Reexamining closed-form formulae for inclusive breakup: Application to deuteron- and ${}^6\mathrm{Li}$-induced reactions

The problem of the calculation of inclusive breakup cross sections in nuclear reactions is reexamined. For that purpose, the post-form theory proposed by Ichimura, Austern, and Vincent [Phys. Rev. C 32, 431 (1985)] is revisited, and an alternative derivation of the nonelastic breakup part of the inclusive breakup is presented, making use of the coupled-channels optical theorem. Using the distorted-wave Born approximation (DWBA) version of this model, several applications to deuteron and ${}^6\mathrm{Li}$ reactions are presented and compared with available data. The validity of the zero-range approximation of the DWBA formula is also investigated by comparing zero-range with full finite-range calculations.

Figure 1

Coordinates used in the breakup reaction.

Figure 2

(a) Experimental and calculated double-differential cross section, as a function of the proton scattering angle, for the protons emitted in the ${}^{93}\mathrm{Nb}(d,pX)$ reaction with an energy of 14 MeV and a deuteron incident energy of $E_d=25.5$ MeV. The dotted, thin solid, and thick solid lines are the EBU (CDCC), the NEB (FR-DWBA), and their incoherent sum, respectively. Experimental data are from Ref. [42]. (b) NEB angular distribution calculated with ZR-DWBA (dotted line), FR-DWBA without remnant (dashed line), and full FR-DWBA (solid line). (c) Convergence of the NEB calculation with respect to the bin width, $\mathrm{\Delta }{k}_b$, used for the $b$ distorted waves. See text for details.

Figure 3

(a) Experimental and calculated angle-integrated proton differential cross section, as a function of the outgoing proton energy in the LAB frame, for the ${}^{58}\mathrm{Ni}(d,pX)$ reaction at $E_d=80$ MeV. The dotted and thin solid lines are the EBU and NEB contributions, calculated with CDCC and FR-DWBA, respectively. The dot-dashed line is the contribution coming from preequilibrium and compound nucleus [50]. The thick solid line is the incoherent sum of the three contributions. Experimental data are from Ref. [43]. (b) Nonelastic breakup calculated with ZR-DWBA (dotted), nonremnant FR-DWBA (dashed), and full FR-DWBA (solid) formulas.

Figure 4

Double-differential cross section of protons emitted in the ${}^{58}\mathrm{Ni}(d,pX)$ reaction at $E_d=100$ MeV in the laboratory frame. (a) Proton angular distribution for a fixed proton energy of $E_p=50$ MeV. (b) Energy distribution for protons emitted at a laboratory angle of $8 {}^{\circ }$ (arrow in top panel). The meanings of the lines are the same as in Fig. 3 and are also indicated by the labels. Experimental data are from Ref. [44].

Figure 5

Elastic scattering of ${}^6\mathrm{Li}+{}^{209}\mathrm{Bi}$ at different incident energies. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Ref. [55]. The experimental data are from Ref. [56].

Figure 6

Angular distribution of $\alpha$ particles produced in the reaction ${}^6\mathrm{Li}+{}^{209}\mathrm{Bi}$ at the incident energies indicated by the labels. The dotted, dashed, and solid lines correspond to the EBU (CDCC), NEB (FR-DWBA), and their sum, respectively. Experimental data are from Ref. [57].

Figure 7

Angular distribution of $\alpha$ particles produced by NEB in the reaction ${}^6\mathrm{Li}+{}^{209}\mathrm{Bi}$ at the incident energies of (a) 24 MeV and (b) 38 MeV. The dotted, dashed, and solid lines are the ZR-DWBA, FR-DWBA without remnant term, and full FR-DWBA calculations, respectively.

Figure 8

Integrated cross sections for the reaction ${}^6\mathrm{Li}$ on ${}^{209}\mathrm{Bi}$ as a function of the incident laboratory energy. The open circles and the squares are the EBU (CDCC) and NEB (FR-DWBA) contributions to the $\alpha$ inclusive cross section. The solid circles are the reaction cross sections, obtained from the CDCC calculations. The arrow indicates the nominal position of the Coulomb barrier.