Fano scattering

Figure 26. Comparison of the resonance scattering from H atoms or H+ obtained by fitting the Fano line shape in HCIO4 (open squares) and H2SO4 (closed squares) with the adsorbed hydrogen coverage (closed circles) and sulfate adsorption (open circles) obtained by cyclic voltammetry. (Reproduced with permission from ref 50. Copyright 2001 The Electrochemical Society, Inc.)...

More recently, Mies and Kraus have presented a quantum mechanical theory of the unimolecular decay of activated molecules.13 Because of the similarity between this process and autoionization they used the Fano theory of resonant scattering.2 Their theory provides a detailed description of the relationships between level widths, matrix elements coupling discrete levels to the translational continuum, and the rate of fragmentation of the molecule. 

To arrive at a more unique way of characterizing autoionizing states in the absence of information about the continuum phase we can recast the QDT equations into an R matrix form which is at the same time similar to the original development of QDT from multichannel scattering theory.2 The relation between the different forms of scattering matrix is discussed by Mott and Massey,11 Seaton,2 and Fano and Rau.12... 

The advances in this field are related with the development of the theory of configuration interaction between different excitation channels in nuclear physics including quantum superposition of states corresponding to different spatial locations for interpretation of resonances in nuclear scattering cross-section [7] related with the Fano configuration interaction theory for autoionization processes in atomic physics [8],... 

As discussed in connection with IR absorption, Raman lines from a discrete transition may also assume Fano type shapes if the transition is coupled to a continuum of scattering states. This continuum may originate from various sources like, for example, electronic or two phonon excitations. In general, Raman scattering makes it even easier to observe Fano lines, since the free carrier response very often covers up details of the line shape in IR reflection. Similarly, such line shapes have so far only been shown by classical semiconductors and most recently by superconductors, and it is a challenge to search for them in other conducting organic systems. 

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 interactions of electromagnetic radiation with the vibrations of a molecule, either by absorption in the infrared region or by the inelastic scattering of visible light (Raman effect), occur with the classical normal vibrations of the system (Pauling and Wilson, 1935). The goal of our spectroscopic analysis is to show how the frequencies of these normal modes depend upon the three-dimensional structure of the molecule. We will therefore review briefly in this section the nature of the normalmode calculation more detailed treatments can be found in a number of references (Herzberg, 1945 Wilson etal., 1955 Woodward, 1972 Cali-fano, 1976). We will then discuss the component parts that go into such calculations. 

Inside the band of two-particle states a channel opens up for polariton decay into two phonons. This process leads to a broadening of the polariton line (see, e.g. (59), (60)), and also a change in the polariton dispersion law. The two most important effects in the latter case are (a) interference of scattering by a polariton and two-particle states (this can lead to drops in intensity of the Fano antiresonance type see (22)), and (b) the presence of singularities in the density of two-particle states (those, in particular, that correspond to quasibiphonons). 

Similarly, quantum defect theory plays a very important role in modern descriptions of atomic physics, and should be included at a less rudimentary level than is found in most texts. Again, its modern developments provide an excellent illustration of many fundamental principles of scattering theory. The principles underlying the Lu-Fano graph are easily grasped, and provide excellent insight into an important aspect of the many-body problem, namely interchannel coupling. Likewise double- and inner-shell excitation are hardly discussed in textbooks, structure in the continuum receives little attention, etc, etc. 

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. 

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