Peak Lorentz

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

There are generally three types of peaks pure 2D absorption peaks, pure negative 2D dispersion peaks, and phase-twisted absorption-dispersion peaks. Since the prime purpose of apodization is to enhance resolution and optimize sensitivity, it is necessary to know the peak shape on which apodization is planned. For example, absorption-mode lines, which display protruding ridges from top to bottom, can be dealt with by applying Lorentz-Gauss window functions, while phase-twisted absorption-dispersion peaks will need some special apodization operations, such as muliplication by sine-bell or phase-shifted sine-bell functions. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

Model Lorentzian Peaks. If all the terms on the right-hand side of Eq. (8.13) can be modeled by Lorentz curves, the integral breadth of the observed peak... 

It should be noted that the Raman-inactive soft mode is observed in the temperature region above Tc. A spectral shape completely different from that of the Lorentz-type peak function indicates the defect-induced Raman scattering (DIRS) in the paraelectric phase of ST018. When centrosymmetry is locally broken in the paraelectric phase, the nominally Raman-inactive soft mode is optically activated locally to induce DIRS in the soft mode. 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

Fig. 4 Effect of nanoclay loading on neat SEBS a Lorentz -corrected SAXS profiles (vertically shifted for better clarity) showing effect of nanoclay arrows indicate peak positions, b Lengths corresponding to first- and second- order scattering vector positions along with the 2D SAXS patterns for each sample of clay-loaded nanocomposites...

Data Processing. Fraser (21) has shown that, with data measured in array form, each array member may be individually given Lorentz (velocity), polarization and absorption corrections. This avoids the problems incurred in the Photometric Peak Center method and Sum Intensity method in which the corrections are applied as if all intensity is recorded at the peak amplitude position. Fraser (22) has also shown that, by choosing data... 

Lorentz corrections. As can be seen, there are significant changes in the resolved parameters, particularly the peak positions and peak heights the effect on peak width is less marked. The standard fibre correction (LOR 1) gives the most reasonable results, the correction (LOR 2), suggested for the equatorial trace of a fibre specimen, is not considered to be realistic. 

The PL-RP approximation allows us to estimate this quantity in another fashion. It is seen in Fig. 6b that near the peak-absorption frequency the contribution of the precessional component Lpp is negligible, so that L = Lpl at A = xm. We shall estimate xm by considering the limit y > 0. The integrand in Eq. (79) comprises the Lorentz-like terms. In this limit their imaginary... 

In Figs. 45 and 46 we see the frequency the dependences typical for resonance-absorption. The isothermal line (at Ft) is characterized by the loss peak, whose intensity is evidently larger and bandwidth is narrower than those pertinent to two other lines. The Lorentz line (at FB) is characterized by (i) a smallest maximum loss and (ii) an absence of the loss shoulder at lower frequencies. This shoulder appears (at Ft and Fq) due to the denominator in Eqs. (370) and (372), pertinent to the self-consistent collision models. Since the plateau-like loss curve is indeed a feature characteristic for water in the... 

In Figs. 66 and 68 the calculated absorption and loss spectra are depicted for ordinary water at the temperatures 22.2°C and 27°C and for heavy water at 27°C. The solid curves refer to the composite model, and the dashed curves refer to the experimental spectra [42, 51]. For comparison of our theory with experiment at low frequencies, in the case of H20 we use the empirical formula [17] comprising double Debye-double Lorentz frequency dependences. In the case of D20 we use empirical relationship [54] aided by approximate formulae given in Appendix 3 of Section V. The employed molecular constants were presented in previous sections, and the fitted/estimated parameters are given in Table XXIV. The parameters of the composite model are chosen so that the calculated absorption-peak frequencies ilb and vR come close to the... 

Figure 6.1 Comparison of 26 — 6 scan profiles obtained by a monochromatized (pure Cu kal) parallel beam configuration (hybrid x-ray mirror) and a conventional parallel beam configuration achieved by divergence slit (ds) module measured at 001/100 (a), 002/200 (b), 003/300 (c), 004/400 (d) of 500nm-thick Pb(Zro.B4Tio.46)03 thin film. Dotted lines represent the second derivative of the profiles, indicating the peak positions. Note that the profiles are simulated fitted profiles for obtained spectrum using pseudo-Voight function (mixed Lorentz and Gauss function).

Fig. 22 Infrared reflectivity of a powder sample of Na2CsC60. A Lorentz Drude peak is shown for 20 and 200 K but at 300 K, the metallic behavior is lost. (From [55])...

There are other factors affecting the intensity of the peaks on a x-ray diffraction profile of a powdered sample. We have analyzed the structure factor, the polarization factor, and the broadening of the lines because of the dimensions of the crystallites. Now, we will analyze the multiplicity factor, the Lorentz factor, the absorption factor, the temperature factor, and the texture factor [21,22,24,26],... 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

Peak No. Energy0 (eV) Term value (eV) Assignment Lorentz width (eV) Peak area (.N) P density (N)... 

Spectral lines are conventionally described in terms of wave number ll = 1, with each line having a peak absorption at wave number % The Lorentz distr-... 

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). 

Figure 2.42. The illustration of Gauss (dash-dotted line) and Lorentz (solid line) peak shape functions. Both functions have been normalized to result in identical definite integrals...

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