Fluctuation of fission observables investigated with a Monte Carlo method

Nuclear fission dynamics described within nuclear energy density functional frameworks (EDF) have seen substantial advances in the last decade. Part of this success stems from projection techniques, which allow the computation f probability distribution functions (pdf) for selected observables such as particle number and angular momentum of the fragments. Predicting the pdf of other observables, such as the total kinetic energy of the fragments, remains undone. This work proposes a method to determine the complete pdf of a new category of observables from a Bogoliubov vacuum projected onto good particle number. It relies on sampling nucleonic configurations in coordinate and intrinsic-spin representation. We assess the feasibility and convergence properties of the method and apply it to states representative of the scission of an actinide. Fluctuations in fragment shapes, inter-fragment Coulomb and nuclear interaction as well as the corresponding torques are analyzed. We find that a significant fraction of the fluctuation of several measured fission observables is already present within the mean-field picture.

💡 Research Summary

The paper introduces a Monte Carlo framework for extracting full probability distribution functions (pdfs) of a broad class of fission observables directly from particle‑number‑projected Bogoliubov vacuum states. Traditional energy‑density‑functional (EDF) approaches have succeeded in predicting mean values and, via projection techniques, pdfs for one‑body quantities such as fragment mass, charge, and angular momentum. However, two‑body observables like the total kinetic energy (TKE) of the fragments have remained out of reach.

The authors formulate the many‑body wave function in the position–intrinsic‑spin representation, denoting each nucleon by a coordinate (r\equiv(\mathbf{x},\sigma)). For a given Bogoliubov state (|\psi\rangle) the probability of a specific unordered configuration ({r_1,\dots,r_N}) is (p(r_1,\dots,r_N)=\frac{1}{N!}|\langle r_1\dots r_N|\psi\rangle|^2). This distribution is invariant under particle permutations and properly normalized. Sampling from this high‑dimensional pdf is achieved with a Metropolis‑Hastings Markov chain: at each step a single nucleon’s spatial coordinates are perturbed by a Gaussian of width (\sigma_{\text{space}}\approx1) fm and its spin is flipped with probability (P_{\text{flip}}\approx0.1). After a burn‑in period and appropriate thinning (jump interval), the chain yields statistically independent configurations.

Only observables diagonal in the position–spin basis are considered, which includes any one‑body operator (\hat T=\int d r, t(r),c^\dagger_r c_r) and any two‑body operator (\hat V=\frac12\iint d r d r’, v(r,r’),c^\dagger_r c^\dagger_{r’} c_{r’} c_r). For a configuration the corresponding kernels are simple sums over nucleons (for (\hat T)) or pairs (for (\hat V)). Because the kernel values are themselves random variables drawn from the sampled configurations, their empirical distribution coincides with the physical measurement pdf. Consequently, the method provides not only expectation values but the complete statistical shape, including higher moments, without encountering the sign problem typical of quantum Monte Carlo.

The probability amplitude (|\langle r_1\dots r_N|\psi\rangle|^2) for a projected Bogoliubov vacuum can be expressed as the squared determinant of a skew‑symmetric matrix (Z) built from canonical‑basis wave functions (\phi_k(r)) and BCS coefficients (u_k, v_k). The authors truncate the canonical basis to retain only states contributing more than (10^{-4}) particles, discretize space on a Lagrange mesh (box size up to 32 fm, cell size 0.5 fm), and approximate (\phi_k) as constant within each cell to accelerate evaluation.

Numerical tests are performed on two systems: the light nucleus (^{20})Ne and the heavy actinide (^{252})Cf near its most probable scission configuration. The HFB equations are solved with the Gogny D1S functional to generate the underlying Bogoliubov states. For (^{20})Ne (≈20 nucleons) the configuration space has ~600 dimensions, while for (^{252})Cf it exceeds 6000 dimensions. After careful tuning of burn‑in length, jump interval, and perturbation widths, the authors obtain thousands of independent samples for the light case and tens of thousands for the heavy case. Convergence of pdfs for fragment shapes (multipole moments), charge distributions, inter‑fragment Coulomb and nuclear potentials, and the associated torques is demonstrated.

The results reveal that sizable fluctuations of many experimentally relevant observables are already encoded in the mean‑field (projected Bogoliubov) wave function. In particular, the spread of the inter‑fragment Coulomb repulsion and the variation of fragment shapes generate a broad TKE distribution comparable to measured widths, indicating that quantum fluctuations at the mean‑field level account for a large fraction of the observed variance.

Finally, the complete algorithm is released as an open‑source C++ package named NucleoScope, enabling the community to apply the method to other fission scenarios, to incorporate additional observables (e.g., neutron or gamma emission), and to explore extensions such as odd‑particle systems or time‑dependent configurations. By providing a practical tool for full pdf extraction, the work bridges the gap between microscopic EDF calculations and the detailed statistical information required for modern fission data evaluation and reactor modeling.