Unified model-independent $S$-matrix description of nuclear rainbow, prerainbow, and anomalous large-angle scattering in ${}^4\mathrm{He}-{}^{40}\mathrm{Ca}$ elastic scattering

Using the evolutionary model-independent $S$-matrix approach, we show that a simultaneous correct description of the pictures of nuclear rainbow, prerainbow, and anomalous large-angle scattering (ALAS) in the ${}^4\mathrm{He}-{}^{40}\mathrm{Ca}$ elastic scattering can be achieved with help of the $S$-matrix moduli and the real nuclear phases exhibiting smooth monotonic dependencies on angular momentum, while the quantum deflection functions have a form characteristic of the nuclear rainbow case. The special role of the surface partial waves in the formation of ALAS is revealed.

Figure 1

(a) Scattering matrix moduli $\eta \left(l\right)$, (b) nuclear phases ${\delta }_r\left(l\right)$, and (c) deflection functions $\Theta \left(l\right)$ for the ${}^{\text{4}}\text{He}{\text{−}}^{\text{40}}\text{Ca}$ elastic scattering at $E=29.05–82.0$ MeV.

Figure 2

(a)–(h) Elastic scattering differential cross sections (ratio to Rutherford) for the system ${}^{\text{4}}\text{He}{+}^{\text{40}}\text{Ca}$ at $E=29.05–82.0$ MeV (solid curves), their farside (dashed curves) and nearside (dotted curves) components, and farside cross-section components calculated without absorption in the scattering matrix (dash-dotted curves). $A1$ and $A2$ denote the Airy minima of first and second orders. The data are from Refs. [24, 25, 26].

Figure 3

Evolution with energy of the angular position of the Airy minima of first $A1$ and second $A2$ orders in the measured differential cross sections for the system ${}^{\text{4}}\text{He}{+}^{\text{40}}\text{Ca}$ (open circles) and in the farside cross-section components calculated without absorption in the scattering matrix (squares), and the same evolution of the nuclear rainbow angle ${\theta }_R$ (solid circles). Straight lines show results of fitting to the data indicated as circles.

Figure 4

Analysis of ALAS formation in the system ${}^{\text{4}}\text{He}{+}^{\text{40}}\text{Ca}$ at $E=42.6$, $36.98$, and $29.05$ MeV. (a)–(c) Differencies $\Delta \eta \left(l\right)$ [$\Delta {\delta }_r\left(l\right)$] between the scattering matrix modulus (the nuclear phase) extracted from the original data (same as in Fig. 1) and the results of its evolution caused by the artificial data that do not contain ALAS. The regions of angular momenta $\Delta l$ where our evolutionary approach has changed the scattering matrix are $\Delta l_1$ for dashed curves, $\Delta l_2$ for dotted curves, and $\Delta l_3$ for dash-dotted curves (Table 2). (d)–(f) Elastic scattering differential cross sections (ratio to Rutherford). Solid curves are the same as in Fig. 2. Dashed, dotted, and dash-dotted curves are calculated with the scattering matrices depicted (in the form of differences) in Figs. 4(a)–4(c) (types of curves correspond).