Coulomb form factors of even-even nuclei described by axially deformed relativistic mean-field models

Background: Combining the relativistic mean-field (RMF) model and distorted wave Born approximation (DWBA) method, Coulomb form factors for elastic electron scattering have been studied for several stable nuclei (${}^{208}\mathrm{Pb}, {}^{40}\mathrm{Ca}, {}^{32}\mathrm{S}$, and ${}^{24}\mathrm{Mg}$) with a methodology that can be extended to exotic nuclei.

Purpose: Previous studies on nuclear Coulomb form factors by the $\text{RMF}+\text{DWBA}$ method were mainly based on the spherical RMF model. This work aims to further extend the studies to the axially deformed RMF model.

Method: The nuclear proton density distributions are first calculated by the deformed RMF model. Next, the axially deformed density distributions are expanded into multipole components. With the spherical ${\rho }_0$ components, the Coulomb form factors of even-even nuclei are calculated by the DWBA method.

Results: For spherical nuclei, the nuclear Coulomb form factors obtained with the deformed RMF model almost coincide with those from the spherical RMF model. For deformed nuclei, Coulomb form factors obtained with the deformed RMF model agree better with the experimental data at the diffraction minima and at high momentum transfers.

Conclusions: Results indicate the proton densities calculated from the axially deformed RMF model are valid and reasonable. The electron-scattering experiments will soon be available for exotic nuclei, and the studies in this paper are helpful to interpret the experimental data of deformed exotic nuclei.

Figure 1

Ground-state proton densities (units of ${\mathrm{fm}}^{-3}$) and nuclear surface shapes of ${}^{208}\mathrm{Pb}$, calculated by the deformed RMF model with FSUGarnet and NL3 parameter sets.

Figure 2

(a) The density multipole components of ${}^{208}\mathrm{Pb}$ for deformed proton distributions in Fig. 1, which are obtained by Eqs. (6) and (7). (b) The proton distributions of ${}^{208}\mathrm{Pb}$, calculated by the spherical RMF model. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 3

Nuclear Coulomb form factors of ${}^{208}\mathrm{Pb}$, where the corresponding proton density distributions are calculated by the spherical RMF model and axially deformed RMF model with two parameter sets, respectively. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 4

Nuclear Coulomb form factors of ${}^{40}\mathrm{Ca}$, where the corresponding proton density distributions are calculated by the spherical RMF model and axially deformed RMF model with two parameter sets, respectively. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 5

Ground-state proton densities (units of ${\mathrm{fm}}^{-3}$) and nuclear surface shapes of ${}^{32}\mathrm{S}$, calculated by the deformed RMF model with FSUGarnet and NL3 parameter sets.

Figure 6

(a) The density multipole components for deformed proton distributions of ${}^{32}\mathrm{S}$ in Fig. 5, which are obtained by Eqs. (6) and (7). (b) The proton density distributions of ${}^{32}\mathrm{S}$, calculated by the spherical RMF model. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 7

Nuclear Coulomb form factors of ${}^{32}\mathrm{S}$, where the corresponding proton density distributions are calculated by the spherical RMF model and axially deformed RMF model with two parameter sets, respectively. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 8

Ground-state proton densities (units of ${\mathrm{fm}}^{-3}$) and nuclear surface shapes of ${}^{24}\mathrm{Mg}$, calculated by the deformed RMF model with FSUGarnet and NL3 parameter sets.

Figure 9

(a) The density multipole components of ${}^{24}\mathrm{Mg}$ for deformed proton distributions in Fig. 8, which are obtained by Eqs. (6) and (7). (b) The proton densities of ${}^{24}\mathrm{Mg}$, calculated by the spherical RMF model. In the figure, FSUGarnet is abbreviated as FSUG.

Figure 10

Nuclear Coulomb form factors of ${}^{24}\mathrm{Mg}$, where the corresponding proton density distributions are calculated by the spherical RMF model and axially deformed RMF model with two parameter sets, respectively. In the figure, FSUGarnet is abbreviated as FSUG.