Intermediate Energy Reactions Versus Heavy-Ion Fusion:Light Particle Emission and Post-Saddle Friction in the Presence of Deformation effects∗

Na Chen(陈娜) and Wei Ye(叶巍)

1School of Applied Mathematics,Nanjing University of Finance&Economics,Nanjing 210023,China

2Department of Physics,Southeast University,Nanjing 210096,China

AbstractA decaying nucleus undergoes a change in deformation when it fi ssions.This a ffects the particle emission in the fi ssion process.Using the dynamical Langevin model,we investigate the role of deformation in the sensitivity of post-saddle neutrons and light charged particles(LCPs)to the post-saddle friction strength(β)for heavy nuclei240Am produced with different initial conditions:(i)a low excitation energy E∗ and a large spin ℓ(provided via a fusion mechanism)and(ii)a high E∗ and a large ℓ as well as a higher E∗ but a small ℓ(provided in peripheral and near-central intermediate energy heavy-ion reactions,respectively).It is shown that deformation obviously enhances the sensitivity of post-saddle neutrons to β at intermediate-energy peripheral collisions and that for case(i),the drop of LCPs emission due to deformation makes post-saddle LCPs to be almost insensitive to β,but for case(ii)LCPs still have a signi fi cant change with β.Furthermore,we find that post-saddle LCPs display a greater sensitivity to β for near-central collisions than for peripheral collisions.These results suggest that given the deformation effects,to better probe post-saddle dissipation properties with neutrons(LCPs)in experiments,it is best to choose those excited heavy nuclear systems populated in peripheral(near-central)collisions at intermediate energies.

Key words:post-saddle friction,deformation,excitation energy,light particles,stochastic model

1 Introduction

Nuclear dissipation plays an important role in the large-scale collective motion,like fusion and quasiif ssion.[1−3]Its critical influence on the fi ssion process of hot nuclei has recently attracted much attention.[4]Dissipation hinders fi ssion and hence increases pre-scission particle multiplicities with respect to the predictions by standard statistical models.[5−6]It has been demonstrated that stochastic approaches based on Langevin models[7−9]have been successfully applied to address dissipation effects in nuclear fi ssion and provided a satisfactory description of different types of fi ssion data.

When applying the Langevin model to handle fi ssion,a key ingredient in it is the deformation dependence of nuclear friction.[10]Currently,a number of works have been made to constrain pre-saddle friction by using evaporation residue cross sections,[11]the first-chance fi ssion probability,[12−13]the widths of fi ssion-fragment charge distributions,[14]etc. As a result,the strength of pre-saddle friction is severely limited.[15]However,these observables only depend on the pre-saddle friction and thereby they are not suited for exploring the postsaddle dissipation effects.Moreover,till now,less e ff ort has been invested to constrain the post-saddle friction strength.[7,16]

different from previously mentioned observables,light particles are evaporated along the entire fi ssion path.They are thus a ffected by post-saddle friction.Also,postsaddle multiplicities are an increasing function of size of the decaying system.So,the particle emission from heavy fi ssioning nuclei is usually used to obtain information of post-saddle dissipation properties.[17−20]Further,when a fi ssioning nucleus evolves from ground state to the scission point,it experiences the change of deformations along the fi ssion trajectory.This a ffects various particle emissions.[21−23]

The nuclear systems formed in intermediate energy heavy-ion collisions and fusion reactions have different excitation energies and angular momenta.However,to date,few have studied the effect of deformation on the evolution of post-saddle neutrons and light charged particles(LCPs)with the post-saddle friction strength for heavy fi ssioning nuclei populated under these different initial conditions.The present work is devoted to this issue.

Our aim is to exploit the favorable experimental condition through which the post-saddle dissipation effects can be better revealed with the particle multiplicity;that is,which experimental approach is more optimal for probing post-saddle dissipation with light particle multiplicity in the presence of deformation effects.To this end,the Langevin model[7−9,22−25]is employed here and it is successfully used to reproduce a volume of fi ssion data for many compound systems over a broad range of the excitation energy,angular momentum,and fi ssility.

2 Theoretical Framework

In the Langevin description of a fi ssion process,the crucial quantity is free energy,which contains a thermodynamic correction.[26]We use the following one-dimensional Langevin equation to perform the fully dynamical trajectory calculations:

Here q is the dimensionless fi ssion coordinate and is deif ned as half the distance between the center of mass of the future fi ssion fragments divided by the radius of the compound nucleus,and p is the conjugate momentum.β and T denote the dissipation strength and temperature,respectively.The inertia parameter m is obtained under the Werner-Wheeler approximation of an incompressible irrotational flow.[27]Γ(t)is a fl uctuating force satisfying⟨Γ(t)⟩=0 and ⟨Γ(t)Γ(t′)⟩=2δ(t− t′).

