Jian Tian·Wei Ye

Investigating nuclear dissipation properties at large deformations via excitation energy at scission

Jian Tian1·Wei Ye1

Using the stochastic Langevin model coupled with a statistical decay model,we study nuclear dissipation properties at large deformations with excitation energy at scission()measured in experiments.It is found that the postsaddle dissipation strength required to fitdata is 12 ×1021s-1for254，256Fm and 6×1021s-1forAu,which has a smaller postsaddle deformation than the former heavy nucleus,showing a rise of nuclear dissipation strength with increasing deformation.

Excitation energy at scission. Langevin model.Nuclear dissipation

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]thatdynamicalLangevin modelsoffissiondescribe 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 theoreticalinvestigations 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)is observable and can be used to survey nuclear dissipation[27].It not only is 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,carries ample information on nuclear dissipation.As an independent information source,thethus constitutes an alternative tool of exploiting postsaddle dissipation properties.

Till now,few researchers have used experimentaldata to pin down postsaddle dissipation.In the present work,thedata from heavy254，256Fm and light189Au 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 deformationdependent level density parameter is used:

where a1=0.073 MeV-1and a2=0.095 MeV-1are taken from Ignatyuk et al.[30].Bsis the dimensionless surface area(for sphere Bs=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 sνis the spin of the emitted particle ν,and mνis its reduced mass with respect to the residual nucleus.The level densities of the compound and residual nuclei are denoted by qc（E*）and qR（E*-Bν-εν）.Bνare the liquiddrop binding energies.ε is the kinetic energy of the emitted particle,and σinv（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,107Langevin 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.Ecollis the kinetic energy of the collective degrees of freedom[20],and Eevap(tsc)is the energy carried away by all evaporated particles by the scission time,tsc.

Forstartingatrajectory,anorbitangularmomentumvalue is sampled from the fusion spin distribution,which reads:

The parameters ecand δe 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 within this work,the presaddle friction strength is set to 3 zs-1(1 zs=10-21s),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*scin16O+238U[36],18O+238U [36],and20Ne+169Tm[37]reactions.

Figure 1a displays a comparison between experimentaldata of the254Fm system and theoretical ones which are calculated with Langevin models.It shows thatis 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(Color online)Theoretical calculations are compared with measured excitation energy at scission for a16O(Elab=288 MeV)+ 238U-→ 254Fm and b18O(Elab=159 MeV)+238U-→ 256Fm reactions.Experimental values[36]are represented by the shaded band.Solid lines are predictions from Langevin models

Overall,the postsaddle 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.

Fig.2(Color online)Calculated and experimental excitation energy at scission in the system20Ne+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

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 thedata from light and heavy fissioning systems can be employed to probe the friction strength at different deformations.To that end,we choose a light189Au system produced in20Ne +169Tm.The comparison between experimental data and calculated results is presented in Fig.2.It is clear thatgets larger with the increase in incident energy as a result of a high initial excitation energy.In addition,a greater β leads to a smaller,because more particles are evaporated at a larger friction,which significantly reduces the excitation energy at scission.By comparing the experimental and calculated,one can notice that the friction strength of～6 zs-1can 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.

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.

Fig.3(Color online)Comparison of Langevin predictions for prescission neutrons with measured values in a18O(Elab=159 MeV)+238U-→256Fm and b20Ne(Elab=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 ofLangevin modeloffission dynamics,we have compared calculated and measured excitation energy at scission from heavy and light fissioning systems.A postsaddle friction value of 12×1021s-1and 6×1021s-1is extracted for248，256Fm and189Au nuclei,respectively.Postsaddle deformation of189Au is smaller than that of248，256Fm,and thus an evident difference in the deduced friction strength for the light and heavy nuclei shows that nuclear friction becomes strong with an increase in deformation.

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10.1051/ anphys:01992001706047100

15 August 2016/Revised:15 August 2016/Accepted:8 October 2016

©Shanghai Institute of Applied Physics,Chinese Academy of Sciences,Chinese Nuclear Society,Science Press China and Springer Science+Business Media Singapore 2016

This work was supported by the National Nature Science Foundation of China(No.11575044).

✉ Wei Ye

yewei@seu.edu.cn

1Department of Physics,Southeast University, Nanjing 210096,China