Resonance Beutler-Fano

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 

Autoionisation is a correlation effect. It occurs for all many-electron atoms in highly-excited configurations which lie above the first ionisation threshold. Many spectra used as illustrations in the present volume provide examples of autoionising lines (see in particular chapter 7). 

In chapter 7, spectra due to inner-shell and double excitations will be discussed. Here, we concentrate on the lineshapes which occur when resonances are embedded in the continuum. 

Autoionisation is one of the most fundamental correlation phenomena. There are different ways of arriving at the Fano lineshape formula for an autoionising resonance. Since these are also alternative approaches to 

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The isolated Beutler-Fano resonance appears, at first sight, to be the simplest situation which can give rise to a resonance in the continuum. In principle, the resonance can appear at any energy above the threshold, since resonance and continuum belong to distinct channels. Thus, there... 

One can argue from the uncertainty principle [211, 225] that there should be a linear relation between energies and widths for giant resonances, and this deduction is obeyed quite well by experiment, as shown in fig. 5.16. This has been termed by Connerade a universal curve for giant resonances. In this respect, giant resonances and Beutler-Fano resonances behave rather differently from each other. 

The widths of Beutler-Fano resonances can vary widely. Their appearance is usually asymmetric, although the degree of asymmetry decreases with increasing energy within a given spectrum, and eventually becomes hardly noticeable in the X-ray range, as other causes of broadening intrude. 

Fig. 6.7. Observed and calculated MOR patterns for a Beutler-Fano resonance in the Ba spectrum (after J.-P. Connerade et al. [296]).

Another way of considering the problem which is perhaps physically more meaningful is that in fact the two bound states 0 > and 1 > are coupled to each other by the two lasers, in one case via a bound virtual state (the Raman path) and in the other via the continuum (the autoionising path) as marked in fig. 8.4. It turns out that, if the Raman channel dominates, the resulting lineshapes tend to become symmetric, while if the autoionisation channel dominates, the characteristic interference asymmetries of Beutler-Fano resonances emerge. ... 

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]... 

For an isolated or Beutler-Fano resonance, q = 0 implies a symmetric window. In the present situation, the presence of other resonances arranged as a Rydberg series, with energy intervals which are not the same on both sides of any given resonance means that the lineshapes are not symmetrical even for q = 0. However, when q = 0 and B = tan7r/x, from the first equation (8.44), the transmission maxima coincide with the resonance energies as they do for isolated resonances. 

A quite different application of the RRPA equag ons is illustrated in Fig. 7 where we compare experimental and theoretical Beutler-Fano resonances in the xenon photoabsorbtion cross section. These resonances occur for photon energies just above the 5p3/2 threshold and are a result of the coupling between nd and ns states converging to the 5pi/2 threshold and the continuum. Since the RRPA automatically provides for the 5p3/2 Pl/2 threshold separation, and includes couplings between the relevant open and closed channels, it is a theory Ideally suited to study such resonances. The comparison in Fig. 7 illustrates how well the theory works in such applications. 

Fig. 9 Beutler-Fano resonances for Xe-like ions. Upper panels Intensities vs. effective quantum number V2 Lower panels Open-chanel phase-shifts 6/tt x(mod 1) vs. V . Results for Xe and Ba are from measurements of Refs. 62 and 68. Results for Cs are from an RRPA calculation.

Resonance studies using RRPA-MQDT can of course be extended from neutral species along isoelectronic sequences. In the upper three panels of Fig. 9 we show the Beutler-Fano resonances in Xe, Cs , ang Ba", resp. as functions of the effective quantum numbers Vg the curves are taken from experimen-... 

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