Coulomb-corrected eikonal description of the breakup of halo nuclei

The eikonal description of breakup reactions diverges because of the Coulomb interaction between the projectile and the target. This divergence is due to the adiabatic, or sudden, approximation usually made, which is incompatible with the infinite range of the Coulomb interaction. A correction for this divergence is analyzed by comparison with the dynamical eikonal approximation, which is derived without the adiabatic approximation. The correction consists in replacing the first-order term of the eikonal Coulomb phase by the first-order of the perturbation theory. This allows taking into account both nuclear and Coulomb interactions on the same footing within the computationally efficient eikonal model. Excellent results are found for the dissociation of ${}^{11}\mathrm{Be}$ on lead at 69 MeV/nucleon. This Coulomb-corrected eikonal approximation provides a competitive alternative to more elaborate reaction models for investigating breakup of three-body projectiles at intermediate and high energies.

Figure 1

Jacobi set of coordinates: $\mathit{r}$ is the projectile internal coordinate, and $\mathit{R}=\mathit{b}+Z\hat{\mathit{Z}}$ is the target-projectile coordinate.

Figure 2

Breakup probabilities as a function of transverse coordinate $b$ for ${}^{11}\mathrm{Be}$ impinging on ${}^{208}\mathrm{Pb}$ at 69 MeV/nucleon. Three energies $E$ are shown: 0.5, 1.274, and 3.0 MeV. The results are obtained within DEA (full lines), CCE (dotted lines), usual eikonal approximation (dashed lines), and first-order perturbation theory (dash-dotted lines). The upper part displays the values at small $b$, whereas the lower part focuses on the asymptotic region.

Figure 3

(a) Breakup cross sections for ${}^{11}\mathrm{Be}$ impinging on ${}^{208}\mathrm{Pb}$ at 69 MeV/nucleon as a function of the ${}^{10}\mathrm{Be}$-n relative energy $E$. The results are obtained within the DEA, the CCE, the usual eikonal approximation with upper cutoff $b_{max}=71$ fm, and the first-order perturbation theory with lower cutoff $b_{min}=15$ fm. (b) Contributions of the $s,p$, and $d$ partial waves.

Figure 4

Breakup cross sections for ${}^{11}\mathrm{Be}$ impinging on ${}^{208}\mathrm{Pb}$ at 69 MeV/nucleon as a function of the ${}^{10}\mathrm{Be}$-n relative parallel momentum $k_{\parallel }$. The figure displays the results obtained within the DEA, the CCE, the usual eikonal approximation with an upper cutoff $b_{max}=71$ fm, and the first-order perturbation theory with a lower cutoff $b_{min}=15$ fm.

Figure 5

Convergence of the multipole expansion in ${\lambda }_{max}$ of the CCE illustrated on the parallel-momentum distribution computed for ${}^{11}\mathrm{Be}$ impinging on ${}^{208}\mathrm{Pb}$ at 69 MeV/nucleon.

Figure 6

Breakup probabilities as a function of transverse coordinate $b$ for ${}^{11}\mathrm{Be}$ impinging on ${}^{12}\mathrm{C}$ at 67 MeV/nucleon. Three energies $E$ are shown: 0.5, 1.274, and 3.0 MeV. The results are obtained within the DEA (full lines), CCE (dotted lines), and usual eikonal (dashed lines) models. The upper part displays the values at small $b$, whereas the lower part emphasizes the behavior in the asymptotic region.

Figure 7

Breakup cross sections for ${}^{11}\mathrm{Be}$ impinging on ${}^{12}\mathrm{C}$ at 67 MeV/nucleon as a function of the ${}^{10}\mathrm{Be}$-n relative energy $E$. Results are obtained within the DEA, the CCE, and the usual eikonal model with an upper cutoff $b_{max}=70$ fm. Contributions of the $s,p$, and $d$ partial waves are shown as well.

Figure 8

Breakup cross sections for ${}^{11}\mathrm{Be}$ impinging on ${}^{12}\mathrm{C}$ at 67 MeV/nucleon as a function of the ${}^{10}\mathrm{Be}$-n relative parallel momentum $k_{\parallel }$. Results are obtained within the DEA, the CCE, and the usual eikonal approximation with an upper cutoff $b_{max}=70$ fm.

Figure 9

Convergence in ${\lambda }_{max}$ of the CCE illustrated on the parallel-momentum distribution for the breakup of ${}^{11}\mathrm{Be}$ on ${}^{12}\mathrm{C}$ at 67 MeV/nucleon.