Lorentzian function profile

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

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[. 

In all three methods, the assumptions of a Lorentzian band profile and of a triangular slit function were made. However, since Methods II and III involve measurement over the complete experimental curve, whereas Method I uses only three points of this curve, the latter is the most sensitive to the first assumption. Method II depends upon fairly small corrections related to the band shape and the slit function. Method HI is almost insensitive to the form of the slit function, but is much more strongly dependent upon the assumed band shape. Consequently, for partially overlapping bands, and in general, Methods II and HI are to be preferred, although Method III has speed in its favor. 

With local information given by INM analysis in mind, we next see the character of rotational relaxation in liquid water. The most familiar way to see this, not only for numerical simulations [76-78] but for laboratory experiments, is to measure dielectric relaxation, by means of which total or individual dipole moments can be probed [79,80]. Figure 10 gives power spectra of the total dipole moment fluctuation of liquid water, together with the case of water cluster, (H20)io8- The spectral profile for liquid water is nearly fitted to the Lorentzian, which is consistent with a direct calculation of the correlation function of rotational motions. The exponential decaying behavior of dielectric relaxation was actually verified in laboratory experiments [79,80]. On the other hand, the profile for water cluster deviates from the Lorentzian function. As stated in Section III, the dynamics of finite systems may be more difficult to be understood. 

The above description is actually a simplified version of reality since a high-resolution analysis of the spectral lines of Cu Koc shows that both the oci and 0C2 peaks are distinctly asymmetric. An understanding of the origin of this asymmetry is important in implementing the so-called fundamental parameters approach to the profile fitting of powder diffraction data peaks, described in Chapters 5, 6, 9 and 13, in which the detailed spectrum of the incident X-rays must be known. A combination of five Lorentzian functions is commonly used to model the peak shape of Cu radiation, though detailed investigations to characterize the X-ray spectrum continue. ... 

The shape fimctions most encountered in the analysis of absorption profiles are Gaussian fimction, Lorentzian function, Voigt function and the damped oscillator model. The mathematical expressions of these fimctions can be described in fimction of the variable v and in function of two parameters, one characterizing the band maximum vq and one describing the bandwidth. For the latter, different descriptions can be used ... 

This represents a Voigt profile, that is, a convolution product of a Lorentzian function with halfwidth y and a Doppler function. A comparison with Vol. 1, (3.33) shows, however, that the Doppler width is reduced by the factor sine = Vx/v = b jld, which equals the collimation ratio of the beam. The collimation of the molecular beam therefore reduces the Doppler width Aa>o of the absorption lines to the width... 

The combined effect of both kinds of collisions gives a line profile with a kernel that can be described by a Lorentzian profile slightly broadened by soft collisions. The wings, however, form a broad background caused by velocity-changing collisions. The whole profile cannot be described by a single Lorentzian function. In Fig. 8.4 such a line profile is shown for the Lamb peak in the laser output Pl(co) at... 

For isotropic liquids the pair correlation function g(r) takes the shape of a wave function, which decays proportionally to exp(-r/ ) (Fig. 4.11c). This is due to the lack of long range order within the liquid. If a is the average distance from one particle to the next, the corresponding scattering profile shows a Lorentzian function with a maximum at In a (Fig. 4.1 Id). The half width of this peak is reciprocally proportional to the correlation length of the short range order within the liquid. 

Identification of dynamics of a particular molecular fractions with diverse mobility Wider frequency range CP SS-NMR eases interpretation Analysis of NMR line width versus delay time (T profiles) using Gaussian-Lorentzian function (Gaussian part crystalline and Lorentzian amorphous fraction) for partially crystallinity Ambiguity about the source of biphasic spin relaxation decay profile (due to dynamic heterogeneity and/or hetero-nuclear coupling)... 

The second term of equation [1], where Li(co) is a Lorentzian function of amplitude Ai(Q), is responsible for the quasi-elastic broadenings of the spectra, related to the kinetics of the molecular motions. The analysis of an experimental spectra thus consists in simulating the profiles as a function of both the energy and momentum transfer variations by means of equation [1]. 

AAS because the analyte concentration is small. However, Lorentz broadening, which involves collisions between analyte atoms and foreign species (Ar atoms in electrothermal atomizers) called perturbers, are significant. Compared to a Doppler spectral profile, the Lorentzian profile is broader, has a lower peak height and is described by a Lorentzian function f(curve L in Figure 5) with the FWHM given by ... 

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

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