1 Introduction

The precise nature of nuclear dissipation remains one of the major problems unresolved in nuclear physics. Dissipation plays a critical role in low-energy nucleus-nucleus collision dynamics [1-7]. It delays fission, resulting in an enhanced emission of prescission light particles and a large evaporation residue cross section with respect to the predictions of standard statistical models [8-10]. Accordingly, information on dissipation in fission is gained by comparing theory and experiment [11-14]. It has been shown [15-19] that dynamical Langevin models of fission describe well a great number of experimental observables, including particle multiplicities and evaporation residue cross sections, for a great number of compound nuclei (CNs) over a broad range of excitation energy, angular momentum, and fissility.

Numerous theoretical investigations indicate that nuclear dissipation is shape dependent [11, 16, 20, 21], and the shape dependence of the nuclear dissipation is identified as a key ingredient [22] in the application of Langevin models to fission of excited nuclei. Currently, intensive efforts are being put on the strength of presaddle dissipation [23-25], and only very few studies focus on the exploration of postsaddle dissipation characteristics.

Light particles are considered to be the main indicators [20, 21, 26] for the dissipation effects. However, they can be evaporated along the whole fission path during the fission process of the CN, which causes an experimental difficulty of distinguishing particles emitted prior to saddle from those of the saddle-to-scission region.

Excitation energy at scission ($E_{\text{sc}}^{*}$) is observable and can be used to survey nuclear dissipation [27]. It is not only related to the number of prescission particles, but also depends on the energy loss taken away by these evaporated particles. Both aspects are connected with the properties of nuclear dissipation. So the quantity, $E_{\text{sc}}^{*}$, carries ample information on nuclear dissipation. As an independent information source, the $E_{\text{sc}}^{*}$ thus constitutes an alternative tool of exploiting postsaddle dissipation properties.

Till now, few researchers have used experimental $E_{\text{sc}}^{*}$ data to pin down postsaddle dissipation. In the present work, the $E_{\text{sc}}^{*}$ data from heavy ^254,256Fm and light ¹⁸⁹Au systems will be employed to probe postsaddle nuclear dissipation.

2 Theoretical model

A brief account of the combination of the dynamical Langevin equation with a statistical decay model (CDSM) [20, 28] is given here. The dynamic part of CDSM is described by the Langevin equation that is expressed by entropy. We employ the following one-dimensional overdamped Langevin equation to perform the trajectory calculations.

Here, q is the dimensionless fission coordinate and defined as half of the distance between the center of mass of the future fission fragments divided by the radius of the compound nucleus. T is the temperature, and ⟨Γ(t)⟩ is a fluctuating force with ⟨Γ(t)⟩ = 0 . M is the inertia parameter [20], and β is the dissipation strength.

The driving force of the Langevin equation is calculated from entropy:

where E^* is the excitation energy of the system. Equation (2) is constructed from the Fermi-gas expression with a finite-range liquid-drop potential [29]. The q-dependent surface, coulomb, and rotation energy terms are included in the potential V(q).

In constructing the entropy, the following deformation-dependent level density parameter is used:

where a₁ = 0.073 MeV^-1 and a₂ = 0.095 MeV^-1 are taken from Ignatyuk et al. [30]. B_s is the dimensionless surface area (for sphere B_s = 1), which can be parametrized by the analytical expression [31],

In the CDSM, light-particle evaporation is coupled to the fission mode by a Monte Carlo procedure [16]. The emission width of a particle of kind ν is given by Ref. [32].

where sν is the spin of the emitted particle ν, and mν is its reduced mass with respect to the residual nucleus. The level densities of the compound and residual nuclei are denoted by ρc(E^*) and ρR(E^*-Bν-εν). Bν are the liquid-drop binding energies. ε is the kinetic energy of the emitted particle, and _inv(εν) is the inverse cross sections [32].

After each emission act of a particle, the intrinsic energy, entropy, and temperature in the Langevin equation are recalculated and the dynamics are continued. Prescission various particle multiplicities are calculated by counting the number of corresponding evaporated particle events registered in the CDSM. To accumulate sufficient statistics, 10⁷ Langevin trajectories are simulated.

Regarding the excitation energy at scission, it is determined by using energy conservation law,

where E^* and V(q) have the same meaning mentioned earlier. E_coll is the kinetic energy of the collective degrees of freedom [20], and E_evap(t_sc) is the energy carried away by all evaporated particles by the scission time, t_sc.

For starting a trajectory, an orbit angular momentum value is sampled from the fusion spin distribution, which reads:

The parameters ${\mathcal{l}}_c$ and $\delta \mathcal{l}$ are the critical angular momenta for fusion and diffuseness, respectively.

3 Results and discussion

During the decay process of a CN, particle evaporation channel competes with fission channel. The nuclear friction retards fission and enhances particle emission, which lowers the excitation energy at scission.

To better explore postsaddle dissipation properties with $E_{\text{sc}}^{*}$ in this work, the presaddle friction strength is set to 3 zs^-1 (1 zs = 10^-21 s), in agreement with recent theoretical estimates and experimental analyses [12, 21, 28, 33-35]. The postsaddle friction strength (β) is determined by reproducing measured $E_{\text{sc}}^{*}$ in ¹⁶O + ²³⁸U [36], ¹⁸O + ²³⁸U [36], and ²⁰Ne + ¹⁶⁹Tm [37] reactions.

