Breit-Wigner

As a result, the energy E of photons emitted by an ensemble of identical nuclei, rigidly fixed in space, upon transition from their excited states (e) to their ground states (g), scatters around the mean energy Eq = E. Eg. The intensity distribution of the radiation as a function of the energy E, the emission line, is a Lorentzian curve as given by the Breit-Wigner equation [1] ... 

Figure 10. Phase lag spectrum of HI in the vicinity of the fe3IIi Breit-Wigner resonance. Panel (a) shows the phase lag between the photoionization of HI and H2S. Panel (b) is the three-photon photoionization spectrum of HI, showing the rotational structure of the two-photon fe3IIi - X1 1 transition. The bottom two panels are the one-photon ionization spectra of HI and H2S. (Reproduced with permission from Ref. 33, Copyright 2000 American Physical Society.)...

Thus in the zero dephasing case, 8s reduces to the Breit-Wigner phase of the intermediate state resonance, elaborated on in the previous sections. In the dissipative environment, it is sensitive also to decay and decoherence mechanisms, as illustrated later. 

In the limit where rp(1 3> T 1, 8s takes the Breit-Wigner shape, observed in the isolated molecule limit, whereas in the limit of fast dephasing T 1, 5s... 

Breit-Wigner phase, two-pathway excitation, coherence spectroscopy energy domain, 180-182 low-lying resonance, continuum excitation, 169-170... 

Lorentz, Cauchy, or Breit-Wigner distribution h(r) = j (nb ( 1 + (r-r0)2 /b2 at r0 with a full width at half-maximum b. 

The capture rate is dominated at thermal energies around 30 keV by 5-waves for which the Breit-Wigner formula gives... 

Because the cross section of a resonance process is expressed by the Breit-Wigner formula, the energy-integrated cross section is written as follows ... 

Again the situation is much simpler when only asymptotic states containing stable particles are considered. Then unstable particles enter neither into the completeness relation nor into the unitary relations of the theory.5 However, in the intermediate states unstable particles may appear. They manifest themselves as poles exactly as in Eq, (16). We may then describe such poles by various approximate formulas of the Breit-Wigner type. But again this approach is severely limited. By definition we have to exclude the production or destruction processes involving unstable particles. It is even not easily seen how this can be done in a consistent manner. 

Let us first consider the case of Y/D 1. This means that at certain values of the compound nucleus excitation energy, individual levels of the compound nucleus can be excited (i.e., when the excitation energy exactly equals the energy of a given CN level). When this happens, there will be a sharp rise, or resonance, in the reaction cross section akin to the absorption of infrared radiation by sodium chloride when the radiation frequency equals the natural crystal oscillation frequency. In this case, the formula for the cross section (the Breit-Wigner single-lev el formula) for the reaction a + A —> C b + B is... 

For reactions involving isolated single resonances or broad resonances, it is possible to derive additional formulas for a(E) [R + R] in the Breit-Wigner form, that is,... 

Equations (15) and (16) are Breit-Wigner s one-level formula for the phase shift. If the pole lies close to the real E axis, i.e., if T is very small, the part 5r of the phase shift increases very rapidly with E by tt/2 within the energy region of width T and centered at Er. It increases by nearly it within several times T. This is a resonance phenomenon. 

However, the assumption of no inelastic scattering off resonance is too restrictive and often unrealistic. Hence, we consider a more general form in the following, which we call the Breit-Wigner S matrix in this article, namely [8,35, 37],... 

Substitution of the Breit-Wigner S matrix (34) into Eq. (48) for the Q matrix and the assumption of an energy-independent B, i.e., an energy-independent background Sb yield the expression... 

A number of closely lying resonances in multichannel scattering is a difficult problem to treat theoretically. Even the representation of the S matrix is very complex for these overlapping resonances as compared with the Breit-Wigner one-level formula. Various alternative proposals are found in the literature, as is reviewed by Belozerova and Henner [61]. This is mainly due to the formidable task of constructing an explicitly unitary and symmetric S matrix having more than one pole when analytically continued into the complex k plane. Thus, possible practical forms of the S matrix for overlapping resonances may be explicitly symmetric and implicitly unitary, or explicitly unitary and implicitly symmetric. 

The derivation of the eigenphase sum for the Simonius S matrix, SSim(E), is straightforward by generalizing the procedure for an isolated resonance shown in Section 2.2.2. Compared with the Breit-Wigner S matrix, Sm(E), the matrix Sr in Eq. (34) is now replaced by SPN, or the product of matrices Sv/ each having the same apparent form as Sr but with different resonance parameters and a different projection matrix. Since the determinant of the... 

C.J. Goebel, K.W. McVoy, Eigenphases and the generalized Breit-Wigner approximation, Phys. Rev. 164 (1967) 1932. 

For energy intervals comprising Nres resonances and which are not too large, detS(E) can be approximated with a product of Nies Breit-Wigner phase factors times a smooth background phase factor ... 

Resonances in half and in full collisions have exactly the same origin, namely the temporary excitation of quasi-bound states at short or intermediate distances irrespective of how the complex was created. In full collisions one is essentially interested in the asymptotic behavior of the stationary wavefunction L(.E) in the limit R —> 00, i.e., the scattering matrix S with elements Sif as defined in (2.59). The S-matrix contains all the information necessary to construct scattering cross sections for a transition from state i to state /. In the case of a narrow and isolated resonance with energy Er and width hT the Breit- Wigner expression... 

These modification are indicative of a chemical mechanism in the SERS process in particular in very thin films for which chemical reactions are optimised. These experiments show in addition that metallic tubes are mainly involved, evidenced by the decrease of the Breit-Wigner-Fano component, as due to a possible direct interaction between nanotubes and C6o leading to the formation of SWNTs+C6o complexes [19]. 

The value of coherent control experiments lies not only in their ability to alter the outcome of a reaction but also in the fundamental information that they provide about molecular properties. In the example of phase-sensitive control, the channel phase reveals information about couplings between continuum states that is not readily obtained by other methods. Examination of Eq. (15) reveals two possible sources of the channel phase—namely, the phase of the three-photon dipole operator and that of the continuum function, ESk). The former is complex if there exists a metastable state at an energy of (D or 2 >i, which contributes a phase to only one of the paths, as illustrated in Fig. 3b. In this case the channel phase equals the Breit-Wigner phase of the intermediate resonance (modulo n),... 

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