Lorentzian lineshapes

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

Raman gain coefficient, whose maximum occurs at exact resonance, - oig = For a Lorentzian lineshape, the maximum gain coefficient is given by... 

More recently, 84 may have been identified by ESR spectroscopy of solutions of Li2S ( >6) in DMF at 303 K. The lithium polysulfide was prepared from the elements in liquid ammonia. These polysulfide solutions also contain the trisulfide radical anion ( 2.0290) but at high sulfur contents a second radical at g=2.031 (Lorentzian lineshape) was formed which was assumed to be 84 generated by dissociation of octasulfide dianions see Eq. (32) [137],... 

In the case of resonance absorption of synchrotron radiation by an Fe nucleus in a polycrystalline sample, the frequency dependence of the electric field of the forward scattered radiation, R(oj), takes a Lorentzian lineshape. In order to gain information about the time dependence of the transmitted radiation, the expression for R(oj) has to be Fourier-transformed into R(t) [6]. 

In order to extract the contributions and dynamics of the ketyl radical and fluoranil anion from the TR spectra obtained with the 416 nm probe wavelength, a deconvolution of the Raman bands were done using a fitting procedure employing a Lorentzian lineshape for the Raman bands of the two intermediates. Figure 3.20 shows a comparison of the best-fit (lines) to the experimental TR spectra (dots) in the left-side spectra and the deconvolution extracted from this best fit for the ketyl radical spectra... 

Lineshape. The two most frequently encountered lineshapes are Lorentzian and Gaussian the latter generally arises from a large number of unresolved hyperfine splittings. In the absence of such complications, epr lines in solution almost always show a Lorentzian lineshape which is defined in the form ... 

Most methods assume an exponential decay for the resonances in the time domain giving rise to Lorentzian lineshapes in the frequency domain. This assumption is only valid for ideal experimental conditions. Under real experimental circumstances multi-exponential relaxation, imperfect shimming, susceptibility variations and residual eddy current usually lead to non-ideal... 

By computer-fitting the observed spectrum was decomposed to a sum of Lorentzian lineshapes, and then the fractions of the a-helix and p-sheet forms for PLA and PLIL were determined. The PLA/PLIL blend sample with a mixture ratio of 50/50 (wt/wt%) corresponds to the molar mixture ratio... 

Fig. 15.11 The K 29s + 27d resonance in the presence of a low frequency rf field. In zero rf field (a), the FWHM is 1.6 MHz. In (b)-(d), a 1.0 MHz field of strength 0.05 V/cm, O.lV/cm, and 0.2 V/cm respectively is present. The solid line in (b) is a numerical integration of the transition probability, and the bold line is the convolution of a Lorentzian lineshape with a sinusoidal shift from resonance. In (e), the rf frequency is 0.5 MHz and its strength is 0.2 V/cm. For these low frequencies, the features are no long frequency dependent but rather are field strength dependent (from ref. 18).

Figure 27.2 XRD data (dots for a) Fe7C3 and (b) cubic oxycarbide Mo2QOr. The solid curve in each figure represent a sum of Lorentzian lineshapes.

It is quite important for the spectra measured to represent the pure absorption mode with neither saturation nor transient effects. They should be measured in a homogenous external magnetic field B0 in order to obtain purely Lorentzian lineshapes for individual spectral transitions. The signal-to-noise ratio should be as high as possible and experimental... 

Typical 13C NMR spectra measured in TDAE-C60 are shown on the first row of Fig. 28. At high temperatures the 13C NMR line is very narrow and has an approximately Lorentzian lineshape. This indicates fast and nearly isotropic rotations of the Qo ions effectively averaging out all anisotropic interactions. Isotropic contact interaction is of course still present and is about 40 ppm (the room-temperature shift of the 13C NMR line with respect to TMS is 188 ppm and the chemical shift is about 145 ppm), which is comparable with the shifts mea-... 

Normalized Lorentzian lineshape function as a function of frequency... 

Fractional parameter giving proportion of Lorentzian lineshape... 

In order to obtain the most satisfactory fit in the presence of noise, it is advantageous to make use of as much prior knowledge about the component lineshapes as possible. In the case of high-resolution solution NMR, the spectra can normally be assumed to be a superposition of Lorentzian lineshapes. In other cases, much more complicated lineshapes can be involved. 

The application of the reference deconvolution technique to overcome the effect of magnetic field inhomogeneity in high-resolution NMR has recently been reviewed by Metz et al.21 In this technique, the observed resonance lineshape from a single resonance line is used to deconvolve the observed lineshapes to produce the Lorentzian lineshapes associated with liquid samples. 

The Lorentzian lineshape is obtained for liquid samples under ideal high-resolution NMR conditions10 and is readily derived from the Bloch equations.31 The normalized Lorentzian lineshape function in the frequency domain is given by... 

Since g(f) is normalized, Mo = G(0) = 1, except for the special case of the unnormalized truncated Lorentzian lineshape. 

Fig. 9. The solid curve depicts the super-Lorentzian lineshape g(u>) for a single spin species in vesicles according to Eq. (57) for Q -= 100. The frequency scale is given in units of I/T2. The wide skirts on which the sharp line is superimposed are illustrated by the plot of 10g(aj) versus u> (dashed curve). A Lorentzian line (x 10) of width 100/72 is shown for comparison (small-dashed curve). Reproduced with permission from M. Bloom, Chem. Phys. Lipids, 1975, 14, 107. 1975 American Chemical Society.

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