Beutler-Fano formula

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) ... 

The photoionisation continuum of H is clean and featureless. Its intensity declines monotonically with increasing energy. Many-electron systems, in general, always exhibit structure embedded in the continuum. Such features are neither purely discrete nor purely continuous, but of mixed character, and are referred to as autoionising resonances. They were discovered experimentally by Beutler [254], and the asymmetric lineshape which they can give rise to follows a simple analytic formula derived by Fano [256]. For this reason, they are often referred to as Beutler-Fano resonances. A typical autoionising resonance is shown in fig. 6.1... 

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. 

Such formulae are used to analyse cases in which two Beutler-Fano resonances overlap in energy. For resonances originating from two different channels, different values of q (qi and qf) are introduced in (8.39) (see, e.g., Heinzmann et al. [427]). Such situations are also treated by MQDT [428, 429]... 

Fig. 8.7. The form of a single resonance in the Rydberg series defined by the Dubau-Seaton formula (a) plotted with different combinations of parameters so that the maximum and minimum in the absorption cross section remain at fixed energies and (b) comparing a Dubau-Seaton profile (curve A) with a Beutler-Fano profile of the same shape near the resonance energy (curve B) (after J.-P. Connerade [413, 414]).

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