Determination of the cluster-decay branching ratio from a near-threshold molecular state in 10 Be
W. Jiang, Y. L. Ye, C. J. Lin, Z. H. Li, J. L. Lou, X. F. Yang, Q. T. Li, Y. C. Ge, H. Hua, D. X. Jiang, D. Y. Pang, J. Li, J. Chen, Z. H. Yang, X. H. Sun, Z. Y. Tian, J. Feng, B. Yang, H. L. Zang, Q. Liu, P. J. Li, Z. Q. Chen, Y. Liu, Y. Zhang, J. Ma, H. M. Jia, X. X. Xu, L. Yang, N. R. Ma, L. J. Sun
DDetermination of the cluster-decay branching ratiofrom a near-threshold molecular state in Be W. Jiang,
1, 2, 3
Y. L. Ye, ∗ C. J. Lin, Z. H. Li, J. L. Lou, X. F. Yang, Q. T. Li, Y. C. Ge, H. Hua, D. X. Jiang, D. Y. Pang, J. Li, J. Chen, Z. H. Yang, X. H. Sun, Z. Y. Tian, J. Feng, B. Yang, H. L. Zang, Q. Liu, P. J. Li, Z. Q. Chen, Y. Liu, Y. Zhang, J. Ma, H. M. Jia, X. X. Xu, L. Yang, N. R. Ma, and L. J. Sun School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Institute of High Energy Physics, CAS, Beijing 100049, China Spallation Neutron Source Science Center, Dongguan 523803, China China Institute of Atomic Energy, Beijing 102413, China School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA RCNP, Osaka University, 10-1 Mihogaoka, Ibaraki, Osaka, 567-0047, Japan RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan (Dated: March 6, 2020)A puzzle has long existed for the α -cluster content in the near-threshold 7.54 MeV state of Be.A new measurement was conducted to measure the cluster-decay partial width of this state, usingthe reaction Be( Be , Be ∗ → α + He) Be at 45 MeV beam energy. Special measures were taken toreduce the strong near-threshold background. The neutron-decay strength was also obtained basedon the three-fold coincident measurement. A cluster-decay branching ratio of (4 . ± . × − isobtained, resulting in a reasonably large α -cluster spectroscopic factor. The present work confirmsthe formation of the σ -bond molecular rotational band headed by the 6.18 MeV state in Be.
It has been well established that, in light nuclei,the quantum states formed near the cluster-separationthreshold tend to possess a large degree of clusterconfiguration [1–5]. In recent years clustering has alsobeen identified in neutron-rich unstable nuclei where theexcess valence neutrons may act as the valence bondssimilar to those in atomic molecular systems [2].The persistence of the clustering effect from stable tounstable nuclei has initially been demonstrated alongthe Beryllium isotopic chain [2, 6–8]. In the case ofneutron-rich Be, the four excited states around 6 MeV(about 2.5 MeV below the 2n separation threshold)can be perfectly explained by the combination of thevalence neutron orbits surrounding the 2- α cores [8, 9].These predominant molecular structures in the excitedstates of Be are also supported by the AntisymmetrizedMolecular Dynamics (AMD) calculations [10, 11], andby various experimental evidences [2]. Based on thesespeculations, the observed excited states in Be weregrouped into several molecular rotational bands headedby states at around 6 MeV and characterized by verylarge moment of inertia [2, 9]. These molecular bandsform a beautiful classification of the excited states in Be [2]. The applicability of this classification depends,of course, on the intrinsic structure of each member statein these bands ([12–15] and references therein).Among the predicted molecular rotational bands in Be, the one formed by the states at 6.18 MeV (0 +2 ),7.54 MeV (2 +3 ) and 10.15 MeV (4 +1 ) has acquired specialattention, owing to its pure σ -bond feature [2, 10, 11],which leads to the longest chain shape correspondingto the largest moment of inertia. In addition to the spin-energy systematics associated with the moment ofinertia, the cluster-decay partial width is of essentialimportance since it determines quantitatively the clustercontent in a resonant state. The cluster-decay strengthof the 10.15 MeV state in Be has been measured inmany experiments [12, 13]. A spin-parity of 4 + and a He + α cluster spectroscopic factor (SF) ranging from0.66 to 2.23, for a channel radius from 1.8 to 1.4 fm,were firmly established, indicating an almost pure clusterstructure in this state [16, 17]. The band head at 6.18MeV (0 +2 ) is below the α -separation threshold at 7.41MeV. It is interpreted as a pure σ -bond molecular statebased on its selective population and γ -decay properties[8], and on the consistent theoretical calculations [2, 11].However, for the 7.54 MeV (2 +3 ) member state, beingjust 0.132 MeV above the α -separation threshold, theexperimental investigations related to the cluster-decayare very limited, due mostly to the extreme difficultiesin the near-threshold measurement and analysis. Thefirst experimental result was obtained from the reaction Li( Li , Be ∗ → α + He) α at 34 MeV beam energy[18]. A branching ratio (BR) of 3 . × − wasreported. Another experiment, using the reaction Li( He , Be ∗ → α + He)t at 18 MeV beam energy,generated only a lower limit of BR ≥ . × − [19]. We notice that the significantly suppressed α -decayfraction is basically attributed to the extremely smallrelative energy against a large Coulomb barrier. Hencethe cluster content inside the mother nucleus may still belarge. Using the above referred BR [18], an unreasonablylarge cluster SF of 51(19) for this 7.54 MeV state wasextracted by Fortune et al. [12]. This puzzle has not a r X i v : . [ nu c l - e x ] M a r been solved so far due essentially to the experimentaldifficulties.We present here a new measurement of the α -decayBR of the 7.54 MeV state in Be, using the reaction Be( Be , Be ∗ → α + He) Be at 45 MeV beam energy.This reaction was chosen after a careful considerationof the beam availability, and the compromise betweenthe optimal detection of the decay fragments andthe necessary to avoid the high-flux elastic scatteringparticles at very forward angles. It would be worth notingthat the π -type orbit in Be(g.s.) may be replaced by the σ -type orbit when expanding the distance between the 2- α cores [2, 8], which is just the case for the well deformed Be(2 +3 ) state [10]. Hence the population of the σ -bondmolecular state, such as the 7.54 MeV (2 + ) state in Be,is possible in the present one neutron transfer reaction,as justified by some previous experiments [20–23]. In thepresent work, some special measures were taken to handlethe near-threshold background and to determine thecontributions from the neutron-decay channel. Finallya much smaller cluster-BR is obtained for the 7.54 MeVstate , being consistent with the theoretical expectations.The experiment was carried out at the HI-13 tandemaccelerator facility at China Institute of Atomic Energy(CIAE) [4]. A 45 MeV Be beam with an intensityof about 1 pnA was used to bombard a self-supporting Be target (166- µ g/cm ). A schematic drawing anddetailed descriptions of the detector setup can be foundin Refs.[4, 24]. The reaction products were detected andidentified by six silicon-strip telescopes, namely U0, U1,U2, D0, D1 and D2, which were placed symmetricallyon both sides of the beam axis [4, 24, 25]. Double sidedsilicon-strip detectors (DSSDs) were employed, providingexcellent two-dimensional position resolutions and theability to record multi-hit events in one telescope. Theforward-angle telescope U0(D0) was centered at 23 ◦ withrespect to the beam direction and at a distance of 140 mmfrom the target. The large-angle telescopes U1(D1) wasat 60 ◦ and 116 mm, respectively. U2(D2) telescope wasinstalled at the backward angle for other purpose [4, 24].The active area of each telescope is 50 ×
50 mm . Energycalibration of the Si detectors was realized by using athree-component α source and the elastic scattering of Be off a
Au target. The typical energy resolution ofthe silicon-strip detector was better than 1.0% for 5.49-MeV α particles [26, 27]. We note that the applicationof the DSSDs with small-size pixels (2 × ) isof essential importance here. Since the targeted 7.54MeV resonance is only 130 keV above the α -separationthreshold, the opening angle in the laboratory systemis small for the decay products, of which the coincidentdetection efficiency depends sensitively on the pixel size.Using the standard energy-loss versus stoping-energy(∆ E - E ) method, excellent particle identification (PID)performance was achieved up to beryllium isotopes(Fig. 1). The overall performance of the detection system was checked by reconstructing the Be energy spectrumfrom the 2- α particles which were coincidentally detectedin one forward-angle telescope [13, 25, 28].The energy released in a reaction, namely thereaction Q -value, is a useful quantity to select thereaction channels [4]. It should be noted that the7.54 MeV resonance in Be has only two possibleparticle-decay channels (n- and α -decay), while its γ -decay is negligible [18]. The purpose of the presentexperiment is then to study two reaction-decay channels:(a) Be( Be , Be ∗ → α + He) Be ( Q ggg = -2.26 MeV);and (b) Be( Be , Be ∗ → n + Be) Be g . s . → α ( Q ggg = -1.57 MeV). Q ggg here means the Q -value forall particles in their ground states. The relative yieldsof these two channels, generated from the intermediate7.54 MeV resonance in Be, allow to deduce theabsolute α -decay BR of this state.For channel (a), due to the interest on the low relativeenergy states, we only use events with α + He pairbeing detected within one forward-angle telescope. Theenergy of the recoil Be can be deduced according togmomentum conservation. As a result, Q = E tot − E beam = Σ E i − E beam , (1)where E denotes the kinetic energy and i runs for FIG. 1. PID spectrum measured by the U0 telescope using the∆ E - E method. The inset shows the projected PID spectrum[27] gated on He as another coincident particle in the sametelescope. -15 -10 -5 0 5 (a) for reaction channel (a) Q ggg Q ggg C oun t s / k e V C oun t s / k e V Q (MeV) (b) for reaction channel (b)
FIG. 2.
