Improvement of one-nucleon removal and total reaction cross sections in the Liège intranuclear-cascade model using Hartree-Fock-Bogoliubov calculations

The prediction of one-nucleon-removal cross sections by the Liège intranuclear-cascade model has been improved using a refined description of the matter and energy densities in the nuclear surface. Hartree-Fock-Bogoliubov calculations with the Skyrme interaction are used to obtain a more realistic description of the radial-density distributions of protons and neutrons, as well as the excitation-energy uncorrelation at the nuclear surface due to quantum effects and short-range correlations. The results are compared with experimental data covering a large range of nuclei, from carbon to uranium, and projectile kinetic energies. We find that the new approach is in good agreement with experimental data of one-nucleon-removal cross sections covering a broad range in nuclei and energies. The new ingredients also improve the description of total reaction cross sections induced by protons at low energies, the production cross sections of heaviest residues close to the projectile, and the triple-differential cross sections for one-proton removal. However, other observables such as quadruple-differential cross sections of coincident protons do not present any sizable sensitivity to the new approach. Finally, the model is also tested for light-ion-induced reactions. It is shown that the new parameters can give a reasonable description of the nucleus-nucleus total reaction cross sections at high energies.

Figure 1

Skin thickness in fm (defined as $|R_n-R_p|$ only to avoid negative values) calculated with the HFBRAD model [31] for nuclei between the proton and neutron drip lines defined according to Ref. [69]. The open squares represent the stable nuclei and the solid contour line indicates the region of known nuclei.

Figure 2

Probability that a proton (a) or neutron (b) hole in ${}^{208}\mathrm{Pb}$ results in an excitation energy smaller than the neutron separation energy, as a function of the distance of the hole from the center of the nucleus. The dotted line corresponds to the HFBRAD calculations. The other lines represent the INCL initial conditions for different values of the fuzziness parameter $f$ defined in Ref. [26], where the standard condition corresponds to $f=0$. One-nucleon removal is dominated by impact parameters to the right of the vertical dotted line.

Figure 3

Same as Fig. 2, but for ${}^{25}\mathrm{Mg}$.

Figure 4

One-neutron (a) and one-proton (b) removal cross sections as a function of the kinetic energy for the reaction $p+{}^{40}\mathrm{Ca}$. Data taken from Refs. [74, 75]. The solid line corresponds to standard INCL calculations and the short- and long-dashed lines represent INCL calculations using the shell model and Hartree-Fock-Bogoliubov calculations for the proton and neutron density profiles together with the surface fuzziness, respectively.

Figure 5

Same as Fig. 4, but for the reaction $p+{}^{208}\mathrm{Pb}$. Experimental data taken from Refs. [76, 77].

Figure 6

One-neutron- and one-proton-removal cross sections as a function of the kinetic energy for different nuclei such as ${}^{12}\mathrm{C}$ [panels (a) and (b)], ${}^{56}\mathrm{Fe}$ [panels (c) and (d)], and ${}^{136}\mathrm{Xe}$ [panels (e) and (f)], respectively. Panels (g) and (h) show the proton-removal cross sections for ${}^{25}\mathrm{Mg}$ and ${}^{74}\mathrm{Ge}$ as a function of the kinetic energy, respectively. Data taken from Refs. [55, 78, 79, 80, 81, 82, 83, 84]. The solid line corresponds to standard INCL calculations. The dotted and dashed lines represent calculations assuming only the Hartree-Fock-Bogoliubov proton and neutron density profiles and considering simultaneously the density profiles and the energy fluctuations, respectively.

Figure 7

One-neutron (a) and one-proton (b) removal cross sections from proton-induced reactions at kinetic energies of 1 GeV as a function of the target mass number for nuclei from ${}^{12}\mathrm{C}$ to ${}^{238}\mathrm{U}$ (solid circles). Data taken from Refs. [55, 75, 77, 79, 81, 84, 85, 86, 87]. The open squares represent the standard INCL calculations and the open triangles correspond to calculations taking into account the proton and neutron density profiles obtained with the HFBRAD model. The open circles correspond to calculations where the two refinements, density profiles and surface fuzziness, are simultaneously applied.

