Fission fragment charge and mass distributions in ${}^{239}\mathrm{Pu}\left(n,f\right)$ in the adiabatic nuclear energy density functional theory

Background: Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the $r$ process to fuel cycle optimization for nuclear energy. The need for a predictive theory applicable where no data are available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics.

Purpose: In this work, we calculate the pre-neutron emission charge and mass distributions of the fission fragments formed in the neutron-induced fission of ${}^{239}\mathrm{Pu}$ using a microscopic method based on nuclear density functional theory (DFT).

Methods: Our theoretical framework is the nuclear energy density functional (EDF) method, where large-amplitude collective motion is treated adiabatically by using the time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). In practice, the TDGCM is implemented in two steps. First, a series of constrained EDF calculations map the configuration and potential-energy landscape of the fissioning system for a small set of collective variables (in this work, the axial quadrupole and octupole moments of the nucleus). Then, nuclear dynamics is modeled by propagating a collective wave packet on the potential-energy surface. Fission fragment distributions are extracted from the flux of the collective wave packet through the scission line.

Results: We find that the main characteristics of the fission charge and mass distributions can be well reproduced by existing energy functionals even in two-dimensional collective spaces. Theory and experiment agree typically within two mass units for the position of the asymmetric peak. As expected, calculations are sensitive to the structure of the initial state and the prescription for the collective inertia. We emphasize that results are also sensitive to the continuity of the collective landscape near scission.

Conclusions: Our analysis confirms that the adiabatic approximation provides an effective scheme to compute fission fragment yields. It also suggests that, at least in the framework of nuclear DFT, three-dimensional collective spaces may be a prerequisite to reach 10% accuracy in predicting pre-neutron emission fission fragment yields.

Figure 1

Potential-energy surfaces obtained with (a) the ${\mathrm{SkM}}^{*}$ and (b) D1S EDF in axial symmetry. The red line separates configurations with $Q_N>4$ from the others. The curvilinear abscissa $\xi$ starts at the symmetric-scission points and runs along the frontier (red line). Values of $\xi$ are indicated along the scission line.

Figure 2

Static properties of the system along the frontier of the domain defined by $Q_N>1$. The position on the frontier is labeled by the curvilinear abscissa $\xi$ starting from the symmetric point $\left(Q_{30}=0\right)$. The ${\mathrm{SkM}}^{*}$ abscissa is shifted by 10 arbitrary units so that the main structures match D1S. The HFB energy is expressed in MeV.

Figure 3

Gaussian neck operator values in the vicinity of the exit of the asymmetric fission valley of the D1S PES. Valley A corresponds to the main asymmetric valley whereas region B corresponds to a different valley of the full Hilbert space projected in between the postscission configurations and valley A. The point P corresponds to abscissa $\xi \simeq 100$ on the frontier defined by $Q_N>1$.

Figure 4

Static properties of the system along a frontier of the domain defined by $Q_N>4$. The position on the frontier is labeled by the curvilinear abscissa $\xi$ starting from the symmetric point $\left(Q_{30}=0\right)$. The ${\mathrm{SkM}}^{*}$ abscissa is shifted by 20 arbitrary units so that the main structures match. The HFB energy is expressed in MeV and the multipole moments in powers of $b$.

Figure 5

Evolution of the ${\mathrm{SkM}}^{*}$ heavy-mass yields as a function of time for three possible fragmentations: $A_H=120$ (symmetric valley), $A_H=137$ (asymmetric peak), and $A_H=145$ (very asymmetric wing of the distribution).

Figure 6

Pre-neutron mass yields for ${}^{239}\mathrm{Pu}\left(n,f\right)$. The ${\mathrm{SkM}}^{*}$ and D1S calculations are compared with two experimental datasets [55, 56]. The data from Nishio et al. are plotted with their statistical uncertainties.

Figure 7

Charge yields for ${}^{239}\mathrm{Pu}\left(n,f\right)$. The calculation obtained with a ${\mathrm{SkM}}^{*}$ functional (plain line) is compared with the European evaluation JEFF-3.1 (dashed line).

Figure 8

Pre-neutron mass yields for ${}^{239}\mathrm{Pu}\left(n,f\right)$ obtained from the two different initial states described in Sec. 2c1. Calculations were performed with both ${\mathrm{SkM}}^{*}$ and D1S potential-energy surfaces. Only the heavy-mass region is represented.

Figure 9

Pre-neutron mass yields for ${}^{239}\mathrm{Pu}\left(n,f\right)$ obtained in the strict TDGCM and quantized ATDHFB formalisms. Calculations were performed with both ${\mathrm{SkM}}^{*}$ and D1S potential-energy surfaces by using the same prescription for the initial state. Only the heavy-mass region is represented.

Figure 10

Sensitivity of the pre-neutron mass yields for ${}^{239}\mathrm{Pu}\left(n,f\right)$ to the width of the Gaussian convolution. Calculations are based on the strict TDGCM formalism.