Fano formula for resonance poles in the Jost--random-phase approximation: Breit--Wigner-type peaks, Fano dips, and Fano-type peaks in $^{16}$O and $^{208}$Pb

Abstract

Fano's formula is derived for a resonance pole of the strength function in the continuum random-phase approximation formulated within the Jost function framework (Jost-RPA). At a pole, the residue of the transition-amplitude vector of the open channels lies along the Takagi vector of the pole for every external field. Of the background, the amplitude vector without the pole term, only the component along the Takagi vector therefore interferes with the resonance, and the strength function can be written exactly in Fano's form for several continuum channels. In this form $q$ is complex and energy dependent, and the modulus $p$ of the cosine of the angle between the Takagi vector and the background serves as Fano's $|ρ|$. In an idealized form of Fano's model, $q$ and $p$ reduce to Fano's $q$ and to the overlap $|ρ|$ of Fano and Cooper. A second exact form separates a Fano term and a Breit-Wigner term, the part of the resonance that does not interfere with the background. Three patterns follow: Breit-Wigner-type peaks for $p^2\simeq0$, Fano dips, and Fano-type peaks for $p^2\simeq1$. A negative real part of the complex transition probability, the residue of the response function at the pole, is equivalent to Fano's condition $|q|<1$ for a dip at the resonance energy, evaluated at the complex conjugate of the pole energy. Earlier Jost-RPA analyses were made only for $^{16}$O. The analysis is applied to $^{16}$O and to $^{208}$Pb, with up to 62 open channels. It attributes the dips found earlier in $^{16}$O to the interference of single poles with their backgrounds, and it shows in both nuclei Fano dips, Fano-type peaks, and poles whose pattern changes with the field. In $^{208}$Pb, narrow poles near the broad poles of the giant dipole resonances produce Fano-type peaks with dips in the field of each resonance. The patterns are not determined by the collectivity indicators of the poles.