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Quasi-fission reactions as a probe of nuclear viscosity

Fission fragment mass and angular distributions were measured from the ^{64}Ni+^{197}Au reaction at 418 MeV and 383 MeV incident energy. A detailed data analysis was performed, using the one-body dissipation theory implemented in the code HICOL. The effect of the window and the wall friction on the experimental observables was investigated. Friction stronger than one-body was also considered. The mass and angular distributions were consistent with one-body dissipation. An evaporation code DIFHEAT coupled to HICOL was developed in order to predict reaction time scales required to describe available data on pre-scission neutron multiplicities. The multiplicity data were again consistent with one-body dissipation. The cross-sections for touch, capture and quasi-fission were also obtained.


I. INTRODUCTION
The mechanism through which energy is dissipated in low energy heavy ion reactions is a topic widely discussed in the last 20 years. The study of deep-inelastic collisions in the 1970s stimulated the development of dissipative theories of nuclear dynamics [1,2]. Following this, the quasi-fission phenomenon was discovered in the 1980s. As well as fusion-fission, it belongs to the class of the completely damped reactions, where all the initial kinetic energy is dissipated, but depending on the entrance channel, the mass asymmetry degree of freedom may be fully or partially equilibrated [3]. Large mass transfer is achieved in a relatively short time ( 2-10 x10 −21 s) [4] and the system quickly re-separates due to the absence of a fission barrier.
In heavy symmetric systems, quasi-fission is the dominant reaction mechanism. A number of systematic studies done at GSI [4][5][6] using 238 U and 208 Pb beams incident on targets ranging from 16 O to 89 Y, have identified the following experimental signatures of quasifission: • Fragment mass distributions wider than the mass distributions resulting from fusionfission reactions.
• Asymmetries in the mass-angle correlations increasing with the target mass.
• Angular anisotropies significantly larger than those in fusion-fission reactions.
Because of their unique nature, quasi-fission reactions can provide a testing ground for different models of dissipative heavy ion collisions. While in fusion-fission reactions, the fragment mass distributions are not informative of the reaction dynamics, in quasi-fission, the mass drift towards symmetry is a sensitive probe of the dissipative forces acting between the reaction partners. The asymmetry in the mass-angle correlations, wherever present, is directly related to the time the system spends in the coalesced state, before re-separation.
In the GSI measurements it was determined, that for all targets heavier than 40 Ca the system made less than one turn before re-separating and the turning angle of the intermediate complex was used to extract the dynamical time scale of the reaction. The data analysis found that the relevant friction mechanism is one-body in nature and is temperature independent. As a result of these works, the extra-push model parameters were established in a systematic way. They were also used in the development of the one-body dissipation model implemented in Feldmeier's code HICOL [7].
Another experimental approach for determining the reaction time scale is the measurement of pre-scission neutron multiplicities. This approach is not limited to quasi-fission reactions, but was also used to deduce fusion-fission time scales [8,9]. When the excitation energy of the coalesced system is high enough, the neutron evaporation times are much shorter than the scission time and thus, the pre-scission neutron multiplicity can be used as a clock. The theoretical treatment of the data involves a statistical model code, which is used to predict the pre-scission neutron lifetimes and to deduce the reaction time. Although the dynamics of the reaction cannot be treated with this model, and a direct comparison with the different dissipation theories has not been done, the quasi-fission time scales derived from pre-scission neutron multiplicities are comparable to the results obtained with mass-angle correlations in similar systems and temperature regimes [4][5][6].
In a recent work, Wilczynski et.al. [10,11] re-analyzed Hinde's et.al. [8,9] pre-scission neutron multiplicity data, using an evaporation code DYNSEQ coupled to the outputs of HICOL. In that way, the dynamics of the reaction was taken into account. The time scales obtained were an order of magnitude longer than the original results of Hinde et.al. Surprisingly large values for the dimensionless friction coefficient γ = 17 − 50 were reported.
These were derived after scaling the one-body dissipation in HICOL by a factor k s = 4 − 12.
