Lineshap Lorentzian

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... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

The vibrational echo experiments yielded exponential decays at all temperatures. The Fourier-transfonn of the echo decay gives the homogeneous lineshape, in this case Lorentzian. The echo decay time constant is AT, where is... 

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 ... 

Figure 16. Rosch and Ratner spectral density (direct damping) Rosch and Ratner lineshapes (lines) Lorentzian fit (circles) Gaussian fit (black dots).

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... 

In many of these descriptions of lineshapes, chemical exchange line-shapes are treated as a unique phenomenon, rather than simply another example of relaxation effects on lineshapes. This is especially true for line-shapes in the intermediate time scale, where severe broadening or overlapping of lines may occur. The complete picture of exchange lineshapes can be somewhat simplified, following Reeves and Shaw [13], who showed that for two sites, the lineshape at coalescence can always be described by two NMR lines. This fact can be exploited to produce a clarified picture of exchange effects on lineshapes and to formulate a new method for the calculation of exchange lineshapes [16, 23]. This method makes use of the fact that lineshapes, even near coalescence, retain Lorentzian characteristics [13] (fig. 3). These lines, or coherences, are each defined by an intensity, phase, position, and linewidth, and for each line in the spectrum, the contribution of that particular line to the overall free induction decay (FID) or spectrum can be calculated. 

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

In principle we could deconvolute the experimental spectrum with the instrumental lineshape, if that were known, to recover the true spectrum. In our example we have some good experimental evidence as to the form of the instrumental lineshape since the acetone signal is (apart from small carbon-13 satellites) a singlet, its experimental shape is just the instrumental lineshape convoluted by a Lorentzian of width l/(7rr2 ), where is the spin-spin relaxation time of the acetone protons. How can we use this experimental evidence to correct the imperfect experimental spectrum The simplest way to deconvolute one function fi uj) by another f2 ( ) is to Fourier transform the ratio of their inverse Fourier transforms ... 

Adjustment) [3], FIDDLE (Free Induction Decay Deconvolution for Line-shape Enhancement) [6] and QUALITY (QUAntification improvement by converting LIneshapes to the Lorentzian TYpe) [5],... 

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