Behavior of the Elastic-Scattering Cross Section σ(E) in the Vicinity of Two Overlapping Levels with Equal Resonance Energies

Abstract

The elastic-scattering process proceeding through two resonance levels that have the same spin j and equal resonance energies, (E₁ = E₂), but different widths (Γ₁ ≠ Γ₂) is considered. It is shown that the energy dependence of the total scattering cross section has two equal maxima at the points E₁ ± (1/2) $\sqrt {{\Gamma _1}{\Gamma _2}}$, the cross-section value at the maxima being 4π (2j + 1)ƛ², where ƛ is the wavelength of the incident particle in the c.m. frame, and that, at the energy E₁, the cross section vanishes, σ (E₁) = 0. The cross section is symmetric with respect to the point E₁.