Resonance Reactions and Anomalous Scattering

The purpose of the present paper is to give a derivation of the resonance formula for nuclear reactions which is free from artificial assumptions. Mathematically, the method used amounts to a Taylor series development of the wave function with respect to the energy. It is assumed that the first (energy independent) term in this development is, within a region of configuration space where all particles are close together, the same, no matter in which way the compound state is formed and that this is, in that region of configuration space, already a good approximation. The second term in the development of the wave function with respect to the energy difference from the resonance energy can then be calculated very easily and this calculation is carried out in Section II for resonance scattering, in Sections III and IV for resonance reactions. It is assumed in both calculations that the colliding particles have zero orbital angular momentum around their center of mass. The third term in the same expansion is estimated for resonance scattering in Section VI and it is shown that, if there were no other resonances in the neighborhood, the effect of the third term would be negligible over a very wide energy range (several hundred kilovolt). The formulae for the cross sections, as obtained, are of greater generality than the customary ones inasmuch as they contain extra terms which could be interpreted as potential scattering and potential reaction. The existence of such terms has been noticed already by Bethe. However, as discussed in Section V, the extra terms are, particularly in the neighborhood of the resonance, much smaller than the resonance terms so that one is led back, in practice, to the ordinary resonance formulae, as given, e.g., by Bethe. In particular, the disintegration probability is, as function of energy, proportional to the velocity with which the reaction products separate if the orbital angular momentum of the separating particles vanishes. It may be worth while to remark that the resonance part of the collision matrix has a particularly simple form and is, e.g., of rank 1. The case of orbital angular mementum $1\hslash$ is discussed in Section VII. In this case, the disintegration probability of the compound state is proportional to the third power of the velocity of the separating particles so that the scattering is, at very low energies, proportional to the square of the energy. The same holds, in this case, of the "potential scattering" also. Section VII also contains an investigation of the region of validity of the formulae in case of angular momentum $1\hslash$ between the colliding particles and shows that this region will extend to the neighboring resonances.