Extension to the high energy spectrum

In this section, we consider the effect of many channels, Auger broadening and the growth in radiative width. 

In previous sections, we have, for simplicity, confined our attention to situations involving one open channel. For isolated Fano profiles, this leads to an exact zero in the cross section at the transmission window, and to a series of exact zeros in the cross section associated with each resonance for a complete Rydberg series. Obviously, if one wishes to study profile shapes in detail and the connection between the bound states and autoionising resonances implied by Seaton s theorem (see previous section) it is desirable to work just above the first ionisation potential, where the number of open continuum channels is at a minimum. In general, this is not possible, and the number of open channels increases rapidly with increasing energy. 

Several complications arise from an increase in the number of open channels. First, the zeros just mentioned are filled in by interactions with many continua. Thus, one sees that the asymmetries of the profiles tend to be less pronounced towards higher energies. Other factors conspire to reinforce this general trend. 

The series limit on which the autoionising resonances converge may or may not be stable. If the corresponding parent ion state can decay ra-diatively, then because radiative widths are (for low excitation energies) usually much smaller than autoionisation widths, the series observed is not much affected until high values of n. On the other hand, if the parent ion state itself lies above the double-ionisation threshold, it becomes susceptible to particle decay and acquires its own intrinsic width through autoionisation. 

Taking b as the autoionisation and c as the Auger channels, we set = 2tt Hauto 2 and T+ = 2tt Hauger 2 and a b = Qaauto 2 whereupon the matrix elements are 

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