Strong absorption and the distribution of zeros of the S-matrix

Abstract

Abstract The only distributions normally included in a discussion of the statistical theory of nuclear resonance reactions are the distributions of the widths (Γ) and spacings ( D ) of the levels of the compound nucleus. However, as the usual Hauser-Feshbach theory makes clear, Γ and D alone are not sufficient to determine the ratio σ cc′ dir σ cc′ fl . In an attempt to determine what further statistical information is sufficient to determine this ratio, in the special limit that it tends to zero for all cc ′, c ≠ c ′ (the “strong-absorption” limit), we study several “picket fence” S -matrix models, as well as a random-residue model exhibiting Ericson fluctuations. These models indicate that the strong-absorption limit ( S cc′ = 0, all cc′) is directly related to the distribution of the zeros of S cc ′ ( E ) in the upper half of the complex E -plane, and that strong absorption is reached only if these zeros are distributed with a high density in the region E → + i ∞. As a by-product, we obtain a generalization of the theorem of Moldauer and Simonius [| det S | = exp (−π Γ / D )] . Our generalization applies to individual optical S -matrix elements (and so to direct-reaction cross sections) rather than just to their determinant.