Line-shape function

Now, it is convention to introduce the so-called "line shape" function I (co) ... 

To summarize, the line shape function I(co) produces the net rate of photon absorption... 

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. 

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 ... 

In fact, the transition cross section can be written in terms of a line-shape function g(v) (with units of Hz ) in the following way ... 

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. 

The line-shape function of a given transition informs us on the particular character of the interaction of the absorbing atom with its environment in the solid. In the most general case, this line shape is due to the combined effect of more than one independent broadening mechanism. In this case, the overall line shape is given by the convolution of the line-shape functions associated with the different broadening mechanisms. 

This result, when substituted into the expressions for C(t), yields expressions identical to those given for the three cases treated above but with one modification. The translational motion average need no longer be considered in each C(t) instead, the earlier expressions for C(t) must each be multiplied by a factor exp(- co2t2kT/(2mc2)) that embodies the translationally averaged Doppler shift. The spectral line shape function 1(G)) can then be obtained for each C(t) by simply Fourier transforming ... 

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 ... 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

The superposition of the different magnetic interactions complicates the interpretation and calculation of the resulting normalized line shape function,/(v). It is therefore advantageous to make use of the so-called second moment M2 as a measure of the linewidth of the solid-state NMR signals. The full-width at halfmaximum of an NMR signal in frequency units, also called the static linewidth, is... 

Fig. 4. The functions a) I(Vt) and b) S(Vk), which are the instrument line shape functions for spectra computed using no apodization and triangular apodization, respectively.

In the Brownian oscillator overdamped model as an attempt to simulate the solvation dynamics, the explicit forms of the line-shape functions gr(t) are... 

As is well known by now, a line shape function 7(co) is the Fourier transform of the correlation function of a physical variable, for example, a dipole moment of a molecule or an atom, or a magnetic moment of a spin. Thus,... 

The Gaussian assumption yields the first equality. Assumption (25) is used in the second expression on the right-hand side. Thus the line shape function /(co) can also be written as... 

It should be mentioned that the line shape function /(co) can also be... 

Here, we shall not go any further into the analysis of this sort. Instead, we show a few examples of line shape functions calculated by Eq. (72) for the Gaussian models, which are characterized by the operator T given by Eq. (52). The Gaussian models are not very adequate for the electric field in a plasma, as is seen by the fact that the Holtzmark distribution is far from Gaussian. Still, it may serve to give one an understanding of the... 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

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