Modeling near-barrier collisions of heavy ions based on a Langevin-type approach

Background: Multinucleon transfer in low-energy nucleus-nucleus collisions is proposed as a method of production of yet-unknown neutron-rich nuclei hardly reachable by other methods.

Purpose: Modeling of dynamics of nuclear reactions induced by heavy ions in their full complexity of competing reaction channels remains to be a challenging task. The work is aimed at development of such a model and its application to the analysis of multinucleon transfer in deep inelastic collisions of heavy ions leading, in particular, to formation of neutron-rich isotopes in the vicinity of the $N=126$ shell closure.

Method: Multidimensional dynamical model of nucleus-nucleus collisions based on the Langevin equations has been proposed. It is combined with a statistical model for simulation of de-excitation of primary reaction fragments. The model provides a continuous description of the system evolution starting from the well-separated target and projectile in the entrance channel of the reaction up to the formation of final reaction products.

Results: A rather complete set of experimental data available for reactions ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt},{}^{208}\mathrm{Pb},{}^{209}\mathrm{Bi}$ was analyzed within the developed model. The model parameters have been determined. The calculated energy, mass, charge, and angular distributions of reaction products, their various correlations as well as cross sections for production of specific isotopes agree well with the data. On this basis, optimal experimental conditions for synthesizing the neutron-rich nuclei in the vicinity of the $N=126$ shell were formulated and the corresponding cross sections were predicted.

Conclusions: The production yields of neutron-rich nuclei with $N=126$ weakly depend on the incident energy. At the same time, the corresponding angular distributions are strongly energy dependent. They are peaked at grazing angles for larger energies and extend up to the forward angles at low near-barrier collision energies. The corresponding cross sections exceed 100 nb for the nuclei located at the border of the known region, which is nearly five orders of magnitude larger than can be reached in the fragmentation reactions.

Figure 1

Example of the nuclear shapes in two-center parametrization and the corresponding potentials $V(\rho =0,z)$ shown for ${\delta }_1={\delta }_2=0.5$ and $\varepsilon =0.5$. The mass asymmetry $\eta =0$ for (a) and $\eta =0.625$ for (b) and (c).

Figure 2

Schematic view of the nuclear degrees of freedom.

Figure 3

Time evolution of the potential energy for the ${}^{48}\mathrm{Ca}{+}^{248}\mathrm{Cm}$ system at zero dynamic deformations. (a) The diabatic potential energy calculated using the double-folding procedure (the first stage). The entrance-channel (b) and fission-channel (c) adiabatic potential energies obtained within the extended macro-microscopic approach. The white arrows schematically show the most probable reaction channels: DI scattering (a); DI scattering, quasifission, and fusion (b); and multimodal fission (c).

Figure 4

The potential at the contact point for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ system calculated at zero values of the deformations. The dashed curve corresponds to the minimum of the potential energy. The contour lines are drawn over each 5 MeV.

Figure 5

A trajectory and its projections showing time evolution of energy (potential + relative motion), elongation, and mass of heavier nucleus for the ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ combination at $E_{\mathrm{c}.\mathrm{m}.}=526$ MeV.

Figure 6

Comparison of experimental (symbols) and calculated (histograms) energy, angular, and charge distributions for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction at energies $E_{\mathrm{c}.\mathrm{m}.}=569$ [panels (a), (d), and (g)], 684 [panels (b), (e), and (h)], and 861 [panels (c), (f), and (i)] MeV. The thick solid histograms show the distributions of final fragments with partially eliminated sequential fission events according to the experimental selection criteria. The thin histograms for energy distributions (a), (b), and (c) correspond to the primary fragments. The solid thin histograms in panels (d)–(i) show the distributions of final fragments including all sequential fission events. The dotted histograms are for the final charge yields with completely excluded sequential fission events. Arrows indicate the projectile charge. The experimental data are taken from Refs. [80, 81, 82].

Figure 7

The angular distribution of Bi-like fragments in the laboratory system for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=569\mathrm{MeV}$. The experimental data (symbols) are from Ref. [80]. The calculated distributions (histograms) are for the primary (dashed) and final (solid) fragments. The cross section of the final Bi-like fragments is 33% less than that of the primary fragments due to sequential fission.

Figure 8

The laboratory-system energy distributions of the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction products for the indicated angles averaged over $3^{\circ }$ bin. The calculation results for primary and final fragments are shown by dotted and solid histograms, respectively. Experimental data (symbols) are taken from Refs. [80, 81].

Figure 9

Experimental [80] (a) and calculated (b) Wilczynski plots ${\mathrm{d}}^2\sigma /\mathrm{d}E\mathrm{d}{\mathrm{\Omega }}_{\mathrm{c}.\mathrm{m}.}$ for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=569$ MeV. The cross section is in $\mathrm{mb}/\mathrm{MeV}\mathrm{sr}$.

