The Fano formula

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. 

Since the subject of autoionisation is quite central, we shall return to it several times from different points of view, giving alternative derivations of the basic profile formula. We begin with Fano s formulation, which forms the most suitable introduction. 

The independent electron model serves as the reference basis. Fano s theory of autoionisation consists in describing the consequence of turning on an interaction between a sharp state and the underlying continuum, which are presumed initially to be devoid of correlations. Of course, the perturbation is a hypothetical one, since it cannot really be turned off. The independent electron atom, as such, does not exist. Hypothetical interactions are familiar in perturbation theory. They carry with them the implication that, if they could be removed, the zero-order Hamiltonian which would result can be solved exactly, providing the basis for a perturbative expansion. For a many-electron atom, this is clearly not so, but the idea is nevertheless convenient. It is a case of pretending that, 

The origin of the perturbation is not of great importance in setting up the theory if the system is an atom, the most usual form for the operator is the interelectronic repulsion e2/ri2, although spin-orbit or spin-spin, etc. interactions are equally possible. For molecules, in addition to electronic and spin-dependent interactions, autoionisation can be driven by rotational and vibrational interactions, and so the possibilities are even wider than for atoms. 

To keep the problem simple, we also assume that only one, isolated excited state (p is present in the energy range of interest, and that the unperturbed continuum would be flat, i.e. would contain no structure at all within the range of the resonance. 

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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]).

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 ... 

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. 

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 [

By assuming a value of 0.1 for the Fano factor, the following formula gives the germanium detector resolution at LN temperature ... 

Numerical values of the quantum Fano factors in comparison to their semiclassical approximations for the fundamental mode, given by Eq. (56), are presented in their dependence on N in Table I and Fig. 8a. Analogously, those values for harmonics are presented in Fig. 8b and Table II as calculated by the numerical quantum method and from analytical semiclassical formula (57). It is seen that the approximate predictions of the Fano factors, according to (56)... 

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... 

Fig. 6.2. Family of curves generated from the Fano lineshape formula for different values of the shape index q. For negative values of q, reverse the abscissa, (after U. Fano [256]).

Techniques which simplify the extraction of the Fano parameters q and r from experimental data for isolated resonances are described by Shore [261]. However, the extraction of a single set of Fano parameters q and T from the formula for an isolated resonance is not reliable when several resonances overlap in energy [262]. 

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.

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. 

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 absorption spectrum containing many bands with partially overlapped contours is typical for the majority of materials [4], The magnitudes of damping of the corresponding IR-active modes can be determined by nonlinear energy transfer processes from the given vibration to other vibrations. The formulas for the dielectric functions in the case of coupled modes were obtained by Barker and Hopfield [85]. The interaction of a (discrete) phonon with a continuous electronic excitation can result in specific band distortions [86] named the Fano resonances. 

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... 

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