Figure 8

(a) Quadruple-differential cross sections for two coincident protons emitted in the reaction $p+{}^{40}\mathrm{Ca}$ at 392 MeV [17]. The values for the detection angles of the second proton ($p{}^{\prime \prime }$) are given close to the corresponding curves. Note the displacement factors for the purpose of display. The experimental data is compared with the standard INCL (solid lines), shell-model (dotted lines), and HFBRAD (dashed lines) calculations. (b) Same as (a) but for the reaction $p+{}^{197}\mathrm{Au}$ at 200 MeV. Experimental data taken from Ref. [88].

Figure 9

Triple-differential cross sections for two coincident protons emitted in the reaction ${}^{197}\mathrm{Au}$($p,2p$)${}^{196}\mathrm{Pt}$ at 200 MeV [88]. Detection angles of the primary and secondary protons were taken around average values of $-{40}^{\circ }$ and 40°, respectively. The experimental data are compared with the standard INCL (solid lines), shell-model (dotted lines), and HFBRAD (dashed lines) calculations.

Figure 10

Isotopic distributions of the $Z=81$ (a), $Z=82$ (b), and $Z=83$ (c) residues produced in $p+{}^{208}\mathrm{Pb}$ at 1 GeV [77, 90]. The experimental data are compared with the standard INCL (solid lines), shell-model (dotted lines), and HFBRAD (dashed lines) calculations.

Figure 11

Isotopic distributions of the $Z=53$ (a), $Z=54$ (b), and $Z=55$ (c) residues produced in $p+{}^{136}\mathrm{Xe}$ at 1 GeV [81]. The experimental data are compared with the standard INCL (solid lines) and HFBRAD (dashed lines) calculations.

Figure 12

Total reaction cross sections for proton-induced reactions as a function of the kinetic energy for different nuclei such as ${}^{56}\mathrm{Fe}$ (a), ${}^{208}\mathrm{Pb}$ (b), and ${}^{238}\mathrm{U}$ (c). The solid line corresponds to standard INCL calculations while the dashed line represents calculations considering the best-fit parameters of the HFBRAD density profile. Experimental data taken from Refs. [55, 77, 91].

Figure 13

Nuclear density probability for ${}^{12}\mathrm{C}$. The open squares show the density calculated with the HFBRAD model. The dashed and solid lines correspond to the best-fit with the modified-harmonic-oscilator (MHO) approximation [see Eq. (1)] and the results from the parametrization implemented in the standard INCL, respectively. The dot-dashed line is the MHO approximation with the experimental parameters taken from Ref. [45]. The vertical dashed and dotted lines are to indicate the average value of the standard INCL distribution and the other ones, respectively.

Figure 14

Total reaction cross sections for different neon (a) and magnesium (b) isotopes impinging on a carbon target at $950A$ MeV. Experimental data taken from Ref. [92]. The solid line corresponds to the standard preparation of light-ion projectiles in INCL while the dashed line represents calculations considering the best-fit parameters of the HFBRAD density profile with the MHO approximation.

Figure 15

Total reaction cross sections for the reactions ${}^9\mathrm{Be}+{}^{12}\mathrm{C}$ (a), ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (b), ${}^{12}\mathrm{C}+{}^{27}\mathrm{Al}$ (c), and ${}^{12}\mathrm{C}+{}^{64}\mathrm{Zn}$ (d) as a function of the projectile kinetic energy per nucleon. Experimental data taken from Refs. [92, 93]. The solid line corresponds to the standard preparation of light-ion projectiles in INCL while the dashed line represents calculations considering the best-fit parameters of the HFBRAD density profile with the MHO approximation.