Experimental Q -value spectra for the two reactionchannels: (a) Be( Be , Be ∗ → α + He) Be ( Q ggg = -2.26 MeV);and (b) Be( Be , Be ∗ → n + Be) Be g . s . → α ( Q ggg = -1.57MeV). all particles in the exit channel. The correspondingspectrum is shown in Fig. 2(a), in which a narrow peakstands for Q ggg and a broader one at lower Q values isassociated with the first excited state of Be (3.03 MeV,2 + ).The relative energy ( E rel ) of the decay products can bededuced according to the invariable mass (IM) method[29]. The associated excitation energy is thus E x = E rel + E th , with E th the cluster separation energy [29]. Sinceboth E x for Be and Q -value for the reaction channel (a)are calculable based on the detected α + He pair, thecorresponding two-dimensional spectrum can be plottedto illustrate their possible correlations (Fig. 3(a)).In order to have a strict constraint on the reactionmechanism, a gate on the Q ggg peak (region G1 inFig. 3(a)) is applied to the projection onto E x , asdisplayed in Fig. 3(b). As shown in the figure, the peak ataround 9.5 and 10.1 MeV agree exactly with the previousobservations [16, 30, 31], indicating the correctness of thepresent measurement and analysis. But for the peak atabout 7.54 MeV, very close to the threshold, cares mustbe taken since there appears a relatively strong bandat around this excitation energy but distributed broadlyalong the Q -dimension, as approximately indicated bythe gate G2 in Fig. 3(a). To analyse its influenceon the event counting for the 7.54 MeV resonance in Be, we project this band onto the Q -value dimension,as displayed in Fig. 3(c). It would be important tocheck the possible contamination to the Q ggg peak bythis background band. We find that this contaminationdepends quite sensitively on the PID selection. In ourcase, as demonstrated in Fig. 1, the identification for He is very clean, whereas that for He might be mixedby some nearby He. We have plotted the Q -valuespectrum gated on the left-side-half or right-side-half ofthe He peak (see the inset of Fig. 1), respectively. Itwas evidenced that the contamination to the Q ggg peakis appreciable with the former gate, but negligible withthe latter gate, similar to the background under the Q ggg peak in Fig. 2(a) and Fig. 3(c), respectively. Thereforethe latter selection of He, together with the normalselection of He, were adopted for the final analysis of theresonance, as presented in Fig. 3. Of course, the efficiencysimulation follows exactly the applied gates at therelevant steps. From Fig. 3(c) it can be seen that the Q ggg peak is well distinguished from the background banddistribution. The remaining minor scattered backgroundexists all over the two-dimensional spectrum, as exhibitedby Fig. 3(d), which can be naturally subtracted fromthe E x spectrum. Actually, from the well standing two-dimensional Q ggg - E x (7.54 MeV) peak (Fig. 3) and bysubtracting the beneath background, we have obtainedthe counts for the resonance as 32 ±
10, taken into accountthe factor 2 for the He selection as mentioned above.The uncertainty here is statistical only, including thebackground contribution. -15 -10 -5 0 57891011 10 20 30 40 7891011 -15 -10 -5 0 505101520
10 20 30 40 7891011 G2 G1r Q ggg G1 E x ( M e V ) (a) with gate G1 E x ( M e V ) (b) Counts
Q (MeV) Q ggg with gate G2 C oun t s Q (MeV) (c)
Counts E x ( M e V ) with gate G1r (d) FIG. 3.
