absorption spectrum. An important extra step — the non-linear fit of (t E) to a Lorentzian line shape — is required, in addition to the extensive dynamical calculations. The procedure is flawless for isolated resonances, especially if the harmonic inversion algorithms are employed, but the uncertainty of the fit grows as the resonances broaden, start to overlap and melt into the unresolved spectral background. The unimolecular dissociations of most molecules with a deep potential well feature overlapping resonances [133]. It is desirable, therefore, to have robust computational approaches which yield resonance parameters and wave functions without an intermediate fitting procedure, irrespective of whether the resonances are narrow or broad, overlapped or isolated. 

To reproduce the inherent broadening of the experimental vibronic spectrum, the stick vibronic lines obtained from the matrix diagonalization calculations are usually convoluted [7] with a Lorentzian line shape function... 

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