Using this new analysis, and results of analysis [12] of the GDR γ-rays in coincidence with fission for the O+Pb, S+W,Pb reactions [13], the authors claimed to observe strong twobody dissipation setting in at nuclear temperature of about 2 MeV, with γ following 1/T 2 temperature dependence. One-body friction, for comparison, yields γ ∼ 4 independent of temperature.
The strength of two-body dissipation is still an open question in the literature. While most authors find that two-body dissipation is weaker than one-body and yields shorter pre-scission time scales [14,15], there are some theoretical works that predict strong twobody viscosity and long time scales associated with this (e.i. [16]). Such strong dissipation, if present, would influence significantly the quasi-fission reaction dynamics and the observables associated with it: fragment mass distributions and mass-angle correlations. It is desirable to compare dissipation results deduced from pre-scission neutron multiplicities and from fragment mass and angular distributions. At present, there are four quasi-fission reactions in which the pre-scission neutron multiplicities have been measured [9]. The data set of massangle correlations is larger, but until now, there was no measurement of both observables in the same reaction at the same center of mass (c.m.) energy, and hence, the same nuclear temperature.
In this work the mass and angular distributions were measured in the 64 Ni+ 197 Au reaction at the 418 MeV and 383 MeV laboratory energies. The mass-angle correlations were also constructed. For the first bombarding energy, pre-scission neutron data exist and a comparison of the two experimental approaches is possible. The second energy was used to investigate the temperature dependence of the friction. With the development of the code HICOL, a detailed analysis of the mass distributions became possible and the relative contribution of the window and the wall friction could be determined from the data. An upper limit to the reaction time, and the friction, was determined by an analysis of the mass-angle correlations using the one-body dissipation model HICOL. An evaporation code DIFHEAT, coupled to HICOL, was developed and applied for analysis of the pre-scission neutron data published in Ref. [9] and a comparison with Wilczynski's results (obtained with a different code) is presented. The deep inelastic, capture and touch cross-sections were also obtained.

II. EXPERIMENT
The experiment was performed at the Stony Brook Nuclear Structure Laboratory, using 64 Ni beam from the FN Tandem Van de Graaff and Super-conducting LINAC accelerators.
The reactions, for which the pre-scission neutron multiplicities were measured and the two existing interpretations for the reaction times, determined in Refs [9] and [10] are given in Table I. To complement the pre-scission neutron multiplicity data, we measured the mass and angular distributions of the fragments produced from 64 Ni + 197 Au at 418 MeV incident energy. Although, in this case the discrepancy shown in Table I is mild compared to the 64 Ni+ 238 U and 40 Ar+ 238 U reactions, the 197 Au target was chosen for its lower fissility.
The measurements were performed in "singles" mode with Silicon Surface Barrier detectors (SSBs) and the event reconstruction was done assuming binary fission. connected to a cooling system and the detectors were cooled to −10 o C, which allowed overbiasing without damaging the detectors. Operating the SSBs in high field strength was crucial, since this minimizes the pulse-height defect and the plasma delay. For the heavy ions studied in this experiment, the plasma delay time was between 0.2−1.5 ns. Corrections were made to compensate for this effect. The pulse height defects were E phd = 5 − 25 MeV.
Although systematic studies of the two effects listed above are present in the literature [17,18], the correct mass reconstruction for ions as heavy as the fission fragments measured in this experiment was difficult. An original procedure for mass reconstruction was developed in this work and is discussed in Ref. [19]. In addition, two small area ion-implanted silicon detectors (EG&G Ortec BU-013-25-300) were placed at ±20 o with respect to the beam axis at a distance 81 cm from the target to monitor the beam quality and for normalization of the measured fission fragment yield to the Rutherford scattering cross section.
