Breit-Wigner function

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] ... 

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),... 

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. 

The complex energy, 2 of Eq. (5) is normally understood in fhe context of resonance scattering theory as the complex pole in the Breit-Wigner amplitude, or in the S-matrix, or in the optical potential of Feshbach s fheory," or in the Green s function, e.g.. Refs. [2,6-8]. 

In Mhssbauer spectroscopy (see Chap. 25 in Vol. 3), the density function of Cauchy distribution is called a Lorentzian curve. In nuclear physics, the same function is also called the Breit-Wigner curve (Lyons 1986). (See also Chap. 2 in this Volume.) This curve is characteristic of the energy uncertainty of excited (nuclear) states, which follows from the fact that excited states have exponential distribution with a finite mean life t. The natural linewidth T, i.e., the FWHM of the Lorentzian energy density, is twice of the parameter y(F = 2y). 

If there is only one open channel, a resonance (of either type) is characterized by a sharp change in the phase shift Sl, which increases sharply by k across the width of the resonance. The phase shift follows a Breit-Wigner form as a function of energy. 

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