A Lorentzian lineshape

For E = Er (the energy including the predissociation-induced energy shift, called the resonance energy), the absorption has its maximum value, aa(Er) = 

In typical experiments, only strong predissociations that result in a nonra-diative lifetime much shorter than the radiative lifetime can be detected by line broadening, since the usual radiative lifetime of 10 8s corresponds to a width of only 5 x 10-4 cm-1. Doppler-free spectroscopic techniques have made it possible to measure extremely small predissociation linewidths (Carre, et al., 1980 Carrington, et al., 1978). 

For a nonradiative lifetime of 10 13s, the linewidth will be 50 cm-1, causing the rotational structure to disappear and the band to become diffuse. For a shorter nonradiative lifetime, 10 15s, the vibrational structure disappears and the spectrum becomes similar to a continuous spectrum. As spectral lines become broader and broader, their intensity is spread out and becomes lost in the background. However, low-resolution techniques, such as photoelectron spectroscopy or electron impact spectroscopy, can enable detection of strongly predissociated bands (see Table 7.2). 

Doppler broadening has a Gaussian lineshape, and its convolution with the Lorentzian natural lineshape yields a Voigt profile. In typical experiments, this effect can be neglected since the Doppler width is usually much smaller than the resolution of the apparatus. Collisional line broadening is also Lorentzian, and the Lorentzian component of measured lines must be carefully extrapolated to zero pressure. 

Measurements by photographic photometry require careful calibration due to the nonlinear response of photographic plates saturation effects can lead to erroneous values. Line profiles can be recorded photoelectrically, if the stability of the source intensity and the wavelength scanning mechanism are adequate. Often individual rotational lines are composed of incompletely resolved spin or hyperfine multiplet components. The contribution to the linewidth from such unresolved components can vary with J (or TV). In order to obtain the FWHM of an individual component, it is necessary to construct a model for the observed lineshape that takes into account calculated level splitttings and transition intensities. An average of the widths for two lines corresponding to predissociated levels of the same parity and J -value (for example the P and R lines of a 1II — 1E+ transition) can minimize experimental uncertainties. A theoretical Lorentzian shape is assumed here for simplicity, but in some cases, as explained in Section 7.9, interference effects with the continuum can result in asymmetric Fano-type lineshapes. 

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

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 ... 

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).

An exponentially decaying FID gives a Lorentzian lineshape upon Fourier transformation. The general form of the absorptive Lorentzian line is IabS = 1/(1 + v2), whereas the dispersive line has the form Idisp = v/(l + v2), where I is the intensity at each point in the spectmm. Far from the peak maximum (v2 >> 1), we have Iabs 1/v2 and Idisp l/y- This is the reason that the dispersive lineshape extends much further from the peak maximum. 

For an optical transition to the middle of the excitonic band, the low-temperature limit width is dominated by phonon spontaneous emission with a lorentzian lineshape, more or less distorted by the density of excitonic states at the final energy — h 2s. With increasing temperature, the line broadens and reaches the high-temperature limit (2.104). 

In C NMR spectroscopy, deviations from a Lorentzian lineshape, which is usually obtained in liquids, can be caused by a chemical shift anisotropy (CSA). If a CSA is present, the position of the resonance line depends on the relative orientation of the molecule with respect to the direction of the magnetic field applied (27,22). The superposition of the individual resonance lines results in typical lineshape patterns that can be described by two parameters the chemical shift anisotropy, AS, and the asymmetry parameter, Tj, respectively. In the case of an axially symmetric CSA tensor, i.e., 17 = 0, the relation between the resonance frequency, w, and the orientation of the molecule is given by... 

The simplest model for the time-dependent dipole correlation function is an exponentially decaying function, (/i(z)/t(O)) exp(—F Z ). This form leads to a Lorentzian lineshape... 

This is a Lorentzian lineshape whose width is determined by the friction. The latter, in turn, corresponds to the rate of energy dissipation. It is significant that the normalized lineshape (characterized by its center and width) does not depend on the temperature. This result is associated with the fact that the harmonic oscillator is characterized by an energy level structure with constant spacing, or classically-with and energy independent frequency. 

We have seen (Section 6.2,3) that a Lorentzian lineshape corresponds to an exponentially decaying dipole autocorrelation function. For the Hamiltonian of Eqs (9.36) and (9.39) this correlation function is C/x(f) = ik g = =... 

It displays a superposition of lines that correspond to the excitation of different numbers of vibrational quanta during the electronic transition (hence the name multiphonon transition rate). The relative line intensities are determined by the corresponding Franck-Condon factors. The fact that the lines appear as <5 functions results from using perturbation theory in the derivation of this expression. In reality each line will be broadened and simplest theory (see Section 9.3) yields a Lorentzian lineshape. 

The resulting absorption rate is proportional to the population difference CTz.ss = Lorentzian lineshape whose width is determined by the total phase relaxation rate kA, Eq. (10.176). 

For very large widths, slight deviations from a Lorentzian lineshape are predicted (Child and Lefebvre, 1978). 

The amplitude of the oscillation is a maximum when the driving frequency equals the natural resonance frequency and the width of the resonance line depends on the degree of damping of the system. If the liquid in which the system is immersed is not viscous, the line will be narrow. On the other hand, the line will be quite broad for a viscous liquid. The line-shape produced by such a damped harmonic oscillator is a Lorentzian lineshape. The full width at half height of such a... 

Fig. 7.23 A portion of the ESR spectrum of DBN triplet excitons atT= 300 K for three neighbouring orientations of the applied magnetic field Bq. The splitting for Boll (o" + 2.2°) indicates the two non-equivalent sublattices I and II. The circles represent a Lorentzian lineshape.

The definition of the convolution product is quite clear like the one of the Fourier transforms, it has a given mathematical expression. An important property of convolution is that the product of two functions corresponds to the Fourier transform of the convolution product of their Fourier transforms. In the context of high-resolution FT-NMR, a typical example is the signal of a given spin coupled to a spin one half. In the time domain, the relaxation gives rise to an exponential decay multiplied by a cosine function under the influence of the coupling. In the frequency domain, the first corresponds to a Lorentzian lineshape while the second corresponds to a doublet of delta functions. The spectrum of such a spin has a lineshape which is the result of the convolution product of the Lorentzian with the doublet of delta functions. In contrast, the word deconvolution is not always used with equal clarity. Sometimes it is meant as the strict reverse process of convolution, in which case it corresponds to a division in the reciprocal domain, but it is often used more loosely to mean simplification. This lack of clarity is due to the diversity of solutions offered to the problem of deconvolution, depending on the function to be deconvoluted, the quality one wishes to obtain, and other parameters. 

Mossbauer Spectroscopy. The Pt Mossbauer spectra for various tetracyanoplatinates at 4.2 K have been reported, Table VIII. The spectra observed were singlets and had a Lorentzian lineshape. The data confirm that the partially oxidized species cannot be considered as a mixture of isolated Pffi and Pffv atoms even on the Mossbauer time scale (lO sec). The results are consistent with the model of an insulator at low temperatures with the electrons essentially delocalized along the Pt chains (360). 

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