The reconstruction of the mass and the energy of the detected fragments, requires an energy and absolute Time-Of-Flight (TOF) calibration of the detectors and the subsequent electronics. The energy calibration was done using a 228 Th α-source and the elastic peak from a 247 MeV 64 Ni beam. At the E lab = 418 MeV energy, additional calibration points from elastic scattering were used for the detectors that were within the reaction grazing angle. The energy resolution was ∼ 2% for the elastically scattered beam and ∼ 1.2% for α-particles. An absolute timing calibration was done using the reference RF signal of the LINAC accelerator and the elastic peak from the 247 MeV run. A thin (100 µg/cm 2 ) selfsupporting Au target was used in the calibration and a 250 µg/cm 2 Au target was chosen for the measurement, compromising between good timing and a reasonably high count rate.
The width of the beam pulse at the target was ≤ 600 ps FWHM. Corrections were made for the plasma delay time of the heavy fragments, and the overall timing resolution for the elastic reaction products measured was ≤ 1 ns.
In addition, the relative solid angles of the SSB's and the monitors were calibrated using the Rutherford scattering yield. Several calibration runs at 247 MeV, bracketing in time the higher energy data, were recorded during the course of the experiment.

A. Mass Reconstruction
Two-dimensional (2D) scatter plots energy versus TOF were made for each detector, and the reaction products were separated from the slit scattered beam and low-energynoise backgrounds by applying 2D gates. An example of such a plot and the gate applied for detector 3, located at 40 o in the lab system, is shown in Fig.2(a). The high energy elastic peak dominates the yield. Quasi-elastic events are seen at an energy lower than the elastics and with similar TOF. The regions of the deep inelastic and the quasi-fission fragments are indicated in the figure. Although the elastic and quasi-elastic channels are not of interest for the quasi-fission reaction studied here, these events (wherever present) provided an important check for the mass reconstruction, so they were included in the first step of the data analysis. Later, when only the damped reaction products were studied, total kinetic energy (TKE) versus mass scatter plots were made for each detector, and a gate was applied to select the deep inelastic and quasi-fission events only. See Fig. 2 After selecting the events of interest, the mass was reconstructed, as described in Ref [19], with overall mass resolution better than 10 mass units.

B. Efficiency Simulations
A Monte-Carlo simulation program was developed to determine the detection efficiency for each mass at each detector position. Events were generated in the c.m. with flat mass distribution and dσ/dΩ c.m. = 1/ sin θ (assuming isotropic distribution in the polar angle θ).
The kinetic energy of the fragments was drawn from a Gaussian distribution with parameters determined from the measured TKE distributions, as discussed later. The simulations included energy loss in the target and in the front Au electrode of the detector, pulseheight-defect and plasma delay, electronics energy thresholds, TOF and velocity cuts. After applying the experimental cuts and the experimental mass reconstruction procedure, the events were transformed back into the c.m. and the "detected" yield was compared to the input. The efficiency curves for the two energies studied and for the two positions of the platform holding the detectors are shown in Figure 3. For small masses, the input angular distribution is recovered from the "data", in the intermediate mass region ( The mass cuts observed in this experiment agree very well with the simulated efficiency curves. This gives confidence that the mass reconstruction in the experimental data was done correctly, and also, that all relevant effects were included in the simulations. Based on the simulated efficiency, correction to the experimental yield was done for the mass bins in which the efficiency was between 0.5 and 1. For lower efficiencies, the centroid of the fragment kinetic energy distribution is not covered, so these data were excluded from the analysis.  [4][5][6]. The variance of the distribution, for which systematic data and empirical formulae do not exist, was determined to be ≈ 15% from the centroid and is again consistent with the measurements in Ref. [6]. The HICOL calculations, discussed later, predict correctly the centroid of the TKE distribution, which shows that the friction mechanism implemented in the code produces the right amount of damping.

Experimental Results
The energy-integrated cross-sections binned in 10 mass units, dσ dθc.m. = 2π sin θ c.m. dσ dΩc.m. , were constructed, after lab to c.m. transformation and normalization to the Rutherford scattering yield.

HICOL Calculations Along the Trajectory
A series of HICOL calculations was performed to investigate the dissipation mechanism responsible for the forward-backward asymmetry in the experimental angular distributions.
