Fano formula

In the field of photoionization, the Fano formula for the cross section has often been used for resonance fitting. Note, however, that the same resonance can sometimes stand out sharply from the background, but can also fail to manifest themselves clearly in the photoionization cross section, depending upon the initial bound state of the dipole transition [51]. Thus, the cross-section inspection might miss some resonances. The asymptotic quantities of the final continuum-state wavefunction, if available, should be much more convenient in general for the purpose of resonance search and analysis. 

Fig. 6.1. A typical autoionising resonance, as observed in the spectrum of Ba by laser spectroscopy. Note the broadening and asymmetry of the observed line profile. The inset shows the 5d5/ resonance on an expanded scale, together with a theoretical fit (smooth curve) based on the Fano formula (see text). Note that the discrepancies between theory and experiment occur mainly in the wings of the line (after J.-P. Connerade [257]).

One can also obtain the Fano formula explicitly from a diagrammatic many-body expansion. There exists a wide variety of alternative theoretical approaches, some of which (e.g. coordinate rotation, projection operator methods, etc) will not even be described in the present monograph. 

In the presence of several continua, the Fano formula still describes the general lineshape of autoionising resonances rather well, but the cross section does not fall to zero near the resonance, i.e. the transmission window is filled in by the presence of several continua. If the cross section falls to zero near the resonance, one can in fact deduce that only one continuum is involved. 

As promised in the Introduction, we now turn to a third method of deriving the Fano formula, namely diagrammatic MBPT (see section 5.26). 

In the present chapter, we have described many aspects of the simplest problem which can arise when an isolated resonance is formed in a single continuum we have shown that autoionisation is an interference phenomenon and compared it with the behaviour of a discrete three-level system. Two different derivations of the Fano formula have been given, and its connection with MQDT has been described. A third approach will be provided in chapter 8. Beutler-Fano autoionising resonances occur in all many-electron atoms, and a number of examples will be provided in the next two chapters. In chapter 8, the interactions between autoionising resonances will be considered, and two further questions will be discussed, namely the influence of coherent light fields on autoionising lines, and the use of lasers to embed autoionising structure in an otherwise featureless continuum. 

One thus obtains the standard form for the Fano formula for an autoionising line, which was obtained by a different route in section 6.4 ... 

Fig. 7. Yield of Li ions vs. photon energy in the ranges 5.04-5.16 eV (Fig. 7a) and 5.39-5.46 cV (Fig. 7b). The Li" " signal is proportional to the partial cross section for photodetachment of Li via the 3 Skp channel see Fig. 6). The experimental data (dots) has been normalized to theory (solid line with scale at the right). The thick solid lines indicate fits of the double Fano formula (Eq. 3) to the data.

Here, the index n represent the parameters for each resonance. Using this method we determined the parameters for resonances c and d in Fig. 7a. The measured positions and widths are shown in Table 1, along with corresponding values calculated by Lindroth [6]. There is a good agreement between the experiment and theory in this case. Lindroth s resonance parameters are derived directly from a complex rotation calculation. The R-matrix calculation of Pan et al. [28] did not explicitly yield the resonance parameters and therefore cannot be used for comparison. Since the Fano formula strictly only applies to total cross sections, the values of the Cj shape parameters are not entirely meaningful in the context of partial cross sections. This parameter is therefore omitted in the table. 

We also attempted, unsuccessfully, to include the resonance labeled e in the fitting procedure. In this case it appears that the resonance is prematurely terminated by the opening of the 42pks channel. The resonance energies in this case depended strongly on the interval of the fit, which was not the case when the c and d resonances were treated as a pair. In a similar manner, the double Fano formula was fit to the two resonances labeled j and k. The parameters obtained from this fit are also included in Table 1. In this case there is no theoretical data available for comparison. 

Figure 7.28 Fano lineshape in H2. The predissociation of the N=2 [R(l) line] and N=1 [R(0) line] levels of the D1ri,ie(u = 5) state by the continuum of B 1is detected by monitoring the Lyman-a emission from one of the fragment atoms. The dots represent the lineshape calculated from the Fano formula [Eq. (7.9.1)] with parameter values Y(N = 2) = 14.5 cm 1,g(N = 2) = -9 r(jV = 1) = 4.8 cm 1,q(N = 1) = —18. These lineshapes should be compared to the symmetric profile of Fig. 7.16 (q = 00). The horizontal dotted line separates the interacting continuum Oi from the noninteracting continua [

In the preceding sections, we have assumed that an absorption line has a Lorentzian shape. If this is not true, then the linewidth cannot be defined as the full width at half maximum intensity. Transitions from the ground state of a neutral molecule to an ionization continuum often have appreciable oscillator strength, in marked contrast to the situation for ground state to dissociative continuum transitions. The absorption cross-section near the peak of an auto-ionized line can be significantly affected by interference between two processes (1) direct ionization or dissociation, and (2) indirect ionization (autoionization) or indirect dissociation (predissociation). The line profile must be described by the Beutler-Fano formula (Fano, 1961) ... 

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