Two-proton sequential decay from excited states of ¹⁸Ne

Two-proton sequential decay from excited states of 18 Ne 1 corresp first-author Corresponding author. E-mail address: yuning_ok@hotmail.com Corresponding author. E-mail address: yuning_ok@hotmail.com yuning_ok@hotmail.com 2 3 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China Dipartimento di Fisica "G.Galilei" , Padova, Italy and Istituto Nazionale di Fisica Nucleare, Padova, Italy Dipartimento di Fisica "G.Galilei" , Padova, Italy and Istituto Nazionale di Fisica Nucleare, Padova, Italy Centro de Física das Interacções Fundamentais, and Departmento de Física, Instituto Superior Técnico, Lisbon, Portugal Centro de Física das Interacções Fundamentais, and Departmento de Física, Instituto Superior Técnico, Lisbon, Portugal Two-proton radioactivity from 18 Ne is discussed in terms of sequential decay. The branch ratios for one-proton emission from excited states are calculated, which including spectroscopic factors, obtained from a Shell-model calculation with realistic interactions. The branch ratios show that the two-proton emission from the 1 - state of 18 Ne at 7.94 MeV is most likely to go through the sequential decay. The same mechanism is discussed for other excited states at higher energy by different interactions. Two-proton radioactivity Nuclear shell model Branch ratios 1 Introduction 1 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 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 45 Fe by Pfutzner, Giovinazzoin experiments at GSI and GANIL [ 2 2 , 3 3 ] . Research in the field flourished after this breakthrough, and to date 54 Zn [ 4 4 ] , 48 Ni [ 5 5 ] , 19 Mg [ 6 6 ] , 16 Ne [ 7 7 ] , 17 Ne [ 8 8 ] , 18 Ne [ 9 9 ] , 10 C [ 10 10 ] , 14 O [ 11 11 ] and 29 S [ 12 12 ] have been found to exhibit two-proton emission. Several theoretical approaches such as Diproton model [ 13 13 , 14 14 ] , R-matrix approach [ 15 15 ] , continuum shell model [ 16 16 ] , adiabatic hyperspherical approach [ 17 17 ] , and the quantum three body cluster approach [ 18 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) 2 He cluster emission where a pair of protons, correlated in a quasi-bound 1 S configuration, breakup, when emitted into two protons (diproton emission). The two protons have strong angular and energy correlations. The 2 He appears as a resonance at 20 MeV/ c in the two-proton relative momentum distribution [ 19 19 ] . The microscopic calculations for the one- and two-proton decays of the 6.15 MeV 1 - state of 18 Ne had been presented in the Ref.[ 20 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 18 Ne in the Ref.[ 9 9 ], it’s strange that some states can be seen in the two-proton emission 18 Ne→ 16 O+2p channel and not in the one-proton emission 18 Ne→ 17 F+p channel. That means that in these states we cannot find the 17 F 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 18 Ne by some different Hamiltonians. 2 Calculation and discussion 1 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 18 Ne. The model space was used, including the 0s, 0p, 1s0d and 1p0f orbits. 