Excitation of soft dipole modes in electron scattering

The excitation of soft dipole modes in light nuclei via inelastic electron scattering is investigated. I show that, under the proposed conditions of the forthcoming electron-ion colliders, the scattering cross sections have a direct relation to the scattering by real photons. The advantages of electron scattering over other electromagnetic probes is explored. The response functions for direct breakup are studied with few-body models. The dependence on final-state interactions is discussed. A comparison between direct breakup and collective models is performed. The results of this investigation are important for the planned electron-ion colliders at the GSI and RIKEN facilities.

Figure 1

Virtual photon spectrum for the $E1,E2,$ and $E3$ multipolarities in electron scattering off arbitrary nuclei at $E_e=100$ MeV and maximum scattering angle of 5${}^{°}$.

Figure 2

Comparison between the virtual photon spectrum of 1 GeV electrons (dashed line), and the spectrum generated by a 1 GeV/nucleon heavy ion projectile (solid line) for the $E1$ multipolarity, as a function of the photon energy. The virtual photon spectrum for the ion has been divided by the square of its charge number.

Figure 3

$\left|{\mathcal{I}}_{s\to p}\right|{}^2$ calculated using Eq. (6), assuming $e_{\mathrm{eff}}=e,A=11,$ and $S_n=0.5$ MeV, as a function of $E_r$. The long dashed curve corresponds to $a_1=-10$ fm${}^{-3}$ and $r_1=-0.5$ fm${}^{-1}$, the dashed curve corresponds to $a_1=-50$ fm${}^{-3}$ and $r_1=1$ fm${}^{-1},$ the solid curve corresponds to $a_1=$ $r_1=0$, and, finally, the dotted curve corresponds to $a_1=-10$ fm${}^{-3}$ and $r_1=0.5$ fm${}^{-1}$.

Figure 4

Jacobian coordinates ($\mathbf{x}$ and $\mathbf{y}$) for a three-body system consisting of a core (c) and two nucleons (1 and 2).

Figure 5

Comparison between the calculation of the response function (in arbitrary units) with Eqs. (9) and (10), using ${\delta }_{\mathit{nn}}=0$ and ${\delta }_{\mathrm{nc}}=0$, (dashed line), or including the effects of final state interactions (continuous line). The experimental data are from Ref. [49].

Figure 6

Hydrodynamical model for collective nuclear vibrations in halo nuclei. The (a) Steinwedel-Jensen (SJ) mode and the (b) Goldhaber-Teller (GT) mode are shown separately.

Figure 7

Contour plot of the nuclear transition density in the hydrodynamical model consisting of a mixture of GT and SJ vibrations. The darker areas represent the larger values of the transition density in a nucleus which has an average radius represented by the dashed circle. The legend on the right displays the values of the transition density within each contour limit.

Figure 8

Hydrodynamical transition densities for ${}^{11}\mathrm{Li}$ and three different assumptions for the SJ+GT admixtures, according to Eq. (13). The dashed curve is for a GT oscillation mode, with the core vibrating against the halo neutrons, with effective charge number $Z_{\mathrm{eff}}^{(1)}=6/11$, radius $R=3.1$ fm, and ${\alpha }_1=1$. The dotted curve is for an SJ oscillation mode, with effective charge number $Z_{\mathrm{eff}}^{(2)}=2/11$, and ${\alpha }_2=1$. The solid curve is their sum.