The code implements a macroscopic description of dissipative heavy ion collisions [7]. A set of collective shape and angular coordinates is used and their evolution is followed, solving the equations of motion. The coupling of the macroscopic and the intrinsic degrees of freedom is considered explicitly in the derivation of the friction and the mass diffusion tensors, in which Fermi gas one-body formulae are used. A typical "along-the-trajectory" calculation, which will be used as an illustrative example is shown in Fig. 6. We will consider a quasi-fission reaction of 64 Ni+ 197 Au at 418 MeV laboratory energy and initial orbital angular momentum L= 50h. The potential energy, the kinetic energy in the shape degrees of freedom T s , in the angular degrees of freedom (T rel and T int ) and the dissipated energy are plotted as a function of time. In panel (a), the actual shapes along the trajectory are also shown. The trajectory starts at time t = 0 when the nuclei are 14 fm apart. All the energy available to the system resides in kinetic energy of the shape and relative rotation. Within 6x 10 −22 s, the nuclei come into contact. At that time, the proximity friction has already produced about 8 MeV of heat. After the touching point, as the window opens, the radial motion is rapidly damped. With this, the shape loses most of its kinetic energy, since the other two shape coordinates -the neck and the asymmetry do not possess any initial momentum. The viscous drag in the window sets into motion the intrinsic rotation. At t = 15x10 −22 s the window has produced 68 MeV of heat and is already widely open. At this point the shape changes from a necked-in to a convex type. The system enters the mono-nuclear regime, and a uniform wall is considered for the subsequent evolution. A long creeping motion with practically no kinetic energy in the shape follows. Without the window friction against the net particle flux, the asymmetry degree of freedom begins relaxing. It has been severely hindered up to now. As we move along the trajectory, the deformation develops again, the window comes back and the total dissipation increases. The potential is rapidly falling, the system gains back kinetic energy in the shape (10 MeV  distributions. This is a typical quasi-fission trajectory, where the system scissions bypassing the compound nucleus stage. The mass asymmetry is not fully relaxed, but almost all the initial kinetic energy is dissipated and the fragments fly apart driven by the strong Coulomb repulsion. In the model implemented in HICOL, there are no free parameters. The only ad hoc procedure used is in the combination of window-plus-wall and mono-wall friction. For shapes with a "fat" neck, there is an ambiguity in the choice of a window plus two walls, or a mononucleus shape without a window. To make a smooth transition between the two regions, a form factor f (x n ) multiplying the rate of energy dissipationQ, is introduced as follows: with x n = R neck /min(R 1 , R 2 ) measuring the ratio between the neck radius and the radius of the smaller nucleus. For "thin" necks (0 < x n < 0.8), f (x n ) = 0 and the full windowplus-wall friction is used. In the region 0.8 ≤ x n ≤ 1, where the shape makes the di-nucleus to mono-nucleus transition, f (x n ) is smoothly rising from 0 to 1 and both widow-plus-wall and mono-wall are used. For x n > 1, the energy dissipation is calculated with the mono-wall formula. In the outgoing part of the trajectory, when the neck develops again, f (x n ) drops from 1 to 0, after which, the full window-plus-wall friction is used until the scission point. The functional form of f (x n ) and the limiting value x n = 0.8 are arbitrarily chosen. The concern when using such a description comes from the fact that the two formulae ( window-plus-wall or mono-wall) give different results, when applied to the same shape. The window-plus-wall formula can produce 4 to 8 times stronger friction against the mass-asymmetry equilibration than the wall formula alone. Fig. 7 (a) illustrates the mass evolution of the projectile-like and the target-like fragment along the E lab = 418 MeV, L= 50h trajectory. The evolution of the form factor f (x n ) is shown below it in Fig.7 (b). One can easily see, that the main mass transfer from the target to the projectile happens during the mono-nucleus part of the trajectory, when the window friction is turned off. In Fig. 7 (c) and (d) we have calculated the mass evolution in the two extreme cases: mono-wall friction along the whole trajectory, or full window-plus-wall everywhere, respectively. Both these cases are clearly unphysical.
