Lorentz function

Here, instead of the usual delta function d(Eg, — Eg — ftv), which supplies only discontinuous spectral lines, a Lorentz function with a line width 29(normalized to one) is introduced for all vibrational levels labeled by n and m of electronic states j and k which contribute to the transition rate of Eq. (9) [58]. M is the transition operator, in general, pertinent to an electric dipole transition. The nuclear wavefunctions are approximated by products of harmonic oscillator functions... 

For evaluating Eq. (9) one rewrites the Lorentz function as a Fourier integral... 

The two simplest peak shape functions (Eqs. 2.49 and 2.50) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Figure 2.42, from which it is easy to see that the Lorentz function is sharp near its maximum but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base but has a rounded maximum. Both functions are centrosymmetric, i.e. G(x) = G -x) and L x) = L -x). 

It can be seen from Eq. (188) that the circular intensity difference is a complicated function of Aw that differs from a simple superposition of Lorentz functions. The temperature dependence is also rather complicated since in vibronic systems the matrix elements m jiK-K]m ) between the states m) and m ) of differing energy (wm + wm ) are nonzero. 

The characteristic emission of X-rays from an X-ray tube, in the strict sense, is not monochromatic (Figure 6.2). The energy distribution of a single characteristic emission line can be described by the Lorentz function ... 

Thus, the transfer function is the complex Lorentz function, and the linear response in the time domain is the inverse Fourier transform of K (w) (cf. Section 4.1),... 

Attempts have been made to simulate the X-ray signal of the ACP in plasma-sprayed hydroxyapatite coatings by a Lorentz function added to the Rietveld refinement algorithm (Hesse et al., 2008). This provides a reasonable fit of the... 

Sliwinski, P. The Lorenz-Lorentz Function of Gaseous and Liquid Ethane,... 

Bmn in Equation 1.17 can be replaced in terms of the sum of the squares of the dipole moment matrix elements (Equation 1.3) and the equation modified by the Lorentz function, yielding the general form of the equation for a the absorption coefficient at any point on the spectral profile ... 

This can be substituted into Equation 1.17 in the same way as was the Lorentz function, to give a value for the absorption coefficient of the Doppler broadened line profile. 

Collisions between molecules are the greatest cause of line broadening at the pressures normally employed for MMW spectrometry. In the Lorentz theory (ref 2, p. 338) the lifetime of the rotational state involved in the transition is ended abruptly by collision with another molecule which stops the rotation. When the molecule starts to rotate again, its phase with respect to the other molecules is random. For an assembly of molecules this will give rise to an absorption line profile with a FWHM of Xjlm, where r is the mean time between collisions. This is the linear sum of two terms, one for the upper and one for the lower state, having the shape of the Lorentz function (Figure 1.4) when Av [Pg.12]

Figure 35 Second derivative of a peak based on a Lorentz function.

ESR lines in solution can almost always be approximated by a Lorentz function. In the solid state the line-shape can in general be reproduced by a Gauss curve. In some instances a so-called Voigt profile can give a better approximation to the experimental line-shape. A Voigt line is a convolution of a Lorentz and a Gauss line. The shape is determined by the ratio ABi/ABg of the respective line-widths. The shapes of the 1st derivative lines of these types are given in Fig. 9.1. 

The method to obtain relaxation times from CW microwave saturation measurements for an inhomogenously broadened line is based on assumptions given in the literature [77, 78], The ESR line shape is then expressed as a convolution of a Gauss and a Lorentz function [81] ... 

Temperature dependence of the (a) spectral linewidth and (b) integrated intensity in UPd2Al3 when the spectra are analyzed by a single-line Lorentz function. Solid curve is a theoretical one calculated by the Debye model. (Modified from Fig. 2 of Ref. 7 with permission of Springer.)... 

When the absorber is thin, the observed transmission Mossbauer spectrum is a superposition of several Lorentz functions. 

Lorentz functions depending on the experimental conditions cf. the section on Peak Decomposition) and the number of components have been tentatively selected. 

Figure 2 Temperature dependent characteristic parameters ofP(S-b-B) during an upward and downward temperature scan, inverse of the normalized scattering peak maximum, I o/l (i o is the peak intensity at the transition), maximum position, q, and peak width, Aq, as obtained from fits of a Lorentz function to the q dependent scattering intensity. The solid lines are guides to the eyes. The data point in brackets represents the peak intensity of the sample directly q/ter the preparation. The dotted line indicates the position of the microphase separation temperature Tmst-...

Since this is a Lorentz function, the half-width at half-height 8v is related to T by... 

Fig. 23 shows Si 2p peak. Table 1 below shows the spin-orbital splitting values of this peak (components ratio in the doublet is 2 1), as well as values of full width at half-height of Gauss and Lorentz functions used in experimental spectra decomposition, energy shifts for surface component of silicon, and its compounds in relation to its bulk component. The values shown in the table 3 were taken from (Olmstead et al., 1986). The shift value for Si-Ba bond was determined in the course of spectrum decomposition, and slightly differed from similar energy shift for Si-Ca bond. 

As the reactant approaches the electrode surface, its valence orbitals start to interact with the metal electrons. Consequently, the energy of the valence orbital is no longer sharp, but is broadened into a density of states (DOS) pa( ), where e denotes the electronic energy. Metal sp bands are broad, and they induce a general, unspecific broadening, which resembles a Lorentz function. They behave roughly in the same way on all metals. The d bands are narrower, and they induce a specific structure into the reactant s DOS. From the DOS, one obtains the occupation probability of the valence orbital and its electronic energy ... 

Voigt function the Voigt function is the convolution of a Lorentz function with a Gauss function ... 

A Lorentz function is smaller than a Gauss function near the peak maximum, but the principal distinguishing characteristic of a Lorentz function is that it has significant... 

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