Examining the enhancement of sub-barrier fusion cross sections by neutron transfer with positive $Q$ values

Background: Significant enhancement of sub-barrier fusion cross sections owing to neutron rearrangement with positive $Q$ values were found for many combinations of colliding nuclei. However, several experimental results on fusion reactions were reported recently in which such enhancement has not been observed in spite of a possibility for neutron rearrangement with positive $Q$ values.

Purpose: We aim to clarify much better the mechanism of neutron rearrangement in sub-barrier fusion reactions to find the other requirements (beside positive $Q$ values) which favor (or prevent) sub-barrier fusion enhancement.

Methods: A channel coupling approach along with the semiclassical model for neutron transfer has been used for analysis of available experimental data on sub-barrier fusion of heavy ions.

Results: The role and interplay of different factors determining the enhancement of sub-barrier fusion (such as $Q$ values for neutron rearrangement, properties of collective excitations, and neutron binding energies) have been studied and clarified.

Conclusions: (1) Only $1n$ and $2n$ transfers with positive $Q$ values have a noticeable impact on sub-barrier fusion. A positive $Q$ value for neutron rearrangement is a necessary but not sufficient requirement for additional sub-barrier fusion enhancement to take place. (2) The “rigidity” of colliding nuclei with respect to collective excitations is important for sub-barrier fusion enhancement due to neutron rearrangement with positive $Q$ values to be clearly visible. (3) The neutron binding energy of the “donor” nucleus has a strong impact only in the case of fusion of light weakly bound nuclei.

Figure 1

(Upper panel) Fusion excitation functions for ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$ (open circles) and ${}^{40}\mathrm{Ca}$ + ${}^{90}\mathrm{Zr}$ (filled circles) [8]. The no-coupling limits are shown by the dotted curves. The dashed curves show the calculations with coupling to surface vibrations and without neutron transfer, whereas the solid line was obtained by accounting for neutron rearrangement in the entrance channel of the ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$ reaction. (Bottom panel) Fusion excitation functions for ${}^{18}\mathrm{O}$ + ${}^{58}\mathrm{Ni}$ (open circles) and ${}^{16}\mathrm{O}$ + ${}^{60}\mathrm{Ni}$ (filled circles) [9]. The no-coupling limit is shown by a dotted curve (which is practically indistinguishable for the two reactions). Other notations are the same as in the upper panel.

Figure 2

Fusion cross section for ${}^{16}\mathrm{O}$ + ${}^{154}\mathrm{Sm}$. The solid and dashed curves correspond to ECC and QCC calculations, respectively. Experimental data are from [38]. The dotted curve shows the result for one-dimensional barrier penetrability. Arrows indicate the barriers for two limit orientations of the deformed ${}^{154}\mathrm{Sm}$ nucleus ($B_1=53.6$ MeV and $B_2=60.7$ MeV) and for its spherical shape with the same volume, $B_{\mathrm{sph}}$. The bottom panel shows the corresponding “experimental” barrier distribution functions, $D\left(B\right)$.

Figure 3

Fusion cross section for ${}^{4,6}$He + ${}^{64}\mathrm{Zn}$. The solid and dashed curves correspond to ECC calculations accounting for and without accounting for the neutron transfer, respectively. The dash-dotted curve shows the ${}^6\mathrm{He}$ + ${}^{64}\mathrm{Zn}$ fusion cross section multiplied by a factor of 0.8. Experimental data are from [25].

Figure 4

${}^6\mathrm{He}$ + ${}^{209}\mathrm{Bi}$: The solid and dashed curves correspond to ECC calculations of the fusion cross section accounting for and without accounting for the neutron transfer, respectively. The experimental data for the fusion cross sections (filled circles) are taken from [18]. The long-dashed curve gives the fusion-fission cross section, and the corresponding experimental data [45] are shown by filled stars. ${}^4\mathrm{He}$ + ${}^{209}\mathrm{Bi}$: The dashed curve is the fusion cross section calculated by using the ECC model without neutron transfer. The dotted curves show the cross sections for $1n$ and $2n$ evaporation channels. The respective experimental data are given by open circles [47] ($1n$ channel) and open squares [48] ($2n$ channel).

