Continuum discretization methods in a composite particle scattering off a nucleus: Benchmark calculations

The direct comparison of two different continuum discretization methods toward the solution of a composite particle scattering off a nucleus is presented. The first approach—the continuum-discretized coupled-channel method—is based on the differential equation formalism, while the second one—the wave-packet continuum discretization method—uses the integral equation formulation for the composite-particle scattering problem. As benchmark calculations, we have chosen the deuteron off ${}^{58}\mathrm{Ni}$ target scattering (as a realistic illustrative example) at three different incident energies: high, middle, and low. Clear nonvanishing effects of closed inelastic channels at small and intermediate energies are established. The elastic cross sections found in both approaches are very close to each other for all three considered energies.

Figure 1

Energy ${\epsilon }_i$ (a) and momentum $k_i$ (b) distributions for the $n\text{−}p$ continuum discretization constructed with a Gaussian basis on the generalized Tchebyshev grid (18).

Figure 2

Comparison of the differential elastic cross sections of the deuteron off ${}^{58}\mathrm{Ni}$-target scattering (as a ratio to the Rutherford cross section ${\sigma }_{\mathrm{R}}$) at $E_{\mathrm{lab}}=80$ MeV calculated with the CDCC (dotted line) and WPCD approaches (full line). The dashed curve corresponds to the single folding model calculation.

Figure 3

Elastic $d{+}^{58}\mathrm{Ni}$ cross section at $E_{\mathrm{lab}}=21.6$ MeV calculated within the CDCC (dotted line) and WPCD (dashed line) approaches for different values of $E_{\max }$: (a) 20 MeV, (b) 38 MeV, and (c) 80 MeV. The full lines correspond to the converged CDCC and WPCD results (with $E_{\max }=139$ MeV), which are almost indistinguishable in these plots.

Figure 4

Converged results (with respect to $E_{\max }$) for the differential elastic cross section (relative to Rutherford) for $d{+}^{58}\mathrm{Ni}$ at $E_{\mathrm{lab}}=21.6$ MeV, calculated with the CDCC (dotted curve) and WPCD (solid curve) approaches. The dashed line corresponds to the folding calculation, in which continuum channels are not taken into account.

Figure 5

Argand plot for the elastic $S$-matrix elements for the reaction $d{+}^{58}\mathrm{Ni}$ at $E_{\mathrm{lab}}=21.6$ MeV calculated within the CDCC (open circles connected by dotted lines) and WPCD (black squares connected by solid lines) methods.

Figure 6

Elastic $d{+}^{58}\mathrm{Ni}$ cross section at $E_{\mathrm{lab}}=12$ MeV calculated within the CDCC (dotted line) and WPCD (dashed line) approaches for different values of $E_{\max }$: (a) 10.2 MeV, (b) 24.5 MeV, and (c) 36 MeV. The full curves correspond to the converged CDCC results ($E_{\max }=80$ MeV).

Figure 7

Converged results (with respect to $E_{\max }$) for the differential elastic cross section (relative to Rutherford) for $d{+}^{58}\mathrm{Ni}$ at $E_{\mathrm{lab}}=12$ MeV, calculated with the CDCC (dotted curve) and WPCD (solid curve) approaches. The dashed line corresponds to the folding calculation, in which continuum channels are not taken into account.