16 O is treated as a s 4 p 12 closed shell, and the low-lying positive parity states of 17 F and 18 Ne are taken as s 4 p 12 (sd) 1 and s 4 p 12 (sd) 2 . The low-lying negative parity states of 17 F and 18 Ne are treated as 1 ħω excitations of the form s 4 p 11 (sd) 2 , s 4 p 12 (pf) 1 and s 4 p 11 (sd) 3 , s 4 p 12 (sd) 1 (pf) 1 . So the emitted protons in the 18 Ne and 17 F 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 21 ] . We use a simply shell-model code by our group, in this code the spurious states are removed by the usual method [ 22 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 Fig.1 . Some states are in reasonable agreement with the energies found in 18 Ne. The low-lying negative states are dominated by the s 4 p 11 (sd) 3 configuration, but the smaller s 4 p 12 (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 18 Ne and 17 F. The decays from the positive states of 18 Ne to the positive states of 17 F and from the negative states of 18 Ne to the negative states of 17 F can go by 0d-shell wave emission or 1s-shell wave emission. The decays from the positive states of 18 Ne to the negative states of 17 F and from the negative states of 18 Ne to the positive states of 17 F can go by 0f-shell wave emission or 1p-shell wave emission. Because the s 4 p 11 (sd) 3 component in 18 Ne is quite larger than that of s 4 p 12 (sd) 1 (pf) 1 , the spectroscopic factors are larger in the channel of positive states in 18 Ne. 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: <math display="block" id="M1"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mi>ℏ</mi><mi>ln</mi><mn>2</mn><mo>/</mo><msubsup><mi>Γ</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup></mrow></math> where the decay width can be found from the relation [ 25 25 , 26 26 ] : <math display="block" id="M2"><mrow><msubsup><mi>Γ</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup><mo>=</mo><msubsup><mi>S</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup><msub><mi>Γ</mi><mi>j</mi></msub><mo>=</mo><msubsup><mi>S</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><mi>k</mi><msubsup><mi>α</mi><mi>j</mi><mn>2</mn></msubsup></mrow><mi>m</mi></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10354879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10354885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10354882&type= 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 . The total width for decay is a sum of partial widths: <math display="block" id="M3"><mrow><msubsup><mi>Γ</mi><mrow><mtext>Tot</mtext></mrow><mi>i</mi></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>j</mi><mi>f</mi></mrow></munder><mrow><msubsup><mi>Γ</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup></mrow></mstyle></mrow></math> The branching ratios are simply the ratio between a partial decay width and the total one: <math display="block" id="M4"><mrow><msup><mrow><mi mathvariant="normal">Br</mi></mrow><mrow><mi>i</mi><mi>f</mi></mrow></msup><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>j</mi></munder><mrow><msubsup><mi>Γ</mi><mi>j</mi><mrow><mi>i</mi><mi>f</mi></mrow></msubsup></mrow></mstyle></mrow><mrow><msubsup><mi>Γ</mi><mrow><mtext>Tot</mtext></mrow><mi>i</mi></msubsup></mrow></mfrac></mrow></math> For the fourth 1 - state at 7.94 MeV in 18 Ne, we find that the spectroscopic factors decaying to the 1/2 - third excited state ( Q 1 p = 0.914 1 MeV) is quite larger than that decaying to the 5/2 + ground state ( Q 1 p = 4.018 4 MeV) of 17 F. The spectroscopic factors and the widths for each of the channels are shown in Table 1 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). 