Lorentzian peak profile

Fig. 15 Small-angle X-ray scattering curves of poly(L-arginine) retinoate (a), poly(L-his-tidine) retinoate (b) and poly(L-lysine) retinoate (c). The dashed lines are fits using a Lorentzian peak profile. Repeat units are 3.62 nm, 3.27 nm, and 3.10 nm. Reprinted with permission from [142]. Copyright 2000 American Chemical Society...

The infra-red measurements were of two types, normal-film measurements with the sample sandwiched between KBr plates, and tilted-film experiments with the sample sandwiched between 45° prisms of KBr, in each case with layers of Nujol to provide optical matching. Whereas the 1616 cm 1 Raman line occurs in a region well clear of other lines so that it was satisfactory to measure peak intensities, the infra-red spectrum of PET shows many overlapping bands. Accurate assessment of absorption intensities therefore requires the computer separation of the spectrum into a set of overlapping peaks (shown to be Lorentzian in profile) and a linear background. The procedures adopted and the band assignments are discussed in detail by Hutchinson et al. 6). 

Fig. 13 Isotopic line splitting of the V3 stretching vibration in single crystalline (see also Fig. 12(a)), after [108, 109], The origin of each absorption band is indicated by an isotopomer present in crystals of natural composition. While the absorption could be fitted by a Lorentzian band profile, the remaining peaks were dominated by the Gaussian contribution in the Voigt band shapes (solid lines below the spectrum). The sum result of fitting the isotopic absorption bands is inserted in the measured spectrum as a solid line...

Fig. 13. X-ray intensity profile with diffi action vector along qx (in the plane of layers) in the smectic X phase for the mixture of polyphilic compounds M70. The solid line through the data points is the smn of three Lorentzian peaks which are shown hy broken curves (Blinov et al. [44])...

From a more general point of view, we can assume that each of the effects related to the grain size or to microstrains leads to Gaussian and Lorentzian contributions to the peak profile [THO 87b]. Furthermore, we can consider the possibility that the shape of the grains and the microstrain tensor are anisotropic... 

The total width, T, is the sum of partial widths, which can be calculated but not observed separately. Only the total width can be observed experimentally. This width does not depend on whether the line is observed in an absorption, photoionization, photodissociation, or emission spectrum because the width (or the lifetime) is characteristic of a given state (or resonance). In contrast, the peak profile can have different line shapes in different channels the line profile, q, is dependent on the excitation and decay mode (see Sections 7.9 and 8.9). For predissociation into H+CT, the transition moment from the X1E+ state to the 3n (or 3E+) predissociating state is zero, consequently q = oo and the lineshape is Lorentzian. In contrast, the ratio of the two transition moments for transitions to the XE+ continuum of the X2n state and to the (A2E+)1E+ Rydberg states leads to q 0 for the autoionized peaks (see Fig. 8.26) (Lefebvre-Brion and... 

Often, it is found that certain data profiles are best described by non-linear bases. A typical example is curves from e.g. infrared spectroscopy [20] which are well described by a linear combination of Lorentzian peaks. Sometimes it is also possible to use Gaussian peaks. The Lorentzian peak function is written as... 

Figure 5 Normalized theoretical Gaussian (G), Lorentzian (L) and Voigt (V) line profiles versus dimensionless frequency. The FWHMs and areas of the Gaussian and Lorentzian peaks are equal the Voigt profile is the convolution of the (G) and (L) profiles.

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

Application. In practice, three or more peak orders must be observed. The shape of the lines should be Gaussian or Lorentzian. On these premises it is promising to carry out line profile analysis. Corresponding graphical separation methods are readily derived from the breadth relations that have just been discussed, after that the breadths B ( (x)) and B (Hdq,) (s)) of a polycrystalline ensemble have been related to structure and substituted in Eq. (8.29) or Eq. (8.30). For the size term the structure relation is... 

Two samples of the same phosphor crystal have quite different thicknesses, so that one of them has a peak optical density of 3 at a frequency of vo. while the other one has a peak optical density of 0.2 at vq. Assume a half width at half maximum of Av = IGHz and a peak wavelength of 600 nm, and draw the absorption spectra (optical density versus frequency) for both samples. Then show the absorbance and transmittance spectra that you expect to obtain for both samples and compare them with the corresponding absorption spectra. (To be more precise, you can suppose that both bands have a Lorentzian profile, and use expression (1.8), or a Gaussian line shape, and then use expression (1.9).)... 

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]. 

Fig. 5.8. Root mean square relative errors of model line shapes fitted to a quantum profile, the quadrupole-induced (XL = 23) component [69], The abscissa gives the ratio of peak intensity and wing intensity of the fitted portion of the exact profile. The superiority of the BC model (lower set of data points) over the desymmetrized Lorentzian (upper set) is evident.

It is a straightforward matter to fit various model profiles to realistic, exact computed profiles, selecting a greater or lesser portion near the line center of the exact profile for a least mean squares fit. In this way, the parameters and the root mean square errors of the fit may be obtained as functions of the peak-to-wing intensity ratio, x = G(0)/G(comax)- As an example, Fig. 5.8 presents the root mean square deviations thus obtained, in units of relative difference in percent, for two standard models, the desymmetrized Lorentzian and the BC shape, Eqs. 3.15 and 5.105, respectively. 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

It can also be noticed in Fig. 1 that spectral features for these three peaks are not symmetrical that is, their spectral shape deviates considerably from a simple Lorentzian line shape. Since the rotational contribution in the peak width in the PHOFEX spectrum is -1 cm-1, which is significantly smaller than the observed peak width, these asymmetrical spectral features are regarded as Fano-type profiles, which can appear in a spectrum for quasibound states. 

The Lorentzian profile (20) is going to appear often later in the resonance theory. It is a symmetric peak with a maximum at E = Er. The full width at half the maximum (FWHM) of this peak is r, and hence, the FWHM of the symmetric peak in Figure 4.1 is T if transformed into a function of the energy E. Therefore, the cross-section structure, i.e., the resonance structure, is narrower for smaller r. A feature of the Lorentzian profile (20) to be noted is that the width parameter T also determines its peak height 4h/ L. This fact will be taken advantage of in the resonance analysis procedure, as will be explained in Section 2.2.5. 

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