Effects of microscopic transport coefficients on fission observables calculated by Langevin equation and its systematics

The Langevin dynamical description of fission observables is inspired by the random evolution of shape parameters across the potential surface. In these work, we shall use mass and friction tensors inspired from Linear Response Theory (microscopic transport coefficients) and obtain the fission observables associated with these calculations. We compare these microscopic results with calculations using hydrodynamical mass tensor and wall-window friction tensor (macroscopic transport coefficients). We are able to calculate the fission product yield, Coulomb kinetic energy and prescission kinetic energy from the Langevin calculation. This allows us to observe the systematic of average light and heavy mass fission product yield calculated using both microscopic and macroscopic calculations. We also compare the results of microscopic and macroscopic calculation total kinetic energy (TKE) with Viola’s TKE systematics. In the case of 236,239U compound nucleus, we do the microscopic calculation for several excitation energy up to 30 MeV and afterwards compare it to the TKE of experimental data and corresponding macroscopic TKE. Reasonable agreement of microscopic TKE to experiment is obtained which shows decreasing TKE with increasing excitation energy. Macroscopic TKE however, is independent of excitation energy and thus contrary to experimental data.


Introduction
Linear response theory [1][2][3][4] allowed us to obtain microscopic mass and friction tensor for a nucleus under perturbation such as in nuclear fission from the response function of these perturbations.The assumption is that the changes of nuclear shapes are due to the evolution of the shape as it travels randomly on a potential surface.Our potential is based on the two centre shell model [5] and the evolution is governed by Langevin equation.
Calculation of the Langevin equation for fission is done by solving the first order differential equations for collective coordinates, q µ = (z o /R 0 , δ, α) and momentum conjugate of the collective coordinates, p µ , The collective coordinates; z 0 is the elongation of the nucleus, R 0 the radius of spherical compound nucleus, δ is the deformation of fragments, and α is the mass asymmetry.We assume both the left deformation, δ 1 and right deformation, δ 2 is similar by setting δ = δ 1 = δ 2 .
a e-mail: usang.m.aa@m.titech.ac.jp; mark dennis@nm.gov.my We also kept the neck parameter constant at = 0.35.The trajectory is governed by the random force, g µν R ν (t).The random force is a product of white noise, R ν (t) and temperature dependent strength factor, g µν .The coefficient g µν is related to friction tensor and temperature via the Einstein relation We call this temperature the local temperature to differentiate it with the transport temperature, T tr which is used in the calculation of microscopic mass, m µν and friction tensor, γ µν .The microscopic mass and friction tensor calculation used in the current calculation are given and described by Ivanyuk [6] incorporates BCS approximations into the mean field Hamiltonian obtained in the current calculation using linear response theory at specific T tr .As for the local temperature, T , it is determined according to the internal energy , E int available to it.E int is related to the excitation energy, E x and level density parameter, a using the relationship, Before any Langevin calculation is done, we calculate microscopic mass [7] and friction tensor [8] at several T tr using linear response theory.The fission product yield for each T tr in the case of 236 U is shown in Fig. 1 several T tr to obtain mass and friction tensor at T tr equal to the local temperature, T .We perform our calculation at E x = 20 MeV and compare them with JENDL 14 MeV incident neutron data except in Sect.3.1 where the calculation is done for a range of excitation energy.
Previously [9], we have shown for 236 U detailed analysis of the fission observables calculated using Langevin equation with microscopic transport coefficients.

Mass yield systematics
In Fig. 2, excellent agreement can be seen of the yield using microscopic transport coefficients in comparison to macroscopic transport coefficients and JENDL fission product yield.The discrepancy at the peak and valley of the yield can be easily removed or at least reduced by adjusting shell corrections.In the current microscopic calculation, full shell and pair correction ( (E int ) sh, pair = 1) are used but we could optionally use Randrup & Möller [10] shell and pair correction formula, The formula above can be reduced to Ignatyuk shell correction [11] by setting e E 1 /E 0 = 0.The parameter E 0 = 15 MeV is interpreted as the shell damping energy, and E 1 is the energy shift, as explained in Ref. [10].As a note, the macroscopic calculation is done using E 1 = 30 MeV for both shell and pairing.The average mass of the light fragments, A L and average mass of the heavy fragments, A H in Fig. 3 shows that for many actinides, our calculation follows the systematic of 14 MeV incident neutron JENDL-4.0 fission product light and heavy mass yield averages with very little deviation, with either microscopic or macroscopic calculation.The current calculation shows that our results are consistent with the bimodal fission argued by Ohtsuki [12].
This mass distribution of light and heavy mass averages is first demonstrated by Flynn [13] but the mass distribution used by Flynn is determined from fission product yield due to thermal neutron, reactor neutron and spontaneous fission.Hence Flynn's mass distribution will have a slightly heavier mass averages at A H ≈ 140, in contrast to the 14 MeV incident neutron evaluated results used in Fig. 3 that has A H ≈ 135.Flynn's A L distribution against compound mass is also slightly steeper.

