Alternative derivation of the Fano formula

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

Consider, as before, an initial state , and let the letter i in Feynman graphs stand for the transition ( — /). We can write the single electron (independent electron) dipole transition as the diagram 

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. 

Finally, we note that autoionisation cannot normally occur in H, because there is only one threshold, which clearly separates bound from continuum states. There is, however, one way of inducing autoionsisation in H if an external field induces new thresholds, with excited states above them, then autoionisation becomes possible even in H. This situation is discussed in chapter 11. 

Many-electron atoms differ from H in an essential respect when they are excited up to and above the first ionisation potential, they exhibit structure which is not simply due to the excitation of one valence electron. The clearest manifestation of this behaviour occurs in the ionisation continuum. For H, the continuum is clean, i.e. exempt from quasidiscrete features. In any many-electron atom, there will be autoionising resonances of the type discussed in chapter 6. Autoionisation is therefore a clear manifestation of the many-electron character of nonhydrogenic atoms. 

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