1 Introduction

Proton radioactivity, experimentally observed as a decay from the ground state, at GSI in 1981, has provided very important information on the structure of nuclei beyond the proton drip-line. The more complicated decaying mode, two-proton radioactivity, proposed 50 years ago in a classical article^[1] opened a new window to investigate nucleon-nucleon correlations and the structure of atomic nuclei. In 2002, the simultaneous emission of two-protons was for the first time observed in the decay of ⁴⁵Fe by Pfutzner, Giovinazzoin experiments at GSI and GANIL^[2,3]. Research in the field flourished after this breakthrough, and to date ⁵⁴Zn^[4], ⁴⁸Ni^[5], ¹⁹Mg^[6], ¹⁶Ne^[7], ¹⁷Ne^[8], ¹⁸Ne^[9], ¹⁰C^[10], ¹⁴O^[11] and ²⁹S^[12] have been found to exhibit two-proton emission. Several theoretical approaches such as Diproton model^[13,14], R-matrix approach^[15], continuum shell model^[16], adiabatic hyperspherical approach^[17], and the quantum three body cluster approach^[18], where the tunneling through the barrier is treated in a dynamical way, were applied to the problem.

There are two different decay modes for simultaneous two-proton emission: (1) three-body direct breakup involving an uncorrelated emission of the two protons, usually referred to as democratic emission. (2) ²He cluster emission where a pair of protons, correlated in a quasi-bound ¹S configuration, breakup, when emitted into two protons (diproton emission). The two protons have strong angular and energy correlations. The ²He appears as a resonance at 20 MeV/c in the two-proton relative momentum distribution^[19]. The microscopic calculations for the one- and two-proton decays of the 6.15 MeV 1^- state of ¹⁸Ne had been presented in the Ref.[20]. It was found that for the two-proton the sequential decay through a ghost of the 1/2⁺ state is within a factor of three of the observed width obtained with the assumption of democratic decay. The calculated width for diproton emission is only about a factor of two smaller than that for sequential decay indicating that the observed decay may be a combination of the two processes. In the excitation-energy spectrum of ¹⁸Ne in the Ref.[9], it’s strange that some states can be seen in the two-proton emission ¹⁸Ne→¹⁶O+2p channel and not in the one-proton emission ¹⁸Ne→¹⁷F+p channel. That means that in these states we cannot find the ¹⁷F in ground state. So the sequential decay for two protons is most likely to occur in these states.

In this paper we present the microscopic shell-model calculations for sequential two-proton decay from excited states in ¹⁸Ne by some different Hamiltonians.

2 Calculation and discussion

The spectroscopic factor is the most important quantity needed to obtain the decay width. In order to calculate it, we perform a shell-model calculation to get the wave functions for ¹⁸Ne. The model space was used, including the 0s, 0p, 1s0d and 1p0f orbits. ¹⁶O is treated as a s⁴p¹² closed shell, and the low-lying positive parity states of ¹⁷F and ¹⁸Ne are taken as s⁴p¹²(sd)¹ and s⁴p¹²(sd)². The low-lying negative parity states of ¹⁷F and ¹⁸Ne are treated as 1ħω excitations of the form s⁴p¹¹(sd)², s⁴p¹²(pf)¹ and s⁴p¹¹(sd)³, s⁴p¹²(sd)¹(pf)¹. So the emitted protons in the ¹⁸Ne and ¹⁷F are coming from (sd)(pf) shells. Two Hamiltonians designed for those types of model space are chosen for calculating the wave functions, namely the WBP and WBT interactions^[21]. We use a simply shell-model code by our group, in this code the spurious states are removed by the usual method^[22] by adding a center-of-mass Hamiltonian to the interaction.

The calculated excited energies of these low-lying states are shown in Fig.1. Some states are in reasonable agreement with the energies found in ¹⁸Ne. The low-lying negative states are dominated by the s⁴p¹¹(sd)³ configuration, but the smaller s⁴p¹² (sd)¹(pf)¹ component is the one responsible for one- and two-proton decay. The shell-model spectroscopic factors are obtained by the wave functions of ¹⁸Ne and¹⁷F. The decays from the positive states of ¹⁸Ne to the positive states of ¹⁷F and from the negative states of ¹⁸Ne to the negative states of ¹⁷F can go by 0d-shell wave emission or 1s-shell wave emission. The decays from the positive states of ¹⁸Ne to the negative states of ¹⁷F and from the negative states of ¹⁸Ne to the positive states of ¹⁷F can go by 0f-shell wave emission or 1p-shell wave emission. Because the s⁴p¹¹(sd)³ component in ¹⁸Ne is quite larger than that of s⁴p¹²(sd)¹(pf)¹, the spectroscopic factors are larger in the channel of positive states in ¹⁸Ne.

According to the scattering theory, the half-life for decay from initial state i to a final state f by one particle emission is given by:

where the decay width can be found from the relation^[25,26]:

is spectroscopic factor which corresponds to the probability that taking away a particle j with angular momentum j from an initial state i, will lead to a final state f. α_j is the asymptotic normalization of the proton single particle wave function in a state of spin j.

Fig.1Full-screen View

WBP and WBT predictions for the low-lying T=1 energy spectrum of ¹⁸Ne. Some levels are labeled by J^π and E_x. The experimental data^[23,24] are presented on the right column. The J^π of levels which are not label are unknown.

The total width for decay is a sum of partial widths:

The branching ratios are simply the ratio between a partial decay width and the total one:

For the fourth 1^- state at 7.94 MeV in ¹⁸Ne, we find that the spectroscopic factors decaying to the 1/2^- third excited state (Q_1p = 0.914 1 MeV) is quite larger than that decaying to the 5/2⁺ ground state (Q_1p = 4.018 4 MeV) of ¹⁷F. The spectroscopic factors and the widths for each of the channels are shown in Table 1. In this table, we can find the branch ratio that decays to 1/2^- state is larger than those decays to the ground state and the first ground state because it has larger spectroscopic factor even though it has smaller single-particle width. We can conclude that the 7.94 MeV 1^- state is most likely to be the best candidate for two-proton sequential decay. That is why it can be seen in the two-proton emission channel and not in the one-proton one. There are other states in this situation, like the 3^- state around 9–10 MeV (9.809 MeV for WBP and 10.099 MeV for WBT) and the 5^- state near 13 MeV (13.412 MeV for WBP and 13.200 MeV for WBT).

3 Conclusion

We have presented some preliminary results of the proton decay branch ratios and decay width in ¹⁸Ne using a shell-model calculation. The results obtained for the branch ratio from the 1^- state at 7.94 MeV in ¹⁸Ne show that this state is most likely to be a candidate for sequential two-proton decay. The 3^- and the 5^- states can also be candidates for the same process.