Sub-barrier fusion of neutron-rich nuclei and its astrophysical consequences

Near-barrier fusion of neutron-rich nuclei was studied within the semiempirical channel coupling model for intermediate neutron rearrangement and within the time-dependent three-body Schrödinger equation. The possibility of neutron transfer with positive $Q$ values considerably increases the barrier penetrability. A huge enhancement of deep sub-barrier fusion probability was found for light neutron-rich weakly bound nuclei (such as ${}^6\mathrm{He}$). This may be quite important for astrophysical primordial and supernova nucleosynthesis.

Figure 1

Potential energy of ${}^{64}\mathrm{Ni}+{}^{132}\mathrm{Sn}$. The proximity potential is used for the nuclear interaction and the surface stiffness parameters are used that reproduce the experimental energies of the $2^{+}$ excited states in the nuclei. (a) Landscape of the potential energy surface. The saddle point and the potential barrier of spherical nuclei ($\beta =0$) are shown by crosses. The ridge of the barrier is shown by the dashed line, whereas the dotted line corresponds to the contact distance of the two nuclei. The incoming flux is shown schematically by the gray-shaded arrow. (b) Potential energy at the ridge of the two-dimensional barrier, i.e., along the dashed line passing through the saddle point. (c) Empirical barrier distribution function $f(B)$ defined by Eq. (1).

Figure 2

Fusion cross sections for ${}^{16}\mathrm{O}+{}^{60}\mathrm{Ni}$ and ${}^{18}\mathrm{O}+{}^{58}\mathrm{Ni}$ [14] (a), ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ [12] (b), ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ [15] (c). Dashed curves are the CC calculations for the ${}^{16}\mathrm{O}+{}^{60}\mathrm{Ni}$, ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, and ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ fusion reactions. Dotted curves are the CC calculations for the ${}^{18}\mathrm{O}+{}^{58}\mathrm{Ni}$, ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$, and ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ fusion reactions, whereas the solid curves are the results of calculations for the same reactions in accordance with formula (2), i.e., taking into account intermediate neutron rearrangement.

Figure 3

Fusion cross sections for ${}^{132}\mathrm{Sn}+{}^{64}\mathrm{Ni}$ and ${}^{134}\mathrm{Sn}+{}^{64}\mathrm{Ni}$ calculated with (solid curves) and without (dashed curves) intermediate neutron rearrangement. Experimental data for ${}^{132}\mathrm{Sn}+{}^{64}\mathrm{Ni}$ are taken from Ref. [1].

Figure 4

Schematic picture of sequential fusion of ${}^6\mathrm{He}$.

Figure 5

Cross sections for the production of Po isotopes in the ${}^6\mathrm{He}+{}^{206}\mathrm{Pb}$ (solid curves) and ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ (short dashed curves) fusion reactions calculated in Ref. [27]. Open circles show experimental yield of ${}^{211}\mathrm{Po}$ in the ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ fusion reaction [38] ($1n$ channel). Solid squares are the cross sections for production of ${}^{210}\mathrm{Po}$ in the ${}^6\mathrm{He}+{}^{206}\mathrm{Pb}$ fusion reaction ($2n$ channel) measured recently in Dubna [7]. Long-dashed curve includes the effect of the energy spread of the ${}^6\mathrm{He}$ beam for the calculated $2n$-channel cross sections. Arrows at the energy axis show the corresponding Coulomb barriers for ${}^6\mathrm{He}$ (left) and ${}^4\mathrm{He}$ (right) which are very close to each other.

Figure 6

Fusion cross sections (a) and barrier distribution function, $d^2(E\sigma )/{\mathit{dE}}^2$ (b) for the fusion reactions ${}^9\mathrm{Li}+{}^{70}\mathrm{Zn}$ and ${}^{11}\mathrm{Li}+{}^{68}\mathrm{Zn}$. Experimental data are for the ${}^9\mathrm{Li}+{}^{70}\mathrm{Zn}$ fusion reaction taken from Ref. [39].

Figure 7

Jacobi coordinates optimal for description of the system consisting of two cores and a valence neutron at large (a) and short (b) distances between the two cores.

Figure 8

Neutron potential energy in the ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ system along the internuclear axis at different distances $R=14\mathrm{fm},R=12\mathrm{fm},R=10\mathrm{fm}$, and $R=1\mathrm{fm}$ (solid, dashed, dot-dashed, and dotted curves, respectively).

Figure 9

Two-center quasimolecular single particle neutron levels in the ${}^{96}\mathrm{Zr}+{}^{40}\mathrm{Ca}$ nuclear system. Solid, dashed, and dotted curves show the states with $\lambda =0,1$, and $2$, respectively. $2d$ is the last occupied level in ${}^{96}\mathrm{Zr}$; $1f$ and $2p$ are unoccupied levels of ${}^{40}\mathrm{Ca}$. Vertical dotted and dash-dotted lines show positions of the Coulomb barrier and the contact point, correspondingly.

Figure 10

Amplitude of the wave function of valence neutron (initially located in the $2d$ state of ${}^{96}\mathrm{Zr}$ nucleus) at the three distances between colliding nuclei shown by the small circles in the upper part of the figure along with the probability to find the neutron in the ${}^{40}\mathrm{Ca}$ nucleus.

Figure 11

Channel nucleus-nucleus interactions (a) and quasimolecular occupation probabilities (b) for collision of ${}^{96}\mathrm{Zr}$ and ${}^{40}\mathrm{Ca}$ at $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=97\mathrm{MeV}$: initial interaction $V_{12}$ (dotted curve), $V_{2d,2d}$ and the corresponding occupation probability $p_{2d}$ (solid curves), $V_{2d,1f}$ and $p_{1f}$ (dashed curves), $V_{2d,2p}$ and $p_{2p}$ (dot-dashed curves).

Figure 12

Probability density $|\Psi (r,R,t)|{}^2$ for collision of ${}^{40}\mathrm{Ca}$ with ${}^{96}\mathrm{Zr}$ at sub-barrier energy $E=B-10=90\mathrm{MeV}$ in the one-dimentional quantum model. Solid curves show equipotential levels for the potential energy $V_{12}(R)+V_n(r;R)$. $t_1={10}^{-21}\mathrm{s},t_2=2\times {10}^{-21}\mathrm{s},t_3=3\times {10}^{-21}\mathrm{s}$. Regions A, B, and C correspond to the scattering (neutron remains in Zr nucleus), transfer and fusion processes, correspondingly.

Figure 13

Barrier penetrability for collision of ${}^{40}\mathrm{Ca}$ with ${}^{96}\mathrm{Zr}$ calculated in the three-body one-dimensional model. The dashed curve shows the result obtained for “frozen” neutron (penetration probability of one-dimensional barrier $V_{12}(R)$).

Figure 14

Part of the nuclear reaction network usually used for the calculations [43].

Figure 15

Near barrier cross sections for the ${}^6\mathrm{He}+{}^{12}\mathrm{C}$ (dashed curve assumes sequential neutron transfer whereas solid curve takes into account $2n$ transfer) and ${}^4\mathrm{He}+{}^{13}\mathrm{C}$ (dotted curve) fusion reactions.