Extracting nuclear sizes of medium to heavy nuclei from total reaction cross sections

Background: Proton and neutron radii are fundamental quantities of atomic nuclei. To study the sizes of short-lived unstable nuclei, there is a need for an alternative to electron scattering.

Purpose: The recent paper by Horiuchi et al. [Phys. Rev. C 89, 011601(R) (2014)] proposed a possible way of extracting the matter and neutron-skin thickness of light- to medium-mass nuclei using total reaction cross section, ${\sigma }_R$. The analysis is extended to medium to heavy nuclei up to lead isotopes with due attention to Coulomb breakup contributions as well as density distributions improved by paring correlation.

Methods: We formulate a quantitative calculation of ${\sigma }_R$ based on the Glauber model including the Coulomb breakup. To substantiate the treatment of the Coulomb breakup, we also evaluate the Coulomb breakup cross section due to the electric dipole field in a canonical-basis-time-dependent-Hartree-Fock-Bogoliubov theory in the three-dimensional coordinate space.

Results: We analyze ${\sigma }_R$'s of 103 nuclei with $Z=20$, 28, 40, 50, 70, and 82 incident on light targets, ${}^{1,2}\mathrm{H}$, ${}^4\mathrm{He}$, and ${}^{12}\mathrm{C}$. Three kinds of Skyrme interactions are tested to generate those wave functions. To discuss possible uncertainty due to the Coulomb breakup, we examine its dependence on the target, the incident energy, and the Skyrme interaction. The proton is a most promising target for extracting the nuclear sizes as the Coulomb excitation can safely be neglected. We find that the so-called reaction radius, $a_R=\sqrt{{\sigma }_R/\pi }$, for the proton target is very well approximated by a linear function of two variables, the matter radius and the skin thickness, in which three constants depend only on the incident energy. We quantify the accuracy of ${\sigma }_R$ measurements needed to extract the nuclear sizes.

Conclusions: The proton is the best target because, once the incident energy is set, its $a_R$ is very accurately determined by only the matter radius and neutron-skin thickness. If ${\sigma }_R$'s at different incident energies are measured, one can determine both the proton and neutron radii for unstable nuclei as well. The total reaction cross sections calculated in this paper are given as Supplemental Material for the sake of future measurements.

Figure 1

Neutron and proton rms radii of Sn isotopes calculated with HF+BCS and HF theory. The SkM* interaction is used.

Figure 2

Matter radius of even-even nucleus as a function of ${(N+Z)}^{1/3}$. Three Skyrme parameter sets are used: (a) SkM*, (b) SLy4, and (c) SkI3.

Figure 3

Neutron-skin thickness of even-even nucleus as a functions of $(N-Z)/(N+Z)$. Three Skyrme parameter sets are used: (a) SkM*, (b) SLy4, and (c) SkI3.

Figure 4

Coulomb breakup cross sections by EPM for projectile nucleus incident on proton at (top) 200 and (bottom) 1000 MeV. The projectile density is obtained with three Skyrme interactions: (Left) SkM*, (Center) SLy4, and (Right) SkI3.

Figure 5

EPM calculation of Coulomb breakup cross sections for (upper) projectile by target and for (lower) target by projectile at 1000 MeV incident energy. Three different targets are employed; (Left) ${}^2\mathrm{H}$, (Center) ${}^4\mathrm{He}$, and (Right) ${}^{12}\mathrm{C}$. The projectile density is obtained with the SkM* interaction.

Figure 6

Glauber model calculation of the total reaction probability and its decomposition to the nuclear and Coulomb breakup probabilities for ${}^{208}\mathrm{Pb}+{}^{12}\mathrm{C}$ collision at 1000 MeV incident energy. The SkM* interaction is used.

Figure 7

Comparison of ${}^{208}\mathrm{Pb}+{}^{12}\mathrm{C}$ Coulomb breakup probabilities calculated by the Glauber and EPM models at three incident energies (a) 1000, (b) 550, and (c) 200 MeV. The cutoff parameter of EPM is $b_{\min }=10.2$ fm. The SkM* interaction is used.

Figure 8

Total reaction cross sections, ${\sigma }_R^{\left(N\right)}+{\sigma }_C$, of various nuclei at 1000 MeV incident energy. The Coulomb breakup contribution is calculated by the EPM. The targets are; (a) proton, (b) ${}^2\mathrm{H}$, (c) ${}^4\mathrm{He}$, and (d) ${}^{12}\mathrm{C}$. The SkM* interaction is used.

Figure 9

Same as Fig. 8 but for a fraction of the Coulomb breakup contribution to the total reaction cross section.

Figure 10

Rutherford ratios of ${}^{12}\mathrm{C}+{}^{208}\mathrm{Pb}$ elastic scattering at 200 and 1000 MeV incident energies. The ratio calculated by ignoring the Coulomb breakup is also plotted for comparison. The cutoff impact parameter $b_{\max }$ is 15.1 and 80.0 fm for 200 and 1000 MeV, respectively. The SkM* interaction is employed. The experimental data are taken from Ref. [69].

Figure 11

Elastic differential cross sections of $p+{}^{208}\mathrm{Pb}$ scattering. Coulomb breakup contribution is ignored. The SkM* interaction is used. Experimental data are taken from Refs. [13, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84].

Figure 12

Total reaction cross sections of ${}^{40}\mathrm{Ca}$, ${}^{58}\mathrm{Ni}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ on a proton target as functions of incident energies. Coulomb breakup contribution is ignored. The SkM* interaction is used. The experimental data with error bars are taken from Refs. [86, 87, 88, 89, 90, 91, 92, 93].

Figure 13

Reaction radii vs point matter rms radii for $A=40–214$ with $Z=20$, 28, 40, 50, 70, and 82 nuclei incident on a proton target at 100, 200, 550, and 1000 MeV. The reaction radii of 550, 200, and 100 MeV are, respectively, shifted by 1, 2, and 3  fm for clarity. The SkM* interaction is used.

Figure 14

Incident energy dependence of the coefficients of the empirical formula for the reaction radius (33) calculated with the SkM*, SLy4, and SkI3 interactions: (a) $\alpha \left(E\right)$ and $\beta \left(E\right)$, and (b) $\gamma \left(E\right)$.

Figure 15

Correlation diagrams of the reaction radii obtained through the HF+BCS densities and by the empirical formula (33) for incident energies of (a) 100, (b) 200, (c) 550, and (d) 1000 MeV. All the data points obtained with the SkM*, SLy4, and SkI3 interactions are plotted in dots. A solid line, $y=x$, is drawn as a guide of eyes.