Breit-Wigner resonance formula

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

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. 

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. 

As described above, time-delay analysis [389] of the energy derivative of the phase matrix 4> determines parametric functions that characterize the Breit-Wigner formula for the fixed-nuclei resonant / -matrix R[N(q e). The resonance energy eKS(q), the decay width y(q). and the channel-projection vector y(q) define R and its associated phase matrix such that tan = k(q)R , where... 

In the resonance region, where the excitation functions exhibit sharp resonances, the cross sections of the individual reactions can be calculated from the line widths of the resonance lines by application of the Breit-Wigner formulas derived in 1936. Emission of neutrons from compound nuclei is preferred over emission of protons and relatively high cross sections are expected for (p, n) and (a, n) reactions if the energy of the incident particles is > 1 MeV. 

This is the Breit—Wigner single resonance formula (Breit and Wigner 1936). The resonance is centred at Cr- The full width at half maximum is... 

However, in order to resolve both the resonance energy E and resonance width T, using the Breit-Wigner formula (24), the propagation has to be carried out for a sufficiently long time so that the uncertainty relation... 

In this relation, Vj is the vibrational quantum number of a non-stable negative iort, and Fq, (in s ) are probabilities of transitions between vibrational states. The cross section of the resonant vibrational excitation process (2-148) can be found in the quasi-steady-state approximation using the Breit-Wigner formula ... 

The lithium plus proton reactions thus exhibit many characteristic features of nuclear reactions as a whole, namely the existence of sharply defined resonant states from which particle break-up is prohibited, the probable influence of direct transitions not involving a compound state, the existence of very broad states constituting a background with which interference can take place in many reactions, the description of the levels by the Breit-Wigner formula, and the implications of the principle of charge independence. The information obtainable on the reduced widths and quantum numbers of the states of Be is discussed in the light of nuclear models in Sect. 76. 

In the case where absorption is not negligible, we note that the resonance only Breit-Wigner formulas for (T bs. nd Uscatt obtained if the potential... 

The capture of neutrons above thermal energies is described rather well by the Breit-Wigner formulas, and an analysis of the case, assuming that one resonance level predominates, is given by Anderson. A more accurate analysis was given subsequently by S. Dancoff and M. Ginsburg. 

Furthermore, () is a constant close to zero (as the resonance is close to zero energy). Thus the first term in the denominator is small compared to the second and, aside from terms that do not depend strongly on energy, the Breit-Wigner formula - O Eq. (3.59) - reduces to a simple 1/v dependence. 

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