Theoretical study of elastic electron scattering off stable and exotic nuclei

Results for elastic electron scattering by nuclei, calculated with charge densities of Skyrme forces and covariant effective Lagrangians that accurately describe nuclear ground states, are compared against experiment in stable isotopes. Dirac partial-wave calculations are performed with an adapted version of the ELSEPA package. Motivated by the fact that studies of electron scattering off exotic nuclei are intended in future facilities in the commissioned GSI and RIKEN upgrades, we survey the theoretical predictions from neutron-deficient to neutron-rich isotopes in the tin and calcium isotopic chains. The charge densities of a covariant interaction that describes the low-energy electromagnetic structure of the nucleon within the Lagrangian of the theory are used to this end. The study is restricted to medium- and heavy-mass nuclei because the charge densities are computed in mean-field approach. Because the experimental analysis of scattering data commonly involves parameterized charge densities, as a surrogate exercise for the yet unexplored exotic nuclei, we fit our calculated mean-field densities with Helm model distributions. This procedure turns out to be helpful to study the neutron-number variation of the scattering observables and allows us to identify correlations of potential interest among some of these observables within the isotopic chains.

Figure 1

Radial dependence of the charge densities of the stable nuclei ${}^{16}\mathrm{O}$, ${}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$. The predictions of the different mean-field models indicated in the legend are compared with the charge densities fitted experimentally by the Fourier-Bessel analysis [6, 7].

Figure 2

Elastic DCS for electron-nucleus scattering in ${}^{16}\mathrm{O}$, ${}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$ as a function of the scattering angle θ, at the energies shown. The results from the mean-field models indicated in the legend are compared with the measured DCS (Exp) [60, 61, 62] and with the DCS calculated in the ELSEPA code from the charge densities fitted experimentally by the Fourier-Bessel analysis [Exp(fit)] [6, 7].

Figure 3

Squared charge form factor for ${}^{16}\mathrm{O}$, ${}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$ as a function of the momentum transfer $q$ at the electron beam energies shown. It has been obtained by applying Eq. (13) as described in the text, both for the mean-field models indicated in the legend and for the experimentally fitted charge densities [Exp(fit)] [6, 7].

Figure 4

Relative differences of DCS in pairs of neighbor nuclei at 250 MeV for Ca and Ti and at 225 MeV for Sn. The results from mean-field models are compared with the measured RDDCS (Exp) and with the RDDCS calculated in the ELSEPA code from the charge densities fitted experimentally [Exp(fit)] in Refs. [61, 63]. Note that in the present figure the vertical scales are linear and that all of them have been magnified by a factor 100.

Figure 5

Squared charge form factor of ${}^{118}\mathrm{Sn}$ and ${}^{208}\mathrm{Pb}$, derived from the covariant mean-field model $G2$, for the indicated electron beam energies. The figure points out that the sensitivity of the charge form factor extracted through Eq. (13) to the beam energy in high-energy elastic electron-nucleus scattering is rather small, even when the atomic number of the target is large as in ${}^{208}\mathrm{Pb}$.

Figure 6

Mass-number dependence of the Helm parameters ${\sigma }^2$ (upper panel) and $R_0$ (lower panel) predicted by the covariant mean-field model $G2$ in the Sn isotopic chain. The average value of ${\sigma }^2$ is depicted by a horizontal dashed line.

Figure 7

Charge densities (upper panel) and charge radii (lower panel) in the Sn isotopic chain, according to the covariant model $G2$ and to the determined Helm distributions.

Figure 8

Results predicted by the $G2$ effective interaction for 500-MeV electron scattering in the Sn isotopic chain. (Upper panel) Mass-number dependence of the momentum transfer at the first inflection point ($q_{\mathrm{IP}}$) of the squared charge form factor in DWBA. (Middle panel) Correlation of the effective momentum transfer at the first minimum of the squared charge form factor in PWBA ($q_{\mathrm{eff},\min }$) with the value of $q_{\mathrm{IP}}$. A linear fit of the results is shown and the correlation coefficient $r$ is indicated. (Lower panel) The change of the charge radii calculated with $G2$ and with the corresponding Helm densities is depicted against the value of $q_{\mathrm{IP}}$.

Figure 9

Correlation along the Sn isotopic chain of the squared charge form factor in DWBA, at its first inflection point ($q_{\mathrm{IP}}$), with the product of the Helm parameter ${\sigma }^2$ and $q_{\mathrm{IP}}^2$. The results are computed with the $G2$ effective interaction for scattering by 500-MeV electrons. Note that each point of the figure corresponds to a different isotope and that the mass number increases from right to left.

Figure 10

Same as described in the caption to Fig. 6 for the Ca isotopic chain.

Figure 11

Same as described in the caption to Fig. 7 for the Ca isotopic chain.

Figure 12

Same as described in the caption to Fig. 8 for the Ca isotopic chain.

Figure 13

Same as described in the caption to Fig. 9 for the Ca isotopic chain.

Figure 14

Comparison in Ca and Sn isotopes between the squared charge form factor in DWBA calculated with the charge densities of the $G2$ mean-field interaction (solid lines) and with the equivalent Helm charge densities (dashed lines). Notice that the vertical and horizontal scales for calcium and tin are different.