Spectra deduced from the detected α + He pairs. (a)two-dimensional plot for E x versus Q -value; (b) The projected E x spectrum for Q -value around Q ggg (gate G1); (c) The projected Q -value spectrum for E x around 7.54 MeV (gate G2); (d) Theprojected E x spectrum for Q -values at the right side of Q ggg (gateG1r). We note that there are other two possiblecontaminating exit channels, namely He + C ∗ ( → α )and α + C ∗ ( → α + Be ∗ ( → α + He)), whichare composed of the same mass combinations as thereaction channel (a). However, in the case of C ∗ → α decay, the simulation shows that having one of these α -particles going into the forward angle telescope whilekeeping another two closely (as Be g.s.) at large anglewould require very high excitation energy in C ( > C high-excitation versus the Be excitation, usingthe real data, does not show any structure correlation.Therefore, this background channel does not affect theactual extraction of the well-distinguished 7.54 MeVpeak in Be. In the case of C production and decay,the two undetected α -particles, one recoiling to a largeangle and another from C-decay emitting to a forwardangle, are separated from each other and cannot fakethe Be(g.s.) as required by the actual Q ggg . Theexclusion of this contamination has also been verified bysimulation and real data analysis [4]. The effect of thetarget impurities, mainly carbon and oxygen contents,were analyzed and eliminated by using the EP-plotmethod [32]. Event mixing was also checked against the Q -value spectrum.To investigate the reaction channel (b), events wereselected by requiring 2-particles being detected by alarge-angle telescope (U1/D1) and one Be nucleuscoincidentally detected by a forward-angle telescope(D0/U0). Again the energy of the missing neutron canbe calculated according to the momentum conservation.Although the two particles detected by a large-angletelescope were not clearly identified due to their very lowkinetic energies, the clear Q ggg peak in Fig. 2(b) assuresthe case since any other mass combination must givemuch lower Q -value. This is further ascertained by gatingon the relative energy of these two nearby particles as theg.s. of Be ( ∼
91 keV). Due to the larger uncertaintiesin detecting these two very low energy α -particles thededuced Q -value spectrum exhibits a larger peak-width(Fig. 2(b)). Remarkably, there is almost no continuousbackground in the lower Q -value region, owing to thethree-folds coincident detection.The presently targeted exit channel is inevitablyaccompanied by a very probable inelastic scatteringchannel Be( Be , Be ∗ → n + Be) Be, which possessesthe same Q -value but does not reflect the Be excitation.In order to reduce this contamination, E x from Be +n reconstruction is plotted against cos ϕ n , as shown inFig. 4(a). Here ϕ n stands for the azimuthal angle of thededuced neutron in the laboratory system, with the 0-degree axis lies in the horizontal plan and points to theside having the Be detection. As presented in Fig. 4(a),cos ϕ n distribution is concentrated around +1 and -1,with the real resonances in E x , such as the one at about7.5 MeV, placed at the +1 end. This plot provides a gooddiscrimination between the targeted reaction channel (b)and the above mentioned contamination channel. Infact, the former tends to emit a neutron close by theforward moving Be, while the latter combined with therecoil Be at the opposite side of Be. Considering boththe signal to background ratio and the signal detectionefficiency, we require cos ϕ n ≥ . E x spectrum for Be (Fig. 4(b)). We note thatFig. 4(a) could be replaced by the Dalitz-plot of Be ∗ versus Be ∗ . However the present method provides abetter performance.Previously the excited states in Be were observedat 7.37 and 7.54 MeV [33], which should be included inour fitting analysis of the relatively broad peak around7.5 MeV. The detection efficiency and energy resolutionas a function of E x were estimated by Monte Carlosimulation taking into account the realistic detector setupand performances [34, 35]. Dotted line in Fig. 4(b) showthe the relative efficiency. The resolution ranges from FIG. 4. (a) Two-dimensional plot for the reconstructed E x ( Be)versus cos ϕ n . ϕ n stands for the azimuthal angle of the deducedneutron, as defines in the text. (b) Excitation energy E x in Bededuced from the n + Be decay channel, subject to event selectionas described in the text. The curves are also explained in the text. ∼
200 keV (standard deviation σ ) at E x = 7.37 MeV upto ∼
240 keV at E x = 7.54 MeV. The intrinsic widthsof the resonances were previously reported as ∼
10 keVfor both states [33]. The data after efficiency correctionwere fitted by using two Gaussian-shape functions with afixed interval of 170 keV between their central positions(dashed lines in Fig.4(b)), together with a smoothbackground function [35] (dot-dashed line in Fig.4(b)).The adopted widths ( σ ) after the variation are 195 keVand 245 keV for the 7.37 MeV and 7.54 MeV peaks,respectively, being consistent with the simulation results.Finally, the counts N n = 19146 ±
138 is determined forthe decay Be(7 .