1 1 1 1 WBP 1 1 WBT 2 1 Expt. 1 1 E x / MeV 1 1 7.648 0 1 1 7.698 6 2 1 7.94 1 2 Channel 4 1 Spectroscopic factor 1 1 WBP 1 1 WBT 2 1 Γ sp / keV 1 1 5/2 + ⊗0f7/2 1 1 0.010 61 1 1 0.003 52 2 1 129 1 1 5/2 + ⊗0f5/2 1 1 0.010 11 1 1 0.007 65 2 1 101 1 1 5/2 + ⊗1p3/2 1 1 0.001 01 1 1 0.004 28 2 1 2 818 1 1 1/2 + ⊗1p3/2 1 1 0.003 34 1 1 0.002 23 2 1 2 239 1 1 1/2 + ⊗1p1/2 1 1 0.000 13 1 1 0.000 13 2 1 2 188 1 1 1/2 - ⊗0d3/2 1 1 0.001 06 1 1 0.005 25 2 1 2 1 1 1/2 - ⊗1s1/2 1 1 0.091 86 1 1 0.240 19 2 1 122 1 1   1 1 Γ WBP 1 1 Γ WBT 1 1 Br WBP 1 1 Br WBT 1 1   1 1   1 1   1 1   1 1   1 1 5/2 + ⊗(0f+1p) 1 1 5 1 1 13 1 1 0.216 1 1 0.277 1 1 1/2 + ⊗1p 1 1 8 1 1 5 1 1 0.32 1 1 0.11 1 1 1/2 - ⊗(0d+1s) 1 1 11 1 1 29 1 1 0.464 1 1 0.613 1 1 Total Γ 1 1 24 1 1 47 2 1 Expt. ≤ 50 keV 3 Conclusion 1 We have presented some preliminary results of the proton decay branch ratios and decay width in 18 Ne using a shell-model calculation. The results obtained for the branch ratio from the 1 - state at 7.94 MeV in 18 Ne 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. References Goldansky V . Nucl Phys , 1960 , 19 : 482 - 495 . Nucl Phys Pfutzner M , Badura E , Bingham C , et al . Eur Phys J A , 2002 , 14 : 279 - 285 . Eur Phys J A Giovinazzo J , Blank B , Chartier M , et al . Phys Rev Lett , 2002 , 89 : 102501 . Phys Rev Lett Blank B , Bey A , Canchel C , et al . Phys Rev Lett , 2005 , 94 : 232501 . Phys Rev Lett Dossat C , Bey A , Blank B , et al . Phys Rev C , 2005 , 72 : 054315 . Phys Rev C Mukha I , Summerer K , Acosta L , et al . Phys Rev Lett , 2007 , 99 : 182501 . Phys Rev Lett Mukha I , Giovinazzo J , Summerer K , et al . Phys Rev C , 2008 , 77 : 061303(R) . Phys Rev C Zerguerras T , Blank B , Blumenfeld Y , et al . Eur Phys J , 2004 , A20 : 389 - 396 . Eur Phys J Raciti G , Cardella G , Napoli M D , et al . Phys Rev Lett , 2008 , 100 : 192503 . Phys Rev Lett Mercurio K , Charity R J , Shane R , et al . Phys Rev C , 2008 , 78 : 031602(R) . Phys Rev C Bain C R , Woods P J , Coszach R , et al . Phys Lett B , 1996 , 373 : 35 - 39 . Phys Lett B Lin C J , Xu X X , Jia H M , et al . Phys Rev C , 2009 , 80 : 014310 . Phys Rev C Brown B A . Phys Rev C , 1991 , 43 : R1513 - R1517 . Phys Rev C Brown B A . Phys Rev C , 1991 , 44 : 924(E) . Phys Rev C Bartlett A J , Tostevin J A , Thompson I J . Phys Rev C , 2008 , 78 : 054603 . Phys Rev C Rotureau J , Okolowicz J , Ploszajczak M . Phys Rev Lett , 2005 , 95 : 042503 . Phys Rev Lett Nielsen E , Fedorov D V , Jensen A S , et al . Phys Rep , 2001 , 347 : 374 - 460 . Phys Rep Grigorenko L , Mukha I , Thompson I , et al . Phys Rev Lett , 2002 , 88 : 042502 . Phys Rev Lett Koonin S . Phys Lett B , 1977 , 70 : 43 - 47 . Phys Lett B Brown B A , Barker F C , Millener D J . Phys Rev C , 2002 , 65 : 051309(R) . Phys Rev C Warburton E K , Brown B A . Phys Rev C , 1992 , 46 : 923 - 944 . Phys Rev C Glockner D H , Lawson R D . Phys Lett B , 1974 , 53 : 313 - 318 . Phys Lett B Bardayan D W , Blackmon J C , Brune C R , et al . Phys Rev C , 2000 , 62 : 055804 . Phys Rev C Hahn K I , Garcia A , Adelberger E G , et al . Phys Rev C , 1996 , 54 : 1999 - 2013 . Phys Rev C Maglione E , Ferreira L S , Liotta R J . Phys Rev Lett , 1998 , 81 : 538 - 541 . Phys Rev Lett Ferreira L S , Maglione E , Liotta R J . Phys Rev Lett , 1997 , 78 : 1640 - 1643 . Phys Rev Lett 10.13538/j.1001-8042/nst.2013.05.017 Supported by the Fundação para a Ciência e a Techologia (Portugal) , Project: PTDC/FIS/68340/2006 and CERN/FP/116385/2010 2013-06-30 2013-10-01 2021-04-11 2013 Nuclear Science and Techniques 5 24