Without the window friction, all trajectories proceed rather rapidly to symmetry, which is in contradiction with the experimental results (Section IV D) , while with the window-pluswall acting everywhere, there are no trajectories that produce symmetric mass-splits. The realistic description of the reaction dynamics needs a mixture window-plus-wall and monowall friction. We consider that the form factor f (x n ) should be used as free parameter and adjusted to match the experimental mass distributions.
In the dynamical analysis of Hinde's et.al. pre-scission neutron data [9], Wilczynski, Siwek-Wliczynska and Wilschut [11] scaled the energy dissipation rateQ by an additional factor k s , which was treated as a free parameter. The factor k s was allowed to obtain values smaller than unity, when a reduction of the full one-body friction was needed, and k s > 1 was interpreted as evidence for two-body dissipation. Along the trajectory, the authors of Ref. [11] divided k s into k in s and k out s . In the early stage of the reaction, when the system is reasonably cold, a factor k in s = 0.5 was used, and after the mono-nucleus stage is reached and the system has converted a significant amount of kinetic energy into heat -k out s = 4 − 12 was applied to make the reaction time long enough to allow the emission of the requisite number of pre-scission neutrons. The justification for using such a description was the expected temperature dependence of nuclear viscosity.  [11]. The quantities plotted are the same as in Fig.6, which was for the standard (k s = 1) HICOL trajectory. By comparing the two calculations, one sees that the shape of the potential energy, kinetic energy and total heat curves is similar, but the time scale is stretched in the large friction trajectory. The relative contributions from window and wall dissipation are different. By scaling down the friction in the early stage of the reaction when the window-plus-wall formula is used, the window contribution is reduced. Then, in the mono-nucleus stage, the weaker wall friction is scaled by a factor of 4. This brings the system into a relatively flat region of the potential with very little kinetic energy (mainly in rotation) and strong friction hindering its motion towards scission. It takes 3.5 times longer to develop the deformation and find its way to the steep slope of the potential that leads to scission. During this long reaction time, the system makes more than two full rotations, as seen in the shape evolution in the figure.
The calculated final scattering angle is −760 o . This will lead to orbiting type (∼ 1/ sin θ) angular distribution of the fragments, instead of the asymmetric distribution expected from the trajectory in Fig.6, which agrees qualitatively with the experimental results. The mass drift in the large-friction trajectory is similar to the one-body case, since the window friction was initially reduced (k in s = 0.5), and later increased, in the final part of the trajectory.

C. Cross-Sections
The experimental angular distributions were extrapolated to 0 o and 180 o as described by Back et.al. in Ref. [20]. For fusion-fission reactions the angular distributions of the fragments are given by [21]: Here I is the total spin of the fissioning nucleus, K is the projection of I on the fission axis, T I is the transmission coefficient for fusion of the Ith partial wave and d I 0K (θ) are the real parts of the Wigner D-functions. Eq. 2 gives angular distributions symmetric about 90 o which approach the 1/ sin θ classical distribution for I >> 0. These are not applicable to describe the asymmetric angular distribution in short-time scale quasi-fission reactions.
However, since a theoretical prescription for this type reactions is not available, Back et.al.
[20] used the following expression to fit the data from a measurement similar to the one done in this work: where β is the angular slope parameter. The same approach was adopted here. The transmission coefficients T I were calculated using the extra-push estimate to the complete fusion cross-section with a program due to Back [22]. The Wigner D-functions for I, K ≤ 25 were calculated using the CERN library routine DDJMNB, and the recursion relations [23] were used for larger spins. The value of K 0 was determined from the mass bin closest to symmetry, and kept fixed in the other mass bins. Best results were obtained with K 2 0 = 25 +25 −10 for the 418 MeV data and K 2 0 = 15 +20 −5 at 383 MeV. The overall normalization and the parameter β were varied to produce best fits to the data. In the systems studied by Back et.al. [20] the clean separation of the quasi-fission and deep inelastic events was possible, since the mass distributions were peaked at symmetry. In the case studied here, the mass distribution has a minimum at symmetry and these two reaction channels could not be separated (see also  Table II.

