Feshbach–Fano R-matrix Method: Application to Potential Scattering

The general Feshbach-Fano R-matrix (FFR) procedure proposed recently by Nestmann [J. Phys. B 31, 3929 (1998)] makes it possible to construct resonance metastable states of transient molecular ions and their coupling elements to the background scattering continuum. These quantities are needed for the study of nuclear dynamics in the framework of the nonlocal resonance model. The performance of this approach is carefully studied and its properties analyzed in the field of potential scattering. An improvement of the Nestmann procedure which makes the calculation more stable and robust is proposed. Introduction Resonances are one of the most striking and ubiquitous phenomena in scattering experiments (e.g. in nuclear, atomic or molecular physics, as well as in the physics of semi-conductors, quantum dots or Bose-Einstein condensates). They are associated with poles, zres = ǫres − iΓres/2, of S-matrix in the complex-energy plane. ǫres and Γres are called the position and width of the resonance. Breit-Wigner formula [Breit and Wigner, 1936] σres = 1 π Γres (ǫ− ǫres) + Γres/4 can usually be fitted to the resonance component of the cross-section. This formula is not valid in the vicinity of fragmentation threshold or in the case of overlapping resonances. It has been generalized by Kapur and Peierls [Kapur and Peierls, 1938], Wigner and Eisenbud [Wigner and Eisenbud, 1947] and by Fano and Feshbach ([Fano, 1961; Feshbach, 1958, 1962]), introducing energy-dependent parameters ǫres(ǫ) ≡ ǫd+∆(ǫ) and Γres(ǫ) ≡ Γ(ǫ) (ǫd is the discrete-state energy, ∆(ǫ) is the energy-dependent level shift, and Γ(ǫ) is the energy-dependent width). The poles of the S-matrix can be found as a solution of implicit equation ǫd +∆(zres)− i 2 Γ(zres)− zres = 0. (1) The Feshbach-Fano (FF) method allows us to associate square integrable wave functions, within a physically relevant linear functional space Q, with the resonances. Once the subspace Q is determined on the basis of the particular physical context, the FF method provides a detailed and intuitive understanding of the resonant process. The basic idea of the Feshbach-Fano R-matrix (FFR) method introduced in Nestmann’s seminal article [Nestmann, 1998] is to define Q as a subset of the scattering states fulfilling the Wigner-Eisenbud boundary conditions on the surface of the R-matrix sphere complemented with set of additional conditions. The major reason why we consider the FFR method is its capability to construct necessary potentials and coupling elements to be used in the application of the nonlocal resonance model (NRM) to electronmolecule scattering. NRM is based on the assumption that a temporary molecular negative-ion state (resonance) is formed in the process of the collision and that this resonance accounts for the coupling of the electronic scattering dynamics with the nuclear motion (see [Domcke, 1991] for comprehensive review). The nonlocal resonance theory yields cross sections for vibrational excitation (VE), dissociative electron attachment (DEA) as well as associative electron detachment (AED) in very good agreement with experimental data describing all the complexity of the problem, i.e. the theory reproduced successfully threshold peaks in VE, Wigner cusps in the process of DEA, isotope effect in DEA, etc. The theory predicted even new features oscillations in VE cross sections below the opening of the DEA channel the existence of which was confirmed subsequently by experiment [Allan et al., 2000]. The purpose of this paper is twofold: first to test Nestmann’s method on clearly defined cases, i.e. to use well-known potentials which support several resonance states and which have been studied by other authors, and to establish the limits the FFR method; second to propose an improvement of WDS'05 Proceedings of Contributed Papers, Part III, 429–435, 2005. ISBN 80-86732-59-2 © MATFYZPRESS