Schematic Treatment of Nuclear Resonances

Abstract

A schematic model of competitive nuclear disintegration processes is examined with special attention to the following features: (a) The definition of the compound state in a manner independent of the introduction of a "nuclear radius" into the definition; (b) the evaluation of the influence of barrier penetration on the parameters entering the dispersion formulas; (c) the comparison of the equivalent disintegration probabilities entering as products in the numerators of the dispersion terms with the resonance widths which occur as coefficients of $i$ in the imaginary part of the denominators in the same formulas. It is found that a definition of the compound state can be given in the case considered without the aid of an arbitrarily assigned nuclear radius. The definition of the compound state is arranged to be such as to be nearly independent of the potential barriers affecting the disintegration products. The damping constants turn out to be expressible primarily through the regular radial function $f$ and are found to depend also on the irregular functions $g$, inasmuch as the latter determine the linear combinations with which the different damping integrals denoted by $I_n^{}{}_{}{}^{\mathrm{}\left(q\right)\mathrm{}}$ combine to give the damping constants $\Gamma$. The cross sections can be expressed in the present model in terms of determinants with a finite number of rows and columns. The answer is also transformed into a "dispersion formula" form and it is found that the relation between the resonance width and the disintegration probabilities can be made to be exact in the "isolated level" form but is only an approximation in the representation of the general case that has been used. At the end of the paper a special case is discussed for which the energy dependence of the answer is worked out in terms of elementary functions rather than infinite series.