Phase-shift parametrization and extraction of asymptotic normalization constants from elastic-scattering data

Abstract

We develop a method to parametrize elastic-scattering phase-shifts for charged nuclei, based on Padé expansions of a simplified effective-range function. The method is potential independent and the input is reduced to experimental phase shifts and bound-state energies. It allows a simple calculation of resonance properties and of asymptotic normalization constants (ANCs) of subthreshold bound states. We analyze the 1 − and 2 + phase shifts of the 12 C+α system and extract the ANCs of the corresponding bound states. For the 1 − state, a factor-3 improvement with respect to the best value available today is obtained, with a factor-10 improvement in reach. For the 2 + state, no improvement is obtained due to relatively larger error bars on the experimental phase shifts.PACS numbers: 03.65. Nk, 03.65.Ge, 25.40.Cm, 25.55.Ci Low-energy nuclear reactions, in particular those relevant to nuclear astrophysics [1], are a fascinating application field for quantum scattering theory [2]. Because of the Coulomb repulsion between nuclei, cross sections are often impossible to measure directly and theoretical extrapolation are indispensable. The simplest of these reactions, elastic scattering, is theoretically described with a partial-wave decomposition, each partial wave l being fully characterized by a nuclear phase shift δ l or scattering matrix S l = e 2iδ l , both functions of the energy. Resonances appear as fast increases of δ l at real positive energies or as poles of S l at complex energies. For some reactions subthreshold bound states at small negative energies also play an essential role. This is for instance the case for the 1 − and 2 + bound states lying just below the 12 C+α threshold which strongly affect the 12 C(α, γ) 16 O cross section, an important reaction for stellar evolution [3]. For these states, energies are generally well-known experimentally but not the ANC of their wave function. Hence, various methods have been proposed to indirectly extract these ANCs from experimental data, like β-delayed α emission [4] or α-transfer reactions [5].A natural way to extract an ANC for a given partial wave is to parametrize the corresponding experimental phase shifts and to extrapolate this parametrization at negative energies, as bound states also correspond to scatteringmatrix poles [2]. The usual tool to do so is the reaction-(R-)matrix method [6], which describes both resonant and bound states as poles characterized by real energies and widths (Mittag-Leffler expansion). This motivated the measurement of high-precision 12 C+α phase shifts [7,8] but led to a loose constraint on the 1 − ANC [7] and to a questionable constraint on the 2 + ANC [9]. The background phase shifts (between resonances) are indeed described in terms of a channel radius and of a high-energy background pole, which adds several parameters with no direct physical meaning to the fit. Hence, simpler parametrizations are necessary.A first option is the effective-range function (ERF) K l of Eq. (2) [10]. Its analyticity properties imply the exis...

Complex-energy analysis of proton-proton fusion

“…To conclude, we have derived an accurate parametrization of the energy dependence of the weak capture matrix element Λ valid up to a few MeVs, that is based on recent effective-range functions [18,19]. This result provides the analytic continuation of Λ to complex energies, and highlights the relationship between its maximum near 0.13 MeV, the broad proton-proton resonance, and the Coulomb sub-threshold singularities.…”

“…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.…”

Connection formulas between Coulomb wave functions

“…These functions are denoted as A( , ) by Seaton. [8][9][10]18 For positive integer , the two functions w ± η are entire in the energy and both reduce to w η , the -order polynomial in given by 3,4,14,15,29,30,37…”

“…27,28 The in-depth survey of these functions has led to recent advances in the theory of effectiverange function for charged particles. [29][30][31][32] The main purpose of this paper is to determine the relations between the regular and the irregular Coulomb functions, respectively denoted as F η (ρ) and G η (ρ), in an easy-toread fashion. These relations are generically referred to as the "connection formulas" in the NIST Handbook.…”

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

“…Theoretical approaches can roughly be divided into two kinds, taking either the ions as degrees of freedom or starting from individual nucleons. The former approach includes a variety of models [2][3][4][5], effective range expansions [6][7][8][9][10][11][12], and effective field theories (EFTs) [13][14][15][16][17][18]; the microscopic approach ranges from simpler models [19] to ab initio computations [20][21][22][23]. Unfortunately, there are still significant uncertainties [1], and data tables for relevant quantities such as asymptotic normalization coefficients (ANCs) or astrophysical S factors may [24] or may not [25,26] contain theoretical uncertainties.…”