Dependence of two-proton radioactivity on nuclear pairing models

Sensitivity of two-proton emitting decay to nuclear pairing correlation is discussed within a time-dependent three-body model. We focus on the ${}^6\mathrm{Be}$ nucleus assuming $\alpha +p+p$ configuration, and its decay process is described as a time evolution of the three-body resonance state. For a proton-proton subsystem, a schematic density-dependent contact (SDDC) pairing model is employed. From the time-dependent calculation, we observed the exponential decay rule of a two-proton emission. It is shown that the density dependence does not play a major role in determining the decay width, which can be controlled only by the asymptotic strength of the pairing interaction. This asymptotic pairing sensitivity can be understood in terms of the dynamics of the wave function driven by the three-body Hamiltonian, by monitoring the time-dependent density distribution. With this simple SDDC pairing model, there remains an impossible trinity problem: it cannot simultaneously reproduce the empirical $Q$ value, decay width, and the nucleon-nucleon scattering length. This problem suggests that a further sophistication of the theoretical pairing model is necessary, utilizing the two-proton radioactivity data as the reference quantities.

Figure 1

Schematic figure of level scheme, in which the correlated 2p emission becomes dominant.

Figure 2

The density distribution for the initial 2p state obtained with the surface SDDC pairing interaction. (i) Top panel: the distribution as a function of $r_{\mathrm{p}\text{−}\mathrm{p}}$ and $r_{\mathrm{c}\text{−}\mathrm{pp}}$. (ii) Bottom panel: the angular distribution as a function of the opening angle, ${\theta }_{12}$, between two protons. This is obtained by integrating $\rho (r_1,r_2,{\theta }_{12})$ for the radial coordinates, $r_1$ and $r_2$. The spin singlet and spin-triplet components are also plotted.

Figure 3

Time-dependent 2p-density distribution plotted as a function of $r_{\mathrm{p}\text{−}\mathrm{p}}$ and $r_{\mathrm{c}\text{−}\mathrm{pp}}$. The surface SDDC pairing interaction is employed.

Figure 4

Time-dependent 2p-density distribution of the decay state ${\rho }_d\left(t\right)$, given by Eq. (16). The surface SDDC pairing interaction is employed. These are plotted in the same manner as in Fig. 2.

Figure 5

(i) Top panel: survival probability, $1-N_d\left(t\right)$, obtained with the surface SDDC pairing interaction. For a comparison, the same result but with the finite-range Minnesota pairing is taken from Ref. [33]. These are plotted in logarithmic scale. (ii) Bottom panel: 2p-decay width of ${}^6\mathrm{Be}$ calculated with the surface SDDC and the finite-range Minnesota pairing interactions. In SDDC case, the spin-singlet ($S_{12}=0$) and spin-triplet ($S_{12}=1$) widths are both plotted. Experimental result, ${\mathrm{\Gamma }}_{2\mathrm{p}}=92\pm 6$ keV, is indicated by the shaded area [38].

Figure 6

Time-invariant discrete energy distribution $d\left(E_N\right)$ obtained with the surface SDDC pairing interaction. Its continuous distribution, plotted in an arbitrary scale, is obtained by smearing $d\left(E_N\right)$ with a Cauchy-Lorentz function, whose full width at the half maximum is 0.1 MeV. The same plot for the Minnesota pairing case is also displayed.

Figure 7

(i) Top panel: density dependence of the SDDC pairing interaction $w\left(r\right)$ in the default (surface), steep, and smooth cases. The core-proton potential is also plotted in the arbitrary scale. (ii) Bottom panel: corresponding result of 2p-decay width.

Figure 8

(i) Top panel: The radial strength $w\left(r\right)$ for three SDDC pairing potentials, $v_{\mathrm{p}\text{−}\mathrm{p}}^{\left(N\right)}({\mathit{r}}_1,{\mathit{r}}_2)=w\left(r_1\right)·\delta ({\mathit{r}}_1-{\mathit{r}}_2)$. (ii) Middle and bottom panels: The one-proton density-distribution at $ct=0$ and 1000 fm, respectively. These distributions are plotted in an arbitrary scale. Confining and original potentials in the $\left(p_{3/2}\right)$ channel of $\alpha \text{−}p$ subsystem are also plotted.

Figure 9

2p-decay width of ${}^6\mathrm{Be}$ obtained with the default, volume, and emitter SDDC pairing models.

Figure 10

Time-dependent 2p-decay width of ${}^6\mathrm{Be}$ obtained with several Minnesota-pairing forces.