Wigner formula

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Interestingly, the correction factor is to a good accuracy approximated by the Wigner formula (2.15) and it is practically independent of friction [Hanggi 1986], by contrast with the relatively strong dependence of on r. ... 

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 ... 

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. 

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... 

For the direct evaluation of the 3/-symbol the Racah formula [8, 9] (equivalent to the Wigner formula [5, 6]) is helpful... 

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. 

A simple expression may be readily obtained for the capture cross section, for 5-wave neutrons. To a first approximation the capture cross section is given by the one-level Breit-Wigner formula ... 

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... 

BREiT-WiGNER-Formel, Breit-Wigner formula 20, 123, 403, 405, 406. BremsprozeB, slowing down process 397, 398. Bremsstrahlung 472f., 476, 509, 524. Bremssubstanz, moderator 398. 

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. 

Some formulas have been proposed for the correlation energy f corr- The Wigner formula... 

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. 

Maximum cross-section resonant absorption, usually some orders of magnitude, exceeds another cross-sections of interaction of y-quanta with matter. That is why the Mossbauer effect is easy observable for Fe and Sn, despite their small content in natural isotope blend ( Fe 2.2%, Sn 8.5%). Cross-section of resonant interaction reaches a maximum value for the energy E = Eq- Dependence of o on E is given by the Wigner formula... 

In the reduction of quadratic matrix elements Vg, the operator part V j represents a component of the basis set that is obtained by a direct product F, Fs. Its representation is, in general, reducible. The decomposition of the operator part may be performed according to the Wigner formula... 

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