Lorentzian shape function

That is, Xw( ) has the Lorentzian shape function. The bandwidth is determined by the damping (or dephasing) constant. 

A comparison between the DOM and the Lorentzian shape function is given in Figure 12. 

SuiQ, E) = SiiQ, E) 0 Sr Q, E), that is, a convolution of the translational dynamic structure factor, Si(Q,E), and the rotational one, 5r(<2, ) In addition, for small Q spectra, Q < 1 A the 5r(<2, E) can be made negligibly small, hence 5 h((2, E) Si(Q, E) and its Fourier transform will give the self-intermediate scattering function F Q, t) that have a stretched exponential FniQj) = exp [ - r (g) r] long-time decay. When the T is above the room temperature, P 1. A situation for which the exponential form Eh(Q, t) exp(—r(g)/) can be approximately used, or equivalently, in frequency domain theSnCg, E) of water is approximated as a Lorentzian shape function [67],... 

If i = i — ik] and H2 = ns — are known as a function of wavelength, Eq. 12 can be used to calculate the entire RAIR spectrum of a surface film. Since transmission infrared spectroscopy mostly measures k, differences between transmission and RAIR spectra can be identified. Fig. 6 shows a spectrum that was synthesized assuming two Lorentzian-shaped absorption bands of the same intensity but separated by 25 cm. The corresponding spectrum of i values was calculated from the k spectrum using the Kramers-Kronig transformation and is also shown in Fig. 6. The RAIR spectrum was calculated from the ti and k spectra using Eqs. 11 and 12 and is shown in Fig. 7. 

We consider the same atom as in Case 1, with a valence electron at an orbital energy of = 12.0 eV above the bottom of the sp band, when the atom is far from the surface. This level is narrow, like a delta function. When approaching the surface the adsorbate level broadens into a Lorentzian shape for the same reasons as described above, and falls in energy to a new position at 10.3 eV. From Eq. (73) for Wa(e) we see that the maximum occurs for e = -i- A(e), i.e. when the line described... 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

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 ... 

When the atom comes closer to the metal surface, the electron wave functions of the atom start to feel the charge density of the metal. The result is that the levels 1 and 2 broaden into so-called resonance levels, which have a Lorentzian shape. Strictly speaking, the broadened levels are no longer atomic states, but states of the combined system of atom plus metal, although they retain much of their atomic character. Figure A.9 illustrates the formation of broadened adsorbate... 

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. 

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

Most peaklike functions become more gaussianlike when convolved with one another. One notable exception of interest to spectroscopists is the Cauchy function, which is the familiar Lorentzian shape assumed by lines in the spectra of gases subject to pressure broadening ... 

Once MOD parameters are obtained a spectrum can be simulated with a suitable choice of the band-shape function f. We have used Gaussian and Lorentzian functions for this purpose. In either case, a width parameter must be chosen. This parameter is generally chosen such that the widths of the peaks in the simulated spectrum are similar to those in the observed 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).

Figure 1.1a shows the Gaussian function. The Lorentzian shape is similar to the Gaussian, but falls off more slowly. The Doppler shift of radiation from an emitting molecule is proportional to its velocity component in the direction of observation. The one-dimensional distribution of speeds in a gas is a Gaussian function. (See any physical-chemistry text.) Hence when Doppler broadening is dominant, we get a Gaussian-shaped line. 

The system used for amplification and detection of an ESR signal is such that the first derivative of the absorption line is recorded. The shape of an ESR line in solution is usually Lorentzian [Equation (3.86)]. The Lorentzian shape resembles a Gaussian (except that it falls off more slowly). Differentiation of the Gaussian shape (3.89) gives — 2cd(v — v0)exp[-d(v - v0)2], which has the form of the u=l harmonic-oscillator function [(1.133) and (1.137)] with x — y v0. Thus the first-derivative of an absorption resembles Fig. 1.1b with the origin at v0. (See also Problem 8.22.)... 

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. 

Here kv is the absorption coefficient at frequency v, Nc is the number of absorbing centres per cubic centimeter, v is the frequency of absorption, and S(v) is the line shape function. For our estimates we shall assume that the line shape is Lorentzian having half width 6. If one evaluates the absorption cross section when the absorption is maximum the above expression takes the form... 

Here, T is the observed line width (Av << F), 7d is the peak-to-valley intensity in the difference spectrum, and To is the peak height of the Raman line. Although this equation is for Lorentzian-shaped bands, the results are approximately the same for Gaussian-shaped bands (the constant 0.385 becomes 0.350). In the case of carbon disulfide-benzene mixtures, the smallest shift observed was -0.06 cm-1, and the associated error was 0.02 cm-1 (77). A convenient rotating system that can be used for (1) difference spectroscopy, (2) normal rotating sample techniques (solid and solution), and (3) automatic scanning of the depolarization ratios as a function of the wave number has been designed (45). 

In the present chapter we shall start from the results obtained in Chapter 3 and treat the Stark effect of a hydrogenic atom or ion with the use of the phase-integral approximation generated from an unspecified base function developed by the present authors and briefly described in Chapter 4 of this book. Phase-integral formulas for profiles, energies and half-widths of Stark levels are obtained. The profile has a Lorentzian shape when the level is narrow but a non-Lorentzian shape when the level is broad. A formula for the half-width is derived on the assumption that the level is not too broad. 

We pointed out in Section 2.5 that the minimum line width as given by the uncertainty principle is proportional to 1/T,. We can now be more precise. Because T2 — T2 S T, we can express the true line width in terms of T2 and the observed line width in terms of T2. If the line shape function g( v) in Eq. 2.16 is Lorentzian, it can be shown that the observed width of the line at halfmaximum is given by... 

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])...

A symmetrical spectral band is described by three parameters position (wavelength or frequency corresponding to the absorption maximum), intensity (absorbance or molar absorptivity at the band maximum) and width (usually the bandwidth at half-height). The band shape functions most commonly used for deconvolution are the Gaussian function and the Lorentzian function. Both are symmetrical functions. UV-visible spectra generally have a Gaussian band shape. The Lorentzian function is useful for the simulation of NMR spectra. The... 

At each th projection point, the ath atom contribution to the yth descriptor is obtained by a Lorentzian function L (but can be any bell-shaped function) ... 

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