Line shape, Lorentzian, absorption

FIGURE 4.4 Line shapes. Lorentzian (broken lines) and Gaussian (solid lines) line shapes and their first derivatives are given. The outermost vertical lines delimit full width at half height (FWHH) of the absorption lines. 

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 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... 

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. 

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 ... 

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).)... 

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). 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

Here kv is the absorption coefficient at frequency v, Nc is the number of absorbing centres per cubic centimeter, v is the frequency of absorption, and S(v) is the line shape function. For our estimates we shall assume that the line shape is Lorentzian having half width 6. If one evaluates the absorption cross section when the absorption is maximum the above expression takes the form... 

The first and third of Eq. (11.21.21) yield a Lorentzian line-shape. Figure 11.53 shows a plot of x" (proportional to absorption) and / (proportional to dispersion). 

The Bloch equations can be solved analytically under the condition of slow passage, for which the time derivatives of Eq. 2.48 are assumed to be zero to create a steady state. The nuclear induction can be shown to consist of two components, absorption, which is 90° out of phase with B, and has a Lorentzian line shape, and dispersion, which is in phase with B,. The shapes of these signals are shown in Fig. 2.10. By appropriate electronic means (see Section 3.3), we can select either of these two signals, usually the absorption mode. 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

These functions describe an absorption with a Lorentzian line shape for which n and k vary, as shown in Fig. 3.19. This is identical to the dispersion illustrated earlier in Fig. 2.4. Quite clearly this model will also apply to molecular vibrations that produce an oscillating dipole. Hence the dispersion associated with vibrations that give rise to infra-red absorption will be of the same form, see Fig. 2.4. 

The transmission throngh an absorber of thickness teff as a ftmction of the relative velocity V between source and absorber is given by the evalnation of the transmission integral discussed, for example, in Ref 2. In case of Fe Mdssbaner spectroscopy for almost all samples which are not enriched in the Mossbauer isotope Fe (the natural abundance of Fe is 2%), the evaluation of the transmission integral leads to Lorentzian line shape with a minimum linewidth of 0.19mms , which is twice the natural linewidth of the Fe Mossbauer transition. The absorption cross section as a ftmction of source velocity is then given as (with... 

In indirect methods, the resonance parameters are determined from the energy dependence of the absorption spectrum. An important extra step — the non-linear fit of (t E) to a Lorentzian line shape — is required, in addition to the extensive dynamical calculations. The procedure is flawless for isolated resonances, especially if the harmonic inversion algorithms are employed, but the uncertainty of the fit grows as the resonances broaden, start to overlap and melt into the unresolved spectral background. The unimolecular dissociations of most molecules with a deep potential well feature overlapping resonances [133]. It is desirable, therefore, to have robust computational approaches which yield resonance parameters and wave functions without an intermediate fitting procedure, irrespective of whether the resonances are narrow or broad, overlapped or isolated. 

Fig. 35. Characteristic line shape for a) Lorentzian and b) Gaussian absorption curve, together with functions for absorption and first derivative curves...

From the application of the recoilless y-absorption technique to MbCO, HbCO, MbC>2, HbC>2, Mb, and Hb Mossbauer results have been derived which are presented in this section. Fig. 7 shows Mossbauer spectra of a frozen solution of MbCO (pH 7.0) at 4.2 °K (50), a typical candidate for ferrous low-spin state. Curve (a) coresponds to a measurement with zero applied magnetic field. Assuming a Lorentzian line shape (see Table 1), a least-squares fit to this spectrum leads to the following values for the Mossbauer parameters J1 (line width) =0.328 0.011 mm/sec, S (isomer shift, relative to iron metal) =0.266 0.010 mm/sec, and AEq (quadrupole splitting) = 0.363 0.006 mm/sec. Curve (b) shows a spectrum taken with a magnetic field of Ho = 47 kOe applied perpendicularly to the y-beam. Both spectra have been found... 

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