Total kinetic energy systematics
We define in our calculation the total kinetic energy (TKE) at scission as the sum of prescission kinetic energy E pre and Coulomb energy E Coul of the fission fragments.
T K E = E Coul + E pre . ( We calculate E Coul by assuming that the point charge of the fission fragments is proportional to its mass and is separated by a distance equal to the distance between the length of these fragments at scission.E pre on the other hand, is the kinetic energy of our trajectory for the elongation, Z z 0 at scission.This allows us to observe the TKE of our calculation and compare them with experiments, evaluated data and available systematics.We are however aware that our definition of TKE is different from Eq. ( 7) in the paper by Straede [15].Terrell [16] argues that if charge divides in the same ratio as the mass ratio, A L / A H , the kinetic energy should be proportional to Z 2 /r 0 A 1/3 .He showed through a linear square fit that the TKE is proportional to 0.121Z 2 A −1/3 .This is further demonstrated by Viola [14] by the linear equation given in 1966 and in 1985, In Fig. 4 we show that our calculations are within this systematic established by Viola and are very close to the TKE of several actinides given by JENDL.

Excitation energy dependence
Currently we are able to show that the TKE of 239,236 U compound nucleus at various excitation energy are roughly consistent with experimental results at various incident energy by assuming that the excitation energy of the compound nucleus for the actinides in experimental is approximately equal to the sum of the incident energy and neutron binding energy of the compound nucleus, B n (A c ) as in the following manner This is demonstrated for 236 U compound nucleus in Fig. 5 and on Fig. 6 in the case of 239 U compound nucleus.Influence and contributions from E pre and E Coul for each calculation to the TKE at each excitation energy can also be seen from the figures.E pre in general, tends to decay with increasing excitation energy while E Coul seems to fluctuate around a fixed constants.This indicates that the decay of TKE at increasing excitation energy is generally due to E pre component.
Our calculated TKE for 236 U slightly overestimates experimental data at low excitation energy but overestimate it at high excitation energy.For 239 U, our calculation underestimates the TKE.These deviations however, are very small as was illustrated in Fig. 5 and Fig. 6.We are also unable to show the onset of pair breaking at 1.5 MeV E n 2.0 MeV as indicated by Straede and his co-workers [15] at the moment but these could be due to insufficient scission events in our calculations to get a reliable enough statistics.

Conclusion
Observables of fission such as mass yield and TKE from the calculation of Langevin equation using microscopic transport coefficient at E x = 20 MeV gives a good approximation of the fission yield in comparison to evaluated data of JENDL-4.0 fission product yield 14 MeV incident neutron.This is demonstrated by the fitness of the microscopic calculation yields alongside evaluated data and the proximity of its heavy and light fragment mass averages to corresponding JENDL's mass fragment averages.The TKE's of these calculations also seem to adhere very well to the systematics of Viola's and JENDL's TKE.
Examination of the TKE's calculated at various excitation energy for 236,239 U shows that our results follow experimental results closely.Additionally, TKE trends for these two actinides to gradually decrease at higher excitation energy seems to be the contribution of E pre .E pre in our calculations experience a consistent decrease at increasingly higher excitation energy while E Coul seems to be fluctuating around a constant value with no clear indication of increasing or decreasing at higher excitation energy.
There are many improvements that could be done to the present result such as adjusting the pair and shell correction for each nuclide for a better fit to fission product yield or improving the statistics of our calculations for example.In other words, we are fairly optimistic that our calculation of the Langevin equation using microscopic transport coefficients could be further improved as more physics are introduced into the calculation.

Figure 2 .
Figure 2. Langevin calculation for compound nucleus 234 U, 236 U and 240 Pu using both microscopic and macroscopic transport coefficients fission fragment mass yield results.

Figure 3 .
Figure 3.Light and heavy mass average systematics for macroscopic and microscopic calculation in comparison to JENDL's 14 MeV incident neutron fission fragment light and heavy mass average.

Figure 4 .
Figure 4. TKE systematics in comparison to the systematic by Viola [14].