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

We introduce a simplified effective-range function for charged nuclei, related to the modified $K$ matrix but differing from it in several respects. Negative-energy zeros of this function correspond to bound states. Positive-energy zeros correspond to resonances and “echo poles” appearing in elastic-scattering phase-shifts, while its poles correspond to multiple-of-$\pi$ phase shifts. Padé expansions of this function allow one to parametrize phase shifts on large energy ranges and to calculate resonance and bound-state properties in a very simple way, independently of any potential model. The method is first tested on a $d$-wave ${}^{12}\mathrm{C}+\alpha$ potential model. It is shown to lead to a correct estimate of the subthreshold-bound-state asymptotic normalization constant (ANC) starting from the elastic-scattering phase shifts only. Next, the ${}^{12}\mathrm{C}+\alpha$ experimental $p$-wave and $d$-wave phase shifts are analyzed. For the $d$ wave, the relatively large error bars on the phase shifts do not allow one to improve the ANC estimate with respect to existing methods. For the $p$ wave, a value agreeing with the ${}^{12}\mathrm{C}({}^6\mathrm{Li},d){}^{16}\mathrm{O}$ transfer-reaction measurement and with the recent remeasurement of the ${}^{16}\mathrm{N}\beta$-delayed $\alpha$ decay is obtained, with improved accuracy. However, the method displays two difficulties: the results are sensitive to the Padé-expansion order and the simplest fits correspond to an imaginary ANC, i.e., to a negative-energy “echo pole,” the physical meaning of which is still debatable.

Figure 1

Simplified effective-range function ${\mathrm{\Delta }}_2$ (red solid line) and renormalized inverse of the modified $K$ matrix (green dashed line) deduced from the elastic-scattering phase shift of the $d$-wave ${}^{12}\mathrm{C}+\alpha$ potential model of Refs. [11, 14]. The $y$-axis scale changes at 1.1 MeV.

Figure 2

${}^{12}\mathrm{C}+\alpha$ elastic-scattering $p$-wave (a) phase shifts ${\delta }_1$, (b) simplified effective-range function (ERF) with resonance zero removed, ${\mathrm{\Delta }}_1/(1-E/E_{\mathrm{res}})$, (c) usual ERF $K_1$, (d) simplified ERF with resonance and bound-state zeros removed, ${\mathrm{\Delta }}_1/(1-E/E_{\mathrm{res}})(1-E/E_B)$, compared with the value (black point with error bar) deduced from the transfer-reaction ANC of Ref. [6] (after a sign change; see text). The experimental data (gray crosses) are from Ref. [9, 10] and the $[N_{\mathrm{fit}}/M_{\mathrm{fit}}]$ Padé approximants are detailed in the text. In (a), (b), and (c), the standard ERF fit of Ref. [19] and the modified $K$-matrix fit of Ref. [4] are also shown. In (c), the three theoretical curves are indistinguishable from the experimental data.

Figure 3

Same as Fig. 2 and 2 for the $d$ wave, with two resonances ($N_0=N_{\infty }=2$) and one subthreshold bound state ($N_B=1$) removed. The $y$ axis changes at 4.1 MeV in (a).