Fano s theory

In Fano s theory the true unperturbed eigenfunction of the system described in the last paragraph is... 

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,... 

In fig. 6.2, we show examples of the line profiles which occur as a result of autoionisation, as predicted by Fano s theory. 

If the probing conditions are appropriate (for further discussion, see section 9.12), the induced structure is expected to possess a Beutler-Fano profile, with the specific attraction that both the resonance energy and the width are controllable, which gives some experimental meaning to the otherwise somewhat elusive concept of prediagonalisation in Fano s theory (see chapter 6). 

Fano s theory (1961) relates the width of a predissociated level to the interaction matrix element of Eq. (7.5.1). It can be shown that the discrete state amplitude, a(E), in the continuum eigenfunction,... 

Similarly, Fano s theory requires that the continuum levels in the interval E to E + dE, where E 7 E°j, contribute an energy shift... 

The problem of resonance sfafes, or of decaying sfafes, normally involves only the continuous spectrum, in which case the energy distribution is, according to the above use of fhe Hermitian I, a real function, given by < To E>I dE = fl(E) dE. The explicit form of fhis disfribution in terms of computable matrix elements is given by Fano s theory or by decaying-state theory, as... 

For example, this form is in harmony with the superposition of energy states in Eq. (2), whose coefficients have been obtained formally by Fano [29]. Although, for the solution of particular problems involving unstable states, we have implemented, in conjunction with the methods of the SSA, the real-energy, Hermitian, Cl in the continuum formalism that characterizes Fano s theory, e.g.. Refs. [78, 82-87] and Chapter 6, in this chapter I focused on the theory and the nonperturbative method of solution of the complex eigenvalue Schrodinger equation (CESE), Eq. (27). 

Real energy hermitian approaches. Extension of Fano s K-matrix, configuration interaction theory... 

For example, Eq. (3a) below, which is a consequence of Fano s elegant and general theory of 1961 [29], expresses the fact that, on resonance, the exact scattering state in the energy continuum is almost the same as the square-integrable (inner) part, Fq, with the energy dependence of the scattering state represented by the coefficient of o-... 

In Fano s [29] formal theory of resonance states, the energy-dependent wavefunctions are stationary, the energies are real, and the formalism is Hermitian. The observable quantities, such as the photoabsorption cross-section in the presence of a resonance, are energy-dependent and the theory provides them in terms of computable matrix elements involving prediagonalized bound and scattering N-electron basis sets. The serious MEP of how to compute and utilize in a practical way these sets for arbitrary N-electron systems is left open. 

Along fhose lines, Cordes and Altick [80] also implemenfed multichannel Cl theory with a basis set of hyperspherical coordinates for the determination of properfies of fhe He (3,3b) resonance. The first Cl multichannel implementation of Fano s formalism was done by Ramaker and Schrader [81] with application to the He (nt ) S autoionizing states. 

Extension of Fano s formalism to the general case of multiple discrete states embedded in multiple continua was first conducted by Mies [54]. Like Fano, Mies assumed a prediagonalized basis. By imposing the asymptotic condition for the continuum state from the scattering theory, Mies derived the complete solution to the total continuum problem and gave formulas for energies and widths of resonances. 

An early theory of the IT value was proffered by Spencer and Fano [44], based on the degradation spectrum. Another method, the Fowler equation, was employed by Inokuti [47] for electron irradiation, based on the approximation that there is only one ionization potential and that the ionization efficiency is unity. These restrictions can be relaxed. The main result of Inokuti s analysis may be given as follows. 

Chang, E.S. and Fano, U. (1972). Theory of electron-molecule collisions by frame transformation, Phys. Rev. A 6, 173-185. 

Fano U (1941) The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld s waves). J Opt Soc Am A 31 213-222... 

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. 

---