Langevin model of low-energy fission

Abstract

Background: Since the earliest days of fission, stochastic models have been used to describe and model the process. For a quarter century, numerical solutions of Langevin equations have been used to model fission of highly excited nuclei, where microscopic potential-energy effects have been neglected. Purpose: In this paper I present a Langevin model for the fission of nuclei with low to medium excitation energies, for which microscopic effects in the potential energy cannot be ignored. Method: I solve Langevin equations in a five-dimensional space of nuclear deformations. The macroscopicmicroscopic potential energy from a global nuclear structure model well benchmarked to nuclear masses is tabulated on a mesh of approximately 10 7 points in this deformation space. The potential is defined continuously inside the mesh boundaries by use of a moving five-dimensional cubic spline approximation. Because of reflection symmetry, the effective mesh is nearly twice this size. For the inertia, I use a (possibly scaled) approximation to the inertia tensor defined by irrotational flow. A phenomenological dissipation tensor related to one-body dissipation is used. A normal-mode analysis of the dynamical system at the saddle point, and the assumption of quasi-equilibrium, provide distributions of initial conditions appropriate to low excitation energies, and is extended to model spontaneous fission. A dynamical model of post-scission fragment motion including dynamical deformations and separation allows the calculation of final mass and kinetic-energy distributions, along with other interesting quantities. Results: The model makes quantitative predictions for fragment mass and kinetic-energy yields, some of which are very close to measured ones. Varying the energy of the incident neutron for induced fission allows the prediction of energy dependences of fragment yields and average kinetic energies. With a simple approximation for spontaneous fission starting conditions, quantitative predictions are made for some observables which are close to measurements. Conclusions: This model is able to reproduce several mass and energy yield observables with a small number of physical parameters, some of which do not need to be varied after benchmarking to 235 U(n,f) to predict results for other fissioning isotopes.

Unitary evolution with fluctuations and dissipation

“…the collective motion over-damped [34,48]. The great advantage of this extension of the TDDFT formalism, as opposed to using classical Langevin equations [2][3][4][5], is that the present formalism is fully quantum, it includes all collective degrees of freedom, the evolution is unitary in spite of including explicitly dissipation, and it allows for all symmetries to be broken during the evolution, as expected for example in a full path-integral description of the dynamics of an interacting many-fermion system [50,51]. Finally, the method is general, and it can be applied to the description of other observables.…”

“…The solution of the quantum Lindblad equation (5) or of its Monte Carlo wave function formulation [11][12][13][14][15][16][17][18][19][20][21][22][23] can turn into a formidable problem in cases of interest in nuclear physics. In the quantum optics the dimensionally of the Hilbert spaces of interest is small ≈ O(1).…”

Analysis of the total kinetic energy of fission fragments with the Langevin equation

“…The potential energy surface U (q) along which the shape evolves according to Eqs. (7) is calculated by the macroscopic-microscopic approach [29][30][31],…”

Energy Dependence of Fission-fragment Neutron Multiplicity in $^{235}$U$(n,f)$

“…Refs. [14,15,16]), which require evaluating a potential energy surface, a collective inertia tensor, and a dissipation tensor. In the idealized limit of strong dissipation, the shape evolution can be simulated by a Metropolis walk [17] on the multi-dimensional potentialenergy surface.…”