Two-proton emission systematics

The simultaneous emission of two protons is an exotic and complex three-body process. It is very important for experimental groups investigating the nuclear stability on the proton drip line to have a simple rule predicting the two-proton decay widths with a reasonable accuracy for transitions between ground as well as excited states in terms of relevant physical variables. In spite of its complexity, we show that the two-proton emission process obeys similar rules as for binary emission processes like proton, $\alpha$, and heavy cluster decays. It turns out that the logarithm of the decay width, corrected by the centrifugal barrier, linearly depends upon the Coulomb parameter within one order of magnitude. On the other hand, the universal linear dependence with a negative slope between the logarithm of the reduced width and the fragmentation potential, valid for any kind of binary decay process, is also fulfilled for the two-proton emission with a relative good accuracy. As a consequence of pairing correlations the two protons are simultaneously emitted from a spin singlet paired state. We evidence that indeed one obtains a linear dependence between the logarithm of the reduced width and pairing gap within a factor of two, giving a good predictive power to this law. It turns out that the two-proton and alpha-cluster formation probabilities have similar patterns versus the pairing gap, while in the one-proton case one has a quasiconstant behavior.

Figure 1

(a) Reduced radius versus Coulomb parameter. (b) Logarithm of the monopole decay width versus the Coulomb parameter. The fit parameters are given in the first line of Table 2.

Figure 2

Logarithm of the experimental reduced width at the geometrical touching radius $R=1.2(A_D^{1/3}+2^{1/3})$ versus the fragmentation potential (a) and pairing gap (b). The fit parameters are given in the second line of Table 2. The rms error in parentheses corresponds to the analysis without considering cases 4 and 13.

Figure 3

Logarithm of the experimental reduced width versus the pairing gap for one-proton emission (a) and $\alpha$ decay from even-even emitters (b).

Figure 4

Logarithm of the experimental decay width versus ${log}_{10}{\mathrm{\Gamma }}_1={log}_{10}{P}_L+a_1{V}_{\mathrm{frag}}+b_1$ (a) and ${log}_{10}{\mathrm{\Gamma }}_2={log}_{10}{P}_L+a_2\mathrm{\Delta }+b_2$ (b). The fit parameters are given in the third line of Table 2. The rms error in parentheses corresponds to the analysis without considering cases 4 and 13.