Beyond the proton drip line: Bayesian analysis of proton-emitting nuclei

Background: The limits of the nuclear landscape are determined by nuclear binding energies. Beyond the proton drip lines, where the separation energy becomes negative, there is not enough binding energy to prevent protons from escaping the nucleus. Predicting properties of unstable nuclear states in the vast territory of proton emitters poses an appreciable challenge for nuclear theory as it often involves far extrapolations. In addition, significant discrepancies between nuclear models in the proton-rich territory call for quantified predictions.

Purpose: With the help of Bayesian methodology, we mix a family of nuclear mass models corrected with statistical emulators trained on the experimental mass measurements. We study the impact of such model mixing in the proton-rich region of the nuclear chart.

Methods: Separation energies were computed within nuclear density functional theory using several Skyrme and Gogny energy density functionals. We also considered mass predictions based on two models used in astrophysical studies. Quantified predictions were obtained for each model using Bayesian Gaussian processes trained on separation-energy residuals and combined via Bayesian model averaging.

Results: We obtained a good agreement between averaged predictions of statistically corrected models and experiment. In particular, we quantified model results for one- and two-proton separation energies and derived probabilities of proton emission. This information enabled us to produce a quantified landscape of proton-rich nuclei. The most promising candidates for two-proton decay studies have been identified.

Conclusions: The methodology used in this work has broad applications to model-based extrapolations of various nuclear observables. It also provides a reliable uncertainty quantification of theoretical predictions.

Figure 1

$Q_{2p}$ values for the five experimentally known $2p$ emitters calculated with the eleven global mass models with statistical correction obtained with GP $\left(\mu \ne 0\right)$ trained on the $\text{AME}2016+$ dataset. Error bars on theoretical results are one-sigma credible intervals computed with GP extrapolation. Theoretical results are listed in the following order: Skyrme models ${\mathrm{SkM}}^{*}$, SkP, SLy4, SV-min, UNEDF0, UNEDF1, and UNEDF2; new models BCPM and D1M; global mass models FRDM-2012 and HFB-24; and model mixing results BMA-0, BMA-I, BMA-II, and BMA-III. Experimental values are shown for comparison.

Figure 2

Uni- and bivariate distributions of the $50000$ MCMC posterior samples for the four-dimensional parameter vector of our Bayesian statistical model [Eqs. (3) and (4)] for UNEDF1 (upper triangle, red) and SLy4 (lower triangle, blue) using the training $S_{2p}$ dataset of even-even nuclei from AME2003. The parameters are plotted relative to their mean values (corresponding to zero on the plots). The diagonal plots show the univariate sample distributions of the GP parameters (histograms) as well as the KDE estimates (lines). Posterior mean and standard deviation are indicated by numbers. The off-diagonal plots show the KDE estimates of the bivariate posterior distributions of the GP parameter samples (color map) as well as the $95\%$ (outer green line) and $50\%$ (inner purple line) HDRs. The numbers mark the posterior correlation coefficients.

Figure 3

The quantified nuclear binding-energy landscape in the proton-rich region obtained in BMA-I (top) and BMA-II (bottom) model averaging calculations. The color marks the probability $p_{\mathrm{ex}}$ that these nuclei are bound with respect to proton decay. For each proton number, $p_{\mathrm{ex}}$ is shown along the isotopic chain versus the relative neutron number $N_0\left(Z\right)-N$, where $N_0\left(Z\right)$, listed in Tables 4 and 5, is the neutron number of the lightest proton-bound isotope for which an experimental one- or two-proton separation energy value is available. The domain of nuclei that have been experimentally observed (both proton-bound and proton-unbound) is marked by open stars; those within FRIB's experimental reach are marked by dots. See text for details.

Figure 4

Probability of true $2p$ emission for the even-$Z$ proton-rich isotopes. The color gives the posterior probability of $2p$ emission, i.e., that $S_{2p}<0$ and $S_{1p}>0$, according to the posterior average models. For each proton number, the relative neutron number $N_0\left(Z\right)-N$ is shown, where $N_0\left(Z\right)$, listed in Tables 4 and 5, is the neutron number of the lightest proton-bound isotope for which an experimental two-proton separation energy value is available. The dotted line represents the predicted drip line (corresponding to $p_{ex}=0.5)$. The domain of nuclei which have been experimentally observed is marked by stars (the experimentally observed $2p$ emitters ${}^{45}\mathrm{Fe},{}^{48}\mathrm{Ni},{}^{54}\mathrm{Zn}$, and ${}^{67}\mathrm{Kr}$ are indicated by closed stars); those within FRIB's experimental reach are marked by dots. See text for details.

Figure 5

$Q_{2p}$ values predicted in BMA-I for even-even isotopes with $16\le Z\le 80$. The thick solid lines mark the lifetime range (14). The mass numbers of selected isotopes are shown. The nuclei with the probability $p_{2p}>0.4$ are indicated by dots. Here, we used this value of $p_{2p}$ rather than $p_{2p}>0.5$ because the criterion (13) of the true $2p$ emission is slightly more restrictive than the energy criterion previously adopted in Ref. [43].