Lorentzian line-shape

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... 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

The bracketed term in Eq. (4-60b) describes a Lorentzian line shape for the NMR absorption band. The maximum in the band occurs at the resonance frequency, wq. Expressed in units of X0W0T2/2, the maximum value of x" s 1 at one-half this maximum peak height we find, by substitution, that (wq — w) = IIT. Using w = 2 ttv to convert to frequency (in Hz) gives (vq — v) = 3-7 T 2. However, the peak width is twice this, or... 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

Fig. 3.3. Lorentzian line shape (solid line) and experimental CARS data (points) of liquid nitrogen (T — 77 K) from [136].

Absorption-mode spectrum The spectrum in which the peaks appear with Lorentzian line shapes. NMR spectra are normally displayed in absolute-value mode. 

Dispersion mode A Lorentzian line shape that arises from a phase-sensitive detector (which is 90 out of phase with one that gives a pure-absorption-mode line). Dispersion-mode signals are dipolar in shape and produce long tails. They are not readily integrable, and we need to avoid them in a 2D spectrum. 

Lorentzian line shape The normal line shape of an NMR peak that can be displayed either in absorption or dispersion mode. 

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

A plot of v vs. T2(a>o co) is shown in Figure 5.1. Equation (5.14) corresponds to the classical Lorentzian line shape function and the absorption curve of Figure 5.1 is a Lorentzian line . The half-width at half-height is easily found to be ... 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

A number of authors have given solutions to Eq. (36) based upon Eqs. (37) and (39). One of the more complete descriptions is reported by Lebedev (37), whose results are shown in Fig. 13 for a Lorentzian line shape. In this figure H and i/p refer to the solutions of Eq. (39) when 0 equals ir/2 and 0, respectively the origin is taken at H . Also,... 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

FIGURE 14-8 (a) Meaning of equivalent width, W (b) Doppler and Lorentzian line-shapes for equivalent half-widths (c) transmission curves for an absorption line for a weak and strong absorber, respectively (adapted from Lenoble, 1993). 

For the Lorentzian line shape, the maximum occurs at v = vQ and equals afb2. At half maximum, v = v0 Av and (3.86) gives... 

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. 

Both Pecora (16) and Komarov and Fisher (17) adapted van Hove s space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the... 

Since NMR spectra are not sequences of lines representing discrete Larrnor frequencies but sequences of Lorentzian frequency distributions f(to) (Fig. 1.9), eq. (2.10) must be replaced by eq. (2.11) M0 sin c is multiplied by the frequency function f(to), where a> represents the difference between the frequency ojx and the Larrnor frequency distribution con + Aw, w = co1 — (w0 + Aro). Further, Mosin0f(tu)e must be integrated over the Larrnor frequency distribution. Given a Lorentzian line shape as in Fig. 1.9, the limits of integration are oo ... 

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