Microscopic calculation of fission product yields with particle-number projection

Fission fragments' charge and mass distribution is an important input to applications ranging from basic science to energy production or nuclear nonproliferation. In simulations of nucleosynthesis or calculations of superheavy elements, these quantities must be computed from models, as they are needed in nuclei where no experimental information is available. Until now, standard techniques to estimate these distributions were not capable of accounting for fine-structure effects, such as the odd-even staggering of the charge distributions. In this work, we combine a fully microscopic collective model of fission dynamics with a recent extension of the particle number projection formalism to provide the highest-fidelity prediction of the primary fission fragment distributions for the neutron-induced fission of ${}^{235}\mathrm{U}$ and ${}^{239}\mathrm{Pu}$. We show that particle-number projection is an essential ingredient to reproduce odd-even staggering in the charge yields and benchmark the performance of various empirical probability laws that could simulate its effect. This new approach also enables for the first time the realistic determination of two-dimensional isotopic yields within nuclear density functional theory.

Figure 1

Potential energy surface for ${}^{240}\mathrm{Pu}$ (top) and ${}^{236}\mathrm{U}$ (bottom) as a function of the axial quadrupole and axial octupole moments.

Figure 2

Probability $\mathbb{P}\left(Z_{\mathrm{R}}|Z,\mathit{q}\right)$ of having $Z_{\mathrm{R}}$ protons in the right fragment given the total number of protons $Z$ as a function of $Z_{\mathrm{R}}$. We show such a probability distribution for a set of scission configurations $\mathit{q}$ in ${}^{236}\mathrm{U}$. The deformations $\mathit{q}$ are in b ($q_{20}$) and ${\mathrm{b}}^{3/2}$ ($q_{30}$), where b = barn.

Figure 3

Charge distribution $Y\left(Z\right)$ in ${}^{240}\mathrm{Pu}$ (top) and ${}^{236}\mathrm{U}$ (bottom) as a function of $Z$. The yields obtained with PNP are compared to a Gaussian convolution of the raw flux with $\sigma =1.6$ and to the experimental data from Refs. [38, 39]; see text for additional details.

Figure 4

Mass distribution $Y\left(A\right)$ in ${}^{240}\mathrm{Pu}$ (top) and ${}^{236}\mathrm{U}$ (bottom) as a function of $A$. Results obtained after PNP are compared to a Gaussian convolution of the raw flux with $\sigma =4.0$ and to experimental data from Refs. [48, 49, 50, 51, 52, 53, 54, 55, 56]. The curve labeled $\mathrm{PNP}+\mathrm{Exp}$. resolution gives the PNP results with an additional convolution by a Gaussian with a FWHM of 5.5 mass units to account for the typical mass resolution of experiments; see text for additional details.

Figure 5

Two-dimensional isotopic fission yields $Y(Z,A)$ for the reaction ${}^{239}\mathrm{Pu}(n,f)$ for an incoming neutron energy of $E_n=1.0$ MeV using PNP in the fragments. Our results predict a deviation from the unchanged charge distribution (UCD) approximation, called the charge-polarization of the fission fragment distributions, in the primary yields. To give a point of reference, we also show the independent two-dimensional yields evaluated in the JEFF-3.3 library. Note that these yields account for the mass number of the fragments after emission of the prompt neutrons (in contrast to our predictions that give the primary fragment yields).

Figure 6

Average neutron excess of fragments produced in the fission of ${}^{240}\mathrm{Pu}$. PNP results at $E_n=2.0$ MeV are compared with the local discrete approach of Sec. 3c and with the experimental data of Ref. [66].

Figure 7

Light peak of the charge distribution $Y\left(Z\right)$ in ${}^{240}\mathrm{Pu}$ as a function of $Z$ for different analytic forms for $\mathbb{P}\left(Z_{\mathrm{R}},N_{\mathrm{R}}|Z,N,\mathit{q}\right)$ as well as from PNP. We only show the curve for the light fragment, since the whole distribution is completely symmetric with respect to $Z/2$; see text for additional details.

Figure 8

Light peak of the fragment charge distribution $Y\left(Z\right)$ for ${}^{240}\mathrm{Pu}$ at $E_n=1.0$ MeV obtained with different values of the frontier definition $q_{\mathrm{N}}^{\mathrm{sciss}}$.

Figure 9

Isotopic yields $Y(Z_{\mathrm{f}},A_{\mathrm{f}})$ for the reaction ${}^{239}\mathrm{Pu}(n,f)$ for different incoming neutron energies $E_n$ using PNP in the fragments. Symmetric fission becomes more probable with increasing excitation energy.

Figure 10

Light peak of the fragment charge distribution $Y\left(Z\right)$ for ${}^{240}\mathrm{Pu}$ obtained with different values of initial energy of the compound nucleus. Results with PNP are compared to the experimental data from Ref. [46].

Figure 11

Evolution of the neutron excess of the primary fission fragments for the reaction ${}^{239}\mathrm{Pu}(n,f)$ according to $E_n$. If the UCD were satisfied, all these quantities would be constant equal to 1.55.