54 MeV) → n + Be , subject to theapplied cuts which will be accounted for by the efficiencysimulation. The uncertainty here is statistical only. Byusing events in this dominating neutron decay channel,together with the integrated incident particle number,the target thickness and the simulated detection solidangle, we obtain a population cross section of 65 µ b/sr atabout 25 lab. (15 µ b/sr at about 55 c.m.) for the 7.54MeV state in Be. Here the statistical uncertainty is lessthan 1% while the systematic uncertainty is estimated tobe about 10% resulted mainly from the beam integration.Monte Carlo simulations were conducted for bothreaction-decay channels (a) and (b), using realisticexperimental setup and event selection configurations.The differential cross section for transferring intothe 7.54 MeV (2 +3 ) state in Be was generatedby the Distorted-Wave-Born-Approximation (DWBA)calculation using the code Fresco [36]. The opticalpotential parameters were taken from Refs. [37, 38].From the simulation the ratio of the acceptances for thetwo reaction-decay channels, ε α / ε n , is determined to be4.14.The BR for α -decay from Be(7.54 MeV) is nowexpressed in the form:Γ α Γ tot = N α /ε α N α /ε α + N n /ε n = N α N α + N n ε α ε n (2)where N α and N n represent the detected α and neutronnumbers from the resonance. Using the known numberswe obtain Γ α /Γ tot = (4 . ± . × − . The errorhere is statistical only. In addition some systematic errorof about 10% is estimated, considering the reasonableparameter variations in the simulation and the functionselection in the fitting procedure. The presentlyobtained cluster-decay BR is significantly smaller thanthe previously measured ones [18, 19]. The existingobserved BR α results are listed in Table I.Based on the single-channel R -matrix approach, the α -decay partial width Γ α can be converted into the reducedwidth γ α according to the formula [34, 39]: γ α = Γ α P l , (3)with P l being the barrier penetrability factor [34, 39]. γ α is generally presented with respective to the WignerLimit γ αW , leading to the dimensionless reduced width θ α = γ α / γ αW with γ αW = 3 (cid:126) /(2 µR ) [16, 40]. Here µ is the reduced mass of the decaying cluster system and R the channel radius given by R = r ( A / + A / ). θ α is also interpreted as the “ α -cluster spectroscopicfactor (SF)” [16], which is sensitively dependent on R .As a matter of fact, extremely large deformation orinter- α -core distance has been predicted for the σ -bondmolecular states in Be, in comparison to its groundstate [3, 10]. Using the known total width Γ tot = 6 . α measured in the present experiment, weobtain θ α ranging from 2 . r from 1.4 to 1.8 fm. This SF α resultis similar to that for the 4 +1 state in the same molecularrotational band, which ranges from 2.23 to 0.66 for r from 1.4 to 1.8 fm [16], and is fairly consistent with themaximum degeneracy of α -particle.We notice that the cluster SF for 2 +3 isobaric analogstate in B (8.894 MeV) was reported to be 0.73(13) [21],when deduced by using the standard reduced channelradius r = 1.4 fm and the Wigner Limit as definedabove [16, 40]. The difference between this value andour results for Be (7.54 MeV) is twice as large as thesummed error bar. The same difference is also evidencedby the BR α in Table I. This kind of cluster-SF differencebetween analog states was observed at other occasions aswell. For instance, the 8.898 MeV (3 − ; T = 1) state in B has a reported large SF α of 0.42, whereas its analogin Be (7.31 MeV, 3 − state) is a well-recognized puresingle-particle state [21]. Indeed, theoretical studiesusing the Gamow shell model have revealed that, forthe weakly bound or unbound systems, the structure ofisobaric analog states varies within the isomultiplet andimpacts the associated particle SF [41]. This variationis mainly related to the large asymmetry betweenproton and neutron emission thresholds which modifythe respective coupling to the continuum. Since thecluster formation occurs most likely at around the clusterseparation threshold and is often mixed with single-particle configuration especially in the case of unstablenuclei, the spectroscopic change within the clusteringisomultiplet might be enhanced. As an example, thiskind of structure change has been demonstrated by the Be - C mirror system. The allowed 2 p -emission fromthe 0 +2 or 2 +3 level of C leads to significant structure
