Higher Angular Momenta and Long Range Interaction in Resonance Reactions

The results of two preceding articles on the energy dependence of the reaction and scattering cross sections are generalized. As before, the configuration space is divided into two parts. In the internal region or reaction zone there is no restriction on the type of interaction between the particles. In the external region, on the other hand, the interaction takes place between the colliding or separating particles without changing their structure (wave function). The present article deals with a more general situation than the preceding ones, first by allowing for an interaction, although only of a restricted type, outside the reaction zone. The most common type of interaction which plays a role outside the reaction zone is the electrostatic interaction of the colliding or separating particles. In addition, the present article does not restrict the angular momentum of the particles to zero, but permits the treatment of arbitrary angular momenta. The cross sections are expressed in well-known fashion in terms of the collision matrix $\mathfrak{U}$ which is, of course, independent of the size of the internal region. $\mathfrak{U}$, in its turn, is expressed by the $\mathfrak{R}$ matrix ((35) and (38)) and the quantities $\omega$, $\mathfrak{B}$, and $\mathfrak{C}$ describing the interaction in the external region. $\mathfrak{R}$ as function of energy is given by (24), and a number of properties of this function are enumerated. None of the quantities $\mathfrak{R}$, $\omega$, $\mathfrak{B}$, and $\mathfrak{C}$ are strictly independent of the size of the internal region, although the combination (38), i.e., the collision matrix, of course, is.