Fano profiles

Durand P, Paidarova I, Gadea FX (2001) Theory of Fano profiles. J Phys B 34 1953... 

Interference effects between allowed and induced phenomena have received attention [246], Most of the work reported to date is for liquids and thus outside the scope of this monograph. We mention, though, that the Fano profiles observed in the HD-X spectra, Fig. 3.36, are interesting examples of such interference. 

A Fano profile was originally derived to interpret an asymmetrical spectral feature of autoionizing atoms [12], but it can also be identified in the electric spectrum of some simple molecules, which indirectly or directly dissociate. It has been known that a transition from an electronic ground state to a resonance state in the excited-state PES, formed through a mixing between zero-... 

Fig. 8.14 (a) Photoionization spectrum of the ground state of Rb in the presence of a 158 V/cm static field for an excitation energy in the vicinity of 33,614 cm-1 and a light polarization parallel to the field, (b) Example of a charcteristically asymmetric profile. Dots represent the best fit to a Beutler-Fano profile. This fit has been obtained by assuming a linear variation of the ionziation background vs the excitation energy (dotted line). 

If we again consider Fig. 21.4, we can see that the cross section vanishes at v2 = 0.32 and that the profile does not match the spectral density, A2, of the autoionizing state. The Beutler-Fano profile of Fig. 21.4 is periodic in v2 with period 1, so the spectrum from the ground state consists of a series of Beutler-Fano profiles. At higher values of v2 the profiles become compressed in energy since dW/dv2 = l/vf. Fig. 19.2 shows two regular series of Beutler-Fano profiles between the Ba+ 6p1/2 and 6p3/2 limits. In this case the absorption never vanishes because there is more than one continuum. 

The original derivation of the Fano profile in Ref. [29] is more general. It applies to any transition matrix for a final state with a QBS interacting with a background continuum. Resonance photoionization cross sections, for example, is describable by Eq. (18) with some background augmented. Fano [29] also generalizes his derivation for either more than one QBS or more than one continuum, but not for both simultaneously, which is the most difficult case and will be discussed in Section 2.3 of this article. 

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.

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]. 

The main difference between (2S-D1ABAT1C) and (IS-GP) results is the appearance of broad Fano profiles on the (2S-D1ABAT1C) transition probabilities, which suggests that the upper adiabatic PES can support resonances which do not exist in the single ground adiabatic surface calculation. This can be investigated further with the lifetime matrix formalism described in Sect. 3.4. Smith lifetime matrices for the (2S-DIABATIC) case differ from the (IS-GP) ones only by the appearance of Lorentzian-shape eigenvalues near and above 4 eV. 

The quantity q is called the shape index and is constant for a Fano profile. In chapter 8, situations in which q varies within an excitation channel will be discussed. However, even in such cases, it can be regarded as a constant over one line. 

Here, we return to the radiative contribution, and consider how the notion of / value can be extended to a Beutler-Fano profile, by taking account only of the discrete part, and we show that this yields the relevant quantity for studies of the refractive index of an autoionising resonance by MOR. 

This generalised definition of the / value for autoionising resonances turns out to be useful in describing the Zeeman and Faraday rotation effects for a Beutler-Fano profile. It also yields a more symmetric form... 

For MOR between bound states, it is in principle possible to determine absolute / values, because the number density N can be eliminated if both a(v) and n(v) are known, as first pointed out by Weingeroff [293]. For a Beutler-Fano profile, this is not possible, essentially because rfano... 

Fig. 6.5. A plot of the rotation angles in a Beutler-Fano profile as a function of detuning, for several values of the shape index in the special case defined in the text. For negative values of q, reverse the abscissa (after J.-P. Connerade [294]).

But, from equation (6.16), the avoided crossing occurs when AE = 0 or Ei = E2, in which case, by equation (6.35), cf = 4- Since it was assumed at the outset that hq2 — fj,01, the case of zero intensity will generally lie to one side of the avoided crossing. This can be compared with the minimum intensity which generally lies to one side of the Fano profile. The analogy with autoionisation is that, in a Beutler-Fano profile, all the possible values of AE are present simultaneously. 

The excited state is hollow, because the Is2 shell is completely empty. It turns out that the observed photoabsorption spectrum of the ls22s 2S — 2s22p2P transition exhibits a broad and asymmetric Beutler-Fano profile... 

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