Systematic model-independent $S$-matrix analysis of ${}^4\mathrm{He}-{}^{40}\mathrm{Ca}$ elastic scattering in going from anomalous large-angle scattering to near-Coulomb-barrier scattering

Applying the evolutionary model-independent $S$-matrix approach, we show that a simultaneous correct description of the ${}^4\mathrm{He}-{}^{40}\mathrm{Ca}$ elastic scattering patterns observed in going from anomalous large-angle scattering to near-Coulomb-barrier scattering (NCBS) can be achieved in a unified way using $S$-matrix moduli and real parts of nuclear phase, which are smooth and monotonic functions of an angular momentum, while quantum deflection functions retain a form characteristic of the nuclear rainbow case. Mechanism of NCBS pattern formation and transformation of the pattern into a pure Coulomb scattering pattern with decreasing energy in the presence of strong nuclear refraction is revealed.

Figure 1

(a) Scattering matrix moduli $\eta \left(l\right)$, (b) nuclear phases ${\delta }_r\left(l\right)$, and (c) deflection functions $\mathrm{\Theta }\left(l\right)$ for the ${}^4\mathrm{He}-{}^{40}\mathrm{Ca}$ elastic scattering at $E=4.5–27.0$ MeV.

Figure 2

(a)–(k) Elastic scattering differential cross sections (ratio to Rutherford) for the system ${}^4\mathrm{He}+{}^{40}\mathrm{Ca}$ at $E=4.5–27.0$ 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 in Figs. 2(a) and 2(b)]. $A2$ and $A3$ denote the Airy minima of second and third orders. The data are from Refs. [3, 6, 24, 25, 26, 27, 28, 29] .

Figure 3

(a) Evolution with center-of-mass energy of the intensities of nuclear refraction $2{\delta }_r\left(0\right)$ and nuclear absorption $2{\delta }_a\left(0\right)$ for the system ${}^4\mathrm{He}+{}^{40}\mathrm{Ca}$ (open circles). Solid curves are only to guide the eye. (b) Evolution with reciprocal center-of-mass energy of the nuclear rainbow angle ${\theta }_R$ (open circles). Straight line shows results of fitting to the data indicated as circles. The results of calculations performed in Ref. [16] are shown as solid circles.

Figure 4

Analysis of the formation of NCBS pattern for the system ${}^4\mathrm{He}+{}^{40}\mathrm{Ca}$ at $E=12.5$ MeV. (a) [(e)] Differences $\mathrm{\Delta }{\delta }_r\left(l\right)\left[\mathrm{\Delta }\eta \left(l\right)\right]$ between the nuclear phase (the scattering matrix modulus) extracted from the data at $E=12.5$ MeV (same as in Fig. 1) and the result of its evolution caused by the data at $E=10.5$ MeV. The angular momentum regions where our evolutionary approach has changed the scattering matrix are $\mathrm{\Delta }l=4–7$ (curves 1–6) and $\mathrm{\Delta }l=4–8$ (curve 7). (b)–(d) and (f)–(h) Elastic scattering differential cross sections (ratio to Rutherford). Solid curves are the same as in Fig. 2. The curves marked with numbers are calculated using the scattering matrices shown (in the form of differences) in Figs. 4 and 4 (numbers of curves correspond). Dashed and dotted curves in Figs. 4–4, 4, and 4 present the farside and nearside components of the cross sections marked as 1–5. Dashed and dotted curves in Fig. 4 show the farside and nearside components of the cross section marked as 7. Vertical arrows in Figs. 4 and 4 mark the positions of the Fraunhofer crossover point.

Figure 5

Analysis of the formation of a pure Coulomb scattering pattern in the presence of strong nuclear refraction for the system ${}^4\mathrm{He}+{}^{40}\mathrm{Ca}$. (a) and (b) Real $\mathrm{Re}{S}_N\left(l\right)$ and imaginary $\mathrm{Im}{S}_N\left(l\right)$ parts of the nuclear scattering matrix $S_N\left(l\right)$. Solid, dashed, and dash-dotted curves are the results for $E=4.5$, 6.5, and 8.5 MeV, respectively (same as in Fig. 1). Dotted curves show the results of the evolution of the nuclear scattering matrix extracted from the data at $E=4.5$ MeV (solid curves) caused by the artificial data that is a pure Coulomb scattering cross section at the same energy.