D. Mass Distribution
After integrating the angular distributions in the different mass bins, the mass distributions for the two energies measured were obtained (see Fig. 9). At both energies, the distributions differ dramatically from the mass distributions of fusion-fission reactions, which are narrower and peaked at symmetry. This behavior can be qualitatively explained since, for this system which has fissility x = 0.915, the symmetric saddle does not exist. The potential energy landscape always favors asymmetric mass division. Theoretical mass distributions were obtained using HICOL's predictions for the final mean fragment masses and their variances for each trajectory that leads to a damped reaction. A weighted sum over angular momentum was constructed, using the measured cross-sections. The resulting mass distributions are plotted together with the experimental data in Fig. 9. The dashed-line histograms give the HICOL predicted mass-distributions. Although the general trend of the data is reproduced, a quantitative agreement is not achieved. The code under-predicts the deep inelastic scattering and over-predicts the yield at the symmetric mass division for both energies. Since the mass drift is governed by the window friction, and in HICOL the relative contributions of window-plus-wall and mono-wall friction are mixed with a form factor determined from the shape of the system along the trajectory, as discussed above, a variation of the form factor was considered in this work, in order to reproduce the experimental results.
Allowing for the window friction to contribute for larger R neck by changing the limits of x n from 0.8 ≤ x n ≤ 1 to 0.95 ≤ x n ≤ 1 gave a satisfactory result for the 418 MeV data and a trend in the right direction for the 383 MeV data (the solid-line histograms in the figure). Further increase in the lower limit of x n still did not reproduce the large deep inelastic cross section measured at E lab = 383 MeV.
The discrepancy between HICOL and the data in the deep inelastic channel was addressed by Feldmeier [7] and was attributed to the fact that the narrowly-peaked phase space approximation is not strictly valid for the deep inelastic trajectories. The mass variance is under-predicted in this region, which limits the partial waves that can contribute to the cross-section in the near-projectile and target mass bins. Including fluctuations in the trajectory model will probably improve the agreement with the data. The code gives a reasonable description of the 418 MeV data, in which the deep inelastic events comprise ≈ 1/4 of the total touch cross-section. At the lower energy, where almost half of the touch cross-section is in the deep inelastic channel, only the shape of the distribution is reproduced by the calculations.
Comparison of the E lab = 418 MeV mass distribution data with other theoretical descriptions from the literature is presented in Fig. 10. In the work by Shen et.al. [6], it was found that the full window-plus-wall dissipation gave reaction times which were too long and inconsistent with the time scales derived from the mass-angle correlations. The authors came to the conclusion that the window friction should not be included in the calculation.
In Fig. 10 (a), a calculation without the window friction is presented. It differs dramatically from the data, since the window friction is the main mechanism that prevents the fast equilibration of the mass asymmetry. Fig. 10 (b), shows a calculation done using Wilczynski's [11] prescription for scaling the friction along the trajectory, which was already discussed in Section IV B 2. Although this calculation reproduces the mass distribution, it does not reproduce the mass-angle correlations, since the reaction times are longer than the rotational period of the system (See Section IV B 2).
A frequently used approach to implement friction in various descriptions of heavy-ion reactions is to use window-plus-wall dissipation during the whole reaction, but scaled by a factor≈ 0.3, which was determined from fitting experimental fission fragment TKE distributions and giant resonance widths [24]. Figure 10 (c) shows the calculated mass distribution using this approach, in comparison to the 418 MeV data. With the window friction present during the whole trajectory, even if it is scaled down, the mass drift towards symmetry is hindered significantly, resulting in disagreement with the data.