Figure 10

Sections of the double differential cross sections for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=569$ MeV. (a) The charge distributions for different values of TKE; (b) the energy distributions for different $Z$ values; (c),(d) the angular distributions for different $Z$ and TKE values, respectively. The symbols are experimental data [80] and the histograms are calculations for final reaction products. The energy and the charge bins are 26 MeV and three charge units, respectively. The centroids of each bin are shown next to the histograms.

Figure 11

Correlations of the interaction time with number of transferred protons (a), TKEL (b), center-of-mass scattering angle (c), and impact parameter (d) for the ${}^{136}\mathrm{Xe}+{}^{209}\mathrm{Bi}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=569$ MeV. The contour lines are drawn over each order of magnitude.

Figure 12

Energy, mass, and angular distributions of primary fragments for the ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ reaction for two collision energies $E_{\mathrm{c}.\mathrm{m}.}=526\mathrm{MeV}$ [panels (a), (c), (e), and (g)] and $E_{\mathrm{c}.\mathrm{m}.}=617\mathrm{MeV}$ [panels (b), (d), (f), and (h)]. Experimental data (symbols) are from Ref. [11]. The angular distributions are shown for light reaction fragments ($A\le 172$) and heavy fragments with masses $200<A<216$. The thin histograms are the calculated mass distributions of final fragments including the sequential fission events. The original experimental data on the mass distributions are multiplied by 2 in order to obtain the same normalization as for the calculated ones.

Figure 13

Calculated primary (dashed histograms) and final (solid histograms) yields of Po, Rn, and Ra isotopes for the ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ reaction in comparison with experimental data (circles, triangles, and diamonds for Po, Rn, and Ra, respectively) taken from Ref. [14] for $E_{\mathrm{c}.\mathrm{m}.}=450$ MeV (a) and Ref. [11] for $E_{\mathrm{c}.\mathrm{m}.}=514$ MeV (b).

Figure 14

Same as in Fig. 13 but for W, Os, and Pt isotopes. The data (triangles, circles, and diamonds are for W, Os, and Pt, respectively) are taken from Ref. [14] for $E_{\mathrm{c}.\mathrm{m}.}=450$ MeV (a) and Ref. [12] for $E_{\mathrm{c}.\mathrm{m}.}=496$ MeV (b). The data of Ref. [12] are multiplied by 10 (see the text).

Figure 15

Comparison of calculated (histograms) and measured [14] (symbols) cross sections for ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{c}.\mathrm{m}.}=450$ MeV.

Figure 16

Mass distribution of reaction products for ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ at $E_{\mathrm{c}.\mathrm{m}.}=450$ MeV. The experimental data (symbols) are from Ref. [14]. The solid and dashed histograms are the calculated mass distributions for final and primary fragments, respectively.

Figure 17

Calculated (histograms) and experimental [13] (symbols) cross sections of PLFs for the ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=643$ MeV.

Figure 18

Cross sections for Os (a) and Hg (b) isotopes integrated over the angular range from ${24}^{\circ }$ to ${34}^{\circ }$ deduced in experiment [13] (open circles) and calculated (solid histograms). Isotopic distributions for different ranges of TKEL from $-25$ to 25 MeV (dash-dotted histograms and filled circles) and from 175 to 225 MeV (dashed histograms and filled squares) are shown.

Figure 19

(a) Calculated (histograms) and experimental (symbols) cross sections for production of isotopes with $N=126$ in reactions ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt},{}^{208}\mathrm{Pb}$. The solid and dashed histograms are for $E_{\mathrm{c}.\mathrm{m}.}=450$ and 643 MeV, respectively. The thin and thick dashed curves are integrated over all angles and over the experimentally covered angles from ${24}^{\circ }$ to ${34}^{\circ }$, respectively. The experimentally deduced cross sections for the ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt}$ system are from Ref. [13] and for ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ are from Ref. [14]. (b) Calculated angular distributions in the laboratory frame for the Ir isotopes for the ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt},{}^{208}\mathrm{Pb}$ reactions. The thick histograms are for the ${}^{203}\mathrm{Ir}_{126}$ nucleus and the thin histograms are integrated over all iridium isotopes. The solid and dashed histograms are for $E_{\mathrm{c}.\mathrm{m}.}=450$ and 643 MeV, respectively. The vertical arrows indicate the corresponding grazing angles of TLFs.

Figure 20

Energy dependence of the production cross sections of neutron-rich nuclei with $N=126$ predicted by the present model. The modeling was performed for the ${}^{136}\mathrm{Xe}+{}^{208}\mathrm{Pb}$ (a) and ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt}$ (b) reactions. The error bars represent statistical uncertainties of the calculated cross sections. The vertical dashed lines show the values of the potential energy at the contact point.

Figure 21

Upper part of the chart of nuclides. The production cross sections in the ${}^{136}\mathrm{Xe}+{}^{198}\mathrm{Pt}$ reaction at $E_{\mathrm{c}.\mathrm{m}.}=643\mathrm{MeV}$ are shown by contour lines drawn over an order of magnitude of the cross section down to 100 nb.