Lorentzian profile

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])...

We approximate the phonon coupling effects by replacing in our spectral sums in Eqs. (41)-(43) and (46)-(49) the discrete summation over different ripplon harmonics by integration over Lorentzian profiles ... 

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. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

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... 

The quantum model of direct damping by Rosch and Ratner [58] leads to the same kinds of profiles as does the semiclassical model of indirect damping by Robertson and Yarwood [84], with the two limit situations of slow or fast modulations leading to Gaussian or Lorentzian profiles. 

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).)... 

In contrast to the narrow signal of H(I), which exhibits a single Lorentzian profile, the feature observed for H(IV) and H(V) was more complex (Fig. 23 lb) with splitting into five lines indicating a superhyperfine structure (line denoted A in Fig. 23 2). This phenomenon is due to the hyperfine interaction between localized spin and two neighboring nitrogen nuclei (nuclear spin... 

This is a Gauss profile with a lorentzian hole, the width of which is determined by the homogeneous linewidth parameters 7ab> T with 7ab " rb depending on the lifetimes 7a, Tb of... 

The importance of the magnetic coupling is easily seen in Fig. 17 which shows two water proton MRD profiles for serum albumin solutions at the same composition (89). The approximately Lorentzian dispersion is obtained for the solution, and reports the effective rotational correlation time for the protein. The magnetic coupling between the protein and the water protons carries the information on the slow reorientation of the protein to the water spins by chemical exchange of the water molecules and protons between the protein and the bulk solution. When the protein is cross-linked with itself at the same total concentration of protein, the rotational motion of the protein... 

MRD profiles are shown in Fig. 19 for Sephadex gels at different pH values 100). The curves appear at first glance to be approximately Lorentzian, but are more accurately represented by a sum of a plateau region and a power law higher field portion. The relaxation rates increase with increasing pH value, which may be expected if proton exchange is an important... 

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]. 

In liquids the interactions between neighboring molecules are considerably more complicated than in gases. The resultant broadening obliterates the fine line structure seen in gas spectra, leaving only broad band profiles. There are many possible contributors to this broadening. In some cases, adequate approximation is obtained by assuming that the band contour is established by collisions. Ramsay (1952) has noted that substitution of appropriate molecular density and collision diameter numbers in the collision broadening formula results in realistic band widths for certain liquid-phase systems. In such systems, the bands typically show an approximately Lorentzian profile. Approximate deconvolution of inherently broadened liquid-phase spectra may therefore be obtained on the basis of the assumption of Lorentzian shape (Kauppinen et al., 1981). 

When pressure broadening dominates, the situation is more complicated because the resulting Lorentzian profile contributes significant area far from the line center. A further complication in this case is that the Lorentzian half-width cannot be accurately calculated and must be measured in other experiments. If both Doppler and pressure broadening are present, however, and if the Lorentzian to Doppler half-width ratio is small, the correction necessitated by pressure broadening is small. In this situation an accurate value of the Lorentzian half-width may not be needed. Line strength in the case of combined Doppler and pressure broadening may be obtained from the equivalent width by the use of tables (Jansson and Korb, 1968). 

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