Effective-range function methods for charged particle collisions

Abstract

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as ∆ [Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of ∆ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton 1 S0 collision. The resonant state of the protonproton collision is successfully computed from the extrapolation of ∆ instead of the standard function. It is shown that interpolating ∆ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the protonproton system). This property is due to the connection between ∆ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because ∆ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.

Low-energy bound states, resonances, and scattering of light ions

“…Here, we employed the Coulomb wave functions F l and H + l . As we employ the Coulomb wave functions at imaginary arguments, some care must be taken in their numerical implementation; we followed Gaspard and Sparenberg [45] and present details in the Appendix. In Eq.…”

“…For the Coulomb wave functions we followed Gaspard and Sparenberg [45]. In that paper, the analytical properties are emphazised.…”

Complex-energy analysis of proton-proton fusion

“…The actual origin of this maximum is revealed by the continuation of Λ to complex energies, as provided by Eq. Furthermore, in contrast to Λ, the function L is holomorphic in the neighborhood of E = 0, because of the properties of the modified Coulomb functions Φ η,0 and Ψ η,0 in the limit E → 0 [18,32]. Therefore, it follows from Eq.…”

“…Therefore, the derivatives of S are likely to be influenced by the relative closeness of this pole. In addition, it is known that the Coulomb interaction between two protons generates a sequence of poles at complex energy which accumulate to the zero-energy point [16,18]. These poles may affect the polynomial extrapolation used in Refs.…”

“…Our parametrization must also describe the resonance pole at (−140 − 467 i) keV and the Coulomb poles. To do so, we resort to a recently introduced effective-range function (ERF), namely the ∆ function [18][19][20], which has only been considered useful for heavier systems until now [21]. This approach is motivated by the efficiency of the ERFs at describing the energy-dependent shape of the proton-proton wave function up to a few MeVs.…”