The Interpretation of Resonances in Nuclear Reactions

The theory of resonances in the one-body problem model is systematized. Energy shifts in the apparent position of levels are found. The apparent position of levels in scattering and in $\gamma$-ray emission is expected to be the same to within $\frac{\sim {\Gamma }^2}{E}$ in the usual notation if the only repulsive barrier is the centrifugal one. In the absence of strong repulsive barriers there may be additional shifts of the order T. Interference between levels is shown to occur for the scattering cross section in the numerator and the denominator of a fraction in accordance with Eqs. (3.3), (3.6). The artificiality of usual formulas with interference for cross sections is discussed in this connection. The construction of Green's function for two-dimensional separable problems is described. The construction works also for some many-dimensional problems. The reduction of the number of essential dimensions is outlined by a reformulation of the problem in one less dimension. Applications are made to the solution of scattering problems which represent schematically idealized nuclear problems. The single particle potential barriers enter the solutions through the regular and irregular solutions $f$, $g$ of the radial equation. It is seen from the solutions that the yield may depend on the $f$ and $g$ for various excited states of the residual nucleus. A simplification occurs if the interaction is highly repulsive and is localized in the two-dimensional space of the distances of the particles from the center. In this case the regular functions $f$ for the incident and final state enter linearly in the wave function as in Eq. (15.5). If the interaction region is made large and if the incident state happens to be especially important in the expansion of Green's function then the factor $g$ of the incident wave occurs in the formula for the cross section as in Eq. (16.4). A resonance model is worked out by assuming the interaction to be highly attractive within a circle and repulsive in a ring surrounding the circle in the two-dimensional diagram. The damping constants in the resultant formula contain $f^2$ as in Eq. (18.8). Here also the single dependence on $f^2$ is a consequence of having localized the interaction to a narrowly defined radius for the incident particle. In the general case each damping constant involves the $g$ of different states. The examples show that quantitative applications with simplified forms of damping constants and with interference types of dispersion formulas have only a limited validity.