Resonance Reactions

The considerations of a previous note are extended to include the possibility of several resonance levels. It is shown, in the case of resonance scattering, that $R$, which is the tangent of the phase shift divided by the wave number, is the sum of the reciprocals of linear functions of the energy (Eq. (12)), each term corresponding to one resonance level. All the coefficients in this expression for $R$ are real, energy independent constants. As a result, it appears most natural to write the cross section (Eq. (12a)) as the square of a ratio of two expressions which are themselves fractional expressions of the energy. It is possible to write the cross section also as the square of a single fractional expression of the energy (Eq. (13)). However, the coefficients of the fractional expression are then slowly varying functions of the energy and are not real but subject to other, more involved limitations. The results are quite similar if, in addition to scattering, a reaction is possible also. The cross sections can be represented, most naturally, by means of the squares of the elements of a matrix (Eq. (33)). However, this matrix is the quotient of two matrices which involve the matrix $\mathfrak{R}$ and only $\mathfrak{R}$ is a simple function (Eq. (33a)) of the energy. The cross sections can be evaluated in a simple closed form only if either there is only one pair of reaction products possible (in addition to the reacting pair) (Eqs. (35), (35a)), or if there are only two resonances present. It is shown, however, that the cross sections can be represented also in the usual form (Eq. (42)) but the "constants" of this form are not strictly independent of energy and not real any more but subject to more involved restrictions. It turns out that the cross section becomes zero between consecutive resonances if only elastic scattering is possible. The elastic scattering cross section does not become zero in general for any value of the energy if a nuclear reaction or inelastic scattering is also possible. If the collision can yield, instead of the colliding pair of particles, only another pair, the cross section for the production of this pair will become zero between successive resonances if the product of certain real quantities has the same sign for both resonances. If the collision may result in any of three or more pairs of particles (e.g., ${\mathrm{H}}^2$ + ${\mathrm{H}}^2$→ ${\mathrm{H}}^1$ + ${\mathrm{H}}^3$, or ${\mathrm{He}}^3$ + neutron, or ${\mathrm{He}}^4+\gamma$ no cross section will vanish, in general, for any value of the energy. The considerations of the present paper are restricted to the case in which the relative angular momenta of the reaction products as well as of the reacting particles vanishes.