Line shape function emission

Here I (co AE) is a line shape function such as those described earlier each of which contains a set of frequencies (e.g., co = C0fv,iv+ Ei,f/fe coj = co + AE/h) at which absorption or emission occurs. 

Whenever the absorbing species undergoes one or more processes that depletes its numbers, we say that it has a finite lifetime. For example, a species that undergoes unimolecular dissociation has a finite lifetime, as does an excited state of a molecule that decays by spontaneous emission of a photon. Any process that depletes the absorbing species contributes another source of time dependence for the dipole time correlation functions C(t) discussed above. This time dependence is usually modeled by appending, in a multiplicative manner, a factor exp(-ltl/x). This, in turn modifies the line shape function I(co) in a manner much like that discussed when treating the rotational diffusion case ... 

Broad bands in optical absorption and emission spectra originating from one or several closely lying electronic transitions are indications of strong vibronic coupling. In the case when the vibrational structure is resolved into a progression of individual bands, a vibronic analysis can be carried out which compares a theoretical line shape function with the intensity profile measured in the... 

For establishing the context to the experiment, we compare the line shape function of a transition, Eq. (29), with the corresponding band profile measured in absorption or emission at low temperature T - 0. The parameters Aa, f a,... 

Fig. 28. Excitation spectra of Pt(2-thpy)2 dissolved in n-octane (a) at zero magnetic field and T = 4.2 K and (b) for different magnetic fields at T = 1.5 K. Concentration = 10 mol/1. For detection, the energy of 16,444 cm with a band width of = 5 cm was used, in order to monitor simultaneously the 713 cm and 718 cm vibrational satellites that correspond to the emissions of the triplet substates I and II, respectively. Under magnetic fields (b), the detection energy is red-shifted according to the size of the field-induced red shift of these satellites. The total excitation spectrum is composed of different spectra. The spectral resolution of the equipment is = 5 cm and = 160 cm for energies below and above the vertical line near 22,300 cm respectively. Note, the halfwidth given in (b) refers to the fwhm of a Lorentzian line shape function which was fit to the red flank of the corresponding peak. (Compare also to the Refs. [74] and [95])...

In this equation, the Franck-Condon factors (the squared Franck-Condmi integrals) and the resonance condition [the delta function in Eq. (4)] have been absorbed into the spectral density 2>eet [135]. It can be factored into the line-shape functions for donor emission and acceptor absorption (this is only possible due to the assumption of local vibrational modes). In addition, the dipole approximation can be made for the electronic coupling matrix element ... 

The line shape function is given for transition in emission from excited state s) to a ground state 0) in the form [92]... 

The hne-shape function gives the profile of the optical absorption (and emission) band and contains important information about the photon-system interaction. Let us briefly discuss the different mechanisms that contribute to this function, or the different line-broadening mechanisms. 

Figure 29. Calculated emission line shapes as a function of delay time, simulating the data presented in Figure 28. [Adapted from (134).]...

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. 

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

Two points should be mentioned here. First, the effect of solutes on the solvent dielectric response can be important in solvents with nonlocal dielectric properties. In principle, this problem can be handled by measuring the spectrum of the whole system, the solvent plus the solutes. Theoretically, the spatial dependence of the dielectric response function, s(r, co), which includes the molecular nature of the solvent, is often treated by using the dynamical mean spherical approximation [28, 36a, 147a, 193-195]. A more advanced approach is based on a molecular hydrodynamic theory [104,191, 196, 197]. These theoretical developments have provided much physical insight into solvation dynamics. However, reasonable agreement between the experimentally measured Stokes shift and emission line shape can be... 

The function Q(o-) is similar to the slit function which distorts lines in spectra collected on dispersion instruments. Q(instrument line shape and can be varied by changing the maximim optical retardation L or by changing the form of a(8). Figure 3 shows several choices for the apodization function and the resulting instrument line shape for each. It can be seen that the width of the instrument line shape is proportional to 1/L. Thus, the larger the optical retardation, the narrower the spectral lines become. For the case where a(8) = 1 for all 8 between 0 and L, the narrowest lines are achieved, but the side-lobes or "ringing" are most severe. When many absorption or emission lines in a spectrim are convolved with this instrument line shape, the spectrum can become difficult to interpret. Therefore, a compromise is usually reached between an apodization function a(8) which produces narrow spectral lines and one which reduces the side-lobes. 

The instrumental resolution of the spectrometers is limited by the combined frequency fluctuation from each CO2 laser (about 15 kilohertz). This, of course, is less than any Doppler limited linewidth and, therefore, does not limit our resolution except for possible sub-Doppler work. This high resolution provides an excellent way of studying pressure shifts and line shape studies of spectral lines. The measurement of OH concentrations in our atmosphere as a function of altitude using absorption and emission measurements requires an accurate knowledge of its linewidth in the atmosphere. 

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

Miyakawa and Dexter (1971) showed that it is still legitimate to write the probability of energy transfer in the form of eq. (142), where p(E) is taken as Ssa, the overlap of the lineshape functions for emission in ion S and absorption of ion A, including the phonon sidebands in the lineshape. It is necessary to consider each partial overlap between the m-phonon emission line shape of ion S and the n-phonon absorption lineshape of ion A. This mathematical assumption has gained experimental credibility through the existence of multiphonon sidebands for trivalent R ions which, in a case of very weak electron-phonon coupling (Auzel 1976) could not be observed directly by usual spectroscopy. 

The shape of a photoelectron line in an XPS spectrum is a convolution of three factors the true shape of the excited peak, the line shape of the x-ray source, and the energy spread function (the resolution) of the electron spectrometer. The true line shape itself may be modified by processes occurring in the specimen as a result of the excitation and emission from the solid of the photoelectron. The line shape is a strong function of the local electronic environment around the photoemitting atom therefore an understanding and appreciation of the factors which govern this important parameter are e.ssential if the maximum information on chemical bonding is to be extracted from the spectrum. 

Second, progress has been made in the theoretical approach to the analysis of DCEMS measurements. The underlying theory of resonance excitation (-> excitation matrix) in the sample by y-quanta including secondary absorption and emission processes, electron transport (- transport tensor), and detection response (—> response function) have been included in a least-squares fit routine [ 103. 105). Adjustable parameters in this fitting are. on the one hand, the hyperfine interaction and line shape variables, and, on the other hand, the variables that give a parametric representation of the depth profiles. The response parameters are also included to allow energy calibration of the experimental apparatus. 

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