The free energy is constructed from the Fermi gas expression of the level density parameter together with a finite-range liquid-drop potential V(q)[28]that contains qdependent surface,Coulomb,and rotation energy terms;that is,

In Eq.(2),the coefficients proposed in Ref.[29]are used to calculate the deformation-dependent level density parameter,which reads as follows:

where A is the mass number of the compound nucleus and Bsis the dimensionless surface area of the nucleus.[30]

In our calculation,prescission particle evaporation along Langevin fi ssion trajectories from their ground state to their scission point has been taken into account using a Monte Carlo simulation technique.The emission width of a particle of kind ν(=n,p,α)is evaluated by Blann’s parametrization[31]

where sνis the spin of the emitted particle ν,and mνits 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).Bνare the liquid-drop binding energies.ε is the kinetic energy of the emitted particle.The inverse cross section is given by[31]

with

where Aνis the mass number of emitted particle ν =n,p,α.

The barriers for the charged particles are[31]

with Kν=1.32 for α,and 1.15 for proton.

The massformula[32]containsthedeformationdependent surface and Coulomb energy terms.The particle binding energy Bi(i=n,p,α)is thus a function of deformation[21−22]and it can be written as

where Mi(i=n,p,α)is the mass of the emitted particles.Mp(q)and Md(q)are the masses of the mother and daughter nuclei,respectively.

We use the formula suggested by Fröbrich and Gontchar[7]to calculate the deformation-dependent chargedparticle emission barriers:

Here the Coulomb energy Bc(q)is evaluated using the method in Refs.[30,33].

When a dynamic trajectory reaches the scission point,it is counted as a fi ssion event.Prescission particles are insensitive to the definition of the scission point(i.e.,zero or a finite neck radius),as they can be emitted along the entire fi ssion trajectory.In our calculation,multiple emissions of light particles and higher-chance fi ssion are taken into account.Prescission particle multiplicities are calculated by counting the number of corresponding evaporated particle events.To accumulate sufficient statistics,107Langevin trajectories are simulated.

3 Results and Discussion

Due to the competition from quasi- fi ssion channels,which become stronger with increasing bombarding energy,heavy compound nuclei(CNs)populated by fusion reaction channels generally have a low excitation energy(<80 MeV)and a high angular momentum(around 40~).However,intermediate-energy(around Fermi energy domain)heavy-ion collisions can deposit more energy into the nuclear systems and yield a variety of fi ssioning nuclei with a different excitation energy and angular momentum.

For example,in near-central collisions the generated nuclear systems have a high excitation energy(∼250 MeV)and a low spin(near 10~).However,in peripheral collisions,the produced fi ssioning systems have an excitation energy over 200 MeV and a large angular momentum(∼ 40~).[34−35]

In the present work,calculations under these three different initial conditions mentioned above for the produced heavy fi ssioning system are carried out and their sensitivities to nuclear friction are compared in the presence of deformation effects.Towards that goal,a heavy240Am was chosen here to investigate post-saddle dissipation characteristics by using light particle multiplicity.To better reveal post-saddle dissipation effects,the presaddle friction strength is set to 4 × 1021s−1,in consistent with recent theoretical estimates and experimental analyses,[8,14,36−37]and dynamical calculations of postsaddle emission are performed considering different values of the post-saddle friction strength(β).

Shown in Fig.1 are the evolution of post-saddle neutrons with β at three different initial conditions of excitation energy and angular momentum for the fi ssioning nucleus240Am with and without deformation effects.

Fig.1(Color online)Post-saddle neutrons versus the postsaddle friction strength β in the absence(a)and in the presence(b)of deformation effects for heavy system240Am calculated for case(i)E∗ =80 MeV and ℓ=40~,case(ii)E∗=250 MeV and ℓ=10~,and case(iii)E∗=200 MeV and ℓ=40~.

We first compare the results of case(i)and case(ii);that is,fusion reactions vs. intermediate-energy nearcentral collisions. Two typical features are observed.First,the calculated post-scission neutrons Mnare larger in case(ii)than in case(i),indicating a stronger effect of dissipation on Mnunder the condition of case(ii).

Another feature is that after incorporating deformation effects into the model calculations(Fig.1(b)),Mnrises,exhibiting a larger influence of dissipation on postsaddle neutrons.A larger Mndue to deformation is that neutron binding energies drop with increasing deformation(Fig.2(a)),enhancing the neutron emission.

A comparison on charged-particle emission(i.e.,protons and α-particles)for case(i)and case(ii)is displayed in Fig.3.First,accounting for the deformation effects decreases Mpand Mαin both cases.The reason is that though deformation lowers emission barriers of LCPs(Fig.2(b)),it increases their binding energies(Fig.2(a)),which is unfavorable for their emissions.As a result of the two opposite factors,the LCPs multiplicity decreases.

Fig.2 (Color online)(a)A change in neutron,proton,and α-particle binding energies of240Am due to deformation with respect to their values at a spherical shape.(b)Emission barriers of protons and α particles of240Am as a function of deformation coordinate q.

