Coulomb and nuclear effects in breakup and reaction cross sections

We use a three-body continuum discretized coupled channel (CDCC) model to investigate Coulomb and nuclear effects in breakup and reaction cross sections. The breakup of the projectile is simulated by a finite number of square integrable wave functions. First we show that the scattering matrices can be split in a nuclear term and in a Coulomb term. This decomposition is based on the Lippmann-Schwinger equation and requires the scattering wave functions. We present two different methods to separate both effects. Then, we apply this separation to breakup and reaction cross sections of ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$. For breakup, we investigate various aspects, such as the role of the $\alpha +t$ continuum, the angular-momentum distribution, and the balance between Coulomb and nuclear effects. We show that there is a large ambiguity in defining the Coulomb and nuclear breakup cross sections, since both techniques, although providing the same total breakup cross sections, strongly differ for the individual components. We suggest a third method which could be efficiently used to address convergence problems at large angular momentum. For reaction cross sections, interference effects are smaller, and the nuclear contribution is dominant above the Coulomb barrier. We also draw attention to different definitions of the reaction cross section which exist in the literature and which may induce small, but significant, differences in the numerical values.

Figure 1

Electric transition probabilities (37) from the $3/2^{-}$ ground state as a function of the $\alpha +t$ continuum energy. Labels represent the final angular momentum $j$. For $E2$ transitions, the contributions of $j=1/2^{-}$ and of $j=3/2^{-}$ are negligible at the scale of the figure. The dots represent the (discrete) pseudostate energies $E_{0,n}^j$. The lines are to guide the eye.

Figure 2

Diagonal amplitudes $|U_{\omega L_{\omega },\omega L_{\omega }}^{J\pi }|$ at $E=26$, 34, and 42 MeV for $J=L_{\omega }+1/2$ and $J=L_{\omega }+3/2$. The curves corresponding to $J=|L_{\omega }-1/2|$ and $J=|L_{\omega }-3/2|$ are similar and therefore not shown.

Figure 3

Convergence of the breakup cross section with $E_{\max }$ (a) ($E_{\max }=15$ MeV and $E_{\max }=20$ MeV are indistinguishable) and with ${\ell }_{\max }$ (b). In panel (a), ${\ell }_{\max }=3$ is used, and $E_{\max }=15$ MeV is used in panel (b).

Figure 4

Breakup cross sections (40) for different partial waves $j$ of the $\alpha +t$ continuum. The ${}^7\mathrm{Li}+{}^{208}\mathrm{Pb}$ energies are 26, 34, and 42 MeV. The dots represent the pseudostate energies, and the lines are to guide the eye.

Figure 5

Contributions of the dominant $\alpha +t$ partial waves $j$ to the total breakup cross section [see Eq. (41)].

Figure 6

Angular-momentum distribution of the breakup cross sections at $E=26,34,42$ MeV. The solid lines represent the full CDCC calculation, while the dashed line are obtained at the Coulomb approximation (34).

Figure 7

Nuclear and Coulomb (a) and interference (b) components of the breakup cross section.

Figure 8

Fraction of Coulomb (a) and nuclear (b) breakup cross sections. The solid lines are obtained as described in the text, and the dashed lines represent the weak-coupling approximation.

Figure 9

Partial-wave decomposition of the Coulomb (a) and nuclear (b) breakup cross section at $E=34$ MeV. The solid lines are obtained as described in the text, and the dashed lines represent the weak-coupling approximation.

Figure 10

Reaction cross section (25) with the Coulomb (28) and nuclear (29) contributions. The dashed line corresponds to the definition (31) of the reaction cross section.

Figure 11

Partial-wave decomposition of the reaction cross section at $E=24$ MeV (top) and $E=42$ MeV (bottom). The Coulomb and nuclear contributions are shown in green and blue, respectively. The solid lines correspond to the full CDCC calculation and the dashed lines to the single-channel approximation (at $E=42$ MeV, the nuclear terms are indistinguishable). The single-channel Coulomb term is negative for $25\le L_{\omega }\le 29$, and the absolute value is shown.