Fano /-parameter

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]. 

The symbols are expressed in terms of mafrix elemenfs [78]. In fhe limif of a bound-resonance transition, Q reduces to the Fano -parameter, while the second term of Eq. (16) cancels the background term. 

This expression contains various (so-called Fano) parameters that are specific for the resonance the peak position Ejt, its width F, the shape factor q, the background non-resonant cross section cTo, and the mixing parameter p. ... 

Table 14 Fano parameters for different ionization resonances for some closed-shell atoms. Except for the calculations marked EXX, the Kohn-Sham single-particle eigenvalues have in all cases been shifted to match the ionization potential. Exact KS represents results with the exact Kohn-Sham potential, GGA those with a gradient-corrected density functional, EXX exact-exchange without correlation, EXX- -LDA the same but with the inclusion of correlation effects with a local-density approximation, and Exp. experimental results. The results are from ref. 91... 

This case is shown schematically in Fig. 5c. In Eq. (50), qj. are generalized y-photon asymmetry parameters, defined, by analogy to the single-photon q parameter of Fano s formalism [68], in terms of the ratio of the resonance-mediated and direct transition matrix elements [31], j. is a reduced energy variable, and <7/ y, is proportional to the line strength of the spectroscopic transition. The structure predicted by Eq. (50) was observed in studies of HI and DI ionization in the vicinity of the 5<78 resonance [30, 33], In the case of a... 

Fano interference, 32, 38 Fast electron distribution, 134 Fast electron generation, 123 Fast electron transport, 125 Fast electrons, 176 Fast-ignition, 124 Femtosecond supercontinuum, 94 Feynman s path integral, 73 Feynman s propagator, 76 Field parameter, 172 Filamentation, 82, 84, 112 Floquet ladder, 11 Fluorescence, 85, 125 FROG, 66 FROG-CRAB, 66... 

At first glance it seems evident that the observed spectrum of Fig. 21.11 is a sequence of Beutler-Fano interference profiles, which reverse in the Fano q parameter across the line. Although the solution may ultimately be expressed in the same mathematical form,1 a simple consideration of the excitation amplitudes... 

Fig. 22.3 Lu-Fano plot of the high lying Ba even parity 7=2 levels above the 6s22d 3E>2 level. The full curve is calculated with the QDT parameters of ref. 8 observed 6snd3E>2 states (+), observed 6snd1D2 states ( ) observed 5d7d perturber (V) (from ref. 8).

Figure 4.1 Fano profile of resonance cross sections normalized by the maximum cross section. The shape parameter q = — cotSb is assumed to be a constant independent of energy E. In actual processes, its dependence on E can alter the profile significantly. The background phase shift 8b is indicated in the parentheses below each value of q.

Table 2 Fit parameters for the isolated shaperesonances. Parameters obtained from non-linear least-squares fit of Fano-profile to the hydrogen-induced shape-resonance in experimental spectra. Applied experimental width Gaussian, 5 eV.

Hermiticity imposes further constraints. For / = 1 there are only five independent multipoles, e.g. (T ), (T/), T ), T ), Tq), with (T/) imaginary and the components of the alignment tensor real (Blum, 1981). When reflection invariance holds in the collision plane the state multipoles can be related to the orientation 0 and alignment parameters A, first introduced by Happer (1972) and Fano and Macek (1973), through... 

Fig. 17.9 Fano asymmetry parameter ( q) as a function of distance from the catalyst for (a) annealed and (b) oxidized Si NWs that were in situ boron doped in the lower half and intrinsic in the upper half (With permission from reference [46]. Copyright (2009) by the American Chemical Society)...

The first term arises from the resonance scattering, the second term, /bg, is due to the off-resonance phase o, while /int describes the interference between the first two. If the background term is small, the cross section reduces to the familiar Lorentzian form and Eq. (7) can be directly used to extract resonance parameters from the experimental or calculated (t(E). On the other hand, if /bg cannot be neglected, one encounters a complicated energy dependence of the cross section known as Fano profiles [112]. 

Whatever the complexity of an entity, it can be associated with a measure of certainty, and in this case a measure due to Fano, who was mentioned in the previous section. For those interested in the details, his important measure may now be described as the Fano s mutual information between any medical parameter A and B, such that... 

A fundamental issue in the description of even the simplest, isolated autoionising resonance in the parametric approach followed by Fano [391] - and further pursued in K-matrix theory - is that the atom cannot be deperturbed, that is one cannot access the so-called prediagonalised states which are imagined to exist prior to autoionisation being included as a perturbative interaction, since the effect is anyway internal to the atom and cannot truly be turned off. This has the disadvantage that the parameters, once they have been obtained, must still be calculated from an ab initio model of the atom for a full comparison with theory. It might seem that the parametric theory cannot really be checked independently of ab initio calculations whose accuracy is hard to ascertain. 

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