Using a shape dependent friction along the whole trajectory, as it was done in HICOL, reproduces in a self-consistent way the mass distributions ( Fig.9(a) The results are plotted in Fig. 11. The contour levels are listed in the figure. A forwardbackward asymmetry, which reveals the short time scale of the reaction, is clearly evident for both energies measured. A two-body friction mechanism with 1/T 2 temperature dependence, as suggested in Ref. [11], would require friction scaling factors of k out the neutron measurement, τ = 18 ± 9 × 10 −21 s [9]. In contrast Wilczynska's calculation [10], extracted τ = 80 +55 −25 × 10 −21 s from the same data. Table III gives the HICOL results obtained for the average trajectories contributing in symmetric quasi-fission for the two energies measured.
The two completely different physics approaches of Ref. [9] and this work have yielded similar results for the reaction time scale, which indicates that the one-body friction implemented in HICOL gives a satisfactory description of the reaction mechanism. However, a dynamical calculation of the pre-scission particle emission is necessary, in order to relate the pre-scission neutron multiplicity to the viscosity. Such calculations were done in this work and are presented in Appendix A. After adjusting the statistical model parameters, so that both the measured pre-scission neutron multiplicities and mean neutron kinetic energies are reproduced, the time scale derived from our dynamical calculation is in agreement with the original "static" results of Ref. [9].

V. ONE-BODY DISSIPATION LIMITS
To determine the experimental limits to the one-body viscosity, implemented in HICOL, calculations were done using scaling factors to the friction tensor ranging from k s = 0.5 to k s = 4 and tested versus the experimental mass and angular distributions. These factors were kept constant along the whole trajectory, thus preserving the original shape dependence of friction, which would be destroyed if different scaling factors were used in different parts of the trajectory.
The results of the calculations are plotted in Fig.12. Scaling factors k s ≥ 1.7 lead to scattering angles ≥ 180 o and orbiting type angular distributions, which are inconsistent with the data. In the lower limit, k s ≤ 0.8, there is considerable discrepancy between the calculated and the measured mass distributions. This analysis sets the upper and lower limits to the one-body friction scaling factors for the temperature range of the quasi-fission reactions studied in this work. The two measurements in the present study cover the temperature range of the pre-scission neutron data from Ref. [9]. In contrast to the calculations in Ref. [11] which found that scaling factors k s = 4 − 12, varying with temperature, were needed to describe the data in Ref. [9], we consider both our measurements are consistent with one-body viscosity. Supporting this conclusion, Feldmeier in [7] analyzed the 238 U+ 48 Ca at E lab =5.9 MeV/u data from Shen et.al. [6], and reproduced the measured mass distributions with his code. A pre-scission GDR γ-ray measurement and analysis [25] in the quasi-fission reaction of 58 Ni+ 165 Ho at E lab = 368 MeV was also consistent with one-body friction.
There have been a number of studies in fusion-fission and fusion-evaporation reactions based on the statistical model, which show the need for dissipation in the fission degree of freedom. The systematic studies by the Stony Brook group (See Ref. [12] and references therein) on pre-scission GDR γ− rays indicates that viscosity increases with temperature and is larger than the one-body value. Some of these results were used in Ref. [11] and together with the pre-scission neutron multiplicity calculations discussed above were interpreted as evidence for strong two-body viscosity setting in at temperature T ≤ 2 MeV.
On the other hand, other studies from pre-scission particle multiplicities (e.g.Ref. [26,27]) and GDR γ-rays(e.g.Ref. [28]) in the same temperature range do not find evidence for such large viscosity. A recent measurement of Morton et.al. [29] of GDR γ-ray multiplicities for the 32 S+ nat W, 208 Pb reactions also found a delay time consistent with one-body dissipation and γ-fission angular correlations that could be described without temperature dependent viscosity. At present, the dissipation mechanisms in fusion-fission reactions and the temperature at which the two-body dissipation sets in is still a subject of considerable debate.

