Nucleus-nucleus cold fusion reactions analyzed with the $l$-dependent “fusion by diffusion” model

We present a modified version of the Fusion by Diffusion (FBD) model aimed at describing the synthesis of superheavy nuclei in cold fusion reactions, in which a low excited compound nucleus emits only one neutron. The modified FBD model accounts for the angular momentum dependence of three basic factors determining the evaporation residue cross section: the capture cross section ${\sigma }_{\text{cap}}(l)$, the fusion probability $P_{\text{fus}}(l)$, and the survival probability $P_{\text{surv}}(l)$. The fusion hindrance factor, the inverse of $P_{\text{fus}}(l)$, is treated in terms of thermal fluctuations in the shape degrees of freedom and is expressed as a solution of the Smoluchowski diffusion equation. The $l$ dependence of $P_{\text{fus}}(l)$ results from the $l$-dependent potential energy surface of the colliding system. A new parametrization of the distance of starting point of the diffusion process is introduced. An analysis of a complete set of 27 excitation functions for production of superheavy nuclei in cold fusion reactions, studied in experiments at GSI Darmstadt, RIKEN Tokyo, and LBNL Berkeley, is presented. The FBD model satisfactorily reproduces shapes and absolute cross sections of all the cold fusion excitation functions. It is shown that the peak position of the excitation function for a given 1$n$ reaction is determined by the $Q$ value of the reaction and the height of the fission barrier of the final nucleus. This fact could possibly be used in future experiments (with well-defined beam energy) for experimental determination of the fission barrier heights.

Figure 1

Theoretical excitation function for the ${}^{208}\mathrm{Pb}$(${}^{58}\mathrm{Fe}$,$n$)${}^{265}\mathrm{Hs}$ reaction calculated with the $l$-dependent FBD model for infinitely thin target (dashed line) and the corresponding curve corrected for the actual target thickness (solid line), compared with experimental data [15, 16]. The height of the theoretical curve, averaged over the beam energy spread due to the target thickness, is fitted to the maximum of the experimental excitation function by adjusting a value of $s_{\text{inj}}$ (the injection point), the only adjustable parameter of the model.

Figure 2

The injection point distances $s_{\text{inj}}$ deduced from fitting predictions of the FBD model to 27 excitation functions of cold fusion reactions listed in Table . The $s_{\text{inj}}$ values are plotted as a function of the excess of kinetic energy above the Coulomb barrier, $E_{\text{c.m.}}-B_0$. A least-square linear fit, $s_{\text{inj}}\approx 2.30\hspace{0.5em}\mathrm{fm}-0.062(E_{\text{c.m.}}-B_0)$ fm/MeV, is shown by the solid line.

Figure 3

Comparison of the measured excitation functions for 12 selected cold fusion reactions (for references see Table ) with predictions of the $l$-dependent FBD model assuming $s_{\text{inj}}$ values given by Eq. (40). Solid curves represent predictions accounting for the effect of the beam energy smearing due to the target thickness. Dashed lines show the theoretical curves shifted in energy and multiplied by a factor to fit the experimental points (see text).

Figure 4

Peak positions of the experimental excitation functions (circles, squares and triangles) compared with the $l$-dependent FBD model predictions and plotted as a function of the Coulomb interaction parameter $z$. The dotted line shows an approximation to the peak position corresponding to the top of the fission barrier of the final nucleus after the neutron emission, given by Eq. (39). The exact theoretical peak position is shown by the solid line.

Figure 5

Complete set of measured cross sections (at the maximum of the excitation function) for cold fusion reactions (see Table ), plotted as a function of the Coulomb interaction parameter $z$, and compared with the $l$-dependent FBD model predictions (dotted line).

Figure 6

Two excitation functions for the synthesis of $Z=114$ nuclei in cold fusion reactions ${}^{208}\mathrm{Pb}$(${}^{74,76}\mathrm{Ge}$,$n$)${}^{281,283}$114, predicted with the $l$-dependent FBD model. The excitation functions are averaged over the assumed beam energy spread corresponding to a thick target of $\Delta E$ $=$ 5 MeV.

Figure 7

Two excitation functions for the synthesis of $Z=115$ nuclei in cold fusion reactions ${}^{209}\mathrm{Bi}$(${}^{74,76}\mathrm{Ge}$,$n$)${}^{282,284}$115, predicted with the $l$-dependent FBD model. The excitation functions are averaged over the assumed beam energy spread corresponding to a thick target of $\Delta E$ $=$ 5 MeV.