Figure 5

Fusion cross sections for ${}^{58}\mathrm{Ni}$ + ${}^{130}\mathrm{Te}$ and ${}^{64}\mathrm{Ni}$ + ${}^{130}\mathrm{Te}$. The experimental data [28] are shown by open and filled circles, respectively. The solid and dashed curves show the ECC calculations with and without neutron rearrangement, respectively.

Figure 6

Fusion excitation functions for ${}^{18}\mathrm{O}$ + ${}^{74}\mathrm{Ge}$ and ${}^{16}\mathrm{O}$ + ${}^{76}\mathrm{Ge}$. The experimental data [29] are shown by open and filled circles, respectively. The solid and dashed curves show the ECC calculations with and without neutron rearrangement, respectively.

Figure 7

Fusion excitation functions for ${}^{60}\mathrm{Ni}$ + ${}^{100}\mathrm{Mo}$ and ${}^{64}\mathrm{Ni}$ + ${}^{100}\mathrm{Mo}$. The experimental data [30] are shown by open and filled circles, respectively. The solid and dashed curves show the ECC calculations with and without neutron rearrangement, respectively.

Figure 8

Fusion excitation functions for ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$. The curves show the ECC calculations with and without (dotted curve) neutron rearrangement channels being accounted for. The solid curve is obtained with real $Q_{xn}$ values; the dashed and dash-dotted curves are the model calculations with $Q_{xn}$ values halved and doubled, respectively.

Figure 9

Fusion excitation functions for ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$. The curves show the ECC calculations accounting for different number of neutron rearrangement channels (from 0$n$ to 6$n$).

Figure 10

The fusion cross sections (a) and (c) and the experimental barrier distribution functions (b) and (d). The solid and dashed curves show the results of the ECC calculations accounting for and without accounting for the neutron rearrangement channels, respectively. Panels (a) and (b) correspond to the ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$ system (with the symbols denoting the experimental data), while panels (c) and (d) are made for the same system but the vibrational properties of ${}^{40}\mathrm{Ca}$ and ${}^{96}\mathrm{Zr}$ are substituted by those of ${}^{60}\mathrm{Ni}$ and ${}^{100}\mathrm{Mo}$, respectively. The dash-dotted curve on panel (d) is the same as the dashed one on panel (b).

Figure 11

Fusion cross sections for the ${}^6\mathrm{He}$ + ${}^{64}\mathrm{Zn}$ reaction. The data (symbols) are taken from Ref. [25]. The curves are the ECC calculations with (solid and dashed) and without (dotted) neutron rearrangement. The solid curve corresponds to real neutron binding energies in ${}^6\mathrm{He}$. The dashed curve shows the model calculations in which larger neutron binding energies in ${}^6\mathrm{He}$ (the same as in ${}^9\mathrm{Li}$) are assumed. The binding energies are shown in MeV.

Figure 12

Fusion cross sections for the ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$ reaction. The data (symbols) are taken from Ref. [8]. The curves are the ECC calculations with (solid, dashed, and dash-dotted) and without (dotted) neutron rearrangement. The solid curve corresponds to real neutron binding energies in ${}^{96}\mathrm{Zr}$. The dashed and dash-dotted curves show the model calculations in which smaller (as in ${}^6\mathrm{He}$) and larger (as in ${}^{16}\mathrm{O}$) neutron binding energies in ${}^{96}\mathrm{Zr}$ are assumed, respectively. The binding energies are shown in MeV.

Figure 13

The nucleus-nucleus potential energies for the ${}^6\mathrm{He}$ + ${}^{64}\mathrm{Zn}$ and ${}^{40}\mathrm{Ca}$ + ${}^{96}\mathrm{Zr}$ systems and classical turning points at $E_{\mathrm{c}.\mathrm{m}.}=B-3$ MeV.