VI. CONCLUSIONS
The present study concentrated on the quasi-fission reaction dynamics, exploiting the unique observables available in this type of reactions: mass-angle correlations and mass distributions. It aimed to clarify and reconcile the data and the theoretical descriptions of nuclear viscosity that were available prior to this. We find that two different experimental In the Wilczynska et.al. [10] approach, HICOL's trajectory was divided into time steps.
A statistical model calculation was applied for each step, replacing the excitation energy with HICOL's dissipated energy minus the energy lost in evaporation from the previous step.
A similar code, was developed independently in this work using a somewhat different approach. Instead of dividing the trajectory into time steps, a continuous evaporation process was assumed, following Refs. [30,31]. The results of Refs. [10,11] could be reproduced with our code DIFHEAT.
Calculating the accumulated neutron multiplicity along the L=40h trajectory in 64 Ni+ 197 Au at E lab =418 MeV, using a level density parameter a = A/9 MeV −1 (the value used in the neutron measurement), one finds that the calculated pre-scission multiplicity does not reach the experimental value. In Refs. [10,11], the authors concluded that the reaction time predicted by HICOL using one-body friction is too short to be able to accommodate the measured pre-scission neutron multiplicities.
A different approach to reconcile the measured pre-scission neutron multiplicity with the dynamical trajectory calculation was applied in this work. It appears that the discrepancy between Hinde's et.al. [9] and Wilczynski's et.al. [10,11] calculations does not stem from the "static" versus "dynamical" approach, but rather reflects the different nuclear temperature achieved in the two cases. In Hinde's calculation the level density parameter was kept fixed at the value a = A/9 MeV −1 , but the excitation energy of the system was treated as a free parameter 1 and adjusted in order to reproduce both the measured pre-scission neutron multiplicity and spectral shapes. Although routinely in statistical model calculations the level density parameter is varied, while the excitation energy is fixed, as Hinde pointed out, both observables are sensitive to the nuclear temperature T = E * /a, which contains the ratio of the two quantities. The authors of Ref. [10] argued that the dynamically calculated excitation energy, as predicted by HICOL 2 should be used. Keeping the level density parameter at the same value of a = A/9 MeV −1 , but using lower excitation energy along the trajectory (E * ∼ 100 MeV) results in lower nuclear temperature. Consequently, this calculation yields longer neutron lifetimes and, in addition, does not reproduce the mean neutron kinetic energy.
The dashed curves in Fig. 13 show the neutron multiplicity and the mean neutron kinetic energy along the trajectory, calculated with a = A/9 MeV −1 and an input heat curve obtained by scaling the friction in HICOL by a factor k s = 4. The shaded regions indicate the experimental multiplicity and mean kinetic energy values reported in Ref. [9].
Since HICOL indeed provides a more consistent description of the excitation energy, than the fitting procedure used in Ref. [9], in order to produce the same temperature (T≈ 2.2 MeV), as derived from the neutron spectra measured by Hinde, we need to vary the level density parameter. From Fig. 6(c) we find that E * ≈ 100 MeV, and we estimate that a = A/12.7 MeV −1 should reproduce both the neutron lifetimes and mean kinetic energy.
The solid lines in the Fig. 13 show the result from DIFHEAT, using a level density parameter a = A/13 MeV −1 and a one-body dissipation heat curve coming from HICOL.
Unlike the conclusion in Ref. [11], that large friction is necessary to reproduce the experimental pre-scission neutron multiplicities, we find that the reaction time extracted from the neutron data ∼ 20×10 −21 s is consistent with HICOL's predictions and with the evaporation calculation using DIFHEAT, if the statistical model parameters are properly constrained.
While HICOL is free of adjustable parameters, in any statistical model calculation one needs at least two observables, to be able to constrain the multi-parameter model calculation. It is beyond the scope of this work to investigate the influence of all statistical model parameters on the results from DIFHEAT. There is a significant deformation dependence both in the particle transmission coefficients and their binding energies [32,26]. Accounting for these effects might change the value of the level density parameter needed to describe the data.
However, we note that values as small as a ∼ A/13 MeV −1 have been previously deduced from particle spectral shapes in heavy hot systems [33,34].