Secondly,when deformation effects are ignored(see triangles connected by blue lines in Figs.3(a)and 3(c)),a variation in Mpand Mαis still discernible as β changes from 0.5×1021s−1to 20×1021s−1,meaning a sensitivity of LCPs to β,though it is quite weak.However,in case(i),as a consequence of a reduced Mpand Mαin the presence of the deformation effects(see triangles in Figs.3(b)and 3(d)),LCPs almost do not vary with a change in β;that is,their sensitivity to friction disappears.In contrast,while deformation effects decrease Mpand Mαin case(ii)(see circles connected by red lines in Figs.3(b)and 3(d)),the LCPs multiplicity shows a signi fi cant sensitivity to the friction strength.

This comparison clearly shows the role of excitation energy in exploring the post-saddle dissipation properties after considering the deformation effects.Further,it suggests that when using LCPs to place a stricter constraint on the post-saddle friction strength,case(ii)is a more optimal experimental condition than case(i).

Unlike fusion reactions which form a CN,intermediate energy collisions generate a variety of excited nuclear systems having a different excitation energy and angular momentum,depending on the collision centralities.Further, fi ssion events and the corresponding information on A,Z,E∗,etc.of fi ssioning sources coming from nearcentral or peripheral collisions can be identi fi ed and obtained experimentally.[34−35,38−39]In these experiments,the folding angle technique was used to measure the correlation angle of the two fi ssion fragments.

Fig.3 (Color online)Post-saddle protons(top panel)and α particles(bottom panel)versus the post-saddle friction strength β in the absence((a)and(c))and in the presence((b)and(d))of deformation effects for heavy system240Am calculated for case(i)E∗ =80 MeV and ℓ=40~,case(ii)E∗=250 MeV and ℓ=10~,and case(iii)E∗=200 MeV and ℓ=40~.

Previously,we compared the calculation concerning post-saddle particles as a function of β for case(i)and case(ii),which represents the conditions provided via fusion and near-central collisions at intermediate energy,respectively.To better employ intermediate energy reactions as a way to probe the post-saddle friction strength,we make a further calculation at E∗=200 MeV and ℓ=40~(case(iii)),which corresponds to conditions available in peripheral collisions which generate a fi ssioning nucleus with a lower E∗and a higher ℓ than that generated in near-central collisions.The calculated results for case(iii)are also plotted in Figs.1 and 3,which are shown by squares connected by green lines.

We note that in the presence of deformation effects,Mndemonstrates an obvious quicker rise with increasing β in case(iii)than in case(ii).This is because while case(iii)contains a lower E∗than case(ii),a higher ℓin case(iii)decreases the fi ssion barrier,which shortens the transient time.Consequently,pre-saddle neutrons are decreased,and more energy is left for post-saddle evaporation,leading to a greater post-saddle multiplicity.This means that case(iii),i.e.,peripheral collisions could provide a more favorable condition to probe β using neutrons than near-central collisions.In addition,we also notice from Fig.1(b)that Mnrises more rapidly with β in case(iii)than in case(i),illustrating the effect of deformation on neutrons as an observable of the post-saddle friction strength.

However,a picture different from neutrons is seen for LCPs;that is,LCPs have a larger value in case(ii)than in case(iii),showing that dissipation has a larger effect on LCPs in case(ii).The reason is as follows.There exists a competition among different decaying channels.A strong neutron evaporation(compare squares and circles connected by the blue and red line in Fig.1(b),respectively)suppresses charged-particle evaporation.While a higher ℓ in case(iii)than in case(ii)raises the multiplicity of post-saddle particles including that of LCPs,the magnitude of excitation energy has a stronger effect than that of angular momentum.This further reveals the important role of E∗in using light charged particles as a tool of the post-saddle friction strength.It implies that when one uses LCPs to better limit β,it is best to choose heavy fi ssioning nuclei populated in near-central collisions.

Putting together all the results calculated for the three cases,as shown in Figs.1 and 3,one can find that intermediate energy reactions are a more preferable experimental approach than heavy-ion fusion,which is mostly adopted in the current experiments,to explore post-saddle dissipation properties with light particle emission,in particular in the presence of deformation effects.

4 Conclusions

In conclusion,we have studied the influence of deformation on probing the post-saddle friction strength(β)with light particle multiplicities of heavy240Am under different excitation energies and angular momenta.It has been found that compared to the fusion approach,the high excitation energy condition provided in intermediate energy reactions apparently enhances the sensitivity of light particles(particularly for LCPs for the case with deformation effects)to β.Furthermore,it has been shown that when using neutrons to constrain β, fi ssioning systems generated in peripheral collisions at intermediate energies are more suitable than those generated in near-central collisions.For LCPs whose emission depends on excitation energy more strongly than on angular momentum,choosing those heavy fi ssioning nuclei from near-central collisions are favorable in experiments for more precisely determining the post-saddle friction strength.