Figure 1(a) displays a comparison between experimental $E_{\text{sc}}^{*}$ data of the ²⁵⁴Fm system and theoretical ones which are calculated with Langevin models. It shows that $E_{\text{sc}}^{*}$ is a decreasing function of β. The reason for this behavior is that dissipation hinders fission, increasing particle emission and hence yielding a small excitation energy at scission. A detailed comparison of model calculations with experimental data reveals that the best-fit value of β is 12.5 zs^-1 (represented by solid square).

Fig. 1.Full-screen View

(Color online) Theoretical calculations are compared with measured excitation energy at scission for (a) ¹⁶O (E_lab = 288 MeV) + ²³⁸U ⟶ ²⁵⁴Fm and (b) ¹⁸O (E_lab = 159 MeV) + ²³⁸U ⟶ ²⁵⁶Fm reactions. Experimental values [36] are represented by the shaded band. Solid lines are predictions from Langevin models.

We also analyze the $E_{\text{sc}}^{*}$ data from another heavy ²⁵⁶Fm nucleus, produced in ¹⁸O + ²³⁸U reaction, and observe from Fig. 1(b) that the best-fit friction strength is ∼ 11.5 zs^-1, an amplitude analogous to that of ²⁵⁴Fm.

Overall, the postsaddele friction strength deduced from the two heavy systems (∼ 12 zs^-1) is stronger than that of presaddle friction, demonstrating a rise of the friction strength with increasing deformation.

A CN undergoes deformation as it fissions. As is well known, a light CN system has a shorter saddle-to-scission distance than a heavy one. It means that the $E_{\text{sc}}^{*}$ data from light and heavy fissioning systems can be employed to probe the friction strength at different deformations. To that end, we choose a light ¹⁸⁹Au system produced in ²⁰Ne + ¹⁶⁹Tm. The comparison between experimental data and calculated results is presented in Fig. 2. It is clear that $E_{\text{sc}}^{*}$ gets larger with the increase of incident energy as a result of a high initial excitation energy. In addition, a greater β leads to a smaller $E_{\text{sc}}^{*}$, because more particles are evaporated at a larger friction, which significantly reduces the excitation energy at scission. By comparing the experimental and calculated $E_{\text{sc}}^{*}$, one can notice that the friction strength of ∼ 6 zs^-1 can provide a satisfactory description of the experimental data. This friction strength is slightly stronger than that of the presaddle region, a result that is consistent with that derived from the heavy Fm systems.

Fig. 2.Full-screen View

(Color online) Calculated and experimental excitation energy at scission in the system ²⁰Ne + ¹⁶⁹Tm ⟶ ¹⁸⁹Au at different laboratory energy per nucleon [37]. Curves represent model calculations at postsaddle friction strengths β = 5, 6, 7, and 8 zs^-1.

It has been noted [21] that, when a modified one-body dissipation strength (which assumes a decreasing function of friction with deformation) was used in the calculation, the theoretical predictions are far below prescission particle multiplicity data of heavy fissioning nuclei with A > 250. For these very heavy nuclei, there is a longer distance between the saddle point and scission and hence a larger deformation is involved. It means that to account for the multiplicity data from heavy decaying systems, it is necessary to introduce a strong postsaddle friction in model calculations.

In addition to $E_{\text{sc}}^{*}$, we further survey particle multiplicity from light and heavy fissioning systems. As an illustration, Fig. 3 shows the comparison between theoretical and experimental prescission neutrons for ²⁵⁶Fm and ¹⁸⁹Au. The best-fit postsaddle friction value required tofit data is found to be 11 zs^-1 for heavy ²⁵⁶Fm and 5.5 zs^-1 for light ¹⁸⁹Au. While the result that a stronger postsaddle friction at a larger deformation is not altered, these best-fit β values obtained from prescission neutron data are slightly different from those obtained from the $E_{\text{sc}}^{*}$ data. This could be due to a consequence of a difference in the sensitivity that different observables to friction.

Fig. 3.Full-screen View

(Color online) Comparison of Langevin predictions for prescission neutrons with measured values in (a) ¹⁸O (E_lab = 159 MeV) + ²³⁸U ⟶ ²⁵⁶Fm and (b) ²⁰Ne (E_lab = 320 MeV) + ¹⁶⁹Tm ⟶ ¹⁸⁹Au reactions. Experimental values [36, 37] are shown by the shaded band, and model calculations are displayed by solid lines.

4 Summary

In the framework of Langevin model of fission dynamics, we have compared calculated and measured excitation energy at scission from heavy and light fissioning systems. A postsaddle friction value of 12 × 10²¹s^-1 and 6 10²¹s^-1 is extracted for ^248,256Fm and ¹⁸⁹Au nuclei, respectively. Postsaddle deformation of ¹⁸⁹Au is smaller than that of ^248,256Fm, thus an evident difference in the deduced friction strength for the light and heavy nuclei shows that nuclear friction becomes strong with an increase of deformation.