TABLE I. Summary of the α -decay branching ratio (BR α ) ofthe 7.54 MeV (2 +3 ) state in Be.Data source BR α Experimental result from [18] (3 . ± . × − Experimental result from [19] ≥ (2 . ± . × − Present experimental result (4 . ± . × − Converted from B analog a (1 . ± . × − adapted from Ref. [21] for a reduced radius r = 1.4 fm. change in comparison to its mirror counterpart in Bewhich has no corresponding 2 n -emission channel becauseof the relatively higher threshold [42]. In addition, theisospin mixing may also result in spectroscopic change.For example, the strong clustering 2 + (8.894 MeV; T =1) state in B has a reported isospin-conservation α -decay width of about 18 keV, together with an isospin-violating decay width of about 12 keV [21]. This isospin-mixing in decay process does not exist in the Be analogstate (7.54 MeV). We notice again that the extractionof the cluster SF depends sensitively on the channelradius of the resonance, which may be changed state bystate for weakly bound or unbound system, as recentlydemonstrated in Ref. [43]. Thus, the valuable comparisonamong SFs of the analog states requires also independentdetermination of the radius. It is obvious that the preciseand independent measurements of cluster-decay BRsand other observables for analog states would provideimportant information for the study of isospin-symmetrybreaking in exotic composite nuclear systems.In summary, a new experimental investigation ofthe cluster structure of the 7.54 MeV (2 + ) resonantstate in Be was performed by using the reaction Be ( Be , Be) Be at 45 MeV beam energy. Specialmeasures were taken to reduce the strong near-thresholdbackground and to assure a reliable extraction of thecluster-decay events from the resonance. The neutron-decay from the same resonance was also analyzedbased on the three-fold coincident measurement. Acluster-decay branching ratio of (4 . ± . × − is determined for the 7.54 MeV resonance in Be.The deduced α -cluster SF is from 2.56(80) to 0.87(27)for the reduced channel radius r from 1.4 to 1.8 fm.The present work, together with the well establishedcluster-description of the 6.18 MeV (0 + ) and 10.15 MeV(4 + ) states, leads to a comprehensive understandingof a perfect molecular rotational band in excited Be.The comparison between the currently obtained clusterSF with that of the analog state in B may stimulatefurther studies of the isospin-symmetry breaking inclustering nuclear systems.The authors wish to thank the staffs of the tandemaccelerator laboratory at CIAE for their excellent workin providing the beams. This work is support bythe National Key R&D Program of China (Grant No.2018YFA0404403) and the National Natural ScienceFoundation of China (Grant Nos. 11535004, 11875074,11875073, U1967201, 11635015, 11775004, 11775013 and11775316). ∗ [email protected][1] K. Ikeda, N. Tagikawa, and H. Horiuchi, Prog. Theor. Phys. Suppl. 68 (1968) 464.[2] W. von Oertzen, M. Freer, and Y. Kanada-En’yo, Phys.Rep. 432 (2006) 43.[3] H. Horiuchi, K. Ikeda, and K. Kat, Prog. Theor. Phys.Suppl. 192 (2012) 1.[4] J. Li, Y. L. Ye, Z. H. Li et al ., Phys. Rev. C 95 (2017)021303(R).[5] M. Freer and H. Horiuchi and Y. Kanada-Enyo and D.Lee, Rev. Mod. Phys. 7 (2018) 035004.[6] Z. H. Yang, Y. L. Ye, Z. H. Li et al ., Phys. Rev. Lett.112 (2014) 162501.[7] B. Buck, H. Friedrich, C. Wheatley, Nucl. Phys. A 275(1977) 246.[8] W. von Oertzen, Z. Phys. A 354 (1996) 37.[9] W. von Oertzen, Z. Phys. A 357 (1997) 355.[10] Y. Kanada-En’yo, H. Horiuchi, and A. Dot´e, Phys. Rev.C 60 (1999) 064304.[11] M. Lyu, Z. Ren, B. 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